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Ind. Eng. Chem. Res. 2010, 49, 11890–11895
Irreversible Anionic Polymerization Kinetics Revisited Isaac C. Sanchez† Chemical Engineering Department, UniVersity of Texas at Austin, Austin, Texas 78712, U.S.
Irreversible anionic polymerization is revisited with an emphasis on determining the dependence of the MW distribution on initial conditions. It is shown that only two parameters govern the distribution: the ratio of initiation to propagation rate constants and the ratio of initial monomer to initiator concentrations. New and simple criteria are developed that predict the breadth of the MW distribution based on these parameters. Even when initiation is slower than propagation, it is still possible under certain conditions to obtain a narrow, Poisson-like MW distribution. Since it impacts self-assembly of block copolymers into microphases, the compositional variation of block copolymers produced by living polymerization techniques is also analyzed. A somewhat unexpected result is the composition variability in block copolymers can be significant (10-15%) even when the blocks have dispersity indices less than 1.01. I. Introduction
II. Kinetic Model 1
In 1940 Flory quantitatively described the ring-opening polymerization kinetics of ethylene oxide. He showed that the molecular weight (MW) distribution should be very narrow and follow Poisson statistics. The ethylene oxide polymerization mimics what is now known as a “living” polymerization that is characterized by the absence of termination or chain transfer reactions. Szwarc2 first coined the expression “living” polymer in 1956 and showed that anionic polymerization could produce “living” polymers. A recent review of anionic polymerization tracing its interesting history is available.3 Flory’s analysis of the ethylene oxide kinetics assumed that the initiation reaction is much faster than any propagation step. In 1958 Gold4 generalized Flory’s treatment to allow for a variable initiation rate and showed that the MW distribution can broaden for slow initiation as has been observed experimentally.5 To quote Szwarc,2 “Polymeric molecules are born in an initiation process, they grow by a propagation process, and finally they ‘die’ in a termination process. This death is regulated by the conditions prevailing in the polymerization process, and thus, the rate of death, the average molecular weight and its molecular weight distribution are well-defined functions of experimental conditions.” The key idea is that at the end of the polymerization, the final MW distribution is a well-defined function of initial conditions. This expectation is not readily apparent in Gold’s analysis. In 1998 Yan6 treated the more general case of a trifunctional initiator with variable initiation rates. Yan focused on the variation in arm lengths caused by nonequal arm initiation rates. Herein an analysis is presented that clearly predicts the dependence of the MW distribution on initial conditions. It is shown that only two parameters determine the distribution. New and simple criteria are developed that predict the breadth of the MW distribution based on initial conditions (see eqs 50-52). Even when initiation is slower than propagation, it is still possible to obtain a narrow, Poisson-like MW distribution. Since it impacts self-assembly of block copolymers into microphases,7 the compositional variation of block copolymers produced by living polymerization techniques is also analyzed. A somewhat unexpected result is the composition variability in block copolymers can be significant (10-15%) even when the blocks have dispersity indices less than 1.01. †
Corresponding author. E-mail:
[email protected].
A. Initiation and Propagation Reactions. kI
I + M 98 p1 kp
p1 + M 98 p2
(1)
kp
p2 + M 98 p3 ... kp
pn-1 + M 98 pn where pn is a polyanion of size n, M is the free monomer, and I is the initiator. All reactions are assumed to be irreversible. The initiator disappears by a second-order rate reaction: d[I] ) -kI[M][I] dt
(2)
d[p1] ) kI[M][I] - kp[M][p1] dt
(3)
as do the polyanions:
d[pn] ) kp[M]{[pn-1] - [pn]} dt
ng2
(4)
[I], [M], and [pn] are the molar concentrations at time t of initiator, monomer, and polyanions of size n, respectively. Since [M] depends on the concentration of the polyanions, [pn], this infinite set of coupled differential equations are nonlinear. Noting that the rate equations contain a common factor of kp[M] suggests a change of variable: dz ) kp[M] dt
(5)
which yields a set of simple linear equations: dP0 + rP0 ) 0 dz
10.1021/ie1003637 2010 American Chemical Society Published on Web 10/20/2010
(6)
Ind. Eng. Chem. Res., Vol. 49, No. 23, 2010
dP1 + P1 ) rP0 dz dPn + Pn ) Pn-1 dz
(7)
[pn(t)] ,n g 1 [I]0
P0 ≡
[I] [I]0
[I] ) [I]0
z 1 - exp[-rz]
r≡
ki kp
P0 ) 1
(9)
lim nj ) lim nj ) 1
tf0+
Pn ) 0 (10)
and for very fast initiation (r f ∞), z f nj - 1. C. A Special Case (r ) 1). If the initiation and propagation rate constants are equal, the set of equations, (7) and (8) can be easily solved by induction. Substituting (16) into (7) yields P1(r ) 1,z) ) ze-z
Pn(r ) 1,z) )
(11)
n
n)1
For monomer, the mass balance requirement is
∑ n[p ] n
n)1
∞
∑ n[p ] n
nj ≡
n)1 ∞
)
∑ [p ]
[M]0 - [M] [I]0 - [I]
(21)
dPn+1(z) + Pn+1(z) ) Pn(z) dz
(12)
From the two mass balances, a general and time dependent relation for the average degree of polymerization nj is obtained:
zn -z e ≡ Pn(z) n!
