Structure factor scaling in aggregating systems

We study the structure factor of evolving two-phase systems such as aggregating colloids and spinodally decomposing fluids. We interpret the total structure factor as described well by the product of cluster-cluster and single-cluster structure factors, each with their own characteristic length, the mean cluster nearest-neighbor separation, and the cluster size, respectively. Both length scales are thus relevant to the total structure factor. For systems with moderate to strong cluster-cluster correlations, this product causes an apparent peak in the structure factor. For compact clusters, i.e., clusters with a fractal dimension equal to the spatial dimension, this peak obeys the experimentally observed scaling law. However, for fractal clusters the two length scales evolve differently, hence scaling cannot occur. Despite this, our simulations show an apparent scaling when the system is dense enough so that the two length scales are comparable in magnitude. When this occurs, each length scale eliminates the individual effect of the other from the total structure factor leaving a peak. These results explain both the lack of scaling early and the scaling observed latter in experiments on aggregating colloids. An important conclusion is that the position of this peak $q_m$ does not represent a true length scale of the system.