162 RAY
Macromolecules
Molecular Weight Distributions in Copolymer Systems. I. Living Copolymers' W. H. Ray Department of Chemical Engineering, State Unioersity of New York at Buffalo, Buffulo, New York 14214. Receioed September 14, 1970
ABSTRACT: An analytical solution to the kinetic equations for living copolymers is presented which allows the easy calculation of the molecular weight distributions (MWD). An example is worked to illustrate the value of the solution.
T
he kinetics of copolymerization has been discussed extensively in the literature?, and several workers4~j have analyzed the molecular weight distribution (MWD) derived from some simple cases. Even though recent contributions6-8 allow easy calculation of the asymptotic behavior for living copolymerizations, there seems to be a lack of both a general formulation of the problem and a calculation procedure which produces the polymer MWD in an efficient fashion. This lack is possibly due to the formidable nature of the equations as well as the great difficulty in experimentally producing copolymer MWD's. In any case, there seems to be a need for a general approach and computational algorithm for producing copolymer MWD's, both because there has been some recent progress in solving the experimental p r ~ b l e m ,and ~ because, by simulation of proposed mechanisms, one can see the effect of parameters on the MWD. This simulation would seem to offer a way to easily test kinetic hypotheses. In this paper we will analyze the copolymer MWD for systems having only initiation and propogation steps (living copolymers). A later paper will treat the case where transfer and termination steps occur. The equations occurring in the steady state in continuous, well-mixed reactors are all algebraic and straightforward to solve; thus we will be concerned here with the more complicated analysis of batch or plug flow tubular reactors. Let us assume that the initiation step is very much faster than the propogation step so that it is a good approximation t o assume that all initiation takes place instantaneously a t time zero. Goldlo has shown that this is often a good assumption for living homopolymerizations. In this case our mechanism is
+ MI 2 P n . m + M? 2Qn,m+1 Pn,m
where P,,, is a polymer (concentration = Pn,,,J with n units of M1 and m units of M P and terminal M1, while Qn,mis a polymer (concentration = Qn,m)with n units of MI and m units of MPand terminal M2. In a batch or plug flow tubular reactor our kinetic equations are d~Pn.m kllMl(Pn-l,m - Prim) dt
+
k21MlQn-1 ,m, - k12Md'n
Qn,m(O) = ~ n , m O j l Q o
where 1 ifn
6a',S
=
= r,m = s
(4)
0 otherwise
By defining the dimensionless quantities
a = -Qo
x =
Po
1
PIP2
and assuming that y, p l , p 2 are maintained invariant with time (e.g., by adjusting M l , M P , or temperature"), we get the dimensionless equations dynm -
dr (1) Part of this work \vas done at the University of Waterloo, Waterloo, Ontario, Canada. (2) (a) H. W. Melville, B. Noble, and W. F. Watson, J . Polym. Sci., 2, 229 (1947); (b) F. R . Mayo and C. Walling, Chem. Reo., 46, 191 ( 1950). (3) G . E. Ham, "Copolymcrization," Wiley, New York, N. Y . , 1964. (4) R. Simha and H. Branson, J . Chem. P h y s . , 12,253 (1944). (5) W. H . Stockmayer, ibid., 13, 199 (1945). (6) F. Horn and J . Klcin, Ber. Bw:se,iges. Phj.s. Chetn., in press. (7) F. Horn and J. Klein, ibid., i n press. (8) F. Horn and J. Klein, ibid., in press. (9) H. Iiiagaki, H . Matsuda, and F. Kamiyama, Macromolecules, 1, 520 (1968). (10) Louis Gold, J . Cheni. Phj.s., 28,91 (1958).
(2)
Pn,m(O) = 6n.m'~oPo
Pn+l,m
(1)
,m
plYn-l,m
+ yZn-l,m - ( p i +
~ i i , m ( O )=
___
dz7L.m
dr
-
YP2Zn.m-1
+
(6)
~ n , v ~ l ~ o
~n,m-l-
zn,m(O) =
l)~n,Tn
+
Y(PZ
1)Znm
(7)
a6n,m0"
Solution to the Kinetic Equations. Through the use of Laplace transforms and generating functions (cf: Appendix), one can get a solution to these equationsin the form (11) W. H . Ray and C. E. Gall, Mucrontolecules, 2,425 (1969).
LIVINGCOPOLYMERS 163
Vol. 4, No. 2 , March-April I971
.......
rn= I rn.12 -.- m.20 LP1
--- rn= 30
I 0-4
n,m 2 1
r
=600
IO-^
n,m 2 1
10-6
(9)
-
E
where
c A
IO-^ and
Io-(
: :
: I : I
I\
/
\
*
\
\
i‘ li
\,
I I
I : 1 :
;1.
I i
Io-‘(
is a confluent hypergeometric function known as a Kummer function.12 The homopolymer is given by the Poisson distribution
*
‘\
1:
where F(a,b,c,x)is the Gauss hypergeometric function. l 2 The function
I
\
I :
I
I
I
3
Figure 1. Distribution of growing chains with terminal styrene, r =
600.
