Thermodynamics of copolymerization

Prog. Polym. Sci., Vol. 17, 917-951, 1992 Printed in Great Britain. All rights reserved.

THERMODYNAMICS

0079-6700•92 $15.00 © 1992 Pergamon Press Ltd

OF COPOLYMERIZATION

RYSZARD SZYMANSKI Polish Academy of Sciences, Center of Molecular and Macromolecular Studies, Sienkiewicza 112, 90-361 Lodz, Poland

CONTENTS 1. Introduction 2. Equilibrium in copolymerization. General outlook 3. Analysis of copolymerization thermodynamics. Previous results 4. Dyad model copolymerization 4.1. Application of equations to real systems 4.2. Recent analogous treatments of other authors 5. Triad model copolymerization 5.1. Application of equations to real systems 6. Chain-ring equilibria in equilibrium copolymerization systems 7. Copolymerization equilibrium in systems with physical interactions 8. Homopolymerizations which can be regarded as copolymerizations References

917 919 921 926 929 929 933 937 938 940 944 950

1. I N T R O D U C T I O N

Studies of thermodynamics of copolymerization processes provide data on equilibrium concentrations of comonomers and microstructure of the resulting copolymer. This allows one to determine the enthalpy (AH) and entropy (AS) of homo- and cross-propagations (other thermodynamic functions can be determined as well). This information, in turn, enables one to predict the values of propagation equilibrium constants for the copolymerization systems considered. Consequently, one can predict whether copolymer can be formed from the given mixture of comonomers at the chosen conditions (provided the appropriate mechanism of copolymerization is available) and also the composition and microstructure of the resultant copolymer. Besides, one can estimate the maximum (equilibrium) degree of comonomer conversion. Polymerization thermodynamics has been reviewed several times ~-6 but thermodynamics of copolymerization was treated in such reviews marginally. The only exception is the review by Sawada z covering papers appearing before 1974. The present review, after introducing general concepts, is mostly based on the papers published during the last decade. Generally, in copolymerization reactions, as in homopolymerization, the molar Gibbs free energy can be defined as the difference between the Gibbs free

917

918

R. SZYMANSKI

energy (in the subsequent text called simply free energy) of 1 base-mole of copolymer and the average molar free energy of the comonomers:

AGx~ = Gy - Gx =

( l / n ) . ,~cop(y)~(")- fAaA'(x) -- faGB(x) . . .

(1)

where x and y describe the standard states of comonomers (x) and copolymer (y), Gy --" ,~copy(~)c:-(") ~,,t"is the free energy of 1 base-mole of copolymer (n = degree of polymerization, G~"o)p~y)= molar free energy of copolymer); Gx = fAGA~x~ + fa GB(~) + • • • is the average free energy of the comonomers (/cA,fB = fractions of different comonomer units in copolymer; GA(x), GB~) = molar free energies of comonomers A and B). Usually the following notation of phase states (assumed as standard states of comonomers or copolymer) is adopted: 1 = liquid, c = condensed, c' = crystalline, s = in solution, etc. Equation (1) is based on the following stoichiometric equation of copolymerization: 1

fA A + fBB + . . . --* - {AfABf, n

fA + f~ + . . . .

. . . }~=)

(2)

1

where n is the average degree of polymerization, {AfABfB... } denotes the composition, while index (~) denotes the specific microstructure (which can be specified) of copolymer. From the view-point of thermodynamics, copolymerization (2) can spontaneously proceed only if AG~y < 0. If AG~y > 0 the reverse reaction (depolymerization) is thermodynamically spontaneous. When reaction (2) is in equilibrium and the forward and reverse reactions are counterbalanced, then =

0.

However, the conditions stated above (AG~y < 0, AGxy > 0 and AG~y = 0) are not sufficient for copolymerization to proceed in the direction indicated above (even if the reaction route is available) or to be at equilibrium. This is an important difference between homopolymerization and copolymerization systems. The new methods of analysis of equilibrium copolymerization systems outlined in this review allow one, on the basis of experimental data (such as comonomer equilibrium concentrations and microstructure of the copolymer), to determine the thermodynamic parameters of copolymerization. On the other hand, if the thermodynamic parameters set of copolymerization are known (e.g. determined in the limited set of starting conditions or calculated on the basis of theoretical considerations), the relations presented in the paper allow one to predict whether the reaction can occur at the conditions considered, and what would be the equilibrium concentrations ofcomonomers and the microstructure of the resultant copolymer. The new treatment of equilibrium copolymerization is more convenient in practical adoption to experimental data than the older methods. Some problems,

THERMODYNAMICS OF COPOLYMERIZATION

919

e.g. those connected with copolymeric macrocycles, have been solved for the first time. 2. E Q U I L I B R I U M

IN COPOLYMERIZATION.

GENERAL

OUTLOOK

The above-mentioned insufficiency of the condition AGxy = 0 for a copolymerization system to be at equilibrium stems from a higher number of degrees of freedom for the systems considered than for homopolymerization systems. The copolymerization-decopolymerization reactions can proceed not only according to the formulated eq. (2) (forward or backward), but also according to an infinite number of analogous reactions of different stoichiometry and/or microstructure of copolymer: ~- {AgA, BT,,. . . }~')

fA, A + UB, B + . . .

(3)

(for the sake of simplicity the degree of polymerization is not indicated). In true equilibrium all these reactions are counterbalanced and all AGxy(,)

=

0(i

=

1,2 . . . .

).

(4)

Thus, in the true copolymerization equilibrium system various copolymer chains and comonomer molecules coexist, and the rate of formation of any kind of copolymer chain is equal to the rate of its decay. If one removes from the equilibrium system some kind of copolymer chain (replacing them, e.g. by solvent), the system will not be in equilibrium any more, although all partial equilibria (3) for remaining copolymer chains will still be maintained, provided that solvent interacts physically with comonomers and copolymer similarly to the removed copolymer chains. Consequently, although for all remaining chains the free energies of copolymerization would be equal to zero (AGx),(i) = 0), similarly to the averaged free energy for all remaining copolymer (AGe> = 0), the system would return to equilibrium only by copolymerization of certain amounts ofcomonomers (or some of them) and decopolymerization of a certain amount of some copolymers. The driving force for such reactions would be the negative free energy of the corresponding complex (composed of several separate processes) reaction, e.g.: XAA + x B B + . . .

AG

=

GLp(y) -

+ Xcop{ArABf,,. XcopGcop(>)

-

. . }(~') ~ {Ar, ABu,,,.. . }(~")

ZAGA(x)

--

xBGBtx)...

< 0

(5) (6)

(reactions of different stoichiometry, e.g. those with comonorners on both sides of the equation, can be considered as well) where {Ay, ABI, B . . . }("> and { A f . A B ~ . B . . . )(~'~ denote the composition and microstructure of copolymers, substrate and product, respectively; G~op(y)and G~op(y) are the free energies of corresponding copolymers; XA + XB + . . . X~op = l;fA +.f~ + . . . . I;JA' + f~' + . . . . I ; XA + X~o~rA ---- fA'; etc. Equation (5) and (6) can be applied to the specified chain types and the whole copolymer treated as one category, as well.

920

R. SZYMANSKI

Thus, the sufficient condition for copolymerization to be at equilibrium is the equality to zero of free energies of allpossible reactions involving copolymerization. Although there are an infinite number of such equilibrium reactions, the adoption of the reversibility-of-microstates rule allows one to decrease the member of considered equilibria to a relatively small number depending on the assumed model of copolymerization. The simplest model of reversible copolymerization is the terminal model not distinguishing the nature of the active centers of propagation: C* + A ~ CiA*; KA C* + B ~- CiB*; KB

(7)

C* = Ci_IA* + Ci_l B* i =

1,2...

where C* denotes the copolymer chain composed of i comonomer units ending in the active center; Kx = [CiX*]/([C*][X]) are the equilibrium constants of propagation (X = A,B; [] denote the equilibrium concentrations, approximating equilibrium activities, of indicated species). This model was originally developed by Tobolsky and Owen 7 for copolymerization of sulfur with selenium. More complicated but much more useful is the terminal model distinguishing the nature of active centers (usually called simply the terminal or dyad model), comprising four equilibria in the case of a binary copolymerization: •

.

,

- AA*; KAA

.

.

,

-

~

.

.

.

-- BA*; KBA

~

.

.

.

-

-A* +A-"

•

.

.

~

•

.

.

•

.

.

-B*

+

•

.

.

-B*

+B-"

- A * + B --~ ~

A --"

AB*; KAn

BB*;

(8)

KBB

w h e r e . . . - X* a n d . . . - XY* denote the copolymer chains with the indicated structure of the active chain-ends; Kxv = [.. • - X Y * ] / ( [ . . . -X*][Y]) is the equilibrium constant of the given kind of propagation• If penultimate (or more remote) units affect the reactivity of active centers, and consequently make the corresponding equilibrium constants different, the models of copolymerization must be expanded to contain more equilibria• For instance, the penultimate-unit effect model of equilibrium binary reversible copolymerization (also called the triad model) comprises eight equilibria described by the following equations: • .. - X Y * + Z ~ . . .

