The official published version of the Article can be accesed from the link below - Copyright @ 2011 APSWe address crystal nucleation and fcc-bcc phase selection in alloys using a multiphase field model that relies on Ginzburg-Landau free energies of the liquid-fcc, liquid-bcc, and fcc-bcc subsystems, and determine the properties of the nuclei as a function of composition, temperature, and structure. With a realistic choice for the free energy of the fcc-bcc interface, the model predicts well the fcc-bcc phase-selection boundary in the Fe-Ni system.This work has been supported by the Hungarian
Academy of Sciences under contract OTKA-K-62588, and by the ESA under PECS Contract No. 98059. Work by JRM has been sponsored by the Materials Sciences and Engineering Division, Office of Basic Energy Sci-
ences, U.S. Department of Energy

Granasy, L

Morris, JR

Toth, GI

Physical Review Letters

English

G. I. Tóth

J. R. Morris

L. Gránásy

Crossref

Ginzburg-Landau-Type Multiphase Field Model for Competing fcc and bcc Nucleation

Brunel University Research Archive

Ginzburg-Landau type multi-phase-field model for competing fcc and bcc nucleationG. I. To´th,1 J. R. Morris,2 and L. Gra´na´sy1, 31Research Institute for Solid State Physics and Optics, P. O. Box 49, H-1525 Budapest, Hungary2Oak Ridge National Laboratory, Oak Ridge, TN 37830, USA3BCAST, Brunel University, Uxbridge, Middlesex, UB8 3PH, UK(Dated: December 23, 2010)We address crystal nucleation and fcc-bcc phase selection in alloys using a multi-phase-field modelthat relies on Ginzburg-Landau free energies of the liquid-fcc (face centered cubic), liquid-bcc (bodycentered cubic), and fcc-bcc sub-systems, and determine the properties of the nuclei as a functionof composition, temperature and structure. With a realistic choice for the free energy of the fcc-bccinterface, the model predicts well the fcc-bcc phase-selection boundary in the Fe-Ni system.PACS numbers: 64.60.Q–, 64.60.My, 64.70.D–, 68.08.–p, 82.60.N–Freezing of undercooled liquids often starts withthe nucleation/formation of metastable (MS) crystallinephases. In agreement with Ostwald’s step rule, atomisticsimulations imply that the first crystal structure to formis the one, whose free energy is the closest to the freeenergy of the liquid [1]. In alloys this represents a multi-phase multi-component solidification problem. To date,the most efficient method used for addressing such prob-lems is the multi-phase-field theory (MPFT) [2]. It is,however, only as accurate as the free energy functionalit relies on. Early versions [2] of the MPFT predictedthat the third phase inevitably appears at the interfacebetween two bulk phases, a behavior originating from thespecific free energy surface assumed. A recent version ofMPFT eliminated the third phase entirely at the interface[3]. This is not always in agreement with real systems:Atomistic simulations for the Lennard-Jones (LJ) systemshow that although the stable phase is fcc, small nucleihave a bcc structure, and even the larger fcc crystalliteshave a bcc-like layer at the solid-liquid interface [1, 4],results also born out by the classical density functionaltheory (CDFT) [5]. These findings accord with the theo-retical prediction of Alexander and McTague that in sim-ple liquids the formation of bcc structure is preferred [6].Further simulations for the LJ system imply that vary-ing the pressure at fixed temperature, the bcc/fcc phaseratio can be tuned in small clusters [7]. Since preferencefor MS phase nucleation is quite