Structure and Properties of Polymers

1 Model Polymers for Materials Science Lewis J. Fetters Exxon Research and Engineering Co., Corporate Research Science Laboratory, Clinton Township, Annandale, NJ, U.S.A. Edwin L. Thomas Massachusetts Institute of Technology, Department of Materials Science and Engineering, Cambridge, MA, U.S.A.

List of Symbols and Abbreviations 1.1 Introduction 1.2 Polymer Synthesis 1.3 Living Polymers 1.4 Model Polymer Requirements 1.5 Functionalization of Model Polymers 1.6 Model Branched Polymers 1.7 Processing 1.8 Model Polymers in Polymer Physics 1.9 References

Materials Science and Technology Copyright © WILEY-VCH Verlag GmbH & Co KGaA. All rights reserved.

2 3 4 5 7 16 18 19 22 29

2

1 Model Polymers for Materials Science

List of Symbols and Abbreviations DPn c dt / GN Mt Mn Mw Mz Mv Nt R Tg wt

number-average degree of polymerization concentration of polymer, g/cm3 tube diameter contour length plateau modulus molecular weight of species i number average molecular weight weight average molecular weight Z average molecular weight viscosity average molecular weight number of moles of species i end-to-end distance of model polymer glass transition temperature weight fraction of species i

[r\\ Q

intrinsic viscosity, cm3/g density of polymer melt, g/cm3

FTIR GPC LC LCP NMR NSE RIS ROMP SANS SAXS SEC TEM

Fourier transform infrared spectroscopy gel permeation chromatography liquid crystal liquid crystal polymer nuclear magnetic resonance neutron spin echo rotational isomeric state ring-opening metathesis polymerization small angle neutron scattering small angle X-ray scattering size exclusion chromatography transmission electron microscopy

1.1 Introduction

1.1 Introduction Materials scientists like to make their measurements on large, pure, single crystal specimens. Such samples afford the opportunity to explore the full anisotropy of physical properties with crystallographic direction and thus more completely characterize structure-property relations. Indeed, often it is the spectacularly different measures of a given property with direction that make a material interesting. Man's first recognition that polymers are useful materials likely derived from the naturally occurring polymers in his environment - the cellulose in wood for housing and plant and animal fibers for his clothing. From the earliest days of polymer science, the novel and diverse behavior of the newly synthesized materials was immediately recognized - for example, after the discovery of the polymerization of nylon, researchers realized that a wide variety of properties were available from a single polymer by careful manipulation of processing conditions. Thus, nylon could be spun into a family of fibers with properties conveniently controlled by extent of draw, draw temperature, post heat setting treatment, etc. The wide range in polymeric behavior stems from the enormous spectrum of structural states that polymers can assume: solutions, melts, glasses, crystals, liquid crystals, gels and rubbers. This in turn stems from the chemical diversity of the macromolecules. Moreover, because of the tremendous conformational diversity of a polymer, a given polymer can yield a huge variety of structures! It is hard to imagine a material replacing rubber's unique physical properties, which essentially derive from conformational diversity (see Chaps. 2, 8, and 9).

In addition to technically relevant properties such as reversible deformation of several hundred per cent (rubber elasticity) and Young's moduli 1.5 times that of steel (rigid rod fibers), polymer molecules offer the scientist a fertile area for basic study. From flexible chains to rigid rods, scientists have been seeking to understand polymeric behavior. This task is very challenging since polymeric materials are highly complex. Polymers are complex because synthesis often does not result in simple, single component species, frequently their organization in the solid state takes place on several length scales, and the relatively long time scales required to reach equilibrium are frequently much longer than experimentally feasible. While a good deal of the complexity of polymers is unavoidable, the difficulties arising from their synthetic complexities has been reasonably addressed via the production of model materials by the modern synthetic polymer chemist. By model material, we mean a species for which the parameters of molecular characterization are very well-represented by the ensemble average. That is, every molecule is essentially identical to every other (monodisperse ideal population) or the collection of molecules corresponds to a valid statistical assembly of species arising from a given stochastic process (ideal random population). Experiments conducted on model materials benefit from the specificity of response to the molecular variables and allow much more precise interpretation and testing of structure-property relationships. Over the past 50 years, polymer chemists have made important advances in the control of macromolecular chain structure. Initially, most polymers were produced by polymerizations in which the chemical linking events were more or less randomly

1 Model Polymers for Materials Science

selected from a large number of possibilities, hence making the product macromolecules highly complex species. The chemist's prime goals were high yields and high molecular weight. This fact of dispersity in composition, in stereochemistry, in sequence distribution and in molecular length meant that the resultant material could only be characterized by average molecular properties. And as became evident, different physical probes were sensitive to different averages or different aspects of the molecular distribution. For example, polydispersity causes the progression of molecular weight averages M z > M w > Mn > M n where Mz is the Z-average molecular weight (size exclusion chromatography), M w is weight average molecular weight (light scattering), Mn is the viscosity average molecular weight (viscosity) and M n is the number average molecular weight (osmometry) (for characterization methods, see Vol. 2 A, Chap. 1). Given that the polymer materials had fairly broad distributions, early workers perforce satisfied themselves with more qualitative measurements than were possible with pure materials composed of a collection of identical macromolecules. Quantitative materials science (akin to that currently practiced in, say, the field of semiconductors) is just beginning to take place for polymers. Careful, precise characterization of highly specific behavior is replacing the former measurements, which were almost always significantly broadened by the spread in the molecular response stemming from the collection of diverse species in a given "pure" sample. The availability of these model materials has facilitated studies of, for example, polymer dynamics, adsorption, block copolymer morphology, diffusion (both tracer and self-diffusion), rheology, unperturbed chain dimensions and their temperature coefficients via small

angle neutron scattering (SANS), which permits the evaluation of those parameters in the realistic environment of the polymer melt. The SANS studies have been made possible by the use of deuterium labelling, either in the form of a perdeutero monomer or by saturation of a hydrogenous polydiene by deuterium. As we approach the 21st century, three activities are especially important in the changeover from such statistically complex materials of the past to the new, precisely defined, materials of the future. Activity in novel synthetic methods, including recombinant gene technology, promises synthesis of uniquely specific macromolecules. Secondly, advances in instrumentation to accurately measure molecular properties of these substances are continually being made. Thirdly, the significant growth of computational power to model, ever more realistically, such molecules provides good prospects for future understanding of structure-property relationships (see Chap. 2). All three of these endeavors, plus the merger of ideas and techniques of the polymer scientist with the molecular biologist and the closer coupling of the polymer chemist with the materials scientist, will contribute in major ways to the success of the "materials science of macromolecules".

1.2 Polymer Synthesis The preparation of homopolymers, block copolymers and functionalized polymers is of both academic and commercial value. In this regard, anionic polymerization (living polymerization) is uniquely positioned since it alone, to date, has the capacity to prepare the above families of model macromolecules in quantitative yield. This is due to the absence, in many cases, of a spontaneous termination event,

1.3 Living Polymers

the consequence of which also allows the preparation of chains of various architecture: for example, linear, graft, H-shaped, rings and stars (see Fig. 1-1). The latter class of materials contains regular armed homo- and block copolymer stars, asymmetric stars, and most recently, mikta (mixed)-arm stars, which are either A2B or ABC type branched stars where the various homopolymer arms radiate from a common junction point (Iatraou and Hadjichristidis, 1992).

1.3 Living Polymers The so-called living anionic polymerization systems are obtainable, in the strict sense, by the use of organolithium initiators and hydrocarbon solvents under conditions that exclude oxygen and protic impurities. Under the proper conditions, initiation can be rapid and near-mono-

Table 1-1. SEC heterogeneity evaluations for low molecular weight polybutadienes. Mwxl0~3

(gmor1) 1.00 1,05 1.05 1.15 2.00 3.60 4.30 5.90 6.50 13.0 18.0 36.0

1L08 0 1L09 8 1L07 4 1L07 4 1L07 4 t.03 5 1L02 9 L02 4 L02 3 1.02 3 L02 4 1.018

1.072 1.078 1.066 1.069 1.066 1.03 3 1.028 1.024 1.023 1.022 1.023 1.018

1.062 1.076 1.059 1.069 1.066 1.03 3 1.028 1.024 1.023 1.022 1.023 1-018

Mz

disperse molecular weight distributions can be obtained. This is shown in Table 1-1, where the size exclusion chromatography (SEC) characterization results are given for 1,4-polybutadienes with molecular weights as low as 103. As expected, the heterogeneity indices decrease as molecular weight increases. tert-Butyl lithium was the initiator with cyclohexane as the polymerization solvent at 20°C. The molecular weights of these materials are controlled by the simple expedient of manipulating the monomer [M] /initiator [/] ratio since each molecule of the latter will generate a growing chain. Thus, from the standpoint of the number-average degree of polymerization, DPn, we have DP-

Figure 1-1. Schematic of macromolecules with various architectures: (a) linear, flexible chain; (b) graft; (c) H-shaped; (d) ring; (e) star.

Mz Mw

Mz+1

Mn

[M]o

(1-1)

where the subscript denotes the initial reactant concentrations. Under normal conditions the reactive chain-ends in hydrocarbon solvents, the so-called "active centers", can retain their reactivity for periods of time that far exceed the time required for complete con-

1 Model Polymers for Materials Science

version. Thus, the various molecular weight moments can be, within experimental error, virtually identical. These are defined as follows: Number average molecular weight: (1-2) Weight-average molecular weight: (1-3) Z-Average molecular weight: Mz =

—l- =

—y

(1-4)

Z + 1-Average molecular weight: (1-5) where wt = cjc and denotes the weight fraction of species i in the polymer and Nt is the number of moles of species i with the molecular weight Mt. The viscosity average molecular weight for a near-monodisperse polymer is virtually identical to M w . As shown in Table 1-1, the ratios of R H R' I I I

H R H R' I I I I C - C = C-C-LI I H H

R"Li

H R I I C-C I I H C = Cv I H

R"Li modifier

H R H R' I I I I

c-c=c-c

I H

I H

H R I I C-C I I / H C = CV I H

these various moments serve as the required pedigree insofar as the length heterogeneity characteristics of a polymer are concerned. Another characteristic of the organolithium-based polymerization systems in hydrocarbon solvents is the control possible in the microstructure of the polydienes (see Scheme 1). The modifiers that can be used to bring about the 1,2 mode of addition are ethers (ROR'), and amines (NR3), or diamines (R 2 N-CH 2 CH 2 -NR 2 ). The saturation of these polydienes leads to polyolefin structures which retain the narrow molecular weight distribution of the parent material. Deuterium can be used in place of hydrogen, if labelling for neutron scattering studies is needed. Some of the polyolefins prepared in this fashion are shown in Scheme 2. The retention of chain-end reactivity has allowed the preparation of model starshaped materials. The most reliable and controllable route involves chlorosilanes, i.e. 4PLi + SiCl4 -> P4Si + 4LiCl, where P denotes the linear polymer. This approach has allowed stars with up to 128 arms to be prepared (Roovers, 1992). Through the se-

1,4 structure

1,2 structure

1,4 and 1,2 structures

Scheme 1

1.4 Model Polymer Requirements

H H I I

H \

H /

c=c-c=c / \

H CH, H H H I I C=C-C=C-CH, / H

Polyethylene Random (ethylene/ butene-1) copolymer

H

H \

H CH3 H I I ' /

c=c-c=c / \ H H

\

H H C9H, H

i r /

c=c-c=c

H

PLi + CH 3 OH -> PH + CH 3 OLi Atactic polypropylene

Alternating (ethylene/ propylene) copolymer

Alternating (ethylene/ butene-1) copolymer

H

Scheme 2 quential polymerization of different monomers, di- or triblock co- or terpolymer stars can be prepared. However, the conditions under which these polymerizations are conducted require attention. Generally an acceptable temperature range is 0° to 50°C. Higher temperatures will lead to active-center deactivation via, for example, lithium hydride elimination or metallation events (Antkoviak, 1971; Kern et al, 1972; Anderson etal, 1972; Roovers and By water, 1973; Neutwig and Sinn, 1980; Broske etal., 1987; Pennessi and Fetters, 1988). Solvents which can cause chain-transfer are also to be avoided. Toluene is often cited as the polymerization solvent. This is an unsuitable choice, since the following occurs (Gatzke, 1969): PLi

nol, will lead to chain coupling (Fetters and Firer, 1977). Ideally, deactivation of the active centers occurs as follows:

LiCH 2 -

Such reactions clearly will distort the molecular weight distribution. Care must also be exercised in the termination of these active centers, since trace amounts of oxygen in, for example, metha-

The presence of oxygen can cause the following side reaction:

The transition state of the chain-ends involves free-radicals which readily couple to yield "doublet" material, with the subsequent distortion of the molecular weight distribution. This dimerization event is also promoted by the fact that the polymers themselves are associated as dimers in hydrocarbon solvents as a consequence of electron deficient bonding present in organolithiums.

1.4 Model Polymer Requirements The high degree of control possible with anionic polymerization can be replicated only partially by other addition polymerization mechanisms. These include cationic (limiting in the main to isobutylene), radical Ziegler-Natta, ring-opening metathesis polymerizations (ROMP) and grouptransfer polymerization. Many of these latter systems have been described as pseudoliving, immortal, quasi-living, livingness enhancement and truly living. As Quirk and Lee (1992) have pointed out, the term "living polymerization" is to be used only for these systems which proceed with absence of the termination or chain transfer steps. The following describes these processes and presents the procedures by which the resultant macromolecules should be characterized in order that they be classified as model materials. The lack of a spontaneous termination event in anionic polymerization systems

8

1 Model Polymers for Materials Science

Table 1-2 a. Effect of polymerization temperature on polydiene chain structure (Young et al., 1984). (1) Polyisoprenea Temp. (°C)

[s-BuLi] 103

46 20 0 25 40 25

Microstructure b

Solvent

0.01 0.01 0.01 0.01 1 1

None None None None Hexane Hexane

[s-BuLi] 103

Solvent

trans-1,4

cis-1,4

3,4

1 1 2 3 18 18

95 95 95 93 76 78

4 4 3 4 6 4

1,2

(2) Polybutadiene3 Temp.

35 20 0

0.007 0.007 0.007

Microstructure b trans-1,4

cis-1,4

9 9 7

85 86 86

None None None

3,4

1,2 6 5 5

Monomer = 2.5 M; b via 300 MHz ^ - N M R .

was recognized in the mid-1930s by Ziegler and his co-workers (Ziegler, 1936; Ziegler and Jacob, 1934; Ziegler et al., 1934, 1936) and by Abkin and Medvedeev (1936). Flory predicted in 1940 that such systems would possess the Poisson molecular weight distribution (Flory, 1940). Approximately twenty years later such polymers were prepared in the case of poly(amethyl-styrene) (Sirianni et al., 1959) and polystyrene (Wenger, 1960). Both polymerization systems utilized tetrahydrofuran as the polymerization solvent at dry-ice temperature, and respectively sodium naphthalene and sodium biphenyl as the initiators. Ethereal solvents and the sodium counter-ion are, though, of limited utility in the preparation of model polymers. The polymerization of styrene is extremely fast requiring attention to heat transfer and

mixing problems - and the use of sodium prohibits the attainment of the high 1,4 enrichment for polydienes. A more subtle problem is the short-term (hours) instability of initiators like sodium naphthalene (Schulz et al., 1970), a facet of this organometallic that went unrecognized by its advocates. Thus, the initiators of choice are the hydrocarbon soluble organolithiums, one of which, n-butyllithium, was used by Ziegler in his work. Organolithium-based hydrocarbon polymerizations readily yield nearmonodisperse molecular weight distributions (Fig. 1-2), along with polydienes of various microstructures (Tables 1-2 a and b) and active center stabilities that permit, in the absence of side-reactions, near quantitative chain-end functionalizations. The advent of SEC (also called gel permeation chromatography, GPC, see

1.4 Model Polymer Requirements

Table 1-2 b. Microstructure of polydienes prepared in solvating media (Young et al., 1984). (1) Polyisoprene Solvent THF THF THF Ether Ether Ether Ether Dioxan TMEDAb TMEDAC DME Radical polymer

Cation

Temp. (°C)

Li free anion free anion Li Na K Cs Li Li Li Li

30 30 -70 20 20 20 20 15

cis-l,4(%)

trans-l,4(%)

3,4 (%)

1,2 (%)

12 22 10 35 a

59 47 45 52 61 43 32 68 55 45

29 31 31 13 22 19 16 18 15 30

1 ?

a

38a 52a 11 30 25

15 63

25

(2) Polybutadiene Solvent

Cation

THF THF THF Dioxan

Li Li Na Na K Cs

Temp.(°C)

trans-lA{%)

80 15 0 15 15 15

Radical polymer

25

25 13 14 15 45 59 50

1,2 (%) 70 87 80 85 55 41 25

a

Cis- and trans-1,4 not separated but predominance of trans expected; b principal solvent was benzene; base:Li ratio was 60:1; c principal solvent was hexane; base:Li ratio was 1:1.

X
A pp

ID _C

\

PEE

CD

•4=

I

j 1

16

v

J II

18

20 22 16 18 Elution volume (ml)

1

1

20

1

1

22

Figure 1-2. SEC chromatographs of atactic-polypropylene and atactic poly(ethylethylene). The former is from the hydrogenation of poly(2-methyl-l,3-pentadiene) and the latter from the hydrogenation of poly(l,2-butadiene).

Vol. 2 A, Chap. 11) allows detailed and rapid insight into molecular weight distributions. However, a cautionary note is in order regarding the oft-cited, and often abused, ratio of Mw/Mn, the so-called polydispersity index, as a hallmark of nearmonodispersity. An elegant experiment by Quirk and Lee (1992) was carried out which involved the programmed methanol-induced termination of polystyrene. The active center concentration was lowered by 5% in a sequential fashion prior to three additions of styrene. The results of that experiment are shown in Fig. 1-3 and Table 1-3. Therein it can be seen that in

10

1 Model Polymers for Materials Science

Table 1-3. Effect of methanol termination (5% per incremental monomer addition) on M n , Mw and Mw/Mn as a function of conversion for seobutyllithium initiated, incremental polymerization of styrene. Sample

% Conversion

Molecular weight (g/mol)

M n (calc) (SEC) (g/mol)

Mn (peak) M n (SEC) (g/mol)

Mw/Mn (SEC)

5.5 xlO 3 13.6 xlO 3 19.4 xlO 3 24.8 xlO 3

5.6 xlO 3 13.5 xlO 3 19.0 xlO 3 25 xlO 3

5.4 xlO 3 13.2 xlO 3 18.0 xlO 3 23.1 x 10 3

1.02 1.03 1.04 1.07

25 58 80 100

A B C D

For the total polymer obtained at the given % conversion.

follows where it is shown that even distinctly bi- and trimodal systems can exhibit Mw/Mn values which appear to suggest a narrow chain length distribution. The calculations in Table 1-4 illustrate that even the relatively small polydispersities of M w / M n = 1.12 and 1.21 disguise the unacceptable character of these materials (see Fig. 1-4). It is not uncommon to find in the literature homopolymers and block copolymers having Mw/Mn ratios of this type described as "monodisperse" (e.g. Puskas etal., 1992). The general equation which describes the relationship between Myv/Mn and the weight fraction, w1? of the lower molecular weight material for a bimodal mixture of monodisperse chains, M n i and M n2 is 35

35 Elution volume (ml)

45

Figure 1-3. SEC chromatographs of polystyrene prepared by incremental monomer addition and deliberate termination - 5% per incremental monomer addition (see Table 1-3). The active center concentration was depleted 5% by methanol addition per incremental monomer addition (see Table 1-3).

spite of a 15% loss in active centers over the duration of the polymerization, a M w / Mn of 1.07 is found, a value which creates the illusion that a "living" system is in play. The possibly misleading nature of the Mw/Mn ratios can be further amplified as

— wr +

(1-6) where n denotes the ratio MnJMni with Mn2>Mni. Thus, for a mixture of two monodisperse polymers where the MnJMni ratio equals 2, the maximum value of Mv,/Mn is 1.125 (Table 1-5) when equal weight fractions of the two components are present. For the case of n equal to three the Mw/Mn ratio is 1.33 for wx of 0.5.

1.4 Model Polymer Requirements

11

Table 1-4. Sample calculations: equal weights of monodisperse polymers. Sample 1: Calculated:

Sample 2:

Calculated:

0.5 wt. fraction M n = 50 x 103 g/mol; 0.5 wt. fraction Mn = 100 x 103 g/mol; Mn = 67 x 103 g/mol; Mw= 75 x 103 g/mol; M w /M n = 1.125 0.33 wt. fraction M n = 50 x 103 g/mol; 0.33 wt. fraction Mn = 100 x 103 g/mol; 0.33 wt. fraction M n = 150 x 103 g/mol Mn = 81.8 xlO 3 g/mol; Mw = 100xl0 3 g/mol; M w /M n = 1.22

Table 1-5. Polydispersity indices for bimodal polymer mixtures. Singlet wt.% 100 95 90 80 70 60 50 40 30 20 10 5 0

Mn -

Doublet wt.% 0 5 10 20 30 40 50 60 70 80 90 95 100

MJMW

Mw/Mn

1.000 1.043 1.074 1.111 1.124 1.122 1.111 1.094 1.072 1.049 1.025 1.012 1.000

1.000 1.000 1.045 1.080 1.105 1.120 1.125 1.120 1.105 1.080 1.045 1.000 1.000

" *; M w = S (w; M^; Mz = X(wt- M?)/

The limitations inherent in the use of the Mw/Mn ratio as a barometer of "monodispersity" have been demonstrated in another study by Quirk and Lee (1992). They prepared synthetic mixtures by combining equimolar amounts of two and three nearmonodisperse polystyrenes as described in Table 1-6. The mixtures were then analyzed by SEC with the resulting chromatographs shown in Fig. 1-3. It is obvious that samples exhibiting the relatively narrow distributions of 1.11 and 1.15 cannot be

I 35

I 40 Elution volume (ml)

Figure 1-4 SEC chromatographs of the polymer mixtures of Table 1-4: dotted curve, sample 1; solid curve, sample 2.

described as monodisperse or even as nearmonodisperse. For these reasons, it is important to rely only on SEC data obtained from a high resolution column set. Examples of SEC data must be presented in view of the potentially deceptive nature of the Mw/Mn ratios as shown in Table 1-6. The term narrow molecular weight distribution must be limited to polymers with M z + 1 / M z , MJMW and Mw/Mn ratios of <1.05. As noted earlier, a high resolution SEC column arrangement must be in place if meaningful distribution profiles are to be obtained. This is particularly true for the case where chain coupling, linking or branching can occur either during polymerization or via a post-polymerization event. An example of the latter is the presence of oxygen during the termination event, the presence of which causes chain

12

1 Model Polymers for Materials Science

Table 1-6. Samples for SEC analysis of mixtures of monodisperse polymers. Sample

M n of component polymers (10 ~3 gmoP 1 )

Mixture molecular weight data (g mol" 1 )

Weight fraction

Mn I II

19.0 34.5 19.0 34.5 48.0

0.5 0.5 0.33 0.33 0.33

coupling (Fetters and Firer, 1977). As has been shown, inadequate resolution will miss the presence of such coupled products (Ambler et al., 1977). A similar situation can be encountered in functionalization reactions. For example, the carboxylation of poly(styryllithium) in benzene by high purity CO 2 at atmospheric pressure produces the carboxylated polymer in only 46-66% yield; the contaminating products include the dimeric ketone (25-30%) and the trimeric alcohol (7-24%). This is shown in Eq. (1-7), where P represents the polystyrene chain (Quirk and Chen, 1992; Quirk et al., 1992). P2CO(A) + P3COH(B)

(1-7)

Procedures which virtually eliminate the ketone (A) and alcohol (B) components have been developed and presented (Quirk etal., 1989, 1992; Quirk and Yin, 1992). The presence of the trimeric alcohol and the dimeric ketone is shown in Fig. 1-5, with the former component unresolved by the /i-Styragel SEC column arrangement. The higher plate-count ultra-Styragel column set reveals the presence of that branched component. The foregoing also applies to AB, ABA or ABC block copolymers. Accurate in-

23.48

25.94

28.72

1.11

29.78

34.29

38.04

1.15

sight into the efficiency and success of each step in the preparation of these multi-component systems can, and must, be evaluated by high resolution SEC. This is shown in Fig. 1-6, where a poly(styrene-isoprene) diblock (Quirk and Lee, 1992) and the polystyrene segment (polymerized first) are shown. The lack of a signal for homopolystyrene in the SEC of the diblock clearly demonstrates that the resulting diblock is free of the homopolystyrene contaminant. A final example of the fallacy of relying only upon an Mw/Mn ratio as an indicator of monodispersity is given for the polyisobutylene sample of Fig. 1-7. This polymer, prepared by "living" polymerization, has bimodal character coupled with a high molecular weight component. The measured Mw/Mn value of 1.11 fails completely in so far as being a quantitative description of distribution profile of that particular material. The preparation of a model polymer must emanate from a polymerization mechanism which exhibits the following characteristics: a) Molecular weight is controlled by reaction stoichiometry: DPn = [M] o /[/] o for the case of a monofunctional initiator such as butyllithium.

13

1.4 Model Polymer Requirements

35 40

42

44

46

48

50

52

40

45

50

Elution volume (ml)

54

Figure 1-6. SEC chromatographs of a poly(styreneisoprene) diblock ( ) and the polystyrene segment (---) (Quirk and Lee, 1992).

Elution volume (ml)

v (gmol"

4x10 5

2x10 5

1

)

105

5x104

i

i

i

I

i

I

50

52

54

56

58

£b 42

44

46

48

50

i

h

PIB

40

2x104

i

J

Elution volume (ml)

Elution volume (ml)

Figure 1-5. SEC chromatographs of tri-modal polystyrene evaluated by a //-Styragel set and an ultraStyragel set. The latter column arrangement reveals the presence of the three-armed star (Quirk et al., 1989).

Figure 1-7. SEC chromatograph of polyisobutylene standard prepared by a "living" polymerization.

14

1 Model Polymers for Materials Science

b) Active center concentration remains unchanged throughout the polymerization (no spontaneous termination events). c) Chain transfer events are absent. d) Polymerization proceeds until monomer is consumed. e) The number-average molecular weight is a linear function of conversion. f) Sequential monomer additions lead to compositionally homogeneous block copolymers. g) Reactions are done under conditions that do not vitiate points (a)-(f). h) Narrow molecular weight distribution polymers are produced. An example of where conditions can vitiate points (a)-(f) is the use of high polymerization temperatures (>50°C) in the anionic polymerizations of dienes and styrenes in hydrocarbon solvents. The use of such temperatures will lead to lithium

75°C/3 h

hydride elimination, metallation, chain branching and active center isomerization (Antoviak, 1971; Kern et al., 1972; Anderson et al., 1972; Roovers and By water, 1973; Neutwig and Sinn, 1980; Broske etal., 1987; Pennisi and Fetters, 1988). Figure 1-8 serves as an example. Therein, SEC chromatographs of a polybutadiene (Pennisi and Fetters, 1988), polymerized at 20 °C, are shown as a function of time after exposure to 75 °C. The onset of branched material is obvious. Other side reactions that can be encountered have been reviewed by Young et al. (1984). Clearly, the mere comment, which is often encountered, that anionic polymerization was used to prepare the polymer in question is an inadequate pedigree. Reaction conditions must be described and their limitations understood. The companion information needed, as has been noted, is proper characterization. An example of what that exercise will reveals is as follows.

10wt.% Linked polymer

0 •a

1 33 wt.% Linked polymer

i

75°C/24 h

a

Figure 1-8. SEC chromatographs of poly(butadienyllithium) as a function of time at 75 °C (Pennisi and Fetters, 1988). 40 SEC counts

15

1.4 Model Polymer Requirements

44

48

52 36

40

44

46

Figure 1-9. SEC chromatographs of two samples of "star" polystyrene (Mays et al., 1988). The large peaks at high elution volume correspond to unlinked arms.

52

SEC counts

The SEC traces shown in Fig. 1-9 are those of polystyrene "star" polymers which were purported to contain less than 5 wt.% unlinked arm (the material exhibiting the longest retention time in the colums). The elementary SEC characterization exercise revealed a markedly different situation: approximately 70% of the total material was unlinked arm! (Mays et al., 1988.) In consonance with proper SEC protocol is the capacity to evaluate molecular weight in an absolute sense via osmometry and light scattering. Of these two methods, the latter is generally more useful - particularly if coupled with SEC evaluations. The importance of absolute molecular weight measurements is highlighted by the following elementary example. Figure 1-10 presents data for the intrinsic viscosities of polystyrene in toluene (measurement temperature not specified). The molecular weights for those samples were taken as

principle, the procedure used would work if the previously listed criteria for model polymer preparation held for the system in question. The following rudimentary evaluation reveals that this is not the case. The solid line denotes the intrinsic viscositymolecular weight behavior of properly characterized polystyrene (Fetters et al., 1992) (where M denotes Mw) of nearmonodisperse molecular weight distributions given as L

(1-8)

y 1.0

o.y

/

.•6

•

oo§-

/

°/

x 104, where M is the

monomer concentration in mol/L and / is the concentration of initiator in mol/L; in other words, the stoichiometric molecular weights were used to describe these polystyrenes prepared in tetrahydrofuran with sodium naphthalene as the initiator. In

0.1

i

10 4

10 5 M

106

Figure 1-10. Intrinsic viscosities of polystyrene as a function of molecular weight. The filled symbol denotes a data point eliminated, without explanation. (From Fig. 1 of Waack et al., 1957).

16

1 Model Polymers for Materials Science

This equation, for a given [rj\9 yields molecular weights larger by more than a factor of two than the one based on the data (Waack etal., 1957) of Fig. 1-10: M =1.6 x 1 0 - 4 M 0 7 4 d L g - 1

(1-9)

Obviously, the erroneous nature of Eq. (1-9) is the result of the failure of the polymerization system to perform as expected, i.e. considerable termination existed, and the failure to characterize the sample in absolute terms. In short, the announcement that anionic polymerization was the synthetic tool does not supersede or replace the necessity for thorough polymer characterization. Unfortunately, encounters of this nature are still possible (e.g. Tsitsilianes and Staikos, 1992).

1.5 Functionalization of Model Polymers In recent years there has been a growing interest in the preparation of well-defined polymers with terminal functional groups.

The primary motivation for this interest in functional groups is their potential for the following applications: - chain extension, branching or crosslinking reactions; - reactions with fundamental groups on other polymer chains; - co-polymerizations with other monomers to form graft copolymers (the macromer concept); - the initiation of polymerization of other monomers; - reversible ionic-association; - endow polymers with surfactant-like properties. Some of the feasible structures are schematically given in Table 1 -7 where X indicates the location of the functional group. As is the case with the model polymers the most well-defined and exploited route to these materials to date has been lithium based anionic polymerizations. Some examples for the functional groups are given in Table 1-8. The foregoing also serves to summarize some of the non-functionalized structures that have been prepared via anionic polymerizations.

Table 1-7. Molecular architecture for functionalized model polymers (Quirk, 1992).

Y Multifunctional

Chain-end

XX X X Random copolymer

Block copolymer (Chain-end)

X X

X X Star polymer (Chain-end)

Heteroarmstar polymer (Chain-end)

X Block copolymer (Block interface)

ex Macrocyclic polymer

1.5 Functionalization of Model Polymers

Table 1-8. Polymer functional groups via anionic reactions (Quirk, 1992). Reactant

Group

CH 9 -CH,

-OH

CH,-CH2 \ / S

-SH

\ / O

Side reactions

O

CO,

so,

-COH

X

0 II -SOH

X

0 0 II

S-OH

x

X

ii

II

0 I

-COOH I -Br

Br2 CH3 N

NCH2CH2CH2Li

/

CH 3 CH3

CH3 C2H5O I CH 3 -C-O(CH 2 ) 5 CH 2 Li

-OH

H

Unfortunately, many of the functionalization reactions which have been described are not well-characterized, as discussed in a recent detailed review by Quirk (1992). Precedent from the reactions of model small molecule analogs is often misleading. Thus, adequate and necessary characterization probes embrace: - chain molecular weight; - functional group titration;

17

- NMR, UV-visible and FTIR spectroscopy; - thin layer and column chromatography; - chemical reactions. The primary need in chain-end characterization is an accurate evaluation of functionalization efficiencies. To this end, knowledge of functional group titration, spectroscopic probes (Vol. 2B, Chap. 11) and chemical reactions can all contribute to determining the efficiency of a given reaction. Chromatography has proven useful (Quirk and Chen, 1992; Quirk et al., 1989, 1992; Quirk and Yin, 1992) for analyzing unfunctionalized polymer mixed with the functionalized product as a consequence of the different migration rates that the two species exhibit. Chain molecular weight, when combined with items b), c) and e), allows a quantitative assessment of the functionalization reaction in question. One of the most useful and widely used functionalization reactions has been the carboxylation of polymeric anions using carbon dioxide. Although intrinsically a simple reaction, a complex mixture of products involving carboxylated polymer, the dimeric ketone and trimeric alcohol are, as noted previously, often obtained. Initially, it was found that the coupling and linking side reactions could be virtually eliminated in systems where the polystyryllithium was complexed with a base such as tetrahydrofuran or tetramethylethylendiamine (Quirk and Chen, 1992; Quirk et al., 1989). An alternative route involved the solid state (achieved by freeze drying the polystyryllithium) exposure of the active centers to CO 2 at atmospheric pressure. In both cases the yield of carboxylated polystyrene was >99% (Quirk etal, 1989, 1992; Quirk and Yin, 1992). The use of 13 CO 2 revealed that approximately 15% of the -COOH groups were present on the aromatic ring. In other

18

1 Model Polymers for Materials Science

PS-CH 2 -CHLi + -^

•

PS-CH2-CHLi+ -<

•

PS-CH2-CHLi+

co2

PS-CH2-CH

PS-CH 2 -CH-CO 2

PS-CH2-CH CO2"

Scheme 3

H

H CO," 1) acid or base 2) H 3 O +

H3O+

PS-CH2-CH-CO2H

6

*

PS-CH2-CH2

words, although carboxylation efficiency was 99% or greater, nonetheless two types of carboxylic acid groups were present. The reaction scheme in question is displayed in Scheme 3. Ring carboxylation was nearly eliminated for the reaction scheme which involved addition of CO 2 to the modifier complexed polystyryllithium, in benzene solution. The foregoing serves to highlight the necessity of a thorough and exhaustive characterization of functionalized polymers in order that confidence is established in the efficacy of the chemistry used. Unfortunately, the functionalization literature is "enriched" with glib assumptions coupled with inadequate, if not nonexistent, characterization (e.g. Park et al., 1991). The review of Quirk (1992) is recommended as a thorough briefing on this topic.

1) acid or base 2) H 3 O +

PS-CH?-CH?

1.6 Model Branched Polymers The preparation of model star polymers is best approached via the application of chlorosilanes as the linking agents and the lithium counter-ion in place on the terminal active center. This approach was first used by Morton and co-workers in 1962 for the preparation of 3- and 4-armed polystyrene stars (Morton et al., 1962). Since that time controlled functionalities as high as 128 have been obtained (Roovers, 1992). Figure 1-11 shows an SEC trace of a 64-armed polybutadiene star having an arm Mw of 6 x 103. The 64-functional chlorosilane linking agent is a multiply branched "dendrimer" type molecule. The chemistry in question has allowed the sequential build-up of star structures containing asymmetric homopolymeric materials (Pennisi and Fetters, 1988), model single segment graft multi-component A2B polymers (Mays, 1992) and the

1.7 Processing

19

F r o m the morphological viewpoint, such materials should exhibit interesting m o r phological characteristics (Mays, 1992). The work of Iatraou a n d Hadjichristidis (1992) is a demonstration of how a model material can be prepared a n d properly characterized.

1.7 Processing

35

40 Elution volume (ml)

45

Figure 1-11. SEC trace of 64-armed polybutadiene star (Roovers, 1992).

recent mikta-armed, ABC, stars (Iatraou and Hadjichristidis, 1992) where A = polyisoprene, B = polystyrene, and C = polybutadiene. These materials rely on the reaction sequences presented in Scheme 4. This approach permits the individual arms to be independently characterized along with the resultant mikta-armed star. The combination of SEC, NMR spectroscopy, refractive index measurements and light scattering provided the pedigree which unequivocally demonstrated that the desired ABC structure was obtained. CH3 I a) PILi + CH3SiCl3 (excess) -> PISiCl2+CCH3SiCl3 (I) Cl I b) I+PSLi-*PISi(CH 3 )PS(II)+LiCl I CH3 (II) PBd I c) II + PBdLi -» PISiPS + LiCl

CH3

Scheme 4

A cornerstone of materials science is processing of materials. Indeed, Vol. 18 of this Series is entirely devoted to polymer processing. It has become increasingly realized that simultaneously combining the steps of synthesis and processing can sometimes lead to better control of microstructure. This combined approach can provide new materials of outstanding performance. Recent nonpolymer examples are chemical vapor deposition and molecular beam epitaxy, wherein precise chemical and structural control of the material is accomplished at the atomic level. In simple "sphere-pack" materials, crystallization proceeds rapidly and organizes the assembly of members at the level of the crystal grain (~one micron). Metallurgists, long ago, developed extensive methods to manipulate sample microstructure via timetemperature-transformation studies with and without the presence of applied fields (mechanical, electrical, etc.). Rational control of microstructure in polymers is a challenge, since one needs to control packing at several length scales simultaneously and because, of their inherent conformational diversity, each molecule has an enormous number of structural states of similar energy to choose from. As indicated, stereochemical and monomer sequence control during species synthesis is the sine qua non for molecular level control of structure and function.

20

1 Model Polymers for Materials Science

The realization of desirable superstructures arising from the self assembly of individual species far more complex than simple spheres requires intermolecular assembly to proceed in a uniform manner in order to allow the carefully built-in specificity of chain structure and the applied environmental conditions to produce the prescribed, desired structural state. With polymers, one discovers even with the most simple macromolecules a wide array of structural states and length scales, e.g. for semicrystalline linear polyethylene, the orthorhombic crystalline unit cell has edge lengths about 1 nm and the all trans chains pack into lamellae typically 20 nm thick. The lamellae are separated by noncrystalline regions wherein a combination of trans and gauche states results in a random chain structure. Moreover, crystallization proceeds from nucleation centers to develop spherulites of average diameter of several microns. Inside a spherulite, the lamellae are radially arranged (see Chap. 3). Such a complex microstructural geometry involving both crystalline and noncrystalline phases makes establishment of structure-property relations difficult; furthermore, the important and practical evolution of the properties with applied force or imposed strain is challenging both to characterize and to model. For this reason, simplified geometries and uniform structures have been produced to enable exploration of fundamental structureproperty relations. Materials scientists have discovered various post-synthesis ways to influence and control the microstructure - from the earliest days of forging, rolling and annealing heat treatments to the use of thermal gradients and single crystal seeds. Kinetic control of microstructure is readily exploited in polymeric systems as in other materials. Spinodal decomposition and nucleation

and growth of phase domains is a major field of investigation (see Chaps. 3 and 7 in Vol. 5 and Chap. 6 in this Vol.). Polymer chemists have had good success in systematically manipulating polymers for liquid crystal (LC) applications. The simple joining-together of mesogens by short flexible spacer units into a long chain results in both nematic and smectic LC materials (see Chap. 5). Liquid crystalline materials can be readily oriented into monodomains via mechanical, electric or magnetic fields. Precise control of the transition temperature from the crystalline into the LC state and from the LC state into the isotropic can be made by variation of the type, length and degree of flexibility of the spacer unit, as well as the attachment of pendant groups from the chain backbone. Side chain LCPs (liquid crystalline polymers) are readily made by the attachment of mesogens to the main chain again by spacer units. These side chain LC materials have the advantage of greatly decreased response times of the mesogens to applied fields and are of increasing interest for optical applications, wherein the polymeric nature of the LC material is exploited for processing and mechanical properties of the final product. It is very easy to produce a polymer glass (see Chap. 2). The atactic distribution of chemical substituents along long, entangled chains in a high viscosity melt, provides even under relatively slow cooling conditions, access to the glassy solid state. Moreover, highly anisotropic glasses are possible. The polymer can be solidified during processing such that the stretch oriented conformations of the molecules are frozen-in by rapid cooling. It is also possible to produce room temperature glassy liquid crystalline materials by quenching a mesogenic polymer with a high glass transition from its high temperature LC state.

1.7 Processing

Self-assembly of block copolymers under an applied bias field can result in materials with single-crystal-like texture. Block copolymers segregate into microdomains composed of the respective blocks. Typical domain sizes are several hundred angstroms and these domains in turn assemble into grains with typical diameters of several microns. The grains of the cylindrical or lamellar microdomain-forming block copolymer have a preferred orientation resulting in local anisotropy of the material properties. However, since the grains are relatively small in comparison to the whole sample and since they are normally in random mutual orientations, the resultant macroscopic sample exhibits isotropic properties. When such materials are examined by scattering or subjected to mechanical testing, the net response is the superposition of the different responses to each of the various grain orientations. Fundamental understanding of structureproperty relationship is thus difficult to achieve. To overcome such limitations, one can employ special processing to convert the isotropic multigranular sample into a single crystal. Such an approach was first taken by Keller and coworkers (Keller etal, 1970; Folkes et al, 1973), utilizing extrusion to produce highly oriented single crystal-like material of a triblock copolymer. This process is akin to the texturing of polycrystalline metals by die drawing. While providing a highly oriented sample, this process suffers from the presence of grain boundary structures and other morphological defects due to the alignment of an already microphase separated material. The influence of such defects and boundaries is unknown at present, but in analogy to work with other materials, one can expect a strong influence on properties. Recently, Albalak and Thomas have devised a roll-casting process which elimi-

21

nates grain boundaries by causing the self assembly of the block copolymer molecules into the microdomains to take place in an orienting flow field (Albalak and Thomas, 1993). The roll-casting process is based on the rolling of an initially homogeneous polymer solution between two counter-rotating cylinders while removing solvent at a controlled rate. The experimental system is schematically pictured in Fig. 1-12 and consists of two adjacent rolls and is similar to conventional set-up of calendars or rollmills. The polymer solution flows between the rolls through a minimum gap, called the "nip" and coats either one or both of the rolls. The detailed velocity profile in such a geometry has been analyzed (Gaskell, 1950) and contains both shear and elongational flow components. The key to this process is that the onset of microphase separation occurs in the applied flow field so that the microdomains form in a globally oriented structure. Figure 1-13 a is a TEM micrograph of a [0001] axial view showing the well-ordered microdomain structure of a PS/PB/PS triblock copolymer containing PS cylinders in the PB matrix. The long-range hexagonal registry of solution

Figure 1-12. A schematic representation of the rollcasting apparatus. A concentrated solution of block copolymer is processed between two adjacent counter-rotating cylinders (Albalak and Thomas, 1993).

22

1 Model Polymers for Materials Science

phology and phase transformation theories and to further the basic understanding of chain dimensions in block copolymer systems (see next section).

1.8 Model Polymers in Polymer Physics

(b)

Figure 1-13. (a) Bright field TEM micrograph of an OsO 4 stained, microtomed section of a roll-cast PS/ PB/PS film. View is along flow direction and shows the two-dimensional hexagonal lattice packing of the PS cylinders, (b) Two-dimensional small angle X-ray scattering pattern of roll-cast PS/PB/PS film with incident beam along the flow direction. Bragg reflections of a well-ordered two-dimensional hexagonal structure are evident (Albalak and Thomas, 1993).

the PS cylinders is quite evident in this projection. Low magnification TEM images show that the single crystal structure is unperturbed for at least many tens of microns. The degree of order over larger distances can also be assessed by small angle X-ray scattering (SAXS). Figure 1-13 b is a two-dimensional SAXS pattern taken with the incident beam along the [0001] direction. The pattern originates from a sample area on the order of 1 mm 2 demonstrating the near single crystal nature of the macroscopic film. Such specimens offer significant advantages in testing phase mor-

There are many examples of the use of model polymers to develop novel materials structure, properties and to test theoretical concepts in polymer physics. We first detail the employment of model polymers in testing the elegant reptation hypothesis of de Gennes, Doi and Edwards and the tube concept in reptation. According to reptation theory, individual chains are assumed to be constrained by their neighbors and are thus compelled to migrate primarily along their own contours in a snake-like manner (see Chap. 9). Thus, a polymer melt is made up of an assembly of chains all simultaneously involved in disengagement from their current cage of strands. In this disentanglement process, each chain creates a new cage and configuration as it undergoes its movement through the mesh. The cage can be visualized as a tube of diameter d and a contour length /, which encloses each chain in the mesh. The chains are assumed Gaussian, with v chains per unit volume. The chain dimensions are expressible as (R2}0/M where
2\ >0

M

n •]["

NJ

(1-10)

23

1.8 Model Polymers in Polymer Physics

where G£ is the plateau modulus (the short time modulus of an entangled polymer liquid). The above equation can be reduced to df = 0.8

r<*2>oi [Me] L M \

(1-11)

where M e denotes the entanglement molecular weight. Thus, a combination of the static [0/M] and the dynamical based [Me] macroscopic parameters allows an evaluation of dt. Until recently the direct measurement of dt had not been accomplished. For that purpose near-monodisperse, amorphous poly(ethylene-propylene) (PEP) was used (Richter et al., 1991,1992; Butera et al., 1991). The route to that polymer was via the hydrogenation saturation of anionically prepared polyisoprene. The analytical tool of choice was the IN-11 neutron spin echo (NSE) spectrometer at the Institut Laue-Langevin in Grenoble. The unique feature of the NSE technique is its ability to evaluate in a direct way the energy changes induced in neutrons during the scattering process. In essence, the dynamical data are gathered by examining the decay in an NSE as a function of an imposed precessional frequency. Therein, a magnetic field of varying strength is applied to the incident polarized neutron beam. The small energy change of each scattered neutron is analyzed as a shift in the processional phase. In view of the limited time resolution (ca. 4 x 10" 8 s) and distance resolution (5 to 100 A) of the IN-11 instrument, only the most dynamically flexible chains with rapid local motion are candidates. PEP and polyethylene (from the saturation of 1,4-polybutadiene) fulfill these criteria. These results of such an NSE investigation on PEP are summarized in Fig. 1-14, which also contains rheological and SANS findings. The tube diameter (NSE) was

300

400

500

600

T[K]

Figure 1-14. Entanglement distance dt, radius of gyration R2G, ratio of G^ax and d? G'J{Q TCJ for alternating poly(ethylene-propylene) as a function of temperature.

found to increase with temperature, while the radius of gyration (SANS) and plateau modulus (rheology) were observed to decrease. The latter in turn signaled an increase in the entanglement molecular weight, which, when combined with the change in chain dimensions, led to an increase in the tube diameter with increasing temperature. It is important to note that dt (NSE) and dt (SANS, G£) were in essential agreement, a picture also found for polyethylene (Richter et al., 1992). These results fortify the elegant Ansatz of de Gennes' reptation model.

24

1 Model Polymers for Materials Science

Flory (1953) predicted that chain dimensions in a 9 solvent where the effect of excluded volume is just balanced by the polymer segment-solvent interaction are identical to those in a polymer melt and that the chain behaves like an ideal random walk. The use of model polymers has also been of great value in the detailed evaluation of rotational isomeric state (RIS) predictions pertaining to the polymer chain dimensions and their respective temperature coefficients (see Chap. 2). The advent of SANS has expanded the range of these evaluations, since both parameters can be measured in the melt state. Some recent findings are presented in Table 1-9. The combination of theta solvent, RIS and SANS results shows excellent agreement with regard to the parameters in question for PE and PEP. However, such agreement is absent for a-PP and a-PEE. It can be seen that neither theory (RIS) nor theta solvent measurements yield chain dimension temperature coefficients that may be related to what is observed in the melt state. From the RIS standpoint there seemingly is a problem in correctly visualizing the role of non-bonded interactions as a function of temperature. The failure of the

RIS approach is duplicated in the failure of the 0-condition results to mimic melt behavior. Thus, the assumption that modeling a polymer in a vacuum (RIS) universally yields its behavior under 6 conditions, which in turn serves as a mimic of melt behavior, is erroneous. Solvent influences on "unperturbed" chain dimensions have been previously predicted (Lifson and Oppenheim, 1960). Our current understanding lacks predictive capacities regarding which solvents can exert so-called specific solvent effects, or, conversely, which polymers are susceptible to such influences. Nonetheless, a partial rationalization as to why solution theta condition is not equivalent to melt behavior is possible. Rotational isomeric states are identified with the minima of the conformational energy maps, minima which can be influenced by local-polymer solvent interactions - the so called "specific solvent effect". This effect can be viewed as one that does not create new rotational isomeric states, but modifies the potential energy of those already existing in the melt. Thus, the location of the various minima remain unchanged while their magnitudes can be altered by the prevailing solvent-

Table 1-9. Polyolefin chain parameters at 298 K. PE = polyethylene; PEB-2 = poly(ethylene-butene) with ca. 2 ethyl branches per 100 backbone carbons. PEP = alternating poly(ethylene-propylene); a-PP = atactic polypropylene with Bernoullian chain statistics; a-PEE = atactic poly (ethyl ethylene) with Bernoullian chain statistics; q = solution intrinsic viscosity or light scattering; SANS = small angle neutron scattering of melt; RIS = rotational isomeric state (theory). Polymer

RG/M

(A mol

PE (PEB-2) PEP

a-PP a-PEE a

0.470 0.381 0.340 0.279

1/2

1/2

g-

aa

C 1/2

(A)

d(ln<£ 2 > 0 )/dT (lO^C- 1 )

)

SANS

*

SANS

0.485 0.397 0.337 0.283

7.9 6.5 6.6 5.5

8.4 7.1 6.1 5.7

Statistical segment length: a2 = 6 Rl/Nw.

SANS 6.1 5.6 5.6 5.1

6.3 5.8 5.3 5.2

-1.1 -1.1 -2.9 -2.3

SANS

RIS

-1.15 -1.16 -0.1

-1.2 -1.0 0 to -1.8 -0.1

0.4

1.8 Model Polymers in Polymer Physics

polymer interaction for any particular state. This scenario requires that the probabilities of occurrence of the most stable conformations - those isomeric states required by the short range intramolecular interactions - can be modified by solvent interaction. Under theta conditions such shortrange interactions can be neglected if solvent discrimination between conformations is absent. This criteria is apparently met for polyethylene and alternatingpoly (ethylene-propylene) copolymer (Table 1-9). Conversely, the difference between the solution and melt behavior for a-PP and a-PEE seemingly demonstrates that their local conformational characteristics are considerably influenced by theta-solvent (n-alkyl alcohols) interactions, interactions which lead to enhanced gauche populations with increasing temperature (and hence a negative temperature coefficient for chain dimensions). It appears that the configuration of the skeletal bonds will lead in some cases to solvent discrimination regarding the attached side groups with gauche sequences being more susceptible to such interactions than their trans counterparts. Given the presence of such favorable interactions, the net effect is a decrease in the trans /gauche ratio with increasing temperature. Similar behavior is evidenced for a-PP in either 2,4-dimethyl heptane or 2,2,4-trimethyl-pentane which gave a d(ln (R2}0)/ dT of -2.4 x 1(T 3 °C x over the temperature range of 10 to 70 °C. The use of those alkanes was based on their assumed capacity to act as athermal solvents, an assumption which the SANS work has shown to be incorrect. Interestingly, like the rc-alkyl alcohol theta solvents, these branched alkanes alter the a-PP trans/gauche ratio with increasing temperature. In a parallel fashion, IR work of Casal et al. (1986) on

25

n-tridecane 7,7-d2 shows that the average gauche population increases from 35% to 60% when the solvent is changed from 7i-heptane to hexadecane. The solution induced behavior of a-PP and a-PEE is strikingly different than that observed in the melt where the former is found to exhibit a virtually invariant population of conformer types while the latter shows that increasing temperature favors the trans sequences. Attempts have been made to account for solvent effects within the framework of the RIS model (Bahar et al., 1986; Brant and Flory, 1965; Yoon et al., 1975 a,b). This involved the adoption of a truncation procedure involving an adjustable parameter, s, which Brant and Flory (1965) took to be the sum of the van der Waals radii of the interacting atoms. Variations of this approach have subsequently led to a procedure (Bahar et al., 1986) which does not utilize adjustable parameters. However, although a-PP and a-PEE apparently acquiesce to the influence of their solvent shells, the nature and identity of these local interactions is not generally understood (Janik etal., 1987). The foregoing SANS studies were facilitated by small heterogeneity indices of the subject polymers, e.g. see Fig. 1-1. Thus, the measured RG (z-average dependent dimension) is equal to the weight-average value. In other words, polydispersity corrections are unnecessary. The types of polyolefins that have been thus far prepared via the polydiene route are shown in Table 1-10 along with the parent material. Table 1-11 lists some of their properties. Model diblock copolymers produced by anionic polymerization and processed into single crystal specimens are quite useful to probe how the microstructure influences chain dimensions. SAXS and SANS measurements, as well as theoretical predic-

26

1 Model Polymers for Materials Science

Table 1-10. Polydiene and polyolefin structure and nomenclature. Common or IUPAC name

Structure

Nomenclature

Polypropylene

CH3 I [-CH 2 CH-]

1,4-Poly(dimethylbutadiene)

CH 3 CH 3 CH3 II' I [_CH2C = CCH 2 ] 9 7 [-CH 2 C-] 3

PP

1,4-PDMBD

C = CH2 I CH3 Head-to-head polypropylene

CH3 CH3 II"

CH3 I "

HHPP

CH2CH - CHCH 2 -] 97 [- CH2C -] 3 HCCH3 1,4-Poly(ethylbutadiene)

CH3 CH3

CH2CH3

1,4-PEB

[-CH 2 CH = CHCH 2 -] 90 [-CH 2 CH-] 10 CCH2CH3 II CH? Poly(ethylene-butene)

CH2CH3 I [-CH 2 CH 2 CHCH 2 -] 90 [-CH 2 CH-] 10

PEB-26 3

HCCH2CH3 I CH3 1,4-1,2-Poly(dimethylbutadiene)

CH3 CH3 CH3 I ' I * I [_CH2C = CCH 2 ] 55 [CH 2 C-] 45

55-PDMBD

CCH3 II CH9 Poly (1,2-dimethyl-1 -butylene-co-1 (1 -methyl-1 -isopropyl)ethylene

CH3 CH3 II"

CH3 I

- CH2CH - CHCH 2 -] 55 [- CH2C - ] 4 5 HCCH3 1,4-Polymyrcene

CH3 - C = CHCH 2 CH 2 -] 90 [- CHCH 2 -] 10 CH? I CH2 I CH II C(CH3)2

C = CH2 CH2 CH II C(CH3)2

1,4-PMYRC

27

1.8 Model Polymers in Polymer Physics

Table 1-10. (continued) Common or IUPAC name Poly( 1 -(4-methy lpenty 1)-1 -butylene-co-1 (1,5-dimethylhexyl)ethylene)

Structure [-CHCH 2 CH 2 CH 2 -] 90 [-CHCH 2 -] 10 CH2 I CH2

CHCH3 I CH2

CH?

CH2 I CH2 I CH(CH3)2

CH(CH3)2

Poly (1 -(4-methyl-3-pentenyl-1 -buteny lene-co-1 (l-methylene-5-methyl-4-hexenyl)ethylene)

Poly (1 -(4-methylpentyl)-1 -butylene-co-1 (1,5-dimethylhexyl)ethylene

Nomenclature

[- C = CHCH 2 CH 2 -] 64 [- CHCH 2 -] 36 ' ' Ln 2 Cri? I I CH2 C = CH2 I CH CH2 II C(CH3)2 CH II C(CH3)2 [-CHCH 2 CH 2 CH 2 -] 64 [-CHCH 2 -] 36 I I CH2 CHCH3 I I CH2 CH2 I I CH2 CH2 I CH(CH3)2 CH2

H21,4-PMRYC

64-PMYRC

H.64-PMRYC

CH(CH3)2 Poly(ethylene-cc>-butene)

PEB-14a

[-CH 2 CH 2 -] 36 [-CH 2 CH-] 14 CH2 CH3

Poly(ethylene-co-butene)

PEB-40a

[-CH 2 CH 2 -] 10 [-CH 2 CH 2 -] 40 I CH2 CH3

Poly(vinylethylene)

[-CH 2 CH = CHCH 2 -],_ 2 [-CH 2 CH-] 98 _ 99

PVE

CH II CH2 Poly(ethylethylene)

[-CH 2 CH 2 -],[-CH 2 CH-] 4 9 I CH2 I CH 3

The number in the PEB series denotes the number of ethyl branches per 100 backbone carbons.

PEE

28

1 Model Polymers for Materials Science

Table 1-11. Glass transition temperatures and density parameters for polydienes and polyolefins. Polymer

ppa

HHPP b PEP 1,4-PEB PEB-26 55-PDMBD H255-PDMBD 1,4-PMYRC H21,4-PMYRC 64-PMYRC H264-PMYRC PVE PEE

Glass transition temp. (°C) 4

1

Peak onset

Peak midpoint

Q at 20°C(gcm~ 3 )

- ci(lnrf/dT(10 K- )

-5 -28 -65 -65 -61 14 5 -71 -57 -6 -46 0 -30

-3 -26 -62 -64 -59 16 10 -68 -54 -65 -45 2 -28

0.857 0.878 0.586 0.891 0.861 0.918 0.892 0.895 0.856 0.897 0.853 0.889 0.870

7.9 7.2 7.0

5.8 6.3 6.7 9.2 6.0 7.5 6.0 6.0

a Tg values of — 6°C (Cowie, 1973) and — 5°C (Burfield and Doi, 1983) have been reported for amorphous polypropylene. b A Tg of — 27°C has been reported for HHPP. That value is via extrapolation to the 0 K m i n ' 1 heating rate. Arichi et al. (1979) gave Tg's of -10°C (HHPP-1) and -17°C (HHPP-2).

tions of the domain spacing in A/B amorphous/amorphous block copolymers, indicate that the spacing varies approximately as M 2 / 3 in the strong segregation regime where the individual phases are essentially pure components and the A/B junction is strongly localized in the narrow interfacial region between domains. The individual component block dimensions should therefore be distorted from the Gaussian behavior of a chain in a single phase melt, where RG scales as M 1/2 . Measurement of the chain dimensions in a phase separated block copolymer is, however, complicated by the strong scattering from the periodic domain microstructure, which occurs in the same angular range as the single chain scattering. To circumvent the domain scattering problem, Hadziioannou and coworkers (Hadziioannou et al., 1979,1982) employed a special shearing process to obtain near-single crystallike specimens of deuterated-PS/PI lamel-

lar diblock copolymers. Because the oriented domain scattering is highly anisotropic, the low angle scattering is dominated by the diffraction from the lamellar lattice only in the direction normal to the lamellae, while only the single chain scattering essentially dominates for the directions parallel to the A/B interface (see Fig. 1-15 a). Using Guinier analysis, Hadziioannou et al. (1982) found the block chain to be contracted to between 60% and 70% of the unperturbed component of the radius of gyration along the parallel direction (Kgll). Hasegawa et al. (1985) attempted to directly determine the chain stretching normal to the A/B interface (via Rgl) by employing both SAXS and SANS to separate out the contributions from single chain scattering and domain scattering. Unfortunately, the subtraction technique is not very precise, making the Rg ± value suspect. Subsequently, Hasegawa and coworkers (Hasegawa et al., 1987) used con-

1.9 References single chain scattering

incident beam

domain

29

scattering

incident beam

(a)

(b)

Figure 1-15. Schematic of oriented lamellar diblock copolymers and SANS patterns, (a) The beam is normal to the lamellae and the scattering results from the deuterium labeled block, enabling Guinier analysis to determine RG)I. (b) The incident neutron beam is parallel to the A/B interface and the small angle scattering is dominated by the periodic lattice structure (adapted from Hasegawa et al., 1987).

trast matching wherein the level of deuterium labelled diblock was adjusted so as to eliminate the domain interference. However, the Rg ± was still indeterminate due to incomplete contrast matching. This problem was believed to arise from nonuniform deuterium concentration within the PS domains from insufficient overlap of the deuterated blocks. The Rg^ value determined was not affected and the result confirmed the chain contraction in the parallel direction previously measured. Most recently, Matsushita et al. (1990) have used anionic synthesis to carefully deuterium label parts of a block. Their focus was on examining the dimension of a selected portion of the chain. Two types of triblock copolymers composed of deuterated PS, PS and poly-2-vinylpyridine (P2VP) were prepared: "end-labelled" (consisting of d-PS, PS and P2VP) and "centered labelled" (consisting of PS, d-PS

and P2VP). Lamellar orientation resulted from the solvent casting process rather than by shearing. The portion of the chain adjacent to the A/B diblock junction was found to be contracted along the direction parallel to the interface to the same extent as previous researchers had found for the entire block (i.e. approximately 70%). The portion of the chain at the free end of the block was localized in the middle of the PS domain and its dimension was unperturbed by the microdomain morphology. Thus, the gradient in chain stretching can be explored in microstructural systems.

1.9 References Abkin, A., Medvedev, S. (1936), Trans. Faraday Soc. 32, 286. Albalak, R., Thomas, E. (1993), J. Polym. ScL, Polym. Phys. Ed. 31, 37. Ambler, M. R., Fetters, L. I, Kesten, Y. (1977), /. Appl. Polym. Sci. 21, 2439.

30

1 Model Polymers for Materials Science

Anderson, J. N., Kern, W. X, Bethea, T. W, Adams, H. E. (1972), /. Appl. Polym. Sci. 16, 3133. Antoviak, T. A. (1971), Polym. Prepr. (Am. Chem. Soc, Div. Polym. Chem.) 12 (2), 393. Arichi, R, Pedram, N. Y, Cowie, J. M. G. (1979), Eur. Polym. J. 15, 107, 113. Bahar, I., Baysal, B. M., Erman, B. (1986), Macromolecules 19, 1703. Brant, D. A., Flory, P. J. (1965), /. Am. Chem. Soc. 87, 2791. Broske, A. D., Huang, T. L., Allen, R. D., Hoover, J. M., McGrath, J. E. (1987), in: Recent Advances in Anionic Polymerization: Hogen-Esch, T. E., Smid, X (Eds.). New York: Elsevier, p. 363. Burfield, D. R., Doi, Y (1983), Macromolecules 16, 702. Butera, R., Fetters, L. X, Huang, X S., Richter, D., Pyckhout-Hiutzen, W, Zirekl, A., Farago, B., Ewen, B. (1991), Phys. Rev. Lett. 66, 2088. Casal, H. L., Yang, P. W, Mautsch, H. H. (1986), Can. J Chem. 64, 1544. Cowie, J. M. G. (1973), Eur. Polym. J. 9, 1041. Fetters, L. X, Firer, E. M. (1977), Polymer 18, 305. Fetters, L. X, Hadjichristidis, N., Lindner, X S., Mays, X W, Wilson, W. W. (1992), unpublished. Flory, P. X (1940), /. Am. Chem. Soc. 62, 1521. Flory, P. X (1953), Principles of Polymer Chemistry. Ithaca, NY: Cornell University Press. Folkes, M. X, Keller, A., Scalisi, F. P. (1973), Colloid and Polym. Sci. 251,1. Gaskell, R. E. (1950), /. Appl. Mech. 17, 334. Gatzke, A. X (1969), /. Polym. Sci. 7, 2281. Hadziioannou, G., Mathis, A., Skoulios, A. (1979), Colloid Polym. Sci. 257, 136. Hadziioannou, G., Picot, C , Skoulios, A., Ionescu, M.-L., Mathis, A., Duplessix, R., Gallot, Y, Lingeler, X-P. (1982), Macromolecules 15, 263. Hasegawa, H., Hashimoto, T., Kawai, H., Lodge, T. P., Amis, E. X, Glinka, C. X, Han, C. C. (1985), Macromolecules 18, 67. Hasegawa, H., Tanaka, H., Hashimoto, T., Han, C. C. (1987), Macromolecules 20, 2120. Iatraou, H., Hadjichristidis, N. (1992), Macromolecules 25, 4649. Janik, B., Samulski E. T., Toriumi, H. (1987), J. Phys. Chem. 91, 1842. Keller, A., Pedemonte, E., Willmouth, F. M. (1970), Nature 225, 538. Kern, W. X, Anderson, X A., Adams, H. E., Bouton, T. C , Bethea, T. W. (1972), /. Appl. Polym. Sci. 16, 3123. Lifson, S., Oppenheim, I. (1960), /. Chem. Phys. 33, 109. Matsushita, Y, Mori, K., Saguchi, R., Noda, I., Nagasawa, M., Chang, T., Glinka, C. X, Han, C. C. (1990), Macromolecules 23, 4387. Mays, X W. (1992), Polym. Bull. 23, 247. Mays, X W., Hadjichristidis, N., Fetters, L. J. (1988), Polymer 29, 680.

Morton, M., Helminiak, T. E., Gadkary, S. D., Bueche, F. (1962), J. Polym. Sci. 57, 471. Neutwig, N., Sinn, H. (1980), Makromol. Chem., Rapid Commun. 1, 59. Olvera de la Cruz, M., Sanchez, I. C. (1986), Macromolecules 19, 2501. Park, I., Barlow, X W, Paul, D. R. (1991), /. Polym. Sci., Polym. Chem. Ed. 29, 1329. Pennisi, R. E., Fetters, L. X (1988), Macromolecules 21, 1094. Puskas, X E., Kaszas, G., Kennedy, X P., Hager, W. G. (1992), /. Polym. Sci., Polym. Chem. Ed. 30, 41. Quirk, R. P. (1992), in: Comprehensive Polymer Science, Supplemental Volume: Eastmond, G. C , Ledwith, A., Russo, S., Sigwalt, P. (Eds.). New York: Elmsford, in press. Quirk, R. P., Chen, W.-C. (1992), Makromol. Chem. 183, 2071. Quirk, R. P., Lee, B. (1992), Polym. Int. 27, 359. Quirk, R. P., Yin, X (1992), J. Polym. Sci., Polym. Chem. Ed., in press. Quirk, R. P., Yin, X, Fetters, L. X (1989), Macromolecules 22, 85. Quirk, R. P., Yin, X, Fetters, L. X, Kastrup, R. V. (1992), Macromolecules 25, 2262. Richter, D., Butera, R., Fetters, L. X, Huang, J. S., Farago, B., Ewen, B. (1992), Macromolecules 25, 6148. Richter, D., Fetters, L. X, Huang, X S., Farago, B., Ewen, B. (1991), J. Non-Crystalline Solids 131-133, 604. Roovers, X (1992), private communication. Roovers, X E. L., Bywater, S. (1973), Polymer 14, 594. Schulz, G. V., Bohm, L. L., Chmelir, M., Lohr, G., Schmitt, B. X (1970), in: IUPAC International Symposium on Macromolecular Chemistry. Budapest: Butterworths, p. 223. Sirianni, A. F , Worsfold, P. X, Bywater, S. (1959), Trans. Faraday Soc. 55, 2124. Suter, U. W, Flory, P. X (1975), Macromolecules 8, 765. Tsitsilianes, C , Staikos, G. (1992), Macromolecules 25, 910. Waak, R., Rembaum, A., Coombe, X D., Szwarc, M. (1957), /. Am. Chem. Soc. 79, 2026. Wenger, R (1960), Macromol Chem. 36, 200. Yoon, D. Y, Sundarajan, P. R., Flory, P. J. (1975 a), Macromolecules 8, 766. Yoon, D. Y, Sundarajan, P. R., Flory, P.J. (1975 b), Macromolecules 8, 784. Young, R. N., Quirk, R. P., Fetters, L. X (1984), Adv. Polym. Sci. 56, 1 (see Tables 17 and 22 for relevant references). Ziegler, K. (1936), Angew. Chem. 49, 499. Ziegler, K., Dersch, R, Willthan, H. (1934), Justus Liebigs Ann. Chem. 511, 13. Ziegler, K., Jacob, L. (1934), Justus Liebigs Ann. Chem. 511, 45. Ziegler, K., Jacob, L., Willthan, H., Wenz, A. (1936), Justus Liebigs Ann. Chem. 511, 64.

1.9 References

31

General Reading Bovey, F. A., Winslow, F. H. (Eds.) (1979), Macromolecules — An Introduction to Polymer Science. New York: Academic. de Gennes, P. G. (1979), Scaling Concepts in Polymer Physics. Ithaca, NY: Cornell University Press. Flory, P. I (1953), Principles of Polymer Chemistry. Ithaca, NY: Cornell University Press.

Mandelkern, L. (1983), An Introduction to Macromolecules, 2nd ed. New York: Springer. Treloar, L. R. G. (1970), Introduction to Polymer Science. London: Wykeham. Wunderlich, B. (1973), Macromolecular Physics, Vols. 1-3. New York: Academic. Young, R. X, Lovell, P. A. (1991), Introduction to Polymers, 2nd ed. London: Chapman and Hall.

2 Amorphous Polymer Microstructure Frank T. Gentile Cytotherapeutics, Inc., Providence, RI, U.S.A. Ulrich W. Suter Institut fur Polymere, ETH Zurich, Switzerland

List of 2.1 2.1.1 2.1.2 2.1.3 2.2 2.2.1 2.2.2 2.2.3 2.2.4 2.2.4.1 2.2.4.2 2.3 2.3.1 2.3.1.1 2.3.1.2 2.3.1.3 2.3.1.4 2.3.1.5 2.3.2 2.3.2.1 2.3.2.2 2.3.3 2.3.3.1 2.3.3.2 2.3.3.3 2.4 2.4.1 2.4.2 2.5

Symbols and Abbreviations Introduction What Is Amorphous? What Is a Polymeric Glass? Why Is There a Need for the Detailed Atomistic Polymer Microstructure? History of Atomistic Modeling (Metropolis) Monte Carlo Methods Static and Quasi-Static Molecular Mechanics Methods Molecular Dynamics Methods What Kinds of Properties Should Be Predictable? Polymer Structure Properties Derived from the Structure Atomistically More Detailed Methods - Structure Prediction A Static Model Representation of Energy Initial Guess Generation Potential Energy Minimizations Example - Atactic Poly(propylene) Extension to Polar Polymers - Atactic Poly(vinylchloride) and Polycarbonates Prediction of Structure Cohesive Energy Density and Solubility Parameter Polymer Microstructure Prediction of Properties in the Quasi-Static Approximation Local Molecular Processes in Polycarbonate Diffusion of Small Molecules in Glassy Polymers Mechanical Properties Atomistically Detailed Methods - Dynamic Behavior Constrained Molecular Dynamics in Cartesian Coordinates Constrained Molecular Dynamics in Generalized Coordinates References

Materials Science and Technology Copyright © WILEY-VCH Verlag GmbH & Co KGaA. All rights reserved.

34 37 37 39 40 40 41 41 42 45 45 46 46 46 47 50 51 51 53 55 55 57 61 62 63 65 69 70 72 74

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2 Amorphous Polymer Microstructure

List of Symbols and Abbreviations a a(Tg) a0 D D (r) E Eij9 Et ft ^conf F inter G q gtj (r) I (Q) hj ( 0 k k^ L Nt iVat Nej p^t P- V- T Q qt qt, qt q^;i r R R1 R2 R* #sd rf
critical radius for Block-Walker dielectric function, D (r) critical ratio of the inter-strophon distance at Tg van der Waals distance between two strophons in neighboring chains which interact according to a Lennard-Jones potential diffusion coefficient distance-dependent dielectric function Young's modulus energies associated with the diffusant on the gate between tetrahedra i and j and internally to tetrahedron i, respectively X-ray scattering power for atom i configurational (intramolecular) strophon force barriers intermolecular strophon force barriers shear modulus gauche minus isomeric state in a vinyl polymer atomic pair distribution functions for atoms i and j elastic X-ray scattering intensity partial interference function for atoms i and j Boltzmann's constant intrinsic torsional potential energy constant Lagrangian function ( = Uk— Up) number of groups of species i in the periodic cube number of distinct atomic types effective number of electrons for atom i a priori RIS probabilities (bond i in state £) pressure-volume-temperature magnitude of the scattering vector point charge magnitude of atom i generalized position, vector coordinate conditional probabilities (bond i in state £ when bond i — \ is in state £) distance between atoms; radial distance cutoff radius for UNB and UNBP radius for the beginning of the quintic spline for UNB radius for the beginning of the quintic spline for [/NBP a length less than or equal to the semidiagonal of the periodic cube semidiagonal of the periodic cube van der Waals radius of atom i root-mean-square end-to-end distance; root-mean-squared displacement of a diffusant locking factor, r = moment arm of a strophon bond directional correlation function bond directional correlation function as a function of distance between bond chords, r displacement vector for atom i in a Delaunay tetrahedron

List of Symbols and Abbreviations

35

Winter (r) S (Q) <52> t T Tg T2

intermolecular part of S (r) elastic X-ray scattering structure factor root-mean-square radius of gyration time; trans isomeric state for a vinyl polymer absolute temperature glass transition temperature (in K ) suggested temperature for which system configurational entropy is supposed to vanish according t o G i b b s et al. Uk kinetic energy Up potential energy U^{(j)) torsional potential energy UC(T) coulombic potential 171 value of the coulombic potential at r = R2 Ul' value of the first derivative of the coulombic potential at r = R2 Ul" value of the second derivative of the coulombic potential at r = R2 UY value of the Lennard-Jones function at r = R1 [/jLJ value of the first derivative of the Lennard-Jones function at r = Rt Li Ui value of the second derivative of the Lennard-Jones function at r = Rx UNB n o n - b o n d e d potential (dispersion forces only) jyNBP n o n - b o n d e d polar potential ^buik bulk polymer energy Ucoh cohesive energy L7par parent chain energy of a polymer chain ^tot, system total energy of bulk polymer R a o function, molar sound velocity increase in long-range interaction energy u p o n addition of the i + 1 skeletal b o n d i a n d its substituents o n a t o m i if b o n d i is in rotational state £ value of the potential tail correction V molar volume; deformed state molar volume Vo molar volume at which the potential energy with respect t o volume at constant T is a m i n i m u m x a r a n d o m n u m b e r d r a w n from a uniform distribution; degree of polymerization x position of a n a t o m inside a tetrahedron X degree of polymerization at A A 5 dffi s a, Eij sn

atomic polarizabilities of a t o m i dimensionless width of potential spline [A = (R — R^/a for l/NB, A = (R — R2)/o for 17NBP] Debye-Waller factor Hildebrandt solubility parameter displacement gradient strain Lennard-Jones potential energy function parameter, for pairs of atoms (ij) bulk dielectric constant

36

2 Amorphous Polymer Microstructure

g0 £ 6 x X pi v0 £ Qt Q0 a, (Ttj a <j y T zt v (/), x//, \j/ coy

permittivity of an electric field in a vacuum rotational isomeric state theory state scattering angle; unperturbed condition; angle between bond chords exponent for the dielectric function wavelength of scattered radiation effective dipole of a repeat unit frequency corresponding to a jump without an energy barrier dimensionless radial distance used in potential splines [£ = (r — RJfcR — R2) for UNB, £ = (r-R2)/(R-R2) for [7NBP]; RIS state number density of atom type i number density of atoms Lennard-Jones potential energy function parameter, for pairs of atoms (ij) internal strain tensor critical value of stress time constant due to molecular vibrations mean residence time of a diffusant in tetrahedron / Poisson ratio torsional angle, torsional angle velocity orientation angle, orientation angle velocity * jump frequency between the tetrahedra i and j

CNDO MC MD NMR PC PP PVC RIS TBPC TMPC

complete neglect of differential overlap Monte Carlo molecular dynamics nuclear magnetic resonance polycarbonate poly(propylene) poly(vinylchloride) rotational isomeric state theory tetrabromo polycarbonate tetramethyl polycarbonate

2.1 Introduction

2.1 Introduction Many of the plastics produced on a large scale for consumer applications exist under application conditions, at least partially, in what is commonly called the glassy state. A polymer glass is a solid that has been described as one marked by a relatively random, or amorphous, arrangement of macromolecular chains (Uhlmann and Kreidl, 1983). All polymer glasses share this definition, but they often have radically different properties. What then is the influence of the molecular structure of the constituent chains on the macroscopic properties of these materials? What degree, or level, of structural information is necessary for the prediction of macroscopic properties? Here we will attempt to address these questions. We will, in part, do this in the context of "modeling". The purpose of modeling, in general, is twofold: on the one hand, the investigator tries to establish a model for a state or process to obtain a "physically correct" functional form as an aid in the interpretation and extrapolation of phenomena; on the other hand, the model is hoped to provide new insights into the phenomena if the predictions prove to be accurate. The eventual goal is the ability to accurately predict the properties of materials that are "still on the blackboard", without going through expensive synthesis or manufacturing trials. Some definitions are relevant.

37

arrangement of atomic positions is disordered such as in a liquid. What is the difference (besides density) between an amorphous solid and a bona fide liquid and which of these two states does a polymeric glass fit into? A distinction can be made between solids and liquids with respect to the microscopic motion taking place in the material and to the response to mechanical deformation (Zahlen, 1983). In a solid, the relative mobility of the atoms is much lower than in a liquid, where atoms are able to translate more freely. Atomic trajectories are schematically shown in Figs. 2-1 (a) and 2-1 (b). One notes that in the solid, an atom is greatly hindered in by its surrounding atoms. It undergoes oscillatory motion, but does not have the strong translational component that an atom has in a liquid. From these microscopic properties one arrives at the standard macroscopic properties: a liquid cannot support a shear stress; a solid has a completely elastic response to an infinitesimally small stress even applied for a long time. An amorphous solid is thus a solid in which there is an absence of long range order or periodicity. This characteristic is most clearly seen in a diffraction experiment. For example, in an X-ray diffraction pattern there exist diffuse halos instead of the sharp Bragg spots or rings that are

2.1.1 What Is Amorphous?

Much has been written about this interesting state of matter (Elliot, 1983; Uhlmann and Kreidl, 1983; Zahlen, 1983; see also Chap. 4 of Vol. 1 of this Series). In general, amorphous species are characterized by the absence of long range order; the

Figure 2-1. Schematic trajectories of atoms in (a) a solid and (b) a liquid.

38

2 Amorphous Polymer Microstructure

produced by crystalline materials. The existence of these diffuse interferences is a prerequisite for characterizing a solid as amorphous (Elliot, 1983). Glasses exhibit glass transitions. The glass transition, characterized by a somewhat elusive characteristic temperature Tg, represents the crossover between the two microscopic situations of atomic motion in Figs. 2-1 (a) and 2-1 (b). Analysis of the nature of the glass is very difficult since the glass structure normally does not equilibrate within the time scale of conventional experiments. Many researchers have attempted to explain the behavior of glasses with strictly thermodynamic or kinetic arguments. Both arguments lead to fallacies. (For example, if Tg were a true thermodynamic transition, such as the melting point, then it should remain independent of different cooling rate, but it does not.) Some attempts to understand glasses are reported below. Kauzmann (1948) investigated the relationship of the glass transition temperature to kinetic and thermodynamic factors. He attempted to identify the molecular motions involved in glass formation from the standpoint of the glass transition being a relaxation process. From a great deal of experimental evidence, he concluded that the relaxation process at Tg involved overcoming a potential energy barrier much higher than the mean kinetic energy of the system. Later, when viewing the degrees of freedom involved in the glass transition, Kauzmann concluded that the degrees of freedom were exclusively configurational and did not involve momenta (i.e., kinetics) at all; glass formation probably was governed by the potential energy term, and not the kinetic term, of the Hamiltonian of the liquid. Thus, it seems that potential energy is a very important parameter when addressing glasses. Following Kauzmann's

work, other researchers have attempted to describe the glassy state. The two best known theoretical treatments to describe the glassy state are based upon equilibrium thermodynamics. The first is the polymer-configuration model of Gibbs and Dimarzio (1958) (and later Adam and Gibbs, 1965) and the second is the free-volume model of Cohen and Turnbull (1959, 1964) and Cohen and Grest (1981). Gibbs and coworkers associated the decrease in molecular mobility with falling liquid temperatures with a decrease in configurational entropy. The configurational entropy was suggested to vanish at a temperature T2 (which is about 50 degrees lower than Tg for most organic materials). The glass thus approaches a state of zero configurational entropy where the major driving force for the configurational changes vanishes. This makes the observed Tg a parameter which, although it can be modified by kinetics, has an underlying transition temperature defined by T2. The temperature T2 was suggested to represent a second order phase transition (unlike the first-order phase transition characteristic of, e.g., the freezing point of a material). Cohen and Turnbull viewed the glassy state as a metastable instead of instable state. In their words, each microscopic structural unit of the glass must lie at a position of static equilibrium, the totality of which is randomly distributed. If one such exists there must be a large number of similar random structures of equal energy. Nevertheless, the entropy of each is zero, because all the structures are mutually inaccessible (Cohen and Turnbull, 1964). Thus, the glass is viewed as a solid which has frozen-in liquid-like disorder. On a microscopic scale, the distinction between a solid glass phase and a liquid

2.1 Introduction

melt of equal density is subtle. Structures in the melt are in dynamic equilibrium and form a statistical-mechanical ensemble which thus requires properly weighted (e.g., Boltzmann-weighted) averages. No such relationship exists in glasses. 2.1.2 What Is a Polymeric Glass? Polymeric glasses are a special subset of glass-forming species that consist of macromolecules. These materials represent an important class of polymers and comprise a large variety of materials, from atactic vinyl polymers to aromatic polycarbonates. All of these materials share the basic characteristics of amorphous solids that were outlined above, however the range of properties and applications is large. Thermodynamics alone is not sufficient to describe the glassy state (a glass is not in global internal thermodynamic equilibrium). Another complication with polymeric glasses is the fact that large macromolecular chains are involved which create entanglements and further highly correlated motions. Nevertheless, many researchers have sought to describe this complicated state of matter. Phenomenological concepts which defy precise definition, such as the "free volume" model of Cohen etal. (Cohen and Turnbull, 1959, 1964; Cohen and Grest, 1981), have been used to explain some of the differences in measured properties between different polymeric species. Recent experimental work on chain packing in the amorphous bulk has also indicated that the chain molecules in the glass assume essentially unperturbed random-coil conformations as Flory had suggested decades ago (1953). However, the complete absence of long range orientational order that is implied by a random coil model has not been completely accepted (Pechhold et al, 1976).

39

The work of Yannas et al. (Yannas, 1975; Yannas and Luise, 1982, 1983) which considers molecular level structure displacement in glassy polymers brought about by mechanical deformation should also be noted here. It was found that deformation of a macromolecular segment occurs primarily by rotation around skeletal bonds, rather than by changes in bond lengths or angles. By extension, it might be suggested that conformation is the primary set of variables to describe polymeric glasses. Recently, Theodorou and Suter (1985 a, 1985 b, 1986) attempted to describe the structure-property relationships in glassy polymers. In their work, a polymeric glass is viewed as a structure trapped in a local minimum of potential energy. This is essentially the same as Cohen and TurnbulFs conceptualization of a glass being in a state of frozen-in liquid disorder. In a static system, the term local minimum of potential energy is synonymous with the term detailed mechanical equilibrium meaning that the sum of all forces and all torques exerted on each atom or bond in the structure is zero. It is important to note that this is a local minimum of potential energy and not a minimum of the relevant thermodynamic parameter, which would be the Helmholtz free energy. Thus, the minimized structures are not in global thermodynamic equilibrium but in a local minimum of potential energy. It is assumed that the aforementioned state of frozen-in liquid-like disorder is due to potential energy barriers trapping the system in the minima. The importance of potential energy is consistent with the work of Kauzmann (1948). It is also assumed that each structure which exists in this local minimum is one microstate and that the totaling of these microstates does not comprise an equilibrium ensemble. An amorphous polymeric glass is best viewed as a solid made up of polymeric

40

2 Amorphous Polymer Microstructure

chains which has a glass transition and exists in a state of frozen-in liquid disorder characterized by local near-equilibrium. There is a total or partial absence of long range order or periodicity. 2.1.3 Why Is There a Need for the Detailed Atomistic Polymer Microstructure?

Structural characteristics of molecular and super-molecular nature greatly affect the relevant properties of macromolecular materials. Which structural characteristics and what level of detail are important for the reliable prediction of material properties? Simple and popular correlations for glassy polymers using group-contribution Table 2-1. Atomistic polymer microstructure: What kinds of properties should be predictable? 1. Polymer structure properties 1. Cohesive energy and Hildebrandt solubility parameter 2. Chain dimensions in the amorphous bulk 3. Bond directional correlation functions (intramolecular and intermolecular contributions) 4. Atomic pair distribution functions (intramolecular and intermolecular contributions) 5. Analysis of empty space 2. Properties derived from the static structure 1. Starting point for a diffusion model to predict diffusion coefficients and permeabilityselectivity 2. Response to deformation (elastic and "finite" plastic deformation) 3. Activation energies and structural changes due to certain dynamic processes 3. Polymer molecular dynamics 1. Simulation of deformation at finite strain rates 2. Correlations of dynamic motion; determination of the frequency spectrum 3. Incorporation of molecular motion in time averaged distributions

methods outlined by Bondi (1968) and Van Krevelen and Hoftyzer (1976) are often very successful for predicting polymer properties but there exist many cases where these correlations fail; they are inherently interpolations. The areas of polymer deformation and diffusion of small molecules in the bulk glassy polymer matrix have also been extensively studied and phenomenological correlations exist. However, they fall short when used to gain insight into or to predict behavior. One can make the jump from molecular structure to properties for glassy polymers by attempting to model the glassy state. Some researchers have attempted to model the structure of the polymer in extreme detail, i.e. on the atomistic level. If a detailed structure existed, they hope, it could be used to simulate properties such as the Hildebrandt solubility parameter or obtaining an X-ray or neutron scattering pattern, or be used to analyze processes such as the diffusion of small molecules in the glassy polymer matrix, or even for studying polymer glass dynamic behavior. These computer experiments, simulations that model polymer structure at the level of determining the position in space of all the atoms in a microstate, are called atomistic modeling.

2.2 History of Atomistic Modeling The earliest models for polymeric glasses were based on lattice methods (Gibbs and DiMarzio, 1958; Alexandrowicz and Accad, 1971; Jagodic et al., 1973; De Vos and Bellemans, 1974; De Santis and Zachmann, 1977; Skvortsov et al., 1977; Wall and Steitz, 1977). Other approaches involved scaling laws to describe the polymer melt-to-glass transition (Simha, 1976, 1977). These models are very general and

2.2 History of Atomistic Modeling

offer little potential of relating detailed chemical structure to macroscopic properties. Modeling of atomistic level detail of amorphous polymeric glasses is rare (mostly due to its computational intensity), and a classification of these attempts is not altogether straightforward, but three main groups can be distinguished: (Metropolis) Monte Carlo models, static and quasistatic molecular mechanics approaches, and molecular dynamics methods. Static and quasi-static molecular mechanics methods and a particular sub-class of molecular dynamics will be discussed in greater detail in Sec. 2.3. 2.2.1 (Metropolis) Monte Carlo Methods Recall the view of a polymer glass being a structure of frozen-in disorder composed of many microstates (Cohen and Turnbull, 1964). All models that attempt to address the amorphous glassy state in detail must contain a stochastic element since the models must select a representative set, i.e. a few of the many possible microstates in the system for detailed analysis. Many of the models employed to analyze dense polymeric systems make extensive use of a catalog of tools that are collectively known as Monte Carlo (MC) methods. Most of this work targets the liquid (or melt) state. The earliest MC simulations of polymers involved only individual chains (Suzuki and Nakata, 1970; Lai and Spencer, 1971). In an attempt to use MC methods for dense systems, a reptation (snake-like movement of the chain) mechanism was employed (De Gennes, 1971; Wall and Mandel, 1975), first for chains on a lattice (Wall and Steitz, 1977), and later to the continuum (Brender and Lax, 1983). MC techniques have been applied to large polymer systems, for example, by Binder (1984)

41

to determine if chain motion in the melt occurs primarily by a reptation mechanism. In another MC approach to a polymeric liquid, the chains were modeled as beads connected by freely rotating, finitely extensible bonds and a reptation mechanism was employed to sample configuration space (Bishop et al., 1980). Vacatello et al. (1980) proposed a detailed model for short linear polyethylene chains in the liquid state; the system consisted of 31 C 30 alkane chains with fixed bond lengths placed in a cubic cell with periodic continuation conditions. The system's initial configuration was obtained by randomly placing trimethylene units on a lattice, whereupon the system was "equilibrated" by a reptation-like MC procedure. In the glassy state, a detailed MC simulation was carried out for isopentane (Yashonath et al., 1985); 54 molecules were placed in a cubic cell with periodic continuation conditions. A realistic potential was used to describe atomic interactions. The system was examined in a temperature range that encompassed the glass transition temperature, and the general characteristics of the glass transition were observed. 2.2.2 Static and Quasi-Static Molecular Mechanics Methods Molecular mechanics methods all involve the minimization of the energy of a system from some previously generated initial state. Usually the structures do not comprise any dynamic components, they are static. Sometimes information can be gleaned on dynamic processes using static energy minimization of a series of perturbed structures and these techniques are termed quasi-static. A feature of molecular mechanics methods considered here is the simultaneous

42

2 Amorphous Polymer Microstructure

consideration of both intermolecular and intramolecular energy during the minimization. This is done by constructing a classical force-field to mimic the behavior of atomic interactions. These usually are determined using experiments in the gaseous or liquid state, quantum mechanical calculations, or crystallographic data on smaller species that are similar to the relevant polymeric fragments. Applications of energy minimization to isolated macromolecules and to crystalline systems are numerous. Among the earliest work involving isolated macromolecules was that of Liquori (1955), and Brant and Flory (1965). Some (among many) consecutive studies are those by Scott and Scheraga (1966a, 1966b), Ooi et al. (1967), Allegra et al. (1970), Tashiro et al. (1977a, 1977b), Suter (1979), and Brisson and Brisse (1986). In the case of packed polymer chains, most of the past work has been performed on polymeric crystals. These efforts have usually fallen into two categories: the prediction of elastic quantities or determination of thermodynamic and structural properties. Anand (1967) applied atom based forcefields to predict the elastic constants of orthorhombic polyethylene. Later, McCullough et al. (1974) developed a matrix technique for representing crystalline assemblies of chains in fixed conformations considering also the existence of packing defects. For flexible polymers, the first prediction of crystal packing at realistic densities by minimizing the intramolecular and intermolecular interaction energy was performed by Tripathy et al. (1981). The minimizations were limited to assemblies of chains of fixed conformation using a row formalism for the single chain. Extension to three dimensions was not possible with this method. Recently Sorenson et al. (1988a, 1988b) have incorporated the

simultaneous minimization of packing energy with respect to intramolecular conformation and intermolecular packing. This type of approach is critical for the prediction of three-dimensional structure and properties. For rigid-rod polymers, computations on chains were performed by Tashiro et al. (1977a, 1977b), Hummel and Flory (1980), and Erman et al. (1980), investigating p-phenylene polyamides and polyesters. Rutledge (1990) and Rutledge and Suter (1991) first showed that the polymeric crystal state of such macromolecules could be modeled in detail; poly(p-phenylene terephthalamide) and ring-substituted poly(p-phenylene terephthalamide) were successfully modeled in the crystal state using a molecular mechanics approach. The term crystalline polymers is somewhat of a misnomer here because these systems have a small degree of disorder which makes them different from extremely ordered systems. The application of molecular mechanics techniques to polymers in the amorphous glassy state will be discussed in more detail in Sec. 2.3. 2.2.3 Molecular Dynamics Methods Molecular dynamics (MD) simulations represent a large category of techniques used by chemists, physicists and engineers for decades, encompassing a broad scope of simulations involving the solution of equations of motion. At one extreme, bulk physical properties of atomic and molecular fluids are studied as the dynamic evolution of a collection of representative particles. At the other extreme, one uses MD to explore possible chemical transformations during collisions between potentially reactive species (e.g., in a dilute gas). In between there are a variety of cases in which

2.2 History of Atomistic Modeling

there is a co-dependence between molecular motion and chemical dynamics. For polymeric glasses, MD is used to explore molecular motion and changes associated with the approach to the glass transition. In this case, MD simulations usually are determined by deterministic equations of motion, but, may contain elements of a stochastic source. Most of the published work relevant to MD simulations of glassy polymers falls into one of three categories: molecular dynamics simulation of the glass transition and glassy state of simple model systems, more realistic polymeric or oligomeric simulations in states other than the glass, and MD approaches which are applied to the glass state. For the simulation of the glass transition and the glassy structure of simple idealized cases such as systems made up of hard spheres (Gordon et al, 1976; Woodcock, 1976, 1978), bistable oscillators (Helfand, 1978; Perchak et al., 1983), continuous potential particles using soft core potentials (Hoover et al., 1971; Cape and Woodcock, 1980), Gaussian core potentials (Stillinger and Weber, 1979, 1980; Weber and Stillinger, 1981), or Lennard-Jones interactions (Baumgartner, 1980; also ref. 7-15 in Angell et al., 1981) have been proposed. Some "real", simple molecular glass forming systems have also been studied (ref. 22-25 in Angell, 1981). Excellent review articles have been written on these simulations by Angell and coworkers (Angell, 1981; Angell et al., 1981), and by Frenkel and McTague (1980). Very few of these simple systems can be compared to real experimental data in a quantitative way. Even the "real" glass forming systems studied, consisting of species such as silicates and molten salts, are radically different from polymeric species. However, these simple simulations have

43

elucidated many qualitative features of the glassy state and the glass transition. For example, what is often considered to be a universal structural feature of a glass state is the splitting of the "second nearest neighbor" peak in the pair distribution functions; this phenomenon has been observed in many radically different simulations under a variety of conditions (Frenkel and McTague, 1980; Angell, 1981; Angell et al., 1981) and was quantitatively analyzed by Wendt and Abraham (1978) with the aim of establishing a convincing structural criteria for the determination of the onset of the glass transition. A characteristic feature of these MD simulations is the extremely high quench rates used, which are necessary for the sake of computational expediency, yet this can produce dramatically premature solidification of the system. Indeed, it is primarily for this reason that some researchers have attempted to study glassy structure in a static or quasi-static fashion first and then use them as a starting point for MD simulations (Ludovice, 1989). The second category of MD simulations consists of larger, more detailed systems. Various MD simulations of liquid amorphous packings of poly(ethylene)-like chains have been carried out by Weber and Helfand (1979). These were detailed atomistic simulations but were void of the important effects of stereochemistry (side groups) and polar interactions that exist in many glassy polymers [e.g., atactic poly(propylene) and poly(vinylchloride)]. Many investigators have carried out MD simulations of polymers in solution and in the melt (Kang et al., 1983; Oh et al., 1983; Khalatur et al., 1985,1986; Grest and Kremer, 1986). Oriented polymer systems (Lukyanov et al., 1985; Brown and Clark, 1986) and polymerizing systems (Yasukawa, 1973; Synek et al., 1977) have also

44

2 Amorphous Polymer Microstructure

been investigated. Simulations of systems of dense polymers have been carried out, but most are lacking realistic detail (Balabaev et al., 1978). Aside from the work of Weber and Helfand, some of the largest and most detailed polymeric simulations have been done on biopolymers (Simon, 1971; McCammon and Karplus, 1980; Van der Ploeg and Berendsen, 1983; Levy et al., 1984; Karplus and McCammon, 1986). However, isolated molecules or bilayer membranes were investigated, rather than dense bulk systems. A common type of molecular dynamics simulation is known as constrained MD. The reason for applying constraints to an MD system is usually to eliminate the high frequency modes of motions, since the time step for MD simulations must be on the order of, or smaller than, the fastest motion in the simulated system. By eliminating these high frequency motions, a larger time step for the integration of the equations of motion may be used. The use of a larger time step in turn makes for a more efficient simulation, which allows for longer simulation times and hence more information on (polymeric) motion. The fastest degrees of freedom in molecular systems are typically bond and angle vibrations. In this context it is relevant to ask if the model is still meaningful if those modes of motion in the polymer glassy state are ignored. Van Gunsteren and Karplus (1982) have investigated the effect of bond length and angle constraints in MD simulations of isolated proteins. It was found that the dynamic properties of the simulated protein system were not significantly altered by the imposition of bond length and bond angle constraints, but the frequency of the torsional motions was reduced. The resulting shift in the frequency spectrum for these completely constrained systems (bond angles and bond lengths were fixed) should be

taken into account when interpreting frequency spectra of similar simulations. This is encouraging; it has been known for some time that even though the fixed bond model may produce a reasonable approximation to the total molecular partition function, the completely flexible model, in the limit of infinitely strong bond and angle bending and stretching force constants, is not identical to the same model with fixed bond lengths and bond angles (Go and Scheraga, 1976; Fixman, 1978 a, 1978 b). Fixman has developed a pseudopotential to correct for this difference in the dynamic behavior between the two models (Fixman, 1978 a, 1978 b). However, one can assume that this complex correction need not be computed, since the dynamic motion in a system as dense as macromolecular glasses will be highly damped and Perchak et al. (1985), in evaluating the Fixman pseudopotential, found that the difference between the two models becomes negligible in the highly damped limit. A means of carrying out constrained MD simulations is to solve several constraint equations at each step in the dynamic trajectory of the system. Lagrange multipliers, which represent the effective forces necessary to satisfy the desired geometry, are used to constrain the system (Ryckaert et al., 1977). In principle, these equations may be solved exactly (Orban and Ryckaert, 1974). This involves an inversion of the constraint equation matrix at each time step and is quite inefficient. Sylvester et al. (1989 a, 1989 b) attempted to adapt this approach to sparse matrix techniques using high speed vector and parallel computers. Because the integration of the equations of motion can only be carried out to a certain degree of accuracy, it is not necessary to solve the constraint equations exactly. Ryckaert et al. developed an approach in which these equations

2.2 History of Atomistic Modeling

are solved only with the accuracy inherent in the integration algorithm (Ryckaert et al, 1977; Ryckaert and Bellemans, 1975, 1978). To avoid a computationally intensive inversion of the constraint matrix, an iterative method that modifies the Lagrange multipliers until they converge to a given tolerance is used for large molecules, such as polymers. This iterative approach has been called the "SHAKE" algorithm (Ryckaert et al., 1977), and is currently the most commonly used constraint algorithm. It is typically applied to the Verlet integration algorithm (Verlet, 1967; Gear, 1971; Berendsen and Van Gunsteren, 1983), but it may also be applied to predictorcorrector algorithms (Van Gunsteren and Berendsen, 1977). Andersen (1983) has applied this complex to a modification of the Verlet integration algorithm (Swope et al., 1982) that potentially reduces round-off error. In analogy to the SHAKE algorithm, Andersen has termed this algorithmic approach "RATTLE". It is possible to recast the equations of motion in a generalized coordinate approach using only the relevant degrees of freedom. This approach goes back to Ryckaert et al. (1977) and was attempted by Ludovice and Suter (1989) for atactic poly(propylene) as a dynamic analogue to the model of Theodorou and Suter for the generation of static structures. This model will be discussed in greater detail in Sec. 2.4.2. 2.2.4 What Kinds of Properties Should Be Predictable? 2.2.4.1 Polymer Structure

In the case of properties derived directly from the polymeric glass structure, one has to analyze the interaction energies of the polymer constituents first. From the inter-

45

action energies of all the atoms in the system, one should be able to determine such properties as the cohesive energy density (due to only intermolecular forces) from which a value for the Hildebrandt solubility parameter can be calculated and then compared to experimental measurements. This provides a first check of whether or not the generated structures are energetically "reasonable". Once an energetically sound set of microstates has been generated, the structural features of individual chains and the bulk state can be studied. For individual chains important properties are the root mean square end-to-end distance (or root mean square radius of gyration) of the polymeric chain and the distribution of rotational angles. Generated microstructures should have global chain dimensions in agreement with known experiments ("lab-bench" or computer). If a static structure in atomistic detail is available, calculations of the quantities are possible and provide a second check into the structure and thus the model. Detailed structural features of the microstates of bulk polymer can be obtained by, for instance, determining the bond directional correlation and atomic pair distribution functions. Bond directional correlations reveal if certain types of bonds have a tendency to mutually orient themselves at preferred angles. Atomic pair distribution functions indicate if certain pairs of atoms have a tendency to reside at specific distances from each other. The intramolecular and intermolecular contributions to these distributions can also be examined. From this information, computer experiments on radiation scattering can be performed and compared with experimental data. Note that properties like scattering curves and global chain dimensions can be easily compared to experiment.

46

2 Amorphous Polymer Microstructure

Other interesting and important concerns, such as the distribution of empty space in the glassy polymer matrix, can also be obtained (Arizzi et al., 1990, 1992). 2.2.4.2 Properties Derived from the Structure Many different types of "experiments" can be envisioned once an ensemble of microstates has been generated. One can simply use the structure as a static scaffolding without attempting any structural changes. The modeling of diffusion of small gas molecules such as helium, oxygen and nitrogen through the glassy matrix may constitute such a use of the static microstructures. Alternatively, one can impose some type of change in the structure and record the structural or energetic consequences. Examples are performing a deformation experiment and recording the effects, and driving a specific bond torsional angle and determining the effect on the structure and energy of the system. These experiments use the static structure to investigate dynamic or time-dependent processes; they are quasi-static. An atomistically detailed molecular technique may be useful for the determination of many structural properties. For example, the correlations of motion in amorphous polymers may be studied to more accurately predict the scattering behavior of the material. Other important motions can be observed by calculation of their frequency spectrum.

2.3 Atomistically More Detailed Methods - Structure Prediction "More detailed methods" computation of glassy structure involve the determination of the location in space of each atom

in a (small) representative sample of a polymer glass. When there is an absence of molecular motion, this type of modeling is known as static atomistic modeling. As was already outlined in the previous section, a detailed structure knowledge can provide a great deal of information and insight into material properties. If the model is constructed with a minimum of adjustable parameters or experimental information, it can also be used as a predictive tool for as yet unsynthesized polymers. The most detailed static atomistic level polymer glass models to date are those based on the work by Theodorou and Suter (1985 a, 1986). This approach employs molecular mechanics methods to generate static microstructures from a "semi-realistic" initial guess configuration. Its first application was to atactic poly(propylene). Later simulations involved polar polymers such as atactic poly(vinylchloride) and polycarbonate of 4,4'-isopropylidene biphenol. 2.3.1 A Static Model The model developed by Theodorou and Suter consists of a single parent chain packed in a cubic cell with periodic continuation conditions well below Tg. One of the first uses of this approach, outside of the prediction of "well relaxed" structure, was the study of polymer response to deformation. Periodic continuation is an expedient frequently used in simulations in which a parallelepiped (a "box") is imagined as a part of an infinite medium. All atoms in the box have other images in space that can be readily obtained with the aid of the continuation vectors of the box. Thus, the box becomes a unit cell of disordered content with which space is filled completely (if a chain exits the box by crossing one of the boundaries, an identical copy enters the

2.3 Atomistically More Detailed Methods - Structure Prediction

47

box from the opposite face). The assumption is a good one for amorphous species provided that the scale of the interatomic interactions is small compared with the size of the box (Theodorou and Suter, 1985 a). One also limits oneself to studying structural features within only a certain range so as not to be confronted by the effect of the artificial periodicity. The use of periodic continuation allows one to fill the space with the images of a limited number of parent chains and permits a clear cut distinction between intermolecular and intramolecular quantities. Theodorou and Suter used a single parent chain, but more can be employed for specific purposes. For example, Mansfield and Theodorou (1989) have modeled thin polymer films based on a system with three parent chains of poly(propylene) in a rectangular box with periodic continuation in two dimensions; the remaining faces of the box represented empty space or substrate layers. Other assumptions and restrictions are:

2.3.1.1 Representation of Energy

1. The model does not incorporate thermal motion (it is completely static). The temperature of the simulation enters only indirectly, through the experimentally determined density. (It is noteworthy that to date, no method of atomistically estimating the "relaxed" density of a polymeric glass exists.) 2. The glassy polymer is pictured as an ensemble of mutually inaccessible states of frozen-in liquid-like disorder. This is similar to the Cohen and Turnbull view of a glass (1964). Estimates of macroscopic properties are obtained by averaging over all generated microstates. 3. The model contains fixed bond angles and bond lengths. Molecular rearrangement takes place only through bond-torsion angles. This is in line with Yannas et al.'s "strophon" model.

The total potential energy of interatomic interactions must be realistically represented before work on atomistic models can commence. For reasons of efficiency, molecular mechanics methods are applied. In calculating the total potential energy of these interactions, incremental additivity is postulated; moreover, non-bonded interactions are assumed to be pairwise additive. Each atom in the system acts as a point center of interaction with no distinction being made between pairs of sites belonging to the same or different chains. Finite range modifications of interchain potentials are required and the LennardJones 12-6 potential is used where the potential tail for r>Rx is substituted by a quintic spline that goes to zero at r > R (the cutoff radius). Thus, the potential energy function, UNB as a function of distance be-

4. In the study of deformation and other properties, entropic contributions are not estimated. Only potential energy effects are considered. As was mentioned in assumption 3, the model contains fixed bond lengths and bond angles. This allows for the use of a coordinate system specific to the relevant degrees of freedom of the system (i.e., mostly the bond torsion angles). Theodorou and Suter used three angles to define the overall orientation (similar to Euler angles) and the bond rotation angles to completely specify the location of each atom in the chain. After an initial guess is generated, a minimization algorithm is used to find the bottom of an energy well of the microstate. This algorithm needs to employ the analytical derivatives of the potential energy with respect to the aforementioned angles to efficiently perform the minimization.

48

2 Amorphous Polymer Microstructure

tween atoms, r, is: r
=

(2-1) e/(7

Rx
2

r> R where £ = ( r - i ^ ) / ^ - ^ ) , C/^, £/;LJ, and UiLi are the value, the first derivative and the second derivative of the LennardJones function at r = Rx (at £ = 0) and A =(R-R1)/a. UNB and its first and second derivatives with respect to r are continuous over the entire range of r and are zero at values of r > R. The values for the parameters atj and stj for the Lennard-Jones potential are calculated for each of the types of pairs of atoms (ij) present in the system from the values of the individual atoms for the atomic polarizabilities, af, effective number of electrons, Nej9 and vanderWaals radii, rf, obtained from the literature (Bondi, 1968). The relevant formulae are: (2-2) 1/2

(2-3)

where af is in A3, rf is in A, and 8tj is in kcal/mol. For reasons of efficiency, two forms of the non-bonded interactions are employed during the course of the minimization, the "soft-sphere" potential using R1=0.94(T and R = 1.04 cr accurately reproduces the repulsive part of the Lennard-Jones interaction, but contains no attractive part. This allows the "relaxation" of the strongest repulsions first. The "full potential" ap-

proximates the entire Lennard-Jones function with Rx = 1.45 o and R = 2.30 o. Electrostatic coulombic interactions between partial charges, pictured to be localized on the atomic nuclei, are represented by a coulombic potential, Uc (r). This electrostatic interaction modified by a distance dependent dielectric constant, D(r) is the effective permittivity of an electric field of the polymer, relative to that Uc(r) =

(2-4)

4nrD(r)80

of vacuum, which is denoted e0 (Ludovice, 1989). The remainder of the potential energy function is simply an electrostatic interaction where qt and q^ are the magnitude of the point charges on atoms i and j respectively, and r is the separation between charges. Typically, this relative permittivity is taken to be one of its limiting values: unity for very short distances, or the bulk value of the dielectric constant for large distances. A refinement employs the BlockWalker approximation for D (r) (Block and Walker, 1973): D(r) =

g B exp{-x/r}, )

r a

(2-5)

(2-6)

where % is the bulk dielectric constant; the parameter a is a critical distance at which the relative permittivity of the system de-

2.3 Atomistically More Detailed Methods - Structure Prediction

creases to unity, its value in vacuo. The value of x is designed such that the two portions of the function converge at r equal to a. This critical value should correspond to the point at which no other polarizable matter, i.e. atoms, may fit between the two charged species (atoms or pseudoatoms). One physically reasonable approach would be the use of a as the sum of the van der Waals radii for the two species in question (Ludovice and Suter, 1990). Another approach would be to simply use the sum of the two largest radii in the system (Hutnik et al., 1991 a). A problem with either approach appears when one attempts to calculate the total potential acting on a given atom due to all surrounding atoms (infinitely) in all directions. This problem will be addressed in Sec. 2.3.3.1. Just as the Lennard-Jones function UNB was truncated and fitted with a quintic spline to make it go to zero at a given distance R, the potential energy function due to non-bonded polar (coulombic) interactions, (7NBP, is also truncated and fit with a quintic spline. The form of (7NBP is shown below:

uc(r) = L/NBP =

49

distance dependent dielectric into the potential. Figure 2-2 shows the similarity between the splined potential function with and without attenuation by the dielectric constant D (r) in dimensionless units for the interactions Cl/methine-H in poly(vinylchloride) using a value of sB = 3.51 (Ludovice and Suter, 1990). In addition to the above potential to describe non-bonded interactions, an intrinsic torsional potential associated with the skeletal bond angle torsion is also used. The intrinsic torsional potential is that portion of the torsional potential commonly argued to include only electron overlap (Borisova and Volkstein, 1961a, 1961 b; Hendrikson, 1961; Scott and Scheraga, 1965) and necessary to bring experiment and molecular mechanics results into agreement for most small molecules. Information on how to construct this potential can be gleaned from experimental evidence (i.e., crystal data) or quantum mechanical calculations on small molecules and will, of course, be specific to the polymer system being studied. For poly(propylene) (like all

4nrD(r)e0'

/

(2-7)

+ I 6 — + 3A - ^ + -z--^ \

a

a/a

)£2\,

2 OL/O2J

J

R2
r> R where l/J, U'°9 and U'^ represent the value, and the first and second derivatives of the coulombic potential at r = R2,

A=(R-R2)/a

and Z =

{r-R2)/(R-R2)

= (r/
unstrained alkanes), the intrinsic torsional potential, L^() is modeled by a standard symmetric three-fold expression: (2-8) where 0 is the bond torsion angle (measured from the trans state using Flory's

50 10

2 Amorphous Polymer Microstructure Electrostatic potential Uc 1.5

Vertical expansion

2.5

0.5

1.0 1.5 2.0 Interatomic separation/a

2.5

Figure 2-2. Quintic spline for the non-bonded polar potential for PVC. The coulombic potential energy function is shown in a dimensionless form.

convention) and fc0 is the height of the energy barrier. Note that more elaborate potentials are needed for more complicated polymeric species (Hutnik et al., 1987, 1991a). 2.3.1.2 Initial Guess Generation

The generation of an appropriate initial guess polymeric structure is the most important part of this class of models. Unrealistic initial guesses lead to unrealistic structures after the minimization is performed and thus the question of how initial guesses are generated is very important. Inasmuch as initial guesses predetermine final structure, they should be representative of "reality" in incorporating the correct chemical structure. For example, in the case of stereopolymers, an appropriate dyad tacticity should be determined. Furthermore, generated chains should express the correct spatial arrangements such as chain dimensions as expressed by the root mean square radius of gyration, <s2>. Unrealistically large gaps in the bulk polymer initial guess should also be avoided. According to Flory's well-established "random coil hypothesis" (Flory, 1953) chain confirmation in the amorphous bulk is essentially identical to the conformation

of chains in a solvent under 9 conditions. Thus, the chain appears effectively statistically unperturbed by long-range interactions and behaves with respect to its global dimensions as if it could self intersect. From the view that glasses are in a state of frozen-in liquid disorder and the fact that chains in the melt are in a 9 state, one can deduce that the conformational statistics of the chains in the glass are not very different from that in the melt. The conformational statistics of unperturbed chains are well described by rotational isomeric state theory (RIS) (Flory, 1969), which rests on a discretization of conformational space into a finite set of rotational states. Using these discrete states, one can compute (using energies of the various states determined by experiments and quantum mechanical methods) the a priori probabilities p^t (bond i in state £) and conditional probabilities q^;i (bond i in state £ when bond f — 1 is in state C) for all bonds in the chain. These probabilities define a Markov process and thus, using a Monte Carlo approach, correctly weighed unperturbed single chain conformations can be obtained. Since RIS theory only incorporates short-range interactions, if a box is filled with chains described by the process above, certain problems will occur: the chains will have unusually high energies and there will be unrealistically large heterogeneities in the initial guess. The incorporation of long-range interactions in the generation of initial guesses is required to solve these problems; it is accomplished by modifying the RIS scheme to include longrange interactions at each step of the chain generation. This is done by redefining the RIS conditional properties to include the total system state probabilities (including long range interactions) (Theodorou and Suter, 1985 a).

2.3 Atomistically More Detailed Methods - Structure Prediction

The use of such modified RIS schemes allows for the generation of initial guess structures that: (1) do not depart strongly from the random-coil hypothesis (i.e., have approximately the correct unperturbed chain dimensions), (2) have lower energies than by using only a simple RIS scheme and (3) have relatively uniform spatial distributions of segments. Thus, satisfactory initial guesses can be obtained with this method. 2.3.1.3 Potential Energy Minimizations Once appropriate initial guesses have been generated, the structure is submitted to a minimization of the total potential energy in order to reach detailed mechanical equilibrium (recall that the term detailed mechanical equilibrium defines a system where the potential energy is a minimum with respect to all degrees of freedom in the system). The total potential energy of the system is the sum of all bond intrinsic torsional energy contributions and all nonbonded interactions. The non-bonded interactions are computed according to the minimum image convention. To date, structures have been minimized using a "quasi-Newton" method algorithm developed by Broyden, Fletcher, Goldfarb, and Shano (Hillstrom, 1976). In order to prevent very large computation times, structure minimization is accomplished in three stages. The stages are designed so as to give the structure an opportunity to relax only the "strongest" interactions during each stage. The exact nature of each stage is, of course, dependent on the polymer being studied. In general the stages make use of different computational expediences: (1) Initial parts of the minimization are performed using the soft-sphere potential discussed earlier. This allows the structure to feel and relax only the highly

51

repulsive interactions. Later, the full potential is used to finish the minimization. (2) Rotational potentials are often not used during the initial course of the minimization. (3) Van der Waals radii of the atoms are shrunk initially and later "blown up" (Vacatello et al., 1980) to provide smaller interactions earlier on. This gives the structure a bit of "elbow room" to minimize properly. The structure is deemed to meet the requirements of a detailed mechanical equilibrium if the derivatives of the potential energy with respect to all degrees of freedom are low (i.e., less than 10 " 4 kcal mol" 1 deg "*), and the exact Hessian matrix for the final stage is positive definite (indicating a true minimum). It should be reiterated that one structure is not nearly enough to study. One requires an ensemble of structures to gather meaningful results. 2.3.1.4 Example - Atactic Poly(propylene) Theodorou and Suter (1985 a) first used the above technique to describe the behavior of amorphous atactic poly(propylene). This polymer was modeled with a degree of polymerization of 76 and containing 48% meso-dyads. The bulk polymer was generated in a cubic cell with a box edge length of 18.15 A. Figure 2-3 shows an example of a model structure in the cube (center) and eight of its neighbors projected on the xy-plane. The parent chain is traced out in a bold line. Hydrogen atoms have been omitted for clarity. Figure 2-4 shows a periodic cube of the same atactic polypropylene). The system size can, however, be significantly larger. Recently, Mott et al. (1989, 1990) have successfully minimized an atactic poly(propylene) structure of degree of polymerization equal to 608 and a cube edge length of 36.8 A.

52

2 Amorphous Polymer Microstructure

Figure 2-3. xy projection of poly(propylene) (X = 76) cube with eight of its nearest neighbors.

Theodorou and Suter used a five-state RIS scheme modified to include long range interactions to generate their initial guess conformations. These conformations had acceptably low energy (10 6 -10 7 kcal/mol) and parent chain dimensions in line with experimentally determined results (Theodorou, 1985). Fifteen structures were minimized, and were used as ensemble members to describe the characteristics of poly(propylene) (see following sections). It is important to note that the model just described contains no freely adjustable parameters. Figure 2-4 Schematic representation of a single polypropylene) microstructure (X = 16).

2.3 Atomistically More Detailed Methods - Structure Prediction

2.3.1.5 Extension to Polar Polymers Atactic Poly(vinylchloride) and Polycarbonates

For poly(propylene), the partial charges that exist on methyl groups, main chain carbon and hydrogen atoms are too small to cause significant polar interactions and can safely be ignored. However, in order to use the technique to predict the properties of polymers with stronger polar interactions, such as poly(vinylchloride) (PVC) or polycarbonate (PC), modifications were required. The modifications involved were in two areas: point charges on the individual atoms were added (and hence, coulombic terms in the calculation of the potential energy), and the stage-wise minimization of the structure also contained scaling of charges. It is possible to estimate partial charges through quantum mechanical calculations on small molecules and chain fragments (ideally, at least one repeat unit of the polymer). A test for the acceptability of these partial charges is to use them to calculate the dipole moment of the small molecule and compare it to the experimental value. The partial charges used in the calculation for polar polymers are considered constant. Although a model that includes polarization effects may be more accurate (Stillinger and David, 1978; Boyd and Kesner, 1980), it would be far too computationally intensive for systems of such high molecular weight. The first polymer investigated was atactic poly(vinylchloride). As with polypropylene) simulation, the initial guess was generated in a step by step fashion employing a modified Monte Carlo method (a RIS scheme modified to take into account long range interactions) based on a 3-state RIS model (Flory and Williams, 1968; Flory and Pickles, 1973).

53

Again, a stage-wise minimization was performed. The minimization stages had to be adjusted due to the presence of electrostatic interactions [coulombic interactions decrease with interatomic separation as 1/r, whereas the Lennard-Jones energies change as 1/r6; use of D(r) causes the coulombic term to depend on r more strongly in an important regime, but still more weakly than the Lennard-Jones term], resulting in a minimization procedure with four stages (Ludovice, 1989). Sets of two different size PVC cubes were simulated, an ensemble of 10 structures of degree of polymerization 76 (cube edge length = 17.84 A), and a single structure of degree of polymerization 200 (cube edge length = 24.64 A) at a density of 1.39 g/ cm3, with 43% meso dyads distributed in a Bernoullian fashion. A rendering of the larger structure is shown in Fig. 2-5. The differently sized cubes were employed for the investigation of size-dependent effects (Ludovice and Suter, 1990). With polycarbonates, a step in the direction of "rigid" chains was attempted. [The terms "flexible" and "rigid" are used here in a vague sense and refer to the size of

Figure 2-5. Schematic representation of a single poly(vinylchloride) microstructure (X = 200).

54

2 Amorphous Polymer Microstructure

Table 2-2. Some properties of polycarbonates of bisphenol A and substituted bisphenol A. Polymer

Density a (g/cm3)

Mechanical failure behavior b

Permeability selectivitya

Oxygen diffusion

PC TMPC TBPC

1.2 1.08 1.95

ductile brittle brittle

5.1 5.1 7.5

5.6 3.8 5.0

a

Data from Koros et al. (1988); b from Freitag et al. (1988); c D is the apparent diffusion coefficient measured by the time lag method in 10~8cm2/s.

kinematically stiff segments of the chain; poly(propylene) and poly(vinylchloride) are flexible in this context.] These species represent a border between "flexible" and "rigid" polymers. The compounds investigated here are the polycarbonate of 4,4'isopropylidene biphenol and substituted analogues. Table 2-2 shows some properties of polycarbonates with different simple substituents on the phenylene ring in ortho positions to the carbonate group (Koros et al., 1988). For the tetra methyl (TMPC) and tetra bromo (TBPC) polycarbonates, both mechanical properties and diffusion of small molecules in the polymer are affected. Two system sizes were simulated at realistic densities at degrees of polymerization of 35 (17 repeat units, cube edge length = 18.5 A) and 151 (75 repeat units, cube edge length = 30 A). Note that the latter is of similar molecular weight (19 800) as some low molecular weight commercial grade PC. In all, thirteen structures with cube edge length equal to 18.5 A and 2 structures with cube edge length of 30 A were minimized satisfactorily. Figure 2-6 depicts one of the larger polycarbonate microstructures. The rigidity of polycarbonate is reflected in the fact that the potential energy hypersurface, on which minimization is attempted, is much more severely corrugated than the flexible chains tested earlier, and

the elements of the Hessian matrix span a much large range. As a result, minimization by stages, as for flexible polymers, usually requires on the order of 30 stages as opposed to 3 and 4 for PP and PVC, respectively. A particularly vulnerable point of this class of models is the "realism" of the produced chain packings. If the structure was sensitive to the path used in its generation it would be a substantial drawback. Which structure, if any, would be "correct"? With the PC system, an attempt at clarifying this question was made: two general minimization paths were chosen, one in which the atomic radii were grown much faster than the partial charges on the atoms, and one where the partial charges

Figure 2-6. Schematic representation of a single bisphenol A polycarbonate microstructure (X = 151).

55

2.3 Atomistically More Detailed Methods - Structure Prediction

Table 2-3. Simulation of structure of densely packed amorphous polymersa. Polymer

Xh

Degrees of freedom

Atoms c

Cube edge length

(A)

Number of microstructures

PP

16 200 608

153 401 1217

455 1199 3647

18.4 25.4 36.8

>20 1 1

PVC

76 200

153 401

455 1199

17.8 24.6

10 1

PC

35 151

109 455

485 2051

18.4 30.0

13 2

TMPC

35

109

485

20.2

12

TBPC

35

109

485

20.5

10

a

See text for references; b degree of polymerization;c number of atoms assumes that all methyl groups are treated as pseudoatoms.

were increased much faster than the atomic radii, during the staged minimizations. Another sensitive point is the accuracy of the force field employed, and one of the most important parameters is the assignment of point charges to the atoms of the polymer. In PC, the point charges used were based upon the results from a semi-empirical quantum mechanical package (AustinMartin Method 1, AMI; AMPAC, 1988). Alternatively, charges based upon CNDO results were also employed. As with path dependence, a significant force-field dependence would also shed doubt on the structures generated. To factor out the possible effects of varying initial guesses, polycarbonate structures were generated in four sets of runs from a single set of initial guesses by the two paths and with the two force fields. The results for a dozen initial guesses showed that no significant changes in the structures occurred through the use of different minimization paths or force-fields (Gentile et al., 1989). The structures of TMPC and TBPC were also simulated. Altogether, twelve structures of TMPC of X = 35 (17 repeat

units, cube edge length = 20.5 A) and ten TBPC structures of X = 35 (17 repeat units, cube edge length = 20.5 A) were obtained (Gentile and Suter, 1992). Note that TMPC and TBPC structures are enclosed in roughly identically sized boxes. This similarity in number density is a coincidence. TMPC and TBPC have very different densities as shown in Table 2-2 (Koros etal, 1988). The results of the prediction of structure and properties for polycarbonates will be discussed, along with PP and PVC, in the following section. Table 2-3 summarizes the polymeric microstructures generated to date. 2.3.2 Prediction of Structure 2.3.2.1 Cohesive Energy Density and Solubility Parameter The cohesive energy, £/coh, is the energy due only to intermolecular forces. It can be estimated from an ensemble of microstructures by taking the difference between the total interaction energy, J7tot>system, and that of the parent chains only, C/par. The Hildebrandt solubility parameter 5, is

56

2 Amorphous Polymer Microstructure

defined as the square root of the cohesive energy density (Van Krevelen and Hoftyzer, 1976). ^par TJ

^tot,systen l/2

(2-9)

where V is the molar volume appropriate to the system. Energy values used in generating bulk structures are based upon range limited Leonard-Jones and, where applicable, coulombic potential energy functions. The values of C/bulk are therefore missing interactions at large distances. In order to correct for the effect of truncating the potentials, a "tail correction" (A(7tails) is performed so that UtoUsysiem =

Uhulk-AUtails

(Theodorou and Suter, 1985 a; Ludovice and Suter, 1990). This tail correction is given by the following expression: Nat

cube, Rsd, using the pair distribution functions obtained from the microstructures. JVat refers to the number of atom species types. Since the generated microstates make use of periodic continuation, pair distribution functions cannot be calculated for values of r greater than the cube semidiagonal, for the integration to r = oo. It is assumed that #0(r) is unity for all values of r greater than K* and that part of the integrals is evaluated analytically. Examination of the atomic pair distribution functions for all cubes confirms that the assignment of gtj(r) to unity beyond Rsd is reasonable. The assumption of gtj(r) = 1 for r > Rsd forces the kernel of the second integral to vanish for all r > R*; this is probably an unrealistically small interaction between charges separated by a large distance.

Nat

7=1

ijir) [U\f(r) - 17™(r) + I7?,(r) y=

r2 dr

(2-10)

min(RUij9R2tij)

where Nt and gt are the number and number density of the atoms or pseudoatoms of species i in the periodic box. The summation indices refer to the different atomic species, i.e. centers with different LennardJones parameters or charges. The different atomic species pair distribution functions are denoted by #l7(r). It should be noted that the calculation of the tail correction is not just useful for the estimation of Ucoh, but also is required when one needs to use the generated microstructures to estimate mechanical properties (Theodorou and Suter, 1986). The integrals in Eq. (2-10) are evaluated numerically from the lower limit to #*, a value less than the semi-diagonal of the

Hence, another (upper) limit of the electrostatic part of the tail correction was estimated using the local electroneutrality of a repeat unit (see below). One should note that making gtj(r) = 1 for all r>R* is tantamount to assuming that the microstructures are completely amorphous on the scale of the cube. However, the slow decay of the coulombic term may contribute a perhaps significant amount of energy due to correlations at larger distances. Since the total charge of this system is zero, the sum of the coulombic interactions of an arrangement of fixed charges decays as 1/r3 for large r (Maitland et al., 1981). When a system is composed of individual dipoles that intially arrange themselves according

2.3 Atomistically More Detailed Methods - Structure Prediction

to a Boltzmann distribution as if in thermodynamic equilibrium (i.e., in a liquid), then the sum of the coulombic interactions decays as 1/r6 (Maitland et al., 1981) but, in a polymeric glass, the high viscosity prevents the arrangement of dipoles according to a Boltzmann distribution. To assess the underestimation of the long-distance coulombic energy by the assumption of total amorphousness, we imagine that liquid-like ordering of the dipoles can indeed occur in the glass. Each charged atom can be assumed to be surrounded by a sea of randomly distributed dipoles oriented with respect to charge according to the Boltzmann distribution. (In reality, these dipoles can probably freely orient themselves only at temperatures above Tg, and one might assume that they are "frozen-in" at Tg during vitrification.) Thus, an upper bound for the long-distance part (r>#*) of the second integral in Eq. (2-10) can be determined. The increment of this liquid-like approximation over the completely amorphous model is given by the expression: 1

Nat

Nat

2.3.2.2 Polymer Microstructure Individual Chain Characteristics The model allows one to study locally the effect of detailed intermolecular interactions on the relaxed glassy polymer

oo

dr = *> ; = i

where \i is the effective dipole of a repeat unit (Boyd and Kesner, 1980). The degree of polymerization is denoted by X, and R* is the distance less than or equal to the cube semi-diagonal from which gtj(r) is assumed to be equal to unity. The temperature is denoted by T and k is the Boltzmann constant, x is given by Eq. (2-6). The potential energy of the parent chains is determined by using non-truncated potential energy functions. Table 2-4 outlines the results for the calculation of the cohesive energy density and Hildebrandt solubility parameter for PP, PVC, PC, TMPC, and TBPC. As shown, for all cases the results are in excellent agreement with those available from experiments or group contribution methods.

7=1

-2KX/I2

(2-11)

3kTV(4neos^)2(R*-x)i^1

Table 2-4. Calculated cohesive energy and Hildebrandt solubility parameter for some amorphous polymers. X (No.)a

Ucoh (kcal/mol)

<5(J/mir

<5exp(J/ml)1/2

76 (15)

174 ±20

14.2±0.8

16.8c

76 (10) 200 (1)

239 + 61 756 + 64

17.0 + 2.2 18.8 + 0.8

20.7 ± 1.5 d

PC f

17 (13) 75 (2)

339 ±39 1514±97

19.4 + 1.1 19.7±0.5

20.2±0.1 c ' 8

TMPC h

17 (12)

304 ±43

15.7 + 1.1

1 9

TBPC h

17 (10)

445 ±68

23.0 ±3.2

21 l

Polymer Atactic PP b Atactic PVC d

a

i

X = degree of polymerization, No. = number of microstructures; b Theodorou and Suter (1985 a); c experimental values from Van Krevelen and Hoftyzer (1976) (for PP this value probably refers to the isotactic form, which would make it somewhat higher than the atactic form); d Ludovice and Suter (1989);e Brandrup and Immergut (1975);f Gentile et al. (1989);g Dr. G. Weymans, Bayer AG (1989), personal communication; h Gentile and Suter (1992); * group contribution calculation (Van Krevelen and Hoftyzer, 1976).

58

2 Amorphous Polymer Microstructure

structure. For PP, PVC and PC, a distribution of rotation angles can easily be determined and compared with RIS values. For PP and PVC, five and three state RIS schemes were used respectively. The simulated structures are much richer in the g state than in the RIS prediction. It is likely that the enrichment in some conformations is due to the steric interactions the chain encounters while "growing", taken into account via intermolecular interactions not present in a standard RIS scheme. Linear correlations along the chain may be quantified by the bond directional correlation function, S, calculated from the chain directions (chords between midpoints of adjacent bonds) separated by a given number of intervening bonds along the parent chain: (2-12) where 6 is the angle between two different chords. S, calculated as a function of the number of backbone bonds separating two given bond chords, i and), is denoted Stj. An example for PVC is shown in Fig. 2-7. The fact that the function deviates significantly from zero only for the smallest value of | i —j | demonstrates that no oriental correlation exists along the chain (Ludovice, 1989); positive values indicate a tendency of parallel alignment of the bond chords, while negative values reflect a tendency toward perpendicular arrangements. Structure of the Bulk Polymer Glass Structural order on a scale smaller than that of roughly half the periodic cube can be observed by analysis of the packed structures. Cubes of different sizes will be considered together here even though there are many fewer structures of larger edge length, since the correlations of inter-

i 0.2 l/f

nn

„ . AA,/V

l\

A h/lM

V

A^-

V \J Y v v v"

-0? 50 100 Bond index difference I i-j I

150

Figure 2-7. Bond directional correlation function, Sij9 as a function of bond order difference for a single poly(vinylchloride) microstructure (X = 200; 26.64 A cube).

est are based on the number of interacting pairs. For example, one PVC microstructure of cube edge 24.64 A (X = 200) has 69% of the number of pair interactions of ten PVC microstructures with cube edge length 17.84 k(X = 76). Hence a few large cubes can be used as a test for convergence: if results from two different sized cubes indicate different spatial correlations, the system size necessary for an unbiased model has not been reached (Ludovice and Suter, 1990). The structure of the bulk polymer can be described by the bond directional correlation function S(r) and pair distribution functions g (r). Bond Directional Correlation Functions The bond directional correlation function discussed in the previous section can also be used to quantify the chain directions (bond chords) separated by a given distance r for both a single chain and different images of the same chain; the separation distance, r, is measured from the midpoints of the respective bond chords. S(r) indicates the correlations at given spatial distances. Of special interest are the intermolecular contributions to S(r), where the two chords are part of different images of

2.3 Atomistically More Detailed Methods - Structure Prediction

-0.5

Figure 2-8. Intermolecular bond directional correlation function, Sinter(r), as a function of radial distance for (a) PP (X = 16), (b) PVC (X = 16; 17.84 A cube), and (c) PVC (X = 200; 26.64 A cube).

59

the chain. Figure 2-8 shows values of ^nter W for PP and PVC (Theodorou and Suter, 1985 b; Ludovice and Suter, 1990; respectively). A strong positive correlation exists in both sized PVC cubes at approximately 4.3 A while PP displays a negative correlation here which has been attributed to the "natural" self-avoidance of the chain at glass densities. Thus, there is some indication of a special tendency towards parallel alignment in PVC. The result implies that a polymer in an amorphous state [Sinter(r) is roughly zero beyond r approximately equal to 7 A], can form some subtle structural order despite an irregular stereochemistry. The strong differences in the two different sized PVC cubes indicates the presence of system-size dependent spatial correlations. This may indicate a tendency towards structural order in "amorphous" atactic PVC (Ludovice and Suter, 1990). In PC, an interesting correlation is that of the mutual alignment of carbonate groups (i.e., the carbonyl, C = O, bond directional function) (Hutnik et al., 1991a). Again, two different sized cubes were studied; and the intermolecular bond directional correlation function for the carbonyl bonds is shown in Fig. 2-9. There is a tendency for carbonyl bonds to orient parallel

13 PC structures (a = 18.5A)

Figure 2-9. Carbonyl bond directional correlation function for PC (X = 35) as a function of radial distance.

-0.5 0

2

A

6

60

2 Amorphous Polymer Microstructure

or anti-parallel at distances of approximately 3 A. The results from the two different sized systems do not differ significantly, suggesting the structural aspects of the system have converged at the level of the smaller cube size. Atomic Pair Distribution Functions Insight into bulk structures can be gained through the atomic pair distribution functions, gtj(r\ for atoms of the type i and j . This function is defined by the probability of finding an i center and a j atom center at a distance in the interval [r, r + dr]. These distribution functions are computed so as to include only those pairs whose distance depends on the degrees of freedom in the system. The portion of the pair distribution function that is independent of conformation would show up as "Dirac spikes" in the pair distribution. gij(r) can also be split into its intramolecular and intermolecular contributions. Typical "amorphous" behavior is almost always exhibited. Interestingly, for PVC (Ludovice and Suter, 1990), both the total and intermolecular pair distribution functions are not markedly different from those computed for PP (Theodorou and Suter, 1985), despite the markedly different interchain alignments (see above). The similarity suggests that the polymer "structure" is largely determined by the steric repulsion of polymer chains, and not much by polar interactions. Atomic pair distribution functions were also computed for PC. Most of these correlations show amorphous behavior comparable to that of poly(propylene), but some degree of intermolecular order does exist as well (Hutnik et al., 1991 a). Specifically, there appear to be not intermolecular "hydrogen-bonding" interactions, but some degree of correlation in the intermolecular

part of gi}{r) for the atoms of the carbonate group, indicating that these species may be important in determining the structure of polycarbonate, leading towards the observed parallel alignment of the carbonate groups. Comparison with Experiment The most direct experimental techniques to study the structure of solids are scattering methods, most commonly involving X-rays or neutrons. It is possible to determine the scattering behavior from the structures described here. The intensity of scattered radiation, I(Q), is related to the structure factor, S(Q), both functions of the magnitude of the scattering vector Q [equal to (4n sin 9)/1, where 9 is the scattering angle and X is the wavelength of the incident radiation] as (Wagner, 1978): /(<2) = < / 2 > - < / > 2 + < / > 2 S ( 0

(2-13)

2

where and denote the averages of the scattering power, ft{Q\ for all scatters i in the sample and are independent of the detailed structure. Expressing the intensity and structure factor as a function of Q only implies that the scattering is completely elastic. The structure factor S(Q) contains all structural information of the material and must incorporate the effect of thermal motion. To account for shortcomings of static models to properly account for dynamic terms, Ludovice and Suter (1990) employed a spherically symmetric Gaussian distribution of thermal motion, so that (Wagner, 1978):

i= 1

rNi

J=l

-i

(2-14)

2.3 Atomistically More Detailed Methods - Structure Prediction

where Nt is the number of atomic species i9 A is the root mean square displacement of the atomic motion taking approximate account of thermal motion (Debye-Waller factor) (Azaroff et al., 1974) and Iis(Q) is the partial interference function: 0

\l

(2-15) where Q0 is the number density of atoms and gij(r) is the pair distribution function which now includes contributions that are independent of conformation. The value of A is usually between 0.3 and 0.4 A for organic molecules (Vainshtein, 1966); 0.3 was used in this case. The overall interference function may be numerically evaluated to yield S(Q) and I(Q). Figure 2-10 shows an application of this method to neutron scattering performed on PC. The experimental curve from Cervinka et al. (1987) is compared to simulations from an ensemble of 13 PC structures in periodic cubes of edge length equal to 18.5 A (Hutnik et al., 1991 a). Intensity is in arbitrary units. The comparison is quite good here, but there are often considerable discrepancies between these simple estimates and experiment. The source of these differences is currently under investigation.

61

Other Structural Features Visual examinations of space filling molecular models yield qualitative information on the degree of homogeneity of the structures. Such structural visualizations showed that there are large heterogeneities in PC in the form of "empty space" in the bulk polymer with characteristics dimensions on the order of 3-5 A in diameter. A more rigorous analysis of the empty space analysis was performed by Arizzi et al. (1990). These researchers investigated the interstitial space in the structures as a prelude to the study of diffusion in glassy polymers. This analysis confirmed the existence of a great deal of volume not occupied by atoms in PP and PC, there being, however, considerably different distributions of "empty volume". 2.3.3 Prediction of Properties in the Quasi-Static Approximation

If a molecular process is slow compared to the local vibrational time scale (i.e., r < c a . 10" 10 s), simulations that rely on the faithful reproduction of molecular motion on an atomistic level are not appropriate. An effective approach can instead be employed which departs from the static,

Experimental

Figure 2-10. Comparison between experimental and simulated PC neutron scattering behavior.

Predicted (a = 18.5 k) 0.0

1.0

2.0

Q in A"

3.0

62

2 Amorphous Polymer Microstructure

fully relaxed structures described on the previous pages ("zero-temperature" structures at some reference density). These "fully relaxed" systems are then subjected to changes in one or a few of their degrees of freedom in small increments with all other degrees of freedom being continuously adjusted to retain a structure of minimal energy. This process "drives" the system along the bottom of a "valley" of potential energy towards a saddle point; the process is quasi-static and reversible in that a reversal of the process lets the system retrace its path in configuration space exactly. The change in potential energy upon these changes is a rough approximation to the mean effect a dynamic system would display for the same changes in driven degree^) of freedom. If the procedure is carried in one direction sufficiently far that a saddle-point is reached, it typically results in an uncontrolled (dissipative) change in structure and a precipitous drop in potential energy; the saddle-point is then viewed as the "activated state" which is still in "equilibrium" with the starting state and the difference in potential energy is a rough approximation to the activation energy of the process being investigated. Rates, finally, can be estimated from this activation energy and estimates of the classical partition functions of the starting and saddlepoint structures can be determined. This method to derive information on the response of detailed structural models to stimuli of various nature is termed quasistatic. In the following, some applications are discussed. 2.3.3.1 Local Molecular Processes in Polycarbonate Hutnik et al. (1989) used PC microstructures to investigate local dynamics of phenylene rings at glass densities. Experi-

ments indicate (Jones et al., 1983; Schaefer et al., 1984; Schmidt et al., 1985; Roy et al, 1986; Bayer AG, 1989) that the phenylene rings in PC undergo oscillation about their mean positions and "ring flip" processes in which the phenylene moieties rotate by about 180 degrees in a relatively short time. These processes are the focus of the computations reviewed here. The energy barrier of a "ring flip" is due to a combination of intramolecular and intermolecular forces. Molecular modeling techniques such as quantum mechanical calculations may be used to determine the intramolecular contribution to this activation energy, but at glass densities, intermolecular contributions are high and could potentially dominate the ring-flip barrier (Hutnik et al, 1989). The frequency range of this dynamic process in PC below the glass transition temperature is on the order of 108 s ~ \ indicating that a quasistatic approach would seem appropriate. To calculate the barrier, one begins with the relaxed static structure. Then, a phenyl ring is incrementally rotated and one of its torsion angles fixed in a specified position while the entire rest of the static structure is allowed to relax. Differences in energy between the "totally relaxed" structure and the newly relaxed structure (with one "fixed" phenyl ring) are recorded. The process is repeated until a flip is actually performed in a laboratory reference frame. Every ring of the microstructure is so treated, yielding a sufficient statistical sample. Microstructures of degree of polymerization of 35 (17 repeat units, cube edge length = 18.5 A) were used in this study. One can make the following observations for the polycarbonate ring flip with respect to the structure: in general, in the "ascent" to the energy "peak", the conformational changes which occur are completely reversible; after, they are not; this

2.3 Atomistically More Detailed Methods - Structure Prediction

behavior corresponds to the unrestrained relaxation from a saddle-point configuration in transition state theory (Hutnik et al., 1989). Furthermore, when viewing the entire structure, one sees that the conformational changes in the chain are far reaching and involve all parts of the packed sample to the borders of the periodic microstructure, indicating that a ring flip actually involves matter in a volume larger than the one modeled. The mean value of the phenyl ring flip activation energy calculated for 34 phenyl ring flips is equal to 10.4 + 6.0 kcal/mol. This is in good agreement with values determined by solid state NMR experiments: 12 kcal/mol determined by proton spin lattice relaxation (Jones et al., 1983); 11 kcal/ mol and 12 kcal/mol determined by 13 C NMR dipolar rotational spin echo experiments (Schaefer et al., 1984; Roy et al., 1986 respectively); and 9.1 kcal/mol by deuterium NMR (Schmidt et al., 1985). [It is also interesting to note that the apparent activation energy of the "^-process" at approximately — 100°C is 12 kcal/mol by dynamic-mechanical spectroscopy (Bayer AG, 1989).] A number of further experimental facts also found their matching counterpart in these simulations, e.g., frequencies of ring oscillations or variation in phenylene-flip axis (Hutnik et al., 1991 b). A similar simulation can be done to determine the changes in the carbonate groups. Hutnik et al. (1991 b) have "driven" 24 carbonate groups by the same method and determined the carbonate group flip activation energy to be 9.9 + 5.5 kcal/mol. This is in good agreement with measurements by dielectric relaxation, 9 kcal/mol (Bayer AG, 1989).

63

2.3.3.2 Diffusion of Small Molecules in Glassy Polymers Unoccupied Space Analysis In an attempt to clarify the phenomena of diffusion and sorbtion of light gases in glassy polymers, Arizzi et al. (1990) started with an analysis of the atomistic interstitial space. This analysis is based on the amount of space available to a hard-sphere "guest" in an atomistic hard-sphere matrix consisting of static amorphous packings of polymer chains, where the spatial coordinates of every atom are known. Given a spherical diffusant with a fixed radius, a quantitative determination of the space available around each polymer atom is required as well as a description of how these local portions of empty space are distributed across the system. First, a Delaunay tessellation (Dirichlet, 1850) is carried out. This is a deterministic filling of space which is dual to the Voronoi tessellation and is defined (in 3-D) as the network of tetrahedra formed by the lines joining nearest neighbors in a Voronoi sense and has the important feature of guaranteeing a unique subdivision of the sample volume into tetrahedral bits that are "empty". A very efficient algorithm of Tanemura et al. (1983) for Voronoi tessellation can be used, a side-product of which is the desired Delaunay tessellation. The space effectively available to sorbtion and diffusion in each tetrahedra is then calculated, by subtracting from the volume of each tetrahedron the pyramidal sectors occupied by the hard-sphere polymer atoms at its vertices. The diffusant (the "guest"), depending on its radius, will be able to move between the tetrahedra in some regions of the static glass (where the faces of the tetrahedra contain unoccupied area sufficient to let the guest pass), whereas its movement beyond these domains will be

64

2 Amorphous Polymer Microstructure

made impossible by denser packings of matrix atoms. This defines a cluster: a part of the polymeric structure where a diffusant of a given size can move from one tetrahedron to a neighboring one. It is obvious that, for a particular structure, the clusters for the diffusants of smaller radii will be larger or more numerous or both. The cluster-analysis allows for a simple characterization of space available to guest molecules and provides opportunity for visualization of that space as well as for simple considerations in relation to diffusion (the motion of a diffusant internal to a cluster can be considered to mimic a process with an activation energy much lower than the one required when moving outside a cluster). Frequency distributions of cluster volume for He, O 2 and N 2 averaged over 10 PC structures of cube edge length 18.5 A are shown in Fig. 2-11 (Arizzi et al., 1992).

diffusion in amorphous metals, one can approach modeling of the diffusion coefficient of small molecules in a glassy polymer based on activated state theory with the static polymer microstructures. The Delaunay tetrahedra are viewed as sites of preferential residence ("ground states") from which the diffusant "jumps" to a neighboring tetrahedron via an activated process. The energy barriers that a diffusant has to overcome in order to move from a position of local energy minimum to a neighboring one are estimated from the simple force fields discussed earlier. A Monte Carlo simulation in which both the random walk direction and the residence time in each local energy minimum are stochastic variables and functions of the energy barriers is then performed. This model can be applied to different gas/polymer systems. The theory of stochastic processes (Chandresekar, 1943) states that in three dimensions

Simulation of Diffusion

D= lim/(6 0

Following Langonet al. (1985), who have developed an interesting method to model

where r2 is the squared displacement of a diffusant, achieved in time t. D is the diffu-

6-i

6

10BPA-PC structures Diffusant :He

5-

(2-16)

5-

10BPA-PC structures

10BPA-PC structures

Diffusant: 0 2

Diffusant: N2

5-

h-

4-

3-

CO

empty

o

o 2 -I

i

1

o.

Jil lii

400 800 Volume in A3

1200

400 800 Volume in A3

1200

400 800 1200 Volume in A3

Figure 2-11. Distributions of empty space for helium, oxygen and nitrogen in 10 microstructures of PC {X = 35).

2.3 Atomistically More Detailed Methods - Structure Prediction

sion coefficient. Given a diffusant in tetrahedron i at time t, the probability of it jumping to the neighbor tetrahedron j at time t + dt is co^dt, where co^ is the jump frequency from site i to site j , and can be written in an activated state theory form as: ij = v 0 exp < -

kT

(2-17)

where v0 is the frequency corresponding to a jump without energy barrier and can be deduced from first principles, while Etj and Et represent the energies associated with the diffusant on the boundary surface (gate) between tetrahedra i and j and internally to tetrahedron i, respectively. These energies are calculated following the approach discussed previously in modeling non-bonded interatomic interactions. The residence time in tetrahedron i is a random variable with an exponential distribution (i.e., T = Tjlnx, where x is uniform in [0,1] and zt = 1/S Wij is the mean residence time j

in tetrahedron i). The jump frequency and the associated probability of a step into any of the neighboring tetrahedra are independent of time, while the residence time distribution is independent of the next tetrahedron to be visited, i.e. the process consists of two random walks, independent of each other, in space and time. These techniques have been used so far to accurately predict the permeability selectivity of oxygen and nitrogen in PC as well as the diffusion constants (Arizzi, 1990). 2.3.3.3 Mechanical Properties

To model the response (e.g., the conformational changes in the polymer matrix) of macromolecular packings to mechanical loading, one can perform simulated "deformation" and "relaxation" experiments on

65

these static microstructures. Strain appears in this treatment as a change in the rules governing translation for spatially periodic continuation conditions. Basically, one can attempt infinitesimal or finite plastic deformation, where the latter will lead to "plastic" events, while the former can be compared to elastic deformation of real materials. The history of predicting mechanical properties of polymeric materials from first principles is not extensive and what little has been done has dealt almost exclusively with the crystalline state, since the existence of a well defined periodic structure allows for drastic simplifications. An early statistical mechanics treatment of crystalline poly(methylene) by Pastine (1968) gave aP-V-T equation of state that compared well with experimental evidence. Later, Tashiro et al. (1977 b, 1978) derived elastic constants for a variety of crystalline polymers to within 35% of observed quantities. Theoretical approaches to determining mechanical properties of amorphous polymers, until recently, were phenomenological or "group-contributional" in character. Haward and MacCallum (1971) proposed that the adiabatic compressibility of a polymeric glass were determined mainly by intermolecular forces. Using a LennardJones relationship for potential energy they showed that the adiabatic compressibility was a function of V/Vo, where V is the deformed state molar volume and Vo is the molar volume in a state where the potential energy with respect to volume at constant temperature is minimal (Theodorou, 1985). This correlation compared well with experimental evidence, indicating that intermolecular forces are relevant variables for "elastic" mechanical properties. Van Krevelen and Hoftyzer (1976) proposed the use of the Schuyler correlation to

66

2 Amorphous Polymer Microstructure

estimate the compressibility for amorphous polymers from the molar voume, V9 and wRao, the Rao function or molar sound velocity. Tables (Van Krevelen and Hoftyzer, 1976) exist for group contribution estimates of V and uRao. From these, the compressibility is obtained which can then be used to determine the Young's modulus, £, and shear modulus, G, once a suitable estimate for the Poisson's ratio has been made. These correlations however fail in many cases. A more sophisticated approach was introduced by Yannas (1975) and Yannas and Luise (1982, 1983, 1991) which considers molecular level displacements brought on by mechanical deformation of the glassy polymer. Assuming that deformation of a macromolecular segment occurs primarily by rotation around skeletal bonds, rather than by bond angle or bond length distortion, Yannas coined the term "strophon" to characterize a three-virtual-bond-long segment as the smallest possible chain section that can change its end-to-end distance during mechanical loading. Forces required to overcome intramolecular and intermolecular barriers to bond rotation in the bulk were estimated based on a simple strophon model. The estimated contributions of both types of forces to the smallstrain modulus was used as a basis to distinguish which was more important during deformation. It was found that both intraand intermolecular forces were of similar significance in both vinyl polymers and non-vinyl polymers (Luise and Yannas, 1990). The calculations of Yannas and his coworkers were performed on many amorphous polymers using only the polymeric repeat unit as a basis. They pinpointed a dimensionless parameter, called the "locking factor" which is equal to the value of the radius of a circle describing the rotation of a characteristic virtual bond around

the chain backbone bond (the moment arm of the strophon), r, divided by the van der Waals distance between two strophons in neighboring chains which interact according to a Lennard-Jones potential, a0. Calculations of this locking factor can point to different mechanisms of deformation; large values of r/a0, it was found, encourage stress transfer to occur during intermolecular pathways during loading. These researchers also proposed that yielding and the glass transition are related through a critical ratio of the inter-strophon distance at Tg, a(Tg), and its value near absolute zero, a0. Both yielding and the glass transition occur at an "equivalent dilatational threshold" which is crossed either by heating a glass above its Tg, or by subjecting it to a yield stress greater than a critical value, F conf , crazing seems to occur. The transition between yielding and crazing which occurs in some polymers as they approach Tg (e.g., in poly(phenylene oxide) and in the polycarbonate of bisphenol A) provided a basis to calculate the critical Finter "barrier" at the distance of closest approach between two strophons (7000 kcal/mol A and suggested a universal criterion for crazing). Atomistically detailed approaches to changes occurring in the polymer microstructure during deformation are few. In crystalline polymers Rutledge and Suter (1991) describe the mechanism of elastic deformation in fibers of poly(p-phenyleneterephthalamide). The first atomistic ap-

2.3 Atomistically More Detailed Methods - Structure Prediction

proach for amorphous polymers at glass densities based on work in amorphous metals by Maeda and Takeuchi (1981) and by Srolovitz et al. (1983) was performed by Theodorou and Suter (1986) on polypropylene) and Hutnik et al. (1991c) on the polycarbonate of bisphenol A. Both these approaches were limited to infinitesimal deformation. The first atomistic level detail model of finite (plastic) deformation in amorphous polymers was performed by Mott etal. (1990). Infinitesimal Deformation At infinitesimal deformation, the structures are changed (deformed) only in a microreversible manner. Time is not taken into account in modeling since the simulated mechanical loading takes place infinitely slowly (quasi-statically). The first use of this approach by Theodorou and Suter (1986) on atactic poly(propylene) involved a thermodynamic and a statistical mechanical analysis which led to neglect of entropic effects and justified the use of the total potential energy of the static structure in place of the internal energy. The simulation for deformation proceeds according to the following steps: (1) Start with an ensemble of static structures which have been fully minimized and are in detailed mechanical equilibrium. (2) Choose the type of deformation to be imposed, at a small degree of deformation 181 <^ 1 (e is the strain). (3) Modify the continuation vectors such that they correspond to the deformation desired. (4) The structure is then reminimized to the criteria of detailed mechanical equilibrium - this minimization is analogous to the last stage of the minimization procedure which generates the static microstructures.

67

(5) To study different degrees of deformation, change the value of s and repeat the procedure. Using the potential energy (approximately corrected for the error introduced due to truncation of potential tails) of the relaxed deformed structure and volume change one can determine the values of most elastic constants and, from here, all moduli (tensile, shear, bulk modulus and Poisson's ratio). The approach outlined above is referred to by Theodorou and Suter (1986) as the energy approach. It is also possible to determine elastic behavior of the static structures by solving the detailed force and torque balance equations for all atoms and bonds in each structure. From this method, called the force approach, one can obtain, by a direct calculation, the internal strain tensor, a, in all states; deformed and undeformed. In short, it yields more information on the deformation process, and equivalent results for the elastic and Lame constants. The comparison for atactic poly(propylene) at degree of polymerization 76 showed quantitative agreement between simulation and experiment. In addition, upon viewing the structures after deformation, it was seen that deformation in the glassy state is accompanied by a concerted displacement of chain segments about ten bonds long. A similar approach was performed on PC by Hutnik et al. (1991 c). Values of the elastic constants were also essentially in agreement with the experimental predictions. Finite (Plastic) Deformation Once the deformation of a glassy polymeric material exceeds a few percent strain, one discovers a rapidly growing number of irreversible changes, i.e. one can observe "plastic events". Mott etal. (1989, 1990)

68

2 Amorphous Polymer Microstructure

is important. Joining all the atoms that share faces of Voronoi tetrahedra produces a network of Delaunay tetrahedra that uniquely tessellates the interstitial regions. The displacement gradient in one of these tetrahedra can be determined in a straightforward manner from the change in atomic positions upon deformation. The displacement gradient of a tetrahedron is shared by each of its four corner atoms, and the gradient is attributed to the four atoms by the proportion of the tetrahedron's volume falling inside each of the four corresponding Voronoi polyhedra. The displacement gradient of an atom is then the volume average of all Delaunay tetrahedra gradients of which it is a corner atom and the atomic level strain is found by subtracting out the rigid body rotation of the atomic displacement gradient. Figure 2-12 depicts the stress-strain curve for a single microstructure of atactic poly(propylene) of degree of polymerization 76 (cube edge length = 18.15 A). Closer inspection of the individual atomic strains reveals that there is little local incremental strain for most deformation steps; at certain steps, however, a deformation

have succeeded in performing plastic deformation using microstructures of atactic poly(propylene). Starting with the minimized microstructures, a small extensional strain step is imposed on the periodic cube. Subsequent reminimization (as was also done in the case of infinitesimal deformation above) causes the bulk polymer to seek a new conformation. When this is done repeatedly, large deformations can be simulated at infinitesimally small strain rates (quasi-static approach). How the atoms of the bulk polymer change during the course of the deformation processes, and especially during plastic events, is monitored by determining the atomic stress tensor of each atom and its Voronoi polyhedra (Tanemura et al., 1983). The concept of "atomic strain" was introduced to explore the kinematics of atomic rearrangement during plastic deformation (Mott et al., 1990). In the macroscopic continuum sense, strain is defined as the change of an infinitesimal line segment, and is a function of position in that body. On the atomic level, however, only the change in the relative position of the atoms

o CL

A

A

a V

• V

"Forward" imposed extension -

\ "Backward" extension y

y y

g 400 y

C 0)

y

y

y

y

y

\y^ ye \

a 3 200 cr
AtM^i^tA**

if) 0)

a y

y

f

u

y y

g

Elastic loading line •

0.05

.

.

f

.

i

0.10 Work equivalent strain

X

\

0.15

•

X

0.20

Figure 2-12. Stress-strain plot of a single configuration of PP(X = 76) showing "plastic" events (strained at constant volume, extending the y-direction).

2.4 Atomistically Detailed Methods - Dynamic Behavior

69

300 •

Change in system pressure

250 '_

Von Mises equivalent tensile stress

i

200 ;

/ ;

/

u) 100 '.

/

&

Elastic loading line

/

50 -

Figure 2-13. Average stressstrain plot of nine simula-

$

•

•

•

•

•

•

•

•

•

•

•

•

/

'

0.05

'

.

'

tions for PP (X = 76, con-

'

0.10

0.15

0.20

stant volume extension).

Work equivalent strain

"event" takes place which causes excessive rearrangement throughout the entire structure. These can be exactly correlated with the sharp "drops" shown in Fig. 2-13 (labeled a-g). As also shown in this figure, if the deformation is reversed before one of these "drops", the structure recovers "elastically". If the deformation is reversed after a "drop", the stress "drop" is not recovered. These precise events thus correspond to a release of the strain energy which is built up in the system during the deformation. A visual examination of the structure also shows that the changes in polymer conformations are very complex and dramatic and go as far as (and probably beyond) the periodic cube edge length. One can use many structures to determine the stress-strain behavior more accurately. Figure 2-13 shows the results from nine simulations. The average von Mises tensile stress follows the elastic modulus line to approximately 5% strain: the apparent yield point. There is also considerable buildup of hydrostatic pressure, which results from performing the simulations under constant volume.

2.4 Atomistically Detailed Methods Dynamic Behavior The major problems with simulations of dynamic behavior in dense multichain systems is that the approach of the system to equilibrium is very slow and that a large part of the range of characteristic frequencies lies outside current methods of deterministic molecular dynamics (MD). This shortcoming plagues especially simulations that try to describe structure in atomistic detail. Nevertheless, some information can be gleaned from such methods as well. Well relaxed static structures, generated by the methods outlined in previous sections, may serve as excellent starting points for simulation. In addition, the method of generating static structures at glass densities can also be used for static bulk structures at temperatures above Tg. Two recent constrained MD techniques which use the atomistically detailed well relaxed static structures as a starting point will be discussed.

70

2 Amorphous Polymer Microstructure

2.4.1 Constrained Molecular Dynamics in Cartesian Coordinates

The reason for carrying out a constrained MD simulation is to eliminate the higher frequency modes of motion; those associated with bond length and bond angle vibrations (see also above). The use of a constrained molecular dynamics technique in Cartesian coordinates involves carrying out the MD simulation by solving several constraint equations during each step of the dynamic trajectory of the system, since there are no implicit constraints when the atoms are allowed to move. Ryckaert et al. (1977) used Lagrange multipliers, which represent the effective forces needed to satisfy the desired constraints. The constraint equations can be solved in an exact fashion by an inversion of the constraint matrix at each point (Orban and Ryckaert, 1980). Sylvester et al. (1989 a, 1989 b) have adapted this approach to sparse matrix techniques on high speed vector and parallel computers; they chose atactic polypropylene) of degree of polymerization 76 in a cube using periodic continuation conditions as a basis for their calculation. Sylvester et al. performed simulations of the dynamic behavior at both constant temperature and constant pressure (or stress). It was assumed that the glass is trapped in a particular region of phase space with minimal potential energy at the locus of the static structure. Temperature, therefore, is defined through the equipartition theorem by the amount of kinetic energy added to the static structure. When the continuation conditions are held constant this system allows a constant volume constant energy simulation. Control of temperature was achieved by the approach of Nose (1984), in which an extrinsic variable force is used to slow down or speed up each atom in order to keep the average

momentum of all atoms constant. A commonly used constant pressure dynamics approach is used to describe the kinetic energy of the system in terms of "effective masses" attached to the periodic boundaries. This technique, first developed for cubic systems by Andersen (1980) and generalized for parallelepipeds by Parrinello and Rahman (1980, 1981) was also employed by Sylvester et al. (1989 a). Sylvester et al. departed from static microstructures like those described in earlier sections of this chapter (rigid bond lengths and angles), with densities corresponding to temperatures above 393 K and below 233 K (Tg = 253K). Bond lengths and bond angles were then associated with harmonic potentials and the structure again minimized statically (this also required adjusting the force field used). Starting from well relaxed polymer structures ("0 K"), the simulated chains were brought to thermal equilibrium at the selected temperatures by carrying out 10-15 ps constant energy constant volume simulations with occasional velocity rescaling to bring the temperature (which drifted for numerical reasons) to the target value, followed by 15-20 ps constant volume simulations, finally followed by 70 ps isothermal-isobaric simulations. The phase space trajectory was studied for intervals of 20 ps and analyzed. Microstructure analysis was performed using Voronoi polyhedra as a probe (Tanemura et al., 1983; Mott et al., 1989) joining the polyhedra of the atoms belonging to the same chain segment [i.e., CH 2 ("achiral") and CHCH 3 ("chiral") in polypropylene)]. The distribution of the volume in each type of segment polyhedra cluster was computed. Figures 2-14 and 2-15 show the distributions of Voronoi polyhedra clusters for achiral and chiral segments in the simulations above and below 71. At both tern-

2.4 Atomistically Detailed Methods - Dynamic Behavior

71

0.025 , Achiral segments Chiral segments

All segments Average volume = 39.0A 3 Std. deviation^.8A 3

0.020

Achiral segments Average volume=33.8A 3 Std. deviation = 6.8A3

0.015

Chiral segments

0.010

Average volum Std. deviation=7.4A3 0.005

10

30

40 50 Volume in

70

80

90

100

Figure 2-14. Segment volume distribution g for PP (X = 76)at233K.

100

Figure 2-15. Segment volume distribution g for PP (X = 16) At 393 K.

0.025 — Achiral segments — - Chiral segments

All segments Average vo!ume = 42.3A3 Std. deviation = 9.6&3

0.020 -

Achiral segments

A*

0.015

1

Average volumex36.9& Std. deviation=7.3& 3

ita} Chiral segments ' ' * Average volume = 47.8A3 J l Std. deviation = 8.AA3

0.010

0.005 -

10

20

30

40 50 60 Volume in A.3

peratures there is an indication that segments with the largest volumes tend to associate with other segments with larger than average volume. However, clustering appears to be less at elevated temperatures. The findings are similar to those found by Deng et al. (1989) who performed computations on a two-dimensional model of an atomic glass which showed that, at temperatures below Tg, there was a greater tendency of atoms representing an excess of "free volume" to segregate into two distinct regions. The figures also show that chiral groups tend to have a Voronoi volume less

70

80

90

than their van der Waals volume while achiral groups tend to have Voronoi volumes larger than their van der Waals volume. This indicates that achiral groups are somewhat freer to move and that the major impediment to large scale motion of the polymer backbone seems to be the chiral groups. As expected, segmental mobility was also found to increase by several orders of magnitude with increasing temperature. A great deal more structural information was deduced through these analyses. However, even with high speed computers,

72

2 Amorphous Polymer Microstructure

the simulated time range is limited to values of the order of magnitude of 10 ~9 s, which converts to frequencies of analyz10 o able processes of roughly 1010 s 2.4.2 Constrained Molecular Dynamics in Generalized Coordinates

The constraint approach used above involves the application of a separate system of constraints. In principle the equations of motion could be cast into generalized coordinates including only the relevant internal degrees of freedom. For a polymer with constrained bond lengths and bond angles these internal degrees of freedom include only the torsional angles along the chain, and the overall orientation angles (Ludovice, 1989). An approach employing only those as variables could be general for all integration algorithms, including variable time step integration schemes (unlike the SHAKE and RATTLE algorithms), and would have the advantage that the integration time step is limited only by the potential energy interactions in the system of study and not by any "convergence window" of the constraining equations themselves. (The Lagrange multipliers in constraint equations act as the effective forces necessary to constrain the system, and if the integration time step is too large, these effective forces may become numerically too large and force the use af a smaller time step.) In their comparison of constrained and unconstrained protein dynamics, Van Gunsteren and Karplus (1982) found that both very small time steps and large convergence tolerances were necessary when the SHAKE algorithm was applied to an isolated protein simulation with constant bond lengths and bond angles. Typical integration time steps for unconstrained systems are on the order of 1 fs (10~ 15 s) (McCammon et al., 1977, 1979; Karplus

and McCammon, 1979; Van Gunsteren and Karplus, 1982). For simulations at equal energy tolerances, Van Gunsteren and Karplus needed to employ time steps of similar orders of magnitude for simulations with constrained bond lengths and bond angles depending on the algorithm. It appears that the window of convergence of the bond angle constraint equations is not particularly large and severely limits the application of simple algorithms in these Cartesian coordinate systems with constrained bond lengths and angles. This convergence window problem does not occur when the equations of motion are cast into generalized coordinates that include only the relevant internal degrees of freedom. This mathematical description is free of separate constraint equations, and hence there is no need to solve these equations to a given tolerance. The constraints are indeed implicit by the choice of coordinate system. Ryckaert and Bellemans (1975) have applied this method to n-butane, but it has been suggested that the technique becomes prohibitively complex when applied to large systems (Van Gunsteren and Berendsen, 1977; Allen and Tildesley, 1987). Despite this, a generalized coordinate approach for polymeric systems was applied to the above-described static model to include the effect of dynamic motion, following a similar approach for simple chains of rigidly linked masses by Pear and Weiner (1979) and using a mathematical formulation to describe the kinematics of rigidly linked bodies by Wittenburg (1977). (Pear and Weiner had analyzed the effect of the Fixman potential on an idealized three bond chain in a viscous medium.) Later, Pear and Weiner (1980) extended this treatment to chains up to 15 bonds in length. Their results indicated the usefulness of the Fixman potential in generalized coordinate constrained

73

2.4 Atomistically Detailed Methods - Dynamic Behavior

dynamics which may also be applied to the Lagrangian approach discussed below. The Lagrangian approach is not nearly as complex as previously believed. Figure 2-16 depicts the generalized coordinate approach (Ludovice and Suter, 1989) for a vinyl polymer with degree of polymerization x of the form CH 3 - (CHR - CH2)JC_ ± — CHR — CH 3 . The initial methyl group (labeled carbon 0) is fixed relative to the periodic cube that contains the chain, and the three orientation (Euler) angles describe the orientation of the polymer chain relative to the periodic cube by determining the orientation and rotation of the vector that lies along bond 1. Because the constraints used in this method are functions of position only (i.e., holonomic) the system may be described with Lagrange's equations of motion for a holonomic system (Landau and Lifshitz, 1976). d 6L At dqt

6L dqt ~

(2-18)

where qt and qt are the generalized position and velocity coordinates, and the Lagrangian function L is equal to the kinetic energy

Uk less the potential energy Up. The proposed system is conservative, which means that its potential energy is a function only of position and not velocity. Taking advantage of the fact that the described system is conservative and using the generalized coordinates, we may rewrite Eq. (2-18) as follows: d dUu At 81/^

dUn

A Wk

AtJft

8*7.

. . . . .

,

, ie{l,2,3,...} (2-19)

7 = *i>

The left hand sides of the above equations are the time derivatives of the angular momenta about a particular rotational degree of freedom while the right hand sides are the effective torques about that degree of freedom. Note that the effective torques include an additional term (the gradient of I7k) besides the gradients of potential energy due to internal changes in the configu-

Terminal Methyl

Terminal Methyl 3 Euler angles: W, . V2 . ^3 3 Euler angle velocities: ^ > 2x-2 Torsion angles: 02 2x-2 Torsional velocities: 02

2

,

02x_., ^2

Figure 2-16. Schematic representation of a MD simulation in generalized coordinates.

74

2 Amorphous Polymer Microstructure

ration of the polymer chain brought upon by the chains lack of rigidity. The derivatives of the potential energy with respect to the degrees of freedom are the same as those determined for minimization of the static structures. Ludovice and Suter (1989) have developed a formulation for the computation of the right-hand sides of the above equation and have carried out the first runs for systems of this type with a thermostat algorithm akin to the Nose method mentioned above. The starting point for this MD approach was a static polymer structure of atactic poly(propylene) of degree of polymerization 20 and 76. Results collected so far indicate that despite the differences between this method and that of the Cartesian coordinate method, this simulation method seems amenable to constant temperature and pressure (or stress) conditions using similar algorithms. For dense systems of long chains, however, this method does not provide significant increase in the time step over methods in Cartesian coordinates using a separate set of constraint equations (like the SHAKE or RATTLE), due to the sensitivity of the derivatives of the degrees of freedom to their relative positions in the chain. (The time derivatives of the torsion angles and angular velocities at the fixed end of the chain are much more sensitive to slight conformational changes than those at the "free" end of the chain; this shows up in a rapid accumulation of truncation error near the fixed end of the chain.) The time step for the simulation is thus limited by the length of the chain. This indicates that this constrained method may not find its best use in systems of long chains at high density, like the polymeric glass, but may be more useful in situations of shorter chains packed at lower densities or individual macromolecules.

These difficulties and limitations notwithstanding, several interesting findings were reported that seem to be of relevance for many dense polymer systems. For example, it was found that the dynamic correlations along the chain between torsion angles is such that the most important cooperative motion, by far, is a WeberHelfand type "crank-like" motion in which a bond and its next-to-nearest neighboring bond move in unison.

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2.5 References

Bondi, A. (1968), Physical Properties of Molecular Crystals, Liquids, and Glasses. New York: John Wiley & Sons. Borisova, N. P., Volkstein, M. V. (1961 a), Zh. Strukt. Khim. 2, 345. Borisova, N. P., Volkstein, M. V. (1961 b), Zh. Strukt. Khim. 2, 437. Boyd, R. H., Kesner, L. (1980), /. Chem. Phys. 72, 2179. Brandrup, 1, Immergut, E. H. (Eds.) (1975), Polymer Handbook, 2nd Ed. New York: John Wiley & Sons. Brant, D. A., Flory, P. I (1965), 1 Amer. Chem. Soc. 87, 2791. Brender, C , Lax, M. (1983), /. Chem. Phys. 79, 2423. Brisson, X, Brisse, F. (1986), Macromolecules 19, 2632. Brown, D., Clarke, I H. R. (1986), /. Chem. Phys. 84, 2858. Cape, J. N., Woodcock, L. V. (1980), J. Chem. Phys. ^ 72, 976. Cervinka, L., Fischer, E. W, Hahn, K., Jiang, B.-Z., Hellmann, G. P., Kuhn, K.-J. (1987), Polymer 28, 1287. Chandresekar, S. (1943), Rev. Mod. Phys. 15, 1. Cohen, M. H., Grest, G. S. (1981), Ann. NY. Acad. Sci. 371, 199. Cohen, M. H., Turnbull, D. (1959), /. Chem. Phys. 31, 1164. Cohen, M. H., Turnbull, D. (1964), Nature 203, 964. De Gennes, P. G. (1971), /. Chem. Phys. 55, 572. DeGennes, P. G. (1979), Scaling Concepts in Polymer Physics. Ithaca: Cornell University Press. DeSantis, R., Zachmann, H. G. (1977), Colloid Polym. Sci. 255, 729. De Vos, E., Bellemans, A. (1974), Macromolecules 7, 812. Deng, D., Argons, A. S., Yip, S. (1989), The Philosophical Transactions of the Royal Society of London, accepted for publication. Dill, K. A., Flory, P. J. (1980), Proc. Natl. Acad. Sci. U.S.A. 77, 3115. Dirichlet, G. L. (1850), Z. Reine Angew. Math. 40, 216. Elliot, S. R. (1983), Physics of Amorphous Materials. New York: Longman Inc. Erman, B., Flory, J. P., Hummel, J. P. (1980), Macromolecules 13, 484. Fixman, M. (1978 a), /. Chem. Phys. 69, 1527. Fixman, M. (1978 b), /. Chem. Phys. 69, 1538. Flory, P. J. (1953), Principles of Polymer Chemistry. Ithaca: Cornell University Press. Flory, P. J. (1969), Statistical Mechanics of Chain Molecules. New York: John Wiley & Sons. Flory, P. I, Pickles, C. J. (1973), J. Chem. Soc, Faraday Trans. 69, 632. Flory, P. J., Williams, A. D. (1968), /. Amer. Chem. Soc. 91, 3118. Freitag, D., Grigo, U., Miiller, P. R., Nouvertne, W (1988), Polycarbonates, in: Encyclopedia of Polymer Science and Engineering, Vol. 11, 2nd Ed. New York: John Wiley & Sons.

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Frenkel, D., McTague, J. P. (1980), Anal. Rev. Phys. Chem. 31, 491. Gear, C. W. (1971), Numerical Initial Value Problems in Ordinary Differential Equations. Englewood Cliffs: Prentice-Hall. Gentile, F. T., Suter, U. W (1992), unpublished. Gentile, F. T., Hutnik, M., Ludovice, P. I , Argon, A. S., Suter, U. W. (1989), AICHE Abstracts, 2nd Topological Conference on Emerging Materials, 111. Gibbs, J. H., DiMarzio, E. A. (1958), /. Chem. Phys. 28, 373. Go, N., Scheraga, H. A. (1976), Macromolecules 9, 535. Gordon, J. M., Gibbs, J. H., Fleming, P. D. (1976), /. Chem. Phys. 65, 2771. Grest, G. S., Kremer, K. (1986), Phys. Rev. A: Gen. Phys. 33, 3628. Haward, R. N., MacCallum, J. R. (1971), Polymer 12, 189. Helfand, E. (1978), /. Chem. Phys. 69, 1010. Helfand, E., Wasserman, Z. R., Weber, T. A. (1981), Polym. Preprints 22(2), 279. Hendrickson, J. B. (1961), J. Chem. Phys. 83, 4537. Hillstrom, K. (1976), Nonlinear Optimization Routines in AMDLIB, Technical Memorandum No. 297, Argonne National Laboratory. Hoover, W. G., Gray, S. G., Johnson, K. W. (1971), /. Chem. Phys. 55, 1128. Hummel, J. P., Flory, J. P. (1980), Macromolecules 13, 479. Hutnik, M., Argon, A. S., Suter, U. W. (1987), Polym. Preprints 28(2), 293. Hutnik, M., Argon, A. S., Suter, U. W (1989), Polym. Preprints 30(2), 36. Hutnik, M., Argon, A. S., Suter, U. W. (1991a), Macromolecules 24, 5970. Hutnik, M., Gentile, F. T., Ludovice, P. I , Suter, U. W, Argon, A. S. (1991b), Macromolecules 24, 5962. Hutnik, M., Argon, A. S., Suter, U. W. (1991c), Macromolecules 24, 5956. Jagodic, F., Borstnik, B., Azman, A. (1973), Makromol. Chem. 173, 221. Jones, A. A., O'Gara, F. F , Inglefield, P. T, Bendler, I T , Yee, A. F, Ngai, N. L. (1983), Macromolecules 16, 658. Kang, H. S., Lee, Y. S., Ree, T. (1983), Bull. Korean Chem. Soc. 4, 223. Karplus, M., McCammon, J. A. (1979), Nature 277, 578. Karplus, M., McCammon, J. A. (1986), Sci. Amer. 254, 42. Kauzmann, W. (1948), Chem. Phys. Rev. 43, 219. Khalatur, P. G., Pavlov, A. S., Marchenko, G. N., Papulov, Y G., Khrapkovskii, G. M. (1985), Dokl. Akad. Nauk. SSR 282, 1186. Khalatur, P. G., Papulov, Y G., Pavlov, A. S. (1986), Mol. Phys. 58, 887.

76

2 Amorphous Polymer Microstructure

Koros, W. I, Fleming, G. K., Jordan, S. M., Kim, T. H., Hoehn, H. H. (1988), Prog. Polym. Sci. 13, 339. Lai, M., Spencer, D. (1971), Mol Phys. 22, 649. Lancon, R, Billard, L., Chambron, L., Chamberod, A. (1985), J. Phys. F: Met. Phys. 15, 1485. Landau, L. D., Lifshitz, E. M. (1976), Mechanics (3rd Ed.). Oxford: Pergamon Press. Levy, R. M., Karplus, M., Kushnick, C , Perahia, D. (1984), Macromolecules 17, 1370. Liquori, A. M. (1955), Ada Cryst. 8, 345. Ludovice, P. J. (1989), Static and Dynamic Atomistic Level Modelling of Polymeric Glasses, Ph.D. thesis, Massachusetts Institute of Technology, Cambridge. Ludovice, P. X, Suter, U. W. (1989), Polym. Preprints 30(2), 13. Ludovice, P. I, Suter, U. W. (1990), Detailed Molecular Structure of a Polar Vinyl Polymer Glass, in: Molecular Level Calculations of the Structures and Properties of Non-Crystalline Polymers, Bicerano, J. (Ed.), in press. Ludovice, P. X, Suter, U. W. (1990), submitted for publication. Luise, R. R., Yannas, I. V. (1991), submitted for publication. Lukyanov, M. N., Darinskii, A. A., Gotlib, Y. Y, Neelov, I. M. (1985), Vysokomol. Soedin., Ser. A 27, 1506. Maeda, K., Takeuchi, S. (1981), Phil. Mag. A 44, 643. Maitland, G. C , Rigby, M., Smith, E. B., Wakeham, W. A. (1981), Intermodular Forces: Their Origin and Determination. Oxford: Clarendon Press. Mansfield, K. R, Theodorou, D. N. (1989), Polymer Preprints 30, 2, 76. McCammon, X A., Karplus, M. (1980), Annu. Rev. Phys. Chem. 31, 29. McCammon, X A., Gelin, B. R., Karplus, M. (1977), Nature 267, 585. McCammon, X A., Wolnyes, P. G., Karplus, M. (1979), Biochemistry 18, 92. McCullough, R. L. (1974), J. Macromol. Sci.-Phys. B9, 97. Mott, P. H., Argon, A. S., Suter, U. W. (1989), Polym. Preprints 30(2), 34. Mott, P. H., Argon, A. S., Suter, U. W. (1990), Macromolecules, submitted for publication. Nose, S. (1984), J. Chem. Phys. 81, 511. Oh, I. X, Lee, Y S., Ree, T. (1983), Bull. Korean Chem. Soc. 4, 87. Ooi, T., Scott, R. A., Vanderkooi, G., Scheraga, H. A. (1967), /. Chem. Phys. 46, 4410. Orban, X, Ryckaert, X P. (1974), Rapport d'activite scientifique du CECAM. Parrinello, M., Rahman, A. (1980), Phys. Rev. Lett. 45, 1196. Parrinello, M., Rahman, A. (1981), J. Appl. Phys. 52, 7182. Pastine, D. X (1968), / Chem. Phys. 49, 3012.

Pear, M. R., Weiner, X H. (1979), /. Chem. Phys. 71, 212. Pear, M. R., Weiner, X H. (1980), J. Chem. Phys. 72, 3939. Pechhold, W, Liska, E., Grossman, H. P., Hagele, P. C. (1976), Pure Appl. Chem. 46, 127. Perchak, D., Yaris, R., Skolnick, X (1983), /. Chem. Phys. 78, 6914. Perchak, D., Skolnick, X, Yaris, R. (1985), Macromolecules 18, 519. Roy, A. K., Jones, A. A., Inglefeld, P. T. (1986), Macromolecules 19, 1356. Rutledge, G. C. (1990), The Solid State Structure and Properties of Stiff Chain Aramids, Ph.D. thesis, Massachusetts Institute of Technology, Cambridge, Ma. Rutledge, G. C , Suter, U. W. (1991), Macromolecules 24, 1934. Ryckaert, X P., Bellemans, A. (1975), Chem. Phys. Lett. 30, 123. Ryckaert, X P., Bellemans, A. (1978), Soc. Faraday Discuss. 66, 95. Ryckaert, X P., Ciccotti, G., Berdensen, H. X C. (1977), J. Comput. Phys. 23, 327. Schaefer, X, Stejskal, E. O., McKy, R. A., Dixon, W. T. (1984), Macromolecules 12, 1479. Schmidt, C , Kuhn, K.-X, Spiess, H. W. (1985), Progr. Coll. & Polym. Sci. 71, 71. Scott, R. A., Scheraga, H. A. (1965), /. Chem. Phys. 42, 2209. Scott, R. A., Scheraga, H. A. (1966a), /. Chem. Phys. 45,2091. Scott, R. A., Scheraga, H. A. (1966b), /. Chem. Phys. 44, 3054. Simha, R. (1976), Ann. N.Y. Acad. Sci. 279, 2. Simha, R. (1977), Macromolecules 10, 1025. Simon, E. M. (1971), Biopolymers 10, 973. Skvortsov, A. M., Sariban, A. A., Birshtein, T. M. (1977), Vysokomol. Soedin 5, 1014. Sorensen, R. A., Liau, W. B., Boyd, R. H. (1988a), Macromolecules 21, 194. Sorensen, R. A., Liau, W. B., Kesner, L., Boyd, R. H. (1988 b), Macromolecules 21, 200. Srolovitz, D., Vitek, V., Egami, T. (1983), Acta Met. 31, 335. Stillinger, P. H., David, C. W. (1978), /. Chem. Phys. 69, 1473. Stillinger, R H., Weber, T. A. (1979), J. Chem. Phys. 70, 4879. Stillinger, R H., Weber, T. A. (1980), J. Chem. Phys. 68, 3837. Suter, U. W. (1979), J. Amer. Chem. Soc. 101, 6481. Suter, U. W, Neunschwander, P. (1982), Macromolecules 14, 523. Suzuki, K., Nakata, Y. (1970), Bull. Chem. Soc. Jpn. 43, 1006. Swope, W. C , Andersen, H. C , Bernes, P. H., Wilson, K. R. (1982), /. Chem. Phys. 76, 637. Sylvester, M. P., Yip, S., Argon, A. S. (1989a), Polym. Preprints 30(2), 32.

2.5 References

Sylvester, M. R, Yip, S., Argon, A. S. (1989b), Macromolecules, accepted for publication. Synek, M., Schieve, W. C , Harrision, H. W. (1977), /. Chem. Phys. 67, 2916. Tanemura, M., Ogawa, T, Ogita, N. (1983), /. Cornput. Phys. 51, 191. Tashiro, K., Kobayashi, M.,Tadokoro, H. (1977a), Macromolecules 4, 731. Tashiro, K., Kobayashi, M., Tadokoro, H. (1977b), Macromolecules 10, 413. Tashiro, K., Kobayashi, M., Tadokoro, H. (1978), Macromolecules 11, 914. Theodorou, D. N. (1985), Molecular Modelling of Polymeric Glasses, Ph.D. thesis, Massachusetts Institute of Technology, Cambridge, Ma. Theodorou, D. N., Suter, U. W. (1985a), Macromolecules 18, 1206. Theodorou, D. N., Suter, U. W. (1985 b), Macromolecules 18, 1467. Theodorou, D. N., Suter, U. W. (1986), Macromolecules 19, 139. Tripathy, S. K., Hopfmger, A. I , Taylor, P. L. (1981), /. Phys. Chem. 85, 1371. Uhlmann, D. R., Kreidl, N. J. (Eds.) (1983), Glass Science and Technology, Vol. 1: Glass Forming Systems. London: Academic Press. Vacatello, M., Avitabile, G., Corradini, P., Tuzi, A. (1980), J. Chem. Phys. 73, 543. Vainshtein, B. K. (1966), Diffraction of X-Rays by Chain Molecules. Amsterdam: Elsevier. Van der Ploeg, P., Berendsen, H. J. C. (1983), Mol. Phys. 49, 233. Van Gunsteren, W. R, Berendsen, H. J. C. (1977), Mol. Phys. 34, 1311. Van Gunsteren, W. R, Karplus, M. (1982), Macromolecules 15, 1528. Van Krevelen, D. W., Hoftyzer, P. J. (1976), Properties of Polymers - Their Estimation and Correlation with Chemical Structure. New York: Elsevier. Verlet, L. (1967), Phys. Rev. 159, 98. Wagner, C. N. J. (1978), /. Non-Cryst. Solids 31, 1. Wall, F. T, Mandel, R (1975), J. Chem. Phys. 63, 4592. Wall, F. T, Steitz, W. A. (1977), /. Chem. Phys. 67, 3722. Weber, T. A., Helfand, E. (1979), /. Chem. Phys. 71, 4760. Weber, T. A., Stillinger, F. H. (1981), Ann. N. Y Acad. Sci. 371, 335. Wendt, H. R., Abraham, F. F. (1978), Phys. Rev. Lett. 41, 1244. Wittenburg, J. (1977), Dynamics of Systems of Rigid Bodies. Stuttgart: Teubner. Woodcock, L. V. (1976), /. Chem. Soc. Faraday Trans. 2 72, 1667.

77

Woodcock, L. V (1978), /. Chem. Soc. Faraday Trans. 2 74, 11. Yannas, I. V (Ed.) (1975), Molecular Interpretation of Deformation in Glassy Polymers - Proceedings of the International Symposium on Macromolecules Rio de Janeiro (July 26-31, 1975). Amsterdam: Elsevier. Yannas, I. V, Luise, R. R. (1982), J. Macro. Sci. B21 (3), 443. Yannas, I. V, Luise, R. R. (1983), The Strophon Theory of Deformation in Glassy Polymers: Applications to Small Deformations, in: The Strength and Stiffness of Polymers, Zachariades, A. E., Porter, R. S. (Eds.). New York: Marcel Dekker. Yannas, I. V, Luise, R. R. (1991), Mechanics of Strophons in Glassy Amorphous Polymers, Unified View of Stiffness, Yielding and Craze Behaviour, in: Molecular Level Calculations of the Structures and Properties of Non-Crystalline Polymers, Bicerano, J. (Ed.), in press. Yashonath, S., Rao, K. I, Rao, C. N. R. (1985), Phys. Rev. B 31, 3196. Yasukawa, T., Takahashi, T, Murkami, K. (1973), Tohoku Daigaku Hisuiyoeki Kagaku Kenkyusho Hokoku23, 155. Zahlen, R. (1983), The Physics of Amorphous Solids. New York: John Wiley & Sons.

General Reading Allen, M. P., Tildesley, D. J. (1987), Computer Simulations of Liquids. Oxford: Clarendon. Bondi, A. (1968), Physical Properties of Molecular Crystals, Liquids, and Glasses. New York: Wiley. Elliot, S. R. (1983), Physics of Amorphous Materials. New York: Longman. Flory, P. J. (1989), Statistical Mechanics of Chain Molecules. Munich: Hanser. Israelachvilli, J. N. (1985), Intermodular and Surface Forces. New York: Academic. Maitland, G. C , Rigby, M., Smith, E. B., Wakeham, W A. (1985), Intermodular Forces. New York: Academic. Uhlmann, D. R., Kreidl, N. J. (Eds.) (1983), Glass: Science and Technology, Vol. 1: Glass Forming Systems. New York: Academic. Uhlmann, D. R., Kreidl, N. J. (Eds.) (1983), Glass: Science and Technology, Vol. 5: Elasticity and Strength in Glasses. New York: Academic. Weiner, J. H. (1983), Statistical Mechanics of Chain Molecules. New York: Wiley. Zahlen, R. (1983), The Physics of Amorphous Solids. New York: Wiley.

3 Structure of Polymer Single Crystals Bernard Lotz and Jean-Claude Wittmann Institut Charles Sadron (CRM-EAHP), Strasbourg, France

List of 3.1 3.2 3.2.1 3.2.1.1 3.2.1.2 3.2.2 3.2.2.1 3.2.2.2 3.3 3.3.1 3.3.1.1 3.3.1.2 3.3.1.3 3.3.2 3.3.3 3.3.4 3.3.5 3.3.5.1 3.3.5.2 3.3.5.3 3.3.6 3.4 3.4.1 3.4.1.1 3.4.1.2 3.4.2 3.4.2.1 3.4.2.2 3.4.2.3 3.4.2.4 3.5 3.5.1 3.5.2 3.5.2.1

Symbols and Abbreviations Introduction Conformation and Packing of Crystalline Polymer Chains Helical Structure of Crystalline Polymers Helical Conformation and Fiber Pattern Conformational Polymorphism Crystal Structure and Crystalline Polymorphism Unit Cell and Crystalline Polymorphism Disorder in Polymer Crystals at the Unit Cell Level Growth and Occurrence of Polymer Single Crystals Growth from Solution The Phase Diagram of Polymer Solutions Self-Nucleation in Polymer Crystallization Other Crystallization Procedures Growth of Lamellar Crystals from the Melt Single Crystals of Biopolymers Polymers Obtained in the Form of Single Crystals Nonfolded or Chain Extended Polymer Single Crystals Solid State Polymerization Crystallization Under High Pressure Inclusion Compounds and Molecular Complexes Thermotropic and Lyotropic Polymers The Crystalline Core of Polymer Single Crystals General Features of Polymer Single Crystals Overview of the Model The Fundamental Experimental Evidence The Lamellar Core and Crystal Defects Edge Dislocations Twins Screw Dislocations Local Defects The Lamellar Surface Overview and Issues The Sharp Fold Computed Models

Materials Science and Technology Copyright © WILEY-VCH Verlag GmbH & Co KGaA. All rights reserved.

81 84 85 85 85 88 89 89 90 91 91 91 92 94 94 95 96 96 97 97 97 97 98 98 98 100 100 101 102 107 107 110 110 Ill Ill

3 Structure of Polymer Single Crystals Bernard Lotz and Jean-Claude Wittmann Institut Charles Sadron (CRM-EAHP), Strasbourg, France

List of 3.1 3.2 3.2.1 3.2.1.1 3.2.1.2 3.2.2 3.2.2.1 3.2.2.2 3.3 3.3.1 3.3.1.1 3.3.1.2 3.3.1.3 3.3.2 3.3.3 3.3.4 3.3.5 3.3.5.1 3.3.5.2 3.3.5.3 3.3.6 3.4 3.4.1 3.4.1.1 3.4.1.2 3.4.2 3.4.2.1 3.4.2.2 3.4.2.3 3.4.2.4 3.5 3.5.1 3.5.2 3.5.2.1

Symbols and Abbreviations Introduction Conformation and Packing of Crystalline Polymer Chains Helical Structure of Crystalline Polymers Helical Conformation and Fiber Pattern Conformational Polymorphism Crystal Structure and Crystalline Polymorphism Unit Cell and Crystalline Polymorphism Disorder in Polymer Crystals at the Unit Cell Level Growth and Occurrence of Polymer Single Crystals Growth from Solution The Phase Diagram of Polymer Solutions Self-Nucleation in Polymer Crystallization Other Crystallization Procedures Growth of Lamellar Crystals from the Melt Single Crystals of Biopolymers Polymers Obtained in the Form of Single Crystals Nonfolded or Chain Extended Polymer Single Crystals Solid State Polymerization Crystallization Under High Pressure Inclusion Compounds and Molecular Complexes Thermotropic and Lyotropic Polymers The Crystalline Core of Polymer Single Crystals General Features of Polymer Single Crystals Overview of the Model The Fundamental Experimental Evidence The Lamellar Core and Crystal Defects Edge Dislocations Twins Screw Dislocations Local Defects The Lamellar Surface Overview and Issues The Sharp Fold Computed Models

Materials Science and Technology Copyright © WILEY-VCH Verlag GmbH & Co KGaA. All rights reserved.

81 84 85 85 85 88 89 89 90 91 91 91 92 94 94 95 96 96 97 97 97 97 98 98 98 100 100 101 102 107 107 110 110 Ill Ill

80

3.5.2.2 3.5.2.3 3.5.2.4 3.5.3 3.5.4 3.5.4.1 3.5.4.2 3.5.4.3 3.5.4.4 3.5.5 3.5.6 3.6 3.6.1 3.6.2 3.6.3 3.7 3.7.1 3.7.2 3.7.2.1 3.7.2.2 3.7.2.3 3.7.3 3.8 3.8.1 3.8.1.1 3.8.1.2 3.8.2 3.8.2.1 3.8.2.2 3.8.2.3 3.8.3 3.8.3.1 3.8.3.2 3.8.3.3 3.9 3.10 3.11

3 Structure of Polymer Single Crystals

Packing of Sharp Folds Tight Fold and Chain Sense Proportion of Sharp Folds The Less Ordered Surface Experimental Techniques Infrared Experiments on Mixed Crystals Neutron Scattering on Mixed Crystals Solid State 13 C Nuclear Magnetic Resonance Microscopical Techniques The Fold Period in Polymer Single Crystals Single Crystal Habits: Influence of Crystallization Conditions and Impact of Internal Structure Single Crystals Produced from the Bulk Polyethylene Oxide Single Crystals Crystallization of Polyethylene Other Polymers Single Crystals of Low M.w. Polymers and Model Compounds Tight Fold Structure Onset of Chain Folding Observation on Polyethylene Oxide Crystals Recent Results on Model Paraffins Onset of Chain Folding and Crystal Morphology Single Crystals Produced Under High Pressure Polymer Single Crystals as Investigation Materials Polymer Crystallography at the Unit Cell Level Electron Crystallography Lattice Imaging of Single Crystals Polymer Single Crystals and Theories of Polymer Crystallization General Outline of Crystallization Theories Kinetics of Polymer Single Crystal Growth Geometry of Growth and Crystal Habit Deformation and Transformation of Polymer Single Crystals Spontaneous and Induced Crystal Phase Transitions Deformation of Polymer Crystals Deformation of Polymer Single-Crystal Mats Conclusion Acknowledgements References

Ill 113 114 115 116 117 117 118 119 123 124 126 127 128 131 131 131 132 132 133 134 135 135 136 136 137 139 139 142 143 145 145 145 145 146 146 146

List of Symbols and Abbreviations

List of Symbols and Abbreviations unit cell parameters chain dimension parallel to growth face chain dimension normal to growth face; thickness of growth layer repeat distance; concentration free energy of formation growth rate lateral spreading rate initial growth rate (at t = 0) growth rate in regime I growth rate in regime II free enthalpy of melting A 0completior l free enthalpy of completion of growth face ^^first stem free enthalpy of deposition of first stem Ah heat of fusion per unit volume AH heat of fusion i deposition rate Bessel function of order n Jn integers k,m,n kB Boltzmann constant lamellar thickness; stem length i L total chain length; length of growth front kinetic length Lk persistence length LP L* critical fold length correction term in equation of critical fold length 5/ molecular weight M.w. number average molecular weight Mn weight average molecular weight Mw number of folds n number of connected segments N rise per residue or repeating unit P P helical pitch R gas constant time; number of helical turns t T temperature crystallization temperature Tc T T temperature of initial final crystallization melting temperature Tm T extrapolated temperature of dissolution (melting) m period of partial dissolution (of a polymer) temperature of partial melting or partial dissolution Ts rotating frame relaxation time T, spin-spin relaxation time T2

a,b,c a b c AF* G Gs Go G, Gn Ag

81

82

3 Structure of Polymer Single Crystals

AT u AU*

extrapolated temperature where free volume ~ undercooling or supercooling number of residues in a helical repeat distance activation energy crystal size

X

V

coefficient angle between unit cell vectors half scattering angle wavelength; drawability maximum drawability difference between the frequencies of the crystal bending field doublet layer height or number lateral surface free energy fold surface free energy correction parameter

AFM Ala D DNA DSC g gly GPC HREM iPP iPS IR LAM LLDPE NIF NMR PE PED PEEK PEK PEO PPX PTFE PVDF P4MP1 R,L sPP t

atomic force microscopy alanine deuterium atom deoxyribonucleic acid differential scanning calorimetry gauche glycine gel permeation chromatography high-resolution electron microscopy isotactic polypropylene isotactic polystyrene infrared longitudinal acoustic mode linear low density polyethylene noninteger folding nuclear magnetic resonance polyethylene deuterated polyethylene polyphenyl ether ether ketone polyphenyl ether ketone polyethylene oxide poly(p-xylylene) poly (tetrafluoroethylene) polyvinylidene fluoride poly(4-methyl-l-pentene) right, left-handed helix syndiotactic polypropylene trans

a

P,y

eX ^max

Avb o

List of Symbols and Abbreviations

TEM TEPD TFAA TMPS TrFE u, d UHMW

transmission electron microscopy total end point dose trifluoroacetic acid tetramethylene-p-silphenylene siloxane trifluoroethylene up-, down-orientation of helices or chains ultra-high molecular weight

83

84

3 Structure of Polymer Single Crystals

3.1 Introduction Polymer single crystals constitute a landmark in the investigation of structure and morphology of crystalline polymers. The possibility of examining individual lamellae has opened up, since the mid 1950s, a whole range of new insights into the structure, the mechanism of growth, the disorder and defects that exist in crystalline polymers. Polymer single crystals have therefore been investigated with a dual purpose: first in their own right, but also whenever possible, as a convenient means of analyzing, in relatively simple systems, the interrelation between the microcrystalline subunit and larger spherulitic texture in bulk crystallization, which had eluded all attempts at understanding. This chapter intends to present some of these dual aspects and will be centered on structural investigations. The modern history of polymer single crystals is associated with the works of Jaccodine (1955), Till (1957), Fischer (1957) and Keller (1957), who observed that, when crystallized on cooling dilute solutions, polyethylene forms regular lozenge shaped flat crystals very similar to paraffine crystals. Selected area electron diffraction demonstrated that the chains are nearly normal to the lamellar surface. From the comparison of the limited thickness of the lamellar crystals (typically 10 nm) with the much larger chain length of the constituting molecules (~ 1000 nm for a PE chain of M.w. 10000, i.e., made of 360 ethylene units) the inescapable concept of chain folding emerged (Keller, 1957). This concept had already been formulated by Storks (1938) when investigating the structure of solution-cast films of trans-1,4poly(2-methyl-butadiene) or gutta percha, but had not attracted sufficient attention. Most importantly, it had not been used as

a means to reanalyze and reassess the various aspects of polymer morphology, and, notably, the structure of bulk crystallized material, where chain folding is also a major factor determining the final morphology. On a structural basis, the chain folding mechanism provides the most elegant and probably the only conceivable means to reconcile crystallographic regularity with the chemical lack of definition of polymers, illustrated by the molecular chain length distribution. Crystallization becomes possible because it involves nearly identical structural subunits (defined by the fold length) of chains of differing length. At the same time, chain folding introduces an added complexity linked with the existence of fold surfaces, that is, of disordered regions which fragment the polymer space into small domains one or a few tens of nanometers thick, a feature which accounts for many of the properties of crystalline polymers. In sharp contrast to low molecular weight materials therefore, polymer single crystals are mainly concerned with subunits of the building (chain) molecules, and possess specific fold regions which deserve particular attention in the structural investigation. Polymer single crystals have been the subject of a number of reviews or books, either on their own, or when considered in the broader aspect of polymer morphology and structure. The first of these is the classical book Polymer Single Crystals by Geil (1963). It describes the early work in, at that time, a developing new field. It constitutes a continuing source of inspiration for further work as it reports a number of experimental observations on various polymers unexplained at that time, of which only a few have received answers in recent years. The series Macromolecular Physics by Wunderlich, and particularly

3.2 Conformation and Packing of Crystalline Polymer Chains

Volume I provides a more recent and well documented account of the field. A noteworthy contribution remains the papers presented at the Cambridge meeting on Organization of Macromolecules in the Condensed Phase in 1979, and grouped in Vol. 68 of the Faraday Discussions of the Chemical Society. These papers represent the state of the art of the experimental techniques and of the debates over the structure of polymers at that time. Several older and more recent reviews on polymer morphology deal specifically with polyethylene (Fava, 1971; Dosiere, 1989), and many rely heavily on this material, which may well remain the most thoroughly studied crystalline polymer. Whereas its molecular and crystallographic simplicity justify this popularity, several crucial issues of polymer structure and morphology are better or only revealed with other systems (importance of helical handedness in polyolefins, influence of chirality of residues on crystal morphology, etc.). For this reason, a wider palette of polymers will be considered in this contribution. The present chapter deals with: - the main characteristics of polymer chain conformation and crystal structure at the unit cell level, insofar as they are a necessary background for the description of polymer single crystal structure; - the different methods developed to grow polymer single crystals from solution, from the melt or directly on polymerization of the monomer, including also natural occurrence of polymer single crystals; - the next two parts deal with structural investigations of the crystalline core and of the fold surface; - finally, selected contributions of polymer single crystals as test materials for the initial discovery or the elucidation of various processes associated with polymer

85

structural behavior, growth mechanisms, etc. are considered. As is usual, a note of warning is needed, regarding some obvious limitations. These arise because the field covered is very active and because the authors have tended to stress their major interests: structure and electron microscopy. This emphasis on morphological and structural aspects of polymer crystals, supported by some relevant electron micrographs, should help the reader to grasp the complexity and originality of polymer single crystals.

3.2 Conformation and Packing of Crystalline Polymer Chains 3.2.1 Helical Structure of Crystalline Polymers Only a brief account of this topic is given here. For further details and experimental techniques the reader is referred to the comprehensive book of Tadokoro (1979). 3.2.1.1 Helical Conformation and Fiber Pattern The conformation of a polymer chain is primarily governed by its chemical sequence and configuration (cf. Chap. 1 of this Volume). Assuming sequence and configuration to be regular, the conformation is mainly governed by intramolecular interactions, which define the general chain geometry, and by intermolecular interactions in the crystal lattice, which may alter the regularity of conformation of the isolated chain, or confer increased stability to any specific chain conformation (e.g., through hydrogen bonding). Due to the existence of covalent links between backbone atoms, the conformation of the chain is mainly defined by internal rotation potentials and nonbonded inter-

3 Structure of Polymer Single Crystals

86

actions. Most helical conformations of vinyl polymers rest on combinations of trans t (180°), gauche g (60°) and g ~ ( - 60°) bonds. However, departures from these conformations do exist. As an illustration, Fig. 3-1 represents the potential energy map of isotactic polypropylene (iPP), which has only two internal rotation angles to the main chain. Whereas the calculated minimum corresponds indeed to the observed helical structure with threefold symmetry, a potential well is obvious. Related polyolefins, but with bulkier side-chains (polybutene, poly-4-methyl-1 -pentene) have structures with internal angles that depart

slightly from this minimum, and therefore result in helices with different symmetries. Repetition of any set of internal rotation angles results in a helical structure. The helix may have crystallographic symmetry (two-, three-, four-, or six-fold screw axis) or not; in the latter case, intramolecular interactions often overweigh intermolecular interactions arising from the crystal lattice and impose the helix structure. Two familiar examples of such helices must be cited. The famous oc-helix of polypeptides and proteins (Pauling and Corey, 1951) has 3.6 residues per turn (185-helix) and was the first noncrystallographic helix to be an-

r 240 >1.2J/ mol m.u

\ 120

K\ \ 120

/

i i

"V )

240 T 2 (degree)

360

(b)

(a)

1/4 -

• 1/4

1/4 •

1/4

1/4 -

(C)

Figure 3-1. (a) Potential energy map of isotactic polypropylene as a function of conformational angles TX and T2 (adapted from Tadokoro et al., 1973) with loci of helices with threefold symmetry, (b) Side view of 3 1 -helix and (c) c-axis view of a-phase unit cell (from Hikosaka and Seto, 1973). Note alternation of right and lefthanded helices in 6-axis direction. Methyl groups shaded.

3.2 Conformation and Packing of Crystalline Polymer Chains

alyzed. Its original geometry results from the existence and optimization of hydrogen bonds between amide groups of successive helical turns. Poly(tetrafluoroethylene) or PTFE has a helical geometry that departs from the conventional planar zig-zag tt conformation of polyethylene due to the bulkiness of the fluorine atoms: the minimum F - F distance is 2.7 A whereas in the tt conformation these atoms would be only 2.55 A away. As a result, PTFE adopts a 136 or a 157 helical structure below or above a solid state transition temperature at ^19°C. Figure 3-2 represents the helical conformation of PE and PTFE chains, respectively. Determination of the helical conformation is most usually achieved by the analysis of fiber patterns. Diffraction of helices was developed initially by Cochran et al. (1952) and has been extended by several authors (cf. Tadokoro, 1979; Spruiell and Clark, 1980). The fundamental information obtained from these fiber patterns for discontinuous helices (i.e., made of repeating units) are the helical pitch P and the rise per residue or repeating unit along the helix axis, p. For a helix with pitch P and rise per residue p, the transform vanishes except on layers of the fiber pattern parallel to the equator with height £ such as m in which m is an integer and n the order of the Bessel function Jn which describes the transform of the continuous helix. If P and p are expressed as a ratio of whole numbers, the crystallographic repeat distance c of the helix is related to the layer number / by

87

Since the number of helical turns t is equal to c/P and the number of residues in the helix repeat distance u = c/p9 it follows that / = tn

um From this selection rule (of layer lines), it follows that information on P and p is therefore given when n = 1, m = 0 and n = 0, m = 1. These situations correspond to the first strong layer line, and the first meridional reflection. Two very illustrative examples are shown in Fig. 3-2 which displays patterns of PE (21 -helix) and PTFE (157-helix). Note that the first layer line of PE becomes split into two layer lines for PTFE (now the 7th and 8th) due to the small difference in chain geometry. Note further the near identity of residue height (0.253 nm/2 and 0.255 nm/2 respectively) corresponding to the 2nd and 15th layer lines in PE and PTFE, respectively. However, a qualitative analysis based on layer line selection is not unambiguous as, for example 83- and 85-helices are expected to have similar layer lines. Detailed analysis of diffraction intensities, helped by conformational energy analysis and vibrational spectroscopy (infrared and Raman), may be required. As an illustration, Tanaka et al. (1974) could settle an earlier dispute over the helical conformation of polyisobutylene [CH 2 -C-(CH 3 ) 2 ] n in favor of an 8 3-helix (Allegra et al., 1970) rather than an 8 5-helix. Most synthetic polymers have single stranded helices, although isotactic poly(methylmethacrylate) was shown to have two strands (Kusanagi et al., 1976). For natural polymers on the contrary, double stranded helices are familiar in DNA (Watson and Crick, 1953 a, b), or some polysaccharides (Atkins, 1979). The most complex helical architecture reported so far is that of collagen, which is made of

88

3 Structure of Polymer Single Crystals

131

(a)

(b)

(c)

(d)

Figure 3-2. (a) Molecular structure of polyethylene and (b) corresponding electron diffraction pattern (b c-plane of the unit cell) with indexing of layer lines. Arrowed reflection corresponds to the monoclinic phase, (c) Molecular structure of polytetrafluoroethylene (136-helix) and (d) electron diffraction fiber pattern of a 157-helix with indication of layer lines. Note splitting of the first layer line in PE into 7th and 8th layer lines in PTFE, linked with small helical twist. (Drawings from Bunn and Howells, 1954.)

three right handed individual helical strands interwoven in a left handed super helix (cf. Fraser and McRae, 1973). 3.2.1.2 Conformational Polymorphism

Different helical conformations, separated by relatively minor differences in energy may be available to certain polymers. One such example, mentioned previously, deals with PTFE, which displays a reversible thermal transition between a 13 6 helix below 19 °C and slightly less twisted one (157) above that temperature. Conformational dimorphism or polymorphism may be enhanced or revealed by using different physical or preparation conditions but can also exist under identi-

cal crystallization conditions. For example, three different helical structures can be produced by solution crystallization of isotactic poly-1-butene (Holland and Miller, 1964; Chau and Geil, 1984). The corresponding helices have 3X-, 134-, and 4Xsymmetries with oaxis repeat distances of 0.65, 2.11 and 0.76 nm, and unit cells with hexagonal, tetragonal and orthorhombic symmetries respectively (Luciani et al., 1988). Note that, although significantly different in terms of helix and unit cell symmetries, all three helices correspond to conformations located near the minimum of the conformational energy maps of isotactic polyolefins shown in Fig. 3-1. Note also that the helix symmetry does not impose the cell symmetry: the 41-helix does

3.2 Conformation and Packing of Crystalline Polymer Chains

not pack in the tetragonal cell of form II, but in an orthorhombic cell. Conformational polymorphism is particularly prominent in poly(vinylidenefluoride)(PVDF):(CH 2 -CF 2 ) n which exists in three different conformations: - an all-trans conformation, form II, which has remarkable ferroelectric properties linked with the orientation of the CF 2 dipoles on one side only of the chain backbone (c-axis repeat distance 0.255 nm); - two forms with different sequences of trans- and gauche-bonds: tgtg" for form I and tttgtttg" for form III (corresponding c-axis repeat distances are 0.462 and 0.923 nm, respectively). An almost caricatural example of conformational polymorphism is provided by poly(DL-benzylglutamate), which has been observed in eight different conformations, i.e., a near extended conformation, three single stranded and four double stranded helices, three of which are stabilized by solvent molecules (Lotz et al., 1976; ColonnaCesarietal, 1977).

3.2.2 Crystal Structure and Crystalline Polymorphism 3.2.2.1 Unit Cell and Crystalline Polymorphism

Parallel packing of polymer helices in a crystal lattice results in a structure in which one cell direction (usually taken parallel to c) is distinctly different from all others since it is characterized by covalent bonding of atoms along the helix backbone. This feature rules out all cubic crystallographic space groups. Packing is governed by optimization of interchain interactions. For example, formation of hydrogen bonds in polyamide structure results in a sheet structure, and therefore introduces essentially three main, and different forces

89

in the structure: covalent forces along the chain axis, interchain hydrogen bonds within the sheets and van der Waals forces for the packing of sheets. Further, completion of hydrogen bond imposes stringent rules on the crystal symmetry, as assessed for example by packing of parallel and antiparallel chains: as an illustration, nearest neighbor chain in sheets of even and odd polyamides [NH-(CH 2 ) n -CO-] m are antiparallel and parallel, respectively (even and odd refer to the total number of carbon atoms, including the carbonyl one, in the chemical repeat sequence). Along similar lines, packing of helices of different hand may be more efficient. The a-phase of iPP is made of an equal number of right and left handed helices; in the monoclinic unit cell (space group P2Jc) any one helix has three nearest neighbors of opposite hand. Crystalline polymorphism is frequent for polymers. It is of course a natural consequence of conformational polymorphism when it exists: this has been illustrated for isotactic poly-1-butene. However, different crystal packings may exist for the same chain conformation. Three examples of crystalline polymorphism for common polymers are as follows: Polyethylene crystallizes in an orthorhombic unit cell with parameters a = 0.747nm, b = 0.494 nm and c = 0.252 nm, space group Pnam-D2hl under stress or by epitaxial crystallization on appropriate substrates, it can form a monoclinic structure with parameters a = 0.804 nm, b = 0.479 nm, c = 0.252 nm, y = 107°9' (Seto et al., 1968); at high temperature and pressure, ( « 3 kbar and 220 °C), it exists in a hexagonal phase with a = 0.488 nm (Bassett, 1982; cf. also Chap. 4 in this Volume, Fig. 4-18). Isotactic polypropylene exists in four different phases based on the same 3 1 heli-

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3 Structure of Polymer Single Crystals

cal conformation: the monoclinic a-phase represented in Fig. 3-1 c, a less well characterized hexagonal jS-phase with parameters a = b = 1.9 nm, c = 0.65 nm and a highly original y-phase in which two sets of chains have their axes at 80° to each other. This is the first and so far only example of a polymer crystal structure with nonparallel chains (Bruckner and Meille, 1989). A socalled "smectic" phase is formed at low Tc; it results from randomization of right and left handed helices positions (Bruckner et al., 1991) linked with fast crystallization. Polyvinylidene fluoride, in addition to the three crystalline a-, /}- and y-phases associated with different chain conformations, can exist in a 5-phase which is not formed spontaneously but results from poling the a-phase in high electric fields: it differs from the a-phase by the similar orientation and sense of dipoles of all chains; the unit cell dimensions of this phase are similar to that of the a-phase (a = 0.496 nm, b = 0.964 nm, c = 0.462 nm) but the space group is P2± en, i.e., the differences are only linked to interchain packing (Lovinger, 1982). Finally, it should be mentioned that PVDF and its copolymers with trifluoroethylene (TrFE) exhibit a high temperature ferroelectric-paraelectric phase transition: at the Curie transition, the polar jS-phase converts to a hexagonal phase in which the polar character is lost, as a result of rotational movements of the chains on their axis.

