Theoretical calculations on Smoluchowski kinetics: Perikinetic reactions in highly aggregated systems

Numerical solutions to the Smoluchowski equations for fast, perikinetic, aggregation have been obtained for highly aggregated dispersions. The temporal dependence of the moments of the distributions both for a size-dependent, Brownian, reaction kernel, over a range of duster fractal dimensions, and also for a constant kernel, was calculated. In the initial stages of the aggregation, the rate of total number decay is dependent on both fractal dimension and average cluster mass. At long aggregation times, the reaction rates reach limiting values, which are dependent solely on the fractal dimension of the clusters. Two theoretical relationships were derived for these two regimes: in the small-aggregate limit (m less than or equal to 10), the overall, perikinetic, rate varies as d(f)-(2.224)log M(0)(-1) at long aggregation times (m greater than or equal to 100), the rate varies as 1/d(f)(2), in the large-aggregate limit. Typically, for diffusion-limited aggregation with d(f) = 1.72, the rate of total number decay has a limiting value (for m > 100) some 20% greater than with a constant aggregation kernel. The average cluster mass, at which the crossover from a size-dependent to a size-independent reaction rate takes place, is given by M(0)(1) = 4.6 + 2.5d(f). For clusters with d(f) = 1.72, this transition takes place when the average radius of the aggregates is 3.6 times that of the constituent particles.