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

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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

574 LOGINOVA et al.: PHASE-FIELD SIMULATIONS OF BINARY ALLOY SOLIDIFICATIONfusion are taken into account. The paper is organizedas follows: Section 1 gives an overview of the modelderivation; numerical aspects are presented in Section2; in the next two sections results of truly non-isothermalcalculations of dendritic growth for two temperatureregimes are presented and discussed.J B and the heat flux J e are given by the linear lawsof irreversible thermodynamicsJ B L A BB δS Lδx A Be δSB δe(6)2. MATHEMATICAL MODELIn this section the phase-field model given in Ref.[1] is considered with modifications accounting fortemporal and spatial evolution of the temperaturefield. The formulation is based on an entropy functionalS s(f,x B ,e) 22 |f|2d (1)where the thermodynamic entropy density s is a functionof the phase-field variable f varying smoothlybetween 0 in the solid and 1 in the liquid, the molefraction x B of a solute B in solvent A and the internalenergy density e, and is a spatial region occupiedby a mixture.Anisotropy is included in the system because thephase change kinetics depends upon the orientationof the interface¯h ¯(1 g cos kb) (2)where ¯ is related to the surface energy s and interfacethickness, g is the magnitude of anisotropy in thesurface energy, k specifies the mode number and theexpression b arctan(f y /f x ) gives an approximationof the angle between the interface normal and theorientation of the crystal lattice.The evolution of the non-conserved phase-fieldvariable is governed byḟ M fδSδf(3)where M f is related to the interfacial mobility. Theevolution of x B and e is governed by the normal conservationlawsJ e L A eB δS Lδx A ee δSB δe(7)L A BB and L A ee are related to the inter-diffusionalmobility of B and A and the heat conduction, respectively.The coefficient L A Be L A eB describes the crosseffects between heat flow and diffusion and will beneglected. The second law requires M f , L A BB and L A eeto be positive. It should be emphasized that the gradients(δS/δx B ) and (δS/δe) are to be evaluated isothermallyand under fixed composition, respectively.The variational derivatives in equations (6) and (7)are given byδSδx B ∂s∂x B m Bm ATV m(8)δSδe ∂s∂e 1 T(9)where T is the temperature and the quantity on theright-hand side of equation (8) is known as the interdiffusionpotential in binary substitutional alloys. Thechemical potentials m A and m B under the assumptionof an ideal mixture have the following formm A °m A (f, T) RT ln(1x B ) (10)m B °m B (f, T) RT ln x B (11)The expressions of the molar Gibbs energy for purematerials are presented as in Ref. [7]°m AV m W A g(f)T [e s A(T A m)c A T A m (12) p(f)H A ]1 T T A mc A T ln T T A mẋ BV m·J B (4)°m BV m W B g(f)T [e s B(T B m)c B T B m (13) p(f)H B ]1 T T B mc B T ln T T B mė ·J e (5)where V m is the molar volume. The diffusional fluxwhere g(f) f 2 (1f) 2 and W A , W B are constants.e s A(T A m) and e s B(T B m) are the energy densities of pure

LOGINOVA et al.: PHASE-FIELD SIMULATIONS OF BINARY ALLOY SOLIDIFICATION575solid A and B at their melting points T A m and T B m,respectively. H A and H B are the heats of fusionper volume, c A and c B are the heat capacities and Ris the gas constant. p(f) is a smoothing function,chosen such that p(f) 30g(f).The phase-field equation (3) in the present modelis used as derived in Ref. [1]whereḟ M f ¯2·(h 2 f)∂xhh ∂ b ∂f (14)∂y∂yhh ∂ b ∂f∂xM f ((1x B )H A x B H B )H A (f, T) W A g(f) 30g(f)H A 1 T 1T A m(15)H B (f, T) W B g(f) 30g(f)H B 1 T 1T B mThe diffusion equation is now obtained by combiningequations (4), (6), (8) and (10)–(13). However, aspointed out earlier, the gradient (δS/δx B ) must beevaluated isothermally, therefore the diffusional fluxmay be writtenJ B L A BBRx B (1x B ) x B (16) (H A (f, T)H B (f, T))fComparison of equation (16) with the normal Fick’slaw of diffusion gives thatL A BB D x B(1x B )RV m(17)where D is the normal Fickian coefficient of interdiffusionof A and B. In this case the so-called thermodynamicfactor is unity because the ideal solution isassumed. The final diffusion equation is then obtainedby combining equations (4), (16) and (17) and takesthe formẋ B ·Dx B V mR x B(1x B )(H B (f, T) (18)H A (f, T))fThe diffusion coefficient is postulated as a functionof the phase-field variableD D S p(f)(D L D S ) (19)where D S , D L are the classical diffusion coefficientsin the solid and liquid, respectively.To complete the derivation of the model, theinternal energy density is postulated according toRef. [5]e (1x B )e A x B e B (20)The internal energy densities of pure materials underthe assumption of equal solid and liquid heatcapacities aree A e s A(T A m) c A (TT A m) p(f)H A (21)e B e s B(T B m) c B (TT B m) p(f)H BInserting equation (20) into equation (5) withL A ee KT 2 implies that the heat equation has the formc¯Ṫ 30g(f)H˜ ḟ Nẋ B ·KT (22)where the following formulae are introducedc˜ (1x B )c A x B c B (23)H˜ (1x B )H A x B H B (24)and N ∂e/∂x B . Similar to the heat capacity and thelatent heat of fusion of the mixture the thermal conductivityof the mixture is approximated by aweighted sum of conductivities of the pure materialsK (1x B )K A x B K B (25)Again, equal solid and liquid thermal conductivitiesof both materials are assumed. Following Ref. [8], theheat equation is simplified by dropping the ẋ B termc¯Ṫ 30g(f)H˜ ḟ ·KT (26)A similar heat equation was derived in Ref. [12] forthe case of a dilute alloy.3. NUMERICAL ISSUESFor convenience, the governing equations (14),(18) and (26) are transformed into dimensionlessform. Length and time have been scaled with a referencelength l0.94d and the diffusion time l 2 /D L ,respectively. The non-dimensional temperature is

