Thermodynamics of Polymerization - articles by Hideo Sawada in Polymer Reviews

AG*/RT =

In

- 0 s ) " 1 In 4

-AG S S /RT = In {4>hUm) + (M s p - Msm)(0s - 4>%) + Mmp^m" is the volume fraction at equilibrium and
336

HIDEO SAWADA

When fiSp = /ism and if either Mmp = 0 or <£m and <£p tend to zero, Eq. (41) reduces t o - A G s s / R T = In ($m/(f>m)- If 4>m/s>

+

Mmp^m

(42)

Deviations from ideal behavior arise from three causes: (1) 0 m is not proportional to the monomer concentration in which case $m/$rh * [MJe! (2) the interaction of solvent with monomer is different from that with polymer, i.e., Msm * f-'sp' $) the monomerpolymer interaction is not negligible, i.e., jimp ^ 0The inequality jnSm ^ f*sp accounts for the dependence of [M]e on the nature of the solvent. The effect of solvent is not eliminated even at an extremely low equilibrium monomer concentration. The last term, Mmp(0m~ 0p)> i s unimportant if m and $ p are small. However, it becomes significant in concentrated polymer solution. Ivin and Leonard [42] considered the effect of a soluble polymer in a liquid monomer on the free energy of polymerization. Consider a homogeneous equilibrium mixture containing a volume fraction <£m of monomer and 0 p of polymer of degree of polymerization n. The free energy of polymerization in an equilibrium mixture is zero and may be expressed as the sum of three terms [-AGj, the free energy change for the removal of 1 mole of liquid monomer from the mixture; AGic, the free energy of polymerization of 1 mole of liquid monomer to 1 base-mole (l/n_moles) of amorphous (liquid or nonglassy solid) polymer; and AG2, the free energy change for the addition of 1 base-mole of polymer to the mixture]: -AG^ + AGlc + AG~ = 0 By inserting the appropriate expressions for AG^ and AG2 from the Flory-Huggins expression, we obtain AGlc = RT{ln ^ m - (In <£p)/n + 1 - l/n + x(p ~ <£m)}

(43)

where x is the polymer-monomer interaction parameter. If n is sufficiently large and $ p is not too small, this can be reduced to AGlc = RT{ln 0 m + 1 + x(0p - 0 m )}

(44)

Equation (44) is applicable to a system such as liquid tetrahydrofuran-

J

THERMODYNAMICS OF POLYMERIZATION. I

337^

dissolved polytetrahydrofuran. Taking the polymer-monomer interaction parameter as 0.3, Ivin and Leonard [42] obtained AHjc = -3.0 kcal/mole and ASjc = -9.8 cal/deg/mole in the case of tetrahydrofuran. References [1] . [2] [3] [4]

F. S. Dainton and K. J. Ivin, Nature, 162, 705 (1948). F. S. Dainton and K. J. Ivin, Trans. Faraday Soc, 46, 331 (1950). F. S. Dainton and K. J. Ivin, Quart. Rev., 12, 61 (1958). K. J. Ivin, in Polymer Handbook (J. Brandrup and E. M. Immergut, eds.), Wiley-Interscience, New York, 1966, pp. 11-363. [5] R. M. Joshi and B. J. Zwolinski, in Vinyl Polymerization (G. E. Ham, ed.), Vol. 1, Dekker, New York, 1967, p. 494. [6] H. N. V. Temperley, J. Res.Natl. Bur. Std., 56, 55 (1956). [7] F. S. Dainton, D. M. Evans, F. E. Hoare, and T. P. Melia, Polymer, 3, 277 (1962). [8] R. W. Warfield and M. C. Petree, Makromol. Chem., 51, 113 (1962). [9] ' F. S. Dainton, D. M. Evans, F. E. Hoare, and T. P. Melia, Polymer, 3, 286 (1962). [10] R. W. Warrield and M. C. Petree, J.Polymer Set., A-2, 5, 791 (1967). [11] G. T. Furukawa and M. L. Reilly, J. Res. Natl. Bur. Std., 56, 285 (1956). [12] F. S. Dainton, D. M. Evans, F. E. Hoare, and T. P. Melia, Polymer, 3, 297 (1967). [13] N. Bekkedahl and H. Matheson, J. Res. Natl. Bur. Std., 15, 503 (1935). [14] R. \V. Warfield and R. Brown, Kolloid-Z., 185, 63 (1962). [15] R. W. Warfield and M. C. Petree,,/. Polymer Set., 4, 532 (1966). [16] T. P. Melia, Polymer, 3, 317 (1962). [17] F. S. Dainton, D. M. Evans, F. E. Hoare, and T. P. Melia, Polymer, 3, 263 (1962). [18] T. P. Melia, D. Bailey, and A. Tyson, J.Appl. Chem., 17, 15 (1967). [19] G. T. Furukawa, R. E. McCoskey, and G. J. King, J. Res. Natl. Bur. Std.,49, 273 (1952). [20] F. S. Dainton, D. M. Evans, F. E. Hoare, and T. P. Melia, Polymer, 3, 271 (1962). [21] F. S. Dainton, D. M. Evans, F. E. Hoare, and T. P. Melia, Polymer, 3, 310 (1962). [22] F. S. Dainton, D. M. Evans, F. E. Hoare, and T. P. Melia, Polymer, 3, 316 (1962). [23] T. P. Melia, G. A. Clegg, and A. Tyson, Makromol. Chem., 112, 84 (1968). [24] T. P. Melia.and A. Tyson, Makromol. Chem., 109, 87 (1967). [25] D. R. Gee and T. P. Melia, Makromol. Chem., 116, 122 (1968). [26] R. W. Warfield and M. C. Petree, J. Polymer Sci., 55, 497 (1961). [27] R. W. Warfield and M. C. Petree, J . Polymer Sci., Al, 1701 (1963).

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HIDEO SAWADA

[28] G. A. Clegg, T. P. Melia, and A. Tyson, Polymer, 9, 75 (1968). [29] F. S. Dainton, K. G. Ivin, and D. A. G. Walmsley, Trans. Faraday Soc. 56, 1784 (1960). [30] T. P. Melia, J.Appl. Chem., 14, 461 (1964). [31] R. S. Jessup, J. Chem.Phys., 16, 661 (1948). [32] K. S. Pitzer, J. Chem.Phys., 5, 469 (1937); 8, 711 (1940). [33] J. H. Baxendale and A. G. Evans, Trans. Faraday Soc, 43, 210 (1947). [34] R. M. Joshi and B. J. Zwolinski, in Vinyl Polymerization (G. E. Ham, ed.), Vol. 1, Dekker, New York, 1967, Chap. 8. [35] R. M. Joshi and B. J. Zwolinski, Macromolecules, 1, 25 (1968). [36] G. S. Parks and H. P. Mosher, J. Polymer Sci., Al, 1979 (1963). [37] T. P. Melia, Polymer, 7, 640 (1966). [38] S. Bywater, Trans. Faraday Soc, 51, 1267 (1955). [39] D. A. Small, Trans. Faraday Soc, 49, 441 (1953). [40] P. J. Flory, Principles of Polymer Chemistry, Cornell Univ. Press, Ithaca, N. Y., 1953, Chap. 12. [41] S. Bywater, Makromol. Chem.,52, 120 (1962). [42] K. J. Ivin and J. Leonard, Polymer, 6, 621 (1965).

Chapter 2. Heat of Polymerization I.

II.

III.

GENERAL ASPECTS A. Breaking a Multiple Bond B. Resonance C. Steric Strain

339 340 340 341

VARIATIONS IN HEATS OF POLYMERIZATION A. Steric Strain in the Polymer B. Conjugation and Hyperconjugation C. Hydrogen Bond and Solvation

341 343 346 347

EMPIRICAL ESTIMATION OF HEAT OF POLYMERIZATION

347

REFERENCES

349

APPENDIX

350

I. GENERAL ASPECTS The heats of polymerization for a number of monomers are shown in the Appendix. Since AHp = AE - P AV, if the volume change can be neglected, the heat of polymerization becomes equivalent to the change in internal energy of the molecule. The internal energy may be reduced in a number of ways, for example, by a release of steric strain or by a loss of IT- electron energy. Here, we consider three factors: (a) breaking a multiple bond, (b) resonance, and (c) opening a ring under strain. 339

340

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A. Breaking a Multiple Bond Polymerization is often accompanied by the opening of a double bond C=C and the formation of a single bond C—C. A double bond has a higher energy than a single bond, and conversion of a ;r-bonded • molecule to a a -bonded framework results in an overall decrease in bond energy. Therefore, the heats of polymerization of vinyl polymers are about 20 kcal, which is the normal difference between the average bond energy of a C=C bond (146 kcal) in a monomer and the sum of the two C—C single bonds (83 x 2 = 166 kcal) formed in the polymer. If no other factors predominate, therefore, polymerization of aldehyde or ketones to polyoxymethylenes should result in an increase in enthalpy (Table 2.1) and becomes thermodynamically impossible because of the obvious negative entropy involved in any association process. However, it may be seen that the predicted values of AHgg, based on the more refined Cox scheme [1] for the compounds mentioned above, are definitely about 5 kcal negative. Table 2.1 Decrease in Bond Energy on Polymerization Bond energies, kcal/mole

Predicted heat of polymerization, kcal/mole

C=C (145.8) to -C-C-(82.6)

-20

C=O (176)

to -C—O—(85.5)

+5

C=N (147)

to -C—N-(72.8)

+1.4

C=N (212.6) to -C=N-(147)

-7.2

C=S (128)

to -C-S-(65)

-2

S=O (104)

to -S-O-(55.5)

-7

B. Resonance It should be pointed out that increased delocalization or resonance stabilization will lower the internal energy and the enthalpy of a molecule. For certain polymerizations, resonance may be entirely responsible for the free energy change in polymerization. If, therefore, the delocalization per repeating unit is different in the monomers and polymers, then those species with the greater resonance stability will be favored in the equilibrium if no other influences predominate.

THERMODYNAMICS OF POLYMERIZATION. I

341

If the polymer is appreciably stabilized by delocalization and the monomer is not, then AHp will be negative, and the position of the final equilibrium will strongly favor the polymer rather than the 'monomer. The bond energies of the C=N, C=N, and C—N groups are simple multiples of each other (in the ratio 3:2:1) so that the opening of these multiple bonds in an addition polymerization will not result in a lower enthalpy and, hence, addition polymerization will not occur. However, the polymerization products -4c(R)=N4are stabilized to a greater or lesser extent by delocalization, and — AH|g should be 10 kcal/mole. This phenomenon is termed resonance-induced polymerization [2]. The hypothetical head-to-head addition of cyclohexene, according to the equation 2n I

I

*- -{(CH2)4—CH=CH—CH=CH-(CH2)44h

results in resonance in the polymer which was not present in the monomer. If the resonance energy of this system is similar to butadiene, 3.5 kcal/-fCH=CH—CH=CH-}- unit, then-AHg g of the polymerization would be 1.8 kcal/mole of cyclohexene. Although the ceiling temperature of such a polymerization will be low, there should be some temperature below which the polymerization would be thermodynamically possible. This polymerization is also an example of resonance-induced polymerization. C. Steric Strain Since steric strain must raise the internal energy, interactions of this type will raise the enthalpy of the ring monomer relative to the linear polymer where the substituent groups become widely spaced. The heat of polymerization, therefore, provides a direct measure of the strain energy in the monomer ring [3-5]. A more detailed discussion is given in the next section. II. VARIATIONS IN HEATS OF POLYMERIZATION Variations in the heats of polymerization of various monomers arise mainly from the following causes: (a) steric strain in the polymer; (b) differences in stabilization energy in monomer and polymer due to conjugation or hyperconjugation; and (c) hydrogen bond and solvation. Some of the factors that affect the heat of polymerization are summarized in an internal energy versus reaction coordinate diagram in Fig. 2.1.

STRAIN

POLYMER

HINDRANCE

COORDINATE

STERIC

CONJUGATION HYPERCONJUSATION H Y D R O G E N BOND SOLVATION

REACTION

MONOMER

STERIC

A

RESONANCE DELOCAUZATION

Fig. 2.1 Internal energy vs reaction coordinate diagram illustrating the factors that affect the heat of polymerization.

DC UJ

z

UJ

z

UJ

O

>

o

m

to

CO

THERMODYNAMICS OF POLYMERIZATION. I

343

A. Steric Strain in the Polymer Steric strain in the polymer tends to make AHp less negative as a result of bond stretching, bond-angle deformation, or interaction between nonbonded atoms. This factor, steric hindrance, is presumably the reason for the low heats of polymerization of the a,a-disubstituted ethylenes, such as vinylidene chloride, isobutene, a-methyl styrene, and methyl methacrylate (Table 2.2), because the crowding together of the groups is greater in the polymer than in the monomer. Table 2.2 Variations in Heats of Polymerization Monomer

Heat of polymerization, kcal/mole

Ceiling temperature in bulk, °C

Styrene a-Methyl styrene

16 7

235 61

Methyl aery late Methyl methacrylate

20 13

164

Formaldehyde Acetoaldehyde Acetone

13 0 -6

Vinyl chloride Vinylidene chloride

32 14

Ethylene Propylene Isobutene

26 21 17

126 -31

407 300 50

Polyoxymethylenes are much more sensitive to side-group steric hindrance, and side groups larger than hydrogen appear to lower T c to room temperature or below. From the earlier discussion, it is clear that if AHp changes from negative toward zero, the ceiling temperature will be reduced accordingly. Since side-group steric hindrance would be expected to decrease the rotational entropy of the polymer, ASp would become even more negative in the presence of bulky side groups. This would serve to lower the ceiling temperature further with increasing steric repulsion of the side groups. Evans and Polanyi [6] showed that, when models were constructed using van der Waals radii for the substituent group, these groups came to within distances smaller than their normal van der Waals

344

HIDEO SAWADA

separation in the final product. This could lead to repulsion energy between substituent groups, and, hence, to a higher energy of the final product. Inspection of models shows that substituents on alternate carbon atoms of a'—C—C—C— chain interfere with each other to a greater extent than those located on adjacent atoms. The construction of scale models of a,/3-disubstituted olefin polymers, such as poly(butene-2) is easier than the construction of scale models of polymers with two substituents alternating on the same carbon as in polyisobutylene. This factor accounts for the lower heat of polymerization of a,a-disubstituted olefins as compared with that of analogous ot,/3-disubstituted olefins. The hypothetical polymer, poly(butene-2), must have two modes of conformation—the isotactic and the syndiotactic—because every carbon in the chain —<5(CH3)H— is asymmetric. The isotactic chain is rather strained and difficult to construct even in the helical configuration, but the syndiotactic is almost completely strain-free. From the written formulas for polymers of a,a-disubstituted monomers, greater steric hindrance might have been expected in the head-to-head or tail-to-tail structure than in the head-to-tail structure. However, the construction of scale models of the head-tohead or tail-to-tail structure is easier than the construction of scale models of the head-to-tail structure. Therefore, the alternate head-to-head or tail-to-tail polymer should be comparatively free from steric repulsions between substituents. As shown in schematically in Fig. 2.2, this is also true in the case of the activated complexes. A polymer radical in which the odd electron is on the —CH2— group [(I) and (II) in Fig. 2.2] will, therefore, prefer to add onto the —CH2 end rather than the —CXY end (n). Similarly, a polymer radical with the odd electron on the CXY group will prefer to add onto the —CXY end of the monomer (IV) rather than onto the —CH2 end (HI). Hence, the alternate head-to-head or tail-to-tail polymer tend to be preferred if there is much steric hindrance. In both the acrylate and methacrylate series the change from methyl to ethyl to n-butyl leads to a steady increase in the heat of polymerization. McCurdy and Laidler [7] have shown that in these series the increasing steric hindrance leads to an increase in the proportion of the alternate head-to-head or tail-to-tail polymer, so that in the polymers there is actually less hindrance with the more bulky substituents than with the smaller ones. In agreement with this, Marvel and Cowan [8] have shown that the monomers

THERMODYNAMICS OF POLYMERIZATION. I INITIAL 4

H

X "<

STATE

345

ACTIVATED

v

/Y X

H .H H

STATE Y

x

A

H

X Y H H

V . A A X

Y

X

i<"/(

-

Y

X

P: //

X

""

Y

vv

vV

A '!

Y

H H

XY

Fig. 2.2 Initial and activated states in polymerization [7]. Cl CH2=C

Br and

CH2=C

where considerable steric effects are expected, lead predominantly to the alternate head-to-head or tail-to-tail polymer. The magnitude of the energy of steric repulsion cannot be calculated, and the difference between observed and calculated heats of polymerization may be considered to represent the energy of steric repulsion between neighboring groups within the polymer chain [9]. For monosubstituted units the difference is small, only for styrene does it definitely exceed 1 kcal/mole. Heats of polymerization of

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HIDEO SAWADA

disubstituted monomers are 3-9 kcal/mole lower (in magnitude) than the calculated values, indicating severe compression of chain constituents. It is difficult to calculate the increase in AHp due to steric repulsions. However, the general principles on which quantitative calculations of steric effect for simple molecules can be performed are outlined in a recent monograph [10]. Attempts have been made to calculate potential functions for organic polymers with the use of appreciable approximations [11], but an accurate solution for polymer molecules is a difficult task. B. Conjugation and Hyperconjugation Differences in stabilization energy in monomer and polymer due to conjugation or hyperconjugation lower the heat of polymerization to an extent equal to the resonance energy of the monomer, which may be evaluated from the difference in heats of hydrogenation of ethylene and the vinyl compound. Conjugation to the ethylenic bond makes an appreciable contribution to the stabilization of styrene and its ring-substituted derivatives, which keeps the heat of polymerization rather low (-AHp = 16.0-16.5 kcal/mole). Similar stabilization of the monomer state by conjugation is apparent in butadiene or isoprene (about 3 kcal), vinyl pyridine (2.5-4 kcal), acrylonitrile, and others. Conjugation with a carboxyl or a carbonyl group in acrylic acid, acrylic esters, acrolein, and methyl vinyl ketone also seem to lower the AHic of these monomers. When the substituent is nonconjugating, as it is in vinyl acetate and higher vinyl ethers, the AHp values would not be much altered. In a series of a-methyl-substituted monomers, such as propylene and ot-methyl styrene, the contribution of hyperconjugation in stabilizing the monomer is apparent. It is rather difficult to account for the particularly high heats of polymerization (exceeding that of ethylene itself) of the derivatives in which there is a strongly electronegative group, such as vinyl chloride (-AHic = 22.9 kcal), nitroethylene (-AH lc = 21.7 kcal), vinylidene fluoride (-AHi c = 3 1 kcal), and tetrafluoroethylene (—AHic = 37 kcal). These increased heats of polymerization may be due to a reduction, on polymerization, of repulsion between nonbonding electrons associated with the electronegative groups. Alternatively, they may arise because of stabilization of the polymer caused by a degree of ionic covalent resonance. In a-methyl styrene, all the factors lowering the AHp have combined; namely, the resonance of the benzene ring, the —CH3 hyper-

THERMODYNAMICS OF POLYMERIZATION. I

347

conjugation stabilizing the monomer, and the a,a-disubstitution destabilizing the polymer. In vinylidene chloride the electronegativity of the chlorine substitution has caused the compensating effect of raising the value, since otherwise the chlorine substituent, being almost the same size as the methyl, should have lowered the AHp to 13 kcal as for isobutene. C. Hydrogen Bond and Solvation It must be emphasized that since hydrogen-bonding and solvation effects are small, the explanations in this section are necessarily tentative. The hydrogen atom can form hydrogen bonds of O • • • H—O or N—H • • -O type. The hydrogen-bonding sites in free monomer molecules, more rigidly fixed than among polymer chains, would lead to higher intermolecular association in the monomer, thus lowering of the heat of polymerization. This point was recognized and stressed by Joshi [12]. Hydrogen-bonded monomers include acrylic acid (-AHic = 16.0kcal), methacrylic acid (-AHic = lO.lkcal), acrylamide (-AHSS= 14.4 kcal in benzene and 13.8kcalin hexane),and methacrylamide (-AHSS = 8.4 kcal in benzene). If a sufficiently dilute aqueous or alcoholic solution of a monomer is polymerized, hydrogen-bonding influences may be appreciably reduced or eliminated almost completely. The AHss (water or methanol) for all these hydrogen-bonded monomers have been found normal in relation to structure and were uniformly higher by about 3 to 5 kcal than those of the associated liquid state. 2-Hydroxyethyl methacrylate and 2-hydroxypropyl methacrylate both exhibit lower heats of polymerization than do the monomers having no hydroxyl groups. The difference seems too large to explain by steric hindrance. The number of hydroxyl groups solvated in the polymer is generally less than in the monomer. The heats of solvation of alcohols are in the range from 2.5 to 4 kcal/mole, so that the effects observed could be explained if only part of the solvation shell is removed on polymerization [7]. Factors, such as heats of emulsification of the monomers or heats of wetting of the polymers, might also play a part. III. EMPIRICAL ESTIMATION OF HEAT OF POLYMERIZATION Heats of polymerization may be either experimentally determined by each of three distinct methods or calculated by empirical methods. Details of the experimental techniques, such as combustion methods, direct-reaction calorimetry, and the thermodynamic equilibrium techniques, have been fully treated in a recent review [13].

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All empirical methods of estimating heats of polymerization of vinyl compounds have been based on empirical relations between the heat of formation and structure of low molecular hydrocarbons. Thus, using the relations deduced by Prosen et al. [14], Flory [15] obtained values of AH|g = —22 to —24 kcal/mole for a number of simple olefins. Jessup [16] has computed values of AH° and AG° for ethylene polymerization over a range of temperatures and chain lengths. Heat of polymerization (assuming no steric hindrance in the polymer) may also be predicted from the parallel reaction of hydrogenation, which involves a change from trigonal to tetrahedral hybridization of the carbon atoms, as in polymerization. Flory [9] has considered the following steps:

CH 2 =CXY

CH3—CH2XY

AH,

(A) The heat of polymerization is given by .

AHp=AHh+AHe

If X = Y = H, AHe is 10.41 kcal. The heat of hydrogenation of ethylene is -32.73 kcal at 25°C; hence, the heat of polymerization for the unsubstituted ethylene AH g g is - 2 2 . 3 kcal/mole. If X is an alkyl group and Y = H, the reference compound (A) is a straight-chain hydrocarbon, and the second process above consists in its transformation to a branched unit (B) having a single substituent. Hence, AH e is 9.6 kcal. If both X and Y are alkyl groups, the reference compound is a single-branched hydrocarbon, and it is converted to a double-branched structural unit in the hypothetical dehydrogenation process. So, AHe is 9.2 kcal. The heats of polymerization of the various hydrocarbon monomers would, therefore, be about 9 to 11 kcal less negative than of the corresponding hydrogenation. Recently Joshi and his associates [17] extended the SomayajuluZwolinski generalized bond-energy scheme to 15 olefinic polymer structures for estimating their enthalpies of formation and enthalpies of polymerization. The agreement with available experimental data is quite satisfactory, provided that allowance is made for the structural energy differences arising from tacticity of the a-olefin polymers.

THERMODYNAMICS OF POLYMERIZATION. I

349

References [1] 12] [3]

J. D. Cox, Tetrahedron, 18, 1337 (1962). R. J. Orr, Polymer, 5, 187 (1964). F. S. Dainton, T. R. E. Deilin, and P. A. Small, Trans. Faraday Soc, 51, 1710 (1955). [4] P. A. Small, Trans. Faraday Soc, 51, 1717 (1955). [5] F. S. Dainton, K. J. Ivin, and D. A. G. Walmsley, Trans. Faraday Soc, 56, 1784 (1960). [6] A. G. Evans and M. Polanyi, Nature, 152, 738 (1943). [7] K. G. McCurdy and K. J. Laidler, Can.J. Chem., 42, 818 (1964). [8] C. S. Marvel and J. C. Cowan, j.Am. Chem. Soc, 61, 3156 (1939). [9] P. J. Flory, Principles of Polymer Chemistry, Cornell Univ. Press, Ithaca, N.Y., 1953, pp. 246-256. [10] F. H. Westheimer, in Steric Effects in Organic Chemistry (M. S. Newman, ed.), Wiley.New York, 1956, Chap. 12. [11] P. DeSantis, E. Giglio, A. M. Liquori, and A. Ripamonti, J. Polymer Sci.,A1, 1383 (1963). [12] R. M. Joshi, J. Polymer Sci., 60, s56 (1962). [13] R. M. Joshi and B. J. Zwolinski, in Vinyl Polymerization (G. E. Ham, ed.), Vol. 1, Dekker, New York, 1967, Chap. 8. [14] E. J. Prosen, W. H. Johnson, and F. D. Rossini, J. Res.Natl. Bur. Std., 37, 51 (1946). [15] P. J. Flory, J.Am. Chem. Soc, 59, 241 (1937). [16] R. S. J e s s u p . J . Chem.Phys., 16, 661 (1948). [17] R. M. Joshi, B. J. Zwolinski, and C. W. Hayes, Macromolecules, 1, 30 (1968).

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Appendix: Heats of Polymerization of Various Monomers aStandard Monomer Acenaphthylene

Acetaldehyde Acetone Acrolein Acrylamide

Acrylic acid

states ss ss cc ss ss lc 1c lc

ss ss ss sc sc lc ss

ss sc sc

Acrylic acid, n-butyl ester Acrylic acid, ethyl ester Acrylic acid, methyl ester

sc Is lc lc lc lc lc lc ss

ss Acrylonitrile

Allyl chloride Biphenyl, p-isopropenyl 1,3-Butadiene

lc lc lc

sc Is ss gg gg lc

-AH, kcal/mole

Temperature, °C

Ref.

23.5 24.0 19.6 17.6 16.9

26.9 26.9 26.9 74.5 74.5

b

0 -6

25 25

d

74.5 26.9 74.5 74.5 74.5 74.5 74.5

c

19.1 19.8 (water) 14.1 (benzene) 16.9 (acetone) 19.5 (water) 13.8 (hexane) 16.0 18.5 (water) 18.4 (water) 17.6 (benzene) 17.2 (carbon tetrachloride) 17.8 (hexane) 18.5 19.1 18.8 18.6 18.7 18.8 18.6 20.2 19.4 17.3 18.3 18.3 18.5 18.5 8.1

17.4 18.7 17.6

20 25

74.5 74.5

b b c c d

b e e e e c f g c

c

74.5 74.5

c e

25 25

g

g

74.5 76.8 74.5

h

25 20

g

74.5 76.8

h i

25

j

74.5 74.5 74.5

k

-15 25 25 25

i h f

k c k 1

1 in

J

THERMODYNAMICS OF POLYMERIZATION. I

351

Appendix (continued) Stan-

Monomer 1-Butene cis-2-Butene trans- 2-Butene Chloral Chloroprene Ethylene

Ethylene, tetrafluoroEthylene, nitroFormaldehyde

1-Heptene 1-Hexene Isobutene

dard states gg lc gg lc gg lc gc lc sc lc gg go gc gc gc gc gc gg lc lc gc gc gc gc gc gg lc gc

lc ss ss

Isoprene

Itaconate, dimethylMaleic anhydride

-AH,

kcal/mole 20.7 20.0 19.1 17.9 18.1 17.0 17 9 8

16.2 22.35 25.4 25.9 24.2 25.5 25.9 41.5 37 39

21.7 17.2 12.2 13

13.2 16.3 20.6 19.8 17.2 11.5 12.8 12.9 12.6 16.9 17.9

lc gg lc Is Is ss

15.7 14.5

Is

14

17

Temperature, °C

Ref.

25 25 25 25 25 25 25 50 50

n

61.3

p

25 25 25 25 25 25 25 25 25 25 25 25 25 25 80 25 25 25 25 25 -50 25 25 25

74.5 34.6 26.9 74.5

n n n n

n o o o q q r

V t

u V

w X

y

z aa aa u bb n n u

r,cc

cc dd t

1 ee ff gg b ff

HIDEO SAWADA

352 Appendix (continued)

Monomer Maleimide

Methacrolein Methacrylamide

Methacrylic acid

Methacrylic acid, benzyl ester Methacrylic acid, n-butyl ester

Methacrylic acid, tert-butyl ester Methacrylic acid, cyclohexyl ester

Standard states ss ss ss ss

74.5 74.5 74.5 74.5 74.5 74.5 74.5

ss ss sc lc

lc

ss ss ss ss lc

Methacrylic acid, n-hexyl ester Methacrylic acid, /3 -hydroxyethyl ester

74.5 74.5 74.5

Ref. hh hh hh hh

c e e e e c

20 25

f

74.5

c

13.4

76.8

ii

lc lc lc Is

13.9 13.5 13.7 14.3

25

g

76.8 74.5 26.9

h

Is

13.0

26.9

lc

12.2 12.7

76.8 26.9

j] b

14.8 13.7

26.9 74.5

b

Is lc

Methacrylic acid, ethyl ester

Temperature," C

16.1 (chlorobenzene) 21.4 (dioxane) 21.2 (acetonitrile) 20.9 (dimethyl formamide) 15.6 13.4 (water) 10.2 (chloroform) 9.4 (acetone) 8.4 (benzene) 10.1 15.8 (water) 13.5 (water) 13.6 (methanol)

Is

Methacrylic acid, p-ethoxyethyl ester

-AH, kcal/mole

g

ii

b

h

120 25

74.5 26.9

h

Is

14.4 13.8 14.2 13.8

lc Is

14.0 14.4

25

g

26.9

b

lc

11.9

25

lc lc lc

kk g b

THERMODYNAMICS OF POLYMERIZATION. I

353

Appendix (continued) Standard

Monomer Methacrylic acid, /3-hydroxypropyl ester Methacrylic acid, isobutyl ester Methacrylic acid, isopropyl ester Methacrylic acid, methyl ester

Methacrylic acid, phenyl ester Methacrylic acid, n-propyl ester Methacrylonitrile

states

Styrene

Temperature, °C

Ref.

lc

12.1

25

g

1c

14.3

74.5

h

lc

14.3

74.5

h

lc

13.9 13.4 12.9 13.6 13.3 13.8 12.9 14.0 13.7 14.0 13.1

76.8 130 20 25 74.5 26.9 130 74.5 74.5 74.5 24

P

lc lc lc lc Is ss ss ss ss Is lc

12.3

76.8

ii

lc

74.5 74.5 130 74.5

h

lc

13.7 13.5 15.3 17.7

ss gg gg sc gc gg lc gg lc lc lc lc lc

8.7

19.2 18.1 16.5 24.9 20.7 20.1 17.8 16.7 16.1 16.5 16.4 17.4

-5 25

lc

ss

Methyl vinyl ketone Naphthalene, 2-isopropenylcis-2-Pentene trans-2-Pentene Propylene

-AH, kcal/mole

(water) (water)

(acetonitrile) (hexane)

f g

d,h b II j mm h h h

25 -78

25 25 25 25 25 76.8 74.5 26.9 127

b mm ,uu

HIDEO SAWADA

354 Appendix (continued) Standard Monomer Styrene (cont'd.)

Styrene, Styrene, Styrene, Styrene, Styrene,

o-ehlorop-chloro2,5-dichloroar-ethyl a-methyl

Styrene, 2,4,6-trimethyl Vinyl acetate

Vinyl benzoate Vinyl n-butyl ether N-Vinylcarbazole Vinyl chloride

states Is Is Is ss Is 1c lc 1c lc

lc lc ss ss ss ss lc lc Is Is Is lc ss ss ss lc lc sc gc gc lc lc

Is Vinylidene chloride

Vinylidene fluoride Vinyl propionate

lc lc lc gc lc gc lc

-AH, kcal/mole 17.5 17.4 17.7 15.9 16.8 16.4 16.0 16.5 16.3 8.4 8.2 8.0 7.0 8.5 8.0

16.7 21.3 20.0 21.4 21.6 21.0 21.5 20.7 20.5 20.2 14.4 15.2 31.5 30.7 17

Temperature, °C

Ref.

25

ss

26.9 26.9

b

-60 24

76.8 76.8 76.8 76.8 25 -20 -20 30 -20 -20

b dd oo tt tt tt tt

w rrnij ww irnijWw XX

k WW

26.9 76.8

b

25 25 24

yy zz

74.5 74.5 74.5 74.5 74.5 40-60 74.5

i

OO

e e e e h aa.i c

25 25 25 25

bbb , ccc bbb,ddd

74.5 76.8

eee

26.7 22.9 14.4 18.0 17.5 24.0 17.7

74.5

35

25

20.5

74.5

25

74.5 25

1 bbb i bbb h bbb c fff e,h

THERMODYNAMICS OF POLYMERIZATION. 1

355

Appendix (continued)

Monomer 2-Vinyl pyridine

Standard states lc

Is sc 4-Vinyl pyridine

lc ss

-AH, kcal/mole 17.1 18.0 17.6 18.7 18.7

Temperature, °C 74.5 74.5 74.5 74.5 74.5

Ref. c c c e e

a Monomers are listed alphabetically; cyclic monomers are listed in Chapter 5 of this review, to be published in this journal in the future. b F . S. Dainton, K. J. Ivin, and D. A. G. Walmsley, Trans. Faraday Soc, 56, 1784 (1960). C R. M. Joshi, Makromol. Chem., 55, 35 (1962). d V. A. Kargin, V. A. Kabanov, V. P. Zubov, and I. M. Papisov, Dokl.Akad. Nauk S.S.S.R., 134, 1098 (1960). e R. M. Joshi, J. Polymer Set., 56, 313 (1962). {A. G. Evans and E. Tyrrall, J. Polymer Set., 2, 387 (1947). 8 Reference [7] h R.M. Joshi, Makromol. Chem., 66, 114 (1963). 1 L. K. J. Tong and W. O. Kenyon, J.Am. Chem Soc, 69, 2245 (1947). J J. H. Baxendale and G. W. Madaras, J. Polymer Sci., 19, 171 (1956). k H. Hopff and H. Lussi, Makromol. Chem., 62, 31 (1963). ID. E. Roberts, J. Res. Natl. Bur. Std., 44, 221 (1950). m R. A. Nelson, R. S. Jessup, and D. E. Roberts, J . Res. Natl. Bur. Std., 48, 275 (1952) r F. S. Dainton, J. Diaper, K. J. Ivin, and D. R. Sheard, Trans. Faraday Soc, 53, 1269 (1957) °W. K. Busfield and E. Whalley, Trans. Faraday Soc,59, 679 (1963). PS. Ekegren, S. Ohrn, K. Granath, and P. O. Kinell, Ada Chem. Scand., 4, 126 (1950). qR. S. Jessup, j . Chem. Phys.,16, 661 (1948). r J . W. Richardson and G. S. Parks, J. Am. Chem. Soc, 61, 3545 (1939). S F. A. Quinn and L. Mandelkern, J.Am. Chem.Soc, 80, 3178 (1958). l G. S. Parks and J. R. Mosely, J. Chem. Phys., 17. 691 (1949). U G. S. Parks and H. P. Mosher, J. Polymer Sci., Al, 1979 (1963). V D. W. Scott, W. D. Good, and G. Waddington, J. Am. Chem. Soc, 77, 245 (1955). W \V. M. D. Bryant, J. Polymer Set., 56, 277 (1962). X F. S. Dainton and K. J. Ivin, in Experimental Thermochemistry (H. A. Skinner, ed.), Wiley-Interscience, New York, 1962, p. 251. VJ. Grodzinski, A. Katchalski, and D. Vofsi, Makromol. Chem., 44-46, 594 (1961).

356

HIDEO SAWADA

Appendix (continued) J

J . B. Thompson, Formaldehyde, 3rd ed., Reinhold, New York, 1964, p. 180. aa F . S. Dainton, K. J. Ivin, and D. A. G. Walmsley, Trans. Faraday Soc., 55, 61 (1959). bb Y. Iwasa and T. Imoto, J. Chem. Soc. Japan, Pure Chem. Sect., 84, 29 (1963). CC A. G. Evans and M. Polanyi, Nature, 152, 738 (1943). dd R. H. Biddulph, W. R. Longworth, J. Penfold, P. H. Plesch, and P. P. Rutherfold, Polymer, 1, 521 (1960). ee R. S. Jessup and A. D. Cummings, J. Res.Natl. Bur. Std., 13, 357 (1934). ff R. M. Joshi, Makromol. Chem., 55, 35 (1962). gg A. A. Korotkov and E. N. Marandzheva, Russ.J.Phys. Chem. (English lransl.),Z7, 135 (1963). "'R. M. Joshi, Makromol. Chem., 62, 140 (1963). 11 L. K. Tong and W. O. Kenyon,,/. Am. Chem. Soc, 68, 1355 (1946). JJ L. K. J. Tong and W. O. Kenyon, J. Am. Chem. Soc, 68, 1355 (1946). • ^ R . E. Cook and K. J. Ivin, Trans. Faraday Soc, 53, 1132 (1957). 11 S. Bywater, Trans. Faraday Soc, 51, 1267 (1955). 111111 S. Bywater, Makromol. Chem., 52, 120 (1962). ™ K. J. Ivin, Trans. Faraday Soc, 51, 1273 (1955). 00 H. Miyama, Bull. Chem. Soc Japan, 29, 711 (1956). PP S . Bywater, Can.J. Chem., 35, 552 (1957). qq C. M. Fontana and G. A. Kidder, J. Am. Chem. Soc, 70, 3745 (1948). " F. S. Dainton and K. J. Ivin, Trans. Faraday Soc, 46, 331 (1950). " D. E. Roberts, W. W. Walton, and R. S. Jessup,./. Res. Natl. Bur. Std., 38, 627 (1947). " L. K. J. Tong and W. O. Kenyon, J. Am. Chem. Soc, 69, 1402 (1947). UU S. Bywater and D. J. Worsfold, J. Polymer Sci., 58, 571 (1962). w D. E. Roberts and R. S. Jessup, J. Res. Natl. Bur. Std., 46, 11 (1962). WW D. J. Worsfold and S. Bywater, J. Polymer Sci., 26, 299 (1957). XX H. W. McCormick, J. Polymer Sci., 25, 448 (1957). ^ W . I. Bengough, Trans. Faraday Soc, 54, 54 (1958). " W. I. Bengough, Trans. Faraday Soc. 54, 1560 (1958). " a M . F. Shostakovskii and I. F. Bogdanov, J.Appl. Chem. USSR, 15, 249 (1942). bbb G. C. Sinke and D. R. Stull, J.Phys. Chem., 62, 397 (1958). ccc J. R. Lacher, E. E. Merz, E. Bohmfalk, and J. D. Park, J. Phys. Chem., 60, 492 (1956). ddd J. R. Lacher, H. B. Gottlieb, and J. D. Park, Trans. Faraday Soc, 58, 2348 (1962). eee R . M. Joshi, Indian J. Chem., 2, 125 (1964). 111 W. D. Good, J. L. Lacina, B. L. De Prater, and J. P. McCullough, J. Phys. Chem., 68, 579 (1964).

Chapter 3. Thermodynamics of Radical Polymerization I.

II.

GENERAL ASPECTS A. Energetics of Radical Polymerization B. Degree of Polymerization C. Activation Energies of Elementary Reactions

358 358 359 361

GENERATION OF FREE RADICALS

361

III.

PROPAGATION REACTION A. The Polanyi Relation B. Reactivity and Heat of Polymerization C. Ceiling Temperature

366 366 368 371

IV.

INTERACTION OF RADICALS A. Combination and Disproportionation Reactions . . . . B. Interaction of Small Hydrocarbon Radicals C. Interaction of Large Hydrocarbon Radicals D. Interaction of Some Large Radicals

371 371 372 373 374

V.

FREE ENERGIES OF FORMATION O F POLYETHYLENE AND POLYTETRAFLUOROETHYLENE

376

A. Free Energies of Polyethylene Synthesis 376 B. Free Energies of Polytetrafluoroethylene Synthesis . 381 ' REFERENCES

385

357

358

HIDEO SAWADA

I. GENERAL ASPECTS A. Energetics of Radical Polymerization Radical polymerization mechanisms can be summarized as follows: 1. Initiation, I

—

2R-

vx = 2kif[I]

R- + M —

P1-

vj =ki[R-][M]

P n -+ M -

Pntl-

v p =k p [P-][M]

3. Termination, Pn' Pm" ~~*

Pn +

2. Propagation,

*n

+

*m

4. Chain transfer, p • + M -*

m

^n+m P n + P, •

v v

td:

tc

v

:

Let us represent graphically the state of the reacting system by plotting its potential energy vs the reaction path. Then the energy of activation of the process is determined by the height of the energy barrier that the reacting system must surmount, as shown in Fig. 3.1. If the reaction is exothermic, the energy of activation is simply the height of the potential barrier E p . The difference between the activation energies of the backward and forward reactions is equal to the heat of the forward process [1]. E d - E p = AHp

(1)

The apparent activation energy E a obtained from the slope of a plot of the logarithm of the rate vs l / T will be related to the individual activation energies as follows [2]: E a = Ed/2 + (Ep - Et/2)

(2)

in which Ed, E p , and Et are the energies of activation for decomposition of initiator, propagation of polymerization, and termination of polymerization, respectively.

THERMODYNAMICS OF POLYMERIZATION. I

ACTIVATED

359

STATE

/

A

I

REACTANTS

PRODUCTS

REACTION

COORDINATE

Fig. 3.1. Energy vs reaction coordinate diagram. The activation energy for the spontaneous decomposition of benzoyl peroxide is about 30 kcal/mole. For many polymerization r e actions, (Ep - Et/2) i s about 5 ~ 6 kcal/mole. The apparent activation energy for initiated polymerization i s , therefore, slightly greater than 20 kcal/mole. This corresponds to a two- to three-fold increase in rate for a 10°C temperature change. B. Degree of Polymerization When the degree of polymerization is large, the fraction of monomer removed by steps other than the propagation step is negligible, so that the rate of polymerization i s given by (3)

and the degree of polymerization P n is given by the rate of removal of monomer divided by the rate of formation of pairs of dead polymer molecule ends, i.e., v

td

+ v

tc

+

•m k p [M] k t [ P - ] + k m [M]

where fy =

+ k{ C .

(4)

3

6

0

H

I

D

E

O

SAWADA

If we assume steady-state conditions, [P •] = Vvj/Rt and VJ = vx = jfp], The degree of polymerization is given by

The rate constants for the reactions can be expressed in an Arrhenius form:

Therefore, / - E m + E D \ _,_ . / - E t / 2 - E./2 + E D \ ,„, m 1/ 7. = a exp ( P I + b exp ( l / P) (6) \ RT / \ RT / where a = A m / A p , b = V2f[I]A1At/([M]Ap). By differentiating this expression with respect to temperature, we obtain _dz dT

=

-a(-Em+Ep) RT2

/ - E m + E p \ _ b(-E t /2 - E,/2 + Ep) \ RT / RT2

x exp ( - E t / 2 - E l / 2 V RT

+Ep\

/

When Pfi has a maximum value the temperature is given by the requirement that z be a minimum with respect to the change in T. Thus, we find at dz/dT = 0 _ E m - E t /2 - Ex/2 a(E m - E p ) Rln ib(E p -E t /2-E 1 /2)(

(8)

Setting TM > 0, we get following relation:

.

E m > Ep > E l±_ E i

(9)

Hence, a maximum in the degree of polymerization vs temperature curve for homogeneous radical polymerization will be observed, if we get the above relation. If the rate of initiation is constant, the relation is expressed as follows: E m > E p > Et/2

(10)

THERMODYNAMICS OF POLYMERIZATION. I

36i_

For thermal polymerization, E m > Ep >

E i

^Et

(11)

where Ej is the energy of activation for initiation. The occurrence of a maximum in the molecular weight vs polymerization temperature curve for polyvinyl acetate and for polymethyl methacrylate seems to demonstrate that the relation E m > E p > E^/2 can be accepted [3,4]. If transfer reactions are negligible, the temperature coefficient of molecular weight will depend upon the initiation process. If an initiator is used, it follows that d In P n dT

=

E p - E t /2 - Ed/2 RT5

(12)

Since E p - Et/2 is 5-6 kcal and E$ is about 30 kcal, this quantity will be negative and the molecular weight will decrease with increasing temperature. The same is true in thermal polymerization. In photopolymerization in the absence of transfer, d In Pn/dT = (Ep - Ed/2)RT2, which is positive. This is the only case in which molecular weight increases with temperature. C. Activation Energies of Elementary Reactions .. Values of AHp, Ep, and Et for several polymerizations are shown in Table 3.1. Except for the dienes, E p is near 7 kcal/mole for most monomers, and Ej. varies from 3 to 5 kcal/mole. The frequency factors Ap vary over a fifty-fold range, which suggests that steric effects may be somewhat more important than the activation energy. For example, Ap is much lower for methylmethacrylate with two substituents on the same ethylenic carbon atom than it is for less hindered monomers. As shown by the data of Table 3.1, Semenov [1] has pointed out that the activation energy E p clearly increases even when the heat of polymerization decreases. However, this relation does not hold in the case of the data collected by Flory [2]. n . GENERATION OF FREE RADICALS This is the type of reaction, which produces radicals, and its occurrence is governed by the strengths of the bonds concerned. In general, the thermal dissociation of a molecule can lead to the formation of free radical fragments. The simplest type of such a reac-

21

19

16

13

17.8

17.3

Methyl aery late

Styrene

Methyl methacrylate

Butadiene

Isoprcne

kcal/mole

AH p ,

Vinyl acetate

Monomer

Table 3.1

12

12

0.51

2.2

10

24

A P . (x 10"T) Flory [2]

5.7

G.9

5.2

4.5

Semenov [1]

9.8

9.3

G.3

7.8

7.1

7.3

Flory [2]

E p , kcal

Heats of Reaction and Activation Energies of Some Monomers

1.2

2.8

1.6

0

Semenov [1]

2.8

2.4

5

5.2

Flory [2]

t, kcal

o

m

ON

THERMODYNAMICS OF POLYMERIZATION. I

363

tion is when only one bond is broken in thermal dissociation; for example, the thermal dissociation of hexaphenylethane leads to two triphenylmethyl radicals, thus,

For a split into radicals, the bond dissociation energy, or heat of reaction, will be equal to the difference in the activation energies of the forward and reverse reactions:

where Ex and E2 are the activation energies of the forward and reverse reactions, respectively, and D is the dissociation energy of the Rx—R2 bond. The reverse reaction involves the recombination of two radicals and, in general, E 2 £ 0 and, hence, Et = D. The activation energy E of the primary bond dissociation is equal to the bond dissociation energy D, conveniently symbolized as D(R, - R2). In this way bond dissociation energies have been measured and the factors influencing these energies have been investigated [5-7]. The dissociation energy of a particular bond type is not a constant but depends upon the molecular environment of the bond. It must be emphasized that the thermochemical bond energy (or simply bond energy) is a mean quantity and that the bond dissociation energy is the energy required to break an individual bond. Bond dissociation energies, or heat of the above reaction at 0°K, have been expressed in terms of heats of formation Qf of molecules and radicals: Qf(R2-) - Qffo - R2)

(13)

364

HIDEO SAWADA

For example, the dissociation energy for the CH3—Clbond,D(CH3—Cl), may be calculated. Thus, for the reaction CH3—Cl

CH3

Cl

D(CH3-C1) = Qf(CH3-) + Qf(Cl •) - Qf(CH3-C1)

(14)

Substituting the appropriate heats of formation, we obtain D(CH3—Cl) = 32.0 + 29.0- (-19.6) =i 80.6 kcal/mole

(15)

Table 3.2 summarizes available data on bond dissociation energies for molecules in organic free radical reactions (taken from the review of Kerr [8]). It is seen that bond dissociation energies are influenced by the molecular environment. The delocalization energy of the radical may confer stability on the radical and lead to a low bond dissociation. Another factor which may influence bond dissociation energies is the relief of steric strain as dissociation occurs. Repulsion forces between large groups in the molecule may give instability to the molecule and lead to a low bond dissociation energy. Table 3.2 Selected Bond Dissociation Energies [8] . Bond CH-H CH2—H CH 3 -H CH2=CH-H HO-H HS-H CH 3 S-H CH 2 =CH 2 (CH 3 ) 2 CH-CH(CH 3 ) 2 CH3—CH3 CH3CO— CH3 CH3-OH C2H5—OH CH 3 O-CH 3

D238, kcal

Bond

D298, kcal

108 104 104 104 119 90 88 167 78 88 82 91 91 80

CH3-C1 CC13—Cl CH3-I HO - O H CH3O—OCH3 CH 3 COO-OOCCH 3 C 2 H 5 COO-OOCC 2 H 5 (CH 3 ) 2 N-N(CH 3 ) 2 (CH 3 )NH-NH(CH 3 ) •CH=CH-H • CH2-H •O-H •S-H

84 73 56.3 51 36.1 30 30 42

47 42 108 101 81

THERMODYNAMICS OF POLYMERIZATION. I

365

In general, D(C—H) values for radicals are low compared to those of saturated molecules. The presence of an odd electron in a molecule has a weakening effect on the surrounding bonds. This phenomenon seems to account for the ease of radical decomposition, rearrangement, and disproportionation reactions. To give radicals by thermal dissociation at relatively low temperatures, a molecule must contain a weak valency bond such as that between oxygen atoms in peroxides. A variety of such basic structures are known, involving chiefly C—C, N—N, O—O, and S—S bonds and their combinations, and containing various substituent groups that decrease the bond dissociation energies by resonance stabilization of the resulting radicals. The basic requirement for the thermal formation of free radicals at ordinary temperatures in the liquid phase is a structure possessing a covalent bond with a dissociation energy of 20-40 kcal. Many thermal sensitizers require an activation energy of roughly 30 kcal/mole for dissociation. As these radicals are unstable and reactive, the activation energy of the primary bond dissociation is equal to the bond dissociation energy. The energies of activation for dissociation also depend upon the nature of solvent—these can be attributed to differences between the extents to which the initial and transition states are solvated. The empirical Morse potential [9] for diatomic molecules, as shown in Fig. 3.2, is given by V=D0{l-exp[-a(r-re)]}2

(16)

in which a is an empirical constant, and r is the interatomic distance. Equation (16) yields when r = «

V = Do

Since the minimum, V= 0 :

when r = r e

Do is, thus, the energy of dissociation D plus the zero-point energy. In calculating Do, we first note that it is the maximum height of the potential curve for large r. If a molecule had this amount of energy, its atoms could just become independent of each other. Therefore, (Evib)max

= D

o

366

HIDEO SAWADA

V

re

r

Fig. 3.2 Morse potential for diatomic molecules. Since the zero-point energy is roughly (£)h!/0l we now have D

= ( E vib)max-

(17)

where h is Planck's constant, and v0 is the fundamental vibrational frequency. HI. PROPAGATION REACTION A. The Polanyi Relation In Fig. 3.3 are shown two types of propagation reactions having different heats of polymerization. Curve A corresponds to a reaction such as AAACH2—CH2- + CH2=CH2, and curve B corresponds to a reaction such as AAACH2—CHX- + CH=CHX, where X represents phenyl group. It is assumed for purposes of illustration that the reactants have the same potential energy. The energy of activation and the heat of polymerization are indicated in the diagram. Because of resonance stabilization, the potential energy of the polystyrene radical is appreciably lower than that of the polyethylene radical. Curve B is lower than curve A. Therefore, the intersection of the potential curves is shifted (only) vertically by a change of Q.

