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Kinetics and Simulation of Solid-State Polymerization for Nylon 6 Jian-jun Xie† College of Chemistry and Chemical Engineering, Xiangtan University, Hunan, Xiangtan, 411105 People’s Republic of China
A model of solid-state polymerization (SSP) for nylon 6 has been proposed in considering the effect of segmental diffusion on the reversible step-growth polymerization with a monofunctional acid regulator. This model is an adaptation of Chiu et al. for characterizing the Trommsdorff effect in free-radical polymerization and of Kaushik et al. for the SSP model of spherical pellets of nylon 6. Then it is verified by using the experimental data ourselves. The results of the simulation are found to show qualitative trends as found by the model of Kaushik and Gupta. The effects of model parameters and the initial values of the individual feed conditions on the process of SSP for nylon 6 are also studied. Introduction
Table 1. Kinetic Scheme of the SSP for Nylon 6a
Solid-state polymerization (SSP) is a process where the polymer solid particles are further polymerized by heating the polymer at a temperature below its melting temperature but above its glass transition temperature in an inert gas stream or under vacuum in order to remove the byproduct, for example, water and/or 1,2ethanediol (glycol), so as to increase the molecular weight of the polymer solid particles. It is one of the most used methods to obtain polyamides and polyesters with better quality and higher molecular weight and is also a subject of special interest in the field of polymers. Two excellent literature reviews of SSP have been reported by Pilati1 and Fakirov.2 A comprehensive effort to understand the postcondensation process started over 3 decades ago. Several experimental studies were reported in the literature. However, attempts on the interpretation of experimental data were quite dispersed. A few workers also tried to develop mathematical models to explain the data. Notable among these is the study of Chen et al.3 Unfortunately, models proposed are semiempirical in nature and often have little molecular basis. Thus, the design of industrial postcondensation reactors is still an art. Sixteen years ago, Chiu et al.4 presented a phonologically model with a sound molecular basis for predicting the effect of the segmental diffusion on the termination rate constant of free-radical polymerization. Kumar et al.5 adapted this model for irreversible stepgrowth polymerization and presented simulation results for the increase in the chain length with time in the presence of segmental diffusion resistances. Kaushik and Gupta6 proposed a molecular model for SSP of nylon 6. They accounted for the low rates of segmental diffusion on reversible step-growth polymerization of ARB (A and B are amino or carboxyl end groups, respectively) polymers and used them to simulate the SSP of nylon 6. An optimal parameter estimation software package is used to obtain the best-fit values of the parameters in the model using experimental data of Gaymans et al.7 The effect of the variation of the individual parameters on the progress of the SSP of nylon 6 is also studied. Later, Kulkarni and Gupta8 also presented an improved molecular model for the SSP of
diffusion-controlled polycondensation
Pn + Pm y\ z Pn+m + W k′
polyaddition
Pn + C1 y\ z Pn+1 k′
†
Fax: 086-0732-8292060. E-mail:
[email protected] ks2 s3
ks3 s3
monofunctional regulator reaction
ks2
Pn + Pm+X y\ z Pn+m,X + W k′ s3
a P , P n n+1, Pm, and Pn+m are polymer molecules of chain length n, n + 1, m, and n + m (n, m ) 1, 2, 3, ...). Pm+X and Pn+m+X are polymer molecules of chain length m and n + m with relative molecular mass regulator (irreactive end group X). W is water. C1 is -caprolactam.
