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Liu et al. J Mater Inf 2023;3:17 https://dx.doi.org/10.20517/jmi.2023.19 Page 7 of 17

Temperature-dependent thermo-physical properties
The investigation of the thermodynamic properties of materials is crucial in the field of high-temperature
engineering applications. In order to gain a deeper understanding of the thermodynamic properties of
(TaZrU)C and (YZrU)C in the temperature range of 0-2,000 K, the Debye-Gruneisen model implemented
[76]
in MFP was utilized in this study. The calculated thermodynamic properties are presented in Table 2,
including the equilibrium volume V , Gibbs free energy, constant volume heat capacity Cv, constant
0
pressure heat capacity Cp, bulk modulus B , and thermal conductivities. The thermodynamic properties of
0
materials can be determined through various experimental techniques, including differential thermal
analysis , chemical analysis , and X-ray diffraction [61,90,91] . Additionally, the hardness of a material is
[89]
[60]
commonly measured using Vickers hardness, which can be determined using a nanometer indentation
method . However, with the advancement of science and technology towards extreme conditions, it
[82]
becomes increasingly challenging to experimentally obtain accurate thermodynamic data. As a result, there
are currently no available experimental or theoretical data on the thermodynamic properties of (TaZrU)C
and (YZrU)C. Therefore, the present findings are of significant value in guiding future research efforts to
develop advanced NTP fuels.

The heat capacity is a fundamental parameter that links the thermodynamic and dynamic properties of
C
materials. The constant volume heat capacity was estimated using an expression C = T(∂S) and
V
v
V
investigated its temperature dependence for (TaZrU)C and (YZrU)C together with the benchmarks of
binary ones, as shown in Figures 3 and 4. In particular, Figure 3 illustrates the constant pressure heat
capacity C and constant pressure heat capacity C of binary carbides, matching well with the available
V
P
reported results in the literature and indicating the precise of our benchmark tests. Moreover, the
thermodynamic properties of (TaZrU)C and (YZrU)C are presented in Figure 4. In addition, the
contributions of ionic and electronic heat capacities are also shown in Figure 4A and B for comparison. The
results demonstrate that the ionic heat capacity outweighs the electronic heat capacity across the entire
temperature range examined. Compared to the binary carbides of each component, (TaZrU)C and
(YZrU)C have larger C and C values due to the mixing of the binary carbides of each component in the
P
V
same temperature range. It is indicated that these two MECs exhibit a sharp increase in at temperatures
C V
below 400 K, which is also followed by a gradual increase toward a constant value at higher temperatures in
line with the Dulong-Petit limit. Notably, the C values of both carbides were very similar at lower
V
temperatures, while the C of (YZrU)C at higher temperatures is slightly higher than that of (TaZrU)C. On
V
the contrary, the C of (TaZrU)C and (YZrU)C is depicted in Figure 4B, and although they exhibit similar
P
trends, their C values are more sensitive to temperature at low temperatures and gradually increase at high
P
temperatures. Furthermore, (YZrU)C displays a larger C than that of (TaZrU)C. It is noteworthy that at
P
very high temperatures, C does not follow the Dulong-Petit law as C does but still shows a small increase.
V
P
This feature may arise from the relationship between C and C with the expression of C = C + 3α BTV.
2
P
P
V
V
Estimating the strength of materials is crucial for their practical applications, and the bulk modulus B is a
measure of resistance to compression of a material. An increase in bulk modulus leads to a stronger
material. The bulk modulus is calculated using B = -V(∂P) , and the results for (TaZrU)C and (YZrU)C
0
T
are displayed in Figure 4C. Both carbides exhibit a significant decrease in bulk modulus with
increasing temperature, indicating a more pronounced softening effect. The bulk modulus of (TaZrU)C is
consistently higher than that of (YZrU)C throughout the entire temperature range of 0-2,000 K, providing
a significant strength advantage of (TaZrU)C in engineering applications.
Entropy is a fundamental thermodynamic parameter that describes the degree of disorder in a crystal
structure. A higher entropy value indicates a higher degree of disorder in the crystal structure. The entropy
value is a crucial factor for defining HEMs, and its variation is related to the formation, stability, and phase