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

Dendritic pattern formation at the interface between liquid andsolid is a commonly observed phenomenon in crystal growth andsolidification process. The theoretical investigation of dendriticgrowth is one of the most profound and highly challenging subjectsin the broad areas of interfacial pattern formation, condensedmatter physics and materials science, preoccupying many researchersfrom various areas. Some longstanding key issues on this subjectfinally gained a breakthrough in the late of last century, via the`{Interfacial Wave} (IFW) Theory' on the ground of systematicalglobal stability analysis of the basic state of dendritic growth.The original form of the IFW theory mainly focus on theinvestigation of various axi-symmetric unsteady perturbed modessolutions around the axi-symmetric basic state of system ofdendritic growth. In reality, the system may allow variousnon-axi-symmetric, unsteady perturbed states. Whether or not thesystem of dendritic growth allows some growing non-axi-symmetricmodes? Will the stationary dendritic pattern be destroyed by some ofsuch non-axi-symmetric modes? Or, in one word, what is the stabilityproperty of the system, once the non-axi-symmetric modes can beevoked? The answers for these questions are important for the solidfoundation of IFW theory. The present work attempts to settle downthese issues and develop a three-dimensional (3D) interfacial wavetheory of dendritic growth. Our investigations verify that dendriticgrowth indeed allows a discrete set of non-axi-symmetric unstableglobal wave modes, which gives rise to a set of multiple arms spiralwaves propagating along the Ivantsov's paraboloid.

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

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.