中国物理B ›› 2009, Vol. 18 ›› Issue (2): 671-685.doi: 10.1088/1674-1056/18/2/046

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Three-dimensional interfacial wave theory of dendritic growth: (I). multiple variables expansion solutions

陈永强1, 徐鉴君2, 唐熊忻3   

  1. (1)School of Mathematical Science, Nankai University, Tianjin 300071, China;Department of Fundamental Subject, Tianjin Institute of Urban Construction, Tianjin 300384, China; (2)School of Mathematical Science, Nankai University, Tianjin 300071, China;Department of mathematics and Statistics, McGill University, H3A2K6 Montreal, Canada; (3)School of Mathematical Science, Nankai University, Tianjin 300071, China;School of materials Science and Engineering, University of Science and Technology Beiji
  • 收稿日期:2008-07-04 修回日期:2008-08-29 出版日期:2009-02-20 发布日期:2009-02-20
  • 基金资助:
    Project supported by the Nankai University, China and in part by NSERC Grant of Canada.

Three-dimensional interfacial wave theory of dendritic growth: (I). multiple variables expansion solutions

Chen Yong-Qiang(陈永强)a)b), Tang Xiong-Xin(唐熊忻)a)c), and Xu Jian-Jun(徐鉴君)a)d)†   

  1. a School of Mathematical Science, Nankai University, Tianjin 300071, China; b Department of Fundamental Subject, Tianjin Institute of Urban Construction, Tianjin 300384, China; c School of materials Science and Engineering, University of Science and Technology Beijing, Beijing 100083, China; d Department of mathematics and Statistics, McGill University, H3A2K6 Montreal, Canada
  • Received:2008-07-04 Revised:2008-08-29 Online:2009-02-20 Published:2009-02-20
  • Supported by:
    Project supported by the Nankai University, China and in part by NSERC Grant of Canada.

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摘要： Dendritic pattern formation at the interface between liquid and solid is a commonly observed phenomenon in crystal growth and solidification process. The theoretical investigation of dendritic growth is one of the most profound and highly challenging subjects in the broad areas of interfacial pattern formation, condensed matter physics and materials science, preoccupying many researchers from various areas. Some longstanding key issues on this subject finally gained a breakthrough in the late of last century, via the `{Interfacial Wave} (IFW) Theory' on the ground of systematical global stability analysis of the basic state of dendritic growth. The original form of the IFW theory mainly focus on the investigation of various axi-symmetric unsteady perturbed modes solutions around the axi-symmetric basic state of system of dendritic growth. In reality, the system may allow various non-axi-symmetric, unsteady perturbed states. Whether or not the system of dendritic growth allows some growing non-axi-symmetric modes? Will the stationary dendritic pattern be destroyed by some of such non-axi-symmetric modes? Or, in one word, what is the stability property of the system, once the non-axi-symmetric modes can be evoked? The answers for these questions are important for the solid foundation of IFW theory. The present work attempts to settle down these issues and develop a three-dimensional (3D) interfacial wave theory of dendritic growth. Our investigations verify that dendritic growth indeed allows a discrete set of non-axi-symmetric unstable global wave modes, which gives rise to a set of multiple arms spiral waves propagating along the Ivantsov's paraboloid.

关键词: Dendritic growth, pattern formation, interfacial waves, selection criterion

Abstract: Dendritic pattern formation at the interface between liquid and solid is a commonly observed phenomenon in crystal growth and solidification process. The theoretical investigation of dendritic growth is one of the most profound and highly challenging subjects in the broad areas of interfacial pattern formation, condensed matter physics and materials science, preoccupying many researchers from various areas. Some longstanding key issues on this subject finally gained a breakthrough in the late of last century, via the `Interfacial Wave (IFW) Theory' on the ground of systematical global stability analysis of the basic state of dendritic growth. The original form of the IFW theory mainly focus on the investigation of various axi-symmetric unsteady perturbed modes solutions around the axi-symmetric basic state of system of dendritic growth. In reality, the system may allow various non-axi-symmetric, unsteady perturbed states. Whether or not the system of dendritic growth allows some growing non-axi-symmetric modes? Will the stationary dendritic pattern be destroyed by some of such non-axi-symmetric modes? Or, in one word, what is the stability property of the system, once the non-axi-symmetric modes can be evoked? The answers for these questions are important for the solid foundation of IFW theory. The present work attempts to settle down these issues and develop a three-dimensional (3D) interfacial wave theory of dendritic growth. Our investigations verify that dendritic growth indeed allows a discrete set of non-axi-symmetric unstable global wave modes, which gives rise to a set of multiple arms spiral waves propagating along the Ivantsov's paraboloid.

Key words: Dendritic growth, pattern formation, interfacial waves, selection criterion

中图分类号:  (Whiskers and dendrites (growth, structure, and nonelectronic properties))

  • 68.70.+w
68.08.-p (Liquid-solid interfaces) 47.35.-i (Hydrodynamic waves) 64.70.D- (Solid-liquid transitions)