3.2.2.2 Disorder in Polymer Crystals at the Unit Cell level

Several types of disorders may affect the crystal structure (at the unit cell level) of crystalline polymers. Many of these are associated with the nearly cylindrical shape of some polymer helices and with the fact that

helices are noncrystallographic. For example, rotational motion of the chain on its axis, as indicated above for PVDF-TrFE, is a dynamical disorder induced by temperature. Comparable, static disorder exists and may be seen as a disorder of rotation or translational displacement or combination of both. In fiber patterns, this disorder manifests itself by discrete reflections on the equator (since lateral packing of chains is not affected) and by continuous scattering in the form of streaks on layer lines (Clark and Muus, 1962). A frequent cause of disorder is linked with the up and down directional disorder of helical chains, as best illustrated with helices of isotactic vinyl polymers. These helices (cf. Fig. 3-1) have both a hand (right or left) and a sense, defined for example in iPP by the orientation of the C-CH 3 bond relative to the helix axis. Reversing the chain sense results in a different orientation of this bond: corresponding up- and down-chains are cataclinic; identically oriented chains are isoclinic. Replacement of an up by a down chain affects packing in the unit cell only marginally since, in the case indicated, outer methyl groups can be restored at their original position (replacement of left by right handed helices has more dramatic consequences, cf. Sec. 3.4.2.2). It follows that unit cells are often statistical ones with occupancy probability at each chain site of 50% for up- and down-chains: the statistical unit cell of a-iPP has C2c symmetry. Annealing close to the melting temperature results in formation of the less symmetrical P2Jc structure of similar dimensions, but different symmetry, as indicated by the appearance of new reflections in the fiber pattern (Hikosaka and Seto, 1973). The structural analysis of this crystal disorder in a-IPP has been carried out in considerable detail by Corradini et al. (1980).

3.3 Growth and Occurrence of Polymer Single Crystals

3.3 Growth and Occurrence of Polymer Single Crystals 3.3.1 Growth from Solution 3.3.1.1 The Phase Diagram of Polymer Solutions

The crystallization of a polymer from a quiescent solution implies several factors:

the formation of a homogeneous solution at high temperature in the chosen solvent, followed by controlled cooling or quenching to some preset crystallization temperature Tc. The fate of the solution is determined by the binary phase diagram, as schematized in Fig. 3-3 (adapted from Richards, 1946). For good solvents (bottom curve) the crystalline phase separates directly out of

Concentration of polymer Polymer

Solvent (a)

(d)

(e)

91

Figure 3-3. (a) Phase diagram of polyethylene in a good (bottom line) and poor solvent (top line), and resulting typical crystal morphologies obtained in concentration and temperature domains indicated (b-f). At high temperature (domain I), homogeneous solution. For good solvents, crystallization of single crystals (b) is obtained at low concentration. Other, more complicated morphologies are dendrites (c) and axialites (e) and, for high polymer concentration or in the bulk, spherulites (f). For poor solvents, a liquid-liquid phase separation takes place (domain II). Crystallization within these droplets on further cooling (i.e., in domain III) results in spherical, crystalline droplets of PE (d). Note that some droplets have rough surfaces; they contain heterogeneous nuclei. Scale bars: (b) 5 urn, (c) 30 jim, (d) 1 um, (e) 5 ^irn, (f) 30 um.

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3 Structure of Polymer Single Crystals

the supercooled solution. For very dilute solutions (typically c < 1 % or even l%o), single crystals (b) and dendrites (c) are grown at high and low Tc. Higher concentrations yield more and more complicated crystalline morphologies, referred to as axialites or hedrites (e), that is, morphologies with levels of complexity intermediate between single crystals and spherulites, and in which the individual lamella and their relative organization is still, although barely recognizable. Spherulites (f) are obtained in the right hand side of the phase diagram. For poor solvents, the solution separates into two liquid phases on cooling (domain II, limited by the binodal QCN, Fig. 3-3), whereas crystallization only takes place on further cooling (domain III). In this case, the crystalline morphology is again quite complicated, even for very low polymer concentrations, as it is the result of an intricate interplay of nucleation and crystal growth in finely dispersed droplets (Fig. 3-3 d). Liquid-liquid phase separation prior to crystallization therefore often results in formation of small crystalline globs, which are a permanent memory of the phase separation, often admixed with single crystals produced in the dilute phase (Schaaf et al., 1987). 3.3.1.2 Self-Nucleation in Polymer Crystallization

Polymer single crystals are generally taken to be those crystals obtained from dilute solutions in good solvents. The terminology of single crystal is not however restricted to monolamellar and well shaped crystals: it also covers multilamellar morphologies induced by screw dislocations or even dendritic habits which may be produced at lower Tc. Besides the solution concentration and cooling rate or crystallization temperature,

a major parameter in deciding the size and complexity of the crystals formed is the nature and number of preexisting nuclei. Heterogeneous nuclei (impurities from the polymer, e.g., catalyst residues, or introduced by the solvent, etc.) often bear several nucleating sites and therefore initiate the growth of multiple lamellae or complicated crystalline entities. Since these heterogeneities resist heating, complicated and irregular morphologies are usually obtained when the polymer solution is heated at high temperatures prior to crystallization, as mostly these nuclei are active. Fortunately, a means specific to crystalline polymers exists to produce, by thermal treatment, a large number of comparatively simple nuclei: the procedure is called self-seeding or following modern terminology self-nucleation. According to the definition initially given by Blundell et al. (1966 a), self-nucleation is the nucleation of a (polymer) solution or melt by crystalline fragments that remain in suspension after partial dissolution or partial melting of previously grown crystals. The technique rests on the wide dissolution or melting temperature range of polymers (cf. Fig. 3-4), which is itself linked to a wide range of thermal stabilities of the crystallites, dependent on the large molecular weight distributions and wide range of crystal sizes and perfections (Sharpies, 1966). A typical procedure applicable to both solution and bulk polymer (Fig. 3-4) involves four steps: - initial dissolution (or melting) at - crystallization in an initial standard state at T cl , - partial dissolution (partial melting) at Ts in the dissolution (melting) range for a period ts, - final crystallization at Tc2.

3.3 Growth and Occurrence of Polymer Single Crystals

Figure 3-4. Principle of self-nucleation experiments and resulting uniform size of single crystals (optical micrograph, PEO-PS block copolymer). Scale bar: 20 um.

Adjustment of ts and most importantly Ts may yield concentrations of nuclei as high as 5 • 10 14 for polyethylene, as determined from the final crystal size (Blundell and Keller, 1968 a). The nuclei have lateral dimensions on the order of or smaller than 10 nm and are made of molecules from the very high molecular weight tail of the molecular weight distribution. Instantaneous growth on these preexisting nuclei under quiescent

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conditions yields mostly single layered crystals of uniform size, as shown in Figs. 3-3 b and 3-4 b. The homogeneity of the crystal population makes nearly every crystal representative of the whole, and any deviation from normality is of physical significance (cf. Sec. 3.4.2.2). The technique is in principle applicable to all crystalline polymers, including copolymers. Its potentialities have indeed been fully exploited in a number of structural studies dealing with twinning, growth kinetics, etc. An extension of the technique - valid only for solutions and less specific to polymers, however - uses the seeding of a virgin solution by crystals grown in a different experiment. In this procedure, a solution previously heated to T ^> Tm (in order to reduce the number of "intrinsic" nuclei, which are essentially heterogeneous nuclei) is taken to Tc and, as soon as it reaches thermal equilibrium, seeded with the crystals grown in a separate experiment. This procedure makes it possible to grow large crystals or mixed crystals made of successive rings of polymers differing in molecular weight or structure. Yet a further variation on this theme has led to the isochronous decoration technique, which is based on abrupt modifications of crystallization conditions of the solution: crystals grown at a given Tc are used as seeds for growth at a different Tc within the same solution. As shown by Dosiere et al. (1986), the crystallization temperature can be changed by 5 °C within 20 s in a cylindrical vessel of 1.5 mm radius and results in a sharp increase or decrease in the lamellar thickness of the single crystals which can easily be visualized by conventional shadowing in electron microscopy. Recently, an original method to produce polymer nuclei has been proposed (Ding etal., 1989). It rests on the fast quench-

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3 Structure of Polymer Single Crystals

ing of a polymer solution [in this case polyethylene oxide)], which produces a liquid-liquid phase separation and crystallization of small polymer globs. The authors claim that, under these conditions, the globs are ideal for further seeding of polymer solutions. The complexity of the resulting morphologies and the general validity and potential improvements over the more conventional self-nucleation technique remain however to be assessed. 3.3.1.3 Other Crystallization Procedures

Crystallization procedures based on thermal treatments as described in the previous sections enable the best control of crystallization conditions and resultant morphologies. A wider palette of different crystallization procedures can however be used and may yield quite perfect crystals: - The method developed by Patel and Patel (1970) is based on the slow extraction of a glass slide from a solution of polymer. Crystallization takes place at constant temperature, in relatively thin layers, while the solvent is evaporating. Although growth conditions are ill defined, the method is suited for a number of polymers, including polyolefins considered above. - Slow change in chemical potential via progressive diffusion of a nonsolvent into the polymer solution [e.g., water or alcohol in a polymer solution in acidic solvent, for example trifluoroacetic acid (TFAA)], by progressive evaporation of the volatile solvent (e.g., TFAA in a TFAA/trifluoroethanol mixture) or by dialysis of an ionic solution (e.g., LiBr solution). These methods are well adapted for polar, strongly hydrogen-bonded polymers, notably polypeptides and polyamides with short methylene segments. To summarize, growth conditions required for the production of polymer single

crystals are essentially similar to those used for low molecular weight materials. They are based on the same overall basic approach, i.e., on a small change in the chemical potential of the solution via drift or change of concentration, temperature, solvent power, etc. The major originalities lie in the possible control of self-nuclei by thermal treatment and in the fact that only the lateral dimensions of the lamellae can be monitored to some extent (usually between ~ 1 Jim and 100 jim, i.e., no more than two orders of magnitude) whereas the third dimension - the lamellar thickness remains in the 10-30 nm range. 3.3.2 Growth of Lamellar Crystals from the Melt

Polymer single crystals can be obtained under specific crystallization conditions that imitate the melt, or directly from the melt. The first approach is due to Keith (1964), who crystallized polyethylene in a low molecular weight paraffinic diluent (typically C 32 H 66 ), which has the same chemical structure as polyethylene, but a much lower melting temperature and viscosity. Crystallization conditions therefore mimic the conditions experienced by the high molecular weight fraction of the polymer in the presence of the still molten lower molecular weight part acting as solvent. The resulting polyethylene single crystals display significant features characteristic of bulk crystallization, e.g., lenticular shapes as opposed to lozenge type and chain tilt in the lamella. However, similar morphologies have been obtained using poor solvents (e.g., n-dodecanol) which also result in rather high Tc (Khoury, 1979; Organ and Keller, 1985, 1987 b). Crystallization of polymers from the melt in the form of single crystals is very uncommon. Indeed, under usual condi-

3.3 Growth and Occurrence of Polymer Single Crystals

tions the supercooling needed to observe nucleation is ten or several tens of degrees, i.e., a temperature range where high growth rates are incompatible with single crystal growth. Furthermore, several disordering processes exist in the bulk crystallization of polymers (e.g., lamellar twisting) which result in complicated, three-dimensional morphologies. Therefore, a combination of slow growth rate and thin film growth are often sought. The first single crystals produced by bulk or thin film polymer growth were for polymers with "inherent" low growth rates because of their complex molecular architecture or size of side chains. Isotactic polystyrene single crystals (Keith et al., 1970) could be extracted from the surrounding melt after quenching and dissolution of the, by then, vitrified melt. Similarly, random block copolymers of poly(butyleneterephthalate) and poly(tetramethylene ether glycol) can form single crystals when grown from the melt in spite of their molecular complexity; fractionation by sequence length of the crystallizable poly(butyleneterephthalate) was demonstrated (Briber and Thomas, 1985). Structures with polygonal shapes reminiscent of single crystals have been reported, e.g., for polytetramethylsiloxane (Magill, 1964). A more systematic investigation of single crystals grown from the bulk had to await the development of the self-nucleation procedure. Controlled production of significant concentrations of nuclei makes it possible to overcome the low nucleation rate at very low supercoolings, i.e., in a temperature range where growth conditions are compatible with single crystal development. The two examples examined so far - polyethylene and poly (ethylene oxide) - do indeed indicate that, however complicated spherulitic textures are produced at

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low Tc, lamellar single crystals grow at the very high end of the crystallization temperature range. In a similar manner, isotactic polypropylene produces at 160°C very extensive lamellae virtually devoid of the lamellar branching (Bassett and Olley, 1984) that occurs more profusely at lower Tc (Olley and Bassett, 1989). Examination of the resulting single crystals is difficult as they are embedded in a surrounding matrix of the same polymer, usually solidified by rapid crystallization on quenching. Staining or etching techniques may be used (Bassett et al., 1988). Conversely, rapid quenching induces crystallization of spherulites and birefringent contours on the crystal edges, which can be observed by optical microscopy in polarized light (Kovacs and Gonthier, 1972; Labaig, 1978) or selectively dissolved (DiCorleto and Bassett, 1990). 3.3.3 Single Crystals of Biopolymers

Chain folded polymer single crystals illustrate a most efficient means of packing long chain molecules. It is not surprising therefore that a number of biopolymers adopt spontaneously a lamellar morphology, or have been obtained in lamellar form. Most characteristic among them are polysaccharides, cellulose and its derivatives, various polypeptides (Padden and Keith, 1965; Keith et al., 1969) including structural models of fibrous proteins (Lotz and Brack, 1979) and antibiotics (Lotz etal., 1977) and fibrous proteins themselves, such as silk of Bombyx mori (Lotz et al., 1982). All these single crystals were obtained using experimental procedures based on melting or dissolution in appropriate (often polar) solvents followed by cooling or dialysis. The most striking illustration of chain folding among biopolymers is provided by

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3 Structure of Polymer Single Crystals

the DNA of bacterial viruses. The head of the E. coll bacteriophage T2 is a hexagonal casing containing a single molecule of DNA - 60 jam long. The DNA can be visualized in its integrity after being released via an osmotic shock (Kleinschmidt et al., 1962). From X-ray diffraction and birefringence studies combined with the shape of the head and the DNA molecular length, the picture of a head made of a chain folded "single molecule single crystal" of DNA was proposed early on (Bendet et al., 1960). Production of hexagonal single crystals of DNA has been repeated in vitro by using fragments of moderate molecular weight, but of sufficient length to establish chain folding in the platelets ~ 1 |im across and ~ 15 nm thick (Giannoni et al., 1969). The key feature of the experimental procedure appears to be heating in a temperature range where the DNA starts to "melt", i.e, where close connectedness of its two strands is partly disrupted by breaking of hydrogen bonds without however reaching the denaturation stage with its associated separation into individual strands. 3.3.4 Polymers Obtained in the Form of Single Crystals

Most crystalline polymers have been obtained in the form of single crystals by one or many of the experimental procedures described so far. A compilation made by Wunderlich (1973, 1976, 1980) describes the experimental conditions, the resulting morphologies and some structural analysis for nearly 50 polymers of various chemical structure: polyenes, polyvinyl derivatives (alcohol, chloride), polyacrylonitrile, polyacrylic acid, polyethers including polyoxymethylene and polyoxyethylene, polyesters, both aliphatic and aromatic [e.g., polyethylene terephthalate)], various polyamides and polypeptides, and cellulose and

its derivatives (e.g., cellulose triacetate). Single crystals of polysiloxanes (e.g., polydipropylsiloxane [Si(C 3 H 7 ) 2 -O] n are also described, as an example of a polymer with a non-carbon backbone. The compilation made by Wunderlich covers roughly the first 15 years after the emergence of polymer single crystals as a research domain in its own right. In the early days, the main purpose was to demonstrate the feasibility and generality of chain folding for polymers of various types. In more recent years however, the scientific motivation when producing single crystals appears to have shifted: whereas the possibility of chain folding is no longer questioned, single crystals are mainly produced as a convenient investigation tool to analyze a number of structural, kinetic, and thermodynamic aspects of polymers. Although similar motivations clearly existed also in earlier studies, this significant shift in emphasis indicates the maturity of the field. Several such applications will be developed in the following sections. The next two sections deal with the structural aspects of the crystalline core and the lamellar surface, and the interactions between these two regions of single crystals. 3.3.5 Nonfolded or Chain Extended Polymer Single Crystals

Besides the conventional chain folded single crystals described above, other types of polymer single crystal morphologies have been produced, using different routes. These morphologies have in common the fact that the chains are either fully or nearly extended, resulting in lamellar thicknesses in the micrometer range or in the absence of lamellar periodicity. Experimental routes to produce such crystals are quite specific and cannot be generalized to all polymers. We will consider in succes-

3.3 Growth and Occurrence of Polymer Single Crystals

sion: the topochemical polymerization of a monomer into extended chain polymers, crystallization under high pressure (notably of polyethylene), and the formation of crystal solvates. 3.3.5.1 Solid State Polymerization

"True" polymer single crystal fibers can be obtained by polymerization of a crystalline monomer when polymerization proceeds along a well-defined crystallographic axis of the monomer crystal and the polymer lattice has parameters close to that of the monomer: this topochemical polymerization can therefore result in polymer single crystals that have the dimensions of the monomer single crystal (which can be macroscopic) and with polymer chain lengths determined by those dimensions. An early example is the polymerization of trioxane into polyoxymethylene (CH 2 -O) n ; it is not ideal however due to differences in lattice spacings of the monomer and polymer, which results in a fiber habit with poor mechanical characteristics. The best known and most studied system is the solid state polymerization of substituted diacetylene monomer crystals to produce the crystal made of extended polydiacetylene chains. Considerable work has been devoted to these systems; the reader is referred to reviews by Wegner (1979), Bloor (1982) and Young (1987). 3.3.5.2 Crystallization Under High Pressure

Crystallization under high pressure of polyethylene produces lamellar crystals with thickness in the micrometer range, i.e, with a much reduced impact of chain folding. This morphology will be examined in more detail in Sec. 3.5.6, in Chap. 4 of this Volume, and in Sees. 2.1.4.1 and 2.1.5,

97

where the role of mobile phases involved in lamellar thickening is developed. 3.3.5.3 Inclusion Compounds and Molecular Complexes

Polymers can form inclusion compounds or molecular complexes with low M. w. organic molecules. Typical examples include PEO and paraffins or aliphatic polyesters with urea or perhydrotriphenylene, PEO with mercuric chloride (Tadokoro, 1979; Farina and Di Silvestro, 1980), and syndiotactic poly(methylmethacrylate) with, e.g., chloroacetone (Kusuyama et al., 1983). The PEO-urea complex can be formed by soaking an oriented PEO sample in a methanolic urea solution. Formation of the complex yields a macroscopic ( « 1 mm) single crystal; on desoaking the complex in a methanolic solution the original PEO sample is restored. The PEO chain conformation in the complex is essentially that of the crystalline PEO (72-helix). In these systems, polymer chains are encaged in a channel created by the crystalline solvent. The systems are ideal to investigate properties, conformational characteristics and mobility of individual chains, in the absence of their traditional surroundings. 3.3.6 Thermotropic and Lyotropic Polymers

Polymers with rigid moieties in their backbone are a class of technologically important systems. They exhibit liquidcrystal behavior either in the presence of solvents or upon heating (lyotropic and thermotropic polymers respectively). These systems are not normally expected to form lamellar, chain folded crystals. However, a lamellar organization develops in a solution of a typical rigid-rod lyotropic polymer, poly(p-phenylene benzobisthiazole)

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3 Structure of Polymer Single Crystals

(Cohen et al., 1982). The polymer slowly coagulates by gradual absorption of moisture in the solution. The crystals have thicknesses comparable to the molecular length and are presumably made of extended chains; furthermore they are most probably crystal solvates, i.e., incorporate solvent molecules bound by hydrogen bonds to the polymer: similar crystal solvates structures are known for a number of rigid rod polymers (Iovleva and Papkov, 1982; Gardner et al., 1983). Thermotropic liquid-crystal polymers have more flexibility in their backbone than lyotropic polymers, often through inclusion of a flexible spacer. When this spacer (for example, a paraffinic segment) is of sufficient length, chain folding can take place and lamellar structures may develop, for example on annealing. This lamellar decoration

technique

of liquid-

crystal polymer films (Wood and Thomas, 1986) can be used to determine director textures in the liquid-crystal structures (Hudson et al., 1990). More recently, single crystals of a related thermotropic liquidcrystal polyester with a main chain C 7 spacer (i.e., based on azeloate) have been reported by Kent and Geil (1991). Lamellar thickness is ^ 1 0 n m whereas chain

Dominant fold orientation

length is « 80 nm, and chain folding is believed to be predominantly reentrant.

3.4 The Crystalline Core of Polymer Single Crystals 3.4.1 General Features of Polymer Single Crystals 3.4.1.1 Overview of the Model

It is most convenient to define the general characteristics of lamellar crystals at the onset. The main features of a polyethylene single crystal grown from solution are represented schematically in Fig. 3-5. The crystal shown here has a lozenge shape, as it is bounded by four lateral {110} growth faces of an orthorhombic unit cell. The crystal is slightly pyramidal, with an overall tentlike morphology. The polymer chains are parallel to the axis of symmetry of the crystal (c-axis of the unit cell) and fold back and forth on the upper and lower fold surfaces (surfaces tangential to the folds), which are thus less ordered than the crystalline core. The schematic representation of the fold structure, conformation and orientation, is more speculative, as only indirect means exist to assess

Growth sector boundaries Unit-cell

Fold surface

Cilia Loops Tight fold

Fold length L (~10nm) Helical or zig-zag chain conformation

{110} Growth faces

Figure 3-5. Schematic representation of a four-sectored polyethylene single crystal with main features of molecular conformation.

3.4 The Crystalline Core of Polymer Single Crystals

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Figure 3-6. Selected area electron diffraction pattern (center) of PE single crystal seen in bright field (a) and three dark fields obtained by selecting diffraction spots indicated. Note sectorization as revealed, e.g., by differences in (c) and (d) when diffracting planes are parallel or not to growth planes. Scale bar: 3 \im. (c)

these features. For reasons developed in some detail later (cf. Sec. 3.5.4.4), folds are represented as being oriented more or less parallel to the growth faces. They may link adjacent stems in the growth face (sharp folds or, in short, folds) or stems further apart either in the same or in nearby growth planes (loops). Chain ends are represented either impinging at or emerging from the lamellar surface in the form of cilia. Whereas some features may vary with crystallization conditions (three-dimensional shape of the crystal, overall geometry defined by growth conditions), several invariants characterize the polymer single crystal morphology and structure: exis-

(d)

tence of a lamellar thickness /, coexistence of crystalline core and less ordered surface layers resulting in a composite structure at a scale of / (~ 1 to 2 nm), preferred orientation of folds within the various growth sectors. Although the structure is more complicated than low molecular weight material single crystals, it provides a very efficient means to reconcile crystalline order with the structural diversity of polymers, as for example their molecular weight distribution. As an illustration, the crystal shown in Fig. 3-6 has lateral dimensions of 8 j^m x 5 jim for a thickness of about 10 nm. Its mass is 4 x 10~ 12 g; it is made of 2 x 108 stems (38 |im2/0.18 nm2, cross-sec-

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3 Structure of Polymer Single Crystals

tional area of a PE chain seen along its c-axis). As a stem of length 10 nm corresponds to a mass of ~ 1000 daltons, the crystal is made, e.g., of ~ 2 x l O 7 chains of molecular weight 104. It therefore comprises ~ 4 x 107 chain ends and ~ 1.6 x 108 folds or loops, divided between upper and lower lamellar surfaces. The above picture and figures indicate that analysis of the structure of polymer single crystals must be made at several levels: whereas the crystalline core structure can be analyzed rather rigorously, at best an average view of the fold surface structure can be achieved. Further, an important part of the structural analysis deals with interplay and mutual interference between crystalline core and fold structure and conformation. 3.4.1.2 The Fundamental Experimental Evidence

Analysis of polymer single crystals and establishment of chain folding rests on a by now familiar approach which supposes determination of chain orientation and comparison of molecular length with stem length in the crystal. Although this experimental evidence can be gained by X-ray diffraction (at wide and low angle) on single crystal mats, the most vivid and illustrative method remains electron microscopy in bright and dark field, and electron diffraction on individual single crystals. The crystal thickness (10-20 nm) is indeed ideal for TEM (transmission electron microscopy) observations. Figure 3-6 is a most classical illustration of electron microscopy on a single crystal of polyethylene. It displays in (a) a bright field micrograph of the crystal. This picture is characterized by dark striations roughly parallel to (130) in the growth sector bounded by the (110) growth face. The stri-

ations correspond to a Bragg contrast, i.e., to areas of the crystals which diffract electrons which are stopped by the objective aperture. Further, the lamellar thickness of the crystal can be determined by the shadow length if the crystal is shadowed with a heavy metal (e.g., Pt-Pd alloy). The single crystal diffraction pattern is represented in the center of Fig. 3-6. It corresponds to the ab reciprocal plane of the unit cell, i.e., it is only composed of reflections which are located on the equator of a PE fiber pattern (a-axis vertical). This pattern unambiguously establishes that the electrons are traveling down the c-axis of the unit cell, i.e., that the chains are normal to the broad faces of the crystal. Combination of this orientation information, of the limited lamellar thickness and of the molecular length as deduced from the molecular weight, which can be orders of magnitude larger, leads to the concept of chain folding. The dark field pictures correspond to enlargements of the diffraction spots formed by the electrons diffracted by (200), (110) and (1T0) planes (Fig. 3-6 b, c, and d respectively). They complement exactly the bright field picture, as they are formed with the electrons "missing" in (a). Information conveyed by the dark field pictures strengthens the case of sectorization (compare c and d); the origin of the diffraction contrast is analyzed below in relation to the overall shape of the single crystal. 3.4.2 The Lamellar Core and Crystal Defects

Single crystals of polymers provide an easy and quite rare opportunity to observe the crystal lattice down the chain axis, i.e., at right angles to the fiber orientation usually produced by mechanical means (stretching, etc.). They provide therefore a

3.4 The Crystalline Core of Polymer Single Crystals

suitable material to investigate most aspects of crystal structure, defects, impact of deformation, etc. The crystallography of polymers at the unit cell level is considered in Sec. 3.7.1. Here we examine: - The various defects that may affect the crystal structure. These are basically of two types: (1) Crystallographic defects such as twins, edge or screw dislocations. These have counterparts in the structure of nonpolymeric materials; original features linked with the polymer nature are underlined when appropriate. (2) Deformation of the crystal lattice induced by the presence of folds on the lamellar surface, or by the specific growth conditions, etc. - The overall conformation of polymer molecules within individual lamellae (e.g., separation of stems belonging to the same molecule in the crystal core).

101

3.4.2.1 Edge Dislocations Edge dislocations correspond to the creation or annihilation of one or several crystallographic planes. A convenient means to assess the existence and concentration of edge dislocations is via moire patterns produced by two overlapping but slightly rotated crystals (Fig. 3-7). Terminating moire fringes indicate the presence of edge dislocations. Concentration of edge dislocations varies of course significantly with crystallization conditions and increases on annealing, but it is found to be remarkably low in some single crystals. Typical figures are 1011 m~ 2 for polyethylene with a majority of dislocations resulting from creation or termination of fold planes. Further, a 103 fold increase takes place on annealing to 95 °C (Holland, 1964). Poly(4-methyl-l-

Figure 3-7. Moire pattern of superposed PEO-PS block copolymer single crystals. Note the virtual absence, in this crystal, of edge dislocations. Terminating fringes can be seen at the crystal center (from Lotzetal., 1966).

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3 Structure of Polymer Single Crystals

pentene) single crystals have comparable concentrations; in polyoxymethylene, fewer dislocations are located preferably in the vicinity of sector boundaries (Bassett, 1964). Moire patterns have been used to investigate structural changes (generation of edge dislocations) in PE single crystals during annealing (Abe et al., 1970). 3.4.2.2 Twins

Twinning is, as in low molecular weight crystals, a most common defect in polymer crystal. Twins are induced during growth or as a result of deformation. It is customary to define reflection twins, in which the twin plane introduces an extra element of symmetry in the twinned crystal, and rotation twins, in which the two parts of the crystal are rotated about an axis which may or may not be an element of symmetry of the crystal lattice (Barrett and Massalski, 1966). Molecular connectedness through successive unit cells along the chain axis seriously limits the number and frequency of possible twinning modes of polymers. Bevis (1978) considered the possibility of twin planes which either avoid or result in a change of molecular orientation (socalled chain-axis invariant and chain-axis rotation twins). This classification has been extended recently by Martin and Thomas (1991) who describe possible grain boundaries in extended-chain polymers and include the notion of axial and lateral planes between grains. It is of course directly relevant to twins in polymers. In this section, growth twins are considered, as illustrated by reflection and rotation twins, and their specificities are analyzed. Twinning induced by deformation is considered in Sec. 3.7.3.2. Reflection twins are the most frequent twins in polymer crystals and indeed the only ones considered so far. They are char-

acterized by a disturbance of the regular packing of molecular stems along the contact surface of the twin components. This disturbance is rather small as the misorientation merely involves a rotation around its axis of a chain that often has a nearly cylindrical shape. As a result, reflection twins in polymers have their composition planes parallel to the c- or helix-axis, i.e., are hkO or chain invariant twins. Single crystals are ideally suited for investigation of these twins, since observation is performed parallel to the twin plane. Investigation is further facilitated when the twin growth habit differs from that of the untwinned crystals (growth conditions which enhance such differences may be purposedly selected). Studies of twinning rely on selected area electron diffraction; twin sectors and even the twin planes may be visualized by dark field imaging. The proportion of twinned crystals can be significantly increased over its natural occurrence by using a variant of the self-nucleation method (Blundell et al., 1966 a). When highly disordered crystals (produced by initial fast quench of the solution) are used as starting material for the self-nucleation procedure, the proportion of twins and multiple twins in the resultant population is significantly increased: up to ~ 50% for polyethylene oxide (PEO, [-CH 2 -CH 2 -O-] M ) as compared to a natural occurrence of less than 5% (Kovacs et al., 1969). Twinned habits have been reported for polyethylene (Blundell and Keller, 1968 a; Wittmann and Kovacs, 1970), for polyethylene oxide (Kovacs et al., 1969), polyvinylalcohol (Tsuboi, 1968), isotactic polypropylene (iPP) (Kojima, 1967), etc. The latter investigation however deals with thin film growth and describes a number of twins which do not correspond to low index twin planes, and are not observed with any frequency in normal growth. The crys-

3.4 The Crystalline Core of Polymer Single Crystals

talline morphology of iPP is actually dominated by another twin mode, of the rotation twin type. The most extensive study deals with twins and multiple twins of PEO. PEO crystallizes in a monoclinic unit cell with, in chain axis projection, 2 a sin p = b. Two twin modes were recognized: (120) twins involving the most densely packed crystal plane, which is also the major growth face, and (100) planes, in which twinned components differ by the mirror symmetry dip of the a-axis. These two twin planes are at 45° to each other and happen to be planes of symmetry for the PEO single crystals which are square or truncated squares. They lend themselves therefore to a systematic classification. It was found that 14 possible twins and multiple twins can exist, comprising up to eight sectors in mirror relationship with their neighbors (Fig. 3-8). Using the modified self-nucleation technique referred to above, 13 out of the 14 possibilities were observed and characterized by diffraction and dark and bright field electron microscopy. A systematic study of twins and multiple twins of polyethylene (Wittmann and Kovacs, 1970) confirms that two twin modes exist for PE: (110) and (310). These twin modes can be rationalized on the basis of FriedeFs rules of twinning by noting the close analogy of the orthorhombic PE subcell and a hexagonal lattice: the two twin planes thus correspond to the two densely packed planes of the pseudo-hexagonal packing (Fig. 3-9). However, the slight departure from hexagonal symmetry of the PE lattice does not allow a systematic classification of the multiple twins as in the case of PEO. Multiple twins comprising up to six different sectors have been observed. (110) twinning leads to highly elongated crystal laths, because the twin plane acts as a continuous nucleation site for the

103

3b

4b

Figure 3-8. The rationale of PEO multiple reflection twins. Untwinned crystal (top, growth sector boundaries represented by dotted lines) and the 14 simple and multiple twins resulting from all conceivable associations of (120) and (100) twin planes (represented by full and broken lines, respectively). The arrow represents the dip of the a-axis in the monoclinic unit cell (from Kovacs et al., 1969).

104

3 Structure of Polymer Single Crystals

67.4°

126.9°

**#<&<&%?•&••

•*%:;;

Figure 3-9. (a) The two twin modes of polyethylene schematized using the hexagonal approximation of the orthorhombic lattice. Simple (b) and multiple (110) twins (c) and (310) twin (d). Note growth rate increase along (110) twin boundary due to existence of a reentrant angle (from Wittmann and Kovacs, 1970). Scale bars: 5 um.

twinned constituents (cf. Fig. 3-9a, b). This feature of (110) twins is used to assess crystallization theories of polymers (cf. Sec. 3.7.2.3). Very elegant work on twins of poly(4methyl-1-pentene) (P4MP1) has been re-

ported recently (Pradere et al., 1988 b). Single crystals of the polymorph III of P4MP1 are square but were found to be composed of narrowly delimited microdomains that give rise to different i.e., symmetrically related electron diffraction patterns. Dark

3.4 The Crystalline Core of Polymer Single Crystals

field images indicate that the shape, size and number of the microdomains may vary from crystal to crystal. The twin mode may imply identical helices but with opposite oaxis (chain axis) orientation, as suggested by the authors. Alternatively, it may imply domains with helices of opposite hand (i.e., right and left handed). In any case twinning occurs spontaneously in these P4MP1 crystals at a much higher frequency than is usual for most polymer crystals. Twinning of the hkO type or "c axis invariant twinning" may lead to three-dimensional structures if the chains are inclined to the surface normal in the crystals. Pradere and Thomas (1989) have reported on the development of such three dimensional structures in twins of a-phase poly(p-xylylene) (PPX). In view of the beam resistance of PPX, HREM on twin boundaries was attempted; (020) lattice fringes were obtained in the sectors and (110) ones near the twin boundary. Reflection twins with twin planes of the hkl type, i.e., which intersect the chain axis at an angle are not normally expected in single crystals grown from solution. Such twin planes have been reported for single crystals of a polydiacetylene derivative (Young, 1987). The change in chain orientation is associated with, for example, introduction of a single gauche conformation in a succession of trans conformations. Staggering of this defect in neighboring chains naturally results in a hkl plane which has all the characteristics of a twin plane. Clearly however, possible existence of such twin planes is strictly limited by chain conformation and chain packing requirements set by the molecular connectedness across the twin plane. Rotation twins are characterized by an w-fold axis of symmetry between twin constituants. In polymer crystals, when the

105

orientation of the axis of symmetry is at an angle to the chain axis, the chain orientations in the twin-related constituants are no longer parallel. Since polymer growth is in lamellar form, rotation twins manifest themselves by the generation of sets of lamellae at a significant, but constant angle. The most telling example of rotation twinning in polymers is the so-called "cross-hatching" of lamellae of isotactic polypropylene in its crystalline a-phase. The morphology of a-iPP in both bulk and solution crystallization is characterized by repeated lamellar branching, the planes of the lamellae making an angle of — 100°. In solution crystallization, the resulting mor phology has been described as a "quadrite" (Khoury, 1966) (Fig. 3-10). Khoury pointed out that the overall morphology is well represented by a two fold rotation twinning along the <101> direction of the unit cell. The molecular analysis of the branching (Lotz and Wittmann, 1986) indicates that it is associated with packing requirements of helices of identical hand in the (010) growth plane of the a-iPP lathshaped crystals. The crystal structure requires alternation of helical hand in successive (040) planes. When this alternation is not respected, the helices of the same hand as the substrate plane are oriented with their axis parallel to the a-axis of that substrate (rather than c) and constitute the start of a new lamellar growth oriented characteristically at the monoclinic fi angle (i.e., 99°) from the parent lamella (Fig. 3-10b and c). The molecular analysis was described as a case of homo- or autoepitaxy; it can equally well, and more simply, be analyzed in terms of rotaton twinning or "lateral chain rotation" in the terminology of Martin and Thomas (1991). The twin plane is (010) and, as indicated by Khoury (1966), the common axes in the

106

3 Structure of Polymer Single Crystals (b) R L R L R R L R

ia;

L

(c)

Figure 3-10. (a) The quadrite structure of isotactic polypropylene made of interwoven lath-shaped crystals seen edge-on, i.e., along b-axis direction (from Khoury, 1966). Scale bar: 0.5 \xm. (b) Molecular origin of branching linked with packing of helices of identical hand [R and L: right and left handed helices; crystallographic directions given in direct (b) and reciprocal («*, c*) space], (c) Interdigitation of methyl side chains in the contact face (shaded and unshaded groups belong to different layers) resulting from rotation twin around <101> axis.

twinned components are <101>. On a structural basis, they correspond to axes of symmetry of strong structural features in the contact planes, i.e. of rows of methyl groups and of a lozenge-type array of methyl groups, depending on the growth faces considered (Fig. 3-10 c). It should be mentioned that repetition with crystallographic regularity of such a packing scheme constitutes the essential feature of an original structure of isotactic polypropylene in its y modification, the first example of a polymer crystal structure with non-parallel chain axes (Meille and Bruckner, 1989).

Rotation twins that result in non-parallel chain axes have seldom been considered in polymer science. Although the a-phase of iPP provides the most evident illustration of repeated lamellar branching, other structures with wide-angle lamellar branching (but admittedly lower frequency) are known in the polymer field (e.g., polyamide crystals) and would deserve renewed examination in the light of this analysis. The above examples have shown two different possible crystallographic responses to packing of helices of different hand: a reflection twin (one explanation for the

3.4 The Crystalline Core of Polymer Single Crystals

twin planes in P4MP1) and a rotation twin for isotactic polypropylene. In both cases, contact between the two phases is limited to a twin plane the surface of which is, for rotation twins, roughly the square of the lamellar thickness i.e., L2. Recent analysis of single crystals of syndiotactic polypropylene has revealed a more diffuse and widespread manifestation of packing of enantiomorphous helices. The chain conformation of sPP is based on a ttgg sequence, and right and left handed helices are isoenergetic. Unit cells including only isochiral helices, or alternate layers of antichiral helices have similar orthorhombic symmetry and parameters, but the former is centrosymmetric, whereas the second is face centered. As a consequence, location of a depositing helix on a growth face depends on the relative hands of the helix and the substrate: two possibilities, located half a unit cell apart are possible (Lotz et al., 1988). At high temperature, only the facecentered unit cell packing is observed, but on lowering the crystallization temperature, a mixture of the two packings is produced, which is vividly manifested by spectacular diffraction effects (Lovinger et al., 1991). 3.4.2.3 Screw Dislocations Screw dislocations are a most important defect in polymer structures. They are the key feature of lamellar proliferation in bulk crystallization. The original characteristic of screw dislocations in polymers resides in the fact that the Burgers vector which defines the amplitude of the dislocation is equal to the lamellar thickness, and thus includes the crystalline core and the two fold surfaces: the whole lamella acts as the structural unit. This is in sharp contrast with low molecular weight materials where the magnitude of the Burgers vector can be

107

as small as one crystallographic repeat distance. Further, fold surfaces of polymer lamellae in the screw forbid crystallographic coherence between successive layers although their surfaces in contact: the only crystallographic cohesion therefore is through the lamellar core, and spirals around the dislocation line vector (which is parallel to the chain axis), at the center of the screw (Fig. 3-11 a). The lack of surface connection may give rise to spectacular rotations of successive layers in the screw which have no counterpart in other systems (Fig. 3-11 b). Rotation is always associated with a pyramidal habit of the basal crystal, and growth of the screw takes place in the concave part (the inside) of the pyramid; as a result, successive layers always wind up in the screw. A detailed analysis of such screw dislocations was made by Keller (1967). 3.4.2.4 Local Defects Defects considered under this heading do not, or only marginally affect the location of chains in the crystal. They include all defects resulting from chemical, configurational or conformational irregularities in the chain. Examples of such defects can be - head to head defects in a head to tail sequence as e.g., in polyvinylidenefluoride (PVDF): -CH2-CF2-CF2-CH2-CH2-CF2- occasional accidents in the strict tacticity of tactic (either iso- or syndiotactic) polymers or departures from conformational regularity. The most classical and thoroughly investigated defect of this class is the so-called kink, which, in a chain based on all trans conformation (Pechhold, 1968): tttttttttttttttttttttt

108

3 Structure of Polymer Single Crystals

(b)

Figure 3-11. Screw dislocations of polymer lamellae, (a) PEO (M. w. = 6000) grown in thin film in a transition region from once-folded to chain-extended growth. The thicker ridges correspond to extended chains. Lower to upper layers have geometries as in Fig. 3-22, Tc = 59.39-59.45 °C (from Kovacs and Straupe, 1980). Scale bar: 5 um. (b) Double screw dislocation in PEO-PS block copolymer crystal grown from xylene solution. Note regular rotation of successive terraces linked with the existence of a disordered interface at the lamellar surface (from Lotz et al., 1975). Scale bars: 1 jam.

is produced by introduction of g and g conformations: ttttgtg~ttt

or

ttttg"tgttt

This kink, of so-called type 2gl is characterized by a shift of chain axes adjacent

to the kink by ~ 0.22 nm and by a contraction relative to the extended stem length by ~ 0.126 nm (considering a polyethylene type structure) i.e., roughly c/2 of the subcell dimension. It is therefore a most likely defect involved in chain translation along its axis, as occurs on annealing etc. Further introduction of gauche bonds can result in significantly larger shifts of chain axes on opposite sides of the defects. Significant among these are the so-called jogs, in which two different segments of the chains occupy two different crystal sites, along either the a or b crystal axes. Jogs may, conceivably, be associated with a chain end located in the crystal core. These defects have been thoroughly described and analyzed by Reneker and Mazur (1988, 1991). Chain ends are potential crystallographic defects typical for polymers, however perfect their chemical structure. If they were systematically incorporated inside the crystalline core, they would dramatically affect the crystalline order. Taking the figures indicated for the PE single crystal (Fig. 3-6), chains of molecular weight 20 000 build up - 20 stems of length 10 nm (or M.w. = 1000), of which two bear chain ends. In other words, every chain would have one or several nearest or second nearest neighbors affected by a chain end. Considering molecular rearrangements associated with such defects (dislocations, etc.) every chain would be affected. Reduction of this impact can, conceivably, take place via pairwise association of chain ends in the crystal or, more realistically, by concentrating chain ends at or above the lamellar surface, in the form of cilia. The importance and impact of location of chain ends at the lamellar surface will become apparent when considering these surfaces. A more drastic point defect is created when incorporating in the crystal lattice

3.4 The Crystalline Core of Polymer Single Crystals

chemical irregularities which affect the chain profile and cross section. The most vivid illustration is provided by ethylenepropylene copolymers, which are in essence polyethylene chains with occasional methyl side branches: -CH2-CH2-CH-CH2-CH2-CH2-

I

CH 3 Methyl branch content is usually low (< 30/1000 C atoms) but such irregularities have spectacular consequences on the crystal morphology and structure (see Khoury and Passaglia, 1976): change in unit cell parameters, with increase of the a-axis, significant alteration of growth habit with development of irregular growth faces and formation of more grainy domains as attested by the spreading of diffraction spots. These manifestations must all be related to incorporation of the chemical defects within the crystal structure, although a number of techniques indicate that the methyl groups are preferably rejected on or in the lamellar surface. The main body of evidence arises from chemical methods aimed at etching or removing the lamellar surface, or selectively modifying it with various reagents (acidic and oxidizing for polyolefins, etc.; for a review see Patel, 1980). Contrary to line defects (e.g., edge dislocations), local defects cannot be detected easily and thus analyzed. Most techniques therefore rely on global characteristics (e.g., deficient morphology) or imperfection of the crystal lattice. The most direct indication for lattice disorder is obtained by the broadening of hkO reflections, which, when analyzed with the Scherrer formalism, indicates lateral dimensions of crystallographic coherence of ~ 30 nm for polyethylene single crystals. Broadening of

109

reflections incorporates however contributions of local and line defects, as well as more important grain boundaries. The small domain size reported for both bulk polymers and single crystals appears however to be contradicted by observation, in single crystals, of much larger domains of crystallographic regularity, as assessed for example by moire patterns and by the lack of direct evidence for a mosaic block structure, which was searched for by dark field electron microscopy (Thomas et al., 1974). Further Harrison et al. (1976) reappraised possible geometric causes of X-ray broadening (chain tilt, lamellar thickness). They conclude that mosaic type X-ray line broadening is virtually absent and thus mosaic structures are very rare in polymer crystallization. Impact of lattice defects on the crystal structure has been analyzed by Hosemann and co-workers (e.g., Hosemann, 1972) in terms of the paracrystalline state, which is based on the blocky nature of matter in general, and polymers in particular. The analysis includes lattice imperfections in directions parallel to the chain axis, linked with the lamellar structure of crystalline polymers, and transverse to the chain axis, linked with the relative weakness of interchain forces in that direction. These studies have found new relevance with the industrial development of the so-called linear low density polyethylenes (LLDPE) in which short chain branches are purposefully introduced in the synthesis by adjunction of propene-, butene-, hexane-, and octene-monomers (see also Chap. 4, Sec. 4.3.4.2). Along a different line, hydrogenated polybutadienes or polyisoprenes provide model polyethylenes of known M.w. and M.w. distribution (with typical MJMn » 1.02) and with precisely known branch content and branch length (see also Chap. 1).

110

3 Structure of Polymer Single Crystals

3.5 The Lamellar Surface 3.5.1 Overview and Issues

The key distinguishing feature of polymer single crystals resides in the existence of the two fold surfaces that sandwich the crystal core. The lamellar surface is a thin layer with virtually no recognizable structure, at least in the conventional, crystallographic sense, although the organization and thickness may and do vary with crystallization conditions, polymer, etc. Very few probes exist to map such local partially organized structures and investigation of the lamellar surface remains a formidable experimental challenge. These difficulties may be partly overcome in the future with the development of atomic force and scanning tunneling microscopy. In view of this lack of clear-cut experimental techniques, most insights on the fold surface structure can only be indirect, and therefore leave ample room for speculation or a priori views. The fold surface structure has been for years in the heart of a heated debate. On one side, the regularity and perfection of polymer crystals strongly suggests a high degree of regularity in the surface. On the other side, the well-known random coil conformation of polymer chains in the melt or in solution seemed irreconcilable with strict order upon crystallization: even if stems in the crystalline core are crystallographically ordered, the various stems of a given chain must be distributed in the crystal in a manner which has a memory of its original random coil conformation; this in turn would imply that the links between these stems (which build up the fold surface) have no azimuthal orientation correlation and no structure, i.e., are essentially amorphous in character. The two fold surface models, in their extreme versions, have become

known as adjacent reentry and switchboard (of the old fashioned type, with cords and sockets). A great deal of research has been devoted to settling this issue. It was indeed a major theme of a meeting of the Faraday Society, which is still considered as a landmark (Faraday Discussions of the Chemical Society, Vol. 68, 1979). Note that this is « 40 years after the initial proposal of Storks (1938) and 22 years after the paper of Keller (1957). A personal account can also be found in Keller (1991). The actual fold surfaces of polymer crystals are a perfectly viable compromise between these two extreme views, with, most probably, significant variations depending on crystallization conditions. The essential arguments, known from the beginning, are as follows: - Single crystals can, conceivably, be made of sharp folds only (with occasional chain ends). Closer understanding requires however the use of appropriate models (e.g., cyclic polymers) and relies heavily on molecular modeling and conformational analysis of possible folds. - Various techniques, including straightforward density measurements, indicate a crystallinity deficit: 10 to 20% of the polymer is not crystalline (figures may vary significantly depending on polymers, crystallization conditions, and experimental probes). The deficit in crystallinity is not located within the lamellar crystalline core, owing to its regularity: it is thus located near the surface, i.e. may be associated with folds. However, on steric reasons alone, the surface of single crystals cannot be made of loops only (i.e., excluding sharp folds between adjacent stems). The most vivid argumentation is given by Frank (1979): overcrowding in the amorphous zone near the crystal surface results from randomization of the chain direction when leaving

3.5 The Lamellar Surface

the crystal, from the avoidance factor (deviations needed to avoid neighbor chains) and from back tracking (chain partly returning towards the crystal surface). These factors make it necessary to introduce some proportion of sharp folds between adjacent stems to relieve steric constraints or they would require a significant chain tilt, i.e., a fold surface oblique to the stem direction, at angles significantly larger than are observed. Having set the stage, we examine now: - The analysis of theoretical models and experimental support for tight folds (i.e., short, compact loops between adjacent stems). Whereas this has been mainly analyzed for polyethylene, additional constraints are imposed on folds when taking into account, for example, the distribution of left and right handed helices in the crystal structure. - The experimental support for a significant fraction of noncrystalline material in single crystals. - Experimental support for structure and organization of the folds, as assessed with recently available model polymers and experimental techniques. Finally, the impact of the fold surface on structure and properties of the single crystals will be analyzed. Only a relatively brief account of classical knowledge (i.e., as can be found in Far. Disc. 1979) is given; more recent results are developed in somewhat more detail. 3.5.2 The Sharp Fold

The perfection of some solution grown crystals of polyethylene suggested that the fold exerts no or very little disturbing influence on the crystal lattice. These observations led to the feasibility of sharp folds and their regular arrangement in the fold surface being considered.

111

3.5.2.1 Computed Models

The early models of sharp folds considered the shortest path of carbon atoms in a diamond lattice, which makes it possible to turn around at a distance of 4.45 A [distance between two stems in the (110) growth plane]. Numerous, more elaborate treatments of the fold structure and conformation have been made using conformational energy analysis techniques (McMahon etal, 1967; Petraccone et al, 1972; Oyama etal., 1973; Dave and Farmer, 1988). The more recent analysis of Dave and Farmer is used here, as it includes various possible modes of deformation (internal rotation, bending, bond stretching and compression), and allows for departures from crystallographically fixed sites of emergence and return in the crystal. Figure 3-12 illustrates the end and side views of the two possible sharp folds between nearest neighbor chain stems, i.e., in (110) and (200) planes of the PE unit cell; they have energies of 1.36 and 1.24 J • mol~x of folds, respectively, relative to the planar zig-zag conformation of PE. The minima found for the energies when relaxing interstem distances differ by less than 0.02 joules and 5 x 10" 3 nm from these values. The minimum surface energies for the 110 and 200 folds correspond to 102.5 and 95.6 erg cm" 2 (1.025 x 10" x and 9.56 x 10~2 J • m~ 2 ), values in general agreement with fold surface energies of (93±8)erg-cm" 2 [-(93±8)xl0" 3 J-m- 2 ] determined experimentally (Hoffmann etal., 1975). 3.5.2.2 Packing of Sharp Folds

Packing of sharp folds in the lamellar surface was also considered early on. The most elaborate and complete models are due to Reneker and Geil (1960), and attempt to account for the three-dimensional

112

3 Structure of Polymer Single Crystals

Figure 3-12. Minimum energy fold conformations for a PE chain in (110) (a) and (010) (b) growth planes (from Dave and Farmer, 1988).

habit of PE crystals, i.e., the slope of the pyramidal facets of the crystals. It is indeed observed that fold surfaces tend to be close to low index (hkl) planes both in (110) and (200) growth sectors. Thus, associating the Miller indices of the surface plane and growth plane in the manner introduced by Bassett et al. (1963 a), lozenge shaped pyramids with {(lll)(110)} and {(312)(110)} have maximum slopes of ^ 30° and make angles of « 20° and 30° with the a-axis of the unit cell (cf. Figs. 3-19 and 3-24). Such habits can be accounted for in an idealized manner by considering the so-called fold packing schemes 1 and 2 characterized by different shifts of neighboring stems in the

a, b and c direction: 0 a, 1 b and 1 c, and a/2, b/2 and c/2, respectively. Fold packing in planes bounded by (200) growth faces is also of interest. Such growth sectors are prominent for solution crystallization at high Tc (e.g., Khoury, 1979; Organ and Keller, 1985) and for thin film and bulk crystallization (e.g., Keith, 1964; Khoury, 1979; Keith et al, 1989). In these sectors chain tilts relative to lamellar normal are close to 0°, 18°, 35° and 45°; i.e., correspond roughly to (001), (101), (201) and (301) fold surfaces. Conformational energy analysis of (200) folds in {(101)(200)} sectors yields however unexpected results: whereas packing energies are reasonable,

3.5 The Lamellar Surface

they are obtained for significantly deformed unit cells with an a parameter increased from « 0.75 to 0.84 nm and b reduced from 0.49 to 0.46 nm. Dave and Farmer question the limitations of their calculation procedure (time and storage requirements make it possible to simulate only the top fold surface) or suggest the possibility of (110) folds in these (200) growth planes; in this situation, the fold would link two growth planes; in an idealized view, the molecule would go back and forth between the two growth surfaces. As examined later on, a similar folding scheme has been proposed for iPP and iPS in order to reconcile the crystal symmetry of the unit cell with the physical requirement of chain folding. Tight folds between adjacent stems in paraffin segments were also established in an extensive set of investigations on aliphatic polyamides, made of paraffinic sequences of different lengths (usually 4 to 10CH 2 ) interrupted by amide groups: CO-NH, which form hydrogen bonds between neighboring chains. Such hydrogen bonds impose some type of crystallographic register between neighboring stems. Further, it was observed (Dreyfuss et al., 1972) that solution-grown lamellae have thicknesses that are multiples of the unit cell length (when taking into account the chain obliquity): e.g., four times for polyamide 6-6. Mats of such lamellae give rise to the strong reflections and subsidiary maxima characteristic of the lamellar and unit cell structure. Their analysis (Atkins et al., 1972) indicates that folding takes place in the diacid part of the polyamide 6-6, and therefore locates very precisely the fold along the chains. The fact that folding occurs in the methylenic segment of the chemical sequence is not unexpected: the original feature however rests in the selection, among two possible methylene se-

113

quences of the shorter C 4 of the diacid, rather than the longer C 6 one of the diamine part. 3.5.2.3 Tight Fold and Chain Sense

The physical requirement of significant folding between adjacent stems may conflict, or may be difficult to reconcile with crystallographic requirements. This would be, for example, the case when the chains have a chemical direction to their backbone, such as polypeptides or proteins (NH-CH(R)-CO) n or polyamides based on the polycondensation of oc-co aminoacids: (NH-(CH 2 ) X -CO) B or, e.g., polypivalolactone: (CH2-C(CH3)2-CO-O)n For polymers of this sort, the only conceivable unit cell symmetries must allow for chain inversion in order to be compatible with chain folding. This requirement is indeed fulfilled for most polyamides and polypeptides. The problem of chain directionality in polypivalolactone has been discussed by Meille et al. (1984) who favor significant proportions of adjacent reentry, which is both permissible and highly favoured in view of the crystal symmetries of its a- and y-modifications with antiparallel chains. A more involved situation exists for polyolefins with a helical structure, as discussed in detail for isotactic polystyrene by Sadler et al. (1984) and isotactic polypropylene by Petraccone et al. (1986). The argument blends crystallographic and conformational constraints and may be summarized as follows: isotactic polyolefins (say 3 x-helices of iPS and iPP) have side chains inclined relative to the helical axis,

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3 Structure of Polymer Single Crystals

as in fishbones (Fig. 3-1), and further may be right or left handed. Folding of a long, regular helix results in alternation of up (u) and down (d) side chain orientations in successive stems (cataclinic helices). Conformational constraints impose that if side chains were all to be oriented up (or down), the helical sense would have to be inversed in alternate stems. Thus folds can only be made between L u -R d , or R u -L d , or R u -R d (and its mirror image L u -L d ). However, the (330) growth faces of hexagonal iPS single crystal and the lateral (010) growth faces of iPP lath shaped crystals are made of successive planes of helices with same hand and side-chain orientation. Even when allowing for some disorder, this crystallographic information is in flagrant contradiction with the above conformational constraints on the fold. Both Sadler et al. and Petraccone et al. (1986) therefore suggest folding schemes that "knit" together two successive layers which are made of cataclinic helices. The scheme is thus based on L u -R d or R u -L d links and results in a zig-zag folding model within a bilayer strip oriented parallel to the macroscopic growth front. For iPS single crystals, neutron scattering evidence produced by Sadler et al. (1984) is broadly compatible with this folding scheme, and indeed does not feature a peak (at q = 0.5k~1) that would be expected if folding had taken place within a single growth plane, between isochiral helices (cf. Fig. 3-13 a). For iPP lath shaped crystals, the folding scheme requires two distinct folds in (140) and (100) planes, the latter being at right angles to the macroscopic lateral growth faces. This folding scheme should be valid for the whole lamellar surface. It is however at variance with results of polymer decoration of iPP laths (Wittmann and Lotz, 1985) that indicate that the laths are sectored (Fig. 3-16);

few1

Figure 3-13. Various possible arrangements of stems of a single (e.g., deuterium tagged) chain in a crystal (here of isotactic polystyrene). Neutron diffraction experiments on iPS and PE favor arrangement d.

the decoration results exclude an earlier folding scheme by Sauer et al. (1965) and indicate that, if unique, the folding scheme and geometries may have variable impacts in different growth sectors. The above examples illustrate both the possibilities and limits of a priori analysis of fold conformations and orientations. Even under the rather favorable condition of sharp folding and simple chemical structure (e.g., polyethylene chain) the fold conformation (if unique) is only known approximately. Even less is known so far on the fold conformation, structure and arrangement of "stiffer" molecules which include bulkier and more rigid groups in the main chain (benzene rings, etc.), not to mention folding of the double helices of DNA. 3.5.2.4 Proportion of Sharp Folds

Besides the problems of chain conformation and trajectory in sharp folds, the problem of their proportion relative to less ordered ones is a crucial issue. Again, most relevant information is available for relatively simple, flexible chains such as polyethylene, and only a few guidelines are given here. Obviously, sharp folds can represent 100% of the folds as occur in cyclic paraffins. The lower limit is more difficult

3.5 The Lamellar Surface

to assess. Flory (1962) pointed out that the "flux" of chains emerging from a crystal surface and entering the amorphous phase must be reduced, and that probably more than 50% of the chains reenter the crystal. As indicated in the introduction of this section, Frank (1979) using purely geometric arguments, arrived at an even higher proportion: ~ 70% of the chains should form tight folds near the surface. The problem has been tackled by theories based on statistical mechanics, including various levels of sophistication (Di Marzio and Guttman, 1980; Guttman and Di Marzio, 1982; Flory et al, 1984; Marqusee and Dill, 1986; Mansfield, 1983; Marqusee, 1989). Whereas initial models considered unperturbed random chains, recent analysis include energy terms for bent conformations (Marqusee, 1989), order favoring tight-fold conformations, and take into account the effect of chain incidence density (Kumar and Yoon, 1989). When these factors are included, the predicted proportion of chains involved in tight folds increases to « 70%, as compared to « 40% for flexible chains (Flory et al., 1984). Although these models do not consider specifically single crystal formation, the underlying principles are of sufficient generality to be applicable. Their main outcome lies in the conclusion that, irrespective of crystallization conditions, a significant majority of chains of the fold surface are involved in tight folds, in general agreement with the early estimate of Frank (1979). 3.5.3 The Less Ordered Surface

Considerably less is known on the conformation of the chains in fold surfaces that are not involved in sharp folds. Such chain segments can be chain ends rejected in the amorphous surface (cilia), long loops connecting non adjacent stems, etc. Most

115

techniques to investigate their structure are global, although some discrimination between the different components has become possible recently. The physical state of the "amorphous" layer in polymer crystals was investigated early on. Fischer et al. (1969) and Fischer and Kloos (1970) measured the absolute value of low angle X-ray scattering intensity for mats of poly-1-butene and of polyethylene single crystals, respectively. They determined the density of the amorphous layer which, when plotted against temperature, showed a break typical of and located at the glass transition of the amorphous polymer. On an overall scale, this layer thus appears as a truly amorphous component. The surface is susceptible to swelling by low molecular weight solvents and may be stained. Swelling of stacks of PE single crystals by decalin was performed early on by Udagawa and Keller (1971) who observed an increase of up to 22 A of the overall lamellar thickness. Similar experiments have been performed recently by Chang and Krimm (1984), using again freeze dried PE single crystals swollen in decalin. Changes in the Raman LAM (longitudinal acoustic mode) were used to study the influence of molecular weight, temperature of crystallization and of measurement. The LAM technique makes it possible to differentiate the responses of terminal defects and stems. It was shown that the cilia are mainly responsible for the swelling in single crystal mats. The susceptibility of amorphous zones to staining is valuable in assessing the extent of the amorphous zone. An illustration is provided by comparing the staining of lamellae of a cycloparaffin and polyethylene: the extent of staining is significantly larger in PE than in the cycloparaffins (Ihn etal, 1990a,b).

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3 Structure of Polymer Single Crystals

Chemical modification and degradation has also been used to evaluate the extent of amorphous phase in lamellar crystals. For PE, early experiments involved degradation with nitric acid (Palmer and Cobbold, 1964). The molecular weight of the residual segments which ultimately were reduced to a somewhat shortened stem length was followed by GPC (Blundell et al., 1966 b). Numerous other methods of chemical modification or degradation of the more easily reachable fold surface have been developed. These studies have been reviewed by Patel (1980). In conjunction with several characterization techniques to determine weight loss, density and crystallinity, heat of fusion, decrease in melting point, amount of mobile fraction by NMR, molecular weight by GPC, lamellar thickness etc., they can measure the amount of amorphous material, but do not assess quantitatively between the different components of the amorphous phase (cilia, loose loops, etc.). Further information on the amorphous phase is obtained from assessment of molecular mobility in this phase as determined by NMR: local environments resulting in different molecular motions can be distinguished by their NMR relaxation times. Many polymers require three components - amorphous, crystalline and "constrained amorphous" - to correctly interpret line width, T2 and 7\ data (spin-spin and rotating frame relaxation times). The mobility of this constrained amorphous phase is accounted for by a model of motion of a fold or cilium with N connected segments (McBrierty, 1979). The concept of a three phase model was further developed by Mandelkern (1979) who introduced the model made of a crystalline core, an interfacial zone associated with the constrained amorphous phase, and an amorphous phase. The interfacial

zone, which may include the sharp folds discussed previously, accounts for 5 to 15% of the material (PE crystals).

3.5.4 Experimental Techniques

A wide variety of experimental techniques has been used to identify the fold structure in polymer single crystals, some of which have already been touched upon in the previous section. As an illustration of the difficulties of the task, we analyze here the reasonings involved, and the contribution of some experimental techniques. The fold surface structure can be inferred from two very different approaches: (a) The more global one rests on the determination of the global chain conformation of one polymer chain in the lamellar crystal. Provided one chain can be differentiated from its neighbors, it is possible to determine average positions and distances of its various crystalline stems. The most straightforward means to differentiate polymer chains is via isotopic labeling. The pioneering work along this line is due to Krimm and collaborators (Tasumi and Krimm, 1967, 1968; Bank and Krimm, 1969; Cheam and Krimm, 1981) who investigated mixed crystals of hydrogenated and deuterated polyethylene by infrared techniques in order to define the local environment of any deuterated stem. Since isotope labeling is the basis of neutron scattering experiments, comparable samples have been used to investigate the chain trajectory and average dimensions of deuterated molecules dispersed in and cocrystallized with a host matrix of the protonated polymer (Schelten et al., 1976; Stamm et al., 1979; Sadler and Keller, 1976, 1977). The above two techniques cannot detail the conformation and orientation of the links

3.5 The Lamellar Surface

between the stems, i.e., the structure of the folds, but they may help differentiate adjacent from nonadjacent stems. (b) More local techniques aim at establishing the fold conformation, orientation and mobility. These issues are now examined via IR, NMR and neutron scattering as well as by the recently developed technique of polymer decoration. Contribution of model, cyclic paraffins is particularly considered in this last section, both in conjunction with decoration techniques, and in a broader context of tight fold conformation. 3.5.4.1 Infrared Experiments on Mixed Crystals

As applied to polyethylene crystals, infrared spectroscopy of mixed crystals is based on the fact that the PE unit cell contains two nonequivalent chains. Vibrational modes of a single chain are split in the crystal due to interactions between these different chains. The magnitude of the splittings depends on the arrangement of hydrogenated and deuterated stems in a mixed crystal. If one of the species (usually the deuterated one) is dilute, the interactions between stems of this species are minimized and its vibrational bands predominantly reflect intramolecular interactions between stems. Specifically, the CD 2 bending crystal field doublet is Avb(CD2) = 4.8 cm" 1 , which is a little larger than the predicted half value of 7.7 cm" 1 observed in pure PED, but is a good indication of extended adjacent re-entry along a single (110) plane. Later work at low temperature, which minimizes the variation in lattice dimensions with morphology and increases the splitting (Krimm and Cheam, 1979; Cheam and Krimm, 1981), using a Fourier trans-

117

form infrared spectrometer (Spells et al., 1984) showed a more complex multiplet band structure. These more complex results were analyzed by using models in which stems of the same molecule have a high probability of adjacency in a given sheet (growth plane), but can be located on several sheets (the so-called superfolding model, cf. Fig. 3-13 d). It should be mentioned that the infrared investigations of Spells and collaborators were performed in parallel with neutron scattering investigations on the same samples (Spells and Sadler, 1984), and their analysis of stem distribution uses the results of the latter study. A second and historically earlier aspect of infrared use to study the fold structure is the characterization and investigation of conformation sensitive infrared bands, which provide a direct insight into the fold structure. Bands specific to folds present in crystals of various polymers have been recognized early on (see e.g., Koenig and Agboatwalla, 1968; Zerbi, 1983). A fairly recent, systematic study is due to Spells et al. (1987). These authors use polyethylene single crystals with a range of different morphologies, i.e., growth planes and therefore, presumably, varying ratios of different fold conformations. A wagging mode peak located at 1342 cm" 1 and already associated with a gg conformation (Zerbi, 1983) present in the fold region of cyclic paraffins was identified. Variations with crystal habit of the wagging mode region were observed, but could not be related in any systematic manner with differences in possible fold types and structures. 3.5.4.2 Neutron Scattering on Mixed Crystals

Neutron scattering covers a wide range in q (q = 4 n sin 6/1 with 2 6 the scattering

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3 Structure of Polymer Single Crystals

angle) and is therefore able, in principle, to yield information on the global chain conformation (at small 9 and intermediate q) down to almost crystallographic detail (i.e., the stem-stem interactions). Experimental difficulties may arise from molecular segregation (isotopic fractionation) between hydrogenated and deuterated species for polyethylene (a fact not observed for isotactic polystyrene). To minimize this effect, investigations on polyethylene single crystals and on the bulk polymer often involve fast crystallization at low Tc. The most significant result obtained in the intermediate q range is that the deuterated molecules form a sheetlike structure consistent with deposition on a macroscopic, flat growth face. However, the radius of gyration determined for crystallized high M. w. PED does not follow the classical Mil2 l a w a n d i s *n fact rather insensitive to M. w. This result was interpreted as being due to superfolding, i.e., to the fact that such high M. w. chains occupy two or more growth layers in a restricted portion of the growth front (Fig. 3-13 d), a feature which does not significantly affect the measured radius of gyration. In addition, the scattered intensity was found to be too low by a factor of two, which is accounted for by a dilution effect illustrated in Fig. 3-13 in which hydrogenated and deuterated stems are intermixed. Based on the above results, Spells and Sadler (1984) ventured to give a very detailed overall picture of molecular conformation in PE single crystals: "a strong statistical preference for adjacent reentry, with 75% of stems occupying adjacent lattice sites. The number of sheets of stems increases with molecular weight, with an average molecular weight per sheet of 21 000" (i.e., 15-20 stems). The above results appear therefore consistent with infrared evidence developed in the parallel investigation (Spells et al,

1984). Before concluding this section based on isotopic effects, it should be stressed again that IR and neutron scattering are sensitive to different features of molecular conformation: as used in this context, IR is primarily sensitive to nearest neighbor (stem) interactions, whereas neutron scattering is sensitive to the spatial arrangement of deuterated stems inside the crystal, which ranges from nearest neighbor lattice sites to the overall conformation. In both cases, however, the crystalline core is investigated, and information on the fold structure features only as consequence of the stem arrangement. The following methods are more directly concerned with the folds themselves. 3.5.4.3 Solid State Resonance

13

C Nuclear Magnetic

Solid state 13 C NMR provides an independent means of observing the morphology of semicrystalline polymers. With appropriate use of magic angle spinning and scalar or high-power dipolar proton decoupling, it is possible to measure independently the chemical shifts and relaxation times of carbon nuclei in the amorphous and crystalline regions, and thus to measure the chain conformation and mobility in the non crystalline regions. In what appears to be the first investigation of this sort (Schilling et al., 1983,1984), solution grown crystals of poly (trans-1,4butadiene) were examined using various NMR techniques. From the similarity of chemical shifts of the olefine and methylene carbons, they conclude that the folds appear to have the same conformations as the 1,4-trans sequences in the amorphous bulk, despite the constraints imposed by the requirement of predominant adjacent reentry established by high resolution NMR (Schilling et al., 1983). The motion in

3.5 The Lamellar Surface

the chain folds and cilia was found to be sufficiently rapid to average the direct 1 H - 1 3 C dipolar interactions and 13 C chemical shift anisotropies. However, upon epoxidation of the double bonds in the fold, the motion is significantly retarded due to the formation of an oxirane link. Kitamaru et al. (1986) describe a detailed high resolution solid state 13 C NMR investigation of bulk and solution crystallized PE and Ando et al. (1985) for PE single crystals. Kitamaro et al. point out a difference in behavior of the two samples; namely the existence of two chemical shifts at 31.3 and 31 ppm for the amorphous component in bulk and 31.3 ppm only in solution crystallized PE. They suggest three phases (crystalline, crystal-amorphous interphase and anisotropic amorphous phase) for the bulk and two for the solution crystallized material (crystalline and amorphous overlayer). Ando et al. (1985) also describe PE single crystals as a two phase system, and further conclude that mostly sharp folds are present. These NMR results rest of course on assignments of phases and tend to partition the structures into wellcategorized phases according to their mobility. Bulk crystallized polymers often yield three phases. Judging from the two known examples of PE and poly (trans-1,4butadiene), only two phases are discerned in solution-grown single crystals, which reflects a difference, if not in nature, at least in character of the fold and amorphous phase for these two crystallization conditions. Further details and analysis of high resolution 13 C NMR spectroscopy can be found in the recent review by Yu and Guo (1990). 3.5.4.4 Microscopical Techniques

A significant contribution to analysis of the fold surface organization rests on the

119

use of techniques that involve physical interactions of these surfaces with other constituants and observation of the resulting pattern. These can be either another fold surface, and the interactions may give rise to the so-called dislocation networks, or can be some material deposited (usually evaporated) on purpose on the fold surface. In the classical experiments (Bassett, 1958), the material of choice has been gold (gold decoration) which reveals topographical details of the fold surface; more recently, polyethylene or paraffins have been vaporized and used to decorate the fold surface of polymer crystals. This polymer decoration technique will be discussed in more detail in view of its novelty and the involvement of the authors in its development. Dislocation networks formed by superposed single crystals of PE were first observed by Holland and Lindenmeyer (1965). They could be produced more systematically by Sadler and Keller (1970 a, b) by using relatively low molecular weight materials (< 15 000) which fold with a small number of stems upon crystallization (Fig. 3-14). The dislocation networks are only observed when two crystals lie on top of each other and when equivalent axes in the crystal are almost, but not exactly parallel (disorientations must be at most a few degrees). In this case, local reorientations of crystal lattices in both crystals can create crystallographic register in small areas of the surfaces in contact, the matching areas being separated by a network of zones where dislocations are concentrated. The crystallographic contact is favoured by low M. w., and explained by surface roughness associated with uneven length of the stems, linked with molecular weight polydispersity and location of chain ends at the fold surface. It should be pointed out that dislocation networks imply interactions between surfaces with some measure of lat-

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3 Structure of Polymer Single Crystals

Figure 3-14. Multilayer crystal of polyethylene with extensive grids of dislocation networks. Note that sectorization becomes apparent. Note further the area with three layers, i.e., two superposed dislocation networks (from Sadler and Keller, 1970 a). Scale bar: 2 urn.

tice regularity; they disappear or are replaced by more conventional moire patterns when only small amounts of high M. w. material are included in the crystals. Gold decoration is normally a technique that reveals surface topographical details (Bassett, 1958). In a noteworthy study, it was used as a means to investigate the fold surface mobility (Blundell and Keller, 1973; Keller and Sadler, 1973). These authors showed that the coarseness and density of gold droplets deposited on PE single crystal fold surface changes (becomes coarser) when the crystals are washed with a solvent. The effect is more pronounced for higher molecular weight, and does not show up if the M. w. is lower than 8000. This difference in behavior, although qualitative, demonstrates different surface mobility, possibly associated with cilia. Polymer decoration (Wittmann and Lotz, 1982,1985) is based on the same principles and experimental set-up as gold decoration. The interacting species are polymer (especially polyethylene) fragments ^ 100 A long which condense from the vapor phase on a substrate at a temperature well below

their normal crystallization temperature. Crystallization is therefore nucleated on any favorable substrate surface feature, and notably on folds in polymer lamellar surfaces. The vaporized molecules form small rods, which are strips of lamellar crystals seen edge-on and with constituting chains normal to the rod and parallel to the substrate (cf. Fig. 3-15). The main novelty of the technique rests in the fact that, the probe being highly anisometric, the rod orientation rather than density and alignment of units becomes the structural marker in the decoration. Although the technique has wider applications (Lotz and Wittmann, 1990), its widespread use in revealing the fold orientation and, to some extent, fold surface structure rests on the fact that it (i) involves short-range interactions, i.e., "sees" the outermost layer, (ii) is a relatively local technique, with lateral resolution ^ 1 0 0 A since the deposited chains interact with only ten sharp folds, and (iii) is applicable to different substrates (heterodecoration, cf. Fig. 3-16). Figure 3-15 a and b illustrate the polymer decoration pattern on a short chain

3.5 The Lamellar Surface

121

Figure 3-15. Single crystals of (a) C 3 6 H 7 4 paraffin and (b) polyethylene decorated with polyethylene vapor (polymer decoration). Inserts: schematic representation of the decorating rod structure. Note uniformly distributed and double orientation of rods on the paraffin crystal, and unique orientation in growth sectors of PE crys-. tals (from Wittmann and Lotz, 1985). Scale bars: 1 um. (b)

paraffin (C 36 H 74 ) and a polyethylene crystal, which differ only in their surface structure: ordered array of methyl chain ends, and chain folds, respectively. Clearly, the selection of only one rod orientation in the growth sectors of the PE crystals must be associated with preferred alignment of the depositing chain stems parallel to the most likely substrate fold

orientation, parallel to the growth face (cf. inset of Fig. 3-15 b). The two rod orientations on paraffin crystals are due to an epitaxial interaction and are consistent with the symmetry of the substrate methyl pattern (inset, Fig. 3-15 a). Comparable orientations were obtained for six-sectored hexagonal polyoxymethylene, and for lath-shaped isotactic poly-

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3 Structure of Polymer Single Crystals

Figure 3-16. Lathlike lamella of isotactic polypropylene decorated with polyethylene vapors. Note the narrow growth sectors on the lateral lath edges (from Wittmann and Lotz, 1985). Scale bar: 1 jam.

propylene crystals (Fig. 3-16). The latter were thus shown to be sectored, ruling out an early, oversimplified folding scheme for such laths. Polymer decoration as in Fig. 3-15 b has often been quoted incorrectly in support of adjacent reentry, i.e., of sharp folding (e.g., Phillips, 1990). Decoration patterns of crystals with curved edges (Wittmann and Lotz, 1985) demonstrate that, for sufficiently high M. w. (e.g., 8000), the decorating rods are always normal to the macroscopic growth front. It follows that the decorating pattern indicates fold orientations that depart from the low index planes (110, 110, and 020) in which tight folds must be confined. This clearly indicates that polymer decoration is more sensitive to the very outer molecules of the surface i.e., long loops, when such loops which link second, third, etc. nearest stems and straddle the sharp folds (Lotz and Wittmann, 1990) are present. The technique has proved particularly valuable for investigating the fold surface of long n-alkane chains and model cyclic paraffins (cf. Sec. 3.6.2.3). Scanning tunneling and atomic force microscopy are the newest and most direct

experimental techniques available. Indeed, most or all of the above decoration techniques are indirect and therefore involve an interpretative step that may be nontrivial. The general (although not unanimous) agreement reached on the main features of the crystal surface structure arises from the physically reasonable character of the various features, and the overall agreement reached by different experimental techniques in spite of their probing sometimes very different parts of the structure (e.g., stem arrangement in the crystal, mobility of the folds, orientation of the folds). These techniques may well be superseded in the near future by progress in nanoprobe scanning techniques. At least two attempts at mapping polymer single crystals by AFM (which is more suited for nonconducting materials) have been reported (Piner et al, 1990; Patil et al., 1990). The published pictures do not have sufficient resolution to reach molecular details of the fold surface. However, the situation is rapidly changing, and results obtained with polymers should shortly approach those obtained with low M. w. systems. At the time of writing, the authors heard of

3.5 The Lamellar Surface

Wunderlich's announcement (1991) of having reached chain resolution in extended chain PE. Almost simultaneously, similar results were obtained by Stocker et al. (1991a). The latter authors also examined the surface structure of n-C 32 , n-C36, cyclic (CH 2 ) 48 and (CH 2 ) 72 . As shown in Fig. 3-17 a, the pattern of methyl groups at the surface of n-alkanes is clearly resolved in unfiltered images. Further, the surface image of the cyclic paraffin, (CH 2 ) 72 , which displays a zig-zag pattern (Fig. 3-17 b), can be directly correlated with the X-ray structure (Fig. 3-17 c): the AFM technique images the very top CH 2 of the folds, which indeed form such a zig-zag pattern. The technique detects small differences in fold arrangement between top and bottom sides of the single crystals, i.e., differentiates (001) and (001) planes. Since methyl group resolution can be achieved, comparable technical achievements should become frequent in the near future and will result in a renewed and more direct approach to the fold surface of polymers.

3.5.5 The Fold Period in Polymer Single Crystals

Lamellar thickness is an important parameter in characterizing polymer morphology, and is a direct outcome of the propensity of polymer chains to fold. The lamellae thickness is measured by classical means: small angle X-ray scattering, elec-

Figure 3-17. (a) Atomic force microscope picture (unfiltered) of the (001) surface of n-C 36 H 74 (size of area: 6 x 6 nm) (b) AFM picture (Fourier filtered from diffraction pattern in inset) of the (001) surface of cyclic (CH2)72 (size of area: 10x10 nm) and (c) corresponding drawing of the fold surface. Note zig-zag pattern of top CH2 groups of the folds, visualized in the AFM (from Stocker et al, 1991). (c)

123

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3 Structure of Polymer Single Crystals

tron microscopy on single crystals by the shadow length, longitudinal acoustic mode in low-frequency Raman spectroscopy. Lamellar thickness may differ from crystalline stem length in view of chain obliquity or the presence of disordered surfaces. The most salient feature of lamellar thickness is its variation with Tc, more precisely its inverse variation with the degree of supercooling AT = T£ - Tc where T£ is the extrapolated temperature of dissolution of an extended chain crystal in solution, or of melting (cf. Sec. 3.7). This relationship is particularly valid for solution-grown crystal morphologies. A recent compilation of data obtained on polyethylene single crystals is schematized in Fig. 3-18 (Keller, 1991) which also demonstrates the continuity with the initial lamellae thickness of bulk crystallized PE (initial as opposed to later, larger values resulting from a process of isothermal thickening (cf. Barham et al., 1985; Barham and Keller, 1989).

Increase of lamellar thickness with annealing temperature, and melting behavior of polymer single crystals is well documented (cf. Wunderlich, 1980). The thin polymer lamellae are metastable from a thermodynamic viewpoint, and reduction of melting temperature relative to that of a crystal of infinite thickness is expressed as T —

where ae is the fold surface energy, / the lamellar thickness and Aft the heat of fusion per unit volume of the crystal. Direct observation of impact on Tm of crystal and fold structure was provided early on by annealing PE crystals with (110) and (200) growth sectors, the latter melting at lower temperature (cf. Geil, 1963). Recent research has mainly dealt with stepwise variation of chain fold length in low molecular weight polymers, as developed in Sec. 3.6.2.

3.5.6 Single Crystal Habits: Influence of Crystallization Conditions and Impact of Internal Structure

Supercooling (°C)

Figure 3-18. Lamellar thickness of PE crystals in bulk (dots and line) and in solution (shaded area). Bulk values correspond to initial fold length (prior to thickening); shaded area encompasses values determined in eight different solvents (adapted from Keller, 1991).

Polymer single crystal morphology may vary significantly depending on crystallization conditions (solvent, concentration, temperature). Whereas crystals produced at low supercooling often reflect the symmetry of the unit cell, development of new growth faces (e.g., 200 in PE single crystals) may blur this correspondence. Crystallization at lower Tc results in dendritic growth often associated with twinning for PE crystals (Fig. 3-3 b). Growth conditions may have a more dramatic impact, as illustrated for PE crystals produced in a phase separated solution in Fig. 3-3 c: the resulting globs are a permanent memory of the phase separated medium. Comparable although less pronouncedly curved mor-

3.5 The Lamellar Surface

phologies have been described by Khoury and Barnes and named bowl-like crystals (cf. Khoury and Passaglia, 1976). A number of other morphologies, often linked with details of growth conditions or complicated geometry of the nuclei, have been described and analyzed: hedrites, axialites, interlocked single crystals (Lotz etal, 1975), etc. A review of these morphologies is not attempted here; rather, several manifestations of impact of the fold and internal structure on crystal morphology and properties are considered. Because of their specific volume, which most likely exceeds that of the corresponding crystal lattice, and because of their preferred orientation, folds are indeed expected to influence crystal morphology. Favorable, regular packing of folds in PE, as analyzed by Reneker and Geil (1960), implies shift of successive folds along the oaxis direction, i.e., leads to three-dimensional, pyramidal single crystal shapes. Figure 3-19 is a most classical representation of pyramidal four- and sixsectored crystals of PE, as well as of corrugated crystals resulting from alternation of fold surface dip in the growth faces. Such corrugated crystals are shown in Fig. 3-3 a and Fig. 3-6. The existence of preferred fold orientations results in slightly different unit cell dimensions in fold and nonfold planes and in more favorable slip of fold planes relative to one another. The latter effect is vividly illustrated by the differences in 110 and lTO dark fields and results from sedimentation of the single crystals on the flat substrate. A number of further, similar effects have been reviewed (e.g., Keller, 1968; Khoury and Passaglia, 1976). The morphological consequences of chain folding in regular, high temperature single crystals are usually limited by the

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(a)

Figure 3-19. Three-dimensional shapes of PE single crystals grown from dilute solution: pyramidal (a) four or (b) six sectored and (c) corrugated single crystals (from Khoury and Passaglia, 1976).

intrinsic symmetry of these crystals. However, when this symmetry is lost, the impact of fold structure may be revealed in a dramatic manner. These more recent aspects are considered mainly in relation to the long standing issue of lamellar twist in some bulk-crystallized spherulites, notably ofPE. Lamellar twist is difficult to establish in bulk spherulites, but has been revealed in a spectacular manner in solution-grown single crystals of Bombyx mori silk fibroin (Lotz et al., 1982). As seen in the stereopair of Fig. 3-20, silk fibroin crystals are twisted

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3 Structure of Polymer Single Crystals

Figure 3-20. Stereopair of twisted single crystal of Bombyx mori silk fibroin in the ^-modification. Twist and sense of twist is linked with the chirality of the peptide residues (from Lotz et al., 1982). Scale bar: 1 jam.

with a period of ~ 1 jam. In this specific example the origin of the twist is not linked with the fold structure, but with the crystal structure: the low energy minimum of the achiral peptide residue alanine departs slightly from the crystallographic 21 -helix conformation. Repetition of this conformation results in slight twist of the chain, of the hydrogen-bonded sheet and ultimately of the single crystal: twist of the latter is related to the configuration of the peptide residue and is inverted for D-, rather than L-amino acids. In PE lamellae, asymmetry is introduced by chain tilt. As seen later in Fig. 3-25, if different fold conformations exist at opposing ends of a stem impinging at an acute and an obtuse angle on the fold surface, such differences should result in uneven surface stresses and ultimately induce twisting (Keith and Padden, 1984). These aspects, relating to polymer single crystals crystallized from the bulk, are discussed in more detail in Sec. 3.6.2.2.

3.6 Single Crystals Produced from the Bulk Growth of polymers in the form of single crystals is not restricted to dilute solutions with low supercooling. Although such single crystals are easy to produce, handle and investigate, the crystallization kinetics, morphologies, etc. may be different from those encountered in bulk crystallization. Conversely, the inherent complexity of spherulites makes it difficult to investigate structural details of their constituent lamellae, and justifies recourse to simpler crystals. Single crystals may be produced in the bulk only in the very high end of the crystallization range, i.e., at low supercoolings. Investigation of lamellar crystals embedded in a matrix of the same material constitutes of course an experimental challenge. The problem can be alleviated in part by resorting to thin film growth (which may introduce extraneous influence in view of

3.6 Single Crystals Produced from the Bulk

constrained geometry and diffusion problems) or the use of media which mimic the molten polymer, e.g., atactic polystyrene as a noncrystallizing molten medium for isotactic polystyrene (Keith et al., 1970) or n-paraffins as a substitute for molten polyethylene (Keith, 1964). When such biases are not used, observation of the melt-grown crystals becomes a crucial issue. Growth is usually arrested by rapid quench at low temperature, which results in crystallization of the molten matrix at low temperature or its vitrification. In the former case, quenching may actually be used as a decoration method to reveal growth faces and any feature susceptible to induce spherulitic crystallization during cooling. This technique is particularly valuable to determine crystal growth rates (Kovacs and Gonthier, 1972; Labaig, 1978). Conversely, replicas of fractured samples, often further etched (e.g., with a permanganic etch for a number of polyolefins, cf. Olley et al., 1979; Olley and Bassett, 1982; Vaughan and Bassett, 1989) can be examined to reveal topographic details. Finally, a most spectacular achievement is the demonstrated possibility (Bassett et al., 1988) to selectively dissolve quenched polyethylene, which has a lower dissolution temperature because of its lower crystallization temperature and reduced lamellar thickness while leaving the high-temperature melt grown crystals virtually intact. This method, of considerable promise, bridges the gap between ease of observation characteristic of solution-grown single crystals as illustrated in the previous sections and truly bulk crystallization conditions. In the following, we give a few case examples of contributions to the general understanding of polymer crystallization by investigations of single crystals grown in the melt.

127

3.6.1 Polyethylene Oxide Single Crystals The most thorough study of growth kinetics, isothermal lamellar thickening and melting of PEO in a range of M.w. fractions (from - 2000 to - 500000) is by Kovacs and collaborators (for a review, see Buckley and Kovacs, 1984). This investigation was extended to cover 6 decades of growth rates by extending the high Tc range and by using self-nucleation combined with the quenchdecoration, ("self-decoration") method. Arlie et al. (1966a,b) and Spegt (1970) showed that low molecular weight PEO (-2000 to -10000) forms lamellae of thickness /, which is always an integer submultiple of the total chain length L (the so-called quantized folding): 1= The number of folds n depends on crystallization temperature Ts and also time ts since folded chain crystals are metastable with respect to extended chain ones. The growth rate data [logG = f(Tc)] (Fig. 3-21) have a notched appearance, resulting from the juxtaposition of independent growth rate branches relative to successive values of n. The micrographs (Fig. 3-22) make it possible to follow morphological changes associated with changes in n, and the crystallographic pathway of chain unfolding (in the present case from n = 1 to full chain extension) within the lamellae. Isothermal melting of such crystals was shown to take place by lateral contraction of the crystals, which is at variance with an earlier model of melting from the fold surface inwards (Kovacs and Straupe, 1980). Parallel work using similar techniques helped establish that integral folding, which implies that the chain ends are essentially located at the fold surface, is not linked with the chemical nature of these chain

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3 Structure of Polymer Single Crystals

Tc in °C

Figure 3-21. Notched appearance of growth rate G versus Tc curves associated with quantized folding of the chains. PEO of different M. w. as indicated; recent experiments have revealed actual minima at some intersections of curves (from Kovacs et al., 1975).

ends, since ethyl terminated (instead of OH) PEOs also display the same behavior. For further details, see e.g., Buckley and Kovacs (1984). Similar investigations have been reported recently in several papers by Cheng and co-workers (for a recent reference, see Cheng et al., 1991). They observe in particular not only notches, but actually local minima in the growth rate curves (Cheng and Chen, 1990), which are associated with the existence of transient nonintegral folding as reported earlier for long chain paraffins (cf. Sec. 3.6.2). 3.6.2 Crystallization of Polyethylene Keith (1964) first revealed a lenticular lamellar morphology significantly different

from the conventional lozenge shaped crystals when PE is crystallized in a paraffinic diluent. The elongated tip is parallel to the fe-axis, and chains are tilted in the crystals at angles of up to 35° in the acplane [fold surface of the type (101) or (201)]. This lamellar morphology is consistent with the observed b-axis radial orientation in spherulites. Similar crystals have been obtained more recently by using poor solvents (typically long chain alcohols). These crystals have been extensively studied by Khoury (1979), who established that chain tilt is uniform throughout the lenticular crystal (with some variations on the outer edges) and by Organ and Keller (1985), in connection with the problem of integer folding.

3.6 Single Crystals Produced from the Bulk

58,50*0 (17 h)

'

'

:

.

• ' . ; • '

•

59.34 °C (48h)

••

59.39 °C (48 h)

59.45°C (72 h)

The technique of self-decoration was used by Labaig (1978) in a systematic study of crystal growth rates of sharp fractions of polyethylene similar to those used later for solution crystallization (Leung et al., 1985). This study established the lenticular shape of crystals grown in the bulk.

129

59.41 °C (48 h)

Figure 3-22. Single crystals of polyethylene oxide (PEO) of M. w. 6000 grown from the melt at the indicated crystallization temperatures and times, and subsequently quenched to reveal the crystal outlines and surface structure. The crystallization domain lies in the transition region from once-folded crystallization (speckled surface decoration) to extended chain crystallization (no surface decoration: the crystals appear featureless). The various morphologies are due to and reveal an intricate pattern of growth rates, crystal growth mechanism and crystal transformation via isothermal thickening (from Kovacs and Straupe, 1980). Scale bars: 20 urn.

61.56°C (288 h)

The recent work by Bassett et al. (1988) on melt grown single crystals of PE confirms the lenticular habit of these crystals grown at high Tc (Fig. 3-23). Electron diffraction on single crystals extracted by selective dissolution establishes that the chain tilt is 35° (fold plane 201) and demon-

130

3 Structure of Polymer Single Crystals

Figure 3-23. Lenticular crystal of PE grown from the bulk, which was subsequently quenched. Permanganic etching reveals the crystal embedded in spherulitic material produced on quenching (from Bassett et al., 1988). Scale bar: 2 |im.

strates, by dark field imaging, the existence of different fold orientations, as in solution grown crystals. Contrary to solution growth, the development of the terraces is limited, often to a single layer: the layers stop growing on one side, leaving a straight edge which corresponds to the growth face that makes an acute angle with the underlying lamella (Fig. 3-24). Bassett et al. (1988) further analyze this observation in connection with possible mechanisms responsible for lamellar twisting in bulk crystallization which leads to banded spherulites (cf. following chapter). Bassett and Hodge (1981), in a landmark investigation of the internal structure of PE spherulites using permanganic etching techniques, have suggested that banding is

a result of changes of orientation occurring sharply around screw dislocations of consistent sign but developing only two or three layers of a spiral terrace. The observation schematized in Fig. 3-24 provides a demonstration of a mechanism of selfselection of dislocations of consistent sign. Whereas the above analysis of banding in spherulites places the emphasis on screw dislocations, Keith and Padden (1984) suggested that lamellar twisting has its origin in differences in fold structure linked with stem inclination (Fig. 3-25). These differences introduce bending moments which are oppositely oriented in the two lamellar halves and thus induce twisting. Polymer decoration of PE single crystals grown at high Tc in thin films provides indeed a clear demonstration that fold surfaces are different on either side of these b-axis spines (Keith et al., 1989). Further, crystals growing on edge, i.e., in the absence of their symmetrical counterpart to which they are usually linked, are systematically curved (with a radius of curvature of ~ 5 |im), thus providing a direct illustration of the bending moments indicated in Fig. 3-25.

(a)

Acute angle

Obtuse angle (b)

Figure 3-24. (a) Sketch of elongated single crystal of polyethylene grown from the bulk at 130°C with asymmetric screw dislocation development. Lateral growth is limited at growth face with acute angle in the tilted chain single crystal shown in cross section in

3.7 Single Crystals of Low M.w. Polymers and Model Compounds

Atf/JM//////////

131

crystals of isotactic polypropylene and polystyrene in monolayer and multilayer forms and revealed by permanganic etching. Contrary to PEO and PE, the growth habits are not significantly different from their dilute-solution grown counterparts. At these low undercoolings, however, elongated iPP laths do not display the characteristic branching habit inductive of quadritic habit in solution growth.

3.7 Single Crystals of Low M. w. Polymers and Model Compounds

Figure 3-25. Postulated surface stresses a in PE lamellae associated with chain tilt (top). Due to asymmetric stresses, the two halves of lamellae, if separated, would tend to bend in opposite directions (middle). In a lamella, these stresses are relieved by twisting (bottom: angle of twist (/> generated over a length s of the lamella) (adapted from Keith and Padden, 1984).

Taken in combination, the observations of Bassett et al. (1988) and Keith et al. (1989) underline the importance of chain tilt in the origin of twist and relate the complicated topological arrangement of lamellae in spherulites to simple features observed in single crystals, illustrating again the role of the latter as model systems. 3.6.3 Other Polymers

Vaughan and Bassett (1989) describe the morphology of bulk crystallized single

Low M. w. and model compounds have been primarily investigated for two purposes: - to characterize the structure of tight folds by using cyclic molecules with chain folds between adjacent stems, - to establish the onset of chain folding as a function of molecular length, i.e., to establish the frontier between paraffinic (parallel collection of straight stems) and polymeric (straight stems connected by folds) behavior in single crystals. 3.7.1 Tight Fold Structure

Cyclic paraffins were from the onset considered as model systems to investigate sharp folding of polyethylene chains. Short cyclic paraffins however crystallize in the monoclinic and triclinic modifications (Kay and Newmann, 1968; Trzebiatowski et al., 1982) in which folds have a slightly deformed ggtgg conformation. Only cyclic paraffins with n > 144 crystallize in an orthorhombic crystal phase similar to polyethylene (Lieser et al., 1988). Out of a number of recent studies on similar compounds, we report on a comparative study on the folded-chain structure and fold orientation in polyethylene and

132

3 Structure of Polymer Single Crystals

cycloparaffins (Ihn et al., 1989,1990 a, b) as revealed by polymer decoration. In accordance with the monoclinic crystal structure, the decoration pattern indicates that for (CH 2 ) 36 , the folds are parallel to the short edge (parallel to the fe-axis) of the rectangular single crystals (cf. Fig. 3-17 c). Since (CH 2 ) 60 displays the same decoration pattern, Ihn et al. (1989) conclude that its (unknown) structure is similar to (CH 2 ) 36 , except for the difference in straight stem length. However, changeover to lozenge-shaped crystals takes place for longer cyclic paraffins: (CH 2 ) 120 produces classical lozenge shaped crystals for which polymer decoration indicates fold orientation in the (110) plane, i.e., parallel to the macroscopic growth front. Interestingly, Ihn etal. (1990) point out that the azimuthal spread of decorating chains deposited on such sharp, crystallographically regular folds is much smaller than when chain folded PE crystals are used as substrates. This observation is of course consistent with the inference of the orienting effect of long loops, and of the model of stem arrangement deduced from isotopic blending experiments (IR and neutron scattering results). The second and on-going research on cyclic paraffins is their examination by atomic force microscopy techniques considered previously (cf. Fig. 3-17). 3.7.2 Onset of Chain Folding

Low M.w. polymers have often been used to establish the onset of chain folding. These samples nevertheless have an intrinsic weakness in that polydispersity unavoidably introduces significant spread of molecular length, and therefore equilibrium melting temperature, crystallization range, etc. which may blur or hide important phenomena, in particular in transition

regions. In spite of these limitations, investigations on fractions, e.g., of PEO, have met with considerable success. Analysis of finer details of the crystallization behavior had, however, to await the synthesis of sufficiently long chains with strictly controlled length. For reasons of commodity and availability of the subunits, and in view of the considerable knowledge already gained on polyethylene, model systems so far investigated are mostly linear paraffins (Lee and Wegner, 1985; Bidd et al., 1987). 3.7.2.1 Observation on Polyethylene Oxide Crystals

As already discussed, the onset of chain folding and the observation of quantized folding to lamellar thicknesses as low as L/4 was first established in bulk crystallization of low M.w. PEO with narrow molecular weight distribution (Arlie et al., 1966 a, b). The chains are either folded an integral number of times, or fully extended, which implies that the chain ends are located in or near the surface layers. "As the crystallization temperature and time increase, the lamellar thicknesses increases in a stepwise manner, due to quantized reduction offt,until full chain extension" (Kovacs and Straupe, 1979): onset of chain folding is clearly linked not only to molecular length, but to crystal growth conditions. Investigation of bulk and thin film grown single crystals as shown in Fig. 3-22 helped establish a number of features of onset and impact of chain folding on crystalline morphology, crystallization kinetics, etc. In particular: - Overall growth rate curves versus Tc have a notched appearance as they result from the succession of independent growth rate branches, corresponding to the different lamellar thicknesses (Fig. 3-21). - The transition from one fold length to the other takes place in a very narrow tern-

3.7 Single Crystals of Low M.w. Polymers and Model Compounds

perature interval which could be defined to better than 0.1 °C. - Extended chain substrates (n = 0) can initiate growth of folded chains (n = 1), but the reverse is not true; as a consequence, growth of extended chain crystals may be arrested by temporary, deliberate or accidental decrease to the growth domain where n = 1 at Tc. - In the transition region from n = 0 to n = 1, composite crystals with both n = 0 and n = 1 can be grown (cf. Fig. 3-22). These spectacular morphologies can be controlled by appropriate selection, or variation of Tc. - The outer border of chain folded crystals experiences considerable molecular rearrangement and structural reorganization. In this region, up to 10 }im wide, crystallization takes place in the (n + 1) folding mode, and reorganizes to the rc-mode. The lower stability of this zone is manifested by its faster melting rate (Kovacs and Straupe, 1980). - Most importantly, a process of isothermal lamellar thickening and chain extension to n = 0 at Tc (also observed upon annealing) could be followed. For most crystals investigated, the process starts at the crystal center, i.e., is initiated by the central, presumably thicker seed produced by self-nucleation. This process results in central, thickened areas made of extended chains which do not self-decorate on quenching and therefore appear dark in Fig. 3-22. This process of lamellar thickening implies that molecules from the surrounding melt are dragged into the crystalline core through the lamellar surface. 3.7.2.2 Recent Results on Model Paraffins

Many of the above features of polymer folding, unfolding and melting have been observed with the recently synthesized long

133

chain paraffins, albeit often in a more global form (i.e., by using averaging techniques). In particular, it was shown that folding occurs for chains made of at least 150 carbon atoms, and that C 3 9 0 may fold up to 4 times. Significant progress was however made regarding several topics: - The initial lamellar thickness may be intermediate between the anticipated values corresponding to L/n, and then evolve to either higher or lower values. This unexpected result is interpreted by a process of initial noninteger folding (NIF), which is in essence visualized as an adsorption process of the chain on the growth face, followed by perfecting in a stage that may be subsequent to growth either by thickening or thinning of the lamella (cf. Keller, 1991). - Thicknesses of solution grown lamellae are 1, 1/2, and 1/3 for a C 1 9 8 paraffin. A crystallization rate minimum is observed at the transition from once folded to extended chain crystallization in the melt (Ungar and Keller, 1987). This highly unusual situation is explained by a mechanism of self-poisoning: on the extended chain growth face, folded-chain molecules deposit frequently, but do not (or rarely) unfold, thus limiting deposition of extended chain crystals which contribute to growth. An essential ingredient in the appearance of the rate-minimum is a substantial difference in the activation barrier between the two processes. Organ et al. (1989) extend the above observations and conclusions to solution crystallization of C 1 9 8 H 3 9 8 . The experimental evidence rests so far on DSC experiments, i.e., on the dissolution behavior of crystals formed in the range 70-85 °C. Crystals formed above and below 77 °C are extended and once folded and have dissolution-temperatures of 90-93 °C and ~ 80 °C respectively. Folded chain crystals are observed to unfold with time at Tc, and

134

3 Structure of Polymer Single Crystals

a significant overall crystallization rate minimum is observed at the transition temperature of 77 °C; this minimum is thought to be due to nucleation more than growth rate effects. Further morphological work is reported to be in progress. Although the above experiments do not clearly distinguish nucleation and growth rate, their connection with reported minima in growth rates for both PEO in the bulk (Cheng and Chen, 1990) and solution crystallization of PE fractions (Leung et al., 1985) appears striking. Cheng and Chen indeed invoke noninteger folding as a possible explanation. In any case, the observation and, under favorable circumstances, the detailed analysis of growth process of molecules susceptible to remaining unfolded or having only few folds provides insights into molecular processes at the growth front which challenge classical crystallization theories. 3.7.2.3 Onset of Chain Folding and Crystal Morphology

Onset of chain folding has of course a significant effect on the internal lamellar structure, and results in the familiar sectorization effects revealed by dark field imaging (cf. Fig. 3-6). Similar effects are not expected for extended chain crystals. It therefore came as a surprise when lozenge shaped crystals of a C 6 0 H 1 2 2 paraffin displayed Bragg diffraction contrast that revealed sectorization, although of a different type than for PE crystals: striations are parallel to 130 rather than 130 (Dorset, 1986; Dorset et al., 1990). C 6 0 appears to be a minimum chain length to reveal this effect. The undulations correspond to the formation of a three-dimensional crystal habit and its sectorization upon collapse on a flat substrate. The three-dimensional habit would result from formation of a spe-

cial, oblique layer form of the orthorhombic subcell, and induce the observed sectorization effects which, although different from PE, blur somewhat the habit transition between the folded and nonfolded crystals. Onset of chain folding may have more spectacular manifestations, notably when chain stiffness precludes formation of sharp, hairpin type folds between adjacent stems as exist in PE. This holds in particular for chains that include aromatic rings, e.g., polyethylene terephthalate, polycarbonate, or the technologically important, heat resistant PEK and PEEK. Polyphenyl ether ketone (PEK) and polyphenyl ether ether ketone (PEEK) have a structure based on benzene rings (cp) and either ether or ketone bridges, thus: [cp-O-(p-CO]M and [cpO-cp-O-cp-CO] n , respectively. When crystallized from solution, these polymers form very imperfect crystals elongated along their b-axis, a few nanometers in size and composed of narrow microcrystallites only ~ 20 nm in dimension with slight mutual misorientation (Lovinger and Davis, 1985, 1986; Tsuji etal., 1989 a). Further, Waddon et al. (1992) report that in PEK this deterioration in crystal perfection (and crystallinity) is not observed for chain extended oligomers. The origin of this behavior rests in the bulkiness and stiffness of the chain folds, which certainly limit or inhibit reentrant folding in favor of more remote folding (and thus irregular fold surfaces). Cumulative stresses exerted on the crystal lattice limit the crystal size as a result of bulging of folds. The stress field in the amorphous layer appears to be anisotropic (probably as a consequence of preferred orientation) and therefore limits crystal size in the aaxis direction, i.e., results in fragmentation along b. Availability of both oligomers and

3.8 Polymer Single Crystals as Investigation Materials

polymers of a stiff chain such as PEK helps to highlight the influence of crystal coresurface interactions in terms of both crystal habit and size. These concepts may be taken over to irregular chains such as PE with short chain branches for which similar irregular crystal habits are systematic (Marks and Carr, 1982). 3.7.3 Single Crystals Produced Under High Pressure

Crystallization of polyethylene under high pressure (3.3 kbar, 220 °C) produces lamellae with thicknesses in the micrometer range (e.g., Wunderlich, 1973, 1976, 1980). Formation of these so-called extended chain crystals is associated with the existence of a pseudohexagonal crystal phase of PE with considerable chain mobility (Bassett et al., 1974; see also reviews by Bassett, 1976). The high pressure phase crystallization mechanism, growth kinetics and thickening processes have been studied in considerable detail, including direct in situ observation, in spite of the experimental difficulties involved (Hikosaka and Seto, 1982,1984). It is observed that crystal thickness may go beyond full chain extension, i.e., involving several molecular chains. Among the various molecular phenomena uncovered, recognition that growth and crystal extension occur while the crystal is in the hexagonal phase and that these processes stop upon conversion to the orthorhombic phase raises fundamental issues, namely the role of metastable transient mesophases in polymer crystallization (Rastogi et al., 1991, also Keller, 1991, and Chap. 4 of this Volume, Sec. 4.2.1.4). The structure of individual anabaric polyethylene lamellae has been studied after extraction from their matrix by selective dissolution (DiCorleto and Bassett, 1990). They are circular discs with diame-

135

ters of about 10 jim and tapering edges which reflect the mechanism of lamellar thickening. They show three sets of striations at 60 °C to each other, which reflect the hexagonal to orthorhombic transition which they experienced, thus confirming that they were first formed in the mobile, hexagonal phase. The rounded habit has been observed for other polymers (Fig. 3-22, Tc = 59A °C) and suggests a growth mechanism with many growth niches of the surface roughening type. Modifications of crystallization theories including lateral growth and sliding diffusion along the chain axis have been proposed (Hikosaka, 1987) but may need be amended to incorporate recent experimental findings. Crystallization of other polymers under elevated pressure have not revealed similar trends: it appears that the specific behavior of PE is linked to the mobile hexagonal phase. Conversely however, the findings on PE have been carried over to low pressure crystallization of poly(£rarcs-l,4-butadiene), which is known to have a high temperature mobile polymorph (Rastogi et al., 1991; Keller 1991, and Chap. 4 of this Volume, Sec. 4.2.1.4).

3.8 Polymer Single Crystals as Investigation Materials Polymer single crystals provide suitable materials for a wide range of investigations on structure, growth mechanisms and behavior of essentially unentangled, although folded, polymer chains. Out of a number of different studies, we examine the contribution of polymer single crystals - in crystal structure determination, i.e., at the unit cell level. Techniques involve electron diffraction and high resolution electron microscopy;

136

3 Structure of Polymer Single Crystals

- in assessing various aspects of crystallization theories; - in investigations of mechanical behavior of crystals upon deformation.

3.8.1 Polymer Crystallography at the Unit Cell Level

Structure determination of polymers remains a challenge in spite of considerable help provided by conformational energy analysis to supplement diffraction evidence. The latter is usually limited to samples with fiber orientation (sometimes double orientation) produced by stretching or rolling, etc. (cf. Fig. 3-1). These samples therefore produce diffraction patterns with frequent overlap of reflections etc. Further, these methods are not suited for crystal modifications which are unstable to mechanical deformation. Information obtained from single crystals has therefore been exploited early on as an aid to solve the crystal structure of polymers. Two main lines of research exist: - conventional structure analysis which makes use of the diffraction intensities; - direct imaging of the crystal lattice, which became possible in recent years due to progress in electron microscope resolution and performance.

3.8.1.1 Electron Crystallography

Electron crystallography has been applied early on in the pionneering work of Vainshtein (1964) on single crystals of paraffins. Structure determination has been considered in some of the earlier publications on polymers (Frank et al., 1959; Claffey et al., 1974). It has been applied rather systematically to paraffins and short-chain biological molecules: phospholipids, etc. (for a review see Dorset,

1989), to linear polysaccharides (for a review see Perez and Chanzy, 1989), to polypivalolactone (Meille et al., 1989), to polyethylene itself in order to work out the finer details of its structure, namely the setting angle made by the plane of the C-C bonds with the crystallographic a- and baxes, an angle that was thought to depend on the fold structure and orientation. Dorset and Moss (1983) find it however to be comparable for PE and paraffins (~ 43°). Advantages of the technique are numerous: - Since the wavelength of the electrons is quite small, the radius of the Ewald sphere is large and it is possible to record simultaneously many diffraction maxima, often down to 0.1 nm" 1 . Tilting stage capabilities make it possible to record reflections on the upper layers (Meille et al., 1989; Brisse, 1990). As a consequence, electron crystallography often uses many more diffraction data than are available to X-ray analysis. - Crystal modifications that are unstable to mechanical treatment or that exist spontaneously only as a small constituent admixed in a more stable modification can be analyzed. For example, single crystals of the polydipeptide (Ala-Gly)w, a model of silk fibroin, yielded crucial information on the sheetlike nature of an unstable structure of silk that had eluded X-ray analysis for over 30 years (Lotz and Keith, 1971). In a similar manner, the a b unit cell projection of poly(l-butene) in its form III determined from single crystals (Holland and Miller, 1964) has been a valuable aid in the later analysis of this structure based on X-ray powder patterns. Also, exploration of nearly the full reciprocal space of y-iPP single crystals provided a combined morphological and diffraction test for its highly unusual crystal structure with nonparallel chains (Lotz et al., 1991).

3.8 Polymer Single Crystals as Investigation Materials

The disadvantages of the technique deal with collection and analysis of intensity data: - Most polymers are degraded under the electron beam. Practical lifetimes, under low magnification (< 5000, preferably < 1000), are usually limited to one or at most a very few minutes, depending on the polymer. Further, during this lifetime, the relative intensities of the various reflections vary significantly, the outermost ones disappearing first (Perez and Chanzy, 1989). - Since polymer crystals have thicknesses in the 10 to 20 nm range and are made of light atoms, the kinematical approximation is usually assumed, although it may yield models of low accuracy (Moss and Dorset, 1983). However, improvements in data collection and analysis as well as quantitative corrections for such factors as crystal deformation (Dorset, 1989) constitute significant advances. At the present stage, electron crystallography of polymers making use of flat-on and tilted single crystals and often aided with epitaxial growth appears as a new tool in structure analysis of polymers especially valuable for unusual or touchy crystal modifications. It is often used in conjunction with more classical conformational analysis or tested with simultaneous analysis of X-ray powder data using the Rietveld method (Meille et al., 1989), but has clear potential for further developments in its own right. Dorset (1989) lists 18 different linear chain polymers for which quantitative electron diffraction structure analysis has been reported. A most recent advance is the application in electron crystallography of the direct phase determination techniques, i.e., phase extension procedures based on the probabilistic methods used in X-ray crystallography (Dorset and Hauptmann, 1976).

137

The first structures examined include polyethylene (Dorset, 1991a) and poly-ecaprolactone (Dorset, 1991b). In the latter case, and using a three-dimensional data set, computed electrostatic potential maps produced a recognizable structure which could be refined by Fourier techniques and atomic displacements. Future developments of this technique, at least for electron beam resistant polymers, may well include determination of the starting set of phases of the strongest diffraction spots by the Fourier transforms of high-resolution transmission micrographs of polymers with molecular resolution (Zhang and Thomas, 1992). 3.8.1.2 Lattice Imaging of Single Crystals High-resolution electron microscopy (HREM) has made considerable progress in recent years. Point to point resolution of 0.2 nm or better can be achieved, i.e., it is in principle possible to discern packing of polymer molecules in lamellar crystals. HREM on polymers constitutes nevertheless a challenge in view of the rapid deterioration of the crystal structure under the electron beam (Keller, 1982). This can be overcome in part by cryoprotection but the low value of the total end point dose (TEPD) which corresponds to the loss of crystallinity requires low dose imaging techniques and highly sensitive photographic plates which however limit the resolution in view of their coarser graininess. These limitations impose relatively low direct magnifications (e.g., 40000 for beam sensitive materials) and the signal to noise ratio is often improved by image processing, either optical (Klug and De Rosier, 1966) or by digital filtering. The first direct imaging of molecular chains in single crystals was reported by Tsuji etal. (1981) for poly(p-xylylene),

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3 Structure of Polymer Single Crystals

[CH 2 -C 6 H 4 -CH 2 -] n , a polymer that is relatively beam resistant owing to the large concentration of benzene rings in the structure. Initial magnification was 100000 and high voltage 500 kV. Individual chains could not be resolved in the image of the a-phase. Molecular resolution or, more precisely, observation in projection of the noncircular cross section of the chains was possible for the /?-phase after optical filtering. Structure analysis of this phase was pursued using electron intensity diffraction data and this information was also used in an X-ray diffraction structure analysis (Isoda et al, 1983). Lattice images of several other polymers have been obtained in recent years including paraffins (Zemlin et al., 1985), polyethylene (Revol and Manley, 1986), polytetrafluoroethylene (Chanzy et al., 1986),

a substituted polydiacetylene both normal and parallel to the chain axis (Young and Young, 1986), poly(tetramethylene-psilphenylene siloxane) or poly(TMPS) (Tsuji et al., 1989 a; Tsuji, 1989; this reference includes a more complete list of polymers). Figure 3-26 represents the results obtained by Tsuji et al. (1989 b) on lamellar crystals of the radiation sensitive poly (TMPS): the high resolution electron micrograph (taken with a total dose on the specimen of 4 x 102 electrons per nm 2 , which corresponds to the TEPD of poly (TM PS)), its optical diffraction pattern, the image processed by optical filtering which is compared to the computer simulated image and the crystal structure obtained by diffraction. As is amply demonstrated by Fig. 3-26, images of the lamellar crystal structure - or

**y-^-

mmm &£MK£& &&&v

(a)

(b)

Figure 3-26. (a) High resolution electron micrograph and diffraction pattern (insert: a* and b* axes in reciprocal space) of a single crystal of poly(tetramethyl-p-silphenylene siloxane) (PTMS). (b) Image in (a) processed by optical filtering. Insert: computer simulated image (from Tsuji et al., 1989 b).

3.8 Polymer Single Crystals as Investigation Materials

more precisely of the stem arrangement and in favorable cases, setting orientation can be obtained for beam sensitive materials. At the present time, their resolution is not sufficient for structural studies at the unit cell level. Since molecular stems are resolved, it would however appear that HREM techniques are suitable for the observation of local features, i.e., crystal defects such as edge-dislocations or twin boundaries. Edge-dislocations and grain boundaries caused by a succession of edge-dislocations have indeed been recorded in original HREM micrographs (e.g., Tsuji et al., 1989b). Pradere et al. (1988a) however point out one pitfall of the use of Fourier filtration performed to enhance the signal to noise ratio. They observe indeed that for original noisy pictures, artifactual edgedislocations and defects may appear in the reconstructed lattice images. These defects appear when the summed amplitude of the noise selected through the mask windows becomes larger than the amplitude of the Fourier transform maxima. In spite of its limitations and potential dangers (in the form of artifacts), the use of HREM opens new perspectives in the analysis of polymer single crystals at a very local scale; as an illustration very recent work made it possible to observe the crystal edges, and therefore gain information on the local roughness of the growth faces (Katayama, 1990; Zhang and Thomas, 1992). 3.8.2 Polymer Single Crystals and Theories of Polymer Crystallization

Considerable theoretical work has accumulated over the years to explain polymer crystallization with folded chains, and account for the numerous experimental observations described in the preceding sec-

139

tions. Analysis of the intricacies of these theories is not attempted here. The reader is referred to Chap. 4, Sec. 1.4 and to reviews, e.g., by Hoffman et al. (1975), and more recently Dosiere (1989), Armitstead and Goldbeck-Wood (1991), who present an up to date and broad coverage of the field, Keller (1991), who devotes one chapter to a shorter version of that review, and Phillips (1990), who discusses in some detail growth regime theories. This section intends, after a general outline of the theory, to review areas in which investigation of single crystal growth kinetics, morphology, structure, has had a significant impact on establishment, or prompted revisions of the theories. Quite expectedly, these are mostly related to important assumptions regarding the mechanism of growth. 3.8.2.1 General Outline of Crystallization Theories

In spite of its specificities, crystallization of polymers has many features in common with that of small molecules. In particular, the inverse temperature relationship of growth rate with temperature is highly suggestive of nucleation controlled growth, in a manner analyzed by Turnbull and Fisher (1949). If we consider only secondary nucleation (i.e., the mechanism involved in crystal growth, as opposed to formation of an initial nucleus), the classical equation of the growth rate is of the form: /-AL/*\ /-AF* G = Goexp( ^ )exp kT kT where G is the growth rate, Go a preexponential factor, AL/* the activation energy of the elementary jump process and AF* the free energy of formation of a nucleus of critical size. The first, transport, term becomes predominant in the bulk at low tern-

140

3 Structure of Polymer Single Crystals

perature, when hindered mobility of the chains near the glass temperature reduces and ultimately arrests growth. As polymer single crystals are always formed near the melting temperature or in solution, the second, nucleation term is predominant. Growth of polymer crystals by secondary nucleation is a sequential process similar to growth of simple substances, with: deposition of an initial chain segment (stem) on the substrate crystal surface (at rate i), attachment of successive stems to complete the newly created face at a lateral spreading rate Gs. Deposition of the first stem results in the creation of new, lateral and end fold surface crystal-solution or melt interfaces. The corresponding free enthalpy is AG

Regime 1: mononucleation

G = bit

Regime 2: polynucleation

f irst stem = 2b I a + 2 ab ae - a b I Ag

where a and b are the dimensions of the chain (in cross section) parallel and normal to the growth face, / the stem length, o and ae the lateral and fold surface free energies and Ag the free enthalpy of melting (Fig. 3-27). Completion of the growth face by attachment of further adjacent stems does not create new lateral faces, thus AGcompletion =

G = b(ig)v

Figure 3-27. Mononucleation and polynucleation regimes in polymer crystal growth with indication of relevant parameters and variables (adapted from Dosiere, 1987).

2ab(Te-ablAg

The most successful theories first proposed by Lauritzen and Hoffman (1960) postulate, as amply demonstrated by later experiments, that the fastest process takes precedence over other slower processes, even if they would result in a more stable crystal: the theories are kinetic rather than thermodynamic in character. Selection of the fastest process is based on the stem or fold length /, which thus introduces the specificity of polymer crystallization. Growth takes place when the initial activation barrier for deposition of the first stem is overcome, and when the bulk free

energy of deposition of a new stem more than compensates the free energy associated with folding. The critical fold length required is then of the form: l

+ d l

AH AT where T£ is the melting temperature, AT the undercooling T£ - Tc, and AH the heat of fusion. The derivation of the equation includes an additional term d /, which has a negligible contribution at low AT but introduces a dramatic upswing of the calculated critical fold length /* at low Tc.

3.8 Polymer Single Crystals as Investigation Materials

This upswing is not observed in practice. Corrections of the theory have taken care of this anomaly, essentially by decomposing the deposition of the first stem into more elemental processes involving subunits, flexibility units (Frank and Tosi, 1961). This was done by Point (1979a, b) and via a W parameter by Lauritzen and Hoffman (1973), which introduces an adsorption of the molecule on the surface prior to crystallization (Hoffman et al., 1975). The above theories do not take into consideration the local dynamics, i.e., the possibility of chain rearrangement as revealed by lamellar thickening observed to occur at the edges of growing single crystals in PEO or demonstrated by the tapering edges of extended chain PE crystals grown under elevated pressure. Hikosaka (1987) has recently developed a crystallization theory which includes sliding diffusion of the chain in the initial stages of the nucleation process. The secondary nucleation approach to polymer crystal growth yields the overall correct dependence of fold length / and growth rates G on AT. However, unsuspected variations of the thermal dependence of log G on 1/(T AT) were observed and led to the introduction of different growth regimes, which differ by the rate constants of secondary nucleation (rate i of deposition of the first stem per unit length of growth front) and tertiary nucleation leading to rate of lateral spreading Gs. At high Tc, the overall linear growth rate G normal to the growth front of length L is controlled by the (low) rate of initial nucleation i [events/(cm • s)]. If b is the thickness of the growth layer:

In this mononucleation regime or regime I, the spreading rate Gs far exceeds

141

the nucleation rate i, L p is a persistence length which interrupts growth laterally at obstacles of undefined nature. Regime II is reached at lower temperature when the nucleation i increases. Many nuclei per growth layer are formed and incompleted layers are themselves sites for additional nucleation. The growth rate in this polynucleation regime is Ga = b(iGs)1'2 A third regime, irrelevant in the present context as it takes place only at very low Tc in the bulk, is reached when the average distance of lateral completion of a layer (through Gs) becomes comparable to the stem separation. Existence of different growth regimes is usually recognized by plotting log G versus 1/(TAT). Indeed, the growth rate is expressed in the three regimes by -At/*

— kbo

o\ )

with k = 2 in regimes I and III, and k = 4 in regime II. A geometrical criterion for differentiating regimes I and II is given by the dimensionless parameter Z (Lauritzen, 1973) 4GS which is well below and above unity for regimes I and II, respectively. The normal growth regime of polymer single crystals is regime I or regime II. Only in rare instances has a break in the logG versus 1/(TAT) curves been observed, e.g. by Organ and Keller (1986, 1987 a) for PE crystals grown at high temperature in good solvents. Point and Dosiere (1989), Dosiere (1990) and Colet (1990), however, critically discuss the essential assumptions regarding

142

3 Structure of Polymer Single Crystals

growth regimes and show that experimental evidence is inconclusive regarding growth regimes in solution crystallization of polymers. 3.8.2.2 Kinetics of Polymer Single Crystal Growth

Growth kinetics are an important issue in the analysis of polymer crystallization theories: variation of linear growth rate with temperature, of growth rate with concentration of the solution, of impact of molecular weight distribution, etc. Numerous investigations along these lines were performed over the years (Blundell and Keller, 1968 a; Cooper and Manley, 1973; Leung et al, 1985), but the experimental procedures which involved sampling at various stages of the crystallization process were tedious. Decisive experimental progress was introduced by Dosiere et al. (1986) with the "isochronous" decoration technique based on rapid temperature jumps (< 20 s) and resultant formation of steps on the fold surface. Typically, 5° C temperature jumps are made at Tc > 80 °C, which result in variation in fold length of 2 nm, easily visualized by conventional shadowing of the single crystals. The method makes it possible to determine quasi-instantaneous growth rates and couple them with instantaneous solution concentration. The concentration dependence of the linear growth rate G is of the form G ~ ca; the coefficient a is usually smaller than one, typically ~ 0.5 for low M.w. fractions and 0.3 for higher M. w. It increases slowly with Tc for a given sample. This molecular weight dependence is accounted for by a theory of self-nucleation due to Sanchez and Di Marzio (1971a, b) which rests on the ability of cilia, i.e., of the chain ends which are not incorporated in a growth

layer upon attachment of the bulk of the chain, to deposit on the first-formed growth front, and thus initiate the development of a new layer. Furthermore, Sanchez and Di Marzio predicted that the concentration dependence of growth rate should become linear at very high dilution. This was experimentally verified by Toda et al. (1986) for concentrations down to 10" 6 , the changeover taking place at c = 10" 3. Variation of growth rate with the degree of completion of crystallization can be measured via the crystal size x. At any time, G(t) = G 0 (l - x2). Point et al. (1986) show that the growth rate decreases faster with x than expected from the concentration dependence and demonstrate that this effect is linked with molecular weight fractionation. This effect is very marked for narrow fractions of rather low molecular weight, but less marked for higher M.w. materials. A considerable body of evidence has accumulated on the M.w. fractionation, e.g., by determining M. w. of crystallized and supernatent fractions of the polymer (Prime and Wunderlich, 1969). Variation of growth rates with crystallization temperature considers the initial growth rate G in view of the above variation due to fractionation. A conventional plot of log G versus 1/(TAT) displays a single linear relationship; earlier departures from linearity appear to be linked with experimental underestimates of initial growth rates at high temperatures. In the most extensive study of growth rates of single crystals using well-defined fractions with polydispersity index ~ 1.1 to 1.3 produced by preparative GPC, Leung et al. (1985) observe that for relatively low M.w. fractions ranging from - 3000 to 11 000, the growth rate versus temperature curves display a minimum, and that for M. w. 3000 to 4000 the overall growth rate curve appears to be composed

3.8 Polymer Single Crystals as Investigation Materials

of two branches, in a manner reminiscent of growth rate curves of low M. w. polyethylene oxide crystals grown from the bulk (Fig. 3-21), which is indicative of quantized folding. 3.8.2.3 Geometry of Growth and Crystal Habit

Investigation of single-crystal growth mechanisms can provide significant insight into parameters of crystal growth theories, as established by the recent works of Point and Dosiere (cf. Dosiere, 1989), and Toda and Kiho (1987), as also discussed in the recent review of Armitstead and GoldbeckWood (1991). An important postulate in regime I is the existence of the so-called persistence length L p . L p is needed in regime I to avoid proportionality of G on the length of the growth face. Initial estimates of L p were on the order of 1 jim, but the smallest growth faces observable with the isochronous decoration technique did not show a reduction in growth rate, i.e., the value of L p must lie in the 200 nm range at most for PE single crystals (Point et al., 1986). Further, Colet (1990) and Dosiere (1990) discuss critically the notion of a persistence length itself. Colet et al. (1986) predicted that PE single crystals with 200 growth sectors might provide a morphological means to assess the control of growth by secondary nucleation. The argument rests on a perfect negative feedback loop if proportionality of L and G is assumed: if growth rate increases with L, increased growth rate results in a smaller growth sector, i.e., reduces L. The steady state growth conditions have been found (Colet, 1990) for a PE of M.w. = 115 000 and c ~ 2 x 10 ~ 3 at 86-92 °C. They result in a single crystal schematized in Fig. 3-28 a

143

with a critical length L* = 250 nm which must be of the order of the kinetic length Determination of Gs and i or at least of their ratio, rests mainly on polymer single crystal morphologies in which a particular growth feature ensures a permanent site for tertiary nucleation, i.e., avoids the initial deposition step L This is for example realized in composite crystals (Fig. 3-22 c) in which the growth rate of the chain extended portion exceeds that of once-folded chain and pushes growth of the latter (compare with Fig. 3-22 b). More frequently, twinned crystals may provide favorable sites for nucleation; for example (110) twins of PE (Fig. 3-9 a) which experience faster growth rate and induce lathshaped morphology (Fig. 3-9 b). In the process, the twin components provide each other with nucleation sites for new layers: in these crystals, the initial nucleation i is an easy process at the twin boundary. As a consequence, the growth plane next to the twin boundary is oblique to the crystallographic one (Fig. 3-28 b), a feature which gives access to the spreading rate Gs, and thus to i (Toda and Kiho, 1987). Figure 3-28 c presents values of Gs and i determined by Colet (1990) for PE crystals grown in xylene solution; they are characterized by a strong variation with temperature. Crystals with curved edges have initiated a revival of investigations on the growth mechanisms and theories of crystal growth in recent years. The classical nucleation theory is suited for growth of flat surfaces, i.e. of crystals with polygonal habit; however, for PE, many crystallization conditions (paraffine diluent, poor solvent at high Tc, bulk, etc.) induce lenticular or even rounded growth faces (Figs. 3-22 and 3-23). Such curved faces are rough on a molecular scale, i.e., have many nucle-

3 Structure of Polymer Single Crystals

144

(a)

(b)

f

Gs (nm/s)

10z

i (events-|jnrf 1 -s" 1 )

82

84

86

(C)

Figure 3-28. Determination of initial deposition rate i and lateral rate of spread, Gs. (a) Steady-state growth regime of (200) microsector (b) (110) twinned crystal of PE with a preferential nucleation site at the reentrant corner. (110) Growth planes adjacent to the twin plane depart from crystallographic (110) plane by an angle a. (c) Variation of Gs and i for single crystals of PE grown in p-xylene, determined with the above procedure (adapted from Dosiere, 1989).

ation sites with no nucleation barrier to growth. Curved edges have been analyzed using various approaches. Sadler (1983) used a rough surface approach, i.e., the supposes a growth regime, normally observed only at high Tc, in which statistical attachment of stems on rough surfaces prevents development of crystallographic facetting. Other approaches are based on variations of the classical secondary nucleation theory and rest on Frank's equations of the growth of crystals by secondary nucleation (Frank, 1974) but with obstruction to the advance of steps by solvent molecules acting as impurities (Toda, 1986). Some consider moving steps (Mansfield, 1988) or incorporate a lattice strain linked with expansion of the lattice which is required to accommodate folds (Hoffman and Miller, 1989). More recently, Point and Villers (1991) account for a whole range of elliptic shapes of polymer crystals essentially only on the basis of Frank's equations of growth with moving boundaries and mediate the theories of nucleation controlled growth as developed by Hoffman et al. (1975) and of rough surface growth advocated by Sadler. To sum up therefore, kinetic theories of crystallization, in their various levels of sophistication and after many years of maturation, account for the major features of polymer crystals: chain folding and fold length dependence on temperature, growth rate and its concentration and temperature dependence, growth morphology. However, many adjustments and revisions were introduced over the years as a response to new experimental findings and a number of more detailed mechanisms or postulates are open to criticism: the theories have had little predictive power. More revisions are under way or can be expected as a response to recent findings on the molecular rearrangements in the initial stages of crystal growth.

3.8 Polymer Single Crystals as Investigation Materials

3.8.3 Deformation and Transformation of Polymer Single Crystals

Single crystals provide ideal model systems for investigating deformation and transformation processes in polymers. Two case examples, illustrative of the possibilities offered, are considered here: - the crystal transformation at the unit cell level taking place either spontaneously or as a response to mechanical stress, - the deformation behavior of polyethylene single crystal mats, which can be regarded as an artificially created, significantly disentangled polymer bulk. 3.8.3.1 Spontaneous and Induced Crystal Phase Transitions

Poly(-l-butene) crystallizes in three different phases which can all be grown as single crystals from solution (Holland and Miller, 1964; Chau and Geil, 1984). Form II, a crystal modification with 11 3 helical conformation and tetragonal cell symmetry, is known to transform spontaneously into form I (3 x -helix, hexagonal symmetry). Holland and Miller observed that this transformation is dependent on the growth face (i.e., fold orientation): in the square crystal of form II, two orientations of the form I unit cell are generated at 90° to each other, corresponding to the different growth sectors. Further, when the folds are solidly affixed to a carbon support film, i.e., are pinned, the transformation is suppressed. These experimental findings, which have not yet been analyzed in molecular terms, deserve further consideration. 3.8.3.2 Deformation of Polymer Crystals

Polymer crystal lattice deformation on drawing may be significantly affected by the existence and directionality of folds. In

145

an early, extensive study of the deformation occurring in PE single crystals, Kiho et al. (1964) were able to detect four different types of deformation, involving different combinations of twinning and phase change (from orthorhombic to monoclinic), depending on the relative orientations of draw and crystal axes and fold directions. Similar studies have been reported for other systems (Wunderlich, 1973, 1976, 1980). Possible dislocation motions which may be involved in the deformation mechanisms of crystalline polymers have been considered (e.g., Keith and Passaglia, 1964). They are severely limited due to chain folding and therefore play a minor role in deformation processes of polymers. 3.8.3.3 Deformation of Polymer Single-Crystal Mats

The recent development of ultra high modulus fibers produced from high molecular weight polyolefins (notably PE, but also iPP, etc.) has underlined the considerable impact of molecular entanglements on the drawability of crystallized polymers. Bulk crystallized polymers with entangled chains have maximum draw ratios on the order of or lower than ~ 10. As a result of more or less sequential deposition during crystallization from dilute solution, molecules in single crystals are less entangled than upon crystallization from the bulk. Sintered mats of single crystals can therefore be considered as bulk polymers in which (within the confines of chain folded crystallization) chains are confined to one lamella and have the minimum achievable number of entanglements with neighboring chains both within their lamella and from nearby lamellae. They therefore can be used as test materials for investigations on drawability, and

146

3 Structure of Polymer Single Crystals

its modification when applying various physical or chemical treatments (annealing, cross linking, etc.). Early studies [e.g., Miyasaka and Ishikawa (1973)] had shown the deformability and improvement of properties of drawn mats of single crystals. Recent studies on such systems have confirmed an extraordinary drawability below the melting temperature (i.e., at « 110-120 °C) of singlecrystal mats, especially when made of ultra high molecular weight (UHMW) material. Deformation ratios of « 250 have been reported for a two-stage deformation process, leading to tensile moduli of « 220 GPa, near theoretical values (Kanamoto et al., 1983). More recent results can be found in reviews. A rapid loss of drawability upon annealing has been reported by (Lemstra and Kirschbaum, 1985; Lemstra et al., 1987 a, b). Annealing UHMW PE single-crystal mats at 142 °C for only 20 s reduces the achievable X from values > 70 to ~ 25. Annealing (melting) for 30 s restores the properties of bulk crystallized material (4max ~ 5). The underlying principles regarding impact of chain entanglement on drawability and possibility to produce fibers with ultra high modulus have been exploited in the gel-spinning process, which includes ultradrawing (Lemstra et al., 1987a,b).

bulk crystallization habits. As a consequence, investigations of polymer single crystals are of considerable help for polymer crystallization in general. Historically the first, and still the most valuable contribution was the recognition of chain folding itself. In the following thirty years, besides some overspecialized research initiated by quarrels on controversial issues, investigation of polymer single crystals have produced a regular flux of results relating to crystal structure and defects, chain fold structure, onset of chain folding, crystallization kinetics and mechanisms, etc. Whereas the main features of polymer single crystals are now well established in spite of several on-going controversies, the advent of new experimental techniques and the uprise of new problems will continue to maintain interest in single crystals considered as the model morphology for crystalline polymer materials.

3.10 Acknowledgements The authors are deeply indebted to numerous colleagues for loan of published or yet unpublished material, for pointing out various works which had escaped their attention, and for helpful comments and discussions. They also wish to thank S. Graff, C. Straupe, and A. Schierer for their contribution to the completion of this work.

3.9 Conclusion Polymer single crystals represent a comparatively simple state of a much more complex, spherulitic morphology normally produced in bulk crystallization. At the same time, they have the main characteristics of bulk polymer crystals, i.e., chain folding, and associated lamellar thickness. Polymer single crystals are of course a simplification, but not an oversimplification of

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4 Crystallization and Morphology of Semicrystalline Polymers Peter J. Barham

H.H. Wills Physics Laboratory, University of Bristol, Bristol, U.K.

List of 4.1 4.1.1 4.1.2 4.1.2.1 4.1.2.2 4.1.2.3 4.1.2.4 4.1.3 4.1.3.1 4.1.3.2 4.1.3.3 4.1.3.4 4.1.3.5 4.1.3.6 4.1.4 4.1.4.1 4.1.4.2 4.2 4.2.1 4.2.1.1 4.2.1.2 4.2.1.3 4.2.1.4 4.2.1.5 4.2.2 4.2.2.1 4.2.2.2 4.2.3 4.2.3.1 4.2.3.2 4.2.4 4.2.4.1 4.2A.2 4.2.4.3

Symbols and Abbreviations Introduction Objectives of Chapter Hierarchies of Structure Molecular Structure Crystals and Crystallinity Lamellae and Chain Folding Crystal Aggregates and Spherulites Experimental Techniques Neutron Scattering X-Ray Diffraction Optical Microscopy Electron Microscopy Spectroscopy Differential Scanning Calorimetry (DSC) Theories and Models Secondary Nucleation Models Entropy Barrier Models Crystallization and Morphology in Bulk Polymers The Crystal Thickness Dependence of Thickness on Crystallization Temperature Annealing Effects Extrapolation to Initial Thickness Extended Chain Crystals Role of Mobile Phases in Determining the Crystal Thickness Development and Structure of Aggregates Isolated Crystals from the Melt Spherulites and Dominant Lamellae Spherulitic Growth Rates and Regimes Measurements of Growth Rates Morphology of Crystals in Different Regimes Nucleation Behavior Outline of Nucleation Theory Experimental Observations of Different Types of Nucleation Chemical Nucleation

Materials Science and Technology Copyright © WILEY-VCH Verlag GmbH & Co KGaA. All rights reserved.

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4.3 4.3.1 4.3.1.1 4.3.1.2 4.3.1.3 4.3.2 4.3.2.1 4.3.2.2 4.3.2.3 4.3.3 4.3.3.1 4.3.3.2 4.3.3.3 4.3.4 4.3.4.1 4.3.4.2 4.3.4.3 4.4 4.5

4 Crystallization and Morphology of Semicrystalline Polymers

Relationships Between Structure and Properties Models Used to Predict Mechanical Properties Simple Models - Utility and Limitations Problems Associated with Realistic Models Some Simple Relations Control of Morphology to Improve Strength Cracking in Poly(hydroxybutyrate) (PHB) Spherulite Expansion and Contraction Prevention of Crack Formation in PHB Control of Morphology to Improve Stiffness Morphological Requirements for Stiffness Extended Chain Fibers Interlocking Shish-Kebab Systems Control of Morphology to Improve Toughness Principles of Rubber Toughening Phase Separation in Polyethylene Blends Control of Phase Morphology and Toughness Concluding Remarks References

194 194 194 195 196 197 197 199 199 200 200 201 204 205 205 205 207 208 208

List of Symbols and Abbreviations

155

List of Symbols and Abbreviations a, b, c A o , A 1 , £ 0 , BX AF g G Gl9 G n , G m Go G\o > G II0 , GIII0 h AH i / /0 k L /, ll912 /min /* §/

crystal axes coefficients defined in the rate equations for backward and forward steps in creating a nucleus free energy change on crystallization rate at which a nucleus once formed spreads across the surface crystal growth rate crystal growth rate in regime I, II, III constant governing the absolute crystal growth rate constants governing the absolute crystal growth rate in regime I, II, III Planck constant enthalpy change on crystallization surface nucleation rate overall nucleation rate constant governing the absolute nucleation rate Boltzmann constant substrate length crystal thicknesses minimum stable crystal thickness initial lamellar thickness mean crystal thickness =/;-/mio

JV0 R S(l) T Tn T^ T^ [/*

n u m b e r of molecules gas constant flux absolute temperature nucleation temperature thermodynamic constant equilibrium melting point thermodynamic constant

p

crs a os Aa A(/>* \jj

transport term representing the rate at which molecules arrive at the surface constant surface free energy surface energy at the ends of a box enclosing a molecule, where the chain emerges or folds interfacial energy between the crystal a n d the foreign surface interfacial energy between the foreign surface a n d the melt = a + <7OS — as critical free energy parameter

BPE DSC

branched polyethylene differential scanning calorimetry

j80 a ae

156

LAM LAXS LLDPE LPE PEEK PET PHB SEM TEM WAXD

4 Crystallization and Morphology of Semicrystalline Polymers

Raman longitudinal acoustic mode low angle X-ray scattering linear low density polyethylene linear polyethylene poly(ether-ether-ketone) poly(ethylene terephthalate) poly(hydroxybutyrate) scanning electron microscopy transmission electron microscopy wide angle X-ray diffraction

4.1 Introduction

4.1 Introduction 4.1.1 Objectives of Chapter

Semicrystalline polymers form a large part of the commercially important thermoplastics. In nearly all applications these materials are melt processed. The properties of manufactured articles are then largely determined by the crystal structure and the arrangement of the crystals grown from the melt during processing. The morphology of semicrystalline polymers and the relations between structure and properties have been studied extensively over the past four decades. This chapter is intended to provide a good grounding in the basics of crystalline polymer morphology. More detailed accounts of specific issues may be found in the texts by Wunderlich (1973, 1976, 1980) and Bassett (1981) and the reviews by Khoury (1976) and Phillips (1990). We start with a brief outline of some of the most important techniques and a description of the sort of information they can provide. Secondly, the reader will be brought up to date with some current issues in the field. Finally, examples of relationships between structure and properties will be described where the morphology has been deliberately modified to improve specific properties. 4.1.2 Hierarchies of Structure 4.1.2.1 Molecular Structure

Polymeric materials exhibit structure at all levels from the atomic to the macroscopic scales. At the molecular level polymers are made up from repeating units, and the chemical nature of these units clearly must have a large effect on the material as a whole. For example, depending on the type of chemical bonds in the chain the molecules may be stiff or flexible. Flex-

157

ible molecules are those in which rotation around the chemical bonds in the backbone of the molecule is possible with activation energies of not more than a few k T. Rigid molecules include both the rigid rod molecules in which such bond rotations are always impossible (these systems are addressed in Chap. 5 of this Volume) and also molecules which are below their glass transition temperatures where the activation energies for bond rotation are prohibitive. In general flexible molecules provide flexible materials (e.g., polyethylene) while stiff molecules provide stiff materials (e.g., Kevlar in the case of rigid rods, or polystyrene in the case of a polymer well below its glass transition temperature). The chemical architecture of polymers has already been described in some detail in Chap. 1. The chemical nature of the molecules, the distribution of monomers along the chain, the strength and flexibility of the chemical bonds between monomers, and any interactions between monomers in the chains all affect properties. However, there are many more subtle ways in which the structure along the chains affects the material. In particular, the chemical composition of the chains has a significant effect on the crystallization behavior. For example, if the chain is made up from several different monomers arranged at random, it is possible that it will be unable to form a stable crystal. The crystal melting temperatures can be strongly influenced by subtle changes in the chemical nature of the molecules. Melting occurs when the free energies of the liquid and solid phases are equal, from which it follows that the melting temperature is given by the ratio of the enthalpy of fusion to the entropy. Specific interactions between chains such as hydrogen bonds can lead to high heats of fusion and hence

158

4 Crystallization and Morphology of Semicrystalline Polymers

to comparatively high melting temperatures. Similarly if the chains are stiff, then the configurational entropy of the melt can be quite low, which also leads to a high melting temperature. 4.1.2.2 Crystals and Crystallinity

The crystals formed by polymers are different from those formed by small molecules as the basis of the crystal (the repeating motif) is a part of the molecule (the monomer) rather than the whole molecule (see e.g., Wunderlich, 1973; Phillips, 1990). Admittedly there are cases of polymers where the whole molecule forms the basis of the crystal, such as in proteins. In such cases all the molecules are strictly identical in their size and chemistry. Although such crystals are interesting and important in their own right, they are not relevant to a description of commercially important semicrystalline polymers. A simple way to begin to understand the difference between small molecule crystals and polymer crystals is to consider the crystals formed by a series of progressively larger molecules. A convenient example is that of paraffins. Short paraffins form crystals in which the whole molecule is clearly the basis. As the paraffins become longer, a sublattice formed by the CH 2 units starts to appear, but all the molecules still align with their end groups in crystallographic register. As the paraffins become even larger, the influence of the ends is diminished, until the polyethylene crystal structure is seen in which the crystal basis is simply that of the sublattice of CH 2 groups. In a polymer crystal one part of the periodic structure is thus built into the molecule and does not come from the crystallization but is forced upon it by the chemistry of the molecules. This connectivity of

monomers leads inevitably to problems of packing in the crystals; for example, where are the chain ends to go? In practice polymer molecules form crystals that are imperfect and there are parts of molecules that are not in any crystals. Such imperfections lead to the concept of the crystallinity of polymeric materials; there are several definitions of the crystallinity (see, e.g., Billmeyer, 1984). These differ in the way in which they define the proportion of material inside the crystal and in the method used to determine it; for example the mass or volume of material can be used. For most practical purposes the differences are not important as the measurement of crystallinity is usually less accurate than any differences between the different definitions. The crystallinity of most melt crystallized polymers lies in the range 30-70%, which should be compared with the slightly higher values (ca. 40-80%) found in single crystals grown from solution (see Chap. 3 of this Volume). At first sight it may appear surprising that the crystallinity is so low that only half the molecules are involved in the crystals; the location of the noncrystalline material is perhaps the most salient issue which is addressed in the following sections where the morphology of the crystals is also described. 4.1.2.3 Lamellae and Chain Folding

The basic unit of polymer crystallization is the lamellar crystal. Individual lamellar crystals (known as single crystals) have been described in some detail in Chap. 3, with a detailed discussion of chain folding. A very brief overview of some of this material most relevant to melt crystallization will be given here for completeness. When flexible chain polymers crystallize they normally form lamellar crystals in

4.1 Introduction

which the chains are approximately perpendicular to the surface. The thickness of these lamellae is very much less than the length of the chains, this thickness being determined by the supercooling at which the crystals are grown. Thus, for isolated crystals, the chains must fold back into the crystals. The manner in which this folding is achieved and its regularity form a major topic in polymer physics and are discussed in detail in Chap. 3. When polymers crystallize from the melt lamellae form but, since they are not usually in isolation, the arguments for chain folding are less compelling. The usiial argument for chain folding in melt crystallized polymers is that the amorphous material between the lamellae is less dense than the crystal and that if all the chains emerge from the crystal, adopt random trajectories and do not fold back into the same crystal, then the density at the surface will be significantly higher than in the crystal. The argument was expressed most eloquently by Sir Charles Frank in his introduction to the Faraday Society Meeting in 1979 (Frank, 1980). The argument, when applied to polyethylene, leads to the conclusion that about one half of the chains emerging from the crystal surface must fold back into the crystal at an adjacent site. The degree of adjacency required depends on the density difference between the crystalline and amorphous phases and on the angle the chain axes make to the lamellar surface. There are some polymers where the arguments do not require chain folding. For example, poly(4-methyl-pentene) crystallizes with the molecules adopting a shallow helical form so that the density of the crystalline phase is lower than that of the amorphous phase. In this case there is no simple geometric reason for chain folding; nevertheless the polymer forms lamellar crystals in the same way as

159

polyethylene. The various models for chain folding and polymer crystallization are described in Sec. 4.1.4 below. The location of at least some of the noncrystalline material should now be apparent. At the surface of each lamellar crystal there must be a layer in which the chains fold back. The thicker this layer is, the lower the overall crystallinity will be. There have been many attempts to distinguish between the thickness of the crystalline core within the lamellae and the overall lamellar thickness or fold period. For polyethylene, which is by far the most studied material, the experiments are largely inconclusive and contradictory. The crystal thickness can, in principle, be measured by X-ray line broadening or Raman LAM frequencies, and the lamellar thickness can be measured by low angle X-ray diffraction or by direct electron microscopy. The detailed analysis of the overall X-ray scattering can also be used to generate the electron density profile across the crystal. (N. B. some details of these techniques are given in Sec. 4.1.3 below.) The data suggest that when the crystal core thickness is measured by Raman scattering, or by X-ray line widths, then the thickness of the amorphous layer at the lamellar surface is quite small and insufficient to account for the observed crystallinity being as low as it is. On the other hand, some of the analyses of the overall X-ray scattering suggest very thin central crystal cores and are consistent with all the noncrystalline material being contained in a thick layer at the lamellar surface. The crystallinity as normally measured is an overall measure, including contributions from differing types of noncrystalline material; imperfections within the crystal lattice contribute to the noncrystalline material. Thus any chemical imperfections along the chains, such as a change of tactic-

160

4 Crystallization and Morphology of Semicrystalline Polymers

ity or inclusion of a side group, may be reflected as a crystal defect and lower the crystallinity. The distinction between different types of imperfections becomes particularly important when considering copolymers where the inclusion of the minority monomer type in the lattice of the major can lead to a significant energy penalty and to an apparently low crystallinity. 4.1.2.4 Crystal Aggregates and Spherulites

When polymers are crystallized from the melt, as is the usual practice for commercial applications, the lamellar crystals usually form aggregates of one form or another. The most common such aggregate is the spherulite. This section is intended to provide some examples of the types of aggregates that occur. Fuller details of the internal structure and the mechanisms involved in their growth are contained in Sec. 4.2.2 below and in Chap. 3 (Sec. 3.6) of this Volume. Figure 4-1 shows some optical micrographs of spherulites from several different polymers. Note that all the spherulites display the characteristic "Maltese Cross" extinction pattern and that some also display the regular concentric banding. Both these features appear because the spherulites are viewed between crossed polarizers, the normal method employed to look at polymer crystals. The Maltese Cross arises from the coincidence of the principal axis of the crystal indicatrix with the extinction direction of the polarizer or analyzer. It can easily be demonstrated that the chains are arranged circumferentially within the spherulites, and hence it follows that the lamellae are arranged radially. The regular concentric bands, when they are seen, arise from a regular twist in the radiating lamellae. Extinction occurs when the lamellae are observed down an optic axis. Figure

4-2 is a schematic illustration of the orientation of molecules and lamellae in a spherulite. Many other types of crystal aggregates are sometimes seen which are mostly related to spherulites. Row structures occur when a series of spherulites are nucleated along a common line, either caused by flow or by the presence of a foreign linear nucleus. The development of the spherulites is restricted to the radial direction away from the nucleus giving rise to the characteristic appearance. A similar morphology can also be observed when a large surface acts as the nucleus. In this case the growth of the incipient spherulites is restricted to the direction normal to the surface. Axialites fall somewhere between single crystals and spherulites; they may, as a crude approximation, be considered to be spherulites in the very early stages of growth. Other aggregates occasionally observed include dendritic and densely branched morphologies. These are often considered as parts in a wide spectrum of morphologies culminating in the spherulitic texture (see, e.g., Phillips, 1990). 4.1.3 Experimental Techniques

There are many techniques that are used to explore the morphology of semicrystalline polymers. It will be useful to the reader to have a grasp of the power and limitations of some of the more common techniques. The following sections outline the principles behind several experimental methods and give some examples of their uses in morphological studies, where appropriate attention is drawn to the limitations of the techniques. 4.1.3.1 Neutron Scattering

Neutron scattering from mixtures of hydrogenated and deuterated polymers is a

4.1 Introduction

161

(b) (d) Figure 4-1. Optical micrographs of spherulites from a variety of polymers, all taken between crossed polarizers: (a) dextran, (b) poly(hydroxybutyrate), (c) polypropylene (showing spherulites of two crystal forms), (d) polyethylene.

particularly powerful technique to examine the overall shape of molecules in the solid state (Ballard et al., 1976; Fischer etal., 1984; Sadler, 1984; Sperling, 1984). The technique relies on the large contrast

in the scattering lengths of deuterium and hydrogen. If all (or some) of the hydrogen atoms along a polymer chain are substituted with deuterium atoms, then the chain will itself have a completely different inter-

162

4 Crystallization and Morphology of Semicrystalline Polymers

Figure 4-2. Schematic illustration of spherulite, showing the crystal orientation for the polyethylene system (Barham and Keller, 1977).

action with neutrons. A basic assumption is that the isotopic substitution, which so strongly affects the neutron scattering behavior, has no significant effect on the other physical properties of the polymer. In particular, it is important that the deuterated and hydrogenous polymers are fully miscible in the melt and in the crystal;

any aggregation of the deuterated species leads to major problems in interpretation of data. Provided that the two species are fully mixed, then the scattering may be considered in the same way as any other scattering and the usual techniques, such as Zimm and Guinier plots, may be used to find radii of gyration. An excellent review of neutron scattering of polymers is given by Sadler (1984) (also see Vol. 2B). In addition to providing information about the size of the random coils, neutron scattering can be used to give information about the shape of molecules in crystals. For example it has been shown (Spells etal., 1980; Spells and Sadler, 1984) that in polyethylene single crystals grown from solution, the molecules are arranged in "super-folded" sheets (see Fig. 4-3). In contrast, in samples quenched from the melt the molecules retain their overall radius of gyration; only local rearrangements occur during crystallization (Stamm et al., 1979; Dettenmaier et al., 1980; Sadler and Keller, 1979). Another area where neutron scattering is invaluable is in determining the overall molecular orientation and extension in ori-

Figure 4-3. A sketch indicating the trajectory of a single chain in a solution grown crystal, after Spells and Sadler (1984).

4.1 Introduction

ented systems such as fibers. For example, it has been shown for drawn fibers of polyethylene that the tensile modulus of the macroscopic fibers depends on the extension of the individual molecules (Sadler and Barham, 1990 c). There is of course one major difficulty with neutron scattering: the two isotopically different species tend to separate during crystallization. There is some evidence for phase separation in the melt prior to crystallization (Sadler, 1986). These problems are overcome by judicious choice of polymers to minimize separation and by rapid crystallization to allow little time for rearrangements. Consequently, with the exception of work on isotactic polystyrene, there is little available data on isothermally crystallized polymers.

163

tfj* •%,

'

* '

••' '' • •''':'"

'W

4.1.3.2 X-Ray Diffraction While neutron scattering provides information about the whole molecule and its trajectory in a crystal, X-ray diffraction (and scattering) provides information about the arrangement of the monomers that make up the chains and the crystals. The most basic information obtained from wide angle X-ray diffraction (WAXD) is the crystal structure, which can usually be deduced from the X-ray diffraction pattern. An example of a diffraction pattern and the corresponding crystal structure is shown in Fig. 4-4. In addition to the crystal structure, wide angle diffraction provides information about the orientation of the crystals with respect to the sample. Low angle X-ray scattering (LAXS) provides information about the size and regularity of packing of the lamellar crystals themselves; the interpretation of low angle scattering is often a difficult matter and great care must be taken when trying to obtain accurate data on the dimensions of the crystals

Figure 4-4. X-ray diffraction pattern (a) and corresponding crystal lattice (b) deduced from it, for a siloxane polymer. A and B show the two possible arrangements of the chains in the lattice. [Gardner et al. (1978), Polymer 19. By kind permission of the authors and of the publishers, © Butterworth Heinemann Ltd.]

(Vonk, 1971; Crist, 1973; Crist and Morosoff, 1973). In addition to these routine "fingerprinting" techniques, wide angle X-ray diffraction is used for several more sophisticated measurements. Perhaps the most impor-

164

4 Crystallization and Morphology of Semicrystalline Polymers

tant of these is the determination of crystallinity. Several methods have been developed for this purpose, all of which rely on evaluating the contributions of Bragg diffraction (from the crystalline part) and of amorphous scattering (from the noncrystalline part) to the total intensity. The most successful and commonly used method is due to Ruland (1971, 1977, 1978). With the availability of synchrotron X-ray sources, new experiments which follow changes in the X-ray scattering as polymers crystallize have become possible (Eisner et al, 1985 a, b; Barham and Keller, 1989; Martinez-Salazar et al., 1985). These experiments have provided a wealth of novel information, some of which will be discussed in further detail in later sections. Work on polyesters at Hamburg (Eisner etal., 1985a,b; Asano et al., 1989) has shown that the crystals when formed have a very low crystallinity with only a thin central crystalline core and thick amorphous layers at the surfaces. On cooling the crystallinity increases primarily by the thickening of the crystalline core at the expense of the amorphous surface layer. Work with polyethylenes has shown that the crystals first form with a small lamellar thickness but in the first minutes at the crystallization temperature they can spontaneously double in thickness (Barham and Keller, 1989). 4.1.3.3 Optical Microscopy

The optical microscope is one of the most useful and versatile tools in the study of the microstructure of semicrystalline polymers. Little, if any, sample preparation is needed and it gives a straightforward view of the texture on the scale of micrometers to the macroscopic object. Examples of optical micrographs feature in many of the figures in this chapter.

Nearly all polymer crystals are highly birefringent so that when viewed using polarized light it is easy to distinguish between crystalline and noncrystalline regions. The size of the crystal aggregates is very easy to measure and from this information, and some knowledge of the crystallization, the overall spherulite nucleation density can be estimated. The optical microscope is commonly used as the starting point for any morphological investigation as a knowledge of the microstructure often enables the best choice of further work. One area where the microscope is particularly useful is in conjunction with a hot stage. For example, the direct observation of crystallization, illustrated for polyethylene in Fig. 4-5, permits the direct measurement of the crystal growth rate. In some cases it also possible to measure crystal nucleation rates and to observe directly the effect of adding nucleants. 4.1.3.4 Electron Microscopy

The transmission and scanning electron microscopes provide information on the microstructure of polymers from the scale of a few molecules up to that covered by the optical microscope. However, for electron microscopy special sample preparation is usually necessary. Nevertheless, electron microscopy has probably provided the greatest amount of morphological information of all the available techniques. A detailed description of techniques in electron microscopy of polymers is to be found in the book by Sawyer and Grubb (1987). Scanning electron microscopy (SEM) traditionally requires that the sample under investigation is conductive. Polymers are usually coated with a metal, for example Au-Pd, to achieve this. The resolution

4.1 Introduction

165

Figure 4-5. A series of optical micrographs showing the growth of polyethylene spherulites from the melt at 123 °C (Olivers etal., 1982).

Figure 4-6. Scanning electron micrograph of the central region of a polyethylene spherulite. The arrows A point to fine lines which are interpreted as steps ca. 70 nm high; these correspond to the thickness of individual crystals in this preparation. Separate sheets of this thickness can occasionally be seen, as for example, at B (Winram et al., 1978).

then depends on how well this coating follows the structure present in the polymer sample. In practice it is possible to obtain resolution down to the scale of the lamellae, although this is not normally needed or achieved. A typical micrograph showing the interior of a spherulite is shown in Fig. 4-6. Recent developments include environmental SFMs. which permit investigations to be carried on poorly conducting materials and in most environments, and low voltage, high resolution SEMs, which allow observations on materials where charging would otherwise cause problems. One of the most remarkable and useful features of SEM is the enormous depth of field, which can be seen to great advantage in Fig. 4-6. Transmission electron microscopy (TEM) provides a much higher resolution but requires very thin samples. Typically

166

4 Crystallization and Morphology of Semicrystalline Polymers

samples are sectioned, using an ultramicrotome, into slices ca. 100 nm thick. The high flux of electrons in the beam damages the polymers under investigation so that the contrast provided by diffraction from crystalline parts disappears very quickly. It is usual practice to provide some enhancement of the contrast by differentially staining the sample with a heavy metal. This is usually done by diffusing a reactive small molecule containing a high atomic number species into the sample. The stain molecules will diffuse more easily into the noncrystalline regions between the crystalline lamellae and so provide mass scattering contrast between crystalline and amorphous regions. Commonly used techniques include the chlorosulfonation method (Kanig, 1973, 1974,1980), which is often combined with a stabilizing treatment with uranyl acetate, and the osmium tetroxide stains which have been pioneered by Voigt-Martin and co-workers (e.g., Voigt-Martin, 1985). The use of staining techniques for quantitative purposes is limited as the stains can distort the crystal sizes leading to incorrect estimates of thickness (Bashir et al., 1986; Hill et al., 1992). Examples of electron micrographs from stained systems can be found, together with full details of the wide variety of stains and the polymers for which they are suitable, in the book by Sawyer and Grubb (1987). An alternative approach is to prepare replicas of an etched surface. In this process the surface detail is provided by a chemical attack. Again this relies on the easier diffusion of the etchant into the noncrystalline regions. The prepared surface has the unetched crystalline parts standing proud. The replicas are usually prepared by evaporating a thin layer of carbon and a subsequent layer of metal onto the surface. The replicas are very stable in the

electron beam. Often small pieces of polymer are fortuitously left on the replica, and can be used in diffraction studies to determine crystal orientation. An example of a replica of an etched surface is shown in Fig. 4-7. Note that while a good deal of detail is discernible the lamellae show most clearly when they are almost normal to the surface. When it is possible to make direct observations of suitable, thin, samples such as solution grown single crystals (see Chap. 3 of this Volume) then the use of diffraction contrast can be very powerful. It is a relatively simple matter to insert an aperture and observe dark field images formed from a particular diffracted beam. As well as distinguishing between crystalline and noncrystalline regions, such images provide direct evidence of the orientation of the crystals. However, when performing direct electron microscopy care must always be taken to avoid problems of damage to the sample from the electron beam. 4.1.3.5 Spectroscopy Infrared Spectroscopy Infrared spectroscopy is able to provide information about the types of chemical species present and, to some extent, their environment. The frequencies of vibration of most chemical groups are tabulated so that by taking an infrared spectrum of a polymer sample it is, in principle, possible to identify the chemical groups from the frequencies at which absorption occurs. This most basic information can be particularly useful in identifying the presence of impurities and additives in a polymer sample, or in determining the relative amounts of two monomers in a copolymer system. However, the real power of infrared techniques in polymer morphology lies in its use to look at the environment sur-

4.1 Introduction

Figure 4-7. Electron micrograph of a replica of the etched surface of a blend of linear and branched polyethylenes, illustrating the detail which this technique can reveal (Hill and Barham, 1992).

rounding a particular species. For example, in a crystal some infrared absorption bands change by splitting into two components due to the crystal field (see the book by Elliott, 1969; and the review by Zerbi, 1984). A measurement of this splitting can provide direct assessment of crystallinity. In copolymer systems the splitting can be used to estimate the degree to which monomers of one species are present in the crystal lattice of the other. Polarized radiation can sometimes be used with oriented samples to identify the orientation of the crystalline and noncrystalline phases through the different degree of intensity of infrared bands associated with each phase. This sort of information can be particularly helpful when trying to assess the contribution from crystal and amorphous phases to the overall mechanical properties. A more specialized, but highly successful, technique is to use mixtures of isotopically different species. This has the same drawbacks associated with phase separation discussed in the section on neutron scattering but can provide detailed information about the folding of the chains in

167

the lamellae. The vibration frequency of a group containing hydrogen atoms will be different from that of the same group containing deuterium atoms. Similarly the crystal field splitting will differ depending on whether the adjacent groups in the crystal are of the same or different species (Bank and Krimm, 1969,1970; Krimm and Ching, 1972; Ching and Krimm, 1975 a, b). It has proved possible using isotopically doped samples to determine the degree to which adjacent reentry folding occurs and even to distinguish between folding along different crystal planes (Cheam and Krimm, 1981; Jing and Krimm, 1982; Spells etal., 1987). Raman Spectroscopy Raman spectroscopy became an important tool for polymer science when it was realized that there was a vibrational mode in polyethylene crystals which depends on the length of the chain in the crystal lamellae. It was initially hoped that it would be possible to associate this vibration with a measurement of the all trans sequence length along the chain, and hence in combination with low angle X-ray diffraction determine the thicknesses to the crystalline core and amorphous surface layers independently in a range of lamellae grown under a variety of crystallization conditions. In practice it has not proved possible to establish a completely clear relationship between the frequency of the Raman LAM mode and the all trans length. The frequency of the mode is best represented in terms of a three block model in which the central block is a rod of high stiffness, representing the crystalline core, and the outer two blocks are rods of lower stiffness, representing the amorphous regions. Several analyses of the Raman LAM mode have been made using such models and it is

168

4 Crystallization and Morphology of Semicrystalline Polymers

possible to draw some conclusions about the thickness of the amorphous layers in a few special cases (Strobl and Eckel, 1977; Strobl and Frentzel, 1984). In the course of these studies Raman spectroscopy has become a routine tool for approximate measurement of crystal thickness, its usefulness being in the ease of making such measurements compared with LAXS. At the same time other uses have been found for Raman spectroscopy. Perhaps the most important of these involve the use of a microscope to focus the laser beam down to a very small spot (ca. a few micrometers in size) so that local information can be obtained. When used in conjunction with stress sensitive Raman bands this provides a most powerful technique for the study of local stress inside a sample. For example it is possible to determine the variation of strain in a single short fiber in a matrix under stress (Galiotis et al., 1985; Day et al., 1987). 4.1.3.6 Differential Scanning Calorimetry (DSC)

Calorimetry is particularly useful in the study of crystallization as it provides direct measurement of thermodynamic properties, heats of fusion, melting temperatures etc. The instrument most commonly used in polymer studies is the differential scanning calorimeter, which uses two pans: one containing the sample and an empty pan acting as a reference. The two pans are heated (or cooled) at a constant rate; they are either kept at the same temperature and the heat difference applied to them monitored or provided with the same heat input and the temperature difference monitored. It is a simple matter to determine heats of fusion, melting, and crystallization temperatures, and glass transition tempera-

tures with these instruments. These quantities depend on the heating rate used and appropriate corrections have to be applied to determine absolute values. More importantly many polymers undergo reorganization during heating, such as the lamellar thickening described in Sec. 4.2.1.2 below. In such cases the character of the observed behavior changes with the heating rate, for example, there may be multiple melting endotherms, the number of which depends on the heating rate. It is common practice to use the measured melting temperatures to infer the crystal thicknesses and the measured heat of fusion to estimate the crystallinity. While such estimates are very easy to make there can be considerable errors associated with them. First it is most important that the melting temperature has been extrapolated to zero heating rate and the heat of fusion to infinite heating rate. Secondly if there are any rearrangements occurring during the heating these must be noted. For example if the lamellar thickness increases then the melting temperature is that characteristic of the thickened lamellae and not of the lamellae before heating (see Wunderlich, 1980).

4.1.4 Theories and Models 4.1.4.1 Secondary Nucleation Models

In order to understand the variety of apparently different models and parameters that have been used in them it is worthwhile reviewing the historical development of the theory of polymer crystallization. There is a strong correlation between the developments in experimentation and in the theories. Sometimes experiments have been designed to test the theories and in other cases new experimental data have forced revision of the theories. In

4.1 Introduction

this section an attempt is made to put the development of the theories of polymer crystallization into context. This is done from a strictly personal viewpoint but I hope an objective one. A detailed review of the theories may be found in Armistead and Goldbeck-Wood (1991). The original question addressed by the secondary nucleation models was, why do chains fold? The observation of polymer single crystals with limited thickness in the chain direction showed that the preferred crystal was not that with the lowest free energy, that is, a thick crystal in which the chains are fully extended. Two important further observations contributed directly to the idea that a secondary nucleation model should be appropriate. First the lateral shapes of the crystals grown at that time all displayed straight edges with well defined low index crystallographic planes making up the crystal sides. Secondly, the thickness of the crystals was found to depend on the supercooling rather than the absolute temperature at which they were grown. The model in its simplest and original form (Lauritzen and Hoffman, 1960; Frank and Tosi, 1961) considers the growth of a layer of crystal on an infinite flat substrate of perfect crystal, Fig. 4-8. The molecule is

169

considered to lie inside a box whose lateral sides have free energy a and whose ends, where the chain emerges or folds, have energy aQ. The model then considers the changes in the free energy of the system as the chain lies down. This can be seen by the illustration in Fig. 4-8. As the stem arrives an end is created, at the energy cost of ab <7e, as it lies down to form a single stem so two new lateral surfaces are formed. These incur an energy penalty a I a but there is a compensating gain of the enthalpy of fusion ablAHAT/T°. Once the chain folds back another end is created with the associated penalty, but as it lies down next to the first stem no new side surface is created so it is energetically downhill all the way until it reaches the end of the first stem and folds back. Provided that ;>2(7 e r m 7(ATAi/) the overall free energy will decrease as more stems are added. Eventually when the free energy is negative a stable secondary nucleus has been formed. The next step in the model is to write down the rate equations for the backward and forward steps in creating a nucleus of thickness /, and find the flux as a function of /. It is then possible to integrate these equations so as to find the mean thickness with which crystals will grow.

Number of stems

Figure 4-8. Diagram illustrating the secondary nucleation process for crystal growth (a) with the corresponding free energy diagram (b).

4 Crystallization and Morphology of Semicrystalline Polymers

170

where 8/ is approximately constant [with a value of ca. k T/(2ba)] for small AF. This simple model showed excellent agreement with the measurements of crystal thickness available at the time, as illustrated in Fig. 4-9. Clearly the model oversimplifies the real situation, in that it assumes that the substrate on which the crystals grow is itself infinite, and that once the first stem is laid down that fixes the thickness for the whole crystal. Modifications to the original model have been made to overcome for these simplifications, but these make no significant differences to the result. There is one feature of the model which, if taken seriously, is surprising; the 5/ term "blows up" to infinity at a finite supercooling. This 5/ catastrophe, which occurs when the gain of free energy on crystallization is so great that there is no maximum in the free energy curve (dotted line in Fig. 4-9), is not observed in practice. To remove the problem, the way in which the free energy is apportioned in the rate equations was modified by introducing a new parameter, \j/. By suitable choice of this parameter it is possible to place the 5/ catastrophe at an arbitrarily large supercooling. Another important feature of the model in this simplest form is that it only allows

The rate equations are 2bla-ablAF kT ablAF\

= P

(4-1)

2abar-ablAF

where AF = AHAT/T£ and p is a transport term representing the rate at which molecules arrive at the surface. The flux is given by =

NOAO[1Ibla

kT ' <"> 2abae-ablAF\l

kf

)]

so that the mean crystal thickness is

2ffe/AF

(4-3)

2ae/AF

which yields = (2aJAF) + 81

20

30 40 Supercooling (*C)

50

Figure 4-9. Lamellar thickness data available when the secondary nucleation model was first proposed, together with the predictions of the model; the dotted line illustrates the "5/ catastrophe".

4.1 Introduction

for completely flat crystal surfaces. Obviously this assumption cannot be true as the surface will have a configurational entropy, which will lead to defects in the surface. It was assumed that this assumption was unimportant in the case of polymers because all the crystals observed had flat low index planes defining their surfaces. This point will be returned to later. The model was developed to describe the crystallization from dilute solution where the inherent assumption that only one chain was crystallizing at the surface at a time was likely to be valid. It was not expected to be immediately applicable to the behavior of polymers crystallized from the melt. Indeed the thickness of crystals grown from the melt did not appear to be well described by the model. However it was noted that the linear growth rate found in melt crystallization varied in the way that might be expected for secondary nucleation. The model was then extended to include calculation of linear growth rates with the result that it was found to give excellent agreement with a wide range of growth rate measurements in many different polymers. The growth rates are calculated for a crystal of the thickness expected at the particular crystallization temperature. There are two important processes whose rates have to be calculated: the rate at which a nucleus once formed spreads across the surface, #, and the rate at which new nuclei are formed, i. If g is sufficiently high with respect to i, then only one nucleus is active on the surface at a time. In this case, called regime I, the growth rate is dominated by the formation of new nuclei and is given by Gl = biL

(4-4)

where L is the length of the substrate. If there are several nuclei active at the same time the crystal growth rate is given

171

by (4-5)

This is known as regime II crystallization. If the rate of formation of new nuclei is sufficiently high that there is little chance for a nucleus to spread out sideways before another nucleus is formed next to it, the growth rate is given by (4-6)

GTTT = aib

The calculation of the growth rate then becomes the calculation of i, which has effectively already been done in the calculation of /. The result is that (4-7)

(4-8)

= GmoiSexp{-

4baae AFkT

(4-9)

where /? is given by

U*

1

(here h represents Planck's constant). The functional dependence of G upon T and the existence of the three regimes have been experimentally verified. The experiments will be discussed further below (Sec. 4.2.4.2). Many of the developments of the model have concentrated on refining its ability to predict growth rates. In particular much effort has been devoted to the absolute value of the growth rate, that is, to the term Go, for each of the growth regimes and on G0's molecular weight dependence. Important though these modifications are, they make little difference to the overall applicability of the model. There have been many different approaches to the best ways in which to de-

172

4 Crystallization and Morphology of Semicrystalline Polymers

velop the theory further (some of these may be found in: Lauritzen and Hoffman, 1973; Point, 1978, 1979; Hoffman et al., 1979; Point and Kovacs, 1980; DiMarzio and Guttman, 1982; Hoffman, 1983; Cheng and Wunderlich, 1986 a, b; and Hoffman and Miller, 1988). Different authors have been concerned to remove assumptions in the basic model that they consider to be incorrect or too simplistic. In general these modifications to the simple theory described above affect the way in which the rates i and g are calculated. In all cases the crystal thickness can be expressed as:

where /min is the minimum stable crystal thickness at the growth temperature and 5/ is a small correction term. Although there is sometimes considerable argument between various authors concerning the detail of these theories they all agree that the secondary nucleation approach gives an excellent agreement with the available experimental results on crystal thickness and growth rates. Thus far the model appears to have been remarkably successful considering the extreme simplicity of its assumptions. However one considerable challenge has been made to its most basic assumptions. As more and more polymer crystals were observed, grown from both solutions and from the melt, it became increasingly apparent that the original crystals, of the first experimental observations, which had flat, low index, surfaces were far from typical. In fact, most crystals have curved or high index faces. It was pointed out that in these cases the surfaces of the crystals should contain many steps. Each of these steps is a perfect site for a molecule to grow without the need to create any lateral crystal surface. In other words the niches, as they have become known, act as sites of growth

and there is no need for the formation of nuclei. Recently there have been several attempts to modify the model so that it is consistent with the observation of crystals with curved faces. Two approaches have been adopted (Mansfield, 1988,1990; Hoffman and Miller, 1989; Toda, 1991 a, b). Both the approaches involve the concept that the lateral spreading rate g decreases as the spreading nucleus grows larger. The first of these models considers that as the chains fold along a surface there is a strain induced in the crystal lattice by the folds. As the number of folds increases the strain accumulates, and induces a mismatch between the growing crystal layer and the underlying substrate, which in turn leads to a reduction in the lateral spreading rate of the growing crystal patch. It can be shown that this reduction can in turn lead to the crystal faces becoming curved. Indeed it is possible to predict the curvature from a knowledge of the variation of g with the size of the spreading patch. The second approach relies on the growth rates along different surfaces differing by a sufficient amount to cause mismatching at the sector boundaries. In this case it becomes necessary to solve the growth equations using moving boundary conditions and one solution provides for curved crystal faces. 4.1.4.2 Entropy Barrier Models

When the problem of curved crystal faces was raised some authors, particularly Sadler, believed that the problem was sufficiently serious to require a different model of polymer crystal growth (Sadler and Gilmer, 1984). The new model was based on the idea of rough surfaces and nucleation-free growth. A surface possesses a configurational entropy so that a perfectly

4.2 Crystallization and Morphology in Bulk Polymers

flat surface is extremely unlikely. A crystal surface should be expected to contain defects such as missing atoms and adsorbed atoms. In the case of small molecule crystals there is a temperature above which this surface entropy becomes so important that the surface can be described as rough. Crystal growth then can occur without the need to form a nucleus. It had been implicitly assumed that in the case of large molecules the roughening transition temperature would be so close to the melting point that this type of growth would not occur. However, Sadler (1987 a-c) showed that it was possible that the roughening transition temperature for polymeric crystals could lie in the range of temperatures used in crystallization studies. In essence three models of rough surface growth in polymer crystals have been developed. The first was computer simulation of a three dimensional crystal growth. Another computer model, which involves growth of a slice through a crystal, provides an essentially two-dimensional problem for which the rate equations can be written down and solved numerically. The final model is a much simplified two-dimensional model, which has been solved analytically. As is the case for the various nucleation theories, all three models provide similar results giving some confidence in the approach. The principal difficulty for these models, which allow growth of the crystal in three directions, is to restrict the growth in the chain direction. This restriction is provided for by the introduction of "pinning rules". In essence these rules state that once a chain has been incorporated into the crystal in a folded configuration it is unable to escape and become extended. The limited growth in the chain direction may then be considered as a reflection of the very low probability of many chains becoming ex-

173

tended at the growing surface before many different parts of them have joined the crystal. The models all produce much the same results as the nucleation theories; the crystal thickness is just a little larger than the minimum stable thickness and the growth rates can also be reproduced, although the rough surface models do not show the different growth regimes. Recent developments of the rough surface models are mostly concerned with their application to copolymers. In this area they have an intrinsic advantage over the nucleation models. The nucleation models tend to consider the chain as a smooth structureless object, or to consist of units each of which has the same heat of fusion and surface energies. On the other hand, the rough growth models that have largely been constructed around numerical and simulation techniques are more readily capable of modification to incorporate different monomers along a chain.

4.2 Crystallization and Morphology in Bulk Polymers 4.2.1 The Crystal Thickness 4.2.1.1 Dependence of Thickness on Crystallization Temperature

When the crystal thickness of melt crystallized polymers is measured no simple relation between crystal thickness and the supercooling is usually found. In most systems there is a broad range of thickness found, with different experiments in which the samples are prepared differently yielding quite widely varying data. Figure 4-10 shows a compilation of data reported in the literature for the crystal thickness of polyethylene as a function of its crystallization temperature. In each of the individual

174

200 116

4 Crystallization and Morphology of Semicrystalline Polymers

Figure 4-10. A compilation of literature data for the crystal thickness as a function of the crystallization temperature for polyethylene crystallized from the melt. 118

120

122 124 126 Temperature (°C)

128

reports the data usually lie on a smooth curve but the overall impression is that there is little correlation between the thickness and the crystallization temperature. In some of the individual reports where a strong correlation between thickness and temperature has been reported the thickness has been found to depend on the inverse supercooling as for the case of crystallization from dilute solutions. This can be seen in the example of poly(hydroxybutyrate) (PHB) illustrated in Fig. 4-11. However the actual thicknesses reported are significantly higher for the polymer crystallized from the melt, rather than from a solution, at the same supercooling. The impression may be gained that it is un-

130

132

likely that the crystallization mechanisms involved in crystallization from solution should be directly applicable to the case of melt crystallization. 4.2.1.2 Annealing Effects

When a crystalline sample is heated up to a temperature close to or above its original crystallization temperature, the lamellar thickness is usually seen to increase (Statton and Geil, 1960). This increase occurs for both solution and melt crystallized materials, the rate of thickening being however usually higher in solution crystallized material. When the sample is held at a constant temperature, the crystal thick -

Figure 4-11. A graph showing the variation in lamellar thickness for poly(hydroxybutyrate) crystallized from the melt as a function of the inverse supercooling. 0.004

0.006

0.008

0.010 0.012 0.014 0.016 Inverse Supercooling (K1)

0.018

0.020

4.2 Crystallization and Morphology in Bulk Polymers

ness increases logarithmically with time. Sometimes two different logarithmic thickening rates are seen, such that at short annealing times there is a faster thickening rate than at long times (Hoffman and Weeks, 1962). In addition to the isothermal annealing referred to above the lamellar thickness can also increase during the heating process. Such thickening is sometimes reflected in the appearance of multiple melting peaks in DSC experiments. It is a common effect in polyethylenes and many other polymers. Similar effects, which have been reported for some polyethers (Cheng etal, 1986; Cebe and Hong, 1986; Blundell, 1987; Bassett etal, 1988) have been ascribed to the existence of two distinct crystal forms and not to the transformation of thin lamellae into thick ones. An example showing the increase of the lamellar thickness, reflected in the changes in the low angle X-ray diffraction pattern during the heating of a PHB sample, is shown in Fig. 4-12. In this case the lamellar thickness increases continuously as the sample is heated from the crystallization temperature to the melting temperature, some 150 °C higher. In most systems the annealing process proceeds in this continuous fashion, how-

175

ever in a few examples where the polymer has a particularly long repeating unit so that there are only a few monomers in each crystalline stem the thickening can occur in sudden discrete jumps (Dreyfuss and Keller, 1970; Mitomo etal, 1978). This type of behavior, which has been reported for nylons, is illustrated in Fig. 4-13. The crystals always contain an integral number of monomer repeats. Since there are usually only a few such repeats there are subsidiary maxima in the wide angle X-ray pattern due to the limited lattice. On thickening an increase of one monomer in the crystal thickness changes the number, and position, of these subsidiary maxima so that the discrete thickening can easily be seen. Several possible mechanisms for this thickening process have been suggested. The approaches are mostly based on the concept of partial melting followed by recrystallization. The models differ in whether the melting is homogeneous affecting all lamellae equally (e.g., Zachmann, 1967; Fischer, 1969) or whether the melting occurs selectively, lamellae of lower stability melting (Kilian, 1969). Strobl et al. (1980) suggest that the amorphous thickness is important in determining when thickening will occur.

250

Figure 4-12. A graph showing how the lamellar thickness of a sample of poly(hydroxybutyrate) increases during heating from the crystallization temperature to the melt (Barker, 1992). 50

100 Temperature (°C)

150

200

176

4 Crystallization and Morphology of Semicrystalline Polymers

30

1 25 CO CO Q)

crystallization do however suggest that there should be some relation between the initial lamellar thickness, /*, before any thickening, and the crystallization temperature. As a result considerable effort has been expended to ascertain this initial crystal thickness.

O

o o

O

O

O

A

A

A

A

o o o °

c 20 A

A

• a

D

O

15

A

A

A

10

6

A

D

a

5

A

•

•

D

Q

a

•

4.2.1.3 Extrapolation to Initial Thickness

0

160

170 180 190 200 Annealing Temperature (°C)

210

Figure 4-13. A graph showing the step-wise increases in lamellar thickness during the annealing of nylon 6,6. The different symbols refer to the three long periods observed (after Mitomo et al., 1978).

It should be obvious that since the lamellar thickness can increase on reheating a sample to a temperature close to its original crystallization temperature, the thickness could also increase during isothermal crystallization itself. This process, which has become known as "isothermal thickening" will inevitably mean that it is impracticable to assign any relationship between the lamellar thickness and the crystallization temperature. The time of crystallization will itself affect the measured thickness. The models of polymer

There have been several attempts to determine whether there is a well defined initial fold length in crystallization from the melt. These have usually taken the form of attempts to measure the thickness as a function of crystallization time and extrapolate back to the initial value, /*. In the case of polyethylene, the most studied polymer, the earliest discussion of the problem, by Hoffman and Weeks (1962) suggested that the measured crystal thicknesses were almost double the initial values. Later workers used fast crystallization techniques, by greatly enhancing the nucleation rates, to obtain large amounts of comparatively young crystals. In these studies there was some evidence for an initial thickness, extrapolated from data such as that in Fig. 4-14. More recently, it has become possible to measure the crystal thickness in very sharp

Figure 4-14. A graph showing the isothermal thickening in polyethylene spherulites. Open symbols are for crystallization at 125 °C and closed symbols are for crystallization at 129 °C. The circles are from Raman data and the squares from X-ray data (Barham etal., 1981). 100 Time (s)

1000

10000

4.2 Crystallization and Morphology in Bulk Polymers

fractions of polyethylene during the process of crystallization itself using the high intensity X-rays provided by synchrotron radiation sources. Some typical results are shown in Fig. 4-15. It can be seen that the lamellar thickness doubles rapidly during crystallization and the lamellae with the doubled thickness continue to thicken by the familiar logarithmic process. Once several such measurements had been made it was possible to measure / * as a function of the crystallization temperature, and to compare the data with that from solution crystallized polymers. Figure 4-16 shows the remarkable coincidence of the data when the crystal thickness is plotted as a function of the supercooling.

123.4 °C

s

Vi V

s

48 s

VL^—

A

\V_3 2

177

s

16 s

^ —

4.2.1.4 Extended Chain Crystals

32.8 nm"1 16.4 nm"1

While the majority of polymer crystals consist of lamellae that are much thinner than the molecules are long there are some cases where the chains are extended in the crystals and the thickness of the crystal in

s—•

Figure 4-15. Selection of low angle X-ray diffractograms, recorded using synchrotron radiation, at various stages in the crystallization of a sharp fraction of polyethylene (Barham and Keller, 1989).

24 n 22

_20

1

f 18

•flf

lie

Figure 4-16. A graph showing the initial fold length, / *, of polyethylene crystallized from a variety of solvents (various symbols) and the melt (closed circles), as a function of the supercooling (Barham et al., 1985).

o

;s 14 E

12 •

105

10

15

20

25 30 35 40 Supercooling (°C)

45

50

55

60

178

4 Crystallization and Morphology of Semicrystalline Polymers

the chain direction may exceed the chain length. These extended chain type crystals can occur in both rigid and flexible polymers. They are not simply a consequence of a difficulty in folding the chains; some polymers can exhibit either chain folded or extended crystals depending on the crystallization conditions. It is clearly of interest to understand the mechanisms that permit crystals to grow in the chain direction. Since such crystals would appear to have no kinetic barrier to growth in the chain direction they seem to contradict the predictions of the nucleation theories outlined above. In the following sections the formation of extended chain crystals from polyethylene by crystallization at high pressure is described, followed by a summary of data on flexible chain polymer systems that can form extended chain crystals at atmospheric pressure.

Figure4-17. An electron micrograph of a fracture surface of polyethylene crystallized at high pressure showing extended chain morphology. The striations running across the lamellae are characteristic of the extended chain morphology. [Bassett (1981), by kind permission of the author and Cambridge University Press.]

Crystallization of Polyethylene at High Pressure When polyethylene is crystallized at high pressures (several kbar) an extended chain morphology is observed (Geil et al., 1964; Bassett and Turner, 1974a,b; Bassett et al., 1974). A typical crystal is shown in Fig. 4-17. Note the striations running parallel to the chain direction across the thickness of the crystals; these are characteristic of extended chain crystals. A significant feature of the crystallization behavior of polyethylene under pressure is that there is a hexagonal crystal phase, which becomes stable at pressures above ca. 4 kbar. The region where this hexagonal phase is stable can be seen from the phase diagram, which is shown schematically in Fig. 4-18. Whenever crystallization takes place in the region where the hexagonal form is the thermodynamically stable phase, extended chain crystals are obtained. This observa-

280

crystal

120

1 2

3

4

5

6

Pressure (kbar) Figure 4-18. Phase diagram for polyethylene under pressure showing hexagonal and orthorhombic phases.

4.2 Crystallization and Morphology in Bulk Polymers

179

tion has led to the concept of "sliding diffusion" (Hikosaka, 1987, 1990; Hikosaka et al., 1992); it is supposed that the lattice forces between crystalline stems in the hexagonal phase are sufficiently low that they provide little barrier to the diffusion of molecules through the crystal along their axes. Thus a crystal that initially grows in a chain folded manner can continue to grow in the chain direction without limit as the folded molecules diffuse through the lattice and new molecules diffuse into the crystal to occupy the resulting empty sites. The process is illustrated in the sketch of Fig. 4-19. When crystals whose growth has been arrested by pressure quenching are examined in the electron microscope they can be seen to have tapered ends (see Fig. 4-20). At the growth tips the thickness of these crystals may be much less than the length of the molecules suggesting that the crystallization does in fact begin in the chain folded form with thickening occurring behind the growth front. Extended chain crystals can also be grown in regions of the phase diagram where the orthorhombic form is the stable

' Thickening growth

U

^ S e c o n d a r y nucleus

growth

Chain sliding diffusion

Figure 4-19. Sketch illustrating process of sliding diffusion (sketch by Hikosaka, reproduced from Keller, 1991).

Figure 4-20. Electron micrograph of a chain extended crystal of polyethylene showing the tapering ends. N. B. the crystal was grown in hexagonal part of the phase diagram (Rastogi et al., 1992).

phase. An example of such a crystal is shown in Fig. 4-21. Recently it has been suggested, based on an optical criterion for distinguishing hexagonal and orthorhombic crystal forms (Bassett, 1982; Rastogi etal., 1992), that all the extended chain crystals actually grew in the hexagonal phase, despite the fact that the orthorhombic phase was the thermodynamically more stable. It is further suggested that

180

4 Crystallization and Morphology of Semicrystalline Polymers

diffuse along their axes, is necessary for the formation of extended chain polymer crystals, and the basic initial crystallization occurs via the more usual chain folded process. Crystallization at Atmospheric Pressure

Figure 4-21. Electron micrograph of an extended chain polyethylene crystal grown in orthorhombic part of phase diagram (Hikosaka et al., 1992).

once a transformation takes place from the hexagonal form to the orthorhombic form, the growth in the chain direction stops and growth normal to the chain direction is dramatically slowed. Figure 4-22 shows a series of optical micrographs of a sample of polyethylene crystallizing at high pressure but in the region of the phase diagram where the orthorhombic phase should be stable. At first, by the optical criterion, all the crystals are judged to be in the hexagonal form; as they continue to grow some transform into the orthorhombic form and growth can be seen to stop. In summary, it appears that although when observed at atmospheric pressure the extended chain crystals of polyethylene are always in the orthorhombic crystal form, they probably grew in the hexagonal form where the weak lattice forces permitted easy diffusion of the chains along their length in the crystal. Thus it may be that a mobile phase, in which the molecules can

There are several polymers that are able to form extended chain crystals during growth at atmospheric pressure. Figure 4-23 shows crystals of polybutadiene grown at atmospheric pressure, which are at least as thick as the molecules are long. In such systems the X-ray diffraction patterns from the crystals often show few discrete Bragg reflections, indicating a poor lattice. This in turn suggests that the lattice forces between adjacent crystalline stems are small so that a process of sliding diffusion may take place during the growth of extended chain crystals. Polybutadiene shows an additional feature: it has two crystal forms, a hexagonal form, which is stable at high temperatures and a monoclinic form stable at lower temperature. When a sample originally crystallized in the chain folded monoclinic form is heated it may transform into the hexagonal form. Once this transformation happens, the lamellae are found to thicken very rapidly and the chains become extended. The temperature at which the hexagonal form becomes more stable than the monoclinic form depends on the thickness of the crystal (due to the high fold surface free energy). Thus a thin monoclinic crystal will transform into a hexagonal crystal at a lower temperature than a thick one. This size dependence of thermodynamic stability has a remarkable consequence. When an initially thin, chain folded, monoclinic crystal is heated it transforms into the hexagonal phase and begins to grow thicker. However, as the crystal grows thicker the

4.2 Crystallization and Morphology in Bulk Polymers

181

Figure 4-22. A series of optical micrographs showing the growth of polyethylene crystals at high pressures, in the orthorhombic part of the phase diagram. Note that when the appearance of a crystal changes the growth stops. The change for the four crystals, A, B, C, and D, is indicated with a change from a solid to a broken arrow. This change in appearance has been associated with the transformation of the crystal from the hexagonal to the orthorhombic form. (From Hikosaka et al., 1992.)

monoclinic phase may become more stable than the hexagonal phase. Once this happens the crystal will transform back into the monoclinic form. At the same time the greater lattice forces of the monoclinic form will prevent further thickening. This remarkable process has, in fact, been observed with the aid of the synchrotron radiation source (Rastogi and Ungar, 1992) and is illustrated in Fig. 4-24. Figure 4-23. An electron micrograph showing polybutadiene extended chain crystals grown at atmospheric pressure (Hikosaka et al., 1992).

182

4 Crystallization and Morphology of Semicrystalline Polymers

Figure 4-24. X-ray diffractograms of thickening in polybutadiene (Rastogi and Ungar, 1992).

4.2.1.5 Role of Mobile Phases in Determining the Crystal Thickness The observations presented in the previous section illustrate that it is possible for crystals to grow in one form and only transform into a more stable form once a critical size has been exceeded. In practice the growth may be faster in such a transient mobile phase, as illustrated by the examples of growth of polyethylene at high pressures, where after the transformation of the crystals into the stable orthorhombic form the growth is arrested. It is possible that there are many other cases where growth occurs through a transient mobile phase, which have not been recognized as yet. If the transient phase exists only as a small zone at the growth front then its detection would not be easy and it is unlikely to have been noticed. In any case where growth does occur through a transient mobile phase the observed crystal thickness may be largely determined not by the thickness at the growth front, but by the thickness at which

the transformation from the transient phase to the stable one occurs. Thus although the kinetic, nucleation based, models of polymer growth may still be expected to apply, they would predict the thickness at the growth front not that at the transition from mobile to stable crystal form. The predictions of growth rates would be unaffected, although the numerical values of the thermodynamic quantities (heats of fusion, surface energies, etc.) should be modified to be those appropriate to the mobile, transient, phase rather than the stable phase. It is possible to draw up some criteria to determine whether growth is possible in a mobile phase. Consider the free energy diagram of Fig. 4-25. Any crystal will decrease its free energy as it grows thicker (since the surface to volume ratio decreases). If there is a mesophase however, which is of intermediate stability between the melt and the normal crystal, then a small crystal may have a lower free energy in the intermediate crystal form. Thus it is possible that growth will begin with a thin crystal of

4.2 Crystallization and Morphology in Bulk Polymers

orthorhombic hexa gonal

183

4.2.2 Development and Structure of Aggregates

O

c

4.2.2.1 Isolated Crystals from the Melt

(D

.

.

.

-

-

-

•

"

" ^ \ melt

1/1 Figure 4-25. Size dependent free energy diagram illustrating how the hexagonal phase may be stable in small crystals. The enthalpy of a solid phase crystalline phase with thickness / has the form H = Ho — 2 a/I. If a is lower for the orthorhombic than the hexagonal form, then the hexagonal form may be more stable in small crystals.

thickness /x (the minimum stable thickness for a crystal of the intermediate form) in the intermediate phase and the crystal will thicken rapidly behind the growth front; once the thickness of the crystal exceeds l2 (the thickness at which crystals of the intermediate and stable form have the same free energy) it will transform into the stable phase. It can be shown that if the ratio of the enthalpy of fusion to the fold surface free energy of the intermediate phase is less than that of the equilibrium stable phase, then there will be a temperature at which the conditions for growth via the mobile intermediate phase will be favored (Goldbeck-Wood, 1991). This concept of crystal growth occurring through a transient mobile phase is comparatively new. Certainly such a mechanism appears to be relevant to the crystallization of polyethylene at elevated pressures and to polybutadiene at atmospheric pressure. However the overall importance of this growth mechanism remains to be determined.

The basic building blocks for the crystal aggregates seen in melt processed polymers are mostly lamellar crystals. There are several instances where it has proved possible to obtain isolated single crystals during melt crystallization. The study of these crystals and the ways in which they branch through dislocations provides the basis for the understanding of the internal architecture of spherulites and other crystal aggregates. The most studied system is probably polyethylene oxide) (Kovacs et al., 1975, 1977; Kovacs and Straupe, 1979, 1980; Cheng and Wunderlich, 1986a,b), which will crystallize from the melt in the form of single crystals that are large enough to examine whilst growing in the optical microscope. These systems are however of a rather low molecular weight and may not be completely typical of high polymer behavior. In most other systems single crystals are only obtained by extraction from the polymer after crystallization is completed, and they can usually only be seen then with the aid of the electron microscope. The detailed description of isolated crystals is given in some detail in Chap. 3 of this Volume. Examples of isolated crystals grown from the melt are shown in Figs. 3-22 and 3-23 of Chap. 3. The important observations for the purpose of this chapter are that these crystals usually contain screw dislocations that provide a site for the branching out of new crystals. It is the manner in which the branching takes place that determines the appearance of the aggregates, which form later.

184

4 Crystallization and Morphology of Semicrystalline Polymers

4.2.2.2 Spherulites and Dominant Lamellae

The underlying architecture of spherulites has been studied in depth by Bassett and co-workers over the past decade (see for example: Bassett and Hodge, 1978 a, b; Bassett, 1981; Bassett and Olley, 1984). The principal observations they have made show that there are two types of lamellae present in the spherulites. Dominant lamellae can be seen to lay down the scaffolding onto which secondary lamellae subsequently crystallize. Figure 4-26 shows a section through a polyethylene spherulite looking down a radius. The thick ridged sheets are the dominant lamellae. The fact that these lamellae are much thicker than the others shows that they are older (they have had longer to thicken at the crystallization temperature). Since they are the oldest lamellae it follows that they form the basic skeleton for the whole spherulite. A selection of micrographs from the work of Bassett et al. is shown in Fig. 4-27,

to illustrate the range of morphologies that can be adopted by these dominant lamellae depending on the precise crystallization conditions. There are two particular questions that should be addressed. First, why do the initially flat lamellae splay out to form approximately spherical objects? And secondly why do the radiating lamellae so often exhibit a regular twist? The answers to both questions may lie in the nature of the screw dislocations present in the crystals. When a crystal branches at a screw dislocation there is a mismatch between the mother and daughter crystals. Thus there may be an energy penalty for them to grow in contact with one another. Consequently the daughter lamellae may tend to splay away from one another. As more and more crystals are formed at the dislocation, a sheaf-like structure can be formed. It has been reported (Patel and Bassett, 1991) that the dislocations are usually all of one sense. It is believed that this is a conse-

Figure 4-26. Electron micrographs showing dominant lamellae looking down the radius of polyethylene spherulites. [Bassett and Hodge (1978 b), by kind permission of the authors and The Royal Society.]

4.2 Crystallization and Morphology in Bulk Polymers

Figure 4-27. A selection of electron micrographs illustrating dominant lamellae in polyethylene spherulites (a)-(c) (taken from Bassett and Hodge, 1978) and polypropylene spherulites, (d), (e) [from Bassett and Olley (1984), Polymer 25; by kind permission of the authors and the publishers, © Butterworth Heinemann Ltd.].

185

quence of the chain tilt within the lamellae. Figure 3-24 of Chap. 3 illustrates how this may arise. As more crystals are nucleated and themselves form screw dislocations so a net twist can be built up. Such a twist in the radiating dominant lamellae will affect the orientation of the secondary, infilling, lamellae and may lead to the periodic banding on the spherulites (Schultz and Kinlock, 1969). An alternative model for the twisting of the lamellae and the consequent banding has been proposed by Keith and Padden (1963,1964 a, b). The model is based on the diffusion of impurities and on the strain at the lamellar surfaces caused by the chain folds. More recently the model has been revised to include the effects of reptation (Keith and Padden, 1986, 1987 a, b). However, the model still is difficult to reconcile fully with all the experimental evidence. Banded spherulites can be grown from high purity polymers such as PHB, and the effects of surface strain should become less pronounced as the lamellar thickness is increased. Also, in some systems the twisting can become more pronounced when the lamellae are thicker.

186

4 Crystallization and Morphology of Semicrystalline Polymers

4.2.3 Spherulitic Growth Rates and Regimes 4.2.3.1 Measurements of Growth Rates

The growth rates of spherulites crystallizing isothermally may easily be measured using direct observation in an optical microscope, provided that the growth rates are not very high, in which case fast recording methods have to be used, or so slow that there are problems of the long term stability of the polymer and the hot stage in use. It is found in all cases studied that the spherulites have a constant, linear, growth rate, the growth rate depending only on the temperature of crystallization. The linearity of the growth may be seen from Fig. 4-5, which shows some polyethylene spherulites at various stages during their growth. Most polymers have a low growth rate at low temperatures; the growth rate is limited by the slow diffusion of molecules to the growth front. The growth rate is also low at high temperatures because the driving force for crystallization is low. The result is that when the growth rate is plotted against the crystallization temperature a bell shaped curve is normally found. A typical example is shown for the case of PHB in Fig. 4-28. The growth rate is also found to depend on the molecular weight of the polymer. The diffusion rates depend strongly on the molecular weight (see, e.g., Chap. 9 of this Volume) so it is not surprising to find that a polymer with a higher molecular weight will crystallize more slowly. The nucleation based, kinetic, models for polymer crystallization give explicit expressions for the growth rate as described in Sec. 4.1.4.1 above, provided that there is a fixed substrate length, L. Without a fixed, invariant substrate length, the growth rate in regime I would not be a constant but

would increase as the size of the crystals increases (Hoffman et al., 1976). There are three growth rate equations referring to the three growth regimes [Eqs. (4-7) to (4-9)]. These may be tested by suitable logarithmic plots. Figures 4-29 and 4-30 show some growth rate data for polyethylene and for PHB plotted in a suitable manner. The data for PHB, Fig. 4-29, show two distinct slopes with a ratio of about 2 between them. This indicates that crystallization occurs in two different regimes depending on the crystallization. Above ca. 120 °C the crystallization is in regime II and below ca. 115 °C it is in regime III. Polyethylene is difficult to crystallize at very high supercoolings, unlike PHB, so that in this case it is usual to observe regime I and regime II behavior. Figure 4-30 shows some data for a polyethylene fraction. The transitions from regime I to regime II are seen by the changes in the slopes of the lines. The transition temperatures as well as the absolute growth rates have been found to depend on the molecular weights (Hoffman et al., 1975). Very few measurements of the growth rate of polyethylene at low crystallization temperatures have been made using special precautions to remove any nuclei and hence reduce the likelihood of crystallization occurring during cooling. The rates observed are very high and consequently difficult to measure and have large errors associated with them. However it can be seen (Fig. 4-31) that the growth rates were significantly higher than would be expected from extrapolation of the lines in Fig. 4-30. The higher growth rates fit in with the idea that regime III crystallization starts for polyethylene at a temperature around 100 °C. The excellent agreement between the nucleation theories and the measured growth rate data, especially the identification of

4.2 Crystallization and Morphology in Bulk Polymers

187

0) DC

O

6

Figure 4-28. A graph showing the variation of spherulitic growth rate with crystallization temperature for PHB (data from Barham et al., 1984). 20

40

60 80 100 120 Temperature (°C)

140

160

180

Figure 4-29. The same data as shown in Fig. 4-28, replotted as log e G+ U*/ [R(T-TJ] against 1/(TAT/). The factor / is used to scale the enthalpy of fusion with crystallization temperature (data from Barham et al., 1984). 0.00E+00

4.00E-05

8.00E-05

1.20E-04

1/(TATf)

Figure 4-30. A graph showing the variation of the growth rate, G, of polyethylene in the form of a plot of In GH vs. l/(TATf). The factor / is used to scale the enthalpy of fusion with crystallization temperature. (After Hoffman et al., 1969.) 1.20E-04

1.40E-04

1.60E-04

1.80E-04

1/(TATf)

three growth regimes, tends to give strong credence to the idea that the nucleation model is appropriate to crystallization from the melt. Furthermore, the data on the initial fold lengths are also found to correspond closely with those predicted by the model.

However a note of caution should be sounded. The fold lengths measured are just above the minimum stable length, which may be accounted for by other mechanisms, for example, the entropy barrier or surface roughness models. The identification of different growth regimes relies

188

10 4.00E-05

4 Crystallization and Morphology of Semicrystalline Polymers

Figure 4-31. The same plot as in Fig. 4-30 but with the addition of the available data on the growth rates of polyethylene at low crystallization temperatures. 8.00E-05

1.20E-04 1/(TATf)

1.60E-04

to some extent on the judicious choice of the thermodynamic constants, U* and T^, used in the growth rate plots. By using different values for these variables, or by using a different way of approximating the heat of fusion at high supercoolings, it is possible to lose the linearity of the plots or to change significantly the ratios of the slopes. In fact it is possible to redraw the graph shown in Fig. 4-29, which shows two straight lines in regimes II and III, so that there is a single continuous curve. It is important therefore to look for independent evidence that there are indeed three growth regimes. An obvious choice is to examine the morphology of crystals grown in apparently different regimes. There should be a difference in the growth surfaces between different growth regimes. For example regime I crystals should have more or less flat surfaces, while regime III crystals should appear ragged. 4.2.3.2 Morphology of Crystals in Different Regimes

In their original work on growth rates in polyethylene, Hoffman et al. (1975) not only found growth rate regimes, but also showed differences in the morphology of the crystal aggregates. In regime I they usually observed axialites and in regime II

spherulites. In the case of PHB cited above the spherulites grown in regime II have no concentric banding while those grown in regime III show clear regular banding. These gross optical features probably arise from some more fundamental differences that remain to be identified. Recently Toda (1991 a, b) has found some more specific differences between individual crystals grown from the melt in regimes land II for polyethylene. In regime I, the crystals have a lenticular shape and in regime II they appear as truncated lozenges. At first sight these morphologies are not what should be expected from the underlying mechanisms. Regime I, which should occur when there is only a single active nucleus, would be expected to produce crystals with flat surfaces. Toda explains this as being the result of growth in regime I occurring along small {110} faces at the ends of the elongated crystals, and shows that the regime transition occurs when the growth rates of the {200} and {110} faces become comparable. 4.2.4 Nucleation Behavior

The nucleation process is of the greatest importance in determining the crystalline morphology. The number and distribution of the nuclei can profoundly affect the ori-

4.2 Crystallization and Morphology in Bulk Polymers

entation and properties of the material after solidification. A trivial example is that if the number of nuclei is very low then in a real production process, the cycle time becomes rather long, the crystallite size becomes very large and the materials are usually brittle. Indeed, in most of the examples where there has been a deliberate attempt to control the properties of polymeric materials through the morphology, it has been achieved through control of the nucleation process. Accordingly an understanding of the various types of nucleation that occur in polymers is necessary for the proper appreciation of the relationships between structure and properties. 4.2.4.1 Outline of Nucleation Theory The nucleation process for polymer crystals is usually described in terms of the balance of free energy between the creation of new surface and new crystal. When a nucleus is formed there is an initial increase in the free energy due to the need to create new surface. As the nucleus grows larger the free energy eventually reaches a maximum value and then decreases until it becomes negative leaving a stable nucleus. Nucleation theories thus depend on being able to determine the free energy of a growing nucleus along any particular path. The simplest case is homogeneous nucleation where no external nucleating particles are involved. In this case we can write down the surface free energies of the growing surfaces (usually 3 different energies are used and an orthorhombic shaped nucleus is considered, but any shape can be accommodated) and the free energy associated with the crystallization in terms of the size of the crystal, a, b, c. The nucleus can grow along any or all of the free directions; different paths lead to different sizes and shapes of nucleus at which the free energy

189

reaches its maximum. The critical nucleus size and shape is then defined as that which passes through the lowest such maximum. It may readily be found by partial differentiation. The rate of nucleation, /, may then be determined from the critical free energy, A0*, that is, the height of the free energy maximum at the critical nucleus: (4-10) where j$ is given by kT

r

£_i

L

R(T-Tj]

or homogeneous nucleation: = 32

AF = AH

a2 a. AF2' AT

where (4-11)

Thus if homogeneous nucleation occurs a plot of log/ against 1/(TAT2) should be linear. In practice it is most unusual to find such behavior. Whenever there is a foreign surface present it will affect the free energy balance, usually so as to increase the rate of nucleation, thus preventing the observation of true homogeneous nucleation. The most significant effect of a surface is to modify the surface free energy. Thus for a nucleus growing on a plane surface we must consider the free energy of the surface with respect to the crystal and to the melt, so that the critical free energy becomes A0* - 16

<7(Te A<7 2

AF

(4-12)

where ACT = o + aos — as, aos is the interfacial energy between the foreign surface and the melt, and as is the interfacial energy between the crystal and the foreign surface.

190

4 Crystallization and Morphology of Semicrystalline Polymers

Note that the same dependence of nucleation rate on the supercooling appears in these equations as in those for homogeneous nucleation, but since the free energy barrier can be much lower, the overall rates are likely to be several orders of magnitude higher for this heterogeneous nucleation, which will thus swamp any homogeneous nucleation and prevent its observation. If the surface of the foreign particle is identical to that of the crystal [or the term A(T, in Eq. (4-12) above, goes to zero] then there is no energy penalty for forming new crystal surface parallel to foreign surface and the above analysis must be modified to a two dimensional approach. This is the same secondary nucleation process described in the section on crystallization and leads to a less strong dependence of nucleation rate on temperature, that is (4-13) AF A surface can also affect the entropy of the melt in polymeric systems. Consider a molecule near the surface: some of the configurations it could adopt in the absence of the surface are now forbidden to it since it cannot cross the surface. This constraint is particularly important if the molecule is adsorbed onto the surface. If the entropy of the melt is reduced then the equilibrium melting point of the crystal is increased (T£ is given by the ratio of the enthalpy and entropy differences between the melt and the crystal). The modified melting temperature in turn leads to a modification to the equations above by providing for an increased apparent supercooling. This change again acts to increase the nucleation rate at any particular temperature. A further effect of the presence of a surface is that if the molecules are adsorbed onto the surface, or have an increased density near to the

surface (which can come about from the entropic repulsion of the molecules from the surface), then the probability of formation of a nucleus is increased. The increase comes from the fact that the molecules are trapped at the surface and lie close to one another; thus the need to bring molecules together to form a nucleus is reduced, that is, the transport term becomes less important. 4.2.4.2 Experimental Observations of Different Types of Nucleation

In order to determine the effectiveness of any additive as a nucleant it is necessary to know the propensity for the polymer to nucleate without the nucleant. In practice it is advantageous if the rate of homogeneous nucleation is known. This is not usually the case since the presence of almost any foreign particle can produce nucleation. A technique involving the division of the polymer into small droplets has been developed (Turnbull and Cormia, 1961; Cormia et al., 1962). The principle behind this technique is that there is only a finite number of foreign particles present in a sample so that if the number of droplets exceeds the number of particles then some of them contain no nucleants and will undergo homogeneous nucleation. The technique is not easy to use, but some data have been obtained for polyethylene (Gornick et al., 1967; Hoffman et al., 1969; Ross and Frolen, 1975; Barham et al., 1987) which appear to show homogeneous nucleation, although it is always possible that nucleation occurs at the interface between the droplets and the medium on which they are supported. It has been assumed that by choosing media with similar properties to the molten polymer, this problem should be minimized. Several media have been used and the one in which

4.2 Crystallization and Morphology in Bulk Polymers

the least nucleation is seen is taken to be the closest to homogeneous nucleation. A plot of the nucleation rate as a function of the crystallization temperature for what is believed to be homogeneous nucleation in polyethylene is shown in Fig. 4-32. The slope of this plot provides a measure of cr2ae, which in combination with growth rate data can be used to estimate a. Such estimates are of some importance in the crystallization theories. The effectiveness of particular materials as nucleants can be investigated using the droplet technique. If the droplets are supported on a substrate made from the potential nucleant then differences in the nucleation behavior due to the substrate can be assessed. A convenient way to determine the nucleating effectiveness of a substrate is by defining the "nucleation temperature", Tn, by the temperature at which those droplets containing no heterogeneous nucleants crystallize during cooling from the melt at a particular rate (Barham etal., 1987). Then substrates that give higher Tn values should be more effective nucleants. In some polymers the background rate of nucleation is sufficiently low that it is possible that homogeneous nucleation may be observed. An example of such a polymer is PHB, which due to its bacterial

191

origin can be exceptionally pure, with no foreign particles present in it. Because of this high purity PHB has proven to be a particularly useful model material for polymer nucleation studies (Barham, 1984). It has been possible to observe several different types of nucleation in PHB. The polymer on its own apparently undergoes homogeneous nucleation; the addition of small particles (e.g., talc) causes nucleation to be favored due to the constraints imposed on the molecules by the surfaces leading to a decrease in the melt entropy. The addition of specific nitrogen containing compounds, which possess a lattice match with the PHB crystal, can lead to secondary nucleation at the interface. Nucleation from a self seeding process where the polymer is not fully melted has also been seen in this system. Figure 4-33 shows the nucleation rates as a function of temperature for PHB without additives and with talc, and saccharin added. The large increase in the rates seen in the presence of the additives gives witness to the extraordinarily low nucleation rate of the pure polymer. In order to examine the underlying causes of the nucleation the data in Fig. 4-33 are replotted in Figs. 4-34 and 4-35, according to Eq. (4-16), above. The data for pure polymer are consistent with the equations for homoge-

Figure 4-32. The nucleation rate for polyethylene, plotted according to Eq. (4-11). This is believed to represent homogeneous nucleation (after Ross and Frolen, 1975). 8.6E-07

8.8E-07 9.0E-07 9.2E-07 1/[T(AT)¥]

9.4E-07

9.6E-07

192

4 Crystallization and Morphology of Semicrystalline Polymers

10000 1000 100

• Pure PHB A PHB + Talc o PHB + Saccharin

Figure 4-33. Three graphs showing the variation of nucleation rate with crystallization temperature in PHB. Circles: pure polymer with no additives; triangles: polymer with 1 talc added; squares: polymer with 1 % saccharin added. (After Barham, 1984.)

10 1 0.1 0.01 0.001 0

20

40

60 80 100 Temperature (°C)

120

140

160

Figure 4-34. A graph of

\ogeI+U*/[R(T-TJ\

against

1/[T(AT) 2 / 2 ] for pure PHB (Barham, 1984). The factor / is used to scale the enthalpy of fusion with crystallization temperature. 2E-07

4E-07 6E-07 1/[T(AT)2f2]

8E-07

1E-06

IT D

12

•

Figure 4-35. A graph of

10

\ogcI+U*/[R(T-TJ]

D

8 D

39(D

+

6 D

4

2 0 5E-08

D

1E-07 1.5E-07 2E-07 2.5E-07 3E-07 3.5E-07 4E-07 1/[T(AT)2f2]

against

1/[T(AT) 2 / 2 ] for PHB to which 1% talc had been added; the supercoolings were calculated using a raised value for T£, 495 K (Barham, 1984). The factor / is used to scale the enthalpy of fusion with crystallization temperature.

193

4.2 Crystallization and Morphology in Bulk Polymers 18 16

D D

-14

i

D

12

D

Figure 4-36. A graph of

logj+U*/[R(T-TJ]

•

^10

i 8

D

I 6 4 2 2E-05

D I

2.5E-05

I

I

3E-05 3.5E-05 1/(TATf)

I

4E-05

neous nucleation, Fig. 4-34, while those for the PHB with talc added fit much better if a higher melting temperature is assumed, Fig. 4-35. When saccharin is added secondary nucleation is indicated as shown by Fig. 4-36. Full details of all these experiments are given in Barham (1984). The same type of nucleation as for saccharin has been observed for other compounds including boron nitride (Organ et al., 1991) and ammonium chloride (Organ and Barham, 1992). Although the exact crystal faces that promote the most nucleation have not been identified in all these cases, it is notable that all these compounds contain nitrogen and have a lattice spacing that matches the c-axis spacing of PHB. Furthermore similar compounds that also have a lattice match but do not contain nitrogen fail to act as nucleants (Organ and Barham, 1992). Thus it has been suggested that there is some specific interaction between the polymer and the nitrogen atoms in the nucleant that serves to promote the epitaxial growth of PHB crystals on the surface of the nucleant crystals. There are very few cases in which epitaxy has been definitely confirmed in the nucleation of polymers. Perhaps the best known example is the epitaxial growth of

against

1/(TAT/) for PHB to which 1% saccharin had been added (after Barham, 1984). The factor / is used to scale the enthalpy of fusion with crystallization temperature.

