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The capacity flow method for vinyl polymerization
The Capacity Flow Method for Vinyl Polymerizathgn A. D. JENKINS The principles o] the capacity flow type of chemical operation are applied to a typical vinyl polymerization system. Equations are deduced for calculating the conversion and the mean degree o] polymerization in terms of the variables including residence time, temperature and reactant concentrations, and it is shown that special cases may arise where the mean degree of polymerization may be almost independent of residence time. A brief discussion of the prestationary period is also given.
THE OPERATIONof chemical processes by the capacity flow technique has a number of advantages with regard to both fundamental studies of reaction mechanism and to commercial practice. As Denbigh 1 has shown, the capacity flow technique enables velocity constants to be obtained from stationary state measurements alone without the necessity of solving differential equations, and on the large scale it may be advantageous economically to operate a reaction continuously in a stirred tank reactor rather than in a tubular reactor or by a batch process. Since 1944 several workers have applied the continuous flow method both theoretically and practically to a variety of reactions but we shall only be concerned here with vinyl polymerization. This subject was treated by Denbigh 2 mainly with respect to molecular weight distributions and he emphasized that a comparatively narrow distribution is obtained by the continuous process because once the stationary state has effectively been reached the monomer concentration remains constant, whereas in the batch or tubular processes the monomer concentration may change by a very large factor if a high conversion is achieved. This principle was later applied in emulsion polymerization by Wall, Delbecq and Florin3. Horikx and Hermans 4 used the continuous flow technique for the polymerization of styrene in toluene initiated by benzoyl peroxide, in order to determine the apparent order of the reaction with respect to monomer concentration. They also investigated the pre-stationary period and found that, under their conditions, the time required for the concentrations of reactants to reach equilibrium values within about 1 per cent was approximately 5 r, where is the average residence time. The present calculations apply the continuous flow principles to vinyl polymerization in solution in order to study the variation of yield and of average degree of polymerization with the variables of the system such as residence time, temperature and initiator concentration. In addition the pre-stationary period is investigated. ASSUMPTIONS
It is assumed that the system consists of a solution containing monomer and initiator flowing at a steady rate into a reaction vessel in which 79
A. D. J E N K I N S
instantaneous perfect mixing is taking place and from which a solution containing polymer, residual monomer and residual initiator is being steadily withdrawn at a rate equal to the rate of inflow. It is further assumed that there is no volume change on reaction, that the temperature in the reactor remains constant and that any reaction occurring in the feed or outflow is negligible. DEFINITIONS
Input
Rate of flow of solution = u 1. min-1 Monomer concentration in feed = m mole 1.- 1 Initiator concentration in feed = [m mole 1.-1 Reactor
Volume of vessel V litres. Concentration of monomer at time t min, M, mole 1.-~; in the steady state, M mole 1.-x. Concentration of initiator at time t min, Ct mole 1.-1; in the steady state, C mole 1.-~. Concentration of radicals at time t rain, Xt mole 1.-~; in the steady state, X mole 1.-1. Concentration of polymer at time t min, Pt base moles 1.-1; in the steady state, P base moles 1.-~. Average residence time, r = V / u min. Output
Concentrations of all reactants are equal to those in the reactor. Rate of flow of solution = u I. min-L POLYMERIZATION
SCHEME
It has been noted that for many vinyl polymerizations the order of reaction with respect to the monomer concentration appears to rise from 1-0 to 1 5 as the monomer concentration tends towards zero. Several mechanisms have been proposed to account for this rather general result and a discussion of this topic has been given previously 5, 6. It now appears that the true explanation of this behaviour lies in the occurrence of termination reactions involving primary radicals 7, s (i.e. those derived directly from the initiator) but for present purposes it is sufficient to write for the rate of initiation I, I = 2 ~kCM x
(1)
where k is the unimolecular rate constant for the decomposition of the initiator, 9 is the efficiency of initiation and 0 ~ x ~ 1, without specifying the precise mechanism underlying the equation. The complete reaction scheme may then be represented as follows. Reaction C + M x > X1
Xr + M Xr+M Xr+T Xr+X.
> > > >
Rate 1 = 2~kCM • k2MX k~MX kjTX k,X 2
Xr+l Pr+X1 P~+X1 P~+, or Pr+P~
In this scheme X = , ~ X'r and T represents a transfer agent. 1
8O
(i) (ii) (iii) (iv) (v)
THE CAPACITY FLOW METHOD FOR VINYL POLYMERIZATION KINETICS
T h e differential equations representing the variations of monomer, initiator, radical and polymer concentrations with time may be obtained at once from the scheme above. The physical model which corresponds to this procedure is that the reaction vessel is filled with the feed mixture which is then instantaneously raised to reaction temperature at the same moment as flow is commenced at the specified velocity. V ~
= mu
-
kzMtXtV
-
Mtu
2q~kCtM~V
-
(2)
l/ dCt = f m u - k C t M ~ V - Ctu
(3)
v d X * = 2 9kCtM~V - k,X~ V - X , u
(4)
vdPt -dt- = k 2 X , M t V - P.,u
(5)
-ffi-
When the steady state has been reached dM dt
--
dC dt
--
dX dt
--
dP dt
= 0
Before proceeding we note that 2 9 k C M x V (the rate of initiation) is negligible compared with k 2 M X V (the rate of propagation) if the average degree of polymerization, P, is high. The mean residence time T is clearly equal to V / u and, therefore, from (3) we have, C=
lm
(6)
m-M k~MT
(7)
krM • + 1
Also, from (2) X= - Substituting (6) and (7) into (4) we have m -M'~ 2 m -M k-~Mr-r / + k2k,M~ ~
29kfmM x k, (krM~+ 1) = 0
(8)
or
2 k~q~klmMx~ 2 rn ~ ( M -z) - 2 m ( M -1) + 1 - k, ( k r M ~ + 1)
(9)
In deriving (9) the additional approximation has been made that This is certainly justified since k ~ / k , for most monomers is of the order of 10-' or less and the working monomer concentration is almost certain to be greater than 0' 1 mole/1. Much controversy has been recorded in the literature over the value of for a number of initiator-monomer systems. Recent determinations ~-11 suggest that where the initiator is azobisisobutyronitrile q~ is considerably less than 1 and probably close to 0"5. Moreover, it now appears H that is independent of monomer or monomer concentration where the latter is greater than 1 mole/l. Since numerical illustrations in this paper are based on such conditions, 9 will be taken to be 0"5. M >~ L_/k4.
