Modeling of Polymer Molecular Weight Distributions in Free-Radical

Eric-Hans P. Wolff*, and A. N. René Bos. Shell Research and Technology Centre, Amsterdam, SIOP-ORTTG/506, P.O. Box 38000, 1030 BN Amsterdam, The ...
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Ind. Eng. Chem. Res. 1997, 36, 1163-1170

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Modeling of Polymer Molecular Weight Distributions in Free-Radical Polymerization Reactions. Application to the Case of Polystyrene Eric-Hans P. Wolff* and A. N. Rene´ Bos† Shell Research and Technology Centre, Amsterdam, SIOP-ORTTG/506, P.O. Box 38000, 1030 BN Amsterdam, The Netherlands

A computational model for free-radical polymerization reactions in an ideally macromixed batch stirred tank reactor has been developed. The polymerization process may take place using an initiator, a chain-transfer agent, and a prescribed temperature program. The model yields, as a function of time, the monomer conversion, the number- and weight-average molecular weights, and a complete molecular weight distribution (MWD) of both the growing and dead polymer chains. A kinetic scheme typical for free-radical polymerization and the method to generate a complete MWD have been adopted from the literature. We included a fundamental approach to account for the “viscosity effects” such as the cage effect, gel effect (Trommsdorff-NorrishSmith effect), and glass effect as proposed by Achilias and Kiparissides. An expression to account for the chain transfer to a solvent or to a chain-transfer agent has been added. The model has been successfully verified with two literature cases and demonstrated with two of our own defined cases. Introduction The modeling of polymerization reactions is often focused on the description of the system evolution in terms of monomer conversion, number- and weightaverage molecular weights, and bulk density. However, a close control of the characteristics of the polymer product, such as the melting point, glass transition temperature, strength, and flow characteristics, is also determined by the molecular weight distribution, MWD (Clay and Gilbert, 1995). This is, for example, the case for polystyrene. Besides the chemical reaction kinetics, mass-transfer limitations of various species are very important in freeradical polymerization (Achilias and Kiparissides, 1992). At high monomer conversions, above roughly 30-50% for the bulk polymerization of styrene (Robertson, 1956; Nishimura, 1966; Horie et al., 1968; Soh and Sundberg, 1982b; Weickert and Thiele, 1983; O’Driscoll and Huang, 1989), most elementary reactions can become diffusion-controlled. Reactions that are influenced by diffusion phenomena include chemical initiation reactions, propagation of growing chains, and termination of these “live” macroradicals. Diffusion-controlled initiation, termination, and propagation reactions have been related to the phenomena of the cage effect, gel effect (or usually called the Trommsdorff-NorrishSmith effect), and glass effect, respectively. These effects arise in this sequence at increasing monomer conversion. Chemical initiation involves the decomposition of initiator molecules to form very active primary radicals capable of initiating new polymer chains. However, due to the very close proximity of the generated radicals, not all of them can eventually escape from their “cage” to react with monomer molecules. Some primary radicals will either self-terminate or react with other nearest-neighboring molecules before diffusing out of the cage. To account for this, an empirical initiator efficiency factor, f, which represents the fraction of all * To whom correspondence should be addressed. † Present address: Shell Chemicalls Europe, SNC, P.O. Box 3005, 3190 GB, Hoogvliet Rt., The Netherlands. S0888-5885(96)00446-0 CCC: $14.00

generated initiator primary radicals leading to the formation of new polymer chains, has been introduced. This parameter may have values between zero and unity. The gel effect has been attributed to the decrease of the termination rate constants caused by a decrease of the mobility of growing polymer chains (Achilias and Kiparissides, 1992; Mita and Horie, 1987). This phenomenon leads to a higher molecular weight and a broader MWD that strongly affects the final polymer properties. For the glass effect occurring at high monomer conversions, roughly above 70-90% for the bulk polymerization of styrene, even diffusion of monomers in the now extremely viscous medium is impeded. The glass effect can be related to the decrease of the propagation rate caused by the decrease of the mobility of the monomer molecules. It appears in polymerization reactions taking place at temperatures below the glass transition temperature of the polymer. For a typical grade of purified polystyrene with a molecular weight larger than roughly 100.000 g mol-1, this is approximately 100 °C (Bueche, 1962). A consequence of this phenomenon is the freezing of the reaction mixture at conversions below 100%, for styrene around 95%. At the limiting conversion, the glass transition temperature of the polymer and monomer mixture becomes equal to the polymerization temperature. We will refer to the combination of cage, gel, and glass effects as to the “viscosity effects”, although this term does not properly reflect the fundamental phenomena. In this work, a computational model for simulating free-radical polymerization reactions in an ideally macromixed batch stirred tank reactor has been developed. We applied the model to the polymerization of styrene, but with modifications, it can also be applied to other free-radical polymerization processes. We assumed one initiator to be initially present, but the model can easily be extended with more initiators fed to the system at any time. Potential applications of the model are in the field of process optimization and product development. During process optimization, the model may help to study the potential reduction of batch time, e.g., the prediction of monomer conversion and the optimal © 1997 American Chemical Society

