Structure function and fractal dimension of diffusion-limited colloidal aggregates

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 df of the DLCA aggregates in the flocculation limit presents a square root type of dependence with the initial colloidal concentration. The df 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^df-3e^-(r/ξ)ᵃ, where $A,$ $a,$ and {ξ} 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 df 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 df calculated from the $g(r)$ agrees well with that obtained from individual clusters. However, caution should be observed to extract a df from the corresponding $S(q).$ Our results indicate that for finite concentrations a df systematically larger than the true value is obtained from such analysis.