HOMOGENEOUS IONIC POLYMERIZATION
May 5, 1963
[COXTRIBUTIOX FROM THE
MELLON INSTITUTE,
PITTSBURGH
1241
19, PENNA.]
11. The Molecular Weight
A Multistate Mechanism for Homogeneous Ionic Polymerization. Distribution BY BERNARD D. COLEMAN AND THOMAS G Fox RECEIVED OCTOBER26, 1962
To explain the “stereoblock” structures which occasionally result from homogeneous anionic polymerizations, we have examined, in paper I of this series, the diastereosequence distribution resulting from a mechanism in which a reactive polymer chain end has two possible states, 1 and 2, in dynamic equilibrium, both capable of adding monomer, but each with its own stereospecificity. Here we consider the molecular weight distribution resulting from such a mechanism when monofunctional initiators are used. We assume no side reactions such as irreversible termination or chain transfer and assume initiation to be instantaneous. As a result, under certain limiting conditions our molecular weight distribution reduces to the Poisson distribution obtained by Flory for the classical one-species polymerization. In general, however, the present mechanism yields a ratio, Y , of the weight to number average degrees of polymerization which is larger than t h a t obtained from the Poisson distribution. Whenever the rate constants for the reactions 1 e 2 are non-zero, our present distribution shares with the Poisson distribution the property t h a t Y approaches unity for long reaction times.
1. Hypotheses.-In an effort to explain the occurrence of “stereoblock’’ polymers in homogeneous2 anionic polymerization of cu-olefins, we recently postulated3 that it sometimes occurs in the course of such polymerizations that the reactive end of a growing polymer molecule exists in two possible states which are in dynamic equilibrium and that each state is capable of adding monomer with its own characteristic rate constant and stereospecificity. On calculating the diastereosequence distribution resulting from such a mechanism we found that this “two-state hypothesis” not only furnishes an alternative to the “Markoffchain h y p o t h e ~ i s ”which ~ ~ ~ investigators6z7 have used to interpret their n.m.r. data, but also yields the first explanation of “stereoblock” structures t o be susceptible to experimental verification through definite predictions of the dependence of n.m.r. spectra on the conditions of polymerization. Here we shall consider the molecular weight distribution resulting from such a mechanism. As in reference 3, we consider that a t a certain time (t = 0) monomer is added to a solution of monofunctional initiator. Polymerization then begins, with initiation being instantaneous, and proceeds without chain transfer, irreversible termination or depolymerization. We postulate that there are four reactions which can take place. -To describe them we indicate the condition of a polymer molecule by the symbol E,”. The superscript n, representing the degree of polymerization, can take on all integral values, while x,signifying the state of the reactive end of the polymer, can take on two values: x = 1 if the reactive end is in state 1 and x = 2 for the alternative state 2. Our postulated reactions are Ab
El” --f Ezn
+ kz Ez” + M +Ezn‘
+
+
$ =
(1.3)
Qi(0) = 1
(1.4)
Since we assume that the reactions l . l a and l . l b are in equilibrium at t = 0, we have PldO)
(l.lb)
=
$,
=
Pn,l(0) = P,,s(O)
m
=
1- $
(1.5a)
>1
(1.5b)
0, for n
m
P,,i(t)
(l.lc)
(1) T h e research reported here was supported in part by the Air Force Office of Scientific Research under Contract A F 49(638)541. ( 2 ) T. G Fox, B. S. Garrett, W. E. Goode, S . Gratch, J. F. Kincaid, A . Spell and J. D. Stroupe, J . A m . Chem. SOL.,8 0 , 1768 (1958). (3) B.D. Coleman and T. G Fox, J . Chem. Phys., 3 8 , 1065 (1963). (4) B . D. Coleman, J . Polymev Sci., 3 1 , 155 (1958). ( 5 ) R. E. Hughes, Abstracts of Papers, 133rd Xational Meeting of t h e American Chemical Society, San Francisco, Calif., April, 1958, p. 10R. (6) U.Johnson, Koiloid Z . , 178, 161 (1961). (7) D . Braun, M. Herner, U. Johnsen and W. K e r n , Makvoiitol. Chem., 51, 15 (1962).