general, it is desirable towork out microscopic models that can handle the struc-tural aspects of phase selection during nucleation.In this Letter, we present such a microscopic modelfor competing fcc and bcc structures. The MPFT is sup-plemented with a free energy that is based on the GLexpansion of the two-phase free energies [8–10], and con-siders thus the structural aspects of multiphase solidifi-cation. Our approach is unique in that it combines crys-tal structure with thermodynamic and interfacial data ofreal systems. In this respect our MPFT is more flexiblethan recent CDFT approaches [11], which, in turn, pro-vide a more detailed description of the solid-solid inter-face. Herein, we apply the GL-free-energy based MPFTto predict phase-selection in the Fe-Ni system.The standard MPFT form of the grand free energy ofa binary system relative to the initial liquid is:∆Ω =∫dr∑i<j2ij2(φi∇φj − φj∇φi)2 + ∆ω(φi, c) .(1)The differential operator on the right hand side has therequired symmetries [2]. In this expression ∆ω is therelative grand potential density and c the concentration.The sum runs over different (φi, φj) pairs of the structuralorder parameters, while∑j φj(r) = 1. When addressingfcc-bcc competition, without loss of generality, we maychose φ1, φ2, and φ3 = 1 − (φ1 + φ2) for the fcc, bcc,and liquid phases, respectively. These order parameterscan be combined to yield formal analogues of the solid-liquid order parameter m that describes crystalline freez-ing, and the solid-solid order parameter χ that monitorsthe fcc-bcc transition (Bain’s distortion) of the crystallattice used in an advanced CDFT of fcc-bcc transition[5]: φ ⇔ ||m|| ∈ [0, 1] and ψ ⇔ ||χ|| ∈ [0, 1], whereφ = φ1 + φ2 and ψ = φ2/φ. The methodology of theMPFT anchors the free energy surface to the free ener-gies of the bulk phases. Specifically, the local grand po-tential density of the multi-phase system is related to thecontributions ∆ωij of the two-phase systems as follows:∆ω(φ, ψ, c) = [1− p12(ψ)]∆ω13(φ, c)++ p12(ψ)∆ω23(φ, c) + a12(c)P (φ, ψ)g12(ψ),(2)where the interpolation functions pij vary monotonouslybetween 0 and 1 so that pij(0) = 0 and pij(1) = 1,whereas P (φ, ψ) = [1 − p12(ψ)]p13(φ) + p12(ψ)p23(φ),which reflects that the solid-solid order parameter is ir-relevant in the liquid state. The first two terms of Eq.2−15 −10 −5 0 5 10 1500.51x [Angstrom]  ||!PFT||||cPFT||||!MC||||cMC||FIG. 1: Interfacial order parameter (solid) and concentrationprofiles (dashed) as predicted by the GL approach for thefcc-liquid interface in the Cu-Ni system. For comparison, thenormalized density peaks (squares) and concentration profile(circles) from atomistic simulations [15] are also shown.(2) interpolate between the fcc-liquid and bcc-liquid freeenergies, while the third term adds a free energy barrierin between the two solid phases that disappears in theliquid. The two-phase contributions can be expressed as∆ωi3(φ, c) = ai3(c)gi3(φ)++ pi3(φ)∆ωi(c) + [1− pi3(φ)]∆ω3(c),for i = 1 or 2, which have the shape of a skeweddouble-well. Here gij are double-well functions, for whichgij(0) = gij(1) = 0, with a maximum in between, whilethe functions ∆ωi represent the grand potential densitiesof the ith phase relative to the initial liquid state.With these definitions, Eq. (2) recovers the relativefree energies of the bulk phases. However, the results forthe non-bulk states depend on the specific choice of thesefunctions. In the usual application of the MPFT, they arechosen intuitively. In contrast, here we use forms deducedfrom