First-principles study of the structural, phonon, elastic, and thermodynamic properties of Al3Ta compound under high pressure
We have investigated the phonon, elastic and thermodynamic properties of L1₂ phase Al₃Ta by density functional theory approach combining with quasi-harmonic approximation model. The results of phonon band structure shows that L1₂ phase Al₃Ta possesses dynamical stability in the pressure range from 0...
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irk-123456789-1570482019-06-20T01:27:51Z First-principles study of the structural, phonon, elastic, and thermodynamic properties of Al3Ta compound under high pressure Leini, W. Zhang, T. Wu, Z. Wei, N. We have investigated the phonon, elastic and thermodynamic properties of L1₂ phase Al₃Ta by density functional theory approach combining with quasi-harmonic approximation model. The results of phonon band structure shows that L1₂ phase Al₃Ta possesses dynamical stability in the pressure range from 0 to 80 GPa due to the absence of imaginary frequencies. The pressure dependences of the elastic constants Ci j, bulk modulus B, shear modulus G, Young’s modulus Y, B/G and Poisson’s ratio ν have been analysed. The elastic constants are satisfied with mechanical stability criteria up to the external pressure of 80 GPa. The results of the elastic properties studies show that Al3Ta compound possesses a higher hardness, improved ductility and plasticity under higher pressures. Further, we systematically investigate the thermodynamic properties, such as the Debye temperature Θ, heat capacity Cp, and thermal expansion coefficient α, and provide the relationships between thermal parameters and pressure. Дослiджено фононнi, пружнi та термодинамiчнi властивостi Al₃Ta у L1₂ фазi методом функцiоналу густини у поєднаннi з квазигармонiчною апроксимацiйною моделлю. Результати структури фононної зони показують, що Al₃Ta у L1₂ фазi володiє динамiчною стiйкiстю у дiапазонi тиску вiд 0 до 80 ГПа завдяки вiдсутностi уявних частот. Проаналiзовано тисковi залежностi Ci j, об’ємного модуля пружностi B, модуля зсуву G, модуля Янга Y, B/G та коефiцiєнта Пуассона ν. Пружнi сталi задовольняють критерiй механiчної стiйкостi аж до зовнiшнього тиску 80 ГПа. Результати аналiзу властивостей пружностi показують, що сполука Al₃Ta володiє вищою твердiстю, кращою тягучiстю i пластичнiстю при високих тисках. Крiм того, систематично дослiджено такi термодинамiчнi властивостi, як температура Дебая Θ, питома теплоємнiсть Cp та коефiцiєнт теплового розширення α, а також встановлено залежнiсть мiж тепловими параметрами i тиском. 2018 Article First-principles study of the structural, phonon, elastic, and thermodynamic properties of Al3Ta compound under high pressure / W. Leini, T. Zhang, Z. Wu, N. Wei // Condensed Matter Physics. — 2018. — Т. 21, № 1. — С. 13601: 1–10. — Бібліогр.: 35 назв. — англ. 1607-324X PACS: 61.82.Bg, 62.20.dc, 71.20.Be, 71.15.Mb DOI:10.5488/CMP.21.13601 arXiv:1803.11412 http://dspace.nbuv.gov.ua/handle/123456789/157048 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| description |
We have investigated the phonon, elastic and thermodynamic properties of L1₂ phase Al₃Ta by density functional theory approach combining with quasi-harmonic approximation model. The results of phonon band structure shows that L1₂ phase Al₃Ta possesses dynamical stability in the pressure range from 0 to 80 GPa due to
the absence of imaginary frequencies. The pressure dependences of the elastic constants Ci j, bulk modulus B,
shear modulus G, Young’s modulus Y, B/G and Poisson’s ratio ν have been analysed. The elastic constants
are satisfied with mechanical stability criteria up to the external pressure of 80 GPa. The results of the elastic
properties studies show that Al3Ta compound possesses a higher hardness, improved ductility and plasticity under higher pressures. Further, we systematically investigate the thermodynamic properties, such as the Debye
