PHYSICAL REVIEW B
VOLUME 53, NUMBER 3
15 JANUARY 1996-I
Ground-state properties and relative stability between the L1 2 and DO a phases
of Ni3Al by Nb substitution
P. Ravindran
Department of Physics, Anna University, Madras 600 025, India
G. Subramoniam
Institute of Mathematical Sciences, Tharamani, Madras 600 113, India
R. Asokamani
Department of Physics, Anna University, Madras 600 025, India
~Received 23 January 1995!
This paper reports the effect of substitution of Nb on the physical properties of Ni3Al using the tight-binding
linear-muffin tin-orbital method. The systematic total-energy studies made on Ni3Al and Ni3Nb in both L1 2
~Cu3Au! and DO a ~b-Cu3Ti! structures successfully explained the structural stability of these two compounds.
In order to understand the relative stability between L1 2 and DO a structures by the gradual substitution of Nb
in Ni3Al, we have performed total-energy calculations for Ni3Al0.5Nb0.5, Ni3Al0.75Nb0.25, and Ni3Al0.25Nb0.75
in both L1 2 and DO a structures at different reduced and extended volumes. The first-principles calculations
clearly show that the cubic symmetry, which favors greater ductility, is retained by the alloy only up to the
composition Ni3Al0.41Nb0.59. The critical e/a ratio corresponding to the cubic→orthorhombic structural transition obtained from our phase stability study ~8.545! is found to be in very good agreement with Liu’s
experimental observation. Further, the heat of formation, cohesive energy, density of states at the Fermi level
N(E F ), equilibrium lattice constants, and bulk modulus, including its pressure derivative as a function of Nb
substitution in Ni3Al, are also calculated and compared with the available experimental values.
I. INTRODUCTION
Ordered intermetallic compounds possess many interesting high-temperature properties which make them useful for
structural applications at high temperatures. However, a major barrier to the widespread use of ordered intermetallics is
that most of them lack room-temperature ductility and
toughness.1,2 The L1 2-type intermetallic compound Ni3Al,
which is the major strengthening phase of Ni-based superalloys, has excellent high-temperature characteristics. Further,
the density of Ni3Al is considerably lower than that of the
other Ni-based superalloys. The single-crystal Ni3Al is a
very good ductile material, but the polycrystalline Ni3Al is
extremely brittle, not only at room temperature but at higher
temperatures also.
The two major factors which have been found to be responsible for brittleness in ordered intermetallics are ~i! an
insufficient number of independent slip systems ~transgranular fracture! and ~ii! grain boundary weakness ~intergranular
fracture!. Ni3Al, which is in the cubic structure, possesses
enough independent slip systems to satisfy von Mises’s criterion for better ductility.3 So, the second factor is mainly
responsible for poor ductility in polycrystalline Ni3Al.
According to Izumi and Takasugi, the Al atom is covalently bonded with the Ni atom in Ni3Al via pd s interaction, and on account of the electronegativity difference, the
Al atom draws charge from the Ni atom.4 As a result of this,
less charge is available to participate in the Ni-Ni bonds in
the grain boundaries and this leads to weakening of the grain
boundaries. On the other hand, if the Nb-like transition-metal
atom is added to the Ni-Al matrix, the covalent character of
0163-1829/96/53~3!/1129~9!/$06.00
53
the Ni-Al bond will be weakened, and hence the suppression
of grain boundary embrittlement can be expected. Furthermore, it has been observed earlier that the materials, which
have isotropic charge-density distribution, must be more ductile than those which have anisotropic charge-density
distribution.5 As mentioned earlier, because of the weakening
of pd s covalent interaction by Nb alloying in Ni3Al, we can
expect more homogeneous charge-density distribution in
Ni3~Al,Nb!. Considerable effort has been made experimentally to increase the grain boundary cohesion and simultaneously maintain the high-symmetry cubic structure upon
ternary addition.