the GL expansion of the two-phase free energies [8–10], which forms contain the structural information:bcc-liquid: g13 = φ2(1− φ)2 and p13 = φ3(4− 3φ),fcc-liquid: g23 = φ2(1− φ2)2 and p23 = φ4(3− 2φ2),bcc-fcc: g12 = ψ2(1− ψ)2 and p12 = ψ3(4− 3ψ).The composition dependent model coefficients are inter-polated as 2ij(c) = (1 − c)2ij,A + c2ij,B and aij(c) =(1− c)aij,A + caij,B , where the constants 2ij,Y and aij,Ycan be expressed in terms of the free energy and thicknessof the equilibrium interface between phases i and j forpure component Y . Unlike the CDFT, where the time-averaged particle density is the order parameter, in ourmodel the solid-liquid transitions are monitored by thereduced Fourier amplitude of the dominant density waves(a single-mode approach), whereas the fcc-bcc transitionby an order parameter related to Bain’s distortion. Thefree energy of the interfaces emerge from bulk and gra-dient contributions associated with a continuous changeof these order parameters across the interface.Since the nucleus represents an extremum of the grandpotential, its properties can be found by solving theEuler-Lagrange (EL) equations [5, 9, 12]: δ∆Ω/δφi =0 10 20 30 4000.51r [Angstrom]  !bcc!fcc0 10 20 30 4000.51r [Angstrom]  !bcc!fcc1300 1450 1600 1750104103102T [K]W*  [kT]  pure bcccomp. bccpure fcccomp. fcc(c)(a)(b)FIG. 2: Crystal nuclei in Fe: (a) composite-bcc nucleus at1449 K [thin lines, square in panel(c)], and composite-fcc nu-cleus at 1441 K [heavy lines, circle in panel (c)]; (b) composite-bcc nucleus at 1300 K [triangle in panel(c)]. (c) Nucleationbarrier vs. temperature.1200 1400 1600104103102T [K]W*  [kT]  pure bcccomp. bccpure fcccomp. fcc0 5 10 15 20 25 3000.51r [Angstrom]  0 5 10 15 20 25 3000.51r [Angstrom]  !bcc!fcc||c||!bcc!fcc||c||(b)(c)(a)FIG. 3: Crystal nuclei in Fe50Ni50: Notations are as for Fig.2. The respective temperatures are T = 1373.5 K and 1183 Kfor panel (a) and 1150 K for panel (b). Note that ||c|| ∈ [0, 1].30 5 10 15 20 2500.51r [Angstrom]  !bcc!fcc0 5 10 15 20 2500.51r [Angstrom]  !bcc!fcc1000 1200 1400 1600 1800104102103101T [K]W*  [kT]  pure bcccomp. bccpure fcccomp. fcc(c)(a)(b)FIG. 4: Crystal nuclei in Ni: Notations are as for Fig. 2.The respective temperatures are T = 1382.5 K and 1050 Kfor panel (a) and 1000 K for panel (b).1200 1400 1600 180000.10.20.30.4T [K]phase fraction  0 0.2 0.4 0.6 0.8 100.10.20.30.4cX bcc  1300 K1400 K1500 K1600 KXbcc − FeXbcc − FeNiXbcc − NiXfcc − FeXfcc − FeNiXfcc − NiFIG. 5: Volume fraction of the third phase (”surface phase”)in composite nuclei: (a) Temperature dependence of bcc frac-tion in composite-fcc nuclei (heavy lines), and of fcc fractionin composite-bcc nuclei (thin lines). (b) bcc fraction vs. Niconcentration at different temperatures.λ(r) and δ∆Ω/δc = 0, where δ∆Ω/δη is the first func-tional derivative of the grand potential difference withrespect to the field η, while the Lagrange multiplier λ(r)ensures the local constraint∑j φj(r) = 1 [13]. The re-spective boundary conditions are as follows: unperturbedliquid properties in the far field and zero field gradientsat the center. We assume isotropic interfacial properties,a fair approximation for metallic systems. This bound-ary value problem has been solved numerically by therelaxation method. The excess free energy of the nucleihas been obtained by inserting the solution into Eq. (1).The two-phase limits of the present model have beentested previously: The GL technique proved successfulin describing (i) the nucleation barrier for fcc