+ Pn.z(t)
Q n ( t ) = Pn.l(t)
For all t > 0
Here h b is a rate coefficient for that reaction which takes a growing chain from state 1 to state 2. In other words, the probability that a polymer molecule, known to be of
(1.2)
Ab)
is the probability that a randomly selected polymer molecule has its degree of polymerization equal to n a t time t . By hypothesis, the degree of polymerization is unity a t t = 0, hence
Of course
(l.ld)
+
Xa/(Aa
Let Pn,Jt),x = 1 or 2, be the probability that a polymer molecule, selected a t random, is in the state Exn a t time t . Then Qn(t)>defined by
(1.la)
Xa
Ez” +El” ki E,” M --f El“+
type Eln a t time t , becomes of type Ezn in the small time interval dt a t t is given by Xbdt f o(dt). Similarly, Xadt o(dt) gives the probability that Ezn + Elnoccurs in dt a t t , given Ezna t time t. The numbers kl and kz are ordinary rate constants for the addition of monomer to polymer chains in states 1 and 2, respectively. For example, the probability that a polymer molecule known to be in the state Eln a t time t makes the transition Eln + Eln+l in dt a,t t is [Mlkldt o(dt) where [If]denotes the monomer concentration. The numbers A,, Ab, k1 and kz will depend on the temperature and solvent used; we assume here that they are independent of the degree of polymerization n. We also make the usual “low conversion hypothesis” and assume that [MI does not change much in the course of a given polymerization.8 It follows from the chemical hypotheses we have made that the probability # that a polymer molecule selected a t random is in the state 1 a t a given moment t is
Pdt)
= $,
=
1- $
(1.6)
n=l
n=l
Furthermore, for n
=
1, t 2 0, we have
dPi,i(t)/dt = - ( [Mlki dPl,z(t)/dt = - ( [Mlkz
+ + X a P i , ~ ( t ) ((1.7a) 1.ib) + A,)Pi,z(t) + Xb)Pl,l(t)
XbPl,l(t)
and for n > 1,t 2 0, we have -dPn’l(t) dt
-
-([Mlki -t Xb)P,L.i(t) f
+ AZn.z(t) [M]kzP,-i.z(t) +
[MlkiPn-i.i(t) dPndt) - - ( [ M l k ~ X,)P,,z(t) dt
+
+
AbPn.l(t)
(1.7C) (1.7d)
(8) Since we are here interested in only degree of polymerization, and not in diastereosequences, we do n o t here differentiate between additions which result in isotactic a n d syndiotactic placements Hence the six reactions Of section 1 of ref. 3 here reduce t o the four reactions 1.1.
BERNARD D. COLEMAN AND THOMAS G FOX
1242
Equations 1.2-1.7 furnish a complete mathematization of our chemical assumptions. On looking a t 1.3 we see that the problem of characterizing the molecular weight distribution is that of investigating the behavior of the solution of the system of differential equations 1.7 subject to the initial conditions 1.6. Our main interest here is in the moments
VOl. 85
merization, which was shown by Floryg t o yield a Poisson distribution for n. It follows from 2.4 and 1.9a that the number average degree of polymerization is given by i i = ~ + e
n2 = 1 1
m
nm =
(1.8)
nmQw(t)
R, =
n=l
of the distribution Qn(t) of the degree of polymerization n. We are particularly interested in the number and weight average degrees of polymerization a(t), aw(t), respectively, and their ratio r ( t ) . These are defined by
(2.5a)
I t also can be shown that
1 = 1
+ 3fi + + 3e + 82
(2.5b)
e2
(2.5~)
l + e 38 82 28 82
+ + + +
(2.5d)
Expressing Y as a function of n we have
m
(1.9c)
J It should be emphasized that in writing our mechanism 1.1 and our equations 1.2-1.7 we make some very strong idealizations : We assume instantaneous initiation, no irreversible termination and no chain transfer or depolymerization. The molecular weight distribution is more sensitive than the previously considered diastereosequence distribution to departures from such idealizations. 2. Special Cases.-Case I : When one of the two A’s in 1.1 is zero, say Ab, and the other, A, is not zero, so that $ = 1, then all of the polymer molecules are in state 1. In this case the four reactions 1.1 reduce to the one reaction Eln
ki +M+ E,“*
(2.1)
I
We have Pn,dt) = 0 ,
P,,,(t)
= Qn(t)
and the system of differential equations 1.7 reduces t o the system
+
(2.2a) (2.2b)
which is now subject to the initial condition for n = 1, Ql(0) = 1 for n > 1, Qn(0) = 0
(2.3a) (2.3b)
The solution of 2.2 subject to 2.3 is known even when [MI or k l , or both, depend on t. For instance, [MI depends on t , i.e., [MI = [ M I 1 ,when an appreciable amount of monomer is consumed in the polymerization, and k1 depends on t , ;.e., k l = k?, if the temperature changes in the course of the polymerization. Let us put [MIrklrds
O(t) =
fin-1
=
( n - I)!
