Gelation and cluster growth with cluster-wall interactions

Horacio G. Rotstein, Amy Novick-Cohen, Rina Tannenbaum

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13 Scopus citations


Metallic cluster growth within a reactive polymer matrix is modeled by augmenting coagulation equations to include the influence of side reactions of metal atoms with the polymer matrix: (Equation Presented) where λ > 0 and where ck denotes the concentration of the kth cluster and p denotes the concentration of reactive sites available within the polymer matrix for reaction with metallic atoms. The initial conditions are required to be non-negative and satisfy ∑j=1 = 1 and p(0) = p0. We assume that Rjk = [djαkα + (j + k)(jα + kα)]/(d + j + k) for 0≤ α ≤ 1, which encompasses both bond linking kernels (Rjk = jαkα) and surface reaction kernels (Rjk = jα + kα). Our analytical and numerical results indicate that the side reactions delay gelation in some cases and inhibit gelation in others. We provide numerical evidence that gelation occurs for the classical coagulation equations (λ = 0) with the bond linking kernel (d → ∞) for 1/2 < ∝ ≤ 1. We examine the relative fraction of metal atoms, which coagulate compared to those which interact with the polymer matrix, and demonstrate in particular a linear dependence on λ 1 in the limiting case Rjk = jk, p0 = 1.

Original languageEnglish (US)
Pages (from-to)119-143
Number of pages25
JournalJournal of Statistical Physics
Issue number1-2
StatePublished - Jan 1998
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Statistical and Nonlinear Physics
  • Mathematical Physics


  • Cluster growth
  • Coagulation equations
  • Gelation
  • Infinite-dimensional dynamical systems
  • Metallic clusters in a polymer matrix
  • Smoluchowski equations with side reactions


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