narrowing molecular weight distribution R O B E R T F. H O F F M A N STANLEY SCHREIEER GEORGE ROSEN
new semibatch method for the preparation A of free radical initiated polymers by bulk or solution polyMization was developed by the
c
application of d i e material a c e s to basic free radical kin The method consists of the continuous iddition:of initiator or monomer to a batch polymerbat+ at such a. rate, that a constant polymer average molecular weight is maintained throughQut the course of the polymerization. The mathematical equations n e w w r y for deswibing the feed rates are uti* of them equations by hand d c u h t i o n s is' Rot practid, therefore a digid,computer prugram yas developed which readily calcdatei the initiator feed schedule for the polymerization. This article represents the cpupling of ds-tial q e t i q s , .,M 'balances, and chemical kinetics., .% remltant od enables one' to' prepa . p j d c t a b l e moleqdq weigh@$t high coxpersions with && weight. than h&=, &ble by batch methods: ' . , Many po&pera are pnpaced by f& radical
m.
I
.
bulk or solution polymerization in which the polymerization takes place in a reactor with the polymer, monomer, initiator, and solvent present as a homogeneous solution. It is a h quite common to conduct such polymerizations in a batch fashion with a single or multiple addition of a free radiqaJ initiator and/or monomer. Initiators that are commonly used are organic peroxides, hydroperoxides, and azo compounds. In the batch polymerizations, it will be shown later that the initiator is consumed at a faster rate than the monomer or vice versa, pmducing wide ranges of molecular weights of polymer during the polymerization. Polymer chemists have shown (7) that improved polymer physical properties (elongation, tensile strength) are obtaimd with polymers having narrower molecular weight distributions. Thdore, it is desirable to conduct polymerizations in a manner such that the normally wide ranges of molecular weights of polymer are not p r c d d during the polymerization. Experimentally, pdymm witb nar-
a
row molecular weight distribution have been prepared by conducting polymerizations to very low degrees of conversion. However, from a practical standpoint, it is more desilrabe to carry out the polymerizations to 50y0 conversion or greater. This paper will show how polymers may be obtained with narrow distributions by feeding initiator or monomer to a batch polymerization a t rates determined from basic kinetic and material balance equations. Essentially, we have sub,jected the basic, well known polymer kinetic expressions to new boundary conditions and applied differential material balances to a batch reactor system to obtain equations describing the feed rates of monomer or initiator to the batch reactor. Because of complexities of developed equations, solutions to the equations were effected on a digital computer. Although other authors (5) have used the digital computer to analyze polymerization kinetics for batch systems and steady state continuous systems, no one to date has considered the kinetic analysis of the semibatch case. Polymerization Theory
The theory of free radical polymerization has been described thoroughly in many references (2, 4) and will thus be briefly reviewed here. The polymerization process essentially takes place in three distinct steps. I n the first, the initiator is decomposed thermally into free radical fragments. These free radicals combine with a monomer molecule to produce a new monomer free radical. This step is commonly known as the initiation step. I n the second or propagation step, the free radical produced in the first step combines sequentially with more monomer molecules to produce new free radicals of increasing chain length. The final or termination step involves the combination of two of the growing free radical chains to form an unreactive polymer chain. An expression for the rate of disappearance of monomer (rate of appearance of polymer) has been developed ( 4 ) and is shown as follows :
The rate of disappearance or decomposition of the initiator has been found in almost all cases (4) to obey a first-order expression so that the initiator concentration can be expressed as follows : -
dC dt
k&
C = C, exp ( - k d t ) (2b) where : C, = the initiator concentration at time = 0 I n addition, an expression has been developed for the instantaneous number average degree of polymerization and is given as follows: (3)
This average degree of polymerization represents the 52
I N D U S T R I A L A N D E N G I N E E R I N G CHEMISTRY
