M. SZWARC AND M. LITT
568
Vol. 62
MOLECULAR WEIGHT DISTRIBUTION OF “LIVING POLYMERS.” PART 11. EFFECT OF IMPURITIES1 BY M. SZWARC AND M. LITT Chemistry Department, State University College of Forestry at Syracuse University, Syracuse 10, New York Received November 6,1967
Polymerization which does not involve spontaneous termination might lead to an unusual distribution of molecular weights of such polymers. The present paper is concerned with a polymerization in which a monomer containing small amounts of impurities is added slowly to a stirred reactor. It is assumed that theseimpuritiesterminate (kill) the “living,” growing polymeric molecules. The expression for the degree of polymerization for N . and N i / N . is derived for such a polymer. It was shown that which measures the degree of dispersion of the polymers formed, varies from unity, when no impurities are pfLese@, to 2 when the amount of impurities introduced is sufficient to terminate (kill) all the growing chains. However, N w / N n is still less than 1.2 when the amount of impurities introduced exceeds slightly 40% of the amount of catalyst present initially.
zw/r.,
An addition polymerization deprived of any termination step leads to the formation of “living” polymers, Le., molecules which could continue their growth if a suitable monomer is added to their solution. The molecular weight distribution of such polymers may be unusual and some aspects of this problem were discussed in a previous communication.2 There it was shown that lack of any termination leads to a dynamic equilibrium between monomer and “living” polymer. Consequently, the molecular weight distribution of the originally formed polymer should continue to change if the equilibrium distribution were not attained initially. The kinetics of such a change was discussed and the rate of this process was shown to depend on kd-the rate constant of the depropagation step, and on the number average degree of polymerization. If the former is sufficiently small and the latter sufficiently high, the non-equilibrium distribution could be maintained for a long time. Particularly, by proper choice of conditions one may produce in such systems polymers characterized by the Poisson molecular weight distribution. The present paper is devoted to the discussion of a situation which is encountered in studies of polymerizations yielding “living” polymers. To simplify the treatment of these problems we introduce a few approximations. We assume that kd is essentially zero, and thus we neglect entirely the depropagation reaction, and therefore we are not concerned with the dynamic equilibrium discussed in the previous paper. Furthermore, we assume that all the “living” polymeric molecules initiated a t any time t would acquire the same size a t time t At, if no termination takes place. The latter assumption means that the Poisson distribution is approximated by mono-dispersed distribution, and such an approximation is permissible since we deal with high molecular weight material. The initiating species are considered formally as zero-mers, the addition of the first monomeric molecule would produce, therefore, a growing monomeric unit, and the degree of polymerization j denotes the number of monomer molecules added to the initiating species,
+
(1) This research was supported by the generous grant of the National Science Foundation, NSF-G.2761. (2) W. B. Brown and M. Sswarc. Trana. Faradnu Soc., 54, 416
(1958).
Slow Addition of a Monomer Containing Impurities.-We will discuss now a polymerization in which the monomer, or its solution in an inert solvent, is added slowly to a well stirred solution of initiating species. The initiation reaction is assumed to be very rapid, and thus the very first portion of added monomer converts quantitatively the catalyst solution into a solution of growing chains. Further addition of monomer causes the uniform growth of the chains formed initially. Let us now consider the effect of impurities present in the monomer which terminate (kill) the “living” polymeric molecules. Say that f denotes the mole fraction of these impurities in the monomer. Hence, the addition of dm moles of monomer a t time t terminates a t the same time f dm moles of “living” polymers degree of polymerization of which is In this equation Co denotes a number of moles of originally present initiating species, while Jf denotes the total amount of monomer added up to time t. When M = Cb/f all the growing species are terminated (killed) and further addition of monomer does not contribute any more to the polymerization. Denoting by dxj the mole fraction of terminated (killed) polymeric molecules, the degree of polymerization of which is confined to the interval j,j dj, we find
+
dxildj = f exp(-fj)
(2)
This expression gives, therefore, the distribution of the killed polymers in terms of their degree of polymerization. If the total amount M of monomer added is less than M,,, = C,/f, then the remaining “living” polymers would be killed eventually by the deliberate addition of a suitable killing agent after completion of the polymerization and the following molecular weight distribution would result in the product formed in such a reaction dxj/dj = f expf-fj) for 0 < j < j,,, zt = (I xj = 0
- fM/Co)
for j = jmax for j > jmax
Here, j,,, denotes the highest degree of polymerization attained in the produced material, and jmax = (-1/f) In ( ( 1 -f[fif/C~l)l
This distribution is illustrated by Fig. 1.