temperature Θ, heat capacity Cp, and thermal expansion coefficient α, and provide the relationships between
thermal parameters and pressure. |
| format |
Article |
| author |
Leini, W. Zhang, T. Wu, Z. Wei, N. |
| spellingShingle |
Leini, W. Zhang, T. Wu, Z. Wei, N. First-principles study of the structural, phonon, elastic, and thermodynamic properties of Al3Ta compound under high pressure Condensed Matter Physics |
| author_facet |
Leini, W. Zhang, T. Wu, Z. Wei, N. |
| author_sort |
Leini, W. |
| title |
First-principles study of the structural, phonon, elastic, and thermodynamic properties of Al3Ta compound under high pressure |
| title_short |
First-principles study of the structural, phonon, elastic, and thermodynamic properties of Al3Ta compound under high pressure |
| title_full |
First-principles study of the structural, phonon, elastic, and thermodynamic properties of Al3Ta compound under high pressure |
| title_fullStr |
First-principles study of the structural, phonon, elastic, and thermodynamic properties of Al3Ta compound under high pressure |
| title_full_unstemmed |
First-principles study of the structural, phonon, elastic, and thermodynamic properties of Al3Ta compound under high pressure |
| title_sort |
first-principles study of the structural, phonon, elastic, and thermodynamic properties of al3ta compound under high pressure |
| publisher |
Інститут фізики конденсованих систем НАН України |
| publishDate |
2018 |
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http://dspace.nbuv.gov.ua/handle/123456789/157048 |
| citation_txt |
First-principles study of the structural, phonon, elastic, and thermodynamic properties of Al3Ta compound under high pressure / W. Leini, T. Zhang, Z. Wu, N. Wei // Condensed Matter Physics. — 2018. — Т. 21, № 1. — С. 13601: 1–10. — Бібліогр.: 35 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT leiniw firstprinciplesstudyofthestructuralphononelasticandthermodynamicpropertiesofal3tacompoundunderhighpressure AT zhangt firstprinciplesstudyofthestructuralphononelasticandthermodynamicpropertiesofal3tacompoundunderhighpressure AT wuz firstprinciplesstudyofthestructuralphononelasticandthermodynamicpropertiesofal3tacompoundunderhighpressure AT wein firstprinciplesstudyofthestructuralphononelasticandthermodynamicpropertiesofal3tacompoundunderhighpressure |
| first_indexed |
2025-07-14T09:23:16Z |
| last_indexed |
2025-07-14T09:23:16Z |
| _version_ |
1837613708037062656 |
| fulltext |
Condensed Matter Physics, 2018, Vol. 21, No 1, 13601: 1–10
DOI: 10.5488/CMP.21.13601
http://www.icmp.lviv.ua/journal
First-principles study of the structural, phonon,
elastic, and thermodynamic properties of Al3Ta
compound under high pressure
W. Leini1, T. Zhang2, Z. Wu3, N. Wei3
1 Anhui Sanlian University, Hefei 230601, P. R. China
2 Chinese Academy of Sciences, Hefei 230031, P. R. China
3 Anhui University, Hefei 230601, P. R. China
Received August 23, 2017, in final form October 9, 2017
We have investigated the phonon, elastic and thermodynamic properties of L12 phase Al3Ta by density func-tional theory approach combining with quasi-harmonic approximationmodel. The results of phonon band struc-
ture shows that L12 phase Al3Ta possesses dynamical stability in the pressure range from 0 to 80 GPa due tothe absence of imaginary frequencies. The pressure dependences of the elastic constantsCi j , bulk modulus B,
shear modulus G, Young’s modulus Y , B/G and Poisson’s ratio ν have been analysed. The elastic constants
are satisfied with mechanical stability criteria up to the external pressure of 80 GPa. The results of the elastic
properties studies show that Al3Ta compound possesses a higher hardness, improved ductility and plasticity un-der higher pressures. Further, we systematically investigate the thermodynamic properties, such as the Debye
temperature Θ, heat capacity Cp , and thermal expansion coefficient α, and provide the relationships betweenthermal parameters and pressure.
