Random graph theory of gelation is a mathematical theory for sol–gel processes. The theory is a collection of results that generalise the Flory–Stockmayer theory, and allow identification of the gel point, gel fraction, size distribution of polymers, molar mass distribution and other characteristics for a set of many polymerising monomers carrying arbitrary numbers and types of reactive functional groups. The theory builds upon the notion of the random graph, introduced by mathematicians Paul Erdős and Alfréd Rényi, and independently by Edgar Gilbert in the late 1950s, as well as on the generalisation of this concept known as the random graph with a fixed degree sequence. The theory has been originally developed to explain step-growth polymerisation, and adaptations to other types of polymerisation now exist. Along with providing theoretical results the theory is also constructive. It indicates that the graph-like structures resulting from polymerisation can be sampled with an algorithm using the configuration model, which makes these structures available for further examination with computer experiments.
Premises and degree distribution At a given point of time, degree distribution u ( n ) {\displaystyle u(n)} , is the probability that a randomly chosen monomer has n {\displaystyle n} connected neighbours. The central idea of the random graph theory of gelation is that a cross-linked or branched polymer can be studied separately at two levels: 1) monomer reaction kinetics that predicts u ( n ) {\displaystyle u(n)} and 2) random graph with a given degree distribution. The advantage of such a decoupling is that the approach allows one to study the monomer kinetics with relatively simple rate equations, and then deduce the degree distribution serving as input for a random graph model. In several cases the aforementioned rate equations have a known analytical solution.
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