Molecular weight distributions in free-radical polymerizations. 1. Model development and implications for data interpretation

Many properties (e.g., film formation) of a polymer depend on the full molecular weight distribution (MWD) as well as on the averages of this distribution. Moreover, the MWD contains the complete kinetic history of a polymerization system, and hence an experimental MWD can yield considerable mechanistic information. Complete equations are developed which enable the MWD to be calculated for a wide range of straight-chain free-radical polymerizations. These equations also give insight into the qualitative form of the MWD, thus providing a method for extracting kinetic and mechanistic information from experimental MWDs. Methods are also described which enable numerical solutions to be obtained for the resulting nonlinear integrodifferential equations. The scheme takes into account initiation, transfer, propagation, and termination, allowing all of these to be dependent upon the degrees of polymerization of the chains involved. The model is applicable to bulk and solution polymerizations and to compartmentalized systems (conventional and micro- and miniemulsion polymerizations); in the latter case, phase-transfer events (radical entry into and exit from latex particles, and the aqueous phase kinetics of the various radical species) are specifically included. The dependence of the termination rate coefficient upon the lengths of both participating chains is especially important in systems where termination is kinetically significant. Approximate analytical solutions are also developed for a number of cases of interest. These show that the instantaneous number MWD at high molecular weights is a single exponential - even when termination occurs to a significant extent - which is a direct consequence of the chain-length dependence of termination rate coefficients, i.e., most termination events involve at least one very short chain. When the concentration of radicals in the system is low, the "decay constant" of the exponential part of the MWD is given by the ratio of transfer to propagation rate coefficients. In addition, the cumulative MWD frequently has the same behavior as the instantaneous MWD. This has the important implication that considerable mechanistic information can be obtained from size exclusion chromatography data by plotting ln(number MWD) against molecular weight. Such plots can reveal, for example, the termination mechanism that controls the MWD, values of transfer constants, and nucleation mechanisms in emulsion polymerization. Illustrative calculations are carried out for a number of systems. These verify the conclusions from the approximate analytic solutions and show also that the MWD in bulk or solution polymerizations at low conversions and high initiator concentrations is sensitive to the mode of termination (combination or disproportionation)