Morphological Changes of a Surface of Revolution due to Capillarity-Induced Surface Diffusion

TLDR

A PDE describing capillarity‑induced surface diffusion on surfaces of revolution was derived assuming isotropic surface tension and diffusion, and a stable, convergent finite‑difference scheme was introduced to solve it for arbitrary geometries, yielding solutions for tip blunting and sphere sintering. The model predicts spheroidization of cylindrical rods, blunting of field‑emission tips below a critical taper and steady‑state shapes above it, and shows that sintering of spheres follows a log‑log relation with an inverse slope ranging from ~5.5 to ~6.5 (instead of 7) that increases without bound at higher x/a.

Abstract

The partial differential equation describing morphological changes of a surface of revolution due to capillarity-induced surface diffusion has been derived under the assumption of isotropy of surface tension and surface self-diffusion coefficient. A stable, convergent finite-difference method has been developed for the general case of an arbitrary surface of revolution and solutions have been obtained for the specific problems of the blunting of field-emission tips and the sintering of spheres. Spheroidization of cylindrical rods, as well as field-emission tips with taper below a certain critical value, is predicted; for tapers above the critical value, steady-state shapes are predicted and equations describing the blunting and recession of the tips are presented. If the sintering results for spheres are represented by a plot of log x/a vs log t, it is found that the inverse slope varies from approximately 5.5 to approximately 6.5 for the range 0.05≤x/a≤0.3, in contrast with the constant value of 7 found by Kuczynski from an order-of-magnitude analysis. At higher values of x/a, n increases steadily and without bound.