A free dendritic growth model accommodating curved phase boundaries and high peclet number conditions
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INTRODUCTION
THE growth of an unconstrained dendrite in an undercooled melt has been a major subject in the theoretical arena of solidification research because of interest in understanding the formation of equiaxed dendritic grains in an ingot and the rapid solidification of a highly undercooled melt. Early models were developed for pure metals and dealt with the steady-state, shape-preserving growth of needlelike crystals under isothermal boundary conditions,[1,2] the curvature effects that require nonisothermal boundary conditions,[3] and the prediction of the dendrite shape as part of the solution.[4] Later models were extended for binary alloys. These account for the coupling of the solutal and thermal effects,[5–12] predict the tip radius from marginal stability criteria,[10–16] and account for kinetic effects at the interface[11,12] that cause nonequilibrium solute partitioning.[17,18,19] The models developed by Lipton et al.[8–11] and by Boettinger et al.[12] have been especially well received because of relative simplicity and the ability to predict the tip radius by using an experimentally verified marginal stability criterion.[13,14] The earliest of these models, referred to as the LGK model,[8,9] is, however, developed with the assumption that the solid-liquid interface is at local equilibrium and as such applies only to free dendritic growth at small undercoolings. This limitation prompted the development of the later models[10,11,12] to account for the large deviations from local equilibrium interfacial conditions that occur in rapid solidification. All of these models concern the steady-state growth of a paraboloidal dendrite tip and thus avoid the complexity encountered in more rigorous treatments. The models presented by Lipton et al.[10] and Trivedi et al.[11] consider only the curvature, constitutional, and thermal undercoolings. A marginal stability criterion appropriate for high Peclet numbers (i.e., high tip velocities)[14] is used to predict the tip radius in both models. The latter ALFRED G. DiVENUTI is former Graduate Student, Department of Mechanical Engineering, Tufts University, Medford, MA 02155. TEIICHI ANDO, Associate Professor, is with the Department of Mechanical, Industrial and Manufacturing Engineering, Northeastern University, Boston, MA 02115. Manuscript submitted March 20, 1998. METALLURGICAL AND MATERIALS TRANSACTIONS A
model[11] also uses an interfacial solute trapping model presented by Aziz[17] that relates the Peclet number to a nonequilibrium solute partition coefficient. It is apparent, however, that assuming a bath undercooling consisting only of the three components does not permit the interface to have nonlocal equilibrium compositions even though these models may mathematically converge when solved at high Peclet numbers. The model developed by Boettinger et al., referred to as the BCT model,[12] solves the latter problem by taking into account the thermodynamic driving force for the advancement of the solid-liquid interface.[20] Another important feature of th
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