16 JOM • September 2001

OverviewComputational Materials

This paper presents a brief overview ofrecent developments in the application offirst-principles calculations to the study ofbulk and interfacial thermodynamic proper-ties and phase equilibria in alloys. Amongthe applications discussed are calculationsof: bulk thermodynamic properties, phaseboundaries, interfacial free energies, and pre-cipitate morphologies. The article concludesby highlighting some further recent develop-ments that are likely to lead to increasingapplications of these modeling techniques.

INTRODUCTION

In recent decades, substantial progresshas been made toward parameter-freemodeling of phase equilibria in metallicalloys.1–8 Fundamental knowledge of themicroscopic factors governing alloyphase stability has increased greatly as aresult of first-principles calculationsbased upon theories of quantum physicsand statistical mechanics. Recent workhas demonstrated substantial quantita-tive improvements in the accuracy ofalloy phase diagrams calculated fromfirst principles. Also, ab-initio methodshave been extended recently to calculateboth bulk and interfacial thermodynamicproperties, which are being used to im-prove the predictive capabilities of mi-crostructural evolution simulations inalloys. The development of a predictivefirst-principles framework for the calcu-lation of alloy-phase stability has ben-efited from enormous increases in com-puting power coupled with significantadvances in the accuracy and efficiencyof computational methods. With theserecent improvements in scope and quan-titative accuracy, ab-initio methods arebecoming a significant tool for alloy de-

A First-Principles Approach toModeling Alloy Phase Equilibria

M. Asta, V. Ozolins, and C. Woodward

velopment. This article presents a briefoverview of these computational meth-ods through a review of recent applica-tions to the modeling of precipitationthermodynamics in Al(Sc) alloys. Theseexamples are intended to represent thecurrent state of the art in applications offirst-principles modeling to the calcula-tion of phase boundaries in binary andmulticomponent alloys, heterophase in-terfacial free energies, and equilibriumprecipitate morphologies. Far more ex-tensive reviews of first-principles alloy-thermodynamic computational methodscan be found in References 1–8; this ar-ticle only highlights a few recent devel-opments in this field.

Because of their high specific strength,aluminum-rich Al-Sc alloys are increas-ingly being considered for applicationsat high homologous temperatures. Scan-dium, which provides the greatest in-crease in strength, per atomic percent, ofany alloying addition in Al,9 is onlyslightly more dense (about 11 percent)than aluminum itself. That strengthen-ing effect is attributed to the small (5–30nm) coherent Al3Sc precipitates that formduring the ageing of supersaturatedAl(Sc) solid solutions9–16 in the tempera-ture range of 300–350∞C. The Al3Sc phaseforms in the face centered cubic (fcc)-based L12 crystal structure.

BULK ALLOYTHERMODYNAMIC

PROPERTIES

The starting point for a first-principlescalculation of phase equilibria is an ac-curate quantum-mechanical calculationof bulk-alloy energetics. Figure 1 showsthe results of such first-principles calcu-lations for the formation enthalpies (DH)of observed intermetallic compounds inthe Al-Sc system. The dark filled sym-bols in Figure 1 represent the results offirst-principles calculations performedwithin the framework of electronic den-sity-functional theory17 (details of thecalculations can be found in Reference18). Open symbols in Figure 1 corre-spond to measured data from high-tem-perature (diamonds)19,20 and reaction-drop (circles)16 calorimetry. For each com-pound, the measured and calculatedvalues are found to agree to within tenpercent (5 kJ/mole) with the level ofagreement between experiment and

theory comparable to that between inde-pendent calorimetry measurements.Similar accuracy has been demonstratedin first-principles calculations of DH fora wide range of metallic alloy systems(see, for example, References 1–3). In theabsence of available calorimetry data,first-principles calculations can be em-ployed readily to calculate compoundformation enthalpies to augment alloythermodynamic databases required inCalphad “computational thermody-namic” modeling21,22 of phase diagrams.By combining quantum mechanics withcomputational thermodynamics, first-principles calculations are finding ap-plication in alloy design.23

