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A microscopic time scale approximation to the behavior of the local slope on the faceted surface under a nonuniformity in supersaturation

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  • 35-Xx Partial Differential Equations

Abstract

A microscopic time scale approximation to the behavior of the local slope on the faceted surface under a nonuniformity in supersaturation Etsuro Yokoyama∗ Computer Center, Gakushuin University Mejiro 1-5-1, Toshima-ku Tokyo 171-8588, Japan Yoshikazu Giga† Graduate School of Mathematical Sciences, The University of Tokyo Komaba 3-8-1, Meguro-ku Tokyo 153-8914, Japan Piotr Rybka‡ Faculty of Mathematics, Informatics and Mechanics, The University of Warsaw ul. Banacha 2, 02-097 Warszawa, Poland November 28, 2007 Abstract The morphological stability of a growing faceted crystal is discussed. It has been ex- plained that the interplay between nonuniformity in supersaturation on a growing facet and anisotropy of surface kinetics derived from the lateral motion of steps leads to a faceted instability. Qualitatively speaking, as long as the nonuniformity in supersaturation on the facet is not too large, it can be compensated by a variation of step density along the facet and the faceted crystal can grow in a stable manner. The problem can be modeled as a Hamilton-Jacobi equation for height of the crystal surface. The notion of a maximal stable region of a growing facet is introduced for microscopic time scale approximation of the original Hamilton-Jacobi equation. It is shown that the maximal stable region keeps its shape, determined by profile of the surface supersaturation, with constant growth rate by studying large time behavior of solution of macroscopic time scale approximation. As a result, a quantitative criterion for the facet stability is given. Key words: facet instability, Hamilton-Jacobi equation, viscosity solution, macroscopic time scale approximation, maximal stable region ∗Email address: [email protected], phone +81-3-5992-1048, fax +81-3-5992-1018 [email protected], [email protected] 1 1 Introduction We are preoccupied with the morphological stability of a growing faceted crystal. In order to put our task in a broader prospectiv

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