Theory of compartmentalised free-radical polymerisation reactions. Part 2

Abstract

Explicit analytic solutions are given for n_r, the number of reaction loci per unit volume containing r radicals, as a function of time, t, for the case of a seeded emulsion polymerisation in which the only significant processes which result in loss of radical activity from reaction loci are kinetically of first order with respect to the concentration of radicals in the loci. The analysis given in this paper is a development of that given in an earlier paper. The result given previously has been generalised to include cases where the parameters σ and k, which characterise the rate of entry of radicals into loci and the rate of loss of radical activity from loci respectively, are time-dependent.

The general expression obtained for n_r is n_r(t)=N{θ(t)}^r//_r!e^–θ(t), where N is the total number of reaction loci per unit volume of reaction system, and θ(t) is a function of t defined by the differential equation dθ(t)}//_dt=σ(t)–k(t)·θ(t) and the initial condition θ(0)= 0. This result is obtained by first deriving the following generating function for the locus populations: ψ(ξ, t)=N e^{(ξ–1)θ(t)} where ξ is an auxiliary variable. Expressions are also derived for the average number of radicals per locus, the total radical population, and the rate of conversion of monomer to polymer. It appears that, whatever the nature of the time-dependence of σ and k, the n_r always form a time-dependent Poisson distribution with respect to the r. The function θ(t) is always the parameter of the distribution.

The general method developed for treating reaction systems of the type described is applied to a few cases of possible practical interest, including, inter alia, that where σ is an exponentially-decaying function of time and k is constant.