Earlier studies on L1 2 intermetallics show that by ternary
alloying of electron donor elements in Ni3Al, one can improve its ductility.6 The recent experimental studies also indicate that the substitutional alloying of Pd or Cu in Ni3Al
results in an increased ductility.7 At this juncture, it is worth
recalling the fact that the small addition of interstitial boron
to Ni3Al has drastically improved its ductility.8 It has been
suggested that the boron segregates mainly at the grain
boundaries and it reinforces the atomic bond at grain boundary regions.9
The ternary alloying of transition metals sometimes
changes Ni3Al from the high-symmetry cubic structure to
low-symmetry structures. Under such condition wherein the
transgranular fracture mechanism predominates, the system
will be more brittle. In the present case, lack of independent
slip systems in Ni3Nb is a major cause for its brittleness
since it crystallizes in the low-symmetry orthorhombic
(DO a ) structure. Therefore, one must investigate in detail
the relative structural stability of different crystal structures
1129
© 1996 The American Physical Society
1130
P. RAVINDRAN, G. SUBRAMONIAM, AND R. ASOKAMANI
as a function of alloying to improve the ductility of intermetallic compounds. A few theoretical attempts have been made
so far to study the stability of transition-metal aluminides
by ternary alloying by means of electronic-structure
calculations.10 So far as this paper is concerned, the main
objective is to study the relative stability between cubic and
orthorhombic phases of Ni3Al as a function of Nb substitution and to determine the critical concentration up to which
the cubic phase is retained. It is worth recalling that the
effective cluster interactions obtained from the total-energy
band-structure studies have recently been used in conjunction with the cluster-variation method to construct the
composition-temperature phase diagram of binary alloys.11
So, the total-energy study as a function of alloying becomes
considerably important to construct the phase diagram theoretically.
From the detailed studies on 84 transition-metal intermetallics, Sinha12 showed that the close-packed layers change
from ‘‘triangular’’ ~related to cubic L1 2 structure! to ‘‘rectangular’’ ~related to orthorhombic DO a structure! at the valence electron per atom (e/a) as 8.65. Further, Liu found a
strong correlation between stacking character and e/a ratio
for the quasiternary system of Ni3V-Co3V-Fe3V such that13
as the e/a ratio increases, the stacking character changes
from purely cubic through different ordered mixtures of hexagonal and cubic layers to purely hexagonal at e/a58.54.
When e/a exceeds 8.54, there is a change from cubic L1 2 to
orthorhombic DO a type. The structural stability study as a
function of Nb substitution in Ni3Al is equivalent to increasing its e/a ratio. So, one of our present motivations is to
understand the relative stability between L1 2 and DO a structures as a function of e/a ratio using our first-principles calculations.
This paper is divided into seven sections. Section II gives
details of the construction of supercells and a brief outline of
the computational details of the tight-binding linear-muffintin-orbital scheme. The first band-structure and density of
states ~DOS! for Ni3Nb and the phase stability studies as a
function of Nb alloying in Ni3Al using the band filling of
bonding states analysis are reported in Sec. III. The results of
the total-energy calculations obtained for cubic and orthorhombic phases of Ni3~Al,Nb! alloys are presented in Sec.
IV. Section V deals with the theoretically calculated cohesive
energy and heat of formation and the comparison of these
quantities with the experimentally available values. In Sec.
VI, the equation of states and the bulk modulus and its pressure derivative determined using the universal equation of
state ~UEOS! analysis for the entire composition range are
reported. The important conclusions arrived from the above
studies are given in the last section.
II. CRYSTAL STRUCTURAL ASPECTS AND METHOD
OF CALCULATION
The crystal structure of single-crystal Ni3Nb has been investigated recently from x-ray-diffraction studies and it was
found that Ni3Nb is stable in the orthorhombic DO a
structure.14 For the band-structure calculations of Ni3Nb in
the orthorhombic structure, we have kept the experimentally
observed b/a and c/a ratios constant and calculated the total
energies for different reduced and extended volumes. From
53
the earlier studies, it has been found that Ni3Al is in the cubic
L1 2 structure15 and the magnetic effect arising from Ni is not
significant to decide the phase stability of Ni3Al ~Ref. 16!;
hence, we have not taken this effect into account in our calculations.
In order to study the structural stability as a function of
Nb alloying, two different supercells for L1 2 structure and
two more for DO a structure are constructed. Takasugi,
Izumi, and Masahashi6 tabulated that the maximum substitutional solubility of Nb in Ni3Al is 7 at. %. But for our model
calculations, we have assumed that Nb is completely soluble
in Ni3Al and constructed our supercells. Even though the
L1 2 structure has a cubic symmetry, the constructed supercells of the L1 2 phase possess tetragonal structure. Hence,
we have made the supercell total-energy calculation of the
L1 2 phase with the c/a52 and c/a54. The preferential site
occupation of substitutional ternary elements in Ni3Al has
been systematically studied previously,17 and from these
studies it is clear that Nb will occupy an Al site in Ni3Al.
