Mathematics of Diffusion-Controlled Precipitation in the Presence of Homogeneously Distributed Sources and Sinks

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Abstract
The growth-rate equations have been derived for diffusion-controlled precipitation while the diffusing species is continually being created and destroyed throughout the material. Three mechanisms were treated: (1) Bulk diffusion from a large spherical region to a small, concentric spherical particle; (2) diffusion in a spherical region to the wall of the sphere and subsequent instantaneous diffusion to a precipitate particle on the sphere's surface; and (3) two-dimensional diffusion in a circular region to a spherical particle located at the center of the circle. Ham's method, in which the concentration is expanded in eigenfunctions of an appropriate eigenvalue problem, was extended to take into account the presence of sinks and sources. The resulting equations were solved with the aid of a digital computer. The results show that, in the short-time approximation, the radius is proportional to the time for case (1), the radius is proportional to the cube root of the time for case (2), and the radius shows a rapid rise at very short times and then a slower, almost linear increase with time for case (3). For long times, the shape of the radius-time curves is more complex.

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