average number of monomer units in a chain of polymer produced a t time t. At an instant of time the actual polymer chains have a statistical distribution of length. This instantaneous value of P,, however, represents the narrowest distribution of chain lengths. Equations 1 and 3 are based upon the polymerization mechanisms used in most polymer chemistry texts (2, 4 ) . This kinetic model assumes a first-order initiator decomposition and second-order polymer propagation and termination. Further, it is assumed that the propagation rate is independent of the length of the propagation chain and termination occurs primarily by combination of chain radicals rather than by disproportionation, Inherent in this last assumption is the necessary absrncr of significant chain transfer with solvent or polymer. Finally, the model assumes that all of the kinetic constants remain truly constant regardless of the degree of polymerization. This latter condition does not hold for certain high molecular weight bulk polymer systems where the termination constant is sometimes reduced by chain immobility, resulting from viscosity effects at high coRversion (Tromsdorf-Korish effect) ( 4 ) . Despite the seemingly idealistic nature of the assumptions of the kinetic model presented above, there is a surprisingly large number of polymer systems that closely follow the model. Most solution polymerizations conducted to conversions of the order of 50Tc and molecular weights less than 100,000 do not have serious variations of the termination constant. Also, many monomers terminate primarily by chain combination. Any solvent chain transfer can be eliminated by the selection of an inert solvent (aliphatic h>-drocarbons, some aromatics, ketones, ethers, etc.). Therefore, we feel that although the model is somewhat restrictive, as noted, there are many useful polymers that can be prepared within the restrictions. It is seen in Equation 3 that the molecular weight or degree of polymerization is proportional to the ratio of the monomer concentration to the square root of the catalyst concentration. Since, by Equation 1 and 2, the monomer and catalyst are depleted a t different rates, this ratio of monomer to catalyst concentration must vzry during the course of polymerization. This means that there ill be a considerable variation of the deyree of polymerization during the process. In many polymerizations the initiator is actually depleted a t a faster rate than the monomer. I t has also been shown by Tobolsky (8) that a limiting degree of conversion is approached at infinite time conditions. By substituting Equation 2b in Equation 1, integrating, and allowing time to approach infinity, he derived the following expressions for the limiting fraction of conversion for monomer to polymer :
- ln(1 -
x
)
=
2 k fW2
liZ
k ll/Zkdl/Z
(4 )
where :
X,
= fraction converted to polymer a t infinite time. A typical variation of per cent conversion and molecular weight with time in a polymerization where a single
charge of initiator is added is shown in Figures 1 and 2. It is readily seen from these figures that the molecular weight of the polymer produced varies considerably with time. The relative increase of the degree of polymerization is an exponential function as given below :
(5) Equation 5 was obtained from Equation 3 for P , by substituting Equation 2b for C and an integrated form of Equation 1 for M in the expression. I t has been common to conduct polymerizations to higher degrees of conversion and to minimize the degree of polymerization changes by making multiple charges of initiator and/or monomer. This improves the molecular weight distribution but does not achieve the narrowest distribution possible. Furthermore this multiple shot of initiator/monomer is a highly empirical method. In the following derivation we will show that the feed schedule can actually be based on true kinetic considerations to produce more uniform polymers than the empirical approach. Development of the Mathematics for the Continuous Feed Process
As was mentioned previously, the minimum molecular weight distribution is obtained for an instantaneous Pn as described by Equation 3. Therefore it is obvious that in order to maintain a minimum molecular weight distribution in an isothermal polymerization, it is necessary that the degree of polymerization described by Equation 3 be maintained a constant throughout the polymerization. Imposing then the boundary condition of constant P,, Equation 3 may be rearranged to Fzgure 7. Rate of conversion in a aolymerization with one charge of initiator
solve for monomer concentration in terms of initiator concentration as shown below :
By substituting Equation 6 into Equation 1, the monomer concentration can be eliminated and one obtains the following differential equation in terms of initiator concentration : (7) Integrating Equation 7, the following expression relating the initiator concentration in a reactor with time is obtained.
Using Equation 8, an expression may be derived for the monomer conversion us. time.