where Pn(z) is the Poisson probability distribution. This result obtains because the Poisson distribution satisfies a recursive differential equation of the same functional form as eq 8:
∞
[M]0 ) [M] +
(20)
Substituting (20) into (8) generates P2 and so on with the general solution
∞
∑ [p ]
(19)
zf0
For polyanions, the mass balance requirement is [I]0 ) [I] +
(18)
The new dimensionless variable z(t) is closely related to the average degree of polymerization, nj(t). Notice that as time approaches zero,
and [I]0 is the initial concentration of initiator. For notational convenience, the fraction of unreacted initiator (P0) is treated as polyanions of size zero. A similar linearization of the more complicated equations for a trifunctional initiator has been employed previously6 and is a mathematical trick with a history.8 B. Initial Conditions and Mass Balance. At t ) 0 [M] ) [M]0
z + (1 - 1/r)(1 - exp[-rz]) 1 )1- + 1 - P0 r
(8)
ng2
where Pn(t) ≡
nj )
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(22)
D. General Solution. Returning to eq 7, a general solution of the rate equations for arbitrary r is obtained by substituting P1 into the propagation equations to generate P2, P3, ..., P2, P3, ..., etc. This inductive procedure is aided by eq 22 and noting that P1 satisfies the following differential equation,
(13)
dP1 + P1 ) rP0(rz) dz
n
(23)
n)1
and has the following solution:
The monomer rate equation is
P1 )
∞
d[pn] d[M] n ) dt dt n)1
∑
(14)
Using the above relations to calculate P2 yields
) -kI[M][I] - kp[M]([I]0 - [I]) The first term in eq 14 represents the disappearance of monomer by reaction with initiator [see eq 2] and the second term the disappearance of monomer by reaction with growing polyanions of concentration [I]0 - [I]. In terms of the variable z it becomes d[M] ) -[I]0{1 + P0(r - 1)} dz
(15)
r 1 P (z) P ) r-1 1 r-1 1 r r 1 P (z) P (z) + rP0(rz) (25) 2 0 r-1 1 (r - 1) (r - 1)2
P2(r,z) )
and then P3 P3(r,z) )
Using the solution to eq 6, P0 ) exp[-rz]
r r {exp[-z] - exp[-rz]} ) [P (z) - P0(rz)] r-1 r-1 0 (24)
(16)
r [P0(z) - (r - 1)P1(z) + (r - 1)3 (r - 1)2P2(z) - P0(rz)]
(26)
Continuing the process yields
and integrating yields [M] ) [M]0 - [I]0{z + (1 - 1/r)(1 - exp[-rz])}
(17) Substituting this result in eq 13 yields a relation between nj and z:
Pn(r,z) )
r {P0(rz) (1 - r)n
n-1
∑ (1 - r) P (z)} k
k
ng1
k)0
(27) In the limit of very fast initiation (r f ∞, z f nj - 1),
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Ind. Eng. Chem. Res., Vol. 49, No. 23, 2010 n-1
lim Pn(r,z) ) (-1)n+1lim
rf∞
rf∞
∑ (-1) r
Pk(z) ) Pn-1(z)
k k+1-n
S(u) )
ru {exp[u - 1] - exp[-rz]} r+u-1
k)0
(28) 1
and the well-known Flory result is recovered:
Note that S(1) ) 1 - exp[-rz] is just the normalization condition. The average degree of polymerization nj is given by ∞
P0 ) 0
1 dS ∑ nP (r,z) ) S(1) du
nj )
zn-1 -z e Pn(rf∞,z) ) Pn-1(z) ) (n - 1)!
