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Acta Materialia 53 (2005) 3225–3252

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First-principles calculation of structural energetics ofAl–TM (TM = Ti, Zr, Hf) intermetallics

G. Ghosh *, M. Asta

Department of Materials Science and Engineering, Robert R. McCormick School of Engineering and Applied Science, Northwestern University,

2220 Campus Dr., Evanston, IL 60208-3108, USA

Received 29 January 2005; received in revised form 29 January 2005; accepted 21 March 2005

Available online 28 April 2005

Abstract

The total energies and equilibrium cohesive properties of 69 intermetallics in the Al–TM (TM = Ti, Zr and Hf) systems are

calculated from first-principles employing electronic density-functional theory, ultrasoft pseudopotentials and the generalized

gradient approximation. This work has been undertaken to investigate systematics in Al–TM alloying energetics, and to aug-

ment available calorimetric data for enthalpies of formation in support of the development of accurate multicomponent ther-

modynamic databases for these technologically interesting systems. The accuracy of our calculations is assessed through

comparisons between theoretical results and experimental measurements (where available) for lattice parameters, elastic prop-

erties and formation energies. The concentration dependence of the heats of formation for all three binary systems are very

similar, being skewed towards the Al-rich side with a minimum around Al2TM. In all three binary Al–TM systems, the cal-

culated zero-temperature intermetallic formation energies generally agree well, within a few kJ/mol, with calorimetric data

obtained by direct reaction synthesis. This level of agreement suggests high accuracy for the calculated enthalpies of formation

reported for structures where no such measured data are currently available. Several intermetallic phases which have previously

been suggested to be stabilized by impurity effects are indeed found to be higher energy states compared to their stable coun-

terparts. It is noted that the CALPHAD model parameters representing alloy energetics vary significantly from one assessment

to another in these systems, demonstrating the clear need for additional enthalpic data for all competing phases to derive

unique thermodynamic model parameters. For the stable intermetallics, the calculated zero-temperature lattice parameters agree

to within ±1% of experimental data at ambient temperature. For the stable phases with unit cell-internal degree(s) of freedom,

the results of ab initio calculations show excellent agreement when compared with data obtained by rigorous structural analysis

of X-ray and other diffraction results. For intermetallic compounds where no such experimental data is available, we provide

optimized unit cell geometries. For most structures we also provide zero-temperature bulk moduli and their pressure derivatives,

as defined by the equation of state.

� 2005 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.

Keywords: Ab initio electron theory; Aluminum alloys; CALPHAD; Crystal structure; Elastic properties

1. Introduction

Intermetallics involving aluminum and early transi-

tion metals (TM) are known to have many attractive

1359-6454/$30.00 � 2005 Acta Materialia Inc. Published by Elsevier Ltd. A

doi:10.1016/j.actamat.2005.03.028

* Corresponding author.

E-mail address: [email protected] (G. Ghosh).

properties, making them desirable candidates for high-

temperature structural applications [1,2]. The properties

include resistance to oxidation and corrosion, elevated-

temperature strength, relatively low density, and high

melting points. The trialuminides, of the type Al3TM

(TM = Ti, Zr, Hf, V, Nb, Ta), have received particular

interest as the basis for advanced engineering materials.Since these intermetallics are inherently brittle,

ll rights reserved.

mailto:[email protected]

3226 G. Ghosh, M. Asta / Acta Materialia 53 (2005) 3225–3252

substantial experimental and theoretical effort has been

directed at identifying the intrinsic and extrinsic factors

limiting their ductility. A strategy that has been exten-

sively explored to increase the ductility of Al–TM inter-

metallics is the use of ternary alloying additions to

stabilize the cubic L12 phase of the Al3TM phases,which form stable tetragonal structures in the binary

systems. Such efforts have given rise to substantial inter-

est in understanding phase stability in these systems,

which are generally characterized by the presence of a

number of often structurally complex stable and meta-

stable intermetallic phases.

Over the last 15 years, there have been significant ad-

vances in the fundamental understanding of both themechanical properties and phase stability of intermetal-

lics provided by the results of quantum-mechanical elec-

tronic structure calculations. Ab initio or first-principles

methods based upon electronic density-functional the-

ory (DFT) (see, e.g. [3]) have been employed to derive

a number of bulk and defect properties including heats

of formation, the relative stability of competing struc-

tures, elastic constants, lattice parameters, and the ener-gies associated with point and planar defects [4,5]. While

many of the ab initio studies in Al–TM systems have

been devoted to understanding the properties of mono-

lithic intermetallics of particular stoichiometry and/or

structure, to design multicomponent and multi-phase

materials it is necessary to model the relative stability

of all competing phases.

To understand processing–microstructure–property–performance links in multicomponent and multi-phase

materials, a framework is needed to address the

dynamics of microstructure evolution [6]. In recent

years computational-thermodynamic methods based

on the CALPHAD framework [7] have become widely

used as the basis for modeling phase stability and

phase-transformation kinetics in complex multicompo-

nent alloy systems [8–11]. The accuracy of the predic-tions derived from these methods depends critically

upon the thermodynamic models that form the basis

for calculations of phase stability and phase-transfor-

mation driving forces. Thus, accurate thermodynamic

and kinetic databases are generally required for suc-

cessful applications of computational thermodynamic

methods in alloy design, and to model the dynamics

of microstructure evolution. For new, relatively unex-plored alloy systems, modeling efforts are often hin-

dered by the need for extensive experimental

measurements required in the development of robust

thermodynamic and kinetic databases. Since ab initio

methods yield calculated thermodynamic properties di-

rectly from first-principles (i.e., with very limited input

required from experiment), they offer the potential for

significantly limiting the extent of costly experimentalmeasurements required in thermodynamic-database

development. Work performed during the past decade

has demonstrated that first-principles methods based

upon DFT yield high accuracy in applications to the

calculation of heats of formation for ordered interme-

tallic compounds in a wide range of alloy systems

(see, e.g. [12–14]). As a result, the integration of such

first-principles with CALPHAD methods is beingincreasingly pursued [15–19].

Here, we present the results of a comprehensive

study of zero-temperature energetics, and the equilib-

rium cohesive properties of Al–Ti, Al–Zr and Al–Hf

intermetallics using ab initio computational tech-

niques. This work has been undertaken to investigate

the systematics in Al–TM alloy energetics, and to aug-

ment the incomplete database of calorimetric data forenthalpies of formation, in support of the develop-

ment of accurate multicomponent thermodynamic dat-

abases for these technologically interesting systems.

Since Ti, Zr and Hf are isoelectronic, it is also of fun-

damental interest to investigate the similarities and

differences in the cohesive properties when these 3d,

4d and 5d elements are alloyed with Al. We note that

ab initio methods are being increasingly used to com-pute not only zero-temperature energetics, but also

finite-temperature contributions to alloy free energies

arising from vibrational entropy (see, e.g. [20], and

references cited therein). The importance of such con-

tributions has been demonstrated in first-principles

calculations of solvus boundaries in Al–Sc [21,22]

and Al–Zr [23], and in ab initio modeling of the

finite-temperature stability of the h phase of Al2Cu[24]. While the present work is limited to calculations

of zero-temperature energetics and cohesive properties,

it nevertheless represents a necessary first-step in the

modeling of phase stability and phase transformations

at finite temperature.

The remainder of the paper is organized as

follows. In the next section we briefly review experi-

mental data of intermetallic compounds in three bin-ary systems, and also the previous studies of phase

stability by ab initio methods and CALPHAD model-

ing. In Section 3 we present the computational meth-

odology employed in the current study. In Section 4

we present calculated equilibrium structural and cohe-

sive properties of the intermetallic compounds. These

include formation energies, bulk moduli, lattice

parameters and (when applicable) cell-internal degreesof freedom for atomic positions. The latter informa-

tion is provided for all structures considered in this

work, since they may represent useful data for com-

parison with future measurements, and also as input

into future ab initio calculations which will reduce

the computation time significantly. The present ab ini-

tio calculated results are compared with available

experimental data, previous ab initio studies, andempirical CALPHAD modeling. Conclusions are sum-

marized in Section 5.

G. Ghosh, M. Asta / Acta Materialia 53 (2005) 3225–3252 3227

2. Literature review

The intermetallics considered are classified into three

types: stable, metastable and virtual. We consider the

stable phases as those which are present in the equilib-

rium phase diagram, irrespective of their temperatureor compositional ranges of stability. Metastable phases

are usually transient in nature, and during annealing/

heat treatment they eventually undergo transformation

to stable phases as governed by the equilibrium phase

diagram. Usually the metastable phases form during

non-equilibrium processing, such as mechanical alloying,

rapid solidification, vapor deposition, etc. Metastable

phases may also be stabilized by extraneous effects, suchas impurity interstitials in an otherwise pure alloy (see

Sections 2.2 and 2.3). Compilation of crystallographic

data for intermetallic compounds in the three binary sys-

tems enables us to define a superset of eighteen stable

structures. Then, for a particular binary alloy, the subset

of these eighteen phases which are not stable are referred

to as ‘‘virtual’’ in that system. The concept of a virtual

phase is a mathematical one in the context of CALP-HAD modeling of intermetallics having a finite homoge-

neity range, using a sublattice model [25].

2.1. The Al–Ti system

The experimental information for Al–Ti [26–44]

available prior to 1984 was assessed by Murray [45].

Since then new experimental data on phase equilibriaand crystal chemistry [46–67] has been reported. Based

on the currently available experimental data, the

crystallographic data for Al–Ti intermetallics is listed

in Table 1.

Al3Ti is known to exist in two stable forms: Al3Ti

(tI32) stable below 950 �C, and Al3Ti (tI8) (D022) stable

between 950 and 1387 �C [67]. Even though the former

Table 1

Crystallographic data of Al–Ti intermetallics

Phase Pearson symbol Strukturbericht designation Spa

Stable

Al3Ti (h) tI8 D022 I4/m

Al3Ti (r) tI32 – I4/m

Al5Ti2 tP28 – P4/m

Al11Ti5 tI16 D023 I4/m

Al2Ti tI24 – I41/a

Al1+xTi1�x tP4 L10 Pmm

Al5Ti3 tP32 – P4/m

AlTi cP2 B2 Pm�3AlTi tP4 L10 P4/m

AlTi3 hP8 D019 P63/

Metastable

Al3Ti cP4 L12 Pm�3Al3Ti tI16 D023 I4/m

Al3Ti (m) tI64 – I4/m

Al2Ti oC12 – Cmc

AlTi3 hP16 D024 P63/

was reported many years ago [37], only recently has its

thermal stability limit been determined. Murray listed

the phase Al5Ti3, but it was not shown in her assessed

phase diagram. Only the recent study by Braun and Ell-

ner [67] clearly shows the phase boundaries involving

AlTi, Al1+xTi1�x, Al11Ti5, Al5Ti3 and Al3Ti. It has beenreported recently that the body-centered cubic (bcc) so-

lid solution may undergo an ordering transition (to cP2

or B2) in the composition range of 60–80 at.% Ti and in

the temperature range of 1150–1400 �C [66]. This sug-

gestion is based on analyses involving the extrapolation

of data from ternary alloys to the binary Al–Ti system.

Although no direct experimental evidence for such an

ordering transition has yet become available, we listB2 AlTi as a stable phase in Table 1 due to its appear-

ance in the equilibrium diagram reported in [66].

As listed in Table 1, we consider five phases as meta-

stable. The formation of cubic-Al3Ti (cP4) has been re-

ported in vapor deposited thin-film [48], mechanically

alloyed [54–58,62] and rapidly solidified [38,59,67] spec-

imens. Tetragonal-Al3Ti (tI16) forms as a metastable

phase, in the temperature range of 495–800 �C, duringheating mechanically alloyed cubic-Al3Ti (cP4) [54].

Above 800 �C, Al3Ti (tI16) transforms to the equilib-

rium Al3Ti (tI8) structure. Another form of Al3Ti, Al3Ti

(tI64), which is considered as a superstructure of Al3Ti

(tI8), has been observed in diffusion couples [42]; a re-

cent investigation of phase equilibria in Al–Ti [67] using

bulk alloy specimens failed to confirm the stability of

this structure. Therefore, we consider Al3Ti (tI64) as ametastable phase, perhaps stabilized by stress effects.

Recently, Al2Ti (oC12) has been observed, in cast alloys,

to transform to Al2Ti (tI24) during annealing [53,64,67].

These results form the basis for our designation of tI24

and oC12 as stable and metastable forms of Al2Ti,

respectively. We note that Murray [45] did not make a

distinction between these two forms of Al2Ti, because

ce group (#) Prototype Reference

mm (139) Al3Ti [35,37,39,40,42–44,47,52–54,65,67]

mm (139) (?) Al24Ti8 [37,67]

mm (123) Al5Ti2 [53]

mm (139) Al11Ti5 [35,53]

md (141) Ga2Hf [33–35,46,49,51,53,64]

m (47) Al1+xTi1�x [53,67]

bm (127) Ga5Ti3 [41,67]

m ð221Þ CsCl [66]

mm (123) AuCu [26–29,44,67]

mmc (194) Ni3Sn [30–32,36]

m ð221Þ AuCu3 [38,48,54–59,67]

mm (139) Al3Zr [54]

mm (139) (?) Al48Ti16 [42]

m (65) Ga2Zr [41,49,53,67]

mmc (194) Ni3Ti [61]


an earlier study [41] did not resolve their relative stabil-

ities. Sahu et al. [61] reported to synthesize AlTi3 (hP16),

listed as a mestable phase in Table 1, under a pressure of

16 GPa.

There have been several studies of phase stability and

electronic structure of Al–Ti intermetallics using ab ini-tio methods. The phase stability of competing structures

(cP4, tI8 and tI16) for Al3Ti has received particular

attention. These previous ab initio studies can be sum-

marized as follows: augmented spherical wave (ASW)

method for Al3Ti [68]; full-potential linearized aug-

mented plane wave (FLAPW) for Al3Ti [69]; linear muf-

fin tin orbital (LMTO) within the atomic sphere

approximation (ASA) for Al3Ti [70–72]; FLAPW forAlTi3 [73]; full-potential (FP) LMTO for Al3Ti [74–

76], AlTi [74,75,77], and AlTi3 [75]; FLAPW for Al3Ti,

AlTi and AlTi3 [78,79]; full-potential linearized aug-

mented Slater-type orbital (FLASTO) for Al3Ti, Al2Ti,

AlTi and AlTi3 [80]. All of the aforementioned studies

made use of the local-density approximation (LDA).