--XYZ*;Kxvz

(9)

where X, Y, Z = A or B; Kxvz = [ . . . - X Y Z * ] / ( [ . . . -XY*I[Z])

(10)

THERMODYNAMICS OF COPOLYMERIZATION

921

(e.g. /(ASS = [. • • -- ABB*]/([... - AB*][B])). As will be shown in the following sections, the equilibrium constants defined in each particular model of copolymerization allow one to calculate the concentrations of any entities present in the equilibrium system, which is equivalent to the knowledge of the equilibrium constant of any copolymerization equilibrium defined by eqs (2) or (3). Consequently, one can state that a few basic equilibrium constants of copolymerization determine all the equilibrium features, such as the equilibrium concentrations of comonomers and the molar fractions of different sequences in the copolymer. 3. A N A L Y S I S

OF COPOLYMERIZATION THERMODYNAMICS. PREVIOUS RESULTS

In this section the results of thermodynamic analysis of equilibrium copolymerization published before 1980 are briefly reviewed. More details, concerning discussed papers that appeared before 1974, can be found in the review by Sawada. 2 Among the first papers in this field were those published by Ivin and c o w o r k e r s , 8-j° w h o studied the first known systems of equilibrium copolymerization, namely copolymerizations of alkenes with sulfur dioxide: O

/

\

l

II I I

SO2 + C = C ~ - 4 ~ S - C - C ) , . / "\ n l l l l

(11)

O They observed that the resulting copolymers are alternating ones, and for a

given temperature the products of the comonomer equilibrium concentrations are constant independently of the ratio of initial comonomer concentrations: [A]e[B]e =

K.

(12)

This relation is in agreement with predictions from thermodynamic analysis o f reaction (11). The Gibbs free energy of the reaction is at equilibrium equal to zero, which fact leads directly to the observed relation: AG AH

=

TAS

=

AH-

TAS

=

0

= T ( A S ° + R ln[A]e + R In [B]e)

[A]e[B]c =

exp(AG°/RT)

(13) (14) (15)

where AG ° = A H - T A S ° is the molar standard free energy of copolymerization. Rearranging eq. (14), one can derive the equation enabling one to calculate the ceiling temperature of alternating copolymerization: Tc

=

A H x / ( A S ° x + R ln[A]o + R ln[B]o)

(16)

922

R. SZYMANSKI

where subscripts "o" indicate initial concentrations while indexes x denote the standard states of copolymer and comonomers (ss, lc, Is, etc). An alternating equilibrium copolymerization is a special case of equilibrium copolymerization which resembles homopolymerization because of the strongly limited number of degrees of freedom of the systems considered. Actually, such a reaction could be treated as homopolymerization of the (AB) complex constituted of the two comonomers A and B. Only for such copolymerizations is it possible to formulate simple equations for the ceiling temperature or for the product of equilibrium comonomer concentrations (similar to analogous equations for homopolymerization). When the equilibrated copolymer is not alternating, the set of several equations has to be considered simultaneously in order to calculate specified features of copolymerization at equilibrium. Several authors studied theoretically the general case of equilibrium copolymerization. Alfrey and Tobolsky 11 were the first ones who derived the equations indicating that at equilibrium in binary copolymerization, if only interactions between neighboring units have to be considered, the ratio of products of numbers of homogeneous and heterogeneous dyads is equal to some thermodynamic constant. However, it was O'Driscoll and coworkers 12who found, not presenting, however, the strict proof of such, that this constant is a simple function of the propagation equilibrium constants of dyad model propagation:

NAANs.

=

KAAK.~

K-

KA. K.A

NAsN.A

(17)

(NAA, NBB, etc = numbers of corresponding dyads in equilibrium copolymer; K ~ , KBB, etc = the propagation equilibrium constants of copolymerization scheme (8)). O'Driscoll and coworkers,12 analyzing a binary equilibrium copolymerization with the assumption of high molecular weight of copolymer derived the following set of equations: [A]¢ = KAsK.A

e/Kaa,

[B]e =

q/KBB

= (1/[Ale -- KAA)(1/[B]e -- Kss) KAsKsA

KAA KBB

(18) (19)

1 -- e 1 -- t1 -

8

r1

(20)

where e = [ - AA*]/[- A*] and r/ = [ - BB*]/[- B*] are the conditional probabilities that the terminal units (A and B, respectively) of active centers will be preceded by another unit of the same kind. It was assumed by the authors, and appeared to be true, 13that the conditional probabilities at the chain end (e and t/) are equal to the corresponding conditional probabilities throughout the main chain ( [ - A A - ] / [ - A - ] = [ - AA*]/[- A'l, etc). Consequently, the addition,

THERMODYNAMICS

OF COPOLYMERIZATION

923

to any three equations from the presented four (eqs (18-20) are not independent), one of the mass-balance equations: [A]o =

[A]~ + XA{[A]o + [B]o -- [A]e -- [B]¢}

(21)

[B]o =

[B]e + Xa{[A]o + [B]o - [Ale - [B]e}

(22)

or

(where XA = [ - A - ] / ( [ - - A - ] + [ - B - - ] ) = (1 - q ) / ( 2 - e - q), X~ = 1 XA; eqs (21) and (22) are not independent, as well), enables us to calculate the equilibrium comonomer concentrations and conditional probabilities (the set of four independent equations with four unknowns - [Ale, [B]~, e, q - can be solved numerically). As can be seen from eq. (19), the relationship between equilibrium concentrations of comonomers and thermodynamic parameters of copolymerization is generally not as simple as found by Ivin and coworkers for copolymerization of sulfur dioxide with alkenes (cf. eqs (13-15)). However, eq. (19) reduces to eq. (12) when one assumes that KAA = /(an = 0. By replacing the propagation equilibrium constants in eq. (19) by the corresponding functions of thermodynamic parameters (Kxv = e x p ( - A H x v / ( R T ) + AShy/R); X, Y = A or B) one can derive the relationship enabling one to calculate numerically the ceiling (or floor) temperature for the initial concentrations of comonomers ([A]e = [A]o, [B]c = [B]o). O'Driscoll and coworkersl4 ~6] studied more thoroughly the equilibrium copolymerization not only of binary but also of multicomponent systems, with dyad as well as triad model propagations. Besides equations describing some relationships at equilibrium, they derived equations for the instantaneous composition of copolymer formed in such copolymerizations. For instance, for binary dyad model copolymerization the copolymer composition equation 12 is given by: d[A] d[B]

a*q(kAA[A] -- ek AA) --

b*~(kaa[B] - r / k a . )

(23)

where a* = [ - A * ] / ( [ - A*] + [ - B * ] ) , b* = [ - B * ] / ( [ - A * ] + [ - B * ] ) ; kxv and k xv (X,Y = A or B) are the rate constants of corresponding propagations and depropagations, respectively (Kxv = kxv/k xv; cf. eq. (8)). For steady-state conditions the following relations are maintained: a* + b* =

1

a*[kAa[B] + (1 - Ok .A] = b'e(1 - q)k

AB

=

b*[kBA[AI + (1 - q)k AB]

a*[e(kAA[A] + k AA -~ kAB[B]) (24)

-- (kAA[A] + e2k-Ag)] a'r/(1 --

e)k

aA =

b*[r/(kan[B] + k_aa + k a g [ A ] )

(kBa[B] + q2k aa)]

924

R. SZYMANSKI

The set of eqs (23) and (24) allows one to simulate numerically the instantaneous composition and microstructure of copolymer (as well as the kinetics of reversible copolymerization without side reactions). By assuming some relations between rate constants, one can reduce the equations derived by O'Driscoll and coworkers to those presented by Lowry 17for copolymerization with reversibility of some steps. Recently, a new method of calculation of the instantaneous composition and microstructure of copolymer in equilibrium copolymerization was proposed by Krueger et al. 18 In a series of papers Sawada and others ~9-21analyzed thermodynamic aspects of copolymerization. It was found that the change of entropy in copolymerization of one mole of copolymer mixture consists not only of the change of entropy because of the chemical process of monomer addition (ASadd) but of the entropy of sequence length distribution (ASo) as well: AS = ASadd

ASadd + ASD

:

ASAAXAA Jr ASBBXBB -I- ASABXAB Jr- ASBAXBA

=

ASAAXA Jr- ASBr~XB -b Z~bXAXB

(25)

(26)

where XAA, XA, etc are molar fractions of different dyads or units in the copolymer; ASAA, /~SBB, etc are the molar changes of entropy in the corresponding comonomer additions; Z = ASAB + ASBA -- ASAA -- ASBB; ~' = XAB/ (XAXB) is the randomness parameter. According to Thei122 ASD

:

- - R { X A [ ( 1 -- P A A ) I n (1 -

PAA) q- PAAInPAA]

+ Xa[(1 - PBB)In (1 - PBB) + PBBlnPBB]}

(27)

where PAA = XAA/XA, PBB = XBB/'r~B are the conditional probabilities that corresponding units will be followed by units of the same kind. Harvey and Leonard 23 were the first investigators to analyze an equilibrium copolymerization from the point of view of the Flory-Huggins theory of intercomponent interactions in polymer solutions. These authors made the assumption that the composition of chain ends is identical to the composition of copolymer chains. As was shown later 13 this assumption is not correct and therefore their treatment gave erroneous final equations. 24 A few years later Mita 24 provided an analogous treatment. Not considering chain ends at all he formulated the condition of overall equilibrium as the equality of corresponding chemical potentials: ~A =

~c¢

~//B =

#~

(28)

where indexes A and B indicate comonomers, and ct and fl corresponding copolymer units. Taking into account the entropy of copolymer sequence distribution and the

THERMODYNAMICS

OF C O P O L Y M E R I Z A T I O N

925

free energy of mixing, he derived equations for chemical potentials of components of the equilibrated copolymerization system. Consequently, he obtained 24 the following set of equations enabling one to predict for any initial conditions the composition of the copolymerization system at equilibrium (for the sake of simplicity molar volumes of solvent, comonomers, and copolymer units were assumed to be equal: Vs = VA = VB = V~ = V~) on the basis of known thermodynamic functions (free energies of copolymerization and interaction parameters): exp[-F/(RT)] ~ao

=

--

[ e x p ( Z A ) - 1][exp(ZB)- 1] exp(--ZA) 1 -- e x p ( - Z B )

~B

1 ---

(I)Ao - - (I) A (I)A o +

(I)B o

=

(29)

(I)A

+ ~B + ~P =

(30)

1 -- ~s

(31)

where Z A

=

AG=°A -

+ Za

=

In (1)A -

I +

ZAp((IDA -- O p )

(~(BP -- XAB)([DB "~- (ZSP -- ~(AS)(DS,

AG~B

-

In ~ B

-

I +

(32)

;(Bp(OB -- O p )

(33)

"-~ (ZAP - - J(AB)(I)A -~- (ZSP - - ~(BS)(I)s,

~A, OB, OP, and ~s are the volume fractions of comonomers, copolymer, and solvent, respectively, where F =

AG~°~+ AG~A - AG~°A - AG~..