z ) nj - 1
Recovering eq 21 as r f 1 is easier if eq 27 is transformed from a finite series to the following infinite series: ∞
∑ z (n(1+ -k)!r) n+k
k)0
k
n
n)1
nj )
∞
∑
(30)
Now it is easily seen that Pn(1,z) ) Pn(z) as it should. As a further check, it is also easy to show that P1(r,z) derived from eq 30 yields eq 24. E. Statistical Properties. Important statistical properties are the average degree of polymerization (nj) and the dispersity index (pI), the ratio of the second moment to the first moment squared. Before calculating these moments a check of the normalization of Pn(r,z) using eq 30 yields
(31)
n
n)1
where S is the double sum, ∞
S)
∞
∑ ∑ z (n(1+ -k)!r) n+k
)
n)1 k)0
∑ z (k (1+ -1)!r) k+1
k)0
∞
∑
z
k
n(n - 1) · · · (n + 1 - k) )
(1 - r) + ... (k + 2)!
k)0
n2 )
k
(32) n2 )
that satisfies the following DE: ∂S ) ∂z
∞
zk(1 - r)k + k! k)0
∑
∞
∑ z (k (1+ -1)!r) k+1
k)0
k
(33)
The solution of this DE is 1 S ) {exp[z] - exp[z(1 - r)]} r
(34)
∞
∑
Pn(z,r) ) 1
(40) u)1
1 d2S S(1) du2
[ |
1 d2S S(1) du2
u)1
+
|
or u)1
dS du
| ] u)1
(41)
(r - 1)(r - 2) z2 - 2z/r + 3z + 1 - exp[-rz] r2
2 pI ≡ n2 ) nj (rz)2 - 2(rz) + 3r(rz) + (r - 1)(r - 2){1 - exp[-rz]} {1 - exp[-rz]}{rz + (r - 1)(1 - exp[-rz])}2 (42)
As a check, the dispersity index is evaluated in 2 limits:
Substituting this result into eq 31 yields P0 +
|
The dispersity index is given by
+ ... )
exp[z(1 - r)] + S
1 dkS S(1) duk
Thus, the second moment is also easily obtained:
+
k+2
(39)
The difference arises because the usual average for the Poisson distribution includes the zeroth component, whereas the definition of the average degree of polymerization does not include the zero component (the initiator) in the average [see eq 13]. Of course, at sufficiently long times (z > 10) both averages become identical since P0(z) f 0. Higher moments are generated from higher derivatives of S(u). In general,
n(n - 1) ) ∞
k
(38)
z z ) -z 1 P0(z) 1-e
∞
∑ P (r,z) ) r exp[-z]S
u)1
and the mass balance requirement for nj in eq 18 is recovered. It is well-known that for the simple Poisson distribution, eq 21, nj ) z, whereas eq 38 yields for r ) 1,
)
n! [z(1 - r)]k rPn(z) (n + k)! k)0
|
1 z ) 1+ 1 - exp[-rz] r
(29)
Pn(r,z) ) r exp[-z]
(37)
(35)
pI(rf1,z>10) )
z(z + 1) (nj + 1)nj 1 ) )1+ nj z2 nj2
(43)
n)1
and it is seen that normalization is satisfied. To calculate the moments of the distribution, the following auxiliary sum is defined: ∞
S(u) )
∞
∑ ∑u k)0 n)1
- r)k (n + k)!
