Exact kinetics of the sol-gel transition

The formation of a gel in a disperse system wherein binary coagulation alone governs the temporal changes of particle mass spectra is studied under the assumption that the coagulation kernel is proportional to the product of masses of coalescing particles. This model is known 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 for a finite total mass of the coagulating system. The evolution equation for the generating functional defining all properties of coagulating systems is solved exactly for this particular kernel. The final output is the exact expression for the single-particle mass spectrum as a function of time. The analysis of the spectrum in the thermodynamic limit shows that after a critical time a giant single particle (the gel) appears. Although the concentration of this giant gel particle is zero in the thermodynamic limit, it actively interacts with smaller particles "eating" them and thus growing in mass. Special attention is given to the transition point, where the gel is appearing. It is demonstrated that the sol-gel transition reminds the second-order phase transition. The time dependencies of the gel mass, the number concentration, and the second moment of the particle mass spectrum are found.