(3)
MOLECULAR WEIGHTDISTRIBUTION OF “LIVINGPOLYMERS”
May, 1958
TABLE I fraction of chains killed in the polymerization
fM/Co
=
fW/Pr,
degree of dispersion of the polymer
I
.5 .6
.7 .8 .9 1.0
The number average degree of polymerization of such polymers is, of course
fin = M/Co
d dxj/dj
1.000 1 028 1.075 1.122 1.168 1.228 1.299 1.382 1.492 1.647 2.000
0.0 .1 .2 .3 .4
569
L Fig. 1.
(4)
while the ratio of weight average degree of polymerization to number average degree of polymerization is calculated to be
i
jmax
concentration in the polymerizing mixture remains constant all the time.4 One finds then that dMj*/dt
kMM*j-I
- (1 +f)kll4Mj*
where M j * is the concentration of “living” j-mers, M is concentration of the monomer, k is the rate constant of propagation, and f is the fraction of impurities in the monomer. Substituting kMdt by d Z and remembering that the initial valde of MI* is Co one derives
We notice that iVw/iVn, which measures the degree of dispersion of the polymers produce_d,is_a function of fM/Co only. This means that Nw/Nndepends only on the per cent. of growing chains killed during Mj* = CoZj-lexp { - ( I +f),Z)/(j - l ) ! the addition of the monomer. Inspection of equation 5 shows that for f M / C o approaching zero the It is obvious, therefore, that in spite of the conratio iVw/iVn approaches unity,3 indicating that the tinual termination by the impurities the remaining produced polymer would be monodisperse in ab- “living” polymers have a Poisson distribution at all sence of impurities. On the other hand, as fM/Co times. approaches unity, i e . , the amount of added monoSince dMj (killed) = fMj*dZ one finds Xjz e r is sufficient to terminate all the growing chains, the mole fraction of “killed” j-mers-to be given N w / N n tends to 2. I n the later case, j varies from by the equation 0 to infinity, and the distribution of polymeric dXj = -f(-ln A)j-lAfdA/(j - l)! molecules is identical with the “normal” distribuwhere -In A denotes Z. On integration tion called by Flory “the most probable distribuj-1 tion.” Xj = [f/(l +f)’) 1 - A l + f Z [ - ( 1 +f)lnA]1/9! Table I gives the values for flW/Rncomputed s=o from equation 5 for various values of fM/Co. It is interesting to notice that even for 40% of impuriAt the end of the polymerization when all the ties in the added monomer, which_ terminate there- ( I living” polymers are “killed” A = 0 and Xj = fore 40% of growing chains, N w / N n is still less than f/(l f)i. Hence 1.2. En= ( I f)/f; Rw= (2 f)/f and We can now examine the effect of the approximaN w / N n = 2 - .f/(l f) tion made at the beginning of this paper, namely, that all the “living” polymers form a mono-dis- For small values of f this result differs insignifiperse system a t any time. Let us choose the rate cantly from that derived previously for f = Co/ of addition of the monomer in such a way that its Mtotal.
l-
{
+
(3) As f M / C o + 0 the last bracket tends t o ‘/a(fM/Cd.’
+
+
+
(4) This choice does not affeot the generality of the solution.
,