THERMODYNAMICS OF POLYMERIZATION. I

367

CO

a. Mi

REACTION

COORDINATE

Fig. 3.3. Intersections of potential curves for processes with heats of reaction. If the pairs of curves (A and B) have very similar shapes and if the effects due to stabilization of the transition state are the same, an increase in heat of polymerization by the amount AQ will cause a smaller decrease in activation energy AE. Hence, AE=-aAQ

0 < a < l

(18)

where a is positive but less than 1, its magnitude depending upon the shapes of A and B near the point of intersection. Evans and Polanyi [10] were the first to find existence of the above relation for exothermic reactions. This relation was proposed and verified experimentally somewhat later using the reactions between Na atoms and a homologous hydrocarbon series. Integration gives E = A - aQ

(19)

where A is a constant for a homologous series. This type of relation, usually referred to as the Polanyi relation, is only applicable to reactions of radicals or atoms with a series of closely related compounds.

368

HIDEO SAWADA

B. Reactivity and Heat of Polymerization The reaction considered here may be of the following type: H H H H \ _ / II R---C, — C« R—C,—Co— / l \ i i H X H X — rx r2— — rx r2— During this reaction the following changes occur: (a) the interaction between the 77 electrons of the double bond is broken and a o bond formed between C1 and R, (b) the u bond between Cx and C2 is extended from a distance of 1.45 to 1.54 A, (c) the C—H links are changed from trigonal to tetrahedral symmetry about the carbon centers. Of these changes, (a) and (b) have been studied, and a simple model involving the free electron on the radical R and the ir electron on the C=C has been used. Evans, Gergely, and Seaman [11] have computed such energy of the system as a function of the distances rx and r 2 . Figure 3.4 shows the results obtained. On these contour surfaces, the point I represents the energy and configuration of the initial state of the system, and F, that of the final state. The problem is to determine the factors influencing the height of the activation energy barrier T. Two methods are available for the treatment of such problems. The main conclusions are as follows: (1) The activation energies of such reactions are low, of the order of 3 to 10 kcal; (2) the weaker the bond formed between the attacking radical and the carbon center of the double bond, the higher the activation energy; and (3) the more exothermic the reaction, the lower the activation energy of the primary radical attack. In a reaction of the type /w\CHY- + CH2=CHX

* - /v^CHYCH2CHX •

the heat of reaction can be expressed as [10] <2°)

Q = Qo - R a + Rf - R ra in which Qo is the heat of the reaction /w\CH 2 - + CH2=CH2

»-

/w\CH2—CH2—CH2 •

and Ra, Rf, and R ra are the resonance energies of the attacking radical, the radical formed, and the monomer, respectively.

THERMODYNAMICS OF POLYMERIZATION. I

369

20

A ^ o

F

.6

)

— . 10

"1-5

.

-

20

T

—

30

25 Tc CH,. A

35

20

B

V

F

* —

T (

\

'

10

10

15

20 < 0H= r l . A

25

30

Fig. 3.4. Energy surfaces showing (A) the potential energy contour of the system CH3- • •CH2=CH2- • -CH2—CH2—CH2— as a function of the internuclear distance and (B) the potential energy contour of the system OH + CH2=CH2— HO—CH2—CH2— as a function of the internuclear distance. The initial transition and final state of the system are represented by the points I and F, respectively; T is the activation energy barrier [11].

The potential energy of initial state Uinit = - ( A c = c + Rm + Ra>

is expressed in the form (21)

where A c = c is the 77-bond dissociation energy of C=C linkage. The potential energy of final state Ufinaj is c-c + Rf)

(22)

where A c _ c is the cr-bond dissociation energy of C—C linkage. Thus, the difference of heat of reaction can be expressed as

370

HIDEO SAWADA

Q = u init ~ ufinal = (2AC_C - A c = c ) - R a + Rf - R m = Qo - R a + Rf - R m

(23)

and, hence, AQ = Q - Qo = - R a + Rf - R m (24) The difference of activation energy is given by the Polanyi relation as (25) AE = E - Eo = - k ( - R a + Rf - R m ) where k is a constant. The activation energy of an initiation or propagation step should vary from one system to another according to E = Eo - k(Rf - R a - R m )

(26)

where Eo is the activation energy for a reaction corresponding to QQ. This treatment leads to the conclusions outlined below. Let us compare activation energies of homopolymerization for monomers Mx and M2, in which the attacking radical is similar to the radical formed: Ex = Eo + kR mi and, similarly, E2 = Eo + kR mz Thus, (E, - E2) = k(R mj - R mz )

.

•

(27)

These considerations are consistent with the empirical observation that the chief factor governing the reactivity of a vinyl monomer is the extent to which the double bond is conjugated with other unsaturated groups. (mj styrene,k p = 176 [12] (m2) methyl methacrylate,kp= 367 [12] (m3) methyl acrylate,k p = 2090 [12] (m4) vinyl acetate,k p = 3700 [12]

THERMODYNAMICS OF POLYMERIZATION. I

371_

Although values of R m are not known accurately, it is very likely that they are in the sequence R m i > R m2 » Rm;j > R m4 . Thus, high reactivity of the nonresonance-stabilized vinyl acetate radical, as compared with the three "conjugated" radicals of styrene, methyl methacrylate, and methyl acrylate, is very evident. C. Ceiling Temperature In 1938, Snow and Frey [13] pointed out that copolymerization of low alkanes and sulfur dioxide would not proceed above a certain temperature, which they termed the "ceiling temperature." This ceiling temperature was found to be a characteristic of the olefin and independent of the catalyst system. There have since been reports in the literature of polymerizations, mainly at high temperatures, which show deviations from normal kinetics. They will not proceed to complete conversion and produce a final equilibrium concentration of monomer which is dependent only on the reaction temperature. Bywater [14] studied the photosensitized polymerization of methyl methacrylate in solution over the range 100-150°C. The reactions did not proceed to completion; at each temperature, the final value of the concentration of monomer was independent of the initial value and corresponded to the concentration of monomer in equilibrium with polymer. At 132.2°C, for example, the limiting value of the concentration of monomer was close to 0.3 mole/liter. For polymers derived from vinyl monomers, the concentrations of monomer in equilibrium with polymer are very low at ordinary temperatures; for example, at 25'C the calculated values for polystyrene and polymethyl methacrylate are, respectively, 10"6 and 10~3 mole/liter. However, at 0°C the equilibrium concentration of a-methyl styrene is 0.76 mole/liter. For many systems, the equilibria cannot be e s tablished at higher temperatures because of side-reactions. The polymerization of a-methyl styrene does not proceed to completion even at -40°C. For a long time it was thought that a-methyl styrene could not be polymerized by a free-radical mechanism. However, in 1958, Lowry [15] proved that this was due to the low ceiling temperature and that this monomer can be polymerized at low temperatures. IV. INTERACTION OF RADICALS A. Combination and Disproportionation Reactions Pairs of radicals can interact by either combination or disproportionation, and the relative importances of these two processes might be afforded by a study of the thermodynamic relations of the system.

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HIDEO SAWADA

As part of a study of this problem, it will be worth while to investigate the changes.in enthalpy, entropy, and free energy accompanying the alternative reactions. Bevington [16] has studied this problem and has been able to make deductions concerning the effect of temperature, concentration, and molecular size on the relative importance of the two modes of termination. B. Interaction of Small Hydrocarbon Radicals The reactions possible for two ethyl radicals are C2H4 + C2H8

(la)

n-C4H10

(lb)

2C2H5

The difference in the changes of free energy accompanying these r e actions is (AGC-AGD)

or

(AHC - A H D ) - T ( A S C - ASD)

The subscripts C and D refer to combination and disproportionation, respectively; (AHC - AHD) may be evaluated either from the heats of combustion of ethylene, ethane, and n-butane or from their heats of formation; the entropy term may be calculated either from the standard entropies of formation. Similar calculations can be carried out for the interaction of other radicals. For 2 moles of ethyl radicals at 25°C and 1 atm reacting at constant pressure, according to reactions (la) and (lb), we have AHC - AHD = - 2 2 . 1 k c a l and ASC - ASD = -33.4 cal/deg The entropy term must be corrected since an increase in entropy occurs when 1 mole of ethane is mixed with 1 mole of ethylene each at 1 atm to give a mixture also at 1 atm. This increase in entropy is -R(N 1 In nj + N2 In n 2 ), where N refers to the number of moles of a substance and n to its mole fraction in the mixture. In this case Nj = N2 = 1 and nx = n2 = \ , so that AS m j x j n g is 2.76 cal/deg and (ASC - ASD) is -36.2 cal/deg. The only effects on (AHC - AHD) are those due to gas imperfections and they are neglected; the entropy

THERMODYNAMICS OF POLYMERIZATION. I

373

change (S2 - Sx) per mole for a change in pressure is R In ( P - J / P J ) . Noting that there are 2 moles of disproportionation products and only 1 of combination product, (ASC — ASD) is changed to—36.2 — R In 106 or -63.6 cal/deg. Combination is more exothermic than disproportionation, but the entropy of the products of the latter reaction is greater than that of the combined product; at 25° C and a pressure of 1(T8 atm, (AGC - AGD) is - 3 . 1 kcal, indicating that the enthalpy term outweighs the entropy term. If (AHC - AHD) and (ASC - ASD) do not vary appreciably with temperatures, there must be a temperature at which (AGC - AGD) is zero; for ethyl radicals this "crucial" temperature is 337°C if the pressure is 1 atm, and 74°C if it is 10"6 atm. C. Interaction of Large Hydrocarbon Radicals The hypothetical case of long straight-chained hydrocarbon radicals interacting at 25°C in the gas phase to give gaseous products is considered first; this corresponds to mutual termination process for the free radical polymerization of ethylene under these idealized conditions. The interacting radicals are, for simplicity, considered to be of the same size; the competing reactions are RCH=CH 2 + RCH2CH3

(2a)

RCH2CH2CH2CH2R

(2b)

where R is a long hydrocarbon chain. There are two distinct methods for evaluating (AHC — AH D ), namely, by extrapolation of data for hydrocarbons of low molecular weight and by comparison with the heat of polymerization. Expressions have been given [17] for heats of formation in kilocalories per mole for gases at 1 atm and 25°C. n-Paraffins: AH = -10.908 - 4.926n ' Olefins with terminal CH=CH 2 : AH = 19.592 - 4.926n where n is the number of carbon atoms, supposed large. Therefore, AHC - AHD = -10.908 - 4.926 x 2n + 10.908 + 4.926n - 19.592 + 4.926n = - 1 9 . 6 kcal

(28)

If differences in strength of C—H bonds of the various types are neglected, disproportionation is equivalent to the breaking of one C—C bond and the making of one C = C bond, and combination to the

374

HIDEO SAWADA

making of one C—C bond. In the propagation step of the polymerization of ethylene, one C=C bond is broken and two C—C bonds are made, therefore (AHC — AHD) ought to equal AHp. The expression (-23.49n + 6.4) cal/deg has been given for the molar entropy of formation of a gaseous normal paraffin with n carbon atoms in the standard state [18], and (-23.4n + 37.2) cal/deg for the similar quantity for a 1-alkene [19]. These expressions lead to (ASC - ASD) = -37.2 cal/deg, but, allowing for the increase in entropy due to the mixing of the products of disproportionation, the final value is approximately-40 cal/deg. If (AHC - AHD) is -21kcal the "crucial" temperature is about 252°C; at 25°C, (AGC - AGD) is 9.1 kcal [16], whereas Bryant [20] gives -8.0 kcal, as discussed in Section V.A below. Considering the difficulties the agreement is good . The term (ASC — ASD) can be calculated by another method if sufficient is known of the molecules concerned. Calculation of (ASC - ASD) is resolved into three distinct parts, namely, estimates of the translational, vibrational, and rotational contributions; restricted internal rotational contributions are included in the vibrational term, and electronic contributions are neglected. To summarize, for reaction of radicals of weight 1015 at 25°C, (ASC - ASD) is equal to -40 cal/deg for the gas phase at 1 atm, and —42 cal/deg for a solution in an ideal solvent when the volume fraction of radicals is 10~5. Further, it may vary with the size of the interacting radicals, the temperature, and the pressure in the gas phase or the concentration in solution. D. Interaction of Some Large Radicals The calculation of the difference in free energy changes for the reactions AAACH=CH(C 6 H 5 ) +

AA/\CH 2 CH 2 (C 6 H 5 )

A^\CHCH(CH)CH(CH)CH

(3a)

(3b)

which are the alternatives for mutual termination in the styrene polymerization, is less satisfactory than the equivalent calculation for polyethylene radicals. The enthalpy term cannot be set equal to the heat of polymerization since combination of radicals is a head-to-head reaction, whereas the growth of polymer is almost certainly head-totail. The reactions used as models for the reactions of the large radicals are given by

THERMODYNAMICS OF POLYMERIZATION. I ,CH 3 CH 2 C 8 H 5 2CH 3 CH(C 8 H 5 )

\

375

CH 2 =CH(C 6 H 5 )

(4a)

(4b)

CH 3 CH(C 8 H 5 )CH(C e H 5 )CH 3

The long polystyrene chains which play no part in the reactions are replaced by hydrogen atoms. The quantity (AHC - AHD) is calculated from the heats of combustion of the three products. Here the heats of combustion are styrene, 1046 kcal/mole (liquid at 18°C); ethyl benzene, 1091 kcal/mole (liquid at 18°C); and dibenzyl, 1811 kcal/mole (crystals at 18°C). Since the molar latent heat of liquid dibenzyl is 4 kcal/mole, the heat of combustion of the liquid at that temperature will be 1815 kcal/mole. Since the substitution of a methyl group for hydrogen raises the heat of combustion by 156.5 kcal/mole, the heat of combustion of dimethyldibenzyl is taken as 2128 kcal/mole. This gives (AHC - AHD) = -9.0 kcal/mole. The calculations of (AHC - AHD) for a series of systems are summarized in Table 3.3. Table 3.3 Values of (AHC-AHD) for Some Monomeric Reactions [16] Heats of combustions, (AHC-AHD), kcal/mole kcal/mole

Monomer

Substances in model reactions

Acrylic acid

CH 2 =CH-COOH CH3—CH2 —COOH CH3CH(COOH) CH(COOH) CH3

327.5 365 675

-17.5

Aci-ylonitrile

CH2=CH-CN CH3—CH2-CN CH3CH(CN)CH(CN)CH3

420.5 456.4 861

-16.0

Methacrylic acid

CH 2 =C(CH 3 )COOH CH 3 -CH(CH 3 )COOH (CH 3 ) 2 C (COOH) C (COOH) (CH 3 ) 2

484 517.4 990

-11.5

Styrene

CH 2 =CH-C 6 H5 CH3 CH2 CgH5 CH3CH(C6H5) CH(C6H5) CH3

1046 1091 2128

-9.0

CH 2 =C(CH 3 )-C 6 H 5 CH 3 -CH(CH 3 )-C 6 H 5 (CH 3 ) 2 C(C 6 H 5 )C(C 6 H 6 )(CH 3 ) 2

1203 1247 2443

-7.0

a-Methyl styrene

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HIDEO SAWADA

The entropy term cannot be estimated with certainty but it is likely to be similar in magnitude for all radicals. If the monomer has an a-methyl group, (ASC - ASD) is made slightly more negative because disproportionation can occur in two ways, giving the unsaturated product as either RCH=C(CH3)X or RCH2CX=CH2. At a given temperature and concentration of radicals, the magnitude of (AGC - AGD) varies appreciably from one system to another. Variations are due mainly to differences in (AHC — AHD), but for radicals derived from monomers with a-methyl groups, the additional contribution to ASD tends to make (AGC - AGD) more positive. V. FREE ENERGIES OF FORMATION OF POLYETHYLENE AND POLYTETRAFLUOROETHYLENE A. Free Energies of Polyethylene Synthesis 1. Heats of Formation of Hydrocarbon Free Radicals. The calculation of the heats and entropies of free radicals has been carried out for the case of hydrocarbons and applied to polyethylene synthesis by Bryant [20]. The heat of formation of a hydrocarbon free radical is most readily calculated from that of the parent hydrocarbon, in combination with the heat of association of atomic hydrogen and the bond dissociation energy. ("t

C

TT

.

nH(2n*2)

H.

f-f

c

..

TT

n H (2nn)

_i_ TT

+

H

1 TT

The heat of formation of methyl free radicals i s , for example, AH3 C + 2H2 — CH4 -17,889 CH4 . HC

— CH 3 - + H— |H2 + f H2 -~ CH 3 -

+102,000 -52,089 +32,000 cal/mole

Heats of formation of primary n-alkyl free radicals are given in column two of Table 3.4 for radicals through n-dodecyl. 2. Molecular Entropies and Entropies of Formation of Hydrocarbon Free Radicals. The entropies of low-molecular-weight gaseous

THERMODYNAMICS OF POLYMERIZATION. I

377

Table 3.4 Free Energy of Formation of Gaseous n-Alkyl Free Radicals [20] Free radical CH 3 . C2H5n-C3H7 • n-C4H9 • n-C 5 H n • n-C 6 H, 3 . n-C,H,5. n-C 8 H 17 • n-C 9 H 19 • n-C 10 H 2 in-C u H 2 3 . n-C12H25-

AH°

AS"

AG°

32,022 25,175 18,091 12,099 6,411 1,151 -3,775 -8,701 -13,627 -18,553 -23,479 -28,405

-2.07 -21.64 -46.94 -69.04 -92.44 -115.83 -139.21 -162.60 -185.99 -209.37 -232.76 -256.14

32,600 31,600 32,100 32,700 34,000 35,700 37,700 39,800 41,800 43,900 45,900 48,000

hydrocarbons can be calculated from spectroscopic data. Methods for the calculation of the entropies of low-molecular-weight gaseous hydrocarbons have been worked out in detail by Pitzer [21,22]. The calculation of the molecular entropies of hydrocarbon free radicals was based on a consideration of the following entropy contributions: translational, rotational, electronic, vibrational, and restricted internal rotational. Translational entropies were calculated by means of the Sackur-Tetrode equation expressed as follows: S t r = 6.8635 log M + 25.9917

(29)

where M is the molecular weight. The rotational effect for the molecule as a whole was calculated by the usual equation for a three-dimensional rigid rotator: S r o t = 2.2878 log I I ' I " - 4.5757 log a + 284.6299

(30)

where I, I', and I" are the principal moments of inertia, and a is the symmetry number. The moments of inertia were estimated approximately by a graphic analysis, assuming a planar arrangement of groups about the trivalent carbon atom with normal bond lengths and •120° bond angles. Except in the case of methyl free radical, the vibrational entropy at 298.2°K was assumed equal to that of the parent hydrocarbon. A

378

HIDEO SAWADA

similar assumption was made in the case of the restricted internal rotational of ethyl, n-propyl, isopropyl, and tert-butyl radicals. The vibrational entropy of the methyl free radical was calculated by means of the Planck-Einstein approximation, assuming the following crude frequency assignments: w = 900(1), 1400(2), and 3000(3), where the unit is the wave number and the numerals in parentheses are the multiplicities. Using the molecular entropies the entropy of formation of methyl free radical is, for example, AS° =S°CH3. -(S' c +|S° H2 ) = 46.11- {1.36 +1(31.21)} = -2.07 cal/deg mole The entropies of formation of several hydrocarbon free radicals at 25°C are given in the third column of Table 3.4 3. Free Energies of Formation of Hydrocarbon Free Radicals. In previous sections we have discussed methods for obtaining AH0 and AS0 at any given temperature. Since AG° = AH0 - T AS0 it is easy to calculate AG° at a given temperature if AH° and AS° are known. The free energy of formation of CH3 • at 25°C is, thus, AG° = AH0 - T AS0 = +32,000 - (298) x (-2.07) = 32,600 cal/mole In the last column of Table 3.4 are given free energies of formation at 25°C for several hydrocarbon free radicals. The magnitudes of the heats and free energies of formation of the short-chain hydrocarbons and radicals are summarized in Fig. 3.5. The free radicals have the largest positive AH0 and AG° values, with the alkenes next, and finally the alkanes, values of which are mostly on the negative side of zero. An interesting feature of the AG° curve for the alkyl radicals is the near equality of the first four members of the series. This is followed by a gradual slope upward toward more positive values. The AH° curves, on the other hand, slope away

THERMODYNAMICS OF POLYMERIZATION. I

379

AG. n-ALKYL FREE RADICALS

o

IS

a ui z u

-20 o UJ X

-40

2

4

6

8

CARBON ATOMS PER MOLECULE

Fig. 3.5. Free energies and heats of formation of homologous hydrocarbon series, including the alkyl free radicals [20].

fairly rapidly with curvature toward the lower positive and higher negative values. In all three series the free energies become more positive with increasing molecular weight, and the heats of formation more negative. 4. Thermodynamics of Polyethylene Synthesis. The average heat and free energy involved in the polymerization of ethylene have been calculated by Jessup [23] for products of increasing degrees of polymerization, x, per mole of monomer. The results with x equal . to infinity are as follows: xC2H4(g) — (C2H4)x(g)

AH° = -22.3kcal,

AG° =-12.2 kcal (31) These values apply to the overall reaction by which polythene is formed but tell very little about the individual reaction steps, since the latter are believed to take place by a series of free radical mechanism. The three following reactions are examples of initiation by free radicals:

380

HIDEO SAWADA

H+ C2H4 = C2H5- AH° = -39.4 kcal, AG° =-33.3 kcal (32) CH3- + C2H4 = n-C3H7- AH0 =-27.0 kcal, AG° =-16.8 kcal (33) C2H5- + C2H4 = n-C4H9- AH" = -25.6 kcal, AG° =-15.2 kcal (34) The large-to-moderate negative free energy changes of reactions (32), (33), and (34) indicate that they are thermodynamically favorable. This type of reaction is also kinetically quite rapid and has a very low activation energy. The highly reactive character of the hydrogen atom, and of the methyl and ethyl free radicals, leaves little doubt that these substances can be active initiators of polyethylene synthesis. There is no fundamental difference between the initiation and chain propagation steps. Both processes involve the addition of a free radical to ethylene to form a second free radical containing two additional carbon atoms. The free energy change for chain propagation apparently reaches a constant value with the formation of the n-octyl free radical. n-C10H21 • + C2H4 = n-C12H25 • AH0 =-22.3 kcal, AG° =-12.2 kcal

(35)

The figures are, of course, the same as those for the overall polymerization step. Chain transfer with hydrocarbons capable of forming secondary or tertiary free radicals are often favorable: n-C12H25- + C3H8 = n-C12H28 + (CH3)2CH • AH° = -4.2 kcal, AG° = -3.9 kcal

(36)

n-C12H25 • + (CH3)3CH = n-C12H2S + (CH3)3C • AH0 = - 7 . 1 kcal,

AG° = - 6 . 6 kcal

(37)

Transfer reactions involving a primary free radical and a closely related alkane have negligible heats and free energies: n-C12H25 • + CI0H22 = n-C12H26 + n-C10H21 • AH0 = 0.0 kcal,

AG° = 0.0 kcal

(38)

It is well known that chain transfer is important in the polyethylene

THERMODYNAMICS OF POLYMERIZATION. I

381

synthesis. Transfer to monomer is, however, unfavorable if the vinyl radical is formed and doubtful if the ethyl radical results: n-C12H25 • + C2H4 = n-C12H28 + C2H3 •

(39)

0

AH = + 12.8 kcal, AG° = + 13.0 kcal n-C12H25- + C2H4 = n-C12H24 + C2H6' AH0 = + 1.6 kcal,

(40) AG° = + 0.4 kcal

The most probable kind of chain transfer is that involving the reaction of a growing primary free radical with a secondary hydrogen from the side of a formed polymer chain. (41) n-C12H28 + C2H5- = n-C6H13CHC5Hu + C2H6 AH° = -10.3 kcal, AG° = -9.7 kcal This indicates a favorable, although not extremely rapid, type of reaction responsible for the start of chain branching. The highly unfavorable character of the monomolecular dissociation to form an olefin and a hydrogen atom is shown below: n-C12H25 • = n-C12HM + H •

AG° = + 33.7 kcal (42) Chain termination by pairs of free radicals, either by combination or disproportionation, is extremely favorable thermodynamically. This thermodynamic advantage is offset by the low concentrations of free radicals usually present in a polymerizing mixture. Two reaction schemes for termination are 2n-C12H25 • = n-C24H50

AH° = + 41.0 kcal,

AH° = -71.8 kcal,

AG° = -59.1 kcal (43)

2n-C12H25 • = n-C12H24 + n-C12H26 AH" = -52.5 kcal, AG° = -51.1 kcal (44) B. Free Energies of Polytetrafluoroethylene Synthesis 1. Heats of Formation of Fluorocarbon Radicals. With both the heats of formation and the C—F bond dissociation energies available for various fluorocarbons, the heats of formation of the corresponding radicals can be calculated. The enthalpy of formation of the trifluoromethyl radical, for example, is available from two independent sequences of reactions [24]. The first is

382

HIDEO SAWADA

AH° CF4(g) - CF,'(g) + F-(g) C(graphite) + 2F2(g) — CF4 (g) F-(g) — £F 2 (g)

+123 kcal -217.8 kcal - 18.9 kcal

C(graphite) + f F2(g) -

-113.7 kcal

CF3 • (g)

and the second is AH3 CHF3(g) C (graphite) + — H-(g) —

CF,-(g) + H-(g) fF 2 (g) + }H2(g) CHF3(g) -|H 2 (g)

C(graphite) + { F,(g)

-

CF3 • (g)

+102 kcal -162.6 kcal - 52.1 kcal -112.7 kcal

The two sources agree within 1 kcal. Enthalpies of formation for a series of fluorocarbon radicals are given in the original paper [24]. 2. Molecular Entropies and Entropies of Formation. Parks and Huffman [25] found that linear equations for S° of fluorocarbons as a function of the number of carbon atoms will fit reliable experimental data within a few tenths of an entropy unit. The following equation of the Parks and Huffman type was derived by Bryant [24]. C n F 2n+2 (g): S°= 46.41 + 16.066n

(45)

where n is the number of carbon atoms in the molecule. Entropies of fluorocarbon free radicals were computed from those of the parent fluorocarbons by considering the following factors: (1) increase in entropy on passing from a singlet to a doublet ground state; (2) change in symmetry number; (3) approximate reduction in entropy due to loss of a fluorine atom. Comparison of entropies of the lower aliphatic fluorocarbons and their radicals lead to the following relationship for monoradicals C n F 2n+1 • (g) derived from perfluoroparaffins:

THERMODYNAMICS OF POLYMERIZATION. I

383

S° = S° (fluorocarbon) + 1.375 (due to greater multiplicity) + 1.375 (due to loss of symmetry) - 4.25 (due to loss of fluorine atom) = S° (fluorocarbon) - 1.50 ± 1.50

(46)

Numerically, for normal perfluoroalkyl radicals, Eq. (46) becomes C n F 2 n + 1 • (g): S° = 44.91 + 16.006n

(47)

for radicals above CF 3 -. The entropies of perfluoroalkyl radicals were calculated with the aid of Eq. (47). Then the molecular entropies of fluorocarbons and their radicals maybe combined withS°c and Sp to yield the corresponding entropies of formation. 3. Free Energy Changes. Where a large number n of monomer units is involved, the overall synthesis may be summarized by the following equation which ignores the individual steps of the mechanism: C2F4(g)-

l/n[C 2 F 4 ] n (g) AH° = -37.10 kcal,

AG° = - 25.32 kcal

(48)

The enthalpy of polymerization for the ideal gaseous state is given in Eq. (48). The corresponding value based on the solid polymer is C2F4 (g) -

l/n [C 2 F 4 ] n (s)

AH0 = -41.12 kcal

(49)

The difference in AH° between Eqs. (48) and (49), 4.02 kcal, is of the correct magnitude for the heat of sublimation. 4. Initiation and Chain Propagation. The first step in a chain of . free radical reactions is that of initiation. This is illustrated by the following propagation sequence: F • (g) + C2F4 (g) — CF3CF2 • (g) AH0 = -79.00 kcal,

AG" =-69.24 kcal

(50)

or CF 3 • (g) + C2F4 (g) 0

CF3CF2CF2 • (g)

AH =-41.90 kcal,

AG° =-29.67 kcal

(51)

384

HIDEO SAWADA

CF3CF2 • (g) + C2F4 (g) -

CF3CF2CF2CF2 • (g)

AH° =-37.10 kcal,

AG° = -25.32 kcal

(52)

n-C10F21 • (g) + C2F4 (g) - n-C12F25 • (g) AG° = -25.32 kcal AH0 =-37.10 kcal,

(53)

The constant values of enthalpy and free energy of propagation in Eqs. (52) and (53) coincide with those for the overall polymerization process if the latter is associated with a product of high molecular weight, since the contributions of initiation and termination become vanishingly small. The free energy of the propagation step is one of the most favorable reactions, although the large enthalpy value suggests that the reaction may become less favorable fairly rapidly with rising temperature. 5. Termination, Chain Transfer Reactions, and Chain Branching. Termination by mutual combination of two free radicals is highly exothermic: 2n-C12F25 • (g) - ' n-C24F50 (g) AH3 =-81.50 kcal,

AG° = -68.50 kcal

(54)

The corresponding termination by disproportionation is also strongly exothermic: 2n-C12F25*(g) — n-C12F28(g)+ n-C12F24 (g) AH0 = -56.80 kcal,

AG° = -55.60 kcal

(55)

Instead of a perfluoroparaffin of double the number of carbon atoms, the products here are a molecule each of perfluoroparaffin and perfluoroolefin. There is no indication of a tendency for a fluorocarbon radical to go over to a perfluoroolefin with the loss of a fluorine atom at ordinary temperature: n-C12F25 • (g) - n-C12F24 (g) + F • (g) AG° = + 56.92 kcal AH3 = + 66.60 kcal,

(56)

' The following examples suggest qualitative similarity to the comparable reactions involving hydrocarbon radicals: n-C12F28 (g) + CF3 • (g) - n-C12F25 • (g) + CF4 (g) AH3 = -0.60 kcal,

AG° = -0.19 kcal

(57)

THERMODYNAMICS OF POLYMERIZATION. I

n-C3F7- (g) + C3F8(g) -

385

i-C 3 F 7 - (g) + C3F8(g)

AH° = -13.40 kcal,

AG= = -13.00 kcal

(58)

Thermodynamically, with respect to two possible reactions, chain transfer with the monomer is even more clear-cut than in the polymerization of ethylene: n-C12H25 • (g) + C2F4 (g) -

n-C12F24 (g) + C2F5 • (g)

AH° =-12.40 kcal,

AG° = -12.32 kcal

n-C12F25 • (g) + C2F4 (g) - n-C12F28 (g) + CF 2 =CF • (g) AG° = + 1.58 kcal AH3 = + 1.60 kcal,

(59)

(60)

However, kinetics may play a more important role here than thermodynamics. It is doubtful whether chain transfer with the monomer is important in the polymerization of tetrafluoroethylene, in view of the customary high molecular weight of the product. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14]

N. N. Semenov, Some Problems in Chemical Kinetics and Reactivity (M. Boudart, transl.), Vol. 1, Princeton Univ. Press, Princeton, N.J., 1958, Chap. 1. P. J. Flory, Principles of Polymer Chemistry, Cornell Univ. Press, Ithaca, N.Y., 1953, Chap. 4. H. Sawada, Chem.High Polymers (Tokyo), 20, 561 (1963). H. Sawada, Chem. High Polymers (Tokyo), 21, 251 (1964). C. Walling, Free Radicals in Solution,Wiley, New York, 1957, Chap. 2. C. T. Mortimer, Reaction Heats and Bond Strengths, Pergamon, London, 1962, Chap. 1. M. G. Evans, in Fibres from Synthetic Polymers (R. Hill, ed.), Elsevier, Amsterdam, 1953, Chap. 3. J. A. Kerr, Chem. Rev., 66, 465 (1966). P. M. Morse, Phys.Rev., 34, 57 (1929). M. G. Evans and M. Polanyi, Trans. Faraday Soc, 34, 11 (1938). M. G. Evans, J. Gergely, and E. C. Seaman, J. Polymer Sci., 3, 866 (1948). M. S. Matheson, E. E. Auer, E. B. Bevilacqua, and E. J. Hart, J.Am. Chem. Soc.,71, 497, 2610 (1949). R. D. Snow and F. E. Frey,hid. Eng. Chem. (Ind. Sect.),30, 176 (1938). S. Bywater, Trans. Faraday Soc, 51, 1267 (1955).

386

HIDEO SAWADA

[15] G. G. Lowry, J. Polymer Sci., 31, 187 (1958). [16] J. C. Bevlngton, Trans. Faraday Soc, 48, 1045 (1952). [17] E. J. Prosen, W. H. Johnson, and F. D. Rossini, J. Res.Natl. Bur. Std., 37, 51 (1946). [18] K. S. Pitzer, Client. Rev., 27, 39 (1940). [19] J. E. Kilpatrick, E. J. Prosen, K. S. Pitzer, and F. D. Rossini, J . Res. Natl. Bur. Std., 36, 559 (1946). [20] W. M. D. Bryant, J. Polymer Sci., 6, 359 (1951). [21] K. S. Pitzer, J. Chem.Phys., 5, 473 (1937). [22] K. S. Pitzer and D. W. Scott, J.Am. Chem. Soc, 65, 803 (1943). [23] R. S. Jessup, J. Chem. Phys., 16, 661 (1948). [24] W. M. D. Bryant, J. Polymer Sci., 56, 277 (1962). [25] G. S. Parks and H. M. Huffman, Free Energies of Some Organic Compounds , Chemical Catalog Co., New York, 1932.

Chapter 4. Thermodynamics of Polycondensation I. II.

GENERAL ASPECTS

387

DEGREE OF POLYMERIZATION

388

III.

EQUILIBRIUM CONSTANT

389

IV.

RING FORMATION IN POLYCONDENSATION

394

REFERENCES

395

I. GENERAL ASPECTS The polycondensation reaction is a random one and the rate constant is independent of molecular size. If the simplest case is considered, i.e., the condensation of an w-hydroxy acid or a dibasic acid and dihydric alcohol in equimolar concentration. In this case, the carboxyl and hydroxy groups are in equilibrium with the ester group and water; i.e., [/w\OH] — [/w\COO/w\] + [H2O] The equilibrium constant for the reaction is

where nH is the concentration of water and p is the extent of reaction. The equilibrium constant for polyester is about one-hundredth of that 387

388

HIDEO SAWADA

for polyamide. Therefore, it is impossible to obtain high-molecularweight polyester by polycondensation without reduced pressure. n. DEGREE OF POLYMERIZATION The number-average degree of polymerization P n is a function [1] of the extent of reaction p:

and, hence, _ Pn =

2K/nH VI + 4K/nH

and if K/nH » 1, then

therefore, JL | 3 In K 2 I 3(1/ T ) The free energy change is given by AG = AH - T AS = -RT In K. If log K is plotted against reciprocals of absolute temperature, a linear relationship holds roughly and AH is given. Hence, at constant water concentration, we have 91ng

" = - a AH (/)

(5)

where a is a constant, and AH is the enthalpy change of the reaction. The formation of condensation polymers in the molten state is exothermic [2], and hence AH < 0. Thus, the molecular weight of the polymer made by a melt process increases as the temperature is lowered [2,3]. However, under the same vapor pressure of water, the degree of polymerization of polycapramide is increased as the temperature , rises [4]. These results show that the change in activity of water with temperature has much greater effect on the degree of polymeri-

THERMODYNAMICS OF POLYMERIZATION. I

389

zation than the change in equilibrium constant with temperature. The heat of polycondensation of polycapramide, AH = - 6 . 8 kcal, when the liquid state is chosen for the standard state of water, seem quite resonable in view of the heats of reactions between acetic acid and ammonia. If the vapor p r e s s u r e of water is employed in place of n w , the value of AH becomes 11.5 kcal (endothermic). Thus, because of the unexpectedly high heat of vaporization of water, the calculated value for the heat of condensation becomes markedly endothermic in spite of the exothermic nature of the amide formation. III. EQUILIBRIUM CONSTANT Wiloth [5] and Hermans [6] have reported that the equilibrium constant K becomes smaller with the increase of water content and the reason is ascribed to using the concentration instead of the activity in kinetics. The reaction in the e-caprolactam polymerization is very complicated, nevertheless the reaction equilibria concerning to chain or ring monomer can be shown as follows: H2O H2O

Yumoto and Ogata [7] have reported that the values of K2 a r e smaller than those of K, and the values of K become smaller with the increase of water content and approach to the mean value of K 2 . This was explained [7] as follows. The equilibrium constants become greater a s the chain length becomes longer; therefore, the equilibrium constant K becomes smaller as the mean degree of polymerization becomes smaller because of the increase of the chain monomer and oligomer contents. The difference among the equilibrium constants means that the reactivity of the functional end-group i s not equal, especially for the chain oligomers. Meggy [8] calculated equations for the activity coefficients of water and polymer in the system water-e-caprolactam. The degree of polymerization is determined by the equilibrium: x - m e r + y - m e r = (x + y)-mer + H2O =

(H2O)([x+y]-mer) '(x-mer) (y-mer)

(6)

390

HIDEO SAWADA

The polymerization does not go to completion; 5-10% of the lactam remains in the equilibrium mixture. The lactam may be regarded as being in equilibrium with 6-aminohexanoic acid and water: K K

(lactam)(H2O) 2

(6-aminohexanoic acid) Lactam is also in equilibrium with polymer: lactam + x-mer = (x + l)-mer _ ([x+ l]-mer) * 3 " (x-mer)(lactam)

l

'

In order to develop these equations, it is necessary to make two assumptions: (a) in Eq. (6) the value of Kx at any temperature is the same for all values of x and y, including x or y = 1 (this is the principle of "equal reactivity of all groups" postulated by Flory); (b) in the presence of a diluent, if a? is the activity of the whole polymer, and m x is the molecular fraction of the x-mer in the polymer, then the activity of x-mer is a2tnx. On the basis of these two assumptions, it is possible to develop Eqs. (6) to (8). Let the activity of water in the system be ax, that of the polymer be a?, that of lactam be a3, and the degree of polymerization be n. The activity of x-mer is a j l - l/n) /n x , the activity of the y-mer is a^l - l/n^'Vny, and of the (x + y)mer is ^ - l/n) x+y "Vn 2 . Substituting in Eq. (6), we have

The molecular fraction of monomer(x = 1) in the polymer is l/n 2 . Hence, K2 = a1a3n2/a2

(10)

The ratio (x + l)-mer/x-mer = ( n - l)/n. K3 = ( n - U / n a ,

(11)

Equations (9), (10), and (11) are not independent; by combining any two, the third is obtained:

THERMODYNAMICS OF POLYMERIZATION. 1

3SM

The composition may be defined conveniently as follows: 1

N2

(mols. of water) ~ (mols. of water + mols. of unit) =

(mols. of unit as polymer) (mols. unit) jls. of water + mols. of ur

(mols. of unit as lactam) N,3 = (mols. of water + mols. of unit) Nx + N2 + N3 = 1

(12)

From the Gibbs-Duhem equation, we have +N2/91nM

+ N3/91naA

=0

(13)

From Eqs. (6) to (13),

where v1 ='a 1 /N 1 , i>2 = ag/Nj, and K^ and K3 are apparent equilibrium constants defined by Kx = (Nx /^I2) P n (Pn ~ 1) and K 3 = ( p n - 1)/ PnN3The values of In u1 and In vz can be evaluated by graphical integration. The value of the true equilibrium constants Kx is given by K

i = ("i/^KT

(16)

Although Kx becomes smaller with the increase of water content, Ky has a constant value at a constant temperature—Kx is 440 at 221.5°C and 379 at 253.5°C. From these values, log Kx is given by logKi = 1.569+ 531/T AH = -2.42 kcal/mole, AS = 7.16 cal/mole deg.

(17)

392

HIDEO SAWADA

Starting from the principle of equal reactivity the entire, reversible polycondensation of polyethylene terephthalate (PET) may be symbolized by E+ E ^ Z+ G where E = — COOCH2CH2OH Z = — COOCH2CH2OOC— G = HOCH2CH2OH The forward reaction proceeds by ester interchange between the alcoholic hydroxyl group of a 2-hydroxyethyl ester end-group, and the ester linkage of another 2-hydroxyethyl ester end-group and linking the ends of two chain molecules under elimination of glycol. The reverse reaction is a glycolytic degradation and proceeds by ester interchange between a hydroxyl group of glycol and an ester link of an ethylene diester group in a PET chain. For comparison with other ester interchange reactions, Challa [9] has expressed the rate constants (k and k') and the equilibrium constant (K) in terms of the elementary reaction between a single hydroxyl group and a single ester link. Therefore, the following rate expression is used: -^=Ke2-4k'gz

(18)

where the corresponding, small letters denote the concentrations of the reactants and products. The rate of the forward reaction is represented by ke2, since one molecule of glycol can be produced by reaction between any hydroxyl group and any ester link belonging to different 2-hydroxyethyl ester end-groups. The rate of the reverse reaction is represented by 4k'gz, since the reaction between any bifunctional ethylene diester group and any bifunctional glycol mole. cule can proceed in four equivalent ways. For dg/dt = 0, Eq. (18) yields the equilibrium conditions:

Replacing the concentrations by the number of equivalents or moles present in the equilibrium mixture results in

THERMODYNAMICS OF POLYMERIZATION. I

393

(2Q)

A more practical form of Eq. (20) can be obtained by introducing the equilibrium extent of end-group reaction p e , i.e., the fraction of original end-groups that has reacted at equilibrium: p e = (2U0 - E e )/2U 0

(21)

E e = (1 - pe)2U0

(22)

or where Uo is the number of moles of repeating units per gram of the sample. Z equals half the number of end-groups E that have reacted. Thus, Z e = (2U0 - E e )/2 = Pe U 0

(23)

Gg is given by Ge = Go + (Eo - E e )/2

(24)

where Go denotes the number of moles of free glycol added before sealing the reaction tube. P^ = 2U0/E0 = 1/(1 - p0)

(25)

where p0 is the well-known Flory extent of reaction. In the case of previous addition of glycol, q is defined as q = Go/Uo

(26)

From Eqs. (22), (25), and (26), Ge is given by Ge = (q + p e - Po) Uo

(27)

Finally, substitution of Eqs. (22), (23), and (27) into Eq. (20) results in K =

Pe(q +Pe-Po) (1-Pe) 2

(28)

From the known starting conditions (p0 and q) and the analytical results, the equilibrium extent of reaction p e can be computed. Then,

394

HIDEO SAWADA

substitution of p0, q, and pe into Eq. (28) yields the polycondensation equilibrium constant K; K increases with the extent of reaction and, thus, also with the average degree of polymerization. Challa [9] proved that the principle of equal reactivity does not fully hold in PET. His result is shown in Fig. 4.1. The influence of temperature on K appears to be of minor importance. Application of the thermodynamic relationships AH = RT2 d In k/dT (29) AS = d(RT In k)/dT (30) yields AH = - 2 kcal/mole and AS = - 5 cal/deg mole.

1-2 .

08-

0-4-

0-2

04 -

0-8

R,

Fig. 4.1. Polycondensation equilibrium constant K at 262.0°C as a function [9] of l - p e .

IV. RING FORMATION IN POLYCONDENSATION Stbckmayer and Jacobson [10] have attempted to formulate a general theory of the polycondensation reaction. Their treatment of the problem is formally identical with that of the thermodynamics . of a perfect Bose-Einstein gas, and the critical phenomena may be compared with the condensation of such a gas. The only arbitrary assumption made in this treatment is that the reactivity of the groups is the same and independent of the molecular

THERMODYNAMICS OF POLYMERIZATION. I

395

size. Furthermore, it is necessary that the elementary steps of the reaction are completely reversible. This theory predicts that the formation of rings depends on the dilution. ACKNOWLEDGMENTS I wish to express my appreciation to Professor K. F. O'Driscoll of State University of New York at Buffalo for his interest and willingness to read and criticize the manuscript, and to state my indebtedness of Mr. M. Dekker of Marcel Dekker, Inc. for his interest and valuable counsel. I am also indebted to the management of Daicel Ltd. for permission to write this review. Thanks are also due Mrs. M. Sakakibara who typed large sections of the manuscript. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]

P. J. Flory, Principles of Polymer Chemistry, Cornell Univ. Press, Ithaca, N.Y., 1953, p. 81. J. Zimmerman, J. Polymer Sci., B2, 955 (1964). H. Sawada, and A. Yasue, J. Chem. Soc. Japan, Ind. Chem. Sect., 67, 1442 (1964). O. Fukumoto,J. Polymer Sci., 22, 263 (1956). F. VViloth, Makromol. Chem., 15, 98 (1955). P. H. Hermans, J. Appl. Chem., 5, 493 (1955). H. Yumoto, and N. Ogata, Makromol. Chem., 25, 91 (1957). A. B. Meggy, J . Chem. Soc., 1956, 4876. G. ChMa, Makromol. Chem., 38, 105 (1960). W. H. Stockmayer and H. Jacobson,J. Chem.Phys., 18, 1600 (1950).

J. MACROMOL. XI.-REVS.

MACROMOL. CHEM., C5(1), 151-174 (1970)

Thermodynamics of Polymerization. II. Thermodynamics of Ringopening Polymerization HIDE0 SAWADA Central Research Laboratory Daicel Ltd. Tsurugaoka, Oi, Irurnagun Saitarna, Japan

I. GENERALASPECTS

.........................

.. . .. .. . .. . . .. . . . . . . . .. . . . . . . . . . . . . .. . . . . . . . . . . . . . . .. . . . . . .. . . . ... . . . . ... . . . HETEROCYCLIC COMPOUNDS . . . . . . . . . . . . . . . . . . A. Cyclic E t h e r s . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. Lactams .... . . . .. . . . . ... . .. .. . . . . . .. .. . C. L a c t o n e s . . .. . . . . . . . .. . . . . . . . . . . . . . . . . . .

11. HOMOCYCLIC COMPOUNDS. A. AngleStrain B. Conformational and Transannular Strain C. Steric Effect of Side Group

III.

165 166 168

. . . . . . ... . SUMMARY.. . .. . .. . .. . . . .... . . .. . ......... ACKNOWLEDGMENTS. . . . . . . . . . . . . . . . . . . . . . . .

170

REFERENCES

172

D. Miscellaneoris Heterocyclic Compounds

IV.

151 152 152 162 163

.............................

170 172 172

I. GENERAL ASPECTS The heat of polymerization for ring-opening polymerization i s affected by the following: (1) angle strain, which is very important for the three- and four-membered rings (i.e., those rings that have the greatest angle strain), (2) conformational strain (crowding of 151 Copyright 0 1970 by Marcel Dekker, Inc.

152

HIDE0 SAWADA

eclipsed adjacent hydrogen atoms) and transannular strain (strain and crowding of hydrogen atoms across the ring); and (3) steric effect of side group. On the other hand, the entropy changes of polymerization do not depend on angle strain, but a r e susceptible to configurational influence. Polymerization of ring compounds to linear polymers may show ceiling temperatures as in the case of tetrahydrofuran and 1,3-dioxolane or floor temperatures as in the case of sulfur and selenium. The equilibrium polymerizations of ring compounds will be discussed in a later review.

II. HOMOCYCLIC COMPOUNDS The cyclic compounds for which most data a r e available a r e homocyclic compounds.

A. Angle Strain Angle strain, defined as the distortion of bond angles from their most favored orientation, serves to increase the internal energy of a molecule. Those molecules in which the skeletal bond angles are closest to the preferred values will, therefore, have the lowest internal energy. Ring-opening polymerization takes place through opening of the ring, and the linear polymers so formed a r e virtually free from strain. The heat of polymerization, therefore, provides a direct measure of the strain energy in the monomer ring [l]. Heats of polymerization for ring monomers a r e summarized in Table 1. Von Bayer [2] pointed out that if the normal angles between the four valencies of a tetrahedral carbon atom a r e assumed to be 109”28‘, i.e., the valencies are equally spaced about the atom, then, in threeand four- membered rings, severe distortion of the normal valency angles must be involved. Thus, in a cyclopropane ring each bond must be distorted from the tetrahedral angle by 24O44’. If all the rings a r e assumed to be planar, the distortions for each ring size a r e shown in Table 2. Small ring compounds are generally reactive due to high strain in their ring structure. The presence of “strain” and hence of decreased stability has been shown by the enhanced values of the molecular heat of combustion per methylene group of cyclopropane and cyclobutane compared t o the values for the larger rings. The next to the last column in the table gives the strain per methylene group, and this quantity is a measure of the distortion present. The last column gives the “strain energy” in the

THE R MODY NAMlCS OF PO LYME R I ZATl ON. I I

153

TABLE 1 Heats of Polymerization for Ring Monomers

Monomer

Cyclopropan e Methylcyclopropane 1,l-Dimethylcyclopropane

No. of atoms Standard -AH, in ring states* kcal/mole

a. Cycloalkanes 3 lc 3 lc 3 lc

Temp., "C

27.0 25.1 23.3

25 25 25

Ref. a a d

Cyclobutane Methylcyclobutane 1,l-Dimethylcyclobutane

4 4 4

lc lc Ic

25.1 23.9 22.3

25 25 25

b

Cyclopentane Methylcylopentane 1,l-Dimethylcyclopentane

5 5 5

lc lc lc

5.2 4.1 3.2

25 25 25

b

Cyclohexane Methylcyclohexane 1.1- Dimethylcyclohexane

6

lc lc lc

-0.7 -2.2 -1.8

25 25 25

b

lc lc

5.1 8.3 11.2 11.5 10.8 3.4 5.3 1.7 2.9 1.9 2.0

25 25 25 25 25 25 25 25 25 25 25

b

24.9 22.6

25 25

e

18.0

25

epf

d

Cycloheptane Cyclooctane Cyclononane Cyclodecane Cycloundec an e Cyclododecane Cyclotridecane Cyclotetradecane Cyclopentadecane Cyclohexadecane Cycloheptadecane

Ethylene oxide

6 6

7 8 9 10 11 12 13 14 15 16 17

IC

lc lc lc lc lc lc lc lc

b. Cyclic Ethers 3 gg lc

h o p y l e n e oxide styrene oxide 3-Nitrostyrene oxide

3 3

lc Ic

24.3 24.1

26.9 26.9

Oxetane

4

ss

19.3

-9

a a

a a

a a

b C

C C C C C C

C

d

d g

154

HIDE0 SAWADA

TABLE

Monomer 3,3-Di (chloromethyl) oxetane 3.3-Di (phenoxymethyl) oxetane 3,3- Dimethyloxetane Dioxolane

Tetrahydrofuran

1 (continued)

No. of atoms Standard -AH, in ring states* kcal/mole

-

Temp. OC

4

lc

20.2

26.9

4 4

ss ss

19.8 16.1

26.9 -9

5

lc 5s

6.2 5.2

20 9

gg gg Ic 1s 1s

5.0 2.9 9.1 4.3 5.3

Is

4.0

lc lc

3.0 5.3

20 25 25 40 40 25 50 25

5

Ref. d

h

i

Tetrahydropyran

6

gg

0.4

20

Trioxane

6

gc cc

3.83 -0.57

25 25

9

0.0

20

h. P

3.6 4.7

-+ 5

r

20

h

6s

1.8 2.1 1.9

26.9 26.9 26.9

cc

0.7

m- Dioxane

j, k , ~

9

- 65 1.3-Dioxepan

7

ss

gg 7

lc ss

Tetraoxane

8

1,3-Dioxocane

a-Pyrrolidone 1-Methyl-a-pyrrolidone

12.8 c. Cyclic Amides 5 lc lc 5 lc

1.1

1.3 0.8

t

20

75 25 25

h

U V

W

THERMODYNAMICS OF POLYMERIZATION. II

155

TABLE 1 (continued)

Monomer ~~

No. of atoms Standard -AH, in ring states* kcal/mole

Temp. "C

~~

a -Piperidone 1-Methyl-a-piperidone E-

Caprolactam

1-Methyl-c- caprolactam 5- Methyl-c- caprolactam

7- Methyl-€- caprolactam E-

Enantholactam

6 6 7

7 7 7 8

1-Methyl-c-enantholactam

q- Capryllactam Selenium Sulfur P-Propiolactone Pivalolactone

Ref. ~

9

lc lc lc

2.2 1.1 -0.5

lc lc 1s 1s

3.8 3.3 3.6 4.5

Is

4.0

1s lc lc Ic

3.3 2.3 3.8 3.8

75 25 250 240 250 230 25 75 75

lc lc Ic lc

5.3 5.7 5.2 3.9

75 25 230 25

bb

lc

7.8

230

bb

400 200

dd.ee

d. Inorganic Monomers 8 1s -2.3 1s -3.2 8

e. Cyclic E s t e r s 4 lc 4 lc

19.2 20.1

75 25 25

25

U

V W

I V

Y L

aa

bb W U

U

U V

W

c,c

ff 66

*Monomer and polymer states: 1. liquid; s, solution; c, condensed; g, gas. aF.S.Dainton,T.RE.Devlin, and P.A.Smal1, Trans, Faraday'Soc., 51, 1710 (1955) bS. Kaarsemaker and J.Coops, R e c . Trau. Chim., 71, 261 (1952). H.van Kamp, J.Coops, W.A.Lambregts, B. J.Visser, and H.Dekker, Rec. Trau. Chim., 79, 1226 (1960). dF.S.Dainton, K. J.Ivin, and D.A.G. Walmsley, Trans. Faraday SOC., 66, 1784 (1960). H.C.Raine, R.B.Richards, and H.Ryder. Trans. Faraday Soc., 41. 56 (1945) 'P.Cray and A.Williams, Trans. Furuduy SOC., 55, 760 (1959). 'J.B.Rose, J . Chem. SOC., p.546 (1946).

