Mathematical Modeling of Bivariate Polymer Property Distributions Using 2D Probability Generating Functions, 1 – Numerical Inversion Methods

Abstract

This is the first of two papers presenting a new mathematical method for modeling bivariate distributions of polymer properties. It is based on the transformation of the infinite mass balances describing the evolution of a two‐dimensional distribution using 2D probability generating functions (pgf). A key step of this method is the inversion of the transforms. In this work, two numerical inversion methods of 2D pgfs are developed and carefully validated. The accuracy obtained with both methods was very satisfactory. The inversion formulas of both methods are simple and easy to implement. A simple copolymerization example is used to show the complete procedure from the derivation of the pgf balances to the recovery of the bivariate molecular weight distribution.magnified image

Modeling of RAFT Polymerization using Probability Generating Functions. Detailed Prediction of Full Molecular Weight Distributions and Sensitivity Analysis

“…This technique consists in transforming the mass balances of polymeric species, characterized by the number of monomer units, to the pgf domain, producing as a result balances for the pgf transform of the MWD. Discrete points of the full MWD are recovered from the pgf domain by applying an appropriate numerical inversion technique . It is important to point out that this method does not require any prior knowledge of the shape of the MWD, and can deal with complex kinetic mechanisms.…”

Graft Copolymers for Blend Compatibilization: Mathematical Modeling of the Grafting Process

“…Univariate pgf are used as transforms of the MWD of the homopolymers, while bivariate pgfs are employed for the CCD of the graft copolymer. A detailed description of the application of this technique is given elsewhere 19, 20, 22. For this system we employ the following pgf definitions.…”

“…The inversion method of univariate pgf used in this work is an adaptation of the method originally proposed by Papoulis18 for the inversion of Laplace transforms 19, 20. Bivariate pgf inversion was carried out by means of the approach recently proposed by Asteasuain and Brandolin 22. The inversion methods and the grid of molecular weights at which the MWD and CCD are to be calculated, determine the number of elements and their values of arrays z , z 1 , and z 2 , at which the pgf transforms are required.…”

“…The complete procedure for inversion to obtain the number fraction MWD, n PE ‐g‐ PS ( x , y ), from ψ 0,0 ( z 1 , z 2 ) is carefully developed in the literature 22. This inversion method recovers the probability mass fractions from their pgf transforms by means of Equation 55–58.…”

Comprehensive Mathematical Modeling of Controlled Radical Copolymerization in Tubular Reactors

“…In this work, the bidimensional MWD of the copolymer is modeled using 2-D probability generating functions (pgf) [Asteasuain and Brandolin, 2010]. Although the maximum chain lengths can, under certain conditions, be set at finite values, the number of equations is still intractable because those values are typically between 10 3 -10 5 .…”