X
= fraction of monomer polymerized. The above derivations, as will be remembered, are based upon a constant molecular weight or degree of polymerization. Therefore, these equations apply to a reactor into which an initiator or monomer solution is being fed to maintain a constant molecular weight. Determination of Feed Stream
It has been mentioned that in some polymerizations the rate of monomer disappearance is extremely great and monomer should be fed to the reactor. With slow polymerizations, initiator should be added. I t is possible to set up a criterion for determining whether or not monomer or initiator should be fed to the polymerization to maintain the constant degree of polymerization. This Figure 2. Effect of time on degree of polymeriiation (molecular weight) in a polymerization with a single mitiator shot
Y
0 Y Y
er
22 0
TIME, t
TIME,? VOL. 5 6
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53
CONCtwTuTIOW OF IUITIATOR IN FEED TANK = S
-
iOlUME OF MIXTURE V CONCEwlRAIlON OF INITIATOR = C Figwe 3. Flow diogrnm for n proms with continuous initidor feed
TIME, t
may be simply done by noting that Equation 7 represents the rate of disappearance of initiator in the reactor that is necessary to maintain. constant molecular weight. However, Equation 2a describes the actual disappearance of initiator due to thermal decomposition. If thio latter rate of disappearance is greater than the rate shown in Equation 7, it would be necessary to add initiator solution. If we call the ratio of Equation 2 to Eguation 7 as ro at time 0, then when r. is greater than one we feed initiator and if r. is less than one we feed monomer. The expression for r,, is shown below:
It is seen that the feeding of monomer or initiator is dependent upon the kinetic constants, the initiator efficiency, and the quantity of initiator in the initial charge. This means that in some monomer systems it would be necessary to feed either monomer or initiator, depending on the molecular weight that one was shooting for.
Figwe 4. COnrimrour initiata f c e w d f e c d cumc input = SRdt consumption = k&Vdt accumulation = d(CV) = C d V VdC and S R d t - k,jCVdt = C d V + V d C
+
Equation l l b is the basic differential material balance equation which demibes the feed rate to the reactor during the continuous initiator feed process. This equation contains four variables and therefore must be modiied for a solution to be dfected. First of all, the volume V i s a function of the amount of initiator which has been added [+ ( R ) ]and the change in volume caused by density change in the conversion of monomer to polymer [+ (x) I.
D M m M Mokrlal Balances
It is now desired to obtain an expression for the rate of initiator or monomer additions to effect this constant ~ 0 1 e c ~ l aweight r condition. Since most of the present work has been done with the feed of initiator, only the development of initiator feed equations will be shown. However, similar development for a monomer feed system is +ble and the final developed equations for this system will be shown later in the paper. To obtain an expression for the feed of initiator it is necessary to apply a material balance differentially for a reactor into which initiator is being fed. Such a process is shown in Figure, 3. As shown in this figure, the process is given as a reactor whose volume is V, with an initiator solution concentration of C and a feed tank which contains an initiator solution with a concentration S,flowing to the reactor at a rate R; The basic material balance equation for initiator for this system (6) is given below: For the time interval dt: 54
INDUSTRIAL A N D ENGINEERING CHEMISTRY
but from differentiating Equation 9
dX -= dt
K'C,,'" (K'Cih
+ 1)'
(14)
The rate of change of initiator concentration in the reactor (dC/dt) is obtained by the differentiation of Equation l? to obtain an expression in terms of time as the only variable. It is now possible to combine Equations l l b , 12, and 13 and the differentiated form of Equation 8 to
F. Hoffman is Pilot Plant Supervisor with Thiokol Chmical Corp., Trenton, N . J . George Rosen is ThiokoPs Bench Scale Laboratory Sufnvisor. Whm this article was prepared, Stanlcy Scheibn was Pilot Plant Supervisor for Thiokol-tu is now with United Technology Corp.