.

.

156

HIDE0 SAWADA

TABLE 1 (continued) ~

hS.M.Skuratov, A.A.Strepikheev. S.M.Shtekher, and A.V.Volokhina, Dokl. Akad. Nauk SSSR. 117, 263 (1957). 'P.H.Plesch and P.H.Westermann, J . Polymer Sci C , 16, 3837 (1968). IS. M.Skuratov, A.A.Strepikheev. and M.P.Kozina, Dokl. Akad. Nauk SSSR, 117, 452 (1957). kR.C.Cass, S . E.Fletcher, C.T.Mortimer, H.D.Springal1, and T.RWhite, J . Chem. SOC.,p.1406 (1958). 'D.Sims, J . Chem. SOC., p.864 (1964). mC.E.H.Bawn, R.M.Bel1, and A.Ledwidth,Polymer, 6, 95 (1965). "K.J.Ivin and J.Leonard, Polymer, 6, 621 (1965). OG.A.Clegg, D.RGee, T.P.Melia. and A.Tyson,Polymer, 9, 501 (1968). 'A.Snelson and H.A.Skinner, Trans. Faraday S O C . , 57, 2125 (1961). T.P.Melia, D.Bailey, and A.Tyson, J . Appl. Chem., 17, 15 (1967) P.H.Plesch and P.H.Westermann, Polymer, 10, 105 (1969). F.S.Dainton, J.A.Davies, P.P. Manning, and S.A. Zahir, Trans. Faraday SOC., 53, 813 (1957). K.Nakatsuka, H.Suga, and S.Seki, J . Polymer Sci. B , 7 , 361 (1969). " A.A.Strepikheev, S.M. Skuratov, 0.N.Kachinskaya. R S . Muramova, E.P.Brildina, and S.M.Shtekher, Dokl. Akad. Nauk SSSR, 102, 105 (1955). "V.P.Kolesov, I.E.Paukov, and S.M.Skuratov. Zh. Fiz. Khim., 36, 770 (1962);Russ. J . Phys. Chem., 36, 401 (1962). WM.P.Kozina and S.M.Skuratov, Dokl. Akad. Nauk S S S R , 127, 561 (1959). * S.M.Skuratov, A.A.Strepikheev. and E.N.Kanarskaya, Kolloidn. Zh., 14, 185 (1952). YA.B.Meggy, J . Chem. SOC., p.796 (1953). 'P.F.Van Velden, G.M.Van d e r Want. D.Heikens, C.A.Kruissink, P.H.Hermanb, and A.J.Staverman, Rec. Traw. Chim., 74, 1376 (1955). aaA.V.Tobolsky and A.Eisenberg, J . Am. Chem. SOC.,81, 2302 (1969). "A.K.Bonetskaya and S.M.Skuratov. Vysokomolekul. Soedin., All, 532 (1969) "A.Eisenberg and A.V.Tobolsky, J . Polymer Sci., 48, 19 (1960). ddF.Fairbrother, G.Gee, and G.T.Merral1, J . Polymer S c i . , 16, 459 (1955). eeA.V.Tobolsky and A.Eisenberg, J . Am. Chem. SOC., 81, 780 (1959). "B.Boyesso, Y.Nakase, and S.Sunner, Acta Chem. Scand., 20. 803 (1966). "H.K.Hall,Jr., Macromolecules, 1. 488 (1969).

.

.

molecule, which is the difference between its observed heat of formation and its value estimated from the group additivity relations using the group values which have been, of course, derived from the unstrained standards [3]. The angle strain is seen to be quite severe in cyclopropane, less severe in cyclobutane, and still less in cyclopentane. Cyclohexane lies at the energy minimum, and for the larger rings the strain increases to a maximum at cyclononane and then falls off as shown in Table 2.

(2) (ethylene)

-

-

-

I

-

-

-

-

-

-

54044" 24"44" 9044" O"44" -5"16" -9"33I' -12"46"

-

-

60" 90" 108" 120" 128"34" 135" -

0"

Angle between valency bonds

Distortion f r o m normal valency angle a

2.0

24.2d 27.0 25.1 5.2 -0.7 5.1 8.3 11.2 11.5 10.8 3.4 5.3 1.7 2.9 1.9

Heat of polymerization AHp,, kcal/mole

168.7 166.6 164.0 158.7 157.4 158.3 158.6 158.8 158.6 158.4 157.7 157.8 157.4 157.5 157.5 157.2

Molecular heat of combustionb p e r CH2 group, kcal/mole

6.4 ( 6.3) 9.9 ( 9.6) 12.8 (12.6) (12.0) (11.0) ( 3.6) ( 5.2) ( 0.0) ( 1.5) ( 1.6) (-3.4)

0.2 ( 0.0)

(22.6) 27.6 (27.6) 26.2 (26.4) 6.3 ( 6.5)

Strain energy,, kcal/mole

aDistortion = (normal valency angle-actual angle between bonds). The distortion caused is assumed to be equally shared between the two bonds. bThe heat of combustion of the gaseous cycloalkane divided by the number of methylene groups. Data from E.L. Eliel, N.L. Allinger, S.J. Angyal, and G.A. Morrison, Conformational Analysis, Wiley-Interscience, New York, 1965,p. 193. Strain energies within parentheses a r e calculated according to the following equation: (the molecular heat of combustion of cycloalkane strain energy = n 157.4) n where n i s the number of methylene groups. Monomer, gas polymer, condensed.

3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

~~~

No. of atoms in ring

TABLE 2 Distortions in Cycloalkanes

HIDE0 SAWADA

158

The calculated heat, entropy, and free energy of polymerization of the liquid cycloalkanes to linear polymer at 25°C are given in Table 3. Using heats of formation of monomer and polymer the heats and entropies of polymerization have been calculated by Dainton et al. [ 4 ] . In Table 3 these A H and AS values show that A H makes TABLE 3 Heats, Entropies, and Free Energies of Polymerization of Cycloalkanes a t 25°C [4]

AH^,, X

kcal/mole

3

27.0 25.1 5.2 -0.7 5.1 8.3

-

AS.9 cal/deg mole

- A G ~c , kcal/mole

L(CHz).d

4 5 6 7 8

16.5 13.2 10.2 2.5 3.8= 0.8a

22.1 21.2 2.2 -1.4 3.9 8.2

CH~~.H(CH~)~-,~H~ 19.1 25.1 20.2 18.8 23.9 17.2 -0.5 4.1 15.3 -4.5 -2.2 7.6 23.3 22.3 3.2 -1.8

-

(CHS),C( CH,)x-zCH, 22.3 18.0 15.7 8.5

16.6 16.0 -1.5 -4.3

'Data f r o m H.L.Finke, D.W.Scott, M. E.Gross, and G.Waddington, J. Am. Chem. SOC., 78, 5469 (1956).

the main contribution to A G for three- and four-membered rings, but the heat and entropy contributions are equally important for five-, six-,and seven-membered rings. These results showed that cyclohexane was the most resistant to polymerization, since AG was positive. For cyclopropane, cyclobutane, cyclopentane, cycloheptane, and cyclooctane, A G for polymerization was calculated to be negative. Thermodynamic feasibility, however, does not always guarantee the practical realization, and no high polymers of cyclopropane and cyclobutane are known [5].

THERMODYNAMICS

OF POLYMERIZATION. II

159

From a qualitative point of view, the angle strain in three- and four-membered carbon ring systems is large: it is small in five- and six-membered rings, and then increases with further increase in ring size, and this is mirrored in the rise and fall of AH>c. On the other hand the entropy changes do not show much marked dependence on angle strain, but are susceptible to configurational influence. The entropy change of polymerization can be given as a function of the probability of ring closure. Thus the entropy change of polymerization is Asp = - b In P - a

(1)

where P is the probability of ring closure and a and b a r e constants, the values of which depend on the nature of the monomer. The probability of ring closure for a chain with n repeating units can be given as a function of the probability that the chain ends will come together. This probability is usually given in t e r m s of a value (?), the rootmean-square of end-to-end distance. Obviously, for ring closure t o occur, r must be less than a few AngstriSm units. The entropy change would have a large negative value for the ring opening of a threemembered ring. For larger rings, as the ends required t o react during the closure go farther and farther apart, the entropy of ring closure should become less and less favorable, leading to a small negative value for the ring opening. When the entropy changes were plotted against the ring size for cycloalkanes (omitting cyclohexanef), a linear relationship was found (Fig. 1). The larger rings had more favorable entropies of ring opening than did the small rings, but very much less favorable enthalpies. From a statistical mechanics treatment, it has been shown that the entropy change of ring closure is [6,7] ASr = R In {PV/2xVsN}

(2)

where P is the probability of ring closure, i.e., the fraction of chains that will close to form ring structures, V is the total volume of the system, V, is the volume of a constrained skeletal atom prior to bond breaking, x is the number of monomer units in the ring, and N is Avogadro's number. Three approaches have been followed for calculating the entropy *The only reason for this omission i s that, if the chain contains six atoms and the bond angle at each is 120', then there i s a high probability that cyclization to a six-membered ring will occur. Thus the entropy change of p o l y m e r ization of cyclohexane is an exceptionally small negative value.

160

HIDE0 SAWADA

3

4

5

6

7

8

X

Fig. 1. Relationship between the entropy changes and the number of ring atoms of cycloalkanes. (Data f r o m Ref. 141.)

change of ring closure: (1)In the Stockmayer-Jacobson development, P was obtained by assuming a Gaussian distribution of end-to-end distances and integrating over the volume element Vs [6] ; (2) P was obtained from the detailed structural model for the chain with fixed bond length, fixed bond angles, and fixed rotational states-ring closure is assumed to occur when the end-to-end distance is less than one bond length (Carmichael-Kinsinger model [7]); (3) another approach [8 J has been employed in that the equilibrium between simple rings and chains is described by means of a partition-function calculation of the equilibrium constant relating each of the variously sized rings to the chain population. The main defect of these approaches is that they assume that the separation of the end group is determined only by statistical considerations. These approaches neglect the very real possibility that the terminal units of a long chain may be held together by ionic charges or by catalysts [9]. It is worth mentioning a case of strain in acenaphthylene [lo]:

161

THERMODY NAMl CS OF POLYMERIZATION. I I

which polymerizes through the double bond of the five-membered ring, with the exceptionally high heat of 24 kcal/mole. If the atomic configuration and valency angles of the naphthalene moiety of this monomer were identical with that of naphthalene, the length of the C-C bond in the ethylenic part would be 2.43 A which is much larger than even the C-C single bond (1.54 A), and, therefore, the angle strain in this five-membered ring must be very large. On polymerization, the double bond in the five-membered ring is converted to a single bond and some of this angle strain is relieved. The large heat of polymerization of maleimide (- 21 kcal/mole) by addition polymerization through the double bond is also attributed to the changes in the internal strain of the monomer ring accompanying polymerization. Calculated values of strain energies for numerous homocyclic and heterocyclic compounds are assembled in Table 4. Cycloalkenes appear to have slightly less strain energies than the corresponding TABLE 4 Strain Energies of Cyclic Compounds [16]

Compound

Strain energy, kcal/mole

Cyclopropane Cyclobutane Cyclopentane Cyclohexane Cycloheptane Cyclooctane Cyclononane Ethylene oxide Propylene oxide Ethylene imine Ethylene sulfide Thiacyclobutane Pyrrolidine Tetrahydrofuran 1,3-Dioxolane

27.5 26.1 6.1 0.1 6.1 9.7 12.5 -28 26.4 23 18.6 18.9 5.5 6.7 7.3

-

Compound

Strain energy, kc a1/mole

Thiacyclopentane Tetrahydropyran 1,3- Dioxane 1,4-Dioxane Cyclopentene Cyclohexene Cycloheptene Cyclooctene Cyclooctatetraene

1.0 2.2 2.9 4.0 4.9 -1 -5 -6 15.2

Benzene Thiophen Furan Pyridine

--22

--16

---19

8

I

cycloalkanes. This finding is surprising at first sight, but it may be that the insertion of a double bond into the cycloalkane ring is accompanied by only a small increase in angle strain. This strain may be outweighed by a reduction in nonbonded interactions between hydrogen atoms, of which there are two fewer in the olefin.

162

HIDE0 SAWADA

However, a small ring, such as cyclopropene, is an example of two trigonal carbons in a highly strained three-membered ring; an estimate of a least 8 kcal/mole has been made for the extra strain energy of cyclopropene over the cyclopropane [ l l ] . The fully conjugated ring compound cyclooctatetraene is seen to be appreciably strained. The strain energy in benzene is negative, that is, - 22 kcal/mole. This is conjugation energy which describes the energy associated with the delocalization of r-electrons. Therefore, the ring-opening polymerization of benzene has not been achieved. Conjugation energies should not be equated with resonance energies, for reasons discussed by Dewar and Schmeising [12]. B. Conformational and Transannular Strain

The cyclopentane ring has a completely eclipsed conformation with ten "oppositions" between adjacent hydrogen atoms:

Although there is almost no angle strain the planar model, there is conformational strain caused by the completely eclipsed conformations. The somewhat higher value for the heat of combustion per methylene group obtained for cyclopentane (as compared with cyclohexane or cycloheptane) is ascribed to the conformational strain in the molecule, as shown in Table 2. In cyclic compounds of more than six members, the strain due to the deviation of the bond angle is relieved because the ring assumes a puckered configuration. However, another steric hindrance arises from the nonbonded intramolecular interaction between the atoms, including interactions between atoms which formally may be regarded as being on opposite sides of the ring in the x = 6-11 range. This intramolecular interaction between atoms attached to different parts of the ring in alicyclic compounds is described as transannular interaction. It gives rise to transannular strain in these molecules. In rings of fifteen or more carbon atoms this effect is absent. In these molecules the ring atoms arrange themselves into two roughly par-

THERMODYNAMICS OF POLYMERIZATION. II

163

allel chains and thus more nearly resemble aliphatic compounds:

It would, therefore, be expected that above x = 6, AH would become more negative, pass through a shallow minimum, and then increase t o a limiting value. In view of the lesser structure sensitivity of AS, the AG values might show a dependence on x which is qualitatively similar t o that of AH. The free-energy changes for rings of more than eight members have not been estimated, but the value might approach zero as the ring size is increased. C. Steric Effect of Side Group

The influence of side-group steric repulsions on the relative thermodynamic stability of polymers and on the relative polymerizability of cyclic monomers will now be discussed in more detail. In an equilibrium between cyclic monomers and linear polymers, steric interference between the side groups o r between side groups and chain atoms will change the equilibrium t o favor the low molecular weight cyclic monomers. Steric repulsions of the b and c types, illustrated in Fig. 2, are more serious in an open-chain polymer

Fig. 2. Changes in side-group repulsion on polymerization.

than in a cyclic monomer. Since steric repulsions must raise the internal energy, interactions of this type will raise the enthalpy of the polymer relative to the cyclic monomer, and so AH will be made more positive, and possibly be changed from negative t o zero. On the other hand, steric hindrance between two side groups on the same carbon atom should cause the external bond angle between those groups to widen. This, in turn, should bring about a narrowing of the opposite skeletal bond angle owing t o a hybridization change.

164

HIDE0 SAWADA

Clearly, narrowing of the skeletal angle will favor the formation of small rings rather than larger rings. It will also force the various components of the polymer closer together. If appreciable intramolecular crowding is present, small changes in the skeletal angle will have a relatively large effect on the enthalpy of the polymer, but less effect on the cyclic monomer. This general effect will be simply called the =gem-dimethyl effect” in this review. The gem-dimethyl effect can be quantitatively interpreted in t e r m s of the thermodynamics of the conformations involved [13,14]. The calculation of AH for the ring-closure reaction will be considered first. The effect of substituents on the enthalpy of ring closure is interpreted in t e r m s of the change in the number of gauche interactions in going from the reactant to the product. The substitution decreases the change in gauche interactions on ring closure. The enthalpy of the ring-closure reaction is reduced for the substituted case and the ring closure is thus favored by substitution. This is reflected in the lower heat of polymerization for the substituted cyclic monomer. To estimate the overall effect of substituents on the ring-closure reaction, the entropy change on cyclization must also be taken into account. A substituent will not alter the entropy of the cyclic compound relative t o the unsubstituted cyclic compound very much. The substituent, however, has a much greater effect on the entropy of the open-chain compound. The substituent restricts rotation of the carbon chain due mainly to the increased height of the barriers t o internal rotations, and thus decreases the entropy of the open-chain compound. The net effect of a substituent is t o make the entropy of ring closure more positive. Bulky substituents which greatly restrict rotation in the open- chain compound will facilitate ring closure through the entropy effect. Thus it can be seen qualitatively that both the entropy and enthalpy effects of a substituent make the free energy of the ring opening reaction more positive relative to the unsubstituted case. The effects of methyl group substitution are to make AHic more positive and ASic more negative, both causing upward displacements of the AG against x curve without great alteration of shape as shown in Fig. 3. The main effect of steric repulsion is t o reduce the rotational entropy of the polymer and to raise the enthalpy of the polymer relative to the cyclic monomer. The larger the dimensions of the side group, therefore, the more the equilibrium should be shifted to the cyclic monomer. Thus, as AH for polymerization changes from negative toward zero, the ceiling temperature will be lowered until depolymerization will occur at any temperature.

165

THERMODYNAMICS OF POLYMERIZATION. II

I

2

3

4

5

6

7

8

X Fig. 3. Free energy of polymerization of liquid cycloalkanes. ( a ) 1.1-Dimethyl substituted; ( b ) methyl substituted; ( c ) unsubstituted; 25'C; x i s the number of atoms in the ring. (Data from Ref. [41.)

III. HETEROCYCLIC COMPOUNDS Small [15] has considered the effect of ring size on A G for heterocyclic compounds. The general shape of the relation between A G and x is expected to be similar to that calculated for the cycloalkanes, particularly if the heteroatom does not differ too much from carbon in size and bond angles (for example, in oxygen and nitrogen compounds but not in sulfur compounds, as shown in Table 5). TABLE 5 Bond Lengths and Bond Angles ~~

C-C-C

angle

109"28'

C-N-C

angle

109"

length

A A 1.44 A

C-0-C

angle

111"

length

1.82

A

c-s-c

angle

100"

C-C

length

C-N

length

C-0 C-S

1.54

1.47

Since bond lengths and bond angles of C-N and C-0 do not differ much from those of C-C bonds, replacement of a carbon atom in a cycloalkane ring by a heteroatom such as nitrogen o r oxygen would

166

HIDE0 SAWADA

not produce a large change in angle strain. The C-S bond length, however, is much larger than that of the C-C bond and would produce a much larger change in structure than replacement of carbon by oxygen o r nitrogen. In addition, a much larger change in vibration frequencies is to be expected for replacement by sulfur than for replacement by oxygen and nitrogen. The strain energies of some cyclic compounds a r e shown in Table 4. The strain energies of ethylene and propylene oxides a r e nearly the same as the strain energy of cyclopropane, whereas the strain energy of ethylene imine is only a little lower; the strain energy of ethylene sulfides is, however, appreciably less than the strain energy of cyclopropane. Amongst the five-membered rings, the order of strain energies [16] is 1,3-dioxolane > tetrahydrofuran > cyclopentane > pyrrolidine > thiacyclopentane. A similar picture is disclosed by the data for the six-membered rings where the oxa compound is slightly strained, the dioxa compounds is much more strained, and the aza and thia compounds are unstrained. The results of the polymerization of cyclic ethers, esters, urethans, ureas, and imides a r e given in Table 6 [17] . In Table 7 are collected a number of AH values for ring compounds. A summary of the results follows: 1. The polymerizability of five- and six-membered cyclic monomers depended markedly on the class of compound. 2. Four-, seven-, and eight-membered rings polymerized in almost every case. 3. Substituents on a ring always decreased polymerizability. 4. Substitution of heteroatoms in the ring had in general little effect on the ease of polymerization as compared to the parent monomer.

A. Cyclic Ethers 1. 3-Membered Rings. Epoxides polymerize readily owing to angle strain. 2. 4-Membered Rings. Oxetanes polymerize smoothly with cationic catalysts. Angle strain and repulsion of adjacent hydrogens furnish the driving force. 3. 5-Membered Rings. Tetrahydrofuran polymerizes readily owing to the repulsions of eclipsed hydrogens, as in cyclopentane. Dioxolane also polymerizes well. Substituted tetrahydrofurans, namely the 2- methyl, 3-methyl, and 2-chloromethyl derivatives, do not polymerize. Substitution in a heterocyclic compound invariably decreases its polymerizability, again as predicted for the cycloalkanes in terms of the gem-dimethyl effect [18].

167

THE RMODY NAMl CS OF PO LYME R IZATlON. I I TABLE 6 Polymerizability of Five- and Six-Membered Cyclic Monomers [17I Polymer iz ability a Class of monomer

5-Ring

Lactam Lactone Urethan Urea Imide Anhydride

+

6- Ring

a Experimental conditions: polymerization was c a r r i e d out at several temperatures between the melting point of the monomer and 250°C. (+) Polymerization; (-) no polymerization. Data f r o m J. Furukawa and T. Saegusa,Polymerization of Aldehydes and Oxides,Wiley-Interscience, New York, 1963, p.19.

TABLE 7 Comparison of Heats of Polymerization AHpc f o r Ring Compounds [l1” ~

Repeat unit in polymer

-(CH2)nOCHzw -(CH2)nCONH-(CHanCON(CHs)(CHJnS+ ( CHz)n-

-

NO. of atoms in ring 5

6

7

8

6.2 1.1 0.8 6.3 5.2

0.0 2.2 -0.5 0.5 -0.7

4.7

3.8 2.3 2.5 5.1

12.8 5.3 3.9 3.8 8.3

=Heats of polymerization expressed in kilocalories per mole.

4. 6-Membered Rings. Tetrahydropyran, 1,3-dioxane, and 1,4-dioxane do not polymerize, in keeping with their strainless chair structures. However, trioxane polymerizes easily owing to the resonance stabilization of the linear, initiated and propagating zwitterions:

168

HIDE0 SAWADA

5. Larger Rings. The seven-membered cyclic formal polymeri z e s easily, The strain in this and larger rings is that caused by repulsion of hydrogens a c r o s s the rings. The eight- membered cyclic ethers, namely tetraoxane and 1,3-dioxocane, also polymerize well. The variation in the heat of polymerization of cyclic monoethers is similar t o that found for the cycloalkanes, in that the values become less negative as the ring size increases, x = 3-5, becoming positive at x = 6. The values a r e shown in Table 8, together with TABLE 8 Changes in Heat of Polymerization Caused by Replacement of CH2 in Cycloalkanes with Ether Linkage

No. of ring atoms

Cycloalkanesa ~~

3 4

5 6