Because of the above fact, we have substituted Nb in the
place of Al in Ni3Al for our total-energy supercell calculations. The supercells constructed with 25, 50, and 75 at. % of
Nb substitution in Ni3Al contain, respectively, 16, 8, and 16
atoms/cell for both L1 2 and DO a structures and their chemical formulas are Ni12Al3Nb, Ni6NbAl, and Ni12AlNb3 , respectively. Due to the complexity of our calculations by considering the supercells, we have maintained the same type
for the atoms as in their basic structures even though the
chemical environment is changed by the Nb substitution.
From the Ni-Al-Nb phase diagram studies, it has been found
that there are three stable ordered ternary phases, namely,
Ni2NbAl, Nb~Alx Ni12x !2 , and Nb10Ni9Al3 .18 Moreover, the
Ni3Al-Ni3Nb phase diagram studies show that the L1 2 phase
is retained up to 9.8 at. % of Nb addition in Ni3Al and from
9.8 at. % up to '20 at. % of Nb addition, a mixture of both
L1 2 and DO a phases is present;19 beyond 20 at. % of Nb
addition, a single DO a phase exists. It is more complicated
to consider the mixed phases in the present type of calculation, and hence we treat both L1 2 and DO a structures independently in our total-energy and electronic-structure calculations.
The band-structure and the total-energy studies are made
within the atomic sphere approximation by means of the
tight-binding linear-muffin-tin-orbital method ~TBLMTO!,20
which is the exact transformation of Andersen’s linearmuffin-tin orbitals21 to localized short-ranged or tightbinding orbitals. The potential is calculated within the
density-functional prescription under the local-density approximation ~LDA! using the parametrization scheme of von
Barth and Hedin.22 The most important relativistic corrections, namely, Darwin’s correction and the mass-velocity
terms, are included, while the spin-orbit coupling term is
ignored. The number of atoms involved in our present calculation is large and hence a fast computation is needed. The
TB ~or screened! representation of the LMTO method makes
the computation fast for mainly three reasons: ~i! the
MTO’s are linear in energy and hence, unlike the augmented
plane-wave or Korringa-Kohn-Rostoker methods, we can get
the eigenvalues within single diagonalization. ~ii! One requires a solution to an eigenvalue equation of size only 939
~for s,p,d electron elements! per atom at each point in re-
53
GROUND-STATE PROPERTIES AND RELATIVE STABILITY . . .
1131
FIG. 1. Band structure of Ni3Nb in the DO a structure.
ciprocal space. ~iii! The screened structure constant for
each atom needs only up to second-nearest-neighbor atoms.
In our calculations, the s, p, and d partial waves have been
used ~i.e., maximum angular momentum l max52!. Apart
from this, the combined correction terms are also included,
which account for the nonspherical shape of the atomic cells
and the truncation of higher partial waves ~l.2! inside the
sphere so as to minimize the errors in the LMTO method. To
exclude any additional freedom in the choice of computational parameters, the same Wigner-Seitz ~WS! radius is chosen for all atoms and the calculated overlaps between the
various atomic spheres in this WS radius are within the allowed range of the atomic sphere approximation. The tetrahedron method for the Brillouin-zone ~i.e., k space! integrations has been used with its latest version, which avoids
misweighing and corrects errors due to the linear approximation of the bands inside each tetrahedron.23 For all our totalenergy calculations, we have chosen 64 k points in the irreducible wedge of the first Brillouin zone ~IBZ! of
orthorhombic structures, 65 k points in the IBZ of tetragonal
structures, and 84 k points in the IBZ of cubic structures. The
total-energy calculation has been made for the constituents in
their respective stable structures to evaluate the heat of formation.
III. BAND-STRUCTURE AND DENSITY OF STATES
STUDIES ON Ni3„Al,Nb…
The band structure of Ni3Al has been extensively studied
earlier.24,25 But the present work is an examination of Ni3Nb
with respect to band-structure and density of states studies.
Among the ternary Ni-based intermetallic compounds, superconductivity has been observed only in Ni2NbA1, Ni2NbGa,
and Ni2NbSn.26 Hence, the electronic-structure studies on
Ni3Nb are considerably significant. The band structure of
Ni3Nb in the DO a structure shown in Fig. 1 depicts the
cluster of bands near 20.15 Ry and this is reflected in the
corresponding DOS curve in Fig. 2. It is interesting to note
that despite the presence of a cluster of bands in the band
structure of Ni3Nb, only two degenerate bands cross the
FIG. 2. The total density of states of Ni3Al as a function of Nb
substitution in the orthorhombic DO a structure ~obtained from the
theoretically estimated equilibrium volumes!.