(34)

The microstructure of the copolymer can then be predicted on the basis of the randomness parameter 0 and the conditional probabilities Po (of unit i to be followed by unit j ) calculated accordingly (0 is defined as P~ + Pe~)24): ~b =

[1 - exp(--ZA)]/X fl P~ P~

=

=

=

[I - exp(--Za)]/x

~b.x~, P~, =

1 - P,~,P~

=

~

(35)

~b.x~

(36)

1 - P~,

(37)

(where x~ is the molar fraction of unit A in copolymer, for the assumed equality of molar volumes equal to (~Ao - (I)A)/((I)Ao -- (IDA -~- (I)Bo -- (I)B); the molar fraction of units B, x~, is consequently equal to (1 - x,)). Mita's treatment is useful in predicting the composition and microstructure ofcopolymer as well as the equilibrium concentrations of comonomers when the corresponding thermodynamic functions are known. When, however, they are not known then their determination from the analysis of the copolymerization experimental data is possible only by numerical fitting of experimentally determined quantities (~i, xj, P0) to eqs (30-33) (ZA and ZB can be calculated from the rearranged eqs (35-37)).

926

R. SZYMANSKI

However, the assumption introduced by Mita that the interactions between copolymer and other components of the copolymerization system (interaction parameters Zie; i = A, B, S) are independent of copolymer composition can be questioned. A new treatment following the Flory-Huggins theory but avoiding this assumption has been provided by the present author 25and is reviewed in one of the succeeding sections of this paper. 4. DYAD MODEL COPOLYMERIZATION A new treatment enabling one to predict all major features of a dyad model copolymerization system, such as equilibrium concentrations of comonomers or composition and microstructure of copolymer, has recently been presented by the present author. 26 It differs from the previously published models by the following features: (1) It is valid for any degree of polymerization of the equilibrium copolymer, (2) it is valid for any number of comonomers and (3) it provides equations which can easily be applied to determination of copolymerization equilibrium constants• The basic set of equations enabling one to predict the elementary properties of an equilibrium copolymerization system consists of 4i equations, where i is the number of comonomers:

-

-

KBAa

--KnAa

(1 -- Kssb)

. . . . .-.

-Kssb

KaHh

] [ YB

(1 - K H . h )

Ao

=

A* YA

no

=

H*Y.

=

Y.

1

(38)

1

(39) (A)

=

Ao + ao -- a

=

A*Y A

(40) (H)

-

-

KAab

-- KAHh

=

1to + ho -

(1 -- KBab)

. . . .

. . . . . .

h

=

H*YH

K.sb

(1 -- KnHh )

B* *

=

B*

(41)

H*.]

where 4i independent unknowns of the equation set are the following (if not stated otherwise brackets [] denote equilibrium concentrations - or more correctly activities - of the indicated entities): a . . . h -- equilibrium concentration of comonomers ([A] . . . [H]), A* . . . H* -- equilibrium concentrations of active centers of the indicated types ([-- A * ] . . . [-- H*]),

T H E R M O D Y N A M I C S OF C O P O L Y M E R I Z A T I O N

927

A * . . . H * - equilibrium concentrations of the shortest active centers (copolymer "chains" containing only one repeating unit) of the indicated types (treated as initiators), YA • YH -- copolymer parameters equal to the ratios of equilibrium concentrations of repeating units and active centers of the given type (Yx = ( X ) / X*). -

•

The other symbols used in the equation set denote: ( A ) . . . ( H ) - equilibrium concentrations of copolymer repeating units ((X) = [ - X - ] + [-X'l), Ao • • • Ho - initial concentration of initiators ([A*]o. • • [H*]o), ao . . . ho - initial concentrations of comonomers ([A]o . . . [H]o), KAA,KAB • • • -- dyad-model copolymerization equilibrium constants. I f initial conditions and equilibrium constants are known, then the solution of the equation set (38-41) enables one to predict the equilibrium concentrations of comonomers (a . . . h) and composition o f the copolymer ((A) . . . (H)) (as well as Yx, X~*, and X* quantities). The other equations provided in the treatment enable one to calculate all details of copolymer microstructure and molecular weight distribution including the equilibrium concentration of chains of any length, composition and microstructure. The simplest are relations concerning overall quantities. For instance, the equilibrium concentration of copolymer sequences of a given type can be calculated from the equation: (PRS.

where

P,R.

. . V = A,B.

. . TUV) . .

=

or H;

(42)

(P)qPRqRS " " " qTuquv

(PRS.

. . TUV)

=

[-PRS.

. . TUV-]

+

is the average conditional probability that unit P will be followed by R (r = JR]). The equations enabling one to calculate other features of the equilibrium copolymerization systems are more complex. Thus, the equilibrium concentrations of living n-mers terminated with active centers of a given type can be calculated from the following equation (formulated previously by Szwarc and Perrin27): [- PRS

. . . TUV*];

qPR = K p R r Y R / YP

1 H*.J

'IA..1 LKAHh

K..h

''"

KH~h.J

(43)

LH~*]

The conditional probability that the given unit will be preceded by a different one depends not only on the nature of the respective units but on the position of them in the chain as well: qRs,m

:

R*-,KRs/(Am •

,KAs +

Bm • -,Kss...

+ H,~,_..,KHs)

(44)

(ORS,,. is the conditional probability that the unit S in the position m of the chain,

928

R. SZYMANSKI

counting from the initiator-orginated repeating unit, will be preceded by a unit R). Taking into account the conditional probabilities of this kind, one can, consequently, calculate the equilibrium concentration of a chain of any microstructure, e.g.: [R-ABHHCA*]

=

A~

•

qCA,6" qHC,5 " qHH,4 " qBH,3 ° qAB,2

(45)

(where R is the chain end derived from initiator R - A * ) . When the difference between the free enthalpies for addition of any two comonomers to the same active center does not depend on its nature, the copolymerization may be treated as ideal (as shown below - ideally random). For such copolymerizations the following relationship is fulfilled: KIR/KIs

=

KKR/KKs

(46)

(for any L K , R , S = A , B . . . H ) . It was shown that consequently, for ideal copolymerizations of i comonomers, the set o f only 2i equations is sufficient to predict, on the basis of known equilibrium constants, the equilibrium comonomer concentrations. However, the set o f 2i equations presented in the paper discussed26: r o -- r =

YRKAR + (1 -+ (ho -

YR = R

=

h)]/(Kgga

1 + (KRAa + . . . A,B...H,r

--KHHh ) [(ao -- a) + . . .

KAAa...

=

+

+

...

(47)

+ KAHh)

KRHh)/(1

-- K A A a . . .

-KHrth)

(48)

a,b...h

is not sufficient, because the equations are not independent. Instead of one of the above equations (any of which is suitable) one has to take an additional independent equation, e.g. the following one: [(ao -- a) + . . . =

[Ao(1 -

+ (ho - h)] (1 -

-KnHh)

KAAa...

1/YA) + . . . + Ho(1 -- 1/Yn)]

(49)

derived from eq. (39) and the following equation for the average degree of polymerization: De

=

{(ao

=

1/(1 --

-- a) + KAAa.

...

+ (ho -

h)}/{(Ao

-- A * )

+

...

+ (11o --

H*)}

• . --Khhh),

derived in the paper discussed. It was proved that copolymerizations fulfilling the relationship (46) are random and the probability of finding a given unit at any position in the chain (other than the starting of final position) is independent of the position and the nature of neighboring units and is equal to the conditional probability that any unit will be preceded by a given unit: PR.... =

~/RS.m =

KRRr/(KAAa + . . .

+ KHHh)

(50)

THERMODYNAMICS OF COPOLYMERIZATION

929

(PR. . . . is the probability that in position m of any n-mer is located an R unit, 1 < m < n; q~s,m is the conditional probability that a unit S in position m of an n-mer will be preceded by a unit R; 2 < m < n + 1). Probabilities of finding a given unit in the starting or final position are different: PR,,,, =

g/,,s,2 --

/ (AoKA.

YR

\

YA

PR.... = KRRr/(KRAa + . . .

+'''

no/,..

+

-~H /

+ KRHh).

(51) (52)

It is worth noting that some relations concerning the microstructure of a copolymer (independently of relations between equilibrium constants) are exactly the same as those derived previously for systems with the assumption of high molecular weight copolymer. Thus, the following simple equations can be used to determine the equilibrium constants of copolymerization from the analysis of a copolymerization equilibrium system, regardless of the molecular weight of the copolymer and the number of comonomers: Kxx = KpRKRp

(53)

(XX)/[(X)x] -

-

(PR) (RP) (P)p(R)r

(54) (55)

KVR = (R*/P*) (PR)/[(R)r]

where the notation is the same as above ((X) = [ - X - ]

+ [ - X * ] , etc.).

