Thermodynamics of Polymerization - articles by Hideo Sawada in Polymer Reviews

LIST OF ARTICLES ON THERMODYNAMICS OF POLYMERIZATION BY HIDEO SAWADA PUBLISHED IN POLYMER REVIEWS Sawada, Hideo (1969) 'Thermodynamics of Polymerization. I', Polymer Reviews, 3, 313 — 338 DOI: 10.1080/15583726908545926 URL: http://dx.doi.org/10.1080/15583726908545926 Sawada, Hideo (1969) 'Chapter 2. Heat of Polymerization', Polymer Reviews, 3, 339 — 356 DOI: 10.1080/15583726908545927 URL: http://dx.doi.org/10.1080/15583726908545927 Sawada, Hideo (1969) 'Chapter 3. Thermodynamics of Radical Polymerization', Polymer Reviews, 3, 357 — 386 DOI: 10.1080/15583726908545928 URL: http://dx.doi.org/10.1080/15583726908545928 Sawada, Hideo ((1969) 'Chapter 4. Thermodynamics of Polycondensation', Polymer Reviews, 3, 387 — 395 DOI: 10.1080/15583726908545929 URL: http://dx.doi.org/10.1080/15583726908545929 Sawada, Hideo (1970) 'Thermodynamics of Polymerization. II. Thermodynamics of RingOpening Polymerization', Polymer Reviews, 5, 151 — 173 DOI: 10.1080/15583727008085366 URL: http://dx.doi.org/10.1080/15583727008085366 Sawada, Hideo (1972) 'Thermodynamics of Polymerization. III', Polymer Reviews, 7, 161 — 187 DOI: 10.1080/15321797208068162 URL: http://dx.doi.org/10.1080/15321797208068162 Sawada, Hideo (1972) 'Thermodynamics of Polymerization. IV. Thermodynamics of Equilibrium Polymerization', Polymer Reviews, 8, 235 — 288 DOI: 10.1080/15321797208068172 URL: http://dx.doi.org/10.1080/15321797208068172 Sawada, Hideo (1974) 'Thermodynamics of Polymerization. V. Thermodynamics of Copolymerization. Part I', Polymer Reviews, 10, 293 — 353 DOI: 10.1080/15321797408076101 URL: http://dx.doi.org/10.1080/15321797408076101 Sawada, Hideo (1974) 'Thermodynamics of Polymerization. VI. Thermodynamics of Copolymerization. Part 2', Polymer Reviews, 11, 257 — 297 DOI: 10.1080/15583727408546025 URL: http://dx.doi.org/10.1080/15583727408546025

Table of Contents Thermodynamics of Polymerization. I Chapter 1. Introductory Survey I. THE CEILING TEMPERATURE CONCEPT A. Thermodynamic Approach B. Kinetic Approach C. Determination of Ceiling Temperatures II. ENTROPY AND FREE ENERGY CHANGES OF POLYMERIZATION A. Entropy of Polymerization B. Determination of Entropy of Polymerization C. Free Energy Changes of Polymerization REFERENCES Chapter 2. Heat of Polymerization I. GENERAL ASPECTS A. Breaking a Multiple Bond B. Resonance C. Steric Strain II. VARIATIONS IN HEATS OF POLYMERIZATION A. Steric Strain in the Polymer B. Conjugation and Hyperconjugation C. Hydrogen Bond and Solvation III. EMPIRICAL ESTIMATION OF HEAT OF POLYMERIZATION REFERENCES APPENDIX Chapter 3. Thermodynamics of Radical Polymerization I. GENERAL ASPECTS A. Energetics of Radical Polymerization B. Degree of Polymerization C. Activation Energies of Elementary Reactions II. GENERATION OF FREE RADICALS III. PROPAGATION REACTION A. The Polanyi Relation B. Reactivity and Heat of Polymerization C. Ceiling Temperature 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 V. FREE ENERGIES OF FORMATION OF POLYETHYLENE AND POLYTETRAFLUOROETHYLENE A. Free Energies of Polyethylene Synthesis B. Free Energies of Polytetrafluoroethylene Synthesis

5 5 6 6 7 11 14 14 16 22 29 31 31 32 32 33 33 35 38 39 39 41 42 49 50 50 51 53 53 58 58 60 63 63 63 64 65 66 68 68 73

REFERENCES Chapter 4. Thermodynamics of Polycondensation I. GENERAL ASPECTS II. DEGREE OF POLYMERIZATION III. EQUILIBRIUM CONSTANT IV. RING FORMATION IN POLYCONDENSATION ACKNOWLEDGMENTS References Thermodynamics of Polymerization. II. Thermodynamics of Ring-Opening Polymerization I. GENERAL ASPECTS II. HOMOCYCLIC COMPOUNDS A. Angle Strain B. Conformational and Transannular Strain C. Steric Effect of Side Group III. HETEROCYCLIC COMPOUNDS A. Cyclic Ethers B. Lactams C. Lactones D. Miscellaneous Heterocyclic Compounds IV. SUMMARY ACKNOWLEDGMENTS References Thermodynamics of Polymerization. III (Cationic Polymerization) I. GENERAL ASPECTS II. FORMATION OF CARBONIUM ION A. Ionization Potential B. Proton Affinity C. Acidity D. Free Energy Change of Formation of Carbonium Ion E. Ions and Ion Pairs F. Energetics of Salvation III. INITIATION OF CATIONIC POLYMERIZATION A. Energetic Consideration of Initiation Reaction by Halogen Acid B. Catalytic Activity in Cationic Polymerization by Lewis Acids IV. PROPAGATION OF CATIONIC POLYMERIZATION A. Energetics B. Heats of Reaction of Cations with Olefins C. Activation Entropy Changes of Propagation D. Thermodynamics of Formation of Zwitterions V. CHAIN TRANSFER AND TERMINATION Acknowledgments References