8l
A. D. JENKINS The problem in practice is to solve equation (9) for M for any given f, m and r, since at constant temperature x, k=/k4 ~, (~ and k are constant and may be assumed to be known. For a given value of m the problem can be simply solved graphically by plotting the left and right sides of equation (9) separately as functions of M and finding the point of intersection for each fixed pair of values of f and 7. Suppose, for instance, that m = 2 mole/l. Then 4~k~k/M~r 2 4 (M -~ - M -1) + 1 - k, (krMx + 1)
(10)
Figure 1 shows the L.H.S. as a function of M where 0 < M < 2 . If x = 0 , the R.H.S. reduces to a number independent of M, and gives a straight line parallel to the M axis. This special case is dealt with further below and can, of course, be solved as a standard quadratic equation. If, however, x has a positive value, the graphical method must be used and this is illustrated for the extreme value, x = 1. The following values of the other constants are taken: k~/k4~=l 1) mole-½ min-~, k = 1 0 -~ min -1, ~=0"5. The value of k : / k , ~ corresponds to e.g. k~=104 1. mole -1 r a i n - l = 166 1. mole -x sec -x and k4= 108 I. mole -~ min-~= 1"66 x 108 1. mole -1 sec -1. T h e decomposition of azobisisobutyronitrile was studied, inter alia, by Bawn and Mellish 12 and by Tal~t-Erben and Bywater ~a, the results of these workers being in good agreement. The values of the velocity constant for the decomposition of the catalyst given in Table 1 are taken from their data. Table 1 T (°C) 103 k (min -1)
60 0"77
{ I
70 2-4
t
80 9"3
90 29"2
For simplicity of calculation we shall now take k = 10 -a min -1, corresponding to a temperature of 63 °C. Four values of the initiator concentration have been chosen: /'=2"5 x 10 -4, 2"5 x 10 -3, 2.5 x 10 -2 and 2"5 x 10 -1 and for each value o f / , r has been assumed to be 100 min and 1,000 min. The R.H.S. has been calculated for each of the eight cases as well as for the Table 2 x=l
x=O
M
Y
1'85 1 "28
100 1000
2.5x I0-2 2.5X I0-I
M
Y
0"075 0"36
1'89 1'33
0-055 0-335
1 "59 0'80
0'20 0'60
0'77
1'65
0"17 0"61
100 1000
1"16 0-41
0'42 0"79
1 '20 0"32
0'40 0'84
100 1000
0"71 0"19
0"65 0"90
0"64 0"12
0-68 0-94
(min) 100 1000
(mole/L)
2.5×I0 -s
2-5 X I0- 4
82
(mole/L)
THE CAPACITY FLOW METHOD
FOR VINYL POLYMERIZATION
corresponding cases where x = 0 and these graphs are plotted in Figure 1. T h e results are shown in Table 2, where Y = ( m - M ) / r n is the fractional conversion. Approximate solutions for intermediate values of f or T can be estimated by inspection of Figure 1.
23f
~
B/* B3
1
~
=
-
A4 B2
-
Figure/---Graphical solution of e.quation (10) for the conditions given in the text: series A, T= 100 min, series B, ~'= 1,000 rain. 1, f = 2 ' 5 × 1 0 - 4 ; 2, f = 2-5xi0-3; 3, f = 2 " 5 x 1 0 - 2 ; 4, f = 2 . 5 × 1 0 - 1 ; --x=0; -x=I
0
n 0
A2 -2 A1 -3 0
I
I
I
2
[M]
mole I.-1
Figure 2 reveals that Y is very insensitive to the value of x, the maximum difference in Y for x=O and for x = l being about 0"2Y. This occurs at low values of Y which are of little practical interest and for high Y the difference is only about 0'04 Y. Very little error will thus be introduced by assuming x = 0, in which case a more convenient solution may be obtained
1.0 ~..