Most recently, the stability of competing structures for

Al3Ti were performed by Colinet and Pasturel [81]employing ultrasoft pseudopotentials (US-PP) within

the generalized gradient approximation (GGA). These

authors performed detailed calculations of the energetics

of additional antiphased structures related to Al3Ti (tI8)

and Al3Ti (tI16), and discussed their results in the

framework of axial next nearest neighbor Ising (AN-

NNI) model; similar calculations were also performed

for Al3 Zr and Al3Hf.The heat of formation of Al3Ti [82–85], Al2Ti [85],

AlTi [82,83,85] and AlTi3 [82,83] has been measured

Table 2

Crystallographic data of Al–Zr intermetallics

Phase Pearson symbol Strukturbericht designation

Stable

Al3Zr tI16 D023Al2Zr hP12 C14

Al3Zr2 oF40 –

AlZr oC8 Bf

Al4Zr5 hP18 –

Al3Zr4 hP7 –

Al2Zr3 tP20 –

Al3Zr5 (h) tI32 D8mAlZr2 hP6 B82AlZr5 cP4 L12

Metastable

Al6Zr oC28 Dh

Al11Zr2 cP39 –

Al3Zr cP4 L12Al2Zr o? –

Al2Zr hP6 B82AlZr cP2 B2

AlZr cF? –

Al3Zr5 (m) hP16 D88AlZr2 tI12 C16

AlZr3 hP8 D019

by calorimetry. CALPHAD modeling of phase equilib-

ria has been reported six times [66,86–90], of which five

are considered to be full-scale CALPHAD optimiza-

tions [66,87–90]. In their phase diagram assessments,

Kaufman and Nesor [86] considered only three intermet-

allics (Al3Ti, AlTi and AlTi3) and treated them as linecompounds; Murray [87] considered four intermetallics

(Al3Ti, AlTi (tP4), AlTi (cP2) and AlTi3); Kattner

et al. [88], Lee and Saunders [89] and Zhang et al. [90]

considered five intermetallics (Al3Ti, Al5Ti2 or Al11Ti5,

Al2Ti, AlTi (tP4), and AlTi3); Ohnuma et al. [66] consid-

ered six intermetallics (Al3Ti, Al5Ti2, Al2Ti, AlTi (tP4),

AlTi (cP2) and AlTi3). To represent a finite homogene-

ity range for Al3Ti, AlTi (tP4), AlTi (cP2) and AlTi3,CALPHAD models of varying degrees of complexity

[66,87–90] have been used. Murray [87] used two differ-

ent models for AlTi (cP2) and AlTi3 (D019).

2.2. The Al–Zr system

As reviewed by Murray et al. [91], the Al–Zr system is

characterized by the presence of ten stable phases whichhave been confirmed many times, and also several meta-

stable phases [54,92–123]. These are listed in Table 2.

Solidification of dilute Al(Zr) alloys [108–112,119], and

also mechanical alloying [54] lead to the formation of

the metastable Al3Zr (cP4) phase. Annealing of super-

saturated Al(Zr) solid solutions prepared by vapor

deposition leads to the formation metastable Al11Zr2,

Al6Zr and AlZr [117,118] phases that eventually trans-form to stable phases. Fecht [121] obtained a metastable

face-centered cubic (fcc) phase at equiatomic

Space group (#) Prototype Reference

I4/mmm (139) Al3Zr [33,92,111,116,122,128]

P63/mmc (194) MgZn2 [33,94,97,116,128]

Fdd2 (43) Al3Zr2 [33,102,116,128]

Cmcm (63) CrB [33,105,106,128]

P63/mcm (193) Ga4Ti5 [33,105,120,128]

P�6 ð174Þ Al5Zr4 [33,96,100,104,128]

P42/mnm (136) Al2Zr3 [33,99,104,128]

I4/mcm (140) Si3W5 [96,104,123,128]

P63/mmc (194) Ni2In [33,98,103,104,123]

Pm�3m ð221Þ AuCu5 [33,93,104,123]

Cmcm (63) Al6Mn [117,118]

Pm�3 ð200Þ Zn11Mg2 [117,118]

Pm�3m ð221Þ AuCu3 [54,108–112,118]

– [118]

P63/mmc (194) Ni2In [118]

Pm�3m ð221Þ CsCl [118]

– [121]

P63/mcm (193) Si3Mn5 [94–96,104,107,115,116,120]

I4/mcm (140) Al2Cu [101]

P63/mmc (194) Ni3Sn [113,114]


composition by mechanical alloying, but did not report

its crystallographic details. Al3Zr5 is known to exist in

two forms: tetragonal W5Si3-type and hexagonal

Mn5Sn3-type. It has been suggested that the hexagonal

form of Al3Zr5 may be stabilized by interstitial by O,

N, C and B [96,116,120]. Similarly, the observation ofAlZr2 (tI12) [101] is believed to have been due to its sta-

bilization by impurities. It has been reported that aging

of Zr(Al) martensite leads to the formation of metasta-

ble AlZr3 (hP8) precipitates [113,114].

There have been several studies of phase stability and

electronic structure of Al–Zr intermetallics using ab ini-

tio methods. The majority of these have been devoted to

the phase stability of competing structures (cP4, tI8 andtI16) for Al3Zr. Previous ab initio studies can be sum-

marized as follows: ASW-LDA method for Al3Zr [68];

LMTO-ASA-LDA for Al3Zr [124]; FPLMTO-LDA

for Al3Zr [23,76] and AlZr [77]; FLASTO and plane-

wave pseudopotential (PWPP) LDA calculations for

all stable and some of the metastable intermetallics

[125]; and US-PP-GGA for competing structures of

Al3Zr [126], as well as several other hypothetical fcc-based ordered compounds [23]. The heats of formation

for Al3Zr [127–129], Al2Zr [127–130], Al3Zr2 [127,128],

and AlZr, Al4Zr5, Al2Zr3, Al3Zr5 [128] have been mea-

sured by calorimetry. CALPHAD modeling of phase

equilibria has been reported twice [131,132].

2.3. The Al–Hf system

The crystallographic details of seven stable and three

metastable intermetallic phases [48,54,133–138,104,139–

143] in Al–Hf are listed in Table 3. Experimentally an

equilibrium transition from r-Al3Hf (tI16) to h-Al3Hf

(tI8) has been established around 650 �C [105]. Accord-

ingly, Murray [144] included both phases in her assessed

phase diagram. However, Srinivasan et al. [54] reported

that when mechanically alloyed cubic-Al3Hf (cP4) isheated, it undergoes a transformation to r-Al3Hf around

750 �C, which remains stable up to 1100 �C. The exis-

Table 3

Crystallographic data of Al–Hf intermetallics

Phase Pearson symbol Strukturbericht designation

Stable

Al3Hf tI16 D023Al3Hf tI8 D022Al2Hf hP12 C14

Al3Hf2 oF40 –

AlHf oC8 Bf

Al3Hf4 hP7 –

Al2Hf3 tP20 –

Metastable

Al3Hf cP4 L12Al3Hf5 hP16 D8

AlHf2 tI12 C16

tence of other stable phases has been confirmed several

times. Like Al3Zr5, it is believed that the observation

of hexagonal Al3Hf5 [134,135] might have been due to

impurity stabilization [105]. Similarly, AlHf2 (tI12), ob-

served in [101,107], is believed to have been stabilized by

Si impurities.Ab initio studies on the phase stability of Al–Hf inter-

metallics are very limited. Carlsson and Meschter [68]

used the ASW-LDA method to calculate the total ener-

gies of Al3Hf with three structures (cP4, tI8 and tI16),

but they reported only the energy differences between

these structures (rather than absolute values for forma-

tion energies). Colinet and Pasturel [145] used US-PP-

GGA to calculate bonding, cohesive properties andphase stability of Al3Hf with competing structures.

The heat of formation of Al3Hf [146,147], Al2Hf [146],

Al3Hf2 [147] and AlHf [146,147] was measured by calo-

rimetry. CALPHAD modeling of phase equilibria has

been reported twice [148,149].

3. Computational methodology

3.1. Ab initio total energy calculations

The ab initio calculations presented here are based on

electronic DFT, and have been carried out using the ab

initio total-energy and molecular-dynamics program

VASP (Vienna ab initio simulation package) developed

at the Institut fur Materialphysik of the UniversitatWien [150,151]. The current calculations make use of

the VASP implementation of ultrasoft pseudopotentials

[152], and an expansion of the electronic wavefunctions

in plane waves with a kinetic-energy cutoff of 281 eV.

For the transition metals the pseudopotentials employed

in this work treated the following states as valence:

Ti-4s, 4p and 3d, Zr-5s, 5p and 4d, and Hf-6s, 6p and

5d. All calculated results were derived employing theGGA for exchange and correlation due to Perdew and

Wang [153]. Brillouin-zone integrations were performed

Space group (#) Prototype Reference

I4/mmm (139) Al3Zr [54,96,98,105,134–137,142]

I4/mmm (139) Al3Ti [98,105,134,139]

P63/mmc (194) MgZn2 [96,98,105,133,139,140,142]

Fdd2 (43) Al3Zr2 [98,105,136,104]

Cmcm (63) CrB [105,138,139,142]

P�6 ð174Þ Al5Zr4 [98,105,134,142]

P42/mnm (136) Al2Zr3 [98,105,135]

Pm�3m ð221Þ AuCu3 [48,54,141,143]

P63mcm (193) Si3Mn5 [134,135]

I4/mcm (140) Al2Cu [101,107]


using Monkhorst–Pack [154] k-point meshes, and the

Methfessel–Paxton [155] technique with a 0.1 eV smear-

ing of the electron levels. For each structure, tests were

carried out using different k-point meshes to ensure

absolute convergence of the total energy to within a pre-

cision of better than 2.5 meV/atom (0.25 kJ/mol). As anexample, the k-meshes for Al3Zr having L12, D022 and

D023 structures were 20 · 20 · 20, 23 · 23 · 10 and

21 · 21 · 5, respectively. Depending on the structure,

up to 518 k-points were used in the irreducible Brillouin

zone. Total energies of each structure were optimized

with respect to the volume, unit cell-external degree(s)

of freedom (i.e., the unit-cell shape) and unit cell-inter-

nal degree(s) of freedom (i.e., Wyckoff positions) as per-mitted by the space-group symmetry of the crystal

structure. Such structural optimizations were iterated

until the atomic forces were less than 4 meV/A in mag-

nitude, ensuring a convergence of the energy with re-

spect to the structural degrees of freedom to better

than 2.5 meV/atom (0.25 kJ/mol). With the chosen

plane-wave cutoff and k-point sampling the reported

formation energies are estimated to be converged to aprecision of better than 5 meV/atom (0.5 kJ/mol).

All results presented below were obtained employ-

ing the computational settings described in the previ-

ous paragraph. However, for the Al–Zr system

several additional calculations were also conducted

with alternative settings to gauge the overall accuracy

of the reported results. Specifically, test calculations

were performed employing the local-density approxi-mation (LDA) [156] rather than GGA, and alternative

transition-metal pseudopotentials which included semi-

core p states as valence. Inclusion of semi-core states

led to increases in the calculated formation energies

(i.e., less negative values) in the range of a few kJ/

mol, and increased computed lattice constants on the

order of one per cent. Switching from GGA to

LDA was observed to have the effect of decreasing(i.e., making more negative) the calculated formation

energies by a few kJ/mol; as has been found in numer-

ous previous calculations for related systems, the

GGA gave rise to significantly better agreement with

experimentally measured lattice parameters as com-

pared to results derived by the LDA, which consis-

tently underestimated equilibrium bond lengths by a

few per cent.

3.2. Equation of state and formation energy

We take the zero-temperature formation energy

(DE/) of an intermetallic, AlmTMn where m and n are

integers, as a key measure of the relative stability of

competing structures (/1,/2,/3 . . .). The formation en-

ergy of AlmTMn per atom is evaluated relative to thecomposition-averaged energies of the pure elements in

their equilibrium crystal structures:

DE/ðAlmTMnÞ¼1

mþnE/AlmTMn

� mmþn

EhAlþ

nmþn

EwTM

� �;

ð1Þ

where E/AlmTMn

is the total energy of AlmTMn with struc-ture /, Eh

Al is the total energy per atom of Al with fcc (h)structure and Ew

TM is the total energy per atom of TM

(=Ti, Zr, Hf) with hexagonal close-packed (hcp) (w)structure.

The equation of state (EOS) generally defines the

relationship between pressure (P), volume (V) and tem-

perature (T). Here we consider only zero-temperature

equations of state, defining pressure–volume relation-ships. Numerous forms for such EOS can be found in

literature, and, in fact, the search for a universal form

of the EOS of solids is still an important problem in

high-pressure physics and geophysics. We have used

the EOS due to Vinet et al. [157] who assumed the inter-

atomic interaction-versus-distance relation in solids can

be expressed in terms of a relatively few material con-

stants. The most commonly used EOSs given by Murna-ghan [158] and Birch [159] work as well as that by Vinet

et al. [157] at low pressures, but at ultra-high pressure

Birch–Murnaghan EOSs, which are based on lower-or-

der Taylor-series expansions, are known to be less

accurate.

