Journal of Statistical Physics

, Volume 31, Issue 3, pp 519–563 | Cite as

Coagulation equations with gelation

  • E. M. Hendriks
  • M. H. Ernst
  • R. M. Ziff


Smoluchowski's equation for rapid coagulation is used to describe the kinetics of gelation, in which the coagulation kernelK ij models the bonding mechanism. For different classes of kernels we derive criteria for the occurrence of gelation, and obtain critical exponents in the pre- and postgelation stage in terms of the model parameters; we calculate bounds on the time of gelationt c , and give an exact postgelation solution for the modelK ij =(ij ω ) (ω>1/2) andK ij =a i+j (a>1). For the modelK ij =i ω +j ω (ω<1, without gelation) initial solutions are given. It is argued that the kernelK ij ij ω with ω≃1−1/d (d is dimensionality) effectively models the sol-gel transformation in polymerizing systems and approximately accounts for the effects of cross-linking and steric hindrance neglected in the classical theory of Flory and Stockmayer (Ω=1). For allΩ the exponents,t=Ω+3/2 andσ=Ω−1/2,γ=(3/2−Ω)/(Ω − 1/2) andΒ=1, characterize the size distribution, at and slightly below the gel point, under the assumption that scaling is valid.

Key words

Smoluchowski equation coagulation polymerization solgel phase transition gelation percolation critical exponents 


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Copyright information

© Plenum Publishing Corporation 1983

Authors and Affiliations

  • E. M. Hendriks
    • 1
  • M. H. Ernst
    • 1
  • R. M. Ziff
    • 2
  1. 1.Institut voor Theoretische FysicaRijksuniversiteit UtrechtThe Netherlands
  2. 2.Department of Mechanical EngineeringState University of New YorkStony Brook

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