Tree models of aggregation: multiple particle and bond-types

This paper considers aggregation processes where particles bond pairwise to form tree-structures. The tree-structures may model chemical polymerization, the precipitin reaction or cell-surface aggregates in immunology, or rouleaux formation in haematology, among other applications. We allow multiple particle and bond-types. Bonding may be directed or undirected. Branching processes specify the aggregate distribution. A theorem for multi-type branching processes gives complete criteria for the existence of an infinite aggregate (a gel). The distribution of bonding in the gel gives the elastic properties of a chemical gel. We use the binomial theorem to extend Good’s (1960) multivariable generalization of Lagrange’s expansion. The extension gives the distribution of finite aggregates (the sol) whether or not a gel is present. Generating-function relations give the mole- and weight-average relative molecular masses of the sol. When certain equireactivity conditions hold, the sol distribution is determined by a combinatorial recursion. Tree models of aggregation require enumeration of labelled trees by partition. Branching processes provide an efficient solution to the problem. Models that incorporate rings into the trees require enumeration by partition of other graphical structures. Whittle’s (1980) enumeration of pseudomultigraphs by partition is not an appropriate model of ringformation for polymer chemistry but Gordon et al . (1971) suggest another model. The model is equivalent to enumerating graphs by partition, and is still an open problem.