nylon 6. They used the free-volume theory of Vrentas and Duda.9,10 These free-volume parameters are difficult to obtain and are not very accurate. So, the present study is to develop a model for the SSP of nylon 6 with a monofunctional regulator based on the model of Kaushik and Gupta,6 and then it is verified by the experimental data ourselves. On the basis of this, the effects of model parameters and the initial values of the affecting factors on the progress of polymerization for nylon 6 are analyzed. Model Development In mathematical modeling of the phenomenon of postcondensation, the following three possible ratedetermining steps should be considered: (1) chemical reaction (inclusive of the micro level diffusion of the polymer molecule); (2) diffusion (micro level) of the condensation byproducts from the inside of the reacting mass/pellet to the surface (this is important for reversible reactions); (3) diffusion of the byproduct molecules from the solid polymer surface to the surrounding inert gas. Griskey and Lee11 and Gaymans et al.7 have concluded on the basis of their experiments that step 1 is the rate-controlling step and the diffusion of byproducts both within the pellet and from the surface of the pellet to the inert gas does not affect the progress of the reaction. The same conclusion is drawn in the theoretical analysis of Chen et al.3 Therefore, according to our experiments and the analysis on the model of Kaushik and Gupta,6 we proposed a model on the basis of the following assumptions. (1) The kinetic scheme for the SSP of nylon 6 is shown in Table 1. The present study incorporates the effect of
10.1021/ie9907775 CCC: $20.00 © 2001 American Chemical Society Published on Web 06/06/2001
Ind. Eng. Chem. Res., Vol. 40, No. 14, 2001 3153 Table 2. Kinetic and Model Equations for the SSP of Nylon 6
(A) Balance Equations
d[C1] ) -ks3[C1]λ0 + k′s3(λ0 - [P1]) dt d[P1] ) -2ks2[P1]λ0 + 2k′s2[W](λ0 - [P1]) - ks3[C1][P1] + k′s3[P2] - ks2λ0X[P1] + k′s2[W](λ0X - [P1X]) dt dλ0 ) -ks2λ02 + k′s2[W](λ1 - λ0) - ks2λ0λ0X + k′s2[W](λ1X - λ0X) dt dλ1 1 ) ks3[C1]λ0 - k′s3(λ0 - [P1]) - ks2λ0Xλ1 - k′s2[W](λ1X - λ2X) dt 2 dλ2 1 1 ) -2ks2λ12 + k′s2[W](λ1 - λ3) + ks3[C1](λ0 + 2λ1) + k′s3(λ0 - 2λ1 + [P1]) - ks2λ2λ0X + k′s2[W](2λ3X - 3λ2X + λ1X) dt 3 6 dλ1X 1 ) ks2λ1λ0X - k′s2[W](λ2X - λ1X) dt 2 dλ2X 1 ) ks2(2λ1λ1X + λ2λ0X) - k′s2[W](4λ3X - 3λ2X - λ1X) dt 6 dλ0X )0 or λ0X ) [R]0 dt d[P1X] ) -ks2[P1X]λ0 + k′s2[W](λ0X - [P1X]) dt
(
)
∂2[W] 2 ∂[W] d[W] ) ks2λ02 + k′s2[W](λ1 - λ0) + Dw + + ks2λ0λ0X - k′s2[W](λ1X - λ0X) dt r ∂r ∂r2 initial conditions at t ) 0, [W] ) [W]0, [C1] ) [C1]0, [P1] ) [P1]0, λ0 ) λ00, λ1 ) λ10, λ2 ) λ20 values from Table 3 boundary conditions at r ) 0, ∂[W]/∂r ) 0; r ) R, [W] ) [W]s closure conditions [P2] ) [P1]; λ3 ) λ2(2λ2λ0 - λ12)/(λ1λ0), λ3X ) λ2X(2λ2Xλ0X - λ1X2)/(λ1Xλ0X) (B) Equations for Rate Constants
ks2,0 ) ζ2(T)λ0; k′s2 ) k′s2,0 ) ks2,0/K2 ks3 ) ks3,0 ) ζ3(T)λ0; k′s3 ) k′s3,0 ) ks3,0/K3
ks2 ) ks2,0 log
{
}
1 - [θt/(D/D0)]ks2[W](1 - λ1/λ0) 1 + [θt/(D/D0)]ks2,0λ0
νf D 1 1 1 1 ) ; ) + D0 A(T) + B(T) νf Tg Tg∞ T 2 µ jn g∞
νf ) 0.025 + (R1 - Rg)(T - Tg)
(micro)diffusion of the reactive end group as well as the diffusion of the condensation byproducts. The diffusion effects are significant only for the forward reaction in the polycondensation and the monofunctional regulator reaction and negligible for the other reactions in Table 1. (2) The SSP process is isothermal, and the reacting polymer solid particles are spherical. (3) The diffusion of byproduct water is obeyed to the first Fick law in the radical direction only, and the (macroscopic) diffusivity of water, Dw, in the pellet is constant. (4) The dependence of the diffusivity of polymer molecules on the reaction temperature and polymer concentration is assumed to be given by the FujitaDoolittle theory. (5) The intrinsic rate constants are proportional to the first moment of all polymer molecules. The catalytic effect of the acid end groups as found for the liquidphase polymerizations has been assumed to be valid for the solid phase as well and is reflected by the presence of the zero moment of all polymer molecules. It is also assumed that the values of the equilibrium constants,