$klr
+ (1 - $)kzT
(2.7)
$Pn,l(t)
+ (1 - $)Pn,n(t)
(2.8a)
where Pn,,= OZn--l
(n
-
e-& l)!’
Oz =
x = 1, 2
[MI X k , ’ d r ,
(2.8b)
Here we again make the low conversion hypothesis and assume [MI to be stationary in time; this saves us from the task of taking into account the competition for monomer between the species 1 and 2. We can, however, continue here to allow kl and kz to depend on time. It follows from 2.8 that n“(t)
= &F(t)
+ (1 - li.)G(t)
(2.9a)
where nzm
m
nmP,,,*,
=
x = 1, 2
(2.9b)
n=l
is the mth moment of the Poisson distribution for species x . Equations 2.8, 2.9 and 1.9 yield
-R
2
=
where $ is given by 1.2. Case IV: A special case of our mechanism which is strikingly different in its behavior from those discussed above is that in which k1 > 0, kz > 0,k l # kz, 0 < $ < 1 but X, = Ab = 0 (exactly). In this case the reactions l . l a and l . l b do not occur a t all and polymerization behaves as though we had a mixture of two non-interacting species, each with its own Poisson distribution
(2.4a)
The solution of 2.2 subject to 2.3 is Qdt)
k;‘
Qn(t)
n2 1
dQi(t)ldt = - [MlkiQi(t) dQn(t)ldt = - [MlkiQn(t) [MIklQn-l(t)
Thus, for reasonably large a,r is very close to 1. For example, when a 2 50, 1 < r < 1.02. Case 11: Similar to case I is that case in which the A’s and k’s are all strictly positive but kl = k2. Clearly, we here again have the Poisson distribution and thus eq. 2.4-2.6. Case 111: Another special, or rather limiting, case is that in which all the k’s and A’s are strictly positive but yet the reactions l . l a and l . l b are both so fast compared to l . l c and l . l d that, on the average, several transitions Eln Ean occur between successive additions of monomer. For a fixed set of non-zero values of the A’s and k ’ s this limit can be achieved by making [MI small. In this limit eq. 2.4-2.6 again describe the molecular weight distribution provided k l r in 2.4a is replaced by
R = $El
(2.4b)
+ (1 - $ ) B z
(2.10b)
i.e., a Poisson distribution in the “reduced time” parameter O(t). This result is not surprising. For, when Ab = 0 our chemical hypotheses reduce to the familiar single-state polymerization with, of course, instantaneous initiation and no termination or depoly-
(2.10a)
(2.10c)
~ _ _ _ (9) P. J. Flory, J . Aril. Chew. Soc., 62, 1501 (1940).
