Makromol. Chem. 191,2795-2799 (1990) 2795

Kinetic and statistical derivation of the copolymerization equation

Mario Farina

Dipartimento di Chimica Organica e Industriale, Universitii di Milano, via Venezian 21, 1-20133 Milano, Italy

(Date of receipt: March 30, 1990)

SUMMARY A critical analysis of the various methods of derivation of the copolymerization equation leads

to the following conclusions: the steady-state assumption is not a necessary condition for the "kinetic" derivation; all known methods of derivation are implicitly or explicitly related to the condition of chain continuity; other methods, simpler than the traditional ones, are reported in the literature and deserve more attention.

Introduction

In spite of the large number of discussions on this subject, a dichotomy still exists between the kinetic and statistical derivations of the copolymerization equation. This paper represents a contribution to the clarification of this problem: as we shall see, the debate has no reason for existence, the two methods being completely equivalent.

Discussion

Alfrey and Goldfinger 2), using kinetic arguments. The copolymerization equation (I) was first derived by Mayo and Lewis ' ) and by

r l - x + 1 (1)

where F, and F2 are the mole fractions of the components I and 2 in the copolymer, Mi and M2 are the concentrations and f l and f 2 the mole fractions of monomers M, and M2 in the feed, and x = fl / f 2 .

In order to obtain this simple equation, use was made of the steady-state assumption, according to which the concentrations of active centers M: and M i remain constant (dM:/dt = 0 and dM;/dt = 0). If we neglect the initiation and termination reactions, this implies that the rate of conversion of M: into Mf should be equal to the rate of conversion of Mf into M:. As a consequence:

F1 Mi r 1 . 4 + M 2 - _ f l r1.f1 + f 2 = x ' n = - = - - F2 M2 r 2 . M ~ + Mi f 2 r 2 . f ~ + f 1 r2 + x

1112 = 1121 and k12*Mf*M, = k21 * M,+*M, (2)

A statistical derivation of Eq. (1) without steady-state assumption was published a few years later by Goldfinger and m e 3 ) . They considered the conditional probabili- ties of sequences M,M2 and M2Ml :

1 X and PZl = r,+x (3)

= r , * x + i 0 1990, Htithig & Wepf Verlag. Basel CCC 0025-1 16X/90/$03.00

2796 M. Farina

and the average lengths of sequences (Mi), and (Md), in the copolymera):

The ratio 4 /& is said to be equal to the molecular composition of the copolymer. It follows:

which is identical to Eq. (1). It is a common opinion that the statistical derivation gives a wider field of

application to the copolymerization equation and frees it from the strictures and lack of generality of the steady-state assumption’). This remark seems however rather weak and somewhat contradictory: the validity of the equation cannot depend on the method chosen for its derivation. This difficulty would be overcomeif we could demon- strate that the steady-state assumption is not really necessary, or that it is implicitly used even in the statistical derivation.

The first point can be discussed in a simple and intuitive way. In a succession of different objects along a chain some necessary or stoichiometric conditions exist, their non-fulfilment inevitably leading to chain breakage. These conditions may be considered as “chain continuity conditions”. More specifically, in a chain containing a statistical distribution of two types of objects 1 and 2, we cannot observe two 12 sequences if we do not interpose a 21 sequence between them. This may be easily verified by examining a generic chain, such as

As a consequence, in a linear chain of infinite length the number of 12 and 21 sequences must be the In a linear chain of finite length, their number differs at most by one; this difference may be larger in branched chains. Analogous conclu- sions are valid for the total number of sequences or runs of equal objects, irrespective of their length’):

S, = n(1) + n(l1) + n( l l1) + . . . = n(2) + n(22) + n(222) + . . . = S,

These rules apply also to the succession of stereochemical and regioisomeric defects ( W Z stereogenic or m/r steric dyads) and direct/inverse units, respectively, as well as to copolymers. For the first two classes, the corresponding equations concerning the number of sequences are:

n(01) = n(I0) (rt I), n(mr) = n ( m ) (* 1) and n(head-to-head) = n(tail-to-tail) (rt 1)

a) Actually, in the original paper3) the authors wrote that 1/pi2 ahd 1/pZi correspond to the “weight” fraction of all Mi and M, sequences (quotation marks in the original paper). Later4), l/fij was defined as “relative weight” (author’s quotation marks again) or better still as number-average sequence length.

