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Liu et al. J Mater Inf 2023;3:17 https://dx.doi.org/10.20517/jmi.2023.19 Page 5 of 17

where E represents the static energy at 0 K, corresponding to the energy of the ground state when the ion is
0
fixed in its lattice position. Additionally, F accounts for the thermal free energy arising from electronic
el
excitation, while F represents the vibrational contribution to the free energy, S is the configuration
conf
ion
entropy of the system. The equilibrium structure of a crystal at any given temperature (T) and volume (V)
can be determined by minimizing the Helmholtz free energy F(V, T). Once the minimum Helmholtz free
energy F(V, T) is determined for a particular T and V, other thermodynamic functions and properties of the
crystal can be deduced as implemented. For instance, entropy (S), isothermal bulk modulus (B ), heat
T
capacity at constant volume , and heat capacity at constant pressure (C ) can be calculated accordingly
(C V )
p
and expressed as :
[75]

All these thermodynamic properties are estimated using the mean-field potential (MFP) method proposed
by Wang et al. and later improved by Song et al. to extend its applicability to more general cases and
[75]
[76]
complex structural crystals [77-79] . This method is suitable for various complex systems and has significant
potential for predicting the thermodynamic properties of multicomponent alloys.

RESULTS AND DISCUSSION
Structural stability
The structures are optimized through full relaxation to obtain the energies E of MECs and their binary
carbides as a function of volumes V. A series of energy-volume data points around the equilibrium volume
were calculated by minimizing the total energy to optimize the crystal structure. Then, the energy-volume
data points of the calculated MECs and their corresponding individual metal binary carbides were fitted
[80]
using the fourth-order Birch-Murnaghan EOS , as shown in Figure 2.

The present results show the effect of TMs on the total energy and volume of MECs. Specifically, the
equilibrium volume for binary carbides is found to increase in the order of TaC < ZrC < UC < YC, which is
roughly consistent with the trend of pure metal atomic size change, i.e., Ta < U < Zr < Y. In addition, the
equilibrium volume of MECs also increases with the increasing volume of TM atoms, in the order of
(TaZrU)C < (YZrU)C. Meanwhile, the total energy of middle entropy carbides, (TaZrU)C < (YZrU)C, also
increases with the improved volumes of TM atoms.

The fitted E-V curves provide valuable information about the equilibrium atomic energy E , equilibrium
0
volume V , bulk modulus B , first derivative of bulk modulus with respect to pressure B ’, and lattice
0
0
0
constants a at zero temperature and pressure for MECs and their corresponding pure carbides. These
predicted properties listed in Table 1 are in good agreement with the experimental and theoretical
properties reported in the literature [81-87] . Specifically, the predicted E and V for (TaZrU)C and (YZrU)C
0
0
are -9.7095 eV, -8.8862 eV and 13.3873 Å , 15.1241 Å , respectively. The lattice constants are a = 9.5097 Å for
3
3
(TaZrU)C and a = 9.9043 Å for (YZrU)C. These values are very close to the average values of the
equilibrium atomic energies E , equilibrium volumes V , and lattice constants a of the three component
0
0