77 79 79 80 81 86 87 87 88 88 89 89 99 100 102 103 105 107 107 109 109 109 111 112 114 114 115 118 119 121 122 124 124 126 127 127 129 132 134 136 136 136

Thermodynamics of Polymerization. IV. Thermodynamics of Equilibrium Polymerization I. POSSIBLE TYPES OF EQUILIBRIUM POLYMERIZATION II. SOME CASE STUDIES OF EQUILIBRIUM POLYMERIZATION A. Vinyl Polymerizations B. Ring-Opening Polymerizations C. Polymerization of Aldehydes III. TRANSITION PHENOMENA IN EQUILIBRIUM POLYMERIZATION IV. MOLECULAR WEIGHT DISTRIBUTION A. Equilibrium Polymerization B. Living Polymerization V, THERMODYNAMICS OF EQUILIBRIUM POLYMERIZATION Acknowledgment References Thermodynamics of Polymerization. V. Thermodynamics of Copolymerization. Part I I. THE GENERAL THEORY OF BINARY COPOLYMERIZATION A. Heat of Copolymerization B. Entropy of Copolymerization C. Equilibrium Sequence Distribution D. Free Energy Change in Binary Copolymerization System E. Equilibrium Monomer Concentration F. Penultimate Unit Effect II. DEGREE OF POLYMERIZATION AND COPOLYMER COMPOSITION OF BINARY COPOLYMERIZATION SYSTEM A. Degree of Polymerization B. Copolymer Composition Equation III. MULTICOMPONENT COPOLYMERIZATION A. Heat of Terpolymerization B. General Theory of Multicomponent Copolymerization Acknowledgment References Thermodynamics of Polymerization. VI. Thermodynamics of Copolymerization. Part 2 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. IONIC COPOLYMERIZATION A. Energetic Characteristics B. Reactivity

138 139 146 147 150 165 167 173 173 179 183 188 189 192 193 193 200 203 209 217 222 224 224 228 243 243 248 250 251 253 254 254 259 263 265 268 272 274 274 279

C. Effect of Polymerization Temperature III. OTHER COPOLYMERIZATIONS A. Ring-Opening Copolymerization B. Miscellaneous Copolymerizations Acknowledgment References

280 282 282 284 290 290

Thermodynamics of Polymerization. I HIDEO SAWADA Central Research Laboratory Daicel Ltd. Tsunigaoka, Oi, Intmagwi Saitama, Japan

Chapter 1. Introductory Survey I. THE CEILING TEMPERATURE CONCEPT A. Thermodynamic Approach B. Kinetic Approach C. Determination of Ceiling Temperatures

;

314 314 315 319

II. ENTROPY AND FREE ENERGY CHANGES OF POLYMERIZATION 322 A. Entropy of Polymerization 322 B. Determination of Entropy of Polymerization 324 C. Free Energy Changes of Polymerization 330 REFERENCES

337

313

314

HIDEO SAWADA

I. THE CEILING TEMPERATURE CONCEPT A. Thermodynamic Approach The Gib'bs free energy of a system at temperature T is defined as G = H - TS

(1)

where H is the enthalpy and S the entropy of the system. The free energy change for any polymerization will be, therefore, AG = Gpoiymer ~ G m o n o m e r = "polymer ~ H m o n o m e r — T^Spoiyj^ej.— S monomer ) = A H p - T AS

(2)

When the polymer has a lower free energy than the initial monomer, a polymerization can occur spontaneously, and the sign of AG is negative. A positive sign for AG signifies, therefore, that the polymerization is not spontaneous. When the system is in equilibrium at a certain critical temperature, there is no tendency for polymerization, and, hence, AG = 0 [1-3]. This temperature is known as the ceiling temperature. These three possible conditions for free energy change of polymerization may be summarized as follows: monomer — polymer AG = - (spontaneous) monomer — polymer AG = + (nonspontaneous) monomer — polymer AG = 0 (equilibrium) At the ceiling temperature, T c , AG is zero, so that T c = AHp/ASp

(3)

where AHp and ASp are the enthalpy and entropy changes per monomer unit. When the polymer chains are long these quantities are identical with the heat and entropy changes of polymerization. If the standard state refers to unit concentration and the monomer behaves ideally, AS = AS° + R In [M]; thus T

AS° + R In [M]

(4)