HOMOGENEOUS IONIC POLYMERIZATION
May 5, 1963
1243
where, by 2.8 and 2.9b
+ ez, x + 38, + OZ2,
= 1
nZz = 1
1, 2
=
(2.1l a )
x = 1, 2
(2.11b)
It follows from 2.11a, 2.10a and 2.8 that ii = 1
+ [MI
(2.12)
k'dr
where KT is again defined by 2.7. When kl and kz are independent of t, 2 . 1 0 ~ can he written in the form Y =
1
1
f = - -z+ n
-
- kz)'$(l
(ki
1 (E)
($kl
+
- $)(E 1)s (1 - $)kZ)ZEZ
m
m
(2.13)
On examining (2.13) for large fi (;,e,, large t ) we find that
m
Hence, our present case is similar to cases 1-111 in that -lim r is independent of [MI but differs in the n+
m
Q(s,t)
snQn(t),
=
m
>s>0
(3.1)
n=l
(3.9)
n = l
Equations 3.6, 3.8 and 3.9 yield
( P -k
(1
-
s)[M]ki f hb)k(S,P)
-
(3.10a)
= S$
Aa&(S,p)
Applying the same process to eq. 3.7b and 3.7d we get
Equations 3.10, being two independent algebraic equations in two unknowns l?, and k2,can be solved. On doing this and adding the results in accordance with 3 . 6 ~we ) find
&,P)
=
+ + ($[MIkz f (1 - $)[MIkl)(s - s2) + [ M I M I - f A,) -(3.11) + [Mlki(l -
~ ( -Ip A,
This function obeys formulas of the type
s'@n.l(p)
= S
n = 2
m
important respect that 1' can now be appreciably greater than 1 for large f i . We remark that eq. 2.14 holds only if ha and Ab are exactly zero; for the analysis of the next section will show that if A a and Ab do not both vanish, then, no matter how small the A's be, the limit t -t must yield r -+ 1. 3. The General Case.-We now return to consideration of the general equations 1.2-1.7, assuming that A, Ab, kl, kq and [MIare all constant in time. Let us consider the function Q(s,t) defined by
m
2 .Yn$n-l,l(p)
(P
Ab)
Ab)($'
s)
hahb
3)
We now apply to 3.11 standard techniques for the inversion of Laplace transforms. The denominator of the fraction on the right in 3.11 is quadratic in p and can therefore be written in the form etc. Hence the determination of Q(s,t) near s = 1 solves the problem of finding the moments G of Qn(t). I t is interesting to note that the generating function Q(s,t) also yields the functions Qn(t) through the formula (3.3)
but we do not use this relationship here. Let us put Rl(S,t)
s"Pn*l(t),
=
s.P,,z(t)
Rz(s,t) =
n=l
11
+ [Mlkl(l - s) +
=
Ri(s,t)
24s) = f
d(1- s)'[MI2(kl
Lm
e-PLPP,,i(t)dt,
=
- kz)'
+ U 2 - 2[M](Aa- hb)(ki - k z ) ( l - s) - (1 - s)[M](kl + kz) - a (3.13a)
1, 2 and A,
a
+
Let us put w = (1 - $)ki
+ Rz(sJ)
(3.5)
i
=
-
elementary calculation yields
(3.4)
=1
Let us define the following Laplace transforms
Sn,$(p) =
s)
(0 - Yl(s))(P - Y Z ( 3 ) ) (3.12) where the roots vl(s) and v2(s) are functions of s. An
Equations 1.3, 3.1, and 3.4 yield Q(s,t)
+ [Mlkz(l - +
Ab)($'
hahb =
where i
m
m
(P
Ab
+ $kz
(3.13b) (3.13~)
Using 3.12 and 3.13, 3.11 may be rewritten in the form
1, 2 (3.6a)
(3.14)
Laplace transforms with the simple dependence on p shown in 3.14 are easily inverted, and a slight rearrangement of the result of such an inversion yields the following expression for Q(slt)
+ + w [ M ] ( l - s))ey2(5)L1(3.15)
(u~(s) u
Using eq. 3.2, 3.15 and 3.13 we can now calculate fi and n2 by differentiation. On doing this straightforward, but lengthy, calculation we find
+ [M]k
(3.16)
+ (1 - $)kz
(3,li)
ii = 1
where
1=
$ki
BERNARD D. COLEMAN AND THOMAS G Fox
1244 and
n 2 = 1 - [ M ] ? b ( l- e - a t ) / u 2
+ ( 3 [ M ] k+ [ M ] ? b / a ) t+ [MI2kZt2 ( 3 . 1 8 )
=
2(ki
- kn)'$(l -
$)
(3.20)
+ ( 1 + [M]Z~/ka)(+i)-~ ( 1 + [ M ] b / & + [ M ] 2 ~ - Z b(l
e-at))(fi)-'
(3.21)
From which we see that whenever a is non-zero lirn 11-
