First Principles Calculations on Elastic, Thermodynamic and Electronic Properties of Co 2 Zr and Co 2 Ti at High Temperature and Pressure

: Co 2 Zr and Co 2 Ti are both cubic crystals with a Cu 2 Mg-type structure. The elastic, thermodynamic and electronic properties of the intermetallic compounds Co 2 Zr and Co 2 Ti are investigated by using ab initio plane-wave pseudopotential density functional theory (PWPDFT) and generalized gradient approximation (GGA) under high temperature and pressure. The partially calculated results are consistent with the available experimental data. The elastic properties of Co 2 Zr and Co 2 Ti under high pressure were ﬁrst studied by ﬁrst principles calculations. The results indicate that the elastic constants, elastic modulus and Poisson’s ratio are functions of pressure, indicating that the e ﬀ ect of pressure on the ductility and anisotropy is signiﬁcant. The thermodynamic properties are also calculated by the quasi-harmonic Debye model. In the range of 0~100 GPa pressure and 0~1500 K temperature, the Debye temperature Θ , the heat capacity C V and the thermal expansion α vary with pressure and temperature. Co 2 Ti has a higher Debye temperature than Co 2 Zr under the same pressure. Decreasing temperature and increasing pressure have the same e ﬀ ects on C V and α . The electron density di ﬀ erence and density of states of Co 2 Zr and Co 2 Ti are ﬁnally investigated. The results show that both Co 2 Zr and Co 2 Ti are typically metal crystals but Co 2 Zr has greater covalence than Co 2 Ti.


Introduction
Intermetallic compounds are widely exploited and utilized due to their distinguished properties among the common metal in mechanical, electrical, magnetic and chemical properties [1][2][3]. For instance, they show high tensile strength, good ductility, high corrosion resistance and thermal stability, which are attributed to the divergences in the chemical bonding of metal atoms [4][5][6][7][8][9][10]. In addition, intermetallic compounds can fill the existing gap between metal and non-metal materials due to their specific properties [11].
In order to make better use of intermetallic compounds as well as exerting their value, it is of importance for us to study their properties. The first principles calculations with ab initio plane-wave pseudopotential density functional theory (PWPDFT) and generalized gradient approximation (GGA) is, in general, considered to be the most effective and convenient method to study the properties of metal materials, since it does not require any experimental parameters but needs some basic physical constants to obtain the basic properties of the ground state of the system. The stability of the crystal structure is the most important properties of intermetallic compounds. First principles calculation with

Calculation Parameters
In the study, the CASTEP code is employed to perform the first principles calculations of all total energy electronic structures. Calculations were performed using norm-conserving pseudopotentials [29] with 3d 7 4s 2 electrons used as valence for Co, 4d 2 5s 2 electrons used as valence for Zr and 3s 2 3p 6 3d 2 4s 2 electrons used as valence for Ti. The exchange-correlation energy of electrons is calculated through GGA-PBE function that is the combination of the Perdew-Burke-Ernzerhof (PBE) system and generalized gradient function (GGA) [30]. It can measure precisely the relaxed lattice constants, ensuring high accuracy of calculation. In our computations, the cutoff energy of plane waves expanding the electronic wave functions that is set for Co 2 Zr and Co 2 Ti is 540 eV, 500 eV for Co 3 Ti [24]. Referring to the conclusion of Monkhorst-Pack [31], the k-points of Co 2 Zr and Co 2 Ti are set as 6 × 6 × 6 and 7 × 7 × 7, and 15 × 15 × 15 for Co 3 Ti [24], respectively, for Brillouin zone integration. Based on these parameters, the total energy of Co 2 Zr and Co 2 Ti is self-consistent converging to 3.4 × 10 −7 ev/atom and 2.0 × 10 −7 ev/atom. These parameters are sufficient to calculate well converged total energy, geometrical configurations and elastic stiffness coefficients.