In the EOS of Vinet et al. [157] the pressure P is

expressed in terms of isothermal bulk modulus

(B0), its pressure derivative ðB00Þ and a scaled quantity

(x):

P ¼ 3B0x�2ð1� xÞ exp½gð1� xÞ� ð2Þwith x ¼ ðV =V 0Þ1=3 and g ¼ 3=2ðB0

0 � 1Þ, where V0 is the

equilibrium volume. Based on Eq. (2) and the relations

between pressure and energy, the total energy (E) and

volume-dependence of the bulk modulus can be ex-

pressed as

EðV Þ � EðV 0Þ ¼9B0V 0

g2f1� ½gð1� xÞ� exp½gð1� xÞ�g;

ð3Þ

BðV ÞB0

¼ x�2½1þ ðgxþ 1Þð1� xÞ� exp½gð1� xÞ�. ð4Þ

Vinet et al. have shown that the second-order pressure

derivative of the bulk modulus ðB000Þ, which is a more se-

vere test of the accuracy of EOS, can be expressed as

B0B000 ¼

19

36� 1

2B00 �

1

4ðB00

0Þ2. ð5Þ

Eqs. (2)–(5) are found to work well for metallic, cova-

lent, ionic and van der Waals bonded solids. As an

example, Fig. 1(a)–(c) shows the E–V plots defining

zero-temperature EOS parameters for Al5Ti3 (tP32),Al3Zr5 (tI32) and Al3Hf2 (oF40), respectively.

-5.642

-5.64

-5.638

-5.636

-5.634

-5.632

-5.63

-5.628

-5.626

0.015 0.0155 0.016 0.0165 0.017

mota/Ve,ygren

E

Volume, nm^3/atom

-7.088

-7.086

-7.084

-7.082

-7.08

-7.078

-7.076

-7.074

0.019 0.0195 0.02 0.0205 0.021 0.0215

mota/Ve,ygren

E

Volume, nm^3/atom

-6.605

-6.600

-6.595

-6.590

-6.585

0.017 0.0175 0.018 0.0185 0.019

mota/Ve,ygren

E

Volume, nm^3/atom

(a) (b)

(c)

Fig. 1. Calculated zero-temperature total energy as function of volume, E(V), for (a) Al5Ti3 (tP32), (b) Al3Zr5 (tI32), and (c) Al3Hf2 (oF40). The filled

circles represent calculated point, and the line is a fit to EOS in Eq. (3).


4. Results and discussions

4.1. Cohesive properties of pure elements

The total energies of Al, Hf, Ti and Zr have been cal-

culated as a function of volume for bcc, fcc and hcp

structures. The resulting zero-temperature cohesive

properties are compared with available experimental

data in Table 4. The lattice parameters of fcc-Al [160]

and hcp-Zr [161] are taken from the measured valuesat 4.2 K. The lattice parameter of bcc-Zr has been re-

ported at 298 K [162], and it is based on the lattice

parameters measured for dilute Zr(U) alloys. The lattice

parameters of bcc-Ti [163], hcp-Ti [163] and hcp-Hf

[164] at 0 K are obtained by extrapolation of corre-

sponding experimental data. In the case of bcc-Ti, due

to large temperature range and for the sake of simplic-

ity, we have used a linear extrapolation. We find that,in general, the lattice parameters agree within ±1% of

the experimental value, while the bulk moduli [165–

168] agree within ±2%, except for fcc-Al where there isa large scatter in the experimental data. Experimental

data of B00 for fcc-Al [169–171] and hcp-Zr [172] also

show some scatter. It has been pointed out [172] that

depending on the measurement technique, ultrasonic

resonance, versus the initial slope of the locus of Hugon-

iot states in shock-velocity particle-velocity coordinates,

the value of B00 may differ even though ideally they

should be the same. It is not uncommon that the B00 pre-

dicted by ab initio techniques differs from the experi-

mental value by as much as 30% see Table 4.

The calculated lattice stabilities of Al, Hf, Ti and Zr

at 0 K are compared with those from the SGTE (Scien-

tific Group Thermo-Data Europe) database [173], as

provided in Thermo-Calc version P [174], in Table 5.

Quantitative differences on the order of a few to sev-

eral kJ/mol are apparent between the calculated andSGTE values for structural energy differences. Such

Table 4

A comparison of selected structural and elastic properties of Al, Hf, Ti and Zr at 0 K

Element Structure (Pearson symbol) Lattice parameter (nm) B0 (·1010 N/m2) B00

Ab initioa Experiment Ab initioa Experiment Ab initioa Experiment

Al BCC (cI2) a = 0.32418 – 6.47 – 4.18 –

FCC (cF4) a = 0.40436 a = 0.40322 [160] 7.42 8.82 [165] 4.11 4.0 [169]

7.94 [166] 5.19 [170]

8.2 [167] 4.42 [171]

HCP (hP2) a = 0.28495 – 7.01 – 4.76 –

c = 0.47486

Hf BCC (cI2) a = 0.35131 a = 0.34342 [164] 10.37 – 3.24 –

FCC (cF4) a = 0.44456 – 10.46 – 3.25 –

HCP (hP2) a = 0.31804 a = 0.31930 [164] 11.03 11.06 [168] 3.43 3.95, 3.28 [172]

c = 0.50208 c = 0.50395

Ti BCC (cI2) a = 0.32398 a = 0.32539 [163] 10.36 – 3.10 –

FCC (cF4) a = 0.40963 – 10.57 – 2.96 –

HCP (hP2) a = 0.29229 a = 0.29443 [163] 10.91 11.0 [168] 3.43 4.37, 3.98 [172]

c = 0.46271 c = 0.46685

Zr BCC (cI2) a = 0.35435 a = 0.35453 [162] 9.06 – 3.74 –

FCC (cF4) a = 0.44935 – 9.49 – 4.06 –

HCP (hP2) a = 0.32084 a = 0.32294 [161] 9.59 9.72 [168] 2.85 4.11, 2.74 [172]

c = 0.51327 c = 0.51414

Calculated lattice-parameter data are listed up to same significant digit as the reported experimental data.a This study [US-PP (GGA)].


discrepancies between ab initio and CALPHAD derived

lattice-stability energies has been discussed previously

by numerous investigators (see [175–177] and references

cited therein). For Ti and Zr, we note that the SGTE

database predicts that bcc-Ti and bcc-Zr are more stable

compared to their hcp states at very low temperatures;

the unexpected zero-temperature lattice stability for

bcc-Ti and bcc-Zr in the SGTE database could be anartifact of their functional representation, since the cor-

rect lattice stability for these elements is predicted at

room temperature. Our calculated results for lattice

Table 5

A comparison of lattice stabilities (kJ/mol) of Al, Hf, Ti and Zr

Property Ab initioa SGTE database [173]

DEFCC!BCCAl 9.4040 10.083

DEFCC!HCPAl 3.2471 5.4810

DEHCP!BCCHf 15.5899 12.3581

DEHCP!FCCHf 6.4942 10.0

DEHCP!BCCTi 9.1053 6.4758b

DEHCP!FCCTi 5.1067 6.0

DEHCP!BCCZr 5.4632 7.3111b

DEHCP!FCCZr 3.1989 7.60

For direct comparison with the calculated results, lattice-stability

values from the SGTE database [173] are all reported at zero tem-

perature, with the exception of DEHCP!BCCTi and DEHCP!BCC

Zr which are

given at room temperature; the values of these lattice stabilities were

found to take large negative values at zero temperature.a This study [US-PP (GGA)].b Values correspond to a temperature of 298.15 K; values at zero

temperature were found to be large negative values.

parameters obtained by the US-PP-GGA method in

Table 4 are in very good agreement with recently re-

ported [177] VASP-GGA calculations based on the pro-

jector augmented wave (PAW) method [178] for Al and

Ti, although the US-PP values are smaller by roughly

1% compared with PAW for Zr and Hf. Similarly, the

present US-PP-GGA calculations agree to within 10%

with the PAW-GGA results for fcc–hcp energy differ-ences, while differences on the order of 1 kJ/mol are ob-

tained for fcc–bcc structural energy differences with the

US-PP results being consistently smaller than those ob-

tained in [177] from PAW.

4.2. Phase stability and cohesive properties of Al–Ti

intermetallics

The results of ab initio calculations for Al–Ti inter-

metallics are summarized in Tables 6–8, and are plotted

in Fig. 2. The crystallographic details of Al–Ti intermet-

allics are known, except for Al3Ti (tI32) [37,67] and

Al3Ti (tI64) [42]. Accordingly, we did not perform ab

initio calculations for either of these two phases. How-

ever, these two structures are thought to be the super-

structures based on Al3Ti (tI8). Colinet and Pasturel[81] studied the effect of anti-phase boundaries (APB)

in stabilizing one dimensional long period superstruc-

tures (1D-LPS), and discussed the energetic results in

the framework of the ANNNI model. They showed that

a number of 1D-LPSs can be stabilized by APBs where

the energy difference between Al3Ti (tI8) and 1D-LPS

lies in the range of 0.2–0.8 kJ/mol.

Table 6

A comparison of heat of formation (DE) of Al–Ti intermetallics obtained by various methods: ab inito calculations (at 0 K), experiment (at different

temperatures), and CALPHAD (at 298.15 K or the standard heat of formation) modeling of Al–Ti Phase diagram

Phase Space group (#) Prototype DE (kJ/mol)

Ab initio [Ref.] Experiment [Ref.] CALPHAD [Ref.]

Stable

Al3Ti I4/mmm (139) Al3Ti �38.895a �36.6 ± 1.1 [82] �34.138 [86]

�41.450b [69] �35.5 ± 1 [83] �39.302 [87]

�40.468c [70,71] �36.6 ± 1.3 [84] �38.849 [88]

�40.500d [72] �39.2 ± 1.8 [85] �32.593 [90]

�41.90e [74,75] �36.148 [89]

�41.443f [76] �44.563 [66]

�39.504g [78]

�39.505h [80]

�39.30i [81]

Al5Ti2 P4/mmm (123) Al5Ti2 �39.398a �38.780 [88]

�32.686 [90]

�43.270 [66]

Al11Ti5 I4/mmm (139) Al11Ti5 �40.18a �38.845 [89]

Al2Ti I41/amd (141) Ga2Hf �42.370a �37.1 ± 0.9 [85] �41.858 [88]

�42.396h [80] �35.730 [90]

�40.500 [89]

�43.694 [66]

Al5Ti3 P4/mbm (127) Ga5Ti3 �41.640a

AlTi Pm�3m ð221Þ CsCl �25.876a �39.422 [87]

�25.052h [80] �37.265 [66]

AlTi P4/mmm (123) AuCu �39.712a �40.1 ± 1 [82] �45.502 [86]

�42.00e [74,75] �36.4 ± 1 [83] �27.583 [87]

�39.505h [80] �35.1 ± 0.5 [85] �30.141 [87]

�37.240j [77] �34.444 [88]

�40.468g [78,79] �41.207 [90]

�39.822 [89]

�43.370 [66]

AlTi3 P63/mmc (194) Ni3Sn �27.395a �25 ± 2.1 [82,83] �23.564 [86]

�27.942c [71] �27.886 [87]

�26.979k [73] �29.522 [87]

�28.70e [75] �28.244 [88]

�26.979i [80] �30.881 [90]

�26.979g [78] �27.520 [89]

�28.447 [66]

Metastable

Al3Ti Pm�3m ð221Þ AuCu3 �36.583a

�38.541b [69]

�35.651c [70,71]

�39.601e [74,75]

�39.569f [76]

�36.614h [80]

�36.907i [81]

Al3Ti I4/mmm (139) Al3Zr �39.656a

�41.819f [76]

�40.100i [81]

Al2Ti Cmcm (65) Ga2Zr �42.013a

AlTi3 P63/mmc (194) Ni3Ti �26.461a

Virtual

Al2Ti P63/mmc (194) MgZn2 �33.361a

Al3Ti2 Fdd2 (43) Al3Zr2 �36.857a

AlTi Cmcm (63) CrB �33.902a

Al4Ti5 P63/mcm (193) Ga4Ti5 �32.023a

Al3Ti4 P�6 ð174Þ Al3Zr4 �37.196a

Al2Ti3 P42/mnm (136) Al2Zr3 �24.780a

Al3Ti5 I4/mcm (140) W5Si3 �25.922a

Al3Ti5 P63/mcm (193) Mn5Si3 �23.716a

AlTi2 P63/mmc (194) Ni2In �30.174a

AlTi3 Pm�3m ð221Þ AuCu3 �25.998a

a US-PP (GGA) [this study]; b,g,k FLAPW (LDA); c,d LMTO-ASA (LDA); e,f,j FP-LMTO (LDA); h FLASTO (LDA); i US-PP with semicore treatment

(GGA).

The reference states are fcc-Al and hcp-Ti.



