Computer Simulations of Liquid Crystals and Polymers

/. Of course, all other parameters of the model (density, value of the mean field parameter A, etc.) should be properly regulated (see Section 2). Starting configurations for the various systems have been generated with rigid groups oriented along the z axis, with all r angles equal to 135° and with all

89 0.8 i 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 -1 20

40

60

SO

100

120

140

160

180


Distribution of tp for polymethylene spacers with the indicated values of

Table 2. The four systems simulated. System

T

To

Unrestricted

n

ET Er ET

T5 T6

Unrestricted Unrestricted E(p(n = 5)

£,,(71 = 6)

conformational changes [45, 46]. A MC cycle consists of 108 attempted moves, approximately 1.5% of which are accepted. The translational and orientational mobilities are large enough for adjacent molecules to exchange their position very frequently in each calculation, while also inverting several times their orientation with respect to the z axis. For instance, the mean square displacement of the centers of mass of the internal rigid groups is as high as 2.5a2/cycle in the isotropic phase of system TQ at 335K, while the internal rigid groups of the trimers in the orientationally ordered phase stable for system T5 at 300K are found to reverse their direction along the z axis every 10 cycles on the average. Therefore, the calculations have been long enough that one can safely assume the attainment of an equilibrium condition in all cases.

90

0

20

40

60

80

100

120

140

160

180

200

Figure 5. Behavior of the order parameter of the rigid groups (s rc ) as a function of Monte Carlo cycles in calculations performed at various temperatures for system T5.

1.2

Thermal behavior

Several calculations have been performed at various temperatures and with various starting configurations for each model system. Fig. 5 plots, as an example, the change of the orientational order parameter of the rigid groups as a function of Monte Carlo cycles in the various calculations performed for system T5. Starting from the left, the initial system with src = 1 was first heated at T = 320K, with the consequence that the order parameter rapidly decreased, reaching a value approximately 0.25 in 30 cycles only. A configuration with src = 0.33 obtained after 26 cycles at 320K was then used as a starting point for new calculations at lower temperatures. When this configuration was cooled at 310K, the system continued to evolve toward the isotropic liquid. On the contrary, cooling the same configuration at 300K caused an increase of the order parameter up to approximately 0.53. This suggests the presence of a transition between isotropic and anisotropic phases in the range 300 — 310K. Calculations performed at other temperatures (Figure 5, right part) indicate that the phase transition is indeed located between 300K and 305K for system T5, and also show the behavior of the order parameter as a function of temperature. The results obtained for all investigated systems are

91 Table 3. Nematic/isotropic transition temperatures and order parameters of the rigid groups in the nematic phase near the transition points for the four systems studied and for system RC. System To

Tb T5 T6 RC

TNI

300 320 300 330 300 -

Src

305K 325K 305K 335K 305K

0.53(300*0 0.57(320tf) 0.53(30010 0.57(330*0 0.53(300*0

summarized in Table 3. Since the distribution of the centers of mass of the rigid groups along the z axis does not show evidence of long range positional order for all systems at all temperatures investigated, the ordered phase can be classified in all cases as nematic. Table 3 shows that the nematic/isotropic transition temperatures of systems To and RC are coincident, i.e. the transition temperature is not influenced by the fact that the rigid groups are connected in sequence by conformationally unrestricted links. The inclusion of the bending potential ET (system T5) has the effect of increasing the transition temperature by approximately 20K. Including both ET and E^ has opposite effects in systems T5 and T&, since the transition temperature decreases in system T5 and increases in system TQ with respect to system T&. It is concluded that the nematic phase is favored by including E^ when n = 6, while it is disfavored when n = 5. Considering Figure 4, the transition temperature is expected to oscillate with increasing n, being higher when n is even and lower when n is odd. However, since the differences between curves for odd and even n become progressively less prominent, the entity of the oscillations is expected to decrease with increasing length of the spacers.

1.3

Orientational order in the nematic liquids

As discussed in Ref. [46], the simulated trimers are complex entities with a wide distribution of molecular shapes. Since different shapes have to be oriented differently in the nematic phase, a complete description of the order in this phase can be given only by taking simultaneously into account orientation and conformation, which is not an easy task. A simplified, but less complete description can be given if orientational and conformational orders are individually considered. Figure 6 shows the orientational distribution of the rigid groups with respect to the nematic director (the z axis) for the nematic phases at temperatures close to the transition points. The plot for system T& refers to the metastable

92 nematic phase found for this system at 325K. It is apparent that the orientational distributions are practically coincident for a given value of 5 rc (the slightly different curve for system TQ is obviously related to the higher order parameter of the nematic phase). They are also remarkably identical to the curves obtained for system RC and for other systems of monomers when src is the same, and not far from the distributions that can be evaluated for rigid rods with this value of src on the basis of the Mayer and Saupe [3-5] and of the Flory-Ronca-Irvine [8-11] theories. It is concluded that the overall nature of the order in the simulated systems is scarcely affected by details of the molecular constitution and conformation.

,T0,*K=0.53, r=30()K

80

90

(9/deg Figure 6. Orientational distributions of the rigid groups with respect to the nematic director for the nematic phases at temperatures close to the transition points.

The orientational distributions plotted in Figure 6 are averages over all rigid groups. They are practically the same for the central and for the external rigid groups in all simulated systems, with the exception of system T5, where the external rigid groups tend to be more ordered than the central ones in the anisotropic phase. For instance, the value src = 0.56 obtained at 295K for system T5 is the average of 0.57 for the external and 0.55 for the internal rigid group. This peculiar behavior, shown by system T5 at all temperatures, is obviously related to the conformational distribution assumed in the nematic phase by the model

93

trimers when (see later).

1.4

is constructed to simulate (C/?2)n spacers with n odd

Conformational changes at the nematic/isotropic transition

Figure 7 shows the distributions of (p in the isotropic phase of the simulated systems at temperatures close to the nematic/isotropic transition points. The distributions are practically featureless in the case of systems To and T& (minor differences for these two systems are explained in Ref. [46]), while all features observed in the curves for systems T5 and TQ are seen to follow closely in position and intensity those present in the corresponding curves of Figure 4. Therefore, the distributions of (p are practically what is expected on the basis of the torsional potentials adopted in the simulations. As far as the angles r are concerned, their distribution (not shown for brevity) is characterized by a broad maximum centered between 80° and 120° in system To, with a second, much less intense maximum at 60° (explained in Ref. [46]). Not unexpectedly, the broad maximum is shifted between 120° and 140° in all other systems, due to the inclusion of the bending potential ET. Overall, it can be concluded that the conformational distribution of the model trimers is scarcely affected by packing effects in the isotropic phase. 1-4 1

N>

- sm''

ir*"""*""*'

20

40

60

80

"m\

100

120

/

140

160

t

180

/d Figure 7.

Distribution of

94

o.o 0

20

40

60

80

100

120

140

160 . _ 180

/d Figure 8.

Distribution of tp (absolute values) in the nematic phases.

On the contrary, the conformational distribution is profoundly altered in the nematic liquids. This is especially true for the torsion angles

95 the nematic phase is fairly obvious, since consecutive rigid groups can be oriented along the director only when

96

0.7 -|

IV 5 1C -0.57. r-33OK

0.6 0.5

uncorrelated rigid groups, sK= 0.53

0.4 -

,:;;,,*^ /-^

,,

Tlr
0.3 0.2 0.1 j 0.0 0

20

40

60

80

100

120

140

160

180

a/deg Figure 9. Distributions of the angle a between consecutive rigid groups in the nematic liquids.

at the transition points. For comparison, the dotted curve shows the distribution of the angle between two vectors taken at random from the orientational distributions with src = 0.53 of Figure 6. Note that a is not an independent conformational parameter, being related to TI, 72 and (p by cos a = sinri sinr2 cos (p - cos T\ COS 72. The distribution of a is seen to be characterized in all nematic liquids by two broad maxima roughly centered at a = 30° and at a = 150°, respectively. If conformations with a < 60° are considered to be compact conformations, while those with a > 120° are considered to be extended conformations, one can say that compact and extended conformations predominate in the nematic liquids. This is far from being unexpected, and is simply due to the fact that the rigid groups are preferentially oriented along the director. In fact, maxima at 30° and 150° are also present in the dotted curve in Figure 9. On the other hand, compact and extended conformations are nearly of the same abundance in system To, while compact conformations are much less than extended conformations in the other cases. Furthermore, Figure 9 shows that the relative abundance of the two types of conformations depends on ET, but is practically independent on E^. As a matter of fact, compact conformations are already reduced in the isotropic liquids in systems including ET, since the distribution curves

97 for the isotropic liquids (not shown here for brevity) are characterized by a single broad maximum, centered at 90° for system To and around 120° for the other systems. It is concluded that the conformational selection taking place at the phase transition tends to favor to the same extent compact and extended conformations, equally compatible with the orientational distribution of the rigid groups characteristic of the nematic liquids. However, for the systems that have been studied, extended conformations are favored by J5r, both in the isotropic and in the anisotropic phases. It is also interesting to point out that the distribution of the angle ol between the two terminal rigid groups of the same trimer molecule in the nematic liquids is intermediate between the distribution of a and the dotted curve in Figure 9. This indicates that the correlation between the orientations of consecutive rigid groups due to the inflexibility of the spacers is propagated through the chain and is still clearly discernible between rigid groups separated by two intervening spacers. Figure 10 plots the distribution of the torsion angle \j) defined by the directions of the three rigid groups in a trimer molecule for system T5 in the nematic liquid at two different temperatures (note that ij) is not equal to the torsion angle
98 0.9 i

0

TV .vrc=0.68. f=270K

0.8

SO 0>

^

0.7 -

^ 0.6

\

\

...... \

uiicoiTclatctl rigid groups, s rc= 0.53

/

0.5 X..

0.4 50

Figure 10.

100

150

^///deg

Distribution of the torsion angle xp.

prominent in systems including £?r, is obviously related to the relative abundance of the various conformational arrangements in the nematic liquids. In particular, conformations with a\ < 60° and a^ < 60° (type A; see Figure 11) or with a\ > 120° and #2 > 120° (type C) are expected to be preferentially found with ij) > 120° in the nematic liquids, while conformations with OL\ < 60° and a 120° or vice versa (type B) are expected to be preferentially found with ip < 60°. The abundance of trimer conformations in the various simulated systems has been classified in Ref. [46] according to the ranges of ai, «2 and if). The proportion of conformations A and C with ij) > 120° in the nematic phase phase of system To has been found nearly twice that of the same types with ip < 60°. The opposite is true for conformations of type B. Conformations of types A and B are obviously less in systems including ET than in system To, while conformations of type C are more abundant (as dictated by the distribution of a in Figure 9). Overall, the proportion of conformations with ip > 120° increases going from system To to systems with ET. It increases even more with increasing the order parameter of the rigid groups, thus explaining the behavior of /^ in Figure 10.

99

Type A Type B

Figure 11.

2,

TypeC

Conformations of types A, B and C (see text).

Dimers of series I

Dimers of series I have been simulated using the methods and parameters described for the trimers, except for the conformational terms. Distributions of r and I/J appropriate to confer on the simulated dimers conformational characteristics similar to those of dimers of series I with n = 6 and n = 7 have been obtained by analyzing the results of atomistic Monte Carlo calculations for the molecular fragments 0 - R - OOC(CH2)nCOO - R - 0 n = 6,7 R being the Ph - C(CH3) = HC - Ph group and Ph the p-phenylene group. These calculations have been performed using a force field in which: a) the methylene groups in the spacers are represented as united atoms; b) all conformational parameters arefixedat literature values [15, 18, 47, 49, 50], except for the bond angles at the methylene groups and for the torsion angles around the bonds CH2 — CH2, R — O and OC - CH2; c) the bending and torsional contributions to the conformational energy are evaluated as in Ref. [47] for the polymethylene spacers, while the torsional contributions for the R - 0 and OC - CH2 bonds are evaluated according to a slightly modified version of the RIS models used in Refs. [15] and [18]; d) 12-6 Lennard-Jones non bonded interactions are evaluated (with the parameters given in Refs. [47, 49] and [50]) only for atoms separated by four bonds. Since the transition temperatures

100 for the dimers of series I with n = 6 and n = 7 are 460K and 426K, respectively, the calculations have been performed at 450K. The resulting distributions of r and (p (absolute value) are shown in Figure 12. The distribution of r is characterized for both dimers by a single maximum centered around 140°, i.e. shifted at angles approximately 20° higher than the maxima of the curves in Figure 3. It is also interesting to note that the maxima for r in Figure 12 are much sharper than those in Figure 3, although the calculations are performed here at a higher temperature (450K, compared to 320K in Figure 3). This indicates that the conformational freedom of oligoesters of series I is substantially restricted, as far as the angles r are concerned, with respect to the trimers with partly idealized polymethylene spacers described in Section 1. The distributions of ip shown in Figure 12 are also peculiar, since they are characterized by the fact that torsion angles near 0° are preferred both for n = 6 and n = 7 over all other conformations. On the other hand, this general trend is modulated according to a behavior reminiscent of that observed in Figure 4 for polymethylene spacers, in the sense that torsion angles near 50° and near 180° are favored and those near 0° and near 110° are disfavored for n = 6 (see the case of even spacers in Figure 4), while the behavior is nearly the opposite for n = 7. Energies ET and E^ to be used in the simulations of oligomers of series I can be easily obtained from the data plotted in Figure 12 with the methods of Section 1.1. However, a realistic description of the conformational distribution of these oligomers cannot be given without taking into account correlations between r and (p. The existence of correlations of this kind is illustrated in Figure 13, showing (as an example) two difference probability maps obtained from the analysis of the molecular fragment with n = 6. The map on the left side shows the difference P(T, (p) — P{r)P{tp), while the map on the right side shows the difference P(juT2) — P(Ti)P{T2), P being the absolute probability that the considered conformational parameter/parameters is/are found in given 5° intervals. A (+) or a (-) sign marks regions in which the differences are positive or negative, respectively. The first map is obviously symmetric with respect to (p = 0°, i.e. P{r,(p) = P(T, —(p) while the second map is symmetric with respect to T\ = T2, i.e. P(TI,T2) = -P(T2,TI). However, symmetry was not imposed in the analysis, and the nearly perfect symmetry of the maps indicates that the calculation error is very small. The map in the right part of Figure 13 indicates a strong tendency of the two angles T\ and r^ to be preferentially in the same range. On the other hand, the left part of figure 13 clarifies that the relatively abundant conformations with torsion angle near 0° are mainly realized with

101 values of T\ and T
150 .,

180

7V/?/deg Figure 12. Distributions of r and

102

180°

i1 \

f

\L \ MM 33

A

80°

P

4 M t.VI!l

Inl •Tnr f • ill 1tmr Ml" win'

Figure 13.

act1/

iHII Bilti


/.

*mi imni

-\

L

•

-) EL

•

•in 1KB

s

IIHSP 1 >$»•'/ \ « 1 •Kill m*W JIUlHk. ^ B l L. J & « simL j fiSB

•• ••

0c

ABML 180°

FT

tw

1 \m Ii w Mnn

( \

\

80°

P

'2

100°

• •

100°

180°

Two difference probability maps obtained for n=6 (see text).

particular, values of a near 20° and near 120° are favored and values near 60° and 160° are disfavored for odd members, while the opposite is true for even members. On the basis of the results obtained for the idealized trimers described in Section 1, it is apparent that oligomers with even spacers will give nematic phases by selecting conformations with a close to 160°, while oligomers with odd spacers will select both conformations with a close to 20° and to 160°. However, the nematic phase is expected to be more stable for oligomers with even spacers than for oligomers with odd spacers. A simple, but approximate method for taking into account these correlations when simulating the behavior of systems of oligomers of series I consists in the introduction of an additive contribution Ea = + C, where faj is the value evaluated in the absence of -RT\n(fa/faj) correlations (dotted curves in Figure 14), T = 450K and C is a constant. One has to remark that this is nothing more than a first approximation, given that a is not an independent conformational parameter. Therefore, adding Ea may alter to some extent the distributions of r and ip with respect to those that would be obtained using only ET and ii^.The preliminary simulations of liquids of dimers of series I with n = 6 and n = 7 described here have been performed utilizing the conformational energy contributions shown in Figure 15. Although the atoms to be considered part of a rigid group in the compounds under study are not perfectly defined, the inspection of molecular models suggests that the rigid groups can be roughly simulated by a sequence of four isodiametric units. Also, by evaluating the volume

103

0.0 30

60

90

150

120

180

a/deg Figure 14-

Distribution of a for dimers of series I with n = 6 and n = 7.

3.0 -

E/RT 2.5 -|

V

2.0

\ / V

/ >

•

,

1.5 ~ 1.0 0.5 0.0 0

30

60

90

120

150

180

r,

104 occupied by various molecular fragments, one can see that the volume fraction of rigid groups is on the order of 0.60 in the monomers and 0.65 in the dimers. Taking into account thermal expansion (the transition temperature of the monomers is approximately 70K higher than room temperature), simulations have been performed for a system of monomers containing 9140 rigid groups in a periodic cubic cell of edge 40cr (i.e. with a volume fraction of rigid groups 0.57). The transition temperature of this system of monomers changed obviously by changing the value of the mean field parameter J4, and was equal to 370K (i.e. in the range of the transition temperatures of monomers of series I) when A = 10 kJ mol"1. Considering that the volume fraction of rigid groups in the dimers is higher than in the monomers, but the transition temperature is 70 — 100K higher, the simulations of dimers have been performed with 4570 molecules in a cubic cell of edge 40cr (fraction of rigid groups 0.57, as for the monomers) and with A = 10 kJ mol"1 (note that A should increase by increasing the fraction of rigid groups). With this set of parameters and with the conformational energy contributions shown in Figure 15, the transition temperature of the dimer with n = 6 turns out to be in the 460 — 465K range, practically equal to the experimental value shown in Table 1. However, this is little more than a mere coincidence, since the corresponding transition temperature of the dimer with n = 7 is as low as 345 — 350K, well below the experimental value (426K) and lower than the transition temperature of the monomers. The latter finding is not unreasonable, since the simulations have been performed with the same density of rigid groups for monomers and dimers, and Figure 14 suggests that the nematic phase of the dimer with n = 7 should be disfavored in this situation with respect to that of the monomer. On the other hand, if one assumes that thermal expansion and higher volume fraction of rigid groups in the dimer molecules compensate exactly for the dimer with n = 6 at 460K, they are obviously unbalanced for the dimer with n=7 at 350 K. Therefore, the density of rigid groups (and the value of the mean field parameter A) should be increased in the latter case, thus increasing the transition temperature to values closer to experiments. A preliminary study of the conformational distributions in the simulated dimers indicates, however, that the situation is more complicated, since the conformational energy contributions shown in Figure 15 turn out to be partly inadequate to represent the conformational characteristics of oligomers of series I, particularly in the case of the dimer with n = 6. For instance, the abundance of torsion angles in the 140 - 180° range in the isotropic liquids is found to be higher than in Figure 12 when n = 6, and lower when n = 7. In other words, the conformational

105 characteristics dictated by the energy contributions of Figure 15 favor the nematic phase more than required for the dimer with n = 6 and disfavor it more than necessary for n = 7. This is a consequence of the fact that correlations between T\ , T2 and cp have been handled in the simplifying approximation of adding the energy contribution Ea (see before), and indicates that realistic simulations of oligomers and polymers of this kind can be only performed by utilizing directly maps like those shown in Figure 13. It also reveals that the behavior of the simulated systems is sensitive to minute details of the conformational characteristics of the model molecules, in good agreement with the experimental finding that oligomers and polymers with slightly different spacers can show quite different behaviors.

3.

Conclusions

The onset of nematic order in liquids of oligomers and polymers consisting of rigid groups separated by semiflexible spacers is an extremely complex phenomenon regulated by a delicate balance of orientational and conformational effects. Since the latter effects play a key role, the behavior of these systems cannot be explained on the basis of theories originally developed for rigid molecules. On the other hand, the conformational characteristics dictated by the presence of a given spacer are specific for that spacer, with the consequence that the predictive capability of any theoretical approach would be scarcely useful for systems of this kind, unless combined with studies performed by conformational analysis and simulation methods. Simulating the occurrence of reversible nematic/isotropic phase transitions in liquids of conformationally flexible molecules requires very large systems. Therefore, the model molecules have to be kept as simple as possible. The large-scale simulations performed until now are based on a substantially idealized model in which the rigid groups are represented by linear sequences of Lennard-Jones centers, the anisotropic attractive forces among rigid groups are modeled as a mean field and the spacers are represented by virtual bonds between consecutive rigid groups. The model takes into account the two factors responsible for the stabilization of nematic order in liquids of rigid molecules (anisometry of the rigid groups and anisotropy of their attractive interactions), and allows to modulate the conformational preferences of the virtual bonds between consecutive rigid groups in such a way to study ideal or realistic systems with semiflexible spacers. The four systems investigated at first consist of trimers in which the conformational characteristics are described in terms of bending energy

106 contributions ET (depending on the various angles r between the rigid groups and the virtual bonds representing the spacers) and of torsion energy contributions E^ (depending on the torsion angles

, while adding E^ has opposite effects in systems T5 and T6, i.e. the transition temperature decreases for system T5 and increases for system TQ with respect to system T5. On the basis of conformational energy calculations performed for (CH2)n spacers of various lengths, the transition temperature of the trimers is expected to oscillate with increasing n, being higher for spacers with n even and lower for spacers with n odd. This behavior originates from the fact that the isotropic/nematic transition is coupled with a conformational selection disfavoring torsion angles in the [60°, 120°] range and favoring those near 0° or 180° in the nematic phase with respect to the isotropic liquids, in such a way that the transition takes place with the assistance of E^ for even members of the (CH2)n series of spacers, while it takes place in contrast to E^ for odd members. The conformational selection taking place at the phase transition is not limited to the torsion angles around the virtual bonds, but is also extended to molecular segments comprising two or more rigid groups. In fact, the distribution of the angle a between two consecutive rigid groups in the same molecule is substantially different in the isotropic and in the nematic phase of systems including ET. This is not simply related to the preferential orientation of the rigid groups along the director, but is also due to an intrinsic preference for extended conformations over compact conformations in the case of the simulated spacers. Longer range effects are also observed for the angle between the two terminal rigid groups of a trimer molecule and for the dihedral angle defined by the directions of the three rigid groups [46]. The calculations performed for the trimers were not meant to simulate specific mesogenic substances. Being the first of this kind, their goal was to elucidate the general aspects of the behavior of mesogenic oligomers by studying a few simple idealized systems. Calculations using similar

107 models, but aimed at explaining the behavior of the well characterized series I of mesogenic oligoesters, are currently underway. A necessary preliminary step for the simulation of real systems of this kind consists in the careful analysis of the conformational distribution of suitable molecular fragments. Once the conformational characteristics to be modeled are known, the various contributions to the total energy of the model oligomers have to be regulated in such a way that the conformational distribution in the isotropic liquids is close to that for the molecular fragments studied. The preliminary results obtained from simulations of monomers and dimers of series I show that this is a critical step, in the sense that the behavior of the model dimers is strongly influenced by approximations eventually utilized to simplify the conformational part of the force field. Calculations in which the conformational distribution of the oligoesters is represented at a more accurate level are currently in progress.

Acknowledgments The financial support of the Ministero delPUniversita e della Ricerca Scientifica of Italy is gratefully acknowledged.

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108 [18] D.Y. Yoon and S. Bruckner, Macromolecules, 18:651, 1985. [19] P. J. Flory, Advances in Polymer Science, 59:1, 1984. [20] P.J. Flory, Mat Res. Soc. Symp. Proc, 134:3, 1989. [21] D.Y. Yoon and P.J. Flory, Mat Res. Soc. Symp. Proc, 134:11, 1989. [22] D.J. Photinos, E.T. Samulski, and H. Toriumi, J. Chem. Soc. Farady Trans., 88:1875, 1992. [23] K. Nicklas, P. Bopp, and J. Brickmann, J. Chem. Phys., 101:3157, 1994. [24] A. Ferrarini, G.R. Luckurst, P.L. Nordio, and S.J. Roskilly, Mol. Phys., 85:131 [10], 1995. [25] H.S. Serpi and D.J. Photinos, J. Chem. Phys., 105:1718, 1996. [26] A. Ferrarini, G. R. Luckurst, P. L. Nordio, S. J. Roskilly, Liq. Cryst, 21:373, 1996. [27] D. Frenkel, H.N.W. Lekkerkerker, and A. Stroobants, Nature, 332:822, 1988. [28] D. Frenkel, Mol Phys., 60:1, 1987. [29] M.P. Allen, D. Frenkel, and J. Talbot, Comput Phys. Rep., 9:301, 1989. [30] D. Frenkel, Liq. Cryst, 5:929, 1989. [31] P.J. Camp, M.P. Allen, and A.J. Masters, J. Chem. Phys., 111:9871, 1999. [32] M.P. Allen, J. Chem. Phys., 112:5447, 2000. [33] G.R. Luckurst, R.A. Stephens, and R.W. Phippen, Liq. Cryst., 8:451, 1990. [34] M.P. Neal, M.D. De Luca, and CM. Care, Mol Simul, 14:245, 1995. [35] R. Berardi, H.G. Kuball, R. Memmer, and C. Zannoni, J. Chem. Soc. Faraday Trans., 94:1229, 1998. [36] J.G. Gay and B.J. Berne, J. Chem. Phys., 74:3316, 1981. [37] S.J. Picken, W.F. Van Gunsteren, P.P. Van Duijnen, and W.H. De Jeu, Liq. Cryst, 6:357, 1989. [38] M.R. Wilson and M.P. Allen, Liq. Cryst, 12:157, 1992. [39] S.S. Patnaik, S.J. Plimpton, R. Pachter, and W.W. Adams, Liq. Cryst, 19:213, 1995. [40] M. Yoneka and Y. Iwakabe, Liq. Cryst, 18:45, 1995. [41] Z. Wang, J.A. Lupo, S. Patnaik, and R. Pachter, Comput. Theor. Polym. Sci., 11:375, 2001. [42] M. Vacatello and M. Iovino, J. Chem. Phys., 104:2721, 1995. [43] M. Vacatell and M. Iovino, Liq. Cryst, 22:75, 1997. [44] M. Vacatello and G. Di Landa, Macromol. Theory Simul, 8:85, 1999. [45] M. Vacatello, Polym. Mat Sci. Eng., 85:442, 2001. [46] M. Vacatello, Macromol Theory Simul, 11:501, 2002. [47] M. Vacatello, D.Y. Yoon, and B.C. Laskowski, J. Chem. Phys., 93:779, 1990. [48] U.W. Suter, E. Saiz, and P.J. Flory, Macromolecules, 16:1317, 1983. [49] M. Vacatello and P.J. Flory, Macromolecules, 19:405, 1986. [50] M. Hutnik, A.S. Argon, and U.W. Suter, Macromolecules, 24:5956, 1991.

MOLECULAR ARRANGEMENTS IN POLYMER-NANOFILLER SYSTEMS Michele Vacatello and Manuela Vacatello Dipartimento di Chimica and INSTM Research Unit, Universitd di Napoli, Via Cintia, 180126 Napoli (Italy) [email protected]

Abstract

Polymers containing randomly distributed spherical filler particles have been simulated by Monte Carlo methods for various particle sizes (4 to 28 times the transverse diameter of the polymer chains) and partial volumes of filler (10% to 50%). The polymer/filler interface consists of densely packed and partly ordered shells of polymer units of thickness nearly twice the diameter of the units. A number of parameters characterizing the molecular arrangements in these systems have been analyzed, leading to a general picture in which the chains are considered to be sequences of interface, bridge and loop segments. The results can be approximately predicted on a quantitative level using a few simple rules. It is also shown that phantom chains can be utilized in the simulations, provided that the interaction energy between chains and filler is modified in order to counterbalance the intrinsic tendency of the chain segments to avoid the filler surfaces. This makes possible to study systems that cannot be simulated at full density (i.e. systems with long chains, and/or with large particles and small filling density).

Introduction Reinforced polymers, i.e. polymers filled with solid particles, are a broad class of materials showing enhanced mechanical properties, with the extremely valuable feature that these properties can be tailored to the needs in a wide range by modifying the chemical nature of polymer and filler, the size and shape of the filler particles and the density of filling. Therefore, they are extensively utilized for a variety of technologically important applications and represent a significant portion of the modern polymer market. It is then quite remarkable that progress in this field has been mainly founded on empirical correlations and that a comprehensive theory explaining the structure/properties relationships of these materials has been not developed or even outlined until now. 109 P. Pasiniet al. (eds.), Computer Simulations of Liquid Crystals and Polymers, 109-133. © 2005 Kluwer Academic Publishers. Printed in the Netherlands.

110 As a matter of fact, the numerous theoretical studies trying to elucidate one or another aspect of the mechanism of reinforcement on one side (see Refs. [1-9], as an example), and the interpretation of the vast body of experimental data accumulated in the last decades on the other side, have been based on a variety of scarcely related assumptions and approximations, due to the absence of a generally accepted physical picture of the mutual arrangements of polymer chains and filler particles. This picture is now emerging from computer simulations of polymer-nanofiller systems carried out in the last few years for various lattice [10-13] and out-of-lattice [14-20] models. One of the problems in simulating filled polymers is the fact that the solid particles are much larger than the transverse diameter of the polymer chains, with the consequence that meaningful simulations with realistic densities of filler and polymer may require the explicit representation of an extremely large number of polymer units. Being simpler and much faster, lattice calculations can handle these large systems, allowing to study the general features of complex phenomena such as the phase behavior of filled polymers, polymer mixtures and copolymers. Out- of-lattice calculations are obviously more demanding in terms of computing resources, but are also better suited for investigating more local properties such as the molecular arrangements of polymer chains and solid particles, the characteristics of the polymer/filler interface or the conformations of polymer segments in contact with the filler. Due to computational limitations, the Molecular Dynamics method has been used only for systems containing one single filler particle [15, 20] while larger systems with randomly distributed filler nanoparticles of various sizes at various filling densities have been simulated [14, 17, 18] using the Monte Carlo method, less limited in this respect. These calculations are reviewed in Section 1. On the other hand, dense systems with realistically long polymer chain and realistically large filler particles cannot be directly simulated, because the equilibration time and the number of polymer units to be represented grow rapidly out of proportion with increasing the chain length and/or the particle size. It has been recently shown that systems with very long chains and large particles can be studied using phantom chains, provided that the interaction energy between chain units and filler particles is properly modified [19]. Calculations performed using phantom chains are described in Section 2.

Ill

1. 1.1

Simulations of dense systems Models and methods

The composition of the dense systems that have been simulated up to now is summarized in Table 1. The polymer chains are modeled as unbranched sequences of 100 isodiametric units connected by links of length a, while the filler particles are modeled as spherical entities with diameter cry. The simulated systems consist of three-dimensionally periodic arrays of cubic cells of edge 40 a containing Np polymer chains and Nf filler particles. The polymer units interact through a 12 — 6 Lennard-Jones potential Euu = e[(a/ruu)12 — 2(a/ruu)6], where ruu is the distance between the interacting units and the minimum distance allowed between units is rUUyTnin = 0.70a. The filler particles interact with the polymer units through the potential Euj = e[(a/r u /) 12 — 2(cr/r a /) 6 ], where ruf is the distance of the given unit from the surface of the particle (with ruf,min — 0.70a). When useful, the filler particles interact with other filler particles through the potential Eff = e[(&/rff)12 — 2(cr/rjj)6], where rjf is the distance of the surfaces of the two particles (rffiTnin = 0.70cr). A similar expression, with the same value of e, is used for all interactions considering that the mutual arrangements of particles and chains in systems with proper density are expected to depend mainly on excluded volume and chain stiffness. The stiffness of the polymer chains is regulated by a bending potential of the form E(9) = (l/2)k$92, with 9 the angle between consecutive links (9 = 0 for collinear links). All interactions are truncated at the distance 2cr, where they are practically negligible. Calculations are performed with e/RT = 0.125 and kg/RT = 1.00 rad~2, in such a way that the softness of the polymer units and the chain stiffness are comparable to those in real systems. The values of tp in Table 1 are evaluated considering that the first layer of chain units in contact with the filler surfaces is found at ruf « a, such that the effective diameter of the filler particles is close to aj + a. System Mo (Table 1), containing no filler particles, represents the reference polymer melt. If the chain units are assumed to be polymethylene isodiametric units (i.e. each unit contains 3.5 CH2 groups and a = 0.45nm, according to Ref. [21], the simulated chains consist of 350 CH2 groups and the overall density (890kg/m3) corresponds to experimental values for long chain alkanes. This system has been initialized by placing the 640 molecules in the base cell at random in such a way that the distance between any two units was not smaller than ^uu^min- It has been then equilibrated in the canonical (NVT) ensemble by Monte Carlo methods, using the reptation technique. At each step, a new isodiametric unit is added to a chain end chosen at random, while

112 Table 1. Composition of the simulated dense systems (Np = number of polymer chains of 100 units; Nf — number of randomly distributed spherical filler particles; aj = diameter of the particles;

System Mo M 4 ,20 M 6 ,20

M 8 ,io M 8 ,20 M 8 ,30 M 8 ,36 M 8 ,50 Mio,2O Mio,36 Mi2,20 Mi6,36 M28,20

Np 640 512 512 576 512 448 410 320 512 409 512 409 512

Nf 0 196 72 17 34 50 60 84 18 34 11 9 1

4a 6a 8a 8a 8a 8a 8a 10a 10a 12a 16a 28a

the unit at the opposite end of the same chain is deleted (the newly generated bond angle is selected at random from the distribution dictated by the value of ke/RT. The trial configuration is immediately rejected when the new unit is closer than ruu^min to any other unit; otherwise, the total energy change is evaluated and the new configuration is accepted or rejected according to the outcome of a standard Boltzmann test. The filled systems have been initialized by placing first the Nf spherical particles in the base cell at random in such a way that the minimum distance between their surfaces was 0.7a. For system Ms,5o, the particles have been placed at the nodes of a face-centered cubic lattice and equilibrated by simple Monte Carlo methods with the potential Eff truncated at Tff = a (only repulsive interactions included). The chains have been subsequently added as in the case of system Mo in such a way that the minimum distance between polymer units and filler surfaces was not smaller than ruf^min. The systems have been then equilibrated by reptation, with the additional constraint that the trial configuration was rejected when the new terminal unit was closer than rufiTnin to the surface of a filler particle. The position of the filler particles was fixed. A MC cycle is defined to consist of 106 attempted reptation moves per chain. Due to the high density, the fraction of accepted moves is only 0.031 for system Mo; it decreases even more with increasing filler content, being as low as 0.014 for system Mg,5o- On the other hand, all units are displaced in 12 cycles in the case of system Mo (i.e., no units are left having the same coordinates of any unit of the same chain before the 12 cycles), with more than 95% of the units displaced in 6 cycles.

113

0.5

0.0 10

12

14

16

l

20

*r/a

Figure 1. The normalized radial distribution function of the polymer units with respect to the filler particles for systems Ms,io, Ms,5o and Mi6,36, (/?, left scale) and the volume fraction of spherical shells not occupied by filler particles in system Ms,50 (/„, right scale) as a function of the distance r of the shell from the center of a filler particle (shell thickness = O.lcr).

The efficiency is progressively diminished in filled systems of increasing content of filler. For instance, it takes 15 cycles to displace 95% of the units in system Ms^o, and more than 75 cycles in system M%^. The results have been averaged over 200 cycles for system MQ and over a number of cycles from 200 to 550 for the other systems, depending on the overall mobility of the chains. This mobility was anyhow large enough to ensure that averages obtained from the MC samples are representative of the equilibrium state for all systems studied.

1.2

The filler/polymer interface

Figure 1 shows the normalized radial distribution function p = pr/(l — ip) of the polymer units with respect to the filler particles for three simulated systems (Msjo, Af^so and Mi6,36; left scale). pris the density of polymer units in a spherical shell of thickness 0.1a at a distance r from the center of a filler particle; for large values of r, pr is expected to be equal to the average density of polymer units in the given system (= 1 —
114 Not unexpectedly, Figure 1 shows that the behavior of p near the surface of thefillerparticles is qualitatively similar to that found for polymer melts near planar solid surfaces [22-24], i.e. it is characterized in all cases by a series of maxima and minima of progressively decreasing intensity with a periodicity approximately 0.8a. This is true in particular for system Ms, io, in which p reaches its bulk value within approximately 2a from the surface of the particles, as found near planar solid surfaces. The behavior is quite different in the other two cases. In fact, the curve for system Mi6,36 shows a monotonous decrease following the initial series of maxima and minima, and the value of p for r = 20a is significantly less than unity, while the curve for system Ms,so shows a complex behavior characterized by the superposition of the series of maxima and minima with periodicity 0.8a with a second series of broad maxima and minima with a periodicity approximately 4a. A correspondingly complex behavior is observed for the other systems studied, such that intrinsically different p curves are obtained for different systems. In particular, the complexity of the curves is found to increase with increasing size and volume fraction of the filler. This behavior is not related to the particular arrangements assumed by the particles in the simulations that have been performed, but is a general feature of systems polymer/nanofiller systems. In fact, the volume of the various shells centered on a given filler particle is partly occupied by other filler particles, with the consequence that p depends on the volume fraction (/„) of each spherical shell which is effectively available to the polymer units. The behavior of /^ as a function of r changes substantially with changing the size of the filler particles and the volume fraction of filler. The full curve in Figure 1 (right scale) plots as an example the values of fv/(l — (p) obtained from Monte Carlo simulations of a large number of configurations of very large systems with composition corresponding to that of system Ms^o- I n order to evaluate /„, the filler particles have been assumed in these calculations to be hard spheres with diameter aj + 1.6a. It is apparent that the curve of p for system M$^o is practically coincident with the corresponding curve for system Ms,io multiplied by the plotted values of /i//(l — (p). The same is obviously true for system Ms,io itself, in which fu, is very close to (1 - (p). for all values of r, and for the other systems listed in Table 1. In conclusion, when the distribution of the space left empty by the filler particles is properly taken into account, the behavior of p is both qualitatively and quantitatively very similar to the behavior observed in polymer melts near planar solid surfaces. As remarked in Ref. [14], the high density of the first shell of units in contact with the filler particles (or with planar solid surfaces) is neces-

115 0.1 7

0.0 -

-0.1 'Mi, -0.2

-•

1

16,36

-0.3 -

-0.4

-0.5 13

15

r/ a

19

Figure 2. The order parameter (s) of chain segments comprising five consecutive units as a function of the distance of their center of mass from the center of the closest filler particle

sarily associated with orientational correlations among spatially neighboring chain segments and/or among these segments and the surface of the filler particles. This is exemplified in Figure 2, plotting for two simulated systems (Mio,36 and Mi6,36) the order parameter of chain segments comprising five consecutive units as a function of the distance of their center of mass from the surface of the closest filler particle. The order parameter is defined as s = (3(cos2 a) — l)/2, with a the angle between the end-to-end vector of a segment and the vector radius of the filler particle through the center of mass of the segment; 5 is 1 for segments parallel to the vector radius, 0 for random disorder and -0.5 for segments perpendicular to the vector radius, respectively. Figure 2 shows that the average value of s in the first high density layers is markedly negative. Similar curves are found for the other simulated systems, indicating in all cases a comprehensible tendency of chain segments in contact with the filler particles to run parallel to the surfaces of the particles. It is concluded that the polymer units at the interface with the filler are arranged in densely packed and partly ordered shells of thickness approximately 2a similar to the layers found for the same model polymers near planar solid surfaces in Refs. [22-24] and

116

70

80

90

100

Figure 3. Plots of R^/ncr2 vs. n, with R^ the mean square distance of units belonging to the same chain and separated by n links, for a number of simulated systems.

for shorter coarse-grained chains near the surface of a single nanoparticle inRef. [15].

1.3

Chain conformation

Figure 3 plots the values of R^/ncr2 vs. n, with R^ the mean square distance of units belonging to the same chain and separated by n links, for the reference polymer melt and for filled systems with various values of

117 obstacles. According to the latter calculations, the chain dimensions should decrease when the particles are large and the chains are short, while they should show a considerable increase for small particles and long chains. More recent single chain RIS calculations [16] do not confirm this trend. Furthermore, it has been shown [14, 16] that calculations performed for single chains in the presence of solid obstacles are inadequate to describe the molecular arrangements and conformations in real systems, since they reveal mainly the behavior of chains running far from the obstacles (see Section 2). It is concluded that the large increase of chain dimensions found in SANS experiments [27] on poly(dimethyl siloxane) filled with polysilicate nanoparticles cannot be explained on the basis of simple excluded volume arguments.

1.4

Molecular arrangements

Figure 4 shows a snapshots of system Mi6,36 at thermal equilibrium. The filler particles have been reduced in size (except for one in the lower right corner) in order to show the arrangements of some selected chains with respect to the particles and their conformations. The chain units in white are interface units, i.e. they are centered in the interface shells 2cr wide surrounding the particles. Approximately 33% of the polymer is constituted of interface units in this system, while the corresponding proportion for the other systems in Table 1 ranges from as low as 8% in system M28,20 to more than 85% in system M^o. Figure 4 shows that each chain visits the surface of several filler particles, and that the chains can be considered to be sequences of interface segments (chain segments totally running in the interface shell of a given particle), bridge segments (sequences of non-interface units with the two adjoining units in the interface shells of two different particles) and loop segments (sequences of non-interface units starting and ending in the interface shell of the same particle). Of course, the chains can have dangling terminal segments. Also, in densely filled systems, the interface shells of two different particles can be directly connected without an intermediate bridge segment (i.e. two consecutive units of the same chain can be placed in the interface shells of two different particles). Note that the minimum length of a bridge segment is one by definition. The mutual arrangements of polymer chains and particles have been investigated by studying the dependence on af and ip of the following parameters: / i , $2 and /3 (the fractions of polymer units belonging to the interface shells of at least 1, 2 or 3 different filler particles, respectively); ff (the fraction of free chains, i.e. chains with no units in the interface shells); Lj, L& and L\ (the average length, in terms of units, of inter-

118

Figure 4- Partial snapshot of system Mi6,36 at thermal equilibrium; the filler particles are reduced in size, except for one in the lower right corner.

face segments, bridge segments and loop segments, respectively); Nj, Nb and Ni (the average number of interface, bridge and loop segments per chain, respectively); Nd (the average number of direct connections between different interface shells per chain); Pc (the average number of different interface shells visited by each chain); Cp (the average number of different chains having at least one unit in the interface shell of a given particle). Table 2 lists the values of / i , /2 / / , Pc and Cp obtained from the various simulations performed for dense systems (note that Cp = Pc * Np/Nf, i.e. Cp is equal to the total number of contacts between chains and filler particles, Pc * iVp, divided by the number of particles), while Table 3 lists length and number of interface, bridge and loop segments per chain and the values of Nd- The behavior of the various parameters is obviously better understood by plotting them as a function of a/ for a given value of (p or as a function of (p for a given value of Of. Plots of this kind are reported and discussed in Refs. [17] and [18]; only two examples are shown here for brevity, and the discussion is limited to a short summary. Not unexpectedly, Table 2 shows that f\ and fa increase with increasing

119 Table 2. The fraction of polymer units belonging to the interface shells of at least 1 or 2 different filler particles (/i and /2, respectively), the fraction of free chains (//), the average number of different interface shells visited by each chain (Pc) and the average number of different chains having at least one unit in the interface shell of a given particle (Cp) obtained from the simulations of dense systems

System

h

M 4 ,20 M 6 ,20

0.58 0.41 0.14 0.29 0.46 0.59 0.86 0.22 0.51 0.18 0.33 0.08

M8,io M 8 ,20 M 8 ,30 M 8 ,36 M 8 ,50 Mio,2O Mio,36 Mi2,20 Mi6,36 M28,20

h 0.13 0.04 0.005 0.02 0.06 0.10 0.35 0.01 0.06 0.01 0.01 0.

// 0. 0.001 0.22 0.01 0. 0. 0. 0.08 0. 0.11 0.005 0.44

Pc

3.4 5.2 1.5 3.3 5.2 6.4 9.4 2.3 4.9 1.7 2.7 0.6

Cp 25 37 51 50 47 44 36 65 59 81 123 290

that the interface shells of adjacent particles are not substantially superimposed when (p is small and the particles are large. Therefore, the molecular arrangements in these systems are expected to be relatively simple. Densely filled systems with relatively small nanoparticles are obviously more complicated. For instance, the overall additive volume of Nf spheres of diameter aj + 4 centered on the particles (i.e. spheres including the interface shells) would be approximately 71% of the total volume is system Ms?3o and greater than the total volume in system Ms,5o- Therefore, the interface shells of adjacent particles are largely superimposed in these cases (as confirmed by the high values of /2), giving rise to a variety of extremely complex local arrangements. It is also interesting to note that nearly 45% of the chains in system M28,20 and more than 20% in system Ms,io are free chains. The abundance of free chains decreases rapidly with increasing (p and with decreasing 07, being only 11% in system Mi2,20 and 1% in system Ms^o- As far as Pc is concerned, it increases as expected with increasing

120 Table 3. The average length of interface, bridge and loop segments (1^, Lb and L/, respectively), their average number per chain (A^, Nb and TV/, respectively) and the average number of direct connections per chain (Nd) obtained from the simulations of dense systems

System M 4)2 o M6,2o M 8 ,io M8,20 Ms, 30 M8,36 M8,50 Ml 0,20 Mio,36 Mi2,20 Mi6,36 M28,20

0.0

Figure 5.

0.1

U 3.3 3.7 4.1 4.0 3.8 3.7 3.2 4.2 4.0 4.2 4.5 4.8

Lb 5.5 8.1 5 12 7.3 4.9 2.4 14 6.6 16 12 42

U 2.8 4.3 7.2 5.6 4.2 3.4 2.1 6.4 4.0 7.3 5.7 9.7

Nj 18 11 3.3 7.4 12 16 27 5.4 13 4.3 7.3 1.6

Nb 2.4 4.4 0.8 2.6 4.4 5.5 4.5 1.6 4.5 1.0 2.4 0.05

Ni 3.1 2.7 1.3 2.4 2.8 2.9 1.4 2.1 3.1 1.7 3.1 1.0

Nd 7.8 2.8 0.4 1.5 4.0 6.5 20 0.8 4.2 0.6 0.7 0.

0.2

Plots of L/, Li and Lb as a function of y> for dense systems with 07 = Scr.

Figures 5 and 6 plot the average lengths of interface, bridge and loop segments as a function of

121 18 16 -

--^r*

1 4 ••!

Lb

12 f 10 i

-

/>"-

i

I II^:::::^

""

L, ^ *—) 8.0 Figure 6.

10.0

crf

12.0

Plots of L/, L\ and Lb as a function of 07 for dense systems with ip = 0.20.

be comparable to that in widely used filled polymers. Figures 5 and 6 show that this length is not far from 4 units for most of the simulated systems. Smaller values are found in extremely crowded systems or when the particles are very small, while larger values are found with large particles (note that the corresponding value for the same model polymer near planar solid surfaces is approximately 5 [22]). In practice, the length of the interface segments is 3.5-5 units for filled polymers of practical interest, and increases within this range with increasing the size of the particles. Considering that f\ decreases with increasing 07 for a given value of y>, it is clear that the number of interface segments per chain is expected to show a corresponding decrease (see Table 3). The average length of the bridge segments shows a marked decrease with increasing

while they are as short as 2.4 units in system Ms,5oThis behavior is rather obvious, since the length of the bridge segments is expected to depend on the average distance between the surfaces of neighboring filler particles. On the other hand, bridge segments are approximately equal in length (12-13 units) in systems Ms,20 and Mi6,36The same is true for systems Mg^o and Mio,36, where the length is nearly 7 units. Therefore, the length of the bridge segments (a fundamental

122 parameter for the elastic behavior of filled polymers) appears to be a function of the ratio af/(p, a n d decreases with decreasing this ratio. Figures 5 and 6 also show that loop segments are always shorter than bridge segments, and that their average length decreases with increasing (p and increases with increasing crj. In practice, with the exception of extremely crowded systems, the length of the loop segments is roughly half that of the bridge segments. The data listed in Tables 2 and 3 can be utilized to extract a general picture of the molecular arrangements in polymer/nanofiller systems, best described in the form of simplified two-dimensional schemes of the kind shown in Figure 7. The distribution of filler particles in these schemes is arbitrary, but consistent with the results of the simulations, and the chains are ideal average chains, in the sense that number and length of the various kinds of segments and all other features correspond approximately to the average values found in each case. The scheme in Figure 7 corresponds to system Ms,3o- The interface shells of adjacent particles are very close in this system, and partly superimposed in most cases (the overall additive volume of Nf spheres of diameter aj + 4cr centered on the particles is as high as 71% of the total volume). In fact, nearly 15% of all interface units belong to the interface shells of two different particles. The average chain is in contact with 5.2 different particles, and contains as many as 12.2 interface segments. These are connected by 4.4 bridge segments, 2.9 loop segments and 4.9 direct connections. The lengths of interface, bridge and loop segments are 3.8, 7.4 and 4.2 units, respectively. The average chain also contains terminal segments of total length 9 units. Accordingly, Figure 7 shows 5fillerparticles arbitrarily distributed in two dimensions in such a way that their interface shells are very close and partly superimposed. The schematic chain is in contact with all these particles and contains 13 interface segments of length 4 units, 4 bridge segments of length 7 units, 3 loop segments of length 4 units, 5 direct connections between different interface shells and two terminal segments of total length 8 units. Similar schemes can be obviously obtained from the data in Tables 2 and 3 for all systems that have been simulated and, by interpolation, for systems with intermediate compositions.

