Kinetic modeling of bulk free-radical polymerization of methyl methacrylate

Abstract

The free-radical bulk polymerization of many monomers is characterized by a sudden rise in the rate of polymerization, a phenomenon called autoacceleration. Many mathematical models have been developed to describe this phenomenon. In this paper, the development of a new kinetic model is described. The model very successfully describes experimental data obtained by differential scanning calorimetry of the bulk free-radical polymerization of methyl methacrylate. The proposed model is composed of two contributions to the conversion of the monomer, one originating from polymerization according to the classical theory of radical polymerization and the other originating from polymerization during the autoacceleration. The rate constant of the autoacceleration (second contribution) is about eightfold higher than the rate constant of the first-order reaction (first contribution).

Introduction

Free-radical bulk polymerization is a versatile process because it can be carried out on many monomers and at a wide range of temperatures.¹ It is well-known that the free-radical bulk polymerization of vinyl monomers (derivatives of acrylic and methacrylic acids, vinyl acetate, styrene, ethylene, and so on) is characterized by the autoacceleration phenomenon. The free-radical polymerization of these monomers can be explained by the classical theory up to a certain monomer conversion. After this critical monomer conversion, the autoacceleration of the polymerization appears. The onset of the autoacceleration is defined as the moment when the polymerization rate departs from the value expected according to the classical theory of free-radical polymerization.^{2, 3, 4} The onset and the intensity of the autoacceleration are determined by the type of monomer, type and concentration of initiator, temperature and other reaction conditions. This phenomenon is particularly apparent in the bulk polymerization of methyl methacrylate (MMA) and is highly undesirable in industrial applications, as it may lead to the thermal runaway of the process, thus causing depolymerization and the plugging of equipment.^{5, 6, 7}

A number of theoretical explanations and kinetic models were developed in order to explain the phenomenon of autoacceleration. The first models^{8, 9, 10} considered only the decrease of the chain termination rate constant as a result of an increased viscosity of the reaction system. Later efforts^{11, 12} focused on investigating changes of both the chain propagation and chain termination rate constants due to changes in the viscosity. Later still, theories^{13, 14} attempted to explain the occurrence of autoacceleration in the rate of polymerization as a result of entanglement of growing macromolecular chains. O’Driscoll introduced the gel effect index as a measure of the severity of the kinetic effect.¹⁵ The model developed by Chiu et al.⁷ used the conversion, temperature and weight-average molecular weight to determine the relative influence of the reaction and the diffusion on the rate of polymerization. Free volume theories^{16, 17} considered the decrease in volume on disposal for the movement of growing macromolecules. Kargin and Kabanov¹⁸ and Korolev et al.¹⁹ suggested that the autoacceleration phenomenon could be explained by the supramolecular organization of the liquid MMA. Roschupkin et al.²⁰ found that poly(methyl methacrylate) (PMMA) grains are formed during the polymerization of MMA and suggested that the autoacceleration is a consequence of the growth of the grain surface.

A model written in terms of the moment generating function and in terms of the moments of molecular weight distribution, and completed with relations that quantify the gel and glass effects was successful in describing MMA bulk polymerization.²¹ A mathematical model was developed for the batch MMA polymerization reactor system by Rafizadeh²² The model includes the complete process, therefore, using the heaters and water valve signals, which make it possible to calculate the process states. Hence, this model is suitable to determine an optimal temperature trajectory during the course of the polymerization and control strategy. More recently, a simple semi-empirical model relating the degree of conversion and the polymerization rate to the time and temperature was developed.¹ The model parameters were calculated from isothermal differential scanning calorimetry (DSC) experiments and then successfully applied to predict monomer conversion in non-isothermal experiments. Sangwai et al.²³ used an empirical model that involves only monomer conversion and temperature, and accounts for the gel and the glass effect to describe the polymerization of MMA in a rheometer-reactor assembly in isothermal and non-isothermal conditions. During the last decade, successful results were achieved through the use of the pulsed laser polymerization technique to determine the values of k_p and k_t for free-radical polymerization.²⁴ Barner-Kowollik et al.²⁵ have presented an extensive review of the experimental methods used to study the dependence of k_t on the conversion and chain length. Buback et al.²⁶ have used this technique to obtain the chain length dependence of k_t for MMA polymerization.

The existing theories, however, have not been completely verified experimentally. Their main shortcoming is that they take only the onset of acceleration as a characteristic point on the polymerization rate vs time curve. We have shown that there are some additional characteristic points in the case of methyl-², ethyl-³, butyl³ dodecylmethacrylates²⁷ and the polymerization of styrene:²⁸ the maximum polymerization rate and the two inflection points before and after that maximum.

