Phase-field modeling of diffusion controlled phase ... - KTH Mechanics

Acta mater. 49 (2001) 573–581www.elsevier.com/locate/actamatPHASE-FIELD SIMULATIONS OF NON-ISOTHERMAL BINARYALLOY SOLIDIFICATIONI. LOGINOVA 1 *, G. AMBERG 1 and J. ÅGREN 21Department of Mechanics, KTH, 100 44 Stockholm, Sweden and 2 Department of Materials Science andEngineering, KTH, 100 44 Stockholm, Sweden( Received 27 June 2000; received in revised form 18 October 2000; accepted 23 October 2000 )Abstract—A phase-field method for two-dimensional simulations of binary alloy solidification is studied.Phase-field equations that involve both temperature and solute redistribution are formulated. The equationsare solved using the finite element method with triangular elements on unstructured meshes, which are adaptedto the solution. Dendritic growth into a supersaturated melt is simulated for two temperature regimes: (a)the temperature is prescribed on the boundary of the computational domain; and (b) the heat is extractedthrough the domain boundary at a constant rate. In the former regime the solute redistribution is comparedwith the one given by an isothermal model. In the latter case the influence of the size of the computationaldomain and of the heat extraction rate on dendritic structure is investigated. It is shown that at high coolingrates the supersaturation is replaced by thermal undercooling as the driving force for growth. © 2001 ActaMaterialia Inc. Published by Elsevier Science Ltd. All rights reserved.Keywords: Phase-field; Kinetics; Diffusion1. INTRODUCTIONOver the last 10 years the phase-field method hasbeen extensively used for simulations of dendriticgrowth. In the phase-field method a new variablef(x, y, t) is introduced to indicate the physical stateof the system at each point. f takes on constant valuesin solid and liquid and changes steeply but smoothlyover a thin transition layer that plays the role of theclassical sharp interface. The governing equationcoupled with modified transport equations are appliedin all of space without distinguishing between thephases. This permits simulations of growing morphologieswithout explicitly tracking the phase boundaries.Solidification of a binary mixture has been studiedby Caginalp et al. [5, 8]. It is shown that the phasefieldequations reduce to the traditional sharp interfacemodels in the limit when the thickness of theinterfacial region d vanishes. At the same time computationsdemonstrate that the phase-field methodsproduce an interface close to the sharp interface problemeven for relatively large d.Despite the great success in predicting qualitativelyrealistic microstructures, the phase-field method has* To whom all correspondence should be addressed. Fax:46-8-796-9850.E-mail address: irina@mech.kth.se (I. Loginova)only been applied to very simple systems. Typicallyheat flow during solidification of pure substances orisothermal diffusion in binary alloys obeying idealsolution thermodynamics have been studied. However,from a technological point of view, it would bevaluable to perform simulations on real alloys, whichare usually multicomponent and have complex thermodynamicinteractions.Warren and Boettinger [1] derived a phase-fieldmodel for isothermal solidification of a binary alloy,applying constant diffusivities within the solid andliquid phases, and performed two-dimensional simulationsof dendritic growth into a highly supersaturatedliquid. This model has been explored in several papers,for example, [9–12]. A desirable extension of themodel is to study the effect of heat flow due to releaseof latent heat. However, the numerical implementationis not trivial since the temperature and soluteevolution occurs on completely different time-scales.A simplified approach was proposed in Ref. [4],where the spatial variation of the temperature is neglectedand the heat equation is replaced by a heat balanceof an imposed heat extraction rate and the latentheat release rate.The present report is part of a project where theultimate goal is to apply the phase-field method tosimulate processing of real alloys. As a first step, thephase-field formulation [1] is applied, with the majordifference being that simultaneous heat flow and dif-1359-6454/01/$20.00 © 2001 Acta Materialia Inc. Published by Elsevier Science Ltd. All rights reserved.PII: S13 59-6454(00)00360-8