Kinetic and statistical derivation of the copolymerization equation 2797

For copolymers having a high molecular weight and a considerable degree of disorder, for which I is negligible with respect to the number of sequences (block copolymers, particularly tri- and pentablock copolymers, are explicitly excluded), the chain-continuity principle may be written as:

x 1 2 = x21

s, = s2 or

or

012 = 021

where Xu is the mole fraction of ij dyads, Si the number of sequences of i units of any length and vii the rate of the reactions that produce the ij dyads. Eq. (8) derives from Eq. (6) and coincides with Eq. (2). However, it does not contain any implication about the concentration of active centers: it merely expresses the observation that at every instant the amounts of dyads 12 and 21 are the same and, as a consequence, their rate of formation must be the same

The kinetic derivation of the copolymerization equation follows that proposed in refs. l ) and ’), the only difference being in the meaning of the condition u12 = uZ1 :

... S e e E q . (1) - - U l 0 1 1 + 021 - 011 + v12

d M 2 022 + V t z (022 * 012/V21) + 012

-- - -

This way of derivation has been recently reported by Hamielec, MacGregor and

As for the second question, Eq. (5 ) is subject to criticism. In a strict sense, this Penlidis @. The same approach was independently developed in our laboratory.

equation should be replaced by:

F, / F ~ = s, . i, /(s2- i;, (9)

i. e., the relative amount of each component in the copolymer is given by the product of the average length of the sequences by the number of sequences of the same monomer. This equation reduces to Eq. (5 ) only if we put S, = S2, i. e., if we apply the principle of chain continuity. The validity of the statistical derivation is therefore related to the acceptance of the same principle used in the kinetic method.

In addition to the classical methods, other ways for obtaining the copolymerization equation are reported in the literature.

Tosi demonstrated that the distribution of repeating units along the chain is such as to yield a copolymer with the maximum constitutional disorder (in other terms, with the highest informational entropy) consistent with the constraints ll). The copolymeri- zation equation was obtained in the form:

by. applying a variational principle to the informational entropy of the chain. In his calculations, the author explicitly poses S, = S2, i.e, he resorts to the chain- continuity condition.

2798 M. Farina

A quite different approach was proposed by It0 and Yamashital2) and, more recently, by Seiner and Litt 13) and Thei15): AU these authors based their discussion on Eq. (6), valid when endgroup effects are negligible (infinitely long chains). Expressions for the concentration of the dyads are written as the product of the expectation of the first event, say Mt , by the conditional probability of the sequence under consideration (MlM3. In a high-molecular-weight binary copolymer, holding the assumption of statistical stationarity 12, 14), expectation of finding a given monomeric unit (M,) equals the mole fraction of that component (F,); hence we write:

From Eqs. (6) and (10) we obtain F, *p12 = Fz -pZ1 and directly Eq. (5). Other methods of derivation, like that reported by P r i c e a s g ) and recommended, in a

slightly different version, by Olaj 15) arrive at the same result using a rigorous mathe- matic approach. In his turn, Harwood ‘6) stresses the usefulness of computer-oriented methods in particular of the matrix-multiplication method for calculating the copoly- mer composition and the unconditional probability of any desired sequence

Conclusions The only condition required for the derivation of the copolymerization equation is

the principle of chain continuity. From this point of view no difference exists between the kinetic and the statistical approaches: both methods start from the same assump- tions and are subjected to the same restrictions. Of course, a steady state for the concen- tration of the active centers could be postulated for kinetic studies, but it does not play any role in the present discussion. The importance given to the steady-state assumption in the early papers can be explained as the result of the scientific environment existing at that time, more sensitive to kinetic than to structural aspects of polymer chemistry.

This work was partly supported by grants of the Minister0 dell’Universitil e della Ricerca Scientifica e Tecnologica (MURST) and of the Consiglio Nazionale delie Ricerche (CNR), Rome, Italy.

‘) F. R. Mayo. F. M. Lewis, 1 Am. Chem. Soc 66, 1594 (1944) ’) T. Alfrey Jr., G. Goldfinger, 1 Chem. Phys. 12, 205 (1944) 3, G. Goldfinger, T. Kane, 1 Polym. Sci. 3, 462 (1948) 4, G. E. Ham, in “Encyclopedia of Polymer Science and Technology’: ed. by H. F. Mark, N. G.

Gaylord, N. M. Bikales, Interscience, New York 1966, vol. 4, p. 165 ’) M. H. The& 1 Polym. Sci, mlym. Chem. M. 21,633 (1983)

A. E. Hamielex, J. F. MacGregor, A. Penlidis, in “Comprehensive Polymer Science’: ed. by G. G. Eastmond, A. Ledwith, S. Russo. P. Sigwalt, Pergamon Press, Oxford 1989, vol. 3, p. 17

7, J. C. Randall, “Polymer Sequence Determination’: Academic Press, New York 1977, p. 43 8, F. P. Price, 1 Chem. Phys. 36,209 (1%2) ’) F. P. Price, in “Markov Chains and Monte Carlo Calculations in Polymer Science’: ed. by G.

G. Lowry, M. Dekker, New York 1970, p. 190-193

Kinetic and statistical derivation of the copolymerization equation

lo) M. Farina, Bp. Stereochem. 17, 1 (1987) 11) C. Tosi, Makromol. Chem. 176, 453 (1975)

13) J. A. Seiner, M. Litt, Macromoleculm 4, 308 (1971) 14) B. D. Coleman, T. G. Fox, 1 Polym. ScL, Part A: 1, 3183 (1963) 15) 0. F. Olaj, personal communication la) H. J. Hanvood, Makromol. Chem., Macromol. Symp. l O / l l , 331 (1987)

K. Ito, Y. ~amashita, J Po@. s c i Part A: 3, 2165 (1965)

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