structure[9], (ii) the properties of the bcc-liquid interfaces [8, 14],and (iii) the transition between the bcc and fcc phases[10]. The chemical part of our model has been testedagainst atomistic simulations for the Cu-Ni system: Theparameter free GL predictions for the order-parameterand nanoscale concentration profiles are in a remarkableagreement with the Monte Carlo results [15] (Fig. 1).Owing to the lack of known equilibrium coexistenceconditions between the bulk fcc and bcc phases, the well-known LJ and hard-sphere systems are not suitable for afull testing of our model. Thus, we have chosen the Fe-Nisystem, where from combined experiments and atomisticsimulations a nearly complete input set is available [16].The least accurate input is the orientation average of thefree energy of the fcc-bcc interface. For Fe, estimatesof γfcc−bcc for different orientations range between 179mJ/m2 [17] and about twice the solid-liquid interfacialenergy (∼672 mJ/m2 [16]), yielding ∼425 mJ/m2 for theaverage of the upper and lower limits, which we take asan estimate of the orientation average. Thus the energycontribution of the defects at the fcc-bcc interface is in-corporated implicitly in a coarse-grained manner.First, we present our results for crystal nuclei in Fe,Fe50Ni50, and Ni (see Figs. 2–4). In panels (a) and (b),the radial phase-field and concentration profiles are dis-played. In all cases we observe at least a small amountof third phase (”surface phase”) at the solid-liquid in-terface. However, the fcc surface layer on bcc nuclei isfar less pronounced than the bcc layer on fcc nuclei [Fig.5(a)]. With increasing undercooling, the volume fraction(X) of the third phase increases [Fig. 5(a)], which is re-flected in the non-monotonic composition dependence ofX [Fig. 5(b)], following from the shape of the respectiveliquidus line in the phase diagram. In Fe, nuclei with abcc core (composite-bcc type) are significantly preferredto fcc core nuclei, whereas in Ni, at temperatures accessi-ble for experiments, composite-fcc nuclei with a bcc sur-face layer dominate [see Figs. 2(c) and 4(c)]. The nucleiobserved at the 1:1 composition behave similarly to thosefor Ni [Fig. 3(c)], however, with some amount of sur-face precipitate of Ni at 20% relative undercooling [Fig.3(a)]. At extremely large undercoolings, composite-bcc40 0.1 0.2 0.3 0.4 0.514001500160017001800cT [K]BCCFCCFIG. 6: Phase-selection in the Fe-Ni alloy system. The greysolid and dashed lines correspond to the liquidus and soliduscurves. The symbols indicate the structure nucleated in theexperiment: squares - bcc; circles - fcc [18]. The fcc-bcc phase-selection boundary predicted for heterogeneous nucleation atthree values of γfcc−bcc are shown: 672 (black dash-dot), 425(black solid), and 179 mJ/m2 (black dashed).nuclei are preferred for all compositions. At all under-coolings we studied, composite nuclei are thermodynam-ically preferable to the respective single-phase nuclei.Next, we use the present MPFT approach to predictthe phase-selection map for Fe-Ni alloys and compare itto experiments [18]. Since in metallic systems homoge-neous nucleation has probably never been realized, weassume heterogeneous nucleation. In the spirit of thehighly successful free growth limited model of heteroge-neous nucleation by Greer et al. [19], the phase-selectionboundary for heterogeneous nucleation is determined bythe condition of equal critical radii for the fcc and bcctype nuclei. The fcc-bcc phase-selection boundary pre-dicted with γfcc−bcc = 425 