4.1. Application of equations to real systems

Equations (53) and (54) were used :s in the analysis of the equilibrium in the cationic copolymerization system of 1,3-dioxolane (Dxl) with 1,3-dioxepane (Dxp) in dichloromethane solution. Microstructure of the equilibrium copolymer was determined by means of 1H N M R spectroscopy (Fig. 1), while the ratio of concentrations of active centers was determined from the 3~P-NMR spectrum of the equilibrium system terminated with tributylphosphine (Fig. 2) (an ion trapping method developed by Penczek and coworkers29). The determined equilibrium constants governing the copolymerization equilibrium are presented in Table 1. Below the table the corresponding thermodynamic parameters are listed. On the basis of the determined equilibrium constants it was possible to simulate the dependence ofcomonomer equilibrium concentrations as well as of copolymer microstructure on initial conditions (Fig. 3). 4.2. Recent analogous treatments o f other authors General treatments of equilibrium in a copolymerization (correct for any molecular weight of copolymer) were proposed by Szwarc and Perrin, 27and Yan

930

R. SZYMANSKI 4

I

L 4.80

I

J

I 4.70

,~

I

1 I 3.55 5.50

I 3.70

H

I 1,70

i

I 1.60

J

.l

I

L

I

I

I

7

6

5

4

.3

2'

'1

0'

~ ppm Flo. 1. t H N M R spectrum of equilibrium copolymer of 1,3-dioxolane (A) and 1,3dioxepane (B). Initial copolymerization conditions: [A]o = 3.15 tool. L - ' , [B]o = 1.09mol. L -I, [(C6Hs)3C+SbF6]o = 5 . 1 0 - 3 m o l • L -I in CH2C12 at 0°C. Assignment of signals: 1, 2, 3:OC_H20 in dyads AA, AB (and BA) and BB, respectively; 4, 5, 6:OCH2C_H20 in triads A_AA, A_AB (and B_AA) and B_AB, respectively; 7: OCH2CH2CH2C_H20; 8: OCH2C_H2C_H2CH20.

and Cal. 3°'31 However, some of the equations following from these treatments

are rather inconvenient to use. For instance, in order to predict, according to Szwarc and Perrin's treatment, the equilibrium comonomer concentrations on the basis of the known initial conditions and known equilibrium constants, one

I 5

I

I

60

I

I

40

I

I

20

I

I

0

I

i

-20

~, p p m

FIG. 2. 31p{i H} N M R spectrum of the equilibrium cationic copolymerization system of 1,3-dioxolane (A) with 1,3-dioxepane (B) reacted with an excess of tributylphosphine. Initial conditions: [A]o = 2.88mol • L-~, [B]o = 1.36mol- L - I , [(C6H5)3C + SbF~- ]o = 2.2 • 10 -2 mol • L-i in CH2CL 2 at 0°C. Assignment of signals: I: P(C4H9)3; 2: OCH2CH2OCH2P+(C4Hg)3; 3: O(CH2)4OCH2P + (C4H9)3; 4: HP + ( C 4 H 9 ) 3 ; 5: O =

P(C4H9)3;

6: H O P + ( C 4 H 9 ) 3.

THERMODYNAMICS OF COPOLYMERIZATION

931

TABLE 1. The equilibrium constants of propagation and the corresponding thermodynamic parameters in the cationic copolymerization o f 1,3-dioxolane (A) with 1,3-dioxepane (B) in dichloromethane solution Temp inK

[A]o mol. L i

[B]o mol. L-I

KAA mol i . L

KBB mol i . L

(KAB" KBA) I/2 mol E . L

273

2.17 2.88 3.15

2.07 1.36 1.09

1.50 1.41 1.38

2.35 2.36 2.52

1.88 1.88" 1.86

295

2.17 2.88 3.15

295

2.17 2.88 3.15 -

-

1.43i"

2.41I

1.871

2.07 1.36 1.09 -2.07 1.36 1.09

0.79 0.78 0.77 0.78t 0.61 0.60 0.60

1.61 1.62 1.70 1.64t 1.25 1.22 1.23

1.13 1.13 1.14 1.13f 0.84 0.85 0.86

-

0.60I"

1.23I"

0.85t

-

--

*KAB = 1.38mol -~ • L; KBA = 2.50mol -t • L. tAverage values of the equilibrium constants of propagation. The corresponding thermodynamic parameters calculated accordingly are the following: AHAA = -- 15.6 + 1.7 kJ • mol ~; ASAA = --54.5 4- 7 . 3 J . m o l - j . K - I ; AHBn = - 1 1 . 9 _+ 1 . 5 k J . m o l i; ASBB = --36.3 _+ 5 . 6 J ' m o l - I K i; (AHAB + AHnA)/2 = - 1 4 . 1 + 1 . 2 k J ' m o l - l ; (ASAB + ASAB + ASBA)/2 = --46.6 + 3.6J • m o l t • K -q .

1.0

1.4

1.2

4

I

0.8

7

1.0 _.-..

0.6

iu 0.8 m oi::

,m,

0.6

0.4

~3 o3

J

,~, 0 . 4

0.2 0.2

0 0

0.2

04

06

0.8

0 1.0

IA]./{ [A]. + EB].I FIG. 3. Simulation o f equilibrium concentrations o f c o m o n o m e r s ([A], [B]) and o f equilibrium fractions o f different dyads 0CAA,etc.) in the copolymer for the cationic copolymerization of 1,3-dioxolane (A) with 1,3-dioxepane (B) in CH2CI 2 at 25°C as a function of initial fraction of A: [A]o/([A]o + [B]o), ([A]o + [B]o = 4.24 mol • L - i ). Simulated curves and experimental points: I, ( ~ ) : [A]; 2, (O): [B]; 3, (o):fAA; 4, ( ~ ) : fBB; 5, (o):fA B + faA-

932

R. SZYMANSKI

has to solve the following set of four equations (written in matrix notation):

[ :]

.:l ~:,

(,,H,)_l

'

bo

L°8,J

]

~o

LOB,

.=~ ,=1 ,,,.=,

(56)

(57)

where the unknowns are: a,b - equilibrium comonomer concentrations (a -[A], etc), and °A~, ° B 1 - - equilibrium initiator concentrations (initiated comonomers are treated as initiators): Ao and Bo are starting initiator concentrations (A o = (°A~)o, etc ); M is the matrix:

VKAAa KsAa] M -- LKAsb KaabJ (KAA, KAa, etc, are the equilibrium constants of dyad model copolymerization); N., m is the matrix: Nn

m

~

'

"

1

' rn, m

Sn, m

where r.,m and S.,m are the conditional probabilities that units A and B, respectively, in the position m of a living n-mer will be followed by units of the same kind. These can be calculated from the relations:

"-'[Koa]/(l

r.m = ,

(1 I) M"

S.m =

(1 1) M ~-m

"

t

KBBb

[(1

1) M . . . . 1 rKAAa ] LKAsbJ

(58)

1) M "-m

(59)

~ LKB.b]

where (1 1) is a one row matrix with two elements equal to 1. Thus, due to the definitions and notations, one must, in fact, solve a set of many equations, the number of which depends on the reasonable approximations of the infinite sums in eqs (56) and (57). Moreover, the Szwarc and Perrin treatment cannot be applied to systems, for which KAAKBB = KABKBA (true for random equilibrium copolymerizations), because the matrices M and N,,m would become singular and consequently N,,~ would not exist. The papers of Yan and Cai 3°'3~ deal with a binary equilibrium copolymerization and the equations presented enable one to calculate such features of the equilibrium systems as equilibrium comonomer concentrations, copolymer composition and microstructure, the degree of polymerization of the copolymer and average lengths of homogeneous sequences.

THERMODYNAMICS OF COPOLYMERIZATION

933

The initial set o f equations allowing one to calculate the most i m p o r t a n t parameters consists o f three equations (reducible immediately to two):

ao

KAA --

bo

Kss

fl

I =

I[rACZ(1 _ fl)2 + rBrKo:fl2 + rcrKO~fl(1 _ fl)] [1 -- a -- fl + ~fl(1 -- rK)] 2 ~t) 2 -f- rArKO~2fl -~- rcrKO~fl(1 -- ~)] [1 -- ct - fl + ~fl(l - r~)] 2

(60)

I[rafl(l --

(61)

Io/{1 + [rAct + raft + ~fl(rcr K -- r A -- ra)]/[1 -- ct -- fl + ctfl(1 -- rK)]} (62)

where ao, bo, Io = initial concentrations o f c o m o n o m e r s and initiator, ~ = KAA[A]e , fl ---- K B s [ B ] e , I -- equilibrium concentration o f initiator, r A ---- KA/ K A A , r s = KB/KBB, rc = (Kn[KAB) A- (KA/KaA), rK = (KABKsA)/(KAAKBB) (KA and Ks = equilibrium constants o f initiation): I + A ~ I - A * ; KA =

[I-A*I/([I][A]), etc.

Unfortunately, due to the complexity o f the equations, this treatment c a n n o t be easily extended to copolymerizations with a higher n u m b e r o f c o m o n o m e r s . K e n n e d y and Wheeler 32 recently provided a new treatment o f copolymerization equilibrium, applying a lattice model o f the copolymerization system in c o m b i n a t i o n with the equations o f the vector model o f magnetism.