0
~ i/~i 1
w=IO00 T'=IO0 min
min
2 /-.,+loglOf
3
Figure 2 - - Y i e l d as a function o f initiator concentration. C o n d i t i o n s as for Figure 1 : • x = 1, O x = 0 83
A. D. J E N K I N S
since (9) now reduces to
m~(M-')-2m(M-1)+l
2 k~cpkfrnr~
k,(k~+l)=0
(11)
Solving the quadratic in M -1 and taking the positive root we find
M = m ( l + r2k~e~kJmr2-] ~ ) -1 t_ k4 (kr+ 1)d
(12)
Collecting together the factors which are constant at constant temperature,
g=2k~gk/k, and defining I"/zImr 2 ] * Q= I+ [ ~ ] we have,
Y = 1 - Q-1
(13) (14)
Since fm is the concentration of initiator in the feed, Y is seen to be independent of the absolute monomer concentration.
DEGREE
OF
POLYMERIZATION
(a) In the absence of transler agents In general, if transfer agents are absent, the degree of polymerization P is given by
1
k~ k4X + -- ks nk2M
(15)
where n depends on the mechanism of the termination step, the extreme values being 1, if this occurs exclusively by disproportionation, and 2, if it occurs exclusively by combination. By substitution for X from (7) and subsequently for m/M from (12) and (13) we find 1
ks
k, (Q2 _ Q)
- k~ +
nk~rm
(16)
(b) In the presence o1 a transfer agent The complete expression for t5 in this case is given by
1 k~ "P= ~
k,X
+~--~2M+
k3"T k2M
(17)
k~" being the velocity constant for transfer to T, and it is therefore necessary to evaluate the concentration of the transfer agent in the stationary state. The appropriate materials balance is given by T o u - k J X T V - T u = O , To being the concentration of T in the feed, or T=ro [1 + ks' (m-My]-x k~M d by substitution for X.
1
(18)
From (17) and (18) we finally obtain
k~ k4(Q~-Q) +k~T°QFI+ kJ" 1)] -1 = ~ + nk~m~ rsm L, -~ ( Q 84
(19)
THE CAPACITY FLOW METHOD FOR VINYL POLYMERIZATION To generalize this expression for a number of transfer agents additional terms similar to the third on the right side would obviously be required. Equations (14) and (19) thus enable both the conversion and the degree of polymerization to be simply calculated for any given reaction conditions provided that the usual kinetic constants of the system: k2/k, ~, ks/k=, k3'/k~ etc. are known.
EXAMPLES
(a) Conversion To illustrate the method some results may be quoted which are based on values of the initiator concentration which probably cover th6 practical range. In a typical case if the weight of initiator present is 0"5 per cent of that of the monomer, then for azobisisobutyronitrile and styrene 0.5 104 f = 100 x 164 =3.2x 10-s Y has been calculated as a function of r for values of [ in the range within 1"0 3.2x10 2
3.2x10k
--
0.=
7
!
0
"
~
3 [Oglo t"
Figure 3--Yield as a function of residence ume for
given temperature and initiator concentration
a factor of 10 either side of this value, taking the feed monomer concentration to be 2 mole/1., and the results are depicted in Figure 3 for 60, 70 and 80°C. It is seen that for high residence times the temperature of the reaction vessel is unimportant. This is because if kr >~ 1, Q becomes
r2 k~÷fmr7
1+ L
k,
J
which is independent of k and only dependent on temperature to a small extent. 85
A. D. JENKINS
A selection of the data for the variation of Y with r, [ and temperature is shown in Figure 4. 1-0
~.
~ ~ 8toO°70 °,~ 60° ~ ]
't'= lO00min "g= 316rain
#if\d F I~\ ,1///.,~~]
~ = 100rain
I 1
I 2
I 3
/--,÷log10f Figure 4--Yield as a function of initiator concentration for given temperature and residence time
(b) Degree of polymerization: Transfer reactions neglected Figure 5 shows the variation i n P with r for the same variety of conditions. For high initiator concentrations P tends to decrease with increasing r at all temperatures, but as t is reduced there is a tendency for this trend to be
(o)
(b)
3000~.__~.~.~
2o0oI
+oor,,
O IOQ ,~, o
I
2
,
,
1000[
3
~
o
,oo
BQ~ i
~,
~
;,
3
4
I 0 0 0 ~ (c) 5 0 0 ~ , 0
?50[
I
2 3 z,
(d)
soo~~.
