Kinetic Reaction Analysis of Gelation: First-Shell Substitution Effects in Step-Growth A2−A2+B2 Thermosets

Abstract

Purely deterministic derivations are reported regarding the internal, network chain structure that contributes to rubber elasticity in polymerizations of A2−A2+B2 monomers. The A2−A2 monomer originally contained two primary sites. Upon reaction, secondary sites and ultimately tertiary moieties form. Therefore, the A2−A2 chain link becomes a fourth-order branch node. Reaction rate constants for primary and secondary sites were assumed to be distinct. The B2 monomer was assumed to be independent of similar first-shell substitution effects. Kinetic reaction analysis based on Smoluchowski type equations yielded analytical descriptions of the molar concentration of molecules in the sol fraction as described by their chemical composition, including the number of primary sites reacted, the number of secondary sites reacted and unreacted, and the number of unreacted B moieties. Chemical reactions within the sol and between the sol and gel are explicitly described in deterministic methodology; reactions within the gel are implicitly described. Conditionally convergent properties of the moments of the population density distribution were used to predict the gel point and to describe the gel fraction. In this phase of the kinetic reaction analysis, branch node distribution dynamics are used to predict physical properties dependent on cross-link distribution dynamics, including the concentrations of elastically active junctions and strands. Because of long-range chain connectivity dependencies within the gel, several authors had stated that this intricate network topology must be based solely on stochastic reasoning. Solutions derived from expectation theory are shown to equal our deterministic solutions. The research clearly illustrates that solutions based on chemical reaction engineering can be evaluated at all levels of conversion. Furthermore, specifics of competing chemical reactions are readily incorporated into models at early stages of development.