1164 Ind. Eng. Chem. Res., Vol. 36, No. 4, 1997 Table 1. Kinetic Scheme (Chaimberg and Cohen, 1990) reaction I f 2R• R• + M f M1• M1• + M f M2• M2• + M f M3• Mn• + M f Mn+1•

reaction rate const Initiationa kd ) kd,0 exp(-Ed/RT) ki ) ki,0 exp(-Ei/RT) Propagation (for n ) 1 to N)b kp ) kp,0 exp(-Ep/RT) kp ) kp,0 exp(-Ep/RT) kp ) kp,0 exp(-Ep/RT)

unit s-1 L mol-1 s-1 L mol-1 s-1 L mol-1 s-1 L mol-1 s-1

Chain Transfer to Monomer (for n ) 1 to N + 1) Mn• + M f Mn + M1• ktr,m ) ktr,m,0 exp(-Etr,m/RT) L mol-1 s-1 Chain Transfer to Solvent or Chain-Transfer Agent (for n ) 1 to N + 1) Mn• + X f Mn + M1• ktr,x ) ktr,x,0 exp(-Etr,x/RT) L mol-1 s-1 Termination by Combination (for n, o ) 1 to N + 1) Mn• + Mo• f Mn+o ktc ) ktc,0 exp(-Etc/RT) L mol-1 s-1 Termination by Disproportionation (for n, o ) 1 to N + 1) Mn• + Mo• f Mn + Mo ktd ) ktd,0 exp(-Etd/RT) L mol-1 s-1 a M • is a radical with an initiator end group and one added 1 monomer unit. If terminated by disproportionation, or chain transfer to M1, it represents a dead polymer with chain length one. b The assumptions made are linear polymer chains and reaction rates independent of chain length.

dosing scheme for additives such as initiators and chaintransfer agents. During product development, the model can be used to determine the influence of the process conditions and additives dosing on the MWD. Model Below, the kinetic scheme, the molar balances, and the inclusion of the viscosity effects are described. The symbols are given in the Nomenclature section. (1) Kinetic Scheme. We have adopted a kinetic scheme typical for free-radical polymerization involving initiation, propagation, chain-transfer, and termination reactions from Chaimberg and Cohen (1990). See Table 1. (2) Molar Balances. The molar balances for an ideally macromixed batch stirred tank reactor (hence no influence of agitation speed) have been described in the Appendix. (3) Viscosity Effects. A fundamental approach of the viscosity effects has been incorporated in the model based on the work of Achilias and Kiparissides (1992). They proposed a theoretical framework for modeling diffusion-controlled free-radical polymerization reactions. Initiator efficiency and termination and propagation rate constants have been expressed in terms of a reaction-limited term and a diffusion-limited one. The latter was shown to be dependent on the diffusion coefficient of the corresponding species (i.e., monomer, primary radicals, and polymer) and an effective reaction radius. The rate constant of chain transfer to monomer has been related to the propagation rate constant. The proposed equations have been based on their previous work and the generalized free-volume theory of Vrentas and Duda (see the references listed in Achilias and Kiparissides (1992)). We added to this an expression to account for the chain transfer to a solvent or to a chain-transfer agent. All parameters appearing in the diffusion-limited part of the kinetic rate constant have a clear physical meaning. The parameters can hence be evaluated in terms of the physical and transport properties of the reacting species. Below the work of Achilias and Kiparissides, as adopted by us, is summarized.

Diffusion-Controlled Initiation Reaction (Cage Effect). The overall initiator efficiency, f, depends on the temperature, primary radical mobility, molecular weight of the (diffusing) species, and composition of the medium. According to the “cage” modeling concept (Figure 1a), the various molecular species in the reaction medium form around the primary radicals R• a cell that inhibits their diffusion outwards. As a result, only a fraction of the generated radicals succeeds in escaping from the cage to initiate new polymer chains. The primary radical diffusion rate depends on the diffusion coefficient of these radicals, DI. The primary radical cage has been approximated by two concentric spheres of radii r1 and r2. It has been assumed that primary radicals are only generated in the initiator reaction sphere of radius r1 placed inside the large diffusion sphere of radius r2. Only radicals escaping from the diffusion sphere of radius r2 can react with monomer to initiate new polymer chains. The equation below accounts for both chemical- (i.e., intrinsic initiator efficiency, f0, that has been assumed to be temperature independent) and temperature-dependent physical phenomena (i.e., primary radical diffusion) affecting the overall initiator efficiency during polymerization:

f)