5. T R I A D

MODEL

COPOLYMERIZATION

The treatment o f the equilibrium system o f m u l t i c o m p o n e n t copolymerization presented in the previous section can be extended to systems with the penultimate unit effect. 26 The equilibrium in the triad-model copolymerization o f i c o m o n o m e r s is determined by i2 elementary equilibria, such as: - A A * + A , ~ ~ A A A * ; KAAA BA* + A z~ ~BAA*; KSAA. etc

(63)

where KAAA,KBAA,etc, are the corresponding equilibrium constants. Because o f the complexity o f equations which can be formulated for the general case o f i c o m o n o m e r s , only relations for a binary triad-model equilibrium copolymerization were derived. 26 The basic set o f twelve equations enabling one to calculate twelve unknowns, parameters YAA, YAB, Yag and YaB defined below, equilibrium concentrations o f c o m o n o m e r s (a,b), initiators (A 1", Bj*) and active centers (AA*, A B*, BA*, BB*;

934

R. SZYMANSKI

A A * = [~AA*]~, etc) is presented below:

Ab0

(1 - K A ~ a )

0 -

-

KSASb

--

KBA A a

0

0 I[A1 Ill

1

0

Ks,Aa

(1 - K . . . b ) J

YSA[

1

Y,,J

1

Ao =

A*(1 + KAAaYAA + K m b Y m )

(65)

o

=

B*(1 + KBAaYBA + KBBbYaB)

(66)

ao -- a

=

AA*YAA + BA*YBA

(67)

bo - b

=

BB*YBs + A B Y m

(68)

A * KAA a =

AA*(1 - KAAAa) -- BA*KBAAa

(69)

A'~ KA~,b

A B * -- A A * K A m b -

BA*KBABb

(70)

AB*KABAa -- BB*KBBAa

(71)

=

B* KaA a =

BA*

B*KBBb

BB*(1 -- KBssb) - A B * K A . . b

=

-

(72)

where KAA, KAS, etc are the equilibrium constants of the equilibria involving initiators (R-A* and R-B*): R-A* + B ~-- R-AB*; /(AS, etc Y~Y

(XY)/XY*

a =

A~

( X , Y = A or B; ( X Y )

[A]e, b =

[R-A*]e, B* =

[O]e, ao =

=

[-XY-]~

[A]o, bo =

[R-B*]¢, Ao =

+ [-XY*k);

[B]o;

[R-A*]o, Bo =

[R-B*]o.

The left-hand sides of eqs (69-72) are equal to the equilibrium concentrations of the corresponding living dimers and can be directly computed when the equation set is solved. The equilibrium concentrations of living n-mers, terminated with the given dyad, can be then computed from the following equation:

A :L B A * [I BB* .]

=

KAA.b

o

K.A.

o

0

Kmg a

0

K.sga

B A * [/

0

Kg.ab

0

K,s,b

BB* .J

(73)

where XY~* denotes the equilibrium concentration of living l-mer terminated with XY dyad. The average conditional probability that dyad PQ will be followed by unit R is equal to: qPQR =

KpQRr YQR[ YPQ.

(74)

T H E R M O D Y N A M I C S OF C O P O L Y M E R I Z A T I O N

935

The equilibrium concentrations of different sequences can then be calculated from the relations: (RS) (PQR...

WXYZ)

= =

(75)

R S * YRs ( P Q ) . qPQR - • - qwxvqxvz

(76)

A or B). The equations for more distinct analysis of the microstructure of the equilibrium copolymer, accounting for the positions of sequences and the length of a chain, are also presented in the cited paper. When the molecular weight of the equilibrium copolymer is high and the equilibrium concentrations of active centers of different kinds can be neglected, there is no need to know the individiual cross-propagation equilibrium constants. The number of parameters governing the equilibrium can be reduced by taking, instead of these constants, some functions combining two or three of the equilibrium constants. Thus, the basic set of equations enabling one to predict the equilibrium c o m o n o m e r concentrations and the copolymer microstructure can be considerably reduced 28 to five equations with five unknowns (a, b, KAA, (P,Q . . . .

g..,/~A, g.x): ao -- a

-

bo - b /(Aa/~',Aab = /(XX = (X

=

KABKBA =

=

a,b;a

=

(77)

1 - /~aaa (1 -- /(AAa)(1 -- /(a.b)

Kx/{Kxxx(Kxxx -

A,B;x

1 -- /(a.b

2(Kxxx -- K x ) x + 1}

Kx)x 2 -

[A],b

=

[B],ao

=

[A]o, bo =

(78) (79) [B]o)

(1 - 2KAAa + KAAAK~, a z) (1 -- 2/£B, b + Ka,a/~Ba b2 ) KABAKBAaa 2b 2 (80)

where Kx

=

KxxvKvxx/Kvxv(X,Y

=

AorB,

X # Y)

is the combined equilibrium constant and /£xx is the apparent equilibrium constant of dyad model propagation computed for triad model copolymerization:

/~××

=

[-

xx

- ]/([-

x - ]lX])

depending (as shown by eq. (79)) on c o m o n o m e r concentration. Analogously, the product o f the apparent equilibrium constants of dyad model cross-propagations (treated as one unknown) is defined: /(A./~'nA =

[-- A B - - ] [ - - B A - ] / ( [ - - A - - ] [ - - B-][A][B]).

It is worth noting that applying the presented equation set to dyad model copolymerization (KAAA = /(BAh, etc) reduces the equation set to two eqs (77)

936

R. SZYMANSKI

and (78) with the bars over the letters K removed. This reduced equation set was proposed previously for systems of dyad model copolymerization with a high molecular weight copolymer. 13 Coming back to a triad-model copolymerization, the microstructure of the equilibrium copolymer can be computed from the following equations: fx

=

(X =

xo -- x ao -- a + bo - b AorB, x =

(81)

aorb)

fxx = fx/~xxX

(82)

fxY = fx -- fxx (Y # X)

(83)

fPQR...wxYz = fr,Q q ~ R . . . qwxYqxYz

(84)

wherefx, fxY, fPQR...wxYzare the molar fractions of different units or sequences in the equilibrium copolymer, and the average conditional probabilities qxyz can be calculated from the equations:

qxxY =

1 -- qxxx (Y # X)

(85) (86)

qYxx =

( _ ~ + ~2 + 4a)/2

(87)

qxxx =

(~ =

KxxxX

Kx/~xx x2/(1 - /~xx x))

qYXY

=

1 - - qYXX

(Y # X).

(88)

T h u s , the basic features of the triad model equilibrium copolymerization systems are governed by only five parameters - two equilibrium constants of homopropagation KAAAand KBBB, and three combined equilibrium constants: KA = KAABKBAA/KBAB, Ks = KBBAKABB/KABA and KABAKBAB. These five constants can be directly determined experimentally from the analysis of the equilibrium copolymerization system applying the following equations: 2s [ - XXX

Kxxx Kx =

- ]

(89)

[ - x x - ] [x]

[ - XXY

[- xx-

- ] [-

YXX

] [- Yxv-

- ]

] IX]

[- ABA - ) [- BAB - ] KABA KBA B

=

[-AS-]

[-BA-]

[AI [BI

(90) (91)

(all concentrations in eqs (89-91) are the equilibrium ones. When, however, a distinct analysis of equilibrium copolymer microstructure (giving access to triads) is not possible, then these equilibrium constants can be determined from a numerical analysis of the dependence of the apparent equilibrium constants of dyad model propagations (cf. eqs (79), (80)) on equilibrium comonomer concentrations. 2s

THERMODYNAMICS

937

OF COPOLYMERIZATION

--q 14 0.8 -'-" "6 E

2I

12

5" 03

2

io

3'

06

-o

's

g

q-~ 0 8 r,rl

o 0.4

~

r ~ 0.6 E

04 b

&

r ~ 0.2

0

0.2

0.4

06

0.8

I0

0

[A]./( [A]. + [B].)

FIG. 4. Simulation of equilibrium concentrations of comonomers ([A]e, [B]e) and copolymer ( [ - A - ] + [ - B - ] ) , and of the equilibrium fraction of 1,3-dioxane (B) units in copolymer (fB) for cationic bulk copolymerization 1,3-dioxolane (A) with 1,3-dioxane (B) at 0°C as a function of initial mole fraction of A: [A]o/([A]o + [B]o) (the assumed equilibrium constants of copolymerization: KAAA = 1.203, K~BB = 0 . 0 5 5 9 , K A = 1.642, Ka = 0.0417, (KABAKBAB) I/2 = 0.226 - all in mol i . L). Simulated curves and experimental points: I, 1', (0~): [A]e; 2, 2', (O): [B]e; 3, Y, (o):JB: 4: [ - A ] + [ - B - ] . Curves 1'. 2' and 3' were simulated taking into account cyclic dimers.

5.1. Application of equations to real systems The a b o v e - o u t l i n e d m e t h o d for the analysis of triad m o d e l c o p o l y m e r i z a t i o n s was a d o p t e d to the cationic c o p o l y m e r i z a t i o n of 1,3-dioxolane (Dxl) with 1,3-dioxane (Dxn) in bulk. The analysis allowed us for the first time, to determine directly the t h e r m o d y n a m i c p a r a m e t e r s of h o m o p o l y m e r i z a t i o n of a n o n h o m o p o l y m e r i z i n g m o n o m e r - 1,3-dioxane. Thus, the e n t h a l p y of p o l y m e r i z a t i o n of this m o n o m e r appeared to be close to zero (AH~°, = - 3.1 kJ • mol ~) while the e n t r o p y of p o l y m e r i z a t i o n was in the range expected for 6 - m e m b e r e d cyclic m o n o m e r s (AS°~ = -- 35.5 J • mol ~ • K '). The small ring strain of D x n is the reason for the n o n h o m o p o l y m e r i z a b i l i t y of this m o n o m e r , O n the basis of e q u i l i b r i u m c o n s t a n t s d e t e r m i n e d for the copolymerization system, the e q u i l i b r i u m c o n c e n t r a t i o n of D x n was estimated to be a b o u t 2 0 m o l • L ~ at r o o m t e m p e r a t u r e a n d a b o u t 1 0 m o l • L ~ at - 7 0 ° while its c o n c e n t r a t i o n in b u l k u n d e r n o r m a l pressure is equal to a b o u t 10.1 tool • L ~. D x l / D x n c o p o l y m e r i z a t i o n experiments allowed us to determine the equil i b r i u m c o n s t a n t s o f c o p o l y m e r i z a t i o n (KAAA, Ka~B, KA, KB, KABAKBAB) a n d c o n s e q u e n t l y to simulate the d e p e n d e n c e o f the e q u i l i b r i u m c o m o n o m e r conc e n t r a t i o n a n d o f the c o p o l y m e r m i c r o s t r u c t u r e o n the feed ratio (cf. Fig. 4).