Recent experimental studies [53,63,67] underscore the

complexity of the Al–Ti phase diagram in the composi-

tion range of 25–50 at.% Ti due to the formation of sev-

eral fcc-based superstructures. These include L10 AlTi

Table 7

A comparison of unit cell-external parameters of Al–Ti intermetallics obta

ambient temperature)

Phase Space group (#) Prototype EOS parameters Unit cell-

V0 B0 B00 Ab initio

a

Stable

Al3Ti I4/mmm (139) Al3Ti 15.929 10.30 4.15 0.38399

11.80 0.37800

12.0 0.38100

11.80 0.37897

0.37600

0.37990

10.23 0.38440

Al5Ti2 P4/mmm (123) Al5Ti2 15.863 10.40 4.01 0.39114

Al11Ti5 I4/mmm (139) Al11Ti5 15.904 10.48 4.02 0.39239

Al2Ti I41/amd (141) Ga2Hf 15.944 10.61 4.02 0.39658

0.39282

Al5Ti3 P4/mbm (127) Ga5Ti3 16.039 10.70 3.98 1.12861

AlTi Pm�3m ð221Þ CsCl 16.161 10.97 3.68 0.31854a

0.31529h

AlTi P4/mmm (123) AuCu 16.181 11.21 3.91 0.39814

12.80 0.39921

0.39530

0.39716

AlTi3 P63/mmc (194) Ni3Sn 16.584 11.19 3.83 0.57372

12.60 0.56496

0.56623

0.56136

Metastable

Al3Ti Pm�3m ð221Þ AuCu3 15.737 10.36 4.12 0.39779a

11.80 0.39200b

15.0 0.39410c

11.80 0.39157e

0.39700f [

0.39345h

0.39820i [

Al3Ti I4/mmm (139) Al3Zr 15.819 10.31 4.08 0.38962

0.38100

10.22 0.38850

Al2Ti Cmcm (65) Ga2Zr 15.950 10.60 4.01 1.21609

AlTi3 P63/mmc (194) Ni3Ti 16.547 11.13 3.65 0.57216

Virtual

Al2Ti P63/mmc (194) MgZn2 16.002 10.93 4.13 0.51329

Al3Ti2 Fdd2 (43) Al3Zr2 16.242 n.d. n.d. 0.92634

AlZr Cmcm (63) CrB 16.538 10.98 4.01 0.30275

Al4Ti5 P63/mcm (193) Ga4Ti5 16.375 n.d. n.d. 0.79181

Al3Ti4 P�6 ð174Þ Al3Zr4 16.057 11.46 3.88 0.51958

Al2Ti3 P42/mnm (136) Al2Zr3 16.902 n.d. n.d. 0.72354

Al3Ti5 I4/mcm (140) W5Si3 16.794 n.d. n.d. 1.03697

Al3Ti5 P63/mcm (193) Mn5Si3 16.894 n.d. n.d. 0.78366

AlTi2 P63/mmc (194) Ni2In 16.544 11.20 3.69 0.45603

AlTi5 Pm�3m ð221Þ AuCu5 16.504 11.03 3.57 0.40416a

Also listed are equilibrium volume (V0, ·10�3 nm3/atom), bulk modulus (B0,a US-PP (GGA) [this study]; b,g,k FLAPW (LDA); c,d LMTO-ASA (LDA

treatment (GGA); n.d.: not determined.

(CuAu prototype), Al1+xTi1�x (tP4), Al11Ti5 (tI16),

Al5Ti2 (tP28), Al2Ti (tI24, oC12) and Al5Ti3 (tP32).

An interesting feature reported in the Al–Ti phase dia-

gram [53,67] is the reported presence of separate AlTi

ined by ab inito calculations (at 0 K) and diffraction experiments (at

external parameters

Experiment

b c a b c

0.86399a 0.38400 to 0.85600 to

0.85100b [69] 0.38537 0.86140 [81]

0.85100c [70]

0.84891e [75]

0.84976f [76]

0.85174h [80]

0.86380i [81]

2.90229a 0.39053 2.91963 [53]

1.65199a 0.39170 1.65240 [35]

0.39230 1.65349 [53]

2.43206a 0.39711 2.43131 [64]

2.40681h [80]

0.40311a 1.12932 0.40381 [67]

[80]

0.40803a 0.40010 0.40710 [44]

0.40400e [75]

0.39925k [79]

0.40510h [80]

0.46825a 0.57750 0.46550 [30]

0.45706 [75]

0.45865k [79]

0.46649h [80]

0.39800 to

[69] 0.40500 [81]

[70]

[75]

76]

[80]

81]

1.66713a 0.38900 1.69220 [54]

1.64592f [76]

1.68230i [81]

0.39322 0.40018a 1.20944 0.39591 0.40315 [53]

0.93462a 0.53120 0.96040 [61]

0.84112a

1.32005 0.53129a

1.04798 0.41688a

0.54286a

0.48039a

0.64572a

0.50081a

0.50825a

0.55078a

·1010 N/m2) and B00 as defined by the equation of state (EOS) at 0 K.

); e,f,j FP-LMTO (LDA); h FLASTO (LDA); i US-PP with semicore

Table 8

A comparison of unit cell-internal parameters (Wyckoff positions) of Al–Ti intermetallics (where applicable) obtained from our ab inito calculations

(at 0 K) and diffraction experiments (at ambient temperature)

Phase Space group (#) Prototype Unit cell-internal parameters (Wyckoff positions)

Ab initio (x,y,z) Experiment (x,y,z) [Ref.]

Stable

Al5Ti2 P4/mmm (123) Al5Ti2 Al1: 2e 0.00000 0.50000 0.50000 0.00000 0.50000 0.50000 [53]

Al2: 2g 0.00000 0.00000 0.14144 0.00000 0.00000 0.14286

Al3: 2g 0.00000 0.00000 0.28498 0.00000 0.00000 0.28571

Al4: 2h 0.50000 0.50000 0.42844 0.50000 0.50000 0.42857

Al5: 4i 0.00000 0.50000 0.07107 0.00000 0.50000 0.07143

Al6: 4i 0.00000 0.50000 0.21365 0.00000 0.50000 0.21429

Al7: 4i 0.00000 0.50000 0.35672 0.00000 0.50000 0.35714

Ti1: 1a 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000

Ti2: 1c 0.50000 0.50000 0.00000 0.50000 0.50000 0.00000

Ti3: 2g 0.00000 0.00000 0.43200 0.00000 0.00000 0.42857

Ti4: 2h 0.50000 0.50000 0.14558 0.50000 0.50000 0.14286

Ti5: 2h 0.50000 0.50000 0.28128 0.50000 0.50000 0.28571

Al2Ti Cmcm (65) Ga2Zr Al1: 2a 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 [53]

Al2: 2c 0.50000 0.00000 0.50000 0.50000 0.00000 0.50000

Al3: 4h 0.17310 0.00000 0.50000 0.17600 0.00000 0.50000

Ti: 4g 0.34476 0.00000 0.00000

Al2Ti I41/amd (141) Ga2Hf Al1: 8e 0.00000 0.00000 0.25009

Al2: 8e 0.00000 0.00000 0.41341

Ti: 8e 0.00000 0.00000 0.07748

Al5Ti3 P4/mbm (127) Ga5Ti3 Al1: 2a 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 [46]

Al2: 2d 0.00000 0.50000 0.00000 0.00000 0.50000 0.00000

Al3: 4g 0.24943 0.74943 0.00000 0.25000 0.75000 0.00000

Al4: 4h 0.37674 0.87674 0.50000 0.37500 0.87500 0.50000

Al5: 8i 0.25378 0.50378 0.00000 0.25000 0.50000 0.00000

Ti1: 4h 0.11752 0.61752 0.50000 0.12500 0.62500 0.50000

Ti2: 8j 0.12023 0.12023 0.50000 0.12500 0.12500 0.50000

AlTi3 P63/mmc (194) Ni3Sn Al: 2c 0.33333 0.66666 0.25000

Ti: 6h 0.83037 0.66075 0.25000

Metastable

AlTi3 P63/mmc (194) Ni3Ti Al: 2a 0.00000 0.00000 0.00000

Al: 2c 0.33333 0.66666 0.25000

Ti: 6g 0.50000 0.00000 0.00000

Ti: 6h 0.83554 0.67108 0.25000

Virtual

Al3Ti I4/mmm (139) Al3Zr Al1: 4c 0.00000 0.50000 0.00000

Al2: 4d 0.00000 0.50000 0.25000

Al3: 4e 0.00000 0.00000 0.37519

Ti: 4e 0.00000 0.00000 0.11875

Al2Ti P63/mmc (194) MgZn2 Al1: 2c 0.00000 0.00000 0.00000

Al2: 6h 0.82840 0.65681 0.25000

Ti: 4f 0.33333 0.66666 0.06406

Al3Ti2 Fdd2 (43) Al3Zr2 Al1: 8a 0.00000 0.00000 0.64554

Al2: 16b 0.18738 0.14018 0.48339

Ti: 16b 0.19299 0.05274 0.00115

AlZr Cmcm (63) CrB Al: 4c 0.00000 0.42744 0.25000

Zr: 4c 0.00000 0.16709 0.25000

Al4Ti5 P63/mcm (193) Ga4Ti5 Al1: 2b 0.00000 0.00000 0.00000

Al2: 6g 0.63746 0.00000 0.25000

Ti1: 4d 0.33333 0.66667 0.00000

Ti2: 6g 0.30436 0.00000 0.25000

Al3Ti4 P�6 ð174Þ Al5Zr4 Al: 3j 0.33333 0.16666 0.00000

Zr1: 1b 0.00000 0.00000 0.50000

Ti2: 1f 0.66666 0.33333 0.50000

Ti3: 2h 0.33332 0.66669 0.30033

Al2Ti3 P42/mnm (136) Al2Zr3 Al: 8j 0.12523 0.12523 0.22908

Ti1: 4d 0.00000 0.50000 0.25000

Ti2: 4f 0.33695 0.33695 0.00000

Ti3: 4g 0.18682 0.81318 0.00000

(continued on next page)


Table 8 (continued)



Al3Ti5 I4/mcm (140) W5Si3 A1l: 4a 0.00000 0.00000 0.25000

Al2: 8h 0.16577 0.66577 0.00000

Ti1: 4b 0.00000 0.50000 0.25000

Ti2: 16k 0.07676 0.22360 0.00000

Al3Ti5 P63/mcm (193) Mn5Si3 A1: 6g 0.60384 0.00000 0.25000

Ti1: 4d 0.33333 0.66666 0.00000

Ti2: 6g 0.23067 0.00000 0.25000


and Al1+xTi1�x phases. The former is the well-known

L10 (CuAu prototype) structure which is often given

the Pearson symbol tP4, although the primitive cell con-

tains only two symmetry-inequivalent atomic sites. The

Al1+xTi1�x (tP4) phase has the same conventional unit

cell structure as L10 and very similar lattice parameters.

Fig. 2. Calculated zero-temperature cohesive properties of Al–Ti intermetallic

mean atomic volume (V0). In (a), the solid line defines the ground-state con

structures that form the ground-state convex hull in (a).

The phase apparently differs from L10 however, by the

absence of a translational symmetry element and the

presence of three independent crystallographic sites.

Specifically, Schuster and Ipser [53] calculated the

X-ray diffraction intensities of Al1+xTi1�x (tP4) by treat-

ing 50% occupancy of Al and Ti at 1g (0,0.5,0.5) site,

s: (a) the formation energy (DE), (b) the Bulk modulus (B0), and (c) the

vex hull, and in (b) and (c) the solid line is drawn through the same


and obtained a good agreement with experimental data.

Thus the Al1+xTi1�x can be viewed as an ordered super-

structure of L10. In the present study we have focused

only on stoichiometric structures, and have not consid-

ered the energetics of the Al1+xTi1�x (tP4) superstruc-

ture phase explicitly. Schuster and Ipser [53] alsoshowed that the X-ray diffraction pattern of Al11Ti5can be reproduced by replacing one of the Al atoms in

the 4e site of the Al3Ti (tI16) structure by Ti. They also

gave the crystallographic details of Al2Ti (oC12) and

Al5Ti2 (tP24), while Miida et al. [41] proposed the struc-

ture of Al5Ti3 (tP32).

Fig. 2(a) plots zero-temperature calculated formation

energies (DE) as a function of Ti content. The ground-state convex hull is asymmetric and skewed towards

the Al side with a maximum in DE at Al2Ti (tI24). A

similar trend was noted by Watson and Weinert [80],

and the electronic origins of this asymmetry have been

discussed by Zou et al. [78]. It is seen that Al3Ti

(tI16), Al2Ti (tI24), L10-AlTi (tP4), Al3Ti4 (hP7) and

AlTi3 (hP8) constitute the ground-state convex hull;

i.e., these are the structures (of the 22 intermetallics con-sidered) that are predicted to be stable alloy phases at

zero temperature. The Al2Ti (tI24), AlTi (tP4) and AlTi3(hP8) phases are known to be stable to low tempera-

tures, and our calculated results, giving these as

ground-state structures, are thus in agreement with

experimental observations. Al3Ti5 (tP32) is experimen-

tally observed at temperatures above 500 �C [67]; in

the calculations the energy of this phase lies very slightlyabove the convex hull, by about 0.2 kJ/mol. The compu-

tational results are consistent with a finite-temperature

stabilization of this phase driven by entropy, since only

a 0.03 kB/atom entropy difference between this phase

and the competing structures would be required to give

rise to its stabilization at 500 �C. Theoretical calcula-tions [21,24] yield vibrational entropy differences be-

tween Al-based intermetallics in the range of severaltenths of a kB/atom. Similarly, the three phases Al11Ti5[53,67], Al5Ti2 [53] and AlTi (cP2) [66] are stable at high

temperatures. The first two lie only slightly above the

ground-state convex hull (by less than 2 kJ/mol) and

the calculations again suggest the possibility of an entro-

pically driven stabilization of these phases at high tem-

peratures. An unexpected result is that Al3Ti4 (hP7) is

predicted to be the ground state even though it hasnot been observed experimentally. It is interesting to

note that in the Al–Zr and Al–Hf systems, both Al3Zr4(hP7) and Al3Hf4 (hP7) are known and also predicted

(see Sections 4.3 and 4.4) to be ground-state structures.

A comparison to the Al–Hf and Al–Zr systems (see Figs.