500[
2 5 0 ~ ,
2
0
1
2
3
4
5 0
(o)
0
~
1
2
Figure 5--Degree of polymerization as a function of residence time for given temperature. Abscissa,/~ ordinate, l°gz°r-'2(a)3 × f = 3 " 2 × I 0 - 4 ; (b) f = l ' 0 2 × 1 0 - a ; (c) f = 10-3; (d) f = 1"02 × 10-2; (e) f = 3 " 2 x 10 -2
86
THE CAPACITY FLOW METHOD
FOR VINYL POLYMERIZATION
reversed, at progressively lower reaction temperatures. This gives rise to some intermediate situations where P varies only slightly with r (see, for e x a m p l e , / = 3 . 2 x 10 -3 at 80°C or l = l . 0 2 x 10 -3 at 70°C). Some of these intermediate cases show minimum values of P and the general conditions which must be satisfied for this to occur will be given now. For ff to have a minimum value the expression (Q2_Q)/r must show a maximum. Differentiating this expression and equating to zero we find
1_(rim/_ 1)
(20)
r'ui"= k k k where rmi. is the residence time for a stationary value of ft. Table 3 gives the values of rmi. calculated from equation (20) for each of the cases considered above. Table 3 f
i
i¸
000032 " 0'00102 0'0032 ' 0'0102 0"032 i
60°C
70°C
80°C
"rmin
IOglo rmin
rmin
logt0 rmin
3,060 12,470 42,300 136.400 422,000
3'486 4"096 4"626 5-135 5625
3'33 1,000 4,090 13,800 44,600
0"522 3"000 3"612 4'140 4"649
rmin [
J
] 1Oglo rmin
N o minimum,
81
191 835 2,980
2"281 2"922 3"474
In most cases high values of P" for given f and temperature are obtained at low r and low values at high r, although this state of affairs may be reversed. THE
PRE-STATIONARY
PERIOD
It is of interest to examine the pre-stationary period in order to discover what length of time is required to reach the steady state. The variation in monomer concentration with time can be found by integration of (2), (3) and (4). This may be carried out numerically for any given set of values of the constants but if certain simplifying assumptions are made an approximate calculation may be performed very easily as follows. Since the reactions of the initiator are independent of the monomer if x = 0 , equation (3) may be integrated directly. The result is In I (k + 1/T) C - [m / T7 = _ (k + 1 / T) t
/mk
d
For 70°C, k = 2 " 4 x 10 -3 min -1 and i f / = 1"02 × 10 -3, m = 2 mole/l. Table 4
lO~ [c]
t
(mole/L)
(min) 0 10 31-6 100 316 1000 3160
r = 1O0 min
7 = 1000 min
2"04 2-00 1"92 1 "76 I "66 1 "65 I '65
2'04 2"00 1'90 1"62 1 '09 0"65 0"60
87
(21) Table 4
A. D. JENKINS
shows how C varies during the pre-stationary period for 7=100 and 1,000 min. It is clear that the stationary initiator concentration is virtually reached in a time equal to one or two multiples of the residence time, and that the half-life for approach to equilibrium, expressed as a multiple of r, is inversely related to r. By integration of (4), assuming that C and M are constant and that x=0, it is easily shown that X attains its steady value very quickly. The integral is given by In { [X/(kr + 1)]t + 2 k , X [ [;t/(kr + ~ ) = [X/(kr + 1)]it
(22)
where A=4kk4fm. For the same reaction conditions X attains its steady value within 10 see so that it will always adjust itself very rapidly to changes in C and M. "T.-:
2'0
e
~
~
~':100min
I'0 Ir=1000 min
!
!
1000 2000 Time rain Figure 6--Variation in monomer concentration during the pre-stationary period: ---approximate calculation assuming constant initiator concentration; - - rigorous calculation Under conditions where the initiator concentration varies but little during the pre-stationary period, that is approximately when ~.-1 > k (the residence time is then so ahort compared with the half-life of the initiator that the steady concentration is close to the feed concentration), it is easy to calculate the dependence o f monomer concentration on time by integration of (2) which gives | f k2mX 1 (23) (k2X + 1/r)[ In L M ( k = X - - ~ - T T ~ ) - m / ~ .~ t, where
]
1 _ { kfm "~~ X = k4 t \ k r + 1 /
88
THE CAPACITY FLOW METHOD FOR VINYL POLYMERIZATION Equation (23) may be shown to be equivalent to the more convenient form
(c[. Young and Hammett 1") (1 - Y , )
In I M
mY8 y j . j7 =t/~
(24)
where Ys is the fractional conversion in the steady state which is ultimately attained. For k = 2 " 4 × 10 -3 min -1, f = 1"02 × 10 -3 and m = 2 mole/L, Figure 6 shows M as a function of t for residence times of 100 and 1,000 min. The figure also gives the rigorous numerical integration allowing for variation in C for the same values of the constants. In the first case ( r = 100min) M attains its steady value within 1 per cent in 3 T and in the second case (r = 1,000 min) in 1"3 T. (Where C has been assumed constant its steady state value has been used.)
The author wishes to thank Dr J. Crank and Mrs Doreen D. Whitmore for carrying out the numerical integrations. Courtaulds Ltd, Research Laboratory, Maidenhead, Berks (Received 9th October, 1959)
REFERENCES I DENBIGH, K. G. Trans. Faraday Soc. 1944, 40, 352 z DENBIGH, K. G. Trans. Faraday Soc. 1947, 43, 648 3 WALL, F. T., DELBECQ, C. J. and FLOmN, R. E. J. Polym. Sci. 1952, 9, 177 4 HORIKX, M. M. and HERMANS, I. J. J. Polym. Sci. 1953, 11, 325 -~JENKINS, A. D. J. Polym. Sci. 1958, 29, 245 6 JENKINS, A. D. Trans. Faraday Soc. 1958, 54, 1885, 1895 7 CHAPIRO, A., MAGAT, M., SEBBAN, J. and WAHL, P. International Symposium on Macromolecular Chemistry 1954. Supplement to La Ricerca Scientifica 1955, 73 s BAMFORD, C. H., JENKINS, A. D. and JOHNSTON, R. Trans. Faraday Soc. 1959, 55, 1451 9 BEVINGTON, J. C. Trans. Faraday Soc. 1955, 51, 1392 10 BAMFORD, C. H., JENKINS, A. D. and JOHNSTON, R. J. Polym. Sci. 1958, 29, 355 ~1 BAMFORD, C. H., JENKINS, A. D. and JOHNSTON, R. In course of publication 12 BAWN, C. E. H. and MELLISH, S. F. Trans. Faraday Soc. 1951, 47, 1216 ~a TAL3,T-ERBEN, M. and BYWATER, S. J. Amer. chem. Soc. 1955, 77, 3713 14 YOUNG, H. H. and HAMMETr, L. D. J. Amer. chem. Soc. 1950, 72, 280
89