1 3 r 1 2 ki0[M] + f0 3f0r1DI

(1)

where ki0 is the rate constant for the initiation reaction and [M] the monomer concentration. To calculate r1, r2, and DI, refer to the expressions in Achilias and Kiparissides (1992). Diffusion-Controlled Termination Reaction (Gel Effect). The effective termination rate, kte, depends on the temperature, growing polymer chain mobility, molecular weight of the (diffusing) species, and composition of the medium. It is given by

kte ) kt + kt,res

(2)

The analytical expression for the effective diffusioncontrolled termination rate constant, kt ()ktc + ktd), has the same functional form as for the diffusion-controlled initiation reaction described above. The equation below accounts for both chemical (i.e., intrinsic termination rate constant, kt0) and diffusional phenomena related to the termination of growing polymer chains (Figure 1b):

kt )

1 rt2λ0 1 + kt0 3Dpe

(3)

where rt is the termination radius coefficient, λ0 the total concentration of macroradicals, and Dpe the effective polymer diffusion coefficient. For the expressions to calculate rt and Dpe, refer to Achilias and Kiparissides (1992). The residual termination rate constant, kt,res, at high monomer conversions, roughly above 50-70% (Soh and Sundberg, 1982a), has been given by

kt,res ) Akp[M]

(4)

where A is a proportionality parameter and kp the rate constant for the propagation reaction (see below). For

Ind. Eng. Chem. Res., Vol. 36, No. 4, 1997 1165

Figure 1. Schematic diagram illustrating the coordinate system used in describing (a) the primary radical diffusion process (cage effect) for the calculation of f, (b) the radical termination process (gel effect) for the calculation of kt, and (c) the propagation process (glass effect) for the calculation of kp.

the expressions to calculate A, refer to Achilias and Kiparissides (1992). Diffusion-Controlled Propagation Reaction (Glass Effect). The overall propagation rate, kp, depends on the temperature, monomer mobility, molecular weight of the (diffusing) species, and composition of the medium. The analytical expression for the diffusion-controlled propagation rate constant has the same functional form as for the diffusion-controlled initiation and termination reactions as described above. The equation below accounts for both chemical (i.e., intrinsic propagation rate constant, kp0) and diffusional limitation of the propagation reaction (Figure 1c):

kp )

1 rm2λ0 1 + kp0 3Dm

(5)

The effective reaction radius, rm, has been assumed to be equal to the termination reaction radius, rt. Dm is the monomer diffusion coefficient. For the expressions to calculate rm and Dm, refer to Achilias and Kiparissides (1992). Chain Transfer to Monomer. The chain transfer to monomer rate constant has been related to the propagation rate constant:

ktr,m ) Ctr,mkp

parameter

value

unit mol L-1 mol L-1 mol L-1 s s-1 L mol-1 s-1 L mol-1 s-1 L mol-1 s-1 L mol-1 s-1 L mol-1 s-1 L mol-1 s-1



0.585 (AIBN) 0.0333 (AIBN) 6.65 2.22 (cyclohexane) 2.16 × 104 4.98 × 10-5 240 2.00 × 106 2.20 × 107 0.152 0 kp (by definition, Yoon and Choi (1992)) -0.112a

([I]V)0 ([R•]V)0 ([M]V)0 ([Mn•]V)0 ([Mn]V)0 ([X]V)0 V0

Initial Conditions 0.0333 (AIBN) 0 6.65 0 (for i ) 1 to N + 1) 0 (for i ) 1 to 2N + 2) 2.22 (cyclohexane) 1 (by definition)

f [I]0 [M]0 [X]0 tb kd kp ktc ktd ktr,m ktr,x ki

mol mol mol mol mol mol L

a Chaimberg and Cohen (1990) do not provide a value for the contraction factor . We calculated  for solution polymerization using the data of pure styrene ( ) -0.147 (Yoon and Choi, 1992)), Fstyr ) 857 kg m-3 (at 75 °C), Mstyr ) 104.14 (g mol-1)) and cyclohexane (Fcyhex ) 726 kg m-3 (at 75 °C), Mcyhex ) 84.16 g mol-1).