938

R. SZYMANSKI

6. CHAIN-RING EQUILIBRIA IN EQUILIBRIUM COPOLYMERIZATION SYSTEMS When back-biting and/or end-biting reaction leading to cyclics can operate in a copolymerization system: R-W . . . . .

X-Y . . . . .

back-biting %

Z*.

end-biting

• R* + ~V. . . . .

R-W . . . . .

X-Y . . . .

X* + 5(. . . . .

;g

Z (92)

(R* - initiator), the chain-ring equilibria have to be taken into account while the copolymerization equilibrium is analyzed. Formally, the macrocycles may be treated as comonomers. Therefore, the relationships derived for an equilibrium copolymerization can be adopted to get equations describing the equilibrium concentrations, composition and microstructure of different cyclics present in the copolymerization equilibrium system. One can define the equilibrium constants of the ring-chain equilibria analogously to the homopropagation equilibrium constants in copolymerization. Consequently, for binary copolymerization, assuming a high molecular weight of the linear equilibrium copolymer, the following equation is obtained. 33

K, AmB =

[ . . . (A, Bm) (AnBm) " . . ] [(A, Bm)(c)] [ ' ' " (A,B,,) '" "1

(93)

where K,~,B is the discussed equilibrium constant (compare eq. (53)) for the macrocycle (A, Bm)(c) composed ofn units A and m units B and (A, Bm) denotes the sequence of the given units exactly the same in the macrocycle as in the involved linear chains (not necessarily of the block microstructure suggested by the notation). K, AmB may differ for macrocycles of the same composition but different microstructure. Rearranging eq. (93) and taking into account eq. (42), one obtains the equation for the equilibrium concentration of different macrocycles in copolymerization systems. 33 [(A, Bm)(c)]

---~

1 rKA"KBA] r K.A,.---~B(KAAa)" (K.B b) m k KAAKs. J

(94)

where KAA, KBB, etc are the propagation equilibrium constants of a dyad model copolymerization, a and b denote the comonomer equilibrium concentrations, and r is the number of A B bonds (half of the AB + BA bonds) in the macrocycle considered. For a triad model copolymerization the analogously obtained equation is the following? 3 1

[(A"Bm)(c)] = K, Am----~(KAAA B a)"(KBBsb)mZ

(95)

THERMODYNAMICS OF COPOLYMERIZATION

939

where

Z

L ~

__1

[ KABnKBBAI'~KsAAKAAB KBBsKABA3 L KAAAK,A,

_

r, S, and t are the numbers o f A B , ABB and BAA (equal to BA, BBA and AAB) sequences in the macrocycles, respectively (if n + m = 2 then s = t = 0, r = 1 for a heterodimer and r = 0 for a homodimer). The equilibrium constants KnAB m a r e the reciprocals of the equilibrium constants of macrocyclization defined in the Jacobson-Stockmayer theory of chain-ring equilibria. 34According to Jacobson and Stockmayer the equilibrium constants discussed in this section for nonstrained cyclics can be calculated from the following equation: KnAmB

=

A/A anAmB/WnAmB(Q )

(96)

where W,A,,B(r) is the function expressing the distribution of end-to-end vector r per unit range in r for a chain of the considered structure, NA is Avogadro's number, and a,a,, R is the symmetry number of the macrocycle. Following Flory and Semlyen 35 eq. (96) may be transformed into another equation: KnAmB

=

N A 0",~An,B ( 2 ~ ( r ~ ) / 3 ) 3/z

(97)

where (r~ > is the mean-square end-to-end length averaged over all configurations of the real chain of size nAmB. Consequently, the equilibrium constants for heteromacrocycles may be related with the equilibrium constants for homomacrocycles (usually known for homopolymerization systems): gnAmB

~-

a.A.,S k',/xt,'m/x. ~n/x ,..rm/x . *XxA ~tXxB U,xA u, xB

W

(98)

w h e r e x = n + m, t

W

~-

/ rx 2 2,A,,B \ \3/2 ~, /j.2~m/x I \rx/xA \rx/.rB /

(99)

O'nAmB, axA and axS are the symmetry numbers for heterogeneous and homogeneous macrocycles composed o f x units, respectively. (r~) with the corresponding index indicates the mean-square end-to-end length of the chain formed of x units (index indicates which units are meant). The coefficient Wfor macrocycles in which A and B do not differ much in size is close to 1 and can be neglected. In other cases W can be calculated assuming, for instance, the freely jointed chain model 36 of the polymer chain. The composition of cyclics may differ from the composition of the linear copolymer. Therefore, while predicting the equilibrium conditions, if cyclics cannot be neglected, the mass balance equations accounting for macrocycles

940

R. SZYMANSKI

have to be included in the equation sets discussed in the previous sections (especially if it cannot be maintained that [ - X - ] = [X]o - IX]). 7. COPOLYMERIZATION EQUILIBRIUM IN SYSTEMS WITH PHYSICAL

INTERACTIONS The Flory-Huggins theory of polymer solutions, when applied to polymerization equilibrium systems, satisfactorily explains the frequently observed dependence of equilibrium monomer concentration on a polymerization system composition. Thus, Ivin and Leonard 37 derived the following equation relating the equilibrium volume fraction of monomer ( ~ ) with volume fractions of polymer and solvent ((I)p and ~ , respectively):

AG~c/RT = In (I) m

1 +

At-

(l)s(Zm s - -

ZspWm/Ws) --[- J~ms((IDp

- - (I)m) ( 1 0 0 )

where AGueis the Gibbs free energy change upon the polymerization of 1 mole of liquid monomer to 1 base-mole of amorphous polymer of infinite chain length, Xij are the Flory-Huggins parameters of interaction of component i with j, letters m,p and s represent monomer, polymer and solvent, respectively, and Viiis the molar volume of component i. The analogous treatment can be applied to copolymerization systems. Unfortunately, Harvey and Leonard, 23who first presented such a treatment, made the assumption, later shown to be invalid, that the composition and microstructure of a copolymer is the same as those of its chain ends. Therefore, their treatment gives incorrect results for some systems. The analogous treatment by Mita z4has been reviewed in one of the previous sections. Recently a new analysis of binary equilibrium copolymerization from the point of view of the Flory-Huggins theory, more general than those by Mita, was presented. 25 The dyad model of copolymerization with four equilibria governed by the propagation equilibrium constants KAA, KBB, KAB and KBA (reaction scheme (8)) was assumed. The additional assumptions of high-molecular-weight copolymer and neglecting the nature of active chain ends enable us to reduce the number of equilibria describing the system to three (cf. eqs (53) and (54)): -A-

+

A

~-

-AA-

;KAA

--B--

+

B

~-

-BB--

;KBB

-A--

+ --B--

+B

+

(101)

-AB" K

A

g A B KBA"

+ -BA-

The changes of the Gibbs free energies corresponding to the above formal reaction equations are in equilibrium equal to zero: AG1 =

AG~,A + A(TA -- A(Ta = 0

(102)

AGn =

AG~B + AdB - A67b =

(103)

0

THERMODYNAMICS OF COPOLYMERIZATION AGIu --- AG~,a + AG~A - AGA + A(~B -- AGa -- AGb --- 0

941 (104)

where indexes I, II, and III indicate the number of the reaction from the equation set (101); AGA,Aaa, AGa,and A(~b denote the partial molar free energies of copolymer repeating units (capital letter indexes) or comonomers (lower case letter indexes); AG~A, AG~B, etc are the standard molar free energies of formation of the corresponding dyads under equilibrium conditions. Each of the free energies of dyad formation contains two terms - one connected with the chemical transformation and another connected with the dyad distribution: , = AG~v + ~'~xv A~(D) AGxv X,Y

(105)

= AorB

where AG~v is the standard molar flee energy of the corresponding comonomer addition while AG~x~ is the partial molar free energy of dyad distribution in respect to the corresponding dyad: AM(D) 0AGD •-'xv = ~nxv

(106)

(the free energy of dyad distribution AGD arises from the fact that copolymers with different dyad distributions can be formed from the same numbers of different dyads; nxv = the number of moles of dyads XY in copolymer). It was shown 25that the equation for the molar free energy ofdyad distribution is the following:

+ n a a l n ( l + nBA~ I n a+ BnnAln( J

+ naa)].naA/d

(107)

Consequently, the partial molar free energies of dyad distribution are the following simple functions of the ratios of the numbers of different dyads in copolymer: AF:.(D) _ VAA AM(D) 'JaB =

R T l n ( I + nAB') HAA /I --

R T I n ( i "4- nBA / nan/

Ar~,.(D) 'JAB _

RTIn(1 + nAA) nAB

A•(D) VBA --

RTln(I

+n~B t nBA ./ "

(108)

942

R. SZYMANSKI

Combining eqs (102-105) with eq. (108) and with the corresponding FloryHuggins equations 36 for the partial free energies of components of a polymer solution (here an equilibrium copolymerization system), the final equations describing an equilibrium in copolymerization, accounting for different molar volumes of components of the system and different interaction parameters for different copolymer units, can be obtained. If a bulk (~s = 0) binary copolymerization is considered then, for a system with parameters of interaction between comonomers and copolymer units equal to each other C( = ;(~A = ZaB = ~(bA = XbB) and molar volumes of comonomers and copolymer units not differing as well (Va = Vb = VA = VB), the general set of equations reduces to the following: AG~,A =