THE OPERATIONof chemical processes by the capacity flow technique has a number of advantages with regard to both fundamental studies of reaction mechanism and to commercial practice. As Denbigh 1 has shown, the capacity flow technique enables velocity constants to be obtained from stationary state measurements alone without the necessity of solving differential equations, and on the large scale it may be advantageous economically to operate a reaction continuously in a stirred tank reactor rather than in a tubular reactor or by a batch process. Since 1944 several workers have applied the continuous flow method both theoretically and practically to a variety of reactions but we shall only be concerned here with vinyl polymerization. This subject was treated by Denbigh 2 mainly with respect to molecular weight distributions and he emphasized that a comparatively narrow distribution is obtained by the continuous process because once the stationary state has effectively been reached the monomer concentration remains constant, whereas in the batch or tubular processes the monomer concentration may change by a very large factor if a high conversion is achieved. This principle was later applied in emulsion polymerization by Wall, Delbecq and Florin3. Horikx and Hermans 4 used the continuous flow technique for the polymerization of styrene in toluene initiated by benzoyl peroxide, in order to determine the apparent order of the reaction with respect to monomer concentration. They also investigated the pre-stationary period and found that, under their conditions, the time required for the concentrations of reactants to reach equilibrium values within about 1 per cent was approximately 5 r, where is the average residence time. The present calculations apply the continuous flow principles to vinyl polymerization in solution in order to study the variation of yield and of average degree of polymerization with the variables of the system such as residence time, temperature and initiator concentration. In addition the pre-stationary period is investigated. ASSUMPTIONS
It is assumed that the system consists of a solution containing monomer and initiator flowing at a steady rate into a reaction vessel in which 79
A. D. J E N K I N S
instantaneous perfect mixing is taking place and from which a solution containing polymer, residual monomer and residual initiator is being steadily withdrawn at a rate equal to the rate of inflow. It is further assumed that there is no volume change on reaction, that the temperature in the reactor remains constant and that any reaction occurring in the feed or outflow is negligible. DEFINITIONS
Input
Rate of flow of solution = u 1. min-1 Monomer concentration in feed = m mole 1.- 1 Initiator concentration in feed = [m mole 1.-1 Reactor
Volume of vessel V litres. Concentration of monomer at time t min, M, mole 1.-~; in the steady state, M mole 1.-x. Concentration of initiator at time t min, Ct mole 1.-1; in the steady state, C mole 1.-~. Concentration of radicals at time t rain, Xt mole 1.-~; in the steady state, X mole 1.-1. Concentration of polymer at time t min, Pt base moles 1.-1; in the steady state, P base moles 1.-~. Average residence time, r = V / u min. Output
Concentrations of all reactants are equal to those in the reactor. Rate of flow of solution = u I. min-L POLYMERIZATION
SCHEME
It has been noted that for many vinyl polymerizations the order of reaction with respect to the monomer concentration appears to rise from 1-0 to 1 5 as the monomer concentration tends towards zero. Several mechanisms have been proposed to account for this rather general result and a discussion of this topic has been given previously 5, 6. It now appears that the true explanation of this behaviour lies in the occurrence of termination reactions involving primary radicals 7, s (i.e. those derived directly from the initiator) but for present purposes it is sufficient to write for the rate of initiation I, I = 2 ~kCM x
(1)
where k is the unimolecular rate constant for the decomposition of the initiator, 9 is the efficiency of initiation and 0 ~ x ~ 1, without specifying the precise mechanism underlying the equation. The complete reaction scheme may then be represented as follows. Reaction C + M x > X1
Xr + M Xr+M Xr+T Xr+X.
> > > >
Rate 1 = 2~kCM • k2MX k~MX kjTX k,X 2
Xr+l Pr+X1 P~+X1 P~+, or Pr+P~
In this scheme X = , ~ X'r and T represents a transfer agent. 1
8O
(i) (ii) (iii) (iv) (v)
THE CAPACITY FLOW METHOD FOR VINYL POLYMERIZATION KINETICS
T h e differential equations representing the variations of monomer, initiator, radical and polymer concentrations with time may be obtained at once from the scheme above. The physical model which corresponds to this procedure is that the reaction vessel is filled with the feed mixture which is then instantaneously raised to reaction temperature at the same moment as flow is commenced at the specified velocity. V ~
= mu
-
kzMtXtV
-
Mtu
2q~kCtM~V
-
(2)
l/ dCt = f m u - k C t M ~ V - Ctu
(3)
v d X * = 2 9kCtM~V - k,X~ V - X , u
(4)
vdPt -dt- = k 2 X , M t V - P.,u
(5)
-ffi-
When the steady state has been reached dM dt
--
dC dt
--
dX dt
--
dP dt
= 0
Before proceeding we note that 2 9 k C M x V (the rate of initiation) is negligible compared with k 2 M X V (the rate of propagation) if the average degree of polymerization, P, is high. The mean residence time T is clearly equal to V / u and, therefore, from (3) we have, C=
lm
(6)
m-M k~MT
(7)
krM • + 1
Also, from (2) X= - Substituting (6) and (7) into (4) we have m -M'~ 2 m -M k-~Mr-r / + k2k,M~ ~
29kfmM x k, (krM~+ 1) = 0
(8)
or
2 k~q~klmMx~ 2 rn ~ ( M -z) - 2 m ( M -1) + 1 - k, ( k r M ~ + 1)
(9)
In deriving (9) the additional approximation has been made that This is certainly justified since k ~ / k , for most monomers is of the order of 10-' or less and the working monomer concentration is almost certain to be greater than 0' 1 mole/1. Much controversy has been recorded in the literature over the value of for a number of initiator-monomer systems. Recent determinations ~-11 suggest that where the initiator is azobisisobutyronitrile q~ is considerably less than 1 and probably close to 0"5. Moreover, it now appears H that is independent of monomer or monomer concentration where the latter is greater than 1 mole/l. Since numerical illustrations in this paper are based on such conditions, 9 will be taken to be 0"5. M >~ L_/k4.