Key words: first-principles, phonon, elastic properties, thermodynamic properties
PACS: 61.82.Bg, 62.20.dc, 71.20.Be, 71.15.Mb
1. Introduction
The L12 type trialuminide compounds Al3M (where “M” represents transition or rare earth elements)
have increasingly attracted attention due to the outstanding mechanical properties such as high specific
strength and elastic moduli [1–4]. Moreover, they also possess low density, high melting points, supe-
rior oxidation resistance, sufficient creep resistance, good thermal stability and conductivity [5–8]. All
these excellent properties enable them as the ideal dispersed strengthening phases for the high-strength
thermally-stable Al based alloys. However, the poor creep-resistance properties and the lack of ductility
have hindered their industrial applications with the elevated temperatures [9–11]. Refractory metals are
alloyed for the purpose of improvement of low temperature ductility, creep resistance, oxidation resistance
and toughness. The most widely used metallic elements are tungsten, molybdenum, rhodium, tantalum
and niobium. In these refractory metals, tantalum is resistant to corrosion from acids, organic chemicals
and aqueous solutions of salts. Tantalum is also used to produce a variety of alloys that have high melting
points, strength and ductility.
There are only a few theoretical and experimental studies dealing with the structural, phonon, elastic,
and electronic properties of Al3Ta in the literature. Asta et al. proposed that Al3Ta is considered to
be a potential electronic material for very large scale integration applications because the compound
reduces the low temperature interdiffusion barrier between aluminium and silicon [11]. Al3Ta is among
trialuminide compounds characterized by amelting point of 1823 K and a density of 6.9 g/cm. Boulechfar
et al. investigated the phase stability and electronic properties in Al3Ta compound using the FP-LAPW
This work is licensed under a Creative Commons Attribution 4.0 International License . Further distribution
of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.
13601-1
https://doi.org/10.5488/CMP.21.13601
http://www.icmp.lviv.ua/journal
http://creativecommons.org/licenses/by/4.0/
W. Leini, T. Zhang, Z. Wu, N. Wei
method [12]. They found that there is observed a characteristic of covalent bonding in Al3Ta compound.
However, it is not clear whether the cubic L12 phase is the phase observed experimentally. Some
researchers suggest that L12 phaseAl3Ta is relatively stable [13, 14]. To our best knowledge, no systematic
experimental and theoretical investigations on the thermoelasticity for L12 Al3Ta are performed up till
now, which could not reflect the high temperature mechanical and elastic behaviour.
Temperature dependent elastic properties are crucial for high temperature applications of alloys.
Moreover, knowledge of thermoelasticity is also essential for predicting and understanding material
response, strength, mechanical stability as well as phase transitions under high temperature [15]. High
pressure and temperature can cause large effects on chemical and physical properties of a solid. As we
know, the fact that high pressure is not easy to reach and control in experiment condition, while adjusting
pressure in theoretical simulations can be accomplished straightly by changing the size of a unit cell.
For elastic constants, they offer a link between the elastic and dynamic behaviours of solid materials and
provide important information on the nature of forces in a material [16, 17]. We can obtain the elastic
parameters such as bulk modulus B, shear modulus G, Young’s modulus Y , B/G and Poisson’s ratio ν by
elastic constants. Moreover, the thermodynamic properties under high pressure and temperature are of
great interest to geophysicists and physicists. To get a better understanding of thermodynamic properties
of L12 phase Al3Ta, more temperature-dependent parameters such as the Debye temperature Θ, specific
heat Cp, and thermal expansion coefficient α are required. In this paper, we firstly focus on investigating
the stability of L12 phase Al3Ta through the lattice dynamics study. Then, we investigate elastic and
thermodynamic properties of Al3Ta under high pressure. We believe this work can help us in designing
and understanding the high pressure behaviour of Al3Ta.