Fermi level in most of the symmetry directions. As a consequence of this, the N(E F ) of Ni3Nb is very small.
The DOS curves of Ni3Al in the L1 2 and Ni3Nb in the
DO a structures with their respective experimental equilibrium lattice parameters are shown in Figs. 2 and 3. The overall features of the density of states curve of Ni3Al in the L1 2
structure are found to be in good agreement with the earlier
studies.24,25 Our N(E F ) value @84.06 states/~Ry F.u.!# calculated for Ni3Al in the L1 2 structure is found to conform with
that obtained from the earlier LMTO study @82.81
states/~Ry F.u.!#.24
There is a strong correlation observed between stability
and the position of the Fermi level in the DOS curve in
binary alloys; that is, if the E F falls on the pseudogap which
separates bonding states from the antibonding/nonbonding
states in a particular structure, the system will be more
stable.27 In other words, the stable structure always has low
N(E F ). As a result of the above fact, the midseries of transition elements stabilizes in the bcc structure and the early
and late transition metals prefer to stabilize in the fcc structure. We have observed recently that the aforesaid correlation
is not always obeyed if the materials have metastability or
martensitic transformation.28 From our more recent studies
on L1 2 superconductors, we have observed that the E F falls
on the peak of the DOS curve, resulting in the metastability
of these systems; thereby we have correlated the metastability with their superconducting transition temperatures.29 It is
worth noting that the glass-forming ability of intermetallics
is larger if the E F falls on a peak of the DOS curve.30
The total DOS of Ni3Al as a function of Nb substitution
for both L1 2 and DO a structures are shown in Figs. 2 and 3,
respectively. The e/a ratio gradually increases with Nb sub-
1132
P. RAVINDRAN, G. SUBRAMONIAM, AND R. ASOKAMANI
FIG. 3. The total density of states of Ni3Al as a function of Nb
substitution in the L1 2 structure ~obtained from the theoretically
estimated equilibrium volumes!.
stitution in Ni3Al and as a consequence, the E F gets shifted
to higher energy states as shown in Figs. 2 and 3 ~dashed
lines! for higher Nb concentrations. On account of the hybridization effect as well as the variation in the topology of
the DOS curve, there is no systematic shifting of E F as expected for both L1 2 and DO a structures as a function of Nb
alloying.
For Ni3Al0.5Nb0.5 in both L1 2 and DO a structures, the
Fermi level falls on the pseudogap in the DOS curve ~Figs. 2
and 3!. This indicates that all the bonding states are filled
with electrons and all the antibonding states are left empty;
therefore, the strongest bonding effect occurs. The N(E F )
value of Ni3Al0.5Nb0.5 in the L1 2 structure is smaller than
that in the DO a structure ~Table II!. As mentioned earlier,
since the stable structure belongs to low N(E F ), we can expect from the above observation a high stability in the L1 2
structure compared to the DO a structure and this is consistent with our total-energy studies.
From the detailed x-ray photoemission spectroscopy studies on Ni-based intermetallic compounds, Fuggle et al. concluded that the Ni d bands become narrower and the d density of states at E F will decrease due to the filled Ni d states,
on account of the presence of an electropositive element in
the alloy.31 Our DOS curves of Ni3Al0.5Nb0.5 in both L1 2 and
DO a structures support the above experimental observation.
It is worth comparing our DOS curve of Ni3Al0.5Nb0.5 in the
L1 2 structure with that of the isoelectronic compound
Ni3Al0.5V0.5 for which the DOS has been obtained by the
all-electron total-energy LMTO calculation.32 In both compounds, the E F falls on the pseudogap but the pseudogap is
deeper in Ni3Al0.5Nb0.5 than in Ni3Al0.5V0.5 and as a result,
the N(E F ) of the former becomes very small ~Table II!. It
53
has been shown earlier that the Ni d-band filling in Ni-based
alloys is due to the hybridization effect which is mainly dependent on the nature of the alloying element.31 Hence, the
low N(E F ) in Ni3Al0.5Nb0.5 compared to that of Ni3Al0.5V0.5
is due to the stronger orbital overlap ~or hybridization! between Ni and Nb than that between Ni and V as the atomic
radius of Nb is larger than V.