In addition to providing accurate totalenergies, density-functional theory hasbeen widely used as the basis for first-principles calculations of bulk and de-fect alloy thermodynamic properties atfinite temperatures. A significant chal-lenge associated with such calculationsis the accurate calculation of both theenergy and entropy contributions to al-loy free energies. For the aluminum-richsolid-solution phase in Al-Sc, the equi-librium solid-solubility of scandium isknown to be very low, less than a fewtenths of an atomic percent at the eutec-tic temperature of 660∞C.11,24–26 Similarly,the equilibrium Al3Sc phase field is ex-tremely narrow,15,16 and the phase isknown to remain highly ordered up tothe melting point.27 Therefore, equilib-rium solution-thermodynamic proper-ties of the Al(Sc) and Al3Sc phases can bedescribed accurately by ideal-solution

Figure 2. The first-principles calculated (solidline)28 and experimentally measured (filledsymbols)11,12,24–26 solid-solubility limits in Al-rich Al-Sc alloys.

Figure 1. The calculated and measured heatsof formation of Al-Sc intermetallic compounds.

172001 September • JOM

models, ignoring interactions be-tween substitutional defects. Thefirst-principles calculation of al-loy free energies in this case canbe reduced to the problem ofcomputing the temperature-de-pendent free energies of pure alu-minum, Al3Sc, and scandium im-purities in aluminum. From theresults of such calculations,phase-boundaries for alumi-num-rich compositions can bepredicted with only the atomicnumbers of the alloy constitu-ents required as input. The re-sults of such a calculation, takenfrom Reference 28, are repro-duced in Figure 2. In this plot,the solid lines are calculated re-sults while the filled symbols rep-resent experimental solubilitylimits derived from resistivitymeasurements.11,24–26 The first-principles calculated phaseboundaries agree to within 50 Kof experimental measurementsto temperatures up to the mea-sured eutectic point. This resultis representative of the highestlevel of accuracy attained to datein a fully first-principles calcula-tion of alloy phase boundaries.

When alloy thermodynamicsare modeled from first principles,the calculations provide detailedinsight into the microscopic fac-tors influencing phase equilib-ria. For example, to attain thelevel of accuracy displayed inFigure 2, the calculations had toincorporate entropic contribu-tions to alloy free energies asso-ciated with atomic vibrations.Specifically, it was found28 thatvibrational entropy accountedfor a 27-fold enhancement of thecalculated solid-solubility limitsfor scandium in aluminum, lead-ing to a 450 K decrease in calcu-lated aluminum solvus phaseboundary at the maximum mea-sured scandium solid-solubilitylimit. The origin of this effectwas traced to a difference in sign be-tween the vibrational entropy per scan-dium atom in the aluminum-solid solu-tion versus ordered Al3Sc. Specifically,the vibrational entropy of formation ofthe Al3Sc phase is found to be stronglynegative, due primarily the stiffness ofthe energetically favorable Al-Sc near-est-neighbor bonds. In an aluminumhost, however, the formation of Al-Scbonds around a scandium solute atomgives rise to perturbations in charge (seeFigure 3) and changes in bond lengths ofdistant Al-Al nearest-neighbors, lead-ing to a weakening of these bonds and anoverall positive contribution from atomicvibrations to the entropy of solution.Thus, calculations lead to the interesting

conclusion that vibrations stabilize thesolution phase while they destabilizethe compound, leading to a significantincrease in computed solid solubility lim-its.

The first-principles calculation of vi-brational entropy in disordered alloyphases remains a significant computa-tional challenge, and such calculationshave been undertaken in relatively fewab-initio studies of alloy thermodynamicproperties to date29–39 (a very recent re-view is given by van de Walle andCeder40). Further work is warranted toestablish the generality of the authors’findings related to the importance ofvibrational entropy for accurate predic-tions of alloy phase boundaries. How-

ever, subsequent to the publica-tion of Reference 28, the authorsbecame aware of work by Ze-ner41 which, more than 40 yearsago, suggested the important roleof vibrational entropy in gov-erning solid solubility limits inalloys. In the Zener analysis, thelogarithm of measured solubil-ity limits is plotted versus 1/T(where T represents the absolutetemperature) and the solubilityenhancement factor associatedwith vibrational entropy can bederived from the intercept of thecurve at 1/T = 0. Zener per-formed such an analysis for awide range of aluminum alloysand derived solubility enhance-ment factors ranging between ten(for zirconium) to 60 (for nickel).The 27-fold increase obtained inthe author's calculations for scan-dium falls in the middle of therange of values found by Zener.