2. Methods
In the present work, all the calculations were performed by using first-principles based on the plane
wave pseudopotential density-function theory (DFT) method, which are carried out on the Quantum
ESPRESSO code [18, 19]. We calculate the elastic constants by ElaStic tool which can be interfaced with
computer packages WIEN2k and Quantum ESPRESSO [20, 21]. We have used the generalized gradient
approximation (GGA) parameterized by Perdew-Burke-Ernzerhof (PBE) for the exchange and correlation
terms in the electron-electron interaction for k-space summation which was 12× 12× 12 Monkhorst and
Pack grid of k-points. The kinetic energy cutoff for wavefunctions is 50 Ry, and the kinetic energy cutoff
for charge density and potential is 500 Ry. The convergence threshold for selfconsistency is 10−8 Ry.
In order to investigate the thermodynamic properties of Al3Ta, we use the quasi-harmonic Debye
model implemented in the Gibbs program [21]. The key procedure for thermoelastic calculations is
to compute the second derivatives of non-equilibrium Gibbs function G∗(V ; P,T) with respect to the
applied strain. For a given volume V and temperature T , non-equilibrium Gibbs function G∗(V ; P,T) can
be written as:
G∗(V ; P,T) = E(V) + PV + Avib(Θ,T), (2.1)
where E(V) is total energy per unit cell of Al3Ta, P is the hydrostatic pressure, Avib(Θ,T) represents the
vibrational Helmholtz free energy which can be taken as:
Avib(Θ,T) = nKT
[
9Θ
8T
+ 3 ln
(
1 − e−
Θ
T
)
− D
(
Θ
T
)]
, (2.2)
where D(Θ/T) is the Debye integral, and n is the number of atoms per formula unit, Θ takes the form of:
Θ =
~
K
(
6π2V
1
2 n
) 1
3
f (ν)
√
BS
M
, (2.3)
where BS is the adiabatic bulkmodulus, M is themolecular mass per formula unit, which can be expressed
in the form:
BS = V
d2E(V)
dV2 . (2.4)
13601-2
First-principles study of properties of Al3Ta compound
And f (ν) is given by:
f (ν) =
3
[
2
(
21 + ν
31 − ν
)3/2
+
(
1
3
1 + ν
1 − ν
)3/2
]−1
1/3
, (2.5)
where ν is Poisson’s ratio. Hence, the non-equilibrium Gibbs function G∗(V ; P,T) as a function of
(V ; P,T) can be minimized with respect to the volume as(
dG∗(V ; P,T)
dV
)
P,T
= 0. (2.6)
By solving equation (2.6), one can get the thermal equation of state V(P,T). After the equilibrium state
of a given V(P,T) has been gained, the isothermal bulk modulus and other thermodynamic properties,
such as the heat capacity, vibrational internal energy, and thermal expansion α can be evaluated in the
appropriate thermodynamic expressions.
CV = 3nK
[
4D
(
Θ
T
)
−
3Θ/T
eΘ/T − 1
]
, (2.7)
CP = 3nK
[
4D
(
Θ
T
)
−
3Θ/T
eΘ/T − 1
]
(1 + αγT), (2.8)
α =
γCV
BV
, (2.9)
where γ is the Grüneisen parameter. This method has already been successfully used to investigate the
thermodynamic properties of a series of compounds [22–25].
3. Results and discussions
3.1. Structural and phonon properties
Firstly, the equilibrium lattice parameters have been computed by minimizing the crystal total energy
calculated for different values of a lattice constant. We calculated the ground-state lattice parameters
of L12 phase Al3Ta (space group: Pm3̄m, No: 221) alloys at 0 GPa. As expected, the optimized lattice
parameter of Al3Ta obtained from GGA is found to be 4.024 Å, there are no experimental results to
compare with, while it is in agreement with the other theoretical value of 4.018 Å [12]. This agreement
provides a confirmation such that the computational methodology utilized in our work is suitable and
reliable. In order to gain an insight into the phase stability of L12 phase Al3Ta under high pressure, the
phonon band structure has been studied. The phonon band structure of L12 phase Al3Ta along some high
symmetry directions in the Brillouin zone at 0 GPa was displayed in figure 1 (a). These curves are very
similar to those obtained for other platinum-based alloys in the same structure. The calculated phonon
dispersion curves do not contain a soft mode at any vectors, which confirms the stability of L12 phase
Al3Ta at 0 GPa. The unit cell of Al3Ta has four atoms, which give rise to a total of 12 phonon branches,
which contains three acoustic modes and nine optical modes. We also provide the phonon curves of L12
phase Al3Ta at 80 GPa in figure 1 (b). It is obvious that Al3Ta is stable under pressure of 80 GPa due to
the absence of imaginary frequencies. Moreover, we can see the phonon band structure at 80 GPa shift to
the high energy on the whole comparing with that at 0 GPa, meaning that the frequencies of the phonon
increase as the pressure increases. Those results indicate that L12 phase Al3Ta possesses dynamical
stability in the pressure range from 0 to 80 GPa and ensure the subsequent study being credible.
