Theory of Compartmentalised Free-Radical Polymerisation Reactions

Abstract

Explicit analytic solutions are given for nr, 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 which fulfils the following conditions: (1) radicals enter the reaction loci from a contiguous external phase at a constant rate; (2) 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; (3) radicals lost by diffusion from loci to the external phase are not available for re-initiation; (4) the volume of the reaction loci is uniform and does not increase significantly as polymerisation proceeds; (5) no nucleation of new loci takes place; (6) no reduction in the total number of reaction loci occurs, e.g., through agglomeration.

The general expression obtained for nr is [graphic omitted] where N is the total number of reaction loci per unit volume of reaction system, σ is the average rate of entry of radicals into a single locus, and k characterises the rate of loss of radical activity by first-order processes. This result is obtained by first deriving the following generating function for the locus populations:[graphic omitted] where ξ is an auxiliary variable. The implications of the result for nr(t) for (1) the approach to the steady state, (2) the average number of radicals per locus, and (3) the rate of polymerisation are discussed. It appears that the nr form a time-dependent Poisson distribution with respect to the r, and that the parameter of the distribution at any instant is the average number of radicals per locus at that instant.