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Polymerization in Two-Phase Systems The Case of Several Growing Radicals STANLEY KATZ and REUEL SHINNAR Department of Chemical Engineering, The City College, C.U.N.Y., New York, Ν. Y. 10031 GERALD M. SAIDEL Department of Biomedical Engineering, Case Western Reserve University, Cleveland, Ohio 44106
A method for obtaining the molecular weight distribution in two-phase polymerization
systems such as emulsion poly
merization is presented. The polymerization
process occur
ring in the dispersed phase is modeled as a Markov process with random arrival and termination tinuous polymer
growth.
Earlier
of radicals
and con
studies in which a dis-
persed-phase particle is assumed to contain (1) zero or one radical or (2) a large number of radicals
are encompassed
as
variation
limiting
cases. The most significant
weight-to-number
average
Xw/Xn
in the
is found to occur when
the mean number of radicals < n > lies between 1/2 and 1. For given rates of arrival and growth, distributions
with
< n > near 1/2 have about the same number average, but they differ markedly in variance.
T n emulsion polymerization and in some suspension polymerizations, free radicals are generated i n a continuous phase and diffuse into a dispersed-phase particle or droplet where polymerization takes place ( 5 ) . The molecular weight distributions or, equivalently, the polymer size distributions of these systems depend on the relative rates of radical arrival and termination. Frequently i n emulsion polymerization the radi cals are terminated so quickly that each particle i n the dispersed phase A
145 Platzer; Addition and Condensation Polymerization Processes Advances in Chemistry; American Chemical Society: Washington, DC, 1969.
146
ADDITION
AND CONDENSATION
POLYMERIZATION
PROCESSES
contains no more than one radical at any time. The opposite situation occurs i n suspension polymerization where the termination rates are so low with respect to the arrival rates that the dispersed phase is considered to contain a virtual continuum of growing radicals. Under each limiting condition a theoretical analysis of the polymer size distribution can be greatly simplified. Experimental studies show, however, that these limiting approxima tions must be used with caution. For example, with some emulsion polymerization systems the mean number of radicals per particle may run from one-half to several depending on the size of the particle ( I ) . Assuming that the polymerization process is stationary with known rates of radical arrival and termination, Stockmayer (6) and O'Toole (3) have shown how to calculate not only the mean number of radicals but the entire number distribution as well. U n t i l now, no methods of the same generality seem to exist for calculating the polymer size distribution. A method is presented here which yields the polymer size distribu tion for arbitrary rates of radical arrival and termination. Furthermore, from this analysis one can see when each of the limiting cases is applicable. The computations are all carried out under stationary conditions with the rates of radical arrival, propagation, and termination constant. Under transient conditions the computations would be much more difficult. F o r the limiting cases, however, the moments of the polymer size distributions under transient conditions can be found (4). Two-phase polymerization is modeled here as a Markov process with random arrival of radicals, continuous polymer (radical) growth, and random termination of radicals by pair-wise combination. The basic equations give the joint probability density of the number and size of the growing polymers in a particle (or droplet). From these equations, suitably averaged, one can obtain the mean polymer size distribution. Polymerization
as a Markov
Process
W e can develop a stochastic model for two-phase polymerization by following the changes i n the number and size of the growing polymers (or radicals) with time in an arbitrary particle of a system. [For a more general discussion of probability methods in particulate systems see Ref. 7.] Let us say that at some time t the particle contains m radicals of sizes Χι, x2, . . . xm ( i n order of their appearance) with probability density pm (xu x2, . . . Xm; t). Since a polymer chain is usually long, we take the chain length or polymer size to be a continuous variable. Now, we assume that in a short time interval [ i , t - f τ] changes in the particle occur by these processes :
Platzer; Addition and Condensation Polymerization Processes Advances in Chemistry; American Chemical Society: Washington, DC, 1969.
8.
ΚΑτζ ET AL.
