Exact kinetics of the sol-gel transition

Abstract

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.

Turbulent inertial gelation and acoustic quasi-gelation of submicron aerosol particles

“…This happens when any higher moment (n 2) in series (2.5) diverges [5], for instance when M 2 diverges (tends to infinity) as time approaches some finite value t g termed the gelation point (Refs. [5,[8][9][10]). As indicated in the Introduction, kernel (1.2) represents a positive homogeneous function of degree λ = 4/3.…”

“…Gelation had been shown to occur in polymerization processes (Drake [3], Ziff [4], Hendriks et al [5], Rotstein et al [6]) and during formation of needlelike aerosol particles in external electric fields (Lushnikov and Piskunov [7], Lushnikov [8]). In both cases the kernel is proportional to the product of masses of coagulating particles,…”

Kinetic percolation

“…This is the well-known, phasetransition-like behavior that occurs when a sol becomes a gel, i.e. at gelation [8][9][10] . This few in number, large in mass population are the superaggregates with their hybrid morphology displayed in Figs.…”

“…Although this phase-transition-like behavior can result from the kinetics of simple binary collisions 8,9 , it has been rigorously connected to a thermodynamic functional 10 . A complete description of gelation would incorporate the relevant aspects of both descriptions.…”

Fractal aggregates in the atmosphere

“…The coagulation in the system with the kernel = was analyzed by Lushnikov [1978Lushnikov [ , 2004Lushnikov [ , 2005 who showed that a gel should form from coagulating sol after a finite interval of time. Experimentally this process was observed by Lushnikov et al [1990 who supposed that the dipoledipole interaction of FA is responsible for this phenomenon.…”