3.2. Mechanical properties
Elastic properties of a solid can provide important information on the mechanical, dynamic and
thermodynamic behaviours of materials. In this work, firstly, we predict the elastic constants of Al3Ta
13601-3
W. Leini, T. Zhang, Z. Wu, N. Wei
−50
0
50
100
150
200
250
300
350
400
P
h
o
n
o
n
f
re
q
u
e
n
cy
(
T
H
z)
(a) P = 0 GPa
0
100
200
300
400
500
600
700
P
h
o
n
o
n
f
re
q
u
e
n
cy
(
T
H
z)
(b) P = 80 GPa
X
X
R
R
M
M
G
G
R
R
X
X
Figure 1. The phonon spectra of L12 phase Al3Ta at pressure of 0 and 80 GPa, respectively.
in the pressure range of 0–80 GPa at 0 K. The elastic constants of solids provide important information
concerning the nature of the forces operating in solids and provide information about the stability and
stiffness of materials. The elastic constants of a material can be obtained using the stress-strain method
by calculating the total energy as a function of lattice deformation. In the method, a small strain should
be loaded on a crystal. The elastic constants are defined by means of a Taylor expansion of the total
energy E(V) with respect to a small strain [26]. The energy E(V) is given as follows:
E(V) = E(V0 , 0) +
1
2
6∑
i
6∑
j
Ci jεiεj , (3.1)
where V0 is the volume at ground state; Ci j represents elastic constants, εi and εj represent the strain. For
L12 phase Al3Ta, there are three different independent elastic constants, C11, C12, and C44. At 0 GPa, the
calculated values of elastic constants C11 = 174.18 GPa, C12 = 94.30 GPa, and C44 = 82.15 GPa. There
is no doubt that the elastic constants of a material are strongly affected by pressure. Figure 2 (a) shows
the pressure dependences of elastic constants Ci j . We can analyse the mechanical stability of L12 phase
Al3Ta based on the elastic constants. The traditional mechanical stability conditions in cubic crystals on
the elastic constants are known as: C11 > 0, C12 > 0, C11 − C12 > 0, and C11 + 2C12 > 0 [27, 28]. It is
obvious that all the elastic constants of Al3Ta in a wide pressure range (0–80 GPa) satisfy these traditional
stability conditions, meaning that L12 phase Al3Ta is mechanically stable under pressure up to 80 GPa.
This result is in agreement with the phonon band structure discussions and results. From figure 2 (a), all
elastic constants of Al3Ta increase almost monotonously with an increase of pressure andC11 has slightly
smaller amplitude. This is attributed to the lattice parameters becoming lower under high pressure.
It is acknowledged that based on the elastic constants, the polycrystalline bulk modulus B, shear
13601-4
First-principles study of properties of Al3Ta compound
50
100
150
200
250
300
350
400
450
500
550
0 10 20 30 40 50 60 70 80
G
P
a
(a) P (GPa)
C11
C44
C12
50
100
150
200
250
300
350
400
450
500
550
0 10 20 30 40 50 60 70 80
G
P
a
(b) P (GPa)
B
G
Y
Figure 2. The calculated elastic constants Ci j (a), the bulk modulus B, shear modulus G and Young’s
modulus Y (b) as a function of pressure for L12 phase Al3Ta.
modulusG andYoung’smodulusY can be estimated based on theVoigte-Reusse-Hill approximation [29].