Polymerization
in Two-Phase
Systems
147
( 1 ) A radical may enter the particle with probability Br Β
m; xl9 x2, . . . xm —* m + 1; xl9 x2, . . . xm,
xm+i
(2) Each radical i n the particle grows or increases its size, deterministically, by G T G
m; xu x2, . . . xm - » m; x x + G , x 2 + GT, . . . xm +
GT
T
(3) A n y two radicals—i.e., growing polymers—may terminate by combination with probability Dr D m;
Χι, x2,
. . . xm
~~* m — 2; χί9
x2,
. . . χ^-ι>
Xi+i, · · · %j-i>
*j + i> · · ·
m
x
Macroscopically, we interpret B, G , and D as the mean rates of radical arrival (birth), propagation (growth), and termination (death) per radi cal pair; for analytical convenience, we assume that B, G, and D are constant. In taking the growth and termination rates G and D to have the same values for radicals of all sizes, we are assuming that no radicals are trapped, all being equally accessible both to monomer and to each other. These rates would otherwise depend on the ages and therefore the sizes of the radicals involved. The assumption that all radicals are equally accessible is quite reasonable i n suspension polymerization and other two-phase polymerizations in immiscible liquids (e.g., see Ref. 2) but might not always apply to cases in which the dispersed particles grow, such as i n precipitation and emulsion polymerization. The size depend ence of G and D can be incorporated into our stochastic model but only at the expense of very considerable analytical and numerical difficulties. Accordingly, we limit ourselves here to the case of constant parameters generally treated i n the literature. Returning then to the random mechanism sketched above, we find that the (conditional) probability density that the "state" of a particle changes from m radicals of sizes xu x2, . . . xm to η radicals of sizes t/i, t/2> . . . y η in short time interval [t, t + τ] is given by: Pmn(xl9
· · · m> ?/l> · · · Vny τ) x
= Βτδ η ,„ ι + ι δ(ΐ/ι -χλ) +
. 2
... S(ym - xm) 8 ( y m + i ) s(t/i - * ι ) . . . B(yt.i
• · ·δ(^.2 +
- Xi-i)
8(yt -
Hyj-1 ~ *J + l) · · · S(t/ m _ 2 -
Chemical Library
< +
i)
X)
[1 - Br - ( m / 2 ) D T ] δ η η ι 8 ( y i - * i - GT) ...S(ym-xm-GT) +o(T) a
American
*
Society
16th St., N.W. Processes Platzer; Addition and1155 Condensation Polymerization Washington, 20036 Advances in Chemistry; American ChemicalDC Society: Washington, DC, 1969.
m
(1)
148
ADDITION A N D C O N D E N S A T I O N
POLYMERIZATION
PROCESSES
where is the Kronecker delta, δ(χ) is the delta function, and the symbol ο (τ) is an order of magnitude smaller than τ . The first term of this expression is the probability density that the m + 1 radical, of vanishingly small size, enters the particle and the other m radicals change negligibly. In the second term, we take into account the probability that any pair of radicals may combine. If neither of the former events occurs, we must consider the continuous growth of radicals as given i n the third term. To find the (absolute) probability that the particle has η radicals of sizes t/i, y2, . . . yn at time t -+- τ , we use the following relation based on the Markov property: p
n
(yu . . . y„; * + τ ) = 2
Γ Pm»(*i> · · · m , y i , · · · y ; τ) x
n
mJ
Pm(*i · · · * W t) dxl9 . . . dxm
(2)
By substituting Equation 1 into Equation 2, we find Pn
(?/!··.
y; n
t +
= Βτρη.1
)
T
(?/i,. . . t / n - i ; t) (y ) - G r 2 ^ (Vu · · · 0
j
+ D 2
f pn+2
ι
r> 5 )
(5)
d r d s
In this study we consider just the stationary process—i.e., we set the time derivatives equal to zero. Size
Distribution
of Dead
Polymer
From the pair-wise combinations of growing polymer, we can get the mean size density f(z) of dead polymers of size [ζ, ζ + dz~\ :
f(
)
^
( " )
f
P
' ' '
n
IT (^)
f
V
n
)
8
+
^n
y
"-y^
~ ~
2
d
y
i
z
)
d
y
" 'd
i
'
"
d
y
n
^
y n
This mean polymer size density is just what would be obtained from the average over all particles i n the system. F o r convenience i n computing f(z) we introduce the following probability functions: on
= Jvn
Φη (yi) =
Pn (*) = Jpn
j
(viy
Pn
· . -y ) n
dyi- · · dyn
(?)
