Please wait a minute...
Chin. Phys. B, 2009, Vol. 18(2): 671-685    DOI: 10.1088/1674-1056/18/2/046
CONDENSED MATTER: STRUCTURAL, MECHANICAL, AND THERMAL PROPERTIES Prev   Next  

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)
a School of Mathematical Science, Nankai University, Tianjin 300071, Chinab 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, Chinad Department of mathematics and Statistics, McGill University, H3A2K6 Montreal, Canada
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.
Keywords:  Dendritic growth      pattern formation      interfacial waves      selection criterion  
Received:  04 July 2008      Revised:  29 August 2008      Accepted manuscript online: 
PACS:  68.70.+w (Whiskers and dendrites (growth, structure, and nonelectronic properties))  
  68.08.-p (Liquid-solid interfaces)  
  47.35.-i (Hydrodynamic waves)  
  64.70.D- (Solid-liquid transitions)  
Fund: Project supported by the Nankai University, China and in part by NSERC Grant of Canada.

Cite this article: 

Chen Yong-Qiang(陈永强), Tang Xiong-Xin(唐熊忻), and Xu Jian-Jun(徐鉴君) Three-dimensional interfacial wave theory of dendritic growth: (I). multiple variables expansion solutions 2009 Chin. Phys. B 18 671

[1] Numerical simulation on dendritic growth of Al-Cu alloy under convection based on the cellular automaton lattice Boltzmann method
Kang-Wei Wang(王康伟), Meng-Wu Wu(吴孟武), Bing-Hui Tian(田冰辉), and Shou-Mei Xiong(熊守美). Chin. Phys. B, 2022, 31(9): 098105.
[2] Phase-field study of spinodal decomposition under effect of grain boundary
Ying-Yuan Deng(邓英远), Can Guo(郭灿), Jin-Cheng Wang(王锦程), Qian Liu(刘倩), Yu-Ping Zhao(赵玉平), and Qing Yang(杨卿). Chin. Phys. B, 2021, 30(8): 088101.
[3] Applying a global pulse disturbance to eliminate spiral waves in models of cardiac muscle
Jian Gao(高见), Changgui Gu(顾长贵), and Huijie Yang(杨会杰). Chin. Phys. B, 2021, 30(7): 070501.
[4] A lattice Boltzmann-cellular automaton study on dendrite growth with melt convection in solidification of ternary alloys
Dong-Ke Sun(孙东科), Zhen-Hua Chai(柴振华), Qian Li(李谦), Guang Lin(林光). Chin. Phys. B, 2018, 27(8): 088105.
[5] Pattern formation in superdiffusion Oregonator model
Fan Feng(冯帆), Jia Yan(闫佳), Fu-Cheng Liu(刘富成), Ya-Feng He(贺亚峰). Chin. Phys. B, 2016, 25(10): 104702.
[6] Numerical simulation and analysis of complex patterns in a two-layer coupled reaction diffusion system
Li Xin-Zheng (李新政), Bai Zhan-Guo (白占国), Li Yan (李燕), He Ya-Feng (贺亚峰), Zhao Kun (赵昆). Chin. Phys. B, 2015, 24(4): 048201.
[7] Control of the patterns by using time-delayed feedback near the codimension-three Turing–Hopf–Wave bifurcations
Wang Hui-Juan (王慧娟), Wang Yong-Jie (王永杰), Ren Zhi (任芝). Chin. Phys. B, 2013, 22(12): 120503.
[8] Controlling the transition between Turing and antispiral patterns by using time-delayed-feedback
He Ya-Feng(贺亚峰), Liu Fu-Cheng(刘富成), Fan Wei-Li(范伟丽), and Dong Li-Fang(董丽芳) . Chin. Phys. B, 2012, 21(3): 034701.
[9] Spatial pattern formation of a ratio-dependent predator–prey model
Lin Wang(林望). Chin. Phys. B, 2010, 19(9): 090206.
[10] Dendritic sidebranches of a binary system with enforced flow
Li Xiang-Ming(李向明) and Wang Zi-Dong(王自东). Chin. Phys. B, 2010, 19(12): 126401.
[11] Three-dimensional interfacial wave theory of dendritic growth: (II). non-axi-symmetric global wave modes and selection criterion of pattern formation
Chen Yong-Qiang(陈永强), Tang Xiong-Xin(唐熊忻), and Xu Jian-Jun(徐鉴君). Chin. Phys. B, 2009, 18(2): 686-698.
[12] Second-order solutions for random interfacial waves in N-layer density-stratified fluid with steady uniform currents
Chen Xiao-Gang(陈小刚), Guo Zhi-Ping(郭志萍), and Song Jin-Bao(宋金宝). Chin. Phys. B, 2008, 17(9): 3387-3393.
[13] Laser-induced pattern formation from homogeneous polyisoprene solutions
Lin Dian-Yang(林殿阳), Li Ming(黎明), Wang Shu-Jie(王淑杰), and Lü Zhi-Wei(吕志伟). Chin. Phys. B, 2008, 17(6): 2156-2159.
[14] Phase-field simulation of dendritic growth in a binary alloy with thermodynamics data
Long Wen-Yuan(龙文元), Xia Chun(夏春), Xiong Bo-Wen(熊博文), and Fang Li-Gao(方立高) . Chin. Phys. B, 2008, 17(3): 1078-1083.
[15] A kind of extended Korteweg--de Vries equation and solitary wave solutions for interfacial waves in a two-fluid system
Yang Hong-Li(杨红丽), Song Jin-Bao(宋金宝), Yang Lian-Gui(杨联贵), and Liu Yong-Jun(刘永军). Chin. Phys. B, 2007, 16(12): 3589-3594.
No Suggested Reading articles found!