Elastic Properties
Elastic properties, indicating the bonding strength of the material atoms, anisotropy information and structural stability, are important parts of the study of metal materials. Elastic constants (C ij ) can help us to understand the mechanical and physical properties of metal alloys, such as ductility, hardness, melting point. Cubic crystal contains only three independent elastic constants (C 11 , C 12 and C 44 ). In addition, other physical quantity such as bulk modulus B, shear modulus G, Young's modulus E and Poisson's ratio σ can be also used to describe elastic properties. In cubic crystals, the following formulas are often used [32][33][34]: where C = (C 11 − C 12 )/2, G V = (2C + 3C 44 )/5, G R = 15(6/C + 9/C 44 ) −1 (2) where G V is the Voigt shear modulus and G R is the Reuss shear modulus, B S is the adiabatic bulk modulus. For cubic crystals, the mechanical stability is judged by Born stability criteria [35,36]: where P is pressure.

Thermodynamic Properties
In order to study the thermodynamic properties of Co 2 Ti and Co 2 Zr, Blanco et al. [37] proposed the quasi-harmonic Debye model (QHDM), in which Debye temperature Θ, heat capacity C V and thermal expansion coefficient α are often adopted [38]: In Equation (5), h is Planck's constant, k B is Boltzmann constant, n represents the number of atoms in a chemical formula, N A is Avogadro's number, ρ stands for crystal density, M means molecular weight, V m is the average sound velocity and is given in the following formula [38]: where V S and V l , respectively, stand for transverse and longitudinal elastic wave velocity, as obtained in Equation (9), namely the Navier equation. In Equation (6), D(Θ/T) represents Debye integral; in Equation (7), γ stands for the Grüneisen parameters, B T is the isothermal bulk modulus, and C V is the heat capacity, while V is the volume [38].
where G * (V; P, T) is the non-equilibrium Gibbs function. In the quasi-harmonic Debye model, where E(V) means total energy, and A Vib (Θ(V);T)is contribution of crystal vibration that can be written as: These formulas are used to calculate the thermodynamic properties of Co 2 Zr and Co 2 Ti.

Elastic Properties
The structural parameters of metal compounds, such as lattice constant a, cell volume V, structure and space group, should be figured out before analyzing their elastic properties. In Table 1, detailed structural information and reference data (Ref. [18]) of Co 2 Zr and Co 2 Ti at zero pressure are given. We can find that the optimized lattice constant, crystal volume and so on are in accordance with the given experimental and other theoretical results. The errors of lattice parameters are 0.47% and 0.25% for Co 2 Zr and Co 2 Ti, respectively. In addition, the values of B 0 , B 0 and B 0 " under 0 GPa pressure are provided, where B 0 is the bulk elastic modulus, B 0 is the pressure derivative of the bulk modulus, and B 0 " is the second pressure derivative of the bulk modulus. It can be seen from the table that the values of B 0 , B 0 and B 0 " of Co 2 Zr crystal derived from the quasi-harmonic Debye model were 192 GPa, 4.370 and −0.031, respectively, while those that are 192 GPa, 4.426 and −0.034 derived from the natural strain equation of state (EOS). For Co 2 Ti, we can get B 0 = 213 GPa, B 0 = 4.533, and B 0 " = −0.033 through the Debye model along with B 0 = 199 GPa, B 0 = 5.247 and B 0 " = −0.045 through natural strain equation of state. Although Co 2 Zr and Co 2 Ti have the same cubic structure, lattice constant a of Co 2 Zr is larger than that of Co 2 Ti while values of B 0 and B 0 are smaller than the counterparts under the same condition. There are three independent elastic constants, C 11, C 12 and C 44 , in cubic crystal, and their values at 0 GPa, 0 K are given in Table 2. We can calculate adiabatic bulk modulus B S , shear modulus G and Debye temperature Θ through Formulas (1), (2), and (5) with their results recorded in the Table 2. It was seen that the calculated values of Co 2 Ti basically coincide with the reference values from Ref. [23]. Meanwhile, the data of other structures, such as CoTi [26] and Co 3 Ti [24], are adopted to make comparison with these values. Unfortunately, there are no data available to compare with our results of Co 2 Zr, thus we only exhibit the data of CoZr [26].