1' =
1
=
$kl
b = 2ki2$(1
We have written ey. 3.16, 3.18 and 3.20 in such a way that the dependence of the averages on [MI is exhibited explicitly. It follows from 3.20, 3.16 and 1 . 9 that ~ 1
k
(3.19)
It follows from 3.18, 3.16 and 1.9b that
7' =
a model for reversible termination: a dynamic equilibrium between two states 1 and 2, one of which, 1, cac add monomer, but the other, 2, cannot. In this case k and b simplify to
-
where b
Vol. 85
(3.22)
m
i.e., in the limit of high number average molecular weight the molecular weight distribution must become perfectly sharp. This does not mean, however, that r is close i o 1 for all degrees of polymerization. If [ M ] b be >> k a , then 3.21 can yield Y >> 1 even for large a, although r - 1 approaches zero with order 0 ( l / n ) . We note that eq. 3.21 becomes identical with 2.6 in the following limits
-+ 0, while AI>, k , , kP, [MI, and t remain fixed a t arbitrary positive values case 11: kl --+ k2, while A,, Ah, k2, [MI, and t remain fixed a t arbitrary positive values case 111: [MI -+ 0, while A,, Ah, kl, k?, and t remained fixed a t arbitrary positive values case I : A,
in agreement with the discussion of section 2. If in 3.21 we take the limit a 40, keeping fixed a t a positive value less than one, and keeping kl, kz and t fixed a t arbitrary positive values (it is important here that t be held finite), we then obtain 2.13. d special case of the present theory is that in which ha, Ab and kl are strictly positive but kz = 0. This gives
-
=
X,kl
+
$) =
(3.23)
Ab
2k1 %aAb ~
(ha
+
(3.24)
Ah)'
but the essential character of our results 3.16, 3.20 and 3.21 is unchanged: In particular, 3.22 still holds. For the experimenter, the outstanding characteristics of the present mechanism are (1) for a fixed set of rate constants such that Xa > 0, A b > 0, and fixed extent of reaction (;,e,, fixed a), r depends on [MI and approaches 1 (a)-' - ( B ) - ~as [MI + 0 ; ( 2 ) for fixed rate constants with Xa > 0, Ab > 0, and fixed [M1, r + l a s t + m. Unfortunately, no systematic experimental studies designed to verify conclusions 1 and 2 have yet been reported. Datalo on the n.m.r. spectra and on the molecular weights of the products of homogeneous anionic polymerization of methyl methacrylate in toluene, and also in mixtures of toluene with ether or tetrahydrofuran, show values greater than one for r and for the persistence ratio p defined in section 3 of ref. 3. Datal'~'* on the products of homogeneous anionic polymerization of a-methylstyrene indicate that values of Y and of p close to 1 are readily obtained for this monomer. Values of r < 1.05 are reported in anionic polymerizations of styrene.13 If our mechanism applies to these systems it appears that, a t the monomer concentrations employed, in methyl methacrylate reactions l a and l b are slower than ICand Id, whereas the reverse seems to be true for a-methyl styrene and possibly also for styrene for which p-values are not yet available.
+
(10) D . A . Glusker, R. A. Galluccio and R. A . Evans, Polymer Division Preprints, 142nd National Meeting of t h e American Chemical Society, Atlantic City, S . J., September, 1902; D . .4.Glusker and R. A . Galluccio, ibid., p , 331. (11) S. Brownstein, S . Bywater a n d L). J. Worsfold, i b i d . , p. 134*. (12) A. F. Sirianni, D. J. Worsfold and S. Bywater, T r a n s . F Q Y Q ~ U Soc., Y SS, 2124 (1959). (13) F. Wenger, M ~ k i ~ m Cdh.e m . , 36, 200 (19GO).