The example in Figure 2 dem-onstrates the first-principles cal-culation of phase boundariesbetween alloy phases with di-lute defect concentrations. It is,therefore, important to stress thatcomparable accuracy has beendemonstrated in several first-principles calculations for alloyswith concentrated compositions.Specifically, the cluster expan-sion framework2–4,42 has foundwide application in the first-prin-ciples calculation of phase dia-grams for metal and semicon-ductor2,43 alloys as well as ce-ramic systems.6,44 The methodprovides a formalism for express-ing the energy of different atomicarrangements on an underlyinglattice framework in terms of anexpansion of pair and many-body cluster interaction param-eters. These parameters are de-rived from electronic pertubationtheories 1,5,45–49 or by fitting to re-sults of first-principles calcula-tions of alloy energetics 2–4. Re-

cently, Zunger, Wolverton, and collabo-rators developed a first-principles meth-odology incorporating a model for com-position-dependent elastic strain energy(associated with atomic size differences)within the cluster expansion frameworkfor alloy energetics described above.2,50–55

INTERFACIALTHERMODYNAMIC

PROPERTIES

Once an accurate parameterization ofthe alloy energetics has been derived,the cluster expansion provides a rapidmethod for calculating energies of alloyphases with arbitrary degrees of chemi-cal short- and long-range order. Specifi-cally, the method provides a first-prin-

Figure 3. The calculated charge density differences in analuminum host resulting from a scandium substitutionalimpurity. The charge density shows characteristic long-ranged oscillations with a functional distance dependencecharacterized by the theory of Friedel.

Figure 4. The equilibrium shape of a coherent Al3Sc precipi-tate in an aluminum matrix, as predicted from first-principlesMonte-Carlo simulations at 600 K. Al atoms are omitted forclarity. Arrows correspond to [100], [010], and [001] crystal-lographic directions. Atoms with five scandium next-nearestneighbors (NNN) are colored blue (corresponding to {100}facets), atoms with four NNN scandium are colored green(corresponding to {110} facets), and atoms with three NNNscandium are colored red (corresponding to {111} facets).Orange and yellow scandium atoms have two and one NNNscandium, respectively.

18 JOM • September 2001

ciples basis for efficient Monte-Carlosimulations that permit the calculationof configurational free energies, short-range order, and phase diagrams foralloys with concentrated compositions.Although applications of the first-prin-ciples cluster expansion framework aretoo numerous to review in the presentarticle, References 2–4, 43, and 44 pro-vide examples and further details. Fol-lowing are examples of recent applica-tions of the approach to calculating in-terfacial free energies 56-57 and modelingcoherent precipitate morphologies52-54.

Figure 4 displays the equilibriumshape of an Al3Sc precipitate obtainedfrom Monte-Carlo simulations at a tem-perature of 600 K.58 These simulationswere based upon a cluster-ex-pansion parameterization of thevibrational free energies58, result-ing in a thermodynamic modelconsistent with the results shownin Figure 2. The simulation con-ditions leading to the structureshown in Figure 4 were chosento produce a precipitate with adiameter of roughly 7 nm. Spe-cifically, simulations were con-ducted with fixed numbers ofaluminum and scandium atoms,with an overall composition be-yond the equilibrium solvusboundary. The simulations wereinitiated with a random arrange-ment of aluminum and scandiumatoms on the sites of an fcc lat-tice, and, after a prolonged equili-bration period, the Al3Sc precipi-tate nucleated, grew, and finallyequilibrated to the shape shownin Figure 4.

The simulation study wasmotivated by a very recent de-tailed transmission-electron-mi-croscopy study of Al3Sc precipi-tate morphologies performed byMarquis and Seidman.13 In Al-0.3 wt.% scandium alloys aged between300∞C and 400∞C, the authors clearlyobserved faceted morphologies for co-herent precipitates ranging between 5nm and 10 nm in diameter. Because fac-ets were observed on 100, 111, and 110precipitate faces in well-annealedsamples, the equilibrium crystal shapefor these small Al3Sc precipitates wasconcluded to be a GreatRhombicuboctahedron.13 In more diluteAl-0.1 wt.% scandium alloys, similar heattreatments were observed to lead to com-plex cauliflower precipitate morpholo-gies (observed also by Novotny andArdell14), interpreted as the result ofgrowth instabilities associated with thelower supersaturation levels.13 The first-principles simulations leading to Figure4 were a basis for establishing the equi-librium crystal shapes of nanometer-scale precipitates in Al-Sc to further vali-date these interpretations of the experi-