4.5E

polyethylene on single crystals of salts, when the polyethylene grows in a modified crystal lattice under the influence of the lattice forces from the salt (Mauritz et al., 1973). Recently the use of the atomic force microscope has permitted the investigation of the details of the interactions between polymer and nucleant at the interface (Lotz et al., 1991; Chap. 3 of this Volume). In the case of polypropylene the importance of the positions of the methyl groups has been stressed. 4.2.4.3 Chemical Nucleation

A further method that has been successfully employed to enhance the nucleation of polyesters in particular, relies on chemical rather than physical processes. Legras and co-workers (Legras et al., 1983, 1984, 1986) have shown that by adding suitable chemicals that apparently are dissolved in the polymer melt it is possible to induce reactions that serve to create crosslinks which are able to hold several molecules together and thus increase the probability of nucleation occurring in PET and more recently in PEEK (Legras et al., 1990). A similar technique has been reported for PHB (Tomka and Maclwain-Sinn, 1977): where two chemicals are added to the poly-

194

4 Crystallization and Morphology of Semicrystalline Polymers

mer, neither of these two materials acts as a nucleant on its own. However there can be a reaction between the two components to produce ammonium chloride, which is known to act as a nucleant. In this case the advantage of the reaction occurring in the polymer melt is probably that it produces particularly small nucleating particles.

4.3 Relationships Between Structure and Properties 4,3.1 Models Used to Predict Mechanical Properties In order to appreciate the influences that the structure and morphology of a material have on its properties it is useful to have some simplified models that can be used to give general guidance as to the mechanical properties for particular morphologies. In practice, as we shall see, there are some very simple models. However, while useful, these can often oversimplify the situation so much that they can actually appear to predict the opposite of the truth. The more detailed models, which allow for proper calculation of stress distribution and transfer, are by contrast usually too complex to be applied and contain so many variables (all the components of the compliance tensors of all the phases are needed) that their practical application is only possible for those deeply immersed in the subject. Accordingly, there is no fully satisfactory method of directly predicting how changes to a particular morphology will affect the mechanical properties, or of predicting what morphology would be needed to achieve particular properties. Nevertheless the simple models available to us do provide a basis for some simple relationships and it is worthwhile having a working knowledge of them.

4.3.1.1 Simple Models Utility and Limitations Semicrystalline polymers may be considered to consist of two distinct phases: the crystalline and the amorphous regions. A simple mechanical model then assigns moduli (or compliances) to these two phases, draws boundaries around them, and calculates the stiffness of the composite material, using simple models such as plane stress or strain to give the conditions of stress transfer. The most commonly used such models are those due to Takayanagi. These models are extensively used in fiber studies where the overall orientation permits the use of only one elastic constant for each phase without too large an error. A detailed description of such models is given in Chap. 7 of this Volume. Figure 4-37 illustrates two simplified Takayanagi models. There are two phases in each model; in one case the crystals are envisaged as lamellar in nature with the amorphous regions connecting in series along the fiber direction, while in the other extreme the crystals are Takayanagi models parallel

series , = c/Ec+(1-c)/Ea

= cEc+(1-c)Ea

amorphous phases

crystalline phases Figure 4-37. Takayanagi models series and parallel, together with the equations giving the modulus, E{, in terms of the crystal and amorphous moduli, Ec and

4.3 Relationships Between Structure and Properties

considered to be fibrous with the amorphous material connecting in parallel. The stiffnesses of the two models are very different; if the crystalline regions have a larger modulus than the amorphous regions then the parallel model will be much stiffer than the series one. These simple models can be useful in understanding why some types of fiber are very much stiffer than others. However they usually fail if more detailed questions are asked of them. For example, they have been used to try to explain the increase in modulus with draw ratio in fibers. In this case a model was proposed by Peterlin (1979) in which there was a mixture of series and parallel coupling of the phases; the idea being that as the fibers are drawn, more "tie molecules" are created between the "fibrils", which are mostly crystalline but have an underlying lamellar structure. Figure 4-38 illustrates the three phase model, which comes from this assignment of phases. The problem in this case is that with three phases it is no longer possible to ignore the actual mechanism of stress transfer across the boundaries be-

Two ways to consider a three phase Takayanagi model

High stiffness

Low stiffness

Figure 4-38. A three phase Takayanagi model, showing how the modulus can be calculated using two different methods, giving different results.

195

tween the phases. As is illustrated in Fig. 4-38 two extreme assumptions may be made that give totally different results. In systems that are not oriented, the Takayanagi models become more difficult to apply, but they can still be useful in giving an overall impression of how the arrangement of crystalline and amorphous phases affects the material properties. The difficult step is to find a unit in the structure that may be treated with a two phase model. Once such a structure has been chosen, then it is possible to imagine the whole material as being made up from such units in all orientations. Thus if we wish to determine the properties of the whole material it would be necessary to calculate the properties of the basic unit in several different directions and find the appropriate average. In practice this would be making too much of the simple model. Nevertheless it is clear that improving the properties of the basic unit would improve the properties of the whole material A simple example is to consider a spherulitic material, where the basic unit might be a stack of lamellae which could easily be represented by a simple two phase model (Fig. 4-37). The stiffness of this unit would be greatest perpendicular to the lamellae (along the chains in the crystals) and least parallel to the lamellae. Improvement in the stiffness of the basic unit could then most easily be achieved by increasing the proportion in the crystalline phase, or by creating more links between adjacent lamellae so as to improve the stress transfer between them. 4.3.1.2 Problems Associated with Realistic Models

To construct a realistic model for the mechanical properties of a semicrystalline polymer it is necessary first to determine

196

4 Crystallization and Morphology of Semicrystalline Polymers

the full stiffness tensor of the crystalline regions and the noncrystalline regions, bearing in mind that there may be more than one type of noncrystalline region present. For example it may be necessary to consider a truly amorphous material (which would be isotropic) and an anisotropic noncrystalline phase at the surface of the crystalline lamellae. Once these stiffness tensors are known the next problem is to decide on the boundary conditions at the interfaces between the various phases. This process is closely related to the calculation of the properties of composite materials and many of the techniques developed in that area have been adapted to semicrystalline polymers, where the crystalline regions are regarded as the reinforcing elements in a matrix of amorphous material. For a more detailed discussion see, for example, the review by Arridge and Barham (1982). In practice it is usual to determine some symmetry in the whole system and set up a unit cell which consists of several phases and whose properties can be calculated. It is usually too difficult to perform such a detailed analysis so that some approximations or assumptions must be made. Several attempts have been made to model spherulitic systems using a variety of assumptions. These usually provide upper and lower bounds on the modulus of the composite material. Such models can be very helpful in the understanding of the basic mechanics of these materials. However, they remain very difficult to apply to real situations, as there are not just a few simple parameters that can easily be related to the morphology; rather the models contain many variables whose dependence on the structure may not be obvious.

4.3.1.3 Some Simple Relations It is worth noting that, despite the apparent difficulty in finding appropriate models to predict mechanical properties in terms of the morphology, there are several straightforward simple relationships, which can easily be seen and are generally true. For example the toughness and the strength of a polymeric material both depend, to some extent, on the number of chemical bonds that must be broken before a crack spreads across the material. This in turn depends on the number of molecules that connect individual crystals together. A crack would usually be expected to propagate through the noncrystalline regions so if there are only a few "tiemolecules" between adjacent lamellae (as in the case of crystals grown from dilute solution) it would be expected that the material would be weak and brittle. This is indeed the case for sedimented single crystal mats. One way to improve the strength and toughness of a polymer is thus to increase the number of tie molecules between lamellar crystals. This could be achieved in various ways; increasing the molecular weight of the polymer is an obvious route. A less obvious route would be to crystallize more rapidly, giving the molecules less chance to evolve from the configurations they had in the melt. Polyethylene which has been crystallized slowly at a high temperature is usually rather brittle compared with the same polymer crystallized on quenching. The stiffness of a product depends of course on the crystallinity, higher crystallinity leading to higher stiffness (provided that the crystalline regions are stiffer than the amorphous regions). However in many cases the stiffness is much more sensitive to the orientation of the molecules. In

4.3 Relationships Between Structure and Properties

general, polymer molecules have a high stiffness in the chain direction so that if a material is suitably oriented its stiffness can be enhanced. A good example is to consider polymer films. Many films are biaxially oriented during processing. This orientation leads to the chains lying mostly in the plane of the film. The in plane stiffness of the film is accordingly enhanced at the expense of the stiffness normal to the plane of the film. However the stiffness normal to the film is not usually important. Whilst the simple relations described above are easy to understand, their interpretation in terms of the gross morphology of a polymeric solid can pose many problems. Consider a material with a spherulitic texture. To predict whether it will be strong and stiff we would need to know, amongst others, the degree to which tie molecules or fibrils span the lamellae. Attempts to provide models that make such estimates possible have been made (Keith and Padden, 1959, 1984, 1986) by modeling the structure of a spherulite. These models of spherulitic growth, which include the incorporation of tie molecules and intercrystalline links, have had limited success in predicting properties. 4.3.2 Control of Morphology to Improve Strength

In the previous section the links between molecular weight and strength have been alluded to. For most polymers there is a minimum molecular weight below which materials processed through the melt are weak and brittle. This is somewhat higher than the entanglement molecular weight. The properties usually show a dramatic improvement once this molecular weight is exceeded but the strength continues to improve as the molecular weight is increased further.

197

An alternative mechanism that is known to improve the strength is to increase the number of interlamellar ties. This usually is achieved with the aid of faster crystallization at lower temperatures and with an increase in the nucleation rate. However there is a further possibility, which in some cases can be important. When the spherulites are rather large, then the interfaces between spherulites often contain few tie molecules (due to the reduction in volume on solidification). These interfaces can become lines of weakness and cracks can propagate around them. In this case simply reducing the spherulite size, by improved nucleation, can lead to a significant improvement in the strength. A polymer that illustrates several of these effects is PHB. In the following the fracture behavior of PHB is described as an example of how planned morphological control can advantageously be used to improve mechanical properties. 4.3.2.1 Cracking in Poly(hydroxybutyrate) (PHB)

When melt processed, PHB is usually a rather brittle, weak, material. Table 4-1 provides some mechanical data illustrating this point. If the polymer is observed in the optical microscope while under strain, a

Table 4-1. Mechanical data for PHB. Sample preparation

Tensile Tensile Breaking modulus strength strain (MPa) (GPa) (%)

Crystallized at 60 °C Crystallized at 130°C Quenched sheet

1.6 1.7 1.5-1.8

<5 <5 8^20

0.7 1.2 0.8-2.0

Crystallized at 130°C Small spherulites (>20um)

1.9

>250

40-100

4 Crystallization and Morphology of Semicrystalline Polymers

198

series of cracks can easily be discerned (Fig. 4-39). Two types of cracks can be observed. In the smaller spherulites crystallized at lower temperatures (in regime III) the cracks tend to run radially across spherulites; while in the larger spherulites grown at higher temperatures (regime II) the cracks appear to run circumferentially around the spherulites. First we shall consider the material crystallized at the lower temperatures which contains predominantly radial cracks. As the strain is increased the cracks can be seen to grow through the spherulites and join together by spreading around the spherulite boundaries (Fig. 4-40). The origin of the cracks has not yet been identi-

0.5mm

Figure 4-39. Optical micrographs of PHB showing the radial and circumferential cracks; (A) crystallized at 60 °C, (B) crystallized at 130°C (Barham and Keller, 1986).

Figure 4-40. Optical micrograph of PHB crystallized at 60 °C, under a strain of 0.2% in the horizontal direction. Two cracks which have grown across this area are indicated by arrows (Barham and Keller, 1986).

fied, but they appear to be related to the process of aging in PHB. The aging in PHB is manifest as a progressive decrease in strength and elongation to break during storage at ambient temperatures for a period of several weeks after the initial processing. When examined under the optical microscope older samples of PHB tend to show a larger number of radial cracks than freshly prepared ones. Thus it is likely that the cracking is caused by the some slow rearrangements, either in the crystalline or amorphous regions, and that it is the presence of more cracks that in turn leads to the degradation in mechanical properties. The material crystallized in regime II does not exhibit the radial cracks (nor does it appear to suffer to the same extent from aging). However all the spherulites seem to be broken up by a series of circumferential cracks (Fig. 4-39). On straining these materials, new radial cracks are formed and grow to join the circumferential cracks and lead to failure. A notable feature is that the central parts (out to a radius of ca. 50 |im) of the spherulites remain crack free. This result indicates that if it were possible to increase the nucleation density sufficiently it should be possible to grow small enough

4.3 Relationships Between Structure and Properties

spherulites, which are completely free of cracks. The origin of these cracks is itself an interesting story that has implications for the processing of most polymers. It is described in detail in the following section. 4.3.2.2 Spherulite Expansion and Contraction

Polymer crystals exhibit strongly anisotropic thermal expansion. Along the chain direction the expansion is usually small and negative, while the expansion coefficients along the other crystal axes are much larger, positive and not normally equal to each other. In a complex crystalline aggregate such as a spherulite this can produce unusual thermal expansion effects. The expansion coefficient along a radius will in general be different from that around the circumference. Either may be larger. In polyethylene the circumferential expansion coefficient is greater than the radial coefficient (Barham and Keller, 1977). On the other hand in PHB and polyethylene oxide) the radial expansion is larger. When a newly grown spherulite is cooled from the crystallization temperature down to room temperature the differential contractions along the radius and circumference will generate strains inside the spherulites. These strains must be accommodated in some manner. In the case of polyethylene (Barham and Keller, 1977) the radiating lamellae wind up on themselves, reducing the periodic band spacing, so as to accommodate the lower radial expansion and preventing the formation of radial cracks. By contrast in the case of PHB the lamellae would need to unwind if the thermal strain was to be accommodated in this way. This does not happen. Instead we observe the formation of the circumferential cracks (Martinez-Salazar etal, 1989).

199

To date, detailed data are only available for PHB and polyethylene; however, similar effects probably occur in all spherulitic systems. They will only become of importance if the spherulite size is sufficiently large that the accumulated thermal strain generates a stress comparable with the breaking stress of the polymer. Of course there are other possible causes for the formation of cracks within spherulitic materials. The heterogeneous nature of the local morphology in any spherulitic system will always lead to local stress concentrations, which may be much larger than any thermally induced stresses. 4.3.2.3 Prevention of Crack Formation in PHB

Since there are two types of cracks, radial and circumferential, it is best to deal with each type separately. The radial cracks pose the greater problem since the underlying cause is not understood. It has been noted that the cracks may be "healed" by a rolling, pressing or heat treatment (Barham and Keller, 1986). However although the strength and elongation to break improve once the cracks have been "healed" (see Table 4-1), the improvement is not permanent and the aging process starts again. To date no permanent solution to the problem has been found. The circumferential cracks are however different. In this case we understand the underlying cause of the cracks and hence it is possible to make appropriate morphological changes to reduce crack formation. The cracks arise from the differential stresses set up during cooling the spherulites, which produce a negative hydrostatic pressure inside the spherulite. The magnitude of this pressure increases with the spherulite size (Barham, 1976). Thus by producing sufficiently small spherulites it

200

4 Crystallization and Morphology of Semicrystalline Polymers

should be possible to limit the magnitude of the thermally induced stresses so that they remain below the failure stress of the material. In order to make PHB strong it is thus necessary to crystallize at high temperatures (in regime II where only circumferential cracks appear) and increase the nucleation density sufficiently that the spherulites are small enough that the internal stresses do not lead to crack formation. The problem may thus be expressed in morphological terms. The achievement of a high nucleation density is itself a problem, but a surmountable one. An example illustrating how a sample of PHB crystallized at high temperatures, but having a small spherulite size, can be a strong, tough material is given in Table 4-1. 4.3.3 Control of Morphology to Improve Stiffness 4.3.3.1 Morphological Requirements for Stiffness

Long chain molecules are inherently anisotropic. Along the chains stress is transferred through covalent bonds, while stress transfer between chains is through weaker interactions such as van der Waals forces, hydrogen bonds or ionic interactions. In any case, the stiffness along the chains will be much greater than that across them. The actual stiffness of a polymer crystal will depend not only on the types of interaction between the chains but also on the conformation of the chains in the unit cell. For example, polymers that have a planar zigzag conformation, such as polyethylene, will have much higher stiffness along the c-axis than polymers that adopt a shallow helical conformation, such as polypropylene, where a stress along the crystal c-axis causes the molecular helix to stretch like a

spring. In the planar zigzag case the stress acts directly on the covalent bonds. To achieve a high stiffness it is necessary to concentrate as much as possible of the stress along the chains. This implies the need for orientation of the chains and suggests that the study of fibers should be a useful area to gain an understanding of the influence of morphology on stiffness. Since fibers are dealt with in some detail later (see Chaps. 7, 10, and 13 of this Volume) only a brief review of some of the salient features will be presented here. A simple calculation shows that the potential modulus of polymer fibers is very much higher than that conventionally achieved. The reason for the failure to obtain routinely high moduli fibers lies in the morphology of the fibers. An examination of the simplest of Takayanagi models (Fig. 4-37) provides a ready explanation. In a conventional fiber, produced by stretching, or drawing the polymer, the crystalline regions become oriented but the lamellar structure is essentially preserved. Thus the fiber may be considered to consist of stiff (crystalline) regions in series with soft (amorphous) regions. The result is a fiber whose properties are dominated by those of the amorphous regions. To achieve a high modulus in the fibers according to this simple model it is necessary to change the morphology so that the amorphous material acts in parallel rather than in series with the crystalline material. Thus it can be concluded that a high modulus requires both a high degree of chain orientation and extension. Many attempts, successful and unsuccessful, have been made to produce such morphologies; some are described in the following section on extended chain fibers. Fuller details can be found in Chaps. 7 and 10. The approximations inherent in the simple Takayanagi models overlook an alter-

4.3 Relationships Between Structure and Properties

native approach to high modulus (Odell et al., 1978), which does not require a high chain extension. If a stack of alternating plates of stiff and soft rubberlike materials is loaded normal to the stack then the stack may be very stiff, provided the constraints at the interfaces do not permit the lateral contraction of the rubbery material under the tensile stress. To stretch such a system the volume of the rubbery region must then increase and the applied stress works against the bulk modulus of the rubbery region. Since rubbery materials, such as amorphous polymers, possess a high bulk modulus, the overall stiffness of such a system can be very large. Thus if a suitable morphology, in which the amorphous regions between lamellar crystals are thoroughly constrained (and contain no voids) can be produced, it should be possible to achieve a high modulus material without the need for high chain extension. Such a morphology has been produced with polyethylene. The use of such morphological control to engineer properties is a most useful example of the importance of structure property relationships and is described in some detail in Sec. 4.3.3.3 below.

4.3.3.2 Extended Chain Fibers

The bulk of the research into the production of high modulus fibers has concentrated on polyethylene and on the achievement of a high degree of chain extension. Two basic approaches have been used: the deformation of already crystallized material either by extrusion or drawing; and crystallization, usually from solution, from previously extended chains. The work on solid state deformation is conveniently divided into two classes depending on whether the polymer was crystallized from the melt or from dilute solution.

201

Fibers from Deformation of Melt Crystallized Material In this section only the most salient observations will be discussed since full details of fiber formation are given in Chap. 10 of this Volume. The first important observation is that when polyethylene samples are deformed, the tensile modulus of the resulting fibers increases with the draw ratio (Cappacio and Ward, 1974; Barham and Keller, 1976). The limitation to the maximum modulus obtainable coming from the maximum achievable draw ratio (providing the drawing is carried out at a sufficiently low temperature to prevent plastic slip, when little or no increase in modulus is observed during drawing). This behavior is illustrated in Fig. 4-41, which shows the modulus as a function of the draw ratio in a typical experiment (Barham and Keller, 1976). Secondly it is found that as the molecular weight of the polymer is increased so the maximum draw ratio that can be achieved decreases, with the consequence that it is difficult to produce high modulus fibers of high molecular weight polymers by deformation of melt crystallized material. There appears to be a simple relationship between the modulus of drawn fibers and the draw ratio as illustrated in Fig. 4-41. Several models have been produced to attempt to explain such structure property relationships (Arridge and Barham, 1978; Gibson et al., 1978; Peterlin, 1979). However whilst these all show some features of the observed modulus draw ratio behavior they all rely on some unmeasurable parameters, such as the number of "taut tie molecules" between "fibrils". Recently, neutron scattering experiments have revealed a more fundamental relationship in drawn fibers (Sadler and Barham, 1990 a -c). The use of neutron scatter-

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4 Crystallization and Morphology of Semicrystalline Polymers

100

3

60

=3 T3 O

Figure 4-41. A graph showing the increase in modulus with draw ratio for melt crystallized polyethylene. 10

15

20 Draw Ratio

25

ing enabled the measurement of the dimensions of individual molecules along and normal to the draw direction. From such data the molecular draw ratio can be calculated as the ratio of the initial to the final size of the molecules along the draw direction. It has been shown that the modulus is a unique function of the molecular draw ratio. This is illustrated in Fig. 4-42, which shows the modulus of polyethylene fibers produced using a variety of different preparation methods as a function of the overall molecular draw ratio. These results reveal a clear relation between molecular configuration and the fiber mechanical properties. Such information will be used to produce new and better models to predict the mechanical properties.

30

35

Gel Spun Fibers and Shish-Kebabs Several different methods which permit the processing of high molecular weight polyethylenes, via a solution state, to produce extended chain fibers, which possess both high modulus and strength, have been described in the literature (Sato and Hirai, 1962; Jurgeleit, 1962; Zwick, 1966, 1967). All describe gel spinning or solution spinning process for polyethylenes, but do not claim high modulus or strengths. It was work by Pennings and his co-workers which showed that fibers with excellent mechanical properties could be obtained via solution processing (Pennings, 1966; Zwijnenburg and Pennings, 1975; Pennings et al., 1977). Later it was realised that

Figure 4-42. A graph showing how the tensile modulus of fibers depends on the molecular extension (after Sadler and Barham, 1990 c). 2 3 4 Molecular Draw Ratio (after neck)

4.3 Relationships Between Structure and Properties

a gellike layer at a surface promoted the production of strong fibers by permitting crystallization of polyethylene at higher than usual temperatures (Zwijnenburg and Pennings, 1975; Zwijnenburg, 1978). The recognition of the importance of a gel-like layer in turn led to the development of gel spinning techniques for polyethylene (Smith etal., 1979; Pennings and Torfs, 1979; Smith and Lemstra, 1980; Kalb and Pennings, 1980; Barham, 1982). The development of these various techniques and their relative merits are described in the reviews, Mackley and Sapsford (1982) and Barham and Keller (1985). Although the details vary between the processes, they all give very similar fibers so that it may not seem important to distinguish between them. There is however one difference that may be of importance; some of the fibers are crystallized directly from flowing solutions in which the molecules have been extended while others are produced by drawing chain folded crystals in the solid state. Some of the processes involve aspects of both processes. In the following a very brief description of the morphological features of these two routes is given. When polymers are crystallized from flowing solutions the typical shish-kebab morphology is obtained, Fig. 4-43. These crystals consist of an extended chain crystalline core onto which chain folded lamellar crystals have grown. The core is formed in the flow field and consists of long chain crystals; at the ends of these crystals some of the molecules are supposed to emerge leaving long hairs or cilia. It is these hairs that later, on cooling under static conditions, form the overgrowths or kebabs. For such fibrous crystals to possess a high modulus they need to maximize the amount of material involved in the extended chain cores and the length of the crystals along the cores. In practice this is

203

Figure 4-43. The structure of shish-kebab crystals: top left a schematic diagram of the arrangement of molecules in the core of a shish kebab; top right a sketch showing the platelet overgrowths on the cores; bottom an electron micrograph of some real shishkebab crystals.

achieved by maximizing the growth temperature of such shish-kebab crystals (Barham, 1982). When polymers are crystallized from static solutions they usually form the typical chain folded lamellar crystals. In these crystals a single chain is usually contained in just one lamella and folds back on itself many times. If the molecular weight is very high then the number of other molecules a particular molecule interacts with is very much lower than it would be in the melt. It is believed that it is this low degree of entanglement in solution prior to crystallization, combined with the high degree of

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4 Crystallization and Morphology of Semicrystalline Polymers

chain folding, that permits very high draw ratios from high molecular weight polymers crystallized from solution. Further the formation of a gel can provide mechanical stability to the system and hence make it easier to handle the material and thus produce fibers on a large scale. However, it has been shown that the best properties are obtained when using dried down mats of single crystals grown from very dilute solutions which would not form gels. In practice for polyethylene of molecular weight of about a million a draw ratio of about 100 can be readily achieved, with a correspondingly high modulus and strength. By contrast the same polymer crystallized from the melt can only be drawn to a draw ratio of ca. 4. 4.3.3.3 Interlocking Shish-Kebab Systems

The alternating stack of stiff and rubbery lamella can have a high modulus as described above if certain conditions are met (Odell et al., 1978, 1984; Keller and Odell, 1978; Bashir et al, 1984). In particular it is important that the surfaces of the stiff lamellae are perpendicular to the fiber direction and the chain axes lie in the fiber direction. It has been found that it is possible to produce such a morphology by first forming fibrous nuclei and then growing lamellar crystals onto these nuclei under conditions of increasing supercooling. The resulting morphology is illustrated in Fig. 4-44. Note the extended chain crystalline nuclei in the background and the tapering, interlocking lamellae, which have grown over them. The tapering nature of the overgrowths allows for a high degree of interpenetration, which leads to a connectedness and helps to prevent fibrillation. These materials have similar, or better, properties than fibers melt drawn from the same polymer. In the drawn fibers the

Figure 4-44. An electron micrograph showing the tapering shish-kebab morphology. [Odell et al. (1978), Polymer 19. By kind permission of the authors and the publishers, © Butterworth and Heinemann Ltd.]

molecules are highly extended (Sadler and Barham, 1991a). However, in these specially prepared interlocking materials, the molecules are actually slightly compressed in the fiber direction (Sadler and Odell, 1980). Several different approaches have been adopted to prepare fibers with this type of morphology, which calculations have shown to give the best modulus and strength. These include the preparation of shish-kebab crystals from dilute solution followed by the melting and recrystallization of the kebabs; and a continuous process of melt extrusion in which a small amount of high molecular polymer, either present naturally in the distribution or deliberately added, forms shish-kebab type crystals in the entrance region of a die (Bashir et al, 1984). A few straightforward requirements for the fibrous nuclei have emerged. These in-

4.3 Relationships Between Structure and Properties

elude a minimum spacing between them. If they are spaced too far apart then the lamellar crystals that grow on them are able to begin to twist (as do spherulitic lamellae) as they grow. If any twisting were to occur then the lamellar surfaces would no longer be strictly normal to the fiber direction and the conditions for the high modulus would be lost. The spacing of the nuclei depends on the actual processing conditions and the molecular weight distribution, which have to be carefully controlled. The understanding of the basic requirements in terms of the morphology has greatly assisted in delineating suitable processing conditions and molecular weight distributions to produce stiff fibers.

4.3.4 Control of Morphology to Improve Toughness

Toughness is a property of considerable practical importance. It is often more important that a product is tough than it is strong or stiff; yet little effort has to date been given to attempts deliberately to increase the toughness through morphological control. Improvements in toughness have usually come through modifications to the molecular weight and the use of additives. However there is a simple rationale that should allow morphological control over toughness in some semicrystalline polymer systems and blends. This has as its basis the concepts of rubber toughening in glassy polymers. The phase behavior of some polyethylene blends is described, illustrating how similar morphologies may be obtained. These are then related to morphologies found in some linear low density polyethylenes (LLDPEs). Finally it will be demonstrated that the control of morphology can, in some cases, lead to enhanced toughness.

205

4.3.4.1 Principles of Rubber Toughening

A discussion of the mechanism of rubber toughening of glassy polymers is given in Chap. 13 of this Volume. In principle, a morphology in which small inclusions of one phase are well dispersed in a matrix of the other can lead to a material that is tougher than either of the phases on their own. The achievement of such morphologies from polyethylene blends is described in the subsequent sections. 4.3.4.2 Phase Separation in Polyethylene Blends

Blends of linear polyethylene (LPE) and branched polyethylene (BPE) have been shown to undergo phase separation both in the melt and during crystallization from a mixed melt (Barham et al., 1988; Hill etal., 1991 a, b; Hill and Barham, 1992). This diversity of phase behavior lends itself to the purposeful design of specific morphologies to obtain required morphologies (see Chap. 6). A generalized phase diagram for a typical LPE-BPE blend system is shown in Fig. 4-45. The determination of such phase diagrams is a somewhat complex matter as

BPE

LPE Blend composition

Figure 4-45. Schematic phase diagram for blends of linear and branched polyethylene.

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4 Crystallization and Morphology of Semicrystalline Polymers

each of the components have similar properties so that conventional techniques such as light scattering may not be applied. In practice a range of indirect methods has been used to obtain information on the state of dispersion of the melt. These rely on the concept that if the melt is quenched rapidly enough then the molecules have insufficient time to move and the phase morphology present in the solid, crystalline, material reflects that present in the melt prior to quenching. Examples of the sort of data on which phase diagrams such as Fig. 4-45 are based are given in Figs. 4-46 and 4-47. Figure 4-46 shows electron micrographs obtained from rapidly quenched blends. Two different morphologies can be seen: one has a homogeneous uniform morphology, indicating that it came from a mixed melt, while the other shows two distinct morphologies, with different lamellar thicknesses, suggesting it arose from a sample that was not mixed in the melt. Figure 4-47 shows DSC traces from the melting of a series of blend samples of differing LPE content all quenched from the same melt temperature (160°C). In the samples with a high LPE content there is a single melting endotherm, indicating only one crystal population; at low LPE contents two endotherms are seen, indicating two distinct crystal populations. Furthermore, the melting temperatures of the two peaks are the same in all the samples that show two peaks, which suggests that the composition of each of the two crystal populations is the same in each case. When there is a single endotherm its peak temperature falls as the LPE content decreases, indicating that the crystals contain an increasing amount of the BPE. Some LLDPEs show thermal behavior and morphologies similar to those exhibited by blends of LPE and BPEs crystal-

Figure 4-46. Electron micrographs of replicas illustrating single and double morphologies seen in blends of LPE with BPE. Top, single morphology seen in a blend containing 50% LPE quenched from 160°C. Bottom, double morphology in a blend containing 10% LPE, the arrow indicates a typical LPE rich area (Barham et al., 1988).

lized from a two phase region of the phase diagram. Figure 4-48 shows a typical electron micrograph showing a double morphology; there are regions with a regularly banded spherulitic texture and a large lamellar thickness imbedded in other regions which display no obvious spherulitic ordering and where the lamellae are thinner. Furthermore there is some evidence that these materials have a bimodal distribution of branch content. It has accordingly been suggested (Debliek and Mathot, 1988) that some LLDPEs separate into

4.3 Relationships Between Structure and Properties

207

Figure 4-48. An electron micrograph of a LLDPE sample showing two different morphologies arising from phase separation in the melt (Deblieck and Mathot, 1988).

4.3.4.3 Control of Phase Morphology and Toughness 70

-t80

90

100

110

120

130

140 C

Figure 4-47. A series of DSC traces from quenched blends of LPE (Sclair 2907) with BPE (CBP PN220) with the indicated proportions of PE. All the samples were prepared by quenching from 160°C (Barham etal., 1988).

two phases in the melt and that this affects their subsequent crystallization behavior.

An important feature of the LLDPEs is their high fracture toughness. It is very tempting to argue that there is a direct link between their phase behavior and the toughness. From the arguments presented above it might be expected that if two crystal phases, one rich in LPE and one rich in BPE, are present and one is well dispersed in the other this could lead to enhanced fracture toughness. Certainly the condition

4 Crystallization and Morphology of Semicrystalline Polymers

Figure 4-49. Fracture toughness of polyethylene blends as a function of blend composition (Hill and Barham, unpublished data). 20

40 60 Blend Composition (%LPE)

80

of good adhesion between the phases should be met, and if the included phase is the LPE rich phase it should be expected to have a higher strength so it might act to reduce the propagation of cracks through the BPE rich matrix. In a series of preliminary experiments it has been shown that the fracture toughness of quenched blends of an LPE with a BPE shows a maximum when the composition is in the range where this sort of morphology occurs, Fig. 4-49. Thus it does appear that at least in this case control of the morphology can lead to a high toughness. Further it lends weight to the speculation that the toughness of the LLDPEs comes about from fortuitous phase separation.

4.4 Concluding Remarks In this chapter we have seen how over the past thirty or so years the morphology of semicrystalline polymers has gradually become clearer; we now have good models for the structure of spherulitic materials, and even some of the more complex morphologies are beginning to become understood. However, although we can interpret the gross morphology in terms of molecules and lamellae, we still are not able to predict

100

in advance the morphology of any polymer crystallized under any given conditions; we are limited in this respect to a few special cases which have been studied in great detail. We have also seen that it is often possible to understand the physical properties of semicrystalline polymers in terms of the morphology. However, we are still unable to make more than some rather general predictions. One of the more difficult, but potentially most rewarding challenges for future studies on semicrystalline polymers will be to change the emphasis of modeling of properties. We should be able to move from the current state where we explain known properties in terms of the observed morphology, to the position where we can predict the properties in terms of the morphology; and further be able to predict the morphology from knowledge of the chemical nature of the polymer and its crystallization conditions.

4.5 References Armistead, K. A., Goldbeck-Wood, G. (1992), Adv. Polym. Sci. 100, to appear. Arridge, R. G. C , Barham, P. J. (1978), Polymer 19, 654-658.

4.5 References

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Keller, A. (1991), Sir Charles Frank, An Eightieth Geil, P. H., Anderson, F. R., Wunderlich, B., Birthday Tribute: Chambers, R. G., Enderby, J. E., Arakawa, T. (1964),«/. Polym. Sci. A 2, 3707-3720. Keller, A., Lang, A. R., Steeds, X W. (Eds.). Bristol, Gibson, A. G., Davies, G. R., Ward, I. M. (1978), U.K.: Adam Hilger, pp. 265-306. Polymer 19, 683-693. Keller, A., Odell, X A. (1978), J. Polym. Sci., Polym. Goldbeck-Wood, G. (1991), private communication. Symp. 63, 155-164. Gornick, R, Ross, G. S., Frolen, L. J. (1967), J. Khoury, F. (1976), in: Treatise in Solid State ChemisPolym. Sci. C18, 70-91. try, Vol. 3: Hannay, N. B. (Ed.). New York: Hikosaka, M. (1987), Polymer 28, 1257-1264. Plenum Press. Hikosaka, M. (1990), Polymer 31, 458-468. Kilian, H. G. (1969), Koll.-Z. Z. Polym. 231, 534Hikosaka, M., Rastogi, S., Keller, A., Kawabata, H. 555. (1993), J. Macromol. Sci., in press. Kovacs, A. X, Straupe, C. (1979), Disc. Farad. Soc. Hill, M. J., Barham, P. J. (1992), Polymer 33, 409968, 225-238. 4108. Kovacs, A. X, Straupe, C. (1980), /. Cryst. Growth 48, Hill, M. J., Barham, P. J., Keller, A. (1991 a), Polymer 210-226. 32, 1384-1393. Kovacs, A. X, Gonthier, A., Straupe, C. (1975), /. Hill, M. X, Barham, P. J., Keller, A. (1991 b), Polymer Polym. Sci., Polym. Symp. 50, 283-325. 52,2530-2541. Kovacs, A. X, Straupe, C , Gonthier, A. (1977), /. Hill, M. X, Bradshaw, D. C , Chevili, R. X (1992), Polym. Sci., Polym. Symp. 59, 31-54. Polymer 33, 874-876. Krimm, S., Ching, X H. (1972), Macromol. 5, 209Hoffman, X D. (1983), Polymer 24, 3-26. 211. Hoffman, X D., Miller, R. L. (1988), Macromolecules Lauritzen, X I., Hoffman, X D. (1960), /. Res. Natl. 21, 3038-3051. Bur. Stand A64, 73-102. Hoffman, X D., Miller, R. L. (1989), Macromolecules Lauritzen, X I., Hoffman, J. D. (1973), /. Appl. Phys. 22, 3038-3054. 44, 4340-4352. Hoffman, X D., Weeks, X (1962), /. Chem. Phys. 37, Legras, R., Mercier, X P., Nield, E. (1983), Nature 1723-1741. Hoffman, X D., Lauritzen, X I., Passaglia, E., Ross, 304, 432-434. G. S., Frolen, L. X, Weeks, X J. (1969), Colloid and Legras, R., Bailly, C , Daumerie, M., Dekoninck, X Polym. Sci. 231, 564-592. M., Mercier, X P., Nield, E., Vichy, V. (1984), PolyHoffman, I D . , Frolen, L. X, Ross, G. S., Lauritzen, mer 25, 835-844. X I. (1975), J. Res. Natl. Bur. Stand. A 79, 671 -699. Legras, R., Dekoninck, X M., Vanzieleghem, A., Hoffman, X D., Davis, G. T, Lauritzen, J. I. (1976), Mercier, X P., Nield, E. (1986), Polymer 27, 109in: Treatise in Solid State Chemistry, Vol. 3: Han117. nay, N. B. (Ed.). New York: Plenum Press, pp. Legras, R., Leblanc, D., Daoust, D., Devaux, X, 497-614. Nield, E. (1990), Polymer 31, 1429-1434. Hoffman, X D., Guttman, C. M., DiMarzio, E. A. Lotz, B., Wittmann, J.-C, Stocker, W, Magonov, S. (1979), Disc. Farad. Soc. 68, 177-197. N., Cantow, H.-X (1991), Polym. Bull. 26,209-214. Jing, X., Krimm, S. (1982), /. Polym. Sci. Polym. Mackley, M. R., Sapsford, G. S. (1982), DevelopPhys. Ed. 20, 1155-1173. ments in Oriented Polymers, Vol. 1: Ward, I. M., Jurgeleit, W (1962), U.S. Patent 3048465. (Ed.). London: Applied Science, pp. 201-224. Kalb, B., Pennings, A. X (1980), J. Mater. Sci. 15, Mansfield, M. L. (1988), Polymer 29, 1755-1760. 2584-2590. Mansfield, M. L. (1990), Polym. Commun. 31, 283285. Kanig, G. (1973), Koll.-Z. Z. Polym. 251, 782-783. Martinez-Salazar, X, Barham, P. X, Keller, A. (1985), Kanig, G. (1974), Kunststoffe 64, 470-485. Kanig, G. (1980), /. Cryst. Growth 48, 303-320. J. Mater. Sci. 20, 1616-1624. Keith, H. D., Padden, F X (1959), J. Polym. Sci. 39, Martinez-Salazar, X, Sanchez-Cuesta, M., Barham, 101-137. P. X, Keller, A. (1989), /. Mater. Sci. Lett. 8, 490Keith, H. D., Padden, F X (1963), J. Appl. Phys. 34, 492. 2409-2421. Mauritz, K. A., Baer, E., Hopfmger, A. X (1973), Keith, H. D., Padden, F X (1964a), /. Appl. Phys. 35, /. Polym. Sci. Polym. Phys. Ed. 11, 2185-2197. 1270-1286. Mitomo, H., Nakazato, K., Kuriyama, I. (1978), Keith, H. D., Padden, F X (1964b), /. Appl. Phys. 35, Polymer 19, 1427-1432. 1286-1296. Odell, X A., Grubb, D. T., Keller, A. (1978), Polymer Keith, H. D., Padden, F X (1984), Polymer 25, 28-42. 19, 617-626. Keith, H. D., Padden, F X (1986), Polymer 27, 1463Odell, X A., Keller, A., Miles, M. X (1984), Colloid 1471. and Polym. Sci. 262, 683-690. Keith, H. D., Padden, F X (1987a), J. Polym. Sci. Organ, S. X, Barham, P. X (1992), J. Mater. Sci., in Polym. Phys. Ed. 25, 229-242. press. Keith, H. D., Padden, F X (1987b), J. Polym. Sci. Organ, S. X, Barham, P. X, Webb, A. (1991), U.K. Polym. Phys. Ed. 25, 2265-2273. Patent Application no. 9012975.0.

4.5 References

Patel, D., Bassett, D. C. (1991), paper presented at: Polymer Physics - A Conference to Mark the Retirement of Andrew Keller. Bristol, U.K., to be published. Pennings, A. I (1966), Proceedings of International Conference on Crystal Growth: Peiser, H. S. (Ed.). Boston: Pergamon Press, pp. 389-399. Pennings, A. X, Torfs, I (1979), Colloid and Polym. Sci. 257, 547-549. Pennings, A. X, Lagaveen, R., Devries, R. S. (1977), Colloid and Polym. Sci. 255, 532-542. Peterlin, A. (1979), in: Ultra-High Modulus Polymers: Ciferri, A., Ard, I. M. (Eds.). London: Applied Science, pp. 279-320. Phillips, P. X (1990), Rep. Prog. Phys. 53, 549-604. Point, X X (1978), Disc. Farad. Soc. 68, 167-176. Point, X X (1979), Macromolecules 12, 770-775. Point, X X, Kovacs, A. X (1980), Macromolecules 13, 399-409. Rastogi, S., Ungar, G. (1992), Macromolecules, in press. Rastogi, S., Hikosaka, M., Kawabata, H., Keller, A. (1992), Macromolecules, in press. Ross, G. S., Frolen, L. X (1975), /. Res. NatL Bur. Stand. U.S. 79A, 701-711. Ruland, W. (1971), J. Appl. Cryst. 4, 70-73. Ruland, W (1977), Colloid and Polym. Sci. 255, 417427. Ruland, W (1978), Colloid and Polym. Sci. 256, 932936. Sadler, D. M. (1984), in: Structure of Crystalline Polymers: Hall, I. H. (Ed.). London: Elsevier PubL, pp. 125-180. Sadler, D. M. (1987a), Nature 326, 174-177. Sadler, D. M. (1987 b), J. Chem. Phys. 87, 17711784. Sadler, D. M. (1987c), Polymer 28, 1440-1454. Sadler, D. M., Barham, P. X (1990a), Polymer 31, 36-42. Sadler, D. M., Barham, P. X (1990b), Polymer 31, 43-46. Sadler, D. M., Barham, P. X (1990c), Polymer 31, 46-50. Sadler, D. M., Gilmer, G. H. (1984), Polymer 25, 1446-1452. Sadler, D. M., Keller, A. (1979), Science 203, 263265. Sadler, D. M., Odell, X A. (1980), Polymer 21, 479480. Sato, H., Hirai, T. (1962), Japanese Patent Sho. 379765. Sawyer, L. C , Grubb, D. T. (1987), Polymer Microscopy. Cambridge: Cambridge University Press. Schultz, X M., Kinlock, D. R. (1969), Polymer 10, 271-278. Smith, P., Lemstra, P. X (1980), J. Mater. Sci. 15, 505-514. Smith, P., Lemstra, P. X, Kalb, B., Pennings, A. X (1979), Polym. Bull. 1, 733-739.

211

Spells, S. X, Sadler, D. M. (1984), Polymer 25, 739749. Spells, S. X, Sadler, D. M., Keller, A. (1980), Polymer 21, 1121-1128. Spells, S. X, Organ, S. X, Keller, A., Zerbi, G. (1987), Polymer 28, 697-704. Sperling, L. H. (1984), Polym. Eng. Sci. 24, 1-21. Stamm, M., Fischer, E. W., Dettenmaier, M., Convert, P. (1979), Disc. Farad. Soc. 68, 263-275. Statton, W. O., Geil, P. H. (1960), J. Appl. Polym. Sci. 3, 357-361. Strobl, G. R. (1983), J. Polym. Sci. Polym. Phys. Ed. 21, 1357-1380. Strobl, G. R., Eckel, R. (1977), Prog. Colloid Polym.

Sci. 62,9-15. Strobl, G. R., Frentzel, K. H. (1984), Proceedings of the IXth International Conference on Raman Spectroscopy. Tokyo, Japan, pp. 112-113. Strobl, G. R., Schneider, M. X, Voigt-Martin, I. (1980), /. Polym. Sci. Polym. Phys. Ed. 18, 13611381. Toda, A. (1991a), Polymer 32, 771-780. Toda, A. (1991 b), Submitted to Colloid and Polym. Sci. Tomka, X, Maclwain-Sinn, N. (1977), U.K. Patent no. 10465046. Turnbull, D., Cormia, R. L. (1961), J. Chem. Phys. 34, 820-831. Voigt-Martin, I. G. (1985), Adv. Polym. Sci. 67, 195218. Vonk, G. (1971), J. Appl. Crystallogr. 4, 340-342. Winram, M. M., Grubb, D. T., Keller, A. (1978), J. Mater. Sci. 13, 791-796. Wunderlich, B. (1973), Macromolecular Physics, Vol. 1. New York: Academic Press. Wunderlich, B. (1976), Macromolecular Physics, Vol. 2. New York: Academic Press. Wunderlich, B. (1980), Macromolecular Physics, Vol. 3. New York: Academic Press. Zachmann, H. G. (1967), Koll.-Z. Z. Polym. 216, 180-191. Zerbi, G. (1984), Adv. Infrared Raman Spectroscopy, Vol. 11, Chap. 6. Zwick, M. M. (1966), Amer. Chem. Soc. Polym. Preprint 7, 814. Zwick, M. M. (1967), Appl. Polym. Symp. 6, 109. Zwijnenburg, A. (1978), Ph.D. Thesis, University of Groningen. Zwijnenburg, A., Pennings, A. X (1975), Colloid and Polym. Sci. 253, 452-461.

General Reading Arridge, R. G. C. (1975), Mechanics of Polymers. Oxford: Clarendon Press. Bassett, D. C. (1980), Principles of Polymer Morphology. Cambridge: Cambridge University Press.

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4 Crystallization and Morphology of Semicrystalline Polymers

Bassett, D. C. (Ed.) (1982), Developments in Crystalline Polymers. London: Applied Science. Hannay, N. B. (Ed.) (1976), Treatise in Solid State Chemistry, Vol. 3. New York: Plenum Press. Phillips, P. X (1990), Rep. Prog. Phys. 53, 549-604. Sawyer, L. C , Grubb, D. T. (1987), Polymer Microscopy. Cambridge: Cambridge University Press. Ward, I. M. (Ed.) (1982), Developments in Oriented Polymers. Vol. 1. London: Applied Science.

Ward, I. M. (1983), Mechanical Properties of Solid Polymers. Chichester: Wiley. Wunderlich, B. (1973), Macromolecular Physics, Vol. 1. New York: Academic Press. Wunderlich, B. (1976), Macromolecular Physics, Vol. 2. New York: Academic Press. Wunderlich, B. (1986), Macromolecular Physics, Vol. 3. New York: Academic Press.

5 Structure of Liquid Crystalline Polymers Matthias Ballauff Polymer-Institut, Universitat Karlsruhe, Karlsruhe, Federal Republic of Germany

List of Symbols and Abbreviations 5.1 Introduction 5.2 Classification, Description, and Theory of Liquid Crystalline Order in Polymers 5.2.1 General Characterization 5.2.2 Definition and Determination of Molecular Order 5.2.2.1 Nematic Phase 5.2.2.2 Smectic Phases 5.2.2.3 Local Distortions in Nematic Phases 5.2.3 Fundamentals of Molecular Statistical Theories of Liquid-Crystalline Order 5.3 Main-Chain Liquid Crystal Polymers 5.3.1 Molecular Architecture of Main-Chain Liquid Crystal Polymers 5.3.2 Calamitic Main-Chain Liquid Crystal Polymers 5.3.2.1 Liquid Crystal Polymers with Flexible Spacers in the Main Chain 5.3.2.2 Liquid Crystal Polymers with Bent or Crankshaft-Like Comonomers . . . . 5.3.2.3 Liquid Crystal Polymers Consisting of Rigid-Rod Polymers with Flexible Side Chains 5.3.3 Discotic Liquid Crystal Polymers 5.3.4 Polymer Blends Containing Liquid Crystal Polymers 5.4 Side-Chain Liquid Crystal Polymers 5.4.1 Calamitic Side-Chain Liquid Crystal Polymers 5.4.2 Liquid Crystalline Elastomers 5.4.3 Conclusions 5.5 Acknowledgements 5.6 References

Materials Science and Technology Copyright © WILEY-VCH Verlag GmbH & Co KGaA. Allrightsreserved.

214 216 219 219 220 220 223 224 225 228 228 232 234 237 239 241 243 244 244 246 247 247 247

214

5 Structure of Liquid Crystalline Polymers

List of Symbols and Abbreviations a ap A, B, C c2 d D /(<9, *P) f(0) F Fi(K T) Hp 7 exp In (s) K /K L M Ma Mw n n n N Pt (cos 0) r RL, U|| 1/2 s Sla{ S"" Tni T* U V F exc x

thickness of the layers in the smectic phase persistence length coefficients in Landau expansion polymer concentration lateral length of the basal plane order parameter to characterize the uniaxial nematic phase distribution function of molecular orientations distribution function for cylindrical molecules free energy ideal gas term of F component of the magnetic field intensity experimental scattering intensity structure (form) factor of a molecule constant depending o n the used radiation K u h n length length of a rod magnetization per unit mass component of the magnetization weight-average molecule weight number of carbon atoms in a spacer ratio of contour length to persistence length unit vector that describes the local direction of alignment number of molecules Legendre polynomials degree of polymerization radii of gyration 1 and || to the preferred direction in the nematic phase z-average of the gyration radius magnitude of the scattering vector Saupe order tensor (order parameter) component of the Saupe order tensor nematic-isotropic transition temperature second order transition temperature positive constant volume volume of the rods axial ratio L/d of the rods

x o Xap Ax Ax W

molecular tensor of diamagnetic susceptibility additional order parameter for the smectic phase magnetic susceptibility diamagnetic susceptibility anisotropy of the nematic phase angle of r o d inclination

List of Symbols and Abbreviations

DSC DTA LC LCP NPL PAA RIS WAXS

differential scanning calorimetry differential thermal analysis liquid crystal liquid crystalline polymer non-periodic layer crystallites p-azoxyanisol rotational isomeric state (model) wide-angle X-ray scattering

215

216

5 Structure of Liquid Crystalline Polymers

5.1 Introduction Usually, matter is thought to exist only in three states: solid, liquid, gas. This statement is not quite correct since certain organic materials exhibit states of intermediate order located between the perfect three-dimensional long range order of a solid and the remaining short-range order in liquids. These phases are called liquid crystals since many of them can flow like an ordinary liquid but display a birefringence and other properties reminiscent of the solid state. Thus a liquid crystalline substance or a liquid crystalline polymer does not directly transform to an isotropic melt when heated above the melting point but to a state where the orientational longrange order is preserved whereas the longrange positional order breaks down. This behavior may be compared to plastic crystals where the orientational long-range order vanishes at the transition point but positional order is still present. In first approximation two classes of liquid crystalline substances can be distinguished: 1. There are thermotropic systems in which the transition is brought about by altering the temperature; since mesophases are mostly turbid, the transition

to the clear isotropic state is called the clearing temperature. 2. Lyotropic systems which show liquid crystalline behavior in solution when a certain critical concentration is exceeded (see Fig. 5-1). For a long time it has been known (Vorlander, 1923) that the stability of a liquid crystalline phase or a mesophase is due to the anisometric shape of the molecules of the mesogenic compound. If the constituent entities are elongated, i.e., rod-like, the long axes of the molecules will tend to lie parallel to each other. Due to thermal motion the resulting order in the mesophase will not be perfect. The determination of the degree of ordering will be discussed in the subsequent section. If, on the other hand, the molecules or entities have a disclike shape, the normals of the discs will approximately align parallel. Figure 5-2 and 5-3 display both types of shape and the ordering occurring in these systems. Rod-like or calamitic liquid crystals (Fig. 5-2) may either form a nematic or smectic mesophase (de Gennes, 1975; Chandrasekhar, 1977). In the former type of ordering only the orientational alignment is preserved, whereas the correlation of the centers of gravity is lost completely. Twisting the average direction of the long molec-

Thermotropic liquid crystals crystalline solid

liquid crystalline phases

melting point

isotropic liquid

clearing point

Temperature

Lyotropic liquid crystals isotropic solution

liquid crystalline solution

critical concentration

Concentration

Figure 5-1. System of liquid crystalline transitions.

5.1 Introduction

nematic

smectic

cholesteric

Figure 5-2. Schematic representation of the molecular order in the different types of phases formed by calamitic (rod-like) liquid crystals.

isotropic

nematic

columnar

Figure 5-3. Schematic representation of the molecular order in the different types of phases formed by discotic (disc-like) liquid crystals.

ular axes results in a cholesteric mesophase (Fig. 5-2). In smectic mesophases the centers of gravity are approximately located within layers. There are many types of smectic liquid crystals indicated as SA, SB, S c ,..., Sj (Vertogen and de Jeu, 1988; Gray and Goodby, 1984). They differ with regard to: 1) The orientation of the preferred direction of the molecules with respect to the layer normal (orthogonal and tilted smectic liquid crystals); 2) the organization of the centers of the molecules within the layers. A detailed classification and description of the various smectic phases may be found in the books by Vertogen and de Jeu (1988) and by Gray and Goodby (1984). A similar classification may be set up for disc-like mesogens forming discotic liquid crystals (see Fig. 5-3). As in the case of calamitic systems there is a nematic phase only characterized by orientational order of the disc normals. An additional correla-

217

tion may be achieved through arranging the discs in columns. Figure 5-3 displays schematically the resulting columnar phase (Chandrasekhar and Ranganath, 1990). From the above discussion of the various types of mesophases it becomes evident that molecular shape is the principal factor responsible for liquid crystalline order. This intuitive insight is now well established by a great wealth of experimental data starting with the classical investigations of Vorlander and his school (Vorlander, 1923). Calamitic liquid crystals thus consist of rigid, linear entities as shown in Fig. 5-4. On the other hand, plate-like or disc-like structures will form discotic phases. Figure 5-5 displays three typical examples of molecular architecture leading to typical discotic mesogens. In most cases the anisometric shape can be deduced directly from the constitution. In some cases, as e.g., groups containing the carboxyl ester moiety, additional structural information is needed to determine the molecular shape. Suitable methods to approach this problem will be discussed in Sec. 5.3.

R

R

Figure 5-4. Molecular constitution of typical calamitic mesogens.

218

5 Structure of Liquid Crystalline Polymers

wards the layer normal and if there is longrange correlation of the tilt direction of the various layers. In order to obtain liquid crystalline polymers (LCP) the rod-like or disc-like objects (mesogenic units) introduced above may be assembled to macromolecular structure as displayed in Fig. 5-7.

Figure 5-5. Molecular constitution of typical discotic mesogens.

All examples shown in Figs. 5-4 and 5-5 refer to entities held together by covalent bonds. In lyotropic systems anisometric objects may result from association of many single molecules to rod- or disc-shaped objects (Rancon and Charvolin, 1988). The discussion of the principal reasons for liquid crystalline ordering up to now tacitly assumed bodies of rotation as rods or discs (see Fig. 5-2 and Fig. 5-3). The foregoing discussion of real molecular shape, however, reveals, that mesogens need not to exhibit perfect rod-like shape but rather have the symmetry of a book. Hence, nematic phases of such objects characterized by a second axis of molecular anisotropy, i.e., book-shaped entities as shown in Fig. 5-6 may exhibit a biaxial nematic phase. Here the parallel arrangement of the "books" lead to two axes of longrange orientational order and is hence called a biaxial nematic phase (Hessel et al., 1987; cf. also Ballauff, 1988). In smectic phases such an optical biaxiality is achieved when the molecules are tilted to-

Figure 5-6. Formation of a biaxial nematic phase of lath-like particles: ordering of the long-molecular axis together with the alignment of second molecular axis.

mesogenic unit

calamitic

discotic

main-chain Icp

side-chain Icp

TTi

liquid crystalline elastomers

Figure 5-7. Schematic representation of the types of liquid crystalline polymers through different arrangement of the mesogenic groups in the main- or in the side-chain.

5.2 Classification, Description, and Theory of Liquid Crystalline Order

They may either be a part of the main chains leading to main-chain LCP or constitute side-chains. The latter class of materials is called side-chain LCP accordingly. In both cases the macromolecules can be cross-linked to yield the respective liquidcrystalline elastomers. The present chapter is devoted to a discussion of the structure of these materials in solid state and in the various mesophases. In Sec. 5.2 we will give a brief general description of the mesophase transition and of the characterization of molecular order. Sec. 5.3 is devoted to main-chain LCP and Sec. 5.4 to side-chain LCP as defined in Fig. 5-7.

5.2 Classification, Description, and Theory of Liquid Crystalline Order in Polymers 5.2.1 General Characterization

As mentioned above liquid crystalline phases are usually very turbid due to the strong scattering of light occurring in these systems. This is caused by the pronounced spatial variation of the refractive index because of the different orientations throughout the sample. The existence of mesomorphic order thus may be inferred directly by visual inspection or by viewing the material between crossed polarizers in the polarizing microscope. The textures thus observed (see below) are very often characteristic of the particular mesophase and may be used for its identification. A useful book with photographs of typical textures has been written by Demus and Richter (1978). A transition from one mesomorphic state to another or to the isotropic phase may be studied by dilatometry and by calorimetry. The former method still requires a larger amount of sample often not available in

219

case of liquid crystalline polymers. Modern calorimetric methods such as differential scanning calorimetry (DSC) or differential thermal analysis (DTA) only need minute amounts of materials and provide a well established tool for studying mesomorphic transitions. Figure 5-8 shows a DSC-recording displaying the various transitions occurring in the low-molar-mass LC p,p'-dinonylazobenzene (Vertogen and de Jeu, 1988). It is clear that the transitions exhibited by this compound are of first order. Transitions between different mesophases, as e.g., the smectic-to-nematic transition, can be of second order too. DSC-analysis often is not sufficient to settle the question for the order of transition since the latent heat often is very small and the usual DSC-recording includes

"C9H19

K 37 S B 40 S A 53 I

40 50 Temperature in °C

Figure 5-8. Thermal analysis of a low-molecular weight liquid crystal. The material first transforms from the crystalline to the smectic B phase, then from smectic B to smectic A, and finally to the isotropic state. All transitions are of first order (Vertogen and de Jeu, 1988).

220

5 Structure of Liquid Crystalline Polymers

the strong pretransition effects (de Gennes, 1975). Here measurements of the specific heat or of the change of volume at the transition point are required. Upon cooling, the transition from the mesophase in the crystalline state can be often supercooled considerably or suppressed altogether. At lower temperature usually a glass-transition intervenes and the order is frozen in to yield a mesomorphic glass. Figure 5-9 gives a typical example recorded for a side-chain LCP (Hessel et al., 1987). A tool of utmost importance for determining the state of order in a LCP is wideangle X-ray scattering (WAXS). Smectic order will reveal itself immediately by the presence of a strong Bragg-peak due to the presence of layering. The intensity of the Bragg-peak and of its higher orders gives direct access to the details of molecular ordering whereas the characteristic shape of the WAXS recording allows assessment of the extent of ordering (Vertogen and de Jeu, 1988). By this method it could be shown (Als-Nielsen et al., 1980) that fluctuations in the smectic layers are responsible for the absence of true long range order. Since many mesophases can be oriented

easily by application of a magnetic field or mechanical flow, WAXS analysis of the resulting oriented specimens very often gives full access to all parameters characterizing order on a molecular scale. Examples obtained from particular LCP will be discussed in Sees. 5.3 and 5.4. Advanced techniques of electron microscopy can give direct images of smectic layers and their characteristic distortions (Voigt-Martin and Durst, 1989). Electron diffraction often is indispensable when dealing with very small crystallites or with a material which cannot be oriented macroscopically (Lieser, 1983).

5.2.2 Definition and Determination of Molecular Order 5.2.2.1 Nematic Phase As outlined in Sec. 5.1 the nematic phase is characterized by an average alignment of the long molecular axes. The local direction of alignment is usually described by a unit vector n9 called the director. The states referring to +/i and — n are indistinguishable, otherwise a ferroelectric phase would

cu,

o

280

300

320 Temperature in K

340

360

Figure 5-9. Thermal analysis of a side-chain liquid crystal polymer. The diagram displays at low temperature the glass transition, the peak at higher temperatures relates to the first-order transition from the liquid crystalline state (biaxial nematic phase) to the isotropic liquid (Hessel etal., 1987).

5.2 Classification, Description, and Theory of Liquid Crystalline Order

result. In the actual sample the orientation of n depends on the boundary conditions as, e.g., applied external fields. Without special boundary conditions the direction of n will vary in space which is followed by differences in birefringence. This gives rise to the characteristic textures when viewing such a sample with the polarizing microscope (cf. Sec. 5.2.1). For the full description of molecular order the local direction of n and the amount of ordering have to be defined. In case of an uniaxial nematic phase we hence need to consider the average projections of the three molecular-fixed axes u, v, w (see Fig. 5-10) onto the preferred axis. Without loss of generality the latter direction may be chosen to be identical with the z-axis of the laboratory-fixed coordinate system. Characterization of molecular order cannot be done, however, by specifying the thermal average of cos 0 since this would violate the equivalence of +n and — n; i.e.,
u,v9w;

a,b: x, y, z

where the ia denote the cosines of the angle formed by the axis a (x, y, z, laboratory frame) and i (u, v9 w, molecule frame). For the present case of an uniaxial phase only the components S™, S™, and S™ are of importance. Thus S™ = f<3sin 2 6>cos 2 Svzvz = f < 3 s i n 2 0 s i n 2 l P - l >

(5-2)

221

Z,i

Figure 5-10. Definition of the two order parameters characterizing the uniaxial nematic phase: The first parameter S(=S™™) determines the average orientation of the long molecular axis with regard to the preferred direction; the second parameter D gives a measure of the molecular biaxiality (see text for further explanation).

Since only two of these three quantities are linearly independent it is expedient to define D = S% - Svzvz = f <sin 2 0

(5-3)

Also, (5-4)

is the so-called order parameter. With the aid of these two order parameters it is easy to show (see, e.g., Buka and de Jeu, 1982) that molecular order can be directly assessed, e.g., by measuring the magnetic susceptibility of the nematic phase. For the uniaxial phase the magnetization M per unit mass follows as In the case of an uniaxial nematic phase with director along the z-axis, x is diagonal Z±,Z||)- Herezii refers direct-

222

5 Structure of Liquid Crystalline Polymers

ly to the component of % along the magnetic field and x± t o the component perpendicular to the field direction. In the isotropic phase the average of x is given by

For the nematic phase it is expedient to define the anisotropy Ax by:

In case of molecular diamagnetism the effect of internal fields can be disregarded in excellent approximation. Hence, we obtain for N molecules (see, e.g., Buka and de Jeu, 1982): Ax/N = Ax-S + Ax+ • D

(5-8)

with l\X

— Xw

"2 V^MM "•"

^w)

ever, there is a biphasic gap of finite width. This reflects the fact that polymers are usually multicomponent systems consisting of distribution of different molecular weights whereas PAA is a true one-component system, of course. Lowering the temperature leads to an increase of Ax pointing to increasing S (and D) through Eq. (5-8). The theoretical description of the dependence of the order parameter S on temperature will be discussed in Sec. 5.2.3. The foregoing considerations demonstrated that the measurable quantity A# (Eq. (5-8)) depends on S and D. In general, the distribution of molecular orientations is described by a distribution function / ( 0 , W). For cylindrical molecules this function depends only on the angle © and

(5-9)

and Ax+ =\(xuu-xvv)

(5-10)

where x is the molecular tensor of diamagnetic susceptibility. For a number of low-molecular-weight liquid crystals D has been shown to be rather small and the second term of (5-8) may be disregarded accordingly (Kiefer and Baur, 1989). Applying special NMRtechniques Esnault et al. (1989) were able to show, however, that D may be of appreciable magnitude in special cases (cf. also Dries et al., 1991). Figures 5-11 and 5-12 show a comparison of measurements of Ax obtained for the typical low-molecular weight LC PAA (p-azoxyanisol) (Dries et al., 1991) and a typical side-chain LCP both forming nematic phases below a transition temperature Tni (Fuhrmann et al., 1992). The strong discontinuity of Ax at Tni is clearly seen and results from the fact that the nematicto-isotropic transition is of first order (see Sec. 5.1). In case of the polymer LC, how-

o 360

380 400 Temperature in K

Figure 5-11. Anisotropy of the diamagnetic susceptibility (Eq. (5-7)) of the low-molecular-weight nematic liquid crystal PAA (p-azoxyanisol) as function of temperature. In the isotropic phase A% is zero by definition, below the transition temperature to the nematic state it assumes a finite value given by Eq. (5-8). Full circles: data of Dries et al. (1991), open circles: data of Buka and de Jeu (1982).

5.2 Classification, Description, and Theory of Liquid Crystalline Order

223

D-5T

Figure 5-12. Anisotropy of the diamagnetic susceptibility Ax as function of temperature for a sidechain liquid crystal polymer (Fig. 5-11). The strong rise at the transition temperature is clearly visible. Since the material represents a multicomponent system, there is a finite biphase region until a homogeneous nematic phase is formed (Fuhrmann et al., 1992).

0.0 340

360 380 Temperature in K

may be expanded into series by:

f(@)=

l)SlPl{cos0)

(5-11)

1=0

where Pl (cos 0) denotes the Legendre polynomials. Here / only assumes even values owing to the equality of states referring to + « and —n. Hence S as defined through Eq. (5-4) is the second moment of f(0):

= j P 2 (cos0)/(0)d(cos<

(5-12)

Experimental determination of the higher moments S4, S6 requires experimental access to quantities governed by and higher powers thereof. For low-molecular-weight LC this has been achieved through employing a special Raman-technique (de Gennes, 1975). In case of LCP the order in the nematic phase can be frozen in

400

by rapid cooling below the glass transition. NMR techniques applied to these solid samples allow the determination of the full distribution function f(0) (Boeffel and Spiess, 1989). In low-molecular weight LC the order parameter S usually is located between 0.6 at the freezing point and 0.35 at the transition point (see, e.g., Kiefer and Baur, 1989). In LCP very often S assumes much higher values. In this case the higher moment of f(0) must not be disregarded since they can be of appreciable magnitude. This fact reflects the poor convergence of the series in Eq. (5-11) (Boeffel and Spiess, 1989). 5.2.2.2 Smectic Phases Smectic phases differ from the nematic state by the organization of the molecules within layers (see Fig. 5-2). Thus it does not

224

5 Structure of Liquid Crystalline Polymers

suffice to quantify the alignment in terms of S and D (see previous section) but the extent of correlation of centers of gravity of the molecules has to be done also. Following McMillan (1971, cf. also Vertogen and de Jeu, 1988) this may be achieved through an additional order parameter a defined by a =
Bend

Splay

(5-13)

where P2 denotes the second Legendre polynomial, 0 the angle between the molecular long axis (w-axis, see Fig. 5-10) and the preferred direction being identical with the z-axis. This direction is also the layer normal and a is the thickness of the layers. The order parameter o thus yields a measure of the strength of the coupling between orientational and positional order. For further details the reader is referred to the treatises of Chandrasekhar (1977) and of Vertogen and de Jeu (1988). 5.2.2.3 Local Distortions in Nematic Phases The previous considerations assumed that the nematic phase can be described by a single order parameter S (when D is neglected) with regard to a preferred direction which does not vary throughout the phase. However, in most practical circumstances this nematic single crystal will not be compatible with boundary conditions imposed by surfaces of the sample or by external fields. There will be some deformation of the alignment. These distortions can be defined in terms of a splay, bend, and twist deformation of the nematic director (Sec. 5.2.2.1) defined in Fig. 5-13 (cf. de Gennes, 1975): As it turns out for nearly all situations of interest in LC and LCP the distance over which significant changes of n occur are much larger than typical molecular dimensions. Therefore these deformations may

Twist

Figure 5-13. Definition of the different modes of static distortion (elastic constants) in a nematic liquid crystal (after de Gennes, 1975).

be described in terms of a continuum theory which disregards the details on a molecular scale. In the frame of this approach the state of order in the system is described in terms of a director field n(r) and three elastic constants refering to the splay, bend and twist distortion of n(r) defined in Fig. 5-13. Because of the difficulties associated with the description of the apolar nematic order by a vector field the fully correct characterization results from specifying the tensor order parameter as defined in Sec. 5.2.2 (Vertogen and de Jeu, 1988). For most practical purposes, however, n(r) suffices to explain the experimental observations. If n(r) varies throughout the sample as dictated, e.g., by the boundary condition (Fig. 5-13) the free energy of the system is only at minimum for certain types of distortion (Frank, 1958; de Gennes, 1975; Kleman, 1991). These distortions are characteristic for a certain type of mesophase order as, e.g., nematic or S^-order. In low-molecular-weight LC the resulting textures viewed under the polarizing microscope between crossed polarizers may therefore be used to identify the respective mesophase (de Gennes, 1975; Chandra-

5.2 Classification, Description, and Theory of Liquid Crystalline Order

sekhar, 1977; Demus and Richter, 1978). For LCP these patterns can be made directly visible by a special decoration technique in the electron microscope (Thomas and Wood, 1985; Hudson and Thomas, 1989, Hudson et al., 1990). Figure 5-14 shows the image of a typical defect as obtained by transmission electron microscopy on a main-chain LCP (Hudson and Thomas, 1989). The director field has been decorated via the controlled growth of small lamellar crystals which are normal to the local director. Such patterns may be used to evaluate the magnitude of the elastic constants of the splay, bend, and twist deformation as defined in Fig. 5-13. Virtually all technical applications of LCP require the fully aligned state, i.e., the nematic single crystals. Monitoring the dissolution of such a distortion under the action of external fields is therefore highly important and of technical interest (Lee and Meyer, 1991; Moore and Stupp, 1987; Fuhrmann et al., 1991).

225

5.2.3 Fundamentals of Molecular Statistical Theories of Liquid-Crystalline Order

In this section we turn to a theoretical description of molecular order. Since this problem has been dealt with in various treaties on low-molecular-weight LC (de Gennes, 1975; Stephen and Straley, 1974; Vertogen and de Jeu, 1988) a short summary of the essentials may be sufficient here. Particular emphasis will be laid on features specifically arising from the macromolecular nature on the LCP. A phenomenological theory of the nematic-isotropic transition has been proposed by de Gennes (1975). It is based on the Landau theory of phase transitions (Landau and Lifshitz, 1971) and considers an expansion of the free energy in powers of the order parameter S in the following form: F = \AS2-\BS*+\CS*

(5-14)

Here the coefficient of the term linear in S vanishes because the isotropic state (S = 0) is the state of equilibrium above Tni. This expression predicts a discontinuous firstorder transition for B + 0. Putting F = 0 and dF/dS = 0 at the transition temperature Tni we obtain for the order parameter S(Tnl):

S(Tni) = 2B/3C

(5-15)

if terms of higher order are neglected in Eq. (5-14). For B = 0, the transition is continuous and A vanishes at the transition point. This is obvious from the fact that a stable disordered phase requires A > 0 for a minimum at S = 0. On the other hand, stability of the ordered phase (S # 0) leads to A < 0 for T
= a(T-T*)

(5-16)

where T* is a second-order transition temperature and a is a constant (cf. the Curie

226

5 Structure of Liquid Crystalline Polymers

temperature in case of a ferromagnet). If B>0, T* is smaller than Tni. For lowmolecular-weight LC {Tni-T*) usually is of the order of 1 K (de Gennes, 1975; Vertogen and de Jeu, 1988); for PLC higher figures are reported (Krigbaum, 1982). In principle, £ + 0 for nematic phases since states S and — S represent entirely different kinds of molecular arrangement. In the other case the molecules are more or less parallel to the preferred axis whereas S<0 refers to a state where the molecules are perpendicular to this axis. Therefore, symmetry requires the inclusion of B 4= 0 for the nematic-to-isotropic transition. This argument shows that this transition is of first order (de Gennes, 1975). From the preceeding Sec. 5.2.2 it became obvious that S may assume high values (0.8-0.9) in the ordered phase. Truncation of the series in Eq. (5-14) the third term thus renders this theory semiquantitative. Nevertheless, the Landau-de Gennes theory has met with gratifying success when used for the description of more complicated transitions in low-molecular-weight LC (Gramsbergen et al, 1986). It has been applied to LCP by ten Bosch et al. (1983) and it is indispensable for the description of networks composed of LCP (Warner et al., 1988). For further details the reader is referred to Sec. 5.3 and 5.4. The aforementioned Landau-de Gennes approach does not take into account molecular features and is entirely phenomenological. Molecular-statistical theories, on the other hand, may give direct insight into the factors governing the stability of ordered phases. To outline the essentials we shall deal mainly with nematic phases; smectic phases may be treated by extensions of the models described herein (Sec. 5.2.2.2). In general, molecular-statistical theories lead to a free energy of the form:

= F{{V,T) FX(V9T)

(5-17)

Here an uniaxial nematic phase has been assumed which is composed of cylindrical molecules. Therefore a description through a distribution f(0) is fully sufficient. (In a mean-field approach, 0 (Fig. 5-10) denotes the angle towards the preferred axis; in the Onsager approach (see below) the angle between two rods. This difference becomes small, however, for small values of 0) The term F{(V, T) is the ideal-gas term of F, the second term takes into account the decreases of entropy due to ordering (Onsager, 1949), and the third term refers to the excluded volume of the rod-like particles as function of molecular orientation. The various theories suitable for the description of LCP may be discussed best by looking at F^T): In the Maier-Saupe (1958, 1959, 1960) approach, F±(V9 T) may be expressed through (de Gennes, 1975): x

=

-±U(V,T)-S2

(5-18)

with U being a positive constant. In the original presentation of Maier and Saupe (1958,1959,1960) it was assumed that Fx is entirely due to anisotropic van der Waals forces. However, from the theory of the liquid state (Hansen and MacDonald, 1986) it is evident that the structure of a liquid is mainly governed by repulsive steric forces. The constant U as determined from experiments therefore contains influences of both, attractive as well as repulsive forces. Eq. (5-17) together with (5-18) may be resolved through application of variational calculus (see de Gennes, 1975) to yield the distribution function f(0) in equilibrium At the transition point, S(Tni) = 0.44, a value indeed being comparable to experimental values found for low-molecular-

5.2 Classification, Description, and Theory of Liquid Crystalline Order

weight LC. Despite its semi-quantitative character and the difficulties associated with the physical interpretation of U the Maier-Saupe approach has been applied frequently. Owing to the simplicity it has been used to treat rather complicated systems successfully (Wang and Warner, 1986). The deductions obtained therefrom turned out to be in qualitative agreement with experimental data (see Sec. 5.4 for a further discussion of this point). Among the theories taking into account the steric, i.e., repulsive forces between the molecules two models have found widespread application to LCP:

227

Figure 5-15. Calculation of the excluded volume for the binary interaction of two rods: At dense packing each rod excludes of volume of the order of Ld2; in dilute solution the excluded volume is of the order of L2d (after Onsager, 1949).

scale as d2 L/L2 d = d/L = l/x with x denoting the axial ratio L/d of the rods. Evaluat(i) The Onsager theory (Onsager, 1949; reing the free energy in Eq. (5-17) with F1 views: Odijk, 1986; Khokhlov, 1991). being substituted appropriately (Onsager, (ii) The Flory lattice model (Flory, 1956; 1949; see also Khokhlov, 1991) the critical reviews: Flory, 1984; Ballauff, 1989 a; volume fraction is indeed found to vary Abe and Ballauff, 1991). inversely to the axial ratio x. The order parameter calculated for the nematic phase Both theories start from the fact, that hard is of the order of 0.8. rods cannot penetrate each other. Starting In principle, the Onsager approach alfrom this premise both models develop an lows only the treatment of diluted systems expression for Fx (Eq. (5-17)) by evaluation because of the restriction to binary interof the partition function for a system of actions. This severe shortcoming can be hard rods. Since extensive reviews are partly remedied by introduction (Khokhavailable on both approaches (see above) it lov and Semenov, 1985) of the "decoupling suffices to delineate the essentials of the approximation" first presented by Parsons underlying models. (1979). A similar approach has been devel(i) Onsager theory: This model treats a oped recently by Lee (1987) and by Colot gas of long thin rods in the limit of the et al. (1988). second virial approximation. The excluded (ii) Another method to circumvent the volume per rod is calculated by the arguuse of the virial series has been presented ment shown in Fig. 5-15: For a dense packing of rods Vexc = d2 L where L denotes the by Flory (1956) through resort to a lattice model. Figure 5-16 displays the main idea length and d the length of the sides of the of this calculation: basal plane (Fig. 5-15). In dilute solution where binary interaction prevails Vexc folConsider a rod inclined by an angle W to lows as L2 d. When the excluded volume is preferred axis (dashed line). The evaluation increased by raising the concentration the of the partition function rests on the calcusteric interaction thus inferred will favor a lation of the number of ways to put a rod more or less parallel alignment over a ranonto cubic lattice already filled to a certain dom arrangement of the rods. From this extent. For this purpose the rod is broken argument the critical concentration should

5 Structure of Liquid Crystalline Polymers

5.3 Main-Chain Liquid Crystal Polymers

r i t!

Figure 5-16. Flory lattice model of the nematic state: Calculation of the partition function in terms of a lattice model. A rod inclined by an angle W towards to domain axis is broken up into submolecules of sequences of lattice sites parallel to the preferred axis (after Flory, 1956).

up into "submolecules", i.e., a sequence of filled lattice sites parallel to the preferred axis as shown in Fig. 5-16. If the angle W is not too high this device gives a reasonable result for the partition function. For a detailed assessment of the merits as well as of the problems of this approach the reader is referred to recent reviews (Ballauff, 1989 a; Abe and Ballauff, 1991). Here it suffices to state that the Flory lattice model has turned out highly useful for the description of LCP. In particular, it has been shown to be indispensable for the treatment of mixtures of LCP (Flory, 1978; Ballauff, 1989 a, b; Orendi and Ballauff, 1992). The latter aspect will be dealt with in more detail in Sec. 5.3.4.

Having dealt with general features of LCP we now turn to a more specific discussion of these materials. As outlined in the introduction LCP may be classified by the connectivity of the mesogenic groups. This has been shown schematically in Fig. 5-7. This section is devoted to the discussion of the first type, the main-chain LCP. Here the polymeric chains consist of mesogenic groups of the calamitic type (Fig. 5-4) or the discotic type (Fig. 5-5). In what is to follow the molecular architecture of mainchain LCP will be reviewed briefly. Special emphasis will be laid upon methods to determine the molecular shape. Subsequent sections will give a summary of the structure and the phase behavior in comparison to the respective molecular architecture of these materials. 5.3.1 Molecular Architecture of Main-Chain Liquid Crystal Polymers

In general, main-chain LCP consist of monomeric units of the calamitic or the discotic type linked together in a linear fashion (Fig. 5-7). In many cases monomeric units of a structure not being common in low-molecular-weight systems have been used to achieve a rod-like shape. Figure 5-17 gives two important examples of such materials, the polyimides based on pyromellitic anhydride and p-phenylenediamine 1, and the poly(p-phenylenebenzobisthiazole)s 2. From the latter material high-modulus fibers may be spun (Kwolek etal, 1987; Northolt and Sikkema, 1990). A rod-like shape of the repeating unit may be also realized without resort to an aromatic structure. Figure 5-17 shows an example, the poly(propellane)s 3 in which the linear shape is effected through a cyclo-

5.3 Main-Chain Liquid Crystal Polymers

—C

Figure 5-17. Examples of rod-like main-chain liquid crystal polymers (Ballauff, 1989 b).

aliphatic structure of suitable geometry (Schluter, 1988). In addition to this, a calamitic shape of the polymer chains can be introduced into flexible polymers by a helical superstructure. This building principle is often found in biopolymers such as DNA Xanthan, or a number of polypeptides like the poly(y-Lbenzylglutamate) (Uematsu and Uematsu, 1984). Through winding up several of these helical strands a nearly ideal rod-like shape may be effected. Finally, an even more complicated molecular structure as present in viruses may lead to a ideal cylindrical particle of colloidal dimensions. The bestknown example of the latter type is the Tobacco-Mosaic virus which has long been investigated and used as a model substance for studying the effect of molecular shape on the structure of the resulting liquid crystalline phases (Fraden et al., 1989, and further references given there). Up to now, discotic main-chain LCP have been mainly

229

realized using the triphenylene unit shown in Fig. 5-5. A moiety used very often for the design of calamitic main-chain LCP is the 4-oxybenzoate unit shown in Fig. 5-18. Here the rod-like shape depends strongly on the coplanarity of the ester bond, i.e., the average fluctuation of the angle x must be rather small (see Fig. 5-18). Owing to the nearly colinear phenyl groups in this unit, the variation of the torsional angles

oFigure 5-18. Calculation of the conformation of stiffchain polymers in terms of the rotational isomeric state model (Flory, 1969, 1974): Definition of the torsional angles for the p-oxybenzoate moiety.

230

5 Structure of Liquid Crystalline Polymers

mine the respective parameters will be given here. The conformation of main-chain LCP may be described in terms of the following models (Brelsford and Krigbaum, 1991): (i) The Rotational Isomeric State model (RIS; Flory, 1969) in which the chain conformation and shape is described using molecular parameters as conformation energies, torsional angles (see Fig. 5-19); (ii) The Kuhn model which idealizes the chain by a sequence of rods having the Kuhn-length /k put together by entirely flexible joints (Tsvetkov, 1989); (iii) The worm-like chain model by Kratky and Porod (Tsvetkov, 1989). Here the rendition of the chain is given a continuously bent stick or worm. Chain conformation is characterized by the persistence length ap; for very long chains 2aP = lk with /k being the Kuhn-length. The experimental determination of parameters necessary for the description in terms of the RIS model requires in most cases involved measurements of the depolarized light scattering of the electric and magnetic birefringence and the knowledge of the dipole moment of carefully chosen oligomers. Very often these data are not available. Therefore in this context we only

discuss methods for determining the persistence length of dissolved macromolecules. For this purpose the following methods are suitable (Tsvetkov, 1989; Ballauff, 1989b; Brelsford and Krigbaum, 1991): 1) Scattering methods (light-, X-ray, and neutron scattering); 2) Depolarized light scattering; 3) Electric and magnetic birefringence; 4) Flow induced birefringence. Among the above methods the scattering methods certainly give the most information using a minimum of assumptions. Therefore a brief description of these methods and their data evaluation for assessment of chain stiffness will be given here (see also Chap. 11 of Volume 2 a of this Series). The intensity of the light-, X-ray- or neutron beam scattered by a solution of macromolecules is determined as a function of polymer concentration c2 and the scattering angle 0. The experimental scattering intensity Iexp(c2,s) where s = (4n/X) sin(0/2) (0: scattering angle) is extrapolated to vanishing concentration. Then the data are plotted in the following way (Kirste and Oberthiir, 1982)

Kratky-Porod

Kc?

1 M W / B (S)

(5-19)

1+

Figure 5-19. Schematic representation of the different models to describe the conformation of polymer chains.

5.3 Main-Chain Liquid Crystal Polymers

where Mw is the weight-average molecular weight, <#g>1/2 the z-average of the radius of gyration, K is a constant depending on the radiation used and the system, and In(s) is the structure- or form-factor of the single molecule. The function In(s) is normalized to 1 for 5 = 0: Jn(0) = l. The radius of gyration 1/2 obtained from the slope of the above plot in the region of small angles gives a measure of the spatial dimensions of the molecule. From the latter quantity the Kuhn length Zk may be calculated if the molecular weight distribution is known (Kirste and Oberthiir, 1982). If the polydispersity is not known with sufficient accuracy this determination of Zk becomes insecure since the molecular weight distribution strongly influences this quantity. As has been demonstrated by Schmidt (1989) this problem may be circumvented in an elegant way by combined measurements of the static and the dynamic light scattering where the latter method yields also the hydrodynamic radius of the chain. Since the quantity only depends on the weight average of the degree of polymerization, the polydispersity as well as the stiffness of the dissolved chains may be inferred in an unambiguous way. Further information is accessible from the form factor In(s). The determination of In(s) very often requires measurement by X-ray and neutron scattering since the range of s accessible in light scattering is too small in most cases. Figure 5-20 shows a reduced Kratky plot In(s • a p ) 2 versus s • ap calculated for wormlike chains differing in chain stiffness (Kirste and Oberthiir, 1982). For a better comparison the reduced variable s • ap has been used instead of s. The parameter n indicated on the curve is the ratio of the total length of the chain L to the persistence length ap. Small values of n = L/ap

231

Figure 5-20. Form factor of worm-like chains in a Kratky-plot. The quantity s is the magnitude of the scattering vector and ap is the persistence length, which is a measure of the stiffness of the chains. The numbers indicated on the respective curves are the ratios of the contour length of the chain to the persistence length; a small n indicates a stiff chain, a large value is characteristic for flexible polymers (after Kirste and Oberthiir, 1982).

in consequence indicate high stiffness, flexible macromolecules are characterized by higher values of n. In the case of flexible chains (n = 150) at first a strong increase of In{s • ap) is observed followed by a plateau like region and finally a practically linear region. Stiff chains, however, are represented by virtually straight lines. A comparison of Fig. 5-20 with experimental form factors may lead to an estimate of the chain stiffness. A problem in this comparison is the finite diameter of the polymer chains leading to additional interferences not being taken into account in the above worm-like chain model of Kratky and Porod. This effect is seen from a downward curvature of I(s-ap)2 in the Kratky-plot at higher values of s. The importance of this problem for a correct interpretation of In(s) has been demonstrated recently by Rawiso, Duplessix and Picot (1987).

232

5 Structure of Liquid Crystalline Polymers

All methods discussed above only allow the determination of the conformation of the single molecules in solution. A determination of the form factor of macromolecules in melt or solid state only can be done by neutron scattering on a mixture of protonated and deuterated molecules (Wignall, 1987). Up to now numerous studies have been performed using the above methods to characterize the shape of rod-like macromolecules. A review may be found in the book of Tsvetkov (1989). In particular, there is quite a number of carefully conducted studies on soluble polypeptides (Schmidt, 1989). On the other hand, there are only a few results available on the oxybenzoate polymers and related macromolecules. The reason for this is the often very limited solubility of these materials in common organic solvents. As pointed out above, nearly all methods discussed above require studies of isolated molecules in solution. We will turn to this problem when dealing with special main-chain LCP. To the author's best knowledge no data are yet available of the conformation of discotic main chain polymers. 5.3.2 Calamitic Main-Chain Liquid Crystal Polymers

As discussed in Sec. 5.3.1 rod-like unities can be used for the design of calamitic main-chain LCP when put together in a linear fashion. In the following section we discuss two well-investigated polymers of this type as examples: the poly(p-phenyleneterephthalamide) 1 (Morgan, 1977) and the poly(p-oxybenzoate) 2 (Kricheldorf and Schwarz, 1990, and further references given therein) (Fig. 5-21). The polyamide 1 only dissolves in strong mineral acids or solvents which can break the hydrogen bonding between the chains.

The polyester 2 is entirely insoluble in any known solvent. Since the extended chains of polyamide 1 lead to a high mechanical strength when spun to fibers, materials of this kind have aroused much technical interest recently (Northolt and Sikkema, 1990). However, the low solubility of typical rod-like polymers makes the processing of these materials very difficult. The high melting point (if any is present at all) and the poor solubility both derive from the fact that dissolution of the crystal structure requires breakage of many enthalpic contacts between the chains. On the other hand, there is only a minor gain in entropy since the stiff chains experience no alteration of shape regardless of the state of aggregation. In contrast to this, flexible macromolecules may assume a vast number of conformations in melt or in solution. Melting or dissolving of a flexible coil is therefore accompanied by a substantial increase of entropy which lowers the transition points accordingly. The regular rod-like shapes of polyamides and polyester like 1 and 2 (Fig. 5-21) clearly lead to a highly ordered and fully crystallized solid state. The structure of polyamide 1 has been elucidated by Northolt (1972) and by Tadokoro (1979). An extended discussion of the structure of the polyamide based on atomistic modelling methods has been given by Rutledge etal. (1990). Fig. 5-22 shows the arrangement of the chains in the unit cell accord-

Figure 5-21. Repeating units of important calamitic main-chain liquid crystal polymers discussed in the text.

5.3 Main-Chain Liquid Crystal Polymers

Figure 5-22. Structure in solid state of the stiff-chain polymer poly(p-phenylene terephthalate) according to Northolt (1974).

ing to Northolt's structure. Two chains pass through the cell, one through the center and the other through the corners. Hydrogen bonds are formed between adjacent chains lying in the (100) plane. The poly(p-oxybenzoate) 2 (Fig. 5-21) has been investigated since a long time (Kricheldorf and Schwarz, 1990; and Ballauff, 1989 b, for references). The first systematic study of the poly(4-oxybenzoate) was conducted by Economy and coworkers (Economy et al., 1976) who also investigated the structure by means of electronand X-ray diffraction. A major improvement of the synthesis and subsequent characterization with regard to molecular weight of the polymers has been achieved by Kricheldorf and Schwarz (1983). Starting from the latter investigations Lieser (1983) succeeded in 1983 to determine the phase behavior and the unit cells of the modifications of the poly(4-oxybenzoate) by electron diffraction. According to Lieser the poly(4-oxybenzoate) forms two orthorhombic modifications I and II at room temperature. Both modifications are transformed through a first-order transition at 330 °C into a pseudohexagonal

233

phase III. Cooling down to room temperature again leads to a mixture of modifications I and II together with structures containing a number of defects. Further transitions at still higher temperatures have been investigated recently by Economy etal. (1988) and by Kricheldorf and Schwarz (1990). From their studies the latter investigators state that poly(p-oxybenzoate) does not melt below 500 °C. Above this temperature rapid degradation and chemical rearrangement sets in. A detailed model of molecular packing was recently suggested by Yoon etal. (1990) and by Hanna and Windle (1988 b). The absence of a melting process of the poly(4-oxybenzoate) renders the processing of this polymer by the usual techniques impossible. Molding may be achieved only through high energy forging. Thus the poly(4-oxybenzoate) has not found yet any technical application. This problem being typical for stiff-chain polymers may be circumvented in some cases, e.g., poly(l,4phenylene terephthalate) which exhibits a sufficient solubility in concentrated sulfuric acid (see above; Northolt and Sikkema, 1990). However, it is obvious that processing from such aggressive solvents causes a number of technical problems. To circumvent such processing difficulties and to shift the melting point below the temperature of decomposition, the following modifications of the structure of LCP have been made (see Fig. 5-23): 1) Insertion of flexible comonomer units ("spacers") such as n-alkylene chains; 2) Inclusion of bent units or units of different size. This leads to a disturbance of the crystal structure and a lowering of the melting point; 3) Appending of flexible side chains onto the stiff main chains. Here the side chains act in a way similar to a "bound

234

5 Structure of Liquid Crystalline Polymers

solvent" which lowers the interaction between the main chains. Also, the side chains lead to a strong gain in entropy when dissolving or melting the polymer. Figure 5-24 gives typical examples of structures used to realize the three concepts. In the following section a brief review on the structure and phase behavior of the materials deriving from the respective concepts will be given. 5.3.2.1 Liquid Crystal Polymers with Flexible Spacers in the Main Chain

In the following we discuss the LCP deriving from the first concept shown in Fig. 5-23. Here the rod-like units are connected through flexible spacers like n-alkyl chains or ethyleneoxy-moieties. The polyesters 1 to 3 of Fig. 5-24 represent typical examples of such materials (cf. Lenz, 1985; Yoon

Flexible Spacers

-I

tWWWl

IWWWl

hA/WM

Bent or crankshaft - like comonomers

Flexible side chains

Figure 5-23. Concepts to overcome the low solubility and the high melting point of stiff-chain polymers (Ballauff, 1989 b).

et al, 1985; Blumstein et al., 1985; Blumstein, 1985; Sirigu, 1991). Here the flexible spacer leads to a strong increase of entropy when going from the solid state to the melt on the solution since the entire chain can assume a vast number of conformations. Thus there polymers are expected to behave like flexible polymers when brought into an isotropic melt or solution. On the other hand, polyesters 1-3 of Fig. 5-24 do form nematic and smectic mesophases. Hence these materials must undergo major conformational changes when going from the disordered to the liquid crystalline state. The obvious consequence of this is the strong change of entropy at the nematic-to-isotropic transition. This is shown for polyester 3 of Fig. 5-25 as function of n, the number of carbon atoms in the spacer (Blumstein, 1985). Also, the order parameter S at T = 0.98 Tni exhibits a strong alternation with n. These data demonstrate that the spacer does not play merely the role of a solvent but takes part actively in the ordering process. Thus it is now well established that the nature of the spacer groups is of great importance the stability of the nematic state. If n is an odd number, the subsequent rod-like entities are not colinear if the spacer assumes an all-trans conformation. Therefore more conformational transgauche inversions have to be introduced for an odd number n of methylene units than for the even member of the series in order to align the calamitic units along the nematic director (Blumstein, 1985). Therefore there is a drastic reduction of AHni and ASni when changing the methylene spacer to the more flexible ethyleneoxy spacer, i.e., when going from polyester 1 to polyester 2 (Fig. 5-25 and Yoon et al., 1985). The influence of the conformational characteristics of the chain on the stability of the ordered phase has been put into

5.3 Main-Chain Liquid Crystal Polymers

235

Main-chain liquid crystal polymers With flexible spacers

C-O-(CH2)rrO-

;

C-O-(CH 2 -CH 2 -O) n -

2

0

CH3

CH3

With bent or crankshaft-like units

With flexible side chains 0

OCnH2lHi \

0

-o-(o)-o-c-(o)—c—

HO

7

OC n H 2 n + 1

COOH

OC n H 2 n + i

\ II \ —NH—(o)— NH—C—(OJ

OC n H 2 n 4 i OC n H 2 n + 1

II C

8

OC n H 2n+1

I 0

10 Figure 5-24. Examples of main-chain liquid crystal polymers deriving from the concepts delineated in Fig. 5-23. References: 1) and 2): Lenz, 1985; Yoon et al., 1985; 3): Blumstein et al., 1985; Blumstein, 1985; 4): Calundann and Jaffe, 1982; 5): Rosenau-Eichin et al., 1988; 6): Matsuda, 1979; Cageao et al., 1990; 7): Ballauff and Schmidt, 1987; 8): Ballauff, 1989 b; 9): Hermann-Schonherr et al., 1986; 10): Stern et al., 1991; 11): Wenzel et al., 1987.

OC n H 2 n + i

11 OC n H 2 n < l

5 Structure of Liquid Crystalline Polymers

236

I

-0.80

Ar!\ i -

M

i\\ A / \\ ;/'// I 1' »\\ /' \\ // I v // I'/ 'V'1 / I V 1/

12 -

/ / /

/ // /

k -

2 0

0

.

i

i

1

2

3

n

\\ Ji

/' 1

ll/l ,\//

\ /,' »\ »\ '

»•' I f

/,

-0.60

1 f i;

If 1/ ti

\4

i

fi

1 A

-0.70

1 I

\i

•

i

l\

1 1*

6 -

1

- 0.50

JL

.

1

I

i

i

i

1

k 5 6 7 8 9 10 11 12 Number n of CH2 groups

=»(H0

from the isotropic to the nematic state (de Gennes, 1984; ten Bosch etal., 1983; Wong and Warner, 1986). A recent study by D'Allest et al. (1988) employing smallangle neutron scattering indeed revealed a strong extension of the chains along the preferred direction in the nematic phase. Figure 5-26 shows as an example the determined values of Rg parallel and perpendicular to the nematic director of polyester 3 (n = 10, Fig. 5-24) dissolved in the low-molecular-weight-LC p-azoxyanisol (PAA). This plot shows furthermore that the alterations effected by nematic order increase markedly with molecular weight. A similar result is found when performing the investigation in melt (D'Allest etal., 1988). Thus these finding clearly reveal the strong coupling between orientational and con-

Figure 5-25. Odd-even effect in LCP with flexible n-alkyl units in the main chain measured for system 3 of Fig. 5-24. Left axis: transition entropy; right axis: order parameter (Eq. (5-4)). After Blumstein (1985).

RG. R/A

150

more quantitative terms by Yoon and coworkers (Yoon et al., 1985) and by Abe and coworkers (Abe and Furuya, 1989). The latter authors studied the order parameter by Deuterium NMR-spectra. These investigations demonstrated that the conformational characteristics of the polymer is very similar to the result found for "twin-dimers", i.e., on model oligomers consisting of two rod-like mesogenic units connected by a single spacer group (Furuya etal., 1991). The foregoing discussion has shown that liquid crystalline ordering in polyesters of the type 1-3 (Fig. 5-24) requires conformational rearrangements as well. Therefore it is expected that the overall dimensions of the chain expressed in terms of the radius of gyration will increase when going

R

±

(A) ISOTROPIC

ANISOTROPIC

R// A-A—A0 : 6 000 A : 18000

100 -

50

100

120

A

A

RG

r

-0

T

I uo

°r

160

•m Figure 5-26. Effect of nematic ordering on the overall dimensions of LCP with flexible units in the main chain: Small angle neutron scattering study of the radii of gyration of polymer 3, Fig. 5-24, dissolved in the low-molecular-weight LC PAA. Measurements have been done in the isotropic and in the nematic state. The orientation due to liquid crystalline state allows the measurement of the radius of gyration parallel (.R||) and perpendicular (R±) to the preferred direction. The difference which depends on molecular weight clearly reveals the extension of the coils along the preferred direction in the nematic phase (D'Allest et al., 1988).

5.3 Main-Chain Liquid Crystal Polymers

formational order in LCP with flexible spacers in the main chain. All systems discussed up to now are homopolymers with a defined sequence of rod-like and flexible units. Stupp and coworkers (Moore and Stupp, 1988; Martin and Stupp, 1988; Stupp et al., 1988) prepared copolyesters from rigid and flexible units and were able to show that the sequence of these units along the chains has a profound influence on the phase behavior. In particular, chemical disorder along the chain leads to species differing with regard to the persistence length. Stupp etal. (1988) advanced the concept of "polyflexibility" to account for the observation of a broad nematic-isotropic biphase range found in these systems. This coupling of the chemical disorder of the chains to the nematic order has been analyzed by Fredrickson and Leibler (1990) in terms of the Landau theory. The deductions from this theory corroborates the analysis developed by Stupp etal. (1988). Thus these considerations lead to the conclusion that the biphasic gap observed at the nematic-to-isotropic transition temperature of these copolymers cannot be explained only in terms of the nonuniformity of molecular weight but must by traced back mainly to the polyflexibility in these systems.

5.3.2.2 Liquid Crystal Polymers with Bent or Crankshaft-Like Comonomers

Due to the strong industrial interest in stiff-chain macromolecules for fabrication of high-strength fibers a great number of copolyesters and copolyamides have been made and patented from fully aromatic hydroxycarbon acids and bisphenols being easy accessible. Figure 5-24 shows three particular examples structures 4-6 which

237

serve for elucidation of the main features of these polymers. The first shown herein is the copolyester 4 consisting of 2-oxy-6-naphthoate and 4-oxybenzoate units ("Vectra", HoechstCelanese, Calundann and Jaffe, 1982). It is an example of the disturbance of crystallization by comonomer units of different size leading to crankshaft-like structure along the polymer chain. The next system 5 built-up from 3- and 4-oxybenzoate units is a series of linear and bent units of approximately (Schwarz, 1981; RosenauEichin et al., 1988) equal size. This copolyester is particularly suited for structural investigations since the phase behavior of the poly(4-oxybenzoate) is well-known. The third example, the copolyamide 6 (Ozawa etal., 1953; Matsuda, 1979) can assume either an extended conformation or a coiled state. This is due to the 3,4'linkage of the diaminodiphenylether in which the 60° bend of the ether bond can be straightened out again by the meta-linkage. Thus the influence of intermolecular interactions on the conformation of the chain can be studied directly. An important question when dealing with these systems is the distribution of the monomer units along the chain. Since the copolymer 5 from m- and p-hydroxybenzoic acid exhibits sufficient solubility in CHC13 up to a contents of 40% of the linear p-component a sequence analysis by means of 13 C-NMR spectroscopy in solution can be obtained (Kricheldorf and Schwarz, 1983; Rosenau-Eichin etal., 1988). This shows that the monomer units are distributed at random along the chain. In case of insoluble products a sequence analysis may be achieved in special cases by means of wide-angle X-ray scattering of solid oriented samples. This method of analysis has been worked out by Blackwell and coworkers (Biswas and Blackwell,

238

5 Structure of Liquid Crystalline Polymers

1987) and by Bonart and coworkers (Bonart 1987; Bonart etal., 1988) and has been applied to a number of copolymer systems. In this treatment the intensity of the meridional reflections of samples with sufficient orientation is compared to results of model calculations which explicitly simulate the deviation from a random sequence of the monomeric units. Using this method it has been shown that copolymer 4 (Fig. 5-24) is a statistical (Biswas and Blackwell, 1987) copolymer. A similar analysis of copolyamide 6 having the 3,4'linkage demonstrated not only the statistical sequence of the monomer units but also the extended conformation in the oriented solid state (Cageao et al., 1990). This result demonstrates the strong steric interactions of the chains in ordered systems which causes an extension of these flexible chains, i.e., an apparent increase of the persistence length when going from solution to the solid state (Sun et al., 1991). Another point of interest is the crystallinity of these copolyesters and copolyamides as well as the kind of distortion exerted from the bent or crankshaftlike comonomer onto the crystal lattice of the linear component. In principle this poses the question of crystallization in rod-like random copolymers. A number of studies of this problem conducted on copolymer 4 (Fig. 5-24) has been done by Windle and coworkers and has led to the concept of the "non-periodic layer crystallites" (NPL) (Windle et al., 1985). Here the formation of small crystallites is modelled by placing together parts of the chains with identical sequences (see Fig. 5-27). In this model, semicrystallinity thus may be brought about through lateral juxtaposition of parts of the chains having the same shape. The probability of finding a given fixed sequence of congruent units strongly diminishes with length of the se-

I I II B BA ABB AAA BBB A AA AAA BBB AAA BBB BBB BBB AAA ABB BBA ABA BAB

ABBA A A BA BABB ABAB AAAA AAAA BBBB AAAA BBBB BBBB BBBB A AAA BABA BAAA ABAA BABB

I I I I I II Figure 5-27. Scheme of the crystallization of stiffchain random-copolymers by formation of "non-periodic layer crystallites" (NPL): Ordered regions are formed through lateral juxtaposition of chains with the same sequence of monomer units (Windle et al., 1985).

quence. Therefore these NPC crystallites will have a rather limited size. These ideas have been further developed by Windle and coworkers (Hanna and Windle, 1988 a). A comparison with results obtained by electron microscopy gives hints that such ordered regions exist indeed (Hanna et al., 1992). The high crystallinity of copolyester 5 (Fig. 5-24) can be explained without the NPL concept since investigations by X-ray and electron diffraction have shown the modifications of the poly(4-oxybenzoate) to incorporate an appreciable amount of bent meta-units (Rosenau-Eichin et al., 1988). In particular electron diffraction studies of these materials can supplement detailed information on the kind of defects. Having discussed the phase behavior of the materials at room temperature, the question is how the disturbance of the comonomer units affects the phase behav-

5.3 Main-Chain Liquid Crystal Polymers

ior at elevated temperature. Investigations of copolyester J (Rosenau-Eichin et al., 1988) have shown that the phase behavior of 5 with a high content of linear units may be compared to that of the poly(4-oxybenzoate): At elevated temperature, modifications I and II transform into modification III which could be shown with electron diffraction. Cooling down to room temperature again leads to a mixture of I and II as observed in case of the homopolymer. Rapid cooling, however, allows to supercool phase III down to room temperature if the disturbance by meta-units is (>30%) is strong enough. These experiments suggest that by judicious choice of the ratio of linear and nonlinear units the degree of order in solid state may be adjusted within wide limits. This may be interesting for possible applications since the phase behavior and the mechanical properties are intimately connected to the crystalhnity of such materials. 5.3.2.3 Liquid Crystal Polymers Consisting of Rigid-Rod Polymers with Flexible Side Chains

The third concept delineated in Fig. 5-23 for improvement of solubility and for lowering the melting points is given by the attachment of flexible side chains onto rigid main chains. In contrast to the foregoing concepts (Fig. 5-23) the rod-like structure of the main chain is fully preserved. During the process of melting or upon dissolution, these side chains can adopt a vast number of configurations as in the case of flexible coils. In first approximation these side chains act as "solvent" affixed to the main chain. By variation of the number and the length of side chains attached on the repeating unit the amount of "bend solvent" may be changed continuously (Ballauff, 1986b). In spite of the

239

high stiffness of the main chain most of these comb-like polymers exhibit a sufficient solubility in organic solvents. The first comb-like polymer of this kind based on stiff-chain polyesters has been introduced by Lenz and coworkers in 1983 (Majnusz et al., 1983). It is a polyester based on terephthalic acid and 2-n-alkylhydroquinone. In accordance with the above considerations the melting point decreases continuously with increasing length of the side chains. This enables the observation of liquid-crystalline phases in a range of temperature being conveniently accessible. Meanwhile a number of these comb-like polymers (s. Fig. 5-24) systems 7-11 have been studied (Ballauff, 1989 b). In all cases a strong increase of solubility as well as a marked decrease of the melting point have been reported. The most important results obtained from these systems will be discussed in the following using polyester 7 (Fig. 5-24). As is obvious from the wide angle X-ray patterns measured at room temperature (Fig. 5-28), polyester 7 exhibits a high degree of crystalhnity. Furthermore the strong Bragg reflections in the diffractogram marked by A and B in the region of small angles together with their higher orders show the existence of two layered modifications termed A and B. Figure 5-29 displays a plot of the layer spacings of A and B as the number of carbon atoms in the side chains (Ballauff and Schmidt, 1987). The linear relationship demonstrates that the length of the side chains governs the packing in solid state. For both modifications the intercept is the same within the limits of error. Its value approximately matches the diameter of the main chains; only the slopes of the curve differ. For the majority of the systems 7-11 gathered in Fig. 5-24 there are three phase

240

5 Structure of Liquid Crystalline Polymers

Figure 5-28. Wide-angle Xray diffractogram of a stiffchain polyester with flexible n-alkyl side chains: The strong Bragg peaks in the low angle region show the existence of two layered structures termed A and B (Ballauff and Schmidt, 1987). 2.0

6.0

10.0

14.0

18.0

22.0

16

Figure 5-29. Dependence of the layer spacing as determined by wide-angle X-ray scattering (Fig. 5-28) on the length of the side chains. The linear relation shows that the side chains govern the width of the layers leading to the model displayed in Fig. 5-30 (Ballauff and Schmidt, 1987).

26.0 . . 30.0

transitions of first order: Starting at low temperatures an increase of disorder of the packing of the side chains occurs with increase in temperature which may be compared to the transition from the crystalline to the rotator-phase in the n-alkanes (Wenzel e t a l , 1987). Further rise of temperature leads to a mesophase and finally to the vanishing of liquid crystalline order and the formation of an isotropic melt. In case of polyester 7 (Fig. 5-24) one observes a nematic mesophase for short side chains (2 < n < 6). If the side chains are long enough (n > 8) modification A already present in the solid state (Fig. 5-28) remains stable also in the liquid crystalline phase. This is obvious from the strong Bragg reflections and their higher orders in the diffractogram of the mesophase (Ballauff and Schmidt, 1987). The present data on the structure being obtained from polyesters suggest an intercalated arrangement of the side chains (Ballauff and Schmidt, 1987). This may be

241

5.3 Main-Chain Liquid Crystal Polymers

argued from the fact that the layer spacing are much smaller than the overall-breadth of the molecule. Figure 5-30 displays the refined version of this packing model taking into account the minimum possible distance between the main chains and the measured densities of polyester 7 (Adam and Spiess, 1990). Only if both phenyl rings of the repeating unit are substituted with side chains, as is the case in polyamide 8, do the layer spacing and the breadth of the molecules (assuming a fully stretched conformation of the side chains) approximately coincide. A similar model has been proposed for polyester 9 (Fig. 5-24) by Hermann-Schonherr et al. (1986). The model depicted in Fig. 5-30 is idealized, of course, since the packing of the side chains will exhibit an increasing number of defects with rising temperature. X-ray measurements conducted on polyimides 11 have shown that there are well-defined intermediate stages in this process (Wenzel et al., 1987).

Form A

Form B

Figure 5-30. Model of molecular packing of stiffchain polymers with flexible alkyl side chains as inferred from the results displayed in Figs. 5-28 and 5-29 (Ballauff and Schmidt, 1987; Adam and Spiess, 1990).

5.3.3 Discotic Liquid Crystal Polymers

The preceding sections have dealt with main-chain LCP consisting of rod-like entities. Liquid crystalline phases may be achieved as well through disc-like units (see Fig. 5-5). This section deals with main chain LCP as well as with the side-chain LCP resulting from this building principle. Figure 5-31 shows two typical structures of these materials [a): Wenz, 1985; b): Kreuder and Ringsdorf, 1983; Hiiser and Spiess, 1988]. Both are based on the triphenylene unit already discussed in Sec. 5.1 (see Fig. 5-5). Polymer b) of Fig. 5-31 has been the subject of recent structural studies (Hermann-Schonherr et al., 1986; Huser and Spiess, 1988; Huser et al., 1989; Hsu et al., 1990). The results may be sum-

(a)

(b)

Jn

Figure 5-31. Structures of typical discotic liquid crystals discussed in the text, a): Wenz, 1985; b): Kreuder and Ringsdorf, 1983; Hiiser and Spiess, 1988.

242

5 Structure of Liquid Crystalline Polymers

marized as follows: All discotic main-chain LCP studied up to now form columnar phases (Fig. 5-3 and the respective discussion). A scheme of the molecular arrangement is shown in Fig. 5-32 (HermannSchonherr etal., 1986). Here the main-chains connect the columns consisting of the discotic triphenylene units. No information is yet available on possible backfolding of the main-chains into the same column (Wenz, 1985). In case of polymer b) Hermann-Schonherr et al. (1985) could show that there is a well-defined distance between the discs within one column. A further point of interest is the orientation of these materials as induced by external fields or by stretching. Hiiser et al. (1989) demonstrated for a particular system based on the triphenylene unit a very interesting difference between discotic main- and side-chain LCP. In the former case stretching leads to an arrangement of the discs parallel to the drawing direction (see Fig. 5-33). On the other hand, if the discotic moieties are affixed as side-chains (see also Sec. 5.4) a perpendicular arrangement results from drawing. Since phenyl groups tend to align parallel to the magnetic field the director axes n are perpendicular to the field. The resulting structure is depicted in Fig. 5-34. Recent investigations (Hsu etal., 1990) have complemented this model by further X-ray and D-NMR measurements. It is thus evident that discotic LCP may be processed easily to yield distinct orientations in the liquid crystalline state which can be preserved by cooling down to room temperature. Therefore these materials are certainly interesting candidates for possible use in optical devices. The stability of the discotic phases depends on the steric interaction between the discs but also on their attractive interac-

tion. As shown by Ringsdorf et al. (1989) and by Bengs et al. (1991) the range of temperature in which the discotic phase exists may be greatly enhanced by adding strong electron accepters like 2,4,7-trinitrofluorenon. In some cases the liquid crystalline phase is only stable when the electron accepter is added, i.e., there is an induction of a liquid crystal phase. The obvious explanation for this is the charge-transfer interaction between the electron-rich discs with the acceptor which decreases the free volume in the system and the distance between the mesogenic unit thus favoring order.

Figure 5-32. Schematic representation of the molecular arrangement in a discotic main-chain LCP as deduced from wide-angle X-ray scattering (HermannSchonherr et al., 1986).

fiber axis

main-chain

side-chain

discotic liquid crystal polymer Figure 5-33. Different ordering of discotic main- and side-chain LCP upon orientation by drawing (Hiiser etal., 1989).

5.3 Main-Chain Liquid Crystal Polymers

243

Figure 5-34. Alignment of a discotic main chain LCP in the magnetic field. The arrangement results from the tendency of the discs to align parallel to the field (Hiiser and Spiess, 1988).

5.3.4 Polymer Blends Containing Liquid Crystal Polymers

Polymer blends represent a highly important class of materials used for a multitude of purposes (see Chap. 6). Up to now virtually all blends used for technical applications are composed of flexible components. In recent years there has been a lot of effort to make blends from stiff-chain polymers with flexible macromolecules (Ballauff, 1990, and further literature cited therein). These "molecular composites" derive from the highly successful concept of fiber-reinforcement of plastic resins. Here, e.g., carbon fibers are embedded in epoxy resins. These composite materials have found wide-spread applications in aircraft industries. The problem encountered, however, is the limited adhesion of matrix and resin. If the fiber has molecular dimensions, i.e., if a rod-like polymer is used for reinforcement, this difficulty does not exist. Molecular composites thus achieved should display the same advantageous properties as the usual composites, i.e., the high modulus and the high strength in the direction of the fibers, without being hampered by the problems of a two-phase system as matrix adhesion, etc. (Hwang et al, 1983; Takayanagi, 1983; and Chap. 13). Numerous experiments have been carried out to achieve this highly interesting

goal. In all cases studied in detail so far there is a strong incompatibility between the flexible and the stiff component whenever a liquid crystalline phase is formed (Flory, 1978, for further references see Ballauff, 1990). This "entropic incompatibility" together with the usual enthalpicdriven demixing of polymers renders the making of a stable, processable molecular composite a virtually impossible task. As an example for the strong demixing upon formation of an ordered phase, Fig. 5-35 displays the ternary phase diagram for the mixture of a rod-like and a coiled polymer in a common solvent (Flory, 1978) calculated with the Flory lattice model. The points representing the composition of the ordered phase lie nearly exactly on the line connecting the rod-like polymer and the solvent in the diagram. This means that the amount of coiled species entering the ordered phase is nearly negligible. Processing of these systems to a (thermodynamically unstable) molecular composite hence requires highly dilute solutions. The Flory lattice calculation can be easily extended to thermotropic systems (Ballauff, 1986 b). Here a similarly strong incompatibility results and the phase diagrams can be predicted in a semi-quantitative fashion by theory (Ballauff, 1990). Experiments conducted on oligomers of flexible polymers dissolved in low-molecu-

244

5 Structure of Liquid Crystalline Polymers Solvent

0.1

0.3

0.3

0.-4

Rod X r = 100

(K

Coil X c =100

Figure 5-35. Ternary phase diagram of a rod-like and a coiled polymer both having 100 segments in a common solvent calculated with the Flory lattice model (Flory, 1978). Dashed lines show the tie lines between the coexisting isotropic and anisotropic phase.

lar-weight LC (Orendi and Ballauff, 1992) compare favorably with the predictions of theory. The occurrence of ordered phases in blends containing liquid crystal polymers thus leads to the observation of "entropic incompatibility" not encountered in isotropic blends.

5.4 Side-Chain Liquid Crystal Polymers 5.4.1 Calamitic Side-Chain Liquid Crystal Polymers

As shown systematically in Fig. 5-7 sidechain LCP may be obtained through lateral fixation of calamitic mesogenic units (Fig. 5-4) to flexible backbones. Up to now a vast number of these materials has been synthesized and investigated (Finkelmann and Rehage, 1984; Finkelmann, 1987; Zen-

tel, 1989 a; Finkelmann, 1991). The main results obtained so far can be summarized as follows: a) The shape and the axial ratio of the mesogenic unit is more or less preserved upon fixation to the main chain. This is in contrast to main-chain LCP where the axial ratio depends directly on the degree of polymerization (Sec. 5.3). The main consequence of the polymeric nature of the material is the decrease of free volume when compared to the low-molecularweight compound. From the considerations given in Sec. 5.2.3 it is directly evident that the denser packing of the anisometric moieties leads to an increased stability of the ordered phase, i.e., to a rise of the transition temperature. Therefore Tni increases with the degree of polymerization until a plateau-value is reached for high molecular weights. Figure 5-36 gives the result of a thorough study of Finkelmann and Rehage (1984) showing that Tni is virtually constant above a certain degree of polymerization r. b) The attachment of the mesogenic unit to a main chain generally stabilizes higher ordered phase. When the alkyl chain ("spacer") used for connecting the mesogenic moiety and the main chain is long enough, smectic phases are observed frequently. In some cases the nematic phase is suppressed altogether and the smectic phase turns directly into the isotropic state (Finkelmann, 1987; Finkelmann, 1991). The fixation of the mesogenic units can be done in two ways: i) The calamitic unit is attached on one end of the spacer (Fig. 5-36), or ii) The fixation of the mesogenic unit is done laterally as shown in Fig. 5-37 (Hessel et al., 1987).

5.4 Side-Chain Liquid Crystal Polymers

In this configuration the rotation around the long molecular axis is more or less suppressed. As a consequence the lath-like mesogenic units can form a biaxial nematic phase as schematically depicted in Fig. 5-6 (Hessel et al, 1987). To the author's knowledge this was the first experimental observation of a thermotropic biaxial nematic mesophase on a side-chain LCP (cf. the survey of literature by Ballauff, 1988). As shown by a number of methods (Finkelmann, 1987) the nematic-to-isotropic transition is of first order. Hence the nematic order parameter S defined by Eq. (5-4) rises strongly at Tni and increases monotonically with lowering of the temperature (Sec. 5.2.2.1, Fig. 5-12). At the glass transition temperature the order

coo—\O/-- 0CH 3

-(CH 2 ) 3 —0

T (K) < isotropic

360-

-

o

320-

/

-

/

nematic

300-

/ 280-

260 -

y

glassy

/

10

100

Figure 5-36. Dependence of the transition temperatures on the molecular weight of a typical side chain LCP. Squares denote the transition from the crystalline directly to the isotropic state (Finkelmann and Rehage, 1984).

245

Figure 5-37. Scheme of the lateral fixation of mesogenic units in side chain LCP (Hessel et al., 1987). The hindered rotation around the long molecular axis stabilizes the biaxial nematic phase (Fig. 5-6).

present in the LCP is frozen in. Thus it is possible to obtain a nematic glass in which the anisotropic properties can be preserved. It is obvious that materials of this kind lend themselves for a great variety of applications, e.g., for storage of information (Anderele et al., 1991) or nonlinear optics. The statistical-mechanical theory of side-chain LCP materials is more complicated than in case of the main-chain LCP since there is a coupling of the alignment of the main and the side chains. The three possible cases are depicted in Fig. 5-38 which has been taken from the work of Wang and Warner (1986). In case of the N, phase the order parameter of the main chain is negative since it is aligned perpendicular to the nematic director. Phases N n and N m are more reminiscent of mainchain LCP since here the main chains are oriented along the preferred direction. Treating the interaction between the different parts of the polymer molecule in terms of a Maier-Saupe model, Wong and Warner (1986) could discuss quantitatively the limits of stability of these phases. The model shown in Fig. 5-38 leads to the conclusion that the nematic order gen-

246

5 Structure of Liquid Crystalline Polymers

nematic director

N Figure 5-38. Schematic representation of the three different possible arrangements of main and side chains in a nematic side chain LCP according to Wang and Warner (1986).

erally should lead to a strong distortion of the backbone conformation. This deformation is expected to be even more pronounced in the smectic state in which the main-chains should be confined between the layers. Recent investigation by smallangle neutron scattering (Ohm et al., 1988; Pepy et al., 1990; Noirez et al., 1991) have corroborated this picture. As an example of the results of such a study Fig. 5-39 displays the radii of gyration of the main chain measured parallel and perpendicu90

r

75 -

+

4

'_ •

•

:50 -

^

...

••

"RH

25 Glassy

0

Smectic

Nematic

80 T/°C

Isotropic

120

160

Figure 5-39. Anisotropy of the shape of a side chain LCP in the various phases: the radii of gyration of the main chain as measured by small-angle neutron scattering parallel (R^) and perpendicular (i?±) to the preferred direction (Fig. 5-26). This result directly demonstrates the oblate shape of the main chain due to its confinement between the smectic layers (Pepy etal., 1990).

lar to the preferred direction (Pepy et al., 1990). The strong anisotropy is directly evident. The oblate shape of the main chains due to the confinement between the smectic layers formed by the side chains can be seen from the fact that R±> Rl{. With increasing temperature this anisotropy is greatly reduced because of an increasing number of defects, i.e., main chains going through the layers ("layer hopping", Renz and Warner, 1986). In the nematic state the deformation of the main chain is much less and the result shown in Fig. 5-39 demonstrates that this phase may be classified as a Nj phase (Fig. 5-38). 5.4.2 Liquid Crystalline Elastomers

In the preceeding section we dealt with side-chain LCP consisting of linear main chains with mesogenic side groups. As shown schematically in Fig. 5-7 crosslinking of the main chains leads to a liquid crystalline network (Finkelmann, 1987, 1991; Zentel, 1989 b). The most prominent feature observed in these systems is the strong coupling between the state of order and the mechanical deformation (see Fig. 5-40). If the crosslinking is achieved in an unoriented nematic sample a liquid crystalline rubber results which does not have a macroscopic preferred direction. Stretching this elastomer leads to an alignment of all mesogenic units (see Fig. 5-40, righthand side): The turbid "multidomain" sample turns into a clear nematic single crystal. As a consequence there is a strong influence of the applied mechanical force on the nematic-to-isotropic transition temperature first predicted by de Gennes (1975). With increasing strain the ordered phase becomes more and more stable until a mechanical critical point is reached. At this

5.6 References

247

5.5 Acknowledgements Financial support by the Bundesministerium fur Forschung und Technologie, Projekt "Steife Makromolekiile", is gratefully acknowledged. The author is indebted to A. Horvath for proof-reading of this chapter.

Figure 5-40. Macroscopic orientation of liquid crystalline elastomer by mechanical forces: Stretching of the elastomer leads from a macroscopically unoriented nematic state to a nematic single crystal (Gleim and Finkelmann, 1987; Finkelmann, 1991).

point the nematic-to-isotropic transition is of second order (Sec. 5.2.3). Experiments conducted by Finkelmann and coworkers showed clearly that this coupling exists indeed (Schatzle etal., 1989, 1991; Finkelmann, 1991). The highly interesting prospects arriving from this coupling of optical to mechanical properties are immediately obvious (Zentel, 1989 b) and can be pursued for a variety of applications. 5.4.3 Conclusions

The foregoing discussion has shown that liquid crystalline polymers present a wide variety of materials with interesting properties and prospects for future applications. Main-chain LCP based on stiff-chain macromolecules are already used for the fabrication of high-modulus fibers whereas side chain LCP certainly will be useful materials for optical applications. The phase behavior can be understood in principle by comparison with low-molecular-weight LC but LCP add a great number of new features not seen in low-molecular-weight systems. Thus these materials represent a highly fruitful area of research both with regard to fundamental science as well as to materials science.

5.6 References Abe, A., Ballauff, M. (1991), in: Liquid Crystallinity in Polymers: Principles and Fundamental Properties: Ciferri, A. (Ed.). New York: VCH Publishers. Adam, A., Spiess, H.W. (1990), Makromol Chem. Rapid Comm. 11, 249. Anderele, K., Birenheide, R., Werner, M. J.A., Wendorff, J.H. (1991), Liquid Cryst. 9, 691. Als-Nielsen, J. etal. (1980), Phys. Rev. B22, 312. Anderson, H.C., Chandler, D., Weeks, J. D. (1976), Adv. Chem. Phys. 34, 105. Ballauff, M. (1986a), Macromolecules 19, 1366. Ballauff, M. (1986 b), Ber. Bunsenges. Phys. Chem. 90, 1053. Ballauff, M., Schmidt, G.F. (1987), Mol. Cryst. Liq. Cryst. 147, 163. Ballauff, M. (1988), Angew. Chem. 100, 775. Ballauff, M. (1989 a), Mol. Cryst. Liq. 168, 209. Ballauff, M. (1989b), Angew. Chem. Intl. Ed. 28, 253. Ballauff, M. (1990), Pol. Adv. Techn. 1, 109. Bengs, H. et al. (1991), Liquid Cryst. 10, 161. Bianchi, E. etal. (1984), Macromolecules 17, 1526. Biswas, A., Blackwell, J. (1987), Macromolecules 20, 2997. Blumstein, A. et al. (1985), Faraday Disc. Chem. Soc. 79, 33. Blumstein, A. (1985), Polymer J. 17, 277. Boeffel, Ch., Spiess, H.W. (1989), "NMR-Methods for Studying Molecular Order and Motion in Liquid-Crystalline Sidegroup Polymers", in: Side-Chain Liquid Crystal Polymers: C.B. McArdle (Ed.). Glasgow: Blackie Publ. Bonart, R. (1987), Makromol. Chemie 188, 1187. Bonart, R., Dietrich, J., Zott, H. (1988), Makromol. Chemie 189, 227. ten Bosch, A., Maissa, P., Sixou, P. (1983), J. Chem. Phys. 79, 3462. Brelsford, G.L., Krigbaum, W.R. (1991), in: Liquid Crystallinity in Polymers: Principles and Fundamental Properties: Ciferri, A. (Ed.). New York: VCH Publishers. Buka, A., de Jeu, W.H. (1982), J. Physique 43, 361. Cageao, R.A., Schneider, A.-L, Biswas, A., Blackwell, J. (1990), Macromolecules 23, 2843. Calundann, G.W., Jaffe, M. (1982), Proc. Robert A. Welch Found. Conf. Chem. Res. 26, 247.

248

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Chandrasekhar, S. (1977), Liquid Crystals. London: Cambridge University Press. Chandrasekhar, S., Ranganath, G.S. (1990), Rep. Prog. Phys. 53, 57. Colot, J.-L., Wu, X.-G., Baus, M. (1988), Phys. Rev. A 38, 2022. D'Allest, J.F., Maissa, P., ten Bosch, A. et al. (1988), Phys. Rev. Lett. 61, 2562. D'Allest, J.F., Sixou, P., Blumstein, A. (1988), Mol. Cryst. Liq. Cryst. 155, 581. Demus, O., Richter, L. (1978), Textures of Liquid Crystals. Weinheim: Verlag Chemie. Doi, M., Edwards, S.F. (1986), Theory of Polymer Dynamics. New York: Oxford University Press. Dries, Th., Fuhrmann, K. et al. (1991), /. Appl. Phys. 69, 7539. Ebert, M., Hermann-Schonherr, O. et al. (1988), Makromol. Chem. Rapid Comm. 9, 445. Economy, J., Storm, R.S. et al. (1976), /. Pol. Sci. Chem. Ed. 14, 2207. Economy, X, Volksen, W., Viney, C. et al. (1988), Macromolecules 21, 2111. Esnault, P., Galland, D. et al. (1989), Macromolecules 22, 3734. Falk, U., Spiess, H.W. (1989), Makromol. Chem. Rapid Comm. 10, 149. Finkelmann, H., Rehage, G. (1984), Adv. Pol. Sci. 60/61, 99. Finkelmann, H. (1987), Angew. Chem. 99, 848. Finkelmann, H. (1991), in: Liquid Crystallinity in Polymers: Principles and Fundamental Properties: Ciferri, A. (Ed.). New York: VCH Publishers. Fischer, E.W. (1987), Makromol. Chem. Makromol. Symp. 12, 123. Flory, P. J. (1956), Proc. Royal Soc. London A 234, 73. Flory, P.J. (1969), Statistical Mechanics of Chain Molecules. New York: Interscience. Flory, P.J. (1974), Macromolecules 7, 381. Flory, P.J. (1978), Macromolecules 11, 1138. Flory, P.J. (1984), Adv. Pol Sci. 59, 1. Fredrickson, G.H., Leibler, L. (1990), Macromolecules 23, 531. Fraden, S., Maret, G., Caspar, D. L. D., Meyer, R. B. (1989), Phys. Rev. Lett. 63, 2068. Frank, F.C. (1958), Disc. Faraday Soc. 25, 19. Fuhrmann, K., Dries, Th., Fischer, E.W., Ballauff, M. (1992), J. Pol. Sci. Phys. Ed. 30, 1199. Furuya, H., Dries, Th. et al. (1990), Macromolecules 23, 4122. Furuya, H., Abe, A. et al. (1991), Macromolecules 24, 2999. de Gennes, P.G. (1975), The Physics of Liquid Crystals. Oxford: Clarendon Press, de Gennes, P. G. (1984), Mol. Cryst. Liq. Cryst. Lett. 102, 95. Gleim, W., Finkelmann, H. (1987), Makromol. Chem. 188, 1489. Gramsbergen, E.F., Longa, L., de Jeu, W.H. (1986), Phys. Rep. 135, 196.

Gray, G.W., Goodby, J.W.G. (1984), Smectic Liquid Crystals: Textures and Structures. Glasgow: Leonard Hill. Hanna, S., Windle, A.H. (1988 a), Polymer 29, 207. Hanna, S., Windle, A.H. (1988b), Polymer Comm. 29, 236. Hanna, S., Lemmon, T. I, Spontak, R. I , Windle, A. H. (1992), Polymer 33, 3. Hansen, IP., MacDonald, I.R. (1986), Theory of Simple Liquids. New York: J. Wiley. Hermann-Schonherr, O., Wendorff, J.H. etal. (1985), Makromol. Chem. Rapid Comm. 7, 97. Hermann-Schonherr, O., Wendorff, J.H. etal. (1986), Makromol. Chem. Rapid Comm. 7, 791. Hessel, R, Herr, R.-P., Finkelmann, H. (1987), Makromol. Chem. 188, 1597. Hsu, T.-C, Hiiser, B. et al. (1990), Makromol. Chem. 191, 1597. Hudson, S. D., Thomas, E. L. (1989), Phys. Rev. Lett. 62, 1993. Hudson, S.D., Vezie, D.L., Thomas, E.L. (1990), Makromol. Chem. Rapid Comm. 11, 657. Hummel, J.P., Flory, P.J. (1980), Macromolecules 13, 479. Hiiser, B., Spiess, H.W (1988), Makromol. Chem. Rapid Comm. 9, 337. Hiiser, B., Pakula, T, Spiess, H.W. (1989), Macromolecules 22, 1960. Hwang, W.F., Wiff, D.R. et al. (1983), J. Macromol. Sci.-Phys. B22, 231. Jung, B., Schiirmann, B.L. (1989), Macromolecules 22, 411. Kiefer, R., Baur, G. (1989), Mol. Cryst. Liq. Cryst. 174, 101. Kirste, R.G., Oberthiir, R.C. (1982), in: Small Angle X-Ray Scattering: Glatter, O., Kratky, O. (Eds.). London: Acad. Press. Khokhlov, A.R., Semenov, A.N. (1985), /. Stat. Phys. 38, 161. Khokhlov, A.R. (1991), in: Liquid Crystallinity in Polymers: Principles and Fundamental Properties: Ciferri, A. (Ed.). New York: VCH Publishers. Kleman, M. (1991), in: Liquid Crystallinity in Polymers: Principles and Fundamental Properties: Ciferri, A. (Ed.). New York: VCH Publishers. Kreuder, W, Ringsdorf, H. (1983), Makromol. Chem. Rapid Comm. 4, 807. Kricheldorf, H.R., Schwarz, G. (1983), Makromol. Chem. 84, 475. Kricheldorf, H.R., Schwarz, G. (1990), Polymer 31, 481. Krigbaum, W. R. (1982), in: Polymer Liquid Crystals: Ciferri, A., Krigbaum, W.R., Meyer, R.B. (Eds.). New York: Academic Press. Kwolek, S. L., Morgan, P. W, Schaefgen, J. R. (1987), in: Encyclopedia of Polymer Science and Engineering: Mark, H.F., Bikales, N.M. et al. (Eds.). New York: Wiley. Landau, L.D., Lifschitz, E.M. (1971), Statistische Physik. Berlin: Akademieverlag.

5.6 References

Lee, S.-D. (1987), /. Chem. Phys. 87, 4972. Lee, S.D., Meyer, R.B. (1991), in: Liquid Crystallinity in Polymers: Principles and Fundamental Properties: Ciferri, A. (Ed.). New York: VCH Publishers. Lenz, R. W. (1985), Faraday Disc. Chem. Soc. 79, 21. Lieser, G. (1983), /. Pol. Sci. Phys. Ed. 21, 1611. Luckhurst, G.R. (1979), in: The Physics of Liquid Crystals: Luckhurst, G.R. (Ed.). London: Academic Press. McMillan, W.L. (1971), Phys. Rev. A4, 1238. Majnusz, J., Catala, J.M., Lenz, R. W. (1983), Europ. Pol J. 19, 1043. Maier, W, Saupe, A. (1958), Z. Naturf. A13, 564; (1959), A14, 882; (1960), A15, 287. Matsuda, K. (1979), Pol. Prep. (Amer. Chem. Soc. Div. Pol. Chem.) 20, 122. Morgan, P.W. (1977), Macromolecules 10, 1381. Moore, J.S., Stupp, S.I. (1987), Macromolecules 20, 282. Moore, J.S., Stupp, S.I. (1988), Macromolecules 21, 1217. Martin, P. G., Stupp, S. I. (1988), Macromolecules 21,

mi. Stupp, S.I., Moore, J.S., Martin, P.G. (1988), Macromolecules 21, 1228. Noirez, L., Moussa, E, Cotton, I P . , Pepy, G. (1991), /. Stat. Phys. 62, 997. Northolt, M.G. (1974), Europ. Pol. J. 10, 799. Northolt, M.G., Sikkema, D. J. (1990), Adv. Pol. Sci. 98, 115. Odijk, T. (1986), Macromolecules 19, 2313. Ohm, H.G., Kirste, R.G., Oberthiir, R.C. (1988), Makromol. Chem. 189, 1387. Onsager, L.A. (1949), Ann. N.Y., Acad. Sci. 51, 627. Orendi, H., Ballauff, M. (1992), Ber. Bunsenges. 96, 96. Ozawa, S., Nakagawa, Y. et al. (1952), Jpn. Pat. Doc. 52-38719; (1953), 53-32838. Parsons, J.D. (1979), Phys. Rev. A19, 1225. Pepy, G., Noirez, L., Keller, P. et al. (1990), Makromol. Chem. 191, 1383. Rancon, Y, Charvolin, J. (1988), /. Phys. Chem. 92, 6339. Rawiso, M., Duplessix, R., Picot, C. (1987), Macromolecules 10, 630. Renz, W., Warner, M. (1986), Phys. Rev. Lett. 56, 1268. Ringsdorf, H., Tschirner, P. et al. (1987), Makromol. Chem. 188, 1431. Ringsdorf, H., Wiistefeld, R. (1989), Angew. Chem. 101, 934. Rosenau-Eichin, R., Ballauff, M. et al. (1988), Polymer 29, 518. Rutledge, G.C., Papaspyrides, C D . , Suter, U.W. (1990), in: Liquid-Crystalline Polymers: Weiss, R.A., Ober, C.K. (Eds.). Washington: American Chemical Society.

249

Saupe, A. (1964), Z. Naturforsch. 19a, 161. Schatzle, I, Kaufhold, W, Finkelmann, H. (1989), Makromol. Chem. 190, 3269. Schatzle, J., Kaufhold, W, Finkelmann, H. (1991), Makromol. Chem. 192, 1235. Schliiter, A.-D. (1988), Macromolecules 21, 1208; (1988), Angew. Chem. 100, 283. Schmidt, M. (1984), Macromolecules 17, 553. Sirigu, A. (1991), in: Liquid Crystallinity in Polymers: Principles and Fundamental Properties: Ciferri, A. (Ed.). New York: VCH Publishers. Spiess, H.W. (1985), Adv. Pol. Sci. 66, 24. Stephen, M.I, Straley, I P . (1974), Rev. Mod. Phys. 46, 617. Stern, R., Ballauff, M., Wegner, G. (1991), Polymer 32, 2096. Sun, Z., Cheng, H.-M., Blackwell, I (1991), Macromolecules 24, 4162. Tadokoro, H. (1979), Structure of Crystalline Polymers. New York: I Wiley. Takayanagi, M. (1983), Pure Appl. Chem. 55, 819. Thomas, E.L., Wood, B.A. (1985), Disc. Faraday Soc. 79, 229. Tsvetkov, V.N. (1989), Rigid-Chain Polymers. New York: Consultants Bureau. Uematsu, I., Uematsu, Y (1984), Adv. Pol. Sci. 59, 37. Vertogen, G., de Jeu, W. H. (1988), Thermotropic Liquid Crystals, Fundamentals. Berlin: Springer Yerlag. Voigt-Martin, I.G., Durst, H. (1989), Macromolecules 22, 168. Vorlander, D. (1923), Z. Phys. Chem. 105, 211. Wang, X.J., Warner, M. (1986), J. Phys. A19, 2215. Warner, M., Gelling, K.P., Vilgis, T.A. (1988), J. Chem. Phys. 88, 4008. Wenz, G. (1985), Makromol. Chem. Rapid Comm. 6, 577. Wenzel, M., Ballauff, M., Wegner, G. (1987), Makromol. Chem. 188, 2865. Wignall, G. (1987), in: Encyclopedia of Polymer Science and Engineering: Mark, H.F., Bikales, N. M. et al. (Eds.). New York: I Wiley. Windle, A.H., Viney, Ch. (1985), Faraday Disc. Chem. Soc. 79, 55. Wolfe, IF., Loo, B.H., Arnold, F.E. (1981), Macromolecules 14, 915. Yoon, D. Y, Bruckner, S. (1985), Disc. Faraday Soc. 79, 41. Yoon, D.Y, Masciocchi, N. et al. (1990), Macromolecules 23, 1793. Zentel, R. (1989 a), in: Comprehensive Polymer Science: Allen, G., Bevington, Z.C. (Eds.). Oxford: Pergamon, Vol. 5, p. 723. Zentel, R. (1989 b), Angew. Chem. Adv. Mat. 101, 1437.

250

5 Structure of Liquid Crystalline Polymers

General Reading Ballauff, M. (1989), Angew. Chem. Intl. Ed. 28, 253. Chandrasekhar, S. (1977), Liquid Crystals. London: Cambridge Univ. Press. Chandrasekhar, S., Ranganath, G. S. (1990), Rep. Prog. Phys. 53, 57. Ciferri, A. (1991), Liquid Crystallinity in Polymers. New York: VCH Publishers.

Edwards, S. F. (1986), Theory of Polymer Dynamics. New York: Oxford University Press. de Gennes, P. G. (1975), The Physics of Liquid Crystals. Oxford: Clarendon Press. Finkelmann, H. (1987), Angew. Chem. 99, 848. Flory, P. J. (1969), Stat. Mechanics of Chain Molecules. New York: Interscience. Weiss, R. A., Ober, C. K. (1990), Liquid Crystalline Polymers. ACS Symp. Series.

Doi? M ?

6 Structure of Polymer Blends Takeji Hashimoto Department of Polymer Chemistry, Kyoto University, Kyoto, Japan

List of 6.1 6.2 6.2.1 6.2.2 6.3 6.3.1 6.3.2 6.3.3 6.4 6.5 6.5.1 6.5.2 6.5.3 6.6 6.7 6.8 6.8.1 6.8.2 6.9 6.10 6.11 6.12 6.13

Symbols and Abbreviations Introduction Unique Features in Polymers Dynamics Phase Diagrams: Large Cooperative Effect in Polymers Formation and Growth of Self-Assembled Patterns via Spinodal Decomposition - A Brief Introduction Early, Intermediate, and Late Stages in Spinodal Decomposition Scaling Postulate, Dynamical Self-Similarity, and Their Experimental Test . . Complex Factors Early Stage Spinodal Decomposition Growth of Self-Assembled Structures in Later Stage Spinodal Decomposition Scaling Analyses of Im(t; T) and qm(t; T) Scaling Analyses of the Structure Factor F(x); Global Character Unique Features of the Scaled Quantities of Polymers Fine Structure in the Scaled Structure Factor F(x) Spontaneous Pinning for Off-Critical Mixtures Interplay with Two Kinds of Phase Transition Interplay with Crystallization Interplay with the Microphase Transition Morphology Control Effects of Applied Fields on the Self-Assembly Summary Acknowledgements References

Materials Science and Technology Copyright © WILEY-VCH Verlag GmbH & Co KGaA. All rights reserved.

252 256 258 258 259 265 265 267 269 270 271 271 275 277 277 283 286 286 287 292 294 296 297 297

252

6 Structure of Polymer Blends

List of Symbols and Abbreviations Ax, Bx A, B, C, X, Y, Z aK B, C, N Dc (T) Dc A(T\ Dc B(T) D a p p (T) /)_,_, £>|| f(T) / F(x; T) F(x; T) AF H AHmix / (q, t; T) I J m (t; T), Im(t\ Im 7 m (T) J mp J, K kB / m Mn Mw n N NK ATA, NB Ne Nl9N2 (9(10, 10 2 ,...) P (NK) q, q q\ q" qc qm(t;T\ qm(t), qm qm(0) Qm(x)

parameter related to the entropic, enthalpic part of %eff constituent polymers statistical segment length branchings self-diffusivity of polymers Dc of constituent polymers A, B collective diffusivity domain size normal, parallel to the interface temperature-dependent monomeric friction constant fraction of A-block chain in A - B block copolymer, / = NA/(NA + NB) scaled structure factor reduced scaled structure factor free energy change upon mixing per lattice site, according to F H single-phase state mixing enthalpy of different constituent polymers scattering intensity state in the miscibility gap peak scattering intensity reduced peak scattering intensity value of Im after pinning index referring to the J-th, K-th polymer Boltzmann constant reduced distance vector order of the scattering maxima number average molecular weight weight average molecular weight Porod scattering exponent degree of polymerization (DP) N of X-th constituent polymer degree of polymerization of constituent polymer A, B D P between the entanglement couplings D P , subscript 1 refers to D P B and subscript 2 refers to H P I rest distribution of NK scattering wavevector, magnitude lower and upper bounds of q crossover q scattering wavevector at time t referring to the dominant mode of concentration fluctuations wavenumber appearing at the early stage SD and predicted by the linearized theory reduced wavenumber

List of Symbols and Abbreviations

253

<2mP r R (q; T) Rm (0 R0,RA, RB Rg Rg0, RgA, RgB S (x) t T tc(T) tc tY (t) Tm tp Ts tl9 t2 To 7\ T2 tu trep AT c (y) AT polym , AT small w, x, y xc x(t; T) z

value of qm after pinning distance vector growth rate mean radius of the interface curvature (persistence length) root of the mean squared end-to-end distance gyration radius of individual polymer coils gyration radius universal scaling function time absolute temperature characteristic time of blends due to concentration fluctuations at T crossover time characteristic interfacial thickness melting temperature "pinning" time spinodal temperature time period initial temperature molten liquid state temperature rapid crystallization process temperature equilibrium value of tY reptation time drop of the critical (or cloudpoint) temperature at y A T (quench depth) for polymer, small molecules fractions of the respective monomeric structures in the polymer chains reduced scattering vector defining crossover behaviour reduced scattering vector defined by x(t; T) = q/qm(t; T) local coordinate normal to the interface

a, /? a b , Ph y, v y ys eT

scaling exponents for qm(t) a n d / m ( 0 block copolymers exponents in scaling laws for D a p p ( T ) , £ ( T ) shear rate scaling exponent for I (t) dimensionless parameter characterizing the thermodynamic state of a system under consideration, regarding a temperature difference from spinodal point mean square fluctuation of the scattering contrast wavelength of radiation, scattering angles Onsager kinetic coefficient wavelength vector, magnitude for characteristic periodicity of d o mains critical value of A defined by 2n/qc periodicity developed by S D (macrophase transition) periodicity of microdomains developed by microphase transition

(r\2{t\ T)> A, 9 X(q) A (t\ A Ac Amacro /l m i c r o

254

6 Structure of Polymer Blends

^polym? Ismail SO,polym> ^ 0 , small QK

0 block

q

X(T) V

V

A l ' A2 Zeff

. Z3

Zs As, macro' As, micro y /NT 1 y \f |

(zJViL macro , micro

Amorph Cryst DOP DP DPS DPB EPR FH GPC

wavelength of the dominant mode of the concentration fluctuations parallel to the compression axis thermal correlation length at T characteristic length intrinsic to polymer molecules £ for polymer, small molecules £0 for polymer, small molecules segmental density of K-th constituent polymer parameter related to the interfacial thickness interfacial area per unit volume reduced time time-independent scaling function for spatial concentration fluctuation volume fraction of one type of lamella critical value of block composition of K volume composition of A, initial and at time t equilibrium compositions of A in two phase state critical composition of A block copolymer volume fraction composition difference between two coexisting domains equilibrium value of A(f> composition fluctuation of the q Fourier mode Flory-Huggins parameter of segmental base interaction between polymers A and B fundamental ^-values effective x between two polymers per monomer unit value of %(T) at the spinodal point %(T) between the K-th and J-th polymer Xs for the macrophase, microphase transitions net interaction parameter beween two polymers XN at the critical point value of xN1 at thermodynamic stability limit for the macrophase transition value of xN± at thermodynamic stability limit for the microphase transition amorphous domain crystalline domain dioctylphthalate degree of polymerization perdeuterated PS perdeuterated PB ethylene-propylene random copolymer Flory-Huggins mean-field theory gel permeation chromatography

List of Symbols and Abbreviations

HPI LC LCST LS PB PCM PET PI POM PP PPO PS PVME RPA S SANS SAXS SBR SD SI TDGL TEM UCST X-7G

protonated PI liquid crystalline domain lower critical solution temperature light scattering polybutadiene phase-contrast light micrograph polyethylene terephthalate polyisoprene polarized light micrograph polypropylene polyphenyleneoxide polystyrene poly(vinylmethylether) random phase approximation solvent small-angle neutron scattering small-angle X-ray scattering styrene-butadiene random copolymer spinodal decomposition polystyrene-polyisoprene diblock copolymer time-dependent Ginzburg-Landau transmission electron microscopy upper critical solution temperature thermotropic liquid crystalline polymer

255

256

6 Structure of Polymer Blends

6.1 Introduction

Molecular Parameters (N K ,P(N K ),X i j ,« K ,p K )

Thermodynamic Variables

Properties of polymer blends depend on their morphology, and control of morphology is the key to controlling their properties. Therefore one of the most important aspects in polymer blend studies is to investigate the effects of fundamental molecular parameters and thermodynamic variables on their morphology. The molecular parameters to be considered are the degree of polymerization (DP) NK and its distribution P{NK\ statistical segment length aK, and segmental density QK for the K-th constituent polymer, and the Flory-Huggins interaction parameter (Flory, 1953) %KJ per segmental base of the K-th polymer with other constituents. The thermodynamic variables are temperature, pressure, volume, and composition c/)K with X*.-l

(6-1)

as summarized in Fig. 6-1. Figure 6-1 describes statistical mechanical or materials science studies on polymer blends. The effect of the molecular parameters and the thermodynamic variables on the morphology and properties is twofold: The parameters and the variables affect (i) the phase diagrams (binodal and spinodal curves) which predict the thermodynamic state of polymer mixtures and the mechanism of phase separation (unmixing), i.e., nucleation growth vs. spinodal decomposition (SD), and (ii) the dynamics of phase transitions (see for example, Gunton etal., 1983), i.e., unmixing or dissolution kinetics involved by quenching from a single-phase state to a two-phase state or by reverse quenching. These parameters and variables affect the time-evolution of the pattern (self-assembled morphology) during the phase transition, and the mechanism of the pattern formation. Control of

Phase Diagram (mechanism of unmixing)

Dynamics (Unmixing, dissolution)

(1) (2) Morphology

Physical Properties or Functionalities

(1) molecular design of morphology (control of selfassembled patterns) (2) molecular design of properties and functionalities (3) characterization of patterns

P(N K );

PK;

T; P;

degree of polymerization of K-th polymer distribution of N K Flory-Huggins interaction parameters between i and j thl components statistical segment length mass density temperature pressure composition of K-th polymer in mixtures

Figure 6-1. Statistical mechanical or materials science studies on polymer blends.

the pattern self-assembled during the phase transition renders a molecular design of the pattern. Basic studies on the characterization of the patterns and on morphology-property (or functionalities) relationships eventually render molecular design of the properties or the functionalities of polymer blends. Such studies are among the most important and challenging themes in statistical physics and materials science. Even for binary blends of polymers A and B, designated hereafter A/B, the patterns formed during the phase transition from a homogeneous state (equilibrium state 1) to a macroscopically phase-separated state (equilibrium state 2) are expected to be greatly varied with many intermediate structures (Fig. 6-2). There will be various processes and mechanisms in

6.1 Introduction

the pattern formation. Temporal and spatial scales in the pattern formations may extend over many decades, typically a time scale longer than a millisecond and a spatial scale ranging from a few ten of angstroms to centimeters. In some systems equilibrium state 2 will never be reached, and the pattern growth will be pinned at some intermediate stage (Sec. 6.9). The variety in the self-assembly of the patterns in polymer blends can be easily imagined from the following examples, (i) There will be interplay of the liquid/liquid phase separation and other types of phase transitions such as (a) liquid/solid phase transition (or crystallization) (Inaba et al., 1986, 1988; Takenaka et al, 1989) if at least one component is crystalline, (b) isotropic liquid/liquid crystal, liquid crystal/solid, or isotropic liquid/crystalline solid phase transitions if at least one component forms a liquid-crystal (Nakai et al., 1986; Hasegawa et al., 1988), (c) sol-gel transition (Kawanishi et al., 1987), and (d) liquid/liquid microphase transition (Hashimoto et al., 1988 a) if at least one component can form microdomains, like block copolymers. In the mixtures which belong to categories (a) and (b) many variations of pattern comprise Cryst/Cryst, Cryst/ Amorph, Cryst/LC, Amorph/Amorph, Amorph/LC and LC/LC where Cryst, Amorph and LC, respectively, refer to crystalline, amorphous and liquid crystalline domains, and Cryst/Amorph refers to the pattern comprising crystalline domains of one component and amorphous domains

Equilibrium State 1

Phase Transition

(Homogeneous Mixture)

Intermediate Structure 1

Intermediate Structure 2

257

of the other component. The binary blends of category (d) comprise A/A-B, C/A-B, (A-B) 1 /(A-B) 2 , A-B/B-C, etc., where A/ A-B denotes binary blends of homopolymer A and block copolymer A-B, A-B/ B - C denotes those of block copolymers A-B and B-C, and (A-B) 1 /(A-B) 2 denotes those of A-B block polymers with different molecular weights and/or compositions of A. Although the blends C/A-B and A-B/B-C comprise three polymer components, they may also be regarded as binary because of the connectivity between A and B or that between B and C. Moreover, fascinating physical effects are expected due to the connectivity, which gives rise to different phase diagrams and pattern formations for the following binary mixtures with the same constituent polymers A, B and C, e.g., C/A-B, A/B-C, B/C-A, A-B/B-C, B-C/C-A, C-A/A-B. These mixtures should have different behaviour from A/B/C ternary blends. The third example of the many patterns imaginable consists of binary blends of linear, branch, star and network polymers. A unique feature of the blends with a network polymer (de Gennes, 1979 a; Bastide et al, 1985, 1990; Briber and Bauer, 1988; Bauer et al, 1989) is that the mixing or unmixing involves the free energy change for elastic deformation of networks (Flory, 1953) as well as the Flory-Huggins free energy of mixing. In this chapter a review will be given on the basis of a series of our recent experimental researches on the dynamics of the

Equilibrium State 2 (Macroscopically Phase-Separated Mixture)

Intermediate Structure N

Figure 6-2. Self-assembly of patterns in polymer blends in phase transition.

258

6 Structure of Polymer Blends

pattern formation of polymer blends. This review will supplement our earlier reviews (Hashimoto, 1987b, 1988a). The readers may refer also to the recent reviews (Nose, 1987; Binder, 1991) or monographs (Han, 1984; Walsh et al., 1985; Utracki, 1989; Fujita, 1990). Tables 6-1 to 6-3 are lists of the characteristics of the samples referred to in this chapter.

6.2 Unique Features in Polymers Here we will briefly summarize unique features in the self-assembly of polymer blend systems in terms of both (i) dynamical (Sec. 6.2.1) and (ii) equilibrium aspects (Sec. 6.2.2). 6.2.1 Dynamics It will be shown later, especially in Sec. 6.5.1 and 6.5.2, that the dynamics and patterns in polymer blends obey some universal features found for low molecular weight mixtures. If this is the case then the selfassembly of polymer blends is unique only in that it has a larger spatial scale and a longer temporal scale than that of low molecular weight mixtures. The spatial scale of polymer blends will be characterized by the thermal correlation length £ in the single phase state. Note that the size of the self-assembling patterns (or growing domains) A (t; T) is scaled also by £(T) in which t is time spent for phase separation at temperature T after quenching from the single phase state and £(T) is assumed to be symmetrical around the spinodal temperature Ts. £(T) is related to the characteristic length £0 intrinsic to polymer molecules and the parameter sT associated with AT given below. In the context of mean-field approximation, :

£o%1/2

(6-2)

(6-3) where £0 is given by (6-4) for the symmetric blends of A and B having equal gyration radii RgQ = RgA = RgB or root of mean squared end-to-end distances Ro = RA = RB for polymer coils in an unperturbed state, x a n d Xs a r e Flory's interaction parameters per segment between polymers A and B at T and Ts, respectively. Thus it follows that ^polym

^O,polym

^ small

^0, small

~iV 1/2

"l/2

(

AT

-1/2

(6-5)

where N is the DP of polymers. Constituent polymers A and B are assumed to be symmetric, having the same DP for simplicity. £polym, £0,poiym and ATpolym are & £0 and AT for polymers, and £small, £OiSmaU and ATsmall are corresponding quantities for small molecules. Thus, under the same AT, ^ and A (t; T) for polymers are greater than those for small molecules by a factor of AT1/2, typically by 10 to 100 times. Thus the light scattering (LS) method becomes important for characterizing the self-assembly in addition to small-angle X-ray scattering (SAXS) and small-angle neutron scattering (SANS) methods conventionally used for metallic alloys and inorganic glasses. The temporal scale of the concentration fluctuations in single phase state and that in the early stage SD are characterized by the characteristic time tc (T) of the systems. The temporal scale for self-assembly in the later stage SD may naturally be related to tc(T). This characteristic time tc(T) is given by (6-6)

6.2 Unique Features in Polymers

where D app (T) is the collective diffusivity given by D app (T) = Dc (T) sT

(6-7)

DC(T) is the self-diffusivity of polymers where again symmetric polymer pairs are assumed for simplicity, DC(T) = Dc A(T) = DcB(T). If polymer melts having N §> iVe, JVe being the DP between the entanglement couplings, obey the reptation dynamics (de Gennes, 1979 b; Doi and Edwards, 1986; Chap. 9 of this Volume), it follows that R20/Dc~f(T)Ne2(N/Ne)3

(6-8)

where trep is the reptation time, and f(T) is the temperature-dependent monomeric friction constant. From Eqs. (6-2), (6-4), and (6-6) to (6-8),

tc(T)~f(T)Ne2(N/Ne)3^

3

o

-2

(6-9)

Equation (6-9) may be useful to compare the temporal scale for polymers systems with that for small molecule systems. Since jVe = O(102) and if N/Ne = O(10), then Ne2{N/Ne)3 = O(107). Thus for a situation in which f(T)sj2 is comparable for the two systems, tc(T) for polymers is longer than that for the small molecules by a factor of 107! In other words, AT of 1 mK in small molecules may roughly correspond to that of 1 K in polymer systems. Up to now the dynamics and patterns in simple liquid mixtures have been extensively studied only near the critical point. This is partly because of the theoretical interest in the critical phenomena, but also because of experimental difficulties in studying the phenomena away from the critical point, i.e., the dynamics can be made sufficiently slow only near the critical point. Furthermore the studies of the timeevolution of the patterns for the simple liquid mixtures have been restricted to those of their global features, as will be discussed

259

in Sec. 6.5.1 and 6.5.2. The long temporal scale of the polymer systems would make it possible to study precisely the phenomena away from the critical points, i.e., at deep quenches and for off-critical mixtures. Moreover, their large spatial scales would make it possible for us to extend the study to time-evolution of local features of the patterns, such as that of interfacial thickness and waviness, as will be discussed in detail in Sec. 6.6. Thus the unique features of the polymers are anticipated to greatly facilitate extremely detailed investigation of time-evolution of the self-assembling patterns via the phase transition. In addition to these unique features on the temporal and spatial scales, as described above, they have specific unique features, as will be described immediately below and in Sec. 6.5.3. 6.2.2 Phase Diagrams: Large Cooperative Effect in Polymers Figure 6-3 shows sketches of some phase diagrams which may be observed in polymer blends; parts (a) and (b) representing phase diagrams with an upper critical solution temperature (UCST) and a lower critical solution temperature (LCST), respectively. A fundamental method to study the self-assembly of the pattern at the phase transition involves the real-time or in-situ analyses of the growing fluctuations at a given temperature T, for example, after a rapid temperature quenching of the mixture with a volume composition of A equal to cf)AO from a single-phase state H to a state I in the miscibility gap. Thus the phase diagrams are the key for the study of the ordering processes or the pattern formation. The simplest, but most fundamental theory predicting the phase diagrams of polymer mixtures is the Flory-Huggins (FH)

260

6 Structure of Polymer Blends LCST

<])A1

<J)AO

(|)A2

(J.A

(a)

(b)

mean-field theory (Flory, 1953). According to FH, the free energy change upon mixing per lattice site is given by AF/kBT = = (1/N) [A In 0A + (1 - A) In (1 - A)] + + Z0A(1-^A) (6-10) for a symmetric pair having JVA = NB = N. Here kB T is the thermal energy. Equation (6-10) predicts the mean-field critical point given by =2

(6-11)

where (#N)C is the net interaction parameter between two polymers A and B at the critical point. Equation (6-11) predicts a large cooperative effect in the critical point or miscibility of polymers, in that a small change of the segmental interaction x induced by a small change of the primary structure (the chemical structure of a monomer) causes a strikingly large change of x N because of a large value of N, i.e., a large cooperation. This is simply because the combinatorial free energy [i.e., the first term of the right-hand side of Eq. (6-10)] is reduced to the order of 1/N, compared with the mixtures of small molecules with N = 1, or because the interaction term [i.e.,

Figure 6-3. Sketches of (a) UCST and (b) LCST phase diagrams, and definitions of A 0 e , (j)A1, A2 and AO.

the second term of the right-hand side of Eq. (6-10)] becomes more significant than the combinatorial term by a factor of N. The FH theory assumes a constant volume upon mixing. In reality a small change of volume, no matter how small it is, significantly affects the miscibility and the critical point if N is large. Thus discussion on polymer phase behaviour requires an extremely high precision in determining the non-combinatorial free energy, i.e., the free energy associated with the interaction energy and the volume change upon mixing (Flory et al, 1964; Flory, 1965; Orwall and Flory, 1967; Sanchez and Lacombe, 1976). The thermodynamic instability at higher temperatures, as shown in the schematic phase diagrams of Fig. 6-3 b, is due to the effect of the volume contribution to the free energy change of mixing, or to the random copolymer effect (ten Brinke et al., 1983; Sakurai et al., 1990 a, 1990 b, 1991) as described below. As an example of this large cooperative effect, we will discuss an effect of the microstructure on the miscibility of diene polymers (Sakurai et al, 1990a, 1990b, 1991; Han et al, 1990), i.e., the polymers used as rubber materials, such as polyisoprene (PI)

6.2 Unique Features in Polymers

and polybutadiene (PB). These polymers are considered to be copolymers composed of monomers of different microstructures (chemical structures) which are generated by a statistical nature of the polymerization reaction. Polyisoprenes have 1,2-, 3,4-, and 1,4-linkages for the isoprene monomers, while polybutadienes have 1,2- and 1,4-linkages for butadiene monomers, as shown schematically in Fig. 6-4 where x, y and u are the fractions of the respective monomeric structures in the polymer chains. A small change in x, y and u causes a small change in the effective x (designated hereafter as xeff) P e r monomer unit between PI and PB, which in turn has a big effect on the miscibility, because of the large N effect. Tacticity of vinyl polymers (Beaucage et al, 1991; Kyu and Lim, 1990 a, 1991) would have a similar effect on its miscibility with other polymers. The effect of microstructure on #eff and miscibility of PI/PB can be quantitatively investigated with high accuracy by using small-angle neutron scattering (SANS) with the deuterium labelling technique (Sakurai et al., 1990a, 1990b, 1991). Table 6-1 gives characteristics of protonated PI (HPI) and perdeuterated PB (DPB). Figure 6-5 shows the temperature dependence of Xeff determined for 10 sets of the HPI/DPB

(-CH-CH2-)y

VH3

(-CH 2 -CH-) U CH I CH2 (d)

where the subscripts 1 and 2 refer to DPB and HPI, respectively. The implications of Fig. 6-5 are as follows. For all the mixtures DPB/HPI, (i) %e{{ is given over the temperature range covered in the experiments by X*f =AX + BX/T

(6-13)

with Bx < 0, and hence (ii) they have an LCST type phase diagram, (iii) Compared with the coefficient Bx, related to the enthalpic part, the entropic part Ax is more sensitive to the microstructure. (iv) For a given HPI, the values Ax or #eff systematically decrease and hence the miscibility is

CH3

(b)

(a)

mixtures by the nonlinear regression fitting of the experimental SANS profiles, measured as a function of temperature in the single-phase state, to the theoretical scattering profiles predicted by the random phase approximation (de Gennes, 1979 b; Warner et al., 1983). All the results shown in Fig. 6-5 were obtained for a given composition of DPB (cf)1 = 0.475). The point marked X at the high temperature end of each straight line in the plot of #eff vs. 1/T designates the value xeff a t the spinodal point (xs) predicted by the mean-field theory,

(- C H 2 - - C - C H - CH 2 -)

CH CH 2

261

-y)

(c)

( - C H 2 - CH = CH -CH 2 -) 1 - u

(e)

Figure 6-4. (a) 3,4-, (b) 1,2-, and (c) 1,4-linkages in polyisoprene and (d) 1,2-, and (e) 1,4-linkages in polybutadiene.

262

6 Structure of Polymer Blends Temperature (°C) 200

-1 T"1 x 10 3 (K" 1 )

Figure 6-5. Effective Flory-Huggins interaction parameters per monomer unit j e f f between polyisoprene and perdeuterated polybutadiene determined by SANS in the single phase state. The value xeff was determined for a given composition of DPB/HPI 47.5 vol.%/52.5 vol.%. The results numbered 1 to 4 are for the mixtures of HPI-7-101 with DPB-12-52, DPB-16-61, DPB-20-103 and DPB-28-71, respectively, and those 5 to 7 are for the mixtures of HPI-12192 with DPB-16-61, DPB-20-103, and DPB-28-71, respectively, and those 8 to 10 are for the mixtures of HPI-15-136 with DPB-16-61, DPB-20-103, and DPB28-71, respectively. The sample codes should be referred to Table 6-1. From Sakurai et al. (1991).

enhanced on increasing the fraction of 1,2butadiene unit. Although the change of / eff with the microstructure is small, of the order of 2 x 10 ~3, the small change causes a big change in the spinodal temperature Ts, because #s is also of the order of 10" 3 for polymers having high molecular weights of

the order of 105. For a given mixture with a given value of #s, the value Ts changes by about 200 °C by merely changing the microstructure of DPB and HPI. (v) The results shown in Fig. 6-5 also imply that, for a given DPB, the values / cff decrease and hence Ts increases, or the miscibility is enhanced upon decreasing the fraction of the 3,4-isoprene unit y, though it is not explicit in Fig. 6-5 (Sakurai et al., 1991). A similar analysis can be applied to isotopic mixtures of DPB and protonated PB (HPB) so that both the isotopic and microstructure effects on miscibility and xeff can be investigated (Bates et al., 1986; Sakurai et al, 1990 a; Han et al, 1990). The isotope effects on the phase transition can be investigated much more purely for mixtures of deuterated (DPS) and protonated polystyrene (PS) (Bates and Wignall, 1986; Green and Doyle, 1986; Yang et al, 1986 b; Schwahn et al, 1990) than for mixtures of DPB and HPI, free from complications due to microstructure effects. In the DPS/ PS mixtures, one may have to keep in mind the effect of tacticity on the phase transition. Table 6-2 gives the characteristics of HPB and DPB used for these experiments. The SANS analyses in the single-phase state mixtures yielded the data on temperature dependence of x eff , as given by Eq. (6-13), but as Bx > 0 for all three mixtures, they thus have a UCST type phase behaviour (Fig. 6-6 a). For a given DPB (e.g., DPB-63-268), #eff is observed to increase, and hence the miscibility is suppressed with decreasing the 1,2-butadiene unit of HPB from 68 to 7 wt.% (cf. the results indicated by the straight lines numbered 1 and 3). For a given HPB (e.g., HPB-7-10 or HPB-11-52 having the nearly equal fraction of 1,2-linkages), #eff tends to increase and hence the miscibility is suppressed with increasing the 1,2-unit of DPB from 11 to 63 wt.% (cf. the results numbered 1

263

6.2 Unique Features in Polymers

Table 6-1. Characteristics of polymers. (a) Protonated polyisoprene Sample code

M

HPI-7-101 HPI-12-192 HPI-15-136

a

Microstructure b in %

xl0~

1.0, 1.03 1.05

100 192 136

3,4

Cis-1,4

Trans-1,4

7 12 15

69 66 63

24 22 22

1,2

(b) Deuterated polybutadiene Sample code

DPB-12-52 DPB-16-61 DPB-20-103 DPB-28-71

M

a

Microstructure b in %

xl0~

49 59 103 71

1.07 I.O3

1.05 1.02

1,2

Cis-1,4

Trans-1,4

12 16 20 28

36 38 36 27

52 46 44 45

a

Number average molecular weight M n and weight average molecular weight M w , determined by GPC equipped with low-angle light scattering apparatus; b microstructure determined by 13 C-NMR.

Table 6-2. Sample characteristics for HPB/DPB blends. Microstructure (%) ( 13C-NMR)

Mnaxl0~3

MJMn>

(GPC)

(GPC)

1,2

Cis-1,4

Trans-1,4

HPB-7-10 HPB-11-52C HPB-68-243

6.9 b 4.9 135

1.5 1.05 1.8

7 11 68

40 36 11

53 53 21

DPB-11-276c DPB-63-268

252 134

1.1 2.0

11 63

36

53 *

Sample code

a

M n : Number average molecular weight, M w : weight average molecular weight; b determined by vapor pressure osmometry; c from Bates, Dieker, and Wignall (1986); * undetectable due to spectral broadening.

and 2). Interestingly, the effect of the microstructure is opposite for DPB and HPB, the origin of which may well be understood in terms of the copolymer effect, as described below (e.g., using Eqs. (6-14) and (6-15), as well as the results shown in Fig. 6-6 b).

DPB and HPB are considered to be random copolymers of 1,2- and 1,4-deuterated and protonated butadiene units, respectively (designated (Dl,2) x -r-(Dl,4) 1 _ J C and (H1,2)y - r - (H1,4) x _ y, respectively, x and y being the weight fractions). Thus in the context of the mean-field theory, %eff

264

6 Structure of Polymer Blends

2,5

2.0

Temperature (°C) 100 50

150 1

30

1

i ^

—

may be expressed in terms of "fundamental" segmental x values between monomers with different microstructures (ten Brinke et al., 1983),

^ ^

1

Xeff

=

^ - - ^

1.5 o

5 i.o

(6-14)


0.5

0 - OR 05

2.3

,—

.

^

-

—

-

^ -

.. -

-

,

- •-

7 1

|

2,5

3.0 MO3 (K"1)

(a)

3,3

where XDI,2/HI,2 represents the "fundamental" x value between deuterated and protonated butadienes with the 1,2-structure. Here we use the word "fundamental" in order to distinguish the specific segmental interaction between different microstructures, such as #DI,2/HI,2 from the average segmental interaction between the two polymers x eff . The effect on xeff due to the difference between cis-1,4 and trans-1,4 structures was ignored for simplicity. If we can further assume, about the isotope effects on %, that ZD1,2/D1,4 — ZH1,2/H1,4 — Xl ZD1,2/H1,4 ~ ZD1,4/H1,2 = Xl ZD1,2/H1,2

=

(6-15)

XD1,4/H1,4 = X3

Xeff is expressed in terms of the three fundamental ^-values, Xi? Xi a n d Z 3 . Since we have three sets of data on #eff (T) for the mixtures having three sets of x and y, we

(b)

Figure 6-6. (a) Temperature dependence of #eff for three sets of HPB/DPB mixtures with the compositions of 70/30 in vol.%/vol.%. The curves numbered 1 to 3 correspond to HPB-7-10/DPB-63-268, HPB11-52/DPB-l 1-276, and HPB-68-243/DPB-63-268, respectively. The sample codes should be referred to Table 6-2. (b) Temperature dependence of the fundamental x values Xi, Xi a n d X3» as obtained from #eff's, measured for the three sets of mixtures HPB/DPB, as shown in Fig. 6-6a. The definitions of Xi to %3 are shown in the figure. From Sakurai et al. (1990 a).

6.3 Formation and Growth of Self-Assembled Patterns via Spinodal Decomposition

can determine X\, Xi a n ^ X3 by solving the simultaneous equations for #eff, the results of which are represented in Fig. 6-6 b. The results indicate Xi, Xi>X?n which is quite natural. The results in Fig. 6-6 b also give the following conclusions: (i) For the isotopic mixture with the same microstructure, i.e., x=y, #eff is minimum when x = 0.5, because Zeff = X3 ~ 2x(l - x ) (X1-X2 + X3) (6-16) Thus the isotope effect depends on the microstructure. (ii) The isotopic mixtures with different microstructures should have a larger %eff and hence are less miscible than those having similar microstructures. Conclusion (i) is confirmed by comparing the experiments of Bates et al. (1986) on DPB-11-276/HPB-11-52 and those of Sakurai et al. (1990 a) on DPB-63-268/HPB68-243.

6.3 Formation and Growth of Self-Assembled Patterns via Spinodal Decomposition A Brief Introduction Here we shall briefly overview general features in the self-assembly of the patterns in polymer blends via spinodal decomposition (SD). More detailed discussions will be carried out in subsequent sections. 63.1 Early, Intermediate, and Late Stages in Spinodal Decomposition The growth of the spatial composition fluctuations and the pattern formation via SD may be classified into at least three regimes; (a) early stage, (b) intermediate stage, and (c) late stage as sketched in Fig. 6-7 (Hashimoto et al., 1986 a). More pre-

265

cise descriptions of the process may obviously require a further refinement of this model by classifying into more stages, as will be discussed later (Bates and Wiltzius, 1989; Hashimoto et al., 1991b). It may even be necessary to add the stage leading to the early stage, as inferred by Strobl (1985) and Schwahn et al. (1990). At this stage, which occurs immediately after a temperature quench from state H to state I and which may be called the very early stage, the initial concentration fluctuations, which existed during state H in the single phase state, change into those relevant to the early stage SD at state I in the two phase state (see Fig. 6-3). In this very early stage time-resolved elastic scattering experiments may show the following timeevolution behaviour: a scattering maximum may appear sometime after the temperature quench at a scattering vector qm. This qm may decrease with time to a value qm(0) relevant to the early stage SD, as will be described below. Investigation of this stage requires future work. In the early stage SD (a), growth of the fluctuations is weakly nonlinear so that it can be well approximated by the predictions based on the linearized theory (Cahn, 1965). The wavenumber qm(t) or the wavelength A(t) = 2n/qm(t) of the dominant mode of the fluctuations is constant, independent of time t. Thus, 2704(0 = qm(t) = qm(O)t~a

(6-17)

with a ^ 0, and the time-independent constant qm(0) is given in the context of the mean-field theory (deGennes, 1980; Binder, 1983, 1991) by „

(c\\

c\ ,"}\l/2 n — 1 c l/2

£ * _ _ 1 U ) — I DA)

^ e O

T

(f. \ Q\ ^u"lOj

for the symmetric polymer blends. The amplitude of the composition fluctuations A<j)q for the g-mode increases exponentially

6 Structure of Polymer Blends

266

(a) Early stage of SD

-A2-

(b) Intermediate stage of SO 0 A (I)

//

H

Ai

II

II 11 1 1

*H

A2

M ,

\ \

f

A<

// ' / ' / /

1

N\

\

\

\

\

i

\

!

\

Figure 6-7. Sketches on time-evolution of the composition fluctuations in the (a) early, (b) intermediate, and (c) late stages of the spinodal decomposition. 0 A1 and 4>A2, respectively, are the equilibrium compositions of A-rich and B-rich domains at temperature 7^ (see Fig. 6-3). From Hashimoto etal. (1986 a).

(c) Late stage of SD

with time and the growth rate R (q), which depends on q, A(l>Jt)~exp[R(q)t]

(6-19)

Hence A(j)(t) for the dominant mode having qm or A, is given by A(f> (t) ~ exp [R (qm) t]

(6-20)

In the intermediate stage (b) the nonlinear effects (Langer etal., 1975; Gunton etal., 1983) on the time-evolution of the spatial composition fluctuations become increasingly important with time. As a consequence, growth of the fluctuations is governed by the mode-mode coupling effects which have been treated by the nonlinear time-evolution equation called the time-

dependent Ginzburg-Landau (TDGL) equation (see for example, Ma, 1976). The pattern growth involves an increase of A or a decrease of qm with t as well as a further increase of A(j)(i). The time change of A(t) can be described by Eq. (6-17), but the time change of Ac/)(t) deviates from that predicted by Eq. (6-20), becoming slower than that predicted by the exponential growth. In the late stage (c) the local compositions of the domains reach equilibrium compositions 0 A1 and 0 A2 (Fig. 6-3) and hence the fluctuation A(j)(t) also reaches the equilibrium value A0e at Tx (Fig. 6-7). The interface between the two coexisting (A-rich and B-rich) domains is expected to be well-defined in the later part of the inter-

6.3 Formation and Growth of Self-Assembled Patterns via Spinodal Decomposition

267

mediate stage or the early part of the late stage (Hashimoto et al., 1991b; Jinnai et al., 1990,1991). In general there must be at least three time-dependent length parameters A{t\ Rm{t\ and t^t) which characterize the pattern growth, as sketched in Fig. 6-8, in which the composition fluctuation A(j)(t) essentially reaches a constant value with t(A0e), during this stage. The interface may be wavy, which is characterized by the mean radius of interface curvature Rm(t). The interface may be diffuse, which may be characterized by the characteristic interfacial thickness defined by £j = A^/ldc^/dzl^

(6-21)

where z is the local coordinate normal to the interface. Up to now the research has been focused on the time-evolution of global aspects of the patterns, and hence on A {t% by reducing or coarse-graining information on the fine details of the patterns, such as tx(t) and Rm(t). Thus the characteristics of the pattern growth have been discussed in terms of the dynamical self-similarity or dynamical scaling hypothesis in the global scale which invokes (6-22)

Figure 6-8. Sketch of the pattern in the late stage spinodal decomposition, and characteristic parameters A(j)(t), A(t% t^t), and Rm(t) describing the time evolution of the pattern. A (t) and A (t) are defined in Fig. 6-7, t^t) and Rm(t) are the characteristic interfacial thickness and the mean radius of the curved interface, respectively. From Hashimoto et al. (1991 b).

interfacial structure, as manifested by t^t) and Rm{t\ should be given more attention. Some discussion along this line will be provided in Sec. 6.6. 6.3.2 Scaling Postulate, Dynamical Self-Similarity, and Their Experimental Test

The pattern growth can be experimentally analyzed in real space and reciprocal a remarkably simplified picture of the pattern growth, where <\> (I) is a time-indepen- space. The real space analysis involves the pattern analysis of optical- or electron-mident scaling function with the reduced veccroscopic images to obtain information on tor l = r/A(t). The dynamically self-similar A(r,0, A(t), Rm{t), etc. (McMaster, 1975). growth in the late stage, which gives rise to The real-time or in-situ analysis of polymer the relation given by Eq. (6-22) or the blends by electron microscopy is in its inscaled structure factor F(x) given by Eq. fancy at present. There are limitations due (6-31) in next section (Sec. 6.3.2), can be to the spatial resolution and the weak imtested experimentally. age contrast in some cases in the real-time Our discussion thus far, both on polyand in-situ analysis by optical microscopy. mer blends and mixtures of small moleThe reciprocal space analysis involves patcules, has been confined to the time-evolutern analysis by scattering methods such as tion of the global structure factor. HowSANS, SAXS, and LS. ever the time-evolution of the detailed

268

6 Structure of Polymer Blends

The reciprocal space analysis has been effectively carried out by the time-resolved SANS and SAXS for most of the small molecular systems (e.g., metallic alloys and inorganic glasses), except in the case of mixtures of simple liquids in the vicinity of critical points. The latter has been carried out by time-resolved LS. The pattern in polymer mixtures have a large length scale as is obvious from a large molecular size Rg0 or £0 [see Eqs. (6-2) to (6-5)]. Thus the LS method and ultra small-angle SANS and SAXS methods, as attained by the Bonse-Hart type camera (Bonse and Hart, 1966, 1967; Schwahn and Yee-Madeira, 1987), are very important, or a combination of the LS, SANS, and SAXS offers complete characterization of the spacetime organization of the patterns (Hashimoto et al, 1989 e; Jinnai et al., 1990,1991). The time change of the pattern is manifested in that of the scattering intensity I(q,t% e.g. for the isotropic patterns in 3-d space, (6-23) with x = q(t)/qm(t)

(6-24)

where q is the magnitude of the scattering vector and is defined by 4 = (4701) sin (0/2) 9 and X being the scattering angle and the wavelength of radiation in the medium, respectively. Equation (6-23) is valid for isotropic patterns. For anisotropic patterns in 3-d space, A and q should be replaced by vectors, I{q9t)^
(6-25)

where AK(t) and qK are, respectively, the component of the vectors A and q along

one of the Cartesian coordinates. {f]2(t)} is the mean square fluctuation of the scattering contrast associated with A(/>A(r,t) = 4>A (r, t) — 4>AO, an order parameter of our system, i.e., time-dependent local composition fluctuation of one component, e.g. A. S(x) is the scaling function which describes the time change in the shape of the pattern. If the shape and detailed statistical nature of the pattern (morphology) change with time, S(x) also changes with time, but if they do not, S(x) is independent of time and is called the universal scaling function. In the latter case the pattern can be scaled by the single time-dependent length parameter A(t) = 2n/qm(t) for isotropic cases. The dynamical scaling hypothesis (Binder and Stauffer, 1974; Binder, 1977) with the single length parameter strictly holds only in the case where the three length parameters have the same time-dependence A (t) ~ Rm (t) ~ tx (t) ~ ta

(6-26)

or two of the length parameters A (t) and Rm(t) have the same time dependence but the remaining parameter is constant (tle) with t9 i.e. Rm{t) ~ t\

tY =

(6-27)

As will be shown later (Figs. 6-12, 6-13, and 6-15 to 6-18 in Sec. 6.5.2 and 6.6), S(x) for the conserved order-parameter systems shows a maximum at x = 1 since I(q) has a maximum at q = qm satisfying Eq. (6-17). It should be noted that in this chapter we are dealing with conserved order parameter systems, satisfying J AK(r,f) dr = 0 (K = A or B) Thus the time change of the wavenumber for the dominant mode of the composition fluctuations can be determined by the time change of the peak scattering vector qm(t). The time change of the peak intensity Im(t)

6.3 Formation and Growth of Self-Assembled Patterns via Spinodal Decomposition

is given from Eq. (6-23) by - tfi

(6-28)

Thus Im(t) contains information on A(t\ (rj2(t)y and S(l). The latter two quantities vary with t in the intermediate stage but become essentially independent of t in the late stage in which the dynamical scaling hypothesis is valid. In this late stage 7m(0 - A3(t) ~ £3a, giving the following relationship between the scaling exponents a and p [Eqs. (6-17) and (6-26)] P = 3a

(6-29) 2

whereas in the intermediate stage (rj (t)} significantly increases with t, so that we expect the following relationship, )8>3a

(6-30)

The scaled structure factor (Binder and Stauffer, 1974; Binder, 1977) at T, F{x; T), is defined by F(x;T) = I(x;T)ql(t;T)

(6-31)

In this case x in Eq. (6-24) depends also on T through qm(t;T). From Eq. (6-23), the structure factor F(x; T) is shown to be proportional to S(x),

F(x;T)~
(6-32)

In the intermediate, (rj2(t;T)} increases with t, and moreover S(x) can change with t. Hence F(x;T) depends on t and thus is not universal with t. However in the late stage where the dynamical scaling hypothesis is valid, (rj2} and S(x) become independent of time and hence F(x; T) becomes universal with t. When F (x; T) is discussed as a function of % it is convenient to reduce F(x;T) further with temperature-dependent

(6-33)

269

where q' and q" are the lower and upper bounds of q, outside of which / (q, t; T) q2 practically goes to zero. The reduced scaled structure factor F(x; T) simply depends on the scaling function S(x% i.e., F(x; T) = F(x; T) I) I(q,t; T) q2 dq ~ S(x) 1 q ' (6-34) and thus becomes universal with t and T if the patterns are scaled with the time- and temperature-dependent single length pa-

rameter

qm{t-T)'\

6.3.3 Complex Factors The pattern growth becomes complex or greatly varied when it is coupled with physico-chemical reactions (Hashimoto, 1992). It will be pinned down at some stages when crystallization (Inaba et al., 1986, 1988) or vitrification (Hashimoto et al., 1984) takes place during the growth, generating locked-in non-equilibrium patterns ("physical pinning", as will be discussed in Sees. 6.8.1 and 6.9). "Spontaneous pinning" will take place for some off-critical mixtures (Hashimoto, 1988 b; Takenaka et al., 1989) even in the absence of the physico-chemical reactions (Sec. 6.7). The growth will be pinned down also by chemical reactions such as crosslinking of at least one of the domains ("chemical pinning", Sec. 6.9) (Hashimoto et al., 1989d). The pattern growth is strongly affected by applied mechanical fields such as shear flow and shear deformation (Sec. 6.10). The spinodal decomposition is coupled with another kind of liquid-liquid phase transition, called the microphase transition, when at least one component is a block copolymer (Mori et al., 1987; Tanaka and Hashimoto, 1988; Hashimoto et al, 1988 a). The interplay between the two kinds of ordering process provides many variations in the pattern growth (Sec. 6.8).

270

6 Structure of Polymer Blends

6,4 Early Stage Spinodal Decomposition

and

Here we focus on the time-evolution of the concentration fluctuations in the small q-regime of q Rg
where qm(O;T) is given by Eq. (6-18). The behaviour of R (q; T) in the large q regime, in which the ^-dependence of A(q) becomes important, was theoretically treated first by Pincus (1981) and Binder (1983), then experimentally by Higgins et al. (1989) and Schwahnetal. (1990). (iv) The mean-field approximation is valid when the Ginzburg criterion (Binder, 1984)

= ao;T)

qG(T) = y/2qm{0;T)

(6-35)

Equation (6-35) takes into account the contribution of the thermal noise to I{q9t;T) (Cook, 1970; Binder, 1983). I(q,t = 0; T) is the initial scattering intensity, and I(q,t = co;T) is the scattering intensity associated with the virtual structure factor at T The linearized theory ignores the contribution of I(q,t = oo; T) to

i(q,nn (iii) The growth rate R(q) for the small q regime is given by R(q;T) = D3pp(T)q2[l-q2/q2AT)]

(6-36)

where D app (T) and q2(T) are the parameters characterizing the early stage SD. These parameters may be obtained from the so-called Cahn's plot in which R(q)/q2 should vary linearly with q2. D app is the mutual (or collective) diffusivity, and qc(T) is the crossover q satisfying R(q;T)>0

for

q
(6-37 a)

R{q;T)<0

for q > qc(T)

(6-37b)

8T > 1/N

(6-38)

(6-39)

is satisfied for symmetric blends with NA = NB = N. Obviously this approximation is good for polymers with large JV's. In this context, D app and q^(0) are given by Eq. (6-7) and 2 q2m (0; T) = q2c(T) = ^2 sT

(6-40)

The crossover from the mean-field behaviour to the behaviour of the kinetic Ising universality class was investigated near the critical point in which eT < 1/N by Schwahnetal. (1987 a, 1987 b, 1991) and by Bates et al. (1990). Okada and Han (1986) and Sato and Han (1988) studied the effects of thermal fluctuations on / (q, t; T) in the early stage SD for PS/PVME mixtures and found that the effect at small q
271

6.5 Growth of Self-Assembled Structures in Later Stage Spinodal Decomposition

reviews by Hashimoto, 1987 b, 1988 a, 1988 b; Nose, 1987; and Han, 1988). However the following specific remarks are worth noting: (a) There is a report (Bates and Wiltzius, 1989) that, even in the early stage, qm(t) directly observed from the scattering peak linearly decreases with t, in contradiction of the linearized theory. Nevertheless, the limit of qm(t) at £ = 0 quantitatively obeys Eqs. (6-4) and (6-40). (b) A number of experimental results reported are that qm (0; T) obtained directly from the scattering peak position or indirectly from the Cahn's plot, in the meanfield regime, was not in good agreement with the theoretical prediction given by Eqs. (6-4) and (6-40) (Sato and Han, 1988; Hill et al, 1985 a, 1985 b). Our experimental values of qm(0;T) were much smaller than the theoretical values by a factor of 2 to 5 (Hashimoto et al., 1986b; Takenaka et al., 1987; Onuki and Hashimoto, 1989). The early stage SD lasts only a small fraction of the entire time span required for blends to achieve two phase equilibrium. In the reduced time % defined by

= t/tc(T)

(6-42)

where tc{T) is a characteristic time for a blend at T [Eq. (6-6)], the early stage occurs approximately up to T = 2 (Hashimoto, 1988 a). Although the early stage SD extends only over a short period in the reduced time scale, it can extend over a very long period in the real time scale for the polymer blends because tc(T) is very large, as discussed in Sec. 6.2.1. This fact greatly facilitates quantitative studies in the very early stage to the early stage SD. Significance of the early-stage studies may also be realized when one envisions the cases where the concentration fluctuations developed during early stage SD may be locked-in by some processes which hin-

der the translational diffusion of polymer molecules. As A^ increases with t, the processes such as vitrification, crystallization, intermolecular interactions such as hydrogen-bonding, gelation etc. may occur, and they hinder the diffusion and hence prevent further growth of the fluctuations (see Sees. 6.8.1 and 6.9, physical pinning).

6.5 Growth of Self-Assembled Structures in Later Stage Spinodal Decomposition In this section we discuss the self-assembling of the structure in the later stage (i.e., the intermediate and late stages) SD for near critical mixtures, i.e., AO ~ 4>AC, the critical composition. 6.5.1 Scaling Analyses of Im(t; T) and qm(t;T) Here we discuss the validity of the scaling postulate proposed by Chou and Goldburg (1979) for mixtures of simple liquids near the critical point. It postulates that the time changes of / m and qm obtained at various temperatures T can be scaled by the temperature dependent correlation length £(T) and mutual diffusivity D app (T). Thus the reduced wavenumber Qm(x) is defined by GmW = «m(T)5(T1)

(6-43)

The data obtained for qm {t; T) at various t and T should fall onto a single master curve when they are plotted as a function of the reduced time i, as defined by Eqs. (6-6) and (6-42). Similarly the reduced scattering intensity Tm{z) is defined by

I{q9t,T)q2dq (6-44)

272

6 Structure of Polymer Blends

Data Tm(r) obtained at various t and T from I(q;t,T) should fall onto a master curve when they are plotted as a function oft. In binary mixtures of simple liquids, t;(T) and D app (T) were determined by static and dynamic light scattering (Chu et al., 1969; Gulari et al., 1972) in the single-phase state as a function of T. Then the values at a given phase separation temperature T were obtained by extrapolating them using the following scaling laws, = fole T r v

critical mixture of SBR1/PB19 with a composition of 50/50 wt./wt. at deep quenches of eT ~ 1 (Izumitani et al., 1990). Figure 6-9 shows the time-evolution of light scattering profiles at 60 °C where the scattered intensity (in arbitrary units) was plotted as a function of the scattering vector g, and the time elapses in the order of (a) to (c). 180000

14-0000 -

(6-45) 100000 -

and (6-46) Equations (6-2) and (6-7) correspond to the mean-field approximation, v = l/2 and y = l. Note that the 3d kinetic Ising model for the critical regime of sT
T(min. 36.99 51.19 82.95 118.94

(a)

(6-47)

600

(6-48)

-400

Hence

where T is given by Eq. (6-42). We follow this practice throughout this review. We should note, however, that £(T) used in the original scaling postulate should correspond to q^T)'1 in the linearized theory [Eq. (6-2)], which gives rise to a difference by a factor of 2 for tc(T) and of 1/2 for T (Hashimoto et al., 1986 b). We show below an experimental investigation of the scaling postulate for the near

©

z X +

TlMin.J 0.07 4.69 9.30 13.92 18.53 23.15 27.76 32.38

t

0.0 0.6 1.2 1.8 2.4 3.0 3.6 4.2

200

10

Figure 6-9. Time-evolution of the light scattering profiles in the early-to-late stage SD for SBR1/PB19 50wt.%/50wt.% mixtures at 60 °C (e T ^l). Time elapses in the order of parts (a) to (c). From Hashimoto etal. (1989b).

6.5 Growth of Self-Assembled Structures in Later Stage Spinodal Decomposition

273

Table 6-3. Characteristics of the polymers referred to in Chapter 6. Sample code

Mwaxl0"4

Microstructure in % c 1,2-

3,4-

Remarks

Cis-1,4

Trans-1,4

SBR1

11.8

1.18

61

16

23

poly(styrener-butadiene), wps = 0.20d

SBR2

19.0

1.2,

46

21

33

poly(styrener-butadiene), wps = 0.15

PB5.8

5.8

1.2

16

-

28

56

polybutadiene

PB5.9

5.9

1.23

9

-

46

45

polybutadiene

PB19

19.0

-

19

35

polybutadiene

54.6

116 1.02 1.07

46

PB55

-

polybutadiene

-

-

-

-

-

-

-

-

polybutadiene

-

PB122

122

PB273

273

-

polybutadiene

PS

21.3

1.25 1.4,

Pill

10.5

1.3

-

7.6

72.3

20.2

polyisoprene

PI55

54.6

1.02

-

8.9

74.9

16.2

polyisoprene

HK-5

23

1.25

38 + 1

59±2

3+1

poly(styreneb-isoprene), wps = 0.29d

BSI-9

22.4e

38±1

59 + 2

3±1

poly(styreneb-isoprene), d Wps = 0.67

HY-12

60.3

38 + 1

59±2

3±1

PP

23.5

4.1

-

-

-

poly(styreneb-isoprene), wps = 0.52d isotactic polypropylene

EPR2

14.9

2.5

PPO

4.3

3.64

-

poly(propylener-ethylene),

—

-

-

polystyrene

-

-

poly(phenylene oxide)

a

Weight average molecular weight; b heterogeneity index; c microstructures of butadiene or isoprene units in polymers; d weight fraction of styrene monomers in the copolymer; e number average molecular weight; f weight fraction of ethylene monomers in the copolymer.

SBR1 is poly(styrene-r-butadiene) and PB19 is polybutadiene (PB), the characteristics of which are shown in Table 6-3. Figure 6-10 shows double-logarithmically the time changes of qm (t; T) (a) and Jm (t; T) (b) for the SD at T = 40 to 110°C. Again

eT ~ 1 in this temperature range. Obviously the higher the temperature, the faster the rate of the self-assembly via SD, giving rise to the faster drop of qm (t; T) and the faster increase of Im(t;T). Figure 6-11 shows double-logarithmically the reduced quan-

6 Structure of Polymer Blends

274 -2.0

,w

X

-2.2 -

X

+

° K

X

f

X

X

X y

\ c X

X

~B ~2'4

X

A

(!)

O

X s

^ - 2 . 6 -_

X X

a

A

©o

V

\

a

J-2.8

—

A

X

3 -3.0 -3.2

_

+

40 °C 5 0 °C 6 0 °C 7 0 °C °C 90 110 X

a

a a a

\ X

1

(a)

6.0

1.5 2.5 LOG (t / min.)

3.5

Figure 6-10. Time changes of the scattering peaks qm (a) and Jm (b) at various T for the SBR1/PB19 58 wt.%/42 wt.% mixture. The arrows indicate the crossover from the intermediate to late stages. From Izumitani et al. (1990).

tities Qm and 7m as a function of the reduced time T. The data obtained for qm{t; T) and Im{t; T) at various t and T are found to fall onto the respective single master curves on the reduced plot. Thus the scaling postulate, which was originally proposed and tested for the binary liquid mixture near the critical point (Chou and Goldburg, 1979), is shown to be valid for the global pattern of the self-assembly, even for the deep quenches far from criticality: temperature affects the temporal and spatial scales for the global growth of the patterns, but the growth mechanism and patterns themselves are invariant to T (Izumitani et al., 1990). The self-assembly for the deep quenches is a subject which remains relatively unexplored because of its being out of the regime covered by critical phenomena and because of experimental difficulties. The scaling analyses require precise determination of, or information on, qm (0; T) and D app (T), both of which, especially the former, are not always easy to assess (see Sec. 6.4). Fujita et al. (1989) proposed a

SBR1/PB19

o

LOG X

Figure 6-11. Reduced Qm and 7m as a function of the reduced time T for the SBR1/PB19 58 wt.%/ 42 wt.% mixture. The arrows indicate the crossover between the intermediate and late stages. From Izumitani et al. (1990).

6.5 Growth of Self-Assembled Structures in Later Stage Spinodal Decomposition 80

new method of scaling which is applicable to systems for which one of the quantities, e.g., qm (0; T) is unknown. The scaling postulate has been similarly proved to be valid by a number of experimental works (Nojima et al., 1982; Snyder and Meakin, 1983; Hashimoto et al., 1986 b; Yang et al., 1986 a; Inaba et al., 1986; Kyu and Saldanha, 1988; Takenaka et al., 1989). However there is a report which clearly claims that the scaling postulate is invalid (Bates and Wiltzius, 1989). Thus it is still important to test the postulate for many polymer blend systems.

60-

4 - 0—*

20 -

+

a

The scaling analysis of the scattering functions / (q, t; T) was also conducted for a number of polymer blends (Hashimoto etal., 1986a, 1989b, 1991b; Takahashi etal., 1986; Tomlins and Higgins, 1989; Bates and Wiltzius, 1989; Nose, 1989; Takenaka et al., 1990; Takenaka and Hashimoto, 1992; Kyu and Saldanha, 1990; Kyu and Lim, 1990 b; Lee and Kyu, 1990) to test critically the dynamical scaling postulate, and the concept and the detailed theoretical functional forms of the scaled structure factor (Binder and Stauffer, 1974; Binder, 1979; Furukawa, 1978, 1979, 1984, 1986). Figure 6-12 shows the time change of the scaled structure factor in the late stage SD at 40 °C for the same polymer blend as in Figs. 6-10 and 6-11 (Takenaka etal., 1990). The time elapses in the order of parts (c) to (a), and the scaled structure factor F(x) is plotted as a function of the reduced scattering vector x on a linear scale. Part (c) covers the time scale t from 64.9 to 220.9 min or T from 1.9 to 6.6, corresponding to the intermediate stage and giving rise to ft > 3 a. Hence F(x) is non-universal with t, increasing and becoming sharp with

No. t(mln) „ 500.61 627.93 D 818.64 882.23 £ 1010.77 < 1238.88 k 1466.47 «. 1693.67

•

*

60-

N o./t

(b) *^>** ^y" * $1 • ) <^ 1

°*

_

6.5.2 Scaling Analyses of the Structure Factor F(x); Global Character

275

(min)

> 220.91 a 250.77 281.63 k 312.77 9 344.24 t < 375.73 > 438.85 h 500.61

3 20 o

/

in

6 0f-

No. /t 1mln ) „ o »

+

(c)

+

o

4-0-

x

_

• *

• +

64.86 73.39 81.92 109.43 137.15 192.45 220.91

1

0

1

2

x = q/qm(t)

Figure 6-12. Scaled structure factors F(x) obtained at 40°C for the SBR1/PB19 58 wt.%/42 wt.% mixture where the time elapses in the order of parts (c) to (a). In part (c), F(x) monotonically increases with an elapse of time but in parts (a) and (b) is essentially independent of time. The reduced time T is obtained by dividing the real time by tc = 2000 s. From Takenaka etal. (1990).

t. The characteristic time tc at 40 °C is 2000 s for this blend. Parts (b) and (a) cover the time scale t from 220.9 to 500 min or T from 6.6 to 15.0 and t from 500.6 to 1693.7 min or T from 15.0 to 50.8, respectively, and hence correspond, respectively, to the stage extending from the final part of the intermediate stage to the beginning of the late stage, and to the late stage. In this

276

6 Structure of Polymer Blends

as discussed in Sec. 6.2.1. Truly unique features of polymers beside these effects will be discussed briefly in Sec. 6.5.3. The detailed functional forms of fm(T) and Qm(x) reflect the ordering mechanisms, information on which is still limited, and should be fully explored in future for many polymeric blends. F(x) uniquely describes the morphology of the growing pattern (domain structure). F(x)'s are also only available for a limited number of systems, and hence they should be determined for various systems in future work. Figure 6-13 shows the time change of the structure factor in the intermediate stage. The intensity increase of F(x) due to the increase of (t](t)2s) is clearly seen in part (a), and the sharpening of F(x) with t is partic-

time scale F(x) becomes essentially universal with t and fits closely with one of the Furukawa's scaling functions (Furukawa, 1989), F (x) = x4/(3/2 + xx °)

(6-49)

A number of similar experimental results were also reported in the literature described in the beginning of this section. Thus the polymer blends obey the universal features found for low molecular weight mixtures (Komura et al., 1984, 1985): (i) Tm{r) and Qm(?) are universal with T, and (ii) in the late stage F(x) is universal with t. In this sense the self-assembly of polymer blends is unique only in that it has a long temporal scale (tc being long) and a large spatial scale [£ ~ l/qm(0) being large],

CD O *X 1

e

2. 8

£0°f 6 xXxxx~ R

7 X*

2. 4 2. 0

Jfc

v

xx

x grtt^

Nt. M a l i . ) 1 » 64.86 2a 73.39 34 81.92 4 • 109.43

5 , 137.15 6 * 250.77

t 1.9 2.2 2.5 3.3 4.1 7.5

xX x ^ ^

""

******* 3

7

/

/

x

x

/

O"

(F(

cr

o o

1. 6

_ •

•

•

•

—

1. 2 i

^

.

(a)

I

i

Nt. H.I..)

2.8

-

1• 2. 3.

J M k x

t

64.86 1.9 73.39 2.2 81.92 2.5

nir^*%( * • 109.43 3.3 1 ^ ^ l L 65xi 250 137.15 4.1 *, " 77 7 ' 5

£~ 2.4

X X

" ^ * Vx^

2.0

\

E c x x

1.6

X

X 6

V\ \\

V

,3

8 1'2 i

-0.5

\

.

-0.3

I

-0.1

.

, +0.1

L0G(q/qm(t,T) )

, (b) +0.3

Figure 6-13. Scaled structure factors F(x) obtained at the intermediate stage SD at 40 °C for the same mixture as in Fig. 6-12 and plotted in double logarithmic scales (a). The profiles in part (b) are shifted vertically and matched at x = l and at T = 7.5. OT is the vertical shift factor. From Takenakaetal. (1990).

6.6 Fine Structure in the Scaled Structure Factor F(x)

ularly obvious in part (b), where F(x)'s numbered 1 to 5 are superposed on number 6 by vertical shifting. 6.5.3 Unique Features of the Scaled Quantities of Polymers

In small molecular systems, qm (t; T) data obtained for various mixtures, with critical compositions, for example, become universal, independent of the particular material when they are plotted using the reduced variables Qm and 7 m . In the case of polymer blends with near critical compositions, Qm(r) showed non-universality when (i) the DP's of constituent polymers were changed (known as "N-branching") (Hashimoto et al., 1986 b; Hashimoto, 1988 a,b), (ii) a small amount of solvent was added to the mixture (known as "C-branching") (Hashimoto, 1988 a), and (iii) a small amount of the block copolymer, corresponding to the constituent homopolymers, is added (denoted as "B-branching") (e.g., A-B type di-block copolymer + A homopolymer -f B homopolymer) (Izumitani and Hashimoto, 1992). In each case, <2m(i) is universal with T but shows branching in the later stage when the DP is changed, or the concentration (C) of the polymers is changed, or the concentration of the block copolymer (B) is changed. There is even a case where Qm(x) shows branching with respect to T (Bates and Wiltzius, 1989). Figure 6-14 a shows an example of Nbranching where Qm at a given T (> 2) tends to increase with increasing molecular weight of PB in the 50/50 wt./wt. mixtures of SBR1/PB, while Fig. 6-14b shows an example of C-branching where Qm(z) at a given T (> 2) tends to increase with increasing total polymer concentration from 50 to 100 wt.%. The early stage SD for the systems whose later stage behaviour is shown in Fig. 6-14 a were discussed in detail by

277

Takenaka et al. (1987). In the experiment shown in Fig. 6-14 b, the polymers used are SBR2 and PB273, and the polymer composition is fixed at 50/50 wt./wt. to which 0 to 50 wt.% dioctylphthalate (DOP) was added as a common solvent. The exponent oc ~1/2 for SBR1/PB273 in Fig. 6-14a and for SBR2/PB273 (in melts) in Fig. 6-14 b may be explained by the theories proposed by Kawasaki and Sekimoto (1987) and Furukawa (1988). Like the Nand C-branchings, the B-branch shows that <2m(T) at T > 2 tends to increase with increasing amounts of the block copolymers (Izumitani and Hashimoto, 1992). It was suggested that the N- and C-branches arose from the entanglement effects of polymer molecules in a condensed phase (Onuki, 1986; Hashimoto et al., 1986b; Hashimoto, 1988 a, 1988 b), while the Bbranch is expected to be associated with the reduction of the interfacial tension due to the localization of the block copolymer in the interface between growing domains. The B-branch may be seen even in small molecule systems by adding a third molecule which can act as a surfactant for the constituent components. Another unique feature of polymers is observed in the fine structure in the scaled structure factor F(x) (Hashimoto et al., 1989 b, 1991b; Bates and Wiltzius, 1989; Nose, 1989; Takenaka et al., 1990) or in the spontaneous pinning (Hashimoto, 1988 b; Takenaka et al., 1989) of the domain growth for off-critical mixtures of high polymers, which will be discussed in Sees. 6.6 and 6.7, respectively.

6.6 Fine Structure in the Scaled Structure Factor F(x) So far the scaled structure factor F(x) has been investigated over a narrow range

278

6 Structure of Polymer Blends

-0.4 -0.8

Temp.:60.0(°C)

-1/2

O:5BR1/PB19 D:SBR1/PB55 A.SBR1/PB122 OISBR1/PB273

(a) -1.2

I

SBR2/PB273=50/50(wt/wt) 70.0°C

O -0.4

81

-0.8

-1/2

-1.2

Figure 6-14. Plots of the reduced wavenumber Qm vs. the reduced time T showing (a) N-branching for mixtures of SBR1 with PB having different N at 60 °C, and (b) Cbranching for SBR2/PB273 in DOP with a composition of 50/50 wt./wt. at 70 °C. The total polymer concentration was changed from 50 to 100wt.% in part (b). From Hashimoto (1988 b).

logr

of x of x ~ 1 both in polymer and smallmolecular systems (Komura et al., 1984, 1985; Hashimoto et al., 1986a; Takahashi etal, 1986; Tomlins and Higgins, 1989; Kyu and Saldanha, 1990; Kyu and Lim, 1990 b; Lee and Kyu, 1990). Taking advantage of studying the pattern growths in polymer systems, with their spatial and temporal scales being much larger and longer than those of small-molecular sys-

tems, it should be possible to study F(x) extremely precisely up to higher x values. Such studies enable us to explore a new aspect of the pattern growth, viz., the time change in the local features of the patterns such as the interface thickness and waviness. This is a subject which has been left essentially unexplored up to now and which has quite recently attracted some experimental and theoretical investigation

6.6 Fine Structure in the Scaled Structure Factor F(x)

(Hashimoto et al., 1989 b, 1991b; Takenaka et al., 1990; Takenaka and Hashimoto, 1992; Bates and Wiltzius, 1989; Nose, 1989; Oono and Puri, 1988 a, 1988 b; Puri and Oono, 1988; Shinozaki and Oono, 1991; Ohta and Nozaki, 1989; Chakrabarti et al., 1989 a, 1989 b; Hayakawa and Koga, 1990; Koga and Kawasaki, 1991; Koga et al., 1992; Kawasaki et al., 1992). Figure 6-15 shows some typical examples of F(x) showing a fine structure at large x (Takenaka et al., 1990). The data were obtained for SBR1/PB19 with 58/ 42wt./wt. or 57.2/42.3 vol./vol. at 40 to 60 °C at which eT ~ 1, i.e., a strong segregation, and in the late stage SD where the value of T was from ca. 10 to 50 at each T. In the small x range of 0.5<x<1.5, F(x)is essentially universal with t and T as shown in Fig. 6-12. The structure factor closely fits the Furukawa's function of Eq. (6-49).

279

However with a large x > 1.5, a number of interesting features are observed: (i) F(x) is not universal with t but increases with t, indicating that the time-evolution of the local structure cannot be scaled with a single length parameter l/qm(t) characterizing the time-evolution of the global structure. (ii) The second-order maximum or shoulder appeared in F(x) at x ~ 2. (iii) At a large x > 2, F(x) has an asymptotic behaviour of x~n where n~A (the Porod scattering) (Porod, 1951, 1952 a, 1952 b). In other words, F(x) exhibits crossover behaviour at xc on increasing x from unity, F(x) 6 for x < xc n ~ 4 for x > xc

(6-50)

0.6

40 °C 2.0

F(x)~-

- F(x) - -

#$ x=9.5 *

50°C 2.0

(3/2)+x

x*>oxx

J*

: ^

^

V ^ 40.6

60 °C

o 1.5 o

0.5

.5

-0.3

-0.1

+0.1 +0.3 LOG(q/qm(t,T) )

+0.5

+0.7

Figure 6-15. Scaled structure factors for SBR1/PB19 with composition of 57.2/ 42.3 vol./vol. (58/42 wt./wt.) in the late stage SD obtained at 40 to 60 °C. They show the fine structure at x = q/qm(t) = 2 and 3 and the dynamical and spatial crossovers from q~n(6
280

6 Structure of Polymer Blends

A similar fine structure in F(x) is also shown in Fig. 6-16 for the late stage SD in DPB/HPI 51/49 vol./vol. mixtures over a wide temperature range where £T changes from 6.68 x 10 ~2 (a weak segregation at a small quench depth) to 7.61X10" 1 (a strong segregation at a large quench depth). The same characteristics as described above for SBR1/PB19 can also be seen for this system, except for a slight modification of the characteristics in (ii), discussed above, i.e., the second-order maximum or shoulder of F(x) is seen at x ~ 3 rather than at x ~ 2. It should be noted that the volume composition of this mixture of DPB/HPI is much closer to 50/50 than the other. If the growing pattern is locally lamella-like, the difference in

the characteristics of (ii) for the two systems is well understood. The lamellar structure gives rise to scattering maxima at (6-51) x — m (m; integers) with the m-th order peak being suppressed when the following condition is satisfied (Hashimoto et al, 1980, 1983), (integer) (6-52) m= Here 4> is the volume fraction of one type of lamella. Thus the second-order peak is suppressed when 0 = 1/2 as in DPB/HPI but not when <j> = 0.57 or 0.43 as in SBR1/ PB19. Thus the results shown in Figs. 6-15 and 6-16 imply a lamellar type pattern with a length scale r < Rm(t) where Rm(t) is the mean radius (or persistence length) of the

2.0

0.6

I

1—1—1 1 1 1

I

"I

0.6 1 I 1111

1

1

r

i

2.0 ~ 1.0

\

2.5

40°C 6.68 x 10 2

60 °C

2.0

a*3

\

'

B

\%

1.5 *

•

1.0

\

80

50°C V

2.0

O •

o

V

1.0

1.0

\

55°C

eT

\ 0.0

°o

\

-

\

4*!

\

0.0

= 7 . 6 1 x 1 0 1 '»

90° c

—

\

\-4 ^

-1.0 ' -0.3

'

' -0.1

'

'1

,

+0.1

'

1

'

+0.3

LOG x

\"4

\

,

'

'i

+0.5

^—» +0.7

°°°r

-1 o *-o.3

I -0.1

. +0.1

I

, +0.3

1

+0 5

+0 7

LOG x

Figure 6-16. Scaled structure factors of DPB/HPI (DPB-20-103/HPI-15-136) with the composition of 51/49 vol./ vol. in the late stage SD at T = 40 to 90°C or £T = 6.68 x 10~2 to 7.61 x 10"1.

6.6 Fine Structure in the Scaled Structure Factor F(x)

wavy interface in the late stage SD. The confirmation of this implication by realspace analysis deserves future work. The crossover beaviour (iii) was proposed to occur at qci = 1/Rm(t) or xc = Qci(t)/qm(t) = A{t)/Rm(t) and was fully discussed elsewhere (Takenaka et al., 1990). At a large q limit, where the Porod law is valid, I(q) is given by (Takenaka et al., 1990) a —• oo

(6-53) Thus the scaled structure factor F(x) at a large x limit is given by lim

exp[-o(t)2qm(t)2x2]

(6-54)

Here I(t) is the interfacial area per unit volume and o(t) is the parameter related to the interfacial thickness. If the interfacial thickness is much smaller than A(t\ [o{t)qm{i)]2 < 1. In this case the non-universality of F(x) at large x in the late stage is primarily determined by the time-dependence of l(t)q^l{t). The increase of F(x) with t suggests that Z(t) decreases more slowly than qm{t\ i.e., if Z(0~rys

(6-55)

then ys a, i.e., the local wavenumber I(t) relaxes to equilibrium much faster than the global wavenumber qm(t), and (ii) a(t) still decreases with t (Hashimoto et al., 1991 b). The former factor decreases F(x) at the large x with t, as I(t)/qm(t) decreases with t, but the latter factor increases F(x) as a(t) qm(t) decreases. The increase of F(x) at the large x with t, which was found in the experiments shown in Fig. 6-15 to 6-17, is a consequence of a (t) outweighing I (t). The previous conclusion

281

of ys < a by Takenaka et al. (1990) was an error caused by neglecting the time change of the interfacial thickness a(t) (Hashimoto etal, 1991b). Quite recently it was also found (Hashimoto etal., 1991b) that, at much a later time in the late stage, F(x) becomes truly universal with time, as shown in Fig. 6-17 for HPB/HPI with 51/49 vol./vol. where parts (a) to (c) represent, respectively, the intermediate stage, the late stage I and the late stage II. In the late stage I, F(x) at large x is still not universal, but in the late stage II it becomes universal so that the pattern growth is scaled with a single length parameter i/qm(t) or 1/S(t), i.e., oc = ys. It was also found that in the late stage II, a(t) reaches temperature-dependent constant values. For this system, a = 1 in the late stage II. The structure factors shown in Figs. 6-15 and 6-16 are therefore considered to be the ones in the late stage I. Thus a series of work suggest that SD is classified into at least four stages: the early, intermediate, late I and late II stage. If we include the very early stage occurring prior to the early stage, as pointed out in Sec. 6.3.1, SD is classified into five stages. Bates and Wiltzius (1989) first proposed a four-stage model and classified the self-assembly via SD into the early, intermediate, transition and late stages. Their four-stage model was compared in detail with the four-stage model described here by Hashimoto etal. (1991b). The structure factor F(x) obtained in the late stage II (solid line) was compared in Fig. 6-18 with the theoretical profile (dashdot line) given by Ohta and Nozaki (1989) (Hashimoto etal., 1991b). Although it is unfair to compare the theoretical and experimental F(x) obtained in different regimes (the former and latter being obtained in the nonhydrodynamic regime of a = 1/3 and the hydrodynamic regime of

6 Structure of Polymer Blends

282

0.6

2.0

1

^

1—r

2

3

T"

I

t (min.)

