Theory of compartmentalised free-radical polymerisation reactions. Part 1

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 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 n_r 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 n_r(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 n_r 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.