# Radially Symmetric Phase Growth Controlled by Diffusion

F. C. Frank

## Abstract

Particular solutions of the diffusion equation, with radial symmetry, in three and in two dimensions, found originally by Rieck, represent the diffusion field around a spherical or cylindrical new phase, growing from a negligible initial radius in an initially uniform medium, maintaining equilibrium conditions at the growing surface. The resulting diffusion field is most simply described by saying that the radial gradient is the same as that of the corresponding potential problem with fixed boundaries, multiplied by a factor exp (-r$^{2}$/4Dt), where r is radius, D diffusivity, t time. The result is applied to phase growth controlled by the diffusion of heat, solute, or both together. It differs appreciably from the static approximation unless the supercooling, or the supersaturation and the solubility, are small.