3 and 4(a)), however, show that this phase is just barely

stable with respect to phase separation to neighboring

compounds in the Al–Ti system, while it is much morestable in the Al–Hf and Al–Zr systems. The calculations

are therefore suggestive that in the Al–Ti system the

Al3Ti4 (hP7) phase is stable at very low temperatures,

but that its stability range may be limited by the pres-

ence of a low-temperature peritectoid reaction giving

rise to its transformation to tP4-AlTi + hP8-AlTi3. If

such a transformation occurs at low enough tempera-

tures it would be inaccessible to experimentalmeasurements.

Focusing now on the composition Al3Ti, of the three

structures considered we find the tI16 phase has the low-

est energy. This result is consistent with the findings of

Amador et al. [76] and Colinet and Pasturel [81] who

emphasized the important role of relaxation energies

(i.e., the energy reduction associated with optimization

of the lattice parameters and cell-internal positions) ingoverning the relative stability of these competing struc-

tures. While there is consensus amongst the different cal-

culations, the theoretical results are in apparent

discrepancy with experimental observations which have

established Al3Ti (tI16) to be a transient phase [54,60,62]

at intermediate temperatures. A possible reason for this

discrepancy could be that entropy differences between

the competing phases in Al3Ti are large enough to re-verse their relative stability at the temperatures where

experiments have been conducted. Indeed, the role of

vibrational entropy in reconciling a similar apparent dis-

crepancy between theory and experiment for the com-

peting phases of Al2Cu was demonstrated by

Wolverton and Ozolins [24]. For Al3Ti, independent cal-

culations of the harmonic vibrational entropies for com-

peting structures have been undertaken [179,180], andboth have shown the D022 (tI8) phase to have a higher

vibrational entropy, by about 0.05 kB/atom, relative to

the D023 (tI16) structure. Due to the small structural en-

ergy differences involved, these small entropy differences

are large enough to lead to a structural transition at

temperatures around 1000 �C. Additional sources of en-

tropy (anharmonic vibrations, electronic and configura-

tional) could lower the transition temperature furtherand possibly reconcile the differences between the zero-

temperature ab initio calculations and experimental

observations at intermediate temperatures. This topic

is one that clearly warrants further investigation,

although due to the small structural energy differences

involved such an effort will require very accurate calcu-

lations of the entropy differences between the competing

structures.Among the metastable phases listed in Table 1, Al2Ti

(oC12) and AlTi3 (hP16) lie just above the convex hull in

Fig. 2(a). While the former has been observed in as-cast

alloys [53,64,65], the latter is stabilized only under high

hydrostatic pressure. As seen in Table 6, the difference in

formation energy for these two structures and their sta-

ble counterparts are 0.35 kJ/mol for Al2Ti and 1 kJ/mol

for AlTi3. The very small energy difference between twoforms of Al2Ti may explain why Al2Ti (oC12), despite

being less stable, is observed in as-cast alloys. It is

Fig. 3. Calculated zero-temperature cohesive properties of Al–Zr intermetallics: (a) the formation energy (DE), (b) the bulk modulus (B0), and (c) the

mean atomic volume (V0). In (a), the solid line defines the ground-state convex hull, and in (b) and (c) the solid line is drawn through the same

structures that form the ground state convex hull in (a).


certainly possible that small differences in the liquid/so-

lid interfacial energy may override the small difference in

chemical driving force during nucleation from the melt.As mentioned in Section 2.1, a variety of ab initio

techniques have been employed to calculate DE of Al3Ti,

Al2Ti, AlTi and AlTi3. From a comparison of the ab ini-

tio DE values, we find that our values agree to within

3 kJ/mol of all previous results, and the agreement is

best (within 0.9 kJ/mol) with the results obtained by

Watson and Weinert [80] using the all-electron FLAS-

TO technique. Furthermore, our results agree to within0.4 kJ/mol with those obtained by Colinet and Pasturel

[81]. The discrepancies between the present results and

previous calculations lie within the range of accuracy

noted in Section 3.1, particularly when it is considered

that most previous calculations made use of all-electron

methods and the LDA, as compared to the present US-

PP-GGA results.

Like the ab initio results, calorimetric data for

enthalpies of formation also show some scatter. One

possible source for the scatter may be associated withthe different temperatures employed in various experi-

ments. The possibility of incomplete reactions, and the

lack of quantification of such effects may also contrib-

ute to the scatter. The heat of formation of Al3Ti (tI8)

was measured by direct reaction synthesis four sepa-

rate times [82–85], and the data represent of spread

of 4 kJ/mol. Among these results, the recent data of

[85] agree very well with the ab initio values. Themeasurements of [85] are, however, in poorer agree-

ment with the calculated results for Al2Ti and AlTi.

The heat of formation of AlTi (tP4) measured by

[82] is in the best agreement with the US-PP and

FLASTO calculations; the data of [82,83] agrees with

the present calculations to within the experimental

uncertainties for AlTi3.

Fig. 4. Calculated zero-temperature cohesive properties of Al–Hf intermetallics: (a) the formation energy (DE), (b) the bulk modulus (B0), and (c) the

mean atomic volume (V0). In (a), the solid line defines the ground-state convex hull, and in (b) and (c) the solid line is drawn through the same

structures that form the ground-state convex hull in (a).


As seen in Table 6, two of the CALPHAD assessments

[66,87] significantly overestimate the magnitude of the

formation energy of B2 AlTi (cP2) phase compared to

both US-PP and FLASTO calculations which agree very

well with each other. The existence of an order–disorder

transition of bcc-(Ti) in the temperature range of 1150–

1400 �C has been proposed only by Ohnuma et al. [66],and warrants further theoretical analysis and experimen-

tal verification. At the time of CALPHAD modeling of

Ohnuma et al. [66], reference to the FLASTO results

for the enthalpy of formation were not considered. The

inclusion of these calculated results would likely have re-

sulted in the prediction of considerably lower order–dis-

order transition temperatures. The BCC-B2 order–

disorder transition at Ti-rich compositions is the subjectof on-going work in our group using both ab initio and

Monte Carlo techniques [181].

Tables 7 and 8 present detailed comparisons between

calculations and measurements for lattice parameters

and atomic coordinates. We find that, in general, the

calculated zero-temperature lattice parameters agree to

within 1% of experimental measurements at ambient

temperature. We also note the good agreement between

calculated and measured (where available) Wyckoffpositions displayed in Table 8: agreement to within

two significant figures is obtained for all non-symme-

try-constrained Wyckoff positions between our calcula-

tions and the measurements where these parameters

were refined. Using the structural model of Al11Ti5 pro-

posed by Schuster and Ipser [53], our ab initio calcula-

tion agrees very well with the measured data,

concerning in particular the magnitude of the expansionof a and contraction of c lattice parameters compared to

those for Al3Ti (tI16), as noted by [35,53].


The large negative heats of formation imply strong

Al–Ti bonds. For the intermetallic phases this inevita-

bly causes an increase in bulk modulus (Fig. 2(b)) and

a decrease in average atomic volume (Fig. 2(c)) rela-

tive to a concentration-weighted average of the pure-

element values. However, it is interesting to note thatthere is no direct one-to-one correspondence between

the lowest-energy phases forming the ground-state

convex hull, and the structures giving rise to maxima

in bulk moduli or a minimum of mean atomic

volume.

4.3. Phase stability and cohesive properties of Al–Zr

intermetallics

The results of ab initio calculations of Al–Zr inter-

metallics are summarized in Tables 9–11, and are plotted

in Fig. 3. Fig. 3(a) plots calculated DE and measured

formation enthalpies as a function of Zr content. As

was observed in the Al–Ti system, the ground-state con-

vex hull is asymmetric and skewed towards Al side with

a maximum in DE at Al2Zr (hP12). Eight intermetallicsare reported to be stable down to low temperatures in

the reported phase diagram. Our calculations yield a

ground-state convex hull defined by five of these struc-

tures: Al3Zr (tI16), Al2Zr (hP12), Al3Zr2 (oF40), Al3Zr4(hP7) and AlZr3 (cP4). The published phase diagrams

also report the stability of AlZr (oC8), Al2Zr3 (tP20)

and AlZr2 (hP6) down to low temperatures. The calcu-

lated formation energies for each of these phases lieabove the convex hull in Fig. 3(a): AlZr (oC8) by 3 kJ/

mol, AlZr2 (hP6) by 1.5 kJ/mol and Al2Zr3 (tP20) by

5.5 kJ/mol. The first two of these structures have ener-

gies lying sufficiently close to the convex hull that their

stability at experimentally accessible temperatures could

arise from entropic contributions differing by a few

tenths of a kB/atom for the competing structures, as dis-

cussed above. The observed stability of Al2Zr3 (tP20),however, is harder to rationalize based on the results

presented in Fig. 3, as structural entropy differences on

the order of 1 kB/atom would be required to stabilize

this structure at temperatures of several hundred Cel-

sius. We note that similar results for the energy of this

structure relative to phase separation between Al3Zr4(hP7) and AlZr3 (cP4) were obtained by Alatalo et al.

[125] employing the all-electron full-potential FLASTOtechnique within the LDA; the theoretical result is thus

one that is apparently not sensitive to the details of the

DFT computational procedures employed. Further

work is clearly warranted to analyze whether finite-tem-

perature contributions to the free energy could be large

enough to bring theoretical predictions in line with the

observed stability of the Al2Zr3 (tP20) structure at

experimentally accessible temperatures. The Al4Zr5(hP18) and Al3Zr5 (tI32) structures are experimentally

observed to be stable only at high temperatures, and

these structures have calculated formation energies that

lie above the convex hull in Fig. 3 by 5–6 kJ/mol.

As mentioned in Section 2.2, several ab initio tech-

niques have been employed to calculate formation ener-

gies of Al–Zr intermetallics. The calculated results in

Table 9 represent a spread of as much as 10 kJ/mol(for Al3Zr (cP4)), which is considerably larger than the

variation between theoretical results found for Al–Ti.

The reason for the larger variation in theoretical results

for the Al–Zr system is unclear. However, we note that

our current results are in very reasonable agreement

(i.e., within the few kJ/mol spread expected based on

the results discussed in Section 3.1) with the most recent

pseudopotential and all-electron full-potential calcula-tions. Plots of DE versus Zr content using the calcula-

tions of Alatalo et al. [125], obtained by both

FLASTO and pseudopotential techniques, show very

similar results compared with Fig. 3(a) in terms of the

shape of convex hull and the phases that describe it; a

single exception is the Al3Zr2 (oF40) phase which lies

above the convex hull in the previous calculations.

For the metastable phases listed in Table 2, the cal-culated DE lie about 1.7–32 kJ/mol above the convex

hull. For the virtual phases considered, the calculated

DE lie about 1.2–4.7 kJ/mol above the convex hull.

Two phases Al3Zr5 (hP16) [94–96,104,107,115,116,120]

and AlZr2 (tI12) [101] were reported as stable phases;

however, it has been suspected they were stabilized

by interstitial impurities [96,116,120] and Si [105],

respectively. Our total energy calculations are consis-tent with the interpretation that these two structures

are metastable in pure alloys; as listed in Table 10

and plotted in Fig. 3(a), DE for both of these phases

lie well above the convex hull.

Standard enthalpies of formation for Al–Zr inter-

metallics have been determined by calorimetry, as re-

ported in three separate publications [127,129,130].

Our ab initio DE values for Al3Zr and Al2Zr agreevery well with the measured DH 298.15

f values of [129]

and [130]; by contrast, the calorimetry values reported

in [127] are significantly smaller in magnitude. Kema-

tick and Franzen [128] measured the equilibrium va-

por pressure of Al over Al3Zr, Al2Zr, Al3Zr2, AlZr,

Al4Zr5, Al2Zr3 and Al3Zr5 in the temperature range

of 1298–1673 K. They derived DH 298.15f values from

the knowledge of the standard enthalpy changes asso-ciated with a particular decomposition reaction, which

in turn were determined by second- and third-law

methods. As noted by Murray [91], Kematick and

Franzen [128] did not account for the change in refer-

ence state of Al (from liquid to solid) when reporting

DH 298.15f values. Therefore, Murray recalculated

DH 298.15f values with respect to solid Al, and also esti-

mated the associated error to be ±4 kJ/mol which islarger than that reported in calorimetric measure-

ments. As seen in Table 9, the DH 298.15f values of

Table 9

A comparison of heat of formation (DE) of Al–Zr intermetallics obtained by various methods: ab inito calculations (at 0 K), experiment (at different

temperatures), and CALPHAD (at 298.15 K or the standard heat of formation) modeling of Al–Zr phase diagram



Stable

Al3Zr I4/mmm (139) Al3Zr �49.106a �44 ± 2 [127] �40.50 [131]

�46.570b [124] �49 ± 4 [91,128] �48.50 [132]

�51.064c [76] �48.4 ± 1.3 [129]

�45.286d [125]

�48.176e [125]

�47.600f [126]

�53.453g [23]

Al2Zr P63/mmc (194) MgZn2 �53.327a �44 ± 2 [127] �45.81 [131]

�53.957d [125] �54 ± 4 [91,128] �52.60 [132]

�54.921e [125] �52.1 ± 1.6 [129]

�51.3 ± 4.3 [130]

Al3Zr2 Fdd2 (43) Al3Zr2 �51.649a �31 ± ? [127] �46.94 [131]

�48.176d [125] �55 ± 4 [91,128] �56.60 [132]

�51.356e [125]

AlZr Cmcm (63) CrB �46.163a �53 ± 4 [91,128] �44.50 [131]

�43.359d [125] �64.95 [132]

�44.322e [125]

�47.120h [77]

Al4Zr5 P63/mcm (193) Ga4Ti5 �42.002a �52 ± 4 [91,128] �41.0 [131]

�38.541d [125] �55.42 [132]

�40.468e [125]

Al3Zr4 P�6 ð174Þ Al3Zr4 �47.555a �58.48 [132]