1.5

Predicting the molecular arrangements

The results obtained from the simulations of dense systems have also enabled to establish a set of simple approximate rules allowing to predict the molecular arrangements in polymer/nanofiller systems, provided that thefillerparticles can be considered nearly spherical and distributed

123

Figure 7. Schematic two-dimensional picture of the mutual arrangement of filler particles and chains in system Ms,30- The dotted circles delimit the interface shells of the various particles; the interface units are shown in white.

at random. These rules, roughly valid for all systems listed in Table 1, with the exception of extreme cases (systems very crowded with small particles or systems containing a large proportion of free chains), are based on a number of considerations that can be briefly summarized as follows. The density of units in a spherical shell between 0.8cr and 2a from the surface of a filler particle is found for all dense systems to be approximately l.l<j~3; therefore, the fraction of interface units is given by h « l.lVsNf/NpLp-(fa + f3 + ...) « lMVs/Vf)ip/(l-
124 particles and 5 units for very large particles, and is roughly given by Li « 5.0 — 6.5£(1 + y>)/V; once Lj is known, the number of interface segments per chain can be evaluated as by Ni « f\Lp/Li. Pc is expected to be equal to the number of filler particles having their center in a sphere of diameter aRg + af + 4a, where Rg is the rms radius of gyration of the chains and a is a constant. In fact, for the dense systems studied, Pc is quite well approximated by Pc «
The number of direct connections per chain is obviously higher in systems where the filler particles are more crowded, and is found to be roughly given by Nj « 100(Ni — l)£ 2 //i- Since on the average in the dense systems simulated the contact of a chain with a given particle consists of two interface segments (separated by a loop or by two consecutive direct connections), the number of bridge segments per chain is approximately JV& « Ni/2 — 1, and the number of loop segments per chain can be evaluated as Ni = N{ — 1 — N^ — N^. Lastly, the length of the bridge segments increases with increasing <jf for a given

, f2 is given by f2 « 40£2 and fx is given by fx « l.l(V3/Vf)(p/{l - (p) - / 2 ; £/ is then given by Li w 5.0 - 6.5f(1 + (f)/
125 Table 4-

Comparison of predicted and found data for three simulated systems. M6,20|

h

h U Lb Li Ni Pc

Nd Nb Ni

Ml2,20

M 8) 30

predicted 0.04 0.41

found 0.04 0.41

predicted 0.01 0.19

found 0.01 0.18

predicted 0.06 0.49

found 0.06 0.46

3.7 8.3 4.1 11 5.4 2.7 4.5 2.7

3.7 8.1 4.3 11 5.2 2.8 4.4 2.7

4.4 17 8.3 4.3 1.8 0.5 1.1 1.7

4.2 16 7.3 4.3 1.7 0.6 1.0 1.7

3.9 7.3 3.7 12 5.0 3.3 5.2 2.9

3.8 7.3 4.2 12 5.2 4.0 4.4 2.8

eral, the agreement tends to be less good in extreme situations, i.e. for crowded systems (due to the large superposition of the interface shells) and for systems with large particles and low

2.

Simulations of phantom chains

The simulations of dense systems have been performed for relatively short chains of 100 units and for relatively small particles. Unfortunately, direct simulations of polymer/nanofiller systems with larger particles and realistically long chains at realistic densities are not feasible, since there are no suitable methods for handling systems of this kind. On the other hand, systems with large particles and smallfillingdensity contain, when the model chains are only 100 units long, a large proportion of free chains and of very long terminal segments. Therefore, they are of little use for predicting the behavior of longer chains. This problem could be solved by simulating long phantom chains in the presence of filler. As a matter of fact, it has been shown in Ref. [16] that the conformational properties of the chains in system Mo are practically unaffected when the interaction energy between chain units (Euu) is turned off and values of 0 higher than 150° are forbidden. In other words, these phantom chains are an excellent model for the unper-

126 turbed chains of the reference melt. However, when this model is applied tofilledsystems with the interaction energy Euf between chain units and particles, the results are very far from those obtained with dense systems. In particular, the fraction of chain units in contact with the filler surfaces turns out to be much smaller for the phantom chains. This behavior is substantially due to the fact that the orientational and conformational freedom of chain segments in contact with the filler surfaces are restricted with respect to those of segments far from the surfaces. As a consequence, the phantom chain units tend to avoid the interface shells, and the chain conformations and the mutual arrangements of particles and chains are strongly perturbed with respect to those observed in dense systems. In order to counterbalance this intrinsic tendency, simulations of phantom chains have been performed [19] by substituting the interaction energy Euf with a similar term E*j = e*[(a/r w /) 12 -2(a/r w /) 6 ], the energy parameter e* being chosen for each system in such a way that the density of units in the shell between 0.8a and 2a from the surface of the filler particles coincides, when properly scaled, with the value l.lcr"3 found in all dense systems. Also, in order to avoid an excessive crowding of chain units in the superposition regions of the interface shells of different particles, E^ is evaluated only for the interaction of the chain units with the closest filler surface. As noted in Ref. [19], the modified energy E^j has no particular merit other than being simple and giving good results for all systems examined; other forms or totally different methods could give equally good results. Systems of phantom chains simulated using Euf or E^ are denoted in the following by the symbols P and P*, respectively. Simulations of phantom chains in the filler environment can be obviously performed by placing one or more chains of the desired length in a large cell containing a large number of particles. In practice, there are no limitations on the chain length and on the size of the particles. Filled systems containing phantom chains with cr/ and

127 particles (100 times the values of Np reported in Table 1). In the case of system Mg^o? this large cell has been also utilized for calculations with chains of 1000 units. The methods used for initializing and equilibrating the systems are practically coincident with those described in Section 1. Figure 8 compares the normalized density of polymer units in spherical shells of radius r and thickness O.lcr centered on the filler particles in three simulated systems with Of = 16a, ip = 0.36 and Lp = 100. The full symbols refer to the dense system Mi6,36, (see also Figure 1), while the open symbols refer to systems Pi6,365 and P*Q^6^ as indicated. The behavior of p for phantom chains with the unmodified energy Euf (system Pi6,36) evidences the said tendency of the chain units to avoid the interface shells. Calculations for P systems have been also performed in a few cases starting from well equilibrated M systems. Sequential snapshots taken during these calculations show [16] that when Euu is turned off the chains, initially distributed in the M systems such to occupy all regions left free by the filler, migrate preferentially to the largest cavities delimited by the filler particles. Although the form given to E^j is totally arbitrary, substituting Euf with J5** is seen to be very effective, in the sense that not only the phantom chains in system Pi6,36 show the correct proportion of units in contact with the filler particles, but the overall shape of the Pi6,36 curve in Figure 8 is quite similar to that of the Mi6,36 curve. Correspondingly good results are obtained for all other P* systems, the differences with respect to M systems concerning mainly the location and intensity of the maxima and minima within the interface shells. These differences are rather obvious, since the packing inside the interface shells is regulated by intramolecular and intermolecular interactions between neighboring chain units. They are also scarcely relevant as far as the mutual arrangement of polymer chains and filler particles is concerned. Substituting Euf with E^j has profound effects on the conformation of the polymer chains. This is clearly seen in Figure 9, plotting R^/ncr2 vs. n for the reference system Mo, for phantom chains of 100 units without filler (system Po) and for the three systems of Figure 8. The curves for systems Mo and Po are nearly coincident, confirming that these phantom chains are a good model for the unperturbed reference melt (see also Figure 3). The curve for system Pi6,36 is rather unusual, showing an initial rise followed by a decrease and by a subsequent tendency to rise again. The same unusual behavior is shown by all P systems, with the intermediate decrease more pronounced for more densely filled systems. This reveals that the chain conformation is strongly altered by the tendency of the phantom chains to avoid the surfaces of the filler particles. On the other hand, the curve for system P*6 36 is seen to be very

128 3.0 i

10

12

14

16

18 r / O

20

Figure 8. The normalized density of polymer units in spherical shells of radius r and thickness O.lcr centered on the filler particles in system Mi6,36 and in large systems of phantom chains of 100 units with a/ = 16a, tp = 0.36.

close to the curve for system Mi6,36, indicating that the conformational distribution observed in dense filled systems is properly restored when the simulations are performed for phantom chains with the arbitrarily modified interaction energy E^*. As a consequence, the average dimensions of the phantom chains of 100 units (characteristic ratio between 2.4 and 2.5 and rms radius of gyration between 6.3<7 and 6.4a) are very similar in P* and M systems having the same composition. Table 5 compares the results obtained for phantom chains of 100 units in P* systems with four different composition with those obtained for the corresponding M systems. The agreement is seen to be truly remarkable, in the sense that all parameters characterizing the mutual arrangements of particles and chains are practically coincident in the two cases (with one exception, see later). Consider for instance a system crowded with relatively small particles like M&jo (see Figure 7). The two calculations agree that an average chain of 100 units consists in this case of 12-13 interface segments of length approximately 4 units, of 4-5 bridge segments of length 6-7 units and of 3 loop segments of length 4 units, with 3 or 4 direct connections between consecutive interface segments; the average chain has terminal segments of total length 8-9 units and is in contact with 5 different particles, while each particle is

129

2.8 2.6 2.4 2.2 2.0 -

1.6 1.4 1.2 •] 1.0 0

10

20

30

40

50

60

70

80

90

100

Figure 9. Plots of Cn = (R^/na2 as a function of n for systems Mo and Mi6,36 and for large systems of phantom chains of 100 units with 07 = 16cr, ip = 0.36.

in contact with about 50 different chains. A similar good agreement is found for the other systems studied, with the exception of the bridge segment length in systems with aj = 28a and (p = 0.20, found to be 42 units in system M28,20 and only 16 units for the corresponding phantom chains. However, one has to consider that system M28,20 is unique when compared to the others, since the base cell contains in this case only one single filler particle (see Table 1). In other words, this system does not correspond to a random arrangement of particles, but to a regular arrangement at the nodes of a cubic lattice with lattice spacing 40a. The phantom chain calculations have been then repeated in Ref. [19] for a cubic arrangement of 8 particles with af = 28a in a cell of edge 80a. The results (in parentheses in Table 5) are in good agreement with those obtained for system M28,20> showing that phantom chain calculations are also very sensitive to the exact arrangement of the particles. It is concluded that simple calculations of this kind can be utilized to study systems that cannot be directly simulated at full density, and that the exact form of the modified interaction energy between particles and polymer units is relatively unimportant, provided that the scaled density of units between 0.8a and 2a from the surface of the particles is forced to correspond to the value (l.la~ 3 ) found in the dense systems.

130 Table 5. Comparison of results obtained for phantom chains of 100 units in large systems of randomly distributed spherical nanoparticles (P* systems) with those obtained for the smaller properly dense systems listed in Table 1 (M systems). The data in parentheses for system P^8 2Q refer to particles arranged at the nodes of a cubic lattice. ^8*20 /l

h ff Ni Li

Nb Lb Ni Li

Nd Pc

CP

M 8 ,20

0.28 0.01 0.007

0.29 0.02 0.009

7.8 3.7 2.9 11 2.8 5.4 1.1 3.4 50

7.4 4.0 2.6 12 2.4 5.6 1.5 3.3 50

-^8*30

M 8 ,30

0.47 0.04

0.46 0.02

0 13 3.7 5.0 6.4 3.3 3.9 3.4 5.2 50

0 12 3.8 4.4 7.3 2.8 4.2 4.0 5.2 50

^16,36

Mi6,36

0.32 0.01 0.005

0.33 0.01 0.005

7.7 4.1 2.4 11 3.3 5.8 1.0 2.7 120

7.3 4.5 2.4 12 3.1 5.7 0.7 2.7 120

p* •»28,20

M28.20

0.08

(0.08)

0.08

0 0.5 1.9 4.4 0.2 16 1.2 8.3 0 0.6 320

(0)

0 0.4 1.6 4.8

(0.4) (1.8) (4.4) (0.06) (42) (1.2) (9.0) (0)

(0.6) (320)

0.05 42 1.0 9.7 0 0.6 290

It is then quite interesting to perform phantom chain calculations for longer chains and to compare the results with those that can be predicted on the basis of the approximate rules described in Section 1. Table 6 indicates that these simple rules, deduced from simulations of dense systems of chains of 100 units and shown to be valid when the systems are not too crowded with small particles or do not contain a large proportion of free chains, can be extended in these cases without modifications to approximate the behavior of long phantom chains. For instance, phantom chains of 500 units in system F8*2o (i.e., with relatively small particles and intermediate filling density) are predicted to consist of 36 interface segments of length 4 units (found: 38 segments of average length 3.8), of 17 bridges of length 11 units (found: 17 bridges of average length 14) and of 11 loops of length 6 units (found: 15 loops of average length 6.1), with 8 direct connections between interface segments (found: 5.4); each chain is predicted to be in contact with 13 different particles (found: 12), and each particle is predicted to be in contact with 37 different chains (found: 35). For the same particles at higher filling density (system ^,30)5 ^he chains are predicted to consist of as many as 62 interface segments of length 4 units (found: 63 segments of average length 3.8), of 30 bridges of length 7 units (found: 27 bridges of average length 6.7) and of 13 loops of length 4 units (found: 18 loops of average length 4.1), with 18 direct connections between interface segments (found: 17); each chain is predicted to be in contact with 20 different particles (found: 17), and each particle is predicted to be in contact with 30 different chains (found: 31). All considered, the

131 Table 6. Comparison of results obtained for phantom chains of 500 units with those predicted a priori according to the approximate rules described in Section 1. F8,30

Pi,20

h h Ni Li

Nb Lb N, Li

Nd Pc Cp

Pi8,20

^16,36

found 0.29 0.01

predicted 0.29 0.03

found 0.48 0.04

predicted 0.49 0.06

found 0.33 0.01

predicted 0.33 0.01

38 3.8 17 14 15 6.1 5.4 12 35

36 4.0 17 11 11 5.5 7.6 13 37

63 3.8 27 6.7 18 4.1 17 17 31

62 3.9 30 7.3 13 3.7 18 20 30

38 4.2 15 14 18 6.4 5 7.5 70

34 4.4 16 12 11 6.1 5.5 6.4 69

found 0.08

predicted 0.08

0 9.3 4.5 1.5 70 6.8 16 0.1 1.8 180

0 7.9 4.7 3 39 3.5 19 0.5 1.5 184

agreement between predicted and found data can be considered quite satisfactory. However, Table 6 also confirms that these rules are inadequate for systems with large particles at low filling density. For instance, the predicted length of the bridge segments is too short and L\ is much less than L&/2 in the case of system Pg 2o i®8 ^so confirmed by calculations with longer chains of 1000 units). This is not unexpected, because the length of the bridge segments is mainly regulated by the average distance between the surfaces of adjacent particles, while the probability that a chain segment circles back to form a long loop decreases with increasing the length of the segment [19]. Since the predictive rules described in Section 1 are based on the results obtained for the few dense systems listed in Table 1, they are obviously open to modifications when results for other compositions become available. Unfortunately, simulations of dense systems with larger particles distributed at random are not feasible today, irrespective of the chain length (the base cell of system Mg^o contains only one single filler particle and more than 50000 polymer units; a cell containing only 20 such particles would also contain more than 106 chain units for the same value of
132

3.

Conclusions

Out-of-lattice simulations of polymer/nanofiller systems with realistic densities of polymer and filler particles can be presently performed only for relatively small particles (diameter up to nearly 20 times the transverse diameter of the chains) and short chains (100-200 isodiametric units). Systems of this kind have been simulated using a simple model in which the polymer chains are represented as linear sequences of 100 isodiametric units and the nanofiller particles as a random distribution of spherical entities with diameter between 4 and 28 times the diameter of the polymer units. The number offillerparticles in the base cell has been regulated such to have a partial volume of filler between 0.1 and 0.5 (depending also on the size of the particles), while the density of the polymer and the stiffness of the polymer chains have been regulated such to correspond to those in filled materials of practical use. The results of these simulations, described in Section 1, indicate that the polymer units at the interface with the filler particles are arranged in densely packed and ordered shells analogous to the layers found near planar solid surfaces, the thickness of the interface shells being approximately twice the transverse diameter of the polymer chains. The polymer chains, slightly reduced in size with respect to the practically unperturbed chains of the unfilled polymer melt, can be considered as sequences of interface, bridge and loop segments. The abundance and length of the various kinds of segments and the consequent overall picture of the mutual arrangements of chains and particles depend obviously on the exact composition of the investigated system. Although the simulated chains are quite short compared to real polymer chains, each chain visits in most systems the interface shell of several filler particles, and each particle is in contact with many different polymer chains. Therefore, even in the absence of specific attractive interactions between filler and polymer, thefillerparticles behave as highly functional physical cross-links. It has been also shown that the relevant characteristics of the simulated systems can be approximately predicted at a quantitative level utilizing a few simple rules. Extrapolating to longer chains the results obtained for the chains of 100 units simulated in dense systems is not straightforward, specially when the particles are large, since these systems contain a large proportion of free chains and long terminal segments. Studies of this kind can be performed using phantom chains, provided that the interaction energy between chain units and filler particles is properly modified in order to counterbalance the intrinsic tendency of the chain segments to avoid the filler surfaces. In fact, although the modified interaction en-

133 ergy used in the calculations described in Section 2 is totally arbitrary, all results obtained for systems of phantom chains of 100 units are nearly coincident with those found for dense systems with identical composition. In particular, this is true for the spatial distribution of the units, for the chain conformations and for the mutual arrangements of chains and particles. Phantom chains can be obviously used for simulating much longer chains and much larger particles. Calculations performed for longer chains of 500 units are practically as expected on the basis of the results obtained for chains of 100 units, with the exception of systems containing large particles at small filling density, where the chains of 100 units are too short to show properties indicative of those of longer chains. On one hand, this confirms that phantom chain calculations can be used to study the molecular arrangements in polymer-based nanocomposites with various compositions. On the other hand, it shows that long chain polymers with large particles and small filling density can be studied only by simulating phantom chains of appropriate length. Similar methods can be likely used for investigating the molecular arrangements and conformations in other systems in which the relevant properties are substantially influenced by the presence of polymer/solid interfaces.

Acknowledgments The financial support of the Ministero dell'Universita e della Ricerca Scientifica of Italy is gratefully acknowledged.

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G. Kraus, Ed., Wiley Interscience

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39:75, 1984.

[3] S. Ahmed and F.R. Jones, J. Mater. Sci., 25:4933, 1990. [4] G. Heinrich and T.A. Vilgis, Macromolecules, [5] C.R. Lin and Y.D. Lee, Macromol.

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Theory SirnuL,

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[6] M. Kluppel, R.H. Schuster, and G. Heinrich, Rubber Chem. Technoi, 1997. [7] C.R. Lin and Y.D. Lee, Macromol.

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109:3285, 1998.

[9] G. Raos, G. Allegra, J. Chem. Phys.,

113:7554, 2000.

[10] E.J. Meijer and D. Frenkel, J. Chem. Phys.,

70:243,

100:6873, 1994.

[II] G. J. A. Sevink, A. V. Zvelindovsky, B.A.C. van Vlimmeren, N.M. Maurits, and J.G.E.M. Praaije, J. Chem. Phys., 110:2250, 1999. [12] J. Huh, V.V. Ginzburg, and A.C. Balazs, Macromolecules,

33:8085, 2000.

DISSIPATIVE PARTICLE DYNAMICS APPROACH TO NEMATIC POLYMERS Antonino Polimeno,1 Alexandre Gomes,1'2 and Assis Farinha Martins2 Universita degli Studi di Padova Dipartimento di Chimica Fisica Via Loredan 2, 35131 Padova, Italy [email protected]

Universidade Nova de Lisboa Departamento de Ciencia dos Materials Faculdade de Ciencias e Tecnologia, 2829-516 Caparica, Portugal

Abstract

We discuss a Dissipative Particle Dynamics (DPD) approach to simulate oligomers and polymeric chains with nematic mesophases. Definition of mesogenic units are discussed, based either on semi-rigid units with standard DPD beads interacting only via soft repulsive potentials and linked by harmonic springs or on corrected DPD potentials including an orienting term between adjacent couples of beads. In the first case oriented phases are presented for systems made of single free semi-rigid units and, as an example of main-chain flexible liquid crystal polymer, by three linked semi-rigid units. In the second case an example of switching system is discussed, in the presence of an external potential.

Introduction Computational treatments of anisotropic fluids are nowadays a highlydeveloped tool, aimed at simulating static and dynamic properties, with varying degrees of accuracy and different spatial and time scales. Local approaches were first developed, starting with the definition of anisotropic interaction potentials by Lebwhol-Lasher [1], applied to the study of the nematic order parameter [2, 3], reorientation process of both nematic and isotropic phases [4], director alignment produced by an external field [5]. The application of Gay-Berne (GB) type potentials to the study of phase diagrams of low molecular weight liquid crystalline phases through molecular dynamics (MD) and Monte Carlo (MC) simulations were first studied, among others, by DeMiguel and Luckhurst [6, 7]. The use of GB potentials was also extended to study 135 P. Pasini et al. (eds.), Computer Simulations of Liquid Crystals and Polymers, 135-147. © 2005 Kluwer Academic Publishers. Printed in the Netherlands.

136 non-equilibrium physical properties, as, in for instance, Smondyrev et a/., who calculated nematic viscosities as function of temperature [8]. MD and MC techniques application to oriented polymers is more recent [9]. Finally, full molecular simulations based on atomistic potentials, are starting to appear in the literature [10]. Despite their highly successful record, MD or MC simulations are still hardly extended to the direct interpretation of complex set-ups, typical of most rheological experiments. In such cases it is preferable to employ mean-field or continuum descriptions, based of the numerical solution of the constitutive equations describing hydrodynamic properties. Such techniques were for instance applied to the prediction of transient director patterns of liquid crystalline nematic samples [11-14]. Hydrodynamic treatments are algebraically complex and computationally intensive, and their implementation is limited mostly to nematic phases. In between the molecular and macroscopic approach several mesoscopic scale techniques are available, like lattice bond fluctuation models [15] and coarse grained techniques, such as smoothed particle hydrodynamics[16], and dissipative particle dynamics (DPD) [17, 18]. The latter is based on the concept of "matter particles", representing a large number of atoms or molecules, interacting via soft potentials and subjected to dissipative and fluctuating forces. Polymeric system can be described by simple bead-spring models, with the advantage that a full chain can be represented by just a few particles, typically from 20 to 50 [18, 19]. The DPD approach was originally developed for describing, essentially, isotropic phases. In this work we explore some means of extending DPD to the study of liquid crystalline polymers. We limit our modelling to main-chain thermotropic nematic polymers. The paper is organized as follow. In the following Section we summarize briefly the DPD approach. Next a discussion is provided on the definition of mesogenic units in DPD simulations and examples of DPD simulations of chains exhibiting ordered phases are discussed.

1.

Dissipative Particle Dynamics

The DPD technique was first proposed by Hoogerbrugge and Koelman [17] in the beginning of the 90's with the intention of studying soft condensed matter, i.e. systems that have both solid and liquid behavior. It has been successfully used for the study of complex fluid systems, like polymeric [18-21] or colloidal [21-23] suspensions, micelles and immiscible mixtures, where one of the main characteristics is the presence of disparate time scales in the dynamics of the system. The technique uses

137 the concept of coarse-grained particles of fluid representing an undetermined but relatively large cluster of molecules, which are described by a soft sphere (usually designated indistinctly as particle or bead) that acts as a center of mass, moves according to Newton laws of motion and interacts with the remaining of the system with simplified force laws. The forces acting on each particle come from the interactions with their neighbors and, if such is the case, from additional body forces, as for instance gravity or external magnetic fields. In addition to pair-wise conservative (purely repulsive) forces the DPD method uses a dissipative force, which damps the relative approaching velocity between particles. The dissipative force is coupled to a random force term to keep the system in thermal agitation. The fact that both the conservative and the dissipative/random forces act pair-wise produces momentum conservation resulting in preservation of macroscopic hydrodynamic behavior [24] at long wavelength and time scales allowing fluid dynamics to emerge from a particle-based simulation in a similar way as Navier-Stokes equations emerge from real molecular fluids. The gap between particles and polymeric chains can be overcome by the use of spring forces chaining beads together, which usually go from simple Hooke springs to Prenkel springs [20]. The architecture of the polymeric chains can be easily drafted through the right linkage of the beads, and it is not limited to linear chains [18, 21].

2.

Methodology

We summarize briefly the DPD methodology in its most common implementation [25]. The DPD particles move according to Newton laws of motion: -dt

=

Vi

(2)

where r^ is the z-th particle position, rrti the mass of the particle, v^ its velocity and f* = ic+^D+^R the force acting over it, which is partitioned in a conservative part, fc, defined with respect to a pairwise potential, a dissipative contribution, fp, and a fluctuating term f#. The particleparticle interaction potential reflects the interaction between patches of fluid, which can interpenetrate each other during the simulation. Typical DPD conservative forces are derived from a soft interaction potential, 'V^'c

( 3 )

138 where a is the interaction parameter, setting the overall repulsion strength between beads, rc is the interaction cut-off radius and r^ the distance between the particles. The interaction parameter is chosen to assure that the compressibility of the system is similar to water [18]. The conservative force ic acting on the i-th bead is then simply given by ciri^Tij

(4)

where iij is the unitary vector Tij/rij and we gives an estimate for the soft repulsive force between beads 1

rc 0

rt3
({J)

Tij > rc

The dissipative force fp is defined as a quantity proportional to the relative velocity between beads

)v|j,.

(6)

vj'- is the projection of the relative velocity on vector f^- and wp = W'Q. Dissipation strength is controlled by 7, and the matching random force is defined as: fl(ry )*« (7) where £(i) is a white noise random term and WR = we- A fluctuationdissipation relation holds for the parameters of the dissipative and random forces a 2 = 2^kBT [25]. A well known advantage of the DPD method is the use of large (relatively to MD) time steps, as a direct results of the soft nature of the potentials employed to describe bead-bead interactions. The pair-wise dissipation and random forces work as a momentum conserving thermostat and consequently preserves the hydrostatic modes, that is, the simulated fluid complies with hydrodynamic equations. Numerical integration of DPD equations is slightly more complicate than MD equations due to the presence of dissipative and fluctuating forces. A standard Verlet leapfrog algorithm, which has second order accuracy, is usually satisfactory. Other algorithms have been proposed, most of them variants of the basic Verlet algorithm, allowing better temperature control [26, 27], more realistic treatment of compressibility and diffusion coefficients [28, 29]. Dissipative Particle Dynamics with energy conservation (DPDE) has been proposed by Bonet Avalos and Mackie [30].

139

Figure 1.

3.

A DPD model for a flexible main-chain LCP with three semi-rigid units

Standard semi-rigid segments

To simulate polymeric chains, one can adopt a standard coarse grained picture in which each chain is obtained from a number of beads linked by elastic forces. For each couple of adjacent beads we define a classical Praenkel spring force fp = kfcj — req)rij, being k the spring elastic constant, r^ the distance between the two connected beads and req the equilibrium distance for the spring. Let us now consider a simple model for liquid crystal polymers (LCP). We report here only preliminary results based on a approach selected to keep the additional computational load at a minimum. Other more complex geometries can certainly be defined, and they will be described elsewhere. First of all we can define semi-rigid segments, made of a varying number n of adjacent beads, as our basic orienting units. Each unit is forced to assume a semi-rigid be-

-

•

•

I

*

13

14

15

0.5-

CL V

nn 10

11

12

n

Figure 2. Equilibrium value (P2) at increasing values of n, i.e. number of beads per units, for a system of 1000 semi-rigid units, at ksT = 0.25.

140 havior by the presence of an extra elastic force between the first and last bead. By changing the elastic constant of this additional spring (called external spring in the following, for brevity), we can control the rigidity of the unit. A DPD model for a flexible main-chain or side-chain LCP can then be described by a sequence of semi-rigid units interspersed with regular DPD beads which forms flexible units, like in Fig. (1), which is made of three semi-rigid units, labelled 1, 2 and 3, with a flexible short unit of two beads between each semi-rigid one. The only additional parameters defining an orienting or semi-rigid unit are n (number of beads per units) and ke, external spring. We can further complicate our model by assuming that the equilibrium distance between adjacent beads in semi-rigid units is different from the same quantity in flexible units, and we can also modify the repulsive potential parameters. Here we shall limit our discussion to the study of systems made up of single semi-rigid units, linked by short flexible units to model main-chain LCP with limited flexibility, like in Fig. (1). For simplicity, we assume the same value for all beads for the parameter of repulsion a = 25 and we fix the internal spring k = 100, while a = 3 as in other standard DPD studies [21]. We first investigate the influence of the number of beads n on the formation of oriented phase. We present here results for systems made up of 1000 semi-rigid units of n beads each, with n between 10 and 15. The temperature factor &jgT has been chosen equal to 0.25 for all cases. The external spring, which assures a semi-rigid nature for each unit, has been chosen ke = 1000. Simulations have been performed with a time step equal to 0.005, for a total number of steps ranging from 105 to 5 x 105, in a periodic box at density p = 1. The system is prepared initially in an aligned configuration with respect to a selected box axis. In Fig. (2) the behavior of an order parameter (P2)) defined with respect to the angle formed by an axis defined by the first and last bead of each units with respect to the alignment axis. For n > 10 an orientational order is observed. Notice that, here and in the following, only nematic phases are observed and discussed: i.e. by using simple orienting units made of standard DPD beads linked semi-rigidly, only orientational order was observed. A similar study has been carried on by changing the thermodynamic temperature ksT, for a given number of beads. In Fig. (3) we present the average order parameter for a system made of 1000 units of 12 beads each, at decreasing values of &#T: a phase transition between an ordered (nematic) and isotropic state, around ksT = 0.5, is observed. A portion of the calculated behavior of (P2) as a function of time is shown in Fig. (4). As a first test of LCP model system, we have considered the case described in Fig. (1). Each chain is defined by 40 beads, arranged as three semi-rigid units of 12 beads each, linked

141

1.0-, •

•

.

0.5-

CL V 0.00.25

0.75

0.50

1.00

-0.5 J

for a system of

Figure 3. Equilibrium value of {P2) at increasing values of 1000 semi-rigid units of 12 beads.

0.25

CL V 0.0

200

400

600

800

1000

t -0.5

Figure 4- (P2) as a function of time at different values semi-rigid units of 12 beads.

, for a system of 1000

142 by two flexible units of two beads each. The overall system is formed by 300 chains, for a total of 12000 beads. The number of total beads is thus equal to a system of 1000 chains of semi-rigid separated units, at the same density, which we use for comparison. The system exhibits a

Figure 5. Snapshot of a model LCP system of 1000 semi-rigid units of 12 beads (only a fraction of the units of the sample is shown).

stable nematic phase, with an equilibrium value of (P2) slightly smaller (0.68) than the corresponding value obtained for the free semi-rigid units (0.74). A snapshot of the model LCP is shown in Fig. (5): the flexible short units are responsible for the decrease of alignment of the longer orienting segments. A critical dependence can be observed in dependence of the number of the length and number of non-orienting units: simulation performed with systems made of only two orienting units of 12 beads, linked by a single flexible unit of 16 beads do not show any order.

4.

An alternative approach

As an example of an alternative approach to the definition of oriented DPD phases, we consider here systems in which mutual alignment is induced by adding to the standard DPD potential an attractive soft potential. The inclusion of ad hoc orienting potentials at least between portions of semi-rigid or flexible units allows a more versatile modelling and the possibility of using shorter units (i.e. with a smaller number of beads per unit). The simplest choice available is given by a simple P2-like potential between couples of adjacent beads. An example is provided in Fig. (6) which shows two semi-rigid units, 1 and 2, each composed of five beads. The units are defined as in the previous case (standard repulsive soft DPD potential between beads, internal springs k between adjacent beads, and an external spring for each unit). However, the couples of

143 beads A\B\ in 1 and A2B2 in 2 are also interacting via the potential

where ri2 and 9 are the distance between centers of mass and the angle between the two couples of beads, respectively. A 'soft' factor is included, with the same quadratic dependence typical of DPD standard repulsive potentials. The additional parameter e measures the force of

V Figure 6.

Examples of two interacting semi-rigid DPD units.

the additional orienting potential. Several set-ups can be easily defined and explored, corresponding to different choices of n, e, ke/k, number or orienting beads per units. As an example, we present here results obtained for a system of 200 chains, each composed by a single semi-rigid unit; all parameters have been chosen as in the previous Section, except that six orienting beads are included, i.e. if we call S a standard bead and AiBj an orienting couple, each chain is defined as A1B1SSA2B2SSA3B3. The temperature factor has been chosen fc#T = 0.25; finally the orienting potential parameter has been taken e = 0.5. The system has a stable nematic phase, and it has been used for a simple numerical experiment in which, by turning on an external potential we have studied the switching between two perpendicular alignment axes. For each couple of orienting beads AiBj whose center of mass and orientation are r and 0, we have defined an external potential as ~ e extP2(^ ~" $ext) ( 1 ~ F~ ) \ 'cj

0

r

< rc

(Q\ yd)

r>rc

where 0ext is the imposed direction direction of alignment. We have chosen eext = 0.5. In the first part of Fig. (7) we show the value of (P2), defined with respect to the external axis (full line) and with respect to an average axis specified by the ensemble of units (dashed line), in the

144

V...-''"'*'"

0.5-

CM

V

0.0 10

-0.5 J

Figure 7. Plot of (P2) as a function of time defined for a system of 200 chains of semi-rigid units of 10 beads with additional orienting potential: the full line represents the order parameter calculated with respect to a fixed axis, while the dashed line represents the order parameter defined with respect to an averaged axis.

presence of an external field parallel to the stable initial director. The two order parameters are relatively close (0.8 vs. 0.7). The central region of Fig. (7) shows the development of the systems under the pull of an external potential, which is instantaneously rotated perpendicular to the initial director: the chains spread, assuming a disordered transient configuration. Finally, in the last part of the simulated switching experiment the system reaches a new equilibrium state, almost completely aligned to the external field. A more pictorial view of this behavior is given in Fig. (8), which shows snapshots of the time evolution of the system at selected times.

5.

Summary

The characterization and analysis of model DPD systems for liquid crystals polymers requires certainly a more thorough investigation than the few examples presented in this communication. Orienting units can be modified by changing the ratio between internal and external spring k/ke, the equilibrium distance between beads, the repulsive factor a and so on. Once a given orienting unit has been defined, it can be used as building block for analyzing different LCP model systems. A potentially

145

Figure 8. Snapshots of a system of 200 chains of semi-rigid units of 10 beads with internal orienting potential, in the presence of a switching external potential (only a fraction of units is shown).

146 relevant field of application is the simulation of rheological observables, which are hardly treated in the framework of standard molecular dynamics studies. Finally, more versatile models can be built by including in the semi-rigid units couples of adjacent beads interacting via an additional orienting potential.

Acknowledgments This work was financed by the Italian Ministry for Universities and Scientific and Technological Research, project PRIN ex-40% and by the Pundago para a Ciencia e a Tecnologia, Portugal, through the research grant SFRH/BD/2982/2000 to A.E. Gomes.

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75(12):2340,

[9] J.H.R. Clarke, Molecular dynamics of amorphous polymers. In The physics of glassy polymers, Edited by R. J. Young, R. N. Haward, London: Chapman and Hall, pag. 33, 1997. [10] M.J. Cook, M.R. Wilson, Mol. Cryst. Liq. Cryst, 363:181, 2001. [11] P. Ziherl, M. Vilfan, S. Zumer, Phys. Rev. E, 52(l):690, 1995. [12] P. Ziherl, S. Zumer, Phys. Rev. E, 54(2):1592, 1996. [13] A. Polimeno, L. Orian, A.E. Gomes, A.F. Martins, Phys. Rev. E, Part A, 62(2):2288, 2000. [14] M. Lukaschek, A. . Gomes, A. Polimeno, C. Schmidt, G. Kothe, J. Chem. 117(9):455, 2002.

Phys.,

[15] K. Binder, W. Paul, J. Polym Sci. Part B - Polym. Phys., 35(1):1, 1997. [16] P. Espafiol, Europhys Lett, 39(6):605, 1997. [17] P.J. Hoogerbrugge, J.M.V.A. Koelman, Europhys. Lett, 19(3):155, 1992; J.M.V.A. Koelman, P.J. Hoogerbrugge, Europhys. Lett, 21(3):363, 1993. [18] R.D. Groot, P.B. Warren, J. Chem. Phys., 107(ll):4423, 1997. [19] Y. Kong, C.W. Manke, W.G. Madden, A.G. Schlijper, J. Chem. 107(2):592, 1997.

Phys.,

[20] A.G. Schlijper, P.J. Hoogerbrugge, C.W. Manke, J. Rheol, 39(3):567, 1995. [21] R.D. Groot, T.J. Madden, J. Chem. Phys., 108(20):8713, 1998.

147 [22] E.S. Boek, P.V. Coveney, H.N.W. Lekkerkerker, J. Phys. 8(47):9509, 1996.

- Cond.

Mat,

[23] E.S. Boek, P.V. Coveney, H.N.W. Lekkerkerker, P. van der Schoot, Phys. Rev. E, 55(3):3124, 1997. [24] P. Espanol, Phys. Rev. E, 52(2):1734, 1995. [25] P. Espanol, P.B. Warren, Europhys Lett, 30(4):191, 1995. [26] W.K. den Otter, J.H.R. Clarke, Europhys. Lett, 53(4):426, 2001. [27] W.K., den Otter, J.H.R. Clarke, Int. J. Mod. Phys. C, 11(6):1179, 2000. [28] G. Besold, I. Vattulainen, M. Karttunen, M. Poison, Phys. Rev. E, 62(6):7611, 2000. [29] P. Nikunen, M. Karttunen, I. Vattulainen, Comput Phys. Commun., 153(3):407, 2003. [30] J. Bonet Avalos, A.D. Mackle, J. Chem. Phys., 111(11):5267, 1999.

SOME THINGS WE CAN LEARN FROM CHEMICALLY REALISTIC POLYMER MELT SIMULATIONS Wolfgang Paul, Stephan Krushev Institut fur Physik, Johannes Gutenberg- University, 55099 Mainz, Germany Wolfgang. [email protected]

Grant D. Smith, Oleg Borodin, Dmitry Bedrov Department of Materials Science and Engineering and Department of Chemical and Fuels Engineering, University of Utah, Salt Lake City, Utah 84112, USA

Abstract

We present in this contribution results from Molecular Dynamics (MD) simulations of a chemically realistic model of 1,4-polybutadiene (PB). The work we will discuss exemplifies the physical questions one can address with these types of simulations. We will specifically compare the results of the computer simulations with nuclear magnetic resonance (NMR) experiments, neutron scattering experiments and dielectric data. These comparisons will show how important it is to understand the torsional dynamics of polymers in the melt to be able to explain the experimental findings. We will then introduce a freely rotating chain (FRC) model where all torsion potentials have been switched off and show the influence of this procedure on the qualitative properties of local dynamics through comparison with the chemically realistic (CRC) model.

Introduction Polymers are complex objects displaying non-trivial structure from the scale of a (typically) carbon-carbon bond (1 A) to the radius of gyration of the coil (Rg « 10-100 A[l]). Here we will be concerned with melts of simple linear polymers where the polymer coils behave as if they were random walks (RG OC N where N is the degree of polymerization of the chains). Connected with the spread in length scales is an even wider spread in time scales: from local bond-length and bond-angle vibrations (10~15 —10~13 s) over conformational transitions between isomeric states 149 P. Pasiniet al. (eds.), Computer Simulations of Liquid Crystals and Polymers, 149-170. © 2005 Kluwer Academic Publishers. Printed in the Netherlands.

150 in the dihedral potential (10~12 -10~ 10 s) to the self-diffusion and overall configurational relaxation of the whole chain which for short chains scales as N2 times the time-scale for conformational transitions and for long chains as N3 times that time-scale. With N ranging from 100 (short chains) up to 10000 and more (long chains) these time scales cover 10~8 s up to 10 s. The local dynamics is naturally strongly dependent on the exact chemical nature and structure of the polymer one studies. The large scale dynamics, however, is largely universal and is described with the Rouse model whereas for longer chains the tube model and reptation concept is believed to describe the chain dynamics [2]. It is easy to see that no single simulation method can capture the physics of polymer dynamics on all these length and time scales [3]. For situations where we can ignore quantum effects (which can, however, be important in polymer crystals [4]) MD simulations with chemically realistic force fields are the method of choice to study local relaxation. Besides the necessity for carefully optimized chemically realistic force fields there is always the question to address whether the simulation is able to equilibrate the model system at a given thermodynamic state point of temperature, density and chain length. In general this means that the simulations are limited to polymer melts well above the glass transition temperature (or density) and to chains not exceeding the entanglement molecular weight, although some newly developed Monte Carlo techniques [5, 6] help to overcome the latter limitation. In the next section we will present quantitative comparisons of MD simulations for 1,4-polybutadiene with data from NMR experiments, neutron scattering and dielectric relaxation. We will also discuss the question of increasing dynamic heterogeneity upon cooling down of the simulated systems as can be observed in the torsional dynamics. In the ensuing section we will then present a comparison of the dynamics in the simulations of the chemically realistic model with that of a freely rotating chain model. This will allow us to address the question of applicability of mode coupling ideas to describe the glass transition in polymer melts.

1.

Quantitative Comparison to Experiment

We will present MD simulations of a chemically realistic united atom model for PB employing a carefully validated quantum chemistry based force-field [7]. We will study in this section a random copolymer of 50% trans 1,4-butadiene, 45% cis 1,4-butadiene and 10% vinyl groups. This polymer was synthesized with an average molecular weight correspond-

151 ing to about 30 repeat units and characterized to possess the above chemical microstructure [8]. We have 40 chains of 30 repeat units in our simulation box and we will be using a united atom model for the CH, CH2 and CH3 groups. The simulations are performed in the NVT ensemble using the Nose-Hoover thermostat [9, 10] after determining the correct density at atmospheric pressure for each temperature. With a total of about 5000 united atoms, these systems can be simulated today over a time range of more than 100 ns (about 2 ns of real time trajectory per day). The ability of the simulation to quantitatively reproduce experimental data on relaxation processes in polymer melts strongly rests on the implementation of the correct force fields for the dihedral angles. These typically possess barriers separating the isomeric states and it is by correlated jumps over these barriers [11-13] that the relaxation processes come about. The frequency of these jumps is exponentially sensitive to the height of the barriers. In order to be able to make a parameter free quantitative comparison with experiment one therefore needs to carefully determine this part of the force field [7], which is done through parameterization against high-level quantum chemistry calculations. The validation of this force field then proceeds through comparison with experiment.

1.1

NMR Experiments

An experimental technique which is very sensitive to the local dihedral barriers is C13 NMR spin lattice relaxation time measurements. For polymers like PB this technique observes the reorientational motion of the CH bonds as quantified through the second Legendre polynomial of the CH bond orientational autocorrelation function. It is sensitive to the local chemical environment giving rise to 12 different resonances which can be identified [14]. A CH bond at the sp2 carbon of the double bond in a cis group next to a trans group relaxes differently from one next to another cis group and differently from the ones in a trans group and differently from the CH bonds at a sp3 carbon. The vinyl group alone gives rise to six different resonances. In a computer simulation one measures the CH vector autocorrelation function and determines the second Legendre polynomial according to

:>CH/

(1)

152 Prom this function one obtains the spectral density by Fourier transformation 1 roo 1 r°

(2)

=5/

The spin-lattice relaxation time finally is given by evaluating the spectral density at the Larmor frequencies of carbon and hydrogen atoms — = K [J(UJU - o/c) + 3J(uc) + 6J(uu + uc)}

(3)

where u;#, uc are Larmor frequencies^ is a constant depending on the hybridization of the nucleus and n is the number of hydrogen atoms attached to the carbon atom under study. To be able to evaluate this function from a stored trajectory of a simulation of a united atom model, we have to reinsert hydrogen atoms into the system. This is done using the information on equilibrium (T = 0) bond lengths and angles which allows the determination of the hydrogen positions from the positions of the carbon backbone atoms (united atoms) alone. In Fig 1 we show a comparison between experiment and simulation for the 12 resolvable resonances for two different temperatures. As one can see, there is a very good agreement (better 273 K 353 K

Q experiment | simulation

1.0

0.1

i

trans-cis trans-tra.ns cis-cis

ris-trira

trtns

cis

v

trtre-QL

(X

vinjl-1 vinyl-^ cis-vinyl

resonance Figure 1. Comparison of the spin lattice relaxation times determined from MD simulations and experiment for two different temperatures. The figure shows data for 12 different resonances (cis-cis for example indicates an sp2 carbon in a cis group next to another cis group, trans an sp3 carbon of a trans group)

than within 20%) between simulation and experiment for all but one

153 resonance. The one that is not reproduced is a rigid body rotation of the side group which does not lead to a conformation relaxation and which we therefore did not try to match more accurately. The temperature dependence of the spin lattice relaxation time for two resonances can be seen in Fig. 2 The spin lattice relaxation time decreases when 3.0 -

H i f f f"

2.5

O cn

• 1.0 0.8 0.6

- 2.0 m

m*

0.4

H • ffi 4$

cis-cis (si lirulation) trans (si mul ation) cis-ci s (experi ment) trans (experiment)

m M

#

• • i

rv i

2.8

_

3.0

3

3.2 10^

3.4 10"

3.6 10^

3.8

1/T Figure 2. Comparison of the spin lattice relaxation times determined from MD simulations and experiment for the cis-cis (sp2) and the trans (sp3) resonance as a function of inverse temperature. Also shown are results for the nuclear Overhauser enhancement which is a measure of the non-exponentiality of the observed relaxation

the autocorrelation time for the observed relaxation increases. For high temperatures, where the nuclear Overhauser enhancement is still close to three, the spin lattice relaxation time traces the temperature dependence of the autocorrelation time of the torsional transitions and the agreement between simulation and experiment is a quantitative validation of the torsional force field [14]. From comparing to the observed torsional autocorrelation function in the simulation one can learn that this identification of the physical motion seen in the NMR experiment breaks down below about 300 K. In Fig. 3 we compare different measures of the local orientational mobility of the chains. The most basic one is the mean time between torsional transitions which we determined with a time resolution of 1 ps along the simulated trajectory. This time scale shows an Arrhenius temperature dependence. It is different for the three relevant torsion potentials along the chain, the allyl bonds next to a cis or trans double bond and the /3 bond at the connection between two butadiene monomers. All these time scales, however, simply follow

154 •trans ally] -cisallyl

io3

IO4 ,.

10*

10

10°

IO1 2.4 10"* 2.6 L0-* 2.8 10"4 3.0 LO"5 3.2 10* 3.4 10* 3.6 1 0 *

3.8

Figure 3. Temperature dependence of several measures of the local orientational mobility of the chain. The lower set of curves pertaining to the right abscissa shows the mean time between torsional transitions for the three relevant torsional angles along the chain. The upper two sets of curves give the integrated autocorrelation time for the second Legendre polynomial of the CH vector orientation, TCH, and the integrated autocorrelation time for the torsion angle autocorrelation function, Trcm-

an Arrhenius law with their energetic dihedral barrier as activation energy in the observed temperature range, so that there are no packing effects observable on the dihedral dynamics in this range. Contrary to the mean time between torsional transitions, the autocorrelation time for the torsions (thick lines in the Fig. 3, defined as the time integral of the following correlation function /(*)

=

(4)

increases in a Vogel-Pulcher like manner. This shows the growing importance of back-jump correlations for the torsional transitions: a torsional transition which occurs with its Arrhenius rate is immediately reversed with an increasing probability upon reducing the temperature thus leading to no decorrelation of the torsional angle. The final set of curves denoted by their resonance name in the figure legend gives the autocorrelation time of the second Legendre polynomial for the CH vectors, i.e., the time integral of (1). These time scales pick up contributions from different torsions adjacent to the carbon atom under study and lie intermediate to the autocorrelation functions of these torsions.

155

1.2

Neutron Scattering Experiments

The comparison between neutron scattering experiments and MD simulations can be done on several length and time scales. Comparing to incoherent scattering as for example measured in time of flight experiments [15-17] one focuses on small length scales (momentum transfers of about one inverse Angstrom) and time scales below 100 ps and the motion of individual atoms. This yields complementary information to the one obtained from the comparison with NMR experiments. On larger length and time scales it is a challenge to reproduce in the simulations the dynamics of a single chain in the melt as observed in neutron spin echo (NSE) experiments [18]. These experiments measure the configurational relaxation of a polymer chain on varying length scales in form of the intermediate coherent scattering function of the chain (5)

Here the sums run over all atoms of the same chain, so one imagines an experiment where one has a few deuterated chains in a protonated matrix, giving rise to strong coherent scattering between the atoms of one single chain.

3 CO

10

15

20

scaled time (us) Figure 4- Single chain coherent intermediate scattering function for PB at 353 K compared between experiment and simulation. Simulation times are scaled by a factor of 0.8 to account for a difference in center of mass diffusion coefficient.

156 This quantity is easily calculated from the computer simulation and Fig. 4 shows a comparison [8] of the results from the simulation (lines) to the experimental data (symbols) for a momentum transfer range of q = 0.05 A" 1 to q = 0.3 A" 1 . It turned out that there is an overall difference in center of mass diffusivity of about 20% between simulation and experiment similar to earlier experience [19], so for the figure the experimental time points are rescaled by a factor 0.8. This makes the whole set of scattering curves, measuring the configurational relaxation of the polymer chain on different length scales superimpose. Polymer melt relaxation on these length and time scales and for chains below the entanglement molecular weight, as in our case, is typically analyzed within the Rouse model. The equation of motion for this chain of phantom beads

(dWna(t)) (dWna(t)dWm/3(t'))

= 0

(6)

= 6nm5ap8(t - t')2<;kBTdt

can be solved analytically and there is a closed expression for the single chain coherent intermediate scattering function

„,

,

S(q,t)

1

( ?n

= jjexp<-—Dnt>

£

1 A

f gV.

