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

1328 I. Loginova et al. / Acta Materialia 51 (2003) 1327–1339tions of the migrating a/g are rapid enough theinterface will be essentially in thermodynamicequilibrium, the so-called local equilibriumhypothesis. In that case the transition to a massivemode of transformation would occur when theundercooling is so large that the composition ofthe parent g falls on the a/a g phase boundaryof the binary Fe–C phase diagram. On the otherhand it has been claimed that the massive transformationmay occur far inside the a g twophasefield. The subject and the different viewpointswere recently discussed in an extensivereview by Hillert [1]. He emphasized that the interactionsbetween solutes and the migrating phaseinterface are essential in order to understand thetransition to the massive mode of transformation.It is thus necessary to analyze the so-called solutedrag effect in detail. Hillert and Sundman [2] werefirst to show by simulation that the solute drageffect during solidification of binary alloys predictsa transition to a partitionless mode of solidificationat high undercoolings. Their approach was to solvea steady state diffusion equation over the interfacialregion and evaluate the part of the availabledriving force that is dissipated by diffusion. Diffusionin the parent liquid was treated analyticallyby a Zener–Hillert type of approach. Ågren et al.[3,4] replaced the diffusion profile inside the phaseinterface with a single representative compositionand were able to model the transition betweenWidmanstätten growth of a into g and a partitionlessmode of transformation at high supersaturations.Over the last decade the phase-field approachhas been tremendously successful in predictingmicrostructures during solidification [5] and solidstate transformations [6]. In this approach the interfacebetween two phases is treated as a region offinite width having a gradual variation of the differentstate variables, i.e. the diffuse interface model.So far, neither the actual properties of the interfacenor the thermodynamic and kinetic properties ofthe alloys have been emphasized. The attention hasmainly been drawn to the capability of the methodto predict very realistic microstructures and thetreatment of the interface has been regarded as amathematical “trick” to solve the difficult movingboundary problem.Nevertheless, the physical pictures behind thesolute drag modeling and the phase-field approachare very similar. Of course, the similarity is usuallyless evident because the interface thickness usedin the numerical phase-field calculations has beenmuch too large to have any physical significance.However, recently Ahmad et al. [7] compared thephase field model with various solute-drag modelsand they found that under steady-state conditionsthe two approaches are indeed very similar and thephase-field calculations will exhibit both solutetrapping (massive growth) and solute-drag effect.Their results thus suggest that the phase fieldapproach is capable of treating the transition tomassive transformation as well as solute drageffects provided that the interface is givenrealistic properties.The purpose of the present report is to apply thephase-field method to the g →a transformation inbinary Fe–C and demonstrate that a transition tomassive transformation is predicted during isothermalgrowth if reasonable properties are givento the a/g phase interface. The predictions will becompared with a newly developed technique tomodel solute drag and we shall investigate underwhat conditions the two approaches are qualitativelyor even quantitatively consistent. No comparisonwill be made with other solute-drag treatmentse.g. sharp-interface models.The main purpose is thus to study the situationat the phase interface and all calculations will bemade for a one-dimensional geometry although thephase-field formulation is readily extended to thefull three-dimensional geometry.2. Phase field formulation of the g →atransformation in Fe–CThe phase-field formulation of the isobarothermalg →a transformation is based on the Gibbsenergy functional:G G m(f,u C ,T) e2V m 2 f2d (1)where G m denotes the Gibbs energy per mole ofsubstitutional atom and V m is the molar volume per