On the concentration dependence of the cluster fractal dimension in colloidal aggregation

We have undertaken the task to calculate, by means of extensive numerical simulations and by different procedures, the cluster fractal dimension (d(f)) of colloidal aggregates at different initial colloid concentrations. Our first approach consists in obtaining d(f) from the slope of the log-log plots of the radius of gyration versus size of all the clusters formed during the aggregation time. In this way, for diffusion-limited colloidal aggregation, we have found a square root type of increase of the fractal dimension with concentration, from its zero-concentration value: d(f) = d(f)(0) + a phi(beta), with d(f)(0) = 1.80 +/- 0.01, a = 0.91 +/- 0.03 and beta = 0.51 +/- 0.02, and where phi is the volume fraction of the colloidal particles. In our second procedure, we get the d(f) via the particle-particle correlation function g(cluster)(r) and the structure function S-cluster(q) of individual clusters. We first show that the stretched exponential law g(cluster)(r) = Ar(df-3)e(-(r/xi)a) gives an excellent fit to the cutoff of the g(r). Here, A, a and xi are parameters characteristic of each of the clusters. From the corresponding fits we then obtain the cluster fractal dimension. In the case of the structure function S-cluster (q), using its Fourier transform relation with g(cluster)(r) and introducing the stretched exponential law, it is exhibited that at high q values it presents a length scale for which it is linear in a log-log plot versus q, and the value of the d(f) extracted from this plot coincides with the d(f) of the stretched exponential law. The concentration dependence of this new estimate of d(f), using the correlation functions for individual clusters, agrees perfectly well with that from the radius of gyration versus size. It is however shown that the structure factor S(q) of the whole system (related to the normalized scattering intensity) is not the correct function to use when trying to obtain a cluster fractal dimension in concentrated suspensions. The log-log plot of S(q) vs. q proportions a value higher than the true value. Nevertheless, it is also shown that the true value can be obtained from the initial slope of the particle-particle correlation function g(r), of the whole system. A recipe is given on how to obtain approximately this g(r) from a knowledge of the S(q), up to a certain maximum q value.