In general, considering the fundamental physics of a solid, the bulk modulus B is usually assumed to
be a fundamental physical property of solids and is used as a measure of the average bond strength of
the atoms for particular crystals. At the same time, the shear modulus G is a measure of resistance to
reversible deformations upon shear stress while the large value of shear modulus G is an indication of
a more pronounced directional bonding between atoms. Young’s modulus Y is defined as the ratio of
the tensile stress to the corresponding tensile strain, and is an important quantity for technological and
engineering applications. We display a relationship between the bulk modulus B, shear modulus G and
Young’s modulusY with the pressure in figure 2 (b). The value of bulk modulus B and shear modulus G is
120.13 GPa and 65.26 GPa, respectively at 0 GPa. This result is in good agreement with Boulechfar et al.
results [12]. It is seen that for Al3Ta, bulk modulus B, shear modulus G and Young’s modulus Y increase
almost linearly with an increasing pressure, where B andY have almost the same variation amplitudes, and
G has a slightly smaller amplitude, indicating that the effect of pressure on these quantities are prominent.
This is attributed to the atoms distance in the interlayers becoming shorter, and the interactions between
these atoms becoming stronger.
To further analyse the mechanical behaviour of L12 phase Al3Ta, the ductile or brittle behaviour
should be discussed. As we know, the bulk modulus B and shear modulus G represent a resistance to
plastic deformation and a resistance to fracture, respectively. According to Pugh formation, B/G ratio has
been proposed to predict a brittle or ductile behaviour [30]. A high B/G value is associated with ductility,
while a low B/G value corresponds to brittleness. If B/G < 1.75, a material exhibits a brittle behaviour.
Otherwise, it exhibits a ductile behaviour [31–33]. Figure 3 presents the calculated B/G value of L12
phase Al3Ta as a function of pressure. We observe that B/G value increases from 1.85 to 1.99 when the
13601-5
W. Leini, T. Zhang, Z. Wu, N. Wei
1.80
1.90
2.00
2.10
2.20
2.30
2.40
2.50
0 10 20 30 40 50 60 70 80
0.20
0.22
0.24
0.26
0.28
0.30
0.32
0.34
B
/G
ν
P (GPa)
B/G
1.80
1.90
2.00
2.10
2.20
2.30
2.40
2.50
0 10 20 30 40 50 60 70 80
0.20
0.22
0.24
0.26
0.28
0.30
0.32
0.34
B
/G
ν
P (GPa)
ν
Figure 3. The value of B/G and Poisson’s ratio ν as a function of pressure for L12 phase Al3Ta.
pressure increases from 0 to 80 GPa. Those suggest that Al3Ta exhibits a ductility characteristic, and
thus its ductility increased under high pressure. Poisson’s ratio ν is consistent with B/G ratio, which is
also related to the brittleness. The critical value which separates ductile and brittle material is 0.26. If
Poisson’s ratio ν > 0.26, the material behaves in a ductile manner. Otherwise, the material behaves in a
brittle manner. In addition, Poisson’s ratio is usually used to quantify the stability of the crystal against
shear. The larger is the Poisson’s ratio, the better is the plasticity. We note that the value of Poisson’s
ratio ν increases from 0.274 to 0.302 as the pressure increases from 0 to 80 GPa in figure 3, suggesting
that L12 phase Al3Ta exhibits a ductility under high pressure and the pressure can improve the ductility
while the existence of an external pressure can improve the plasticity.
