April, 1943
715
ON THE KINETICS OF POLYMERIZATION REACTIONS
out. The dependence of the course of the reaction, of the final average chain length and of the chain length distribution upon the rate constants, in particular upon the rate of initiation and upon the initial concentration, is shown. Two limiting cases can be distinguished. If the ratio between the rate of cessation and that of initiation is large, and the ratio of the rate of propagation and that of cessation is also large, then the average molecu-
[CONTRIBUTION FROM THE
lar weight will be large at the end and remain approximately constant during the last stages. If these ratios are small, then the average molecular weight will increase continuously and reach a small final value. It is shown how the individual rates may be derived from a knowledge of these quantities. The theory of Schulz, and Norrish's and Brookman's results, appear as special cases. WASHINGTON, D. C.
DEPARTMENT OF CHEMISTRY,
RECEIVED OCTOBER3. 1942
POLYTECHNIC INSTITUTE OF BROOKLYN]
On the Kinetics of Polymerization Reactions. 11. Second and Combined First and Second Order Initiation Reactions. Mutual Stabilization of Growing Chains' B Y ROBERT GINELLAND ROBERT SIMHA~
In a previous publication3 polymerization reactions have been considered in which growth and termination are brought about by the interaction of stable monomer molecules with the growing polymer. The initiation of active chains was represented by a first order reaction. For processes which occur. with the aid of a true catalyst or in dilute solution, a first order initiation represents a possible mechanism. Second order activation, for instance, between monomers may be found in uncatalyzed chain polymerizations. In pure phase as well as in solution, first and second order processes may appear simultaneously, the first probably becoming more predominant as the concentration of monomer decreases. In this paper relations will be sought which allow a differentiation on the basis of experimental data between the two cases mentioned above. As pointed out in (I), toward the end of the reaction i t is probable that growth and terrnination will proceed by mutual interaction of the growing chains as well as by interaction with monomer. The complexity of the rate equations does not permit a rigorous treatment of such cases. An exact solution can be given, if the monomer-polymer interaction alone contributes to the growth and cessation. Approximate expressions will be developed which allow an estimation of the polymer-polymer interaction. (1) Presented at the One-Hundred-Fourth meeting of the American Chemical Society held in Buffalo, New York. September 7th to 11th. 1942. (2) Present address, Department of Chemistry, Howard University, Washington, D. C. (3) R. Ginell and R . Simha, THIS JOURNAL, 66, 706 (1943), in the following designated by (I).
In regard to the general approach and the notation used, (I) may be consulted. Second Order Initiation of Nuclei.-The calculations will be based on the following mechanism for the three elementary processes kiz + NI + nl + N I kza nj + N1 +nt+l kaz nj + N1 +Nj+1
Activation: N I
Growth: Cessation:
j = 1,2 ...
j = 1,2 ...
The possibility that the initiation of a chain leads directly to the formation of a dimer would cause only minor changes in our final expressions. The same holds true for cessation without inclusion of monomer. Chain transfer can be treated by the same method, but has been omitted here. The rate equations assume now the following form m
m
with the additional equations
d
m
Ni = -kleNP
- kz&
m
ni 1
which follow directly from (1). As in (I) we
ROBERTGINELL AND ROBERT SIMHA
716
Vol 65
divide through by iVl and introduce the variable
linearity of the above differential equation, the solution will consist of three terms. Each of these will satisfy the equation resulting from the In 6 our equations are linear. The boundary introduction of the respective Pi-term for nl in conditions are: for t = O , $ = 0, nj = 0, f o r j 2 1; the above differential equation. It may be verified by insertion, that the P3-term leads to the folN j = 0 , j 2 2 , N1 = N l ( ' ) . Fort = m, 4 = 'VI = 0. The solution of system (2) and (1) lowing term in nj will evidently consist of exponential functions. kZ2j-1 -4 j - 1 P 3 e - ( k n + k a ) + The secular determinant of system (2) has the solution The contribution of the PI-term on the other hand can be shown by induction to be XiXz =
kiz(kzz
+ 2kaz);
Xi,
XP
> k l z . The solution of (2) has then the form m
Enl =
hlugleW
+ lszulzeXld
1 J-1
= IL~~.&+
f hzaszeh
It, = Nl(Q)klzk2d-le-fknr+kn)b
.
