The phase-field method: simulation of alloy dendritic solidification during recalescence
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INTRODUCTION
IN a recent series of articles, a new method for computing solidification morphology in binary alloys has been developed for single-phase growth E~5j and coarsening}6] and for eutectic growth.t7 10] These articles extend research on pure materialsV~ 17] that employ a phase-field variable, ~b, which is a function of position and time, to describe whether the material is liquid or solid. An extra equation is developed to govern the behavior of this new variable in addition to equations for heat and solute transport. Interfaces between liquid and solid are described by smooth but highly localized changes of this variable between values that represent liquid and solid (often 0 and 1, respectively). Realistic simulations of dendritic growth in pure materials have been obtainedE~s~31that naturally include the selection of tip radius and growth rate for prescribed values of the supercooling far from the growing dendrite. For a binary alloy, Warren and Boettinger~4~have computed growth shapes and solute microsegregation patterns for two-dimensional dendrites growing into a highly supersaturated liquid using an isothermal model. The isothermal approximation necessarily ignores the release of latent heat. The present article gives an overview of the phase-field method and also gives new results for alloy dendritic growth that include a simplified model of the latent heat release. An overall heat balance in the computational domain that is subject to external cooling is used to govern the temperature, which is assumed to vary with time but not with position. Thus, recalescence behavior is obtained.
WILLIAM J. BOETTINGER, Metallurgist, and JAMES A. WARREN, Physicist, are with the Metallurgy, Division, Materials Science and Engineering Laboratory, National Institute of Standards and Technology, Gaithersburg, MD 20899. This article is based on a presentation made at the "Analysis and Modeling of Solidification" symposium as part of the 1994 Fall meeting of TMS in Rosemont, Illinois, October 2-6, 1994, under the auspices of the TMS Solidification Committee. METALLURGICAL AND MATERIALS TRANSACTIONS A
While the essential ingredients of phase-field methods have existed for many years, recent progress is directly related to the increased availability of computational resources. This increase has also lead to renewed attempts to perform solidification calculations using more traditional sharp interface methods as well. These methods require a solution to the difficult moving boundary problem and the application of boundary conditions at the interface. Techniques that treat the moving boundary problem with Green's functions,t241 front tracking,~25~or adaptive finite elementst261have produced simulations of dendritic growth for pure materials that show much of the same detail obtained with the phase-field method. The phase-field method, while computationally intense, requires only the application of finite-difference techniques to the relevant equations. The location of the interface is obtained from the solution for
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