Sol-gel transition in a source-enhanced coagulating system

Abstract

This paper considers the time evolution of disperse systems in which binary coagulation and a source of fresh particles govern the temporal changes to the particle mass spectra. The source is assumed to produce fresh particles at a constant rate. The Smoluchowski equation describing the time evolution of the particle mass spectrum is solved exactly for the coagulation kernel proportional to the product of masses of two coalescing particles. It is shown that after a critical time tc a gel forms in the system and the sol spectrum becomes an algebraic function of the particle mass at t=tc. It begins to shrink after the critical time due to the mass loss supporting the growth of the gel mass. The pre- and post-critical behavior of the mass spectrum and its integral characteristics (total particle number and mass concentrations) are investigated for the source productivity I(g) dropping down algebraically with the particle mass g as I(g) proportional variant g -gamma. The critical particle mass spectrum is proved to be a universal function of (it drops down as g -5/2) if the third moment of I(g) is finite (gamma>4). Otherwise (3<gamma< or =4)this and other critical exponents begin to depend on . Still the mass spectrum remains self-similar, i.e., it depends on a combination of g and t. At smaller gamma the gelation process is shown to begin at t=0. All critical characteristics of the particle mass spectrum are determined for this case.