(6)

where Ctr,m is the chain transfer to monomer constant. Chain Transfer to Solvent or Chain-Transfer Agent. We added the rate constant of chain transfer to a solvent or a chain-transfer agent that has been related to the propagation rate constant:

ktr,x ) Ctr,xkp

Table 2. Constants for Literature Case 1, at T ) 348.15 K (Chaimberg and Cohen, 1990)

(7)

where Ctr,x is the chain transfer to a solvent or a chaintransfer agent constant. Verification of Four Cases Verification of Literature Case 1. To test the model implementation, the batchwise solution freeradical polymerization of styrene at constant temperature (75 °C) using the initiator AIBN as taken from

Chaimberg and Cohen (1990) has been evaluated. In this case, the presence of viscosity effects during the polymerization process was avoided by carrying out the batch experiments in solution with cyclohexane (Blavier and Villermaux, 1984). Chain transfer to the monomer, but not to the solvent, has been taken into account. Termination both by combination and disproportionation takes place. The total batch time was 6 h. Table 2 gives an overview of the constants used. Verification of Literature Case 2. To verify the model, the batch free-radical polymerization of styrene at constant temperature (60 °C) and using the initiator AIBME, taken from Achilias and Kiparissides (1992), has been evaluated. Chain transfer to the monomer has been taken into account. It has been assumed that no termination by disproportionation takes place. The numerical values of the terms Mw,0, i, λ0 (at t ) 0), and

1166 Ind. Eng. Chem. Res., Vol. 36, No. 4, 1997 Table 3. “Constants” for Literature Case 2, at T ) 333.15 K (Achilias and Kiparissides, 1992), and for the Polymerization Process with a Chain-Transfer Agent at T ) 333.15 K (Case 3) or with a Prescribed Temperature Program; See Text (Case 4) parameter f0 [I]0 [M]0 [X]0 tb kd kp0 ktc0 ) kt0 ktd ktr,m ktr,x ki 

([I]V)0 ([R•]V)0 ([M]V)0 ([Mn•]V)0 ([Mn]V)0 ([X]V)0 V0

value

at T ) 333.15 K

unit

0.4 (AIBME) 0.01 (AIBME) 8.36 0.001 (1-butanethiol)a 1.2 × 105 exp(36.8 - 16100/T) 1.09 × 107 × exp(-29.50 × 103/RT) 1.70 × 109 × exp(-9.49 × 103/RT) 0 1.0 exp(-3212/T)kpb 22kpc kp (by definition, Yoon and Choi (1992)) -(0.137 + 4.4 × 10-4(T - 273.15))

0.4 0.01 8.36 0.001 1.2 × 105 9.86 × 10-6 258.2

mol L-1 mol L-1 mol L-1 s s-1 L mol-1 s-1

5.53 × 107

L mol-1 s-1

0 0.0168 5.68 × 103 258.2

L mol-1 s-1 L mol-1 s-1 L mol-1 s-1 L mol-1 s-1

-0.163

Initial Conditions 0.01 (AIBME) 0.01 0 0 8.36 8.36 0 (for i ) 1 to N + 1) 0 0 (for i ) 1 to 2N + 2) 0 0.001 (1-butanethiol)a 0.001 1 (by definition) 1

mol mol mol mol mol mol L

a We chose a typical concentration for 1-butanethiol of 0.001 mol L-1 (Mahabadi and O’Driscoll, 1977). During the simulation of case 2 and case 4, the 1-butanethiol concentration was zero. b Chain transfer to monomer with C tr,m ) 1.0 exp(-3212/T). c Chain transfer to 1-butanethiol with C tr,x ) 22, at 60 °C (Mark et al., 1989).

Mw1/Mw, as used by Achilias and Kiparissides (1992), could not be traced in their publication. We have made the following estimates: Mw,0 ) 150.000 g mol-1 (from ref 10 in Achilias and Kiparissides (1992)), i ) 1 (Yoon and Choi, 1992), λ0 ) x2f0kd[I]0/kt0 ) 3.77 × 10-8 mol L-1 (for t/tb < 0.001, λ0 has been derived from eqs 9 and 11 assuming quasi-steady state for all radicals and excluding chain transfer) and Mw1/Mw ) 1. The total batch time was 33.33 h. The constants have been summarized in Table 3. Achilias and Kiparissides (1992) calculated the monomer conversion, total radical concentration, and polydispersity (or heterogeneity) index, Mw/Mn, using the so-called “momentum method”. We not only reproduced their results but calculated the complete MWD as well. Verification of the Chain-Transfer Agent (Case 3) and Nonisothermal Polymerization Process (Case 4). The influence of the chain-transfer agent 1-butanethiol, a mercaptan that is used in some commercial systems (Mark et al., 1989), on bulk styrene polymerization similar to case 2 has been demonstrated (case 3). As some industrial polymerization processes, such as for expandable polystyrene, operate at nonisothermal conditions, the effect of a prescribed temperature program was illustrated (case 4): At t ) 0, the temperature was increased with 5 °C h-1 from 20 to 70 °C. A temperature plateau was maintained at 70 °C for 13.33 h, and then the temperature was decreased with 5 °C h-1 from 70 to 20 °C (Figure 8). This program has been chosen around 60 °C because some parameters, such as the initial initiator efficiency (f0), the initial diffusion coefficient (Dio), and the initial monomer diffusion coefficient (Dm0), have been estimated at this temperature (Achilias and Kiparissides, 1992). They have been assumed to be temperature independent in