3.3. Thermodynamic properties
The data of thermodynamic properties under high pressure and temperature can provide a valuable
information for industrial applications of materials under extreme conditions. We employ the quasi-
harmonic Debye model to obtain the thermodynamic properties of L12 phase Al3Ta at various temper-
atures and pressures from the energy-volume relation. As one of the important physical quantities for
a solid, the Debye temperature Θ is a an important parameter describing the material thermodynamic
properties in solid state physics. It is closely related to specific heat, bond strength, elastic stiffness
constants and melting temperature. Figure 4 displays the dependence of the Debye temperature of Al3Ta
on temperature and pressure. Variations in Debye temperature Θ versus temperature at different fixed
pressures, which are P = 0, 40, and 80 GPa, are shown in figure 4 (a). From the figure, it is obvious that
Debye temperature Θ in the range of temperature from 0 K to 900 K remains approximately unaltered,
suggesting that Debye temperature Θ is insensitive to the temperature. In figure 4 (b), the Debye tem-
perature Θ increases almost linearly with applied pressures at a given temperature, which indicates the
change of the vibration frequency of atoms under pressure. This may be attributed to the influence of the
isothermal bulk modulus which is independent of pressure. Hence, the pressure has a more significant
effect on the Debye temperatureΘ of L12 phase Al3Ta comparing with the temperature. These results are
consistent with the general behaviour of a decreasing Debye temperature with an increase of temperature
in other L12 structures [34, 35].
The heat capacity is an important parameter of amaterial. Knowledge of the heat capacity of a solid not
only provides an essential insight into its vibrational properties but also offers some instructions for many
applications. In figure 5 (a), we show the heat capacity Cp as a function of temperature T at the pressures
of 0, 40, and 80 GPa. It is realized from the figure that when T < 300 K, the Cp increases very rapidly
with the temperature; when T > 300 K, the heat capacityCp increases slowly with the temperature, and it
almost approaches a constant value referred to as Dulong-Petit limit for this compound. Additionally, the
heat capacity Cp rapidly approaches zero while the temperature approaches absolute zero. In figure 5 (b),
13601-6
First-principles study of properties of Al3Ta compound
450
500
550
600
650
700
750
800
0 100 200 300 400 500 600 700 800 900
Θ
(
K
)
(a) Temperature (K)
0 GPa
40 GPa
80 GPa
400
450
500
550
600
650
700
750
800
850
0 10 20 30 40 50 60 70 80
Θ
(
K
)
(b) P (GPa)
300 K
600 K
900 K
Figure 4. (Colour online) The Debye temperature Θ as a function of temperature at different pressures of
P = 0, 40, and 80 GPa for L12 phase Al3Ta (a). The Debye temperature Θ as a function of pressure at
different temperatures of T = 300, 600, and 900 K for L12 phase Al3Ta (b).
0
20
40
60
80
100
120
0 100 200 300 400 500 600 700 800 900
C
p
(
J/
m
o
l*
K
)
(a) Temperature (K)
0 GPa
40 GPa
80 GPa
0
20
40
60
80
100
120
140
0 10 20 30 40 50 60 70 80
C
p
(
J/
m
o
l*
K
)
(b) P (GPa)
300 K
600 K
900 K
Figure 5. (Colour online) The heat capacity Cp as a function of temperature at different pressures of
P = 0, 40, and 80 GPa for L12 phase Al3Ta (a). The heat capacityCp as a function of pressure at different
temperatures of T = 300, 600, and 900 K for L12 phase Al3Ta (b).
we depict the variation of heat capacity with pressure at a fixed temperature of 300 K, 600 K, and 900 K.
It is obvious that the heat capacity keeps almost unchanged with an increase of the pressure and the
further compression slows down this trend. This reveals that the heat capacity Cp is mainly dependent
on the temperature, and a higher temperature almost does not affect the heat capacity Cp.