K2 and K3, reported by Tai and Tawaga12 for the liquidphase polymerization are valid for the SSP of nylon 6. (6) The water concentration in the gas phase and at the pellet surface is taken as constant. (7) The glass transition temperature, Tg, is assumed to be a function of the chain length and to increase as the chain length of the polymer increases. (8) The carboxyl and amine end-group concentrations and water concentration are assumed to be uniform within the pellets before the SSP of nylon 6. According to the above assumptions and referring to the model for the SSP of Kaushik and Gupta6 and for polymerization in the liquid phase of Kumar and Gupta,13 we obtain the final equations for the SSP of nylon 6 by mass balance and moment operation (Table 2). The equilibrium constants for the liquid-phase reaction were those reported by Tai and Tagawa12 (also see eq 1).
Ki ) exp[(∆Si - ∆Hi/T)/Rg] (i ) 2 and 3)
(1)
The thermodynamical parameters, the other model parameters, and the other initial conditions are also
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Table 3. Initial Values of Solid Pellets of Nylon 6 Polymer for SSP parameters
sample 1
number-average degree of polymerization number-average molecular weight regulator concentration [R]0 (wt %) [COOH] (mequiv/kg) [NH2] (mequiv/kg) pellet size (mm)
sample 2
µn ) 153.631
µn ) 164.654
Mn ) 17 385
Mn ) 18 627
0.02
0.03
59.83 37.74 rectangle: ca. 5 × 3 thick: ca. 1
57.46 52.20 rectangle: ca. 5 × 3 thick: ca. 1
given in Tables 3-5.6,14,15 The finite-difference technique in the r direction has been applied to reduce the partial differential equation for [W] (Table 2) to a set of coupled ordinary differential equations. These equations for all species (Table 2) are solved numerically by using Runge-Kutta-Hamming’s method for solving a set of coupled, nonlinear, ordinary differential equations.
Figure 1. Comparisons between the model calculation values and the experimental values. 1: ζ2 ) 60, ζ3 ) 20, A(T) ) 0.05, B(T) ) 0.03. 2: ζ2 ) 100, ζ3 ) 20, A(T) ) 0.06, B(T) ) 0.03. (s) Simulation: b, µn ) 153.63; 9, µn ) 164.65.
Experiment Nylon 6 polymer is made from melt polymerization of -caprolactam under dry nitrogen gas flow according to the usual method16 in a aluminum-made fixed bed, which has four glass tubes intubated with an automated temperature-controlled system thermostated to the accuracy of (0.5 °C. The tube reactor is 2.5 cm in diameter and 300 cm in length. The SSP was conducted in the same apparatus under dry nitrogen flow. The relative viscosity of nylon 6 for SSP is determined in 95.7% sulfuric acid as a solvent at 20 ( 0.1 °C by using a Ubbelohde-type viscometer. The concentration for the measurement of the relative viscosity of SSP nylon 6 samples is 0.01 g/mL. The number-average molecular weight Mn, which can be used to determine the numberaverage degree of polymerization, µn, is determined by the following equation.17
M h n ) 11500(ηr - 1)
(2)
Table 3 gives the parameters of nylon 6 polymer solid particles for the above polymerization product, which is also a nylon 6 sample for the SSP, and the other initial values (see columns 2 and 3 in Table 4). Results and Discussion In the model calculation for SSP of nylon 6, the initial values for all polymer solid particles are given in Table 4 (the water concentration is 10 times less than that of the melt polymerization, but the concentrations for other parameters are the same as the values of the melt polymerization). Otherwise, all of the parameter values in the model calculation are based on Table 5 and column 2 in Table 4 except for Figure 5. Figure 1 is the comparison of the experimental data ourselves and the simulation results for the spatial average value of the number-average chain length, µn, vs time. The results have shown that the model values basically agree with the experimental values by regulating model parameters. Effect of Model Parameters. The effect of model parameters on µn and polydispersity index, Q, of the SSP for nylon 6 is analyzed. The results are as follows (referring to the figures and results of Gupta et al.:6,8 (1) Increasing either A(T) or B(T) slows down the
Figure 2. Effect of polymerization temperatures on µn and Q: (a) µn; (b) Q. T/°C: 1, 225; 2, 220; 3, 200; 4, 190 (ref); 5, 180; 6, 160; 7, 130; 8, 100.
