INTRODUCTION:
Among
advanced ceramic materials of transition metal, the transition metal carbides are
known as refractory compounds. These materials have high melting points and
are of high hardness, are brittle, show metallic luster and metallic conductivity,
consequently, they present a large interest for theoretical investigations and technological
applications1-3.
Carbides
of transition metal with MX stoichiometry are usually cubic, where the metallic
atoms (M) form the face –centered cubic sublattice, and nonmetallic atoms (X) occupy
interstitial positions, forming B1-type structure4. Understanding
the macroscopic mechanical properties require information obtained from accurate
calculation of elasticity5. MnC in B1 structure is a metastable carbide.
The cohesive energy 
studied
by J. Haglund et al. using the LMTO method shows low value. This is probably
related to the symmetric nature of the ground-state valence configuration of the
Mn atom6. The thermodynamic properties of Mn-C compounds have been assessed
by Djurovic et al. using CALPHAD (calculation of phases diagram) method and
the results for MnC
( ranging
from 0.26 to 1) show that for MnC the enthalpy of formation is positive7.
A.N. Arpita Aparajita et al synthesize for the first time novel manganese
monocarbide (MnC)and the structure of MnC was found to be ZnS phase with a lattice
parameter
. Gutsev
et al.8 identified the X state of MnC as of4Σ-symmetry
with 
, dissociation
energy 
, 
depending
on the functional (BPW91, BLYP and B3LYP) used9. As far as we know there
is no report related to the ab initio study of the great interest compounds7,10-19.
In the present work, monocarbide of manganese MnC in B3 structure is investigated
under high pressures. This study also includes the ground state properties,
such as lattice parameter
, bulk modulus
, and pressure
derivative of bulk modulus 
of MnC in B3
structure.
Debye
temperature has also been calculated from the average sound velocity.
CALCULATION
METHODS:
This Computational study was carried out using
Abinit package. A plane wave basis set with energy cut-off 60 Ha is
applied for all compounds. Accurate Brillouin zone integrations are performed using
the standard special k point’s technique of Monkhorst and Pack (MP)20.
10×10×10 MP meshes is used for the B3 phase. In order
to calculate the total energies, basic ground state properties (equilibrium lattice
constant 
, bulk
modulus 
, derivative
pressure 
and the
elastic constants 
and 
) generalized
gradient approximation (GGA) exchange-correlation functional was used. The investigation
of thermodynamic properties was performed using the quasi-harmonic Debye model implemented
in the Gibbs2 program21.
RESULTS
AND DISCUSSIONS:
Structural
properties:
Figure
1shows the total energy curve as a function of unit cell volume for MnC. The total
energy is computed using the scheme of Perdew Burke and Ernzerhof (PBE).The lattice
constant at equilibrium is obtained computationally by minimizing the total energy
as a function of cell volume and obtained in the present investigations on MnC in
B3 structure is 4.314Å. Equilibrium lattice constant, bulk modulus and pressure
derivatives of bulk modulus computed by minimizing the crystal total energy for
different values of the lattice constant by fitting third-order Birch-Murnaghan
equation of state22-24 are tabulated in Table 1.The equation is obtained
by the following expression:
The
calculated lattice constant(a0=4.4151Å) at equilibrium
(P=0 GPa and T=0 k) is in good agreement with experimental result (a0=4.4294 Å)
obtained by
A.N. Arpita Aparajita et al.7.
Table
1. Calculated lattice constant 
, bulk modulus
, and its
pressure derivative B0’were compared
to experiment and other theoretical works for MnC in B3 structure.
|
Compound
|
a0
(Å)
|
B0
(GPa)
|
B0’
|
|
MnC
B1
Present
Exp.
Theory
|
4.4151
4.42947
4.2907
|
209.33
-
-
|
3.95
-
-
|
Figure
1. Energy versus volume for MnC at P=0 and T=0 K in B3 structure.
Elastic
properties:
For a cubic crystal the elastic constants
, 
and can be
determined by computing the stress generated by forcing a small strain to an optimized
unit cell25.The calculated elastic constants of MnC in B3 structure
in its ground state (P=0GPa and T=0k) are presented in Table 2.