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where AS0 is the entropy change accompanying polymerization at the standard state when the concentration of monomer is unity. Therefore, T c can be raised by increasing the concentration of monomer when a solvent is present. Equation (4) emphasizes that T c is characteristic of monomer-polymer equilibrium only and is quite independent of the monomer or the nature of the active centers in the system; for a given value of [M], the ceiling temperature should, therefore, be the same whether the active centers are radicals or ions. Many polymers are stable even above the ceiling temperature only because of the difficulty of initiating degradative centers on the polymer molecule. In practice, terminated polymer appears stable at temperatures above the ceiling temperature, being in a state of metastable equilibrium. Therefore, the polymer cannot depolymerize spontaneously but can do so under appropriate conditions. Catalyst residues that are not removed during the purification of a polymer may also cause depolymerization reaction. There are four important possibilities of polymerization as follows: a. In addition polymerization, AH and AS are usually both negative, and so AG becomes positive above the ceiling temperature of the system. Thus, the high polymer cannot be formed above the ceiling temperature. b. If the polymerization is endothermic (AH > 0) and AS is greater than zero, no polymer can exist below a floor temperature, above which AG becomes negative. The phenomenon of floor temperature is exhibited by the polymerization of S8 rings. c. When AH is positive and AS negative, AG is always positive; therefore, polymer cannot exist at any temperature. d. When AH is negative and AS positive, AG is always negative; therefore, polymer can exist at any temperature. B. Kinetic Approach It is interesting to consider the ceiling temperature phenomenon from a kinetic point of view [1,3]. At ordinary temperatures the rate constant for depolymerization is small. However, the activation energy of this rate constant is quite high (10-26 kcal/mole) compared to that for propagation, and at high temperature the depolymerization can become important compared with the polymerization. Let us consider the propagation reaction p k^

AAAAR

n*i

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The rate constants for propagation and depropagation reactions can be expressed as k p = A p exp(-E p /RT),

kd = Ad exp(-E d /RT)

where Ap and Ad are the collision frequency factors which approximate to the entropy of activation, and E p and E d are the activation energies for polymerization and depolymerization, respectively. If the degree of polymerization is large, E p - E d = AHp. For long chains, AHp is equal to the heat of the overall polymerization. The rate of propagation is essentially the same as the overall rate of disappearance of monomer, since the number of monomers used in chain transfer and initiation must be small compared to that used in propagation if the polymer chains are long. Therefore,

^I

(5)

where [M*] is the concentration of propagating species, and [M] is the concentration of monomer. Depolymerization may now be considered to be the reverse of propagation, then v

d = k dt M nl

(6

Thus the overall rate of polymerization is

The degree of polymerization is given by the rate of polymerization divided by the rate of termination, i.e., _

(k d -k p [M])[M n ]

(8)

f([Mn]) where f([M*]) is a function of the number of active centers, M*, present. AHp is usually negative (the polymerization is exothermic), and so E d is usually much larger than E p . Therefore, although k d may be negligible compared with kp[M] at ordinary temperature, it will increase more rapidly with increasing temperature. At the ceiling temperature T c the rate of depropagation becomes

THERMODYNAMICS OF POLYMERIZATION.

equal to that of propagation, regardless of the variations of [MJ^] and ) with temperature, kp[M*][M] = which can also be written in the form: Ap exp(-E p /RT c )[M] = Ad exp(-E d /RT c )

(9)

and, therefore, T

Ep - E d AHp c - R In (Ap[M]/Ad) ~ R In (Ap[M]/Ad)

U0)

At this temperature, the extrapolated Rp vs T and DP vs T curves will cut the temperature axis. This is illustrated in Fig. 1.1. At temperatures that are not far below the ceiling temperature, only polymers of low molecular weight form. It seems to be impossible to predict the variations of Rp and DP with temperature right up to T c from Eqs. (7) and (8). When kd approaches kp[Mx], DP becomes small and consumption of monomer in the initiation process is no longer negligible. Then, k p and k d may show a dependence on DP. Nevertheless the limiting slope of dRp/dT as T approaches T c may be numerically so large that such effects are of very minor importance and operate only over the last fraction of a degree below T c . The limiting slope for the rate vs temperature curve can be obtained by differentiating Eq. (7) with respect to temperature: dR p /dT = [M*](k p [M]E p /RT*-k d E d /RT 2 ) + (k p [M]-k d )d[M*]/dT and substituting kp[M] = kd when T = T c : T lim c (dR p /dT)

= kp[M][M*] (Ep - E d )/RT C

( n )

= k p [M][M*]AH p /RT c Then kp[M][Mj![] is the rate that would have been observed at T c in the absence of depropagation. From transition state theory the frequency factor is given by A = (kT/h) exp(AS*/R). Therefore,

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HIDEO SAWADA

TEMPERATURE

Fig. 1.1. Expected shapes of rate vs temperature and DP vs temperature graphs, without chain transfer to monomer. X = lim TT

AS0 = AS* - ASd = - R In A d /A p

(12)

Substituting Eq. (12) into Eq. (10), we find c

AS3 + R In [M]

(4)

Polymerization of monomer and depolymerization of the polymer may take place under various conditions. These are indicated in Table 1.1 for polymerization (for depolymerization, numerical values have the opposite sign). When the polymer is partially or wholly crystalline this may be denoted by c'. The superscript (°) for the

THERMODYNAMICS OF POLYMERIZATION. I

319

Table 1.1 Thermodynamic Definitions for Polymerization-Depolymerization Equilibria Notation

State of monomer

State of polymer

gg

Gas

Gas (usually hypothetical)

gc

Gas

Condensed (liquid or amorphous solid)

