Dynamics of molecular weight distributions for polymer scission

Source: American Institute of Chemical Engineers (AIChE) Journal
Publication date: 2001-11-01
Arrival time: 2001-12-11

The dynamics of molecular weight distributions (MWDs) for polymer degradation is of interest to various applications. The time evolutions of MWDs can be determined by solving the governing population balance equations, which are generally solved by moment techniques wherein the initial distribution is represented by a gamma distribution. The evolution of MWD is determined by the time dependence of the gamma distribution parameters. The population balance equations (PBEs) can also be solved numerically by converting them to partial differential equations (PDEs). The degradation rate coefficient in the PBE depends on the molecular weight x as (x - x^sub 0^)^sup lambda^, or as a quadratic polynomial in x The solutions obtained with the moment technique, which are inaccurate for certain cases, are compared with the solutions determined by solving the PDEs. The utility of the numerical scheme is also discussed for cases where the initial distribution cannot be represented satisfactorily by a gamma distribution.

Introduction

The study of the degradation of polymers is an important aspect of polymer science and engineering, the applications of which include polymer recycling (Miller, 1994) and characterization (Flynn and Florin, 1985). Practical problems like low heat-transfer rates and high viscosity of melting polymers have shifted the focus of degradation of polymers by pyrolysis to the degradation of polymers in solution, which promises to be a better technology for controlled degradation of polymers (Sato et al., 1990).

A polymer is a mixture of molecules with varying sizes and, therefore, a molecular weight distribution (MWD) is needed to describe the polymer. MWDs can be characterized by the moments of the distribution of the molecules. A finite number of such molecules implies that the moments are the sums over the molecules in the polymer that constitute the distribution. However, sums are difficult to evaluate compared to integrals and, therefore, continuous distribution kinetics is generally employed to analyze the time evolution of MWDs of reacting polymers (McCoy and Madras, 2001). Continuous distribution models have been previously used to model polymer degradation (Aris and Gavalas, 1966; McCoy and Wang, 1994; Madras and McCoy, 1998). Rate coefficients are measured for the degradation of polymers by examining the time evolution of the MWDs of the reacting polymers. (Madras et al., 1997; Wang et al., 1995). The rate coefficients of the degradation reactions depend on the molecular chain length and thus on the molecular weight x. For small conversions, the rate coefficients can be assumed to be independent of the molecular weight of the reacting polymer (McCoy and Wang, 1994). However, this assumption fails when the change in the molecular weight of the reacting mixture is significant (Madras et al., 1997). McCoy and Madras (1998) assume the molecular weight dependence of the rate coefficient as x^sup lambda^, so that the rate coefficient can be constant, linear, or quadratic depending on the value of lambda. A more general form of the dependence of the rate coefficient on molecular weight dependence is k^sub 0^ + k^sub 1^x + k^sub 2^x^sup 2^ (Madras et al., 1997).

For polymer degradation, the population balance equations (PBEs) govern the behavior of the MWDs of the reacting polymer (Aris and Gavalas, 1966; Ramkrishna, 1985). Depending on the type of the polymer and operating conditions, polymers fragment to yield a random or parabolic distribution of the binary daughter products or a specific distribution of products (McCoy and Madras, 1997). The stoichiometric kernel in the PBE incorporates the information on the distributions of products of the fragmentation process. Ternary or higher scission events are usually not considered as they can be described as multiple sequential binary scissions (McCoy and Wang, 1994).

Several articles have discussed the mathematical solutions for the governing population balance equations for a variety of special cases (Aris and Gavalas, 1966; McCoy and Wang, 1994; Wang et al., 1995; McCoy and Madras, 1998; Vigil and Ziff, 1989). McCoy and and Madras (1998) present the solutions to the population balance equations in the context of similarity solutions or self-preserving solutions (Ramkrishna, 1985) by using the moment solutions techniques for a general rate dependence of the form x^sup lambda^.

The governing PBEs, in the form of integro-differential equations, are generally solved by moment techniques by converting them to ordinary differential equations for the molecular-weight moments. The initial MWD is assumed to be a gamma distribution, which is a versatile representation of distributions (Abramowitz and Stegun, 1968). The gamma distribution fit can be constructed by knowing the zeroth, first, and second moments. The moments are then expressed in terms of the generalized gamma distribution parameters alpha(t) and beta(t) (McCoy and Madras, 1998). Therefore, the time evolution of these parameters provides the dynamic behavior of the MWDs of the reacting polymers.

The gamma distribution is actually the first term in the generalized expansion of the MWD function in orthogonal Laguerre polynomials with higher-order terms related to the third and the higher moments of the distribution. Higherorder moments are hence neglected in considering a gamma distribution. The moment solutions are, therefore, approximations to the original solutions that can be made more accurate by considering further terms in the expansions of the orthogonal polynomials.

In the case of quadratic polynomial dependency of the rate coefficients (Madras et al., 1997), other simplifying assumptions are made to solve the ordinary differential equations that result from the application of the moment operations to the population balance equations.

The objective of the current investigation is to formulate a general numerical solution for the evolution of MWDs for a random scission stoichiometric kernel such as for the case where the degradation products are randomly distributed. Continuous distribution kinetics provide the governing PBEs, and these equations are converted to partial differential equations (PDEs). The molecular weight dependency is taken as k^sub s^(x) = (x - x^sub 0^)^sup lambda^, which provides the generalized rate coefficient and reduces to the rate coefficient of k^sub d^x^sup lambda^ assumed by Madras and McCoy (1998) for x^sub 0^ = 0. Solutions are also obtained for the case of a rate coefficient of the form k^sub 0^ + k^sub 1^x + k^sub 1^x^sup 2^. The results obtained by solving the PDEs are compared with the appropriate moment solutions. The solution obtained by the method of moments is an approximation and, as shown in this study, this approximation is not valid in some cases. We also investigate the time evolution of MWDs for cases wherein the initial distribution cannot be represented by a single gamma distribution.

Acknowledgments

The author thank the Department of Science and Technology, India for financial support.

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Manuscript received October 17, 2000, and revision received May 4, 2001.

Khamir Mehta and Giridhar Madras

Dept. of Chemical Engineering, Indian Institute of Science, Bangalore 560 012 India

Correspondence concerning this article should be addressed to G. Madras.

Copyright American Institute of Chemical Engineers Nov 2001

Publication date: 2001-11-01

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