The Sol-Gel Process Simulated by Cluster-Cluster Aggregation

The pair-correlation function $g(r,t)$ and its Fourier transform, the structure factor $S(q,t)$, are computed during the gelation process of identical spherical particles using the diffusion-limited cluster-cluster aggregation model in a box. This numerical analysis shows that the time evolution of the characteristic cluster size $\xi$ exhibits a crossover close to the gel time $t_g$ which depends on the volumic fraction $c$. In this model $t_g$ tends to infinity when the box size $L$ tends to infinity. For systems of finite size, it is shown numerically that, when $t<t_g$, the wave vector $q_m$, at which $S(q,t)$ has a maximum, decreases as $S(q_m,t)^{-1/D}$, where $D$ is an apparent fractal dimension of clusters, as measured from the slo pe of $S(q,t)$ . The time evolution of the mean number of particles per cluster $\bar {n}$ is also investigated. Our numerical results are in qualitative agreement with small angle scattering experiments in several systems.