1c

Liquid

Condensed

Is

Liquid

Solution in monomer

ss

Solution

Solution

sc

Solution

Condensed

thermodynamic quantities indicates that the standard state is specified (e.g., ASgS refers to the standard state of 1 mole/liter of monomer). If a mixture of different monomers is the starting material, the standard state is unit concentration for each monomer. In general, a pure liquid monomer that gives an insoluble polymer will have a single well-defined ceiling temperature, given by T c = A H I C / A S I C . A pure liquid monomer that gives a soluble polymer will have a series of ceiling temperatures corresponding to different percentage conversions of monomer to polymer. The ceiling temperatures for a number of monomers in bulk are given in Table 1.2. The ceiling temperatures at equilibria involving monomers in solution have been extensively investigated and the results are summarized by Ivin [4]. C. Determination of Ceiling Temperatures The ceiling temperature for a monomer in its standard state of unit concentration is given by the term, T c = AHp/ASp. Thus, if the ceiling temperature cannot be measured directly, it may be estimated from AHp and ASp. Experimentally, the ceiling temperature may be estimated from plots of the rate of formation of polymer (or average molecular weight of the high polymers) against T [2]. As the ceiling temperature is approached, the rate decreases steeply. Extrapolation to zero rate gives, therefore, the required ceiling temperature. The accuracy of the method depends on the steepness of the descending line and usually it is good to about 2-3°C. Comparing Eqs. (4) and (10), we may write ASp = R In (A p /A d ) + R In [M] = ASp + R In [M]

320

HIDEO SAWADA

Table 1.2 Ceiling Temperatures of Pure Liquid or Gaseous Monomers in Bulk

Monomer Acetaldehyde n-Butyraldehyde Chloral Ethylene Formaldehyde Methacrylic acid, ethyl ester Methacrylic acid, methyl ester Methacrylonitrile Propionaldehyde Propylene Selenium Styrene Styrene, a-methyl Sulfur . Sulfur trioxide Tetrahydrofuran Trioxane a

Standard states

T

c

°C

Mole fractione (monomer at equilibrium)

Ref.

1

g

1 1

g h

a

Is Is Is gc gg gc gc

-31

126 133

760 760 760 760

gc

173

760 mm Hg f

m

gc

164 177

760 mm Hg f

n o

Is Is gg Is gg gc Is

-31a -39b 300C

1 1

g g

760 mm Hg f

j P

61

760 mm Hg f : 760 mm Hgf 760 mm Hg f

Is

159 d 30. 4 70 ± 5 80 ± 3

1 1 1 1

lc Is Is gc

-3& -16 96

407

83

c

d

235= 275C

36=

mm mm mm mm

Hg f Hg f Hg f Hg f

1

1

i j k 1

j j q r s t u

•

V

Atactic polymer. Isotactic polymer. c Calculated value. d Floor temperature. e Based on total monomer units. 1 Equilibrium pressure. g A. M. North and D. Richardson, Polymer, 6, 333 (1965). h O. Vogl, J. Macromol. Set., Al, 243 (1967). 'W. K. Busfield and E. Whalley, Trans. Faraday Soc,59,679 (1963). ••Reference [2]. k F. S. Dainton, K. J. Ivin, and D. A. G. Walmsley, Trans. Faraday Soc, 55, 61 (1959). b

THERMODYNAMICS OF POLYMERIZATION. I

321

Table 1.2 (continued) 'Reference [37]. R. E. Cook and K. J. Ivin, Trans. Faraday Soc, 53, 1132 (1957). n K. J. Ivin, Trans. Faraday Soc, 51, 1273 (1955). °S. Bywater, Canadian J. Chem., 35, 552 (1957). PA. Eisenberg and A. V. Tobolsky, J. Polymer Set., 46, 19 (1960). q j . G. Kilroe and K. E. Weale, J. Chem. Soc, 1960, 3849. r F . Fairbrother, G. Gee, and G. T. Merrall, J. Polymer Set., 16, 459 (1955). S D. C. Abercromby, R. A. Hyne, and R. F. Tiley, J. Chem. Soc, 1963, 5832. 'C. E. H. Bawn, R. M. Bell, and A. Ledwidth, Polymer, 6, 95 (1965). "Reference [42]. v Reference [18]. m

Therefore, an equilibrium concentration of monomer at a given temperature is given by

Rewriting Eq. (4), we obtain ln[M]=^P-^!p eq RTC R

(13)

Equation (13) gives the equilibrium monomer concentration as a function of the ceiling temperature. Thus, if the monomer is volatile, T c may be estimated from measurements of the equilibrium concentration (pressure) of the volatile species over the polymer at different temperatures. Since both AHp and ASp are functions of temperature, the correct method of evaluating T c is to plot AHp and T ASp against temperature and to identify the ceiling temperature with the points of intersection of these curves. The enthalpy and entropy of polymerization are generally not very dependent on temperature if phase changes do not take place. Thus, it is usually accurate enough to operate with values given for a definite temperature, e.g., 25°C. Since both AHp and ASp change in the same direction with temperature, their ratio changes only slightly.

322

HIDEO SAWADA

II. ENTROPY AND FREE ENERGY CHANGES OF POLYMERIZATION A. Entropy of Polymerization The entropy of a system is a measure of the statistical probability or degree of disorder of that system. The depolymerization of one polymer molecule to many monomer or oligomer molecules is accompanied by an increase in translational entropy since it is a dissociative process, whereas polymerization results in a decrease in translational entropy. The growth of oligomers should probably lead to a smaller change in the entropy of the system than the growth of analogous high polymers. The entropy values of polymerization for most monomers are quite similar. In vinyl polymerizations, for instance, the entropy of polymerization is within the limits of 25 to 30 eu for a wide variety of monomers, although the monomers have very different ceiling temperatures. The ceiling temperature should reflect the binding forces in a polymer, since depolymerization introduces disorder into the polymer, and a high binding energy between monomer units would tend to oppose the introduction of disorder. The higher the ceiling temperature, the stronger the binding forces between monomer units in the polymer. Thus, ASp = AH p /T c = 25-30 cal/°K/mole

(14)