�43.359d [125]

�45.286e [125]

Al2Zr3 P42/mnm (136) Al2Zr3 �39.298a �49 ± 4 [91,128] �38.43 [131]

�38.541e [125] �55.18 [132]

Al3Zr5 (h) I4/mcm (140) W5Si3 �37.599a �48 ± 4 [91,128] �36.25 [131]

�33.723d [125] �55.48 [132]

�35.657e [125]

AlZr2 P63/mmc (194) Ni2In �36.753a �33.37 [131]

�33.723d [125] �48.36 [132]

�35.651e [125]

AlZr3 Pm�3m ð221Þ AuCu3 �31.088a �27.0 [131]

�28.906d [125] �36.16 [132]

�29.869e [125]

Metastable

Al6Zr Cmcm (63) Al6Mn �22.035a

Al11Zr2 Pm�3 ð200Þ Zn11Mg2 �12.718a

Al3Zr Pm�3m ð221Þ AuCu3 �46.418a

�41.816b [124]

�50.064c [76]

�43.358d [125]

�45.286e [125]

�44.600f [126]

�51.195g [23]

Al2Zr P63/mmc (194) Ni2In �21.982a

AlZr Pm�3m ð221Þ CsCl �29.995a

�26.015d [125]

�27.942e [125]

Al3Zr5 (m) P63/mcm (193) Mn5Si3 �35.816a

�33.723e [125]

AlZr2 I4/mcm (140) Al2Cu �31.318a

�28.906d [125]

�29.869e [125]

AlZr3 P63/mmc (194) Ni3Sn �29.691a

Virtual

Al3Zr I4/mmm (139) Al3Ti �46.552a

�42.907b [124]



Table 9 (continued)



�47.520c [76]

�42.395d [125]

�46.249e [125]

�46.300f [126]

�51.248g [23]

Al5Zr2 P4/mmm (123) Al5Ti2 �47.545a

Al11Zr5 I4/mmm (139) Al11Ti5 �47.590a

Al2Zr Cmcm (65) Ga2Zr �51.660a

Al2Zr I41/amd (141) Ga2Hf �52.104a

�51.067e [125]

Al5Zr3 P4/mbm (127) Ga5Ti3 �49.956a

AlZr P4/mmm (123) AuCu �44.891a

�41.432d [125]

�43.359e [125]

a US-PP (GGA) [this study]; b LMTO-ASA (LDA); c,g,h FP-LMTO (LDA); d FLASTO (LDA); e PW-PP (LDA); f US-PP with semicore treatment

(GGA).

The reference states are fcc-Al and hcp-Zr.


[91,128] agree reasonably well with ab initio DE values

for Al3Zr, Al2Zr and Al3Zr2. A systematic deviation

from our ab initio values is noted for intermetallic

phases with increasing Zr content. Considering the

overall better agreement between our calculations

and calorimetry, we conclude that heat of formation

values obtained by direct reaction synthesis in a calo-

rimeter are far more reliable than those obtained bythe second- and third-law methods.

A seen in Table 10, the calculated lattice parameters

of stable phases agree within 1% of the experimental val-

ues at ambient temperature. Even though Alatalo et al.

[125] reported DE values for fully optimized unit-cell

geometries by PW-PP method, they did not report

cell-external and cell-internal parameters. Among the

metastable phases, except for Al11Zr and Al3Zr, theagreement between experiment and theory varies from

reasonable (Al2Zr and AlZr2) to poor (Al6Zr, AlZr

and Al3Zr5). In particular, for Al6Zr and AlZr a discrep-

ancy of up to 15% is noted. The lattice parameter of

AlZr (cP2) reported by [118] is 0.29 nm, which is very

small compared to the value that would be derived from

a weighted mean of the bcc-Al and bcc-Zr values (see

Table 4). The lattice parameters of this phase was mea-sured by electron diffraction [117,118], and it is uncer-

tain if the discrepancy is associated with the

calibration of camera constant in a transmission elec-

tron microscope. Table 11 presents detailed compari-

sons between calculations and measurements for

atomic coordinates.

Ma et al. [122] performed a rigorous structural

analysis of Al3Zr (tI16) using X-ray and large angleconvergent beam electron diffraction techniques. Both

single crystal and powder specimens were used. They

reported cell-internal parameters of Al (4e) and Zr

(4e) up to five significant digits, and we note a very

good agreement between these measured and our cal-

culated Wyckoff positions. For other phases, earlier

experimental data were not obtained by as rigorous

analysis of X-ray diffraction data as by Ma et al.

[122] in the case of Al3Zr (tI16), and for these phases

the agreement between experiment and theory can be

considered as only reasonable.

Fig. 3(b) and (c) shows the variation of B0 and V0,

respectively, as a function of Zr content. As expectedthey show positive and negative deviation, respec-

tively, from the ideal behavior shown by the dotted

line. As noted in the previous section, there is no

one-to-one correspondence between DE of phases

forming the ground-state convex hull and the relative

B0 and V0 properties.

4.4. Phase stability and cohesive properties of Al–Hf

intermetallics

The results of ab initio calculations for Al–Hf inter-

metallics are summarized in Tables 12–14 and are plot-

ted in Fig. 4. Fig. 4(a) shows the plot of DE as a function

of Hf content. As in the Al–Ti and Al–Zr systems, the

ground-state convex hull is asymmetric and skewed to-

wards the Al side with a maximum in DE at Al2Hf.Six intermetallics are reported to be stable at low tem-

peratures in the equilibrium phase diagram [144], as

indicated in Table 3 (Al3Hf (tI8) is a high-temperature

phase). Of these, three appear on the calculated

ground-state hull, namely Al3Hf (tI16), Al2Hf (hP12)

and Al3Hf4 (hP7). The phases Al3Hf2 (oF40), AlHf

(oC8) and Al2Hf3 (tP20), which are also observed at

low temperatures, lie about 0.2, 1 and 5 kJ/mol abovethe convex hull, respectively. As discussed in detail

above, entropic terms could very likely lead to the stabil-

ization of the first two structures at experimentally

accessible temperatures, while for the latter structure

the 5 kJ/mol energy difference seems relatively large to

Table 10

A comparison of unit cell-external parameters of Al–Zr intermetallics obtained by ab inito calculations (at 0 K) and diffraction experiments (at


Phase Space

group (#)

Prototype EOS parameters Unit cell-external parameters

V0 B0 B00 Ab initio Experiment

a b c a b c

Stable

Al3Zr I4/mmm (139) Al3Zr 17.366 10.25 3.98 0.40082 1.72969a 0.40074 1.72864 [128]

0.39100 1.70476c [76] 0.39993 1.72832 [122]

0.40200 1.73600f [126]

10.00 0.39185 1.70375g [23]

Al2Zr P63/mmc (194) MgZn2 17.559 11.36 3.99 0.52773 0.87349a 0.52807 0.87491 [128]

0.52820 0.87480 [94]

Al3Zr2 Fdd2 (43) Al3Zr2 18.574 10.86 4.05 0.96949 1.38994 0.55705a 0.96173 1.39343 0.55842 [128]

0.96010 1.39060 0.55700 [102]

AlZr Cmcm (63) CrB 19.508 10.88 4.02 0.33603 1.08770 0.42696a 0.33590 1.08870 0.42740 [106]

0.33621 1.08923 0.42742 [128]

Al4Zr5 P63/mcm (193) Ga4Ti5 19.679 10.33 3.77 0.84344 0.57477a 0.84322 0.57912 [128]

0.84470 0.58100 [33]

Al3Zr4 P�6 ð174Þ Al5Zr4 19.361 10.76 3.86 0.54479 0.52724a 0.54330 0.53900 [100]

0.54300 0.53890 [96]

Al2Zr3 P42/mnm (136) Al2Zr3 20.250 10.72 3.89 0.76313 0.69541a 0.76334 0.69962 [128]

0.76301 0.69981 [99]

Al3Zr5 (h) I4/mcm (140) W5Si3 20.301 10.33 3.63 1.10429 0.53272a 1.10432 0.53922 [128]

1.10490 0.53960 [96]

AlZr2 P63/mmc (194) Ni2In 20.277 10.52 3.77 0.48882 0.58798a 0.48939 0.59283 [103]

0.48820 0.59180 [33]

AlZr3 Pm�3m ð221Þ AuCu3 20.629 10.14 3.33 0.43536a 0.43720 [93]

10.77 0.42879g [23] 0.43740 [33]

Metastable

Al6Zr Cmcm (63) Al6Mn 18.774 n.d. n.d. 0.81415 0.81443 0.79278a 0.74890 0.65560 0.89610 [118]

Al11Zr2 Pm�3 ð200Þ Zn11Mg2 17.337 8.54 4.37 0.85459a 0.85000 [117]

Al3Zr Pm�3m ð221Þ AuCu3 17.190 10.31 3.93 0.40968a 0.40500 to

10.0 0.40730b [124] 0.40930 [126]

0.41100f [126]

9.96 0.40099g [23]

Al2Zr P63/mmc (194) Ni2In 18.224 9.03 4.00 0.47662 0.55559a 0.48820 0.59180 [118]

AlZr Pm�3m ð221Þ CsCl 19.201 10.32 4.56 0.33738a 0.29000 [118]

Al3Zr5 (m) P63/mcm (193) Mn5Si3 20.373 9.75 3.57 0.83393 0.54098a 0.82800 0.56900 [107]

0.81840 0.57020 [95]

AlZr2 I4/mcm (140) Al2Cu 20.708 n.d. n.d. 0.68227 0.53383a 0.68540 0.55010 [101]

AlZr3 P63/mmc (194) Ni3Sn 20.072 10.09 2.79 0.61604 0.50447a

Virtual

Al3Zr I4/mmm (139) Al3Ti 17.581 10.18 4.05 0.39479 0.90214a

11.0 0.39500 0.88200b [124]

0.39600 0.90400f [126]

9.97 0.38752 0.88432g [23]

Al5Zr2 P4/mmm (123) Al5Ti2 17.601 n.d. n.d. 0.40435 3.01419a

Al11Zr5 I4/mmm (139) Al11Ti5 17.797 n.d. n.d. 0.40699 1.71901a

Al2Zr Cmcm (65) Ga2Zr 17.999 n.d. n.d. 1.27847 0.40597 0.41615a

Al2Zr I41/amd (141) Ga2Hf 17.988 n.d. n.d. 0.41125 2.55256a

Al5Zr3 P4/mbm (127) Ga5Ti3 18.339 n.d. n.d. 1.18295 0.41936a

AlZr P4/mmm (123) AuCu 19.093 10.54 3.91 0.42811 0.41654a

Also listed are equilibrium volume (V0, ·10�3 nm3/atom), bulk modulus (B0, ·1010 N/m2) and B00 as defined by the equation of state (EOS) at 0 K.

a US-PP (GGA) [this study]; b LMTO-ASA (LDA); c,g FP-LMTO (LDA); f US-PP with semicore treatment (GGA); n.d.: not determined.


be overcome by structural entropy differences at several

hundred degrees Celsius. This apparent discrepancy be-

tween experimental observations and calculations again

warrants further theoretical investigations focusing on

calculations of the finite-temperature free energies of

these competing structures.

For all virtual phases considered, the calculated DElie about 0.4–25 kJ/mol above the convex hull, except

for AlHf3 (cP4) which lies on the convex hull. Even

though it has not been experimentally observed, the

prediction of AlHf3 (cP4) as the ground state is not

surprising given that AlZr3 (cP4) is also observed and

Table 11

A comparison of unit cell-internal parameters (Wyckoff positions) of Al–Zr intermetallics (where applicable) obtained from our ab inito calculations




Stable

Al3Zr I4/mmm (139) Al3Zr Al1: 4c 0.00000 0.50000 0.00000 0.00000 0.50000 0.00000 [122]

Al2: 4d 0.00000 0.50000 0.25000 0.00000 0.50000 0.25000

Al3: 4e 0.00000 0.00000 0.37502 0.00000 0.00000 0.37498

Zr: 4e 0.00000 0.00000 0.11851 0.00000 0.00000 0.11886

Al2Zr P63/mmc (194) MgZn2 Al1: 2c 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 [94]

Al2: 6h 0.82902 0.65804 0.25000 0.83333 0.66666 0.25000

Zr: 4f 0.33333 0.66666 0.06546 0.33333 0.66666 0.06525

Al3Zr2 Fdd2 (43) Al3Zr2 Al1: 8a 0.00000 0.00000 0.61831 0.00000 0.00000 0.62500 [102]

Al2: 16b 0.18170 0.13514 0.49351 0.18500 0.11600 0.50000

Zr: 16b 0.18270 0.05266 0.00234 0.18200 0.06800 0.00000

AlZr Cmcm (63) CrB Al: 4c 0.00000 0.42842 0.25000 0.00000 0.43000 0.25000 [106]

Zr: 4c 0.00000 0.15947 0.25000 0.00000 0.16000 0.25000

Al4Zr5 P63/mcm (193) Ga4Ti5 Al1: 2b 0.00000 0.00000 0.00000

Al2: 6g 0.63032 0.00000 0.25000

Zr1: 4d 0.33333 0.66667 0.00000

Zr2: 6g 0.29085 0.00000 0.25000

Al3Zr4 P�6 ð174Þ Al5Zr4 Al: 3j 0.33331 0.16669 0.00000 0.33333 0.16666 0.00000 [96]

Zr1: 1b 0.00000 0.00000 0.50000 0.00000 0.00000 0.50000

Zr2: 1f 0.66662 0.33334 0.50000 0.66666 0.33333 0.50000

Zr3: 2h 0.33332 0.66669 0.26651 0.33333 0.66666 0.25000

Al2Zr3 P42/mnm (136) Al2Zr3 Al: 8j 0.12101 0.12101 0.21835 0.12500 0.12500 0.25000 [100]