8l
A. D. JENKINS The problem in practice is to solve equation (9) for M for any given f, m and r, since at constant temperature x, k=/k4 ~, (~ and k are constant and may be assumed to be known. For a given value of m the problem can be simply solved graphically by plotting the left and right sides of equation (9) separately as functions of M and finding the point of intersection for each fixed pair of values of f and 7. Suppose, for instance, that m = 2 mole/l. Then 4~k~k/M~r 2 4 (M -~ - M -1) + 1 - k, (krMx + 1)
(10)
Figure 1 shows the L.H.S. as a function of M where 0 < M < 2 . If x = 0 , the R.H.S. reduces to a number independent of M, and gives a straight line parallel to the M axis. This special case is dealt with further below and can, of course, be solved as a standard quadratic equation. If, however, x has a positive value, the graphical method must be used and this is illustrated for the extreme value, x = 1. The following values of the other constants are taken: k~/k4~=l 1) mole-½ min-~, k = 1 0 -~ min -1, ~=0"5. The value of k : / k , ~ corresponds to e.g. k~=104 1. mole -1 r a i n - l = 166 1. mole -x sec -x and k4= 108 I. mole -~ min-~= 1"66 x 108 1. mole -1 sec -1. T h e decomposition of azobisisobutyronitrile was studied, inter alia, by Bawn and Mellish 12 and by Tal~t-Erben and Bywater ~a, the results of these workers being in good agreement. The values of the velocity constant for the decomposition of the catalyst given in Table 1 are taken from their data. Table 1 T (°C) 103 k (min -1)
60 0"77
{ I
70 2-4
t
80 9"3
90 29"2
For simplicity of calculation we shall now take k = 10 -a min -1, corresponding to a temperature of 63 °C. Four values of the initiator concentration have been chosen: /'=2"5 x 10 -4, 2"5 x 10 -3, 2.5 x 10 -2 and 2"5 x 10 -1 and for each value o f / , r has been assumed to be 100 min and 1,000 min. The R.H.S. has been calculated for each of the eight cases as well as for the Table 2 x=l
x=O
M
Y
1'85 1 "28
100 1000
2.5x I0-2 2.5X I0-I
M
Y
0"075 0"36
1'89 1'33
0-055 0-335
1 "59 0'80
0'20 0'60
0'77
1'65
0"17 0"61
100 1000
1"16 0-41
0'42 0"79
1 '20 0"32
0'40 0'84
100 1000
0"71 0"19
0"65 0"90
0"64 0"12
0-68 0-94
(min) 100 1000
(mole/L)
2.5×I0 -s
2-5 X I0- 4
82
(mole/L)
THE CAPACITY FLOW METHOD
FOR VINYL POLYMERIZATION
corresponding cases where x = 0 and these graphs are plotted in Figure 1. T h e results are shown in Table 2, where Y = ( m - M ) / r n is the fractional conversion. Approximate solutions for intermediate values of f or T can be estimated by inspection of Figure 1.
23f
~
B/* B3
1
~
=
-
A4 B2
-
Figure/---Graphical solution of e.quation (10) for the conditions given in the text: series A, T= 100 min, series B, ~'= 1,000 rain. 1, f = 2 ' 5 × 1 0 - 4 ; 2, f = 2-5xi0-3; 3, f = 2 " 5 x 1 0 - 2 ; 4, f = 2 . 5 × 1 0 - 1 ; --x=0; -x=I
0
n 0
A2 -2 A1 -3 0
I
I
I
2
[M]
mole I.-1
Figure 2 reveals that Y is very insensitive to the value of x, the maximum difference in Y for x=O and for x = l being about 0"2Y. This occurs at low values of Y which are of little practical interest and for high Y the difference is only about 0'04 Y. Very little error will thus be introduced by assuming x = 0, in which case a more convenient solution may be obtained
1.0 ~..
0
~ i/~i 1
w=IO00 T'=IO0 min
min
2 /-.,+loglOf
3
Figure 2 - - Y i e l d as a function o f initiator concentration. C o n d i t i o n s as for Figure 1 : • x = 1, O x = 0 83
A. D. J E N K I N S
since (9) now reduces to
m~(M-')-2m(M-1)+l
2 k~cpkfrnr~
k,(k~+l)=0
(11)
Solving the quadratic in M -1 and taking the positive root we find
M = m ( l + r2k~e~kJmr2-] ~ ) -1 t_ k4 (kr+ 1)d
(12)
Collecting together the factors which are constant at constant temperature,
g=2k~gk/k, and defining I"/zImr 2 ] * Q= I+ [ ~ ] we have,
Y = 1 - Q-1
(13) (14)
Since fm is the concentration of initiator in the feed, Y is seen to be independent of the absolute monomer concentration.