An empirical relation between the energy of vaporization and the boiling temperature is known as Trouton's rule. Equation (14) is analogous to Trouton's rule. Steric hindrance in the polymer, which markedly affects AHp, has comparatively little effect on ASp. Steric hindrance may decrease the rotational entropy of the polymer chain by increasing its rigidity and, hence, it could increase the value of -AS p . The loss of internal rotational entropy in the polymer as a result of steric hindrance seem to outweigh the gain of internal vibrational entropy. However, the observed changes in entropy of polymerization are relatively insignificant, e.g., - A S p for styrene polymerization to a solid polymer amounted to ~ 25 eu, whereas the corresponding value for a-methyl styrene was found to be ~ 26 eu. This is, indeed, a small change when compared with a 10 kcal/mole decrease in the heat of polymerization. Therefore, side-group steric hindrance serves to lower the ceiling temperature further with increasing steric repulsion of the side groups.

THERMODYNAMICS OF POLYMERIZATION. I

323

Monomer resonance implies higher bond energies and relatively large vibration frequencies of the bond within the monomer. For example, a bond which is increased in strength from 80 to 88 kcal and in vibration frequency from 1000 to 1100 cm' 1 will change its entropy contribution at room temperature from 0.094 to 0.065 cal/ deg mole. The total effect of monomer resonance will always be a reduction of the internal entropy of the monomer. Therefore, relatively small entropy changes of conjugated vinyl monomers and styrene are probably due to the decreased total internal entropies of these monomers. The entropy contribution to the free energy change at 25°C will vary only within 7.4 to 9.0 kcal/mole. Structural influences on AG, thus, operate mainly through the enthalpy terms. Dainton and Ivin [2] analyzed AS|g in terms of the component entropy changes, translational ASj., external rotational AS r , vibrational ASV, and internal rotational ASj r , in order to try to understand why the entropy is an approximately additive property, leading to approximately constant values of ASgg. Values of St, S r , and S v for the monomer can be calculated from standard formulas, since its molecular weight, moment of inertia, and vibrational frequencies are known. Sj r is found by difference (S^ + S r + Sv) and the experimental (third-law) value for Sg (monomer). For the polymer, it is readily shown from standard formulas that S v + Sj r » St + S r regardless of molecular shape, so Sj + S r may be neglected. Thus, S v + Si r is the standard entropy of the gaseous polymer, which must be found semiempirically. The numerical values for ethylene, isobutene, and styrene are summarized in Table 1.3. These values show that on polymerization the loss of external rotational entropy nearly balances the gain in vibrational and inTable 1.3 Analysis of AS g g for Vinyl Polymerization 3 (cal/deg/mole)

Substance Ethylene Isobutene Styrene

M

St

,, Monomer Sv Sr

Polymer . unit, + Sir -ASgg Si g =Sv

Sir

S|

0.6

0

52.4

18 .4

34.0

28. 05

35 .9

15.9

56. 10

38 .0

23.1

-

9.1

70.2

29 .2

41.5

104. 14

39 .8

27.9

10.1

4.7

82.5

47 .0

35.5

"Values from F. S. Dainton and K. J. Ivin, Quart. Rev., 12, 81 (1958).

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HIDEO SAWADA

ternal rotational entropy, so that -AS|g has a value quite close to the monomer's translational entropy; this is fairly insensitive to the molecular weight or structure of the monomer. Joshi and Zwolinski [5] have observed that the variation of the polymerization entropy bears some relationship to the molar volume change of polymerization and may be empirically fitted to a linear relationship, approximately given by the equation AS = 25 + \ R In (Vi/V-j), where Vx and V2 are the unit volumes of monomer and polymer, respectively. Entropy of stereoregularity is discussed later. B. Determination of Entropy of Polymerization The entropy change can be calculated from the measurement of equilibrium monomer concentration, from the entropies of monomer and polymer, from the ratio of the kinetic frequency factors Ap and Aj, and from empirical and statistical calculations [1-3]. 1. Measurement of Equilibrium Monomer Concentration. Equation (13) gives the equilibrium monomer concentration as a function of the ceiling temperature. Instead of saying that a monomer at concentration [Mx] has a ceiling temperature T c , it will frequently be convenient to reverse the viewpoint and say that at temperature T the monomer concentration in equilibrium with the long-chain polymer is [ M j e , where t = T c and [ M j e = [Mj = exp {(AH°- T AS°)/RT}. Measurements of equilibrium monomer concentrations at two different temperatures, at least, permit evaluation of AH and AS from the slope and intercept, respectively, of the plot of In [M] vs l / T as related by Eq. (13). 2. Enthalpy Change and One Equilibrium Temperature. This method is based on the relation AS = AH/T, and it is only limited by the accuracy of the measurements of heats of polymerization and ceiling temperatures. 3. Entropies of Monomer and Polymer. Standard entropies of monomers can be determined from specific and latent heat data, and the standard entropies of some gaseous monomers could be derived from spectroscopic data. The estimation of entropies from specific heat data is based on the third law of thermodynamics which states that the entropy of perfect crystal is zero at 0°K. Since polymer molecules have a residual entropy at 0°K, polymer molecules cannot achieve perfect crystallinity at the absolute zero. Temperley [6] points out that the four most relevant considerations regarding this problem are that: (a) a polymer is a mixture of chains of many different lengths; (b) since crystallization i n a polymer

