Random graph theory of gelation
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 late 1950's, 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 formula_1, is the probability that a randomly chosen monomer has formula_2 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 formula_1 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. One type of functional groups. In the case of step-growth polymerisation of  monomers carrying functional groups of the same type (so called formula_4 polymerisation) the degree distribution is given by: formula_5 where formula_6 is bond conversion, formula_7 is the average functionality, and formula_8 is the initial fractions of monomers of functionality formula_9. In the later expression unit reaction rate is assumed without loss of generality. According to the theory, the system is in the gel state when formula_10, where the gelation conversion is formula_11. Analytical expression for average molecular weight and molar mass distribution are known too. When more complex reaction kinetics are involved, for example chemical substitution, side reactions or degradation , one may still apply the theory by computing formula_12 using numerical integration. In which case, formula_13 signifies that the system is in the gel state at time t (or in the sol state when the inequality sign is flipped). Two types of functional groups. When monomers with two types of functional groups A and B undergo step growth polymerisation by virtue of a reaction between A and B groups, a similar analytical results are known. See the table on the right for several examples. In this case, formula_14 is the fraction of initial monomers with formula_15 groups A and formula_16  groups B. Suppose that A is the group that is depleted first. Random graph theory states that gelation takes place when formula_10, where the gelation conversion is formula_18 and formula_19. Molecular size distribution, the molecular weight averages, and the distribution of gyration radii have known formal analytical expressions. When degree distribution formula_20, giving the fraction of monomers in the network with formula_21 neighbours connected via A group and formula_22 connected via B group at time formula_23 is solved numerically, the gel state is detected when formula_24, where formula_25 and formula_26. Generalisations. Known generalisations include monomers with an arbitrary number of  functional group types, crosslinking polymerisation, and complex reaction networks.