It has been strongly believed that the ductility of TiAl can
be improved by enhancing its d-d interaction through the
transition-metal substitution.33 Even though the d-d interaction in Ni3Al is enhanced by substitution of Nb or V, it has
been experimentally observed that its ductility is not improved due to intergranular fracture.34 Our theoretical studies
show that the cubic phase which favors good ductility is
retained even with 50 at. % of Nb substitution in Ni3Al. The
above observation rules out the possibility of fracture arising
from the slip incompatibility at the grain boundaries. So, the
possible reason for brittleness even after substitution of Nb
or V in Ni3Al is the strong covalent hybridization ~directional bonding! effect.
In Ni3Al and Ni3Al0.75Nb0.25, the E F falls within the bonding states in both L1 2 and DO a structures. The value of
N(E F ) of both compounds in DO a structures is lower than
that of the compounds in L1 2 structures. Hence, we can expect DO a structures to be more stable than the L1 2 structures. But the experimental studies on Ni3Al and our totalenergy studies on the above two compounds show the
opposite. This may be due to the E F of DO a structures
~Table II!, which is in a higher energy state than in L1 2
structures for both Ni3Al and Ni3Al0.75Nb0.25. It is interesting
to note from the DOS curve of both Ni3Nb and
Ni3Al0.25Nb0.75 in L1 2 structures that the E F falls on the antibonding states and hence the system will be unstable in that
structure. So, we infer from our DOS studies of Ni3Nb and
Ni3Al0.25Nb0.75 in both L1 2 and DO a structures that both
compounds will stabilize in the orthorhombic structure. This
observation is consistent with the experimental results of
Ni3Nb ~Ref. 14! and our total-energy studies on both Ni3Nb
and Ni3Al0.25Nb0.75.
IV. TOTAL-ENERGY CALCULATIONS
The total energies for Ni3Al, Ni3Al0.75Nb0.25,
Ni3Al0.5Nb0.5, Ni3Al0.25Nb0.75, and Ni3Nb in the L1 2 and
DO a structures which are calculated for different reduced
and extended volumes are shown in Figs. 4~a! to 4~e!. From
the above figures, the cohesive energy, the heat of formation,
and the equilibrium lattice constants of the above-mentioned
systems in both L1 2 and DO a structures are calculated. From
Table I, we understand that the theoretically obtained equilibrium lattice constants are underestimated compared to the
corresponding experimental values. The theoretical lattice
constants are always underestimated28 and this is partly ascribed to the local-density approximation used in our calculations. To minimize these deviations, it is argued that zeropoint vibrations have to be included in the calculation.30
From Table I, we have found that the lattice parameter of
Ni3Al as a function of Nb alloying in both structures gradually increases when we go from Ni3Al to Ni3Nb.
Our total-energy curves of Ni3Al in the L1 2 and DO a
structures @Fig. 4~a!# confirm the experimental observation
53
GROUND-STATE PROPERTIES AND RELATIVE STABILITY . . .
1133
FIG. 4. Total energy as a function of volume for ~a! Ni3Al, ~b! Ni3Al0.75Nb0.25, ~c! Ni3Al0.5Nb0.5, ~d! Ni3Al0.25Nb0.75, and ~c! Ni3Nb in
the cubic ~L1 2! and orthorhombic (DO a ) structures. The equilibrium volume ~V 0! used in each composition is given inside the figures.
that Ni3Al is indeed in the L1 2 structure although the energy
difference between the L1 2 and DO a structures is very small
~0.002 337 Ry/F.u.!. Figure 4~b! clearly shows that the 25
at. % of Nb substitution in the place of Al in Ni3Al does not
change its crystal structure and even after 50 at. % of Nb
substitution, the same structure is retained @Fig. 4~c!#. But, for
the ordered intermetallic of the composition Ni3Al0.25Nb0.75,
the orthorhombic phase is favored as compared to the cubic
phase @Fig. 4~d!# and subsequently the total-energy studies
made for Ni3Nb @Fig. 4~e!# show a greater stability for the
orthorhombic DO a phase which is in conformity with the
experimental observation.14 These total-energy calculations
performed for various Nb concentrations lead to the important conclusions that the cubic phase, which has greater ductility and hence may be of potential use for structural applications, will be possessed by Ni3~Al,Nb! up to the
composition of Ni3Al0.5Nb0.5. Thereafter, it goes over to the
less symmetric orthorhombic phase for higher Nb concentration. Xu, Oquchi, and Freeman have done phase stability
studies on Ni3Al0.5V0.5 and extended their arguments predicting that Ni3Al0.5Nb0.5 will also be stabilized in the L1 2 structure. Our detailed calculations confirm their prediction.32
P. RAVINDRAN, G. SUBRAMONIAM, AND R. ASOKAMANI
1134
53
TABLE I. The experimental and theoretical equilibrium volume ~V 0 in Å3! and lattice constants ~a, b, c in Å! of Ni3~Al,Nb!. The
experimental values are taken from Refs. 14 and 15.