R T In 1 + OAA,] + ln~A + (1 -- I/DP) -- (2Oa + 2 ~ b -- I)X +

bZabJ

AG~ = R T In 1 + OB~/ + lnq)b + (1 -- I/DP) -- ( 2 ~ a + 2Ob -- 1)X +

AG~B + AG°A =

(109)

(llO)

~.XabJ

R T In 1 + ~ABJ + In 1 + ~BA} + l n ~ + lnOb + 2(1 -- 1/DP)

(111)

- 2(20~ + 2Ob -- 1))~ + ( ~ + Ob))~b ] where O~, ~b, OA, and OB are volume fractions of comonomers and comonomer units, respectively, Oxv = Vxnxv/(naV~ + nbVb + nA VA + nsVB); nxv is the number of moles of dyads XY in the copolymer; n~, nb, na, and nB are the numbers of moles of comonomers and copolymer units, respectively; V~is the molar volume of component i, and DP is the number average degree of polymerization of the copolymer. The set of eqs (109-111) combined with the mass balance equations: • ° = O~ + OAA + OAB

(112)

O~', =

Ob + ~BB + ~BA

(113)

(1)BA

(114)

(1)AB

=

(where • ° and ~ are the initial volume fractions of comonomers; (IDAA "~ OAB = Og, ~BB + OBA = OB) is sufficient tO be solved, predicting in that way

T H E R M O D Y N A M I C S OF C O P O L Y M E R I Z A T I O N

943

the equilibrium conditions of a copolymerization system for any initial copolymerization system composition, provided the standard molar free energies of comonomer additions (AG~,A, AG]B, AG~,B + AGOg) are known. It is worth noting that these thermodynamic functions refer to pure liquid comonomers and pure amorphous copolymer as standard states (being in fact AGue type parameters). Choosing as standard states for copolymer units the solutions of infinite dilution in the corresponding comonomers results in an alternate set of equations for bulk copolymerization: 25 /~AA --

(I)AA --

KAAeXp[~bZab + 2(1 -- (l)a - ~bb).~]

(115)

KB~exp[(1)aZab + 2(1 -

(116)

(I)A (/) a

/~BB =

/~AB/~BA

~BB

OSOb =

=

(1)AB(1)BA

=

dog -

Ob).~]

K A B K B A e x p [ ( q b a -4- (l)b)Za b

OA~SOaOb +

4(1

(1 17)

- - (IDa - - (I)b)~]

(the mass balance equations remain unchanged) where K'AA, etc, are the apparent equilibrium constants defined by the equations, while Kxv can be considered as the corresponding absolute equilibrium constants: Kxv =

exp(-AG*v/RT)

(118)

(AG*y is the AGes type standard molar free energy of the corresponding comonomer addition in comonomer y (as solvent);/~xv = Kxv when Z = Zab = 0.) The set of eqs (115-117) can be reduced (by putting ;~ = lab = 0) to the equation set equivalent to that derived for copolymerization without intercomponent interactions (cf eqs (53) and (54)). It is worth noting that independently of the equilibrium system composition and intercomponent interactions, the simple relationship between fractions of copolymer dyads is maintained: (I)AB*BA (I)AA(I)BB

fAafBA

--fAAfBS -

/~AB/(aA KAAI~sB =

KAa KBA KAAKsB "

(119)

The above relationship, which does not hold in the equilibrium copolymerization system of 1,3-dioxolane with 1,3-dioxane, 2s allowed us to choose the penultimate unit effect (not intercomponent interactions) as the reason for the dependence of the apparent equilibrium constants of dyad model homopropagation (/~AA, RSS) on the system composition. Simulations, based on the set of eqs (112-117) (Figs 5-8) showed that although both comonomer-copolymer (D and comonomer-comonomer (Xab) interaction parameters, while positive, decrease the equilibrium comonomer

944

R. SZYMANSKI 1.0

- 0.5

-e-

3"

0

I

0.5

2

1.0

FiG. 5. Equilibrium volume fractions ofcopolymer units A (~A) (curves 1"-3') and B (~a) (curves 1"-3") in bulk copolymerization as a function of copolymerization system composition ( ~ - initial volume fraction of comonomer A), computed for different parameters of interaction between comonomers Zab: 1 (curves 1', 1"), 0 (2', 2"), -- 1 (3', 3"), and for parameters of interaction between copolymer and comonomers (Z) equal to zero; KAA = 2, KBs = 8, ( K A B K B A ) 1[2 = 4. concentrations, the dependence o f equilibrium conditions on X is m o r e pronounced. W h e n c o m o n o m e r s do not h o m o p o l y m e r i z e (KAA, KSB < 1; equilibrium constants defined by volume fractions), the c o p o l y m e r in the system without i n t e r c o m p o n e n t interactions (Z = ;tab = 0) can be formed only when KABKBA > 1 (heterodyads prevail). W h e n KABKBA >>KAAKBBthe equilibrium c o p o l y m e r is approximately an alternating one (cf. Fig. 7). When, however, interaction parameters are positive, then for certain copolymerization systems o f nonhomopolymerizing c o m o n o m e r s even r a n d o m copolymer can be expected (for KAItKaA = KAAKBB < 1; cf. Fig. 8). 8. H O M O P O L Y M E R I Z A T I O N S WHICH CAN BE R E G A R D E D AS COPOLYMERIZATIONS W h e n in h o m o p o l y m e r , existing in equilibrium with m o n o m e r , two or more kinds o f repeating units can be distinguished, such a system can be treated as a copolymerization system. The simplest system o f that kind, if one disregards the real copolymerization

THERMODYNAMICS OF COPOLYMERIZATION

945

1.0

i ,I 2"

3"

113 -

I~

0.5

0

0.5

l0

o

FIG. 6. Equilibrium volume fractions ofcopolymer units A ((IDA) (curves 1'-3') and B (~B) (curves 1"-3") in bulk copolymerization as a function of copolymerization system composition ( ~ - initial volume fraction of comonomer A), computed for different parameters ofcopolymer-comonomerinteraction: Z: 0.3 (curves 1', 1"), 0 (2', 2"), -0.3 (3', 3"), and for parameters of interaction between comonomers (Xab)equal to zero; K A A = 2, KBB = 8, (KABKBA) U2 = 4. systems o f enantiomeric monomers, is a polymerization o f a prochiral monomer. M o n o m e r addition o f such m o n o m e r s can lead to two isomeric (chiral) repeating units. One can formulate the following set o f equilibria: - A* + M ,-~ - AA*; KAA

--A* + M ~- - A B * ;

KAB

-B*

-BA*;

KBA

-BB*;

KBB

(120) + M-~

--B* + M ~

where A and B denote p o l y m e r repeating units o f different chirality. The same set o f equilibria can be formulated for different kinds o f isomerism o f polymer repeating units, e.g. directional isomerism leading to head-to-tail and head-to-head connections. A p p l y i n g the relationships derived for equilibrium copolymerization systems leads to equations describing the equilibrium in the discussed homopolymerizations. Both o f the above mentioned systems, with chirality o f repeating units and directional isomerism o f m o n o m e r addition,

946

R. SZYMANSKI

0.10 nr~

¢PA

"e-~. o,o8 ,m -e•-e-

0.06

0.04

0.02

O

i

1

O

0.5

1.0

Fro. 7. Equilibrium volume fraction of copolymer units A (~A) and B (OB), and of units A in dyads AB (~AB)in bulk copolymerization as a function of copolymerization system composition (~, - initial volume fraction of comonomer A), computed for Z = Zab = 0, KAA = 0.5, KBB = 0.2, (KABKBA)I/z = 2.

have o n e c o m m o n feature: n a m e l y , the h o m o p r o p a g a t i o n e q u i l i b r i u m c o n s t a n t s (KAA a n d KaB) are equal. This e q u a l i t y h o l d s b e c a u s e o f the e n a n t i o m e r i s m o f chains f o r m e d b y h o m o p r o p a g a t i o n s in p r o c h i r a l systems a n d the i n d i s t i n g u i s h a b i l i t y o f chains f o r m e d b y h o m o p r o p a g a t i o n s in d i r e c t i o n a l i s o m e r i s m systems (only h - t o r t - h connections are possible, a n d c h a i n ends c a n be neglected for l o n g chains). C o n s e quently, the final e q u a t i o n s are the same: fA qAA = qAB =

qBA =

1-

= f~

q~a =

fnB + fBA = =

0.5

(121)

KAA[M]

KAA[M] =

f a n + fBB =

[M]

=

(KA~K~A)~/2[M]

(122) (123)

K~,[M]

(124)

(KABKBA)'I2[M]

(125)

(KAA + (KABKaA)'/2)- '

(126)

THERMODYNAMICS OF COPOLYMERIZATION

0.2

947

CB

m

-O-O-

~t -o-

o.ML

0

/

/

*As

O.5

I0

¢° 0

FlG. 8. Equilibrium volume fractions of copolymer units A (~A) and B (~B), and of units A in dyads AB (~AB) in bulk copolymerization as a function of copolymerization system composition ( ~ - initial volume fraction of comonomer A), computed for Z = Zab = 0.5, KAA = 0.98, KSB = 0.9, (KABKBA) 1/2 = 0.94.

where fA and fB are the molar fractions of the corresponding units in polymer (chiral or directional isomers); qgA, etc, are the conditional probabilities that one unit will be followed by another, faA + fBB is the molar fraction of homodyads (isotactic dyads in prochiral systems, and h-t + t-h connections in directional isomerism systems), f m + faA is the molar fraction of heterodyads (syndiotactic dyads in prochiral systems, and h-h + t-t connections in directional isomerism systems). It is worth noting that according to eq. (126) the relationship In[M] versus l/Tcan be nonlinear (namely when AHAa + AHaA :/: 2AHAA). A quite different example of homopolymerization which can be treated as copolymerization is a polymerization of cyclic monomers composed of repeating units redistributable in the polymer. For instance, by polymerizing 1,3,5trioxepane, one obtains polymer containing sequences which cannot be derived

948

R. SZYMANSKI

directly from the structure of the monomer:

o~CH2CH2~o

I

n

I ,-----"

CH 2 ....