AUTHOR Robnt
RslC EQUATIONS FOR CONTINUOUS FEED OF INlTlAT~
(k,
-
2 K'C'")
(2
R._I
- +)
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FigUn 5. Cbtnpmm of &a with plcdiction. Tk points show naadrrriw chain dislribulions fw plynu prcparcd by calinuow
obtain a rigorous material balance equation from which the rate of feed can be calculated.
{
R [ S - C] = C Vo[k,(l W;'C'"(l
- X*Ap)]
- XaAp) - aAp
(3-
+ [k, - 2 K r 0 " ] 1 Rdt )
(15)
Equation 15 gives the relationship of rate of feed with time. The initiator concentration, the conversion, and the rate of c o n d o n in chis equation can be exprerwd in terms of time by Equations 8,9, and 14, respectively. Sollllion ofh.d w o n
The solution of Equation 15 and its related equations would be extrrmely long and involved by normal hand calculation. Therefore, it was decided that solution of the equation and determination of feed rates would be made by means of a digital computer. An B M 650 digital wmputer was available for a solution. However, in d e r to solve the equation by the computer, it was mea8ary to rearrange the equations in terms of finite
dilTerenCes. The first such change that was necessary was the a p k n of the integral of ( R dt) in Equation 15 in terms of finite diffaences. This integral was changed to the fdowing summation :
In addition, all of the expressions involving the time t were replaced by the total interval time i&. W g these substitutions, it was +ble to solve direetly for R, (the rate of initiator feed solution at any time t = iAt) in terms of the constants for the system. The equations used for final computer solutions are shown in Table I. In Table I, Equations F through I have been added to represent the initial conditions which are determined from the physical conditions and properties of the polymer, monomer, and solvent, respectively. The rates of feed for the system are computed by sequential solution of the process equations beginning with Equation I. The Equations F through I determine the secondary constants of the problem-namely, V, the initial volume of the polymerization; In the initial charge of initiator required to give the desired molecular weight; and a, the volume fraction of the monomer initially in the system. The secondary constants are then sequentially substituted in Equations C, D, B and finally in Equations la and A to obtain the feed rate for the interval. The computer program was arranged so that we would have a printout of the conversion, rate of feed, and TABLE 111 F'ndktd
The interval At is chosen depending on the accuracy
70 70 55 55
desired.
63
56
I N D U S T R I A L A N D ENGINI3ERING C H E M I S T R Y
P.
Adual
P.
68 70.5 58 55.5 64
cumulative feed added with respect to time. The shorter the time interval chosen for At, the more accurate will be the values of the feed rate R; however, the computer running time will increase proportionally. For most polymerization systems with high degrees of conversion our running time on the computer was 1 to 5 minutes. This usually included 40 to 50 At steps. As mentioned previously, we will not attempt to go through the similar development of the material balance equations for the feed of monomer to a polymerization. The development is quite similar to the initiator feed solution except the material balance is made differentially for the monomer and not for the catalyst, as done before. When this is done, expression for the feed rate of monomer will be obtained as shown briefly in Table 11. Results
Polymerizations have been conducted in our laboratories for a number of different polymer systems with initiator and monomer feed schedules based on the calculation method described above. Since the exact nature of the polymer-initiator systems are within the realm of proprietary information, it is not possible at this time to describe the systems in detail. However, it can be stated that for a certain vinyl-type monomer, polymers were prepared and fractionated to check the validity of the assumption that narrow molecular weight polymers would be produced by our feed schedule. Also polymers of various molecular weights were prepared to determine if the calculation method could predict the desired average molecular weight. Table I11 shows typical results comparing predicted and experimental average degrees of polymerization for polymers prepared with about 50y0 conversion. The experimental degrees of polymerization in the table were calculated based upon end group analyses and ebulliometric measurements