Analytical results on the polymerisation random graph model

TL;DR

This paper models step-growth polymerisation as a time-continuous random graph process with variable degree bounds, providing exact analytical results for phase transition timing and component size distribution, validated by simulations.

Contribution

It introduces a novel random graph model for polymerisation with degree bounds, deriving closed-form expressions for phase transition and component sizes, and offering efficient computation methods.

Findings

Closed-form expressions for phase transition timing.

Exact size distribution up to convolution power.

Validation through Monte Carlo simulations.

Abstract

The step-growth polymerisation of a mixture of arbitrary-functional monomers is viewed as a time-continuos random graph process with degree bounds that are not necessarily the same for different vertices. The sequence of degree bounds acts as the only input parameter of the model. This parameter entirely defines the timing of the phase transition. Moreover, the size distribution of connected components features a rich temporal dynamics that includes: switching between exponential and algebraic asymptotes and acquiring oscillations. The results regarding the phase transition and the expected size of a connected component are obtained in a closed form. An exact expression for the size distribution is resolved up to the convolution power and is computable in subquadratic time. The theoretical results are illustrated on a few special cases, including a comparison with Monte Carlo simulations.