DEGREE
OF
POLYMERIZATION
(a) In the absence of transler agents In general, if transfer agents are absent, the degree of polymerization P is given by
1
k~ k4X + -- ks nk2M
(15)
where n depends on the mechanism of the termination step, the extreme values being 1, if this occurs exclusively by disproportionation, and 2, if it occurs exclusively by combination. By substitution for X from (7) and subsequently for m/M from (12) and (13) we find 1
ks
k, (Q2 _ Q)
- k~ +
nk~rm
(16)
(b) In the presence o1 a transfer agent The complete expression for t5 in this case is given by
1 k~ "P= ~
k,X
+~--~2M+
k3"T k2M
(17)
k~" being the velocity constant for transfer to T, and it is therefore necessary to evaluate the concentration of the transfer agent in the stationary state. The appropriate materials balance is given by T o u - k J X T V - T u = O , To being the concentration of T in the feed, or T=ro [1 + ks' (m-My]-x k~M d by substitution for X.
1
(18)
From (17) and (18) we finally obtain
k~ k4(Q~-Q) +k~T°QFI+ kJ" 1)] -1 = ~ + nk~m~ rsm L, -~ ( Q 84
(19)
THE CAPACITY FLOW METHOD FOR VINYL POLYMERIZATION To generalize this expression for a number of transfer agents additional terms similar to the third on the right side would obviously be required. Equations (14) and (19) thus enable both the conversion and the degree of polymerization to be simply calculated for any given reaction conditions provided that the usual kinetic constants of the system: k2/k, ~, ks/k=, k3'/k~ etc. are known.
EXAMPLES
(a) Conversion To illustrate the method some results may be quoted which are based on values of the initiator concentration which probably cover th6 practical range. In a typical case if the weight of initiator present is 0"5 per cent of that of the monomer, then for azobisisobutyronitrile and styrene 0.5 104 f = 100 x 164 =3.2x 10-s Y has been calculated as a function of r for values of [ in the range within 1"0 3.2x10 2
3.2x10k
--
0.=
7
!
0
"
~
3 [Oglo t"
Figure 3--Yield as a function of residence ume for
given temperature and initiator concentration
a factor of 10 either side of this value, taking the feed monomer concentration to be 2 mole/1., and the results are depicted in Figure 3 for 60, 70 and 80°C. It is seen that for high residence times the temperature of the reaction vessel is unimportant. This is because if kr >~ 1, Q becomes
r2 k~÷fmr7
1+ L
k,
J
which is independent of k and only dependent on temperature to a small extent. 85
A. D. JENKINS
A selection of the data for the variation of Y with r, [ and temperature is shown in Figure 4. 1-0
~.
~ ~ 8toO°70 °,~ 60° ~ ]
't'= lO00min "g= 316rain
#if\d F I~\ ,1///.,~~]
~ = 100rain
I 1
I 2
I 3
/--,÷log10f Figure 4--Yield as a function of initiator concentration for given temperature and residence time
(b) Degree of polymerization: Transfer reactions neglected Figure 5 shows the variation i n P with r for the same variety of conditions. For high initiator concentrations P tends to decrease with increasing r at all temperatures, but as t is reduced there is a tendency for this trend to be
(o)
(b)
3000~.__~.~.~
2o0oI
+oor,,
O IOQ ,~, o
I
2
,
,
1000[
3
~
o
,oo
BQ~ i
~,
~
;,
3
4
I 0 0 0 ~ (c) 5 0 0 ~ , 0
?50[
I
2 3 z,
(d)
soo~~.
