Determining Polymer Chain Length Distributions Using Numerical Inversion of Laplace Transforms

Abstract

The kinetic expressions for a chain growth polymerization mechanism lead to an infinite set of ordinary differential equations that describe the material balance behaviour of living and dead polymer molecules of arbitrary length. One approach for solving these equations is to make a continuous variable approximation in the chain length dimension, thereby converting the ordinary differential equations to a finite set of partial differential equations. The set of partial differential equations can be solved by taking the Laplace transform with respect to the chain length, yielding ordinary differential equations in time, parameterized by the Laplace variable s. This system of ordinary differential equations can be numerically integrated over the desired reaction time with appropriate boundary conditions and the chain length distribution can be recovered by inverting the Laplace transform. Practical application of this methodology for calculating chain length distributions requires numerical solution of the ordinary differential equations and numerical inversion of Laplace transforms, since analytical inverses can be obtained for only a few simple cases. In this article, two representative algorithms for numerical inversion of Laplace transforms (Talbot's method and Weeks' method) are used in the solution of molecular weight distributions problems and guidelines are presented for their use. The analysis is illustrated using several published polymer reaction problems of varying complexity. The proposed technique shows great promise for calculating molecular weight distributions in branched systems because it does not require the stationary state hypothesis for growing polymer chains.