(Vi, .· - yn) dy2. . . dyn
(f/i, · · . * / « )
δ(ί/ι + y2 - z) dyt. .. dyn
(8) (9)
where 6n is the probability that a particle has η growing polymers, and Φη (yi) dy1 is the joint probability of choosing a growing polymer of size lyu Vi + dyxl i n presence of η of them. I n terms of the above functions, the polymer size distribution is
f «
=τ(4)'-•*[-(#)"(£)·] for which 2 G / Β \1/2
Platzer; Addition and Condensation Polymerization Processes Advances in Chemistry; American Chemical Society: Washington, DC, 1969.
8.
ΚΑτζ ET AL.
Polymerization
in Two-Phase
153
Systems
Again we write the cumulative distribution of the tail on a normalized scale as
Γ
F (t) =
Jv(t)
f (z) dz = (2t + 1) e-2t
where 2G /
Discussion
of
Β \
1
/
2
Results
From a numerical solution of Equations 19 and 20 we obtain the polymer size distribution for different values of the ratio D/B of termina tion to arrival rates. Figure 1 shows two distributions that essentially encompass the range of variation as plots of the fraction F (t) of polymer having size greater than X t, where X n is the number average. The values of D/B for which the limiting cases provide suitable approximations can be seen in Figure 2, where the number average is in the form (B/2 G) X „ = < n>, the mean number of growing polymers, together with the corre sponding weight average ( B / 2 G ) X and their ratio X /X are plotted n
w
ol 0
w
ι 4
ι .8
1
Figure
1.
Limiting
polymer
F(t) Xn Β D
Polymer fraction with size greater than X n t Number average Arrival rate Termination rate per radical pair
= t= == =
1
n
1.2 1.6 NORMALIZED POLYMER SIZE , t size
1
2.0
1 2.4
distributions
Platzer; Addition and Condensation Polymerization Processes Advances in Chemistry; American Chemical Society: Washington, DC, 1969.
154
ADDITION
A N D CONDENSATION
POLYMERIZATION
PROCESSES
as functions of y/D/B. We see here that in order to make the leading moments Xn and Xw agree well with the limiting case D/B Î oo, we must still take D/B of the order of 100, while to get good agreement in X w and Xw at the other end of the range, we need only go down to D/B of the order of 0.1.
zo
Figure Xn Xw Β G
= = = :=
2.
Moments
of the size
distribution
Number average Weight average Arrival rate Growth rate
D c= Termination rate per radical pair
t= —
Xn =
Mean number of growing radicals
From Figure 2 we may also make an illuminating comparison be tween the behavior of Xw/Xn and that of < η >. For values of < η > larger than 5 or so, the value of Xw/Xn is very close to the limiting value, 1.5, for infinite < η >. Even for < η > = 2, the difference is small. As far as the leading moments go, we may confidently treat by continuum
Platzer; Addition and Condensation Polymerization Processes Advances in Chemistry; American Chemical Society: Washington, DC, 1969.
8.
KATZ ET AL.
Polymerization
in Two-Phase
155
Systems
methods the two-phase heterogeneous polymerizations for which < η > is larger than 2. The most significant change in Xw/Xn occurs when < η > lies be tween 1/2 and 1. As a consequence, relatively small deviations from < η > = 1/2 cause the variance of the molecular weight distribution to be substantially different from that of an exponential distribution. Accord ing to Gerens ( J ) , emulsion polymerization w i l l often be in this range of < η >, and the results presented here might give one a qualitative insight into the effect of such deviations on the molecular weight distri bution. Unfortunately, data on molecular weight distributions or even the moments of the distribution with respect to D/B do not appear in the literature. It should, however, be pointed out that i n the case of a non-ideal emulsion polymerization, D and therefore < η > change slowly with time owing to the growth of the particle. Under these conditions our calculations do not give the final molecular weight distribution of the product but only the distribution for the incremental amount of dead polymer formed during a short time interval in which D can be con sidered constant. A more complete dynamic analysis of emulsion poly merization is given by Saidel and Katz (4) for the limiting cases.