progress of the reaction, though the effect of A(T) is more drastic; the spatial average value of the number-average chain length, µn, decreases with an increase of A(T) and B(T), the spatial average value of polydispersity, Q, increases with an increase of A(T) and attains the equilibrium value in the final, and the effect of B(T) on Q is neglected. (2) As the value of the parameter ζ2, characterizing the forward rate constant ks2, is increased, the progress of the reaction becomes faster, µn increases, and Q becomes smaller. The effect of ζ2 above a value of about 150 kg2/mol2‚h becomes relatively insignificant. The effect of ζ3 is slower as compared to that of ζ2. The progress of the reaction is observed to be enhanced; that is, µn is increased and Q decreased as the value of ζ3 is increased. (3) The effect of the characteristic migration time, θt, on µn and Q shows that, for θt varying from about 0 to 100 h, the effect is almost insignificant. As the value of θt is increased from about 100 to 5000 h, the progress of the reaction is affected significantly. At a value of θt above 10 000 h, the effect on the progress of the reaction is relatively insignificant. That is to say, µn is decreased and Q is increased as the value of θt is increased. Effect of the Initial Conditions in SSP of Nylon 6. Figure 2 gives the relation to µn and Q vs time at different polymerization reaction temperatures. The results show that µn increases and Q decreases with an increase of the reaction temperature. This plot also
Ind. Eng. Chem. Res., Vol. 40, No. 14, 2001 3155 Table 4. Initial Values for the Polymer Solid Particles and the Values of the Model Parametersa parameters
µn ) 164.654b
µn ) 153.631
µn ) 23.122
µn ) 35.80
µn ) 55.52
µn ) 72.35
µn ) 95.60
[C1]/[mol/kg] [P1]/[mol/kg] [λ0]/[mol/kg] [λ1]/[mol/kg] [λ2]/[mol/kg] [W]/[mol/kg] [λ0X]/[mol/kg] [λ1X]/[mol/kg] [λ2X]/[mol/kg] [P1X]/[mol/kg] µw Q ζ2/[kg2/mol2‚h] ζ3/[kg2/mol2‚h] A(190 °C) B(190 °C)
1.21877 0.00035 0.04347 7.16158 2389.741 0.011654 9.7859 × 10-5 0.39246 130.32 9.7859 × 10-5 333.605 2.0261 60 20 0.05 0.03
1.55263 0.00040 0.04461 6.86705 2153.608 0.011539 1.6802 × 10-4 0.35698 110.73 1.6802 × 10-4 313.4456 2.04025 100 20 0.06 0.03
7.97256 0.00079 0.03332 0.81922 28.94464 0.012669 2.0037 × 10-3 0.00685 0.18996 2.0037 × 10-3 35.26921 1.52535 200 40 0.04 0.03
7.14440 0.00072 0.04375 1.63408 91.038 0.011625 1.7090 × 10-3 0.01858 0.84795 1.7090 × 10-3 55.59904 1.55292 200 40 0.04 0.03
5.79967 0.00065 0.05153 2.94545 269.6899 0.010848 1.2941 × 10-3 0.04913 3.9196 1.2941 × 10-3 91.3681 1.64567 200 40 0.04 0.03
4.76344 0.00060 0.06327 3.94149 498.1104 0.010674 9.9925 × 10-4 0.08674 9.947 9.9925 × 10-4 126.1244 1.74325 200 40 0.04 0.03
3.58201 0.00055 0.05204 5.05219 908.448 0.010796 6.776 × 10-4 0.15376 26.075 6.776 × 10-4 179.5106 1.8777 200 40 0.04 0.03
a Initial conditions in liquid-phase polymerization to obtain the above values: T ) 250 °C, [C ] ) 8.80 mol/kg, [P ] ) [λ ] ) [λ ] ) [λ ] 1 1 0 1 2 ) 0.0 mol/kg, [W] ) 0.16 mol/kg, [P1X] ) 0.002411 mol/kg. b The values of column 2 are the reference values for the following calculation.