2 ^ exp i

.

2Nq2a2

—\n-m\

( ) c- (!=!) (1 - e--^)} .

(7)

This function depends only on the length scale a of the model and the segmental friction £ entering the center of mass diffusion coefficient Z?N and the longest relaxation time of the chain TR. These quantities can be measured independently in the simulation so that we get a parameter free comparison to the prediction of the Rouse model (Fig. 5). The simulation data clearly show a stronger stretching compared to the Rouse model prediction, although there is an astonishingly good overall agreement considering the simple nature of the model. The differences are, however, significant and can be traced to the failure of the dynamic Gaussian assumption [20]

<exp{i(f • (rn(t) - fm(0))}) = exp | - ^ ( ( f n ( i ) - f m (0)) 2 ) j underlying the calculation of the scattering function. For the Rouse model this assumption is correct because it models the segmental motion

157

10

time (ITS)

Figure 5. Single chain coherent intermediate scattering function for PB at 353 K (full lines) compared to the Rouse model prediction (dashed lines). Momentum transfers are q = 0.05, 0.08,0.1, 0.14, 0.2, 0.24,0.3 A - l .

as a Gaussian process, but in reality this assumption is not fulfilled due to interactions between a segment and its environment leading to non-Gaussian, heterogeneous mobility on the relevant time scales. This heterogeneity in the local mobility can be found also in the dielectric data we will discuss next.

1.3

Dielectric Relaxation Experiments

Dielectric relaxation measurements couple to the dynamics of the dipole moment of the sample. The dielectric permittivity is the FourierLaplace transform of the dipole moment autocorrelation function. J

I iJf

^-^Ae

TOO

= l-iu

Jo

$(t)e-iojtdt

(8)

with $(£) defined as *(*) =

(M(t) • M(0)) (M(0) • M(0))

The strength of the dielectric relaxation has been calculated using [21] — e r — eu —

eu3Ve0kBT

158

where eu and er are the unrelaxed and relaxed dielectric constants of the material, respectively. The dipole moment of the box is the sum of local dipole moments along the chains so that
0.6

s CO

0.4 0.2

°'°io Figure 6. frequency.

6

10" "To" io 13 io 7 To8 i o 9 co(s!)

Real and imaginary part of the dielectric permittivity as a function of

For PB the dipole moments are located on the vinyl side group and the cis group. The partial charges, however, are small and that was the reason we did not include Coulomb forces into our MD simulation. Consequently, there can be no correlations due to the local dipole moment occurring in our simulation trajectory. We then have to see, how well our results for the dielectric permittivity agree with the experiments [22]. To this end we reinsert partial charges into the simulated trajectory. We know that we have to find (M2(0)) = iVchains(Mc2hain(0)) and we also find that the autocorrelation function of the box dipole moment can be calculated through the autocorrelation function of the chain dipole moment $(*) = *chain(<)

159

12 10 8

£4 5

2

O Exp. Arbe et al. • Exp. Aouadi et al. A Simulation

0 -2 2x10""

3x10

4x10

5x10"

6x10"

Figure 7. Position of the peak in the dielectric loss as a function of temperature for the simulation (filled triangles) and two sets of experimental data (open circles and open squares). The experiments show some scatter, but the in the overlap temperature regime experiment and simulation agree.

which has a 40 times better statistics. In Fig. 6 we show the real and imaginary part of the dielectric permittivity as obtained from the simulation. The frequencies of maximum loss can now be compared with experimental results. Experimentally the maximum loss frequency is typically measured for lower temperatures [23, 24] to study the temperature dependence of the structural glass transition or a process. Two sets of experiments in the literature show some discrepancies in an intermediate temperature window but agree with each other and with our simulation data at higher temperatures where we have an overlap temperature window between simulation and experiment. Prom this we can conclude, that also in the experiment one sees no correlations between the dipole moments of different chains. When we furthermore compare the time scale given by the maximum loss frequency with the time scales of the Rouse modes for our chains we can obtain from the simulation, we can say that the dielectric measurements on PB see the relaxation of a chain segment of about 6 backbone carbons, which is exactly the length of a statistical segment of the chains.

160

In the temperature regime accessible to our MD simulations so far we are actually observing the so-called combined a — f3 process, i.e., there has been no separation of time scales yet between the two processes which can be observed at lower temperatures. For a single type of relaxation mechanism underlying the dielectric spectrum we should be able to see time-temperature superposition if we scale the dielectric loss data in Fig. 6 by the peak position and peak height, respectively, which is shown in Fig. 8. Our results in Fig. 8 show that the time temperature

t

emerging but unresolved J3 process

500 combined (merged) a and p processes

10

10 10° co/co max

1

103

Figure 8. Time temperature superposition applied to the dielectric loss data obtained from the computer simulation in a temperature range from 500 K down to 253 K.

superposition is only borne out for temperatures above 300 K. Below this temperature the scaling breaks down, indicating a change in the molecular motion mechanisms underlying the dielectric data observed. We interprete this as the emergence of the /3 process which becomes clearly resolvable as an independent feature in the loss spectrum only at lower temperatures. Let us come back now to the question of increasing heterogeneity in the local mobilities upon decreasing the temperature. We have already identified a tendency for immediate back jumps after one torsional transition as the reason for the different temperature dependencies of the mean waiting time between torsional transitions (tWait) a n d the torsional autocorrelation time {TTACF)- I n & homogeneous system, where every chemically identical torsion shows identical dynamics on the time scales of observation the probability distribution of waiting times should be

161 Poissonian. As we can see in Fig. 9 which shows the waiting time distri-

10 ts Figure 9. Distribution of waiting times for a total of 10 torsional transitions per dihedral degree of freedom to occur plotted versus 10 times t/(twa,\t). The thick line is the Poisson distribution. Upon lowering the temperature the deviation from the Poisson distribution increases.

bution for the 10th jump to occur for temperatures T = 500 K, 450 K, 400 K, 353 K, 323 K, 293 K, 273 K and 253 K, at high temperatures the curves approach the Poisson distribution indicated by the fat line in the figure. At lower temperatures two features emerge. There is an increasing probability of very short waiting times which captures the increasing amount of direct back jumps for the dihedral transitions. Furthermore the distribution develops a fat tail towards very long waiting times indicating the increasing heterogeneity between the different torsion angles. For the high temperatures this heterogeneity will also show up on the distribution for the average waiting time until the next jump occurs, but on the scale of 10 jumps the heterogeneity at high temperatures has been smeared out: slow angles have had time to become fast and vice versa. At low temperatures this homogenization time becomes much longer than the time scales observed for the jump distribution and the two extremes of the mobility distribution show up clearly. When we characterize the distribution of waiting times for several values of n, the number of transitions observed, by their normalized dispersion D{n,T) =a2{n,T)/n

162 this means that at fixed temperature D(n,T) becomes smaller with increasing n approaching the Poisson limit of 1/n. For fixed n, D(n,t) 3.5

i4

wiat

3.0

V -

£ 2.5

-

-

2.0

m

TSSB

o

1.5

1 .u M 3 2.0 IO"

Figure 10. (see text).

2.5 KV*

3,0 10"*

Relaxation times £D(n)=i> (*wait),

3.5 \<S*

TTACF

4.0

as a function of temperature

increases dramatically with decreasing temperature due to the fat tails developing in the waiting time distribution. We can define a measure for the homogenization time of the dihedral mobility by determining the number of jumps n*(T) (this will be a real value determined by interpolation) after which D{n,T) reaches a fixed value, for example D(n*,T) = 1. This corresponds to a time scale t*(T) = n*(T)(twait)(T). The temperature dependence of this time scale is compared in Fig. 10 with that of the average waiting time and the torsional autocorrelation time. It is clear from this comparison that the torsional autocorrelation time is sensitive to this homogenization process of the torsional mobilities and that the slowing down of this homogenization leads to the stronger than Arrhenius-like increase in relaxation time as compared to the average waiting time between transitions.

2.

Changing the model Hamiltonian

So far we have presented MD simulations of a chemically realistic united atom model for PB employing a carefully validated quantum chemistry based force-field. We have shown that MD simulations employing this force-field are able to quantitatively reproduce experimental results on the structure and dynamics of polybutadiene melts at elevated

163 temperatures. For the following study we will slightly modify the chemically realistic model to one without vinyl side groups, i.e. it will be random copolymer of 55% trans 1,4-PB and 45% cis 1,4-PB. We will again have 40 chains of now 29 repeat units (115 backbone carbons) in our simulation box and will be using the same united atom force field. To bring out the effects of the torsional barriers on the local dynamics of the atoms most dramatically, we use the ability of the simulation approach to modify parts of the force field selectively. We will compare the results from the CRC model simulations to those for a freely rotating chain (FRC) version of this chemically realistic chain (CRC) model, where we switch off all torsion potentials [25]. When we discuss the

10' o

10

— f D(15.06*q) FRC 273 K CRC 273 K CRC 240 K CRC 353 K

-2

10"'

10

-2

10

1

10u

10'

q [A" ]

Figure 11. Single chain structure factor for the CRC model at different temperatures indicated in the legend and for the FRC model at 273 K. Also included is the Debye function which fits the behavior for small momentum transfers.

static structure of an amorphous polymer melt we always have to take into account two different measures of that structure. One of these is the chain structure as described by the single chain structure factor (9)

where the average is meant to include a spherical and a thermal average. This function is shown in Fig. 11 for T = 240,273 and 353 K

164 for the CRC model and for 273 K for the FRC model. The behavior for small momentum transfers agrees well with the Debye function /£>(#) = P"( e ~ x + ^ 2 — 1) where x = qRg, which describes the scattering of a Gaussian coil. The obtained value for the radius of gyration agrees well with the directly measured value. The single chain structure factor shows no temperature dependence in the depicted temperature range and agrees perfectly between the FRC and CRC models. This is a peculiarity of PB since for this polymer all minima in the different dihedral potentials are iso-energetic, which also explains the lack of temperature dependence of the single chain structure factor of the CRC model.

3-

O CRC 273 K FRC 273 K

q [A"1] Figure 12. Melt structure factor for PB as obtained from simulations of the CRC model and the FRC model at 273 K.

The other quantity characterizing the melt structure is the liquid structure factor. This is shown over a wide momentum transfer range in Fig. 12 at 273 K. For the calculation we have used the united atoms as scattering centers of equal scattering strength, calculating in this way the structure of the actual simulated systems. A quantitative comparison to the structure factor of PB would be improved by reinserting the hydrogen atoms into their mechanical equilibrium positions [26] and explicitly using the scattering lengths of the carbon and hydrogen atoms in the system. It is gratifying, that even so the position of the amorphous halo at q = 1.47 A" 1 agrees nicely with the experimental results [27]

165 and also the behavior at higher momentum transfers is comparable (to as large q as there are experimental data available). We have performed both sets of simulations at the equilibrium density of the CRC model and this result shows that under these conditions the liquid structure is the same in both models. The relevance of these findings becomes obvious when we think about the mode-coupling theory of the glass transition. In this theory one assumes that the relevant slow variables for describing the glass transition are density fluctuations and one tries to describe the arrest of the structural relaxation. Starting from the Liouville equation and using the Mori-Zwanzig projection operator formalism one arrives at the following formally exact equation [28]

4>q(t) + n2q(j>q(t) - uj>q(t) + n2q f

dt'mq{t - t')$q(t') = o ,

where cf>q(t) = (6pq{t)6p*(0))

(11)

is the correlator for density fluctuations or intermediate scattering function. ttq is a microscopic frequency scale and mq(t) is a memory kernel containing the essential physics of the problem. In the idealized version of MCT this kernel is again expressed in terms of coupled density fluctuations and the coupling constants are completely determined by the static structure of the melt, and here again mainly by the two-point correlation function of the liquid structure factor. Upon lowering the temperature towards the glass transition or increasing the density this coupling induces a qualitative change in the dynamics. In the supercooled liquid regime a two-step relaxation develops consisting of the final a or structural relaxation and a plateau or MCT-/3 relaxation regime intervening between the microscopic dynamics and the structural relaxation. This /3-regime is the time regime of caging. Upon lowering the temperature, the life-time of the plateau (cage) increases until it is infinite at Tc and all correlation functions only decay onto their plateau value. For the incoherent density correlations this plateau value is the Debye-Waller factor of the glass. For the glass transition in the bead spring model of Bennemann et al. [29, 30] this picture was essentially confirmed. In the preceding paragraphs we have shown that we have two models at hand which show the same static structure on the level of the twobody correlation functions. Do they have the same dynamics? The high temperature behavior of the CRC model (curve at T = 353 K) and the behavior of the FRC model agree. One observes a crossover from short time ballistic and vibrational motion to a subdiffusive Rouse-like regime determined by the connectivity of the chains. For the CRC model at

166

10°

t[ps] Figure 13. Mean square displacements of the sp3 carbons along the chain back-bone as a function of time for several temperatures for the CRC model and for T = 273 K for the FRC model.

273 K, however, one observes a plateau intervening between the short time motion and the Rouse-like regime and this plateau becomes more pronounced upon lowering the temperature to 240 K. It starts at around t = 1 ps and extends almost to 100 ps for 240 K. Within a mode coupling picture treating only the density fluctuations as slow variables this finding is unexpected. Also, the simulation temperatures are still in the high temperature liquid-like regime of PB (the experimental estimate for the temperature Tc where mode-coupling effects should be observable is 220 K). This slowing down is not due to packing effects which mode-coupling theory tries to capture but obviously due to the presence of intramolecular barriers against dihedral rotation. On the time scale of 1 ps the fast vibrational dynamics of the bond angles and torsion angles is damped out and this time scale is not strongly dependent on temperature. The mean time between torsional transitions, however, as we have seen increases in an Arrhenius-like fashion with decreasing temperature. Consequently we are observing a separation of time scales between the vibrational dynamics and the relaxational dynamics governed by the torsional transitions. At 240 K the mean waiting time between torsional transitions has reached about 100 ps and this is exactly the time scale of the break-up of the plateau. For shorter times the

167 mean-squared displacement curves only pick up contributions from the fast-moving torsions in the waiting time distribution and upon lowering the temperature these become fewer and fewer (at a fixed time). In principle the memory kernel in the mode-coupling equation contains contributions from three particle correlations, however, for all systems studied so far in computer simulations, these only slightly modified the predictions of the theory and helped improve agreement between simulation and theory [31]. Also, there has been an extension of the theory taking chain connectivity into account [32] which improved agreement with the simulations of the bead-spring model, but it remains to be seen whether an application of this theory to the two models presented here can account for their strongly different dynamic behavior.

0.0

Dt [A2] Figure 14- Comparison of the single chain coherent intermediate scattering function for the CRC (full lines) and FRC (dashed lines) models at 353 K. The difference in segmental friction is absorbed into a rescaling of the time axes by the chain center of mass diffusion coefficient.

A different conclusion, however, has to be drawn when we look at the dynamics on larger length and time scales [34], i.e., at the regime which is typically described by the Rouse model. These are length scales larger than the statistical segment length of the chain and time scales where connectivity dominates which are several times the typical time scale between torsional transitions. On these time scales the difference in local dynamics observed in Fig. 13 gets absorbed into just one effective rate constant, the segmental friction £. One would therefore conclude

168 that on these time and length scales the dynamics is the same after rescaling time scales for the difference in segmental friction and this is exactly what we find in Fig. 14. Note that here we really have the same chemical polymer and the same local packing for the CRC and FRC models which might explain the difference in behavior compared to the experimental results for polyisobutylene and polydimethylsiloxane [33] which have about the same statistical segment length but different local packing.

3.

Summary

In this contribution we have discussed chemically realistic MD simulations of 1,4-polybutadiene melts to exemplify which physical questions can be addressed by this technique. These simulations are targeted at understanding molecular structure and relaxations on small to intermediate length and time scales. They rely on carefully validated force fields which are able to quantitatively reproduce experimental data on well defined model systems. These force fields become available today thanks to a combination of experiments, high-level quantum chemistry calculations and simulations of simple model systems. Having established the ability of the simulation to reproduce available experimental data without any adjustable parameters one can then proceed to exploit the strong points of the simulation approach. Where the range of thermodynamic parameters coverable in the simulation is often strongly limited, for those thermodynamic state points where a simulation in full equilibrium is possible one gets the complete information on the system under study down to every coordinate and momentum of every particle. This allows for measuring properties and correlations not available to experimental techniques, like for instance the distribution of waiting times between torsional transition, which are then instrumental in understanding and explaining effects of dynamic heterogeneity in polymer melts. Another strong point of the simulation approach is its ability to selectively change parts of the model Hamiltonian. In this way one can compare a chemically realistic model of PB with a freely rotating chain version of the same polymer and does not have to switch to a completely different polymer with some of the same properties like is unavoidable in experiments [33]. With this approach we could establish that identical structure on the two-body correlation function level (single chain and liquid structure factors) does not imply identical dynamics which raises questions on the applicability of the mode-coupling theory of the glass transition to polymer melts.

169

Acknowledgments This is a paper on results from Molecular Dynamics simulations, but obviously much of this work would not have been possible without the close collaboration with the following colleagues from the experiment side: M. D. Ediger, M. Monkenbusch, X.H. Qiu, D. Richter, L. Willner. The authors acknowledge funding from the German Science Foundation under grant PA473/3-l,2, BMBF under grant 03N6015 and the American Chemical Society under grant ACS-PRF-3321AC7.

References [I] K. Binder editor, Monte Carlo and Molecular Dynamics Simulations in Polymer Science, Oxford University Press, 1995. [2] M. Doi and S.F. Edwards, The Theory of Polymer Dynamics, Oxford University Press, 1988. [3] S.C. Glotzer and W. Paul, Annu. Rev. Mater. Res. 32: 401, 2002. [4] R. Martonak, W. Paul and K. Binder. Phys. Rev. E, 57:2425, 1998. [5] V.G. Mavrantzas, T.D. Boone, E. Zervopoulou and D.N. Theodorou, Macromolecules, 32:5072, 1999. [6] N.C. Karayiannis, V. G. Mavrantzas and D. N. Theodorou. Phys. Rev. Lett., 88:105503, 2002; N.C. Karayiannis, A.E. Giannousaki, V.G. Mavrantzas and D.N. Theodorou, J. Chem. Phys., 117:5465, 2002. [7] G. D. Smith and W. Paul, J. Phys. Chem. A, 102:1200, 1998. [8] G.D. Smith, W. Paul, M. Monkenbusch, L. Willner, D. Richter, X.H. Qiu and M.D. Ediger, Macromolecules, 32:8857, 1999. [9] S. Nose, Progr. Theor. Phys. SuppL 103:1, 1991. [10] W.G. Hoover, Phys Rev A, 31:1695, 1986. [II] R.H. Boyd, R.H. Gee, J. Han and Y. Jin, J. Chem. Phys., 101:788, 1994. [12] G.D. Smith, Do Y. Yoon, W. Zhu, M.D. Ediger, Macromolecules, 27:5563, 1994. [13] W. Paul, G. D. Smith and Do Y. Yoon. Macromolecules, 30:7772, 1997. [14] G.D. Smith, O. Borodin, D. Bedrov, W. Paul, X. Qiu and M.D. Ediger, Macromolecules, 34:5192, 2001. [15] G.D. Smith, W. Paul, D.Y. Yoon, A. Zirkel, J. Hendricks, D. Richter and H. Schober, J. Chem. Phys., 107:4751, 1997. [16] K. Karatasos, F. Saija, J.-P. Ryckaert, Physica B, 301:119, 2001. [17] O. Ahumada, D.N. Theodorou, A. Triolo, V. Arrighi, C. Karatasos and J.-P. Ryckaert, Macromolecules, 35:7110, 2002. [18] G.D. Smith, W. Paul and D. Richter, Chem. Phys., 261:61, 2000. [19] W. Paul, G.D. Smith, Do Y. Yoon, B. Farago, S. Rathgeber, A. Zirkel, L. Willner and D. Richter, Phys. Rev. Lett, 80:2346, 1998. [20] G.D. Smith, W. Paul, M. Monkenbusch and D. Richter, J. Chem. Phys., 114:4285, 2001.

170 [21] H. Frolich, Theory of Dielectrics, Oxford University Press, 1958. [22] G.D. Smith, O. Borodin and W. Paul, J. Chem. Phys., 117:10350, 2002. [23] A. Arbe, D. Richter, J. Colmenero and B. Farago, Phys. Rev. E, 54:3853, 1996. [24] A. Aouadi, M.J. Lebon, C. Dreyfus, B. Strube, W. Steffen, A. Patkowski and M.R. Pick, J. Phys. Condens. Matter, 9: 3803, 1997. [25] S. Krushev and W. Paul, Phys. Rev. E, 67:021806, 2003. [26] W. Paul, Do Y. Yoon and G.D. Smith, J. Chem. Phys., 103:1702, 1995. [27] D. Richter, B. Prick and B. Farago, Phys. Rev. Lett. 61:2465, 1988. [28] W. Gotze and L. Sjogren, In Transport Theory and Statistical Physics, S. Yip and P. Nelson, eds., Marcel Decker, 1995, pp 801. [29] C. Bennemann, J. Baschnagel and W. Paul, Eur. Phys. J. B, 10:323, 1999. [30] M. Aichele and J. Baschnagel, Eur. Phys. J. E, 5:229, 2001; 5:245, 2001. [31] F. Sciortino and W. Kob, Phys. Rev. Lett, 86:648, 2001. [32] S.-H. Chong and M. Fuchs, Phys. Rev. Lett, 88:185702, 2002. [33] A. Arbe, M. Monkenbusch, J. Stellbrink, D. Richter, B. Farago, K. Almdal, and R. Faust, Macromolecules, 34: 1281, 2001. [34] S. Krushev, W. Paul and G. D. Smith, Macromolecules, 35: 4198, 2002.

MONTE CARLO SIMULATIONS OF SEMI-FLEXIBLE POLYMERS Wolfgang Paul, Marcus Muller, Kurt Binder Institut fur Physik, Johannes Gutenberg-University, 55099 Mainz, Germany Wolfgang. [email protected]

Mikhail R. Stukan, Viktor A. Ivanov Physics Department, Moscow State University, Moscow 119992, Russia Abstract

We present Monte Carlo simulations on the phase behavior of semiflexible macromolecules. For a single chain this question is of biophysical interest given the fact that long and stiff DNA chains are typically folded up into very tight compartments. So one can ask the question how the state diagram of a semiflexible chain differs from the coil-globule behavior of aflexiblemacromolecule. Another effect connected with rigidity of the chains is their tendency to aggregate and form nematically ordered structures. As a consequence one has two competing phase transitions: a gas-liquid and an isotropic-nematic transition potentially giving rise to a complicated phase diagram.

Introduction The physics of semi-flexible macromolecules has received increasing attention over the last two decades. Imagine a polymer with local bonded interactions giving rise to a substantial stiffness of the chain, i.e., the persistence length p or Kuhn length (statistical segment length) is much larger than the length of a chemical bond. When one looks at very long chains of this nature L 3> p, where L is the contour length of the chains one recovers normal Gaussian chain behavior and all the scaling properties [1] theoretical physics has been emphasizing for a long time after early work on worm-like chain molecules. In the biophysical area and also for many applications of stiff macromolecules it is important not to study the limit L —» oo but to look at the properties of chains with L ~ p. A most significant problem from biology in this context is the foldability of long and stiff DNA chains 171 P. Pasini et al. (eds.), Computer Simulations of Liquid Crystals and Polymers, 171-190. © 2005 Kluwer Academic Publishers. Printed in the Netherlands.

172 into very tight compartments in the cells [2]. This folding is often in the form of tightly wound up toroidal structures. What is the region of stability of these structures in the state diagram of a semi-flexible chain of length L (or equivalently degree of polymerization N)l Note that we will refer to a state diagram as opposed to a phase diagram because no thermodynamic limit is taken here. The existence of equilibrium toroidal structures [3-9] as well as the kinetics of the folding process [10-13] have found much interest in recent years and have been studied in 2 as well as 3 dimensions using off lattice models as well as lattice models. We will look at this state diagram and its dependence on chain length in the next section. When one studies solutions of very stiff chains one always has to deal with aggregation and ordering effects, even at very low concentrations. This has to do with the fact, that these chains have little conformational entropy and very little translational entropy to lose upon aggregation, but much interaction energy to gain by forming ordered aggregates. So the important physics of these solutions is a competition and interaction between a liquid-gas and an isotropic-nematic transition. Both transitions have been studied separately in great detail by computer simulations. The liquid crystalline behavior in solutions of semiflexible polymers has been addressed in many simulation studies over the years [14-22] the interplay between both transitions, however, has received relatively little attention [23, 24]. Which transition will prevail in the thermodynamic phase space spanned by density and temperature for which choices of stiffness of the chains? We will address this question in section 1.2.

1.

State Diagram of a Semi-flexible Chain

In all of the work we will present in the following we will employ the bond-fluctuation lattice model [25-28] a rendering of which is shown in Fig. 1. The repeat units of the polymer chain (monomers) are represented by unit cubes on a three dimensional simple cubic lattice. They are connected by a set of allowed bonds (108 in total) comprising 5 different bond lengths (2, \/5, \/6, 3 and \/l0 ) and allowing for 87 different bond angles between consecutive bonds. The properties of this model, for single chains as well as for solutions or melts, are intermediate between classical lattice random walks and continuous space polymer models like the bead-spring model. In the wide range of relevant length and time scales in polymer physics [29, 30], Monte Carlo simulations of such coarse-grained lattice modelshave proven to be particularly efficient for

173

Figure 1. model.

Rendering of the three-dimensional version of the bond-fluctuation lattice

the examination of the generic phase behavior as well as generic relaxation processes of polymer systems. For the simulation of the state diagram of a semi-flexible chain we will employ the following Hamiltonian H

-1

N

(1)

where di is the angle between two consecutive bonds k and Zj+i, b is the stiffness energy in units of T, N is the chain length and ?7LJ is the nonbonded attractive interaction which we take in the form of a discretized Lennard-Jones potential =

f - e / T [2(r - 2)3 - 3(r - 2)2 + 1] r = 2, >/5, I 0 other (2)

174 We will vary the parameters b (stiffness) and (3 = e/T (inverse temperature in units of the interaction strength). For the simulation we will employ a mixture of local random hopping moves, where one tries to move a randomly chosen monomer into a randomly chosen lattice direction subject to bond and excluded volume constraints and slithering snake moves, where one tries to attach a randomly chosen bond at one end of the chain and cut off the first bond at the opposite end. The first of these types of moves is efficient in equilibrating local structure and the second one in equilibrating large scale structure because it displaces the center of mass of a chain with each successful slithering snake move by a distance 0(1), whereas the local hopping moves only lead to a displacement O(l/N). To characterize the shape of the globules occurring in the simulation upon decreasing the temperature we measured the three principle moments of inertia M^M^M^ (eigenvalues of the gyration tensor) and constructed the following shape parameters from them M3

M

M1 + M3 and K

^W

,. (3)

A sphere is characterized by K\ = l,i^2 = 1, an ideal rod by K\ = 0, K2 = 1 and an ideal disk by K\ — 0.5, K2 = 0.5. We can furthermore characterize a tendency of the chain to wind up, like we would expect for a toroidal structure, by measuring the following quantity

a=

X^ ^ x k

h ISIIS h

(4)

where /$, i = 1 , . . . , iV — 1 denote the bond vectors along a chain. For a random coil, disordered globule or nematic globule with hairpins into different directions this quantity will be small, whereas it will be large for a chain winding around (an object or an empty volume) in a consistent direction.

1.1

Mean Field Scaling Theory

Looking at the coil-globule transition for chains of finite length we have to consider several ways to define the transition point which all agree in the thermodynamic limit but show different finite size behavior. The true transition temperature would be the ^-temperature where the second virial coefficient of the chains vanish. In a simulation one would use a finite size scaling plot to determine this temperature and look for the temperature, where R%/N (neglecting logarithmic corrections) is

175 independent of N (see Fig. 2). One can also define a transition temperature for instance as the temperature of the maximum offluctuationsin the measurements of R2, which would correspond to the temperature of vanishing free energy difference between the coil state and the globule state (see Fig. 3). For the finite size effects on this temperature theory [31, 32] predicts

Ttr/e - I ~ - p 3 / 4 i v 1 / 2 .

Rearranging for the stiffness at the transition as function of temperature (and ignoring constants) we obtain a prediction for the coil-globule transition line in our state diagram: p ~ (8* — #~~1)4/3JV2/3

(5)

To predict the coil-torus and torus-globule transition lines we have to make an ansatz for the excess free energy of the collapsed states with respect to the coil state: ** = -^elastic ~r -^attraction • -^surface •

The elastic bending free energy describes the loss of conformational entropy of a stiff macromolecule inside a cavity of size R ^elastic ^ NTp ( ^ '

where a is the segment size. The polymer volume fraction inside the collapsed state (torus or sphere) can be written as ~ Na3/Rr2 where R = r for a sphere and r < R for a torus. The attraction based part of the free energy is then Na3 ^attraction =

N j

N

where nmax is the number of neighbors in the densely packed collapsed state. In analogy, the surface contribution to the free energy is given by the fact that a monomer on the surface misses some neighbors

_ _

RrNo?

^surface — - e

where n m a x is the number of neighbors on the surface of a densely packed collapsed globule or torus. Assuming now 0 = 1 (which relates R and r) and neglecting the numbers of neighbors as irrelevant prefactors, we obtain the following ansatz for the scaling behavior of the excess free energy

^ - e N

+ e^.

(6)

176

Minimizing with respect to R gives the following scaling relations for big and small radii and the excess free energy T?

R ^ CLN^^TP1^

B~~^^

'

T rsj n N2 ^ T)~^~ ^ & ^

•

— ^"Nf^l^r^-I^R~^L/^>— (7)

Prom t h e condition F = 0 we obtain t h e coil-torus transition line

p ~ f3N2

(8)

and from the condition r = R we obtain the torus-sphere transition line p ~ pN1/s .

1.2

(9)

State Diagram

Let us begin this section on results by providing the determination of the ^-temperature of our model [7] Prom Fig. 2 we can read off a 2.0 G B O

ON = 120 QN = 160 ON = 200

1.5

0.59

v

0.63

0.67

0.71

G—ON = 20 Q

0.5

0.0 0.0

E3N = 4O

O ON = 60 A—AN = 80 <JN = 100 <

0.5

1.0

1.5

2.0

1/T Figure 2. Determination of the 9 temperature from the finite size scaling plot of ^ versus temperature (for stiffness 6 = 0).

^-temperature of 0.64 in the flexible chain limit. This temperature is

177 expected to show a stiffness dependence of 9 ~ p/ ln(p) [31] which would lead to a negligible effect in the range of stiffnesses we study, so that we will work with 6 = 0.64 in the following. 1.0 G ON = 80 Q 0N = 12O <3> ON = 160 A—AN = 200

0.8

1/T Figure 3. Transition temperature as determined by the maximum of fluctuations in the measurement of the radius of gyration (for stiffness 6 = 2).

The transition temperatures can be determined from the maximum of fluctuations in the radius of gyration as shown in Fig. 3. This maximum in the fluctuations corresponds to the state point of equal weight of two coexisting phases. Employing the equal weight rule we identified in the simulation the transition lines "1" and "2" in Fig. 4, shown by open circles (for N = 80) and filled circles (for N = 40). Line "3" is determined by following the shift of the unimodal distribution for the radius of gyration from the extended coil state to the collapsed globular state and determining the mid-point of the shift interval. Since we are working here for finite chain lengths, all regimes in this state diagram are, of course, metastable, and moving around in the state diagram changes the relative stability of the possible structures. In the left part (high temperatures) the coil is the predominant structure, in the lower right part (low temperature and stiffness) the globule is the predominant structure

178

0.8

1.0

1.2

1.4

1.6

1.8

2.0

Figure 4- State diagram of stiff macromolecules of length AT = 40 (filled circles) and N = 80 (open circles). Lines are fits using the theoretical predictions for the transition lines (see [8]).

and in the upper right part (low temperatures but high stiffness) the torus is the predominant structure. Especially close to the transition lines the structures are easily interconvertible. At the transition line between torus and globule other intermediate structures like disks and rods are also found at a significant percentage. This agrees well with recent findings from atomic force microscopy studies of collapsed DNA and polysaccharide chains [33]. A typical configuration of a toroidal globule looks like depicted in Fig. 5. To identify these structures in the course of the simulation one has to partition the simulated equilibrium structures into sub-ensembles, for instance according to their value of the winding measure a. The distribution of a-values is bimodal, allowing for the definition of two sub-ensembles with a < 150 and a > 150, respectively, for chain length N = 40. We expect the toroidal structures to be contained in the subensemble with a > 150. All the structures in this sub-ensemble are characterized by a shape parameter K\ around 0.5. Looking now at the density profiles obtained for the structures in this sub-ensemble in Fig. 7 we can identify one set of structures with a hole in the middle (toroids) and another one which is densely packed in the middle. Since these structures also have K\ ~ 0.5 they have to be disks, which can be verified

179

Figure 5. Snapshot figure of a toroidal globule for a chain of length N = 240 at stiffness 6 = 15 and inverse temperature j3 = 1.

by inspection of selected configurations. There is a rich state diagram of these semi-flexible chains and whether there are other possible shapes of the globules occurring at larger stiffnesses and for longer chains is an open question.

2*

Solutions of Semi-flexible Chains

When we now go from the regime of dilute solutions and basically isolated chains to the regime of semidilute and dense solutions of semiflexible chains we can expect two main effects. Due to the stiffness of the chains and at sufficiently large aspect ratio the chains will prefer a nematic alignment and we expect to see an isotropic to nematic transition. In the limit of almost rigid rods this transition happens at very dilute concentrations. When the chains become moreflexiblethe transition density increases. This behavior was studied for a lyotropic system in [22]. The Hamiltonian employed in this simulation of the bond-fluctuation model was given by H{b, $) = J2tb(bb

b0)2 + Y,e# ti

cos

#(! + co c o s #o)

180

2500 -I

(a)

20001500-

500-

0

50

100 150 200 250 300 350 400 450 500

a Figure 6. Histogram of the values for the winding measure a obtained in the simulation for chain length N — 40. One can partition the measurements into subensembles with a < 150 and a > 150, respectively. The line is a fit using two Gaussians.

(a)

Figure 7. Density profiles for the structures found in the sub-ensemble with a > 150 for chain length N = 40. The left panel shows the toroidal structure and the right panel is the profile for disk-like structures.

where the sums run over all bonds and all bond angles in the system, respectively. The parameters were chosen as €& = 1, €# = 0.67, 6o = 0.86 and Co = 0.03. Performing a simulation at constant density, (j) = 0.5, which is a melt density in this model, one observed an isotropic-nematic transition at a temperature of about T = 0.25 for chains of length N =

181 20 (the temperature and chain length together determine the aspect ratio of the molecules and this in turn determines the location of the isotropic-nematic transition). When one now introduces an attraction between the monomers leading to a coil-globule transition for the isolated chain, on the one hand the tendency for nematic alignment is increased and on the other hand a liquid-gas critical point is introduced into the model. For flexible chains a Hamiltonian of the form H = -en where n is the number of contacts, i.e., the number of monomer pairs within the neighbor shell 2 < Ar < \/6, introduces a liquid-gas critical point at Tc/e = 1.79 [34]. We combine stiffness and attraction into the model Hamiltonian H = -en + Y, t'biP - bo)2 + J2 e'# c o s tf(! + co cos

tf0)

(10)

where e'b = e6/0.25 = 4 and e'o = e^/0.25 = 2.68. When we set T = 1 the intramolecular part keeps the chain stiffness constant at the value it had at the isotropic-nematic transition for e = 0. Increasing e from zero we expect a shift and a widening of the coexistence region between isotropic dilute or semidilute liquid and nematic dense melt. Within mean field theory the possible shapes of the phase diagrams for such a model system have been studied theoretically in reference [35]. Depending on the chain stiffness the liquid-gas critical point can be either observable or buried in the two-phase region of the isotropicnematic transition (see Fig. 8). One can have a coexistence between dilute/isotropic and dense/nematic in the simplest case, but also three phase coexistence regions are possible between isotropic gas, isotropic liquid and nematic liquid or isotropic gas, nematic semidilute liquid and nematic dense liquid. We therefore expect to have to look for phase transitions in two coupled order parameters, density and orientation. To vary the density in the simulation we will employ a grand-canonical simulation technique using the configurational bias scheme for chain insertion and deletion [36]. We performed simulations for chain length N = 20 in a box of linear size L = 90. The linear dimension of the box is therefore 2.5 times larger than the ground state length of the polymer chains (all bond lengths equal to 2 and all bond angles equal to 0). We can therefore fit 2 — 3 nematic domains into the linear dimension of the box (some test runs also used L = 50 to look for finite size effects). In addition to the grand-canonical configurational bias moves we will also perform slithering snake moves of the chains (one suggests a new bond at one

182

— stiff chains, p=20 stiffer chains, p=50 rod-like chains, p=500

3-

». VI

r i\V___

\

1-

0- : i . •' 1 • i • 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

1

•

1

1

1

1

1

1

1

1

1

Figure 8. Theoretical phase diagram for the solutions of semi-flexible chains of different stiffnesses, parameterized by the persistence length p.

chain end and, if successful, cuts off the first bond at the opposite end) and local random hopping moves [23]. By increasing the chemical potential we will start to fill the box and at some point observe a gas-liquid transition either with or without isotropic-nematic transition accompanying it. The nematic order can be monitored with the help of the Mayer-Saupe tensor x

Qa/? =

iVp(JV-l) j

NP(N - 1) §

2

(ii)

where the sum runs over all bond vectors in the system, Np is the number of polymer chains, N the degree of polymerization and a and (3 are Cartesian coordinates. The largest eigenvalue of this tensor is the orientational order parameter, S. When one limits the sum in this definition to only run over the bonds in one single chain and then averages the largest eigenvalue of the so-defined tensor over all chains, one can define a chain nematic order parameter, 5'chainThe transition from a high-density nematic state to an isotropic or nematically ordered (semi-)dilute state can be observed when we start from a columnar crystal completelyfillingthe lattice and then reduce the chemical potential to decrease the density. A typical curve for the density

183

O-O 0.00 D-D 0.08 AAQ.15 OO0.20 v v 0.30 >>0.40

15 20 (a)

25

30

35 40 |Ll

45

50

55 60

Figure 9. Density as a function of chemical potential for two paths: increasing the chemical potential (filled symbols) and decreasing the chemical potential (open symbols). The legend gives the values of the attractive interaction strength in the simulations.

as a function of chemical potential for both paths is shown in Fig. 9. For e = 0 their seems to be a continuous curve from the dilute to dense regime with a kink around /J, ~ 52. With increasing e, however, a stronger and stronger hysteresis loop starts to develop indicating an increasingly strong first order transition. The first order nature of this jump in the density goes along with a jump in the nematic order parameter shown in Fig. 10. Again the curve seems to be continuous for e = 0 indicative of the weakly first order nature of the isotropic-nematic transition in this case. Increasing the attractive interaction a jump in the order parameter develops which is especially pronounced for stronger attraction and the path of decreasing chemical potential. Let us indicate two points which are worrisome in these last two figures. Firstly we can observe that the density of the nematic phase is around 0.9 which means that we have to work with an almost filled lattice with extremely long relaxation times for all chemical potentials larger than the one for which the jumped occurred. Nevertheless, both for increasing and decreasing chemical potential path the densities of the

184

1.0

jQ

0.8 0.6 0.4 o o o.oo

0.2 0.0

20

D-D 0.08 A-A0.15 O-O0.20 VV 0.30

25

30

35

40

45

50

55

60

Figure 10. Nematic order parameter as a function of chemical potential for two paths: increasing the chemical potential (filled symbols) and decreasing the chemical potential (open symbols). The legend gives the values of the attractive interaction strength for the simulations.

nematic dense phase nicely match up with each other. This is not the case for the order parameter leading us to the second worrisome point. Upon increasing the chemical potential we do not jump to the almost perfectly ordered nematic state (5 < 0.9), to which the columnar phase quickly relaxes, but to a state with a much reduced order parameter of S ~ 0.6. The reason for this reduction in order can be seen in Fig. 11. The model systems form nematic domains separated by domain walls running across the simulation box. These domain walls are extremely stable (there is little enthalpic driving force for their dissolution) and they may also be stabilized by finite size effects of the simulation box, due to a size mismatch between the simulation box and the preferred equilibrium domain size. For a small simulation box (for instance for linear size L — 50) one can avoid the multidomain state, however, at the cost of artificially enhancing the ordering tendency due to this finite size effect which leads to a larger error in the determination of the transition chemical potential. Ideally, the bimodal line in density and nematic order as a functionof temperature and chemical potential would be determined by applying the equal weight rule to the probability distributions of density and ne-

185

Figure 11. Snapshot of a multidomain state of our model system which is reached by increasing the chemical potential (density) from the dilute side.

matic order observed in the simulations. At the first order phase transition one should observe a bimodal structure of (semi-)dilute and dense in the density distribution and isotropic and nematic in the order parameter distribution. This bimodal structure can be observed for the density, however, we were not able to observe it in the order parameter. For the small simulation volume L = 50 we could observe two peaks and also observe a single system tunneling between the two states in the course of the simulation. For the large system L = 90 no bimodal structure could be obtained because the multidomain states fill up the order parameter histogram obtained in the simulation between the isotropic and nematic values. Employing expanded ensemble techniques to allow for a frequent tunneling between dilute and dense states in the density histograms was possible, however, due to the weak coupling between density and nematic order parameter this also did not lead to an improvement in the order parameter distribution. Having only the chemical potential at our disposal as the control variable for the simulation we can directly manipulate density but we were not able to improve the sampling for the order parameter. An alternative way to determine the location of the bimodal in the phase diagram consists of finding the loci of equal pressure between the isotropic and nematic states. For a lattice model pressure determina-

186 tion is more difficult than for a continuum model and the best working method for this task is the repulsive wall method of Dickman et al. [37] which has been successfully applied to the bond fluctuation model before [38]. It turned out, however, that this method contained finite size effects which had not been taken into account before. After quantifying these effects and establishing a method how to avoid them [39] we were able to calculate the pressure in our simulation volume for the isotropic state over a large range of densities to high accuracy. The unsolved problem here is the question what is actually meant by the osmotic pressure of the system and whether one should not rather consider a pressure tensor (the repulsive wall method anyhow measures only the normal component of the pressure tensor in the system and assumes that the bulk of the system is homogeneous and isotropic to relate that measurement to the osmotic pressure.

°'°00.0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Figure 12. Phase diagram for the isotropic-nematic transition. The determination of the coexistence densities and error bars is described in the text.

Having excluded the applicability of more precise methods of locating the binodal lines in the phase diagram we resorted to a very simple minded determination. From the simulation results shown in Fig. 10 we can determine the limits of stability coming from the disordered as well as from the ordered side, i.e., we are getting the loci of the spinodal lines. We then assume that the binodal value for the chemical potential

187 lies in the middle between these two spinodal values (which would be true only for a symmetric phase diagram) and with this value determine the coexisting densities using the results shown in Fig. 9. We then use a conservative error estimate by giving the error through the values for the spinodal densities. The resulting phase diagram is shown in Fig. 12. We can identify a type of phase diagram predicted from the theory for intermediate stiffnesses. We do not see a coexistence between two nematic states as was predicted for very large stiffness and also no liquid-gas coexistence a lower densities. The liquid-gas coexistence point seems to be buried within the two-phase region of the isotropic-nematic transition and we have indications [23] that it may become observable when we slightly reduce the intrinsic chain stiffness.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Figure 13. Change of the persistence length of the chain upon going from the isotropic to the nematic state.

We also note that the stiffness of our chains is lower by a factor of 15 compared to the theoretically predicted region. Some numerical discrepancy was expected due to the mean-field nature of the theory. May be, however, the theory also underestimates the stiffening the chains undergo upon ordering. When we measure the persistence length in the simulation defined by the decay of orientational correlation between

188 bonds along the chain

N-l-l

C(l) =

we observe a very strong stiffening effect shown in Fig. 13. Upon approaching the isotropic-nematic coexistence around 0 = 0.5 there is a small increase in chain stiffness and upon transition into the nematic phase the chains practically stretch out completely.

3,

Summary

We have discussed in this contribution some results on the phase behavior of semi-flexible polymers. The results we presented were taken exemplarily from simulations of the bond-fluctuation lattice model. For single chains the introduction of chain stiffness imparts much complexity onto the state diagram of a semi-flexible chain of finite degree of polymerization. It is this additional complexity which, for instance, manifests itself in the existence of a region in the stiffness-temperature plane, where a toroidal globule is the thermodynamically most stable structure which makes these chains interesting model systems for biopolymers like DNA and their packing properties in the cells. With increasing chain length at constant stiffness (i.e., the ratio of contour length over persistence length increases) the region of stability of the toroidal globule becomes smaller and it vanishes in the thermodynamic limit. In that limit, however, there is a region of stiffnesses where a nematically ordered dense globule is stable [40]. When one leaves the single chain limit and goes to semidilute solutions of semi-flexible chains one immediately has to face their tendency for aggregation. Stiff chains lose practically no conformational entropy and very little translational entropy upon aggregation but gain much in interaction energy. For semi-flexible chains this introduces a liquid-gas coexistence with a very low density in the gas phase into the phase diagram. At the same time, the chains stiffen upon aggregation and one observes an isotropic-nematic transition. For the range of stiffnesses we introduced into our model Hamiltonian there was just one simultaneous liquid-gas and isotropic-nematic coexistence. We did neither find two coexisting isotropic phases of different densities nor two coexisting nematic phases of different densities. This situation may, however, change upon reducing the stiffness or changing the length of our chains.

189

Acknowledgments The authors profited from helpful discussions with A.R. Khokhlov, A. Yu Grosberg, V. V. Vasilevskaya, P. G. Khalatur, A. N. Semenov, I. A. Nyrkova and J. Baschnagel. We also acknowledge funding by the Deutsche Forschungsgemeinschaft, grant no. 436 RUS 113/223, INTAS grants 01-607 and YSF 2001/1-174 and the Russian foundation for Fundamental Research, grant 03-03-32773.

References [I] P.G. De Gennes, Scaling Concepts in Polymer Physics, Cornell University Press, Ithaca, New York, 1979. [2] A. Yu. Grosberg, T.T. Nguyen, and B.I. Shklovskii, Rev. Mod. Phys., 74:329, 2002; V.A. Bloomfield, Biopolymers, 44:269, 1998. Chap. 7 at http://biosci.umn.edu/biophys/BTOL/supramol.html. [3] H. Noguchi, S. Saito, S. Kioaki and K. Yoshikawa, Chem. Phys. Lett, 261:527, 1996. [4] K. Yoshikawa, M. Takahashi, V.V. Vasilevskaya and A. R. Khokhlov, Phys. Rev. Lett, 76:3029, 1996. [5] H. Noguchi and K. Yoshikawa, Chem. Phys. Lett, 278:184, 1997. [6] H. Noguchi and K. Yoshikawa, J. Chem. Phys., 109:5070, 1998. [7] V.A. Ivanov, W. Paul and K. Binder, J. Chem. Phys., 109:5659, 1998; V.A. Ivanov, M.R. Stukan, V.V. Vasilevskaya, W. Paul and K. Binder, Macromol. Theory Simul., 9:488, 2000. [8] M.R. Stukan, V.A. Ivanov, A. Yu Grosberg, W. Paul and K. Binder, J. Chem. Phys., 118:3392, 2003. [9] Yu. A. Kuznetsov and E.G. Timoshenko, J. Chem. Phys., I l l : 3744, 1999. [10] H. Noguchi and K. Yoshikawa, J. Chem. Phys., 113:854, 2000. [II] T. Sakue and K. Yoshikawa, J. Chem. Phys., 117:6323, 2002. [12] B. Schnurr, F.C. MacKintosh and D.R.M. Williams, Europhys. Lett, 51:279, 2000. [13] B. Schnurr, F. Gittes and F.C. MacKintosh, Phys. Rev. E, 65: 061904, 2002. [14] T.M. Birshtein, A.A. Sariban and A.M. Skvortsov, Polymer, 23: 1481, 1982. [15] P.G. Khalatur, Y.G. Papulov and S.G. Pletnava, Mol. Cryst Liq. Cryst, 130:195, 1985. [16] A. Baumgartner, J. Chem. Phys., 84:1905, 1986. [17] A. Kolinski, J. Skolnik and R. Yaris, Macromolecules, 19:2560, 1986. [18] M.R. Wilson and M.P. Allen, Mol. Phys., 80:277, 1993. [19] M. Dijkstra and D. Frenkel, Phys. Rev. E, 51:5891, 1995. [20] F.A. Escobedo and J.J. de Pablo, J. Chem. Phys., 106:9858, 1997. [21] A. Yethiraj and H. Fynewever, Mol. Phys., 93:693, 1998. [22] H. Weber, W. Paul and K. Binder, Phys. Rev. E, 59:2168, 1999.

