Exact kinetics of sol-gel transition in a coagulating mixture.
The formation of a gel in a two component disperse system wherein binary coagulation governs the temporal changes to particle composition spectra is studied under the assumption that the coagulation kernel is proportional to m1n2+m2n1, with m,n being the numbers of monomers of the first and the second component in the coalescing pair of particles. This model is shown to reveal the sol-gel transition, i.e., the formation of one giant cluster with the mass comparable to the total mass of the whole system. This paper reports on the exact solution of this model within the Marcus-Lushnikov stochastic scheme. The evolution equation for the generating functional of the probability to find in the system a given set of occupation numbers (the numbers of particles containing m and n monomers of each component) at time t is formulated and solved exactly. The expression for the particle composition spectrum is derived and analyzed in the thermodynamic limit. It is shown that after a critical time a giant single particle (the gel) appears. The time evolution of its composition is found. Special attention is given to the transition point, where the gel is appearing. The time dependencies of the gel composition, the number concentration, and the second moments of the particle composition spectrum are found.
Lushnikov
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Department of Physical Sciences, University of Helsinki, P.O. Box 64, FIN-00014, Helsinki, Finland. alex.lushnikov@helsinki.fi