~~~

Cyclic ethersa

Change in AH, kcal/mole

~~~

27.0 (Ic)

25.1 (lc) 5.2 ( l c ) -0.7 (Ic)

22.6 ( l c ) 19.3 ( 6 s )

3.0 ( l c ) 0.4

(a)

4.4 5.8 2.2 -.0.3

aMonomer and polymer states a r e denoted in parentheses: 1, liquid; s , solution; c, condensed (liquid o r amorphous solid); and g, gas.

the difference between the heats of polymerization of cycloalkanes and cyclic e t h e r s having the same number of atoms in the ring. The difference is large for the three- and four-membered rings where the effect of introducing an oxygen atom into the ring is to reduce strain considerably. For the five-membered ring the difference is small, and almost zero for the six-membered strainless rings. The heats of polymerization of the formal also follow the same pattern as the cycloalkanes, becoming increasingly more negative as the ring size increases, x = 6-8. B. Lactams The heat of polymerization of the lactams becomes steadily more negative as the ring size increases from five- , through six- and seven-, to eight-membered rings. The change is much less than for the corresponding cycloalkanes series. Bonetskaya and Skuratov [19] discussed thermodynamics of polymerization of lactams with 5 to 13 atoms in cycle. In Fig. 4 the AG& values are plotted against x, the number of atoms in the lactam ring, and from positive A G it is seen that it is thermodynamically impossible t o polymerize pyrrolidone at 25” C. The effects of substituents on the enthalpy of ring closure of capro-

THERMODYNAMICS OF POLYMERIZATION. II

169

20 -

5

7

9

II

13

X Fig. 4. Free energy of polymerization of lactam. x is the number of atoms in the ring. (Data from Ref. 1191.)

lactam were discussed in t e r m s of the change in gauche interactions and configuration of the amide bond [14]. Substituents can also favor cyclization through the entropy effect. Thus, substituents have the maximum effect on the five- and six-membered rings, completely preventing polymerization. Substituted four- membered rings polymerize as do substituted seven- and eight-membered rings [14]. According to Table 9, AH makes the main contribution to A G for TABLE 9 Heats, Entropies, and Free Energies of Polymerization of Lactams [19]

-AG -AS,

(T= 298.16"K),

cal/deg mole

kcal/mole

5.6

7.3 6.6 -1.1 -4.0 - 10 -15

2.8

-20

-2.3 -0.1 3.6 6.6 10.8 10.1 8.8 7.0 9.0

No. of ring atoms

-AH, kcal/mole

5 6 7

-0.1 1.7 3.3 5.4 7.8

8 9 10 11

12 13

-0.5

0

- 25 -30

170

HI DEO SAWADA

seven- and eight-membered rings, but the heat and entropy contributions are equally important for nine-, ten-, and eleven-membered rings. On the other hand, AS makes the main contribution to AG for other lactams with more than twelve-membered rings [19].

C. Lactonee In the cyclic ester series, four-ring propiolactone polymerizes, but Carothers [20] found that neither y -butyrolactone nor ethylene carbonate could be polymerized; all unsubstituted six- ring esters polymerized, as well as the seven-membered €-caprolactone ; thus in this series the maximum in AG& occurs at five atoms in the ring. Brown et al. [21] suggested that cyclic carbonyl compounds with six ring atoms were markedly less stable than the corresponding compounds with five ring atoms. They cited the polymerizability of &valerolactone, as opposed to the nonpolymerizability of y -butyrolactone, in support of this thesis. D. Miscellaneous Heterocyclic Compounds

The polymerizabilities of a variety of atom-bridged bicyclic lactams, lactones, carbonates, ureas, urethans, imides, ethers, and anhydrides were found to be dependent on ring strain [22]. The bicyclo [2:2:2] octane and bicyclo [3:2:2] nonane series, in which the cyclohexane ring occurs in the boat form, underwent polymerization readily. Monomers of the bicyclo [3:2:11 octane group, which consists of a chair cyclohexane fused to a cyclopentane ring, underwent polymerization. The bicyclo [3:3:1] did not polymerize, in keeping with the general stability of two fused chairs. These results a r e summarized in Table 10. The polymerization of a variety of spiro-oxetanes has been described [18] :

The behavior of such spiro ethers resembles that of the oxetane

ring.

THERMODYNAMICS

OF POLYMERIZATION. I I

171

TABLE 10 Polymerizability of Bridged Bicyclic Monomers [ 18,22] a Bicyclic System [ 2:2: 21

M.P

P

Bicyclic System [3: 2: 11

Bicyclic System [ 3: 3: l]

Bicyclic System [ 3: 2: 21

aM

and P denote monomeric and polymeric forms, respectively.

172

HIDE0 SAWADA

IV. SUMMARY The heat of polymerization AH to an open-chain polymer affords a direct measure of the strain energy of the ring. On the other hand, the entropy change of polymerization AS does not depend on the ring strain but is susceptible to the configurational influence. In small rings, AH much affects AG, whereas in medium rings the effects of AH and AS are comparable although both have small values. However, AS makes the main contribution to AG for large rings.

ACKNOWLEDGMENTS I wish to express my appreciation to Professor K. F. O’Driscoll of the State University of New York at Buffalo for his interest and willingness t o read and criticize the manuscript. I a m also indebted t o the management of Daicel Ltd. for permission t o write this review. Acknowledgment is a l s o due Miss Y. Nishikawa who typed large sections of the manuscript. References [l] F. S. Dainton, K. J. Ivin, and D. A. G. Walmsley. Trans. Faruday SOC., 56, 1784 (1960). [Z] A. von Bayer, B e r . , 18, 2277 (1885). 131 S. W. Benson, Themochemical Kinetics, Wiley, New York, 1968, p. 48. [41 F. S. Dainton, T. R. E. Devlin, and P. A. Small, Trans. Faraday SOC., 51, 1710 (1955). 151 J. Furukawa and T. Saegusa, Polymerization of Aldehydes and Oxides, Wiley-Interscience, New York, 1963, p. 16. (61 B. Jacobson and W. H. Stockrnayer, J . Chem. Phys., 18, 1600 (1950). [7]J. B. Carmichael and J. B. Kinsinger, Can. J . Chem., 42, 1966 (1964). [8] R M. Levy and J. R. van Wazer, Polymer Preprints, 7(2), 938 (1966). [9] N. R. Allcock, Heteroatom Ring Systems and Polymers, Academic, New York, 1967, p. 88. (101 R. M. Joshi, Makromol. Chem., 62, 140 (1963). 1111 R M. Joshi and B. J. Zwolinski, in Vinyl Polymerization (G. E . Ham, ed.), Vol. 1, Dekker, New York, 1967, p. 487. [12] M. J. S. Dewar and H. N. Schmeising, Tetrahedron, 5 , 166 (1959); 11, 96 (1960). [13] N. L. Allinger and V. Zalkow, J . Org. Chem., 25, 701 (1960). [14] R. C. P. Cubbon,MakromoZ. Chem., 80, 44 (1964). [15] P. A. Small, Trans. Faraday SOC., 51, 1717 (1955). [16] J. D. Cox, Tetrahedron, 19, 1175 (1963). [17]H. K. Hall, Jr., and A. K. Schneider, J . A m . Chem. S O C . , 80, 6409 (1958).

THE RMODY NAMlCS OF POLYMER I ZATl ON. I I

I73

[la] H. K. Hall, Jr., Polymer P r e p n ' d s , 6 ( 2 ) , 535 (1965). [19]A. K. Bonetskaya and S. M. Sukuratov, Vyskomolekul. Soedin., A l l , 532 (1969). [20] H. Mark and G . S. Whitby, eds., Collected Papers of Wallace H . Carothers on Polymerization, Wiley-Interscience, New York, 1940, p. 107. [Zl] H. C. Brown, J. H. Brewster, and H. Schechter, J . A m . Chem. SOC., 7 6 , 467 (1954). [22] H. K. Hall, Jr., J . A m . Chem. SOC., 80, 6412 (1958).

J. MACROMOL. SCI.-REVS. MACROMOL. CHEM., C7(1), 161-187 (1972)

Thermodynamics of Polymerization. HIDE0 SAWADA* Central Research Laboratory Daicel Ltd. Tsurugaoka, Oi,Irumagun Saitama, Japan

I. GENERAL ASPECTS

. . . .. . . . . . . .. . . . .. .. .. . . . .... . . . . ..

11. FORMATION O F CARBONIUM ION . . . . . A. Ionization Potential , .. . . .... B. Proton Affinity , . . C.Acidity D. F r e e Energy Change of Formation of Carbonium Ion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E. Ions and Ion P a i r s . . .. . . . F. Energetics of Solvation. . . . . . . . . . .. . . . . . .

. . .. . . .. .. . .. .. .. .... . . ...... .... . . .. . ....

. . . . . . . . . . . . . . . . . . . . . . . . . . . ..

111.

IV.

. ... .. . .. .. . . . . ... ... . INITIATION O F CATIONIC POLYMERIZATION . . . . . . A. Energetic Consideration of Initiation Reaction by Halogen Acid . , , . . , . . . . . . . . . . . . . . . . . . . . B. Catalytic Activity in Cationic Polymerization by Lewis Acids . . . . . . . . . . . . . . . . . . . . . . . . . . . PROPAGATION O F CATIONIC POLYMERIZATION . . . . A. Energetics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. Heats of Reaction of Cations with Olefins . . . . . . . . . .

162 164 164 165 168 169 171 172 174 174 176 177 177 179

*Present address: Filter Laboratory, Daicel Ltd., Teppocho, Sakai, Osaka, Japan. 161 Copyright 0 1972 by Marcel Dekker, Inc. NO PARTof this work m y be reproduced or utilized in ony form or by ony meuns. electronic or mechanical, including xerography, photocopying, microfilm, and recording, or by any information storage and retrieval system, without the written permission of the publisher.

H. SAWADA

162

........

......

182 184

...........

186

C. Activation Entropy Changes of Propagation D. Thermodynamics of Formation of Zwitterions

V. CHAIN TRANSFER AND TERMINATION REFERENCES

.............................

186

I. GENERAL ASPECTS

The polymerizations considered in this review will be restricted to the cationic reactions, i.e., those in which the growing polymer chain carries a positive charge-whether a s a free cation o r as the positively charged member of a partially dissociated ion pair. In general, ionic polymerizations initially involve the dissociation of a particle into a positive and a negative ion in an organic environment. Propagation then occurs through successive additions of monomeric units to the charged o r reactive ends of the growing chains. The polymerization steps a r e then analogous to a free radical mechanism, The most important reactions of a growing carbonium ion a r e (a) propagation, in which further monomer units a r e added to the chain; (b) chain transfer, in which the positive charge is transferred to another molecule (usually monomer), thus stopping the growth of one polymer molecule and initiating another; and (c) termination, in which the carrier ion is destroyed. These three reactions determine the course of a cationic polymerization. For a cationic polymerization, the general steps may be written: Initiation: ABH

AB'H'

AB-H' + M

ki

%

HM'AB'

Propagation:

Termination MiAB-

kt : M,+AB-H'

Transfer: MiAB-+ M kmt =- M, + HM'AB-

THERMODYNAMICS OF POLYMERIZATION. Ill

163

The main characteristics of cationic polymerization are as follows [ I - ~ I :

1. The over -all energies of activation for cationic polymerizations usually fall within the range -10 to +15 kcal/mole, a s shown in Table 1. If negative, then the reaction rate decreases as the temperature rises. The activation energy of initiation is 5-7 kcal/mole, and this value is very low compared with that of the usual radical polymerization (20-30 kcal/mole). Although the cleavage of a covalent bond of an initiator needs a large amount of energy in the radical initiation, the ionic initiator is polarized and the initiation reac tion in the cationic polymerization may not need high energy. Therefore cationic polymerization reactions themselves are very rapid and can take place at low temperatures-often almost instantaneously. High molecular weight polymers a r e usually obtained at low temper atures when propagation is much more rapid than termination and transfer reaction.

-

Table 1 The Over-all Energies of Activation for Cationic Polymerization [ 7 1 Monomer Propylene Isobutene

Styrene

a-Methyls tyr ene

Initiator AlBq TiC14 Tic14 SnC14 SnC14 SnC14 SnC14 SnCl, Tic14 Tic14 AlClS SnC14

Cocatalyst

CCIsCOzH

Solvent

E (kcal/mole) 10.3 8 - 7.5 7 3 3 4.5 5.5 - 8.5 -1.5 1.2 -3.5

2. The dielectric constant and solvating tendency of the medium can have a profound effect upon the course taken by a given ionic polymerization. A medium of low dielectric constant and low solvating power for the ions will favor the formation of a covalent bond. A medium of high dielectric constant and high solvating power will favor the solvent-separated ion pair. 3. The monomers susceptible to polymerization by acids and

164

H. SAWADA

Friedel-Crafts halides are those olefins, cyclic amines, and oxides where the active intermediate in chain growth is a carbonium ion, an ammonium ion, o r an oxonium ion, respectively. R'

+ CH,=CHY

RCH,-CHY - R H N ~ c '~

R+ + H)N ::

)

R+ + 0 : )

The first group has been most extensively studied, but there is increasing interest in the possibilities of polymerization of the cyclic compounds and these were dealt with in P a r t 11 of this review. The olefins and vinyl compounds susceptible to cationic polymer ization are those having substituents tending to induce an electronrich double bond, e.g., isobutene, styrene, and the vinyl alkyl ethers. Those vinyl monomers having electrophilic substituents, e.g., vinyl acetate, vinyl chloride, and the acrylic derivatives, do not respond to cationic initiators, indeed vinyl chloride is so inert that it has been used as a solvent for the low-temperature cationic polymerization of isobutylene. 4. The substances able to initiate the cationic polymerization a r e all strong Lewis acids, i.e., powerful electron acceptors. They can be systematically classified as: (a) Simple protonic acids; (b) Friedel-Crafts halides plus cocatalyst; (c) Carbonium salts (mainly perchlorates and perfluorides); (d) Cationogenic substances (e.g., triphenylmethyl chloride). II. FORMATION OF CARBONIUM ION

A. Ionization Potential The ionization potential, I, is defined as the energy required to remove an electron from a molecule o r atom in the dilute gas phase. R -R++e:

AH=I

In principle, any number of ionization potentials correspond to various energies of the liberated electron and to different states of the resulting cation. Unless otherwise qualified, the ionization potential

THERMODYNAMICS OF POLYMERIZATION. Ill

165

is assumed to refer to the most weakly bound electron in the molecule. The ionization potentials of alkyl radicals are listed in Table 2. The comparison of methyl and ethyl radical ionizations is shown in Fig. 1. Carbonium ion stabilization by a methyl group is equal to 35 kcal/mole, in comparison to radical stabilization by the methyl substituent of 8 kcal/mole. Table 2 Ionization Potentials of Alkyl Radicals [13 1 Ionization potential (kcal /mole)

Radical

229.4 f 0.7 202.5 f 1.2 200.4 i 1.2 182.2 f 1.2 199.2 f 1.2 182.9 f 1.2 171.8 f 1 . 2 164.2 f 2.3 181.3 f 1 . 2 164.2 f 2.3 192.1 f 2.3 217.9 f 1 . 2 188.2 f 0.7 185.2 f 1.2 177.8 f 1.2 190.2 i 1.8

B. Proton Affinity [8] It is important to estimate the basicities of various olefins in terms of proton affinities and carbonium-ion affinities as defined in the following equations:

H' + M Mi + M

-

HM'

P = proton affinity

Mi+l

C

=

carbonium-ion affinity

The proton affinities of olefins will be determined by the following four steps [g], the proton in this example being added to carbonl.

H. SAWADA

166

202 Kcal/molo

- -T - - A 0

-CHaCHg.

8 Kcal/mols

Flg. 1. A comparison of the ionization potentials for methyl and ethyl radicals. The stabilization energy of the carbonium ion i s shown a s AC+ and that of the radical a s AC..

(1) H+ + e

(3) Ha (4)

+

R,’

-

AH, = +i

H-

t>c -CH,

-

-

C-CH,

R 1 l

R,/

R1lC-CH, R,’ C +-CH,

+e

AH3 = +D

AH^ = -I

The quantity i is known as the electron affinity and is determined by the reverse procedure of measuring the ionization potential of a proton, fl is the energy of opening the second half of the double bond, D is the carbon-hydrogen bond strength, and I is the ionization potential of the radical formed by addition of hydrogen atom to olefin. For any nonsymmetrical olefin the proton affinity for addition to carbon 1 will be different from that for addition to carbon 2. In Table 3 a r e listed values of the proton affinities for ethylene, propylene, and isobutene. The carbonium ion affinities (C, and C,

THE RMODY NAMlCS OF POLYMERIZATION. III

167

Table 3 Proton Affinities (kcal/mole) and Atom Affinities (kcal/mole) of Olefins [81

CH2CH2 CH$2H(CH,) CH2C(CHs)2

57.5 52.5 52.0

97.5 95 94

97.5 89 86

200 179 165

200 180 178

152 175.5 189

152 168.5 168

40 42.5 42

40 36.5 34

corresponding to P, and P,) will follow the same sequence as the proton affinities. P, (and therefore C,) increases markedly from ethylene to isobutene. This would account for the increase in the ease of polymerization as the olefin changes from ethylene, which is unaffected by cationic catalysts, to isobutene, which is highly reactive. As shown in Table 3, a decrease of the ionization potential of the radical causes an increase of the proton affinity. Figure 2 shows the linear

9.01

-

90

95

10.0

I, (monomer)

105

(ev)

Fig. 2. Comparison of the ionization potential of radical with that of monomer [lo]. Monomer (radical); (1)C€12=CH2 (-CH2CHS), (2) CH2=CHCH3 ( . C H ~ C H ~ C H S )( .3 ) CHFCHCH2CHs (*CH2CH2CH2CHS), ( 4 ) CH2zC- (CH3)2 (.CH?CH(CHJ),), ( 5 ) CHJCH=CHCH~(*CH(CHs)CH,CH,) (6) ( 7 ) CHsCH=C(CH3)2 (*CH(CHs)CH(CHs)2).

(.o),

168

H. SAWADA

relation between the ionization potential of the radical and that of the monomer [lo]. Therefore the proton affinity in cationic polymerization increases a s the ionization potential of the monomer decreases. It is also seen from Table 3 that P, is much greater than P, for isobutene. This means that a proton will add to the isobutene double bond more readily at the CH, than at the C(CH,), end. Thus if the addition of HX to a double bond involves an initial proton attack, then the H will add onto the carbon with the greater number of hydrogen atoms on it. We may thus interpret Markovnikov's rule in terms of bond strengths and ionization potentials. C , Acidity Brbnsted [ll] defined acids as species which had a tendency to lose a proton, and bases as species which had a tendency to gain a proton. An acid-base pair BH' = B

+ H+

(1)

consists of the acid (BH') and its conjugate base (B) or, alternatively, of the base (B) and its conjugate acid (BH'). The equilibrium of Eq. (1) cannot be observed; all reactions of an acid involve the transfer of the proton to a base which may be the solvent:

Aqueous solution is an important and convenient solvent for the study of acids, with the water molecule acting a s a base for the removal of the proton from the acid: BH' + H,O

=

H,O+ + B

(3)

Although the equilibrium constant for the reaction shown in Eq. (3) is formally given by

it is usual to omit the water term because in dilute aqueous solution, the change in concentration of the water due to the formation of the hydronium ion, H,O+, is negligible. The commonly used expression for the acid dissociation constant (the acidity constant of acid BH'), K,, is given as:

THERMODYNAMICS OF POLYMERIZATION. I l l

169

The reciprocal of the acidity constant of acid BH+ is called the basicity constant of the conjugate base B. The wide range over which Ka varies makes it more convenient to express the information in logarithmic form. Thus pKa = log Ka. Only strong acids can protonate the molecules of weakly basic solvents. Perchloric acid, hydrogen bromide, sulfuric acid, ptoluenesulfonic acid, and hydrogen chloride form a series of decreasing acid strength in acetic acid, although they are apparently of equal strength in the more basic solvent, water. Meaningful comparisons of acid strengths must be made in a single solvent o r under such conditions that medium effects a r e minimized. The acidity function H,,[12] is defined by

It depends, therefore, upon the activity of hydrogen ion aH+and also upon the ratio of the activity coefficients of a neutral base f, and its conjugate acid f B H t . By combining Eq. (5) with Eq. (6) and remembering that f = a/c, one obtains an alternative expression for H,

The acidity function is not identical with the pH, it merely becomes equal to it in dilute aqueous solution. D. Free Energy Change of Formation of Carbonium Ion

We shall briefly discuss some general aspects of the stability of carbonium ions. The stability of an ion must be associated with the free energy change accompanying its formation. However, for the formation of carbonium ions from alkyl halides in the gaseous phase, we only know the values of the enthalpy of formation of carbonium ions which a r e determined from electron impact measurements and hence the derived values of the heterolytic bond dissociation energy. Since the corresponding entropy changes a r e usually small, it is possible to discuss the variations in the free energy change in t e r m s of the variations in the corresponding enthalpy change. The results a r e listed in Table 4. The energy required to dissociate an alkyl halide RX into the ions R + and X - can be considered to be composed of the energy changes in the three hypothetical steps:

H. SAWADA

170

Table 4

-

R+ + Br- 1131 Enthalpies of Gaseous Ionization R-Br (Heterolytic Bond Dissociation Energies)

R

D(R-Br) (kcal/mole)

I(R) (kcal/mole)

AHga (kcal/mole)

Methyl Ally1 Benzyl -p-c1 -0-Me -m-Me -p-Me -m-NOp -m-CN -p-CN

67 45.4 50.5 50.1 48.5 50.5 49.1 48.4 48.1 49.7

229.4 188.2 178.9 183.3 175.5 176.4 172.0 197.4 197.9 192.8

214 152 147 151 142 145 139 164 164 161

--

aCalculated a s AHg = D(R-Br) + I ( R ) - E(Br), with E(Br) = 82 kcal.

(1) RX

Re

(2) R. + X. (3) R'

+ e + X.

+ XR'

AH, = D

+ e + XR'

+ X-

AH2 = I AH3 = E

The quantity AH, is the homolytic bond dissociation energy, and the term AH2 is called the ionization potential of the alkyl radical. The quantity AH3 is the electron affinity of the X atom and so is independent of the nature of R. The ionization potential of the alkyl radical and the enthalpy changes for formation of carbonium ions a r e given for a series of alkyl halides in Table 4. These figures all refer to completely separated ions in the gas phase. As can be seen from Table 4, the dissociation energies of alkyl compounds are not expected to be as sensitive to the nature of R as are the ionization potentials. The formation of gaseous carbonium ions by heterolysis of neutral molecules requires high energies; heterolytic reactions in the gas phase a r e not easily observed. Organic reactions involving carbonium -ion formation in solution are much more frequently found. More detailed discussions a r e given in Section 11-F.

THERMODYNAMICS OF POLYMERIZATION. I I I

171

E. Ions and Ion Pairs Winstein [14] has developed the notion of distinguishable stages of ionization where, between the covalent compound RX and the fully dissociated (and solvated) ions, 111, there a r e two types of ion pairs: I, the initimate ion pair, and 11, the solvent-separated ion pair. RX

[R'X-]

R'X-

I

11

R'

+ X111

Griffiths and Symons [I51 make the distinction more specific by a slightly different terminology (e.g., species I is called a contact ion-pair) and by proposing two subspecies in stage II-solventshared (IIa) and solvent -separated (IIb) ion-pairs. The heat of dissociation of an ion-pair may be determined directly by heat of dilution measurements, o r may be estimated by application of the equation d In K/dT

=

AH/RT2

where K is the dissociation constant of the ion-pair, and AH is the heat absorption corresponding to the dissociation of 1 mole of the ion-pair into ions at infinite dilution. Most of the published figures were obtained by the second method. However, the calorimetric determination of AH values should be much more reliable. For a system of spherical, nonpolarizable ions in a structureless dielectric medium (dielectric constant, D), the probability of finding an ion B at a distance r from ion A is

where c is the ionic concentration, 4 r r 2 d r is the volume of shell considered, Q is the work of separation, k is Boltzmann's constant, N is Avogadro's number, and T is the absolute temperature. The work of separation Q is given, according to Coulomb's law, by

4

= -Z,Z,e2/Dr

(11)

where Z, and Z B are the ionic charges and e is the electronic charge. For ions of like sign, P is very small: for oppositely charged ions P passes through a minimum value when the ionic separation is

H. SAWADA

172

At this point it will be seen that the work of separating the ions is equal to 2kT, i.e., four times the mean kinetic energy per degree of freedom. This result was first obtained by Bjerrum [16]. At separations greater than rmIn the thermal energy of ions is greater than the electrostatic energy and the ions can be regarded as free. The converse is true for separations less than rmln,the two ions behaving a s essentially a single species o r ion-pair. Bjerrum [16]proposed that ions separated by a distance smaller than rmlnshould be treaked a s ion-pairs. Typical values of rmIn for 1:l electrolytes a r e 3.6 A in water, 45 8, in acetic acid, and 120 8, in benzene. In media of low polarity two ions can be regarded as an ion-pair by this definition even though they a r e separated by one o r more solvent molecules. A free energy vs. reaction coordinate diagram for the formation of carbonium ion-pair intermediate is shown in Fig. 3.

t (3

a w

z

W

W W

a LL

I

I I

Tmin

REACTION

COORDINATE

Fig. 3. Free energy vs. reaction coordinate diagram for the formation of a carbonium ion-pair intermediate.

F. Energetics of Salvation Carbonium -ion reaction in solution inevitably involves solvated ions; the carbonium ion and its counter-ions are stabilized by interaction with the solvent. Since such stabilization is primarily associated with the presence of electric charges, it follows that the ionized form will be favored relative to the un-ionized form by increased solvation. All solvents exert some stabilization, relative to the gas phase.

THERMODYNAMICS OF POLYMERIZATION. Ill

173

The ionization of an organic molecule RX in the gas phase (g) and in solution (s) is represented in Fig. 4. R+. .X’ is the transition state of ionization and R+ and X- are the infinitely separated ions. It is clear that AGO, the standard free energy of formation of ions in solution from gaseous alkyl halide, must be the sum of AG; and AGO,. From Fig. 4 it is evident that AG:

Flg. 4. Free energy cycles for heterolysis of RX.

AG:

= AG;

AGf = AG;

+ AG; + AGO,

(13)

+ AG; + AG:

(14)

This simple relation (often referred to as the Born equation) has been used to calculate the free energy of solvation of the ion. The free energy change on transferring the ion of radium r and charge e from the gas phase to the solvent (D)is thus:

According to Eq. (15) the solvatingenergy of an ion willbe greater in solvents of high dielectric constant. Although the dielectric constant is generally used a s an indication of the solvating power of a solvent, it is not necessarily a quantitative measure. Specific solvation effects and polarizability a r e also of importance. Furthermore it would seem that ions of small radius would be most strongly solvated, and this is indeed true for inorganic cations. Therefore, the

174

H. SAWADA

interaction between the ion and the solvent should be strongest for small r and large D. Generally speaking, the solvation energies of carbonium ions will therefore be smaller than those of single charged metallic cations. Thermodynamic quantities of activation for the ionization of tert butyl chloride in various solvents are shown in Table 5. It is clear

-

Table 6 Thermodynamic Quantities for the Ionlzation of tert-Butyl Chloride in Various Solvents a t 25°C [17]'

Solvent

c

€3 2 0 HC02H HCONH2 CHsOH CHsCOOH CzH50H

78.5 56 100 32.6 6.2 24.3

AG" AH" ASa (kcal/mole) (kcal/mole) (e.u.) 19.5 21.5 23.5 25.8 26.5 27.1

23.2 21.0 22.4 24.9 25.8 26.1

12.2 - 1.7 - 3.8 - 3.1 - 2.5 - 3.2

AGG AGg (kcal/mole) (kcal/mole) 6.7

- 135

2.64 2.50 2.42

- 125 -124 -123

aAGi = 155 kcal/mole.

that AGO, is numerically very much smaller than AGO,in all solvents, and solvents such a s water o r acetic acid may contribute about 125 kcal to the dissociation process. The solvation energies in three hydroxylic solvents with the exception of water are very similar; in this case (as in many others) the bulk dielectric constants are a very poor measure of solvating power. Finally, it must again be emphasized that carbonium ions in solution a r e accompanied by an equal number of counter-ions. It is therefore impossible to study carbonium ions in solution in isolation: for example, the free energy of ionization of a molecule Rx will include the solvation energies of both R* and X-. III. INITIATION OF CATIONIC POLYMERIZATION

A. Energetic Coneideration of Initiation Reaction by Halogen Acid Let us consider the reaction of halogen acid HA with olefins. \

HA + ,C=C,-

0

\

CH-C,

/

+/ *

A-

THERMODYNAMICS OF POLYMERIZATION. I l l

175

The process of formation of cation may be divided into three steps:

+ A-

(1) HA-H' \

\

(3) ,CH-C,

\

0

(2) Ht + ,C=C,+/

+ A'

+El

,CH-C,

-

\

,CH-C,

+/

+/

-

* *

A-

€2

- €3

The energy for process (1) is the heterolytic bond dissociation energy of HA, and the quantity E , is the proton affinity of olefin. The potential energy between ion pairs separated at a distance r is

c3 = e2/rD

(17)

where e is the unit of electronic charge and D the dielectric constant. According to our convention, the negative sign refers to unlike ions (attraction) and thus E, is negative. The total energy change of the system will be

The enthalpy change of initiation of cationic polymerization depends upon the acid strength of the initiator E,, the proton affinity of the monomer c,, and solvating powers of the solvent E ~ . The free energy change of initiation is AG, = AH

- TAS

and upon substitution AG, = e l

- E, - (ez/rD) - TAS

(19)

The entropy change, of course, is negative, and hence AH must be negative in order to make AG, a negative value. Let us now consider the initiation of propylene by HC1 [18]. Enthalpy of gaseous ionization of HC1 is about 330 kcal mole and proton affinity of propylene is 175.5 kcal/mole. If r = 2.5 and D = 1, e3 is 130 kcal/mole. Thus AG, must be positive and no polymerization occurs. The formation of gaseous carbonium ions by heterolysis of neutral molecules requires high energy (Table 4). However, E , must

d

176

H. SAWADA

be greatly reduced by solvation. The high energy required for heterolysis seems to be offset by interaction between the ions and their

surroundings, in particular solvent molecules and other ions. is found to be about 25 kcal/mole for HC1 in H,O. Thus, for example, the halogen acids HC1 and HBr have been reported as effective initiators for olefins in solvents. The olefins and vinyl compounds found susceptible to cationic polymerization a r e those having substituents tending to induce an electron-rich double bond, e.g., alkyl, alkoxy, and aryl groups. For these monomers in which E , are large positive quantities, AG, is negative and hence polymerization will readily occur. Thus electron-releasing substituents favor the formation of a cationic site:

It is concluded that the rate of initiation increases with increasing

electron-density in ethylene double bond by the electron-donating effect of the substituent. When E, is small (corresponding to a positive free energy change), polymerization will not occur. Those vinyl monomers having electrophilic substituents, e.g., vinyl acetate, vinyl chloride, and the acrylic derivatives, do not respond to cationic initiators. According to Eq. (17) carbonium ions will be less stabilized and thus more difficult to form as the dielectric constant of the medium is decreased. A medium of low dielectric constant and low solvating power for the ions will favor the formation of a covalent bond, and only the addition of an initiator to monomer will occur. Therefore polymerization does not occur well by cationic methods in such cases.

B. Catalytic Activity in Cationic Polymerization by Lewis Acids In the cationic polymerization of styrene catalyzed by Lewis acids, the cationic activity of Lewis acids is in the following order [19]: SbC1, >> TiCl, > AlBr, > SnC1, > FeCl, > ZnC1, > HgC1, (252) (175) (163) (133) (141) (57) (52) Kagiya et al. [20]used the shift of the characteristic band of the carbonyl group in the IR spectrum of the Lewis acid-xanthone complex and reported that an increase in the shift showed 'an increase in

177

THERMODYNAMICS OF POLYMERIZATION. I l l

the acidity of Lewis acid. The values in parentheses in the above relation represent the magnitude of the carbonyl shift. There is a good correlation between the catalytic activity and the carbonyl shift in the Lewis acid-xanthone complex. Therefore it is expected that an increase in the cationic activity of Lewis acid will arise from an increase in its activity. Similarly, the activity of cocatalysts in the polymerization sys tem of the same monomer and the same Lewis acid catalyst, e.g., in the cationic polymerization of isobutene catalyzed by tin chloride, decreases in the order [21]: CC1,COOH

> CH,ClCOOH > CH,COOH > CH,CH,NO, >

(pKa = 0.64) CH,NO,

(2.87) > C,H,OH

(11)

(10)

(4.76)

(9)

> H,O (15)

As the pKa value of cocatalyst decreases, its activity increases.

W . PROPAGATION OF CATIONIC POLYMERIZATION A. Energetics Cationic polymerizations a r e quite exothermic since the reaction involves the conversion of n-bonds to o-bonds. The heat of polymerization for any particular monomer is essentially the same irrespective of the mode of initiation (if the monomer can be polymerized by both radical and cationic catalysts). The apparent activation energies a r e given by

Erst, = -2.303Rd log (rate)/d(l/T)

(20)

EDp = -2.303Rd log (DP)/d(l/T)

(21)

The over -all energies of activation for cationic polymerizations Erate usually fall within the range -10 to +15 kcal/mole. ED, is always negative, with values from -3 to -7 kcal/mole. These quantities are empirically convenient ways of representing the temperature dependence, but must be treated with caution because they a r e composite quantities and do not necessarily have any exact theoretical significance. In the cationic polymerization the counter-ion derived from an initiator exists near the growing chain end and gives an effect not

H. SAWADA

178

only on the initiation reaction but also the propagation reaction. If the polymerization proceeded via an ion-pair mechanism,, the value of kp would be changed by changing the initiator and solvent, even with a fixed monomer and polymerization temperature. Also, in this model the monomer molecule is highly polarized in the transition state of the propagation reaction, and it is expected that the activation energy and frequency factor would be smaller than that of radical polymerization. The activation energy of kp in cationic polymerization decreases with increasing polarity of solvent. The activation energies EDp and Eratofor the degree and rate of polymerization, respectively, are obtained as

EDp = E,

- Et

where Ei, Ep, and Et are the activation energies for the initiation, propagation, and chain termination steps, respectively. Et will be replaced by Etr when termination occurs by a transfer reaction. The values of Ei and Et are greater than Ep in most cases. For many polymerization systems Eratois negative and one observes the rather unusual phenomenon of increasing polymerization rates with decreasing temperatures. The sign and value of Erst, vary from one monomer to another. Even for the same monomer, the value of Eratomay vary considerably depending on the catalyst, cocatalyst, and solvent employed. The variations in Eratoare a consequence of the differences in Ei, Ep, and Et caused by the differences in the catalyst and the solvating power of the reaction medium. It should be noted that, irrespective of sign, the values of Eratoare generally smaller than in radical polymerizations. The rates of cationic polymerizations do not quantitatively change with temperature as much as those of radical polymerizations. The activation energy EDPfor the degree of polymerization is always negative because Et is greater than Ep for all cases irrespective of the mode of termination. This means that the degree of polymerization decreases as the polymerization temperature is increased. EDPhas greater negative values when termination is by transfer reactions than when termination is by spontaneous termination or by combination since the transfer reactions have greater activation energies. As the polymerization temperature is increased, the mode of chain breaking will shift from termination to transfer. Figure 5 shows the dependence of log D P of polyisobutene on the reciprocal temperature [22]. The plot starts to deviate from linearity

THERMODYNAMICS OF POLYMERIZATION. I l l

I

179

1

4.0

5.0 6.0

7.0

8.0

9.0 10.0

Fig. 5. Temperature dependence of DP of polyisobutene in propane solvent 1221.

around -lOO"C, then bends over to assume a lesser slope. The corresponding over-all activation energies a r e -3.54 and -0.22 kcall mole, respectively. This has been attributed to a change in the termination step from chain transfer to monomer below -100°C to chain transfer to solvent above -100°C. B. Heats of Reaction of Cations with Olefins

The propagation may be regarded as an electrophilic attack by the ion at the n-electrons of the olefin; accordingly, the ease with which it occurs is governed by the stability of the ion and by the basicity of the olefin. In particular, information on the energetics of the reaction is sparse: an indication of this affinity should be obtained from the calculated values of AHo for the gas-phase reaction as follows: \

R' + /C=C,-

/

I

I

R-C-C' I I

H. SAWADA

180

These values are given in Table 6 for some simple ions and olefins. In the propagation reaction the olefin monomer will be attacked by the carbonium carbon of the growing polymer chain. By analogy with the initiation reaction we might expect that at each step in the propagation the attack of the carbonium ion on the monomer is at its tail end, so that+at the end of the growing po+lymer chain there is always a head CH,C(CH,) and not a tail C(CH,)CH,. This would result in a head-to-tail propagation process, and would exclude the tail-to-head, the head-to-head, and tail-to-tail types of propagation. The nucleophilicity of monomer increases with a decrease in its ionization potential on the propagation reactions. The same is the case with the initiation reactions. Figure 6 shows the relation between the rate constant of propagation in the cationic polymerization of the styrene derivatives and their ionization potentials. Evans and Polanyi [8] can give a more detailed calculation of the propagation process as follows. For the head-to-tail addition they [8] have calculated this value for the head-to-tail step as 19.5 kcal. The tail-to-tail addition step was calculated as about 40 kcal exothermic, and the head-to-head step as slightly endothermic.

R1,

R

+

C -CH,

+ CH,=C,

/

RZ

R 2 ’

Rl\

RZ’

I

+

CH-CH,

+ CH,=C\ /

R‘ RZ

7 1 7 1

-+

Q = 0 kcal

CH,-C-C-CHi I I

Rz Rz -+

Rl\

R Z ’

CH -CHz-CHz-C

+2

1

\

RZ Q = 40 kcal

Thus, of the three possible steps for the carbonium-ion mechanism, the tail-to-tail addition is most probable, the head-to-tail addition is still quite probable, but the head-to-head addition is very improbable. Since the tail-to-tail addition can only occur in chain propagation if the head-to-head addition is also involved, this discussion indicates that for a carbonium-ion mechanism, however the chaip starts, it will always proceed with the head a s the positive end CH,C(CH,),, and this will lead to the head-to-tail type of addition

THERMODYNAMICS OF POLYMERIZATION. Ill

181

Table 6 Heat of Reaction of Cations with Olefins in the Gas Phase [5] A Ha (kcal/mole)

Ion

E thy1ene

Propylene

Isobutene

Methyl Ethyl n-Propyl n-Butyl Isopropyl sec-Butyl tert- Butyl

-69.5 - 35 - 22.5 - 25 -8.5 0 +7.5

-90.5 61 -45 -47 -30.5 - 22 -14.5

- 103 - 71 -56 -58 -42 -33 - 24

-

with a constant heat of reaction of 19 kcal at each step. This is shown in Fig. 7 [8]. In this figure we have also included the heats of reaction for the consecutive steps of the head-to-head, tail-totail mechanism, which alternate between slight endothermicity and compensating strong exothermicity. I

r

-

0,

0

Flg. 6. Relation between the rate constant of propagation in cationic polymerization of styrene derivative and its ionization potential [20]: (1) styrene, (2) p-chlorostyrene, (3) p-methylstyrene, (4) p-methoxystyrene.

H. SAWADA

182

z 0

a a +10

v)

5 +30 W

+40 1

2

3

4

5

STEPS

Fig. 7. Heats of reaction for the steps involved in different types of propagation mechanisms [8]. ( A ) Head-to-tail radical o r carbonium mechanism, when no steric hindrance is present.. (V)Head-to-tail radical o r carbonium mechanism, when ateric hindrance is present. ( x ) Head-to-head, tail-to-tail carbonium mechanism. (0)Head-to-head, tail-to-tail radical mechanism.

The reverse of the propagation reaction steps, i.?., depolymerization or depropagation, is generally not important in cationic polymerization because the reactions are normally carried out well below the ceiling temperatures. At higher temperatures, monomerpolymer equilibrium would be difficult to detect because of the instability of the chain ion, the importance of transfer processes, and the fact that decomposition products are likely t o be the monomer themselves. No cationic vinyl polymerization has yet been found which conforms to equilibrium polymerization. However, equilibrium polymerization arising after cationic initiation of tetrahydrofuran has been reported [23]. @-Methylstyrene, at room temperature, is near its ceiling temperature, so that the depropagation reaction might be expected to be fast. Worsfold and Bywater [24] found that it does not show typical equilibrium behavior when polymerized by BF,-Et,O. These authors suggested a s explanation that the depropagation reaction is too slow. C. Activation Entropy Changes of Propagation Pepper and Reilly [25] report that the change of activation entropy in the propagation reaction is based on two phenomena: (a) the entropy change of immobilization of a free-moving monomer into a polymer chain, and (b) the difference of entropy in the solvation between the reactants and the transition state.

183

THERMODYNAMICS OF POLYMERIZATION. Ill

AS'

(immobilization of monomer) + ASf (solvation)

=

AS'

=

-28 eu + AS' (solvation)

(24)

ASf (immobilization of monomer) is estimated from the radical polymerization. ASf (solvation) is a positive term, because in the transition state the electric charge is more diffuse and the degree of a solvation is lower than that of the more compact dipolar species corresponded to reactants. In general the activation energy becomes smaller and the frequency factor becomes larger as the dielectric constant of the solvent increases. The value of Ap of cationic polymerization is much smaller than that of radical polymerization.

Ap (radical polymerization)/Ap (cationic polymerization) = 104-s

(25) According to Eyring's theory, the propagation constant (kp) is expressed as kp = Ap exp (-AEP/RT) f = (kT/h) exp (ASf/R) exp (-AEg/RT)

(26)

and activation entropy AS* is the function of f 7 and f i , which are the partition functions of the transition and initial states of the propagation step, respectively.

AS: = R In (f,/fi)

+ RT

a In (f,/fi)/aT

(27)

f r and fi can be reasonably assigned to the partition function of the monomer in the transition state (f,(m)) and that of the monomer in the initial state (fi(m)) because other parts of the growing chain are common in the transition and the initial states of the propagation reaction. Furthermore, the second term of Eq. (27) is negligible compared with the first term. So the following equation can be derived:

AS: = R In (f,(m)/fi(m))

(28)

then Ap (radical polymerization)/Ap (cationic polymerization) z

f,(m) (radical polymerization)/f,(m) (cationic polymerization)

(29)

184

H. SAWADA

The vibrational and the rotational modes in the transition state of the radical propagation and that of cationic propagation are considered to be

radical propagation

-

* *

C*

.

rotation M vibration

- -

/

cationic propagation 1 vibration

...c+.. .M'--'A-

/ vibration vibration

rotation

In the case of cationic propagation the rotation of the monomer unit in the transition state is greatly restricted by the counter-anion. Therefore, Ap (radical polymerization)/Ap (cationic polymerization) f 7(m) (radical poly merization)/f 7(m) (cationic polym -

erization) = frot2fvl,,/fri,,3

= frotz/fvIbz =

where k and h are Boltzmann's and Planck's constants, respectively, I is the moment of inertia, and vi is the vibration frequency of the monomer unit. I = g cm2 and v 1013 sec" are inserted in the above calculation. This result coincides rather well with experimental results [26]. It is therefore concluded that A of the cationic polymerization by the ion-pair mechanism is mu,! smaller than that of the radical polymerization since the counter-ion restricts the mobility of an entering mohomer in the transition state of the propagation reaction. On the other hand, in the polymerization of styrene by perchloric acid, Ap is as large as that of the radical polymerization; that is, the propagation reaction proceeds through a free end. Ap of the free-ion mechanism is larger than that of the ion-pair mechanism, and Ap decreases with an increase in the polarity of a solvent. The polymerization in a polar solvent shows an Ap value with the same order as the radical polymerization. D. Thermodynamics of Formation of Zwitterione

In cationic polymerization initiated by uncharged Lewis acids, the formation of zwitterions in the initiation process has been postulated [27]. Monomer addition to the cationic chain ends may lead to macrozwitterions, i.e., polymer chains with a negative charge at

one end and a positive charge at the other. However, thermodynamic

185

THERMODYNAMICS OF POLYMERIZATION. Ill

consideration during propagation steps has been used as an argument against this type of zwitterionic polymerization mechanism. Considering the cationic polymerization of isobutene by AlCl,, we may write the over -all reaction as AICl,(sol) + C,H,(sol)

-

XlC1, - CH,~(CH,),(SOI)

AHA

Our reaction of interest will consist of the following five steps, all at the same temperature. (1) AlCl,(sol) + C,H,(sol) (2) C4H,(g)

- AlCl,(g) + C,H,(g) AH,,

- CH2C(CH,)2(g)

= 15 kcal/mole

p, = 50 kcal/mole

- eH,6(cH3)2(g) I, + A, = 140 kcal/mole (4) AlC1, + cH,6(CH,),(g) - a C 1 , CH,t(CH,),(g) -D, = -60 kcal/mole (3) CH,C(CH,),(g)

*

(5) fiCl,CH,6 (CH,),(g)

- aCl,CH26(CH,),(sol) -AH82

The t e r m s AH,,, D, I, and A correspond to the heat of solvation, the bond dissociation energy, the ionization energy, and the electron affinity, respectively; these terms were calculated theoretically by Plesch [28]. Since AH,, is supposed to be much smaller than 130 kcal/mole, the formation of macrozwitterions in cationic polymerizations is easily disturbed by side reactions. However, such simple energetic considerations are not in general a valid argument against the formation of macrozwitterions [29]; propagation macrozwitterions need not be linear chains (A) with an increasing distance between the two ionic epd groups. They may be cyclized to form ion-pairs from the two ionic chain ends (B) (Fig. 8). If the chain ends form ion-pairs, macrozwitterions may greatly enhance the first propagating step.

(A)

( B)

Flg. 8. Macrozwitterions a s ( A ) linear chains and ( B ) ion pairs.

186

H. SAWADA

V. CHAIN TRANSFER AND TERMINATION It is very difficult to get a high molecular weight polymer above room temperature in cationic polymerization. This is due to the frequent transfer reaction to transfer agents, monomers, and impurities. This transfer is the most important process governing the molecular weights in carbonium ion polymerizations. The ratio kp/km is somewhat temperature dependent; the activation energy of the transfer reaction is higher than that of propagation by 10 kcal/mole at most, but usually is about 2-3 kcal/mole. Therefore the polymerization must be carried out at the'lowest possible temperatures in order to obtain products of the highest molecular weights. However, this requirement does not present the practical difficulties which might be expected; because the overall activation energies for most cationic polymerizations are in the range -5 to +10 kcal/mole, the polymerization rates do not change rapidly with temperature. We can also see that the carbonium mechanism offers a possibility for the occurrence of polymerization at the lowest temperatures, whereas a free-radical process would be inevitably slowed down by extreme cold. Since the spontaneous termination reaction cannot be distinguished from the termination reaction by impurities, the termination constant ratio (kt/kp) is only an apparent value and quantitative discussion has not been carried out. Moreover, the existence of a true termination reaction is observed in only a few systems, and many systems seem to have no termination reaction [30]. Acknowledgments I wish to express my appreciation to Professor K. F. O'Driscoll, University of Waterloo, for his interest and willingness to read and criticize the manuscript. I am also indebted to the management of Daicel Ltd. for permission to write this review. Acknowledgment is also due Miss Y. Nishikawa who typed large sections of the manuscript. References [ l ] M. L. Burstall and F. E. Treloar, in The Chemistryof Cationic Polymenzation (P. H. Plesch. ed.), Pergamon, London, 1963, Chap. 2. [2]F. S. Dainton, in Cationic Polymerization and Related Complexes ( P . H. Plesch. ed.), Heffer, Cambridge, 1953. [3] T. Higashimura. in Kindai Kogyo Kagaku,Vol.16 (R.Oda et al., eds.). Asakura, Tokyo, 1966, Chap. 4.

THERMODYNAMICS OF POLYMERIZATION. I l l

187

[4] D. C. Pepper, in Friedel-Crafts and Related Reactions'(G. A. Olah, ed.), Wiley (Interscience), New York, 1964, p. 1293. [5] A. M. Eastham, in Encyclopedia ofPolymerScience and Technology, Vol. 3, Wiley (Interscience), New York, 1965, p. 35. [6] T. Higashimura, in Structure and Mechanism in Vinyl Polymerization (T.Tsuruta and K. F. O'Driscoll, eds.), Dekker, New York, 1969, Chap. 10. [7] Ref. 2, p. 158. [ 8 ] A. G. Evans and M. Polanyi,J.Chem. SOC., 1947, 252. [9] A. G. Evans, in F i b r e s from Synthetic Polymers (R. Hill, ed.), Elsevier, New York, 1953, Chap. 3. [ l o ] T. Kagiya, Y. Sumida, and T. Nakata, Bull.Chem. SOC. J a p . , 41, 2239 (1968). [ l l ] J. N. B r h s t e a d , Rec.Trav. Chim. Pays-Eas.42, 718 (1923). (12) L. P. Hammett, Physical Organic Chemistry, McGraw-Hill, New York, 1940, p. 267. [13] D. Bethel1 and V. Gold,Carbonium Ions, Academic, New York, 1967, p. 67. [14] A. Ledwith, M. Hogo, and S. Winstein, Proc. Chem. Soc., 1961, 241. [15] T. R. Griffiths and M. C. R. Symons,Mol. Phys., 3, 90 (1960). [16] N. Bjerrum, Kgl. Danske Videnskab. Selskab, Mat. F y s . Medd., 7, (9), (1926). [17] Ref. 1, p. 26. [18] Ref. 3, p. 12. [19] S. Okamura, T. Higashimura, and H. Sakurada, Kogyo Kagaku Zasshi 61, 1640 (1958). [20] T. Kagiya, Y. Sumida, and T. Nakata, Bull. Chem. SOC. Jap., 41, 2247 (1968). [21] K. E. Russell, T r a n s . Faraday SOC.,48, 114 (1952). [ 2 2 ] J. P. Kennedy and R. M. Thomas, International Symposium on Macromolecular Chemistry, Montreal, Canada, 1961. [23] M. P. Dreyfuss and P. Dreyfuss,Polymer, 6, 93 (1965). [24] D. J. Worsfold and S. Bywater, J. A m e r . Chem. SOC.,70, 4917 (1957). [25]D. C. Pepper and P. J. Reilly,Proc. Roy. SOC., S e r . A , 291, 41 (1966). [26] N. Kanoh, T. Higashimura, and S. Okamura, Makromol. Chem., 56, 65 (1962). [27] W. Kern and V. Jaacks, J . Polym. Sci.,48, 399 (1960). [28] Ref. 3, p. 19. [29] V. Jaacks and N. Mathes, Makromol. Chem., 131, 295 (1970). [30] Ref. 2, Chap. 30.

J. MACROMOL. XI.-REVS. MACROMOL. CHEM., C8(2), 235-288 (1972)

Thermodynamics of Polymerization. IV. Thermodynamics of Equilibrium Polymerization HIDE0 SAWADA Filter Laboratory Daicel Ltd. Teppo-cho, Sakai, Osaka, Japan

I. 11.

111. IV.

V.

POSSIBLE T Y P E S O F EQUILIBRIUM POLYMERIZATION

238

SOME CASE STUDIES O F EQUILIBRIUM POLYMERIZATION . . .. . . .. A. Vinyl P o l y m e r i z a t i o n s . . . B. Ring-Opening P o l y m e r i z a t i o n s . C. P o l y m e r i z a t i o n of Aldehydes . . ..

243 244 247 262

TRANSITION PHENOMENA IN EQUILIBRIUM POLYMERIZATION . . . . . ...

264

..................................

. .. .. . . .......... .. . .. .... ... . . . ... . . .... . ... .. . . . . . . .. . . . .. .. .

. .... . ... . MOLECULAR WEIGHT DISTRIBUTION . . , . , . . , . . . . A. E q u i l i b r i u m P o l y m e r i z a t i o n . . . . . . . . . . . . . . . . . B. Living P o l y m e r i z a t i o n . . . . . . . . . . . . . . . . . . . . .

270 270 276

THERMODYNAMICS O F EQUILIBRIUM POLYMERIZATION.....

280

. . . .. . .

............................

. . ... , . ... . . .. .. . .... . ............................

ACKNOWLEDGMENT..

285

REFERENCES

286

235 Copyright ill, 1972 by Marcel Dekker, Inc. All Rights Rejerved. Neither this work nor any part may he reproduced o r transmitted in :my form o r by any means, eleclronic or mechanical, including photocopying. microfilming. and recording, or by any information storage and retrieval system, without permi\sion in writing from the publi\her.

SAWADA

236 -

I. POSSIBLE TYPES OF EQUILIBRIUM POLYMERIZATION

In many cases polymerization proceeds under conditions of equilibrium between polymer and monomer. In some cases an initiator enters into the polymerization equilibria. A general theory of equilibrium polymerization was originally developed by Tobolsky in a long series of papers [l-61. The particular approach used in the treatment presented here is that developed by Tobolsky and Eisenberg [51. Three basic types of possible equilibrium polymerizations are: Case 1 Kl

XY + M LXMY XMY

K2 +M A XM,Y

... XMn-,Y

Kn

+M

XMnY

. .. Case I1 Kl

M & M* M*

+M

K2 L M,*

(The asterisk represents an activated state such as a diradical o r a zwitterion.) Case I11 K1

M + M -M2 K2 M, + M

...

M,

THERMODYNAMICS OF POLYMERIZATION. IV

237

Polymerization proceeds under conditions of equilibrium between polymer and monomer in Cases I1 and 111. In Case I an initiator enters into the polymerization equilibria. The experimental problem is to place monomer (or monomer plus initiator) of known concentration [M,,] (and XY, where present) in a sealed system. The system is kept a t constant temperature until equilibrium is attained. At this point the system is quenched, and the sample analyzed for the M, P, and X Y . Let u s consider the detailed mathematical s t e p s in the treatment of Case 111. The overall kinetic scheme is then, in the absence of transfer

...

.

where K , , K,, , . , K, are the equilibrium constants for the initiation and succeeding propagation reactions respectively. All the activities appearing in the equilibrium equations are defined in moles per kilogram. The problem of volume contraction may, however, be circumvented by expressing all the concentrations in moles p e r unit m a s s of solution. The subscript e denotes that all the relevant concentrations have the equilibrium values. The equilibrium concentration of n-mer is

The total concentration of polymer molecules, N, will be given by n-1

N = ll Kk[M,Ien k =I

The assumption will be made that the constants K,, K 3 , . are all equal and can be represented by K,. Thus

.., K,, .. .

238

SAWADA

and hence the total equilibrium concentration of polymer molecules is N =

2KIe1[Ml]t n=z

= K1[M1],2(1 + K,[M,], + K12[M1],2 +

-

* *

(4)

K,[M, 1," 1 - K,[M1l,

and the total equilibrium concentration of monomer segments incorporated in polymer, W, is given by

=cnK,'''[M,],n m

W

n=a

Consequently the number-average degree of polymerization, P is

P = W/N

-

1 1 - KI[M,I*

. ..

Let us assume that K, = K,,, = = K m for some value of j ->I+,. It has been pointed out by Szwarc [7] that when the system attains its state of equilibrium, the following relation is obeyed.

where [M,,] denotes the total amount of the monomer introduced into a unit volume of the solution, [MI, the equilibrium concentration of the monomer, [Po*] the concentration of all the polymer molecules, [Q,*] the concentration of those polymers which have a degree of polymerization i < j, and [R,]the amount per unit volume of the monomer incorporated in these polymers (i < j). For a high number-

THERMODYNAMICS OF POLYMERIZATION. IV

239

average degree of polymerization the ratio ([P *I - [Q,*])/([M,,] [R,] [MI,) approaches zero and then K, = l/[M],. Tobolsky [5] introduced two physically reasonable approximations

-

K, = K

& = K-3 = & = .

s.

= Kn =

.. . = K-3

K, = K, = K-3 = .

* *

= K, =

*

- .= K-3

(a) (b)

Whether a given equilibrium falls under approximation (a) o r (b) can sometimes be determined from chemical intuition. The detailed mathematical steps in the treatment of Case IIIa are as follows: M +M M, M,

+M +M

--

K

-

K3

M, M,

Kg

& M,

The total concentration of polymer molecules is

N

=

2 [MnIe n 2

and hence the total equilibrium concentration of monomer segments incorporated in polymer, W is given by