LOGINOVA et al.: PHASE-FIELD SIMULATIONS OF BINARY ALLOY SOLIDIFICATION577when a0. The steady-state growth velocity of thedendrite tip was chosen as a reliable quantity to definetime-step. It was found that with t0.2 the numericalscheme gives an error of the tip speed ca 3%.Solute redistribution obtained for isothermal calculationsis given in Fig. 1.4. TEMPERATURE FIXED ON THE BOUNDARYTo complete the model describing cooling of a meltwith growing nuclei, thermal boundary conditions areto be specified. Different types of boundary conditionsdetermine different thermal regimes affectingthe temperature variation in space. In this sectionFig. 1. Concentration field, isothermal model. Time is 2.75 ms.Fig. 2. Concentration field given by the non-isothermal modelwith temperature fixed on the boundary. All the physical andphase-field parameters are the same as in isothermal calculations.Time is 2.75 ms. The color scheme uses yellow–redpalette in grey scale for low and high concentration values,respectively.equations (14), (18) and (26) are solved withT1574K as initial and boundary conditions for theheat equation. This is done in order to fit the nonisothermaltemperature regime to the isothermal case.The results of this simulation are shown in Figs 2and 3.As one would expect, the temperature does notvary much, neither in time nor in space, for example,the spatial temperature difference does not exceed1.6K all the time (Fig. 4). However, comparison ofthe solute patterns (Figs 1 and 2) shows that the compositionbehaviour is very sensitive to even smallchanges in the temperature field. Increased melt temperaturereduces sources of instability, which leads toless developed structure of the non-isothermal dendrite.The length of the primary arms is about 6% lessthan the corresponding one calculated under isothermalconditions. The solute diffusion length islarger compared to the isothermal growth, as well asspacing of interdendritic liquid pockets. The value ofthe composition at the solid/liquid interface isdecreased since the operating point in the phase diagramhas moved to the left due to the increased temperature.Figure 3 demonstrates spatial redistribution of thetemperature field with imposed interface location attime 2.75 ms. As predicted by the Gibbs–Thomsoncondition, the tips are the coldest parts of the dendritedue to the large curvature here. The location of thehottest point during the crystal growth varies and ingeneral corresponds to one of the tips of those sidebrancheswhich grow towards each other and form aclosed liquid pocket. Melt in this pocket is of highesttemperature due to release of latent heat by thesegrowing sidebranches. The less sharp the sidebranchtip, the higher temperature of the sidebranch.The maximal value of the system temperature as afunction of time is given in Fig. 4. The curve isslightly oscillating because the temperature is takenat different grid points wherever the maximum valueoccurs. The local peaks occur when two or more sidebranchesgrowing towards each other merge and stopgrowing and consequently to produce the latent heat.Then another sidebranch surrounded by hot meltstarts to release more latent heat than the others andbecomes the hottest place in the system. Fig. 5 showsthe area of the dendrite tip with imposed isotherms,which are refracted over the interface. This reflectsthe change of the temperature gradient owing to therelease of latent heat.The results presented in this section indicate clearlythat for a supersaturated binary alloy the developmentof dendritic patterns is governed mainly by solute diffusion,but the temperature variation alters significantlythe morphology of the microstructure.5. RECALESCENCE CALCULATIONSThis section represents the results obtained forsimulations of dendritic solidification in the thermal

578 LOGINOVA et al.: PHASE-FIELD SIMULATIONS OF BINARY ALLOY SOLIDIFICATIONFig. 3. Temperature field corresponding to the solute redistributionin Fig. 2. The lighter color, the higher the temperature.The solid/liquid interface is shown as a band where0.1f0.9. Time is 2.75 ms.Fig. 4. Maximal temperature of the system vs time, minimaltemperature T1574K is kept on the boundary all the time.regime when a heat flux is extracted from the domain.This regime is modeled in two ways: (a) the isothermalmodel is coupled with a heat balance [4]which neglects temperature variations in spacedTdtṪ H˜c˜ddt 1 f(x, y, t)dS (30)SSwhere Ṫ is the cooling rate, S is the area of thedomain, H˜ and c˜ are evaluated for the compositionfar away from the interface; (b) non-isothermal modelwith Neumann boundary conditions for the heat equation.The heat flux Q imposed on the outer boundaryis related to Ṫ by Q c˜Ṫa , where a is the side length4of the computational domain.Fig. 5. Location of isotherms near the dendrite tip. The picturecorresponds to the run in Fig. 3.There is a lot of numerical and physical issues,which should be investigated for these types of simulations.In these calculations the effect of the coolingrate and the size of the domain is analyzed. The computationsare performed in two square domains6.910 5 and 2.2810 5 m on a side (which will bereferred to as “large” and “small” boxes), for threecooling rates: 2.110 3 , 3.410 4 , and 1.310 5 K/s.In order to keep the same non-dimensional geometry,d1.6210 8 m is chosen for the “small” box.Figure 6(a and b) shows the temperature–time historyfor all cooling rates, in the “big” and “small”box, respectively. Three curves represent each simulation:two dashed lines show the minimal and maximalvalue of the system temperature calculated by thenon-isothermal model, the solid line is T(t) obtainedthrough the heat balance [equation (30)]. It is necessaryto note that T(t) calculated by the isothermalapproach differs significantly (2K as the worst) fromthe short-term T(t) dependence in Fig. 6 in Ref. [4].T(t) is very sensitive to the choice of numerical aswell as physical parameters which were not unambiguouslydefined in Ref. [4].In general, both the models give a similar T(t)behaviour. Initially, at the highest cooling rates(3.410 4 and 1.310 5 K/s) the temperature fallsdown because the composition needs more time to bechanged and to cause solidification. As a crystal startsto grow, the latent heat release increases the temperature,that is, recalescence occurs. For the lowest coolingrate, the initial growth of the nuclei is fast enoughto overcome the imposed heat extraction rate and,hence, the temperature–time curve initially has a positiveslope. T(t) given by the isothermal model initiallyapproximates an average temperature, but later thelatent heat of fusion is released faster, and therefore,for all three cooling rates the time evolution is faster.As a result, the isothermal model gives an overestimatedvalues of the temperature.Comparison of the dashed curves in Fig. 6(a and