The variation of the thermal expansion coefficient α of L12 phase Al3Ta with temperature and various
pressures is shown in figure 6. It is clearly seen that α exhibits a similar trend for all isobars, and at
0 GPa, the thermal expansion coefficient α increases exponentially with T in the low temperature region
and becomes flat at high temperatures in figure 6 (a). The thermal expansion coefficient possesses the
highest values for lowest (P = 0 GPa) pressure in all the temperature range considered. Moreover, as
seen from figure 6 (b), the thermal expansion coefficient α increases with an increasing pressure at a
13601-7
W. Leini, T. Zhang, Z. Wu, N. Wei
0
0.5
1
1.5
2
2.5
3
0 100 200 300 400 500 600 700 800 900
α
(
1
0
−
5
/K
)
(a) Temperature (K)
0 GPa
40 GPa
80 GPa
1
1.5
2
2.5
3
0 10 20 30 40 50 60 70 80
α
(
1
0
−
5
/K
)
(b) P (GPa)
300 K
600 K
900 K
Figure 6. (Colour online) The volume thermal expansion coefficient α as a function of temperature at
different pressures of P = 0, 40, and 80 GPa for L12 phase Al3Ta (a). The volume thermal expansion
coefficient α as a function of pressure at different temperatures of T = 300, 600, and 900 K for L12 phase
Al3Ta (b).
given temperature under 10 GPa and then decreases with an increasing pressure. This suggests us that
L12 phase Al3Ta will have the largest thermal expansion coefficient α value at about 10 GPa.
4. Conclusions
In this work, the structural, phonon, elastic and thermodynamic properties of Al3Ta compound under
high pressures were theoretically investigated by performing GGA calculations based on DFT method.
The phonon band structure of L12 phase Al3Ta indicates that it possesses a dynamical stability in the
whole studied pressure range from 0 to 80 GPa due to the absence of imaginary frequencies. The pressure
dependence of the elastic constants Ci j , bulk modulus B, shear modulus G, Young’s modulus Y , B/G,
and Poisson’s ratio ν are successfully calculated. The results of the elastic properties studies show that
Al3Ta compound is mechanically stable and possesses a higher hardness, improved ductility and plasticity
under higher pressures. Moreover, we have studied the thermodynamic properties, such as the Debye
temperature Θ, heat capacity Cp, and thermal expansion coefficient α using the quasi-harmonic Debye
model in the range of temperatures from 0 K to 900 K.
Acknowledgements
The calculations were performed in Center for Computational Science on the ScGrid of Supercomput-
ingCenter, ComputerNetwork InformationCenter ofAnhui university andChineseAcademy of Sciences.
This work was supported by the National Science Foundation of China under Grants Nos. 11774284 and
the Anhui Provincial Natural Science Foundation of China (1508085SME219).
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W. Leini, T. Zhang, Z. Wu, N. Wei
Першопринципнi дослiдження структурних, фононних,
пружних та термодинамiчних властивостей сполуки Al3Ta
при високому тиску
В. Леiнi1, Т.Шенг2,Ш. Ву3, Н. Вей3
1 Аньхойський унiверситет Саньлянь, Хефей 230601, Китай
2 Китайська академiя наук, Хефей 230031, Китай
3 Аньхойський унiверситет, Хефей 230601, Китай
Дослiджено фононнi, пружнi та термодинамiчнi властивостi Al3Ta у L12 фазi методом функцiоналу густи-
ни у поєднаннi з квазигармонiчною апроксимацiйною моделлю. Результати структури фононної зони
показують, що Al3Ta у L12 фазi володiє динамiчною стiйкiстю у дiапазонi тиску вiд 0 до 80 ГПа завдяки
вiдсутностi уявних частот. Проаналiзовано тисковi залежностiCi j , об’ємного модуля пружностi B, модулязсуву G, модуля Янга Y , B/G та коефiцiєнта Пуассона ν. Пружнi сталi задовольняють критерiй механi-
чної стiйкостi аж до зовнiшнього тиску 80 ГПа. Результати аналiзу властивостей пружностi показують,що
сполука Al3Ta володiє вищою твердiстю, кращою тягучiстю i пластичнiстю при високих тисках. Крiм того,
систематично дослiджено такi термодинамiчнi властивостi, як температура ДебаяΘ, питома теплоємнiсть
Cp та коефiцiєнт теплового розширення α, а також встановлено залежнiсть мiж тепловими параметрами
i тиском.
Ключовi слова: першi принципи, фонон, пружнi властивостi, термодинамiчнi властивостi
13601-10
Introduction
Methods
Results and discussions
Structural and phonon properties
Mechanical properties
Thermodynamic properties
Conclusions
|