240

SAWADA

Thus the number-average degree of polymerization, P, is

When equilibrium concentrations of monomer and initiating species a r e attained in a polymerization system, the degree of polymerization is dependent upon one equilibrium constant only, K,, and is independent of the rate of initiation. This is in direct contrast to the conditions in an irreversible polymerization (or in a polymerization with some depropagation but in which equilibrium conditions are not attained) when the degree of polymerization depends upon the rate of initiation. Consequently the initial monomer concentration, [M,,], is

Combining Eqs. (10) and (11) and setting P - 1/P = 1 (valid for P >> l ) , it is possible to eliminate [MI, and obtain

From Eq. (lo), if P >> 1, we find K,

G

l/[MI,

(13)

The above results are equally applicable to the Case 11% where all constants a r e identical by assumption (b). Thus we have ([MolK,)”2 = ([%]/[M]e)1’2

P

(14)

Let us consider the detailed mathematical steps in the treatment of Case Ia. XY

+M

K

XMY

K3

XMY + M 2 XM,Y K3

XM,Y + M & XM3Y

...

[XMY], = K[XYI,[MI, [XM,Y], = K3[M],[XMYle [XM,Y], = (K,[M1,)2[XMY1,

THERMODYNAMICS OF POLYMERIZATION. IV

24 1

The total concentration of polymer molecule is

The total equilibrium concentration of monomer segments incorporated in polymer is given by

and hence the number-average degree of polymerization is W 1 = N 1 - &[MI,

p =-

Consequently the initial monomer concentration is

Therefore the initial concentration of initiator is

In addition, the following derived relationship is also useful p =-

[%I - [MI, - [xyle

(19)

[ ~ y o ]

In Table 1 a r e summarized the necessary relations for various cases. Let us now examine the occurrence of maxima in the molecular weight-temperature curve for equilibrium polymerization [8]. Sub-

242

SAWADA

Table 1 Types of Equilibrium Polymerization [ 51 Case

P

[M,

1

stituting the expression for [XY], from Eq. (18) into Eq. (19), inserting 1/& for [MI, (which is valid P >> 11, and simplifying, we find

If P > &/K (this is true for Case Ib in which K, = K), maxima are impossible since essentially only a propagation reaction is involved; in a case in which P z K,/K a maximum height might possibly occur, but to ascertain the fact, Eq. (20)would have to be solved for P and differentiated with respect to T. Finally, if P/& << 1/K, and this is the most important case since here we are most likely to observe maxima, we can neglect P/K, with respect to 1/K, and obtain the simplified expression

Differentiating this expression with respect to temperature after writing K = exp(AS'/R) exp(-AH'/RT) and the equivalent for K, and setting d(P2)/dT = 0, which is an acceptable criterion for the occurrence of a maximum in P vs. T, we obtain

It should be stressed here that a close relationship exists between the temperature at which the maximum in P occurs, T,, and the transition, i.e., 'ceiling" o r "floor" temperature, T,. Based on a previous development of expressions for transition temperatures,

THERMODYNAMICS OF POLYMERIZATION. IV

243

this relationship can be written a s T m _ -

Tt

AS,"

AS,"

+ R In[&]

+ R In

[&](AH'

- AH,') AHo

If AH: is much smaller numerically than AHO, these two temperatures a r e seen to be very close together. Elsenberg [a] has considered this problem in more detail, and have applied it to equilibrium polymerization of elemental sulfur shown in Fig. 1. &

1

D

TEMPERATURE ( O K )

Fig. 1. Degree of polymerization vs. temperature for the equilibrium polym: Calculated values. erization of sulfur [3,6].(0):Experimental points. (-)

11. SOME CASE STUDIES OF EQUILIBRIUM POLYMERIZATION

The establishment of an equilibrium between a monomer and its addition polymer is not usually an easy matter. This is because the active centers involved in the formation of the polymers a r e usually short-lived. To establish a reasonably mobile equilibrium between monomer and polymer, it is necessary (a) that a certain concentra-

244 -

SAWADA

tion of active centers is continually present in the polymerization system, and (b) that in such a system only long-chain polymer is present at equilibrium and there is no danger of side reaction; in addition, (c) it i s desirable that the polymerization proceeds homogeneously. However, the fundamental principles of equilibrium polymerization have been presented and applied somewhat concisely to several cases. It might be profitable to consider some actual cases of equilibrium polymerization. The particular cases to be discussed a r e chosen so as to illustrate a variety of equilibrium polymerization type.

A. Vinyl Polymerizations In treating equilibrium polymerization of vinyl monomers, particularly methyl methacrylate, Dainton and Ivin [9] and Bywater [lo] implicitly treated the problem as a one-constant case, and derived the following relation by other methods:

However, Tobolsky [6]determined P in this system and treated the problem as a two-constant case where

In fact, from the measured value of P, a value of K, can be determined which i s about 3 X 10“ (cf. K, = 3). Equation (13) has been used with success for methacry lonitrile [ l l ]as well as methyl methacrylate, in both of which initiation of either polymerization o r depolymerization proceeded by a radical mechanism. The anionic homopolymerization of tert-butyl vinyl ketone appeared to be an equilibrium polymerization and the effect of temperature on the equilibrium monomer concentration was investigated [12]. The studies of McCormick [13] and of Worsfold and Bywater [14, 151 illustrate applications of Eq. (13) to such systems as a-methylstyrene-poly(a-methylstyrene) and styrene-polystyrene. a-Methylstyrene in tetrahydrofuran reaches equilibrium with its polymer in the presence of sodium naphthalene used as an anionic initiator at all temperatures [13]. On the other hand, styrene-polystyrene system initiated by butyllithium reaches equilibrium with the polymer in the temperature range of 100-150°C [15]. The results are shown graphically in Fig. 2 as a plot of log K against 1/T. The equilibrium

TH E R MODY NAM I CS OF

PO LY M E R I ZAT ION. I V

245

9c

-

+ 8.C

10

5 Q I

7c

6.(

24

2 5

26

1031~

Fig. 2. F r e e energy functions plotted against re ciprocal t emper at ur es f o r the equilibrium polymerization of s tyrene [15]. (0):Benzene. (+) Cyclohexane.

concentrations were found to differ in the two solvents benzene and cyclohexane, but the calculated free energies of the polymer-monom e r systems were the same after corrections had been applied for the different heats of solution. F r o m the free energy data the heat and entropy of polymerization of styrene were calculated and found to agree with the combustion data and many other calorimetric estimations. Tobolsky, Rembaum, and Eisenberg [16] investigated the equilibrium polymerization of a-methylstyrene a t 0°C initiated by naphthalenesodium and diphenylacethylenesodium. With both initiators the extrapolated equilibrium monomer concentration has been found to be equal to 0.89 mole/kg. Initiation by naphthalenesodium was shown to be an equilibrium electron-transfer reaction with a high equilibrium constant. In the sodium diphenylacethylene system the initiation consists of the equilibrium step involving bond formation between the monomer and initiator, and the equilibrium constant has a sufficiently low value to be determined practically. When the initiation equilibrium was taken into account, good agree-

SAWADA

246

ment was found between theoretical calculations and experimental results, With both initiator at O"C, as well as when cesium was used in place of sodium, the same equilibrium monomer concentration was found. Polymerizations of substituted a-methylstyrenes were carried out using anionic initiator [17]. Data of heats and entropies of polymerization from equilibrium studies are given in Table 2. Table 2 Heats and Entropies of the Polymerization of Substituted a-Methylstyrenes Monomer ~~

a-Methylstyrene

p-tertButyl a-methylstyrene p-Diisopropenyl benzene m-Diisopropenylbenzene 1,3,5-Triisopropenylbenzene 1,2,4-Triisopropenylbenzene

6.96 8.02 7.47 8.50 7.1 7 .O 8.2 5.9 4.7

24.8 28.75 26.5 30.6 25.5 23.0 25.8 17.1 13.6

13 14 a

b

17 17 17 17 17

aA. Vrancken, J. Smid, and M. Szwarc, Trans. Faraday Soc., 58,2036 (1962).

bH.Hopff and H. Liisi, Makromol. Chem., 45,169,183 (1960). The equilibrium pressure of methyl methacrylate vapor over its polymer has been determined by Small [18] and Ivin [19]. This press u r e is given by In P

=

(AH/RT)

- (AS/R)

(24)

Small [18] measured the equilibrium pressure of monomer at temperatures between 100 and 160°C. The attainment of equilibrium was induced by radicals derived from the photodecomposition of benzoin, the equilibrium partial pressure of monomer being measured by the dew-point method. On the other hand, Ivin [19] used a kinetic method to obtain the equilibrium pressure of monomer over polymer at temperatures between 96 and 142"C, the approach to equilibrium being induced by direct absorption of ultraviolet light. Values of AH are AHgc = -19.5 kcal/mole [18], AH,, = -13.4 kcal/mole [19], and the

THERMODYNAMICS OF POLYMERIZATION. IV

247

corresponding entropy value ASlc = -27.8 eu [19]. Equilibrium pressures of ethyl methacrylate over its polymer have been determined from 102 to 138.5"C [20]. The equilibrium pressures are slightly lower, and the derived heat and entropy changes slightly higher than for the methyl ester. It should be stressed that the results of such studies as outlined above are independent of the mechanism of the reaction, and although they were derived from studies of anionic o r radical polymerizations, they apply equally well to any polymerization process involving the investigated monomer and polymer. The nature of the solvent does, however, affect the results since it modifies the f r e e energy of the initial and final states. B. Ring-Opening Polymerizations

1. Tetrahydrofuran. Meerwein [21] observed that the maximum possible degree of conversion to polymer at room temperature was around 70% and therefore suggested that the propagation reaction involved oxonium ions with a monomer-polymer equilibrium(Scheme 1).

3 4" 1

CHj-CH,

OdCH

24

0 \

CH,-CH,

CH2-CH,

+ rFC>o CH,-CH,

S~O-(CH,$I< n.1

1

CH2-CH,

Scheme 1.

Several papers have recently appeared concerning the equilibrium polymerization of tetrahydrofuran under the influence of cationic catalysts such as phosphorus pentafluoride [22], triphenylmethyl hexachloroantimonate [23], trialkyloxonium salts [24,25], o r benzenediazonium hexafluorophosphate [26]. The polymerization process is reversible and an equilibrium between monomer and polymer may be established in the undiluted polymer [22-24,271 o r in solution [25,28]. The data in Table 3 show clearly that the degree of conversion of monomer to polymer reached a constant (equilibrium) value at 25°C independent of initiator concentration [23]. The conversion a t equilibrium is given by conversion

=

([M,]

- [M],)/([M],)

(25)

Tobolsky [25] found that [MI, i s indeed a constant, namely 2.62 mole/l at 0°C provided that [M,,] is larger than 5-6 mole/l. Above an initial monomer concentration of 5-6 mole/l, agreement between the calculated value from Eq. (25) for [MI, = 2.62 mole/l and actual

SANADA

248

Table 3 Polymerization of Tetrahydrofuran at 25'C' [ 231

M

Weight of polymer recoveredb (9)

(dug)

1.37 2.06 2.74 3.53 4.93 8.00

6.44 6.85 6.35 6.71 6.57 6.29

3.55 2.88 2.70 2.06 1.76 1.64

[ ph,C+SbCl,-] X l o 3

[VlC

'10.0 mi of THF used for each experiments. bEach reaction left for approximately 48 hr. cMeasured in benzene at 25.0"C.

equilibrium conversions i s within experimental error. When equilibrium conversions at various temperatures are plotted against temperature (Fig. 3), extrapolation gives a ceiling temperature of 84°C [26].

0

30

60

90

TEMPERATURE VC) Fig. 3. Ceiling temperature in tetrahydrofuran polymerization 1261.

THERMODYNAMICS OF POLYMERIZATION. IV

249

The equilibrium concentrations [MI, at different temperatures have been used to derive the heat and entropy of polymerization using the simple formula ln[M], = (AH/RT)

- (AS/R)

(26)

A plot of ln[M], vs. 1/T should be a straight line of slope AH/R according to Eq. (26). Figure 4 shows t h i s plot for data obtained in Dreyfuss's work [26], and the value of AH obtained in this manner is -4.58 kcal/mole and the corresponding entropy change AS is -17.7 eu. Results of thermodynamic parameters calculated from equilibrium concentrations of tetrahydrofuran by several workers a r e shown in Table 4. 24-

i

-

16-

I 08' 26

I

I

\

32

29

lo3 I T Fig. 4. Equilibrium monomer concentration a s a function of t e m p e r a t u r e for the equilibrium polymerization of tetrahydrofuran [ZS].

2. c-Caprolactam. The process of polymerization of c-caprolactam initiated by water may be divided into three steps: ring opening (K), condensation (K2),and addition (K3).Both ring opening (the hydrolysis of caprolactam to the linear €-amino caproic acid) and addition (the direct attachment of a molecule of c-caprolactam to an endgroup of a polymer molecule) are the principal reactions according to which €-caprolactam is converted, whereas the condensation

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Table 4 Thermodynamic Constants of THF Polymerization Standard statesa

-AH (kcal/mole)

-AS ( 4

TC ("C)

Ref.

Is Is Ic Is Ic

4.28 f 0.2 4.58 2.97 5.3 f 1.0 5.5

17.0 f 0.6 17.7 9.75

83 84 80 f 3 60 - 70

22 26 27 23 C

20.8

aMonomer and polymer states: 1, liquid; s, solution; c, condensed. bCeiling temperature in bulk polymerization. CB.A. Rosenberg, E. B. Ludvig, A. R. Gantmakher, and S. S. Medvedev,J. Polyrn. Sci.,Part C,16,1917(1967).

reaction results in an interlinking of linear species by the reaction between an amino end group and a carboxyl end group forming an amide group and a molecule of water (Scheme 2).

co

NH

K HOOC(CH,),NH,

4-

HOOC(CH,

NH,

1,

HOOC(CH,),

n

NHCO(CH,),

0, 1, 2, 3, 4, - - - - - - -

Scheme 2.

NH2

-k

H,O

25 1

THERMODYNAMICS OF POLYMERIZATION. I V

At a given temperature, for a given value of [q] and [&I, P and [MI, a r e available from data in the literature [29]. Referring to Eq. (10) and Table 1, K, was obtained from P = 1/(1 K,[M],). [XI, was obtained from P = ([%I - [MI,)/([%] [XI.). Finally, K was obtained from [&] = [X],(l + K[M],P). The constants K and K, may be found by application of the van't Hoff equation with the result [5]

-

-

K = exp(ASo/R) exp(-AH"/RT),

AS" = -6.8 eu,

AH" = +2.24 kcal/mole

K,=exp(AS,"/R) exp(-AH,"/RT),

AS," = -7.0 eu,

AH,"

-4.03 kcal/mole

=

corresponding to the initiation and propagation reactions, respectively. The overall reaction of c-caprolactam initiated by an alkyl amine o r an organic acid can be written as in Scheme 3. To illustrate the problem, let us consider the reaction K3 RNH, + CL LR(CL)NH,

R(CL),NH,

t

K3

CL & R(CL),+,NH,

n

=

1, 2, 3, 4, . . .

where CL represents caprolactam and (CL) represents -NHCO(CH,),-. In this case we take K = 1(3 aa a good approximation since the same functional groups a r e involved in the equilibria. No satisfactory equilibrium data exist in the literature for this case. Tobolsky [4]ventured to predict that this equilibrium could be described by using the results of Case Ia by merely replacing K by KS.The first iml/[M],. portant prediction is obtained from Eq. (131, namely K, Hence, in the range of high degrees of polymerization, the equilibrium concentration of monomer is a function of temperature only and independent of the nature of the initiator.

Scheme 3.

It should be pointed out that in theory, a s outlined above, the presence of cyclic oligomers is not taken into consideration. These a r e present in the equilibrium mixture to an extent of at most 5% [30],and since the size of the rings varies just as does the length

252

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of the open chains, they a r e assumed not to change the value of P which is normally obtained for the straight chains only. Also, condensation reactions of the type M,* + M,* * M$+, are assumed not to change the number-average molecular weight of the polymer at equilibrium since equilibrium depends only on the initial and final state, not on the reaction path o r reaction mechanism. The AH, for the condensation reaction -NH,

+ -COOH

K2 ---L -NHCO-

+ H,O

was reported to lie between -1400 and -600 cal/mole. Meggy [31] found AH, = 3560 cal/mole for the propagation reaction. A calorimetric study of the polymerization sets the total heat of reaction Q = 28.5-29.0 cal/deg, and Fukumoto [32] found the heat of condensation between linear polymers to be -6.8 kcal/mole. The temperature dependence of K, of a nylon 6 is given by the equation [33] log K, = 1,342.5/T + 0.2033 AH, = -6,140

cal/mole

AS, = 0.9 eu When E-caprolactam polymerizes at 257'C, about 7% of the lactam monomer remains unchanged on account of the ring-chain equilibrium. If one of the hydrogen atoms is substituted by a methyl group, the amount of the lactam monomer unchanged increases, that is to say, the equilibrium i s shifted to the side of ring molecules, If two of the hydrogens are substituted by two methyl groups, the polymerization does not occur. These shifts in chain-ring equilibria a r e explained by the change in the thermodynamical properties caused by the existence of rotational isomers [34]. The effect of the alkyl group substitution on the amount of the lactam ring remaining at the equilibrium is shown in Table 5. 3. Laurolactam. The equilibrium polymerization of laurolactam was studied between 260 and 320°C using organic acids and amines as initiators, and it was treated as a two-constant case by Elias and Fritz [35]. The heat of polymerization was found to be AH, = -3.1 kcal/mole, the corresponding entropy AS, = -3.75 eu, the enthalpy and entropy for the formation of N-lauroylamino lauric acid to be AH = -3.35 kcal/mole, and AS = -4.5 eu. The ceiling temperature is approximately 550°C. 4. 1,3-Dioxolane. Equilibrium polymerization of 1,3-dioxolane has been studied with cationic catalysts in methylene dichloride [36]

TH E R MODY NAM ICS OF PO LYM E R I ZAT ION. I V

253

Table 5 Chain-Ring Equilibria in the Polymerization of Substituted eCaprolactams [ 341 Substituted ECaprolactam

Equilibrium monomer concentration Iwt%)

ECaprolactam 3-Methyl4-Methyl5-Methyl6-Methyl7-MethylN-Methyl5-Ethyl5-Propyl3,6Dimethyl-

~

0 bserved

Calculated 16 23 23 23 16 50 60 100

7 13 27 28 28 16 100 67 100 100

o r in benzene [37]. The living nature of the polymerization of 1,3dioxolane was confirmed by the following evidence: A second portion of monomer added to a polymerization mixture in which reaction had ceased gave a reaction rate of the same order of magnitude a s the first reaction and the yield of polymer and its degree of polymerization corresponded to what was obtained when a quantity of monomer equal to the sum of the two portions was polymerized in one step. The degree of polymerization of the polymer and the position of equilibrium could be adjusted to any desired value by adjusting the final temperature of the reaction mixture, irrespective of the temperature at which the polymerization had been carried out. This evidence shows that termination is unimportant, and that the polymerization of 1,3-dioxolane is a true equilibrium polymerization [36,37]. From the measured yields of polymer the equilibrium constant K, for the monomer-polymer equilibrium o r the equilibrium monomer concentration, which is the same, was calculated by Plesch et al. [36] and Yamashita et al. [37]. From the variation of K, with temperature the following values of AH,, and ASs, have been obtained: kcal/mole

AS,, = 18.6 f

1.2 eu

0.6 kcal/mole

AS,, = -14 f

2 eu

AH,, = -5.1 i 0.2 AH,, =

-3.6

f

by Plesch et al.

byyamashitaetal.

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254

Direct calorimetric measurements in the adibatic reaction calorimeter gave AHam= -5.2 f 0.1 kcal/mole, and depolymerization experiments gave for the enthalpy of depolymerization AH,, = +5.2 f 0.6 kcal/mole [36]. A plot of AG,," against temperature gave AS,," = -18.6 f 1.2 eu and a ceiling temperature T," = 1.5"C for the solution in the standard state 1 mole/liter. Since AH was found to vary neither with monomer concentration nor with the degree of polymerization of the polymer, it is possible to assume that the observed AH,, = AH,," and, from this and AS,,", Tf = 1.O"C. For the 1.22 M solution T, is found to be 7°C. In Fig. 5 is shown the variation of the degree of polymerization with temperature. It is observed from this figure that the curve extrapolates to P = 1 at 8"C, the ceiling temperature for the 1.22 M solution, in good agreement with the calculated value. The shape of the curve indicates, according to Dainton and Ivin [38], that in this reaction there is transfer to monomer. The fact that this curve appears not to become asymptotic to the temperature axis, although the P's range from 70 down to 7, and the close agreement between the experimental and calculated values of T,", both indicate that AG,," cannot vary much with P [36]. The ceiling temperature of the bulk polymerization of 1,3-dioxolane was found to be

3

TEMPERATURE ( O K ) Fig. 5. Dependence of degree of polymerization on temperature for the equilibrium polymerization of 1,3-dioxolane 1361.

THERMODYNAMICS OF POLYMERIZATION. IV

255

about 150°C [37]. This is much higher than the ceiling temperature for the polymerization of tetrahydrofuran. 5. 1,J-Dioxepan. The first thermodynamic information on the polymerization of this monomer, which is an equilibrium reaction, came from the semiempirical calculation of AHgg= -4.7 kcal/mole by Skuratov and his co-workers [39]. A brief exploratory study of the monomer-polymer equilibrium of this and some related compounds was made by Strepikheev and Volokhina [40]. 1,3-Dioxepan was polymerized under vacuum in methylene dichloride solution by anhydrous perchloric acid. The reaction involves a perfectly clear monomer-polymer equilibrium. The oligomers and polymers a r e cyclic and a r e formed by a ring-expansion mechanism (Scheme 4). This system was studied by Plesch and Westermann [41] who obtained AH,," = -3.5 f 0.3 kcal/mole

Scheme 4.

from the van't Hoff plot of the results; this showed that there is no significant variation of AH,," with temperature. From the plot of AG,," against temperature, AS,," = - 11.7 f 1.5 eu, and AS,," was also found to be invariant with temperature. From Fig. 6 the standard ceiling temperature T," at which AG,," = 0 for the 1 M solution (strictly, the ideal 1 M solution) is found to be +27"C. The value calculated from AH,," and AS,," is T, = +26"C. Under these conditions departures from ideal behavior seem to be unimportant. For comparison, polymerization of 1,3-dioxepan and 1,3,6-trioxocane were examined in benzene solution with BF,,Et,O [371. From the relation between equilibrium monomer concentration and polymerization temperature the heats and entropies of these polymerizations were evaluated as follows: AHs8 = -3.2 f 0.5 kcal/mole 1,3-Dioxepan

AS,, 1,3,6-Trioxocane

= -9.3

AH,, = -5.3

* 1.4 eu * 0.8 kcal/mole

AS,, = -9.3 f 1.4 eu

256

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I I I

12 200

I

I

I

220

240

260

TEMPERATURE

I

280

I

300

(OK)

Fig. 6. Graphical determination of standard ceiling temperature of 1,3dioxepan [4 11.

6. Trioxane. In the cationic polymerization of trioxane and tetraoxane near room temperature, the equilibrium trioxane concentration is not negligible during polymerization [42,43]. Tetraoxane was polymerized with BF,Et,O in various solvents, and the equilibrium concentration of trioxane produced by the polymerization of tetraoxane and equilibrated with the growing polyoxymethylene chain was determined [44]. The equilibrium trioxane concentrations were 0.05, 0.13, and 0.19 mole/l in benzene, ethylene dichloride, and nitrobenzene a t 30°C, respectively, and 0.20 mole/l in ethylene dichloride at 50°C. The values in ethylene dichloride showed that the approximate values of AH3 and AS, were -4.2 kcal/mole and -9.7 eu, respectively [44]. Equilibrium pressure of gaseous monomers a r e measured spectroscopically over the solid polymer at different temperatures. The heat and entropy of polymerization of gaseous trioxane to the crystal line polymer were calculated from the formula

RT In P

= AH

-

TAS

Thermodynamic constants of polymerization of trioxane were calculated for different phase conditions of monomers and polymers [45]. Results of calculation are indicated in Table 6 with those of tetraoxane Values of the standard enthalpy and standard entropy of polymerization of solid trioxane can be calculated using t h e data of Bus-

.

THERMODYNAMICS OF POLYMERIZATION. IV

257

Table 6 Thermodynamic Constants for Polymerization of Trioxane and Tetraoxane Standard states

-AH" (298°K) (kcallmole)

-AS" (298°K) (eu)

-AGO (298°K)

Monomer

(kcal/mole)

Ref.

Trioxane

gc

15.2' 9.9b

4.2 1.45 0.1 0.9 1.5 2.38 1.3 0.42 0.45 1.03

45 45 45 45 45 46 43

Tetraoxane

Ic cc cc sc gc cc cc

37.0 25.3 0 11.7 1.6 -4.3 3.8 9.7 37.2 -7.23 -0.82 0.01

O.lb

5.4b 2.0b 1.1 0.93b 4.2a 11.5b 1.71b 0.79 f 0.04'

*

_+

*

d d

48

'Data from equilibrium studies. bData from various kinds of thermal data from different sources. 'Data from measurement of the heat capacity of monomer and polymer and heat of polymerization. dT. P. Melia, D. Bailey, and A. Tyson, J. Appl. Chem., 17,15 (1967).

field and Merigold [46] and that obtained from formaldehyde/trioxane equilibrium experiments [47]. Thus for the reaction (c,crystalline)

,-(CH,0)37 AH,," = 1.1 AS,,"

=

__t

3/n ---(CH201n---

* 0.93 kcal/mole

-(4.3

f

3.8) eu

Recently the free energy of polymerization of crystalline tetraoxane h a s been accurately determined by Nakatsuka and others [48]. Their results are compared with the corresponding thermodynamic functions for the polymerization of crystalline trioxane in Table 6. For both monomers both AH,," and AS,," a r e close to zero and the standard free energy of polymerization, AG,,", is negative at all temperatures. Neither system therefore exhibits either a ceiling or a floor temperature and polymerization is thermodynamically possible at all temperatures. However, in the case of trioxane, the

258

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precision is such that the sign of both AH,," (298°K) and AS,," (298°K) is a little uncertain. The effect of temperature on these quantities also makes extrapolation to other temperature ranges uncertain without accurate heat capacity data. 7. Sulfur. The first relatively complete theory of the equilibrium polymerization of liquid sulfur was given by Gee [49,50]. This theory was characterized by the use of two distinct treatments: one valid below the transition temperature and one valid above the transition temperature. Tobolsky and Eisenberg [3] later presented a unitary theory valid above and below the transition temperature. This theory was perfectly consistent with the results of Gee. The conditions at equilibrium at any temperature for the polymerization of sulfur may be derived as

Let N and W represent the total concentration of polymer molecules and the total concentration of monomer segments (S, units) incorpo rated in the polymer, respectively.

THERMODYNAMICS OF POLYMERIZATION. IV

259

It is obvious that W/N = P, where P is the number-average chain length (in terms of S, units) P = W/N = 1/(1

- %[MI,)

(10)

Thus

Equation (10) may be substituted into Eq. (29) and

[%I

=

P-1

PK, + Z P ( P -

1)

(30)

For the temperature region in which high polymer is stable, i.e., above the floor temperature, P is >> 1, from Eq. (30) we see that p 2 (

[%IK,--1 K

)

It must be emphasized that in deriving Eq. (30), no assumptions were made that restrict the validity of the formula to any temperature region; it should be applicable in the entire liquid range. Only a knowledge of K, and & for any temperatures is required for the determination of P and [MI,at that temperature and, conversely, if P and [MI,are known, K and K, can be determined. From Eqs. (10) and (29), one can calculate the value of K and I& at a given temperature from the experimental values of [MI, and P. This was done for two temperatures above 159"C, and by use of the van't Hoff equation the following relations were obtained for the equilibrium polymerization of sulfur:

In K = -AH"/RT + AS"/R; In K,= -AH,"/RT

+ AS,"/R;

AH" = 32.8 kcal/mole, AH," = 3.17 kcal/mole,

AS" = 23.0 eu AS," = 4.63 eu

The linearity of the plots of In K vs. 1/T and In K3vs. 1/T is striking evidence for the validity of the theory, At this point K and K, were known at all temperatures and P and [MI, could also be evaluated at all temperatures. Remarkably this led to a complete prediction of the P vs. T curve and the [MI, vs. T curve, as shown in Figs. 1 and 7, including the prediction of the sharp transition. The

SAWADA

260

I 450

1

I

I

500

55 0

TEMPERATURE (OK) Fig. 7. Equilibrium monomer concentration a s a function of temperature for the equilibrium polymerization of sulfur [3l. (0):Experimental points. (-) : Calculated values.

entropy change on polymerization is positive, as the larger rings had more favorable entropies of ring opening than did the small rings. The standard entropy change involved in converting cyclohexane to hexene-1 is 25 eu, which agrees well with the value of AS" = 23 eu which was obtained for the opening of the S, ring to the S, diradical. The enthalpy change is also positive in sulfur polymerization because of the greater stabilization of the S-S bonds in the ring, due to resonance and bond interactions, than in the opening- chain polymer. The Tobolsky-Eisenberg theory was based on the chemical reac tions

K S,

+

S,*,

n-1)

3

_8 , *

7'

LS8n*1

=

K3[S81[S$(n-1)l

(33)

where the asterisk indicates chain molecules and the only nonpolymer molecule is supposed to be the 8-membered ring. With the help of these equations and a straightforward application of the theory of chemical equilibria, the weight concentration of polymers as well as the mean chain lengths were expressed in terms of K, and I(3. It is obvious that Eq. (33) can be replaced by Eq. (34) with-

26 1

THERMODYNAMICS OF POLYMERIZATION. I V

out any change whatever in the theory o r its results: s8p*

+

s8q*-

s,*(p+q)

cs,*(DW)l

(34)

= K5[SED*I[SE,*l

(p,q=1,2, 3,4,.

. .)

The relation between K,, I(3, and K, is

K, = K,K,

(35)

Though this theory gave an essentially correct picture of the polymerization of sulfur, its basis as given by Eqs. (32) and (33) o r (34) contains the assumption that the number of atoms in all the molecules can only be multiples of eight. It is possible to write alternative equations where this assumption is avoided [51]. Let u s consider the following relations Sn* + Sm*-

s,

Sfn+m)

(n , m = 1, 2, 3, 4 ,

. . .)

-

s,*

Equation (38) accounts for the possibility of other than the 8-membered ring. In order to compare the consequences of Eqs. (36)-(38) with Eqs. (32)-(35), the following definition f o r & is introduced, in analogy with Eq. (35):

Equation (36) still contains a simplifying assumption, namely that

Z, is the same for all values of n and m. While this is true for large values of n and m, it is doubtful for values of n and m = 1, 2, etc. The equations resulting from these equilibria are identical with equations resulting from the theory based upon Eqs. (32) and (33) after substitution: K, = 8E1,

K, = l/SE,,

I(3

=

&

(40)

The equilibrium polymerization of selenium follows a scheme

262

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mathematically identical to that of sulfur. The results for the various thermodynamic quantities for selenium a r e [51] AH" = 25 kcal/mole,

ASo = 23.0 eu

AH,' = 2.27 kcal/mole,

AS," = 5.47 eu

Although AH and AS are positive, the volume change on polymerization (- AV = 6.15 ml/mole) is negative, as in a vinyl polymerization. The floor temperature for the appearance of polymer in the liquid state is calculated as 83°C. C. Polymerization of Aldehydes Anionic polymerization of chloral at low temperature was found to be living, and an equilibrium between polymerization and depolymerization was reached easily provided that the degree of polymerization of polychloral was not too high. True equilibrium monomer concentration and polymer yields were obtained only when an endcapping reaction was carried out at the polymerization temperature because the nonend-capped living polymer was depolymerized immediately at room temperature. When the polymer chain grows longer, polymerization is stopped not by termination but by occlusion of the active end. Consequently a true equilibrium did not hold. From the temperature dependence of the equilibrium monomer concentration, the change in enthalpy and entropy as well as the ceiling temperature for the polymerization of chloral in tetrahydrofuran were determined as -3.5 kcal/mole and -12 eu and 1l0C, respectively [52]. On the other hand, Busfield and Whalley [53] have obtained for monomer-polymer equilibrium of chloral in pyridine -8.0 kcal/mole, -28 eu, and 125°C for the change in enthalpy, entropy, and ceiling temperature, respectively. Chloro- and/or methyl-trisubstituted acetaldehydes and isobutyraldehyde were polymerized anionically in tetrahydrofuran at low temperatures, All the polymers were living and an equilibrium between polymerization and depolymerization was reached easily provided that high catalyst concentrations were used. From the temperature dependence of the equilibrium monomer concentration, the heats and entropies of polymerization and the ceiling temperatures were determined. The ceiling temperatures of the polymerization of trisubstituted acetaldehydes decreased in the order CC1,CHO

> CCl,(CH,)CHO > CCl(CH,),CHO > C(CH,),CHO

TH E R MO DY NAM ICS OF PO LY M E R I ZAT I ON. I V

263

and the heats of polymerization decreased in the reverse order, showing that chlorine substitution for methyl groups in aldehydes increases the stability of a polymer relative to its monomer and that the change in entropy has a stronger influence on the order of the ceiling temperatures of the series than the change in enthalpy [ 541. Formaldehyde is one of the few aldehydes where an equilibrium with the addition polymer can be established fairly readily without the deliberate addition of catalysts. Therefore, equilibrium pressures of gaseous formaldehyde over solid polyoxymethylene, p, have been measured by several workers [55,56]. The heat and entropy of polymerization were calculated from R T l n p = AH

- TAS

These results are compared in Table 7 with some of the more recent published data [57]. Table 7 Thermodynamic Constants for Polymerization of Formaldehyde from Equilibrium Studies

-AH Standard states

(kcaVmole)

15.9 12.2 16.3 14.3 17.0 13.1a .-

16.7 ll.lC 14.gC

40.7 30.6 41.8

-

41Bb 42.6 20.7' 38.7c

46 55 56 d e

57 f

45 45 45

"Data from combustion method. bData from third law. 'Data from various kinds of thermal data from different sources. dJ. F. Walker, Formaldehyde, 3rd ed., Reinhold, New York, 1964. eJ. B. Thompson and W. M. D. Bryant, Polym. Preprints, 11(1), 204 (1970). fF. S. Dainton, D. M. Evans, F. E. Hoare, and T. P. Melia,Polymer, 3,263 (1962).

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111. TRANSITION PHENOMENA IN

EQUILIBRIUM POLYMERIZATION In certain cases there exists a sharply defined “ceiling temperature’’ above which high polymer is thermodynaniically unstable with respect to monomer; in other cases there exists a sharply defined “floor temperature” below which polymer is unstable with respect to monomer. For temperatures above t h e ceiling temperature, since polymer is thermodynamically unstable with respect to the monomer, the equilibrium monomer concentration equals approximately the original monomer concentration, i.e., [&I

[MI,

(41)

and therefore P

1/(1 - KJM,,])

(4 2)

At the ceiling temperature, Tobolsky [58] obtained the approximations P,

(K,[%])-’”

([&I

- [MI,),

(43) (44)

= ([%I/K3)1’2

the subscript t denotes the transition (floor o r ceiling) temperature.

The steepness of the transition can be indicated by the value dP/dT at the ceiling temperature

(3

AH,”

AH,O/RT*

~

, = RT2{2K3[%I - (K3[%13”)I

- KJX,I{2

- (K3[%I)1’2}

(45) We add another equation which defines the sharpness of the transition as a function of [&]in the range of the critical [M,,] concentration at constant temperature (dP/d[M,,]),

[%1/2

If there is a sharp transition, [MI, should be close to [M,,] and P should be large at the transition temperature. F r o m Eq. (42) we see that the transition temperature is defined by K,[&]

=

1

(4 6)

THERMODYNAMICS OF POLYMERIZATION. I V

265

This will be a ceiling temperature if AH,' i s negative, and a floor temperature if AH," is positive. And thus

[M,,] exp(AS,"/R) exp(-AH,'/RT,)

=

1

or

I~[M,,] = -AS,O/R

+

AH,O/RT,

Therefore

Thermodynamically, Dainton and Ivin [38] conceived the ceiling temperature to be that temperature at which the free energy of polymerization (for long-chain polymers) passes from a negative to a positive value a s the temperature i s raised, i.e., T, = AHJAS,

o r , defining AS," as the entropy change for [MI = 1 mole/l of monomer, T, =

A H, AS," + R ln[M],

(49)

which i s equivalent to Eq. (47) of Tobolsky. It i s interesting to compare Dainton's definition of ceiling temperature with that of Tobolsky. Both have the same mathematical form. Dainton's definition refers to the single chemical change free monomer

-

monomer segment of a high polymer

and takes no account of the building process leading to a macromolecule. On the other hand, Tobolsky refers to the whole stepwise building reaction and gives full consideration to the initiation processes. The factors affecting the sharpness of the transition should be examined. The expression giving [dP/dT], shows clearly that the transition is sharper the greater AH,' (the heat of polymerization) and the smaller X,, (the number of initiating species). Increase in AH value narrows the temperature range in which K varies from values much larger than 1/[M,,] to those much smaller. For lower X,,, the amount of polymerized monomer is distributed among fewer

266

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chains, giving a greater increase in P. Similarly, the degree of polymerization, P, as a function of [q] for constant temperature and constant [X,,]may be calculated, The sharpness of the transition is defined by ( d P / d [ q ] ) in the vicinity of the critical concentration [M,,] and is given approximately by equation (dP/d[M,,],) = $[%I. It should be pointed out, however, that if only one equilibrium constant is involved, o r stated alternatively, if the constants for initiation and propagation were identical as Dainton [38] assumed, and as is presented in P a r t I of this review [59], the transition would be s o diffuse that any indication of the sharpness of the transition would not be recognizable [60]. All of the kinetic studies of Dainton's group [38] showed that the rate of polymerization does not drop abruptly to zero at the ceiling temperature, but that in its vicinity the curve, giving rate as a function of temperature, becomes asymptotic to the T axis. This again points to a broadening of the transition arising from a decrease in the molecular weight of the resulting polymer. Equations (42) to (45) were applied numerically to a-methylstyrene. Using the data of Worsfold and Bywater [14] and also of McCormick [13], the following result for I(3 (the concentration units a r e still moles/kg and the standard state is 1 mole/kg): K, = exp(AS,'/R) exp(-AH,"/RT) AH,' = -7.2 kcal/mole

AS," = -27.6 eu For an initiator concentration of 0.001 mole/kg and for two values of [M,,], namely, 1.0 and 2.5 moles/kg, Tobolsky and Eisenberg [58] have computed P vs. T according to Eqs. (42) and (43) as shown in Fig. 8. In these same conditions the total weight concentration of polymer ([M,,] - [MI,) will appear as shown on Fig. 9 as a function of temperature. In Fig. 10 P is plotted vs. [M,,] at two different temperatures, +5.6 and -7.6"C., for the same initiator concentration. The numerical values for the transitions are: For

[M,,] = 1 (Figs. 8 and 9) T, = 5.6"C P, = 31.6 (Point A) ([M,,] - [MI,),= 0.0306 mole/kg (Point B) (dP/dT), = -24.7 units/degree (d([M,,] - [M],)/dT), = -0.0247 mole/kg/degree

THE R MODY NAM I CS OF POLYME R I ZAT I ON. I V

26 7

TEMPERATURE PC) Fig. 8. Dependence of degree of polymerization on temperature for the polymerization of a-methylstyrene for Mo = 1 and 2.5 moles/kg. Xo = 0.001 mole/kg [58].

For

[q]= 2.5 T, P,

(Figs. 8 and 9) = 25.6"C = 50 (Point C)

-

([%I [MI,), = 0.0490 mole/kg (Point D) (dP/dT), = -53.1 units/degree (d([M,,] [M],)/dT), = -0.0531 mole/kg/degree

-

For

K, = 1 (T = 5.6"C, Fig. 10) P, = 31.6 (Point A) ([M,] - [MI,), = 0.0306 mole/kg (dP/d[w]), = +500 units/degree

For

K, = 2 (T = -'7.6"C, Fig. 10) P, = 22.4 (Point E) ([q] - [MI,), = 0.0214 mole/kg (dP/d[w]), = 500 units/mole/kg

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TEMPERATURE ('C)

-

Flg. 0, ( Mo Me) VB. temperature for the polymerlzatlon of cu-methyletyrene for Mo = 1 and 2.6 molee/kg. Xo = 0.001 mole/kg [68].

For bulk sulfur [w]= 3.90, and this transition occurs at 159"C, a temperature at which the liquid suddenly seems to acquire a very high viscosity. The AH,' is positive in this case, so we have a "floor temperature." At this transition point, the following relationships hold: T, = 432'K = 159°C P, = 1 / ~ 1 / 3= 1.2 x 10' units

([%I - [MI,),= K1/,/KS = 3.24

3RTa

(50)

lo4 mole/kg

(51)

= 1.35 x loBunits/degree

(52)

X

From Eqs. (50) and (61) it is clear that at the transition temperature the relative amount of polymer is very small, but the degree of

THE R MODY NAM ICS OF PO LYME R I ZAT ION I V I

-

209

Flg, 10. Dependence of degree of polymerleatlon on the lnltlal monomer concentratlon for the polymerlaatlon of rY-methylatyrene for T = 8.6 and -7.B"C. Xo = 0.001 mole/kg [68].

polymerization is very high. Furthermore, at the transition temperature, the rate of increase of degree of polymerization with temperature is very great, as shown by Eq. (52). In summary, for temperatures below the floor temperature, the following approximations can be used:

For the temperature region in which high polymer it3 stable, Le., above the floor temperature, P is >> 1, and the following approximations can be used:

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IV. MOLECULAR WEIGHT DISTRIBUTION A. Equilibrium Polymerization Polymeric materials display a distribution in molecular sizes. For example, Flory [61] developed the theory for size distribution in condensation polymers by a statistical treatment. From a consideration of the kinetics of condensation polymerization, it has been shown that the molecular size distribution in linear condensation polymers containing equal numbers of the two cooperating functional groups (e.g., OH and COOH) is given by N, = NpX-'(l - p)

(54)

where N, is the number of molecules composed of x monomer units, N is the total number of molecules, and p is the extent of reaction, i.e., the fraction of the functional groups which has condensed. The distribution of Eq. (54) for the equilibrated linear polymer can be derived [62] in a manner analogous to the derivation of the Maxwell-Boltzmann energy distribution law:

2~~= N

(55)

x =1

2 x N x = n,,

x =1

where n,, is the total number of units and N=

- P)

(57)

Both n,, and N are constant under the conditions, and Eqs. (55) and (56) are analogous, respectively, to the conditions of conservation of matter and energy in the Maxwell-Boltzmann derivation. A macrostate is defined merely by the numbers of molecules of the various size, i.e., by N,, N,, N,, etc. For a given macrostate there are

w = N!/TN~! microstates. Solution of Eq. (58) for the maximum value of W consistent with Eqs. (55) and (56) by the usual variation method yields Eq. (54) for the most probable macrostate. Thus, under the assumption that the thermodynamic stability of a given interunit bond is independent of the size of the molecule and its position along the chain, the equilibrium distribution is ultimately attained by random synthesis. The difference in entropy between a mole of a hetero-

THERMODYNAMICS OF POLYMERIZATION. I V

27 1

geneous polymer and a mole of the single species of molecular weight equal to the number-average molecular weight for the heterogeneous polymer is given by the entropy of mixing expression: = -RC(N,/N)

ln(N,/N)

(59)

which may be called the molar entropy of heterogeneity. For the most probable distribution, substitution of Eq. (54) in Eq. (59) gives =

R[ln(p/l

- p) - (In p)/(l - p)1

(60)

which also can be obtained directly from the Boltzmann relation S = k In W, where W is taken to be W, in Eq. (58). For a highmolecular-weight polymer, p is near unity and AS,

= ~ [-iIn(1 - p)]

AS,

= R[1 + ln(P,)]

or

since the number-average degree of polymerization is given by P, = 1/(1 - P)

(63)

For the entropy of heterogeneity per mole of structural unit, we have ASh’ = AS,/P,

(64)

These equations express the maximum entropy of heterogeneity, o r entropy of mixing, for a given degree of polymerization. Any distribution other than Eq. (54) will yield a lower entropy of heterogeneity. The entropy per mole of polymer molecules AS,, increases without limit as p approaches unity and P, approaches infinity; the entropy per mole of structural units mh’, after reaching a maximum at a very low degree of polymerization, decreases asymptotically toward zero as P, increases. At any given temperature there should be an equilibrium distribution i n sizes, corresponding to a condition where entropy and heat are balanced so as to give a minimum free energy, The logarithm of the number of configurations SZ available to a mixture of n, solvent molecules and nI molecules of monomer, n, molecules of dimer, n, molecules of x-mer, etc., is [63]

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where v is the coordination number of lattice and u is the symmetry number of the long-chain molecule (usually 2). If equilibrium properties of polymer sizes is known, Eq. (65) gives the number of available configurations. The energy of each distribution with specified values of the n,'s can easily be written in terms of the free energy change of polymerization per molecule, f: F = -xn,(x 1)f (66)

-

The equilibrium properties of a system containing n, solvent molecules and N monomeric units (which may be linked together in any fashion into polymeric units) can be obtained in theory by evaluation of the partition functions:

where the subscript i refers to a particular set of the n,'s corresponding to a definite distribution in sizes among the polymer molecules. The macrostate corresponding to 'equilibrium" is the state for which the free energy is a minimum and can be found by maximizing the expression In Sat

- FI/kT

with respect to the variables n,, n,, tions that xxn, = N

(68)

.., , n,, . ,, subject to the condi(69)

Xm1

The problem of maximizing Eq. (68) subject to Eq. (69)can be best solved by the method of Lagrange multipliers. In other words, multiply Eq. (69) by -a/kT and add to Eq. (68) to give, after substitution, Eq. (65). Hence, it is easy to show that the resulting distribution [64] will be identical with the one described by Eq. (54). The same distribution results when random scission occurs to infinitely long chains [65]. Flory [66] has derived the above molecular size distribution equation by a similar procedure based on the same equations, To describe the properties of a polymerizing system at equilibrium, the change in free energy of polymerization must be estimated, Polymerization from monomer to the equilibrium polymer consisting of a mixture (solution) of monomer, dimer, trimer, n-mer, etc., may be considered to be the sum of two processes, the polymerization of

TH ER MODY NAM ICS OF PO LY ME R I ZAT ION. IV

273

monomer to pure homopolymers and the mixing of the pure homopolymers in their polymerization system, The free energies of formation, and hence, the free energies of polymerization, of pure gaseous monomer to form pure gaseous n-mer are expected to be proportional to molecular size, i.e., chain length, for an homologous series such as the polymerization of ethylene gas to the gaseous 1-alkanes [67]. This is a consequence of assuming that vibrational, rotational, and translational motions of a molecule are independent of one another and that partition functions describing these motions may be written as the product of partition functions, each describing one type of motion [68]. The free energies of formation, and polymerization of gaseous o r liquid monomer to form pure liquid n-mer homopolymer, may o r may not be linear functions of molecular size depending upon the degree of order present in pure liquids, The difference in entropy between a rigid (crystalline) n-mer molecule and a flexible n-mer molecule is approximately [69]

where k is Boltzmann’s constant, n is the degree of polymerization, and z is the coordination number of a lattice approximating the structure of the liquid polymer. Equation (70) is derived by counting the number of ways that n links of a flexible n-mer molecule can be arranged in n subvolumes. A consequence of Eq. (70) is the expectation that free energies of formation and polymerization of monomer to pure liquid n-mer homopolymer may be expected to be linear functions of chain length n if the liquid is highly random, approximating the disorder of pure, dense gaseous n-mer, and to depart from linearity by approximately kT ln(n) per molecule if the liquid is highly ordered, approaching the regularity of a crystal. That real liquid homopolymers a r e somewhere between these two extremes seems likely. The two limiting cases for polymerization equilibri.um arise as follows: (1) Assume the free energies of formation and polymerization of monomer to pure, crystalline (perfectly oriented) n-mer homopolymers are linear functions of molecular size. Mix the monomer, dimer, trimer, n-mer, etc., to form the equilibrium polymer. This gives a “most probable” distribution at polymerization equilibrium [661.

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(2) Assume the free energies of formation and polymerization of monomer to pure, unoriented (random oriented) n-mer homopolymers are linear functions of molecular size. Mix the monomer, dimer, trimer, n-mer, etc., to form the equilibrium polymer. This gives a very broad molecular weight distribution at polymerization equilibrium [TO].

The free energies of formation and polymerization to form liquid alkanes (or 1-alkanes) may be expected to be linear functions of chain length. If normal alkanes are disoriented liquids, Case 2 leading to a very broad molecular weight distribution at equilibrium is the more likely. If normal alkanes are highly ordered liquids, Case 1 leading to a “most probable” molecular weight distribution at equilibrium may be correct in spite of the expectation that entropy of polymerization of monomer to pure, oriented polymer might be nonlinear with chain length by the logarithm of chain length according to the estimate of disorder of a random polymer chain (Eq. 70). Consider the polymerization of n-gaseous monomer units at unit fugacity to form liquid n-mer in the equilibrium polymerization mixture under a total pressure P: n-monomer (g,f = 1)

n-mer (L,P)

The activity product for this reaction may be written as

where f , = fugacity of monomer, a, = activity of n-mer, Nu = mole fraction of n-mer, y(n) = activity coefficient of n-mer, and AF,O(n) = standard change in free energy of polymerization of gaseous monomer at unit fugacity to pure, unoriented, liquid, n-mer homopolymer at pressure P. A bulk polymerization is a solution (mixture) of monomer, dimer, trimer, n-mer, etc. In such a system in the liquid phase, the activity coefficient y(n) is given by In y(n) = In(n/P,) + 1 - (n/p,) where P, is the number-average degree of polymerization. The entropies of homologous series of liquid hydrocarbons are proportional to molecular size [71]. The heats of combustion (and formation) of real polymers are proportional to molecular size [72]. Thus the standard changes in free energies of polymerization to pure liquid homopolymers may be expected to be given by

THERMODYNAMICS OF POLYMERIZATION. IV

275

+ pn

(73)

AF,”(n) =

(Y

where cy and /3 a r e constants (dependent upon pressure and temperature) and n is the number of monomer units. For the mole fraction of n-mer N, = exp {n In f,

- ___ c y + m- l n -n- l + RT

P,

(74)

For the number-average degree of polymerization P, - 1 P, = In P,

(75)

For the weight-average degree of polymerization P, = 1 + Pi In P,

The molecular weight distribution of an equilibrium polymer as given by Eq. (74) is very broad. The difference between the treatments that lead to the “most probable” and the very broad molecular weight distributions at polymerization equilibrium arises in whether the entropies (and, hence, the free energies of formation and polymerization) of an homologous series of molecules are proportional to molecular size in the liquid state o r in the solid state. For relatively disordered liquids, such as polymerizing, never-frozen monoolefins, the liquid is perhaps best described as a dense gas a s far a s configurational entropy is concerned. In this case a broad molecular weight distribution at equilibrium might be expected. On the other hand, in a polymerizing system involving ionic species in a melt, a high degree of order is present and the system is probably best described a s a crystalline system. In this case, a narrow molecular weight distribution, perhaps approaching a “most probable” distribution, might be expected. For Case 2, assuming the entropies of pure, unoriented homopolymers a r e linear functions of degree of polymerization, the free energies of polymerization may be estimated empirically using ideal gas state free energy data and measured vapor pressures and molar volumes for the 1-alkenes. If observed molecular weight, weightaverage/number-average ratios, and molecular weight distributions

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for ordinary polyethylenes a r e compared with these properties calculated for equilibrium polyethylene using published thermodynamic data for the 1-alkenes, reasonable agreement is found. This indicates that polymerizations of ethylene, particularly by the “high pressure” process, may be controlled by energetics, at least in part, rather than by kinetics alone, A broad molecular weight distribution is predicted for an equilibrium polymer. Thus, if ethylene polymerizations are energetics-controlled, the observed broad molecular weight distributions of polyethylenes are not anomalous but are to be expected for vinyl polymerizations in bulk if the reactions are carried out in such a way as to approach equilibrium. In conclusion, the Lundberg [TO] distribution mentioned above is a result of extreme deviations from ideality, Thus the Schulz-Flory distribution can be derived not only from the ideal solution theory at infinite dilution, but also from the Flory-Huggins theory at finite concentrations. In fact Harris [73] points out that the Schulz-Flory distribution results when deviations from ideality are expressed in terms of the second virial coefficient, provided that the latter is independent of molecular weight, However, the distribution law is not independent of the thermodynamics of the system and, indeed, may vary significantly at higher concentrations from the idealized expression. Eisenberg and Tobolsky [58] found that the distribution in equilibrium polymerization, even in the presence of an initiator, was the random distribution. For example, the expression for the mole fraction of x-mers obtained by dividing the expression for the absolute concentration of x-mers by the total concentration of polymer is N,/N = n, = (&[M],)a-l(l

- K,[Ml,)

(77)

which is the Schulz-Flory equation with p = &[MI,. Furthermore,

-

Pa = 1/(1 &[MI,) P, = (1 + &[MI,)/(1 - KJMIJ and the heterogeneity index is P,/P, = 1 + %[MI, = 2 - 1/P,

(10) (78) 2

(79)

B. Living Polymerization It was shown [74] that the polymerization of vinyl monomers may be carried out under conditions which exclude the termination step. The resulting polymers a r e referred to as “living” polymers, since

277

THERMODYNAMICS OF POLYMERIZATION. IV

their ends retain the activity required for further propagation of the polymerization process. Since the molecular weight distribution of the product yielded by such a polymerization is unusual, this distribution will be examined in some detail. Let us consider now the molecular weight distribution of a living polymer produced in a system in which all termination processes are excluded. The polymerization which interests us is initiated by some anionic species denoted by MI*. The propagation of the polymerization involves the reactions MI* + M M,*+M