..... rn= I
-
rn: -.--- mm.
These solutions are readily calculated on the computer and provide a means of determining explicitly the influence of the parameters a, y, p l , pn, and T on the shape of the MWD. To illustrate this, a simple example is worked. Example. The system styrene-butadiene has the following anionic rate constants at 29” in benzene with Li+ counterion: l 3 kll = 0.06, k12 = 1.2, kll = 0.006, kZ2= 0.018. It perhaps is a suitable system for caliculation of the copolymer MWD. Equations 8, 9, 13, and 14 were calculated for a grid of chain lengths n and m for various times in the life of the copolymer. The dimensionless concentrations of both types of growing chains are shown in Figures 1-6 under the assumption that equal numbers of the two types of growing chains were initiated at t = 0 ( L e . , a = 1) and the monomer concentrations are maintained equal (Le., M l / M 2= 1). From the figures it can be seen that the very sharp peak in the MWD surface moves in the direction of increasing n and rn with time, and more and more of the growing chains have butadiene as the terminal group (due to the very small value of y).
z
12 20
30
r.600
Ec M
Summary
An analytical expression for the MWD of living copolymers in batch or plug flow tubular reactors has been presented. The result is straightforward to use (contact the author for (12) A. Erdelyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, “Higher Transcendental Functions,” McGraw-Hill, New York, N. Y., 1955. (13) M . Szwarc and J. Smid, Progr. React. Kine&.,2, 218 (1964).
,
1 ; o-81 :
I
IO
I;!, 20
I
I
I
30
40
50
n
60
__t
Figure 2. Distribution of growing chains with terminal butadiene, r
=
600.
Macromolecules
164 RAY
IO+
IO+
IO-'
E , c %
IO-^
10-'O
lo-"
Figure 3. Distribution of growing chains with terminal styrene, = 1800.
7
Figure 5 . Distribution of growing chains with terminal styrene. = 3600.
r
......... -m.49
......... -
rn= 19
m.31
-.e
m.73 rn= I03
lo-*
IO-^
I 0-3
Io
IO-^
-~
E
E , c
I
c KI
r+l
IO-^
IO+
Io-6
IO-^
IO-^ IO+
-
I O-e
n
n
Figure 4. Distribution of growing chains with terminal butadiene, 7 =
1800.
0 ____t
Figure 6 . Distribution of growing chains with terminal butadiene, = 3600.
r
LIVINGCOPOLYMERS165
Vol. 4, No. 2, March-April 1971 information about the computer program) and allows very detailed information about the MWD surface of the copolymer. The requirement that the parameters pl, pz, and y be invariant with time means that the results are best applicable to systems where M l / M 2is maintained nearly constant either through addition of monomer or by operation at nearly azeotropic conditions. If one wished to compute total chain length distributions or composition distributions from these detailed surfaces, it would be very straightforward to compute a small grid of yF,,, z,,,, (enough to define the surface) and use interpolation and summation to produce the other distributions of interest. Our experience with this approach has been very favorable. Acknowledgments. The author is indebted to Mr. B. S. Jung for carrying out thl? numerical calculations and to the National Research Council of Canada for research support. Very helpful discussions with Dr. F. Horn and Dr. J. Klein are gratefully acknowledged.
Now making use of the binomial expansion we get D-1
5 ,f(D)(b2ru2)p-nuln[bll + cu21n (A-11) n
=
p=On=O
Then letting I results
p - n and using another binomial expansion
=
Finally, letting m
=
1
+ k gives
By collecting coeffjcients of ulnuzrnand noting that c/bl1bx2= x-1
Appendix
We shall here derive the solution to eq 6 and 7. Let us first take the Laplace transform to yield ~
xn,rn(s> = A x n , m ( O )
+ Blxn-l,m(s) +
BZn,m-l
(-4-1)
where
E(m
+n - k - 2)(nk
b >O
1)
n-1
(bii)"-1(b22)rn-1(X- l ) k
(A-14)
L B2 =
[by] b y ]
=
o
o
[
s-t
0
0
1
YP2
Y(P2
(A-15)
J
By making use of the identities
1
+ 1 ) s + Y(P2 + 1 )
('4-2)
and the overbar denotes a transformed variable. Now let us define the generating function
F(-a,-b,l,x)
E k20
("
+
; ">(:)(. -
-
=
1)k
where F is the Gauss hypergeometric function,12 and defining f(a,b,x) E F( - U , - b,l,x)
(A-17)
we get
f(n
where &n,m,s)
=
(+ s
P1 Pl
1
- 1,m - 1,x)
+ 1)"( + Y(P2 + 1) yp2
)rn
(A-19)
(A-20)
Now inverting the Laplace transform and making use of the identity m f(n,m- 1,x) = --f(n - l,m,x) n
+
( 1 - x ) f ( n - 1,m - 1,x) (A-21) we get the solution given in eq 8-10.
