Equilibrium at stationary solid-liquid interface during phase-field modeling of alloy solidification
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I. INTRODUCTION
THE numerical calculation of solidification microstructure using the conventional sharp interface model involves the free boundary Stephan problem. In the model, diffusion equations for solute and heat are found in both phases separately, with boundary conditions at the interface. Therefore, the solid-liquid (SL) interface position needs to be tracked at each time-step, a time consuming process. In the phasefield model, phase state (solid, liquid) is defined by a phasefield variable as a function of position x and time t. At the interface region, the varies continuously from one value to the other value through a finite thickness of interface. By defining the phase-field variable and a corresponding governing equation of the phase field, the diffusion equations for solute and heat can be solved without tracking the SL interface. In phase-field modeling, all the governing equations, e.g., phase-field equation and mass or heat diffusion equation, can be written in unified forms in the whole system space without distinguishing the interface between mother and new phases. Therefore, the interface position tracking is unnecessary during calculation. Phase-field modeling is very efficient in simulating pattern formation during phase transformation. The evolution of phase-field distribution, the microstructural change, can be obtained by solving the governing phase-field equation. Recently, many phasefield models have been developed to study structural evolution during solidification of pure materials[1–9] and binary WON TAE KIM, Associate Professor, is with the Center for Noncrystalline Materials and Department of Physics, Chongju University, Chongju 360-764, Korea. SEONG GYOON KIM, Professor, is with the Department of Materials Science and Engineering, Kunsan National University, Kunsan 573-360, Korea. JAI SANG LEE, Researcher, formerly with the Department of Metallurgy, The University of Tokyo, is with the Plate, Rod and Welding Research Group, Technical Research Laboratories, POSCO, Pohang, Kyungbuk 790-785, Korea. TOSHIO SUZUKI, Professor, is with the Department of Metallurgy, The University of Tokyo, Tokyo 113, Japan. Manuscript submitted November 29, 1999. METALLURGICAL AND MATERIALS TRANSACTIONS A
alloys with single[10–13] or two solid phases.[14,15,16] The phase-field model,[10] is widely used. In this model, the free energy density at a point within the interface region is assumed to be a mixture of solid and liquid phases with the same composition c, where c is the concentration varying continuously from the equilibrium concentration cSe in the solid cLe in the liquid across the interface. In order to match the phase-field parameters and material parameters,[10,17] two different types of relationships were developed at two different conditions, i.e., sharp and thin interface conditions. Although the phase-field model is very efficient in describing various kinds of phase transformation, it is difficult to extend the model to a large system due to the interface thickness limit. In phase-fie
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