n+k (1 nz
(36)
Using methods similar to those used to derive eq 34 yields
pI(rf∞) )
z2 + 3z + 1 nj - 1 1 )1+ =1+ nj (z + 1)2 nj2
(44) Both results check with those obtained directly from eqs 21 and 29. F. Late Stages of Polymerization. In the late stages of polymerization (t f ∞), there are 2 distinct possibilities [cf. eq 13]:
lim nj ) nj∞ ) tf∞
{
Ind. Eng. Chem. Res., Vol. 49, No. 23, 2010
[M]0 ≡ n0 [I]0
fast initiation
n0 [M]0 ) slow initiation [I]0 - [I] 1 - P0
(45)
In the first case, monomer and initiator are completely consumed (vanishingly small amounts remain), whereas in the second case, initiation is so slow that the initiator is only partially consumed (but all monomer is consumed). For fast initiation nj∞ f n0
z f z∞ ) n0 - 1 + 1/r
rz∞ > 10 (46)
The condition rz∞ > 1 0 guarantees that the initiator is completely consumed; see eq 16. For fast initiation the dispersity index (42) now becomes pI ) 1 +
n0 - 1 2
+
n0
1 (rn0)2
(47)
But in addition, if rn0 > >1, then pI = 1 +
1 n0
(48)
and a narrow Poisson-like distribution is expected in the late stages of the polymerization for fast initiation. Slow initiation is defined as the condition where all monomer is consumed, but the initiator is only partially consumed. This situation only occurs if r ,1 and rn0 < 5. In the extreme limit of r f 0, nj∞ )
z∞ + 2 ; 2
n∞2 )
2z∞2 + 9z∞ + 6 ; 6 2 4 z∞ + 9z∞ /2 + 3 pI ) 3 (z + 2)2
[
∞
]
(49)
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and pI quickly asymptotes to 4/3 as z∞ increases. Actually, these relations hold not only at the end of the polymerization but also at any time t in the limit of r f 0. Typically, initial conditions are such that n0 > 102; on the basis of this expectation for n0, a classification of the dispersity index at the completion of the polymerization can be made (t f ∞): pI ) 1 +
pI ) 1 +
1 n0
1 1 + n0 (rn0)2
rn0 . 1
(fast initiation)
rn0 > 10
(50)
(moderate initiation)
(51) 1+
1 4 < pI e n0 3
rn0 < 5
(slow initiation)
(52) The effects of slow initiation are illustrated in Figure 1. Also see Figure 2. An example of slow initiation is one in which 10% of the initiator remains when the polymerization is complete: rz∞ ) ln 10 ) 2.302 ... nj∞ )
n0 10n0 1 10 ln 10 )1+ -1 ) 1 - P0 9 r 9
pI ) 1 +
[
]
(53)
1 - 10[ln 10/9]2 + r[10 ln 10/9 - 1] 10 ln 10/9 + r - 1
which corresponds to the condition where 90% of the initiator has been consumed (P0 ) 1/10). The behavior is illustrated in Figure 2. As less and less initiator is consumed, corresponding to slower and slower initiation, pI f 4/3, characteristic of a uniform distribution (all chain lengths equally probable). By the time 99.99% of the initiator is consumed, the dispersity starts to
Figure 1. Initial conditions determine the final distribution at the end of the polymerization. Both distributions yield an average degree polymerization of 100. The narrow distribution (rn0 . 1) follows the simple Poisson distribution given by eq 29 and has a dispersity index of 1.01, whereas the broad distribution (rn0 ) 1.417) has a dispersity index of 1.15. For the broad distribution, 10% of the initiator remained at the end of the polymerization. Equation 27 was used to create the broad distribution with r ) 0.01574 and rz∞ ) ln 10.
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Ind. Eng. Chem. Res., Vol. 49, No. 23, 2010
Figure 2. Statistical behavior at the end of the polymerization for slow initiation conditions where rz∞ ) ln 10. The actual degree of polymerization possible is determined by initial conditions; here nj∞ ) (10/9)n0 where n0 ) [M]0/[I]0. Under fast initiation, nj ) 500 would produce pI ) 1.002, whereas under the slow initiation conditions defined here, the same nj ) 500 yields pI ) 1.14.
resemble that obtainable by moderate initiation and eq 47 determines the dispersity index.
fi )
ni
≡
m
∑ nj
ni nj
jf i )
and
nji nj
(56)
i
i
III. Composition Variation in Block Copolymers Anionic living polymers are often used to create block copolymers. Since it affects the microphase diagram,7 a key issue is the compositional variation within a block copolymer. As can be seen in Figure 1, even though the dispersity index is 1.01, the width of the distribution (2σ) is still a significant 20% of the average degree of polymerization: 2σ ) 2/ √nj ) 2√pI - 1
(54)
In general, the fractional composition variation (δf) in a multiblock copolymer containing m blocks is given by δf )
∑ δf ≡ ∑ (f i
i
m
δf 2 )
∑
(fi2 - jf i2) )
i
m
∑ jf
i
- jf i)
(55)
i
where fi is the mole fraction of the ith block of size ni in the copolymer and jfi is its average value:
(pIi - 1)
2 i
(57)
i
where pIi is the dispersity index of block i. Equation 57 is very general and applicable to any block copolymer architecture and statistics. But if we now require that each block satisfy Poisson statistics, then δf 2 )
m
jf i2
i
i
∑ nj
)
1 ) nj
1 m
∑ i
m
m
Squaring eq 55 and then averaging yield
(58)
1 pIi - 1
In the special case where all blocks have the same dispersity index:
√δf 2 ) (√(pI - 1)/m
(59)
Ind. Eng. Chem. Res., Vol. 49, No. 23, 2010
11895
Figure 3. Expected variation in composition of a diblock copolymer where one block has pI ) 1.01 and the second block has the indicated dispersity index. Note that even when the second block has pI ) 1.005 there is still an 11% composition variation. For pI ) 1.005, the mole fraction of the second block is 2/3. Equation 58 was used to make the plot.
