Gelation and Cluster Growth with Cluster-Wall Interactions

Abstract

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: $\left\{ \begin{gathered} {\kern 1pt} \dot c_j = \frac{1}{2}\,\,\sum\limits_{k\, = \,1}^{j\, - \,1} {\,\,R_{j\, - \,k,\,\,k} c_k c_{j\; - k} - c_j } \,\sum\limits_{k\, = \,1}^x {\,\,R_{jk} } c_k - \delta _{1j} \lambda c_j p,{\text{ }}j = 1,\,2,... \hfill \\ \dot p = - \lambda c_1 p \hfill \\ \end{gathered} \right.$ where λ > 0 and where c _k 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 $\sum\nolimits_{j\, = \,1}^\infty {\,jc_j } (0) = 1$ and p(0) = p ₀. We assume that $R_{jk} = [dj^\alpha k^\alpha + (j + k)(j^\alpha + k^\alpha )]/(d + j + k)$ for 0≤α≤1, which encompasses both bond linking kernels (R _jk = j ^α k ^α) and surface reaction kernels (R _jk = j ^α + k ^α). 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<α≤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 R _{= jk} , p ₀=1.