(217dg) Statistical Thermodynamics of Irreversible Gelation

Gelation refers to the formation of an infinite chain that represents a finite fraction of the mass of the system but an infinitesimal fraction of the total number of clusters. In the Smoluchowski treatment of the problem, the onset of gelation coincides with the divergence of certain moments, rendering post-gel solutions impossible without ad-hoc corrections of the coagulation equation. We present here an exact solution of coagulation for finite systems based on the ensemble theory of populations, which we have recently formulated. We construct the discrete tree that represents binary reactions between a population of M monomers starting as discrete units and ending up as a single chain of molecular weight N. We show that the emergence of a giant cluster (gel-phase) is a formal phase transition akin to condensation that produces two coexisting populations, the "sol" and the "gel," both of which are characterized by the same temperature and pressure. We present exact solutions for the pre- and post gel stages and confirm the results by Monte Carlo simulation. We further the stability of power-law kernels using standard thermodynamic stability criteria and confirm that gelation occurs if the degree of homogeneity of the kernel exceeds 1, in agreement with the results of Ziff [1], who reached the same conclusion by studying the stability of the Smoluchowski equation. Morevover, we obtain the complete phase diagram of power-law kernels including the position of the gel point and the evolution of the gel fraction. Finally, we discuss the Flory [2] and Stockmayer [3] models and show that only Flory's model is consistent with binary reactions between clusters.

[1] R. M. Ziff, E. M. Hendriks, and M. H. Ernst. Critical properties for gelation: A kinetic approach. Phys. Rev. Lett., 49:593-595, Aug 1982. doi: 10.1103/PhysRevLett.49.593.

[2] Paul J. Flory. Molecular size distribution in three dimensional polymers. II. Trifunctional branching units. Journal of the American Chemical Society, 63(11):3091-1941. doi: 10.1021/ja01856a062.

[3] W. H. Stockmayer. Theory of molecular size distribution and gel formation in branched-chain polymers. The Journal of Chemical Physics, 11(2):45-55, 1943. doi: 10.1063/1.1723803.