• 42.6 n 53.7 * 61.4 0 92.0 x 107.5 x 210.4

1.0

x 5.3 5.2 6.0 9.0 10.5 20.6

0.0 2.0

1.0

• Q * o

t (min.)

T

210.4 249.2 392.6 853.0

20.6 24.3 38.4 83.4

x 0.0 O

2.0

1.0 • 0 * 0 x x + v • x

0.0

-1.0

t (min.) t 948.9 92.7 1004.2 98.1 1041.9 101.8 1098.2 107.3 1136.0 111.0 1191.7 116.4 1228.9 120.1 1420.4 138.8 1611.3 157.5 1280.9 176.1

Figure 6-17. Scaled structure factor F(x) for HPB/HPI (PB 5.8/PI 11) with a composition of 51/49 vol./vol. in (a) the intermediate stage (T<10.5), (b) the late stage I ( 1 0 < T < 1 0 0 ) ,

and (c) the late stage II (T>100). From Hashimoto

-2.0 -0.3

+0.

-0. 1

1

0-

+0.5

+0.3

+0.7

etal. (1991b).

Logx

I

\

/

Experimental curve -—— Theoretical curve (Ohta-Nozaki)

\

- 1 /-

\ \

O

V

-2-

\

(a) I

-0.2

I

1

0

1

1

0.2 Log x

1

1

0.4

0.6

Figure 6-18. Comparisons of experimental (solid line) and theoretical scaled structure factors by Ohta et al. (dash-dot line). The experimental structure factor was obtained for the HPB/HPI mixture in the late stage II (Fig. 6-17(c)). From Hashimoto etal. (1991b).

6.7 Spontaneous Pinning for Off-Critical Mixtures

a = 1, respectively), the two profiles are surprisingly in good agreement. It is quite impressive that both results tend to show a higher order maxima at x ~ 3, implying the locally lamellar pattern. The experimental profile also fits quite well with that obtained by a computer simulation by Chakrabarti et al. (1989) and by Shinozaki and Oono (1991). Bates and Wiltzius (1989) also reported a good fit of the experimental F(x) with that of Ohta and Nozaki, though they claimed a higher-order maximum at x ~ 2.

6.7 Spontaneous Pinning for Off-Critical Mixtures A unique feature in polymers was also observed in the ordering process of off-critical mixtures consisting of high molecular weight polymers. Figure 6-19 shows the growth of the patterns as observed by the time changes of (a) qm and (b) Jm for the SBR1/PI55 mixtures with the off-critical compositions of 20/80 and 30/70 wt./wt., as well as the near critical composition of 50/50 wt./wt. (Takenaka etal., 1989). The

SBR1/PI55 at 60° C

20

'4-

-2.2

tc

-2.4o o

283

C

for 30/70

30/70

mmmm

jj&°©CX!X!fc tC

-2.6 -

20/80

X

X

for 50/50 \

-2.8 50/50 -3.0 -

z I

^ 0

(a)

1

6.0 t i

5.0l

o o

tc

4-.0-

Z

l

E*

*••

3.0-

30/70 ^

^M»

20/80

2.0-

° 1.0

tp 1

i

2

3 LOG t

(b)

Figure 6-19. Time change of (a) qm and (b) Im for the near critical (50/50 wt./wt.) and off-critical mixtures (20/80 wt./wt. and 30/70 wt./wt.) of SBR1/PI55 at 60°C tp and tc refer, respectively, to the time at which the pinning, and the crossover between the intermediate and late stages, occur. From Takenaka et al. (1989).

284

6 Structure of Polymer Blends

concentration fluctuations for the near critical mixture keep growing with time, giving rise to a - 0.77, 0 ~ 2.4 and jS/a = 3.1 in the late stage SD at t>t = 110 min (see the vertical arrow marked tc for the 50/50 mixture, tc being the crossover time between the intermediate stage and the late stage). In contrast to the near critical mixture, the off-critical mixtures show the pinning of the growth such that at t > tp both qm and Im hardly change with t. For example the time change of qm for the off-critical mixtures is given by qm(t) = qmp(t/tp)~cc

(6-56)

where a ^ 0 for t/tp < 1 and a = 0 for t/tp > 1 and the value of qm after the pinning (qmp) is given by <2mp =

> tp)

(6-57)

The peak scattered intensity Im is found to be essentially pinned down at the level of

Impatt>tp. It is observed that the more the composition is biased from the critical composition, the earlier the stage at which the pinning occurs, yielding the higher value of qmp, the lower intensity level of Imp, and the shorter time tp. It was also found that the higher the temperature, the earlier the stage of the pinning (Hashimoto, 1988 b). The pinning was found to occur in the intermediate stage of ttc (e.g., the 30/70 mixture at 60°C). The pinning phenomenon, which is unique to the ordering process for an offcritical mixture with high molecular weights is proposed (Takenaka et al., 1989) to originate from the "dynamical percolation-to-cluster transition" occurring during the coarsening process of SD (Hasegawa et al., 1988). The phase separation via SD develops a bicontinuous periodic domain structure. The domain structure can ini-

tially maintain the macroscopic percolation, i.e., both types of the domains being continuous through the whole sample space. However, as the domains grow, one type of domain, composed of the component with a minor volume fraction, cannot maintain volumetrically the macroscopic percolation, resulting in a break-up into fragments with local percolations only. The fragments eventually degenerate into spherical droplets due to interfacial tension, generating a cluster of spheres. Although we do not have a direct realspace pattern to support this idea of the percolation-to-cluster transition for our particular mixtures SBR1/PI55, we have it for mixtures of X-7G/PET where X-7G and PET stand for a thermotropic liquid crystalline polymer and poly(ethylene terephthalate), respectively. Figure 6-20 shows the time-evolution of an unmixing structure through SD for the 50/50 wt./wt. blend at 270 °C (Hasegawa et al., 1988). At 270 °C, X-7G and PET by themselves are, respectively, anisotropic and isotropic liquids. The mixture is thermodynamically unstable and undergoes phase separation between the anisotropic liquid domains of X-7G and isotropic liquid domains of PET through the SD mechanism. The anisotropic and isotropic domains appear to be bright and dark under the crossed polarizers, respectively. The percolated domain structure self-similarly grows with time in the time domain I where the structure evolution relevant to the late stage SD is observed. As the domain grows as a consequence of the network instability, the anisotropic liquid domains can no longer maintain the macroscopic percolation, resulting in the formation of fragments of the network with local percolation only, as seen in the beginning of the unmixing structure in the time domain II. The fragmented networks then degenerate

6.7 Spontaneous Pinning for Off-Critical Mixtures

285

Figure 6-20. Time-evolution of the unmixing structure showing the dynamical percolation-to-cluster transition. In regime I the percolated structure grows with dynamical self-similarity. The transition takes place in regime II and clusters are formed in regime III. From Hasegawa et al. (1988).

into spherical droplets to minimize interfacial free energy (in the time domain II) and result in clusters of droplets (in the time domain III). The growth of the droplets occurs extremely slowly for this particular mixture of X-7G/PET. The external potential exerted by the unwetting phenomenon of X-7G gives a driving force of the diffusion-coalescence of the spheres (Nakai etal, 1992). Once the cluster structure is formed in our off-critical mixtures of SBR1/PI55, ac-

cording to the mechanism similar to that shown in Fig. 6-20, the growth of the droplets is again expected to be very slow or essentially pinned. The growth occurs as a consequence of the diffusion-coalescence of the droplets. The diffusion of the droplets, in turn, involves, the dissolution and condensation (Lifschitz and Slyzov, 1961) of the molecules forming the droplets, i.e., as shown schematically in Fig. 6-21, the molecules A in domain A dissolve into Bmatrix being rich in B molecules, translate

286

6 Structure of Polymer Blends State I

State II

B-matrix

© o

I

II

State

Figure 6-21. Schematic illustration of the kinetic energy barrier for the diffusion-coalescence process. Since state I has a higher free energy than state II, a transformation from state I to II is thermodynamically favoured. However it must overcome the barrier associated with the heat of mixing of unlike chains in the diffusion-coalescence process. From Hashimoto et al. (1992 a).

over a distance, and condense in a different position in domain A. Such fundamental molecular motions cause Brownian diffusion of the droplets, and when they coalesce, they grow into larger droplets. However this transport process involving dissolution, diffusion and condensation has to overcome a kinetic barrier associated with the enthalpy of mixing of unlike constituent polymers AiJmix. AHmix per chain is proportional to % N kB T where N is the DP, NA = NB = N being assumed here for the sake of simplified arguments. Thus when N is very large, as in the case of SBR1/PI55, xN is very large, and hence the kinetic barrier is too large to be overcome (i.e., very low solubility of A molecules in the B matrix). Thus the molecular diffusion process is kinetically frozenin, causing pinning of the growth of the droplets. Thus pinning is a unique feature associated with the ordering process of polymer mixtures consisting of high molecular weights. A similar pinning effect was

observed (Shibayama et al., 1983) for the growth of the spherical microdomains of block copolymers induced by increasing the segregation power. In the mixtures with low molecular weights for which Aifmix is relatively small (AHmix < kB T), growth of the droplets due to the diffusioncoalescence can always take place, and hence pinning will not be observed. Even for mixtures with very large molecular weights, coarsening and hence the decrease of qm with t can occur as long as the mixtures maintain the percolated structure, i.e., a continuous network structure. In this case the coarsening takes place by partial disruption of the networks and degeneration of the disrupted part of the networks into the existing networks (Nakai et al., 1986, 1992). Thus the pinning results from two effects, i.e., (i) the dynamical percolation-to-cluster transition and (ii) the high molecular weight effect which causes the kinetically freezing-in of the molecular diffusion. A detailed discussion of this effect will be given elsewhere (Hashimoto et al., 1992 a).

6.8 Interplay with Two Kinds of Phase Transition 6.8.1 Interplay with Crystallization

When the spinodal decomposition is coupled with other kinds of phase transition, such as crystallization (liquid-solid phase transition) and microphase transition of block copolymers, the resulting morphology is expected to be very rich. An interesting morphology generated by coupling SD with crystallization was reported for binary mixtures of polypropylene (PP) and poly(ethylene-r-propylene) (ethylene-propylene random copolymer, EPR) (Inaba et al., 1986, 1988). It was

287

6.8 Interplay with Two Kinds of Phase Transition

found that the pattern developed by SD in the molten liquid state at 7\ > Tm was conserved during a rapid crystallization process of PP at T2 < Tm, and that the crystallization effectively pinned down further growth of the pattern due to SD at T2, Tm being the melting temperature of PP. The coupling generated patterns composed of dual morphological entities of spherulites and periodic bicontinuous domain structures, as shown in Figs. 6-22 and 6-23. The two pictures in Fig. 6-22, which were obtained on the same field for the same sample, show the morphology of the volume-filling spherulites (a) which contain, as their internal structure, the periodic and bicontinuous structure of PP-rich and EPR-rich domains (b). The bicontinuous percolated structure was first developed by SD at Tx for a time period t1, and the spherulitic structure was developed by a subsequent lowering of the temperature from Tt to T2. At T2 rapid spherulitic crystallization occurred in the interconnected PP-rich domains, without significantly disrupting the pattern originally developed by SD. The polarized light micrograph in Fig. 6-23 shows that the pattern typically developed by the SD at Tx, i.e., a bicontinuous percolated structure of PP-rich and EPRrich domains with a characteristic periodicity of A ~ 60 jam. In this case the time spent for SD is very long, and the domains are large enough so that the PP-rich domains contain the volume filling spherulites and hence have high optical anisotropy. The EPR-rich domains have less crystallinity and hence smaller optical anisotropy. 6.8.2 Interplay with the Microphase Transition

In the case where blends contain a block copolymer at least as a component, they undergo in principle both "macrophase

Figure 6-22. (a) Polarized light micrograph (POM) and (b) phase-contrast light micrograph (PCM) for PP/EPR mixture unmixed at 200 °C for 5 min and subsequently crystallized at 125 °C for 5 min. The POM which was obtained by setting polarizer and analyzer in vertical and horizontal directions clearly shows the volume-filling spherulites, while the PCM clearly shows the fine structure developed by SD. The two pictures were obtained on the same field for the same sample. From Inaba et al. (1988).

POM

100 Figure 6-23. Polarized light micrograph showing the bicontinuous percolated structure developed by SD for a PP/EPR mixture unmixed at 200 °C for 20 min and subsequently crystallized by quenching the mixture in an ice-water bath (polarizer and analyzer being set in vertical and horizontal directions). Note that the percolated PP-rich domains contain the volume-filling spherulites and have high optical anisotropy. From Inaba et al. (1988).

288

6 Structure of Polymer Blends

and microphase transitions" (Tanaka and Hashimoto, 1988; Hashimoto et al., 1988 a; Ijichi and Hashimoto, 1988), interplays of which also generate a rich variety in their patterns. For example, the blends such as A-B/A, A-B/B, A-B/C, (A-B)J(A-B)2, A-B/B-C etc., as described in Sec. 6.1, belong to this class of blends. Figure 6-24 shows schematically the "microphase transition" of a pure A-B type block copolymer (a) and the "macrophase transition" of A/B blends between the two homopolymers (b), as well as equilibrium structures both in the single-phase state (upper half) and the two-phase state (lower half) (Hashimoto et al., 1988 a). In the two-

Polymer Blend

Block Polymer

...r t •U ,

Disordered '

State

no connectivity between A 8

molecular connectivity between A 8 B

B

( X A B N )C

State

Q

Or*^

£•••'

Dx/Rg Dn/Rg

Dx/Rg -*oo

D I I / Rg - » oo

(a) (b) Figure 6-24. Comparison of concentration fluctuations of (a) A-B block copolymers and (b) A/B polymer blends in the single-phase (disordered) state and two-phase (ordered) state at thermal equilibrium, and the phase transitions between the two states, i.e., order-disorder transition ("microphase transition") for the block copolymers vs. liquid-liquid phase transition ("macrophase transition") for the polymer blends. From Hashimoto et al. (1988 a).

phase state (or ordered state in the sense that it has a lower entropy) the A/B blends separate into two macroscopic phases where the domain sizes parallel and normal to the interface, D\\ and D ± respectively, are much larger than the radius of gyration Rg of individual polymer coils. On the other hand an A-B block copolymer cannot separate into the macroscopic phases owing to a covalent bond between A and B polymers; instead it separates into submicroscopic domains with D± nearly equal to Rg (Inoue et al., 1969; Meier, 1969, 1987; Helfand and Wasserman, 1976,1978, 1980; Hashimoto et al., 1980, 1983; Noolandi and Hong, 1980; Hong and Noolandi, 1981; Leibler, 1980; Hadziioannou and Skoulios, 1982; Ohta and Kawasaki, 1986; Kawasaki et al., 1988,1990). At the segregation limit the domains have to be arranged periodically in space with a long range order characteristic of "crystals" with a super-large lattice spacing D±. The variety in the domain morphology depends on the composition / of one of the components A or B (see for example, Hashimoto et al., 1983; Hashimoto, 1987 a; Thomas et al., 1986; Hasegawa et al., 1987; Hashimoto et al., 1992 b). The sketch in the bottom left hand corner of Fig. 6-24 shows only one type of domain morphology (i.e., lamella). In contrast to the ordinary liquid-liquid phase transition of polymer blends, we denote this order-disorder transition of block copolymers (Hashimoto, 1987 a) as a "microphase transition", and in contrast to this microphase transition, we denote the ordinary phase transition as a "macrophase transition". The thermodynamic stability limits for macrophase and microphase transitions, Zs, macro and zs, micro (i-e., x-parameters at the spinodal points) were discussed for blends containing block copolymer(s), based upon the mean-field random phase approxima-

6.8 Interplay with Two Kinds of Phase Transition

tion (RPA). Figure 6-25 shows an example of the phase diagram for the stability limits of A-B/A, in which A-B has N = N1 and / = NJ(NA + NB) = NJN = 0.5, and A has N = N2 = NJ2 (Hashimoto etal., 1988 a). The stability limits faiVi)s,maCro (solid line) and fa Nx)St micro (broken line) are evaluated as a function of the block copolymer volume fraction biock. (f>c is the critical value of block below which only the stability limit for the macrophase transition exists. It is needless to say that this phase diagram cannot predict the equilibrium structures; it can predict only the stability limits. The equilibrium structures may be predicted by the approach developed by Hong and Noolandi (1983) and Whitmore and Noolandi (1985). First consider the case of block > c where stability limits exist for both the microphase and macrophase transitions. In the region where XNX faNX macro , both microphase and macrophase transitions can take place. The mechanism and ordering processes of the mixtures quenched inside this stability limit (e.g., quenched to point R) deserve future studies. The ordering processes of the mixtures to achieve equilibrium at point R should also depend on the point

289

block

Figure 6-25. Phase diagrams showing the thermodynamic stability limits for microphase and macrophase transitions, (xNJs, micro and (x^i) s , macr o, respectively, for the binary mixture of A-B/A. A-B has N = N1 and / = l/2, and A has N = N2 = NJ2. From Hashimoto etal. (1988a).

from which they are quenched (e.g., point P or Q). Study of such a memory effect might produce quite fascinating problems for future work. In the regime where block < c the mixtures are in the single-phase state and mixed at molecular level if xNx faiV1)s,macro]. However, as the amplitude of the concentration fluctuations grows with time after quenching the mixture from points S to T, the regions rich in block polymer begin to have locally ^biock grater than c, and hence begin to undergo a microphase transition. This is

290

6 Structure of Polymer Blends

the phenomenon of the microphase transition induced by the macrophase transition (Tanaka and Hashimoto, 1988), and this process will develop locally ordered microdomains with the periodicity of Amacro. The ordered microdomains have the internal periodicity of ^ m i c r o , as discussed earlier in Fig. 6-24. However, the regions rich in the homopolymer stay in a disordered state in which a minor fraction of the block polymer is mixed at molecular level with the homopolymer. This ordering process generates a truly modulated structure or a superlattice. It should be noted that the phase diagram is extremely sensitive to the variables /, N1 and N2 (Hashimoto et al., 1989 c). Figure 6-26 shows the microdomains of pure SI block copolymers HK-5 and BSI-9 (Table 6-3) and of their 50/50 blend observed under transmission electron microscopy (TEM) on ultrathin sections stained by O S O 4 (Hashimoto et al., 1988 a). The HK-5 clearly has hexagonally packed PS

cylinders in the matrix of PI, while the BSI-9 has hexagonally packed PI cylinders in the matrix of PS. The 50/50 blend shows the single microdomain morphology of alternating lamellae of PS and PI with a long range order, which is believed to occur as a consequence of uniform mixing of the two block copolymers in the microdomain space. This pattern formation presumably occurs in the regime in which only the microphase transition can occur. Figure 6-27 sketches a possible pattern formed in the ordering process of the two SI block copolymers ab and /?b in the regime where both microphase and macrophase transitions can occur (Hashimoto et al., 1988 a). If the two block polymers ab and /?b first undergo the macrophase transition according to SD, the large scale concentration fluctuations of ab and jSb with a characteristic wavelength vi macro , of the order of 0.1 to 1 Jim, will be developed in the system (Fig. 6-27 a). If this macrophase transition is followed by a microphase

Figure 6-26. Microdomain morphology of solution cast films of the pure SI block polymers of HK-5 and BSI-9 and their 50/50 mixture. Toluene is used as a solvent. From Hashimoto et al. (1988 a).

6.8 Interplay with Two Kinds of Phase Transition

S

I

Macrophase Transition

(a) Amacro = 0 ( 0 > l M m )

Microphase Transition

I Cylinder

I/S

S

I/S

I

Lamella Cylinder Lamella Cylinder H h Amicro = O(IOOA)

transition, and if the two polymers ab and /?b are mixed at molecular level to form a microdomain, as in the case shown in Fig. 6-26, the microphase transition may develop a spatially varying morphology of microdomains, as shown in Fig. 6-27 b, with a large scale periodicity of AmacTO within which a small scale periodicity of the microdomain structure of AmiCTO of the order of 100 A is superposed. Here we assume that ab is the PS-rich block copolymer, hence possibly giving rise to the microdomains of Pi-cylinders in the matrix of the PS in the ab-rich domains. Similarly, ph is the Pi-rich block copolymer so that the microdomain of PS-cylinders in the matrix of the PI may be formed in the /?b-rich domains. In the domains where ab and /Jb are present at nearly the same concentration, the alternating lamellar microdomains may be formed. An example of such modulated patterns is shown in Fig. 6-28 for the 80/20 mixture of an SI block copolymer (HY-12 in Table 6-3) and poly(phenyleneoxide) (PPO) prepared by slow evaporation of its dilute toluene solution (Hashimoto et al., 1991 a). Pure SI has a lamellar microdomain. During the solvent evaporation process, SD

(b)

291

Figure 6-27. Possible modulated structure (superlattice structure) for a mixture of two different SI block polymers ab and /?b with a large-scale characteristic spacing Amacio of the order of 0.1 to 1 (im resulting from spinodal decomposition (a), and a small-scale characteristic spacing ^micro °f t n e order of 100 A resulting from the microphase separation (b) which could follow the spinodal decomposition. Note that the microdomain morphology itself could spatially vary with the spacing AmacTO. From Hashimoto et al. (1988 a).

occurs, which generates the spatial concentration fluctuations of SI and PPO, as shown in Fig. 6-27 a, where ab and /Jb can be regarded as SI and PPO, respectively, in this case. This macrophase transition is followed by the microphase transition, which then forms the microdomains. Since PPO is immiscible with PI but miscible with PS, PPO is selectively solubilized into PS microdomains, increasing the volume fraction of the phase comprising PS and PPO relative to that of the PI phase. Thus the Si-rich domains form the lamellae (region A), PPO-rich domains form the PI spheres in the matrix of PPO and PS (region D), and the domains intermediate between the two form PI cylinders in the matrix of PS and PPO (regions B and C), as shown in Fig. 6-28. The modulated pattern shown in Fig. 6-28 has Amacro ~ 5 jum and ^micro — 100 nm. The microdomain morphology in region B, the immediate neighbour of region A, may be regarded as a "mesh structure" (Hashimoto et al., 1992 b) in which there are catenoid channels of PS and PPO traversing through the PI lamellae, and interconnecting the lamellae comprised of PS and PPO.

292

6 Structure of Polymer Blends

HY12/PPO (80/20) Blend

Figure 6-28. Example of modulated structure obtained by coupling between the macrophase and microphase transitions in a solvent cast film of the 80/20 wt./wt. mixture of an SI block copolymer (HY-12) and PPO. Toluene is used as a solvent. From Hashimoto et al. (1991 a).

6.9 Morphology Control Information obtained on the various ordering processes, as described in the preceding sections, serves as a key to controlling the morphology of polymer blends. Here we summarize the processes leading to morphology control, the basis of which was discussed in the preceding sections in this chapter and which involves freezing-in of the growing patterns by means such as (i) spontaneous pinning, (ii) physical pinning, and (iii) chemical pinning. Spontaneous pinning was discussed in Sec. 6.7 and is applicable to off-critical mixtures. The morphology is controlled by the composition and the unmixing tempera-

ture for a given pair. Physical pinning involves (a) crystallization when at least one component is crystallizable and (b) vitrification. This type of pinning, as well as chemical pinning, involves pinning the domain growth by freezing-in the translational diffusion of the center-of-mass of the molecules. Physical pinning due to crystallization was discussed in Sec. 6.8.1 and the morphology control is achieved by the time and temperature for SD and crystallization. An example of a pattern controlled by vitrification is shown in Fig. 6-29 (Hashimoto et al., 1984) for the solvent cast-film of a PS/PB5.9 mixture with 20/80 wt./wt. composition, in which parts (a) and (b)

6.9 Morphology Control

show the light scattering pattern and the light micrograph of the solvent-cast film. During the solvent evaporation process, the concentration of the polymers increases, which induces the phase separation due to the SD mechanism (Sasaki and Hashimoto, 1984). The pattern developed in the solution will be frozen-in at some stage of SD at which the domains rich in PS were vitrified, causing freezing-in of further growth of the pattern. This frozen-in pattern in the solution is essentially con-

Solvent Cast Film

Light Scattering Pattern

(a)

••>mxmr: » #**

:••:••••

:

t'^;\

photomicrograph

10H (b)

Figure 6-29. Light scattering pattern (a) and photomicrograph at the solvent-cast film of PS/PB5.9 with a composition of 80/20 wt./wt. stained by O S O 4 . Toluene was used as a solvent. From Hashimoto et al. (1984).

293

served in the solvent-cast film. The scattering maximum was found to correspond to the average spacing of the PI spheres but not to the average size of the spheres themselves. Presumably the percolation-to-cluster transition, as discussed in Fig. 6-20 of Sec. 6.7, may have happened before the vitrification, generating the clusters of the spheres. Hence the reciprocal-space and real-space patterns in Figs. 6-29 a and b reflect the memory of the periodic pattern developed by the SD process (Hashimoto, 1992). Chemical pinning involves the chemical crosslinking of polymer chains of one component or both components in binary blends, for example. It was found (Hashimoto et al., 1989d) that peroxide crosslinking is quite effective at pinning down the pattern growth for the binary polymer blends of SBR2/PI55 (in Table 6-3) and DPB/HPI (DPB-20-103/HPI-15-136 in Table 6-1) both in the long and short length scales, as studied by time-resolved light and small-angle neutron scattering methods, respectively. Figure 6-30 sketches a controlled pattern for elastomers composed of two component rubber-like polymers such as SBR2/PI55. The elastomer contains bicontinuous and periodic domains with a given average spacing A, and level of unmixing A<j> (see Fig. 6-7), and with a given crosslinking density. When an applied field, such as an electric (or magnetic) field, deformation, shear (or extensional) flow, is imposed on the systems, their self-assembly and morphology control become extremely intriguing both conceptually and for practical applications. There are a large number of reports on a class of multicomponent polymer systems called "interpenetrating network" (see for example, Manson and Sparling, 1976). Their patterns are controlled by a subtle and complex balance of chemical-reaction

294

6 Structure of Polymer Blends

Figure 6-30. Schematic illustration of bicontinuous and percolated domains of rubber-like polymer blends formed by SD and subsequent crosslinking reaction. Note that both domains are chemically crosslinked.

induced phase separation, chemical pinning, vitrification, etc. Real-time and insitu analyses of the self-assembly during the chemical reaction badly need to be carried out as a fundamental study.

6.10 Effects of Applied Fields on the Self-Assembly There are a number of interesting and fundamental observations for binary polymer mixtures of A and B without a solvent (A/B) or with a solvent (A/B/S), and for a solution of A (A/S) under shear flow, where S stands for solvent. Homogeneous solutions (A/S) undergo phase separation under shear. The "shear-induced phase separation" or the "shear-induced concentration fluctuation" has not been reported so far for small molecular systems and has been attributed to the elastic deformation of polymer coils (Ver Strate and Phillippoff, 1974; Schmidt and Wolf, 1979; Rangel-

Nafaile et al, 1984; Kramer and Wolf, 1985; Tirrell, 1986; Vrahopolou and McHugh, 1987; Pistoor and Binder, 1988; Helfand and Fredrickson, 1989; Onuki, 1989,1990; Milner, 1991; Hashimoto and Fujioka, 1991; Wu et al., 1991; Yanase et al., 1991). In contrast to the above phenomenon, phase separated A/B/S and A/B systems become homogeneous under shear (Silverberg and Kuhn, 1954; Hashimoto et al., 1988b, 1990a; Cheikh Larbi et al., 1988; Takebe and Hashimoto, 1988; Takebe etal., 1989, 1990; Nakatani et al., 1990). This "shear-induced homogenization" was also found for a binary critical mixture of simple liquids (Beysens et al., 1979; Beysens and Gbadamassi, 1979). However, the size of the drop in the critical temperature or cloud-point temperature due to shear ATc(y) was found to be three-to-four order of magnitudes bigger in the polymeric systems than in the small molecular systems (Hashimoto etal., 1988b, 1990a; Takebe et al., 1989, 1990). The drop ATc(y) for the small molecular systems was theoretically treated on the basis of critical dynamics and explained in terms of a suppression of the critical fluctuations by shear (Onuki and Kawasaki, 1979 a, 1979 b; Onuki et al., 1981; Onuki, 1989). On the basis of this theory, the strikingly large effect of shear on ATc(y) in polymers is attributed to an enormously long molecular relaxation time for the polymer system compared with that for the simple liquid system (Takebe etal., 1989; Hashimoto etal., 1990 a). Such a shear effect on the critical dynamics of simple liquids may also be important for the polymer systems. In addition to this effect, however, one cannot overlook a shear effect on molecular deformation, the contribution of which is insignificant on the concentration fluctuations in the small molecule systems but may become significant in polymers.

6.10 Effects of Applied Fields on the Self-Assembly

295

Figure 6-31. Light scattering pattern for the as-cast film (a) and the change of the pattern with a number of repeats (n) of the 1/2 compression. Patterns (b)-(f) correspond to n = 1, 2, 4, 6 and 12, respectively. The angle mark shows 6 = 10° in air. SBR1/PB19. From Hashimoto et al. (1989 a).

In recent years an increasing number of reports have become available on quantitative small-angle light scattering (Hashimoto et al., 1986c, 1988 a, b; Takebe et al., 1989,1990; Hashimoto and Fujioka, 1991; Wu et al., 1991) and small-angle neutron scattering (Nakatani et al., 1990) for polymer systems under shear flow, which are believed to provide key information to understanding rheology and phase transition through shear-rate dependent structure factors. Deformation also has a strong influence on the pattern of the concentration fluctuations developed by SD. Phase-separated, immiscible, rubber-like polymer blends with higher molecular weights are reported to be homogenized by applying a repetitive uniaxial compression, as shown in Fig.

6-31 (Hashimoto et al., 1989 a). The unmixing rate of the high molecular weight blends is much slower than the applied deformation rate, although the latter rate itself is very slow, so that the compression did not involve significant molecular orientation. In this case, the deformation caused by repetitive uniaxial compression is cumulative, i.e., the pattern hardly grows and changes during the time period between the successive deformations. After a number of compression the wavelength of the dominant mode of the concentration fluctuations parallel to the compression axis A{1 becomes shorter than the critical value Ac = 2n/qc. Here qc is the crossover wavenumber predicted in the linearized theory of SD by Cahn (1965), the modes of the fluctuation with the wavelength A greater

296

6 Structure of Polymer Blends

than Ac being able to grow but the modes with A
6.11 Summary Some key problems and future work are listed below as a short summary of this chapter. (1) The stage immediately after quenching, from the very early stage to early stage SD, should be thoroughly studied (Sec. 6.3.1). This kind of investigation may be effectively done by using polymer systems having a very long temporal scale as a model system. The wavenumber qm(0\ having the maximum growth rate in R(q), should be accurately determined and rigorously compared with existing theories for a greater number of systems (Sec. 6.4). (2) In the early stage SD, the growth of fluctuations with wavenumber q>i/Rg should be more throughly investigated, because this is a feature unique to polymeric systems. In this regard the ^-dependence of the Onsager kinetic coefficient X(q) should be further investigated (Sec. 6.4). (3) The scaling postulate on the time change in the global patterns at various temperatures T, as observed by qm (t; T) and / m (t; T), have been checked only for a limited number of polymeric systems. In

order to firmly check its validity in polymeric systems, the postulate should be further checked for a greater number of polymeric systems with different molecular weights, compositions, quench depths, and concentrations of total polymers (if the solvent is added) (Sec. 6.5.1). The non-universal behaviour of Tm(x) and Qm{x) as observed in the N-, C-, and B-branches should be clarified in future (Sec. 6.5.3). Also relationships between the coarsening mechanism and the scaling exponents a and ft should be made clear, for which realspace analyses are badly required (Sec. 6.5). (4) The scaled structure factor F(x) should be investigated further over a wide x-range and for a greater number of systems (Sec. 6.6). Investigation over a wide x-range is made possible by combining light scattering and small-angle neutron or X-ray scattering. Further study should be focused on the local feature of F(x), and the origin of higher order maxima or shoulders at x ~ 2 and/or 3 should be clarified. Again real space analysis is important for this. (5) Unique features in the phase diagram of polymers are attributed to a negligible contribution of the combinatorial entropy to the net mixing free energy. The role of the non-combinatorial part on the phase diagram should be further studied both theoretically and experimentally. Dependence of x o r Zeff o n the microstructure, tacticity, composition (f)AO, molecular weight, etc. should be further investigated (Sec. 6.2.2). (6) Self-assembly under external fields, such as a simple shear flow, should be further investigated (Sec. 6.10). It is hoped that sound statistical mechanical theories will be established in order to interrelate the structure factor, rheology, and phase transition. It is especially important in polymer science to explore the role of polymer coil deformation on the structure fac-

6.13 References

tor, rheology and phase transition, since this effect of the molecular deformation is essentially absent, and hence unexplored, in the science of small molecules. (7) It may be extremely challenging and interesting in future problems to investigate the variations in self-assembly caused by interplays or couplings of two kinds of phase transition (Sec. 6.1 and 6.8). (8) In this chapter we dealt with selfassembly via SD, and no discussion was made of self-assembly via nucleation growth. This is simply because not many quantitative studies have been reported so far on nucleation growth in high polymer mixtures (Binder, 1983). This type of investigation deserves future work.

6.12 Acknowledgements The author is grateful to Dr. Charles C. Han for his collaboration, upon which part of this review is based. This work is supported in part by the Grant-in-Aid for Scientific Research in Priority Areas "New Functionality Materials-Design, Preparation and Control" from the Ministry of Education, Science and Culture, Japan.

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Nakatani, A., Kim, H., Takahashi, Y, Matsushita, Y, Xakano, A., Bauer, B. J., Han, C. C. (1990), /. Chem. Phys. 93, 795. Nojima, S., Ohyama, Y, Yamaguchi, M., Nose, X (1982), Polym. J. 14, 907. Nose, X (1987), Phase Transitions 8, 245. Nose, X (1989), in: Space-Time Organization in Macromolecular Fluids: Xanaka, E, Doi, M., Ohta, X (Eds.). Berlin: Springer, pp. 40-50. Ohta, X, Kawasaki, K. (1986), Macromolecules 19, 261. Ohta, X, Nozaki, H. (1989), in: Space-Time Organization in Macromolecular Fluids: Xanaka, E, Doi, M., Ohta, X (Eds.). Berlin: Springer, pp. 51-57. Okada, M., Han, C. C. (1986), J. Chem. Phys. 85, 5317. Onuki, A. (1986), J. Chem. Phys. 85, 1122. Onuki, A. (1989), Phys. Rev. Lett. 62, 2472. Onuki, A. (1990), J. Phys. Soc. Jpn. 59, 3427. Onuki, A., Hashimoto, X (1989), Macromolecules 22, 879. Onuki, A., Kawasaki, K. (1979a), Ann. Phys. (N.Y.) 121, 456. Onuki, A., Kawasaki, K. (1979 b), Phys. Lett. 72A, 233. Onuki, A., Yamazaki, K., Kawasaki, K. (1981), Ann. Phys. (N.Y.) 131, 217. Oono, Y, Puri, S. (1988a), Phys. Rev. A 38, 434. Oono, Y, Puri, S. (1988 b), Mod. Phys. Lett. B2, 861. Orwall, R. A., Flory, P. J. (1967), /. Am. Chem. Soc. 89, 6814. Pincus, P. (1981), J. Chem. Phys. 75, 1996. Pistoor, N., Binder, K. (1988), Colloid & Polym. Sci. 266, 132. Puri, S., Oono, Y (1988), Phys. Rev. A 38, 1542. Porod, G. (1951), Koll. Z. 124, 83. Porod, G. (1952a), Koll. Z. 125, 51. Porod, G. (1952b), Koll. Z. 125, 108. Rangel-Nafaile, C , Metzner, A. B., Wissbrum, K. F. (1984), Macromolecules 17, 1187. Sakurai, S., Hasegawa, H., Hashimoto, X, Glen Hargis, I., Aggarwal, S. L., Han, C. C. (1990a), Macromolecules 23, 451. Sakurai, S., Jinnai, H., Hasegawa, H., Hashimoto, X, Han, C. C. (1990b), Polym. Prepr., Div. Polym. Chem., Am. Chem. Soc. 31, 147. Sakurai, S., Jinnai, H., Hasegawa, H., Hashimoto, X, Han, C. C. (1991), Macromolecules 24, 4839. Sanchez, I. C , Lacombe, R. H. (1976), J. Phys. Chem. 80, 2352, 2568. Sasaki, K., Hashimoto, X (1984), Macromolecules 17, 2818. Sato, X, Han, C. C. (1988), / Chem. Phys. 88, 2057. Schwahn, D., Yee-Madeira, H. (1987), Colloid & Polym. Sci. 265, 867. Schwahn, D., Mortensen, K., Yee-Madeira, H. (1987 a), Phys. Rev. Lett. 58, 1544. Schwahn, D., Mortensen, K., Springer, H., YeeMadeira, H., Thonas, R. (1987b), /. Chem. Phys. 87, 6078.

300

6 Structure of Polymer Blends

Schwahn, D., Hahn, K., Streib, I, Springer, T. (1990), J. Chem. Phys. 93, 8383, and private communication. Schwahn, D., Springer, T., Janssen, S., Hadicke, E. (1991), /. Appl. Cryst. 24, 685. Schmidt, R., Wolf, B. A. (1979), Colloid & Polym. ScL 257, 1188. Shibayama, M., Hashimoto, T., Kawai, H. (1983), Macromolecules 16, 1434. Shinozaki, A., Oono, Y. (1991), Phys. Rev. Lett. 66, 173. Silverberg, A., Kuhn, W. (1954), J. Polym. ScL 13, 21. Snyder, H. L., Meakin, P. (1983), J. Chem. Phys. 79, 5588. Strobl, G. R. (1985), Macromolecules 18, 558. Takahashi, M., Horiuchi, H., Kinoshita, S., Ohyama, Y, Nose, T. (1986), /. Phys. Soc. Jpn. 55, 2687. Takebe, T., Hashimoto, T. (1988), Polym. Commun. 29, 261. Takebe, T, Sawaoka, R., Hashimoto, T. (1989), J. Chem. Phys. 91, 4369. Takebe, T, Fujioka, K., Sawaoka, R., Hashimoto, T. (1990), J. Chem. Phys. 93, 5271. Takenaka, T, Hashimoto, T. (1992), J. Chem. Phys. 96, 6177. Takenaka, M., Izumitani, T., Hashimoto, T. (1987), Macromolecules 20, 2257. Takenaka, M., Tanaka, K., Hashimoto, T. (1989), in: Multiphase Macromolecular Systems: Culbertson, B. M. (Ed.). New York: Plenum, pp. 363-384. Takenaka, M., Izumitani, T., Hashimoto, T. (1990), /. Chem. Phys. 92, 4566. Tanaka, H., Hashimoto, T. (1988), Polym. Commun. 29, 212. Tanaka, H., Hasegawa, H., Hashimoto, T. (1991), Macromolecules 24, 240. ten Brinke, G., Karasz, F. E., MacKnight, W. J. (1983), Macromolecules 16, 1827. Thomas, E. L., Alward, D. B., Kinning, D. X, Martin, D. C , Handlin, Jr., D. L., Fetters, L. I (1986), Macromolecules 19, 2197. Tirrell, M. (1986), Fluid Phase Equilib. 30, 367. Tomlins, E., Higgins, J. (1989), J. Chem. Phys. 90, 6691. Utracki, L. A. (1989), Polymer Alloys and Blends. Munich, Vienna, New York: Hanser. Ver Strate, G., Phillippoff, W. (1974), J. Polym. ScL, Polym. Lett. Ed. 12, 267.

Vrahopoulou, E. P., McHugh, A. J. (1987), J. NonNewtonian Fluid Mech. 25, 157. Walsh, D. J., Higgins, J. S., Maconnachie, A. (Eds.) (1985), Polymer Blends and Mixtures, Nato ASI Series. Warner, M., Higgins, J. S., Carter, A. J. (1983), Macromolecules 16, 1931. Whitmore, M. D., Noolandi, J. (1985), Macromolecules 18, 2486. Wu, X-L., Pine, D. I, Dixon, P. K. (1991), Phys. Rev. Lett. 66, 2408. Yanase, H., Moldenaers, P., Mewis, I , Abetz, V., van Egmond, J. W, Fuller, G. G. (1991), Rheol. Acta 30, 89. Yang, H., Shibayama, M., Stein, R. S., Shimidzu, N., Hashimoto, T. (1986 a), Macromolecules 19, 1667. Yang, H., Stein, R. S., Han, C. C , Bauer, B. I , Kramer, E. J. (1986 b), Polym. Commun. 27, 132.

General Reading Binder, K. (1991), in: Materials Science and Technology, Vol. 5: Phase Transitions in Materials: Haasen, P. (Ed.). Weinheim: VCH Publishers, Ch. 7. de Gennes, P. G. (1979), Scaling Concepts in Polymer Physics. Ithaca, London: Cornell Univ. Press. Doi, M., Edwards, S. F. (1986), The Theory of Polymer Dynamics. Oxford: Clarendon Press. Flory, P. J. (1953), Principles of Polymer Chemistry. New York: Cornell Univ. Press. Guinier, A., Fournet, G. (1955), Small-Angle Scattering of X-rays. New York: Wiley; London: Chapman & Hall. Gunton, J. D., San Miguel, M., Sahni, P. S. (1983), in: Phase Transitions and Critical Phenomena, Vol. 8: Domb, C , Lebowitz, J. L. (Eds.). London: Academic Press, pp. 269-466. Hashimoto, T. (1987), in: Thermoplastic Elastomers: Legge, N. R., Holden, G., Schroeder, H. (Eds.). Munich, Vienna, New York: Hanser, pp. 200-242. Hashimoto, T. (1988), Phase Transition 12, 47. Hashimoto, T., Shibayama, M., Fujimura, M., Kawai, H. (1983), in: Block Copolymer Science and Technology: Meier, D. J. (Ed.). New York: MMI Press, Harwood Academic, pp. 63-108. Ma, S. K. (1976), Modern Theory of Critical Phenomena. London, Amsterdam: Benjamin/Cummings.

7 Elastic Properties of Crystalline Polymers David T. Grubb Department of Materials Science and Engineering, Cornell University, Ithaca, NY, U.S.A. List of 7.1 7.1.1 7.1.2 7.1.3 7.1.4 7.1.4.1 7.1.4.2 7.1.4.3 7.2 7.2.1 7.2.2 7.2.3 7.2.4 7.3 7.3.1 7.3.2 7.3.3 7.3.4 7.4 7.4.1 7.4.2 7.4.3 7.5 7.5.1 7.5.2 7.6 7.6.1 7.6.2 7.6.3 7.6.4 7.6.5 7.6.5.1 7.6.5.2 7.6.5.3 7.7 7.8

Symbols and Abbreviations Introduction General, Scope of Chapter Definitions and Terms in Elasticity Units Non-Ideal Elasticity Viscoelasticity Nonlinear Elasticity Large Strains Experimental Determination of Elastic Properties Quasi-Static Measurements Dynamic Measurements Ultrasonic Measurements High Speed Testing Experimental Determination of Crystal Modulus X-Ray Diffraction Raman Spectroscopy Inelastic Neutron Scattering Macroscopic Polymer Crystals Theoretical Modeling of Elastic Constants General Introduction Molecular Dynamics - Empirical Force Fields Molecular Orbitals - Quantum Mechanical Calculations Interpretation of Results for Crystal Modulus Experimental Data Theoretical Calculations Interpretation of Macroscopic Modulus Aggregate Models Takayanagi Models Tsai-Halpin Equations Lamellar Assembly Modeling Models of High Modulus Fibers Fiber Composite Model Crystalline Bridge Model Molecular Models Concluding Remarks References

Materials Science and Technology Copyright © WILEY-VCH Verlag GmbH & Co KGaA. All rights reserved.

302 305 305 307 309 309 310 310 311 311 311 315 316 318 319 319 323 326 326 328 328 328 332 334 334 337 337 338 341 346 346 348 348 349 351 352 352

302

7 Elastic Properties of Crystalline Polymers

List of Symbols and Abbreviations A b B C C1,C2,C3 cpq d D £, Et Ea Ec Ef £app ^matrix ED E\, £3 E\, £3 £|l £± Ea1,Ec1 / £ /p /x G, Gtj Gm G|| ij J Jo.J^ K kx fca,fcp L Lc Ll9L ± L 3 ,L|| / p P Pa9Pc p, q

chain area; constant fraction of fibrous crystals bulk compressibility matrix of elastic constants carbon atoms in the all-trans backbone of P E elastic constants spacing of the crystal lattice planes tensile compliance Young's moduli modulus of disordered component in semi-crystalline polymer crystal modulus in the chain direction (short form of £3) modulus of reinforcing fibrils apparent modulus matrix modulus (fiber composite model) modulus of entanglement defect region amorphous moduli crystalline moduli tensile modulus in fiber direction tensile modulus transverse to the fiber direction transverse moduli of the anisotropic units of the amorphous, crystalline phase fractional length force transverse force in bond deformation stretching force in bond deformation shear moduli matrix shear modulus (fiber composite model) torsional modulus about the fiber axis integer (1 < ij < 3) shear compliance immediate elastic value, long term value of J bulk modulus C - C bond stretching force constant C 1 - C 2 - C 3 bond bending force constant length of molecular chain unit length of crystalline fibril transverse ultrasonic modulus longitudinal ultrasonic modulus bond length hydrostatic pressure; probability that a crystalline chain traverses the amorphous region and enters the next lamella elastic property (general form of Tsai-Halpin equations) P referring to the amorphous, crystalline phase integer (1 < p, q < 6)

List of Symbols and Abbreviations

rc S S' spq sfpq s 'pq>spq v V{ ^o

radius of crystalline fiber compliance matrix sample compliance matrix compliance constants, elements of S elements of S' sample compliance elements of a two-phase model velocity of s o u n d fibril fraction velocity of s o u n d in immersion liquid

a ytj e a ,e c sp rj 29 9L,9T X lm vm

a a ,(7 c (jp <JS Tpq 4> (j)a, (j)c \j/

b o n d angle; degree of crystallinity shear stress tensile strain in the a m o r p h o u s phase, crystal lattice tensile strain component parameter in Tsai-Halpin equations diffraction angle critical angle of incidence for longitudinal, transverse ultrasonic wave X-ray wavelength; width of the crystalline region (Takayanagi model) wavelength of chain LAM frequency of chain LAM Poisson's ratios Poisson's ratios in the case of fiber symmetry geometric parameter in Tsai-Halpin equations density density of the molecular chain tensile stress in the amorphous, crystal phase tensile stress components sample tensile stress shear strain inclination of object from the fiber axis <j) referring to chains in the amorphous, crystalline phases length of the crystalline region (Takayanagi model)

ASTM CNDO DMTA iPP ISO LAM LCAO LVDT MNDO NMR PBO PBZT

American Society for Testing and Materials complete neglect of differential overlap dynamic mechanical thermal analysis isotactic polypropylene International Organization for Standardization longitudinal acoustic mode linear combination of atomic orbitals linear variable differential transformer modified neglect of differential overlap nuclear magnetic resonance poly(p-phenylene benzobisoxazole) poly(p-phenylene benzobisthiazole)

v

tj

Q

303

304

7 Elastic Properties of Crystalline Polymers

PE PET PEO POM PTFE SAXS

polyethylene poly(ethylene terephthalate) poly(ethylene oxide) polyoxymethylene polytetrafluoroethylene small angle X-ray scattering

7.1 Introduction

7,1 Introduction 7.1.1 General, Scope of Chapter

Most uses of solid polymers depend to some extent on the mechanical properties of the material, and the basic mechanical properties of a solid are its stiffness, strength and toughness. Stiffness represents the resistance of a body to elastic deformation, the ratio of load over deflection. This depends on the material and on the shape of the body. The material properties themselves are expressed as a modulus, a ratio of stress over strain, or as a compliance, a ratio of strain over stress. A deformation is elastic if the sample recovers its orginal shape as soon as the load is removed. The strength of a material is the stress at which the material fails by fracture or by irreversible deformation. Toughness is a measure of the work required to fracture a material. The complex topics of plastic deformation and failure are dealt with in Chaps. 10 and 15 of this Volume. In traditional materials science, where metals are always taken as typical, the elastic properties of a solid are not very interesting. The modulus is seen as structure insensitive, while strength and toughness are the mechanical properties affected by microstructure. Modulus is very basic; it relates to the bond strength and thus to the melting point, and there are few surprises. The elastic constants can be obtained by direct measurement on a single crystal sample. They can also be calculated from bond force functions, using the known crystal structures. The elastic properties are often nearly isotropic and not sensitive to temperature, so a single number called "the modulus" is used to describe the elastic properties. In the world of polymers, the elastic properties of a solid sample are equally

305

basic, but they may be strongly structure sensitive, anisotropic, and rate or temperature dependent. Thus different samples of the same polymer tested in different ways may show moduli that are different by orders of magnitude. Most solid polymers are viscoelastic. The formal description of viscoelasticity is given in Chap. 9 of this Volume. It is sufficient here to say that viscoelastic solids show some time dependent deformation even at low stresses. Measurement of elastic properties must take this into account; in particular, the modulus will be a function of time. The slower the test, the more deformation, so the larger the strain and the softer the material appears to be. Generally the time dependent deformation is thermally activated, faster at high temperature, so that the modulus will be strongly temperature dependent as well as time dependent. Real samples of crystalline polymers are not simply polycrystalline, they contain many disordered interfaces between the small polymer crystals so that they are partially ordered at best. Crystalline polymers are usually considered to consist of two phases, being a mixture of perfect crystals and fully disordered material. They are often called "semi-crystalline" for that reason. Generally the crystals are denser and stiffer than the amorphous phase, so the crystal modulus is an upper limit for the modulus of a real specimen. Comparison of sample modulus with crystal modulus then allows a judgement of how much remains to be achieved, if the aim is a stiff material. Polymer crystals are ordered assemblies of parallel molecular chains, with bonds between chains that are much weaker than the covalent main chain bonds. The crystals are therefore intrinsically anisotropic. It is not normally possible to obtain macroscopic samples of a perfect polymer crystal, so its mechanical properties cannot

306

7 Elastic Properties of Crystalline Polymers

be obtained by direct mechanical testing. Indirect measurements using X-ray diffraction or vibrational spectroscopy need careful interpretation and often do not agree. Theoretical estimates of the elastic constants of the crystals are now quite easily obtained but may not be accurate. If the chain conformation in the crystal is such that bending or stretching of covalent bonds is required to extend the chain, it will be very stiff and the crystal anisotropy will be very high. If the conformation is a helix that can be stretched by uncoiling, which only involves rotation of the bond angles, the chain will be much softer. Modeling of the molecular stiffness will give a good first approximation to the stiffness of the crystal in the chain axis direction. This type of modeling has been done for some time. It is now possible to model the full crystal structure using semiempirical bond properties, giving a better estimate of the modulus in the chain axis direction, and a full set of elastic constants for the crystal. Computational requirements limit quantum mechanical calculations to a single chain. Amorphous polymers by definition contain a wide range of polymer chain conformations; most will be able to deform by bond rotation. The stiffness of glassy polymers then depends on the constraints and barriers to bond rotation imposed by neighboring chains. These constraints are difficult to model in a random system. The properties of amorphous polymer solids are dealt with in Chap. 2 of this Volume. In some cases, bulk samples of amorphous material can be obtained and tested, but there is no guarantee that the disordered material in the semi-crystalline sample has the same properties. In particular, the disordered material in the semi-crystalline state is constrained by the surrounding crystals and is often limited to

very small regions. This extra constraint will affect the molecular mobility and thus the stiffness. When the sample is drawn and the crystalline phase becomes oriented, there will be orientation of the amorphous phase too. Thus the word "phase" is used loosely here and in later discussion of mechanical models, and does not imply a thermodynamic equilibrium phase with well defined properties. A semi-crystalline polymer sample approximated by a two-phase model is thus a composite material where the mechanical properties of the two components are not well known. For modeling mechanical properties there is the further serious difficulty that the internal arrangement of the two phases may be unknown. Even when the arrangement is known, constructing a self-consistent mechanical model for a complex structure is not straightforward. An important case of this type is the spherulitic lamellar structure described in Chap. 4 of this Volume. Whatever model is used, the elastic properties of a real semi-crystalline polymer cannot be predicted from the crystalline and amorphous elastic properties alone. Structural parameters are required, and the most important of these are orientation and crystallinity. Clearly, if the molecular or crystal properties are highly anisotropic and the chains or crystals in a sample are aligned, then the elastic properties of the sample will also be anisotropic. As the chain direction is generally the stiffest, alignment of chains by flow or colddrawing will increase the sample stiffness in the draw direction. Generally the modulus of semi-crystalline samples will also increase with crystalline content. These are not the only structural parameters that affect modulus, as samples of the same degree of orientation and crystallinity may still have very different moduli. In high per-

307

7.1 Introduction

formance polymer fibers the fiber modulus approaches that of the perfect crystal. In this case measurement of elastic properties as a function of processing conditions and comparison with those of the perfect crystal may be used to investigate the mechanical defects that relate to imperfections in the structure of the sample. 7.1.2 Definitions and Terms in Elasticity

The oriented chains in crystalline polymers make the crystals anisotropic. To describe the elastic properties of anisotropic materials Hooke's law is generalized to three-dimensional Cartesian axes x, y, and z. The 3-D set of equations is conventionally given in a contracted form: q=6 —

X^ C q=l

q—6 P

—

P '

""S"* '

(7-1)

planes. Thus <J4 may also be written as T 2 3 , and g4 as y23. Care is required to derive the exact form of these components, as factors of 2 arise in some of them (Nye, 1957; Arridge, 1975; Ward, 1979). External stability requires that cpq == cqp and this reduces the number of independent components from 36 to 21 (Hearmon, 1961). If the material has orthotropic symmetry (sometimes called orthorhombic, as in crystallography) the number is reduced to 9, with a compliance matrix of the form n

S

12

s13

12

S22

•S23

13

S23

«33

0 0 0

0 0 0

0 0 0

0 0 0 S44

0 0

0 0 0 0 S55

0

0 0 0 0 0

(7-2)

S £L

66

q=l

C is the matrix of elastic constants, components cpq. Each cpq is the ratio opjzq when all other strain components are zero. S is the compliance matrix, components spq. Each spq is the ratio &p/oq when all other stress components are zero. Indices 1, 2, and 3 refer to normal stress and extensional strain along x, y and z respectively. 4, 5, and 6 refer to shear in the yz, xz, and xy-

Figure 7-1. Axes and compliance component directions for a polymer sheet with orthotropic symmetry.

A common case of anisotropic elasticity in polymers where orthotropic symmetry exists is a sheet biaxially drawn or drawn at constant width. Conventionally, in this case the z-axis is taken as the draw direction of the sheet, x is transverse to the draw direction and in the plane of the sheet. The y-axis is then normal to the sheet, as in Fig. 7-1. The polymer molecules tend to lie in the plane of the sheet, but more along z than along x. The extensional compliances in the machine and transverse directions in the plane of the sheet are s 33 and s11 respectively. They will be different and both will be less than that of an isotropic material, while the compliance perpendicular to the sheet, s 22 , is higher. The out-of-plane compliance is difficult to measure in real thin sheets, and is also of little practical importance in many cases. If the drawing is uniaxial and the sample dimension along x and y not too different, then the x- and y-axes become equivalent. This is called transverse isotropy or fiber symmetry, and such samples then require

7 Elastic Properties of Crystalline Polymers

308

shear experiment in the yz-plane of the orthotropic material will give the shear modulus G 2 3 = I/544. The compliance terms along the diagonal of the compliance matrix [Eq. (7-2)] are the inverses of the three Young's moduli and three shear moduli. The off-axis terms relate to the Poisson's ratios. In full for the orthotropic material

only 5 independent elastic constants. An expert in mechanics would naturally choose the x- or 1-axis as the unique (fiber draw) axis, but in polymer fibers it is the z or 3-axis. This is because the molecular chain axis becomes oriented along the fiber axis by drawing and the chain axis is conventionally labeled the c, z- or 3-axis in poly-

1/E, — v12/— v13/-

-

0 0 0

0 0 0

mer crystals* so that the compliance matrix becomes Eq. (7-3). Unless a detailed study of anisotropic elastic behavior is being made, the results of mechanical experiments are expressed not as elastic constants spq or cpq but as engineering constants. These are Young's moduli, shear moduli and Poisson's ratios. (7-3) Ll 9

s=

13

0 0 0

5

12

5

13

0

0

0

51t

S^

0

0

0

S

13

S33

0

0 0 0

0 0 0

0 0

0 0 0

S

44

0 0

S

44

0

:2(s,,

-

>12.)

Consider a material with orthotropic symmetry that has a tensile stress as applied along z. ol = os, all other stresses are zero and there will be strains e l5 e2 and e 3 . The tensile or Young's modulus in the 3-direction, E 3 , is the ratio aje3, so s 33 = 1/E3. The Poisson's ratios for this experiment are defined as v31 = — s1/s3 and v32 = — s2/s3. Thus s 31 , which by definition is ej(rs when other stresses are zero, is equal to — v 3 1 /£ 3 . This relation may also be expressed as v31 = — s 31 /s 33 . A

V32AE

0 0 0

0 0 0 VG23 0

0 0 0 0

0 0 0 0 0

(7-4)

0

since spq = spq, vtj = v^EJEj). Isotropic materials have single values of E, G and v. There are only two independent elastic constants, so for example the shear modulus may be calculated as G = E/[2(1 +v)]. The bulk modulus K = [pressure/^! + e2 + £3)L the tensile compliance D = l/£, the shear compliance J = 1/G or the bulk compressibility B = l/K may also be used to describe the material. Returning to anisotropic materials, it may be difficult to obtain all the compliance components experimentally; theoretically derived components are often used for perfect crystals, and estimates for amorphous or semi-crystalline materials. If all the compliance components are known, the elastic constants cpq can be obtained by inverting the matrix, C = S~1. For orthotropic materials the shear constants are simple, but the extensional constants are not (McCullough, 1977). For example, C44 = IA44 = G 2 3 but

1 Except in some monoclinic crystal structures, such as poly(vinyl alcohol) where crystallographic conventions confusingly require that the chain axis is b.

7.1 Introduction

^33

=

- 2(E3/Ex)v12 v23 v13 -

"?2

For fiber symmetry with unique axis 3, as in Eq. (7-3), EX=E2 = E±, E3 = £,,, = G 2 3 = G|,,

and

=

V23=V,,

v

i2 — V2i = v i - This simplifies the expressions for the elastic constants considerably. Using the same example as above,

"

l-2(£||/£1)v2

(7-6)

In a fiber used as a primary mechanical component such as a tire cord, the most important elastic constant is the tensile modulus in the fiber direction, £j ( . The tensile modulus in the transverse direction, E l9 may also be important, but all transverse properties are difficult to measure. The torsional modulus about the fiber axis G|| = l/s 4 4 is the only other property likely to be determined. Mechanical properties may be measured by the determination of longitudinal ultrasonic wave velocity, as the speed of sound depends on the density and modulus of the material. In this case the transverse strains are zero, so the experiment does determine a component of C, the elastic constant or modulus matrix, directly. In the case of fiber symmetry with z the fiber axis [Eq. (7-3)], the longitudinal ultrasonic modulus along the fiber direction, L 3 or L|( = c 33 and Lx or L± = c11. Note that Ly =(= l/s 3 3 , the usual E j|. 7.1.3 Units The SI unit of modulus and stress is the pascal, which is 1 newton • meter" 2 . Materials are likely to have moduli over 109 Nm~ 2 , or 1 GPa. Other units that may be used are dyne • cm" 2 , and in more practical work psi (lb-in" 2 ), kg mm" 2 , or

309 (7-5)

kg • cm 2. The relation between them is: l G P a = 10 1 0 dyne-cm" 2 = = 101.9 kg • mm" 2 - 145000 psi = = 145 kpsi The units of compliance are the inverse of these, m 2 /N and so on. The "practical" units use units of mass confusingly, using them to express the force "lb wt." or "kg f" that results from gravity acting on the mass. When the mass of the product is important to the application, the modulus of the material is often expressed as a specific modulus, (modulus/density). This measure makes polymers look better than metals or ceramics. The unit for this is J - k g " 1 , which may be reduced to (m/s)2. The specific modulus is unfortunately often quoted in terms of inches. This is due to writing it as modulus in units of lb • in" 2 divided by density in units of lb • in" 3 , and falsely cancelling a force in pounds weight with a mass in pounds. Specific modulus is also used in fiber science because of the difficulty in determining the cross sectional area of fine fibers. The specific modulus units are N • tex" 1 and g • denier" *. Denier and tex are units of mass per unit length, the product of density and cross sectional area and they are obtained simply by weighing a known length of fiber. The tex is 1 0 " 6 k g m or l g - k m " 1 , while 1 g • denier" 1 is 0.9 N • tex" 1 . Thus: 1 M J - k g " 1 = 1 km 2 s~ 2 = = 106m2s"2 = 1 N-tex"1 7.1.4 Non-Ideal Elasticity The definition of the modulus given above assumes a well defined ratio between stress and strain that does not depend on

310

7 Elastic Properties of Crystalline Polymers

load, that is, the elasticity is linear. The derivations use engineering stress and strain, which are related to the original cross-sectional area and original length of the specimen. All assume that the strains are small. More fundamentally, any discussion of elasticity assumes that elastic and plastic strain can be distinguished clearly. The mechanical deformation of real polymers may not meet any of these assumptions of ideal elasticity. Most important of these non-ideal behaviors is the viscoelastic nature of the semicrystalline materials, due to the disordered components in the material. It is assumed that the polymer crystals themselves are simple elastic bodies with a well defined plastic yield point. 7.1.4.1 Viscoelasticity Viscoelastic properties are of practical importance, and give insights into the molecular mechanisms of deformation (see Chap. 9 of this Volume). Here the main point is that elastic measurements and definitions must take into account the unavoidable plastic flow component of displacement. All data for sample stiffness will depend on the time scale of the experiment. Compliances measured at fixed load during creep, at fixed extension during stress relaxation and during oscillatory loading may all be different. Strictly, all are better defined than the compliance determined by the usual simple tensile test at constant cross-head speed. It is not generally possible to make useful comparisons unless the exact conditions of testing are known. For example, materials suppliers who quote a modulus for their product without giving the strain rate used or the ASTM standard test number are not being helpful. If stiffness is a selling point, they can be assumed to have used a high strain rate or ultraso-

nic measurement. This gives a high value of modulus that may or may not be appropriate to the end use being considered. Elastic deformation is immediately recovered on unloading, and plastic deformation is permanent and unrecoverable. What then of a polymeric material that deforms under load, and when the load is removed, slowly recovers the deformation so that say 24 hours later is in its original form? This behavior is quite common, as you may see when you move furniture on a carpeted floor. The weight crushes the fibers, but they should recover over a period of hours or days so that the marks disappear. This is called a delayed elastic or anelastic response. In general terms it is the long range interactions and the long relaxation times of polymer chains that cause this behavior. If a metal wire is kept under constant load and slowly extends then it is creeping. This is a plastic deformation, and the elastic modulus is a constant. A polymer slowly extending under constant load may be creeping or it may be showing how its compliance increases from the immediate elastic value Jo to the larger long term value J^. The only way to distinguish these two cases is to allow the sample to relax for long times and see how much of the deformation is recoverable (Turner, 1966). 7.1.4.2 Nonlinear Elasticity Simple materials are often linear elastic; at small strains, stress is a linear function of strain, so that the modulus or compliance is constant as the load increases. In many polymer systems this is not true, and the compliance generally increases with load. Thus measurements of modulus should also quote the maximum strain used in determining the slope of the stressstrain curve. A high modulus is obtained

7.2 Experimental Determination of Elastic Properties

by using precise and well aligned testing machines, where grip motion and joint alignment do not spoil the very first part of the test. For viscoelastic systems where the compliance will already be a function of time and temperature, the situation rapidly becomes very complex. The stiffness may change during a mechanical test because of structural changes in the material, or because the time that the material has been under stress is increasing. In most cases tests are made under standard conditions, and the nonlinear behavior is ignored. Studies that take nonlinearity into account do so largely on an empirical or curve fitting basis (Arridge, 1975) as the equations for anisotropic materials contain many unknowns. The main aim of such studies is usually to predict elastic properties under some conditions, such as creep or cyclic loading, from tests made under other conditions. For some high modulus fibers, nonlinear elastic behavior may be predicted or explained by changes in the orientation or geometry of chains under stress (Northolt and Van der Hout, 1985; Wierschke et al., 1992). Modulus has been found to increase during creep of aramid fibers (Rogozinsky and Bazhenov, 1992). 7.1.4.3 Large Strains When strains are not small, many factors in Hookean elasticity have to be altered, and so do the testing methods. Consider a tensile test in a testing machine with a constant rate of motion of cross-head. At large strains the gauge length and the cross-sectional area of the specimen do not remain constant. This means that the strain rate falls and the load variations do not follow changes in stress. To have a test at a constant rate of strain, the cross-head must

311

accelerate. As metals and most other materials fail at small elastic strains, this is not a practically important problem for them. The theory of linear elasticity at large strains has been developed (Green and Adtkins, 1970; Arridge 1975) and in polymer science is mostly used to describe rubber elasticity (Chap. 8 of this Volume). In that case extensions of several times the original length must be considered and the unstrained material is isotropic. When semi-crystalline polymers are made into very strong fibers, they can undergo elastic deformations of 1-2%. While this is far from the extensions possible in elastomers, it can be enough to cause a change in the molecular chain orientation. For highly oriented systems with very stiff molecular chains this can increase the apparent tensile modulus (Northolt, 1980).

7.2 Experimental Determination of Elastic Properties 7.2.1 Quasi-Static Measurements Quasi-static refers to all tests where the measurement is made over a comparatively long time scale, commonly in the range 1 to 100 s. There are two classes of test used. In one class the sample dimensions and all other conditions of testing are defined by some standards organization, such as ASTM or ISO. Most of these tests use standard mechanical tension/compression testers, where a cross-head is moved by screw or hydraulic control at a fixed speed. Since the sample gauge length, the crosshead speed, and the strain at which the modulus is to be determined are all defined, the time scale of testing is also standardized. ASTM D 2990-77, D 638-77 a, D 882-75 b are examples of standard tests for solid polymer samples.

312

7 Elastic Properties of Crystalline Polymers

The strain rates in such tests are generally in the range of 10~ 4 to 10" 1 s" 1 . In this range the modulus of many practical semi-crystalline polymers does not vary strongly with strain rate, but only increases slowly as the testing rate increases. Thus these tests give reliable and reproducible measurements of the elastic properties. Most of them are concerned only with tension or torsional shear and with largely isotropic samples such as compression molded sheet or injection molded bars. If the materials being tested are in short supply, the tests may be modified to use smaller samples at some cost of precision. In the other class of test, a more fundamental measure of elastic properties is sought. A constant load is applied for a given time, and the sample strain is then measured. If the time was 10 s, the result would be the 10 s creep compliance. Applying different loads allows one to build up an isochronous stress-strain curve, Fig. 7-2. This may be used for the study of nonlinear elasticity, or more simply to ensure that the loads and extensions measured are small enough to remain in the linear elastic region. In principle a static modulus or the long term compliance J^ can be obtained by testing over a range of time scales and extrapolating to infinite time; this is not often done. Testing machine design for tensile tests of this type is not very demanding. The load is applied by attaching a weight to the sample; polymer samples are not usually very stiff, so that machine compliance is not a problem. Displacement measurement by linear variable differential transformer (LVDT) is sufficiently accurate for most purposes. More accuracy may be obtained from direct extensometer measurement of sample length, but extensometers with knife edges may deform the sample locally unless the polymer is stiff. Optical

0.5

7

0.4

r

Increasing time under load

O 0.3

.2 'E. o 0.2 o PH


o 0.1 - Linear

Region 0.0

200 300 400 500 600 Load, g Figure 7-2. Isochronous creep compliance as a function of load for isotactic polypropylene monofilaments. The times of loading were 15, 40, 100, 300, 1000, and 9300 s. (From Ward and Onat, 1963.) 100

extensometers or devices using noncontact transducers should instead be used. For temperature control the sample may be mounted in a tube inside a temperature controlled oil bath (Ward, 1964) or an annular tube through which cold gas flows. The shear modulus of isotropic samples may be measured by the torsion of a rod or tube, but quasi-static measurements are not common. Torsional strain measurement by deflection of a light beam is simple and accurate (McCrum and Morris, 1964). The general relations between E, G, K and v can be used to derive Poisson's ratio and bulk compressibility; for example v = E/(2G) + 1. Poisson's ratio may also be measured directly as lateral shrinkage on tensile deformation. Many special mechanical tests have been designed for special purposes. Saimoto (1986), for example, describes how a rapid change of strain rate may be applied dur-

7.2 Experimental Determination of Elastic Properties

ing a tensile test. The resulting stress may be used to derive the elastic properties of a sample on a time scale of a second during plastic deformation. The idea is to detect changes in properties caused by deformation that may relax faster than more conventional tests can be performed. Quasi-static measurement of elastic constants of anisotropic samples is discussed in detail in Ward (1983, Sec. 10.3). It is important to remember that the mechanical anisotropy in well oriented crystalline polymers may be 10:1 or 100:1, much higher than in other classes of materials. Normal intuition may fail in these circumstances. In tensile tests parallel to the stiff direction, nonuniform stresses due to clamping will not die away until a length many times the sample diameter, so long (or very thin) specimens are needed (Folkes and Arridge, 1975). The extensional compliances in the plane of a sheet, s 33 and sll9 are easily determined by cutting samples out in the correct directions. Further progress is not so easy, and defining a full set of nine elastic constants for an anisotropic specimen is a long and complex task. The determination for drawn polyethylene terephthalate) (PET) sheet is described in a series of papers by Ladizesky and Ward (1973), Wilson etal. (1976 a, b, c) and Lewis and Ward (1980 a). The earlier paper demonstrated that the sample has orthotropic symmetry and that time dependance of modulus is very small for this specimen, eliminating one set of experimental problems. The extensional compliance perpendicular to the plane of the sheet s22 requires compression of the sheet. The problem of friction between the sheet and the faces of the compression cell will make the apparent compliance depend on sample size. A detailed analysis of the problem is given in Wilson et al. (1976 c), where it is

313

shown that for the case of PET and steel, friction prevents any lateral motion in the compressed film. As there is no lateral displacement, this measurement is actually one of c33 and not s 33 . Calculation involving the other elastic constants is required to get back to the compliance, and this reduces the accuracy. An alternative has been described by Akindayini et al. (1986), where the sample was put under hydrostatic pressure p, and at the same time a tensile stress of 2p was applied at 45° to x and z. The result is shear in the plane of the film and a tensile compressive stress of p along y. The change in thickness of a 0.5 mm thick film was measured with a Hall effect transducer to give 5 33 with an accuracy of 5-10% at a compressive strain of 0.1%. Figure 7-3 shows the apparatus used. The shear compliances s 44 , s 55 and s 66 are normally determined by torsion of the sheet. The torque required to produce a given twist about z depends on s 44 and s 55 , and the formula depends on the shape of the specimen. Twisting several different shapes about x and about z can then give the shear compliances (Raumann, 1962; Ladizesky and Ward, 1971). Again there are serious end-effects, which can be dealt with by using several sample lengths and extrapolating to infinite length (Folkes and Arridge, 1975). The in-plane Poisson's ratio compliance s 13 was measured optically, by depositing a fine grid on the sheet surface, and measuring its displacement with an optical microscope (Wilson et al., 1976 a). The accuracy of this measurement was 10 |im, a strain of 10~ 4 in a 10 cm sample. Changes of the sheet thickness must be measured to obtain the out-ofplane Poisson's ratio compliances s 23 and s 21 . If the sheet is transparent, the interferometry can be used (Wilson et al., 1976 b). This measures optical thickness, the prod-

314

7 Elastic Properties of Crystalline Polymers

Matrix (1) is an example of S, in units of G P a ~ \ for PET sheet drawn at constant width (Lewis and Ward, 1980 a). Note that as expected for this type of specimen, s22 > Sii > 533- The anisotropies are high, with s22/s33 = 15 and the maximum ratio of compliances s66/s13 almost 1000. The accuracies of measurement also range widely, from 2 - 3 % for the "easy" s339s1± and s 44 to 17% for the difficult s 22 . When the sample is a fiber, any measurement except s 33 and the torsional compliance s 44 can be a problem. The transverse stiffness is measured by compressing a fiber between two plates. The analysis requires that the fibers are circular in cross-section, and the surface constraint gives plane strain conditions. With glass plates and comparatively large monofilaments optical microscopy gives the width of the contact zone and thus the strain under a constant load, Fig. 7-4 (Pinnock et al., 1966). PoisFigure 7-3. Exploded view of the apparatus used to measure the in-plane compliance s22 of oriented sheet, from Akindayini et al. (1986). (A) piston, (B) sample, (C) clamps, (F) Hall effect magnetic field sensor, (G) permanent magnets with like poles adjacent, (H) to (K) locating and alignment devices.

uct of thickness and refractive index, so the stress-optical coefficient must also be determined. Otherwise, sensitive extensometers can be used (Darlington and Saunders, 1979; Richardson and Ward, 1978). These use capacitance or Hall effect transducers to measure displacements of less than

Microscope objective

Monofi lament

Glass blocks

Figure 7-4. Measurement of transverse fiber modulus E±. A fiber is compressed between two glass plates and the contact area is measured in an optical microscope (Pinnock et al., 1966).

0.36 ±0.01 -0.38 ±0.04 -0.018 + 0.001 0 0 0 -0.38 ±0.044 0.9 ±0.16 -0.037 + 0.005 0 0 0 -0.018 ±0.001 -0.037 ±0.005 0.066 + 0.001 0 0 0 0 0 0 9.6 ±0.3 0 0 0 0 0 0 0.56 ±0.02 0 0 0 0 0 0 14 + 0.

(1)

315

7.2 Experimental Determination of Elastic Properties

Driver

Strain Gauge

Load Cell I

Sample I

=H3

D "i

Chamber ' I

son's ratio can also be obtained from the increase in lateral fiber diameter in this case. Using optically flat steel plates to compress fibers laterally in an Instron testing machine, transverse modulus and failure strain could be measured for fibers as small as 8 jim in diameter (Phoenix and Skelton, 1974). Naturally, alignment in this compression test must be very good indeed. 7.2.2 Dynamic Measurements

Dynamic measurements of elastic properties involve oscillatory motion of the specimen. There are many methods and many commercial devices for the purpose. The commonest is where the specimen is put into forced oscillations by a driver similar to a loudspeaker coil at a frequency usually in the range 0.01 -100 Hz. The time scale of the deformation that is being measured is now controlled by this frequency. Figure 7-5 is a schematic of a dynamic mechanical analyzer where the driver and transducers are mounted on an optical bench. In this case the specimen is always tested in tension, but other instruments have a range of sample mounts for tensile, bending (a) or shear deformation (b), Fig. 7-6. The applied displacements and the resulting load are measured in both amplitude and relative phase, giving both an elastic storage modulus and a viscous loss modulus. The procedure is called dynamic mechanical analysis, or since the tests are

Adjust Tension

Figure 7-5. A schematic diagram of the Rheovibron instrument for dynamic mechanical testing in tension.

often run over a range of temperatures, dynamic mechanical thermal analysis, DMTA. In a modern instrument, automatic measurements at several frequencies will be made during the continuous heating or cooling of the sample. When a film or tape is tested in tension, there must be a static tensile load as well as an oscillatory one, to keep the sample in tension. End effects must again be corrected for, and this is done by using several samples of different length. There are many possible sources of error in obtaining a reliable absolute measurement of modulus, including machine compliance, sample slippage and misalignment that also affect static tests. In a dynamic tensile test the static load may change the sample structure, and inertia and resonance in the testing machine have to be considered. Massa

(a)

Ib)

Figure 7-6. Sample mounting arrangements for a more modern DMTA (Polymer Laboratories PLC). These are for tests in dual cantilever bending mode (a), and tension (b). The supports and measurement devices are all on one side, making it easier to design a good temperature controlled environment.

316

7 Elastic Properties of Crystalline Polymers

(1973) and Wedgewood and Seferis (1981) discuss these errors, and methods of dealing with them particularly for the models of the Rheovibron viscoelastometer available at that time. The shear attachments for DMTA instruments are generally designed for the testing of soft elastomers or foams. A comparatively stiff film of crystalline polymer will have insufficient displacement. The dynamic shear of such samples is measured with a free vibration torsion pendulum. One end of a long sample is clamped, while the other end is attached to an inertia bar. The sample is held in light tension, by the weight of the bar or other counterweight and the bar is set into oscillation, Fig. 7-7. The frequency gives the elastic shear modulus and the decay of amplitude gives the loss modulus. The frequency is commonly set to be about 1 Hz by choice of the inertial weights, but can be from 0.01 to 50 Hz. At higher frequencies the wavelength of the stress wave may become comparable to the sample dimension, producing resonant effects.

Counter balance

Inertia rod

Specimen

rflinin

Figure 7-7. Torsion pendulum apparatus for the measurement of G,,. The inverted arrangement makes it simple to place the sample in a temperature controlled bath.

7.2.3 Ultrasonic Measurements Ultrasonic modulus measurements are simple in principle and have many advantages. For unoriented materials and for oriented materials when the acoustic waves propagate down symmetry axes of the sample, the relevant modulus component is simply the density times the wave velocity squared. In an unoriented material longitudinal waves give £, transverse waves give G and so any of the other elastic properties can be calculated. Table 7-1 shows some typical results for semi-crystalline polymers obtained at room temperature and 2 MHz by Hartmann and Jarzynski (1974). Continuous wave propagation methods have been used for the measurement of wave velocity (Kolsky, 1958; Ward, 1983, Sec. 6.5). There the phase of a stress wave of frequency 1-10 kHz is measured as a function of distance along a sample. Alternatively, scanning frequency to find resonance peaks in rod shaped samples relates frequency to wavelength (Davidse et al., 1962). Today pulse propagation at 0 . 1 10 MHz is used, and the modulus may be derived from the group velocity determined from the pulse transit time. Dispersion is low, and phase and group velocity are treated in the same way; difficulties may arise in highly anisotropic materials if the pulse direction is not well defined. Higher frequencies allow faster pulse rise times and so more accurate pulse timing, but attenuation increases with frequency. The sample thickness must be kept low, negating the improvement. In the case of oriented materials, for longitudinal waves propagating down z, c33 = Q v2. If the sample has fiber symmetry and thus transverse isotropy, the velocity of transverse waves along z gives c 44 ; waves traveling along x give c11 and

7.2 Experimental Determination of Elastic Properties

317

Table 7-1. Elastic properties of unoriented semicrystalline polymers, measured ultrasonically (Hartmann and Jarzynski, 1974). Polymer

Nylon 66 Nylon 6 Polypropylene Polyoxymethylene High density polyethylene Poly(vinylidene fluoride)

Bulk modulus (GPa)

Shear modulus (GPa)

Young's modulus (GPa)

Poisson's ratio

6.53 6.45 4.37 6.59 4.54 5.18

1.43 1.43 1.54 1.43 0.91 1.07

3.99 4.00 4.13 4.01 2.55 3.00

0.40 0.40 0.34 0.40 0.41 0.40

[through c 66 , Eq. (7-3)] c 12 . The fifth elastic constant, c 13 , can be obtained from a velocity measurement at a known angle to z. Disks a few millimeters thick may be cut from an oriented sample rod and bonded between pairs of transducers (Rider and Watkinson, 1978; Leung et al., 1987), Fig. 7-8. This requires a large initial specimen, and in a general anisotropic material defines the direction of wave propagation rather than the direction of pulse propagation. Alternatively, a rectangular sample block of known dimensions is immersed in a liquid such as water or silicone oil. Then the transducers are also immersed and do not need to be bonded directly to the sample, so many arrangements are possible (Rawson and Rider, 1974; Read and Dean, 1978; Leung et al., 1980). In one setup, the transmitter and receiver are pointed directly at one another, and the sample is mounted between them on a turntable. The extra pulse delay when the sample is inserted in the path gives the velocity. Rotating the sample on the turntable allows continuously varying propagation directions, and thus curve fitting with the elastic constants as variables rather than their derivation from a few results. Refraction at the sample surface has to be taken into account in the calculation. The arrangement can be further simplified

with a reflecting surface behind the sample, so that one transducer acts as transmitter and receiver. A large number of measurements may be required, but automated systems may be used. An automated ultrasonic system was used by Habeger (1990) to measure all in-plane moduli as a test for orthotropic symmetry. If the velocity of sound in the immersion liquid is known, the velocities in the sample can be determined without any timing measurements (Treloar, 1970). The transducers are set to measure reflectivity. For this the transmitter and receiver move so

Figure 7-8. Schematic diagram showing samples cut from a polymer rod or tube with transverse isotropy for the ultrasonic measurement of moduli. A is used for c 33 , B for c 11? c 44 and c 66 , and C for c 13 (Leung etal., 1987).

318

7 Elastic Properties of Crystalline Polymers

that the angle of incidence and the detected angle of reflection from a sample surface remain equal as they vary (as in an X-ray diffractometer). At the critical angle of incidence the reflected wave reaches a maximum constant amplitude. This would be 100% of the incident amplitude, if there were no losses in the system. There are two such angular positions 6L and 9T corresponding to longitudinal and transverse wave propagation in the specimen. At each position, v/v0 = 1/sin 9, where v0 is the velocity of sound propagation in the liquid. In optical terms, which may be more familiar, this is equivalent to the determination of refractive index by measuring the critical angle for total internal reflection. More recently methods have been developed which do not require sectioning or immersion of the sample; pulsed lasers are used as point sources of ultrasonic waves. Lasers can also be used as interferometric detectors of the displacements due to ultrasonic pulses (Hutchins, 1988; Sachse and Hsu, 1989). These optical noncontact techniques remove the need for immersion of the specimen. In anisotropic sheet specimens the pulse propagation direction is defined by the vector from the point source to a small transducer. The laser is scanned over the sheet to give a range of directions and as both longitudinal and shear waves are detected, the result is a full set of elastic constants (Castagnede et al., 1991), Fig. 7-9. These techniques have been applied to polymer matrix composites (Sachse et al., 1990, 1991), but not yet to oriented crystalline polymers. Measurement of pulse propagation through the thickness of fine fibers or films less than 1 mm thick requires too much accuracy in timing, but the longitudinal properties of fibers can be obtained. Again a pulsed laser beam is focused onto the specimen and the thermal pulse creates a

Composite Specimen Receiver Scanned Ultrasonic Point-source

Pulsed .' X Laser Source

(on rear surface)

fx 2

Figure 7-9. Schematic of the use of a laser beam as a moving point source of ultrasonic pulses. The detector is of small area, and at a fixed position on the rear surface of the sample (from Sachse et al., 1991).

shock wave that moves along the fiber (Smith et al., 1987). Timing the shock wave arrival at the end of the long fiber gives the fiber modulus. The sample may be held at high temperature or in a controlled atmosphere during the test. 7.2.4 High Speed Testing

High speed testing machines use hydraulics or compressed air to move the cross head and electronic data storage to record the data. They can be used for tensile tests on specimens at strain rates of 10-100 s" 1 . In ballistic impact testing a projectile is fired at the sample, and its deformation is recorded by high speed photography or other means. If the geometry is simple, the modulus can be calculated from the record of the deformation of the sample. One example of simple geometry is a semicylindrical projectile incident perpendicular to a fiber under tension (Smith etal., 1960; Prevorsek et al., 1991). Strain rates of over 1000 s" 1 are achieved, roughly equivalent to strains of 0.01% at a frequency of 10 MHz in ultrasonic testing. High speed testing and ballistic testing are both a lot more trouble than ultrasonic

7.3 Experimental Determination of Crystal Modulus

testing, so they are used primarily to obtain data relating to failure, strength and toughness, and not elastic properties.

7.3 Experimental Determination of Crystal Modulus 7.3.1 X-Ray Diffraction Wide angle X-ray diffraction spots appear at diffraction angles 2 0 given by Bragg's Law, 2 d sin 9 = A, where X is the X-ray wavelength and d is the spacing of the crystal lattice planes. When the crystal is elastically strained during diffraction, d-+d + §d and differentiating Bragg's Law gives

^ = e

= - c o t 0-50

(7-7)

Thus measurement of the change of the diffraction angle of reflections under load gives £c, the crystal lattice strain, directly. If possible a reflecting plane that has its normal along the chain axis, such as (002) in polyethylene (PE), is chosen to give the strain in the chain direction. Other reflections such as (200) in PE can be used to investigate strains perpendicular to the chains. In principle these measurements can be made on either isotropic or oriented material with any stress state, but simple tension on a highly oriented material is almost always used. Parallel alignment of chains makes interpretation of the results feasible, and tension along the chain axis has the important practical advantage that the sample is stronger. It can then withstand larger stresses without fracture or plastic flow, and the shift of the X-ray reflection is larger and easier to measure. To determine the crystal modulus in the chain direction E% (simply called Ec from

319

now on) we need the crystal stress ac as well as sc. Experimentally the macroscopic sample stress as is measured and in principle oc is not known. Some structural model for the sample structure or some assumption must be used to calculate oc from the sample stress as. The simplest possible assumption is that these stresses are equal. In that case the ratio oj&c is the crystal modulus. The structure required to give this result of uniform or homogeneous stress is one where the crystals and any disordered material are mechanically in series. The literature is strongly divided on the validity of this assumption. On one hand some authors use the uniform stress assumption and say that X-ray diffraction determines the crystal modulus. On the other are those who do not accept the assumption and take (jjsc as an apparent modulus E app , which gives information about the structure of the material. They may use values of Ec determined by other methods, or combine X-ray diffraction with other data to determine Ec from Eapp. The basic argument in favor of the uniform stress assumption is that it gives consistent values for Ec even when samples with very different macroscopic moduli are used. There is further discussion of this point in Sec. 7.5. To measure crystal modulus by X-ray diffraction a drawn sample or bundle of fibers is mounted in a straining frame horizontally on an X-ray diffractometer, Fig. 7-10. Measurements of this type were first made by Dulmage and Contois (1958). Many materials have since been studied by Sakurada and his co-workers (Sakurada et al., 1962,1964,1966; Sakurada and Kaji, 1970) and more recently by Clements et al. (1978, 1979), Nakamae et al. (1987 a, 1990, 1991) and Matsuo and Sawatari (1986, 1988 a, 1988 b, 1990). The polymer sample is stretched under dead loading, using a pulley to transfer the load to the horizontal

320

7 Elastic Properties of Crystalline Polymers

(a)

— to Sealer

(b)

Figure 7-10. (a) Top view of a horizontal diffractometer with stretching frame mounted on it, for the measurement of X-ray modulus. The load is applied from a weight W through a pulley and cable to the sample C. (b) Simple stretching frame used in (a). Clamps (1) hold fiber sample (2); extension indicated on length scale (3) (Sakurada et al., 1962).

sample. The more recent papers listed measure the crystal modulus as a function of temperature. The accuracy of strain measurements is high. The error in determining 2 9 was first given as one minute of arc, ±0.0166° (Sakurada et al., 1962, 1966). In a more recent paper an error of + 0.005° is quoted for the case of PE (Nakamae et al., 1991). The PE (002) reflection, used to determine the strain parallel to the molecular chain,

has a lattice spacing d = 0.127 nm. With an X-ray wavelength of the usual value, X = 0.154 nm, 2 9 = 74°, 2 tan 9 = 1.5 and a 0.5% lattice strain gives a change of diffracted angle of 7.5 mrad or 0.4°. A 0.005° error in 2 9 then corresponds to a strain error of ± 5 x 10~5. The line width of the (002) reflection is normally from 0.1 to 1°, depending on the size and perfection of the crystals in the sample and the collimation and wavelength spread of the X-ray beam. Thus the position of the center of the reflection is measured to an accuracy 2 0 200 times better than the width of the reflection. The total shift is only of the same order as the linewidth. Using a double focused monochromatic X-ray beam from a powerful synchrotron source, the same type of experiment has been conducted with a single high modulus fiber as the sample. Poly(p-phenylene terephthalamide) [PPTA or Kevlar (DuPont)] and high modulus polyethylene have been used, with sample diameter as low as 12 jum (Prasad and Grubb, 1990; Li etal, 1991). Figure 7-11 shows the (002) reflection from a Spectra 1000 (Allied Signal) polyethylene sample at low and high stress. In this case the deconvoluted linewidth at low stress was less than 0.1°, and curve fitting defined the centroid of the peaks to better than ±0.001°. However, this corresponds to a positional accuracy for the reflection of ± 3 jim, and if the sample were to move, so would the reflection. Thus the true accuracy of strain measurement depends on the stability of the sample and its holder under load. This also holds for the experiments carried out on larger samples using a normal diffractometer. Matsuo and Sawatari show in Fig. 1 of Matsuo and Sawatari (1988 a) that the observed crystal strain is proportional to the applied stress, even when there is some misalignment or instability of sample posi-

7.3 Experimental Determination of Crystal Modulus

321

2.05 GPa 1.62 GPa 1.14 GPa

0.56 GPa

Figure 7-11. Polyethylene (002) reflection from a single Spectra 1000 fiber (28 (im in diameter) showing the shift to lower 2 0 and broadening as stress increases up to 1 GPa, then no change (Prasad and Grubb, 1989).

0.00 GPa V9 73.0

73.5

74.0

74.5 2©B, degrees

75.0

75.5

tion under load. Unexpected changes in diffracted intensity signaled this misalignment, and allowed the authors to discard the poor data. Without this care, values for the crystal modulus ranging from 86 to 374 GPa could have been derived, an error range of ± 60%. Not all materials are as convenient as PE. PET, for example, has no strong reflection parallel to the chain direction in the crystals. Instead the (105) reflection must be used, which is 11 ° away from the chain axis (Sakurada et al, 1966; Thistlethwaite etal., 1988). The situation is similar in f-polypropylene (iPP), where the reflection nearest to the meridian is the (T13). The modulus is measured along the (113) plane normals, and a complicated mathematical treatment is used to recover the chain modulus (Matsuo and Saw atari, 1986; Sawatari and Matsuo, 1986). The ratio ojsc varies as a function of temperature in some specimens and this may be interpreted directly as a real change in Ec. Significant temperature variations in £ c , more than that merely caused by thermal expansion of the lattice, may be expected in high modulus polymers. These

76.0

materials, PPTA, for example, contain chains extended to their maximum stable end-to-end distance, so that any bond rotation reduces the end-to-end distance of the chain. On heating, thermal excitation of torsional vibrations makes the chain become more of an open helix, on average. The thermal expansion coefficient in the chain direction of such an extended chain is thus negative. Some low energy bond rotations can extend these thermally contracted chains, so their extensional modulus is less than that of the original extended chains and Ec falls as temperature increases. This interpretation has been made for several materials, for example PE (Nakamae etal., 1987b; Nishino etal., 1992), PPTA (Nakamae etal., 1986a; Tadaoki et al., 1987), and polyoxymethylene (POM) (Nakamae et al, 1990); results are summarized in Nakamae and Nishino (1991). In polytetrafluoroethylene (PTFE) changes of Ec with temperature are more specifically associated with phase transitions of the chain. As the temperature increases the PTFE chain conformation in the crystals changes from a 13/6 helix to a 15/7 and then to an irregular helix. Fig-

322

7 Elastic Properties of Crystalline Polymers

200

150 -

100 -

- 1.300

ttf)

'o

-100

0

100

Temperature,

200

1.295

c

Figure 7-12. Modulus in the chain direction for polytetrafluoroethylene as determined by X-ray diffraction. The changes as a function of temperature are associated with changes of chain conformation, indicated by the meridional d-spacing changes shown in the lower plot (Nakamae et al., 1986 c).

ure 7-12 shows the modulus changes and the changes in meridional X-ray spacing for PTFE (Nakamae et al., 1986 c). For para-linked aromatic polymer chains where every main chain single bond is collinear, bond rotations have no effect on chain length. In principle, thermal excitation of chain bending vibrations can contract the chain and reduce the extensional modulus. As bond angle changes require a lot more energy than bond rotations, these effects will be small, and Ec will be affected by heating primarily through transverse thermal expansion of the interchain distance. If the uniform stress assumption is not made, temperature variation of ajgc may be due to changes of crystallinity or to changes in £ a , the stiffness of the disordered component. Increase of £ a p p at low temperature is then associated with an increase of Ec and Ea in PE (Clements et al.,

1978,1979) and in POM (Brew et al., 1979; Jungnitz et al., 1986; Wu et al., 1989). The ratio
7.3 Experimental Determination of Crystal Modulus (a)

2.0

0.15

r /•

-

GPa

• Loading O Unloading

0.10

1.5 -

w" CD /

1.0 _

/ / • y^

S

led

co

323

Hi* 0.5

0.00

100

200

300

400

500

Stress a, MPa

nn

a

/

0.0

i

i

i

i

0.1

0.2

0.3

0.4

0.5

Crystal strain, % Figure 7-14. Crystal strain vs. sample stress for a high modulus PE fiber (Spectra 1000, Allied Corp.). This plot of X-ray crystal strain is plotted as a normal stress-strain curve, where a modulus is a slope (opposite to the plots of Fig. 7-13). Derived from data like that in Fig. 7-11, it shows no change in apparent crystal modulus until a sample stress of about 1 GPa. Above this stress and up to 2 GPa, ec remains at 0.5%. This would imply an infinite crystal modulus on the uniform stress assumption. The transition is less sharp on unloading (Prasad and Grubb, 1989).

(b) 1.5

1.0

o

0.0

100

200

300

400

500 600

Stress a, MPa Figure 7-13. Plots of crystal strain vs. sample stress for (a) PE and (b) POM. (a) Results and fit from Sakurada et al. (1966); + and o are from different specimens. The apparent crystal modulus increases from 235 to almost 400 GPa at a stress of about 150 MPa. (b) Combined data from Sakurada et al. (1966) (+) and from Nakamae et al. (1990) (o). The apparent crystal modulus decreases from 52 to 29 GPa at a stress of about 250 MPa.

tion. Although the samples are then much weaker, the transverse crystal moduli are also much smaller, so that the lower loads produce measurable crystal strains. Interpretation using the homogeneous stress

assumption is more doubtful, as there is neither a series structure in the samples, nor sufficient information to use consistency of results as an argument. Poisson's ratio measurements of the changes in lateral crystal dimensions under an axial load are straightforward but rarely reported (Slutsker et al., 1986; Li et al., 1991); such measurements are difficult to interpret quantitatively. 7.3.2 Raman Spectroscopy

In Raman spectroscopy monochromatic light is scattered by the specimen. The spectrum of the scattered light shows that most has the same wavelength as the incident light, and a small fraction has a slightly different wavelength. The energy

324

7 Elastic Properties of Crystalline Polymers

change of this inelastic or Raman scattered light is a quantum of molecular vibrational energy, so the vibrational frequency spectrum is the result. Most Raman spectrometry is used for chemical analysis by recognition of the spectra. One vibrational mode of a polymer chain that can be excited is the longitudinal acoustic mode (LAM), frequency vm, where all motion is parallel to the chain axis. Figure 7-15 shows the first mode, m — \ (LAM-1) where both the ends move toward the center and then both move away from the center, which remains stationary. The figure also shows the 2nd mode, m = 2 where there are two stationary points. The wavelength Xm of the first mode is twice the chain length. If the frequency is known from spectroscopy and the wavelength from the chain length, the velocity of sound v is known. This gives £ c , since if the chain is free of interchain interactions and QC is the chain density,

Raman bands corresponding to the vibration of methyl groups and other small components of the molecule may be hundreds or thousands of wavenumbers from the exciting line. Because the wavelengths of LAM vibrations are so much longer, peaks may be only 10 cm" 1 from the exciting line if Ec is not very high, as in the example shown in Fig. 7-16. With modern laser sources and monochromators, these peaks can be measured even when they are very weak. The polymer chain must have a regular conformation if it is to contribute to a well defined LAM peak and the effective chain lengths in the specimen must all be similar. Thus a well defined Raman LAM peak indicates that the chains included in the measurement are chains within crystals of uniform size. How is the effective length of the vibrating chain defined? In polymers it is the length of the chain passing through the crystals. The problem of defining the exact

(7-8)

295K

^

m=l A=2L

•

_

_

—

:

—

—

^

^

^

—

o\ tT No' No^ scf V \ P\ P\ P\ P\ P\ P

c re

ra CC

m=2

:

•

^

cAA

"

^

"

^

:

) Q p. p pip o c O : o ;o o o 6 o o

Figure 7-15. The longitudinal acoustic modes of vibration of a polyethylene chain. The figure shows a schematic of the chain of carbon atoms, the displacement and the distorted chain for the first and second modes. In the first mode there is one stationary point at the center of the chain. In the second mode there are two at 1/4 and 3/4 of the length, and the chain length remains constant.

25

15

5 1

Frequency (cm~ )

Figure 7-16. First order longitudinal acoustic mode (LAM-1) Raman spectrum from polyoxymethylene, at about 12cm" 1 . There is little or no change in frequency on cooling from room temperature to 173 K (Fanconi and Rabolt, 1985).

325

7.3 Experimental Determination of Crystal Modulus 600

0.4

Figure 7-17. Frequency of LAM Raman bands for n-alkanes. The variable on the x-axis, m/n, is proportional to 1/(wavelength), m is the number of nodes along the chain, length L, so X = 2 L/m. n is the number of carbon atoms in the alkane, and is thus proportional to L. The slope at long wavelength (small m/n) gives the chain modulus (Shauffele and Shimanouchi, 1967).

position of the surface or correcting for the partially disordered surface layers is a difficult one (Hsu and Krimm, 1976). In oligomers the chain length in the crystal is the total length of the chain. The best method of determining chain modulus Ec is thus to use a range of monodisperse oligomers, if possible. Using a range of nalkanes (and higher modes of the LAM vibration) Shauffele and Shimanouchi (1967) obtained 358 GPa as the chain modulus of PE, Fig. 7-17. This result was obtained by use of Eq. (7-4), which takes the chains to be free rods. Later work showed this to be inaccurate. Strobl and Eckel (1976) corrected for the effects of the adjacent molecular layers in n-alkanes and obtained 290 GPa. The calculation was refined by Kobayashi et al. (1983) who used band splitting in a particular crystal form of n-alkanes to determine

the interaction strength, and arrived at 281 GPa. Oligomers of polyethylene oxide) (PEO) have also been used to determine the chain modulus of the polymer, Fig. 7-18 (Viras etal., 1983; Song and Krimm, 1990). Applying Eq. (7-4) to the oligomer Raman frequencies gives Ec = 72 GPa. Corrections for perturbations to the free rod reduce this to 22 GPa (Viras et al., 1983), in agreement with results using the lamellar thickness of the polymer (Hartley et al., 1976). Using detailed molecular modeling of the interactions and normal mode analysis Song and Krimm (1990) obtain 13 GPa for a free chain. This is in agreement with theoretical results for a free chain (Matsuura and Miyazawa, 1969) and close to the X-ray results of 10 GPa (Sakurada et al., 1966). 60

r

0.0

0.2

0.4

0.6

0.8

1/Length, nm"1

Figure 7-18. Raman LAM frequency vs. reciprocal chain length for polyethylene oxide), (o) Experimental data for oligomers, 9-mer to 35-mer. (+) Calculated frequencies for free chains, using force fields that fit the Raman data when full interchain interaction in the crystal is included. The slope of the line through the calculated data gives a modulus close to that determined by X-ray diffraction. [Data from Viras et al. (1983), diagram and calculations from Song and Krimm (1990)].

326

7 Elastic Properties of Crystalline Polymers

Raman scattering in the LAM mode from long chains becomes very close to Brillouin scattering, where acoustic modes of the whole sample are excited. To measure these very small frequency shifts from the incident beam frequency, a FabryPerot interferometer is used as a dispersion device, in place of the gratings used in Raman spectroscopy. Brillouin spectroscopy has been used to measure polymer chain modulus (Rabolt and Wang, 1983) and to measure the anisotropic elastic properties of films less than 1 jam thick (Krueger et al., 1987).

500

r

'0.0

0.2

0.4

0.6

0.8

1.0

Reduced Wavevector along [001]

7.3.3 Inelastic Neutron Scattering

This is a spectroscopic method for determining modulus, similar in basic principle to the Raman method (White, 1970). A monochromatic collimated beam of neutrons is allowed to fall on a well oriented polymer sample. The energy spectrum of the scattered neutrons is determined by spectrometry or by time-of-flight measurement at a number of scattering angles. Some neutrons lose energy by exciting lattice vibrations (phonons). The energy loss gives the energy and thus frequency of the phonon, while the angle of scattering relates to the momentum transfer and thus the wavelength of the phonon. If phonon energy is plotted as a function of 1/wavelength, Fig. 7-19, the slope as (1/wavelength) goes to zero gives the modulus. This is simply because infinite wavelengths corresponds to a uniform extension of the material. Coherent inelastic scattering is used, and this is strong when the conditions for elastic crystalline diffraction are met. The results then relate to the crystalline component and not to the whole sample. This very short description of the principle of the technique avoids many issues of

Figure 7-19. Dispersion curve, phonon energy plotted as a function of reduced wavevector (oc 1/wavelength), for deuterated polyethylene. The sample is oriented and the wavevector is along c, the chain axis. (After Twistleton and White, 1972.)

measurement. For example, the high incoherent scattering cross-section of hydrogen means that deuterated PE must be used in these experiments. PTFE does not have this problem. For more detailed information, see the reviews of Holliday and White (1971) and White (1970), or the original papers where the crystal modulus was determined for PE (Feldkamp et al., 1968; Twistleton and White, 1972) and for PTFE (Koenig and Boerio, 1969; La Garde et al., 1969). These results and a comparison with other methods are shown in Table 7-2 and Fig. 7-22. All neutron scattering measurements concerned moduli in the chain direction except that Twistleton and White (1972) also determined a transverse modulus for PE to be 6 GPa. 7.3.4 Macroscopic Polymer Crystals

There is one class of polymers for which the crystal modulus is more accessible experimentally. These are single crystal poly-

7.3 Experimental Determination of Crystal Modulus

327

Table 7-2. Numerical values for the Young's modulus in the chain direction for crystals of various polymers. Polymer

Polyethylene

PTFE Polyethylene oxide) Polyoxymethylene i-Polypropylene

Polydiacetylene

Crystal modulus Ec (GPa)

X-ray

Raman

235a 235 d 266 h 220l

340 b 358 e 290* 281m

156°

203 p 22 r

10°

Neutron scattering

Theory

329 329 j

182C 340 8 285 k 351 n

222 q

160 g

f

13s

73 w

189U 146X

150 v 109 y

35 h

88V

30°

Direct

Brillouin

Theory

43 q

47«

50 p

54°

a

Sakurada et al. (1962); b Miszushima and Shimanouchi (1949); c Treloar (1960a); d Nakamae et al. (1991); Shauffele and Shimanouchi (1967);f Feldkamp et al. (1969);8 Shimanouchi et al. (1962);h Clements et al. (1979); 1 Strobl and Eckel (1976); j Twistleton and White (1972); k Frank (1970); l Matsuo and Sawatari (1986); m Kobayashi et al. (1983); n Sorenson et al. (1988); ° Sakurada et al. (1966); p Rabolt and Fanconi (1977a); q La Garde et al. (1969); r Hartley et al. (1976); s Matsuura and Miyazawa (1969); l after removing chain interactions: Song and Krimm (1990); u Rabolt and Fanconi (1977 b); v Asahina and Enomoto (1962); w Nakamae et al. (1990); x Runt et al. (1987);y Tashiro et al. (1988);z Wu et al. (1989); a Enkelmann et al. (1980); p Batcheldor and Bloor (1979). e

mers produced by topotactic polymerization of a single crystal of monomer. Production and properties of these have been reviewed by Young (1987). Most are poly(diacetylenes) and the single crystals obtained by solid state polymerization are of size up to a few cm long and 100 |im wide (Wegner, 1977). The molecular chains run the length of these needle shaped crystals, so macroscopic mechanical testing can give Ec directly (Galiotis et al., 1984; Wu et al., 1989). The results are in good agreement with theory (Batcheldor and Bloor, 1979) and with measurement using Brillouin scattering (Enkelmann et al., 1980)

or ultrasonic methods (Rehwald et al., 1983). This is confirmation of the theoretical methods used, but is not much help in understanding other polymers with a more complex microstructure. There is no question of inhomogeneous stress here, as the sample is truly a single crystal. The spectroscopic result is slightly higher than the quasistatic measurement (see Table 7-2). These polymer single crystals have been used as fibers in model polymer composites, where Raman spectroscopy is used to determine the strain distribution along the fiber (Galiotis et al., 1984; Robinson etal., 1987, 1991).

328

7 Elastic Properties of Crystalline Polymers

7.4 Theoretical Modeling of Elastic Constants 7.4.1 General Introduction

In principle the elastic constant matrix of a material can be derived from the fundamental atomic properties of the constituent elements using quantum mechanics. Until recently the rigorous calculations for this were impractical if there were more than a few atoms in the system. The first theoretical calculations of modulus used empirical values for the forces between atoms, not values derived from quantum mechanics. Methods of this type are now part of molecular dynamics. Next used were approximate quantum mechanical methods where certain interaction integrals are ignored. These are called "semi-empirical" calculations. Improvements in computing power and in methods that reduce the effective number of independent atoms now allow the rigorous "ab initio" quantum mechanical calculations to be applied to polymer chains, but derivation of elastic constants in this way is not common. Nonbonded intermolecular interactions cannot be dealt with by any of the quantum mechanical methods. Thus a molecular dynamics method, or some hybrid, has to be used if the elastic properties of a crystal are to be predicted. The quantum mechanical calculations are restricted to a single polymer chain, and thus to the chain modulus. What is more, most of the moduli derived from quantum mechanics are too large, outside the range of experimental results. For both of these reasons the empirical valence force field methods are still in active use, not yet superseded by the more fundamentally based methods.

Simple qualitative considerations can give a very approximate Young's modulus, predict comparative moduli between similar polymers, or indicate the level of mechanical anisotropy expected in a polymer crystal structure. They should not be forgotten during the discussion of detailed quantitative calculations that follows. These basic considerations are (1) Bond strengths increase strongly in the order van der Waals bonds (or dispersion forces), hydrogen bonds, covalent bonds. (2) The force required to distort a bond increases as the distortion changes from bond rotation to bond bending (valence angle changes) to bond stretching; roughly by a factor of ten at each step (Dulmage and Contois, 1958). (3) The two factors above control chain stiffness. The polymer modulus in the chain axis direction will also be proportional to the number of chains per unit area; that is, a chain with side groups having a large cross-sectional area will have a smaller Young's modulus. Thus polymers with high modulus in the chain direction require at least a planar chain (to eliminate bond rotation) and a small cross sectional area. If the main chain bonds are close to the chain direction (as in p-linked aromatics) then bond bending becomes less effective in extending the chain, and it will be stiffer. Hydrogen bonding between chains increases the transverse modulus in linear chain crystals. 7.4.2 Molecular Dynamics Empirical Force Fields

This common method of calculation of elastic constants is firmly based in experimental data. A crystalline unit cell geometry experimentally derived from X-ray dif-

7.4 Theoretical Modeling of Elastic Constants

fraction is normally used, if available, and interatomic potentials are derived from the infrared and Raman spectra of model compounds. The frequencies of bands in these vibrational spectra depend on the restoring force to a particular local motion of the polymer. Thus if the bands are correctly assigned to stretching, wagging or twisting motions of various groups of atoms, the force required to stretch, bend or rotate the relevant bonds is known. Strictly these are not really independent theoretical calculations of the elastic properties at all, as the vibrational spectra are also elastic properties, they are transformation of one measurement into another. Complex calculations are needed to turn the measured vibrational results into crystal moduli. If the crystalline unit has high symmetry, mathematical techniques that originated in spectroscopy may be used to reduce the number of independent variables. The geometrical structure may be found by optimization to a minimum energy form using the same interatomic potentials. In complex systems such as polymers with a large number of atoms in the unit cell, the minimization of potential energy is itself a difficult problem. A calculation may fail to give the correct configuration because it has got stuck in a local energy minimum. It is common then to use some combination of an approximate initial chain geometry with final refining by minimization. The early derivations of modulus considered single polymer chains, and initially just the carbon atom backbone. When the chain conformation is all-trans as in polyethylene, or some other planar arrangement of backbone bonds, then bond rotation will not be effective in producing a tensile strain. The simplest calculations of Young's modulus then ignored all but main chain bond stretching and main chain bond bending. There are only these two

329

types of force constant required, so the method can be described as a two-constant valence force field. This was first used by Meyer and Lotmar (1936) for cellulose. For polyethylene, there is only one type of bond, so the deformations considered are only the C-C stretch and C - C - C bend. Treloar (1960 a) has a particularly clear presentation of the simple analysis. This is summarized below, as it serves as a model for all the later more complicated analyses. Figure 7-20 shows the geometry of the carbon backbone of part of an all-trans polyethylene chain. The C - C - C bond and angle a is 112° so that each bond is at 0 = ( 1 8 0 - a)/2 = 34° to the chain axis. The applied force F acts along the chain axis and thus at the mid points of the bonds. k±, the C-C stretching force constant, is defined as /x/5/, where fx is the force acting along the bond and 5/ its change of length. The C 1 - C 2 - C 3 bending force constant may be expressed in angular terms, fca = torque/5a, or as fcp = / p /5/ p where fp is the force applied at C 3 , perpendicular to C 2 -C 3 and in the plane of C x C 2 - C 3 , and 5/p is the perpendicular displacement of C 3 . For a bond length /, the torque is lfp and 5/p = Ida so ka = I2kp. The component off acting to extend the bonds fx= F cos , and the increase in

Figure 7-20. C x , C 2 and C 3 are three carbon atoms in the all-trans backbone of polyethylene. It is being extended by a force F acting along the chain axis. The C - C - C bond angle is oc. /, and fp are forces acting along the bond and perpendicular to it, used in defining the force constants. (After Treloar, 1960 a.)

330

7 Elastic Properties of Crystalline Polymers

chain length along F, SL l9 due to this will be 8L1 = 81 cos (j) = ^— k

(7-9)

The couple on the valence angle is (F//2)sin0, since F acts half way along the bond. The change in angle, 8a, is [F//(2fca)]sin(/>. The change in bond inclination, 5c/>, is Sa/2 and the effect of 80 on the chain length, 8La is Fl2 8La = 8(j) I sin 4> = —— sin2 <j> = 4/c (7-10) The total chain extension 8L is just the sum of the two expressions above and Young's modulus of the chain is given by E=

F/A 5L/L 4/Ci/CpL

A (4 fcpcos2 4> + k1 sin 2

(7-11)

where A and L are the chain area and unit length, obtained from X-ray diffraction. With A = 0.182 nm2, L-0.127 nm (per bond), fc1 = 436N/m and fcp = 35 N/m (from Rasmussen, 1948), Treloar obtained Ec = 182 GPa (Treloar, 1960 a). The simple two-constant method was also applied to Nylon 66, PET (Treloar, 1960 b), cellulose (Treloar, 1960 c) and poly(phenylene sulfide) (Unwin and Ward, 1988). Having to sum over rings and include different atoms in the main chain and different bonds rapidly increases the complexity of even the simplest calculations for planar chains. The limitations of this method were realized, and soon more complex force fields were used. Urey-Bradley force fields including effects of bond twisting and interatomic repulsions were used for polyethylene and

a number of nonplanar chains such as i-polypropylene and polyoxymethylene (Shimanouchi et al., 1962; Asahina and Enomoto, 1962). Holliday and White (1971) contains a summary of this work. The result obtained for polyethylene was 340 GPa, now recognized as high, but at the time agreeing with Raman LAM data. A major advance was the inclusion of intermolecular interactions, initially for PE with the methylene groups treated as single units (Enomoto and Asahina, 1964; Miyazawa and Kitagawa, 1964; Anand, 1967) and then with full H - H and C - H interactions for bonded and nonbonded atoms. Odajima and Maeda (1966) used Lennard-Jones and other potentials for the intermolecular interactions to derive the complete elastic constant matrix C for polyethylene. Ec was found to be 257 GPa, and intermolecular effects on this parameter were only 0.2%. Recent advances have been in techniques for dealing with crystals containing large numbers of atoms. Keeping track of co-ordinate system transformations between the bond motions, the unit cell and Cartesian axes is a problem in complex systems. As in normal mode analysis of vibrational spectra, the use of symmetry relations is important in simplifying the treatment. Tashiro et al. (1978 a) use the B matrix method (Shiro and Miyazawa, 1971) but enter the space group symmetry of the crystal at the beginning of the calculations. An orthorhombic system with an elastic constant matrix as in Eq. (7-2) is factorized into four blocks, one 3 x 3 and three l x l blocks for the remaining diagonal terms. The result is that it is not necessary to consider all the atoms in a unit cell, as one asymmetric unit implies all the others. Applied to polyethylene the result for the compliance matrix is given by Tashiro etal. (1978 b) as

7.4 Theoretical Modeling of Elastic Constants

0.145 0.0478 0.00019 0 0 0

-0.0478 0.117 -0.00062 0 0 0

-0.00019 -0.00062 0.00317 0 0 0

The a form crystal structure of nylon 6 contains 76 atoms in one asymmetric unit, and using this method would require matrices of order 230. Tashiro and Tadokoro (1981) found it necessary to approximate the crystal structure, in a manner chosen to leave nearest neighbors relatively unaffected. For a qualitative view of the transverse moduli, a graphical representation as in Fig. 7-21 is more effective than a matrix of elastic constants or compliances. The large transverse modulus associated with hydrogen bonding can be seen in nylon and in poly(vinyl alcohol), while polyethylene is closer to transverse isotropy. As computer power increases, more complex systems can be dealt with, and recently full elastic constant matrices for cellulose I and cellulose II have been reported (Tashiro and Kobayashi, 1991). If the unit cell is too large for a full crystal calculation or if the polymer lacks full three-dimensional order a molecular dynamics calculation of single chain stiffness

0 0 0 0 0 0 0.313 0 0 0.618 0 0

0 0 0 0 0 0.2

331

(7-12)

is very useful. Crystal symmetry or the helical symmetry of the chain can be used to simplify the calculation, even when intermolecular effects are neglected (Manley and Martin, 1973; Tashiro et al., 1977 b). Calculations of single chain properties often refer to high modulus materials such as aromatic polyamides (Tashiro et al., 1977 a) or poly(p-phenylene benzobisoxazole) (PBO) (Tashiro and Kobayashi, 1991). It is technologically interesting to know how close practical high modulus materials are to the theoretical limit, and for these materials intermolecular effects on the modulus along the chain direction are likely to be small. The methods outlined in the references above, which once involved a considerable effort in computation are becoming available in commercial packages running on inexpensive workstations. With a given molecular configuration, crystal parameters are rapidly derived by energy minimization and a touch of a button will gen-

PVA

nylon 6

P

'

0

IOGPQ

Figure 7-21. Graphical representation of the transverse modulus of crystals of polyethylene, poly(vinyl alcohol) and a and y forms of Nylon 6, to the same scale, superimposed on drawings of the projected molecular chains of the crystals (Tashiro and Tadokoro, 1981).

332

7 Elastic Properties of Crystalline Polymers

erate an elastic constant matrix (Yang and Hsu, 1991). The atomic force constants used are derived from a wide range of spectroscopic studies. However, there is a choice of constants and intermolecular potentials to be made by the programmer or the user. If these empirical choices were made on one set of compounds and the calculations are made on a different type, there may be doubts about the accuracy of the result. The uncertainty may be hidden to the casual user by the precision and computer presentation of the results. Interatomic potentials can be used to calculate many properties of the material, so if experimental data is available it is possible to check the calculations. The observed chain conformation and crystal structure should correspond to an energy minimum for the system if the correct potential function is being used. The vibrational spectrum of the polymer should be accurately predicted by the same potentials as are used for the modulus. The heat capacity at low temperature can also be derived from the long wavelength phonon spectrum. 7.4.3 Molecular Orbitals Quantum Mechanical Calculations

The accuracy required of the quantum mechanical calculations of modulus is very great. The energy of a covalent bond is very small compared to the total energy binding the electrons in an atom. Mechanical displacement corresponds to phonon excitation, a small perturbation of the covalent bonds. Molecular orbital calculations are more often concerned with bond stability and with electronic excitations, than with mechanical properties. A basis set of atomic orbitals is chosen, and these are combined to form the total electronic configuration (LCAO - linear

combination of atomic orbitals). For a given geometry of the chain, the combination that gives the wave function with the lowest energy is found. The geometry is altered and the energy recalculated. This gives the geometry of minimum energy. If a crystalline chain conformation is known, it acts as a check on the procedure so far. The minimum energy conformation had better be very close to that found experimentally. Next the length of the chain can be increased by some amount - by a small amount if Young's modulus is sought, by a large amount if it is the breaking strength that is to be calculated. All other geometrical parameters of the chain are again allowed to vary, and a new minimum energy found. The increase in energy on extension is the work done, and so the restoring force is the first differential of energy with respect to length, and the modulus is the second differential. If a number of length steps are made, on either side of the equilibrium position, nonlinear elasticity and differences between tension and compression can be investigated. There are many different ways to perform these calculations. All use the Hartree-Fock self-consistent field oneejectron approximation. Some use semiempirical procedures to approximate and simplify the calculations, setting some unknowns to values that fit experimental data. The methods of this type that have been used for modulus calculation are known by the acronyms CNDO and MNDO for complete or modified neglect of differential overlap. Methods without these approximations are generally called "ab initio" methods. An important check on these calculations, as for the molecular dynamics method, is to derive as many properties as possible from the same set of input data.

7.4 Theoretical Modeling of Elastic Constants

Boudreux used the more approximate CNDO method to determine Ec for polyethylene as 297 GPa (Boudreux, 1973). This is an unusually low value for molecular orbital calculations. Dewar et al. (1979) used MNDO on n-alkanes and polyethylene to obtain chain geometry, vibrational spectra, electronic band structure and £ c , which was found to be 490 GPa. According to Hong and Kertesz (1990), who obtained a value of 369 GPa by a similar method, incomplete relaxation of carbon atom position after deformation was the reason for this exceptionally high value. Klei and Stewart (1986) obtained 360 GPa for Ec in PE with MNDO. Dewar et al. (1979) regarded the MNDO method as a "useful stop-gap until such time as effective ab initio methods become feasible, which unfortunately, seem very remote at present". The first ab initio calculation of polyethylene modulus came later the same year (Crist etal., 1979). While the previous semi-empirical calculations had used cyclic boundary conditions to calculate the properties of an infinite chain, Crist et al. (1979) calculated the energy as a function of length for three linear n-alkanes, C 3 H 8 , C 5 H 12 and C 7 H 1 6 . Using these very small clusters allowed them to do geometrically optimizing ab initio calculations at that early date. The effects of the end groups were removed by subtracting the energies of adjacent n-alkanes, to give the energy as a function of extension for the -(CH 2 ) 2 unit. The result was the same for both alkane pairs, and using X-ray data for the chain cross-sectional area gave a chain modulus of 405 GPa. The calculations were later extended to C 9 H 2 0 with about the same result (Brower et al., 1980).

333

As in empirical calculations, crystal symmetry can be used to limit the number of independent atoms in the system considered. Karpfen (1981), as part of a series of ab initio calculations on polymers, used the S2 screw axis along the PE chain to reduce the atomic basic of the chain set to a single CH 2 group. Karpfen (1981) used two different basis sets of orbitals. Although the effect of basis choice on chain geometry was very small, the C-C bond stretch force constant changed. The chain modulus was 405 GPa with the minimal (STO-3G) basis and 345 GPa with a larger set. Suhai (1983) used similar methods and basis sets but added the refinement of correcting for electron correlation effects using perturbation theory. The uncorrected results for the chain modulus of polyethylene were very close to those of Karpfen (1981) and the corrections reduce the moduli by about 10%, to 303 GPa for an extended basis set. Karpfen remarked that his methods typically gave force constants about 10% too high, compared to studies including electron correlation and accurately predicted a "best value" of 300 GPa from his higher result (Karpfen, 1981). All the quantum mechanical calculations so far referred to have been on polyethylene. This is taken as the typical polymer, and it is also a simple hydrocarbon with few atoms in its basic repeat unit. Combining the theoretical results mentioned here and listed in Table 7-2 with the experimental results from that table and Sec. 7.3, and plotting as a function of year of publication gives Fig. 7-22. There are no clear trends with time, but excluding values below 200 and above 400 GPa, the rolling average has remained in the range 298 ± 7 GPa since 1963. This is close to the best ab initio value and not far from the

334

7 Elastic Properties of Crystalline Polymers

450 O A • 400 - +

X-ray Raman Neutron Scattering Mol Dynamics

x MNDO 0 ab initio

cd

\ vAvA

I !

•. .

. .

350

§ 300 250

ex, 200

150

1950

1960

1970

1980

1990

Year Figure 7-22. Theoretical and experimental values for the crystal modulus Ec of polyethylene (the single chain modulus, for ab initio and MNDO calculations), plotted as a function of the year of publication.

current Raman data. The uncorrected MNDO results are too high, and the X-ray results at room temperature are low; at 77 K values are higher. The ab initio calculations should give the elastic constants at 0 K. The empirical calculations are based on X-ray diffraction and spectroscopy performed at room temperature. Thermal vibrations have then been taken into account in some way, but small molecule compounds may not model the polymer behavior well. Theoretical modeling of thermal expansion for polymer chains remains a problem, even at the level of quasiharmomic force fields (Baughman, 1973; Kobayashi, 1979; Barron and Rogers, 1989). There are a few quantum mechanical calculations on other polymers. MNDO moduli for PBO and PBZT were obtained

by Klei and Stewart (1986), but the minimum energy geometry pedicted is not the experimental structure. Wierschke et al. (1992) have made semi-empirical calculations using a different method (AMI) that predicts the correct structure. They predict the chain modulus of ds-PBO to be 690 GPa and that of trans-PBZT to be 620 GPa, both about twice the experimental values. The calculations predict that chain stiffness drops rapidly on compression, and the chains undergo a buckling failure at 2 - 3 % compressive strain. Hong and Kertesz (1990) have made MNDO and hybrid calculations of the modulus of polyacetylene in different chain conformations. The hybrid calculations used chain geometries obtained from energy minimization with MNDO and force fields obtained from spectroscopy for PE and polyacetylene (Schachtschneider and Snyder, 1963; Takeuchi et al., 1988). The advantage over using the empirical force fields to optimize chain geometry was not made clear. As MNDO calculated force constants are too high (Fig. 7-22) Hong and Kertesz (1990) say that the alternative to empirical force fields is to "scale" the forces, that is, reduce them by some arbitrary factor to agree with experiment (Pulay et al., 1983). This process removes most of the attraction of a fundamental calculation of materials properties.

7.5 Interpretation of Results for Crystal Modulus 7.5.1 Experimental Data The modulus in the molecular chain direction is the only crystal elastic constant where there is sufficient experimental data to be concerned with interpretation. In that case, however, there is considerable

7.5 Interpretation of Results for Crystal Modulus

controversy. The reader may choose a review concluding that spectroscopy is correct and X-ray diffraction wrong (Fanconi and Rabolt, 1985) or one with the opposite conclusion (Nakamae et al., 1991). The methods do give different results, and the first question to ask is whether they measure the same thing. Song and Krimm (1990) point out that they do not. In the quasi-static X-ray measurement, each stretched chain is surrounded by other stretched chains and interchain distances and positions are altered to new equilibrium values. Raman spectroscopy measures the stiffness of a vibrationally excited molecule which moves in a crystal that remains in the zero stress form. Raman spectroscopy relates to a single chain, but it is not an isolated chain; the interchain forces will be larger in Raman than in the X-ray experiment as the neighboring chains have no time to relax. According to Song and Krimm (1990) this means that the X-ray experiment is much closer than the Raman experiment to stretching an isolated chain. Although continuum concepts do not really belong at this molecular level, the lack of lateral adjustments in Raman makes the measurement more like one of c 33 , while the "X-ray modulus" is l/s 3 3 . This analogy shows that the results need not be identical, and if there is a difference due to this effect, the Raman modulus will be the larger. Initial Raman experiments gave values much larger than those obtained from Xray diffraction. Later corrections to the simple theory have reduced the Raman values, to 20-30% greater than the X-ray values in the cases of PE and PEO described above. In some polymers disagreement is still large, as shown in Table 7-2, but in these cases less detailed consideration of corrections to the Raman results has been made. The table is far from com-

335

300 1 2 3 4 5 6 7 8 9 10 11 12 13 14

O 250

200

0 cd

-

Silk PPS i-PP PEK PEEK Technora PET Vectran Ekonol Cellulose I / Kevlar 1 4 9 / Nylon 6 /© PE / PVA1 /

in

g 100

a

(ft)

o

0

50

100

150

200

250

300

Crystal Modulus, GPa Figure 7-23. Comparison of the maximum specimen modulus that has been reported and the crystal modulus in the chain direction for a number of polymers. (After Nakamae et al., 1991.)

plete, and understates the complexities of interpretation of the X-ray data by putting the results into a single column. Only for the single crystal samples of polydiacetylenes is the direct mechanical testing result comparable with determination of crystal modulus. However, measured values for highly oriented, highly crystalline samples are a lower limit for the possible values of crystal modulus. In a number of cases the largest observed specimen modulus does approach the X-ray or other value for the crystal modulus Ec, Fig. 7-23. Experimentally the difficulty with the Raman LAM is that monodisperse oligomers or samples with a well defined lamellar structure are needed. Small angle X-ray scattering (SAXS) gives a long period in most cases, which is some measure

336

7 Elastic Properties of Crystalline Polymers

of lamellar thickness. Samples with a high fiber modulus have no long period, or an extremely weak SAXS peak that may not be representative (Adams et al., 1985). To get chain length from the long period it must be corrected for the crystallinity (the thickness of the amorphous layers) and for tilt of the molecular chains in the crystal. These factors will not be accurately known in many cases. The difficulty in interpretation of the Raman results is seen to be that various corrections can have a very large effect, at least on helical chain polymers. Confidence in the result demands confidence in the molecular dynamics used for the correction. The same molecular dynamics can be used to predict the modulus directly, and this should at least be consistent with the Raman result. The difficulty with the X-ray results remains that of determining ac, the stress on the crystals. The uniform stress model has been used for a long time. Sakurada et al. (1962,1964,1966) and then Nakamae et al. (1987 a, 1991) have found for polyethylene and other materials that the ratio ojsc remains the same, even though the structure and the macroscopic modulus vary widely. There seemed to be only two possibilities: (1) All the samples have a series structure, with uniform stress on crystal and amorphous components. They have different sample moduli because of different amounts of the components. (2) All the samples have different more complex structures which for an unknown reason happen to give the same ratio of sample stress to lattice strain. The second possibility seemed extremely unlikely to Sakurada and Nakamae, so they have taken the first to be correct, and describe ajsc as the lattice modulus Ec in every case.

Several factors have now caused the uniform stress assumption to be questioned. In some cases fibers have been prepared with a modulus E{1 higher than the original X-ray value of Ec (see Chap. 13 of this Volume; also Ward, 1985). SAXS and electron microscopy show that most well oriented samples contain parallel lamellar sheets of crystals separated by disordered material. The crystal/amorphous sandwiches are stacked along the draw direction, so the samples are largely series structures and the series model will be generally correct. Samples with modulus Ey comparable to Ec show little lamellar structure, and appear fibrous. SAXS shows a declining intensity in the meridional two-spot pattern that indicates lamellar structures as the stiffness of fibers increases (Ohta, 1983; Adams etal, 1985). Dark field and high resolution transmission electron microscopy shows that the fibrils in this type of material are not long perfect single crystals, but contain many short (< 100 nm) crystalline regions (Grubb and Hill, 1980; Sherman et al., 1982). Mechanical models for such structures are discussed in Sec. 7.6.5, but clearly a series model is no longer likely to be correct. There are also independent indications that the stress is not homogeneous at all times in all samples. As was mentioned in Sec. 7.3.1, the (002) X-ray reflection in polyethylene becomes broader as it shifts under stress, and has been observed to split into two peaks. Similarly, Raman bands (regular vibrational bands, not LAM) shift in position when the crystal is strained. The shift is proportional to molecular strain, so when the ratio (shift/<7s) is found to be different for different samples, this implies a different £ app (Tashiro et al., 1988). A change of band shape indicates a distribution of different stresses. The shape of bands under stress has been studied in iPP

7.6 Interpretation of Macroscopic Modulus

(Roylance and DeVries, 1971; Wool and Statton, 1974; Wool, 1980), PET (Fina et al, 1988), POM (Tashiro et al, 1988) and PE (Wool et al., 1986; Tashiro et al, 1988; Prasad and Grubb, 1989; Van Eijk et al, 1989; Kip et al, 1991). The bands generally increase in width and may become asymmetric or form a doublet. In iPP, PET and some PE samples an asymmetric tail was assigned to a small fraction of the molecules in the samples being stressed to much higher than average values. In high modulus PE fibers the bands split when stress is applied at low temperatures, implying that there are two different regions each under a different uniform stress (Van Eijk et al, 1989; Kip et al, 1991; Grubb and Li, 1992). During creep large changes occur in the shift of these bands and also in the shift of the X-ray reflections (Prasad and Grubb, 1989, 1990; Matsuo and Sawatari, 1988 a, b). These changes are associated with plastic deformation, but if homogeneous stress applies to all drawn fibers it should apply equally well to undeformed and to deformed fibers. Differences in a s /e c between different specimens of the same material at the same temperature are difficult to explain as changes of £ c . In the case of POM, Nakamae et al. (1990) find a difference of 50% in Ec between samples, and explain it as a difference in molecular mobility. The crystal structures are the same, with a difference in the c-axis lattice spacing of only 0.05%. Jungnitz et al. (1986) find that £ app varies as a function of draw ratio. This led them to a structural model for POM with three phases, and a value for Ec not equal to £ a p p . Both £ app and Ec vary with temperature in this material. Wu et al. (1989) combine a two-phase model with data for £ a to obtain Ec as a function of temperature. In these more complex models, variation of Ea with degree of orientation affects

337

the derived value of E c . In the uniform stress model, Ea affects only the sample strain <js and the sample modulus E^ (Sec. 7.6.2). 7.5.2 Theoretical Calculations Theoretical methods are discussed in Sec. 7.4, but two points should be made here. One is that disagreement on the interpretation of experimental data is fueled by uncertainty in the theoretical values. Table 7-2 shows that for many materials the proponent of any reasonable experimental value can find theoretical support. Resolution and recognition that different experiments measure different properties may come from advances in computation that give reliable values for interatomic forces in specific situations. The other point is that current methods of calculation of the elastic properties of a large assembly of atoms are difficult to follow even in principle. The practice necessarily depends on computer programs for their execution. Like much simulation, these are really equivalent to experiments. The reader must rely on the author to do the work correctly, and the results become more reliable when they are repeated by others. When a body of results using the same input data for atomic interaction and methods of calculation can be built up, relating to mechanical, thermal and spectroscopy properties of a range of polymers, then the range of acceptable values for the results will diminish.

7.6 Interpretation of Macroscopic Modulus The modulus of isotropic crystalline polymers is on the order of 1 GPa, and most oriented fibers and sheets are no more than ten times stiffer. Most crystals

338

7 Elastic Properties of Crystalline Polymers

are highly anisotropic, with moduli in the chain direction of 40 to 280 GPa, dropping to 5-10 GPa for polymers with a broad helical chain conformation. For "normal" oriented specimens, interpretation in terms of some structural model is required to explain the effect of orientation and why the specimen modulus remains low. In high modulus fibers, where the fiber modulus approaches the crystal modulus, the high anisotropy of the polymer chains makes even small degrees of misorientation important. It is for these materials that the uncertainties in crystal modulus are more important in modeling. The models described below fall into three groups: (1) Models that consider the effects of orientation distribution in detail, but ignore all details concerning crystal and amorphous phases - these are aggregate models. (2) Models that consider the interaction of crystal and amorphous phases, neglecting shear and assuming that the orientation is perfect or random - Takayanagi models. (3) Models that are based on more sophisticated composite mechanics, and combinations of the above. 7.6.1 Aggregate Models Aggregate models ignore detailed structural features of the material; only the orientation distributions in the sample are considered important. The one-phase aggregate model has been used in a wide range of polymers to model the partially oriented or unoriented moduli in terms of the elastic properties of a well oriented sample (Ward, 1962; Hadley et al., 1969). The model is applied to amorphous, crystalline and liquid-crystalline polymers. Any partially oriented sample is treated as

an aggregate of anisotropic units with transverse isotropy. The orientation distribution of the aggregate is that of the sample, and the elastic properties of the anisotropic units are those of a fully oriented material. The elastic properties of an aggregate can be calculated in many ways. The simplest are to assume either: (1) Uniform strain throughout, when the modulus of the aggregate will be the average of the moduli of the units. (2) Uniform stress through the sample, when the compliance of the aggregate will be the average of the compliances of the units. These are both simplifications, because a uniform stress gives strain discontinuity at the interfaces between units, and uniform strain would leave the interfaces out of stress equilibrium. The aggregate modulus obtained by averaging modulus is often called the parallel or Voigt average (Voigt, 1928) since components at the same strain are effectively in parallel. The inverse of the compliance average is the series or Reuss average (Reuss, 1929) since components at the same stress are effectively in series. Analysis shows that the properties of any real aggregate must lie between the Reuss and Voigt limits. For a random aggregate giving an isotropic material, closer limits can be defined (Hill, 1963; Peselnick and Meister, 1965; Boyd, 1983). In the case of uniform stress, the strain of each unit can be calculated from the compliance of the unit, 5, and the angle 0 between the direction of the applied stress and the unique (3) axis. The average strain and thus the sample compliance Sf will be controlled by average values of the angular functions sin4

ss 33 = — = 33

E

(7-13)

(25 13

7.6 Interpretation of Macroscopic Modulus

Clearly if the stress is applied in the draw direction and_the sample becomes fully oriented, then 0 -• 0 and the sample compliance becomes that of the unit. That is, 5 33 -* 533- For a random aggregate, forming averages in spherical polar coordinates, 1_ E

8

1

+ (7-14)

A similar equation defines the Reuss average for shear compliance of the isotropic aggregate (Hadley et al., 1969; Ward, 1983, Sec. 10.6) 1 _ 14 15^ +

4 \5S32

S

44

(7-15)

Using these equations and analogous ones derived by Voigt averaging, the isotropic elastic constants can be predicted from those of a highly oriented sample. If the experimental results lie between the two predictions, then the model may apply. Using this test, it was found that one-phase aggregate model could be used for low density PE and PET, but not for iPP, high density PE, Nylon 6 or Nylon 66 at room temperature (Hadley et al., 1969; Ward and Hadley, 1988). For this model to work, the structure of the unit in the aggregate must not change during orientation of the sample. It is not surprising, therefore, that the model fails on several semi-crystalline polymers when used to cover the range from unoriented samples to samples drawn 4 to 6 times. The model does work well on low density PE, indicating that the structure of the bulk melt crystallized material, containing small lamellae, is maintained in the drawn samples that were used. Orientation distributions predicted from the birefringence

339

(which gives sin 2 0) and models of the drawing process were used first in the model. Compliance averaging then gave a good qualitative fit, predicting the observed and initially unexpected dip in £(, in low density PE samples extended up to 150% (Gupta and Ward, 1967). When experimental orientation distributions derived from X-ray diffraction or broad line NMR were used, this fit became quantitatively accurate (McBrierty and Ward, 1968; Gupta and Ward, 1970), Fig. 7-24. These orientation experiments relate to the crystalline fraction of the material. It seems that even in the one-phase model of low density PE, a material that is 50% amorphous, it is the crystalline orientation that is important. Deviations from the model predictions at low draw ratios may be due to reversible stress induced twinning (Frank et al., 1970). This mode of de-

400

2

3

Draw ratio Figure 7-24. The modulus of oriented low density polyethylene as a function of draw ratio. Triangles are experimental values for E^, crosses experimental values of E±. The two dashed curves are the aggregate model predictions when orientation functions from broad line NMR are used. (After McBrierty and Ward, 1968.)

340

7 Elastic Properties of Crystalline Polymers

formation changes the sizes of the aggregating units, so it acts to change the structure, which is not covered by the aggregate model. Kausch (1967, 1970) found that an aggregate model with compliance averaging fitted the extensional modulus as a function of draw ratio for Nylon 66 and a number of other polymers. These included poly(vinyl chloride) and polyacrylonitrile, polymers of low crystallinity. The fits were over a narrower range of orientations. Unit properties were obtained by fitting the model to the sample modulus at a range of draw ratios instead of using only the properties of the most highly drawn sample. The one-phase aggregate model has also been shown to work very well for systems containing fully extended and largely rigid rod polymer chains, such as aramids and thermotropic liquid crystals (Northolt and Van Aartsen, 1978; Green et al., 1990). Writing Eq. (7-13) in terms of sin as

misorientation of units which are fibrous crystals. Even if (f) is only 1° or less, the effect of misorientation on modulus cannot be ignored, as the crystals are so highly anisotropic. The extrapolation to <> / = 0, a completely aligned fiber, in Fig. 7-25 gives a modulus in agreement with Ec as measured by X-ray diffraction. Although there is evidence for larger scale structures in the aramid fibers, (e.g., the pleated sheet structure) these structures are apparently not important for elastic properties, except as they affect (/>. The extended chains generally have high chain moduli, so s 33 is small and can be neglected in the second term and third terms of Eq. (7-16). Often s 1 3 <^s 4 4 and particularly if s 44 and s 13 are not accurately known, s 13 can also be neglected without loss of real accuracy. For PE,

(2s 13 + s 4 4 -2s 3 3 )sin 2 (/> •(2s 13

— 2s33)sin4

(7-16)

Northolt and Van Aartsen (1978) measured the modulus of a range of aramid fibers ultrasonically, and measured the orientation distribution by X-ray diffraction. They find a good fit to Eq. (7-16) when using accepted values for the transverse compliance components. In particular, when sin2 <j> is less than 0.05 ( < 12°) the third term in Eq. (7-16) can be neglected, giving 1

1 = -^ + A sin2

(7-17)

Figure 7-25 shows that for well oriented fibers, the fiber compliance does follow Eq. (7-17). Thus the modulus of different samples can be entirely explained by the

0.00 0.01

0.02

0.03

0.04 0.05

Figure 7-25. Plot of experimental axial modulus E ^ for a range of poly(p-phenylene terephthalamide) fibers, as a function of the crystal misorientation. The aggregate model prediction is a straight line, with Ec as the intercept at sin2 4> -• 0. The dashed lines are the limits of predictions using reasonable values for the transverse elastic constants (Northolt and Van Aartsen, 1978).

7.6 Interpretation of Macroscopic Modulus

Eq. (7-12) gives the ratios s 4 4 :s 3 3 :s 1 3 as 313:3.2:0.5. For well oriented materials where the third term in Eq. (7-16) is negligible, the expression then becomes: 1

sin2

1

(7-18) £ E G This has been used to model highly oriented PE by Powell etal. (1990). G,, was measured independently, so the excellent fit of the data to Eq. (7-19) allowed an accurate estimation of sin2 for the aggregates. This was found to be consistently some 30% lower than sin2 (j) for the crystals as measured by X-ray diffraction. The implication of this is that the mechanical units are made up of a number of crystals, imperfectly ordered within the unit. Then the extrapolated value of E ~ 250 GPa should correspond to the modulus of the unit, and this should be a little less than that of the PE crystals. A recent study of iPP shows that the one-phase aggregate model fits the data for £|l measured at a temperature of — 95 °C for unoriented samples, and samples drawn up to 18 x (Unwin et al., 1990). In this case the crystal compliances obtained by X-ray diffraction were used as the properties of the anisotropic unit. These authors use a two-phase aggregate model for the E^ data at 45 °C, where the sample extension compliance / 3 3 is given by __

I

y33 = asr3% + (1 - a)s'i3

(7-19)

Here a is the degree of crystallinity, and the compliances of the two phases s'33 and s'33 are obtained by averaging over an orientation distribution as in Eq. (7-13). An obvious problem with this approach is that the elastic properties of the highly oriented but amorphous material, s*pq, are generally not known. The modulus will increase with chain orientation in an unknown manner, and the increase may be substantial. In se-

341

ries Takayanagi models (Sec. 7.6.2) values of up to 80 times the unoriented amorphous modulus are used. It is also difficult to determine the amorphous orientation function, to obtain terms such as cos4 0 a . If the orientation of the sample is not too high, and the chain is stiff so that s33 and s13 are small, the terms s33 cos 4 <j> and s13 sin2 (j) cos2 <> / in Eq. (7-14) will be small and so / 3 3 = sxl sin 4 4> + s44sin2 0 cos 2 4> (7-20) then

If

(sin4 (/> + sin2 (/> cos2 (j)) = s11 sin

£33

In this approximation the sample modulus in the orientation direction is simply the lateral modulus of the aggregating unit divided by the second moment of the orientation distribution of the units. This approximation is reasonable but not very accurate (Ward, 1983, p. 291). It can be seen as the basis for a much earlier twophase aggregate model for iPP (Samuels, 1974). In that model a

-r-^—

(1 — oc) 2 a sin (/> E\

(7-22)

Here
342

7 Elastic Properties of Crystalline Polymers

well described by a lamellar structure of alternating crystalline and amorphous regions. It is therefore natural to model their elastic properties as that of a two-phase composite. The Takayanagi type of model considers the sample to be made up of blocks of the phases that make up the composite. Each block has a uniform tensile stress and strain, and is connected to the other blocks in series or in parallel. There is no consideration of any transverse tractions or any shear in the material. Generally this model is applied to well oriented materials, and the crystalline phase is assigned the elastic properties of a single crystal. In some studies the amorphous phase is considered isotropic; in others it is allowed to have an orientation and different parallel and perpendicular moduli. In general the first treatment of any property of a semi-crystalline polymer is in terms of a mixture of two phases - perfect crystal and ideal amorphous. This is known to be only a first approximation, but when information is lacking it is used in preference to giving arbitrary properties to intermediate structures. Similarly, if no reliable data is available for the amorphous properties in the samples under consideration, a complex model using an ideal isotropic amorphous phase properties may be made. It will be as useful as a simpler model with arbitrary amorphous moduli obtained by fitting only that particular model to that sample. As mentioned before the simplest composite arrangements are to have either uniform stress on all components, or uniform strain. Uniform strain throughout the material is represented as a parallel model of two blocks, one for each phase, Fig. 7-26 a. In parallel the two blocks must have the same extension and the same length. The sample load is the sum of the loads on each block, so the sample stress as is

(

1-a

A

(b)

(a)

Figure 7-26. Two-component Takayanagi models of semicrystalline polymers, (a) The components are in parallel, so their extensions and strains are the same, while loads and stresses add. (b) The components are in series, so their loads and stresses are the same, while extensions and strains add.

+ (1 — a)cra. Dividing by the uniform strain gives the Voigt average modulus

VLOC

E = a Ec + (1 - a) Ea

(7-23)

Similarly uniform stress is represented as a series arrangement of the crystalline and amorphous phases, as in Fig. 7-26 b. In series the two blocks must have the same load, and as they have the same cross-section the stress is the same. The sample extension is the sum of the extensions of each block. The sample strain es is then as c + (1 — (x)sa. Dividing by the uniform stress gives a Reuss compliance averaged extensional modulus for the sample of 1

1-a

(7-24)

Generally neither the series nor the parallel model is fully satisfactory for the prediction of modulus. For most samples the parallel model predicts far too high a modulus parallel to the chain direction, E^. The series model gives much better results, but changes in crystallinity, for example, may not produce the predicted changes in mod-

343

7.6 Interpretation of Macroscopic Modulus

ulus at constant £ a . Logically, if the series model is used to explain E^ for an oriented sample, then the parallel model should be used for E ±. When data for E ± is available it rarely fits this simple model exactly. The series model was used by Takayanagi et al. (1966) to explain the temperature dependance of E^ and E ± in drawn and annealed high density PE. E^ is higher than E± at very low temperatures, but it is strongly affected by the fall in Ea at the glass transition temperature. E L is modeled as a parallel or Voigt arrangement and falls much less as Ea falls, so there is a reversal of anisotropy, a cross over as the temperature rises (Fig. 7-27), in agreement with experimental data (Takayanagi et al., 1966). A "Takayanagi model" normally refers to these parallel and series models, or to more complex arrangements that he developed to represent one phase dispersed in another (Takayanagi et al., 1963). For most samples, the modulus is quite low and is dominated by the modulus of the amorphous region. The crystalline phase is therefore dispersed in the amorphous phase. Figure 7-28 shows the basic model and the two possible ways of dividing up the continuous amorphous phase. The models were originally devised to explain the viscoelastic response of crystalline polymers; predictions of complex moduli were made from them. We shall restrict ourselves here to the elastic response. In arrangement Fig. 7-28 b the elastic modulus will be 1

(7-25)

and in Fig. 7-28 c it will be (7-26)

12

(a)

Tiaa)

Ttac)

TEMPERATURE ' 1

I i i i

j 1

i i

11

12

i

i |1

10

1i i i

A Ttta)

Ttarc)

12 10

(C)

T(crd T
As for the series and parallel models, if one arrangement such as Fig. 7-28 b is used for £|l then the other should be used for £ ± . An example of this analysis is the work of Lewis and Ward (1980b, 1981) on oriented a and y phase Nylon 6 sheets. The 10 s in-plane extensional compliances s 33 and s11 were determined for well oriented sheets of nylon 6 in dry and wet conditions. The comparison of wet and dry is useful because moisture should affect only the amorphous moduli. Both crystalline and amorphous phases were treated as transversely isotropic. Literature values were used for crystalline moduli E% and E\ ob-

344

7 Elastic Properties of Crystalline Polymers

switched to c, the results for xj/, X and Ea (wet) are hardly affected. Lewis and Ward (1980 b, 1981) also determined the shear compliances s 44 , s 55 and s66 of the Nylon 6 sheets. As shear is excluded from Takayanagi type models, interpretation of these compliances used entirely different concepts. The relative shear compliances were explained in terms of the geometry of the hydrogen bonds in the two crystal structures. This discussion therefore does not belong in this section. It is mentioned as a reminder that no single model and no single approach is likely to be most suitable for the interpretation of all aspects of elastic properties. A series model may be satisfactory for explaining E^ in high modulus fibers and tapes (Matsuo and Sawatari, 1988 a; Nakamae et al., 1991), but a or E\ may then be far from the usual values. Nakamae et al. (1991) find that a PE fiber with £„ = 207 GPa has £ c = 235 GPa. With a series structure, either a is extremely high at over l-X A 99.9%, or there is a very high amorphous modulus, 65 GPa at a crystallinity of 95%. A The data of Matsuo and Sawatari (1988 a) for PE fibers at 140 °C is more specific: El, = 130 GPa, a-0.75 and £ c = 211 GPa, ( so £ ^ - 8 0 GPa. Although the normal value for Ea is only 1 GPa, these materials (a) have been extended to very high draw ratios, which will orient and extend the A amorphous chains. In the extreme case of I A fully extended amorphous material, PE chains will be in the all-trans conformaC A C tion. The modulus of such a chain will be essentially the same as that of the crystal at 250 GPa or more. It is possible that a num(b) (c) ber of defects preventing crystallization do Figure 7-28. (a) Schematic of an oriented crystalline not drop the modulus by more than half, phase C dispersed in a continuous amorphous phase so any value for E% up to 120 GPa or so A. The width of the crystalline region is X and its may be feasible. "Amorphous" may no length is \\i so the crystallinity oc = X \j/, (b) and (c), the longer be a good description of this matetwo alternative Takayanagi models for arranging this as three components (Takayanagi et al., 1963). rial.

tained from X-ray diffraction (Kaji and Sakurada, 1974), and for the oriented amorphous moduli in dry conditions, E% and E\ (Prevorsek et al., 1973). The series model for E{1 gives a = 0.61, and E% (wet) = 0.93 G P a 1 . However, using the series model on the draw direction means that the parallel model should be used on the perpendicular direction. When this is done, the calculation gives a = 0.46 and E\ (wet) is negative, which is an unsatisfactory result. Using the series-parallel arrangement of Fig. 7-28, with (b) for the draw direction and (c) for the perpendicular direction, works much better. The extra degrees of freedom allowed in choosing if/ and X permit a consistent value for a = \jj X = 0.65 to be obtained. (Note that i// for one direction is X for the other.) Both E\ (wet) and E% (wet) remain positive, as required for a realistic model. If the arbitrary choice of model to use for the draw direction is

7.6 Interpretation of Macroscopic Modulus

The alternatives to very high values of E% are some parallel components of continuous crystal, or a completely different type of model. The simplest Takayanagi models, where the crystalline phase is continuous, are as described above in Fig. 7-28 and Eqs. (7-27) and (7-28), but with the phases reversed. There is then a block of amorphous material surrounded by crystal. Then a = (1 — \//) (1 — X) and unless (1 — X) or if/ is very small, the sample modulus £|| will be close to (1 — X)EC. Take for example the case of "shishkebab" structures in PE. These contain fibrous crystals with their long axes parallel to the alignment direction and lamellar overgrowths. The natural Takayanagi model for this material has a crystalline component (the fibrous crystals) in parallel with a series arrangement of crystalline and amorphous (the lamellae). The fibrous crystals have a higher melting point in the differential scanning calorimeter, so the amount of each crystalline component may be measured, as well as the total crystallinity. The model predicts that E^ will be close to directly proportional to the fraction of fibrous crystals. It is easy to create more complex models of this type, by adding more components. The situation is the same as in the modeling of viscoelasticity, where complex models can be created by adding springs and dashpots in series and parallel. The problem is also the same; unknowns are added to the expression for modulus along with every component. With sufficient unknowns any model may fit the data, and unless the modeling is based on structural information, the result is no more helpful than any numerical fit to the data. More complex models in the literature are therefore generally based on structural information or concepts. Takayanagi (Takayanagi and Kajiyama, 1973) used a

345

modified series model for high density PE, similar to that in Fig. 7-28 a, but the material in parallel with the crystalline material was a third phase, not amorphous. This phase had properties between those of the crystalline and amorphous phases, and was meant to represent the defect structure that broke up the crystalline lamellae into small mosaic blocks. Unfortunately it has since been shown that it was a misinterpretation of X-ray data that led to the idea that mosaic blocks are always present (Harrison et al., 1976). A four-component model has been used for £|| of PE fibers to give two series arrangements in parallel, as in Fig. 7-29 (Grubb, 1983; Rossignol et al., 1988). The justification for this was again structural; the extra series component of fractional length / represented the incomplete continuity of the fibrous crystals. There are a large number of variables in this model, but if Ec $> Ea and / <^ 1, the effect of the right hand side of the model on E^ can be neglected. The extra series component is due to defects in the highly stressed fibrous

1 1

I

Ec

l-f

1

-—b

— l-b —

Figure 7-29. A four-component Takayanagi type model that has been used for E^ of PE fibers, b is the fraction of fibrous material, (1 — b) the fraction of lamellar material. In the cases considered, /, the fraction of defect material in the fibrils, is small and Ec P Ea so that the right hand series arrangement has little effect on the sample modulus (Grubb, 1983; Rossignol etal., 1988).

346

7 Elastic Properties of Crystalline Polymers

crystals and these were associated with entanglements in the molecular network. Network theory was then used either to predict that the fraction of this material, /, should vary with draw ratio as I/A2, or to interpret nonlinear elastic properties in terms of / (Rossignol et al., 1988). The predict fiber modulus is

bEn

,

bET

(7-27)

Where ED is the modulus of the entanglement defect region, assumed to be equal to Ea by Rossignol et al. (1988), and b is the fraction of fibrous crystals. If b is taken to be a constant, the variation of E^ with draw ratio agreed well with this prediction. The resultant fit was quite satisfactory, but the model has the serious flaw of ignoring all shear interactions between the components. The important small fourth component is modeling tensile stress transfer at the ends of the fibrous crystals. In composites containing long fibers, transfer of stress from fiber to matrix by shear will be important and it is the tensile loads at the fiber ends that are more usually neglected (Cox, 1952).

7.6.3 Tsai-Halpin Equations The Tsai-Halpin equations are semiempirical mixing rules to determine the modulus of a composite from the moduli of the components. They have been applied to many heterogeneous systems, but the interest here is their application to semicrystalline polymers (Halpin and Kardos, 1972; Kardos, 1973). For a mixture of crystalline and amorphous phases with crystallinity a the general form of the equation is: 1 — f]OL

P is an elastic property such as Young's modulus or shear modulus and £ is a geometric parameter. Some specific geometric interpretations of £ have been made, for example that for the Young's modulus of a material reinforced with short parallel fibers £ is twice the aspect ratio of the reinforcing elements (Halpin and Kardos, 1976). It may be better to see £ as a more general parameter which indicates where the mechanical response lies, between the Reuss and Voigt limits. Expanding the above equation, and using E as the property, l

}

With this formula it is easier to see that if £ -> 0 then the Tsai-Halpin equation becomes the Reuss uniform stress limit. If £ -• oo, it becomes the Voigt uniform strain limit, so any possible result must correspond to some positive value of £: ocEa

-a)Ec

(7-30) = oo

(7-31)

McCullough (1977) has used the TsaiHalpin equations to compare various theoretical models, by calculating the equivalent £. The equation continues to be practically useful for its convenience and simplicity. For example Crist et al. (1989) use the equation to show that no model of constant structure (constant £ and constant E a ) can explain the variation of E with a for linear polyethylene-butene-1 copolymers. Modeling the modulus with £ = 1 then gives the variation of £ a with crystallinity a.

7.6.4 Lamellar Assembly Modeling Boyd (1983) has been able to predict close limits to the tensile and shear moduli of

7.6 Interpretation of Macroscopic Modulus

lamellar semi-crystalline polymers. The input data required are the elastic properties of each phase scpq and spq and the volume fraction of crystallinity a. The method uses composite mechanics to predict the elastic properties of a stack of parallel lamellae. This stack is then used as the anisotropic unit in an aggregate model. Both components are assumed to have at least orthotropic symmetry, with one axis (3) normal to the lamellar planes. Since the lamellae are large and the layers are thin, a uniform strain-additive stress response (parallel model) is assumed for in-plane deformation, and a uniform stress-additive strain (series model) for deformation normal to the lamellar plane. Unlike in the simpler models described above, shear is not neglected in considering the combination of these models, and a full 9 component stiffness matrix is calculated for the lamellar stack, C. Inversion of this matrix gives the compliance matrix S. The sample contains lamellar stacks in a range of orientations and complex organization. Its elastic properties cannot be calculated exactly, but the average response of the lamellar stack may be obtained either by averaging S or C over all orientations. These are equivalent to the Reuss and Voigt limits mentioned earlier in the aggregate model section, and the correct values must lie between them. For simplicity, the sample is assumed to be transversely isotropic. Then as before the elastic constants of the sample depend on the second and fourth moments of the orientation distribution sin2(/> and sin4(/> (Ward, 1962; McCullough, 1977). For both isotropic and oriented samples of PE, the results of this analysis are generally much closer limits on the possible tensile and shear moduli. In the isotropic case the limits are close to the lower Reuss bound for modulus obtained from a series model (Fig. 7-30).

347

Figure 7-30. Calculated bounds for the shear modulus of isotropic lamellar polyethylene as a function of crystallinity, using 300 MPa as a value for Ga. The outer dashed lines are the simple Voigt and Reuss bounds. The much closer solid lines were obtained by Boyd (1983).

From the assumptions about the lamellar stack, the fully oriented material with lamellar normals all along z must be at the low series limit of modulus. If the lamellar normals are perpendicular to z then the specimen modulus along z be at the high parallel limit. Even with a crystallinity of 70% and an anisotropy in the crystal phase of c33/cXi — 32, the prediction is that the axial tensile modulus E^ will fall when the isotropic material is first oriented. This is in agreement with the initial drop in £ ( | shown in Fig. 7-23, and based on this model the later increase of modulus with orientation implies changes in the lamellar structure or amorphous orientation.

348

7 Elastic Properties of Crystalline Polymers

7.6.5 Models of High Modulus Fibers

The use of both aggregate models (Northolt and Van Aartsen, 1978) and series or more complex Takayanagi models for high modulus fibers (Matsuo and Sawatari, 1988 b; Tashiro et al., 1988; Nakamae et al., 1991; Grubb, 1983) has already been mentioned. The other models described here generally attempt to bring more structural information or concepts to bear on the problem of understanding fiber modulus. Further discussion of the mechanical behavior of high modulus fiber can be found in Chap. 13 of this Volume. Because almost all the data refers to high molecular weight linear PE, the models also refer mostly to this material. "Fibers" is here taken to include samples in the form of tapes and sheets as well as fibers. Most of these samples have fiber symmetry, though very highly drawn samples may approach a single crystal texture. 7.6.5.1 Fiber Composite Model

High modulus fibers appear to contain fine fibrils. The long axis of the fibrils is parallel to the fiber axis on average, and they are unlikely to be continuous throughout the specimen. The relevant composite structure is then one of discontinuous parallel fibers in a matrix. The reinforcing fibers are taken to have the crystal modulus, and the matrix is some less well ordered and more compliant material. Without detailed information, the crystalline fibrils are assumed cylindrical. Barham and Arridge (1977) used the shear lag theory (Cox, 1952) to model high modulus PE as a fiber composite. Arridge and Barham (1978 a, 1978 b) extended this work to include iPP. The shear lag theory makes several simplifying assumptions, for example, there is no slippage at the interface and that the state of stress in the matrix is one

of simple shear. It also sets the tensile stress across the fiber ends to be zero and ignores problems of stress concentrations. Nevertheless it is found that in many macroscopic short fiber composites this simple theory works as well as much more complicated ones. The primary result of the theory is that the composite modulus £(j is given by:

£,, = VfEf[l -

+ (1 - Kf)£matrix

V{ is the fibril fraction, the crystallinity a in this case, and £ f the modulus of the reinforcing fibrils, here EG. Ec P Ematrix so that the second term can be neglected. The parameter x depends primarily on the aspect ratio of the reinforcing fibers, LJrc, and on the matrix shear modulus, Gm as follows: x=

Ec In (3.6 a)

(7-33)

When the fibrils have a large aspect ratio, x is large and the applied load is efficiently transferred by shear of the matrix. Then £|l = a £ c , just as in a parallel model. Since the interfibrillar matrix material is likely to be oriented and partially ordered, Gm is unknown. Barham and Arridge (1977) calculated the fibril aspect ratio from the observed £||, with a sensible and constant value of Gm for a range of highly drawn melt crystallized PE samples. They found that the aspect ratio was proportional to the draw ratio after necking, raised to 3/2 power (Fig. 7-31). This led to the idea that the isotropic structure of the original sample was broken down and fibrils were formed in the neck. The fibrils then affinely deformed with the sample in subsequent drawing, increasing in length and reducing in width so that the aspect ratio increases as the 3/2 power of draw ratio.

7.6 Interpretation of Macroscopic Modulus

0

2

4

6

(taper draw ratio)3/2

8

Figure 7-31. The short fiber composite parameter x, proportional to the aspect ratio of the reinforcing elements, as a function of the post-neck draw ratio, t3/2. The samples are highly drawn polyethylene fibers. The model predicts that x oc t3/2; the solid line is the best linear fit to the data, and it passes accurately through the origin, x is derived from the measured modulus; a particular value of Ec must be chosen (after Barham and Arridge, 1977).

One problem with this model is that it seems unreasonable for a stiff reinforcing element to deform affinely with the softer sample. The fibrils could increase in aspect ratio during the plastic deformation of drawing by breaking up and reforming, but then there would be no clear connection between aspect ratio and draw ratio. The fibril aspect ratio is important for this model, and there are problems in determining it. The mean width of the fibrils may be measured by X-ray diffraction, but the length is generally unknown. In many cases of fiber processing, such as gel spinning, it is not possible to identify any necking, so an analysis that relies on the postneck draw ratio cannot be applied. Unwin et al. (1990) avoid these problems when applying the model to iPP. They

349

treat the shear-lag parameter x as the variable to be fitted and count the model a success if x oc (total draw ratio)3/2. Using total draw ratio instead of the draw ratio after the neck assumes that the extension ratio at the neck is a constant. Unwin et al. (1990) find that the model fits for draw ratios over 8, but they also find that aggregate models fit the data well. High modulus gel-spun fibers may still have a comparatively small mean crystal length, as determined by (002) linewidth of 40-70 nm (Smook and Pennings, 1984; Grubb and Prasad, 1992). The width of fibrous crystals in PE is in the range of 10-30 nm, so the aspect ratio of these crystals is apparently only 1-4. Unless Gm is very high, this is not consistent with the shear lag theory. A possibility is that the crystal continuity is disrupted as far as Xrays are concerned, by some defect that has no significant effect on the stiffness. Then a long fibril, made of many crystals, could be the effective reinforcing agent. Recent Xray data shows the fibril length to be 150200 nm for high modulus gel-spun polyethylene fibers (Grubb and Prasad, 1992), giving an aspect ratio nearer 10. 7.6.5.2 Crystalline Bridge Model

The crystalline bridge model combines aspects of a Takayanagi model with fiber composite ideas. It is based on a structural observation on a range of highly drawn melt crystallized PE fibers. The lamellar thickness found by SAXS in these fibers is less than the mean crystal length in the molecular chain direction found from the width of the (002) wide angle X-ray reflection (Clements et al., 1978; Gibson et al., 1978). As the fiber draw ratio and £j, increase, the crystal length increases more than the lamellar thickness. In a purely lamellar material the crystal size and

350

7 Elastic Properties of Crystalline Polymers

lamellar size are the same. If there are fibrous or acicular crystals present as well as lamellae, then the mean crystal length will refer to both components (weighted toward the longer, as a weight average). The SAXS signal depends on having a regular stack of crystals of similar size close together, so will continue to reflect only the lamellar size. Gibson et al. (1978, 1982) assumed that the longer crystals are very small in width and distributed randomly throughout the lamellar structure. The probability p that a crystalline chain traverses the amorphous region and enters the next lamella is taken to be independent of the history of that chain. This leads to a most probable distribution of crystal lengths. Figure 7-32 shows

a schematic drawing of this concept, but note that it is only a schematic. The bridging elements cannot be single chains, as they would not appear crystalline in an X-ray experiment. Detailed analysis including dynamic properties led to a mean diameter of 1.5 nm (Gibson et al., 1982), but even this is a very small "crystal" that would not lead to simple interpretation of the X-ray data. From the X-ray data for drawn PE, 0.05 < p < 0.4, the probability of traverse increases with draw ratio. Any crystal longer than one lamellar thickness was taken to be a reinforcing fiber element. The fraction of this fibrous crystalline material is a p (2 — p). The modulus of the fiber reinforced sample was then calculated on the basis of the shear lag theory, as described above. The sample modulus was found to be

[l-ap(2-p)]£matrix

Figure 7-32. Schematic model for the structure of high modulus polyethylene. Only crystalline chains are shown. Much of the material in this diagram is lamellar, but a significant number of crystalline chains bridge the amorphous regions between the crystalline sheets (Gibson et al., 1978).

(7-34)

This can be seen as a parallel model result, averaging the modulus of the reinforcing fibers with that of the matrix. An average shear lag factor [1 — (tanh x)/x] modifies the effective stiffness of the crystalline fibers. The matrix component was modeled as a series arrangement of lamellae and amorphous material, so the overall model is close to a Takayanagi model with a continuous crystalline component. It was found that for temperatures above — 50 °C the second (matrix) term in Eq. (7-34) could be ignored. As with the simpler fiber composite model (which took all the crystals to be in the form of fibers), a difficulty with using this model is in determining the aspect ratio of the reinforcing elements. The fibers of highest modulus often have no SAXS long spacing, little indication of the existence of

7.6 Interpretation of Macroscopic Modulus

regular lamellae, so the usual X-ray methods cannot be applied. The lack of long spacing is not in disagreement with the model; as E^ rises, the lamellar fraction diminishes and the SAXS signal is predicted to fall. The SAXS peak will also fall due to changes in the density of the amorphous and crystalline material during processing, and due to lack of regularity in the remaining lamellae (Adams et al., 1985). Wu et al. (1980) and Ohta (1983) point out that the only reliable structural indicator of high modulus and high strength in PE fibers is a fall in the integrated SAXS intensity. 7.6.5.3 Molecular Models

The aggregate models do not distinguish between crystalline and amorphous phases, but the basic unit used is assumed to be large compared to individual molecules. The unit properties are derived from macroscopic tests, or are assumed to be those of the crystalline phase. All the other models so far described are two (or more) phase models, where the sample to be modeled is notionally divided into distinct phases with different mechanical properties. This leaves models based directly on molecular properties. Termonia et al. (1985, 1986a, 1986b) have performed numerical modeling of an assembly of oriented molecular chains. The model is simplified, but retains many features not explicitly dealt with elsewhere. These include the shear and slippage of adjacent chains and the existence of chain ends due to the finite molecular weight of the initial polymer. Breakage of chains during deformation is also allowed. The main thrust of this work was to predict strength and extension to failure, but the model also calculates a modulus, the nonlinear elastic behavior at high strains and

351

the molecular stress distribution. As might be expected for molecules that have a reasonable shear strength and do not slip past each other immediately, the small strain properties of a fully aligned molecular array do not depend strongly on the molecular weight. For PPTA Smith and Termonia (1989) have calculated the modulus as a function of molecular weight and chain end segregation. They find that when the molecular weight is only 10000 the modulus is reduced by just 10% from that for the infinite chain. That is when the chain ends are randomly placed; it drops by another 10% if there is segregation of chain ends. Irving and Smith (1986) and Postema and Smith (1990) also use a molecular approach in modeling oriented semi-crystalline polymers. They relate the axial modulus to the draw ratio in a simple manner that ignores structural features such as crystals and measured orientation functions. The assumptions are that molecular coils in the original unoriented material deform affinely with the sample. This gives an orientation distribution that is modeled by a series arrangement of fully aligned chains and completely unoriented chains. The amount of fully oriented material is set to make cos2 <j) the same as for the affine deformation. The result is a complicated expression for the axial modulus as a function of draw ratio, which simplifies at high draw ratios to: 1 _ 1 A

(7-35)

where A is a constant that depends on the modulus chosen for the unoriented chains. Data for PE (Irving and Smith, 1986) and other materials including PET and POM (Postema and Smith, 1990) follows this relation, with a single value for A even when preparation methods differ (Fig. 7-33). This

352

7 Elastic Properties of Crystalline Polymers

0.06

o.o/,

0.02

0.04

Figure 7-33. Axial modulus E^ for polyoxymethylene samples with a range of draw ratios X from 10 to 33. The plot is l/£ vs. /T 3 / 2 , and the straight line is the prediction of the molecular model of Irving and Smith (1986). The intercept at A~3/2 = 0 is the (inverse) modulus of a perfectly aligned system.

good agreement is the support for this simple model. It should be noted that when JE|I <^ Ec, the result is £(( oc X3/2 which is close to the prediction of the fiber composite model described above. Also, the 4element Takayanagi model predicts a modulus [Eq. (7-27) with / oc X~2] that at high values of X (when / is small) is the same as the above, with the power of X increased from 1.5 to 2. Much of the data cannot distinguish these cases, when Ec is allowed to vary within reasonable limits.

7.7 Concluding Remarks The contents of this chapter should show that the elastic properties of semicrystalline polymers are not simple, and are relevant both to polymer engineering and polymer science. The complexities of interpreting experimental results on real samples are due to the close linkage be-

tween sample stiffness and sample structure. Modeling the elastic properties often uses or implies structural models, and such study is in a way an investigation into the material structure. Theoretical and experimental investigations into the elastic properties of the perfect crystalline phase are continuing, and there are strong proponents of several different methods that often give conflicting answers. However, in systems where a lot of work has been done, such as linear polyethylene and polyethylene oxide), there has been some convergence of results and explanation of differences. It is to be hoped that this trend extends to other materials. Although the continuing increase in available computer power makes theoretical prediction of polymer moduli faster and easier, the values obtained are not yet necessarily reliable. This too may change.

7.8 References Adams, W. W, Briber, R. M., Sherman, E. S., Porter, R. S., Thomas, E. L. (1985), Polymer 26, 17. Akindayini, O. O., Nix, E. L., Ward, I. M. (1986), /. Phys. E19, 911. Anand, J. N. (1967), /. Macromol. Sci. Phys.Bl, 445. Arridge, R. G. C. (1975), Mechanic of Polymers. Oxford: Clarendon Press. Arridge, R. G. C , Barham, P. J. (1978 a), Polymer 19, 654. Arridge, R. G. C , Barham, P. J. (1978b), J. Polymer Sci. B; Polymer Phys. Ed. 16, 1297. Asahina, M., Enomoto, S. (1962), J. Polymer Sci. 59, 101. Barham, P. X, Arridge, R. G. C. (1977), J. Polymer Sci. B: Polymer Phys. Ed. 15, Mil. Barron, T. H. K., Rogers, K. X (1989), Molecular Simulation 4, 27. Batcheldor, D. N., Bloor, D. (1979), /. Polymer Sci. B: Polymer Phys. Ed. 17, 569. Baughman, R. H. (1973), /. Chem. Phys. 58, 2976. Boudreux, D. S. (1973), J. Polymer Sci. B: Polymer Phys. Ed. 11, 1285. Boyd, R. H. (1983), /. Polymer Sci. B: Polymer Phys. Ed., 21, 493-504. Brew, B., Clements, X, Davies, G. R., Jakeways, R., Ward, I. M. (1979), J. Polymer Sci. B: Polymer Phys. Ed. 17, 351.

7.8 References

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353

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Nishino, X, Ohkubo, H., Nakamae, K. (1992), /. Macromolec. Sci. B31, 191. Northolt, M. G. (1980), Polymer 21, 1199. Northolt, M. G., Van Aartsen, J. J. (1978), /. Polymer Sci. C 58, 283. Northolt, M. G., Van der Hout, R. (1985), Polymer 26, 310. Nye, J. F. (1957), Physical Properties of Crystals. Oxford: Clarendon Press. Odajima, A., Maeda, X (1966), J. Polymer Sci. C 15, 55. Ohta, X (1983), Polymer Eng. and Sci. 23, 697. Peselnick, L., Meister, R. (1965), /. Appl. Phys. 36, 2870. Phoenix, S. L., Skelton, J. (1974), Textile Res. J. 44, 934. Pinnock, P. R., Ward, I. M., Wolfe, J. M. (1966), Proc. Roy. Soc. A 291, 267. Postema, A. R., Smith, P. (1990), Macromolecules 23, 3296-3301. Powell, A. K., Craggs, G., Ward, I. M. (1990), /. Mater. Sci. 25, 3990-4000. Prasad, K., Grubb, D. X (1989), J. Polymer Sci. B: Polymer Phys. Ed. 27, 381. Prasad, K., Grubb, D. X (1990), /. Polymer Sci. B: Polymer Phys. Ed. 28, 2199. Prevorsek, D. C , Harget, P. I , Sharma, R. K., Reimschussel, A. C. (1973), /. Macromolec. Sci. B 8,127. Prevorsek, D. C , Chin, H. B., Kwon, Y D., Field, J. E. (1991), J. Appl. Polymer Sci., Appl. Polymer Symp., 47, 45-66. Pulay, P., Fogarasi, G., Pongor, G., Boggs, J. E., Vargha, A. (1983), J. Am. Chem. Soc. 105, 7037. Rabolt, J. F., Fanconi, B. (1977 a), Polymer 18, 1258. Rabolt, J. F, Fanconi, B. (1977b), /. Polymer Sci., Polymer Lett. Ed. 15, 121. Rabolt, J. F., Wang, C. H. (1983), Macromolecules 16, 1698. Rasmussen, R. S. (1948), J. Chem. Phys. 16, 712. Raumann, G. (1962), Proc. Phys. Soc. 79, 1221. Rawson, F. F., Rider, J. G. (1974), J. Phys. D7,A\. Read, B. E., Dean, G. D. (1978), The Determination of Dynamic Properties of Polymers and Composites. Bristol: Adam Hilger. Rehwald, W, Vonlanthen, A., Meyer, W (1983), Phys. Stat. Sol. (a) 75, 219. Reuss, A. (1929), Z. Angew. Math. Mech. 9, 49. Richardson, I. D., Ward, I. M. (1978), J. Polymer Sci. B: Polymer Phys. Ed., 16, 667. Rider, J. G., Watkinson, K. M. (1978), Polymer 19, 645. Robinson, I. M., Young, R. I, Galiotis, C , Batcheldor, D. N. (1987), /. Mater. Sci. 22, 3642. Robinson, I. M., Galiotis, C , Batcheldor, D. N., Young, R. J. (1991), / Mater. Sci. 26, 2293. Rogozinsky, A. K., Bazhenov, S. L. (1992), Polymer 33, 1391. Rossignol, J. M., Seguela, R., Rietsch, F. (1988), Polymer 29, 43-48.

7.8 References

Roylance, D. K., DeVries, K. L. (1971), J. Polymer Sci. Lett. 9, 442. Runt, J., Wagner, R. R, Zimmer, M. (1987), Macromolecules 20, 2531. Sachse, W, Hsu, N. N. (1989), in: Physical Acoustics, Vol. XIV: Mason, W. P., Thurson, R. N. (Eds.). New York: Academic Press, Chap. 4. Sachse, W, Castagnede, B., Grabec, I., Kim, K. Y, Weaver, R. L. (1990) Ultrasonics 28, 97. Sachse, W, Every, A. G., Grabec, I. (1991), in: Enhancing Analysis Techniques for Composite Materials: Reddy, J. N. (Ed.). New York: ASME-AMD. Saimoto, S. (1986), J. Mater. Sci. 21, 3686. Sakurada, I., Nukushina, Y, Ito, T. (1962), J. Polymer Sci. 57, 651. Sakurada, I., Ito, T., Nakamae, K. (1964), Makromol. Chem. 75, 1. Sakurada, I., Ito, T., Nakamae, K. (1966), J. Polymer Sci. C15, 75. Sakurada, L, Kaji, K. (1970), /. Polymer. Sci. C31, 57. Samuels, R. J. (1974), Structured Polymer Properties. New York: Wiley. Sawatari, C, Matsuo, M. (1986), Macromolecules 19, 2653. Schachtschneider, J. H., Snyder, R. G. (1963), /. Polymer Sci. C 7, 99. Seferis, J. C , Samuels, R. J. (1979), Polymer Eng. Sci. 19, 975. Shauffele, R. R, Shimanouchi, T. (1967), /. Chem. Phys. 47, 3605. Sherman, E. S., Porter, R. S., Thomas, E. L. (1982), Polymer 23, 1069. Shimanouchi, T., Asahina, M., Enomoto, S. (1962), /. Polymer Sci. 59, 93. Shiro, Y, Miyazawa, T. (1971), Bull. Chem. Soc. Jpn. 44, 2371. Slutsker, A. L, Savitskii, A. V, Ismonkulov, K., Sidorovich, A. A. (1986), Vyskomol. Soyed. A 28, 978; english translation in: Polymer Science USSR 28, 1091. Smith, J. I, Jiang, H., Eby, R. K., Adams, W. W. (1987), Polymer Comm. 28, 14. Smith, J. C , Blandford, X M., Scheifer, H. R (1960), Text. Res. J. 30, 752. Smith, P., Termonia, Y (1989), Polymer Comm. 30, 66-68. Smook, J., Pennings, A. J. (1984), Coll. Polymer Sci. 262, 712. Song, K., Krimm, S. (1990), /. Polymer Sci. B: Polymer Phys. Ed. 28, 63-69. Sorensen, R. A., Liau, W. B., Kesner, L., Boyd, R. H. (1988), Macromolecules 21, 200. Strobl, G. R., Eckel, R. (1976), J. Polymer Sci. B: Polymer Phys. Ed. 14, 913. Suhai, S. (1983), J. Polymer Sci. B: Polymer Phys. Ed. 21, 1341. Tadaoki, I., Tashiro, K., Kobayashi, M., Tadokoro, H. (1987), Macromolecules 10, 731.

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8 Rubber Elasticity R.Ullmanf Department of Nuclear Engineering and Macromolecular Research Center, The University of Michigan, Ann Arbor, MI, U.S.A.

List of Symbols and Abbreviations 8.1 Introduction and Historical Background 8.2 Rubber Processing 8.2.1 Vulcanization of Natural Rubber 8.2.2 Some Aspects of Rubber Chemistry 8.3 Theory of Rubber Elasticity 8.3.1 Thermodynamics of Rubber Elasticity 8.3.2 The Statistical Theory of Rubber Elasticity 8.3.3 The Mooney-Rivlin Model of Rubber Deformation 8.3.4 Other Phenomenological Models of Rubber Deformation 8.4 Experimental Procedures in Rubber Deformation 8.4.1 Hysteresis 8.4.2 Some Experimental Investigations of Elastomeric Deformation 8.4.3 Some Swelling Experiments 8.5 Orientation of Chain Segments 8.5.1 The Statistics of Highly Stretched Chains 8.5.2 Optical Birefringence of Elastomers 8.5.3 Fluorescence of Labeled Elastomers 8.5.4 Quadrupole Splitting in NMR Spectra of Swelling Agents by Oriented Rubber 8.6 Small Angle Neutron Scattering (SANS) of Elastomers 8.7 Rubber-Glass and Rubber-Crystal Transitions 8.8 Thermoplastic Elastomers 8.9 Ultimate Strength 8.10 References

Materials Science and Technology Copyright © WILEY-VCH Verlag GmbH & Co KGaA. All rights reserved.

358 361 362 362 362 363 363 365 369 370 370 371 371 374 375 375 376 377 378 379 383 384 385 386

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List of Symbols and Abbreviations A A, B at atj Ao AAQl AAm A^4S A^.A'y b Bl9B2 c c C C C1, C 2 / F G H / (q) h> In 13 k k0 kB L L S£(y) m M Mo Mc Mn Mw n n Hi n2 N NA Nv

Helmholtz free energy constants depending on molecular weight, concentration, and scattering contrast factors exponents in the Ogden equation constant in the generalized Rivlin equation cross sectional area of undeformed sample change in elastic free energy change in free energy on mixing Nx solvent molecules with N2 polymer molecules change in free energy of swelling parameters of experimental front factors length of a statistical chain segment constants in a birefringence deformation equation specific heat crack length stress optical coefficient constant in the strain energy function constants in the Mooney equation network functionality magnitude of force Gibbs free energy enthalpy scattering intensity symmetric invariants of deformation unit vector in the direction of the incident neutron beam unit vector in the direction of the scattered neutron beam Boltzmann's constant length interlamellar distance Langevin function parameter in Tschoegl-Gurer equation molecular weight monomer molecular weight molecular weight between crosslinks number average molecular weight weight average molecular weight number of chain segments in a polymer chain mean refractive index, n = {n1-\-2n2)/Z refractive index parallel to stretching direction refractive index perpendicular to stretching direction number of chains in a specimen Avogadro's number, iV A = 6.023 x 10 2 3 number of chains per unit volume

List of Symbols and Abbreviations

N1

359

i?ll R± S Ss(q) ST(q) T To u U V vx W W w(X) z

number of solvent molecules (Flory-Huggins theory of polymer solutions) number of polymer molecules (Flory-Huggins theory of polymer solutions) vapor pressure of solvent in polymer vapor pressure of pure solvent vapor pressure of solvent in polymer solution vapor pressure of solvent above crosslinked polymer probability function pressure 2nd Legendre polynomial P2 (cos 0) = (3 cos 2 9 -l)/2 wave vector magnitude of q value of q at a maximum in a plot of q2 S(q) versus q vector connecting chain ends mean value of r value of r for an undeformed chain vector connecting monomers ij on same or different polymer chains vector connecting monomers i,j on the same polymer chain ensemble average of r2 fluctuation in r ideal gas constant radius of gyration apparent radius of gyration measured at an azimuthal angle

= 90° entropy single chain scattering function total scattering function temperature threshold tearing strength mean energy of a chemical bond internal energy volume molar volume of solvent work strain energy function function in the Valanis-Landel equation number of monomer units in a statistical segment

a1 a2 P y

parallel polarizability of a statistical chain segment perpendicular polarizability of a statistical chain segment quantity in the theory of highly stretched chains, p = ^'x (r/(n b)) shear gradient

N2 Pi p? pu pc P(r,r + dr, n) P P 2 (cos 9) q q qpeak r r r0 rtj r'ij
360

8 Rubber Elasticity

Q

angle between stretching axis and C - D bond scattering angle, equal to twice the Bragg angle polar angle polar angle of segment i parameter in the Flory-Erman theory neutron wavelength deformation ratio deformation ratios in x-, y-, and z-directions chemical potential of solvent splitting frequency in deuterium NMR spectrum osmotic pressure density stress principal stresses Flory-Huggins interaction parameter azimuthal angle function in the Ogden equation volume fraction of solvent volume fraction of polymer angle between stretching axis and magnetic field

GPC MEK NMR PB PS PTMI PTMO PTMT SANS TPE

gel permeation chromatography methylethylketone nuclear magnetic resonance polybutadiene polystyrene poly(tetramethylene isophthalate) poly(tetramethylene oxide) poly(tetramethylene terephthalate) small angle neutron scattering thermoplastic elastomers

y 9 9

Av n a

X cp (pi

cpx cp 2

8.1 Introduction and Historical Background

361

8.1 Introduction and Historical Background Rubber elasticity is the property of reversible deformability of large magnitude exhibited by a number of polymeric materials, chief among which is natural rubber, a substance prepared from the coagulated sap of a tropical tree, Hevea brasiliensis. Under certain conditions, noncrosslinked polymers will show rubbery behavior due to chain entanglements and/or some type of physical crosslinks such as crystalline microdomains (see Chapter 9). True rubber elasticity requires a permanent network (see Fig. 8-1). Upon cooling, rubbers become glassy, and some rubbers partially crystallize when cooled. Natural rubber becomes glassy at -70°C before vulcanization, and vulcanized natural rubber is a glass at -60°C. Conversely, crosslinked polymeric glasses become rubbery when heated. An example is polystyrene crosslinked by divinylbenzene, which is glassy below 80 °C, but is a rubber at 150°C. In the absence of deformation, natural rubber has an X-ray pattern at room temperature similar to that of a low molecular weight liquid. Katz (1925) showed by Xray diffraction that natural rubber crystallizes at ordinary temperatures when stretched. Crystallization of natural rubber occurs slowly at lower temperatures (0°C and less), in the undeformed rubber as well. The shear modulus of a typical rubber is of the order of 20 N/cm 2 , rubber has an elastic deformability of several hundred percent, and a bulk modulus thousands of times greater than the shear modulus. The molecular mechanism of rubber elasticity is distinctly different from the elasticity of metals and other hard solids. The relationship between molecular and macroscopic deformation in rubbers is central to the understanding of elastomeric materials.

Figure 8-1. (a) Schematic depicting entangled linear polymer chains. Such materials exhibit rubbery behavior for short times at temperatures above their glass transition, (b) Schematic depicting a crosslinked network with tetra-functional junctions. Such samples can show nearly ideal rubber elasticity.

The unique mechanical properties of elastomers account for their use in a wide variety of industrial products such as pneumatic tires, elastic foams and rubber bands. Much of the scientific interest in rubber is the consequence of the broad applicability

362

8 Rubber Elasticity

of elastomeric substances for commercial use. Michael Faraday (1826) worked on the chemical composition of rubber. He found a weight ratio of carbon to hydrogen of 6.8 (the correct value is 7.5). On this basis, he suggested that rubber contained a ratio of eight carbon atoms to 7 hydrogen atoms. Only a few years after this, Goodyear (1844) in America and Hancock (1844) in England, reported that heating rubber with sulfur improved its properties in several ways. Tackiness is eliminated, stiffness of the rubber can be increased if desired, and solubility in a variety of solvents is eliminated. In 1805, Gough (1805) discovered some highly important thermoelastic properties of rubber. He found that rubber which is rapidly stretched, increases in temperature. He also found that a strip of rubber from which a small weight is suspended shrinks on being heated. These results, obtained on unvulcanized rubber, were confirmed by Joule (1859) on the vulcanized product a half century later. Thomson (1857) (later Lord Kelvin) provided a thermodynamic analysis of rubber which yielded a clear explanation of the results found by Gough and by Joule. A number of researchers in the last decades of the nineteenth century and first decade of the twentieth century studied the addition of fillers in rubber. A report by Mote of the India Rubber, Gutta Percha and Telegraph Works of Silverton, England on the reinforcement of rubber by carbon black was never published, but laid the basis for the use of carbon black as a reinforcing filler in rubber tires. Several interesting synthetic rubbers were developed in the two decades after 1920. Butadiene-styrene copolymers known as Buna S were synthesized, and these were rubbery at room temperature if the butadiene content was sufficiently high. Rubbers

made from copolymers of styrene and acrylonitrile, known as Buna N, were resistant to attack by oil and grease, chloroprene (2-chlorobutadiene) could be made into a rubber with excellent properties. Thiokol rubber was prepared from sodium polysulfide and 1,2-dichloroethane. Polymerization of isobutene led to a tacky elastomer known as Vistanex, and copolymerization of isobutene with a small quantity of butadiene yielded the vulcanizable rubbery product known as butyl rubber. Today, a great variety of rubbery materials exist. All derive their favorable physical properties from their molecular network structure.

8.2 Rubber Processing 8.2.1 Vulcanization of Natural Rubber

An elastomeric material has a molecular structure of an interconnected network of polymeric chain molecules. The network is usually based on primary chemical bonds between the chains, but can also be formed by strong attraction between the chains (e.g., by functional groups on these chains or by crystallization). As mentioned above, the process of vulcanization was first developed by Goodyear (1844) and Hancock (1844) by heating natural rubber with sulfur. Chemical bonds are formed by chains of sulfur atoms which react with the unsaturated bonds of the primary macromolecules. Modern elastomers utilize a wide variety of chemicals to produce the permanent network. 8.2.2 Some Aspects of Rubber Chemistry

Natural rubber poly(ds-isoprene) contains one double bond per monomer before vulcanization, and still contains many unreacted double bonds after vulcaniza-

8.3 Theory of Rubber Elasticity

tion is completed. These unreacted double bonds react slowly with oxygen from the air, which can lead eventually to degradation of the rubber. The unsaturation of the rubber insures that rubber is chemically active with a large variety of substances both before and after vulcanization. Hydrogenation of rubber to saturate the residual double bonds brings about chemical stabilization. Rubbers can also be chlorinated, and these chlorinated products have many applications. Many other chemical reactions can be brought about with natural rubber. Network formation usually takes place by chemical bonding of the primary macromolecules of which the network is formed (for a detailed discussion, see Vol. 18 of this Series). Synthetic rubbers, prepared from hydrocarbons, contain unsaturated bonds in the chain which can be crosslinked by reaction with organic peroxides. A good example is the preparation of butyl rubber by reaction of an isobutenebutadiene copolymer with dicumyl peroxide. The mixture of polymer and peroxide is heated, the peroxide decomposes to form free radicals, and these react with the double bond in the polymer to form a stable molecular network. Polydimethylsiloxanes are usually prepared from the cyclic tetramer by ringopening polymerization. The product is a linear polymer chain which is liquid at room temperature. Addition of a small quantity of trifunctional or tetrafunctional monomer to the initial reactants leads to formation of a three dimensional rubbery network. Crosslinking can also be brought about by reacting a vinyl terminated linear polysiloxane with a source of free radicals which, for example, are generated by thermal decomposition of an organic peroxide. Silicone rubbers are more heat-resistant than hydrocarbon rubbers, and are often

363

used when stability at high temperatures is an important requirement and at very low temperatures where flexibility is important. Polyurethanes are condensation products of polyhydric alcohols and di- or triisocyanates. Linear polymers are formed by reaction of stoichiometric quantities between dihydric alcohols and diisocyanate. Addition of a small amount of tri- or tetrahydroxy alcohol or of a triisocyanate leads to the formation of a three-dimensional rubbery network. Fluorocarbon rubbers are prepared by crosslinking a copolymer of tetrafluoroethylene and hexafluoropropylene. These rubbers are thermally stable and have very low coefficients of friction. They are resistant to oxidative degradation and to swelling by hydrocarbons. They are excellent materials for a variety of specialized applications.

8.3 Theory of Rubber Elasticity 8.3.1 Thermodynamics of Rubber Elasticity

If the work done by the system is purely mechanical, either elastic or dilatational, dW = PdV — FdL, P being pressure, V9 volume, F, force, and L, length. In most experiments on elastomers, PdV is thousands of times smaller than FdL, since rubber volumes are largely unchanged when the rubber specimen is deformed. In such circumstances, the PdV term is neglected so that the derivative of the Gibbs free energy is

In applications in which pressure - volume changes are small, VdP and PdV are omitted in thermodynamic calculations. Then dU = dH and the Helmholtz free en-

364

8 Rubber Elasticity

ergy and Gibbs free energy are related as dA = dG. From Eq. (8-1), it follows that

2.0

if

(8-2) 1.6

//

and using Maxwell's relations PLT

'

(8-3)

d¥

Equations (8-2) and (8-3) are based on the constant volume assumption for rubber. Using these equations, changes in entropy, free energy, and internal energy are derived from uniaxial force elongation experiments as a function of temperature.

My//

1.2 ro Q_ _c b

0.8

0.4 /

&.A = J F dL;

T is constant

(8-4 a)

Li

(c)

0 .

\

(8-4 b) AU =

TAS

(8-4 c)

Studies of changes in free energy, entropy and internal energy of elastomers have been performed by many researchers. The actual experimental process may be difficult because of hysteresis in stretching and relaxation. Also, real rubbers may creep or exhibit stress relaxation if crosslinking is incomplete, or if the substance undergoes slow chemical degradation while under stress. In Fig. 8-2, experiments on changes in entropy, free energy, and inter* nal energy of natural rubber as a result of stretching (Anthony et al., 1942) are shown. It is clear from this study that changes in internal energy upon deformation of rubber are very small by comparison with changes in free energy and the entropy component of free energy. Changes in internal energy can become substantial at high deformations in some elastomers, but this is a result of crystallization. The small change in internal energy at moderate de-

Figure 8-2. Tensile stress-deformation measurements on natural rubber, (a) G versus A; (b) — T(dcr/dT)x versus A; (c) o— T(da/dT)^ versus X [curve (a) — curve (b)]. The areas under each of these curves yield A^4, - TAS, and A U, respectively. (Anthony et al., 1942.)

formation is akin to the small change of internal energy of a dilute gas under compression. Accordingly, an ideal rubber is similar to an ideal gas. Changes in internal energy upon isothermal deformation are generally small, though easily measurable. If a rubber specimen is rapidly stretched there is an increase in temperature. The specific heat is roughly constant during elongation. Data on rubber by Joule (1859) and by James and Guth (1943) on adiabatic heating of rubber are shown in Fig. 8-3. Note that AT is negative at very low elongation, which arises from an initial increase in entropy which takes place before the large decreases in entropy occur at higher deformations. The minimum in the

8.3 Theory of Rubber Elasticity

0.20

J

0.16

— C

o in

>

>

c

c

0.08

u

/

t_

L_

A/

A /

tic

0.12

o

sion

,o

/

(0

JQ

E 0.04

o

(0 TJ <^

/

A

(8-5)

A*

0.00

1.0

1.2

links is almost never fully extended, but is coiled to a considerable degree. In an assembly of such polymer chains, the ends of the chains are separated by a distance r which varies with time and differs from one chain to another. In the absence of external forces, the probability that the end-to-end vector lies between r and r + &r is given by

/ /

— In H

|

365

I

I

I

1.4

1.6

1.8

Figure 8-3. Temperature rise of rubber in adiabatic extension, A: Joule (1859); o: James and Guth (1947).

AT versus deformation plot is known as the isometric inversion point, and the point at which AT = 0 is known as the adiabatic inversion point.

8.3.2 The Statistical Theory of Rubber Elasticity

The fundamental ideas for a statistical mechanical analysis of rubber appear in works by Meyer et al. (1932) and also by Karrer (1933). The latter was aimed primarily on a theory of muscle relaxation. In 1934 a statistical theory of rubber elasticity was put forth (Guth and Mark, 1934) which developed the ideas of Meyer and co-workers by formal statistical mechanical methods. The Guth-Mark theory was based on what is known today as a Gaussian polymer chain. In this model, a rubber is considered as an assembly of polymer molecules, each composed of n links of length b connected to each other by covalent chemical bonds. These links rotate about each other in space. The chain of n

Since the mean square distance between the ends of such chains is given by (r2y = nb2, it is easy to verify that a = (3/2) nb2. This provides the background for calculations of molecular dimensions of a polymer chain in a rubbery polymer described by a Gaussian model. A rubbery material composed of an ensemble of Gaussian chain molecules undergoes a partial orientation and elongation when subjected to an applied external force. In the theory of rubber elasticity, the deformation of the macroscopic specimen is identified with the molecular deformation of the polymer chains of which the elastomer is composed. Two classical pictures play prominent roles in this description, one is known as thefixedjunction or junction-affine model, and the other is called the phantom network. In these models, three or more Gaussian polymer chains are joined at their ends by primary chemical bonds to form a three dimensional network with crosslink junction points (see Fig. 8-1 b). In the fixed junction model, these crosslink junctions are immobile with respect to the deformed Cartesian coordinates, which transform affinely with the macroscopic deformation. If the macroscopic deformation is given by XY, X2, and A3 in the x-, y-, and z-directions, it follows from the fixed junction model that the mean square Cartesian components of the

366

8 Rubber Elasticity

deform according to

end-to-end vector deform as

nb:

f

nb2^

(8-6 a) (8-6 b) <2 2 >

nb: =

(8-6 c)

The subscript '0' signifies the value of the quantity before deformation of the polymer chain. The angle brackets indicate an ensemble average. In the phantom network, the name currently in use for the model of James and Guth (James, 1947; James and Guth, 1947), the crosslink junctions fluctuate about mean positions. When the rubber is deformed and the mean values of the components of the end-to-end vector of each chain are deformed by factors /l 1? A2, and A3, r fluctuates about that mean owing to Brownian motion. These fluctuations are calculated using the theory of Brownian motion of harmonically bound particles. It is assumed that the fluctuations are not influenced by the presence of neighboring chains in the rubber, and are independent of r, the mean value of r. r = r + 5r 2

2

(8-7 a) 2

r = r + (Sr)

(8-7 b)

The network functionality, /, is defined as the number of polymer chains joined at a crosslink. In the phantom network 2 <(5r) 2 >=y

(8-8)

One of the interesting characteristics of the theory is that the magnitude of the fluctuations is independent of the deformation of the network. Since = nb2, <(5r)2> =

(2/f)nb2, and = (1 -2/f)nb2.

Upon

deformation, the Cartesian components

7

(8-9)

2

Similar equations hold for and . Since the crosslink functionality is commonly three or four, the difference between predictions of the fixed junction and phantom network models is numerically large, as a comparison between Eqs. (8-5) and (8-9) shows. The theoretical models are used not only for estimating chain deformation resulting from stretching a rubber, but also to yield predictions of thermodynamic changes which occur as a result of sample deformation. Consider a network composed of N polymer chains, each containing n chain segments of length b. A chain segment characteristically contains a few (3 to 10) monomer units. The entropy of a chain in an undeformed specimen with chain ends separated by a distance r is given by S(r) = const + kB In P{r) = = const — kBar2

(8-10)

Here, kB is Boltzmann's constant. If the specimen is deformed by factors Al9 A2, and X3 in the three principal directions, the contribution of the deformation along the x-direction to the entropy is ix)

=

const — kBaAf2x2

(8-11)

where Xf = X1 for the fixed junction model, and k%2 = X\ (1 - 2//) + 2 / / for the phantom network. For the entire network, the entropy is calculated by averaging over all arrangements of the unperturbed chain multiplied by the number of chains. The entropy change is given by

8.3 Theory of Rubber Elasticity

Equation (8-12) for the phantom network is just obtained by multiplying the fixed network result by a factor of (1—2//). The point of view taken in developing these equations is that the internal energy of the elastomer is unchanged by specimen deformation, at constant temperature, and therefore AA = - TAS (8-13) The number of network chains N equal the number of chains per mole, NA, which is Avogadro's number, multiplied by density and sample volume and divided by the molecular weight between crosslinks, M c . Equation (8-13) may therefore be written (8-14) In uniaxial extension, kx = k and A2 = A3 = X~1/2. In this case, the free energy change of the fixed junction model is given respectively by (8-15) The free energy change is the reversible work of stretching. Therefore, the force exerted on a specimen in uniaxial deformation is given by

RTQA0

(8-16)

Ao is the area of cross section of the specimen before deformation. Using the constancy of rubber volume upon deformation, the stress on the sample in the deformation direction, a = F/A9 is given by

-('-?)(*--

(8-17)

In many experimental studies, researchers work with rubbers swollen by

367

solvents. The free energy changes calculated above yield the elastic part of the free energy only. In addition, there is a term in the free energy arising from solvent-polymer interaction, and this is part of the total free energy change of swelling. The calculation of the total free energy of swelling is usually performed by treating the elastic and solution contributions as independent additive components. The free energy of swelling is based on the Flory-Huggins theory of polymer solutions (Flory, 1942; Huggins, 1942, 1943). In the Flory-Huggins theory, the free energy of mixing of Nx solvent molecules with N2 polymer molecules is given by (8-18) In this theory, the volume of a solvent molecule is taken to be equal to that of one monomer unit on a polymer molecule, and cp1 and q>2 are the volume fractions of solvent and polymer respectively. % is known as the Flory-Huggins interaction parameter, and its magnitude characterizes the strength of the polymer-solvent interaction energy (see Chap. 7, Vol. 5 of this Series). In applying Eq. (8-18) to a swollen elastomer, there is only one polymer molecule, that is, N2 equals one, and the term JV2 lncp2 is negligibly small. However, q>2 is a number usually between 0.1 and 1.0, and, accordingly, the final term in Eq. (8-18) is substantial in magnitude. The solvation part of the free energy of swelling of a network is then given by A^ s = /cBT(iV1ln(p1+ZiV1(p2)

(8-19)

The elastic free energy of swelling, A^4e, is given by Eq. (8-13) with the linear swelling ratios equal in all directions A =— A-£ -— A 2 ~~~ A 3

(8-20)

368

8 Rubber Elasticity

The total free energy of swelling becomes AA = (8-21) The number of polymer chains in the rubber specimen is AT, and these are interconnected by crosslinks to form a single gigantic macromolecule. The factor 1 — 2 / / appears in the phantom network result, but is omitted in the formula for the fixed junction model. Many studies on swollen elastomers are conducted by deforming the isotropic gel by stretching or compressing. The additional change in free energy arising from uniaxial deformation is given by - - 3

(8-22)

for the phantom network. In Eq. (8-22), X is the linear deformation ratio of the already swollen elastomer. In general, the chemical potential of the solvent in a swollen elastomer differs from that of the pure solvent, and this accounts for changes in vapor pressure of the solvent and accounts for an osmotic pressure of the swollen gel. The chemical potential of the solvent is calculated by differentiating AA with respect to N1. In doing this, it is essential that the interrelations between 2, X, and Nx be included specifically in the expression for AA. Working from Eq. (8-21) for AA, the difference in chemical potential between the pure liquid and the solvent in the swollen elastomer is Aft i = kB T (In cp! +
(8-23)

To the extent that the solvent in the vapor phase may be treated as an ideal gas (8-24)

where px is the vapor pressure of the solvent above the swollen elastomer and p\ is the vapor pressure of the pure solvent. By measurement of these vapor pressures, it becomes possible to ascertain whether Eq. (8-23) correctly describes gel swelling. The osmotic pressure, n, of the solvent in the swollen gel is linked to the chemical potential of the solvent by An1 = -nvl

(8-25)

where v1 is the partial molecular volume of the solvent. Measurement of osmotic pressure provides another useful method for checking the theoretical equations. The equilibrium swelling ratio of an elastomer occurs at a point where the chemical potential of the solvent in the gel equals that of the pure solvent, that is, A / ^ ^ 0 . The Flory interaction parameter, X, can be extracted from Eq. (8-23) by setting Afix to zero, by use of measurements of the maximum swelling of the elastomer. This procedure has been adopted in a number of investigations. The thermodynamics of network deformation by applied mechanical forces or by solvent interaction as described above treats polymer chains as ideal random flight species taking no account of attractive and repulsive interactions between chain segments or of entanglements of the polymer chains. This has been long recognized as a serious defect of the elementary models, can in some recent theories, corrections for these omissions have been introduced. There is no general agreement among specialists how to make these corrections, and some current approaches will be considered next. One major effort to modify the elementary model was initiated by Flory (1976, 1977), and was further developed by Erman and Flory (Flory and Erman, 1982; Erman and Flory, 1982). In their model,

8.3 Theory of Rubber Elasticity

the effect of entanglement and other interchain interactions is to change the mean position of the crosslink junctions and to reduce the magnitude of the junction fluctuations. Flory and Erman introduced new parameters in the theory to account for the effects. The principal quantity, x, is the ratio of the mean square junction fluctuation in the phantom network to that of the real network. The fluctuations in this model are smallest in the undeformed network, but become larger as the network is stretched. The phantom network limit is approached as the deformation becomes very large. Networks tend to be more like the fixed junction model in the undeformed state. Another approach to a corrected theory of rubber elasticity is based on tube models. This idea, introduced by de Gennes (1971), has been extended by Edwards and co-workers (Ball, 1981; Edwards and Vilgis, 1987). In the Edwards-type model, a polymer chain in an elastomeric network can be treated as a long worm encased in a tube composed of neighboring molecules, and the presence of this tube constrains the motion of the chain and limits the number of conformations available to the chain. The equations for chain deformation and chain thermodynamics take into account the tube structure, and provide predictions of elastomeric behavior of a stretched or swollen network. Some specialists in rubber elasticity have used two-network models to represent the behavior of real rubbers. Chief among the proponents of this point of view have been Ferry and Graessley. In a two network model, contributions to the modulus arise from retractive forces imposed by the primary chemical network plus retractive forces originating in the entanglement structure. These contributions are assumed to be additive, with the actual elastic modulus taken as the sum of the moduli of the

369

two networks. Experiments by Ferry and associates and by Graessley and co-workers tend to be in good agreement with this view. Details of some of these studies will be discussed later. 8.3.3 The Mooney-Rivlin Model of Rubber Deformation The molecular theory of rubber elasticity based on the unmodified Gaussian structure of a polymer chain leads to the result that the energy stored in deformation of an elastomer may be written as W=C(kl + kl + kl-3)

(8-26)

where C is a constant derived from the molecular theory and W is known as the strain energy function. Equation (8-26) is clearly inadequate for explaining many experiments on rubber deformation. In 1940, Mooney (1940) proposed a more general equation which satisfies Hooke's law for a sheared elastomer. Mooney's equation for the strain energy function is

(8-27) A9

Aa

This result is consistent with a variety of experiments on rubber over a fairly wide range of deformation. A few years later, Rivlin (1948) developed this model much further. According to Rivlin, the stored energy could be written as a general function of the simplest symmetric invariants of the deformation. These invariants are (8-28 a) 12 ~~ A^ A 2 T

_

;2 ;2

1 3 — Ax

1 A^ A 3 ;2

A2A3

1 A 2 A3

(8-28 b) (8-28 c)

and since, for rubber at ordinary pressures, / 3 is approximately equal to unity

370

8 Rubber Elasticity

(due to its small compressibility), Mooney's equation is the linearized general expansion of the free energy in these invariants,

strain proposed that W=1£ci
(8-31 a)

i

W=I.aij(Ii-3)i(I2-3y

(8-29)

In fact, Rivlin and his associates often use the Mooney equation, Eq. (8-27), in interpreting experiments, though they recognized that higher terms are often required to provide a fit to experiments over a wide range of deformation. The Mooney-Rivlin analysis is a purely phenomenological model, and is not based on molecular theory. The lack of a molecular basis does limit the usefulness of the analysis, but nevertheless, the equations provide an interesting framework for treating elastomeric deformation. It is implicit in the model that a particular set of constants used for a given material is expected to apply regardless of the mode of deformation. Experiments, for example, in uniaxial deformation should yield the same Cx and C 2 constants in Eq. (8-27) as those obtained in shear, torsion, or biaxial stretching. 8.3.4 Other Phenomenological Models of Rubber Deformation Valanis and Landel (1967) proposed that the strain energy function could always be decomposed into a sum of functions of the deformation in each of the three principal directions according to W= w ^ J + w(A2) + w(X3)

(8-30)

This is consistent with the Mooney equation, Eq. (8-27), though it is not in accord with the Rivlin generalization, Eq. (8-29). The Valanis-Landel hypothesis is also consistent with other proposals of a number of different researchers. Ogden (1972), in an effort to account for rubber deformation over a broad range of

(Pi^A) = A1 + A 2 + A3

(o-31 D)

In this formulation, the quantities at are constants but are not necessarily integral, and are not necessarily the same for different materials. A two-network model of the Ogden type was adopted by Tschoegl and Gurer (Tschoegl and Gurer, 1985; Gurer and Tschoegl, 1985). It was applied to bulk elastomers and also to swollen networks. Agreement between experiment and theory was achieved by determining the parameters at at low strains, and applying the result to data collected over a much wider range of strain. Tschoegl and Gurer found ax equal to two and a2 equal to 0.34 for the experiments they analyzed.

8.4 Experimental Procedures in Rubber Deformation Rubber deformation can be studied in many ways. The procedure most commonly used is that of uniaxial stretching. In that case, the strain energy function of a Gaussian network reduces to (8-32) The strain energy function for uniaxial deformation in the Mooney equation is

(8-33) The stress is given by k

TV

- x

(8-34)

8.4 Experimental Procedures in Rubber Deformation

It follows from Eq. (8-34) that a plot of a/(A2-l/X) versus 1/1 yields both C1 and C 2 from which the modulus is obtained as a function of deformation. One common experimental technique is to measure the response of an elastomer to homogeneous biaxial deformation. In such experiments, it is customary to vary the deformation ratios in the two principal directions. Two cases are of particular importance. One of these is the pure shear experiment in which Ax is varied, k2 is kept equal to unity, and A3 adjusts to the constant volume requirement, /L3 = l/^i. The second important case is that of equal biaxial deformation where kl = X2 and A3 = l/Af. In the case of pure shear, the Mooney equation yields (8-35 a) (8-35 b)

= 2(C1-C2)

where u1 and <x2 a r e the principal stresses. The equal biaxial stretch result is

A? '

i = 1,2

(8-36)

Simple shear is not a pure strain. In simple shear X2 equals unity, X3 = l/X1 and Mooney's equation gives (8-37) The shear gradient y, is Xx — 1/Al5 from which it follows that W = (C1 + C2)y2. The corresponding shear stress, ak equals The strain energy function has been used in other deformation modes for calculation of stresses, which are worked out as needed. The stresses given in the formulae presented above are the stresses induced by

371

applied tensions or pressures. In reality, the principal stress in any direction is equal to that imposed by the applied forces minus the pressure. In experiments carried out at high pressures or at extremely low applied forces, the pressure should be included in the stress equations. 8.4.1 Hysteresis The analysis of elastomeric deformation discussed above is based on relations between stress and strain of a material in equilibrium. In fact, in many experiments in which applied force and sample deformation are measured, the forces required to produce a particular deformation change with time, and depend on whether the successive increments in stress are positive or negative. In order to obtain equilibrium results, it is necessary that experiments be carried out slowly or by raising and lowering temperature successively. The lag in attaining equilibrium is more common with bulk elastomers than with swollen systems, and it is often difficult to establish when a stretched rubber is at equilibrium. One of the problems is that some rubbers crystallize slowly during deformation. In some materials, oxidation is hastened by applied stress, and permanent chemical changes may obscure stressstrain equilibrium of the specimen. In fact, many experimental studies have been reported where there is doubt that equilibrium was attained, and this problem often fuels controversies over theoretical models of rubber deformation. 8.4.2 Some Experimental Investigations of Elastomeric Deformation

Changes of internal energy of rubber are small at low deformation, but may become large when deformations are large as is

372

8 Rubber Elasticity

commonly observed for natural rubber and other crystallizable elastomers. Large changes of internal energy in stress-strain behavior of rubber at high deformation were observed by Wood and Roth (1944) as shown in Fig. 8-4, while experiments on GR-S, a polybutadienepolystyrene rubber which does not crystallize, exhibit relatively modes changes in internal energy up to high deformations (Roth and Wood, 1944). In these investigations, both hysteresis and permanent distortion were evident in some examples. The elementary Gaussian statistical theory of rubber elasticity provides a basis for analysis of rubber mechanics at small deformations, but when deformations become substantial, the corrections to that model become significant. Experiments

8 Figure 8-4. Stress-deformation measurements on natural rubber, (a) o versus X\ (b) —T(da/dT)k versus L (Wood and Roth, 1944.)

with both bulk and swollen rubber by Gumbrell, Mullins and Rivlin (Gumbrell etal., 1953) provide an excellent example for the need for higher terms in the stored energy function. The Gaussian theory leads to a stored energy function given by Eq. (8-26) with the constant C equal to Q VR T/(2MC) for the fixed junction model and [QVRT/ (2MC)] [1 -(2//)] for the phantom network. In real rubbers dangling ends of chain molecules are not crosslinked, and thus are not part of the network. In addition, chains which are not crosslinked may be incorporated in the rubber matrix. Owing to these effects, the constant C obtained experimentally would be expected to be lower than the theoretical results of the models. Contrary to this expectation, it is often found that C or the front factor, as it is known, is larger than theoretical predictions. In the investigation of Dossin and Graessley (1979) on polybutadiene crosslinked by radiation, experiments were conducted on both bulk and swollen specimens. The data were analyzed by the Mooney equation, and the authors found that the constant Cx of Eq. (8-27) accounts for the entire chemical network and half of the topological entanglement contribution to the modulus. The constant C 2 accounts for the other half of the contribution from entangled chains. According to Dossin and Graessley, fluctuations of crosslink junction points are largely suppressed and the entanglement contribution to the modulus is predominant for those networks crosslinked in bulk. Ferry and co-workers (Carpenter et al., 1977; Ferry and Kan, 1978) carried out stress-strain studies on elastomers crosslinked by radiation while in a state of uniaxial extension. The crosslinking was performed in most cases below the glass transition temperature. After crosslinking and

373

8.4 Experimental Procedures in Rubber Deformation

rewarming to room temperature, the specimen length of the material in this so-called state of ease is greater than that of the undeformed material before crosslinking, but less than that of the stretched material during crosslinking. The authors explain this result as a balance between forces of the chemical network with those of the entanglement network. The model treats the chemical network with simple Gaussian statistics, and the entanglement network as one of the Mooney type. In the study of Carpenter etal. (1977), crosslinking was performed by y-radiation, and the initial elongation during crosslinking was varied from 1.2 to 2.8. The elongation in the state of ease after rewarming varied from 1.05 to 1.48. While most crosslinking experiments were carried out below Tg, those carried out above Tg sometimes showed a smaller fraction of trapped entanglements. Radiation times were varied. Some data are assembled in Table 8-1. A considerable body of experimental studies of deformation of rubbery networks has been carried out by Mark and associates, and they have analyzed the results of these experiments generally by application of the Flory-Erman theory. One good example is given in Llorente and Mark's investigation of end-linked polydimethylsiloxane (Llorente and Mark, 1980). Experiments on uniaxial extension and compression were carried out, and data plotted to yield constants of the Mooney equation. The modulus as a function of elongation is obtained, and therefore, the parameters of the theory. In this study, the authors estimate the degree to which the networks approach the fixed junction model in the unstretched network, and how closely the system approaches the phantom network at high elongation. Table 8-2 contains some of those results.