500[
2 5 0 ~ ,
2
0
1
2
3
4
5 0
(o)
0
~
1
2
Figure 5--Degree of polymerization as a function of residence time for given temperature. Abscissa,/~ ordinate, l°gz°r-'2(a)3 × f = 3 " 2 × I 0 - 4 ; (b) f = l ' 0 2 × 1 0 - a ; (c) f = 10-3; (d) f = 1"02 × 10-2; (e) f = 3 " 2 x 10 -2
86
THE CAPACITY FLOW METHOD
FOR VINYL POLYMERIZATION
reversed, at progressively lower reaction temperatures. This gives rise to some intermediate situations where P varies only slightly with r (see, for e x a m p l e , / = 3 . 2 x 10 -3 at 80°C or l = l . 0 2 x 10 -3 at 70°C). Some of these intermediate cases show minimum values of P and the general conditions which must be satisfied for this to occur will be given now. For ff to have a minimum value the expression (Q2_Q)/r must show a maximum. Differentiating this expression and equating to zero we find
1_(rim/_ 1)
(20)
r'ui"= k k k where rmi. is the residence time for a stationary value of ft. Table 3 gives the values of rmi. calculated from equation (20) for each of the cases considered above. Table 3 f
i
i¸
000032 " 0'00102 0'0032 ' 0'0102 0"032 i
60°C
70°C
80°C
"rmin
IOglo rmin
rmin
logt0 rmin
3,060 12,470 42,300 136.400 422,000
3'486 4"096 4"626 5-135 5625
3'33 1,000 4,090 13,800 44,600
0"522 3"000 3"612 4'140 4"649
rmin [
J
] 1Oglo rmin
N o minimum,
81
191 835 2,980
2"281 2"922 3"474
In most cases high values of P" for given f and temperature are obtained at low r and low values at high r, although this state of affairs may be reversed. THE
PRE-STATIONARY
PERIOD
It is of interest to examine the pre-stationary period in order to discover what length of time is required to reach the steady state. The variation in monomer concentration with time can be found by integration of (2), (3) and (4). This may be carried out numerically for any given set of values of the constants but if certain simplifying assumptions are made an approximate calculation may be performed very easily as follows. Since the reactions of the initiator are independent of the monomer if x = 0 , equation (3) may be integrated directly. The result is In I (k + 1/T) C - [m / T7 = _ (k + 1 / T) t
/mk
d
For 70°C, k = 2 " 4 x 10 -3 min -1 and i f / = 1"02 × 10 -3, m = 2 mole/l. Table 4
lO~ [c]
t
(mole/L)
(min) 0 10 31-6 100 316 1000 3160
r = 1O0 min
7 = 1000 min
2"04 2-00 1"92 1 "76 I "66 1 "65 I '65
2'04 2"00 1'90 1"62 1 '09 0"65 0"60
87
(21) Table 4
A. D. JENKINS
shows how C varies during the pre-stationary period for 7=100 and 1,000 min. It is clear that the stationary initiator concentration is virtually reached in a time equal to one or two multiples of the residence time, and that the half-life for approach to equilibrium, expressed as a multiple of r, is inversely related to r. By integration of (4), assuming that C and M are constant and that x=0, it is easily shown that X attains its steady value very quickly. The integral is given by In { [X/(kr + 1)]t + 2 k , X [ [;t/(kr + ~ ) = [X/(kr + 1)]it
(22)
where A=4kk4fm. For the same reaction conditions X attains its steady value within 10 see so that it will always adjust itself very rapidly to changes in C and M. "T.-:
2'0
e
~
~
~':100min
I'0 Ir=1000 min
!
!
1000 2000 Time rain Figure 6--Variation in monomer concentration during the pre-stationary period: ---approximate calculation assuming constant initiator concentration; - - rigorous calculation Under conditions where the initiator concentration varies but little during the pre-stationary period, that is approximately when ~.-1 > k (the residence time is then so ahort compared with the half-life of the initiator that the steady concentration is close to the feed concentration), it is easy to calculate the dependence o f monomer concentration on time by integration of (2) which gives | f k2mX 1 (23) (k2X + 1/r)[ In L M ( k = X - - ~ - T T ~ ) - m / ~ .~ t, where
]
1 _ { kfm "~~ X = k4 t \ k r + 1 /
88
THE CAPACITY FLOW METHOD FOR VINYL POLYMERIZATION Equation (23) may be shown to be equivalent to the more convenient form
(c[. Young and Hammett 1") (1 - Y , )
In I M
mY8 y j . j7 =t/~
(24)
where Ys is the fractional conversion in the steady state which is ultimately attained. For k = 2 " 4 × 10 -3 min -1, f = 1"02 × 10 -3 and m = 2 mole/L, Figure 6 shows M as a function of t for residence times of 100 and 1,000 min. The figure also gives the rigorous numerical integration allowing for variation in C for the same values of the constants. In the first case ( r = 100min) M attains its steady value within 1 per cent in 3 T and in the second case (r = 1,000 min) in 1"3 T. (Where C has been assumed constant its steady state value has been used.)
The author wishes to thank Dr J. Crank and Mrs Doreen D. Whitmore for carrying out the numerical integrations. Courtaulds Ltd, Research Laboratory, Maidenhead, Berks (Received 9th October, 1959)
REFERENCES I DENBIGH, K. G. Trans. Faraday Soc. 1944, 40, 352 z DENBIGH, K. G. Trans. Faraday Soc. 1947, 43, 648 3 WALL, F. T., DELBECQ, C. J. and FLOmN, R. E. J. Polym. Sci. 1952, 9, 177 4 HORIKX, M. M. and HERMANS, I. J. J. Polym. Sci. 1953, 11, 325 -~JENKINS, A. D. J. Polym. Sci. 1958, 29, 245 6 JENKINS, A. D. Trans. Faraday Soc. 1958, 54, 1885, 1895 7 CHAPIRO, A., MAGAT, M., SEBBAN, J. and WAHL, P. International Symposium on Macromolecular Chemistry 1954. Supplement to La Ricerca Scientifica 1955, 73 s BAMFORD, C. H., JENKINS, A. D. and JOHNSTON, R. Trans. Faraday Soc. 1959, 55, 1451 9 BEVINGTON, J. C. Trans. Faraday Soc. 1955, 51, 1392 10 BAMFORD, C. H., JENKINS, A. D. and JOHNSTON, R. J. Polym. Sci. 1958, 29, 355 ~1 BAMFORD, C. H., JENKINS, A. D. and JOHNSTON, R. In course of publication 12 BAWN, C. E. H. and MELLISH, S. F. Trans. Faraday Soc. 1951, 47, 1216 ~a TAL3,T-ERBEN, M. and BYWATER, S. J. Amer. chem. Soc. 1955, 77, 3713 14 YOUNG, H. H. and HAMMETr, L. D. J. Amer. chem. Soc. 1950, 72, 280
89