CHAIN-LENGTH-DEPENDENT TERMINATION RATE-PROCESSES IN FREE-RADICAL POLYMERIZATIONS .1. THEORY

A complete set of rate equations describing the kinetics of free-radical polymerization is deduced, in which termination rate coefficients are allowed to depend on chain length. The rate equations are applicable to bulk, solution, and heterogeneous (e.g., emulsion) polymerization systems. They incorporate the following kinetic processes: initiation, propagation, transfer, chain-length-dependent termination, and, for emulsion systems, exit and reentry (i.e., aqueous-phase kinetic processes are taken into account). It is shown that, despite their apparent complexity, the full set of population balance equations can, to a good approximation, be reduced to a single first-order differential equation. This equation and that for the overall rate of polymerization form a pair of coupled differential equations that can be solved easily and are thus suitable for routine modeling of experiment. The key parameters required in the model are the rate coefficients for transfer and propagation and the diffusion coefficient of the monomer as a function of the polymer weight fraction (conversion), this variation being used in expressions for the dependence of the termination rate coefficient on the length of a growing chain. In accord with previous experiment and theory (Adams, M. E.; Russell, G. T.; Casey, B. S.; Napper, D. H.; Gilbert, R. G.; Sangster, D. F. Macromolecules 1990, 23, 4624), the present theory indicates that, at intermediate conversions, termination is dominated by interactions between short chains formed by transfer and entangled long chains. In the regime in which propagation is diffusion-controlled, most termination events involve two highly entangled chains, whose ends move by the "reaction-diffusion" process (Russell, G. T.; Napper, D. H.; Gilbert, R. G. Macromolecules 1988, 21, 2133). The mathematical treatments of this paper are of very general applicability and should inter alia be useful in addressing practical problems such as minimizing the residual monomer content of polymer products.