a
Alloy
Type
b
Theory
c
a
b
Experiment
c
4.234
4.505
Ni3Al
L1 2
DO a
3.505
4.908
4.057
4.358
L1 2
DO a
3.530
4.955
4.096
4.400
L1 2
DO a
3.572
5.010
4.141
4.448
L1 2
DO a
3.625
5.062
4.184
4.494
L1 2
DO a
3.640
5.073
4.194
4.505
3.566
Ni3Al0.75Nb0.25
Ni3Al0.5Nb0.5
Ni3Al0.25Nb0.75
Ni3Nb
In order to understand the variation in the unit cell volume
of Ni3Al by ternary alloying of Nb, we have drawn the unit
cell volume as a function of Nb substitution in Fig. 5 for both
L1 2 and DO a structures. The continuous line in Fig. 5 represents the volume variation of stable phases in Ni3~Al,Nb!.
The crystal structures of Ni3Al and Ni3Nb are different and
hence we have used Zen’s law instead of Vegard’s law for
understanding the volume variation with Nb substitution in
Ni3Al.35 Zen’s law gives the linear variation of volume with
respect to substitution and is represented by the dashed line
in Fig. 5. Our calculation shows that the unit cell volume of
Ni3Al increases with the Nb substitution and this trend is in
agreement with Zen’s law. But the theoretical volume is always lower than the experimental volume and the main reason for this deviation is due to noninclusion of thermal effects as well as the use of LDA in our calculations.
FIG. 5. The variation of unit cell volume of Ni3Al as a function
of Nb substitution. The dashed line represent Zen’s law, the continuous line indicates the theoretical volumes of stable phases, and
the circle and square represent the equilibrium volume of L1 2 and
DO a phases, respectively.
5.122
V. HEAT OF FORMATION AND COHESIVE
ENERGY STUDIES
The heat of formation for Ni3~Al,Nb! compounds is estimated from the energy difference between the compound and
the weighted sum of their constituents, i.e.,
DH5E Nix Aly Nbz 2 ~ xE Ni1yE Al1zE Nb! ,
~1!
where x, y, and z are the chemical composition of Ni, Al,
and Nb, respectively.
In order to understand the relative stability between L1 2
and DO a structures of Ni3Al as a function of Nb substitution, the heat of formation of Ni3Al as a function of Nb
substitution is plotted for both L1 2 and DO a structures in
Fig. 6. From the figure, it is clearly seen that the L1 2 structure is retained up to 14.75 at. % of Nb substitution in Ni3Al
and the corresponding chemical composition is
Ni3Al0.41Nb0.59. As there is an empirical correlation between
the e/a ratio and the DO a →L1 2 structural transition in in-
FIG. 6. The variation of heat of formation as a function of Nb
substitution in Ni3Al for L1 2 ~continuous line! and DO a ~dashed
line! structures.
53
GROUND-STATE PROPERTIES AND RELATIVE STABILITY . . .
1135
TABLE II. The theoretically calculated cohesive energy ~E coh in eV/F.u.!, heat of formation. ~DH in kcal/mol!, bulk modulus ~B 0 in
Mbar!, and its pressure derivative (B 80 ) for Ni3~Al,Nb!.