Table 8-1. Data collected on poly(butadiene) crosslinked in a state of strain. i 0 is the initial elongation at which the specimen was crosslinked. Xe is the elongation in a state of ease after crosslinking and rewarming. Tc is the temperature at which the specimen was crosslinked. (Results of Carpenter et al., 1977.)

1.400 1.565 1.620 1.749 1.242 1.275 1.306 1.309 1.508 1.826 1.839

K

TC(°C)

1.108 1.202 1.185 1.251 1.078 1.106 1.107 1.127 1.193 1.231 1.240

-15 -15 -15 -15 -10 -10

-io -10 -10 -10 -10

11.888 11.993 :1.782 1L.345 11.702 11.478 :>.499 11.330 11.553 \1.774

K

Tc (°C)

1.355 1.346 1.482 1.148 1.296 1.187 1.527 1.138 1.254 1.366

-10 -10 -10 -5 ^

-5 -5 0 0 0

Table 8-2. Stress-strain behavior of end-linked polydimethylsiloxane networks. / is network functionality. Ay is obtained from the constant 2Ct of the Mooney equation applied to experiment. If the Erman-Flory theory applies, Av should equal 1 — 2//. A'y is obtained from 2C 1 +2C 2 using the Mooney equation, and according to Erman and Flory should equal unity. (Data from Llorente and Mark, 1980.) /

1-2//

A.

A'

3.0 3.0 3.0 4.0 4.0 4.0 4.6 5 5 5 6 6 8 8 11 11 37 37

0.33 0.33 0.33 0.50 0.50 0.50 0.56 0.60 0.60 0.60 0.67 0.67 0.75 0.75 0.91 0.91 0.97 0.97

0.88 0.85 0.79 0.94

1.32 1.32 1.37 1.34 1.37 1.32 1.42 1.39 1.09 1.29 1.43 1.39 1.43 1.43 1.44 1.48 1.17 1.16

0.82 0.96 1.07 1.07 0.87 1.09 1.16 1.15 1.33 1.30 1.39 1.42 1.14 1.69

374

8 Rubber Elasticity

Thirion and Weil (1983) examined equilibrium stresses on a rubber prepared by crosslinking ds-polyisoprene with dicumyl peroxide. The data were analyzed in two ways, one by use of the Mooney equation, and the other by a variant of the tube model developed by Ball et al. (1981), in which slip-links characterize the restrictions of the polymer chains in the tube. A slipping coefficient needed in the theory was taken from earlier studies on deformation in pure shear. The authors come to the conclusion that the analysis of Ball et al. works well for rubber elasticity. The experiments in this study are in good accord with earlier work of Ferry and Kan (1978) and also with that found by Dossin and Graessley (1979). Tschoegl and Gurer (Tschoegl and Gurer, 1985; Gurer and Tschoegl, 1985) analyzed elastic deformation of a variety of elastomers using a two-term expression for the strain energy function. The first term was appropriate for a network of Gaussian chains, and the second was of the Ogden type, C2/[rn2(A™ + ^ + ^)]

with m = 034.

The coefficients were obtained using uniaxial extension measurements at small deformation, and then applied to uniaxial measurements over a much broader range of deformation. Elastomers studied included natural rubber, styrene-butadiene rubber, polydimethylsiloxane, poly-(ethyl acrylate) rubber swollen in bis(2-ethoxyethyl) ether, polybutadiene, and butyl rubber. Some experiments were carried out on the rubber in bulk, and others on specimens prepared by swelling in a solvent. The term quadratic in the quantities kt in the strain energy, according to Tschoegl and Gurer, arises from chemical crosslinks, the other term contains the contribution from the entanglement network.

8.4.3 Some Swelling Experiments Gee (1946) studied the maximum swelling of rubber in certain swelling agents, and also studied the vapor pressure of some of these swollen rubbers. From this investigation and earlier work, he obtained a set of X values for several liquids. Results are shown in Table 8-3. The numbers obtained from measurement of the maximum swelling ratio agreed well with that obtained from vapor pressure measurements in most cases. Brotzmann and Eichinger (1983) investigated the vapor pressure of cyclohexane above swollen crosslinked polymer and uncrosslinked solutions of polydimethylsiloxanes. The data are presented in a graph of kln(pc/pu) versus A2, p c , being the vapor pressure of the swollen crosslinked polymer, and pu, the vapor pressure above the uncrosslinked material at the same solvent concentration. Data are found in Fig. 8-5. The maximum in the figure is not found for the unmodified Gaussian model of rubber, but a maximum, less steep than that in the measurement, is given by the Flory-Erman analysis. No reasonable choice of parameters in the Flory-Erman

Table 8-3. Values of /, the Flory interaction parameter for natural rubber. (Data from Gee, 1946.) Swelling agent

CC14 CHC13

cs2

Benzene Toluene rc-Propyl acetate Ethyl acetate MEK Acetone

X Swelling

Vapor pressure

0.29 0.34 0.425 0.395 0.36 0.62 0.78 0.94 1.37

0.28 0.37 0.49 0.43 0.43-0.44

8.5 Orientation of Chain Segments

6.0 -

(a)

4.0 2.0

1

00

1

1

1

6.0 o Q.

(b)

4.0 2.0

1

00

1

1

1

6.0

1 (c)

4.0 2.0

1 1.0

1 2.0

1^-—1—-t— 3.0

375

where chain extension is limited such that the end-to-end distance of a chain is less than 30% of the length of the fully extended chain. For greater chain extensions, deviations from the Gaussian model become considerable. A careful investigation of the statistics of extended chains was put forth by Kuhn and Griin (1942). The polymer chain contains n segments, each of length b, with a mean square end-to-end distance, (j2y = nb2 in the unstretched state. The angle made by the zth segment initially with a particular direction, later to be identified with the stretch direction, is 9f. The fraction of segments at a polar angle 9 and azimuthal angle cp measured with respect to the stretch direction is given by

4.0

Figure8-5. ^ln(p c /p u ) versus X2 for swollen polydimethylsiloxane gels and solutions. Three specimens: Specimen (a): Mc = 11 300, / = 3; specimen (b): Mc = 11 300, / = 4; specimen (c): M c = 18 500, / = 4. (Brotzmann and Eichinger, 1983.)

dn n

g.38 a )

4 7i sinh P

where r/(n b) = 5£ (j5), and equivalently p = <£~x{rl(nb)\ Z£{p) is the Langevin function defined by if (j8) = coth(jS) - l/p

theory, however, can provide a quantitative fit to the experiment. Brotzmann and Eichinger have stated their opinion that part of the discrepancy between theory and experiment arises from the assumption in the theoretical development that the elastic and the swelling contributions to the free energy are additive, an assumption which has been widely adopted by theorists, but which remains unproved.

8.5 Orientation of Chain Segments 8.5.1 The Statistics of Highly Stretched Chains

A Gaussian model of polymer chain statistics may be applicable to polymer molecules or polymer chains in rubber

(8-38 b)

and ^?~1(r/(nb)) is the inverse function. The probability density for r is given by

-Ol. ,

cexp 1 - p- , .

b

\smh P) J

(8-39)

which reduces to the Gaussian result [Eq. (8-5)] for small r/(n b) by using the leading terms of the expansion of /} in terms of r/(nb). Roe and Krigbaum (1964) working with the Kuhn-Griin equations, and a closely related model of Treloar (1954), calculated values of w(cos0), which is the fraction of segments between 9 and (9 + dO), for chains of 50 and 150 segments (3-10 monomers), for several values of L Their results are shown in Fig. 8-6. Roe and Krigbaum de-

376

8 Rubber Elasticity

veloped a general formula for which is

n=50 1.0

(8-40 a)

^s:li-a where F2 is given by

0.8

" X-4 0.6

\

36

"" X-2 ^ ^ ^ O ^ :ir -^

0.4

-

0.2

-

2

5n 875 n 108 6125 n3

X 3

(8-40 b) (a)

I 60°

30°

90°

The orientation of polymer segments in a stretched rubber can be calculated from this analysis and is of interest in studies of optical birefringence, fluorescence, and NMR quadrupole splitting.

n = 150

8.5.2 Optical Birefringence of Elastomers

1.0 ^ v .

51=10

0.8

0.6

k^

0.4

0.2 I 30°

I 60°

90°

0 Figure 8-6. Plots of w (cos 6) versus 0 for (a) n = 50, A = 2, 4, 6; and (b) n = 150, A = 2, 4, 6, 8, 10. (Roe and Krigbaum, 1964.)

When a transparent rubber is uniaxially stretched, it becomes birefringent.The polarizability of a chain segment is normally different along the segment axis from the polarizability perpendicular to the segment axis. After stretching, the segmental orientation becomes anisotropic, and the overall polarization perpendicular to the stretching direction differs from that parallel to the stretch. This is manifested in a difference in refractive index in these two principal directions. The birefringence is calculated from the polarizability difference by use of a suitable equation. The Lorentz formula is normally adopted for this purpose. The optical birefringence ni — n2 for a Gaussian network takes the form 2 71 ( * '

— n7 =-45 (8-41)

8.5 Orientation of Chain Segments

<xl5 and a 2 are the polarizabilities of the polymer segment in the parallel and perpendicular directions respectively, and iVv is the number of polymer chains per unit volume; n is the mean refractive index of the rubber plus swelling agent. The uniaxial true stress in a swollen Gaussian network is also proportional to A2 —A"1, and

Co = n1—n2

(8-42 a)

where C, the stress-optical coefficient is given by

°" 45

(8-42 b)

HkBT

This simple stress-birefringence relation does not apply at extremely high orientations for which the Kuhn-Grun model of chain deformation is required. The optical birefringence for a highly stretched polymer depends on higher terms in the deformation, and is given by _2TT Hl

~n2~j5~

Stress optical coefficients have been determined for a number of different elastomers. It is generally noted that the values obtained experimentally for the constant C in Eq. (8-42 a) are higher than expected for the unswollen rubber on theoretical grounds, but the magnitude is very much reduced by swelling. The reduction is greatest if the solvent molecule is not very anisotropic. In Table 8-4, data collected by Ishikawa and Nagai (1969) show that C decreases upon swelling. Similar results were obtained by Gent (1969), who analyzed data collected earlier by Saunders (1956, 1957). As part of their analysis of their own birefringence studies of poly(l,4butadiene), Fukuda, Wilkes and Stein (Fukuda et al., 1971) discuss the possibility that the Lorentz-Lorenz equation is an inaccurate representation of the relation between birefringence and polarizability. 8.5.3 Fluorescence of Labeled Elastomers

(f

30/?

70 nz

377

(8-43)

In many experiments on birefringence of elastomers, deformation is not very great, and the Gaussian model is adequate. Simultaneous measurements of stress and birefringence lead to values of the stressoptical coefficient. The instrument of choice for the measurement of birefringence is a Babinet compensator. The data are often plotted with Mooney type equations for both stress and birefringence (8-44 a) (8-44 b)

In an elastomer, the orientation of chain segments can be measured by putting a fluorescent group in the polymer chain, and by observing the polarization of fluorescence in a stretched sample. The theoretical point of departure is taken from the Roe-Krigbaum analysis (Roe and Krigbaum, 1964) which yields the orientation of chain segments with respect to the stretchTable 8-4. Polarizabilities calculated from stress-optical coefficients of 1,4-ds-polybutadiene radiation crosslinked by 12 Mrad electrons. (From Ishikawa and Nagai, 1969.) OL± - a 2 ( 1 0 ~ 3 0 m 3 ) 7.5 6.1 5.5 5.9 4.9

Vi

1 0.15 0.12 0.14 0.21

(benzene) (CC14) (tetralin) (decalin)

378

8 Rubber Elasticity

ing direction for model chains, as a function of the number of segments per chain and the degree of deformation. In the experiment, the polarization of fluorescence is measured as a function of elongation. The mean second moment of the cosine of the angle between the transition moment of the fluorescent group and stretching direction is given by

(8-45 a) which, if the Gaussian model for stress is valid, yields (5QkBT(p2)

(8-45 b)

At high elongations, it is not expected that Eq. (8-45 b) would apply. In order to obtain the mean orientation of polymer segments with respect to the stretching direction, it is necessary to know the orientation of the fluorescent group with respect to the segments, and the direction of the fluorescent excitation relative to the molecular axis of the fluorophore. This is more of a problem if the fluorescent group can vary in conformation relative to the chain. Jarry and Monnerie (1978,1980) studied orientation in polyisoprene labeled with dimethyl anthracene in the backbone of the polymer chain. Some measurements were performed on dry networks and others on networks slightly swollen (
tends to disappear. Examples of results from the studies of Jarry and Monnerie (1979) are presented in Fig. 8-7. 8.5.4 Quadrupole Splitting in NMR Spectra of Swelling Agents by Oriented Rubber

An elastomer swollen in a deuterated solvent of high symmetry and stretched in an NMR spectrometer at right angles to the steady magnetic field will exhibit deuterium NMR spectral lines which are split. The stretched elastomer creates an anisotropic electric field gradient in the swelling agent which causes the splitting. The governing equation is Av =

:— P2 (cos Q) P2 (cos 9{) P2 (cos y) (8-46) where P2 is the second Legendre polynomial, Av is the difference in frequency arising from quadrupole splitting, e2 q Q/h, the gradient factor, is of the order of 200 Hz for a C-D bond, Q is the angle between the 2

0

5

10 15 o in MPa Figure 8-7. Second moment,
8.6 Small Angle Neutron Scattering (SANS) of Elastomers

379

1.0

0.8

lay N

0.6

/

>

0.4

0.2

^*TI

1

I

1

1

1

8

10

12

14

16

10 0.3

0.5 0.6 0.70.80.91.0

Figure 8-8. Quadrupole splitting of deuterobenzene in oriented rubber versus X2 — X~x. (a) (p2 = 0.93; (b) cp2 = 0.80; (c) cp2 = 0.52; (d) (?2 = 0.32. (DeLoche and Samulski, 1981.)

Figure 8-9. Quadrupole splitting of deuterobenzene in rubber. Av/(A2 — X~ *) versus q>2. Slope of line = 4/3. (DeLoche and Samulski, 1981.)

stretching axis and the magnetic field, 0t is the angle between the ith C - D bond and the molecular axis of the elastomer, and y is the angle between the stretching axis and the C-D bond. Experimental splittings of the order of 100 Hz can be measured, thus allowing determination of orientations with P2 less than 0.01. An example, found in the splitting of hexadeuterobenzene in swollen stretched sulfur-crosslinked polyisoprene, was obtained by DeLoche and Samulski (1981). Some results are exhibited in Fig. 8-8. Figure 8-9 shows how Av/{l2~rl) varies with polymer concentration in a swollen specimen.

8.6 Small Angle Neutron Scattering (SANS) of Elastomers The SANS method (see Chap. 20, Vol. 2B of this Series) for studying the shape and size of polymer chains has been applied to measurement of polymer chains which make up elastomeric networks. The coherent elastic scattering is proportional to the Fourier transform of the segmental distribution of labeled chains in the network. The labeling is achieved by replacing ordinary hydrogen with deuterium in some of the chains which make up the network. The intensity of coherent elastic scattering is given by q)

(8-47)

380

8 Rubber Elasticity

where A and B are known constants depending on polymer molecular weight, concentration, and scattering contrast factors. Ss(q) is a single chain scattering function, ST(q) is known as the total scattering function and contains both single chain and interchain contributions to the scattering. The interchain contributions are often referred to as intermolecular, though they arise from pairs of monomer units on different chains on the same molecular network. For a polymer chain, Ss(q) may be written as (8-48) UJ

Here r[j is a vector connecting monomers i and j on the same polymer chain, q is the scattering wave vector, defined by q = (2 n/X) (k — k0), where k0 is a unit vector in the direction of the incident neutron beam, k, a unit vector in the direction of the scattered beam, and X is the wavelength of the neutrons, normally between 0.3 nm and 1.5 nm in this sort of experiment. The magnitude of q is given by \Q\=
(8-49)

Here rtj is the vector connecting monomer i with monomer j but now, these monomers

may be on the same chain or on different chains. The average is taken over all possible arrangements of the chains in the specimen. In a bulk network containing no solvent, and in which labeled and unlabeled chains are chemically identical, the constant B of Eq. (8-47) equals zero. In a swollen network, B is, in general, not equal to zero. B is also nonzero if the labeled and unlabeled chains are chemically different, or if they differ in molecular weight. It is possible to cause B to vanish by adjusting the fraction of labeled chains, or by adjusting the deuterium-hydrogen ratio in the solvent. These are procedures occasionally adopted to insure that the coherent elastic scattering arises from single chain scattering only. For an isotropic sample, Ss(q) may be written as a power series in q (8-50) Rg is the radius of gyration of a polymer chain, and can be determined from a low q plot of [/(g)]"1 versus q2 whenever ST(q) = 0. The measured scattering intensity is given by (8-51) K is an instrumental constant which can be determined independently. Rg can be obtained by plotting I(q)-KBST(q)~1 versus q2 if ST(q) is nonvanishing. By determining how Rg varies with solvent concentration and molecular weight of the polymer chains, it becomes possible to determine changes in chain geometry under different conditions of network preparation and swelling. In a stretched network, Ss(q) may be written as

Ss(«) = l - V * +

(8-52)

381

8.6 Small Angle Neutron Scattering (SANS) of Elastomers

where R^ is the apparent radius of gyration measured at an azimuthal angle

is at 90° to the direction of stretch R^ = R\. Theoretical values of R^ for deformed elastomers are easy to derive. In the phantom network model

&/

70 c

y^ 60 a

(8-53a)

\--A + =,

cr 50

'/<

^ "

(8-53 b)

a

*''

c

,-

*"*** — —Cr**"

40

a

(8-53 c) l

In the nonstretched swollen network [Eq. (8-53 a)], X is the linear swelling ratio, while in Eqs. (8-53 b) and (8-53 c) k is the uniaxial stretching ratio. In the fixed junction model, the calculated values of R2 are obtained by dropping the quantity 2 / / in Eqs. (8-53 a, b,c). There have been many SANS studies of elastomeric networks, and these have been directed, in part, to discovering which of the molecular models, if any, of an elastomeric network applies to the material. The earliest published experiments on SANS of rubbery networks are those of Benoit et al. (1976) on anionic polystyrene networks crosslinked with divinylbenzene and swollen in selected solvents. The polymerization procedure was such that the functionality of these networks was not determinable. The polymerization was carried out in 10% solution, the radius of gyration of the uncrosslinked polymer was measured as well as the radius of gyration of the chains in the unswollen network. Values of Rg versus degree of swelling for three networks are shown in Fig. 8-10. On the same figure, calculated values of Rg from a phantom network model with / = 4 is also given.

I

1

2.5 3.0 2.0 X Figure 8-10. SANS measurement of Rg of labeled chains in swollen polystyrene networks. Rg versus X for three networks, all with M c = 26000 in different solvents: (a) cyclohexane, (b) hydrofurfuryl alcohol, and (c) benzene. The theoretical prediction for a phantom network with / = 4 is shown in curve (d). (Benoit etal., 1976.) 1.0

1.5

Calculations of the scattering function of labeled polymer chains in a deformed network were obtained by Pearson (1977), by Warner and Edwards (1978) and by Ullman (1979, 1982). These calculations have been used for interpretation of experimental studies by many researchers. Duplessix (1975) and Bastide, Duplessix, Picot, and Candau (Bastide etal., 1984), have measured SANS spectra of crosslinked polystyrene networks swollen in benzene and subsequently partially deswollen osmotically. Some of their results are shown in Fig. 8-11. Beltzung and co-workers (1982, 1983, 1984) have applied SANS to an investigation of polydimethylsiloxane networks. These networks were prepared by crosslinking linear deuterated chains with chains of the protonated species using vi-

8 Rubber Elasticity

382

(b)

90 ^- '—

80 —

•E DC

^~~

1.3

_

Fixed junction/ •'if Phantom ° o °

1.2

70

1.1

60

1.0

US

o

o

X Phantom

0.9

50

**''''

1.0

I

I

1.2

1.4

I 1.6

o

1.8

-o—

rx,

2.0

.o

2.2

40 30 20 10 I

I

I

10

15

Figure 8-12. Ratio a of the radius of gyration parallel to (l?(|) or perpendicular to (R±) the stretch direction to the unperturbed radius of gyration {Rg) in stretched polydimethylsiloxane networks with various M c . •: M c = 3 x l 0 3 ; +: Mc = 6 x 103; x: Mc = 104; o: Mc = 2.5 x 104. (Beltzung et al., 1984.)

X3 Figure 8-11. Radius of gyration of chains in osmotically deswollen polystyrene networks versus X3. Curve (a) phantom network; curve (b) fixed junction model; data: n, •, •. experiments of Duplessix (1975).

nyl or allyl crosslinking agents designed to create either tetrafunctional or hexafunctional crosslink junctions. Radii of gyration of the uncrosslinked chains as well as radii of gyration of the chains in the crosslinked but unswollen network were determined. Crosslinking caused no change in the scattering of the unswollen rubber. i?t, values of the stretched network were equal to or less than that predicted by the phantom network, this being a function of the molecular weight of the chains. R± was close to that predicted for the phantom network. Figure 8-12 shows some results. These same networks, swollen in cyclohexane showed that Rg increased with molecular weight between crosslinks in accord with phantom network predictions. It also turned out that Rg values in these swollen networks varied with molecular weight in the same way as do Rg values of dilute solutions of uncrosslinked polymer in the same solvent. Some data are shown in Fig. 8-13.

4.0

4.5

5.0

logM z Figure 8-13. Radius of gyration of chains of polydimethylsiloxane networks swollen in cyclohexane (*), and Rg in dilute solutions of polydimethyl siloxane in cyclohexane (•), versus Mz (Z: average molecular weight measured by GPC). (Beltzung et al., 1983.)

Tsay and Ullman (1988) studied SANS spectra from polystyrene crosslinked by reaction with benzene meta disulfonylazide. Measurements were performed on both stretched and swollen networks. In these experiments, the molecular weight of the chains before crosslinking was 105, and networks were synthesized to obtain an average of 3, 4, or 5 crosslinks per chain. The results were compared with calculations

8.7 Rubber-Glass and Rubber-Crystal Transitions

made by Ullman (1982) on SANS from polymers with several crosslinks per chain. Plots of R\\/R% and RJR% for stretched networks lie between the fixed junction and phantom models, and are closer to the phantom network at higher elongation (2 = 1.6 to 1.9). The chain dimensions were changed much more by swelling in toluene than by swelling in cyclohexane. It was found that Rg(X)/Rsg in toluene was approximately equal to Rg(X)/R\ in cyclohexane where #g stands for radius of gyration in dilute solution. The radius of gyration of a polymer chain is determined from scattering at very low q values. SANS measurements at higher q yield further information on chain shape. These data are often examined by plotting q2 Ss(q) versus q, a technique originally used by Kratky for studies of small angle X-ray scattering of polymers. Bastide, Herz, and Boue (Bastide et al., 1985) investigated SANS from high molecular weight (>10 6 ) deuteropolystyrene crosslinked by y-radiation in cyclopentane solution with lower molecular weight protonated polystyrene. The Kratky plot of the deswollen network with a molecular weight between crosslinks of 35000 with a labeled chain of 2.6 x 106 molecular weight exhibited a broad peak at q&0.15 nm'1 (see Fig. 8-14). The peaks calculated for the crosslinked phantom and fixed junction networks were greater than that found in the experiment. The authors suggest that this peak could be accounted for in part by interchanges in spatial positions of geometric near neighbors with chemical near neighbors upon deswelling. In the Tsay and Ullman experiments, Kratky plots on stretched samples showed a broad peak at q « 0.5 nm" 1 for scattering in the perpendicular direction. Scattering in the parallel direction showed a slight maximum in the same q range.

i

I

383

I

I I I i i 0.4 0.6 1 q in nrrf Figure 8-14. Kratky plot comparing data obtained from a dry gel (A) and an isotropic melt (•) of deuterium labeled polystyrene to calculated form factors for the same deformation ratio: isotropic Gaussian (plain line below); junction affine model, Mmesh = 35000 (plain line above); phantom network model, Mmesh = 35OOO and Mmesh = 50000 (below), dashed lines. (Bastide et al., 1985.)

There have been many other SANS studies on rubbery networks which have not been discussed here. It has not yet been possible to determine from these studies which of the various models of rubber elasticity is most realistic. The Flory-Erman model and model of Edwards and associates have been applied to the SANS problem, but it is not possible at this stage to tell which view of rubber elasticity is most nearly in agreement with SANS measurements.

8.7 Rubber-Glass and Rubber Crystal Transitions When a rubber is cooled, it becomes a hard solid. If the rubber has an irregular chemical structure such as polystyrenebutadiene), the solid is a glass. If the elas-

384

8 Rubber Elasticity

tomer has a regular chemical structure, it may partially crystallize upon cooling. Lightly crosslinked natural rubber is in the latter category. The crystalline structure which is formed on cooling can also be induced by stretching the rubber above the crystalline melting point. Strain induced crystallization of rubber was first shown by Katz using X-ray diffraction (Katz, 1925) on natural rubber. One of the consequences of strain-induced crystallization is an increase in modulus and also in ultimate strength. Crystallinity in an oriented polymer is accompanied by a crystalline X-ray pattern at wide angles and, often, a two-point pattern in the small angle domain. When a polymer crystallizes, its density usually increases substantially since the chains pack more efficiently in the crystal than in the liquid or glassy states. In addition, a sharp endotherm is evidenced during heating since the polymer crystals melt. If a rubber is crystallized in part by stretching, the crystalline fraction can be melted by heating, in which case the modulus decreases and the density drops. The assumption that the volume of a rubber is largely unchanged by stretching, used widely here and in other studies of rubber elasticity, is only valid under circumstances where no crystallization takes place. Rubbers which are normally crystallizable can be prevented from crystallizing by increasing the extent of crosslinking. An increased density of crosslinks reduces the regularity of the chemical structure of the rubber and places severe constraints on the chains, resulting in a serious impediment to the formation of a crystalline structure. Crystallization of rubber without strain has been investigated by many research groups. In one interesting case, crystalline morphology of fractionated, uncrosslinked

natural rubber was studied by Phillips and Vatansever (1987).

8.8 Thermoplastic Elastomers Thermoplastic elastomers (TPEs) are multiblock copolymers, comprising hard segments and soft segments, which microphase separate into hard domains in a soft segment matrix (Legge et al., 1987). The crosslinks in TPE materials are physical in nature arising from aggregation of the hard segments into domains, which physically constrain the rubbery soft segments between them. The hard segment domains can be both glassy or crystalline, depending on the chemical structure of the hard segment. TPE materials such as polystyrene - polybutadiene - polystyrene triblock copolymers, now in large volume commercial use, depend for their elastomeric properties in the anchoring of the rubber chains by the hard segment domains. Polyurethanes, polyesteramides, polyetheramides, and oc-olefin TPE materials all exhibit interesting stress-strain behavior. The anionically prepared PS-P.BPS triblock materials are a useful model system since domain size, shape, and spacing are very well defined. Keller and coworkers (Folkes et al., 1973; Keller and Odell, 1985) have examined the deformation behavior at both low and high strain. For strains below about 50%, affine deformation takes place, but at higher strain levels, complications arise from hard segment domain reorientation and domain breakup. Stevenson and Cooper (1988) studied the changes which occur with deformation of the elastomeric block copolymer of poly(tetramethylene oxide) (PTMO) with poly(tetramethylene isophthalate) (PTMI) or with mixtures of PTMI and poly (tetra-

8.9 Ultimate Strength

methylene terephthalate) (PTMT) as the hard segment. The PTMO block had a number average molecular weight of 1000. The material prepared by transesterification of poly(tetramethylene ether glycol), dimethyl isophthalate, dimethyl terephthalate, and 1,4-butanediol had a molecular weight of 25000 to 30000. The ester block forms crystalline hard segment domains, while the ether block imparts flexibility to the polymer. The weight fraction of the hard segment (PTMI + PTMT) was approximately 60%. Samples were films prepared in a press by heating 30 °C above the melting point and then cooling. The crosslinks in this system are physical in nature, are localized in the hard segment aggregates, and change from glassy to partially crystalline as the system is annealed. Stress-strain measurements during crystallization at room temperature exhibit stresses which decrease markedly with time at a given elongation.

8.9 Ultimate Strength Elastomers tear when subjected to high stresses. The stress required to bring about this mechanical failure depends on a number of factors. The result may depend on viscoelastic response of the rubber, molecular weight between crosslinks, the temperature at which the stress was applied, the chemical makeup of the material, and the existence of flaws in the specimen. Rubbers which crystallize tend to be stronger than materials which remain amorphous upon stretching. In 1967, Lake and Thomas (1967) provided an analysis of strength of elastomers based on stored energy, and how this energy is released when the elastomer is torn. This work was extended by Gent and Tobias (1982) who performed many experi-

385

ments and incorporated results of earlier researchers in their studies. They derived a formula for the threshold tearing strength, a quantity which represents the smallest value of the tensile failure stress that would be measured in a real experiment. In many circumstances, the energy stored in a rubber on the verge of tearing is composed in part of stored energy which would relax viscoelastically if given sufficient time. As a result, the measured strength is greater than that expected in a threshold model. The Gent and Lake-Thomas equations take the form T0 = KM^/2c

(8-54 a)

To is the threshold tearing strength, M c , the molecular weight between crosslinks, and c is the length of a flaw perpendicular to the tearing direction. K is given by K=

3V /2 QU ~

71/2

M1'2

(8-54 b)

Here Q is density, u is the mean energy of a main chain bond, z is the number of main chain monomers per statistical segment, M is the molecular weight of a polymer chain, M o is the monomer molecular weight. K is predicted to be about 0.3 J/m2, measured values tend to be a factor of 3 higher. Threshold tearing strengths for polybutadiene and polyisoprene rubbers lie between 40 and 100 J/m 3 for a series of materials where Mc is bounded by 3 x 103 and 104. Polydimethylsiloxane networks have slightly lower threshold tear strengths. The data on siloxanes include both end-linked and randomly crosslinked materials. A number of measurements were performed on swollen specimens of the siloxanes, and, as expected, X2 To for the swollen specimens equalled To for the bulk network.

386

8 Rubber Elasticity

8.10 References Anthony, R. L., Caston, R. H., Guth, E. (1942), /. Phys. Chem. 46, 826. Ball, R. C , Doi, M., Edwards, S. E, Warner, M. (1981), Polymer 22, 1110. Bastide, X, Duplessix, R., Picot, C , Candau, S. J. (1984), Macromolecules 17, 83. Bastide, X, Herz, X, Boue, F. (1985), /. Phys. 46, 1967. Beltzung, M., Picot, C , Rempp, P., Herz, X (1982), Macromolecules 15, 1594. Beltzung, M., Herz, X, Picot, C. (1983), Macromolecules 16, 580. Beltzung, M., Picot, C , Herz, X (1984), Macromolecules 17, 663. Benoit, H., Decker, D., Duplessix, R., Picot, C , Rempp, P., Cotton, X P., Farnoux, B., Jannink, G., Ober, R. (1976), J. Polymer Sci., Polymer Phys. Ed. 14, 2119. Brotzmann, R. W. Jr., Eichinger, B. E. (1983), Macromolecules 16, 1131. Carpenter, R. L., Kramer, O., Ferry, X D. (1977), Macromolecules 10, 111. de Gennes, P. G. (1971), J. Chem. Phys. 55, 572. De la Condamine, C. M. (1751), Mem. Acad. Roy. Sci. 17, 319. DeLoche, B., Samulski, E. T. (1981), Macromolecules 14, 575. Dossin, L. M., Graessley, W. W. (1979), Macromolecules 12, 123. Duplessix, R. (1975), Thesis, Doctor d'Etat, L. Pasteur University, Strasbourg. Edwards, S. R, Vilgis, T. A. (1987), Reports on Progress in Physics 51, 243. 'Erman, B., Flory, P. X (1982), Macromolecules 15, 806. Faraday, M. (1826), Quart. J. Sci. 21, 19. Ferry, X D., Kan, H. C. (1978), Rubber Chem. Tech. 51, 731. Flory, P. X (1942), J. Chem. Phys. 10, 51. Flory, P. X (1977), /. Chem. Phys. 66, 5720. Flory, P. X, Erman, B. (1982), Macromolecules 15, 800. Folkes, M. X, Keller, A., Scalisi, F. P. (1973), Colloid and Polymer Sci. 251, 1. Fukuda, M., Wilkes, G. L., Stein, R. S. (1971), /. Polymer Sci. A2 9, 1417'. Gee, G. (1946), Trans. Faraday Soc. 42B, 33. Gent, A. (1969), Macromolecules 2, 262. Gent, A. N., Tobias, R. H. (1982), /. Polymer Sci., Polymer Phys. Ed. 20, 2051. Goodyear, C. (1844), U.S. Patent 3633. Gough, X (1805), Mem. Lit. Phil. Soc. Manchester I, 288. Granick, S., Ferry, X D. (1983), Macromolecules 16, 39. Gumbrell, S. M., Mullins, L., Rivlin, R. S. (1953), Trans. Faraday Soc. 49, 1495.

Gurer, C, Tschoegl, N. W. (1985), Macromolecules 18, 687. Guth, E., Mark, H. (1934), Monatsh. Chem. 65, 93. Hancock, T. (1844), British Patent. Huggins, M. L. (1942), /. Amer. Chem. Soc. 64,1712. Huggins, M. L. (1943), /. Phys. Chem. 46, 151. Ishikawa, T, Nagai, K. (1969), /. Polymer Sci. A2 7, 1123. James, H. M. (1947), /. Chem. Phys. 17, 651. James, H. M., Guth, E. (1943), J. Chem. Phys. 11, 455. James, H. M., Guth, E. (1947), /. Chem. Phys. 17, 669. Jarry, X P., Monnerie, L. (1978), /. Polymer Sci., Polymer Phys. Ed. 16, 443. Jarry, X P., Monnerie, L. (1979), Macromolecules 12, 316. Jarry, X P., Monnerie, L. (1980), J. Polymer Sci., Polymer Phys. Ed. 18, 1879. Joule, X P. (1859), Phil. Trans. Roy. Soc. 149, 91. Karrer, E. (1933), Protoplasma 18, 475. Katz, X R. (1925), Chem. Z. 49, 353. Keller, A., Odell, X A. (1985), in: Processing, Structure and Properties of Block Copolymers: Folkes, M. (Ed.). New York: Elsevier. Kramer, O., Carpenter, R. L., Ty, V., Ferry, X D. (1974), Macromolecules 7, 79. Kuhn, W., Grim, F. (1942), Kolloid Z. 101, 248. Lake, G. X, Thomas, A. G. (1967), Proc. Roy. Soc. A 300, 108. Legge, N. R., Holden, G., Schroeder, H. E. (1987), Thermoplastic Elastomers — A Comprehensive Review. Munich: Hanser Publishers. Llorente, M. A., Mark, X E. (1980), Macromolecules 13, 681. Meyer, K. H., von Susich, G., Valko, E. (1932), Kolloid Z. 59, 208. Mooney, M. (1940), /. Appl. Phys. 11, 582. Ogden, R. W. (1972), Proc. Roy. Soc. A 326, 565. Pearson, D. S. (1977), Macromolecules 10, 696. Phillips, P. X, Vatansever, N. (1987), Macromolecules 20, 2138. Rivlin, R. S. (1948), Phil. Trans. Roy. Soc. A 241, 379. Roe, R. X, Krigbaum, W. R. (1964), /. Appl. Phys. 35, 2215. Roth, F. L., Wood, L. A. (1944), J. Appl. Phys. 15, 749. Saunders, D. M. (1956), Trans. Faraday Soc. 52, 1414, 1425. Saunders, D. M. (1957), Trans. Faraday Soc. 53, 860. Stevenson, X C , Cooper, S. L. (1988), Macromolecules 21, 1309. Thirion, P., Weil, T. (1983), Polymer 25, 609. Thomson, W. (1857), Quart. J. Appl. Math. 1, 57. Treloar, L. R. G. (1954), Trans. Faraday Soc. 50, 881. Tsay, H. M., Ullman, R. (1988), Macromolecules 21, 2963. Tschoegl, N. W, Gurer, C. (1985), Macromolecules 18, 680.

8.10 References

unman, R. (1979), J. chem. phys. 7i, 436. Ullman, R. (1982), Macromolecules 15, 1395. Valanis, K. C , Landel, R. F. (1967), /. Appl. Phys. 38, FF

2197

Warner, M., Edwards, S. F. (1978), /. Phys. A. 11, 1649. Wood, L. A., Roth, F. L. (1944), J. Appl. Phys. 15, 781.

387

General Reading fe

Treloar, L. R. G. (1975), The Physics of Rubber Elasticit y- O x f o r d : Clarendon Press.

9 Viscoelastic and Rheological Properties Masao Doi Department of Applied Physics, Faculty of Engineering, Nagoya University, Nagoya, Japan

List of 9.1 9.2 9.2.1 9.2.2 9.2.3 9.2.4 9.2.5 9.2.5.1 9.2.5.2 9.2.5.3 9.3 9.3.1 9.3.2 9.3.3 9.3.4 9.4 9.4.1 9.4.2 9.4.3 9.4.3.1 9.4.3.2 9.4.4 9.4.4.1 9.4.4.2 9.4.4.3 9.4.4.4 9.4.4.5 9.5 9.5.1 9.5.2 9.5.3 9.5.4 9.5.5 9.5.6

Symbols and Abbreviations Introduction Linear Viscoelasticity Significance of Linear Viscoelasticity Shear Flow Stress Relaxation Boltzmann's Superposition Principle Parameters and Functions Characterizing Linear Viscoelasticity Steady Shear Flow Oscillatory Experiments Creep Experiments Characteristics of the Viscoelasticity of Polymeric Liquids Time-Temperature Superposition Rule Behavior of Relaxation Modulus Linear Polymers with Narrow Molecular Weight Distribution Effect of Molecular Weight Distribution and Branching Non-Linear Viscoelasticity Stress Tensor Basic Principles in Continuum Mechanics Non-Linear Viscoelasticity in Rheometrical Flows Shear Flow Uniaxial Elongational Flow Constitutive Equation Second Order Fluid Convective Maxwell Model Integral Form of the Convective Maxwell Model Non-Linear Convective Maxwell Model Single-Integral Constitutive Equation Molecular Theory I: The Rouse Model Polymer Motion in Concentrated Systems Basic Equation of the Rouse Model Normal Coordinates Self-Diffusion and Segmental Motion in Equilibrium Molecular Expression for the Stress Tensor Constitutive Equation for the Rouse Model

Materials Science and Technology Copyright © WILEY-VCH Verlag GmbH & Co KGaA. Allrightsreserved.

391 393 393 393 394 395 396 397 397 397 398 399 399 399 401 403 404 404 405 406 406 408 408 409 409 409 410 410 410 410 411 413 413 414 414

390

9.6 9.6.1 9.6.2 9.6.3 9.6.4 9.6.5 9.6.6 9.6.7 9.6.8 9.7

9 Viscoelastic and Rheological Properties

Molecular Theory II: The Tube Model Characteristics of the Polymer Motion in Strongly Entangled Systems . . . . Reptation and Fluctuations Reptation Model Self-Diffusion Constant Stress Relaxation Fluctuations Constitutive Equation Effect of Branching and Molecular Weight Distribution References

415 415 416 418 418 419 421 422 423 424

List of Symbols and Abbreviations

391

List of Symbols and Abbreviations B (t, t') E (£, f) f{P)

Finger-strain tensor deformation gradient tensor force exerted on the material below the plane j8 by the material above the plane /? G (t) relaxation shear modulus G' (co), G" (co) storage a n d loss modulus GN plateau modulus Go instantaneous relaxation modulus H (T) relaxation spectrum J(t) creep compliance Je steady state compliance kB Boltzmann constant M molecular weight Mc characteristic molecular weight Me molecular weight between the entanglement ML, M s molecular weight of long and short polymers Mx molecular weight of a polymer segment between the cross link Nx (y) first normal stress difference Qf^l tensor relating the stress and the formation tensor E R gas constant Rg center of mass of the molecule Sap orientational order parameter of a tube segment T torque T temperature Tg glass transition temperature t time U (t) orthogonal tensor representing the rotation of a frame u(s, t) unit vector tangent to the tube axis v velocity W elastic potential Z number of entanglement points per chain M/Me y (t) y (t) e & C rj rj (y) rj0 %(&) Y\ + (t, y) rj* (co), rj' (co)

shear strain shear rate elongation strain (Hencky strain) elongational rate friction constant of a Rouse bedd viscosity steady state viscosity zero shear rate viscosity elongational viscosity shear stress growth function complex viscosity

392

9 x xittP) k Q, QS a aE oa(i T rd Te TR Q co

9 Viscoelastic and Rheological Properties

gap angle velocity gradient tensor velocity gradient tensor evaluated at time t, at a material point P dimensionless material constant density at temperature T and Ts stress tensor elongational stress a,/? component of the stress tensor relaxation time reptation time relaxation time Rouse relaxation time angular velocity of the cone angular frequency

9.2 Linear Viscoelasticity

9.1 Introduction This chapter deals with mechanical properties of polymers in the liquid state. When a plastic is heated or when it is dissolved in a solvent, it becomes a liquid. The liquid state of polymers, generally called polymeric liquids, have unusual mechanical properties. Firstly, polymeric liquids have elasticity. This is demonstrated by chewing gum. Chewing gum is a liquid since it does not have any macroscopic equilibrium shape and can be molded into any shape. For slow deformation, chewing gum flows like usual liquids and shows viscosity. On the other hand, for quick deformation, it responds like an elastic material: for example, if we stretch the gum and release quickly the gum shrinks as if it remember the original shape. Thus chewing gum has both viscosity and elasticity. Such a material is called viscoelastic. Polymeric liquids are a typical example of viscoelastic material. Secondly, polymeric liquids have nonlinear mechanical properties. For example, the viscosity of polymeric liquids is not constant but decreases with the shear rate. This phenomena, called shear thinning, is quite dramatic: the viscosity changes over several orders of magnitude. The nonlinearity also appears as various curious flow phenomena: for example when a polymeric liquid is sheared between concentric cylinders, the liquid crawls up the inner cylinder. This effect is called the normal stress effect. The complex flow properties of polymeric liquids are important in various practical situations. Industrial processes of polymer technology involves flow of polymers: injection molding, fiber spinning, film blowing, surface coating are a few examples. In such processes, understanding

393

the flow properties is essential in designing and controlling the processes. The unique flow properties of polymers are also utilized in the industries of paints, glues, surface coating etc. The science of studying the flow and deformation of materials is called rheology. Although these problems have been discussed in the two well established sciences of fluid mechanics and elasticity theory, there are many materials which cannot be dealt with by classical continuum mechanics. Polymeric material is a typical example. Since it has both elasticity and viscosity, it can be dealt with neither by fluid mechanics nor elasticity theory. Rheology is concerned with such materials. Because of their importance in industry, polymeric liquids have been most intensively studied in rheology. Many examples of their unusual flow behaviors are given in the textbooks on rheology (Bird et al., 1987; Barnes et al., 1989). In this chapter, an introductory description of the rheology of polymeric liquids is given: in particular, it will be discussed (i) how the complex flow properties can be characterized, and (ii) how they are related to molecular structures.

9.2 Linear Viscoelasticity 9.2.1 Significance of Linear Viscoelasticity

The rheological properties of a material are characterized by the relationship between stress and strain. The stress represents the force applied to the material and the strain represents the deformation caused by it. The full description of such relations becomes rather complex for polymeric liquids, and will be postponed to a later section. Here we shall start with a simple case, called linear viscoelasticity.

394

9 Viscoelastic and Rheological Properties

When the stress level is low, a linear superposition principle, the precise meaning of which will be given later, holds in the relationship between the stress and strain. Such regime is called linear. In this regime, the viscoelasticity can be characterized by a single material function. In many practical applications, the stress level is not so small for linear viscoelasticity to be directly applied. However, linear viscoelasticity is important because of the following reasons: (i) Much experimental work has been done in the linear regime since material characterization is easy in this regime. Indeed most information on the relation between molecular structure and viscoelastic properties has been obtained through the study of linear viscoelasticity. (ii) Linear viscoelasticity is useful in predicting the behavior in the nonlinear regime. There are many empirical relations which predict the nonlinear behavior from linear viscoelasticity.

In shear flow, a material is sandwiched between two parallel plates, and the top plate is moved parallel to the bottom (see Fig. 9-1). If the material is homogeneous, and if the end effect is negligible, the velocity v (r, t) of the material at position r and time t is given by

9.2.2 Shear Flow

y (t, t') = J dt" y (t'f)

vv = vz = 0

(9-1)

The constant (9-2) is called the shear rate. To induce such a flow, a force has to be applied to the top plate. The shear stress a is the x component of this force per unit area. Actually, a is the x-y component of the stress tensor, and should be written as oxy. However, as we discuss only the shear stress in the next first two sections, we shall for now abbreviate it as o. The integral of y from time t' to t t

(9-3)

t'

The material function characterizing linear viscoelasticity can be obtained by various types of flow, but most commonly, shear flow is used. Therefore we shall describe linear viscoelasticity taking shear flow as an example.

Figure 9-1. Shear flow between parallel plates.

is called the shear strain applied between time t' and t. In general, if the velocity field is described by Eq. (9-1) in a certain reference frame, which may be moving with the fluid element, it is called a shear flow. Thus the flows shown in Fig. 9-2 are also shear flows: (i) Flow between concentric cylinders (see Fig. 9-2 a). Here two concentric cylinders with the radii rl9 and r2 (r1 < r2) are rotated relative to each other. The flow can be regarded as a shear flow if the gap distance (r2 ~ rt) is much less than rx. If the inner cylinder is rotated with the angular velocity Q relative to the outer cylinder, the shear rate is given by y = '1

r, Q ^ —

' 1

(9-4)

9.2 Linear Viscoelasticity

Figure 9-2. Examples of shear flow. Side and top views of (a) concentric cylinders and (b) cone and plate.

(a)

The shear stress is related to the torque T on the inner cylinder: T G = (9-5)

2nr\L

where 2nr1L is the surface area of the inner cylinder dipped in the liquid. (ii) Flow between the cone and plate (see Fig. 9-2 b). Here the liquid is sandwiched between a fixed plate and a rotating cone. The shear rate is constant throughout the fluid, and is given by y = Q/9

2nr3

the shear stress G (t). Alternatively, we can control the shear stress G(t) and measure the shear rate y (t). A particularly simple experiment is the stress relaxation, in which we apply a shear strain y0 in a very short time, and measure the stress while keeping the shear strain constant (see Fig. 9-3).

y(t-)

(9-6)

where Q is the angular velocity of the cone, and 6 is the gap angle. The shear stress is related to the torque T acting on the plate by G =

395

(a)

(9-7)

time, /•

a(t)

where r is the radius of the plate. 9.2.3 Stress Relaxation To characterize the viscoelasticity of a material, we apply a certain shear flow controlling the shear rate y {t\ and measure

(b)

time, t

Figure 9-3. The shear strain (a) and the shear stress (b) in stress relaxation.

396

9 Viscoelastic and Rheological Properties

If the material is an elastic material, the deformation creates a shear stress proportional to the applied shear strain y0, and the ratio G = a/y0 is called the shear modulus. If the material is viscoelastic, the flow creates a shear stress which decays with time. The stress eventually becomes zero if the material does not have an intrinsic shape. Such a material is called a viscoelastic liquid. Polymer melts and solutions belong to this class. On the other hand, for gels or rubbers, the stress approaches a constant non-zero value. Such a material is called a viscoelastic solid. In this chapter, we shall deal only with viscoelastic liquids. Chapter 7 deals extensively with the viscoelastic solids. If the shear strain y0 is small, the shear stress is proportional to y0, and is written as a(t) = G(t)y0

(9-8)

called Boltzmann's superposition principle. The principle states that provided the stress level is low, the effect of strain on stress can be added. Let us consider a shear flow characterized by the shear strain: (9-12) Suppose that when a shear strain y1(t) is applied to a material, the stress cr1(t) appears and that for a shear strain y2(0> a stress a2 (t) appears. Then the Boltzmann's principle states that when the shear strain is yi(t) + y2(i), the shear stress will be

MO + MOIf this principle holds, the stress for any strain history y (t) can be expressed by the relaxation modulus G (t). To see this, notice that any time-dependent shear strain y(i) can be regarded as a sum of step strains (see Fig. 9-4). The step strain applied be-

G (t) is called the relaxation shear modulus. For polymer melts with narrow molecular weight distribution, G (t) can be approximately written as G(0«Goexp(-t/T)

(9-9)

A/ (t')

Go is called the instantaneous relaxation modulus, and T the relaxation time. In the general case, G(t) has many relaxation times, and can be written as t/zp)

> t

(9-10)

or G(t)=

j dlnTH(T)exp(-r/T)

(9-11)

— oo

The function H (T) is called the relaxation spectrum. 9.2.4 Boltzmann's Superposition Principle We shall now explain the meaning of the linear superposition principle, which is also

r

t

Figure 9-4. The principle of calculating the stress for arbitrary strain.

9.2 Linear Viscoelasticity

397

tween t' and t' + At' is

9.2.5.2 Oscillatory Experiments

Ay (t<) = y(t' + At1) - y ( 0 = y (?) At1 (9-13)

In this experiment an oscillatory shear strain

If this strain is applied independently, it will create the following shear stress at a later time t: (9-14)

= G(t-t')Ay(t')

According to the Boltzmann's principle, the effect of these strains can be added. Thus the stress at time t is the sum of the contributions of all the previous step strains. Thus (9-15)

a(t)= } dt'G(t-t')y(t')

(9-16)

Equation (9-16) is the basic equation in the linear viscoelasticity. 9.2.5 Parameters and Functions Characterizing Linear Viscoelasticity

9.2.5.1 Steady Shear Flow A steady shear flow is realized when the shear rate is kept constant. The steady state viscosity rj0 is the ratio between the shear stress and the shear rate. For constant f, Eq. (9-16) gives

oo

is applied to a material. If the material is an ideal solid, the stress is proportional to the shear strain, and is written as a(t) = Gy(t) = G y0 cos (co t)

(9-20)

On the other hand, if the material is a Newtonian fluid, the stress is proportional to the shear rate, and is given by a(t) = rjy(t)= -riy0sin(cot)

(9-21)

<* (t) = Jo [G> M cos (co t) - G" (co) sin (co t)\ (9-22) G'(co) and G" (co) are called the storage modulus and loss modulus respectively. Often they are combined to give G* (co) = G' (co) + i G" (co)

According to Eq. (9-16), the shear stress in various strain histories can be calculated by G (t). Thus, in principle, all the parameters and the functions characterizing the linear viscoelasticity can be expressed by G (t). Here we shall give a few examples.

-

(9-19)

If the material is viscoelastic, the behavior is intermediate between solid and fluid:

or in the continuous limit


= y0cos(cot)

= y $dt'G(t') 0

ri*(co) =

(9-23) (9-24)

ICO

which are called the complex modulus and the complex viscosity respectively. From Eqs. (9-16) and (9-19), it can be easily shown that G' (co), G" (co) and G* (co) are related to G(t) as G'(co) =co] dtG(t)sin(cot) o

(9-25)

00

G" (co) =co\dtG o

(t) cos (co t)

G*(co) = ico JdtG(Oexp(-icoO o

(9-26) (9-27)

(9-17)

From Eqs. (9-18) and (9-26), rj0 can be expressed by G" (co) or rj' (co) also:

(9-18)

nQ = lim G-{^

Hence rj0 is given by

= lim rj'(co)

(9-28)

398

9 Viscoelastic and Rheological Properties

If G (0 is given by Eq. (9-9), G' (co), G" (co) and G* (co) are given by G'(co) =

G n CO 2 T 2

measured. In the linear regime, the shear strain is proportional to
G 0 COT 2

2

1+CO T ' 1COT

'1 + icor

(9-29)

(9-30)

The function J {t) is called the creep compliance. For a viscoelastic liquid, the material will eventually flow with constant shear rate (J0/rj0, so that J (t) approaches to 07 0 for large t. From Eq. (9-16), it is easily shown

When COT §> 1, G*(co) = Go, which means that the polymeric material behaves as an (9-31) elastic material for quick deformation. On the other hand, for co T <^ 1, G* (co) = i co Go T, The intercept of the asymptotic line of J (t) which indicates that the material behaves with the vertical axis gives the steady state as a viscous liquid of viscosity rj0 = Go T for compliance J e . slow flow. t

(9-32)

9.2.5.3 Creep Experiments

Je = lim

The creep experiment is shown in Fig. 9-5. Here a constant shear stress a0 is applied for a sample at equilibrium, and the time evolution of the shear strain y(t) is

It can be shown by Eq. (9-16) (Bird et al., 1987, p. 270) that J e is related to G(t) as

no

\<\tG{i)t o

(9-33)

(a

or by Eqs. (9-25) and (9-26), = lim

G"(co)2

(9-34)

Je can also be obtained by setting the stress to zero after the steady shear flow is attained (see Fig. 9-5). When the stress is set to zero at a time t l 5 the elasticity of the material causes a reverse flow, and J e is related to the recovered strain = -lim 0".0 t-> oo

Figure 9-5. The shear stress (a) and the shear strain (b) in the creep experiments.

(9-35)

399

9.3 Characteristics of the Viscoelasticity of Polymeric Liquids

9.3 Characteristics of the Viscoelasticity of Polymeric Liquids 9.3.1 Time-Temperature Superposition Rule

The mechanical properties of a polymeric material change dramatically with temperature. For example, polystyrene appears as a glassy solid at low temperature, but as the temperature is raised, it becomes soft and finally becomes a fluid. The effect of temperature on the rheological behavior can be well described by a useful empirical relation called the time temperature superposition rule. The time-temperature-superposition rule states that well above the glass transition temperature Tg, changing the temperature is approximately equivalent to changing the time scale. For example, let G (t, Ts) and G (t, T) be the relaxation moduli measured at two different temperatures Ts and T. The time-temperature superposition rule states that they can be related to each other as =

bTG(t/aT,Ts)

(9-36)

where aT and bT are constants independent of t. From Eqs. (9-18), (9-33) and (9-36), it follows

Empirical equations are available for aT and bT\ aT =exp[

-

(9-38) l

s

"T" ^ 2

and (9-39) where Q and QS are the densities of the material at temperature T and Ts respectively, and Cx and C 2 are constants. If Ts is taken as the glass transition temperature Tg, Q

and C 2 become independent of the material (Ct « 17 and C 2 ~ 52). Examples of C1,C2 and Ts can be found in the literature (Ferry, 1980). For the other viscoelastic functions, the time temperature superposition rule gives the following relations: = bTG' (coaT,Ts); (9-40) and =

l/bTJ(t/aT9TJ

(9-41)

A similar superposition rule exists for the effect of pressure = fcPG(t/flF,Ps)

(9-42)

These superposition rules indicate that there is only a single fundamental time scale which governs the dynamics of polymers, and that changing the temperature or the pressure only affects this fundamental time scale. 93.2 Behavior of Relaxation Modulus The relaxation modulus G(t) shown in the literature covers an extremely wide range of time scale; typically of the order of 1010. Data for such a wide time scale is obtained by doing the experiments at different temperatures and then synthesizing a master curve by using the time temperature superposition rule. Figure 9-6 shows a sketch of the relaxation modulus of polymer melts. The time dependence of the relaxation modulus can be broadly divided into four regions, the glassy region, transition region, rubbery region and flow region. The glassy region corresponds to the behavior at low temperature, where polymeric materials have a rather high modulus of the order of 1010 Pa. The relaxation modulus decreases in the transition region, and then becomes flat

400

9 Viscoelastic and Rheological Properties

flow

Figure 9-6. Sketch of the relaxation modulus of a polymeric liquid. 10-8

10-6

10-4

10-2

1

in the rubbery region, where the polymeric liquids behave like a soft rubber (G « 105 - 106 Pa for polymer melts). In the flow region, which corresponds to the time scale larger than the longest relaxation time, the polymeric materials behave as a viscous fluid. Generally speaking, the relaxation behavior at longer time scale reflects the molecular motion on larger length scale.

The behavior in the glassy and transition region is determined by the local structure of polymers, such as the chemical structure of monomeric units or the short side branches etc. On the other hand, the behavior in the rubbery and flow regions is determined by the global characteristics of polymers, such as molecular weight, molecular weight distribution, and the structure of any long branches. As the molecular weight is increased, the rubbery region extends towards longer times, but the glassy and the transition regions remain unchanged. Figure 9-7 shows a sketch of the storage and loss moduli. The above characteristics can also be seen in this graph. The flow region corresponds to the region of low frequency where G' (co) and G" (co) have a linear portion on the log-log plot with respective slopes of 2 and 1 (see Eqs. (9-24) and (9-34)) Gf(co) = rjlJeco2

Figure 9-7. Sketch of the storage modulus G' (co) and the loss modulus G" (co) of a polymeric liquid.

and Gff (co) = rj0co (9-43)

The rubbery region corresponds to the plateau of G' (co), and the transition region corresponds to the increase of G' (co) and G" (co) at higher frequency.

9.3 Characteristics of the Viscoelasticity of Polymeric Liquids

9.3.3 Linear Polymers with Narrow Molecular Weight Distribution

For linear polymer melts with narrow molecular weight distribution, the relaxation modulus in the rubbery and flow regions can be approximated to a single exponential curve. This is shown in Figs. 9-8

-6

-5

-4

-3

-2

401

and 9-9 in the curves of G'(co) and G"(co). According to Eq. (9-29), G' (co) becomes constant for COT ^ 1 and G"(co) shows a peak at COT = 1. Such a plateau and peak are indeed seen in experimental data for polystyrene in Figs. 9-8 and 9-9. The flat region of G' (co) is called the rubbery plateau, and its height GN, called the plateau

Figure 9-8. Storage modulus of polystyrene with narrow molecular weight distribution. (From Onogi, S., Masuda, T, Kitagawa, K. (1970), Macromolecules 3, 109; reproduced by permission of American Chemical Society.)

-1

PS 160 °C

x

6

1w yr-

J

7/.

V

-6

-5

-k

-3

Y>

-2

-1 0 log UQ T ,sec"

y

F

Figure 9-9. Loss modulus of polystyrene with narrow molecular weight distribution. (From Onogi, S., Masuda, X, Kitagawa, K. (1970), Macromolecules 3, 109; reproduced by permission of Americal Chemical Society.)

402

9 Viscoelastic and Rheological Properties

modulus, is independent of the molecular weight, GN oc M°

(9-44)

while the relaxation time is strongly dependent on the molecular weight; T oc M 3 - 4 (9-45) The plateau modulus GN is used to estimate M e the molecular weight between the entanglement junctions. As it is described in Chap. 8, the rigidity modulus G of a rubber can be related to the molecular weight Mx of a polymer segment between the crosslinks: M =

QRT

(9-46)

where Q is the density of the polymer, R the gas constant and Tthe absolute temperature. Likewise, from the height of the rubbery plateau, one can define a molecular weight, M =

QRT

M 3 4

M -

(M < MJ (M > Mc)

(9-48)

The characteristic molecular weight M c is about two or three times larger than M e . Table 9-1. Characteristic molecular weights for selected polymers. Polymer species Polystyrene Poly(a-methyl styrene) 1,4-Polybutadiene Polyvinyl acetate Poly(dimethyl siloxane) Polyethylene 1,4-Polyisoprene Polyisobutylene

Me 19 500 13 500 1900 12000 10000 1300 6 300 8 900

Mc

M'c

35 000 130000 28000 104000 5000 13 800 24 500 86000 24400 61000 3 800 12000 10000 35 000 15 200 —

101*-

1012-

(9-47) 1010-

which is supposed to characterize the spacing between the entanglement junctions. In the early theory, the physical entity of the entanglement junction was not clarified. However M e turns out to be an important quantity in the modern theory of entanglement (see Sec. 9.7). Values of M e for some typical polymers are listed in Table 9-1. In the region where G(co) shows a plateau, the steady state viscosity depends on M as rj0 oc M 3 4 . As the molecular weight is decreased, the plateau disappears, and the molecular weight dependence of rj0 changes (see Fig. 9-10). When the correction for the chain end effect on the monomeric friction constant is accounted for, the viscosity is proportional to M. Thus the molecular weight dependence of the viscosity can be approximately expressed by

a 10* -

106 -

10* -

102 -

100 -

107

Figure 9-10. Molecular weight dependence of the viscosity. The filled circles indicate data corrected for the chain end effects on the monomeric friction constant. The open circles indicate unadjusted result. (From Colby, R. H., Fetters, L. J., Graessley, W. W. (1987), Macromolecules, 20, 2226; reproduced by permission of American Chemical Society.)

9.3 Characteristics of the Viscoelasticity of Polymeric Liquids

/ DO

°V /

-o

9.3.4 Effect of Molecular Weight Distribution and Branching

oo o

?•

8-7-

/ o

ft-

LOG

5 Mw

Figure 9-11. Molecular weight dependence of the steady state compliance of polyisoprene with narrow molecular weight distribution. (From Odani, H., Nemoto, N., Kurata, M. (1972), Bull Inst. Chem. Res. 50, 117.)

Figure 9-11 shows the molecular weight dependence of the steady state compliance. The change in the molecular weight dependence is also seen in J c . For M < M'c, Je is proportional to M, while for M > M'c, Je becomes independent of M. Values of M c and M' are shown in Table 9-1.

PS 160

7

^ fit\^&T

5-

A?

o L 15 •

/

r

/ /

G

2-

1 -

-5

-4

-3

If the polymer has broad molecular weight distribution, or long side chains, the relaxation spectrum becomes broad, and the clear plateau region disappears. An example for polystyrenes is shown in Fig. 9-12. As a result of the broad relaxation spectra, the steady state compliance becomes much larger than that of narrow distribution polymers. On the other hand, the viscosity is known to be described by the empirical Eq. (9-48) even for a system with broad molecular weight distribution provided the weight averaged molecular weight M w is used for M. The effect of branching has been studied in some detail for star shaped polymers which consists of / arms of equal length connected at the central node. Figure 9-13 compares the viscosity of such polymers with that of linear polymers. With fixed molecular weight, the viscosity decreases with the increase in the arm number /,

°C

6-

3-

403

-1

-2

log

sec- 1

PS 7

s>

Figure 9-12. G'(co) and G" (co) of polystyrene of narrow molecular weight distribution (unfilled circles) and broad molecular weight distribution (filled circles). (From Masuda, T., Kitagawa, K., Inoue, T, Onogi, S. (1970), Macromolecules 3, 116; reproduced by permission of the American Chemical Society.)

404

9 Viscoelastic and Rheological Properties

tion. To do that, it is essential to introduce the stress tensor. The stress tensor describes the force acting within the material. The a-j? component of the stress tensor at a point P in the material is defined as follows (see Fig. 9-14). Consider a small region of area AS on a plane which includes the point P and is normal to the jff-axis. Let / (/?) be the force which the material above the plane exerts on the material below the plane through the region AS. For small AS, the force is proportional to AS. The stress tensor aap is defined by the ratio (9-49) Figure 9-13. Viscosity of linear and star branched polyisoprene in normal tetradecane: circle: linear, square: 4 arm star and hexagone: 6 arm star. (From Masuda, X, Onogi, S. (1973), Annual Rep. Res. Inst. Chem. Fibers. 33 9.)

since the molecule is becoming more compact. However with the increase of the molecular weight at fixed arm number, the viscosity of star polymers increases more rapidly than linear polymers, and eventually exceeds it at high molecular weight. This tendency is seen for other types of branched polymers. This is because at high molecular weight molecular disentanglement in branched polymers takes place much more slowly than in linear polymers.

If there is no electric or magnetic field, the stress tensor is symmetric. Gap = Gpa

(9-50)

Also since the polymeric liquids may be regarded as incompressible, the stress tensor is written as °ap =
(9-51)

where p is an arbitrary function which is determined by external conditions (such as the external pressure). Therefore as to the diagonal components, GXX9 Gyy, GZZ, only

9.4 Non-Linear Viscoelasticity 9.4.1 Stress Tensor

So far, we have restricted the discussion to a special case, i.e. the linear viscoelasticity in shear deformation. We shall now discuss the general case, i.e., the nonlinear viscoelasticity for general types of deforma-

Figure 9-14. Definition of the stress tensor. The case of p = z is shown.

405

9.4 Non-Linear Viscoelasticity

their difference oxx — oyy, oyy-azz need to be characterized by rheological experiments. 9.4.2 Basic Principles in Continuum Mechanics

Rheological properties of a material are described by the constitutive equation. The constitutive equation relates the stress tensor at each material element with the past history of the deformation of the material. There are two basic principles which the constitutive equation has to obey. (1) Principle of local action: The stress tensor at a material element depends only on the previous history of the material element and not on the state of the neighboring material elements. Let us consider a stress tensor at a material element P. Let r(t,P) be the position vector of P at time t. This is determined by the flow field v (r, t) of the material: d

dV

= v(r(t,P),t)

(9-54)

where we have dropped the dependence on P assuming that both the stress and the velocity gradient refer to the same material element. (2) Principle of frame indifference: The constitutive equation must be independent of absolute motion of the material element. Any superposed rigid body motion (i.e., translation and rotation) does not affect the mechanical response of the material. That the constitutive equation is independent of the translation of the material element has already been accounted for in Eq. (9-54) since a is independent of v. The independence of the rigid rotation imposes a restriction on the functional form for the constitutive equation. Suppose that the material is rotated around the material element P. The rotation changes the vector dr connecting P and its neighboring material element to (9-55)

(9-52)

where U is an orthogonal tensor satisfying:

Due to the principle of local action, the stress tensor of the material element P is determined by the deformation experienced by the element P. The principle of local action states that the deformation at the material element P is described by the history of the velocity gradient tensor at P, i.e., %a

o(t)=

(t\p) = —^

(9-53)

^ P at point P, time t'

and that the higher order derivatives of the velocity field are irrelevant. Given the previous values of x (?, P) for ? < t, the stress tensor a (t, P) is determined uniquely. Thus the general form of the constitutive equation is that a (t, P) is a functional of x (?, P) for ? < t. In a symbolic form,

U r (t')U(O = l

(9-56)

Then the velocity gradient tensor changes as

(9-57) T

T

W (?) = U (?) * (?) U (?) + U (?) U (?)

The principle of frame indifference states that the stress tensor & (t) in the rotating material must be equal to the geometrical rotation of a (t), i.e., (t) = UT(t)<j(t)U(t)

(9-58)

Thus the constitutive equation must satisfy:

= U r (t)#[x(t')]U(0

(9-59)

for arbitrary orthogonal tensor U (t).

406

9 Viscoelastic and Rheological Properties

Due to the principle of frame indifference, we may choose an arbitrary frame in the study of the constitutive equation. Also due to the principle of local action, we may disregard the spatial dependence of the velocity gradient. Thus without loss of generality, we can assume that the flow field is given by (9-60)

v(r9t) = x(

In this case, each material point moves according to (9-61)

dt

Since this is a linear equation for r(t\ the general solution is written as r(t) = E(Ut')r(t')

9.4.3.1 Shear Flow The shear flow is shown in Fig. 9-1. It has a reflection symmetry with respect to the xy plane. Therefore the components °xz = Gzx a n d °yz = Gzy a r e always zero. Thus there are three stress components to be characterized, axy = ayx, N1 = axx — ayy, N2 = Vyy — °zz- Ni anc * N2 a r e called the first and second normal stress differences respectively. In the steady shear flow, the shear stress, and the normal stress differences depend on the shear rate y only. Due to the symmetry of the flow, the shear stress is an odd function of y, while the normal stress differences are an even function of y. Thus for small f, these functions must behave as

(9-62)

and

NuN2ccf

f

(9-66)

where E (t, t ) is a solution of the equation 8 dt

(9-63)

The ratio between the shear stress and the shear rate

with the initial condition E(t\t') = \

(9-64) r

The tensor E (t, t ) is called the deformation gradient tensor. The deformation of the material is expressed either by the velocity gradient tensor or the deformation gradient tensor. The constitutive Eq. (9-59) may be written as (9-65)

a(t) =

(9-67)

y

r
9A3 Non-Linear Viscoelasticity in Rheometrical Flows We shall now briefly describe the characteristic nonlinear viscoelasticity of polymeric liquids in simple flows. Two types of flows are often employed in the characterization of the rheological properties.

is called the steady state viscosity, and the ratios

f

f2

(9-68)

are called the first and second normal stress coefficients. An example of the shear stress <Jxy(y) and the first normal stress difference N± (y) are shown by the solid lines in Fig. 9-15. Notice that the normal stress difference is quite large: indeed at high shear rate, the normal stress difference is much larger than the shear stress. In Fig. 9-15 2G'(co) and G" (co) are also plotted for comparison. The curves axy(y) and G" (co) coincide with each other on the left side of the graph since they are proportional to rj0 y or rj0 co respectively for small y or co. The curves Nx (y) and 2 G (co) also agree with each

9.4 Non-Linear Viscoelasticity

y/s"\ Figure 9-15. Shear stress
other on the left side. This agreement is not fortuitous. By the retarded motion expansion (Bird etal, 1987, p. 329), it can be shown that for any isotropic material, lim Nt (y)y~2 = 2 lim G'(co)co - 2 co-*

407

Fig. 9-16, where the solid line is \rj*(co)\, and the circles denote rj (y). In this case, the Cox-Mertz law works very well. Although there is no theoretical justification, the Cox-Mertz law is known to work well for many polymeric liquids. The second normal stress difference is negative and its magnitude is small compared to the first normal stress difference (usually 10% to 30% smaller than the first normal stress difference). Figure 9-17 shows the growth of the shear stress when a shear flow of constant shear rate y is started. Here rj+ (t, y) is defined by

n+(t,y) =

(9-72) y It is seen that for large y the shear stress reaches a maximum before reaching the steady state value. This phenomenon is

(9-69)

0

or (9-70) Notice that the curves start to deviate from the straight lines at about the same y and co, which indicates that the nonlinearity and the viscoelasticity become important at the same characteristic time. This is an example of the correlation between the linear and nonlinear viscoelasticity. The correlation between the linear and nonlinear viscoelasticity is even more clearly demonstrated by an empirical relation

which is called the Cox-Mertz rule. An example of the Cox-Mertz law is shown in

JO"' 102 y/s~\

Figure 9-16. Steady state viscosity r\ (y) and the first normal stress difference coefficient ^ ( y ) of 10% polystyrene solution in Kaneclor is plotted against the shear rate. Je (y) is defined by ^ (y)/2 rj (y)2. (Data reproduced from Takahashi, M., Masuda, T., Onogi, S. (1977), Rheol. Soc. Jpn. 5, 72; reproduced by permission of Rheological Society of Japan.)

408

9 Viscoelastic and Rheological Properties

isotropic material

105H

lim rjE (e) = 3 lim rj (y)

(9-75)

10*-

103-

102 10-2

10-2

100

101

102

Msec)

Figure 9-17. Shear stress growth function n + (t,y) = (Txy(t, y)/y of poly butadiene is plotted for several shear rates. (From Menezes, E. V., Graessley, W. W. (1982), J. Polym. Sci. Phys. 20, 1817; reproduced by permission of American Physical Society.)

In Fig. 9-18, the steady elongational viscosity is plotted together with the steady shear viscosity. As the elongational ratio is increased, rjE (e) increases slightly and then decreases. For other polymers, it is sometimes difficult to achieve the steady elongational flow since the material ruptures before it reaches the steady state.

called the stress overshoot. Sometimes the overshoot is observed for the first normal stress difference at high shear rate. 9.4.3.2 Uniaxial Elongational Flow

The velocity field of the uniaxial elongational flow is given by vy=-\ey\

VX=-\E\

vz = az (9-73)

where s is called the elongational rate. Such a flow is realized when a cylindrical specimen is stretched in the z direction. Due to the symmetry of the flow, axx is equal to ayy, and all the off-diagonal components of the stress tensor
G

zz ~

G

y

which is called the elongational stress. When s is constant, aE approaches a steady state value. The ratio (9-74) is called the elongational viscosity. By symmetry it can be shown that for any

-2 -1 log / ; logf( log s~1)

0

Figure 9-18. Shear viscosity rj(y) and elongational viscosity nE (e) of near monodisperse polystyrene are shown as a function of y and s respectively. (From Takahashi, M, Masuda, T., Onogi, S. (1984), Polymer Preprints, Japan 33, 871.)

9.4.4 Constitutive Equation

The constitutive equation is a mathematical model for the rheological properties of a material. Since the nonlinear viscoelasticity is a complex phenomena, it is very difficult to have a simple mathematical equation that describes all features of the material property. A variety of constitutive equations have been proposed, but none of them are complete. If excessive accuracy is pursued, many parameters and a complicated set of equations are needed, and it will make the model difficult to use. Thus which type of constitutive equations one should use depends on one's purpose,

9.4 Non-Linear Viscoelasticity

and the types of the flow to be analysed (see the recent literatures, Tanner, 1985; Crochet et al., 1984; Tucker et al., 1989). Here I shall describe a few constitutive equations which frequently appear in the literature. Detailed discussions on the constitutive equation for the polymeric liquids are given in the books by Bird et al. (1986) and Larson (1988). 9.4.4.1 Second Order Fluid The constitutive equation for the Newtonian fluid is (9-76)

ear effect, and holds rigorously for any fluid if the acceleration and the velocity gradient are small. However, the model cannot describe the viscoelastic effect. 9.4.4.2 Convective Maxwell Model The simplest model which accounts for the viscoelastic effect is the convective Maxwell model, whose constitutive equation is 5a_ . (9-81) bt This gives a relaxation modulus G{t) for the viscoelasticity

where

(9-82)

(9-77) is the rate of strain tensor. Equation (9-76) does not describe the shear rate dependence of the viscosity. An obvious way to account for the shear rate dependence is to add a higher order term of the strain rate tensor. From the principle of frame indifference, it can be shown that such an expansion must have the form *,/, = M , / . + & 2 ^ + & 3 W , ,

(9-78)

where, bl, b2 and b3 are material constants and the operator 5/81 is defined by dt

409

The distribution of the relaxation f;mes can be accounted for by writing v = H°k

(9-83)

k

where each ak obeys Eq. (9-81) with material constants xk and rjk. The convective Maxwell model predicts that in the steady shear flow, the first normal stress difference is positive. However, the model does not predict the shear thinning behavior: the calculated steady shear viscosity is independent of the shear rate. 9.4.4.3 Integral Form of the Convective Maxwell Model

at (9-79)

for any tensor Aafi. This model is called the second order fluid. By considering the shear flow, the coefficients bl9 b2 and b3 can be related to the first and the second normal stress difference:

The differential equation for the convective Maxwell model can be put into an integral form by using the deformation gradient tensor E(t, f). The solution of Eq. (9-81) is written as (9-84)

h = rj0; b2 = - \ W± (0); b3 = W2 (0) (9-80) The second order fluid model describes the lowest order correction for the nonlin-

where = E(t,t')-E(t,t')

+

(9-85)

410

9 Viscoelastic and Rheological Properties

is called the Finger-strain tensor. For the system with a relaxation spectrum, the integral form becomes: = \

(9-86)

This constitutive equation has the same limitation as the original Maxwell model; it fails to describe the shear thinning.

pends on B (t, t'). (9-89)

a(t)=

f d

where the tensor O (B) is a certain non-linear function of B. Often it is furthermore assumed that this function is derived from a potential function W which is a scalar function of B. Such a scalar function is generally written using the two invariants of the tensor B:

9.4.4.4 Non-Linear Convective Maxwell Model

The shear thinning effect can be accounted for by assuming that the coefficients x ox r] are the functions of V or a. A well known example is the combination of the convective Maxwell model and the second order fluid model 5a

(9-87)

where x' is a material constant with the dimension of time. This model is called the Oldroyd fluid B. Another model proposed by Gieskus is Xx c 6 + —a

5a hi

—

(9-88)

where X is a dimensionless material constant. These models can describe many characteristic features such as the shear thinning, the normal stress effect and the viscoelasticity. However, for quantitative description, one has to introduce the relaxation spectra as has been done for the convective Maxwell model. 9.4.4.5 Single-Integral Constitutive Equation

A simple generalization of Eq. (9-86) is to assume that the kernel in Eq. (9-86) de-

(9-90) Then

B

(9-91)

The form of the single integral equation (9-89) and (9-91) is called the factorized BKZ model. With an appropriate choice of the function W{Il9I2), the BKZ model describes the rather complex rheological behaviors of polymeric liquids. Examples of the potential function W are ,Q ^ Woe log [1 + | ( / i - 3)] or

WocTr(Ba)

where a is a constant between 0 and 1.

9.5 Molecular Theory I: The Rouse Model 9.5.1 Polymer Motion in Concentrated Systems

We shall now discuss the rheological properties from a molecular viewpoint. In polymer melts or concentrated solutions, polymers entangle with each other as is shown in Fig. 9-19. At equilibrium, the configurations of polymers are quite random; there is essentially no correlation in the position of the centers of mass, and in the direction of polymer segments.

9.5 Molecular Theory I: The Rouse Model

411

9.5.2 Basic Equation of the Rouse Model

Figure 9-19. Polymers in concentrated system.

The molecular origin of the viscoelasticity of such a system can be qualitatively understood as follows. Consider the stress relaxation experiment. When the sample is stretched, the polymers are stretched and create an elastic force which tends to recover the original shape. The molecular origin of the restoring force is the same as for the rubber elasticity, i.e., the loss of the configurational entropy. In the case of rubbers, the polymers cannot recover the equilibrium configuration as long as the sample is stretched since the polymer chains are cross-linked. In the case of polymer melts, however, the polymers can recover the equilibrium configuration even if the macroscopic shape is held fixed since the polymers in liquid can diffuse freely. To describe such a process, we need a dynamical model of polymer chains. There are two models for the molecular dynamics, the Rouse model and the tube model. The Rouse model describes the motion of unentangled polymers, or the motion of entangled polymers over a short time. On the other hand, the tube model describes the slow motion of strongly entangled polymers. These models are for the general dynamics of polymers, and can be used, for example, for the calculation of the diffusion constant.

The Rouse model represents a polymer by a set of beads connected by springs (see Fig. 9-20). The idea behind the model is as follows. Suppose we divide the backbone atoms of a polymer into N — 1 groups, each consisting of ATS consecutive backbone atoms. Each group is called a submolecule. Let Rt (i= 1,2,... N) be the position vector of the backbone atom at the boundary of the submolecules. If Ns is taken to be large, the distribution of the end-to-end vector of each submolecule at equilibrium is Gaussian (993) Weq({R{}) oc

This distribution is equivalent to the Boltzmann distribution exp (— U/kB T) of the bead-spring model whose potential energy is given by (9-94) Thus the bead-spring model represents the equilibrium configuration of the polymer accurately. To introduce dynamics for the beadspring system, we assume that each bead experiences a drag as it moves through the surrounding polymers and that the drag is described by Stokes law. Let us consider a macroscopic flow v(r,t) = x(t)r

(9-95)

Figure 9-20. The Rouse model (left) and a polymer (right).

412

9 Viscoelastic and Rheological Properties

then the drag is proportional to the difference between the segment velocity Rt and the average velocity at the point of the segment x i?,.. Thus the equation of motion is (9-96)

m Rt =

where £ is the friction constant of the bead, and the last term represents the random force due to the thermal motion. The time correlation function of ft is given by the fluctuation dissipation theorem (see Doi and Edwards, 1986, Chap. 3): a ( W ) > = 2UkBT5ijd{t - t')

(9-97)

Usually in the time scale relevant for rheological properties, the inertia term is very small compared to the frictional term, and may be neglected. Thus from Eq. (9-94) and (9-96), we have

i_1

- 2/?,) + jft

Equations (9-98) to (9-102) are the basic equations for the Rouse model. Historically, the Rouse model was originally proposed for a polymer in dilute solution. However, the model turned out to be inappropriate for the dilute solution since it neglects the hydrodynamic interaction. On the other hand, the model has turned out to be useful for polymer melts or concentrated solutions if the molecular weight is smaller than the entanglement molecular weight M e . Why the model works so well in polymer melts is not clearly understood, but many experimental results can be fitted by the Rouse model. For convenience of calculation, a continuous version is employed for the Rouse model. This is obtained by regarding i as a continuous variable ranging from 0 to JV, and using the following transformation rule from a discrete model to a continuous one:

dR

where Cb2 3kBT

(9-99)

corresponds to the characteristic relaxation time of a submolecule. For the end bead, the equation of motion is slightly different: for i = 1, the equation becomes (9-100) This equation can be regarded as a special case of Eq. (9-98) if we define Ro by (9-101)

Ro =

Similarly, the equation for RN is obtained from Eq. (9-98) if RN + 1 is defined by = RN

(9-102)

1

di2

(9-103)

Equation of motion for the continuous Rouse model is (9-104) 8 1 d2R 1 — R(i, t) = — - ^ r + -f(i91) + x J?(i, t) 2 dt TS di C where the time correlation function for the random force is given by ,g . ^ <J{i, 0/0', t)> = 2t;kBTId (i -j) 5 (t - f) The differential equation (9-104) is supplemented by the boundary conditions which are derived from Eqs. (9-101 and 9-102): — = 0 at i = 0 and i = N

(9-106)

9.5 Molecular Theory I: The Rouse Model

9.5.3 Normal Coordinates The Rouse model gives a linear equation for R (i, t). A standard way of treating such a system is to use normal coordinates each capable of independent motion. In the present problem, the normal coordinates are given by

9.5.4 Self-Diffusion and Segmentai Motion in Equilibrium At equilibrium, a polymer molecule moves around by thermal motion and its speed is characterized by the self-diffusion constant defined by - RG(0))2}

= lim

(p = 0,1,2,...)

(9-107)

It then follows from Eq. (9-104), 8 dt

P

Cp

P

P

(9-114)

iv o

P

(for p = l,2,...)

(9-108)

(9-113)

where RG(t) is the center of mass of the molecule. For the Rouse chain, RG (t) is given by

1 p

413

F r o m

where

(9-115) and CP =

for p = l,2,...

and 6n2kBT Nb2

J

= 0,1,2,...)

(9-109)

Substituting this into Eq. (9-96) and using Eq. (9-110), we have (t) - RG(0))2} =

(9-116)

and/ p 's are the random variables satisfying = 0 p = 0,1,2,...;

(9-110)

Comparing this with Eq. (9-113), we have (9-117)

According to Eq. (9-108), the characteristic relaxation time of Xp is Cp/kp = TR/p2, where 3 n2 kn T

(9-111)

TR corresponds to the longest relaxation time of the Rouse model, and is called the Rouse relaxation time. Since N is proportional to the molecular weight M, Eq. (9-111) indicates that TROCM2

(9-112)

Thus the self-diffusion constant is proportional to the inverse of the molecular weight: Z)Goc Af"1

(9-118)

Although the motion of the center of mass obeys the simple diffusion law, the motion of a Rouse bead does not follow this law. It can be shown that the mean square displacement of a Rouse bead is given by (de Gennes, 1979) 6DGt

for

tp

IOr

t ^ T

TR

1/2

(9-119)

414

9 Viscoelastic and Rheological Properties

Notice that in a short time scale the mean square displacement increases at t1/2. 9.5.5 Molecular Expression for the Stress Tensor Whe shall now discuss the viscoelastic properties of the Rouse model. To do this, first we have to know the molecular expression for the stress tensor. This is obtained as follows: As it is explained in Sec. 9.4.1 the stress tensor aafi represents the a component of the force acting through the plane normal to the /? axis. In polymeric materials, the force has two origins: one is the molecular potential (such as the van der Waals potential) which acts between the atomic groups in proximity, and the other is the chemical bond which connects the backbone atoms. The stress arising from the molecular potential is essentially the same as for usual liquids, and will not be important for viscoelastic properties. Thus we shall consider the stress arising from the forces acting through the backbone atoms. Now if a chain is passing through the plane at a polymer segment i, the upper plane exerts a force ,Q . ^ ™..x

3fc

B^

„

,

The isotropic term represents the contribution from the molecular potential. In terms of the normal coordinates, the stress tensor is written as (9-122)

*.,(') = v £

kp<Xpa(t)Xpfi(t)y-P8afi

9.5.6 Constitutive Equation for the Rouse Model Given the molecular expression for the stress tensor, we can obtain the constitutive equation for the Rouse model. First from Eq. (9-108), we can show (see Doi and Edwards, 1986, p. 112) (f

(t)X

(t)y = k TS

(9-123)

Using the relation and Eq. (9-108), we have

±<x,MXpf(t» =

Kll(t)Xpfl(t)

Xpx(t)[

\Xpf(t))-

_ J

3kBTdR(Ut)

on the lower plane (see Fig. 9-21). If there are v chains per unit volume, the probability that the segment i of any polymer passes through the plane is

xXfl(Xpfl(t)Xpll(t)) (9-124) where

where AS is the area of the plane. Thus the stress tensor is given by

= v-

(9-125)

2K

Notice that Eq. (9-124) is similar to the convective Maxwell model. Indeed if we define (9_12g) = vkp(^Xpa(t)Xpfi(t)>

-

9.6 Molecular Theory II: The Tube Model

415

Since the Rouse relaxation time TR is proportional to M 2 , Eq. (9-132) indicates that rj0 oc M;

(9-133)

Je oc M

This is in agreement with the experimental relation Eq. (9-48) behavior for M < Mc or M<M'C (see Figs. 9-10 and 9-11). For t < TR? the sum in Eq. (9-131) may be replaced by an integral for p. Thus

AS,

Figure 9-21. Molecular origin of the stress tensor for the Rouse model.

G{t)&vkBT J dpexp

/

2n2 —t

1/2

(9-134) we can rewrite Eq. (9-124) in the form of the convective Maxwell model

B(afi

fi

(9-127)

The complex modulus is obtained from the Fourier transform of G{t). Especially for CDTRP>

1, G*(co) is given by T

\l/2

- )

<JP + *P^t°P

=

vk

(9-128)

BTTpy/

The total stress is written as oo

(9-129)

* = Z Op

As an example, let us consider the linear viscoelasticity in the shear flow. From the above constitutive equation, we obtain the relaxation modulus:

expf-

(9-130)

p=i

The factor vkBT can be rewritten as Q R T/M (Q being the weight of polymers per unit volume). Hence the relaxation modulus is written as (W31) The viscosity and the steady state compliance are then calculated by Eqs. (9-18) and (9-33). The result is K2QRT

1

"

2M J

(

tj

ll2

KvNkBT(toTs) (l

+0

(9-135)

Thus G' (to) = G" (to) at high frequency and increases as co1/2. These features are seen in Figs. 9-8 and 9-9 at high frequency period.

9.6 Molecular Theory II: The Tube Model 9.6.1 Characteristics of the Polymer Motion in Strongly Entangled Systems

The basic idea of the tube model is as follows (de Gennes, 1979; Doi and Edwards, 1986). Consider the motion of a certain test polymer in a strongly entangled system (see Fig. 9-22 a). If the test polymer attempts to move perpendicularly to its own contour, it will create a large scale motion of many surrounding polymers and encounter large resistance. On the other hand, if the polymer tends to move along its own contour, it will feel much less resistance. Thus dynamically the polymer

416

9 Viscoelastic and Rheological Properties

Figure 9-22. Entangled polymer system (left) and the tube model (right).

is considered to be confined in a tube (Fig. 9-22 b). The tube represents a mean field potential created by the other polymers, and gives a constraint to the polymer in it. In a simple version of the model, the tube is assumed to be fixed in the material. The motion of the polymer perpendicular to the tube is restricted by the tube wall, but its motion along the tube is free. This situation is represented by the Rouse chain confined in a tube (Fig. 9-23). At equilibrium, the tube takes a random configuration. The persistence length of the tube would be of the order of the tube diameter. To specify the model, it is usually assumed that the central axis of the tube consists of straight segments (called the tube segments) of equal length a, connected in random directions. In this model, only one new parameter is introduced. This is the length a, which characterizes the entanglement effect of the surrounding polymers. The rest of the parameters are the same as for the Rouse

model, i.e., N, b and £. All quantities in the tube model can be expressed by these parameters. For example consider the contour length L of the tube axis. At equilibrium, the tube axis is a randomly connected tube segment of length a. Since there are Z = L/a tube segments, the mean square end-to-end distance of the tube axis is Za2. This must be equal to the mean square end-to-end distance of the polymer Nb2. Thus b2 N Z = N-^ = —

and

L = Za

(9-136)

where we have introduced N =

(9-137)

As it will be shown later, JVe corresponds to the entanglement molecular weight M e . The above model is consistent with the time temperature superposition rule. Among the parameters a, N9 b and £, only C is considered to depend on the temperature or pressure sensitively. If the other parameters are assumed to be independent of temperature, the time-temperature rule is derived from the model by dimensional analysis. 9.6.2 Reptation and Fluctuations

Figure 9-23. Rouse chain in a tube.

From the model shown in Fig. 9-23, one would realize that there are three characteristic types of motion.

9.6 Molecular Theory II: The Tube Model

417

1) The lateral fluctuation

(3) Reptation

In a very short time scale, each Rouse bead can move around freely without feeling the constraint of the tube wall. This picture would be valid provided the mean square displacement of a Rouse bead is less than a, or from Eq. (9-119)

In the time scale longer than TR, only the mode of p = 0 is effective. Thus the Rouse chain may be regarded as moving along the tube with a diffusion constant

~

1/2

— I

b


t

(9-138)

where 'a

T

N2

(9-139)

This time characterizes the fluctuations of the Rouse bead in the length scale a. (2) Longitudinal fluctuation On the time scale longer than Te, the polymer motion perpendicular to the tube axis is hindered, but the motion along the tube is free. The motion along the tube can be studied by using the normal coordinates taken along the tube. The modes corresponding to Xp with p > 0 represent the fluctuations of the segment density along the tube. The characteristic time of this longitudinal fluctuation is the Rouse relaxation time :JV 2 T C

c

T

Nt;

As the polymer moves along the tube, one end moves out of the tube, then a new part of the tube is created in a random direction such that the length of the new tube segment is a. When a tube segment is created on one end, the tube segment on the other end becomes empty of the polymer. Such a tube segment will not impose any constraint on the polymer, and can be regarded as destroyed. Thus the one-dimensional diffusion of the polymer is accompanied by the creation and destruction of the tube (see Fig. 9-24). Such a model was first considered by de Gennes, who called the motion reptation.

(9-140)

As a result of the longitudinal fluctuation, the contour length of the polymer along the tube fluctuate around L. The magnitude of the fluctuation has been estimated (see Doi and Edwards, 1986, p. 206) AL « N1'2 b

Z l/2

(9-141)

Thus for a very long polymer, the relative fluctuation AL/L becomes small, and the tube may be regarded as having a fixed contour length.

(9-142)

Figure 9-24. Reptation motion.

418

9 Viscoelastic and Rheological Properties

The characteristic time of reptation can be estimated by the time needed for the polymer to disengage from a certain tube, or the time needed for the polymer to move over the distance L along the tube. Thus L2

(9-143)

This is called the reptation time. Using Eqs. (9-136) and (9-142), this can be written as (9-144) Notice that the ratio of these characteristic times is written as (9-145) Thus for a very long polymer of Z > 1, the time scales of these modes are quite different from each other.

moves out of the tube, and can go in a random direction, whence:

r{L,t + At) = r(L9t) + v{t)

(9-148)

where v (t) is a random vector, whose mean and variance are given by <» = 0; (v2(t)}=aAs

(9-149)

Similarly if As < 0, r (0, t) changes as r (0, t + At) = r(0,t)-v (t)

(9-150)

Equations (9-147) to (9-150) describe the reptation motion. 9.6.4 Self-Diffusion Constant As an application of the kinetic equation, let us consider the mean square displacement of the center of mass: (9-151)

9.6.3 Reptation Model

From Eq. (9-147), we have

The analysis of the reptation model becomes simple if the fluctuation modes are neglected. In this case, the polymer is regarded as having a fixed contour length L and moving only along itself with the diffusion constant Dc. The equation of motion for reptation is obtained as follows. Let r (s, t) be the position of a polymer at the contour length 5 (0 < s < L) and time t. Consider that the polymer moves a distance As along itself in a time interval At. Clearly, As takes positive or negative values randomly, its mean and variance are given by =

(9-146)

Since the polymer moves along itself, the time evolution of r (s, t) is given by r(s,t + At) = r(s + As,*)

(9-147)

Special consideration is needed for the chain end. If As > 0, the segment at s = L,

(9-152) 1 r G (t + At) - vG (t) = - (r (L, t) - r (0, t)) As where we have neglected the terms of order v/L. Taking the average of the square of Eq. (9-152) and using the fact that As is independent of r (s, t\ we have <(rG(t + At)-r G (t)) 2 > = = - ^ « v { L , t) - v(0, t))2} (9-153) The first average ((r(L,t) - r(0,t))2} is equal to Nb2, and the second average is given by Eq. (9-146). Hence (9-154) Nb2 2 = C At Thus the self-diffusion constant of the polymer is (see Eq. (9-113)), Afh2

Af b J1

3L2

7

(9-155)

9.6 Molecular Theory II: The Tube Model -6-

According to Eq. (9-155), DG is proportional to M~ 2 . Figure 9-25 shows the self-diffusion constant of monodispersive polystyrene in melts. The results can be described as

DGx

2

M~

for for

M<MC M>MC

419

monodisperse PS Tn +125 °C

-8-

(9-156) ^ -10-

This is in agreement with Eqs. (9-118) and (9-155).

© Bueche 6 Bachus+Kimmich

en o

-o Fleischer

-12-

9.6.5 Stress Relaxation Now we shall discuss the viscoelastic properties using the reptation model. First we have to know the expression for the stress tensor. This is obtained from Eq. (9-121) if we assume that the polymer is stretched homogeneously along the tube. According to this assumption, the Rouse segment i is located at the point s = L(i/N). Hence dR/di is evaluated as

dR(i,t)

(9-157)

where u(s, t) is the unit vector tangent to the tube axis, i.e.,

u(s,t) =

(9-158)

8s

Substituting this into Eq. (9-121), we have 3k T

L

^ L j d ^ f c O - P S . , (9-159)

J

-14-

I log Afw

Figure 9-25. Self-diffusion constant of linear polystyrene. (From Watanabe, H., Kotaka, T. (1987), Macromolecules 20, 530; reproduced by permission of American Chemical Society.)

but since the contour length of the polymer is relaxed, the polymer occupies the central part of the tube of contour length L (see Fig. 9-26). The initial value of Sa)8 can be calculated by geometrical considerations. If a tube segment is in the direction of a unit vector u before the deformation, it will be in the direction E //

(9-162)

u = IE-ill

where (9-160) Let us consider the stress relaxation experiment. Suppose that at t = 0, a step deformation is applied for a system in equilibrium. Let E be the deformation gradient. Under the deformation, the shape of the central axis of the tube deforms as rf =

(9-161)

B'

Figure 9-26. Deformation of a tube in shear. When the shear is applied, the points A and B are transformed to A' and B', but since the tube length is fixed, the ends of the tube come to A" and B".

420

9 Viscoelastic and Rheological Properties

after the deformation. The length of the tube segments becomes a|Eif|. Now the probability that an arbitrary chosen polymer segment is located on the tube axis parallel to u is proportional to the step length of the tube segment a|E«|. Thus SaP(s,+0) =

u (s, t) is isotropic at s = 0 and L. Hence Safi vanishes at s = 0 and L: Safi (s, t) = 0 for s = 0 and s = L

(9-167)

Equation (9-166) can be solved under the initial condition (9-163) and the boundary condition (9-167). The result is: (9-168)

Safi(s,t) =

sin where

(E-»),

= (ME)

(9-163)

where <...>„ denotes the average of u for the isotropic distribution of u\ <•••>« = J-$du--

(9-164)

For t > 0, w(s, t) satisfies u(s,t + At) = = u(s + As,t) (see Eq. (9-147)). Hence Sajg (s, t + At) is written as SaP(s,t + At) = (Safi(s + As,t)>

(9-169) = 3ZTR n2D, Substituting this into Eq. (9-159), we have 3vL 2 . Nb2 (9-170) where (9-171)

SaP{s,t + At) = Sap + < As >^ s «/> +

/

(2p-l)2 \

For shear strain, the deformation gradient E isj given by

(9-165)

where the average is taken for the random variable As. Expanding the right hand side of Eq. (9-165) with respect to As, and using Eq. (9-146), we have

l

8 y n2 p=

E=

1 y 0 0 l 0 0 0 1

(9-172)

For small shearing strain y, the tensor Q is expanded with respect to y as 4

~ ' "

(9-173)

The shear stress is then calculated as 4 vL2 ;kBTy
'a?

Thus the relaxation modulus is given by

or

^1

(9-166)

6s 8fL The equations at the chain end (Eqs. 9-148 and 9-150) indicate that the distribution of :

2

G(t) = Go0(t) where 4 vL2 5 Nb2

B

(9-175) 4viV 5 AT

B

(9-176)

421

9.6 Molecular Theory II: The Tube Model

Notice that

4vN Me J i\ or M 4N

4 QRTN

MNe

e

The molecular weight dependence of the viscosity rj0ccM3 is not completely in agreement with the experimental results (Eq. 9-48). The discrepancy, however, can be removed if the fluctuations are accounted for (see Sec. 9.6.6). On the other hand, the molecular weight in dependence of J e is in agreement with the experimental result for polymer melts with narrow molecular weight distribution (see Fig. 9-11).

(9-177) 9.6.6 Fluctuations

(9-178)

Thus ATe corresponds to Me, and a2 = Ne b2 corresponds to the mean square end-toend distance of the polymer with the molecular weight M e . The viscosity and the steady state compliance are calculated from G (t) by the formula (9-18) and (9-33). The result is (9-179)

The analysis in the previous section is valid for a very long polymer with Z > 1. In reality, however, Z is not very large, typically less than 100. Therefore, the coupling between the reptation and longitudinal fluctuations can be important. An important effect is in the reptation time r d . If the contour length is fluctuating rapidly, the polymer can move out of a tube when it moves the distance L — AL along the tube (see Fig. 9-27). Hence xd * (L - AL)2/DC

and

6Me

(9-180)

5QRT

(a)

L2

D

*

(9-181) 2

(l-Z^ocM3 1-

^

(b)

Figure 9-27. The Brownian motion of a polymer with (a) fixed contour length, and (b) fluctuating contour length. The oblique lines denote the region that has not been reached by the end of the polymer. The length of this region decreases faster in (b) than in (a). (From Doi, M. (1981), J. Polym. Sci. Lett. 19, 265.)

422

9 Viscoelastic and Rheological Properties

For the region of the molecular weight 10 < M/Mc < 100, the molecular weight dependence of Eq. (9-181) is close to the empirical relation r d oc M3<4. If this correction is introduced in Eq. (9-179), it explains the experimental relation for the viscosity rj0 oc M3-4. The effect of the lateral and the longitudinal fluctuations can be seen in the rheological properties of short time scales. Let us consider the stress relaxation. For t < xe, the behavior of the relaxation modulus of the tube model should be the same as that of the Rouse model. Thus G(t)

'T\

vNknT[-\

1 / 2

(for t
Therefore at t « Te N G(xe)xv — kBT

(9-183)

This is the same order of magnitude as Go, the initial value in the reptation model (Eq. (9-176)). Hence G(t) for the Rouse model crosses over smoothly to the reptation regime at t « r e . Thus the relaxation due to the longitudinal motion does not appear in G (t). This is because the contour length is unaffected by small shear strain. The longitudinal mode starts to contribute to the stress relaxation as the shear strain becomes large. Figure 9-28 shows the non-linear stress relaxation modulus defined by G (t, y) = oxy (t, y)/y. As the shear strain increases, a new relaxation process appears. The characteristic time of the process zk is proportional to M 2 , which indicates that zk corresponds to TR and that the process corresponds to the relaxation of the contour length. For t > TR, the stress is given by Eq. (9-170). Hence the nonlinear relaxation modulus is given by (9-184) ) = —Qxy(y)G

102-

a.

1{

100-

10-1-

100

102 103 t/s Figure 9-28. Nonlinear relaxation modulus G (t, y) for solution of polystyrene in chlorinated biphenyl. Magnitude of shear y are < 0.57, 1.25, 2.06, 3.04, 4.0, 5.3 and 6.1 from top to bottom. (From Osaki, K., Nishizawa, K., Kurata, M. (1982), Macromolecules 15, 1068; reproduced by permission of American Chemical Society.) 101

where (9-185) Thus for t > TR, G{t,y) can be factorized into two parts; one depends on shear strain only, and the other on time. This prediction has been confirmed by experiments (Fig. 9-29). The function h(y) is called the damping function. Figure 9-30 shows a comparison between the theoretical damping function and the experimental one. The agreement is very good considering that h (y) includes no adjustable parameter. 9.6.7 Constitutive Equation

It is not possible to get a simple analytic constitutive equation for the reptation model described above. However, there is a

423

9.6 Molecular Theory II: The Tube Model

100 -

\

ifl-1 -

\8"

V

\ \

10-2-

100

10-1

101

Figure 9-29. Reduced nonlinear relaxation modulus G{t,y)/h(y). Each curve for y > 1.25 in Fig. 9-27 is shifted vertically by an amount — log [h (y)] so that it superposes on the top curve in the long time region. TJ indicates the longest relaxation time, and xk the characteristic time below which the superposition is not possible. (From Osaki, K., Nishizawa, K., Kurata, M. (1982), Macromolecules 15, 1068; reproduced by permission of American Chemical Society.)

Figure 9-30. The damping function h (y) obtained by the procedure explained in Fig. 9-29. Filled circles represent polystyrene of molecular weight 8.42 x 106 and the unfilled circles of 4.48 x 106. Directions of pips indicate concentrations which range from 0.02 gem" 3 to 0.08 gem" 3 . The solid line represents the theoretical value. (From Osaki, K., Nishizawa, K., Kurata, M. (1982), Macromolecules 15, 1068; reproduced by permission of American Chemical Society.)

useful approximation, called the independent alignment approximation which enables us to derive a simple constitutive equation. In the presence of a flow, the equation of motion for u (s, t) becomes

culate the stress tensor for general flows. The result is (9-188)

u (s,t + At) = u{s + As, t) + Awflow (9-186) The first term represents the reptation motion, and the second term the effect of the macroscopic flow. The independent alignment approximation assumes that Anflow is given by the change of a unit vector embedded in the material. Atffiow = (x(t) • u - (*:uu)u)At

(9-187)

This is not consistent with the deformation model shown in Fig. 9-26, but the error of the approximation has been shown be small as long as the direction of the flow is not changed. If the independent alignment approximation is used, it is possible to cal-

°*,(t) = G'o \ — 00

dtd(P{t~tf)QW(E(t,t>)) or

where G'o = 5/4 Go, &(t) is given by Eq. (9-171), and Q^(E) is given by

Equation (9-188) is the BKZ type constitutive equation explained in Sec. 9.4.4.5. It reproduces major characteristic aspects of the non-linear viscoelasticity. Detailed comparison of the theory with experiments has been done by Osaki and Doi (1984). 9.6.8 Effect of Branching and Molecular Weight Distribution

The crucial assumption of the model described above is that the tube can be re-

424

9 Viscoelastic and Rheological Properties

garded as fixed in the material. For linear polymers with narrow molecular weight distribution, this simple picture appears to hold: at least many experimental results can be accounted for qualitatively or sometimes quantitatively by this simple model. However, the simple picture breaks down for polymers with broad molecular weight distribution. The failure of the fixed tube model can be demonstrated by a simple example. Consider a mixture of short (S) and long (L) polymers. If the molecular weight of the short polymers M s , is close to that of the long polymers M L , the fixed tube assumption will be valid. However as M s decreases, the constraints imposed by the short polymers become weaker. In the extreme case of M L > M s , the constraint imposed by the short polymers will be negligible for the long polymers. Thus in a system with broad molecular weight distribution, the assumption of the fixed tube will be invalid. A mechanism for the motion of the tube is illustrated in Fig. 9-31. The topological constraints imposed on the polymer A is

Figure 9-31. Constraint release process.

released and recreated if the polymer C moves as shown in Fig. 9-31. Such a process, called the constraint release, causes the motion of the tube. The constraint release is considered to be important also for branched polymers. If the polymer has long side branches, the reptation motion is severely suppressed. In such a case, the contour length fluctuation, and the constraint release are considered to be the dominant mechanism of stress relaxation. However, it is difficult to account for the coupling between reptation, constraint release and contour-length-fluctuation, and quantitative theory is not yet available. The constraint release is discussed in detail by Graessley (1982) and Marrucci (1985).

9.7 References Barnes, H. A., Hutton, X E, Walters, K. (1989), An Introduction to Rheology. Amsterdam: Elsevier. Bird, R. B., Armstrong, R. C. Hassager, O. (1987), Dynamics of Polymeric Liquids, 2nd ed. volume 1. New York: John Wiley. Crochet, M. I, Davies, A. R., Walters, K. (1984), Numerical Simulation of Non-Newtonian Flow. Amsterdam: Elsevier. de Gennes, P. G. (1979), Scaling Concepts in Polymer Physics. New York: Cornell Univ. Press. Doi, M., Edwards, S. F. (1986), The Theory of Polymer Dynamics. Oxford: Oxford University Press. Ferry, I D. (1980), Viscoelastic Properties of Polymers, 3rd ed. New York: Wiley. Graessley, W. W (1982), Adv. Polym. Sci. 47, 67. Larson, R. G. (1988), Constitutive Equations for Polymer Melts and Solutions. Stoneham: Butterworths Publishers. Marrucci, G. (1985), Adv. Transport Processes 5: Mujumdar, A. S., Mashelkar, R. A. (Eds.), Wiley Eastern Ltd., New Delhi. Osaki, K., Doi, M. (1984), Polym. Eng. Rev. 4, 35-72. Tanner, R. I. (1985) Engineering Rheology. Oxford: Clarendon Press. Tucker III, C. L., (Ed.) (1989), Computational Modeling for Polymer Processing. Munich: Hanser Publishers. Walters, K. (1975), Rheometry. London: Chapman & Hall.

9.7 References

General Reading Astarita, G., Marrucci, G. (1974), Principles of NonNewtonian Fluid Mechanics. London: McGraw Hill. de Gennes, P. G. (1975), The Physics of Liquid Crystals. Oxford: Oxford Univ. Press. Edwards, S. F. (1976), The Configurations and Dynamics of Polymer Chains in Molecular Fluids: Balian, R., Weill, G. (Eds.). London: Gordon and Breach. Flory, P. F. (1953), Principles of Polymer Chemistry. Ithaca, N.Y.: Cornell Univ. Press.

425

Janeschitz-Kriegl, H. (1964), "Polymer Melt Rheology and Flow Birefringence", Elastic Liquids: Springer, A. S., Lodge, T. P. (Eds.). London: Academic Press. Lodge, T. P., Rotstein, N. A., Prager, S. (1990), "Dynamics of Entangled Polymer Liquids: Do Linear Chains Reptate?", Adv. Chem. Phys. 79, 1-132. Osaki, K., Doi, M. (1984), "Nonlinear Viscoelasticity of Concentrated Polymer Systems", Polymer Engineering Reviews 4, 35-72. Tirrell, M. (1984), Rubber Chem. Tech. 57, 523. Tobolsky, A. V. (1960), Properties and Structure of Polymers. New York: John Wiley.

10 Plastic Deformation of Polymers Buckley Crist Department of Materials Science and Engineering, Northwestern University, Evanston, IL, U.S.A.

List of 10.1 10.1.1 10.2 10.3 10.4 10.4.1 10.4.2 10.4.3 10.5 10.5.1 10.5.2 10.5.3 10.6 10.6.1 10.6.2 10.6.3 10.7 10.8 10.9

Symbols and Abbreviations Introduction Basic Concepts of Plastic Deformation Mechanical Testing and Definitions Criteria for Yielding and Crazing Yielding and Deformation Behavior Glassy Polymers Semicrystalline Polymers Blends and Block Copolymers Fundamental Nature of Polymer Yielding Glassy Polymers Semicrystalline Polymers Crystalline and Liquid Crystalline Polymers Post-Yield Deformation and Modeling Viscoelastic Models Molecular Models Continuum Mechanics of Necking Summary Acknowledgements References

Materials Science and Technology Copyright © WILEY-VCH Verlag GmbH & Co KGaA. All rights reserved.

428 430 430 432 436 437 438 440 444 445 446 451 457 460 460 461 464 465 467 467

428

10 Plastic Deformation of Polymers

List of Symbols and Abbreviations a A, Ao b C E (t) Eo F G H AH k /, l0 Za lc L Lk m n P r0 R* t t' T Td 7^ Tg Tm TR w* A I/* v* w z*

radius of molecule or bundle in a double kink instantaneous loaded area, initial loaded area modulus of the Burger's vector of a dislocation exponential factor in nonlinear viscoelastic model tensile stress relaxation modulus elastic (unrelaxed) tensile modulus force elastic shear modulus hardness activation energy in absence of stress Boltzmann's constant instantaneous length, initial length thickness of amorphous layer thickness of lamellar crystal long period of semicrystalline polymer length of a kink band number of cooperatively rearranging conformers width parameter of distribution of relaxation times pressure radius of dislocation core critical radius of dislocation loop time relaxation time or average relaxation time absolute temperature temperature of tensile drawing temperature at which yield stress vanishes glass-liquid transition temperature melting temperature reference temperature for stress-strain behavior critical size of region sheared by screw dislocations activation barrier for forming nucleus of critical size activation volume height of kink band in loading direction critical size of kink pair

a 7, y 7y f a, g n , £t e, £ n , £t £p £y

fractional crystallinity shear strain, shear strain rate shear yield strain preexponential frequency factor in Arrhenius equation uniaxial strain, nominal strain, true or logarithmic strain uniaxial-strain rate, nominal-strain rate, true- or logarithmic-strain rate plastic strain uniaxial yield-strain

List of Symbols and Abbreviations

e* critical strain for onset of plastic flow r\ viscosity 9 fictive temperature A uniaxial stretch ratio fi pressure coefficient of shear yield-stress v Poisson's ratio °2> °3 principal stresses T shear stress Ty shear yield-stress Ty* shear yield-stress at zero pressure T0 yield strength in absence of thermal fluctuations 4> angle between kink b o u n d a r y a n d loading direction i// angle between b o u n d a r y normal a n d loading direction co angular rotation of chain segment in double kink model WLF SANS KWW

Williams-Landel-Ferry equation small-angle neutron scattering Kohlrausch-Williams-Watts function

429

430

10 Plastic Deformation of Polymers

10.1 Introduction Plastic deformation is closely associated with the technology of polymers. Thermoplastics and thermosets in engineering applications ultimately fail by yielding, that is, by the onset of plastic deformation. This may be caused by large scale ductile failure, or by local crazing which precedes "brittle" fracture. Yielding thus limits the conditions under which polymer materials can be used in load bearing situations, but at the same time provides for toughness by ductile rather than truly brittle fracture. The study of this process is essential for understanding the strength of solid polymers. From a different viewpoint, processing to impart macroscopic anisotropy to chain-like macromolecules is routinely employed to modify the behavior of polymers. A good example of property enhancement by plastic deformation is fiber drawing, which increases tensile strength and elastic modulus in the draw direction by factors of 5 or more, compared to the isotropic material. Films are also subjected to biaxial stretching to modify mechanical, optical, and permeation properties. Despite the importance of yielding and post-yield deformation in design, processing, and utilization of plastics, films, or fibers, our fundamental understanding of these phenomena in polymers is not so advanced as in other classes of materials. There are many reasons for this situation. The principal one is that physical microstructures of polymer solids are more irregular and heterogeneous than those of polycrystalline metals or ceramics. This complicates the establishment of structureproperty relationships. Mechanical behavior of polymers is furthermore sensitive to such variables as temperature, strain rate, hydrostatic pressure, and environment. Presented here is an overview of plastic

deformation and the theoretical approaches that have been developed to account for observed behavior. Emphasis is on shear yielding, which occurs at essentially constant volume. Crazing, very local plastic deformation with considerable void volume dispersed among microscopic fibrils, has been reviewed by Kramer and Berger (1990). The relation of crazing to deformation and fracture is treated in Chap. 15 of this Volume. Further articles treating mechanical properties of solid polymers are those on elastic behavior of crystalline polymers (Chap. 7, this Volume) and elastomeric networks (Chap. 8, this Volume). 10.1.1 Basic Concepts of Plastic Deformation A solid body responds to an applied stress field by changing shape and, in most cases, volume. Simple elastic deformation depends only on the instantaneous stress field; elastic strains are established immediately upon imposition of stress and are independent of the history of the material. In practical terms, the dimensions of an elastic body are time-independent in a constant stress field and revert to their original values when the stress is removed. For an ideal elastic-plastic material, permanent or irreversible shape changes occur at a particular stress, which is defined as the yield stress; there is an associated (elastic) yield strain imposed before the onset of plastic deformation. The yield stress and yield strain define the yield point. Within a solid that has yielded, the local packing of atoms, ions, or molecules has been perturbed to a new configuration which is stable in the absence of the stress. Perhaps the clearest example is the shear deformation of adjacent planes of spherical atoms depicted in Fig. 10-1. Two features

10.1 Introduction

oooo

DQOD (a)

431

(b) y = 0 . 5

of this simple atomic model are quite general and apply to polymers: incremental plastic deformation occurs by local shear strains on the order of unity, and there is no appreciable change in volume or density. It is obvious that this model does not describe the local arrangement of matter in polymers, particularly in glassy ones. Computer simulation of the deformation of polymer glasses indicates that smaller shear strains may be distributed over relatively large volumes (Chap. 10, Vol. 6; Mott etal., 1993). Nevertheless, the basic idea that structural elements (repeat units of polymer chains) are mutually displaced by local shear to new, perhaps metastable, equilibrium positions is appropriate. At this point one feature which is unique to plastic deformation of chain-like macromolecules can be identified. The conformations of polymer chains are altered by the deformation process, that is, the chains are stretched. This leads to the well-known phenomenon of "reversion" or essentially complete recovery of the undeformed macroscopic shape when the material is heated beyond an appropriate temperature

(c)

Figure 10-1. Model for elementary shear-displacement; rows of circles represent planes of atoms or polymer chains. Energies of undeformed (a) and plastically deformed (c) states are the same. The maximum resistance i y (derivative of energy with respect to shear) occurs at yy = 0.25, midway between the states of lowest (a) and highest (b) strain energy.

(the glass transition temperature Tg for amorphous polymers, or the melting temperature Tm for semicrystalline polymers). The significance of plastic deformation in polymer processing derives from chain orientation and its effect on properties. Interatomic bonding is conspicuously anisotropic for polymers in either glassy or crystalline states. Local shear is accomplished by disruption and re-establishment of weak intermolecular bonds by a process analogous to the sketch in Fig. 10-1. Strong covalent bonds along the polymer chain are unaffected (aside from their orientation), and the collective effect is distortion of the chain conformation as shown in Fig. 10-2. Macroscopic chain orientation leads to anisotropic macroscopic properties within the plastically deformed polymer (Ward, 1975). If, on the other hand, a polymer is deformed in the liquid state at a temperature so high that microbrownian motion can rapidly relax the chain elongation, shear flow occurs by mutual displacement of essentially undistorted coils; macroscopic shape change is achieved without substantial chain orientation or

432

10 Plastic Deformation of Polymers

Figure 10-2. An isotropic polymer chain (a) is stretched and partially oriented by local shear (b). This conformation is retained in the unloaded glass (or crystal) (c) because new intermolecular bonds have been established as in Fig. 10-1. Entropic forces drive recovery of original conformation (a) when the sample is melted.

attendant anisotropy of properties. Melt processing of polymers by viscous flow is in this sense similar to plastic deformation of metals or other atomic solids. Viscoelastic and rheological properties of melts and solutions are discussed elsewhere (Chap. 9, this Volume). The model in Fig. 10-1 can be used to estimate the conditions at which plastic deformation will commence (see e.g., Bowden, 1973). The result is a yield point defined by a critical shear stress i y « G/6, where G is the elastic shear modulus, and a critical shear strain yy « 0.25. Experimental shear yield strengths at low temperature are about a factor of 2 below this estimate (Bowden, 1973; Brown, 1986). Glassy and semicrystalline polymers thus have strengths which are reasonably close to the theoretical limit established by the simple model; this subject is treated in more detail in Sec. 10.5. Characterization of plastic deformation requires definition of the yield point (the stress and strain at which irreversible strains are first observed). Another quan-

tity of interest is the maximum plastic stretch that can be imposed before fracture; this is a measure of the ductility or toughness of the material. Related to postyield deformation are strain softening and strain hardening, that is, changes in the material's susceptibility to additional plastic deformation. Since polymers are viscoelastic, one expects mechanical responses to be rate and temperature dependent. There are also pronounced effects from pressure and certain chemical environments. Experimental and theoretical aspects of plastic deformation are covered in this chapter. Sections 10.2 and 10.3 outline procedures used for mechanical testing and the relation between yield under different stress fields. Descriptions of yield and deformation behavior of isotropic polymers, both glassy and semicrystalline, are presented in Sec. 10.4. The emphasis is on homopolymers, though copolymers and blends are treated briefly. Theoretical developments for yield in glassy, semicrystalline, and crystalline polymers are discussed in Sec. 10.5. More phenomenological models are used to describe the flow of polymers after the yield point; these are reviewed in Sec. 10.6.

10.2 Mechanical Testing and Definitions Mechanical properties are derived from observation of dimension changes in a solid body subjected to uniaxial or multiaxial stress fields. Yielding does not occur under uniform hydrostatic stress; loading geometries of lower symmetry are needed to achieve plastic deformation. This statement does not apply to anisotropic or porous materials, which are not considered here. The most common procedure is uni-

10.2 Mechanical Testing and Definitions

axial tensile deformation of a specimen having a gage section of reduced area Ao and length l0 between two clamping regions of larger area as sketched in Fig. 10-3 a. The clamped ends are separated at a constant rate and the force F is recorded as a function of this separation. Samples are usually rectangular or circular in cross section. The true tensile stress is force F divided by instantaneous area A; at=F/A. One frequently employs the nominal stress on = F/Ao, where ^40 is the area before deformation. Strain is the displacement per unit length. Most commonly used is the engineering or nominal strain: A/

L

(10-1)

where / is the instantaneous gage length, generally calculated from displacement of the clamps or loading points. Alternatively, strain gages may be used to define nominal strain over a small section of the gage length (Rutherford and Brown, 1980). True uniaxial strain is the integral of infinitesimal nominal strains: d/

(10-2)

Since properties of viscoelastic polymers are rate dependent, the nominal strain rate sn is reported as the quotient of separation rate (dl/dt) over Zo. The true strain rate et can be controlled or determined with procedures discussed at the end of this section. Uniaxial compression (Fig. 10-3 b) employs relatively short samples with uniform circular or rectangular cross sections. Stress and strain are defined as above. Difficulties can be encountered with compressive buckling (alignment problems) and with frictional end effects. An alternative arrangement for compressive yield testing

(c)

433

(d)

Figure 10-3. Mechanical tests used to study yield in polymers: (a) tension; (b) uniaxial compression; (c) plane strain compression; (d) simple shear. (After Bowden, 1973.)

is plane strain compression, one arrangement for which is sketched in Fig. 10-3 c (Bowden, 1973). An obvious advantage is that the area Ao between the loading surfaces is constant, though friction between the dies and the polymer must still be reduced by a lubricant. Plane strain compression can be thought of as pure shear, as only two of the three sample dimensions are changed. Simple shear is more difficult to achieve and may be complicated by coordinate rotations at large strains. One scheme for imposing shear deformation is shown in Fig. 10-3 d. Shear stress is given by x — FjA, where the force F is applied in the plane of area A. Pure shear deformation does not affect A, so no distinction is required between true and nominal shear stress. Shear strain is defined as y = Ax/y, where Ax is the displacement of planes separated by a distance y; Ax is in the direction of the applied force and is perpendicular to y. Note that, as with plane strain compres-

434

10 Plastic Deformation of Polymers

sion, there is no change of sample dimension in the third (z) direction. The shear strain rate (time derivative of y) is written as y. In most instances tensile or compressive plastic deformation in polymers is macroscopically inhomogeneous, complicating evaluation of stress and strain from overall sample dimensions or changes in load point separation. For the most commonly encountered tensile experiment, multiple shear bands inclined ~ 45° to the loading direction coalesce to form the familiar "neck". Compression testing of rectangular samples frequently results in a single large shear band, while higher symmetry cylindrical samples bulge or "barrel" along the gauge section. Strains within these yielded regions are obviously larger than those calculated from overall sample dimensions. Uniaxial stress is larger in the necked down portion of a tensile specimen (a t >(7 n ). There is local thickening in compressive (shear) yield, leading to a lower true stress in the deformed section (at < an). True stress and true strain are required, however, to describe the material response in post-yield flow. There is no way to avoid inhomogeneous yielding and neck formation in tensile deformation; intrinsic material response dictates this behavior as described below. Before or up to the yield point the relative difference between true and nominal tensile-stress is of the order of the strain 2 < sy « 0.1. Thus the yield point can be determined with reasonable accuracy from force-displacement data; see for example Fig. 10-6 below. By careful control of sample and loading geometry to minimize stress concentrations, it is possible to achieve homogeneous uniaxial compression to / « 0.4 /0 (Boyce and Arruda, 1990). In this range it is straightforward to calculate true stress (and true strain) from overall sample dimensions. With properly

designed pure shear experiments, the stress T and strain y are uniform throughout the deformed section of the sample (G'Sell etal, 1983; G'Sell, 1988). Representative tensile stress-strain behaviors are given in Fig. 10-4. Curve A shows the case where yielding is followed quickly by failure; the yielded volume element has a reduced area A on is large enough to cause fracture. The yield point can nevertheless be defined by the yield stress (7y (dashed line in Fig. 10-4) and the yield strain ey. A large number of semicrystalline and glassy polymers that deform inhomogeneously with a stable neck in uniaxial tension display the behavior shown by curve B. The neck is formed at the yield point designated by oy and sy. A load drop, which may or may not indicate a decrease in true stress at, accompanies neck formation and stabilization. The neck travels along the gage section at a fairly constant flow or propagation stress. Strain hardening is manifested as an increase in load or
Figure 10-4. Representative tensile stress-strain curves. (A) neck formation and fracture; (B) stable neck propagation; (C) uniform drawing without a neck. Dashed lines indicate magnitude of yield stress oy. Plastic strain ep is not recovered after load is removed.

10.2 Mechanical Testing and Definitions

yield and undergoes a reasonable amount of plastic strain sp before failure. This type of behavior is rare for reasons which will be considered below. Neck formation obscures the intrinsic response of the material because of the abrupt decrease in loaded area A of the deforming volume element. Considered construction (Fig. 10-5) can be used to understand some essential features (Vincent, 1960; Haward, 1973; Ward, 1983). True tensile stress at is plotted against nominal tensile strain en for cases sketched in Fig. 10-4. Material A has a yield point but no strain softening (decrease in true stress) and little strain hardening (increase in true stress). Throughout this chapter "strain softening" refers to a decrease in true stress with increasing strain (negative tangent modulus). Material Bx yields without strain softening but strain hardens thereafter, while B 2 displays both strain softening and strain hardening. Materials A, Bx and B 2 stretch inhomogeneously after neck formation at the maximum nominal stress on which is conventionally named yield stress crv. The condition for a maximum in

-1

o

Figure 10-5. Considered construction for tensile deformation. Conditions for maxima or minima in on are given by tangent lines from en = — 1. Material A forms unstable neck, Bt and B2 form stable necks, C deforms without a neck.

G

435

n = 0tO + O is easily shown to be represented by the first point of tangency of at with a line from e n =—1; this extrinsic yield point corresponds to the maximum load F or stress
436

10 Plastic Deformation of Polymers

are seldom applied. Some consequences of variable strain rate on neck formation are discussed in Sec. 10.6.3. Strain softening can be studied by yield behavior in compression or shear, where loaded area A cannot decrease. A load maximum in either of these geometries is sufficient to establish strain softening and intrinsic yield behavior.

300 -

10.3 Criteria for Yielding and Crazing Large scale or macroscopic plastic deformation occurs in polymers by shear displacements which are usually in the direction of maximum resolved shear stress (45° from a principal stress axis). A good example is afforded by polycarbonate at 77 K, well below 7g = 418 K (145 °C). As shown in Fig. 10-6, this glassy material yields in either tension or uniaxial compression at a yield stress a y « 220 MPa and at a yield strain sy« 0.1. Yield strength is somewhat higher in compression than in tension because of the pressure dependence of shear strength discussed below. Fracture occurs more readily in tension than in compression, as suggested by the data in Fig. 10-6. Tensile failure almost invariably involves craze formation and the growth of a crack through the craze (Chap. 15, this Volume). Crazes cannot be formed in pure compression or shear fields, so shear yielding always occurs in those geometries. A polymer may yield in uniaxial or biaxial tension by shear yielding or crazing, or both mechanisms may coexist. Conditions for shear yielding in isotropic polymers are best summarized by the pressure modified von Mises criterion (Raghava etal., 1973; Argon, 1975): a2)2

{a2 - <73)2

= 6T2

(10-3)

Figure 10-6. Stress-strain curves for polycarbonate at T = 11K in tension and uniaxial compression. Nominal stress on is given by dashed lines; true stress at by solid lines. The material fractures soon after yielding in tension. From Imai and Brown (1976); reproduced with permission. © John Wiley & Sons, Inc.

Here o-1? a2, and o3 are the principal stresses and ry is the stress required for yield in pure shear. Equation (10-3) corresponds to yield occurring when the elastic shear strain energy density in the stressed material achieves a critical value. The absolute value of Ty depends, of course, on temperature, strain rate, and pressure. Given the fact that polymers are for the most part van der Waals solids, it is not surprising that their mechanical properties are subject to hydrostatic pressure effects. The yield stress ry in Eq. (10-3) is observed to be pressure dependent: (10-4)

10.4 Yielding and Deformation Behavior

Here xy is the (strain rate dependent) yield stress at zero pressure and P = — (GX + G2 + <73)/3 is the negative of the hydrostatic part of the stress tensor. In a compressive-stress field (hydrostatic pressure P > 0), the strain energy terms on the left side of Eq. (10-3) must increase; yield is more difficult at higher pressures. The pressure coefficient /x has been determined by Bowden and Jukes (1972) to be of the order of 0.1-0.2 for most glassy polymers. One often employs a uniaxial stress field (G± = Gy, G2 = G3 = 0) for which Gy = ^3zy

(10-5)

It should be recalled that ry (and hence Gy) is strain rate and pressure dependent. For a representative pressure coefficient /i = 0.15, the uniaxial yield stress Gy will be 19% larger in compression than in tension. An example of this is seen in Fig. 10-6 for polycarbonate where the ratio of true yield stresses is, in fact, 1.19. For macroscopically isotropic polymers, yielding or plastic deformation of any sort cannot occur in purely dilatometric or hydrostatic stress fields. The shear yield surface is thus a tapering cylinder, centered on the GX = G2 = G3 axis, which broadens as applied pressure P becomes larger (increasing hydrostatic compression). A two-dimensional cut through this surface produces the elliptical line in Fig. 10-7. It should be kept in mind that the shear strengths depicted by such a line are for a set of pressures P (determined in part by
437

\ \ \

/A \

Figure 10-7. Section of the yield surface for the von Mises criterion, indicating yield stresses for uniaxial tension (A), biaxial tension (B), simple shear (C), uniaxial compression (D), and biaxial compression (E). Crazing will not occur below or to the left of the dashed line al = — o2.

discussed by Sternstein (1975) and reviewed by Kinloch and Young (1983) and Yee and Narisawa in Chap. 15 of this Volume. For the purposes of the present discussion, we note that crazing may occur in uniaxial or biaxial tension; this is the prime reason for limited tensile ductility of many polymers. Crazing cannot occur in pure shear (the line corresponding to G1 = — G2 in Fig. 10-7) or in any stress field to the lower left of that line.

10.4 Yielding and Deformation Behavior In this section we describe some important phenomenological aspects of plastic deformation of polymers. More complete reviews have been presented by Bowden (1973), Haward (1973), Ward (1983), Matsuoka (1986), and Brown (1986). Throughout this article uniaxial yield strength Gy and shear yield strength T are assumed to

438

10 Plastic Deformation of Polymers

be related by Eq. (10-5) for isotropic materials. This condition does not apply for the oriented systems treated in Sec. 10.5.3. 10.4.1 Glassy Polymers Uniaxial stress-strain behavior of polycarbonate depicted in Fig. 10-6 is typical of polymer glasses. Yielding in tension is followed almost immediately by fracture at this low temperature, while considerable post-yield flow occurs in compression. Other typical aspects are negative temperature dependence of yield strength (Fig. 10-8) and positive strain-rate dependence of ay (Fig. 10-9). Note that the tensile modulus E is not sensitive to either temperature or strain rate en for testing well below Tg. For polycarbonate the tensile yield stress drops to 45 MPa at 352 K (80°C), less than

one quarter of its value at liquid nitrogen temperature. Increasing 8n by four orders of magnitude increases oy by 35%, and the yield strain £y is increased by less than 20%. While the sensitivity of oy to temperature and strain rate varies from material to material, these features are common to all glassy thermoplastics and thermosets well below Tg. Bulk viscoelastic properties become more evident when the glass transition temperature is approached, as demonstrated by results in Fig. 10-10 for plane strain compression of polystyrene. The elastic modulus [E/(l — v2), where v is Poisson's ratio] is conspicuously temperature and strain rate dependent at 75-80°C because the material is fairly close to Tg = 100°C. The drop in true stress at after yield shows that polystyrene yields intrinsically

100 r-

80

60

5% per minute

20

12 Strain (%)

15

18

Figure 10-8. Tensile stress-strain curves for polycarbonate as a function of temperature, sn = 0.083/s. Lines are calculated from nonlinear viscoelastic models described in Sec. 10.6.1. From Matsuoka (1986); reproduced with permission of Carl Hanser Verlag.

10.4 Yielding and Deformation Behavior

5 min"

12 Strain (%)

Figure 10-9. Tensile stress-strain curves for polycarbonate as a function of strain rate at room temperature. Lines are calculated from nonlinear viscoelastic models described in Sec. 10.6.1. From Matsuoka (1986); reproduced with permission of Carl Hanser Verlag.

Strain rate Is'1)

60-

1.5 > 50

•£30

75 °C

20 80 °C 10

0.05

0.10 0.15 True strain

0.20

Figure 10-10. True stress-true strain curves for polystyrene determined in plane strain compression. From Bowden and Raha (1970); reproduced with permission of Taylor and Francis Ltd.

439

with strain softening, though the amount or intensity of strain softening is moderated at higher temperatures and lower strain rates. Note that no strain softening is observed at 80 °C for the lowest et. Virtually all glassy polymers strain soften after yielding in tension, compression, or shear (Haward, 1973; G'Sell et al., 1983). This lowered resistance to further plastic deformation leads to inhomogeneous stretching in regions where stress concentrations, etc., trigger the first yield events. The more compliant, strain softened regions are thus subjected to larger (local) strains and strain rates than the unyielded portions, resulting in the formation of microshear bands or shear patches (Bowden, 1973; Argon, 1975; Kinloch and Young, 1983). The true stress in Fig. 10-10 is seen to increase after a certain amount of strain softening. This strain hardening is essential to stabilize a region of plastic instability, particularly in tensile deformation; see Considered construction, Fig. 10-5. If the material did not strain harden, those regions that yielded first would continue to deform until a stretched zone failed by one of many possible mechanisms. It should not be surprising that strain hardening is seen in most polymers of commercial importance. Those without this characteristic would be difficult to process and unreliable in service. Haward (1973) has discussed strain hardening in detail, associating it with uncoiling polymer chains or sections of chains in the stretch direction. The chain orientation process is reversible, leading to recovery of macroscopic sample shape when the plastically deformed glass is heated above Tg (see Fig. 10-2). It should be remembered that the glassy polymer is still globally isotropic at, and just beyond, the yield point. Chain stretch and macroscopic

440

10 Plastic Deformation of Polymers

anisotropy are generated when post-yield flow is continued. Strain or orientation hardening of this sort can be treated in terms of the finite extensibility of chains, by analogy with rubber elasticity theory, and is considered in Sec. 10.6.2. Qualitative support for such an approach is given by the strain hardening characteristics of glassy epoxies crosslinked to different extents (Lohse et al., 1969). Increased crosslink density leads to more pronounced strain hardening at lower strains, the expectation if "network" strands in the glass were defined by crosslinks. The major mechanical distinction between glassy thermoplastics and thermosets is in strain-hardening characteristics. Crosslinking by itself has little or no fundamental influence on yielding or strain softening, though ay at a fixed temperature may be affected if the glass transition temperature and elastic modulus (E or G) are changed. This effect has been demonstrated nicely by tensile-yield behavior of (uncrosslinked) poly (methyl methacrylate) reported by Hope et al., 1980. The presence of 6% monomer plasticized the glass, lowering Tg from 110°C to 90°C and decreasing
amorphous regions between lamellar crystals. The crystals, particularly those for chains preferring extended or all trans conformations such as polyethylene, polyamides (nylons) and poly(ethylene terephthalate), are very anisotropic. Melt crystallized polymers generally exhibit a spherulitic morphology; ribbon-like lamellae are disposed radially in the polycrystalline aggregate (Peterlin, 1987; Chap. 4, this Volume). Spherulite diameters are normally in the range of 2-20 jim. Other relevant sizes are 10-25 nm for crystal thickness /c in the chain axis direction, ~ 0.1 - 1 |im for the transverse dimensions of the lamellae and 5-10 nm for amorphous layer thickness /a. A deviatoric-stress field causes the spherulite to change shape in an affine manner at low (elastic) strains. Yielding is inhomogeneous within a spherulite (Samuels, 1974; Schultz, 1974) and requires irreversible shear deformations in both the amorphous and crystalline regions. Lamellar bundles oriented at 45° to the principal stress direction experience the largest resolved shear stress, deforming by interlamellar shear (stretching of the amorphous regions between crystals) or intracrystalline shear (sliding of polymer chains along planes containing the covalently bonded molecular axis). Deformation continues until the original spherulites are transformed to well oriented microfibrils with transverse dimensions of the order of 100 A (Peterlin, 1971,1987). Recent studies on nylon 6 (Galenski et al., 1991) and polyethylene (Galenski et al., 1992) provide good summaries of techniques and models used to evaluate morphology changes which occur during deformation of semicrystalline polymers. As with glassy polymers, plastic deformation causes chain axes to become aligned in the stretch direction. Besides this orientation effect, there are a number of

10.4 Yielding and Deformation Behavior

other morphological changes. We concentrate here on the "cold drawing" regime above Tg of the amorphous regions and usually not too far below the melting temperature Tm of the crystals. Here tensile deformation proceeds with a stable neck giving a "natural draw ratio" A = exp(at), where et is the true tensile strain after necking. Structural changes resulting from this type of deformation have been studied and described many times (Samuels, 1974; Heisse et al., 1977; Peterlin, 1987); here we review the most salient features. The isotropic, usually spherulitic, microstructure is converted to microfibrils, generally with no significant change in crystallinity. The crystalline-amorphous long period L = lc + /a changes to a value determined by the draw temperature, independent of the original L in the undrawn polymer. Chain orientation in the stretch direction is greater for the crystalline segments, though amorphous orientation continues to improve as X is increased at a fixed temperature. Transverse crystal dimensions (perpendicular to the chain axis) are decreased. Macroscopic sample dimensions as well as most aspects of the original isotropic structure can be recovered by heating the drawn polymer above Tm. The significance of these effects is considered in Sec. 10.5.2. There are two important temperatures for semicrystalline polymers, Tg for the amorphous fraction and Tm for the crystals. A major distinction applies above and below the glass transition temperature; below Tg both components are solids, while above Tg the amorphous component is liquid-like. For Tg
441

sponse of the liquid-like phase is sensitive to temperature and strain rate, particularly in the range just above Tg. When T < Tg the mechanical properties of the two phases (glassy and crystalline) are relatively similar. Here properties are less dependent on degree of crystallinity and, to a first approximation, are comparable to those of glasses. An example of the latter type of crystalline-glassy polymer is poly(butylene terephthalate), for which T g =87°C and Tm=33O°C. The temperature dependent tensile behavior in Fig. 10-11 is similar to that seen for polycarbonate in Fig. 10-9; oy and sy both increase as the temperature is lowered, whereas the tensile elastic modulus is virtually independent of temperature. Note the "glassy" yield strain values of sy « 0.05-0.10. The load drop after yield is more pronounced at lower temperatures. All these features are generally consistent with the behavior of polymer glasses (Bowden, 1973; Haward, 1973; Matsuoka, 1986). As a rule, the amount of tensile ductility in semicrystalline polymers below Tg is small, though considerable post yield deformation can be achieved in compression or shear. It is important to keep in mind that chains in both glassy and crystalline regions are involved in the yield process. Semicrystalline polymers above Tg behave quite differently. Tensile results for high-density polyethylene are given in Fig. 10-12. One of the most conspicuous features is the temperature dependence of the tensile modulus, a result of linear viscoelastic processes in both phases (Boyd, 1985). Note that the yield stress is rather small; oy« 30 MPa at room temperature and drops to zero as Tm is approached. For comparison the (compressive) yield stress of poly(methyl methacrylate) is 50 MPa at 100°C, only 10° below Tg (Bowden, 1973). Yield strain ey, defined at the maximum of

442

10 Plastic Deformation of Polymers

120

100 -

80 -

Figure 10-11. Tensile stress strain curves for poly(butylene terephthalate) as a function of temperature, sn = 0.086/s. Lines are calculated from nonlinear viscoelastic models described in Sec. 10.6.1. From Matsuoka (1986); reproduced with permission of Carl Hanser Verlag. 10 Strain (%)

load or <jn, can be as large as 0.2 at sufficiently high temperature or low strain rate. Lowering the temperature causes the (extrinsic) yield stress ay to increase, the sharpness of the load drop to increase, and ey to decrease. The first two of these effects are common to glassy polymers, and the "negative" shift in yield strain is seen in glasses near Tg (see for instance Fig. 10-10). Tensile 160

1

1

1

1

I

1

1

77

140

1 -

120

2

IOO

u 80

• A~

-

172 "^^^.^^ ^215

CD

£ 60

= ^ >

40 20

-

294

^ 3TT-

wr^K-'^v-—r-

~0

2

4

6

81 10 1 12 14 Strain (%)

16

18 20

Figure 10-12. Tensile stress-strain curves for polyethylene as a function of temperature, en = 3.3 x 10~ 4 /s. Material fractures before yield at T<100K. From Kamei and Brown (1984); reproduced with permission. © John Wiley & Sons, Inc.

ductility vanishes as Tg (about 160 K for linear polyethylene) is approached from above. From the curves in Fig. 10-12, "cold drawing" with stable neck propagation is achieved at temperatures above 240 K. Strain-rate effects (Fig. 10-13) generally mimic temperature effects, with a higher en corresponding to a lower temperature. In semicrystalline polymers there is the opportunity to vary crystallinity by either changing cooling rate or incorporating comonomer or other irregularities in the chains. Crystallinity effects have been studied extensively in polyethylene and its random copolymers, for the most part at room temperature. Increasing crystallinity increases the yield stress (Fig. 10-14) and the sharpness of the load drop which accompanies neck formation (Popli and Mandelkern, 1987; Crist et al., 1989). Tensile yielding in polyethylene is extrinsic; in contrast to glassy polymers, there is no strain softening or drop in true stress (Meinel and Peterlin, 1971; G'Sell and Jonas, 1979; Coates and Ward, 1980). Strain hardening is obviously present to stabilize the neck.

10.4 Yielding and Deformation Behavior

J

I

L_

443

Figure 10-13. Tensile stressstrain curves for polyethylene as a function of strain rate, T = 25 °C. Lines are calculated from nonlinear viscoelastic models described in Sec. 10.6.1. From Matsuoka (1986); reproduced with permission of Carl Hanser Verlag.

10 Strain (%)

One interesting observation is that the strain-hardening rate is greatest in copolymers of low crystallinity, as shown in Fig. 10-15. A similar effect is seen in ultrahigh

0.90

0.92

0.94

0.96

p (g/cm3)

Figure 10-14. Tensile yield stress as a function of density Q of polyethylene and random copolymers of polyethylene. Crystallinity a varies from 0.35 and 0.70 and crystal thickness lc varies from 4 to 27.5 nm over the reported density range. •, o linear polyethylene; o, • polyethylenes with short chain branching; A, A, • polyethylenes with short long chain branching. From Crist etal. (1989).

molecular weight polyethylene, which also has a crystallinity less than 50% (Meinel and Peterlin, 1971). Strain hardening in semicrystalline polymers is discussed in Sec. 10.6.2. Strain hardening in semicrystalline polymers also depends on the type of deformation, again in contrast to glasses. G'Sell etal. (1983) have measured shear stressshear strain relations for a large number of polymers shown in Fig. 10-16. Each glassy polymer yields in shear with a stress drop (strain softening) followed by strain hardening. Semicrystalline polymers have no strain softening in shear, which is consistent with tensile behavior (see Fig. 10-15). Note, however, that no strain hardening is observed in post-yield shear deformation for either polyethylene or polypropylene. Since both of these materials draw in tension with stable neck propagation, there is considerable strain hardening in tension; this is shown explicitly for polyethylene in Fig. 10-15. Different strain hardening characteristics in tension and shear are discussed in Sec. 10.6.2.

444

10 Plastic Deformation of Polymers r

LU 0.5

(r r = 2.6) (r r = <

150 LU 7

100

50

5

10

2 3 Shear strain

Figure 10-15. True stress-true strain curves for polyethylene and copolymers at room temperature. Sample designations give number of short chain branches/1000 carbon atoms; density ranges from 0.918 g/cm 3 (LU34) to 0.961 g/cm 3 (LU0.5). From Meinel and Peterlin (1971); reproduced with permission of Pergamon Press pic.

Figure 10-16. Shear stress-shear strain curves for various polymers at room temperature. (1) Polyoxymethylene, (2) poly (methyl methacrylate), (3) nylon 66, (4) polycarbonate, (5) poly (vinyl chloride), (6) polybutene-1, (7) polypropylene, (8) polyethylene. From G'Sell et al. (1983); reproduced with permission of Chapman and Hall.

Plasticizers may have an appreciable effect on the deformation of semicrystalline polymers if the system is tested near the glass transition region of the amorphous component. A good example is nylon 6, for which Tg = 75 °C when dry and Tg = 20°C when equilibrated at 50% relative humidity (water content is about 3%). Tensile yield-stress at room temperature is decreased from 80 MPa to 44 MPa by the addition of this amount of plasticizing water (Bonner et al., 1973). Adding a plasticizer to such a polymer is thus equivalent to raising the deformation temperature. (See the strong influence of temperature on yield strength in polyethylene for ^ r g , Fig. 10-12.)

as extensively as the single polymer systems described above. A good review of earlier work can be found in Manson and Sperling (1974). The classic example of a multiphase polymer-polymer system is rubber toughened polystyrene. Incorporation of 0.1 volume fraction of ~ 1 jim droplets of elastomer (e.g., polybutadiene) decreases the yield stress and increases toughness remarkably. Unmodified polystyrene fractures in tension by crazing at very small strains, while rubber modified polystyrene can be elongated by ~ 50%. Argon and Cohen (1989) have shown that this macroscopic strain comes from a large number of crazes originating at interfaces between the glassy matrix and the more compliant inclusions; crazing efficiency is calculated in terms of particle size and relative elastic moduli. The crazing strain acts as a stress relief mechanism, preventing or retarding fracture. Another important aspect of the polystyrene-polybutadiene system is plas-

10.4.3 Blends and Block Copolymers

Multicomponent polymer systems, most of which have more than one phase, are of great technological importance. These complex materials have not been studied

10.5 Fundamental Nature of Polymer Yielding

ticization of the matrix by the rubber molecules, which can dissolve in polystyrene under hydrostatic tension. The behavior of other binary systems is not so well understood. Most polymerpolymer blends are macroscopically phase separated with limited mutual solubility at the interface. Generally the yield (or fracture) stress of such blends is below that predicted by linear combination of component properties. The critical role of the interface in such blends can be modeled in terms of thermodynamic parameters. Pukansky and Tiidos (1990) treat the ability of stress to be transmitted from phase to phase in terms of interfacial surface energy. This approach appears to work well for the composition dependence of strength in immiscible systems, for which it was designed. It accounts for the properties of a single phase (miscible) blend as well, probably because the macroscopic sample is implicitly treated as an "interface". Chow (1990) has adopted a different method for treating the composition dependence of ay in glasses of miscible polymers. A nonequilibrium interaction between chains is invoked for the glassy state; this excess attraction is manifested as a minimum in specific volume or a maximum in yield stress at intermediate compositions. Chow's scheme effectively correlates nonadditivity of volumes with enhanced yield strength. Interface strength is not an issue in triblock copolymers with long enough block lengths, wherein covalently bonded chains traverse microdomains having dimensions ~10nm, governed by the size of blocks or subchains. It is possible to combine "glassy" and "rubbery" blocks over a large range of volume fractions and morphologies (Chaps. 1 and 6 of this Volume). Of most interest are thermoplastic elastomers which contain a large amount of rubbery material. These have some unique mechan-

445

ical properties which are exemplified by the styrene-butadiene triblock system studied by Kawai and Hashimoto (1979). Solvent-cast block copolymer has well formed alternating lamellae of polystyrene (Tg = 100°C) and polybutadiene (Tg = — 90 °C), each of thickness ~ 25 nm. This material yields in tension, oy = 5 MPa, sy = 0.05, with neck formation qualitatively similar to glasses or semicrystalline polymers. Strain is largely reversible, however, recovering from en = 3.5 to sn = 0.2 (94% strain recovery) with significant hysteresis. This retraction is attributed to the elastomeric component. The yield mechanism was not identified, but post-yield stretch involved rotating and shearing the "rigid" polystyrene lamellae into smaller blocks. Original morphology (and presumably mechanical properties) was recovered by annealing at 60°C for a few minutes. The tensile behavior of block copolymers approximates that of nonlinear elastic models used to treat inhomogeneous plastic deformation (Sec. 10.6.3).

10.5 Fundamental Nature of Polymer Yielding Macroscopic yielding requires local shear displacements and establishment of new intersegmental bonds as sketched in Fig. 10-1. These occur in both glassy and semicrystalline polymers in times as short as seconds, as seen from the stress-strain curves presented in Sec. 10.4. There are two broad categories of explanations for rearrangements at temperatures where (stress free) chains are effectively immobile. One proposes that the stressed material is converted to the liquid state, after which the molten chains respond quickly to the applied stress, then solidify. The second invokes various molecular mechanisms to

446

10 Plastic Deformation of Polymers

account for rapid chain rearrangement in the stressed solid. 10.5.1 Glassy Polymers Yielding mechanisms in glassy polymers have been reviewed by Bowden (1973), Ward (1983) and Kinloch and Young (1983). The earliest ideas were based on mechanical work raising the sample temperature to Tg, or on free volume from dilatometric strain bringing Tg down to the test temperature. In either case chain stretching would occur quickly in the stressed "glass". While both these effects may be important under certain conditions, they cannot be the fundamental reason for yield in polymer glasses. The mechanical work done on a glass is generally insufficient to raise the sample temperature to Tg, even in the adiabatic limit (Haward, 1983). Yielding furthermore occurs at low strain rates or with thin samples under conditions which are isothermal (T
solute rate theory (Bowden, 1973; Argon, 1975; Ward, 1983). Yielding is described as viscous flow in which the activation barrier for local shear displacements (analogous to those in Fig. 10-1) is decreased by the applied stress. For an experiment performed at a shear rate y under a shear stress T, which corresponds to the flow stress or yield stress i y in this case, the strain rate is given by (,0-6) Here f is a fundamental rate factor, AH is the activation energy for the flow process (in the limit of T = 0), v* is the "activation volume" and k is Boltzmann's constant. It is assumed in Eq. (10-6) that D * T > / C T , meaning that reversal of shear steps (back reaction) is neglected. This analytical scheme describes the temperature and strain-rate dependence of yield stress and the (phenomenological) shear strength T 0 at 0 K: • = Tn +

d In y

2kT

2kT v*

2 AH tn

=

In

(10-7 a)

V

(10-7 b) (10-7 c)

The activation volume v* is readily evaluated from the experimental strain-rate dependence of Ty through Eq. (10-7 b). The remaining parameters f and AH are obtained from the temperature dependence of Ty. Note that athermal strength T 0 is proportional to AH, the barrier for intersegmental rearrangements. The Eyring model accounts for many features of yielding in glassy polymers, notably the negative temperature dependence [Eq.(10-7a); ln(y/T)<0] and positive strain rate dependence [Eq. (10-7 b)] of the

447

10.5 Fundamental Nature of Polymer Yielding

yield stress. The basic concept is that yielding and plastic deformation of the glass occur by viscous flow which is activated by the combination of applied stress and thermal fluctuations. Interpretation of the kinetic parameters in physical terms is difficult, though the values of v* are found to be the size of a few repeat units. These and other aspects of the Eyring model have been discussed by Bowden (1973) and Ward (1983). A more specific model for yield of glassy polymers, also based on applied stress and fluctuations of thermal energy, has been developed by Argon (1973, 1975). It is proposed that local shear displacements for glassy, interpenetrating polymer molecules can be represented by the rotation of a chain segment of length z into the direction of principal stretch. Rotation is accomplished by nucleation of a pair of kinks, each characterized by a bending angle co « 2 radians, in a chain segment. This value of w was chosen to correspond to valence bond angles of about 115°, but rotation about single bonds in the backbone is a more feasible mechanism for changing the shape of a chain. In this sense the kink pair is analogous to a "crankshaft" defect. The resultant strain is proportional to 71 a2 CD2 z, where a is the radius of one chain or a cooperatively deforming bundle of chains. Argon calculates the net energy A U of a kink pair of size z, formed under a stress T in an elastic medium having shear modulus G and Poisson's ratio v. Resistance to kink formation is intermolecular, as is the case with the Eyring model. The expression for A U includes the self-energy of the kink pair and the kink-kink interaction energy, reduced by the (plastic) work done on the system by the stress T in forming a kink pair of size z. AC/ goes through a maximum with respect to z as the kink pair increases in size. The maximum value

AC/* is the activation barrier for nucleation of a stable kink pair, i.e., one which grows beyond the critical value z*. Yielding occurs when thermal fluctuations over the stress dependent activation barrier AC/*(T) create kink pairs at a rate which provides the macroscopic strain rate y: y=Texp

H^]

(10-8)

In this expression the activation energy AC/*(ry) is made smaller by the applied stress (here equated to the yield stress) in a manner similar to the Eyring formulation in Eq. (10-6). The shear stress required to achieve a particular strain rate is 16/5

(10-9) where the athermal shear strength T0 is 0077 G u

1-v

,10-lOJ

Note that Argon's kink pair nucleation model [Eq.(10-9)] and the Eyring treatment [Eq. (10-7 a)] have similar dependencies of i y on both temperature T and strain rate y; the exponents are 1.2 and 1.0, respectively. The kink pair nucleation model has been used by Argon (1973, 1975), Argon and Bessonov (1977), and Yamini and Young (1980) to interpret the temperature dependence of Ty in a number of glassy polymers. Equations (10-9) and (10-10) require only two adjustable parameters, the frequency factor f and the product co2 a3 (or a3, since co = 2), provided that G and v are known as functions of temperature. Fits of the model to experimental data are generally excellent. Derived parameters are reasonable, with f & 10 13 s" 1 and the effective chain

448

10 Plastic Deformation of Polymers

radius a ranging from about 0.8 to 3 times the single chain radius a0. The model also predicts absolute shear strength in terms of the elastic shear modulus G. Evaluating Eq. (10-10) with a representative value of Poisson's ratio v « 0.38 (Brown, 1983), the athermal strength at T = 0K becomes 0.077 G ;0.12G 1-v

(10-11)

This affords another check on the kink nucleation model, provided that low temperature values of ry and G are available. A third approach to stress and thermal activation of yield was proposed by Bowden and Raha (1974). The energetics of elementary shear displacements are treated in terms of a dislocation loop of radius R forming and growing under the combined effects of shear stress T and thermal energy kT The (unknown) radius of the dislocation core, r0, is replaced by y/3 b/e (b is the modulus of the Burger's vector of the dislocation and e is the base of natural logarithms) to conform to the maximum shear strength: = 0.184 G

(10-12)

predicted for a close packed atomic crystal. The analysis proceeds by evaluating the nucleation barrier

.__, Gb2R* (2R* AU*= In bJ3

(10-13a)

for creating a dislocation loop of the critical size R*: R* =

Gb 4 TIT,

HSH

(10-13 b)

Equations (10-13 a) and (10-13 b) cannot be rearranged to solve for ry analytically. The model is fitted by adjusting the Burger's vector b and the ratio AI7*/(feT) to con-

form to the experimental temperature dependence of i y at a particular strain rate y. The critical activation barrier is frequently chosen as 50 kT (Bowden and Raha, 1974), which leaves only one disposable parameter, b. This choice of At/* establishes the strain rate dependence of r y , as well. It should be emphasized that Bowden and Raha (1974) do not contend that dislocations of the classical sort exist in polymer glasses. Implicit, however, is that elementary shear displacements occur in a spatially correlated linear domain which can close on itself to form a loop. Description of the energy of this configuration by a dislocation model is not based on micromechanics which are unique to crystals. Dislocation theory is quite amenable to continuum treatments; only far field elastic effects are considered in terms of modulus G and Burger's vector b, the nature of the dislocation core being ignored aside from its radius r0. In crystals the Burger's vector b has a well defined length imposed by structural periodicity. The Burger's vector in Eq. (10-13) should be thought of as an average step size for shear displacements in the glass. The derived values of b are consistent with the size of the repeat unit in the chains (Bowden and Raha, 1974; Yamini and Young, 1980). The three thermally activated yield models are compared in Fig. 10-17. Calculated strength i y is normalized by the athermal yield stress r 0 in Eq. (10-10). This ratio exceeds 1 for the Bowden-Raha model because the normalization factor is smaller than that defined in Eq. (10-12). Parameters are based on compressive yielding of poly (methyl methacrylate) at e «10~ 3 /s, f = 1013/s, v* = 1.75 x 10" 2 7 m 3 (Bowden, 1973), and T 0 = 0.43 GPa (Brown, 1983) in Eq. (10-7 a); r = 10 13 /s, co2 a3 = 1.95xlO" 2 8 m 3 (Argon, 1973), G = 3.0 GPa, and v = 0.34 (Brown, 1983) in Eq.

10.5 Fundamental Nature of Polymer Yielding

(10-9); b = 2Jx 10" 1 0 m, At/* = 50 fcT(Yamini and Young, 1980), and G = 3.0GPa (Brown, 1983) in Eqs. (10-13 a) and (10-13 b). The format of Fig. 10-17 compares only the intrinsic temperature dependence associated with thermal fluctuations over activation barriers. Actual temperature dependence of Ty will be more pronounced for the kink pair and dislocation loop models because the shear modulus G decreases at larger T. Similarly, the activation volume y* in the Eyring analysis appears to become smaller at higher temperatures, giving rise to a larger decrease in ry than indicated in Fig. 10-17. As mentioned above, experimental distinctions between the Eyring and Argon treatments are too subtle to permit discrimination between them. It should be appreciated, however, that the kink pair model establishes both the absolute value of Ty and its temperature dependence with measured elastic constants and explicit molecular parameters. The fitting parameter a = 3.7 x 10 " 1 0 m is identical to the ra-

o

200 300 Temperature (K)

Figure 10-17. Temperature dependence of yield stress calculated by the Eyring model ( ), the kink pair nucleation model (—) and the dislocation loop nucleation model (--). Parameters used are appropriate for compressive yield of poly (methyl methacrylate).

449

dius of a single poly (methyl methacrylate) chain, though other relative sizes are seen with different polymer glasses (Argon and Bessonov, 1975). The Eyring model accounts for a nearly identical temperature dependence of ry in terms of an "activation volume" which corresponds to the size of about 12 monomer units in poly (methyl methacrylate); the physical significance of v* has not been established. It is clear that the temperature effect is much different for the dislocation model, which has a large, negative slope of the plot at low T. This characteristic feature has not been observed, though it is difficult to obtain data at low temperatures where the effect is most pronounced. Yield stress ry near 300 K is appreciably lower than that of the other models, but this results from the temperature dependence of G having been ignored when establishing b = 2 . 7 x l 0 ~ l o m (Bowden and Raha, 1974). The high temperature value of Ty/T0 can be adjusted to ~ 0.6 if the Burger's vector b is increased by 40%, which is still about the size of a monomer unit in poly (methyl methacrylate). An additional comparison can be made on the basis of strain rate dependence expressed as dT y /dlny. The Eyring analysis gives 4.6 MPa [which is the experimental value, see Eq. (10-7 b)], the kink pair model 3.2 MPa, and the dislocation loop model 2.0 MPa, all at 300 K. Agreement between Eyring and Argon is improved if the shear modulus G is dropped to the room temperature value. The Eyring treatment is strictly phenomenological and it predicts that the deformation is homogeneous; there is no mechanism for intrinsic yield or strain softening. Argon's model is molecularly based. As presented here, deformation is also globally homogeneous and incapable of strain softening. Once a segment of critical size z* has rotated, no further strain results

450

10 Plastic Deformation of Polymers

from separation of kinks (z>z*); additional strain increments are achieved by nucleating additional kink pairs with the same energy penalty. The dislocation approach, on the other hand, does predict that plastic strain is inhomogeneous, being confined to dislocation loops which can grow to micrometers in diameter. The amount of strain per nucleus is not determined explicitly. Argon (1973, 1975) and Bowden and Raha (1974) both consider how strain softening may occur in the context of their respective models, though these concepts have not been used to account for experimental data. The three treatments of yielding in glasses discussed above are based on the applied stress modifying intermolecular interactions. An entirely different approach has been offered by Robertson (1966). This is a uniform deformation model, in the sense of the Eyring treatment, but the applied stress is considered to increase the intramolecular energy of chains, which have high energy (flexed) and low energy (unflexed) rotational states. The ensemble of stressed chains with this additional energy can flow at a rate which would be achieved by stress-free chains at a fictive temperature 9 > Tg. At this fictive temperature the solid behaves as a Newtonian fluid: T=rj(6)y

(10-14)

The 9 dependence of viscosity rj is calculated from the empirical WLF equation (Williams et al., 1955) with no adjustable parameters; universal constants Cx = 17.44 and C2 = 51.6° are employed together with the characteristic viscosity rj(Tg) = 1013'6 Pas at the glass transition temperature. This model is remarkably successful, accounting for the absolute value of Ty within a factor of 2, as well as its temperature and strain rate dependencies with even better

accuracy. Such agreement is seen within about 80 °C of the glass transition temperature Tg; Argon and Bessonov (1977) have discussed the shortcomings of Robertson's treatment at lower temperatures. From the comparisons in Fig. 10-18, Robertson's model seems superior to the double kink model as Tg is approached, while Argon's kink model captures the behavior at lower temperatures where proximity to Tg is unimportant. These behaviors stem from the "reference states" used in the two treatments. Argon's model is based on shear strength T 0 at T = 0 K, while Robertson's model uses the flow stress at Tg. Brown (1983) has considered the maximum shear strength T 0 at 0 K of polymer glasses having three elementary displacement mechanisms operating in parallel. The dominant one is concluded to be conventional intersegmental shear as represented schematically by Fig. 10-1. The critical stress to achieve flow or yield is estimated to be T 0 = (0.064-0.092) G, which compares favorably to the experimental average of T 0 = (0.074 ± 0.03) G for 6 polymers for which ry was extrapolated to OK. It can be concluded that polymer glasses yield at stress levels which are remarkably close to theoretical estimates of strength. Experimental athermal yield-stress T 0 « 0.074 G agrees with Brown's estimate and is only 40% below the prediction from the more comprehensive kink pair model, Eq. (10-11). The dislocation model is based on a rather arbitrary definition of T 0 [Eq. (10-12)] which is about 2.5 times larger than experiment; this discrepancy could be moderated, without effect on other aspects of the analysis, by assigning a larger value to the radius r0 of the dislocation core. It is believed, however, that the extreme temperature dependence of that model (see Fig. 10-17) makes it an unlikely candidate

10.5 Fundamental Nature of Polymer Yielding

for correctly describing yield in polymer glasses. Confining our attention to the Argon and Eyring treatments in Fig. 10-17, thermal activation decreases i y by only about 45% at T = 350 K, thus yield occurs at more than half the maximum theoretical strength. Larger drops in Ty, seen for instance in Fig. 10-18, result from the temperature dependence of material parameters G or AH and v*. Another indication that glasses yield near theoretical values is the magnitude of the yield strain e y ^0.1 (Sec. 10.4.1). Converting this to an equivalent shear strain one obtains yy = ^/3 8 y ^0.17, which is of the order of yy = 0.25 for the basic shear model sketched in Fig. 10-1. The discussion in the preceding paragraph is based on stress (and thermal energy) overcoming intersegmental interactions to cause shear deformation. Elastic energy barriers drop very rapidly as Tg is approached, leading to underestimation of ry. In this region Robertson's intramolecular model accounts better for observed behavior.

451

10.5.2 Semicrystalline Polymers The complexity of the deforming system - a representative region contains an anisotropic lamellar crystal, amorphous chains, and two crystalline-amorphous interfaces - has frustrated efforts to define the basic yield mechanism or mechanisms in semicrystalline polymers. Most experiments have been done on polyethylene and polypropylene for which the amorphous regions lie above Tg at room temperature; in this range one achieves very large stretch ratios {sn > 4) which cause useful increases in stiffness and strength. To a first approximation there is no yielding in the liquidlike amorphous component of such composite materials; stresses there are accommodated by elastic or viscoelastic strains. This simplification is consistent with the observation that tensile and shear elastic moduli of polyethylene are dominated by the compliance of the amorphous component, meaning that the mechanical response at small strains is approximated by a series model (Boyd, 1983; Crist et al., 1989, Chap. 7, this Volume).

0.12

fc 0.10

-

Figure 10-18. Normalized shear yield strength for poly (methyl methacrylate) and polystyrene as a function of temperature. The shear modulus G at 0 K is represented by the symbol (x. Lines are calculated from kink-pair (Argon, 1973) and intramolecularflex-energy (Robertson, 1966) models. Fom Argon and Bessonov (1977); reproduced with permission of Taylor and Francis Ltd.

."5 0.08

0.06

0.04 PMMA o Argon o

PMMA • Robertson 0.02

PS O Argon PS a Robertson i

1

100

200 Temperature (K)

300

400

452

10 Plastic Deformation of Polymers

It has long been recognized that the yield stress increases with crystalline fraction a when deformation is done at Tg < T
Evidence supporting the partial melting and recrystallization mechanism for yielding has been reviewed thoroughly by Popli and Mandelkern (1987). Here we emphasize one of the most compelling facts, the observation that the long period L=lc+la is a function of deformation temperature Td, essentially independent of L in the isotropic polymer before drawing (Peterlin, 1971). This is observed for plane strain compression (Bessell and Young, 1974) and solid state extrusion (Chuah et al., 1986) as well as uniaxial tensile drawing. The decrease in long period L seen in such experiments is a particularly strong argument for the deformation stress mobilizing chain segments in a manner very similar to melting. Chuah et al. (1986) term this "quasimelting", and note that it does not occur in polyethylene which was crystallized in extended chain (as opposed to lamellar) morphology. Something akin to quasi-melting can be seen in the classic model of Peterlin (1965) for the transformation to oriented fibrils (Fig. 10-19). Phillips and Philpot (1986) have reported electron microscopy evidence for a molten zone some 5-10 |im in extent for polyethylene drawn at room temperature. This is consistent with the partial or local melting hypothesis of Popli and Mandelkern (1987). Additional support for quasi-melting is provided by small-angle neutron scattering (SANS) from mixtures of conventional and deuterated polyethylene. These experiments rely on the tendency of the two isotopic forms of polyethylene to segregate during crystallization, though statistical cocrystallization can be achieved by rapid quenching from the melt. Wu and Wignall (1985) and Wu et al. (1992) have examined intentionally clustered mixtures before and after plane strain compression at room temperature. They find that the amount of isotopic segregation, determined from the

453

10.5 Fundamental Nature of Polymer Yielding

absolute intensity scattered at zero angle, is reduced by as much as a factor of five after plastic deformation. This intensity drop commences at yield and increases continuously with further deformation. From considerations of intensity and "cluster size" it is argued that chain mixing of this sort must result from melting during the deformation process. Sadler and Barham (1990 a, b) have done related SANS experiments with isotopic mixtures of polyethylene subjected to tensile drawing at or above room temperature. Their analyses consider changes in the radius of gyration or cluster size, not the absolute scattered intensity. For k^l they find evidence for melting and recrystallization when drawing is done above a fairly well defined temperature in the range from 65°C to 95°C; this critical temperature increases with molecular weight (64000-350000 g/mol). It is concluded that melting and recrystallization do not occur during tensile drawing to X K, 7 at room temperature or on further stretching to X « 40 at Td < 118°C. Thus two sets of SANS experiments each support somewhat disparate concepts of melting and recrystallization. It is not known whether these differences result from unlike deformation modes, or from peculiarities of the SANS experiments. Alternatives to the melting or phase transformation model are based on more conventional approaches to crystal plasticity. These include stress induced crystalcrystal transformations, twinning and slip which have been reviewed by Bowden and Young (1974), Young (1979), and Saraf and Porter (1988). Slip of covalently bonded chains on (h k 0) planes in either the chain axis direction [001] or transverse directions [u v 0] have received the most attention. It is assumed that the crystal structure has been defined conventionally with the chain axis in the crystallographic [001] direction.

(hkO) [001] slip is the most likely mechanism for achieving large macroscopic strains with conversion to a fibrillar morphology as suggested in the Peterlin model, Fig. 10-19. Furthermore, this shear mechanism can be used to provide quantitative estimates of the yield stress t y , which can be measured with considerable precision for various slip systems using macroscopically textured samples (Bartczak et al., 1992 b; Lin and Argon, 1992). The relation between (hkO) [001] slip and the experimental yield stress of about 5-30 MPa in polyethylene can be established with a dislocation nucleation model. Shadrake and Guiu (1976) calculated the energy of a [001] screw dislocation in a lamellar crystal of thickness lc interacting with a free (hkO) surface or another [001] screw dislocation. A sketch of the system is shown in Fig. 10-20. Under the influence of a shear stress T, the net energy AU for creating a shear region of size u is written in terms of T, lc, elastic modulus G for shear on (h k 0) planes, Burger's vector b (equal to the crystallographic c axis in the chain direction), and dislocation core radius r0. This model is developed similarly to the

(a)

(b)

Figure 10-19. Model for transformation from lamellar (a) to fibrillar (b) morphology on drawing a semicrystalline polymer. From Peterlin (1965); reproduced with permission. © John Wiley & Sons, Inc.

454

10 Plastic Deformation of Polymers

Figure 10-20. Sketch of [001] screw dislocation nucleus of size u in a lamellar crystal of thickness /c. Magnitude of the Burger's vector b is exaggerated. From Crist et al. (1989).

Bowden-Raha analysis of the dislocation loop in Sec. 10.5.1. The nucleation barrier for creating a shear region of critical size u* is AU* =

2TT

10 d (nm)

Figure 10-21. Dependence of yield stress of polyethylene crystals on crystal thickness lc (here represented by d) at room temperature. • linear polyethylene; • branched polyethylene. Line is calculated from Eqs. (10-15 a) and (10-15 b). From Young (1988); reproduced with permission. © Institute of Metals and Materials Australasia.

(10-15 a)

The yield stress at which a stable nucleus of size w* is formed by thermal fluctuations is Ty

Gb 2nu*

(10-15 b)

where w* is defined through Eq. (10-15 a). Young (1988) and Crist et al. (1989) have applied this dislocation nucleation model to the experimental yield stress of polyethylene. The only assumption is that the nucleation barrier A I/* is set at 50 kT to 60 kT so laboratory scale strain rates of ~ 10~ 3 / s c a n be achieved when the fundamental frequency is ~ 10 13 Hz. All structural and elastic parameters are known with reasonable accuracy; in neither study were model parameters adjusted to fit observed results. One sees immediately that the absolute magnitude of ry is quite good, though the model has too large a dependence of Ty on crystal thickness (Fig. 10-21) and too small a dependence on temperature (Fig. 10-22). The steep rise in the experimental strength ry for T < 175 K is due to strengthening from the vitrifying amor-

200

300

400

7"(K)

Figure 10-22. Temperature dependence of resolved yield stress i y for polyethylene under uniaxial tension (o) and plane strain compression (•). Lines are calculated from Eqs. (10-15 a) and (10-15 b) for lamellae of indicated thicknesses (nm). From Crist et al. (1989).

phous fraction as Tg is approached (Brown, 1986); the purely crystalline model is not applicable at such low temperatures. Some additional features can be obtained from the dislocation model. The

10.5 Fundamental Nature of Polymer Yielding

athermal strength at T = 0 K for a crystal yielding by (hkO) [001] slip via screw dislocations is Gb Tn =

= 0.015 G

(10-16)

where the core radius r0 has been replaced by 4 b. Note that this theoretical strength for a crystal (which is independent of thickness /c) is about one tenth the value for glasses which yield by either kink pairs [Eq. (10-11)] or dislocation loops [Eq. (10-12)]. The screw dislocation [Eqs. (1015 a) and (10-15 b)] can be compared in detail to the dislocation loop [Eqs. (10-13 a) and (10-13 b)] for glasses. A factor of 2 in T0 arises from different strain energies, and the balance comes from evaluation of the dislocation core radius r0. Bowden and Raha (1974) somewhat arbitrarily set r 0 = v / 3 b / e = 0.64 b for glasses, whereas computer simulation of the [001] screw dislocation core in polyethylene (Bacon and Tharmalingam, 1983) gives a more reliable result r o = l n m = 4fc. Using G = 3.1 GPa for the low temperature shear modulus of the polyethylene crystal, T 0 = 47 MPa, which is appreciably lower than experimental strength at low temperature where the vitrified amorphous regions have reinforced the composite (see Fig. 10-22). This confirms the idea of Brown (1986) that yield below Tg is governed by the stronger glassy component. The strain rate dependence of ry for yield by nucleation of [001] screw dislocations can also be calculated from Eqs. (10-15 a) and(10-15b): dlnT y _ 2nkT dlny ~~Gb*Tc

(10-17)

Evaluating this with parameters for polyethylene at room temperature, i y is predicted to increase by about 10% for a 103

455

increase in strain rate. This is much smaller than the - 100% increase in Fig. 10-13 or the more modest 50% increase reported by G'Sell and Jonas (1979) from a series of experiments done at constant true strain rates et which conform to the theoretical framework. This shortcoming of the model is related to underestimation of the temperature dependence of Ty mentioned above. It is probable that the amorphous regions, particularly the fold surfaces or "interphases", influence the energy of the dislocation nucleus. Boyd (1985) describes the a-relaxation process in polyethylene in terms of coupled displacements of fold surface segments and crystalline stems. A crystal-amorphous coupling of this sort would make the effective shear modulus G more temperature and strain-rate dependent, leading to better agreement between model predictions and experiment. The dislocation model was motivated primarily by the positive dependence of ay (or Ty) on crystal thickness /c. Balta Calleja (1985) has used microhardness measurements on polyethylene (hardness H « 3 ay) to consider the same issue. Those experiments led to cry~ v // c , which was interpreted as a decrease in resistance to chain slip resulting from less dense packing of chains in thinner crystals. In a different analysis of essentially the same data (Balta Calleja and Kilian, 1985), it was concluded that the increase in oy is caused by a reduction in the number density of (hkO) shear planes in lamellar aggregates with a large lc. Neither of these schemes accounts for the absolute value of
456

10 Plastic Deformation of Polymers

dence of xy. Flory and Yoon (1978), among others, contend that the irregular fold surface in melt crystallized polymers precludes substantial shear displacements of the sort sketched in Fig. 10-19, which is based on adjacent reentry or a regular fold surface. Nevertheless, X-ray diffraction shows clearly that [001] slip is the mechanism by which yielding and plastic deformation occur in plane strain compression of melt crystallized polyethylene (Young et al., 1973). Detailed pole figure studies by Krause and Hosford (1989) reach the same conclusion for uniaxial extension, uniaxial compression, and plane strain compression. These findings have been confirmed for plane strain compression at different temperatures by Bartczak et al. (1992 a, b). Elegant electron microscopy experiments on specially textured melt crystallized polyethylene deformed in tension have been done by Adams et al. (1986). Here [001] slip is directly seen to occur on both (100) and (010) shear planes. These experiments all demonstrate that melt crystallized polyethylene deforms by [001] or caxis slip, regardless of constraints imposed by chain reentry at the lamellar surfaces. Most of the experimental evidence, together with the reasonable success of dislocation nucleation as a quantitative model, lead to the conclusion that intracrystalline shear by [001] slip is the dominant yieldmechanism. This statement applies to linear polyethylene deformed at or near room temperature, though it is thought to be more general. The change of the semicrystalline long period L is inconsistent with this mechanism. Increases in the long period at high draw temperatures can, however, be ascribed to normal lamellar thickening by annealing at Td, perhaps accelerated by the applied stress. Indeed, the increase of L when drawing polypropylene at 20°C < Td < 150°C can be duplicated by

annealing at the respective draw temperatures (Peterlin, 1971). The more problematic decrease of L on drawing has been addressed in the electron microscopy studies of polyethylene deformation by Brady and Thomas (1989), where well developed lamellae are seen to be transformed into fibrils in a manner generally consistent with Fig. 10-19. After the lamellae have sheared by [001] slip, block-like crystals are observed to "decrystallize" (become less ordered as judged by electron diffraction) by the generation of a large number of unspecified internal defects. These structurally irregular regions become thinner in the chain axis direction. This process can be thought of as "melting" of crystals containing sufficient defects to lower Tm to the deformation temperature Td (room temperature in this case). But destruction of the original crystals is done by mechanical work, not by the increase of thermal energy from viscoelastic effects. The mechanically mobilized chains respond to applied stress and are reordered into crystals with a thickness characteristic of Td. Galenski et al. (1992) have proposed a related explanation. They envision (amorphous) defects to rearrange, permitting a "virtual shape change" of highly sheared crystal fragments to establish a new structural periodicity L. For the best studied case of polyethylene, it appears that yield (the onset of plastic deformation) at or near room temperature is controlled by crystal plasticity via dislocation nucleation. Post-yield flow, involving neck formation and stabilization in tension, involves "quasi-melting" or "decrystallization" at subsequent stages in the deformation process.

10.5 Fundamental Nature of Polymer Yielding

457

10.5.3 Crystalline and Liquid Crystalline Polymers

Highly oriented polymers display a unique yielding mechanism when deformed in compression along the chain axis direction. The strain is localized in bands a few nanometers thick which are inclined by an angle ^ « 5 0 ° - 7 0 ° to the compression direction. These may be isolated or occur in bundles as large as 100 jrni in the loading direction. This yield process is important because it limits the compressive strength of polymer preparations which are, by virtue of good chain axis orientation, very strong in tension. DeTeresa et al. (1988) have considered a number of highly oriented fibers of liquid crystalline polymers. These have (brittle) tensile strengths of about 2.5-3.2 GPa but fail by yielding at compressive stresses only — 0.1 as large. The macroscopic strain for this compressive yielding is quite small, typically less than 0.5% (|ey| < 5 x 10~3). Continued deformation increases the number and size of the bands. Axially oriented cracks form at the boundaries between sheared and unsheared regions, ultimately leading to fracture. Examples are shown for a liquid crystalline polymer fiber (Fig. 10-23) and a single crystal of substituted polydiacetylene (Fig. 10-24) deformed in compression. The applied stress first causes the chain axes to shorten by elastic strain. Conventional plastic deformation by [001] slip cannot be activated because the resolved shear stress is zero on (h k 0) slip planes. At some structural defect, frequently on the surface, local stress concentrations induce cooperative shear displacements of chain sections which evolve into the shortened "steps" in Figs. 10-23 and 10-24. The driving force for this local shear is the reduction in overall strain energy in the

Figure 10-23. Dark field electron micrograph of a poly(para-phenylene benzobisoxazole) fiber fragment. A single kink of length Lk « 500 nm and height w « 40 nm is toward the top of the figure. A large group of kinks is in the center of the figure. From Martin and Thomas (1991); reproduced with permission of Chapman and Hall.

compressed body which now has a smaller length in the loading direction. For polymer single crystals the boundary between sheared and unsheared regions is often a low index crystallographic plane [e.g., (012) in Fig. 10-24], and the strain is described as chain axis rotation deformation twinning (Bevis, 1978; Young et al., 1979). In liquid crystals and in oriented polycrystalline materials such as extended chain polyethylene one sees noncrystallographic boundaries indicative of kink bands (Martin and Thomas, 1991). A common feature is that the chain axes are bent severely and cooperatively at two well defined bound-

458

10 Plastic Deformation of Polymers (b)

la) <

ao 0=10.9° SDjun

Figure 10-24. (a) Scanning electron micrograph of a deformation twin in a polydiacetylene single crystal. Bending of the chain molecules at the (012) twin boundary is illustrated schematically in (b). From Young et al. (1978); reproduced with permission of Chapman and Hall.

aries which may extend entirely across a 10 jam diameter crystal or fiber. Each boundary is a mirror plane relating the chain directions in sheared and unsheared regions. A sketch of the atomic arrangements is presented in Fig. 10-24 where the angle of the twin boundary <> / = 90° — x//. The chains are sheared by a strain y = 2 tan 0 = 2 cot \jj in creating the band. Note that this shear strain is neither parallel nor perpendicular to the [001] direction of the chain axes. For deformation twinning these boundaries are low index crystallographic (hkl) planes which are true mirror planes, while there is no such geometric restriction for kinking. Hence the angle x// or 0 must have fixed values for deformation twins, while it may vary continuously for kinks. Twin or kink bands have some characteristics of crazes formed under tensile stress fields; plastic strain is localized in planar regions and fracture originates from these regions in both cases. Crazing, however, transforms originally isotropic material into oriented microfibrils with appreciable void formation (Chap. 15, this

Volume). This is quite distinct from the bands here, in which only the local orientation direction of the polymer chains is changed. Some theoretical work has been done on these deformation modes. Possible (hkl) twin planes (deformation twin boundaries) for bands of the type illustrated in Fig. 10-24 are established from crystallographic symmetry (Bevis, 1978). This analysis ignores energy penalties at the twin boundary, aside from the requirement that chains are not broken or excessively distorted. It is usually assumed that those twin deformation modes having the smallest shear displacements (smallest rotations of chain segments) are favored, though exceptions have been noted (Young et al., 1979). The precondition of no excess energy at twin boundaries is an oversimplification when chains at the boundary are "bent" by rotation about single bonds or by deformation of intramolecular bonds (see Fig. 10-24). A more realistic description would include energy required to deform the chains. This has been done by Pertsev et al. (1981), though their rather cumbersome theoreti-

10.5 Fundamental Nature of Polymer Yielding

cal results have not been applied to experiments. The concept of a twin plane describing the relation between sheared and unsheared sections of chains is less appropriate in liquid crystalline polymers within which the chains lack strict crystallographic registration. Martin and Thomas (1991) have applied the treatment of kink bands developed by Frank and Stroh (1952). The model uses arrays of dislocations of opposite sign to define the two boundaries between kinked and unkinked regions. The strain fields of these dislocation arrays provide a formal method for calculating the energy of the boundary between sheared and unsheared material. The energy At/ of a kink band of length L k and height w is evaluated as a function of the shear strain y in the kink band, the applied shear stress T, elastic constants G and v, Burger's vector b and dislocation core radius r0. AU goes through a maximum with respect to L k , permitting definition of the critical kink nucleus of size L\ which can grow by generation of additional dislocation pairs, thus lowering the energy of the system by increasing the length L k of the kink band. With some simplifications the result is

2L*J

(10-18)

This nucleation model differs from those in Sees. 10.5.1 and 10.5.2 in that the critical size L\ is not evaluated explicitly in terms of At/*, meaning that absolute shear strength xy (and its temperature and rate dependence) is not obtained. Furthermore, the relation between shear stress T and compressive stress a is determined by the unspecified angle \j/ (related to local strain by y = 2 cot \j/) between the boundaries and the axial loading direction.

459

Martin and Thomas (1991) have used the kink nucleation model to interpret the kink bands seen in poly(para-phenylene benzobisoxazole) fibers. They find that the experimental height of a single kink (w « 30 nm), is in good accord with the predicted value [w = L k T/(Gy) « 24 nm]. Tensile and compressive hydrostatic stress fields caused by the dislocation arrays are suggested to account for the relative sharpness of the top boundary and the slight downward curvature of the kink band as it grows in from the surface towards the center of the fiber (see Fig. 10-23). Thus the Frank-Stroh model accounts for the approximate size as well as certain morphological features of kink bands seen in liquid crystalline polymer fibers. Absolute strength i y is not given by the Frank-Stroh model, though this could be obtained in a fairly straightforward manner by invoking thermal activation, as was done for the kink pair and dislocation models described in Sees. 10.5.1 and 10.5.2. Mechanical response of single poly (paraphenylene benzobisoxazole) and other rigid chains has been modeled with semiempirical molecular orbital calculations by Wierschke et al. (1992). That work shows the chain "kinks" under compression by distortion of covalent bond angles in one of the heterocyclic rings. Such molecular yielding bears a possible relation to kink deformation in fibers, though the calculated (athermal) stress for this process is about —10 GPa, considerably larger than experimental strengths, which are about - 0 . 3 GPa. DeTeresa et al. (1985) have presented a rather phenomenological model of fiber buckling by a mechanism similar to kink-band formation. Their model predicts that the compressive stress required to cause buckling is oy = G (the shear modulus of the fiber parallel to the chain axes), while experiments give ay « G/3 (DeTeresa

460

10 Plastic Deformation of Polymers

et al., 1988). This discrepancy is attributed to various defects, though no consideration is given to thermal activation of the local yielding process.

10.6 Post-Yield Deformation and Modeling Glassy or semicrystalline polymers generally exhibit considerable ductility or flow before fracture. Stress-strain curves can be represented by various mechanical models which serve to correlate data taken under different conditions and, in some cases, provide insight on the nature of the deformation process. 10.6.1 Viscoelastic Models Much work deals with tensile deformation at a constant strain rate e, for which the simplest linear viscoelastic model is (10-19) Parameters are the unrelaxed modulus Eo and the relaxation time t'. The stressstrain curve has an initial slope Eo. The slope decreases conspicuously at a strain s « s t\ after which o approaches the limiting (Newtonian) flow stress Eo 81'. No distinction is usually made between nominal stress and true stress because strains are modest in the region of interest. This model can be modified to mimic the behavior of plastics deforming in tension. Matsuoka (1986) has considered nonlinear viscoelastic expressions of the following type: a{8) = £ o e e x p ( - Ce) exp( - — 1 (10-20) The first exponential factor involving C 8 is invoked to describe yielding, i.e., the presence of a maximum in o versus 8. Recall,

however, that a maximum in the nominal tensile-stress always results from extrinsic yield, not from strain softening as implied in Eq. (10-20) (see Sec. 10.2). The yield stress has a strain-rate dependence dominated by the second exponential factor. The single relaxation time has been replaced by an empirical distribution of relaxation times having the average value t' and a width parameter n. The response function for this distribution [sometimes called the Kohlrausch-Williams-Watts or KWW function (Matsuoka, 1992)] is a "stretched exponential" R{t) = exp[-(t/t')n], which is seen to reduce to a simple exponential for n = 1. The width parameter n is obtained directly from the time dependence of the elastic modulus in a relaxation experiment; n approaches 1 as the width of the distribution vanishes. This model predicts that ay ~ £n, where n is of the order of 0.05 for glassy polymers (Matsuoka, 1986). Examples of fits to experiments on polycarbonate are given by the solid lines in Fig. 10-9. The model does not include strain hardening, so the constant flow stress implied in Fig. 10-9 is arbitrary. A similar treatment, with one additional parameter to account for the strain dependence of t\ will account for the positive shift of yield strain £y with strain rate. The two factors relating stiffness and relaxation time to strain can be associated with modified entropy and free volume in the deforming solid (Matsuoka, 1986). A related analysis of tensile deformation which accounts for yield with intrinsic strain softening has been developed by Knauss and Emri (1987). The interesting idea here is that a tensile strain, assumed for the moment to be constant, causes free volume and chain mobility to increase in a time dependent manner. This is because the stress-volume response function, i.e., the bulk compliance, changes with time from a

10.6 Post-Yield Deformation and Modeling

low glassy value to a higher liquid-like value with a time constant which is itself decreased by the evolving free volume. When this concept is incorporated in an otherwise linear viscoelastic model for tensile stretching, a stress maximum followed by strain softening appears due to the time delayed, autocatalytic increase in free volume and mobility. Strain softening is more pronounced at high strain-rates, in agreement with experimental observations. The same model handles very nicely nonlinear viscoelastic stress relaxation, and accounts for time-dependent effects of hydrostatic pressure, for example, physical aging. Shay and Caruthers (1990) have developed a related treatment in which yield occurs when the entropy of the deformed body is increased to that of the unstretched material at Tg. Entropy also governs the time evolution of the system relaxation time t\ largely through the volume increase in tensile stretching. A basic feature is that nonlinear effects are predicted in terms of measurable linear viscoelastic properties. With input data for poly (vinyl acetate) the model predicts extrinsic yield (no decrease in true stress) in tension. While there is no stress drop, strain hardening is observed after the yield point. This interesting behavior comes from a decrease in volume after yield, though the origin of this densification is not understood. Temperature effects can be treated by time-temperature superposition in viscoelastic models. Matsuoka (1986) has developed an analysis based on Arrhenius activation for the average relaxation time t' which reduces to a particularly simple relation. All stresses and strains obtained at a temperature T are related to those at a reference temperature TR by the coefficient (Tc - T)/{TC-TR\ where Tc is the temperature at which the yield stress of the glass extrapolates to zero. Application of this

461

scheme to polycarbonate is illustrated in Fig. 10-8 for temperatures above and below 7 R =23°C. The analysis requires that the basic shapes of the curves do not change with temperature, a feature which is observed sufficiently below Tg. Semicrystalline polymers below Tg can be treated as if they were glassy; see Fig. 10-11. Those above Tg have initial moduli which are temperature and strainrate dependent. A fairly simple empirical scheme for correlating data has been developed by Matsuoka (1986). The strain rate dependence for polyethylene is fitted in Fig. 10-13. These treatments of tensile stress-strain relations ignore the fact that nominal stress on will drop after the extrinsic yield point because a neck is formed. A simple linear viscoelastic model [Eq. (10-19)] will generally result in a maximum load or an if treated according to Considered construction; see curve A in Fig. 10-5. Nevertheless, virtually all glasses yield with intrinsic strain softening as accounted for correctly, if phenomenologically, by nonlinear viscoelasticity. To account for strain softening, these analyses rely explicitly or implicitly on a free volume increase during tensile deformation. That assumption is necessary, in this context, for yield and plastic strain. The free volume approach is conceptually difficult to implement for shear or compression where macroscopic volume is constant or decreasing. 10.6.2 Molecular Models Kinematics of deformation, appropriate for describing large plastic stretches, has been developed by Boyce et al. (1988). In that framework they analyze the response of an elastic-plastic constitutive model which is based on molecular parameters for polymers. The yield stress is described

462

10 Plastic Deformation of Polymers

close to T g =110°C, in Fig. 10-25. The model captures most elements of the observed behavior, though the calculated yield strains are about 0.2 of the observed values, for reasons which are not clarified. Especially impressive is the agreement in the strain hardening region. Matsuoka (1991,1992) has recently proposed a model for viscoelastic-plastic behavior which eliminates the need for free volume to activate flow in a glass. It is postulated that yield or plastic strain occurs by the cooperative rearrangement of domains composed of m "conformers"; a conformer is a repeat unit containing a carbon-carbon single bond in the polymer backbone. Conformational entropy of a domain having m units is utilized in a transition-rate theory to establish the relation between the average relaxation time t' and applied stress a. According to this model, linear viscoelasticity prevails until the

by Argon's kink pair model [Eq. (10-9)], which predicts the temperature and strain rate dependence of ry without recourse to free volume. Pressure effects are described by the von Mises criterion [Eq. (10-4)] and strain softening is introduced by an empirical modification of T 0 [Eq. (10-10)] after the yield point. Strain hardening is treated by back stresses generated as Langevin chains with finite extensibility are stretched. The idea that strain hardening in polymers results from increased entropic resistance had been proposed by Haward (1973). Back stress or strain hardening rate depends linearly on temperature T, but is assumed to be independent of strain rate. All model parameters except those for pressure effects and strain softening are based on molecular theories. Calculated curves for tensile deformation at constant (true) strain rates are compared to experiment for poly (methyl methacrylate) at T= 90 °C,

i

•

i

i

|

•

i

i

i

|.;,

; • / /

:[!

100.0

/

50.0 s"1

- Exp. i-\

Mod. f = 1 s-1 %

>

Jlt\ * -%? '* M ^^. " * - - - : - - * ^

- •• Exp. fr0.1 S"1

<** f^

- Mod. fr0.1 s"1 - Exp. f=0.01 s-1 - Mod. fr0.01 s-1

n

r

0.50

,

i

i

1.00

•

1

I

I

1

1.50

Figure 10-25. Tensile true stress-true strain curves for poly (methyl methacrylate), Td = 90 °C, at constant strain rates indicated. The molecularly based constitutive model accounts quite well for observed behavior. Exp.: experimental; Mod.: model. From Boyce et al. (1988); reproduced with permission of Elsevier Science Publishers B.V.

10.6 Post-Yield Deformation and Modeling

strain energy reaches a critical value at which point the mechanical behavior shifts to classical plasticity. This viscoplasticity approach is distinct from linear (or nonlinear) viscoelastic models in which elastic deformation evolves into viscous flow after a certain time t&t' [see Eq. (10-19)]. Matsuoka simplifies the energy criterion for yield in terms of a critical strain 8* = ay/E0. A spectrum of relaxation times is required for the model to have the observed dependence of <7y (or e*) on strain rate. The distribution of relaxation times is described by a width parameter n and an average t' as discussed for Eq. (10-20). Equating 8* with the yield strain ey, the result is

= —{e*y-n{ety

(10-21)

Here E (t) is the relaxation modulus, written for uniaxial deformation, either tensile or compressive. A comparable expression can be obtained for shear deformation, as volume changes have not been invoked. This model again predicts that the yield stress is proportional to 8n, but without recourse to free volume effects. Plastic deformation of the glass results from stress enhanced rate of intramolecular rearrangement in a manner related in some ways to the approach of Robertson (1966). As the viscoelastic-plastic model represented in Eq. (10-21) extends only to the yield point, the issue of strain softening is not addressed. Valuable insight into the plastic deformation of isotropic crystalline polymers has been provided by the analysis of Parks and Ahzi (1990). They calculate stressplastic strain relations for an initially isotropic array of polyethylene crystals which deform by shear. Earlier crystal plasticity theories rely on the presence of at least 5 independent (hkl) [uvw] slip systems to permit plastic flow of arbitrarily oriented

463

crystals without stress singularities. Polymer crystals are deficient in the number of slip systems because of constraints imposed by unbreakable covalent bonds. The authors develop the kinematics of deformation of these deficient systems with a "constrained hybrid" model for an incompressible rigid-viscoplastic material; no volume change occurs, small elastic strains are ignored, and the critical resolved shear stress for slip has a positive strain rate dependence. The model provides numerical solutions for stress, strain and texture [orientation of (hkl) planes] for an initially isotropic set of 248 crystals. This analysis was applied to polyethylene having 4 slip systems ([001], [100], [010], and [110] slip directions on appropriate (hkO) planes). The model is constructed for 100% crystalline polyethylene; no account is taken of the amorphous component. Calculated results for tensile and shear deformations are presented in Fig. 10-26. The equivalent macroscopic stress <req = ert = T y/3 is normalized by the critical resolved shear stress for [001] slip and equivalent (true) strain is 8eq = st = y/y/3. One sees immediately that plastic deformation commences at a common equivalent stress after which strain hardening develops much more rapidly in tension than in shear. This calculated response is very similar to the constant true strain rate results of G'Sell et al. (1991) for polyethylene (crystallinity a « 0.7) in Fig. 10-27. The difference in strain hardening rates can be attributed to development of a c axis or [001] texture and rotation of the (hkO) slip planes into the stretch direction (Parks and Ahzi, 1990). For tensile deformation this leads to "textural hardening" as the resolved shear stress on active slip planes becomes smaller, requiring ever increasing tensile stress at for additional strain. In shear, rotation of [001] and slip planes to-

464

10 Plastic Deformation of Polymers

0.5

o

1.0

1.5

Figure 10-26. Calculated stress-strain curves for isotropic crystalline polyethylene. Strain hardening is pronounced in tension, negligible in shear (cf. Fig. 10-27). From Parks and Ahzi (1990); reproduced with permission of Pergamon Press pic.

200 HDPE

150

and polyoxymethylene are qualitatively similar to polyethylene, yielding abruptly at a shear strain of y y « 0.3 followed by essentially flat flow curves with very modest shear strain hardening. Each of these polymers strain hardens in tension with stable neck propagation. The other two semicrystalline polymers, nylon 66 and polybutylene-1, have rather diffuse yield points followed by strain hardening in shear. The reason for this different behavior is not understood. In summary, glassy polymers display strain hardening in tension and shear, which is accounted for by finite extensibility of chain segments in the entangled network of interpenetrating chains (see Fig. 10-25). No fundamental reason for ubiquitous strain softening in polymer glasses has been proposed. Semicrystalline polymers behave differently, yielding with no strain softening in either tension (Fig. 10-15) or shear (Fig. 10-16). For these materials the presence of strain hardening in tension and the absence of strain hardening in shear is attributed to texture evolution.

100

10.6.3 Continuum Mechanics of Necking 50

Simple shear

0

0.5

1.0 Equivalent strain

1.5

2.0

Figure 10-27. True stress-true strain curves for polyethylene deformed at constant strain rate in tension and shear. From G'Sell et al. (1991); reproduced with permission of the author.

ward the stretch direction still allows most of the deformation to occur by easy chain slip without pronounced textural hardening. This type of behavior seems fairly general, as seen in Fig. 10-16. Polypropylene

Knowledge of intrinsic mechanical response of the material, that is, the true stress - true strain behavior, is essential for understanding the response of a polymer to an arbitrary stress field. Considerable attention has been paid to the plastic deformation of polymers in uniaxial tension when, as described in Sec. 10.2, plastic instability often leads to neck formation. Interest derives from two factors. From an engineering viewpoint, "cold drawing" by neck propagation is used to achieve high orientation and increased stiffness and strength (Ward, 1975). Description of the necking process is, furthermore, a challenge to those interested in the continuum

10.7 Summary

and molecular approaches to polymer deformation. Necking phenomena are based on postyield deformation, as introduced in Sees. 10.2 and 10.4. Intrinsic mechanical response (at versus et) is needed as a function of strain rate and pressure because of the strain rate gradient and hydrostatic tension associated with a nonuniform cross section. These combine to determine the profile of the neck and the stress at which it propagates. Coates and Ward (1980) have described the relations between true stress, true strain, strain rate, and neck profiles in polyethylene. More pronounced strain hardening stabilizes the neck, leading to a smaller stretch ratio in the necked portion and a more gradual thinning of the sample in the neck region. Materials with small strain hardening rates undergo large stretches with sharp necks. Most viscoelastic polymers have a positive dependence of yield stress and post-yield flow stress on strain rate. This strain rate sensitivity also stabilizes the neck, leading to less deformation and more gradual neck profiles. Hutchinson and Neale (1983) provided the first kinematic description of steady state neck propagation in materials like semicrystalline polymers with strain hardening but without intrinsic strain softening. They first modeled the true stresstrue strain behavior by a nonlinear elastic solid (this is permissible if the true stress does not decrease in the process). A "jump" analysis based on initial and final states permits a Maxwell line solution for the nominal or engineering draw stress and the "natural" draw ratio X in the necked portion. Elastic-plastic constitutive models were analyzed with J2-flow theory for a more complete description of the propagating neck with multiaxial local stresses in a cylindrical rod. The effects of strain-hardening rate and strain-rate sensitivity of

465

flow resistance were considered. Increasing either of these material characteristics stabilizes the neck, i.e., decreases both the draw ratio and the sharpness of the neck. Necking phenomena can be examined in somewhat more detail with finite-element analyses of nonlinear-elastic models, as first done by Neale and Tugcu (1985). They found that the analytical approach of Hutchinson and Neale (1983) was generally adequate for neck propagation in axisymmetric fibers. A similar study of plane strain neck propagation was performed by Fagler and Bassani (1986) and later extended to include plastic anisotropy by Batterman and Bassani (1990). Boyce and Arruda (1990) have done a revealing finite element study of necking in polycarbonate using the molecularlybased constitutive model described in Sec. 10.6.2. The model reproduces the observation that the strain hardening rate is greater in tension than in uniaxial compression because the amount of chain stretching, which leads to back stresses, is smaller in uniaxial compression (or biaxial extension). The same model accounts for the shape of the neck which evolves in tension and the variable strain rate of the material in the neck.

10.7 Summary The yield strength of polymer glasses absolute value and temperature and strainrate dependence - can be accounted for by nucleation models of either kink pairs (Argon, 1973) or dislocation loops (Bowden and Raha, 1974) with reasonable molecular parameters. Within about 100° of Tg, the intramolecular energy model (Robertson, 1966) is a viable, even preferable, alternative. The shear strength of polymer

466

10 Plastic Deformation of Polymers

glasses is described quantitatively by two or three distinct theories, which is an unusual situation. If one accepts the kink-pair model, it follows that glassy polymers yield at, or very near, their theoretical strength. Most of the drop in ry (or oy) that occurs when the temperature is raised toward Tg comes from the decrease of shear modulus G. Fundamental understanding of shear strength and its temperature dependence hence reverts to the temperature dependence of G. Theodorou and Suter (1986) have developed a semiempirical method to calculate the structure and elastic constants of an isotropic polymer glass, though it has not been applied to temperature effects. This simulation technique has been extended to the study of plastic strain in a polymer glass (Mott et al., 1993); an account is presented in Chap. 10, Vol. 6 of this Series. Structural information from such simulations may permit discrimination between existing theories or provide insight to generate better ones. Semicrystalline polymers at temperatures below Tg of the amorphous component yield similarly to glasses. However, the yield stress ay decreases much more rapidly in the range just above Tg. This "transition region" over which the amorphous regions become liquid-like had not been studied in detail, but it is probable that most of the effects are associated with easier shear in the noncrystalline portions of the composite (Brown, 1986). At still higher temperatures, where amorphous strength is negligible, yield stress is found to correlate with thickness lc of lamellar crystals. This has been modeled successfully with nucleation of screw dislocations for the case of polyethylene (Young, 1988; Crist etal., 1989), taking advantage of known structures of the crystal and the postulated defect. Calculations of equilibrium structures, elastic constants, and their

temperature dependencies are well established for crystals (McCullough and Peterson, 1973; Sorensen etal., 1988; Chap. 7, this Volume), being much more straightforward than those for glasses. Static defects such as dislocations have been modeled by Bacon and Tharmalingam (1983). It is hoped that these methods will be extended to other polymer crystals. Complementing the mechanical studies are investigations of morphology, generally before and after considerable plastic deformation. Some of these have led to the suggestion that yielding and plastic deformation involve melting of the crystallites (Popli and Mandelkern, 1987), though even semiquantitative predictions have yet to be made. While a large number of models have been proposed to account for yield strength, relatively little consideration has been given to descriptions of plastic deformation beyond yield. The important issue of strain softening, observed in glasses under almost all conditions, has been treated phenomenologically without much attention to its cause. One exception is "delayed dilatancy", which can account for strain softening in tension with a nonlinear viscoelastic model (Knauss and Emri, 1987). An interesting development in this area has been reported by Hasan et al. (1993). Positron annihilation lifetime measurements indicate that the size of free-volume packets increases irreversibly after compressive yielding of poly (methyl methacrylate). This implies that free volume ideas may be more general than thought previously; additional experiments to probe local free-volume (as opposed to macroscopic volume changes) should be quite important. It would be desirable to have structural or dynamic measurements of polymers under stress to confirm these ideas. Semicrystalline polymers do not display strain softening in tension or shear,

10.9 References

perhaps because they are usually tested above Tg of the amorphous component. Virtually all polymer solids, glassy or semicrystalline, display conspicuous strainhardening in tension. Glasses strain harden in compression and shear, as well. This has been modeled successfully by entropic resistance to flow as chain segments of finite extensibility are stretched. Reasonable molecular parameters describe observed behavior, and the model accounts for the strain-hardening rate being greater in tension than in compression (Boyce and Arruda, 1990). Quite different explanations apply to semicrystalline polymers, at least between Tg and Tm. Texture development by [001] slip leads to pronounced strain hardening in tension, and very little strain hardening in shear (Parks and Ahzi, 1990). Computer simulations of large strain behavior have been extended by Lee et al. (1993) to include deformations of amorphous and crystalline regions. This approach accounts for many features of postyield flow of polyethylene, and should become a powerful method for unraveling the complex response of other semicrystalline polymers. The past decade has seen a number of significant advances in our understanding of yielding and plastic deformation in polymers. Experimental innovations and refinements have provided intrinsic responses which are essential input for theories and models. Sophisticated theories and powerful numerical methods have been applied with success. Future developments will certainly involve advanced microscopies, spectroscopies and simulations to describe how structure and dynamics on the molecular scale respond to stresses beyond the elastic limit.

467

10.8 Acknowledgements The author is indebted to B. Moran and many other colleagues who assisted in the preparation of this article through discussions and helpful comments. Partial funding was provided by the Gas Research Institute, Physical Sciences Department, Contract No. 5090-260-2066.

10.9 References Adams, W. W, Yang, D., Thomas, E. L. (1986), /. Mater. Sci. 21, 2239. Argon, A. S. (1973), Phil. Mag. 28, 839. Argon, A. S. (1975), in: Polymeric Materials: Baer, E., Radcliffe, V. S. (Eds.). Metals Park, OH: Am. Soc. Met., pp. 411-486. Argon, A. S., Bessonov, M. I. (1977), Phil. Mag. 35, 917. Argon, A. S., Cohen, R. E. (1990), Adv. Polym. Sci. 90/91, 301. Bacon, D. I , Tharmalingam, K. (1983), / Mater. Sci. 18, 884. Balta Calleja, F. J. (1985), Adv. Polym. Sci. 66, 111. Balta Calleja, F. J., Kilian, H. G. (1985), Coll. Polym. Sci. 263, 697. Bartczak, Z., Cohen, R. E., Argon, A. S. (1992a), Macromolecules 25, 4692. Bartczak, Z., Argon, A. S., Cohen, R. E. (1992b), Macromolecules 25, 5036. Batterman, S. D., Bassani, X L. (1990), J. Polym. Sci. 30, 1281. Bessell, T. J., Young, R. J. (1974), J. Polym. Sci., Polym. Lett. Ed. 12, 629. Bevis, M. (1978), Colloid Poly. Sci. 256, 234. Bonner, R. M., Kohan, M. I., Lacey, E. M., Richardson, P. N., Roder, T. M., Sherwood, L. T. (1973), in: Nylon Plastics: Kohan, M. I. (Ed.). New York: Wiley-Interscience, Chap. 10. Bowden, P. B. (1973), in: The Physics of Glassy Polymers: Haward, R. N. (Ed.). New York: Wiley, Chap. 5. Bowden, P. B., Raha, S. (1970), Phil. Mag. 22, 463. Bowden, P. B., Raha, S. (1974), Phil. Mag. 29, 149. Bowden, P. B., Jukes, J. A. (1972), /. Mater. Sci. 7, 52. Bowden, P. B., Young, R. J. (1974), J. Mater. Sci. 9, 2034. Boyce, M. C , Parks, D. M., Argon, A. (1988), Mech. Mater. 7, 15. Boyce, M. C , Arruda, E. M. (1990), Polym. Sci. Eng. 30, 1288. Boyd, R. H. (1983), J. Polym. Sci., Polym. Phys. Ed. 21, 493.

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10 Plastic Deformation of Polymers

Boyd, R. H. (1985), Polymer 26, 323, 1123. Brady, J. M., Thomas, E. L. (1989), /. Mater. Sci. 24, 3311. Brown, N. (1983), Polymer 18, 2241. Brown, N. (1986), in: Failure of Plastics: Brostow, W., Corneliussen, R. D. (Eds.). New York: Hanser, Chap. 6. Chow, T. S. (1990), Macromolecules 23, 4648. Chuah, H. H., Lin, J. S., Porter, R. S. (1986), Macromolecules 19, 2732. Coates, P. D., Ward, I. M. (1980), J. Mater. Sci. 15, 2897. Crist, B., Fisher, C. X, Howard, P. R. (1989), Macromolecules 22, 1709. DeTeresa, S. X, Porter, R. S., Farris, R. J. (1985), /. Mater. Sci. 20, 1645. DeTeresa, S. X, Porter, R. S., Farris, R. X (1988), J. Mater. Sci. 23, 1886. Fagler, L. O., Bassani, X L. (1986), Int. J. Solids Struct. 22, 1243. Flory, P. X, Yoon, D. Y. (1978), Nature 272, 226. Frank, F. C , Stroh, A. N. (1952), Proc. Phys. Soc. B65, 811. Galenski, A., Argon, A. S., Cohen, R. E. (1991), Macromolecules 24, 3953. Galenski, A., Bartczak, A., Argon, A. S., Cohen, R. E. (1992), Macromolecules 25, 5705. G'Sell, C. (1988), Rev. Phys. Appl. 23, 1085. G'Sell, C , Jonas, X X (1979), /. Mater. Sci. 14, 583. G'Sell, C , Boni, S., Shrivastava, S. (1983), J. Mater. Sci. 18, 903. G'Sell, C , Hiver, X M., Dahoun, A., Phillipe, M. X, Esling, C. (1991), in: Abstracts of Plasticity 91 Symposium, Grenoble.. Hasan, O. A., Boyce, M. C , Li, X. S., Berko, S. (1992), /. Polym. Sci., Polym. Phys. Ed. 31, 189. Haward, R. N. (1973), The Physics of Glassy Polymers. New York: Wiley, Chap. 6. Heisse, B., Kilian, H. G., Peitrella, M. (1977), Progr. Colloid Polym. Sci. 62, 16. Hope, P. S., Duckett, R. A., Ward, I. M. (1980), /. Appl. Polym. Sci. 25, 1373. Hutchinson, X W, Neale, K. W. (1983), J. Mech. Phys. Solids 25, 1373. Imai, Y, Brown, N. (1976), J. Polym. Sci., Polym. Phys. Ed. 14, 723. Juska, T, Harrison, I. R. (1982), Polym. Eng. Sci. 22, 166. Kinloch, A. X, Young, R. X (1983), Fracture Behavior of Polymers. New York: Applied Science, Chaps. 4,5. Kamei, E., Brown, N. (1984), /. Polym. Sci., Polym. Phys. Ed. 22, 543. Kawai, H., Hashimoto, T. (1979), in: Contemporary Topics in Polymer Science: Shen, M. (Ed.). New York: Plenum Press, pp. 245-266. Knauss, W G., Emri, I. (1987), Polym. Eng. Sci. 27, 86. Kramer, E. X, Berger, L. L. (1990), Adv. Polym. Sci. 91/92, 1.

Krause, S. X, Hosford, W. G. (1989), J. Polym. Sci. B, Polym. Phys. 27, 1853, 1867. Lee, B. X, Argon, A. S., Parks, D. M., Ahzi, S., Bartczak, Z. (1993), Polymer, in press. Lin, L., Argon, A. S. (1992), Macromolecules 25, 4011. Lohse, G., Schmid, R., Batzer, H., Fisch, W. (1969), Br. Polym. J. 1, 110. Manson, X A., Sperling, L. H. (1974), Polymer Blends and Composites. New York: Plenum. Martin, D. C , Thomas, E. L. (1991), J. Mater. Sci. 26, 5171. Matsuoka, S. (1978), Polym. Sci. Eng. 21, 907. Matsuoka, S. (1986), in: Failure of Plastics: Brostow, W, Corneliussen, R. (Eds.). Munich: Carl Hanser, Chap. 3. Matsuoka, S. (1991), in: Proc. MRSSymp., Vol. 215: Roe, R. X, O'Reilly, X M. (Eds.). Pittsburgh, PA: Mater. Res. Soc, pp. 71-80 Matsuoka, S. (1992), Relaxation Phenomena in Polymers. New York: Carl Hanser, pp. 111-120. McCullough, R. L., Peterson, X M. (1973), / Appl. Phys. 44, 1224. Meinel, G., Peterlin, A. (1971), Eur. Polym. J. 7, 657. Mott, P., Argon, A. S., Suter, U. W. (1993), Phil. Mag., in press. Neale, K. W, Tugcu, P. (1985), /. Mech. Phys. Solids 33, 323. Parks, D. M., Ahzi, S. (1990), J. Mech. Phys. Solids 38, 701. Peterlin, A. (1965), /. Polym. Sci. C9, 61. Peterlin, A. (1971), /. Mater. Sci. 6, 490. Peterlin, A. (1987), in: Encyclopedia of Polymer Science and Engineering, Vol. 10: Mark, H. F., Bikales, N. M., Overberger, C. G., Menges, G., Kroschwitz X I. (Eds.). New York: Wiley, pp. 72-94. Pertsev, N. A., Romanov, A. E., Vladimirov, V. I. (1981), /. Mater. Sci. 16, 2084. Phillips, P. X, Philpot, R. X (1986), Polymer Commun. 307. Popli, R., Mandelkern, L. (1987), /. Polym. Sci. B, Polym. Phys. 25, 441. Pukansky, B., Tudos, F. (1990), Makromol. Chem., Macromol. Symp. 38, 221. Raghava, R., Caddell, R. M., Yeh, G. S. Y. (1973), J. Mater. Sci. 8, 225. Robertson, R. E. (1966), J. Chem. Phys. 44, 3950. Rutherford, X L., Brown, N. (1980), in: Methods of Experimental Physics: Polymers: Physical Properties, Vol. 16C, Fava, R. A. (Ed.). New York: Academic, pp. 117-135. Sadler, D. M., Barham, P. X (1990 a), Polymer 31, 36. Sadler, D. M., Barham, P. X (1990 b), Polymer 31, 46. Samuels, R. X (1974), Structured Polymer Properties. New York: Wiley. Saraf, R. F., Porter, R. S. (1988), J. Polym. Sci. B, Polym. Phys. 26, 1049. Schultz, X (1974), Polymer Materials Science. Englewood Cliffs, NJ: Prentice-Hall; Chaps. 3, 11. Shadrake, L. G., Guiu, F. (1976), Phil. Mag. 34, 565.

