DETERMINING POLYMER-CHAIN LENGTH DISTRIBUTIONS USING NUMERICAL INVERSION OF LAPLACE TRANSFORMS

The kinetic expressions for a chain growth polymerization mechanism le ad to an infinite set of ordinary differential equations that describe the material balance behaviour of living and dead polymer molecules o f arbitrary length. One approach for solving these equations is to mak e a continuous variable approximation in the chain length dimension, t hereby converting the ordinary differential equations to a finite set of partial differential equations. The set of partial differential equ ations can be solved by taking the Laplace transform with respect to t he chain length, yielding ordinary differential equations in time, par ameterized by the Laplace variable s. This system of ordinary differen tial equations can be numerically integrated over the desired reaction time with appropriate boundary conditions and the chain length distri bution can be recovered by inverting the Laplace transform. Practical application of this methodology for calculating chain length distribut ions requires numerical solution of the ordinary differential equation s and numerical inversion of Laplace transforms, since analytical inve rses can be obtained for only a few simple cases. In this article, two representative algorithms for numerical inversion of Laplace transfor ms (Talbot's method and Weeks' method) are used in the solution of mol ecular 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 branc hed systems because it does not require the stationary state hypothesi s for growing polymer chains.