Zr1: 4d 0.00000 0.50000 0.25000 0.00000 0.50000 0.25000

Zr2: 4f 0.33994 0.33994 0.00000 0.34000 0.34000 0.00000

Zr3: 4g 0.19576 0.80424 0.00000 0.20000 0.80000 0.00000

Al3Zr5 (h) I4/mcm (140) W5Si3 A1l: 4a 0.00000 0.00000 0.25000 0.00000 0.00000 0.25000 [96]

Al2: 8h 0.16337 0.66337 0.00000 0.16666 0.66666 0.00000

Zr1: 4b 0.00000 0.50000 0.25000 0.00000 0.50000 0.25000

Zr2: 16k 0.07902 0.21992 0.00000 0.07700 0.21800 0.00000

Metastable

Al6Zr Cmcm (63) Al6Mn Al1: 8e 0.24001 0.00000 0.00000

Al2: 8f 0.00000 0.24022 0.00001

Al3: 8g 0.25001 0.25001 0.25000

Zr: 4c 0.00000 0.49999 0.25000

Al11Zr2 Pm�3 ð200Þ Zn11Mg2 Al1: 1b 0.50000 0.50000 0.50000

Al2: 6e 0.23069 0.00000 0.00000

Al3: 6g 0.16171 0.50000 0.00000

Al4: 8i 0.22104 0.22104 0.22104

Al5: 12k 0.50000 0.23478 0.34433

Zr: 6f 0.30859 0.00000 0.50000

Al3Zr5 (m) P63/mcm (193) Mn5Si3 A1: 6g 0.60653 0.00000 0.25000 0.59000 0.00000 0.25000 [95]

Zr1: 4d 0.33333 0.66667 0.00000 0.33333 0.66667 0.00000

Zr2: 6g 0.23712 0.00000 0.25000 0.23000 0.00000 0.25000

Zr: 8h 0.15049 0.65049 0.00000

AlZr3 P63/mmc (194) Ni3Sn Al: 2c 0.33333 0.66667 0.25000

Zr: 6h 0.82832 0.65665 0.25000

Virtual

Al5Zr2 P4/mmm (123) Al5Ti2 Al1: 2e 0.00000 0.50000 0.50000

Al2: 2g 0.00000 0.00000 0.14187

Al3: 2g 0.00000 0.00000 0.28561

Al4: 2h 0.50000 0.50000 0.42834

Al5: 4i 0.00000 0.50000 0.07302

Al6: 4i 0.00000 0.50000 0.21446

Al7: 4i 0.00000 0.50000 0.35702

Zr1: 1a 0.00000 0.00000 0.00000

Zr2: 1c 0.50000 0.50000 0.00000

Zr3: 2g 0.00000 0.00000 0.43235

Zr4: 2h 0.50000 0.50000 0.14680

Zr5: 2h 0.50000 0.50000 0.28168



Table 12

A comparison of heat of formation (DE) of Al–Hf intermetallics obtained by various methods: ab inito calculations (at 0 K), experiment (at different

temperatures), and CALPHAD (at 298.15 K or the standard heat of formation) modeling of Al–Hf Phase diagram



Stable

Al3Hf I4/mmm (139) Al3Zr �39.632a �40.6 ± 0.8 [146] �41.818 [148]

�40.000b [145] �44.7 ± 2.4 [147] �41.077 [149]

Al3Hf I4/mmm (139) Al3Ti �38.649a �38.077 [149]

�38.900b [145]

Al2Hf P63/mmc (194) MgZn2 �43.289a �43.8 ± 1.3 [146] �48.307 [148]

�41.673 [149]

Al3Hf2 Fdd2 (43) Al3Zr2 �41.796a �40.8 ± 2.6 [147] �47.512 [148]

�42.885 [149]

AlHf Cmcm (63) CrB �39.028a �39.9 ± 2.0 [146] �46.298 [148]

�36.1 ± 4.3 [147] �45.203 [149]

Al3Hf4 P�6 ð174Þ Al5Zr4 �38.616a �44.372 [148]

�47.868 [149]

Al2Hf3 P42/mnm (136) Al2Zr3 �31.702a �43.535 [148]

�48.908 [149]

Metastable

Al3Hf Pm�3m ð221Þ AuCu3 �36.828a

�37.300b [145]

Al3Hf5 P63/mcm (193) Mn5Si3 �28.218a

AlHf2 I4/mcm (140) Al2Cu �25.189a �41.023 [148]

�41.244 [149]

Virtual

Al5Hf2 P4/mmm (123) Al5Ti2 �38.207a

Al11Hf5 I4/mmm (139) Al11Ti5 �38.274a

Al2Hf Cmcm (65) Ga2Zr �42.313a

Al2Hf I41/amd (141) Ga2Hf �42.811a

Al5Hf3 P4/mbm (127) Ga5Ti3 �41.199a

AlHf Pm�3m ð221Þ CsCl �20.132a

AlHf P4/mmm (123) AuCu �35.270a

AlHf3 P63/mmc (194) Ni3Sn �22.891a

Al4Hf5 P63/mcm (193) Ga4Ti5 �32.320a

Al3Hf5 I4/mcm (140) W5Si3 �28.847a

AlHf2 P63/mmc (194) Ni2In �27.829a

AlHf3 Pm�3m ð221Þ AuCu3 �24.288a

The reference states are fcc-Al and hcp-Hf.a US-PP (GGA) [this study]; b US-PP (GGA) [145].

Table 11 (continued)



Al2Zr Cmcm (65) Ga2Zr Al1: 2a 0.00000 0.00000 0.00000

Al2: 2c 0.50000 0.00000 0.50000

Al3: 4h 0.17281 0.00000 0.50000

Zr: 4g 0.34711 0.00000 0.00000

Al2Zr I41/amd (141) Ga2Hf Al1: 8e 0.00000 0.00000 0.25012

Al2: 8e 0.00000 0.00000 0.41379

Zr: 8e 0.00000 0.00000 0.07646

Al5Zr3 P4/mbm (127) Ga5Ti3 Al1: 2a 0.00000 0.00000 0.00000

Al2: 2d 0.00000 0.50000 0.00000

Al3: 4g 0.25128 0.75128 0.00000

Al4: 4h 0.37832 0.87832 0.00000

Al5: 8i 0.25222 0.50222 0.00000

Zr1: 4h 0.11629 0.61629 0.50000

Zr2: 8j 0.11950 0.11950 0.50000


Table 13

A comparison of unit cell-external parameters of Al–Hf intermetallics obtained by ab inito calculations (at 0 K) and diffraction experiments (at


Phase Space group (#) Prototype EOS parameters Unit cell-external parameters

V0 B0 B00 Ab initio Experiment [Ref.]

a b c a b c

Stable

Al3Hf I4/mmm (139) Al3Zr 17.093 10.53 3.87 0.39898 1.71719a 0.39190 to 1.71390

10.48 0.39870 1.71790b 0.40100 1.76530 [145]

Al3Hf I4/mmm (139) Al3Ti 17.251 10.54 4.15 0.39439 0.89103a 0.39280 0.88800 [134]

10.50 0.39310 0.89300b 0.39830 0.89250 [98]

Al2Hf P63/mmc (194) MgZn2 17.163 11.83 3.95 0.52346 0.86720a 0.52880 0.87390 [96]

0.52300 0.86510 [98]

Al3Hf2 Fdd2 (43) Al3Zr2 18.052 11.43 4.02 0.95133 1.37692 0.55119a 0.94740 1.37370 0.55010 [98]

0.95230 1.37630 0.55220 [136]

AlHf Cmcm (63) CrB 18.809 11.74 4.11 0.32505 1.08215 0.42742a 0.32560 1.08320 0.42810 [134]

0.32520 1.08220 0.42800 [138]

Al3Hf4 P�6 ð174Þ Al5Zr4 18.922 11.61 3.84 0.53481 0.53519a 0.53430 0.54220 [98]

0.53310 0.54140 [134]

Al2Hf3 P42/mnm (136) Al2Zr3 19.514 12.30 3.97 0.75439 0.68574a 0.75490 0.69090 [98]

0.75350 0.69060 [135]

Metastable

Al3Hf Pm�3m ð221Þ AuCu3 16.988 10.38 4.13 0.40807a 0.40480 to

10.30 0.40910b 0.40800 [145]

Al3Hf5 P63/mcm (193) Mn5Si3 19.988 10.99 2.79 0.80937 0.56376a 0.80660 0.56780 [134]

AlHf2 I4/mcm (140) Al2Cu 20.209 11.21 4.96 0.68026 0.52106a 0.67760 0.53720 [101]

Virtual

Al5Hf2 P4/mmm (123) Al5Ti2 17.313 n.d. n.d. 0.40262 2.99047a

Al11Hf5 I4/mmm (139) Al11Ti5 17.288 n.d. n.d. 0.40538 1.70272a

Al2Hf Cmcm (65) Ga2Zr 17.622 n.d. n.d. 1.26367 0.40354 0.41468a

Al2Hf I41/amd (141) Ga2Hf 17.611 n.d. n.d. 0.40925 2.52363a

Al5Hf3 P4/mbm (127) Ga5Ti3 17.927 n.d. n.d. 1.17105 0.41833a

AlHf Pm�3m ð221Þ CsCl 18.867 10.92 3.33 0.33434a

AlHf P4/mmm (123) AuCu 18.621 11.10 3.35 0.42286 0.41618a

AlHf3 P63/mmc (194) Ni3Sn 20.118 11.09 2.86 0.61095 0.49824a

Al4Hf5 P63/mcm (193) Ga4Ti5 19.097 10.53 3.87 0.83231 0.57297a

Al3Hf5 I4/mcm (140) W5Si3 19.968 n.d. n.d. 1.08442 0.53550a

AlHf2 P63/mmc (194) Ni2In 19.638 n.d. n.d. 0.48185 0.58601a

AlHf3 Pm�3m ð221Þ AuCu3 20.003 11.14 3.56 0.43109a

Also listed are equilibrium volume (V0, ·10�3 nm3/atom), bulk modulus (B0, ·1010 N/m2) and B00 as defined by the equation of state (EOS) at 0 K.

GA: a US-PP (GGA) [this study]; b US-PP (GGA) [145]; n.d.: not determined.


predicted (as discussed in Section 4.3), to be a stable

phase in the Al–Zr system. The fact that this structure

has a formation energy lying essentially on the convex

hull could indicate that it is in fact stable at very lowtemperature, but that it transforms to the experimen-

tally observed phase-separated mixture of Hf and

Hf2Al through a low-lying peritectoid reaction. Two

phases Al3Hf5 (hP16) [134,135] and Al2Hf (tI12)

[101,107] were reported as stable phases; however, sub-

sequent investigations suggested that they were stabi-

lized by interstitial impurities [105,140] and Si [105],

respectively. Our total energy calculations are consis-tent with the interpretation that these two structures

are metastable in pure alloys; as listed in Table 12

and plotted in Fig. 4(a), DE for both of these phases

lie well above the convex hull. Furthermore, both of

these phases are energetically less favorable compared

to the virtual counterparts, Al3Hf5 (tI32) and AlHf2(hP6), considered in this study.

In the only other theoretical work for this system,

Colinet and Pasturel [145] used the same US-PPs as inour study, and reported DE of Al3Hf. Their DE values

agree very well, within 0.4 kJ/mol, of our values for all

three competing structures of Al3Hf. Two sets of enthal-

py of formation measurements have been reported.

Meschel and Kleppa [146] measured the standard en-

thalpy of formation of Al3Hf, Al2Hf and AlHf using

high-temperature direct synthesis calorimetry, and re-

ported a maximum uncertainty of 2 kJ/mol. As seen inTable 12, our ab initio DE values are in very good agree-

ment with their measurements, lying within the reported

measurement error bars in each case. Balducci et al.

