Morphological stability of two-dimensional nucleus

Abstract

Morphological stability of a disk-shaped nucleus growing on a substrate by surface diffusion of adsorbed atoms is analysed for two physically distinct cases: (a) a conservative system, in which a nucleus grows without addition or removal of mass from the system; (b) a non-conservative system: nucleus growing on a substrate in presence of a vapour source. The system is idealized by considering such a nucleus surrounded by a supersaturated infinite matrix of diffusing adsorbed atoms. Following the method of Mullins & Sekerka (1963), to examine the stability of the shape of the nucleus a periodic small fluctuation represented by circular harmonics is introduced into the disk shape. The flux of adsorbed atoms at any point of the nucleus interface is determined by solving the Laplace equation (conservative system) and the Helmholtz equation (non-conservative system) together with appropriate boundary conditions. For the conservative system, it is found that a perturbation will grow monotonically as soon as the nucleus size is bigger than a certain limiting value. For the non-conservative system, the result is totally different. As long as the ratioR*/xs(whereR* is the critical nucleus size for nucleation andxsthe mean diffusion distance of an adsorbed atom on the substrate) is smaller than a limiting value (R*/xs)c, a selective amplification of a perturbation is possible only when the nucleus sizeRlies between two valuesRc1andRc2. AtRc1the perturbation begins to grow and atRc2it begins to attenuate. The two systems are compared and discussed; it is seen that a two-dimensional nucleus growing in a non-conservative system is inherently more stable and shape-preserving than it is in a conservative system.