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Acta Materialia 53 (2005) 3225–3252
www.actamat-journals.com
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.
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]
3228 G. Ghosh, M. Asta / Acta Materialia 53 (2005) 3225–3252
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]
G. Ghosh, M. Asta / Acta Materialia 53 (2005) 3225–3252 3229
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]
3230 G. Ghosh, M. Asta / Acta Materialia 53 (2005) 3225–3252
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).
G. Ghosh, M. Asta / Acta Materialia 53 (2005) 3225–3252 3231
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)].
3232 G. Ghosh, M. Asta / Acta Materialia 53 (2005) 3225–3252
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.
G. Ghosh, M. Asta / Acta Materialia 53 (2005) 3225–3252 3233
3234 G. Ghosh, M. Asta / Acta Materialia 53 (2005) 3225–3252
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)
G. Ghosh, M. Asta / Acta Materialia 53 (2005) 3225–3252 3235
Table 8 (continued)
Phase Space group (#) Prototype Unit cell-internal parameters (Wyckoff positions)
Ab initio (x,y,z) Experiment (x,y,z) [Ref.]
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
3236 G. Ghosh, M. Asta / Acta Materialia 53 (2005) 3225–3252
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
G. Ghosh, M. Asta / Acta Materialia 53 (2005) 3225–3252 3237
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).
3238 G. Ghosh, M. Asta / Acta Materialia 53 (2005) 3225–3252
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).
G. Ghosh, M. Asta / Acta Materialia 53 (2005) 3225–3252 3239
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].
3240 G. Ghosh, M. Asta / Acta Materialia 53 (2005) 3225–3252
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
Phase Space group (#) Prototype DE (kJ/mol)
Ab initio [Ref.] Experiment [Ref.] CALPHAD [Ref.]
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]
(continued on next page)
G. Ghosh, M. Asta / Acta Materialia 53 (2005) 3225–3252 3241
Table 9 (continued)
Phase Space group (#) Prototype DE (kJ/mol)
Ab initio [Ref.] Experiment [Ref.] CALPHAD [Ref.]
�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.
3242 G. Ghosh, M. Asta / Acta Materialia 53 (2005) 3225–3252
[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
ambient temperature)
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.
G. Ghosh, M. Asta / Acta Materialia 53 (2005) 3225–3252 3243
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
(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
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
(continued on next page)
3244 G. Ghosh, M. Asta / Acta Materialia 53 (2005) 3225–3252
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
Phase Space group (#) Prototype DE (kJ/mol)
Ab initio [Ref.] Experiment [Ref.] CALPHAD [Ref.]
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)
Phase Space group (#) Prototype Unit cell-internal parameters (Wyckoff positions)
Ab initio (x,y,z) Experiment (x,y,z) [Ref.]
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
G. Ghosh, M. Asta / Acta Materialia 53 (2005) 3225–3252 3245
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
ambient temperature)
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.
3246 G. Ghosh, M. Asta / Acta Materialia 53 (2005) 3225–3252
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
(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
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
(continued on next page)
G. Ghosh, M. Asta / Acta Materialia 53 (2005) 3225–3252 3247
Table 14 (continued)
Phase Space group (#) Prototype Unit cell-internal parameters (Wyckoff positions)
Ab initio (x,y,z) Experiment (x,y,z) [Ref.]
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
3248 G. Ghosh, M. Asta / Acta Materialia 53 (2005) 3225–3252
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
G. Ghosh, M. Asta / Acta Materialia 53 (2005) 3225–3252 3249
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
3250 G. Ghosh, M. Asta / Acta Materialia 53 (2005) 3225–3252
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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