Structure function and fractal dimension of diffusion-limited colloidal aggregates

Abstract

On a three-dimensional lattice and at different concentrations we perform extensive numerical simulations of diffusion-limited colloidal aggregation (DLCA). In a previous work, we showed that the fractal dimension ${d}_{f}$ of the DLCA aggregates in the flocculation limit presents a square root type of dependence with the initial colloidal concentration. The ${d}_{f}$ was obtained from the slope of a standard log-log plot of the number of particles versus size of the formed aggregates. In this work we confirm the concentration dependency using the particle-particle correlation function $g(r)$ and the structure function $S(q)$ of individual aggregates. We demonstrate that the ${g(r)=Ar}^{{d}_{f}\ensuremath{-}3}{e}^{\ensuremath{-}(r/\ensuremath{\xi}{)}^{a}},$ where $A,$ $a,$ and \ensuremath{\xi} are parameters characteristic of the aggregates, and $a>1.$ This stretched exponential law gives an excellent fit to the cutoff of the $g(r).$ The structure function reveals the ${d}_{f}$ from the slope of a log-lot plot of $S(q)$ versus $q$ for high $q$ values. We also analyze $g(r)$ and $S(q),$ at different times during the reaction, for the whole aggregating system composed of many clusters of different sizes. We observe that the ${d}_{f}$ calculated from the $g(r)$ agrees well with that obtained from individual clusters. However, caution should be observed to extract a ${d}_{f}$ from the corresponding $S(q).$ Our results indicate that for finite concentrations a ${d}_{f}$ systematically larger than the true value is obtained from such analysis.