(k3Z
(Xi
+
-
(kzz
+ kaz +
X2)
(kzz
+ kan1 +
{
ex16 - e - ( k n + k a ) d
c
s=o
[(RIP
+ s!ksz + Xi)+]*
A n analogous term results from the Pt-term. In this way we find the final expression for nj by remembering the definition of the Pi and transforming the sums
$
X:)1 > E j - ,
the hi and aiRare constants, to be determined from (2) and the boundary conditions. In this manner we obtain, if the relations between XI and As are taken into account
Xdj-1
j-'
__._ .
Xl)
Pi
kpzj-l
[(kzs
+ kaz+
Xi)418-
'1
S! 1
(Sa) gives the size distribution of the polymer. A comparison with the analogous formula (4a)
of (I) shows that (5a) contains two terms of the same type as (4a), however with more complex
and therefrom with the aid of (1)
I
We can now find the nj f o r j 2 2. They obey the equation dn,/d$ = k,tn,-l
- (kw +
k3diz1
together with the above expression for nl. In order to solve this system of equations, we Start with the equation for n2. Because of the
expressions in the rate constants. The third term is characteristic for the second order initiation process. With increasing 4 the distribution approaches more and more the one found in (I). For the second sum becomes smaller than the first, because X T < A,, a n d this holds d l the mort
April, 1913
O N THE KINETICSOF
71 7
POLYMERIZATION REACTIONS
j 2 2) = for the third term. The parameters of the zd+; I nl 1---- N Ni(0) distribution curve, however, remain different from those determining its shape in equation bar kir [ ( h a Xl)ek* - (ksr XI)eb*] -$) I XI- XZ (4a) X1 Xs . , of (I). The distribution of the stable kzz 2ksa (7b) Nj is best found for the first phases of the reaction by numerical integration. Before dis- The equation for the weight average molecular cussing the conditions prevailing during the last phases we must consider the range of variation of weight requires a knowledge of the sum 2 j 2 the variable 6. Its connection with the time t is (nj Nj). The final expressions are rather complicated. It may suffice, therefore, to indiestablished by means of the relation cate in the appendix the method of derivation. Before examining the relationships obtained, we will derive the analogous expressions for the case of complex roots XI and A2 of the secular from which 6 can be expressed in terms of t by in- determinant. This situation will be encountered version. As shown in the appendix, this integral if we start to increase k12 at constant ktz and ksz. can be evaluated by series expansion The solution will then be of a different type, but certain essential features will remain unchanged as shown in the following. We write [I - e - M e - r ( h - ~ c ) + ] (6a)
+
+
+
+
W
+
= X
X1,z
* ip
+
It is not possible, in general, to solve this equak12(ks2 2ksz) - (kis ks2)' tion for 4 analytically. In an evaluation of experi- A = - ( k l z 2 ksz); p = 4 mental data, the value of 4 corresponding to a (4') Instead of ( 5 ) we obtain now given value of t can be found from a plot of the weight of the polymerized ma- N 1 = N 1 ( 0 ) p4 + k l z ) sin p g ] e ~ P terial versus the time elapsed, = - Nl(0) sin p (6- - 4 )ex*; p+- = tan-' in the manner described in (I). sin p + kiz It is easy however to invert Ni (0) (sa) for sufficiently large Valnj = -k12 sin p& ex4 P ues of 4. First it may be seen j a l Ni(O)kls nl = from (5) or (6) that 4w = [ ~ Z Z cos PQ (5') kzdksz kao) kizksr Evidently for our Purpose we ksz(k2~ 832) X(kz2 2k.d sin p + k22e-(Ra+krr14 need consider the first term P only in the sum because E N i = - -Nl(O)kaz X(kt2 k d sin Nl(O)eX* [ K l t ( k 2 ~ 2 k d (XI-XXZ) > 0. The one in the . kes aka2 ksz 2ks2 I.