The values of C 11, C 12 and C 44 of Co 2 Zr and Co 2 Ti at 0 K and 0~100 GPa are provided in Tables 3  and 4. It may be seen from this table that our values agree well with other theoretical and similar structural results, with the errors of C 11 , C 12 and C 44 being, respectively, 10%, 14.9%, 3.9% for Co 2 Ti compared with Ref. [23]. According to the well-known Born stability criteria [35,36,39,40], if the three independent elastic constants satisfy Formula (4), the crystals are stable. Therefore, we can find that Co 2 Zr is always stable under the pressure from 0 to 100 GPa, whereas Co 2 Ti can satisfy the Born stability criteria only under pressures smaller than 80 GPa, which indicates that Co 2 Ti is unstable when the pressure is more than 80 GPa. What is more, it is possible that a transition phrase exists under the pressure of 85 GPa.    Table 4. The C ij (GPa), B (GPa), G (GPa), E (GPa), Θ, Poission ratio σ, anisotropy factor A and G/B of Co 2 Ti at various pressure. The shear modulus G represents the plastic deformation resistance, while the bulk modulus B represents the fracture resistance. In 1954, the ratio between shear modulus G and bulk modulus B G/B was given by Pugh [41] to evaluate the ductility of given materials. A smaller G/B value indicates toughness while a larger G/B value shows brittleness, and G/B = 0.57 is the critical value between toughness and brittleness. The G/B values of Co 2 Zr and Co 2 Ti that change by pressure (0~100 GPa) at 0 K are given in Tables 3 and 4. For Co 2 Zr crystal, the G/B values are abnormal at 10 and 30 GPa, which were 0.718 and 0.635, respectively, both over 0.57, which illustrates the Co 2 Zr crystals are brittle at 10 and 30 GPa. Under other pressures, we can find that the G/B decreases continuously and is far less than 0.57 with the increase in pressure, indicating the toughness of Co 2 Zr increases with the increment of pressure except at 10 and 30 GPa. For Co 2 Ti, G/B = 0.528, Co 2 Ti is apt to ductile at 0 K and 0 GPa. The G/B is in proportion to the pressure between 10 and 60 GPa, and the G/B is equal to 0.559 at the pressure of 60 and 70 GPa. When the pressure is greater than 70 GPa, the G/B is inversely proportional to the pressure, which manifests in the brittleness of Co 2 Ti increasing with the increase in pressure when the pressure is less than uniaxial 60 GPa while the brittleness decreases with the pressure when the pressure is higher than 70 GPa. Comparing Tables 3 and 4, the G/B of Co 2 Ti is always larger than that of Co 2 Zr except when the pressure is 10 and 30 GPa, which illustrates that the toughness of Co 2 Ti is weaker than that of Co 2 Zr. Additionally, with its G/B value fluctuating around 0.55, the ductility of Co 2 Ti varies little with pressure, being situated between toughness and brittleness.
Young's modulus presents the resistance of materials against tension. However, Poisson's ratio is defined as the ratio of transverse contraction strain versus longitudinal extension strain when materials are stretched. Poisson's ratio makes known the bonding properties within materials. Haines et al. [42] put forward that Poisson's ratio for pure covalent compounds is small (usually 0.1); for typical metals, the value is 0.33; for the ionic covalent crystal, it is between 0.2 and 0.3. The values of Co 2 Ti are all over 0.3, which also confirms the metallic characteristic of Co 2 Zr in the pressure range from approximately 0 to 100 GPa, whereas, the values of Co 2 Zr are larger than 0.33, expect for at 10 and 30 GPa, indicating the Co 2 Zr is metallic but is an ionic covalent crystal at 10 and 30 GPa.