In the current paper, we have focused on determining the existence of these characteristic points, as well as on testing a mathematical model of the free-radical bulk polymerization developed earlier by our research group and successfully tested on the polymerization of styrene.²⁸ Experimental data were obtained by using DSC to follow the bulk free-radical polymerization of MMA at different temperatures.

Materials and methods

MMA monomer (MMA, Tokyo Chemical Industry Co., Ltd., Tokyo, Japan) was washed two times with 10% sodium hydroxide solution to remove the inhibitor. Then, the MMA was washed two times with distilled water, dried over anhydrous calcium chloride and vacuum distilled. Initiator 2,2′-azobisisobutyronitrile (AIBN, Merck, Darmstadt, Germany) was recrystallized from methanol before usage. A solution of 0.5 wt.% AIBN in MMA was prepared. Approximately 5–10 mg of the solution was placed in a hermetic aluminum DSC pan and sealed with an aluminum lid. The bulk polymerization of MMA was performed in a Q20 DSC (TA Instruments, New Castle, DE, USA) under isothermal conditions at 60, 70, 80 and 90 °C. Every experiment was repeated three times. The temperature and heat flow scales were calibrated using the melting of high-purity indium. Nitrogen was used as a purge gas with a flow rate of 50 cm³ min⁻¹.

Results and Discussion

Figure 1 shows the DSC thermograms of MMA free-radical bulk polymerization at different temperatures. Three characteristic moments can clearly be observed in Figure 1: the onset of acceleration (point M), the maximum of the polymerization rate (point S) and the end of polymerization (point K).^{2, 3} Point M was determined as the minimum and point S as the maximum in the DSC thermogram. Point K was determined as the moment when the isothermal DSC curve becomes horizontal.

Figure 1

DSC thermograms of MMA free-radical polymerization at different temperatures.

To find the monomer conversion degree the following formula was applied:

where X is the monomer conversion, X=(C_M0−C_M)/ C_M0; C_M0 and C_M are the initial monomer concentration and concentration after time τ, respectively; dH is the heat evolved by polymerization during an infinitesimal time (dτ); τ_K is the time required to achieve point K; H_D is the heat evolved during polymerization of an unreacted monomer left after point K as determined by dynamic DSC measurement. The conversion of MMA at different temperatures is shown in Figure 2.

Figure 2

MMA conversion vs time at different temperatures.

The conversion vs time curves of the MMA polymerization exhibit portions that have an ‘S’ shape, characteristic for autoacceleration. The final conversion of MMA has higher values at the higher temperatures.

The rate of polymerization, R_pol=dX/dτ, is equal to the slope of the conversion vs time curve. The acceleration, dR_pol/dτ=d²X/dτ, is equal to the slope of the R_pol vs time curves (Figure 3). The polymerization of MMA exhibits the following characteristic points:² the onset of acceleration (M), the maximum acceleration (P) and the maximum polymerization rate (S), proceeded by the deceleration stage with minimum (R) and final conversion (K).

Figure 3

Characteristic points on the conversion (X), rate of polymerization (r_pol) and acceleration (dr_pol/dτ) vs time (τ) curves obtained by transformation of the DSC curve of MMA polymerization at 60 °C.

The time and MMA conversions, required to achieve the characteristic points, are given in Table 1. The conversion X_M at the onset of the autoacceleration increases as the polymerization temperature increases, in accordance with published data.²

Table 1 Time and MMA conversion, necessary to achieve characteristic points (determined from DSC curves)

Kinetic model

Looking at Figure 2, three regions can be observed during the polymerization of MMA. The first is up to point M. The second is the acceleration portion from point M to point S. The third is the deceleration from point S to point K. A potential kinetic model should fit the conversion vs time curves, but it should also exhibit the derivatives (dX/dτ) and (d²X/dτ) that resemble the shape of the corresponding curves derived from the experimental results (Figure 3). A potential model should fulfill the following conditions: conversion starts from zero (τ=0, X=0), the first derivative (dX/dτ) has a local minimum (M), then reaches the first inflection point (P) followed by a maximum (S) and second inflection point (R), becoming horizontal (K) at the end.

It is generally accepted that the first part of the conversion vs time curve can be explained by the classical free-radical polymerization. Hence, the polymerization rate R_pol in that part can be expressed by the well-known equation (2a).²⁹ The reaction rate is first order with respect to the monomer and an order of 0.5 with respect to the initiator concentration.

Here, X₁ is the conversion of a part of the monomers that polymerize according to the classical theory of polymerization and k_pol,1 is the corresponding polymerization rate constant; C_I is the initiator concentration after time τ. It is usually supposed that there is a negligible change of the initiator concentration during the polymerization, that is, C_I=C_I0=const. Hence, the rate of polymerization is described by the equation (2b):

where k₁=k_pol,1 C_I^1/2 is the rate constant for the first-order reaction.