... MI* + M

-

Ma* M,*

-

___c

M:+1

Let us assume the rate constants of these reactions to be independent of j and equal to k,*. The principle of microscopic reversibility demands the occurrence of the reserve reactions, namely, decomposition of an active j-mer into a monomer and an active (j - 1)-mer, and the unimolecular rate constant of this depropagation, assumed again to be independent of j, is denoted by k,. Since there is no termination, the system described above must eventually reach an equilibrium state, the equilibrium concentration [MI, of the monomer being determined by the equation

where [MI,, etc, denote concentrations. If the initial concentration of all the active species is denoted by C, then g[M,*], = C I

and

g[M,*], = C 2

- [MI*],

(81)

where [MI*Ie denotes the equilibrium concentration of the active species [MI*]. For a polymerization proceeding to a high molecular weight product, [MI*], is negligible compared with C, and we obtain

This result is obtained from Eq. (80) by putting d[M]/dt = 0. The equilibrium concentration of the monomer is independent of the ini-

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tial concentrations of the monomer and of the catalyst (i.e., of the active species), and is determined uniquely by the temperature and by the nature of the monomer, of the polymer, and of the solvent. The equilibrium molecular weight distribution of ‘living” polymers may be calculated from the following set of equations:

The subscript e denotes that all the relevant concentrations have the equilibrium values. It is essential to realize that in solving the above set of equations, one must not apply the approximate Eq. (82), but must use the exact expression C-’d[M]/dt = -k,[M],

+ kd(1

- [M,*],/C)

=

0

(84)

which leads to [MJ*Ie = [MI *],(I

-

[Mi *Ie/C)’-’

(85)

Thus the equilibrium molecular weight distribution of “living” polymers is given by the usual function named by Flory “the most probable distribution” “751, and leads to P, = M, /C and P, /P, = 2. (M,-the total amount of polymerized monomer-is given by the equation M, = [&] - [MI,.) Although the equilibrium molecular weight distribution of “living” polymers is the same as that characterizing polymers formed by many conventional processes, an entirely different molecular weight distribution may be formed immediately after completion of the polymerization process. Whenever a high-molecular-weight polymer is produced, the propagation rate constant, k,, must be much greater than kd, the depropagation constant, and the half-life time of the depropagation is long compared with the time of the experiment. Furthermore, the initial concentration of monomer, [&I, must be substantially larger than its equilibrium concentration [MI, and [&]/C >> 1. Polymerization in such a system usually leads to a Poisson molecular weight distribution when nearly all the polymeric molecules have a degree of polymerization close to [&]/C. Although such a system is not yet in its true equilibrium state, the polymeriza-

THERMODYNAMICS OF POLYMERIZATION. IV

279

tion is essentially completed, and the concentration of the monomer differs only insignificantly from that attained at the ultimate equilibrium. On the other hand, the higher averages, e.g., P,-the weight-average degree of polymerization-vary in time. A s was pointed out by Flory [76], when initiation is a t least as fast as propagation, a Poisson distribution of polymer chain lengths is obtained

N, = N exp(- v)F1/(x

- l)!

where v = P, - 1. The entropy of heterogeneity for this distribution, obtained by substituting Eq. (86) in Eq. (59), is given to a close approximation when the degree of polymerization is large by

R

Ash = y{1 + ln(2nPJ)

The distribution of Eq. (86) covers a much narrower range than the equilibrium distribution. Correspondingly, its entropy of heterogeneity is less than for the equilibrium distribution a s given by Eq. (62). Therefore, if interchange between polymer molecules occurs, the distribution of Eq. (86) will be broadened toward the equilibrium distribution of Eq. (54). Miyake and Stockmayer [77] have investigated the mathematical problem of the reversible living polymer system without transfer and termination reactions, concentrating particularly on the molecular weight distribution. This problem is identical to that for the kinetics of BET adsorption, and a complete analytical solution is obtained when the monomer concentration is kept constant. The time required for the initial narrow Poisson distribution to go over into the final “most probable” equilibrium o r Schulz distribution is proportional to the square of the average chain length. For polystyrene at room temperature, the Poisson polymer is produced in a few seconds, but the final equilibration requires of the order of 100 years, although the unreacted monomer concentration reaches its equilibrium value very early. Thus the equilibrium constant for polymerization can be accurately measured long before complete equilibrium is attained in the system. For purposes of illustration, let us discuss the polymerization in essentially three stages as has been indicated by Miyake and Stockmayer “771. The material formed in the first stage has a Poissonian character and for all practical purposes it may be taken as monodisperse if the average chain length is large. In the

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second stage the statistical character does not alter appreciably while the free monomer concentration acquires its equilibrium value, In the third stage there is redistribution of monomers over the chains. The number-average chain length stays constant while the weight-average chain length changes to the most probable value. Under the assumption k,[&] >> kd, the first two stages are completed in a time which is negligible compared with the time involved in the third stage. Thus, to investigate changes in the statistical character of the polymer due to depropagation, we may effectively assume that at t = 0 the polymer sample has a Poisson distribution of chain lengths with a number-average chain length equal to [w]/C, Nanda and Jain [?a] have obtained an expression for size distribution which is applicable to the earlier stages of widening of the distribution, V, THERMODYNAMICS OF EQUILIBRIUM POLYMERIZATION

The effects of solvents on the equilibrium constants were interpreted on the basis of the thermodynamic theory of solutions in Section I of Ref. 60. Let us now examine the effect of polymer concentration on the thermodynamic activity of the monomer from a thermodynamic point of view, The yield and the degree of polymerization of equilibrium polymerization are independent of initiator concentration, the polymer content of the system o r the solvent, and depend only on the initial monomer concentration and the temperature. This ie strictly true only if the polymer is insoluble in the reaction medium and the activity of the monomer is equal to its concentration, Ivin and Leonard "791 have considered this problem in detail, and have applied it to the case of the equilibrium anionic polymerization of a-methylstyrene in tetrahydrofuran where, at certain temperatures, the equilibrium monomer concentration decreased linearly with increasing concentration of polymer, The polymer is soluble in tetrahydrofuran and its concentration therefore influences the thermodynamic activity of the monomer. This effect waa first noted experimentally by Vrancken, Smid, and Szwarc [80] who found a fall in the equilibrium concentration from O,? M to about 0.46 M at 0" as the polymer concentration increased from 0 to 2.8 base-mole/l. The free energy change for the conversion of 1 mole of liquid monomer to 1 base-mole of amorphous polymer, AQlo, will be given by AQ,,

9

AU,,,

- AUn

(88)

28 1

TH E R MODY NAM ICS OF PO LY ME R I ZAT I ON. I V

where A& is the partial molar free energy of the monomer (per mole) in the equilibrium mixture relative to that of the pure liquid monomer, and ha, is the partial molar free energy of the polymer (per base-mole) in the equilibrium mixture relative to that of the amorphous polymer. Expressions for A&, and AGD in a three-component system (polymer-monomer-solvent)are quoted by Flory [el] as

1 AGJRT = ;{ln +

+p

+ (1

(xDm$m

+

- + J - + m ( X J J x J - +,(XJX,)

xpn$a)($m

+

$8)

- xmn(XJXrn)+m~J

(go)

where CpI is the volume fraction of component i, XIis the number of segments per molecule for component i, n is the degree of polymerization, R is the gas constant, T is the temperature, and x is the free energy parameter between any two components, the subscripts m, 8 , and p referring to monomer, solvent, and polymer, respectively. Putting X,/X, = vI/v,, the ratio of the molar volumes, assuming vJv, = n, and expressing xpmand xpl in terms of the molecular weight- independent quantities xmDand xnpthrough the relationships , find [81I Xpm = Xmg(VJVm) = Ymfl and pa = X ~ ~ ( V m / V a ) we AG,,*/RT = In

CPm

+1 + (xmm$m

+ xmD@D)(+#

+

6,)

-

x ~ D ( v ~ ~ v 8 ) ~ E ~ D

+ Xrn,9m@m

(91)

where +, now represents the equilibrium volume fraction of monomer in the presence of a volume fraction of polymer equal to $D, Neglecting terms in l / n when n is large, and replacing ($a + $,) by (1 $m) and ($m + $,I by (1

-

-

AG~RT = In + m + 1 +

+a(Xmm

-

XapVm/Va) + X m s ( + p

-

+m)

(g2)

Equation (B2) is more general than that deduced by Bywater [62], who made the approximation v, = v,. AG,, represents the freeenergy upon the polymerization of 1 mole of liquid monomer to 1 base-mole of liquid amorphous polymer of infinite chain length,

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In their work on the polymerization of a-methylstyrene, Ivin and Leonard [79] found that the monomer concentration varies linearly with the polymer concentration. Experimentally, it is found that the variation of 9, can be expressed by

where (I, is the value of +,,,when 9, approaches zero and B is the slope obtained from a plot of 9, against Qq. In all cases n is quite large. A linear dependence of (I,,, on 9, may be obtained from Eq. (92) as follows [83]. First, In (Im is expanded in a series; that is,

-

In Q,,,= III a + ( ( I ~ a)/a

- ((I,, - aY/2a2 + (@,,, + aI3/3a3 - -.(94)

where 0 < @,, < 2a. If the value of constant is such that ( Q m - a)'/2a2 is negligible with respect to the first two t e r m s of Eq. (941, one can then write

-

-

(I, and substituting Eq. (95) for (Im, after Then, with ( I 8 = 1 (I, rearranging Eq. (92), the following expression for the variation of @,-with @, is obtained:

-

where p = xms xsp(v,/vs). With the exception of (I, and $I,, all the terms appearing in Eq. (93) a r e assumed to be constant at a given temperature. Comparing Eqs. (93) and (96), it then follows that

B = (Xmp

-

-

+

-

Xmp

- l/a)

(97)

Since (I,,, (Imas 9, 0, the first t e r m on the right-hand side of Eq. (96) is equal to (I,,. From this relation the variation of the equilibrium monomer concentration with the polymer concentration may be explained in terms of the constant a and the interactions between the components of the system measured by the thermodynamic parameter p and xmp. Since 9 , is a constant for a given temperature, a can therefore be replaced by 9 , in Eq. (95).

TH E R MOD Y NAM ICS 0 F PO LY ME R I ZAT ION. I V

283

From the above considerations and comparing Eqs. (93) and (96), one finally obtains @ml

=

-(AG,JRT)

P

+

Xmp

+ In a + p

- l/a

=a

(98)

Using this definition of a, Eq. (96) may be rewritten to give

9,

=

a

+

(x,,,~- P ) / @

+ Xmp

- l/a)QP

(99)

If (AG,JRT), 0 and xmPare known, and Eq. (98) can be solved for a. Once a is known, it is then possible to calculate the variation of @, with 9, using Eq. (99) and to compare it with experimental results. Table 8 gives experimental values of Q m and B obtained for the anionic polymerization of a-methylstyrene in tetrahydrofuran a t various temperatures and the corresponding values calculated by means of Eqs. (98) and (99). The variation of $,,,with QP is shown in Fig. 11 where a good agreement is obtained between the experimental curve and curves computed using Eqs. (98) and (99).

,

0.0 0.0

0.1

aP

0.2

0.3

Fig. 11. Variation of bm with for the equilibrium polymerization of amethylstyrene in THF at 20 and -20°C 183). (Q): Experimental points. (-) : Values calculated using xmp = 0.3

0.100

0.279

0

+ 20

-0.299

-0.133

-0.044

aExperimental data from Ref. 79. bWith xmp= 0.3. 'With xmp= 0.4.

0.030

-20 1.34

2.68

4.08 -1.38

-1.52

-1.65 0.282

0.102

0.031 -0.363

-0.165

-0.058

Thermodynamic Parameters Used for the Computation of a,,, [83]

Table 8

-+.I77

-0.400 0.288

-0.061 0.103

0.031

P

m

N

THERMOClYNAMlCS OF POLYMERIZATION. IV

285

As can be seen, the effect of monomer-polymer interactions is not predominant, and the same assertion can be made on the solventpolymer interactions. On the other hand, a change in monomersolvent interactions can bring about a pronounced change in the position of equilibrium for the polymerization of a given monomer. This can be shown by comparing the polymerization of cy-methylstyrene in tetrahydrofuran and cyclohexane at a given temperature. We find @,

= 0.60

- 0.61@,

(in tetrahydrofuran)

and similarly @,,

= 0.20

+ 0.051@,

(in cyclohexane)

Here, the solvent effect on the equilibrium position is obvious. Changing the solvent from cyclohexane to tetrahydrofuran should increase @a threefold. From the above results it can be seen that the monomer concentration will decrease rapidly upon the addition of polymer in tetrahydrofuran, whereas it will have little o r no effect in cyclohexane, except for a slight depolymerization which will occur upon the addition of polymer. Recently it has been shown [84] that for a given temperature, the equilibrium monomer concentration in p-dioxane is approximately 20% lower than the corresponding monomer concentration in tetrahydrofuran, together with the variation of @,, with @., The effect is explained in terms of a solvent-monomer and solvent-polymer interaction parameter. The equilibrium polymerization of dioxolane has been studied over a wide range of initial monomer concentrations at temperatures from 20 to 60°C in methylene chloride, benzene, and 1,4-dioxane. In all the solvents, over a wide range of initial concentrations, the equilibrium monomer concentration falls with an increase in the polymer content of the system [85]. A measure of the effect of the solvent on the equilibrium concentration of dioxolane is given by the value of x (the apparent Flory-Huggins parameter taking account of polymer-solvent interaction). Thus the dependence of the equilibrium monomer concentration on the quantity of polymer in the system (or the initial monomer concentration), and on the solvent, can also be explained on the basis of the thermodynamic theory of polymer solutions [85].

Acknowledgment The author wishes to thank Prof. K. F. O’Driscoll of University of Waterloo for h i s continuing interest and helpful advice.

286_ _

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References A. V. Tobolsky, J. Polym. Sci., 25,220 (1957). A. V. Tobolsky, J. Polym. Sci., 31, 126 (1958). A. V. Tobolsky and A. Eisenberg,J. Amer. Chem. Soc., 81,780 (1959). A. V. Tobolsky and A. Eisenberg,J. Amer. Chem. Soc., 81,2302 (1959). A. V. Tobolsky and A. Eisenberg,J. Amer. Chem. SOC., 82,289 (1960). A. V. Tobolsky, Properties and Structures of Polymers, Wiley (Interscience), New York, 1960, p. 266. [ 71 M. Szwarc, Proc. Roy. Soc., Ser. A, 279,260 (1964). [ 8 ] A. Eisenberg, Makromol. Chem., 65,122 (1963). [ 9 ] F. S. Dainton and K. J. Ivin, Nature, 162, 705 (1948). [ l o ] S. Bywater, Trans. Faraday Soc., 51,1267 (1955). [11] S. Bywater, Can. J. Chem., 35,552 (1957). [12] C. G. Overberger and A.M. Schi1ler.J. Polym. Sci., Part C, 1, 325 (1963). [13] H. W. McCormick,J. Polym. Sci.,25,488 (1957). [14] D. J. Worsfold and S. Bywater, J. Polyrn. Sci., 26,299 (1957). [15] S. Bywater and D. J. Worsfold,J. Polym. Sci., 58,571 (1962). [ 161 A. V. Tobolsky, A. Rembaum, and A. Eisenberg,J. Polyrn. Sci., 45,347 (1960). [ 171 R. Asami, in Kobunshi No Tenbo 1970 ( Y .Iwakura, ed.), Maruzen, Tokyo, 1970, p. 53. 1181 P. A. Small, Trans. Faraday Soc., 49,441 (1953). [19] K. J. Ivin, Trans. Faraday Soc., 51,1273 (1955). [20] R. E. Cook and K. J. Ivin, Trans. Faraday Soc., 53,1132 (1957). [21] H. Meerwein, D. Delfs, and H. Morschel, Angew. Chem., 72,927 (1960). [22] D. Sims,J. Chem. Soc., 1964,864. [ 2 3 ] C. E. H. Bawn, R. M. Bell, and A. Ledwith,Polymer, 6,95 (1965). [24] B. A. Rosenberg, 0. M. Chekhuta, E. B. Ludving, A. R. Gantmakher, and S. S. Medvedev, Vysokomol. Soedin., 6,2030 (1964). [25] D. Vofsi and A. V. Tobolsky,J. Polym. Sci.,Part A, 3, 3261 (1965). [26] M. P. Dreyfuss and P. Dreyfuss,J. Polym. Sci., Part A-Z,4,2179 (1966). [27] K. J. Ivin and J. Leonard,Polymer, 6,621 (1965). [28] D. Sims, Makromol. Chem., 98,235 (1966). [29] F. Wiloth.2. Phys. Chem., 4,66 (1955). [ 301 P. F. Van Velden, G. M. Van Der Want, D. Heikens, Ch. A. Kruissink, P. H. Hermans. and A. J. Staveman, Rec. Trau. Chim. Pays-Bas, 74,1376 (1956). [31] A. B. Meggy,J. Chem. Soc., 1953,796. (321 0. Fukumoto,J. Polym. Sci., 22,263 (1956). [ 3.71 H. K. Reimschuessel, in Ringopening Polymerization (K. C. Frisch and S. L. Reegen, eds.), Dekker, New York, 1969, Chap. 7. [ 3 4 ] H. Yumoto, J. Chem. Phys., 29,1234 (1958). 1351 H A . Elias and A. Fritz, Makromol. Chem., 114,31(1968). [ 361 P. H. Plesch and P. H. Westermann, J. Polym. Sci., Part C, 16,3837 (1968). [37] Y. Yamashita, M. Okada, K. Suyama, and H. Kasahara, Makrornol. Chem., 114, 146 (1968). [38] F. S. Dainton and K. J. Ivin, Quart. Reu., 12,61(1958). [ 391 S. M. Skuratov, A. A. Strepikheev, S. M. Shtekher, and S. V. Volokhina, Dokl. Akad. Nauk SSSR, 117,263 (1957). [ 11 [ 21 [ 31 [4] [5] [ 61

THERMODYNAMICS OF POLYMERIZATION. I V

28 7

A. A. Strepikheev and A. V. Volokhina, Dokl. Akud. Nuuk SSSR, 99,407 (1954). P. H. Plesch and P. H. Westermann, Polymer, 10,105 (1969). T. Miki, T. Higashimura, and S. Okamura, J. Polym. Sci., Purt A - I , 5,95 (1967). T. Miki, T. Higashimura, and S. Okamura, J. Polym. Sci., Purt A - I , 5,2997 (1967). T. Miki, T. Higashimura, and S. Okamura, J. Polym. Sci., Purt A - I , 8,157 (1970). A. A. Berlin, S. A. Vol’fson, E. F. Oleinik, and N. S. Yenikolopyan, Vysokomol. Soedin., A12,443 (1970). [46] W. K. Busfield and D. Merigold, Mukromol. Chem., 138,65 (1970). [47] W. K. Busfield and D. Merigold, J. Chem. Soc., 1969, A 2975. [48] K. Nakatsuka, H. Suga, and S. Seki, J. Polym. Sci., Purt B, 7,361 (1969). [49] G. Gee, Trans. Furuduy Soc., 48,515 (1952). [50] F. Fairbrother, G. Gee, and G. T. Merrall, J. Polym. Sci., 16,459 (1955). [51] A. Eisenberg and A. V.Tobolsky,J. Polym. Sci., 46,19 (1960). [52] I. Mita, I. Imai, and H. Kambe, Mukromol. Chem., 137,143 (1970). [53] W. K. Busfield and E. Whalley, Trans. Furuduy Soc., 59,679 (1963). [54] I. Mita, I. Imai, and H. Kambe, Mukromol. Chem., 137,155 (1970). [55] F. S. Dainton, K. J. Ivin, and D. A. G. Walmsley, Trans. Furuduy SOC., 55,61 (1959). [ 561 Y. Iwasa and T. Imoto, J. Chem. SOC. Japan, Pure Chem. Sect., 84,29 (1963). [ 571 G. S. Parks and H. P. Mosher, J. Polym. Sci., Purt A , 1, 1979 (1963). [ 581 A. V. Tobolsky and A. Eisenberg, J. Colloid Sci., 17,49 (1962). [ 5 9 ] H. Sawada, J. Mucromol. Sci., C3(2), 313 (1969). [60] A. V. Tobolsky, Polym. Preprints, 11,1,165, February 1969. [61] P. J. Flory, J. Amer. Chem. SOC.,58, 1877 (1936). [62] P. J. Flory, J. Amer. Chem. Soc., 64,2205 (1942). [ 6 3 ] P. J. Flory, J. Chem. Phys., 12,114 (1944). [ 6 4 ] A. V. Tobolsky, J. Chem. Phys., 12,402 (1944). [65] W. Kuhn, Chem. Ber., 63,1503 (1930). [ 6 6 ] P. J. Flory, J. Chem. Phys., 12,425 (1944). [67] R. S. Jessup,J. Chem. Phys., 16,661 (1948). [68] M. Born and R. Oppenheimer,Ann. Phys. (Leipzig), 84,457 (1927). [ 691 P. J. Flory, Principles of Polymer Chemistry, Cornell Univ. Press, Ithaca, New York, 1953, p. 502. [70] J. L. Lundberg,J. Polym. Sci., Purt A , 2,1121 (1964). [ 711 H. L. Finke, M. E. Gross, G. Waddington, and H. M. Huffman,J. Amer. Chem. Soc., 76,333 (1954). [ 721 F. S. Dainton and K. J. Ivin, in Experimental Thermochemistry, Vol. 2 (H. A. Skinner, ed.), Wiley (Interscience), New York, 1962, p. 253. [ 731 F. E. Harris, J. Polym. Sci., 18,351 (1955). [ 741 M. Szwarc, Nature, 178,1168 (1956); M. Szwarc, M. Levy, and R. Milkovich, J. Amer. Chem. Soc.. 78,3590 (1956). 54,416 (1958). [75] W. B. Brown and M. Szwarc, Trans. Furuduy SOC., [ 7 6 ] P. J. Flory,J. Amer. Chem. SOC., 62,1561 (1940). [ 771 A. Miyake and W. H. Stockmayer, Mukromol. Chem., 88,90 (1965). [ 78 1 V. S. Nanda and S. C. Jain, Eur. Polym. J., 6,151 7 (1970). [79] K. J. Ivin and J. Leonard,Eur. Polym. J., 6,331 (1970). [80] A. Vrancken, J. Smid, and M. Szwarc, Trans. Furuduy Soc., 58,2036 (1962). [81] Ref. 69, p. 549.

[40] [41] [42] [43] [44] [45]

288

SAWADA

[82] S.Bywater, Makromol. Chem., 62,120(1962). [83] J. Leonard, Macromolecules, 6,661 (1969). (841 J. Leonard and S. L. Malhotra,J. Polym. Sci., Part A-1, 9,1983(1971). [85) L. I. Kuzub,M. A. Markevich, A. A. Berlin, and N. S. Enikolopyan, Vysokornol. Soedin., 10,2007(1968).

J. MACROMOL. X I . - R E V S . MACROMOL. CHEM.,

C10(2),293-353(1974)

Thermodynamics of Polymerization. V. Thermodynamics of Copolymerization. Part I HIDE0 SAWADA Filter Laboratory Daicel Ltd. Teppo-cho, Sakai, Osaka, Japan

I. THE GENERAL THEORY OF BINARY COPOLYMERIZATION . . .294 A. Heat of Copolymerization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294 B. Entropy of Copolymerization . . . . . . . . . . . . . , . . . . . . . . . . . . . . . 301 C. Equilibrium Sequence Distribution . . . . . . . . . . . . . . . . . . . . . . . . . 304 D. Free Energy Change in Binary Copolymerization System. . . . . . . . . 310 E. Equilibrium Monomer Concentration . . . . . . . . . . . . . . . . . . . . . . . 318 F. Penultimate Unit Effect. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323 11. DEGREE OF POLYMERIZATION AND COPOLYMER COMPOSITION OF BINARY COPOLYMERIZATION SYSTEM . . . . . 325 A. Degree of Polymerization.. . . . . . . . . . . . . . . . . . . . . . . . . . . , . . . . 325 B. Copolymer Composition Equation. . . . . . . . . . . . . . . . . . . . . . . . . . 329 111. MULTICOMPONENT COPOLYMERIZATION . . . . . . . . . . . . . . . . . . . 344 A. Heat of Terpolymerization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344 B. General Theory of Multicomponent Copolymerization . . . . . . . . . . 349

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351 ........................................... 352

ACKNOWLEDGMENT REFERENCES

293 Copyrighf 0 1974 by Marcel Dekker. Inc. All Rights Reserved. Neither this work nor any part may be reproduced or transmitted in any form or by any means. electronic or mechanical, including photocopying, microfilming, and recording, or by any information storage and retrieval system. without permission in writing from the publisher.

294

SAWADA

We will discuss the main features of copolymerization, with particular emphasis on the reversibility of copolymerization. The treatment is a general one, and heat of copolymerization, entropy of copolymerization, and free energy change of copolymerization a r e applicable to all types of copolymerizations. Before proceeding to a discussion of some specific types of copolymerization, it will be worthwhile to discuss the general aspects of thermodynamics of copolymerization.

I. THE GENERAL THEORY OF BINARY COPOLYMERIZATION A. Heat of Copolymerization In the case of copolymerization involving monomers M, and M,, there a r e four distinct propagation steps. Each of these will have a characteristic molar heat of reaction associated with it, H,,, H,,, H,,, and H,,, respectively. WM: + M,

+ M, WAM; + M, 'M MZ + M, %M:

--

--+

--A

M,-M:

H,,

M,-M;

Hl2

M,-M;

H,,

wM,-M:

H21

Let us consider low conversion copolymers of high degree of polymerization containing N, monomer units of type M,, and N, monomer units of type M,. There will be four types of bonds to consider: 1) those between two monomer units of type M,, 2) those between two monomer units of type M,, 3) those between two monomer units M, and M,, and 4) those between two monomer units M, and M,. In these copolymers made from 1 mole of monomer mixtures there will be N,, bonds of type M,-M,, N,, bonds of type M,-M,, N,, bonds of type M,-M,, and N,, bonds of type M,-M,. The total number of bonds in these copolymer is No. Thus the molar heat of copolymerization will be given by

N

N

N

AH = 2 H,, + A H z 2 + -H1, NO NO NO

N

+ ANOH , ,

A s the reactions between M: and M, and M,* and M, forming M,-M$ and M,-M:, respectively, occur with equal frequency during the polymerization, Nl, = N,,

(2)

THERMODYNAMICS OF POLYMERIZATION. V.

295

The next step is to calculate Nll, N,,, and Nl,. Before attempting this, the treatment may be simplified by elimination of N,, and N,, as follows. The copolymers contain N, monomer unit of type M, and each of these has two bonds. Each M,-M, bond contains one monomer unit of type M,, whereas each M,-M, bond contains two monomer units of type M,. Further, each monomer unit is shared with two bonds, thus the number of monomer units of type M, involved in M,-M, and M,-M, bonds is (N,, + N,,)/2, and the number in type M,-M, bonds is N,,. Since there are N, monomer units of type M,, + Nzl

N, =

2

+ N,,

= N,,

+ N,,

(3)

Similarly, for monomer units of type M, N,

=

N,,

+

N21

2

+

N,, = N,,

+

N,,

Thus N,, and N,, can be expressed in terms of N,,, N,, and N,, From EqS. (1)-(4):

Equation (5) gives the molar heat of copolymerization. If the two types of monomer units a r e arranged completely at random in the copolymer chain, N,, can be readily evaluated from simple statistical considerations. Let us consider copolymers made from 1 mole of monomer mixtures. When the molecular weight of copolymers is high, No is given by No = N, + N,

and hence the mole fraction of N, groups in the copolymer is

and similarly

X, = N,/No The probability that a monomer unit of type M, will be on a given site is XI in the case of a random copolymer. The probability that a

SAWADA

296

monomer unit of type M, will be on a neighboring site is X,, and the probability that both will be on their respective sites simultaneously is X,X,. The total number of bonds in the copolymers is No; therefore, the number of MI-M, bonds will be equal to the total number of bonds multiplied by the probability that a bond will be of the M,-M, type. Thus

The copolymerization parameter 51 is defined a s 51 = H I 2

+ H,,

- (Hll + H,,)

(8)

We find that the molar heat of copolymerization is

Since 51 is independent of composition, XIX,s2 is shown to be a parabolic function of composition (Fig. 1). In the case H,, = Hzz, the function is symmetric about X, = X, = 0.5 (Fig. 2). (Note that any gain in AH on the part of a system is considered to be positive and any loss negative, but this is not considered here for simplicity.) Let us consider an M, monomer unit in the copolymer surrounded by two neighbors. If the copolymer is completely random, the probability that an M, monomer unit would occupy a site adjoining an M, monomer unit in this copolymer is simply X , . For a copolymer exhibiting regular alternation o r long sequences, the probability of occupancy will be P,, f XI. The randomness parameter JI is defined a s ill

For complete randomness, J, = 1; for some regular alternation, J, will be greater than unity; for long sequences, JI will be less than Unity. The number of MI-M, pairs, N12, may be readily expressed i n terms of JI. Rearranging Eq. (10) we find

THERMODYNAMICS OF POLYMERIZATION. V.

- 0 0

0.2

0.4

0.6

297

0.8

1.0

XI Fig. 1. Heat of copolymerization as a function of composition. H,, = 13 kcal/mole; H22= 18 kcal/mole. !J = 20 kcal/mole (1);10 kcal/mole (2); 0 kcal/nlole (3); -10 kcallmole (4);and -20 kcal/mole (5) [ 1 1.

The average number of M, monomer units which surround a given M, monomer unit is 2P,,, so therefore this is the number of MI-M, and M,-M, bonds per M, monomer unit on the average. The number of M, monomer units in the copolymer is NoX2, and thus the total number of MI-M, pairs is N,, = N,, = N,X,P,, = N,X,P,,

(12)

where Pi, is the conditional probability that an M, monomer unit selected at random will be followed by an M, monomer unit. Substitution of Eq. (12) into Eq. (5) yields [l, 21

298

SAWADA

20

-

a,

-. 0

E

3

x

10

I

a n 0

I

I

I

I

0.2

0.4

0.6

0.8

1.0

XI Fig. 2. Heat of copolymerization as a function of composition. H,, = Hzz = 15 kcal/mole. a = 20 kcal/mole (1); 10 kcal/mole (2);0 kcal/mole (3);-10 kcal/mole (4);and -20 kcal/mole (5) [ 11.

In the case of radical copolymerization, Alfrey and Lewis [2] obtained an expression for X2P2,as XZP,I =

1 - [l

- 4X2(1 - X2)(1 - rlr2)]1’2 2(1 - rlrJ

The reactivity ratios rl and r2a r e defined as kl,/kI2 and k,,/k,,, respectively. Substituting Eq. (11)into Eq. (13), we find AH = HllX,

+ H,&, + X1X,J1S2

(15)

Since the reaction is proceeding at constant temperature and pressure, AH is the enthalpy change of the copolymerization. However, the reaction is assumed to take at moderate pressure with negligible change in volume. Therefore, the enthalpy and the internal energy of copolymerization can be considered equivalent. Equation (15) is applicable to all types of copolymerizations (e.g., radical

THERMODYNAMICS

OF POLYMERIZATION.

299

V.

o r ionic copolymerization). For radical copolymerization at low conversion, close agreement between experimental values and calculated values from Eq. (15) was obtained, a s will be described in a later section. For all copolymerizations except azeotropic copolymerizations, the comonomer feed and copolymer compositions a r e different. The comonomer feed changes in composition a s one of the monomers preferentially enters the copolymer. When a high conversion copolymer is obtained, it is a blend of copolymers made from a continuously varying charge ratio. Therefore, Eq. (15) cannot be applied to copolymers at high conversion. In the case of high conversion copolymers, it is necessary to divide the polymerization into an arbitary number of successive stages, each of sufficiently short duration that [X,]/[X,] may be considered constant [3]. By combining Eqs. (11) and (14) it is found that

and if 1 - r,r, $J

2

0, Eq. (16) will reduce to

1 + X,X,(1

- r,r,)

(17)

The calculated values of J ) a r e shown a s a function of XI in Fig. 3. It is observed that the function is symmetric about XI = X, = 0.5.

1

\\

l

0

l

l

02

l

l

04

l

l

06

l

l

08

l

l

10

Fig. 3. Randomness parameter as a function of composition [ 11.

If r1r2= 1, the two types of monomer units a r e arranged completely at random in the chain, so that J, will be unity. If rlrz is less

SAWADA

300

than unity, J, will be greater than unity, and the structure deviates from random in the direction of regular alternation. If r1r2is greater than unity, J, will be less than unity and the self-propagation reactions will be favored over the alternation reactions. In the case r, = r, = 0, the monomers alternate regularly along the chain, regardless of the composition of the monomer feed. The mole fraction X, is then one-half for all monomer compositions. Hence, $ will be greater than unity (11, = 2). Expressing the rate constant according to the Arrhenius equation, we have k = A expi-E/RT}

(18)

where E and A are the activation energy and the frequency factor, respectively. We may write each of these rate constants in the form given by:

The reactivity ratio r l is

The relation between the heat of polymerization and the activation energy has a quantitative form [4],viz., -AE = CYAH 0 < a

<1

(20)

This equation says that in a related series of reactions, the decrease in the activation energy AE is a fraction a of the increase in the heat of polymerization AH. Equation (20) can be integrated to give

where C is a constant. Substituting Eq. (21) into Eq. (19), we find rl = (A,,/Al,)

and similarly

exp((H,,

- H,,)dRT}

THERMODYNAMICS OF POLYMERIZATION. V.

301

A s a result

is a function only of the r1r2product rather than of the individual values of r, and r, a s we will discuss in P a r t 11. This is an interesting result, since the r,r, product has been shown to be an index of the alternation tendency in copolymerization. In Part I1 we will also discuss the quantitative data concerning the heat of radical copolymerization It should be noted that configurational irregularities such as head-to-head and tail-to-tail modes o r stereoirregular placements of monomers are not examined, and, in addition, the formation and possible effects of more than one phase are not considered here.

.

B. Entropy of Copolymerization In the case of copolymerization involving monomers M, and M,, there a r e four distinct propagation steps. Let S,,, S,, S,,, and S,, represent the corresponding entropy changes, per mole, for the reactions M,-M,, M,-M,, MI-M,, and M,-MI, respectively. It is considered that the contribution of the growing polymer chain to entropy changes was determined mainly by the nature of the terminal unit. Thus the entropy change per mole of monomer polymerized is given by [5]

where

x

= (S,,

+ S,)

- (S,,

+

S22)

(24)

In addition, there a r e some other entropies in copolymers to be considered as follows: (1) Residual entropy at 0°K by the glass state. (2) Entropy due to chain configuration. (3) Entropy due to copolymerization randomness.

O r r [3] concluded that the contribution to the entropy at 0°K by the glassy state was negligible. The contributions from the con-

SAWADA

302

figurational entropies were discussed by Temperley [6] who decided they were quite small. Let us now compute the entropy of copolymerization randomness, which is an entropy of mixing. The probability of a sequence of M, units L units in length is given by P =Py(1

- P,,)

(25)

If the number of sequences per chain is 2n (of which half must be MI sequences and half M, sequences), the number of MI sequences of length L is

In such a case, let there be a, sequences of one M, unit, a, sequences of two M, units, . . ., and ay sequences of yM, units, thus n = a , + a , + a, +

- - - + ay

(27)

The number of permutations (W) open to these MI sequences is W

=-

"I!

%!a,! a,!

- - - a,!

In dealing with entropy calculations, we shall be interested in values of In W and hence be concerned with the logarithm of factorials. There is a convenient approximation, useful for this situation, known as Stirling's formula which is applicable only when n is very large. Stirling's approximation is given by Inn! g n l n n - n Thus, for large values of a, In W = n

In n

- a,

In a,--.

- . -ay In ay

The number of M, units in the copolymer is a,

+ 2a, + 3a,

-I.

- - - + yay

The entropy of mixing per mole is given by AS,,,,== R In W = n l n n - a, l n a , a, + 2a, + 3a, + -

a

- - - aY In - . + yay

a Y

THE R MODY NAMlCS OF POLYMERIZATION. V.

303

If y is very large, Eq. (31) may be simplified by expanding in t e r m s of P,, o r P,,. It may be readily shown that a, + 2a,

+ 3a, + .

1 . + yay 2 1 PI1

Therefore I. Y

C a, ~n a, I=l

[

n+ n ~n

- pl1)+-

In P,,]

(33)

These equalities become exact only when y is infinite. Appropriate substitution into Eq. (31) then yields AS,,,

=

-R [PI, In PI, + (1 - Pll) h ( 1

- PI,)]

(34)

Equation (34) is only exact if the permitted number of units in the longest sequence is infinite and PI, = P,,. However, there was no significant difference in values of copolymerization entropy calculated from Eq. (31), taking n = 100, and those calculated from Eq. (34), assuming infinite molecular weight. We see that the entropy from Eq. (34) has the proper limiting behavior: in the case of PI, = X I , AS,,, = -R(X, In XI + X, In X,), which is the entropy of random copolymers. In the case of P,, = P,, = 1, ASmIx= 0 in agreement with what is expected for block copolymers. F o r alternating copolymers, PI, = P,, = 0; so AS,,, = 0. Thus the total entropy change per mole involved in the copolymerization reaction, AS,,,, is the sum of entropy changes associated with copolymerization reactions and the entropy of copolymerization randomness. AS,,, = -(S,,X, + s,zx,

- R [(I - $J x,)

+ XWIX,)

(1

- J, x , )

+ $J

x,

JI

x,l

(35)

O r r [3] and North and Richardson [7)indicated the entropy of copolymerization randomness to be 0.5 to 1.5 cal/deg mole of repeating units, therefore the contribution of this entropy will seldom be responsible for more than 5% of total entropy change. Theil [8]expressed the entropy of sequence length distribution per mole in the copolymers a s AS = -R(X,[(l

+ XJ(1

- Pll) + PI, In PI,] - p22) In (1 - P Z Z )+ p,,

Pzzl)

(36)

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Equation (36) differs from Eq. (34) which does not recognize that the entropy of sequence length distribution includes the contributions of the separate arrangements of both the MI and the M, sequences. To define mathematically the concept of copolymer Urandomness,” Tosi 191 introduced a new quantity, which h e proposed be called %formational entropy.” It is defined a s the ratio between the natural logarithm of the number of ways of disposing of the homosequences of a copolymer and the number of monomer units contained in the copolymer; it is calculated by applying the methods of information theory. Information theory originally arose as an application of probability theory and statistical methods to problems connected with the translation of messages into codes. When applied to copolymers, information theory allows one to calculate the information gained in knowing whether a certain position in the chain is occupied by monomer unit MI o r by monomer unit M,. It is apparent that some analogy between information and entropy must exist. Information is a negative t e r m in the entropy of a system. On the other hand, entropy measures the lack of information; it gives us the total amount of missing information on the ultramicroscopic structure of the system.

.

C Equilibrium Sequence Distribution A s pointed out earlier, it is clear that equilibrium polymerization-depolymerization may govern the molecular weight distribution at high temperatures. By the same token it is expected that if a copolymer were treated in this manner, depolymerization-polymerization would determine both the molecular weight distribution and the structural sequence distribution of the copolymer. We can assume equilibrium polymerization-depolymerization will be achieved at quite high temperatures and will determine the copolymer structure and composition. Alfrey and Tobolsky [ l o ] first treated the case of an infinite copolymer molecule mathematically to determine the sequence distribution along the chain.

=

exp(AG,,/RT)

=K

(37)

where K is an equilibrium constant which is expressible in terms of vibrational partition functions f,,, fZ2, and f,, and the energy change

305

THERMODYNAMICS OF POLYMERIZATION. V.

AE,,. The partition functions f,, and f,, include a symmetry factor of 2. The quantity AE,, is defined as 2E,, - Ell - E,,, where El, is the energy per mole of an MI-M, bond, E,, is the energy per mole of an M,-M, bond, and El, is the energy per mole of an MI-M, bond. The energy Ell is approximately the heat of polymerization p e r mole of monomer MI and similarly for E,, and El,. AG,, is 2G,, - GI, - G,,, and GI, is the molar free energy of formationof an M,-M, bond, etc. The distribution in sequence lengths is found to be given by the formula nk = N,Pf;l(l

- PI,),

n: = N,P,X;'(l

- P,,)2

where n: is the number of MI sequences of length x in the copolymer chain, n: is the number of M, sequences of length x in the copolymer chain, and PI, and P,, a r e defined as (N,

- N,,)/N,

(38)

P2, = (N,

- N,J/N,

(39)

PI,

=

Combining Eqs. (12), (38), and (39) with Eq. (37), the following equation is obtained: P,,P,l -- exp(- AG/RT) p,,p,,

A similar conclusion was drawn by Theil [8] and by IZUand O'Driscoll [ll]. Consider the copolymerization reaction as occurring at some constant temperature, T. For the sake of illustration, let u s consider the copolymers which are 50% MI and 50% M,, and the change in free energy for this reaction, AG, is given by

AG

- TAS

=

AH

=

-1/2 [HI, + H,, + (W/2)1 + RT{[1 + (T/2)[S,, + S,

+

(Jlx)/Zl

- (J1/2)1

- (Jl/2)1) (41)*

*Instead of AF, it is more convenient theoretically to use AG, the Gibbs free energy, and thus speak of a constant pressure process. Note that any gain in a quantity o n the part of a system is considered to be positive and any loss negative. Since heat is evolved and randomness for the polymer decreased relative to the monomer during the polymerization, copolymerization reactions are exothermic (negative AH) and exoentropic (negative AS).

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306

Interchange reactions occurring in copolymer chains will ultimately lead to copolymer equilibrium; that is, an equilibrium distribution of compositional sequence lengths along the chains. This equilibrium sequence is determined as a function of temperature by the requirement that the change in free energy AG be a minimum with respect to the randomness parameter J,.Thus the condition aAG/aJ, = 0 gives [5]

9

= 2/[1 + exp{(-n/aRT)

(42)

+ (x/~R)}I

Equation (42) determines the equilibrium J, as a function of T, provided 51 and x a r e given. In Fig. 4, J, is plotted as a function of parameter X = 2RT/S1 for t h e case where x = 0 and 52 > 0. Regardless of the sign of s2 terms, J, will always be 2/(1 + ex&/2R}) at sufficiently large value of T, since 3 2/(1 + exph/2R)) a s T 00. Thevalues of entropy changes for most polymerizations a r e quite similar [12]; thus x will be very close to zero. If x = 0, the copolymer will be random at sufficiently elevated temperatures since 3 1as T 03. When a, x a r e zero, at any temperatures JI is unity, and hence the two types of units will be arranged a t random along the polymer chains. When 52 > 0, at low temperatures J, is very close to 2, and hence the polymer structure deviates from the random in the direction of regular alternation.

-

-

-

-

2.0

1.5

1.0

0

2

4

6

8

10

12

X = 2RT/Q Fig. 4. Randomness parameter as a function of 2RT/Cl[ 51.

THERMODYNAMICS OF POLYMERIZATION. V.

If x > 0, there must be a temperature at which --S1/2RT zero. This temperature is given by

307

+ x/2R is (43)

T = S2/x

At this temperature the copolymer will be random. Let us consider the energetics of attack of radical on a double bond. The weaker the bond formed between the attacking radical and the carbon center of the double bond, the higher the activation energy [13]. The more exothermic the reaction, the lower the activation energy. Therefore, the more exothermic the reaction between MI and M,, the stronger the bond formed between M, and M,. If there is an attractive interaction between unlike monomer units (large electrostatic interaction of charges on the radical and monomer for example), the MI-M, bond will be stronger than an M,-M, bond o r M,-M, bond, hence a will be positive in sign, yielding regular alternation. On the other hand, Joshi [14] found experimentally that the monomer units alternate regularly along the chain only when S2 > 0. Conversely, if there is repulsive interaction between unlike monomer units, -S1 will be negative and repeating units occur in blocks. An alternative approach [ 151to the problem is based on Boltzmann statistics, and almost similar conclusions are reached. Consider the copolymers with equal number of MI and M, monomer units, XI = X, = 0.5. If the average enthalpy change on formation of like bonds -(Hll + H,,)/2 is greater than that of unlike bonds -(HI, + H2,)/2, we then have the simple energy level system indicated in Fig. 5. If

I AH'

Fig. 5. Energy relation between unlike snd like bonds. [ 5 ]

the change in entropy for the reaction is zero in thermal equilibrium, the relative probability of occupancy of the high energy state to that

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308

of the low energy state is given by Boltzmann statistics: energy state/Plow

energy state

= exP{-AH’/RT}

(44)

The probability of occupancy of high energy state is

Similarly the probability of the low energy state is

Substitution of Eqs. (45) and (46) into Eq. (44) yields

JI

= 2/[1

+ exp{--Sl/2RT}]

(47)

where !2 = 2AH’ = H,,

+ H,,

- (H,,

+ HZ2)

F ro m Eqs. (15) and (47) we obtain: 2x,x2n

1 + exp(-n/2RT}

for X, = X, = 0.5. If CZ is positive at low temperatures, AH will approach -(Hi, +H,,)/2, yielding alternating copolymers. If 0 is negative at low temperatures, AH will approach -(Hll + H,,)/2, yielding block copolymers. Regardless of the sign of $2t e r m s at high temperatures, AH will approach -(H,, + H,, + H,, + H,,)/4, yielding random copolymers. If G? is zero, the copolymer will be random at any temperatures and hence AH is given by

AH

= -(HI1 + H,J/2

= -(HI2 + H,,)/2

= -(HI2 + HZ1 + Hi, + H22)/4

THERMODYNAMICS 0 F POLYMER I ZATl ON. V.

309

Copolymerization reactions will exhibit ceiling temperatures [16]. A t the ceiling temperature T,, AG is zero, so that Q HI, +

+

H,,

+

1 + exp{-52/2RTJ

In(' + exP{-52/2RT3) 1 + exp{G?/ZRTJ

+

- T,(S,,

2RTc +

ln(1 + exp{-Q/2RT,$ 1 + exp{-52/2RTJ

S,,)

=

0

(49)

Equation (49) does not lend itself readily to the calculation of T,. However, T, is found by solving Eq. (49) when HI1, H2,,SI1,and S,, a r e known and f2 = 0. Thus we obtain the ceiling temperature for the copolymer where X, = X, = 0.5 and x = G? = 0. The ceiling temperature for t h i s case is T,

=

(H,, + H,,~/~Sll+ S,,

- 2R In 2)

= [(HI, + H,, + H,, + H,,)/2]/(S1,

+ S,,

- 2R In 2)

This is the ceiling temperature for random copolymers. The number-average sequence length of MI monomer units is given by the ratio of the probability of randomly selecting an MI monomer unit, X I , to the probability of occurrence of an M,-M, bond, X,P,,. Thus

w,,= UPl2 = 1/(1 - PI,) = m x , ) Similarly

w,

=

U P , , = 1/(1

- P2,) = l / ( W , )

The variation of the number-average sequence lengths of the monomer units in a binary copolymer is shown a s a function of temperature in Fig. 6 [8]. When 52 is negative (corresponding to a block copolymer), both W,, and W, increase as the temperature decreases. The effect of varying 52 on the relationships between temperature and sequence length is shown in Fig. 7 [8]. Negative values for Q cause the sequence length in Curves l a and l b to increase a t low temperature and to remain consistently higher than that for random copolymer which is formed at all temperatures when 52 = 0 kcal/mole (Curve 2). When 52 is a positive quantity, the sequence length decreases with temperature until the MI monomer units a r e nearly isolated from one another at -100°C (Curve 3).

310

SAWADA

-100

0

100

2 00

300

TEMPERATURE("C)

Fig. 6. Number-average sequence lengths vs copolymerization temperature for a copolymer for which XI= 0.2,52 = 2.0 kcal/mole, and x = 0.0 eu [ 81.

D. Free Energy Change in Binary Copolymerization System

In the first place, let us consider the general shape of a free energy change vs composition curve for a binary copolymerization system in the vicinity of either end of the binary [17]. The free energy change of copolymerization a t some particular composition will be given by the equation AG = AH

- TAS

To simplify the following treatments, it is assumed that JI is independent of X,. This assumption can be regarded only as a first

THERMODYNAMICS 0 F POLYMER I ZATl ON. V.

31 1

20.0

10. 0

\

8.0

2

5.0

3. 0

2.0 2

1.0 -100

0

100

2 00

300

TEMPERATURE ("C)

Fig. 7. Number-average sequence lengths of M I monomer units vs copolymerization temperature for a copolymer for which XI = 0.4: (la) Sl = 2.0 kcal/mole, x = 0.0 eu; ( l b ) Sl = 2.0 kcal/mole, x = 2.0 eu; (2) f2 = 0.0 kcal/mole, x = 0.0 eu; (3) f2 = -2.0 kcal/mole, x = 0.0 eu [ 81.

approximation, since there i s no doubt that $ depends on X, with the exception of random copolymers, alternating copolymers, and block copolymers. However, in the vicinity of the end of binary, $ is independent of X, and $ = 1. Differentiating AG with respect to X, at constant temperature, we find

312

SAWADA

In the vicinity of the end of binary, we may say that (1 - JlX,) Thus

+ RTJl In JlX,

-

1.

(52)

-

If J, is not zero, so that In JlX, -03 a s X, 0, aAG/aX, will always be negative at sufficiently small values of X,, regardless of the sign of any term. Thus the introduction of the first comonomer unit into a homopolymer will always result in a decrease in free energy of the system; hence, a pure homopolymer in the presence of a comonomer is always thermodynamically unstable. This conclusion has important ramifications in connection with the important process of copolymerizing impurities such as oxygen from its environment [18]. If 51 is positive and x is negative o r zero, AG a s a function of X, will be concave downward at all compositions below the ceiling temperature a s shown in Fig. 8. Let us consider a general case for which 51 is a large positive value. It is assumed that x = 0 from experimental data of O r r [3] as well as from his theoretical discussion. It therefore allows that x has comparatively little effect on AG although 0 affects AG markedly. At a temperature T, well below the ceiling temperatures of either component, the free energy change vs composition curves would appear schematically a s shown in Fig. 8.

u -TIAS

0 0

f

0 Q

1.01

-

Fig. 8. Free energy change vs composition for a copolymer when i2 is positive and T = T , [17].

THERMODYNAMICS OF POLYMERIZATION. V.

313

Above the ceiling temperature of both components at temperature T,, the curves would appear a s shown in Fig. 9. This gives rise to a ceiling temperature maximum and a modified phase diagram a s shown in Fig. 10.

t (3

Q

Fig. 9. Free energy change vs composition for a copolymer when i2 is positive and T = T, [ 171.

When 51 is a large positive quantity, the introduction of two types of monomer unit into a polymer chain raises the ceiling temperature for polymerization. Therefore the maximum ceiling temperature occurs at a polymer structure of the regular alternation. The alternating copolymerization of sulfur dioxide and isobutene might appear to be a typical example of maximum ceiling temperature [19]. A positive value of s1 reflects on attractive interaction between unlike neighbors. Since this attraction would be more pronounced in the copolymer, and because of the strong bonding resulting from this attraction, the ceiling temperature of the copolymer relative to both homopolymers would be increased. It is easily shown that the maximum ceiling temperature also occurs when 51 0. This result is consistent with the experimental data of Richardson and North [ 203 who found the maximum in the ceiling temperature for the copolymerization of acetaldehyde and propionaldehyde.

314

SAWADA

T

I

1

I

I-

II : I

M1 a x 2

-

I I

I

M2

Fig. 10. Modified phase diagram corresponding to free energy curves of Figs. 8 and 9 (at T, the free energy diagram is given by Fig. 9.): (1) monomer; (2) copolymer; T, = ceiling temperature [ 171.

Let us now consider the case of a large negative value of Q. A t some temperature T, below the ceiling temperature of the homopolymers, the curves would appear a s shown in Fig. 11. The free energy curve for the copolymerization would exhibit minima since f2 < 0. The most stable situation at various compositions is: (a) 0 < X, < a, polymer is stable; (b) a < X, < b, monomer is stable; (c) X, > b, polymer is stable. This gives rise to a ceiling temperature minimum and results from a repulsive interaction between unlike nearest neighbors and hence a weakening of the copolymer structure. In the case of a repulsive interaction, s2 will be negative and clustering will occur [7]. The modified phase diagram associated with this type of system is shown in Fig. 12, with Fig. 11 being associated with TI. There will be a repulsive interaction between unlike nearest neighbors in the copolymerization of vinyl monomer with cyclic monomer, and hence the cross-propagation reaction occurs with difficulty [21]. In this copolymerization must be negative, indicating the production of block copolymers [71. Tsuda and Yamashita[21]

THERMODYNAMICS OF POLYMERIZATION. V.

315

Fig. 11. Free energy change vs composition for a copolymer when i2 is a large negative value and T = T, [ 171.

Fig. 12. Modified phase diagram corresponding to free energy curve of Fig. 11 (at

T, the free energy curve is shown by Fig. 11): (1) monomer; (2) copolymer; (3) two copolymers of different compositions; T, = ceiling temperature [ 171.

found the block copolymer for the copolymerization of styrene and P-propiolactone. If a case is considered for which 51 is l e s s negative than in the former case, the ceiling temperature minimum is not realized.

316

SAWADA

This modified phase diagram i s shown in Fig. 13. A s the temperature is lowered, however, the AH term will begin to dominate in the free energy expression. A s discussed above, near the ends of the binary ACi is always negative, so the complete curve will be shown as in Fig. 14.

Fig. 13. Modified phase diagram for a copolymer when S2 is a small negative value (at T, the free energy curve is given by Fig. 14.):(1) monomer; (2) copolymer; (3) two copolymers of different compositions; T, = ceiling temperature [ 171.

Fig. 14. Free energy change vs composition for a copolymer when 52 is a small negative value [ 17 1.

317

THERMODYNAMICS OF POLYMERlZATl ON. V.

Consider now a copolymer of composition a. It will have a value of AG equal to AGa and negative. Therefore this copolymer is stable relative to pure homopolymers Nil and M,. The same is true for copolymer b. Consider copolymer x, however. It might have a negative value of AG but we have to question whether o r not AG could be lower by dissociating the copolymer. If it dissociates into two copolymers of different compositions, the lowest stable values of AG which can be attained by the decomposition is AG at composition y and z, respectively. These copolymers a r e in equilibrium with one another since component M, would have the same chemical potential in both. This is because for S l < 0, (H12 + HZl) < (Hll + H,,) and MI-M, and M,-M, pairs are energetically favored over MI-M, pairs. Hence at low enough temperatures, two copolymers having different compositions will be formed. It is important to note that compositions y and z are not necessarily associated with the minima of the AG vs X, curve. The free energy changes at composition y and z may be different from the minimum values, since the important criterion for equilibrium is that the chemical potential of a given component is the same in both copolymers, not that the free energy change of each copolymer be a minimum. Thus copolymers to the left and the right of y and z, respectively, are stable, whereas copolymers of composition between y and z a r e unstable and will dissociate into two different types of copolymers. In connection with Fig. 14 it will be observed that two inflection points exist between y and z. Let us calculate the maximum critical temperature To where decomposition will occur and the composition of the copolymer Xg, at this temperature. These points are shown in Fig. 13. Above the temperature To the curve of AG vs X, is everywhere concave downward. Below this temperature two minima will occur, as already discussed, and there will be two inflection points where a2AG/aXz = 0. A s the temperature is raised toward To,the minima will move closer together in composition and as a result so will the inflection points. At To both minima and the inflection points will coincide at the same temperature. Thus To is the temperature where both a2AG/aXE and a3AG/aXl equal zero at the critical composition X i [22].

a3AG/aX,3 = -RTo$(l

- Z$X,O)/[Xi(l - $Xi)]'

=0

(54)

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318

Hence we find

xg = 1/2#

(56)