190 [23] V.A. Ivanov, M.R. Stukan, M. Muller, W. Paul and K. Binder, J. Chem. Phys., 118:10333, 2003. [24] W. Hu, D. Prenkel and V.B.F. Mathot, J. Chem. Phys., 118: 10343, 2003. [25] I. Carmesin and K. Kremer, Macromolecules, 21:2819, 1988. [26] H. P. Wittmann and K. Kremer, Comp. Phys. Commun., 61:309, 1990. [27] H.-P. Deutsch and K. Binder, J. Chem. Phys., 94:2294, 1991. [28] W. Paul, K. Binder, D.W. Heermann and K. Kremer. J. Phys. (France), 1:37, 1991. [29] K. Binder editor, Monte Carlo and Molecular Dynamics Simulations in Polymer Science, Oxford University Press, 1995. [30] S. C. Glotzer and W. Paul. Annu. Rev. Mater. Res., 32:401,2002. [31] A. Yu. Grosberg and A. R. Khokhlov, Adv. Poylm. Sci., 41:53, 1981; A. Yu. Grosberg and A. R. Khokhlov, "Statistical Physics of Macromolecules", Americal Institute of Physics, 1994. [32] A.Yu. Grosberg and D. V. Kuznetsov, Macromolecules, 25: 1970, 1992; 25:1980, 1992; 25:1991, 1992; 25:1996, 1992. [33] B.T. Stocke, presentation at the Europolymer Congress, Stockholm, 2003. [34] N. Wilding, M. Muller and K. Binder, J. Chem. Phys., 105:802, 1996. [35] A.R. Khokhlov and A.N. Semenov, J. Stat Phys., 38:161, 1985. [36] B. Smit, Moi Phys., 85:153, 1995. [37] R. Dickman, J. Chem. Phys., 87:2246, 1987. [38] H.-P. Deutsch and R. Dickman, J. Chem. Phys., 93:8983, 1990. [39] M.R. Stukan, V.A. Ivanov, M. Muller, W. Paul and K. Binder, J. Chem. Phys., 117:9934, 2002. [40] U. Bastolla and P. Grassberger, J. Stat. Phys., 89:1061, 1997.

MACROMOLECULAR MOBILITY AND INTERNAL VISCOSITY. THE ROLE OF STEREOREGULARITY Giuseppe Allegra Dipartimento di Chimica, Materiali e Ingegneria Chimica "G. Natta" Via L. Mancinelli 7, 20131 Milano, Italy giuseppe.allegraOpolimi.it

Sergio Bruckner Dip. di Scienze e Tecnologie Chirniche, Universita Via Cotonificio 108, 33100 Udine (Italy).

Abstract

Macromolecular dynamics at the scale of a few chain bonds is largely controlled by the " internal viscosity" effect if the energy barriers hindering the skeletal rotations are sufficiently large. In an extensive spin-echo neutron scattering analysis, Richter and co-workers (Macromolecules, (2001), 34, 1281) investigated by spin-echo neutron scattering the dynamic properties of polyisobutylene (PIB) and polydimethylsulfoxide (PDMS) in toluene solution, the latter polymer being currently assumed to have very small rotational barriers. Analysis of the data according to a theory proposed by one of us (G.A.) enabled them to obtain realistic values both of the rotational barrier around C-C bonds (w 3kcal/mol) and of the natural frequency of the rotational jumps for PIB. - A problem related to chain internal viscosity concerns the iso- and syndiotactic forms of polystyrene (respectively i-PS and s-PS). After a careful conformational analysis it is shown that i-PS has very large effective energy barriers due to interactions between phenyl rings. This effect is compounded with that of the intrinsic rotational barrier and helps explaining the kinetic difficulty to crystallise of i-PS as compared with s-PS.

Introduction The motion of a long polymer chain is bound to take place via rotations around chain bonds, see Figure 1. In turn, each rotation implies that a conformational energy barrier must be surmounted, see Figure 2; 191 P. Pasini et al. (eds.), Computer Simulations of Liquid Crystals and Polymers, 191-201. © 2005 Kluwer Academic Publishers. Printed in the Netherlands.

192 consequently the process entails energy dissipation for reasons analogous to those discussed by Eyring in his viscosity theory for a system of heavy spheres [1]. The dynamical-statistical description of this phenomenon has been the subject of a large body of investigations, and in the last few years several experimental results have been made available [2-4]. In the present paper we shall discuss the results in the light of a theoretical approach due to the present author and to Fabio Ganazzoli [5-8]. The main ideas will be concisely presented in this paper.

\ isKw^.,

Figure 1.

J

/

.2k

Any deformation of a polymer chain implies chain rotations.

CH,,CH,

0° Figure 2.

60°

120°

180°

240°

300°

6

The internal energy for the rotation around the central bond in n-butane.

193

1.

Internal viscosity

It is useful to decompose ideally the chain conformation into Fourier sinusoidal components. Each component consists of the alternation of a compressed chain strand comprising a large amount of gauche skeletal rotations, and of an extended strand with a larger proportion of trans rotations, thus creating a wave-like distribution of compressed/extended regions. Each wave moves towards either chain direction, and the superposition of these travelling waves produces the overall chain conformational dynamics. The propagation of each travelling wave along the chain is controlled by internal viscosity, i.e., the dissipation effect produced by the energy barriers hindering the rotations. The resulting dynamics may be described through the Langevin dynamical equation, where each chain atom experiences four forces, namely: 1) the elastic force

K[R(k + 1, t) - 2R(*, t) + R(fc - 1, t)] « kd^t];

(k =

where Coo is the polymer characteristic ratio, / is the C-C bond length, ks Boltzmann's constant, T the temperature and the position vector of the fc-th chain atom (k = 1,2...AT) at time t (see Fig.3);

K=3 K=5 ^

^

\

K=4

^

K=2

K=l LJ

/

R(K,t)

Figure 3. Scheme showing the vector representation of the chain atoms from an arbitrary origin (point 0).

2) the friction force

194 dR(k,t) ^ dt 3) the Brownian force X(fc, t) (purely stochastic). 4) the polymer internal viscosity, to be identified with an additional force proportional to *R(k,t)

dkdt

{k

3kBT

' -~cJ*}

The sign ambiguity is related with the direction of propagation of the deformation. The rotational characteristic time To is AE where AE is the average rotational barrier, see Fig.2, whereas A~l is proportional to the rotational frequency.

2,

Recent experimental investigations

A.Arbe, D. Richter and coworkers [4] carried out careful neutron spinecho investigations on the dynamics of monodisperse chains of PDMS (a polymer with a small value of the rotational barrier, therefore of TQ) and of PIB with a similar length, in toluene solution (see Figure 4). They had previously studied PIB melts with the same technique [3]. The classical Rouse-Zimm theory, with no internal viscosity, accounts satisfactorily for the data from PDMS. Conversely, the results from PIB could only be interpreted after adding the internal-viscosity, according to the approach proposed by Allegra and Ganazzoli [5-8] (see Figures 5,6). In the case of PIB, the numerical results for TO were (30 < T < 100°C, dilute toluene solutions) r0 = 1.27 x H r 1 2 e x p ( - | ^ )

[s]

{RT in kcal/mol)

Analogous calculations carried out for type-I inversions yield a much lower energy barrier, i.e. around 4kcal/mol. The result for AE (= 3.1kcal/mol) is in a reasonable agreement with the current value of the rotational barrier for alkyl polymers. The prefactor is consistent with the rotational frequency around C-C bonds, in one of the limiting cases considered by Kramers [9]; this is currently referred to as the natural oscillation frequency, whereby the medium viscosity produces a very small friction, however sufficient to provide the Brownian energy at constant temperature. In the case of PIB in

195

\ CH3

CH 3

CH 3

CH3

CH3

Figure 4-

Scheme showing the structure of PIB (above) and PDMS (below).

the melt the theoretical interpretation is qualitatively similar but the value of AE is much larger (« lOkcal/mol instead of « 3kcal/mol, in the temperature range 100 - 200° C). It is reasonable to assume that in the molten state the skeletal rotations of adjacent chains are strongly coupled, the degree of coupling becoming larger and larger as the glass transition temperature Tg is approached.

3. 3.1

Steric hindrance to rotational propagation Isotactic Polystyrene (i-PS)

In addition to internal viscosity, arising from intrinsic rotational potentials, essentially due to interactions among electron pairs on adjacent atoms, the chain conformational motion may be hindered by steric interactions among side groups belonging to the same chain. A significant

196

Figure 5. Chain dynamic structu factor of PDMS (empty symbols) and PIB (full symbols) in toluene solution at 300K (a) and 378K (b). The corresponding <J(= 4TT sin 6/X ) values are indicated. Lines through the points are guides to eye.

example is provided by isotactic polystyrene (i-PS). With isotactic polymers, left- and right-handed chain strands with a threefold helical conformation follow one another with alternating types of conformational inversions. The resulting model is qualitatively shown in the following Scheme, where T and G respectively stand for a trans and a gauche rotation around a skeletal bond (see Fig.2).

197

0.2

2.8

3.2 3.6 1000/T(K)

Figure 6. T-dependence of the solvent viscosity (dashed line) and the characteristic time TO deduced for the conformational transitions in PIB (diamonds). The solid line through the points is the best fit to an Arrhenius law.

Scheme:

While in the conformational inversion of type I side groups of adjacent monomer units point away from the inversion site, in type II they point towards the inversion site. We carried out a computational project to evaluate the internal energy profile involved in the displacement of both inversion points from one monomer unit to the next [10]. Our conformational calculations were performed with the CERIUS computer program. By far the most critical conformational path is the one involved with type II-inversion. In this case a significant steric hindrance arises from interactions between 4-th and 5-th neighbouring phenyl rings, see Fig.7. As it may be seen in Fig.8, skeletal rotations around 5 consecutive chain bonds were driven together in the search for the lowest-energy path that shifts transition point II by one monomer unit, see Scheme.

198

Figure 7. Molecular models of i-PS involved in a conformational transition shifting the inversion point (II in Scheme 2) by one monomer unit. The starting point is (a), the final model is (c), the intermediate model (b) is recorded near the energy maximum.

As shown in Fig.8, the resulting energy barrier turns out to be about 15kcal/mol; it effectively increases the natural barrier producing TQ. The resulting energy plot is given in Figure 10. The energy barrier is in the vicinity of 6kcal/mol.

3.2

Syndiotactic Polystyrene (s-PS)

In this case the transition shown in Fig.9 was investigated. As in the isotactic case, it implies the displacement of the inversion point by one monomeric unit. The transition does not involve strong conflicts among side groups, and it was performed by driving four chain bonds one at a time without co-operative changes of other bonds.

199

Figure 8. Energy (black circles) as a function of the conformational path in transition II for i-PS (see SCHEME). Open circles represent the torsional contribution to the total energy. The reaction coordinate along the abscissa corresponds to changing by less than 10° one of five consecutive rotational states at a time, along the minimum-energy path (see ref.[10] for details).

4,

Some concluding remarks on internal viscosity and steric rotational hindrance

Generally speaking, within a polymer chain in a low-viscosity solvent strain propagation takes place via rotational rearrangements and its velocity cannot exceed the limiting value r~h (bonds/s along the chain contour), where reff is the characteristic time of the effective rotational barriers. In a polymer like PIB, with a relatively small steric hindrance to skeletal rotations, we have refj = TO, or the characteristic time due to the intrinsic energy barrier (mainly due to electron-pair interactions). The resulting linear rate is around lcra/s, for a coiled chain with N = 100 skeletal atoms at 300K. The same limiting rate of strain propagation should hold in a highly swollen polymer network. In the case of i-PS, a polymer with a strong steric hindrance to skeletal rotations, chain dynamics in general - and crystallisation rate in particular may be strongly depressed in comparison with other polymers having a chemically similar but stereochemically different structure. In this case the effect of the intrinsic energy barrier is compounded with the steric effect, and we may write as a first approximation r

e//

= T

0{intrinsic)

200

Figure 9. Molecular models of s-PS: the starting model (a), the final model (c) and an intermediate model recorded at the relative minimum corresponding to eight consecutive trans states.

It is interesting that isotactic polystyrene (i-PS) crystallises more slowly and to a lower degree than the syndiotactic polymer (s-PS), in qualitative agreement with the results reported. These considerations suggest new possible experimental verifications of internal viscosity in highly swollen polymer networks. We point out that in an unswollen (bulk) polymer sample, strain propagation takes place at a much larger rate via direct interatomic contacts (the sound velocity). At lower temperatures, in the bulk polymer the effective rotational energy barrier (E increases, as an increasing number of adjacent chains tend to couple their skeletal rotations. At the glass temperature Tg the effective barrier AEeff is expected to go to very large values. As a last remark, we point out that friction phenomena occurring within a bulk, amorphous polymer sample sliding on a hard surface [11] are basically controlled

201

E 12 (kcal/mof) 8 6 4 2 -

Figure 10. Energy as a function of the conformational path for s-PS. See also the caption to Fig. 8.

by large-frequency dynamic effects; a quantitative investigation on this issue is under way in our laboratory [12].

References [I] H. Eyring, J. Chem. Phys., 4:283, 1936. [2] D. Richter, M. Monkenbusch, J. Allgeier, A. Arbe, J. Colmenero, B. Farago, Y. Cheol Bae and R. Faust, J. Chem. Phys., 111:6107, 1999. [3] D. Richter, M. Monkenbusch, W. Pykhout-Hintzen, A. Arbe and J. Colmenero, J. Chem. Phys., 113:11398, 2000. [4] A. Arbe, M. Monkenbusch, J. Stellbrink, D. Richter, B. Farago, K. Almdal and R. Faust, Macromolecules, 34:1281, 2001. [5] G. Allegra , J. Chem. Phys., 61:4910, 1974. [6] G. Allegra and F. Ganazzoli, Macromolecules, 14:1110, 1981. [7] G. Allegra, J. Chem. Phys., 84:5881, 1986. [8] G. Allegra and F. Ganazzoli, Advances in Chemical Physics, I. Prigogine and S.A. Rice (Eds.), 75:265, 1989. [9] H.A. Kramers, Physica, VII:4, 1940. [10] S. Bruckner, G. Allegra and P. Corradini, Macromolecules, 35:3928, 2002. [II] N. Maeda, N. Chen, M, Tirrell and J.N. Israelachvili, Science, 297:379, 2002. [12] G. Allegra, paper in preparation.

PROTEIN ADSORPTION ON A HYDROPHOBIC GRAPHITE SURFACE Giuseppina Raffaini and Fabio Ganazzoli Dipartimento di Chimica, Materiali e Ingegneria Chimica "Giulio Natta", Politecnico di Milano Via L. Mancinelli 7, 20131 Milano, Italy [email protected]

Abstract

We review here recent atomistic simulations of the adsorption of some protein fragments on a hydrophobic graphite surface. Fragments of unlike secondary structure containing either Q-helices or /3-sheets and of unlike hydropathy were taken into account. The simulations were carried out with simple energy minimizations to describe the initial adsorption on a bare surface, and with molecular dynamics runs to study the final and most stable adsorption geometry in a dielectric medium. Large conformational changes were found, involving complete denaturation of the fragments and a large spreading on the surface, with some degree of bidimensional ordering. The kinetics of surface spreading is also briefly reported. Finally, the statistical hydration of the isolated and adsorbed fragments was also investigated by explicitly accounting for the solvent.

Introduction Biomaterials are synthetic or natural materials intended to interact with biological systems. The success of an implant in human body is conditioned by the interactions between the biological system and the biomaterials [1], usually mediated by proteins [2, 3]. There are many experimental techniques for measuring the amount of adsorbed proteins on solid surfaces and for studying their conformational changes [4]. Examples of these techniques are depletion methods, quartz crystal microbalance measurements, ellipsometry, total internal reflection fluorescence, Fourier transform infrared reflection, neutron and X-ray reflectivity, Atomic and Scanning Force Microscopy, and circular dichroism spectroscopy. In spite of a large number of experimental investigations, the mechanism of adsorption and binding of proteins on solid surfaces is still only partially understood. In recent years, computer simula203

P. Pasini et al. (eds.), Computer Simulations of Liquid Crystals and Polymers, 203-219. © 2005 Kluwer Academic Publishers. Printed in the Netherlands.

204

tions have added significant new insights to our understanding of these phenomena, complementing both theory and experiments. In this paper, we discuss some recent advances in this field, focusing in particular on our simulation results. At first, we briefly review both theoretical approaches and coarse-grained simulations with an emphasis on the relevance of atomistic simulations. Afterwards, we discuss our simulation methodology and the protein fragments we took into account, and then we report our results for the initial and the final adsorption stage on a hydrophobic surface in a dielectric medium. Later on, we give a preliminary overview of the kinetics of spreading of the adsorbed protein fragments, and then we discuss the simulations in the explicit presence of water. Finally, we give an outlook to further issues we are currently tackling.

1.

Short background of theoretical and simulation methods

Early theoretical approaches to study protein adsorption on a solid surface adopted semimacroscopic models derived from colloid science, providing encouraging results when electrostatic interactions with a charged surface are dominant. For instance, Roth and Lenhoff [5, 6] modelled the protein as a charged sphere having the same volume as the protein under consideration, and including also the van der Waals interactions through the Hamaker approach calculated the equilibrium constant for the protein- surface interaction. Later, the full protein structure with the actual charge distribution at its envelope was accounted for within the same methodology [7], or using Brownian Dynamics simulations [8], but still assuming throughout a rigid molecular structure equal to the crystallographic one. On the other hand, simulations with coarse-grained models on a lattice or in continuous space are often used in polymer science to describe the general, often universal, large-scale behaviour. Accordingly, similar models have been used to investigate certain aspects of protein folding [9], or the kinetics of absorption and denaturation [10], considering an appropriate copolymer simply formed by hydrophilic and hydrophobic units (an amphiphilic copolymer) on a lattice [9, 10]. In addition to oversimplify the molecular structure of proteins, these approaches can be plagued by lattice artefacts, which favour locally ordered structures and strongly constrain the conformational degrees of freedom, which is a crucial limitation in particular for compact structures. The lattice models were also improved upon by considering the possibility of directional interactions among the units that mimick the formation of

205 antiparallel /3-sheets [11], or by adding further terms that penalise the formation of kinks, thus effectively enhancing the backbone stiffness [12]. The latter model was employed to investigate adsorption on attractive surfaces, providing a rough description of the large molecular rearrangements involved in the process. In this way, it was claimed that bundles of a-helices, coarsely modelled on a cubic lattice, may undergo denaturation and subsequent surface refolding to a /3-sheet structure [12]. We feel such claims to be largely unsubstantiated, being obviously affected, and most likely dictated, by the lattice structure. It must be stressed therefore that the above-mentioned approaches, though fairly satisfactory from certain viewpoints, cannot describe the specific rearrangements and possible denaturation of real proteins maximizing the interaction energy, in particular on neutral or on hydrophobic surfaces [13-16]. Therefore, progress in the understanding of these important biological phenomena must account for the specific protein structure, including also the atomistic details. Only in this way one can achieve a realistic description of molecular rearrangements, conformational changes and possible ordering after adsorption on a solid surface. On the other hand, atomistic molecular dynamics simulations are lacking, to the best of our knowledge. Therefore, we recently tackled this problem using molecular mechanics (MM) and molecular dynamics (MD) methods, studying both the intramolecular rearrangements on a surface and the energetics and kinetics of the adsorption process for a possible comparison with the experimental data. Our methodological approach and some results were already reported in Refs. [17], and [18], indicated in the following as paper I and paper II for brevity.

2.

Simulations details

In this paper, we discuss our recent atomistic simulations of the protein adsorption on a hydrophobic surface, considering a flat graphite surface chosen because of its simplicity and of its rigidity, so that it can be treated as a fully rigid body. This surface may be viewed as a zero-order approximation of pyrolytic carbon, widely used in certain body implants such as cardiac valves. As proteins, we considered human serum albumin, the most abundant blood protein, and fibronectin, a protein present in the extracellular matrix and in body fluids that is involved in the early stages of blood clotting. Due to the very large size of these proteins, we selected two albumin fragments, the A and E subdomains, and the fibronectin type I module shown in Figure 1. Thus, we chose protein fragments with unlike secondary structures, a-helices or antiparallel /3-sheets. As starting geometries we used the experimen-

206

tal structures deposited with the Protein Data Bank [19] (human serum albumin, 1AO6, and fibronectin type I module, 1FBR, Ref. [19]), while the graphite planes were prepared from scratch.

subdomain A

subdomain E

X<

Figure 1. Experimental geometry of the albumin A and E subdomains (above) and of the fibronectin type I module (below). For simplicity, a-helices are indicated through cylinders, and /5-sheets through solid arrows.

More details on the amino acids comprised within these protein fragments, and on their hydropathy index calculated according to the KyteDoolittle scale [20] are reported in paper I and in paper II. In Figure 2 we show the frequency of residues of the albumin subdomains and of the fibronectin module as a function of their hydropathy index. It should be noted that a positive hydropathy value indicates hydrophobicity and a negative one hydrophilicity. We point out that the albumin A subdomain has the largest fraction of hydrophilic residues, as shown by Figure 2, while the fibronectin module shows the largest fraction of residues with a null hydropathy index, thus being neither markedly hydrophilic nor hydrophobic. However, the amino acid distribution within the a-helices of the albumin subdomains is uneven. Thus, for instance, the A subdomain contains a strongly hydrophobic and a very hydrophilic a-helix. On the other hand, the E subdomain is more representative of the whole albumin, because it contains amino acids with unlike hydropathy index in all a-helices. Moreover, we note that in the fibronectin module all the hydrophobic residues are buried inside the molecule, while the outer en-

207

velope comprises only hydrophilic, or at least non-hydrophobic residues, reflecting a general pattern of globular proteins. 0.5

Albumin subdomainl \A | module ig§ n Fibronectin 0.4J

I

I

0.2 0.1 0.0 .5 .4 -3 .2 -1 0 1 2 3 hydropathy index

4

5

Figure 2. Frequency of residues of the two albumin sub domains and of the fibronectin module as function of their hydropathy index (see text).

We studied the adsorption of these protein fragments on a hydrophobic graphite surface adopting a two-step strategy in an effective dielectric medium: 1 - Direct optimizations of different interaction geometries corresponding to different directions of close approach to the flat graphite surface. This stage is important to understand the initial adsorption stage on a bare surface. 2 - MD runs of selected geometries and optimizations of many instantaneous snapshots to study the final adsorption state, the best adsorption geometry and the kinetics of spreading on the surface. Short MD runs with a few thousands of water molecules were also carried out to analyze the average hydration of the protein fragments, to detect possible conformational changes due to the solvent and to assess the stability of the geometries found. All simulations were carried out with the Insightll/Discover 2000 package, distributed by Accelrys Inc. [21] (San Diego, CA), using the Consistent Valence Force Field [22]. The MD simulations were performed in the dielectric medium or in water with periodic boundary conditions at T = 300K with a time step of lfs. In all cases, the long

208

MD runs in the dielectric medium lasted for at least Ins, until eventually an equilibrium state was achieved. More details of our simulation methodology are reported in paper I and in paper II.

3-

Initial adsorption stage in the dielectric medium

As mentioned before, we first minimized the energy of the protein fragments close to the surface to study the initial adsorption stage. In this stage, the protein strands locally optimize the interaction with the surface by loosing in part the secondary structure. In particular, the albumin subdomains display a local loss of their a-helical structure, whereas the fibronectin module shows lesser changes thanks to its (3sheet structure. Typical examples of these rearrangements are shown in Figure 3.

Figure 3. The most stable initial adsorption geometry of the albumin A subdomain at left showing large rearrangements of the most hydrophobic a-helix in contact with the surface, while minor changes are observed for the fibronectin module at right.

For the optimized geometries, we calculated the interaction energy, EinU defined as Eint = (Efree + Epianes)-Etot, where Efree is the energy of the free, isolated molecule in the energy minimum and Epianes is the energy of the carbon planes defining the graphite substrate. According to this definition, Eint > 0 is the energy required to desorb the molecule and bring it back to the free, optimized state. We also calculated the where Efrozen strain energy, Estrain, defined as Estrain == Efrozen-Efree, is the energy of the molecule in the frozen geometry it adopts upon adsorption. A larger interaction energy and a greater strain energy take place when a larger number of amino acids are close to the hydrophobic surface. Therefore, we determined the number of amino acids in contact with the surface, n °, taking conventionally 5A as the upper limit for the contact distance, and found a significant positive correlation between or Estrain and ?\AO &s shown in Figure 4. Considering separately 5A

209

the data points for the protein fragments, the best-fit lines through the origin are given by [17, 18] 71(3) • n « kJ mol l for the albumin subdomains 54(1) • n ° kJ mol"1 for the fibronectin module

{

^'

17(1) • n ° kJ mol x1 for the albumin subdomains 13(1) - n o kJ mol" for the fibronectin module

(2)

5A

with the figure in parentheses giving the standard error on the last significant digit.

• fibronectin • albumin o 10

20

30

40

n5A

Figure 4- Interaction energy Eint (left) and strain energy E'strain (right) plotted as a function of nsoA- The results obtained for the two albumin subdomains in different orientations are stained for the two albumin subdomains in different orientations are shown with empty symbol, and for the fibronectin module with full symbols. The solid lines are the best-fit lines through the origin given by eqs. (1) and (2).

For all protein fragments, the driving force for adsorption, locally modifying the secondary structure, consists of the favourable van der Waals interactions with the surface, mainly due to the hydrophobic or at least the less hydrophilic residues. We remark that according to eqs. (1) and (2) Eint increases with the number of residues in contact with the surface faster than Estrain- By extrapolation, we infer that the fragments may undergo much larger deformations so as to maximize their interaction with the surface. We also point out that the initial optimizations in the dielectric medium show many widely different energy minima. Therefore, the configurational phase space of adsorbed proteins

210 displays a rugged energy landscape, reminiscent of glassy states. However, it turns out that the energy barriers separating the local minima are often not prohibitively large, and can be easily surmounted through a suitable kinetic energy input in the MD runs.

4.

Final adsorption stage by molecular dynamics in the dielectric medium

Selected geometries, in particular the lowest- and the highest-energy states obtained in the previous section, were subjected to MD runs in search of the best adsorption geometry after optimization of many instantaneous snapshots. Interestingly, when equilibrium was achieved in the MD runs, further energy minimizations produced only relatively minor readjustments, mainly involving local features with modest energy changes. We report in Figure 5 the optimized geometries obtained in the final adsorption stage. We found a very similar behaviour for both albumin subdomains and therefore we report just one case.

Figure 5. Side and top view of two best adsorption geometry after the MD runs and subsequent energy minimizations. At left we show the albumin A sub domain, and at right the fibronectin module. Note the lack of any secondary structure in both cases.

Both albumin subdomains show an extensive denaturation with the formation of a monolayer of amino acids. Such a large molecular spreading on the surface is consistent with experimental data obtained for the whole protein on a hydrophobic surface [15]. On the other hand, for the fibronectin module it is more difficult to form a monolayer on the graphite surface because it contains four disulfide bridges acting as intramolecular crosslinks. We studied how such crosslinks affect or even hinder the conformational changes during the MD runs by per-

211 forming similar MD simulations after replacing the disulfide groups with thiol moieties. Preliminary results indicate that indeed eliminating the crosslinks allows the molecule to achieve a much larger flattening on the surface, suggesting that eventually a monolayer can also be formed by the modified (i.e., crosslink free) fibronectin module in the long run. It can also be observed in Figure 5 that the most stable geometries found for the final adsorption stage display a bidimensional order, whereby the backbone trajectory roughly tends to give an antiparallel arrangement. This feature can be seen more clearly in the albumin subdomain (see Figure 5 at left), but it is also largely present in the fibronectin module. Such ordering is due to strong dipolar interactions that involve the side groups of the facing amino acids, and therefore it does not lead to the formation of true /3-sheets. From the energetic viewpoint, we note that the interaction energy per amino acid in contact with the surface is equal to 56 mol"1 for the albumin subdomains and to 57 kj mol"1 for the fibronectin module, solely due to the dispersion (or van der Waals) forces. Not surprisingly, these values are very similar, since the protein fragments are made up of the same 20 natural amino acids. Note that for the albumin subdomains the value of this interaction energy is lower than that extrapolated from eq.(l), namely 71 kJ mol"1, because the initial adsorption is dominated by the hydrophobic aminoacids, whereas in the most stable state all residues interact with the hydrophobic surface, including also the hydrophilic ones. The strain energy per residue in contact with the surface in the most stable state amounts to 17 kJ mol"1 for the albumin subdomains and to 15 kJ mol"1 for the fibronectin module. These values are surprisingly close, but we believe such result to be accidental, in view of the different secondary structure of the two protein fragments. The large interaction energies just discussed strongly suggest that protein adsorption on hydrophobic surfaces is usually irreversible. Such conclusion is consistent with experimental results, which show that the slow spreading of albumin over a hydrophobic surface eventually leads to a smaller coverage in terms of the adsorbed mass of protein per unit surface due to the lower bare surface that remains available for other molecules [15, 16]. Unfortunately, analogous data for the fibronectin module are lacking. The conformational changes can also be investigated using the radius of gyration Rg and its components, or the principal axes of the molecules as geometric descriptors. During the adsorption process, we found a pronounced increase of Rg compared to the isolated fragments [17, 18] For the isolated albumin A subdomain, Rg measures 12.1 A , increasing to 21.5 A in the final state, while Rg is equal to 16.2 A for the isolated

212 fibronectin module, and to 28.5 A in the final state. Upon spreading, we also found that the systems become also much more anisotropic. In fact, the principal axes of the isolated albumin A subdomain are equal to 9.6 - 5.6 - 4.9 A, and become equal to 19.3 - 9.3 - 1.0 A in the final state, while for the isolated fibronectin module they are equal to 14.4 - 5.9- 4.4 A, and in the final state to 23.2 - 16.1 - 3.5 A. Needless to say, in all cases the shortest axis is orthogonal to the surface. For all the protein fragments, the very large interaction energy above reported and the great anisotropy, in particular the very small molecular thickness, suggest that in the final adsorbed state the protein fragments cannot be easily removed under a shear stress, such as due for instance to the flow of the surnatant solution.

5.

Kinetics of surface spreading

We are currently analysing the kinetic data for the molecular spreading on the surface, and therefore here we only report a few preliminary and qualitative results. The MD runs show that the final adsorption state is attained through a liquid-like spreading of the protein fragments, often accompanied by tilting of the whole molecule, as reported in Figure 6, or by multi-stage changes with sequential disruption of the secondary structure, as reported in Figure 7. In Figure 6 we report the kinetics of the fibronectin module from the least stable initial adsorption geometry to the final best adsorption stage, shown through the time evolution of the potential energy Epot and of the distance d of the molecular centre of mass from the surface. The process is best described in terms of an initial backbone tilting towards the surface, which leads to a faster decrease of d compared to Epot. Similar plots are shown in Figure 7 for the albumin E subdomain, where the multi-stage changes involve the sequential, and quite abrupt, adsorption of denatured strands. For the fibronectin module in different orientations, we often found also an interesting equilibrium state with a continuous breaking and reforming of /3-sheets through minor displacements of the backbone trajectory. Note that in some cases the overall conformation remains quite similar to the isolated molecule. On the contrary, starting from the lessstable initial adsorption geometry, we found larger rearrangements and a complete denaturation with a stronger adsorption after the backbone tilting above discussed in connection with Figure 6. In the most stable state, found after optimization of many selected snapshots, the fibronectin module and the albumin subdomains optimize the surface interaction by spreading as much as possible, thus maximiz-

213 2000

i 100

200

'(ps)

Figure 6. The less stable initial adsorption geometry (upper picture) leading to the most stable final adsorbed state (lower picture), at left, and plots of the potential energy and of the distance of the molecular centre of mass from the surface as a function of time at right.

ing their footprints. Quantitative estimates of the footprints and of the exposed surface can be found in paper I and paper II. Here we limit ourselves to briefly mention that the changes in molecular size and in the overall anisotropy that take place upon surface spreading can also be easily monitored as a function of time. We used as geometrical descriptors the radius of gyration, Rg, and in particular its components parallel and perpendicular to the surface, and the principal axes to characterize the molecular anisotropy. The overall shape of the isolated protein fragments in a globular conformation is essentially isotropic, and it does not show major changes upon the initial adsorption. On the contrary, during the surface spreading the molecular shape becomes strongly anisotropic, as already anticipated in the previous section for the final adsorption state. We report in Figure 8 an example of the time change of Rg and of its components during the kinetic process already shown in Figure 6. From the right-hand- side plot of Figure 8, it can be seen that the decrease of the perpendicular component yielding the molecular thickness is somewhat faster than the full spreading that maximizes the molecular footprint, monitored in turn through the parallel components.

214 1600

1 0

200

400

600

800

'(PS)

Figure 7. The same as in Figure 6 for the albumin E sub domain. Note the different time scale.

0

250

500

750

1000

'(ps)

Figure 8. Radius of gyration and its squared components calculated during the MD runs of the fibronectin module already shown in Figure 6.

A more detailed analysis of the kinetics of spreading on the graphite surface shall be reported in a future paper [23].

215

6,

Hydration of the adsorbed protein fragments

The simplest way to investigate the hydration of the adsorbed fragments is through the accessible surface area exposed to the solvent. However, although this area can be taken as proportional to the dispersive interaction energy with the solvent, it is not indicative of the full solvation energy because it does not account for the electrostatic contribution, in particular for the hydrogen bonds. Nevertheless, we note that the fragment spreading on the surface enhances both the molecular footprint and the accessible area to the solvent. In this way, it is possible to maximize both the hydrophobic interactions with the graphite surface and the interactions with the solvent, hence the molecular hydration, at least on the exposed side. Quantitative results for our protein fragments can be found in papers I and II. A more lengthy, but also a more powerful tool for studying the hydration of the protein fragments consists on the explicit inclusion of a very large number of water molecules in the simulations. Therefore, we considered the protein fragments in different adsorption geometries introducing them in a box containing a few thousand of water molecules with periodic boundary conditions, and saved many different frames during the MD runs to calculate the pair distribution function (PDF) of the water molecules around the protein backbone. This function yields the probability density of finding the water molecules (or its oxygen atoms in the present case) at a distance r from the backbone. In Figure 9 we report such PDF for all the fragments we studied.

0.3

final 0.2

i initial adsorption

isolated 4 K#
Figure 9. Pair distribution function (PDF) of the oxygen atoms of the water molecules around the backbone of the two albumin subdomains in the initial and final adsorption state (left), and around the backbone of the fibronectin module as an isolated molecule and in the final adsorption state (right).

216 For the isolated molecules and in the initial adsorption state, we found a first peak at about 2.7 - 2.8 A corresponding to the first hydration shell, and a second maximum at about 4 A due to the second shell. Therefore, the initial adsorption states are hydrated as the isolated fragments, whereas significant changes are found upon final adsorption. For the albumin subdomains the peaks corresponding to the first hydration shell keep the same position, but their heights become much larger. In other words, upon spreading the subdomains optimize both the surface interaction and the molecular hydration, because all residues are well exposed to the solvent, including in particular the hydrophilic ones initially hidden inside the subdomain. For the fibronectin module in the final state, the two peaks are shifted to shorter distances, namely 2.5 A and 3.6 A (see Figure 9 at right), but the peak heights become shallower. Therefore, the final adsorption geometry leads to fewer water molecules more strongly bound to the backbone compared to what found for the isolated module. This pattern is quite different from what obtained for the albumin subdomains because in the fibronectin module the molecular rearrangement undergone upon adsorption exposes the hydrophobic residues, which are mostly buried within the isolated molecule. Moreover, one side of the adsorbed molecule is not exposed to the solvent being shielded by the surface, so that in conclusion there is a somewhat poorer overall hydration.

7.

Conclusions and outlook to future work

In the present paper we review our recent work on the adsorption of two albumin subdomains and a fibronectin module on a graphite surface by atomistic simulations through energy minimizations and molecular dynamics runs. We adopted a simulation strategy in two-steps to study the initial and the final adsorption state on a bare surface in a dielectric medium and in the explicit presence of the solvent. In the initial adsorption stage obtained upon simple energy minimization, the protein fragments show only local rearrangements of the backbone close to the hydrophobic surface. Both the interaction and the strain energy of the fragments approaching the surface in different orientations are well correlated with the number of amino acids in contact with the graphite surface. However, the interaction energy increases faster than the strain energy. Therefore the systems can display much larger deformations to maximize the adsorption. During the MD runs leading to the final adsorption state, we found a very large molecular spreading with maximum coverage of the surface. In this way, the protein fragments optimize both the surface interactions and their hy-

217 dration. Cooperative effects are important in bringing the hydrophilic residues as close to the surface as the hydrophobic ones, while being still exposed to the solvent. Analysis of the kinetics of spreading taking place within the MD runs is currently in progress. Through the time evolution both of the potential energy, in particular of the van der Waals contribution, and of the distance of the molecular centre of mass from the surface, we observed one or more fast stages. At first, there is a rapid liquidlike spreading qualitatively independent from the nature of the approaching fragment, namely of its hydropathy. This spreading is often accompanied by a backbone tilting towards the surface. Later, a complete denaturation takes slowly place with the possible formation of a monolayer. In the albumin subdomains, all a-helices become fully denatured, and all amino acids are eventually in contact with the hydrophobic surface, forming indeed a monolayer coating the surface. On the other hand, a smaller loss of the secondary structure is shown by the fibronectin module, thanks to its /3-sheet structure, often with an interesting equilibrium of disruption and formation of /3-sheets through minor changes in the backbone trajectory. In all cases, the isotropic molecular shape shown by the isolated fragments and by the initial adsorption geometry becomes strongly anisotropic after the final spreading. Thus, the component of the radius of gyration perpendicular to the surface greatly decreases, with a marked increase of the parallel components. Therefore, the system shows a smaller cross section to a flow, hence a larger resistance against shear stresses, which is often of great practical relevance. We point out that in the explicit presence of the solvent we did not find significant conformational changes compared to the simulations in the dielectric medium, apart from some readjustments of the side groups. However, the spreading enhances the accessible surface area exposed to the solvent and the molecular footprint so as to maximize both the hydrophobic interaction with the surface and the interaction with the solvent. The molecular hydration is best described through the distribution of the water molecules around the protein backbone. In our cases, the hydration of the isolated proteins fragments and of the systems in the initial adsorption stage are very similar, whereas upon spreading the albumin subdomains are better hydrated than the fibronectin module. As a conclusion of the present review, we note that we only discussed the adsorption of globular protein fragments on a hydrophobic surface. We are currently investigating the adsorption of the same fragments on a hydrophilic surface of amorphous, glassy poly(vinyl alcohol) to assess the relevance of the surface nature for protein adsorption. Pre-

218 liminary results indicate that the initial adsorption is much weaker than on graphite, and grossly independent on the hydrophilic or hydrophobic character of the approaching strands. Thus, the interaction energy is 5 to 10 times smaller compared to what obtained on the hydrophobic graphite surface. We are also planning additional simulations to investigate the adsorption of fibrous proteins such as collagen on a biomaterial surface. Finally let us mention another challenging open issue that can be addressed with atomistic MD simulations on state-of-the-art workstations, namely what happens when different proteins may simultaneously adsorb on a surface. The problem may be tackled by considering the interaction of a protein with a covered surface. In this case, the newcomer may adsorb only at the exposed graphite "islands" still available, or on top of another protein previously adsorbed. In both cases, the initial layer modifies the surface by modifying its hydrophobicity or hydrofilicity and by decreasing the available bare surface. Alternatively, it would be most interesting to consider what happens when unlike proteins may adsorb in order to understand whether the adsorption process is competitive or simultaneous and whether it is under kinetic or thermodynamic control, thus reproducing in the computer the experimental protocol adopted for instance in Ref. [16].

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[2] S. Ramakrishna, J. Mayer, E. Wintermantel and K.W. Leong, Composites Science and Technology, 61:1189, 2001. [3] J. Black, Biological Performance of Materials: Fundamentals of Biocompatibility, Marcel Dekker (New York), 1992. [4] K. Nakanishi, T. Sakiyama and K. Imamura, J. of Biosc. Bioeng., 91:233, 2002. [5] C M . Roth and A.M. Lenhoff, Langmuir,

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[8] S. Ravichandran, J.D. Madura and J. Talbot, J. Phys. Chem. B, 105:3610, 2001. [9] K.A. Dill, S. Bromberg, K. Yue, K.M. Fiebig, D.P. Yee, P.D. Thomas and H.S. Chan, Protein Sci., 4:561, 1995. [10] V.P. Zhdanov and B. Kasemo, Proteins: Struct, 1998. [II] V.P. Zhdanov and B. Kasemo, Proteins: Struct,

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219 [15] C.F. Wertz and M.M. Santore, Langmuir, [16] C.F. Wertz and M.M. Santore, Langmuir, [17] G. Raffaini and F. Ganazzoli, Langmuir,

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[18] G. Raffaini and F. Ganazzoli, Langmuir, (submitted) [19] H.M. Berman, J. Westbrook, Z. Feng, G. Gilliland, T.N. Bhat, H. Weissig, I.N. Shindyalov and P.E. Bourne, The Protein Data Bank Nucleic Acids Research,

28:235, 2000; See also the URL http://www.rcsb.org/pdb/. [20] J. Kyte and R.F. Doolittle, J. MoL BioL, 157:105, 1982. [21] Accelrys Inc. Insightll 2000, Accelrys Inc.: San Diego, CA,, 2000; See also the URL http://www.accelrys.com. [22] P. Dauber-Osguthorpe, V.A. Roberts, D.J. Osguthorpe, J. Wolff, M. Genest and A.T. Hagler, Proteins: Struct, FuncL, Genet, 4:31, 1988. [23] G. Raffaini and F. Ganazzoli, paper in preparation.

MULTISCALE SIMULATION OF LIQUID CRYSTALS Applications in the modeling of LC-based biosensors Orlando Guzman, Sylvain Grollau, Evelina B. Kim, and Juan J. de Pablo Department of Chemical and Biological Engineering University of Wisconsin, Madison 53706-1619 USA [email protected]

Abstract

Nematic liquid crystals are characterized by the occurrence of disclination lines, topological defects where the average molecular orientation changes abruptly. Recent experiments have shown that, in addition to their application in displays, liquid crystals permit the detection of ligand-receptor binding by optical amplification. The optimal design of LC-based biosensors requires an understanding of the effects of the presence of biomolecules on the structure and dynamics of nematic liquid crystals. We present a multiscale approach that combines molecular simulations and mesoscale modeling: Monte Carlo simulations are used to study the interactions of diluite colloidal particles, as well as the structure of topological defects; these results compare satisfactorily with the corresponding theoretical calculations at the mesoscale level. The mesoscale modeling of a multi-particle sensor shows that adsorbed biomo- lecules modify the relaxation dynamics in the device: at low surface-coverage densities, the equilibrium structure is characterized by a slightly perturbed uniform nematic order; at a critical density, the dynamics exhibits a slowdown at late stages, characteristic of the inability of the nematic to achieve a uniform order. These results are compared with experimental observations of the nematic response in biosensors.

Introduction Consider a device capable of transducing and amplifying the binding of proteins or viruses at surfaces into optical signatures that could be easily read with the naked eye. In order to do so, the device would have to bridge length scales over four orders of magnitude: from the size of a protein (lOnm) to patterns large enough for the eye to see (0.1mm). 221 P. Pasini et al. (eds.), Computer Simulations of Liquid Crystals and Polymers, 221-247. © 2005 Kluwer Academic Publishers. Printed in the Netherlands.

222

Such devices have in fact been in existence for half a decade. The basic elements are (see Fig. ) a thin film of a liquid crystal (LC) sandwiched between ligand-conjugated self assembled monolayers (SAMs), supported by semitransparent gold films on glass substrates [1, 2].

Gold film ~10nm ^ Ligand-conjugated monolayer - 2nm LC ~ 2-20 |mm

Figure 1. A schematic view of a liquid crystal-based biosensor: a thin film of a nematic liquid crystal is confined between parallel walls. The walls are assembled using self-assembled monolayers (SAMs) formed from ligand-conjugated thiols and alkanethiols on semitransparent gold films supported on a thin layer of titanium supported by glass. (For clarity, the titanium and glass layers are not shown.)

The principle of operation of this kind of biosensors relies on several characteristic properties of LCs [1]: first, they exhibit long range orientational order, which means that LCs can report events and orientations to regions that are macroscopic length scales away (e.g. 0.1 mm). Second, because LCs are fluids, the changes induced by binding events at the surface can propagate throughout the medium relatively rapidly. And finally, the optical anisotropy caused by the local preferred orientation of the mesogens is easily transduced into an optical signal that can be read with ambient light. The anchoring behavior of the LC at the substrates is controlled by the surface chemistry [3] or topography [4] at the nanometric scale, so that in the absence of binding events the orientation of the mesogens is uniform throughout the sensor. But when ligand bounds to the receptors, the preferred orientation at the surface is no longer uniform. This is translated into an optical image that shows a proliferation of multidomains and topological defects as the concentration of ligand is increased [2]. However, the design and optimization of liquid crystal-based biosensors would benefit from a fundamental understanding of the structure and dynamics of the domains and defects that are present in the de-

223

vice. One would like to anticipate the defect structures associated with nanoscopic particles, to relate the concentration of adsorbed particles to the optical textures that are observed experimentally, and to determine whether additional, useful information can be inferred from the dynamic behavior of the liquid crystal. In order to achieve these goals, we have adopted a multi-scale approach that comprises molecular and mesoscopic models for the liquid crystal. The molecular description is carried out in terms of Monte Carlo simulations of repulsive ellipsoids (truncated and shifted Gay-Berne particles), while the mesoscopic description is based on a dynamic field theory[5] for the orientational tensor order parameter, Q. l The mesoscopic model, being phenomenological, requires an appropriate selection of parameters in order to reproduce experimental data. This is achieved in this work by determining a length scale factor and an energy scale factor that translate the theoretical results into simulation units. Molecular simulations are used to study defect structures around a colloidal particle at the upper end of the nanometric scale (5-10nm). A field-theoretical formalism is then used to calculate and reproduce those defect structures, extracting the appropriate length and energy mappings between both approaches. The agreement between molecular simulations and field theory is verified by predicting the defect structures for other systems, and the coarse-grained formalism can subsequently be used to study the kinetics of defects over large distances (~lmm), where molecular simulations cannot currently be used. This chapter is organized as follows. In section 1.1, we introduce our notation and present the details of the molecular and mesoscale simulations: the expanded ensemble-density of states Monte Carlo method,and the evolution equation for the tensor order parameter [5]. The results of both approaches are presented and compared in section 1.2 for the cases of one or two nanoscopic colloids immersed in a confined liquid crystal. Here the emphasis is on the calculation of the effective interaction (i.e. potential of mean force) for the nanoparticles, and also in assessing the agreement between the defect structures found by the two approaches. In section 1.3 we apply the mesoscopic theory to a model LC-based sensor and analyze the domain coarsening process by monitoring the equal-time correlation function for the tensor order parameter, as a function of the concentration of adsorbed nanocolloids. We present our conclusions in Section 1.4.

1 We are currently adding another level of detail to this multiscale model, by performing atomistic simulations of mesogens.

224

1. 1.1

A multiscale model for LC-based sensors Molecular simulations

In this section we describe the molecular simulations that we use to study the liquid crystal at the molecular length scale. The main components of these simulations are the interaction potentials and the sampling scheme. Liquid crystal molecules are represented as repulsive Gay-Berne ellipsoids, confined between two parallel walls (see Fig. 1.1.1).

Figure 2. A snapshot from a molecular simulation, showing a nanoscopic colloid immersed in a Gay-Berne nematic. The fluid is confined in the vertical direction by parallel walls, and periodic boundary conditions are applied in the other directions.

The interaction between a pair (i, j) of molecules is a truncated and shifted, and it is given by Uij = 4e0 (-if - \ ) + eo, (Hj < 2 1 / 6 \PPij PJ

(1) (2)

where p^ is a function of the separation r^ and its orientation with respect to the unit vector u along the largest axis (ai) of each ellipsiod: Pij = {rij - Oij + CTO)/ao

with

225

In these equations, fy = r-y/ry (i.e., the unit vector along the line joining the centers of the ellipsoids), while parameter x is given by (5)

x=

The elongation K is defined as the ratio of the length to width of the ellipsoids: K = o\jo§. The interaction of the mesogens with the surface of the colloidal spheres and the confining walls is also described by the shifted Gay-Berne potential given in Eq. (1), but defining pijSpn for the particle-sphere interaction as Pi,sph = (\ri ~ rsph\ -

(6)

(R is the radius of the colloidal particle), and setting Pi,wall = (\zi - zwall\

(7)

for the interaction with the walls. The two walls, located at z = ±zwau, are separated by a distance equal to 34ao; in what follows, zsph = \z{ — zwau\ is the normal distance between the surface of the i-th colloidal particle and the wall (see Fig. 2). +Zwall

sphj

sph

1-z Figure 3. Schema of the system illustrating the notation used in the definition of the confining wall potentials.