Figure 3. Effect of the initial water concentration in the interior of pellet on µn and Q in SSP: (a) µn; (b) Q. [W]/[mol/kg]: 1, 0.0001; 2, 0.001; 3, 0.011 654 (ref); 4, 0.05; 5, 0.2; 6, 0.5; 7, 2.0.
shows that µn increases with an increase of the time and that while at first Q increases rapidly, it then decreases and finally reaches an equilibrium value as the time increases. The effect of the initial water concentration [W]0 on the progress of the reaction is shown in Figure 3. The plot shows that µn and Q decrease with an increase of the initial water concentration. This agrees with the effect of [W]0 on µn and Q for the hydrolytic polymerization of caprolactam. We can also see from Figure 3 that µn and Q appear to be different at the value of [W]0 above about 0.2 mol/kg. The reason may be because the higher [W]0 promotes the reversible polycondensation reaction moves toward the left. The effect of the diameter of the polymer particle dp on µn and Q is shown in Figure 4. The results show that µn increases and Q decreases with a decrease in the diameter of the polymer particle and that µn and Q is not affected because dp is greater than 0.06 mm. The effect of the initial polymerization degree of the polymer µn0 on µn and Q is shown in Figure 5 (for all parameters, see columns 4-8 in Table 4). The
Figure 4. Effect of the polymer particle diameters on µn and Q in SSP: (a) µn; (b) Q. dp/[mm]: 1, 0.01; 2, 0.1; 3, 0.6; 4, 1.2 (ref); 5, 2.0. Table 5. Initial Values for the Other Model Parameters Tg∞ ) 350 K ag ) 4.5 × 10-4 K-1 al ) 10-3 K-1 Dw ) 1.0 × 10-11 m2/h
θt ) 1000 h dp ) 1.2 mm T ) 190 °C Kg/Tg∞2 ) 0.03 K-1
∆H2 ) -2.4877 × 104 J/mol‚K ∆H3 ) -1.6919 × 104 J/mol‚K ∆S2 ) 3.9846 J/mol‚K ∆S2 ) -29.06 J/mol‚K
polymerization rate is faster and µn and Q obviously increase as µn0 increases. We also obtain that if µn0 only is changed while the other parameters are not changed, µn and Q will not be changed. The effect of the concentrations of caprolactam on the progress of the reaction is shown in Figure 6. The results show that µn increases and Q decreases with an increase of the concentrations of caprolactam. Figure 7 shows the effect of the concentration of the relative molecule mass regulator on the progress of the reaction. This plot shows that µn decreases and Q increases with an increase of the concentration of the relative molecule mass regulator. There is a gradual changing process as the concentration of the regulator increases, that is, the effect of the concentration of the regulator [R]0 on µn and Q is insignificant at the value of [R]0 below about 0.01 mol/kg.
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Figure 5. Effect of µn0 of the polymers on µn and Q in SSP: (a) µn; (b) Q. µn: 1, 23.12; 2, 35.80; 3, 55.52; 4, 72.35; 5, 95.60.
Figure 7. Effect of regulator concentrations in the interior of the particle on µn and Q in SSP: (a) µn; (b) Q. [R0]/[mol/kg]: 1, 9.79 × 10-5 (ref); 2, 0.001; 3, 0.005; 4, 0.01; 5, 0.1.
Notation
Figure 6. Effect of the initial caprolactam concentrations in the interior of the particle on µn and Q in SSP: (a) µn; (b) Q. [C1]/[mol/kg]: 1, 0.0; 2, 1.22 (ref); 3, 4.0; 4, 6.0; 5, 8.0.