this work. Further, using this prescribed temperature program approximately the same final monomer conversion will be obtained compared to case 2. No chaintransfer agent has been assumed to be present in case 4. The constants have all been taken from the literature and are summarized in Table 3. Model Implementation and Numerical Solution The highly coupled nonlinear nature of the system, and “irregular initial conditions”, precludes an analytical solution of the model equations. To solve the equations numerically, the maximum length of the growing polymer chain has been assumed to be N + 1. There is no propagation to higher chain length growing polymers. The selected value for the maximum chain length must be large enough so the presence of chains larger than this maximum can be neglected. The restriction on the maximum growing polymer chain length sets an upper limit on the chain size of the dead polymer to be 2(N + 1), due to the possibility of a combination of the largest growing polymer chain. Thus, the solution of the differential equations requires the numerical integration in the order of 3(N + 1) + 5 equations. A rigorous solution requires roughly 10 00050 000 (Bandrup and Immergut, 1989) stiff differential equations to be solved simultaneously. To make the numerical solution more tractable, the growing polymer chains have been lumped into groups. The lumping method is similar to the scheme presented by Chaimberg and Cohen (1990). For typical free-radical polymerization rate constants, even the lumped system of equations is large and stiff (Skeirik and Grulke, 1985). The software package SimuSolv, developed by The Dow Chemical Company (Steiner et al., 1989), has been used to solve the set of ordinary differential equations. The package runs on a SUN Microsystems workstation (640 Mbyte). The integration method used is an implicit “Gear method” with adaptive step-size control. The model is run without any quasi-steady-state approximation assumptions. To check our results, we defined an overall monomer mass balance (OMMB; see Nomenclature) that should theoretically always be 100%, but minor deviation is allowed for computing inaccuracies. In all cases described below, we found the deviation to be less than 2%. Results and Discussion Verification of Literature Case 1. Figure 2 shows the agreement of our computed cumulative dead polymer MWDs for various reaction times with the literature. The maximum peak height of the computed MWD is 11% higher than the MWD as presented in the literature. A possible reason for this discrepancy could be enclosed in the value of the volume contraction factor, . Chaimberg and Cohen (1990) do not provide a value for this factor. Our estimation ( ) -0.112; see Table 2) could differ from what has been used by them. It is remarkable that the discrepancy is negligible if we assume  ) 0, so probably, they neglected the volume contraction. The required computer run time was 8 min. The styrene conversion profile and average molecular weight as a function of reaction time, as calculated by the model, are in excellent agreement with the literature (not shown; see their Figures 2 and 3). These results give us confidence in our model implementation. Verification of Literature Case 2. Figure 3 shows the cumulative dead polymer MWDs at various times

Ind. Eng. Chem. Res., Vol. 36, No. 4, 1997 1167

Figure 4. Product MWD distribution at various dimensionless reaction times for case 3 with chain-transfer agent 1-butanethiol, tb ) 33.33 h (ζ ) 94.7%, Mn ) 2.15 × 105 g mol-1, Mw ) 1.08 × 106 g mol-1, Mw/Mn ) 5.03).

Figure 2. Reproduction of the product MWD distribution at various reaction times of literature case 1. Top figure is from Chaimberg and Cohen (1990), and bottom figure is from this work (ζ ) 69.2%, Mn ) 1.78 × 104 g mol-1, Mw ) 3.66 × 104 g mol-1, Mw/Mn ) 2.05).

Figure 3. Calculated product MWD distribution at various dimensionless reaction times of literature case 2, tb ) 33.33 h (ζ ) 94.8%, Mn ) 2.94 × 105 g mol-1, Mw ) 1.13 × 106 g mol-1, Mw/Mn ) 3.85).