Alloy
Type
E coh
2DH
B0
B 80
e/a
N(E F )
EF
Ni3Al
L1 2
DO a
L1 2
DO a
L1 2
DO a
L1 2
DO a
L1 2
DO a
2.3437
2.2783
2.3463
2.3437
2.4589
2.4454
2.4948
2.4466
2.6037
2.6531
50.74
50.00
37.97
37.18
39.23
35.00
16.40
1.26
16.45
31.97
1.462
1.435
1.449
1.440
1.443
1.356
1.419
1.408
1.483
1.508
2.455
2.631
2.853
2.366
2.338
2.620
2.320
2.291
2.248
2.247
8.25
8.25
8.375
8.375
8.5
8.5
8.625
8.625
8.75
8.75
78.35
48.08
53.86
20.00
7.34
26.50
50.31
41.47
213.52
33.27
20.0397
20.0349
20.0327
20.0108
0.0057
20.0225
0.1663
0.0373
0.0323
0.0000
Ni3Al0.75Nb0.25
Ni3Al0.5Nb0.5
Ni3Al0.25Nb0.75
Ni3Nb
termetallic compounds,12,13 we can confirm the relation
through our present quantum-mechanical calculations, and
here we have estimated the critical e/a ratio for the above
composition, which is 8.545. This is in good agreement with
the earlier observations.12,13
Even though our calculations lead us to the conclusion
that the cubic structure can be retained in the Ni3Al system
up to 14.75 at. % of Al replaced by Nb, according to the
experimental phase diagram, the system is in the cubic phase
only up to 9.6 at. % of Nb substitution; thereafter, it goes to
the mixed phase up to 20 at. % of Nb substitution and finally
goes over to the orthorhombic phase. The first-principles
band-structure calculations, which have been used here, will
not be able to explain the presence of mixed phases. However, in principle, by introducing the effect of temperature
via entropy, it is possible to explain the mixed phase regime
even though it involves additional calculation at this stage.
But it should be noted that our calculated value of the heat
of formation for Ni3~Al,Nb! alloys given in Table II
shows that DH~Ni3Al12x Nbx ! is always greater than
(12x)DH~Ni3Al!1x~DH~Ni3Nb!! and this means that
Ni3Al12x Nbx always segregates into two phases, namely,
Ni3Al and Ni3Nb at the ground state.
The calculated value of the heat of formation for Ni3Al
obtained from our total-energy study ~50.74 kcal/mol! is
found to be comparable with the experimental values @36.6
kcal/mol ~Ref. 36! and 37.5 kcal/mol ~Ref. 37!# and is in
good agreement with the earlier theoretical total-energy studies ~44.8 kcal/mol!.16 Similarly, the heat of formation theoretically obtained for Ni3Nb is 31.9 kcal/mol and is in very
good agreement with the experimental value of 30.4
kcal/mol.36 The value of the heat of formation for
Ni3Al0.25Nb0.75 is small for both L1 2 and DO a structures,
indicating that the system is likely to go to the disordered
state more easily, as has been observed by Xu, Lin, and
Freeman in the case of the Fe3Ti system.38 From the detailed
studies on L1 2 intermetallics, it has been found that the intergranular brittleness of the ordered states of the material is
larger than that of the disordered state.6 Our heat of formation study on Ni3~Al,Nb! shows that the ordering energy decreases by Nb substitution in Ni3Al. This is a positive indication of the improvement of ductility of Ni3Al by the Nb
substitution. Compared to Ni3Al, the calculated heat of formation for Ni3Al0.75Nb0.25 and Ni3Al0.5Nb0.5 is lower in the
equilibrium volume of their stable structures and this is one
of the reasons for the existence of the mixed phase in the
Ni3Al-Ni3Nb phase diagram19 around this composition.
In order to understand the intergranular fracture mechanism of structural intermetallics, the estimation of the energy
absorbed by bond stretching and breaking off along grain
boundaries is necessary. The cohesive energy is the indirect
measure of the bond strength of solids. In this connection,
the cohesive energy of Ni3Al as a function of Nb substitution
is calculated and given in Table II. It has been found from
Table II that the cohesive energy gradually increases from
Ni3Al to Ni3Nb as a function of Nb alloying. The trend in
cohesive energy from Ni3Al to Ni3Nb as a function of Nb
substitution shows that the bond strength will gradually increase with the Nb substitution and this is consistent with the
experimental studies in the sense that the solid to liquid
transition takes place at 1390 °C for Ni3Al and at 1430 °C
for Ni3Nb.19 Further, the electrical resistivity studies on
Ni-Al and Ni-Nb systems show that the later system has
more resistivity than Ni-Al.39 This may be due to the present
observation of the increasing trend in cohesive energy in
Ni3Al by the Nb substitution.40
It is well known that the atomic size, electron concentration, and electronegativity are the three major factors which
determine the stability in intermetallics. The equilibrium
volumes/F.u. for both L1 2 and DO a structures are very close
to each other for Ni3~Al,Nb! alloys in Figs. 4~a! to 4~e!. As
discussed by Xu, Lin, and Freeman, the above fact implies
that the atomic-size effect may have negligible influence on
the structural stability of these alloys.38 The electronegativity
difference between Nb and Al is very small ~0.1! and hence
the structural transition from L1 2 to DO a by Nb substitution
in Ni3Al is not due to electronegativity. So, we have inferred
that the changeover from cubic to orthorhombic structure by
Nb substitution in Ni3Al is mainly due to the variation in the
e/a ratio, and this observation is consistent with Sinha’s and
Liu’s conclusions.
