Relation of total volume to total surface area for a suspension of particles undergoing growth

Abstract

For a suspension of spherical particles with radii initially distributed normally or lognormally, and growing without nucleation at rates proportional to their surface areas, expressions are derived for the sum (R) of the radii, the total surface area (A) and the total volume (V) of the particles, in terms of a parameter θ=(kt+₀)/σ, where k is the growth rate constant, t the time, ₀ the initial mean value, and σ the standard deviation of the radii; for V in the lognormal distribution, an additional parameter γ, the skewness coefficient, is also involved. For the plausible value ₀/σ= 4, and for at least a 50-fold increase in V, it is shown that V∝Aⁿ where n= 1·475, 1·472, 1·468, 1·463, and 1·453 (error in V within 0·72, 0·87, 1·03, 1·29, and 1·74% respectively) for γ= 0 (i.e., normal distribution), 0·778, 1·750, 3·263, and 6·185 respectively. This is to be compared with n= 1·5, valid for a monodisperse suspension.