LOGINOVA et al.: PHASE-FIELD SIMULATIONS OF BINARY ALLOY SOLIDIFICATION579Fig. 6. Temperature–time dependence for the boxes of size2.2810 5 m (a) and 6.910 5 m (b). Results for the coolingrates 2.110 3 , 3.410 4 , and 1.310 5 K/s are shown from topto bottom.b) obtained for the same cooling rates shows theinfluence of the domain size. In outline, at the sametime the temperature of the melt in the “large” boxhas a lower value than the temperature in the “small”domain. This feature can be explained by the fact thata larger box contains fewer nuclei per area, and thus,less latent heat is produced in the box. It is interestingto notice that a larger computational domain andhigher heat extraction rate cause greater spatial variationsof the temperature field. The maximum temperaturealteration is ca 9° in the “large” box and 1.2°in the “small” one. The tendency of a slightlydecreasing difference of maximum and minimumtemperature as time goes is due to reducing fractionof melt. Therefore, the assumption made in Ref. [4]about absence of the temperature gradient in spacecan be accepted for a small computational domainwith a low imposed heat extraction rate, but in othercases spatial variation of the temperature field shouldbe taken into account.The calculated growth morphologies vary significantlyamong the three cooling rates and the computationaldomains. Solute patterns and correspondingtemperature fields are presented in Figs 7 and 8. Comparingthese figures one should remember that the realside sizes of the boxes differ by a factor of three.At the lowest cooling rate [Figs 7(e) and 8(e)] thetemperature rise reduces the melt undercooling whichgives the least developed crystal. In this case the solidificationis driven by solute change in both phases,solid and liquid. The higher the heat extraction rate,the greater the influence of the size of the computationaldomain on the crystal growth. While the dendriticmorphology calculated in the “small” box [Fig.7(c)] consists only of primary stalks with no perturbationon the interface, the corresponding morphologyin the “large” box [Fig. 8(c)] has welldeveloped secondary arms. It should be noted, thatdue to the high value of heat extraction rate the interfacestays planar (which gives the crystal a diamondshape) before perturbations appear at the interfaceand secondary sidebranches start to develop.The most intriguing morphology is presented inFig. 8(a) and obtained for the highest cooling rate inthe “large” domain. The morphology growth exhibitsan almost circular shape with cells developing at thelater times. The explanation is that for very high coolingrate, the solidification of a binary alloy is governedmainly by the heat transfer. When a meltfreezes quickly, composition does not have time tobe changed and to cause instability. The presence ofsolute trapping in this simulation is observed whenthe interface velocity V reaches its maximum valueof 0.03 m/s, which corresponds to the largestundercooling of the melt. The partition ratio at themoment is c S /c L 0.97, as opposed to an equilibriumvalue of 0.85.The solute trapping effect in the phase-field modelswas studied in Ref. [15]. It was shown that for aplanar interface the solute trapping occurs when thesolute diffusion length ahead of the interface is comparablewith the interface thickness D I /Vd whereD I D(f 0.5). For the d used in the calculationsthis gives V0.01 m/s. The simulations of directionalsolidification [16] demonstrate that the interfaceremains stable for velocities above 0.024 m/s. Hence,qualitatively, the growth behaviour of the dendrite isconsistent with the significant solute trapping.It should be concluded that a crystal morphologydepending on the heat extraction rate varies fromsmooth primary stalks without secondary arms (lowcooling rate) to well-developed dendritic structure toplanar or cellular shape at extremely high coolingrate.6. CONCLUSIONSThe presented results are thought to be a firstattempt to model non-isothermal dendritic solidificationof a binary alloy. Removing the isothermalassumption makes the computations much more timeconsuming, due to the large difference between thermaland mass diffusivities.

580 LOGINOVA et al.: PHASE-FIELD SIMULATIONS OF BINARY ALLOY SOLIDIFICATIONFig. 7. Solute and temperature redistributions (top and bottom rows, respectively) obtained in the “small” box(2.2810 5 m), for the cooling rates 1.310 5 , 3.410 4 and 2.110 3 K/s varying from the left column to theright one. Time is 1.4 ms. The concentration and the temperature fields employ the same color scheme as inFigs. 2 and 3, respectively. The black line shown with the temperature field represents the location of thesolid/liquid interface.Fig. 8. Solute and temperature redistributions (top and bottom rows, respectively) obtained in the “large” box(6.910 5 m), for the cooling rates 1.310 5 , 3.410 4 and 2.110 3 K/s varying from the left column to theright one. (a) and (b) are given at time 1.7 ms, while the others are at 2.5 ms.

LOGINOVA et al.: PHASE-FIELD SIMULATIONS OF BINARY ALLOY SOLIDIFICATION581For low cooling rate and many nuclei the spatialtemperature variation is small and may be neglected,and thus the isothermal approach is applicable. However,the non-isothermal effect becomes visible forhigher cooling rate and fewer nuclei, when the spatialtemperature difference is not small compared to thedifference between solidus and liquidus in the phasediagram.On increasing the cooling rate, the growth eventuallybecomes governed by thermal diffusion ratherthan redistribution of solute. Due to limitations on thewidth of the diffuse interface the results for the highestcooling rate, showing strong solute trapping, maynot be quantitatively correct. However, qualitativelythe predicted behaviour is in agreement with what isexpected at high cooling rates.Acknowledgements—This work was supported by the SwedishResearch Council for Engineering Science (TFR).REFERENCES1. Warren, J. A. and Boettinger, W. J., Acta metal., 1995,43, 689.2. Amberg, G., http://www.mech.kth.se/~gustava/femLego/.3. Amberg, G., Tönhardt, R. and Winkler, C., Math. Comput.Simulation, 1999, 49, 257.4. Boettinger, W. J. and Warren, J. A., Metall. Mater. Trans.,1996, 27A, 657.5. Caginalp, G. and Jones, J., Ann. Phys., 1995, 237, 66.6. Tönhardt, R. and Amberg, G., J. Crystal Growth, 1998,194, 406.7. Wang, S. -L., Sekerka, R. F., Wheeler, A. A., Murray, B.T., Coriell, S. R., Braun, R. J. et al., Physica D, 1993,69, 189.8. Caginalp, G. and Xie, W., Phys. Rev. A, 1993, 48,1897–1909.9. Warren, J. A. and Murray, B. T., Modeling SimulationMater. Sci. Engng, 1996, 4, 215.10. McCarthy, J. F., Acta mater., 1997, 45, 4077.11. Warren, J. A., IEEE Computational Science and Engineering,1995, 2, 38.12. Conti, M., Phys. Rev. E, 1997, 55, 765.13. Pavlik, S. G. and Sekerka, R. F., Physica A, 1999, 268,283.14. Karma, A. and Rappel, W. J., e-print cond-mat/990201.15. Ahmad, N. A., Wheeler, A. A., Boettinger, W. J. andMcFadden, G. B., Phys. Rev. E, 1998, 58, 3436.16. Boettinger, W. J. and Warren, J. A., J. Crystal Growth,1999, 200, 583.