THERMODYNAMICS OF POLYMERIZATION. I

325

is never complete, at least some of the polymer is present in the amorphous or disordered state—the communal entropy problem; (c) because of "freedom at the joints," any one chain may assume a wide variety of configurations in space; and (d) in a copolymer the number of ways in which the two sets of monomer can be linked to form a chain is infinitely great. Temperley [6] concluded that for a hydrocarbon-like polymer the contribution to S from (a) and (b) is no more than 3- 5 times k per polymer chain and the contribution from (c) is no more than k/5 per link; for a rubberlike polymer the entropy associated with chain configurations is no more than k per monomer unit. The general validity of these conclusions is borne out by studies on natural rubber and isotactic and atactic polypropylene, which indicate that the residual entropy of these polymers is less than 1 cal/deg/mole. Data have become available on the entropies of the following polymers by the specific heat method: polythene [7,8], isotactic and atactic polypropylene [9], isotactic polybut-1-ene [9,10], polyisobutene [11], isotactic and atactic polystyrene [9], cis- and trans-1,4polybutadiene [12], polyisoprene [13], polyvinyl alcohol [14], poly(vinylidene chloride) [15], polymethyl methacrylate [16], polyoxymethylene [17,18], polytetrafluoroethylene [19], poly-3,3-bis(chloromethyl) oxacyclobutane [20], propene, but-1-ene and hex-1-ene polysulfones [21], polycarbonate [22], ethylene-propylene copolymers [23], and poly(-4-methyl-l-pentene) [24]. These results are summarized in Table 1.4. •The entropy of polymerization can be obtained by subtracting the entropy of the monomer from that of the polymer. All available results obtained by this method are summarized in Table 1.5. Second-law entropies of polymerization can be obtained from measurements of equilibrium pressure (or concentration) or ceiling temperature as discussed in Sec. II. B.I. The only systems for which both second- and third-law entropies of polymerization are available are shown in Table 1.6. To obtain a direct comparison between a second- and third-law entropy of polymerization, it is essential that specific heat measurements on a given polymer should be combined with monomer-polymer equilibrium studies on the same polymer. From such data it would be possible to estimate the residual entropy of the polymer. However, in no cases were the second- and thirdlaw entropies of polymerization evaluated for the same monomerpolymer system. Thus, although the small difference between the two values for styrene may be due to the possession of a residual entropy by the polymer, it may also result from different tacticities or crystallinities of the polymers used in the two sets of measurements [30].

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HIDEO SAWADA

Table 1.4 Entropies of Linear Polymers (cal/deg/mole, 25°C) Polymer Polyethylene

Isotactic polypropylene Atactic polypropylene Isotactic polybut-1-ene Isotactic polystyrene Atactic polystyrene Polyisobutene cis-1,4- Polybutadiene trans-1,4- Polybutadiene Polyisoprene Polymethyl methacrylate Polyoxymethylene Polytetrafluoroethylene Poly-3,3-bis(chloromethyl) oxacyclobutane Propene polysulfone But-1-ene polysulfone Hex-1-ene polysulfone Poly(4,4'-dioxydiphenyl2,2-propane carbonate) Ethylene-propylene copolymer (containing 31 mole % propylene) Isotactic poly(4-methyl1-pentene) Polyvinylidene chloride a

At 317°K. At 280°K. c At 200°K. b

% Crystallinity 0 58 79 100 0 48 100 0 16 100 0 44 100 0 43 100 0 100 0 -

Highly crystalline Highly crystalline

Ref.

14.8 12.4 11.7 11.0 19.4 17.2 14.8 19.6 18.9 15.0 27.6 24.6 20.7 32.0 31.4 30.7 32.2 30.9 22.9 27.5 23.2a 30.6 34.2 10.3 10.6 24.1b

[9]

m 19] [9] [9] [9] [9] [9] [9] [9] [9] [9] [9] [9] [9] [9] [9] [9]

[11] [12] [12] [13] [16] [17,18] [19]

-

45.4 32.1 37.2 52.8

[20] [21] [21] [21]

—

75.3

[22]

-

15.1

[23]

65 —

36.2 17.0c

•

[24] [15]

THERMODYNAMICS OF POLYMERIZATION. I

327

Table 1.5 Entropies of Polymerization (cal/deg/mole, 25°C)

Monomer Ethylene Ethylene Propylene Propylene Propylene But-1-ene But-1-ene Isobutene Styrene Styrene Styrene • Styrene Buta-l,3-diene to the cis-l,4-polymer Buta-l,3-diene to the • cis-l,4-polymer Isoprene Methyl methacrylate Formaldehyde Trioxane Tetrafluoroethylene 3,3-Bis (chloromethyl) Oxacyclobutane Vinylidene chloride 4- Methyl- 1-pentene

Standard states

-AS

Ref.

17]

lc lc

41.5 37.7 49.0 44.4 45.7 52.3 45.4 28.8 24.9 25.2 26.7 26.5

lc

21.2

[3]

lc lc lc gc gc lc lc

20.1 24.2 9.6a 41.8 37.2 26.81> 19.9

gc gc

21.2= 51.7

gc gc gc gc gc gc gc lc lc lc

[V] [9] [9]

[25] [9] [9]

[11] [3] [9]

[26] [9]

[12] [3]

[27] [17] [28] [3]

[20] [15] [24]

"At 210°K. Atl98°K. = At 200°K. b

4. Ratio of the Frequency Factors of the Propagation and Depropagation Reactions. By comparing Eqs. (4) and (10), Dainton and Ivin [2] state that the entropy of polymerization is given by AS° = R In (Ap/Ad)

(15)

where Ap and A^ are the frequency factors for the propagation and depolymerization reactions, respectively. Assuming that A^ is 1013 sec' 1 , i.e., the reaction is a normal unimolecular one, they have com-

HIDEO SAWADA

328 Table 1.6

Comparison of Entropy Data for Polymerization 8 (cal/deg/mole, 25°C)

states

Second-law entropies

Third-law entropies

Ref.