Anisotropy refers to a property that all or part of the physical or chemical properties of materials varies in direction, exhibiting the divergence in different directions. As an important property of crystal, anisotropy has significant value as it performs specifically in the elastic modulus, hardness, fracture resistance, yield strength, thermal expansion coefficient, thermal conductivity and so on. In the study of the anisotropy of crystals, we often use a dimensionless parameter A to represent it, that is, A = [(C 12 + 2C 44 )/C 11 ] − 1 [38]. The anisotropic parameter A of Co 2 Zr and Co 2 Ti at 0 K and various pressures (0-100 GPa) has been given in the Tables 3 and 4. Table 3 shows that when the pressure is 20 GPa, there is a abnormal anisotropic parameter A = 0.219 in Co 2 Zr. Except for 20 GPa, we can find that the anisotropy reaches its highest degree at the pressure of 30 GPa. When less than 30 GPa, the degree of anisotropy increases with the increasing of pressure while the trend is contrary when the pressure is more than 30 GPa. In addition, the degrees of anisotropy are equal at 20 and 70 GPa. The anisotropy of Co 2 Ti varies regularly without any abnormal conditions. From 0 to 70 GPa, the degree of anisotropy increases with the increasing of pressure, whereas that is quite contrary above 70 GPa. We know that the anisotropy of Co 2 Zr is larger than that of Co 2 Ti only at 10 and 30 GPa by comparing Tables 3 and 4. Finally, Figure 1a,b exhibit variation curves of elastic constants C ij , elastic modulus E, shear modulus G and adiabatic bulk modulus B S with pressure of these two crystals at 0 K. The elastic constant C 11 stands for the elasticity in length and the longitudinal strain causes changes of it while C 12 and C 44 dominated by transverse strain, respectively, represent the elasticity in shape [31]. Figure 1a shows that in Co 2 Zr, the changes in C 11 and C 12 with pressure are not monotonic, with the figures under 10 and 30 GPa lower than adjacent pressure. So, it is possible that the physical properties of Co 2 Zr become abnormal, the reason for which is unknown under the present study. In addition, we can find that C 44 increases monotonously with pressure. In all, however, as the pressure increases, C 11 rises faster than C 12 and C 44. Hence, C 11 of Co 2 Zr is more sensitive to pressure than C 12 and C 44 . However, the C 11, C 12, C 44 of Co 2 Ti are nearly insensitive to the pressure because the change curves of Appl. Sci. 2020, 10, 2097 7 of 15 these three elastic constants are approximately the same, and when the pressure is greater than 20 GPa, C 11 , C 12 , C 44 tend to be stable, where C 11 is much larger than C 12 and C 44 . So, the effect of pressure on the shape and length of Co 2 Ti is very small.
Co2Zr become abnormal, the reason for which is unknown under the present study. In addition, we can find that C44 increases monotonously with pressure. In all, however, as the pressure increases, C11 rises faster than C12 and C44. Hence, C11 of Co2Zr is more sensitive to pressure than C12 and C44. However, the C11, C12, C44 of Co2Ti are nearly insensitive to the pressure because the change curves of these three elastic constants are approximately the same, and when the pressure is greater than 20 GPa, C11, C12, C44 tend to be stable, where C11 is much larger than C12 and C44. So, the effect of pressure on the shape and length of Co2Ti is very small.