Conclusions A model for SSP of nylon 6 has been proposed in this work, and it is verified by the experimental data ourselves. We explore the effect of model parameters on the results of the SSP of nylon 6. Then the effect of the affecting factors on µn and Q for the SSP of nylon 6 such as the temperature and time, the diameter of the particle, the initial number-average degree of polymerization (µn) of the polymer, the initial concentrations of water and monomer (caprolactam), and the regulator of the relative molecular weight is also analyzed. It is observed that, using the model, one can predict the qualitative trends observed experimentally on the SSP of nylon 6 polymer solid particles. More and comprehensive experimental data on the SSP of nylon 6 are required to obtain the model parameters ζ2, ζ3, θt, A(T), and B(T) at different temperatures.
A(T) ) parameter in the free-volume equation B(T) ) parameter in the free-volume equation C1 ) -caprolactam dp ) particle diameter, m D ) overall (micro)diffusivity of a polymer molecule, m2/h D0 ) overall diffusivity of a polymer molecule at some reference condition, m2/h Dw ) (macro)diffusivity of water inside the reacting pellet, m2/h ∆Hi ) change in enthalpy of the ith reaction in the liquid phase (i ) 2 and 3), J/mol ksi ) overall forward rate constant of the ith reaction in the solid phase (i ) 2 and 3), kg/mol‚h k′si ) overall reverse rate constant of the ith reaction in the solid phase (i ) 2 and 3), kg/mol‚h ksi,0 ) intrinsic forward rate constant of the ith reaction in the solid phase (i ) 2 and 3), kg/mol‚h k′si,0 ) intrinsic reverse rate constant of the ith reaction in the solid phase (i ) 2 and 3), kg/mol‚h kp ) overall forward rate constant for the ARB polymerization, kg/mol‚h kp0 ) intrinsic overall forward rate constant for the ARB polymerization, kg/mol‚h Ki ) equilibrium constant of the ith reaction (i ) 2 and 3) Pn ) polymer molecule of chain length n (n ) 1, 2, 3, ...) [P]m ) concentration of polymer molecule P at radius r, mol/kg Q ) polydispersity index (average value in the radius), R 2 2 Q ) ∫R 0 r (λ2 + λ2,x) dr/∫0 r (λ1 + λ1,x) dr r ) distance depart from the center of the pellet at any point in the radius, m R ) radius of the solid pellet, m Rg ) gas constant in eq 1, J/mol‚K [R]0 ) concentration of the relative molecular mass regulator, mol/kg ∆Si ) change in entropy for the ith reaction in the liquid phase (i ) 2 and 3), J/mol‚K t ) reaction time in nylon 6 SSP, h, or flowout time in viscosity measurement, s t0 ) flowout time for the 95.7% sulfuric acid in viscosity measurement, s T ) reaction temperature, K or °C Tg ) glass transition temperature, K
Ind. Eng. Chem. Res., Vol. 40, No. 14, 2001 3157 Tg∞ ) glass transition temperature at infinite chain length, K νf ) free-volume fraction inside the reacting pellet [W]s ) surface concentration of water, mol/kg [ ] ) concentration symbol Greek Letters Rl ) coefficient of thermal expansion of polymer in the liquid state, K-1 Rg ) coefficient of thermal expansion of polymer in the glassy state, K-1 ζ0 ) parameter used in the forward rate constant for the ith reaction in the solid phase, kg2/mol2‚h ∞ λk ) kth moment of all polymer molecules, λk ) ∑n)1 nk[Pn] (k ) 0, 1, 2), mol/kg λ1b ) bulk value of the first moment, mol/kg ηr ) relative viscosity of SSP nylon 6 sample, ηr ) t/t0 θt ) parameter denoting the characteristic time of migration, h µn ) number-average chain length or number-average polymerization degree (average value in the radius), µn R 2 2 ) ∫R 0 r (λ1 + λ1,x) dr/∫0 r (λ0 + λ0,x) dr µw ) weight-average chain length or weight-average polymerization degree (average value in the radius), µw ) R 2 2 ∫R 0 r (λ2 + λ2,x) dr/∫0 r (λ1 + λ1,x) dr