mJ/m2 is in a fair agreementwith the experiments (Fig. 6). For comparison, resultsfor the upper and lower limits are also shown, which enve-lope the experimental fcc-bcc phase-selection boundary.Summarizing, we have presented a microscopic multi-phase-field theory of competing fcc and bcc nucleationthat is anchored to measurable physical properties. Ourstudy indicates that composite nuclei are preferable tosingle-phase nuclei. With a reasonable choice of modelparameters, the GL-free-energy based MPFT predictsthe phase-selection map fairly well for Fe-Ni alloys.This work has been supported by the HungarianAcademy of Sciences under contract OTKA-K-62588,and by the ESA under PECS Contract No. 98059. Workby JRM has been sponsored by the Materials Sciencesand Engineering Division, Office of Basic Energy Sci-ences, U.S. Department of Energy.[1] P. R. ten Wolde and D. Frenkel, Phys. Chem. Chem.Phys. 1, 2191 (1999).[2] I. Steinbach et al., Physica D 94, 135 (1996).[3] R. Folch and M. Plapp, Phys. Rev. E 72, 011602 (2005).[4] P. R. ten Wolde et al., Phys. Rev. Lett. 75, 2714 (1995).[5] Y. C. Shen and D. W. Oxtoby, Phys. Rev. Lett. 77, 3585(1996).[6] S. Alexander and J. McTague, Phys Rev. Lett. 41, 702(1978); W. Klein, Phys. Rev. E. 64, 056110 (2001).[7] C. Desgranges and J. Delhommelle, Phys. Rev. Lett. 98,235502 (2007); J. Am. Chem. Soc. 128, 10368 (2006).[8] W. H. Shih et al., Phys. Rev. A 35, 2611 (1987).[9] L. Gra´na´sy and T. Pusztai, J. Chem. Phys. 117, 10121(2002); G. I. To´th and L. Gra´na´sy, J. Phys. Chem. B113, 5141 (2009).[10] V. P. Dimitriev et al., Phys. Rev. B 44, 7248 (1991).[11] K. R. Elder et al., Phys. Rev. Lett. 88, 245701 (2002);A. Jaatinen et al., Phys. Rev. E 80, 031602 (2009).[12] L. Gra´na´sy et al., Phys. Rev. Lett. 88, 206105 (2002);98, 035703 (2007).[13] The EL equations are: M(Φ, c)∆ΦT = b(Φ, c,∇Φ,∇c)and 2∂c∆ω +∑i<j ∂c2ij(c) (φi∇φj − φj∇φi)2 = 0. HereΦ = (φ1, φ2), ∆Φ = (∆φ1,∆φ2), superscript T denotestransposition, bi = Ei − E3, whereas Ei = ∂φi∆ω +∑i 6=j[∂cij(c)2∇c+ 2ij(c)2∇φj](φi∇φj − φj∇φi), andthe elements of the 2 × 2 matrix M are: M11 = 213(1 −φ2)2 + (212 + 223)φ22; M22 = 223(1− φ1)2 + (212 + 213)φ21;and M12 = M21 = 213φ1(1−φ2)+223φ2(1−φ1)−212φ1φ2.[14] K.-A. Wu et al., Phys. Rev. B 73, 094101 (2006).[15] H. Ramalingam et al., Interf. Sci. 10, 149 (2002).[16] Free energy of bulk phases: Y. Y. Chuang, K. C. Shieh,and Y. A. Chang, Metall. Trans. A 17, 1373 (1986). Inter-face thickness: 1 nm for all interfaces, [∼ 3.5 atomic dis-tances: B. B. Laird and A. D. J. Haymet, Chem. Rev. 47,2491 (1993)]. Solid-liquid interface energy of Fe and Ni atthe melting point (T f ): L. Gra´na´sy and M. Tegze, Mater.Sci. Forum 77, 243 (1991). Interface energy of MS phases:The ratio αfcc/αbcc ≈ 1.72 deduced for Turnbull’s coef-ficients from atomistic simulations for Fe [D. Y. Sun, M.Asta, and J. J. Hoyt, Phys. Rev. B 69, 174103 (2004)]was used to estimate it: γMS/γS ≈ (αMST fMS)/(αST fS ).We utilized Richard’s rule and neglected the density dif-ference of the crystalline phases. Here α = γL−1ρ−2/3,γ the interface energy, L the heat of fusion, and ρ thesinglet density. Subscript S denotes the stable phase.[17] J. K. Chen et al., Acta Mater. 45, 4415 (1997).[18] A. Zambon et al., Acta Mater. 46, 4657 (1998).[19] A. L. Greer et al., Acta Mater. 48, 2823 (2000).

Ginzburg-landau-type multiphase field model for competing fcc and bcc nucleation

Ginzburg-landau-type multiphase field model for competing fcc and bcc nucleation

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