When x = 0 o r x < 0, 0 must be negative in order for To to be positive. A negative value of 0 indicates a repulsive interaction between unlike components, and block copolymer will occur “71. Thus, from Eq. (55), the larger this repulsive interaction, the higher the temperature will be where decomposition begins. A s a generalization we might say that virtually all systems with negative f2 and x should give two copolymers of different compositions. Thermodynamically, of course, copolymers of composition in the region y < X, < z a r e unstable at this temperature, and two copolymers of different compositions y and z will be favored. This has not been found experimentally. The reason for this lack of agreement with theory is related to the fact that unless 51 is a sufficiently large negative value, To will be so low that the kinetics of the decomposition in the copolymer will be infinitesimally slow, and thus a copolymer will be generally found which is actually metastable. More recently, Harvey and Leonard [23] have considered the complete reversibility in equilibrium copolymerization in solution, taking into account nonideal behavior of the solution through the use of Flory’s expression for the partial molar free energy of mixing of monomers and copolymer [24]. The value of P,, is set equal to K exp(-AG,,$RT), where K is a numerical constant found to be 1 and AG,,, is the free energy change upon the addition in solution under conditions prevailing at equilibrium of 1 mole of monomer M, to copolymer chains in order to form MI-M, bonds. Specific equations for equilibrium bulk copolymerization were derived in the presence of monomers a s the solvent of copolymers.

E, Equilibrium Monomer Concentration Izu and O’Driscoll [ l l ] treated the system where the monomers and the copolymer coexist. Since the equilibrium state will occur near the ceiling temperature in the case of addition copolymerization, it is desirable to develop a theory giving an equation for the equilibrium monomer concentrations.

THERMODY NAMl CS OF POLYMER IZATl ON. V.

319

The possible reactions and associated rate constants are as follows :

- k‘, wM,-M: kl,

dM:+Ml

k,2

k’i w M 1 - W

- WM,-M: ““‘w Ml k,?

-M:

+ M2-

k21

+

I

.“.%

+

M

k22

X mM2-M:

Z

This model is often referred to as the diad model. The constants kll, k12, kl, and $, are the rate constants of the propagation steps; k c , kfi, k z , and k, are the rate constants of the depolymerization

steps. It is assumed that the constants are independent of the last number of the chain. The following probability parameters were defined [25] to express the relative amount of various chain ends:

([Mil),

=a

(57)

([MiIn-,/[M,In)t = E

(58)

( [ ~ z I n - l / [ ~ z l nt )= 77

(59)

where ([M,]), is the probability of finding M, monomer unit at the chain end, ([Ml]n-l/[Ml]n)t is the conditional probability of finding M, monomer unit at the penultimate unit given that the chain end is occupied by M, monomer unit, and ([M,]n,/[M,]n)t is the conditional probability defined analogously. The following relationships are obtained. ([MZI),

=1-a

(60)

([MzIn-1/[MJn)t

=

1- 6

(61)

- 77

(62)

( [ ~ 1 l n - l / [ ~ , l n ) t= 1

According to the principle of detailed balancing at equilibrium, we find for a chain of infinite length: k,,[M,],a

- k,aE

=0

(63)

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320

- kid1 - a)(l - q) = 0 k,,[M,],(l - a) - k,a(l - E ) = 0 &z[M,Ie(1 - a) - k& - a)s = 0 klz[Mzlea

(64) (65) (66)

where [MI], and [M,], are the monomer concentrations at equilibrium. Since we have four independent equations, we can determine all the variables if a condition which gives another equation between variables is specified. F r o m Eqs. (63)-(66) we see that

where the equilibrium constants are K, = kll/kfi, 6 = kl,/kE, K3 = k&,, and & = kzz/kE. Assuming that there is no side-reaction in the system, we obtain the following constraints which state that the monomer structure is conserved in the system. [MI],

+ X,{[MIo

[~zl+ e xzff ~

1

.- ([Mile + [MzIe)) = fl[MIo

(70)

-0 ([Mile + [ ~ z l e ) }= f J ~ 1 0

(71)

The Monomer State + The Polymer State = The Total

(72)

where [MIo is the total concentration of both monomer units including both the monomer and polymer states, X, and X, a r e the mole fractions of M, and M, monomer units in the copolymer, and f, and f, a r e the mole fractions of M, and M, monomer units in the monomer mixture. When the conditional probability parameters at the chain end are equal to those on the main chain, it is possible to express the copolymer composition and diad fractions in t e r m s of E and q.

THERMODYNAMICS OF POLYMERIZATION. V.

321

where Fll, F12,FZ2,and F,, a r e diad fractions. A l l eight rate constants are not necessary for solving the equations; the sufficient values are K,, &, and (K&)/(Kl&) which are expressed in t e rm s of the standard free energy changes of the elementary steps:

w = *(AG:,

+ AGX,

- AG;, - AG;,)

(81)

where AG;,, AG;,, AG;,, and AG& are the standard free energy changes of the corresponding elementary steps. Equation (80) is identical in form to Eq. (40),with 2w being identified with AG,,, which represents the standard free energy of formation of a heterogeneous bond from homogeneous bonds. Thus we can predict theoretically the equilibrium state of an addition copolymerization which obeys the diad model if x,, K,, &, and w are given. For the case of the critical condition beyond which the monomer mixture fails to be polymerized, we know that

[ML,

[MI],

+ [ ~ z l e=

[MI,

(82)

Thus, if we specify the value of [MI,, the values of [M,], and [M,], at the critical condition can be calculated from Eqs. (68) and (82). Modified phase diagrams a r e calculated using Eqs. (67), (73), and (74) from the values of [M,], and [M,],. Izu and O'Driscoll [ll], using such an approach, have calculated the monomer and polymer curves for some hypothetical cases and obtained the results shown in Figs. 15, 16, and 17. In these figures a r e shown the results of calculations for the cases where there is 1) no interaction, 2) positive interaction, and 3) negative interaction between unlike monomer units. The monomer curves represent the condition above which the

322

SAWADA

POLYMER

0

I

I

I

0.2

0.4

0.6

MOLE FRACTION

I

0.8

1.0

MI

Fig. 15. Equilibrium copolymer and comonomer compositions computed as a function of temperature for [MI, = 1.0, AS;, = AS:z = -28 eu, AH;, = -10 kcallmole, AH& = -16 kcal/mole, w = 0 [ 111.

350

k

I

MONOMER

I

h

Y

v

300 3 +-Q LI

w

POLYMER

a w

I-

250

t MOLE F R A C T I O N

Mi

Fig. 16. Equilibrium copolymer and comonomer compositions computed as a function of temperature for [MI, = 1.0, AS:, = AS"z = -28 eu, AH;, = -15 kcal/mole, AH& = -16 kcal/mole, w = -1.5 kcal/mole [ 111.

323

THERMODYNAMICS OF POLYMERIZATION. V.

3 50

h

Y

v

W

5

300

tQ [L

W

a I

I

W k

I

POLYMER

250

0

0.2

0.4

0.6

MOLE FRACTION

0.8

1.0

Mi

Fig. 17. Equilibrium copolymer and comonomer compositions computed as a function of temperature for [MI, = 1.0, AS:, = AS& = -28 eu, AH;, = -17 kcal/mole, AH& = -16 kcal/mole, w = 1.5 k c a h o l e [ 111.

monomer state is more stable. The polymer curves represent the copolymer composition which is expected to be formed reversibly at the critical condition. It is observed from Figs. 16 and 17 that either maximum o r minimum critical temperature for the monomer mixture is observed when there is a positive or negative interaction, respectively

.

F. Penultimate Unit Effect The behavior of some comonomer systems indicates that the reactivity of the propagating species is affected by the next-to-last o r penultimate monomer unit. This effect is often referred to as the penultimate unit effect. Silberberg and Simha [26] have discussed the kinetics of reversible processes on linear lattices that would include copolymerization with penultimate unit effect. O’Driscoll et al. [ll, 271 have used a kinetically based approach to treat reve r sible chain- growth copolymerization with penultimate unit eff ect , which will be discussed in a later section. Simha and Zimmerman [28] have enumerated grouping and sequences in a binary copolymer for which the penultimate unit effect is considered. There are eight conditional probabilities which describe the sequence length distribution of the chain, and those are

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324

interrelated by five equations. The probabilities a r e of the kind Pljk, which may be defined a s the probability that an i - j pair is followed by a k where i, j , and k can be either 1 o r 2. There are four equations of the type Pijl

1 In addition, we find + pijz

=

(83)

Theil [29] has extended the thermodynamic treatment of the penultimate effect to the case of the infinite molecular weight binary copolymers where the penultimate effect may he operative. Together with Eq. (84), Theil [29] has suggested that four equations expressed in terms of four conditional probabilities define the equilibrium state.

In cases where the nearest-neighbor approach is sufficient, the following will be true in the present treatment. Pijk will reduce to P,, and thus P j j l + Pijz = 1 and Pi,, + Pjj2= 1 where i, j , k = 1 o r 2. Also & i j k will then equal A E , ~and ~ , klj,will vanish. In such a case Eq. (85) will reduce to the equilibrium expression for the nearest neighbor case. p,,p,, ~-

- exp{2AG:,/RT}

P1,PZl

where AG;, is the average free energy of formation of an M,-M, and M,-M, bond. Equations (86) and (87) become identities with zero in this more special case. Equation (88) is identical in form to Eq. (40), with 2AG:, being identified with AG,,.

THERMODYNAMICS OF POLYMERIZATION. V.

325

11. DEGREE OF POLYMERIZATION AND COPOLYMER COMPOSITION OF BINARY COPOLYMERIZATION SYSTEM A. Degree of Polymerization

The equations derived by Tobolsky [30] for the various cases of equilibrium copolymerization may well serve a s a simple means of predicting degree of copolymerization. Tobolsky and Owen [30] neglected the interaction between monomer units in the polymer chain so that the equilibrium sequence distribution of these models should always be random. The concentrations of polymer chains containing n monomer units of any distribution o r proportion of MI and M, is designated a s C,.

M,

K;

(89)

M,*

-

Equation (82) applies for n = 1 a, The total equilibrium concentration of copolymer molecules N is given by

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326

The total equilibrium concentration of monomer segments W incorporated in the copolymer is found to be

+...

The number-average degree of copolymerization at equilibrium is given by P = W/N

The amount of monomer MI entering the copolymer is governed by Conc MI entering copolymer Conc M, and M, entering copolymer

Therefore

THERMODYNAMICS OF POLYMERIZATION. V.

327

Similarly

F r o m Eq. (95) and by analogy with previous results for equilibrium homopolymerization, a transition tsmperature (floor temperature) will be encountered where K,[M,],

+

&[MzI, = 1

(103)

Below the transition temperature the equilibrium concentrations [M,], and [M,], are equal to [M,], and [M&. The degree of polymerization of the copolymer is P = 1/[1 - (Kl[MIIo + %[M,I,)I

(104)

Above the transition temperature, the degree of polymerization is large and the following approximation is valid:

K ~ C M ~+I %~ E ~ z l e = 1

(105)

From Eqs. (97) and (98) we find ([MI],

- [M,Ie)/([MzI, - [MZIe) = K1[M,Ie/Kz[MzIe

(106)

A t any temperature above the transition temperature, [MI], and [M,], can be computed from Eqs. (105) and (106). By rearranging Eq. (99) we obtain the following expression for calculating P above the transition temperature:

SAWADA

328

Results of calculations were tested experimentally using the copolymerization system of sulfur and selenium [30]. The following nomenclature will be used in the derivations: M, = S, monomer unit, M, = Se, monomer unit, M: = S, diradical,

and M: = Se, diradical.

Calculations were made for an initial monomer composition of 90% sulfur and 10% selenium by weight, using Eqs. (103) through Eq. (107), and the calculated values of P a r e plotted in Fig. 18 together with the experimental results obtained by Schenk [31]. Experimental values agree well with the theoretical results.

1

350

I 400

I

1

4 50

500

TEMPERATURE

550

( O K )

Fig. 18. Degree of polymerization vs copolymerization temperature for sulfurselenium copolymers in the case of an initial monomer composition of 90%sulfur and 10%selenium by weight. Solid line, experimental; dashed line, calculated [ 311.

Kang and O’Driscoll [32] have studied the effect of copolymerization reversibility on molecular weights by calculating leading moments by the generating function technique. Results of calculations of average molecular weights from the resulting equations were tested experimentally using the radical copolymerization system of styrene and a-methylstyrene at 60 and 100°C by gel permea-

THE RMODY NAMl CS OF POLY ME R IZATION. V.

329

tion chromatography or membrane osmometry. The effect of copolymerization reversibility on molecular weights mainly occurs 'indirectly," i.e., from an increasing termination rate constant, as a-methylstyrene feed mole percent increases. The 'direct effect," i.e., decrease of net rate of polymerization from the participation of depropagation steps, will be observed in the temperature range higher than 100°C. This implies that reversibility need only be considered for the homopropagation reaction of cu-methylstyrene in the temperature range 60 to 100°C. B. Copolymer Composition Equation We will now discuss the copolymer composition equation in the presence of depolymerization reactions. There a r e two principal approaches to the interpretation of the composition of equilibrium copolymerization. The first approach is to consider the kinetics of copolymerization with depropagation. The second approach is to relate the composition of copolymer to the probability of producing a certain sequence of monomers, and the approach used is to consider the relative probabilities of the various propagation and depropagation steps.

1. Kinetic Approach. The binary copolymerization is generally considered in terms of four propagation steps involved in the formation of a chain, and these a r e irreversible:

and the copolymer composition is then given by

Equation (108) is known a s the copolymerization equation o r the copolymer composition equation. The copolymer composition, d[M,]/ d[M,], is the molar ratio of the two monomer units in the copolymer. The parameters of copolymerization r, and r2 a r e termed the monomer reactivity ratios, and a r e given by the ratio of the rate constants of the homopolymerization (kll and h2)to the rate constants of the cross-propagation (klz and k,,).

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330

Ham [33] has generalized the original penultimate unit treatment of Mertz, Alfrey, and Goldfinger [34], and has found satisfactory agreement with the data in certain systems a t given temperature. Ham’s approach 1331 emphasizes the kinetic nature of the chain growth and the impossibility of particular reactions. In contrast to Ham’s approach, Lowry [35] has treated the deviations from the copolymer composition equation from the point of view of the reversibility, and has derived copolymer composition formulas on the basis of three different sets of assumptions (I, II, and m)about the depropagation ability of different radical structures, and Hazel1 and Ivin [36] added a fourth set, Yamashita e t al. [37] added a fifth set, and Wittmer [38] added a sixth set. These assumptions a r e summarized in Table 1 [39]. Table 1 Copolymerization Mechanisms with Depropagation Terminal structureapb

111*

Case

211*

121*

221*

I

I1

+

I11 IV

v v I +

+

+

222*

122*

+ + + + + +

+

212*

35,39 35,39

+ +

+

The copolymer composition for Case I is given by

(Y

defined by

Ref.

35,39

alll*denotes-M,M,M, *. + denotes that depropagation is assumed to occur.

with

112*

+

36,39 31 38

THERMODYNAMICS OF POLYMERIZATION. V.

33 1

The qualitative predictions of Case I a r e as follows: 1) Temperature will have a marked effect on the shape of the copolymer composition curve in the region of the ceiling temperature of M,, 2) dilution of the monomers by an inert solvent will have a marked effect on the shape of the copolymer composition curve in the region of the ceiling temperature of M,, and 3) the limiting mole fraction of M, in a copolymer prepared at high temperature will be one-half. These effects a r e shown by curves in Figs. 19 and 20 for the hypothetical copolymer system assumed below: (a) rl = r, = 1 for all temperatures, (b) ceiling temperature = 50°C, (c) AE* for k,, = 2 kcal/mole, and (d) AE* for kE = 1 2 kcal/mole. The copolymer composition for Case I1 is given by

where a is defined by Eq. (110) and v by U =

K[M,l + K[M,]/r,

-a

K[M,I where K is the equilibrium constant for the equilibrium expressed in the form -M,M,M,M,+

mM,M,M?

+ M,

This leads to the same qualitative predictions as before but the limiting mole fraction of M, in the copolymer prepared at high temperature is two-thirds rather than one-half. The qualitative predictions of Case III a r e the same as those of the previous two. The quantitative differences between the second and third cases involve so many independent parameters that it may not be possible t o choose between the two by copolymerization experiments alone. The results which support the assumption of Lowry were obtained by several workers [40-421. O'Driscoll and Gasparro [40] have studied the copolymerizations of styrene (M,) with a-methylstyrene (M,) and methyl methacrylate (M,) and of acrylonitrile (MI) with a-methylstyrene (M,) in the approximate regions of the ceiling temperatures of a-methylstyrene and methyl methacrylate. In all cases the copolymerizations were carried out over a'wide range of monomer feeds, and Lowry)s Case I1 was found to be quite adequate in describing the results.

332

SAWADA

0

0.2

mole fraction

0.4

M2

0.6

0.8

1.0

in monomer mixture

Fig. 19. Effect of temperature on copolymer composition at [MI] + [M2I (1) 0°K;(2) 0°C; (3) 50°C (T,.); and (4) 100°C [35].

L

=

1:

1.0

0,

-E, 0

a

0.8

0 U

.-C

0.6

s

0.4

C 0 .-c U

L

0.2

.4-

0

0

E

n

"

0

0.2

mole fraction

0.4

M2

0.6

0.8

1.0

in monomer mixture

Fig. 20. Effect of [ M, ] + [ M, ] on copolymer composition at the ceiling temperature: (1) [MI] + [M, ] = 10; (2) [M,] + [M,] = 1.0;and (3) [MII + [M,I = 0.1 1351.

THERMODYNAMICS OF POLYMERIZATION. V.

333

The influence of reversal of propagation reactions on copolymer composition has been studied for the following systems [41]: 1) anionic copolymerization of vinyl mesitylene (M,) and a-methylstyrene (M,) in tetrahydrofuran, and 2) radical copolymerization of styrene (M,) and methyl methacrylate (M,) in o-dichlorobenzene. F o r these systems Lowry's Case I1 provides a reasonably satisfactory interpretation of the variation of copolymer composition with [M,] at constant feed composition. Lowry's Case I was studied by the cationic copolymerization of styrene (M,) with a-methylstyrene (M,) as the simplest mechanism of depropagation [42]. The equilibrium constant of M, homopolymerization thus obtained is much smaller than the literature value. The deviation from the literature value becomes larger the higher the polymerization temperature. The equilibrium constant is dependent on the sequence length of a-methylstyrene and becomes larger with an increase in the sequence length. On the other hand, it has been suggested by Johnston and Rudin [43] that the simple copolymer equation, Eq. (108), was clearly an adequate model for the radical copolymerization of a-methylstyrene with styrene and with methacrylonitrile at 60"C, although the ceiling temperature of a-methylstyrene is 61°C [43]. Depropagation effects were not important in these copolymerizations because the mean sequence lengths of a-methylstyrene in the copolymers were short. Such short sequences have higher polymerization enthalpies and hence higher ceiling temperatures than higher molecular weight homopolymers. The radical terpolymerization of methacrylonitrile, styrene, and a-methylstyrene was also studied at 60°C in toluene solution [44]. Again the behavior of a-methylstyrene can be described by a simple model without reference to ceiling temperature o r penultimate effects because the sequence lengths of this monomer in the terpolymer a r e short. Kang and O'Driscoll [45] interpreted the physical significance of reactivity ratios obtained by applying the simple copolymer equation to composition data from systems which a r e expected to polymerize with some depropagation. Thus it is possible to represent composition data for a reversible copolymerization by Eq. (108) and to put a correct physical interpretation on the parameters derived from that equation. The radical copolymerization of styrene (M,) with a-methylstyrene (M,) in the temperature range from 60 to 150°C can be described with the assumption that addition steps of a-methylstyrene may be reversible [46]. For the temperature range from 60 to 110°C it could be demonstrated that it would be sufficient to consider only

SAWADA

334

the reversibility of the addition steps of a-methylstyrene to a radical end with a-methylstyrene as the terminal unit (Lowry’s Case I). A t a reaction temperatures of 150°C the addition of a-methylstyrene to a styryl chain end is also reversible (Lowry’s Case IV). Olefins and sulfur dioxide copolymerized in the liquid phase under the influence of radicals to form 1 : l copolymers as olefin polysulfones. A good example of Case IV is provided by the system cyclopentene-isobutene-SO, which gives a regular 1:1 olefin-SO, copolymer [36]. For the cationic copolymerizations of 3,3-bischloromethyl oxacyclobutane (M,) and tetrahydrofuran (M,), Yamashita et al. [37] have considered two models in relation to the known mechanism. Model 1, which corresponds to Case IV in Table 1, assumes a reversible addition of M, monomer at both M, and M, ends, and Model 2, which corresponds to Case V in Table 1, assumes reversible addition of both M, and M, monomers at the M, end only. In both models the rates of propagation and depropagation reactions are assumed to be independent of the composition of the residual polymer chain. With the steady-state assumption applied to the concentration of living ends, a copolymer composition equation is derived by Lowry’s procedure. Yamashita et al. [37] came to the conclusion that Model 2 expresses the mechanism of copolymerization rather better than Model 1. In a copolymerization which includes a comonomer with a ceiling temperature near room temperatures such as tetrahydrofuran, the depropagation reaction considerably influences the copolymer composition, especially at the higher feed ratios of this monomer. In the case of the depropagation of unsaturated compounds, it is sufficient to remove the last unit to reconstruct an active center. For heterocyclic compounds, however, it is necessary to consider the removal of the last unit and cyclization of the penultimate one. Penczek [47] derived equations for these compounds based on this model. The copolymer composition equation with reversibility of all propagation reactions was derived [38, 481. The reaction scheme is the previously described diad model. To treat the problem quantitatively, let us define two new parameters, q, and q, [38]:

The rate constants kE and kE can be replaced by the equilibrium constants K, and K, according to K, = kE /kll and K, = kZ&.

The copolymer composition is given by

335

THERMODYNAMICS OF POLYMERIZATION. V.

where x, is the mole fraction of the reactive chains ending with monomer M, with a sequential length 1.

and similarly

Knowing the constants and the monomer concentrations in Eqs. (115) and (1161, we can calculate the numerical values for x, and y,. This can be done graphically, but it is more practical to use a computor. Using Eq. (1141, we can then calculate the composition of the polymer formed. F o r the special case where K, = 0, q, = 0, and q, = 0, we can write the simpler equation

1 - x , = r,([Ml] + K,) + [M,] 2r,K,

- , / r ( [ M l l + K,)

+

[M,]

2r1K1 (118)

This case corresponds to Case ZI in Table 1, where only the homopolymerization of the monomer M, is reversible.

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336

F o r the special case where K, f 0, q, = 0, and q,

=

0, we know that

x, is given by Eq. (118), and y, by Eq. (120):

This case corresponds to Case VI in Table 1, where the two homopolymerization reactions a r e reversible and the alternative steps a r e irreversible, If the equilibrium constants depend on the length of the monomer sequence (penultimate effect), further changes must be introduced into the equations. Wittmer [38] considered the simple case where the sequence of two monomer units of M, does not depolymerize, and longer sequences of M, a r e subject to equilibria with a constant K,. And for the special case where K,, q,, and q, a r e zero, we find

d[M,l/d[M,l =

(121) 1 + r,[M,l/[M,l

u corresponds to Eq. (118):

1 r,([M,l + K,) + [M,]

(J=-

2

r,K,

(122) Equation (117) can be applied to the radical copolymerization of a-methylstyrene (M,)-methyl methacrylate (M,) at 20 to 100°C. Between 100 and 150°C the reversibilities of the homopolymerization step of methyl methacrylate and of the alternating steps had to be considered, i.e., Eq. (114) in connection with Eqs. (115) and (116).

THERMODYNAMICS

OF POLYMERIZATION. V.

337

Equation (122) can be applied to the system a. methylstyrene (M,)-acrylonitrile (M,) where the sequence of two monomer units of a-methylstyrene is stable and does not depolymerize. The reversibility of the polymerizations of a-methylstyrene and methyl methacrylate can be explained by sterically induced strain in the polymer chain [35]. In the copolymer a-methylstyrene-methyl methacrylate this strain involves the whole polymer chains whereas in the a-methylstyrene-acrylonitrile system the strain is broken by the acrylonitrile sequences and is built up again by a-methylstyrene. This explains the differences in the depolymerization tendencies of sequences of two units of a-methylstyrene and longer sequences in this system. The same idea might be applied to the results of Johnston and Rudin [43]. 2. Probability Approach. In contrast to the kinetic approach, O’Driscoll and his co-workers [25, 27, 491 have treated the derivations of the copolymer composition equation using a probability approach. The copolymer composition equation for the diad model is given by [25]

where a is the transient probability that a chain ends in a [M,] unit and b is the transient probability that a chain ends in a [M,] unit, and q and E represent the transient probabilities of Eqs. (52) and (53), respectively. The following relations must be considered:

= bb,, + (1 a k , , + (1 - ~)h1

- 17)k~l

bE(1

- V)k,

aq(1

- E)kc = b[~)l(k,,+ k z + k,,) - (k2, + V%z)1

= a[E(k,,

(125)

+ kz + k,,) - (k,, + e2k;)]

Equations (124)-(127) a r e the set necessary to solve for a, b, q in t e rm s of the eight kinetic rate constants.

(126) (127) E,

and

By letting k z = kE = k z = &, = 0, it is easy to demonstrate that Eq. (123) reduces to the simple copolymer equation, Eq. (108). Equation (123) can also be shown to reduce to Lowry’s Case I which, in t er m of O’Driscoll’s model [25], occurs when k z = k z = k z = 0. The

338

SAWADA

next three equations give the equivalence between O’Driscoll’s notation and Lowry’s: k,, /kZI = [Mzlrz /[MI1

(128)

kzz/b

(129)

= [MZIP

(130)

q=(Y

Using the notation of the right-hand side of Eqs. (128)-(130) and Eq. (127), we find azrz/([MIIP)

- d l + rz[MzI/[MII + rz/([MIIP)I

+ rz[MzI/[M,I = 0

(131) Upon rearrangement,

- a(1 + [MZIP + [M,IP/rJ

+ [MJP = 0

(132)

which is identical to Eq. (4) in Lowry’s paper [35]. Furthermore, Eq. (123) becomes d[M,I/d[MJ = ~ ( +1rJ/[r2

- (arZ/P)I

(133)

which is identical t o Eq. (12) in Lowry’s paper [35]. Equations (125) and (126) also simplify to

and

The Aentity of results w-th respec- -3 composition in O’Driscoll’s treatment and Lowry’s is consistent with expectation. Further, the probability approach has yielded new sequence distribution equations. The mathematical treatment of the penultimate effect in a binary copolymerization involves the use of the sixteen r a t e constants, for there a r e eight reversible reactions to consider: wMIMf +M ,

-

-M,M:+MZ

- 1;,

kl

k3

k,

,W

M,M,M:

-M,M,M,*

THERMODYNAMICS OF POLYMERIZATION. V.

339

In a manner similar to that used in deriving Eq. (123), the copolymer composition equation for this model, often referred to a s the triad model, is given by [25]

In this equation, four transient probabilities for antipenultimate units a r e given by

The following relations must be considered:

340

SAWADA

The foregoing equations can be shown to reduce to Lowry's Case I1 i f we assume the following: k, = k4 = k, = k, = k,, = k12,=k14 = 0, k, = k9, k, = kll, k, = k12, and k, = k15. Equation (136) is the most general copolymer composition equation which has ever been de-

rived [25]. Computor simulation of typical copolymerizations with depropa gation by means of a Monte Carlo technique were made on hypothetical cases [ 271. The composition-feed relationships for copolymerization of the monomer pairs having those hypothetical values of thermodynamic and kinetic quantities shown in Table 2 were simulated a t various temperatures. It should be noted that the kinetic quantity is dependent on a kinetic mechanism whereas the thermodynamic quantity is independent of it. Cases A to D correspond to copolymerizations with monomer pairs having the same ceiling temperature. In Cases A-1 and A-2 we assume that all the elementary steps have the same thermodyTable 2 Typical Cases Assumed for Computation [ 271

-AH;, , kcal/mole -AHy2, kcal/mole -AHil, kcal/mole -AH;, ,kcal/mole -AS", cal/(deg)(mole) 11 = 12

k,l/kl2

= k22/klI

Sl = k,,/k21

A-1

A-2

B

C

D

E-1

E-2

E-3

15.0 15.0 15.0 15.0

15.0 15.0 15.0 15.0

15.0 12.0 18.0

15.0 20.0 20.0

28.0 0.1 10.0 1

15.0 28.0 1 1 1

15.0 28.0 1 1 1

20.0 15.0 15.0 10.0 28.0

20.0 20.0

28.0 1 1 1

20.0 10.0 10.0 20.0 28.0 1 1 1

1 1 1

20.0 10.0 20.0 10.0 28.0 1 1 1

10.0 10.0 28.0 1 1 1

THERMODYNAMICS 0 F POLYMER I ZATl ON. V.

34 1

namic quantities and that the values of rl and rz a r e different. The results for Case A-1 in Fig. 21 show the polymer composition is identical to that of the feed at 100 and 200°C. It is impossible to obtain high polymer at 300°C. L

-: 1.0 0

( I

3! 0.8 .-C

2

0.6

C

0.4

z

*-

u

0.2

E

0

0

0.2

mole fraction

0.4

M2

0.6

0.8

1

in monomer mixture

Fig. 21. Results of Monte-Carlo calculations of copolymer composition curves for Cases A-1 and A-2 of Table 2: (1) A-2, 100°C; (2) A-2,2OO0C;(3) A-2, 250°C; and (4) ( 0 ) A-1,lOO"C; ( 0 ) A-l,20OoC [ 271.

The curve at 100°C in Fig. 21 for Case A-2 is similar to that obtained by plotting Eq. (108) with the same rl and r, because very few depropagation reactions occurred. A s the temperature rises, it becomes similar to Case A-1, which shows that thermodynamic control dominates the kinetic control. Case B, shown in Fig. 22, is different from Case A-1 only in the values of -AH;z and -AH&; the sum of terms is 30 kcal/mole in both cases. In Fig. 22 this did not make Case B different from Case A-1 at the lower temperature (100°C). A t the higher temperature (200°C) the curve appears similar to a curve with rl < 1.0 and rz > 1.0 in Eq. (108). This again shows that thermodynamic factors controlled the reaction at the higher temperature. When both of -AHlz and -AHzl are less than the heat of homopolymerization (Case D, Fig. 221, the curve approaches one which corresponds to the case where both rl and r, a r e larger than unity in Eq. (108), and thus the copolymer formed under this condition is similar to a block copolymer. When both -AH,, and -AH,, are larger than the heat of homopolymerization (Case C, Fig. 22), the curves resembles one obeying Eq. (108) with rl and r, having values

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342

.-C

EN o.6 C

U

0.4

2 c

0 mole fraction

M,

i n monomer mixture

Fig. 22. Results of Monte-Carlo calculations of copolymer composition curves for Cases B, C, and D of Table 2: (1) D, 300°C; (2) C, 300°C; (3) B, 100°C; C + D, 200°C; and (4) B, 200°C [ 271.

smaller than unity. The deviations from Eq. (108) at the higher temperature are due to the occurrence of the depropagation reaction and are apparent in Figs. 21 and 22. The general trend of the behavior of the composition-feed curve in Cases E-1 to E-3 i s that the shape of the curves become flatter with increasing temperature (Fig. 23).

,5 0.6

zN

s

0.4 .c

-u

2

0.2

0

E

0 0

0.2

mole fraction

0.4

0.6

Mzin

monomer mixture

0.8

1.(

Fig. 23. Results of Monte-Carlo calculations of copolymer composition for Case E-1: (1) 100°C; (2) 200°C; and (3) 300°C [27].

THE RMODY NAMl CS 0 F POLYMER I ZATION. V.

343

O'Driscoll and his associates [49] reported the effect of reaction temperature on the copolymer composition curve in the radical copolymerization of a-methylstyrene and methyl methacrylate, which have relatively low ceiling temperatures. The diad model was adopted a s a simple explanation. The relative values of the eight rate constants of the diad model a s estimated by the seven constants are

The values of K, and & are available from the thermodynamic data of homopolymerization by using the relation -RT In K, = AG; = AH:

- TASY

(149)

Since no data a r e available on K , and &, Eq. (150) has been assumed throughout the calculations: AG, - AG: O

=

1/2(AG," + AG:)

(150)

The compositions of the polymer is expressed by Eq. (151) a s parameters 17 and E : =

(1

-M

1

-4

(151)

The values of q and E a r e determined by solving Eqs. (124)-(127) simultaneously. Since these equations a r e nonlinear, we can solve them by a numerical method. The solid lines in Fig. 24 show the theoretical prediction obtained by this procedure. The theoretical curves fit well to the experimental points. Since the temperature range and its effect a r e quite large, the diad model appears to be quite adequate for accurately describing the reversible copolymerization of this morlomer pair.

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344

0

0.2

0.4

0.6

0.8

1.0

mole fraction M2 in monomer mixture

Fig. 24. Dependence of temperature and monomer feed composition on the copolymer composition in the radical copolymerization of MMA(M, -a - MS (M, ): ( X ) 60°C;( 0 ) 114°C;( 0 ) 147°C;and (-) theoretical prediction [ 491.

III. MULTICOMPONENT COPOLYMERIZATION A. Heat of Terpolymerization Terpolymerization, the simultaneous polymerization of three monomers, has become increasingly important from a technological point of view. Since no data are available for heats of terpolymerization systems and such data are more difficult to obtain than data of binary systems, it is desirable to develop a method for the prediction of heat of terpolymerization systems from binary data [50]. The quantitative treatment of terpolymerization is quite complex since nine propagation reactions are involved. Reaction Heat of reaction M: + M, M: + M, M: + M3 M: + M, M,* + M, M,* + M3 M: + M I M: + M, M: + M3

H23 H31

H32

H33

THERMODYNAMICS OF POLYMERIZATION. V.

345

The expression for the molar heat of reaction of each of the propagation reactions is shown above. An expression for the heat of terpolymerization a t low conversion can be obtained by a treatment similar to that used in binary copolymerization. The molar heat of terpolymerization is given by AH, = L N' H , ,

+A NH z z+ N X H 3 3+ A N H l z + SN H 2 ,

NO

NO

NO

N

N

NO

NO

NO

NO

N

+ A H l 3 + A H 3 1 + S H Z 3+ S H 3 2 NO

NO

where the number of nearest-neighbor pairs of i and j is N i , , and the total number of pairs in the copolymer is No. The number of M, monomer units in the copolymer is N S , , and thus the total number of M,-M, pairs is Nll = (NdC1)Pll

(153)

where X, is the mole fraction of the monomer unit M,, and P,, is the conditional probability that the M, monomer unit will follow the M, monomer unit in the presence of the M, monomer unit. Similarly, we have

where Xi is the mole fraction of M i monomer unit, and Pi, is the conditional probability that the Mi monomer unit selected at ran-

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346

dom will be followed by the M, monomer unit. The value of Pi, may be calculated from rate constant k by kij[M:I[MjI = kij[M:l[Mjl + kii[M:l[Mil

+ kIk[M:IIMkI

(162)

where i, j, and k have values of 1, 2, o r 3. Combining Eqs. (153)(161) with Eq. (152), we obtain AH, = XlP11H1, + X1P13H13

+ X2PZ2H22 + X R P 3 3 H 3 3 + X3P31H31

+

+ X2P23H23

XlPl2Hl2 iX*P2,%1

+ X3P32H32

(163)

Ham [51] has treated the problem of terpolymer composition by making the simple assumption that the probability of producing a certain sequence of monomers is the same as producing the exact reverse sequence. This relationship of product probabilities is p12p23p31

=

p13p32p21

(164)

A similar concept may be expressed as [52]

X,/Xj =

Pjl/Pl,

(165)

Equations (164) and (165) have undergone some debate in the literature [53-561, but they a r e anecessary consequence of the Q-e scheme and can be derived from it. Let us now evaluate the heat of terpolymerization for monomers which do obey the Q-e scheme. Since we a r e dealing with monomers which obey the Q-e scheme, we find XIPI2

= XZPZl

(166)

x1p13

= x3p31

(167)

'2'23

= '3'32

(168)

Substitution of Eqs. (166), (167), and (168) into Eq. (163) yields

THERMODYNAMICS OF POLYMERIZATION. V.

347

In the present case for an infinite molecular weight terpolymer, P1, + P,, P,,

+ P13= 1

+ P,, + P,,

=1

P3, + P3, + P3, = 1

Thus Eq. (169) can be expressed in t e r m s of the probabilities of cross-propagation:

Upon substitution of Eqs. (166)-(168) into Eq. (173), we find

We define QIj

QIj

=

as

(H,j

+ HjJ

-

+ Hjj)

(175)

Consequently, the heat of terpolymerization is given by =H l A

+

HZZX,

+ xlp12Q2,Z

+

H33X3

+ x1p13Q13

+ x2p23Q23

(176)

The heat of terpolymerization for the ternary system acrylonitrile (AN = 1)-styrene (ST = 2)-methyl methacrylate (MMA = 3) calculated from Eq. (176) is presented in Fig. 25. (Note that any

SAWADA

348

gain in AH on the part of a system is considered to be positive and any loss negative, but this is not considered here for simplicity.)

a 0

0.2

0.4

0.6

0.8

1.0

mole fraction AN Fig. 25. Heat of terpolymerization for the acrylonitrile-methyl methacrylate-styrene system: (mole MMA)/( mole ST)= 1 [ 501.

Values in Fig. 25 a r e obtained by using [57]

rlz = 0.18,

rZ1= 1.35,

HI, = 16.74 kcal/mole,

r13= 0.04,

rZ3= 0.46

H,, = 13.42 kcal/mole,

H,, = 16.29 kcal/mole

and a,, are shown in Table 3. Ham [58] has found that Eq. (164)can be applied to this ternary system. Thus it is possible to predict the heat of terpolymerization in this system from binary data. P1,,and P,, In the case of binary copolymerization, X,, aI3,aZ3, will vanish. In this case Eq. (176)will reduce to Eq. (177), which is identical to Eq. (13), with a,, being identified with 51. AH = HllXl

+

Hz&z

+ PlZ~12xl

(177)

349

THERMODYNAMICS OF POLYMERIZATION. V.

Table 3 Parameters Used in Prediction of Heat of Terpolymerization (kcal/mole) Binary system AN-MMA AN-ST MMA-ST

Hij + Hji

Hii + Hjj

Qi j

28.4a 34.67a 32.74b

30.4a 32.76a 29.74b

-2.00 +1.91 +3.00

aH. Miyama and S. Fujimoto, J. Polym. Sci., 54,s33 (1961);M.Suzuki, H. Miyama, and S. Fujimoto, Bull. Chem. SOC., Japan, 35,57,60(1962). bM. Suzuki, H. Miyama, and S. Fujimoto, J. Polym. Sci., 31,212(1958).

Using the same procedure as for the heat of terpolymerization in Eq. (176),the following equations are found in the case of multicomponent copolymerization containing n monomers which obey the Q-e scheme.

+ X,P,,a,, + X3P3,Sz,, + *

* *

+ X,P,,a,, + *

'

*

Equation (178) gives the heat of multicomponent copolymerization with n monomers, and this equation is only applicable to multicomponent systems for which copolymerization follows the Q-e scheme. Similar expressions hold for entropy and free energy changes with n monomers which obey the Q-e scheme.

B. General Theory of Multicomponent Copolymerization Let us consider a terpolymerization system which consists of the components M,, M,, and M,. The reaction which occur during equilibrium terpolymerization process a r e shown in Eqs. (179)-(181).

SAWADA

350

where AGIj is the characteristic molar free energy of formation for the nearest-neighbor pairs of i and j. From the quasi-chemical approximation [59] based on nearestneighbor interactions, the following three coupled equations are obtained.

--%3

- eXp(- AG,,/RT)

N22N33

~N321

- exp(- AG,,/RT)

N33N11

Since the formation o r disappearance of an i-j bond is concomitant with the like process for a j-i bond, it is reasonable to call N,, the number of the nearest-neighbor pairs of i and j along the chain, and to express it as N,, = N,, and similarly AF,, = AF,,. If Eqs. (182), (183), and (184) a r e multiplied number by number, it can be shown that

Values of N,, a r e given by Eqs. (153)-(156), (159), and (160). Substitution of these equations into Eq. (185) yields pC2p23p31 pllp22p33

= exp[-(AG,,

+ AG23 + AG3,)/2RT1

THE RMODY NAMl CS OF POLYMERIZATION. V.

35 1

where the enthalpy changes of formation for ii, j j , i j , and j i bonds a r e denoted by Htt, H j j , Hi,, and Hji, respectively, and the entropy changes of such bonds by S II,S j J , SiJ, and SJI. Equation (186) may also be extended to a general treatment of multicomponent copolymerization. The general form of multicomponent copolymerization with n monomers is obtained as follows:

i. 1

i. 1

O’Driscoll et al. [60] suggested that the reversible, multicomponent copolymerization can be solved by the probability approach. The resulting equations were compared with Monte-Carlo simulations and were also tested experimentally using the terpolymerizing system acrylonitrile-methyl methacrylate- a-methylstyrene over a wide temperature and concentration range, The effect of reaction temperature and total monomer feed concentration on the terpolymer composition curve in the radical terpolymerization of acrylonitrile, methyl methacrylate and a-methylstyrene was studied. Acknowledgment The author wishes to thank Prof. K. F. O’Driscoll for many valuable criticisms of this manuscript and for providing reprints of papers and an advance copy of h i s manuscript.

352

SAWADA

References H. Sawada, J. Polym. Sci., A , 2,3095(1964). T. Alfrey and C. Lewis, Ibid., 4,221 (1949). R. Orr, Polymer, 2,74 (1961). M.G.Evans, J. Gergely, and E. C. Seaman, J. Polym. Sci., 3,866(1948). H.Sawada, Ibid., A, 3,2483 (1965). H. N. V.Temperley, J. Res. Nut. Bur. Stand., 56,55(1956). A. M.North and D. Richardson, Polymer, 5,227 (1964). M.H.Theil, Macromolecules, 2,137 (1969). R. Corradini and C. Tosi, Eur. Polym. J., 4,227 (1968). T. Alfrey and A. V. Tobolsky, J. Polym. Sci.,38,133(1959). M. Izu and K. F. O’Driscoll, Polym. J., 1,27 (1970). J. C. Bevington, Radical Polymerization, Academic, New York, 1961,p. 13. M.G. Evans, in Fibres from Synthetic Polymers (R. Hill, ed.), Elsevier, New York, 1953,Chap. 3. R.M. Joshi, Makromol. Chem., 66,114(1963). H. Sawada, J. Polym. Sci.,B, 1,659 (1963). F . S.Dainton and K. J. Ivin, Trans. Faraday Soc., 46,346(1950). H. Sawada, J. Polym. Sci., A-1, 5, 1383 (1967). J. C. Bevington, Radical Polymerization, Academic, New York, 1961,p. 166. R. E. Cook, K. J. Ivin, and J. H. C’Donnell, Trans, Faraday Soc., 61,1887( (1965). A. M. North and D. Richardson, Polymer, 6,333(1965). T. Tsuda and Y. Yamashita, Makromol. Chem., 86,304(1965). H. Eyring, D. Henderson, B. J. Stover, and E. M. Eyring, Statistical Mechanics and Dynamics, Wiley, New York, 1964,p. 394. P. E. Harvey and J. Leonard, Macromolecules, 5,698 (1972). P. J. Flory, Principles o f Polymer Chemistry, Cornell Univ. Press, Ithaca, New York, 1953,p. 549. J. Howell, M.Izu, and K. F. O’Driscoll, J. Polym. Sci.,A-1, 8,699 (1970). A. Silberberg and R. Simha, Biopolymers, 6,479(1968). M.Izu and K. F. O’Driscoll, J. Polym. Sci.,A-1, 8,1675(1970). R. Simha and J. M. Zimmerman, J. Theoret Biol., 2,87 (1962). M.H. Theil, J. Polym. Sci.,C, 31,1 (1970). A. V. Tobolsky and G. D. T. Owen, Ibid., 59,329 (1962). J. Schenk, Physica, 23,325 (1957). B. K.Kang and K. F. O’Driscoll, To Be Published. G. E. Ham, J. Polym. Sci.,45,169,177,183(1960). E. Mertz, T. Alfrey, and G. Goldfinger, Ibid., 1,75(1946). G. G. Lowry, Ibid., 42,463(1960). J. E. Hazel1 and K. J. Ivin, Trans. Faraday Soc., 58,176 (1961). Y. Yamashita, H. Kasahara, K. Suyama, and M. Okada, Makromol. Chem., 117,242(1968). P. Wittmer, in Multi Component Polymer Systems (N. A. J. Platzer, ed.), American Chemical Society, Washington, D. C., 1970,p. 140. K. J. Ivin, Pure Appl. Chem., 4,271 (1962).

THERMODYNAMICS OF POLYMERIZATION. V.

353

K. F. O’Driscoll and F. P. Gasparro, J. Macromol. Sci.-Chem., A1(4), 653 (1967). K. J. Ivin and R. H. Spensley, Ibid., A1(4), 653 (1967). Y. Inaki, S. Nozakura, and S. Murahashi, Ibid., A6(2), 313 (1972). H. K. Johnston and A. Rudin, Macromolecules, 4,661, (1971); J. Paint Technol., 42,435 (197O);Ibid., 42,429 (1970). A. Rudin, S. S. M. Chiang, H. K. Johnston, and P. D. Paulin, Can. J. Chem., 50,1757, (1972). B. K. Kang and K. F. O’Driscoll, J. Macromol. Sci.-Chern., A7(6), 1197 (1973). J. P. Fischer, Macromol. Chem., 155, 211 (1972). S. Penczek, Bull. Acad. Pol. Sci., Ser. Sci. Chim., 20,437 (1972). A. A. Durgaryan, Vysokomol. Soedin., 8,790 (1966). M. Izu and K. F. O’Driscoll, J. Polym. Sci., A-I, 8,1687 (1970). H. Sawada, J. Macromol. Sci.-Chem., A7(4), 905 (1973). G. E. Ham, J. Polym. Sci., A , 2,2735 (1964). G. E. Ham, J. Macromol. Chem., A1(2), 403 (1966). F. R. Mayo, J. Polym. Sci., A, 2,4207 (1964). K. F. O’Driscoll, Ibid., B, 3,305 (1965). P. W. Tidwell and G. A. Mortirner, Zbid.. B, 4,527 (1966). K. F. O’Driscoll, P. W. Tidwell, and G. A. Mortirner, Ibid., B, 5,575 (1967). H. Miyama and S. Fujimoto, Ibid., 54, s33 (1961); M. Suzuki, H. Miyama, and S. Fujirnoto, Bull. Chem. SOC. Japan, 35, 57, 60 (1962). G. E. Ham, in Copolymerization (G. E. Ham, ed.), Wiley, New York, 1964, p. 42. E. A. Guggenheirn, €’roc. Roy SOC.,A, 183,213 (1944); Trans. Faraday Soc., 44,1007 (1948). B. K. Kang, K. F. O’Driscoll and J. A. Howell, J. Polym. Sci., Polym., Chem. Ed., 10,2349 (1972).

J. MACROMOL SCI.-REVS. MACROMOL. CHEM., C11(2), 257-297 (1974)

Thermodynamics of Polymerization.VI. Thermodynamics of Copplymerization. Part 2 HIDEO SAWADA

Filter Laboratory DaicelLtd. Teppo-cho, Sakai, Osaka, Japan

I. RADICAL COPOLYMERIZATION A. Heat of Copolymerization B. Ceiling Temperature C. Q-e Scheme D. Substituent Effect E. Effect of Polymerization Temperature F. Effect of Solvent II.

III.

258 258 263 267 269 272 276

IONIC COPOLYMERIZATION A. Energetic Characteristics B. Reactivity C. Effect of Polymerization Temperature

278 278 283 284

OTHER COPOLYMERIZATIONS A. Ring-Opening Copolymerization B. Miscellaneous Copolymerizations

286 286 288

ACKNOWLEDGMENT

294

REFERENCES

294

257 Copyright O 1974 by Marcel Dekker, Inc. All Rights Reserved. Neither Ihis work nor any part may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, microfilming, and recording, or by any information storage and retrieval system, without permission in writing from the publisher.

258

H. SAWADA

The discussion in Part 1 [l] is largely concerned with thermodynamics, and is quite general without any specification as to whether copolymerization occurs by radical or ionic propagation. Consider now some of the specific types of copolymerization. Thus, much discussion will concern the free energy of the transition state which is closely related to the mechanism and the reactivity of copolymerization. I. RADICAL COPOLYMERIZATION A. Heat of Copolymerization As described in the previous paper, the total heat of copolymerization is given by [2] 1 - [1 - 4X2(1 - X 2 )(l zr : + X,HU + X2H22

—

" 221

(1)

where the X's a r e the mole fractions of each monomer in copolymer, the r ' s are the reactivity ratios of each monomer, and the suffixes 1 and 2 denote the values of each monomer. If the values of the X's and r ' s a r e known, measurements of the heat of copolymerization and the heat of homopolymerization of each monomer give the value of H12 + H 21 . However, H12 and H21, the heats of reaction for the alternation reactions, are not experimentally measurable. From the fact that the values of H12 + H21 are almost constant (Table 1), the application of Eq. (1) to the methyl methacrylate-styrene system seems reasonable [3]. As shown in Fig. 1, the values of AH in Table 1 agree with the full line expressing Eq. (1). In the case of the styrene-diethyl fumarate system [4], the r e markable difference between r t and r 2 greatly affects the calculation of Xj and X2, and this makes the factor 1 - [1 - 4X2(1 - X 2 )(l -

rtr2)>^

2(1 - r t r 2 ) in Eq. (1) zero. Thus the following equation can be obtained by a s suming negligible heats of cross-propagation: AH = XXHU + X2H22

(2)

THERMODYNAMICS OF POLYMERIZATION. VI

259

• Table 1 The Values of AH and H I2 + H2I in the Methyl Methacrylate-Styrene System at 24°Ca

x,

AH (kcal/mole)

HI2 + H2I (kcal/mole)

1.00

13.19

-

0.79

14.25

32.09

0.65

15.35

33.25

0.57

15.60

32.84

0.50

15.57

31.74

0.46

15.92

32.39

0.35

16.39

32.89

0.21

16.79

34.01

0

16.55

av 32.74

a

Reprinted from Ref. 3 by permission of John Wiley & Sons, Inc.

Figure 2 shows the comparison between AH values derived from Eq. (2) and values obtained by direct thermochemical experiments [4]. It is observed that the experimental points follow the curve derived from Eq. (2). It is seen that Eq. (1) must be used in the acrylonitrile-styrene [5] and methyl methacrylate-styrene [3] systems in which the difference between reactivity ratios of each monomer is small. On the other hand, in acrylonitrile-methyl methacrylate [5, 6], vinyl acetate-methyl methacrylate [3], styrene-vinyl acetate [3], vinyl acetate-acrylonitrile [5], and styrene-diethyl fumarate systems [4], in which this difference is remarkable, it is not necessary to use Eq. (1), but it is sufficient to use Eq. (2). Tong and Kenyon [7] have derived Eq. (3), which is the special case of Eq. (1) where one of the monomers does not homopolymerize by a radical mechanism: AH = 2X(AHCOP - H u ) + H u

(3)

260

H. SAWADA

0-8

0-6

0-4

02

X,(MMA) Fig. 1. Comparison between heat of copolymerization in methyl methacrylate-styrene system calculated from Eq. (1) and experimental values: (o) experimental value; (—) calculated value. Data from Ref. 3.

10

0-8

0-6

0-4

0-2

0

Xt (styrene) Fig. 2. Comparison between heat of copolymerization in styrene-diethyl fumarate system calculated from Eq. (2) and experimental values: (o) experimental value; (—) calculated value. Reprinted by permission of John Wiley & Sons from Ref. 4.

THERMODYNAMICS OF POLYMERIZATION. VI

261

where —AH = observed heat of polymerization per mole of monomer unit, X = mole fraction of 1,2-disubstituted monomer which does not homopolymerize in the copolymer, — H u = heat of homopolymerization per mole of vinyl monomer, and — AHC0P = heat of copolymerization per mole of mixed monomer in 1:1 ratio. Equation (3) requires AH to be linear with respect to X and equal to AHC0P at X = 0.5. The polymerization reaction can be interpreted as proceeding in two stages: The first stage is mainly the copolymerization of two components in a 1:1 ratio, and the second stage is homopolymerization of excess vinyl monomer. A summary of the heats of 1:1 copolymerization systems where one of the monomers is considered to be difficult to homopolymerize is given in Table 2. Joshi [8] has made an experimental study of heats of alternating copolymerization systems. The heats of cross-propagation in these copolymerizations were found to be high, exceeding in many systems the heat of homopolymerization of either of the two monomers. Ivin et al. [9] studied heats of copolymerization of solid olefin polysulfones from liquid SO2 and liquid olefins. These reactions are radical-catalyzed and will only proceed if the temperature is below a ceiling temperature Tc given by T c = AH2/AS2, where AH2 and AS2 are the heat and entropy changes, respectively, which accompany the formation of polymer from a given reaction mixture. The value of AH2 will differ from that of AHX to a small extent depending on 1) heat of mixing of the monomers AH3 and 2) the heat of solution of the polymer AH4. The relation between these quantities is shown in the following cycle, where M denotes olefin and N denotes SO2, present in excess, and the AH values are per mole of olefin [9]: N(liquid) + M(liquid) AH, I

AH3 >~ mixture (liquid) I

AH2

polymer (+ N + M) £_ polymer(solution) (solid) AH2 = AH2 + AH3 - AH4

(4)

The values of AH2, AH3, and AH4 will depend on the initial relative amounts of M:N and on the percentage conversion to polymer. The values of AH3 have been determined for some olefins using an initial ratio M:N = 1:10, which is the same composition used for the determination of AH2 by direct calorimetry [10]. Precise values of

262

H. SAWADA

Table 2 Heats of Copolymerization Monomers Sulfur dioxide: -Cyclohexene -Propylene -Butene-1 -cis-Butene-2 -trans-Butene-2 -Isobutene -Hexene-1 -Hexadecene -Cyclopentene Maleic anhydride: -Vinyl acetate -Isopropenyl acetate •Styrene -a-Methylstyrene -Allyl chloride -Vinyl n-butyl ether Diethyl fumarate: -Vinyl acetate -Styrene Diethyl malate: -Vinyl acetate Maleimide: -Styrene -a-Methylstyrene Fumaryl chloride: -a-Methylstyrene -Styrene a

Standard statesa

-AH b (kcal/mole)

ss sc ss lc ss Ic ss Is lc lc ss ss ss

9 10.1 10.6 10.0 10.07 8.93 9.35 8.43 7.41 9.4 10.4 10.0 9.6 10.8

ss ss ss ss ss ss

20.2 17.8 19.3 17.3 17.7 21.5

7 7 8 8 8 8

ss lc

18.6 16.4

7 6

ss

20.0

7

ss ss

20.9 17.2

8 8

ss ss

17.1 19.1

8 8

£S

Refs. c

10 10 9 10 9 10 9 9 15 10 10 d

10

Monomer and polymer states: 1, liquid; s, solution; c, condensed AH represents the heat liberated by the reaction of 0.5 mole of each monomer to form one mole of mixed monomer units. cj. E. Hazell and K. J. Ivin, Trans. Faraday Soc, 58, 342 (1962). d F. S. Dainton, K. J. Ivin and D. R. Sheard, Trans. Faraday Soc, 52, 414 (1956). b

THERMODYNAMICS OF POLYMERIZATION. VI

263

AHj were derived from the heats of combustion of the polysulfones. Thus from Eq. (4) we can find AH4 in these cases. It was impossible to measure directly the heat of copolymerization of SO2 with isobutene by the fusion calorimeter at 26.9°C because the ceiling temperature for this polymerization is at about 5°C. However, isobutene polysulfone is insoluble in the reaction medium so that we may set AH4 equal to zero and derive AH2. The values are summarized in Table 3 [9]. Table 3 Heats in kcal per Base-Mole Polymer or per Mole Olefina Olefin

-AH t

-AH 2

-AH 3

-AH 4

Butene-1

20.06 ± 0.52

21.2 ± 0.1

1.43

0.3 +0.6 b

Isobutene

14.82 + 0.40

15.65 ± 0.4b

0.82

0

cis-Butene-2

17.85 ± 0.36

20.15 ± 0.1

0.64

-1.65 ± 0.5 b

trans-Butene-2

16.86 ± 0.35

18.7 +0.1

0.66

-1.2 + 0.5b

a

Reprinted from Ref. 9 by permission of the Faraday Division of the Chemical Society. Values underlined are derived by means of Eq. (4).

b

Correlations between the heats of copolymerization of SO2 with olefins and other heats of addition to olefins are evident. In Table 4 are listed the heats of hydrogenation of gaseous and liquid olefins, heats of bromination, and heats of self-polymerization [10]. It can be seen that cyclopentene has the lowest heat of hydrogenation of any of the listed olefins but the highest heat of copolymerization with SO2. The reason for this high heat of copolymerization might be that the presence of the five-membered rings in some way enables the cyclopentene polysulfone to take up energetically more favorable conformations which are denied to other polysulfones [ll]. B. Ceiling Temperature The ceiling temperature effect was first discovered in the formation of olefin polysulfones [12] through its thermodynamic explanation was not realized for many years [13]. In a comprehensive study of ceiling temperature of olefin polysulfones, Cook, Dainton, and Ivin [14] attempted to determine the ceiling temperatures T c (at which the free energy change for the reaction is zero) for the formation of polysulfones from a wide

264

H. SAWADA

Table 4 Heat of Addition to Olefinsa Selfpolymerization (calculated per one mole of olefin) (g) (1)

+H2 at 25°C

+S0 2 at 26.9°C (1)

(g)

(1)

Propylene

20.2

29.72

29.8

20.6

20.05

Butene-1

21.2

30.12

30.29

20.6

20.0

Hexene-1

20.7

30.00

30.2

20.5

19.8

Hexadecene-1

19.9

30.00

-

-

trans-Butene-2

18.7

27,48

27.76

18.1

17.0

cis-Butene-2

20.2

28.48

28.91

19.1

17.9

Cyclopentene

21.65

26.33

26.44

-

Olefin

a Reprinted from Ref. 10 by permission of the Faraday Division of the Chemical Society. Heats of reaction expressed in kilocalories per mole.

variety of olefins and allyl compounds under standard concentration conditions ([M][s] = 27 mole 2 /liter 2 , M = unsaturated compound, S = SO 2 ). The r e s u l t s a r e discussed in t e r m s of s t r u c t u r a l effects on the heat and entropy changes and on the monomer activity coefficients and polymer solubilities. Some of ceiling t e m p e r a t u r e s for polysulfone formation a r e shown in Table 5 [14]. When the poly(olefin sulfone) is insoluble in the reaction mixture, the expected relation between T c and the activities of olefin and SO2 i s readily deduced from the following cycle [15]: AG,,

