Steady needle growth with 3-D anisotropic surface tension

Abstract

The effect of the anisotropic interfacial energy on dendritic growth has been an important subject, and has preoccupied many researchers in the field of materials science and condensed matter physics. The present paper is dedicated to the study of the effect of full 3-D anisotropic surface tension on the steady state solution of dendritic growth. We obtain the analytical form of the first order approximation solution in the regular asymptotic expansion around the Ivantsov's needle growth solution, which extends the steady needle growth solution of the system with isotropic surface tension obtained The solution is expanded in the general Laguerre series in any finite region around the needle-tip, and it is also expanded in a power series in the far field behind the tip. Both solutions are then numerically matched in the intermediate region . Based on this global valid solution, the dependence of Peclet number Pe and the interface's morphology on the anisotropy parameter of surface tension as well as other physical parameters involved are determined. On the basis of this global valid solution, we explore the effect of the anisotropy parameter on the Peclet number of growth, as well as the morphology of the interface. ½ ÁÒØÖÓ Ù Ø ÓÒIn studying free dendritic growth, the effect of anisotropic interfacial energy on pattern formation and selection bas been an important subject, which has preoccupied many researchers in the field of materials science and condensed matter physics for quite a long time now [1][2][3][4]. During the past decades, most theoretical investigations on this subject were focused on the system of 2-D dendritic growth with anisotropy of surface tension, or the system of 3-D axi-symmetric dendritic growth with axial anisotropy of surface tension, neglecting the three dimensionality of the anisotropy of surface tension [5][6][7][8][9][10][11][12][13][14][15]. With these efforts, the effect of the anisotropy of surface tension on the morphology of the steady state, interfacial stability mechanisms and selection of dendritic pattern formation for 2-D systems are now well explored analytically.A realistic system of dendritic growth is always three dimensional. The anisotropy of surface tension at the interface is also three dimensional. Up to now in literature, there are very few analytical works on dendritic growth in systems with full 3-D anisotropic surface tension. As a consequence, the effects of three dimensionality of anisotropy on the steady state solution as well as the stability mechanisms of dendritic growth are still unclear.In order to explore the effect of 3-D anisotropic surface tension on the stability mechanisms and the selec-

Mechanism of Dendritic Branching

“…[7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24] These efforts included precepts and hypotheses on dendritic interfacial physics as varied as the following: Take, for example, item (a), the concept of maximum velocity, which would occur, hypothetically, for a paraboloidal dendrite, the solid-liquid interface of which has interfacial energy, i.e., capillarity. As all interfaces have excess free energy, or surface tension, this idea could, conceivably, have provided a reasonable supposition about the operating state of dendrites: namely, that dendrites grow steadily when they achieve their maximum velocity, as allowed by the thermal conduction field and the level of supercooling specified in the melt.…”

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

“…However, in order to understand the selection mechanism and essence of pattern formation of dendritic growth, one is only interested in the behaviour of the system in a finite region behind the dendrite-tip away from the root, not near the root. It is just in that region, that for a given (ξ, η, θ), as ε → 0 the basic state of dendritic growth can be expanded in the following regular perturbation expansion (RPE): [14,22,24,25]…”

“…[22]- [24], and Ref. [25]). Especially, for the system with isotropic surface tension, the temperature in the liquid is…”

Three-dimensional interfacial wave theory of dendritic growth: (II). non-axi-symmetric global wave modes and selection criterion of pattern formation

“…From the far field condition, we derive D 1 = 0, and we also have D 3 = D 3 . The connection condition between the coefficients {D 1 , D 3 = D 3 } in sector (S 1 ) and {D 1 = 0, D 3 } in sector (S 2 ) is to be derived by matching the outer solution (8) with the inner solution near the singular point ζ c .…”

“…However, when the parameter of anisotropy of surface tension α 4 is large, the ratio Pe 0 Pe may be quite large (refer to Refs. [8], [9] and [10]. Therefore, it is not surprising, that there is the case for dendritic growth, in which the ε * -criterion may be significantly different from the σ * -criterion.…”