[147] also reported the standard enthalpy of formation

of Al3Hf, Al3Hf2 and AlHf. Their experimental

Table 14

A comparison of unit cell-internal parameters (Wyckoff positions) of Al–Hf intermetallics (where applicable) obtained from our ab inito calculations




Stable

Al3Hf I4/mmm (139) Al3Zr Al1: 4c 0.00000 0.50000 0.00000

Al2: 4d 0.00000 0.50000 0.25000

Al3: 4e 0.00000 0.00000 0.37538

Hf: 4e 0.00000 0.00000 0.11920

Al2Hf P63/mmc (194) MgZn2 Al1: 2c 0.00000 0.00000 0.00000

Al2: 6h 0.82909 0.65818 0.25000

Hf: 4f 0.33333 0.66666 0.06413

Al3Hf2 Fdd2 (43) Al3Zr2 Al1: 8a 0.00000 0.00000 0.62176 0.00000 0.00000 0.63000 [104]

Al2: 16b 0.18236 0.13574 0.49257 0.18500 0.12900 0.50000

Hf: 16b 0.18461 0.05291 0.00156 0.18500 0.05400 0.00000

AlHf Cmcm (63) CrB Al: 4c 0.00000 0.42953 0.25000 0.00000 0.42500 0.25000 [138]

Hf: 4c 0.00000 0.16297 0.25000 0.00000 0.16700 0.25000

Al3Hf4 P�6 ð174Þ Al5Zr4 Al: 3j 0.33331 0.16669 0.00000

Hf1: 1b 0.00000 0.00000 0.50000

Hf2: 1f 0.66666 0.33333 0.50000

Hf3: 2h 0.33333 0.66666 0.26055

Al2Hf3 P42/mnm (136) Al2Zr3 Al: 8j 0.12112 0.12112 0.21139 0.12500 0.12500 0.21000 [135]

Hf1: 4d 0.00000 0.50000 0.25000 0.00000 0.50000 0.25000

Hf2: 4f 0.34392 0.34392 0.00000 0.34000 0.34000 0.00000

Hf3: 4g 0.20133 0.79867 0.00000 0.20000 0.80000 0.00000

Metastable

Al3Hf5 P63/mcm (193) Mn5Si3 A1: 6g 0.60357 0.00000 0.25000 0.61500 0.00000 0.25000 [134]

Hf1: 4d 0.33333 0.66666 0.00000 0.33333 0.66666 0.00000

Hf2: 6g 0.23635 0.00000 0.25000 0.24000 0.00000 0.25000

AlHf2 I4/mcm (140) Al2Cu Al: 4a 0.00000 0.00000 0.25000

Hf: 8h 0.15086 0.65086 0.00000

Virtual

Al5Hf2 P4/mmm (123) Al5Ti2 Al1: 2e 0.00000 0.50000 0.50000

Al2: 2g 0.00000 0.00000 0.14189

Al3: 2g 0.00000 0.00000 0.28515

Al4: 2h 0.50000 0.50000 0.42867

Al5: 4i 0.00000 0.50000 0.07254

Al6: 4i 0.00000 0.50000 0.21437

Al7: 4i 0.00000 0.50000 0.35701

Hf1: 1a 0.00000 0.00000 0.00000

Hf2: 1c 0.50000 0.50000 0.00000

Hf3: 2g 0.00000 0.00000 0.43203

Hf4: 2h 0.50000 0.50000 0.14635

Hf5: 2h 0.50000 0.50000 0.28206

Al2Hf Cmcm (65) Ga2Zr Al1: 2a 0.00000 0.00000 0.00000

Al2: 2c 0.50000 0.00000 0.50000

Al3: 4h 0.17297 0.00000 0.50000

Hf: 4g 0.34567 0.00000 0.00000

Al2Hf I41/amd (141) Ga2Hf Al1: 8e 0.00000 0.00000 0.25035

Al2: 8e 0.00000 0.00000 0.41373

Zr: 8e 0.00000 0.00000 0.07715

Al5Hf3 P4/mbm (127) Ga5Ti3 Al1: 2a 0.00000 0.00000 0.00000

Al2: 2d 0.00000 0.50000 0.00000

Al3: 4g 0.25099 0.75099 0.00000

Al4: 4h 0.37833 0.87833 0.00000

Al5: 8i 0.253122 0.50312 0.00000

Hf1: 4h 0.11727 0.61727 0.50000

Hf2: 8j 0.12032 0.12032 0.50000

AlHf3 P63/mmc (194) Ni3 Sn Al: 2c 0.33333 0.66667 0.25000

Hf: 6h 0.82993 0.65987 0.25000

Al4Hf5 P63/mcm (193) Ga4Ti5 Al1: 2b 0.00000 0.00000 0.00000

Al2: 6g 0.62773 0.00000 0.25000

Hf1: 4d 0.33333 0.66667 0.00000



Table 14 (continued)



Hf2: 6g 0.29011 0.00000 0.25000

Al3Hf5 I4/mcm (140) W5Si3 A1l: 4a 0.00000 0.00000 0.25000

Al2: 8h 0.16450 0.66450 0.00000

Hf1: 4b 0.00000 0.50000 0.25000

Hf2: 16k 0.07761 0.22062 0.00000


approach relied on measurements of the Al vapor pres-

sure for these compounds in the temperature range of

1280–1680 K using Knudsen cell-mass spectrometry;

from these measurements the standard enthalpy of for-

mation was derived employing second- and third-law

methods. Their values are found to differ by about ten

per cent from those derived from calorimetry by

Meschel and Kleppa, although the errors associatedwith Balducci�s data are reported to be much higher

than those of Meschel and Kleppa. For AlHf the values

of DH from the two groups are thus within the estimated

uncertainties. Further, our ab initio calculated value of

DE of Al3Hf2 also agrees with the measurement of Bald-

ucci et al. [147] to within the reported error bar.

As seen in Table 13, the calculated lattice parameters

of all stable phases agree within ±1% of the experimen-tal values at ambient temperature. In contrast to the sit-

uation for Al–Zr, the calculated lattice parameters for

the metastable Al3Hf5 (tI32) and AlHf2 (hP6) phases

also agree to within ±1% of the experimental values.

Table 14 presents detailed comparisons between calcula-

tions and measurements for atomic coordinates. Except

for Al3Hf, there is no recent study of crystal structures

of Al–Hf intermetallics by diffraction method. However,where comparisons between experiment and calcula-

tions are possible, we typically find (as in both the Al–

Ti and Al–Zr systems) very good agreement, at the level

of two significant figures.

Fig. 4(b) and (c) shows the variation of B0 and V0,

respectively, as a function of Hf content. As expected

they show positive and negative deviation, respectively,

from the ideal behavior shown by the dotted line. Onceagain, there is no one-to-one correspondence between

DE of phases forming the ground-state convex hull

and the ordering of their B0 and V0 properties.

4.5. Ab initio phase stability and CALPHAD modeling:

comparison and implications

As mentioned in Section 2, ab initio phase stabilitieshave been calculated using various techniques, such as

LMTO-ASA, FP-LMTO, FLAPW, FLASTO and US-

PP. In comparing the ab initio phase stability and

CALPHAD model parameters, it is worth mentioning

several points. First, we note that DE of stable phases

obtained by the ab initio techniques (see Tables 6, 9

and 12) typically agree within ±3 kJ/mol (a notable

exception being Al–Zr where such agreement is only

found between the most recent results). Second, in the

majority of the cases ab initio DE values agree well,

within the experimental uncertainties, with heats of for-

mation measured directly by calorimetry. Further,

DH 298.15f values obtained from the CALPHAD assess-

ments are found to vary over a wide range of in thesystems Al–Ti and Al–Zr. Specifically, considering the

Al–Ti assessments [66,87–90] we find that DH 298.15f val-

ues of Al3Ti show a spread of 12 kJ/mol, while for

Al5Ti2 [66,88,90] and Al2Ti [66,88,90] the variations

are 10.5 and 8 kJ/mol, respectively. The spreads of

DH 298.15f of Al3Ti and AlTi are much larger than the

experimental uncertainties (see Table 6) in the measured

enthalpies of formation. These results are noteworthygiven that most of these CALPHAD assessments are

based on same experimental thermodynamic and phase

diagram data. Similarly, considering Al–Zr assessments

[131,132] we find that DH 298.15f values of AlZr, Al2Zr3,

Al3Zr5 and AlZr2 differ by 20, 17, 19 and 15 kJ/mol,

respectively. Consistent with this large variation, the

maximum difference between ab initio DE and CALP-

HAD DH 298.15f is about 20 kJ/mol.

A common practice in the CALPHAD community is

to use the heat of formation values predicted by Mie-

dema�s semi-empirical model [182] when there is no calo-

rimetric data. It is important to note that Miedema�soriginal goals were to predict the sign of heat of forma-

tion (rather than the absolute value) and to investigate

the chemical trends associated with alloying. The limita-

tions of the Miedema model for deriving quantitativevalues for enthalpies of formation have been discussed

in detail previously (e.g. [183]). The current ab initio cal-

culated values of DE differ from Miedema�s prediction

[183] by as much as 30 kJ/mol. In the case of the Al–

Hf system, due to lack of sufficient calorimetric data,

Wang et al. [149] used Miedema�s values for CALPHAD

optimization. This can be seen in Table 12 to have given

rise to a large discrepancy between the ab initio DE andCALPHAD-optimized value for DH 298.15

f .

It has become widely recognized that for predictive

modeling of multicomponent phase stability and kinet-

ics, as relevant to design and processing of engineering

alloys, computational thermodynamics and kinetics

based on the CALPHAD approach represents the only


viable option. It is equally important to note, however,

that the accuracy of the predictions derived from such

models depend critically on the accuracy of the thermo-

dynamic models underlying the computational formal-

ism. In the context of CALPHAD modeling of phase

diagrams, one difficulty arises in the construction ofaccurate free energy functions due to the fact that inde-

pendent entropic and enthalpic contributions cannot be

determined uniquely from common-tangent construc-

tions alone. The non-uniqueness of the thermodynamic

functions derived by fitting only to phase-boundary

information alone is a cause for concern in applying

CALPHAD free energies in the calculation of thermo-

dynamic driving forces for phase transformations, aswell as the prediction of multicomponent phase equilib-

ria. Both applications rely upon extrapolations of the

CALPHAD thermodynamic functions into regions of

the phase diagram away from where they have been

fit, and thus depend upon accurate predictions of the

composition and temperature dependencies of the calcu-

lated free energies. In CALPHAD modeling, the phase

boundaries are determined by free energy functions,not just DH 298.15

f . Therefore, if the same phase bound-

aries are represented by widely differing DH 298.15f values,

then entropic parameters will also vary widely. In such

situations the creation of multicomponent thermody-

namic and kinetic databases by combining correspond-

ing databases for binary systems originating from

different sources can lead to substantially different pre-

dictions for multicomponent systems. The need foraccurate thermodynamic data, in addition to phase-

boundary information, to derive unique and predictive

free energy model parameters is clear in such applica-

tions. In the absence of adequate experimental measure-

ments, ab initio methods present a viable means for

substantially augmenting the databases required in the

generation of accurate thermodynamic model

parameters.

5. Conclusions

A systematic and comprehensive study of phase sta-

bility of intermetallic phases in Al–TM (TM = Ti, Zr,

Hf) systems has been carried out using electronic den-

sity-functional theory. The total energies of 69 interme-tallic compounds have been calculated using the US-PP

approach and the GGA. By combining the crystallo-

graphic data of intermetallic phases in three binary sys-

tems, we define a superset of 18 crystal structures. Then,

the intermetallic compounds in three binary systems are

classified as stable, metastable and virtual types. The fol-

lowing conclusions are drawn:

(i) The zero-temperature cohesive properties of Al,Ti, Zr and Hf are calculated for the bcc, hcp and fcc

structures. We find that, in general, the lattice parame-

ters agree within ±1% and B0 agree within ±2% when

the calculated values are compared with the correspond-

ing experimental values at low temperatures (either mea-

sured or extrapolated). The calculated lattice stabilities

of these elements in three structures are also provided.

(ii) The zero-temperature formation energies of Al3Ti(tI 8), Al2Ti (tI 24), AlTi2 (oF40) and AlTi3 (hP8) agree

reasonably well with measured heats of formation, par-

ticularly considering the scatter of calorimetric measure-

ments. The predicted ground-state structures are

consistent with those known to be stable at low temper-

atures. Al3Ti4 (hP7) is predicted to be a stable ground-

state structure, although it has not been observed

experimentally to date. The calculated lattice parametersof all intermetallic phases agree within 1% of the exper-

imental values.

(iii) The zero-temperature formation energies of

Al3Zr (tI16), Al2Zr (hP12) and Al3Zr2 (oF40) agree well

within the uncertainty associated with calorimetric mea-

surements. For AlZr (oC8), Al4Zr5 (hP18), Al2Zr3(tP20) and Al3Zr5 (tI30), a significant discrepancy (up

to 10 kJ/mol) is noted between ab initioDE and heat offormation obtained by second- and third-law methods.

Consistent with an earlier assertion that Al3Zr5 (hP16)

and AlZr2 (tI12) may be stabilized by impurity effects,

and are thus metastable in pure alloys, we find that their

DE values lie about 7 kJ/mol above the ground-state

convex hull. The calculated lattice parameters of all

intermetallic phases agree within 1% of the experimental

values.(iv) The zero-temperature formation energies of

Al3Hf (tI16), Al2Hf (hP12) and Al3Hf2 (oF40) agree well

within the uncertainty associated with calorimetric mea-

surements. Consistent with an earlier assertion that

Al3Hf5 (hP16) and AlHf2 (tI12) may be stabilized by

impurity effects, and are thus metastable in pure alloys,

we also find them, like Al–Zr system, to lie well above

the calculated convex hull. AlHf3 (cP4) is predicted tobe a ground-state structure, even though it has not been

experimentally observed. The calculated lattice parame-

ters of all intermetallic phases agree within 1% of the

experimental values.

(v) Considering the stable phases in three binary sys-

tems we find that the formation energies predicted by

various ab initio techniques, such as FP-LMTO,

FLAPW, FLASTO and US-PP, agree within ±3 kJ/mol (a noteable exception was found between the most

recent and older values for Al–Zr). On the other hand,

the CALPHAD model parameters, representing alloy

energetics, vary significantly from one assessment to an-

other where the maximum spread is noted to be 12 kJ/

mol in Al–Ti, 20 kJ/mol in Al–Zr and 5 kJ/mol in Al–

Hf system. The maximum difference noted between

our ab initio DE and the reported CALPHAD modelparameters is 12 kJ/mol in Al–Ti system, 21 kJ/mol in

Al–Zr system and 17 kJ/mol in Al–Hf system. These


discrepancies underscore the point that CALPHAD

modeling generally requires accurate thermodynamic

data over the entire composition range to derive unique

parameter sets; in the absence of such complete dat-

abases the parameters will depend on judgments made

by the assessor in optimizing the thermodynamic modelparameters. In such cases, the present results demon-

strate how first-principles calculations can be employed

as a viable framework for greatly augmenting available

thermodynamic data for intermetallic phases in the con-

struction of accurate thermodynamic databases.

Acknowledgments

This research was supported by the US Department

of Energy, Office of Basic Energy Sciences, under Con-

tract Nos. DE-FG02-02ER45997 (GG) and DE-FG02-

01ER45910 (MA). Supercomputing resources were

provided by the National Partnership for Advanced

Computational Infrastructure at the University of Mich-

igan. One of us (GG) would like to thank Prof. J.C.Schuster of University of Vienna for clarifying crystal

structure of some Al–Ti intermetallics.

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