Mixing of components in two-component aggregation

The problem of binary component aggregation with kernels that are independent of composition is considered. The bivariate distribution as the product of two distributions is studied, one that refers to the size of the aggregates, and one that describes the distribution of the component of interest (solute), and obtain the governing equations for all three. The distribution of solute within aggregates of size v has a steady-state solution, that is independent of the size distribution: it is a Gaussian function whose mean and variance are both proportional to the aggregate size v. To quantify the degree of blending, the sum-square X2, of the deviation of the amount solute from its mean, is studied. Two cases are identified for which X ² is constant during aggregation: (a) "partially mixed" seeds regardless of kernel; and (b) sum-type kernels regardless of seed distribution. Simulations confirm the results for these two cases, and further indicate that in the general case, X ² is nearly constant. The degree of mixing is determined solely by the initial distribution of components, but does not depend on the kernel. Optimum initial conditions that minimize the time required to reach a desired level of homogeneity between components are identified.