The simulations are performed at constant temperature and density. The reduced density p* = pa^ is fixed at p* = 0.335, while the temperature is set to T* = fc^T/eo = 1. One of the quantities of interest is the effective interaction between a colloidal particle and a wall (or between two colloidal particles) mediated

226

by the liquid crystal. This can be quantified through a potential of mean force. The potential of mean force (PMF) measures the difference in free energy between two states of a system, as a function of one of its degrees of freedom. This degree of freedom is referred to as a "reaction coordinate". In this work the reaction coordinate £ refers to the distance between the particle and the wall, or the distance between two particles. The method employed here to compute the corresponding potentials of mean force has been described in the literature [6], and only a brief account is provided here. The potential of mean force A^ r (() (measured with respect to some reference state Co) is related to the probability P(C) of finding the system in the state C through the reversible work theorem [7], )

(8)

In standard Metropolis Monte Carlo simulations, the probability of visiting high energy states is low, and their sampling is generally poor. To improve the sampling of these regions of phase space, it is possible to constraint the reaction coordinate or to introduce a biasing potential. Another strategy is to use expanded ensembles, where artificial states are introduced to facilitate transitions between states separated by large energy barriers [8]. The EXEDOS method that we use combines an expanded ensemble formalism with a density-of-states [9, 10] scheme for the determination of the potential of mean force. In the expanded ensemble method, besides the canonical variables (number of mesogens JV, volume V, and temperature T) the state of the system is also labelled by the value of C coordinate: we consider M intervals of width S of the reaction coordinate, then the system is said to be in state m if m - 1 < C/S < m. For sufficiently narrow intervals, we can associate the midpoint Cm = {jn — 1/2)£ as the representative value. The partition function of the expanded ensemble is given by M

M

Z(N, V, T, m; w) = £ Z(N, V, T, m)wm = £ Zmwm. 771=1

171=1

Here Zm = Z(N,V,T,m) is the canonical partition function for the system with C = £m> while w is a set of positive parameters that are used as weighting factors. In the expanded ensemble, the probability of observing the system in the state m is ^

(10)

(9)

227

By setting wm = a/Zm

(11)

with a an arbitrary constant, Eq. (10) becomes P(m\w) = ^

(12)

The converse is also true, that is, if we have a uniform distribution P(m\w) = 1/M, then wm oc Zml. The PMF AT is the free energy of the system in state ra, relative to that of state fc, Af = F(m) - F(k) = -kBT log (Zm/Zk)

(13)

^From Eq. (10), we obtain

When one has achieved uniform sampling, Eq.(12), the PMF reduces to = F(m) - F(k) = -kBT(\ogwm

- logwk)

(15)

The PMF can be computed in a simple way from a set of weights w that results in uniform sampling in the expanded ensemble. We now describe a practical way to determine such a set. The reader is referred to previous publications for a more detailed description [6, 11]. An initial set w° is proposed, and gradually updated on the fly on the basis of the number of visits to each state. In analogy to the original implementation of the DOS method, a histogram of the visits to each state is monitored. Every time the system visits a state m, the weight for this state is changed to Wm -> Wm/f

(16)

and the corresponding entry of the histogram is incremented. Number / is a convergence factor. Monte Carlo moves from a state o to state n are accepted with probability P(o -* n) = min(l,exp(-/3(C/n - Uo) + \ogwn - \ogwo)) Once the histogram is deemed flat (e.g., when the entry is greater or equal than 80% of the average), tor / is updated (e.g., / —> y/J). After this, the but the set of current weights is retained for the

(17)

minimum histogram the convergence fachistogram is cleared, next iteration. The

228

simulation proceeds until log/ falls below a given threshold. In ordinary DOS simulations, this threshold is typically set to 10~7 or 10~8 in order to achieve sufficient accuracy in the calculation of the density of energy states, g(E). However, in this application, the set of weights (that corresponds to a "density of separation states") can be accurately estimated with much less stringent convergence factors. Typically, it is sufficient to set the threshold to log fthr ~ 3 x 10~3, corresponding to five successive updates of the convergence factor. One of the benefits of the EXEDOS method is that it provides an additional test for convergence: in addition to the estimate of the PMF obtained from the weights, one can also integrate the average force on the colloidal particles measured at each state. The average force F(£m) is a simple average of the sum offerees acting on a given colloidal sphere, along the direction of increasing reaction coordinate. Thus, for the sphere-wall interaction, it is the average total force (between the colloid and all the mesogens, and also with both walls) along the z-axis. For the sphere-sphere interaction, F((m) is the total force (between the given colloid and all mesogens, the walls and the second sphere) acting along the line connecting the centers of the spheres. In both cases, a second estimate of the PMF may be obtained as F(C)dC

(18)

Both estimates, Eq. (15) and Eq.(18), must agree if the method has truly achieved uniform sampling. Therefore, in addition to the condition / < /thr? the agreement of both estimates can be viewed as another criterion of convergence. After both methods agree, either one can be used. In this work we have chosen to present the estimates obtained from integration of the forces as these curves have a smoother appearance. To extract local static properties of the system, such as the density p and the tensor order parameter Q, a separate series of NVT runs were performed with spheres fixed at a few selected values of the reaction coordinate. The corresponding simulation configurations were analyzed by binning the LC molecules in a rectangular grid and computing the quantities of interest for each bin. In particular, for each bin with a volume VB the density was found as p = NB/VB, where NB is the number of molecules within this bin; the tensor order parameter was computed as 1

NB

229 where I is the identity tensor. Prom the tensor order parameter one can compute the scalar order paramete S (sometimes denoted also as P2) and the director, n. If we denote the largest eigenvalue of Q as Ai, then (20) and n is the unit eigenvector associated with Ai.

1.2

Dynamic Field Theory

In some instances, it is possible to describe the configuration of the LC just by specifying the director field n(r); this is commonly referred to as the "continuum theory". The free energy of the system is a functional of the spatial derivatives of n : T = I / Ki(V • n) 2 + K2(n • (V x n))2 + K3(n x (V x n))2dr

(21)

However, since this approach assumes a constant, homogeneous value of the scalar order parameter, it fails to describe the core of topological defects, where P2 deviates from its bulk value on length scales of the order of £. The continuum theory for n also fails to describe multidomain systems because it cannot handle the variation of P2 across domain boundaries. As an alternative to the continuum theory, the mesoscopic approach can be based in a dynamic field theory for the tensor order parameter, Q. This tensor can be viewed as a coarse-graining of the microscopic probability distribution function ^(u,r,t) for the molecular orientation u. In this sense, Q corresponds to the symmetric, traceless part of the tensor of second moments of ip at the point r and time t:

Q(r, t) = I (uu - i l ) V(u, r, *)du.

(22)

The local state of the liquid crystal is therefore described by five degrees of freedom. In this model, the properties of the liquid crystal are described by a Landau-de Gennes free energy that contains two contributions: first, a contribution of the form

Ts

11

= 11 ii ~ ff ))

t tr (( QQ 22) )

t r {t rQ{ Q3 3) ) + " ir T^Q 2 ) 2 d r -

(23)

which describes the excluded volume effects that drive the first order transition from the isotropic to the nematic phase. The coefficients A

230

and U are phenomenological parameters that depend on the liquid crystal of interest. These (and the other phenomenological parameters of this theory) can be selected by comparison with empirical data. They can be assigned a microscopic interpretation [5]: A corresponds to pfc#T, while U is proportional to <JI<JQP. In this model, that transition occurs at U = 2.7. The second contribution to the Landau-de Gennes free energy describes the long range elastic forces dominant in the nematic phase:

(24) The coefficients Li, L2, L3 are related to the splay, twist and bend elastic constants K\^K2^K^ through the equations [5]

L2 = ^ = r ^ ,

(25b)

^ = ^ ,

(25c)

In the one constant approximation [K\ = K2 = K$ = JFC), L\ = K/(2S2) while L2 and L3 become zero; the long range elastic contribution simplifies to

)dr

(26)

J '

and the total free energy is jr = jr5

+

jPe.

(27)

The relative strength of the contributions from Eq. (23) and Eq.(26) depends on the type of liquid crystal that one studies: for polymeric liquid crystals, short-range interactions are expected to be dominant, while long-range interactions are dominant for low-molecular weight liquid crystals. The evolution equation for Q that we use in this work corresponds to a particular case of the Beris-Edwards dynamical formalism [5] and is given by

f(MS))

The coefficient T = 6£>*/(l - 3tr(Q 2 )/2), where D* is the rotational diffusivity coefficient for the mesogens.

231 In Eq. (28) it is understood that | £ represents the symmetric part of the functional derivative of T with respect to Q: 2

Inspection of Eq. (28) shows that Q will remain symmetric and traceless as it evolves. When the functional derivatives in Eq. (28) are evaluated, one obtains a system of partial differential equations for the components of the tensor Q. We solve this system with a finite difference method over a rectangular grid. One can construct characteristic length and time scales from the parameters of the theory. As an instance, it is convenient to introduce the quantity £ = ^18Li/AU as a characteristic length for changes of the order parameter. This length corresponds to a few times the molecular length. For the time scale, we choose the quantity T =

(6D*A(1-U/3))~1.

2.

Clusters of particles

2.1

Mapping of simulation and field theory length scales

We first consider the insertion of one spherical particle into a nematic liquid crystal. The particle is fixed in the middle of the cell (both in simulations and in the field theory). Prom past theoretical studies, it is known that defect structures that arise in the proximity of an interface [16], as well as the nature of their interactions [17], depend on the anchoring properties of the liquid crystal. Experimentally, the orientation and strength of anchoring at various interfaces depend on the system under study. As an instance, it is possible to control the anchoring properties at the surface of water droplets immersed in a liquid crystal through the use of amphiphilic molecules adsorbed at the droplet interface [15]. For the case of solid surfaces, the anchoring properties can be controlled using self-assembled monolayers (SAMs) of alkanethiols or other compounds [1]. In molecular simulations, the anchoring properties are determined directly through the choice of interaction potentials. For the Gay-Berne fluid, Andrienko et al. have shown that a colloidal particle immersed in the bulk nematic phase exhibits strong homeotropic anchoring (i.e. the 2

It turns out that, for this model, the indicated functional derivative is already symmetric.

232

Figure 4- Two of the possible defect structures associated with a strong hometropic sphere: (a) Hedgehog point defect (b) Saturn ring disclination line.

orientation of the molecules is perpendicular to the surface) [18]. In our simulations, we observe strong homeotropic anchoring at the surface of the colloids and also at the confining walls; hence, in the theory we only consider this type of anchoring by imposing the corresponding boundary condition to the field Q. Two possible defect structures that can occur for a homeotropic colloidal particle are the Saturn ring disclination line and the Hedgehog point defect. (See Fig. 1.2.1) In this work we consider particles with a radius R of the same order of £ (which corresponds to the characteristic length for changes of the order parameter). For such small particles, the Saturn ring configuration is expected to be more stable than the Hedgehog configuration [12, 13]. Our Monte Carlo simulations and field theory agree when they indicate that the particle is surrounded by a Saturn ring. To map the characteristic length scale of the theory (£) into the units of the Monte Carlo simulations (CTQ), we can estimate the ratio (30)

by analyzing the dependence of the Saturn ring radius (a) as a function of the particle size (R). One can start by plotting the Monte Carlo simulation values for the ratio a/R as a function of R/GQ. In Fig. 4 one can observe that a/R —» 1.2 for large particles. This limit compares well with the value obtained using continuum theory (a/R ~ 1.236) [14]. We can overlay the field theory results by plotting a/R as a function of /?/(£/&£,); the best match is obtained by setting kL = 2£, and this provides us with the desired mapping of scales.

233

Figure 5. Size of the Saturn ring (in units of the particle radius) as a function of the particle size.

2.2

Sphere/substrate interactions

In this section we consider the interaction between a particle and a wall. We first present the results from the theory. Figure 1.2.2 shows the director profile and scalar order parameter maps for a particle of radius R = 0.36£ located at the surface of a wall. Comparing these maps with those for a particle further away, (Fig. 1.2.2 ), we observe that the disclination line is no longer at the equatorial plane, but is shifted towards the wall. Similar deviations of the position of the ring are observed in the Monte Carlo simulations. We define the potential of mean force AF(s) as the free energy difference when the center of the particle is located at s + R and when it is at the center of the cell. (Hence, s is the separation between the wall and the surface of the sphere.) We present AF(s) for three different radii in Fig. 6. The theory predicts a positive, repulsive potential between the particle and the wall. The intensity of this repulsion increases with the radius of the particle; the simulations predict a global repulsive force, with the occurrence of oscillations and local extrema. The presence of these minima and maxima is due to the formation of smectic-like modulation of the LC density by the confining wall. (See Fig. 7) A comparison of the density profile and the PMF reveals that configurations with the particle's center between two density peaks are more stable. These features, originated in the molecular structure of the LC, cannot be captured by the coarse-grained approach of the field theory because it assumes a constant density for the liquid crystal.

234

5

10

1$

Figure 6. Contour map of the scalar order parameter for a particle of radius R = 6 in the proximity of the wall

1

Li

250 Vi

{-,

200

3

150

[

'

—

R-3o 0 (MO)

_

R-6a*0^)

„

•A • RHoJciiiooiy) 4, R-5a0Crbeacy) A

1

100

1

1

5\D Vi • A

50 0-

-

A

1

Figure 7. Potential of mean force as a function of the radius of the sphere, from simulations and field theory. See the text for an explanation of the mapping of length and energy scales between these two approaches.

235

Figure 8. Contour map of the density of mesogens, for a particle of radius R = 6 in the vicinity of the wall

It is interesting to compare quantitatively both approaches. In order to do that, we need to find a scaling factor so that = A/e.

(31)

This factor is obtained by matching the maxima of the PMF obtained from the theory and the simulations for R = 4<7o: kE = AF(0; R = 4a0, FT)/AF(0; R = 4
(32)

In Fig. 6 we observe that the agreement between both sets of families is quantitative: the results obtained from the field theory constitute a smooth average of the free energy curves obtained from the molecular simulations. It is remarkable that the particle size effect and the range of repulsion are captured quantitatively by the theory down to just a few molecular lengths. The quantitative agreement observed between both approaches indicates that the field theory used here provides a good description of the system's behavior down to a few molecular diameters, provided that appropriate boundary conditions for the tensor field are imposed. The caveat is that the fine structure of the interactions cannot be captured by the theory, which neglects the layered character of the density profile

2.3

Two particle systems

Having established the agreement between the theory and simulations in the case of a single sphere, we turn to the problem of two spheres. In this case we consider a system of two identical spherical with radius R

236

Figure 9. A snapshot from the molecular simulation of two nanoscopic particles immersed in a confined nematic.

suspended in a nematic liquid crystal. This is illustrated in Fig. 1.2.3 with a snapshot from one of the molecular simulations. The liquid crystal is confined once more between parallel walls. The walls are separated by a distance Zws\\ and the centers of the sphere are constrained to the plane z = Zwa\\/2. The surface of the walls and spheres impose strong homeotropic anchoring to the liquid crystal molecules. Once more, the EXEDOS method can be applied to find the potential of mean force AF(s) as a function of the separation of the two spheres. For convenience, we define the reaction coordinate s as the separation between the spheres' surfaces s = 11*121 — 2x2,

(33)

where ri2 is the vector joining the centers of the spheres. The states of the expanded ensemble correspond to M intervals of width 6 : s/S G [m - 1, ra), and we associate to the ra-th state the midpoint value sm = (m - 1/2)6. We can visualize these states in the plane z = Zwa\\/2 as concentric tracks of width 6. The density-of-states updating of the weights proceeds as in the previous case, except for one detail: since the area associated with each track increases in proportion to 5, the histogram of visits to each track (i.e. state) must be divided by the area of the track before testing for flatness. Otherwise, the potential of mean force would present a repulsive contribution AF geo (5) = — Iog(ri2), simply because geometrically there are more "separation states" at large separations. If we examine the system at various separations, we observe that for large s each sphere is surrounded by its own Saturn ring. However, for s < R we observe that the two disclination rings interact and form a new type of defect structure, shown in Fig. 9: two incomplete equatorial Saturn rings connected to a third ring in the plane normal to ri2.

237

.-•"' y

Figure 10. range.

Three ring structure observed for two homeotropic particles at close

In Fig. 10, we compare the structure of the three-ring defect obtained from simulation and theory for s = 0.3R with R = 3
238

-1.5

Figure 11. Contour maps of the scalar order parameter in the #2-plane: the results from the field theory are shown above those of molecular simulations.

239

--L-T-.-I-.-I.-.-.-.-T-.---T-I

separation (s/R)

Figure 12. Radius of the three-ring structure as a function of the separation between the spheres. The radius of the equatorial ring remains constant and the third ring shrinks as the separation increases.

the vertical ring in the z direction. Plotting the ratios a/R, b/R, and c/R as functions of s/R (see Fig. 11) we observe that a > b > c. The same figure indicates that the excentricity of the third ring decreases as it shrinks, when the particles are separated. Galatola et al. have analyzed a related system using field theory: two spheres immersed in the isotropic phase at the onset of the isotropicnematic transition [19, 20]. In that case, the general, nonlinear equations of the theory simplify to a linear form, which permits solution by analytical, linear superposition methods. They have predicted a single-ring structure, also in the plane normal to the equators. We have reproduced this structure with our field theory and Monte Carlo simulations. In contrast to the nematic system, however, the size of the single ring increases with separation: it is easier to expand the ring in the isotropic case. This can be understood by considering the line tension of the rings, that is, their average free energy per unit length. In the isotropic case, the ring is observed to lie at the border of the region where S becomes uniformly zero; therefore, the ring's free energy contribution will be small and so will its line tension. It is of interest to discuss what type of experiments may be able to detect the new defect structure. A first comment is that this structure is not expected to arise for large particles in the bulk [21]. Saturn rings are not stable for large particles; instead, hyperbolic hedgehogs are the stable structures. However, experimentally it is known that Saturn rings can be stabilized by confinement [12]. In this case, it would not be

240

advantageous to search for the three-ring defects in randomly quenched systems with many particles, because in such systems the bulk director n and the vector ru joining a pair of particles would only rarely be orthogonal, and also because the separation between particles would be difficult to control. At least two different approaches could be of some use: thefirstone would be to use optical tweezers to bring two particles close to each other, in a liquid crystal host confined between two plates [22]. Operating the tweezers in a birefringent material and minimizing the effect of the strong electromagnetic field of the lasers on the structure of the LC are challenges to overcome. The second approach could be based on the magnetic manipulation of spherical droplets filled with ferrofluid in order to prepare the required particle configurations and also for measuring the attractive forces between them.

3.

Ordering kinetics in a LC-based biosensor

In this section we examine the time dependent response of a model LC-based detector of nanoscopic particles adsorbed at solid surfaces. As we have explained, these sensors are made by confining a thin film of liquid crystal between two parallel solid surfaces. The liquid is injected in its isotropic phase, and then quenched into the nematic phase. The confining surfaces' topography and chemistry are engineered to promote uniform planar anchoring of the LC[1, 21, 23, 4]. The confining surfaces also present binding sites that are specific to a ligand (e.g. a protein or a virus). When particles attach to the surfaces, they distort the local orientational order of the liquid crystal and this manifests as a proliferation of defects that can be detected by optical means. In general, when a system is quenched from a disordered phase into an ordered one, it undergoes a phase ordering process where the nonequilibrium state slowly evolves with multiple domains of ordered phase growing with time. The dynamics of phase ordering occurs in a wide variety of physical systems such as fluid mixtures, metal alloys, magnets, and also systems with more complicated order parameters, such as nematic liquid crystals. An important issue in the dynamics of phase ordering is to determine whether the behavior of the system obeys scaling, that is, whether at late stages the system enters a regime characterized by a single length scale L(t). In addition, the characteristic length L(t) may obey scaling laws that depend on the dimensionality of the system (d), the number of components n of the order parameter, and the absence or presence of a conservation law [24]. Due to the relative complexity of the nematic order parameter, its coarsening dynamics has been the subject of numerous experiments and

241 numerical simulations [25-31]. In this section we focus on the coarsening dynamics of a two dimensional macroscopic nematic film confined between two parallel substrates (see Fig. 1.3). It is important to note that, even though the system is bi-dimensional, the orientation of the LC is not restricted to the rcy-plane. Hence, the order parameter is still a 3x3 symmetric, traceless tensor. We restrict the boundary conditions to strong anchoring: homeotropic at the surface of the adsorbed particles and planar (along the x axis) for the confining substrates. The size of the system along the horizontal and vertical dimensions is large compared to the nematic coherence length £ = y/18Li/AU. In particular we present results for Lx = 290£, Ly = L x /6.

(34) (35)

covered by a rectangular grid of 1024 x 171 points. The radius of the particles is R — 2.4£, which corresponds to a typical biological particle (e.g. a virus). ^___

.

0.8jim

s' N.

\

mm

\ \

/

R=40nm

Surface properties: planar anchoring Periodic Boundary Conditions in the ^-direction

4.8fim

Figure 13.

Schematic view of a model LC-based sensor.

The initial configurations exhibit a random orientation of the liquid crystal distributed uniformly over the unit three-dimensional sphere. The system is equilibrated in the isotropic phase at the nematic potential U = 2 for a time t = 20r (r = 1/(6D*A(1 - 17/3)) is a characteristic relaxation time). The system is then quenched into the nematic phase at U = 6. In order to analyze the coarsening dynamics, we compute the orientational correlation along the ^-direction in the middle of the cell:

242

(a) -

-0.5,

100

150

200

Figure 14- The orientational correlation function in the absence of adsorbed particles tends to unity at long times.

The correlation function is normalized so that (7(0, t) = 1 at all times. We define the correlation length Lc(t) as the solution to the equation C(Lc,t) = 1/2.

(37)

Figure 13 shows the correlation function in the absence of particles. There is a first regime 0 < t < lOOr where the size of the domains grows in a similar way as it does in the bulk, but at later times the orientation imposed by the surfaces is transmitted to the middle of the cell and the growth rate is increased. Finally, at the latest stage C(x,t) -» 1, indicating a perfect uniform orientational order dictated by the surface anchoring conditions. In contrast, Fig. 14 shows C(x,t) when the adsorbed particles cover a fraction 0 = 0.33 of the surfaces. The presence of the particles affects the dynamics by slowing down the domain growth; in particular, for this value of (j) the correlation function at large separations never becomes greater than 0.5. Fig. 15 shows two snapshots of the computation cell. In these pictures we simulate the optical appearance of the system between crossed polarizers: the intensity of light is proportional to (nxny)2. Hence, white zones correspond to regions where the director's projection in the xyplane is oriented at ±45° with respect to the a>axis or the y-axis. The upper panel shows the system with = 0.33 at an early time (t = 5.6r), while the lower shows the system at t = 315r. The structure of the nematic at the later stage is characterized by the presence of two groups

243

(b) -

-0.5,

100

150

200

Figure 15. The orientational correlation function, for concentrations larger than a critical value 0* 0.33, does not go to a value of unity at large separations, but remains below a value of 0.5.

of defects: on the one hand those defects attached to the adsorbed particles, and on the other the isolated defects that move slowly through the cell. As indicated by the alternating bright and dark bands bridging both substrates, the structure of the nematic after relaxation is that of a disordered nematic induced by the presence of the particles at the substrates. Finally, by plotting the correlation length as a function of time for / (see Fig. 16), one sees that the effect of a small different values of <> amount of particles (cf) = 0.08 or 0.16) is to slow down the relaxation dynamics. In these cases, surface properties impose a quasi-uniform (long range) orientational order that is only slightly perturbed by the presence of a few topological defects: the growth of the correlation length Lc(t) when <> / is small is similar to the case 0 = 0, with the exception that the time at which the system departs from the scaling law Lc(t) ~ t1/2 occurs later for larger values of . At a critical value <j>* ~ 0.33, the perturbations created by the presence of the particles are so big that the surface anchoring cannot induce planar uniform order throughout the cell anymore: even at very late stages, C(x, t) seems to be frozen into a shape that does not go to unity at long times. The two-dimensional nature of the model prevents a quantitative comparison with experiments; the optical signatures observed in our calculations, however, are qualitatively similar to those observed experimentally: at low concentrations of adsorbed proteins, a quasi-uniform order

244

-150

-100

-50

Figure 16. Snapshots of the system with = 0.33 of the substrates' surface covered by adsorbed particles. The upper image corresponds to 5.6 times the characteristic time r; the lower image corresponds to 315r. Even at this relatively late stage, the system does not exhibit a uniform orientational order.

600

Figure 17. Size of the domains (represented by the correlation length Lc(t)) as a function of time for different values of 0, the fraction of substrate covered by the particles: = 0 (circles), 0.08 (squares), 0.17 (triangles), 0.33 (stars)

245 is observed, with only a small amount of topological defects. At high concentrations of adsorbed particles, the optical texture is indicative of a multi-domain liquid crystal texture. This comparison between experiment and calculations suggests that, in the experiments, additional information may be obtained from the time dependent response.

4.

Conclusion

We have shown that a multi-scale approach to the modelling of liquid crystal based biosensors provides useful insights into the mechanisms that control the behavior of such systems. At the level of a single particle immersed in a confined liquid crystalline host, both field theory and molecular simulations are able to reproduce the experimentally observed defect structures (i.e. Saturn rings). For the case of a pair of spherical particles, both methods agree in predicting a new defect structure: two incomplete Saturn rings fused with a third ring (perpendicular to the usual ones). In comparison with the findings of earlier studies, which assumed linear superposition of the order parameter field (be it n or Q), we observe that in the full nonlinear problem the defect structures are able to rearrange into novel structures. A molecular simulation of a macroscopic biosensor is still beyond our computational means. We have therefore used only the field theory to model its dynamical behavior. We have found that the effect of the presence of adsorbed particles at the confining walls is to slow down the coarsening of domains in the middle of the cell. When the concentration of adsorbed particles reaches a critical value, the correlation length seems to be frozen and the system never orders into a single cell-encompassing domain, as it does in the absence of adsorbed particles. Hence, the time dependent response of the sensor encodes additional information regarding the amount (and perhaps also the distribution) of adsorbed particles. It is possible to add more levels of description to the multi-scale approach considered here. As an instance, the Gay-Berne potential used to represent the interactions may be regarded as a coarse graining of an atomistic-detailed interaction potential, which may be computed through ab initio calculations. Or, on the field theory side, we may want to add the density modulations by adding a second field p(r, t) to the mesoscopic description.

246

Acknowledgments This work was funded through the NSFC Materials Science and Engineering Research Center (MRSEC) on Nanostructured Interfaces, at the University of Wisconsin-Madison.

References [I] V. K. Gupta, J. J. Skaife, T. B. Dubrovsky, and N. L. Abbott, Science, 279:2077, 1998. [2] J. J. Skaife and N. L. Abbott, Langmuir, 16:3529, 2000. [3] J. J. Skaife and N. L. Abbott, Langmuir, 17:5595, 2001. [4] Y.-Y. Luk, M. L. Tingey, D. J. Hall, B. A. Israel, C. J. Murphy, P. J. Bertics, and N. L. Abbott, Langmuir, 19(5):1671, 2003. [5] A. N. Beris and B. J. Edwards, Oxford University Press, 1994. [6] E.B. Kim, R. Faller, Q. Yan, N. L. Abbott, and J. J. de Pablo, J. Chem. Phys., 117(16):7781, 2002. [7] D. Chandler, Oxford University Press, New York, 1987. [8] A. P. Lyubartsev, A. A. Martinovski, S. V. Shevnukov, and P. N. VorontsovVelyanov, J. Chem. Phys., 96(3):1776, 1992. [9] F. Wang and D. P. Landau, Phys. Rev. Lett., 86(10):2050, 2001. [10] F. Wang and D. P. Landau, Phys. Rev. E, 64(5):056101, 2001. [II] S. Grollau, E. B. Kim, O. Guzman, N. L. Abbott, and J. J. de Pablo, in preparation, 2003. [12] Y. Gu and N. L. Abbott, Phys. Rev. Lett, 85(22):4719, 2000. [13] S. Grollau, N. L. Abbott, and J. J. de Pablo, Phys. Rev. E, 67(l):011702, 2003. [14] J. Fukuda and H. Yokoyama, Eur. Phys. J. E., 4:389, 2001. [15] P. Poulin and D. A. Weitz, Phys. Rev. E, 57(1):626, 1998. [16] R. W. Ruhwandl and E. M. Terentjev, Phys. Rev. E, 55(3):2958, 1997. [17] M. Tasinkevych, N. M. Silvestre, P. Patricio, and M. M. Telo Da Gama, European Physical Journal E, 9(4):341, 2002. [18] D. Andrienko, G. Germano, and M. P. Allen, Phys. Rev. E, 63(4):041701, 2001. [19] P. Galatola and J.-B. Fournier, Phys. Rev. Lett, 86(17):3915, 2001. [20] P. Galatola, J.-B. Fournier, and H. Stark, Physical Review E, 67(3):031404, 2003. [21] J. A. van Nelson, S. R. Kim, and N. L. Abbott, Langmuir, 18(13):5031, 2002. [22] K. Lin, J. C. Crocker, A. C. Zerio, and A. G. Yodh,

Phys. Rev.

Lett,

87(8):088301, 2001. [23] R. R. Shah and N. L. Abbott. Langmuir, 19(2):275, 2003. [24] A. J. Bray. Soft and fragile matter: Nonequilibrium Dynamics, Metastability and Flow, chapter Coarsening dyanmics of nonequilibrium phase transitions, pages 205, Institute of Physics Publishing, 2000. [25] A. P. Y. Wong, P. wiltzius, and B. Yurke, Phys. Rev. Lett, 68:3583, 1992.

247 [26] A. P. Y. Wong, P. wiltzius, R. G. Larson, and B. Yurke, Phys. Rev. E, 47:2683, 1993. [27] R. E. Blundell and A. J. Bray, Phys. Rev. A, 46:R6154, 1992. [28] M. Zapotocky, P. M. Goldbart, and N. Goldenfeld, Phys. Rev. E, 51:1216, 1995. [29] A. Bhattacharya, M. Rao, and A. Chakrabarti, Phys. Rev. E, 53:4899, 1996. [30] N. V. Priezjev and R. A. Pelcovits, Phys. Rev. E, 64:031710, 2001. [31] N. V. Priezjev and R. A. Pelcovits, Phys. Rev. E, 66:051705, 2002.

POLYMER CHAINS AND NETWORKS IN NARROW SLITS Highlights of recent results Giuseppe Allegra, Guido Raos and Carlo Manassero Dipartimento di Chimica, Materiali e Ingegneria Chimica "G. Natta" Via L. Mancinelli 7, 20131 Milano, Italy [email protected]

Abstract

We review some recent results obtained by our group, on the general subject of polymers confined within narrow slits. First, we present a derivation of the free energy of compression of two- and three-dimensional networks between parallel walls. Then, we consider the problem of the adhesion between two parallel surfaces, produced by an ensemble of chains forming irreversible and randomly distributed bonds with the walls. We evaluate the free energy change and the elastic moduli of the polymer layer, corresponding to both tangential (shear) and normal (elongation and compression) deformations. Both calculations adopt different variants the phantom chain model, whereby polymer-polymer interactions are neglected.

Introduction A better understanding the static and dynamical properties of polymers confined within narrow slits would have important implications for a number of seemingly unrelated phenomena. These include polymermediated adhesion and lubrication, the elasticity and failure of polymerbased composites or particle-filled rubbers, size-exclusion chromatography, colloid aggregation and stabilization, and nanofabrication of materials[l-4]. In view of the experimental difficulties and the number of variables affecting the possible outcome of an experiment, some of these phenonema have been traditionally considered as "engineering" subjects, with little room for "basic science". However, the situation has changed considerably in recent years. In particular, the development of the surface force apparatus has represented a major step forward on the experimental frontfl]. Different variants of the instrument are now operative 249 P. Pasini et al. (eds.), Computer Simulations of Liquid Crystals and Polymers, 249-268. © 2005 Kluwer Academic Publishers. Printed in the Netherlands.

250 in several laboratories, allowing the accurate measurement of both normal (compression and extension) and tangential (shear) forces, under both static and dynamic consitions, for confined polymeric films and brushes at wall-to-wall distances of the order of a few nanometers. See for example refs.[5-ll] for further reviews and recent applications. These have often highlighted dramatic modifications in the polymer properties, which may turn for example from rubbery to solid-like on going from the bulk to the confined state. The present article summarizes two recent theoretical contributions by our group, respectively addressing the elastic properties of compressed polymer networks[12] and of an adhesive polymer layer bridging two parallel surfaces[13]. Hopefully, the general methods and the specific results described here will provide a useful complement to scaling arguments [14, 15], which form the backbone of much polymer physics and have been largely invoked in the interpretation of the experimental results. In both cases, we adopt the phantom chain model, whereby chain-chain interactions (excluded-volume and entanglements) are neglected. The chains are thus expected to adopt random-walk conformations, as in polymer melts, dry cross-linked rubbers and 9 solutions[14, 15]. Within this model, the system's elasticity is entirely entropic in nature and arises from changes in the large-scale conformational state of the chains. If desired, a first-order estimate of enthalpic contributions to the system's elasticity may be recovered separately by a mean-field Flory-Huggins free energy term. Before we go into greater detail, and considering general scope and purpose of the present volume, it seems appropriate to discuss briefly computer simulation methods. As in every other field of contemporary chemistry and physics, their importance has been growing steadily and is bound to increase even further in the future. Here we simply mention some recent publications, with the aim to provide some entry points to the literature and give a flavour for the variety of approaches which have been successfully adopted. Yoon, Vacatello and Smith[16] have given a useful summary of the situation up to 1995, with a special emphasis on their Monte Carlo simulations on united-atom models of confined polymer melts. More recently, Vacatello has addressed the effect of chain stiffness on packing and orientational order at the polymer-wall interface[17]. Binder and coworkers have applied on- and off-lattice Monte Carlo simulations to a number of problems, including for example the scaling properties of confined polymer solutions[18] or the glass transition of polymer melts in narrow slits[19]. The former problem has also been investigated by mesoscale field-theoretic simulations[20]. Ref. [21] describes a representative application of the molecular dynamics

251 method to the equilibrim properties of confined polymer melts. Simulations have also acted as a benchmark for integral equation and density functional theories of polymeric liquids at solid surfaces[22]. Finally, the molecular and Brownian dynamics methods have been applied to steadystate non-equilibrium situations, such as the sliding of interpenetating polymer brushes under shearfll, 23]. All the studies mentioned above adopt some kind of coarse-grained description of system. The simulation of adhesion and friction between atomistically detailed models of self-assembled alkylsilane monolayers[24] may be cited as an interesting exception. The rest of the paper consists of two separate Sections, respectively on the compression of polymer networks and on polymer-mediated adhesion. The two may be read independently of each other. We sketch the mathematical derivations, which are given more fully in refs.[12, 13], and try to emphasize the most significant physical results.

1. 1.1

Compressed polymer networks A Gaussian chain in a harmonic potential

Assuming that the confining walls are perpendicular to the x axis and are located at x — ±L/2, we introduce a harmonic potential of the form: kBT

2

\l

where ksT is the thermal energy and / is the root-mean-square (r.m.s) length of the polymer segments. These are the natural units in the present problem, and they will be omitted in the following equations in order to simplify the notation. Both the compression energy {EH) and the mean-square (m.s) distance of the network elements from the center of the slab (A#) may be derived from the partition function ZH• An increase of H has the obvious effect of increasing EH and diminishing A2H. The square root of the latter may be identified with the slab width L (to within a factor of the order of unity). We shall then use H as a "driving parameter" in order to establish the relationship between compression energy and slab width. We consider a long Gaussian chain made up of n + 1 beads connected by n harmonic springs, each of elastic constant n = 3 (this is actually ft = 3&#T//2, but takes the simpler form because of our choice of the energy and length units). Its Hamiltonian is:

i=\

l

t=0

252 T

(3)

where x is a row vector containing the bead coordinates, I is the (n + 1) x (n + 1) unit matrix and R is the Rouse matrix for an open linear chain[29, 30]: 0 0 1 -1 -1 2 -1 0 2 -1 0 -1

o

0

... ... ...

0 - 1

0

0 0 0

(4)

1 J

The first term in Eq.(2) accounts for the chain connectivity, the second one for the confining potential. Note that we consider explicitly only one spatial dimension (x) and neglect the other two (y and z). The reason is that these components give rise to additional terms of the Hamiltonian which, however, are irrelevant since they are independent of the external field V(x). In other words, the statistics of the chain along y and z are not affected by confinement along x (this would not be true for a more complex model with excluded-volume interactions among the beads, where "squeezing" along x is expected to produce an expansion in the orthogonal directions). The partition function of an unconstrained chain, where all beads are free to fluctuate under the quadratic field, is: (5) t=o

This may be evaluated by changing integration variables according to the orthogonal transformation which diagonalizes (3R + HI) and using the standard formulae for Gaussian integrals: 1/2

zH= u

2?r

+H

(6)

The Ap's are the eigenvalues of R: = 2 1 - cos pn (7) n+1 The m.s distance of the beads from the median plane is obtained by differentiation of the partition function:

Al =

1

jxj] exp[-H{xj}]

253

~

n + 1 3H •

In the large-n limit, this may be approximated as:

where we have introduced the reduced variable a:

•*-=£• To a good approximation, A2H may be identified with the m.s distance of the beads from the center-of-mass of the chain. This is actually the chain's m.s radius of gyration along x (S^). The two quantities are related by:

A% = Sl + 6l

(11)

where 5% is the m.s fluctuation of the center-of-mass from the middle of the slab. The latter may be safely neglected for a very long chain under a sufficiently strong confining potential, when J > 1 or n > l/^/H. The reason is that a fluctuation of the center-of-mass off the median produces a minor entropic gain, which however has to be balanced against a large increase in potential energy. The case of a constrained chain, with the two end beads fixed at coordinates XA and X#, will be useful for the following treatment of networks. Its partition function is now defined to be:

ZH (XA, XB) =[••[ ft dxi exp [-KM] 6(x0 - XA)S(xn - XB). J

J

i=o

(12) Note that we use similar symbols for the unconstrained and constrained partition functions. The dependence of the latter on the coordinates of the fixed beads will always be explicitly indicated, thus allowing the reader to distinguish between the two. The two 6 functions are conveniently represented by the Fourier integrals: I r+oo 8{x-X) = — exp {iu{x - X)} du. (13) After some algebra, the final result may be cast in the form:

ZH(XA,XB)

= Cffexp [-**1L(X2A + X2B) - ^(XA - X B ) 2 ] , (14)

254 where, in the large-n limit: QH

=

Una ,

BH

=

n 6atanha

CH

=

Ti

(15) ,

3a (coth a - tanh a) Eq.(14) has a physically appealing interpretation. After integration over n the internal bead coordinates, the partition function for the constrained chain is formally identical to the statistical weight of a dumbbell (two beads connected by a spring) in an external harmonic potential. However, both the "external potential" and the "spring" terms BE and CH depend on the confining potential through a. In the limit of a vanishing field, we recover the usual result for a Gaussian chain of n links (BE —>• 0 and CH -> 3/n). Interestingly, CH is a decreasing function of cr, so that the effective spring becomes looser as the confining potential becomes stronger. This suggests interesting effects upon the spectrum of relaxation times of the chain.

1.2

The two-dimensional network

Let us consider a regular network of chains with square connectivity. Topologically, it has a dimensionality of two. All the chains comprise n springs, n - 1 internal beads and two terminal ones. The latter represent the junctions and are shared with the adjacent chains. Each junction is labelled with two integer indexes, j running from 1 to J and k from 1 to K (J,K ^> 1). Letting Xj^ be the x coordinate of a junction, the statistical weight of the network with fixed cross-link coordinates is J

K

K

j=lk=l

(16) The statistical weights Z#(Xy,Xfc,/) of the individual chains are given by Eq.(14). It is convenient to assume periodic boundary conditions along both topological directions, so that (V j,k).

(17)

255

The partition function of the unconstrained network is obtained from Eq.(16) by integration over the coordinates of the cross-links: J

ZH

= J

K

fflf[dXj>kZH{XJ!k}

(18)

j=\k=\ j=\k=\

where X is a row vector containing the cross-link coordinates

and U is a block-cyclic matrix with K x K square blocks •

U =

s

-I

-I 0

s

0 -I

-I

s ...

0

-I

0

...

S

... - I ... 0 -I

(19)

Each block of U is a J x J matrix, I being the indentity and r 4 cosh 2a -1 0 -1 4cosh2cr -1 0 -1 4cosh2cr -1

1 L

...

0 0

0

(20)

— 1 4cosh2cr

Integration of Eq.(18) is carried out in analogy with single-chain case ofEq.(6): v J

zH={c,H?

jK

K

„

-. 1/2

n n TT^—

•

(21)

All that is required are the eigenvalues of U (AUjV, for u = 1,..., J and v = 1,..., if). The cyclic nature of U and of its S blocks allows these to be obtained in closed analytical form: = 4 cosh 2<J — 2 cos —

2 cos

K

(22)

Again, the m.s size of the network may be obtained from the derivative of lnZ [see Eq.(8)]:

nJK

da dH

(23)

256 The final result is:

[j.oa(!kr)

(24)

+

n 48a L a n where K is the complete elliptic integral of the first kind: K

= Jof1

- q2x2)

The energy of compression EH is given by

\n*ty = \ VnJK] H (A2H) .

(25)

Remembering from Eq.(lO) that H = 12a 2 /n 2 , the energy per c/iain is then:

gL

1.3

mn

(26)

Numerical results

Eqs.(24) and (26) are the main results of the previous section on twodimensional networks. The m.s distance of the network elements from the minimum of the confining potential (proportional to the square of the slab width L2) and the network energy per chain are expressed as a function of a compression parameter a. The latter may be eliminated graphically, by driving it from a very small up to a very large value, and plotting EH and A^/n against each other. The result is given in Figure 1, alongside analogous plots for: • a one-dimensional "network", made up of several chains connected head-to-tail to form a very large ring • a three-dimensional network with cubic topology • some intermediate cases, constructed by superposition of a finite number (2, 4 or 8) of two-dimensional layers, giving a three-dimensional network with finite thickness along one topological direction. Mathematical details for these additional cases may be found in ref.[12]. Figure 1 demonstrates a very similar behaviour of the one-, two- and three-dimensional networks at strong compressions. When the external field is strong to the point of producing a compression of the individual strands, their topological connectivity becomes irrelevant. On the other

257

1-

.___chain(1D) —— "

0.1 -:

H

I

:

0.01

1

\

1E-3-

\

\

\

1 layer (2D)

\

\

\

infinite layers (3D) 1E-40.1

•

i

•

0.2

i

03

0.4

2

[/n]

05

0.6

1/2

Figure 1. Reduced free energ per chain SH VS. the reduced r.m.s distance of the jn (~ reduced slab width), for the one-, two- and beads from the slab center three-dimensional networks.

hand, significant differences are observed under weak compression. In the limit of a vanishingly small applied field, Eq. (24) gives A ^ - S2X = n [a - b In a]

(a, b > 0)

(27)

and we have a slow, logarithmic dependence of S% on a (-> 0). The analogous result for the three-dimensional case is: A2H ~ 5^ = n x 0.1218..., as found long ago by Ronca and Allegra[26]. We see the interesting feature that the three-dimensional network has an unperturbed m.s size which, in the absence of any squeezing force, is comparable to that of a single chain strand. However, the energy vs. network size curve is infinitely steep in this case, so that further compression of this network requires the application of infinitely strong forces.

2. 2.1

Polymer-mediated adhesion The model

We are interested in modelling adhesion phenomena produced by a thin layer of polymer chains confined between two rigid parallel walls.

258 Our aim is to derive the small-strain tangential and longitudinal moduli of the adhesive layer. We will not address its large-strain behaviour and ultimate failure properties, which are also very important for applicative purposes. The chains are assumed to be very long, so that their unperturbed radii of gyration are much larger than the wall-to-wall distance L. As a matter of fact, we consider a single infinitely long chain, compressed by the walls to melt-like densities (see Figure 2). We assume that, at a given instant in time, all the chain monomers in contact with the walls react and form permanent chemical bonds with it. The original infinite chain is thus broken up into shorter subchains, with their two end monomers either on the same or on opposite surfaces ("loops" and "bridges", respectively). The distribution of lengths for the loops and bridges, as well as their relative probabilities, is not given a priori, but will be derived from statistical principles. Thus, formation of irreversible polymer-surface bonds corresponds to taking an "instantaneous snapshot" of an equilibrium configuration the system — an idea pioneered some time ago by Deam and Edwards in the context of polymer networks [27]. Unlike the previous section, now there are no cross-links between the chain(s). Note that the present system corresponds to the "Model B" of ref. [13]. In the same paper, we also investigated the properties of a so-called "Model A", consisting of monodisperse polymer chains bridging the two surfaces. We adopt a lattice model, following earlier treatments by Di Marzio [31], Scheutjens and Fleer[32], Silberberg [33] and one of us[34]. The statistical segments of the polymer lie on the edges of cubic cells. The lattice model is in many ways artificial, but it has also some distinct advantages. In particular, it allows the enumeration of the chain conformations and we implement it in a way which accounts exactly for finite chain extensibility (unlike Gaussian models). The more realistic representation of the interaction with a solid wall, compared with the harmonic potential of the previous section, is an additional bonus. The polymer is assumed to be in a rubbery amorphous state above its glass transition temperature, with a perfectly uniform density across the slab. The latter constraint is implemented by application of a suitable shortrange polymer-wall attractive potential.

2.2

The transfer matrix

We consider two walls at a distance L, in cubic lattice units (equal to the length of the chain segments). Thus, L is also the number of lattice "layers" between the bottom and the top wall. We denote by P^ the

259

Figure 2. Representation of a very long chain filling the space between two parallel walls, forming loops, bridges and trains of variable length. The slab width is in this case L = 12. All the beads in contact with a wall are irreversibly bonded to it, after reaching configurational equilibrium. L x L transfer matrix: 4 1 0 0 0

1 4 1

0 1 4

0 0 1

0 0

1 0

0 0 0 4 1

(28)

1 4

Its (i, j) element represents the number of ways of placing one bond on the lattice, so as to connect a bead on layer i with a bead on layer j . Its eigenvalues A^ and eigenvectors a^ = [ai^,a2fc,... ,a>Lk] a r e : =

4 + 2 cos

(29) (30)

We illustrate the usage of the transfer matrix by writing the partition functions — equal to the total number of allowed conformations, since there are no energetic terms associated with chain bending or bead-bead

260

interactions — of a chain of n segments, whose ends are constrained to lie either on the same ("loop", LP) or on opposite walls ("bridge", BR). In the first case we have: 1 0 0

Z LP (n,L) = [ 1 0 0

L k=l

whereas in the second case:

0

0

0 0 0

L

(32) jfe=l

2.3

Statistical population of loops and bridges

As mentioned above, we consider a infinitely long chain, which is initially unconstrained and free to reach statistical equilibrium. The chain meanders between the walls, forming loops and bridges of all possible lengths n. We label their partition functions by ZLP and ZBR. These differ from the partition functions of the previous section since: a. In the present context, a bridge/loop presesents a chain travelling from one to the other/same surface, without ever touching either wall in the intermediate steps (otherwise, the chain would be further broken down into shorter bridges and/or loops). b. We wish to model a situation with a uniform polymer density across the wall-to-wall slab. However, a phantom chain interacting with purely repulsive walls would have a density maximum at the middle, since it suffers from "entropic repulsion" [15, 34]. This is understood by recognizing that, on a simple cubic lattice, a segment within the bulk has six possible orientations. Instead, when one of its end beads is in contact with a wall, the segment has only five possible orientations. This can be remedied by placing a suitable "premium" on the monomers which are directly in contact with a wall. In practice, the above requirements are implemented in the following way:

261 i. The two terminal bonds are constrained to be orthogonal to the surface, so that the effective number of free bonds in a subchain of n segments is n — 2. ii. We use a matrix P of order L - 2 instead of L as in Eq.(28), to fordid the internal beads from visiting the upper and lower layer. Hi. The premium for the atoms at either wall is obtained by multiplying the partition function of each subchain by a factor of | . There is only one such factor, even though each subchain has two terminal beads, to avoid double counting of the chain-wall contacts. These points may be summarized as: (33) Z x (n, L) = \zx{n - 2, L - 2) (n, L > 2; X = LP or BR). o Finally, one should not forget that the chain may also form "trains" of segments in contact with the walls (see again Figure 2). These will be dealt with implicitly, since a train may be viewed as a sequence of "one-segment loops". All that is required is the additional rule: iv. A one-segment loop (a chain bond lying flat on a wall) has the partition funtion: 24

ZLP(1,L) = —,

(VL)

(34)

given by the | premium times the number of possible orientations of the segment (four). The relative populations of loops and bridges are expressed by their probabilities: % ^ ,

(35)

where 6n is the partition function of a generic unconstrained n-segment section of an infinitely long chain. It ensures the fulfilment of the normalization condition: f > L p ( n , L ) + p B R ( n , £ ) ] = l.

(36)

71=1

These quantities are given graphically in Figures 3 and 4, for some selected strand lengths or slab widths. We observe that, for a given strand length, the loop and bridge probabilities are virtually identical up to wall-to-wall distances of the order of the unperburbed radius of gyration, i.e. up to L ~ y/n. Afterwards, the loop probability tends to

262

\n=25

10

15

20

25

30

35

Figure 3. Probabilities of loops (solid lines) and bridges (dashed lines) of selected lengths n, as a function of the wall-to-wall distance L.

a constant value, whereas the bridge probability drops to zero. Short loops are always more more likely to occur than long ones. Instead we observe a crossover for bridges, since a short bridge is more probable than a long one for narrow slits, and less probable for wider slits. For a given wall-to-wall distance L, the most probable bridges have a strand length n~ L2. The average strand length is defined as oo

— y n

»,£)].

(37)

n-\

It has a simple expression, which may be obtained by the following argument. We know from ref.[34] that, if the density on the internal layers is equal to p, the density on the terminal ones is | p . The total number of beads within the slab is thus: (38)

where S is the slab area (->• oo). Since the infinite superchain is broken into two separate strands whenever one of its beads touches one of the

263

10

50

100

150

200

250

300 350

Figure 4- Probabilities of loops (solid lines) and bridges (dashed lines) for some selected wall-to-wall distances L, as a function of their length n.

walls, we may obtain (n) from the ratio of the total number of beads and the number of beads on the terminal layers: 3. 1 N (39) <») = 2|P5 The average strand length is nearly proportional to the slab width. It is actually smaller than it, since loops are always more probable than bridges (for a given strand length).