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Acta Materialia 51 (2003) 1327–1339www.actamat-journals.comThe phase-field approach and solute drag modeling of thetransition to massive g → a transformation in binary Fe-CalloysI. Loginova a , J. Odqvist b,∗ , G. Amberg a ,J.Ågren baDepartment of Mechanics, KTH, 100 44 Stockholm, SwedenbDepartment of Materials Science and Engineering, KTH, 100 44 Stockholm, SwedenReceived 18 April 2002; accepted 25 October 2002AbstractThe transition between diffusion controlled and massive transformation g →α in Fe–C alloys is investigated bymeans of phase-field simulations and thermodynamic functions assessed by the Calphad technique as well as diffusionalmobilities available in the literature. A gradual variation in properties over an incoherent interface, having a thicknessaround 1 nm, is assumed. The phase-field simulations are compared with a newly developed technique to model solutedrag during phase transformations. Both approaches show qualitatively the same behavior and predict a transition toa massive transformation at a critical temperature below the T 0 line and close to the a/a + g phase boundary. It isconcluded that the quantitative difference between the two predictions stems from different assumptions on how theproperties vary across the phase interface yielding a lower dissipation of Gibbs energy by diffusion in the phase-fieldsimulations. The need for more detailed information about the actual variation in interfacial properties is emphasized.© 2003 Acta Materialia Inc. Published by Elsevier Science Ltd. All rights reserved.Keywords: Diffusion; Thermodynamics; Gibbs energy; Dissipation; Interfacial properties1. IntroductionIf austenite (g) in low-carbon iron alloys isquenched to sufficiently low temperatures, but stillabove the martensite start temperature M s , it willbe decomposed by a massive transformation thatyields a characteristic blocky or massive microstructure.The massive transformation is partitionlesslike the martensitic transformation, i.e. it∗Corresponding author. Fax: +46-8-100411.E-mail address: joakim@met.kth.se (J. Odqvist).does not involve any composition change, and thuslong-range diffusion is unnecessary. The growth ofmassive ferrite (a) occurs with a constant growthrate that is more or less independent of crystallographicorientation relationships in contrast to themartensitic transformation. Thermodynamically apartitionless transformation g →a is possiblebelow the T 0 temperature, at which a and g of thesame composition have same Gibbs energy. However,at what temperature the massive transformationreally becomes kinetically possible has beena matter of considerable controversy over theyears. It may be argued that if the interfacial reac-1359-6454/03/$30.00 © 2003 Acta Materialia Inc. Published by Elsevier Science Ltd. All rights reserved.doi:10.1016/S1359-6454(02)00527-X

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

I. Loginova et al. / Acta Materialia 51 (2003) 1327–13391329substitutional atom and will be approximated asconstant, is the phase-field variable taken 0 ina and 1 in g. The u-fraction u C is defined from thenormal mole fraction of C, x C asu C x C(2)1x CAs already mentioned, the temperature T isassumed constant all over the whole system. However,it may of course vary in time but heat conductionis assumed so rapid that all temperaturegradients can be neglected. The molar Gibbsenergy G m is postulated as a function of the phasefieldvariable:G m (1p(f))G a m p(f)G g m g(f)W (3)whereg(f) f 2 (1f) 2 (4)p(f) f 3 (1015f 6f 2 ) (5)and the choice of the parameter W will be discussedlater. G a m and G g m denote the normal Gibbsenergy functions of the a and g phases and aretaken from the assessment of Gustafson [8]. Thecomplete expressions are given in Appendix A. Itshould be mentioned that g(f) and p(f) have beenchosen so that dp/df 30g(f).The evolution of the phase-field variable isgoverned by the Cahn–Allen equation [9](6)ḟ M fdGdf M f 1 V m∂G m∂f e2 2 fThe kinetic parameter M f is related to the interfacialmobility as will be shown later.The evolutionof the concentration field is governed by the normaldiffusion equation. When u-fractions are usedand the molar volume is approximated as constantthe normal diffusion equation can be rewritten asu˙ C·JVC (7)mThe diffusional flux of carbon J C is given by theOnsager linear law of irreversible thermodynamics:J C L d Gd u c (8)The quantity dG/du C is the normal chemicalpotential of C denoted by m C . If the so-called gradientterms are neglected we have∂ G m /∂ u C m Cand Eq. (8) may be expanded in terms of the concentrationand phase-field gradientsJ C 1 DV C u C L ∂2 G mf (9)m ∂u∂fThe first term corresponds to the normal Fick’s lawand we may thus identify the normal diffusioncoefficient of C asD C V m L ∂2 G m∂u 2 C(10)The second-order derivative corresponds to Darken’sthermodynamic factor and the parameter L is relatedto the diffusional mobility [10] M C by means ofL u CV my Va M C (11)where y Va denotes the fraction of vacant interstitials,i.e. 1-u C for g and y Va 1u C /3 for a. Fora given C content the fraction of vacancies wouldthus depend on the character of the phase, i.e. itwill depend on the phase-field variable. We havepostulatedu C y Va (1p(f))u C (1u C /3) p(f)u C (1u C )(12)The diffusional mobility in the two phases coulddiffer by several orders of magnitude, therefore wehave chosen the following combination:M C (M a C) 1p(f) (M g C) p(f) (13)For substitutional solutes the mobility in thecenter of the interface is most probably muchhigher than in the any of the crystalline phases. Forinterstitial solutes like carbon it may not be muchhigher and the approximation represented by Eq.(13) may not be too crude. The mobilities of carbonin a and g were taken from Ågren [11,12].The complete expressions are given in AppendixB.