VI. EQUATION OF STATES OF Ni3„Al,Nb…
The total energy of Ni3~Al,Nb! alloys has been calculated
for different reduced and extended volumes and is fitted with
the sixth-order polynomial @the dotted and continuous lines
in Figs. 4~a! to 4~e!#. From the first derivative of the poly-
P. RAVINDRAN, G. SUBRAMONIAM, AND R. ASOKAMANI
1136
53
calculated from our total-energy study is 1.435 Mbar and this
is found to be in good agreement with the previous highpressure studies by Ono and Stern, who observed the value
to be 1.503 Mbar, whereas the elastic constant studies by
Frankel et al. gave a higher value of 1.740 Mbar.42,43
It has been both experimentally and theoretically observed that the ternary alloying of vanadium with Ni3Al enhanced its hardness.44,32 Our present studies show that the
hardness of Ni3Nb should be slightly larger than that of
Ni3Al as Nb is isoelectronic with V. However, the intermediate composition of Nb substitution in Ni3Al shows no significant variation in the hardness.
VII. CONCLUSION
FIG. 7. The equation of states of Ni3~Al,Nb! in the stable structures.
nomial, the P-V data of Ni3~Al,Nb! in their stable structures
are generated. The P-V curves for Ni3Al as a function of Nb
substitution are shown in Fig. 7.
Vinet et al. have proposed a universal equation of state
which is valid for all the classes of solids under
compression.41 The UEOS is expressed as
P5
3B 0 ~ 12x !
exp@ h ~ 12x !# ,
x2
~2!
where x denotes (V/V 0 ) 1/3 and B 0 refers to the bulk modulus. If one defines H(x)5x 2 P(x)/[3(12x)], then the
ln[H(x)] versus 12x curve should be nearly linear and obey
the relation
ln@ H ~ x !# 'lnB 0 1 h ~ 12x ! ,
~3!
From our theoretical total-energy studies on Ni3~Al,Nb!,
we have arrived at the following conclusions:
~i! The critical e/a ratio, above which the stabilization of
DO a structure obtained from our calculation ~8.545! is found
to be in very good agreement ~8.54! with Liu’s experimental
observations.
~ii! We have successfully explained the structural stability
of Ni3Al and Ni3Nb from our total-energy studies. The theoretically estimated equilibrium lattice parameters are found
to be in good agreement with the experimental values.
~iii! The replacement of 75 at. % of Al by Nb in Ni3Al
shows that the system prefers DO a structure rather than the
cubic L1 2 structure. The heat of formation of Ni3Al0.25Nb0.75
shows that the alloy will prefer to be in the disordered solid
solution rather than the ordered state.
~iv! The theoretically estimated heat of formation and
bulk modulus are found to be in good agreement with the
available experimental values.
where the slope of the curve ~h! is related to the pressure
derivative of the bulk modulus (B 80 ) by
h 5 32 @ B 80 21 # .
~4!
Using the above equations, the bulk modulus and its pressure derivative of Ni3~Al,Nb! alloys in the L1 2 and DO a
structures are estimated as in the case of Zr3Al ~Ref. 28! and
are given in Table II.
From Fig. 7, it can be noted that the P-V curves do not
vary much for Ni3~Al,Nb! alloys in the entire pressure range
and this trend is reflected in the bulk modulus and its pressure derivatives estimated by the UEOS analysis ~Table II!
for the different compositions. The bulk modulus of Ni3Al
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ACKNOWLEDGMENTS
The author ~P.R.! wishes to thank the Council of Scientific
and Industrial Research ~CSIR!, India and the authors ~P.R.
and R.A.! thank the Department of Atomic Energy ~DAE!,
India for their financial support to carry out this work. The
author ~P.R.! is indebted to Professor S. Sankaralingam for
fruitful discussions and critical reading of the manuscript.
The authors are grateful to Dr. R. V. Ramanujan, Metallurgy
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Raju, PMS, IGCAR, for their useful discussions. The computational facilities provided by the Institute of Mathematical
Sciences, Madras have been acknowledged.
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