Mean molecular size distributions and the sol–gel transition in finite, polycondensing systems

Abstract

A new theory is presented for the molecular size distributions and the sol–gel transition in a model, nonlinear, polycondensing system. The theory differs from the earlier theories of Flory and of Stockmayer in that the molecular size distributions are calculated as averages over the ensemble of all states of the finite system rather than as the ’’most probable’’ distribution in the limit as the size of the system goes to infinity. This modification is necessary because the mean molecular size distribution for a finite system is not described adequately over the range of large molecular sizes, at any extent of reaction, by the correctly normalized most probable distribution. This is clearly indicated by the failure of the most probable distribution to describe the gel explicitly. Numerical evaluations of the mean and the most probable distributions are shown to agree over the range of smallest molecular sizes at all extents of reaction. Beyond the gel point a second peak, clearly descriptive of the gel, is present over the range of larger molecular sizes in the mean molecular weight distribution. This peak is well‐separated from the one describing the sol, and reflects thereby the disparate molecular sizes distinctive of the coexisting sol and gel. It is noted that in view of the analogy between gelation theory and Mayer vapor condensation theory, which was originally observed by Mayer and by Stockmayer and was recently established more firmly by Cohen et al., the difficulties in condensation theory can reasonably be expected to disappear if that theory is similarily reformulated.