throughout the reaction. Prior to the onset of the gel effect, the weight concentration of polymer chains of length of approximately 20 × 103 and higher is negligible. The gel effect results in a substantial weight fraction of polymer up to a chain length of approxi-

mately 80 × 103. Furthermore, the cumulative MWD displays a tendency to a bimodal distribution for reaction times larger than 0.7 × 33.33 h ) 23.33 h. The computer run time was 31 h. This is significantly longer than for case 1 due to the increased amount of differential equations to be handled and the time to proceed through the equations describing the viscosity effects. Both the monomer conversion and total radical concentration as a function of reaction time and the polydispersity index as a function of conversion as presented by Achilias and Kiparissides (1992) were successfully reproduced (not shown; see their Figures 12 and 13). The final monomer conversion was 94.8%. Verification of the Chain-Transfer Agent (Case 3) and Nonisothermal Polymerization Process (Case 4). Figure 4 shows the dead polymer MWD results of the presence of the chain-transfer agent 1-butanethiol. It has been concluded that initially the MWD has been shifted to lower chain lengths, as expected. At increasing reaction times, however, the MWD shifts to longer chain lengths due to the depletion of the chain-transfer agent. Using the chain-transfer agent also broadens the MWD. The polydispersity index of the final MWD increased from 3.85 (case 2) to 5.03. The styrene conversion profile is shown in Figure 5. The conversion increases steadily up to a value of approximately 50%. Subsequently, due to the diffusional limitations imposed on the growing polymer chains at the onset of the gel effect, the rate of conversion accelerates rapidly. The final monomer conversion was 94.7%. Correspondingly, there is a sharp rise in the average molecular weight of the polymer. This is for the weight-average molecular weight more profound than the number-average molecular weight, as may be seen by the polydispersity index shown in Figure 6. The required computer run time was 28 h. Figures 7-9 show the results with the prescribed temperature program (case 4). The calculated dead polymer MWD is slightly shifted to lower chain lengths (Figure 7) when compared with the standard case 2 (Figure 3). Until t/tb ) 0.3, the polymerization process proceeds slowly but the concentration dead polymer increases fast afterwards. Within the 33.33-h batch time, a complete monomer conversion could not be reached. It levels off at 95.8% due to the lowered

1168 Ind. Eng. Chem. Res., Vol. 36, No. 4, 1997

Figure 8. Monomer conversion profile for case 4 with a prescribed temperature program; the dimensionless reaction time is t/tb, where tb ) 33.33 h. Figure 5. Monomer conversion profile for case 3 with chaintransfer agent 1-butanethiol; the dimensionless reaction time is t/tb, where tb ) 33.33 h.

Figure 9. Polydispersity index profile for case 4 with a prescribed temperature program. Figure 6. Polydispersity index profile for case 3 with chaintransfer agent 1-butanethiol.

continuation of the polymerization process, as expected. The polydispersity index shown in Figure 9 shows a sharp rise at 0.5 × 33.33 h ) 16.67 h. The required computer run time was 7 h. For case 4, we used a different integration algorithm as in cases 1-3, resulting in a relatively short required computer run time. Although case 4 included a more or less arbitrarily chosen temperature program and some parameters have been assumed to be “unjustly” temperature independent, it shows that the modeling results can be used in the field of process optimization and product development. Conclusions

Figure 7. Product MWD distribution at various dimensionless reaction times for case 4 with a prescribed temperature program, tb ) 33.33 h (ζ ) 95.8%, Mn ) 2.32 × 105 g mol-1, Mw ) 8.87 × 105 g mol-1, Mw/Mn ) 3.82).

temperature that freezes the reaction mixture (Figure 8). Lowering the temperature in the region where the viscosity effects are strongly present is fatal for the

A model for free-radical polymerization reactions in an ideally macromixed batch stirred tank reactor has been developed. The model yields, as a function of time, the monomer conversion, the number- and weightaverage molecular weights, and a complete MWD of both the growing and dead polymer chains. The model includes a fundamental approach of viscosity effects such as the cage, gel, and glass effects. The model has been successfully verified with two literature cases and demonstrated with two of our own defined cases. Case 1 (from Chaimberg and Cohen (1990)) describes the batch free-radical solution polymerization of styrene in cyclohexane at 75 °C without using a chain-transfer agent. The cage effect has been