( bracket can be neglected. (kzz ksz)COS P+ Theref ore Explicit expressions for the nj may be omitted. Ni(')(ksz + Xi)t(+) FS - (" - ") e-Xp$ (&) A1 They can be found by collecting the real and as may be also found directly from ( 5 ) by omitting imaginary terms in (5a). The end-point of the the second exponential in the integration. reaction corresponds to a finite value for &,, The next step is the computation of the number found by setting the two terms in Nl equal to and weight average chain length.6 The defini- each other.' The general integral (6) leads again tion of Z,z6 leads, with the aid of equations (5) to to complicated expressions. It is possible however to expand sink (&, +) and retain the f i s t member only, if k s and therefore p are small enough. (6) then reduces to an exponential integral. The first approximation valid for the (4) The alternative of making N i = 0 by setting the two expreslast phases of the reaction, when A(&, - +) > k32, we find an over-all second order rate with a constant (k12k22)(k82)-1. In the previous publication we found under similar approximations the same expression for a first order over-all rate. If, however, equations (5’) and (6b’) ought to be consulted, the above procedure does not yield a simple order for the rate of decrease of N1. I t appears therefore that the over-all reaction is approximately of the first or second order in its later stages, as defined by the nature of the approximation, according to whether the rate of initiation of active nuclei follows a first or second order process. The over-all rate is in both cases given by the product of the velocities of initiation and growth, divided by that of the breaking of chains, if kzz >> k82. However, this conclusion can be reached only if the rate of initiation is negligibly small in comparison with that of cessation. Because of the conditions imposed
+
by equation (4), this latter restriction is much more stringent in the case treated a t present. Naturally this result depends also upon the order of the second and third step of the chain process. Figures 1 and 2 show for one particular example the progress of the polymerization as given by equations (5') and (7b'), respectively. The molecular weight used was that of styrene and the initial concentration was assumed to be 8.74 mole per liter. As in the previous paper an induction period occurs. It depends here in a more complicated manner upon all three rates.
20 30 40 Time in hours. Fig. 1.-Percentage polymerization versus time: kie = 10-8 mole-' liter see.-'; ksz = 10+ mole-' liter sec.-l; k32 = mole-1 liter sec.-I.
The reason, however, for its appearance in our theory is the same, namely, the nature of the boundary condition assumed a t t = 0 for the active nuclei. Figure 3 shows the dependence of the final average chain length on k 1 2 a t constant kn and h32. These curves have the same general shape as those for the corresponding plot for kn in (I) and hence the same general conclusions hold. However, it should be noticed that a t any given being the same) the value of k1 (the K z 2 and value of 2, is lower in the case of second order initiation than in the case of first order initiation as previously discussed.
10
I
i~ 1
I
I
20 30 40 Time in hours. Fig. 2.--Change of number average chain length with time: klt = 10-8 mole-' liter see.-'; kPZ = mole-' liter sec -l; k,: = 117- mole-' liter ~ l r c--I
10
io-% 10-710-6 ki2
-
10-6
number average chain length versus rate = 10-2 mole-' liter sec.-l; kas = mole-' liter see.-'; B, kse = mole-' liter sec.-l, kar = 10-4 mole-' liter see.-'. Fig. 3.-Final
of initiation: A,
k22
Figure 4 gives the dependence of the final number average chain length on k 3 2 a t constant k 1 2 and k 2 2 . Here the curves are very similar to those in Fig. 6 in (I) and the same general conclusions may be drawn. It should be noted, however, that the scale in Fig. 6 in (I) is different from that in Fig. 4. This choice of scale was the result of the choice in values of kle. They have been chosen smaller throughout than those in Fig. 6 of (I) since this brings about a case of more marked dependence of 2, on kaz. Finally the size distribution of the growing polymer as given by equation (5a) or a corresponding expression found therefrom for complex XI and X 2 , may be briefly discussed. As in paper (I) we propose merely to show that Schulz'sY expression for the number of stable chains o k 8)
(>
V Schulz, Z physrk Ciivnz B30
(7'1 ' 1 9 j i )
ON THE KINETICS OF POLYMERIZATION REACT~ONS
April, 1943
0.012 -
)
I
'
72 1
1
I
I
I
24000.
