Microscopic theory of gelation: distribution of cyclic species and theory of critical point

Abstract

A matrix method is introduced to derive the distribution functions of cyclic species in network formation. The new method presents a concise solution for the A_g–R–B_ƒ–g model which otherwise requires a harder calculation. A critical point is given as a solution of λ= 1, where λ is an eigenvalue of a matrix. It is shown that the distribution function of chains in a network corresponds to a generalization of the Flory distribution for the linear model. The asymptotic behaviour of chain fluctuation near the critical point is very different from that of cluster sizes; i.e. the former remains finite at the critical point, while the latter diverges.

An explicit relation is introduced between the critical extent of the reaction D_C and the inverse of the initial monomer concentration 1/C: D_c=D_C0+Γ′(24) where Γ′ is a measure of the number fraction of rings and a function of 1/C, with 0 denoting a limiting case without rings. By means of the power series expansion of Γ′, we derive an approximate solution whose general behaviour is in accordance with Wile's experiments