Markovian approach to nonlinear polymer formation: Free-radical crosslinking copolymerization

Abstract

A Markovian model is used to extend the Flory/Stockmayer gelation theory to nonequilibrium reaction systems, by taking free-radical crosslinking copolymerization of vinyl and divinyl monomer as an example. Free-radical polymerizations are kinetically controlled; therefore, each primary polymer molecule experiences a different history of crosslinked structure formation. By assuming that the primary chains with identical birth time conform to the same chain connection probabilities, the nonlinear structural development can be viewed as a system in which the primary chains formed at different birth times are combined into nonlinear polymers in accordance with the first-order Markov chain statistics. According to the present Markovian model, the weight-average chain length, P̄w is given by a matrix formula, P̄w = Wp(E — Q)−1 l where Wp is the row vector that concerns the weight contribution of a primary chain, E is a unit matrix, Q is the transition matrix representing the chain connection statistics, and I is a column vector whose elements are all unity. For an equilibrium system, Wp = P̄wp (weight-average chain length of the primary chains), E = 1, Q = ρP̄wp (ρ is the crosslinking density), and I = 1; therefore, the present formula reduces to the Flory/Stockmayer equation, P̄w = P̄wp/(1 − ρP̄wp). The criterion for the onset of gelation is simply stated as a point at which the largest eigenvalue of the transition matrix Q reaches unity, i.e., det(E − Q) = 0. The present Markovian approach elucidates important characteristics of the kinetically controlled network formation, and provides greater insight into nonequilibrium gelling systems.