Molecular weight distribution (MWD) by method of moments

Simulation of the MWD with time by the method of moments

Introduction

Pyrolytic thermal degradation is generally a two-phase process, with the higher molecular weight species remaining in the condensed phase and the lower molecular weight products evaporating. In the continuous distribution kinetics approach, the molar mass of the polymer is assumed to be a continuous variable and the polymer is characterised by a distribution function of this continuous variable. The degradation and aggregation rate coefficients are considered to be dependent on the molecular chain length. The temporal nature of the molecular weight distribution (MWD) is described by time dependent moments.

Theoretical model

To represent the molar mass distribution of the initial polymer, a gamma distribution is used, which reduces to the Gaussian or Poisson distribution as special cases. The molar mass being a random variable x follows a Gamma distribution of parameters α and β (strictly positive), if its probability density function which represents its gamma distribution can be put in the form :

With 𝑦= (𝑥-𝑥0)/𝛽 and x0 the constant molar mass of the small molecule in the polymer MWD.

Model assumptions

o Random degradation reactions are first order and all rate coefficients are independent of MW.

o The MWD of the reaction mixtures can be described by a gamma distribution whose parameters depend on the residence time, and are therefore different from those of the starting polymer. These parameters can be determined by moment methods.

o All products of the binary degradation processes are dissolved in solution, and no cross-linking or repolymerisation reactions occur in the experiments.

The moment solutions correspond to the molar concentration (moment zero), mass concentration (moment one) and variance (moment two) of the MWD of the polymer. The number average molecular weight (Mn), weight average molecular weight (Mw) and polydispersity index of the polymer can be calculated as 𝑝(1)/ 𝑝(0), 𝑝(2)/𝑝(1) and Mw/Mn, respectively.