10.9 References

Shay, R. M., Caruthers, J. M. (1990), Polym. Eng. Sci. 30, 1266. Sorensen, R. A., Liau, W. B., Boyd, R. H. (1988), Macromolecules 21, 194. Sternstein, S. S. (1975), in: Polymeric Materials: Baer, E., Radcliffe, S. V. (Eds.). Metals Park, OH: Am. Soc. Met., pp. 369-410. Theodorou, D. N., Suter, U. W. (1986), Macromolecules 19, 139. Vincent, P. I. (1960), Polymer 1, 7. Ward, I. M. (1975), Structure and Properties of Oriented Polymers. New York: Wiley. Ward, I. M. (1983), Mechanical Properties of Solid Polymers, 2nd ed. New York: Wiley, Chap. 11. Wierschke, S. G., Shoemaker, J. R., Haaland, P. D., Pachter, R., Adams, W. W. (1992), Polymer 33, 3357. Williams, M. L., Landel, R. R, Ferry, J. D. (1955), /. Am. Chem. Soc. 77, 3701. Wu, W, Wignall, G. D. (1985), Polymer 26, 661. Wu, W, Wignall, G. D., Mandelkern, L. (1992), Polymer 33,4131. Yamini, S., Young, R. J. (1980), J. Mater. Sci. 15, 1814.

469

Young, R. J. (1979), in: Developments in Polymer Fracture-1, Andrews, E. H. (Ed.). London: Applied Science, pp. 223-261. Young, R. J. (1988), Mater. Forum 11, 210. Young, R. I, Bowden, P. W, Ritchie, J. M., Rider, J. G. (1973), J. Mater. Sci. 8, 23. Young, R. I, Bloor, D., Batchelder, D. N., Hubble, C. L. (1978), /. Mater. Sci. 13, 62. Young, R. I, Dulniak, R., Batchelder, D. N., Bloor, D. (1979), J. Polym. Sci., Polym. Phys. Ed. 17, 1325.

General Reading Haward, R. N. (Ed.) (1973), The Physics of Glassy Polymers. New York: Wiley. Peterlin, A. (Ed.) (1971), Plastic Deformation of Polymers. New York: Marcel Dekker. Schultz, J. (1974), Polymer Materials Science. Englewood Cliffs, NJ: Prentice-Hall. Ward, I. M. (1983), Mechanical Properties of Solid Polymers. New York: Wiley.

11 Dielectric Properties of Polymers Graham Williams Department of Chemistry, University College of Swansea, Swansea, U.K.

List of Symbols and Abbreviations 472 11.1 Introduction 476 11.2 Phenomenological Theory 478 11.2.1 General Frequency-Time Relations 478 11.2.2 Empirical Relations for Permittivity 479 11.2.3 Temperature and Pressure Dependence of the Average Relaxation Time . 481 11.3 Molecular Theory 482 11.3.1 Equilibrium Theory 482 11.3.2 Dynamic Theory 485 11.4 Experimental Methods 487 11.5 Experimental Data for Polymer Systems 488 11.5.1 Introduction 488 11.5.2 Experimental Data for Liquids and Solutions 489 11.5.2.1 Flexible Chains 489 11.5.2.2 Rod-Like Chains 492 11.5.3 Experimental Data for Amorphous Solid Polymers 494 11.5.4 Experimental Results for Crystalline Polymers 501 11.5.4.1 Introduction 501 11.5.4.2 Polymers of High Degree of Crystallinity 501 11.5.4.3 Polymers of Medium Degree of Crystallinity 503 11.5.5 Experimental Data for Liquid Crystalline Polymers 507 11.5.5.1 Introduction 507 11.5.5.2 Non-Chiral Liquid Crystalline Side-Chain Polymers 509 11.5.5.3 Chiral Liquid Crystalline Polymers 517 11.6 Concluding Remarks 521 11.7 Acknowledgements 523 11.8 References 523

Materials Science and Technology Copyright © WILEY-VCH Verlag GmbH & Co KGaA. All rights reserved.

472

11 Dielectric Properties of Polymers

List of Symbols and Abbreviations A a, b a, b A, B, To, T^ aktk±j

material constant (Arrhenius equation) Landau coefficients Havriliak and Negami relaxation parameters parameters of Vogel equation coefficients which add the contributions to from each of the dipoles + j and — j removed from the reference dipole k ah,as relaxation strengths due to backbone and side chain motions, respectively A\\,AL geometric quantities for dielectric permittivity function in LC state B magnitude of the directing magnetic field c concentration Cb (t\ Cs (t) dipole moment correlation functions for backbone and side chain motions, respectively Cel electrode capacitance due to electrode polarization Co geometrical capacitance of electrodes Ci,C2 WLF parameters D dissipation factor ( = e"/e') Dr rotational diffusion coefficient Dpm Wigner rotation matrix element E(t),Ez(t) electric field, in z-direction Eo amplitude of E (t) A£ trans-gauche energy difference / frequency (Hz) phase space distribution function ^ one-sided Fourier transform, Fourier sine transform and Fourier cosine transform, respectively / c ,/ c ' crossover frequencies for real permittivity and loss factor curves, respectively, for a LC material fd frequency of directing (aligning) electric field fuft local field factors in a uniaxial liquid crystal fm frequency of maximum loss factor [= (2 n < T » " X] Fj (t) the i / t h relaxation function in time for a LC material F|] (co), JF|j (co) longitudinal and transverse components of relaxation functions, respectively, measured parallel to the LC director Fl± (co), F\ (co) longitudinal and transverse components of relaxation functions, respectively, measured perpendicular to the LC director Q Kirkwood g-factor G constant of proportionality flf|l ,g j_ Kirkwood g-factor in LC phase gN,gS, S,N, I glassy nematic, glassy smectic, smectic, nematic and isotropic phases, respectively H, Hs, P homeotropic, homeogeneous and planar alignments, respectively, of a uniaxial LC phase

List of Symbols and Abbreviations

473

h H, H o k £ M M(0),M(t)

relaxation function Hamiltonian, unperturbed Hamiltonian Boltzmann constant Liouville operator molecular weight spontaneous polarization of macroscopic volume V at times 0 and t, respectively Mc critical molecular weight for entanglements Mz (t) average dipole moment along laboratory z-axis n director axis in L C phase n refractive index Nr number of dipole groups per unit volume JV V concentration of dipolar molecules p (co) internal field factor [ = x (co)/x (0)] P pressure V\,CL\ conjugate position and momentum of particle i Pz (t) macroscopic polarization along laboratory z-direction °p r probability of obtaining environment V Q apparent activation energy for relaxation (Arrhenius equation) <2P,<2v constant pressure and constant volume apparent activation energies, respectively gpr strength factor for r'th component of P relaxation process R G a s constant r• (0), r• (t) end to end vector for chain i at times 0 and t, respectively S molecular order parameter for a uniaxial liquid crystal Sd macroscopic director order parameter S ^ S J j S ^ S f ^ S x chiral-smectic A, C, F, J, and X phases, respectively t time T temperature Tc clearing temperature Tg glass transition temperature Tt transition temperature v,v[9vb volume, free volume, bound volume, respectively V volume W(t) time-dependent transition probability X degree of crystallinity (0 < X < 1) /? p (0), /? (t) y1 yLC £(/) S(CD) sn, sY

K W W relaxation parameter polar angle between molecular dipole moment fi{ and laboratory z-axis in a LC material effective rotational viscosity of a liquid crystal rotational viscosity of a ferroelectric liquid crystal complex dielectric permittivity relative to free space dielectric permittivity tensor defined by Eqs. (11-48 a, b)

474

11 Dielectric Properties of Polymers

sl],8h 2V e el e 1 ,e 2 SQ^S^ s\ e" e^ efT9 e" £|l (co), £±(co) e

O||>

fi

°o||

ds As, i ( A£ p

n

9C 9kk> 90 x fi, fi0

fi0 lik,\ik> ju1?iut [ir, fik /i s ,ju b /A-J (0), \xA (t) <JUJ|>,
Q o T r b ,T s (j) (T) 4>

(0) (j)i

upper and lower bounds of measured permittivity permittivity of free space complex permittivity due to electrode polarization permittivities of the amorphous a n d crystalline phases, respectively limiting low and high frequency permittivities, respectively real a n d imaginary parts of dielectric permittivity, respectively maximum loss factor real and imaginary parts, respectively, of relaxation part of total permittivity s principal permittivities parallel, perpendicular to the uniaxial liquid crystal director axis n principal low frequency permittivities of a uniaxial liquid crystal principal high frequency permittivities of a uniaxial liquid crystal dielectric anisotropy (= fiy — £±) dielectric relaxation strength, for individual processes relaxation strengths of the a and (3 relaxation processes, respectively internal rotation parameter (=
List of Symbols and Abbreviations

(t) (t)

CO

time-correlation functions for the oc and Pr relaxation processes total dipole moment correlation function for macroscopic volume V time correlation function of order J and indices "Pm" for dipole motion in a liquid crystal anisotropy of magnetic susceptibility

coc

angular irequency (= znj) orientation of body in Euler space cut-off angular frequency

DGEBA DOBAMBC DRS ITO KWW LC LCD LS MW NM NMR PEMA PET PHIC POM POE PVC SRT VDCN/VAc VHF, UHF WLF WVWP

particular epoxy resin (Sec. 11.6) particular liquid crystal mixture (Sec. 11.5.5.3) dielectric relaxation spectroscopy indium/tin oxide Kohlrausch-Williams-Watts liquid crystal liquid crystal display (high frequency) local segmental mode M ax well - Wagner (low frequency) normal mode nuclear magnetic resonance poly(ethyl methacrylate) poly(ethylene terephthalate) poly(hexyl isocyanate) polyoxymethylene polyoxyethylene poly(vinyl chloride) single relaxation time vinylidene cyanide/vinyl acetate very, ultra high frequency Williams, Landel and Ferry Warchol, Vaughan, Wang and Pecora

Q,Q0

475

476

11 Dielectric Properties of Polymers

11.1 Introduction Synthetic polymer materials as amorphous or crystalline thermoplastics, as elastomers, thermosetting resins or composites find diverse applications in modern society. The commercial products take the form of moulded or extruded articles, spun fibres, foams, films, tapes, laminates or adhesive layers. The practical uses of such materials are determined mainly by their mechanical and/or optical properties but in many important applications, e.g., for electrical cable insulation, film capacitors, components in electrical or electronic circuitry, the electrical properties are of great importance and interest. For such applications the polymer material must be a good electrical insulator at the frequencies and voltages of operation, i.e., the in-phase electric current should be as small as possible, thus avoiding dielectric heating or electrical breakdown or minimizing the attenuation of signals in submarine telephone cables. During World War II, Professor Arthur von Hippel and his group at M.I.T. and Professor Willis Jackson and his group at London University developed experimental methods for measuring the dielectric properties of different materials in the range extending from power frequencies (10-10 3 Hz), to audio and radio frequencies (10 2 -10 5 Hz), UHF and VHF frequencies (10 5 -10 8 Hz) and microwave frequencies (K^-IO 1 1 Hz). Data were obtained for the complex dielectric permittivity e ( / ) measured at different frequencies, /, and different temperatures, T, where the dielectric properties are expressed in terms of the complex relative dielectric permittivity £(/), where

(11-1) and e'(/) is the real permittivity and c" (/)

is the dielectric loss factor, both measured at a frequency of / = co/2 n. The additional quantity D = tan£ = s" (f)/sr (f) is also commonly used as a measure of the energy dissipation in a dielectric material. Extensive data for a number of materials, including synthetic polymers were published in von Hippel's 1954 monograph. Knowledge of the dielectric properties at different frequencies and different applied voltages allowed the applications of the materials to be assessed without any need for molecular interpretations of the origins of the particular values of the permittivity and loss factor and their dependencies on temperature and frequency. However, the molecular origins of dielectric relaxation spectroscopy had been established much earlier for dipolar molecular liquids and rotator-phase solids by Debye (1927) who had shown that the applied electric field perturbs the orientational distribution function for the dipolar molecules, leading to a static relative permittivity s0 greater than n2, where n is the optical refractive index, and a dispersion for £'(/) accompanied by a peak in e"(/). Dielectric relaxation due to classical molecular reorientational motions is a form of pure absorption spectroscopy whose frequency range of interest for materials, including synthetic polymers, is 10~6 to 10 11 Hz. Experimental data for different dipolar systems, mainly of low molar mass, were reviewed by Smyth (1955), Frohlich (1958) and later by Hill et al. (1969). It is fair to say that the static permittivities of simple dipolar liquids and rotator-phase solids are well-understood in terms of the molecular structure and organization of the phase while the dielectric relaxation behaviour of such systems is well-understood in terms of the classical reorientation motions by rotational diffusion or motion in barrier systems (for solids).

11.1 Introduction

In parallel with the studies of von Hippel and Willis Jackson, the American scientists, R. M. Fuoss and J. G. Kirkwood, carried out some of the first systematic studies of the dielectric behavior of polymer materials, e.g., poly(vinyl chloride) in its unplasticized and plasticized forms. They showed that extensive motions of the dipolar chain segments occurred above the apparent glass transition temperature Tg and that these motions, whose average rate was strongly dependent on sample temperature, could be investigated by dielectric relaxation spectroscopy. They documented the dielectric properties of several polymers as a function of frequency and temperature and gave a simple empirical function for the loss factor which fitted, approximately, the broad experimental curves. They also proposed molecular theories for £0 and s(f) for linear polymers (Fuoss and Kirkwood, 1941; Kirkwood and Fuoss, 1941). e0 was related to the dipole moment of a chain segment and a g-factor, the Kirkwood correlation factor, which expresses the angular correlations between dipolar groups along a chain and which arise from chain connectivity and chain conformation. Their studies laid the foundation of our understanding of the dielectric properties of polymers. As new polymers were prepared during and after the war, their dielectric properties were investigated and the literature expanded at a remarkable rate. Of the many publications, that by Reddish (1950) for polyethylene terephthalate, which featured three-dimensional contour diagrams for permittivity and loss, and that by Scott and coworkers (1962) for polychlorotrifluoroethylene, which covered a range of sample crystallinity and particularly wide frequency and temperature ranges, were of special importance in establishing molecular structure-dielectric property relations

477

for solid polymers. Multiple dielectric relaxation processes were observed including one associated with the dynamic glasstransition process (a relaxation) and another relating to limited motions of groups in the glassy state (P relaxation). Further systematic studies were made by Ishida and his group for several amorphous and crystalline polymers (see Ishida, 1969, for a review). By the early 1960's many of the conventional polymers used today were available and their dielectric properties had been investigated. The dielectric studies complemented those made using the techniques of dynamic mechanical relaxation and nuclear magnetic resonance. Comparing the results of such studies it was found possible to assign the origins of the multiple relaxation regions and to suggest specific mechanisms for motion. In 1967 McCrum, Read and Williams published a research text in which the dynamic mechanical and dielectric relaxation properties of solid polymers were compiled, were described phenomenologically, and were discussed, in a comparative way. The origins of the different relaxations were discussed in terms of the underlying molecular relaxation (or motional) processes (McCrum et al., 1967). Since that time several reviews and texts have been published which are concerned with the dielectric properties of polymers (Williams, 1972; van Turnhout, 1975; Hedvig, 1977; Williams, 1979; Jonscher, 1983; Williams, 1989). Thus, there already exists a substantial literature and review of both the experimental and theoretical aspects of the dielectric properties of polymers. The present account seeks to outline certain essential aspects of the phenomenological theory and molecular theory of the dielectric properties of polymers, to indicate some of the modern methods used to measure the

478

11 Dielectric Properties of Polymers

dielectric permittivity and loss factor and to describe representative results for different systems. It is worth noting that although the dielectric properties of polymers have been studied for over fifty years, (i) the general theory relating the complex permittivity to molecular behaviour in terms of the time-correlation functions of the motion is fairly recent (Cole, 1965; Cook et al., 1970; Williams, 1972); (ii) that there has been a marked improvement in dielectric measurement techniques during the past ten years as a result of improved instrumentation and computer control; (iii) the precise nature of the different relaxation processes seen for amorphous, crystalline and liquid crystalline polymers is still a matter of debate. A similar comment applies to the equivalent processes observed using dynamic mechanical relaxation, NMR relaxation, dynamic light scattering and the time-dependence of fluorescence depolarization.

11.2 Phenomenological Theory

where the subscript "r" refers to the complex admittance (relaxation component) and a is the dc conductivity (in Q~x cm" 1 ) of the sample. £,(/) and sj.'(/) depend on the frequency of measurement, £,(/) showing dispersion generally (a falling-off of its value with increasing frequency) and e" (/) showing an absorption peak (see Fig. 11-1 below). Experimentally, s(f) is the steadystate complex permittivity for the sample in the presence of alternating field of frequency / An alternative experiment is to apply a step-on or step-off field to the sample and to measure the effective permittivity as a function of time. The normalized decay function
(11-3)

— fin

where e0 and are the values of sr (t) at t = 0 and t -+ oo, respectively. These are also known, respectively, as the relaxed and unrelaxed permittivities (McCrum 1.2-1

11.2.1 General Frequency-Time Relations

The basic definitions of the permittivity and complex permittivity of a dielectric material are well known and will not be developed here (see, e.g., Smyth, 1955; Hill et al., 1969; Bottcher and Bordewijk, 1978; McCrum et al., 1967). It is usual to regard a dielectric sample as being a complex admittance shunted by a purely resistive element - which represents the steady dc conductivity component of the sample. The complex relative permittivity s(f) with respect to the free-space permittivity ev is given by 1.8xlO12<7

7

(11-2)

0.2-

Figure 11-1. Plots of the normalized permittivity (£r-eoo)/(eo-eoo) a n d l o s s factor £?/(£()-£«,) against log 10COT for a single relaxation time process, indicated as points o and • respectively.

11.2 Phenomenological Theory

479

etal, 1967). The connection between dielectric measurements made in the frequency domain and in the time domain is given by the Fourier transform relation below (McCrum etal., 1967; Williams, 1972).

curves in the frequency domain means that the relaxation function
-"^—^=l-io)j4>(t)-exp[-io)t]dt S £ °- " ° (11-4) where co = 2nf and where s0 and s^ now take the meaning of the limiting low and high frequency permittivities, respectively. Separating real and imaginary parts of Eq. (11-4) we may write

Several empirical functions have been proposed which give broadened loss curves in the frequency domain. The usual approach is to modify the SRT equation empirically, as was done by Cole and Cole (1941), Davidson and Cole (1950) and Havriliak and Negami (1966). The general equation which includes as special cases all three empirical functions is as follows

£r ( f) —

(11-6) where J^ and #~c indicate the one-sided Fourier sine and cosine transforms, respectively. For the special case where # (t) follows the single exponential decay function 0 (t) = exp (— t/x)

(11-7)

then Eqs. (11-5) and (11-6) give the familiar single relaxation time expressions

£

0 ~~ COT

b0

fc^

1 - f CO T

Figure (11-1) shows plots of the normalized permittivity and loss against log COT for this single relaxation time (SRT) process. The dispersion and absorption curves are complementary. The loss curve has a half width of 1.14 units in log! 0 COT and is symmetrical about log COT = 0. For solid polymers the dispersion and absorption curves are generally far broader, and often unsymmetrical, compared with that for a SRT process. Such broadening of loss

11.2.2 Empirical Relations for Permittivity

1 [l+(icoT)a]b

(11-9)

For a = b = 1 we have the SRT relation. For b = 1, 0 < a < l , Eq. (11-9) becomes the Cole-Cole function, for a = l, 0 < b < l , Eq. (11-9) becomes the Davidson-Cole function while for 0 < b < 1, 0 < a < 1, Eq. (11-9) becomes the Havriliak-Negami equation. The Cole-Cole function gives a loss curve which is broad but symmetrical about log COT = 1 while both the Davidson-Cole and Havriliak and Negami equations give unsymmetrical loss peaks which are skewed to the high-frequency side. For solid polymers in the amorphous state two dominant dielectric relaxation processes are observed. The a process (dynamic glass transition process) is well-fitted by the Havriliak-Negami equation (see, e.g., data for polyvinyl acetate by Mashimo et ah, 1982 b) while the p process (local motions of chain segments in the glassy state) is well-fitted in many cases by the Cole-Cole equation (see, e.g., data for polyethylene terephthalate in McCrum, Read and Williams, 1967). Other functions which represent empirical modifications of the single relaxation time function in the

480

11 Dielectric Properties of Polymers

where 0 < /? < 1. This function was used originally by R. Kohlrausch in 1854 to describe charge decay in the Leiden jar (Kohlrausch, 1854) but has subsequently found wide applications for creep and stress relaxation in glass-forming solids (see, e.g., Williams, 1985, for a review). The frequency-dependence of the dielectric permittivity corresponding to Eq. (11-12) was determined by Williams and Watts (1970) and Williams et al. (1971), by use of the (11-10) = J 0(r)exp(transform relation Eq. (11-4). It was shown o that broad, asymmetric dispersion and abwhich on insertion into Eq. (11-4) gives sorption curves were obtained for different values of /?, and these were very similar to those exhibited by the oc-process in amorphous polymers (see Williams et al., 1971; Experimental dielectric data may always Williams and Watts, 1971 a, b,c). It is also be fitted approximately by a suitable choice found that a wide range of organic and of the distribution function (T)inorganic glass-forming materials exhibit may be determined directly from loss data relaxation phenomena (dielectric, mechanin the frequency domain using methods deical, NMR, specific heat, quasi-elastic light scribed by Ferry (1970), McCrum et al. scattering, electric field relaxation) which (1967) [see, e.g., data for the a relaxation in conform, at least approximately, with the poly(methylacrylate) obtained by Williams KWW representation in the time or fre(1964 b)]. Since the approach is purely emquency domains (see, e.g., Williams, 1985, pirical within the phenomenological the1989, for reviews and further references). ory of relaxation no physical significance Lindsey and Patterson (1980) have shown need be attached to the experimentally-dethat the KWW function, Eq. (11-12) may termined function 0(T); rather it may be be represented formally in terms of a conregarded as a convenient vehicle in relating tinuous distribution of exponential decay time-dependent and frequency-dependent functions as expressed in Eq. (11-10). Sevdielectric relaxation behaviour through eral models have been proposed which Eqs. (11-10) and (11-11). Molecular models lead to the KWW functional form (see for chain relaxation may lead, however, to Williams, 1985, 1989, for reviews; Dorfspecific forms for (T) or its discrete anamiiller and Williams, 1987). Most of the logue (Ferry, 1970; McCrum et al., 1967). models include cooperativity explicitly A further empirical representation which (e.g., kinetic Ising models) or implicitly has been found useful for dielectric relax(e.g., diffusion models) as a requirement for ation behaviour and related phenomena, is molecular motion of representative moleto express the function $ (t) as a particular cules or chain-segments in a glass-forming analytical function of time. One such repsystem. resentation is the "stretched exponential" Up to this point we have considered (or Kohlrausch-Williams-Watts) function dielectric relaxation in terms of the time or (t/T)f (11-12) frequency dependence of the permittivity

frequency domain are due to Fuoss and Kirkwood (1941), Kirkwood and Fuoss (1941) and Jonscher, Hill and Dissado (see Jonscher, 1983). An alternative approach to empirical representations of dielectric relaxation is to express the relaxation function as a discrete or continuous distribution of single exponential decay functions. For the latter case we write

11.2 Phenomenological Theory

or loss factor. An alternative method of presentation of dielectric data is the Argand diagram in which s" is plotted against e!t which, for a single relaxation time process, is a semi-circle. For the Cole-Cole function a depressed semi-circle is obtained while for the Havriliak-Negami, Davidson-Cole and KWW functions a skewed arc is obtained which resembles the semi-circular arc only at the lowest frequencies. Such diagrams are a convenient and compact way of representing dielectric data for polymers, but plots of permittivity or loss against log frequency are to be preferred since they indicate clearly all three items of information which may be obtained from the dielectric experiments, namely (i) the relaxation strength(s), Asa, A^p, etc., (ii) the average relaxation time = (2Tc/m)~1 where fm is the frequency of maximum loss factor and (iii) the shape of relaxation function (t) [via Eqs. (11-5) and (11-6)] for the individual relaxation regions. 11.2.3 Temperature and Pressure Dependence of the Average Relaxation Time

The temperature dependence of for the a relaxation in amorphous polymers is usually found to follow the Vogel equation

= A - exp

B T-Tr

In

applications of the WLF equation to dielectric, dynamic-mechanical and NMR relaxation data in the a relaxation region for amorphous polymers are widespread (Ferry, 1970; McCrum et al, 1967; McKenna, 1989). The common feature of the a relaxation in amorphous polymers is that the plot of log/ m vs. [T(K)]" 1 is strongly curved as Tg is approached from higher temperatures. Several theories have been proposed to account for Vogel-type (or WLF) behaviour, and those involving free-volume concepts are prominent (Ferry, 1970; McCrum et al., 1967; McKenna, 1989). The secondary (or P) relaxations for amorphous polymers give extremely broad loss curves but usually is found to follow the Arrhenius equation

where A and Q are material constants and Q is known as the apparent activation energy for relaxation (McCrum et al., 1967). The temperature dependence of reflects the fact that dielectric relaxation is a thermally-activated process. Pressure also has a marked effect on relaxation frequencies for motions in polymers. Constant pressure and constant volume apparent activation energies are defined by the relations

and are related according to

-RT'< C 2 + (T2 -

(11-14)

where subscripts 1 and 2 refer to temperatures Tx and T2, respectively, and where C1 = B/(T1 - To) and C2 = TY-

(11-15)

= . 4 - e x p ( ^

(11-13)

where A, B and To are material constants. The familiar Williams-Landel-Ferry equation (WLF equation) follows from Eq. (11-13) as

481

T o . The

/81nr\

pTjv{

dP )T

(11-16)

(11-17)

(see Williams, 1964a,b). Here (8P/8T)K is the thermal pressure coefficient and is given by the ratio of the thermal expansion coefficient and the isothermal compress-

482

11 Dielectric Properties of Polymers

ibility of the material. It follows from Eq. (11-17) that Qv < QP in practice. According to the simplest free-volume theories for relaxation in the Tg range (oc process) the value of is determined by the free volume vf where vf = v — vh and v and vh are total volume and "bound" volume, respectively (Ferry, 1970; Williams, 1964ab; McKenna, 1989). If vb is independent of temperature and pressure then raising the temperature of a sample at constant volume would not lead to a change in v{ and hence QY would be expected to be zero. If vh increases with temperature under constant volume conditions then v{ would actually decrease and QY < 0 would be predicted. In practice QY/QP = 0.7 to 0.8 for the dielectric a relaxation in amorphous polymers which is inconsistent with the simple free volume theory and emphasizes the fact that the rate of relaxation is determined by the thermal energy, k T, of the material. Raising the sample temperature at constant macroscopic volume or constant macroscopic pressure increases the rate of relaxation through the increase in thermal energy and not, primarily, through a change in free volume.

For polymers in solution and for unoriented amorphous polymers the theory for Ae is essentially that due to Kirkwood and to Frohlich (see McCrum et al., 1967, for a historical account). For an ensemble having N polymer molecules in a macroscopic spherical volume V the desired relation between Ae and molecular quantities is (Cook etal, 1970) (11-18) 4TI 3e o (2e <M(0) • M(0)> 3kT (2e0

11.3 Molecular Theory


where M(0) is the instantaneous dipole moment of the macroscopic sphere at the arbitrary time t = 0, and where M(0) may be expressed as the sum of the dipole moments, suitably defined, for the polymer molecules which are contained in the sphere. Through thermal motions the dipole moment vectors change their direction continually in time leading to <M(0)> = 0 but to a finite value of the macroscopic mean square moment <M(0)-M(0)>. Here "<>" indicates a statistical average over all configurations of the dipole moments in the sphere. For the special case of rigid dipolar rodlike molecules each having a dipole moment fi then (11-19)

11.3.1 Equilibrium Theory

The theory of the static permittivity of a polymer material is different for amorphous, crystalline and liquid crystalline systems. In each case a mean square dipole moment, suitably defined, is determined from the incremental dielectric permittivity Ae = e0 — e^ where s0 and e^ are the limiting low frequency and high frequency permittivities so that Ae contains the dispersion magnitudes of all dielectric relaxation processes for the material at the given sample temperature and pressure.

where iVv is the number of dipolar molecules per unit volume "//' is a "liquid" dipole moment which is related to the vacuum dipole moment fi0 according to the relation (11-20)

3(2e o

Combination ofEqs. (11-18), (11-19) and (11-20) gives the familiar Onsager equation _4nNv ~ 1kT

3e 0 1 P -UP

fe^ + 2^2

483

11.3 Molecular Theory

which allows JH0, the molecular dipole moment of the rod-like molecule to be determined from a knowledge of ATV, e0 and 8^. For flexible polymer chains in bulk or in solution it is convenient to consider as the representative reference unit any dipole group, k say, in any chain where the group is sufficiently removed from the chain ends. It follows that (Cook et al, 1970) <M(0) • M(0)>

(11-22)

and where

Nr and \ir are the number of dipole groups per unit volume and the dipole moment of a repeat unit, respectively. The terms represent the angular correlations between the reference dipole moment fik and further dipole moments \ik, along the same polymer chain. As the chain conformation changes through internal thermal motions the projection of dipole moment \ik on dipole moment \ik, will vary, so
lowing Kirkwood) (11-24)

g = \+ E

is composed for the example of a model chain. A particularly convenient case is that of the polyether series [(CH 2 ) p _ 1 -O] n where for p = 2 we have polyoxymethylene and for p = 3 we have polyoxyethylene. Cook et al. (1970) adapted the theory of the dipole moments of these polyethers given by Read (1965) to the form equivalent to Eq. (11-24). Table 11-1 shows the results of these model calculations. Here <j> = 0 for the trans conformation and 4> = ± 120° for the gauche conformations. The coefficients akk±j add the contributions to from each of the dipoles +j and —j removed from the reference dipole k. Consider first the case of free internal rotation, i.e., rj = 0. For POM the next nearest neighbors to the reference dipole group contribute —0.222 to the g-factor [Eq. (11-24)] but further neighbors give only a small contribution (relative to 1). Thus for POM for the special case rj = 0 the dielectric permittivity increment (and relaxation) for amorphous phase material may be regarded as arising from group ether dipoles [autocorrelation, leading term of unity in Eq. (11-24)] and nearest neigh-

Table 11-1. Values of the coefficients akJi±j for polyoxymethylene (p = 3) for a tetrahedral chain for chosen values of the internal rotation parameter ?7 = <(cos(/>>. Polyoxyethylene

Polyoxymethylene

n -0.5 -0.25 0 + 0.25 + 0.50 + 0.75

a

k,k±i

-1.00 -0.639 -0.222 + 0.250 + 0.778 + 1.361

a

k,k±2

+ 0.250 + 0.084 -0.025 + 0.016 + 0.300 + 0.926

a

k,k±3

+ 0.062 + 0.013 -0.003 -0.007 + 0.117 + 0.630

a

k,k±i

-0.50 -0.183 -0.074 -0.187 -0.537 -1.137

a

k,k±2

+ 0.062 + 0.013 -0.003 -0.007 + 0.113 + 0.630

a

k,k±3

+ 0.024 0.000 0.000 -0.004 -0.038 -0.361

484

11 Dielectric Properties of Polymers

bor correlated dipoles. For POE all cross correlation terms are small for rj = 0 so in this case the dielectric permittivity increment is well-approximated by the contribution from the autocorrelation terms alone. As the ether groups are more-widely spaced along the chain [POM -> POE -> polytrimethylene oxide etc. (see Cook et al., 1970)] then the higher order terms are of decreasing importance if the chains have r\ around zero. However, when the chains take up preferred conformations leading to \r\\ quite different from zero then the cross correlation terms retain their importance over large separations of the groups, e.g., for POM for rj = 0.75 (trans-preferred conformation) the cross-correlation terms are large and positive while for POE for rj = 0.75 the cross-correlation terms are large and alternate in sign. These calculations demonstrate that for the special case of chains with free internal rotation and with well-separated dipolar groups, we may approximate the dielectric behavior as arising from the motions of the individual equivalent dipolar groups. However, for chains with strongly preferred internal conformational states, as is the case at low temperatures for most polymer chains, the value of g is an algebraic sum of autocorrelation (1) and cross-correlation terms «cos0 kk ,(O)». The theory of the dipole moment of polymer chains is well-developed through the work of Volkenstein (1963) and Flory (1969), and many examples have been treated quantitatively. Although it is possible to determine the dipole-correlation parameters using chain conformation models for bulk amorphous polymers together with experimental dielectric data, this is seldom done (McCrum et al., 1967). An interesting example is that of polyacetaldehyde (Williams, 1963) where the energy difference AE between trans and gauche conformations was found

to be ~ 3.9 kJ mol 1 using static dielectric permittivity data. The theory of the static permittivity of crystalline dipolar polymers is less well-developed. The anisotropy of the crystalline phase necessarily means the static permittivity becomes a tensor, having different values along different crystallographic directions. Since single crystals of polymers are extremely difficult to prepare, little interest has been attached to measurements of their dielectric properties. However, the average dielectric permittivity of unoriented and oriented specimens of lightly-oxidized polyethylene has been discussed by Boyd (1983) taking into account the anistropy of the permittivity of the individual crystals. Liquid crystalline polymers are a recent addition to the dielectrics literature and the theory of the static dielectric permittivity in this case has received scant attention. However, the theory of the static permittivity of low molar mass liquid crystals is well-developed through the works of Meier, Martin and Saupe, de Jeu and Bordewijk (see Bottcher and Bordewijk, 1978; de Jeu, 1980). For the case of a nematic (axially-symmetric) liquid crystal phase the principal components of the static dielectric permittivity tensor are eO|| and £ 0 1 where "||" and " ± " indicate measurements made parallel with or perpendicular to, respectively, the director n of the liquid crystalline phase. The generalization of the Onsager-Kirk wood-Frohlich equation for isotropic media to the nematic mesophase is outlined by Bordewijk (Bottcher and Bordewijk, 1978) who writes To

_ o

L

£

1

0y J

[8 Oy

3&r

(11-25)

485

11.3 Molecular Theory

where y is || or _L and hence gy is g^ or g ±, which correspond to the longitudinal and transverse Kirkwood g-factors and which contain terms relating to the (anisotropic) angular correlations between molecular dipole groups in the LC phase. Ny is the dipole group concentration, {eO||, s0 _,_} and {£oo||>£oo±} a r e the limiting permittivities measured parallel and perpendicular to n. The factors A^ and A ± are geometric quantities and (11-26) - S) • /j

S/2) • ft (11-27)

where fx and / t are local field factors and /j.l9 Ht are the dipole moment components of the molecular (or group) dipole moment parallel with or perpendicular to, respectively, the (assumed) principal axis of the molecule (or group). S is the local order parameter of the principal molecular axis with respect to the LC director axis and is typically in the range 0.4 < S < 0.8 for low molar mass liquid crystals. Although Eqs. (11-25) to (11-27) have not been applied, as yet, to liquid crystalline polymers, they have been used for low molar mass liquid crystals with some success (Bottcher and Bordewijk, 1978; Dunmur and Miller, 1980). Parallel correlations (gy > 1) and antiparallel correlations (gy < 1) between dipoles in the semi-ordered LC phase may be deduced from static permittivity data. These equations, Eqs. (11-25) to (11-27), may be applied to LC polymers having the dipole groups in the side chain. It is clear from the form of the above equations that experimentally-determined values of eO|| and 80 ± reflect, in part, differences in the terms g ^ and g ± (jU^)- In the absence of orientational correlations between dipole groups (i.e., g{l = g ± = 1) then e0 ,| and

a0 _L are seen to differ due to the presence of the order parameter S 4= 0 in Eqs. (11-26) and (11-27) and the dielectric anisotropy term (s0^ — s0 ±) is proportional to S. 11.3.2 Dynamic Theory The theory of the dielectric relaxation of dipolar media is well-documented (Hill et al., 1969; Bottcher and Bordewijk, 1978), but is made complicated by local field factors. For a detailed assessment the reader is referred to the accounts of Deutch and coworkers (Titulaer and Deutch, 1974; Sullivan and Deutch, 1975). The modern approach to dielectric relaxation theory, in which the complex permittivity is related to molecular dipole moment quantities independent of any assumed model for relaxation, is the linear-response theory of Kubo as developed for dielectric relaxation by Glarum (1960) and Cole (1965). Williams (1978) has outlined their theory for an ensemble of N equivalent dipolar molecules for the simple case of a medium of low permittivity, i.e., e0 ~ e^, as follows. The phase space distribution function f(p,q) obeys the Liouville equation of motion (11-28)

dt

where £ is the Liouville operator and

_ " [ 6 H 8/

8H 8/"

and H is the Hamiltonian of the system, pt and qt are the conjugate position and momentum of molecule (or group) i. For an electric field Ez (t) applied in the laboratory z-direction

H(p9q9t) = H0(p9q)-MM' EM (U-30) where Mz(q) - Ez(t) = - Z / i ^ ) • Ez(t) and is the energy of interaction between the

486

11 Dielectric Properties of Polymers

dipole moments and the electric field. After considerable manipulation, the field-induced average macroscopic dipole moment <MZ (t)} is obtained as the superposition relation (Williams, 1978) <M(0) • M(0)> =

JkT

• } EAn-^t-ndf

(n-31)

— oo

where <M (0) • M(0)> has its previous significance [see Eq. (11-18) above] and the total dipole moment correlation function <^(T) = <M(0)

• M ( T ) > / < M ( 0 ) • M(0)>

has

Eq. (11-34) we write

Le

(a)) = l - i

(11-36) ° where p (co) = x (co)/x (0). Comparing Eqs. (11-4) and (11-36) we see that for the special case p (co) = 1 that (^ (t) =

appeared. Equation (11-31) is the general result of the linear response theory for the arbitrary form for Ez (t). Three cases are of special interest. (i) Eo applied as a step at t' = 0

i k l

(11-32)

(ii) Eo applied at t -+ — oo, removed as a step at t' = 0

<M, W ) =

J

(11-33)

(iii) Steady state: E (t') = Eo exp (i co t') 3kT (11-34) where #" indicates the Fourier transform. Since the macroscopic electric polarization along the z-direction is given by 'Ez(t)'x-8y

(11-35)

where x is an internal field factor and av is the permittivity of free space, it follows that Eqs. (11-30) to (11-35) give the time-dependent or frequency-dependent permittivities for prescribed E(t). For example, from

(11-37) where dipoles /c and k' belong to the same chain and the strengths of the cross-correlation terms decay with increasing separation of dipoles along the chain, as discussed above (see also Cook et al., 1970). For flexible polymer chains it is convenient to think of dielectric relaxation as arising from motions of a reference dipole group k, giving the autocorrelation term for dipoles (k, k') along the chain. Since Tg. Note that the linear response theory predicts that the transient rise and decay functions for stepon and step-off electric fields, respectively, are given by [1 - #„(£)] and (p^t) and that

11.4 Experimental Methods

the time-domain and frequency-domain behaviors are related through a Fourier transform [see Eqs. (1 l-32)-(l 1-34)]. Note also that the theory relates the dielectric properties to ^ ( t ) , the molecular property, and that several relaxation mechanisms may contribute to its decay from unity to zero. The theory of the dielectric relaxation of crystalline and liquid-crystalline systems will not be detailed here but will be described below when particular systems are considered.

11.4 Experimental Methods The conventional transient, lumped circuit and distributed circuit methods for measuring the dielectric properties of polymer systems are well documented, and cover, in total, the frequency range 10 ~6 to 1010 Hz (McCrum et al., 1967; Hill et al, 1969). During the past five years or so, modern instrumentation together with computer control of data-acquisition and processing has made a marked improvement in the speed with which accurate data for polymer systems may be obtained. We summarize briefly some of the techniques which are available for particular frequency bands. In the range 10 ~6 to 1 Hz, time-domain or frequency domain methods are used conveniently. Mopsik (1984) has described a particularly wide-ranging instrument for measuring polymer samples. Both permittivity and loss factor data have been obtained in the range 10 ~4 to 104 Hz by using measurements in the time-domain and carrying out a numerical Fourier transform to determine s(co). One inherent difficulty with the time-domain method is the enforced truncation of the Fourier integral since the data are band limited, but Mop-

487

sik (1984) has described methods for overcoming this difficulty. An alternative transform procedure to numerical integration is that due to Brather (1979) which has been applied to data for solid polymers by Ribelles and Calleja (1985) and by Attard etal. (1987 c). An advantage of the transient method is its speed but truncation and electrode-polarization effects make it difficult to use in practice. A low frequency bridge, the Solartron 1250 Frequency-Response Analyzer with a special high impedance input attachment allows measurements to be made in the range 10" 3 to 65 kHz at selected frequencies, and is particularly attractive for studies of polymers in the Tg range. Perhaps the most-used frequency range for studying the dielectric properties of solid polymers is 10-10 5 Hz. This may be covered with high precision and accuracy using commercial apparata such as the DETA instrument (Polymer Laboratories), the Wayne Kerr-B221 meter, the HewlettPackard 4272 LCR meter or the Gen Rad 1689 Digibridge. With computer control such instruments may be arranged to acquire accurate dielectric permittivity and loss spectra over this frequency range and over a wide temperature range and the data may be displayed on the screen and as hard copy. The Hewlett-Packard Model 4192A LF Impedance Analyzer provides a convenient method for measuring permittivity and loss over the range 5 Hz to 13 MHz. The range 10 5 -10 8 Hz connects lumped circuit and distributed circuit methods and has always been a difficult range experimentally. The Hewlett-Packard Model 4191A RF Impedance Analyzer with suitably-designed cells may be used for polymer systems in the range 10 6 -10 9 Hz. The method of time-domain-reflectometry, which involves measuring the time-de-

488

11 Dielectric Properties of Polymers

pendent reflexion coefficient of a sample contained in a coaxial line, has been developed in recent years by Cole and coworkers (Cole, 1977; Mashimo et al., 1983), to operate in the equivalent frequency range 10 7 -10 9 Hz. The method uses commercially-available equipment and has been used to obtain accurate dielectric data for polymer solutions (Mashimo et al., 1983). However, considerable expertise is required to obtain reproducible accurate data (Cole, 1977; Clarkson et al., 1977). Studies of the dielectric properties of insulating polymers (e.g., polyethylene, polytetrafluoroethylene) in the microwave range K^-IO 1 1 Hz continue to be made using conventional microwave methods operated at single frequencies, using methods described briefly by McCrum et al. (1967). However, the measurements are time-consuming and it is difficult to obtain data points over the entire range for a given material and to cover a range of sample temperatures. Consequently, few studies have been made for polymer systems in recent years. With regard to the geometry of samples, parallel-plate or coaxial configurations are frequently-used for lumped circuit measurements in the range 10" 6 to 108 Hz. In recent years, the parallel plate samples used would be typically 1 cm diameter, 100 jum to 3 mm in thickness and would be contained in a three-terminal electrode system (see, e.g., McCrum et al., 1967). A particularly useful, but inexpensive, cell for solids has been described by Jones et al. (1976).

11.5 Experimental Data for Polymer Systems 11.5.1 Introduction

As indicated above, the dielectric permittivity and loss data for dipolar polymer systems give information on chain conformation and structure through the mean square dipole moment (Volkenstein, 1963; Flory, 1969) and on the reorientational-dynamics of chains (McCrum et al., 1967; Williams, 1979). The existing literature for the dielectric behavior of different polymer systems is far too extensive to be reviewed in detail in the present account. In consequence, we shall seek to summarize briefly the kind of information on polymer behavior which can be deduced from studies of selected systems, and we shall attempt to give interpretations which reflect current models for chain dynamics. In a typical dielectric experiment the permittivity and loss curves for a polymer system are measured over a set frequency range (say 10-10 6 Hz) for different sample temperatures and, possibly, applied pressures. For polymer solutions a range of solute concentration c would also be studied. The information obtained experimentally is in three parts (i) relaxation strengths for individual processes, As{(T,P, c\ (ii) average relaxation time = [inf^T.P.c)]'1 and (iii) the shape of the loss peak, expressed in terms of parameters of empirical functions discussed above or distribution of relaxation times, where these depend generally on (T,P) and c, where appropriate. In the following we shall show experimental data, where appropriate, and we shall emphasize the essential information obtained from dielectric studies. Since the motions of chains leading to dielectric relaxation may be very different in different phases, i.e., solution,

11.5 Experimental Data for Polymer Systems

liquid, bulk amorphous solid, crystalline solid, liquid crystalline solid, we give brief accounts of selected studies in all such systems. 11.5.2 Experimental Data for Liquids and Solutions 11.5.2.1 Flexible Chains

The segmental motions of flexible polymer chains in solution or in the bulk liquid state at elevated temperatures would normally occur on a time-scale < 1 jas thus leading to dielectric absorption in the frequency region above 1 MHz. Such studies are difficult to perform experimentally so there is not a substantial body of dielectrics literature in this range. However, in recent years studies have been made up to 3 x 109 Hz for poly(methyl vinyl ketone) in dioxane (Mashimo et al., 1983), copoly(methyl methacrylate-methyl acrylate) in benzene (Mashimo et al., 1982 a) and for polypropylene oxides in benzene (Mashimoto et al., 1984). The results for poly(methyl vinyl ketone) are interesting because two distinct dielectric absorption peaks are observed, centred around 60 MHz and 1 GHz and are due to skeletal motions of chain segments in the backbone (low frequency process) and the independent motions of the side chains (high frequency process). This is a case where the dipole moment /i of the chain segment has components fih and /is associated with backbone and side group motions and the autocorrelation function for dipole relaxation may be written as = l*2[a*Ch(t) + asCs(t)] (11-38 a) where ah = ^ / / x 2 , as = \i\j\i1,

as + ah = l

and Cb (t) and Cs (t) are the normalized autocorrelation functions for the relaxation of \i\ and /x2, respectively. Thus,
489

decays in time with two component relaxations, giving two fairly-well-resolved loss peaks in the frequency domain. In physical terms this means that the motions of the side groups relaxes fi2 at short times (high frequencies) and the segmental motions of the chain relax the remainder of \x2, i.e., /i b . The correlation times for each process [Ti = (2nfmi)~1, z' = b,s] are, in this case, sufficiently different to allow each process to be observed independently as a loss peak. In many polymer systems we would expect Tb ~ TS, leading to a single broad loss peak which may not be deconvolved into the component relaxations since with at least six adjustable parameters (three for each process) the uncertainties in their determination become large. Note that if the side group motions are far slower than the backbone motions, fi2 will be relaxed along with [i\ so Eq. (11-38 a) would become, in this instance = \i2 Cb (t)

(11-38 b)

i.e., side group motions may not be observed at a lower frequency than backbone motions. Although segmental motions of the kind discussed above are usually very fast, it is possible to slow-down such motions by going to low temperatures in a glass-forming solvent, as was described by Adachi and coworkers (Adachi et al., 1975,1977; Adachi and Ishida, 1976). Several relaxation peaks were observed at low frequencies and were attributed to different kinds of motion of chain segments and solvent. Dielectric relaxation spectroscopy has been particularly informative with regard to the long-range motions of polymer chains in solution or in the bulk state for the special case where a polymer chain has a cumulative dipole moment along its contour. In the original work of Baur and Stockmayer (1965) it was shown that

490

11 Dielectric Properties of Polymers

bulk polypropylene glycols of low molar mass exhibited two dielectric relaxation processes: (i) a high-frequency molecular weight-independent process due to local chain segment motions, (ii) a low frequency molecular-weight-dependent process due to motions of the cumulative dipole moment along the chain. The low frequency (NM) process could be attributed to Bueche-Rouse-Zimm-type normal modes of motion (see Ferry, 1970). The low frequency dielectric process in polypropylene glycols was amply confirmed by Alper et al. (1976) and by Beevers et al. (1979), for materials having low molar mass ( M ~ 2 - 6 x l O 3 ) . Beevers et al. (1979, 1980) also observed a low frequency process in dynamic Kerr-effect experiments for polypropylene glycols. The frequency-temperature locations for this process were consistent with those for the dielectric NM process for each molecular weight when the differences between second and first-order time correlation functions are considered (Beevers et al., 1980). More recently Johari and Monnerie (1986) showed that a sample of polypropylene oxide having a molecular weight of 4000 gave, for a 47% solution in toluene, two relaxation processes above Tg (due to local motions and NM process) and one broad process below Tg (due to limited local motions of segments). The NM process is only observed in polymers for which there is a cumulative dipole moment along the chain contour. Polypropylene glycol, through its chain structure, has such a dipole moment (Baur and Stockmayer, 1965; Stockmayer, 1967). Cis-polyisoprene also has this special property and in recent series of papers Adachi and coworkers have demonstrated that dielectric absorption peaks may be observed for local segmental motions (LS process, say) and the NM process for this

polymer in its undiluted (bulk) state (Adachi and Kotaka, 1983,1984,1985) and in solution (Adachi and Kotaka, 1988; Adachi et al., 1989 a, b, c and references therein). Figure 11-2 shows permittivity and loss data for ds-polyisoprenes of different molecular weight in the bulk liquid state. At high frequencies the LS process is observed whose frequency location is independent of molecular weight. The NM process is observed in the range 10 4 -10 6 Hz for these samples at this temperature and its frequency of maximum loss decreases rapidly with increase in molecular weight. These authors showed that the frequencytemperature loci for the LS and NM processes both followed the Vogel equation [see Eq. (11-13) above] for different samples having molecular weights in the range 103 to 105. Figure 11-3 shows the molecularweight dependence of the relaxation time for the NM process for samples at a temperature of 320 K. The striking feature of this figure is the observation of two linear regions with a change in slope occurring around M = 104. Below this molecular weight the NM process may be interpreted as being due to the normal mode motions of the chains as envisaged by Rouse and Zimm and as was applied to the mechanical relaxations of polymers (Ferry, 1970) and the dielectric relaxations of polymers (Baur and Stockmayer, 1965; Stockmayer, 1967). Above M ~10 4 the slope of the plot log T vs. log M is raised to 3.7 and is similar to that for the dependence of the maximum mechanical relaxation time in an entangled polymer system, known as the "3.4 power law" (Ferry, 1970; Graessley, 1974; Adachi and Kotaka, 1985). It therefore appears that the NM process has different mechanisms below and above the critical molecular weight M c for the formation of entanglements. For M > Mc a number of models may be considered, prominent among

11.5 Experimental Data for Polymer Systems

-0.01

rn

Figure 11-2. Frequency dependence of e' and e" at 320 K for bulk ds-poly(isoprenes) of molecular weight Mn equal to approximately 103, 3.3 x 103 and 6.6 x 103, indicated as points o, • and A respectively. (After Adachi and Kotaka, 1985, reproduced with permission.)

• i • i 4 6 log [/(Hz)]

which is the reptation model of motion of a chain in a virtual tube, as outlined by de Gennes (1971, 1979) and by Doi and Edwards (1978). The dielectric NM process for ds-polyisopropene in its bulk state for M > M c and its dependence on molecular weight may be explained semi-quantitatively by the tube theory. It is apparent from these studies that dielectric relaxation spectroscopy can provide important information on the long range motions of • Figure 11-3. Molecular weight-dependence of the relaxation time for the NM process in ds-poly(isoprene) at 320 K. The line - • • - represents the maximum mechanical relaxation time, the line corresponds to the maximum relaxation time of the Rouse theory. (After Adachi and Kotaka, 1985, reproduced with permission.)

491

logM

492

11 Dielectric Properties of Polymers

chains. The marked dependence of T on M for M > M c implies that it may be difficult to observe the NM process in suitable polymers. For example, for M ~ 104 for ds-polyisoprene, Adachi and Kotaka (1985) found log/ m to be 8.5 for the LS process and 0.5 for the NM process, at 320 K. Lowering the temperature would quickly remove the NM process to inaccessibly low frequencies. Adachi and Kotaka (1988) showed that the autocorrelation function for dipole motion of an entire chain may be written as

and in a 9 solvent, dioxane (Adachi and Kotaka, 1988). Concentrated solutions (20-100% solute) of essentially monodisperse polymer samples were shown to give a dielectric NM process which was approximately twice as intense as the higher frequency LS process. It was shown that the exponent m in the relation TNM ~ Mm was 2.0 ± 0.2 below the entanglement region, which is to be compared with m = 2 for the Rouse theory and m = 1.5 for the Zimm theory. For concentrations above the entanglement condition the experimental value of m = 4.3 was obtained for 20, 30 + Z Z , and the mechanism of this process changes on going from 11.5.2.2 Rod-Like Chains M < Mc (where the Rouse-Zimm modes apply) to M > Mc, where entanglements Synthetic polypeptides, such as poly-ylead to the M 3 7 power law for the relaxbenzyl L-glutamate, and certain polyalkyl ation time. isocyanates are rod-like molecules in soluRecently Adachi and coworkers have tion. They also possess cumulative dipole observed both LS and NM processes for moments along the chain contour, thus it is solutions of ds-polyisoprene in the good expected that they would give a low fresolvents benzene and toluene (Adachi and quency dielectric relaxation process in soKotaka, 1988; Adachi et al, 1989a, b, c) lution due to the small-step rotational dif-

493

11.5 Experimental Data for Polymer Systems

fusion of the molecules. The dielectric studies of polypeptides in solution were made by A. Wada several years ago, while similar studies of polyalkyl isocyanates were made some years ago by Yu and coworkers (Yu et al., 1966) and by Bur and Roberts (1969). We shall only discuss, briefly, the work with polyalkyl isocyanates. In the early studies of Bur and coworkers it was shown that poly(n-butyl isocyanate) in solution gave a well-defined dielectric absorption at low frequencies due to the reorientation of the molecules. It was shown that the molecules are rod-like up to M ~ 105 but a higher molecular weights the chains tend to form a rigidworm-like configuration. Subsequently, several dielectric and dynamic Kerr-effect studies were made for poly-n-butyl and poly-rc-hexyl isocyanate in solution and these are discussed in detail by Beevers et al. (1977). The essential results support the original conclusions of Bur and coworkers. Of special interest is the formation of lyotropic liquid crystalline (LC) solutions at high concentrations of the rod-like solute molecules. Aharoni had shown that polymers and copolymers of the higherpoly-alkyl isocyanates gave LC solutions at concentrations exceeding about 30% solute. In a series of papers Moscicki, Aharoni and Williams showed that solutions of copoly(n-butyl/rc-nonyl) isocyanate and poly(rc-hexyl isocyanate) exhibited a complex pattern of dielectric behavior on going from isotropic solution to lyotropicnematic LC solution via a bi-phasic range (Moscicki et al., 1981, 1982; Moscicki and Williams, 1983 a, b). As one example, Fig. 11-4 shows dielectric absorption data for poly(rc-hexyl isocyanate) (PHIC) (molecular weight 7.6 x 104) in toluene at 292 K and for concentrations ranging from 4.9% to 40% solute. In the initial range 4.9%-

24-

A

2016-

128-

//

/

1

/

^vK

k-

0-

i

1

i

2

i

i

i

3 log [MHz)]

4

5

Figure 11-4. e" against log [/(Hz)] for PHIC in toluene at 292.2 K. Curves 1-8 correspond to c equal to 4.9, 10.0, 14.8, 21.4, 25.0, 30.6, 35.3 and 40.4% polymer (w/w) respectively. (After Moscicki et al., 1982, reproduced with permission.)

21.4% a single broad dielectric absorption peak is observed and is due to the reorientational motions of the rod-like molecules in isotropic solution. The peak height is proportional to concentration in this range, as expected, and the frequency of maximum loss decreases due to the increasing viscosity of the solution. Above c = 21.4%, after a slight increase in height (at 25%), the loss peak magnitude falls markedly, broadens and moves to higher frequencies. Figure 11-5 shows plots of log / m , the static permittivity s0 and the loss-peak height e^ as concentration is varied throughout the entire range studied. It is apparent from the data of Figs. 11-4 and 11-5 that the bi-phasic range, which is a mixture of isotropic and lyotropic LC phases, forms for c > 25% and that the homogeneous LC phase exists beyond c ~ 35%. In the LC phase the rod-like molecules are unable to reorientate totally,

494

11 Dielectric Properties of Polymers

the rod-like molecule. Equation (11-40) shows that \x2 D\ is not relaxed, since the motions are limited to the cone (flip-flop motion is excluded) and that the strength of the dielectric absorption is proportional to D\. The effective relaxation time is

\oq[fm/[Hz))

T = [vJ(vi + l)D r ]- 1

c(wt.%) Figure 11-5. (a) Log [/m(Hz)], (b) s0 and (c) s'^ against solute concentration (% w/w) for PHIC in toluene at 292.2 K. (After Moscicki et al, 1982, reproduced with permission.)

their motions are restricted to angular fluctuations up to several degrees in an effective cone imposed by the LC nematic potential field. A convenient model for such motions is due to Warchol and Vaughan (1978) and Wang and Pecora (1980) (see also Williams, 1983 b) who deduced the dipole moment correlation function for a rod-like molecule undergoing free, smallstep rotational diffusion in a cone WVWP show that if the cone angle 90 < TT/3 then v{(vi + l)-D r t]}

(11-40)

where D°, D\ and v\ depend upon Qo, and Dr is the rotational diffusion coefficient of

(11-41)

For isotropic rotation v} = 1 so T = (2!),.)"1, but for motion in a cone v\ increases rapidly with decreasing 90 (Wang and Pecora, 1980), giving a marked decrease in T. Hence on going from isotropic solution to the LC phase, the loss peak is predicted to decrease in magnitude and move to higher frequencies and thus provides an explanation of the behavior shown in Figs. 11-4 and 11-5. Moscicki and Williams also calculated the phase behaviour of polydisperse rods in solution using the Flory model (see Moscicki and Williams, 1983 a and references therein) and, in combination with the WVWP model for dielectric relaxation for rods in a cone and the Perrin-Doty model for isotropic diffusion of rods in isotropic solution, were able to rationalize most aspects of the dielectric behavior of PHIC in its isotropic, biphasic and LC phases (Moscicki and Williams, 1983 b). 11.5.3 Experimental Data for Amorphous Solid Polymers

The dielectric properties of amorphous polymers have been reviewed extensively and in great detail (McCrum et al., 1967; Ishida, 1969; Williams, 1979,1985; Pethrick and Richards, 1982; Williams, 1989 and references therein). All amorphous polymers exhibit two relaxation processes, one, the a process, is due to the large scale microbrownian motions of chains and becomes prohibitively slow for temperatures below the operational glass transition Tg

11.5 Experimental Data for Polymer Systems

and the other, the (3 process, is due to the limited motions of chains and is normally observed in the glassy state. Certain general features of the dielectric relaxation of all amorphous solid polymers are apparent, irrespectively of chemical structure, distribution of molecular weight, tacticity or whether the material is a homopolymer or a random copolymer. The relaxation map which gives the loci of the different relaxation processes in all amorphous polymers is indicated schematically in Fig. 11-6. The oc-process follows the Vogel (or WLF) equations described in Sec. 11.2.3 above and as Tg is approached its frequency moves rapidly to very low values. The (3 process, which is usually, but not always, a small broad process is normally observed below Tg and its locus follows the Arrhenius equation also described earlier. As the temperature is raised through Tg the p process is continued to be observed until coalescence with the oc process leads to the formation of the oc (3 process which relaxes all of the mean square dipole moment of the chain.

10—,

5N X

' 0-

-5-

495

For many polymers, e.g., poly(vinyl chloride), poly(vinyl acetate), isotactic poly(methyl methacrylate), aromatic polycarbonates, polyethylene terephthalate) and high molecular weight poly(ethylene terephthalate) and high molecular weight poly(propylene oxide), Aea > AB^. However, there are notable exceptions for which Asp > A£a, as is the case for poly(ethyl methacrylate) and atactic and syndiotactic poly(methyl methacrylate). In the latter polymers the relative magnitudes of a and P processes may be reversed by applying pressure to the polymer. This leads to a marked decrease in Aep accompanied by a corresponding increase in Asa showing that oc and (3 processes are inter-related although the mechanisms for motion are entirely different. [For pressure studies see Williams (1979), for a review, Williams (1966 a, b), Sasabe and Saito (1968).] Without the need to discuss data for particular polymers, it appears that most of the features of the dielectric relaxation behavior of amorphous polymers may be understood in terms of the partial reorientation (P process) and total reorientation (a process) of a representative dipolar repeat unit of a chain (Williams and Watts, 1971 a, b, c; Williams, 1979). The group is supposed to have the possibility of a range of local environments (given by the probability °pr) and undergoes, at short times, partial relaxation in a given environment V , say, characterized by a relaxation strength \x2 • q^r and a correlation function Pr(0- Summing over all partial relaxations, the remaining relaxation strength Aa = ^2YJ°pr(i — q$r) is relaxed by microbrownian motions, with the assoiated correlation function a(t)The overall correlation function for group motion is given by

1/r Figure 11-6. Log [/m(Hz)] against reciprocal temperature for the a, p and (a P) absorptions in an amorphous polymer (schematic).

(H-42)

496

11 Dielectric Properties of Polymers

The relaxation map, Fig. 11-6 is consistent with this picture. For T < Ty9 a(£) is so slow that only the P process can be observed. For T ~ Tg, a and P processes coexist in plots of e" vs. log / while for higher temperatures 0a(f) decays faster than the Pr relaxations, giving Aga for certain polyalkyl methacrylates merits further ex-

planation. In these polymers the ester group pendant to the chain undergoes motion fairly independently of the backbone, in the local environments prescribed by neighbouring chains in the solid state, hence most of \x2 is relaxed by this process under normal conditions. On raising pressure, the extent of this motion, but not the average rate of motion, is changed appreciably (Williams, 1966 b) thus As^ decreases and Aea increases. The fact that isotactic polymethyl methacrylate exhibits Aea > As^ indicates that the local motions of the ester group are extensively blocked in this polymer. Pressure is found to have a large effect on the frequency location of the oc process in amorphous polymers. As one example, Fig. 11-7 shows the permittivity and loss

Figure 11-7. e' and e" as a function of log (/// m ) for poly(ethyl methacrylate) at 273.2 K and different applied pressures. Curves 1, 2, 3, 4 correspond to 1, 340, 690 and 1020 bar respectively. (After Williams and Watts, 1971a, reproduced with permission.)

e"xio

log

[f/fj

11.5 Experimental Data for Polymer Systems

curves for the a relaxation in polyethyl acrylate (Williams and Watts, 1971a). As pressure is increased the loss curve moves rapidly to lower frequencies with a slight change in height and no change in shape. The quantity (81og/ m /8P) T lies in the range l - 4 k b a r - 1 , depending on sample temperature and pressure. The constant volume and constant pressure apparent activation energies, Qy and QP lie in the range 100-160 and 160-240 kJ mol" 1 , respectively, giving Qy/Qp ^ 0.70 and raising doubts regarding the applicability of simple free volume theories to such data (see Sec. 11.2.3 above). The master curves for permittivity and loss factor are shown in Fig. 11-8. The loss curve is broad and asymmetrical and is typical of that observed for the dielectric a relaxation in

497

amorphous polymers (see Williams et al., 1972, for a discussion). Although such a master curve may be fitted approximately by Davidson-Cole or Havriliak-Negami functions, a reasonable fit was obtained using the transform of the KWW function Eq. (11-12) above. The continuous curves in Fig. 11-8 were calculated for P = 0.38 and a satisfactory fit is noted. Williams and coworkers found that the a relation in a range of amorphous polymers including poly(vinyl acetate), poly(methyl acrylate), poly(propylene oxide), polyethylene terephthalate), poly(vinyl octanoate), acrylonitrile-butadiene copolymer (40:60), styrene-p-chlorostyrene copolymers and styrene-acrylonitrile copolymers could all be fitted using the frequency transform of the KWW function, with /? values in the

Figure 11-8. Normalized master curves for permittivity and loss for poly(ethyl acrylate) as obtained from the data of Fig. 11-7. •, • and • correspond to 1, 340 and 690 bar respectively. The continuous curves were calculated using the KWW function with 0 = 0.38. (After Williams and Watts, 1971a, reproduced with permission.)

498

11 Dielectric Properties of Polymers

range 0.4-0.7 but with most around 0.5 (Williams et al., 1971; Williams and Watts, 1971 a, b; Williams et al., 1972; Cook et al., 1975). The fact that the KWW function gave a reasonable representation of the dielectric a relaxation in polymers of such different chemical structures was intriguing and became the object of much attention (see, e.g., Williams, 1982 a, b, 1985; Ngai et al., 1986; Ngai et al., 1987 a, b; Rendell and Ngai, 1985). In addition, it was found that the KWW function also gave a reasonable representation of other relaxation phenomena for amorphous polymers in the glass-transition region including mechanical, NMR and volume relaxations, quasi-elastic light scattering and fluorescence depolarization (see, e.g., Williams, 1982a, b, 1985, 1989; Ngai et al., 1986; Matsuoka et al., 1985; Rendell et al. 1987; and references therein). The common occurrence of the KWW form for different relaxation processes studied in amorphous polymers also extends to studies of smallmolecule glass-forming systems (see, e.g., Williams, 1975; Williams and Hains, 1971, 1972; Williams and Crossley, 1977) and to ionic systems (see, e.g., Moynihan et al., 1976). The ubiquity of the KWW (or stretched exponential) function in molecular dynamics is presently a major challenge for the theorist. The cooperative nature of motions in polymeric and nonpolymeric systems which are glass-forming provides a common basis for all models of the dynamics of low frequency processes but the precise formulations which lead to KWW behavior is still a matter of discussion [see, e.g., Williams (1985); Blumen (1987), for a consideration of different models including the defect-diffusion models, kinetic Isinglattice models and the continuous timerandom walk models of Shlesinger and Montroll (1984)]. An interesting approach is taken by Ngai and Rendell (Ngai et al.,

1986, 1987 a, b; Rendell and Ngai, 1985) who take as the starting point the following rate equation for a relaxing quantity h9 say, (11-43) For W(t) equal to a constant, a single relaxation time process occurs. For W(t) having the following form W(t)=

Wo(coct)-n

(11-44)

i.e., a time-dependent rate coefficient with parameters coc and n, then for the isothermal case integration of Eq. (11-43) using Eq. (11-44) gives the KWW form (f/r)1""]

(11-45)

where (11-46) and T 0 = Wo *. According to Ngai and coworkers as n decreases away from unity, the degree of complexity of the relaxation increases. The application of Eqs. (11-43) to (11-46) to isothermal and non-isothermal relaxations in glass-forming polymers has met with success. Ngai and coworkers have reasoned that Eq. (11-43) should be regarded as the starting point for their "coupling model", and they have outlined different models which may lead to this equation (Ngai et al., 1986, 1987a). We note that Ngai and Rendell (1987) have used the KWW approach to fit dielectric data obtained by Adachi, Okazaki and Kotaka for poly(2,6-dichloro-l,4-phenylene oxide) in chlorobenzene. As the polymer concentration is raised from 2.5 to 56.9% the loss peak moved from ~ 107 Hz to ~10 3 - 5 Hz and broadened in the KWW sense. The polymer concentrations were below the entanglement condition for all solutions studied. Using a deconvolution method the KWW parameter /? was deter-

11.5 Experimental Data for Polymer Systems

mined for each solution and was found to vary systematically from 1.0 in dilute solution to 0.38 for the most concentrated solution. Of the many studies of the dielectric oc process in amorphous polymers that of Mashimo et al. (1982 b) for poly(vinyl acetate), which covered the range 10" 6 to 10 + 6 Hz, is perhaps the most comprehensive. In that case the loss curves were wellfitted by the Havriliak-Negami equation.

499

The permittivity values of dipolar amorphous polymers lie in the range 2-20. Recently, it was shown (Furukawa et al., 1986) that amorphous copoly(vinylidene cyanide/vinyl acetate) (VDCN/VAc copolymers) have exceptionally high permittivities due to the high polarity of the paired nitrile groups in the chain. Figures 11-9 and 11-10 show the permittivity and loss spectra for the alternating 1:1 copolymer. At the lowest frequencies the rising

190° C \ \ 195°C 3-

185°C \ 180

Iog6' \

\

\

•A\\

V

2-

175°C V

Figure 11-9. Log e' against log [/(Hz)] for the VDCN/ VAc copolymer at different temperatures. (After Furukawa et al., 1986, reproduced with permission.)

170°C 1-

-2

-1 log [MHz)

log 6

Figure 11-10. Log e" against log [/(Hz)] for the VDCN/ VAc copolymer at different temperatures. After Furukawa et al., 1986, reproduced with permission.)

01

1

I

CNI

-1

0

I

1 log If (Hz)]

I

2

I

3

500

11 Dielectric Properties of Polymers

loss factor indicates the presence of a conductivity process (the slope of the plot is — 1) while the rising permittivity indicates electrode polarization due to charge migration. Treating the sample as a complex capacitance shunted by a resistance which is in series with an equivalent capacitance Cel due to electrode polarization, with £e! oc (i co)~m where m is an empirical exponent, Furukawa et al. were able to resolve the different contributions to the observed permittivity and loss. Figure 11-11 shows, as one example, the result at 190 °C and the fit is extremely satisfactory. From Fig. 1111 we see that the static permittivity at this temperature is ~ 120 which is far larger than the values obtained for other dipolar polymers [poly(vinyl acetate) ~ 9.5, poly(acrylonitrile) ~43]. The loss curve may be fitted using the Havriliak-Negami equation with a = 0.8, b = 0.6 [see Eq. (11-9) above]. The frequency-temperature location of this process (a process) followed the WLF equation described above, with Tg = 170°C, C± = n.09 C2 = 5i. These data for a copolymer serve to illustrate how a dipole relaxation process which is partly obscured by conductivity and electrode processes may be extracted from the

experimental permittivity and loss curves. We note that Furukawa et al. (1986) comment that this copolymer exhibits a strong piezoelectric activity which is proportional to the remanent polarization induced by a poling field when the latter is very large due to the large static permittivity of the material. Up to this point we have discussed, mainly, the a relaxation in amorphous polymers. The (3 relaxation is always extremely broad with a half-width in the frequency plot of the loss-factor of 4 - 6 decades. It seems almost certain that this unusual broadness arises from individual local motions occurring with very different rates in a variety of local environments [see Eq. (11-42)]. A surprising feature of the P relaxation is that it follows the Arrhenius law, Eq. (11-15), in all polymers. Pressure is found to have only a small effect on log fm for the P process in all cases studied (see, e.g., Williams, 1966 a, b, 1979, Williams and Watts, 1971c). The effect of pressure on the strength Aep is more interesting. For polyethylene terephthalate (Williams, 1966 a) a 20% decrease was observed in e^ for 2kbar applied pressure while for poly(vinyl chloride) (Williams and Watts,

Figure 11-11. Comparison of the observed data (o, •) with the fitted curves for the VDCN/ VAc copolymer at 190 °C. The line - • - • - • - indicates the curve for the dipole relaxation process. (After Furukawa et al., 1986, reproduced with permission.) 1

2 log [//(Hz)]

3

11.5 Experimental Data for Polymer Systems

1971c) a 50% decrease was observed for 2 kbar applied pressure. For poly(ethyl methacrylate) (Williams, 1966 b) a decrease similar to that for PVC was noted for T < Tg but for T ~ Tg, Aep decreases by a factor of three for 2 kbar applied pressure. In all cases, the extent of motion is being decreased by application of pressure. The qualitative difference between the response for PEMA and PVC reflects the fact in PEMA the motion of the ester side group is being suppressed by application of pressure. With regard to a detailed mechanism for motion for the P process, simple models are precluded since they invariably lead to single relaxation time processes. The (3 process is also observed in non-polymeric glass-forming liquids, as was first made clear by Johari and Goldstein (Johari and Goldstein, 1970; Johari, 1987, and references therein) so its mechanism is not unique to polymer glasses. It seems likely that the dielectric (3 process in all glassforming systems arises due to limited reorientational motions in a variety of temporary local environments (Williams and Watts, 1971a) as expressed by Eq. (11-42) above. 11.5.4 Experimental Results for Crystalline Polymers 11.5.4.1 Introduction

Partially-crystalline polymers may broadly be classified into two categories, (A) those which have a degree of crystalline, X say, which does not exceed 50% and, (B) those which have X above about 80%. Melt crystallized polyethylene terephthalate, nylons 6-6 and 6-10, fall into category A while high density polyethylene, isotactic polypropylene and polyoxymethylene fall into category B. For materials of type A the amorphous phase may be glassy or mobile, depending on sample temperature,

501

and would be expected to exhibit multiple relaxations of the kind observed for amorphous polymers. For materials of type B, the amorphous phase is of a different kind from that in bulk amorphous polymers and is sometimes regarded as being disordered regions on the surface of crystals and within crystals. It is therefore expected that the dielectric relaxation behavior of these two categories of crystalline polymer will be quite different. The dielectric properties of several crystalline polymers have been reviewed and discussed in detail (McCrum et al., 1967; Williams, 1979,1982 a, b, 1989) and the reader is referred to those accounts for comprehensive documentation and analysis. The interest in the dielectric properties of crystalline polymers stems from their applications as high quality electrical insulation [poly(ethylene), poly(tetrafluoroethylene)], as piezo and pyroelectric materials [poly(vinylidene difluoride) and related copolymers] and as the plastic base for magnetic tapes (polyethylene terephthalate). We summarize briefly the essential dielectric properties of some of these polymers. 11.5.4.2 Polymers of High Degree of Crystallinity

Polyethylene occupies a special position among crystalline polymers owing to the simplicity of its chemical structure and its wide application for cable insulation (e.g., submarine telephone cables). Since the CH 2 group has only a very small dipole moment, pure polyethylene samples may have loss factors down to the practical limit of measurement (£"<10~ 5 ). However, practical materials are always slightly oxidized, with carbonyl groups decorating the chain and revealing intrinsic motions which occur in the disordered and crystalline regions. These motions may be studied

502

11 Dielectric Properties of Polymers

using dielectric relaxation spectroscopy and extensive accounts of the early experimental work are available (McCrum et al., 1967; Ishida, 1969; Wada, 1977; Hoffman etal., 1966; Ashcraft and Boyd, 1976). Chlorinated poly(ethylenes) (Matsuoka et al., 1972; Ashcraft and Boyd, 1976) and copolymers of ethylene and carbon monoxide, containing 0.5-1% carbon monoxide comonomer have also been studied. In all cases, three relaxation regions, a, P and y in decreasing order of temperature are observed. The definitive studies are those of Ashcraft and Boyd (1976) which gave extensive data for lightly oxidized or chlorinated low density and high density polyethylenes. In all cases, the carbonyl or C-Cl groups act as a probe on the motions which occur even in the pure polymer. The essential findings are as follows. (i) The occ process is assigned to a rotation-translation of chains in the crystalline lamellar. Support for this mechanism is provided by the observation that the a process disappears on melting oxidized linear polyethylene. A rotation-translation model has been modelled by Frohlich (1958) and by Hoffman et al. (1966) and it is shown that for short alkane chains, up to ~ 40 A long, the motion is that of a rigid rod while at higher chain lengths, corresponding to those for lamellar crystals, the chain reorientation is assisted by chain twisting, leading to a constant apparent activation energy for the process. Subsequently, Mansfield and Boyd (1978) made conformational energy calculations for the twistassisted reorientation process. They find that reorientation is accomplished by a twisted region ~ 1 2 C H 2 units long which propagates smoothly along the chain and hence across a crystal lamellar. In view of the detailed experimental data available for the a relaxation in the different polyethylenes, and the success in representing

the molecular-length dependence of the relaxation parameters (Hoffman et al., 1966; Ashcraft and Boyd, 1976; Mansfield and Boyd, 1978), the mechanism for this process is perhaps the best understood of all those studied in crystalline polymers. (ii) The P process is assigned to the motions of chain segments in the "amorphous" regions of the polymer, and is analogous to the a process observed in amorphous polymers, its activation energy being characteristically large (220 kJ mol" 1 ). The p process is more intense in low density polyethylenes. (iii) The y process is thought to be a composite process containing contributions arising from the amorphous and crystalline regions (Hoffman et al., 1966). Support for this possibility is provided by Kakizaki et al. (1985) whose studies of the dielectric, mechanical and NMR relaxations in linear polyethylene indicate that the y process arises from motions in an interlamellar amorphous region and at the lamellar surface. However, Ashcraft and Boyd (1976) conclude that the dielectric y process originates in the amorphous phase. As is the case for the P-process in amorphous polymers, the y process in oxidized and chlorinated polyethylenes is extremely broad in the frequency domain. Ashcraft and Boyd (1976) suggest that this arises because the chain segments in the amorphous phase find themselves in a variety of local environments, each with its local barrier system to reorientation. As a result a broad distribution of activation energies is presented for the y relaxation leading to a very broad dielectric absorption. This interpretation is very similar to that given above for the P relaxation in amorphous polymers (see Eq. (11-42). The dielectric properties of polymers having high degrees of crystallinity, such as polyoxymethylene, polytrimethylene oxide

11.5 Experimental Data for Polymer Systems

and polytetramethylene oxide, have been studied extensively (see McCrum et al., 1967; Wetton and Williams, 1965). Two relaxation processes, termed p and y in descending order of temperature, are observed. In a classic study, Ishida et al. (1965) showed that a single crystal mat laminate of polyoxymethylene exhibited only the y process while the same specimen, on melting and recooling to form a melt-crystallized bulk sample, exhibited both y and (3 processes. This may be taken as evidence that the y process arises from motions of the ether dipoles in defect regions associated with the crystals and in the disordered regions on the lamellar surface. The P process is due to motions of the amorphous regions beyond the crystal surfaces; but it is likely that such motions are strongly hindered by the presence of the crystals, as is the case for polyethylene terephthalate (see below). As the sample temperature was lowered from 35 °C to — 75 °C the P-loss peak in bulk polyoxymethylene broadens markedly, indicating that the motions in the amorphous regions are increasingly constrained by the crystalline regions. These studies demonstrate that different motional processes in crystalline polymers may be detected using dielectric relaxation spectroscopy and that the influence of the crystalline regions on the motions in the amorphous regions are manifested as a broadening of the loss process as sample temperature is decreased. The effect of pressure on the dielectric a and y relaxations has been described by Sayre et al. (1978), while Gilchrist (1978) and Yano et al. (1977) have given dielectric data for poly(ethylenes) at liquid helium temperatures. The important feature of the latter results is that facile motions of hydroxyl groups present as additives in the pure polyethylene are detected as low fre-

503

quency-low temperature dielectric absorption peaks. 11.5.4.3 Polymers of Medium Degree of Crystallinity Of the many polymer materials which may be crystallized from the melt to give samples of only medium degree of crystallinity, the nylons and polyethylene terephthalate are of particular interest due to their continued commercial importance. The dielectric properties of these polymers have been described by McCrum et al. (1967). For the nylons, multiple relaxation regions are observed, the a process being due to motions of the amide groups in the amorphous regions, and the shape of this process broadens markedly as sample temperature is decreased which indicates that the crystalline regions hinder the motions in the amorphous regions, as we have discussed above. In an important study Boyd and Porter (1972) found that the dielectric oc process in nylon 6-10 was continuous on going from the crystalline solid into the melt, demonstrating that the process is occurring in the amorphous regions in the solid. The strength of the dielectric a process increased by a factor of two on melting the polymer which is predicted since more amorphous phase is formed at the expense of the dielectrically-inactive crystalline phase. Further work by Yemni and Boyd (1979) for the odd-numbered nylons, 77 and 11, showed that the dielectric strengths of the a processes in these materials were larger than those for the even-numbered nylons, which is consistent with the fact that the dipole moment components of the chain perpendicular to the chain-axis are parallel to each other in the odd nylons and oppose each other in the even nylons. Such studies demonstrate that the dielectric relaxation behavior is sensitive to

504

11 Dielectric Properties of Polymers

chain structure and conformation in a way which is not found for dynamic mechanical relaxation. The motions of the amide groups in the nylons are made complicated by the inter-molecular effects of hydrogen bonding. One linear polymer which is of medium crystallinity but does not have hydrogen bonding is polyethylene terephthalate (PET). It was first studied over wide ranges of frequency and temperature by Reddish (1950) who found that a melt crystallized sample gave well-defined a and (3 relaxations where the a process is due to the large-scale microbrownian motions (dynamic Tg process) of the chains in the amorphous regions and the P process is due to local segmental motions of the chain in the glassy amorphous phase and is observed below the Tg of the polymer. A full account of the early studies of PET, for different degrees of crystallinity are given in McCrum et al. (1967). Essentially, it is found that the degree of crystallinity (X) may be varied in the range 0-0.5 by quenching the melt to room temperature, giving amorphous material, then heating to different temperatures to induce different degrees of crystallinity (quench-anneal method). Alternatively, by cooling from the melt slowly into the crystalline state, melt-crystallized materials is obtained. The dielectric data show that as X is raised from 0 to 0.5, the dielectric oc process broadens, moves to lower frequencies and decreases in its relaxation strength (due to removal of amorphous phase). However, the a process in the wholly-amorphous material and the a process in the crystallized material (X ~ 0.5) have been shown to be qualitatively different as a result of the following experiment. Tidy and Williams (1978) (see Williams, 1979) prepared an amorphous sample of PET by quenching a sample from the melt. Rapid heating to 106.7 °C gave an amorphous material

above its Tg whose loss spectrum could be measured in the a relaxation range. Keeping the sample at this temperature resulted in its crystallization over a period of time. At this temperature approximately six hours were required to crystallize the sample to the extent that the material appeared fully-spherulitic in the optical microscope. The dielectric spectra were recorded for different times of crystallization. It was found that as the normal a peak reduced in magnitude, due to crystallization, its frequency, location and shape did not change. In parallel with the disappearance of the a peak, a broad, lower frequency process appeared (a' say) and grew as crystallization proceeded. In the later stages of crystallization, the magnitude of this peak decreased with time, indicating secondary crystallization. The peak of the a' process occurred at ~ 2 units of log 10 frequency lower than that for the normal a peak. This experiment suggests that the motions of the chains in the crystallized specimen (a' process) are hindered by the presence of the crystallites, giving a lower frequency (larger average relaxation time) for the motion and a marked increase in the breadth of the loss peak where the latter indicates that the motions occur in a variety of local amorphous states influenced by the crystallites. It is interesting to note that the crystallized sample appears to be all spherulites, as observed in the optical microscope, so the "abnormal" amorphous phase is contained entirely within the spherulites. The dielectric data show there is an intimate connection between the disordered and ordered regions within spherulites. More recently, Coburn and Boyd (1986) studied nine specimens, ranging from X = 0 to X = 0.62, over wide ranges of frequency and temperature. The partially crystalline samples were prepared by heat-

11.5 Experimental Data for Polymer Systems 1

4.7

1

1

1

1

I

I

I

I

I

I

I

I

I

I

I

505

I

O 1hz 0 10hz a 100hz ^ 1khz O 10khz a 100khz

-

4.5 ~

~~4.3 a

Jillrr

o

£3.9 o .« 3.7 Q 3.5 3.3 I

-200

I

-160

I

I

-120

[

I

-80

I

-40

I

I

I

I

0

I

I

80

I

I

120

I

I

160

Temperature (°C)

(a)

0.12

I

-

0.08 =r

Figure 11-12. (a) Curves of permittivity e' and (b) curves of loss factor e" against temperature at given frequencies for a partially crystalline sample of polyethylene terephthalate (X = 0.38). (After Coburn and Boyd, 1986, reproduced with permission.)

0.04 -

0.00 -200

(b)

-160

-120

-80

-40 0 40 Temperature (°C)

ing the glassy amorphous film to different temperatures above Tg and allowing crystallization to go to completion. The samples of the highest degree of crystallinity were prepared by isobaric crystallization from the melt at high applied pressure, followed by annealing at 100 °C stabilise their density. Plots of e" vs. T at chosen frequencies clearly revealed the a and (J relaxations in each specimen. Figure 11-12 shows, as

120

160

one example, the permittivity and loss data as a function of temperature for a sample having X = 0.38. The entire set of data for all samples was fitted numerically using a Havriliak-Negami function for each process. Such data-analysis provides an example of what can be presently achieved using modern numerical methods. The strength factor Ae, the relaxation frequency fm and the Havriliak-Negami parameters, a, b,

506

11 Dielectric Properties of Polymers

[Eq. (11-9) above] were obtained for a and (J processes for all nine samples over a wide range of temperature, thus allowing conclusions to be drawn based on quantitative data. The essential observations are as follows. The relaxation strengths Asa and Aep were shown to decrease approximately linearly with increase in sample density (Q in the range 1.34 to 1.4 gem" 3 ). Zero relaxation strengths for both processes would appear to occur at crystallinities, X, comparable with that for the single crystal, hence both processes are considered to arise from the amorphous regions of the materials (Coburn and Boyd, 1986). Using a composite model for the dielectric permittivity of a partially-crystalline polymer, as developed by Boyd (1983), it was possible to calculate the permittivity of the amorphous phase in each sample, and hence use the Onsager-Kirkwood equation to calculate the correlation factor [see Eq. (11-24) above]. The composite model leads to upper and lower bounds for the measured permittivity according to the equations By + 2 8H

1 1 / 1 eL 3 \ e v

(11-47 a)

2 ,

(ll-47b)

where s^ and eL are the upper and lower bounds of the measured permittivities defined by — v)&2 1

v

1-v

—=—+——

(11-48 a) (11-48 b)

where &x and s2 are the dielectric permittivities of the amorphous and crystalline phases, respectively, and v is the volume fraction of the amorphous phase.

Proceeding in this way, the experimental values for the g-factor for the (3 relaxation were in the range 0.05-0.07 for all nine samples while those for the combined a and (3 processes were in the range 0.230.29, depending upon the sample and its temperature. The low value for g for the (3 process seems not to be due to orientational correlations between dipoles, but is due to the possible inapplicability of the Onsager-Kirkwood equation to the glassy state. The derivation of this equation assumes that individual dipoles may access 471-solid angle of orientation space, whereas in the glassy state the dipole motions are spatially limited - see Sec. 11.5.3 above for a discussion of oc and (3 processes in amorphous polymers. The consistency of the g-values for the combined oc and (3 processes is impressive. According to Coburn and Boyd, the relatively low value of the g-factor (-0.31) for the a and (3 processes in all samples arises from intramolecular correlations where the dipole moments of the two ester groups on each ring partially cancel, the degree of cancellation depending on the conformation (cis or trans) of the two groups with respect to each other. The g-factors for the semicrystalline specimens were all significantly lower than that for the amorphous sample (# a ~ 0.33-0.29 in the range 60-100°C). This property seems to arise from the restriction on conformations and spatial freedom for the chains due to the presence of crystals to which they are directly connected. Thus, the static permittivity data of Coburn and Boyd indicates that the amorphous phase in the partially crystalline PET is influenced in its equilibrium properties by the presence of the crystallites. The a relaxation in the amorphous material is far narrower, and more skewed, in the Davidson-Cole or KWW sense, than the fairly symmetrical and broad a' process

11.5 Experimental Data for Polymer Systems

observed in the partially crystalline specimens. There is a marked difference in the relaxation map for the a process between that for the amorphous sample (a) and those for the crystalline samples (a'). However, according to Coburn and Boyd, as X is raised, the locus is displaced to lower temperatures, indicating significant changes in the nature, and hence the dynamics, of the amorphous regions within the spherulites as the degree of crystallinity is raised. The samples prepared by crystallization under pressure, which have the highest crystallinities of all samples measured, give relaxation maps for the a' process which are rather different from those for all the other partially-crystalline samples in that the slope of the plot log/ m vs. 1/T is lower than that for the others. It seems possible that the amorphous regions in these pressure-crystallized samples are less constrained by the crystals than are the corresponding amorphous regions in the isothermally-crystallized specimens (Coburn and Boyd, 1986). Further support for this suggestion is obtained from the earlier mechanical and structural work of Illers and Breuer (1963). These detailed studies of polyethylene terephthalate show that the dielectric method enables information to be obtained on the nature, structure and the dynamics of the amorphous regions within a partially crystalline polymer. It is evident that the amorphous phase is abnormal in the sense that the crystallites limit and slow-down the motions in the amorphous regions and also broaden the loss process, indicating that there are a wide range of environments in which the cooperative motions of the chains, which lead to the a' process (and hence the Tg process), occur. The dielectric data may be complemented by NMR data, which also may give information on different ranges of mobility in

507

the amorphous phase in polyethylene terephthalate, as has been described by English (1984) and discussed by Coburn and Boyd (1986). 11.5.5 Experimental Data for Liquid Crystalline Polymers 11.5.5.1 Introduction

Thermotropic liquid crystalline (LC) polymers having the mesogenic (liquidcrystal-forming) groups in the main chain and/or in the side chain of a linear polymer have been the object of intense interest during recent years. The interest in main chain LC polymers arises from the fact that they provide anisotropic solid materials with a high modulus in the direction of orientation and which have a low coefficient of expansion. Their applications are in the form of sheet material, or film or fibre material, with good mechanical, electrical and thermal properties. From the dielectrics point of view, the interest would be to study the different thermal transitions, but only a small amount of work has been done (Laupretre et al., 1985) for the main chain polyester. On the other hand, the interest in side chain LC polymers concerns their possible use as materials, in the form of films (1-100 jam thick) for optical information storage (Shibaev et al., 1983; Coles and Simon, 1983, 1985; Eich etal., 1987; Kaempf, 1987; Schmidt, 1989; McArdle, 1989), for nonlinear optics (Williams, 1983 a; McArdle, 1989; Chemla and Zyss, 1987) and for optical elements such as diffraction gratings, optical waveguides and Fresnel zone plates (Eich et al., 1987; McArdle, 1989). All applications of LC side chain polymers require a material to be aligned macroscopically in the homeotropic (H), homogeneous (Hs) or planar (P) forms. Figure 11-13 illustrates the H- and Hsaligned materials by showing the orienta-

508

11 Dielectric Properties of Polymers

(a) Homogeneous Alignment

(b) Homeotropic Alignment Figure 11-13. Schematic illustration of the organization of the mesogenic groups in homeotropic and homogeneous alignments of a nematic LC material, n indicates the director axis for the LC phase. The main chains of the polymer are not shown.

tion of the director n, for a non-chiral liquid crystal, with respect to the geometry of the film. Both H and Hs samples are monodomains of the LC phase (shown as nematic phase in the figure) and differ in that n || x y-plane for the Hs sample and nixy-plane for the H sample. The planar arrangement (not shown) has n\\xyplane, like the Hs form, but the local directors are random in the x y-plane, unlike the Hs form. All three forms may be achieved for a film of a LC side chain polymer using methods which have been developed for low molar mass liquid crystals (see Clark etal, 1980; Clark, 1981; Pranoto and Haase, 1983; McArdle, 1989). These involve either surface treatment of the substrate to which the LC material is attached or application of dc or ac electric fields or magnetic fields. Having achieved a film having a particular orientation, full advantage may be taken of its anisotropic optical properties. Thus, for example, data storage may be achieved with H- or Hs-aligned films using a focussed laser beam which

heats the material locally, producing a scattering texture. This is known as the thermorecording method (see Shibaev et al., 1983; Coles and Simon, 1985; Simon and Coles, 1986; McArdle etal., 1987; McArdle, 1989; Schmidt, 1989). An alternative method is to include photoactive groups in the side chain (as mesogenic groups) or as guest molecules in the LC polymer host. In these cases, a polarized focussed laser beam acting on an aligned film will produce a birefringent, dichroic image on the film through the process of photobleaching. This optical recording method for LC side chain polymers has been described by Eich et al. (1987) (Eich and Wendorff, 1987; and by Anderle et al., 1989). The optical recording process appears to involve photoisomerization of groups via angular dependent photoselection and is applicable to both LC polymer films and glassy polymer films. The precise details of the mechanism of this optical recording method are still to be established for LC films but the mechanism of angular dependent photoselection has been demonstrated for a photochromic glassy polymer (Jones etal., 1989; Kozak and Williams, 1989). A feature of both the thermorecording and optical recording methods, as applied to LC polymers, is that reversible digital, analogue or holographic optical storage at high densities may be achieved (see Schmidt, 1989). It is important to remember that the electric-field induced switching of a LC material, of low molar mass or polymeric, is caused by the dielectric torque, which is proportional to 5s = s^ — s±9 where £|l and e ± are the dielectric permittivities measured parallel with and perpendicular to, respectively, the director axis w, of the liquid crystal (assumed to be axially-symmetric for simplicity). e(| and £ ± are frequency dependent for a LC polymer mate-

11.5 Experimental Data for Polymer Systems

rial and a dielectrically-positive material will give dispersions and absorptions as indicated schematically in Fig. 11-14. For such a material the crossover frequency / c for the permittivity will determine the macroscopic alignment achieved by the mesophase when a strong ac electric field of frequency fd is applied. H-alignment is obtained if fd / c . This is the "two-frequency addressing principle" which is used extensively in display technology for low molar mass liquid crystals. The dielectric relaxations of £|| and e ± show that the dipolar groups in the LC phase undergo anisotropic molecular orientations within the local LC potential. The nature of such motions may be studied if H and P (or Hs) films can be prepared. Therefore, the dielectric properties of LC side chain polymers are of interest since they form a part of the origin of the macroscopic orientational processes (switching processes) and they give information on molecular motion of the dipolar groups. In addition to this, there has recently been an interest in preparing thin films of ferroelectric materials based on low molar mass or polymeric liquid crystals which contain

log [MHz)]

Figure 11-14. Schematic illustration of the frequency dependence of the real and imaginary components of £|l (co) and s ± (co) for a dielectrically-positive LC material. Note the crossover frequencies fc and / c ' for permittivity and loss factor, respectively.

509

chiral groups. Such films show promise as fast optical switches for display technology. In the account below we summarize briefly some of the features of the dielectric properties of selected LC polymer materials. 11.5.5.2 Non-Chiral Liquid Crystalline Side-Chain Polymers

The earliest studies of the dielectric properties of LC side-chain polymers were made by Kresse and co-workers (Kresse and Talrose, 1981; Kresse, 1982; Kresse and Shibaev, 1983, 1984) for polymers of the structure

(1)

where ,R = H (acrylate polymer), R = (methacrylate polymer). For ,R = H the polymer formed a nematic (N) LC phase and had the transitions gN323N360I, where gN indicates glassy nematic. For R = CH3 a smectic (S) phase was formed. The transition temperatures were gS333S397I, where gS and I indicate glassy smectic and isotropic phases, respectively. All transition temperatures are in Kelvin. A well-defined relaxation process was observed for unaligned or partially aligned samples of both polymers, with relaxation frequencies in the range 1 to 105 Hz and, for a given temperature, the motion in the (smectic) methacrylate polymer is substantially slower (~ 3 decades) than that for the (nematic) acrylate polymer. A further feature of their data is that log fm shows only a slight increase on going from N -> I for the acrylate polymer, whereas Alog/ m = 0.85 for the acrylate polymer at the S -> I transition. Subsequently, Zentel et al. (1985) studied a num-

510

11 Dielectric Properties of Polymers

ber of unaligned acrylate LC polymers and observed, 5, a, p, yx and y2 processes in their dielectric experiments. They have assigned each process to particular motions of the dipolar groups. Further studies were made by Attard et al. (1986) for a siloxanechain polymer which had the following structure

(2)

gN 273 N 3261. Similar results to those obtained by Kresse and Talroze (1981) and Kresse et al. (1982) were obtained in that one broad process was observed in the LC state which narrowed on going into the isotropic state, and the process showed a slight increase in relaxation frequency at the transition. However, it is not possible to analyze such data to reveal the underlying relaxation processes. That may only be achieved by studying the dielectric properties of materials prepared in different states of macroscopic alignment. Systematic studies of the dielectric properties of LC side-chain polymers in different states of alignment have been made by Haase and co-workers (see Pranoto et al., 1984; Haase and Pranoto, 1985; Haase et al., 1985; Pranoto et al., 1986; Bormuth and Haase, 1987, 1988; Bormuth et al., 1987; Haase and Pfeiffer, 1990, and references therein) and by Williams, Attard and co-workers (see Attard et al., 1986; Attart and Williams, 1986a-e; Attard et al., 1987a-d; Attard, 1986; Araki et al., 1988; Kozak et al., 1989 b; Attard et al., 1988, 1989; Williams et al., 1990; Attard and Araki, 1986; Araki and Attard, 1986; and references therein). It is not possible to give a complete description of all these studies here. Instead, we consider representative examples and

initially describe briefly the study by Attard and Williams (1986d) and Attard et al. (1987 c, 1988) of the following smectic LC side-chain polymer CH 3

si-of O-(CH 2 ) 8 -O-<^

(3)

where n ~ 35. The glass transition temperature Tg = 274 K and the clearing temperature Tc = 360 K. A 100 jim thick sample was aligned homeotropically using 300 V (root mean square) at 3 kHz applied to the melt sample followed by cooling into the LC phase with the voltage maintained. The H-aligned material was optically transparent, the unaligned (U) sample was turbid. Figure 11-15 shows plots of G/co(= &" Co) against log 10 [/(Hz)] for the U and the Haligned samples. Here C o is the geometrical capacitance of the sample. Several features of the relaxation curves are apparent: (i) The U-sample spectra are broad with evidence of a bimodal feature while the Haligned sample spectra are close to those for a single relaxation time process (halfwidth ~1.4 compared with 1.14 for the SRT process). (ii) a^ax is increased by a factor of ~ 2.3 on going from U -• H-aligned sample but the integrated areas below both curves, at a given temperature, are not very different. (iii) log/ max (observed) for each sample increases rapidly with increasing temperature at a rate comparable with that for the oc process in amorphous polymers (see Sec. 11.5.3 above). The apparent activation energy depends on temperature but is near 130kJmol - 1 for both samples in the higher temperature region just below Tc. The variation in loss spectra and the origins of the loss processes may be under-

11.5 Experimental Data for Polymer Systems 15-

10GT

/

2

^

/

Q_

30-{

u 1

I

I,

•Q

/

T

511

3 ,--—\ 20-

343.2K

t

328.2K

3r

320.2K

j |.

309.2K

02

3 log 10 lf(Hz)]

log 10 lf(Hz)]

Figure 11-15. E" - C0( = G/co) plotted against log[/'(Hz)] for unaligned (U) and homeotropically-aligned (H) samples of LC polymer (3) of the text, for different sample temperatures. Curves 1 - 4 correspond to 309.2, 320.2, 328.2 and 343.2 K, respectively. (After Attard and Williams, 1986d, reproduced with permission.)

stood in terms of the results of earlier work with low molar mass liquid crystals (Maier and Meier, 1961; Nordio et ah, 1973; Luckhurst and Zannoni, 1975; Bottcher and Bordewijk, 1978; de Jeu, 1980; Kelker and Hatz, 1980). The dielectric permittivity for a uniaxial liquid crystal (e.g., nematic, smectic A, which are also the simplest forms of thermotropic liquid crystals) is a tensorial quantity s(co) whose principal permittivities are s^{co) and s±(co) where subscripts "||" and "_L" mean measured parallel to or perpendicular to, respectively, the LC director axis n. From Fig. 11-13 if s(co) is measured in the z-direction then the H-aligned sample gives s^(co) and the Hs-sample gives £L(co). A generalization of the theory of the static permittivity of a nematic LC material by Maier and Meier (1961) to the dynamic situation has been made, independent of any assumed model for motions of the mesogenic groups, by Attard et ah (1987d) and by Araki et ah (1988). They obtain the following relations for the principal permittivities

+ (1 - S)

(ll-49a)

G

3kT

[(!-.

(ll-49b)

where e^y and 8 ^ are the limiting high frequency permittivities measured parallel or perpendicular, respectively, to the local director n. G is a constant, S is the local order parameter for the LC phase, /JLX and jnt are the longitudinal and transverse components of the dipole moment fi of the mesogenic group and Fj(w) = l-ico^

[Fj (t)]

(11-50)

where ^ indicates a one-sided Fourier transform [see Eq. (11-4)] and the Fj(t) are

512

11 Dielectric Properties of Polymers

linear combinations of certain time-correlation functions #p m (0 for the orientational motions of the mesogenic groups and are given by

(11-51)

where the Dpm (Q) are Wigner rotation matrix elements expressed in the laboratory frame. Equations (11-49) indicate that four orthogonal dielectric relaxation modes will occur, two for e^ and two for s ±. We denote the modes as 00, 01, 10 and 11 and their relative strengths are given (1 + S/2) • /it2. Pictorial representations of the motions of the axes in Euler space (a, /?, y) of the mesogenic grouping have been given by several workers (see, e.g., Attard, 1986). The 00 mode has a relaxation function given by (\\ S7\ where /? is the polar angle in between the molecular z-axis and the laboratory Z-axis. The well-defined sharp loss peak seen for the H-aligned sample in Fig. 11-15 arises from the 00 relaxation mode, i.e., it is due to the motions of fi{ about the short axis of the mesogenic group. The high-frequency tail to the overall absorption is assigned to the 01 mode. The loss curve for the U-sample is due to a superposition of the four component relaxation modes, suitably weighted. The permittivity of the U-sample corresponds to the average permittivity E(co) which is given by the relation

So from Eqs. (11-49) and (11-53) we see how the four modes contribute to &(co). The broad loss curves for the U-sample (Fig. 11-15) cannot be resolved into its four components without making further assumptions. Attard and Williams (1986d) assumed that the 01, 10 and 11 relaxation

modes all had the same relaxation functions, allowing the loss peak for the Usample to be decomposed into a low frequency process (01, 10 plus 11). The fits of the observed loss data for H-aligned, partially-aligned and U-samples of polymer (3) two component loss curves were satisfactory, and allowed calculations to be made for (S,/iJiid giving the values (0.63, 1.43). Thus the dielectric loss curves could be analyzed using a molecular theory for the anisotropic motions of the dipolar mesogenic groups, without specifying the precise nature of those motions, e.g., small-step diffusion vs. large step-diffusion of mesogenic group reorientations in the LC local potential. Note that the main source of the dipole moment (and hence incremental permittivity) in polymer (3) resides in the mesogenic group, the siloxane backbone having a much smaller dipole moment per repeat unit. However, for the majority of LC side chain polymers this is not the case, e.g., acrylate and methacrylate polymers have ester dipole moments which relax with the motions of the chain backbone and may give an important contribution to dielectric relaxation in addition to the four relaxation modes associated with the anisotropic motions of the mesogenic group. In such cases, an additional term, isotropic in nature, is added to the equations for £|l (co) and s ± (co) as has been explained by Haase and co-workers. Further words of caution are appropriate at this point. Any analysis of the dielectric data for aligned LC polymers should take into account the Kirkwood g-factors, 011 and g ± [see Eq. (11-25)] for the orientational correlations between mesogenic groups, but this has not yet been done. Furthermore, the molecular theories for dipolar group motion in nematic and smectic A LC phases assume that these groups undergo 180° rotations (or "flip-

11.5 Experimental Data for Polymer Systems

flop" motions) in the local LC potential allowing /xx to reverse, and contributing to the 00 relaxation mode. However, it seems physically unlikely that the dipolar mesogenic groups in LC polymers [see, e.g., structures (l)-(3) above] can achieve such a reorientation since the tail of the alkyl group is attached to the polymer chain and, furthermore, it would be necessary for all mesogenic groups to undergo such motions while preserving the director axis. It seems difficult to imagine that such largescale motions, which would of necessity be co-operative, would not lead to a disalignment of the director and hence to the disalignment of a macroscopically-aligned sample. The precise molecular origin of the different relaxation modes observed in dielectric experiments for LC side chain polymers is still a matter of discussion and, in the writer's view, also raises similar questions for the analogous dielectric relaxation processes observed for low molar mass liquid crystals. We note that for the latter materials large static dielectric per-

513

mittivities may be obtained for the nematic phase and for the different smectic phases and their dielectric relaxations occur at low frequencies (de Jeu, 1980; Bat a and Buka, 1981; Kresse, 1982). The molecular structures of the smectic phases indicate well-ordered arrangements of the molecules, especially for the higherorder smectics (Goodby and Gray, 1984) so it seems unlikely that large scale motions such as "flip-flop" motions can occur. It is possible that angular fluctuations of the mesogenic head groups are responsible for the observed anisotropic dielectric permittivities and relaxations. While such motions would not relax all of the available <^2>, the relaxation strengths would be enhanced if the Kirkwood g-factors were greater than unity. Attard et al. (1987c) showed that the plots of log fm vs. 1/T for the resolved 8 and oc processes for polymer (3) were curved, Fig. 11-16, and each obeyed, approximately, the Vogel equation with (B, To) values of (1800 K, 28 K) and (946 K, 238 K) for the

Figure 11-16. Log [fm (Hz)] against 103 T ^ K ) for LC polymer (3). Unaligned material: observed peak (£<•), resolved 8-peak (o), resolved oc-peak (A, £). Homeotropically-aligned material: observed peak (•, |). (After Attard et al., 1987 c, reproduced with permission.) 2.7

2.9

i 1 3.1 i o 3 r -1 IK)

I

3.3

1

I

3.5

3.7

514

11 Dielectric Properties of Polymers

8 and a processes, respectively. The loci tend to merge as T -> Tg, i.e., at the lowest frequencies 1 0 ~ 2 t o l 0 ~ 4 Hz, which means that the complex anisotropic motions of the mesogenic head groups, which couple to motions of main chain and spacer groups, undergo a critical slowing-down as Tg is approached. For T < Tg the relaxational modes are effectively frozen in this the glassy LC phase. Similar information has been obtained from studies of related siloxane-chain LC polymers. It was shown (Araki and Attard, 1986; Attard and Araki, 1986; Araki et al., 1988) that the following polymer CH3 (CH 2 ) 6 -O-/

(4)

[n~35, Tg = 275K, TC(N - • / ) = 314 K] could be aligned homeotropically (H) or planarly (P) by cooling from the melt in the presence of a strong low frequency or high frequency electric field, respectively. Figure 11-17 shows the loss curves for the three forms of the same sample in the dielectric cell. Several points may be noted. (i) The loss curve for the U-sample is broad and featureless with no indication of a high frequency shoulder, in contrast to that obtained for polymer (3) in its unaligned state. (ii) The 5 process obtained for the Haligned sample is similar in shape to that obtained for the equivalent process in polymer (3). This is assigned to the 00-relaxation mode. (iii) The loss curves cross at a common frequency (~ 900 Hz) which is an isosbestic frequency analogous to the crossover frequency for the real permittivity (Fig. 11-14). Assuming that each loss component has a Fuoss-Kirkwood form, the data for the H-

and P-aligned samples were analyzed and the loss curve for the U-sample was calculated using Eq. (11-53). Excellent agreement between the experimental and calculated curves was obtained (Araki et al., 1988, Fig. 9 therein). Attard et al. (1987d) showed that the complex permittivity for a partiallyaligned LC sample of uniaxial symmetry was given by

+ |(1 -Sd)'8±(co)

(11-54)

where Sd is the macroscopic director order parameter which describes the average orientation of the local directors with respect to the measuring field (z) direction S = |<3cos20nz-1>

(11-55)

S = 0,1 and -(1/2) for unaligned, Haligned and P-aligned samples, respectively. For Sd = 0 Eq. (11-54) reduces to Eq. (11-53). Taking the real part of Eq. (11-54) we see that the condition £|( (co) = s\(co) defines the crossover frequency / c and that the permittivity curves for samples of different alignment — 1/2 < Sd < 1 all cross at fc for a given sample temperature. The crossover frequency is an isosbestic frequency. Similarly the condition s'l'l (co) = e"± (co) defines the crossover frequency / c ' for the loss curves for samples of different alignment [e.g., Fig. 11-17 for polymer (4)]. The existence of isosbestic frequencies have been demonstrated by Attard et al. (1987 a), Araki and Attard (1986), Attard and Araki (1986) for LC polymers, and dielectric measurements with the use of Eq. (11-54) have been shown to be a simple and direct means of determining the extent of macroscopic ordering in a partially-aligned LC polymer. We note that fc depends upon sample temperature in a manner similar to that for log fm for the 5

11.5 Experimental Data for Polymer Systems

515

Figure 11-17. e" • Co plotted against log [/ (Hz)] for a H-aligned, P-aligned and unaligned (U) sample of LC polymer (4) at 309.2 K. (After Araki et al., 1988, reproduced with permission.) log [f (Hz)]

and a processes. The ability to achieve Hand/or P-alignments using directing electric fields of chosen frequencies is determined by the value of the crossover frequency at a given sample temperature. In practice the crossover frequency for a LC polymer may be too high to be accessible experimentally from conventional power sources so that while H-alignment may be obtained, P-alignment may not be obtained in cooling from the melt in the presence of an ac field. Further difficulties in achieving H- or P-alignment stem from (i) dielectric heating due to dielectric dipolar losses or conduction losses at the frequency of the ac directing field, (ii) electrohydrodynamic instabilities and flow of material in the directing ac field, (iii) the high viscosity of the polymeric LC phase which slow-down the alignment process. As a result of (ii), homopolymers (3) and (4) may not be aligned in the LC state by directing ac electric fields. However, they may be aligned by cooling from the melt in the presence of a strong ac electric field. Copolymers having siloxane backbones

(Kozak et al., 1989 a; Williams et al., 1990) and a malonate polymer (Kozak et al., 1989 b) may be aligned in the LC state using ac electric fields. The rate of alignment decreases markedly as the temperature is reduced below Tc (Kozak et al., 1989 a). The dielectric relaxation behavior of a siloxane polymer

(5) and its mixtures with the low molar mass liquid crystal. C 6 H 1 3 O-

(6)

have been described recently by Seiberle et al. (1990). The samples were aligned in a directing magnetic field and s^(co) and 8 ± (co) were determined over a wide range of temperature in the frequency range 102 to 107 Hz. A well-defined 8 process was observed in all the mixtures with no evidence for two peaks despite the fact that

516

11 Dielectric Properties of Polymers

log/m((5) values are ~ 104 different for the pure materials at a given temperature. This demonstrates that each kind of mesogenic group [see structures (5) and (6)] moves, on average, at the same relaxation rate in the mixtures. The a process was also observed and, in accord with the result for the siloxane LC polymer, structure (3) above, the 5 and a processes tend to coalesce on lowering the temperature. The dynamics of realignment of the aligned LC from one orientation to another, 10° distant, was also studied using the dielectric method, and was found to follow the relation As(t)

As(oo)

x

/

= tan

t

(11-56)

where
constants of the LC phase. Similarly, Araki etal. (1989b) used DRS to demonstrate the recovery of H-alignment in a partiallyhomeotropic sample of polymer (4) when it was cooled, in the absence of an electric field from the biphasic region back into the nematic LC state. The theory of alignment recovery, and the loss of its efficiency by repeated cycling, was outlined by Attard (1989) and involves the template recrystallization of LC material on an aligned substrate. Williams and coworkers (1990) have studied the realignment behavior of a copolymer having longitudinally-attached and transverselyattached mesogenic groups in the side chains. In this case, a sample is realigned H -> P using a high frequency field, the field is removed and the sample is found to realign P -> H as judged by the change in the dielectric loss spectrum with time. Such memory effects (Araki etal., 1989b; Attard, 1989; Williams etal., 1990) are readily studied by DRS, in a quantitative manner and are not easily studied using other methods, e.g., polarized spectroscopy. While dielectric studies of aligned LC polymers give more information on component relaxations than those for unaligned polymers, considerable information is obtained from the latter studies, e.g., for polymers of the structure (7)

:OO(CH2)m-C

-COCH

(7)

with (Rl9 m, R2) being (H, 2, CH3), (H, 6, CH 3 ) and (CH 3 , 6, C 4 H 9 ) (Vallerien et al., 1989 a). These authors observed 5, a, p and y dielectric processes in these polymers where (i) the 5 process is assigned to motions of the mesogenic group around the polymer main chain, and (ii) the a process is assigned to the dynamic Tg process (cf. oca

11.5 Experimental Data for Polymer Systems

process) of main chain segments, including the resolved dipole moment component of the ester group attached to the main chain, (iii) The |3 process is assigned to motions of the mesogenic group about its long axis and, (iv) the y process is assigned to motions of the terminal butoxy group since it was only observed in the polymer containing that group. Both p and y processes were only observed for T
Broad-band dielectric relaxation spectroscopy has been used to study the molecular dynamics of combined main-chain and side-chain LC polymers (Kremer et al., 1989). One material studied was structure (8) below:

(CH2)6-O

517

€H 2 .CH.C 2 H 5 CH3

(8) The unaligned materials (nine different materials were studied) all exhibited an a process, which was assigned to the dynamic Tg process, and one or more |3 processes, labelled |3m or |3S and which were assigned to rotational motions of the mesogenic groups in the main chain (m) or side chain (s) about their individual long axes. For polymer (8) the (3m and Ps processes are clearly seen in the plots of e" vs. T(K), with peaks evident at - 250 K and 180 K, respectively, i.e., the local motions of main chain mesogenic groups are, on average, slower than those of the side chain mesogenic groups. A feature of these studies is that there is no apparent discontinuity or change in the dielectric loss peak as the oc process moves to higher frequencies through the transformation range from LC phase to isotropic phase (see Fig. 4 of Kremer et al., 1989). If samples were aligned by electric fields or magnetic fields, as we have discussed above in Sec. 11.5.5.2, then marked changes in the dielectric spectra would be anticipated and a 5 process, presently not observed for their materials, might be detected. The polymers studied by Kremer et al. (1989) have a chiral mesogenic head group in the side chain. The interest in chiral LCforming molecules of low or high molar mass stems from the observation that ferroelectric films may be prepared in thin surface-stabilized LCD cells which are subjected to strong electric fields. The LC materials are aligned macroscopically in this way and being a chiral-smectic phase, fast optical switching may be induced by ap-

518

11 Dielectric Properties of Polymers

plied electric fields, leading to many applications in display technology and optical data storage devices. A full description of the dielectric properties of ferroelectric LC materials is beyond the scope of this account, so reference is made to the following recent accounts by Martinot-Legarde and Durand (1981), Osipov and Piken (1983), Osipov (1984), Beresnev et al. (1984, 1988), Levstik et al. (1987), Carlsson et al. (1988), Filipic et al. (1988), and Gouda et al. (1989). These experimental and theoretical studies demonstrate that ferroelectric LC materials are formed by alignment of a chiral smectic phase (S£, S£) in a "bookshelf" geometry [see Vallerien et al. (1989 c) for an illustration of this arrangement] and that their dielectric properties are characterized by: (i) a large spontaneous polarization for the sample, (ii) large static permittivities and two large low frequency dielectric loss processes due to director motions which are described as the "soft" mode and the "Goldstone" mode. These relaxation modes, which are properties of a macroscopicallyaligned film ( ~ l - 1 0 jum thick), have a complicated, but well-defined dependence on temperature and applied external biasing field. As examples of studies of the dielectric behavior of low molar mass ferroelectric liquid crystals we may refer to the works of Gouda et al. (1989), Biradar et al. (1989 a c), Wrobel et al. (1989), Biradar and Haase (1989, 1990), and Vallerien etal. (1989c). The latter authors studied the material C 8 H 1 7 O-

VooC.CH-CH-C2H5

(9)

Cl CH*

which has two chiral centres (indicated as "*") and exhibits chiral smectic A (S*) and chiral-smectic C (S*) phases (crystal

322KSg328KS£338KI). The ferroelectric film was formed between ITO conducting glass plates (separation, 10 |im) coated with polyimide. Parallel rubbing of the glass surface led to the desired bookshelf geometry. Figure 11-18 shows the dielectric loss data, in the vicinity of the S* <-> S* transition, for the sample subjected to a biasing dc voltage of 5 kVcm" 1 . Two loss processes are observed, the high frequency process being the soft mode, the low frequency process being the Goldstone mode. On lowering temperature towards T(S% - Sg) the relaxation strength of the Goldstone mode increases dramatically. Below the transition the soft mode and Goldstone mode both contribute to the total loss curve. The Goldstone mode may be effectively removed by application of a strong biasing dc field, as is shown in Fig. 11-19. According to theory the biasing field unwinds the helix of the chiral smectic phase [see Fig. 1 in Vallerien et al. (1989 c), for an illustration of this effect] and thus quenches the strength of the Goldstone mode and allows the soft mode to be clearly observed. Figure 11-20 shows how the mean relaxation frequency and reciprocal relaxation strength depend upon temperature for the soft mode as the transition region S*<->S£ is traversed. According to continuum theory the soft mode corresponds to the fluctuations of the tilt angle
11.5 Experimental Data for Polymer Systems

519

20 • 328.9 K • 327.9 K © 326.9 K

16

Soft mode

Figure 11-18. e" plotted against log [/ (Hz)] for a low molar mass ferroelectric LC material, structure (9), at three different temperatures; sample subjected to a dc biasing field of 5 kVcm" 1 . (After Vallerien et al., 1989 c, reproduced with permission.) U 5 log [f (Hz)]

200 160-

\

\ 0

o

120-

o oo

o

o oQ

80-

o

o

40-

o (

0

1

i

1

i

2

4

6

€

E(kV cm" 1

Figure 11-19. e" plotted against log [/(Hz)] for a ferroelectric LC material, structure (9) at 326.4 K; sample subjected to dc biasing electric fields from 0 to 8 kVcm" 1 . The insert shows the field dependence of e" at 2.51 kHz. (After Vallerien et al., 1989 c, reproduced with permission.)

520

11 Dielectric Properties of Polymers 400320(a)

fm (Hz)

240-

160-

Figure 11-20. (a) Mean relaxation frequency vs. temperature for the soft mode and Goldstone mode, for different values of the external biasing field, for the ferroelectric LC material, structure (9). The Goldstone mode (x) is only present for the bias field <5 kVcm" 1 , o, n, • and
8 0 - Goldstone mode \

0.16-

(b)

0.12-

0.08-

0.04-

320

324

I 332

328

336

340

T{K)

product fm • Ae is expected to be approximately independent of temperature on both sides of the transition, and this has been shown to be the case for the mixture "DOBAMBC" (see Gouda etal, 1989, Fig. 3 therein). Also it may be shown that for the soft mode 1 f

(11-58)

where a and b are Landau coefficients and yLC is a rotational viscosity coefficient. Hence yLC may be obtained from the experimental data for fm{T) and l/Ae(T). We have discussed the behavior of low molar mass LC ferroelectric materials in some detail since they form the basis of our understanding of the behavior of poly-

meric LC ferroelectric materials. It is difficult to make ferroelectric films of low molar mass chiral LC materials and it may be more difficult to prepare ferroelectric films of their polymeric analogues. Despite the many difficulties of a chemical and physical nature, films of ferroelectric polymers have been prepared and studied (Vallerien et al., 1989d; Pfeiffer et al., 1990). Vallerien et al. (1989d) have shown that ferroelectric films of the chiral main chain-side chain LC polymer, structure (8) above may be prepared between parallel-rubbed polyimide-coated ITO glass plates (separation 10 jam) to which an electric field of 35 kV cm" 1 was applied for 14 hours in the Sg phase directly below the S% -> Sg transition temperature. The dielectric loss data

11.6 Concluding Remarks

(Fig. 3 in Vallerien et al., 1989d) show that the soft mode is observed in the S*, S* and Sf phases and that the Goldstone mode is observed in the Sg phase. Further dielectric work with ferroelectric LC polymers has been conducted with the following material, (10) (Vallerien et al., 1989e).

l CH3

(10) It was shown that the Goldstone mode could be observed in the aligned S£ phase, and the soft mode in the S£ of this material and that a strong biasing dc electric field decreases the strength of the Goldstone mode by at least a factor of four. In a further study Vallerien et al. (1990) measured the dielectric properties of a ferroelectric material whose structure was similar to (10) except the biphenyl group was replaced by an azoxy group in the side chain [cf. structure (8) above]. In this material the Goldstone mode was sufficiently suppressed to allow a soft mode to be observed in both the S% and S$ phases, and it was also shown that fm for this mode exhibited Curie-Weiss behavior around T(S% — S$) (cf. Fig. 11-20 for a low molar mass LC). However, whereas the range of critical slowing-down of the soft mode is ~ ± 5 K about the transition for the low molar mass LC, it is ~ + 40 K for the ferroelectric polymeric LC. Also, whereas the relaxation strength was strongly-dependent on temperature in the transition region for the low molar mass LC, only a small dependence of this quantity on temperature was observed for the polymeric material. These first studies of ferroelectric LC polymers are of great interest and in addi-

521

tion to the use of such materials in storage and display systems, it is envisaged that new applications may emerge, such as piezomaterials from cross-linked elastomers, pressure sensors and an optical molecular gyroscope, as have been mentioned by Kremer et al. (1990). Very recently, Dumon et al. (1990) showed that aligned siloxane-chain LC copolymers which have a mesogenic group in the sidechain terminated by two chiral centres, as in structure (9) above exhibit ferroelectric behavior. They report the fastest switching times yet reported for ferroelectric LC polymer films (300 JIS using 2 V/jim at 73 °C). It is apparent that such materials, despite their chemical complexity and the difficulties associated with the preparation of well-aligned ferroelectric films, show considerable promise, especially as multifunctional materials which may exhibit different dielectric, piezoelectric, pyroelectric and electro-optical effects [see e.g., Buckley et al. (1990) for accounts of multifunctional polymer materials].

11.6 Concluding Remarks It is apparent from the above account that dielectric relaxation spectroscopy (10" 3 to 107 Hz) provides detailed information on the molecular dynamics and physical structure of polymers in solution and in the bulk amorphous, crystalline and liquid crystalline states. Such information complements that obtained using other spectroscopic, scattering and relaxation methods. The modern instrumentation allows accurate measurements to be made quickly, as has been indicated above and is further illustrated by the work of Liu et al. (1990) for trans- 1,4-polyisoprene in the ranges 100-320 K and 10~2 to 104 Hz. We have not been able to discuss some of the

522

11 Dielectric Properties of Polymers

most recent work in detail, e.g., the dielectric study of Boese et al. (1990) of multiarmed star polymers of bulk ds-l,4-polyisoprene of narrow molecular weight in which both normal mode and segmental mode relaxations were observed (cf. Adachi and Kotacka studies described above) and which provides a test of models for chain motions both below and above the entanglement condition. Similarly, we have not discussed the recent dielectric study of the plasticization of poly(ethylmethacrylate) by the non-polar gas carbon dioxide at pressures up to 60 atm by Kamiya et al. (1990). In this study is was shown that remarkable changes in loss behavior occur as the gas pressure is raised, whose interpretation may relate to the earlier studies of this polymer in which oc, p and (a |3) relaxations were demonstrated (Williams, 1966 b). It is evident from these studies that dielectric relaxation spectroscopy is able to provide important and detailed information on the behavior of novel polymer systems. It is also important to emphasize that modern instrumentation allows dielectric studies to be undertaken which were not possible previously. We have indicated above (Sec. 11.5.4.3) that the crystallization process of amorphous polyethylene terephthalate can be followed by dielectric spectroscopy. In those studies, manual balancing of the transformer bridge, and noting the times of measurement, allowed a cross plot to be made which gave loss factor against frequency at different times during the isothermal crystallization of the sample. Such measurements could now be made automatically thus allowing accurate and comprehensive dielectric data to be obtained. The speed of measurement would allow more comprehensive experiments to be conducted (i.e., at different temperatures of crystallization). Thus, it is envisaged that dielectric studies of the crystallization, in

time, will be a new application of the dielectric method. Similarly, it is expected that the dielectric method will be extensively used to study polymerization and cross-linking reactions as they proceed in real time. For example, it is known that bisphenol-A (DGEBA) epoxy resins exhibit dielectric relaxation properties which depend on the structure and molecular weight of the resin (Sheppard and Senturia, 1989). Mangion and Johari (1990 a, b) have shown that the Tg relaxations and sub-Tg relaxations of bisphenol-A-based thermosets cured with diaminodiphenyl methane or diaminodiphenyl sulfone vary markedly as the materials are cured. In a particular important paper, Mangion and Johari (1990 b) have shown how the permittivity and loss factor vary with time as the curing reaction proceeds. sf(co) decreased from a large value, characteristic of a liquid dipolar medium, to a smaller value (ef ~ 4) as the material transformed from a liquid, through a viscoelastic range to a glassy solid. In parallel, sff(co) decreased from a plateau, through to a minimum and then gave a peak at the time when &'(<JO) exhibits an inflexion. These data show the course of the curing reaction can be monitored in real time using dielectric relaxation spectroscopy. Mangion and Johari note that space-charge losses (MW losses) do not appear in their data for their materials, in contrast to studies of other systems where the short-time behavior of s' and e" is mainly concerned with the MW process (Sheppard and Senturia, 1986). Mangion and Johari show how the curves of s' (co) and s"(co) vs. curing time vary systematically with the measuring frequency / = co/2 71 and they discuss the observed time-temperature variations of s'(co) and e" (co) in terms of the relaxation behavior of a system which changes from a mobile liquid to a glass through a viscoelastic range

11.8 References

523

as the curing reaction proceeds. One obsersimilar to that used for integrated circuit vation is that the complex-plane (Colechips. In these connections polyimides feaCole) diagram of e" (CO) VS. e' (co), where co is ture strongly as a result of their low dielecfixed and s' and s" vary as the reaction tric permittivity and loss, their processiproceeds, has a contour which is well-fitted bility into accurately-defined geometrical using the KWW function with exponent in structures and their excellent thermal and the range 0.3-0.35, i.e., the scaling of the mechanical properties. Again the polymer relaxation behavior of a chemically-changhas a passive role in the final product. ing system with time rather than with freTechnical problems with the polyimides inquency gives behavior of the type found for clude the mismatch between the thermal chemically-stable systems in their glassexpansion of polymer with ceramics or transition regions. These studies show that metals, which affect adhesion of elements dielectric relaxation spectroscopy provides in an assembly, and the permeation of a powerful direct method of following the moisture into the polymer, which affects its curing reaction of a system composed of insulation characteristics. These and other dipolar molecules. aspects of polymers as electrical insulation materials are discussed in the articles by Finally we note that a major use of synTummala, Keyes, Grobman and Kapur in thetic polymers is to provide electrical inTummala and Rymaszewski (1989) and in sulation in electrical cables, electrical and Barfknecht et al. (1989). electronic components (e.g., capacitors), circuit boards and integrated circuits. It is fair to say that synthetic polymers are in11.7 Acknowledgements dispensible in such applications, the principal materials being the different polyethylThe author acknowledges the support of enes, poly(vinyl chloride), poly-(styrene), the Air Force Office of Scientific Research polytetrafluoroethylene and different poly(research programme sponsored by SDIO/ imides. Ultra-low loss polyethylene, for 1ST and managed by AFOSR under Conexample, is used in submarine telephone tract F49620-87-C-0111). The United States cables while polyimides may be used for Government is authorized to reproduce circuit boards. In all such applications the and distribute reprints for governmental role of the polymer is a passive one, the aim purposes notwithstanding any copyright being to provide a material of low dielecnotation hereon. tric loss, of high dielectric strength (for high field situations), of accurately defined dimensions and having good thermal and mechanical properties. The traditional needs of the electrical and electronic industries are well-met by the polymers indicated above. In recent years there has been an increasing demand for polymers to be used in circuit boards of all descriptions and for thin-film packaging of electronic circuits in which the conductors and insulators are fabricated together in a manner

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524

11 Dielectric Properties of Polymers

Adachi, K., Hattori, M., Ishida, Y. (1977), / Polym. Sci., Polym. Phys. Ed. 15, 693. Adachi, K., Imanishi, Y, Kotaka, T. (1989a), /. Chem. Soc, Faraday Trans. 1, 85, 1065. Adachi, K., Imanishi, Y, Kotaka, T. (1989b). /. Chem. Soc, Faraday Trans. 1, 85, 1075. Adachi, K., Imanishi, Y, Kotaka, T. (1989c), /. Chem. Soc, Faraday Trans. 1, 85, 1083. Alper, T., Barlow, A. I , Gray, R. W. (1976), Polymer 17, 665. Anderle, K., Birenheide, R., Eich, M., Wendorff, J. H. (1989), Makromol Chem. Rapid Commun. 10, 477. Araki, K., Attard, G. S. (1986), Liq. Cryst. 1, 301. Araki, K., Attard, G. S., Kozak, A., Williams, G., Gray, G. W, Lacey, D., Nestor, G. (1988), J. Chem. Soc. Faraday Trans. II, 84, 1067. Araki, K., Aoshima, M., Namiki, N., Ujiie, S., Koide, N., Iimura, K., Imamura, Y, Williams, G. (1989a), Macromol. Chem. Rapid Commun. 10, 265. Araki, K., Attard, G. S., Williams, G. (1989b), Polymer 30, 432. Ashcraft, C. R., Boyd, R. H. (1976), / Polym. Sci., Polym. Phys. Ed. 14, 2153. Attard, G. S. (1986), Molec Phys. 58, 1087. Attard, G. S. (1989), Polymer 30, 438. Attard, G. S., Araki, K. (1986), Mol. Cryst. Liq. Cryst. 141, 69. Attard, G. S., Williams, G. (1986a), Chem. Britain 22, 919. Attard, G. S., Williams, G. (1986b), Polymer 27, 2. Attard, G. S., Williams, G. (1986c), Polymer 27, 66. Attard, G. S., Williams, G. (1986d), Liq. Cryst. 1, 253. Attard, G. S., Williams, G. (1986e), /. Molec Electronics 2, 107. Attard, G. S., Williams, G., Gray, G. W, Lacey, D., Gemmel, P. A. (1986), Polymer 27, 185. Attard, G. S., Araki, K., Williams, G. (1987a), /. Molec. Electronics 3, 1. Attard, G. S., Moura-Ramos, J. J., Gray, G. W, Lacey, D., White, M., Nestor, G., Toyne, K . I (1987b), Makromol. Chem. RapidPubl. 188, 2769. Attard, G. S., Moura-Ramos, J. J., Williams, G. (1987c), /. Polym. Sci., Polym. Phys. Ed. 25, 1099. Attard, G. S., Araki, K., Williams, G. (1987d), Brit. Polym. J. 19, 119. Attard, G. S., Araki, K., Moura-Ramos, J. I, Williams, G. (1988), Liq. Cryst. 3, 861. Attard, G. S., Araki, K., Moura-Ramos, J. J., Williams, G., Griffin, A. C , Bhatti, A. M., Hung, R. S. L. (1989), in: Polymer Associated Structures, Microemulsions and Liquid Crystals: El Nokaly, M. A. (Ed.). Washington: ACS Symp. Series, No. 384. Barfknecht, A. T., Partridge, J. P., Chen, C. X, Li, C. J. (1989), Advanced Packaging Materials, Mat. Res. Soc Symp. Proc, Vol. 167. Bata, L., Buka, A. (1981), Mol. Cryst. Liq. Cryst. 63, 307.

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11.8 References

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11 Dielectric Properties of Polymers

General Reading

Dorfmuller, T., Williams, G. (Eds.) (1987), Molecular Dynamics and Relaxation Phenomena in Glasses, in: Springer Led. Notes in Physics, Vol. 277. Berlin, Heidelberg: Springer Verlag.

McCmm, N. G., Read, B. E., Williams, G. (1991), Anelastic and Dielectric Effects in Polymeric Solids. New York: Dover. Williams, G. (1979), Adv. Polym. Sci. 33, 60. Williams, G. (1989), in: Comprehensive Polymer Science: Allen, G., Bevington, J. C. (Eds.); Vol.2: Booth, C , Price, C. (Eds.). Oxford: Pergamon Press, Chap. 18.

12 Optical Properties of Polymers Wolfgang Knoll

Frontier Research Program, The Institute of Physical and Chemical Research (RIKEN), Wako, Saitama, Japan and Max-Planck-Institut fur Polymerforschung, Mainz, Federal Republic of Germany

List of Symbols and Abbreviations 12.1 Introduction 12.2 Theoretical Concepts for the Description of the Interaction of Light with Matter 12.2.1 Wave Propagation in Free Space 12.2.2 Polarization of Light 12.2.3 Maxwell's Equations in a Medium 12.2.4 Linear Dielectric Media 122 A A Linear, Nondispersive, Homogeneous, and Isotropic Media 12.2.4.2 Inhomogeneous and Anisotropic Media 12.2.4.3 Absorption 12.2.4.4 Dispersion 12.2.4.5 Dielectrics with Resonances 12.2.5 Nonlinear Optics 12.2.5.1 Second-Order Nonlinear Optics 12.2.5.2 Third-Order Nonlinear Optics 12.3 Polymers with Specifically Tuned Optical Properties 12.3.1 Polymers as Transparent Substrates for Optical Discs 12.3.2 Solid Polyelectrolytes as Glassy Polymers with Tunable Refractive Indices and Dispersion Characteristics 12.3.3 Magneto-Optical Polymers 12.4 Polymers in Special Configurations 12.4.1 Ultrathin Films 12.4.1.1 Langmuir Monolayers 12.4.1.2 Langmuir-Blodgett-Kuhn Multilayer Assemblies 12.4.1.3 Self-Assembly Mono- and Multilayers on Solid Supports 12.4.2 Integrated Optics 12.4.2.1 Planar Waveguide Structures 12.4.2.2 Channel Waveguides 12.4.2.3 Polymer Optical Fibers (POFs) 12.4.3 Polymers for Optical Recording 12.4.4 Polymers for Molecular Opto-Electronics Materials Science and Technology Copyright © WILEY-VCH Verlag GmbH & Co KGaA. Allrightsreserved.

531 535 536 536 537 538 538 539 539 540 540 541 542 543 544 545 545 546 547 549 549 550 553 560 561 562 569 570 572 575

530

12.5 12.5.1 12.5.2 12.5.2.1 12.5.2.2 12.5.2.3 12.5.3 12.5.3.1 12.5.3.2 12.5.3.3 12.5.4 12.6 12.7 12.8

12 Optical Properties of Polymers

Nonlinear Optical Properties of Polymers General Concepts for the Functionalization of Polymers for Nonlinear Optics Second-Order Nonlinear Optics Control of Orientational Order Second-Harmonic Generation (SHG) Electro-Optic (EO) Modulation Third-Order Nonlinear Optics Third-Harmonic Generation (THG) Degenerate Four Wave Mixing (DFWM) All-Optical Switching Photorefractive Polymers Outlook Acknowledgements References

576 576 581 581 585 587 588 590 591 592 592 593 593 593

List of Symbols and Abbreviations

List of Symbols and Abbreviations A A, B c Co

D d d0 e E E(0) Eo Ely

f F FB

FR G H Ho h I k k~ k0 k0 kiz

*fP k° k°

^sp /

L Lc Ld m m! M n N ND nD, nF, n c

area per molecule amplitude of Eiy in medium i magnetic induction speed of light in a medium speed of light in free space electric displacement film thickness thickness per monolayer magnitude of electron charge electric field static electric field electric field amplitude y-component of E in medium i local field factor force electric force restoring force grating vector magnetic field magnetic field amplitude Planck constant intensity wavenumber complex wavenumber wavenumber in free space wavevector in free space z-component of k in medium i fc^p including a shift caused by thin dielectric coating parallel component of photon wavevector wavevector of plasmon surface polariton at metal-air interface waveguide propagation vector component penetration depth; sample length device length coherence length conjugation length mass of electron; order of waveguide mode order of second harmonic mode magnetization density refractive index number of charges per unit volume density of (active) dye molecules refractive indices at three standard wavelengths (blue, yellow, red) refractive index of medium i

531

532

12 Optical Properties of Polymers

np nz n0 n2 H|l? n± An p P PNL r R rp rs R, 5 t T Tg Tm u V JC X x y

refractive index of prism refractive index of a multilayer normal to its plane background refractive index optical Kerr coefficient refractive index parallel, perpendicular to film plane refractive index change pressure polarization density nonlinear component of the polarizability spatial coordinate (position vector) reflectivity reflectivity coefficient for p-polarized light reflectivity coefficient for s-polarized light chain segments, i = 1, 2, 3, 4 variable (Kramers-Kronig relations) time absolute temperature glass transition temperature figure of merit variable representing Ex y z or Hx y z volume; Verdet constant displacement vector functional group unit vector in the x-direction unit vector in the y-direction

a a0 a± /? PD P2 y yD 7xxxx y0 Th F{ A g s 8tj s0 3 6

absorption (extinction) coefficient; exponential factor extinction coefficient prior to poling extinction coefficient normal to film after poling propagation constant molecular second-order nonlinear optical polarizability nonlinear absorption coefficient surface tension molecular third-order nonlinear optical polarizability c o m p o n e n t of y D surface tension of w a t e r - a i r interface homogeneous linewidth inhomogeneous linewidth ellipsometric phase angle electric permittivity of a medium complex electric permittivity of a medium electric permittivity tensor element (second rank) electric permittivity of free space distribution angle incident angle

List of Symbols and Abbreviations

K

X A

K

x0

Av 77 T

z z Xu

i z" z(2) Xo

00

CO

ATR a.u. BAM CCD CD DFWM DMPE DRAW DUV EO

critical angle for total internal reflection resonance angle resonance angle including a shift caused by thin dielectric coating elastic constant of a spring wavelength of light groove spacing laser light wavelength wavelength in free space spectral position of linear absorption magnetic permeability frequency of light Abbe number central frequency spectral width or line width lateral surface pressure switching time Faraday rotation angle electric susceptibility complex electric susceptibility electric susceptibility tensor element (second rank) real part of (complex) susceptibility imaginary part of (complex) susceptibility coefficient of second-order optical nonlinearity coefficient of third-order optical nonlinearity macroscopic averaged nonlinear susceptibility susceptibility constant in dielectrics with resonances phase phase factor (internal) resonance angle order parameter nonlinear phase shift ellipsometnc amplitude angle photon frequency frequency of laser light attenuated total internal reflection arbitrary unit Brewster angle microscopy charge-coupled device compact disc degenerate four wave mixing dimyristoylphosphatidylethanolamine direct read after writing deep ultraviolet electro-optic(al)

533

534

12 Optical Properties of Polymers

esu GI IOC IR LBK LC NIR NLO OWM PC PM-co-OLG PPA PPV ROM s, p SAM SH(G) SI SPM THG UV

electrostatic unit graded index integrated optical circuit infrared Langmuir-Blodgett-Kuhn liquid crystalline near infrared nonlinear optics optical waveguide microscopy poly(carbonate) poly(y-methyl-L-glutamate-co-y-octadecyl-L-glutamate) poly(phenylacetylene) poly(p-phenylenevinylene) read-only memory polarization of light (s: perpendicular, p: parallel) self-assembly monolayer second-harmonic (generation) step index surface plasmon microscopy third-harmonic generation ultraviolet

12.1 Introduction

12.1 Introduction Without polymers and their optical properties our daily life certainly would be by far less colorful: just think about the pictures in a magazine, all the paint in and around our buildings, or our clothing (be it a Japanese kimono or a western T-shirt). However, in most of these cases it is not the polymer itself that shows the desired coloring effect through the interaction with light but rather some pigments and dyes that are dispersed into the material, more or less loosely connected to the macromolecular matrix. These obvious optical properties are not within the scope of this contribution. We will rather concentrate on the intrinsic optical properties of polymers, that is, focus on the presentation of those phenomena and properties of the materials that describe their response to visible light. A rough classification of these properties could start, for example, with transparency and birefringence, which determine the intensity decrease and change of polarization, respectively, of light being just transmitted through the sample. The response to external fields is also an intrinsic property, for example, the change of the optical properties by applying a (strong) magnetic field is described by the magneto-optical effect. Some polymers are designed so as to strongly interact with light through specific electronic excitations, giving rise to a broad range of phenomena: spectral changes due to the resonance character of absorption, or secondary processes (following the excitation) like luminescence phenomena (fluorescence, phosphorescence, etc.) are examples. Many systems show changes in other properties when photoexcited, for example, become photoconductive. The general term photoresponsive was coined to summarize the nearly unlim-

535

ited list of property changes that can be induced by a purposeful design of the material, typically by the proper functionalization: this list includes photoisomerization and photochemical reactions resulting in a change of optical, mechanical, structural, morphological, rheological, or (meso-)phase properties. Finally, nonlinear optical properties, relevant for the description of the response of the material to ultrastrong optical fields, that is, to a giant laser pulse, define the most active field of research in recent years related to optical properties of polymers. In what follows, we will be able to cover only a few selected examples to try to highlight some of the improvements that have been achieved by extensive research and development that has mostly been focused on two nearly opposite objectives: one goal was to minimize the interaction with light, that is, to reduce the losses in a polymer optical fiber (POF) used in (local) data communication networks or to decrease the noise for optical recording and reading in information storage on optical discs. The other strong effort was to enhance the interaction with light, for example, to increase the light intensity-dependent refractive index change n2 in selected polymers needed to switch "light by light" in a future photonics device application. This limitation to some examples from the above list also means that many other aspects of the interaction of light with polymers will be ignored. These include, in particular, the interaction with light of wavelengths other than in the visible [X = 400 to 800 nm) with the only exception to this being some considerations related to the near infrared (NIR) losses of POFs. No IR vibrational spectroscopy will be considered. Another area important at least for diagnostic purposes in polymer research is the scattering of light (either

536

12 Optical Properties of Polymers

elastic, quasi-elastic, or inelastic, i.e., Brillouin- or Raman-spectroscopies) that we have to ignore. (But see Chap. 14 and Chap. 7 of this Volume.) And finally, all versions of optical labeling techniques, including absorption or fluorescence labels in steady state or time-resolved experimental configurations will not be covered. However, for some of the optical phenomena and properties described we will also give some consideration to the experimental techniques that had to be developed in order to be able to quantitatively determine the relevant material parameters. Finally, some of the discussed improvements of materials properties need to be seen in the context of their present and future application in optical device configurations. Here, the emerging technology of the 21st century, photonics, is a major driving force for research and development in polymer materials science and technology. This chapter is organized as follows: In Sec. 12.2 we give a brief summary of the basic theoretical concepts, mostly based on the classical electrodynamic description of the interaction of light with matter given by Maxwell's theory, also including a short introduction to nonlinear optics. Section 12.3 concentrates on some "passive" materials properties related, for example, to its transparency or its dispersion properties. Section 12.4 is called "polymers in special configurations" and describes polymers in ultrathin films from monomolecular layers to multilayer assemblies, integrated optics with planar and channel waveguide structures and fibers, and optical recording principles. Section 12.5 finally is devoted to nonlinear materials and device concepts based on second (x(2)) and third (#(3)) order nonlinear optical effects.

12.2 Theoretical Concepts for the Description of the Interaction of Light with Matter Light is an electromagnetic wave phenomenon which is described by the same theoretical concepts that govern all forms of electromagnetic radiation. Figure 12-1 displays the (unlimited) spectrum of electromagnetic waves. Optical frequencies occupy a band of frequencies, v, and wavelengths, k, from the infrared (v = 3x 1011 Hz, l = lmm) to the ultraviolet (v = 3 x 1016 Hz, X = 10 nm) with the visible spectrum being limited to the narrow range between X = 400 and 800 nm. The appropriate theory, which describes the interaction of light with matter and hence is also the background for the understanding of the optical properties of polymers, is based on Maxwell's equations. We will give in the following only a very brief outline of the concepts that are needed to understand some of these properties and to see the specific requirements of the material for some functional applications discussed in this contribution. 12.2.1 Wave Propagation in Free Space We start with the basic principles describing the propagation of light in free space (Born and Wolf, 1980). These are summarized in the following partial differential equations, known as Maxwell's equations: V x H(r, t) = 80

\xE(r,t) =-

dE(r,t) dt dt

(12-1) (12-2)

V-E(r9t) =0

(12-3)

V-H(r,t) =0

(12-4)

with the electric permittivity e0 « 1/(36 TC) x l 0 ~ 9 and the magnetic permeability

12.2 Theoretical Concepts for the Description of the Interaction of Light with Matter

and

Frequency / Hz 1018

1015

109

1Q12

1Q6

X

1mm

1m

(12-9)

1Q3

CD

1nm

537

1km Wavelength

Figure 12-1. The electromagnetic spectrum and the range of "light" (after Saleh and Teich, 1991).

k0 is called the wavevector with its magnitude the wavenumber k0 = 2 n/l0 = 2n v/c0. From Maxwell's equation it follows that k0, E and H are mutually orthogonal. Since E and H lie in a plane normal to the propagation direction given by k0, the wave is called a transverse electromagnetic (TEM) wave. 12.2.2 Polarization of Light

fi0 = 4TC x 10 7 (in MKS units, in SI units: 80 = 8.8542 x 1(T 12 As V" 1 m" 1 , fi0 = 1.2566 x 10 ~6 Vs A ^ m " 1 ) of free space. Vx and V- are the usual curl and divergence operations, respectively. E(r, t) is the electric field and H(r,t) is the magnetic field that describe the electromagnetic wave as a function of position r and time t. A direct consequence of Maxwell's equations, (12-l)-(12-4), is the so-called wave equation r2

(12-5)

c)t2

where u represents any of the three components (£ x , E y , Ez) of E or the three components (ifx, ify, Hz) of H, V2 is the Laplace operator given by 8 8x

8 2• + •

' 8y

6 2

' 8z2

(12-6)

and

« 3xlO 8 m/s

(12-7)

is the speed of light in vacuum. One particular solution to the above wave equation is a monochromatic, electromagnetic wave whose electric and magnetic field components are plane waves: E{r,t) =

(12-8)

Consider a monochromatic plane wave of frequency v travelling in the z-direction with velocity c0. The electric field lies in the x-y plane and can be described by (12-10)

with ^ x = ^ x O COS

(12-11)

and Ey = Ey0 cos

(12-12)

x and j> are unit vectors along the x- and y-directions, Ex and Ey are the £-field components in x- and y-directions, respectively, and cpx and cpy take into account that the two orthogonal partial waves, in general, can have different phases. If the components also differ in £ x 0 and E y0 , one is dealing with elliptically polarized light because at a fixed value of z, the tip of the electric field vector E rotates periodically in the x-y plane, tracing out an ellipse. The shape of this ellipse depends on the ratio of the two amplitudes Ex0/Ey0 and on the phase difference (p = cpx — cpy. This is schematically depicted in Fig. 12-2 a. If one of the components vanishes (Ex0 = 0, for example) the light is linearly polarized in the direction of the other component (the y-direction). The wave is also

538

12 Optical Properties of Polymers

This wave is called right circularly polarized. For cp = — n/2 the rotation is counter clockwise and the light is said to be left circularly polarized. These two cases are sketched in Fig. 12-2c. 12.2.3 Maxwell's Equations in a Medium

In a dielectric medium with no free charges or currents Maxwell's equations read —^

V xE(r,t)

(c) right

\4

8 / > M ) J± \l , I)

r! I /////// K\ ! /\! z

WVJWl y/'

(12-13)

dt

dB(r,t)

(12-14)

dt

V •D(r,t) = 0

(12-15)

V •B(r,t) = 0

(12-16)

The electric displacement D (r, t) is related to the electric field by

left

f \>¥\/\i\!Pr\!\: n

where P(r,t) is the polarization density of the dielectric medium. Similarly, the magnetic induction B(r, t) is related to the magnetic field by , t) -

Figure 12-2. Trajectories of the endpoint of the electric-field vector at a fixed position z = const, (right side) and a snapshot at a fixed time t = const, (left side) for an elliptically polarized wave (a), a linearly polarized wave (b), and for left and right ciruclarly polarized light (c) (after Saleh and Teich, 1991).

r, t)

M(r9

(12-18)

with M(r,t) being the magnetization density. Since we have to deal only with nonmagnetic materials Eq. (12-18) is replaced by (12-19)

linearly polarized if the phase difference

12.2.4 Linear Dielectric Media

The nature of the dielectric medium is exhibited in the relation between the polarization density P(r, t) and the electric field E(r,t\ called the medium equation. This relation then also determines the optical properties of polymers.

12.2 Theoretical Concepts for the Description of the Interaction of Light with Matter

12.2.4.1 Linear, Nondispersive, Homogeneous, and Isotropic Media

with a speed of light in the medium c =

Let us first recall a few definitions (Saleh and Teich, 1991). A medium is said to be linear if the polarization density P(r,t) is linearly related to the electric field E(r,i). Then the principle of superposition holds. The medium is said to be nondispersive if its response is instantaneous. Clearly, this is an idealization, because any system, no matter how fast it may be, has a finite response time. It is said to be homogeneous if the relation between P and E is independent of the position r. The medium is called isotropic if the relation between P and E is independent of the direction of E, that is, the material looks the same from all directions. Then, P and E must be parallel. And finally a medium is said to be spatially nondispersive if P at each position is induced only by E at the same position. For such a medium the relation between P and E is given by p(v A — p v F(v i\ A If , Li — OQ /

JL* It , Li

(\ ?-7(W \±Z*Z*\Jj

a

where x is scalar constant called the electric susceptibility. Substituting Eq. (12-20) into Eq. (12-17) gives /)(r, t) = eo(l +x) E(r, t)

(12-22)

Hence it follows that also D and E are parallel and proportional. The constant is the electric permittivity of the medium. The ratio s/e0 = (1 +x) *s the dielectric constant. The wave equation (12-5) then is changed to

y2u

-^w=°

The ratio of the speed of light in free space to that in the medium is called the refractive index n: (12-24)

n=—

and is related to the dielectric constant by

« = (—Y=(l+Z)*

(12-25)

12.2.4.2 Inhomogeneous and Anisotropic Media In an inhomogeneous dielectric, for example, a graded index polymer (cf. Sec. 12.4..2.3), which otherwise is linear, nondispersive, and isotropic, the relations P = soxE and D = &E remain valid but the proportionality constants x a n d s a r e now functions of the position x = x(r) a n d £ = £ (r). Likewise, the index of refraction n = n(r) is position dependent. For materials with slowly varying inhomogeneities (on the length scale of the wavelength of light) one can show that the wave equation still holds, however, with a locally varying speed of light c (r): d2E

\2E-

=0

(12-26)

(12-21)

which we write as D(r,t) = eE(r,t)

539

(12 23)

"

where c(r) = co/n(r). A material with a certain profile of the index of refraction, therefore, has a correspondingly varying speed of light. In an anisotropic medium the relation between the vectors P and E depends on the direction of E and the two vectors are not necessarily parallel. If the medium is linear, homogeneous, and nondispersive, each component of P is a linear combination of the three components of E p _ y^ F

y

F

(\ 2-21)

540

12 Optical Properties of Polymers

where ij = x, y, z. Thus, the polarization response of an anisotropic material to an electric field is described by the secondrank susceptibility tensor Xij- This is schematically characterized in Fig. 12-3. Similarly D and E are related by the electric permittivity tensor stj n — vP U^

— 2^ ^ij j

l

(12-31)

(\i ?R^

F *-*\

\VL-LQ))

By a suitable rotation of the coordinate system one can always make the off-diagonal elements to go to zero, a situation which then defines the principal axes of the medium. 12.2.4.3 Absorption Dielectric materials that absorb light can be phenomenologically represented by a complex susceptibility X = X' +

A plane wave propagating in such a medium in the z-direction is described by a complex amplitude proportional to Q~^Z. Since £ is complex it is useful to write k in terms of its real and imaginary parts

'n"

(12-29)

corresponding to a complex permittivity £ = £O(1+X). The wavenumber then also becomes complex: (12-30)

Then the intensity of the wave is seen to be attenuated by the factor |e^ z | 2 = e~ az so that the coefficient a represents the absorption coefficient (or extinction coefficient). /? is the propagation constant P = n-k0

(12-32)

with n the effective refractive index of the medium. Substituting Eq. (12-32) into Eqs. (12-31) and (12-30) we obtain an equation which relates the refractive index n and the absorption coefficient a to the real and imaginary parts of the susceptibility / and x", n —l

2fc0

'

*

•

A. ,

For weakly absorbing media (/, obtain

(12-33) <^1) we (12-34)

- Kf

(12-35)

12.2.4.4 Dispersion

Figure 12-3. For an anisotropic medium the relation between the E-field and the polarization P is characterized by nine elements of the susceptibility tensor

Dispersive media are characterized by a frequency-dependent (and therefore wavelength-dependent) susceptibility x(v), refractive index n(v), and speed of light c(v). This phenomenon, well-known from inorganic materials like glasses, can be either the basic concept for the function of an optical component, for example, the desired dispersion of white light into its different colors by a prism, or can be the source of a limited performance, for example, the chromatic aberration of a lens.

541

12.2 Theoretical Concepts for the Description of the Interaction of Light with Matter (a)

(b)

blue

red

Figure 12-4. The dispersion of light in a dielectric medium with a wavelength dependent refractive index (a) is responsible for the separation of the light's "colors" by a prism (b), but also for chromatic aberration of a lens (c) (after Saleh and Teich, 1991).

This is schematically shown in Fig. 12-4. A quantitative measure of the dispersion of the material is the Abbe number vD, defined by Vn =

(12-36)

- nc)

with nD, nF, and nc being the refractive indices at three standard wavelengths (blue: A = 486.1nm, yellow: 589.2 nm, and red: 656.3 nm, respectively). A direct consequence of the relation between the polarization density P(t) and the applied electric field E(i) given by theory for a linear system are the so-called Kramers-Kronig relations:

n I s2 - v2

ds

(12-37)

niv2-s2

ds

(12-38)

A practical consequence of these equations which relate the real and the imaginary part of the susceptibility and hence relate absorption and dispersion is that once the absorption spectrum of a given material is known (measured) over a sufficiently broad wavelength range (including all relevant excitations) the index of refraction of the material can be calculated.

12.2.4.5 Dielectrics with Resonances

An instructive way to visualize the optical properties of a medium near an electronic transition (excitation) is to consider it as an assembly of forced harmonic oscillators according to the model due to Lorentz (Zernicke and Midwinter, 1973). The electrons are treated as charges, e, attached to the nuclei by a spring of elastic constant K. In an isotropic approximation the force FE exerted by the electric field on the electron FE = eE

(12-39)

then induces a displacement x from equilibrium position which in turn induces a restoring force FK on the electron by the spring FR=

(12-40)

-KX

The resulting force produces an acceleration of the mass m of the electron and the equation of motion reads — F — OITV

)2 x —

—

(\ 2-41 ^

where 2nv0 = KJm is the resonance frequency of the (free) oscillator electronnucleus. Introducing a damping term, — 2 7i Av • dxjdt, then gives the familiar differential equation describing the forced,

542

12 Optical Properties of Polymers

a (v)

•n 0

Av

damped harmonic oscillator: — ^ + 2n Av - ^ + (2TT V 0 )22JCJC== — dt d m ' (12-42) From the electron displacement x results a polarization density P = N - e - x, N is the number of charges per unit volume. A monochromatic wave with an £-field oscillating with frequency v,E = Eo ei27CVt, satisfies Eq. (12-42) if =

(2nvo)2soxoE

(12-43)

with 2

eN mso(2nvo)2 Comparing Eq. (12-43) with Eq. (12-20) and Eq. (12-29) then gives the real and imaginary parts of x (v), v2o(v2o-v2) (12-45) V AV

= ~Xo

Figure 12-5. Absorption coefficient a (v) and refractive index n (v) of a dielectric medium (background index n0) near an electronic resonance (after Saleh and Teich, 1991).

This frequency dependence is illustrated in Fig. 12-5. Typical dielectric media contain multiple resonances. The overall susceptibility is the sum of contributions from these resonances. An example for a material that exhibits resonance absorptions in the ultraviolet and in the infrared is given in Fig. 12-6. Shown are the absorption coefficient and the refractive index as a function of the wavelength. 12.2.5 Nonlinear Optics In nonlinear media the relation between P and E is nonlinear (Shen, 1984). However, Maxwell's equations still can be used to derive a nonlinear partial differential equation that describes the behavior of .Visible

(12-46)

If these resonant oscillators are sufficiently dilute in a host medium of (background) refractive index n09 then the overall refractive index and absorption coefficient are 2nc

«(v)«-('^V(v) noco

(12-47) (12-48)

0.01

0.1

10

100

X//xm

Figure 12-6. Typical wavelength dependence of the absorption coefficient a and the refractive index n of a material with several resonant transitions (after Saleh and Teich, 1991).

543

12.2 Theoretical Concepts for the Description of the Interaction of Light with Matter

electromagnetic waves in nonlinear media. For a homogeneous and isotropic medium one obtains d2E

d2P

(12-49)

It is convenient to write P as the sum of linear and nonlinear parts as they may be obtained by an expression in a Taylor series about E=0: (12-50)

der terms are needed to describe the polarization induced by an E-field. One has to bear in mind, however, that these deviations are typically extremely small and it was not until the invention of the laser in 1960 that optical field intensities could be generated that could compete with interatomic electric fields (typically 105 to 108 V/m) so that the experimental investigation of nonlinear optical effects could begin.

with (12-51)

12.2.5.1 Second-Order Nonlinear Optics

(12-52)

We examine briefly the optical properties of a nonlinear material that exhibits a second-order nonlinear susceptibility

Equation (12-49) then becomes

V2E l 2 v h 2 dt ~ c

dt2

which is a wave equation in which the term 92 P L — /x0 ^ acts as a (nonlinear) source of radiation from the material. Equation (1252) is the basic equation that underlies the theory of nonlinear optics. Figure 12-7 may illustrate graphically the situation. Figure 12-7 a gives the polarization for a material responding to an electric field in a linear way: doubling the £-field doubles the polarization. Figure 127 b shows the situation (highly exaggerated) for a nonlinear material. Higher or-

(a)

(b)

]

\E\2

(12-53)

If a sinusoidal E-field E(z,t) = E o COS(2TTV£ — kz) induces a nonlinear polarization in such a medium we find

= y Xi2) E2

COS(4TIV£- 2kz)]

=

(12-54) This process is illustrated graphically in Fig. 12-8 a. PNL(0) describes the generation of a DC voltage, a process called optical rectification. The term PNL(2 v) is the source for radiation at twice the input frequency, a process called second-harmonic generation (SHG). If we apply an optical field E(v) together with a DC (or low frequency AC) electric field E (0) then the sum of the two fields E = E(0) + Eo cos(2nvt-

kz)

(12-55)

induces a polarization Figure 12-7. The relation between the polarization P and the electric field E for a linear (a) and a nonlinear medium (b).

E0 cos(2nvt-

kz)]2 (12-56)

544

12 Optical Properties of Polymers

dotted line. The polarization at frequency v

(a)

PNLW

PNL(0)

= 2s0X(2> £(0) Eo coS(2nvt-

kz) (12-57)

can be rewritten in the form PNL(v) = e 0 A Z E(v)

(12-58)

where AX = 2X(2)E(0)

(12-59)

represents a change in the susceptibility proportional to the electric field E(0). The nonlinear nature of the medium creates a coupling between the electric field £(0) and the optical field E(v), causing one to control the other. This effect is called the linear electro-optic effect (Pockels effect).

time t

(b)

12.2.5.2 Third-Order Nonlinear Optics PNi_(t)

In media with centrosymmetry the second-order term in the expansion of PNL is absent. The dominant nonlinearity is then of third order E(v)

time t

Figure 12-8. (a) Scheme demonstrating the effect of a sinusoidal input field exciting a nonlinear optical medium: a DC-polarization PNL (0) ("optical rectification") and a polarization component at twice the input frequency (giving rise to second-harmonic generation) are obtained, (b) Linearization of the secondorder relation |P NL | = s0x(2)\E\2 in the presence of a strong electric field E (0) and a weak optical field E (v). (After Saleh and Teich, 1991.)

which contains components at frequencies 0, v, and 2 v. If the optical field is substantially smaller than the DC field \E(v)\2 <^ \E(0)\2, the 2 v-polarization component can be neglected. This is equivalent to the linearization of PNL as a function of E at E = E(0) as depicted in Fig. 12-8b, dash-

ID I _ p v (3) I J7|3 I NLI — 0 /C I I

(IJ-fiCW v W)

and the material is called a Kerr medium. A monochromatic optical wave at frequency v induces a nonlinear polarization which contains contributions at the frequencies v and 3v. The latter process is called third-harmonic generation (THG). The polarization component at v corresponds to an incremental change of the susceptibility A% at frequency v given by A* = 4

# = TX ( 3 ) |£(V)I 2

(12-61)

This change in susceptibility is equivalent to a change in refractive index An. Both are proportional to the square of the field amplitude. The overall refractive index is, therefore, a linear function of the optical intensity /, n{l) = n + n2l

(12-62)

12.3 Polymers with Specifically Tuned Optical Properties

where v<3)

(12-63)

This effect, called the optical Kerr effect, is a self-induced effect in which the phase velocity of the wave depends on the wave's own intensity. This intensity-dependent refractive index change n2 is the basis for a broad range of nonlinear optical phenomena observed also in polymers, the most important being the degenerate four wave mixing. This form of real-time holography is particularly helpful for the determination of ultrafast nonlinear response times needed for future photonic device configurations. This technique will be discussed in Sec. 12.5. Other phenomena based on the optical Kerr effect are, for example, self-phase modulation, self-focusing, optical phase conjugation, and spatial solitons.

12.3 Polymers with Specifically Tuned Optical Properties Following the outline given in the introduction we will concentrate in this section on the discussion of the remarkable progress that has been made in recent years in the development of polymeric materials with a specific, tailor-made property profile. Depending on the various applications, these synthesis- and processing-optimizations were aimed at reducing as much as possible any interaction with light so as not to change its intensity, state of polarization, etc. Clearly, these are the applications where polymers are to replace inorganic glasses as windows, shields, eye glasses, substrates etc. A more recent example, the use of polymers as substrate materials for optical storage discs, will be briefly discussed below. A second rapidly growing area for the use of highly transpar-

545

ent polymers as polymer optical fibers will be presented in Sec. 12A.23. Another important optical property of polymers is their dispersion behavior: The isotropic or anisotropic refractive index and its dependence on the wavelength of light are key parameters that need to be tunable to certain values in all areas of polymers for integrated optics. As an example, solid polyelectrolytes with a broad range of refractive index values and dispersion characteristics are discussed below. Next, a short introduction to some recent developments of magneto-optical applications of polymers is given. Probably the most important application of polymers as resist materials in microelectronics is treated in Vol. 18 of this Series. These resist materials are optimized with respect to their absorption of photons (and, of course, with respect to other property changes such as solubility following the excitation by the exposure to light). 12.3.1 Polymers as Transparent Substrates for Optical Discs

The availability of low cost lasers has made optical information storage an attractive alternative to existing mass storage concepts. The largest volume use of polymers for optical recording, however, is still in the disc substrates. The material used for compact discs (CD) and CDROMs (read-only memory) and video discs has to fulfill extreme requirements (Kampf, 1985): • optical homogeneity (low birefringence); • high transparency in the visible, in some cases also in the UV (2 = 280-380 nm); • high shape stability (low thermal expansion, water uptake less than 0.3%); • isotropic expansion upon temperature or humidity increase; • good mechanical properties;

546

12 Optical Properties of Polymers

• constant thickness, planarity; • constant rheological properties for the molding process of the "pits" (cf. Fig. 12-41). Currently, mostly poly(methylmethacrylate) (PMMA) and poly(4-methyl-l-pentene) as well as bisdiallylpolycarbonates (PCs) are used for the fabrication of substrate discs. These polymers offer the optical quality of glass. Some selected physical properties of PMMA and PC are summarized in Table 12-1 (after Pearson, 1988). 12.3.2 Solid Polyelectrolytes as Glassy Polymers with Tunable Refractive Indices and Dispersion Characteristics

The various available polymers can have very different refractive indices and dispersion properties. However, Fig. 12-9 shows that these materials exhibit typically much lower refractive indices than inorganic glasses due to the absence of highly polarizable units like, for example, the heavy metals with their many electrons in flint

s

"D

1.80

s

/

1.60-

p^

uo-.

^ ^

^

m

\

organic polymers

A 80

60

Physical property

PC

PMMA

Transmittance (%) at 830 nm Refractive index at 830 nm Tg (°C) Thermal expansion coefficient

90 1.58 150 6-7

92-93 1.49 100 7.6

Rockwell hardness (Mscale) Izod impact strength (kg cm/cm) H 2 O absorption (%)

75 1-2 0.25

90 1.6 0.54

glasses. One extreme case, for example, is poly(tetrafluorethylene) with nD = 1.345 (data point A in Fig. 12-9). A material with a substantially higher refractive index such as poly(iV-vinylcarbazole) with nD = 1.675 has a very high dispersion vD = 19 (data point B in Fig. 12-9) [note that the higher vD the lower the dispersion, cf. Eq. (12-36)]. One successful attempt to extend the available polymeric materials towards higher refractive indices was recently reported for solid polyelectrolytes, also called ionenes (Simmrock et al., 1989). Given their general structure [R1-N + (R3)2-R2-N+(R4)2]X2-

/

s

- inorganic glasses

Table 12-1. Selected physical properties of PMMA and PC substrates (after Pearson, 1988).

20

Figure 12-9. Refractive indices nD and optical dispersion, vD, of inorganic glasses (dotted area), organic polymers (dashed area) and a series of selected ionenes (numbered I through VII according to Table 12-2). A: poly(tetrafluorethylene), B: poly(JV-vinylcarbazole).

with the main chain segments Rx and R2, and R3 and R4 being organic moieties, for example aliphatic groups, there is an almost unlimited range for tuning their solid state properties, for example, their crystallinity or their ability to solidify as glasses, while on the other hand their optical properties can be largely varied by the introduction of suitable counterions X 2 ~, both monovalent and divalent. In particular, by the introduction of structural irregularity into the main chain (R± / R2) glassy materials with very different polymeric properties, e.g. the glass transition temperature, can be synthesized. Table 12-2 indicates the progress in extending the selec-

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oo O

m

o

1.62*

m

L6U l.58f

Q

vo

in

1.591

0

5.0 9.7 1.0 7.6 4.6

12.3 Polymers with Specifically Tuned Optical Properties


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547

tion of polymers towards higher refractive index with a wide range of Abbe numbers (cf. Fig. 12-9). The interrelation between high refractive index and high dispersion is demonstrated in Fig. 12-10 for two different ionenes (Mathy et al., 1991). The first one, polyelectrolyte I (see Table 12-2), with ZnBr4 as counterions has its strongest absorption in the DUV range not exceeding l = 220nm (see Fig. 12-10 a). The Kramers-Kronig analysis [cf. Eqs. (12-37) and (12-38)] with a background refractive index of no = 1.516 shows that as a result the dispersion in the visible is very low despite a relatively high refractive index (Fig. 12-10 b). On the other hand, the introduction of Hgl 4 ions into the polyelectrolyte VI extends the absorption of the material into the near UV range (Fig. 12-10 c). This causes a consequently much higher (resonance enhanced) refractive index but also a very strong wavelength dependence, that is, low Abbe number (Fig. 12-10d). It is clear that this flexibility for tuning the linear optical properties of polymeric materials is a very promising feature for future device configurations, for example, in integrated optics. We should also mention that the introduced counterions, of course, may also have high nonlinear polarizabilities, allowing the preparation of highly functionalized materials, for example, for x(3)-applications (Meyer et al., 1991).

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12.3.3 Magneto-Optical Polymers Certain materials act naturally as polarization rotators, a property known as optical activity: waves with right- and left-circular polarization (cf. Fig. 12-2) propagate with different velocities. This property is found in media in which the molecular units have an inherently helical character

548

12 Optical Properties of Polymers

1.70

1.62

180

260

340

420

1.54

250

350

450

550

650

X/nm

X/nm

n

A

1.9-

1.8-

1.7-

(d)

Figure 12-10. Absorption spectra of ionene I (cf. Table 12-2) (a), and VI (c) and the corresponding Kramers-Kronig analysis (b), and (d), respectively, with a background refractive index, n0 = 1.515.

\

A

V

1.6-

180

260

3Z.0 Z.20 X/nm

250

350 £50 550 X/nm

(quartz, selenium, tellurium, but also many organic/polymeric materials). Materials that have different absorption coefficients for right- and left-circularly polarized light are said to exhibit circular dichroism. An effect related to this, the Faraday effect, describes the polarization rotation of light passing through a material placed in a static magnetic field. The Faraday rotation angle £ is given by Z = V-B-l

(12-64)

where B is the magnetic induction in the direction of the light wave propagation, / is the length of the sample, and V is a material parameter, called the Verdet constant. This constant can be relatively high in polymers containing phenyl groups [e.g., V « 0.1 min/(Oe • cm) (« 7.5 s/A) for poly(styrene), Muto and Ito (1992)]. In a Faraday rotator the sense of the rotation is governed by the direction of the magnetic

650

field and, therefore, does not change with the reversal of the direction of propagation of the wave. This is the basic concept for optical isolators based on this phenomenon. Its principle is outlined in Fig. 12-11: A linearly polarized wave passing the Faraday rotator may experience a polarization rotation of, say, 45°. It passes another polarizer B tuned to this angle. After reflection it can again go through the polarizer, but is then rotated by the rotator by another 45° in the same direction. This wave is now orthogonal to the incident polarizer A and hence is blocked. Muto and coworkers (1991) have used a poly(amethylstyrene) rod of I = 19 mm to construct a 45° rotator for A = 488 nm light at 4 kOe (« 3.2 x 105 A/m) magnetic field. Its transmission characteristic is shown in Fig. 12-12. Inserting the isolator into the laser beam causes a loss of about 5 dB, but the isolation obtained is approximately

12.4 Polymers in Special Configurations

549

A 45°

(a)

Polarizer B

y

Faraday rotator

Polarizer A

Figure 12-11. An optical isolator based on the Faraday effect transmits light in one direction (a) but blocks it in the opposite direction (b) (after Saleh and Teich, 1991).

15 dB. Further improvements of the device performance are expected for materials with lower birefringence.

12.4 Polymers in Special Configurations The foregoing discussion was focused on polymers where only the bulk optical properties were of interest, specifically tailored for selected applications. In this section we concentrate on optical properties of polymers prepared in a very special configuration, where this configuration and the fabrication protocol are an integral aspect of their optical properties. 12.4.1 Ultrathin Films Ultrathin polymer films, in the extreme case only one monomolecular layer thick, are of special interest from an applied point of view for all kinds of surface modifica-

tions, be it for lubrication, isolation, adhesion, or any other possible purpose, and provide a special challenge for the experimentalist who has to develop methods that are sensitive enough to allow for an optical characterization of coatings only a few

insertion loss

-90 -Lb 0 LS 90 Faraday rotation angle/deg

Figure 12-12. Transmitted light intensity through a polymeric Faraday rotator measured as a function of the analyzer angle (polarizer B in Fig. 12-11) (after Muto etal., 1991).

550

12 Optical Properties of Polymers

nanometers thick. In the following we will summarize just a few of the techniques developed in recent years for that purpose concentrating on a few special types of monolayers: Langmuir films prepared at the water-air interface, Langmuir-Blodgett-Kuhn multilayer assemblies prepared by the multiple deposition of monolayers onto solid substrates, and self-assembled monolayers obtained by controlled adsorption from a solution to a solid support. 12.4.1.1 Langmuir Monolayers

All surface-active molecules like soaps, lipids, etc. can be prepared as monomolecular layers at a water-air interface. Driven by the reduction of the surface free energy of water these molecules spread when applied to the surface, for example, from a volatile solvent (Fig. 12-13). The physical properties of these monolayers were first investigated in the 1940s by I. Langmuir. A so-called Langmuir trough (cf. also Fig. 12-14) filled with water defines an exactly known area for the spread molecules (Fig.

12-13). At low lateral density these molecules behave like a quasi-two-dimensional gas. If the area for the molecules is reduced by a movable barrier, this lateral compression will eventually lead to a measurable lateral pressure 77 (force F per unit length of barrier). It can be measured by a socalled Wilhelmy balance (Kuhn et al., 1972), which is the difference between the surface tension of the free, y0, and the layer-covered water surface, y: =

yo-y

(12-65)

If the lateral pressure is plotted as a function of the area per molecule A (cf. Fig. 12-13), one obtains a U-A isotherm, which is the equivalent of the three-dimensional pV diagrams of real gases: Here, too, any change in the compressibility indicates changes in the 2D phase behavior of such a monolayer at a given temperature. What is sketched for a low molecular mass amphiphile in Fig. 12-13 was extended in recent years also to the characterization of polymeric amphiphiles.

area/(molecule • nm 2 )

Figure 12-13. Pressure-area (77 — A) isotherm for a hypothetical amphiphile organized as a monomolecular layer at the water-air interface showing a phase transition from a fluid-expanded to a solid-condensed state upon compression. The insets show cross-sectional diagrams of the Langmuir trough with the compressed monolayer in different phase states.

12.4 Polymers in Special Configurations

It is clear that for the investigation of the optical properties of these highly organized polymer films extremely sensitive techniques are necessary. Ellipsometry has proven to be also applicable for the characterization of monolayers at a water-air interface. Figure 12-14 shows schematically the set-up used in a so-called compensating configuration: laser light is passed through a polarizer and a compensator to become elliptically polarized. After the reflection from the water surface typically at an angle of incidence close to the Brewster angle for maximum sensitivity, the then linearly polarized light passes through an analyzer at orthogonal polarization and the minimum intensity is detected (null-ellipsometry). Polarizer and analyzer angles at compensation can then be converted to the two ellipsometric angles A and V which describe the complex ratio of the reflectivities for s- and p-polarized light, that is, for light with its plane of polarization being perpendicular (s) and parallel (p) to the plane of incidence, respectively, = eiA tan

551

Wilhelmy-system

Teflon-trough

barrier

subphase

Figure 12-14. Ellipsometric set-up integrated into a Langmuir trough.

186 -

(12-66)

One difficulty associated with ellipsometric measurements of polymer monolayers at the water surface is the limited sensitivity of V. This is exemplified in Fig. 12-15, where we have plotted A and W for increasing monolayer thicknesses calculated with different refractive indices as indicated. Given the experimental accuracy of the ellipsometric angle determination of about 0.005°, it becomes evident that for monolayer thicknesses below approximately 2 nm only A can be reliably measured. This means then, however, that no independent determination of the (geometrical) thickness of the polymer layer and its refractive index (not even in an isotropic approximation) can be obtained.

5.08 r| M M | I i i i | I i i i | i i i i | i i i I | I I

0

0.5

1.0

1.5

2.0

2.5

thickness / nm

Figure 12-15. A and "Pasa function of n1, dx for a monolayer at the air-water interface, calculated with A = 594nm, 0 = 50.00°, no = 1.333. Different curves correspond to different values of n1 as indicated in the figure. Above the Brewster angle, 6 « 53°, an increase in A indicates a thicker layer; below the Brewster angle, it is the reverse.

552

12 Optical Properties of Polymers

Nevertheless, even by recording only A as a function of the molecular area (ellipsometric isotherm) helpful information about polymer monolayers can be obtained. One example measured with poly(y-methylL - glutamate -co-y- octadecyl - L - glutamate) (PM-co-OLG) is given in Fig. 12-16. The n~A isotherms during compression and decompression are compared with the A-A isotherms from ellipsometry. One can see, for example, that the first break in the compressibility at approximately 0.20 nm 2 per repeat unit is caused by a change in the intermolecular interaction whereas the overall film thickness still increases monotonically. On the other hand, the plateau region beyond the second break in the TI-A isotherm seen for areas per repeat unit smaller than about 0.14 nm 2 is not associated with a collapse of the monolayer everywhere on the trough. It is rather an instability just in front of the barrier: Only if the material piled up by this "snowplow" effect reaches the laser spot of the ellipsometric measurement do the A -values increase strongly, indicating the increasing film thickness. These ellipsometer configurations typically average monolayer properties over approximately 1 mm2. In some cases, however, for example during first-order phase transitions with coexisting domains, higher lateral resolution of the optical data is needed. Two microscopic techniques were recently introduced that give qualitative information about optical properties on the jLim2 scale. One is the Brewster angle microscope (BAM), which operates in a reflection mode at the Brewster angle (of the pure water surface) with p-polarized light (Honig and Mobius, 1991; Henon and Meunier, 1991). The dark image of the surface changes its intensity when covered by a monolayer (Fig. 12-17). In this way, details of the lateral structure formation of

0.05

0.10

0.15

0.20

0.25

area / repeat unit (nm2)

Figure 12-16. II — A diagram and simultaneously measured \A — AU2O\ isotherm of PM-co-OLG during compression and decompression. Measurements were carried out at T = 20 °C and I = 633 nm at an angle of incidence of 0 = 54.07°.

-V >

*

i

«t

++ •••

4#+:++*++,+.*• Figure 12-17. Brewster angle microscopic image of an azobenzene-derivatized liquid-crystalline Langmuir monolayer at the water-air interface.

12.4 Polymers in Special Configurations

polymer monolayers during compression can be resolved that were previously observable only by fluorescence microscopy after adding a suitable dye. Clearly, the BAM has the advantage that no label is necessary. In passing we note that also the ellipsometric set-up can be converted easily into a microscope that allows one to take pictures of laterally heterogeneous monolayers. Figure 12-18 shows an example obtained from a phospholipid monolayer in the coexistence region of fluid and condensed domains. By changing the polarizer settings contrast inversion can be induced and analyzed. This gives information about the optical thickness of the different phases just as ellipsometry does but with a high lateral resolution (Reiter et al., 1992).

553

Figure 12-18. (a) Image of domains of the phospholipid DMPE at the air-water interface, compressed at a temperature of 20 °C into the coexistence region. The dark parts correspond to the crystalline analogous domains and the bright ones to the fluid matrix, (b) Contrast inversion by changing the extinction settings with respect to the fluid matrix.

12.4.1.2 Langmuir-Blodgett-Kuhn Multilayer Assemblies The above-mentioned monolayers can be transferred to a solid support by dipping and withdrawing a suitable substrate through this highly organized film at the water-air interface. Thus, multilayer assemblies with a precise control of their thickness can be obtained, as is schematically sketched in Fig. 12-19. First reported by K. Blodgett (1935), this technique has regained worldwide interest as an excellent tool for the build-up of supramolecular architectures (Roberts, 1990) with tailormade molecules ever since the pioneering work of H. Kuhn in the early 1970s. These systems are therefore called LangmuirBlodgett-Kuhn (LBK) layers. Various techniques have been developed in recent years sensitive enough to allow a quantitative evaluation of the optical properties of these ultrathin coatings (Swalen, 1986). One group of techniques uses evanescent waves as highly specific

Langmuir

—

Blodgett

—

Kuhn

Figure 12-19. Schematic representation of the Langmuir-Blodgett-Kuhn technique: The Langmuir monolayer prepared at the water-air interface at a selected surface pressure is transferred during the withdrawal of the solid substrate. Supramolecular architectures can be built up by choosing different functional units in the individual layers.

sensitive interfacial light probes (Knoll, 1991 a, b). The simplest case for the existence of an evanescent wave is the well-known total internal reflection of a plane electromagnetic wave at the base of a glass prism (index of refraction nx) in contact with an optically less dense medium (with n2
12 Optical Properties of Polymers (b)

etector

Glass Prism

Detector

Glass Prism •Metal

Evanescent Field

0

Figure 12-20. Two configurations for evanescent wave optics, (a) Top: total internal reflection of a plane wave at the base of a glass prism; bottom: the reflectivity R recorded by a detector as a function of the angle of incidence shows the increase to unity at 6C, the critical angle for total reflection, (b) ATR setup for the excitation of surface plasmons (PSPs) in Kretschmann geometry. Top: a thin metal film (d « 50 nm) is evaporated onto the base of the prism and acts as a resonator driven by the photon field; bottom: the resonant excitation of the PSP wave is seen in the reflectivity curve as a sharp dip at coupling angle 60.

proaches the critical angle, 0C, for total reflection (Fig. 12-20 a, bottom). Closer inspection of the E-field distribution in the immediate vicinity of the interface shows that above 9C the light intensity does not fall abruptly to zero in air, but there is instead a harmonic wave traveling parallel to the surface with an amplitude decaying exponentially normal to the surface. The depth of penetration / defined by the 1/e attenuation is given by \

...

.

(12-67)

and is found to be on the order of the wavelength of light. This type of wave is called an evanescent wave.

One way of introducing surface plasmons [plasmon surface polaritons or PSPs (Burstein et al., 1974; Raether, 1977)], especially in the experimental set-up called the Kretschmann configuration (Kretschmann, 1972) (see Fig. 12-20b, top), is to note that the nearly free electron gas in the thin (^50nm) metal film evaporated onto the base of the prism acts as an oscillator that can be driven by the electromagnetic wave impinging upon that interface. Therefore, we are dealing with the resonant excitation of a coupled state between the plasma oscillations and the photons, that is, the "plasmon surface polaritons" (Raether, 1988). This resonance phenomenon can be clearly seen in the ATR scan (attenuated

12.4 Polymers in Special Configurations

total reflection; see Fig. 12-20b, bottom): Below 6C the reflectivity is very high because the metal film acts as a mirror with little transmission. Above 6C for total internal reflection, however, a relatively narrow dip in the reflectivity curve at 90 indicates the resonant excitation of such a PSP wave at the metal-air interface. The coupling angle is given by the energy and momentum matching condition between photons

PSP U

PSP1

555

and surface plasmons: CO

{

.

= nx — sm< c

(12-68)

with fc°p being the magnitude of the PSP wavevector, k°h the parallel component of the photon wavevector, co the photon energy, c the speed of light, and (p0 the (internal) coupling angle (see Fig. 12-20b). Again, we are dealing with an evanescent wave propagating along the interface with a penetration depth into air of the order of the wavelength. The resonance character of this excitation gives rise to an enhancement of the £-field at the interface by more than a factor of 10, which is the origin of the remarkable sensitivity enhancements obtainable, for example, in Raman spectroscopy when working with PSP light (Ushioda and Sasaki, 1983; Knobloch etal., 1989). Since we are dealing with well-defined modes obeying a known dispersion relation, co versus fc°p, each photon of energy h coL allows the excitation of exactly one PSP mode. This is schematically depicted in Fig. 12-21 a. The solid curve represents the dispersion of the surface plasmons at a

8i

50 nm Ag or Au Langmuir-Blodgett-Kuhn Multilayer Assemblies

Figure 12-21. (a) Dispersion relation, co vs. ksp, of plasmon surface polaritons at an Ag-air interface (PSP0, solid line) and at an Ag-dielectric coating-air interface (PSP1, broken line). Laser light of energy hwL couples at angles 60 and 61, respectively, given by the energy and momentum matching condition (see the intersection of the horizontal line at coL with the two dispersion curves), (b) Schematic of the experimental set-up for surface plasmon spectroscopy.

556

12 Optical Properties of Polymers

(a) Hydrocarbon Polyglutamate Hydrocarbon Silver Glass

(b)

PSPs at the base of the prism which is first covered with the thin metal layer by evaporation and then coated, for example, by LBK layers of increasing thickness. The reflected intensity is monitored as a function of the angle of incidence by a photodiode. The thin dielectric coating causes a shift of the dispersion curve (PSP) to higher momentum klp = k°p + Aksp

Z.0

55

6/deg

Figure 12-22. (a) Schematic drawing of the architecture of the thin film: 50 nm Ag is evaporated onto the glass-prism and then a double layer of polyglutamate is laid down by the Langmuir-Blodgett-Kuhn technique. The long, oriented rods are the polypeptide oc-helices. Hydrocarbon represents the disordered alkyl sidechains. (b) Reflected intensity (AL = 633 nm) as a function of the (external) angle 6 for various film architectures: Solid circles are bare Ag (eAg = —16.35 + i0.6, thickness 5 = 50.8 nm), solid triangles are two layers of polyglutamate on top, solid squares: 4; crosses: 6; open triangles: 10, and diamonds: 20 layers. Solid lines are Fresnel calculations.

metal (e.g., Ag)-air interface (PSP0). The horizontal line at coL intersects the dispersion curve at /c°p and thus defines the coupling angle 60. A typical experimental setup is schematically given in Fig. 12-21 b. p-Polarized photons from a laser excite

(12-69)

which, according to Eq. (12-68), shifts the resonance to a higher angle 81. From this shift and Fresnel's equations, one can calculate the optical thickness of the coating (Gordon and Swalen, 1977). This is demonstrated quantitatively for LBK multilayer assemblies prepared from polyglutamate monolayers. A schematic drawing of a transferred double layer is presented in Fig. 12-22 a. It shows the quasi-two-dimensional nematic structure of these "hairy rods" - called stiff polyglutamate-helices with long, flexible alkyl sidechains (Duda et al., 1988). Figure 12-22b gives the results of ATR scans obtained from the bare metal and 2, 4, 6, 10, and 20 layers of polyglutamate, respectively (Hickel et al., 1990). The symbols are the experimental data points, the solid lines Fresnel calculations. All these layer preparations can be described by a constant index of refraction, nz = 1.486, and the multiple of a constant thickness per monolayer of d0 = 1.75 nm.

12.4 Polymers in Special Configurations

The sensitivity of this technique to extremely small refractive index changes is also demonstrated in Fig. 12-23. Figure 12-23 a gives the angular scan (full circles) of the PSP resonance found for a sample consisting of Ag/8 layers of polyglutamate/ 6 layers of an azobenzene derivatized (and hence photoreactive) liquid-crystalline polyacrylate (structural formula is given in the inset) which was processed by the LBK technique (Sawodny et al., 1992). The spacer layers were necessary in order to decouple (electronically) the azobenzene chromophores from the metallic acceptor states in the substrate. Illuminating the sample with UV light [1 = (360 ± 30) nm] shifts the dark-adapted all-trans chromophores through trans-cis isomerization to a new photostationary equilibrium with a

557

high ds-isomer content. As a result the optical refractive index anisotropy is changed and the index normal to the layers is reduced. This shifts the surface plasmon resonance to smaller angles - the LBK multilayer is now optically thinner (open triangles in Fig. 12-23 a). If this change of the optical thickness is followed on-line (during illumination) by recording the reflected intensity at a fixed angle of incidence (e.g., 0 = 47.2°, cf. Fig. 12-23 a) a kinetic analysis of the refractive index changes can give information about the reaction rates, the equilibrium changes, the reversibility, etc. The latter important aspect is shown in Fig. 12-23b. The first rapid decrease of nz upon UV illumination can be partly restored by switching to visible light [X = (450 ± 30) nm], which isomer-

-VIS-

4?*

1.57 (b)

100

1.56 •

1.55

46

48 6 /deg.

1.54

0

10 20 exposure time / min

Figure 12-23. (a) Reflected intensity as a function of the angle of incidence for a sample consisting of Ag 8 layers PM-co-OLG/6 layers of the photoreactive liquid-crystalline polymer given in the inset. Full circles: layers as prepared (in the dark-adapted trarcs-state); open triangles: after illumination of the layers with U V light (A = 360 nm) for 10 min. Full curves are Fresnel fit calculations, (b) Refractive index change as obtained by recording the reflected intensity of the same sample at a fixed angle of incidence [0 = 47.2°, cf. (a)] while first illuminating with UV light, then after 6 min switching to visible light (VIS), etc.

558

12 Optical Properties of Polymers

izes the azobenzenes back into the transstate. These reaction cycles can be conducted many times. It is important to note that, given the signal-to-noise level of these data, index changes as small as Anz = 0.0001 can be monitored in reactive layers that are only 15 nm thick! The above described angular shift of the resonance condition for different coating thicknesses is also the basic mechanism that generates the high contrast achievable in surface plasmon microscopy (SPM) of heterogeneous thin films (Rothenhausler and Knoll, 1988). Here, the reflected, scattered, and diffracted plasmonic light is Fourier-backconverted by a lens to form an image of the interface in real space on a TV camera in our case (see Fig. 12-24 a). Only those areas that are at resonance appear dark, whereas all other domains according to their relative shift of the coupling condition are more or less bright. If the laser light is coupled in at 90 the bare metal surface is at resonance for PSP excitation, and hence almost no light is reflected. Any coated region, however, is still offresonance, and hence reflects the full laser intensity. The different areas, therefore, show maximum contrast in reflection. In previous work we demonstrated that variations of the thickness of different areas as small as a few tenths of a nanometer are enough to generate sufficient contrast for an image (Hickel et al., 1989). Illumination of the sample is typically done with a He-Ne taser beam of approximately 1 mm2 spot size. The lateral resolution of this microscopic technique has been shown to be better than 5 jim (Hickel and Knoll, 1990 a). If SPM pictures are taken (and stored on magnetic tape) as a function of the angle of incidence, image analyzing computer routines allow quantitative evaluation of the reflected intensity from areas as small

as 5 x 5 }im2. The information obtained, again reflected intensity versus angle of incidence as in the usual ATR scan, can be fit to Fresnel's formulas as discussed above to yield the optical thickness, but this time with lateral resolution. The example given concerns the SPM of laterally structured subtle thickness changes caused by UV-photon-induced scission of Si-Si bonds in polysilane films, which leads to a volatilization of the illuminated material (Sawodny et al., 1991). This class of substances recently attracted considerable interest as self-developing photoresists, materials which do not require additional wet etching after exposure to convert the illumination pattern into a surface relief structure (Miller and Michl, 1989). Figure 12-24b shows a series of SPM pictures taken at various angles of incidence from approximately 40 nm thick poly(methylphenylsilane) film spin-coated onto a Cr/Au substrate and then illuminated for 15 min in an Ar atmosphere (10~ 3 MPa) through an electron microscope copper grid. The exposed areas tune into resonance first - a clear indication that they are optically thinner. A quantitative analysis is possible if one plots the mean gray values of preselected areas obtained by an image analyzing program that takes the intensity histograms from the pixels in each of these local frames as a function of the angle of incidence, 6. This is given in Fig. 12-24c for both illuminated and unexposed areas. Assuming that the index of refraction for both areas is the same (n = 1.635), we can obtain from a Fresnel fit to the data the two respective thicknesses, as indicated in Fig. 12-24c. Given the shift of the resonance curve upon ablation of only 2 nm, one can estimate that the thickness resolution of this quantitative microscopic technique is a few angstroms.

12.4 Polymers in Special Configurations

559

(a)

Laser, wavelength X

Screen/Camera

Evanescent Field of PSP

(b)

185°

66

68 9/deg

Figure 12-24. (a) Scheme of SPM set-up in the Kretschmann configuration. The scattered and out-coupled plasmon light is Fourier transformed by a lens to give an image of the interface in real space, (b) Series of SPM pictures taken at various angles of incidence from a structured polysilane film partly ablated by irradiation for 15 min in an Ar atmosphere through an electron microscopy grid (a line grating in this case). First, the thinner areas are tuned into the PSP coupling resonance: Between 66° and 67° the contrast inversion is reached, and then the unexposed areas are darker, (c) Quantitative analysis of the average gray value (a measure of the reflected intensity) taken from the two different areas (irradiated: open circles, unexposed: solid circles) pictured in (b) as a function of the respective angle of incidence. Solid and broken lines are Fresnel fits to the data points. From their relative shift, one calculates an ablated thickness of Ad = 2 nm.

560

12 Optical Properties of Polymers

12.4.1.3 Self-Assembly Mono- and Multilayers on Solid Supports

An extremely attractive alternative to transferred Langmuir monolayers are monomolecular films prepared by a selforganization process: Suitable substrates are just immersed into a solution that contains the surface-active molecules. These then self-assemble at the solid-solution interface to a well-organized monomolecular layer with a structural (positional and orientational) order absolutely comparable to Langmuir films. This is schematically shown in Fig. 12-25. The first examples, introduced by Sagiv (1980) were based on the specific interaction between the silyl-headgroup of the amphiphiles (e.g., of octadecyltrichlorosilane) and the polar silanol groups on glass- or quartz-substrates. More recently, a major research activity has focused on the physics and chemistry of self-assembly monolayers (SAMs) based on the interaction of thiol-, disulfide-, and sulfide-groups with Au and Ag surfaces. The possibility for chemical manipulations at the co-position of long chain thiols [X-(CH 2 ) n -SH with X: functional group] opened a wide field of tailor-made surface functionaliza-

Figure 12-25. Scheme of the self-assembly process that leads to a (polymer) monolayer at a solid substrate.

tions. This molecular engineering of surface properties has wide implications for lubrication, patterning, (bio-)sensing, cellsurface interactions, etc. One example of a polymeric SAM with multifunctionalities is presented in Fig. 1226. The structural formula of this copolymer (Fig. 12-26 a) shows the disulfide-containing "stitching"-units that accomplish the covalent and hence stable binding to the (noble) metal surface (Spinke et al., 1992). The long alkyl-chains in another comonomer play the role of the hydrophobic tails of "classical" amphiphiles while the short alcohol moieties are introduced to balance the polar and apolar parts of the monolayers. The self-assembled monolayers can be analyzed again very sensitively by PSP spectroscopy. This is demonstrated in Fig. 12-26b. The resonance angle recorded for the bare Au substrate (full circles and full curve) is shifted to higher angles by A9 = 0.5° (open circles). From the Fresnelfit calculations (dashed curve) one obtains an (optical) thickness of the monolayer of d = 1.9 nm (assuming an index of refraction of ft = 1.55). From this result (and from other measurements) the structural model that is depicted in Fig. 12-26c has emerged: a well-ordered monomolecular layer of amphiphilic moieties are coupled to the rough and irregular substrate by a polymeric buffer layer. This decoupling of the perfectly organized monolayer from the imperfect solid support not only results in a polymer coating with interesting viscoelastic properties but is also believed to allow for the build-up of fluid membrane systems, an extremely relevant aspect for many membrane biophysical problems. In passing we note that recently the formation of multilamellar structures by the self-assembly technique was also reported (Maoz etal, 1988; Tillmann et al., 1989):

561

12.4 Polymers in Special Configurations

(a) CH2 I ^CH2(CH2)16CH3 CH,-C-COO-CH, CH,-NH-CO-N V •" -CH2(CH2)16CH3 CH2 H-OCOO-CH 2 CH 2 -OH

CH 3 -S-S-CH 2 CH 2 -OOC-C-CH 3

Figure 12-26. (a) Structural formula of a ter-polymer with different functionalities that leads to a well-ordered self-assembly monolayer, (b) PSP-resonance of the bare gold substrate (full circles) and after coating with a self-assembly monolayer of the ter-polymer (open circles), (c) An artist's view of the polymer-supported monolayer prepared by the self-assembly process on a rough substrate.

(b)

(c) Monolayer

Polymer Support

Substrate 6/deg

After the formation of the first monolayer, suitable processing of the endgroups of the molecule (e.g., the oxidation of terminal double bonds to carboxyl-groups) transforms the SAM layer into a substrate for further self-assembling of the next monolayer. Even thick multilayers of more than 100 layers have been demonstrated. 12.4.2 Integrated Optics

Since the concept of "integrated optics" emerged in the late 1960s a significant growth of the areas of optical communication, signal processing, etc. has stimulated

a major research and development effort in the field of organic and polymeric materials aimed at optimizing their functional properties by molecular design and synthesis (Stegeman et al., 1987; Prasad and Williams, 1991; Messier et al., 1989). This concerns not only their nonlinear optical properties (i.e., primarily their second- and third-order hyperpolarizabilities, see Sec. 12.5) but also their linear optical response to light and, equally important, also other chemical and physical properties of the material and processing parameters like stability (against chemical attack, temperature, light, aging, delamination, etc.), ease

562

12 Optical Properties of Polymers

of preparation (compatible with technologies developed for microelectronic device fabrication), cost-effectiveness and others. One very crucial requirement of a material for the envisaged integrated optics application is that it can be prepared as a structure capable of guiding light waves with acceptable losses. The three corresponding basic configurations (Lee, 1986; Chang, 1991) are schematically depicted in Fig. 12-27: The planar waveguide structure (Fig. 12-27 a) given in the so-called asymmetric slab configuration with a solid substrate and air as superstrate, the various channel waveguides (Fig. 12-27 b, strip, embedded strip, and buried strip, from top to bottom) prepared by starting in some cases with a planar waveguide and then further structuring and etching the sample, and the optical fiber (Fig. 12-27c) with a

Planar Waveguide Substrate

high index core and a lower index cladding. Such waveguide structures are discussed in the literature not only with respect to their role as passive interconnects between different integrated optical circuit (IOC) components but also in the context of various design concepts for active IOC modules such as directional couplers, phase and amplitude modulators, and second harmonic generators for laser diodes (Stegeman and Stolen, 1989). In this section we first describe very briefly the basic theoretical approaches (within Maxwell's theory) needed to understand the phenomenon of guiding light in confined geometries. We limit the discussion to the case of a planar waveguide structure but the general ideas allow for a straightforward extrapolation to more complicated designs. After a short presentation of various schemes for coupling plane electromagnetic waves, for example, photons from a laser, into such structures we give a few examples of waveguides fabricated from novel polymeric materials with promising optical properties that can be tailored to some extent for future device applications.

Channel Waveguides

12.4.2.1 Planar Waveguide Structures

Guiding Light in an Asymmetric Slab

c) Fiber Cladding Core

Figure 12-27. Basic waveguide structures, (a) Planar waveguide (shaded) on a substrate, (b) Channel waveguides: strip, embedded strip, buried strip (from top to bottom), (c) Optical fiber with the core and the cladding of lower refractive index.

We consider the asymmetric slab configuration sketched in Fig. 12-28. The thin waveguide layer (film with index of refraction n2 and thickness d) is bound by the substrate (n3) and the superstrate (e.g., air n1 = 1). In order to obtain total internal reflection at each interface the waveguide material must fulfill the requirement n1
12.4 Polymers in Special Configurations

563

Air Film Substrate n3

Figure 12-28. The geometry of a planar waveguide with substrate (index of refractionrc3),film (thickness d, n2), superstrate, e.g., air (n^). Given is also the coordinate system used to derive the mode equation. For illustration only note the ray optic description of the guided light.

schematically in Fig. 12-28). Note the evanescent wave extension giving rise to the Goos-Hanchen shift (Goos and Hanchen, 1947). Within Maxwell's theory it is straightforward to show that for the threelayer system depicted in Fig. 12-28 one can write the optical field in each of the media (omitting the time dependence and the propagation in the x-direction). z>d 0
^3y —

(12-70) (12-71) (12-72)

Matching both electric and magnetic fields at each interface we obtain at z = 0 (12-73)

A3 = A2 and fc

3z^3 = fc2z^2 s i n ( £

(12-74)

Therefore, 0 = tan" 1 p23 with p23 = k3jk2z

(12-75)

and (12-76)

= A3 At z = d one finds A, = A2 cos (k2zd + )

(12-77)

and — klzA1 =

-k2zA2 sin(k2zd + 4>) (12-78)

(12-80) The phase factor in Eqs. (12-75) and (12-79) must be the same within a multiple of 7i. Therefore, one arrives for the nonabsorbing case considered here at the wellknown eigenvalue equation for waveguide modes of order m (Tien, 1969): k2zd = tan" 1 jS21 + tan" 1 j8 23 + mn (12-81) For a given set of materials and hence refractive index combination (which determines t a n " 1 / ^ ! and tan~ 1 j8 23 for a fixed frequency) the number of eigenmodes depends only on the thickness of the waveguide layer. The detailed analysis of Maxwell's equations can yield the optical field distribution for each eigenmode (Tamir, 1979). Figure 12-29 shows the optical field distributions for the first three modes (with their E-field being parallel to the film along the y-axis) calculated with the following indices of refraction: substrate w3 = 1.46, waveguide film n2 = 1.55, cladding (air) n1 — \. Note that the optical field extends as an evanescent wave into both the cladding and the substrate where it decays exponentially as a function of distance from the interface, though for this asymmetric slab in an asymmetric way. Coupling Light into Waveguides

Then (j) = —k2zd -f tan"ip21

and

with jg 21 = ' C l z / ' C 2 z (12-79)

The three most important techniques for coupling light into a waveguide are

564

12 Optical Properties of Polymers Air

Film

Substrate Figure 12-29. Optical field distribution in a planar waveguide structure of substrate/film/air configuration for the first three s-polarized modes (m = 0,l, 2). The higher index of the substrate (n3 = 1.46) compared to the superstrate (air, n1 = \.O) is the reason for the much stronger evanescent field penetrating into the substrate material.

schematically sketched in Fig. 12-30. The grating (Fig. 12-30 a) and prism coupling method (Fig. 12-30b) are commonly used for planar waveguides. Efficient coupling can be achieved only under wavevector matching conditions. For grating coupling this requires that the photon wavevector component parallel to the surface plus or minus a multiple of the grating vector magnitude G = 2 n/A with A being the groove spacing of the grating matches to the waveguide propagation vector

= kon1sin9

±mG

(12-82)

a) Grating

Prism

c)

End-Fire

Figure 12-30. The three basic coupling geometries, (a) Grating coupling, (b) prism coupling. Both methods require the tuning of the angle of incidence, 9. (c) End-fire coupling, e.g., by a microscope objective.

In the case of prism coupling the projection of the photon wavevector at the base of the prism must be equal to the guided wavevector: /cx?m = /corcpsin6>

(12-83)

with np being the index of refraction of the prism. This requires a prism with an index higher than the waveguide material. The end-fire coupling (e.g., from a microscope objective to a channel waveguide) matches the incident field spatial distribution to that of the guided wave (Fig. 1230 c). For a given waveguide, for example, a planar structure of fixed thickness, the number of modes and their angular positions when excited by a grating or prism [see Eqs. (12-82) and (12-83)] depends on the wavelength employed and on the index of refraction of the various layers. For a thin film of an unknown waveguide material this allows for the determination of the corresponding materials parameters, as we will show next. Planar Waveguides from Novel Polymeric Materials For the examples presented in the following, a modified prism coupling technique as schematically depicted in Fig. 1231a was used. The waveguide modes were launched with the help of a high index

12.4 Polymers in Special Configurations

,7) Detector

Glass Prism Metal Waveguide

(b) CH 3

CH 3

100

:

\

2

i

50

t

_J

20

1

i *

1

1

0/deg

Vv

f

1; 80

Figure 12-31. (a) Attenuated total reflection set-up for coupling surface plasmon - as well as waveguide modes into a planar film structure. Coupling is achieved through a thin metal gap at the base of the prism, (b) Reflected intensity recorded as a function of the angle of incidence for a thin film of a solid polyelectrolyte (structure formula given on top) spincoated onto the Ag-coated (s = 50 nm) base of a high index prism (SF57) [cf. (a)]. The dots are the experimental data points; the full curve is the theoretical fit. Excitation with p-polarized light from a He-Ne laser (A = 633nm).

prism by coupling through a metal (Au or Ag) gap of 50 nm thickness excited with p-polarized light. In this geometry the m = 0 mode is the surface mode, the surface plasmon. Its optical intensity peaks at the metal-waveguide interface and decays (for

565

films of typical waveguide thicknesses of approximately 1 jim or more) completely within the film. Its polarization is predominantly parallel to the film normal. Its angular position, therefore, depends only on nz, not on the waveguide thickness. The other waveguide modes can be excited with either s- or p-polarized light and will, in general, depend on all three indices, that is, HX, tty, and nz as well as on the film thickness [cf. Eq. (12-81)]. Figure 12-31 b shows a series of waveguide modes obtained with p-polarized excitation of X = 633 nm in a spin-coated film of a solid polyelectrolyte whose structure formula is given in the inset. The various modes were measured in reflection. The broad dip in the reflected intensity at the high angle side (6 « 73°) is the surface plasmon. The crosses are the experimental data points, the full line (for better clarity somewhat shifted) is a theoretical fit based on a Fresnel calculation (Wolter, 1956). From this one obtains the thickness of the waveguide film d = 1.85 jwm as well as its index of refraction n = 1.63. By recording the reflectivity as a function of the angle of incidence for s-polarized light (not shown) it was found that an isotropic approximation of the refractive index was sufficient in order to characterize the whole waveguide mode pattern. Nevertheless, a closer comparison between the experimental and the theoretical curve of the angular reflectivity shows some distinct differences. First of all, the experimental modes are somewhat broader than calculated with an ideal slab geometry for the waveguiding film. This indicates some thickness and lateral refractive index heterogeneities within the spin-coated film. In addition, not all calculated angular positions for the various modes fully coincide with the experimental ones. Given the different field intensity distributions of the different modes (cf. Figs. 12-29 and 12-31)

566

12 Optical Properties of Polymers

this discrepancy points to a certain perpendicular refractive index variation (an index profile) within the waveguide. Both deviations from ideality within the plane and normal to the plane need to be carefully avoided in an IOC structure because such heterogeneities clearly would limit the optimum performance of the device.

Screen/Camera

Waveguide

Optical Waveguide Microscopy (OWM) One way of optically characterizing the quality of planar waveguide structures, for example, spin-coated films, is the recently introduced optical waveguide microscopy (OWM) (Hickel and Knoll, 1990 b) which was developed in analogy to surface plasmon microscopy. Both techniques employ bound optical waves to illuminate the sample instead of using normal photons. The basic principle of OWM is schematically depicted in Fig. 12-32 a. Laser light is coupled to the waveguide at an angle of incidence 9. The nonideal planar structure of the film leads to an angular broadening of the resonance condition. Now, the spe-

Figure 12-32. (a) Scheme of the experimental configuration for taking microscopic pictures of the local resonances in planar waveguide structures. Laser light is coupled to the waveguide at an angle of incidence 6. The scattered and out-coupled light is Fourier-backconverted by a lens to form an image of the waveguide on a TV camera, (b) Series of optical waveguide microscopic pictures taken from a planar film of a solid polyelectrolyte spin-coated onto the base of a Ag-coated prism [cf. (a)]. The images taken at different angles of incidence (as indicated) show the local variation of the resonance by the changing pattern of dark areas. Each picture represents 570 urn x 420 jam on the waveguide, (c) Topographical map of the "planar" waveguide structure investigated by OWM. The lines were obtained from pictures like those shown in (b) by simple image analyzing routines indicating the local waveguide resonances. The numbers give the corresponding local thickness in nanometers of the waveguide as derived from Fresnel calculations.

345.0

12.4 Polymers in Special Configurations

cial prism coupling configuration used for these experiments ensures that the propagation lengths of the guided modes do not exceed circa 10 jim because guided light intensity constantly couples out through the metal gap and the prism. This light reflected and scattered and out-coupled is Fourier backconverted by a simple lens to form an image of the thin film structure on a screen or on a TV camera (CCD) in our case, just as discussed above for surface plasmon microscopy. As a result we are dealing with a microscope that uses the guided light itself as the illumination light source for the imaging of the sample. Figure 12-32b shows a series of OWM pictures taken with p-polarized light (k = 633 nm) at different angles of incidence as indicated. The sample was a spun-on film of the ionene polymer shown in Fig. 12-31 b. The dark regions correspond to the local resonant coupling to waveguide modes. As one varies the angle of incidence, different areas tune into resonance corresponding to the respective local fulfillment of the eigenvalue equation [see Eq. (12-81)]. The reduced propagation length of the modes in this experimental configuration allows for the rather local analysis of this resonance pattern. If the index of refraction of this thin film material is known, a Fresnel calculation allows for the conversion of the angular position of the resonance into a local thickness. A simple image analyzing computer routine gives for each frame, the line(s) of lowest intensity which - for a series of pictures taken at different angles of incidence - can be used to construct a topographical map of the thickness variations of the waveguide slab. This is shown in Fig. 1232 c for our ionene sample. That we are dealing, in fact, with thickness and not refractive index variations was checked independently by performing the same analysis

567

with s-polarized light (not shown), which gave essentially the same thickness data. It is very obvious that such a film would not meet the requirements for device applications. This example demonstrates, however, that experimental techniques are available that allow a comprehensive testing of polymer optical properties and processing steps. Waveguide Loss Measurements Typically, waveguides are designed to propagate light over long distances. This requires, first of all, the absence of any absorptive losses, either by molecular, electronic or vibronic transitions or by dissipation, for example, in metallic materials used as electrodes. Equally important is the control of any scattering losses, which would also result in an enhanced damping of the guided modes. This includes not only the minimization of bulk (volume) fluctuations of the optical density but requires also the ultimate control of surface scattering processes, for example, at interfacial roughness caused by processing steps. One way of characterizing waveguide structures along these lines is to measure the losses of the guided waves directly. Figure 12-33 a shows a scheme of an experimental set-up designed for loss measurements in planar structures. Laser light is coupled to the waveguide by a grating (or prism). A coherent fiber bundle is used to transmit the light scattered off the guided streak in the planar structure which is a measure of the local mode intensity. A photomultiplier tube with a slit diaphragm reads this scattered intensity with high spatial resolution and hence allows the quantitative analysis of the (exponentially) attenuated optical field in the waveguide. A data acquisition and analysis system gives the

568

12 Optical Properties of Polymers

translation stage PMT

waveguide substrate

aspects are noteworthy: (i) For most of the wavelength range investigated the data seem to follow a l ~ 4 power law (straight line in Fig. 12-33 b) which suggests Rayleigh scattering as the predominant attenuation mechanism, (ii) For wavelengths X > 650 nm losses in the range of 1 dB/cm or less seem to be feasible with these materials, which makes them also technologically interesting. Supramolecular Design of Waveguide Films by the Langmuir-Blodgett-Kuhn Technique

500 600 Wavelength X/nm

700

Figure 12-33. (a) Scheme of the set-up used for the determination of mode losses in waveguide structures. Laser light is coupled to the planar waveguide by means of a grating or prism. A fiber bundle records the local level of stray light, which is a measure of the field intensity along the guided streak, (b) Loss data for s-polarized (m = 0) modes (given as attenuation in dB/cm) for various wavelengths as obtained for a planar waveguide structure fabricated by spin-coating from a solid polyelectrolyte. The straight line is a X ~ 4 power law. Data from Mathy et al. (1991).

loss-values typically in dB/cm (1 dB corresponds to an intensity attenuation to 79%). Figure 12-33 b gives the results obtained from a polyelectrolyte thin film prepared by spin-coating onto a low-index glass substrate measured in air with prism coupling. Data points taken from Mathy et al. (1991) were measured with different laser lines and plotted correspondingly as a function of the wavelength employed. Two

Finally we should briefly discuss possibilities for the ultimate control of a planar waveguide structure at the molecular level as it is in principle possible by the LBK method. Here, multilayer assemblies capable of carrying guided modes are built-up, layer by layer, by dipping and withdrawing a substrate many times through a monomolecular layer prepared at the water-air interface (Swalen et al., 1978). The number of possible molecular structures of the individual layers is near infinity and so are the functionalities that can be tailored into these systems. Even if it is unlikely that the tedious preparation process will ever make it into a large-scale production line, these LBK films nevertheless are extremely helpful as model systems to develop and test molecular structural concepts, functional units (nonlinear chromophores, etc.), supramolecular assemblies, and device concepts. As an example of a simple passive planar waveguide structure prepared by this LBK technique we present in Fig. 12-34 results obtained with a multilayer assembly of the polyglutamates previously presented. Figure 12-34 demonstrates the good optical quality of a thick multilayer assembly (150 layers) capable of guiding modes with different polarizations: the upper frame was

12.4 Polymers in Special Configurations

569

n

\\ ~ni = 0.01 and demonstrates the sensitivity of the coupling conditions to subtle refractive index variations. 12.4.2.2 Channel Waveguides

8/deg

Figure 12-34. Waveguide mode pattern as obtained from a polyglutamate (PM-co-OLG) multilayer assembly of 150 layers for p- (upper frame) and s-polarized light (lower frame). Full circles are obtained for modes propagating perpendicular to the dipping direction, which is the preferred axis of the hairy rods, crosses are from modes propagating parallel to the rods. The full curves are Fresnel calculations showing the corresponding in-plane anisotropy of the index of refraction.

taken with p-polarized light, the lower frame shows data with s-polarization. The full circles are experimental points measured with waveguide modes propagating perpendicular to the dipping direction, the crosses are for parallel propagation. The full lines are Fresnel fits and show the excellent theoretical description of these structures as planar slabs. The optical inplane anisotropy that can be analyzed quantitatively amounts to only An =

The theoretical description of guiding light in a two-dimensional waveguide structure (optical confinement in two dimensions as in the channel waveguides shown in Fig. 12-27b) adds nothing new, and just requires more complex numerical techniques for the calculation of the exact dispersion relation and hence the various modal structures. These channel waveguides are perhaps the ultimate structures needed for any IOC. These include, in particular, active devices based on nonlinear optical effects. Two basic structures are shown in Fi^. 1235: the (electro-optical) coupler, in which two channel waveguides are sufficiently close so that they can interact like two weakly coupled mechanical oscillators, is shown in Fig. 12-35 a. The strength of the interaction which determines, for example, the flow of light intensity from one channel (a)

Electrode / \Channel Waveguides

(b)

Figure 12-35. Basic structures for integrated optical circuits (with special emphasis on electro-optical control): (a) Channel waveguide coupler, (b) (integrated) Mach-Zehnder interferometer.