Ind. Eng. Chem. Res., Vol. 36, No. 4, 1997 1169

modeled by a fixed efficiency factor. This literature case excludes the influence of the gel and glass effects. Case 2 (from Achilias and Kiparissides (1992)) describes the bulk styrene polymerization at 60 °C and without a chain-transfer agent. The fundamental approach to account for the viscosity effects has been included in this case. The influence of the chain-transfer agent 1-butanethiol (case 3) or a prescribed temperature program around 60 °C (case 4) has been illustrated on bulk styrene polymerization based on case 2. Acknowledgment We thank G. W. Colenbrander and H. Ho¨lscher (SRTCA) for their useful comments on the manuscript. Nomenclature A ) proportionality constant (L mol-1) C ) chain-transfer constant D ) diffusion coefficient (m2 s-1) E ) activation energy (J mol-1) f ) initiator efficiency I ) initiator species or its molar concentration (mol L-1) j ) chain length coordinate k ) reaction rate constant (variable; see Table 1) M ) monomer or its molar concentration (mol L-1) Mn ) dead polymer species of length n or its molar concentration (mol L-1) Mn• ) growing polymer species of length n or its molar concentration (mol L-1) Mn ) number-average molar mass ) ∑n)12N+2([Mn] × VnMstyr)/∑n)12N+2([Mn]V) (g mol-1) Mstyr ) molar weight of the styrene monomer, 104.14 g mol-1 Mw ) weight-average molar mass ) ∑n)12N+2([Mn] × V(nMstyr)2)/∑n)12N+2([Mn]VnMstyr) (g mol-1) Mw1 ) weight-average molar mass at a weight fraction polymer close to zero (g mol-1) N ) maximum chain length growing polymer n ) chain length coordinate R ) universal gas constant, 8.314 J mol-1 K-1 R• ) primary radical or its molar concentration (mol L-1) r ) effective reaction radius (m) T ) absolute temperature (K) t ) time (s) V ) volume (L) X ) solvent or chain-transfer agent or its molar concentration (mol L-1)

m ) monomer n ) chain length coordinate o ) chain length coordinate p ) propagation or polymer ref ) reference res ) residual termination at high monomer conversion styr ) styrene t ) total termination (summation of combination and disproportionation) tc ) termination by combination td ) termination by disproportionation tr,m ) chain transfer to monomer tr,x ) chain transfer to a solvent or to a chain-transfer agent Abbreviations AIBME ) 2,2′-azodiisobutyrate AIBN ) 2,2-azobis(isobutyronitrile) MWD ) molecular weight distribution OMMB ) overall monomer mass balance ) no. of units M in (growing) polymer - no. of units X transferred to M)/ no. of units M consumed × 100 ) ([∑n)1N+1(n[Mn•]V) + ∑n)12N+2(n[Mn]V) - (([X]V)0 - [X]V)]/[([M]V)0 - [M]V]) × 100 (%)

Appendix I. Molar Balances The molar balances for an ideally macromixed batch stirred tank are

initiator d[I]V ) -kd[I]V dt initial condition

[I]V ) ([I]V)0 primary radicals d[R•]V ) (2fkd[I] - ki[R•][M])V dt initial condition

Subscripts 0 ) initial conditions 1 ) effective reaction radius coordinate 2 ) effective reaction radius coordinate b ) batch cyhex ) cyclohexane d ) initiator decomposition e ) effective I ) initiator i ) monomer initiation

(9)

[R•]V ) ([R•]V)0 monomer N d[M]V ) (-ki[R•][M] - kp[M]∑[Mj•] dt j)1 N+1

ktr,m[M] ∑ [Mj•])V (10)

Greek Symbols  ) fractional volume contraction at 100% monomer conversion i ) ki/kp ) ki0/kp0 ζ ) percentage monomer conversion ) [(([M]V)0 - [M]V)/ ([M]V)0]100 (%) λ0 ) total concentration of macroradicals ) ∑j)1N+1[Mj•] (mol L-1) ξ ) fractional monomer conversion ) ζ/100 F ) density (kg m-3)

(8)

j)1

initial condition [M]V ) ([M]V)0 polymer radicals (for n ) 1 to N + 1) d[Mn•]V

• • ) (ki[R ][M] + kp[Mn-1 ][M] dt (n ) 1) (n ) 2 to N + 1) • kp[Mn ][M] - (ktr,m[M] + ktr,x[X])[Mn•] + (n ) 1 to N) N+1

(ktr,m[M] + ktr,x[X]) ∑ [Mj•] j)1

(n ) 1) N+1

(ktc[Mn•] + ktd[Mn•]) ∑ [Mj•])V (11) j)1

1170 Ind. Eng. Chem. Res., Vol. 36, No. 4, 1997

initial conditions [Mn•]V ) ([Mn•]V)0 dead polymers (for n ) 1 to 2N + 2) d[Mn]V dt

N+1

1 • • ) ( /2ktc ∑ ([Mj ][Mn-j ]) + j)1

(n ) 2 to 2N + 2) N+1

(ktr,m[M] + ktr,x[X])[Mn•] + ktd[Mn•] ∑ [Mj•])V (12) j)1 (n ) 1 to N + 1) (n ) 1 to N + 1) initial conditions [Mn]V ) ([Mn]V)0 solvent or chain-transfer agent N+1 d[X]V ) -ktr,x[X] ∑ [Mj•]V dt j)1 initial conditions [X]V ) ([X]V)0

(13)

volume V ) V0(1 + ξ)

(14)

dξ dV ) V0 dt dt

(15)

d[M]V dξ ) -1/([M]V)0 dt dt

(16)

Thus,

and

Substitution of (16) in (15) gives

d[M]V dV ) -(V0/([M]V)0) dt dt initial condition V ) V0

(17)

By using the summation procedure in eq 12, all terms that undergo this summation have been counted twice. Therefore, the factor “1/2” has been placed here. However, if Mj• ) Mn-j•, then the factor “1/2” must be omitted.