on the polymer. Table 111 shows that there was excellent agreement between actual and predicted values. It should be noted that the kinetic constants (k,/kt1l2) used to predict the molecular weights were experimentally determined in the laboratory by running polymerizations with single initiator charges but with various initial initiator concentrations, then measuring the conversion us. time in each of the polymerizations. Such a kinetic scheme for determining these constants is described by Flory (4) on page 122. The initiator decomposition constants were determined by independent measurements of initiator rates. The cumulative molecular weight distributions on a typical polymer (predicted to have a P , of 70) are shown in Figure 4 on normal probability scales. This polymer was made to a 50% conversion. Also presented in Figure 4 is a theoretical cumulative distribution for an instantaneously produced polymer with a P , of 70. This theoretical curve was computed from an equation presented by Flory (4) (page 336, Equation 30) which is based upon the model of termination by coupling and negligible chain transfer. I t is seen that up to about
70 to 80% of the polymer fractions, the theoretical curve closely represents the data points. The deviations at higher molecular weights-Le., experimental polymer has less of the very high pn values-are believed to be due to inability to closely fractionate the polymer in this high range. That is, the experimental fractionation combined fractions in the high molecular weight range. Clearly, though, the feed schedule does produce polymer with very narrow molecular weight distribution which is far less than the distributions for a single initiator charge. Conclusions
A method of calculation of a schedule for addition of monomer or initiator to a batch polymerization has been presented. This method is based upon well founded polymerization reaction kinetics for predicting the feed rates. I t has been shown that the method not only permits one to produce a polymer with a predicted average molecular weight but minimizes the molecular weight distribution of the polymer. The method is applicable to a variety of bulk and solution polymerization systems but is restricted by any limitations of the theoretical polymerization model. NOMENCLATURE C C, C,
= Concentration of initiator, moles/liter =
Initial concentration of initiator, moles/liter
= Concentration of monomer in feed solution for continuous
feed of monomer, moles/liter
F,
= Volume fraction of monomer in polymerization vessel = Feed rate of monomer in continuous monomer feed proc-
ess, liters/hr. Initiator efficiency = Number of At steps = Decomposition constant for initiator, hr.-l kd = Propagation constant of polymerization, liter/mole hr. k, = Termination constant of polymerization, liter/mole hr. kt K‘ = kpkd1/2f1/2/kt1’2, liter1~2/(mole)1~z hr. L = Volume of solvent in initial charge per mole of initiator, liters/mole W,vl = Molecular weight of monomer Wl = Molecular weight of initiator M = Monomer concentration in reactor, mole/liter M , = Initial monomer concentration, mole/liter P , = Degree of polymerization R = Rate of feed of initiator, liters/hr. R, = Initial rate of initiator feed = Initial charge of initiator to polymerization, moles I, S = Concentration of initiator in feed solution, moles/liter t = Time, hours V = Volume of polymerization mixture in reactor, liters = Initial reactor volume V, = Volume of polymerization mixture for the nth At time inV, terval X = Fraction of monomer converted to polymer = Density of initiator, g./liter p1 = Density of polymer, g./liter pp = Density of monomer, g./liter p~
f i
=
REFERENCES (1) Alfrey, T., Jr., “Mechanical Behavior of High Polymers,” pp, 495-8, Interscience, New York, 1948. ( 2 ) Billmeyer, F. W., “Textbook of Polymer Chemistry,” Part 111, pp. 171-207, Interscience, New York, 1957. (3) Doehnert, D. F., Mageli, 0. L., Mod. Plustics 36, No. 6, 142 (1959). (4) Flory, P. J., “Principles of Polymer Chemistry,” Chap. IV, Cornell Univ. Press, Ithaca, N. Y . ,1953. (5) Liu, S., Amundson, N. R., Rubber C h e m . Technof. 34, No. 4 (1961). (6) Micklev H. S. Sherwood T. K. Reed C. E , “Applied Mathematics in Chemical Engin&ring,” 2nd Ed.,’Chap.’3, MCGrawjHill, New York, 1957. (7) Rohm & Haas Co., Philadelphia, Pa. “Catalysts for the Polymerization of Acrylic Monomers,” B d l . No. SP-159 (1659). (8) Tobolsky, A. V., J.A m . Chem. Soc. 80,5927 (1958).
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