570

12 Optical Properties of Polymers

to the other can be controlled by the application of a DC electric field through the Pockels effect provided the polymeric waveguide material has a correspondingly large second order susceptibility x(2)- The integrated Mach-Zehnder interferometer shown in Fig. 12-35 b allows for a phase shift of the light in one branch relative to that propagating in the other channel. This then leads to constructive or destructive interference at the output. Again, this interaction can be controlled by either #(2)-active materials through the application of an electrical field (as shown in Fig. 12-35 b) or by a material with an intensity-dependent refractive index n2 [cf. Eq. (12-62)]. The fabrication of these channel waveguides in many cases starts with a polymeric thin film. In addition to the mechanical embossing of the channel structures, a technique called photolocking, schematically sketched in Fig. 12-36, has been widely used to generate a local confinement for guided light: a photoreactive polymer film is exposed to light through a suitable mask in a way similar to photoresist patterning. Heating the film leads to an evaporation of the unreacted dopant whereas the cross-linked areas define the channel waveguide. Refractive index changes of approximately 1%, sufficient for most configurations, can thus be obtained. Another technique has recently been demonstrated as an easy way to also generate passive waveguide structures. A polymeric multilayer composed of buffer layer, a nonlinear core layer and a top buffer layer was used as a matrix for the fabrication of a 1 x 2 channel waveguide splitter by a simple bleaching of the chromophores in the core layer. A 6 jim wide channel was split in a Y-configuration of two channels, each 6 jim wide and 50 |im apart (Mohlmann et al, 1990). The intensity output

(a)

I1 i

11 ! (b)

(c)

Figure 12-36. The photolocking process: (a) Doped polymer film, spin cast onto a substrate; (b) polymerization reaction following (local) irradiation with light; (c) heating the film evaporates the unreacted dopant.

profile of the 1 x 2 splitter shown in Fig. 12-37 demonstrates the good optical isolation between the two channels. There is certainly still a long way to go before a more complex integrated optical chip is a commercial reality but the feasibility in principle of all basic structures has been demonstrated. 12.4.2.3 Polymer Optical Fibers (POFs) Single-mode glass optical fibers are widely used as long-distance and high-

6|im guides

Figure 12-37. Intensity distribution at the output of a 1 x 2 passive channel waveguide splitter (see insert). Data from Mohlmann et al. (1990).

12.4 Polymers in Special Configurations

bandwidth communication media because of their excellent transparency. In shortdistance networks, however, where many junctions between different fibers are necessary, already slight displacements of two coupling cores each with a diameter of only about 5 jim would cause substantial coupling losses. Polymer optical fibers (POFs) with their considerably larger core diameter (ca. 1 mm) are in this respect much easier to handle. In addition, they offer other advantages, such as low cost, easy processing and flexibility. However, all commercially available POFs have been of the step-index (SI) type (Fig. 12-38 a). For these multimode fibers the different guided waves all propagate with the same phase velocity but along different paths within the core, a phenomenon which is called modal dispersion. This is schematically depicted in Fig. 12-38 a. Due to internal mode coupling, even a single input pulse (cf. Fig. 12-39 a) is greatly distorted and considerably broadened. An example is given in Fig. 12-39 b. This means that two consecutive input pulses need to be separated in time in order to be resolved at the output as two individual pulses. The transmission bandwidth is thereby greatly reduced and currently amounts for SI-POF to about 5 MHz km. A solution to this problem is the graded index (GI) fiber. Here the refractive index profile with the highest values in the center of the core results in a higher phase velocity of those "rays" propagating in the outer periphery of the core. Their longer pathways are, therefore, compensated by a faster propagation speed. As a consequence the input pulse is far less broadened, which results in a higher bandwidth. An example that was recently reported for GI-POFs prepared by a special copolymerization process (Koike, 1991) is given in Fig. 12-39: Fig. 12-39 c shows the refractive

571 n=1

(a)

(b)

Figure 12-38. Typical optical rays and refractive index profile of a multimode step-index (SI) fiber (a), and a graded-index (GI) fiber (b) (after Saleh and Teich, 1991).

1ns/div (a)

Input pulse

SI POF , ^ %

(b)

Put pulse

Out

GI POF (d)

ft Output

pulse

(c)

-0.005 "

-0.010 " -0.015 0.0

Figure 12-39. (a) Short input pulse in a POF; (b) output pulse after traveling through a step-index fiber; (c) index profile of the graded-index fiber that transmits the input pulses nearly unbroadened as shown in (d) (after Koike, 1991).

572

12 Optical Properties of Polymers

index profile, Fig. 12-39 d compares the input pulse with the output pulse. The bandwidth obtained was shown to be 2 GHz • km (Koike, 1991). In the development of polymer optical fibers another improvement concerns the intrinsic losses arising from absorption processes. This is indicated in Fig. 12-40 for a fiber made of poly(methylmethacrylate) (Groh et al, 1989). The higher overtone vibrations of the C - H groups in the material are responsible for attenuation losses at A^680nm of more than about 200dB/km. Perdeuteration results in a substantial reduction of the band strengths but also in a shift of their spectral position. Both effects reduce the loss in these fibers considerably (Fig. 12-40). Other attempts

to reduce the absorption coefficient at relevant communication wavelengths successfully introduced fluorocarbon moieties into the material with the additional improvement of the maximum operating temperature. This is an important issue, for example, for the implementation of POFs in automobiles (Groh et al., 1989). 12.4.3 Polymers for Optical Recording

With the development of low-cost solidstate laser diodes, the optical storage of information has become an attractive alternative to, for example, magnetic storage concepts. The basic principle is schematically depicted in Fig. 12-41. The information is stored on the disc as a succession of

200 -|

J

150

X

OSS

DQ 100

50

PMMA-d8 structural imperfections

Figure 12-40. Spectral loss data for a regular PMMA POF (dashed curve) and for a fiber made from perdeuterated PMMA (full curve). Some vibrational band assignments are given (after Groh et al., 1989).

I

500

600

700

800

900

X (nm)

Pre-Embossed Data Pits Reflecting Layer Disk Substrate Laser Read Beam

Figure 12-41. Some details of the pit sequence in a CD-ROM. Schematically given is also the laser read beam.

12.4 Polymers in Special Configurations

small pits of varying length and repetition frequency (Bowden, 1988). Read-out is accomplished by focusing a laser beam (typical diameter ~ 1 |Lim) onto the surface of the rotating disc and monitoring the reflected light by a photodiode. The reflectivity of the pits differs from that of the surface or land area, so that the intensity (or polarization) of the reflected light and hence the electrical output of the photodiode will be modulated according to the pattern of the pits. Various schemes have been developed for optical data storage: the read-only memories (ROMs), the "write-once-readmany times" (DRAW: direct read after write) memories, and the erasable memories. The first category follows the consumer products such as the optical video disk and the digital audio disc (CD: compact disc) where the pit sequence is produced already at the disc fabrication step, for example, by injection molding into a master mold containing all the pit structures (CD-ROM). Some requirements of the polymer material for the disc substrates were discussed already in Sec. 12.3.1. For the write(-once) process different mechanisms have been introduced. They can be categorized according to the schematic picture given in Fig. 12-42. Ablative processes produce marks in the form of shallow pits (Fig. 12-42 a) or deep pits (Fig. 12-42b). Another thermal process may lead to bubble formation (Fig. 12-42c) or to any other change of the optical proper-

573

ties of the thin film recording medium (Fig. 12-42d). In all cases, light typically from a near IR laser is (locally) absorbed by the film and converted into heat. This requires a functionalization of the polymer by a suitable absorber which can be an inorganic additive (carbon black, metal powders, or y-Fe 2 O 3 pigments) or an organic dye either mixed or copolymerized with the polymer matrix. The low thermal conductivity of polymers leads to very high local temperatures within the irradiated spot, causing the polymer material to degrade and volatilize. For bubble formation this degradation of the material occurs at the interface to the (reflecting) substrate so that the trapped vapor deforms the hot absorber layer to reproduce the desired microstructure. Some of the mechanisms suggested for phase-change storage in polymer films (Fig. 12-42d) include photobleaching and photochromism of dyes or the thermal conversion of amorphous into crystalline domains. By far less developed are erasable optical storage concepts. Examples of organic polymer based reversible media are few. One class consists of liquid crystalline (LC) side group polymers (see also Chap. 5 of this Volume). The LC polymer in a homogeneous, transparent meso-phase, for example, in a smectic state (Fig. 12-43 a), frozen below the glass transition temperature is locally distorted by the heat generated by the absorption of a laser pulse (Fig. 12-43b). This distortion can be read,

Figure 12-42. "Marks" produced during different laser writing processes: (a) Shallow pit, (b) deep pit, (c) bubble deformation, (d) optical property change.

574

12 Optical Properties of Polymers

(a) mmwmmm

t Figure 12-43. Schemes of the data storage process in liquid-crystalline polymer matrices doped (by mixture or copolymerization) with photo-addressable dyes, (a) In the smectic mesophase the matrix is highly ordered and hence transparent; (b) a focused laser pulse locally heats the sample into a disorded state; the bit is written; (c) the disorder remains frozen in and scatters the light if the reading beam is scanned across the polymer; (d) erasure of the stored information is accomplished by heating the sample into the nematic, which allows the disorder to anneal.

for example, by its ability to scatter light (Fig. 12-43 c). Since the matrix is in a glassy state back-relaxations are extremely slow and the data point is sufficiently long-lived. Erasure of the stored information can be accomplished by heating the film to a temperature above the glass transition temperature (but below the transition temperature to the isotropic phase) (Fig. 12-43 d). Another reversible storage concept is based on the bubble formation mentioned above. Here, a double layer structure, called the expansion and the retention layers, are the basic unit (Fig. 12-44). The expansion layer is an elastomer containing a dye absorbing the writing laser light at 840 nm. Its expansion deforms the retention layer - a surface bump that can be detected during the read cycle. Erasure is effected by exposure to a second laser beam at 780 nm, the wavelength which can be absorbed by the dyes in the retention layer but not in the expansion layer. The thermal softening of the retention material then allows the relaxation of the elastic energy stored in the deformation, and planarity is restored. The aforementioned two-dimensional storage concepts are limited in storage

density by the spot-size of the focused laser beam and reach currently about 108 bit/ cm2. Future availability of blue laser diodes emitting in the wavelength range of X = 400 nm will allow a further enhancement of the bit density. However, other

- Retention layer - Expansion layer - Substrate

-Write beam

- Read beam

Erase beam

Figure 12-44. Erasable dye-polymer double layer storage scheme based on bubble formation (for details see text).

12.4 Polymers in Special Configurations

concepts involving three- (or even four-) dimensional storage principles will compete with the "classical" 2D versions. It was suggested that the use of volume holograms would be one way to enhance the storage density by including the third spatial coordinate of the storage medium as well. One possibility is the use of photopolymers which change their refractive index irreversibly (An < 10 ~2) upon irradiation. This way, the interference pattern between a reference beam and an object beam both from a coherent (laser) light source can be stored persistently in the material. Read-out is performed just with the reference beam. Suitable polymers with a residual monomer content of approximately 10% are polyacrylates (Franke et al., 1984), polyacrylamides, polystyrenes, and polycarbonates. Theoretical storage densities of up to 10 14 bits/cm3 should be possible. A completely different principle for introducing a third dimension is based on photochemical hole-burning. The schematic representation of this storage process in the frequency domain is shown in Fig. 12-45. The absorption spectrum of a dye in a polymer matrix shows a rather broad band, reflecting the broad distribution of sites, each of them defining a slightly different microenvironment for the incorporated dye molecules. As a result each dye molecule with a narrow homogeneous

absorption band Fh appears in the absorption spectrum with a slightly different solvent shift thus generating the broad inhomogenous absorption spectrum (linewidth /]) of the dye ensemble (Fig. 12-45 a). A laser with a narrow line profile (narrower than the homogeneous line width of the dye molecule) at a (stabilized) wavelength, i L , within the absorption band is able to interact only with those dye molecules whose absorption profile overlaps with that of the laser. If this absorption leads to a photochemical process with a photoproduct absorbing at a distinctly different wavelength, a narrow hole is burned into the broad absorption spectrum (Fig. 1245 b). If the laser wavelength is tuned across the inhomogeneously broadened line, different subsets of molecules can thus be addressed within the same illuminated laser spot. With rjrh = 103 to 104 a storage density of 1011 bits/cm2 is then possible. The main drawback of this storage technology is the required low temperature: Only at T < 20 K are the holes sufficiently narrow and stable. 12.4.4 Polymers for Molecular Opto-Electronics

In this last section we just touch on a research area with still rather fuzzy contours: molecular electronics. The ultimate goal is the realization of logic operations at

(b)

(a)

575

Laser

O ) p LO

Figure 12-45. Photochemical hole-burning into an inhomogeneous absorption band composed of many individual homogeneous absorptions (a). Irradiating with a narrow laser line at coL burns a population hole which may appear as a photoproduct at a different spectral position &)p (b).

576

12 Optical Properties of Polymers

the molecular level. This requires molecular wires, bistable molecules that can store information, etc. Although it is completely unsolved how, in general, to address such molecular devices from the outside macroworld, some concepts include special optical properties of selected polymers. An example is given in Fig. 12-46. The photochromic isomerization of a salicylideneaniline unit incorporated into a polyacetylene chain could be used as a switch in the molecular wire from an on-state to an offstate. It is well conceivable that with the availability of tunnel microscopy tips and their ability to control and manipulate systems at the molecular level we will see a rapidly growing research activity towards molecular electronics and bio-chips.

12.5 Nonlinear Optical Properties of Polymers Polymers with large nonlinear optical susceptibilities are likely to play a major role in the rapidly growing area of photonics. This can be deduced from the many conferences and workshops in this area, from a tremendous increase in the number of publications both in journals and as books and, finally, from the fact that all major funding agencies have substantial programs to stimulate research in this interdisciplinary field of physics, chemistry, and materials science. Many excellent reviews have been devoted recently to the

off - s t a t e

topic so that we can concentrate in the following on some general fundamental device concepts, materials design considerations, and on some specific polymer issues relevant for nonlinear optics. We first give a few remarks on the functionalization of polymers for their use in nonlinear optical device configurations (Zyss, 1985). We then focus on some applications related to the second-order nonlinear optical susceptibility x(2) and finally summarize some third-order / ( 3 ) effects and how they could be implemented in future technologies for information processing. 12.5.1 General Concepts for the Functionalization of Polymers for Nonlinear Optics

The nonlinear optical response of polymers developed for future IOCs needs to be optimized through a suitable functionalization of the matrix by chromophores with correspondingly large second- or third-order susceptibilities. Both processes need, however, different strategies for an optimized material (Kajzar, 1992). For the second-order processes (cf. Sec. 12.2.5.1) the relation between the macroscopic {average) nonlinear susceptibility X(2) and the molecular nonlinear polarizability pD is determined by the following relation (Williams, 1987; Singer et al, 1987): = f. pD . ND .

(12-84)

where ND is the density of the active molecules, / is a local field factor of the

Figure 12-46. Switching unit (salycilideneaniline) incorporated into a molecular wire (polyacetylene) that can be switched optically from an on-state to an off-state (after Kampf, 1985).

12.5 Nonlinear Optical Properties of Polymers

optical field, and is a measure of the order parameter describing the average orientational distribution of the chromophore's nonlinear optical axis relative to the applied field. Maximum x(2) is obtained if all factors in Eq. (12-84) are optimized. The first step is the choice of a chromophore system with a large /?D. Most promising candidates are dyes of the chargetransfer type with a strong donor- and acceptor-group linked together by a conjugated 7i-electron system (Singer et al., 1989). This leads to a strong and highly asymmetric polarizability and hence to a large nonlinearity (cf. Fig. 12-7). By now, hundreds of different chromophores have been synthesized and tested, some of which are given in Table 12-3 (cf. also Burland et al., 1991). This compilation demonTable 12-3. Molecular control of /?D. Structure

j5D at 1.9 urn (xl(T 3 O esu) 5.7

N(CH

2

)

2

^C^

C

NC'"

_

N

21.4

.CN 41.8

20.1

50.7

23.4

61.6

111.2

577

strates that by a systematic variation of the donor- and acceptor-strengths and of the conjugation lengths substantial improvements in /?D can be achieved. The next step, the maximal level of functionalization, concerns the question of how to dope the polymer matrix. Three strategies have been suggested. The first is the mere mixing of a suitable dye as the functional guest in a polymeric host material. One example of a suitable dye is presented in Fig. 12-47 (Levenson et al., 1991). The dye's structure formula is given in the inset. The addition of 15wt.% dye to the PMMA matrix changes the linear optical properties of the polymer as discussed already in Sec. 12.2.4.5 (cf. also Fig. 12-5): The relatively flat background index of refraction of PMMA (full curve in Fig. 1247 a) shows for the mixture the resonant dispersion behavior (broken curve) of a strong chromophore. The corresponding behavior is seen for the extinction coefficient OL (Fig. 12-47b). This material showed (after poling, see next section) decent x(2) values and could be successfully implemented into an electro-optic phase modulator. A major drawback of the host-guest systems is the limited solubility of the dyes in the polymer: typically only 15-20 wt.% can be mixed without phase segregation and crystallization. Another problem, in particular in view of device applications, arises from the typically low temporal stability of the noncentrosymmetric orien, tation distribution induced by the poling process (see below). Depending, of course, on the storage temperature, decay times of a few hours to days have been reported (Page et al., 1990). However, certain hostguest systems based on polyimide matrices have been demonstrated with stable %(2) values for hours even at temperatures as high as T = 300°C (Wu et al., 1991).

578

12 Optical Properties of Polymers 0

0 II II 2 -0-C-CH 3

1

0

16

(a) 1 ^

doped PMMA

-

\ \

PMMA ^

/

\

1.6 -

/

/

/

|

.075-

>^

^'

i

v/ — —

(b)

Figure 12-47. Example of a host (PMMA)-guest (TP, structural formula given in the inset) system, (a) The wavelength dependence of the refractive index n of pure PMMA (full line) and after addition of 15 wt.% TP (dashed line), (b) The corresponding absorption coefficient. (After Levenson et al., 1991.)

doped PMMA /

/// V\ v\ /

.05 -

PMMA

025s

0

300

400

500

600

700

800

900

X / nm

The aforementioned problems have led to the development of polymers where the chromophores were either an integral part of the main chain (Kohler et al., 1990) or, more widely reported, were attached to the host polymer as a functional side chain. A small selection of the many systems presented in the literature is given in Fig. 1248, only examples based on acrylates or methacrylate polymers are added. All these attempts were aimed at fabricating materials that can be easily processed, for example in the form of a glassy film or channel waveguide, that show a high /?D-value of the employed chromophores, that allow a high NLO moiety content, and that give a stable configuration when poled (Man and Yoon, 1992). In fact, these copolymers allowed a much higher functionalization of the material than the host-guest systems. Yet another approach recently reported (Eich etal., 1989; Jungbauer et al., 1991; Swalen et al., 1991; Chen et al., 1991) currently seems to be the ultimate solution for

a stable chromophore orientation distribution: The incorporation of the NLO moieties into the polymer network as an integral part of the cross-linked matrix. One of the systems is shown in Fig. 12-49 (Swalen et al., 1992): An epoxy monomer (Fig. 1249 a) is reacted with £-field oriented NLO active amine monomers (Fig. 12-49b) to give a highly functionalized cross-linked and hence stable x(2)-material (Fig. 12-49 c). Very good long-term stabilities have been achieved (see below). For the third-order nonlinear optical processes polymeric materials with a high degree of delocalization of the conjugated electron cloud seem to be the most promising candidates. This leads to a large component yxxxx of the molecular hyperpolarizability tensor yD along the polymer chain (conjugation) direction. Some examples, currently being studied in many laboratories, are given in Fig. 12-50 (see, e.g., Jenekhe etal., 1992, and references therein), other systems are summarized in Table

12.5 Nonlinear Optical Properties of Polymers

579

(a)

(b)

Q CH3-C-C-O-CH3 CH2_ CH3-C—C-O—CH2—CH2N -N=NCH3— CH 2 7

CH2

(c) CH 3 -C—C—0— CH2 o I II Ho C C 0

—CH3

(CHo)ov

CHo

> CH3

(d) CH2 0 HC-C-O-(CH2)2-OH CH20 HC-( I !-(OCH 2 CH 2 )3

o

Figure 12-48. Examples of different dye-substituted sidechain polymers. After: (a) Herman etal. (1991), (b) Shuto et al. (1991) and Sasaki (1992), (c) Verbiest etal. (1991), (d) Penner et al. (1991), and (e)Tomono etal. (1992).

CH3-C-C-O-CH3 CH2 CH 3 -C-C-O-CH 2 -CH 2 ^

CH2

I

"tm

CH3CH2

12-4. According to (12-85) this is one prerequisite for a high value of the macroscopic nonlinear susceptibility X(3). Another important issue also for thirdorder materials is the orientational order

of the chromophores described by . Although not as crucial as for the x(2) devices, which require a noncentrosymmetric arrangement of the NLO active groups, here, too, the efficiency of the material depends on a good orientation: = 0.2 for a three-dimensional dis-

580

12 Optical Properties of Polymers

1

(a)

H

trans-Polyacetylene

H

Polydiacetylene NH2

(b) INK*1

R1

o

/"V

Poly(para)phenylene

NO 2 Amine Monomer

M

Polythiophene

n

(c)

"Rigid rod polymer"

p, Figure 12-49. Examples of a crosslinkable system composed of bifunctional matrix monomer (a) and a trifunctional /(2)-active chromophore (b). (c) Scheme of the network obtained by curing these two constituents (after Swalen et al., 1992).

Poly(para)phenylenevinylene from precursor

Figure 12-50. Structural formulas of some selected X(3)-active polymers (cf. also Table 12-4).

Table 12-4. Some conjugated polymers and their linear and nonlinear optical properties.

PPV

460

1.2 + 0.6x10" 1 0

PPA-1

R: Si(CH3)3

525

1.9 + 0.4x10" i i

PPA-2

R: CH 2 CH 3

450

8.4+1.7x10"

PPA-3

R: CH,

450

1.5 + 0.4x10" 1 1

order whereas = l for a system where all the polymer chains are parallel to a given direction. This means that the internal order can be responsible for substantial enhancements of the device performance. The packing density of the chro-

12

mophores which determines ND [cf. Eq. (12-85)] is another factor that needs to be optimized for a large #(3) but has to be balanced against other materials and processing parameters. For example, most of the chromophores presented in Fig. 12-50

12.5 Nonlinear Optical Properties of Polymers

are nearly intractable, that is, they are largely insoluble in most solvents and cannot be prepared, for example, in thin film form. This has led to many more or less successful routes to a final product via some soluble and hence processable precursor materials. One example is given in Fig. 12-50: here, a tetramethylene sulfonium chloride precursor polymer is converted as a spin-cast thin film by a thermal treatment to the final poly(p-phenylenevinylene) system (Lenz et al., 1988) which showed decent yD-values (Bubeck et al., 1989). Another trend in current x(3)-materials research concerns the incorporation of molecular structures with a two-dimensional 7i-electron delocalization. Some examples are given in Fig. 12-51 (Sasabe

581

et al., 1990; Schrader et al., 1991). The obtained nonresonant #(3)-values, however, are not too promising as yet (Schrader etal., 1991). 12.5.2 Second-Order Nonlinear Optics

Among the numerous device configurations based on #(2)-processes that have been realized and demonstrated in various laboratories we will only focus on two very basic concepts, that is, a polymeric waveguide structure designed for second harmonic generation (SHG) and an electrooptic modulator which is the key unit for many switching devices. Both modules require that the NLO active chromophores are highly oriented in a noncentrosymmetric narrow distribution. 12.5.2.1 Control of Orientational Order

(a)

T

CH3XCK

CH3 CH3

Annulene

Figure 12-51. Some examples of systems with a near two-dimensional 7r-configuration. (a) After Schrader et al. (1991), (b) and (c) after Sasabe et al. (1990).

For polymeric materials two strategies have been developed for obtaining oriented samples. The first is based on the LBK technique. For the usual Y-type deposition where one monolayer (A) is transferred on the down-stroke and the other (B) on the up-stroke this requires a double trough arrangement with the /(2)-active layer being transferred, for example, only on the upstroke whereas an optically inert material is deposited on the down-stroke (Beckerbauer, 1990; Penner etal., 1991). A different approach to the problem of generating noncentrosymmetric structures by the LBK method has recently been reported for molecules (2-docosylamino-5-nitropyridine) whose chromophores are arranged in a plane parallel to both dipping directions with a net polar orientation giving rise to a large x(2)-activity (Bosshard et al., 1991). If two different types of chromophores with opposite pD components are used for the A- and for the B-layer, respectively, then it is possible to prepare

582

12 Optical Properties of Polymers

LBK multilayer assemblies in the Y-mode with a very high chromophore density (Cresswell et al., 1990). In passing we should note that the self-assembly multilayers with their intrinsically noncentrosymmetric arrangement have also been successfully introduced for x (2) ~ act i ve structures (Allan et al., 1991). All these systems, however, have in common that in addition to the NLO active chromophores highly specialized molecular moieties need to be incorporated in order to tune the molecule's properties to the desired character (e.g., to allow for a self-assembly process) (Bubeck et al., 1991). From a technical point of view, it is also highly unlikely that such a time-consuming process as the LBK deposition of multilayers will ever make it to a commercially competitive device level. Therefore, the alternate strategy, that is, starting from a functionalized polymer with random chromophore orientation spin-coated onto a solid substrate and then subsequently oriented by a poling process, is certainly the more promising and hence more widely applied approach. In the following we will briefly review some aspects of the poling process and its stability. Some of the basic experimental configurations used for the static E-field induced noncentrosymmetric orientational order of the chromophores in functionalized polymer thin films are summarized in Fig. 12-52. The complete EO waveguide structure shown in Fig. 12-52 a includes in addition to the NLO active core the two buffer layers that isolate the guided mode from any absorption losses in the (metallic) electrodes. The latter will be used in the final device as the control electrodes but can also be used to apply a DC field of the order of some 10 MV/cm during the poling process at elevated temperatures (see below). A simplified version optimized for mi-

croscopic characterization of planar waveguide structures (cf. Sec. 12.4.2.1 and Fig. 12-31 a) is given in Fig. 12-52b. These two versions for typically out-of-plane orientation of the chromophores are complemented by an in-plane poling set-up schematically depicted in Fig. 12-52c. And finally, the top electrode free poling process based on a corona discharge set-up is sketched in Fig. 12-52d. At present, it is not yet clear which approach - electrode or corona poling - will yield the better results, that is, produce a higher degree of orientation, better homogeneity, tolerance against defects, etc. (Herminghaus et al., 1991). In any case, the poling process requires an orientational (rotational) flexibility of the chromophores in the polymer matrix. This is achieved by heating the sample to a temperature close to the glass transition temperature Tg of the material. Details of the actual temperature-electric field (applied voltage) protocol differ from laboratory to laboratory but certainly depend also on whether one is dealing with a hostguest, main chain or side chain polymer or a crosslinkable system that can be cured by a thermal process. For the latter case, a typical protocol is given in Fig. 12-53 (Swalen et al., 1991). Without any voltage applied, the system is heated to a temperature where crosslinking is initiated leading to a slight increase of Tg. This precuring period is followed by the actual poling process which in the presence of the orienting field further crosslinks the polymer-chromophore network with a concomitant increase of the glass transition temperature. The poling process can be followed online by recording the SHG or electro-optic efficiency or, very simply, by monitoring the linear absorption data. This is exemplified in Fig. 12-54. The extinction coefficient a is plotted as a function of the wavelength measured for a thin polymer film in trans-

12.5 Nonlinear Optical Properties of Polymers

583

(a)

Top Electrode

- Buffer Layer EO-active Polymer -Buffer Layer -Electrode -Substrate

Detector

Electrode EO -active Waveguide Electrode

(c)

top view

side view

Electrodes Polymer Film Substrate

(d)

Needle Electrode

Metallic Grid Ugrid

Polymer Film Conducting Layer • Substrate Heated Copper Block

Figure 12-52. (a) Schematic cross-section of a polymer layer with electro-optic activity sandwiched between two buffer layers and two electrodes. This configuration can be used for (out-of-plane) poling and for (simultaneous) waveguide spectroscopy. (b) Simplified version of (a), sufficient for many characterizations of the EO active polymer, (c) In-plane poling, (d) Scheme of a set-up used for corona (discharge) poling.

584

12 Optical Properties of Polymers

Tg CJ

cure

Temperai

CD

1 *

Vol tage

precure

*cure

•

*

-

•

Figure 12-53. The temperature-voltage protocol for a dye-polymer system that undergoes a thermally induced cross-linking reaction with a simultaneous increase of the glass transition temperature Tg (after Swalen et al., 1991).

time

mission prior to (Fig. 12-54 a) and after the poling process (Fig. 12-54b). The decrease of the optical density normal to the film is caused by the out-of-plane poling that orients the chromophores preferentially normal to the film. This optical anisotropy can be used to derive information on the orientational distribution function characterized by an order parameter 0S (Page et al., 1990) & = 1-—

(12-86)

0.4"

a 0.30.2" 0.1 " 0.0 400

500

600

700

X / nm Figure 12-54. Change of the absorption of a thin dyedoped polymer film measured in transmission before (a) and after (b) out-of-plane poling (after Chen et al., 1991).

with a± being the extinction normal to the film after poling and a0 the extinction prior to the poling assuming an isotropic orientation distribution of the chromophores. Once the polymer sample is cooled down again to room temperature, this polar anisotropy and hence the noncentrosymmetric x(2)-activity is frozen in, however, not completely, depending on the system. For device purposes this is, of course, an issue of utmost importance and hence has been the subject of many studies. With some exceptions (discussed above already), the general behavior can be classified as shown in Fig. 12-55: the host-guest systems show the fastest back-relaxation to a more or less isotropic distribution. Much better results are obtained with chromophores covalently attached to the polymer backbone although clearly this reduced flexibility also may result in a reduced net orientation obtainable through poling. The best stability is found for the crosslinked system. It should be pointed out that most of these results are obtained, at least to some extent, by a trial-and-error approach.

585

12.5 Nonlinear Optical Properties of Polymers

(a) 100 poled + crosslinked

o i

0)

copolymers

50-

host - guest

Poled Polymer time

Figure 12-55. Qualitative stability behavior of poled polymer films. Typically, the host-guest mixed system randomizes very fast; the copolymer system has a high residual orientation even after long times; the cross-linked system has a long-term stability.

(b)

7.9 5.9-

ZJ CD

O

3.9-

CO

1.9-

Many of the details are not understood yet and in many cases only very crude models have been treated. What is certainly lacking is a better understanding of the correlation of these linear and nonlinear optical properties with other polymer-specific characteristics given by the structural and dynamical features of these functionalized materials. 12.5.2.2 Second-Harmonic Generation (SHG) One of the key device applications of X(2)-active polymers is second-harmonic generation. The driving force is the search for efficient blue light sources, in this case through frequency doubling of red laser diode emission, for example, for the use in optical storage concepts (cf. Sec. 12.4.3). The basic configuration is given in Fig. 1256 a. Laser light of frequency co, incident on the sample of thickness d at an angle 9 with a fixed polarization, is refracted at the surface of the nonlinear medium and propagates through the film. The second harmonic wave follows the fundamental beam and as such is called the bound wave (Jer-

-0.1 -70

-40

-10

20

50

80

0/deg

(c) ZJ CD

O CO

-70

-50

-30

-10 0 10

30

50

70

0/deg Figure 12-56. (a) Schematic representation of the propagation vectors whose interference leads to the Maker fringe pattern, (b) Maker fringes for a thick polymer film, (c) One Maker fringe for a thin poled polymer film. (After Swalen et al., 1992.)

phagnon and Kurtz, 1970). In addition, a second harmonic wave is generated at the surface of the film which is, however, refracted at a different angle because the refractive index n(2co) of the NLO material is different from the value at the fundamental n(co). This wave is called the free wave.

12 Optical Properties of Polymers

Both waves interfere to give the overall SH intensity with an oscillatory pattern as one rotates the sample called the Maker fringes (Ledoux et al., 1987; Verbiest et al., 1991). Depending on the thickness of the sample several minima and maxima can be observed [Fig. 12-56 b for a thick film, Fig. 12-56c for a thin film, after Swalen et al. (1992)]. Now, the required high laser light intensity for efficient nonlinear conversion is best realized in a waveguide configuration with strong optical field confinement. The ultimate structure, therefore, will be the second harmonic generating channel waveguide (Norwood and Khanarian, 1990). However, the dispersion of the material and of the waveguide structure (modal dispersion) would result in a useless device because after a certain propagation length of the fundamental and the second harmonic beam (called the coherence length Lc), their relative phase difference would lead to destructive interference of the second harmonic light. Several concepts have been put forward to overcome this problem called the phase-matching problem. The first approach is based on the modal dispersion of waveguide structures: This term accounts for the fact that different modes (modes of different order) propagate with different phase velocities, hence can be described by different effective refractive indices. Phase-matching in such a concept is achieved by tuning the dimensions of the waveguide, for example, the thickness of a planar structure, in such a way that the dispersion of the material (cf. Fig. 12-6) is compensated by the modal dispersion. This then results in a configuration in which the guided fundamental mode of order m propagates with the same velocity as the second harmonic mode of order m'. Problems arising from the need to optimize the field intensity overlap integrals

have so far hampered a major breakthrough of this approach. The second concept called quasi-phase matching is schematically depicted in Fig. 12-57. After the first coherence length in the poled polymer waveguide the x(2)-activity is either destroyed (by bleaching or randomization of the chromophores) or reversed in its polar direction (by the use of correspondingly patterned and poled electrodes) (Fig. 12-57 a). The first configuration is called a unidirectional poled waveguide, the second, bidirectional; both overcome the problem of destructive interference of the SHG light, with the bidirectionally poled structure showing a twice as efficient build-up of the 2co signal (Fig. 1257 b). Finally, we should mention a third version called the Cerenkov frequency doubler. Here, the refractive indices of waveguide material and substrate are such that the fundamental light is guided in the film, but the SH light is constantly coupled out through the substrate (because the condi(a)

Poled Waveguide, unidirectional

\J|f

\\f\^\l\\H\\&^\lV\/^J.\\if bidirectional

n\\l\\/[tHVJlff\ f\JV/\\ffi\Urt\\f f (b)

bidirectional SHG Build-up

586

/ ^ /*

/ ^

unidirectional

Figure 12-57. (a) Scheme of unidirectional and bidirectional poling of waveguides for quasi-phasematching and (b) the resulting build-up of secondharmonic light in the waveguide.

12.5 Nonlinear Optical Properties of Polymers

587

Detector

Transparent Substrate Transparent Electrode Polymer Film Reflecting Electrode

tion for total internal reflection at the filmsubstrate interface is not fulfilled at 2co). In this way, also no destructive interference limits the SHG efficiency. 12.5.2.3 Electro-Optic (EO) Modulation The second major class of x(2)-based devices use the Pockels effect, that is, the manipulation of the refractive index of the material by an applied electric (DC) field. The ultimate structure, again, is certainly a channel waveguide (Shuto et al., 1991) with an architecture schematically shown in Fig. 12-52 a. If only materials parameters are to be measured various simplified architectures are sufficient. One set-up operating in a reflection mode similar to ellipsometry is given in Fig. 12-58 (Teng and Man, 1990). The highly reflecting Au or Al electrode on the rearside of the EO active thin film is used to measure differences of the reflectivities for s- and p-polarized light propagating through the poled polymer in the absence and presence of an applied E-field. The differences of the reflected light intensities can be quantitatively analyzed in terms of the two / ( 2 ) tensor components, x?3 anc * X33> needed to describe the nonlinear (Pockels) response of a uniaxial film (Gadret et al., 1991). The advantage of such a simple set-up for NLO materials characterization relative to a SHG set-up are obvious: No high power

Figure 12-58. Set-up for EO coefficient measurement in reflection.

laser is needed, and, therefore, wavelengthdependent measurements are much easier. A more detailed analysis of the linear and nonlinear optical properties of planar thin film samples is possible by surface plasmon (Cross et al., 1988; cf. also Sec. 12.4.1) and by waveguide spectroscopy outlined in Sec. 12.4.2.1. This has been demonstrated for many samples of poled polymers. In particular, if electro-strictive effects are also present in the investigated material only the waveguide technique allows a separation of the different processes (Dumont and Levy, 1989). It is straightforward to demonstrate the potential of the method for a waveguide structure prepared by the LBK technique. The sample consisted of 240 buffer layers of an inert material on each side of the EO active A-B layer system of 20 layers each. The structural formula of the chromophore-containing LBK systems is given in the inset. The experimental set-up was as in Fig. 12-52b. Figure 12-59 a shows the three waveguide modes (m = 0, 1, 2 as indicated) measured with s-polarized light {X = 593 nm). Figure 12-59b gives the electro-optic response of the sample under the influence of an applied voltage of 30 V peak-to-peak. Most remarkable is the alternating magnitude of the effect: The m = 1 mode shows only a very small modulation. The interpretation is obvious on the basis of the field intensity distribution given in Fig. 12-29.

588

12 Optical Properties of Polymers

according to yDocLad ~

0.5

(a)

CH3-(CH2)17-0-<5>-CH=N-NH-<5>-N02

Different theories give for ID systems oc = 5 to 7 (Rustagi and Ducuing, 1974; Agraval et al., 1978; Chopra et al., 1989; Grossman et al., 1989). For example, treating the conjugated polymers as one-dimensional systems with a 7i-electron delocalization length L d , Agrawal et al. (1978) and Flytzanis (1987) derived a scaling law for the third-order susceptibility X(3)ocL6d

8/deg

60

Figure 12-59. Waveguide mode pattern (a) and electro-optic response by applying a 30 V peak-to-peak voltage (b) across a LBK multilayer assembly composed of 240 buffer layers/40 A-B layers (with B being the #(2)-active material)/240 buffer layers. Structure of the NLO active dye is given in the inset (Aust et al., 1993).

The m = 1 mode has a node in the middle of the waveguide right where the EO active material is located. Any refractive index modulation by the applied electric field influences, therefore, the overall index configuration for this mode only very little. The even-indexed modes, however, with their intensity maximum in the middle, are influenced much more strongly. The quantitative analysis gives for all three modes the same electro-optic coefficient x{il = 14pm/V. 12.5.3 Third-Order Nonlinear Optics The molecular hyperpolarizability yD of one-dimensional (ID) conjugated polymers depends on the conjugation length L d

(12-87)

(12-88)

If L d is taken to be inversely proportional to the optical gap then x{3) should scale with A^ax with Amax being the spectral position of the linear absorption. This is qualitatively found for a wide range of polymers (Fig. 12-60) although for certain systems true nonresonant #(3)-values would be required for a rigorous testing of this theoretical approach (Zhao et al., 1988; Neher et al., 1990). In view of possible applications of these materials for all-optical information processing where (nonresonant) X(3)-values in the range of 10" 9 esu or larger are estimated to be required, it is, however, highly doubtful whether linearly polyconjugated molecules allow this nonresonant #(3)-value to be reached at all. At present, it is an open question whether materials with an improved nonlinear response and/or an optimized figure of merit will be available in the future. These figures of merit are defined such as to scale the enhancement of a desired nonlinear susceptibility, for example, the intensity dependent refractive index change in the sample n2 [cf. Eq. (12-62)], against other linear or nonlinear response parameters. For example, in a nonlinear directional coupler (cf. Fig. 12-35 a) the (desired) maxi-

12.5 Nonlinear Optical Properties of Polymers

589

10-12 700 /nm

Figure 12-60. Survey of #(3)-values as determined from THG experiments. In addition to values taken from Table 12-4, data from the literature are plotted: polydiacetylene (PDA) and polyacetylene (PA) from Neher et al. (1990); poly(3-decyl thiophene)s (PTs) are from Salcedo et al. (1987). x(3)-values are plotted as a function of the spectral position Xmax of the linear absorptions. The higher %(3)-values found for PPV are explained as a consequence of the higher 7i-electron density due to the lack of bulky substituents (Neher et al., 1990).

mum phase shift ANL is given by (12-89) with L being the interaction (device) length. For a given n2 of the material, the required phase shift of the device could be obtained simply by choosing L correspondingly (high). However, the maximum value of L is limited by the absorption in the polymer: max {L} < a" 1 . Since a = a 0 + /J2 • /, that is, a contains not only the linear absorption term a0 but also the nonlinear, for example two-photon, absorption term P2 -1 which is also intensity dependent, one may define (even in the absence of linear absorption oc0 < /?2 • I) a figure of merit Tm = 2k^

(12-90)

which must be smaller than 1 for a material to be used in a nonlinear directional coupler. On the other hand, many of these parameters are wavelength dependent. This

defines the need for a rigorous materials characterization and optimization program for those research activities committed to the development of photonic devices based on polymeric systems. Firstly, of course, X(3) has to be determined. One experimental technique for this purpose is third-harmonic generation (THG), which will be briefly presented in the next section. Of interest are not only the contributions from electronic or thermal processes or from excited states but, in particular, the spectral dispersion behavior of #(3). This must also be known for the linear and, equally important, for the nonlinear absorption coefficients. In fact, it is currently one of the promising strategies to find spectral windows where the device performance is not limited by the two-photon (or three-photon) absorption in the materials. Since one is interested in fast data processing devices, with switching times T < 0.1 ps, the nonlinear recovery time is of vital importance. One experimental technique for its determination is degenerate

590

12 Optical Properties of Polymers

four wave mixing (DFWM) briefly outlined below. We should point-out, once again, that in addition to these (linear and nonlinear) optical parameters, other material properties, for example stability and processibility into device configurations such as waveguides, are equally important for a successful application. In the last section we then give an example of a nonlinear directional coupler fabricated from polydiacetylene as a basic unit for the photonic concept of all-optical switching. 12.5.3.1 Third-Harmonic Generation (THG) Third-harmonic generation describes the process in which an incident photon field of frequency co generates, through nonlinear polarization in the medium, a coherent optical field at 3 co. It gives directly the electronic hyperpolarizability with ultrafast response time. The typical experiment performed with thin film polymer samples prepared on suitable substrates either in transmission or reflection geometry monitors contributions from the substrate as well as from the film. Rotating the sample, as outlined above for the SHG experiments, leads to a coherent superposition of the free and bound waves, giving rise to a fringe pattern similar to the one shown in Fig. 12-56 for the SHG signal. If one knows the substrate's cubic susceptibility, the experimental data can yield both the real and imaginary parts (or the modulus and the phase) of the third-order susceptibility of the polymer material. If suitable laser systems are available these data can be obtained covering a wide range of frequencies, allowing the search for suitable spectral windows off the one-, two-, and three-photon resonances. One example is given in Fig. 12-61 for a polydiacetylene system (after Guo et al., 1992). The experi-

1000

1500 X/nm

2000

Figure 12-61. Modulus (a) and phase (b) of x(3) measured for polydiacetylene (cf. Fig. 12-50, with Rx = R2 = {(CH2)4OCONHCH2COO(CH2)3CH3}) as a function of the fundamental wavelength. The full curves are theoretical calculations based on a model with 4 essential states (after Guo et al, 1992).

mental data show the strong spectral dependence of both the modulus and the phase of #(3). The full curve is a theoretical description of the nonlinear susceptibility of the polymer on the basis of an "essential states" picture with four levels (Guo et al., 1992) which gives for these and other data satisfying agreement between experiment and theory. 12.5.3.2 Degenerate Four Wave Mixing (DFWM) Degenerate four wave mixing is a holographic technique in which the output of a laser system is split into three beams. Two of them coherently interfere within the thin film sample, thus generating a phase grating (refractive index grating) through the intensity-dependent refractive index change n2 in the Kerr medium. The third

12.5 Nonlinear Optical Properties of Polymers (a)

sample

3

©

LBK - film (50 layers) of polymer

*)

®

LBK - film (80 layers) of monomer

Cl

©

Polystyrene - film doped with monomer

Cl

®

Poly (styrene - co - phthalocyanine) 0 - C H 3

R

R2

R

3

591

R' R

3

Ri = O-CH3, R 2 = 0 - CsHi7 in all permutations

CH 3

CH 3

Si ~
(c)

A.

A.

3

•—^_

2

A. 600

700

Intens ity/a.u.

(b)

CH 3

r ..K

2

1 . 800

0

1 10

20

Delay Time/ps X/nm Figure 12-62. (a) Chemical structure of phthalocyanine derivatives and sample configurations used for DFWM experiments, (b) Absorption spectra and (c) decay-curves of transient grating of some phthalocyanine thin films: (1) LBK film of 50 layers of a polymeric sample; (2) LBK film of monomeric dye; (3) polystyrene, doped with monomer; (4) styrenephthalocyanine-copolymer.

beam probes this grating: Its diffraction efficiency (the 4th wave!) is monitored during the experiment. If short laser pulses are used, time-resolved experiments can be performed: If the third pulse is slightly delayed relative to the two interfering pulses the diffracted intensity is a measure of the grating decay. Thus, information about the important nonlinear recovery time can be obtained. (By comparison with a material

of known #(3) also the magnitude of the third-order nonlinear susceptibility can be obtained.) Examples are given in the literature of DFWM experiments performed with 60 fs pulse duration, showing the ultrafast response times possible with polymeric systems (Prasad, 1991). Figure 12-62 summarizes some data aimed at demonstrating the influence of electronic coupling between different chromophores. The close interrelation between the detailed molecular structure of the chromophore, the supramolecular architecture of the matrix into which it is incorporated, and the linear and nonlinear optical properties of the resulting functionalized material is demonstrated for various thin film samples of phthalocyanines. These dyes are of general interest because of their excellent light and temperature stability. The chemical structure and a few details of the different systems used in these studies are given in Fig. 12-62 a. In system (4) the phthalocyanine moieties were incorporated into a copolymer with styrene. Figure 12-62b shows the linear absorption spectra and (Fig. 12-62c) the decay curves as obtained from DFWM experiments at X « 650 nm. Remarkable differences for the various systems are found (Kaltbeitzel et al., 1989). An isolated phthalocyanine

592

12 Optical Properties of Polymers

molecule as, for example, in system (4) gives a sharp absorption spectrum. This leads to a very slow decay time of the transient grating beyond the resolution of the set-up. Increasing aggregation eventually leads to an inhomogeneous broadening of the absorption band and a concomitant reduction of the transient time (by orders of magnitude!). Obviously, the electronic coupling between individual resonators affects both the linear and the nonlinear response of a chromophore system. Various contributions to the observed phenomena may include (i) fluorescence life time reduction by aggregational quenching, (ii) energy migration (Salcedo et al., 1987), for example exciton diffusion into the nonilluminated area, (iii) energy trapping at lower states, (iv) bimolecular exciton quenching (Ho and Peyghambarian, 1988), etc. The response times of the conjugated polymers described above are presumably also related to the relaxation processes (iii) and (iv). The extended 7i-conjugation with a tight electronic coupling via intra- and interchain energy migration leads to the ultrafast transient times of these systems, even near resonances. 12.5.3.3 All-Optical Switching

In this last section we just touch on a still very speculative subject, namely controlling light by light (through x(3)-active materials). Various device concepts have been proposed, some of which have also been experimentally tested. A very basic unit is the nonlinear directional coupler. Its realization was reported by Townsend et al. (1989). However, it turned out that the intensity-dependent switching of the input laser pulse from one output channel to the other was based on a thermal effect attributed to a three-photon absorption. Recently, a similar structure was presented by

Kaneko et al. (1992) who claimed on the basis of their experiments with Ti: A12O3 laser pulses of 100 fs duration at X = 825 nm an electronic mechanism for the observed switching behavior. The high pulse repetition rate still means a substantial power deposition in the sample and it will take single-pulse experiments to show unambiguously the fast electronic nature of the all-optical switching phenomena seen in these polymer waveguide samples. 12.5.4 Photorefractive Polymers

The photorefractive effect was first established in LiNbO 3 (Chen, 1967). It arises when charge carriers, photogenerated by a spatially modulated illumination, separate and become trapped to produce a nonuniform space charge distribution. The resulting internal field then modulates the local refractive index via the linear electro-optic effect, which results in a phase image or hologram of the illumination pattern. A photorefractive material, therefore, needs to combine some functional units that act as photoionizable charge carriers, some that transport the photogenerated charges until they are trapped at certain sites, and it has to show a nonlinear electro-optic change of its refractive index upon a spacecharge field. Many possible applications have been suggested, including high-density optical data storage, various imageprocessing techniques, phase conjugation, and programmable interconnection (Giinter and Huignard, 1988). Polymers again would be ideal materials for photorefractive devices because of their excellent processability and potentially high electro-optic coefficients (see above). Also, photoconductivity in functionalized polymers is a well-established phenomenon because of its importance for the photocopy process (Burland and Schein, 1986)

12.8 References

and for laser-printing. It was nevertheless not until very recently that the first reports on photorefractive polymers were published (Schildkraut, 1991; Ducharme et al., 1991; Tamura et al., 1993) showing indeed very promising features which certainly will stimulate a very dynamic research activity in this field.

12.6 Outlook Polymers with their broad variety of optical properties have already their established place in modern products and technologies. The needs of our information society and the emerging technology of the next century, photonics, means a rapidly growing demand, chance, and challenge for ever better performing polymers with specifically tuned optical (and other) properties. Optical data storage for consumer products and in information technologies will be a growing area for the use of more and more sophisticated materials. Closely linked to this development is the requirement for a cheap yet powerful blue laser light source. At present it is an open race whether the blue solid state laser diode or a device based on a red laser diode with an integrated frequency doubling (polymeric) material will be available first. Integrated optics, in general, will have a growing share in data transmission, distribution, and processing. Here, a great deal will depend on whether improved nonlinear optical materials will be available. These ends will be met only if, by an interdisciplinary effort, a better understanding of the various materials parameters at the molecular level can be obtained. The close relation between macroscopic properties and functions on the one hand and the microscopic aspects of structure, dynamics, and order on the other, will re-

593

main a continuous challenge for modern materials science and technology.

12.7 Acknowledgements It is my pleasure to thank many colleagues and friends for their support and for many helpful discussions. In particular, Y. Koike, H. Kuhn, Y. Levy, W. H. Meyer, V. Mizrahi, M. R. Philpott, H. Ringsdorf, H. Sasabe, K. Sasaki, G. I. Stegeman, X Stumpe, I D. Swalen, C. Urban, G. Wegner, T. Wada, and J. Yang contributed also to this chapter through their stimulating comments. As to our own results, I am particularly grateful to my coworkers E. F. Aust, C. Duschl, W. Hickel, H. Motschmann, M. Sawodny, R. Reiter, and B. Rothenhausler. Financial support came from the Deutsche Forschungsgemeinschaft, the Bundesministerium fur Forschung und Technologie, the Volkswagen Stiftung and the Leonhard-Lorenz-Stiftung.

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12 Optical Properties of Polymers

Organic Molecules for Nonlinear Optics and Photonics: Messier, X, Kajzar, R, Prasad, P., Ulrich, D. (Eds.). Dordrecht, Netherlands: Kluwer Academic PubL, p. 433. Swalen, J. D., Fleming, W, Jurich, M., Moerner, W. E., Smith, B. A., Herminghaus, S., Bjorklund, G. C. (1992), Mater. Res. Soc. Symp. Proc. 228. Boston, MA: Mater. Res. Soc, p. 101. Tamir, T. (Ed.) (1979), Topics in Applied Optics, Vol. 7: Integrated Optics. Berlin: Springer-Verlag. Tamura, K., Padias, A. B., Hall, H. K., Jr., Peyghambarian, N. (1993), Appl Phys. Lett., in press. Teng, C. C , Man, H. T. (1990), Appl. Phys. Lett. 56, 1734. Tien, P. K. (1969), Rev. Mod. Phys. 49, 361. Tillmann, N., Ulman, A., Penner, T. L. (1989), Langmuir 5, 101. Tomono, T., Nishikata, Y, Pu, L. S. (1992), Sen-i Gakkai Symp. Preprints B-102. Townsend, P. D., Jackel, J. L., Baker, G. L., Shelburne III, J. A., Etemad, S. (1989), Appl. Phys. Lett. 55, 1829. Ushioda, S., Sasaki, Y. (1983), Phys. Rev. B27, 1401. Verbiest, T., Persoons, A., Samyn, C. (1991), Nonlinear Optical Properties of Organic Materials IV, Proc. SPIE 1560, 353. Williams, D. J. (1987), in: Nonlinear Optical Properties of Organic Molecules and Crystals: Chemla, D. S., Zyss, J. (Eds.). Orlando, FL: Academic Press. Wolter, H. (1956), in: Handbuch der Physik: Fliigge, S. (Ed.). Berlin: Springer-Verlag. Wu, J. W., Valley, J. R, Stiller, M., Ermer, S., Binkley, E. S., Kenney, J. T., Lipscomb, G. F , Lytel, R. (1991), Nonlinear Optical Properties of Organic Materials IV, Proc. SPIE 1560, 196. Zernicke, F , Midwinter, J. E. (1973), Applied Nonlinear Optics. New York: John Wiley and Sons.

Zhao, M. T., Singh, B. P., Prasad, P. N. (1988), /. Chem. Phys. 89, 5535-5541. Zyss, J. (1985), J. Mol. Electron. 1, 2.

General Reading Bowden, M. J., Turner, S. R. (Eds.) (1988), Electronic and Photonic Applications of Polymers, Advances in Chemistry Series, Vol. 218. Washington, DC: ACS. Emerson, J. A., Torkelson, J. M. (Eds.) (1991), Optical and Electrical Properties of Polymers, Materials Research Society Symposium Proceedings, Vol. 214. Pittsburgh, PA: MRS. Kai Chang (Ed.) (1991), Fiber and Electro-Optical Components, Vol. 4 of Handbook of Microwave and Optical Components. New York: Wiley. Kampf, G. (1985), Ber. Bunsenges. Phys. Chem. 89, 1179. Knoll, W. (1991), Mater. Res. Bull 16, 29. Prasad, P. N., Williams, D. J. (1991), Introduction to Nonlinear Optical Effects in Molecules and Polymers. New York: Wiley. Raether, H. (1988), Surface Plasmons on Smooth and Rough Surfaces and on Gratings, Springer Tracts in Modern Physics, Vol. 111. Berlin: Springer. Roberts, G. G. (Ed.) (1990), Langmuir-Blodgett Films. New York, Plenum. Saleh, B. E. A., Teich, M. C. (1991), Fundamentals of Photonics. New York: Wiley. Singer, K. D. (Ed.) (1991), Nonlinear Optical Properties of Organic Materials IV, Proc. SPIE 1560.

13 High Performance Polymer Fibers Hao Jiang Lawrence Associates Incorporated, Dayton, OH, U.S.A. W. Wade Adams Materials Directorate, Wright Laboratory, Wright-Patterson Air Force Base, OH, U.S.A. Ronald K. Eby Institute of Polymer Science, University of Akron, Akron, OH, U.S.A. List of 13.1 13.1.1 13.1.2 13.1.3

Symbols and Abbreviations Introduction Definition of High Performance Polymer Fibers Ideal Structure of High Strength and High Modulus Polymer Fibers Historical Review of the Development of High Performance Polymer Fibers 13.2 Fiber Processing 13.2.1 Gel Spinning of Flexible and Semi-flexible Polymers 13.2.2 Melt Spinning of Thermotropic Liquid Crystalline Polymers 13.2.3 Dry-Jet Wet Spinning of Lyotropic Liquid Crystalline Polymers 13.2.3.1 Polymer Synthesis 13.2.3.2 Lyotropic (Liquid Crystal) Solutions 13.2.3.3 Fiber Spinning 13.2.4 Solution Spinning 13.2.5 Solid State Extrusion 13.3 Fiber Morphological Structure 13.4 Fiber Properties 13.4.1 Tensile Properties 13.4.2 Compressive Properties 13.4.3 Thermal Properties 13.4.4 Miscellaneous Properties 13.4.5 End Uses of High Performance Polymer Fibers 13.5 Other High Performance Fibers 13.5.1 Fibers for Medical Uses 13.5.1.1 Surgical Fibers 13.5.1.2 Biological and Chemical Membrane Fibers 13.5.1.3 Fiber Carriers for Biologically Active Agents 13.5.2 Polymer Fiber Precursors to Superconductors 13.5.2.1 Fibers from Mixtures of Superconducting Oxides and PVA 13.5.2.2 Conducting Polymer Fibers 13.6 The Future 13.7 References Materials Science and Technology Copyright © WILEY-VCH Verlag GmbH & Co KGaA. All rights reserved.

598 600 600 602 603 605 605 607 609 610 611 613 615 616 617 626 626 631 635 636 638 639 639 640 642 644 644 645 645 646 647

598

13 High Performance Polymer Fibers

List of Symbols and Abbreviations / /0 m Mw P

Hermans-Stein orientation factor test length of the fiber Weibull modulus molecular weight cumulative failure probability function

Tg Tm TNI

glass transition temperature melting temperature isotropic-nematic transition temperature

M O

intrinsic viscosity stress normalization factor for Weibull model

ABA ABP AFM ANA AS BF CSA CTE DABDO DABTO DF DMSO DP ED FESEM GPC HMPA HPF HPOPF HREM HT IA LM MSA MTM NMP PAN PBZO PBZT PDA PE

p-acetoxy benzoic acid p,p-diacetoxy biphenyl atomic force microscopy acetoxy naphthoic acid as-spun bright field chlorosulfonic acid coefficient of thermal expansion 4,6-diamino-1,3-benzenediol dihydrochloride 2,5-diamino-l,4-benzenedithiol hydrochloride dark field dimethyl sulfoxide degree of polymerization electron diffraction field emission scanning electron microscopy gel permeation chromatography hexamethylphosphoramide high performance fiber high performance organic polymer fiber high resolution electron microscopy heat treated isophthalic acid light microscopy methane sulfonic acid micro-tensile testing machine rc-methylpyrrolidone polyacrylonitrile poly( p-phenylene benzobisoxazole) poly( p-phenylene benzobisthiazole) polydiacetylene polyethylene

List of Symbols and Abbreviations

PET PGA PMMA POM PP PPA PPD PPTA PSF PVA SAED SAXS SEM SSF STM SWAXS TA TCI TDT TEM TFA TGA THF TLCP TMA UHMW UV WAXS

poly(ethylene terephthalate) polyglycolic acid polymethyl methacrylate polyoxymethylene polypropylene polyphosphoric acid p-phenylene diamine poly( p-phenylene terephthalamide) polysulfone poly(vinyl alcohol) selected area electron diffraction small-angle X-ray scattering scanning electron microscopy spin stretch factor scanning tunneling microscopy synchrotron WAXS terephthalic acid terephthaloyl chloride thermal distortion temperature transmission electron microscopy trifluoroacetic acid thermogravimetric analysis tetrahydrofuran thermotropic liquid crystalline polymer thermal mechanical analysis ultra-high molecular weight ultraviolet wide-angle X-ray scattering

599

600

13 High Performance Polymer Fibers

13.1 Introduction "A civilization is both advanced and limited by the materials at its disposal" Sir George Padget Thompson 13.1.1 Definition of High Performance Polymer Fibers High performance fibers (HPFs) have excellent mechanical properties or other superior features, for example, thermal stability, chemical resistance, electrical properties, or unique biological behavior. They can be divided into two groups: the inorganic and the organic polymers. The representatives of the inorganic group are carbon, boron, silicon carbide (SiC), high strength glass, alumina (A12O3), aluminum and titanium alloy fibers. The typical high performance organic polymer fibers (HPOPFs) are aromatic aramid fibers such as poly(/7-phenylene terephthalamide) (PPTA) under the Kevlar trademark, including Kevlar 29, Kevlar 49 and Kevlar 149; aromatic heterocyclic polymer fibers such as poly(/>-phenylene benzobisthiazole) (PBZT) and poly(j?-phenylene benzobisoxazole) (PBZO); aromatic polyesters such as polyester-polyarylate fiber (Vectran); and ultra-high molecular weight (UHMW) extended chain polyethylene (PE) fibers such as Spectra, including Spectra 900 and Spectra 1000 (Table 13-1). These broad classes of high performance fibers make giving a unique definition difficult, if not impossible, and indeed many names have been used, such as "high-tech fibers", "strong fibers" and "super fibers". In this chapter, our discussion is confined to high performance organic polymer fibers. A detailed discussion of carbon and other inorganic high performance fibers is presented in Volume 13 of this Series.

High performance organic polymer fibers are unique because they possess many excellent properties that significantly exceed those of traditional textile fibers, such as nylon and polyester (Fig. 13-1). High stiffness and strength combined with low density make the HPOPFs very competitive among fibrous materials, Table 13-1. Properties of fibers. Materials (Fibers)

Tensile Tensile Compres- Density modulus strength sive strength (GPa) (GPa) (g/cm3) (GPa)

Steel 200 71 Al-Alloy Ti-Alloy 106 350-380 Alumina Boron 415 SiC 200 S-Glass 90 Carbon PI00 725 (pitch-based) Carbon M60J 585 (PAN-based) Kevlar 49 125 Kevlar 149 185 PBZT 325 PBZO 360 Spectra 1000 172 Vectran 65 Technora 70 Nylon 6 Textile PET 12

2.8 0.6 1.2 1.7 3.5 2.8 4.5 2.2

_ 6.9 5.9 3.1 >1.1 0.48

3.8

1.67

1.94

3.5 3.4 4.1 5.7 3 2.9 3.0 1.0 1.2

0.39-0.48 0.32-0.46 0.26-0.41 0.2-0.4 0.17 _ 0.1 0.09

1.45 1.47 1.58 1.58 1.0 1.4 1.39 1.14 1.39

7.8 2.7 4.5 3.7 2.5-2.6 2.8 2.46 2.15

'Glass

6

12

Strain (%) Figure 13-1. Stress-strain curves of various fibers.

18

601

13.1 Introduction 4.0 T"

T1000 O PBZO

O Spectra 1000

3.0 - -

Kevlar 49 QSpectra 900 Technora O Kevlar 149 OVectran OAS4

c 0)

2.0

Kevlar 29

"c/5 c

S-glass

O M60J

°T300

(D

O Boron

T50 ;

OsiC

O Nylon

O E130

OGY70

P100

Figure 13-2. Specific tensile strength versus specific tensile modulus of various fibers.

OP25

Q. CO

Steel Al-alloy

0.0

0

70

140

210

280

350

420

Specific Tensile Modulus (MJ/Kg)

especially when compared on the basis of specific properties, such as specific strength and specific modulus (Fig. 13-2). These specific properties are defined as strength or modulus divided by density. Table 13-2 lists the unit conversions. Other useful characteristics of HPOPFs vary somewhat from fiber to fiber and include good chemical, corrosion, moisture, and high-temperature resistance, dielectric

properties, easy processing and (sometimes) low cost. High performance can also refer to unique biological properties, such as resistance to biodegradation, ability to form hollow fibers with selective permeability, and ability to act as carriers for drugs. The development of HPOPFs has created a new generation of materials. They are an enabling technology for many present and future high-technology prod-

Table 13-2. Unit conversions.a

g/Tex

g/d

g/Tex

kg/mm2

dynes/cm2

1

0.111 1

QX 8.995

#x 8.818 xlO 8

QX 8.818 x 107 e x 1.281 xlO 4

^x 80.96

QX 7.936 x 109

QX 7.936 x 108 QX 1.152 x 105

1

9.803 xlO 8

9.803 xlO 7

1.281 x 104

xlO" 1 0 1.019 x l 0 ~ 8

1

0.1

6.889 X 104

xlO" 9

10

1

1.451 X 10" 4

6.889 x104

6.889 x103

1

9.000

dynes/cm2

xl0~9 Q

Pa

0.0124

0.111

kg/mm2

lb/in2

Q

xlO~

lb/in2 a

Pa

Q is the density of the fiber in g/cm3.

1.019xl0~ 8 7.013

602

13 High Performance Polymer Fibers

ucts, especially as the backbone materials of the advanced composites used in the aerospace industries. Composites with high performance polymer fibers provide advances in stiffness and weight savings (every kilogram saved in the structure of an airplane saves hundreds of dollars over its lifetime, and for a spacecraft, tens of thousands of dollars) (Adams and Eby, 1987). Therefore, not only performance but also cost effectiveness can benefit from their increased utilization. 13.1.2 Ideal Structure of High Strength and High Modulus Polymer Fibers

Research to date has concentrated on the one-dimensional architecture, i.e. linear molecular chain polymers, for high performance fibers. Staudinger predicted the structure of the ideally crystalline polymer in the early 1930s when he sketched the "continuous crystal" model (Fig. 13-3) (Staudinger, 1932). His crystal model corresponds to the current ideal structure of a high performance polymer fiber: all molecular chains fully extended to form crystals with no or very few chain-end defects. All the molecules are well-oriented and densely packed in good lateral order to create a structure with very low free energy. The modulus of the fiber is thus able to approach the modulus of the crystals, which in turn approaches the intrinsic molecular chain stiffness. Therefore, as the precursor materials of high performance fibers, polymers must meet the following four requirements: bonding in the molecular backbone should be very strong and stiff; molecular weight should be high to guarantee long molecular chains; molecular conformation should be linear (or close to linear); and the crosssection of the molecular chain should be small. Table 13-3 lists bonding energies for

Figure 13-3. Schematic diagram of Staudinger's "continuous crystal" model for polymers.

Table 13-3. Bonding energy of different bonds. Bond

Bonding energy (kcal/mol)

C-C

83 146 200 86 73 147 65 128

c=c c=c c-o C-N C=N

c-s c=s

different types of connections. Polyethylene, ^ra^-polyacetylene and polyene are three good examples, since their bonding angles are 112°, 122° and 180°, respectively (for polyene, the sp-sp connection is completely linear), and the bond length between carbon atoms decreases and the bonding energy (as seen in Table 13-3) increases in this sequence. Figure 13-4 is a plot of theoretical axial modulus [measured values in the cases of polydiacetylenes (PDAs), diamond and graphite] against the effective density. The effective density of a polymer has been defined as the density of the proportion of atomic mass being actually in the backbone (Donald and Windle, 1992). For example, the effective density of PE is taken as the actual density x 24/28, which is the ratio of the molecular weight of the backbone atoms to that of the total molecular weight per chain repeat, (CH 2 -CH 2 -) n . As shown in Fig. 13-4, the calculated values of ultimate stiffness for PE, PPTA and PBZO, as well as the measured values for PDAs,

13.1 Introduction 1500

Diamond < 1000 — -

Graphite

• PBO 500 -

PE • PPTA « • PDAs | 0

1

I 2

3

Effective chain density (g/cm3)

Figure 13-4. Plot of calculated axial modulus versus effective density (which only takes into account the mass of the backbone atoms) for different materials (Donald and Windle, 1992).

diamond and graphite (in the basal plane), lie approximately in a line, which implies that the backbone packing density affects the ultimate fiber stiffness (Donald and Windle, 1992). Thus, the goal in high strength/stiffness fibers is, from suitable precursors and through optimum processing, to manufacture highly ordered polymer fibers that have structure and properties close to those of the ideal model. 13.1.3 Historical Review of the Development of High Performance Polymer Fibers

Fiber technology is ancient, with the first fibers being from natural products which were abundantly available. These can be divided into three groups: vegetable (cellulose), e.g. linen and cotton; animal (protein), e.g. silk and wool; and mineral, e.g. asbestos and some metal filaments. Nature has provided man with abundant spinners of fine organic fibers, for example, spiders (silk). Earliest records of linen prepared from flax appeared during the early Egyptian dynasties nearly 6000 years ago. Silk, with the cultivation of bombyx

603

rnori, began to attract the attention of Chinese royalty during the 3rd millennium B.C. and gold threads were used in India to decorate saris thousands of years ago (Delmonte, 1985). As in the development of synthetic polymers, the first step toward the development of synthetic fibers was to modify natural materials. This step took place in the nineteenth century, when filaments were spun from solutions of cellulose derivatives, such as cellulose nitrate. In the early part of the twentieth century, viscose rayon, cellulose acetate, and others followed. Since the pioneering work of Carothers which led to the introduction of nylon fibers by DuPont in the 1930s, polymer scientists have pursued the development of high performance fibers to replace natural or metallic products with products of improved mechanical properties and reduced weight. Strong motivation for the invention of new polymer fibers has come from the aerospace industry, which seeks fibers to use in reinforced composite structural materials. Development was relatively slow and evenly paced until DuPont revolutionized the field with the release of Kevlar. In the 1970s and 1980s, the field boomed with new developments. Table 13-4 lists in chronological sequence of publication some authors and literature that reported either new fiber processing technologies or new high performance polymer fibers. This table is given only as an example since many new processing methods developed in companies have not been published yet, or were published much later. In the past 20 years, as shown in Fig. 13-5, polymer fibers have shown great improvements in strength, compared to some inorganic fibers, such as glass and boron fibers, whose properties have remained largely unchanged. Today, several high performance polymer fibers are com-

604

13 High Performance Polymer Fibers

Table 13-4. Literature survey of high performance polymer fibers. Fiber

Reference

PE Kevlar PE PE Nylon, PET PE PBZT POM PDAs POM PE

Southern and Porter (1970) Kwolek (1972) Pennings et al. (1972) Capaccio and Ward (1974) Kunugi etal. (1979) Smith and Lemstra (1980 b) Allen etal. (1981b) Nakagawa et al. (1983) Galiotis and Young (1983) Ishida (1987) Kanamoto et al. (1987)

Method Solid-state extrusion Liquid-crystal spinning Surface growing Super drawing Zone drawing Gel spinning Liquid crystal spinning Microwave heating-drawing Solid-state polymerization High-pressure thermo-drawing Reactor powder two-stage drawing

mercially available with properties that compete with the best inorganic fibers and are far superior to metal fibers. There are, at present, two primary technical routes to HPOPFs: (1) For the flexible or semiflexible polymers, new technologies are used, such as gel spinning, to make fibers with extended molecular chains that form crystals highly oriented along the fiber axis direction (Pennings, 1977; Smith and Lemstra, 1980a, b; Lemstra et al., 1986; Hoogsteen et al., 1988 a, b; Igachi and Kyotoni, 1990).

(2) For the rigid or semi-rigid polymers, because of the stiff aromatic or heterocyclic structural units in their molecular backbone, liquid crystal spinning solutions are used in dry-jet wet-spinning techniques to produce fibers with very high orientation and crystallinity. Some of these anisotropic polymers can also be melt or solution spun followed by high-draw and heat treatment to obtain high performance fibers (Watt and Perov, 1985; Adams and Eby, 1987; Ohta et al., 1988; Yang, 1989; Yamada, 1991; National Materials Advisory Board, 1992).

Extruded liquid crystal f i b e r s .

Melt-spun nylon

Wet-spun cellulose Natural fibers \ 1900

1910

1920

1930

Figure 13-5. Strength of typical commercial organic polymer fibers developed over a number of years (After National Materials Advisory Board, 1992).

1940

1950

Year

1960

1970

1980

1990

13.2 Fiber Processing

13.2 Fiber Processing Advances in the performance of polymer fibers have come about because of continuing progress in new fiber-forming precursors and fiber processing research and development. A more detailed account of fiber processing is presented in Volume 18 of this Series. Due to breakthroughs in processing techniques and new materials, especially since the 1970s, the mechanical properties of polymer fibers have improved dramatically (Fig. 13-5). The new, nearly perfectly oriented fibers can now be spun from liquid-crystal polymers through melt-spinning and dry-jet wet-spinning techniques, as well as from linear flexible chain polymers through gel spinning. The selection of a suitable process depends largely upon the solution or melt behavior of a polymer. For example, gel spinning may not be satisfactory for rigid polymers because of limited gel drawability. Similarly, semi-rigid polymers which do not form liquid crystal solutions or melts may be converted to high performance fibers from their isotropic solutions followed by drawing at elevated temperature. Therefore, such process selection is significantly dependent on polymer structure. Various processes are now reviewed. 13.2.1 Gel Spinning of Flexible and Semi-flexible Polymers

As detailed in Chapter 4, in the 1970s and 1980s, the pioneering work of Pennings (Pennings et al., 1972; Pennings, 1976) developed the idea of making high performance fiber from solution spinning of flexible chain PE, and later improvements made by Smith and Lemstra (Smith and Lemstra, 1980 a, b) involved the drawing of a gel. Because the linear molecular chains are arranged parallel to the fiber

605

axis, the tensile strength and modulus of gel-spun PE fiber can reach 7 GPa and 200 GPa, respectively (Ward, 1985 a, b). This technique has created a new route to produce high strength and high modulus fibers from flexible chain polymers. The decade of the 1980s was the primary stage for the industrialization of "gel spinning". Up to now, three commercialized PE extended-chain fibers have been introduced to the world market, Spectra manufactured by Allied-Signal, Dyneema by DSMToyo and Tekmilon by Mitsui (Chang and Weedon, 1986; C & E News, 1987). Interest in this area is now growing and many other flexible or semi-flexible chain polymers, such as nylon, polypropylene (PP), poly(vinyl alcohol) (PVA) and polyethylene terephthalate) (PET) are being studied in relation to this technique to produce high performance fibers (EP 83108258; EP 84114872; Peguy and Manley, 1984; Takahashi et al., 1991a, b). In the so-called "gel spinning" of polyethylene fibers, the key feature is to use ultra-high molecular weight PE (Mw > 1 000 000) dissolved in a solvent, such as paraffin, xylene, naphthalene or decalin at a low concentration. When the dilute solution of ultra-high molecular weight PE emerges from the spinneret, it is first quenched by an air gap and then enters a cooling bath. A thermodynamically reversible gel is created. Owing to the very long polymer chains in the dilute solution, entanglements between molecules are formed and in the procedure of making gel, the entanglement-network is maintained. It is very interesting that even after removal of the solvent the gel can still be highly drawn. The reason for this is that during cooling, the as-spun fiber forms small chain folded lamellae with a significant amount of solvent between them so that the as-spun fiber keeps the physical

606

13 High Performance Polymer Fibers

gel formation, namely, the swollen network in which crystallites form the junctions, as shown in Fig. 13-6 (Ward, 1987). In the drawing, entanglements act as temporary physical crosslinks and play an important role. This feature allows the asspun fiber to be drawn to very high ratios, even over 200 times. The maximum draw ratio is closely related to the average distance between the entanglements, i.e. the "crosslinking" density and the solution concentration (Fig. 13-7) (Smith and Lemstra, 1980 a, b; Iguchi and Kyotani, 1990). Upon hot drawing of the as-spun fiber, the lamellae are gradually transformed into smooth fibrils. If the draw ratio is high (typically over 20), "super-drawn" fiber is produced, as shown in Fig. 13-8 (Ohta etal., 1988). In the super-drawn fiber, there are long arrays of extended-chain crystals interrupted by disordered domains that originate from the presence of topo-

i i i i

I I i i I

i i i i i i i i i i i i

i i i i

I I I I I

i i i i i i i i i i i i

i i i i

I I I I I

i i i i i i i ii

f\

i i i

r\

i i

I I I

I I I I I I

I i

I I I

I I I I I I

i i

I I i

I II

i i

I I I

I I I I I I

w

I I i

M M I i

I i

I I I

I I I I I I I I I I I I

I I I

I I I I I I I I I I I I

I I I

I I I I I I I I M

I I

I I I

I I I I I I I I I I

I I I

I I I I I I I I '

i I I

II I II

Figure 13-6. Structural model of gel-like fibers (Ward, 1987).

100

10 Concentration of PE (%)

Figure 13-7. Relation between maximum draw ratio and UHMW PE solution concentration at 120 °C (circles) and room temperature (squares) (Iguchi and Kyotani, 1990).

Figure 13-8. Structural model of super-drawn UHMW PE fibers (Ohta et al., 1988).

logical defects, such as chain entanglements. Adjacent crystalline blocks are thought to be connected by taut tie molecules traversing the disordered domains (Penning et al., 1991; Agosti et al., 1992) (see Chap. 7). Therefore, this structure results in a high strength and high modulus fiber. The so-called gel-spinning process is in fact sol-spinning and gel-drawing. A typical spinning process employs a solution of 5-15 wt.% UHMW (generally 5-7 million) PE in paraffin oil which is extruded through an air gap into the nhexane bath. Figure 13-9 (Lemstra etal., 1986) is a schematic diagram of the gelspinning process. The as-spun fiber is dried under vacuum and then drawn in a tube

13.2 Fiber Processing

607

Suspension UHMW PE


Quenching/extraction bath

Oven

containing silicon oil with a temperature gradient of 100-150°C. The maximum draw-ratio is normally 30-100 times (Barham and Keller, 1985; Ohta et al., 1988; Hoogsteen et al., 1988 a, b; Iguchi and Kyotani, 1990). The cost of solvent recovery and the controlled low-speed spinning and drawing increase the cost of the fiber to a level comparable to other high performance fibers, despite the lower polymer cost (National Materials Advisory Board, 1992). 13.2.2 Melt Spinning of Thermotropic Liquid Crystalline Polymers

Incorporating aromatic cyclic or heterocyclic rings in the molecular backbone to build stiff chain fiber-forming polymers is another commercially successful way to improve fiber mechanical and thermal properties. Following this synthetic approach from the 1950s, many new polymers have been produced and processed into high performance fibers, typified especially by the aromatic polyesters. One attractive feature of these polymers is that they can form a liquid crystalline phase

o

Fiber

Figure 13-9. Schematic diagram of gel spinning (Lemstra et al., 1986).

over a certain temperature range in the melt, i.e. without the need for a solvent. Such polymers are called thermotropic liquid crystalline polymers (TLCP). Therefore, they can be spun with conventional melt-spinning techniques to achieve high performance properties. Generally, their melting points are around 300-400°C, near their isotropic-nematic transition temperature (TNI). Unfortunately, very few of the aromatic polyesters are suitable for fiber-making because of their tendency to degrade at high temperature and the poor tractability of the polymer melts (Yang, 1989; MacDonald, 1991). The commercialized thermotropic liquid crystal copolyesters can be divided into three general groups, all of which are made by condensation polymerization (Kyotani, 1990). The first group has low melting temperatures, and their thermal distortion temperatures (TDTs) are around 60120 °C. An example from this group is polyester X7G, manufactured by Kodak. X7G was the first thermotropic liquid crystalline fiber on the world market. Polyester X7G is a copolymer of polyethylene terephthalate (PET) and />-acet-

608

13 High Performance Polymer Fibers

oxy benzoic acid (ABA) with a melting point (Tm) of about 230 °C. The comonomer ratio of PET to ABA is in the range of 40:60 to 20:80. The chemical synthesis of X7G is as follows: O

O

O II -C-OH

o

--O-CH 2 -CH 2 -O-C-/(J)VC--

CH.-C-C

-CH.COOH

ABA

PET

O-CHo-CHo-O-CO-

-CO X7G

The second group of TLCPs has intermediate melting temperatures with TDTs of about 180-240°C, as represented by Vectran, marketed by Hoechst Celanese Co. Vectran is a polyester-polyarylate fiber with 7^ around 270 °C. Its synthesis, with a comonomer ratio of about 30:70 ANA (acetoxy naphthoic acid): ABA, is as follows:

COOH

O

O

II

/p^\

II

CH 3 -C-OVrjVC-OH

-CH3COOH

ABA

ANA

Vectran

The third group of TLCPs has high melting temperatures with TDTs around 250350 °C. Ekonol, which was developed at Carborundum and is now produced by Sumitomo Chemical Co. and Nippon Exlan Co., is representative of this group. Its melting temperature is above 400 °C. The general synthesis for the base polymer is o

o

CH3~CO-O VQWQVO-CO-CH3 + ABP O O II II HO-C-r^^rC-OH IA

TA

o o CH 3 -C-OVQVC-OH

-CH3COOH

ABA

Ekonol

Its comonomer ratio of (ABP: TA: IA: ABA) is 5:4:1:10, where ABP is />,//-diacetoxy biphenyl; TA, terephthalic acid; and IA, isophthalic acid.

13.2 Fiber Processing

The spinning process is similar to conventional PET melt spinning. The polymer is first washed with water, alcohol or acetone to remove impurities. Then it is heated to about 300-400°C, near its isotropic-nematic transition temperature, to form the thermotropic liquid crystalline spinning melt. The spinning melt with a typical intrinsic viscosity of 1.5-3 dl/g, is extruded at 280-350°C through spinneret holes to form the fiber. The spinning speed is in the range of several hundred to several thousand meters per minute. The as-spun fibers are heat treated at elevated temperatures (normally around 250- 300 °C) for several hours, which is much longer than in a conventional annealing process. During this time a solid state polymerization occurs which is an important factor in increasing the fiber strength. The spinning and heat treatment conditions are very critical to the fiber formation and quality. These processing conditions are largely selected from experience. The temperature of melt spinning must be above the crystalto-nematic transition temperature to guarantee formation of the liquid crystalline phase. The higher the processing temperature, the faster the melt degrades; but the lower the temperature, the higher the melt viscosity and the poorer the processing. Molecular weight control is of special importance. In the melt, increasing the molecular weight leads to a sharp increase in the melt viscosity, which can even be sufficient to prevent processing. Thus the upper limit of the molecular weight for the melt spinning of thermotropic liquid crystal polymers is relatively low, which results in lower fiber tensile strength. However, the possibility of these fibers undergoing a solid state polymerization during the heat treatment can cause a significant increase in the molecular weight, and thus the tensile properties. This process thus requires

609

much longer heat-treatment times than conventional ones. It was observed that after this type of heat treatment there were new connections between the original microfibrils, but the overall fibril structure and orientation did not change (Matsuda, 1985a, 1985b; Yang, 1989). Heat-treatment technologies with shorter times have been evolved. It was reported that by attaching substituents (such as the phenyl group) to the hydroquinone moieties of aromatic polyesters, even with short heattreatment times (30 min), the fibers had satisfactory tensile properties (Jackson, 1980). In general, these fibers have tensile strengths of 2 - 6 GPa, tensile moduli of 60-130 GPa and an elongation to break of 2 - 4 % (Ohta et al., 1988; Yang, 1989; Kyotani, 1990; Koide, 1990). Compared with the solution spinning of lyotropic liquid crystal polymers (which will be discussed in the next section), melt spinning of thermotropic liquid crystal polymers has the following benefits: there are no solvent disposal and recycling problems; it is easy to obtain small diameter fibers; and there is less difference between the outer layer of the fibers, "skin", and the center or "core". Many major chemical companies have been involved in research on thermotropic liquid crystal polymer fibers, driven by a potential market for new industrial applications and tempered by the realities of the high cost and poor availability of the TLCP precursors, as well as their processing difficulties. 13.2.3 Dry-Jet Wet Spinning of Lyotropic Liquid Crystalline Polymers

Most high performance polymers decompose at high temperatures without melting and can be dissolved in very few solvent systems due to their aromatic structure and to their rigid molecular

610

13 High Performance Polymer Fibers

backbone. Therefore, conventional meltspinning and solution-spinning technologies cannot be used. Fortunately, it has been found that certain rod-like polymers such as aromatic polyamides, when dissolved in an appropriate solvent, form a liquid crystalline solution. High performance fibers can be processed from these lyotropic (liquid crystal) systems by means of the dry-jet wet-spinning technique. Since the late 1960s, many lyotropic polymers have been reported and very active fiber research has occurred all over the world, including the U.S., Japan, the Netherlands, France, and the former Soviet Union. Examples are poly(ra-phenylene isophthalamide), known as Nomex from DuPont; poly(/?-benzamide), called Fiber B (DuPont); polyamide hydrazide, X-500 studied by Monsanto in the early 1970s;

-NH-r^rNH-CO-

13.2.3.1 Polymer Synthesis -CO-

Nomex

CO-NHH-CO Polyamide hydrazide, X-500

Kevlar

PBZT

the first commercial and the most successful high performance fiber, poly(/?-phenylene terephthalamide) (PPTA), DuPont's Kevlar; poly(/?-phenylene benzobisthiazole) (PBZT), and, poly(/?-phenylene benzobisoxazole) (PBZO), which are two of the strongest high performance fibers today. However, in spite of all these efforts to date, only Kevlar (introduced to the market in 1972 by DuPont) and Twaron (later by Akzo) are produced on a large scale (Adams and Eby, 1987). The reasons for this are difficult polymer processibility, few precursor sources, high manufacturing cost, small market size and the long time required to evaluate fiber behavior in industrial usage. In this chapter, we focus on the most successful aramid fiber, Kevlar, and the strongest polymer fibers, PBZT and PBZO, as models of HPOPFs.

The synthesis of rigid-rod polymers appears to be deceptively simple condensation chemistry. In practice, it is technologically difficult, to which the great time lag in commercialization testifies. PPTA is synthesized by the condensation of terephthaloyl chloride (C1CO-C 6 H 4 -COC1) (TCI) and /?-phenylene diamine (H 2 NC 6 H 4 -NH 2 ) (PPD) in a mixture of hexamethylphosphoramide (HMPA) and nmethylpyrrolidone (NMP) solvents (Matsuda, 1985a, 1985b; Yang, 1989). The polycondensation reaction is normally kept at a low temperature (preferably at 10-20°C) to avoid degradation and side reactions. The reactants are vigorously stirred and the products are subsequently ground in water, filtered, and the resulting solid polymer thoroughly washed. The degree of polymerization (DP) of PPTA is relatively low (about 7-264), with a Mw in the range of 1700-63 000 measured using

611

13.2 Fiber Processing

gel permeation chromatography (GPC) (Yang, 1989; Fitzgerald and Irwin, 1991). Among the aromatic heterocyclic polymers, the most attractive for potential use as high performance fibers at present are PBZT and PBZO. These two polymers offer excellent fiber tensile and thermal properties unsurpassed by other aromatic fibers. Polymer chemists at the U.S. Air Force's Materials Laboratory, working along directions parallel to the DuPont workers, were exploring aromatic heterocyclic polymers for high-temperature applications in the 1960s. These polymers have higher thermal stability than the polyesters or the aramides. In the 1970s1980s, an extensive development program at both Celanese and DuPont led to successful fiber preparation from these rigidrod polymers. Recently, Dow Chemical Co. has been considering the commercialization of PBZO fibers, because of their excellent mechanical properties and cheaper synthetic route compared to PBZT (Reisch, 1987; Arnold, 1989). trans-PBZT is polymerized from 2,5-diamino-l,4-benzenedithiol hydrochloride (DABDT) and terephthalic acid (TA), while cis-PBZO is based on 4,6-diamino-l,3-benzenediol dihydrochloride (DABDO) and TA, as shown in Fig. 13-10. Both reactions are carried out in polyphosphoric acid (PPA) solvent. Technological requirements include small particle size, crystalline TA and very pure PPA, DABDT and DABDO. The synthesis of DABDO and DABDT and subsequent polymerization were originally developed by Wolfe (Wolfe, 1985 a, b,c, 1988, 1989; Lysenko, 1988). 13.2.3.2 Lyotropic (Liquid Crystal) Solutions

In 1956 Flory predicted that concentrated solutions of rigid-rod polymer mol-

TA

DABDT

HCl-H2N

NH2-HCl HOOC^QVCOOH

HO

OH TA

DABDO

PBZO Figure 13-10. Syntheses of PBZT and PBZO.

ecules readily form liquid crystals (Flory, 1956). The development of liquid crystalline solutions depends on the molar mass, the length-to-diameter ratio and the flexibility of the molecules, the solvent and the temperature (also see Chapter 5 of this Volume). However, the most important parameter is the polymer structure, which must exhibit some rigidity. The necessary rigidity of many of the polymers usually makes them decompose before melting. The important consequence of solvent addition is that liquid crystallinity can be obtained with rigid-rod polymers which would not otherwise exhibit the mesophase at temperatures below that of thermal decomposition. Because they are much less soluble in common solvents than more flexible polymers, their dissolution often requires the use of strong, protonating acids (such as H 2 SO 4 , CF 3 SO 3 H and CH3SO3H) or aprotic solvents (such as

612

13 High Performance Polymer Fibers

dimethylacetamide or TV-methyl pyrrolidone in conjunction with LiCl or CaCl2 in small percentages). The persistence length of a molecule in solution is considered a useful measure of its rigidity, and is defined as the average projection of the end-to-end distance of an infinite chain in the direction of the first link. The longer the persistence length of the polymer, the lower the concentration necessary to form a liquid crystal (Ohta etal., 1988; Donald and Windle, 1992). Persistence lengths of a few polymers are listed in Table 13-5. In the stiff chain polymers there is significant uncertainty associated with the persistence length measurement, because of the difficulty of finding suitable good solvents. An additional problem is that the presence of solvent may change the preferred conformation of the molecule and thus its rigidity (Berry, 1989; Brelsford and Krigbaum, 1991). Therefore, the data in Table 13-5 should be considered only for relative comparison.

The results for PBZT and PBZO are in qualitative agreement with the results of molecular dynamics simulations (Zhang and Mattice, 1992; Farmer and Chapman, 1993). Such results can be subject to uncertainties resulting from questions of force fields, sampling of all of phase space, etc. As polymer is added to the solvent, the viscosity increases [Fig. 13-11 (Won Choe and Kim, 1981)] but the solution remains isotropic and clear. With increasing polymer concentration, the rod-like polymers tend to cluster together in bundles of quasi-parallel rods and to form "domains". These domains are anisotropic

2.1 AT 70 °C 90

80

70

Isotropic

Anisotropic

60 Table 13-5. Measured persistence lengths for various polymers. 50

Polymer trans-PBZT ds-PBZO PPTA

Experimental persistence length (A) >500 (Crosby etal., 1981) >640 (Wong etal., 1978; Aharoni, 1985) 100-175 (Wong etal., 1978) 435 (Erman etal., 1980) 200-400 (Arpin, 1976; Arpin and Strazielle, 1977)

PBA

410 (Erman etal., 1980) 200-400 (Arpin, 1976; Arpin and Strazielle, 1977)

X-500

75 (Ciferri, 1975)

Cellulose PE

42 (Kamide, 1983) 5.75 (Erman etal., 1980)

30

20

1

1

V

2

i

i

i

3

k

5

6

7

8

9

10

% PBZO in 100% H2SO4

Figure 13-11. Critical concentration curve for PBZO (Won Choe and Kim, 1981).

13.2 Fiber Processing

and within them there is nematic order of the chains. However, there is little or no directional correlation among these domains. At this stage, even though the solution changes from monophasic to biphasic, its viscosity still increases with concentration. At a critical concentration, the solution becomes opaque and anisotropic. There is a sharp decrease in the viscosity with further increase in polymer concentration. This decrease reflects the formation of the oriented liquid crystal domains in which the chains can be aligned parallel to the flow direction, thereby reducing the frictional drag on the molecules. The critical concentration is a function of the solvent and the temperature. It tends to decrease with increasing molar mass and with increasing length-to-diameter ratio of the polymer. Theoretical predictions of the behavior of viscosity around the critical concentration have been developed (Kawai, 1980; Doi, 1981). Another characteristic of the liquid crystal solutions is that their viscosity is a function of the alignment of the molecules. In other words, all external fields, such as flow, stress, and even electric and magnetic fields, which can influence the orientation of the molecules will affect their viscosity (Collings, 1990; Donald and Windle 1992). For instance, in the anisotropic liquid crystal solutions the directors readily align with the shear direction, and with increasing shear rate the viscosity drops (Hermans, 1962; Baird, 1978). Although the rheological behavior of the lyotropic solutions has been studied extensively, especially for those technologically important materials, it is still not well understood (Donald and Windle, 1992). For PPTA in concentrated H 2 SO 4 the intrinsic viscosity [rj] correlates with the weight average molecular weight (Mw) (Schaefgen et al., 1976) as

613

M = 7.9xlO- 5 Mi/ 06 (13-1 a) (for Mw> 12000) [rj] = 2.8 x

10~7Ml7

(for Mw< 12000) (13-1 b) The fiber strength increases with increasing viscosity (Matsuda, 1985 b). When the viscosity increases from 4 dl/g to 6 dl/g, corresponding to a molecular chain length of 220 nm, the strength improves linearly from 2.6 GPa to 3.7 GPa. Therefore, to make satisfactory fibers, the viscosity of the polymers should be greater than 4 dl/g (i.e., molecular weight > 27 000) and normally polymers with a viscosity around 5.1 dl/g are used (Ohta et al., 1988). PBZT and PBZO are soluble only in strong aprotic acids such as polyphosphoric acid (PPA), methanesulfonic acid (MSA), chlorosulfonic acid (CSA), trifluoroacetic acid (TFA), and 100% sulfuric acid. For PBZT and PBZO in MSA the intrinsic viscosity-weight average molecular weight relations are as follows (Wolfe, 1988): for PBZT (13-2) 7

[rj] = 2.77 x 10" x M

for PBZO (13-3)

By dissolving isolated trans-VBZT in MSA (about 10%) or using PBZT/PPA solution (5-6%), a nematic spinning dope is made for dry-jet wet spinning. PPA solutions offer significant advantages of improved properties and process simplicity, since one less step is involved, as the polymer does not have to be isolated and redissolved (Wolfe, 1988, 1989). 13.2.3.3 Fiber Spinning

The superior mechanical properties of Kevlar, PBZT and PBZO fibers have been achieved with an optimized combination of high molecular weight polymer, liquid crystal spinning dope, spinning process

614

13 High Performance Polymer Fibers

and final tensioned heat treatment at high temperature. It has been demonstrated that for lyotropic polymer solutions, dryjet wet spinning can produce excellent fibers (Won Choe and Kim, 1981; Allen etal., 1981a, b, 1985; Allen and Farris, 1989; Yang, 1989). As schematically shown in Fig. 13-12, this technique involves the extrusion of the liquid crystalline solution under heat and pressure through an air gap into a coagulation bath, followed by washing, drying, tensioning and heat-setting. As shown in the phase diagram [Fig. 13-13 (Kikuchi, 1982)] of the PPTA/H 2 SO 4 system, a solution with a concentration of about 20% reaches the nematic liquid crystal phase at ca. 80 °C and the isotropic

Air gap —

Winder Washing bath

Coagulation Bath

Figure 13-12. Schematic diagram of dry-jet wet spinning.

phase at ca. 140 °C. Thus the spin dope is extruded at 70-90°C into the air and then immediately passed through a coagulation bath. The air gap serves two purposes: to allow the spin dope to be at a considerably higher temperature, enabling more concentrated spinning solution and higher spinning rates to be used; and to provide a region of extensional flow in the fiber while it is still liquid crystalline in order to promote the development of good orientation of the molecular chains across the full cross-section of the fiber. Normally the spinning speed is around several hundred meters per minute. It is interesting that a spinning speed of 2000 m/min can be achieved (Yang, 1982). It has been reported that the spin stretch factor (the ratio of filament wind speed to extrusion speed) (SSF) is an important parameter for the spinning of lyotropic polymers (Kwolek et al., 1977). Fig. 13-14 shows the SSF effect on the tensile strength of Kevlar as-spun fibers. Below a SSF of 3, the fiber tenacity is considerably lower (Ohta et al., 1988). For Kevlar, cool water at about 5°C is used as coagulant, and for PBZT other coagulants such as dilute phosphoric acid, methanol, and N H 4 0 H have also been used, as well as a room temperature water coagulation (Yang, 1989; Rakas and Farris, 1989). During coagulation, the acid diffuses out of the fiber, which crystallizes

140 120 Isotropic

100

H 2 S0,

80 60

4.8

98%

40

1.3 1.3

96%

Solid

20 0

5

10

15

20

25

Polymer concentration (wt.%)

30

98%

Figure 13-13. Phase diagram of PPTA/H2SO4 system (Kikuchi, 1982).

615

13.2 Fiber Processing

3

Figure 13-14. Relation between the tensile strength of as-spun Kevlar fibers and the spin stretch factor (SSF) (Ohta et al., 1988).

in a highly oriented manner. The coagulation bath may not be a simple one-step process if strong acids are used (as in the production of Kevlar), which themselves need to be neutralized without hydrolyzing the fiber. The as-spun fiber is then washed to remove the residual solvent and dried. Subsequent heat treatment is often used to further improve fiber mechanical properties and it is commonly done under tension. For PBZT, the tensioned fiber passes through a tubular oven in a nitrogen atmosphere. Heat treatment temperatures of 500-700 °C with a residence time of a few seconds to several minutes are typically used (Pottick and Farris, 1985; Rakas and Farris, 1988, 1989; Allen and Farris, 1989; Chenevey and Timmons, 1989; Ledbetter et al., 1989). After tensioned heat treatment, the fibers have an improvement in their overall axial orientation, crystal perfection and lateral order, which results in an increase in their mechanical properties. Table 13-6, taking PBZO as an example,

shows the enhancement in orientation (measured from azimuthal scans of the Xray equatorial reflections), axial and lateral crystallite size (determined from the equatorial diffraction patterns), and the tensile modulus and strength through the tensioned heat treatment (Martin and Thomas, 1991a; Krause et al., 1988). Other novel lyotropic polymer fibers have also been spun by using the dry-jet wetspinning techniques (Kumar, 1990 a, 1991). However, several difficult hurdles are encountered during the development of the spinning process for the polymer lyotropes, such as environmental problems, waste disposal, toxicity and corrosiveness of ingredients, and handling of hazardous materials. Multi-disciplinary chemical and engineering skills are required to solve these problems and thus, costs can be high. 13.2.4 Solution Spinning

High performance polymer fibers can also be manufactured by solution spinning from isotropic spinning dopes of aromatic copolyamides. Technora fiber, commercialized by Teijin Ltd. in the late 1980s, is an example. Technora is based on copoly(/7-phenylene/3,4'-diphenyl ether terephTable 13-6. Changes in structure and property parameters of PBZO fibers through heat treatment at 600 °C (Krause et al., 1988; Martin and Thomas, 1991a). As-spun Heat treated PBZO PBZO Average lateral and (axial) crystallite size (nm) Misorientation angle Modulus (GPa) Tensile strength (GPa) Breaking elongation (%)

2.6

3.9

(3.0)

(3.9)

6°

4°

165

317

4.3

4.9

2.8

1.7

616

13 High Performance Polymer Fibers

thalamide). The polymer is synthesized by the low-temperature polymerization of pphenylenediamine, 3,4'-diaminodiphenyl ether, and terephthaloyl chloride in an amide solvent with a small amount of alkali salt (calcium chloride or lithium chloride) (Shimada et al., 1982). The polymerization reaction is carried out at 0-80°C in 15 hours at a polymer concentration of 6 12% and is terminated with the addition of neutralizing agents (calcium hydroxide, calcium oxide). The reaction mixture is filtered to give a pure isotropic polymer solution. The solution is then spun from spinnerets into an aqueous coagulating bath to form fibers. The coagulant contains 3 5 50% CaCl2 or MgCl2 to control the coagulation rate. The as-spun fiber is washed and extensively drawn (6-10 times) at a high temperature (485 °C). As a consequence, Technora exhibits properties that resemble those of conventional fibers in some aspects and those of rigid chain aramid fibers in other aspects. In addition to high tensile strength (3 GPa) and modulus (70 GPa), it offers some unique fiber properties, such as chemical and impact resistance (Yang, 1989). 13.2.5 Solid State Extrusion Extraordinary fibers have been produced from flexible polymers such as PE by modifying the method of solid state extrusion (Zachariades and Porter, 1983; Ciferri and Ward, 1979; Igushi and Kyotani, 1990). A bulk polymer in the solid state (i.e. at an elevated temperature but below its melting point) is forced through a narrowing die at very high pressures to form a transparent fiber. In the original process, melt crystallized polyethylene was extruded to draw ratios of up to 48, and the extrudate had a modulus of up to about 40 GPa with a degree of crystallinity of

about 85 % (Adams et al., 1985). However, by extruding (typical extrusion draw ratios of about 5) a sedimented mat of single crystals of ultra-high molecular weight PE (up to about 8 million Daltons) followed by post drawing the extrudate at temperatures near the melting point (typical draw ratio of 40-50), very high overall draw ratios, up to 280 times, were achieved (Kanamoto et al., 1983; Porter, 1992). This drawing process, starting with few chain entanglements, produced extremely efficient conversion of the lamellar crystals into fibrillar crystals, with a measured degree of crystallinity of up to 96 % in the fibers. The modulus of the fibers depends primarily on the draw ratio and increases with draw ratio. On the other hand, the tensile strength appears to depend much more on the polymer molecular weight (Yang, 1989). It has been reported that the highest achieved modulus for PE is 220 GPa, which is a significant fraction of the theoretical estimated modulus of about 320 GPa (see Chap. 7) (Kanamoto et al., 1983; Ward, 1985 a; Iguchi and Kyotani, 1990). Although this two-step method appears to be rather difficult to commercialize at present, it has set the current aim for a flexible chain polymer fiber tensile modulus. Many other polymers have been extruded from the solid state, and a great deal of interest currently centers around poly(vinyl alcohol) (PVA). PYA has theoretically high mechanical properties, comparable with those of UHMW PE (Sakurada, 1985; Kanamoto et al., 1990), and it may be possible for its intra- and interchain hydrogen bonds to act as junction-points in forming a highly crosslinked network which may be beneficial for enhanced extension (Sakurada, 1985; Shibayama et al., 1990) and for reducing the likelihood of fiber plastic deformation by creep (Conibeer et al., 1990).

13.3 Fiber Morphological Structure

13.3 Fiber Morphological Structure The morphological structure of high performance polymer fibers has been extensively investigated using wide-angle X-ray scattering (WAXS), small-angle X-ray scattering (SAXS), light microscopy (LM), scanning electron microscopy (SEM), and transmission electron microscopy (TEM), including bright-field (BF), dark-field (DF), and selected-area electron diffraction (SAED) techniques (Yabuki et al,, 1975, 1976; Northolt and Ven Aarsten, 1977; Dobb, 1979; Adams et al., 1979; Roche et al., 1980; Minter et al., 1981; Blumstein, 1984; Gutierrez and Blackwell, 1984; Tashiro et al., 1987, 1988; Krause et al., 1988; Odell et al., 1989; Martin and Thomas, 1989, 1991a, 1991b; Fratini etal., 1989; Shimamura and Hashimoto, 1991). It is very interesting that there are several common structural features in high performance polymer fibers. Most of the fibers have round cross-sections, except for a few which have specialized shapes for specific applications, for example, triangular cross-sections for higher surface area.

617

The typical fiber diameter varies from 10 to 25 microns. Most solution-spun high performance polymer fibers have a distinct skin-core structure. As seen in Fig. 13-15, the TEM micrographs and SAED patterns of Kevlar 149 exhibit an apparent difference in morphological structure between skin and core regions (Young et al., 1992). The existence and magnitude of the skin/ core differences are believed to be the result of processing conditions, such as solvent diffusion and fiber coagulation. There are also voids in these fibers. The SAXS pattern of heat-treated PBZT (Fig. 13-16) shows strong equatorial scattering attributed to voids elongated parallel to the extrusion direction. One of the most prominent structural characteristics of high performance polymer fibers is their oriented fibrillar microstructure. Fig. 13-17 displays LM and SEM surface micrographs of UHMW PE fibers. The fiber surfaces appear rather uniform and smooth but with a fibril-looking texture, which is the consequence of high-drawing. Sometimes the fibers show kinks, which are the result of axial com-

Figure 13-15. TEM micrographs and SAED patterns (inset) of skin (a) and core (b) regions of Kevlar 149 fibers (Young et al., 1992). (a)

(b)

618

13 High Performance Polymer Fibers

Figure 13-16. SAXS pattern of heat-treated PBZT showing strong equatorial scattering caused by voids (Minter, 1982).

pression. A hierarchical structure for high performance polymer fibers has been proposed (Sawyer et al., 1993). As shown in Fig. 13-18, a fiber is envisioned to be comprised of macrofibrils, fibrils, and microfibrils, the typical diameters for these being 5 microns, 0.5 micron and 0.05 micron, respectively. The microfibrils consist of crystallites in combination with amorphous or disordered regions. Although a fibril structure is the common feature of high performance polymer fibers, the dimensions cannot always be discerned from the ratios indicated in the above model. In other words, different high performance polymer fibers have different fibril structures because of the various molecular structures and processing conditions. The PPTA molecule has a semi-rigid and extended chain structure. This rigidity is mainly attributed to the structure of aromatic rings connecting with the planar amide groups O

H

50

Figure 13-17. SEM surface morphology of UHMW PE fibers showing fibril structure, (a) optical micrograph; (b) SEM micrograph (DeTeresa, 1985).

which is resistant to rotation. The crystal structure of PPTA fiber is a monoclinic unit cell (modification I) with pseudoorthorhombic symmetry and two chains per cell {a = 0.773-0.784 nm, b = 0.5150.522 nm, c (repeat distance) = 1.28 1.29 nm, and y = 90°), as depicted in Fig. 13-19 (Northolt and Sikkema, 1991). Interchain hydrogen bonding occurs only along the (100) planes whereas weak Van der Waal forces operate in other lateral directions. The fiber density observed is 1.44 g/cm3 and the calculated (crystal) density is 1.52-1.54 g/cm3. By changing the spinning and coagulation conditions, different crystal unit cell structures are obtained. Spinning from sulfuric acid solu-

13.3 Fiber Morphological Structure

619

Oriented fiber

^lacrofibri ^ — 5.0u.m

\

Fibril 0.5|im Microf ibril Enlarged view of microfibrils

Figure 13-18. A hierarchical structural model for high performance polymer fibers, with more detail of the micro fibril sizes, shapes and order (Sawyer et al., 1993).

(a)

(b)

Figure 13-19. Crystal unit cell structure (modification I) of PPTA, (a) the view is perpendicular to the c-axis and parallel to the a-axis, (b) the view is down the c-axis (Northolt and Ven Aartsen, 1973).

620

13 High Performance Polymer Fibers

tions with a concentration of less than 9 % yields crystal modification II (Haraguchi et al., 1979a, b). It has almost the same unit cell dimensions, except the hydrogenbonded plane through the center of the unit cell is shifted along the 6-axis by a distance of 6/2. For the spinning dope with a concentration of 9-15%, both crystal modifications I and II coexist. When the concentration > 15 %, only modification I forms (Northolt, 1981). It has also been found that if the coagulant changes from methanol to water, the PPTA crystals change from modification I to II (Northolt and Ven Aartsen, 1973; Haraguchi et al., 1979a, b; Shen et al., 1983). The broadening of the equatorial reflection in the PPTA X-ray diffraction pattern is mainly caused by the lateral crystallite size distribution. The apparent crystallite size in various directions for Kevlar 49 is 6.4, 5.3, and > 10.9 nm measured from the (110), (200), and (006) reflections, respectively (Kumar, 1991). TEM imaging of the PPTA crystal lattice exhibits a very high degree of parallel alignment of the chains inside the crystallites (Dobb et al., 1975). The electron diffraction (ED) patterns demonstrate that within a domain size of about 0.5 jim the crystallites are almost perfectly oriented (Dobb et al., 1977 a, b). A distinctive structural feature of PPTA and some other aramid fibers is the pleated sheet structure, which was observed using the techniques of dark field TEM imaging from meridional reflections and polarized light microscopy. As modeled in Fig. 13-20 (Dobb et al., 1977a, b), the pleated sheets consist of parallel-oriented chains, which are hydrogen-bonded along the direction of the crystallographic 6-axis. They are usually oriented perpendicular to the surface of the fibers, i.e. in the circular crosssection the 6-axis is directed preferentially along the radius. The angle between adja-

Figure 13-20. Model of radial pleated-sheet structure of PPTA (Dobb et al., 1977).

cent planes of the pleat is about 170° and the distance between two pleats is about 150-250 nm, but both values may vary with fiber processing conditions. However, the pleat structure can be diminished by heat treatment under tension, especially with a super-heated stream (Fitzgerald and Irwin, 1991), and in the new, high modulus Kevlar product, Kevlar 149 (which has a different heat treatment process from the other Kevlar products), the pleated structure has apparently been straightened (Krause et al., 1989). Several different structural models have been suggested for the morphology of PPTA fibers (Fig. 13-21). They all share common structural features. (1) As mentioned above, the fiber is characterized by a pleated structure along the fiber axis and the pleated sheets are composed of fibrils. However, the highly oriented heat-treated fibers do not exhibit any lateral banding (Krause et al., 1989; Sawyer et al., 1993). (2) By means of TEM, field emission scanning electron microscopy (FESEM) and scanning tunneling microscopy (STM), it has been found that fibrils about 1 jim across consist of microfibrils about 10 nm

621

13.3 Fiber Morphological Structure

fill I

Core

a

]_J200-250nmf

\

I1

/

f \ \ \ \

111 11 \11 •

Chain with length 220nm Fibril axis

a

—| |—60nm Skin 0.1^m thick random chain end distribution

Shish-Kebab

70A

70A

"Lr

H600h nm

Longitudinal cleavage plane Zigzag fibril

Transverse cleavage plane (Spacing- 0.3/jm) \ Tangential

Epidermis Cylindrical lamella (Spacing 0.2-0.3yL/m) 10-15/im — .

Figure 13-21. Different models for PPTA morphological structure: (a) Northolt (Northolt and Ven Aartsen, 1977); (b) Morgan (Morgan et al., 1983); (c) Panar (Panar et al., 1983); (d) Manabe (Manabe, 1980); (e) Yabuki (Yabuki et al., 1975); (f) Horio (Horio, 1984) and (g) Li (Li et al., 1983).

622

13 High Performance Polymer Fibers

across. The smallest microfibrils are 3 to 30 nm wide and 2 to 5 nm thick. Both fibrils and microfibrils appear to be flat or tape-like, and readily form a layered structure (Sawyer et al., 1992, 1993). Most of the microfibrils are well aligned in the fibrils, but some are twisted and/or slightly tilted in relation to each other (Weng et al., 1986; Sawyer et al., 1992, 1993). It is considered that the radially oriented crystallites form the main, highly ordered fibrillar structure which constitutes the basic load bearing element of a fiber. (3) There is a skin-core differentiation in PPTA fibers. The chain ends in the skin are arranged essentially randomly relative to one another, but become progressively more clustered in the fiber interior (Northolt, 1981; Northolt and Sikkema, 1991). Elongated voids with their long axis approximately parallel to the fiber axis are sometimes formed in the core. In addition, kink bands observed on fiber surfaces are also seen in the bulk (Sawyer et al., 1992). PBZO and PBZT are inherently stiff molecules. Measurement of the orientation distribution indicates a highly uniform orientation of the molecules parallel to the fiber axis, but certain instances of local misorientation have been observed. Figure 13-22 shows 3-4° misorientation between adjacent crystallites within 20 nm in the high resolution electron microscopy (HREM) image of a PBZO fiber (Martin and Thomas, 1991a). SEM and TEM show that the ordered molecular chains comprise crystallites, which form fibrils/ microfibrils, and that the fibrils/microfibrils are predominantly lath-shaped (Allen et al., 1985). For example, PBZO consists of ribbon-like fibrils with widths from about 30 nm up to a few micrometers. The crystallites are about 5-20 nm wide and 5-10 nm long. Occasionally, crystallites of up to 30 nm long and 40 nm in lateral ex-

Figure 13-22. HREM image of PBZO showing local misorientation between adjacent crystallites (Martin and Thomas, 1991a).

tent were observed (Krause et al., 1988). Comparison of the crystallite structures for Kevlar, PBZO and PBZT shows that all the molecules, crystallites and microfibrils/fibrils are well aligned along the fiber-axis. The monoclinic crystal unit cell and lattice parameters of PBZO are similar to those of PBZT [Fig. 13-23 (Fratini et al., 1989)]. For PBZO, a = 1.120nm, b = 0.354 nm, c = 1.205 nm and y = 101.3°. The Z?-axis length is roughly the perpendicular distance between the faces of heterocyclic rings between chains, the a-axis length is roughly the distance between equivalent edges of heterocyclic rings adjacent in the chains, and the oaxis is the distance along the chain of a single repeat

13.3 Fiber Morphological Structure

unit; for PBZT, a = 12.2 nra, b = 3.65 nm, c = 12.5 nm and y = 106.2°. The difference in lattice parameters between PBZO and PBZT is that in PBZT the a-axis is slightly larger than in PBZO, possibly due to the non-planarity of PBZT molecules and the larger diameter of the sulfur atoms. The calculated crystal densities for both the fibers are 1.66 g/cm3, while the measured heat-treated fiber densities are 1.58 g/cm3 (Krause et al., 1988; Martin and Thomas, 1991 a). Crystallites of PBZO fibers are 3dimensionally ordered and exhibit limited long range order, as determined from WAXS and SAED patterns. PPTA fibers have greater three-dimensional (3D) order of their crystallites and also greater long range order. In PBZO, the planar molecules have improved chain registry, thus promoting 3D crystallinity. However, the PBZO crystalline structure is not as well ordered as that of PPTA. In WAXS and SAED measurements, only first order (hkl) reflections are clearly present in PBZO (as shown in Fig. 13-24), whereas many higher order (hkl) reflections are observed in PPTA. In PBZT, the non-planar molecules appear to have only 2D crystal order with less long range periodicity. Be-

(b)

Figure 13.23. C-axis view of the molecular packing of PBZT (a) and PBZO (b), the a-axis is horizontal (Fratini et al., 1989).

623

Figure 13-24. SAED patterns of PBZO (Adams et al., 1986).

cause of the axial translational disorder, no off-axis (hkl) reflections are observed in WAXS measurements of PBZT. Another feature of the crystallites is that through heat treatment, the crystallites in PBZO grow in the (hkO) directions by a factor of about 2 while in PPTA and PBZT, the crystallites grow in the (00/) directions by a factor of roughly 2. Therefore, the transverse dimensions of heat-treated PBZO crystallites are generally equal to or greater than those of PPTA and PBZT, while the axial dimensions are less than those of PPTA and PBZT by a factor of one half to one eighth (Krause et al., 1988). A significant skin-core difference is also found for PBZT and PBZO fibers (Krause et al., 1988; Young et al., 1992), which is similar to that for the PPTA fibers. For PBZO, this effect is on a scale of 10-100 nm (Krause et al., 1988). By measuring the intensity spread of (hkO) electron diffraction reflections, the skin material is found

624

13 High Performance Polymer Fibers

to have a better defined structure and a higher degree of molecular orientation than the core material (Young et al., 1990, 1992). The special features of Technora fiber are that its fibrils are composed of ordered crystallites and highly oriented amorphous regions, and that it exhibits a uniform, dense fiber morphology. However, because of the presence of amorphous regions and the absence of lateral crystalline order, Technora cannot attain as high a modulus as Kevlar (Yang, 1989). Gel spun HPOPFs also fit the "standard" morphological model for high performance fibers, where the oriented fibrils/ microfibrils contain a number of crystallites in series. In Spectra 1000, the fibrils are about a thousand angstroms long and 5-20 nm wide, as measured by synchrotron X-ray diffraction (Grubb and Prasad, 1992 b). By means of atomic force microscopy (AFM), Sheiko suggested that the elemental fibril unit in UHMW PE has a diameter of about 20-90 nm and named it a "nanofibril". The fibril structure is formed by the nanofibrils. The diameter of the fibrils observed by electron microscopy and X-ray diffraction is strongly affected by the draw ratio and processing conditions, but the diameter of nanofibrils remains fairly constant (Sheiko et al., 1992). One of the unique structural characteristics is that a shish kebab morphology can be obtained in UHMW PE fibers, except in the case of extremely high draw ratios (> 200). The shish kebab structure consists of core microfibrils with nearly extended chains and a number of chain folded platelets (lamellae) attached to the cores [Fig. 13-25 (Pennings, 1977)]. At draw ratios of greater than about 200, UHMW PE fibers consist mainly of smooth microfibrils with nearly extended molecules, while the amount of lamellar platelets is

Figure 13-25. A schematic representation of a shish kebab model showing the intimate connections of crystal platelets with the central core (Pennings, 1977).

negligible (Hofmann and Schulz, 1989). No long period or only a weak long period was detected by X-ray diffraction of the fully drawn fibers, which indicates that chain folding would be rare in fibrillar units (Van Hutten et al., 1985; Grubb and Prasad, 1992 a). The microfibrils of UHMW PE fibers have a finite length of about 1000-2000 nm, with an aspect ratio of (length/diameter, L/£>) 250-500 (Schaper et al., 1989). In addition, the presence of a weak intensity, two-spot S AXS pattern indicates a weak long period of about 150-200 nm (Grubb and Prasad, 1992 b). The structure of microfibrils is thus not uniform in density, but contains alternating regions of different density attributed to perfect crystals and domains containing a high concentration of crystal defects. A WAXS study of Spectra 1000 PE fibers revealed that the crystal unit cell is essentially the same as that reported originally (Bunn, 1939): orthorhombic with a = 0.741 nm, b = 0.495 nm and c = 0.255 nm [Fig. 13-26 (Busing, 1990)]. With increasing draw ratio, the crystallite length increases from 30 to 70 nm and the lateral size of the crystallites is 5.2-5.9 nm (Murthy, 1990; Grubb and Prasad, 1992 a). Measurements of Raman spectra and synchrotron WAXS (SWAXS) on single fibers show that Spectra 1000 is not perfectly

13.3 Fiber Morphological Structure

625

(a)

Figure 13-26. Crystal unit cell structure of Spectra 1000 PE fiber, (a) the view down the fiber axis, (b) the view perpendicular to the c-axis and parallel to the 6-axis (Busing, 1990). (b)

crystalline, but generally contains crystallites and "amorphous" regions. The crystallinity, as normally measured, is in the range of 85-95%. The "amorphous" (or disordered) region is not detectable by WAXS but by the Raman band at 1063 cm" 1 . The order within the "disordered" region is similar to that of a nematic liquid crystal. This partially ordered region is generally associated with tie

molecules, which connect the crystallites in and between the microfibrils (Prasad and Grubb, 1990). It has been reported that only a minor portion of the intercrystalline tie molecules are taut (Hofmann and Schulz, 1989). Electron microscopy also indicates that these disordered domains, whose longitudinal dimension appears to be 4 - 5 nm, are "amorphous" domains that may contain a substantial amount of

626

13 High Performance Polymer Fibers

chain ends. This "amorphous" domain is covalently bonded to the adjacent, nearly perfect, needle-like crystalline domains whose L/D is about 40 (Prevorsek et al., 1992). Both crystallites and microfibrils show very high orientation along the fiber axis direction. A recent WAXS study of a bundle of Spectra 1000 fibers has given a variation for the axial misorientation of only 0.69° (Busing, 1990), the corresponding Hermans-Stein orientation factor, defined a s / = {3-l}/2, is 0.9998, where a is the angle between the fiber axis and the meridian reflection (Hermans et al., 1946; Stein, 1958). However, the average misorientations derived from synchrotron radiation on a bundle of Spectra 1000 filaments, after correcting for the instrumental and sample-size effects, are 1.4° (the corresponding / = 0.9991) for the crystallites (WAXS) and 2.8° (the corresponding / = 0.9964) for the fibrils (SAXS); on a single filament, 0.9° is found for the crystallites (the corresponding/= 0.9996) (see Chap. 3 for more detailed information). The simplest explanation for the difference between the orientations of crystallites and fibrils is that individual interfaces are being detected by SAXS. If the fibril has a non-uniform cross-section, then the interface will be tilted further away from the fiber axis than the chain axis (Grubb and Prasad, 1992 a, b).

13.4 Fiber Properties All high performance organic polymer fibers display outstanding ultimate tensile strength and elastic modulus, two of the most elementary fiber properties. Most high performance fibers also exhibit excellent hydrolytic and chemical resistance. In addition, aromatic fibers show good ther-

mal stability and are suitable for use in the temperature range of 300 to 450 °C, depending upon the environment and exposure time. Most of the high performance organic polymer fibers have high radiation resistance, but they are not as resistant as carbon fibers. A prolonged exposure to light and ultra violet (UV) radiation is detrimental to the mechanical properties of aramid fibers such as Kevlar. Normally, the HPOPFs have good dielectric properties, but poor electrical conductivity. Some conjugated fibers also exhibit interesting non-linear optical behavior. One of the weak points for most of the high performance polymer fibers is their comparatively poor axial compressive strength. If this weakness could be overcome, it would be a breakthrough in their market share compared to their primary competitors, the carbon fibers. 13.4.1 Tensile Properties

Table 13-1 shows the tensile properties of various fibers. Tensile moduli of the high performance polymer fibers are at least four times higher than for polyester, at least nine times higher than for high tenacity nylon, and at least two times higher than for steel and glass fibers. Figure 13-2 is a plot of specific tensile strength versus specific tensile modulus of various fiber materials. It is interesting that all the high performance polymer fibers rank high, especially UHMW PE due to its low density and PBZO due to its superior tensile properties. Typical stress-strain curves of some fibers are given in Fig. 13-1. In general, their elongation at break is relatively low, about 1-4%. Fig. 13-4 shows the predicted theoretical chain moduli of some fibers (Donald and Windle, 1992). Recent advances in computational materials science allow the full quantum me-

627

13.4 Fiber Properties

chanical calculation of the intrinsic axial molecular modulus of polymer chains (Wierschke, 1989, 1992), as well as its strain and temperature dependence (Klunzinger et al., 1991 a, b). These calculations, while still at early stages of refinement and validation, serve to establish upper limits for the achievable tensile modulus. To date, few fibers in practice exceed 50 % of these theoretical values. The theoretical tensile strengths of Kevlar, PBZT, PBZO and UHMW PE extended-chain fibers are all more than 20 GPa (Ohta et al., 1988). If the polymer chains in the test fiber were as long as the gauge length and perfectly parallel to the fiber axis, the value of the measured stress at break would be the same as the theoretical one. This stress, which is solely determined by the covalent bonds and lateral packing density of the main chains, is estimated to be 10-15% of the theoretical chain modulus (Kelly and MacMillan, 1986). However, calculation of the tensile strength of real fibers is rather complicated due to many factors such as chain length distribution, the misorientation of the chain with respect to the fiber axis, different connections between atoms (covalent bonds in the main chain and secondary forces between chains), the non-uniformity and anisotropy of the fiber structure and morphology, as well as the presence of impurities and voids. Different ones or combinations of these are important in different situations. We shall present here some of the considerations and hypotheses that have been put forward. The Weibull model, which is based on the "weakest link theory", has been widely used for the statistical description of the fiber tensile strength (Weibull, 1951). This model assumes that fiber tensile strength is governed by pre-existing strength-limiting defects, which are randomly distributed

throughout the fiber. Thus the cumulative failure probability function, P, which represents the fraction of fibers that fail at or below a stress a is given by (13-4) where l0 is the test length of the fiber,
65""o

4-

_j

c

—

3-

CL

i

2-

^

1-

jy

y

oL0=25mm -13

3.5 4 4.5 Strength (GNrrf2)

5

Figure 13-27. A Weibull model fitting of the fracture stress distribution of a PPTA (Twaron) fiber with m = 10.6 (Northolt and Sikkema, 1991).

628

13 High Performance Polymer Fibers

strength and lifetime under constant stress of single PPTA fibers using Weibull statistics discloses that after short creep times, the fiber failure mechanism is transverse crack propagation, but after long creep times, it changes to splitting and fibrillation (Wu et al., 1988). It is suggested that because of the fibril morphology, if the shear forces at a discontinuity exceed the shear strength between the fibrils, the fiber tensile strength should be proportional to the fiber shear strength (Knoff, 1987). There is another argument that for the high performance organic polymer fibers, flaws might have a relatively small detrimental effect on the tensile strength (Donald and Windle, 1992). Due to their highly oriented fibril structure and comparatively weak lateral interchain bonding, the transverse cracks are either effectively blunted by intermolecular shear, or deflected to run along the fiber axis direction. The study of fracture morphology plays an important role in understanding the factors controlling the fiber strength. Disregarding the effect of stress concentration due to impurities, it may be expected that brittle fracture, characterized by a small area fracture surface oriented normal to the fiber axis, probably results from normal stress on the crystallite (directed along the c-axis), while a fibrillated, large area fracture surface is caused by shear stress (Northolt and Sikkema, 1991). All the aromatic polymer fibers produced by the dryjet wet-spinning technique display a fibrillated fracture surface. The possible reason is that due to the pre-existing impurities and inhomogeneities and a strong anisotropy in the axial and lateral moduli, a large stress concentration arises near a crack tip. The maximum shear stress is parallel to the direction of the largest modulus value (Kelly and MacMillan, 1986), which would ultimately cause a fibrillated

fracture surface. Thus, for highly oriented fibers containing impurities or inhomogeneities the tensile strength might not depend on the fiber modulus (Northolt and Sikkema, 1991). It has also been found that no apparent differences in the fracture morphology are seen in PPTA fibers broken under simple tensile, fatigue and creep conditions (Yang, 1989). The tensile properties, especially the tensile strength, of the high performance polymer fibers generally depend on their molecular weight. For UHMW PE, tensile strength and modulus increase approximately with the logarithm of the molecular weight. When the molecular weight reaches 3 x 106, the experimental tensile strength is about 14% of the theoretical value and the experimental modulus is 65% of the theoretical one (Ohta et al., 1988). Comparison of experimental data published by different authors shows that the highest modulus achievable for UHMW PE fibers appears to depend mainly on the molecular weight (Wang et al., 1990). As mentioned in Sect. 13.2.3.2, the tensile strength of Kevlar fibers increases with increasing intrinsic viscosity of the spinning dope (increasing molecular weight). Molecular weight also affects the mechanical properties of PBZO fibers (Ledbetter et al., 1989). A PBZO/PPA solution with an intrinsic viscosity of approximately 30 dl/g gave fibers with the highest tensile modulus and strength. For heattreated Vectran fibers, there is a correlation between the tensile strength, the molecular weight and the molecular weight distribution (Yoon, 1990). Based on the theoretical study of the influence of the molecular weight (M w ) on the tensile properties of polymer fibers, it has been suggested that for Kevlar and UHMW PE fibers, the molecular weight drastically affects the tensile strength, but only moder-

13.4 Fiber Properties

ately affects the Young's modulus (Termonia et al., 1985 a, Termonia and Smith, 1986, 1987 a, b; Smith and Termonia, 1989). In particular, for the UHMW PE fibers the modulus tends to be constant for M w > 1 0 4 (Termonia et al., 1985b). It is also predicted that the nature of the fiber failure process would gradually change from chain slippage to chain fracture with increasing molecular weight (Termonia et al., 1985b). Figure 13-28 shows the significant effect of orientation with respect to the direction of applied stress on the effective Young's modulus for a PBZT single crystal (Jiang, 1989). It is clear that for small angles of orientation, a small change in angle can cause a large change in modulus. Thus the orientation produced irreversibly by the processing of a highly oriented crystalline polymer can have a strong effect on the tensile modulus. The orientation produced by processing can also be increased reversibly by the application of tensile stress

2

4

6

8

10

12

14

16

18

or(Degree)

Figure 13-28. A schematic diagram of the effect of misorientation angle a on the effective Young's modulus for a PBZT single crystal (Jiang, 1989).

400

800

1200

629

1600

Stress (MPa) Figure 13-29. Sonic modulus at room temperature as a function of static tensile stress for PBZT fibers: as-spun, heat-treated at 525 and 650 °C. The tested fibers were, under static tension, subjected to pulses from a Q-switched Nd: YAG laser. The resultant ultrasonic wave speed was used to calculate the sonic modulus (Jiang et al., 1989).

at room temperature. There is a concomitant reversible increase in the tensile modulus of the fiber (nonlinear elasticity). The effects of processing conditions and tensile stress applied at room temperature on the modulus are shown in Fig. 13-29 (Jiang et al., 1989,1990). The corresponding changes in the orientation are shown as the halfwidth at one-half of the maximum intensity of the orientation distribution. The change of orientation with applied stress at room temperature accounts for most, but not all, of the observed change in the modulus (Jiang, 1989). A stiffening of the

630

13 High Performance Polymer Fibers

molecules themselves with increasing stress also plays a role (Klungzinger et al., 1992 a). Possibly there are some other factors involved which are not well understood yet (Jiang et al., 1989, 1990). As will be discussed later, temperature also affects the nonlinear elasticity. Nearly all of the HPOPFs show creep and stress relaxation phenomena, which are indicators of viscoelastic behavior. However, the rates of creep and stress relaxation for HPOPFs are extremely low compared with the conventional textile fibers like nylon and polyester. For PPTA, the observed creep strains amount to less than 20 % of the initial elastic strain after several years under stress (Walton and Majumdar, 1983). Presumably this low creep is caused by the combination of rigid chains and high crystallinity. One of the unique characteristics of Vectran is its lack of creep under static loads of up to 50% of the breaking strength (Beers and Ramirez, 1990) (also see Chapter 5 of this Volume). Figure 13-30 (Beers and Ramirez, 1990) shows the results of stress-relaxation tests for Vectran HS (high strength Vectran fibers), Kevlar 29 and Spectra 900 ropes. It is clear that no relaxation is oberved for the Vectran rope. By comparison, UHMW PE fiber has the highest relaxation. The fatigue behavior of high performance fibers has been a major concern in light of their poor compression behavior. In the fatigue test of cyclic bending over a freerunning spindle, it is found that Kevlar 29 and 49 fibers exhibit rapid strength loss in the first few cycles, but a more gradual decline afterwards. Kevlar 49 has a greater strength loss than Kevlar 29 (Dobb, 1985). During cyclic axial compression, after 100 cycles, the strength loss of Kevlar 49 was only about 10%, even though the compressive strain of 1.2 % exceeded the strain required for kinking (DeTeresa et al.,

Load, (lbs.) 7000

6000 -

5000 -

4000 -

3000 -

2000 10 100 Log time, (hours)

1000

Figure 13-30. A stress relaxation test for different fibers. The tested samples were tensioned to a fixed load and the tension was recorded periodically from load cells at each end of the samples. As creep occurred, the load decreased and the samples were subsequently retensioned back to the original load (Beers and Ramirez, 1990).

1984) (see next section for discussion of compressive kinking). The tensile properties of the high performance polymer fibers are temperature dependent. At 250 °C, heat-treated PBZT fibers retained 80 % of their room temperature tensile modulus and about 70% of their room temperature tensile strength (Allen et al., 1985, Allen and Farris, 1989; Im et al., 1989; Kumar, 1990a, 1991). However, the fiber which was heated to 300 °C for 65 hours in air and then tested at room temperature did not indicate any loss of tensile strength and modulus (compared to the fiber at room temperature) (Kumar, 1990). For some fibers, such as PBZT, the nonlinearity increases with increasing temperature. In the 300-400°C range, PBZT exhibits a mechanical relaxation that is associated with a structural

13.4 Fiber Properties

change in the same range (Feldman et al., 1987; Jiang et al., 1989, 1990; Klunzinger et al., 1992a; Macturk et al., 1993).

631

5-

ID

13.4.2 Compressive Properties

Compared with the superior tensile properties, the axial compressive strength of high performance polymer fibers is disappointingly low. The ratio of compressive strength to tensile strength is only about 10 to 20%. This is one of the obstacles preventing high performance polymer fibers being even more widely used in structural composites. While composites can fail in compression by a variety of modes, an important one results from the weak axial compressive strength of the fiber itself. Compressive strengths are listed in Table 13-1. It is clear that the compressive strength of highly oriented polymer fibers is low, while isotropic boron, alumina and silicon carbide fibers are among the highest compressive strength fibers available. PAN-based carbon fibers exhibit compressive strengths superior to pitchbased carbon fibers. The compressive behavior of fibers has been extensively studied in the last decade (DeTeresa et al., 1982, 1984, 1985a, 1988; Drzal, 1986; Allen, 1987; Kovar et al., 1989; Kumar et al., 1988; Kumar and Helminiak, 1989b; Kumar, 1989, 1990a, b, 1991; Kumar and Adams, 1990; Martin and Thomas, 1989, 1991a, b; Huh et al., 1990; Jiang et al., 1991 a, b). There is also nonlinearity in compressive behavior (Macturk et al., 1991; McGarry and Moalli, 1991 a, b; Klunzinger et al., 1991 b, 1992 b). Contrary to nonlinear behavior in tension, with increasing axial compressive stress the compressive modulus decreases, the so-called "modulus softening" phenomenon. Figure 13-31 shows a typical stress-strain curve for the high perfor-

32-

-2

-1

3

4 5 Strain (%)

Figure 13-31. Typical tensile (a-c) and compressive (d) stress-strain curves of PBZO fibers: (a) as-coagulated (wet), (b) as-spun (dried), (c) and (d) heattreated (Martin and Thomas, 1989; Kumar, 1990 a).

mance polymer fibers under tension and compression (Martin and Thomas, 1989; Kumar, 1990 a). The compressive modulus (Ec) may be less than the tensile modulus (Et), and some theoretical estimates predict Ec to be 70-90% of Et (McGarry and Moalli, 1991a). However, it was also reported that the compressive modulus appears approximately equal to the tensile modulus (Fawaz et al., 1992; Klunzinger et al., 1992 b). Under compression, the fiber birefringence decreases corresponding to a crystal disorientation away from the fiber axis. This disorientation would contribute to the decreasing modulus and could play a role in compressive failure (Klunzinger et al., 1992b). There is no simple correlation between the axial compressive behavior and tensile properties. For most of the high performance polymer fibers, an improvement in their tensile modulus does not change their compressive strength sig-

632

13 High Performance Polymer Fibers

nificantly (Van der Zwaag and Kampschoer, 1988; Van der Zwaag et al., 1989). On the other hand, the formation of kink bands during compressive deformation causes only a small loss in tensile strength (DeTeresa et al., 1984,1985, 1988; Van der Zwaag and Kampschoer, 1988; Van der Zwaag et al., 1989). These results imply that the compressive failure process depends upon different structural factors from those controlling tensile strength. Several HPOPFs, such as PPTA, PBZT and UHMW PE show a linear relation between compressive strength and shear modulus. However, the compressive strength of most high performance polymer fibers is in the range of about V* to % of their shear modulus (Kumar, 1992). Compressive failure of high performance polymer fibers is always associated with the formation of kink bands (shown in Fig. 13-32). Kink bands are nucleated in a localized region, normally initiated somewhere near the fiber surface, at a certain critical stress, and then propagate away from the initiating point and con-

tinue to grow in towards the center with the other side of the fiber under compression. For different fibers, the strain at which the first kink band occurs is different (Van der Zwaag and Kampschoer, 1988; Van der Zwaag et al., 1989) due to their variation in structural order. For PBZT fibers observed by optical microscopy, there are no kinks formed up to about 0.1 % compressive strain. When the strain increases above 0.1%, kink bands start to appear and the number of kink bands increases suddenly. Most new kink bands occur near the initial kink bands, showing some periodicity in the concentration of kink bands (Huh et al., 1990). For a certain range of compressive strain, the kink band density is proportional to the applied compressive strain, but at a larger compressive strain, the kink band density tends to become saturated (Takahashi et al., 1983; Huh et al., 1990). Upon the application of axial tensile strain after compressive kinking, kink bands have been reported to disappear, at least on the scale observed by LM, but this "reversibility"

Figure 13-32. SEM image of kink bands on Kevlar 49 fiber surface (a) as-received; (b) after 3 % axial compression, kink bands form on the fiber surface; (c) after > 3 % axial compression, more kink bands form. Compression due to nylon matrix shrinkage (DeTeresa, 1985b).

13.4 Fiber Properties

phenomenon has not been confined yet by higher resolution techniques (Huh et al., 1990). It was reported that the kink band angle related closely to the fiber structure and properties (Van der Zwaag and Kampschoer, 1988; Van der Zwaag et al., 1989). However, it is easy to obtain incorrect information by using optical microscopy to observe the angle of the kink bands, because of insufficient resolution for clearly distinguishing individual kinks (Martin and Thomas, 1991; Vezie et al., 1992). Furthermore, only in the correct position, namely, the in-plane position, would one obtain the correct angles: a, the angle of change in molecular orientation across the kink boundary, and /}, the kink boundary angle [shown in Fig. 13-33 (Vezie et al., 1992)]. There is clearly a correlation between the structure and the two angles. If p = a/2, the boundary is symmetric and the kink is isovolumetric. This is the case for compressive kinks in the crystallites of PBZO fibers (Martin and Thomas, 1991 a, b). If p > a/2, the boundary is nonsymmetric and the volume inside the kink increases (tension between the structural units). Most cases, such as kinks in the fibrils and microfibrils of PBZO fibers, belong to this category (Martin and Thomas, 1991b). If /?
633

a = change in molecular orientation across kink boundary P = kink boundary angle = angle between normal to kink boundary and molecular axis x0 = distance between chains microfibrils or width of whole fiber in unkinked region x1 = length of kink boundary W = perpendicular distance between chains or microfibrils within the kink, or the transverse width of the kink ff=

cos/? P ^ 90°, a - p # 90° (a - 90°, p = 0 => not physical) Figure 13-33. Measurement of kink band angles (Vezie et al., 1992).

(Martin and Thomas, 1989, 1991b). A possible mechanism for this reorientation has been investigated by computational modeling (Wierschke, 1989; Klunzinger and Eby, 1992). There are several different arguments about the mechanisms of the kink formation and the compressive failure. Argon pointed out that for composite materials, kink band formation occurs when a compressive stress (a) along the orientation direction causes locally misaligned elements to experience a relative shear stress which depends on the misorientation angle, # 0 . The compressive stress would be given by o =

(13-5) 0

634

13 High Performance Polymer Fibers

where TS is the plastic shear strength of the matrix (Argon, 1972). DeTeresa proposed that the kink formation relates to an elastic buckling instability in high performance polymer fibers and there is a correlation between the compressive strength and the longitudinal shear modulus of the fiber (DeTeresa et al., 1982, 1984, 1985, 1988). Huh found that the compressive strength of high performance polymer fibers is not limited by fiber buckling, but by fibril buckling according to the Euler buckling model (Huh et al., 1990). Van der Zwaag argued that the kink bands are formed before elastic instability occurs, so they may therefore be attributed to a plastic deformation process (Van der Zwaag and Kampschoer, 1988; Van der Zwaag et al., 1989). Northolt suggested that compressive deformation and the glass transition have common origins in chain flexibility and intermolecular interactions; hence an empirical relation holds between the compressive strength and (Tg)2 (Northolt, 1981). This result can also be derived quantitatively by assuming that the work for compressive yielding is proportional to the activation energy of the glass transition (Northolt and Sikkema, 1991). Attenburrow and Basset investigated the kinking of chain-extended PE and found that the extended chain deformation facilitates "fine" shear (Attenburrow and Bassett, 1979). Shigematsu proposed that the uniform oaxis shear and the intercrystalline slip mechanisms are both involved in forming kink bands (Shigematsu et al., 1985). It is also noted that kink formation is related to crystalline slip (Takahashi et al., 1983; Miwa et al., 1991a, b). For UHMW PE, (110X001 > and (100)<001> slip systems (Takahashi et al., 1983) are associated with kink formation; for Kevlar, the (200)<001> (Takahashi et al., 1991 a b); for PBZT the (010)<001> (Roche

et al., 1980) and for the aromatic polyester fiber, the (110)<001> and the (100)<001> (Takahashi et al., 1991a, b; Xiao and Takahashi, 1990). However, many unanswered questions still remain in this field. Up to now an outstanding problem in the study of compressive behavior of a single fiber is that there is no satisfactory technique to measure the axial compressive strength of a single fiber owing to its easy buckling. Several methods, most being indirect ones, have been developed, such as the bending beam (DeTeresa, 1982, 1984, 1985), elastica loop (Sinclair, 1950; Greenwood and Rose, 1974), tensile recoil (Allen, 1987), single fiber embedded composite (Drzal, 1986), fiber broken pieces in composite (Ohsawa et al., 1990; Miwa etal., 1991a, b) and the composite test. Direct methods are single fiber compression using a micro-tensile testing machine (MTM) (Fawaz et al., 1992) and "nanocompressometer" (Macturk et al., 1991). Little has been done to establish the correlation between the data obtained on the same fiber and these different tests. Therefore, data from different methods must be compared with caution. Significant research efforts have focused on attempting to enhance the compressive properties of the high performance fibers. Changing process conditions, morphological structures and modifying synthetic routes, increasing shear strength by bonding molecular chains more strongly together, coating the fibers as well as infusing monomer into the fibers then curing, are all ways to improve compressive behavior. In some cases, compressive strengths of 500-700 MPa were reported (Bhattacharya et al., 1989; Chuah et al., 1989), but most were in the range of 2 5 400 MPa (Dang et al., 1989; Kumar and Helminiak, 1989 a). Due to lack of reliable test techniques for the single fiber com-

635

13.4 Fiber Properties

pressive strength and the nonlinearity of the compressive behavior, the validity of these results may be questionable. It was also reported that the presence of glass within PBZT/sol-gel microcomposite films could increase the resistance of the film to compression (Kovar et al., 1989). Similar treatment of PBZT and PBZO fibers may be expected to produce PBZT/sol-gel and PBZO/sol-gel microcomposite fibers with improved compressive strength (Kumar, 1990 a). Unfortunately, no significant success has yet been reported. 13.4.3 Thermal Properties

All the aromatic high performance fibers have demonstrated great thermal stability because of their aromatic chemistry, rigid molecular chains and high crystallinity. Kevlar does not undergo decomposition until 480 °C in N 2 and about 380 °C in air (Yang, 1989). For PBZT and PBZO, the thermal behavior is even better. The thermogravimetric analysis (TGA) measurements show that the degradation temperature for PBZO in air is 620 °C. At 371 °C in air for 200 h, the weight retention in PBZO is reported to be 78% and in

PBZT, 70%. Kevlar, PBZT and PBZO fibers do not melt below the decomposition temperature and do not support combustion, but will char at high temperature. They generate only low quantities of smoke when burning (Goldfarb et al., 1989). HPOPFs as a group have negative and small axial coefficients of thermal expansion (CTE). For example, for Kevlar, ac = - 2 x 10" 6 /°C and for PBZT, - 1 to - 2 . 5 x l O " 6 / ° C (Yang, 1989). For the latter, the coefficient extrapolates to — 7 x 10 ~ 6 at perfect orientation (Lusignea, 1989). The negative values arise from transverse vibration of the molecules, as has been demonstrated by molecular dynamics simulations and molecular orbital calculations (Klunzinger et al., 1991a, 1992 a; Klunzinger and Eby, 1993; Macturk et al., 1993). Therefore, these fibers have very low shrinkage at high temperature and good thermal dimensional stability. Table 13-7 lists the CTE data for wet and dry PBZO and Kevlar 49 fibers (at two loads) for various heat cycles. The CTEs for a given tension were approximately the same for all heat cycles. The results from wet PBZO at low tension were also similar to those of their dry counterpart. For wet

Table 13-7. Axial CTEs (K" 1 x 10" 6) of Kevlar 49 and PBZO fibers from thermal mechanical analysis3 (Im et al., 1989). Fiber

PBZO PBZO PBZO Kevlar 49 Kevlar 49 Kevlar 49 a

Condition

Dry Dry Wetb Dry Dry Wetb

Applied load

Thermal cycles

(MPa)

1

2

3

4

1.24 62.1 1.24 0.62 31.0 0.62

-10.8 -7.6 -10.3 -11.0 -7.8 -1.5

-10.5 -8.2 -10.8 -11.8 -7.8 -9.0

-10.6 -7.6 -11.3 -11.4 -7.7 -9.6

-10.8 -7.4 -11.9 -11.3 -7.7 -10.3

Heating rate of 5 °C/min under a nitrogen environment; Wet fiber specimens were prepared by soaking them in water for 1 min, followed by conditioning at 100% relative humidity at room temperature for 24 hours. b

636

13 High Performance Polymer Fibers

Kevlar 49 the first scan was significantly different from the subsequent scans, from - 1 . 5 to 9.0 x 10~6/°C. This difference between wet PBZO and Kevlar 49 is due to the difference in their moisture retention characteristics. When exposed to a temperature range of 25-250°C for 24 hours, Vectran HS has a strength retention of about 85%, better than Kevlar 29. However, when treated at elevated temperature Vectran does not retain its strength as well as the solutionspun aramid fibers because of its thermoplastic characteristics (Beers and Ramirez, 1990). Figure 13-34 shows the stressstrain curves of UHMW PE fiber (Dyneema SK60) at different temperatures. With decreasing temperature (down to — 30 °C), its tensile strength and modulus increase. At 130°C for 3 hours, the retention of tensile strength and modulus is about 80% (Ohta et al., 1988). 13.4.4 Miscellaneous Properties High performance polymer fibers are hydrolytically stable and resistant to most

solvents. For evaluation of hydrolytic stability, PBZT fibers exposed at 120°C to a pH 5 buffer solution were found to be extremely stable compared to other reinforcing organic fibers. After the fibers were dried at 250-300°C in a vacuum, the moisture regain was < 1 wt.%. These results are to be expected as PBZT will not undergo hydrolysis. While most organic solvents have no effect on Kevlar fiber, it is attacked by strong acids or bases at high temperature and at high concentrations. Aromatic polyester fibers are less resistant to solvents than UHMW PE. Table 13-8 shows that after being immersed in various chemical solutions for a period of six months, Spectra fiber retained almost all of its original strength (Allied Signal Fiber Data Sheet, 1990). The study of radiation stability for a number of polymers disclosed that high performance polymer fibers are the most electron radiation-resistant (Kumar and Adams, 1990b). However, Kevlar is degraded by UV radiation. The affected surfaces are self-screening so that thick fabrics

Table 13-8. Strength retention (%) after immersion for six months (Allied Signal Fiber Data Sheet, 1990).

-30°C

Agent

25°C 80°C

Strain (%)

Figure 13-34. Effect of temperature on stress-strain behavior of UHMW PE fiber (Dyneema SK60) (After Science and Technology Center of Osaka, 1987).

Sea water 10% Detergent solution Hydraulic fluid Kerosene Gasoline Toluene Perchlorethylene Glacial acetic acid 1 M Hydrochloric acid 5 M Sodium hydroxide Ammonium hydroxide (29%) Hypophosphite solution (10%) Clorox bleach (5.25% Sodium hypochlorite)

Spectra 900

Aramid fiber

100 100 100 100 100 100 100 100 100 100 100 100 91

100 100 100 100 93 72 75 82 40 42 70 79 0

637

13.4 Fiber Properties

retain most of their original strength, although thin fabrics may lose 50% of their strength on exposure to Florida sunlight for 5 weeks (Fitzgerald and Irwin, 1991). PBZT fiber has good tensile strength retention after exposure to UV radiation. Both as-spun (AS) and heat-treated (HT) PBZT fibers exposed to UV radiation in the carbon-arc weather-o-meter (GM test) exhibited little or no strength loss over the course of the experiment. Figure 13-35 is a comparison of strength retention of different fibers under light radiation. UHMW PE fiber exhibits excellent light radiation stability (Allied Signal Data Sheet, 1990). For most high performance polymer fibers except UHMW PE, dimensional stability is excellent. For example, Kevlar shows negligible dimensional change in dry air at 160°C, in boiling water or at high relative humidity, even though moisture pick-up may be as high as 7.5% (Fitzgerald and Irwin, 1991). High performance polymer fibers also have very good dielectric properties. The high dielectric strength coupled with elevated temperature resistance and outstanding moisture resistance provide electrical efficiency in prevention of current leakage. This combination along with dimensional stability and low CTE make the high performance polymer fibers useful for specialized electronic uses. The dielectric constant and loss tangent of composite materials are generally considered to be a measure of radar transparency. The type of fiber selected would dominate the resulting composite's dielectric properties because composites usually have a fiber-to-resin ratio of 60:40. In practice, most transmissivity problems encountered in radome materials are caused by reflection of the electromagnetic energy at the interface of two materials having different dielectric constants. The smaller

HP-PE

100 300 500 700

1000

1500

Time (hr)

Figure 13-35. Comparison of light stability for different fibers (Ward, 1987).

Table 13-9. Dielectric constants of some composite materials (DuPont Product Literature, 1988; Anderson and Nguyen, 1992; Cordova and Collier, 1992; Harper et al., 1992).a Fiber/matrix

Spectra 1000/polyester Kevlar 49/polyester Quartz/polyester Glass/polyester Polyester resin only Spectra 1000/epoxy Kevlar 49/epoxy Quartz/epoxy Glass/epoxy Epoxy resin only a

Dielectric constant

Loss tangent

2.4 3.5 3.5 4.3 2.7-3.2 2.7 4.1 3.5 4.2 3.0-3.4

0.007 0.025 0.017 0.015 0.005 0.02 0.017 0.017 0.017 0.018 0.01-0.03

Data for 10 GHz, at 23 °C.

the difference, the smaller the reflection. Table 13-9 lists the dielectric constants of some composite materials (Anderson and Nguyen, 1992; Cordova and Collier, 1992; Harper et al., 1992). Apparently, the advantages of high performance organic polymer fibers used in radome construction lie in their low dielectric constant and low loss tangent. Compared to carbon fiber composites, composites with high performance polymer fibers are better suited for low radar observability. Another interesting feature is that PBZT and PBZO

638

13 High Performance Polymer Fibers

polymers are good candidates for third-order nonlinear optical materials because of their conjugated 7r-structure. They have a very high optical damage threshold, higher than 10 GW/cm2 with picosecond and shorter pulses (Rao et al., 1986; Prasad, 1989; Goldfarb and Medrano, 1989; Goldfarb et al., 1989; Lee et al., 1991). 13.4.5 End Uses of High Performance Polymer Fibers

After about 20 years of production, Kevlar plays a key role in the high performance fiber market. Many applications have been developed, which include composites, anti-ballistics, fabrics, tires, ropes, cables, and pulps. Almost all of them characteristically utilize the great strength, modulus, toughness, thermal stability, and light weight of Kevlar fibers. Kevlar composites can generally retain a large amount of tensile strength under static and cyclic conditions. They also provide significant advantages in composite density, a good balance of lamination properties and impact damage tolerance. Although their compressive strength is relatively poor, Kevlar 49/epoxy composites do not suffer catastrophic failure at high bending, as do graphite and glass composites. The environmental stability of Kevlar composites is generally good except in extremely hostile conditions (Yang, 1989; Fitzgerald and Irwin, 1991). In light of their tensile properties, thermal-oxidative stability, chemical and hydrolytic resistance, dielectric and nonlinear optical behavior, PBZT and PBZO fibers have great potential to be widely used for composite materials and a variety of special uses, such as space and marine applications, ballistic and fire protection, athletic equipment, cables and multilayer circuits and non-linear optical applica-

tions. It was reported that PBZT and PBZO composites compared favorably in tensile behavior to other composites reinforced with Kevlar 49, S-glass as well as aluminum (Yang, 1989). For lower temperature usage, UHMW PE fiber offers good impact, cutting and abrasion resistance. The impact energy of Spectra is of the order of 20 times that of glass, aramid and graphite. Therefore, it is recognized as the most ballistic-resistant fiber and has been widely used in shield materials, cut-resistant clothing and gloves. Since it has light weight, high tensile strength and modulus, low moisture regain and superior chemical resistance, it is suitable for strings, ropes, cables, cordage, sails and composite materials in marine and aerospace applications (Ladizesky and Ward, 1986; Ohta et al., 1988; Allied Signal Fiber Data Sheet, 1990). Vectran and Technora fibers can be used in an intermediate temperature range as composite-enhanced materials, ropes, cables, chemical resistant packing and sporting equipment, because of high strength with very low creep, excellent abrasion, chemical and moisture resistance, and good property retention over a broad range of temperatures (Yang, 1989; Beers and Ramirez, 1990). An interesting application area for rigidrod polymers is their use in "molecular composites". A molecular composite is the molecular-level analog of fiber-reinforced composites. The goal is to form a moderately homogenous mixture of a rigid-rod polymer and a flexible chain matrix polymer. The concept of forming molecular composites originated in the early 1970s with scientific literature appearing in the early 1980s (Takayanagi et al., 1980; Hwang et al., 1983). There are several approaches: (1) Takayanagi finely pulverized and mixed rigid crystalline polymer mi-

13.5 Other High Performance Fibers

crodomains (such as PPTA) and various flexible polymer matrices (such as nitrilebutadiene rubber, nylon 6 and nylon 66) (Takayanagi et al., 1980; Takayanagi, 1984); (2) using guidance from solution thermodynamics [theoretical (Flory, 1978) and experimental (Hwang et al., 1983)], Hwang, Helminiak and coworkers codissolved the rigid-rod (such as PBZT) and various flexible polymer matrices (such as aromatic heterocyclic semiflexible polymers or coil-like nylon polymers) in strong acids (such as methane sulfonic), and then quickly coagulated the isotropic solution to "quench" in the dispersed state (Helminiak etal., 1980; Kozakiewicz, 1986; Hwang etal., 1986); (3) Lenke and Wiff polymerized rigid-rod molecules (such as polyazomethine) in a matrix precursor (such as for nylon 6), then polymerized the matrix precursor to form an "in-situ" molecular composite (USP 5,068,291; Wiff and Lenke, 1992). Fibers and films made from molecular composites display outstanding mechanical and thermal properties, toughness, chemical and environment resistance. In addition, some of the detrimental interface problems with conventional fiber-reinforced composites can be avoided, such as mismatch of CTE between fibers and matrix (Wiff et al., 1988; Adams and Kumar, 1992). Unless care is taken, the rigid-rod molecules are not completely dispersed, but form small domains (Maclachlan et al., 1987). Many new molecular composite systems are under development and research activity is concentrated on improving processability (Kovar et al., 1989; National Materials Advisory Board, 1992; Wiff and Lenke, 1992; Adams and Kumar, 1992).

639

13.5 Other High Performance Fibers As fiber science and technology have progressed, the definition of high performance has been gradually extended. Some specialty polymer fibers which have unique functions to fulfill specific requirements are also considered to be high performance fibers. In addition to their use in construction materials, sports equipment, auto and airplane parts, polymer fibers have found wide use in biomedical, pharmaceutical, food and other physics and chemistry related areas in the last decades. More and more novel fiber products have been developed, and at the same time, more and more research efforts have been made to meet the extreme competition. Since these fields make use of so many different kinds of fiber materials and a great variety of new fiber products continue to emerge, we confine ourselves to only the relatively mature and actively studied areas to give a concise introduction. 13.5.1 Fibers for Medical Uses The biomedical area is one of the newest and most promising fields for further HPF applications. Fiber materials used in these areas can be divided into three main groups: (1) materials in surgery, such as suture threads, artificial skin and tendon, gauze for alloplasty of thoracic and abdominal walls and man-made blood vessels; (2) bio- and chemical membranes with the functions of separation, condensation and purification (for example, artificial kidney and lung); (3) carriers of bioreactors for immobilization of enzymes and microorganisms (see Volume 14).

640

13 High Performance Polymer Fibers

13.5.1.1 Surgical Fibers Two kinds of sutures are currently available, the assimilated (digestible) type and the non-assimilated. An ideal suture should be a monofilament with the necessary mechanical properties and a smooth surface which can pass through the skin easily. It should be biocompatible, nontoxic, and be absorbed in the living body within a predetermined time. Compared to natural sutures, polymer fibers possess greater mechanical strength, uniformity, and resistance to microorganisms. Synthetic polymer fibers also allow the control of physical and chemical properties through fiber composition and structure (Lyman, 1991; Chu, 1991) (see relevant chapters in Volume 14). For a long time, catgut was the only choice for an assimilated suture. The first successful polymer suture designed to meet the above requirements was polyglycolic acid (PGA), Dexon, developed by the American Cyanamid Co. in the 1960s-1970s. Its molecular structure is [-O-CH 2 -CO-] n , and it is synthesized through a polycondensation of hydroxyacetic acid (HO-CH 2 -COOH). PGA fibers are made by a melt-spinning process. They can have high molecular weight (M w : 10 000-100 000) and high crystallinity with a Tm of 224-226 °C and a density of 1.501.64 g/cm3. PGA is insoluble in common solvents. Laboratory and clinical studies have demonstrated that Dexon has compatibility, predictable absorption time, flexibility, ease of handling, knot security and high tensile strength. It is one of the strongest suture materials available [see Table 13-10 (Privalova and Zaikov, 1990)]. It normally takes about 60-90 days to be absorbed by the body, but in some cases, it was not fully absorbed by the body after one to two years (Privalova and Zaikov,

Table 13-10. Knot-pull strength of different sutures (with standard USP 3-0 gaugea) (Privalova and Zaikov, 1990). Suture material Silk Catgut Cotton Dacron (PET) Nylon PE Polypropylene (PP) Dexon (PGA) Stainless steel

Knot-pull tensile strength (kg) 1.54 1.73 1.23 1.63 1.54 1.41 1.50 2.09 2.36

a In the standard of United States Pharmacopoeia (USP), the unit of knot-pull strength is lb.

1990; Hongu and Philips, 1990). The applications of PGA have expanded considerably recently. A very interesting application is bio-active fibers produced by the processes of adding drugs and/or bio-additives into the spinning dope. The bio-active fibers include antimicrobial, anesthetic, anti-inflammatory, homeostatic, anticoagulant, radioactive, enzyme-containing and X-ray contrast fibers. Along with suture threads, PGA has also been used in woven gauze and felt-like sponges for packing the surface of a bleeding organ. There are some shortcomings in using PGA sutures. Since the fibers are a little too stiff to be used as a monofilament suture, they are commonly used as a bundle, which is more prone to infection. In spite of great experience acquired in the last decade, the chemical, physical and medicobiological behavior during treatment in the living body is not yet clearly understood. PGA suture is not recommended in cases where the tissues should be kept stretched. Much research has gone into improving PGA-based sutures, including Maxon, the product of Johnson and Johnson. Most of the efforts are concentrated

13.5 Other High Performance Fibers

on copolymerization with lactones, which have a lower Tm and less crystallinity than the homopolymer. These attributes favor a decrease in the absorption time, an increase in the flexibility, and a lower processing temperature. The latter favors making bio-active fibers because of the low decomposition temperature of most drugs and bio-active additives. Artificial skin is used mostly in the case of burns. Normally, there are three choices for the skin treatment: cultured self-skin from the patient; natural materials; and polymer materials. Cultured human skin is a good choice but since grafting of other people's skin has not been successful in the case of large area burns, artificial skin is necessary. Nonwoven polymer materials made from polyurethanes and Kevlar fibers are used as artificial skins. The former have advantages of compatibility, elasticity and gas-permeability; the latter have good mechanical properties (Takakuda, 1991). One promising material for artificial skins is the nonwoven fiber product made

641

from chitin and chitosan compounds, which belong to the polysaccharides. Chitosan is a linear (1-4) linked 2-amino-2-deoxy-/?-Z)-glucan and chitin is its 7V-acetyl product. As shown in Fig. 13-36, both structures are similar to cellulose. Chitin and chitosan are widely distributed in nature and can be made into fibers with solution-spinning techniques. It was reported that the fiber tensile strength was 1.22.6 g/d when dried, and decreased to 0.151.2 g/d under wet conditions. Its knot strength was 0.1-2 g/d and its breaking elongation, 3-35%. Both chitin and chitosan are almost non-toxic, biodegradable, very compatible to and able to be absorbed by the body. Laboratory safety and clinic tests show that they are suitable as medical materials (Hirano and Tokura, 1982; Hirano e t a l , 1986; Shimahara, 1990; Sakurai, 1990; Iizuka, 1990; Mochizuki and Yamashita, 1990). Up to now, they have been used as powders and tablets for the controlled release of drugs and nutrients, dialysis membranes, digestible sutures and artificial skins for wound healing and

OH (a) 0 —

NHCOCH, (b)

H H

NHCOCH3

0—

H CH2OH

(0

Figure 13-36. Chemical structure of (a) cellulose, (b) chitin, and (c) chitosan.

642

13 High Performance Polymer Fibers

dressing. However, the barrier which prohibits their wide use is the absence of suitable solvents to dissolve them, due to the strong resistance of their micelle structure to chemical agents. In light of their affinity to the growth of new cells during the healing of human surface skin and other excellent bio-behavior, they could become one of the main sources of artificial skin. Furthermore, if the processing and mechanical properties of their fibers were improved, they would be ideal materials for suture and artificial tendons. Artificial tendons require strong fibers with good dynamic mechanical properties. In addition to bio-compatibility, they should have good anti-fatigue and antifriction behavior, good interfaces with bones, excellent adaptability to the geometry of their position and the ability to assist the growth of human organisms on their surface, in vivo. Commonly used polymer fiber materials such as polypropylene (PP), polyethylene terephthalate) (PET), Kevlar, Teflon, PE, fiber-reinforced polyurethane and carbon fiber composites have been used as artificial tendons. The bio-absorbed polymer fibers are also used, such as the PGA and chitin families (Snyder, 1991).

out each year all over the world and the number of the patients using artificial kidneys is increasing (Hongu and Philips, 1990). Some people have survived more than 25 years depending on an artificial kidney. The artificial kidneys are not yet able to perform all the renal functions, but the removal of waste, toxic metabolic products and excessive water from the blood, to avoid organ toxicity, can be carried out. It is well known that hollow fibers are used as purification membranes because of their great specific surface areas and optimum packing density (Lloyd, 1985; Ikada, 1991). Generally, the diameter of the fibers is about 200 \\m. They are arranged parallel to each other or crossed at a small angle. Blood passes through the center of the hollow fibers while the dialysate solution flows around the fibers. By using the pressure difference, excessive water is filtered out and by means of the concentration differences, the small solutes in the blood and in the dialysate solution, such as electrolytes and urea proteins, can permeate

blood in

13.5.1.2 Biological and Chemical Membrane Fibers

The kidney is one of the chief excretory organs of the human body, along with the respiratory system and the skin. As kidney function decreases, various substances which are normally excreted by it will accumulate, thus leading to morbidity and disability. The loss in excretory capacity has to be matched by an artificial kidney in order to keep the overall mass balance in the body. More than 20 million artificial kidney procedures are successfully carried

electrolytes

water, urea etc.

t out

in

dialysate solution

Figure 13-37. Schematic presentation of the function of an artificial kidney.

13.5 Other High Performance Fibers

through the membrane (as shown in Fig. 13-37). Among the hollow fiber membranes, cellulose and its derivatives have about 80% of the market share (Ikada, 1991). Cellulose nitrate (Collodion) was the first membrane used in hemodialysis. Cuprophane hollow fibers, a regenerated cellulose made by the cuoxan process, are the most widely used membranes today. Cellulose acetate hollow fibers, which can be made by melt-spinning or solutionspinning techniques, are also used in great amounts in commercial products. It appears that even though a number of new synthetic polymer membranes have been developed over the last few years, Cuprophane is still considered the hollow fiber of choice for an artificial kidney, due to its good bio-behavior, permeability, mechanical properties and low cost. Research is now mainly focused on socalled selective permeability. The blood contains proteins of molecular weight between 10000 and 30000, which can cause kidney trouble, and also beneficial substances, such as albumin, of molecular weight around 70 000. Therefore a suitable membrane must have the ability to pass proteins of molecular weight around 20000, but to keep those around 70000. Modification of cellulose hollow fibers is one way to make high performance fibers with more and bigger open pores in order to raise the ultrafiltration rates for middlesized molecules. Coating or grafting functional materials on the surface of the cellulose can also improve its properties (Ikada, 1991; Corretge et al., 1988). As an alternative, new synthetic polymer hollow fibers made from polysulfone (PSF), polyacrylonitrile (PAN), polycarbonate and poly(methyl methacrylate) (PMMA) have also been commercialized to provide adequate clearance for molecules of low hydrophilicity. In particular, PSF has been

643

extensively studied for membrane application because of its good chemical resistivity (Pusch and Walsch, 1982). The lung is a form of gas exchanger and supplies oxygen to the blood and removes carbon dioxide. The artificial lung was first developed to take over the lung's functions during heart surgery, and it is now extensively used for this purpose. It is even used as a supplementary respiratory device over a longer term, to assist the breathing of patients (about 250000 U.S.A. patients per year) (Hongu and Philips, 1990). A successful artificial lung must provide efficient gas-blood contact in order to oxygenate up to five liters of blood per minute to nearly 100% saturation. Simultaneously, it should remove a proper amount of CO 2 without causing respiratory acidosis (CO2 retention) or respiratory alkalosis (CO2 depletion). In addition, it should eliminate any harmful side-effects. The breakthrough in this field was the development of heterogeneous hydrophobic membranes, such as microporous PP hollow fibers and Teflon membranes. The gases can freely pass through the hydrophobic pores in the membranes, but blood cannot because blood cells and most components of blood are hydrophilic. The PP microporous membranes can offer gas transfer rates for both O 2 and CO 2 of about four times those of Cellophane. However, it was found that the microporous membranes damage blood plasma components and suffer a progressive reduction in the gas transport, because the gas channels in the membrane flood with fluid during long term use. Most artificial lung membranes only last about a week, due to a drop in their ability to remove CO 2 . In order to overcome these drawbacks, several interesting approaches have been proposed, such as treating the fiber surface with bioactive agents, e.g. albumin.

644

13 High Performance Polymer Fibers

Hollow fibers or woven tubes made from PTFE and polyester are two wellknown artificial blood vessels. The former exhibits both good biocompatibility and anticoagulant activity, but cannot be used for small diameter vessels. The latter is biocompatible and has good mechanical properties, but poor anticoagulant activity. A very fine polyester blood vessel has been successfully developed recently (Hongu and Philips, 1990; Ikada, 1991). 13.5.1.3 Fiber Carriers for Biologically Active Agents

A relatively new and developing field in biotechnology is to utilize enzymes or micro-organisms which are capable of producing a great variety of antibiotics, vitamins, amino acids and hormones for different applications. Over 2000 enzymes have been found suitable for use in a bioreactor, but only a few are now commercialized. It has been realized that there are some drawbacks when the biologically active entities, such as enzymes, are employed in free form. They are not sufficiently stable, especially in extremes of both temperature and pH. They are watersoluble and hence difficult to separate from substrates and products, which prevents them from being used repeatedly. In clinical applications, the free form would usually cause immunological reactions, rapid removal and inactivation. Immobilization is a way to solve the above problems. Fibers, especially porous, non-circular cross-section and hollow fibers, have already been employed in bioreactors for bio technological applications. They are generally used in two immobilization techniques: the enzymes are covalently bound to the fiber carriers; or the enzymes are adsorbed ionically or physically to the fiber support materials. The polymer fibers

can be used, not only as a carrier, but also as a barrier to separate one bulk solution from another. Bioactive agents may be attached to the surface of the fibers or entrapped within the fiber network. The substrate molecules can freely permeate through the membrane and react with enzymes or micro-organisms. The product molecules are transported freely in opposite directions, but the membrane is substantially impermeable in either direction to the much higher Mw species. For example, penicillinacylase can be linked onto a reduced porosity PAN fiber to produce 6amino penicillin, which serves as a raw material for reactive penicillin. Several thousand tons are manufactured annually. In another commercial application acetobacter are adsorbed onto cotton-like PP fibers. The bioreactor is used in acetic acid production, and the carrier fibers can be regenerated by replacing the inactive enzyme. The fibers can also be made in the form of filter paper or non-woven fabrics and used as biosensors. 13.5.2 Polymer Fiber Precursors to Superconductors

Since the development of new oxides which exhibit superconductive behavior at temperatures above that of liquid nitrogen, a new horizon has been opened in materials physics. Making superconductive fibers is of great interest to meet the many potential new applications requirements (Goto, 1987a, b, 1988a, b, c). Research into making superconducting fibers is now world-wide. There are several spinning processes for making superconducting fibers, such as suspension-solution spinning, sol-gel spinning and melt spinning. Among them, the suspension-solution spinning methods are associated with polymer fiber technology, because of its

13.5 Other High Performance Fibers

maturity and the great advantages in process versatility. 13.5.2.1 Fibers from Mixtures of Superconducting Oxides and PVA

The primary process at present is a suspension-solution spinning method (Goto, 1987a, 1987b, 1988a, 1988b, 1988c). Powders of the oxides and dispersers are put into a PVA solution to make a spinning dope, which is extruded through spinnerets into a coagulation bath. The asspun fibers are heat-treated to produce superconducting fibers. The spinning dopes can be of two types, aqueous and nonaqueous solutions. For the former, the powders of Y 2 O 3 , BaCO 3 , and CuO are used as precursors to make superconducting Y 1 Ba 2 Cu 3 O Jc . Then the powders of the superconductor and the dispersers (nonionic type + anionic type) are placed in a water solution, and added to the PVA solution to form the spinning dope. The spinning solution is extruded into the coagulation bath (Na 2 SO 4 + NaOH) to produce fibers. After washing and drying, the asspun fibers are heated in O 2 atmosphere at 980 °C for five minutes, then gradually cooled to room temperature at 100°C/h. In this process, the PVA decomposes and the superconducting fibers are formed. For the latter type (non-aqueous solution), the solvent for PVA is dimethyl sulfoxide (DMSO), the coagulant is methanol and the superconductor is Ln 1 Ba 2 Cu 3 O :c (where Ln = Er, Ho, Dy). The degree of polymerization (DP) of PVA has a great effect on the fiber spinning and the fiber properties. If the DP < 2450 or > 16 000, spinning cannot be carried out. If the DP is kept in the range of 3300 to 12100, the superconducting properties are improved as the DP is increased. Unfortunately, the superconducting properties of these fibers have not yet met practical requirements.

645

13.5.2.2 Conducting Polymer Fibers

As early as the 1930s, theoretical studies suggested that it was possible to have conductive polymeric materials (Jones, 1937). In the last two decades, research on conducting polymers has become gradually more and more active. Now several polymers are of great interest, including polyacetylene, polythiophene, polypyrrole, and poly(/?-phenylene vinylene). The common characteristic of all these conducting polymers is conjugation along the backbone of the polymer chain, which appears to be a prerequisite for a conducting polymer. However, due to this high conjugation, most of the conducting polymers are very difficult to process. Several methods have been proposed for making conducting polymer fibers. The so-called "Durham" route is a three-step method (Feast and Winter, 1985). The first step is to use a soluble polymer precursor to produce fibers, then the second step is to convert the fiber into the conjugated polymer fiber, and the third step is to dope for conductivity. For example, 3,6-bis(trifluoromethyl) pentacyclo-[6.2.0.0]dec-9-ene was used as a monomer to synthesize an intermediate polymer that is relatively stable at room temperature and is conveniently processed. Then the intermediate polymer is converted to polyacetylene by heating it to 75 °C to eliminate hexafluoro-6>-xylene. Poly(2,5-thienylene vinylene) and poly(2,5-furylene vinylene) can also be prepared by this route (Jen, 1987; Yamada, 1987). The main problem with this method is that all the precursor routes lead to a polymer which cannot be reprocessed, and this technique is, consequently, of limited use. Sato used 3-alkyl thiophene which can be dissolved in common organic solvents (such as CHC13 and tetrahydrofuran,

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13 High Performance Polymer Fibers

THF) for polymerization. The polymer can be spun into fibers by means of the melt-spinning technique (Sato, 1987). Polyheterocyclics synthesized within the mesoscopic pores of a host membrane is another way of making conductive polymer fibers. The membrane contains linear and cylindrical pores and separates a solution of the monomer from a solution of a chemical oxidizing agent. The monomer and oxidizing agent diffuse toward each other through the pores to accomplish the synthesis. Polypyrrole and poly(3-methylthiophene) fibers were made according to this method (Cai and Martin, 1989). PBZT and PPTA have been spun from dopes containing metallophthalocyanine for making conducting fibers. The mechanical properties are adversely affected as the concentration of metallophthalocyanine increases, but process optimizations have not yet been carried out. Some ladder polymers are also possible candidates as conducting polymer fibers produced by solution spinning (Little, 1964; Blythe, 1979; Wynne et al., 1985; Polis et al., 1989). All these activities are pushing research in this field to higher levels although it appears now that there is still a long way to go to reach the target of conductive polymer fibers for practical applications. Conducting polymeric materials are very attractive high performance materials for science and technology development in the 21st century and we can reasonably predict breakthroughs in the next decade.

13.6 The Future This cursory glance at a few representative high performance polymer fibers is not intended to exhaustively survey this field, or even to delineate all the areas of current

interest in these materials. Instead a few areas of real or potential new levels of performance have been identified in order to interest the materials scientist in the vast scope of performance of polymers. A tenfold growth in consumption of high performance fibers is predicted by the year 2000, and the real potential for replacement of higher density materials has probably not yet been fully identified. Certainly the aerospace industry has driven the search for new, lighter, stronger, stiffer, durable, tough, processable, and even radar-invisible fibers to be used in reinforced composites for airplanes and spacecraft. Achievements such as the non-stop, round-the-world, unrefueled flight of the Voyager and massive research projects such as the "Orient Express", the Mach 15 National Aerospace Plane to travel from New York to Tokyo in two hours, are glowing testament to progress in composites. There remain tremendous scientific problems to be solved in the understanding of fibers. It is only by understanding the mechanisms of strength, stiffness, compressive failure, and transport in these high performance materials that we can design new molecules or new processes to overcome current limitations. The rewards for success in these endeavors should be considerable, both scientific and technological. Projections of world demand for advanced composites indicate that by the early decades of the 21st century industrial and other applications will have grown to 55% of the market share while the aircraft/aerospace market share will drop to 45 % (National Materials Advisory Board, 1992). Therefore, when the fiber quality meets the requirements, reducing manufacturing and processing costs and even making new low-cost high performance

13.7 References

polymer fibers are critical. It is forecast that if the cost of these fibers could be reduced to a few dollars per pound, the demand would be a factor of 10 higher (National Materials Advisory Board, 1992). In addition, the tailorability of polymers, with resultant enormous variations in properties make them very attractive for research programs aimed at specific achievement of property targets. The specialty polymers used in the high efficiency separation and conductivity fields are new materials in the expanding catalog of high performance materials. With the ongoing rapid progress in the science and technology of materials and processing, more new polymer fibers will emerge to advance the available technology of high performance fibers. "A civilization is both advanced and limited by the materials at its disposal" Sir George Padget Thompson.

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Northolt, M. G. (1981), Br. Polym. J. 13, 64. Northolt, M. G., Sikkema, D. J. (1991), in: Advances in Polymer Science 98. Berlin: Springer-Verlag, p. 115. Odell, J. A., Keller, A., Atkins, E. D. T, Nagy, M. J., Feijoo, J. L., Ungar, G. (1989), in: The Materials Science and Engineering of Rigid-Rod Polymers: Adams, W. W, Eby, R. K., McLemore, D. E. (Eds.). Pittsburgh, PA: MRS Symp. Proc. Vol. 134, p. 223. Ohsawa, T, Miwa, M., Kawade, M. (1990), J. Appl. Polym. Sci. 39, 1733. Ohta, T, Kunugi, T, Yobuki, K. (1988), High Tenacity and High Modulus Fibers: Society of Polymer Science, Japan (Ed.), Kubunshi Shinsozai One Point-9. Tokyo: Kyoritsu Shuppan Co., Ltd. Panar, M., Avakian, P., Blume, R. C , Gardner, K. H., Gierke, T. D., Yang, H. H. (1983), J. Polym. Sci., Polym. Phys. Ed. 21, 1955. Peguy, A., Manley, R. S. J. (1984), Polym. Commun. 25, 39. Penning, J. P., Dijkstra, D. I, Pennings, A. J. (1991), J. Mater. Sci. 26,4121. Pennings, A. J., Schouteten, C. J. H., Kiel, A. M. (1972), J. Polym. Sci. C38, 167. Pennings, A. J. (1976), Colloid Polym. Sci. 253, 452. Pennings, A. J. (1977), /. Polym. Sci., Polym. Symp. 59, 55. Polis, D. W, Dalton, L. R., Vachon, D. J. (1989), in: The Materials Science and Engineering of RigidRod Polymers: Adams, W. W, Eby, R. K., McLemore, D. E. (Eds.). Pittsburgh, PA: MRS Symp. Proc. Vol. 134, p. 679. Porter, R. S. (1992), private communication. Pottick, L. A., Farris, R. J., Thomas, E. L. (1985), Polym. Eng. Sci. 25, 284. Prasad, K., Grubb, D. T. (1990), /. Polym. Sci., Pt. B, Polym. Phys. 28, 2199. Prasad, P. N. (1989), in: The Materials Science and Engineering of Rigid-Rod Polymers: Adams, W. W, Eby, R. K., McLemore, D. E. (Eds.). Pittsburgh, PA: MRS Symp. Proc. Vol. 134, p. 635. Prevorsek, D. C , Chin, H. B., Kwon, Y D., Kavesh, S. (1992), Polym. Prepr. 33, 368. Privalova, L. G., Zaikov, G. E. (1990), Polym.-Plast. Technol. Eng. 29(5&6), 455. Pusch, W, Walsch, A. (1982), Angew. Chem. Int. Ed. Engl. 231, 660. Rakas, M. A., Farris, R. J. (1988), Polym. Mater. Sci. Eng. 58, 1024. Rakas, M. A., Farris, R. J. (1989), in: The Materials Science and Engineering of Rigid-Rod Polymers: Adams, W W, Eby, R. K., McLemore, D. E. (Eds.). Pittsburgh, PA: MRS Symp. Proc. Vol. 134, p. 277. Rao, D. N., Swiatkiewicz, I, Chopra, P., Ghoshal, S. K., Prasad, P. N. (1986), Appl. Phys. Lett. 48(18), 1187. Reisch, M. S. (1987), C & E News 65(5), p. 9.

13.7 References

Roche, R. I , Takahashi, T., Thomas, E. L. (1980), in: ACS Symp. Ser. 141, 303. Sakurada, I. (1985), Polyvinyl Alcohol Fibers. New York: Marcel Dekker. Sakurai, K. (1990), Sen-i Gakkaishi 46(12), P-553 (in Japanese). Sato, M. (1987), Synth. Met. 18, 229. Sawyer, L. C , Jaffe, M. (1992), Mater. Res. Soc. Symp. Proc. 255, 75. Sawyer, L. C , Chen, R. T., Jamieson, M. G., Musselman, I. H., Russell, P. E. (1992), /. Mater. Sci. Lett. 11, 69. Sawyer, L. C , Chen, R. T., Jamieson, M. G., Musselman, I. H., Russell, P. E. (1993), /. Mat. Sci. 28, 225. Schaefgen, J. R., Foldi, V. S., Logillo, F. M., Good, V. H., Gulrich, L. W, Killian, F. L. (1976), Polym. Prepr. (Am. Chem. Soc, Div. Polym. Chem.) 17, 69. Schaper, A. (1989), Phys. Stat. Sol. (a)116, 179. Science and Technology Center of Osaka (1987), in: Report on High Strength Polymers', p. 108 (in Japanese). Sheiko, S. S., Magonov, S. N., Moller, M. (1992), Polym. Prepr. (Am. Chem. Soc, Div. Polym. Chem.) 33, 788. Shen, D. Y, Molis, S. E., Hsu, S. L. (1983), Polym. Eng. Sci. 23(10), 543. Shibayama, M., Kurokawa, H., Nomura, S., Roy, S., Stein, R. S., Wu, W.-L. (1990), Macromolecules 23(5), 1438. Shigematsu, K., Imada, K., Takayanagi, M. (1985), /. Polym. Sci., Polym. Phys. Ed. 13, 73. Shimada, K., Mera, H., Sasaki, N., Aoki, A. (1982), U.S.P. 4355151. Shimahara, K. (1990), Sen-i Gakkaishi 46(12), P-547 (in Japanese). Shimamura, K., Hashimoto, A. (1991), Sen-i Gakkaishi 47, P-325 (in Japanese). Sinclair, D. (1950), /. Appl. Phys. 21, 380. Smith, P., Lemstra, P. J. (1980 a), Colloid Polym. Sci. 7, 258. Smith, P., Lemstra, P. J. (1980b), J. Mater. Sci. 15, 505. Smith, P., Termonia, Y (1989), Polym. Commun. 30, 66. Southern, J. H., Porter, R. S. (1970), J. Macromol. Sci., Phys. B4, 541. Staudinger, H. (1932), in: Die Hochmolekularen Organischen Verbindungen: Berlin: Springer, p. 111. Stein, R. S. (1958), J. Polym. Sci. 31, 327. Snyder, R. W. (1991), in: High-Tech Fibrous Materials, ACS Symp. Ser. 457: Vigo, T. L., Turbak, A. F. (Eds.). Washington, D.C.: ACS, p. 124. Takahashi, T, Miwa, M., Sukurai, S. (1983), J. Appl. Polym. Sci. 28, Takahashi, T, Xiao, C. R, Sakurai, K. (1991 a), Sen-i Gakkaishi 47, P-397. Takahashi, T, Suzuki, K., Aoki, T, Sakurai, K. (1991b), J. Macromol. Sci., Phys. B30(l&2), 101.

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Takakuda, K. (1991), Sen-i Gakkaishi 47(3), P-142 (in Japanese). Takayanagi, M., Ogato, T, Morikawa, Kai, T (1980), J. Macromol. Sci., Phys. B17, 591. Takayanagi, M. (1984), in: Interrelations between Processing Structures and Properties of Polymeric Materials: Seferis, J. C , Theocavis, P. S. (Eds.). Amsterdam: Elsevier. Tashiro, K., Nakata, Y, Li, T, Kobayashi, M., Chatani, Y, Tadokora, H. (1987), Sen-i Gakkaishi 43, P-627 (in Japanese). Tashiro, K., Nakata, Y, Li, T, Kobayashi, M., Chatani, Y, Tadokora, H. (1988), Sen-i Gakkaishi 44, P-7 (in Japanese). Termonia, Y, Meakin, P., Smith, P. (1985a), Macromolecules 18, 2246. Termonia, Y, Meakin, P., Smith, P. (1985b), Polym. Prepr. (Am. Chem. Soc, Div. Polym. Chem) 26 (2), 158. Termonia, Y, Smith, P. (1986), Polymer 27, 1845. Termonia, Y, Smith, P. (1987a), Macromolecules 20, 835. Termonia, Y, Smith, P. (1987b), Polymer Commun. 28, 60. Van der Zwaag, S., Kampschoer, G. (1988), in: Integration of Fundamental Polymer Science and Technology, 2nd Ed. London: Elsevier Chap. 2, p. 545. Van der Zwaag, S. (1989), J. Test. Eval. 17, 292. Van der Zwaag, S., Picken, S. J., Van Sluijs, C. P. (1989), in: Integration of Fundamental Polymer Science and Technology, 2nd Ed. Lemstra, P. J., Kleintjes, L. A. (Eds.). London: Elsevier Appl. Sci., Vol. 3, p. 199. Van Hutten, P. R, Koning, C. E., Pennings, A. J. (1985), /. Mater. Sci. 20, 1556. Vezie, D., Speck, J. S., Thomas, E. L. (1992), APS Symp. Bull. 37(1), 516. Walton, P. L., Majumdar, A. J. (1983), /. Mater. Sci. 18, 2939. Wang, L.-H., Porter, R. S., Kanamoto, T. (1990), Polym. Commun. 31, 457. Ward, I. M. (1985 a), in: Key Polymers: Properties and Performance, Advances in Polymer Science. New York: Wiley-Interscience, p. 1. Ward, I. M. (1985b), Adv. in Polym. Sci. 70, 1. Ward, I. M. (1987), Development in Oriented Polymers - 2. Amsterdam: Elsevier, p. 39. Watt, W, Perov, B. V. (1985), Strong Fibers, Handbook of Composites Series: 1. Amsterdam: Elsevier Science Publishers. Weibull, W. (1951), /. Appl. Mech., 18, 293. Weng, T, Hiltner, A., Baer, E. (1986), /. Mater. Sci. 21, 744. Wierschke, S. G. (1989), in: The Materials Science and Engineering of Rigid-Rod Polymers: Adams, W. W, Eby, R. K., McLemore, D. E. (Eds.). Pittsburgh, PA: MRS Symp. Proc. Vol. 134, p. 313. Wierschke, S. G., Shoemaker, J. R., Haaland, P. D., Pachter, R., Adams, W. W. (1992), Polymer 33(16), 3357.

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Wiff, D. R., Helminiak, T. E., Hwang, W.-F. (1988), in: High Modulus Polymers: Zachariddes, A. E., Porter, R. S. (Eds.). New York: Marcel Dekker, Chap. 8, p. 225. Wiff, D. R., Lenke, G. M. (1992), in: 37th Int. SAMPE Symp., p. 991. Wolfe, I (1985 a), U.S.P. 4533692. Wolfe, J. (1985b), U.S.P. 4533693. Wolfe, J. (1985c), U.S.P. 4533724. Wolfe, J. (1988), in: Encyclopedia of Polymer Science and Engineering: Kroschwitz, J. I. (Ed.), //: 11. New York: Wiley, p. 601. Wolfe, J. F. (1989), in: The Materials Science and Engineering of Rigid-Rod Polymers: Adams, W W, Eby, R. K., McLemore, D. E. (Eds.). Pittsburgh, PA: MRS Symp. Proc. Vol. 134, p. 83. Won Choe, E. W, Kim, S. N. (1981), Macromolecules 14, 920 Wong, C. P., Ohnuma, H., Berry, G. C. (1978), /. Polym. Set 65, 173. Wu, H. E, Phoenix, S. L., Schwartz, P. (1988), /. Mater. Sci. 23, 1851. Wynne, K. I, Zachariades, A. E., Inabe, T., Marks, T. J. (1985), Polym. Commun. 26, 162. Xiao, C. E, Takahashi, T. (1990), Sen-i Gakkaishi 46, T-49. Yabuki, K., Ito, H., Oda, T. (1975), Sen-i Gakkaishi 31, T-524. Yabuki, K., Ito, H., Oda, T. (1976), Sen-i Gakkaishi 32, T-55. Yamada, S. (1987), /. Chem. Soc, Chem. Commun. 309, 1448. Yamada, S. (1991), Sen-i Gakkaishi 47, P-24 (in Japanese). Yang, H. H. (1982), U.S.P. 4340559. Yang, H. H. (1989), Aromatic High-strength Fibers. New York: Wiley. Yoon, H. N. (1990), Colloid Polym. Sci. 268, 230. Young, R. X, Day, R. X, Zakihhani, M. (1990), J. Mater. Sci. 25, 127.

Young, R. X, Lu, D., Day, R. X, Knoff, W. E, Davis, H. A. (1992), /. Mater. Sci. 27, 5431. Zachariades, A. E., Porter, R. S. (1983), The Strength and Stiffness of Polymers. New York: Marcel Dekker. Zhang, R., Mattice, W L. (1992), Macromolecules 25, 4937. European Patents: 83108258; 84114872.

General Reading Adams, W W, Eby, R. K., McLemore, D. E. (Eds.) (1989), The Materials Science and Engineering of Rigid-rod Polymers. MRS Symp. Proc, 134. Baer, E., Moet, A. (1991), High Performance Polymers. New York: Hanser (distributed by Oxford University Press). Donald, A. M., Windle, A. H. (1992), Liquid Crystalline Polymers. Cambridge: University Press. Hongu, T, Philips, G. (1990), New Fibers. New York: Ellis Horwood. Kelly, A., Macmillan, N. H. (1986), Strong Solids. Oxford: Oxford Science Press. Northolt, M. G., Sikkema, D. X (1991), in: Advances in Polymer Science 98. Berlin: Springer-Verlag. Ohta, T., Kunugi, T, Yobuki, K. (Eds.) (1988), High Tenacity and High Modulus Fibers. Society of Polymer Sciences, Japan, Tokyo: Kyoritsu Shuppan Co. Ltd. Vigo, T. L., Turbak, A. F. (Eds.) (1991), High-Tech Fibrous Materials. Washington, DC: ACS Symp. Series 457. Watt, W, Perov, B. V. (Eds.) (1985), Strong Fibers. Amsterdam: Elsevier Science Publishers. Yang, H. H. (1989), Aromatic High-Strength Fibers. New York: Wiley.

14 Polymer Surfaces and Interfaces with Other Materials Matthew Tirrell and Edward E. Parsonage Department of Chemical Engineering and Materials Science, University of Minnesota, Minneapolis, MN, U.S.A.

List of Symbols and Abbreviations 14.1 Introduction and General Considerations 14.1.1 Motivation 14.1.2 Scope and Aims 14.2 Computer Simulation 14.2.1 Monte Carlo and Molecular-Dynamics Results 14.2.2 Implications and Interpretations 14.3 Surface Tension of Polymeric Materials 14.3.1 Introduction 14.3.2 Experimental Techniques for Measuring Surface Tension or Surface Energy 14.3.2.1 Polymer-Liquid Surface Tension 14.3.2.2 Polymer-Solid Surface Energy 14.3.3 Experimental Results for the Surface Tension of Polymeric Materials 143 A Theoretical Aspects of Polymer Surface Tension 14.4 Techniques to Examine Polymer Surface Properties 14.4.1 General Considerations 14.4.2 Imaging Methods 14.4.3 Reflection Methods 14.4.4 Spectroscopic Methods 14.4.5 Ion-Beam Methods 14.5 Materials Science Issues Concerning Polymer Surfaces 14.5.1 Overview 14.5.2 Surface Composition 14.5.3 Surface Ordering 14.5.4 Thin Films on Solids: Wetting, Spreading, Interactions and Dynamics 14.6 Conclusions and Remarks on Future Directions 14.7 References

Materials Science and Technology Copyright © WILEY-VCH Verlag GmbH & Co KGaA. Allrightsreserved.

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14 Polymer Surfaces and Interfaces with Other Materials

List of Symbols and Abbreviations a

K b D E F AF M AF C T / k K L M

Mn n "i

N P P q R,R\,R2

s s SB

Sc T Tg T* U Vs

z Y 7 7C A £ bb 0 K X /Hi

radius of contact area (between two solids) area (surface region) site size (in a lattice) thickness of surface region internal energy external force, applied load tensile force to separate two solids Flory-Huggins free energy of mixing surface free energy per segment intensity Boltzmann constant bulk elastic modulus period repeat-unit molecular weight number-average molecular weight Macleod exponent number of moles of /th component number of segments in a polymer chain length, or degree of polymerization pressure "parachor" wave vector radii of elastic spheres (JKR theory) radius of gyration entropy surface area segmental-order parameter chain-order parameter temperature glass-transition temperature reduced temperature energy volume (surface region) number of chain segments surface tension (excess free energy per unit area) solid-surface energy in vacuum critical surface tension phase difference of components of an electric field bead-bead (or segment-segment) interaction parameter contact angle, incident angle, or take-off angle (in XPS) compressibility wavelength, or mean free path (in XPS) chemical potential of /th component

List of Symbols and Abbreviations

655

E, ;re p a 0 0e 0 X W

correlation length spreading pressure density, or density of chain ends segment diameter average monomer density surface-excess composition volume fraction of segments in a lattice Flory-Huggins parameter used in tan*F, where this term represents the amplitude ratio of components of an electric field

AFM B-PE d-PS DSIMS ELLI ESCA FRES HREELS IETS IR-ATR IR-GIR JKR L-PE NR NRA PCP PDMS PEO PEP-PEE PET PIB PMDA-ODA PMIM PMMA PnBMA PP PPO PS PS-PMMA PYAC PVME PVP RBS rms

atomic-force microscopy branched poly(ethylene) deuterated poly(styrene) dynamic secondary-ion mass spectroscopy (spectrometry) ellipsometry electron spectroscopy for chemical analysis forward-recoil spectroscopy (spectrometry) high-resolution electron-energy-loss spectroscopy (spectrometry) inelastic electron-tunneling spectroscopy (spectrometry) infrared-attenuated total-reflection spectroscopy (spectrometry) infrared grazing-incidence reflection spectroscopy (spectrometry) Johnson-Kendall-Roberts (theory) linear poly(ethylene) neutron reflectometry (reflectivity) nuclear-reaction analysis poly(chloroprene) poly(dimethylsiloxane) poly(ethylene oxide) poly(ethylene propylene)-poly(ethyl ethylene) block copolymer poly(ethylene terephthalate) poly(isobutylene) poly(pyromellitic dianhydride oxydianiline) phase-measurement interference microscopy poly(methyl methacrylate) poly(n-butyl methacrylate) poly(propylene) poly(propylene oxide) poly(styrene) poly(styrene)-poly(methyl methacrylate) diblock copolymer poly(vinyl acetate) poly(vinyl methyl ether) poly(vinyl pyridine) Rutherford backscattering root mean square

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SCMF SEM SIMS SSIMS STM TEM VASE XPS XR

14 Polymer Surfaces and Interfaces with Other Materials

self-consistent mean field scanning electron microscopy secondary-ion mass spectroscopy (spectrometry) static secondary-ion mass spectroscopy (spectrometry) scanning tunneling microscopy transmission electron microscopy variable-angle spectroscopic ellipsometry X-ray photoemission spectroscopy (spectrometry) X-ray reflectometry (reflectivity)

14.1 Introduction and General Considerations

14.1 Introduction and General Considerations 14.1.1 Motivation Surfaces are the regions through which materials connect and interact with their surroundings. Transmission of stress, adhesion, friction, abrasion, permeability to gases and liquids, compatibility with biological or harsh, corrosive environments are all properties of polymeric materials which are dominated by the properties of the material within tens of nanometers of the surface. This statement is not unique to polymeric materials. However, surfaces of polymeric materials have several special features, relative to surfaces of other materials, lending particular scientific interest to their study. Large molecular size is the sine qua non of polymers, which at surfaces means that an individual molecule at the surface may also extend relatively deeply into the bulk of the material, thus effectively enlarging the zone of material affected by the surface. Molecular connectivity of macromolecules effectively links the surfaces to the interiors of polymer materials. Surface regions of polymers are usually less sharply delineated than those in other materials. The gradients in structure resulting from the influences of the surface are typically less abrupt in polymers and consequently have a larger characteristic dimension. Furthermore, the three-dimensional coil configurations of macromolecules must arrange themselves in special ways to accommodate the spatial restriction incurred by the surface. This point must be comprehended in order to understand and effectively manipulate the surface properties of polymers to the fullest extent. Polymer materials are usually partially or completely amorphous solids, or viscous or

657

rubbery liquids, so that familiar ideas in ordered solids, of surface crystal structure, reconstruction or defects seldom apply in a simple way to polymers. Near surfaces, polymeric materials are sometimes found to be more ordered than in the bulk (Factor et al., 1991; Menelle et al., 1992), in contrast with other kinds of materials, owing to the influence of the surface constraints on the packing of macromolecules. At the same time, macromolecules are frequently multicomponent molecules, either by design or by accident. This means that an individual molecule may contain more than one (possibly several) distinct chemical functionalities. Polymer molecules that are amphiphilic (meaning multiple chemical affinities within the same molecule) in character, such as block copolymers, can be synthesized (see Chap. 1 of this Volume). Oxidation or other environmental exposure (or intentional chemical modification) can modify portions of individual macromolecules, thus chemically also imparting different surface affinity to those segments. In either case, one finds that certain molecules, or certain portions of molecules, segregate to the surfaces of the material and thereby can assume a dominant role over the average bulk composition in determining how the material interacts with its surroundings. This behavior could be thought of as the counterpart to surface reconstruction of inorganic crystalline materials (Allen and Gobeli, 1962). In technological practice, the tendency toward surface segregation in multicomponent polymer systems provides many real and potential advantageous opportunities for interfacial engineering of polymers, since properly designed macromolecules will self-assemble at the surface where they are desired. Processing can also be used to accomplish what pure thermodynamics only does reluctantly or sluggishly (for example, quenching by cooling or solvent remov-

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14 Polymer Surfaces and Interfaces with Other Materials

al to trap useful non-equilibrium structures, shearing to induce surface ordering, or surface chemical reactions to tailor interfaces (Bergbreiter et al., 1984; Ward and McCarthy, 1987)). Surfaces are the regions through which polymers make contact with other materials. The large size of polymer molecules, and the related long-range surface gradients, imply that surface polymer molecules often intermingle or entangle substantially with adjacent material in their surroundings. This characteristic is important inter alia in adhesion, in the surfaces of polymer materials exposed to solvents that swell or partially dissolve them, and in the interdiffusion or welding of polymer materials placed in contact with one another.

14.1.2 Scope and Aims The broad area of polymer surfaces and interfaces is too large to cover effectively in a single chapter, so this chapter is intentionally focused, and some arbitrary decisions have been made about the scope of the coverage. Bulk polymer materials are dealt with exclusively, leaving out many interesting aspects arising from interactions with solvent or in solvent-swollen, adsorbed layers of polymers on other solids. Two comprehensive reviews on polymer adsorption have recently appeared (Ploehn and Russel, 1990; Kawaguchi and Takahashi, 1992); surface modification due to grafting chains on to solids has been covered in another review (Halperin et al., 1992). Surface chemistry of polymer materials is discussed in a cursory fashion here; an excellent review is available elsewhere (Ward and McCarthy, 1987). Techniques of experimental characterization and analysis of polymer surfaces are surveyed mainly by describing the kind of information they provide on

polymer surface properties. Thorough and informative reviews of polymer surface analysis methods have recently been published (Russell, 1990,1991;Gardella, 1989; Gardella and Pireaux, 1990; Stamm, 1992). Polymer surfaces and interfaces are frequently discussed in tandem. In the parlance we adopt for this article, "surface" means an external interface of a piece of polymer material, most often in contact with air or another impenetrable solid, whereas "interface" is used to mean internal or buried interfaces which can result from microphase separation in multicomponent polymers, or from the contacting of two different polymers. This chapter deals more with external surfaces than with internal interfaces. Thin films of polymers, where the entire sample experiences restriction in the dimensionality of space, merit special, concentrated attention which they will not fully receive here; however, some additional aspects of thin films, which have two external surfaces, will be brought out in the general discussion of surfaces. The aim of this chapter is to bring out clearly the aspects of polymer surfaces relating most directly to their physical structure and properties. All of the available information on the configurations and surface arrangements of macromolecules in polymeric materials is discussed. Some of the most informative work, albeit on very idealized systems, has been done by computer-simulation experiments, so that this is the point of departure for the chapter. More readily, and therefore more extensively, characterized by laboratory experiments than surface configurations are surface energies and tensions so this is the next topic presented, along with the theoretical approaches that have been used in this area. This is followed by a summary of the arsenal of experimental techniques that are currently being used to study polymer surfaces.

14.2 Computer Simulation

The final section presents information on a range of materials-science issues pertaining to polymer surfaces.

14.2 Computer Simulation 14.2.1 Monte Carlo and MolecularDynamics Results Macromolecules near the surfaces of bulk polymeric materials experience conflicting influences. An undiluted polymer material attempts to maintain its bulk density right up to the surface. An enthalpic penalty must be paid to incorporate void space and reduce the density below the equilibrium bulk value. On the other hand, the constituent macromolecules feel increasing constraints on the number of configurations that they can adopt near the surface, and thus they pay an entropic penalty in trying to maintain bulk density near the surface. Surfaces and interfaces are, of course, zones of finite width over which the transitions in density and polymer configuration are made. The thickness of these zones is governed by the factors just mentioned. For multicomponent polymers, the surface affinities of the various components also play a key role. For a single component, homopolymer material, full knowledge of the state of a polymer surface region would comprise information on the profile of density change through the surface, including its characteristic width, and information on the polymer configuration as a function of position near the surface, including the positions of the centers of mass of the molecules, the distribution of dimensions of the macromolecules, measured both parallel and perpendicular to the surface, and the positions of the chain ends.

659

Over the past several years, a considerable number of computer-simulation experiments have been conducted to probe both the equilibrium and dynamical properties of polymer surfaces. These have the advantage over the current state of laboratory investigations of polymer surface structure in that they produce completely detailed, visualizable and quantifiable pictures of polymers at surfaces. However, in assessing the results of simulations it must always be borne in mind that they are only as good as the rules and representations of interaction from which they are built. Madden opened this line of inquiry, performing extensive lattice Monte Carlo simulations for a film adsorbed against a hard wall and exposed to vacuum (Madden, 1987, 1988). The film thicknesses were greater than several tens of segment diameters, i.e. not so thin that the two surfaces interacted with one another. This work has the feature of comparing directly two kinds of polymer surfaces of interest, i. e. hard, impenetrable solid surfaces and interfaces with vacuum (or air or vapor). In one case, a sharp boundary is imposed, whereas in the other, the density profile can broaden by the incorporation of density defects in the vacuum interfacial region. At most temperatures, the similarities between these two types of surfaces far outweigh the differences. In general, lattice-type simulations may be expected to yield reliable information on large-scale properties where the detailed structure of the lattice is not important. Some representative density profiles from the lattice Monte Carlo simulations of Madden (Madden, 1987) for a melt-vacuum interface are shown in Fig. 14-1. Results are shown as a function of the reduced temperature T* for the simulation of a polymer film having an average degree of polymerization of 100. (Madden also investigated the effects of polydispersity; Figure 14-1

660

14 Polymer Surfaces and Interfaces with Other Materials

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Z/o Figure 14-1. The results of a Monte Carlo simulation of a layer of polymer melt between an impenetrable solid (left) and vacuum (right), shown for several reduced temperatures, 7* (as defined in text): +, 7* = 1.5; #, T* = 1.75; x, T* = 2.0; D, T* = 2.5; V, 7* = 3.0. Z is distance from the solid surface and o is the segment diameter. The density in the middle of the film "reaches that of the bulk" and decreases with increasing temperature. The solid wall enforces an essentially constant total density right up to the surface. At the vacuum surface, the density is also constant and the density profile is then quite sharp, except at the highest temperature (Madden, 1987).

corresponds to a rather low value of polydispersity, i.e. 1.08.) The dimensionless temperature is given by T* = kT/ebb, where k is Boltzmann's constant, T is temperature and £bb is the bead-bead (or segment-segment) interaction parameter on the lattice. In this lattice model, the corresponding interaction energy parameters for segmentvacuum and vacuum-vacuum were set to zero. The results indicate a narrowing of the interfacial region with decreasing temperature and a corresponding increase in the bulk density. Measured in terms of the width of the density gradients, the surface region is narrow, except at high temperature, and covers only several segment diameters. Other results, which are not shown here (Madden, 1987, 1988), demonstrate, inter alia, that there is a peak in the density of chain centers of mass, which is located at a distance comparable to the radius of gyration away from each surface. No such inhomogeneity shows up in the total polymer density (Fig. 14-1).

Lattice Monte Carlo simulations were pursued by ten Brinke et al. (1988), who confirmed this latter feature of Madden's work for a polymer melt (actually a rather sparse one at an average polymer volume fraction of 0.8, which is comparable to Madden's case where T* = 2.0) in contact with a solid wall. Centers of mass are depleted from the immediate vicinity of the hard wall. It was noted in this work that there was a distinct enrichment of chain ends within a few segment diameters away from the wall. For a series of chain lengths (N = 40, 60 and 80 segments), the average degree of enrichment over the bulk density of chain ends was 8 + 2%, apparently independent of chain length, though as we shall discuss directly, this conclusion of a molecular-weight independence appears to be incorrect. Off-lattice Monte Carlo simulations were subsequently pursued in order to examine local structural details (Dickman and Hall, 1988; Yethiraj and Hall, 1990;

14.2 Computer Simulation

Kumar et al., 1988, 1990; Vacatello et al., 1990). We focus on the results of Kumar et al. (1988, 1990) here, as they bring out a range of interesting general properties of polymer surfaces. Kumar and co-workers confirmed that there is enrichment of chain ends near a neutral solid wall. In contrast to the work of ten Brinke et al. (1988), the enrichment found by Kumar et al. was stronger, reaching about 40% for long chains, and was dependent upon the molecular weight. The difference in these findings does not apparently relate to the specific, underlying lattice in the simulations carried out by ten Brinke and his co-workers, since molecular-weight dependent chain-end enrichment at solid surfaces has also been seen in very recent lattice Monte Carlo simulations by Wang and Binder (1992). In all of the discussions of chainend enrichment thus far, we are considering cases where there is no enthalpic factor favoring chain-end adsorption, i. e. there is no extra adsorption affinity of the chain ends, a factor which can be very important if it occurs. Enrichment of chain ends occurs in immediate proximity to the surface, within a distance of about one segment diameter (cr).

661

This produces a depletion in the density of chain ends over the next few segment diameters of distance from the surface. The profile of chain-end density settles down to the homogeneous bulk level beyond about 2Rg (where Rg is the radius of gyration). An example of the density profile of chain ends is shown in Fig. 14-2. An empirical relationship suggested by Kumar et al. (1990) for the molecular-weight dependence of the chain-end density within one segment diameter of the surface [pe(cr = 1)] is pe(<7= 1) = a + b exp(-c/N)

(14-1)

where a, b and c are fitted constants. The density profile of chain centers-of-mass, shown in Fig. 14-3, is peaked at about 3(7 from the wall, less than the bulk Rg for chains of this size, and achieves its bulk value at a distance of about 2 Rg from the wall. The work of Kumar et al. (1990) brings out clearly several additional effects of surfaces on the configurations and shapes of molecules in undiluted polymer materials. The average dimensions of the chains measured in the direction normal to the wall are reduced relative to those measured in the bulk, for those chains where the center of mass is located within Ro of the wall.

1.50

Pi

Figure 14-2. Density of chain ends in a polymer, relative to that in the bulk material, as a function of the distance £ (= Zlti) away from a solid wall, measured as distance divided by the segment diameter, with the chain lengths A, N = 50; O,N= 100; O, N = 200 (adapted from Kumar et al., 1990).

662

14 Polymer Surfaces and Interfaces with Other Materials

Figure 14-3. Density of chain centers of mass in a polymer, relative to that in the bulk material, as a function of the distance £cm of the center of mass (4m ~ Zcm/o) away from a solid wall, measured as distance divided by segment diameter, with chain lengths A, N = 50; O: N - 100 (adapted from Kumar et al., 1990).

Furthermore, as illustrated in Fig. 14-3, the reduction in dimensions normal to the surface scales with Rg. Chain dimensions parallel to the surface are greater on average than those in the bulk. The total radii of gyration appear to be independent of position. Asymmetry appears in the chain configuration for chains within 2Rg of the surface, owing to the fact that the "wall-side", of macromolecules close to the wall, is flattened. These latter two points will be taken up again later in this section and in Section 14.2.2. Chain segments are strongly oriented, but only very near (< 2a) to the wall, where they exhibit a segmental order parameter, SB of -0.2 (SB = [ 3 < c o s 2 0 x > - l]/2, with < cos 2 0 x > being the average square of the cosine of the angle made by all the segments with the x-axis normal to the surface) (Kumar et al., 1990). (As the definition of SB indicates, SB = -0.5 would correspond to perfect orientation in the plane of the surface.) At a > 2, the segmental order parameter is zero. Whole chains are, on average, more weakly oriented than segments. A chain order parameter (defined by Sc = [3 < cos2Oxi > - l]/2, where the average is taken over all the segments of the ith chain)

indicates a weaker parallel tendency at ~ -0.08, but is detectably different from zero out to distances of 2Rg of the center of mass from the wall. These latter features are elaborated more fully in the work of Mansfield and Theodorou (1989). They studied a polymer-vacuum interface, so that the general similarity of their configurational results to those of Kumar and his co-workers for polymersolid interfaces, confirms the essential connection between these two situations. Mansfield and Theodorou extended previous work to investigate the question of whether the flattening that they observed, and which also can be seen in Fig. 14-3, is due to a change in the intrinsic shape of the chains, or whether it is caused by a tendency for chains as whole entities to assume orientations parallel to the surface. To investigate this, they examined two quantities that measure the intrinsic shape of the macromolecules independently from the orientation, namely the spans of the chains and the eigenvalues of the radius-of-gyration tensor. These are quantities (defined precisely by Mansfield and Theodorou) which measure the shape of a macromolecule relative only to its internal coordinates.