Table
2. The elastic constants in GPa for MnC in B3 structure
|
Material
|
Ref.
|
C11
|
C12
|
C44
|
|
MnC (B)
|
Present
|
259.8
|
187.7
|
119.6
|
This study shows that the B3 structure of MnC
is mechanically stable, thusit obeys the following elastic stability criteria for
a cubic crystal:
, 
and 
.
In
order to calculate the shear modulus 
, the following
equation according to Voigt and Reuss approximations is used:
Where:
,
Also,
the Young’s
modulus 
was calculated
as follows:
where
The
derived elastic properties of MnC in B3 structure from elastic constants, such the
bulk modulus, shear modulus and Young’s modulus are listed in Table 3.
Table
3. Bulk Modulus, Shear Modulus, Young’s Modulus and 
)Value of
MnC in B3 structure.
|
Crystal
|
|
 (GPa)
|
 (GPa)
|
 (GPa)
|
|
|
MnC
(B3)
|
Present
|
211.73
|
74.12
|
199.13
|
2.86
|
Thermodynamic
properties:
The
quasi-harmonic Debye model, in which the phononic effect is considered, is applied
to investigate the thermodynamic properties of MnC. In the quasi-harmonic Debye
model, the non-equilibrium Gibbs function 
is given
in the following form:
Where
is the
total energy per formula unit, 
corresponds
to the constant hydrostatic pressure, 
is the
Helmholtz free energy including the contributions of the lattice vibration on the
internal energy and entropy change, which is generally obtained by the lattice dynamics
calculations of quasi harmonic approximation.
Where
is the
state density distribution
Considering
the heat effect and the internal relation of the thermal properties with 
and 
can be changed
to the following form:
and
the vibrational Helmholtz free energy 
can be written
as:
Where
is Debye
temperature, 
represents
the unit molecular formula of atomic number,
is Debye
function, which is written as follows:
For
an isotropic solid, 
can be
expressed as:
where
is the
molecular mass, 
is the
adiabatic bulk modulus, which is approximated by static compressibility, 
can be
converged as:
The
Poisson ratio 
is taken
from the calculated elastic constants, the non-equilibrium Gibbs function 
as a function
of 
can be
minimized with respect to volume:
By
solving this equation, one can get the thermal equation of state (EOS). The isothermal
Bulk modulus, the heat capacity 
and the
thermal expansion coefficient 
are given
by:
where
is the
Gruneisen parameter, which is defined as:
The
Debye temperature is the important parameter for analyzing the thermal characteristics
of materials. The Debye temperature may be estimated from the average sound velocity
.
where 
is Planck’s
constants, 
is Boltzmann’s
constant, 
is Avogadro’s
number, is the
number of atoms per formula unit, 
is the
molecular mass per formula unit,
is the
density, and is obtained
from
Where
and 
are the longitudinal
and the transverse elastic wave velocities, respectively, which
are obtained from Navier’s equation as follows:
Where
is the
shear modulus and is the
adiabatic bulk modulus26-32.
The
curves in(Figure 2) are identical and in the form of almost linearly increases with
the pressure increasing. But, the parameter (a0) varies from 4.47 to
3.71Åforthe pressure
interval going from 0 to80 GPa. The effect of temperature on the lattice parameter
is little, and when the pressure increases from 0 to 80 GPa, the variation in the
value of the lattice is less than 1.5 %.
Figure
3 shows the Bulk modulus 
versus temperature under different pressures for MnC in B3 structure. It is
found that decreases
gradually as pressure increases, which means that MnC becomes more and more
difficult to be compressed with increasing pressure. We remark that the bulk modulus
is inversely proportional from 0 to 1400 K and the influences of temperatures
are important. Then, parameter B equals 50 GPa at zero pressure but equals 300
GPa at 80 GPa.
When temperature changes from 0 to 1400 K, the Bulk modulus of MnC by 22%,
4.74%, 2.82% and 1.55% respectively. Where the Bulk modulus represents to
plastic deformation.