Styrene

lo

25.0

25.8

[3,9,26]

Formaldehyde

gc

41.8

30.7

[17,29]

Standard Monomer

"Values from Ref. [30].

puted the change in entropy in passing fro"m the liquid monomer to a molar solution of the polymer in the monomer. For vinyl acetate this figure is-26.2 cal/deg mole (Ap = 1.65 x 108 mole"1 sec"1) and for styrene, -27.9 cal/deg mole (Ap = 1.0 x 106 mole"1 sec"1). 5. Empirical Estimation. The methods available for evaluating AS|g empirically are in principle the same as those for the estimation of AH|g. Thus, Jessup [31] has utilized a relation connecting AGf for the formation of the mono-olefins and the number of carbon atoms to calculate AG™ for the polymerization of ethylene. The values of AS|g may then be obtained in the usual way. The calculated entropies of polymerization show better agreement with the experimentally found values than do the predicted heats of polymerization, because the entropy change in polymerization is not as sensitive to monomer structure. 6. Statistical Calculations. The factors that determine the magnitude of AS|g may be appreciated by considering the translational, rotational, and vibrational contributions separately [2]. For the dimerization, the translational contribution ASgg(t) is given by AS gg(t) = - | J | R In 298.1-2.298 J + | R In (2m) - | R In (m) = -11/967 - 3.4305 log (m) cal/deg mole

(16)

where m is the molecular weight of the monomer. The external rotational contribution AS| g ( r ) is given by AS

ee(r) = - 5 - 8 0 + 2 - 2 8 7 log

(T

T

UAiB

T

U/S

*c^

n-l/2

- 4.575 \og-?—

(17)

where a denotes symmetry number, IA moment of inertia in atomic

THERMODYNAMICS OF POLYMERIZATION. I

329

mass A2 units, and the subscripts 1 and 2 refer to monomer and dimer, respectively. The vibrational contribution AS° (r) is

(18) where a = 4.827 x 10' 3 cm, v denotes a vibration frequency, and the summations are taken over all vibrations of the two molecules concerned. Owing to the lack of values of the moment of inertia, symmetry numbers and vibration frequencies of the commonly used monomers and their dimers, expression (16), (17), and (18), do not seem to be useful; Moreover, all internal rotational contributions to the entropies of the monomer and dimer have been included as vibrations in [18]. Nevertheless, analysis of AS|g for the dimerization in this way has advantages. Thus, a consideration of the vibration frequencies assigned to aliphatic hydrocarbons by Pitzer [32] immediately indicates that the total vibrational, including internal rotational contributions, of most monomers or dimers at room temperature is only about 10-15% of the total entropy. Also there will be little change in external rotational entropy on dimerization for those systems in which a! =I

+ ? £ (An) - R E (AX)

(19)

330

HIDEO SAWADA

where RA is the contribution to the entropy of one vibration and y and z denote the number of such terms appropriate to polymer and monomer, respectively; it will be noted that

When n is very large AS° = -S° of the monomer plus the sum of the vibrational and internal rotational entropy of the monomer unit in the polymer. In ethylene, AS| g = -34.1. whereas S | for the monomer = 52.5 cal/deg mole, and we, therefore, conclude that this latter quantity is of the order of 18 cal/deg mole. The same problem has been treated by Evans and Baxendale [33]. Their treatment leads to a value for the frequency factors of propagation and termination given by log A p = 1 - log (fm / f p ) - log w + log kT/h

(20)

log At = 1 - log wr + log kT/h

(21)

and the ratio of the two frequency factors is, thus, found to be A p /A t = fp/fm

(22)

The entropy of polymerization ASJS is given by ASJS = R{1 - log i/o- - log (f m /f p )}

(23)

where v is the coordination number of the space lattice, cr is the symmetry number of the polymer molecules, and fm and fp are the respective partition functions of the free monomer units and of the monomer segments in the polymer. This expression refers to bulk polymerization and a standard state of 1 mole/liter. The same difficulty arises, namely, that the vibration frequencies of the monomer units in the polymer chain are unknown, and, therefore, fp and ASjs cannot be evaluated. C. Free Energy Changes of Polymerization Since AG° = A H 0 - T AS D

it is easy to calculate AG° at a given temperature if AH" and AS° are known. The results of such calculations for all cases in which the

THERMODYNAMICS OF POLYMERIZATION. I

331

necessary data are available are shown in Table 1.7. We get for AG° of a polymerization reaction at equilibrium, AG° = - R T In K a

(24)

where K a is the thermodynamic equilibrium constant of polymerization. Under the equilibrium conditions the rates of forward and reverse reactions occurring at all active centers M*, M* M , etc., may be equated: =kdf)

p

[M*]

(25)

For a high-molecular-weight polymer

£ tMn] = S [Mn] n+1

n

and, hence,

• ^

= K a =[M e r

(26

>

where [M] e is the equilibrium monomer concentration. If polymer solutions were ideal, the equilibrium condition may be established at the ceiling temperature T c . Thus, AG° = - R T C In K a = RTC In [M]e = AH°-T C AS°