14.2 Computer Simulation

They show conclusively that, within the precision of their simulations, the overall shapes of the macromolecules are practically unperturbed, even within one radius of gyration of the surface. Macromolecules in bulk are generally found in a distribution of anisotropic configurations (Sole and Stockmayer, 1971); spherical symmetry is a property only of the average configuration over the entire distribution. The implication of the results of Mansfield and Theodorou (1989) is that the surface region selects from the bulk distribution those molecules with a tendency to align their long axes parallel to the surface. This conclusion is in accord with and illuminates the observation of Kumar et al. (1990) that the overall radius of gyration of the molecules varies little from the bulk values even through the surface region. Carton and Leibler (1990) have shown that the conformations of anisotropic objects adapt in a predictable and general manner to regions of inhomogeneous density, such as polymer surfaces. Another significant piece of work contributing to the understanding of polymer surfaces via computer simulations is the molecular-dynamics study by Bitsanis and Hadziioannou (1990). This is both the most rigorous and the most elaborate means for simulating realistically the properties of polymer-melt surfaces. It is necessary to follow this route if one is interested in dynamic properties (e.g. diffusion near surfaces), as these authors were; structural information, of the type we have discussed here, is also produced by the simulation but it is not necessarily superior to that obtained via Monte Carlo methods. All of the structural features described above were also observed by Bitsanis and Hadziioannou (1990). These workers, along with Wang and Binder (1991, 1992), also studied the effects of preferential surface attraction or

663

repulsion by certain segments. This is important in situations where certain segments are chemically modified, thus producing surface activity. Predictably, preferential attraction of chain ends brings them to the surface (Wang and Binder, 1992); somewhat less obvious is the fact that modifying all of the segments so that they have an attraction for the surface diminishes the enhancement of chain ends at the surface. In retrospect, this is an effect to be anticipated, since a chain of TV segments has only two ends; if 7V»2, as usual, the interior segments will win the competition for the surface. Bitsanis and Hadziioannou (1990) showed that, near the surface, the molecular mobility parallel to the surface increased while that measured perpendicular to the surface decreased; surface attraction reduced the mobility still further. Chain connectivity appears to reduce the surface ordering induced by a solid wall relative to that which would appear in a fluid of disconnected, but otherwise identical, segments (Horn and Israelachvili, 1981). 14.2.2 Implications and Interpretations The overall picture of the facts emerging from computer simulation experiments is rather clear and consistent over a variety of simulation methods. The low compressibility (at all but quite high temperatures) of dense bulk polymer forces the average polymer density (over distances comparable to the segment diameter) near a surface to assume a value very close to the bulk density. The influence of a surface on chain conformations is essentially exclusively exerted on those segments that are within approximately 2crof the surface. These segments have a propensity to align parallel to the surface. This produces some local asymmetry of polymer configurations but does not distort the overall shapes of the macro-

664

14 Polymer Surfaces and Interfaces with Other Materials

molecules very much. The average shapes of macromolecules near the surface are similar to those in bulk but in this case they have their long axes preferentially aligned with the surface. Chain ends are enhanced in density within 2<7 of the surface by as much as 40%, depending somewhat on possible attractive interactions with the surface (de Gennes, 1987). Consequences for adhesion with other materials are to be anticipated (Prager and Tirrell, 1981; Brochard and Pincus, 1992). In the absence of specific interactions with the wall, this can be understood (Helfand and Tagami, 1971) by an argument based on random walks. Simply put, placement of a middle segment of a chain near an impenetrable surface requires that two random walks emanate from the point of that placement without crossing the impenetrable barrier. This is demonstrably (Helfand and Tagami, 1971) more difficult entropically, and therefore of lower probability, than the accommodation necessary for the single, albeit longer, random walk resulting from placing a chain end segment near the surface. Owing to the connectivity of chain molecules, some of the effects exerted on the segments in direct propinquity to the surface are transmitted over distances of the order of the size of the polymer coils in depth away from the surface. Both the density of chain centers of mass and the alignment of anisotropic chain configurations exhibit surface enhancements within 2Rg of the surface. Thus, 2(7is the length scale for surface perturbation of segments, whereas 2Rg is the length scale for surface modification of whole chain properties. Polymer materials exhibit more nonlocal effects of their surfaces than other materials, and these are of some importance in the understanding of polymer surface tensions (Rabin, 1984). However, most features at the segment level of polymer-surface structure are indepen-

Overlappmg polymers in bulk

Creation of a surface by configurational swap maintaining constant density Figure 14-4. Schematic illustration of the idea of configurational swap. In the top picture, two chains are overlapping as in the bulk polymer. To create a surface along the dashed line, while not changing the density of the polymer segments, those portions of the lightly and heavily drawn chains that cross the line are folded, or reflected, back on to their sides of the surface, the effect of which is to leave unaffected the population of rotational states about each bond, but to compress the overall extent of the macromolecules normal to the surface (Silberberg, 1988).

dent of the molecular weight of the entire chain. (Surface chain end enrichment is an exception to this statement; see Eq. 14-1.) Figure 14-4 illustrates schematically a simplified way to think about how and why polymer-surface configurations differ from configurations in the bulk. Essentially all of the above characteristics obtain for both polymer melts and glassy solid surfaces in contact with either hard solid walls or vacuum (or air) (Theodorou, 1988, 1989). The only significant difference between these closely related situations is that in free surfaces the total density profile has somewhat more freedom to accommodate than at solid boundaries (see Fig. 14-1); at many temperatures of interest, the vacuum interface behaves effectively like a solid boundary for the polymer. Crystalline surfaces of polymers have not yet been studied extensively by computational methods. Theoretically, most of the phenomena seen in simulations are understood to a rea-

14.2 Computer Simulation

sonable degree. Self-consistent mean-field (SCMF) theories (in a variety of related forms; see Scheutjens and Fleer, 1980; Freed, 1987; des Cloiseaux and Jannink, 1990) can predict many aspects of the chain configurations observed in simulations, such as the enhancement in density of chain ends near the surface. SCMF theory models the polymer chain as a random walk, distorted by a potential field arising from interactions among neighboring segments (Helfand and Tagami, 1971). Thus the positions of the segments and the effective potential is determined self-consistently. Lattice versions of SCMF theories are good vehicles to predict the observed segment order parameters (Helfand, 1976; Helfand and Weber, 1976). For example, a model due to Helfand (1976), predicts the segment order parameter in the first lattice layer near the surface to be -0.22, then decaying to zero beyond the second layer, in excellent agreement with the results of Kumar et al. (1990). A particularly illuminating and appealing, if somewhat simplistic, theoretical view of polymer configurations at surfaces of undiluted polymers is that offered by Silberberg (1988). He has elaborated a concept known as "configurational swap", which is essentially coincident with the SCMF theory for polymer configurations under the imposition of a reflecting boundary condition at the polymer surface, but is simpler computationally. [It is comparable to the limit of infinitely strong incompatibility and zero compressibility in the model of Helfand and Tagami (1971)]. Imbedded in this analysis is the idea that the surface exerts no influence on the local random orientation of segments, or on the random placement of chain ends, right up to the surface. This, we have seen from simulations, is not true in detail but is valid beyond 2crfrom the surface. "Configura-tional swap" is thus a

665

reliable means to model the global configurational properties of the chain as a whole. The procedure, following Silberberg (1988), is to imagine an isotropic, homogeneous polymer melt in thermodynamic equilibrium through which a surface is then laid in such a way that one infinite plane of segments is traversed by it. If this surface is to mimic an effectively impenetrable solid or vacuum surface, the cutting of the surface by chains must be removed. "Configurational swap", or the reflecting boundary condition, which eliminates crossing of the surface while maintaining bulk density, proceeds by moving along the surface and uncoupling all chains that lie across the surface according to the following procedure. For each chain, or part of a chain, on one side of the surface there is a corresponding mirror-image chain, or part of a chain, on the other. Such chains are uncoupled by removing all parts of the chain from one side of the surface and putting in their place the mirror-image parts from the mirror-image chains. Correspondingly, into the mirror-image parent chains are coupled those parts of the original chains that had been uncoupled. Thus, total density is preserved, as all bonds are replaced by ones that are energetically equivalent; thus, there is no change in the internal energy of the system, although there is an entropy decrease. This idea is depicted in Fig. 14-4. The methodology of the analysis is illustrated in Fig. 14-5. In the absence of a surface, the contribution of the average monomer density 0, in the x-direction from those chains having their center of mass between x0 and x0 + dx0 is given by (Silberberg, 1988) 0 (x, x0) = 0o exp [-p (x - x0 )2]

(14-2)

where 0O is a constant and /?= 3/2 No2. This function arises from the basic Gaussian dis-

666

14 Polymer Surfaces and Interfaces with Other Materials I

0L

Figure 14-5. Segment distribution in the polymer melt. A macromolecule, with its center of mass at JC0, has the segment distribution E'ACD, while a macromolecule with its center of mass at -x0 has the segment distribution D'C'AE. If the plane at x - 0 is a surface, the portions of the distributions E'AO and EAO are, respectively, disallowed and swapped. This gives the distribution OABCD. The center of mass is shifted from x0 to xG, and the mean radius of gyration normal to the surface is reduced from Rgx to Rg x (adapted from Silberberg, 1988).

tribution governing the chain configurations in the bulk polymer. When the polymer is cut by a surface, the configurational statistics of the nearby polymer chains will adjust with the constraint of maintaining bulk density, while not crossing the surface. The resulting density profile for those chains originally at x0 is given by [-/?(*- x0)2]

(x, x0) = exp [-

(14-3)

As illustrated in Fig. 14-5, the net effect is to shift the chain center of mass from x0 to xG = x0 erf (x0 /31/2) + (TT/T 172 exp (-/k 2 ) (14-4) and to reduce the Jt-component of the radius of gyration: Rg,x =Rg,Xo-(*G-Xo)

(14-5)

where Rgx is the x-component of the radius of gyration in the "uncut" polymer. The overall effect is illustrated in Fig. 14-6, where the segmental density contribution is plotted for chains at various distances from the surface. In addition to the local density gradients discussed earlier, it is evident that a surface can exhibit nonlo-

cal or long-range effects on a length scale comparable to the unperturbed size of the polymer coils. The overall effect of Eqs. (14-3) - (14-5) is such that polymer molecules in the near-surface region take on relatively flattened configurations in a direction parallel to the surface, just as seen in the simulations described earlier (Mansfield and Theodorou, 1989). Furthermore, the modified density distribution [see Eq. (14-3)] produces a maximum in the centerof-mass density at a distance corresponding approximately to Rg 0. Figure 14-6 shows clearly that chains which have their centers of mass located at or beyond about 2Rg are essentially unperturbed in their Gaussian distribution of segments found in bulk. These effects, which were seen in many of the simulation results discussed earlier, produce important effects on surface properties, such as non-local contributions to the surface energy, with further ramifications on properties such as adhesion and wetting yet to be fully explored.

14.3 Surface Tension of Polymeric Materials

667

*O/31/2=2.O|

1.0

x o /?1 / 2 = 1 1.0 -

Reduced Distance from Surface

Figure 14-6. Segment distribution at different distances from the surface of a polymer melt. The numbers in boxes refer to the positions of the centers of mass of the unperturbed chains, JC0, scaled by the average size of those chains, with the distance from the surface, x, being scaled in the same way: see text for the definition of /3; , location of center of mass; , radius of gyration, Rg (Silberberg, 1988).

14.3 Surface Tension of Polymeric Materials 14.3.1 Introduction Surface energetics are arguably the most important characteristics of material surfaces. Surface tensions, or for solids, more appropriately, surface energies, express the work that must be done to create surface. In turn, they largely dictate many aspects of how material surfaces interact with one another macroscopically. We will use the terms surface tension and surface energy essentially interchangeably in this chapter, although the latter is more general and the former should strictly be applied only to

liquids. Tension refers to forces arising from the stresses associated with the reversible creation of new surface area, a process which is often experimentally infeasible for solids. Here, however, we will use both surface tension and surface energy to characterize the reversible work required to create unit surface area. In the previous section, attention was focused on the molecular aspects of polymer surfaces. It was observed that although the density drops precipitously from bulk to essentially zero over a distance comparable to several segment diameters, the average conformations of the chains were affected on a much larger scale comparable to the unperturbed dimensions of the coils. The de-

668

14 Polymer Surfaces and Interfaces with Other Materials

crease in configurational entropy associated with these non-local effects, along with the increase in internal energy associated in part with the steep density gradients, as well as new energetic interactions experienced by the segments in the interfacial region, give rise to an excess free energy for the system as a whole. In terms of thermodynamic variables, the excess free energy (per unit area), % is given by the relation y= (E-TS + PVS - X Nt

(14-6)

where E is the internal energy, T absolute temperature, S the entropy, P the pressure, Vs the volume, As the area, and Nt and fa are the number of moles and chemical potential, respectively, of the /th component, all of these quantities referring to the surface region. The first three terms on the right hand side of Eq. (14-6) represent the free energy of the system with the surface. X NiHi represents the free energy of a hypothetically identical mixture of the same components without the surface. Thus, the surface tension, y, represents the excess free energy, per unit area, associated with the presence of the surface. Surface tension can be modeled theoretically by means related to the SCMF methods mentioned in the preceding section. However, before discussing theory, we examine experimental methodology and results for surface tension measurements. 14.3.2 Experimental Techniques for Measuring Surface Tension or Surface Energy The type of experimental technique employed for the measurement of surface energy depends in large part on the state of the polymer sample. Comprehensive reviews of procedures and results have been given by Wu (1982) and by Koberstein (1987). For surfaces of polymers in the liquid state, sur-

face tension is most conveniently measured by one of a variety of methods that applies forces to the surface and measures the resultant distortion of the surface area. Recently invented methods of this type that measure surface distortions by natural capillary waves (Sauer et al., 1987), or by measuring small surface distortions from electromagnetic forcing (Ito et al., 1990), are finding considerable application for polymer solutions, but as yet have not been applied effectively to bulk polymers. For solid polymers, procedures based on the reversible creation of surface by application of force, which work well for liquids, are virtually impossible to implement. For example, cross-linked polymers would resist surface deformation by (frequently irreversible) mechanisms that have nothing to do with surface energy (e.g. viscoelasticity). Therefore, several categories of indirect methods have been employed for solids. Most common among these are the extrapolation methods and the wetting methods. More recently, determination of surface energies of solid polymers by direct measurement of the molecular level contact adhesion has been explored (Merrill et al., 1991). 14.3.2.1 Polymer-Liquid Surface Tension The most versatile, and accurate if done with proper precaution, methods for measuring surface tension of polymer liquids are drop (or bubble) profile methods that are based on the principle that the equilibrium shape of a drop is governed by a balance between the surface-tension forces, acting to minimize surface area, and the effects of gravity, which distort and expand the surface area. Drops are termed either sessile or pendant, depending on whether gravity acts to pull the drop towards or away from a supporting surface, respectively. The

14.3 Surface Tension of Polymeric Materials

force balance for a pendant drop is described by the Bashforth-Adams equations (Bashforth and Adams, 1892; Hartland and Hartley, 1976; Huh and Reed, 1983): dfl _ 2 | gzAp ds ~ R 7 dx -r- = cos o9 ds dz

= sin 6

sin Q x

(14-7 a) (14-7b) (14-7c)

where 0 is the angle measured between the tangent to the profile at the drop apex and the tangent at Cartesian coordinates x and z along the two-dimensional projection of the drop profile (assumed in three dimensions to be axisymmetric), R is the radius of curvature at the apex, g is the acceleration due to gravity (for gravity along the z-direction), Ap is the density difference between the drop and its surroundings, and s is an arclength coordinate, measured from the apex along the projected drop profile. Boundary conditions on the three unknowns, 0, x, and z are that they are equal to zero when s is equal to zero. Surface tension is determined by comparison of a digitized image of the drop profile with the shape of the solution to the set of Eqs. (14-7a, b,c), varying / a s a parameter until good correspondence is achieved (Anastasiadis et al., 1988). Advancements in image-analysis methods are improving the accuracy and convenience of these methods. Other drop (or bubble) configurations can be used, with the physics, if not the details, of the analyses being identical to those described for the pendant-drop method. The choice made among the other drop configurations depends to a large extent on the materials involved (i.e. their physical properties, such as densities); this is a particularly important issue when the methods are used to measure interfacial tension between two polymers and the density differences are small.

669

Another category of drop-profile methods, operating on different physical principles, is that involving spinning- or rotatingdrop methods (Princen et al., 1967). A liquid drop or air (or gas) bubble is suspended in a horizontal cylindrical tube. If one is interested in determining the tension between a polymer material and air, this is done by placing an air bubble in the midst of the polymer fluid. Rotation of the tube at angular frequency (O causes the lower-density material to lie along the axis of rotation and to deform axially to some equilibrium length L under the action of surface tension, buoyancy and centrifugal forces. The surface energy is calculated from (14-8)

AC

where C is determined from 4(Cr 3

3

r

l)

(14-9)

with r being the initial radius of the drop. All drop-profile methods require, and are therefore plagued in their execution by the need for, insurance that the drop configurations being measured actually represent equilibrium conditions. This can necessitate long measurement times with the concomitant need to insure stable, controlled experimental conditions of temperature, humidity, etc., over this period. The other principal way to measure surface tension by physical distortion of the surface is to insert an object, such as a plate or a ring (Wilhelmy, 1863; Dettre and Johnson, 1966) through the surface, and to use this as the means of applying a known force of distortion (usually by lifting the surface against gravity). When a solid plate (in practice, glass or platinum is frequently used) is immersed vertically in a polymer

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14 Polymer Surfaces and Interfaces with Other Materials

fluid, that is, when it is inserted across the interface between polymer and air (or other controlled environment), if the polymer wets the solid preferentially to air, the polymer fluid will be entrained on the plate, producing a resultant force in the direction of gravity, given by F = pycosO-

pgAd

(14-10)

where p, A and d are the perimeter, crosssectional area and immersion depth, respectively, of the solid plate, and 9 is the contact angle of the wetting polymer fluid on the plate. Surface tension can be determined from Eq. (14-10) by measuring the force on the plate as a function of the immersion depth (Dettre and Johnson, 1966). Insurance of equilibrium is an important consideration here, as well. 14.3.2.2 Polymer-Solid Surface Energy Indirect methods have been applied most often to the determination of polymer-solid surface energies. Extrapolation methods are discussed in detail by Wu (1982); these involve measuring the surface tensions of chemically identical polymers under conditions other than the one of immediate interest, such that the polymer of interest is liquid under those other conditions. The extrapolation involved is the systematic variation of the other conditions back toward the conditions of interest. For bulk polymers, two such approaches have been used and give reasonably consistent results where they have been examined. Variations of both molecular weight and of temperature have been used to create polymer samples in the liquid state; extrapolation to the solid-state condition of interest then proceeds by measuring surface-tensions as temperature decreases or as molecular weight increases. The liquid surface tension methods discussed above are

employed (Wu, 1982). Concentration extrapolation, that is, dissolving a polymer solid in a solvent and measuring the surface energy as a function of solvent content, and extrapolating to bulk, is not advisable since, in binary mixtures where one component can segregate preferentially to the surface, extrapolation is very unreliable. Extrapolation through a glass transition temperature can be carried out reasonably accurately if the slope of the extrapolating line is changed at Tg by the ratio of the thermal-expansion coefficients of the liquid and the glass (Wu, 1974). Some results concerning extrapolation methods will be described briefly in subsequent sections; it is appropriate here, however, to mention some general considerations which affect their utility. Chief among these considerations is that any extrapolation from the liquid state cannot capture the effects ofcrystallinity on the surface energy of the solid (Schonhorn, 1968a, b). Many polymers are at least partially crystalline; furthermore, the state of polymer crystallinity at a surface may be different from that in the bulk, owing both to possible effects of the surface in contact with which the polymer solidified, and to the effects of some of the surface-orientation phenomena (discussed earlier) on crystallization near surfaces. The density of a material is known to be an important factor in determining surface tension [see Eq. (14-15] below] and, for example, the density of crystalline polyethylene is 17% greater than that of amorphous polyethyl-ene at 20°C. These comments on the effects of crystalline order on surface tension also apply, in a modified form, to other microstructured polymer materials. Estimation of the surface tension of an ordered block copolymer material, obtained from measurements made in a disordered liquid state, would be inaccurate, because both sur-

14.3 Surface Tension of Polymeric Materials

face ordering and segregation under the conditions of interest could not be adequately comprehended by an extrapolation from high temperatures. An additional consideration in the use of extrapolation methods is a knowledge of the form of the variation of surface energy with the parameter that is being varied. Whereas there is reasonable theoretical and experimental justification for the idea (see next section) that surface energy varies as the reciprocal of the temperature, the form of the molecularweight dependence of polymer-liquid surface tension is more poorly understood, particularly from a theoretical point of view. This uncertainty makes extrapolation potentially unreliable. Wetting methods are the other major category of indirect routes to the determination of the surface energies of solid polymers. These involve the measurement of the contact angle 0 for sessile drops of various liquids in contact with the surface. Fox and Zisman pioneered this approach and conceived the most widely practiced method of this type, which has come to be known as the measurement of the critical surface tension (Fox and Zisman, 1950). This quantity, yc, is determined by measuring the equilibrium contact angles of a series of liquids of different liquid-vapor surface tensions, 7LV, and extrapolating to the condition of complete spreading, i.e. when 6=0: LV

(14-11)

Young's equation, i. e. ^ v cos 0 = y s v + ySL, relates the liquid-vapor surface tension of the sessile drop to the surface energy of the solid and the solid-liquid interfacial energy via the contact angle 6. Using Young's equation in the limit of zero contact angle gives 7c = 7 - 7 s L - ^ e

(14-12)

where KQ is known as the spreading pressure and is equal to the difference between

671

the solid-vapor surface energy, y sv , and the solid surface energy in vacuum, y. Thus, yc will be a good estimate of /when ySL and 7iQ are small. The former can be minimized by using liquids of polarities similar to that of the solid; the latter becomes larger as the liquid approaches complete wetting of the solid (the implication here is the rather paradoxical conclusion that one should not extrapolate from too close to zero contact angle). This wetting method is a particularly convenient and simple technique to estimate the surface energy of a solid, but is most usefully and appropriately regarded only as an estimate. Direct comparison with the other methods, for example the more rigorous liquid-state methods discussed above, shows that yc is almost always of an order 10% lower than the values of /measured by other methods, which is understandable when based on the considerations embodied in Eq. (14-12). Difficulties in selecting an appropriate series of liquids, and in reliably extrapolating to zero contact angle, beset this method. The best series of liquids are those that are similar in polarity, both to one another and also to the solid in question. Wu (1979) has proposed an alternative, but related, wetting method, known as the "equation of state" method, which mini-mizes the inaccuracies caused by some of these effects. Questions of how the liquid interacts with the solid, i.e. in ways other than through the surface energetics, also need to be recognized when using these techniques. A sessile drop will exert a mechanical tension normal to the solid and possibly induce molecular rearrangement. Furthermore, while immiscibility with the solid is an essential characteristic of the liquid that is to be used, it is sometimes difficult to insure that absolutely no liquid permeates the solid.

672

14 Polymer Surfaces and Interfaces with Other Materials

Recently, progress has been made (see Merrill et al., 1991; Chen et al., 1991; Chaudhury and Whitesides, 1991; Horn and Smith, 1992) in determining the surface (or interfacial) energies of solids by measuring the contact-adhesion forces and shapes of surfaces of identical (or different) solids brought into well-defined, intimate, molecular contact. These methods are based on the Johnson, Kendall, Roberts (JKR) theory (Johnson et al., 1971) of solids in contact with each other, which is an approximate theory in that it assumes that the intermolecular interactions giving rise to the contact adhesion are very shortranged and so act only through those portions of the surfaces in contact (and not through those elements of the surface that are close to one another but not in contact). While this assumption is not defensible in detail, the testable predictions of the JKR theory have been examined thoroughly (Horn and Israelachvili, 1987; Merrill et al., 1991) and appear to be reasonably quantitatively reliable. From the point of view of determination of surface energies of solids, the core results of the JKR theory for contact between two elastic spheres of radii R{ and R2 [where R =RlR2(Ri +^2)] and identical properties, including bulk elastic modulus K, and surface energy, 7 under the action of an external force F, are those for the radius of the contact area between the two solids, a: 3 a

= ±

Table 14-1. Comparison of surface-energy values for PET determined by different methods.

(14-13) and for the (tensile) force required to separate the two solid surfaces: = 3nRy

termination of /by two independent means. In the first method, a is measured as a function of the applied load F, and 7 is determined by fitting the data to Eq. (14-13). The first term is for nonadhesive elastic spheres, whereas the other three terms embody the additional contributions of surface energy to the deformation of the solids. Chaudhury and Whitesides (1991) have examined the validity of Eq. (14-13) for the case of crosslinked spheres of poly(dimethylsiloxane) (PDMS), with a microscope apparatus enabling the accurate measurement of a(F), and found it gave an excellent fit to their data plus a good value of the surface energy (y= 22 mJ/m2, well-known from earlier measurements) (see Wu, 1982). Merrill et al. (1991) have used the surface force apparatus techniques, developed by Israelachvili (1991), to investigate the contact between two identical surfaces of polyethylene terephthalate) (PET). This method involves making thin, smooth sheets of the PET and using the optical interferometry of the method to determine the magnitude of the "jump apart" from adhesive contact, which is directly related to Fs. They, too, found that the JKR theory gave an excellent account of their observations. Measurement of the force to separate [Eq. (14-14)] gave a value of 7 of 61 ± 2 mJ/m2. Table 14-1 compares this with

(14-14)

Measurement of these two quantities (for known values of K and R) enables the de-

7 (mJ/m2) 43 44 45 61 a d

Method

Ref.

Critical surface tension Wu's "equation of state" Melt extrapolation Surface-force measurement

[a] [b] [c] [d]

FoxandZisman(1950); b Wu(1979); c Wu(1982); Merrill et al. (1991).

673

14.3 Surface Tension of Polymeric Materials

values of 7 for PET determined by other methods. The difference between the surfaceforce-determined value and those obtained from the other methods raises some interesting issues. It is possible that this higher value is the most accurate, given the tendencies of wetting methods to underestimate surface energies, as described by Eq. (14-12), and of the inability of extrapolation from the melt to capture influences of the crystallinity in PET on 7. However, there are subtle effects to consider in the application of surface-force methods. The force to separate the surfaces may be elevated by dissipative effects arising from surface rearrangement and/or interpenetration on contact, possibilities it shares with wetting methods. Chen et al. (1991) and Chaudhury and Whitesides (1991) have seen, from examination of Eq. (14-13), that there is an hysteresis effect for some surfaces, i.e. a different contact radius is seen for the same applied force, depending on whether the applied force is in the process of being increased or decreased. They attribute these effects to dynamic rearrangement processes occurring in the zone of contact between the two surfaces.

14.3.3 Experimental Results for the Surface Tension of Polymeric Materials Comprehensive compilations of experimental data for the surface tension of polymer materials are available (see Wu, 1989). Typical experimentally measured surfacetension values for a variety of commercially important polymers are listed in Table 142. Surface tensions for polymers generally range from 20 to 50 mJ/m2; these values are larger than their corresponding small-molecule equivalents. This is in part due to the increased density of the higher molecularweight materials, as surface tension is a sensitive function of the density of the material. Group contribution techniques have been applied for the approximation of surface-tension values, the basis of these being the empirical Macleod relation (Macleod, 1923) connecting surface tension with density r= 7o pn

(14-15)

where 7 is the surface tension, p is the density and 70 and n are constants. Values for the Macleod exponent, n, are generally between 3 and 4, and several of these experimentally measured values are also listed in

Table 14-2. Surface tensions and their temperature coefficients for some common polymers (adapted from Wu, 1989) Polymer Poly(propylene) (atactic) Poly(styrene) (molecular weight, 9300) Poly(tetrafluoroethylene) Poly(methyl acrylate) (molecular weight, 25000) Poly(methyl methacrylate) Poly(ethylene terephthalate) Polycarbonate (bisphenol A) Nylon 66 (molecular weight, 17000) Poly(dimethyl siloxane)

y(mJ/m2)

- d//dr (mJ/(m2K))

Macleod's exponent

29 39 24 41 41 45 43 47 21

0.056 0.065 0.058 0.070 0.076 0.065 0.060 0.065 0.059

3.2 4.0 4.2 3.5

674

14 Polymer Surfaces and Interfaces with Other Materials

Table 14-2. In turn, the constant y0 can be approximated by (14-16) where the quantity P is the combined "parachor" for a repeat unit and M is the repeat unit molecular weight. Extensive lists of the parachors for various atoms and chemical groups are available (Quayle, 1953; Van Krevelen, 1976; Wu, 1982). Surface tension values, calculated with this group-contribution technique, are often in reasonable agreement with experimental values. In general, the surface tension is dependent on temperature, molecular weight and chemical composition. The functional form of these dependencies can be anticipated, in part, by the semi-empirical Macleod equation [Eq. (14-15)]. The chemical composition affects both the constant y0, as described above, and is also a factor in determining the bulk density of the material. However, for a given polymer, the effect of molecular weight and temperature on the surface tension parallel the effects of these variables on the bulk density p. Although detailed predictions for the surface tension of polymers are available (see later), certain semi-empirical relations for the molecular weight and temperature dependence are useful. A semi-empirical expression which adequately describes the experimental molecular weight dependence of surface tension is given by (LeGrandandGaines, 1969, 1973): k M 2/3

(14-17)

where Mn is the number-average molecular weight. Polymers increase somewhat in

density as the average molecular weight increases; this effect has been elucidated by equation-of-state treatments of compressible polymers (Sanchez and Lacombe, 1976, 1977, 1978; Sanchez, 1983 a). By the use of Eq. (14-15), the increase in surface tension with molec-ular weight can be interpreted in terms of the enhanced density of the bulk polymer. Equation (14-17) is most appropriate for lower molecular weight polymers (Wu, 1982), and usually, the magnitude of the constant £e in this equation is such that the surface tension y reaches, within ex-perimental error the limiting value y,, at fairly low molecular weights, of the order of 104, over the same range where the bulk density is also becoming molecular-weight independent. For higher-molecular-weight polymers, the surface tension is rather insensitive to molecular weight. Temperature is also a key variable affecting the surface tension of pure-component polymers. Experimentally it is observed that, to a high degree of approximation, the surface tension decreases linearly with increasing temperature. y=7o-kTT

(14-18)

Some experimentally measured surfacetension values, as a function of temperature, are shown in Figs. 14-7 and 14-8 for a variety of common polymers. Polymeric materials generally have a positive thermal-expansion coefficient. Similar to the effect of molecular weight, the decrease in density with increasing temperature, produces a corresponding decrease in the surface tension of the material [see Eq. (14-15)]. The constant, kT- -dy/dr, varies between 0.05 and 0.07 mJ/(m2 K) for a wide variety of polymers: several values for this temperature coefficient are also listed in Table 14-2. The temperature coefficient for small molecules is generally larger [~ 0.1 mJ/(m2

14.3 Surface Tension of Polymeric Materials

50

200 100 150 Temperature (°C) Figure 14-7. Surface tension as a function of temperature for some polymer melts: PEO, polyethylene oxide); L-PE, linear poly(ethylene); B-PE, branched poly(ethylene); PP, poly(propylene) (Wu, 1992).

675

high-density polymers (Koberstein, 1987). Another contribution is the nonlocal entropy effect. The decrease in configurational entropy of the "flattened" polymer molecules in the near-surface region produces a positive contribution to the temperature coefficient (Silberberg, 1988). However, due to the sensitivity of the surface tension to density, as indicated by Eq. (14-15), the decrease in density with increasing temperature is the dominant effect. The net result is a decrease in the surface tension at elevated temperatures. The details of the origin of this will become clearer in the following section.

14.3.4 Theoretical Aspects of Polymer Surface Tension

100 150 Temperature

200

Figure 14-8. Surface tension as a function of temperature for some polymer melts: PCP, poly(chloroprene); PMMA, poly(methyl methacrylate); PVAc, poly(vinyl acetate); PIB, poly(isobutylene); PnBMA, polyO-butyl methacrylate); PDMS, poly(dimethyl siloxane) (Wu, 1982).

K)] (Padday, 1969). Several factors contribute to the smaller temperature coefficients found for polymers, with perhaps the most important of these being the lower thermalexpansion coefficient associated with

Before describing some theoretical approaches to predicting the surface tensions of polymers in detail, it is useful to obtain some approximate conceptual ideas of how certain of the relevant factors come into play. We will follow closely the line of argument developed by Helfand and Wasserman (1982). For this purpose, we first need a model for the free energy of the material whose surface tension we wish to understand. For this purpose here, and in the subsequent discussion, we will adopt essentially the representation of Sanchez and Lacombe (1978) (though not their detailed lattice-fluid model, which is much more careful and intricate). This approach models a bulk polymer as a mixture of polymer segments and holes in the Flory-Huggins manner (Sanchez, 1983 a). A lattice, of site-size b, is used to compute the free energy of mixing the connected segments and the holes which represent the density, or more exactly, the compressibility due to the free volume of the material. The Flory-Huggins free energy of mixing per unit volume, for a vol-

676

14 Polymer Surfaces and Interfaces with Other Materials

ume fraction of segments 0, with a fraction (l-O) of holes, is (14-19) -
where N is the degree of polymerization of the polymer; AUis the total cohesive contact energy per segment and is (one half of) the increase in internal energy when a segment of size b3 is moved out of the bulk at 0 = 1 to 0= 0 (Silberberg, 1988). AU/kT could be thought of as a Flory-Huggins, ^-parameter for the mixing of the two components, holes and segments. We will use this notation part of the time in this discussion since it enables the discussion of surface tension to be made in a virtually identical way to that of the interfacial tension between two immiscible polymers with interaction parameter x that form an incompressible interface (Helfand, 1975). Consider (following Helfand and Wasserman, 1982) a surface with a sharp density profile, such as the curve with T* = 1.5 shown in Fig. 14-1. Assume that the polymer-melt segment number density falls abruptly from p to 0 over a short distance. Consider further a section of a polymer chain of length Z segments lying in this surface region. A first question to discuss is that concerning the typical length of such a sequence of segments broaching the surface region. A segment entering the surface region will pay an energy cost of order X^T; thus, a sequence of Z segments will suffer an energy penalty of ZxkT. The energy required to pay this penalty to drive the chain segments into the surface region of a melt can come from but one source at equilibrium, namely, thermal fluctuations of

energy kT. The consequence of this is that ZxkT^kT, resulting in Zoc x~l ~ ALT1. If the configuration of the sequence of Z chain segments is roughly that of a random walk [see Eq. (14-2)], then it will span a characteristic spatial dimension of the order bZl/2. This length characterizes the thickness of the surface region, D, which from these considerations must vary as D ^ bx~l/2 ^ AC/""2. If we interpret x a s the energetic cost of introducing holes into, or lowering the density of, the surface region, then we conclude that the surface-region thickness should vary as D °c K~1/2, that is, inversely with the square root of the compressibility of the polymer. For the interface between immiscible polymers, it is the ^-parameter that controls the interfacial thickness (D °c X~1/2)- In order to estimate the surface tension, we observe that the density variation in the surface region of the melt occurs in a volume of DS, where S is the surface area of the material. In this region, there are pDS segments, each with a typical energy of the order of ^fcT, so that the total energy of the region, calculated as the energy of mixing of holes and segments, is

yS=pDSxkT

(14-20a)

or

y=pbxuzkT

(14-20b)

Following the line of argument above, these equations suggest an inverse relationship between surface tension and the square root of the compressibility, or likewise, a proportionality to the square root of the cohesive contact energy between segments. This connection has been demonstrated in an extensive examination of experimental data, and via more detailed theory, by Sanchez (1983b). The serious limitation of this style of argument is its fundamental inconsistency. A

14.3 Surface Tension of Polymeric Materials

sharp interface is assumed, but random-coilsize dimensions of the surface region are found. A more consistent, detailed theory of surfaces and interfaces is necessary and is available (Helfand, 1980; Helfand and Wasserman, 1982; Poser and Sanchez, 1979). There is more than one way to approach the development of this theory but the essential physics is similar. The polymer molecule lays itself out in the surface region in a manner that minimizes the free energy, with the latter containing contributions from the effects of the surface on the configurations of the macromolecules as well as interactions with other molecules in the surface region. Here, we discuss the basic approach, based on the Cahn-Hilliard theory (Cahn and Hilliard, 1958) using the free energy model of Eq. (14-19). The alternative SCMF theory has been worked out only in the limit of infinite molecular weight (Helfand and Wasserman, 1982; Anastasiadis et al., 1988). The detailed theories of surfaces and interfaces (Poser and Sanchez, 1979; Silberberg, 1988) begin by a more detailed calculation of the density profile of segments in the surface region than that given in the simplified argument above. The dependence of 0 on the position JC, normal to the surface in the surface region, is determined by minimizing the Cahn-Hilliard free-energy integral: ^

ck

(14-21)

where A F ^ = AFM() (0/0M)AF(0M) and 0M is the bulk volume fraction [~O(1)]. K is known as the influence parameter, containing information on intermolecular interactions and is calculable from a combination of derivatives of AF aM with respect to O. Using Eq. (14-19) to determine AFoM leads to K = AW2b. Some authors (Poser

677

and Sanchez, 1979) have treated K as a parameter to be derived by fitting to experimental data. AFO is the surface free energy per segment. The two terms in Eq. (14-21) represent, first, the free energy of the material, were it homogeneous at the local density, and second, the contribution to the free energy of the surface region due to the fact that the density is varying. This form of the theory has come to be known as "square-gradient theory" and has been a very successful approximation (Davis and Scriven, 1982), despite the fact that one could have imagined that, especially for very sharp interfaces, higher-order gradients of the density profile would have to be considered. Surface tension in this theory is determined by minimizing the free energy of Eq. (14-21) with respect to variation in the profile 0{x) of density of segments through the surface region. This minimization yields AF

- K

(14-22)

dx

so that the surface tension derived from the surface free energy of Eq. (14-21) can be written, with a change of variables, as 7 = 2 J [KAFaM(0)]V

d0

(14-23)

Equations (14-22) and (14-23) provide the means to calculate surface (or interface) density profiles, and surface (or interfacial) tension theoretically. The former can be tested against simulations such as those of Fig. 14-1, or against the growing body of recent experimental results, to be discussed subsequently. The theory is versatile in that different models for the free-energy expression can be employed. For bulk polymer melts at most temperatures of interest, the widths of the surface

678

14 Polymer Surfaces and Interfaces with Other Materials

profiles predicted by square gradient theory are very narrow, of the same order as the segment size b. This is entirely consistent with the conclusions drawn from the simulations discussed earlier. More important are the comparisons with experimental data for surface tension, and its temperature coefficient, which have been explored most extensively by Silberberg (1988) and by Poser and Sanchez (1979). Figure 14-9 shows a comparison of experimental data and theory for several polymers, presented in terms of reduced variables, with K constant at AU/2b. Clearly the agreement is good in several respects; the magnitude of /could be matched nearly perfectly by adjusting K to 0.55 AU/b (Poser and Sanchez, 1979). The negative temperature dependence of y is also captured reasonably accurately. The origin of this is physically illuminating. Several authors (Hong and Noolandi, 1981; Rabin, 1984; Silberberg, 1988) have shown that there is an additional contribution to the surface tension, not included in Eq. (14-23). This results from the nonlocal effects of chain connectivity near the surface; in other words, the effect, captured in simulations but not in square-gradient theory, that

connected segments, by attempting to pack into constrained surface regions, propagate the interactions to distant regions along the chain. Rabin (1984) showed that this could be incorporated into Eq. (14-23) by modifying K to be AU/(2b) + kT/(24®b). This modification adds a positive contribution to the temperature coefficient of the surface tension. Therefore, whereas it may be significant quantitatively, it is not the dominant effect in determining the temperature dependence of polymer surface tensions. This positive effect comes from the fact that, as temperature increases, the constraints on the chain configurations are increasingly important, thus raising the surface free energy with temperature. The dominant effect comes from the fact that, as temperature increases, more holes are introduced into the surface region, so reducing its density. The density profile of the surface broadens, as seen in Fig. 14-1, and this creates increased freedom for the arrangement of segments in the narrow constrained surface region as temperature increases. Sanchez further refined the generalized gradient approach to polymer surfaces and derived a universal relationship between

Figure 14-9. Reduced surface tension y [= yb2/(AU)] as a function of the reduced temperature f [= £77(At/)] shown for several polymers: experimental data compiled by Poser and Sanchez (1979), and compared with theoretical results obtained from a model based on Eq. (14-23), developed by Silberberg (1988). The points are data for the different polymers; the solid line is the theory.

Qt 0.6

0.7

0.8

0.9

Reduced Temperature,!"

1.0

14.4 Techniques to Examine Polymer Surface Properties

surface tension y, bulk density p and isothermal compressibility K (Sanchez, 1983b):

14.4 Techniques to Examine Polymer Surface Properties 14.4.1 General Considerations

1/2

^{K

679

(14-24)

The surface tension is predicted to vary inversely with the square root of compressibility, as we anticipated earlier in this section when based on a qualitative argument (Helfand and Wasserman, 1982). The constant Ao is independent of both temperature and molecular weight, and its value is invariant for certain classes of materials. Figure 14-10 illustrates the universal behavior in experimental data suggested by Eq. (14-24) using surface-tension values for PDMS determined by Bhatia et al. (1985) at different molecular weights and temperatures. For most polymers, as well as other organic liquids, the constant AQ2 varies only slightly between 2 - 3 x 1CT4 (erg cm2/g)1/2. Equation (14-24) is a simple and accurate correlation for estimating the surface tension of polymers.

if)

(p/K-) 1/2 x10- 8 (Nkg/m^)

Figure 14-10. Experimental test of the semi-empirical correlation [Eq. (14-24)] proposed by Sanchez (1983 b) using the data obtained by Bhatia et al. (1985) for PDMS of molecular weights 75000 (O) and 3900

The battery of experimental methods now used to study the characteristics of polymer surfaces, in addition to surface energy, has expanded rapidly in recent years. There is, of course, an intimate relationship between the molecular structure and properties of a surface region and the fundamental thermodynamics discussed earlier. The effect on the surface tension of molecular ordering and surface enrichment in multicomponent materials has been recognized (Wu, 1982, 1989; Bhatia et al., 1985). Surface texture, or roughness, is another important aspect of surface structure which may be probed directly. Consistent with the aims of this chapter, the focus is on those techniques suitable for the study of the external surfaces of polymers, or their interfaces with other materials. For the additional, and interesting, subject of buried interfaces in polymers, the reader is referred to the recent review by Stamm (1992). Objectives in the characterization of polymer surface properties include measurement of roughness, mass density and composition profiles, information on the configuration of the molecules in the surface region, and where appropriate, information on the mechanical, or other material properties. There are two length scales associated with polymer surfaces, which make them particularly challenging experimentally. Certain properties, such as mass density and roughness may vary over the size of a repeat unit; other properties such as chain configuration and compositional variations, may vary over length scales associated with the polymer molecule taken as a whole. In many cases, several different techniques will be required to cover the entire polymer sur-

680

14 Polymer Surfaces and Interfaces with Other Materials

face region. Several excellent reviews of techniques are available for both polymer surfaces and interfaces (see Briggs, 1987; Feast and Munro, 1987; Koberstein, 1987; Russell, 1990, 1991; Stamm, 1992). Table 14-3 summarizes many of the surface-sensitive techniques which are used to study polymer surfaces. The various forms of microscopy are listed first and offer the advantange of a direct image of surface structure and composition. On the other hand, X-ray and neutron reflection yield Fourier- space information on surface structure with very high resolution. Various surface-sensitive spectroscopic techniques can give detailed information on average chemical composition and composition profiles. In some cases, the non-conducting or radiation-sensitive nature of organic polymers requires special precautions. On the other hand, the ability to synthesize polymer molecules bearing isotopic or chemical labels can offer great advantages. Surface sensitivity for many of the techniques is derived from either the limited penetration depth of an incident probe beam, or the stimulated emission of information-carrying particles having very small escape depths. Advances in all these techniques has pushed the resolution into the angstrom range. However, to achieve such resolution, sample quality becomes extremely important, and often some theoretical underpinning is required for complete interpretation of the data. 14.4.2 Imaging Methods Several techniques are available which can produce images of polymer surfaces and enable one to characterize the surface structure at a molecular level. The most conventional, but nevertheless very powerful, imaging technique is scanning electron microscopy (SEM) (Goldstein et al., 1981;

White and Thomas, 1984; Sujata and Jennings, 1991). Nanometer focusing of the incident electron beam offers very high resolution for mapping surface topography. In addition, surface chemical analysis with lateral resolution in the millimeter range may be obtained through energy analysis of Auger electrons (Auger spectroscopy) and X-ray fluorescence (Goldstein et al., 1981). One must take care to avoid specimen damage at higher accelerating voltages, and samples are often coated with small amounts of a heavy element (e.g. platinum) to improve resolution. Environmental scanning electron microscopes that can image at high pressure (20 Torr) offer the exciting possibility of directly studying the effect of changing ambient conditions on polymer-surface structure (Sujata and Jennings, 1991). Transmission electron microscopy (TEM) has also been applied to the study of polymer thin films and surfaces (Sujata and Jennings, 1991; Hasegawa and Hashimoto, 1985a, b; Henkee et al., 1988; Ishizu and Fukuyama, 1989). Both chemical staining and microtoming are required sample preparation for TEM. The technique can offer images of two-dimensional composition profiles with nanometer resolution (Russell, 1991). Atomic resolution under environmental conditions can be achieved with scanning tunneling microscopy (STM) (Rabe, 1989; McMaster et al., 1991; Rabe and Buchholz, 1991). In this method a conducting point probe is rastered across the surface (typically) at a constant tunneling current to give a direct three-dimensional image of the surface topography. Conductivity requirements limit the experimental studies to isolated molecules or thin monolayers on conductive substrates such as graphite or gold. Atomic fofce microscopy (AFM) can image non-conductive polymer surfaces by measuring the force between a fine tip and

14.4 Techniques to Examine Polymer Surface Properties

681

Table 14-3. Techniques for polymer surface analysis. Technique

Acronym Sampling Depth Lateral depth resolution resolution

Scanning electron microscopy

SEM

10 nm

Transmission electron TEM microscopy Scanning tunneling STM microscopy

10 nm

Information content

Comments

5 nm

Direct image of surface topography

3 nm

Two-dimensional profile Molecular imaging and surface topography Molecular imaging and surface topography Composition profiles and roughness Composition profiles

Auger and X-ray mode can give chemical composition Special sample preparation required Conductive substrate required

0.1 nm

0 nm

AFM

0.1 nm

0.5 nm

X-ray reflectometry

XR

1 nm

-

Neutron reflectometry

NR

1 nm

-

Ellipsometry

ELLI VASE

1 nm

-

X-ray photoemission spectroscopy

XPS ESCA

10 nm

0.1 nm

10 Jim

HREELS

1 mm

1 nm

1 jam

Surface composition, vibrational spectra

1 urn

Surface composition

-

Composition profile

Atomic-force microscopy

High-resolution electron-energyloss spectroscopy

Static secondary-ion SSIMS mass spectrometry Dynamic secondary- DSIMS ion mass spectroscopy Forward-recoil FRES spectrometry Rutherford backRBS scattering spectrometry Nuclear-reaction NRA analysis

1 nm

Infra red attenuated total-reflection spectroscopy

IR-ATR

|um

Infra red grazingincidence-reflection spectroscopy

IR-GIR

Film thickness, refractive index profile Surface composition, composition profiles

Interpretation requires model predictions Interpretation requires model predictions Limited contrast Radiation damage and sample charging are problems Radiation damage and charging are problems Complex spectra Etch rate is composition dependent

°°

13 nm

Jim

80 nm

Composition profile

|Ltm

30 nm

Composition profile of marker

Requires heavy elements

jixm

13 nm

Hydrogen and deuterium composition profile Vibrational spectra of polymer surface adjacent to ATR-crystal surface Vibrational spectra of thin films and adsorbed layers

Resolution is depth dependent

682

14 Polymer Surfaces and Interfaces with Other Materials

the surface (Hansma et al., 1988; Meyer et al., 1990; WittmanetaL, 1991; Collinetal., 1992). At present, AFM cannot match the resolution of STM entirely; however, its ability to image dielectric surfaces makes it an invaluable tool in studying polymer surface topography and measurement of surface physical properties. A complementary imaging technique is phase-measurement interference microscopy (PMIM) (Biegen and Smythe, 1988; White et al., 1990; Smith et al., 1991; Stamm, 1992). Here, the interference pattern from a laser beam reflected from the sample surface and a piezoelectrically driven reference surface is monitored with an area detector. The interference pattern can then be related to the thickness and/or topology of the sample film. Although the lateral resolution for PMIM is of the order of microns or greater, height-contour maps with a vertical resolution of 0.6 nm are possible. 14.4.3 Reflection Methods Two related reflection methods which can offer excellent depth resolution in the study of polymer surface structures are X-ray reflectometry (XR) and neutron reflectometry (NR) (Heavens, 1976; Born and Wolf, 1980; Lekner, 1987; Russell, 1990, 1991). Both of these measure the scattered intensity of an incident beam at grazing angle to the polymer surface as a function of the wave vector q = (27t/A)sin0 where A and 0 are the wavelength and incident angle, respectively. For XR, q is scanned by measuring intensity as a function of the angle 6, whereas both X and 0 may be varied in NR, in ways that depend on the neutron sources (Russell, 1990, 1991). Contrast for neutron reflectometry is provided by the scattering-length densities of the constituent nuclei (Felcher et al., 1987).

Deuterons are considerably more efficient scatterers of neutrons than protons, making NR a sensitive technique for evaluating multicomponent systems where one component has been selectively deuterated. On the other hand, X-ray reflectometry is sensitive to differences in electron density, and is amenable to the study of multicomponent systems where there is a difference in electronic structure between the components (Bilderback, 1981). Reflectivities can be measured to as low as 10~6 for neutrons and 10"10 for X-rays (Russell, 1990). Both XR and NR, particularly the former, can give accurate measures of surface roughness (Braslaw et al., 1985; Foster et al., 1990). Although the lateral sampling area for both these techniques is large, the ideal Fresnel reflectivity curve, which decreases as q~4 for an ideal smooth surface, is damped exponentially by the effect of surface roughness (Russell, 1990; Stamm, 1992). The equations relating the measured intensity profiles for both XR and NR to the composition gradients within the sample are very similar, and highly non-linear. As a result, measured intensity or reflectivity profiles cannot be uniquely and directly inverted to obtain composition profiles. Instead, experimental reflectivity profiles are compared with calculated profiles based on an assumed structure. Iterative comparison leads to a best-fit profile, but uniqueness of such a profile cannot be proven. Ellipsometry (ELLI) is a light-reflection technique which has been used to study thin film and surface behavior of polymers. In all ellipsometric measurements, it is the change in polarization state of polarized light after passing through a sample which is studied, either by reflection or transmission (Azzam and Bashara, 1977; Debe, 1987). The use of polarization distinguishes ELLI from XR and NR and confers on it high depth resolution, despite the long-

14.4 Techniques to Examine Polymer Surface Properties

wavelength radiation employed. The polarization state is characterized by the measurement of the amplitude ratio, tan % and the phase difference, A, for the two Cartesian components of the electric field. Thus, at fixed angle and wavelength, two pieces of information characterizing the sample are obtained, and are usually an average thickness and refractive index for an assumed uniform film. The ability to extract much more detailed information has come with the development of variableangle spectroscopic ellipsometry (VASE) (Kimetal., 1989; Arwin and Aspnes, 1986). VASE, at multiple angles, produces additional independent data on the sample and so is particularly useful in examining multilayer surface regions. As with XR and NR, direct inversion of the measured profiles to obtain the spatial variation of optical constants is usually not possible, and model predictions are required to interpret the data. 14.4.4 Spectroscopic Methods Spectroscopic methods of most use for polymer surfaces are either electronic or vibrational in character (Garbassi and Occhiello, 1987). The electron spectroscopic techniques all require high vacuum and so the surface structure to be examined must survive this state. A common-high vacuum surface analytical technique which can usefully measure surface composition and composition profiles at polymer surfaces is X-ray photoemission spectroscopy (XPS), which is also commonly known as electron spectroscopy for chemical analysis (ESCA) (Clark et al., 1973; Clark et al., 1974). In XPS, the polymer surface is irradiated with X-rays from an anode or synchrotron source. The X-rays produce photoelectrons within the sample (the photoelectric effect) which are collected at a take-off angle (6)

683

and the energy is analyzed. Since the kinetic energy of these core-level electrons is directly related to the binding energy, energy analysis of the emitted electrons produces a semi-quantitative spectrum of polymer-surface atomic composition. XPS is not able to detect the presence of extremely light elements, i.e. H and He. XPS is a surface-sensitive tech-nique because the mean free path (for inelastic collision) A, of photoelectrons in a polymer matrix is usually less than 5 nm. Thus, even at normal incidence (6 = 90°), maximum sampling depths are generally not greater than 100 A (Fadley et al., 1974). Furthermore, angle-resolved XPS has depthprofiling capability in that the required escape depth of a photoelectron emitted at depth z for the surface decreases as l/sin#, where 6 is the take-off angle. The measured intensity I*(0) for an electronic core level of a particular nucleus is related to the concentration profile within the sample by (Fadley et al., 1974): (14-25)

7» w (The intensity I*(q) is normalized by the 90° value). The integral prefactor l/sin0, represents the increase in acceptance area with decreasing take-off angle, and should not be included when the entire specimen is being sampled (Fadley et al., 1974). Although faced with an inversion problem similar to XR and NR, when using the above equation, XPS is able to study composition gradients of species which have binding energies which are resolvable in the XPS spectrum. It does, however, require knowledge of the mean free path X. Several values of A have been reported in the literature for a variety of electron kinetic energies and host materials (Clark and Thomas,

684

14 Polymer Surfaces and Interfaces with Other Materials

1977; Bain and Whitesides, 1989; Parsonage et al., 1991). Small-spot sampling techniques have also been used to study lateral variations of surface composition with a resolution of around 10 |im (Kelley, 1991). Some disadvantages of XPS include the inability to probe beyond several tens of angstroms into the sample. Furthermore, angle-resolved XPS measurements require relatively long exposure times, and certain polymers will suffer radiation damage. Charging of the dielectric polymer samples can also cause peak shifts and broadening of spectra. Charging can be partly compensated for by the use of electrostatic flood guns and internal standards. Another high-vacuum technique for surface analysis is high-resolution electronenergy-loss spectroscopy (HREELS) (Wandass and Gardella, 1985; Gardella and Pireaux, 1990; Pireaux et al., 1986, 1988). In addition to XPS-like information on surface composition, HREELS can also probe surface vibrational spectra as well. In this case, the energy loss for a beam of low-energy electrons is analyzed after interacting with the surface. The technique is thus able to probe both vibrational and electronic transitions depending on the electron energy. It is less quantitative than XPS for compositional information, although it does allow for the detection of light elements. Both Raman and IR-active modes are detected in the vibrational spectra. The technique is very surface sensitive, with a sampling depth of about 1 nm (Wandass and Gardella, 1985; Gardella and Pireaux, 1990). However, both charging and radiation damage are problems for polymer samples. Another highly specialized form of vibrational spectroscopy which has found application for polymer surfaces in contact with metals is inelastic electron-tunneling spectroscopy (IETS). In this method, a thin film of polymer is incorporated into a sol-

id/polymer/lead junction and the tunneling current is measured. Inelastic effects in the tunneling through the junction can be related to the vibrational spectrum of the polymer. Colletti et al. (1987) used this method to determine the reactivity and orientation of segments of acrylic polymers in contact with aluminum oxide. There are many other analytical spectroscopic techniques for the study of surfaces and thin films which can be directly applied to polymers under the right circumstances (see, for example, Debe, 1987). Included in these are the various surface and thin-film specific vibrational spectroscopic techniques such as infrared attenuated total reflection spectroscopy (IR-ATR) and infrared grazing-incidence reflection spectroscopy (IR-GIR). IR-ATR measures the vibrational spectrum in a thin surface region of the polymer sample which has been placed in intimate contact with a highrefractive-index material such as silicon, germanium or zinc selenide (Ulman, 1991; Harrick, 1967; Harrick and du Pre, 1966; Haller and Rice, 1970). The technique is not particularly surface sensitive, with a sampling depth comparable to optical wavelengths (> 0.5 jam). It does provide a measure of overall composition in the near-surface (glass-interface) region, and sampling depths can be varied somewhat through the refractive index of the coupling material. IR-GIR measures the vibrational spectrum of a thin film by external reflection(s) from a supporting substrate (Ishitanti et al., 1982). The technique is not in itself surface specific, but it can provide information on the orientation of transition dipoles adjacent to the surface for very thin films and monolayers (Ulman, 1991).

14.4 Techniques to Examine Polymer Surface Properties

14.4.5 Ion-Beam Methods Ion-beam methods require an apparatus for acceleration of the ions, generally operate at high vacuum (as with electron spectroscopic methods), and can be damaging to samples. On the other hand, they have a versatile capacity for probing many aspects of polymer surfaces. An important ion-beam technique for chemical analysis and depth profiling of polymer surfaces is secondary-ion mass spectroscopy (SIMS) (Stamm, 1992; Coulon et al., 1989). In static SIMS (SSIMS), a primary-ion beam (e.g. Ar, Cs, Ga, Xe, etc.) is focused on the surface, where it reacts specifically to produce secondary-ion fragments which are then analyzed by a mass spectrometer. SSIMS spectra for polymers can be very complicated, and chemical identification is accomplished by comparison with spectra of known materials. SSIMS can be very surface sensitive, with a sampling depth of around 1 nm, and in addition, the ability to focus the primary-ion beam allows for two-dimensional studies of chemical composition with a lateral resolution of 1 mm (Kelley, 1991). The basis of dynamic SIMS (DSIMS) is the etching of the sample surface by rastering of a primaryion beam (e.g. 3 keV O2+) across the surface and continuous monitoring of the resulting secondary ions by mass spectroscopy. The SIMS experiment is quite sensitive to the mass difference between the proton and deuteron, and selective deuteration of one component allows for depth profiling of that species (Coulon et al., 1989; Russell et al., 1989). The advantage of DSIMS over angle-resolved XPS is the ability to monitor composition profiles at large depths into the polymer sample (albeit by destructive techniques). However, DSIMS has considerably lower depth resolution than XPS, with a minimum resolution of approximate-

685

ly 125 A. The low resolution is in part due to the necessity of calibrating the etch rate as a function of composition. In addition, problems in obtaining a steady-state etching process demand special preparation of the polymer surfaces (Coulon et al., 1989). Less destructive ion-probe techniques include forward recoil spectroscopy (FRES), Rutherford backscattering (RBS) and nuclear reaction analysis (NRA). These techniques were originally developed for semiconductors but have been transferred to polymer applications with great success. The incident beam for both FRES and RBS consists of helium ions accelerated to a few MeV. (4He is often used to minimize beam damage.) FRES measures the energy spectrum for protons lH and deuterons 2H ejected in the forward direction (Mills et al., 1984; Green and Doyle, 1990). At a fixed acceleration voltage and take-off angle, the energy of the ejected particles is related to its mass and the depth at which it originated within the sample. The ability to differentiate the mass difference of protons from deuterons allows for the study of composition profiles where one component has been selectively deuterated (Mills et al., 1984, 1986; Green et al., 1986; Green and Kramer, 1986a, b; Green and Doyle, 1990; Sokolov et al., 1989; Kausch and Tirrell, 1989). The depth resolution is approximately 80 nm and is limited by both the inelastic scattering processes which relate energy to depth, and the use of a stopper foil to eliminate accessory 4He particles (Stamm, 1992). Recent advances using a time-of-flight technique and removing the stopper foil have considerably improved the resolution (Sokolov et al., 1989). In the case of RBS, it is the 4He ions, backscattered from a heavy-element marker within the sample, which is analyzed to produce composition-profile information (Green and Doyle, 1990; Green et al., 1985). The depth

686

14 Polymer Surfaces and Interfaces with Other Materials

resolution for RBS is around 30 nm and has been used to study polymer interdiffusion by the movement of a gold marker located at a polymer-polymer interface (Green et al., 1985) A new method with improved resolution over FRES and RBS is nuclear reaction analysis (NRA) (Lanford, 1978; Payne et al., 1989; Chaturvedi et al., 1989; Steiner et al., 1990). Like FRES, NRA has the ability to examine composition profiles for partially deuterated polymer samples. Both *H and 2H can be depth profiled, depending on the incident beam: for example, by using a 0.7 MeV 3He beam, protons and 4He resulting from the reaction of the incident beam with deuterium can be energy-analyzed to give a 2H depth profile. The resolution of 3He-NRA varies between 14 and 30 nm, depending on the sampling depth (Stamm, 1992). Alternatively, hydrogen depth profiling can be achieved by using a 15 N beam. In this case, the 15N projectile reacts with protons at a resonance energy of 6.4 MeV to produce a 4.4 MeV y-ray. Hydrogen depth profiling is thus achieved by y-ray detection as a function of the 15N penetration depth, which is controlled through the acceleration voltage. The resolution in this case can vary between 4 and 12 nm, depending on the sampling depth (Stamm, 1992).

14.5 Materials Science Issues Concerning Polymer Surfaces 14.5.1 Overview Four principal areas of materials-science interest concerning the properties of polymer surfaces can be delineated: (1) surface chemistry, including modification and functionalization; (2) surface composition, as in multicomponent materials, where one

species can be enriched at the surface; (3) surface structure and organization and; (4) dynamics of polymer chains near surfaces. As explained in Sec. 14.1, surface chemistry will not be dealt with here in any detail, as it is a large subject and there are other excellent sources of information available (see Ward and McCarthy, 1987). We shall discuss problems that exist in each of the other three areas and in so doing illustrate the methods applicable to their study and explain the current state of knowledge on polymer surfaces. 14.5.2 Surface Composition Multicomponent polymers, including blends, copolymers and additive-containing polymers, are the norm in engineering practice. Indeed, they provide, in many cases, physico-chemical (as opposed to reactivechemical) routes to modify surface properties. Multiple components are often present in materials in order to control bulk properties and in so doing exert adventitious surface effects. The components used have a wide range of mutual miscibilities, from completely miscible to essentially immiscible. One component of a multicomponent mixture will segregate to the surface of the material if it lowers the energy of the system, and in most cases, this means that the component of lower surface energy will be enriched in the surface region. The characteristic dimensions of the depth of the surface-enriched zone will be of the same order as the size of the segregating macromolecules (several hundred angstroms), not just a few segments in depth (or several angstroms), as found for the surface regions of pure polymeric materials. The effects of segregation to surfaces in miscible systems were first observed some time ago during measurement of surface tensions of mixtures and copolymers (Ra-

14.5 Materials Science Issues Concerning Polymer Surfaces

stogi and St. Pierre, 1969). Nonlinear-mixing rules with varying compositions for the surface tensions of blends and block copolymers of poly(ethylene oxide) (PEO) and poly(propylene oxide) (PPO) were measured, whereas random copolymers exhibited linear mixing rules for y. The difference was attributed to the inability of the lower-surface-energy segments (i.e. PPO) to segregate effectively to the surface in random copolymers, although this was possible for blocks and blends. Bhatia et al. (1988) have observed a similar reduction of the surface tension in blends of poly (styrene) (PS) with poly(vinyl methyl ether) (PVME), suggesting segregation of PVME to the surface. Silicone polymers, such as poly(dimethylsiloxane) (PDMS) - are often employed in this way to lower the surface energies of materials. XPS has been used frequently (see Thomas and O'Malley, 1981; Schmitt et al., 1986, 1989; Clark et al., 1989) to detect the presence of surface segregation, relying on its chemical specificity to detect the presence of a characteristic element or chemical functionality (Briggs, 1987). (XPS is employed similarly to study surface contamination and to analyze purposeful surface modification.) XPS was employed very early on to pick up surface enrichment in multicomponent materials containing PEO (Thomas and O'Malley, 1979), and has become a standard tool in assessing the surface composition in many applications, for example in putative biocompatible materials such as polyurethanes (Gardella, 1989). Bhatia et al. (1988) were the first to correlate surfacetension reduction with the direct evidence of segregation provided by XPS. Recent efforts have been aimed at more quantitative measurements of surface enrichment than XPS can provide, and at the determination of the composition profiles in the near-surface region. Forward recoil

687

spectroscopy (FRES), dynamic SIMS and neutron reflectivity have all been used. All of these techniques can detect isotopic substitution and have been applied to blends of hydrogenated and deuterated poly(styrene) (PS and d-PS, respectively). The slightly lower polarizability of the the C-D bonds, relative to the C-H bonds, confers on d-PS a slightly lower surface energy, although, except at high molecular weights (Bates, et al., 1985), PS and d-PS are miscible in all proportions at the temperatures of interest. The first surface-enrichment experiments carried out on this system using FRES (Jones et al., 1989; Sokolov et al., 1989) were able to determine quantitatively the integrated surface excess (expressed as the thickness, z*, of a layer of pure d-PS containing an amount equal to the integrated amount of d-PS in the surface composition profile, using the bulk average composition as baseline). Surface excesses of tens of angstroms are seen and surfacevolume fractions of d-PS can be inferred to exceed their bulk values by a factor of about three for macromolecules with molecular weights of the order of 106, after equilibration for several days at 184°C. Owing to the depth resolution of FRES (~ 500 A), the complete composition profile, decaying as it does from the surface into the bulk over a distance of about the same length, cannot be determined by this method. Jones et al. (1990 a) have applied the higher spatial resolution methods of SIMS and neutron reflectivity. Figure 14-11 shows the reflectivity data for a PS blend containing 15% of the d-PS component, plotted in a format designed to emphasize the quality of the fit near the critical angle (the maximum in the figure). As can be seen in the inset, the fit can be made very good by a slight adjustment of the profile from the one predicted by a mean-field theory (Schmidt and Binder, 1985).

688

14 Polymer Surfaces and Interfaces with Other Materials

100

0.007

0.014

0.021

0.028

0.035

1

q (A" ) Figure 14-11. Reflectivity data obtained for a d-PSPS blend (5% d-PS), plotted as reflectivity times the fourth power of the neutron wavevector, and shown as a function of the wavevector q (Jones et al., 1990 a): the dashed line represents the best fit achievable by using the mean-field theory of Schmidt and Binder (1985); and the solid line is an empirically adjusted fit, constrained to have the same surface and bulk compositions, indicating that some improvement over the mean-field prediction is possible. The real-space forms of the composition profiles are shown in the inset.

The data of Fig. 14-11 pertain to two polymers of relatively high, and approximately equal, molecular weights. Several recent studies have examined the effects of equal molecular weight on this surfaceenrichment problem (see Composto et al., 1989, 1990; Hariharan et al., 1991). From this work, it is clear that lower-molecularweight polymers are enriched in composition near surfaces, all other things being equal. Thus, for two polymers, with different surface affinities and different molecular weights, the two surface-enrichment tendencies could be antagonistic, providing

an opportunity to balance the two effects against each other. Shull et al. (1990) have used FRES to determine the surface and interface excesses of a d-PS-poly(vinyl pyridine) (PVP) block copolymer mixed with PS, both at the PS surface and at the interface with homo-PVP. In this case, the surface-enrichment phenomenon is modified in several ways by the fact that the segregating species is a block copolymer. The d-PS would tend to segregate to the surface of the PS mixture, as is the case for a homopolymer. However, were the d-PS chains to go directly to the surface, they would then expose immiscible PVP to the PS bulk. Thus, what is inferred to happen, which is supported by microscopic evidence, is that micelles of d-PSPVP form, and these are preferentially attracted to the surface. Had the block copolymer been miscible with PS, for example, as in an block copolymer of d-PS and PS, the surface segregation would still have been different from the surfaceenrichment process of a homopolymer blend. Every d-PS block segregating to the surface entrains a tethered block of PS at the surface. The net result is that a brush of PS chains would form at the surface (Halperin et al., 1992); the energetics of this brush must be accounted for when modelling the assembly of the surface layer. This surface-enrichment phenomenon is connected theoretically to the subject of "wetting transitions", on which there has been an enormous body of work for simple fluids and binary alloys. The term "wetting transition" refers to the situation where a binary mixture, which has a miscibility gap, exists in a state of two-phase coexistence, and has a surface region that favors one of the phases. In this case, the surface can be coated with either a microscopic or macroscopic layer of one phase, even if the bulk is in the other phase. The phenomena

14.5 Materials Science Issues Concerning Polymer Surfaces

of transitions from microscopic to macroscopic layer thickness are called wetting transitions and are thought to occur universally when sufficiently close to the critical point. Several variants of mean-field theories are available for polymer mixtures (Nakanishi and Pincus, 1983; Schmidt and Binder, 1985; Carmesin and Noolandi, 1989; Hariharan et al., 1991). The first two are phenomenological, while the latter two are based on self-consistent mean-field (SCMF) theory, which enables a more accurate accounting of polymer configurations. The data of Fig. 14-11 are accounted for reasonably well by the theory of Schmidt and Binder (1985). Recent simulations of wetting transitions in polymer mixtures by Wang and Binder (1991) agree reasonably well with the theory of Carmesin and Noolandi (1989). The data of Fig. 14-11 show a monotonic composition profile, resulting from a longterm annealing of a surface layer to reach equilibrium. Jones et al. (1990b) have recently shown that for a homogeneous mixture, quenched into the two-phase coexistence region below the spinodal, a time-dependent sequence of oscillatory composition profiles develops. In this case, the surface enrichment phenomenon directs the spinodal decomposition phase-separation process to occur preferentially in a direction normal to the surface. The practical implications and possibilities for manipulating surface properties by these means still remain to be explored. 14.5.3 Surface Ordering The state of organization of a polymer surface can be the main determinant of many properties of polymer materials, not only those properties that are directly surface sensitive, such as adhesive or tribological

689

properties, but also properties that depend on transport or stress transmission through the surface. Surface ordering is a very important issue for semicrystalline polymers, and since the properties of these materials often depend strongly on the degree of crystallinity, the local degree of crystallinity at the surface could be expected to be significant for surface properties. In many cases, the degree of surface crystallinity is different from that in the bulk. Surfaces of semi-crystalline polymers often contain a high percentage of the amorphous component (Wu, 1982), and lower surface tension values for amorphous materials promote migration of that component to the surface. The degree of surface crystallinity can be systematically varied by using substrates which promote surface nucleation to varying degrees (Schonhorn, 1968a,b;Gray, 1974). Increasing the degree of surface crystallinity increases the surface energy, understandably in the light of Eq. (14-15), but not necessarily concomitantly with an increase in all surface properties. For example, surface ordering may create a brittle or weak boundary layer in the material. Determining the degree of surface crystallinity or ordering is difficult (AlsNielsen, 1987; Jark et al., 1989). Factor et al. (1991) have studied the surface ordering of the aromatic polyimide, poly(pyromellitic dianhydride oxydianiline) (PMDAODA), a material of significant technological importance as an interlevel dielectric in integrated microelectronics. PMDA-ODA does not crystallize in bulk, though it has been reported (Takahashi et al., 1984) to have local paracrystalline or liquid-crystalline order. X-ray reflectivity was used (Factor et al., 1991) to determine that a thermally cured film from a spin-cast precursor, of ellipsometrically determined thickness of 2600 A, had an rms surface roughness of

690

14 Polymer Surfaces and Interfaces with Other Materials

8 ± 2 A. Using a synchrotron x-ray source, grazing-incidence X-ray diffraction was measured. In this method the photons are incident on the surface of the sample at an angle less than the critical angle for total external reflection. An evanescent wave penetrates the sample and is diffracted by structural elements near the surface of the film. Varying the angle around the critical value can alter the depth of penetration, thus probing order as a function of depth. Near the surface of the PMDA-ODA material, it was found that there were strong reflections corresponding to interchain packing distances, which were not observed in the bulk polyimide. The authors concluded that the surface of this polymer was very highly crystalline, despite the absence of crystallinity in the bulk. Similarly, side-chain liquid-crystalline polymers forming smectic-A layers have been found by X-ray reflectivity to have the layers organized parallel to the surface (Stamm, 1992). STM has shown PEO and other polymers to adopt helical structures in thin layers or solids (Yang et al., 1990). This surface ordering has been explained in part through statistical-mechanical considerations, including segment-surface attractions (Chanetal., 1991). Surface ordering can occur in other microstructured polymers, even in the absence of crystallinity. Henkee et al. (1988) observed by TEM that thin-film samples of microphase-separated block copolymers, of different bulk-phase symmetries (lamellar, cylindrical, spherical), stacked in distinct ways, reflecting their accomodation to the contraints of the thin film. As in previous work by Hasegawa and Hashimoto (1985 a, b), Henkee et al. (1988) found that in the thinnest regions of the film, the formation of the bulk microdomain symmetry was inhibited, owing to the desire of the lower-surface-energy component of the block co-

polymer to occupy the surface preferentially. Russell and co-workers (Green et al., 1989; Coulon et al., 1989, 1990; Anastasiadis et al., 1989, 1990; Russell et al., 1990; Menelle et al., 1992), in a pioneering series of studies using neutron reflection, have studied the thin-film microstructure of poly(styrene)-poly(methyl methacrylate) (PSPMMA) diblock copolymer on silicon substrates. At higher molecular weights, this is a strongly segregating system, due to the relatively large repulsive interaction between the two blocks {% « °- 0 4 a t 25°C). Ordered lamellae were shown to be oriented parallel to the free surface at 25°C, over macroscopic lateral distances, after annealing above or near the glass transition temperature of this material (~ 100°C). Lamellar spacings measured for four different values of (symmetric) block degree of polymerization, N, were found to be comparable to those in bulk samples, all being greater than or equal to values calculated from theories rigorously applicable to the strong segregation limit OfAf» 10) (Helfand, 1975). For conditions in the vicinity of the bulk order-disorder transition (i.e. for lower molecular weights, where %N approaches 10), Menelle et al. (1992) inferred from the reflectivity, not dis-ordering but, rather, that weakly damped oscillatory composition profiles set in, where the bulk block copolymer would have been completely disordered. In this system, PMMA has a strong adsorptive affinity for the silicon substrate, whereas PS preferentially occupies the surface with the air. For very thin films, Menelle et al. (1992), as Henkee et al. (1988), found that the order in the block copolymer was disrupted, in this case probably because of the difficulty in satisfying the opposite surface affinities of the two blocks, while still maintaining order in the film.

14.5 Materials Science Issues Concerning Polymer Surfaces

Work by Foster et al. (1992) on symmetric poly(olefin) diblock copolymers in thin films on silicon enlarges on these observations. The materials in this study were poly (ethylene propylene)-poly(ethyl ethylene) (PEP-PEE) diblock copolymers. Being similar hydrocarbons, they are more weakly segregating than PS-PMMA and do not exhibit strong affinity for either surface; PEE is preferred at both the air and the silicon surface. Figure 14-12 shows the composition profiles across the thin film for the polymer with N = 975, which has an order-disorder transition temperature in the bulk of 125°C. Clearly the thin film maintains a discernible structure well above this temperature, crossing at higher temperatures into a regime of damped oscillatory profiles, as suggested by Menelle et al. (1992). For symmetric diblock copolymers near the order-disorder transition (Leibler, 1980), Fredrickson has developed a theory which predicts oscillatory and exponentially damped composition profiles, with the low-surface-energy sequence having a maximum concentration near the sur-

face (Fredrickson, 1988). Such predicted profiles have been tested experimentally via neutron reflectivity on thin films of PSPMMA diblock copolymers in the disordered state (Anastasiadis et al., 1989). These experiments revealed the free surface of the copolymer film to be enriched with PS. In agreement with the predictions of Fredrickson, the PS composition varies as cos

7=130°C 1 0.5

" = 75°C

0 500

1000

1500

2000

zA)

2500

3000

(2nz —jr-

(14-26)

where £, is the correlation length, L the period, 0e the surface-excess composition, 0 the average composition in the bulk and z the distance from the surface. For nearly symmetric PS-PMMA diblock copolymers, values of L = 150 A and | = 95 A, obtained experimentally, were in good agreement with theoretical predictions for these same quantities (Anastasiadis et al., 1990). A similar enrichment of the PMMA component at the silicon-substrate interface was also observed. In further support of these results, enrichment of the free surface by the PS sequence of a symmetric PS-PMMA

7"=U0°C

0

691

3500

£000

Figure 14-12. Composition profiles across a thin (~ 3600 A) film of PEP-PEE block copolymer shown for several temperatures spanning the bulk order-disorder transition (~ 125°C) (from Foster et al., 1992). Order in the film clearly persists above the bulk disordering temperature. The data obtained at 140°C approximately correspond to the damped oscillatory form suggested by Eq. (14-26) (Fredrickson, 1988).

692

14 Polymer Surfaces and Interfaces with Other Materials

diblock copolymer has been measured using angle-resolved XPS (Green et al., 1989). The surface excess of the PS block increased with increasing molecular weight of the copolymer. Another important observation concerning surface ordering of symmetric (lamellar) block copolymers is that of the appearance of "islands" in thin films (Ausserre et al., 1990; Collin et al., 1992). These islands occur because thin films are typically produced (for example by spin-coating) in arbitrary thicknesses; however, their internal microstructure dictates that they span this thickness in a quantized number of layers (the quantum number depends on whether or not the same or different blocks wet the two surfaces of the thin film). If the total film thickness does not equal the product of the quantum number times the individual lamellar dimension, there is a deficit or surfeit of material that can manifest itself as holes or islands on the surface of the film. The existence of these patches, and detailed studies of their time evolution, have been investigated by both PMIM and AFM methods (Ausserre et al., 1990; Collin et al., 1992). 14.5.4 Thin Films on Solids: Wetting, Spreading, Interactions and Dynamics Dynamics in surface regions of polymeric materials have been little explored, except for the case of interdiffusion between two polymers (Stamm and Majkrzak, 1987; Russell, et al., 1989), a subject that is reviewed thoroughly elsewhere (Kausch and Tirrell, 1989; Russell, 1991; Stamm 1992). Information is beginning to emerge in another area of polymer surface dynamics concerning the processes by which thin films of polymer spread on and wet the surfaces of other solids. These are important

processes for bulk polymer melts used as lubricants, coatings and adhesives. Polymer-melt films confined between mica sheets have been studied recently through force measurements (Horn and Israelachvili, 1988; Montfort and Hadziioannou, 1988; Van Alsten and Granick, 1990). A principal observation in common among these studies is the measurement of a strong repulsive force as the film is compressed down toward the molecular dimensions of the polymer molecules. The consensus of opinion among these investigators is that these long-range repulsive forces are manifestations of nonequilibrium effects, namely that the confined polymer material in contact with the surface is very slow to relax. On the theoretical side, there have been suggestions (Kremer, 1986), and computer simulations in support of them (Bitsanis and Hadziioannou, 1990; Chakraborty et al., 1991; Chakraborty and Adriani, 1992), that polymers in contact with solids can exhibit slow, glass-like relaxations, arising from strong adsorptive interactions between the macromolecules and the solid (Chakraborty et al., 1990; Konstadinidis et al., 1992). Silberzan and Leger (1992) have determined the diffusion coefficient of PDMS in contact with silica, both by observation of the rate of spreading of a molecularly thin film on the surface and also by pattern photobleaching of dye-labelled molecules (Kausch and Tirrell, 1989). They found that a PDMS sample (with a molecular weight of approximately 5 x 104) had a diffusion coefficient on the surface about 30 times smaller than that found for this same molecule in the bulk-PDMS melt. Furthermore, they found that a droplet of this polymer, during spreading on silica, exhibited not a smooth hemispherical cap-like shape but a plateau region [referred to as a "bump" or "step" (Brochard and de Gennes, 1984; Bruinsma, 1990)] around its periphery. This

14.6 Conclusions and Remarks on Future Directions

step has a thickness of about the radius of gyration of the macromolecules (found for several different molecular weights). The step occurs because the monomolecular film appears to have a greatly increased viscosity, owing to the attractive interaction of the polymers with the wall. The state of entanglement in a thin layer of bulk polymer interacting strongly with a wall is poorly understood but is likely to be an important factor in determining the thin-film viscosity. That entanglement may be an important issue is indicated by the observation of Silberzan and Leger (1992) that the radius of the edge of this plateau expands diffusively [r(t) oc t1/2], and, for different molecular weights the apparent diffusion coefficient varies as the inverse square of the molecular weight, as seen for polymer-melt diffusion (Kausch and Tirrell, 1989). Dewetting processes, in which homogeneous thin films of polymers that are trapped by nonequilibrium processing conditions such as spin coating on surfaces spontaneously rupture, form holes and de-wet, have also been observed recently by PMIM investigations (Reiter, 1992; Foster et al., 1992).

14.6 Conclusions and Remarks on Future Directions Polymer surfaces and interfaces are regions where large molecules experience constraints on their configurations and interactions that differ from those that they experience in the bulk. A considerable amount of experimental data and theoretical understanding have been developed over the last decade on the ways in which polymer surfaces affect structure and order in polymer materials. Surfaces are preferentially occupied by those constituents of the polymer materials that can best accomo-

693

date the constraints and interactions. Chain ends tend to be enhanced near surfaces, and the configurations of polymers near surfaces tend toward an orientation of the macromolecule and its segments parallel to the surface, driven largely by the incompressibility of bulk polymers. In multicomponent polymers, the surface region is generally preferentially enriched in the component that lowers the surface energy of the material. Surface energies of polymer fluids have been measured extensively, although assuring equilibrium in viscous materials is always an issue, and can be theoretically modelled reasonably accurately by the square-gradient version of Cahn-Hilliard theory. Existing techniques for measuring surface energies of polymer solids are all indirect and merit further development. Owing to the preference of some components for surfaces, segregation tendencies can produce ordering processes at surfaces, affecting crystallinity and microphase separation, and therefore the physical properties of polymer surfaces. Conversely, the constraints of near-two-dimensional thin films can frustrate the assembly of ordered structures. The dynamic processes at polymer surfaces have been studied extensively only in the context of interdiffusion, wetting and spreading processes. Computer simulations have preceeded experiments in their ability to bring out detailed structural features of polymer surfaces and may be expected to be similarly valuable in exploring polymer surface dynamics. The substantial progress that has been made on polymer surfaces in the last decade derives largely from advances in instrumental methods with new and specific kinds of surface sensitivity. Further new developments in techniques will be necessary for the outstanding problems of polymer-surface science. Particularly

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14 Polymer Surfaces and Interfaces with Other Materials

lacking are surface- sensitive methods for assessing the physical properties of polymer surfaces. The needs include quantitative and theoretically understandable methods for studying solid surface energy, surface defects, adhesion, hardness, friction and permeability, among others. Beyond that lies the possibility of making connections between the impressive amount of new information on molecular surface structure and surface properties.

14.7 References Allen, F. G., Gobeli,G.W. (1962),Phys.Rev. 127, 150. Als-Nielsen, J. (1987), in: Structure and Dynamics of Surface II. Phenomena, Models and Methods: Schommers, W., vonBlanckenhagen, P. (Eds.). Berlin: Springer, pp. 181-222. Anastasiadis, S. H., Gangarz, I., Koberstein, J. T. (1988), Macromolecules 21, 2980. Anastasiadis, S. H., Russell, T. P., Satija, S. K., Majkrzak, C. F. (1989), Phys. Rev. Lett. 62, 1852. Anastasiadis, S. H., Russell, T. P., Satija, S. K., Majkrzak, C. F. (1990), /. Chem. Phys. 92, 5677. Arwin, H., Aspnes, D. E. (1986), Thin Solid Films 138, 195. Ausserre, D., Chatenay, D., Coulon, G., Collin, B. (1990), J.de Phys. 51,2571. Azzam, R. M. A., Bashara, N. M. (1977), Ellipsometry and Polarized Light. Amsterdam: NorthHolland. Bain, C. D., Whitesides, G. M. (1989), /. Phys. Chem. 93, 1670. Bashforth, S., Adams, J. C. (1892), An Attempt to Test the Theory of Capillary Action. London: Cambridge University Press. Bates, F. S., Wignall, G. D., Koehler, W. C. (1985), Phys. Rev. Lett. 55, 2425. Bergbreiter, D. E., Chen, Z., Hu, H. (1984), Macromolecules 77, 2111. Bhatia, Q. S., Chen, J. K., Koberstein, J. T., Sohn, J. E., Emerson, J. A. (1985), /. Colloid Interface Sci. 106, 353. Bhatia, Q. S., Pan, D. H., Koberstein, J. T. (1988), Macromolecules 21, 2166. Biegen, J. F, Smythe, R. A. (1988), Proc. SPIE-Int. Soc. Opt. Eng. 897, 207. Bilderback, D. H. (1981), Proc. SPIE-Int. Opt. Eng. 315, 90. Bitsanis, I., Hadziioannou, G. (1990), /. Chem. Phys. 92, 3827. Born, M., Wolf, E. (1980), Principles of Optics, Vol. 6. Oxford: Pergamon.

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15 Crazing and Fracture of Polymers Ikuo Narisawa

Department of Materials Science and Engineering, Yamagata University, Yonezawa, Japan Albert F. Yee

Department of Materials Science and Engineering, University of Michigan, Ann Arbor, MI, U.S.A.

List of Symbols and Abbreviations 15.1 Introduction 15.2 Crazing: Initiation and Growth 15.2.1 What is Crazing? 15.2.2 Morphology and Structure of Crazes 15.2.2.1 Amorphous Glassy Polymers 15.2.2.2 Semicrystalline Polymers 15.2.3 Initiation of Crazes 15.2.3.1 Initiation Model of Crazes 15.2.3.2 Mechanical Criteria of Crazing 15.2.3.3 Crazing and Environment 15.2.3.4 Crazing and Polymer Structure 15.2.3.5 The Effect of Molecular Weight 15.2.4 Crazing: Growth and Theory 15.2.4.1 General Characteristics of Craze Growth 15.2.4.2 The Growth of Crazes at a Crack Tip 15.2.4.3 Microscopic Mechanisms of Crazing 15.2.5 Mechanical Properties of Crazes 15.2.5.1 Stress-Strain Behavior 15.2.5.2 Micromechanics of a Craze 15.2.6 Craze Thickening and Fracture 15.2.6.1 Kramer's Model for Craze Thickening 15.2.6.2 Ductile-Brittle Transitions 15.2.6.3 Craze Fracture Process 15.3 Fracture 15.3.1 Theoretical Strength of Polymers 15.3.2 Fracture Initiation in Uncracked Polymer Specimens (Micromechanism of Fracture) 15.3.2.1 Microvoid Generation due to Tension 15.3.2.2 Chain Scission in Oriented Polymers (Fibers) 15.3.2.3 Relationship Between Chain Scission and Macroscopic Fracture Materials Science and Technology Copyright © WILEY-VCH Verlag GmbH & Co KGaA. Allrightsreserved.

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15.3.3 15.3.3.1 15.3.3.2 15.3.3.3 15.3.4 15.3.4.1 15.3.4.2 15.3.4.3 15.3.5 15.3.5.1 15.3.5.2

15 Crazing and Fracture of Polymers

Ductile Fracture Fracture of Shear Deformation Bands Plastic Fracture Failure Envelope Nonuniform Fracture of Blunt-Notched Polymers Brittle Fracture of Ductile Materials Stress Distribution Around a Local Plastic Deformation Zone Fracture of a Notched Specimen due to Plastic Constraints Fracture of a Cracked Specimen An Application of Linear Fracture Mechanics The Strain Energy Release Rate gc and Fracture Toughness Kc of Glassy Polymers 15.3.5.3 Plastic Deformation at a Sharp Crack Tip 15.3.5.4 Crazing at a Sharp Crack Tip 15.3.5.5 Various Factors Influencing Fracture Toughness 15.4 Concluding Remarks 15.5 Acknowledgements 15.6 References

745 745 746 747 748 748 750 753 756 756 756 758 758 759 762 762 762

List of Symbols and Abbreviations

List of Symbols and Abbreviations a A , B, C a0 b B c g ,c r C d E F AF GN gc 0craze glc AG (T, t) h /x k k! Kc KY Kcl Kc2 ^craze Ko XIc / /c /e /0 m Mc Me Mn Mo n n' NA Npu p pc

average diameter of micro voids; crack length (temperature-, time-dependent) constants initial crack length average distance between micro voids plate thickness compressibilities of the glassy, rubbery state material-dependent parameter average distance between entanglements tensile modulus force activation energy rubbery plateau modulus critical strain energy release rate critical strain energy release rate for the appearance of crazing gc of mode I fracture activation enthalpy active zone in Kramer's model of craze thickening first stress invariant Boltzmann's constant; constant depending on the polymer species constant critical stress intensity factor mode I stress intensity factor fracture toughness in the plane strain rate fracture toughness in the plane stress state critical stress intensity for the appearance of crazing stress intensity factor Kc of mode I fracture length period of a surface arc in the meniscus instability model; craze length in the environment craze length contour length between entanglements length of the repeat unit constant critical value of the molecular weight entanglement molecular weight number average molecular weight molecular weight of each repeat unit constant constant Avogadro number number of cavities at the ends of microfibrils hydrostatic pressure or mean stress; dilative stress dilative stress for crazing

701

702

15 Crazing and Fracture of Polymers

R rc r0 S s s* t T th Tg U vf Vo x x, y , z X, Z Y

gas constant; chain-scission rate craze length at the crack tip intermolecular distance craze fibril flow stress length of the plastic zone maximum extent of the logarithmic slip-line field time absolute temperature time to fracture glass transition temperature backbone covalent bond energy volume fraction of fibrils molar volume of a solvent crystallinity Cartesian coordinates directions normal to stress direction parallel to stress

a (xg,(xr P P{ P fl0 y F yp ys Ys 3 (5a,(5d,(5h <5 p , (5 S 5t <5pa><5pd <*sa><*sd e ea eb £c e x ,e y ,e z e1 Asy X Amax v

constant; activation volume thermal expansion coefficients of the glassy, rubbery state volume fraction of microvoids initial volume fraction of microvoids microvoiding rate microvoiding rate constant van der Waals surface energy craze surface tension energy of plastic deformation per unit area surface energy obtained surface energy solubility parameter (SP) SP contributions to polarity, dispersion force, hydrogen bonding SP of a polymer, solvent total SP ^ a A of a polymer ^ a A o f a solvent (overall) strain; creep strain strain of amorphous regions strain to fracture critical crazing value of tensile strain tensile strain components first respective principal strain strain distribution around crazes extension ratio maximum extension ratio Poisson's ratio

List of Symbols and Abbreviations

ve vx Q o <jb

o1,o2,o?> T (j) co co0

density of entanglement points density of chemical cross-links density; radius of the notch tensile or external stress stress bias (difference of m a x i m u m a n d m i n i m u m principal stresses); stress to fracture critical tensile stress for craze initiation stress in the radial direction in an U-notched bar axial tensile stress in a notched b a r average axial tensile stress tensile stress components uniaxial yield stress characteristic tensile stress behavior in uniaxial direction after yield maximum of Gy for x < s* critical stress for craze growth respective principal stresses octahedral shear stress crack tip opening displacement notch angle constant

ABS CED DZ ESR HIPS PA PAMSS PBT PC PE PEEK PEI PEOB PES PET PMMA POM PP PPO PS PSAN PSMLA PSMMA PSU

acrylonitrile-butadiene-styrene copolymer cohesive energy density deformation zone electron spin resonance high impact polystyrene polyamide poly(a-methylstyrene styrene) poly(butyl terephthalate) polycarbonate polyethylene poly(ether ether ketone) poly(ether imide) poly[p-(2-hydroxy ethoxy) benzoic acid] poly(ether sulfone) poly(ethylene terephthalate) polymethylmethacrylate poly(oxymethylene) polypropylene poly(phenylene oxide) polystyrene poly(styrene acrylonitrile) polystyrene maleic anhydride) polystyrene methyl methacrylate) polySulfone)

GC GT Gt (it Gx,Gy,Gz GY G'Y 'y(max)

703

704

PTBS PVC PVK PVT PaMS SAN SAXS SBR SEM SLF SP TEM WLF

15 Crazing and Fracture of Polymers

poly(ter£-butylstyrene) poly(vinyl chloride) polyvinylcarbazole polyvinyltoluene poly(a-methylstyrene) styrene acrylonitrile small angle X-ray scattering styrene-butadiene rubber scanning electron microscopy slip-line field solubility parameter transmission electron microscopy Williams-Landel-Ferry equation

15.2 Crazing: Initiation and Growth

15.1 Introduction Polymers are being used in an ever increasing range of applications. Whereas they were initially used as inexpensive substitutes for metal, ceramics and wood, polymers are now most often used because of their intrinsic properties. These include their optical, electric, magnetic, chemical, or specific strength properties, and their processability. However, in these applications, polymers often fail from the mechanical stresses that they are subjected to. It is therefore essential that we are familar with their mechanical behavior, and understand their physical and chemical bases. This might allow us to develop mechanically better polymers on the one hand, and to make allowance for their properties in the design process. The viscoelastic behavior (Chap. 9 of this Volume) and the yielding behavior (Chap. 10) of polymers are treated elsewhere in this volume. The present chapter is concerned with the failure, usually by fracture, of polymers. To understand the failure process, it is essential for us to realize that the mechanical behavior of a given polymer depends strongly on such factors as the molecular weight, its distribution, the degree of crystallinity, the thermal and processing history, the existence of anisotropy, etc. It also depends on such extrinsic factors as the flaw content, and the environment that the polymer is exposed to. Without a thorough knowledge of these materials aspects, it is virtually impossible to understand or predict the failure behavior of a polymer. Some of these aspects are discussed in this chapter. The reader interested in further details should consult the relevant chapters. In all high molecular weight, thermoplastic polymers, fracture is always preceded by crazing. We therefore begin our

705

discussion with this most important topic in Sec. 15.2. We will then proceed to discuss fracture itself in Sec. 15.3. The fundamental concepts of fracture mechanics, which are extremely important for analyzing and quantifying fracture, are treated in Vol. 6 of this Series.

15.2 Crazing: Initiation and Growth The word "craze" originally meant a pattern of fine cracks often found in the glaze on pottery. The phenomenon of craze generation is called crazing. Crazes in polymeric materials, however, are different from those observed on the surface of pottery. It is a characteristic of thermoplastic polymers. It represents an intermediate state between microscopic fracture and yielding, yet it can be considered a precursor to macroscopic fracture. Crazing plays important roles in the fracture of thermoplastic polymers, especially in such cases as brittle fracture, environmental degradation, and fatigue failure. One can say without exaggeration that the fracture of the majority of polymers cannot be explained without mentioning crazing. 15.2.1 What is Crazing?

Crazes exhibiting the pattern shown in Fig. 15-1 are often observed on the surface of glassy polymers such as polystyrene (PS) and polymethylmethacrylate (PMMA). These materials become whitened under additional stress and lose their transparency, the result of the generation of a large number of crazes that are much finer than those in Fig. 15-1. It was believed, due to their appearances, that crazes were microcracks occurring on the surface of a sample. Sauer and co-workers (Sauer et al., 1949; Hsiao and Sauer, 1959) investigated crazes

706

15 Crazing and Fracture of Polymers

Figure 15-1. Crazes occurring on the surface of PMMA.

in PS by optical and electron microscopy and X-ray diffraction techniques and demonstrated that crazes are not simply tiny cracks. However, crazing as a phenomenon was not extensively investigated until about a decade later. Despite extensive investigations, some controversy still exists among scientists about the underlying mechanisms to crazing. The difficulty in uncovering the physical basis of crazing is due to the complexity of the phenomenon, some aspects of which are outlined below: (1) Crazes are observed in high molecular weight glassy polymers except thermosets. They also occur in crystalline polymers such as polypropylene (PP) (Olf and Peterlin, 1974; Friedrich, 1977; Garton et al., 1977, 1978) and polyamide (PA) (Garton et al., 1978). (2) Crazes can be found on the surface and in the interior of a material as well as at crack tips. In glassy polymers, their microstructures are generally independent of where they occur. However, because the mechanical state at the free surface of a material is different from that in the interior, there is ambiguity in the mechanical criteria for craze initiation.

(3) For a given mechanical state, the occurrence of crazing is stochastic in both location and time. It is strongly affected by the distribution of microscopic defects on the sample surface and in its interior; consequently, a statistical treatment is often required in its analysis. (4) Crazes occurring on the surface, in particular, are affected strongly by the environment. Generally speaking, crazing is significantly accelerated by the presence of organic liquids. This phenomenon is called solvent crazing. It is, however, different from stress cracking. The latter refers to the situation where cracking occurs by stress acting alone, usually for a fairly long period of time. Since an induction period is necessary for the generation of a craze, it is affected by the absorption and diffusion of environmental agents. (5) In most cases, crack propagation (subor supercritical) is preceded by crazing ahead of the crack tip. Crazing at the crack tip requires a large amount of energy and therefore it can relax the stress concentration there. It is difficult, however, to relate quantitatively the large amount of energy consumption accompanying crazing to factors such as the polymer species, stress condition, and the environment. 15.2.2 Morphology and Structure of Crazes 15.2.2.1 Amorphous Glassy Polymers

Depending on where crazing occurs, the morphology of crazes can be classified into three types: (1) Surface crazes: This type of crazes often occurs on the surface of a specimen, as seen in Fig. 15-2. They grow in a direction normal to the maximum tensile stress. The length of the crazes can reach about 10 mm, while the thickness

15.2 Crazing: Initiation and Growth

This is illustrated by a schematic sketch and by an interference photomicrograph of a craze (Fig. 15-3). Surface crazes are almost certainly the result of some surface environment. (2) Crazes at a crack tip: Crazes can initiate at a crack tip and then propagated in the direction normal to the maximum tensile stress (Fig. 15-4). The difference between crack tip crazes and surface crazes is that the former often penetrate deep into the sample ahead of the crack, while surface crazes do not. It is also possible, depending on loading conditions, for a craze to grow until it traverses the entire specimen. The thickness of such crazes depends on the crack tip opening displacement and the material and may range from a few to over 10 |iim. (3) Internal crazes: Crazes can occur in the interior of a material. Such crazes are different from surface and crack tip crazes in that their occurrence is due purely to mechanical conditions, i.e., the environment probably plays little or no role in their initiation or growth. Figure 15-5 shows an internal craze in front of a plane strain plastic deformation zone in a notched sample of polycarbonate (PC). The craze plane is normal to the tensile stress direction.

i! '!

'.II f ! i . i ? ' *ii ; I.

I '

;

707

«

Figure 15-2. Surface crazes on a PS film after immersion in ethanol. The stress direction is perpendicular to the craze planes.

(in the stress direction) is extremely small, and is often in the range of 0.1 to several |im. They are also very sharp. The depth perpendicular to the surface is usually in the range of 0.1-0.5 nm.

Figure 15-3. A schematic crosssection of a craze (a) and a corresponding interference photomicrograph of a craze on the surface of a PS film (b). (a)

Ib)

708

15 Crazing and Fracture of Polymers

| Stress direction

0.1mm Figure 15-4. Crazes occurring at a crack-tip in PMMA.

Internal crazes can occur as isolated crazes or in multitude. The whitening of toughened glassy polymers - high impact polystyrene (HIPS), poly(acrylonitrile-butadiene-styrene copolymer) (ABS), and other styrenics - and sometimes toughened semi-crystalline polymers under stress is often the result of the occurrence of a large number of crazes inside the material. Multiple crazing is therefore an important toughening mechanism. The fact that a craze is not a crack can easily be demonstrated. Even if a craze occurring at a crack tip extends through the

Figure 15-5. A craze formed at the tip of a plane strain plastic zone in front of a notch in PC.

entire cross-section of a specimen, the specimen will still exhibit some load bearing capacity. Crazes are also different from shear localization in that they contain voids. Experimental techniques for the study of the structure of crazes include optical methods, which can estimate the fraction of voids by measuring the refractive index, X-ray and electron beam diffraction methods, which can measure the size of voids in a craze and characterize the orientation of the craze fibrils, and TEM and SEM observation methods, which can directly observe the structure of crazes in thin films and in bulk. Observations on the structure of crazes are too numerous to be described in detail here. The interested reader is referred to the list of General Reading given at the end of Sec. 15.6 if more details are desired. The following deductions can be made regarding the typical craze structure: (1) Crazes consist of elongated voids and fibrils with orientation in the major principal stress direction. This internal structure is therefore like that of a sponge. Surface crazes, internal crazes, and crazes at the crack tip all have this spongelike feature in common. This

15.2 Crazing: Initiation and Growth

means that the craze structure does not depend significantly on the environment. (2) The void content is usually in the range of 50-80%. The diameter of each void is about 20 nm. The voids are interconnected, forming a continuous network interrupted only by craze fibrils. (3) Craze fibrils are highly oriented along the deformation direction when under load. Their diameters are about 6 45 nm.

709

An example of surface crazes formed in PC that has been in contact with ethanol 1 is shown in an SEM micrograph (Fig. 15-6). This micrograph of a section perpendicular to the craze plane was obtained by breaking the specimen in liquid nitrogen. Figure 15-7 is a direct transmission electron micrograph of a craze in a PS thin film which has been immersed in methanol, which shows clearly the characteristic craze features described above. 15.2.2.2 Semicrystalline Polymers In contrast to crazes in amorphous polymers, the morphology and structure of crazes in semicrystalline polymers have received attention only recently. One of the basic problems posed by crazes in semicrystalline polymers is their complicated microstructure: a combination of crystalline and amorphous phases which constitutes many weak spots for craze nucleation. Another problem is that at room

Figure 15-6. SEM micrograph of a section of a craze in PC.

1 Loss of solvent may change the microstructure of a craze. The present examples, however, are adequate representations of both dry and solvent-induced crazes at these scales.

Figure 15-7. Ethanol-induced crazes in a PS thin film.

710

15 Crazing and Fracture of Polymers

temperature many crystalline polymers are above their glass transition and they tend to fail macroscopically in a ductile manner. Consequently, it has long been believed that crazing does not play an important role in the deformation and fracture of crystalline polymers. The appearance of crazes formed above glass temperature tend to be indistinct because of the lack of transparency; they resemble fine cracks because the thickness of individual crazes is usually more than 10 times larger than those observed for amorphous polymers. However, the appearance of the crazes is strongly affected by the crystalline texture which causes the direction and magnitude of local stresses to differ from that of the average far field stress. Figure 15-8 shows typical crazes formed in PP above its glass temperature. The length of individual crazes is shorter than that typically observed in amorphous polymers. These crazes grow preferentially along trans-spherulitic paths which run through the centers of the spherulites. The internal structure of these crazes are shown in SEM micrographs in Fig. 15-9. They are basically similar to those of crazes found in

amorphous polymers, although in this case the craze fibrils are much thinner. They span and interconnect the walls of what would otherwise be a microcrack. When crazes are formed in crystalline polymers composed of much finer spherulites at temperatures above their glass transitions, such as polyethylene (PE) at room temperature, they are short and irregular because the growth of these crazes is confined within a few spherulites and are more affected by local stress directions and the local spherulitic structure (Narisawa and Ishikawa, 1990). The short and irregular crazes are also found in the higher molecular weight crystalline polymers in which the amount of entanglement of amorphous tie chains in interlamellar and interspherulitic regions increases with reducing crystallinity and with increasing molecular weight (Friedrich, 1983). The crazes formed at temperatures below the glass transition temperature in semicrystalline polymers, such as nylon-6 at room temperature, are usually very few in number and are long compared with those formed above the glass transition temperature. They grow normal to the

Figure 15-8. Crazes formed in crystalline PP above its glass transition temperature.

15.2 Crazing: Initiation and Growth

711

Figure 15-9. SEM micrographs showing the internal structure of a craze in PP.

Figure 15-10. The effect

of temperature on crazes formed at the notch tip of a notched specimen of nylon-6 subjected to three point bending. Tg is between 313 K and 323 K. 303 K

313 K

323. K

major principal stress direction, ignoring the spherulitic structure as if it were in an amorphous polymer. Nevertheless, their internal structure is quite similar to those formed above the glass transition temperature. The effect of temperature on the craze pattern in nylon-6 is shown in Fig. 15-10. The glass transition temperature of nylon6 is between 313 K and 323 K. Thus, in general, the morphology and structure of crazes in crystalline polymers are affected by the crystalline structure, but the interspherulitic boundaries or the centers of the spherulites are not necessarily the favored sites for craze nucleation.

333 K

15.2.3 Initiation of Crazes 15.2.3.1 Initiation Model of Crazes

The structure of crazes indicates that its occurrence is accompanied by localized cavitation and fibrillation of molecular chains along the deformation direction. By way of contrasting with shear yielding, which involves no cavitation, Sternstein and Meyers (1974) termed crazing "normal stress yielding". Furthermore, they point out that strong constraints must exist in both of the normal directions (X and Z) in order to generate cavitation. These two types of yielding are illustrated in

712

15 Crazing and Fracture of Polymers

Fig. 15-11. In the case of internal crazes in which the plane strain condition is satisfied, it is easy to understand that crazing is generated due to normal stress yielding. However, on the surface of a material, since there is no such constraint in the Z-direction (which is perpendicular to the surface), one needs to consider a mechanism that includes, firstly, a certain type of deformation occurring on the surface of a material, and secondly, normal stress yielding occurring at a point immediately beneath it. The initial deformation on the surface could be microscopic yielding initiated by surface flaws and defects. The sequence of events leading to surface crazing can be illustrated by an experiment by Wellinghoff and Baer (1975) on the deformation of a polystyrene (PS) thin film by electron microscopy. They made the thin film of PS by casting a dilute solution of it onto a thicker, oriented poly(ethylene terephthalate) (PET) film. This sandwich was then deformed in a jig. The deformation was transferred to the PS film via the stronger PET film. The PS film was then stripped off and observed in the TEM. They found that micronecking oc-

curs first in regions about 30 nm in size, which are randomly dispersed in the material. These regions are in a dilated state although no voids greater than 1.5 nm could be found. Since further deformation of the micronecks was not possible because the elastic regions surrounding the micronecked regions impose constraints in the X- and Z-directions, cavitation and fibrillation of these regions then followed. These observations are depicted schematically in Fig. 15-12. Subsequent cavitation occurs in those regions where plastic deformation has proceeded to the greatest extent. Under such conditions the voids could grow to 10 nm in diameter. A similar observation has been made earlier by Wyzgoski and Yeh (1974). We note here the critical role played by flaws or microdefects in initiating the micronecks. We note also that localized shear plasticity is considered to be a key mechanism in the model on craze initiation proposed by Argon and coworkers which will be discussed in a later section. The observations by Wellinghoff and Baer appear consistent with Argon's model. An alternative mechanism that does not require the occurrence of local shear plasticity has been proposed by Kausch, which will also be discussed in a later section. 15.2.3.2 Mechanical Criteria of Crazing

(a)

(b)

Figure 15-11. A schematic representation of the deformations involved in shear yielding (a) and normal stress yielding (b).

A number of proposals on the criterion for crazing have been reported. They are summarized as follows: (1) Critical stress criterion: This criterion states that crazing occurs when the tensile stress reaches a certain value ac. The critical stress can be measured by optical techniques in a constant tensile stress experiment, i.e., in creep. In reality, the crazing stress depends on the strain rate and temperature, and can

15.2 Crazing: Initiation and Growth PET film Stress concentration region

|E| Shear yielding |

Voids of 10 nm in average

H I Porous micro-neck voids <1.5 nm

Figure 15-12. A schematic representation of shear yielding and crazing in a thin film of PC deformed by stretching it on a PET thick film substrate (Wellinghoffand Baer, 1975).

also be affected by molecular orientation and the environment. The usefulness of this criterion is therefore mainly macroscopic and empirical. (2) Critical strain criterion: This criterion states that crazing occurs when the tensile strain approaches a certain value ec. The critical strain can be obtained either by observing the crazing process in a sample strapped to a specimen holder which has the shape of a quarter ellipse and is immersed in a chemical liquid (or air), or by carrying out a bending test in a chemical liquid. The critical strain criterion is also macroscopic and empirical, but it is preferable to use this criterion to explain why crazing occurs even though the stress is gradually relaxed (Bernier and Kambour, 1968).

713

(3) Criterion based on fracture mechanics parameters: This criterion is useful primarily for crazing at a crack tip, which can be obtained by determining the critical stress intensity for the appearance of crazing Kcmzc or strain energy release rate gcraze. In addition, X craze and gCTaze are sometimes used as parameters determining the growth rate of crazes (Andrews and Bevan, 1972). (4) Dilative stress criterion: This criterion proposes that the mechanical criterion for crazing should include the dilative stress component since it is required if cavitation does occur in crazes. Sternstein and co-workers (Sternstein and Meyers, 1974; Sternstein et al., 1963) proposed that the segmental mobility of the polymer will increase due to dilative stresses and that at a certain point this activated mobility triggers the cavitation and causes molecular segments to orient along the maximum stress direction; as a result, a characteristic structure is formed. The criterion can be expressed as B

(15-1)

where oh is the stress or stress bias required for orienting the fibrils in crazes, It is the first stress invariant and i\ = GX + G2 + G3 > 0, where al9 a2, 0 (where p is the hydrostatic pressure or mean stress) becomes a criterion for crazing. In a plane stress condition ( <72,<73 = 0), ah = a1 — o2. The above equation thus becomes (T1 — <72 = A

+

B

(15-2)

714

15 Crazing and Fracture of Polymers

The regions of crazing and shear yielding obtained under torsion-tension stress conditions for PMMA are shown in Fig. 15-13. The agreement is seen to be excellent. However, a difficulty in the criterion expressed by Eq. (15-2) is that <7b = o1 — o2 is in fact proportionate to the shear stress, and that the strain direction is the maximum shear stress direction. Yet, crazes grow in the direction normal to the maximum principal stress. This obviously cannot explain the normal stress yielding which was originally proposed by Sternstein himself. The crazing criterion proposed by Sternstein was further criticized by Breuer (1979, 1985), who pointed out that it does not yield a significant failure surface in the principal stress space or a significant failure curve in the octahedral stress diagram. Its validity is therefore limited to the plane

(15-1) Crazing and shear yielding

4.0 ^NPure shear \ \

\

Crazing

A >

\ *r 3.0

o

Shear yielding

V N

= 2.0

\ y^—Von Mises shear yielding locus

1.0 -

No crazing no shear yielding \ \ \ \ \ \ \ \

0

-3.0

-2.0

-1.0

a2 d0" 9 N/m2)

Figure 15-13. Effect of biaxial stress state on the mode of deformation of PMMA (Sternstein and Meyers, 1974). (o) No crazing or shear yielding; (•) crazing only; (A) shear yielding only; (A) crazing and shear yielding. The temperature is 60 °C.

stress case. Therefore, Breuer considers this criterion to be useful for describing surface crazing only. Noting some of these difficulties, Oxborough and Bowden (1973, 1974) performed experiments on PS in the tension-compression quadrant (a3 = 0). They found that the data could be fitted to an equation of the form —V(7j

=

(15-3)

where v is Poisson's ratio, and A and B are temperature- and time-dependent constants. Since a1 — va2 = Eel9 Oxborough and Bowden proposed that the craze criterion can be written as Y'

(15-4)

Although Eq. (15-4) has the same form under plane stress as that proposed by Sternstein (Fig. 15-13), the physical implication here is more explicit, i.e., it is a critical strain criterion. Furthermore, the strain direction is normal to the craze plane which is consistent with experimental observations. Another advantage of this criterion is that it is three dimensional. However, it should be noted that Breuer (1979) also objects to this criterion on the basis that a critical strain criterion cannot be circumscribed by a unique three-dimensional failure boundary. Using the same assumption that cavitation from dilative stress is the primary cause of crazing, Gent (1970, 1973) proposed a quite different criterion. In this criterion, the dilative stress decreases the glass transition temperature of materials to the level of the testing temperature. He further proposed that it is the transition, i.e., from the glassy state to the rubbery state, that makes cavitation possible. The dila-

715

15.2 Crazing: Initiation and Growth

tive stress for crazing pc is given by

- No crazing

_20

o

Pc = P(Tg-T)

with p =

cr

^ ^ (15-5) cg

2

(15-6)

Given the experimental conditions described, Zhurkov probably detected scattering from crazes that had already been formed, rather than from craze nuclei. Recent results from Kramer and co-workers using synchrotron radiation found no evidence of precursor voiding.

•o

o-

18 \

where ag, and ar are the thermal expansion coefficients of the glassy and rubbery states, respectively, and cg, and cr are the compressibilities of the two states. In ordinary polymers, fi is about 5 MPa • °C~l, thus for materials with glass transition temperature of 100-150 °C, pc is in the range 400-500 MPa. According to Gent's proposed criterion, crazing should be suppressed if an external hydrostatic pressure of 400-600 MPa is applied to a material. Such an experiment was performed by Ishikawa and Narisawa (1983). Their experimental data, given in Fig. 15-14, indicate that crazing can be eliminated at a much lower pressure. This observation is obviously inconsistent with the calculation based on Eq. (15-5). Argon and co-workers (Argon, 1974; Argon et al., 1977; Argon and Hannoosh, 1977) proposed that voids with diameters of about 10 nm may first occur in the material because of chain breakage due to the mechanical stress. Their assumption is based on SAXS results obtained by Zhurkov 2 (Zhurkov et al., 1969 b). Furthermore, he regarded voiding as a stressdependent kinetic process, and he assumed the voiding rate dependence on temperature to obey Arrhenius' equation. That is, the microvoiding rate /? can be expressed by: kT

Quenched sample

/

crazing / — S l o w cooled sample

o

0

s

10 20 30 40 50 60 External hydrostatic pressure (MPa)

70

Figure 15-14. Effect of external hydrostatic pressure on internal crazing in PC at 23 °C. Crazing is completely suppressed when sufficiently high pressure is applied (Ishikawa and Narisawa, 1983).

Elastic area

Surrounding area of microvoids

Figure 15-15. Argon's model for craze initiation assumes plastic dilatation of microvoids (Argon et al., 1977).

where fi0 is a constant, AG (T, p) is the activation enthalpy, T is the temperature, and k is Boltzmann's constant. AG depends on the dilative stress p and the octahedral shear stress t. Consequently, the number of microvoids increases linearly with time, and will become, after time t,

[-^TT4]

(15-7)

Accordingly, once the fraction of voids in a material reaches a certain value, the microvoids (shown in Fig. 15-15 by a circle with a radius r) will plastically expand themselves as they unload the elastic stress in

716

15 Crazing and Fracture of Polymers

the surrounding area. By assuming a material to be strain-hardening, the characteristic behavior after yield can be given by <7v = <7Y + A ( X2 — y

(15-8)

where oY stands for the uniaxial yield stress, A is the extension ratio, and A is a constant. On the basis of the Mises criterion, the stress in the radial direction can be obtained by solving the equation of equilibrium, which gives the stress p required for these microvoids to plastically expand and transform into the nuclei of crazes: (15-9)

As defined in Fig. 15-15, if the average diameter of the microvoids is a, and the average distance between the microvoids is b, then the volume fraction P of the microvoids is P = (a/b)3. ft{ is the initial volume fraction of the microvoids at the instant of plastic expansion. In the case of no strain hardening, Eq. (15-9) can be simplified as follows: 2(7 Y

(15-10)

In reality, the actual dilative stress required for the nucleation of crazes, when a material is in a state near yield, is much lower than the value estimated from Eq. (15-10) (Argon and Hannoosh, 1977). Once they arrive at a certain size, the nuclei that have homogeneously expanded in the initial stage will deform preferentially in the direction normal to the tensile stress axis, because of the greater stress concentration in this direction. Fibril orientation will also become easier. The overall shape of the craze will become oblate and grow in the normal direction. The Argon criterion is quite complicated because many param-

eters need to be determined when it is to be used for prediction. However, it is the first theory that gives an explanation for the time-dependence of crazing. In addition, the concept that the voids can plastically expand as the surrounding stress is elastically unloaded is based on the observation of void expansion in ductile metallic materials. The difference between the two cases lies in the addition of the mechanism, in the case of polymers, wherein the voids do not evolve into a porous patch but instead become crazes by collective deformation. This mechanism is possible due to the strong strain hardening capability of high molecular weight polymers. Kausch proposed an initiation mechanism that is similar to that of Argon but does not require the use of plasticity concepts (Kausch, 1976). This model takes into account molecular structure, chain rigidity, conformational changes, and intermolecular interactions. The initiation process (Fig. 15-16) itself is proposed to consist of three sequential steps, viz.: (1) Decohesion of molecular coils in regions of low entanglement (B1 in Fig. 15-16), which forms unstable microvoids, i.e., these microvoids could collapse due to elastic strain. (2) Anelastic deformation of adjacent molecular coils which are more entangled (B2). (3) Transmission of strain to adjacent coils which stabilizes the microvoids. With sustained or increased load, the microvoids coalesce and leave fibrils which connect the opposite craze surfaces. Narisawa and co-workers (Ishikawa et al., 1977; Narisawa et al., 1980 a) considered more simple reasons for cavitation. They proposed that cavitation before crazing must be preceded by some type of shear yielding. For example, if yielding occurs in the localized area in front of a notch in a

15.2 Crazing: Initiation and Growth *

•

•

^

-

—

"

—

*

—

.

-

-

"

• '

717

-

•

v

>

I X

(

[

5I

^ \

j

/

Y

/

I

-

\

—-r

r -4—/

•

'

-

"

i*

J

>

X

\

>

-

/

•

3« *

\

•k

10 nm IPMMA M = 60000 g/mol.

-

.

nm. a = 4.A nml

-" ,

-

-

Figure 15-16. Kausch's model of craze initiation and fibril formation assumes that in an elastically strained region A, there exist regions B, which have insufficient interpenetration of molecular coils 1 and 2. B : are then possible nucleation sites. B2 are regions where there is sufficient interpenetration of molecular coils 3. The stresses in Bt are transferred to B 2 . This process continues to propagate to surrounding areas. (Kausch, 1976.)

material, the elastic area that surrounds the yielded area will act in such a way that the plastic shear yield stress will be raised. If this occurs in a sufficiently thick specimen (the plane strain state), a large dilative force in front of the yielded area will act to initiate microvoids. (For a detailed calculation of the dilative force, refer to Hill's slipline field theory described in the section on ductile fracture.) Most of the criteria proposed so far pertain to crazing on the surface. The effects of dilative stresses are implicit in these criteria. However, since no dilative stress can physically exist on the outermost surface of a material, experimental observations would seem to be contradictory to most of the proposed criteria. But, if one considers the flaws and defects on the surface as small notches, then a dilative stress can still be obtained. This is possible because local-

ized shear yielding can occur first in front of flaws and defects on the surface of a specimen, and if the front radius is small enough so that a triaxial stress field is generated inside the specimen, then this stress field induces a stress concentration and eventually leads to cavitation (voiding). The time-dependence of crazing can also be explained on this basis if we consider the fact that shear yielding that must precede voiding is a kinetic process. 15.2.3.3 Crazing and Environment

Crazing can be accelerated significantly under the influence of organic liquids. The effects of environmental liquids can be classified inter two categories in relation to the crazing mechanism: (1) the environmental liquids accelerate both the shear yielding process, which precedes crazing,

718

15 Crazing and Fracture of Polymers

and the fibril orientation process in crazes; (2) the environmental liquids decrease the surface energy during cavitation. The first category describes the role of environmental liquids as plasticizers due to absorption. Generally speaking, the glass transition temperature Tg of polymers will decrease because of the swelling due to the absorption of chemical liquids; consequently, it may be possible for molecular segments to move at lower stresses than if there was no swelling. Figure 15-17 shows the decrease of Tg of PS after being swollen to equilibrium in various chemical liquids versus the critical strain of crazing gc (Kambour et al., 1973). The ec for PS, which has been plasticized by chlorobenzene in advance, is also plotted in the same figure. The results clearly indicate that lowering the Tg will have identical effects on crazing, no matter what method is used to swell the sample. It has also been shown for materials other than PS that accelerated crazing is a combined effect of the decrease in Tg due to swelling and the interaction between the material and chemical liquids. The crazing behavior at crack tips exhibits exactly the same behavior.

100 90 80 70 60 50 40 30 20 10 0-10-20-30

Figure 15-17. Decrease in Tg of PS due to solvent absorption, and its effect on the critical strain for crazing (Kambour et al., 1973). (o) Swollen samples in various chemical solvents; (•) extended in air mixed with PS-dichlorobenzene.

It takes a long time to swell a material until equilibrium is reached; moreover, it is difficult to simultaneously measure the Tg. For those chemical liquids that have slow absorption rates, the time to reach equilibrium becomes especially long. Therefore, in order to evaluate accelerated crazing by a chemical liquid, it is convenient to quantify the interaction between the material and the chemical liquid by the solubility parameter (SP). The solubility parameter is often denoted by 5, and 3 = ^JCED where CED is the cohesive energy density of the material or the solvent. Figure 15-18 schematically depicts the relationship between the critical crazing strain sc and the SP value. Let the SP values of polymer and solvent be 3p and 3S, respectively. When 3p becomes nearly equal to 3S, the polymer will very likely become dissolved in the liquid or, if not dissolved, will be cracked intensely. With the increase in the difference between 3p and (5S, crazing begins to occur, and sc will also increase and the chemical-acceleration effect becomes less significant. The relationship depicted in Fig. 15-18 is especially useful for the situation in which a nonpolar material and a nonpolar solution are involved. In a situation where polarity or hydrogen bonding is involved, however, a different relationship might be exhibited. In Fig. 15-19, for instance, the crazing ability of the solvents are expressed in terms of the SP values and the hydrogen bonding parameter with respect to a polar polymer, PMMA, in a two-dimensional plot (Vincent and Raha, 1972). If the SP value contains contributions from the dispersion force, polarity, as well as hydrogen bonding, denoted as 3d, 3a and 3h respectively, then it is useful to consider the effective total solubility parameter 3t (Henry, 1974): <5? = <5d + <5a2 + <5h

(15-H)

15.2 Crazing: Initiation and Growth

719

We also need to consider the enthalpic contribution from each of these components, which is proportional to:

Crack or dissolution Craze

V0(Sp-Ss)2

Solubility parameter $s (\/J/mJ

Figure 15-18. Schematic relationship between the critical crazing strain of a polymer and the solubility parameter SP of solvents.

U

(15-12)

where Vo is the molar volume of a solvent. In the case of polymers with polarity, it has also been proposed (Jacques and Wyzgoski, 1979) that it is reasonable to consider the contributions from dispersion 3d and polarity (5a separately and express them as Void,* ~
22 20 3 18

w w

'V

v

o en

•-§

o _o
1.5

u 12

gMO

10 11 12 SP value

13 U

15 16

Figure 15-19. The combined effect of hydrogen bonding parameter and SP on crazing behavior of PMMA (Vincent and Raha, 1972). (•) Dissolution, ( x) crack, (v) small crazing ec, (A) large crazing sc.

Figure 15-20. A crazing strain map for a polar polymer (PC-) polar solvent system (Jacques and Wyzgoski, 1979).

720

15 Crazing and Fracture of Polymers

crazes on the surface of a material at sufficiently high pressure (Matsushige et al., 1975). It is therefore possible that results from some previously reported mechanical experiments conducted under superimposed hydrostatic pressure may contain artefacts. The structure of crazes generated also depends on the environment itself. A combination of a material and an environmental agent that can lower the Tg significantly will exhibit larger volume fraction of voids (Yaffe and Kramer, 1981).

melt state and the melt as if it were a rubber network, the rubbery plateau modulus GN can be related to the entanglement molecular weight Me by

15.2.3.4 Crazing and Polymer Structure

where NA is Avogadro's number. The contour length le between entanglements is

The effect of molecular structure on crazing behavior is of utmost interest, and has been explained in detail by Kramer and his coworkers in terms of molecular entanglements. Although the direct relationship between the chemical structure and intermolecular interactions which result in networklike behavior, i.e., entanglements, is not explained by these workers, the microscopic model produced by them is both intuitive and consistent with experimental observations. Kramer's model is most clearly explained by two reviews written by himself (Kramer, 1983; Kramer and Berger, 1990). Due to space limitations, only a brief summary of the part of the model that deals specifically with molecular entanglements is presented in this and a later section. The reader interested in more details and other aspects of the model should consult the reviews and original references cited therein. A key feature of Kramer's model is that, in order for a craze to be stable, there must be molecular entanglements. Thus, the deformation of a stable craze can be related to the deformation of the molecular segment between entanglements. For polymers with a sufficiently high molecular weight so that interactions occur in the

QRT

(15-13)

where Q is the density. In such a network, the density of chains between entanglement points, ve, is ve = g^

le = l0—

(5-14)

(15-15)

where /0 is the length of the repeat unit and M o is the molecular weight of each repeat unit. If the polymer chain can be treated as a Gaussian coil, then the average distance between entanglements, d, is =

C(Me)112

(15-16)

where C is a constant. It is clear from these definitions that the maximum extent that the chain between entanglements can be stretched, Amax, is X

™ = -d

(15-17)

The actual extension of craze fibrils X can be calculated from densitometry measurements on transmission micrographs of crazes formed in thin polymer films. The results of such experiments on a number of polymers and copolymers are shown in Fig. 15-21 plotted against the theoretical Xmax. These results show that the general trend of X agrees very well with the Xmax calculated. The deviation at high X is suggested by Kramer to be due to chain scission, a process to be explicitly considered in a later section. As the entanglement den-

15.2 Crazing: Initiation and Growth

721

10

4 -

3 ~

2 ~ PSAN2 1 -

sity increases (i.e., as Amax becomes smaller), the polymers tend to deform by shear yielding (at temperatures well below Tg) because the crazing stress at the craze tip increases beyond the shear yielding strength of the material. The craze to shear yield transition can then be understood in terms of the relative ease of shear yielding vs. crazing as the entanglement density is changed. This has been demonstrated in a series of miscible blends of polystyrene and poly(phenylene oxide) (PPO) (Donald and Kramer, 1982) where the transition to shear yielding behavior when the fraction of PPO is increased is explained in terms of a gradual increase in the average entanglement density. 15.2.3.5 The Effect of Molecular Weight

Since entanglements are so important in the deformation of crazes, it is not surpris-

Figure 15-21. Experimental extension ratio of crazes in various homo- and copolymers plotted against the theoretical maximum extension ratio (Kramer, 1983).

ing that the molecular weight strongly influences the number and configuration of crazes. At low molecular weights, both the width and length of crazes are large but their number density is small. The number of fibrils inside such a craze is also very low (Brady and Yeh, 1973). In contrast, at high molecular weights, a number of thin crazes are generated (Sato, 1965; Fellers and Kee, 1974; Rudd, 1973). It was generally thought that the more chain entanglements there are, i.e., the higher the molecular weight is, the more difficult it is for fibrils to form within a craze. The crazing stress might therefore be expected to increase with increasing molecular weight. Nevertheless, an examination of the surface crazes of PS indicates that there is no molecular weight dependence as long as Mn > 2M e (Fig. 15-22). When Mn < 2M e , no crazing was observed and only load-in-

722

|

15 Crazing and Fracture of Polymers

30

S 20 Zn 10

-M.

0

5 6 7 8 9 10 1 1 1 2 J 3 U Number average molecular weight Afnx10~*

15

Figure 15-22. Molecular weight dependence of crazing and fracture stress of PS (Fellers and Kee, 1974). (•) Failure stress, (•) crazing stress.

18 9-"

from crazing to shear yielding (Henkee and Kramer, 1984; Glad, 1986; Berger and Kramer, 1988). The fact that crazing stress is independent of molecular weight if M n > 2 M e has been interpreted to mean that crazing is not accompanied by any large scale segmental movement. This is consistent with Kramer's model presented in the last section. An important corollary of this observation is that craze growth must take place primarily by drawing fresh chains from the uncrazed surrounding into the craze once Amax is approached. A discussion on craze growth is the subject of the next section. 15.2.4 Crazing: Growth and Theory 15.2.4.1 General Characteristics of Craze Growth

Length (mm)

Figure 15-23. The growth of both length and thickness of crazes at a crack tip in PMMA (Kramer et al., 1978).

duced fracture occurred. Where crazing occurs, failure starts from these crazes, and the failure stress increases as the molecular weight increases. This implies that the stress required for extending the fibrils to their critical strains increases with molecular weight. The effect of molecular weight and entanglement density can be extended to the case of chemically cross-linked polymers. If a linear polymer is gradually cross-linked, the deformation mode can gradually shift

Both the length and the thickness of crazes increase with loading time, as illustrated by results on PMMA (Kramer et al., 1978) in Fig. 15-23. Since the thickening process eventually leads to fracture, this process is discussed in conjunction with craze fracture in a later section. In the present section we focus on some general characteristics of craze growth, especially in the longitudinal direction. Unless otherwise indicated, the term growth rate refers in this section specifically to longitudinal growth. Surface crazes do not continue growing indefinitely; instead, their dimensions will approach a certain equilibrium as time elapses. It has also been found that crazes evolve with a constant length to thickness ratio (Verheulpen-Heymans, 1979). The growth behavior of crazes is related to the stress state around them, especially near the tips of growing crazes. Therefore, it can be expected that the stress state of the surface craze is much different from that of the craze at the crack tip. Figure 15-24 shows

15.2 Crazing: Initiation and Growth 50°

40°

30°

20°

10° 0=0

/?=!0

(a) 50° 60°

70°

/,0°

/

30°

20° >^

10° 6-

<> Ay x h >
--——

2/

^--—

A

\

\

80°

/

1I

/

/

723

stress direction coincides with the maximum strain direction, it is not possible to distinguish which one is related to the growth direction of the crazes. To clarify this, one can use an anisotropic specimen in which the molecular chains are oriented in a certain direction, and carry out tensile tests at different directions to determine whether the principal stress or principal strain direction is related to the growth of crazes. Figure 15-25 gives the results of one such experiment (Beardmore and Rabinowitz, 1975). It indicates that crazes do not necessarily grow in the direction normal to the principal stress. The crazes grown in the direction normal to the maximum strain either cease to grow if their tips encounter a shear deformation band (Fig. 1526), or continue to grow, but in a different direction, i.e., along the shear deformation band. We now consider briefly the growth kinetics of crazes in the thickness direction.

(b)

Figure 15-24. Principal stress directions and growth directions of crazes in PC with (a) a hole and (b) a rigid cylinder in the specimen. R is the radius of the hole or cylinder (Miltz et al., 1978).

how the crazes in a PC specimen, which contains a hole or a rigid cylinder, evolve. The maximum principal stress direction is indicated in the figure in order to relate the growth of crazes with the stress direction. It is apparent that crazes grow in the direction normal to the maximum principal stress (Miltz et al., 1978). The same results have been obtained for PMMA (Wang et al., 1971) and PS (Sternstein et al., 1963). From these results, it seems sufficient to simply conclude that crazes grow in the direction normal to the maximum principal stress. However, in the case of isotropic materials, since the maximum principal

Isotropic

65 0

90

Figure 15-25. The effect of orientation on the craze growth direction with respect to the stress axis (Beardmore and Rabinowitz, 1975). a is the angle between the extension direction and the craze plane, 6 is the angle between the orientation direction and the extension direction.

724

15 Crazing and Fracture of Polymers

Figure 15-26. A craze in PC which has ceased to grow due to the formation of a pair of shear bands ahead of it (Narisawa, 1982).

The time dependence of the increase in thickness of surface crazes in PC under a constant load is shown in Fig. 15-27 (Verheulpen-Heymans, 1979). Two possible mechanisms for the growth of crazes in the thickness direction have been proposed: (1) creep of craze fibrils, and (2) gradual transformation of the uncrazed material in the interfacial area into fibrils. Although the kinetics suggests that the creep of craze fibrils is dominant (Verheulpen-Heymans

and Bauwens, 1976 b), electron microscopy observations of craze structure favor the second mechanism for both crazes on the surface and at the crack tip (Chan et al., 1981; Donald et al., 1981; Lauterwasser and Kramer, 1979; Trent et al., 1981). In other words, the fibrils in a craze are stretched at first, stabilized by the entanglements; this process, however, will become more difficult because of the strainhardening effect induced by the advanced

MPa

A

+ & + A

o DA DA

o log t (h)

Figure 15-27. Time dependence of thickness d of surface crazes in PC formed under different conditions (VerheulpenHeymans, 1979). ( + ) No treatment; (A) annealed, (o) thickness after unloading.

725

15.2 Crazing: Initiation and Growth

orientation. As a result, the material at the two ends of the fibrils must be drawn out to relax the stress until it eventually approaches a new equilibrium. Figure 15-28 shows the extension ratio of the craze fibrils at a crack tip in PS and the variation of stress there. In this figure, the increase in the thickness of the craze away from the crack tip is not accompanied by a concomitant increase in the extension ratio. This is consistent with the entanglement model proposed by Kramer and his coworkers (Kramer, 1983). The growth of surface crazes in the longitudinal direction has been reported to be proportional to time (case 1), or proportional to the logarithm of time (case 2). For the surface crazes of PS at room temperature in case 1, the observation results indicate that, except in the very initial period, the growth rate of the crazes can be expressed by (Sauer and Hsiao, 1953) m

(15-18)

where m is a constant, and a0 is the critical stress for craze growth and has a value close to that of the crazing stress. The growth behavior of crazes has also been investigated in a microscopic view for crazing in PMMA and PS at room temperature or lower (Argon and Salama, 1977). In case 2, the notion that craze length is proportional to the logarithm of time can be expressed as = fclogt or d/c

k

df

t

(15-19)

(15-20)

Equation (15-20) shows that the growth rate decreases with time. This kind of behavior has been observed in PMMA (Regel, 1956) and PC which has been plas-

Craze front m

200

150

100 Craze rear (crack tip)

50

0

1

2

3 4 Extension ratio

5

6

Figure 15-28. The extension ratio of craze fibrils at a crack tip in PS and their associated fibril stresses (Lauterwasser and Kramer, 1977).

ticized to various extents (Sato, 1966; Verheulpen-Heymans and Bauwens, 1976 a). k is a constant that depends on the polymer species, the extent of plasticization, and the stress as well as the temperature. According to Kambour (1973), the fact that the growth rate of crazes decreases with time is due to creep of the overall specimen during the growth of crazes. That is, assuming that the growth rate of crazes is inversely proportional to the increase of the creep strain, we have k

(15-21) +e where k is a constant which depends on stress only, and s is dependent on both the stress and time. If we denote k' and e by k = fc'V, and e = Aea°t", Eq. (15-21) becomes df

d/.

dt

1+Ae«atn

(15-22)

where A, a, and n are constants obtained from creep experiments. Since the approximation log(l + Ad1" f)~logt can be made over a wide range of temperatures,

726

15 Crazing and Fracture of Polymers

Eq. (15-22) can be reduced to Eq. (15-21). Indeed, the reason for the growth rate of crazes being inversely proportional to the loading time is that most experiments carried out so far have been either on materials which exhibit significant creep deformation or in conditions under which creep deformation easily occurs. The growth of crazes at a crack tip can be treated by using the stress intensity factor as a parameter since the stress at the crack tip is the driving force. Specifically, for the growth of crazes in environmental liquids, a critical Kc exists below which the growth of crazes will cease. From many experimental results, the length of the crazes in this case (ignoring the very initial period) can be expressed as follows (Miltz et al., 1978; Marshall et al, 1970; Narisawa and Kondo, 1973; Kitagawa, 1972; Kitagawa and Motomura, 1974,1976; Andrews and Levy, 1974; El-Hakeem et al., 1975): lc = A tn>

60

90 120 Time (min)

180

150

(a)

(15-23)

where n' is a constant that is independent of the stress intensity factor, n' ~ 0.5 in most cases for reasons given in the following sections and is related to the mechanism for the growth of crazes.

30

60

90 120 Time (min)

150

180

(b)

15.2.4.2 The Growth of Crazes at a Crack Tip

Consider the growth of crazes which occurs at a crack tip in a liquidlike environmental agent. In this case, growth of the crazes as a whole is governed by the stress intensity factor Ko = a sjnao due to an initial crack with length aQ. The growth curves for the crazes in PMMA and PC in kerosene are presented in (a) and (b) of Fig. 15-29, respectively. When KO>KC, growth continues until the specimen fails. Marshall and co-workers (Marshall et al., 1970) proposed a simple mechanism for the

Figure 15-29. Craze length vs. time for (a) PMMA (b) PC exposed to kerosene and subjected to various levels of stress intensity factor (Narisawa and Kondo, 1973).

growth of crazes due to an environmental fluid in terms of fracture mechanics:

= AknJt

(15-24)

where A is a constant which depends on the viscosity of the fluid and the stress required for craze initiation. The driving force is considered by them to be atmospheric pressure.

15.2 Crazing: Initiation and Growth

Equation (15-24) explains why n' = 0.5 in Eq. (15-23), that is, why the flux rate of the environmental agent determines the growth of crazes. Another mechanism, which resembles that described above, has been proposed. It states that the flow of the environmental agent into the crazes is driven by the much higher capillary pressure arising from the size of the craze voids (Kramer and Bubeck, 1978; Kambour and Yee, 1981). The pressure gradient due to capillary pressure is about 100 times larger than one atmosphere. Consequently, the flow of the environmental agent into the tip of crazes through the voids could be much faster.

727

(a)

(b)

15.2.4.3 Microscopic Mechanisms of Crazing

The ability of fluids to flow into the tips of crazes must be because the voids are interconnected. Two-dimensional electron microscopy observations sometimes give the impression that a craze grows by nucleating single voids at its tip. The reason for the voiding was proposed to be the lowering of the glass transition temperature of the region around the craze tip to a level below the testing temperature due to the dilative stress (Gent, 1970,1973). However, the model proposed by Argon et al. (Argon and Salama, 1976, 1977; see also Vol. 6, Chap. 10 of this Series) shown schematically in Fig. 15-30 provides a more reasonable explanation for the formation of the characteristic structure of crazes and their growth. The basis of the Argon model is the phenomenon known as meniscus instability (Saffman and Taylor, 1958; Fields and Ashby, 1976). It is observed when a regular arced surface is created between air and liquid, or when two rigid plates with a liquid in between are pulled apart, or when a

(c)

(d)

Figure 15-30. A model of craze growth based on meniscus instability (Argon and Salama, 1977).

liquid climbs up a capillary tube. In Argon's model, instead of the liquid, the material at the craze tip is considered to be plastically deformed. Because the extent of the instability is related to the surface energy of the interfaces, when an environmental liquid is

728

15 Crazing and Fracture of Polymers

allowed to flow into the craze in place of air, the surface energy of the interface is lowered and this effectively accelerates the meniscus instability. The mechanism proposed by Argon can therefore explain the environmental effect quite well. The length period of these arcs, /, is determined by the viscosity and the surface energy in the case of liquids, and by the viscoelastic characteristics of the unfibrillated areas and the surface energy of the interfaces in the case of crazes. / at the craze tip is in fact the gap between the fibrils, which, according to Argon's calculation, is about 140 nm with fibrils of 50 nm diameter for PMMA and PS. Electron microscopy observation of the craze tip in PS with specimens of varied thicknesses by Kramer and co-workers (Chan etal., 1981; Donald and Kramer, 1981) found no independent voids at the craze tip, i.e., all the voids are interconnected. Moreover, the latter authors actually observed the fingerlike structure of voids at the craze tip as predicted by the meniscus instability model. Such results led Kramer to conclude that Argon's microscopic model is probably the most reasonable mechanism for the growth of crazes (Kramer, 1983). 15.2.5 Mechanical Properties of Crazes 15.2.5.1 Stress-Strain Behavior The reason why a craze, even though it is about 50% in void content, can still carry load is explained by the fact that the fibrils, which are oriented in the loading direction, can bear stress. Figure 15-31 shows the stress-strain relationship of crazes in PC measured after the specimen has been removed from ethanol (Kambour and Kopp, 1969). The result indicates that the longitudinal modulus of craze fibrils is almost the same as that of uncrazed fibrils.

0

k

12 16 20 2k 28 32 36 40 U 48 52 Strain (%)

Figure 15-31. Stress-strain behavior of crazes in PC formed after exposure to ethanol, then removed. The stress-strain behavior of the uncrazed material is also shown for comparison. (Kambour and Kopp, 1969.)

Moreover, the stress-strain behavior undergoes a yieldlike process after which the strain shows a drastic jump. As can be seen from the loading and unloading curves, the fibrils in crazes have much higher ductility than the uncrazed materials. The fact that the hysteretic loss is larger in crazes than in uncrazed materials should also be noted. A similar stress-strain curve has also been obtained for the crazes in PS (Hoarse and Hull, 1972). In the measurement of the stress-strain relationship of crazes in PC, we note, however, that the removal of ethanol from crazes will cause them to shrink, and plasticizing effects will definitely be involved in the crazes generated under the influence of environmental liquids. Therefore, the results obtained from this experiment do not necessarily reflect the real mechanical behavior of the craze as it goes through the processes of growth and fracture, especially in the absence of an environmental liquid.

15.2 Crazing: Initiation and Growth Monochromatic light

15.2.5.2 Micromechanics of a Craze

One of the methods for measuring the displacement of crazes at the crack tip is the interference pattern method, which is shown in Fig. 15-32. The distribution of displacement of the crazes at the crack tip obtained from the interference pattern measurement is given in Fig. 15-33 for PMMA (Brown and Ward, 1973; Morgan and Ward, 1977; Frazer and Ward, 1978; Weidman and Doll, 1978; Israel et al, 1979; Andrews and Levy, 1974; Young and Beaumont, 1976; Shirrer and Goett, 1980). Similar data have been obtained for PS (Kreuz et al., 1976), in HIPS and ABS (Hoffman and Richmond, 1976; Newmann and Williams, 1980), and PC (Pitman and Ward, 1979; Kambour and Miller, 1976). The patterns in these materials are quite consistent with the prediction from Dugdale's model. An extremely sensitive method - laser holographic interferometry - has been used to directly measure the strain distribution, Ae^, around crazes (Peterson et al., 1974; Krenz et al, 1976). Figure 15-34 illustrates the distribution of strain in a craze at a crack tip in PS and the corresponding stress distribution calculated from the relation oy = E sy. This figure shows crazing in (a) methanol, and (b) rc-heptane. One can see that the shapes of the two distributions are quite different. In the case of rc-heptane, the stress decreases sharply to a value comparable to that in the vicinity of the crack tip. This means that n-heptane has a significant plasticizing effect on the craze fibrils in PS. The stress distribution in this case is also quite different from that predicted by Dugdale's model. A contrasting situation exists for methanol wherein the stress distribution obtained deviates only slightly from that predicted by Dugdale's model. For this reason, the stress at the craze tip

729

Reflected light

tn)

Figure 15-32. (a) A schematic of the interference technique used to measure the thickness of crazes, w is the thickness of the craze at the position of the beam drawn in the figure, (b) An interference fringe pattern of a craze at a crack tip in PMMA. (Doyle, 1975.)

1.00

0.80

0.60

0.40

0.20

0

0

0.20

0.40

0.60

0.80

1.00

Figure 15-33. The normalized distribution in a PMMA craze determined from interference fringe data such as those shown in Fig. 15-32. The abscissa is the position X along the craze normalized with respect to the length of the craze lc. The ordinate is the thickness of the craze W (x) at position X normalized with respect to the width at the crack tip Ao. (Morgan and Ward, 1977.)

730

15 Crazing and Fracture of Polymers

culated stress compares well with that from a finite elements method (Bevan, 1982), although the latter cannot yet give as detailed a profile as from the technique used by Kramer and co-workers. These results show that there is a significant increase in stress at the tip of the craze, depending on the extension of the interconnected fibrils within the craze.

(a) 10 x 10"3

-Before crazing

Q

~ m-3

*•> After crazing

10

2 x 10-

H«-Cra

0 /A

i "0 02 ( U 0 6 08 Distance from crack tip (mm)

(b)

After crazing—1 / •

15.2.6 Craze Thickening and Fracture

\

\

\

-Before crazing /

7 x 10"'

\

Crazes eventually transform into cracks, given sufficient time and stress. To understand this transformation process, we must focus our attention on how the craze thickens and eventually breaks down. Some phenomenological aspects of longitudinal craze growth have been presented in a previous section. We discuss here the phenomenology of craze thickening specifically in the context of their behavior just prior to their transformation into cracks.

10" 0

06 1.2 1.8 IX Distance from crack tip (mm)

Figure 15-34. The distribution of strain in a craze at a crack tip in PS and the corresponding stress distribution. Crazing occurred in (a) methanol and (b) nheptane. (Krenz et al., 1976.)

has to be divided into two steps to calculate the distribution of the displacement; as a result, a modified Dugdale's model has been proposed (Graham et al., 1976). The stress distribution along a single craze can be calculated using either the Fourier transform procedure (Lauterwasser and Kramer, 1979) or the distributed dislocation method provided that the strain distribution along the craze can be determined. Kramer and co-workers were successful in this endeavor using TEM (Lauterwasser and Kramer, 1979). The cal-

15.2.6.1 Kramer's Model for Craze Thickening

A recent model proposed by Kramer and co-workers on craze growth is presented here because of its direct relationship to craze failure and its ability to explain ductile-brittle transitions. A schematic of Kramer's model is shown in Fig. 15-35. In this model, the craze thickens by drawing molecular matter from a certain layer h at the craze interface into the fibrils. The value of the width h can be determined from TEM of the thin polymer film decorated with gold particles, the average distance between which is changed by deformation. It was observed that the active zone h is quite narrow, on the order of the fibril diameters, and is dependent on the local strain rate and the temperature. This experiment further reveals that the stress transforms the material in the thin

15.2 Crazing: Initiation and Growth

731

polymer glass;

Figure 15-35. A schematic drawing of the craze-bulk polymer interface in Kramer's model. The hydrostatic tensions (cro)s and (cro)m form a pressure gradient. (Kramer and Berger, 1990.)

zone h so that it "flows" into the fibrils. While the craze as a whole thickens, the volume fraction of fibrils, vf, remains a constant, hence the extension ratio X also remains constant. Starting from this model, a detailed calculation involving assumptions of power-law fluid behavior, pressure gradients between the top of the fibril and the top of the void surface, and the surface tension there yields a craze fibril flow stress, i.e., the crazing stress of the polymer, S, which is approximately proportional to the square root of the craze surface tension per unit width of the active zone, F/h, viz., 05-25) and = y+ ^

v = v e = vx

(15-26)

where y is the van der Waals surface energy and U is the backbone covalent bond energy, and vx is the density of chemical cross-links. C" is a material dependent parameter that is sensitive to temperature and strain rate. This model allows us to understand the role played by the entanglement density on the crazing stress. Increasing v raises the crazing stress. If craze growth involves chain scission, e.g., when a strand crosses an interface, then the second term becomes important. On the other hand, if the kinetics of disentanglement is very rapid and there are no chemical crosslinks, then the surface tension is entirely

due to the van der Waals force involved in disentanglement, viz., F=y

(15-27)

Kramer calls this "van der Waals crazing". Note that since the term F/h is the energy density near the craze surface, it follows that its square root is the solubility parameter. Equation (15-27) then relates the crazing stress to the solubility parameter of the polymer. A solvent which reduces the van der Waals term would reduce the stress required for crazing to occur. This is, of course, consistent with experimental observations of solvent crazing. An important consequence of differentiating between scission dominated crazing and van der Waals crazing is that it allows us to understand the brittle-to-ductile transitions when chemical cross-link densities are increased and when the temperature is increased. Surprisingly, these concepts also allow us to understand ductileto-brittle transitions with increasing temperature observed in certain polymers. 15.2.6.2 Ductile-Brittle Transitions

Brittle to ductile transitions with crosslink density in the case of normally brittle thermoplastics are easy to explain in view of the model presented above. Some results collected by Kramer's group are presented in Fig. 15-36 to illustrate such transitions. We note that care should be taken in extrapolating such results to thermosets which begin as monomers and eventually

15 Crazing and Fracture of Polymers

732

1 PTBS PSMLA 3 PVT 4 PAMSS 5 PSAN1 6 PMMA 7 PSAN1 8 PSAN2 9 PPO 10 PC CVJ

1.5

_ \ E

1.0

[3 T * Crazes

0.5 Crazes-

; Crazes DZ's

0

1

1

10 I/U10 2 5

20 chains/m3)

form networks with very high cross-link densities without going through the thermoplastic stage. At very high cross-link densities, such thermosets are usually quite brittle. Kramer's results should then be taken to mean that, for low entanglement density polymers, strengthening the network with chemical cross-links will shift the deformation mode to shear yielding. However, if the cross-link density is sufficiently high, then shear yielding once again becomes difficult, and brittle fracture without crazing occurs. In this case, chain scission clearly dominates the fracture mechanism. The effect of temperature on ductilebrittle transitions is somewhat more complex. One normally expects a polymer to shear yield more easily at temperatures near the Tg. Thus, a polymer that is brittle at temperatures well below Tg would undergo a brittle-to-ductile transition as the Tg is approached. In such a polymer, the low temperature craze growth should involve a certain amount of chain scission,

#-DZ'S

Figure 15-36. The true strain ratio £R in crazes and deformation zone (DZs) in various polymers and copolymers plotted against the network strand density v = ve 4- vx. The open triangles and hexagons represent blends of PS and PPO. The filled triangles and circles represent crosslinked PS. (Henkee and Kramer, 1984.)

30

while at high temperatures, disentanglement should dominate. This situation is illustrated by the experiments of Plummer and Donald (1989). In their experiment, crazing stresses for previously undeformed and previously crazed but "healed" (at 130°C for lOmin) PS having molecular weights of 260000 and 1 150000 were determined. The ratio of these two stresses is plotted against the temperature in Fig. 15-37. To explain their results, Plummer and Donald argue that at low temperatures, the initial crazing causes a significant amount of chain scission in the craze, and the subsequent healing is insufficient to reestablish the network. Crazing of the healed glass therefore involves only the van der Waals term. Hence, the initial crazing stress is higher. At high temperatures, crazing is primarily by disentanglement, therefore the deformation history makes little difference to the crazing stress. For the higher molecular weight PS, the scissiondominated crazing continues to higher temperatures. At these high temperatures,

15.2 Crazing: Initiation and Growth

shear yielding becomes relatively easy, therefore a transition to this mode of deformation occurs. In the case of the low molecular weight PS, a transition to disentanglement crazing requiring lower stresses occurs at low temperatures; therefore, a transition to shear yielding does not occur. Some ductile glassy polymers have been reported to exhibit a reverse transition with temperature, i.e., undergo crazing as their Tg is approached (Wellinghoff and Baer, 1978). On the basis of the entanglement model described above, Donald and Kramer (1982) suggested that this phenomenon is due to a transition to disentanglement-dominated crazing at high temperatures. At lower temperatures, crazing would require stresses sufficiently high to cause extensive chain scission, therefore shear yielding would be preferred. This explanation was given strong support by a series of experiments by Plummer and Donald (1989). Their results on poly(ether sulfone) (PES) illustrate these ideas beautifully (Fig. 15-38). In Fig. 15-38 a, the tensile strain and the corresponding deformation mechanisms are plotted against tempera1.6 1.5

_, n.

U

8

O

r 1.3 1.2 -

\

1.1 1.0 20

i

30

40

50

60

70

80

90

100

rra Figure 15-37. A plot of the ratio r of the initial crazing stress to the crazing stress for van der Waals crazing as a function of temperature for PS having Mn of 260000 (•) and 1115 000 (o) (Plummer and Donald, 1989).

733

ture for two PES resins with molecular weights of 47000 and 69000. In Fig. 15-38b, the corresponding craze fibril extension ratios are plotted. At low temperatures, the thin films deform by forming deformation zones (DZs), which is an indication of ductile behavior in the bulk state. The DZs form instead of crazes because the crazing stress is too high - a result of high entanglement density. At higher temperatures, disentanglement becomes easier, and eventually the crazing stress becomes lower than the yielding stress, and brittleness results. The temperature at which the transition to disentanglement crazing occurs increases with the molecular weight because chain-scission dominated crazing persists to higher temperatures in this situation. Similar results have been obtained by this group on PC. 15.2.6.3 Craze Fracture Process

Usually, in the tension-induced fracture of polymers, especially those that craze easily, a number of crazes already exist on the surface of the specimen before catastrophic failure occurs. Fracture then begins with one of these crazes. Two fracture initiation mechanisms are possible in such a situation: One is that fibril breakage inside a craze occurs and a void forms; the void thus formed then expands by joining up with other voids, leading to fracture; another is that the craze itself acts as the cause for creating the stress concentration around it, just like surface flaws and cracks. This is considered to be an initiation mechanism for relatively brittle fractures that occur at low temperatures. It has generally been believed that fibril breakage occurs in the midrib of the craze. However, recent TEM observations by Kramer and co-workers (Yang et al., 1986 a; Yang etal, 1986 b; Kramer and Berger,

734

15 Crazing and Fracture of Polymers

0.05

20

40

60

80

100

120

140

160

180

200

Figure 15-38. (a) Tensile strain at the onset of plastic deformation is PES as a function of temperature for samples with an Mn of 47 000 (o) and 69000 (•). (b) The craze extension ratios corresponding to the specimens in (a). (Kramer and Berger, 1990.) 20

(b)

40

60

80

100 120 T(°C)

140

160

1990) show that without exception, all fibril breakdowns initiate at the craze-bulk polymer interface, i.e., at the active zone /z, whether or not it is initiated by a flaw (Fig. 15-39). In these TEM micrographs, the midribs are clearly below the lower end of the pear-shaped voids and hence are not involved in the initial breakdown process. Such pear-shaped voids have also been observed in bulk specimens (Behan et al., 1975; Hull, 1970). Similar breakdown morphologies have been observed in other vinyl polymers such as PMMA, poly(a-methyl styrene) (PaMS), and poly(styrene acrylonitrile) PS AN (Kramer and Berger, 1990).

180

200

With the exception of these recent observations, the craze breakdown process has been described by Murray and Hull (1969, 1970 a, b, c). Their observations are modified here to reflect the results of Kramer and co-workers, and summarized as follows: (1) As the craze grows in the thickness direction, fibril breakage initiate at the craze-bulk interface, and expand to form larger voids. The initiation usually occurs around heterogeneities in the craze such as foreign matter and impurities. The initiation stress can be described by Weibull statistics.

15.2 Crazing: Initiation and Growth

(b) Figure 15-39. TEM micrographs of craze fibril breakdown in the absence of dust inclusion (a) and associated with a dust inclusion (b) (Kramer and Berger, 1990).

735

(2) Hach of these voids initially grows independently, but they eventually impinge on each other. Finally, a crack is formed in a direction normal to the stress. (3) Once the cracks reach a certain size, they pass through the midrib or the interfacial regions between crazed and uncrazed materials along the craze plane and propagate by breaking more and more fibrils. (4) Several cracks, which have undergone the above three steps, become connected to each other, and this leads to macroscopic fracture. Ductile fracture of craze fibrils can also occur when the time between crazing and fracture initiation is long enough to allow the craze fibrils to be plasticized by environmental agents, which could facilitate the disentanglement process. Likewise, high temperatures could facilitate the disentanglement process. Although most of the observations described above have been made on carefully crazed specimens and then fractured in quasistatic loading or at low strain rates, in more practical loading conditions crazing

736

15 Crazing and Fracture of Polymers

always precedes cracking as long as the cross-link density is not too high. This can be identified by the interference pattern found at the propagating crack tip or the interference color on the fracture surface (Murray and Hull, 1970 a; Graham et al., 1976; Kambour, 1965). The green and pink interference color is a characteristic indication that the fracture surface is covered by a thin broken craze layer which has a lower refractive index (Higuchi, 1958) than that of the uncrazed material (Kambour, 1964 a, b). Even in dynamic fracture, crazing is always found (Takahashi et al., 1978). For different polymers the thickness of the craze layer on the fracture surface varies measurably with the crack propagating velocity and temperature, and therefore the fracture surface often exhibits complicated features. The evidences presented above demonstrate that, apparently without exception, for thermoplastic glassy polymers over a wide range of temperatures, crazes are always created at the crack tip and the crack then propagates through the body of the craze.

15.3 Fracture Polymeric materials can be classified according to their fracture type and characteristics into four categories: amorphous polymers, crystalline polymers, highly oriented polymers such as fibers, and elastomeric materials such as rubber. In terms of the macroscopic distribution of internal stress and strain, polymer solids in the glassy state are more homogeneous compared with those possessing crystallinity. It is often convenient to regard crystalline polymers as a two-phase system wherein two components of different mechanical properties are appropriately combined. In this case, fracture will always oc-

cur in the weakest phase. The fracture of fibers, however, can be considered essentially elastic because of the high degree of anisotropy and the small capacity for plastic deformation. In the fracture of elastomers, the strain to failure is considerably larger, and a particular relationship exists between the strain to failure and the strength at that point. This distinguishes the fracture of elastomers from that of any other material. In this chapter, we shall first review the microscopic mechanisms of polymer fracture. In later sections, we shall focus on the fracture of materials with a crack or a flaw in terms of fracture mechanics. 15.3.1 Theoretical Strength of Polymers

There are a number of observations that help to elucidate the microscopic mechanisms that cause polymers to fracture. (A) Whether a polymer exhibits brittle or ductile behavior depends on how the molecular chains respond to the external force, either in concert or individually. The molecular response of a polymer to stress is determined by its chemical (primary) structure (molecular species, molecular weight, molecular weight distribution, whether crystalline or amorphous, cross-linking, entanglement density, etc.), secondary structure (crystallinity, morphology and the dimensions of crystals, orientation, etc.), the state of the material (with respect to the glass transition temperature and other secondary or higher transition temperatures), and conditions of the external force (the type of loading, loading rate, etc.). (B) When a material in which the molecular chains are randomly oriented is subjected to a tensile stress, the chains tend to orient along the stress direc-

15.3 Fracture

tion as they are drawn. In this process, chain slippage and separation occur preferentially to chain scission since the covalent bonds which constitute the backbone chain are much stronger than those due to intermolecular interactions such as van der Waal's force. (C) Crystals and chemical cross-linking prevent the molecular chains from slippage. At temperatures below the glass transition temperature, physical entanglements among molecular chains have a similar, but smaller, effect than that of cross-links, even though they are not permanent bonds that can prevent chain slippage. For this reason, higher stress is needed to induce sufficient chain slippage in the glassy state than in the rubbery or melt states. (D) When the fraction of crystals and cross-links in a material is large, it will be difficult for chain slippage to occur and the molecular chains will be subjected to tension elastically from the onset of deformation. Some of the chains may begin to break. A similar situation occurs when drawn materials are subjected to an external force in the direction of drawing. Such materials usually exhibit macroscopically brittle fracture. (E) In those materials where chain slippage is a main cause of fracture and chain scission is a minor effect, either crazing, where the deformation is usually localized, or plastic (shear) deformation, where the deformation often involves a substantial portion of the sample, will occur. Macroscopically, these materials are ductile. Even in the case of plastic deformation, only those molecular chains which have been oriented in advance will bear the external force. Therefore, a nonuniform stress

737

distribution on a molecular scale has already been generated in the material at this point, what we shall call the first microscopic stress concentration. Similarly, the molecular chains in crystalline and amorphous phases also differ in the amounts of stress they bear. (F) When the stress is further increased, some of the chains that are under tension will begin to break. This chain scission is generally regarded as a stress-dependent kinetic process in which the rate R can be expressed by AF — a a = co0 exp —

kT

(15-28)

where AF is the activation energy, a is the activation volume which includes the stress concentration factor, a is the external stress, k is Boltzmann's constant, T is the absolute temperature, and co0 is a constant. (G) Upon further loading, more and more "newly" stretched chains break as chain scission progresses. When the number of broken chains reaches a certain density, a microvoid can be formed; consequently, more slippage occurs because of freer molecular motion around the void. This process will further accelerate chain scission or will lead to the merging of the voids, which eventually generates a crack. We shall call stress concentration due to microvoids the secondary microscopic stress concentration. (H) To initiate macroscopic fracture, a single large crack is sufficient. The crack will propagate by breaking either molecular chains or intermolecular interactions at its tip. The microvoids that have already formed in (G) can render crack propagation easier. (I) If there exist some foreign particles or impurities that lead to macroscopic

738

15 Crazing and Fracture of Polymers

stress concentrations in the material, processes (E) through (G) will proceed around them locally. If there is any crazing, the deformation will be concentrated in the crazes, in which case processes (E) through (G) will occur inside the crazes. 15.3.2 Fracture Initiation in Uncracked Polymer Specimens (Micromechanism of Fracture) 15.3.2.1 Micro void Generation due to Tension

In polymers, even if there are no cracktype defects such as voids and flaws generated during processing or service, inhomogeneities in the structure will still cause microscopic stress concentration and could lead to the formation of Griffith-type defects. This will give rise to a considerable difference between the theoretical and the practical strengths. For specific examples of structural inhomogeneities, one can cite the disordered lamellar crystals which constitute the microcrystals or spherulites, the amorphous region between lamellae and between lamellar bundles, and the interface between spherulites, etc. Other examples are found in glassy polymers such as PS and PMMA, etc., which often exhibit brittle fracture. The strength increase in these materials will eventually disappear as the size of the intentionally induced crack is decreased; this can be regarded as evidence for the existence of an inherent flaw in the material (Berry, 1961 a, b, c). The size of this inherent flaw calculated from Griffith's theory depends on the species of amorphous polymer, but it is usually in the range of 0.1-1.0 mm, which is unreasonably large. Because inherent flaws of this size have never been observed, stress concentration due to structural inhomogeneities (much

smaller in scale than the inherent flaws) must be present, which could lead to and accelerate the initiation and growth of larger defects such as crazes, etc., around them. Defects grown in this way would act as a quasicrack, and could eventually lead to macroscopic brittle fracture. This hypothesis explains why, in the brittle fracture of glassy polymers, it is difficult to achieve the theoretical strength since crazing is generally involved. In polymers that do not exhibit brittle fracture at the initial stages of deformation, molecular chains in locations where microscopic stress concentrations exist due to structural inhomogeneities probably rearrange themselves to relax the stress. Once this process ceases and plastic deformation has proceeded to a sufficient extent, microvoids generated from chain scission due to the structural flaws will grow and eventually lead to the catastrophic failure of the material as described previously. Microscopic voids generated during the initial stages of failure were first discovered by Zhurkov and co-workers (Zhurkov and Kuksenko, 1965; Zhurkov et al., 1969 a). They investigated void generation during deformations such as simple tension and creep for various oriented polymers, and estimated the number and size of the microvoids as a function of load or time. The results in Fig. 15-40 show that the number of microvoids in nylon-6 increases sharply as the load or time is increased. 15.3.2.2 Chain Scission in Oriented Polymers (Fibers)

If no chain slippage occurs, microvoids are generated by chain scission. The generation of these microvoids plays an important role in fracture initiation, particularly for fibers in which the molecular chains have been well-oriented with little room

15.3 Fracture

739

Figure 15-41. Fiber structure model in terms of crystal lamellae and amorphous regions (Peterlin, 1972). A, B are tie-molecules. 0.5 1.0 Creep t (min) 0

20 40 Constant strain rate a (kg/mm2)

0

5 10 Constant load rate £ (%)

3

Figure 15-40. The number density of micro voids generated in nylon-6 under various loading conditions.

for plastic deformation. Several fine structure models for fibers have been proposed so far. We shall consider in the following the model proposed by Peterlin (Peterlin, 1969, 1971a, b, 1972, 1974 b). The fiber structural models proposed by Peterlin for PE and PA-6 are shown in Figs. 15-41 and 15-42. In Fig. 15-41 a fibrous structure is depicted as an alternating combination of crystal lamellae composed of folded chains with orientation in the fiber axis and tie-chain regions. The tiechain regions incorporate chain ends and loops that are not included in the crystals and taut chains that connect the crystals. In other words, the amorphous area is repeated alternatively along the fiber axis to form a number of microfibrils with widths of 10-20 nm and lengths of 5-20 |im. These microfibrils will further aggregate to

Figure 15-42. Microfibrils and the associated fiber structure model (Peterlin, 1974 b).

form fibrils as depicted in Fig. 15-42 with widths of 0.1-1.0 |im. The stress in the fiber direction will be transmitted through tie-chains, which are thought to comprise 10-30% of the number of chains in the lamellae. Tie-chains, therefore, are subjected to quite a large force. The tie-chains between microfibrils are much fewer, and in PE amount to about 10% by volume

740

15 Crazing and Fracture of Polymers

(Meinel and Peterlin, 1968; Meinel et al, 1968). In PA-6, the fraction of tie-chains is also about 10%. However, this number generally depends on the polymer species as well as the type of processing that the material has undergone (Peterlin, 1974 a). The stress is distributed through microfibrils to the serially connected sandwich structures consisting of the lamellaamorphous regions. The amount of deformation in the lamellae is negligible compared with that in the amorphous regions. Let x be the crystallinity. When an overall strain s is applied, the strain of amorphous regions sa can be expressed by a

(15-29)

1-x

For example, if we suppose that the crystallinity is 50%, a strain twice as large as that of the overall strain will be applied to amorphous regions, according to Eq. (15-29). Since one cannot expect that all the tiechains have the same length, it is easy to imagine that shorter chains will be tightly stretched while longer ones are almost unstrained. In other words, even if the strain in the amorphous regions is constant, a distribution can still form within the region due to the distribution of lengths of initial tie-chains. In fact, Zhurkov (Zhurkov et al, 1969 a) has demonstrated that a stress ten times as large as the average stress is applied to some tie-chains in PE and PP. Therefore, if the strain applied to the amor-

a

b

• t

'/I •

B

phous regions increases, as shown in Fig. 15-43, some of the tie-chains will begin to break (Peterlin, 1972). The chain scission will generate free radicals which can be detected by ESR. Since Zhurkov's first reports (Zhurkov and Kuksenko, 1965; Zhurkov et al., 1964; Zhurkov and Tomashevskii, 1966), the generation of free radicals under tension has been extensively investigated by Peterlin and co-workers (Campbell and Peterlin, 1968; Verma and Peterlin, 1970), Becht and co-workers (Becht and Fischer, 1969; 1970), Kausch and co-workers (Kausch and Becht, 1970; Kausch, 1970) and Takayanagi and coworkers (Nagamura et al., 1973; Nagamura and Takayanagi, 1974). The following is a summary of the results: (1) Free radicals are not generated immediately after the application of tension. They appear upon reaching a stress level that is about 40-60% of the fracture stress, and increase sharply thereafter (DeVries et al, 1971) (Fig. 15-44). (2) In creep, free radicals increase sharply immediately after loading. At low stresses the rate of free radical generation is almost constant, but increase again up to near fracture at high stresses (DeVries et al, 1971) (Fig. 15-45). (3) In cyclic tension, free radicals can be accumulated (DeVries et al, 1971) (Fig. 15-46). (4) In step-wise tension, free radicals can vary with the step, and as a result, a

Figure 15-43. Schematic representation of interlamellar chain scission which proceeds according to the length distribution of tie chains (Peterlin, 1972). a, b, and c are undeformed, slightly deformed, and greatly deformed amorphous regions, respectively. A, B, and C are unstressed, stressed, and ruptured tie chains, respectively.

15.3 Fracture

741

Stress i

Radicals Time

Figure 15-46. Free radical accumulation due to cyclic loading (DeVries et al., 1971).

Time

Figure 15-44. The time dependence of stress and strain and the number of free radicals generated (DeVries et al., 1971).

-Radicals -Stress

High stress Strain

Radicals

Time

Strain

Figure 15-45. Radical generation due to creep (DeVries et al., 1971).

Figure 15-47. Frequency distribution of free radical generation due to stepwise tension (DeVries et al., 1971).

frequency distribution of free radical generation corresponding to strain can be obtained (DeVries et al, 1971) (Fig. 15-47). By combining the fiber structure with the occurrence of chain scission, the microscopic mechanism for fracture initiation can be modeled in two ways. Model I: DeVries et al. attempted to quantitatively relate the characteristics of the free radical generation described above to the stress-strain curve of PA-6 by changing the temperature and rate of deformation under various loading con-

ditions (constant stress rate, constant strain rate, creep, and cyclic deformation) (DeVries et al, 1971; Lloyd et al, 1972; DeVries and Farris, 1970; DeVries, 1971). By assuming that amorphous regions in the microfibrillar structure deform uniformly, and that the results illustrated in Fig. 15-48 reflect the distribution of lengths of tie-chains in the interlamellar regions, they proposed that the stress generated when a certain strain is applied is equal to the sum total of the stresses to which each stretched tie-chain is subjected. They considered the breakage of an individual tie

742

15 Crazing and Fracture of Polymers

Figure 15-48. The strain (a/E), the number of free radicals (R/R^) and the rate of generation of free radicals (dR/dX) as functions of amorphous region extension ratio X when a normal distribution of tiechains (W/l) is assumed.

chain to occur as a result of a thermally activated stochastic process before it reaches its critical strain. The length distribution of tie-chains obtained by ESR is further approximated by a normal distribution (with a standard deviation of 0.01 0.02); when an arbitrary strain is applied to the amorphous region, a distribution of strain corresponding to that of the lengths of the tie-chains is assumed. In practical calculations, the continuous distribution of strain is obtained by splitting it up into discontinuous blocks and taking the summation. The calculated results are in good agreement with experimental results with the exception of the initial period of tension. Similarly, Kausch et al. (1972) also assumed that stress is uniformly transmitted in amorphous regions in a microfibril.

However, they found that the detected radicals due to chain scission could account for only about 0.3% of all the tie-chains. Of the remainder, only 1.1% were stretched but unbroken, and the other 98.6% of the tie-chains were only slightly stretched. They concluded that a major part of the stress is borne by only a small fraction of the tie-chains. Takayanagi and co-workers (Nagamura etal., 1975) also carried out a calculation similar to that done by DeVries but for poly[p-(2-hydroxy ethoxy) benzoic acid] (PEOB) fibers. The difference between the two calculations is that the effect of a crystal-crystal phase transition (which occurs during tension), the dangling chain ends, the broken chains, or any unstrained chains were all taken into consideration in the case of PEOB. Since tie-chains were assumed to break at their critical strains, the effect of time was ignored. Moreover, all the above effects were added independently. The calculated stress-strain curve was found to be quite consistent with the experimental results, although the curve itself was somewhat jagged. Model II: The early models proposed by Peterlin and co-workers (Peterlin, 1971 a,b; Verma and Peterlin, 1972) were the same as those postulated by DeVries and Kausch. Later, however, Peterlin proposed a new model. Using Model I, he first obtained the stress-strain curve, the frequency distribution and the accumulation of the free radicals (Fig. 15-48) by assuming that the length distribution of tie-chains follows the normal distribution, and that each tiechain breaks at its critical strain. He then examined and critically analyzed this model in light of these observations: (1) In an ideal model, the stress will gradually decrease to zero after reaching the maximum. In DeVries' model, because the microvoids are generated immedi-

15.3 Fracture

ately after the stress had reached its maximum, the right side of the void generation frequency distribution could not be observed, and therefore the point of maximum stress coincides with the center of the distribution. In actuality, however, the point of maximum radical generation shifts significantly to the right side of the point of maximum stress. (2) In the experiments determining the stress-strain curve, failure occurred at a stress level beyond the maximum stress because a bundle of fibers was used. If a single fiber had been used, the maximum stress would have been completely equal to the fracture stress. In the case of fiber bundles, fluctuations in the mechanical properties and in the lengths of single fibers should be taken into consideration. (3) The number of radicals generated varies significantly with the amount of heating and the mechanical history of the fiber. In Model I, it was shown that the strength grows with the width of the distribution of radical generation frequency when the strain is increased. This, however, ignored the effect of mechanical history. (4) According to Model I, if a once-fractured sample is retested, its strength must be lower than that of the original sample since most of the tie-chains have already been broken. No such experimental results, however, have been observed. (5) According to calculations based on Model I as well as observations, the number of radicals generated until fracture occurs is about 1% of the tiechains. Peterlin considered the above problems to be caused by the inappropriate assumption of uniform tension in the amorphous

743

regions within microfibrils, and proposed an alternative which we shall call Model II. In this model, which addresses this difficulty, the ends of microfibrils are thought to give rise to two types of defects, i.e., cavities and dislocations, and amorphous regions around these cavities are assumed to be under greater strain than those regions away from the fibril ends. This model is depicted in Fig. 15-49. The frequency distribution of radical generation detected by ESR actually reflects the nonuniformity of the stress rather than the length distribution of tie-chains. Indeed, according to Model II, the tensile strength of once-fractured samples or the number of radicals occurring until fracture may be explained since the tie-chains in the amorphous regions other than the ends of microfibrils may remain unbroken. However, it is not clear how far-reaching the effect of cavities at the ends of microfibrils might be along the microfibrils and it is therefore difficult to carry out a quantitative analysis. Because Model II cannot provide a quantitative calculation as can Model I, it is still unclear which model is more reasonable. Nevertheless, the idea in Model II that cavities can be formed at the ends of microfibrils thereby introducing a nonuniform stress field in the material, may be considered as a new direction in understanding the mechanical properties of fibers. Since Model II cannot explain the initial Young's modulus of fibers, a revised model has recently been proposed (Peterlin, 1975) in which tie-chains with lengths shorter than the average lamellar thickness are assumed to make the surface of the lamellae uneven. According to this model, the number of extended tie-chains in the initial stage will be large and this leads to a higher Young's modulus.

744

15 Crazing and Fracture of Polymers Cross section DD'

Cross section CC

B1

Cross section AA1

-4f-4f-4C'

C4-4f-t

(a) Cavity

•ODD D1

an ana

Figure 15-49. Defects caused by the ends of microfibrils: (a) cavities and (b) dislocations (Peterlin, 1971b).

D'

(b) Dislocational defect

15.3.2.3 Relationship Between Chain Scission and Macroscopic Fracture

Macroscopic fracture does not necessarily occur even if all the tie-chains in the cross-section have been broken. It is reasonable to imagine that some sort of Griffith-type microcracks, which can also lead to macroscopic fracture, can be generated during the tension. This process can be described by the following sequence of events: (1) Stress concentrations arise in the structural heterogeneities in fibers. (2) Chain scission occurs due to the stress concentration, leading to the formation of microvoids. (3) Microvoids grow and evolve into cracks by aggregation. Zhurkov and co-workers (Zhurkov et al., 1969 a) investigated how microvoids grow and become a crack through coales-

cence, and obtained a parameter for the aggregation state of microvoids at which catastrophic fracture can occur. This type of microvoids is thought to be generated throughout the amorphous regions. This is deduced from the fact that the surface area of a microvoid estimated from the number of tie-chain scission, 10 6 -10 7 m 2 /m 3 , is roughly consistent with that calculated from the size and the number of microvoids, 10 5 -10 6 m 2 /m 3 . On the other hand, Model II argues that the number of cavities at the ends of microfibrils, Npu = 1021/m3, which is in agreement with the number of microvoids calculated above. If this is true, then the number of microvoids is determined in the beginning, i.e., it depends only on the structure. However, experiments demonstrate that the number of microvoids increases with extension, which is inconsistent with this postulation of Model II.

15.3 Fracture

Whatever the case may be, catastrophic fracture of a sample will occur once the transformed microvoids grow into a large crack. This fracture may proceed along the path of least resistance as shown in Fig. 1550. In the case of PA-6, which has a large interfibrillar adhesion because of the relatively strong hydrogen bonding, this path can be the ends and amorphous regions of the microfibrils. For PE, where the microfibrils are held together by van der Waal's force only, the path can be the interfibrillar areas. This theory would perhaps explain why the number of radicals generated during fracture is fewer for PE than for PA-6. 15.3.3 Ductile Fracture

In the previous section, we focused on the micromechanism of brittle fracture initiated from chain scission mainly for crystalline polymers such as fibers in which sufficiently uniform orientation has already been established by processing. If the starting material for deformation is unori-

n n

n

(a)

(b)

Figure 15-50. A schematic representation of the fracture path in PE (a) and PA-6 (b) (Peterlin, 1975). A crack propagates between the ends of microfibrils (A), parallel between two microfibrils (inter-microfibril) (C), and through a microfibril (intra-microfibril) (B).

745

ented, then fracture occurs after it has undergone macroscopic plastic deformation. Although this fracture is ductile, it is different from plastic flow in which molecular chains slide past each other. In this section, we shall concentrate on ductile fracture again in the sense of chain slippage, but in the context of a slightly different phenomenon. 15.3.3.1 Fracture of Shear Deformation Bands

When the ductile deformation is constrained, shear deformation bands (slipbased) occur as a nonuniform plastic deformation. Two different configurations of this type of shear deformation bands can be observed. One is called coarse shear bands in which the width of the deformation band is large, and the boundaries between the sliding and the surrounding areas are distinct. The other configuration is called the diffuse shear band in which a shear band is composed of a number of fine shear bands. In the former case, shear fracture along the deformation band often occurs; in the latter case, a large kink will appear in the specimen rather than fracture (Chau and Li, 1976). Although shear fracture occurs when the deformation band completely traverses the specimen, in the interior of the deformation and, deformations such as strands (fibril-like deformation bands) and void sheets may have already formed during the growth of the shear deformation bands (Chau and Li, 1980) (Fig. 15-51), and catastrophic fracture will proceed along the boundary between the void sheets and the undeformed areas (Chau and Li, 1981). The phenomena described above occurs in compression. If we reverse the compression, i.e., apply a tensile load before compressive shear fracture occurs, fracture will

746

15 Crazing and Fracture of Polymers

Figure 15-51. A shear deformation band in PS which was formed in compression (Chau and Li, 1980).

still occur along the band, but the strands inside the band are now stretched and exhibit a remarkable fracture morphology (Fig. 15-52) where the strands are formed in steps (Chau and Li, 1981). When deformation is not constrained, plastic deformation due to tensile loading will begin with necking which is due to postyield softening. Thus, necking unloads the surrounding unyielded material and the elastic strain energy there is released. In materials with a significant orientation-induced strain hardening effect, a neck will propagate regardless of the tempera-

ture elevation; however, in materials where the strain hardening effect is not strong, temperature elevation will lower the strength in the neck area. In the neck itself heat is generated by the plastic deformation. At sufficiently high rates of deformation (if yielding is still possible) fracture may result from localized heating. This type of fracture has been reported for poly(vinyl chloride) PVC and PC, and is usually called thermal fracture (Cross and Haward, 1973). The molecular basis for the reason why various plastics have different tendencies for strain hardening is still not known. Even at moderate rates of loading, neck formation may cause fracture in the neck itself, probably because the neck induces a higher triaxial tensile stress due to the plastic constraint. 15.3.3.2 Plastic Fracture

At the beginning of tensile deformation, crazes sometimes occur on the surface of the specimen, but do not cause fracture. A

(b) Figure 15-52. Fracture surface (a) and side view (b) of a shear band in PC which was initially formed in compression, then fractured in tension (Chau and Li, 1980).

15.3 Fracture

possible mechanism for this is that the craze is blunted by a pair of shear bands. In this case the crazes may persist in the necking area. Once the neck has propagated it is difficult for the crazes to grow either along the chain orientation or in the normal direction. They may then remain in the neck area without growth. When the stress in the specimen is further increased, cavities may occur in some of the crazes (Fig. 15-53). In accordance with their appearance, they are called diamond cavities. With increasing tensile stress, a diamond cavity will grow first along the tensile direction until it stabilizes and then in the transverse direction until the specimen fails. Examples for this type of fracture after plastic deformation are found in PVC (Cornes and Haward, 1974; Walker et al., 1979 a), PC (Cornes et al., 1977; Walker et al, 1981), PSU (Cornes et al, 1977), PET (Walker et al, 1979 b), and at higher temperatures, in PMMA and PS (Cornes et al, 1977; Smith and Haward, 1977). The generation of diamond cavities may not always be due to crazes in plastic deformation. The reason for the formation of this particular type of cavity is generally thought to be due to the pure shear deformation in the orientation direction at the tip of the cavity (Walker et al, 1979 a). However, since it occurs in anisotropic materials after plastic deformation, it is difficult to estimate the states of stress or strain around it. 15.3.3.3 Failure Envelope

The fracture of elastomeric materials such as vulcanized rubber, etc, in uniaxial tension may exhibit an extremely large strain to failure, although it does not involve permanent plastic deformation, i.e., ductility. In the tensile deformation of this type of materials, an important experimen-

747

Figure 15-53. A SEM micrograph of a diamond cavity in PMMA (Cornes et al., 1977).

tal result has been found, i.e., changing the deformation rate at a constant temperature has the equivalent effect on the failure strain as changing the temperature at a constant deformation rate. A master curve, shown in Fig. 15-54, which is called the failure envelope, can be constructed for any elastomer using the scheme described (Smith, 1958, 1969). The phenomenological basis of the failure envelope is the time-temperature superposition principle well-known for describing the viscoelastic behavior of polymers. In this particular application of the principle, the stress and time to fracture, i.e., <jb and th of elastomeric materials under uniaxial tension at various temperatures, can be collected into a single master curve by using a shift factor based on the WLFequation. Similarly, the strain to fracture £b and the time to fracture can be collectively described by using the same shift factor. Thus, the relationship ah — th — eh, i.e., ah — £b, can be obtained through the master curve. At constant rates of extension,

748

15 Crazing and Fracture of Polymers

Symbol

Temp. °C

X

140 100 70 40 25 10 -5 -20 -35 -45

•

1.5

Q

O

1.0

©

a 9

0.5

15.3.4 Nonuniform Fracture of Blunt-Notched Polymers

99

a y a jf a

/

/ a

0

a a|

-

8^r

o

0.5 i

-0.2

0

0.2

i

0.4 log CVb-1)

1

1

0.6

0.8

Figure 15-54. The failure envelope of SBR rubber (Smith, 1969).

th = sb/s; therefore, altering the strain rate is equivalent to altering the temperature. The failure envelope is valid if the structural change prior to fracture is kept in the equilibrium or nearly equilibrium state. Moving the curve towards the upper righthand direction (see Fig. 15-54) corresponds to ah — sh behavior at higher strain rates or lower temperatures. At the top of the curve, the experimental data begin to deviate from the envelope. This is due to deviations from equilibrium (Smith, 1965). For example, when an elastomer is deformed at low strain rates or high temperatures, a sudden increase in the strain rate during extension (Knauss, 1965) or the occurrence of a fracture due to cyclic deformation (Valanis and Yilmazer, 1978) results in deviations from the envelope due to departure of the deformation mode from the equilibrium state.

Structural components often contain notches of various geometries. Brittle failure sometimes occurs as a result of the presence of such notches, especially under impact conditions. Notched impact strength testing has therefore become an important practical tool for evaluating the fracture resistance of structural components containing blunt notches or abrupt changes in geometry. However, the apparent ductile or brittle behavior is highly dependent on the geometry of the notches, the thickness of the specimen, the rate of deformation, and the temperature. The origin of macroscopic behavior governing the fracture of notched polymers and the causes of geometric effects are discussed in this section. 15.3.4.1 Brittle Fracture of Ductile Materials

Fracture can occur in an unnotched (smooth) specimen in tension after uniform plastic deformation. Such phenomena have been discussed in a previous section. The same material with a notch in it may fracture in a brittle fashion. This is generally called notch (-induced) brittleness and occurs in many different types of polymers. Figure 15-55 shows the stress-strain relationship of a PE bar with a U-shaped circumferential notch for various notch depths. When the notch depth exceeds a certain value, a ductile-brittle fracture transition occurs (Narisawa et al., 1979). The phenomenon of notch brittleness in ductile has been explained by Orowan (Orowan, 1949). When a U-notched bar is pulled in tension, an extension occurs in the pulling direction and at the same time a contraction occurs in the transverse direction as the pulling stress is increased (Fig. 15-56). In elastic deformation, the

15.3 Fracture

p- 1.0 mm

749

elastic compared with that at the base of the notch since the cross-sectional areas here are larger, hence the stresses are lower, moreover, plastic deformation has not yet occurred at these locations. Consequently, the material in the notch will experience a tensile stress in the radial direction, and simultaneously a tensile stress of nearly the same magnitude will arise in the circumferential direction. As a result, in spite of the fact that the stress originally applied was uniaxial tension, a triaxial stress is actually set up in the cross-section of the notch. Since all the normal stresses are tensile, a negative hydrostatic pressure (dilative stress) is in effect exerted. When the axial stress is increased, and if the material is not so brittle as to fracture almost immediately, then plastic deformation will proceed inward along the notch cross-section, and the stresses in the radial and circumferential directions will also increase accordingly. When the maximum stress or dilative stress exceeds the critical value for

Strain (%)

Figure 15-55. The stress-strain behavior of a PE specimen with circumferential notches of various depths d: (A) 27 mm, (B) 21mm, (C) 15 mm, (D) 0.9 mm, (E) 0.7 mm, (F) 0.2 mm, (G) 0 mm. (Narisawa etal., 1979.)

contraction is due to the Poisson's ratio effect. If the tensile stress continues to be increased, plastic deformation will occur first at the base of the notch as a result of stress concentration. If the plastic deformation resulting from the longitudinal stress is shear yielding instead of crazing, an even larger contraction will occur in the transverse direction since the volume in the area of plastic deformation must remain approximately constant. On the other hand, in the upper and lower ends of the notch, the lateral contraction is very small and

Figure 15-56. A schematic illustrating how a radial stress
750

15 Crazing and Fracture of Polymers

cohesive failure of the material, the specimen will fail at the notch. In this case, the plastic deformation is limited only to part of the cross-section of the notch; therefore the fracture is still macroscopically brittle. When shear yielding occurs at the tip of the notch, the axial stress becomes as high as (7t == aY + ox (GT is the stress in the radial direction) where is accordingly called the mean plastic constraint coefficient. The occurrence of a triaxial stress state in a specimen that is nominally under uniaxial tension is not limited to the situation described above. For instance, a bar with a local neck in which a soft region is sandwiched between two rigid regions could give rise to a similar stress state. This is the reason why a thin elastomeric disk perfectly bonded between two rigid surfaces and subjected to tension perpendicular to the plane of the disk fractures brittlely, exhibiting none of the high extension that we generally associate with elastomers. But, if

a;

p

°z

a notch exists in a specimen and is not in a shape that can maintain sufficient constraint in the radial and circumferential directions, then plastic yielding in the notch area will proceed until the entire cross-section has yielded. Fracture in this case can be regarded as ductile in a real sense. 15.3.4.2 Stress Distribution Around a Local Plastic Deformation Zone

We now consider a plate with a blunt notch, as shown in Fig. 15-57. If it is in the elastic range, it will contract in the z-direction due to the Poisson's effect. If its thickness is sufficient to maintain a plane strain state during the deformation, i.e., sz = 0, then a tensile stress oz which is to resist the contraction will appear in the thickness direction of the specimen. Also, a contraction in the width (x-direction) will occur and for the same reason, a tensile stress ax is generated in the notch, because sx inside the notch is constrained by the smaller £x outside the notch. oz has a maximum in the center of the plate and is expressed by G = v z (°x + Gy) (v i s Poisson's ratio). On the surface of the specimen, the stress oz is, of course, zero. The stress ax increases from zero at the base of the notch, reaches

1

Figure 15-57. Elastic stress distribution in the interior of a thick plate with a notch whose radius is Q.

15.3 Fracture

a maximum, and will decrease again farther away from the notch. Thus, inside the region near the base of the notch, ay> az> ax and the maximum shear stress T max = iGy — °x)/2 if the plane strain condition (ez = 0) is satisfied. If shear yielding occurs at the base of the notch, the slip planes will be normal to the x y-plane and propagate at some angle to the notch direction, as illustrated in Fig. 15-58b. This mode of deformation is said to be plane strain. On the other hand, if the specimen is thin, constraint will not exist in the z-direction, thus GZ will be zero throughout the thickness of the specimen. In this case, shear yielding will occur through the thickness, once again because Tmax = (
751

Figure 15-58. Plane stress (a) and plane strain (b) deformation (Hahn and Rosenfeld, 1965).

tion by Nimmer (1991) has shown that the analytical solution provided by the SLFtheory is surprisingly accurate for two ductile glassy polymers, PC and poly(ether imide) PEL According to the SLF-theory, the slip lines, i.e., lines of maximum shear stress, at the notch tip can be described as a logarithmic spiral, and ay inside the plastic zone (x < s) is given by Gy = (jY 1 + l n M + - j

(15-30)

Figure 15-59. Plastic deformation zone at a blunt notch tip in a thick PC specimen.

752

15 Crazing and Fracture of Polymers

Figure 15-60. Plastic deformation and stress distribution according to the slipline field theory. The notch tip radius Q and the thickness are such that the specimen is in plane strain, s is the extent of the plastic zone.

where Q is the radius of the notch, and oY is the uniaxial yield stress. If we further assume, for the sake of simplicity, that the material obeys Tresca's yield criterion, then, from the relationship oy — aY = ax, we have

i\

(15-31)

The maximum extent of the logarithmic slip-line field s* is determined by the geometry of the notch tip, and is given by S* = Q

ex

PU - ^

-

(15-32)

where co is the notch angle. As the load is further increased, i.e., s > s*, the stress in the range s* < x < a is given by K

CO

(15-33)

The value of ay is the maximum value of the stress in the tensile direction when a plastic zone exists at the notch tip, and depends on the opening angle of the notch CD only. When x < s*, the maximum of ay is at the elastic-plastic boundary x = s, i.e., = (7,

L

1 + In ( 1 + -

V Q.

(15-34)

Since there is little volume change during the plastic deformation, the relationship (15-35) holds. We therefore have az = ^ |~1 + 2 In (1 + -Yl (15-36) 2 L \ QJA The hydrostatic stress p due to the triaxial tensile stress at the notch tip is P=

(15-37)

and is consistent with the value of Eq. (15-36). In materials with cohesive strength substantially higher than shear yield strength, e.g., metals, the plastic deformation at the notch tip will simply grow with the applied load. In bending, a plastic hinge will develop. However, isotropic thermoplastics generally possess cohesive strengths (i.e., crazing strengths) not substantially different than the shear yield strength. Consequently, the formation of a plastic hinge under plane strain conditions is seldom, if ever, observed. Instead, fracture due to cohesive failure truncates the plastic deformation process. This phenomenon is described in the next section.

15.3 Fracture

15.3.4.3 Fracture of a Notched Specimen due to Plastic Constraints

In deformation under plane strain, if plastic deformation at the base of a notch proceeds, a large dilative stress expressed by Eq. (15-36) will occur at the tip of the notch. In Fig. 15-5 we showed a micrograph of the plastic zone at the root of a notch in a thick PC specimen after the deformation has proceeded to a sufficiently large extent. One can see that cracks are generated in the direction normal to the tensile axis (Ishikawa et al, 1977; Narisawa et al., 1980 c). The corresponding fracture surface is presented in Fig. 15-61. There are three significant features on this surface: (a) a spot that is the nucleus of fracture at the tip of the plastic zone where cracking started, (b) a smooth, flat zone that surrounds the nucleus formed by the fracture of a craze that had grown relatively slowly, and (c) a zone which clearly indicates that a rapid propagation of cracks has proceeded from the smooth area, (c) Is in part caused by unstable fracture as a result of the use of a notched 3-point-bend specimen.

Figure 15-61. Fracture surface of a thick PC specimen with a blunt notch.

753

The type of fracture surface in PC described above has been observed in thick plates of U-notched PVC and PMMA as well. In thick plates of PE and PP, a number of cracklike features can be observed starting at the root of the notch (Fig. 15-62) rather than in the interior as is the case of most glassy polymers. This behavior is observed in many semicrystalline polymers (Narisawa and Ishikawa, 1990). Figure 15-63 shows the fracture surface in PE

0.1 mm Figure 15-62. Multiple crazing at the notch root of a thick PE specimen subjected to bending (Narisawa etal., 1979).

Figure 15-63. Fracture surface of a PE specimen broken as in Fig. 15-62 (Narisawa et al., 1979).

754

15 Crazing and Fracture of Polymers

(Narisawa et al., 1980 c) obtained by bending a thick, notched specimen. Although no fracture nucleus can be clearly identified, it is apparent that fracture began ahead of the base of the notch rather than at its base. Again, in this case, plastic constraint became the cause of fracture, which is clearly demonstrated in the stress-strain curves of notched PE specimens in Fig. 15-55. When the location of a fracture nucleus can be identified on the fracture surface, its distance from the bottom of the notch is equal to s* in Eq. (15-33). From this, one can obtain the critical dilative stress during fracture from Eq. (15-36). Figure 15-64 shows an example in which the critical stress of PC so obtained is plotted against the strain rate. Although the yield stress generally increases with increase in strain rate, the critical dilative stress exhibits almost no change (Narisawa et al., 1980 a). In thermoplastics, the fracture nucleus has been shown to begin as a craze (Narisawa etal, 1980c; Ishikawa et al., 1981). Table 15-1 gives the critical dilative stresses of various polymers determined in this manner. We have shown that the use of a critical dilative stress criterion is quite reasonable for craze initiation. The dilative stress at the tip of the plastic zone in plane strain first generates crazes. The crazes will at first grow thicker in the normal direction, and upon further thickening the crazes evolve into cracks. It has been argued that crazing is the result of a strong tensile stress concentration like that due to the intersection of the slip lines in metals (Chau and Li, 1981; Kramer, 1975; Camwell and Hull, 1973). However, the slip line field at the notch tip does not always involve distinct shear bands. For example, in quenched PMMA, the plastic zone exhibits only diffuse slip lines (Fig. 15-65). This clearly

Table 15-1. Critical dilative stress due to plastic constraint. Polymer (20 °C)

Critical dilative stress (MPa) 37 40 56 69 74 67 87 96

PS (slow cooled) PS (quenched) PSAN (slow cooled) PSAN (quenched) PMMA (quenched) PVC (quenched) PC (slow cooled) PC (quenched)

100

Q_

50

OVA

-I*

-3

-2

Strain rate (s"1)

Figure 15-64. Critical dilative stress for crazing for PC and its dependence on the strain rate (Narisawa etal., 1980 a).

Figure 15-65. Polarized (A) and normal (B) light micrographs of a thin section of quenched PMMA from a thick notched specimen deformed to initiate a craze.

755

15.3 Fracture

demonstrates that even when there are no well-defined slip lines in plastic deformation, crazing can still occur. The existence of slip lines is therefore not essential for crazing. Instead, the dilative stress due to plastic constraint is the real cause for crazing (Orowan, 1949; Ishikawa et al., 1977; Narisawa et al., 1980 a, b; Ishikawa et al., 1981; Grade and Weiss, 1972; Mills, 1976). Since a dilative stress is necessary for craze initiation, it stands to reason that if this dilative stress is counteracted by an externally imposed hydrostatic pressure, then crazing should be suppressed. In Fig. 15-66, the bending moment at which crazing occurs in a thick notched PC specimen under an external hydrostatic pressure is shown as a function of displacement. The bending moment required to cause crazing increases with the hydrostatic pressure until yielding occurs through the entire cross-section of the specimen, i.e., a plastic hinge is formed (Ishikawa and Narisawa, 1983). The testing temperature also changes the location of the craze nucleus. Figure 15-67 shows such an example for PMMA. By extrapolating to the point where the nucleus is at the surface of the notch, the ductile-to-brittle transition temperature can be found. Generally speaking, small flaws with a large tip radius in a specimen can be safely ignored in considering the strength of polymer products. This is particularly true in those polymers which exhibit ductile behavior. However, as we have indicated, even a small plastic deformation will generate plastic constraints, thus crazing or cracking. It is therefore important to realize that the dilative stress at the tip of the plastic zone will increase due to these constraints and will lead to macroscopic brittle fracture of the specimen.

20 ^ S l o w cooled

29.4 MPa/

16 --

1 ^^-Quenched

9.8 MPa/ 12 /

A

t 5.9 MPa 3.9 MPa

V

7.8 MPa 39.2 MPa

MPa

MPa

f

i 1.0

i

i

4.0

2.0 3.0 Displacement (mm)

Figure 15-66. Effect of hydrostatic pressure (numbers in the figure) on bending moment at which crazing occurs in slow-cooled and quenched PC (Ishikawa and Narisawa, 1983).

o [Bending

•

•"I

X

speedi

1 mm/min

1.0

o

-

x

o o

•

•

x

10 mm/min o

X

•

•

X X

•

0.5 -o/

o

100 mm/min X

/

X /

i

n

300

320 Temperature (K)

i

340

Figure 15-67. Effect of temperature and testing rate on the location of the craze nucleus relative to the notch radius, R/Q for PMMA (Ishikawa et al., 1981).

756

15 Crazing and Fracture of Polymers

15.3.5 Fracture of a Cracked Specimen 15.3.5.1 An Application of Linear Fracture Mechanics

According to Griffith's criterion (Griffith, 1921), fracture occurs when the elastic strain energy released per unit increment area of a crack with length a exceeds the increase in surface energy ys, i.e., 2KG2

a>4ys

and can be rewritten as (15-39)

and for a three-dimensional crack, it can be expressed by G >

-v2)na

(15-41)

(15-38)

Thus, the stress for fracture is

/2ysE

tionship between the radius of the crack tip and the interatomic distance. This can be seen if we let the radius of the crack tip be Q, the stress at the crack tip then becomes
(15-40)

where v is Poisson's ratio. Equations (1539) and (15-40) give a necessary condition for fracture of a perfectly elastic body in terms of energy conditions. Another necessary condition is the stress. We can estimate the level of stress by considering what happens when the crack tip radius approaches the interatomic distance. Equations (15-39) and (15-40) imply that the maximum stress almost reaches the level of the interatomic bond strength and is thus a sufficient condition for fracture in terms of the mechanics of continuous media. This consideration suggests to us that one more condition must be satisfied other than Griffith's condition, i.e., the local stress at the crack tip must exceed the interatomic bonding force (Yokobori, 1975). The failure strength should be obtained by satisfying both conditions, and determined by the one which gives a larger critical stress. Which condition should be applied in a particular situation depends on the rela-

o>

(15-42)

Comparing Eq. (15-39) with Eq. (15.42), one can clearly see that when 7tg(8r0) < 1, i.e., Q < 3ro/n, the stress for Griffith's criterion Eq. (15-39) becomes larger, and is therefore the critical stress, since the local stress condition will be automatically satisfied; while if Q > 8ro/7i, the local stress in Eq. (15-41) will become larger, and thus will be required instead as the fracture criterion. In practice it would be difficult to find Q < 3ro/7i. Thus, the stress for fracture is always greater than the value determined by Eq. (15-39). Furthermore, the energy required to grow the crack exceeds the surface energy by a considerable amount due to the plastic energy that must accompany the crack growth. These considerations are described in the following sections. 15.3.5.2 The Strain Energy Release Rate gc and Fracture Toughness Kc of Glassy Polymers

In early studies, polymers which tended to exhibit brittle fracture, such as PMMA (Irwin et al., 1952; Benbow and Roesler, 1957; Benbow, 1961; Svensson, 1961; Berry, 1961a, 1962, 1963; Broutman and McGarry, 1963; Berry, 1964), and PS

15.3 Fracture

(Irwin etal, 1952; Benbow, 1961; Berry, 1961 b, 1963; Murray and Hull, 1971) were investigated to see if Griffith's criterion was applicable or not. This was accomplished by preparing specimens with cracks of various lengths and by determining if Eq. (15-39), i.e., (Th~l/y/a9 holds. Recently, with the development of testing methods for fracture toughness in metals, more attention has been paid to the preparation of cracks, the method of loading, and the shape of the specimen (Marshall et al., 1973), and more exact measurements of the critical strain energy release rate gc or the critical stress intensity factor Kc has become possible since then. Table 15-2 summarizes the values of Klc for a number of polymers. The additional subscript I is used to indicate Mode I (opening mode) fracture. In general, the value given is for plane strain fracture. Because the values of glc or Klc are affected by: (1) the thickness of the specimen,

757

(2) strain rate, (3) temperature, (4) molecular weight, (5) cross-linking, (6) whether oriented or not, and (7) for crystalline polymers, the morphology, it is difficult to directly compare the values in Table 15-2 on the basis of molecular structure alone. Nevertheless, some generalizations could be made. For example, thermosetting materials which have a cross-linked structure usually exhibit lower glc and Klc than thermoplastic materials. Among thermoplastic materials, vinyl polymers such as PMMA and PS show lower glc and Klc; by contrast, PC and PSU, and toughened materials such as ABS, exhibit high glc and X Ic . The theoretical surface energy ys for polymer solids is less than 0.1 J/m2. Even for brittle thermosetting materials the measured ys is higher by at least two orders of magnitude. The reason is that the fracture of glassy polymers does not occur in an elastic fashion; instead, there are microdeformation mechanisms which can act as

Table 15-2. X Ic of various polymers. Polymer

Specimena

Temperature b (°C)

Crosshead speed0 (mm/min)

References

SENB SENB SEN4B CT SENB SENB SENT SENB SENB SENB SENB CT SENB

23 23 RT RT 20 20 RT -60 -60 20 -40 RT RT

1.0 5.0 0.076 0.03 0.5 0.5 SS 0.1 0.5 0.1 0.1 0.03 0.254

Misaki et al. (1989) Chan and Williams (1981) Gupta etal. (1986) Serrano et al. (1982) Hashemi and Williams (1985) Hashemi and Williams (1985) Hobbs and Bopp (1980) Hashemi and Williams (1985) Fernando and Williams (1980) Hashemi and Williams (1985) Hashemi and Williams (1985) Serrano et al. (1982) Wu and Schultz (1989)

(MPa m1/2) Epoxy HDPE PS PMMA PVC PBT PP POM PA66 PPO PEEK a

1.12 1.18 1.29 1.00 1.97 2.81 2.73 3.37 4.1 4.05 3.84 4.85 7.0

SENB; single edge notch bending (3-point bending), CT: compact tension (tension), SENT: single edge notch tension (tension), SEN4B: single edge notch bending (4-point bending); b RT: room temperature; c SS: slow speed.

758

15 Crazing and Fracture of Polymers

energy sinks before the propagation of cracks. Even when the fracture appears brittle by macroscopic observation, the contributions from these mechanisms are still significant. Therefore, the experimentally obtained surface energy y's = ys + yp (7P > 7s) (Orowan, 1949; Irwin, 1948), where yp is the energy of plastic deformation per unit area in a thin layer ahead of a crack tip. 15.3.5.3 Plastic Deformation at a Sharp Crack Tip

The occurrence of plastic deformation at and ahead of a sharp crack tip has been identified as the main source for a large strain energy release rate gc. When a thin PC specimen with a crack (hence it is in the plane stress state) is extended (Fig. 15-68), a plastic deformation zone that increases with the increasing load will appear before the crack propagates (Narisawa et al., 1976, 1978 a, b). By cutting a thin section from this well-developed plastic zone in the direction normal to the growth direction and observing it under a polarized microscope, one can obtain the type of pattern shown in Fig. 15-69 (Ishikawa et al., 1976), where the symmetric slip of the deformation zone in the maximum shear stress direction as well as their intersection zones

/Cx=3.56

4.93

are plainly visible. This is a typical pattern of plastic deformation in the plane stress state as seen in Fig. 15-58. Such a characteristic shape and structure of plastic deformation at a crack tip has also been observed in other materials such as PVC (Narisawa et al., 1977). 15.3.5.4 Crazing at a Sharp Crack Tip

Another cause for the very large gc and Kc values compared with theoretical predictions is crazing that could occur at the crack tip. When crazing already exists at a crack tip, the glc is found to be 40 ~ 50% lower than if no crazing occurred (Narisawa and Kondo, 1972; Narisawa, 1972 a, b). When the specimen thickness is large (plane strain state) crazing usually occurs at the crack tip before crack propagation. For the most part glc is consumed in the formation and deformation of the crazes if account is taken of the energy required for the deformation of fibrils within the crazes until failure (Kambour, 1966 b). The shape of the crazed zone that occurs at the crack tip in a thick plate (plane strain) is generally consistent with that calculated by the Dugdale model. In practice, the lengths of crazes rc are often such that rc <^ a, where rc = n Kf/(& al). Thus, the crack opening displacement
6.20

0.2 mm

Figure 15-68. Plastic zone growth under plane stress conditions in a thin specimen of PC with a sharp crack.

15.3 Fracture

759

tained from the shape of the crazes is less than half of the value actually measured (Mills and Walker, 1976; Brown and Chin, 1980; Brown and Stevens, 1978). 0.20

15.3.5.5 Various Factors Influencing Fracture Toughness

The following is a summarized list of the various factors that influence gc or Kc. (1) Effects of specimen geometry and crack initiation. If one compares the plastic 0.79 deformation zone and the crazes that occur at a crack tip in the plane stress state with 1.12 those in the plane strain state, one would observe that the former requires more energy than the latter simply because the vol1.40 ume of material involved in the plastic deformation is larger. The glc measured for Figure 15-69. Polarized light micrographs of crossPMMA and PS exhibit almost no thicksections of the plane stress deformation zone ahead of ness effect (Berry, 1964); in the case of ducthe crack in a thin PC specimen. The numbers inditile polymers such as PC (Brown and cate distances from the crack tip. Ward, 1973; Parvin, 1981; Frazer and Ward, 1978; Parvin and Williams, 1975a,b; proximated by <j> = K?(l — v2)/(£ ac) and Pitman and Ward, 1979; Kambour and the calculation can be treated linearly. This Miller, 1976; Kambour et al., 1978), PPO has been demonstrated in a number of ex(Kambour and Smith, 1979), PSU (Ting, periments on PC (Frazer and Ward, 1978; 1981), and PE (Bandyopadhyay and Pitman and Ward, 1979; Kambour et al., Brown, 1982), the fracture toughness was 1978), PMMA (Murray and Hull, 1971; found to be dependent on the thickness of Brown and Ward, 1973; Weidmann and the specimen. This is because the specimen Doll, 1978; Doll and Weidmann, 1979), surface deforms in plane stress, while the PVC (Weidmann and Doll, 1978), and ABS interior deforms in plane strain if sufficient (Doll and Weidmann, 1979). The relationconstraint exists. If the specimen is very ship between KY and (j) thus becomes thick, then the bulk of the specimen deforms in the plane strain state, and a more brittle type of behavior dominates the de(15-43) 1 8 r c ( l - v 22\2 ) formation. However, if the specimen is so thin that the plane strain region never Thus the fracture toughness can be obforms because of the lack of sufficient contained if we know the shape of the crazes straint, then the specimen could deform in immediately before fracture. For PVC, it is a very ductile manner. At intermediate known that the opening displacement is thicknesses, a mixed mode deformation smaller than that calculated from the Dugcould occur, especially near the notch tip dale model and the plastic zone tends to be where some elastic constraint could exist. long and sharp. The fracture toughness ob0.29

760

15 Crazing and Fracture of Polymers

The fracture surface of a PC specimen that fractured in a mixed mode is shown in Fig. 15-70. Remnants of shear deformation that started from the two side surfaces (shear lips) can be seen. The crack initiated by crazing in the mid-thickness of the specimen near the crack tip where the elastic constraint prevented the shear lips from forming. Consequently, the apparent fracture toughness, Kc, of such specimens decreases as the thickness of the specimen increases, and approaches a constant plateau value (Fig. 15-71). This plateau value should be cosnsidered as the true plane strain fracture toughness of the material.

Plane stn zone

Figure 15-70. Mixed mode fracture of a PC specimen.

It has been shown that if the thickness of the plate is B, and if half of the thickness is greater than twice the dimension of the plastic deformation zone, then crack propagation will occur in the plane strain state in the middle of the plate (Irwin et al., 1958); that is, (15-44) 7i

aY

The dimension of the plastic zone is thus given by the Dugdale model as s = 7TKC2/(80"Y) f° r the surface and s = Kl/(2n GY) for the interior where a state of plane strain exists. Kc2 is the fracture toughness in the plane stress state. Assuming that the plane stress plastic zone (shear lips) and the plane strain deformation zone (crazes or cracks) form independently of each other, the fracture toughness of a specimen of arbitrary thickness may be estimated by assuming that it is a simple linear combination of the two contributions. With such an assumption, experimentally obtained fracture toughness in the case of a mixed plane stress and plane strain deformation can be expressed by (Parvin, 1981; Parvin and Williams, 1975 a, b) n Gy B

Ductile fracture Plane stress

Thickness

Figure 15-71. The thickness dependence of apparent fracture toughness of ductile polymers.

(15-45)

where Kcl is the fracture toughness in the plane strain. In the case of PC, for example, Kc2 = 2.5 Kcl (Parvin and Williams, 1975 a). If the plane stress deformation zone is able to grow to a significant extent then the plane strain deformations may be prevented from taking place. When interaction between the two modes occurs, the additivity assumption underlying Eq. (1545) may not be valid. Clearly, then, it is important for the person performing the test to note at what thickness the fracture toughness is evaluated.

15.3 Fracture

In the experimental evaluation of the toughness of polymers, for the same reason that specimen thickness affects the apparent fracture toughness, the manner with which the crack is introduced into the specimen could also significantly affect the measured apparent toughness. For example, pressing a sharp razor blade into the specimen could create a compressive yield zone and produce artificially high apparent toughness even if the crack does appear to be very sharp. The use of fatigue loading could also create a damage zone at the crack tip, once again increasing the apparent toughness. By the same token, blunt notches will usually produce higher apparent toughness than sharp notches. Therefore, in fracture toughness testing, it is essential that a sharp crack free of the above-mentioned defects be used. The failure of many investigators to exercise caution in this respect has caused a certain amount of confusion regarding the true fracture toughness of polymers. There is no set rule on how to produce a sharp crack, except that the crack tip should be examined to see if plastic deformation has been introduced by the crack initiation technique, and that the specimen dimensions are such that the measured toughness is independent of the geometry. In the case of extremely ductile polymers such as PE and highly toughened plastics, nonlinear fracture mechanics techniques must be used. A discussion of such techniques is beyond the scope of this review. (2) The effect of strain rate. We note that, although fracture toughness values are sometimes reported as a function of testing rate, the effective strain rate at a crack tip is usually much higher than the nominal strain rate. In general, as Fig. 15-72 indicates, glc decreases as the strain rate or testing rate increases (Narisawa and Kondo, 1972; Narisawa, 1972; Ting, 1981;

761

Yamani and Young, 1977, 1979; Phillips etal., 1978; Cherry and Thomson, 1981a; Arad etal., 1973). The strain rate effect arises because the toughness depends strongly on whether there is sufficient time for the crazes or other forms of plasticity at the crack tip to develop fully; the larger the amount of plasticity, the higher the toughness will be. (3) The effect of molecular weight. gc is relatively independent of the molecular weight above a critical value Mc = 2M e , where M e is the entanglement molecular weight, but diminishes below this value. For PMMA it has been found that a relationship between the number average molecular weight Mn and gc holds: gc = A — (B/Mn) (Berry, 1964) where A and B are constants. The reason why gc is low when Mn is less than Mc was thought to be the increased probability of finding molecular chain ends inside the crazes; thus, it is more difficult to form fibrils bridging the entire thickness of the crazes (Kramer, 1978; Haward and Daniels, 1978). Kramer's work, which is described in Sec. 15.2, shows that crazes are stabilized by entanglements (Kramer and Berger, 1990). For those polymers with low entanglement densities, having a high Mn causes the polymer to preferentially fail by crazing which involves chain scission, thus increasing the crazing strength. For those polymers that have high entanglement densities, having a high enough M n prevents crazing from becoming the easier deformation process, so shear yielding occurs instead. Therefore, regardless of the mode of deformation preferred by a given polymer, satisfying the relationship Mn > Mc always results in a higher toughness.

762

15 Crazing and Fracture of Polymers

15.4 Concluding Remarks We have attempted to outline for the reader in a limited amount of space the extremely complicated phenomena of crazing and fracture of polymers. Although we have attempted to give a generally accepted overview of these phenomena, some controversies regarding both the phenomenology and the underlying mechanisms undoubtedly still exist, and many specific experimental observations pertaining to different materials, morphologies, environmental conditions, stress or strain histories have been omitted. Certain aspects of the microscopic mechanisms of crazing have been carefully explained via experiments and theories by Kramer and his outstanding collaborators at Cornell. But much of the remainder of this vast subject still awaits clarification and the development of models. The reader who is interested in exploring this subject further is encouraged to consult the general reference listed at the end of this Chapter.

15.5 Acknowledgements The authors are indebted to Dr. Liang Bao Liu for rendering into English a first draft of this review from the Japanese original (Narisawa, 1982). They are also grateful to Ms. Sonia Janich and Ms. Susan Wright for their help in editing the manuscript.

15.6 References Andrews, E. H., Bevan, L. (1972), Polymer 13, 337. Andrews, E. H., Levy, G. M. (1974), Polymer 15, 599. Arad, S., Radon, J. C , Culver, L. E. (1973), / Appl. Polym. Sci. 17, 1467. Argon, A. S. (1974), The Solid State of Polymers: Geil, P. H., Baer, E., Wada, Y. (Eds.). New York: Marcel Dekker, p. 573.

Argon, A. S., Hannoosh, J. G. (1977), Phil. Mag. 36, 1195. Argon, A. S., Salama, M. M. (1976), Mater. Sci. Eng. 23, 219. Argon, A. S., Salama, M. M. (1977), Phil. Mag. 36, 1217. Argon, A. S., Hannoosh, J. G., Salama, M. M. (1977), Fracture 1977, ICF4, Waterloo, Canada, Vol. 1. New

York: Pergamon, p. 445. Bandyopadhyay, S., Brown, H. R. (1981), /. Polym. Sci. Phys. 19, 749. Beardmore, P., Rabinowitz, S. (1975), J. Mater. Sci. 10, 1763. Becht, I, Fischer, H. (1969), Roll. Z. Z. Polym. 229, 167. Becht, I, Fischer, H. (1970), Roll. Z. Z. Polym. 240, 776. Behan, P., Bevis, M., Hull, D. (1975), Proc. Roy. Soc. A 343, 525. Benbow, J. J. (1961), Proc. Roy. Soc. 78, 970. Benbow, J. X, Roesler, F. C. (1957), Proc. Roy. Soc. 70 B, 201. Berger, L. L., Kramer, E. J. (1988), J. Mater. Sci. 23, 3536. Bernier, G. A., Kambour, R. P. (1968), Macromol. 1, 393. Berry, J. P. (1961a), J. Polym. Sci. 50, 107. Berry, J. P. (1961b), /. Polym. Sci. 50, 313. Berry, J. P. (1961c), SPE J. Trans. 1, 3. Berry, J. P. (1962), J. Appl. Phys. 33, 1741. Berry, J. P. (1963), /. Appl. Phys. 34, 62. Berry, J. P. (1964), /. Polym. Sci. A2, 4069. Bevan, L. (1982), J. Appl. Polym. Sci. 27, 4263. Brady, T. E., Yeh, G. S. Y. (1973), /. Mater. Sci. 8, 1083. Breuer, H. (1979), Angew. Makromol. Chem. 78, 45. Breuer, H. (1985), in: Polymer Blends and Mixtures: Walsh, D. I, Higgins, J. S., Maconnachie, A. (eds.). Dordrecht: Martinus Nijhoff, p. 375. Broutman, L. J., McGarry, F. C. (1963), J. Appl. Polym. Sci. 9, 589. Brown, H. R., Chin, T. H. (1980), J. Mater. Sci. 15, 611. Brown, H. R., Stevens, G. (1978), J. Mater. Sci. 13, 2373. Brown, H. R., Ward, I. M. (1973), Polymer 14, 469. Campbell, D., Peterlin, A. (1968), /. Polym. Sci. B6, 481. Camwell, L., Hull, D. (1973), Phil. Mag. 27, 1135. Chan, M. K. V., Williams, J. G. (1981), Polym. Eng. Sci. 21, 1019. Chan, T, Donald, A. M., Kramer, E. J. (1981), J. Mater. Sci. 16, 676. Chau, C. C , Li, J. C. M. (1976), J. Mater. Sci. 11,434. Chau, C. C , Li, J. C. M. (1980), /. Mater. Sci. 15, 1898. Chau, C. C , Li, J. C. M. (1981), J. Mater. Sci. 16, 1858. Cherry, B. W, Thomson, K. W. (1981a), J. Mater. Sci. 16, 1913.

15.6 References

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General Reading Kambour, R. P. (1973), A Review of Crazing and Fracture in Thermoplastics, J. Polym. Sci.: Macromol. Rev. 7, 1. Kausch, H. H. (Ed.) (1983), Crazing in Polymers, Vol. 52/53 of Adv. Polymer Sci. Berlin: SpringerVerlag. Kausch, H. H. (Ed.) (1990), Crazing in Polymers, Vol. 2, Vol. 91/92 of Adv. Polymer Sci. Berlin: Springer-Verlag. Kausch, H. H. (1978), Polymer Fracture. Berlin: Springer-Verlag. Kinloch, A., Young, R. (1984), Fracture of Polymers. London: Applied Science.