lo-' 10-610-6 krs
-
-
lo-'
Fig. 4.-Final number average chain length versus rate of termination: A, k1a = 10-8 mole-' liter set.-'; kIp = lo-* mole-' liter sec.-l; B, R l t = lo-' mole-l liter sec.-l; ktt = 10-2 mole-' liter sec.-l.
polymerization degree 'y) is the limiting value for kn + 0. Only in this case do Schulz's elementary statistical considerations and our theory lead to the same result. We can for this purpose restrict ourselves to the case treated in (5a). It has already been pointed out after the derivation of this equation that it assumes the same functional type as (4a) in (I), if the value of 4 is chosen large enough to permit an omission of the second and third term in the brace of (sa). As shown previously by Dostalg we can approximate the sum by a step function equal to exp. { (k22 K32 AI)+], if j is smaller than the exponent and equal to zero if the reverse inequality holds. In between, there will be a transition region, the contribution of which can however be neglected, if C#J approaches 4 - = m . Accordingly (sa) reduces to
+ +
-I
0.006
I
I
4
8 PdJ
.
12
16
20
Fig. 5.-Total number of polymer molecules us. synthetic time: k l t = 10-8 mole-' liter sec.-l; kzs = mole-' liter sec.-l; ka, = 10-6 mole-' liter set.-'.
With decreasing values of the rate of initiation, furthermore, the final average chain length and the final distribution of polymer sizes is reached sooner in the course of the reaction. Schulz's result, however, as already pointed out by himself, is obtained only under special conditions. It may be seen by integration of the above limiting form for njw1,that for larger values of kI2, the expression for Nj as function of the degree of polymerization "j" will be
This expression can now be integrated from zero to infinity because of the nature of the step function, and gives in this fashion the limiting value where the function f depends upon all three for Nj according to (5a). Then we neglect the velocity constants. For the mechanism disterms of higher than the first order in klz in cussed in the preceding publication, we obtain equation ( 4 ) for XI and Az. In the result so for larger values of k11 a formula of a similar type derived for N j we set throughout k a = 0 and obtain finally, if ktz >> ks2 lim Nj
kn+O
a
IJ
Nl(O) (1
kzz/(ks
-
+ ksd
(9) H. Dostal, Monalsh., 67, 1 (1935).
as can be seen from (4a) in (I). The approximations involved in the derivation have been pointed out in the discussion of Schulz's distribution in (I),
ROBERT GINELLAND ROBERTSIMHA
722
VOl. 65
, first order initiation of nuclei should be admitted. The further alternative of activation by
(kzn
,
lim i+
z,
+
Fs
m
k32
kzz 2kaa kn(k22 ksz) + n'1(0)(k3Z hi)
Ni =
- kzz
1
m
nj
-
kizN1
- kli
1
J
with the solution
gni = 1
(NlCo)+ 2) A1
(ah@
- eW) 10
Because for C#I = 0, n~ = 0 ; c N j = N1('), it 1
follows from (la) and (5b) that nl will turn out as in (5) except for the replacement of the factor
.
The same will be true for m
the rest of the nj,j 2 2 .
N j will also contain 1
this factor and furthermore an additive constant -k l l / k l z . XI and XZ retain their original meaning. As is to be expected, the end-point of the reaction, Nl (+") = 0, corresponds now to a finite value of C#I' The ratio of kn to k12 would probably be of the order of magnitude of one or less. As an estimation of all quantities in (5b) shows, will have a very large value. It can (10) See page 718
7- e?)Xz]
(eh;
kia
'
(8)
This latter equation holds if the second exponential can be neglected throughout. Because of the smallness of klz and the usually large value of (pm this is permissible. This equation may be compared with equation (6c') in (I) describing a first order initiation, if k11