< 27 >

Alternatively, this may be written T

• c

AH

P

AS^+ R l n [ M ] e

and [M] e = exp(AG 0 /RT c )

(28)

Equation (29),known as the van't Hoff reaction isobar,defines the temperature coefficient of K a in terms of the heat of reaction AH° and the temperature T: 3 l n K a _ AH° 3T RT2

(29)

332

HIDEO SAWADA

Table 1.7 Heats, Entropies, Free Energy (changes , and Equilibrium Constants for Polymerization at 25°C Standard Monomer Tetrafluoroethylene Ethylene Vinyl acetate Propylene

Butadiene Butene-1

Isoprene Styrene Methyl methacrylate Ethyl methacrylate Isobutylene a-Methyl styrene Vinylidene 'chloride Ac enaphthy lene Formaldehyde

Tetrahydrofuran Trioxane a b

At 265°K. At 200'K.

state

-AH"

-AS'

-AG»

1c

37

29

gg lc lc lc lc gg lc gg lc lc lc lc gg

22.2 25.9 21.2 19.5 24.9 20.7 17.6 19.1 19.0 20.7* 17.9 16.7 18.1

26.8 34.0 41.5 26.2 27.8 49.0 39.9 20.5 39.8 26.8 29.8 a 24.2 25.0 35.5

lc lc

13.2 13.8

28.0

4.8

9.6

11.8

—

lc lc lc gg lc

13.8 12.9 17.2

29.7 28.8

4.9 4.3 3.9

5.0 x 104

lc gg gc gc gc lc

14.4 b 20.4 17.2 13.2 13.2 5.3

18

7.0 4.1 0.7 0.9 0.0

gc

3.83

12.39

0.14

8.1 8.4

35

24.8 21.17 b 45

43.8 41.8 41.8

12.1 13.5 13.4

Ka

Ref.

3.0 x IO22

[34] [35] [7,36] [34] [34] [9,36] [35] [34] [35] [34] [10] [34] [34] [35]

—

7.1 x 1010 2.0 x io 9

8.8

-

11.5

2.9 x 109

7.2 -

1.3 x io 9

10.3

12.8 a 10.7 9.2 7.5

-

7.5 x 108 5.0 x io 7 -

3.5 x 104

-2.3 1.0

4.1 x 10

10.2 b

1.0 x 10

[34] [27] [34] [34] [36] [35] [34] [15] [35] [37] [17] [36] [34] [18]

THERMODYNAMICS OF POLYMERIZATION. I

333

For polymerization reactions, when K a = [M]"1, Eq. (29) may be written

9T

RT2

Polymer solutions are, however, rarely ideal, and equations more complex than Eq. (28) are usually needed to derive heats and entropies of polymerization from ceiling temperatures. Bywater [38] and Small [39] have given expressions for nonideal solutions. Considering the polymerization of 1 mole of monomer to 1 base mole of polymer, the change per mole in free energy is given by AG = AG° + RT In (a p /a m )

•

(31)

where a p is the activity per base mole of the polymer and a m that of the monomer; AG° is the standard free energy change with the compounds at unit activity. This corresponds to AGic of Dainton's notation if the activities of monomer and polymer in the pure liquid and solid states, respectively, are taken as unity. If the standard state is defined as a 1 M solution, the relation becomes AG = AGSS + RT In (a p /a m ) + RT In (a m /a p )

(32)

where a p and a m are the activities in the standard 1 M solution and AGSS is the change in free energy with reactants and products in 1 M solution. Since polymer solutions do not behave ideally, it is necessary to modify the above equations by relating activities to concentrations using the Flory-Huggins-type equations. In order to make the equationsusable, it is necessary to use simplifying assumptions. Thus, for this purpose, it is assumed that the ju has the same value for the monomer-polymer and solvent-polymer interactions and the heat of mixing of solvent with monomer will be neglected. With these assumptions, the equations relating activity and volume fraction 6 become formally equivalent to the simple Flory-Huggins equations [40]: In a m = In P

(33)

In a p = (l/x) In p + (l/x - 1)(1 - 6p) + M(1 - p)2

(34)

where x is the ratio of the molar volumes of polymer and solvent, but $ m + $ p is no longer equal to unity as in the simple case. Graph-

334

HIDEO SAWADA

ical plots of these functions in the concentration ranges of a large excess of solvent show that these equations reduce with sufficient accuracy to In a m = In 0 m ;

In a p = /i - 1

(35)

Equations (35) are a good approximation for x > 100 and give a reasonable approximation for x values as low as 10. Equations (31) and (32) become, on substitution of Eq. (35), AG = AGlc + RT In Ox - 1) - RT In 0 m

(36)

AG = AGSS + RT In (l/[M])

(37)

where [M] is the molar concentration of monomer. At equilibrium, AG = 0 and concentrations become equilibrium concentrations, hence, -AG l c /RT = (M - 1) - In a n ^ Msm c o r r e s P ° n d i n g to the pairs solvent-polymer, monomer-polymer, and solvent-monomer. At equilibrium, -AG/RT = ln a p - ln a m

where ap and a m are activities of polymer and monomer, respectively. The activities refer to liquid monomer and solid polymer as their respective standard states: -AG l c /RT = (fj sp - Msm)0s ~ 1 ~ l n

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 —

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.

372

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

376

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

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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).

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[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:

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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.

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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,, .. .

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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

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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-

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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/& 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

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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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270

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

270

SAWADA

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-

278

SAWADA

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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SAWADA

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,

SAWADA

282

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.

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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

>•>

(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 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