mental observations. The simulation re-sults represented in Figure 4 predict anequilibrium precipitate morphology thatis clearly faceted on 100 planes. Further-more, the rounding of cube corners andedges in the simulated precipitate shapescan be interpreted as reflecting a ten-dency towards faceting on 110 and 111planes as well. These simulation resultsclearly support the interpretation13 thatequilibrium crystal shapes of 5–10 nmAl3Sc precipitates are highly faceted.Furthermore, these first-principles cal-culations confirm continuum-model pre-dictions59 that the morphologies of Al3Scprecipitates with diameters in the rangeof 5–10 nm are negligibly affected byanisotropies in the elastic strain energyof precipitate and matrix phases. Rather,the equilibrium shapes of these smallprecipitates are found to be a reflectionof the anisotropy of Al/Al3Sc hetero-phase interfacial free energies.

Figure 5 plots the temperature depen-dence of first-principles calculated ex-cess free energies (g) for coherent Al/Al3Sc interfaces with 100 and 111 crystal-lographic orientations. These results rep-resent an extension of previous work60,61

devoted to the calculation of interfacialfree energies in this system. Figure 5exceeds previous work by incorporatingvibrational entropy in the calculation ofexcess free energies58 (using the samefirst-principles free energy model lead-ing to the results in Figure 4). The calcu-lated temperature dependence is foundto be appreciable between 400–800 Kwith 100 and 111 free energies decreas-ing by roughly 25 percent over this tem-perature range. From the standpoint of

precipitate morphology, the im-portant feature of the curvesshown in Figure 5 is the appre-ciable anisotropy between 100and 111 interfacial free energies.Using previously published,first-principles-calculated zero-temperature interfacial energies60

in a Wulff construction, Marquisand Seidman13 established thatthe degree of anisotropy is suffi-cient to lead to faceting of thetype observed experimentally.These authors also consideredthe magnitude of the first-prin-ciples calculated values of g inlight of results obtained for coars-ening kinetics, finding reason-able agreement between theo-retical and measured coarseningrate constants in this system.

TERNARY SYSTEMAPPLICATIONS

To date, first-principles calcu-lations of alloy phase equilibriahave been applied primarily tobinary systems, with relativelyfew such studies undertaken forternary systems.62–73 Because

technologically relevant alloys generallypossess many more than two compo-nents, there is considerable motivationto extend the range of applicability offirst-principles methods to multicompo-nent systems. At present, accurate first-principles calculations of phase diagramsfor concentrated ternary systems remaina significant challenge, with applicationsto systems with four or more compo-nents largely intractable. However, forsystems such as Al-Sc, in which the equi-librium defect concentrations are dilute,the first-principles computational frame-work described in the previous para-graph can be readily generalized to mul-ticomponent systems, as detailed in Ref-erence 74.

For example, consider the partition-ing of magnesium in dilute two-phaseAl-Sc-Mg alloys. Experimentally, it hasbeen determined that magnesium ter-nary additions partition strongly to the

Figure 6. The calculated distribution of magnesium ternaryadditions across a planar (100) coherent interface in a two-phase Al-Al3Sc alloy at 600 K. Blue and red circles in the tophalf of the figure denote sites occupied by aluminum andscandium atoms, respectively, at zero temperature. In theplot black symbols represent calculated magnesium con-centrations in the pure-Al (010) planes, while green arecompositions on mixed Al-Sc (010) planes. In agreementwith experimental observations,75 magnesium is predictedto partition strongly to the aluminum phase. First-principlescalculations also predict equilibrium segregation of magne-sium to {100} Al/Al3Sc interfaces, with preferred sites corre-sponding to second neighbors of interface scandium atoms.

Figure 5. The first-principles calculated inter-facial free energies for coherent Al/Al3Sc in-terfaces. Square and diamond symbols cor-respond to first-principles calculated resultsfor {100} and {111} interface orientations,respectively.