1330 I. Loginova et al. / Acta Materialia 51 (2003) 1327–13393. Solute drag modeling of the g→atransformation in Fe–CIn the solute drag modeling the thickness (δ) ofthe interface is considered small enough comparedto the curvature of the interface to approximate thediffusion field inside the interface as planar, seee.g. in ref. 2. The steady-state solution of Eq. (7)will be a good approximation inside the interfacebecause it is so much thinner than the distance ittravels. Eq. (7) takes the formv(uVC u a C) J C (14)mwhere v is the interface migration rate. The flux isgiven by Eqs. (8) or (9) and is set to 0 in the growinga phase. For a given combination ofv and u a C the solution of Eq. (14) yields a concentrationprofile across the interface and thus also theC content on the g side of the interface, i.e. u g/aThe dissipation of Gibbs energy due to diffusioninside the interface, expressed per mole of Fe anddefined as positive, is given byG diffm V mvdC .dm CJ C dz (15)dzBy combining Eqs. (14) and (15) and makinguse of ∂ G m /∂ u C m C one obtainsG diffm (u C u a C) d ∂ G mdz (16)dz∂ u CdIn order to perform calculations the variation inthermodynamic properties and mobility inside thephase interface must be known. In the solute dragtheory G m is postulated as a function of both distanceand composition. Several choices are possible.One choice is to use Eqs. (3) and (13) insidethe interface but rather than Eqs. (4) and (5) onesimply postulates that g(f) 0 and p(f) z/d.This approach was taken by Hillert and Sundman[2] and will be used in the present report.The interface has a finite mobility due to interfacialfriction and often a linear relation betweeninterface migration rate and driving force isobserved experimentally, i.e.v M V mG i m (17)where G i m is the driving force needed to movethe interface and M is the interfacial mobility.G i m is defined positive for a spontaneous reactionand is the Gibbs energy dissipated by the interfacefriction. The total Gibbs energy dissipated in theinterface is thus given by the sum ofG diffm and G i m. The dissipation must be suppliedfrom the total driving force available over the interface.It is given per mole of atoms and expressedin terms of the individual chemical potentials onp. 152 in ref. [13]. By instead introducing theGibbs energy per mole of Fe, G m , and its firstderivative we obtainG totm G g mG a m(u g/aC u a C) ∂Gg m∂ u g C(18)where G totm is defined as positive for the consideredreaction to occur. For a given migrationrate v we may, by combining Eqs. (9), (14) and(16)–(18) find the u a C and u g/aC that makes the dissipatedGibbs energy exactly matchG totm . In thelimit of low migration rates u a C and u g/aC willapproach the local equilibrium values predicted bythe phase diagram but at high velocities they willapproach each other.The solute-drag modeling of the interfacial reactionsmay be combined with a treatment of C diffusionin g ahead of the interface. By such anapproach it is possible to describe the gradual deviationfrom local equilibrium as the interfacemigration rates increases. In principle such modelingcould be based on numerical methods as inthe DICTRA software [14] or semi-analyticalmethods as the Green function formalism usedrecently by Enomoto [15]. For the sake of simplicitywe will here take a simpler approach basedon the linear-gradient approximation. For the thickeningof a grain-boundary precipitate it yields thefollowing expressionv D (u g/aC u gC ) 2(19)2(u a CuC g/a )(u a CuC g )where is half the thickness of the grain-boundaryprecipitate and u gC is the carbon content in g faraway from the interface, i.e. the initial content of

I. Loginova et al. / Acta Materialia 51 (2003) 1327–13391331g. By combining Eq. (19) with the previous resultwe may calculate the thickness corresponding to agiven velocity. It should be emphasized that thesolute drag calculations may be performed independentlyof the alloy composition uC g . As alreadymentioned, we then obtain u a C and uCg/a as functionsof migration rate v at a given temperature.For a particular alloy content u gC we may use Eq.(19) to establish a relation between v and .4. Approximate equivalence of the twoapproaches for Fe–CIn order to demonstrate the similarity betweenthe phase-field and solute drag approaches Ahmadet al. [7] considered a planar case and the steadystate formulation of the phase field, i.e.ḟ v df(20)dzwhere ḟ is given by Eq. (6). Their approach willnow be applied to the g →a transformation and itwill thus be slightly modified. However, for theconvenience of the reader the derivation will nowbe given in some detail.By multiplying both sides of Eq. (20) withdf/dz and integrating across the interfacial region,with a thickness d, we obtainM df 1∂G m df fdfV m ∂f dz e2d2 (21)dz dzdz 2v df 2dzdzdApplying integration by parts we find that the lastterm inside the left-hand side integral vanishesbecause df/dz 0 outside the interfacial region.The first term may be expanded becausedG mdz ∂G mdf∂f dz ∂G mdu C(22)∂u C dzand Eq. (21) thus becomes1M fVdm dG mdz ∂G mdu C∂u C dzdz (23)v df 2dzdzdi.e.1M fV m(G g mGm) ∂G a mdu C∂u C dzdz (24)dv df 2dzdzddIntegrating Eq. (16) by parts and rearranging wefind ∂G mdu C∂u C dzdz (u g/aC u a C) ∂Gg mG∂u diffm (25)CInserting Eq. (25) in Eq. (24) and dropping theminus sign on both sides yieldm (26)1M fVmGm(G g a m)(uC g/a u a C) ∂Gg m df dz vd2dz∂u CG diffBy comparing Eqs. (18) and (26) we find1M f [GV totm G diffm df 2dz (27)dzm ] vdThe quantity inside parentheses on the left-handside is clearly G i m in Eq. (17) and it only remainsto evaluate the integral on the right-hand side ofEq. (27). It should be emphasized that the left-handside of Eq. (27) comes out as an exact result whensteady state is assumed. The right-hand side ismore difficult but Ahmed et al. assumed it wasmore or less independent of the quantities on theleft-hand side and set1M f [GV totm G diffm ] va (28)mComparing Eqs. (28) and (17) we may thus identify.M 1 a M f (29)

1332 I. Loginova et al. / Acta Materialia 51 (2003) 1327–1339where a (df/dz) 2 dz. For the simple case wheredwe approximate the variation in f as linear insidethe interface we obtain a 1/d. However, itshould be emphasized that generally the interfacethickness is less well defined for a diffuse interfacethan in solute drag modeling. In principle the interfaceextends over the whole region where f varies,which is strictly z for the diffuse interface.Thus, the integration should be performedover that region. However, in practice it may yieldsufficient accuracy to extend the integration over afinite region somewhat thicker than the thicknessused in solute drag modeling. In the present casea more realistic variation in f is given by the equilibriumsolution of Eq. (6):f 1 21 tanhz/2d (30)where d e/√W/V m . By numerical integrationover a the region 3d z 3d and adopting Eq.(30) one obtains a 0.235/d.5. Physical parametersAs already mentioned, the complete set of thermodynamicand kinetic parameters for the Fe–Csystem is given in appendices A and B, respectively.The phase-field mobility M f is related to theconventional interfacial mobility M by means ofEq. (29). In the present study different choices ofa will be tested. The interface thickness will bechosen as d = 10 9 m. For a pure element the parametersW, , d and the surface energy s are relatedas follows [16]:W s d V 6m(31)2Ws e (32)18V mi.e. e 2 3√2sd. With the surface energy s 1J/m 2 and V m 710 6 m 3 mol 1 we obtain W =29.698 10 3 J/mol e 2 4.24 10 9 J/m. We shallassume that W and are independent of temperatureand composition.6. Numerical details6.1. Phase-field simulationsThe standard second order central difference inspace and first order in time transform Eqs. (6) and(7) into a discrete problem. Zero Neumann boundaryconditions are applied for both variables. Theresulting non-linear systems of equations aresolved using the Newton–Raphson method.From the experimental observations, we expecttwo different regimes: “slow” growth, controlledby C diffusion in g, and “fast” massive growth controlledby the interfacial reactions. During slowgrowth, we thus expect a parabolic behavior withan interface velocity v that is essentially proportionalto 1/√t except for the later stages whenimpingement sets in and the system finallyapproaches the state of equilibrium. During the“fast” or partitionless growth, the solution shouldyield a single concentration spike traveling withconstant velocity v until all the initial γ is transformedinto a. In order to simulate these regimes thetime step is adjusted to the interface dynamicsaccording tot ch (33)nwhere h is the space resolution in a uniform gridand c is the Courant number taken as 0.01. Withthis approach simulations of massive growth areperformed with constant time step equal to itsinitial value 3.5· 10 10 s. In the case of parabolicgrowth, the time step is gradually increased andat the final time, when the equilibrium is nearlyachieved, t is 10 6 times larger than initially. Thechoice of the grid resolution was verified based onthe known content of carbon in both phases at theequilibrium. With h 0.1125 nm the relativeerror of equilibrium u C is 1.2%. Another test is thatthe slope of the interface position vs √t is a constantexcept for the later stages.6.2. Solute drag simulationsThe steady-state diffusion equation, i.e. Eq. (14),reduces to an ordinary differential equation for the