Thus, in a 50:50 diblock copolymer where, for example, each block has a dispersity index of 1.005, there is still an expected 10% variation in the overall composition. In Figure 3 the compositional variation in a diblock is illustrated. It should be noted that, if every block satisfies a simple Poisson distribution, then the average composition of the copolymer is well-defined by the dispersity indices:
jf i )
nji ) nj
pIi m
∑ i
1 -1
some time must elapse before the distribution starts to resemble a narrow Poisson type distribution. The gradual sharpening of the MW distribution as the polymerization proceeds has been experimentally reported.9 Memory loss of initiation conditions is characteristic of a Markov chain, which irreversible anionic polymerization closely resembles at sufficiently long times. Finally, the composition variability in block copolymers synthesized from Poisson distributed polyanions can still be significant (10-15%) even when the blocks have dispersity indices less than 1.01.
(60)
1 i pI - 1
IV. Summary Criteria have been developed that predict the final MW distribution from initial conditions. There are only two parameters that are relevant: the ratio of initiation to propagation rate constants (r ≡ ki/kp) and the ratio of initial monomer to initiator concentrations (n0 ≡ [M]0/[I]0). The latter parameter represents the limiting value of average degree of polymerization when all initiator is consumed. Even when r < 1, a narrow, Poissonlike distribution at the end of a polymerization is obtained if rn0 . 1. When the latter condition is not strictly satisfied, eq 51 governs the dispersity index. A necessary, but not sufficient, condition for a narrow MW distribution is that the initiator must be exhausted. As long as the initiator is present and producing monomeric anions, a low MW tail continues to be created that broadens the distribution. Figure 1 illustrates this effect. As a general rule, if the polymerization is terminated before the initiator is completely consumed, a broad MW distribution will be obtain. As r f 0, less and less initiator is consumed, and the dispersity index approaches 4/3, characteristic of a uniform distribution (all chain lengths equally probable). Even after the initiator is consumed,
Literature Cited (1) Flory, P. J. Molecular Size Distribution in Ethylene Oxide Polymers. J. Chem. Phys. 1940, 62, 1561–1564. (2) Szwarc, M. Living Polymers. Nature 1956, 178, 1168–1169. (3) Baskaran, D.; Muller, A. H. E. Anionic vinyl polymerization - 50 years after Michael Szwarc. Prog. Polym. Sci. 2007, 32 (2), 173–219. (4) Gold, L. Statistics of Polymer Molecular Size Distribution for an Invariant Number of Propagating Chains. J. Chem. Phys. 1958, 28, 91–99. (5) Szwarc, M.; Vanbeylen, M.; Vanhoyweghen, D. Simultaneity of Initiation and Propagation in Living Polymer Systems. Macromolecules 1987, 20 (2), 445–448. (6) Yan, D. Y. Graphical method for kinetics of polymerization. 4. Living polymerization initiated by trifunctional initiator with nonequal initiation rate constants. Macromolecules 1998, 31 (3), 563–572. (7) Lynd, N. A.; Meuler, A. J.; Hillmyer, M. A. Polydispersity and block copolymer self-assembly. Prog. Polym. Sci. 2008, 33 (9), 875–893. (8) Falkovitz, M. S.; Segel, L. A. Some Analytic Results Concerning the Accuracy of the Continuous Approximation in a Polymerization Problem. Siam J. Appl. Math. 1982, 42 (3), 542–548. (9) Lee, W.; Lee, H.; Cha, J.; Chang, T.; Hanley, K. J.; Lodge, T. P. Molecular weight distribution of polystyrene made by anionic polymerization. Macromolecules 2000, 33 (14), 5111–5115.
ReceiVed for reView February 16, 2010 ReVised manuscript receiVed June 30, 2010 Accepted September 14, 2010 IE1003637