(127)

CH2

OCH2OCH2OCH2OCH2CH2OCH2OCH2CH20 . . . .

(e.g. the -CH2CH2OCH2OCH2CH2- grouping presented above). However, the equilibrium system of such a polymerization can be treated as a copolymerization of simpler comonomers containing units present in the considered monomer. Thus, the equilibrium polymerization system of 1,3,5trioxepane is identical to the equilibrium copolymerization system of 1,3dioxolane with formaldehyde in 1 : 1 molar ratio. 38 Taking into account equations derived for ring-chain equilibrium in copolymerization systems (Section 6), it was possible to derive the equation enabling us to determine the equilibrium constant of formation of uniform polymer from 1,3,5-trioxepane. 33

KT

=

1 [- EMM - ]2 [T] [ - E - ) ( [ - M - ] - [ - E - ])

-

(128)

-

where E and M denote (OCH2CH2) and (OCH2)O, respectively, [T] is the equilibrium concentration of 1,3,5-trioxepane, and KT is the equilibrium constant of its homopropagation: (-CH2OCH2CH2OCH 20)nCH2* + T ~ (-CH2OCH2CH2OCH 2O)-.+ ~CH*; (T: 1,3,5-trioxepane).

(129)

The analogous treatment of thermodynamics of 1,3,5-trioxepane polymerization was previously provided by Szwarc and Perrin. 39However, their equations were based on the assumption of random distribution of different repeating units in the equilibrium polymer. The last example of a homopolymerization, which can be analyzed using relationships derived for copolymerization, is a polymerization in the presence of a reagent forming equilibrium complexes with the polymer: --A* + A ~ - A A * ; KAA

(130)

M ....

AAnA . . . .

+ M ~--- . . . .

A-,~n-A ....

; KI

(131)

where n denotes the size of a sequence of polymer repeating units participating in formation of a single complex with reagent M.

THERMODYNAMICS OF COPOLYMERIZATION

949

Due to reversibility of microstates the equilibrium in the homopolymerization system is the same as in the copolymerization system with hypothetical (or true) c o m o n o m e r B (complex of reagent M with n molecules of A): ....

A* + A ~ . . . .

AA*; KAA

....

A*

+

....

AB*; KAB

. . . .

B •

+ A ~

....

BA*;

. . . .

B •

+ B ~- . . . .

B ---~

M

(132)

KBA

BB*; KBB

M A -- A. - A - , etc.

where B = ?~., -- ABA

The only difference between the discussed homopolymerization system and a hypothetical copolymerization system is the presence of c o m o n o m e r B. Nevertheless, the analysis of the copolymerization system gave the following set of equations, enabling one to predict the equilibrium conditions in a homopolymerization system in the presence of reagent M on the basis of known initial composition: [A]o-[A] Mo

( =

K,x ) (1 - x ) [ K l x + /(2(1 - x)] + n (1 - ~)

(1 - x)/{Mox" [K~x + /£2(1 - x)]}

=

x

=

(133) (134)

KAA[A]

(135)

(unknowns: [A], ~, x; ct = [M]/Mo) where Mo = [M]o (for [B] = 0; if[B] # 0 then Mo = [M]o + [B]o - [B] and because of a new unknown, [B], an additional equation is needed: [B] = KBeMo[A]" where Ka is the equilibrium constant for formation of B from M and A); K~ is the equilibrium constant of the reaction (131), while K, is the equilibrium constant of the similar reaction involving the sequence of polymer units adjacent to the sequence complexing another molecule of reagent M: M ....

M

•~ . - A . - A

....

+ M ~

....

M

A.-?~.-A

....

;/(2. (136)

It is worth noting that the equilibrium constant (/£3) of the similar reaction involving the sequence of polymer units lying between two sequencies complexing reagent M is related to K~ and Kz :4o M

....

M

t~.-A.-~,.-

M

...

+ M ~ ....

K , / K 2 = K2/K 3.

M

M

,~.-A.-'~.-

"'';

K, (137) (138)

950

R. SZYMANSKI

Although the set ofeqs (133-135) has not yet been applied to real polymerization systems, it was used to estimate the equilibrium constant of formation (in methanol solution) of the complexes between sodium tetraphenylborate and poly(oxydimethylene), 4° and between the same salt and polymeric trisubstituted esters of phosphoric acid. 41 For such systems the equation set is reduced to two eqs (133) and (134), and [A]o - [A] is replaced by the total concentration of polymer repeating units. REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33.

R . M . JOSHI, Thermodynamic properties, Encyclopedia of Polymer Science and Technology (H. F. MARK and N. G. GAYLORO Eds), Vol. 13 p. 788, Interscience, New York (1970). H. SAWADAThermodynamics of Polymerization (K. F. O'DRISCOLL Ed.), Marcel Dekker Inc., New York (1976). K.J. IVIN and T. SAEGUSA,Ring Opening Polymerization, Elsevier Applied Science, London (1984). S. PENCZEK, P. KUmSA and K. MATYJASZEWSKI,Cationic ring-opening polymerization, Chapter 2, Adv. Polym. Sci. 68/69 (1985). S. PENCZEK, Cationic ring-opening polymerization: Thermodynamics, Comprehensive Polymer Science (G. ALLEN and J. C. BEVlNGTON Eds), Vol. 3, p. 719, Pergamon Press, Oxford (1989). K.J. IVlN and W. K. BUSFIELD, Polymerization thermodynamics, Encyclopedia of Polymer Science and Engineering (H. F. MARK, N. M. BIKALES, C. G. OVERBERGERand G. MENGES Eds), Vol. 12, p. 555, Wiley-Interscience, New York (1989). A.V. TOBOLSKYand G. D. T. OWEN, J. Polym. Sci. $9, 329 (1962). K.J. IvlN, W. A. KEITH and H. MACKLE, Trans. Faraday Soc. 55, 262 (1959). E.S. DAINTON, J. DIAPER, K. J. IVIN and D. R. SHEARO, Trans. Faraday Soc. 53, 1269 (1957). F.S. DAINTON, K. J. IVIN and D. R. SHEARD, Trans. Faraday Soc. 52, 414 (1956). T. ALFREY and A. V. TOBOLSKY,J. Polym. Sci. 38, 133 (1959). J . A . HOWELL, M. Izu and K. F. O'DRISCOLL, J. Polym. Sci., Part A-1 8, 699 (1970). R. SZYMANSKI,Makromolek. Chem. 187, 1109 (1986). B . K . KANG, K. F. O'DRISCOLL and J. A. HOWELL, J. Polym. Sci., Part A-1 10, 2349 (1972). M. Izu and K. F. O'DRISCOLL, Polym. J. 1, 27 (1970). M. Izo and K. F. O'DRISCOLL, J. Polym. Sci., Part A-1 8, 1687 (1970). G . G . LOWRY, J. Polym. Sci. 42, 463 (1960). H. KRUEGER, J. BAUER and J. REUBNER, Makromolek. Chem. 188, 2163 (1987). H. SAWADA,J. Polym. Sci., Part B 1, 659 (1963). H. SAWADA,J. Polym. Sci., Part A 2, 3095 (1964). H. SAWADA,J. Polym. Sci., Part A 3, 2483 (1965). M . H . THEIL, Macromolecules 2, 137 (1969). P.E. HARVEYand J. LEONARD, Macromolecules 5, 698 (1972). I. MITA, J. Macromolec. Sci., Chem. AS, 1273 (1974). R. SZYMANSKI,Makromolek. Chem., Theory Simul. l, 129 (1992). R. SZYMANSK1,Makromelek. Chem. 188, 2605 (1987). M. SZWARC and C. L. PERRIN, Macromolecules 18, 528 (1985). R. SZYMANSKI,Makromolek. Chem. 192, 2943 (1991). K. BRZEZINSKA,W. CHWIALKOWSKA,P. KUBISAand S. PENCZEK, Makromolek. Chem. 178, 2491 (1977). D.-Y. YAN and G.-F. CAI, Macromolec. Chem. 186, 2133 (1985). G.-F. CAI and D.-Y. YAN, J. Macromolec. Sci., Chem. A24, 869 (1987). S.J. KENNEDY and J. C. WHEELER, J. Phys. Chem. 88, 6595 (1984). R. SZYMANSKI, Makromolek. Chem. 190, 2903 (1989),

THERMODYNAMICS OF COPOLYMERIZATION

34. 35. 36. 37. 38. 39. 40. 41.

951

H. JACOBSONand W. H. STOCKMAYER,J. Chem. Phys. 18, 1600 (1950). P.J. FLORY and J. A. SEMLYEN,J. Am. Chem. Soc. 88, 3209 (1966). P.J. FLORY, Statistical Mechanics o f Chain Molecules, lnterscience Publishers, New York (1969). K . J . lVIN and J. LEONARD.Eur. Polym. J. 6, 331 (1970). R . C . SCHULZ, K. ALBRECHT, C. RENTSCH and Q. V. TRAN Tin, ACS Syrup. Ser. 59, 77 (1977). M. SZWARCand C.L. PERRIN, Macromolecules 12, 699 (1979). R. SZYMANSKI,Makromolek. Chem. 192, 757 (1991). R. SZVMANSKIand S. PENCZEK, to be published.