I. Loginova et al. / Acta Materialia 51 (2003) 1327–13391333concentration of each solute over the interface. Avariable order backward differentiation formula,usually referred to as Gear’s method [17], is usedto solve this differential equation. We have usedthe algorithm of Gear’s method that isimplemented in the HARWELL SubroutineLibrary [18].7. Results7.1. Phase-field simulationsPhase-field simulations were performed at anumber of temperatures and alloy contents. Thesize of the system was 10 µm, corresponding to anaustenite grain size of 20 µm if ferrite is formedat the austenite grain boundaries. As mentionedearlier, d was chosen to be 10 9 m. Thus both thesize of the system and the interface thickness weregiven physically realistic values. In a first set ofcalculations the parameter a in Eq. (29) was chosenas 1/d, i.e. M f M/d was applied in the phasefieldequation.The initial state was always homogeneous g witha very thin layer of a ( 4.5 nm) formed at theleft side of the system. The composition of theinitial layer of a was taken as 0.1·uCg if parabolicgrowth was expected. However, the compositionof this layer does not affect the calculation becauseit is adjusted automatically during the first few timesteps. In case of massive growth the initial compositionwas uniform over the domain.As an example, Fig. 1 shows the concentrationprofile for an alloy with u gC 0.01at T 1093 K and different instancest 1, 10, 20,...,60 s. Fig. 2 shows the half thickness as a function of √t. As can be seen, thegrowth is indeed parabolic up to t 10 s whereimpingement sets in. These results are in excellentagreement with DICTRA [14] simulations usingthe same set of data. The DICTRA simulations arebased on a sharp interface model and local equilibriumat the interface. At the same temperature acompletely different behavior is found for an alloywith uCg 0.001. In this case the massive growthoccurred with a constant growth rate of ca. 0.1 m/sFig. 1. Phase-field calculation of carbon concentration profilesat different instances, t = 1, 10, 20,… 60 s. u gC 0.01 at T =1093 K.Fig. 2. Phase-field calculation of half-thickness of ferrite precipitateas function of √t.until all g was transformed into a. The concentrationprofile, a traveling wave, is shown in Fig. 3.By performing phase-field simulations for alarge number of alloy compositions at each tem-

1334 I. Loginova et al. / Acta Materialia 51 (2003) 1327–1339Fig. 3. Phase-field calculation of carbon concentration profileat interface during massive growth of ferrite, v 0.1 m/s, foru gC 0.001 at T = 1093 K.perature it is possible to establish a critical compositionbelow which the massive growth occurs atthat temperature. Of course the critical compositiondepends on the value of M f . In Fig. 4 we havesuperimposed, on a part of the calculated Fe–Cphase diagram, curves a and b representing thecritical composition calculated for two differentchoices of M f . The T 0 line, where the Gibbs energyof the g and a phases have the same value for thesame composition, has also been included in thediagram. The simulations also reveal that the interfacevelocity as well as the non-equilibrium partitioncoefficient k decrease when the alloy compositionu gC is increased and approaches the criticalvalue. This behavior is in agreement with the velocitydependence on k discussed in [7].7.2. Solute-drag simulationsSolute-drag simulations were now performedaccording to the approach outlined in section 3.For a given temperature and composition of thegrowing a Eq. (14) was solved for a series ofinterfacial velocities. For each velocity the C concentrationprofile across the phase interface as wellas the C content of g at the g-side of the phaseFig. 4. Calculated Fe–C phase diagram with the T 0 line superimposed.Curves a and b are the critical compositions fromphase-field simulations using different choices of M f :a)M/dand b) 0.235M/d. Curve c) is the critical composition from solute-dragsimulations. M is given in Appendix B.interface were obtained and the Gibbs energy dissipationdue to diffusion, G diffm , across the interfacewas subsequently obtained by integration of theconcentration profile and adding the contributiondue to interfacial friction, G i m. From the C contenton both sides of the interface the available drivingforce, G totm , was calculated using Eq. (18). Atgiven temperature and composition of the growinga the total dissipated Gibbs energy, G diffm G i m, and the available driving force may be plottedas functions of interfacial velocity. The generalappearance of the two curves is shown in Fig. 5.The total dissipation, solid curve, starts from zeroat low velocities, and grows due to dissipationcaused by diffusion inside the interface, i.e. solutedrag. After a maximum it decreases at high velocitiesbut at very high velocities there is an increasedue to the interfacial friction. If the interfacialmobility is low in comparison with the diffusivityinside the interface the dissipation caused by diffusionand friction may overlap and there will beno minimum. The available driving force derivesfrom composition differences across the interface.At low velocities they will yield a negative value

I. Loginova et al. / Acta Materialia 51 (2003) 1327–13391335Fig. 5. Dissipated Gibbs energy by diffusion inside the interfaceas a function of interface velocity obtained by solute-dragtheory. The solid line represents the dissipation inside the interface.The dashed line denotes available driving force over theinterface.if the a phase is chosen inside the equilibrium a+ g phase field. It stays constant until the compositionon the g side starts to decrease due to limiteddiffusion inside the interface. Then it increases andturns positive if it started from a negative value.At very high velocities it will approach the differencein Gibbs energy between a and g when evaluatedfor the same C content, uC g/a u a C.Figs. 5 and 6 are based on u a C values inside thea + g phase field where the driving force startsfrom a negative value. From Fig. 5 it is evident thatthe curves must then intersect at an even number ofpoints. We have found that two cases occur. Eitherthe curves intersect at two points, one at a highvelocity corresponding to partitionless growth andone at low velocity corresponding to slow growthcontrolled by C diffusion in g, or they do not intersectat all. We have not found any cases wherethere are four but that could very well occur if carbonhas a tendency to segregate to the interface.At the intersections the dissipation exactly matchesthe available driving force. On the left side of thelow velocity intersection and on the right side ofthe high velocity intersection the available drivingFig. 6. Carbon contents on the α and γ side of the interface,u a C and uCg/a , as functions of growth rate corresponding to theintersections in Fig. 5. The dot denotes the highest C contentthat a could grow with.force is lower than required by the interfacial reactions.In the range between the two intersectionsthe available driving force exceeds what is requiredfor the interfacial reactions and this range thus representsphysically possible states. Moreover, if weassume that Gibbs energy is only dissipated by diffusionalprocesses and interfacial friction then thetwo intersections represent the only combinationsof growth rate, u a C and u g/aC that are physicallypossible. As we increase the C content of the growinga the two intersections move towards eachother until they finally meet in a point of tangencybetween the two curves. At higher C contents ofα no solution at all is obtained. If u a C values insidethe a one-phase field had been chosen the dashedcurve in Fig. 5, denoting the driving force, wouldhave started from a positive value at low velocities.However, since we are primarily interested in thepossibility of partionless growth in the two-phasefield no such u a C values were considered. In Fig. 6we have combined the information on growth rate,u a C and u g/aC for the intersections from a largenumber of calculations. The parts stemming fromthe high-velocity intersections have been dashed.At low growth rate u a C and u g/aC correspond to the