2.4

Free energy, elastic forces and moduli

The Helmholtz free energy of a single chain strand may be evaluated according to the usual statistical-mechanical recipe Ax(n,L) = - lnZx(n, L) (where X=LP or BR). On the other hand, the free energy of an ensemble of chains whose terminals are irreversibly bonded to the walls is obtained by averaging the free energies of the strands, weighting them by the loop and bridge probabilities calculated in the previous section: oo

A(L) = £ [pLp(n,L0)^Lp(n,L) +pBR(n,L0)ABR(n,L)). 71 = 1

(40)

264

It is important to observe that the weighting coefficients are fixed at the time of the polymer-wall reaction(they are "quenched variables"). In general, they may correspond to the equilibrium configuration of the system at a slab width Lo ^ L, Thus, the calculated properties are bound to depend on the "preparation conditions". There is a strong analogy with the elasticity of cross-linked rubbers, as discussed by Deam and Edwards[27]. The elastic forces are given by the first derivatives of the free energy A(L) with respect to diplacements of the two walls, referred to a unit surface. Since the loop and bridge probabilities are constant, all that is required are the derivatives of ALP and ABR for all possible strand lengths. The evaluation of normal (logitudinal) forces is in principle straightforward: dAx(n,L) /x(n,L) = —m— =-

1 2 { n L )

dZx(n,L) dL

•

(41)

According to our sign convention, / x is positive for attractive wall-wall forces (i.e., adhesion) and negative for repulsive ones. We computed these derivatives by numerical differentiation, using quadruple precision arithmetic (necessary for accurate summation of Eqs.(31) and (32)). The result is as expected on simple physical grounds. Loops and bridges exert virtually identical repulsive forces up to L ~ n (the unperturbed strand size). At larger distances, the loop forces vanish, whereas the bridge forces turn into attractive. They are linear at moderate extensions, and they take the inverse Langevin form [35] as the chain approaches its maximum elongation. Since we are neglecting entanglement effects, only the strands which bridge the two walls contribute to shear (tangential) forces. Evaluation of these forces is slightly more complicated, but luckily the results may be summarized in a very simple and concise way. In the limit of small lateral displacements a, the force is linear and given by: /BR(n,L,a) = —.

(42)

Here n\\ is the average number of bonds parallel to the surfaces, which is in turn given by: ZBR(nl,L)

= 4n—-^r—,—rr~^-

,. v

(43)

The elastic moduli K\0lig and «tan a r e obtained by further differentiation of the forces per unit area (they are second derivatives of the free energy). We skip the mathematics and go directly to the final results in

265

NT1!

longitudinal

10" 2 -

v tangential ^

10-3,

1

(D

1

5

10

15

20

25

30

L

Figure 5. Tangential and longitudinal moduli, as a function of the wall-to-wall distance L. The symbols represent the computed points, the straigth lines are leastsquare fits.

Figure 5. The plots give the two moduli as a function of the wall-to-wall separation. They have been obtained for L = Lo> at wall-to-wall distances identical to those at cross-linking [see eq.(40)]. The longitudinal modulus is roughly one order of magnitude larger than the tangential one at small L's. Both moduli decay to zero in a roughly exponential way at larger L's: K(L) = R 1(T L / L . (44) Fits of the log-linear plots within the range 4 < L < 30 give: Kl0ng = 0.376 , «tan = 0.078 ,

Llong = 9.86;

The basic reason for the fast decay of the force constants with inceasing thickness is that fewer and fewer short chains connect the two walls (see again Figs. 3 and 4), whereas the long chains contribute much less to the modulus. It should also be borne in mind that, in the absence of the walls and of a polymer-surface reaction, the polymer would be a simple rubbery liquid with a null zero-frequency shear modulus.

266

As a final note of caution, we point out that the conformational entropy of the chains is only one of the contributions to the elastic modulus. In particular, under extension the polymer layer must undergo substantial volume changes (instead, there is roughly no volume change for shear deformations). This may be associated with important enthalpic effects, whose contribution to the elongational modulus may be estimated by a Flory-Huggins energetic term incorporating a suitable x parameter[14, 15, 28]. Once more, see ref. [13] for details.

3.

Conclusions

We have summarized two recent theoretical papers on the elasticity of polymer networks and chains confined within narrow slits[12, 13]. The neglect of polymer-polymer interactions (phantom chain assumption) allows an essentially exact — although sometimes complicated — mathematical description. The presentation has been didactic in spirit and we have strived to highlight the essential physics of these systems. Hopefully this will stimulate more theoretical, experimental and computational studies. The problem of polymer-mediated adhesion would certainly deserve further scrutiny. We are not aware of any experimental or computational studies of systems directly related to our model (very long chains, irreversibly bonded to both surfaces of two confining walls). All the experimental work by the surface force apparatus [1, 5-10] has concentrated on confined polymer melts and solutions or interpenetrating polymer brushes (chains bonded at one end only). Presumably, one of the reasons for this is that irreversible chemical bonding at both surfaces is not easily reconciled with reversible and easily repeatable experiments (at the end of each experiment, a large extensional force would have to be applied, in order to break the polymer film and bring the surfaces apart). It may be possible to circumvert this problem by exploiting easily hydrolizable polymer-surface bonds, such as those of the ester or amide functional groups. Computer experiments are also expected to provide important qualitative insights and quantitative results. The simulation of irreversible chemical bonding at the polymer-wall interface and of the mechanical properties of the resulting adhesive films may be expected to be very challenging steps, in analogy with computer simulations of chemically cross-linked networks[36, 37].

267

Acknowledgments We thank Paolo Pasini and Claudio Zannoni for inviting us to contribute this paper. We have benefited from the financial support of the Italian MIUR through COFIN2003.

References [1] J. Israelachvili, Intermolecular London, 1994.

and Surface Forces, 2nd ed., Academic Press,

[2] S. Granick (editor), Polymers in Confined Environments, Springer, Berlin, 1999.

Adv. Polym. Sci., 138,

[3] R.G. Larson, The Structure and Rheology of Complex Fluids, Oxford University Press, Oxford, 1999. [4] A.V. Pocius, Adhesion and Adhesives Technology, An Introduction, lishers, Munich Vienna and New York, 1997. [5] P.F. Luckham and S. Manimaaran, Adv. Colloid Interf

Hanser Pub-

Sci., 73:1 1997.

[6] G. Luengo, F.-J. Schmitt, R. Hill and J. Israelachvili, Macromolecules, 30:2482, 1997. [7] M. Ruths and S. Granick, J. Phys. Chem. B, 103:8711, 1999. [8] H.S. Kim, W. Lau and E. Kumacheva, Macromolecules, 33:4561, 2000. [9] U. Raviv, R. Tadmor and J. Klein, J. Phys. Chem. B, 105:8125, 2001. [10] S. Yamada, G. Nakamura and T. Amiya, Langmuir, 17:1693, 2001. [11] P.A. Schorr, T.C.B. Kwan, S.M. Kilbey II, E.S.G. Shaqfeh and M. Tirrel, Macromolecules, 36:389, 2003. [12] G. Allegra and G. Raos, J. Chem. Phys., 116:3109, 2002. [13] G. Allegra, G. Raos and C. Manassero, J. Chem. Phys. 119:9295, 2003. [14] P.-G. de Gennes, Scaling Concepts in Polymer Physics, Cornell University Press, Ithaca, 1985. [15] A.Yu. Grosberg, A.R. Khokhlov, Statistical Physics of Macromolecules, can Institute of Physics, NewYork, 1994.

Ameri-

[16] D.Y. Yoon, M. Vacatello and G.D. Smith, in Monte Carlo and Molecular Dynamics Simulations in Polymer Science, K. Binder ed., ch. 8, p. 433, Oxford University Press, New York, 1995. [17] M. Vacatello, Macromol. Theory Simul.,

11:53, 2002.

[18] A. Milchev and K. Binder, Eur. Phys. J. B, 3:477, 1998. [19] C. Mischler, J. Baschnagel and K. Binder, Adv. Colloid Interf. Sci., 94:197, 2001. [20] A. Alexander-Katz, A.G. Moreira and G.H. Fredrickson, J. Chem. 118:9030, 2003. [21] T. Aoyagi, J. Takimoto and M. Doi, J. Chem. Phys., 115:552, 2001. [22] A. Yethiraj, Adv. Chem. Phys.,

121:89, 2002.

[23] T. Kreer, M.H. Miiser, K. Binder and J. Klein, Langmuir, 17:7804, 2001.

Phys.,

268 [24] B. Park, M. Chandross, M.J. Stevens and G.S. Grest, Langmuir, 19:9239, 2003. [25] H.M. James and E. Guth, J. Chem. Phys. 10: 455, 1943; H.M. James, J. Chem. Phys. 15:651 1947. [26] G. Ronca and G. Allegra, J. Chem. Phys., 63:4104, 1975. [27] R.T. Deam and S.F. Edwards, Philos. Trans. R. Soc. London, Ser. A, 280:317, 1976. [28] B. Erman and J.E. Mark, Structure and Properties of Rubberlike Networks, Oxford University Press, New York and London, 1997. [29] P.E. Rouse, J. Chem. Phys., 21:1272, 1953. [30] M. Doi and S.F. Edwards, The Theory of Polymer Dynamics, Clarendon Press, Oxford, 1986. [31] E.A. Di Marzio, J. Chem. Phys., 42:2101, 1965; E.A. Di Marzio and R.J. Rubin, J. Chem. Phys., 55:4318, 1971. [32] J.M.H.M. Scheutjens and G.J. Fleer, J. Phys. Chem., 83:1619, 1979; ibid., 84:178, 1980. [33] A. Silberberg, J. Coll. Inter}. Sci., 90:86, 1982. [34] G. Allegra and E. Colombo, J. Chem. Phys., 98:7398, 1993. [35] P.J. Flory, Statistical Mechanics of Chain Molecules, Hanser, NewYork, 1989. [36] K. Kremer and G.S. Grest, in Monte Carlo and Molecular Dynamics Simulations in Polymer Science, K. Binder ed., ch. 4, p. 194, Oxford University Press, New York, 1995. [37] F.A. Escobedo and J.J. de Pablo, Phys. Rep., 318:85, 1999.

ROTATION AND DEFORMATION OF POLYMER MOLECULES IN SOLUTIONS SUBJECTED TO A SHEAR FLOW Siegfried Hess Institut fur Theoretische Physik, Technische Universitat Berlin, PN 7-1, Hardenbergstr. 36, D - 10623 Berlin, Germany [email protected]

Gary P. Morriss School of Physics, University of New South Wales, Sydney NSW 2052, Australia [email protected]

Abstract

The complex rotational and deformational behavior of polymer molecules in dilute solutions subjected to a shear flow as studied in non-equilibrium molecular dynamics computer simulations can be understood qualitatively within a simple dumbbell model. It allows a numerical test of a conjectured relation between the average angular velocity of a flexible polymer molecule and a ratio of components of the gyration tensor. The model involves a pseudo-friction coefficient which is chosen such that the peculiar kinetic energy is kept constant: Gaussian thermostat. The orbits show rotation, wagging and tumbling, depending on the shear rate, combined with radial stretching and compression. The angular velocity divided by minus the shear rate is equal to 1/2 at small shear rates, corresponding to a solid body like behavior. At high values of the shear rate the angular velocity decreases strongly with increasing shear rates. In both these regimes, the conjectured relation holds true. For intermediate shear rates, however, this relation between the true angular velocity and the corresponding expression inferred from the gyration tensor is violated. The behavior of the dumbbell is highly irregular for these shear rates, a sensitive dependence on the initial conditions and on the shear rate are noticed. The largest Lyapunov exponent is positive, indicating chaotic behavior for certain values of the shear rates. For certain shear rates, no unique assymtotic state exists. At some inermediate and at high shear rates, stable periodic orbits with long periods are observed. The irregular behavior of the angular velocity at intermediate shear rates persists when the Gaussian thermostat is replaced by

269 P. Pasini et al. (eds.), Computer Simulations of Liquid Crystals and Polymers, 269-293. © 2005 Kluwer Academic Publishers. Printed in the Netherlands.

270 a Nose-Hoover thermostat and even when an additional thermostat is applied which controls the configurational temperature.

Introduction The rotation of a flexible polymer molecule in a dilute solution, subjected to a simple shear flow, as observed in non-equilibrium molecular dynamics computer simulations [1] is quite complex and intriguing due to the coupling between the rotation and the deformation of the molecule. In real experiments, information on the average angular velocity can be obtained indirectly via an analysis of the behavior of components of the gyration tensor averaged over many molecules [2]. This estimate for the angular velocity relys on a theoretical relation [3] based on approximations. The limits of applicability of such a relation have yet to be tested by experiments as reported in [4] where it was demonstrated that it is also possible to observe directly the effect of a shear flow on the shape of an individual molecule. Both the conformational changes as characterized by the gyration tensor and the actual rotation of molecules can and have been extracted from non-equilibrium molecular dynamics (NEMD) computer simulations [1, 5, 6]. An analysis of the angular velocity gave the impression that regular and chaotic types of motions occur [1], Here a model is introduced which contains the features characteristic for the coupling between rotation and deformation and it is simple enough to allow, in particular in its two-dimensional version, a detailed study of the rotational dynamics and its relation with the conformational changes. Rather smooth as well as irregular rotations and radial oscillations, wagging, stretching and contracting tumbling motions are found. The chaotic behavior and the periodic orbits occuring at certain shear rates analysed. This article proceeds as follows. Starting from the angular momentum balance, a relation between the average angular velocity of a polymeric chain and the ratio of components of the gyration tensor is derived in section 1. The simple dumbbell model is introduced in section 2. Section 3 is devoted to the discussion of the ratation and deformation of of the dumbbell and a first test of the relation between the shear-induced angular velocity and the expression based on the gyration tensor. The chaotic behavior observed at intermediate shear rates as well as the occurrence of stable periodic orbits are presented in section 4. In section 5, finally, some results obtained for the test of the angular velocity relation are mentioned briefly for the cases where the Gaussian thermostat is replaced by a Nose-Hoover thermostat and where an additional thermostat is applied which controls the configurational temperature.

271

1.

Angular Velocity and Deformation

A stiff particle immersed in a fluid undergoing a shear flow picks up an average angular velocity equal to the local vorticity which, in turn, is one half of the rotation (curl) V x v of the velocity field v. Flexible chain molecules are deformed, in particular stretched, by the shear and their average angular velocity, given by the ratio of their angular momentum and their moment of inertia, decreases with increasing shear rate. This was conjectured on the basis of a relation between this angular velocity and a ratio of components of the gyration tensor, as first discussed within the framework of a theory associated with the key word "internal viscosity". Here, as in [1], a simpler derivation of this relation is given, in the spirit of the ideas put forward by Debye [7]. Consider a chain molecule composed of N beads with mass m located at positions r% with i = 1,2,... N. Each bead is assumed to obey an equation of motion of the form m ^£- = F* — mCf^gr — v ( r *)) • The term involving the friction coefficient £ is the friction force the bead experiences when it moves relative to the flow velocity v the solution would have in the absence of the polymer chain. The quantity Fl stands for all other forces acting on the bead i. The time change of the total angular momentum L = m YliY% x Tgr is given by (1)

The first term on the r.h.s. corresponds to a torque and it vanishes for central forces. In the presence of non-central forces the time average of the torque is assumed to vanish. Also the time change of the angular momentum is assumed to approach zero when averaged over long times. Then (1) leads to /

\

(2)

The angular bracket indicates the time average. Notice that the friction coefficient £ has disappeared from this relation although its existence was of crucial importance for the derivation of (2). The angular velocity a) is defined as the ratio of (L) and of the relevant component of the (time averaged) moment of inertia tensor. To be more specific, a plane Couette geometry is considered with the flow in ^-direction and the gradient of the velocity in y-direction, viz. vx = 7 y, vy = vz = 0, where 7 = dvx/dy is the (constant) shear rate. Then (2) is equivalent to (Lz) = -jm(Gyy).

(3)

272 Here Gyy is the yy-component of the gyration tensor

Greek subscript indicate cartesian components. It is understood that the center of mass of the polymer chain corresponds to r = 0, i.e. one has 5^ rl = 0. For the present geometry, the relevant moment of inertia is m (Gxx + Gyy). Thus the expression for the angular velocity, inferred from the average angular momentum is _=

(Lz) ((GXX) + (Gyy))

rn

The relation (3) becomes u) = U)Q ,

(6)

with the expression UQ based on the geometry of the polymer coil given by y^xx/

~r \S*yy)

* \

Y^xx)

\ yJyyl J

In an undeformed equilibrium state, the coil is spherical, on average, then one has GJQ = — 7/2. The same applies for small shear rates. At intermediate and at high shear rates, the polymer molecule is substanially deformed such that GxxGyy, on average. This implies that the ratio £IG = ^ G / ( " ~ 7 ) becomes significantly smaller than 1/2 which is its small shear rate limiting value. A relation of the form (6) between the rotational angular velocity a) and the quantity UQ associated with the deformation of the chain molecule was first proposed by R. Cerf [3]. Nonequilibrium molecular dynamics (NEMD) computer simulations can and have provided a test of this relation [1, 6]. The rotational and elongational behavior of alkanes was studied in [8]. Here a test is conducted for a simple two-dimensional model which, however, displays a rather complex dynamics. Before this model is introduced, it is mentioned that in microscopic calculations one may also test the conjectured relation involving averages of ratios rather than ratios of averages, viz. <w> = With

L

(WG>

,

(8) G

This relation has also been tested in NEMD simulations [1] where both the polymer and the solvent molecules where treated microscopically, i.e.

273

no friction coefficient was used explicitely. In the NEMD simulations the relation (8) is well obeyed at all shear rates studied. This is surprising since no friction is used in the underlying dynamics. To understand the basic physical processes, we introduce a simple model with 3/2 "degrees of freedom" rather the few thousand involved in the NEMD simulations.

2.

A Simple Model

In the spirit of dumbbell models [9], we mimic the dynamics and the shape of polymer molecule by that of a particle at position r where r = 0 corresponds to its center of mass. A force F acting on this particle can be chosen such that the time average of r 2 coincides with the mean square radius of gyration of the polymer coil in equilibrium. Furthermore, the effect of the flow of the background fluid is taken into account via a "friction" force proportional to the difference between the velocity of the particle and the flow velocity v(r). For this model, reduced (dimensionless) variables are used, but they are denoted by the same symbols as the corresponding physical quantities. The reduced mass is put equal to 1. Then the equation of motion reads

£='-<(!-*>)•

<«»

The force is chosen as Prr-rr"1*', -$' = r - 3 - r 3 ,

(11)

where <&' is the derivative of the potential function <3>(r) = (1/2) r~ 2 + (1/4) r 4 - 3/4 with respect to r. The potential is of nonlinear elastic type with a minimum at r = 1, in reduced units. The equilibrium radius of gyration TQ is somewhat larger, depending on the temperature T. In particular one has ro ~ 1.03 in two dimensions and for T = 0.1, the temperature used here. The quantity ( used in (10) is a time-dependent pseudo-friction coefficient which, as specified below, does not violate the time-reversability of the equations of motion. First, we introduce the "momentum" variable

P=ft~ v(r),

(12)

which is the peculiar velocity, and use -^ = ^ f — ^f • Vv(r), for the case of a time independent flow velocity v. Then (10) is rewritten in terms of two equivalent differential equations of first order in time

^ at

= F - C p - ( p + v(r))-Vv(r).

(14)

274

When £ is identified with a real friction coefficient, one has to include fluctuating Brownian forces in the expression for F, in addition to a deterministic force, in order to have the particle moving such that it has a (peculiar) kinetic energy in accord with the temperature of the fluid. As an alternative, we use here the Gaussian constraint to ensure that p2 = DT = const.,

(15)

where D is the space dimension and T stands for the (reduced) temperature. More specifically, we choose £ as a dynamic variable determined by C = p . ( F - ( p + v(r)).Vv(r))/p2. (16) Insertion of this expression in either (10) or (14) results in equations which still are invariant under time reversal, in contradistinction to the case where one uses a true friction coefficient. On the other hand, the equations of change are not of Hamiltonian type. Next, we consider the special case of a plane Couette flow with the velocity given by v(r) = 7J/e x , where e x is a unit vector in ^-direction and 7 = dvx/dy — const is the shear rate. Furthermore, for simplicity, the motion of the particle is restricted to the rry-plane. Then the equations of motion correspond to the (two-dimensional version) of the "SLLOD" algorithm used in NEMD simulation studies of the viscous properites of fluids [10]: dx

dy

The expression for the coefficient ( now is \

(19)

which guarantees that p2 = 2T = const. The value of the temperature T is fixed by the initial condition. The time average of £ vanishes for 7 = 0.

3.

Rotation and Deformation

The results to be presented next are based on the solutions of the equations of motion which were integrated, by one of the authors (SH) with "NDSolve" of Mathematica (mostly with the Runge-Kutta method), both as differential equations of second and of first order, corresponding to (10) and (17,18), as well as with a fourth order Runge-Kutta method

275

50

100

150

200

50

100

150

200

50

100

150

200

Figure 1. The angular momentum Iz and the quantity —jGyy (dashed curve) as functions of the time t for the shear rates 7 = 0.1 (upper graph), 7 = 0.3, and 0.5 (lower graph). The horizontal lines indicate the averages of these quantities.

programmed in Fortran by the other author (GM). To analyse the shear flow induced effects, in particular the change of the angular momentum or of the angular velocity due to a finite shear rate 7, the results of two simulation runs are compared. More specifically, two (independent)

276

200

50

100

150

200

50

100 t

150

200

Figure 2. The angular momentum Iz and the quantity —~iGyy (dashed curve) as functions of the time t for the shear rates 7 = 1.0 (upper graph), 7 = 1.3, and 1.6 (lower graph). The horizontal lines indicate the averages of these quantities.

particles start at the same position, e.g. rj = r"2 = (1,0) but have initial velocities such that the sum of their angular momenta vanishes initially, e.g. V l - (TV2, - T 1 / 2 ) , v2 = V ^

277

For 7 = 0, the sum of the angular momenta remains zero at all times. For 7 / O , the total angular momentum and the z-component of the shear-induced angular velocity a;, obtained by the ratio of the sum of the angular momenta and of the relevant moment of inertia, cf. (9) are non-zero. In Fig.l the shear induced angular momentum lz, i.e. one half the total angular momentum of the two particles is plotted as function of the time t for the shear rates 7 = 0.1 (upper graph), 7 = 0.3, and 0.5 (lower graph). In Fig.2, the corresponding quantities are presented for 7 = 1.0 (upper graph), 7 = 1.3, and 1.6 (lower graph). For comparison, lG = -jGyy (dashed curves) is also displayed, where Gyy is the mean value of the component of the gyration tensor of the two particles. The horizontal lines indicate the time averages of lz and lG. As expected, the dependence of lz and lG on time is qualitatively similar, with considerable quantitative differences. Their time averages agree closely, in most cases, in accord with relation (3). Of course, to test this and the other relations stated above, a systematic study for more shear rates and averages over longer times are required. Furthermore, different types of motions are encountered for the different ranges of shear rates. This point shall be discussed later.

~2

Figure 3. The rotational velocity as function of the shear rate. The black dots mark the rotational velocity evaluated from the angular momentum, the gray dots stand for the values inferred from the gyration tensor. The large and small symbols are associated with the average of ratios and with the ratio of averages, respectively.

278 01

01 1.5 1 0.5

1.5 1 0.5 0 .0.5 . 1 . 1.5

j

y,?

o .0.5 .1 . 1.5

(A

1.0.50 0.5 1 1.5

'1.5. 1.0.50 0.5 1 1.5 Xn

0.3 1.5 1 0.5 Q

1.5 1

Sx M

0.5

V

0 .05 .1 . 1.5

2

.0.5 . 1 . 1.5 .1.5. 1.0.50 05 1 1.5 X

.1.5. 1.0.50 0.5 1 1.5 X

.1.5.1.0.50

0.5 1 1.5

x2

.1.5. 1.0.50 0.5 1 1.5 X

2

Figure 4- The orbits for the shear rates 7 = 0.1, 0.3 and 0.5. The trajectories start from the same point and with the same magnitudes of the velocities, but the y-component of the initial velocity of the left and right orbit is negative and positive, respectively.

279 In Fig.3 the reduced angular velocity, i.e. the averaged shear induced angular velocity, divided by minus the shear rate, is displayed as function of the shear rate. The averages were taken over 500 time units. The black dots mark the rotational velocity evaluated from the angular momentum, c.f. (5) and (9), the gray dots stand for the values inferred from the gyration tensor, c.f. (7) and (9). The large and small symbols are associated with the average of ratios and with the ratio of averages, c.f. (5) and (7), respectively. At small and at very high shear rates, the black dots overlap with the gray ones which means that relations (6) and (8) are obeyed. Furthermore, where the small dots cannot be seen in the graph, in particular for shear rates less than about 0.3, the averages of ratios practically coincides with the ratios of the averages. At intermediate shear rates, however, fluctations are larger which implies that the small and large dots deviate significantly. The overall behavior of the average shear induced rotational velocity is as expected. For small shear rates the data are close the theoretical value 0.5, indicated by the dashed horizontal line. At intermediate and high shear rates, the angular velocity decreases considerably with increasing shear rates. The overall behavior of the reduced angular velocity found is qualitatively similar to that one found in the NEMD simulations. The decrease of u/{—7) and of (OJ)/(—7), at high shear rates, however, is much stronger than that one found in [1, 6]. This may be associated, on the one hand, with the fact that the model presented here mimics a rather stiff chain. For the three-dimensional version of the present model interaction, the ratio of the equilbrium average of r 4 and of the square of the average of r 2 is equal to the corresponding value of a worm-like chain [11] whose total length is just about 4 times its persistence length. On the other hand, the Gaussian iso-kinetic constraint may be rather severe for a system with just two degrees of freedom. A more realistic behavior at high shear rates is achieved, as indicated later, when an additional thermostat is introduced, which controlls the configurational temperature [12]. A closer look at Fig.3 shows that the relations (6) and (8) are violated in certain ranges of shear rates. This is expected in real physical systems and for data inferred from NEMD simulations as presented in [1, 6], as well as for the model considered here since the use of a constant friction coefficient in the equations of motion discussed above is an approximation. Physical reality is closer to a time dependent effective friction coefficient as used here. Assuming, as above, that the long time average of the time change of the angular momentum vanishes for a stationary state and that there is no torque associated with the force F, (3) has to be replaced by Gyy) = -{Q-1

(6C(6Lz+7m6Gyy))

.

(20)

280 7

7.5 0.5 -7

-2-10

1 2 3 X

Figure 5.

2

The orbits for the shear rates 7 = 1.0, 1.3 and 1.6.

Here 6X — X — (X) indicates the deviation of a dynamical variable X from its time average. Similar corrections apply to (6) and (8). The physics underlying a situation where there are stronger fluctuations and where the shear induced angular velocity cD is rather small compared with a ficticious angular velocity UQ inferred from the components of the gyration tensor is revealed by the orbits shown in Figs.4,5, over about 100 time units. The trajectories start from the same point, viz. r = (1.0,0.0) and with the same magnitudes of the velocities, but the y-components of the initial velocities of the left and right orbits

281

co -

CO

60

80

100

40

60

80

100

40

60

80

100

-

0

co

20

Figure 6. The angular velocity u> as function of the time for the shear rates 7 0.1,0.3 and 0.5. The dashed horizontal line indicates the reference value —7/2.

point downwards and upwards, respectively. This means, the left particle starts it rotational motion with the sense of rotation imposed by the shear flow, the right one against it. For the smallest shear rate, both particles move happily in a rather regular fashion, combining a rotation

282

CO

0.2 0 -0.2 -0.4 -0.6 20

CO

40

60

80

100

80

100

80

100

1.3

0.2 0 -0.2 -0.4 -0.6 0

20

60

40

1.6 0 0.2 co -0.4 0.6 -0.8 20

40

60 t

Figure 7. The angular velocity w as function of the time for the shear rates 7 1.0,1.3 and 1.6. The dashed horizontal line indicates the reference value —7/2.

with an oscillation in radial direction. For the shear rates 0.3 and 0.5 (middle and lower graphs of Fig.4), rather irregular orbits are observed. The question whether this type of motion is chaotic will be addressed later. At the shear rates 1.0, 1.3 and 1.6, cf. Fig.5, rather different types of orbits occur. For the shear rate 1.0, the particles spend a consider-

283 0.5

0.2

0.1 0.05

0.02

0.01

reduced angular velocity black, from angular momentum gray} from gyration tensor

o.i

0.2

0.5 shear rate

1.0

Figure 8. The reduced rotational velocity, as inferred from the angular moentum (black) and from the components of the gyration tensor (gray) as functions of the shear rate. The horizontal dashed line marks the limiting small shear rate value 0.5, the inclined dashed line shown for high shear rates corresponds to a power law exponent —6.

able time in a "wagging" mode close to the #-axis where the effect of the shear flow is hardly felt. A rotational occurs just occasionally. Thus the rotational velocity inferred from the angular momentum is considerably smaller than UJQ based on the components of the gyration tensor. At the shear rate 1.3, the orbit is considerably stretched in the flow direction, both wagging and tumbling is observed. At higher shear rates this tendency for stretching increases and the time spent in the wagging state decreases. For 7 > 1.6 practically no wagging occurrs anymore and one has a stretching-contracting tumbling motion. The wagging mode seen here is more complex than that one described for "tumbling nematic liquid crystals" [13] since it is coupled with radial oscillations. The different types of rotational behavior can also be seen in plots of the angular velocity versus the time as displayed in Figs.6, 7 for several shear rates. The dashed line indicates the reference value —7/2. A mere inspection of the curves reveals that the magnitude of the time average of the angular velocity is smaller than 7/2, for shear rates larger than 0.1. For shear rates between about 0.3 and 1.2, the dependence of

284

the shear induced angular velocity OJ on the shear rate is rather eratic and a sensitive dependence of actual orbits on the initial conditions is observed. These are indicators for a chaotic behavior which is looked into in more detail. For specific values of the shear rate simple and also rather complex periodic orbits are found. To demonstrate the sensitive dependence of the rotational behavior on the shear rate, the averages as shown in Fig.3, (where the shear rate and the reduced angular velocity varied between 10~2 and 10 and between about 1 and 10~~6,) are now presented in Fig.8 with a finer resolution where these variables are between about 10"1 and 2, and between about 1 and 10~2. The data needed for the averages were extracted over a runtime of about 500 time units, for each shear rate. The black and gray dots stand for the reduced angular velocity inferred from the angular momentum and from the gyration tensor, respectively. In both cases, ratios of averages are displayed, i.e. the expressions (5) and (7) are used (and divided by - 7 ) . For some shear rates and some initial conditions, the average angular velocity (5) may even have the wrong sign, compared with the value —7/2 observed for small shear rates. For this reason the logarithm of the absolute value of Qj(-7) (black dots) is shown in the graph. The quantity U>G/(—7) (gray dots) is always non-negative, by definition, cf. (7). Although its dependence on the shear rate is less irregular than that of o)/(—7), significant deviations from a smooth curve behavior is also seen. Comparison of the black and gray points shows again that the relation (6) is violated at shear rates between about 0.3 and 1.2. The dashed horizontal line in Fig.8 marks the theoretical value 0.5, expected to apply for small shear rates. The dashed line seen for high shear rates has the power law exponent - 6 , i.e. the reduced angular velocity decreases inversely proportional to the 6th powerer of the shear rate for 7I.3.

4.

Shear-Induced Chaotic Behavior and Periodic Orbits

The observations mentioned so far, in particular the sensitive dependence of the shear induced angular velocity on the external controll parameter "shear rate" and the violation of the relation (6) are strong, but only qualitative indicators of a chaotic behavior. For a quantitative analysis, one should determine the spectrum of Lyapunov exponents A* and check whether and when at least one of them is positive. The four Lyapunov exponents were determined by using the standard Benettin algorithm [18] running the simulation for 50 x 106 timesteps where each timestep is A = 0.001 (thus a total time elapsed time of

285

Figure 9. The Lyapunov exponents as a function of shear rate 7 in the regions where they appear to be independent of initial conditions. For small 7 the four exponents are all equal to zero. Prom about 0.35 to 0.71 there are two positive exponents and two negative exponents. From about 1.17 there is one positive exponent and three negative exponents. At 1.24 the smallest exponent is less than —5 and continues to become more negative. At 1.5 this exponent is around —30.

50000) discarding the results, then re-running for the same elapsed time to calculate exponents. Various time step sizes were used but the most reliable results were those obtained with A = 0.001. The exponents were calculated twice for shear rates between 0 and 1.5, first in increasing steps of 0.005 beginning at 7 = 0, and second in decreasing steps beginning at 7 = 1.5. The first numerical observation was that for some ranges of shear rates different results were obtained from these two series of simulations, indicating that the calculated Lyapunov exponents depend upon the initial condition. If the largest Lyapunov exponent is positive this suggests that there are at least two disjoint sets of initial conditions that separate the available space into distinct ergodic components. The

286 200

Figure 10.

The periods of the stable closed orbits between 7 = 1.1 and 1.5.

results for the Lyapunov exponents in the ranges where they appear to be independent of initial conditions are given in Fig. (9). A number of points deserve special mention. 1) For shear rates between 0 and 0.8 the results appear robust and reproducible. There is a steady change in the largest exponent with shear rate with Xmax = 0 from 7 = 0 to 0.18, and then returning to 0 just before 0.8. Between 0.18 and 0.35 the largest Lyapunov exponent is very close to 0. There are small windows near 7 = 0.34,0.42 and 0.52 where the largest exponent appears to be zero, but there is no obvious change in the trajectory at these places. 2) There is a regular closed stable trajectory near a shear rate 7 = 0.8 which must have no positive Lyapunov exponents. For this window of periodic orbits the period varies smoothly with 7 and the results are summarised in the Table(l). The periodic orbit is similar to that for 7 = 1.0 shown in Fig. (5) but with many less loops associated with wagging s.

287

0.5

0.2

0.1

•

V

0.05

0.02

0.01

reduced angular velocity black} from angular momentum gray, from gyration tensor

0.1

0.2

0.5 shear rate

**• \ 1.0

V

2.0

Figure 11. The reduced rotational velocity, as inferred from the angular moentum (black) and from the components of the gyration tensor (gray) as functions of the shear rate for calculations with a Nose-Hoover thermostat. The horizontal dashed line marks the limiting small shear rate value 0.5, the inclined dashed line shown for high shear rates corresponds to a power law exponent —6.

Table 1. The periods of stable trajectories in the range 0.7825 < 7 < 0.8135 as a function of shear rate 7 7 period

0.7825 0.7850 0.7900 41.250 40.545 40.145

0.7950 0.8000 0.8050 0.8100 0.8125 39.725 39.350 38.975 38.605 38.550

0.8135 38.505

3) Between 0.82 and 1.075 the results appear to depend on the initial condition, and different initial conditions lead to different stabilities. Beginning at 0.82 and adding increments gives trajectories for which all the exponents are zero, whereas beginning at 1.075 and subtracting increments give trajectories which can have one positive exponent. Here we also find a periodic orbits around 7 = 1.0. This periodic orbit has a long period (compared with those near 7 = 0.8, but the orbit at 0.995 remains on the left-hand side of the y-axis (thus it is termed 'onesided'). This orbit is also observed for a shear of 1, so there appear to be at least four different types of trajectories for a shear of 1, three periodic (including the one-sided orbit and its symmetric pair) and one

288 ~~~7 period

0.9950 42.270

1.0000 117.225

1.0050 114.820

Table 2. Periods of stable trajectories near 7 = 1 as a function of shear rate 7

weakly chaotic. This suggests that the phase separates into at least four independent ergodic components. 4) There is a stable orbit of period 186.5 at a shear of 1.08, similar in shape to the orbit at 1.0 but with more loops associated with wagging and radial oscillations on each side. 5) From 1.085 onwards the results appear to be largely independent of the initial conditions. Initially, there is a small region with one positive exponent but after 1.13 the orbits are almost all stable. There is a change in orbit structure with fewer and fewer wagging loops as the shear rate is increased, and the possibility of small unstable regions where the number of loops changes, but most often there is a simple steady change in the orbit shape with shear rate. The graphs of orbits at 1.2 and 1.3 give an indication of the change in the number of loops. 6) For a number of shear rates the trajectory is not symmetric with respect to rotation through n (in the range 1.19 to 1.235, although not everywhere as 1.22 and 1.225 are exceptions). This implies that the rotated orbit must also be a solution to the dynamics, so once again the steady state is not unique. This occurs for 1.215, 1.23 and 1.235 (and probably also for other orbits between 1.19 and 1.235). We emphasise that this is only a preliminary study of the stability properties of this system and questions related to the observed existence of different ergodic components in the phase space at particular values of the shear rate has not been characterized. One may wonder whether the complex dynamics observed at intermediate shear rates is linked with the use of the Gaussian thermostat. Extensive computations with other thermostats imply that the answer to this question is "no", although actual orbits are, of course, affected by the particular thermostats. Some brief remarks on this point are made next.

5.

Other Thermostats

Instead of the Gaussian thermostat, which keeps the total kinetic energy of the system constant, the Nose-Hoover thermostat [19] is also frequently used in NEMD computer simulations [10], [14-17]. For this thermostat, the kinetic energy fluctuates around its average value determined by the prescribed temperature T, and in equilibrium, its dis-

289 0.5 0.2

0.1 0.05

0.02

reduced angular velocity black, from angular momentum gray, from gyration tensor

0.01

0.1

0.2

0.5 shear rate

1.0

2.0

Figure 12. The reduced rotational velocity, as inferred from the angular momentum (black) and from the components of the gyration tensor (gray) as functions of the shear rate for calculations with a Gaussian and an additional configurational thermostat. The horizontal dashed line marks the limiting small shear rate value 0.5, the inclined dashed line shown for high shear rates corresponds to a power law exponent —0.75.

tribution is canonical. For the present problem, equation (16) for the pseudo-friction coefficient is replaced by the differential equation (21) Here D is recalled as the dimension of the system. The "relaxation frequency" v is an additional system parameter which has to be specified. For large values of v a behavior close to that of the Gaussian thermostat is approached. We computed the shear rate dependence of the angular velocity for values of v ranging from 100 to 1. The reduced angular velocity is diplayed in Fig. 11, as function of the shear rate, similar to Fig.8. The data shown are for the relaxation frequency v — 10. The overall behavior is rather similar to that obtained with the Gaussian thermostat. The angular velocity at high shear rates decreases very strongly with the same power law exponent —6. This is different for v = 101/2 « 3.162 and v — 1 where the power law exponents are - 4 and —2.75, repectively. Furthermore, the intermediate regime where an irregular behavior is seen is shifted towards smaller shear rates for v <

290

10. In addition to the Nose-Hoover thermostat which controls the second power of the momentum, a "p4-thermostat" was used which controls the forth power of the momentum, as suggested by [20]. More specifically, a term -^4^(p • p/T)p is added on the r.h.s. of (14). The dynamic variable f obeys the equation ^ = 1/4 ((P • P/T) 2 - (2 + D)p • p / T ) .

(22)

It involves an additional relaxation frequency v± which has to be specified. The irregular behavior found at intermediate shear rates is not suppressed by this additional thermostat. At high shear rates the decrease of the reduced angular velocity with increasing shear rate becomes even somewhat faster, e.g. onefindsa power law exponent —3.5, -4.0 for v = 1.0 and v± = 0.316, 1.0 whereas this exponent is -2.75 for v = 1.0 and ^4 = 0.0. For a system with few degrees of freedom, it seems advisable also to use a configurational thermostat which ensures that the configurational part of the phase space density has the canonical form in equilibrium, i.e. that it is proportional to exp(—$/T). Extending the ideas of Nose and Hoover, a whole class of appropiately modified equations of motions was proposed in [21]. An appealing special case involves the (instantaneous) "configurational" temperature Tconf defined by Tconf = F • F/A [12]. The square of the Einstein frequency UE is related to the Laplacian A applied on the potential by D UJ\ = A$, with the dimension D = 2 or 3. The equation of change (13) for the position vector is replaced by -

=

p + v(r)+TaaF.

(23)

The additional dynamic variable a obeys the differential equation ^

(24)

The "relaxation time coefficient" ra is an additional system parameter which has to be chosen appropiately. Notice that one has F-F/T—A<& = A$(Tconf/T - 1), i.e. a does not change when the configurational temperature matches the prescribed temperature T. It must be stressed that the modified equations of motion still are invariant under time reversal, though they are not of Hamiltonian type. The configurational thermostat has been combined with the Nose-Hoover thermostat, all combinations of v — 1,10,100 with ra = 0.1,0.01 and 0.001 were tested. For increasing values of r a , the decrease of the reduced angular velocity with increasing shear rates in the high shear rate regime gets considerably weaker because the configurational thermostat prevents the extreme

291 stretching of the orbits. Values of ra larger than about 0.1 lead to unphysical results for small shear rates. There is always an intermediate range of shear rates where the angular velocity shows an irregular dependence on the shear rate. This intermediate regime is shifted towards smaller shear rates for smaller values of v, as aleady noticed for the pure Nose-Hoover thermostat. The combination of the Gaussian with the configurational thermostat seems to be preferable since it involves just one additional dynamic variable and one additional system parameter, viz. ra. As an example, the reduced angular velocities, both calculated from the angular momentum and the gyration tensor, for the case ra = 0.003, are displayed in Fig. 12, similar to Fig.8 which corresponds to ra = 0. For small shear rates the magnitude of the shear-induced angular velocity is somewhat larger than the expected value 0.67. The rather weak decrease of the angular velocity at high shear rates has the power law exponent -0.75. This is the value observed in the NEMD simulations of polymer chain molecules [1]. The shear induced velocity as presented here is computed from the difference of the angular momentum of two dumbbells. The corresponding information can be inferred from the motion of a single dumbbell when one applies, in addition to a Gaussian or a Nose-Hoover thermostat, a "twirler" [23] which, in equilibrium, essentially randomizes the direction of the angular velocity. The regular behavior at small and large shear rates and the complex dynamics at intermediate shear rates seems to be a genuine property of the sheared and thermostated system, irrespective of the particular time-reverible thermostat used.

6*

Concluding Remarks

The (two-dimensional) model for a relatively stiff molecule subjected to a simple shear flow, on the one hand, shows many features observed in NEMD simulations of finitely extendible nonlinear elastic (FENE) chain molecules. On the other hand, the dynamics found for the simple model is intriguingly complex and it deserved a careful study on its own. It seems appropriate also to analyse the system at higher temperatures. Furthermore, the model provides a convenient test bed for various thermostats; other and additional thermostats, e.g. based on deterministic scattering [22] should be tested. Obvious extensions of the present model may involve other potential functions of nonlinear elastic type such as $ = ( V 2 ) r 2 + (1/4) r 4 or $ = (1/4) (1 - r2)2 as well as FENE potentials. A treatment of a plane Poiseuille flow similar to the plane Couetteflowconsidered here is feasible. The influence of confining

292 walls on the polymer rotation and on the deformational changes deserves special attention for both flow geometries. Furthermore, the three dimensional case deserves an analysis similar to that one presented here. Due to the similarities with the tumbling nematic liquid crystals seen for the wagging mode in the twodimensional case, one may wonder whether kajaking, kajaking-tumbling and kajaking-wagging modes known to occur for liquid crystals [24, 25] can also be found for the rotating polymer molecule in three dimensions. It must be stressed, however, that the equations governing the alignment of liquid crystals subjected to a shear flow are of dissipative type in contradistinction to the time reversible equations used here. For the nematic liquid crystal, the chaotic motions are only found when all five components of the alignment tensor are taken into consideration [25] which corresponds to the three dimensional case for the polymer problem.

Acknowledgments This work has been conducted under the auspices of the Sonderforschungsbereich SFB 448 "Mesoskopisch strukturierte Verbundsysteme" of the Deutsche Forschungsgemeinschaft (DFG). Financial support is gratefully acknowledged. Part of this work was done while one of us (SH) stayed at the Institute for Theoretical Physics, Santa Barbara, California with support in part by the National Science Foundation under grant no. PHY99-07949. Furthermore, we thank Bill Hoover, Dennis Isbister, Rainer Klages and Martin Kroger for helpful discussions. We acknowledge the Max Planck Institut fiir Physik Komplexer Systeme in Dresden for providing a stimulating atmosphere for high level discussions during a workshop in August 2002.

References [1] C. Aust, S. Hess, and M. Kroger, Macromolecules, 32:8621, 2002. [2] P. Lindner, Physica A, 174:74, 1991; A. Link and J. Springer, Macromolecules, 26:464, 1993; E.C. Lee, M.J. Solomon, and S.J. Muller, Macromolecules, 30:7313, 1997. [3] R. Cerf, J. Chem. Phys., 68:479, 1969; E.R. Bazua and M.C. Williams, J. Polym. Sci. 12:825, 1974 [4] D.E. Smith, H.P. Babcock, and S. Chu, Science, 283:1724, 1999. [5] C. Pierleoni and J.P. Ryckaert, Macromolecules, 26:5097, 1995; S. Hess, C. Aust, and M. Kroger, Cah. Rheol. (Prance), 15:1, 1996. [6] C. Aust, Molecular Dynamics Investigations of Polymer Solutions Undergoing Shear Flow, Shaker, Aachen, 2000; C. Aust, Kroger, and S. Hess, Macromolecules, 32:5660, 1999.

293 [7] P. Debye, J. Chem. Phys., 14:636, 1946. [8] R. Edberg, G.P. Morriss and D.J. Evans, J. Chem. Phys., 86:4555, 1987; G.P. Morriss, P.J. Daivis and D.J. Evans, J. Chem. Phys., 94:7420, 1991. [9] R.B. Bird, C.F. Curtiss, R.C. Armstrong, and O. Hassager, Dynamics of Polymeric Liquids, Volume 2, Kinetic Theory, Wiley, New York 1987; G.K. Praenkel, J. Chem. Phys., 20:642, 1952. [10] D.J. Evans and G.P. Morriss, Statistical Mechanics of Nonequilibrium Liquids, Academic Press, London, 1990. [11] J.J. Hermans and R. Ullman, Physica, 18:951, 1952; H. Yamakawa, Modern theory of polymer solutions, Harper & Row, New York, 1971. [12] H.H. Rugh, Phys. Rev. Lett, 78:772, 1997; O.G. Jeps, G. Ayton, and D.J. Evans, Phys. Rev. B, 62:4757, 2000. [13] G. Rienacker and S. Hess, Physica A, 267:294, 1999. [14] M.P. Allen and D.J. Tildesley, Computer Simulation of Liquids, Clarendon, Oxford, 1987. [15] G. Ciccotti, D. Prenkel and I.R. McDonald, eds. Simulations of Liquids and Solids, North Holland, Amsterdam, 1987. [16] G. Ciccotti and W.G. Hoover, eds. Molecular Dynaamics Simulations of Statistical Mechanical Systems, North Holland, Amsterdam, 1986. [17] W.G. Hoover, Molecular Dynamics, Springer, Berlin, 1986; Computational Statistical Mechanics, Elsevier, Amsterdam, 1991. [18] G. Benettin, L. Galgani and J.-M. Strelcyn, Phys. Rev. A, 14:2338, 1976. [19] S. Nose, J. Chem. Phys., 81:551, 1984; W.G. Hoover, Phys. Rev. A, 31:4757, 1985. [20] W.G. Hoover, Phys. Rev. A, 40:2814, 1989; B.L. Holian, W.G. Hoover and H. Posch, Phys. Rev. Lett, 59:10, 1987; K. Rateitschak, R. Klages, W.G. Hoover, J. Stat Phys., 101:61, 2000. [21] A. Bulgac and D. Kusnezov, Phys. Rev. A, 42:5045, 1990. [22] R. Klages, K. Rateitschak, G. Nicolis, Phys. Rev. Lett, 84:4628, 2000; C. Wagner, R. Klages, G. Nicolis, Phys. Rev. E, 60:1401, 1999; K. Rateitschak, R. Klages, Phys. Rev. E, 65:036209-1, 2002. [23] S. Hess, Z. Naturforsch., 58 a:377, 2003. [24] R.G. Larson and H.C. Ottinger, Macromol, 24:6270, 1991. [25] G. Rienacker, M. Kroger, and S. Hess, Physica A, 315:537, 2002; Phys. Rev. E, 66:040702 (R), 2002.

REGULAR AND CHAOTIC RHEOLOGICAL BEHAVIOR OF TUMBLING POLYMERIC LIQUID CRYSTALS Siegfried Hess Institut fur Theoretische Physik, Technische Universitat Berlin PN 7-1, Hardenbergstr. 36, D - 10623 Berlin, Germany [email protected]

Martin Kroger Polymer Physics, Materials Science, ETH Zentrum, ML H18, CH-8092 Zurich, Switzerland

Abstract

The rheological properties of nematic liquid crystalline polymers are strongly affected by the dynamic behavior of the molecular alignment. Starting from a closed nonlinear inhomogeneous relaxation equation for the five components of the alignment tensor which, in turn, can be inferred from a generalized Fokker-Planck equation, it has recently been demonstrated (G. Rienacker, M. Kroger, and S. Hess, Phys. Rev. E 66, 040702(R) (2002); Physica A 315, 537 (2002)) that the rather complex orientation behavior of tumbling nematics can even be chaotic in a certain range of the relevant control variables, viz. the shear rate and tumbling parameter. Here the rheological consequences, in particular the shear stress and the normal stress differences, as well as the underlying dynamics of the alignment tensor are computed and discussed. For selected state points, long-time averages are evaluated both for imposed constant shear rate and constant shear stress. Orientational and rheological properties are presented as function of the shear rate. The transitions between different dynamic states are detected and discussed. Representative examples of alignment orbits and rheological phase portraits give insight into the dynamic behavior.