Literature Cited Achilias, D. S.; Kiparissides, C. Development of a General Mathematical Framework for Modeling Diffusion-Controlled FreeRadical Polymerization Reactions. Macromolecules 1992, 25, 3739. Blavier, L.; Villermaux, J. Free Radical Polymerisation Engineering-II (Modelling of Homogeneous Polymerisation of Styrene in a Batch reactor, Influence of Initiator). Chem. Eng. Sci. 1984, 39 (1), 101.

Brandrup, J.; Immergut, E. H. Polymer Handbook, 3rd ed.; John Wiley & Sons: New York, 1989; Vol. 83. Bueche, F. Physical Properties of Polymers; Interscience: New York, 1962; p 114. Chaimberg, M.; Cohen, Y. Kinetic Modeling of Free-Radical Polymerization: A Conservational Polymerization and Molecular Weight Distribution Model. Ind. Eng. Chem. Res. 1990, 29, 1152. Clay, P. A.; Gilbert, R. G. Molecular Weight Distributions in FreeRadical Polymerizations. 1. Model Development and Implications for Data Interpretation. Macromolecules 1995, 28, 552. Horie, K.; Mita, I.; Kambe, H. Calorimetric Investigation of Polymerization Reactions. I. Diffusion-Controlled Polymerization of Methyl methacrylate and Styrene. J. Polym. Sci., Part A-1 1968, 6, 2663. Mahabadi, H. K.; O’Driscoll, F. Evaluation of the Rate Constant for Primary Radical Termination in Free Radical Polymerization. Makromol. Chem. 1977, 178, 2629. Mark, H. F.; Bikales, N. M.; Overberger, C. G.; Menges, G.; Kroswitz, J. I. Encyclopedia of Polymer Science and Engineering; John Wiley & Sons: New York, 1989; Vol. 16, p 29. Mita, I.; Horie, K. Diffusion-Controlled Reactions in Polymer Systems. J. Macromol. Sci., Rev. Macromol. Chem. Phys. 1987, C27 (1), 91. Nishimura, N. Kinetics of Diffusion-Controlled Free-Radical Polymerizations. II. Bulk Polymerizations of Styrene and Methyl Methacrylate. J. Macromolec. Chem. 1966, 1 (2), 257. O’Driscoll, K. F.; Huang, J. The rate of Copolymerization of Styrene and MethylmethacrylatesI. Low Conversion Kinetics. Eur. Polym. J. 1989, 25 (7/8), 629. Robertson, E. R. Diffusion Control in the Polymerizations of Methyl Methacrylate and Styrene. Trans. Faraday Soc. 1956, 52, 426. Skeirik, R. D.; Grulke, E. A. A Calculation Scheme for Rigorous Treatment of Free Radical Polymerizations. Chem. Eng. Sci. 1985, 40, 535. Soh, S. K.; Sundberg, D. C. Diffusion-Controlled Vinyl Polymerization. II. Limitations on the Gel Effect. J. Polym. Sci., Polym. Chem. Ed. 1982a, 20, 1315. Soh, S. K.; Sundberg, D. C. Diffusion-Controlled Vinyl Polymerization. IV. Comparison of Theory and Experiment. J. Polym. Sci., Polym. Chem. Ed. 1982b, 20, 1345. Steiner, E. C.; Blau, G. E.; Agin, G. L. Introductory Guide SimuSolv, Modelling and Simulation Software (A Guided Tour of SimuSolv in the Realm of Math Modelling), revised 2nd printing; The Dow Chemical Company: Midland, MI, March 1989. Weickert, G.; Thiele, R. Modell zur Beschreibung des Geleffekts fu¨r die Auslegung von Massepolymerisationsreaktoren. Plast. Kautsc. 1983, 30 (8) 432 (in German). Yoon, W. J.; Choi, K. Y. Free-Radical Polymerisation of Styrene with a Binary Mixture of Symmetrical Bifunctional Initiators. J. Appl. Polym. Sci. 1992, 46, 1353.

Received for review July 22, 1996 Revised manuscript received October 21, 1996 Accepted October 24, 1996X IE960446H

X Abstract published in Advance ACS Abstracts, February 15, 1997.