1336 I. Loginova et al. / Acta Materialia 51 (2003) 1327–1339contents given by the phase diagram, i.e. localequilibrium is established, but they approach eachother at higher growth rates. The point of tangencybetween the dissipation and driving force curves inFig. 5 appears as a maximum on the curve foru a C denoted by the dot in Fig. 6 and the two curveshave then got very close to each other. Thismaximum represents the highest C content that acould grow with. With a mobility according toAppendix B, the maximum C content of a fellclose to the equilibrium phase boundary for a.However, the maximum C content and the correspondinggrowth rate depend on the properties ofthe interface. The higher the interfacial mobility,compared to the diffusivity, the higher themaximum content and the corresponding growthrate. In order to demonstrate clearly the generalbehavior the calculations presented in Fig. 6 weremade using an interfacial mobility ten times largerthan the one given in Appendix B. These calculationsgave an a well inside the a + g phase fieldand a growth rate about ten times higher. A calculationwith the mobility 100 times larger than theone in Appendix B yields a maximum even closerto the thermodynamic limiting value, i.e. the T 0line.as functions of growth rate from Fig. 6, or similardiagrams calculated using different values of theinterfacial mobility. For a given alloy we may thusestablish a relation between half thickness andgrowth rate. The result of a series of such calculationsis given in Fig. 7. The alloys havinguCg 0.008 and 0.005, respectively, are representedby the two curves at the bottom left cornerof the diagram. When is close to zero in the veryearly stages, the assumption of local equilibriumpredicts an infinitely high growth rate. From Fig.6 and Eq. (19) it is obvious that this could neveroccur because as the growth rate becomes veryhigh u g C/a will approach u a C and the numerator inEq. (19) becomes very small. It should be emphasizedthat Eq. (19) is only physically meaningfulfor u g/aC uCg when a grows with equal or lowerC content than g. In these two alloys ferrite willthus grow under partitioning with a very high butfinite rate in the very early stages. For the alloyhaving u gC 0.003 there is a similar curve to theleft in Fig. 7 but, in addition, there is now a possibilityof having a partitionless transformationbecause u gC is lower than the maximum C contentof a. The partitionless growth could then occur7.3. Effect of diffusion in g and the transition tomassive growthWe would now like to study the transition frompartitional to partitionless growth and it will benecessary to investigate how g of a given compositionwill transform to a. In the phase-field simulationssuch a transformation occurs when g has acontent below a critical limit, presented by curvesa and b for different M f values, in Fig. 4. In orderto investigate the predictions of solute drag theoryit is necessary to account for C diffusion in g aheadof the migrating g/a interface. It should first of allbe realized that the calculations for the interfacewere made under steady-state growth, whichstrictly implies that the transformation is partitionless.However, it has already been mentionedthat the results could be used as a good approximationfor non-steady state conditions, i.e. for partitionaltransformations.We now apply Eq. (19) and take u a Cand u g/aCFig. 7. Relation between half thickness and growth rate predictedfrom Eq. (19) and Fig. 6. Each curve holds for a differentalloy content as indicated. The hatched area represents velocitieswhere no physical solutions are possible.

I. Loginova et al. / Acta Materialia 51 (2003) 1327–13391337with two different growth rates represented by thetwo sides of the hatched area in Fig. 7. However,as shown by Hillert [1] only the case of the highestgrowth rate represents a stable situation. The criticalC content below which the transformation g →a is partitionless thus is the maximum of the u a Ccurve in Fig. 6. That value and correspondingvalues at other temperatures are plotted as curve cin Fig. 4.7.4. Comparison between phase-field and solutedragsimulationsThe critical C content was calculated as a functionof temperature by means of the solute-dragtheory and phase-field simulations. With themobility taken according to Appendix B solutedragtheory predicts a critical composition close tothe a/a g phase boundary, see curve c in Fig.4. Phase-field simulations using M f M/d gavea critical composition almost in the middlebetween the T 0 temperature and the phase boundary,see curve a. However, it was argued in section4 that a more realistic variation of f should betaken into account when evaluating M f . Therelation M f 0.235 M/d was then derived and thecorresponding critical compositions are given bycurve b. The discrepancy between solutedrag andphase-field simulations is much smaller for thischoice of M f but is still not acceptable. However,even though Eq. (28) was obtained as an exact resultit does not necessarily imply that the twoapproaches must give the same result because thediffusional mobilities and the thermodynamicproperties have a different variation across the phaseinterface in the two approaches. In addition, in thephase-field method all thermodynamic and kineticproperties as well as all their derivatives withrespect to distance are continuous over the wholeregion where φ is defined ( z ). Theinterface is truly diffuse in this case. However, inthe solute drag theory these properties vary piecewiselinearly over the interface, i.e. already the firstderivatives with respect to distance are discontinuous.Thus, both the concentration profile and thequantity G diffm evaluated from the concentrationprofile using Eq. (16) will differ and as a consequencethe part of driving force available to overcomethe interface resistance will differ. In Fig. 8the concentration profile has been plotted at 1000K for the velocity 0.0122 ms 1 and u C 0.00201. The dashed curve is obtained from thesolute-drag calculations and the solid curve fromphase field. In this case phase field and solute draggave a diffusional dissipation of 150 and 232 Jmol -1 , respectively. The total driving force, i.e.including the spike ahead of the interface, is 270Jmol -1 in both cases. That driving force leaves 120to overcome the interface friction in the phase-fieldsimulation. From M 7.2 10 10 m 4 J 1 s 1 at 1000K and V m 710 6 m 3 mol 1 we obtain by meansof Eq. (17) v 0.0123 ms 1 in very good agreementwith the phase field result. On the other hand,as can be seen from Fig. 8, this growth rate is notenough to make the transformation partitionlessaccording to the solute-drag simulations and thedriving force across the interface, excluding thespike in g, is 265 Jmol -1 leaving only 33 Jmol -1to overcome the interface friction. However, evenif we neglect that slight decrease in driving forceit is obvious that not enough driving force is leftto overcome interface friction and we conclude thatFig. 8. Phase-field calculation of the concentration profile atT = 1000 K, v 0.0122 m s -1 , u C 0.00201 usingM f 0.235M/d (solid curve). Solute-drag calculation of theconcentration profile at the same temperature and u a C 0.00201 (dashed curve). M is given in Appendix B.