Appendix Limiting
Polymer
Size
Distribution
Following Ref. 4, the polymer size distribution i n the Smith-Ewart limit D/B Î oo can be developed from the probability equations: dp (t) = [f dt 0
Pi (*>t)dx-p
B
^r
dJ
1
+
G
(t)
0
"N^ = » 1
B[p
*> "
(i)
8 (
Ρ ι
*'
(
i)]
where p (t) is the probability that a given particle contains no radical at time t, and pi (x, t) dx is the probability that the particle contains one radical of size [χ, χ - f dx]. Here again Β and G are the rates of radical arrival and growth. Under stationary conditions, where the time de pendence disappears, we find 0
Po =
J
Pi
M dx=
γ
Platzer; Addition and Condensation Polymerization Processes Advances in Chemistry; American Chemical Society: Washington, DC, 1969.
156
ADDITION AND CONDENSATION POLYMERIZATION PROCESSES
and dpi(x)
Β
Γ 1
, ν
Λ
G [ - 2
S
(
x
, ν Ί -
)
P
i
{
Χ
)
\
Since this equation is equivalent to
(*)
dpi
dx
=
Β , G Pi (*) -7Γ
-
ν
. X
Λ
> °
with U=o
Pi
2G
:
we see at once that
P
^
X
=
)
Y
G
e
x
p
[
-
-
^
]
X
X
>
°
Since the situation is stationary, we may simply identify the size distri bution of dead polymer with that of growing polymer:
f (x)dx
L
G
Pl
G
J
To find the polymer size distribution in the opposite limiting case, where D/B I 0, we start with phenomenological equation
+ G *Lgl = Β δ (,) - Dr (Χ, t) fr (y, t) dy where r (x, t) dx is not a probability but rather the number of growing radicals of size [χ, χ + dx] at time t. Under stationary conditions, where the time dependence disappears, this becomes an ordinary differential equation i n r (x) which can again readily be solved. Letting
I ^
n=
r
dy
the total number of growing radicals present, we find dr (χ) - B T =
Β
,
Λ
G
^
fiD
x
X
)
-
-
G
. R
(
x
X
)
This has the solution
from which, by integration / Β \
1 / 2
Platzer; Addition and Condensation Polymerization Processes Advances in Chemistry; American Chemical Society: Washington, DC, 1969.
8.
KATZ ET AL.
Polymerization
in Two-Phase
Systems
157
so that r(«)=*-.«p Conditions being stationary, we may calculate the size distribution of dead polymer by catching two growing radicals at the moment they terminate each other and accounting, with suitable weighing, for the sum of their sizes. W e find / / r(x) r(y) δ ( * + y - ζ) dxdy
= ——rr
f { z )
J J
J*r(x)
r(x) r(y)
dxdy
r(z — x) dx
( J * r(x)d y x
I
B
Y
= { G )
Γ
D
Ύ [--α{-Β) ζβχρ
Β
I
D \
Ί
1 / 2
\
Z
Acknowledgment
The work reported here was supported in part under N S F Grant N o . GK-943. Literature
(1) (2) (3) (4) (5) (6) (7)
Cited
Gerens, Η., Z. Electrochem. 60, 400 (1956). Goldstein, R. P., Amundson, N. R., Chem. Eng. Sci. 20, 195 (1965). O'Toole, J. T., J. Appl. Polymer Sci. 9, 129 (1965). Saidel, G. M., Katz, S., J. Polymer Sci., in press. Schildknecht, C. E., "Polymer Processes," Interscience, New York, 1956. Stockmayer, W. H., J. Polymer Sci. 24, 314 (1957). Katz, S., Shinnar, R., Ind. Eng. Chem., in press.
RECEIVED April 1,
1968.
Platzer; Addition and Condensation Polymerization Processes Advances in Chemistry; American Chemical Society: Washington, DC, 1969.