192001 September • JOM

Al phase75 in two-phase Al-Al3Sc alloys.Figure 6 shows the results of a first-principles calculation of the magnesiumsolute distribution across a (100) Al/Al3Sc interface in two-phase, Al-rich, Al-Sc-Mg alloys. The plotted symbols, whichcorrespond to calculated compositionprofiles across the interface, clearly showthe magnesium partitioning to the alu-minum phase, in agreement with ex-perimental observations. Additionally,these calculations suggest that magne-sium is enhanced at the interface. Inother words, equilibrium segregation ofmagnesium solute to the interfacialboundary is predicted, leading a non-negligible interfacial excess concentra-tion (adsorption) of this ternary alloyingspecies. The magnesium atoms are foundto prefer crystallographic positions thatare second-neighbors to interface scan-dium atoms. The calculations leading tothe results in Figure 6 were based on theformalism of dilute solution thermody-namics as described in Reference 74. Dueto the relatively concentrated composi-tions of magnesium near the interfacethe approach is not expected to be quan-titatively precise. However, the meth-odology clearly provides valuable in-sight into trends related to phase parti-tioning and interface adsorption, impor-tant information from an alloy-designperspective that is often difficult to probeexperimentally.

CONCLUSIONS

In a recent article, Chen et al.76 discussa strategy for linking the types of first-principles methods discussed in this ar-ticle with higher length-scale phase-fieldmodels of microstructural evolution.This combination of atomic and meso-scale models provides a novel hierarchi-cal multi-scale framework for predictivemodeling of microstructural evolutionaccompanying solid-state phase trans-formations occurring in the processingof alloys. An important recent develop-ment is the formulation of a cluster-expansion framework for modeling in aconsistent first-principles frameworkboth thermodynamic and kinetic prop-erties of alloys.77 Specifically, van derVen et al.77 recently described how first-principles calculations of migration en-ergies for vacancy diffusion can be in-corporated within a cluster-expansionmodel of alloy thermodynamics, form-ing the basis for ab-initio kinetic Monte-Carlo studies of diffusion and orderingkinetics in alloys. This approach shouldprove to be of practical value in aug-menting kinetic databases employed inthe modeling of solid-state phase trans-formations in alloys. Also, an effort isunderway to automate the first-prin-ciples methods described above for thecalculation of alloy phase diagrams.78

It is increasingly recognized that first-principles computational techniques,

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M. Asta is with the Department of Materials Science andEngineering at Northwestern University. V. Ozolins iswith Sandia National Laboratories. C. Woodward is withthe U.S. Air Force Research Laboratory, Materials andManufacturing Directorate.

For additional information, contact M. Asta,Northwestern University, Department of Ma-terials Science and Engineering, 2225 NorthCampus Drive, Evanston, Illinois 60208-3108;telephone (847) 491-5940; fax (847) 491-7820;e-mail m-asta@northwestern.edu.

when combined with complementarymodeling methods, including computa-tional thermodynamics and meso-scalemodels of microstructure evolution, canplay a role in reducing the time and costassociated with the process of materialsdevelopment. With continued develop-ments in accuracy, computational power,and automation, the first-principlesframework outlined in this article is an-ticipated to be applied more and more asa computational modeling tool in thescience and engineering of alloys.

ACKNOWLEDGEMENTS

Research by M. Asta has been supportedby the National Science Foundation undergrants DMR-0080766 and DMR-0076063.Research by V. Ozolins was supported by theU.S. Department of Energy, Office of BasicEnergy Sciences, Materials Science Divi-sion, under contract number DE-AC04-94AL85000. The research of C. Woodwardwas supported by the U.S. Air Force Office ofScientific Research and was performed at theU.S. Air Force Research Laboratory, Mate-rials and Manufacturing Directorate,Wright-Patterson Air Force Base under Con-tract No. F33615-01-C-5214. This researchwas also supported in part by a grant ofcomputer time from the U.S. Department ofDefense High Performance Computing Mod-ernization Program, at the AeronauticalSystems Center-Major Shared Resource Cen-ter, on the IBM-SP2 and IBM-SP3. Use wasalso made of resources at the National En-ergy Research Scientific Computing Center,which is supported by the Office of Science ofthe Department of Energy under ContractNo. DE-AC03-76SF00098.

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