1338 I. Loginova et al. / Acta Materialia 51 (2003) 1327–1339according to solute drag theory the alloy u C 0.00201 cannot grow partitionless at 1000 K.We may thus conclude that the discrepancy inthe predicted critical composition for massivetransformation stems from a different variation inthermodynamic properties and mobilities acrossthe interface in the two methods and that the interfaceis not as diffuse in the solute-drag theory asin the phase-field method. It should be emphasizedthough, when the interface velocity is so high thatall the dissipation of Gibbs energy is due only tointerfacial friction the two approaches give exactlythe same results, provided that we choose theinterfacial mobilities in a consistent way. This issimply because in Eq. (17) no assumptions aboutthe variation of properties across the interfaceenter.the detailed variation is unknown and thus one cannotsay that one assumption is better than another.One thus needs to evaluate the variation in propertiesthat best fit the experimental information onthe critical temperature. Another possibility is tocalculate the variation ab initio. Both theseapproaches will be explored in the near future.AcknowledgementsThe authors would like to thank professor MatsHillert for valuable advice and stimulating discussions.This work was financed by the SwedishResearch Council for Engineering Science and theSwedish Foundation for Strategic Researchthrough the Brinell Centre.8. ConclusionsAs shown recently by Ahmad et al. isobarothermalphase-field simulations predict a transition topartitionless transformation when the supersaturationis high enough. Here we have consideredformation of ferrite from austenite in binary Fe–Calloys and compared a novel method to simulatethe solute-drag effect with phase field simulationsusing an interfacial thickness in the order of atomicdimensions. The methods predict qualitatively thesame behavior although the quantitative agreementis less good. However, if the kinetic parametergoverning the evolution of the phase field is chosenconsistent with the actual variation of the phasefieldvariable the discrepancy is much reduced. Infact, in the limit of no dissipation by diffusionexact agreement is obtained between the phasefieldsimulations and conventional modeling basedon interfacial friction and an interfacial mobility.The reason that a discrepancy remains when thereis diffusional dissipation is that different assumptionsare made for the variation of properties acrossthe interface in phase-field and solute-drag modeling,respectively. In the phase field method thevariation is expressed as a function of the phasefield variable, obtained from the solution of thephase-field equation, whereas in solute-drag modelinga function of distance is postulated. At presentAppendix A. Thermodynamic description ofFe–C system, from Gustafson [8].G a m 0 G a m u C3 (0 G a FeC 0 G a Fe) 3RT u C3 ln u C3 1 u C3ln1 u C3 (A1) u C31 u C3L a Cva G momG g m 0 G g m u C ( 0 G g FeC 0 G g Fe) RT{u C lnu C (1u C )ln(1u C )} (A2) u C (1u C )LCvagThe quantities introduced in the expressions aboveare given functions of temperature:0G a Fe 1224.83 124.134T23.5143TlnT0.00439752T 2(A3)5.89269·10 8 T 2 77358.5T 10G a FeC 0 G a Fe 322050 75.677T (A4)L a Cva 190TG momif t 16507.5 t410 t14315 1500 t24(A5)(A6a)

I. Loginova et al. / Acta Materialia 51 (2003) 1327–13391339G mom 9180.5 9.283T9309.8 t46 t10135 600 t16Diffusional mobility in g [12]:RTMC g 4.529·10 exp 2.221·104 1995;43:689.[17] Gear CW. Numerical initial value problems in ordinary1 differential equations. Englewood Cliffs, NJ: Prentice-7 (B2)Hall, 1971.T[18] HARWELL Subroutine Library Specification, ComputerScience and Systems Division, HARWELL Laboratory,(17767u C 26436)m 2 s 1 Mobility of a/g interface [19]:(A6b)M 0.035exp 17700Tm 4 J 1 s 1 (B3)if t 1where t T/T C and T c 1043 K is theCurie temperature.References0G g Fe 237.57 132.416T24.6643TlnT0.00375752T 2(A7)[1] Hillert M. Metall. Mater. Trans. 2002;33A:2299–308.[2] Hillert M, Sundman B. Acta Met. 1977;25(11).5.89269·10 8 T 3 77358.5T 1[3] Ågren J. Acta Met. 1989;37:181.[4] Liu Z-K, Ågren J. Acta Met. 1989;37:3157.0G g FeC 0 GFe g 7720715.877T (A8) [5] Loginova I, Amberg G, Ågren J. Acta Mater. 2001;49:573.[6] Artemev A, Jin Y, Khachaturyan AG. Acta Mater.L g Cva 34671(A10)2001;49:1165.V m 710 6 m 3 mol 1(A11)[7] Ahmad NA, Wheeler AA, Boettinger WJ, McFadden GB.Phys. Rev. E 1998;58:3436–50.[8] Gustafson P. Scand. J. Metall. 1985;14:259.[9] Cahn JW, Allen SM. J. Phys. (Paris) Colloque1977;C7:C7–C51.Appendix B Kinetic parameters for Fe–C[10] Andersson J-O, Ågren J. J. Appl. Phys. 1992;72:1350.[11] Ågren J. Acta Met. 1982;30:841.Diffusional mobility in a [11]:[12] Ågren J. Scripta Met. 1986;20:1507.RTM a C 0.02·10 4 exp 10115[13] Hillert M. Phase equilibria, phase diagrams and phasetransformations—their thermodynamic basis. Cambridge,Texp0.5898 (B1)UK: Cambridge University Press, 1998.[14] Borgenstam A, Engström A,Höglund L, Ågren J. J. Phase1 2 p arctan 14.985 15309Equilibria 2000;21:269.Tm 2 s 1[15] Enomoto M. Acta Mater. 1999;47:3533.[16] Boettinger WJ, Warren JW. Acta Metall. Mater.Oxfordshire OX11 0RA, UK.[19] Hillert M. Metall. Trans. A 1975;6A:5.

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