Bicomponent aggregation with composition-dependent rates and the approach to well-mixed state

We consider the problem of aggregative mixing of components from a theoretical standpoint. We formulate equations for the evolution of the bivariate size distribution, the classical size distribution, and the compositional distribution, and introduce a segregation index to quantify the degree segregation and blending in the population. We consider composition-dependent kernels and examine their effect on the degree of blending of components by aggregation. To systematically study the effect of the kernel, we introduce a new kernel that allows us to adjust the strength of cross-aggregation over that of self-aggregation. Kernels that promote cross-aggregation lead to more efficient blending of components, as judged by the rapid decrease of the segregation index, while kernels that promote self-aggregation inhibit blending and may even result in segregation during the earlier stages of aggregation. In all cases, however, the segregation index ultimately scales inversely with the mean particle mass. Thus, regardless of kernel details, the segregation index at long times becomes arbitrarily small and follows the same scaling as with kernels that are independent of composition.