Dendritic growth with fluid flow in pure materials
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I. INTRODUCTION
DENDRITIC growth is important because it is the basic microstructural pattern in solidified metals. The pattern selected during solidification of a pure material depends on the existing thermal field during freezing. Once this pattern is set, it is difficult to change in the solid state without substantial effort, e.g., through mechanical deformation and heat treatment. The evolution of dendritic microstructures is reasonably well understood for pure materials growing into undercooled melts under purely diffusive conditions, i.e., when fluid flow is absent. The tip of the dendrite approximates a paraboloid of revolution. Ivantsov[1] determined the solution for the thermal field surrounding an isothermal dendrite, neglecting surface tension, and determined a relation between undercooling and the (constant) tip velocity, Vtip , and tip radius, tip , given by ⌬ ⫽ I (Pev) where ⌬ is the dimensionless undercooling (Tm ⫺ T⬁)/(Lf /cp), with Tm as the equilibrium melting point and T⬁ as the far-field temperature, scaled by the characteristic temperature, Lf /cp , with Lf as the latent heat of fusion and cp as the specific heat. The term Pev is the Pe´clet number based on the tip radius, defined as Pev ⫽ tip Vtip /␣, and ␣ is the thermal diffusivity. The function I is a combination of exponentials and error integrals, whose form is well known and not important for the current discussion. Because the tip radius and velocity appear only as a product, the diffusion solution does not uniquely determine the shape. The pattern-selection problem is resolved by including surface tension and its anisotropy in the boundary condition for the temperature of the dendrite and by relaxing the assumption that the shape is known.[2,3] This body of theory is known as “microscopic solvability.” In two-dimensions, it gives a correction to the shape of the interface near the tip, converging to the Ivantsov solution far away from the tip where curvature becomes negligible. The theory provides a second relation for the dendrite-tip velocity and radius, given by * ⫽ 2d0␣ / 2tipVtip, where * is known as the JUN-HO JEONG, Research Scientist, is with the Nano Mechanisms Group, Korea Institute of Machinery and Materials, Daejon 305-343, Korea. JONATHAN A. DANTZIG, Professor, Department of Mechanical and Industrial Engineering, and NIGEL GOLDENFELD, Professor, Department of Physics, are with the University of Illinois, Urbana, IL 61801. Contact e-mail: [email protected] This article is based on a presentation given in the symposium “Fundamentals of Solidification,” which occurred at the TMS Fall meeting in Indianapolis, Indiana, November 4-8, 2001, under the auspices of the TMS Solidification Committee. METALLURGICAL AND MATERIALS TRANSACTIONS A
selection constant, and d0 is the capillary length, a material constant. The problem is somewhat more complicated in three-dimensions, where corrections to the Ivantsov shape are large. Brener[4] provides a treatment of this problem. Recent numerical calculations using the phase-field me
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