Introduction A time dependent, periodic response of a physical system to a time independent, steady stimulus is an intriguing phenomenon. Such a behav295 P. Pasini et al. (eds.), Computer Simulations of Liquid Crystals and Polymers, 295-333. © 2005 Kluwer Academic Publishers. Printed in the Netherlands.

296 ior occurs in 'tumbling' nematic liquid crystals subjected to a stationary shear flow. It has been observed in low molecular weight thermotropic liquid crystals [1, 2], in polymeric liquid crystals [2, 3], as well as in lyotropic liquid crystals and micellar solutions (living polymers) [4, 5]. Even more dramatic is a chaotic behavior found in theoretical studies [6-8] of the orientational dynamics. It is the purpose of this article to present results for the rheological behavior of such a liquid crystal, for a few state points, as function of the shear rate, as it follows from the dynamics of the alignment analyzed previously in [6, 8]. At the same time, the theoretical approach is outlined here and the applications to flow birefringence, flow alignment and to the computation of the viscous properties are discussed. Point of departure are coupled equations for the second rank alignment tensor specifying the orientation of nonspherical particles and for the friction pressure or stress tensor. These equations have been derived from irreversible thermodynamics quite some time ago [9], for extensions to spatially inhomogeneous systems and for some further extensions see [10] and [11]. The non-linear relaxation equation for the alignment tensor has also be derived from a generalized FokkerPlanck equation for the orientational distribution function [12-14]. Indications for rheo-chaos, i.e., for chaotic rheological behavior of fluids have recently been found in experiments [15], and theoretical studies were performed [16], where the underlying dynamics, however, differs from the case studied here. It is hoped that the computations presented will help to detect rheo-chaos and identify its properties for tumbling nematic polymeric and lyotropic liquid crystals. For simplicity, liquid crystalline monodomains are treated where the velocity gradient can be considered as an external variable. The full polydomain problem where the hydrodynamic equations for the flow field have to be solved simultaneously with those for the alignment tensor is possible, but computationally more demanding; for first studies of this kind which indicate also the existence of spatial chaos see [17]. Nevertheless, a detailed analysis of the spatially homogeneous case is needed to find the range of parameters where a regular, stationary or periodic and where a more complex and even chaotic temporal behavior occurs. Tumbling of isolated ellipsoidal particles in a shear flow was described by Jeffrey over 80 years ago [18]. Within the frame work of the EricksenLeslie theory [19], the director specifying the direction of the average orientation in a nematic liquid crystal shows a similar dynamic behavior or goes to a flow aligned state when the 'tumbling parameter' is below or above 1, respectively [2]. The Ericksen-Leslie theory is a limiting case of the more general tensorial theory provided that the alignment tensor remains uniaxial, as in equilibrium, and that the degree of order is not

297 affected by the flow. For intermediate and large shear rates, however, the biaxiality of the alignment and the change of the order parameter do play a role and the dynamic behavior of tumbling nematics is by far richer than what the Ericksen-Leslie theory can describe. Numerous theoretical studies of the behavior of nematics in a shear flow, based on a tensorial description and also on the dynamics of the orientational distribution function have been performed, some key references and reviews are [2, 9, 20, 23, 27]. The rheological properties in the chaotic regime have not yet been presented previously. This article is organized as follows. In section 1.1, the basic theory is outlined. The nonlinear relaxation equation for the second rank alignment tensor is formulated and the constitutive relation for the friction pressure or stress tensor is stated. These equations involve characteristic relaxation time coefficients pertaining to the relaxation of the alignment and to its coupling with the stress tensor or with the velocity gradient. At the same time, the stress contains a contribution associated with the deviation of the alignment from its equilibrium value. Scaled variables are introduced for all physical quantities of interest. The essential model parameters are firstly, a temperature or density variable which determines, for an equilibrium situation, whether the system is in an isotropic or nematic state. Secondly, the value of the tumbling parameter decides whether a stationary or a time dependent response results from a small applied steady shear rate. This quantity, of course, is also an essential control parameter. With the help of 5 basis tensors, the alignment tensor equation is rewritten in terms of its 5 components. In the case of a plane Couette flow, 3 of these components suffice when the alignment has a symmetry in accord with that of the velocity gradient tensor. Spontaneous symmetry breaking, however, can occur for certain ranges of parameters. Thus the full set of components is taken into consideration. Similarly, the stress tensor is characterized by 5 components, the symmetry adapted ones are linked with the shear stress and the first and second normal stress differences. Stress components violating the Couette symmetry can also be nonzero. A brief survey of the nomenclature used to describe the different types of stationary states and of symmetry adapted time dependent solutions, like 'tumbling' and ' wagging ', as well as of symmetry breaking ones, like ' tumbling' and 'kayaking wagging' is given. Section 1.2 is devoted to a presentation of the rheological properties inferred from time averages of the dynamic equations for selected state points. In addition to the imposed shear rate, also the case of an imposed shear stress is considered. Graphs are shown for the shear stress and for the viscosity as functions of the shear rate for the isotropic phase, for a shear flow induced transition into the

298

nematic phase and for a flow aligned nematic. The main emphasis is on a tumbling nematic where, with increasing shear rates, transitions between different dynamic states occur, e.g. from kayaking tumbling, to tumbling, to kayaking wagging, to kayaking tumbling, to a complex and chaotic behavior, and to a flow aligned state. Graphs of the shear stress and of normal stress differences, as well as of properties of the alignment tensor give evidence of these transitions. The nature of the dynamic states, however, is more clearly revealed by 'orbits' or 'phase portraits' where components of the alignment tensor are plotted against each other, as well as the first normal stress difference is displayed vs. the shear stress. Representative examples of these curves are displayed and discussed in section 1.3.

-0.5

0.5 1 alignment

1.5

Figure 1. The Landau de Gennes potential as function of the alignment order parameter for tf = 2.0 (top), 1.125, 1.0, 0.5, and 0.0 (buttom).

1. 1.1

The model equations Relaxation equation for the alignment tensor

The alignment of effectively uniaxial particles with a molecular unit vector u is characterized by an orientational distribution function /(u, t) [12, 28-30]. The appropriate order parameter for a nematic is the 2nd

299

0.5 temperature Figure 2.

The order parameter as function of the temperature or concentration

variable $

rank alignment tensor

= ff(u,t) Jfuud\

(1)

which is the anisotropic second moment characterizing the distribution. The symbol 5? indicates the symmetric traceless part of a tensor x. Frequently, the alignment tensor is also referred to as 'Q-tensor', sometimes as 'S-tensor'. The symmetric traceless part of the dielectric tensor which gives rise to birefringence is proportional to the alignment tensor. Thus the shear flow induced modifications of the alignment can be detected optically [31]. For the special case of uniaxial symmetry (uniaxial phase) the alignment tensor a can be parameterized by a scalar order parameter a and the director n, i.e., a = a(3/2)1//2 rm, such that a2 — a : a, and -\/5/2 < a = (3/2)l/2 a : nn < y/E. The parameter a is therefore proportional to the Maier-Saupe order parameter S2 = ffiiu-n)) = a/y/b, where P2 denotes the second Legendre polynomial. We start revisiting the underlying original nonlinear relaxation equation for the alignment tensor, based on irreversible thermodynamics. The equation involves characteristic phenomenological coefficients viz. the relaxation time coefficients r a > 0 and r ap , a dimensionless coeffi-

300

-1

-0.5

0.5 temperature

Figure 3. The tumbling parameter as function of the temperature or concentration variable d for AK = 1.45,1.25 and K = 0 (upper and lower thin curves) as well as AK = 1.25,1.05 and K = 0.4 (upper and lower thick curves). The dashed horizontal line marks the limit between the flow aligned (Aeql) and the tumbling (Aeq < 1) states.

cient (shape factor) K, the pseudo-critical temperature T*, the nematicisotropic transition temperature TK with Tk > T*, and the positive of a Landau-de Gennes poparameters Ao,B,C (with C<2B2/(9AQ)) tential $ ( a ) = (1/2)A(T)a

: a - (l/3)\/6B(a

-a) :a + (C/4)(a

: a)2

with A(T) = A0(l - T*/T). The value of Ao depends on the proportionality coefficient chosen between a and (uu). The choice made in (1) implies Ao = 1, cf. [9]. The coefficients, on the one hand, are linked with measurable quantities and, on the other hand, can be related to molecular quantities within the framework of a mesoscopic theory [1214]. The equation of change for alignment tensor in the presence of a flowfieldv reads [9, 11]:

^ - 2 ^ r S - 2 K r . a + ra-1$(a) = - V 2 ^ r ,

(2)

where 4£(o) =E u*& I ud =z A CL — v 6 B a • a ~\~ C a CL : a is symmetric traceless. The symbols V and u; denote the symmetric traceless part of the velocity gradient tensor (strain rate tensor) V = Vv,

(3)

301 and the vorticity w = (V x u)/2, respectively. In the case of a simple shear flow v = jyex in ^-direction, gradient in y-direction, and vorticity in ^-direction, to be considered throughout the following analysis, these quantities simplify to T = /yexey and u> = — (1/2)7 e*. The unit vectors parallel to the coordinate axes are denoted by ex,ey,ez. Equation (2) has been extended to inhomogeneous systems by changing the time derivative from a partial to a substantial one, and by adding a term oc Aa characterizing inhomogeneous systems [10], see also [32] for related works. For lyotropic liquid crystals the concentration c of nonspherical particles in a solvent rather than the temperature determines the phase transition, i.e., in this case one has A oc (1 - c/c*), where c* is pseudo critical concentration [13]. In Ref. [33] similar equations have been used to study the flow-alignment and rheology of semidilute polymer solutions, where c* denotes the overlap concentration. In Ref. [26] some mathematically rigorous results were obtained using geometric techniques concerning a Landau-de Gennes model of a liquid crystal in a uniform flow. An analysis based on the dynamics of the orientational distribution function is presented in [24]. In Refs. [8, 11, 25] the symbol a was used instead of K. The special values 0 and ±1 for the coefficient n in (2) correspond to corotational and codeformational time derivatives. From the solution of the generalized Fokker-Planck equation one finds, for long particles, « « 3/7 « 0.4.

1*2

Constitutive relation for the pressure tensor

The alignment is not only influenced by the flow but the flow properties as characterized by the friction pressure tensor are affected by the alignment. The full pressure tensor p consists of a hydrostatic pressure p, an antisymmetric part, and the symmetric traceless part p referred to as friction pressure tensor [9]. The latter splits into a 'isotropic7 contribution as already present in fluids composed of spherical particles or in fluids of non-spherical particles in an perfectly 'isotropic state' with zero alignment, and a part explicitly depending on the alignment tensor: (4) with [11] - 2K a • *(aj) .

(5)

In equilibrium one has $(a) = 0 and consequently p^J1 = 0. The occurrence of the same coupling coefficient r ap in (5) as in (2) is due to an

302

Onsager symmetry relation. For results on the rheological properties in the isotropic and in the nematic phases with stationary flow alignment, following from (2) and (5) see [9-11, 25] . Here, the main attention is focused on the rheology in the tumbling and in complex dynamic states.

1.3

Scaled variables: alignment tensor and relevant parameters

The equations (2) and (5) can be rewritten in scaled variables [911, 25] which are convenient for the theoretical analysis. At the same time, the essential parameters in the system of differential equations are identified and their physical meaning is discussed. Firstly, the alignment tensor is expressed in units of the value of the order parameter at the isotropic-nematic phase transition, 2# = (6) "* 3C occurring at the temperature TK > T*. With the temperature variable

9AC^ l - r y r 2 B2

{ }

1 - T*/TK

the temperature dependence of the uniaxial equilibrium alignment is aeq = 0 for *& > 9/8 (isotropic phase) and aeq/aK = \ (3 + V9-8tf),

for d < 9/8 (nematic phase).

(8)

Notice, that # = 1 corresponds to the equilibrium phase coexistence temperature. The values $ = 9/8 and d == 0 are the upper and lower limits of the metastable nematic and isotropic states, respectively. The quantity 5K = 1 — T*/TK which sets a scale for the relative difference of the temperature from the equilibrium phase transition is known from experiments to be of the order 0.1 to 0.001. On the other hand, it is related to the coefficients occurring in the potential function according to 6

(9)

The scaled Landau-de Gennes potential as function of the scaled order parameter and the scaled equilibrium alignment as function of the variable d are displayed in Figs. 1,2. The derivative 3> of the potential function in (2) can be written as 2 B2 **(a*),

$*(a*) =

tfa*-3\/6

$ re f = a K - — = aKSKAQ,

a* • a* + 2a* a* : a*.

a* =

(10)

303

Clearly, the variable # suffices to characterize the equilibrium behavior determined by * = 0. It should be mentioned that d can be also be interpreted as a density or concentration variable according to $ = (1 — c/c*)/(l - CK/C*) where c stands for the concentration in lyotropic liquid crystals. For the full nonequilibrium system, times and shear rates are made dimensionless with a convenient reference time. The relaxation time of the alignment in the isotropic phase is raAQl(l — T*/T)~x showing a pre-transitional increase. This relaxation time, at coexistence temperature Tk, is used as a reference time rref = TO(1 - T ' / T K ) " 1 V

= T^K1 V

= ra —

- raaK^{.

(11)

The shear rates are expressed in units of Tr~£. The scaled shear rate, being a product of the true shear rate and the relevant relaxation time, is also referred to as 'Deborah-number'. Instead of the ratio T^/ra^ the parameter AK = - ( 2 / 3 ) ^ ^ 0 ^ (12) is used. As was shown previously [10-12, 25], the coefficients ra and r ap are proportional to the Ericksen-Leslie [19] viscosity coefficients 71 and 72, respectively. The present theory applies both for the isotropic and for the nematic phase. The Ericksen-Leslie theory follows from the present approach when the alignment tensor is uniaxial and when the effect of the shear flow on the magnitude of the order parameter can be disregarded. Then it suffices to use a dynamic equation for the 'director' which is a unit vector parallel to the principal axis of the alignment tensor associated with its largest eigenvalue. This is a good approximation deep in the nematic phase and for small shear rates. For intermediate and large shear rates and, in particular, in the vicinity of the isotropic-nematic phase transition, the tensorial description is needed. The 'tumbling coefficient' A = -72/71 = A(aeq) is given by Aeq = AK

+ ~ «, Q>Qq

(13)

O

where aeq is recalled as the equilibrium value of the alignment in the nematic phase. Thus Aeq is equal to AK at the transition temperature provided that « = 0. In the limit of small shear rates 7, the tumbling parameter is related to the Jeffrey tumbling period [18], see also [25]. Within the Ericksen-Leslie description, theflowalignment angle x in the nematic phase is determined by cos(2X) = -7i/72 = 1/Aeq.

(14)

304

A stable flow alignment, at small shear rates, exists for |Aeq|l only. For |Aeq| < 1 tumbling and an even more complex time dependent behavior of the orientation occur. The quantity |Aeq| - 1 can change sign as function of the variable #, cf. Fig. 4. For |Aeq| < 1 and in the limit of small shear rates 7, the Jeffrey tumbling period [18] is related to the Ericksen-Leslie tumbling parameter Aeq by (15) for a full rotation of the director.

-1

0.5

-0.5 temperature

Figure 4- The tumbling parameter as function of the temperature or concentration variable $ for AK = 145,1.25 and n = 0 (upper and lower thin curves) as well as AK = 1.25,1.05 and K = 0.4 (upper and lower thick curves). The dashed horizontal line marks the limit between the flow aligned (Aeql) and the tumbling (Aeq < 1) states.

In the following, both AK and K are considered as model parameters. The first one is essential for the coupling between the alignment and the viscous flow. The second one influences the orientational behavior quantitatively but does not seem to affect it in a qualitative way. If one wants to correlate the present theory with the flow behavior of the alignment in the isotropic phase, on the one hand, and in the nematic phase, on the other hand, for small shear rates where the magnitude of

305 the order parameters is practically not altered, it suffices to study the case AK 7^ 0, K = 0 in order to match an experimental value of A by the expression (13). Mesoscopic theories [12, 14, 23] indicate that n ~ AKThus we also study the case K ^ 0, in particular K = 0.4.

1.4

Scaled variables: stress tensor

The stress tensor (5) associated with the alignment is related to the relevant quantities expressed in terms of scaled variables by

m

B

K

r e f

^r I

K

,

$ = #* + ^V6 OAK

statt3a* • **,

(16) where a* = CL/CLK and 3>* = $/$ r ef i n (16). The dimensionless shear stress S a l associated with the alignment is defined by

Then, Eq. (16) is equivalent to

\ ^

W

h

(18)

where Gai is a shear modulus associated with the alignment contribution to the stress tensor, and the product Ao a^ is essentially one parameter entering the theoretical expressions. The quantity 7?ref = GaiTai serves as a reference value for the viscosity. With the scaling used here, the dimensionless (first) Newtonian viscosity, in the isotropic phase, is ^New = * + 77fso w i t t l ^fso = ^iso/^ref- For high shear rates the dimensionless viscosity rf approaches the second Newtonian viscosity rj*so. The total deviatoric (symmetric traceless) part of the stress tensor, in units of G a i, is denoted by a. In terms of the quantities introduced here it is given by , cf. (4), a = -p = 2 77iS0r - £d = 2r,[soT + V2G al S a l .

(19)

In the following, we will denote quantities in reduced units by the same symbols as the original ones, unless ambiguities could arise.

1.5

Basis tensors and component notation

The symmetric traceless alignment tensor has five independent components. It can be expressed in a standard [34] ortho-normalized tensor

306

basis as follows: 4

a = £ aKTk,

T° = fi/2 ezez,

T1 = 0 7 2 (exex - eyey),

k-0

(20) where Tl with i = 1,..,5 are the basis tensors by which a is uniquely expressed. The orthogonality relation and the expression for the coefficients ax are given by T% : Tk = ^ and ax = a : Tfc. Using these basis tensors, from (2) we obtain a system of five ordinary differential equations 1 0 3 \/3 AK7 2 1 2

1 VSn'yao , 3

(21)

m

'

where $0

=

(^ ~ 3ao + 2a )ao + 3(ax + a2) — - ( a 3 + a 4 ) ,

$i

=

($ + 6ao + 2a )ai — - v 3 ( o 3 — a 4 ) , (22)

and a2 = ag + a2 + a2 + a2 + a4. The parameters ft, 7?, AK were introduced in the foregoing section. Prom their definition we see, that the order parameters satisfy |a*| < (15/4) 1 / 2 for i = 1,2,3,4, and - 5 1 / 2 / 2 < a0 < V2 The corresponding expansion with respect to the basis tensors and the component notation can be used for the other second rank irreducible tensors. Prom equations (17) and (16) one deduces expressions for the (dimensionless) shear stress axy, and the normal stress differences N\ —
307

with ( i-1

,

(24)

- k and k = 2«/(3AK).

1.6

Characteristic solutions for the orientational dynamics

Depending on the relevant model parameters 7, AK, «, #, the solutions of the dynamic equations (21,22) for the alignment tensor, in the case of an imposed stationary shear rate either approach a steady state or are time dependent. Furthermore, solutions which, for long times, maintain the symmetry of the plane Couette type velocity gradient and where the tensor components a% and a± vanish have to be distinguished from symmetry breaking solutions where these components are non-zero. The latter ones are also referred to as 'out-of-plane' solutions, in contradistinction to the 'in-plane' states where the 'main' director, i.e., the axis associated with the largest eigenvalue of the alignment tensor is in the flow plane determined by the direction of the flow and its gradient. The following types of orbits have been found [8]: • Symmetry adapted states with as = a^ = 0: A Aligning: stationary in-plane flow alignment with ao < 0. Furthermore, one may distinguish states A+ and A- pertaining to positive and negative values for the flow alignment angle %. For nematics composed of rod-like particles the first case occurs for small, the latter one for very large shear rates. T Tumbling: in-plane tumbling of the alignment tensor, the main director is in the flow plane and rotates about the vorticity axis. W Wagging: in-plane wagging or librational motion of the main director about the flow direction. L Log-rolling: stationary alignment with a\ = a2 = 0 and aoO. This out-of-plane solution is instable, in most cases. • Symmetry breaking states with as ^ 0, a\ ^ 0:

308

SB Stationary symmetry breaking states which occur in pairs of a3,a4 and — <23,— a^.

KT Kayaking-tumbling: the projection of the main director onto the flow plane describes a tumbling motion. KW Kayaking-wagging: a periodic orbit where the projection of the main director onto the flow plane describes a wagging motion. C Complex: complicated motion of the alignment tensor. This includes periodic orbits composed of sequences of KT and KW motion with multiple periodicity as well as aperiodic, erratic orbits. The largest Lyapunov exponent for the latter orbits is positive, i.e., these orbits are chaotic. For a given choice of parameters, in general, only a subset of these solutions are found by increasing the shear rate 7. The T and W states can be distinguished in a plot of a\ vs. a
2. 2.1

Rheological behavior Solutions for imposed shear rate and shear stress

In the following, results are presented for the rheological properties calculated from numerical solutions (with a Runge-Kutta method, e.g. via NDSolve of Mathematica) of the equations (21,22) governing the dynamics of the alignment and with the expressions (23) for the relevant components of the stress tensor. As initial conditions a state close to a random orientation is chosen with small (0.01 to 0.1), but non-zero values for the components of the alignment tensor. Time averages of

309

0.1

o.i 0.2

0.5 1.0 2.0 shear rate

0.2

5.0 10.0

0.5 1.0 2.0 shear rate

5.0 10

Figure 5. The shear stress (left) and viscosity (right) as functions of the shear rate for AK = 1.25 and K = 0, in the isotropic phase, at the temperature fi = 2. The large gray dots which practically form a continuous line stem from calculations with constant shear rates, with a maximum shear deformation 150. The smaller black dots are for calculations with imposed shear stress.

0.1 0.5 1.0 2.0 shear rate

5.0 10.0

0.2

0.5 1.0 2.0 shear rate

5.0 10

Figure 6. The shear stress (left) and viscosity as functions of the shear rate for AK = 1.25 and K = 0, in the isotropic phase, at the temperature d = 1.3, where a shear-induced transition into the nematic phase takes place. The large gray dots stem from calculations with constant shear rates, with a maximum shear deformation 150. The smaller black dots are for imposed shear stress.

the components of the alignment tensor and of the stress tensor are

310

I

0.1

0.1

0.05

0.05 0.1 0.2

0.5 1.0 2.0 shear rate

0.2

5.0 10.0

0.5 1.0 2.0 shear rate

5.0 1C

Figure 7. The shear stress (left) and viscosity as functions of the shear rate for AK = 1.25 and K = 0, in the nematic phase, at the temperature fl = 0.8, where the flow alignment angle, at small shear rates, is about 10 degrees. The large gray dots stem from calculations with constant shear rates, with a maximum shear deformation 150. The smaller black dots are for imposed shear stress.

evaluated for the relevant range of shear rates. The total run time trun is such that the total shear deformation 7 = jtrun 3> 1. The desired data are computed as the average of the values of the tensor components evaluated at time intervals St < 1. The initial transient behavior is disregarded, i.e., the data are extracted beginning at times between t r u n /3 and tTun/2. For those model parameters where a steady state solution exists, stationary solutions are approached rather quickly and the averaging procedure actually would not be needed for long times. In the case of a non-steady response of the alignment, however, the evaluation of time averages is essential in order to compare the theory with non-time-resolved rheological measurements where the time average is preformed automatically. So it is advantageous to use a method which works in the isotropic phase and for flow aligned as well as for tumbling nematics. Experiments are not only made for imposed shear rates but also for imposed shear stress. Calculations intended to give long-time averages for the latter case are performed by replacing the constant shear rate 7 in (21) by the dynamic shear rate g = g(t) which obeys the equation 9 = -(<7xy(t)

-

Here axy is the instantaneous shear

(Timp)/(

(25)

311 stress as given by (23) with (23), <7imp is the imposed constant shear stress, 77iS0 is the (second Newtonian) viscosity discussed above and rg is a relaxation time coefficient determining the speed of shear stress control. Since no information on physical values for rg is available, computations with imposed shear stress are presented here just for time averages with trun > rg. The value rg = 1, in reduced units, has been chosen. The transient and the dynamic behavior of the alignment and the stress tensor is only analyzed for calculations with an imposed shear rate.

2.2

Isotropic phase and flow aligned nematic

The shear stress and the viscosity in the isotropic phase are presented in Figs. 5, 6 as functions of the shear rate. Here and in the following, the parameters AK = 1.25, n = 0 and 7?iS0 = 0.1, in reduced units, have been chosen. In the graphs, the large gray dots which in some cases form a continuous line stem from calculations with constant shear rates. The smaller black dots are for calculations with imposed shear stress. Notice that a double logarithmic scale is used. Figure 5, pertaining to the reduced temperature $ = 2, shows the behavior typical for the isotropic phase, viz. a first Newtonian viscosity, r) = 1.1 in reduced units for small shear rates, a strong shear thinning for intermediate ones and the approach to the second Newtonian viscosity rjiso = 0.1 for high shear rates. Calculations with imposed shear rate and imposed shear stress give equivalent results. Still in the isotropic phase, but closer to the phase transition temperature, a shear induced transition to the nematic phase occurs, see Fig. 6. Based on the equations presented here, such a behavior has been predicted theoretically quite some time ago [20, 21]. This phenomenon has been observed in lyotropic liquid crystals, in particular with wormlike micelles [5] and in side-chain liquid-crystalline polymers [35]. In Fig. 6, results are presented for d = 1.3. For comparison, the highest temperature for which a metastable nematic phase exists, # = 9/8 = 1.125, is included. For imposed shear rates, shear stress and consequently the viscosity jump smaller values at the induced phase transition. For imposed shear stress there is a jump to higher shear rates.

2.3

Tumbling nematic

Survey. Next, results are presented for the time-averaged rheological behavior and the time averaged alignment in the nematic phase at a state point where no stable flow alignment is possible. In particular, the temperature d = 0 and the parameters AK = 1.25 and K = 0 are chosen. In Fig. 8 the shear stress and the viscosity are displayed as functions

312 of the shear rate, over two decades, in double logarithmic scales. The large gray dots stem from calculations with constant shear rates, with a maximum shear deformation 600. The smaller black dots are for imposed shear stress. Differences between the two sets of data are more pronounced than in the previous cases. The position of the effective step seen in the curves for shear rates 7 < 0.5 moves to the left for longer run times. Thus it is associated with a transient behavior which is not discussed in the following. Some peculiarities are more clearly visible in Fig. 9 where data from calculations with constant shear rate, between 1 and 5, are shown as linear plots. There are discontinuities at shear rates between 1.6 and 1.9 and an even stranger behavior between 3.6 and 4.1. As will be pointed out in the following, there are transitions between different types of periodic behavior in the first of these shear rate ranges, in the second one, an irregular chaotic behavior is found.

0.1

0.1 0.1 0.2

0.5 1.0 2.0 shear rate

5.0 10.0

0.1 0.2

0.5 1.0 2.0 shear rate

5.0 10

Figure 8. The shear stress (left) and viscosity as functions of the shear rate for AK = 125 and K = 0, in the nematic phase, at the temperature d = 0, where no stable flow alignment is possible. The large gray dots stem from calculations with constant shear rates, with a maximum shear deformation 600. The smaller black dots are for imposed shear stress.

Transitions between periodic states. In Fig. 10, the shear stress and the viscosity are shown for shear rates between 1.5 and 2. Now it is obvious that the single point in Fig. 8 or the few points in Fig. 9 in the vicinity of 7 « 1.75, which are 'off the curve', are not the computational version of a measuring error but are associated with transitions between

313

0.5 shear rate

Figure 9. The shear stress (left) and viscosity as functions of the shear rate for AK = 1.25 and K = 0, in the nematic phase, at the temperature d = 0, in a linear plot for shear rates between 1 and 5. All data points stem from calculations with constant shear rates, the maximum shear deformation is 750.

different types of dynamical behavior mentioned above which can only be detected when the shear rate is varied in smaller steps.

0.3 1.5

1.6

1.7 1.8 shear rate

1.9

1.5

1.6

1.7 1.8 shear rate

1.9

Figure 10. The shear stress (left) and viscosity as functions of the shear rate for AK = 1.25 and K = 0, in the nematic phase, at the temperature d = 0, in a linear plot for shear rates between 1.5 and 2. All data points stem from calculations with constant shear rates, the maximum shear deformation is 750.

314

1 7.5

1.6

7.5

1.7 1.8 1.9 shear rate

1.6 1.7 1.8 1.9 shear rate

0.9 ^ 0.8 | 0.7 .§> 0.6

"S 0.5 0.4 1.6

1.7 1.8 1.9 shear rate

1.5 1.6

1.7 1.8 1.9 shear rate

Figure 11. The first normal stress difference (upper left), the zz-stress difference (upper right), the average flow alignment angle (lower left), and the magnitude of the in-plane alignment (lower right) as functions of the shear rate for AK = 1.25 and n — 0, in the nematic phase, at the temperature d = 0, in a linear plot for shear rates between 1.5 and 2. All data points stem from calculations with constant shear rates, the maximum shear deformation is 750.

The first normal stress difference N\/2 = Si (upper left), the zzstress difference (upper right), i.e., the quantity So ~ <*zz - {pxx + ayy)/2, cf. (23), the average flow alignment angle (lower left), and the magnitude of the in-plane alignment, i.e., {a\ + a^)1!2 (lower right) are displayed as functions of the shear rate in Fig. 11. The first normal stress difference changes sign twice (positive to negative to positive) in the shear rate interval considered. The same applies to the 'average flow angle' x which is inferred from a ratio of time-averaged components di of the alignment tensor according to sin(2x) = #2/(^1 + a i ) ^ / 2 ) • The quantity So, referred to as zz-stress difference, is practically zero, see the upper right of Fig. 11. Thus one has N2 = -JVi/2, cf. (23). This is a peculiarity associated with the use of K = 0. For the shear rates where N\ is negative, the magnitude of the in-plane alignment is significantly larger compared with those where N\ is positive. Obviously, the main director has larger in-plane components in the first case. The

315

0.15 0.15

g 0.125 |

0.1

2* 0.075 |

0.05

0.05 0.025 0

1.5

1.6

1.7 1.8 shear rate

1.9

7.5

2

1.6

1.7 1.8 1.9 shear rate

Figure 12. The magnitude of the symmetry-breaking alignment (left) and stress components (right) as functions of the shear rate for AK = 125 and K = 0, in the nematic phase, at the temperature # = 0, in a linear plot for shear rates between 1.5 and 2. All data points stem from calculations with constant shear rates, the maximum shear deformation is 750.

0.7 0.6

/

\-##

.V.

^0.5

•

I 0.4

V

*

> 0.3

•

0.2 3.6

3.7

3.8 3.9 shear rate

3.6

3.7

3.8 3.9 shear rate

4.1

Figure 13. The shear stress (left) and viscosity as functions of the shear rate for AK = 1.25 and n = 0, in the nematic phase, at the temperature $ = 0, in a linear plot for shear rates between 3.5 and 4.1. All data points stem from calculations with constant shear rates, the maximum shear deformation is 1500.

data presented so far indicate two transitions, with increasing shear rate, between the different dynamic states as discussed above. A likely guess

316

o

0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 3.6

• • 5.7

7.5 5 2.5 Q

5.5 3.9 shear rate

/

•• •

.

4

4.1

0.0075 0.005 0.0025 0 -0.0025 •• -0.005 -0.0075

•

•

• .• . • #

3.6 3.7 3.8 3.9 4 shear rate

4.7

* • • • •

•

• 1

•

' •

-2.5 -5

•

5.6

3.7

5.5 3.9 shear rate

5.5 3.9 shear rate

Figure 14The first normal stress difference (upper left), the zz-stress difference (upper right), the average flow alignment angle (lower left), and the magnitude of the in-plane alignment (lower right) as functions of the shear rate for AK = 125 and K = 0, in the nematic phase, at the temperature $ = 0, in a linear plot for shear rates between 3.5 and 4.1. All data points stem from calculations with constant shear rates, the maximum shear deformation is 1500.

is KT —> T and T -> KT. However, plots of the magnitude of the time averaged symmetry-breaking components of the alignment tensor, viz. (a§ + a|)(l/2) and of the stress tensor, as displayed in Fig. 12, indicates that additional transitions occur around 7 « 1.8. In more detail, the magnitude of these components is small, but definitely non-zero for 7 < 1.68, it is zero for 1.69 < 7 < 1.8, it is large for 1.8 < 7 < 1.82, and again small, but non-zero for 1.83 < 7. A closer inspection of the orbits, examples are given later, indicates the sequence of dynamic transitions KT -> T -> KW -> KT in the range of shear rates considered right now. At higher shear rates, to be discussed in the following, the further transitions KT —> C -> A are observed. The first normal stress difference N\/2 = Ei (upper left), the zzstress difference (upper right), i.e., the quantity So ~ ozz — (axx + ayy)/2, cf. (23), the average flow alignment angle (lower left), and the magnitude of the in-plane alignment, i.e., (a\ + a\)1/2 (lower right) are

317

1.2

* • •%

0.6 1

<*) 0.5

m

.op 0.4

•

•

s 0.3

•

0.2

• • • •

1

O.I

•

3.6

3.7

• #

• •• •

| as

••

• • • •

3.8 3.9 shear rate

•

% #

•*

•

^0.6

• •

I 0.4

•

u

0

•

• ••• 4.1

•

•

#

0.2

0

3.6

3.7

3.8 3.9 shear rate

4.1

Figure 15. The magnitude of the symmetry-breaking alignment (left) and stress components (right) as functions of the shear rate for AK = 125 and K = 0, in the nematic phase, at the temperature # = 0, in a linear plot for shear rates between 3.5 and 4.1. All data points stem from calculations with constant shear rates, the maximum shear deformation is 1500.

displayed as functions of the shear rate in Fig. 11. The first normal stress difference changes sign twice (positive to negative to positive) in the shear rate interval considered. The same applies to the 'average flow angle' x which is inferred from a ratio of time-averaged components a,i of the alignment tensor according to sin(2x) = ^ / ( a f + aD^l/2) . The quantity Eo, referred to as zz-stiess difference, is practically zero, see the upper right of Fig. 11. Thus one has N2 = —iVi/2, cf. (23). This is a peculiarity associated with the use of n = 0. For the shear rates where N\ is negative, the magnitude of the in-plane alignment is significantly larger compared with those where JVi is positive. Obviously, the main director has larger in-plane components in the first case. The data presented so far indicate two transitions, with increasing shear rate, between the different dynamic states as discussed above. A likely guess is KT —>- T and T -> KT. However, plots of the magnitude of the time averaged symmetry-breaking components of the alignment tensor, viz. (a| + ^ ( 1 / 2 ) and of the stress tensor, as displayed in Fig. 12, indicates that additional transitions occur around 7 « 1.8. In more detail, the magnitude of these components is small, but definitely non-zero for 7 < 1.68, it is zero for 1.69 < 7 < 1.8, it is large for 1.8 < 7 < 1.82, and again small, but non-zero for 1.83 < 7. A closer inspection of the orbits, examples are given later, indicates the sequence of dynamic transitions KT —> T —> KW —>• KT in the range of shear rates considered right

318 now. At higher shear rates, to be discussed in the following, the further transitions KT -> C —> A are observed. Irregular and chaotic behavior. The shear stress and the viscosity, for shear rates between 3.5 and 4.1 are displayed in Fig. 13. A rather irregular behavior, i.e., a sensitive dependence on the value of the imposed shear rate is seen. This is indicative of a chaotic behavior. Indeed, the computation of Lyapunov exponents revealed that the largest one is positive for shear rates in intervals which have a are rather fractal character [8]. In Figures 14, 15, the first normal stress difference N\/2 = Si, the zz-stress difference So, the average flow alignment angle, the magnitude of the in-plane alignment, as well as the magnitude of the symmetrybreaking alignment and stress components are presented as functions of the shear rate, analogous to the data shown in Fig. 11 and Fig. 12. Again, a rather sensitive dependence on the shear rate is seen. Notice that the first normal stress difference and the average flow angle are positive for 7 < 4.0 and negative for 7 > 4.0. A smooth, curve-like dependence on the shear rate is seen for 7 > 4.1 where a flow-aligned state is reached. The zz-stress difference is practically zero, the inplane alignment increases strongly between the shear rate values 4.0 and 4.1. The patterns seen in the symmetry-breaking components of the alignment and stress tensors are strikingly similar although these quantities are not just proportional to each other.

2.4

Nonzero n

So far, results have been presented for the case K = 0. Since the derivation of the basic dynamic equation for the alignment tensor from a generalized Fokker-Planck equation implies « ^ 0 , one may wonder what are the effects associated with a non-zero value for n. The brief answer to this question is: some quantitative but practically no qualitative changes in the dynamic behavior when the parameters AK and n are chosen such that the equilibrium tumbling parameter Aeq has the same value, cf. (13) and Fig. 4. In particular, the irregular and chaotic behavior for shear rates in the vicinity of 4 is observed both for n = 0 and K — 0.4, although the actual values for the shear stress and the onset of the irregular behavior as well as the transition to the flow aligned state happen at slightly different values of the shear rate, see Fig. 16. There, the shear stress is plotted versus the shear rate for AK = 1.25 and a = 0, (large gray dots) and AK = 1.05 and n = 0.4, (smaller black dots), at the temperature 6 = 0. The tumbling parameter is Aeq = 0.833, in both cases.

319

4.4

Figure 16. The shear stress versus the shear rate for AK = 1.25 and a = 0, (large gray dots) and AK = 105 and n = 0.4, (smaller black dots), at the temperature d = 0. The tumbling parameter is Aeq = 0.833, in both cases. The curves have been recorded over a time t corresponding to the shear deformation 7^ = 1500.

Figure 17 shows the first and the zz normal stress differences (top row), the flow alignment angle and the in-plane alignment (bottom row) as functions of the shear rate for AK = 1.05 and K = 0.4, at the temperature # = 0. Comparison should be made with Fig. 14. The main qualitative difference is in the plots for the zz normal stress difference ~ So which is non-zero for K ^ 0 and rather similar to the in-plane alignment. As a consequence, the magnitude of the second normal stress difference N2 becomes smaller than 0.5 |iVi|.

320

0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 -1

8 aj £ 0.2 1 0.7 •

§ 0 § -0.7 8 -0.2

4.1 4.15 4.2 4.25 4.3 4.35 4.4 shear rate

•«••/ v

4.1 4.15 4.2 4.25 4.3 4.35 4.4 shear rate

1 0.9 0.8 ,0.7 0.6 0.5 4.7 4.75 4.2 4.25 4.3 4.35 4.4 shear rate

4.1 4.15 4.2 4.25 4.3 4.35 4.4 shear rate

Figure 17. The first and the zz normal stress differences (top row), the flow alignment angle and the in-plane alignment (bottom row) as functions of the shear rate for AK = 105 and K — 0.4, at the temperature $ = 0. The tumbling parameter is Aeq = 0.833. The total run time t corresponds to the shear deformation 7$ = 1500.

3-

3.1

Orbits General remarks, flow aligned state

Instead of analyzing the components of the alignment and stress tensors as functions of the time, it is more instructive to produce 'orbits' or 'phase portraits' where one component is plotted versus another one. The true alignment orbit is a curve in five-dimensional space, the twodimensional orbits to be shown are projections on various planes. Usually, the notion 'phase portrait' is applied to plots of a 'velocity' (time derivative) versus a 'coordinate'. Here a plot of a normal stress difference versus the shear stress is referred to as ' rheological phase portrait'. In a stationary state, the alignment orbits consist of a single point, relatively simple curves indicate the transient behavior during the approach towards the asymptotic state. Examples are shown in Fig. 18 and Fig. 19 for the approach to a high shear rate flow aligned state. More specifically, the components ai,ao,a3 of the alignment tensor vs. 0,2 and a± vs. a3 are plotted in Fig. 18, and Fig. 19 shows the first normal stress dif-

321 ference N\/2 = Ei versus the shear stress. Notice that Ni < 0, whereas one has Ni > 0 in the low shear rate flow aligned state as it occurs for higher temperatures. The model parameters are AK = 1.25, K = 0, 9 = 0, as before, and the shear rate is 7 = 4.2. The initial values for ao,. • • ,04 were -0.1,0.1,0.1,0.1, -0.1. The symmetry adapted components 00,01,02 of the alignment tensor as well as the shear stress and the normal stress differences approach single non-zero values whereas the symmetry-breaking components 03,04 tend to zero. The orbits look drastically different for periodic and for chaotic solutions to be presented next, in an order as encountered with increasing shear rates.

3•2

Kayaking-1 umbling

As already mentioned above, for AK = 1.25, n = 0, and 9 — 0, periodic solutions of KT type are found for shear rates below « 1.68 as well as between 1.83 and about 3.65. Typical orbits are shown in Fig. 20 and Fig. 21 for 7 = 1.6. Here limit cycles are approached after a relatively long and complex transient behavior. The instantaneous magnitude of the symmetry-breaking components 03 and 04 can be rather large. Their time averages, however, are relatively small since these quantities are approximately symmetric about zero. This is different for the KW solutions to be discussed later.

3-3

Tumbling

In-plane tumbling solutions occur for shear rates between approximately 1.7 and 1.8, for the same model parameters as above. Typical orbits, analogous to the previous ones, are shown in Fig. 22 and Fig. 23 for 7 = 1.75. Notice that the symmetry-breaking components of the alignment tensor vanish in the long time limit as it is typical for inplane periodic solutions. The component ao is also a periodic function oscillating about a constant, negative value. The rheological phase portrait Fig. 23 looks more complex than the corresponding plot of a\ vs. a2 for the alignment tensor, cf. the upper left diagram of Fig. 22. The instantaneous values of the first normal stress difference can be positive and rather large despite the fact that their time average yields a negative value. Also negative values occur for the shear stress, during a small fraction of the periodic orbit. The time average of the shear stress is positive, and thus in accord with the second law.

322

-0.1

1.4 1.2 / ' 1 (

-0.2

\

6)

-0.3

V

al 0.8

\ \

-0.4

\

aO

-0.5

0.6

y

0.4 0.2

-0.2

0

//

(Q) -0.2

0.2 0.4 0.6 0.8 a2

/

J 0

0.2 0.4 0.6 0.8 a2

0

0.08

-0.02 \

0.06

-0.04

\ a4

\

a3 0.04

\

\\

\

-0.06 \

0.02

-0.02

^—\

-0.7 -0.8

0.1

0

-0.6

J_ -0.2

J 0 0.2 0.4 0.6 0.8 a2

-0.08

-0.1 -0.02 0 0.02 0.04 0.06 0.08 0.1 a3

Figure 18. Orbits showing the components ai,ao, CLZ of the alignment tensor vs. a
3A

Kayaking-wagging

For the model parameters under consideration, the KW type of dynamic behavior occurs in a narrow range of shear rates slightly above 1.8. In Fig. 24 the components 01,00,03 of the alignment tensor vs. 02 and 04 vs. 03 are plotted for AK = 1.25 and K = 0 at the temperature d = 0. The shear rate is 7 = 1.82. Notice that the asymptotic orbit of the symmetry-breaking components 03,04 does not enclose the point

323

Nl

-0.5

-1

0.6

0.8 shear stress

1.2

Figure 19. Phase portrait showing the first normal stress difference N\ vs. the shear stress for AK = 125 and K = 0, in the nematic phase, at the temperature d = 0. The shear rate is 7 = 4.2. The curves have been recorded over a time t corresponding to the shear deformation 7^ = 4000.

a3,a4 = 0,0. This is a feature characteristic for the KW mode. It implies that the time average of these components is distinctively different from zero. Furthermore, the asymptotic state reached in the KW case is not unique. Also the solutions with -as, —04 exist and will be attained from different initial conditions. The corresponding rheological phase portrait is shown in Fig. 25.

3.5

Chaotic behavior

A stringent proof for a true chaotic behavior does not follow from an inspection of the time dependence of the tensor components or of the phase portraits. However, it is instructive to compare the previous

324

al

aO

-0.5 -0.5

1

0.5

a4

0

-0.5 -1 -1

-0.5

0 a3

0.5

1

Figure 20. Orbits showing the components ai, ao, a$ of the alignment tensor vs. at and a A vs. 03 for AR = 1-25 and n = 0, in the nematic phase, at the temperature 0 = 0. The shear rate is 7 = 1.6. The curves have been recorded over a time t corresponding to the shear deformation 7^ = 1000.

periodic asymptotic orbits with those for which it is known [8] from the computation of the Lyapunov exponents that the behavior is chaotic. Examples for the alignment and the rheological properties are shown in Fig. 26 and Fig. 27.

325

0.75

0.25 Nl

-0.25

-0.5

0.5 shear stress

Figure 21. Phase portrait showing the first normal stress difference N\ vs. the shear stress for AK = 1.25 and K = 0, in the nematic phase, at the temperature 6 = 0. The shear rate is 7 = 1.6. The curves have been recorded over a time t corresponding to the shear deformation 7^ = 1000.

4.

Conclusions

In this article, results are presented for the rheological behavior and for the underlying dynamics of the alignment of tumbling nematic liquid crystals. The basic theory is outlined. For selected state points and model parameters, rheological properties, orbits and phase portraits are displayed graphically, as obtained from numerical solutions of the relevant equations (20)-(23). The analysis was restricted to a situation with a spatially homogeneous velocity gradient as in an ideal plane Couette flow. In general, one has to deal a spatially inhomogeneous alignment and with the onset of a secondary flow as typical for a non-newtonian fluid. Thus the equation for the alignment tensor has to be amended and

326

-0.5-0.25

0 0.25 0.5 0.75 1 a2

-0.5-0.25 0 0.25 0.5 0.75 1 a2

-0.5-0.25 0 0.25 0.5 0.75 1 a2

0

0.02 0.04 0.06 0.08 0.1 a3

Figure 22. Phase portraits showing the components ai,ao,a3 of the alignment tensor vs. a 2 and a A VS. az for AK = 125 and K = 0, in the nematic phase, at the temperature 9 = 0. The shear rate is 7 = 1.75. The curves have been recorded over a time t corresponding to the shear deformation 7* = 1000.

the full hydrodynamic problem has to be solved. The first issue is related to the tensorial treatment [36] of the Prank elasticity of nematics. In the one-coefficient approximation, this requires an additional term [10] - l 2 A o , with a characteristic length £, on the left hand side of (2). For the second task, it is desirable to apply and test, in addition to grid based methods, Smooth Particle and Dissipative Particle methods in the spirit of [37], i.e., the stress and alignment tensors should be used as local dynamic variables. The tensorial rheological model used in [38] to

327

0.75

0.25 Nl

-0.25

-0.5

-0.75 -0.2

0

0.2

0.4

0.6

0.8

1

Figure 23. Phase portrait showing the first normal stress difference N\ vs. the shear stress for AK = 1-25 and K = 0, in the nematic phase, at the temperature 0 = 0. The shear rate is 7 = 1.75. The curves have been recorded over a time t corresponding to the shear deformation jt = 1000.

treat fluids which show both shear thinning and shear thickening is also expected to imply chaotic behavior which deserves a similar analysis. Furthermore, it seems feasible to treat the shear flow induced changes seen in side chain polymeric liquids [35] by a two alignment tensor theory, one tensor specifying the alignment of the mesogenic side groups, the other one characterizing the orientation of the back bone. Again, the dynamics of these systems is expected to be even more complex than what has been detected so far.

328

-0.5-0.25

-0.5-0.25

0 0.25 0.5 0.75 1 a2

0 0.25 0.5 0.75 1 a2

-0.5-0.25

0 0.25 0.5 0.75 1 a2

0.05 0.1 0.15 0.2 0.25 a3

Figure 24- Orbits showing the components oi, ao, az of the alignment tensor vs. at and a,4 vs. az for AK = 125 and K = 0, in the nematic phase, at the temperature 0 = 0. The shear rate is 7 = 1.82. The curves have been recorded over a time t corresponding to the shear deformation 7$ = 1000.

329

0.75

0.25

Nl

-0.25

-0.5

-0.75 -0.2

0

0.2

0.4 0.6 shear stress

0.8

Figure 25. Phase portrait showing the first normal stress difference iVi vs. the shear stress for AK = 1-25 and n = 0, in the nematic phase, at the temperature 6 = 0. The shear rate is 7 = 1.82. The curves have been recorded over a time t corresponding to the shear deformation jt = 1000.

330

aO

-0.5 -0.2

0.2 0.4 0.6 0.8 a2

0

-0.2

0

-1

-0.5

0.2 0.4 0.6 0.8 a2

-0.4 -0.2

0

0.2 0.4 0.6 0.8 a2

0

0.5

1

Figure 26. Orbits showing the components a\, ao, a>z of the alignment tensor vs. a2, and a4 vs. 0,3, for AK = 125 and n — 0, in the nematic phase, at the temperature i9 = 0. The shear rate is 7 = 3.7. The curves have been recorded over a time t corresponding to the shear deformation 7^ = 4000.

331

NI

-0.5

-1

1.5

2.5

3.5

shear stress

Figure 27. Phase portrait showing the first normal stress difference Ni vs. the shear stress for AK = 125 and K = 0, in the nematic phase, at the temperature 0 = 0. The shear rate is 7 = 3.7. The curves have been recorded over a time t corresponding to the shear deformation 7^ = 4000.

