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September  2004, 3(3): 527-543. doi: 10.3934/cpaa.2004.3.527

Asymptotic theory for disc-like crystal growth (II): interfacial instability and pattern formation at early stage of growth

Jian-Jun Xu 1, and  Junichiro Shimizu 2,

1. 

Department of Mathematics, McGill University, Montreal QC H3A 2K6
2. 

National Space Development Agency of Japan (NASDA), Tsukuba Space Center, Tsukuba

Received  March 2003 Revised  February 2004 Published  June 2004

This paper is a continuation of the paper ([1] ), which is called paper (I) afterward. In the present paper, which we shall call paper (II), we study the interfacial instability property of the side interface of a growing disc-like crystal at the early stage of growth by using the approach developed in the interfacial wave (IFW) theory of dendritic growth. Our analysis show that the system allows two types of unstable modes over the side-interface: (1). The axi-symmetric $(m=0)$ modes. The most dangerous axi-symmetric mode is the base mode $A_0$, which is responsible for formation of the axi-symmetric pattern over the side-interface, anti-symmetric about the central plane of the disc; (2). The non-axi-symmetric modes, which are responsible for non-axi-symmetric pattern formation around the edge of the disc. The growth rates of these non-axi-symmetric modes are much smaller than the growth rate of the base mode $A_0$. During the course of disc growth, the unstable $A_0$-mode merges first. It leads to the formation of anti-symmetric pattern about the central plane over the side-interface. Following the onset of unstable base mode $A_0$, a set of non-axi-symmetric growing modes also appear. However, due to the smallness of growth rate of these unstable modes, the non-axi-symmetric pattern around the edge of the disc becomes observable, only after a sufficiently long time. Our theoretical predictions are in good agreement with the available experimental data.
Keywords: Interfacial Instability, quantization condition, surface tension, unstable eigen-modes, kinetic attachment., disc-like crystal growth.
Mathematics Subject Classification: 78M35, 82C24, 82C2.
Citation: Jian-Jun Xu, Junichiro Shimizu. Asymptotic theory for disc-like crystal growth (II): interfacial instability and pattern formation at early stage of growth. Communications on Pure & Applied Analysis, 2004, 3 (3) : 527-543. doi: 10.3934/cpaa.2004.3.527
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