In
this way, (Figure 4) shows that as pressure increases, the three elastic
constants
, and increase.
It is noticed that when pressure is smaller than 21 GPa,is more
sensitive to change of pressure than and , but
if pressure is greater than 21 GPa,is more
sensitive to change of pressure than and .
Figure
4. Elastic constants 
and
Bulk modulus B versus pressure at zero temperature for MnC in B3 structure.
The
calculated values of density longitudinal sound velocity,transverse
sound velocity
,
average sound velocity, and
Debye temperature
are
given in Table 4.
Figure
5 shows the relationships between the heat capacity 
for
MnC versus
temperature under different pressures in B3 structure. The heat capacity
exhibits a strong dependence on temperature and pressure. increases
with temperature increasing at a given pressure and decreases with pressure
increasing at a given temperature. The heat capacity is
close to Dulong-Petit limit. In this study, the heat capacity is
49.6 J/mol.K at zero pressure and zero temperature.
The
Cv graphs are confused, we distinguish for zero GPa two regions, the first of
0-100K is an almost vertical increase and the second an almost horizontal and
stable increase which reaches up to 45 for all pressures studies. While under
the influence of other pressure values, the Cv is slightly inclined. For
example, for 240 GPa, we distinguish three regions: the first is of 0-250 K,
the second is of 250-600 K and the third is of 600-1400 K.
Figure
5. Heat capacity 
versus
temperature under different pressures for MnC in B3 structure.
On
the other hand, Fig.6 shows the variations of the volume expansion coefficient
versus
temperature under different pressures for MnC in B3 phase.
At a given temperature increases
rapidly with temperature rising at low pressures (0 pressures). The calculated
values of the volume expansion coefficient (α), when T= 400 K and P=0GPa,
the coefficient α is 3.2 for P=20GPa, α is 1.1, for P=40GPa, α
is 0.6 and for P=80GPa, α is less than 1×10-4/K.
We
notice drops of α between 0 and 50 K, for example at 100 K α=1.25, at
300 K α=3.2, at 600 K α=3.5, at 800 K α=3.7, at 1000 K α=4,
at 1200 K α=4.5, they reach towards values lower than 0.510-3/K.
Figure
6. Volume expansion coefficient
versus temperature
under different pressures for MnC in B3 structure.
We
have plotted the variations of the volume expansion coefficient
with pressure
at different temperatures for MnC in B3 phase to determining the thermodynamic
nature of the phase transition for MnC in (Figure 7).
At
a given pressure increases
rapidly with temperature rising at low temperatures, then increases slowly at higher
temperatures.
Figure
7. Volume expansion coefficientwith pressure
at different temperatures for MnC in B3 structure.
CONCLUSIONS:
In
the present paper, we studied the structural, elastic and thermodynamic properties
of manganese
monocarbide MnC
in B3 structure under high pressure using the density functional theory with GGA-PBE
functional. The effects of the pressure and temperature on the above properties
were also investigated in details. The calculated equilibrium lattice constant in
our work is in good agreement with experimental data and other theoretical values
calculated in several studies obtained for this time. This study shows that
the B3 structure of MnCis mechanically stable and the heat capacity obtained
is 49.6 J/mol.K. For the compound MnC when pressure is smaller than 21 GPa,is more
sensitive to change of pressure than and , but if
pressure is greater than 21 GPa,is more
sensitive to change of pressure than and .According
the thermodynamic calculations, we found out that the effect of temperature and
pressure on Cv are apposite, and the effect of temperature is larger than pressure,
which is consistent with a compression volume rate. The Debye temperature
is also
calculated for this system. Finally, the curves of a prove an
exponential increase at lower temperature and gradual linear increases at higher
temperatures.
ACKNOWLEDGEMENTS:
This
study was supported by the Algerian Ministry of Higher Education and Scientific
Research.
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) and Debye parameter
were also calculated at the different temperatures. The pressure effects on the
above parameters were computed and their values are compared with the experimental
results.