C4H8(liquid) + SO2(liquid)

(l/n)[C 4 H 8 SO 2 ] n (solid) /

mixture (liquid)

AG = 0 atT c

(filtration)

mixture(liquid) + polymer(solid)

The condition for equilibrium at the ceiling temperature is AGIC = AGm + AG3 = RT C In a m + RT C In a s

THERMODYNAMICS OF POLYMERIZATION. VI

265

Table 5 Ceiling Temperatures of Polysulfone Formation3 Olefin Straight chain 1-olefins Ethylene Propylene 1-Butene 1-Pentene 1-Hexene 1-Hexadecene Branched 1-olefins Isobutene 2-Methyl-l-pentene 3-Methyl-l-pentene 3-Methyl-l-butene 2-OIefins and cycloolefins trans-2-Butene cis-2-Butene Cyclopentene Cyclohexene Allyl compounds Allyl alcohol Allyl ethyl ether Allyl formate Allyl acetate Allylacetic acid a

Tcb (°C) >135 90 64 63

60 69 4.5 -34 36 14

38

46 102.5 24 76 68 45

45 66±4

Reprinted from Ref. 14 by permission of John Wiley & Sons, Inc. Values are corrected to [M] [S] = 27 mole2/liter2.

b

where am and a3 represent the activities of olefin and SO2, respectively. At equilibrium AGlc = AH1C - TcASlc and therefore AH1C - TcASlc = RTC In amas where AHlc and ASIC are the enthalpy and entropy changes for the conversion of 1 mole of pure liquid olefin and 1 mole of pure liquid SO2 to 1 base-mole of insoluble 1:1 copolymer. It can then be shown that the ceiling temperature T c for the copolymer is given by [15]

266

H. SAWADA

AHle ASlc + R l n a m a 3

=

AH lc AS l c -+Rlnf

(5)

where fm and fs are the mole fraction activity coefficients of olefins and SO2, respectively, and xm and x3 are the mole fractions of two monomers in the monomer mixture, respectively. Ceiling temperatures have been measured over the complete composition range for isobutene [15] and for 3-methyl butene-1 [16]. Both of these olefins are completely miscible with SO2 in all proportions, and poly(olefin sulfones) are insoluble in all reaction mixtures. Figure 3 shows the results obtained by Brady and O'Donnell [16], in which plots of ln(frofsxmx3) against 1/T consist of two arms in each case, one arm corresponding to SO2-rich mixtures and the other to olefin-rich mixtures. It is probably the swelling of the polymer in SO2-rich mixtures that results in the formation of crystalline polymer (for a further discussion, see Ref. 16).

Fig. 3. Ceiling temperature equation plots for the copolymerization of 3-methyl butene-1 (A) and isobutene (B) with sulfur dioxide showing the discrepancies between sulfur dioxiderich mixtures (lower arm) and olefin-rich mixtures in each case. Reprinted by permission of Microforms,International Marketing Corp. from Ref. 16.

Irradiation of mixtures of butene-1 and SO2 in the gas phase can produce poly(butene-l-sulfone). The rapid decrease in polymerization rate with increasing temperature is good evidence for a ceiling

THERMODYNAMICS OF POLYMERIZATION. VI

267

temperature effect. It was later recognized by Brown and O'Donnell [17] that the ceiling temperature is lowered by about 60cC compared with the liquid phase. C. Q-e Scheme Alfrey and Price [18, 19] proposed that the rate constant for a radical-monomer reaction, e.g., for the reaction of M^ radical with M2 monomer, is given by k12 = P,Q2 exp(—e,e2)

(6)

where Px is a measure of the general reactivity of the growing radical, Q2 is a measure of the general reactivity of the monomer, and e1 and e2 are measures of the polarities of the radical and monomer, respectively. The P and Q terms primarily define the resonance effects in the radical and monomer. If the same e values applies to both a monomer and its radical (that is, et defines the polarities of Mt and M^, while e2 defines the polarities of M2 and M2-), then according to Eq. (6), rj and r2 for the system are found to be

rx =-§iexP{-e1(e1 - e2)}

(7)

r 2 =-^exp{-e 2 (e 2 - e i )}

(8)

Schwan and Price [20] have developed a modified copolymerization equation by utilizing the parameters q and e in place of Q and e. Their equations for r1} r2, and r x r 2 are: rx = exp{-( qi - q2)/RT}exp{-7.23 x 10aoe1(e1 - e2)/RT}

(9)

x 102oe2(e2 - eJ/RT}

(10)

r 2 = exp{-(q2 -

qi )/RT}exp{-7.23

r x r 2 = exp{-7.23 x I020(e1 - e2)2/RT>

(11)

The advantage of this modification is that the parameter q and e are temperature independent (whereas Qand e depend upon temperature), and consequently the above equations will hold for any temperature range. Evaluation of the activation energy for the reactions of radical additions to vinyl compounds was studied as an extension of the theory employing two Morse functions for the initial and the final

268

H. SAWADA

systems [21]. The Evans-Polanyi and Semenov rules, the Hammetttype rule, and the Q-e scheme in the radical copolymerization were derived by using this approach and have been discussed in detail [21]. As has already been discussed [22], the activation energy of the propagation reaction can be expressed by E = Eo - k(Rf - Ra - Rm)

(12)

where Eo is the activation energy of propagation reaction involving polyethylene radical and monomeric ethylene, and the factor k has a value less than unity, and Rf, Ra, and Rm are the respective resonance energies of the initial radical, the monomer, and the radical formed after addition of the monomer. Thus, E t (corresponding to k n ) = Eo - k(Rtj - Ra - Rm} and, similarly, E2 (corresponding to k12) = Eo - k(R,2 - Ra - Rmi) Therefore, Ex - E2 = k[(Rmi - Rfi) - (Rma - R,2)] If the preexponential factors for both reactions are the same, we have r, = i i i = exp{k[(Rm2 - Rf2) - (Rmj - Rfl)]/RT}

(13)

K 12

This gives [23] Q = exp[-(Rm - Rf)/RT]

(14)

Several authors [24-27] have attempted to give a theoretical basis to the Q-e scheme, even though Alfrey and Price [18] pointed out that this was strictly an empirical approach. Indeed, in many cases calculated Q and e values do not agree with literature data or, conversely, relative reactivity ratios calculated from published Q and e values do not agree with experimental data.

r THERMODYNAMICS OF POLYMERIZATION. VI

269

D. Substituent Effect Hammett [28] has proposed that the effect of meta- or para-substituents of side-chain reactions of benzene can be expressed as log(k/k0) = po

(15)

where kp and k are the rate or equilibrium constants for the reaction of the unsubstituted and substituted compound, a is a parameter having a single value for each substituent, and p is a constant for any particular reaction. The parameters a and p are interpreted as measures, respectively, of the ability of the substituent to withdraw electrons or donate them to the site of reaction and the effect of such electronavailability on the reaction. Since Walling et al. [29] reported that the reactivities of nuclearsubstituted styrenes toward a polymer radical were correlated with Hammett's a constants, similar relationships were observed on phenyl vinyl sulfides [30] and vinyl benzoates [31]. Imoto et al. [32] carried out the mutual copolymerization of p-substituted styrenes, and found that log(k/k0) = pa + R

(16)

where R is a parameter of the resonance stabilization in the transition state. This can be written in terms of the resonance substituent constant ER and the effect of such resonance availability y on the reaction so that logft/ko) = po + yER

(17)

which is proposed by Yamamoto and Otsu [33]. When the polar effect is significant, as often occurs with ionic reaction, y is equal to zero and Eq. (17) reduces simply to Eq. (15). However, when only resonance effects play a significant role, as in certain radical reactions, the value of p becomes zero and Eq. (17) reduces to [34]: logtk/k,,) = r E H

(18)

On the other hand, Bamford et al. [35] proposed that rate constants for the reactions between polymer radicals and a number of

270

H.SAWADA

substrates, including monomers, are given by log k = log kT + aa + /3

(19)

where a and j3 are constants characteristic of a given monomer, a is the algebraic sum of Hammett's a constants for the substituents in the terminal monomer unit of the radical, and kT is the rate constant for chain transfer to toluene (taken as a nonpolar measure). Equations (16), (17), and (6) can be written in the form of the following equations, respectively: log k = log k,, + pa + yER

(20)

log k = log kg + per + R

(21)

log k12 = log Px - exe2 + log Q2

(22)

Comparison of Eqs. (20)-(22) and Eq. (19) indicates that in each case the first term of the right-hand side stands for the general reactivity of the attacking radical, the second term relates to the polar effects of both attacking radicals and reactants, and the third term relates to the resonance factor of the reactants. The factor e of the scheme increases almost linearly with Hammett's a constant [20, 36]. Cammarata et al. [37] have found that with substituted styrenes as the monomer, the parameters E R , R, and log Q2 are correlated by the square of Hammett's a constant. Thus log k = acr + bff2 + c

(23)

where a, b, and c are constants. Chernobai et al. [38] derived a relation between Hammett's a constant, activation energies, and thermodynamic characteristics in polymerization of vinyl monomers comprising substituted styrenes. Equations were derived to relate a with activation energies, propagation constants, and heat of polymerization. Let us consider the reactions of series of alphatic compounds where the reaction sites are not in conjugation with the substituents. Since resonance effects are only of minor importance, steric effects are often as important as polar effect. The value of log(l/r 1 ) = Iog(k12/kn) is considered to represent the activity of monomer M2 relative to the reactive center of the reference monomer Mx. Thus it was found that [39] log(l/r1)=p*CT* + 6Es

(24)

THERMODYNAMICS OF POLYMERIZATION. VI

271

where a * is Taft's polar substituent constant, Es is Taft's steric constant, and p* and 6 are the reaction constants. In the copolymerization of styrene and a-substituted methyl acrylates, the reactivities of the esters toward the styryl radical were influenced by both the steric and polar nature of the a-substituent and followed Eq. (24) [40]. Chikanishi and Tsuruta [41] found a linear relationship between log^l/rj) and Taft's steric constant in the case of the relative reactivities of methyl a-alkyl acrylate toward attack of styryl radical. Thus = 6ES

(25)

In the case of 2-vinyl-4-substituted-l,3-dioxolanes (M2) with acrylonitrile (Mj), a linear relationship was obtained by plots of logU/rJ with Taft's steric constant, but not with the a * value in spite of the remote distance of the substituent from the reaction center [42]. Yamashita et al. [42] suggested that the substituent exerted a steric effect on the double bond in the transition state. Fairly good linear relations were also obtained when the relative reactivities of a-, cis-/3-, and trans-/3-alkylstyrenes toward the polyacrylonitrile radical are plotted against values of E s in Eq. (25) for alkyl groups [43]. On the other hand, Otsu et al. [44] reported that the relative reactivity of alkyl esters of methacrylic acid was influenced only by the polar effect and not by Es. Values of p* thus obtained are usually small and depend on the polar character of the attacking radicals. Since ester alkyl groups in alkyl methacrylates are well away from the reactive double bond, steric effects are negligible. The reactivities of 1,1-disubstituted monomers such as a-substituted acrylonitrile and acrylic esters toward polystyrl radical were given by [45] log rt = A log Qx + O.83CTP

(26)

where A log Q, and crP are the resonance and polar substituent constants of a substituents, respectively. Consequently, it seems reasonable to assume that log(k/ko) can be expressed as a sum of polar, resonance, and steric factors. Thus log(k/ko) = F P + F R + F s

(27)

where F P , F K , and F s represent the polar factor, the resonance fac-

272

H. SAWADA

tor, and the steric factor, respectively. For p-substituted styrenes, we have F P + FR » F s = 0 For a-substituted acrylates where the substituents are not in conjugation with the reactive center, we have F p + F s » FR s 0 For a-substituted acrylates where the substituents are located on the reactive center, we have F s » F p + FK = 0 The treatment in this section is at present best considered as an empirical approach to placing monomer reactivity on a quantitative basis. Any theoretical approach has not yet been quantitatively . successful. E. Effect of Polymerization Temperature Lewis and co-workers [46] reported copolymerizations of styrene in which activation energies of reactivity ratios were slight and there was essentially no difference in the activation entropies of the competing propagation reactions. Only in the reaction of the diethyl fumarate radical with styrene made the difference in entropies of activation clearly differ from zero by more than experimental error. Goldfinger and Steidlitz [47] and Johnston and Rudin [48] also found a preponderance of negative activation energies. Since a reactivity ratio is the ratio of two rate constants, it may be expressed as [46] r, = exp|""»

R

"»+ - " " 7 T n " * |

(28)

where S u t, Sjjt, H u t, and H,,t are, respectively, the entropies and heats of activation for reaction of a radical ending in monomer i with monomers i and j . For the binary copolymerization system, rxr2 is given by r x r 2 = exp{[(H12* + H21*) - (Hu* + H22*)]/RT - [(Su* + S21*) -(Snt+Sj)]/R} = exp{ni2VRT = exp{AG12VRT}

XUVR}

(29)

THERMODYNAMICS OF POLYMERIZATION. VI

273

where * (H12* + Hgl*) - (Hxx* + H 22 t)

oj Xl2 *

= (Sxa* + Sax*) - (S xl * + S22»)

AG12* = flja* — Tx12* Therefore from Eq. (29) we find that 9

to

r r

i 2 _ _ ^

'

(30)

Whether or not rxr2 increases or decreases as the temperature increases depends upon the sign of O12t for the copolymerization system. Taking the logarithm of both sides of Eq. (30), i

/

g

(31)

According to Eq. (31), a plot of In r ^ vs 1/T should give a straight line with slope equal to n i 2 t/R, and the intercept of In r x r 2 at 1/T = 0 is equal to —x12VR. In Fig. 4 In r^., values are shown as a function of 1/T for styrene-methyl methacrylate. If x12t > 0 and fl12t > 0, or \ l2 * < 0 and S212t < 0, there must be a temperature at which AG12t is zero. This temperature will be given by

At this temperature, r x r 2 = 1 and therefore the copolymer is completely random. The effect of temperature on the monomer reactivity ratio has been examined experimentally for styrene-methyl methacrylate [49], aerylonitrile-methyl methacrylate [50], and methacrylonitrilestyrene [51]. A list of the values of n i2 * and X12* of these binary copolymerization systems is given in Table 6. It is observed from this table that all of the values of J212t are negative, indicating that r x r 2 increases as the temperature increases. It will be noted that all values of X12* a r e nearly equal to zero, indicating that the values of entropy change for most polymerizations are quite similar. Thus the completely random copolymer appears to fail for these copolym-

H. SAWADA

274

Fig. 4. Least-square fit of literature values of In ifo vs 1/T for the copolymerization of styrene and methyl methacrylate: In r ^ = 0.17 — 521/T. Data from Ref. 49 and H. Mark, B. Immergut, E. H. Immergut, L. J. Yong, and K. I. Boynon, in Polymer Handbook (J. Brandup and E. H. Immergut, eds.), Wiley-Interscience, New York, 1966, p. 11-142.

erization systems. For these systems, however, the tendency toward random copolymerization is enhanced by an increase in temperature. A plot of In rx vs l/T affords a measure of the difference in entropies and enthalpies of activation. O'Driscoll [52] assumed that the entropies of activation for the two competing propagation reactions are approximately equal, and thus In r, = - H u t -

(33)

THERMODYNAMICS OF POLYMERIZATION. VI

275

Table 6 Values of fi, 2 * and X12 * for Binary Copolymerization Systems Binary systems

(kcal/mole)

X»* (eu)

Refs.

STa/MMAb

-1.03

0.33

49

-0.79

-

e

AN=/MMA

-1.38

0.46

50

MANd/ST

-1.08

-0.85

51

a

ST = Styrene. MMA = Methyl methacrylate. C AN = Acrylonitrile. d MAN = Methacrylonitrile. e S. Russo, B. M. Gallo, and G. Bonta, Chim. Ind., 54, 521 (1972). b

The temperature dependence of the reactivity ratio is H»t - H n t djnri_ d(l/T) " ~ R

(34)

•

From Eqs. (33) and (34) we find &T1-

(35)

The term (—RT In r1) = (H11t — H u ^ i s the apparent activation energy for r. Since the value of r itself appears in its activation energy, the only reactivity ratios which will exhibit significant temperature dependence will be those which are very large or very small. Equation (35) predicts that a reactivity ratio greater than unity will decrease with increasing temperature and vice versa. The value of r will always approach unity with increasing temperature. Therefore, the tendency toward random copolymerization is enhanced by an increase in temperature. On the other hand, the tendency toward alternating copolymerization is reduced by an increase in temperature. For the styrene-methyl methacrylate system, the reactivity ratios decrease with decreasing temperature, and this effect leads to a greater degree of alternation in copolymers produced at lower temperatures [53],

276

H. SAWADA

Yamada and Yanagita [54] have proposed the product relation of the copolymerization reactivity ratios: Txr/=a

or

(k n /k 12 )(k 22 /k 21 ) s = a

(36)

where a and/3 are the thermodynamic parameters that will be defined in the following description. In terms of transition state theory, the reactivity ratios can be expressed by r, = exp[(Snt - S 12 t)/R]exp[-(H u t - H12*)/RT] r2 = exp[(S22t - S21t)/R]exp[-(H22* - H21*)/RT] and substituting into Eq. (36), we find

= exp

p j RT

- H 21 *)1 J

If Eq. (36) is independent of temperature, there is the relation

The parameters /3 and a can be defined as P = -(H u * - H12*)/(H22t - H21t)

•

a = exp{([S11t - S12* ] + /3[S22t - S21t ])/R}

(38) (39)

The radical copolymerization systems were classified into two groups corresponding to the negative and the positive sign of /3 by Yamada and Yanagita [54]. They proposed two types of mechanisms which influence interactions between a polymeric chain end and the comonomer. F. Effect of Solvent The Alfrey and Price scheme [18, 19] predicts that the dielectric constant of the solvent affects the monomer reactivity ratios. However, it is well known that the solvent effect is not as significant in radical copolymerization [55]. For example, Lewis et al. [46] and

THERMODYNAMICS OF POLYMERIZATION. VI

277

Price and Walsh [56] indicated that solvents are without effect on the monomer reactivity ratios of the radical copolymerization of styrene and methyl methacrylate. Solvents which are capable of forming hydrogen bonds with monomers, e.g., acrylic and methacrylic acids and numerous monomers containing nitrogen, have a strong influence on these copolymerizations. Joshi [57] first pointed out that the copolymerization anomalies (e.g., deviation from the ideal composition equation) are better explained by the association of the monomeric state through hydrogen bonding and possibly even by dipole-dipole interactions of lesser magnitude which stabilize the monomer state and lower the heat of polymerization. Kerber [58] correlated changes in the reactivity ratios of styrene and acrylic acid with associations of the solvent with acrylic acid through hydrogen bond. For acrylamide [59] and methyl acrylamide [60], which are nonionizable monomers, the existence of the following equilibrium was assumed: ©

CH,=CH—C—NH, - t II

»*- CH,= CH—C=NH, =p=^= 2 2 le

o

o° CH2=CH—C=NH

OH The strong influence of the solvent on the copolymerization of acrylamide should be correlated with the possibility of an amide-enol equilibrium because such an equilibrium could be influenced by solvents of different polarities and dielectric constants. For the N,N-dimethylacrylamide [61], the enolic form cannot be present because no amidic hydrogen is available: /CH 3 CH2=CH—C—NQ -« CH 3 o'

© .CH3 >- CH 2 =CH-C=N(' eo CH 3

Thus, in this case no influence of the solvent on copolymer compositions was found. Minsk et al. [62] indicated quite conclusively that both polar effects and hydrogen bonding affect the copolymerization of acrylamide and styrene. Ito and Otsu [63] found that the copolymerization parameters

278

H. SAWADA

changed significantly with the polarity of solvents, and the relative reactivity of methyl methacrylate toward the polystyryl radical was correlated with the E T values of solvents, which are empirical values of the proton-donating ability of solvents. The reactivity of vinylpyridine depends to a large extent on the acidity of the solvent used [64]. This effect was tested using styrene—2-vinylpyridine, styrene—4-methyl-2-vinylpyridine pairs. With a reduction in solvent acidity, the tendency to alternation decreases as a consequence of the lower polarization of vinylpyridine in the H-complex and a slight difference in the polarities of the styrene and vinylpyridine bonds. The influence of solvent and temperature has been analyzed for the methacrylic acid-methyl methacrylate system. According to type of solvent used, methacrylic acid can exist either as an individual molecule (I), or be associated with itself (II), with the solvent (III), or with the carboxyl groups of the copolymer formed (IV). Forms II and IV mainly appear in petrol and benzene, and are more reactive than Form III which appears in isopropanol and tetrahydrofuran [65]. In principle, solvents should have an effect on radical copolymer composition only in special cases such as the copolymerization of acid or basic monomers or ionizable monomers where the solvents modify the free energy of the initial and final state. II. IONIC COPOLYMERIZATION A. Energetic Characteristics If we consider the propagation step in the case of ionic copolymerization, it is clear that, from the point of view of polarity, there is no possible difference between the attacking ions. Therefore, it is reasonable to assume that the competition between two monomers toward the attacking carbonium ion or carbanion will be independent of the nature of the monomer last added to the chain. The propagation rate constant k u is identical with k 2l , and k12 is the same as k22: Ki = k2i

.

k22 = k12

.

(40) (41)

Therefore k n A 1 2 = k 21 /k 22

(42)

THERMODYNAMICS OF POLYMERIZATION. VI

279

and the product of the reactivity ratios is unity: rxr2 = 1

(43)

Consequently, the equation derived for free radical propagation becomes [66] d[Mj/d[M 2 ] = r i [Mj/[M 2 ]

(44)

Equation (44) will hold only for the case of monomers of similar polarities [67]. Equations (40) and (41) have been denied by some experimental results of Okamura et al. [68], who showed that the polymer cation derived from a less reactive monomer had a larger reactivity. However, the same conclusion was reached in anionic copolymerization by Kuntz and O'Driscoll [67] who assumed the more general equations k u = xk21

(45)

k12 = xk22

(46)

It is to be noted that Eqs. (40) and (41) are special cases of Eqs. (45) and (46) where x has a value of unity. O'Driscoll [69] developed the kinetics of copolymerization of dissimilar monomers. It was shown that the ratio of monomers in the initial copolymer d t M j M M j is directly proportional to the square of the initial monomer ratio ([Mj^Mj) 2 when the monomers are dissimilar: d[Mj/d[M 2 ] = (kx /k 2 )(k u /k22)([Mj/[M2])2

(47)

The above derivation assumes that the initial homopropagation is fast relative to the initiation, and that the crossover reaction is a negligible term at the beginning of the polymerization. The conclusion is drawn that exponent values of 1.0 and 2.0 represent the extremes of behavior for monomers of very similar and very dissimilar polarities. The system styrene-methyl methacrylate was considered as an example of Eq. (47), while styrene-isoprene served as an example of Eq. (44). The reactivity ratios of several vinyl monomers such as styrene, acrylonitrile, methacrylonitrile, and methyl methacrylate have been determined during their copolymerization initiated with phenyl mag-

H.SAWADA

280

nesium bromide [70]. With couples of monomers of very different electronegativities, such as the systems styrene-methyl methacrylate and acrylonitrile-methyl methacrylate, their copolymerization equation will be given by Eq. (47). An increase of temperature generally increases a cross-propagation reaction, and consequently Eq. (44) will hold for the system acrylonitrile-methyl methacrylate [70]. In terms of absolute reaction theory, the propagation rate constant k u for the polymer chain (P t ), which ends in monomer i adding monomer j (Mj), is given by k,, = (RT/Nh)exp{-AFtyRT} FBj*)/RT}

(48)

where F , ^ refers to the free energy of the activated complex formed between P, and Mj. Substituting the four forms of Eq. (48) into Eq. (43), we have •Til

r

—*•>>•>

(49)

•>

Figure 5 illustrates schematically the specific case of anionic copolymerization of butadiene-styrene [67]. The uppermost curve represents the addition of styrene to a butadiene chain end; the one beneath it represents the addition of butadiene to a butadiene chain end. The difference between the peaks of these two curves is the same as the difference between the peaks of the two lowest curves TRANSITION STATE ,12

O

cr

UJ LU LU LU

cr

MONOMER

REACTION

21

POLYMER

COORDINATE

Fig. 5. Illustration of possible transition state energy levels. Reprinted by permission of John Wiley & Sons from Ref. 67.

THERMODYNAMICS OF POLYMERIZATION. VI

281

which represent styrene adding to a styrene end and butadiene adding to a styrene end. Since the enthalpy change AH* of monomers of similar polarities should be the same, we may approximate to Sia*-S11t=S22t-S21*

(50)

where S12$, S u t, S22t, and S21t are the corresponding entropy changes. From Eq. (50) it is seen that there is a large change in entropy involved in the transition state during the addition of a diene monomer to a chain end as compared with addition of a styrene unit. This equation holds for the propagation step in anionic polymerizations in nonpolar solvents [67]. Let us now consider thermodynamics of the transition state more closely when r x r 2 < 1. Using the four versions of Eq. (48) where i,j = 1, 2, we find [71] RT In r.r, = - ( F u * + F22t) + (F12* + F21*) < 0 F 12 t + F 2 1 t < F u t + F 2 2 t

(51)

' H a * + Hia* - T(S21* + SX2t) < H u * + HJ - T(S u t + Sj) (52) In an earlier section we found that the observed values of entropy of polymerization are generally in the narrow range between 25 and 30 eu [72], and that the entropy of activation differs little from entropy of polymerization [73]. The entropies of activation are quite similar as pointed out by some experimental results of Lewis et al. [46]. Thus H l2 J +H 2 X t
(53)

and hence

Equation (53) is the usual case in free radical copolymerization and is responsible for the observed alternation tendency. Assuming a linear free energy relationship [74], AF,, can be expressed in terms of a free energy of activation AF,,$: AF,, = -aAF,jJ + /3

(54)

where a and /3 are constants. From Eq. (54) we see that, when r x r 2 < 1, H u + H22 < H12 + H2l

(55)

In this case we find fi = (H12 + H21) - (H u + H22) > 0 This case was discussed earlier where the treatment of Sawada was presented [75]. For the case where r t r 2 ^ 1, . H18t + H21t % H u t + H22t

(56)

and similarly, H12 + H21
(57)

and 0 < 0. Wall [76] suggested that the electronegativity of the monomer, e,, could be distinguished from that of the chain end, e ^ . Thus r x r 2 = exp{(e, - e,)(e1* - e,*)}

(58)

It is reasonable to consider that the difference in carbonium or carbanion electronegativities may be quite small, and thus [71 ] e,*-ej*=0

(59)

In ionic copolymerization, monomer reactivity ratios are significantly affected by solvent and catalyst, and systematic interpretation is difficult even in the simplest case. It is possible only to speculate on the reasons for the apparent anomalities. Since the mechanism of ionic propagation is more complex than free radical propagation, e,* would be expected to be a function of the solvent and counterion. Thus, in ionic copolymerization, other factors than the stabilization energy, especially the electrostatic interaction between monomer and ion, should be taken into consideration to explain the reactivity [77]. As a result, it is important that the reactivity ratio products change from approximately unity to values much greater than unity as the solvent is changed [71]. Good examples of this come from the experimental work of Overberger and his co-workers [78].

THERMODYNAMICS OF POLYMERIZATION. VI

283

If there is a linear relationship between the ground state free energy and that of the transition state, we have F l f t = (1 + a iJ )Fp 1 + (1 + /3u)FMj

(60)

where a and p are constants. Thus k u = P.Q, expC-e^e,) = -§£ exp{-(a n F P i +

ftjF^/RT}

(61)

If no synergistic effects are encountered (i.e., an = ctu and /3U = i = ft), we find [71] r.r, = expi-^F^ -

ftF^/RT}

(62)

Therefore, r t r 2 can be expected to be precisely unity only for quite similar monomers. If their contributions to the transition state are slightly different, r,r 2 will be smaller than unity. If there are synergistic effects, r x r 2 may be much greater or less than unity [71]. The dependence of relative reactivity of cationic copolymerization on solvents was explained in terms of the difference in the ability of complex formation between monomer and growing carbonium ion [79, 80]. The solvent effect in anionic copolymerization was also studied by O'Driscoll [81]. B. Reactivity Walling et al. [29] correlated monomer reactivity of substituted styrenes with structure in radical copolymerization systems using Hammett's op concept. The same concept was extended for cationic copolymerizations. Overberger et al. [82] studied the effect of para-substitution in the cationic copolymerization of a-methylstyrene with several parasubstituted styrenes. It was found that the logarithm of the monomer reactivities gave a good fit to the straight line when plotted vs Hammett's a-values for the various substituents with the exception of p-methoxystyrene. Tsuruta [83] and Brown and Okamoto [84] have suggested that linearity was improved by the use of a* values. A similar treatment of the data obtained from the three para-substituted a-methylstyrenes studied by Dunphy and Marvel [85] resulted in a straight line. Similarly, Tobolsky and Boudreau [86] found that 1) the electrophilic substituent constants, a*, of Brown are superior to Hammett's a values in correlating reactivity ratios in ionic copolymerizations of substituted styrenes, and 2) the reac-

284

H. SAWADA

tion constant is positive for anionic, nucleophilic copolymerizations of substituted styrenes. Mizote et al. [87] proposed Eq. (63) in their study of steric effects of p-methyl group on monomer reactivity in cationic copolymerization of styrene (Mx) and p-methylstyrene (M2). F s = log r 2 - log(l/r 1 )

(63)

The relative reactivities of vinyl aromatic monomers in coordinated anionic copolymerization were interpreted in terms of polar and steric effects of the substituents [88]. The relative reactivity, a, of each monomer was defined as the ratio between its overall rate constant kmon and the correspondent constant of styrene at the same temperature: a

=

^mon/Kstyr

For halogen- or p-n-alkyl-substituted styrenes and for some other monomers, a linear relation may be considered to hold between log a and the a constant values as shown in Fig. 6. On the other hand, a few other monomers (p-isopropylstyrene and m-methylsubstituted styrenes) have a lower value than those which would be predicted from the above linear relation. The behavior of these last monomers could be due to steric effect. The most interesting result is the negative slope of the straight line of Fig. 6(p= —1.0), indicating that the reaction is favored in the case of monomers having an electron-releasing substituent. The greater the electronic density on the double bond, the more reactive the monomer is, as if an electrophilic attack on the double bond was kinetically determinant. This might be at first sight appear as a typical feature of cationic polymerization in which a positive growing ion attacks the monomeric double bond, but there is another interpretation in the mechanism proposed for a "coordination anionic polymerization" by Natta [89]. Szwarc et al. found that Hammett's relation is obeyed by the rate constants of anionic copolymerization of p-substituted styrenes with living polystyrene [90]. For anionic polymerization p = 5.0, while for radical polymerization p = 0.5 and for coordination polymerization p = —0.95. C. Effect of Polymerization Temperature Limited data for some radical copolymerization systems show that Arrhenius plots of log r against 1/T passes a zero, or nearly zero, intercept. Extrapolation of the plots in cationic copolymeriza-

THERMODYNAMICS OF POLYMERIZATION. VI

285

P- -0-95 (c)O

0

cn

(k)

O

(0

O

-0-5

O

(m) O

-

1 -03

i -0 2

1

-0-1

1

i

1

0-1

02

0-3

Fig. 6. Plots of relative reactivities (k m o n /k s t ) of substituted styrenes with regard to the styrene. Effect of substituents:' (a) p-0CH3; (b) p-CH3; (c) p-C2Hs; (d) H; (e) p-F; (f) 0-C4H4; (g) p-Br; (h) p-Cl; (i) m-F; (j) m-Cl; (k) 3,4-(CH3)2; (1) p-iC3H7; (m) m-CH3; and (n) 3,5(CH3)2. Reprinted by permission of Huthig & Wepf Verlag from Ref. 88.

tion of isobutylene and styrene gives intercepts, the logarithms of which are considerably greater than zero [91]. This can be interpreted to mean that the elementary growth steps of an ion undergoing cationic copolymerization possess much larger differences in entropies of activation (S u t — S12i) of their transition state complexes than do those of growing free radicals. From the plot of log r t vs 1/T, the linear relation was obtained in the cationic copolymerization of acenaphthylene Mx and n-butyl vinyl ether M2 [92]. Differences in the energy and the entropy of activation evaluated from the slope and the intercept are - H12t = 2.6 kcal/mole

- S12t = 6.5 eu

and

Since the products of rx and r2 are approximately unity, the corresponding differences must be nearly equal to the values mentioned above, but of opposite signs: - H21* = -2.6 kcal/mole

and

S

t -

= - 6 . 5 eu

In the propagation steps, the energy of activation, H u , is higher by 2.6 kcal/mole than H^t, but the entropy of activation, S u t, is more

286

H. SAWADA

positive than S^*. Assuming that H12$ = U21$, we obtain H u t — H.,.,* = 5.2 kcal/mole. This value corresponds to a difference in overall activation energy for the two homopolymerizations. Thus the difference in activation energies of initiation and termination reactions are almost negligible [92]. The reactivity ratios in CEVE (2-chloroethyl vinyl ether) [Mj— a-methylstyrene [M2] copolymerizations have also shown an appreciable temperature coefficient [93]. The logarithm of r : is approximately linear with 1/T, corresponding to a difference in activation energy for the two addition reactions to the CEVE cation, H u t — H12t = 2.5 kcal/mole. For addition to a-methylstyrene carbonium, r 2 has a very low value at 25°C due to depropagation since a-methylstyrene has its ceiling temperature in this region. However, we should not expect depropagation to be important at low temperatures, and here we find that the temperature dependence of l/r 2 is similar to that of Rowing to the relation of r l r 2 = l,i.e.,H 22 t — H^t, is also about 2 kcal/mole. Masuda and Higashimura [94] suggested that in the ionic copolymerization of monomers with different structures, fit and x* in Eq. (29) are usually different from zero and have negative values. On the other hand, in the ionic copolymerization between homolog, Ot and x* are almost zero; an example of such a copolymerization would be the a-methylstyrene—p-chlorostyrene system where the difference in entropy of activation is very small [95]. In the cationic copolymerization of styrene and p-tert-butylstyrene, AHxt = H u t - H12t, AS^ = S u t — S12*, AH.,* = AH22t — H21$, and AS2$ = S22t — S21t were evaluated [95]. The following relations were derived between AH^ and AStt: H,,* =aSit$ +b AH,* = H u * - H,,* = a(S u t - S,,*) = aAS,* where a and b are constants for a homologous series. Therefore, these copolymerizations must have proceeded through the same mechanism regardless of the type of solvent and catalyst used [96], m . OTHER COPOLYMERIZATIONS

A. Ring-Opening Copolymerization Yamashita et al. [97, 98] indicated that the basicity of cyclic ethers, pKb, is a factor in the cationic copolymerization of cyclic ethers; the plot of log(l/r t ) vs pKb for monomer M2 gave a straight

THERMODYNAMICS OF POLYMERIZATION. VI

287

line in the copolymerization of 1,3-dioxolane [99]. The linear plot of l / r r vs pKb means that the proton affinity or basicity toward the Lewis acid is a measure of carbonium ion affinity. Hence the nucleophilic attack of cyclic ethers on the cationic active chain end is assumed to be one of the driving forces in this copolymerization. The slope of the plot shows the selectivity of the chain end. Aoki et al. [100] and independently Tanaka [101] investigated the effects of both ring strain and basicity of a-monosubstituted cyclic ethers on their reactivity. The relative reactivities of cyclic ethers to the poly[3,3-bis(chloromethyl) oxetane] cation are empirically expressed by [100] logUAi) = -0.086ARS - 0.31(pKb) + 0.57 where ARS and pKb are characteristic of the ring strain and basicity of a-monosubstituted cyclic ethers arid are calculated from the difference in free energy of polymerization and in basicity between Mt and M2, respectively. The pKb of the substituted cyclic ethers are easily obtained from the equation pKb = (pKb)x + 2.0u*. The relative reactivities of the cyclic ethers whose pKb values are not available may therefore be predicted from this equation whenever the a * value of the substituent is obtainable from the literature. A linear relationship can also be observed between the a * value and the free energy of polymerization of the cycloparaffins calculated by Dainton et al. [102]. Kagiya et al. [103] pointed out that in cationic copolymerization of cyclic ethers the relative reactivity can be expressed as a function of two energetic values of the nucleophilic coordination power of monomer and the radical bond-dissociation energy of the broken bond. Thus + 6'ADt)

(65)

where AAFp is the increase in the free energy change between two monomers, AD, is the radical dissociation energies of broken bonds, and cpy and 6' are constants. The anionic coordination copolymerization of a number of monosubstituted ethylene oxide derivatives with propylene oxide as the standard was studied in the presence of a 1:1 diethyl zinc:water catalyst. The value of logU/rJ is given by [104] + 0.267Es - 0.145

(66)

288

H. SAWADA

where ax is the polar constant and E s is the steric constant of the substituent on the a-oxide. The relative reactivities of the o-oxides during anionic copolymerization in the presence of sodium phenolate as catalyst were also found to depend on the polar as well as on the steric influence of the substituents [105], and thus logU/rj) = 2.09O"! + 0.154Es + 0.11

(67)

Yamashita et al. [106] studied the cationic copolymerization of dioxolane and trioxane, mainly by examining the microstructure of the copolymers by NMR. The incorporation of dioxolane in the copolymers decreased with increasing conversion and polymerization temperature and with increasing dilution by solvent. This may be attributed to the lower equilibrium concentration of formaldehyde for trioxane polymerization and the higher equilibrium monomer concentration found for the polymerization of dioxolane. The results showed that dioxolane is kinetically more reactive but thermo. dynamically less polymerizable owing to a remarkable ease of depolymerization of poly(dioxolane). Yamashita et al. [107] also studied the cationic copolymerization of dioxolane and tetroxane. The very reactive tetroxane was consumed at an early stage, whereas dioxolane was consumed slowly until an equilibrium conversion was reached. Thus the incorporation of the less reactive dioxolane increased with polymerization time. After constant composition was reached, there appeared to be no more polymerization of the remaining dioxolane. Yamashita et al. [107] concluded that tetroxane is far more reactive than dioxolane from both the kinetic and thermodynamic points of view. However, one should note that the thermodynamics of these copolymerizations are very complicated because of the number of side reactions possible, such as polymer transfer (trans-acetalization) and the backbiting reactions. B. Miscellaneous Copolymerizations The effects of monomer composition on the ceiling temperature are shown in Fig. 7 for the copolymerization of acetaldehyde and propionaldehyde [108]. The maximum ceiling temperature occurs at a polymer composition of 74 mole % acetaldehyde. The copolymerization of o-phthalaldehyde and styrene was studied at 0 and -78°C by Aso et al. [109]. At -78°C, o-phthalaldehyde preferentially polymerized to yield living cyclopolymers until an equilibrium concentration with o-phthalaldehyde monomer was reached. Thus styrene propagated from the living terminal rather slowly. The

289

THERMODYNAMICS OF POLYMERIZATION. VI -30

-35

-

75

25

100

MOLE •/. ACETALDEHYDE

Fig. 7. Ceiling temperature as a function of monomer feed composition. Reprinted by permission of the publishers, IPC Science and Technology Press from Ref. 108.

block structure of the copolymer was confirmed by chemical and spectroscopic methods. In the copoly me rization at 0°C, the ophthalaldehyde unit in copolymer consisted both of cyclized and uncyclized units. This copolymer seemed to contain short o-phthalaldehyde sequences. Since this polymerization temperature is much higher than the ceiling temperature (—43CC) of the homopolymerization of o-phthalaldehyde, depropagation of the o-phthalaldehyde unit should be extensive, and the presence of long sequences of o-phthalaldehyde units is improbable [109]. The presence of a donor-acceptor charge transfer complex during the copolymerization of a strong donor monomer (D), such as styrene, with an acceptor monomer (A), such as methyl methacrylate or acrylonitrile, in the presence of a metal halide or other Lewis acid (MX) has been verified by UV spectroscopy [110]. It has been MX] proposed [ i l l ] that the homopolymerization of the [D + ... :A complex is responsible for the formation of an alternating copolymer. The concentration of the complex is dependent upon the monomer, metal halide, and solvent concentrations, the A/MX ratio, and the reaction temperature. (1) A + MX --, (2) D + A (3) x[D+

MX "A

^- A * MX]

MX ^

[D+

"A

MX]

»- -(DA)X— + xMX

290

H. SAWADA

At low temperature, the equilibrium in Process (2) should shift to the right, resulting in a higher complex concentration. Increasing the temperature may shift the equilibria in Processes (1) and (2) to the left, resulting in a decrease in the concentration of the complex and an increase in the concentration of free donor monomer as well as acceptor monomer [112]. Condensation copolymerization proceeds in the same way as in the formation of the simple homopolymers, but the chains now consist of mixed units. It is often assumed that at first the reagents condense less randomly according to the relative reactivities of the groups, and that the resulting low or medium molecular weight products rearrange as the reaction proceeds, finally giving a randomly arranged copolymer. To treat the problem quantitatively, we shall now introduce a new parameter v [113]: v = [P12 - (P I2 ) rlad ]/[(P 12 ) BM - - ( P 1 2 ) r a J

(68)

where P 12 represents the number of Mj-M2 pairs in the actual copolymer; (P12)rMd represents the number of Mx-M2 pairs which would exist in a copolymer of the same composition if the copolymer were random; and (P 12 ) mai represents the maximum number of Mi-M., pairs which would exist if complete order occurred. (P 12 ) raad is given by (P 12 ) rmd = 2X1X2N0

(69)

where Xx is the mole fraction of M t in the copolymer, X2 is the mole fraction of M2 in the copolymer, and No is the total number bonds. In the case of the alternating copolymer, where X1 = X2 = 1/2, (Pi 2 )»« = No

(70)

From Eq. (69) for this copolymer (P i a ) r i a d = N o /2

(71)

and hence from Eqs. (68), (70), and (71) we obtain i>=2P12/N0-l

(72)

Then the number of Jv^-M., pairs can be expressed P 12 = N0(l + v)/2

(73)

THERMODYNAMICS OF POLYMERIZATION. VI

291

Out of a total number of bonds No, the fraction of bonds of the -Mj type is (74)

Consequently, the fraction of bonds which are like bond is fu + f22 = 1 - fia = (1 - v)/2

(75)

Assume that the enthalpy change on bond formation is the same in an interchange reaction and in a polycondensation; then the enthalpy changes on formation of M^^ M2M2, M2Mlt and bonds are denoted by H n , H22, H21, and Hl2, respectively, and the numbers of such bonds by P u , P22, P 21 , and P 12 . For a system with an equal number of IV^ and M2 monomer units, P u = P22. If the average energy of like bonds (Hu + H22)/2 is greater than that of unlike bonds (H12 + H 2I )/2, we have the simple energy level system indicated in Fig. 8. It should be noted that any gain H on the part of a system is considered to be positive and any loss negative. [Since heat is evolved during the polymerization, polycondensation reactions are exothermic (Hn < 0, H u < 0).]

AH

Fig. 8. Two level systems with probabilities P,, P2. Reprinted by permission of John Wiley & Sons from Ref. 113.

If the change in entropy for the reaction is zero, in thermal equilibrium, the relative probability of occupancy of the upper energy state Px to that of the low energy state P 2 , is given simply by Boltzmann statistics P 1 /P 2 = exp(-AH/kT) where k is Boltzmann's constant.

292

H. SAWADA

The probability of occupancy of the upper energy state is Pi = (P» + P22)/NO = fu + f22 Similarly the probability of the low energy state is *2

=

"l2 /"o

~

12

From Eqs. (74) and (75), (1 - v)/{l + v)= exp(-AH/kT)

(76)

where AH = (Hlt + H22)/2 - (H12 + H21)/2 and so v = {1 - exp(-AH/kT)}/{l + exp(-AH/kT)}

(77)

In Fig. 9, v from Eq. (77) is plotted vs temperature. If AH is positive, v will be positive and the monomer units will tend to alternate in the polymer. If AH is negative, v will be negative and the monomer units will tend to cluster in 'blocks" in the copolymer. If AH is positive at 0°K, v is predicted to be equal to —1, and at high temperatures to approach zero. Therefore, in any case, it is observed that randomness becomes more predominant as the temperature is elevated. In general, melt polymerization methods yield condensation copolymers with random distributions. In contrast, the method of interfacial polycondensation yields condensation copolymers ranging in order from alternating to block arrangements, since no redistribution of groups will take place because of the low polymerization temperature (as shown in Fig. 9). In the case of equal numbers of Mt and M2 monomer units and equal probabilities for the formation of M ^ and M2M2, and M1M2 and M-JMJ bonds, the molar heat of condensation copolymerization is given by [114] H = (H u + H22 + H12 + H 21 )/4 - fAH/2 where AH = {(H u + H22) - (H12 + H 21 )}/2

(78)

THERMODYNAMICS OF POLYMERIZATION. VI

293

• 1-0

T.'K

-10

Fig. 9. 7-Parameter vs temperature. Reprinted by permission of John Wiley & Sons from Ref. 113.

H u , H22, H12, and H21 are the enthalpy changes per mole of elementary reactions. Substituting Eq. (77) into Eq. (78), we obtain H = (H u +H22 + H12 + H21)/4 - AH{1 - exp(-AH/RT)}/ 2{1 + exp(-AH/RT)} = Ho - vAH/2

(79)

where Ho =

+H

22

In Fig._10, H from Eq. (79) is plotted vs temperature. _ If AH is positive at 0°K, H is predicted to be equal to (H1Z + H2i)/2 and at high temperatures, to approach_Ho. In this case v is 1, indicating an alternating copolymer. If AH is negative at a temperature of 0°K, v will be —1 and clustering will occur. Therefore, at 0°K, H is predicted to be equal to (H u + H22)/2 and, at high temperature, to approach Ho. If a copolymer is cooled rapidly from a high to a low temperature, a metastable state may be produced in which a nonequilibrium disorder is ^frozen" in the structure. In such a case H has a large value, even at low temperatures.

294

H. SAWADA

Fig. lO^Molar heatof. condensation copolymerization as a function of temperature. A = (Hu + H22)/2; B = (HI2 + H2,)/2. Reprinted by permission of John Wiley & Sons from Ref. 114. Acknowledgment The author wishes to thank Prof. K. F . O'Driscoll for many valuable criticisms of this manuscript. References [1] H. Sawada, J. Macromol. Sci.—Revs. Macromol. Chem., CIO, 293 (1974). [2] T. Alfrey, / Polym. Set, 4, 221 (1949). [3] M. Suzuki, H. Miyama, and S. Fujimoto, Ibid., 31, 212 (1958);./. Chem. Soc. Japan, 79, 607 (1958). [4] K. Horie, I. Mita, and H. Kambe, J. Polym. Sci., A-l, 7, 2561 (1969). [5] H. Miyama, and S. Fujimoto,Ibid., 54 s32 (1961);M. Suzuki, H. Miyama, and S. Fujimoto, Bull. Chem. Soc, Japan, 35, 60(1962). [6] M. Suzuki, H. Miyama, and S. Fujimoto, Ibid., 35, 57 (1962). [7] L. K. J. Tong and W. 0. Kenyon, J. Amer. Chem. Soc, 71,1925 (1949). [8] R. M. Joshi,Makromol. Chem., 66,114 (1963). [9] K. J. Ivin, W. A. Keith, and H. Mackle, Trans. Faraday Soc, 55, 262 (1959). [10] F. S. Dainton, J. Diaper, K. J. Ivin, and D. R. Sheard, Ibid., 53,1269 (1957). [ 11 ] C. T. Mortimer, Reaction Heats and Bond Strengths, Pergamon, New York, 1962, p. 95. [12] R. D. Snow and F. E. Ftey, Ind. Eng. Chem., 30,176 (1938); J. Amer. Chem. Soc, 65, 2417(1943).

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295

[13] F. S. Dainton and K. J. Ivin,Nature, 162, 705 (1948);Proc. Roy. Soc, A, 212, 207 (1952). [14] R. E. Cook, F. S. Dainton, and K. J. Ivin, J. Polym. Sci., 26, 351 (1957). [15] R. E. Cook, K. J. Ivin, and J. H. O'Donnell, Trans. Faraday Soc, 61,1887 (1965). [16] B. H. G. Brady and J. H. O'Donnell, Eur. Polym. J., 4, 537 (1968); Trans. Faraday Soc, 64, 29 (1968). [17] J. R. Brown and J. H. O'Donnell, J. Polym. Sci., 10,1997 (1972). [18] T. Alfrey, Jr. and C. C. Price, Ibid., 2, 101 (1947). [19] T. Alfrey, Jr. and L. J. Young, in Copolymerization (G. E. Ham, ed.), WileyInterscience, New York, 1964, Chap. 11. [20] T. C. Schwan and C. C. Price, J. Polym. Sci., 40, 457 (1959). [21] T. Kagiya and Y. Sumida, Polym. J., 1,137 (1970). [22] H. Sawada, J. Macromol. Sci-Revs. Macromol. Sci., C3, 370 (1969). [23] M. G. Evans, J. Gergely, and E. C. Seaman, J. Polym. Sci, 3, 866 (1948). [24] M. Chorton and A. J. Capata, Ibid., A, 2,1321 (1964). [25] M.C.Shen,/6irf.,B, 1,11(1963). [26] N. Kawabata, T. Tsuruta, and J. Furukawa,Makromol. Chem., 51, 80 (1962). [27] T. Yonezawa, K. Hayashi, C. Nagata, S. Okamura, and K. Fukui, J. Polym. Sci., 14, 312 (1954); K. Hayashi, T. Yonezawa, C. Nagata, S. Okamura, and K. Fukui, Ibid., 20, 537 (1956); G. S. Levinson,Ibid., 60,43 (1962). [28] L. P. Hammett, J. Amer. Chem. Soc, 59,96 (1937);Pftysica/ Organic Chemistry, McGraw-Hill, New York, 1940. [29] C. Walling, E. R. Briggs, K. Wolfstern, and F. R. Mayo, J. Amer. Chem. Soc, 70,1537 (1948). [30] K. Tsuda, S. Kobayashi, and T. Otsu, J. Polym. Sci.,A-l, 6, 41 (1968); J. Macromol. Sci.-Chem., Al, 1025 (1967). [31] M. Kinoshita, T. Irie, and M. Imoto,Makromol. Chem., 110, 47 (1967). [32] M. Imoto, M. Kinoshita, and M. Nishigaki,/&;
296

H. SAWADA

[45] B. Yamada and T. Otsu, J. Macromol. Sci.-Chem., A3,1551 (1969). [46 ] F. M. Lewis, C. Walling, W. Cummings, E. R. Briggs, and F. R. Mayo, J. Amer. Chem. Soc, 70,1519 (1948). [47] G. Goldfinger and M. Steidlitz, J. Polym. Sci, 3, 786 (1948). [48] H. K. Johnston and A. Rudin, J. Paint Technol, 42,435 (1970). [49] V. E. Meyer, J. Polym. Sci.,A-l, 5,1289 (1967). [50] R. M. Joshi and S. L. Kapur, J. Sci. Ind. Res., 16B, 379 (1957). [51 ] A. Rudin and R. G. Yule, J. Polym. Sci., A-l, 9, 3009 (1971). [52] K. F. O'Driscoll, J. Macromol. Sci.-Chem., A3, 307 (1969). [53] A. D. Jenkins and M. G. Rayner, Eur. Polym. J., 8, 221 (1972). [54] A. Yamada and M. Yanagita, J. Macromol. Sci.-Chem., A4,1841 (1970). [ 55 ] C. Walling, Free Radical in Solution, Wiley, New York, 1957, p. 35. [56] C. C. Price and J. G. Walsh, J. Polym. Sci., 6, 239 (1961). [57] R. M. Joshi,Ibid., 60, 556 (1962); J. Sci. Ind. Res., 18B, 279 (1959). [58] R. Kerber.il/aftromo/. Chem., 96, 30 (1966). [59] G. Saini, A. Leoni, and S. Franco,Ibid., 144, 235 (1971). [60] G. Saini, A. Leoni, and S. Franco,Ibid., 147,213 (1971). [61 ] G. Saini, A. Leoni, and S. Franco, Ibid:, 146,165 (1971). [ 61 ] L. M. Minsk, C. Kotlarchik, and R. S. Darlak, J. Polym. Sci., Polym. Chem. Ed., 11,353(1973). [63] T. Ito and T. Otsu, J. Macromol. Sci.-Chem., A3,197 (1969). [64] A. V. Ryabov, Y. D. Semchikov, L. A. Smirnova, N. N. Slavitskaya, N. L. Khvatova, and V. N. Kashayeva, Vysokomol. Soedin., A, 13,1414 (1971). [65] V. G. Markert and H. Pennewiss, Angew. Makromol. Chem., 11, 53 (1970). [66] Y. Landler,/ Polym. Sci., 8, 63 (1952). [67] K.F. O'Driscoll and I. Kuntz,/6jtt, 61,19 (1962). [68] S. Okamura, N. Kanoh, and T. Higashimura,Makromol. Chem., 47, 35 (1961). [69] K. F. O'Driscoll,,/. Polym. Sci., 57, 721 (1962). [70] F. Dawans and G. Smets,Makromol. Chem., 59,163 (1963). [71] K. F. O'Driscoll, T. Higashimura, and S. Okamura, Ibid., 85,178 (1965). [ 72 ] H. Sawada, J. Macromol. Sci.-Revs. Macromol. Chem., C3 322 (1969). [73] R. M. Joshi, in Vinyl Polymerization, Part 1 (G. E. Ham, ed.), Dekker, New York, 1967, p. 500. [ 74 ] P. R. Wells, Linear Free Energy Relationships, Academic, New York 1968, p. 1. [75] H. Sawada, J. Polym. Sci., A-l, 5,1383 (1967). [76] L. A. Wall.TSitt, 2, 542 (1947). [77] N. Kawabata, T. Tsuruta, and J. Furukawa,Bull. Chem. Soc. Japan, 36, 905 (1963). [78] C. G. Overberger, L. Arond, and D. Tanner, J. Amer. Chem. Soc, 73, 5541 (1951); C. G. Overberger, R. Ehrg, and O. Tanner, Ibid., 76, 772 (1954); C. G. Overberger and V. G. Kamath,Ibid., 85, 446 (1963). [79] T. Masuda and T. Higashimura, J. Macromol. Sci.-Chem., A5(3), 549 (1971). [80] K. F. O'Driscoll, T. Yonezawa, and T. Hogashimura,Ibid., 1,17 (1966). [81] K. F. O'Driscoll, J. Polym. Sci., A, 3,2223 (1965). [82] C. G. Overberger, L. H. Arond, D. Tanner, J. J. Taylor, and T, Alfrey, Jr., J. Amer. Chem. Soc, 74,4848 (1952). [83] T. Tsuruta, J. Japan. Chem. (Kagaku no Ryoiki), 8, 209 (1954);BuH. Inst. Chem. Res. Kyoto Univ., 32,149 (1954).

THERMODYNAMICS OF POLYMERIZATION. VI

[84] [85] [86] [87 ] [88] [89] [90] [91] [92] [93] [94] [95] [96] [97] [98] [99] [100] [ 101 ] [102] [103] [ 104 ] [105] [ 106] [107] [108] [ 109] [110] [Ill] [112] [113] [114]

297

Y. Okamoto and H. C. Brown, J. Org. Chem., 22,485 (1957). J.F.Dunphy and S. C. Marvel, J. Polym. Sci., 47,1 (1960). A. V. Tobolsky and R. J. Boudreau, Ibid., 51, s53 (1961). A. Mizote, T. Tanaka, T. Higashimura, and S. Okamura, Ibid., 3, 2567 (196&). G. Natta, F. Danusso, and D. Sianesi, Makromol. Chem., 30, 238 (1959). G. Natta,Experientia Suppi, 7, 21 (1957). M. Shima, D. N. Bhattacharyya, J. Smid, and M. Szwarc, J. Amer. Chem. Soc, 85,1306 (1963). J. Rehner, Jr., R. L. Zapp, and VV. J. Sparks, J. Polym. Sci., 11, 21 (1953). M. Imoto and K. Saotome./fciU, 31, 208 (1958). G. R. Brown and D. C. Pepper, J. Chem. Soc, 1963, 5930. T. Masuda and T. Higashimura,Polym. J., 2, 29 (1971). C. G. Overberger, L. H. Arond, and J. J. Taylor, J. Amer. Chem. Soc, 73, 5541 (1951). J. Furukawa, T. Taniguchi, and E. Kobayashi, Presented at the 21st Symposium on Polymer Science, Society of High Polymers, Japan, 3A11,1972. S. Iwatsuki, N. Takigawa, M. Okada, Y. Yamashita, and Y. Ishii, J. Polym. Sci., B, 2, 549 (1964). Y. Yamashita, T. Tsuda, M. Okada, and S. Iwatsuki,Ibid., A-l, 4, 2121 (1966). M. Okada, N. Takikawa, S. Iwatsuki, Y. Yamashita, and Y. Ishii,Makromol. Chem., 82,16 (1965). S. Aoki, Y. Harita, Y. Tanaka, H. Handai, and T. Otsu, J. Polym. Sci., A-l, 6, 2585 (1968). Y. Tanaka, J. Macromol. Sci.-Chem., Al(6), 1059 (1967) F. S. Dainton, T. R. E. Devlin, and P. A. Small, Trans. Faraday Soc, 51,1710 (1955). T. Kagiya, Y. Sumida, and T. Inoue,Polym. J., 1, 312 (1970). V. A. Ponomarenko, A. M. Khomutov, S. I. I'lchenko, A. V. Ignatenko, and N. M. Khomutova, Vysokomol. Soedin.,A, 13,1551 (1971). V. A. Ponomarenko, A. M. Khomutov, S. I. I'lchenko, A. V. Ignatenko, Ibid., A13, 1546 (1971). Y. Yamashita, T. Asakura, M. Okada, and K. Ito, Makromol. Chem., 129, 1 (1969). Y. Yamashita, T. Inoue, G. Hattori, and K. Ito,/6iU, 151, 91 (1972). A. M. North and D. Richardson,Polymer, 5, 333 (1964). C. Aso, S. Tagami, and T. Kunitake, J. Polym. Sci., A-l, 8,1323 (1970). T. Ikegami and H. Hiral, Ibid., A-l, 8,195, 463 (1970). N. G. Gaylord and A. Takahashi, J6/d.,£, 6, 743 (1968). N. G. Gaylord and B. Patnaik,/&!