All the theories of the autoacceleration in the rate of polymerization assume that the polymer produced during the autoacceleration stage has a catalytic effect on the polymerization. Hence, we propose that, in that stage, the rate of polymerization depends not only on the concentration of the residual monomer M but also on the amount of the created polymer P, that is, equation (3a). This equation can easily be transformed to equation (3b).

where X₂ is the conversion of a portion of the monomers that polymerize according to the autoacceleration and deceleration mechanisms of polymerization and k₂=k_pol,2C_M0C_I^1/2 is the corresponding polymerization rate constant. It should be noted that equation (3b) is a parabola that has a maximum. Hence, it can be used to describe both the acceleration and deceleration portions of the polymerization rate vs time curves.

After integration and rearrangement, equations (2b) and (3b) become (4) and (5), respectively:

where τ_2max is the time required to achieve the maximum rate of the autoacceleration stage.

Equation (4) and equation (5) are conversions according to classical first-order polymerization and acceleration, respectively. At the end of the polymerization, the achieved final conversion X_K consists of two fractions: a is the monomer fraction polymerized by autoacceleration (that is, equation 5) and (X_K−a) is the monomer fraction polymerized by a first-order reaction (that is, equation 4). Based on these assumptions, the dependence of the conversion on time can be presented by a mathematical model (equation 6). The first addend corresponds to the first-order reaction and the second addend to the autoacceleration reaction. The model takes into account the overlap of these two contributions from the beginning of the polymerization. This assumption was tested by fitting the model to the experimental data.

In equation 6, the values for X and τ were obtained from the experimental DSC data, while k₁, k₂, a and τ_2max were calculated using the method of least squares. The values of all calculated parameters are given in Table 2. The proposed mathematical model (equation 6) describes the experimental dependence of the monomer conversion degree on the polymerization time (Figure 4.) The low values of the s.d. and the high correlation coefficients (Table 2) confirm this conclusion. The proposed model includes both addends from the beginning to the end of polymerization. The first addend in equation 6 gives the main contribution to the value of the conversion, in comparison with the negligible contribution of the second addend (Figure 4). The contribution of the second addend becomes important after a certain amount of time after which it increases and becomes dominant in the second portion of the polymerization.

Table 2 The values of the parameters in equation 6 determined for MMA free-radical polymerization at different temperatures (obtained using the method of least squares)

Figure 4

MMA conversion vs time at 60 °C.

As shown in Table 2, the values of k₂ are seven to eightfold higher than the values of k₁. This means that monomers that react by autoacceleration react much faster. The apparent activation energies and pre-exponential factors for the two reaction rate constants were determined according to the Arrhenius law (Figures 5 and 6). The values of the two apparent activation energies are very close but the pre-exponential factor of autoacceleration is approximately fourfold higher than the first-order reaction.

Figure 5

The dependence of k₁ on temperature for MMA polymerization.

Figure 6

The dependence of k₂ on temperature for MMA polymerization.

The conversion (X), rate of polymerization (dX/dτ) and acceleration (d²X/dτ) vs time (τ) curves calculated by our mathematical model (Figure 7) have the same trends as the corresponding curves obtained from the experimental data (Figure 3). The model fulfills all the conditions set during model development and mentioned earlier.

Figure 7

Conversion (X), rate of polymerization (dX/dτ) and acceleration (d²X/dτ) vs time (τ) curves obtained by our mathematical model (equation 6 using parameters from Table 2, describing MMA polymerization at 60 °C).

Conclusion

The polymerization of MMA exhibits the same characteristic points as the polymerization of other lower alkylmethacrylates^{2, 3} and styrene:²⁸ the onset of acceleration (M), the maximum acceleration (P), and the maximum polymerization rate (S), proceeded by a deceleration stage with a minimum (R) and a final conversion (K). A kinetic model that has been tested earlier on styrene polymerization²⁸ was adjusted and applied to describe MMA free-radical bulk polymerization. The model is composed of two contributions, one from the first-order reaction and the other from the autoacceleration reaction. Experimental data of MMA conversion dependence on reaction time are well described by the proposed kinetic model. The proposed model provides values for the four parameters. Model parameter a is the monomer fraction polymerized by autoacceleration. Parameter τ_max corresponds to the time required to achieve the maximum rate of polymerization. Parameter k₁ is a compound constant that incorporates three constants: k_i (initiation rate constant), k_p (propagation rate constant) and k_t (termination rate constant), that is, k₁=k_p··(k_i·C_I/k_t)^1/2. It is not possible, however, to obtain separate values for these three constants. Parameter k₂ is also a compound constant that describes the polymerization of the monomer fraction by autoacceleration. It can be ascribed to the polymerization of an organized fraction of monomers.^{18, 19} This possibility fits into the description of the model discussed previously.^{2, 28, 30}