Literature Cited (1) Pilati, F. Comprehensive Polymer Science. In Step Polymerization; Eastmond, G. C., Ledwith, A., Russo, S., Sigwalt, P., Eds.; Pergamon: New York, 1989; Vol. 5, pp 201-216. (2) Fakirov, S. Solid State Behavior of Linear Polyesters and Polyamides. In Solid State Reactions in Linear Polycondensates; Schltz, J. M., Fakirov, S., Eds.; Prentice-Hall: Englewood Cliffs, NJ, 1990; Chapter 1, pp 1-74. (3) Chen, F. C.; Griskey, R. G.; Beyer, G. H. Thermally Induced Sold State Polycondensation of Nylon 66, Nylon 6-10, and Poly(ethylene terephthalate). AIChE J. 1969, 15, 680-685. (4) Chiu, W. Y.; Carratt, G. M.; Soong, D. S. A Computer Model for the Gel Effect in Free-Radical Polymerization. Macromolecules 1983, 16, 348-357. (5) Kumar, A.; Saxena, K.; Foryt, J. P. Effect of Segmental Diffusion on Irreversible, Step Growth Polymerization of ARB Monomers. Polym. Eng. Sci. 1987, 27, 753-763.
(6) Kaushik, A.; Gupta, S. K. A Molecular Model for Solid-State Polymerization of Nylon 6. J. Appl. Polym. Sci. 1992, 45, 507-520. (7) Gaymans, R. J.; Amritharaj, J.; Kamp, H. Nylon 6 Polymerization in the Solid State. J. Appl. Polym. Sci. 1982, 27, 25132526. (8) Kulkarni, M. R.; Gupta, S. K. Molecular Model for SolidState Polymerization of Nylon 6. II. An Improved Model. J. Appl. Polym. Sci. 1994, 53, 85-103. (9) Vrentas, J. S.; Duda, J. L. Diffusion in Polymer-Solvent Systems. I. Reexamination of the Free Volume Theory. J. Polym. Sci., Part B: Polym. Phys. 1977, 15, 403-416. (10) Vrentas, J. S.; Duda, J. L. Diffusion in Polymer-Solvent Systems. II. A Predictive Theory for the Dependence of Diffusion Coefficients on Temperature, Concentration, and Molecular Weight. J. Polym. Sci., Part B: Polym. Phys. 1977, 15, 417-431. (11) Griskey, R. G.; Lee, B. I. Thermally Induced Solid-State Polymerization in Nylon 66. J. Appl. Polym. Sci. 1966, 10, 105111. (12) Tai, K.; Tagawa, T. Simulation of Hydrolytic Polymerization of -Caprolactam in Various Reactors. A Review on Recent Advances in Reaction Engineering of Polymerization. Ind. Eng. Chem. Prod. Res. Dev. 1983, 22, 192-206. (13) Kumar, A.; Gupta, S. K. Simulation and Design of Nylon 6 Reactors. J. Macromol. Sci., Rev. Macromol. Chem. Phys. 1986, C26 (2), 183-247. (14) Achilias, D. S.; Kiparissides, C. Modelling of DiffusionControlled Free-Radical Polymerization Reactions. J. Appl. Polym. Sci. 1988, 35, 1303-1323. (15) Achilias D. S.; Kiparissides, C. Development of a General Mathematical Framework for Diffusion-Controlled Free-Radical Polymerization Reactions. Macromolecules 1992, 25, 3739-3750. (16) Tang, Z. L.; Lin, J.; Huang, N. X. Simulation of the Hydrolytic Polymerization of -Caprolactam With Bifunctional Regulators. Angew. Macromol. Chem. 1997, 250, 1-14. (17) Fantoni, R. F. Polymide 6sBasic Chemistry of Caprolactam Polymerization; Noywallesina Engineering Ed.: Parre (BG), Italy, 1990; p 68.
Received for review October 28, 1999 Revised manuscript received November 29, 2000 Accepted March 21, 2001 IE9907775
