Stability of the Shape of a Solid Cylinder Growing in a Diffusion Field

The stability of the shape of an infinite cylinder undergoing radial growth controlled by diffusion is studied by a method originated by Mullins and Sekerka (MS). It is found that the circular cross section of a cylinder is stable when its radius is less than and unstable when its radius is greater than a certain radius Rc. This result is analogous to the MS result that a sphere is stable below and unstable above a certain radius Rc, which is seven times the critical radius R* of nucleation theory. However, in the present case, the ratio Rc/R* is not equal to seven, but is a function of S=(c∞-cs)/(C-cs), where c∞, cs, and C are the concentrations of the solute at infinity, at the surface of the cylinder, and in the precipitate, respectively. The case of perturbations in the radius along the length of the cylinder is also treated. Potential application of the result to such problems as the growth of branches on dendrites is discussed.