332

Acknowledgments This research was supported in part by the National Science Foundation under grant No. PHY99-07949 via the program 'Dynamics of complex and macromolecular fluids' at the ITP, Santa Barbara, and it has been performed under the auspices of the Sonderforschungsbereich 448 'Mesoskopisch strukturierte Verbundsysteme' (Deutsche Forschungsgemeinschaft).

References [I] C. Gahwiler, Phys. Rev. Lett, 28:1554, 1972); P. Pieranski and E. Guyon, Phys. Rev. Lett, 32:924, 1974. [2] R.G. Larson, The Structure and Rheology of Complex Fluids, Oxford University Press, Oxford, UK, 1999. [3] G. Kiss and R.S. Porter, J. Polymer Sci. Polymer Symp., 65:193, 1978; P. Moldenaers and J. Mewis, J. Rheol., 30:567, 1986; J. Mewis, M. Mortimer, J. Vermant, and P. Moldenaers, Macromolecules, 30:1323, 1997. [4] P. Fischer, Rheol. Ada, 39:234, 2000. [5] V. Schmitt, F. Lequeux, A. Pousse, and D. Roux, Langmuir, 10:955, 1994; A. S. Wunenburger, A. Colin, J. Leng, A. Arnedeo, and D. Roux, Phys. Rev. Lett, 86:1374, 2001; J.F. Berret, D.C. Roux, G. Porte, and P. Lindner, Europhys. Lett, 25:521, 2002. [6] G. Rienacker, Orientational dynamics of nematic liquid crystals in a shear flow, Thesis TU Berlin, 2000; Shaker Verlag, Aachen, Germany, 2000. [7] M. Grosso, R. Keunings. S. Crescitelli, and P.L. Maffettone, Phys. Rev. Lett., 86:3184, 2001. [8] G. Rienacker, M. Kroger, and S. Hess, Phys. Rev. E, 66:040702(R), 2002; Physica A, 315:537, 2002. [9] S. Hess, Z. Naturforsch., 30a:728, 1975. [10] S. Hess and I. Pardowitz, Z. Naturforsch., 36a:554, 1981. [II] C. Pereira Borgmeyer and S. Hess, J. Non-Equilib. Thermodyn., 20:359, 1995. [12] S. Hess, Z. Naturforsch., 31a:1034, 1976. [13] S. Hess, Flow alignment of a colloidal solution which can undergo a transition from the isotropic to the nematic phase (Liquid crystal), in: Electro-optics and dielectrics of macromolecules and colloids, ed. B.R. Jennings, Plenum Publ. Corp., New York, 1979. [14] M. Doi, Ferroelectrics, 30:247, 1980; J. Polym. Sci. Polym. Phys. Ed., 19:229, 1981. [15] R. Bandyopadhyay, G. Basappa, and A. K. Sood, Phys. Rev. Lett, 84:2022, 2000; R. Bandyopadhyay and A. K. Sood, Europhys. Lett., 56:447, 2001. [16] M.E. Cates, D.A. Head, and A. Ajdari, Phys. Rev. E, 66:025202, 2002. [17] M. G. Forest, Q. Wang, and R. Zhou, Rheol. Acta, 2004) in press; S. Fielding and P. Olmsted, preprint: arXiv.org/abs/cond-mat/0310658; B. Chakrabarti, M. Das,

333 C. Dasgupta, S. Ramaswamy, and A.K. Sood, preprint: arXiv:cond-mat/0311101 vl. [18] G. B. Jeffrey, Proc. R. Soc. London Ser. A, 102:171, 1922. [19] J. L. Ericksen, Trans. Soc. Rheol., 5:23, 1961; F.M. Leslie, Arch. Ration. Mech. Anal, 28:265, 1968. [20] S. Hess, Z. Naturforsch., 31a:1507, 1976. [21] P. D. Olmsted and P. Goldbart, Phys. Rev. A, 41:4578, 1990; Phys. Rev. A, 46:4966, 1992; H. See, M. Doi, and R. Larson, J. Chem. Phys., 92:792, 1990. [22] G. Marrucci, Macromolecules, 24:4176, 1991; G. Marrucci and P. L. Maffettone, ibid., 22:4076, 1989; J. Rheol, 34:1217, 1990; P. L. Maffettone, A. Sonnet, and E. G. Virga, J. Non-Newtonian Fluid Mech., 90:283 , 2000; P. L. Maffettone and S. Crescitelli, ibid., 59:73 , 1995; J. Rheol, 38:1559, 1994; Y. Farhoudi and A. D. Rey, ibid., 37:289, 1993; Q. Wang, ibid., 41:943, 1997; N. C. Andrews, A. J. McHugh, and B. J. Edwards, ibid., 40:459, 1996; P. Ilg, I.V. Karlin, M. Kroger, and H.C. Ottinger, Physica A, 319:134, 2003. [23] J. Feng, C. V. Chaubal, and L. G. Leal, J. Rheol, 42:1095, 1998. [24] M. G. Forest, R. Zhou, and Q. Wang, J. Rheol., 47:105, 2003; [25] G. Rienacker and S. Hess, Physica A 267:294, 1999. [26] D. R. J. Chillingworth, E. V. Alonso, and A. A. Wheeler, J. Phys. A: Math. Gen., 34:1393, 2001. [27] M. Kroger, Phys. Reports, 390:453, 2004. [28] A. Peterlin and H. A. Stuart, Hand- und Jahrbuch der Chemischen Physik, Vol. 8, 113, Ed. Eucken-Wolf, 1943. [29] C. Zannoni, Liquid crystal observables: static and dynamic properties, in Advances in the computer simulations of liquid crystals, P. Pasini and C. Zannoni, eds., Kluwer Academic Publisher, Dordrecht (2000. [30] W. Muschik, H. Ehrentraut, and C. Papenfuss, J. Non-Equilib. Thermodyn., 22:285, 1997; W. Muschik and B. Su, J. Chem. Phys., 107:580, 1997. [31] G. G. Fuller, Optical Rheometry of Complex Fluids Oxford University Press, New York, 1995. [32] G. Marrucci and F. Greco, Mol. Cryst. Liq. Cryst., 206:17, 1991; M. Kroger and H.S. Sellers, in: Complex Fluids, ed. L. Garrido) Lecture notes in physics 415, Springer, NY, 1992) pp. 295-301; G. Sgalari, G. L. Leal, and J. J. Feng, /. Non-Newtonian Fluid Mech., 102:361, 2002; [33] C. Schneggenburger, M. Kroger, and S. Hess, J. Non-Newtonian Fluid Mech., 62:235, 1996. [34] P. Kaiser, W. Wiese, and S. Hess, J. Non-Equilib. Thermodyn., 17:153, 1992. [35] C. Pujolle-Robic, P.D. Olmsted and L. Noirez, Europhys. Lett., 59:364, 2002. [36] I. Pardowitz and S. Hess, J. Chem. Phys., 76:1485, 1980. [37] M. Ellero, M. Kroger, and S. Hess: J. Non-Newtonian Fluid Mech., 105:35, 2002. [38] O. Hess and S. Hess, Physica A, 207:517, 1994.

PARALLEL COMPUTER SIMULATION TECHNIQUES FOR THE STUDY OF MACROMOLECULES Mark R. Wilson and Jaroslav M. Ilnytskyi Department of Chemistry, University of Durham, South Road, Durham DEI 3LE United Kingdom [email protected]

Abstract

This article will review some of the progress made recently in developing parallel simulation techniques for macromolecules. It will start with simple methods for molecular dynamics, involving replicated data techniques; and go on to show how parallel performance can be improved by careful load-balancing and reduction of message passing. Domain decomposition MD methods are then presented as a way of reducing message passing further, so that effective parallelisation can occur with even the slowest of communication links (ethernet). Finally, parallel techniques for conducting Monte Carlo are reviewed, and ways of combining parallel methods are presented. The latter looks like becoming an effective way of using massively parallel architectures for macromolecules, without the need to simulate huge systems sizes.

Introduction In recent years two important developments in computing have occurred. At the high-cost end of the scale, supercomputers have become parallel computers. The ultra-fast (specialist) processors and the expensive vector-computers of a few years ago, have largely given way to systems which combine extremely large numbers of processors with fast inter-processor communications. At the low-cost end of the scale, cheap PC processors have started to dominate the market. This has led to the growth of distributed computing, with clusters of individual PCs linked with slow (but very cheap) communications such as simple ethernet. For both types of computer system, effective parallel simulation 335

P. Pasini et al. (eds.), Computer Simulations of Liquid Crystals and Polymers, 335-359. © 2005 Kluwer Academic Publishers. Printed in the Netherlands.

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techniques are essential if simulators are to utilise these machines for macromolecular simulations. This article reviews some of the progress made in using parallel processor systems to study macromolecules. After an initial introduction to the key concepts required to understand parallelisation, the main part of the article focuses on molecular dynamics. It is shown that simple replicated data methods can be used to carry out molecular dynamics effectively, without the need for major changes from the approach used in scalar codes. Domain decomposition methods are then introduced as a path toward reducing inter-processor communication costs further to produce truly scalable simulation algorithms. Finally, some of the methods available for carrying out parallel Monte Carlo simulations are discussed.

1. 1.1

Parallelisation: the basic concepts Types of parallel machine

There are many types of possible computer architecture, including parallel architectures [1, 2]. However, by far the most common one for simulators is the so-called MIMD architecture (multiple instruction stream - multiple data stream). On MIMD machines, each processing element is able to act independently (unlike some other specialist parallel machines). This provides maximum flexibility to the programmer, and unsurprisingly MIMD machines make up most of todays parallel systems. Within the MIMD class of machines, programmers have to deal with shared memory and distributed memory machines. As the name implies, on shared memory machines, each processor is able (in principle) to see the whole of the physical memory. This is tremendously useful, and makes for efficient parallelisation (as will be seen below). However, the technology required to achieve this is complicated, and as a consequence shared memory over many processors has significant financial costs and therefore belongs to the regime of todays supercomputers. In contrast on a distributed memory machine, each processor will have access to its own memory. This is easy to implement, in the sense that a parallel system can be built from an independent array of computers of similar specification, such as a PC cluster. (Actually even this is not a requirement for a distributed machine today, as it is perfectly reasonable to built a distributed memory parallel machine from a set of workstations with different specifications and different operating systems.) The drawback of a distributed memory machine, is that, unless the parallel application is embarrassingly parallel (such as the same application running different starting conditions on different processors) message passing will be

337

required between processors at points in code execution where a chunk of data is required that is not stored in local memory. This dramatically slows down a distributed memory system, and for this reason, applications on distributed systems often scale poorly when large numbers of processors are used.

1.2

Message Passing

For distributed memory machines a mechanism is required to pass data from one processor to another. In term of hardware this can be done by communication channels between separate processors. In the crudest of parallel machines, such as a PC cluster, the message passing hardware would simply consist of an ethernet connection between the separate PCs. Also, in terms of software, a set of communication routines are required to carry out the communication. Fortunately for the programmer, a number of public domain software libraries now exist, which have routines that can be linked with a simulation program to handle the communications. The most commonly used are the interfaces PVM [3] (parallel virtual machine) and MPI [4] (message passing interface) both of which work with Fortran or C code. Most MIMD systems will have an implementation of one or both of these libraries. This includes PCs, workstations, specialist distributed memory computers with highspeed communication links and even shared memory machines, where transfer of data between separate distributed memory segments is not required. Consequently, an investment in programming using (say) MPI can be rewarded with a portable code that runs over anything from a PC cluster to a supercomputer.

1.3

Typical parallel programs for distributed memory machine

The two most common types of parallel programs are the master-slave program, and the iV-identical workers approach. In the master-slave approach, one master program calls the shots, and will spawn a set of slave processes (on separate processors), which will be responsible for doing the work.The master will farm out work to each of the processors and gather in the results at the end. This algorithm is easy to implement and has a range of applications. However, the second approach of Nidentical workers (fig. 1), is better suited to molecular simulation. In this approach iV-identical programs are started, one on each processor. Each process is given a separate task ID (TID), and a list of TIDs for the other processes that are running. This process is known as enrolment. Each processor then executes the same code, up to the point where a

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parallel operation is possible. At that point each processor does separate work, and the results are combined at the end. In this stage of program execution, message passing may be needed to share data between processes, or indeed to combine the results at the end of a parallel section. Processors make use of their own TIDs to know which part of the code they are responsible for executing, and to know which process they may need to pass or receive data from. There may be just one, or many sections of the code where parallelisation is possible in this execution stage. Finally, each process should terminate and free the processor that it is running on. Many specialist parallel machines, start from the assumption that everything can be programmed within the iV-identical workers framework; so they work simply by starting N copies of the same program, on on each processor. Hence the alternate name Single Program Multiple Data (SPMD) is often used for this form of parallel program execution.

Stage 1 Spawning

=— Stage 2 Enrolment

(host machine spawns //-identical processes on separate processors)

Stage 4 Termination

(each processor gets unique TID)

Stage 3 Execution (each processor executes the same program) task id used e.g. IF( TID .EQ. 0) THEN ELSE ENDIF

Figure 1. Schematic diagram illustrating the four stages of a parallel program using the iV-identical workers approach.

1.4

The global sum operation

One of the most important type of message passing operation is the global sum operation. This often occurs in molecular simulation where a

339

quantity is required, which represents the sum of a set of independently calculated quantities. For example, ,

(1)

where the quantities a may be (say) the energy of particle i, and where different values of e* have been calculated on different processors. At the end of the calculation each processor will carry out its own sum of ei values, but to complete the answer each processor will need to know the complete sum of ei values from every other processor. The can be achieved in several ways [5]. The simplest way (but not necessary best way) involves each processor sending their partial sum to one single nominated processor. That processor adds up all the answers and then broadcasts the final result back to each processor. At the end of this "global sum" operation, each processor knows the correct sum, E, over all values of i. In message passing interfaces such as MPI, a global sum operation (called a "reduce" operation in MPI) can be carried out by calling a single routine that will carry out the global sum for the programmer. In specialist parallel machines, fast routines are available to do this operation using the quickest possible means, often taking advantage of the architecture and connectivity of the machine itself.

1.5

Pointers to successful parallelisation

There are two key factors leading to successful parallelisation. These apply to any parallel application not just to molecular simulation. The first involves successful load balancing. This simply means that each processor must have a roughly equal share of the work. It stands to reason that if some processors do more work than others, some processors will be left idle for some of the time and parallel efficiency will be reduced. The second important consideration is minimisation of communication costs. If processors have to wait for an inter-processor communication to receive data, then they remain idle during that period and parallel efficiency again suffers. In many parallel algorithms the main technical problem to be faced simply involves finding ways of minimising the ratio of communication cost to computational cost. In molecular simulation, both rise with the size of the system. However, computational costs (which usually depend on the number of pair interactions in the system) tend to rise quicker than communication costs as the size of the system increases. Consequently, most parallel algorithms are reasonably efficient for huge system sizes. The challenge however is to make algorithms work well in parallel for the typical systems that are

340

tackled usually on scalar machines. Only then, will parallelisation be able to tackle important problems such as speeding up equilibration in simulations of macromolecules.

2. 2.1

Parallel molecular dynamics: the replicated data approach The replicated data concept

The replicated data method [6, 7, 8], uses the AMdentical workers strategy described above. The same molecular dynamics program is run on each processor, and each processor has a copy of the coordinates, velocities and forces. At a point where parallelisation is possible, work is split between the different processors. Afterwards a global sum operation is usually required to restore a copy of the correct data summed over each process. The name replicated data comes from the fact that the main simulation data needs to be duplicated on each processor. As will be shown below, this is a relatively easy parallel strategy to implement, requiring only minor changes to scalar code. The drawbacks of the approach are that memory usage is high (due to duplication of data), and communication costs are quite high. The latter limits scalability to large numbers of processors unless the system itself is very large, while the former can prevent really large systems being studied if the memory limit available to individual processors is reached.

2.2

Application to atomic simulation

In a typical molecular dynamics program, the forces, f, on particles are computed from the potential Uij, ry);

(2)

and the equations of motion are integrated using a finite difference algorithm, such as the leap frog algorithm [9] (equations 3 and 4), to update the velocities, v*, and the positions, r^, at successive time-steps, St. Vi(* + ^ t ) = v i ( « - i f t ) + (J«^-, I

I

(3)

rri

Yi(t + St) = Ti{t) + Stvi(t + ^5t).

(4)

Analysis of the time taken for this algorithm is quite informative. For small systems (« 256 particles) at least 80% of CPU time will be spent

341 in the force loop (equation 2). The next largest use of time is the integration stage (equations 3-4), which will take up around 10% of the time. However, when the system gets larger the time for force evaluation grows as N2 and the time for integration grows as JV. This means that for relatively large systems, or complicated potentials, the force calculation totally dominates in terms of CPU time. Consequently, if the force loop can be parallelised, then the algorithm should work well in parallel regardless of how the rest of the program parallelises.

loop over all j DO I:

* of interactions 8-processor example

atoms

DO J = I+1,N

,node 0 1 9 ... /node 1 2 10 ... force /^node 2 3 11 ... calculation f -node 3 4 12 ... \ U o d e 4 5 13 ... Vnode 5 6 14 ... Vnode 6 7 15 ... Viode 7 8 16 ...

calculate pair distance evaluate forces fy sum the forces

ENDDO 5UM(F,N)

Figure 2. Example showing the parallelisation of a simple force loop from a molecular dynamics program. IDNODE ranges from 0 to NODES-1 and represents the processor number. Each step in the loop is taken by successive processors, as shown for an eight processor system on the right of the figure. The call to the routine GDSUM at the end of the loop, will ensure a global sum for each one of the force vectors ft.

Figure 2 shows how a simple force loop can be parallelised using the replicated data approach. Instead of looping from I = 1, N-l, as in a normal force loop, the loop jumps forward in steps of size NODES, representing the number of processors. Consequently, each processor in turn takes successive values of I as shown on the right hand side of the figure for an eight processor example. At the end of the loop each processor will have summed up the forces for each atom f^, but each sum will be incomplete. So a final global sum operation is required (GDSUM), which is equivalent to the sum A:=NODES-1

£ fc=O

ft,

(5)

342

for each atomic force vector f^. The simplicity of the parallel method in figure 2 explains why it is now possible to parallelise loops such as this automatically on a shared memory computer by means of a parallelising compiler. The compiler is able to identify where execution of a loop involves independent operations for each trip round the loop, and then parallelise accordingly. A de-facto standard for shared memory computing OpenMP [10], is now available, which allows the user to add compiler directives to their program to help the compiler parallelise loops efficiently. OpenMP is a simpler approach to parallelisation than PVM or MPI message passing, where the user must organise data transfer and processor synchronisation themselves. However, only the latter two allow portability of code between shared memory and distributed memory machines.

2.3

Improved load balancing

A close look at the example of figure 2 will show that as the values of I increase then the number of pair forces required to be generated drops for that I. This means that processor 0 will always end up doing more work than processor 1, which in turn does more work than processor 2 etc. Better load balancing can therefore be achieved if this workload is spread more evenly. This can be done by using Brode-Ahlrichs decomposition [11] and is shown schematically in figure 3. Code to implement this scheme is given in reference [5]. In practise, Brode-Ahlrichs decomposition is usually combined with a Verlet neighbour list for each atom. One nice feature of this is that it is only necessary for each node to hold the neighbourlist for the atoms it is responsible for. This means that the full neighbourlist, which can take up large amounts of memory, can be distributed over each processor in the system, with each one storing only a partial neighbourlist.

2.4

A practical example for a Gay-Berne liquid crystal

The leap-frog algorithm of equations 3 and 4 can easily be extended to an anisotropic system, such as the Gay-Berne mesogen, by introducing equations for rotation about the centres of mass of a linear molecule with long axis vector, e^, and with u^ = e^ (subject to the constraints et • e^ = 1, and e* • U{ = 0) [12, 9]

StXi(t) = -2iii(i - -6t) • ei(t)

(6)

343

1 2 3 4 5 6 7 8

9

X• • • • X• • • • X• • • • X• • • • X• • • • • X• • • •• X• • ••• • •••• X

1 2 3 4 5

1 2 3 4 5 • • • • • • • • • •

X

X

X

X

X

6 7 8 9

• •• ••• 10 • • • •

6 • • • • •

X

7 8 • • • • • • • • •

X• X

9 10

• • • • •

X

• • • • •

X

Figure 3. Brode-Ahlrichs decomposition for a 9 atom and 10 atom system. The rows represent atom i and the columns represent atom j . Each circle represents an interaction to be calculated. Reading across a row gives the interactions to be computed for atom i. Note that Brode-Ahlrichs decomposition is slightly more efficient for systems containing odd numbers of atoms, as it gives prefect load-balancing in this case.

\- 6t) = et(t) + 6tUi(t +-zSt); axe the so-called gorques, calculated from

where the quantities

d

,. _

(8)

V

U

= 8i~ (gt • <*)<%,

(9)

Si —

(10)

-

(11)

Figure 4 plots results for the force evaluation (including anisotropic parts) of a system of Gay-Berne particles on a Cray T3D. Increasing

344

the number of particles leads to a reduction in the ratio of communication costs/computational costs, leading to an improvement in parallel performance with system size, as shown in the figure. It is clear for the smallest system studied, that beyond 128 processors, increasing the number of processors actually leads to the program running slower. This is termed parallel slowdown, and arises when an algorithm becomes communications bound. This will always occur at some point as the number of processors increases, and represents the limits of parallelisation. It should be noted from figure 4 that the replicated data algorithm scales pretty well. This is because fast communication links were available for the Cray system used. However, for a workstation cluster with slow communications over ethernet, parallel slowdown can occur using only a few processors (often < 8).

1000

.65536 16384 2048 O~256

1

3

6

10

30

60 100

number of processors Figure 4Results for parallel force evaluation (including anisotropic terms) for systems of Gay-Berne particles as described in reference [8]. The results use standard PVM calls on a Cray T3D. Improved performance over these results is possible by using cache-cache data transfers for the global sums at the end of the force evaluation.

The integration part of the algorithm can also be parallelised. However, here there is only a single loop over the number of particles i using equations 3, 4, 6-8, and this must be followed by a global sum for coordinates, velocities, orientations, e*, and orientational derivatives, u;. Not surprisingly the ratio of communication costs/computational costs tends to be poor, and so for many systems parallel integration is slower than scalar integration [8].

345

2,5

Extension to macromolecular systems

The replicated data approach is readily extendable to macromolecules. Many MD studies of polymers have used the simple bead-spring model [13], where Lennard-Jones sites are linked together by simple springs. Alternatively, combinations of isotropic Lennard-Jones and anisotropic Gay-Berne particles can easily be linked together to form an oligomer [14], polymer [15] or dendrimer [16], or a full molecular mechanics force field can be used to represent the polymer [17, 18]. Evaluation of simple intramolecular bond stretching, bond bending and torsional terms can be parallelised easily by analogy to the force loop shown above. Because data is replicated, each processor has a copy of the connectivity list for each type of molecules, so parallelisation is straight-forward. As an example, parallelisation of a loop over bond angles would require a single loop of the form: DO NUM = IDNODE, NANGLES, NODES !find atom numbers defining angle i = 1, angl(num) j = 1, ang2(num) k = 1, ang3(num) !find energy and forces for angle i-j-k

ENDDO This would replace a scalar do-loop starting: DO NUM = 1, NANGLES. It should be noted that no additional global sum is required for the forces at the end of this angle loop, because a global sum of the forces will be occurring anyway for nonbonded terms. The only additional global sum required is for the total angle interaction energy, and this can be done at the same time as the other intra- and intermolecular energies. In this sense parallelisation of all intramolecular forces comes virtually free of charge in terms of additional communication costs. The simplicity of implementing this parallelisation scheme means that a large number of replicated data MD programs are available for macromolecules, including DLPOLY (a general purpose parallel MD program [19] that is well-suited to large molecules), and the GBMOL program from our own laboratory [20, 5].

346

3. 3.1

Parallel molecular dynamics: the domain decomposition approach The domain decomposition concept

Figure 5 illustrates the domain decomposition concept. The simulation box (domain) is divided into a series of regions, each of which is assigned to a separate processor. Within the regions the box is further subdivided into cells, which have dimensions slightly greater than the cut-off for the nonbonded potential. The idea here is that great savings can be made, in terms of both memory usage and communication costs, by avoiding the need to replicate all data across all the system. Each processor can be responsible for calculating the interactions of its own particles, and integrating their equations of motion. Communication between processors should only be required to calculate the interactions for particles near the boundary between different regions. Such communications are obviously minimised when the boundary regions are small in relation to the size of the simulation box. For short range potentials, domain decomposition (DD) has already shown itself to be effective in a large number of studies [6, 8, 21, 22, 23, 24, 25, 26, 27]. Load balancing is generally good for this algorithm provided that, for equal sized regions, the density remains homogeneous throughout the simulation box.

3.2

The force evaluation strategy

There are two possible force evaluation strategies for domain decomposition [6]. These are illustrated in figure 6. Prior to the force calculation data about the coordinates and (in the case of anisotropic sites) orientations, is exchanged between processors. In the first strategy all data in the boundary cells of each region is exchanged between neighbouring nodes. This must be done with three separate message passing events. First data must be passed in the ^-direction, then the y-direction and then the ^-direction. The passing of data in the + and — directions can be done simultaneously. In between each data passing event, the arriving coordinates must be sorted in case any of the arriving coordinates need to be passed on to another node. This is essential to ensure that a region receives coordinates from neighbouring regions that connect only via the edge of a cell, or only via a cell vertex. Figure 7 illustrates how, for a 2d example, a single region receives data from 8 surrounding cells, and now contains a set of particle data from other cells as well as from resident atoms. The forces can then be computed. It should be noted that the force evaluation involves some duplication for the boundary atoms. Obviously, duplication can

347

Domain - whole system

Regions - for each node

Cells -within a region

-3'-

Figure 5. Diagram illustrating the decomposition of a 3d simulation box into separate regions, which are to be controlled by separate processors. The 2d example shows the situation for a 4 processor system where each region is subdivided into cells which are slightly larger than the cut-off for the nonbonded potential. (Note that for the small number of processors shown here, the linear dimensions of the whole simulation box are spanned by just two processors, so one processor will have the same neighbour to its right and left).

be reduced by using large regions. This also means being careful not to count contributions to the total energy twice. An alternative force evaluation strategy involves only sending coordinate information in one direction (i.e. + direction for #, y and z), prior

348

Force evaluation -alternative strategy (1)

:: :.::• <

I

f

m 0

t

cr

1 ?

4

_i

:

3

1

—i

1" IFF v im

*

Force ievaJuatie>n -allternative sitrat<

i rw K

#

T .V

t T Figure 6.

#

1

1 0 ¥ t

T

T

3

1

4T

h

^

The two alternative force evaluation strategies for domain decomposition.

to the force calculation, as shown in the second force evaluation strategy diagram in figure 6. This removes duplication in the force calculation, but necessitates a second transfer of forces (in the opposite direction to the transfer of coordinates shown in figure 6); so that each processor

349

tail

1 2 Export of boundary sites

3

4

6/ 7

absolute atomic number

Index

Figure 7. Diagram showing how a central region receives data from 8 surrounding cells prior to the force calculation, when force evaluation strategy 1 is used for a 2d example. On the right of the diagram the boxes correspond to nodes, the dark circles are resident particles and the lighter circles are particles transferred from another node. In 3 dimensions the central box must receive data from 26 other boxes. This is handled in three sets of overlapped +/— communications with neighbouring boxes in the -He/ — x, +y/ — y and +z/ — z directions.

can have a copy of the true forces for the particles in its boundary cells. The best strategy for any particular simulation will depend on system size, speed of communications and cost of force evaluation. Small system sizes, fast communications, and costly force calculations favour the second strategy.

3.3

Integration and reallocation

The beauty of the DD method is that integration can occur perfectly in parallel. Each node integrates the equations of motion for its resident particles. However, at the end of the simulation time-step, it may be the case that integration has led to particles moving outside the region considered by their host node. In this situation, reallocation of particles must occur, and the data stored by the host node on behalf of that particle, (absolute atom number, coordinates and orientations) must be transferred to another node. Again, this must occur in the same 3 stage process, as needed prior to the force calculation.

350

3.4

A practical example for the Gay-Berne liquid crystal

Trials for a Gay-Berne system using the domain decomposition strategy on a Cray T3D have yielded successful linear speeds-ups for up to 256 processors [8] as shown in figure 8. Interestingly, for the fast interprocessor communications available on the Cray, small systems (256 and 2048), are slower with domain decomposition than replicated data on 8 nodes. This reflects the duplications in the force calculation for the DD case. This finding is totally reversed if the fast Cray communications are replaced by standard ethernet, or if the algorithm is scaled up to more than 8 processors. The latter is shown in figure 9, where performance improvements of 13+ are achieved for DD over RD for a system of 65536 particles running on 256 processors of a Cray T3D.

256

64

I 96

128

160

192

224

256

number of processors Figure 8. Results showing the parallel performance of a domain decomposition molecular dynamics program using the first force evaluation strategy on a Cray T3D. The results are for a system of 16384 Gay-Berne particles using standard P VM calls on a Cray T3D. Improved performance over these results is possible by using cache-cache data transfers for the global sums at the end of the force evaluation.

3.5

Extension to macromolecular systems

The basic problem to be faced when DD is applied to molecular systems is illustrated in figure 10. When a molecule is broken over several separate regions, a major problem occurs when it comes to computing

351

30

60 100

number of processors Figure 9. Results showing the ratio of parallel performance of a domain decomposition molecular dynamics program compared to a replicated data molecular dynamics program. The results are for a system of Gay-Berne particles using standard PVM calls on a Cray T3D. (Data are taken from reference [8]).

intramolecular interactions such as bond stretching, bond angle bending and torsional potentials. This is because, in general, each processor only knows about the atoms in its own region. There have been many elegant attempts to solve this problem for linear chains [22, 25, 28, 26] and for more general molecules [29, 30, 31, 32, 33]. It should be stressed that some of these methods are quite complex, and consequently DD MD is much harder to implement than RD MD. The authors of this article have implemented a very simple scheme, which can be applied to molecules of arbitrary topology composed of spherical or nonspherical sites. Each node keeps a copy of data which does not change during the simulation, such as the topology of each different type of molecule, and each atom is given a unique atom number, which is stored by its resident processor. The unique atom numbers of all neighbouring atoms involved in intramolecular interactions (bonds, angles, dihedrals) with a resident atom can be generated from this stored information. To be successful this scheme relies upon the fact that the distance between four atoms connecting any dihedral angle (and by implication the distance between atoms for each angle and each bond also) is always less than the cut-off distance on the potential. This means that the usual message passing stage prior to the force calculation guarantees

352

Figure 10. Sketch of a molecule overlapping several regions, illustrating the basic problem when domain decomposition is applied to molecular systems.

that each node has a copy of all the atomic coordinates it needs to carry out the force evaluation for any intramolecular force field term. This scheme has being quite successful and is described in detail in references [34] and [35].

4. 4.1

Parallel Monte Carlo Why does standard Monte Carlo perform so badly?

The standard metropolis Monte Carlo approach works very badly in parallel. Such an approach involves: • calculating the energy of a particle, • moving the particle, • recalculating the energy • deciding whether to accept or reject the move. A standard replicated data approach to parallelising this algorithm would involve each processor taking part in the energy evaluation, followed by a global sum operation (so that each processor could have the total energy

353 summed over each processor). The problem with this idea is immediately apparent. Unlike molecular dynamics, where all pair interactions in the system must be computed each step, the energy evaluation is usually comparatively cheap, involving only a few interactions. So the ratio of communication time to calculation time is high. (Indeed, in the worse case scenario, for a hard particle system, finding one overlap would be sufficient for the move to need rejecting without further checking of other particles.) In some cases, the potential may be sufficiently expensive that the communication/calculation ratio improves, but the author does not know of any systems where parallelisation in this way has usefully been extended to more than a few processors.

4.2

Embarrassingly parallel Monte Carlo

A simple parallel method that can often be applied (for MD as well as MC) is to setup N independent simulations and then combine the statistics. Of course this is often easier said that done! Statistically independent starting configurations are required, and for macromolecular simulations there is often a major problem in simulating for long enough times to reach equilibration, and such an approach cannot be applied to speed up the equilibration process. However, within its area of applicability it represents a popular use of parallelisation.

4.3

Parallel configurational-bias Monte Carlo

In configurational-bias Monte Carlo, the whole, or part of, a polymer chain is deleted and regrown [36]. For a linear chain n trial new replicas of an atom are generated according to the Boltzmann weight associated with its bonded interactions (bond length, bond angle and dihedral angle), and one of these is chosen according to the Boltzmann weight associated with its nonbonded interactions. This ensures that the total probability of accepting the new position of an atom is given by its Boltzmann factor (that includes all interactions) and that the regrown chain follows a "low energy route". After the new trial configuration has been generated it is accepted or rejected based on the ratio of Rosenbluth factors for the new and old configurations, which corrects for the bias introduced in the regrowing process. Two forms of parallelism are possible. Each processors can generate the n trial replicas of a new atom in parallel. However, a communication step is then required to choose one trial position and update the coordinates on each processor. This synchronisation must occur before a processor can proceed to the next atom. A second parallel strategy involves allowing each processor to grow one or several copies of a new chain.

354 One chain can then be accepted based on its Rosenbluth weight. This strategy of multiple chain growth has been tried on a scalar machine, and has been shown to improve sampling efficiency within configuration bias MC [37]. It should also translate to parallel machines because a reasonable amount of computational time is required to generate a chain (or multiple chains) on each processor, relative to the inter-processor communication required at the end of chain regrowth. 4.4

Multi-move Monte Carlo

Multi-move Monte Carlo relies on dividing the simulation box up into domains slightly larger in dimension that the cut-off (figure 11). Small molecules separated by more than the cut-off on the potential can be moved independently in parallel without affecting each other. Or two segments of a polymer chain that fulfil the same requirement (as shown in figure 11) can also be moved independently. Parallelisation enters because the energy changes for these moves can be calculated on different processors. Obviously, this approach relies on the site-site potentials being short-range potentials, and is quite difficult to implement for long chain molecules. It has been however been successfully applied by the Mainz group in simulations of the bond fluctuation model [38], and a simpler variant has been used for off-lattice simulations of a linear polyethylene chains [39].

c) V

Figure 11.

c

A domain decomposition approach for use with multi-move Monte Carlo.

355

4.5

Hybrid Monte Carlo

Hybrid Monte Carlo is generally taken to mean the combination of Monte Carlo and molecular dynamics methods. This can be done in a number of different ways. For example, long molecular dynamics steps can be taken (much longer than ones which would conserve energy in normal molecular dynamics), and these steps can then be accepted or rejected with the usual acceptance criteria. Or a series of smaller MD time-steps can be taken, and this sequence of steps can be treated as a Monte Carlo trial move. Doing Monte Carlo in this way allows for all the parallel methods that are applied to molecular dynamics simulations to be applied to the hybrid MC scheme. There are however drawbacks, the most obvious one being that in several studies with scalar algorithms, no big advantages have been seen over standard MD. Of course it is possible that there may be some advantages available, if this approach can correctly be combined with other types of Monte Carlo move. There have been no systematic studies of hybrid MD/MC methods with different mixes of moves, so the true effectiveness of this method is still an open question.

4.6

Parallel tempering

In parallel tempering [40, 41, 42] several simulations are carried out at the same time, and Monte Carlo moves are made to swap systems. This can be done by running several different temperatures at the same time, or by softening the potential and running different potentials at the same time [42]. The idea is illustrated in figure 12, where four temperatures are simulated at the same time, and MC moves are carried out to swap systems. Over a period of time, the simulation at the lowest temperature is able to sample states from higher temperatures.

450K

j"

i

35OK-—-——r—

i

- ---

r

•

Figure 12. Idealised diagram showing parallel tempering over four temperatures. The dashed lines between temperatures represent Monte Carlo moves swapping systems.

356 Parallel tempering is geared towards improving phase space sampling. So sampling with the correct Boltzmann statistics from higher temperature simulations can speed up the path through phase space and overcome problems caused by bottle-necks. In a similar way, softening the potential can remove barriers to equilibration. In the most extreme case, the potential can be softened to such an extent that the particles are able to pass through each other altogether removing all barriers. However, it is sometimes difficult to move between this potential and a repulsive potential of the R~12 form, even with many steps in between. Parallel tempering can, of course, be done in parallel with almost 100% efficiency. It can also be used with separate molecular dynamics simulations as well as with Monte Carlo. However, here it is often difficult to move between systems, without sometimes causing problems in the solution of the equations of motion that can only be solved by using a small time-step.

5.

Summary

This article summarises the current state-of-play in parallel simulations of macromolecules. Molecular dynamics works well, both with replicated data and domain decomposition methods. The former is easiest to implement, requiring only minor changes from a scalar code. However, the latter uses less memory, and is most efficient for large numbers of processors and large system sizes. Parallel Monte Carlo is still in its infancy, and is not yet proven for large numbers of processors. However, there are several interesting methods that have been proposed recently, and the possibility of combining parallel methods, (such as parallel tempering, parallel configuration bias, hybrid MC/MD and multi-move MC) looks attractive. Finally, a note of caution should be sounded. It is easy to quote spectacular performance figures for parallel computing, by showing the results from MD simulations of large systems. Here the computational costs far outweigh any communication costs and parallelisation works extremely well. However, the practising polymer simulator will know that the most common practical problem faced is not going to larger system sizes, but rather going to longer simulation times (or in MC terms, sampling more of phase space). For many problems, system sizes of around 10000 sites are all that is required; and the practical problem here involves using parallel methods to speed up phase space sampling. It is here that newer methods, such as parallel tempering, parallel configurational bias MC and parallel hybrid MC/MD methods are likely to make the biggest impact over the next few years.

357

Acknowledgments The authors wish to thank the UK EPSRC for funding High Performance Computers at the University of Durham, proving computer time on a Cray T3D, and for providing funding for JML MRW and JMI thank NATO for providing funds towards attending this Erice meeting on polymers and liquid crystals. They also wish to thank Profs Paolo Pasini, Claudio Zannoni and Slobodan Zumer for invitations to attend the workshop and for splendid organisation that made this an excellent meeting.

References [I] T. J. Fountain, Parallel Computing principles and practice, chapter 2. Cambridge University Press, 1984. [2] K. Hwang, and F. A. Briggs, Computer Architecture and Parallel Processing. McGraw-Hill Book Company, 1985. [3] PVM, Excellent on-line references for PVM can be found at http://www.csm.ornl.gov/pvm/, (PVM source code is also available free from here), 2002. [4] MPI, Free implementations of MPI and excellent on-line MPI references can be found at http://www-unix.mcs.anl.gov/mpi/ or by following links from this page. Two freely available portable implementations of MPI are MPICH (http://wwwunix.mcs.anl.gov/mpi/mpich/) and LAM-MPI (http://www.lam-mpi.org), 2003. [5] M. R. Wilson, Parallel molecular dynamics techniques for the simulation of anisotropic systems. In P. Pasini and C. Zannoni, editors, Advances in computer simulation of liquid crystals, volume 545 of Series C: Mathematical and Physical Sciences, chapter 13. Kluwer Academic Publishers, 2000. [6] W. Smith, Comp. Phys. Comm., 62:229, 1991. [7] W. Smith, Comp. Phys. Comm., 67:392, 1992. [8] M. R. Wilson, M. P. Allen, M. A. Warren, A. Sauron, and W. Smith, J. Comput. Chem., 18:478, 1997. [9] M. P. Allen and D. J. Tildesley, Computer Simulation of Liquids. Oxford University Press, Oxford, 1987. [10] OpenMP, The home page for OpenMP, is http://www.openmp.org, most manufacturers of shared memory machines will produce their own implementations of the OpenMP standard, 2003. [II] S. Brode and R. Ahlrichs, Comp. Phys. Comm., 42:51, 1986. [12] D. Fincham, CCP5 Quarterly, 12:47, 1984. [13] K. Binder, Monte Carlo and Molecular Dynamics Simulations in Polymer Science. Oxford University Press, 1995. [14] M. R. Wilson, J. Chem. Phys., 107:8654, 1997. [15] A. V. Lyulin, M. S. A. Barwani, M. P. Allen, M. R. Wilson, I. Neelov, and N. K. Allsopp, Macromolecules, 31:4626, 1998.

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Index

adhesive polimer layer, 250, 258 albumin fragments, 205 subdomains, 206 atomistic simulations, 205, 340 models, 59 Barker-Watts technique, 4 bead-bead interaction, 138 bead-spring model, 61, 62, 136 Benettin algorithm, 284 Beris-Edwards dynamical formalism, 230 biosensor LC based, 222, 224 ordering kinetics, 240 birefringence, 46, 50 bond-fluctuation model, 62, 172, 179 boundary conditions bipolar, 5, 11, 23 homeotropic, 32 planar, 31, 37 radial, 5, 6, 11, 13, 23 toroidal, 5 bridge, 260 Brode-Ahlrichs decomposition, 342, 343 Brownian force, 194, 274 chain, 86 arrangement, 117 chemically realistic model, 163 configurational bias moves, 181 configurational relaxation, 156 conformational dynamics, 193 conformational entropy, 266 connectivity, 165, 252 deformation, 272 elastic forces, 263 extensibility, 258 flexible model, 181, 271 free energy, 263 freely rotating model, 163 gaussian, 251 insertion-deletion, 181

internal viscosity, 193 isodiametric units, 111 local random hopping moves, 182 moduli, 263 longitudinal, 264 tangential, 264 nematic order parameter, 182 orientational mobility, 153 persistence length, 187 phantom, 110 polymer, 270-272, 291 rigid state diagram, 176 scattering function, 155 semi-flexible, 172 aggregation, 188 state diagram, 179 snake moves, 181 stiffness, 111, 171, 187 structure factor, 163 unconstrained, 252 worm-like, 279 chain molecules, 271 chemical potential, 182 coarse-grained lattice model, 172 coarse-graining, 61, 75 coarsening dynamics, 241 coil, 175 coil-globule transition, 174 coil-torus transition, 175 colloidal particle, 225, 228 defect structure, 223, 232 homeotropic anchoring, 232 compression energy, 256 parameter, 256 connectivity, 252, 254 continuum theory, 229 Couette, 307 flow, 274, 291, 297, 325 geometry, 271 symmetry, 297

362 Debye, 271 defect structure, 223, 232 correlation length, 243 three ring, 237 dendrimer, 69 liquid crystal, 57, 59, 69 deuterium quadrupolar splitting, 10, 34 dielectric permittivity, 157 relaxation, 157 dissipative force, 137, 138 dissipative particle dynamics (DPD), 63, 135, 136, 138 DNA chains, 171 domain decomposition, 346 elastic constants, 61 electric capacitance, 45 Ericksen-Leslie, 296, 297, 303 tumbling paramter, 304 viscosity coefficient, 303 evolution equation, 230 excluded volume interaction, 252 EXEDOS method, 226 expanded ensemble method, 226 reaction coordinate, 226 FENE (finitely extendible nonlinear elastic), 61, 291 fibronectin module, 206, 216 kinetics, 212 flexible units, 142 fluctuation model, 136 fluctuation-dissipation relation, 138 Fokker-Plank, 295, 296, 301, 318 force field, 59 force loop, 341 Fourier transform, 9, 35 Frenkel springs, 137 friction, 200 coefficient, 269, 271, 273, 274, 279, 289 force, 193, 271, 273 pressure, 296, 297, 301 Gaussian chain, 251 constraint, 274, 279 thermostat, 269, 270, 288, 289, 291 Gay-Berne, 1, 60, 135, 224, 342, 350 global sum operation, 338 globule, 175 shape, 174 toroidal, 178 grand-canonical simulations, 181

graphite surface adsorption, 207 gyration radius, 177, 273 tensor, 269-272, 277, 279, 280, 283, 284, 287, 289, 291 Hedgehog point defect, 232 structure, 15 Hooke springs, 137 hybrid model, 63, 69 hydration adsorbed protein fragments, 215 shell, 216 hydropathy index, 206 integration algorithm, 342, 344, 349 Jeffrey period, 303, 304 kayaking-tumbling, 297, 298, 308 kayaking-wagging, 297, 308 Kuhn length, 171 Landau-deGennes, 298, 300-302 lattice model, 3, 23, 61 coarse-grained, 172 Lebwohl-Lasher, 1, 29, 62, 135 Legendre polynomial, 3 Lennard- Jones potential, 85, 173 light transmission, 46, 50 load balancing, 342 Lyapunov exponent, 269, 284, 308, 318, 324 macromolecules, 345 semiflexible, 171 stiff, 177 magnetic susceptibility, 15, 20 Mayer-Saupe tensor, 182 mean field scaling theory, 174 message passing, 337, 338 MPI, 337, 339 OpenMP, 342 PVM, 334, 337 molecular dynamics, 9, 135, 150, 207 molecular mechanics, 59 Monte Carlo, 2, 3, 9, 23, 29, 87, 111, 135, 172, 223 hybrid, 355 Metropolis, 4 multi-move, 354 parallel, 352 simulation steps, 9 time scale, 11 motion

INDEX chaotic, 270, 282, 292 rotational, 281 tumbling, 270, 283 wagging, 286 multiscale simulation, 223 nanoparticles, 110 nematic biaxial, 61 confined, 8 droplet, 2, 10, 12 droplets, 11 order parameter, 4, 8, 17 ordering, 3, 16, 17, 182 tumbling, 283 uniaxial, 8 nematic-isotropic transition, 4, 6, 51 NEMD (non-equilibrium molecular dynamics), 270, 272-274, 279, 288, 291 network, 200 one-dimesional, 256 three-dimesional, 256 two-dimensional, 254 neutron scattering, 155 spin echo, 155 NMR, 4, 34 cycle, 10M 11, 13 lineshapes, 3 magnetic field, 11 polybutadiene, 151 spectra, 10, 16, 23, 35, 48, 52 spectrometer field, 15, 18 spectrum, 8 spin lattice relaxation time, 153 time scale, 10 Nose-Hoover, 290 thermostat, 270, 287, 290, 291 odd-even oscillation, 87 Onsager, 302 order parameter, 229 biaxiality, 30, 34 field, 17, 20, 39, 43 nematic, 4, 8, 17, 30, 34 orientational, 90 radial, 6, 18 tensor field theory, 229 ordering kinetic, 240 ordering matrix, 30, 34 Pake-type, 16, 18, 21, 22 powder spectrum, 11, 12 parallel computer distributed memory, 336, 337, 342

363 MIMD, 336 shared memory, 336, 337, 342 parallel tempering, 355 paranematic ordering surface-induced, 32, 36, 49 phase diagram bimodal line, 184 spinodal line, 186 polybutadiene, 150 NMR, 151 polymer internal viscosity, 193 liquid crystal, 59 main chain, 57, 66 side chain, 57, 64, 311, 327 melt dynamic heterogeneity, 160 relaxation, 156 semiflexible, 188 polymer coil, 272, 273 polymer fiber irregular array, 41 regular array, 30, 37 surface roughness, 30, 31 polymer networks aligning ability, 29, 31 dispersed in nematics, 27 topography, 37, 41, 43 polymer substrate, 16 polymer-filler interface, 110 polymer-nanofiller, 110, 114 polymer-wall potential, 258 reaction, 264 polymerization degree, 182 polymethylene, 111 program master-slave, 337 N-identical workers, 337, 340 protein adsorption, 203, 205 final stage, 210 initial stage, 208 protein fragments, 208 adsorbed hydration, 215 interaction energy, 208 strain energy, 208 surface spreading, 212 radial distribution function, 113 random force, 138 reaction coordinate, 226 relaxation dynamics, 243 replicated data strategy, 340 reptation, 111 rheological phase portrait, 320 rigid group, 86 attractive force anisotropic, 83

364 shape anisometry, 83 ring disclination line, 232, 237 rotational barrier, 194 characteristic time, 194 frequency, 194 rotational viscosities, 61 Rouse matrix, 252 model, 156 Runge-Kutta, 308 method, 274 segmental friction, 167 self-avoiding walk (SAW), 62 semi-rigid units, 135, 139-146 shape parameter, 174 shear, 271 flow, 271, 275, 281, 283, 292 rate, 271-275, 277-279, 283-285, 288, 289, 291, 309, 311 finite, 275 limiting value, 272 slab widths, 261 spacer distribution, 87 polymethylene, 87 semi-flexible, 88 spin model, 1, 29 strand length, 261 stress shear, 297, 298, 305, 308, 311, 319321 tensor, 296, 297, 305, 308, 309, 311, 316-318, 320 surface boundary conditions, 3 hydrophobic, 205 planar, 114 switching external field-induced, 37 structural phase diagram, 40 tensor diffusion, 13 quadrupolar, 8 threshold Preedericksz, 38, 39, 43, 45, 46

saturation, 38, 39 topological charge, 32 defects, 32 structure, 34 torsional barrier, 163 force field, 153 torus-globule transition, 175 transfer matrix, 258 transition torsional, 153 translational diffusion, 35, 48, 52 trimer, 86 conformation, 86 conformational distribution, 93 links, 86 phase transition, 90 tumbling, 269, 283, 296, 297, 300, 302, 304, 308 coefficient, 303 in-plane, 307, 321 Jeffrey period, 303, 304 motion, 270, 308 nematic, 283, 292, 295-298, 310, 325 parameter, 295-297, 300, 303, 304, 318-320 polymeric liquid crystals, 296 rotational, 283 twist grain boundary phase, 60 united atom model, 60, 87, 150 velocity angular, 270, 271, 284 shear-induced, 280, 284, 291 flow, 271, 273 gradient, 271, 296, 297, 300, 307, 325 Verlet algorithm, 138 viscosity, 193 internal, 194, 271 Newtonian, 305, 311 non-Newtonian, 308 wagging, 297, 308 Zeeman splitting, 8