Calculations on the Collisional Coalescence of Gas Bubbles in Solids

Abstract

A largely analytical study is made of the collisional coalescence of spherical gas-filled pores similar to those involved in the swelling of nuclear fuel materials. Physical assumptions are essentially the same as adopted by earlier workers, and major attention is given to situations where pore mobility results from surface diffusion. The calculations are based on the integrodifferential equation for the distribution in radius ψ (r, t) and emphasize problems for which similarity (or self-preserving) solutions can be obtained readily. For these cases, ψ is a power of the time t multiplied by a function of a single variable G (η), where η = rtβ. It is found immediately that the pore density varies as t2β, and the moments 〈rn〉 as t−nβ. For two important problems, one finds β = −⅕ (random-migration coalescence) and β = −1 (biased-migration coalescence). Convenient procedures lead with little computation to estimates of the magnitudes of the lower moments. The estimates are good, as comparisons with the numerical studies of Gruber show. (A large discrepancy appears in an example, but is clearly the result of an incidental arithmetical error.) Limited results are also obtained on how a simple initial distribution goes over quickly to the similarity form. A few results are given for related problems, such as the coagulation by Brownian motion of liquid droplets in colloidal suspension. It is pointed out that some problems of interest for pore coalescence appear not to lead to solutions of the similarity type.