Shape stability of a two-dimensional nucleus

Abstract

Stability of the lateral interface of a disk shaped nucleus is reviewed when such a nucleus is growing parallel to the basal plane by surface diffusion of ad-atoms and their eventual attachment at the interface. Two physically distinct systems have been considered: (a) a conservative one in which the nucleus grows without addition or removal of matter from the system (b) a non-conservative one, in which the nucleus is growing in the presence of a vapour source. The system is idealised by considering a single nucleus surrounded by an infinite matrix of ad-atoms and quantities like edge-free energy, interface attachment coefficient as isotropic.Behaviour of a small fluctuation represented by circular harmonics has been determined in the usual way of Mullins and Sekerka, by solving the steady state diffusion equation (Laplace equation for conservative case, Helmholtz equation for non-conservative system) with appropriate boundary conditions.For the conservative system, the results are similar to the 3-dimensional case, a perturbation will grow as soon as the nucleus size is greater than a limiting value. The effect of non infinite interface attachment coefficient is to push this radius at which instability sets in at a higher value. For the non-conservative system the situation is radically different: the instability is governed by the parameter R∗/xs(R∗ critical nucleation radius, xn mean diffusion distance) and the interface attachment efficient β. When the interface attachment mechanism is too slow (β smaller than a critical value βc) the nucleus is stable, but when it becomes sufficiently rapid (β . βc) instability may occur provided that R∗/xs is smaller than a critical value [R∗/xs] e; in this case amplification of the perturbation is possible only when the size of the nucleus lies between two limiting values. The interface attachment mechanism appears to be an important stabilizing factor.The influence of temperature and impurity concentration at the interface on stability are discussed. In connection with experimental evidence, a model is developed for the instability of thin discs of ice crystals growing on the surface of slightly undercooled water.