Thermodynamic Properties
The thermodynamic properties of Co2Zr and Co2Ti are analyzed with plane wave pseudopotential density functional theory (DFT) based on quasi-harmonic Debye model (QHDM) [37] and generalized gradient approximation (GGA) method, concentrating on their Debye temperature Θ, Heat capacity CV and thermal expansion coefficient α. The quasi-harmonic Debye model (QHDM) is used to study the physical properties of intermetallic compounds under high temperature [24,38]. Figure 2 shows that the V/V0 of Co2Zr and Co2Ti are functions of pressure at 0, 500, 1000, and 1500 K. Under a certain temperature, pressure increases while the V/V0 decreases monotonously. When the pressure is over 10 GPa, Co2Zr decreases faster than Co2Ti. When P<10 GPa, the two crystals have the same V/V0, and will not be affected by temperature. When P > 10 GPa, V/V0 decreases with increasing temperature at a certain pressure but the amount of variation is very small. What is more, the V/V0 of Co2Ti is always larger than that of Co2Zr, which indicates the pressure has a greater impact on volume of Co2Ti than that of Co2Zr. Figure 3 illustrates the relationship between Bulk modulus B and pressure at a certain temperature (0 and 1500 K), as well as the relationship of bulk modulus B and temperature at 0 GPa. As shown in Figure 3, under the same temperature and pressure, the B of Co2Ti is always larger than Co2Zr. With the increasing of pressure, the B of these two crystals increases in the same trend under a certain temperature. At a certain pressure, the B value is inversely proportional to the temperature, and Co2Ti fells more drastically than Co2Zr does within the same temperature difference. Therefore, for both Co2Zr and Co2Ti, increasing the pressure and decreasing the temperature have the same effect on the B value.

Thermodynamic Properties
The thermodynamic properties of Co 2 Zr and Co 2 Ti are analyzed with plane wave pseudo-potential density functional theory (DFT) based on quasi-harmonic Debye model (QHDM) [37] and generalized gradient approximation (GGA) method, concentrating on their Debye temperature Θ, Heat capacity C V and thermal expansion coefficient α. The quasi-harmonic Debye model (QHDM) is used to study the physical properties of intermetallic compounds under high temperature [24,38]. Figure 2 shows that the V/V 0 of Co 2 Zr and Co 2 Ti are functions of pressure at 0, 500, 1000, and 1500 K. Under a certain temperature, pressure increases while the V/V 0 decreases monotonously. When the pressure is over 10 GPa, Co 2 Zr decreases faster than Co 2 Ti. When P<10 GPa, the two crystals have the same V/V 0 , and will not be affected by temperature. When P > 10 GPa, V/V 0 decreases with increasing temperature at a certain pressure but the amount of variation is very small. What is more, the V/V 0 of Co 2 Ti is always larger than that of Co 2 Zr, which indicates the pressure has a greater impact on volume of Co 2 Ti than that of Co 2 Zr. Figure 3 illustrates the relationship between Bulk modulus B and pressure at a certain temperature (0 and 1500 K), as well as the relationship of bulk modulus B and temperature at 0 GPa. As shown in Figure 3, under the same temperature and pressure, the B of Co 2 Ti is always larger than Co 2 Zr. With the increasing of pressure, the B of these two crystals increases in the same trend under a certain temperature. At a certain pressure, the B value is inversely proportional to the temperature, and Co 2 Ti fells more drastically than Co 2 Zr does within the same temperature difference. Therefore, for both Co 2 Zr and Co 2 Ti, increasing the pressure and decreasing the temperature have the same effect on the B value. Appl. Sci. 2020, 10, x FOR PEER REVIEW 8 of 15  In the Debye model, the vibration of the elastic wave of the crystal is the main factor contributing to the heat capacity that reflects the ability to absorb heat from the surrounding environment. Figure  4a-c, respectively, show how the CV of Co2Zr and Co2Ti vary with temperature under certain pressures (0, 50, and 100 GPa). Due to the anharmonic approximations of the Debye model used, the CV values of Co2Zr and Co2Ti rely on temperature and pressure when T < 1300 K and a small amount of temperature difference will cause a great change in heat capacity. At a high temperature, the curve tends to be horizontal, which means that the heat capacity is close to constant, and the effects of temperature and pressure can be ignored. The heat capacity increases with the increasing temperature, but decreases with the increasing pressure, which indicates that the decrease in temperature and the increase in pressure have the same effect on the heat capacity of CV. In addition, under the same conditions, the heat capacity of Co2Zr is always greater than Co2Ti.  In the Debye model, the vibration of the elastic wave of the crystal is the main factor contributing to the heat capacity that reflects the ability to absorb heat from the surrounding environment. Figure  4a-c, respectively, show how the CV of Co2Zr and Co2Ti vary with temperature under certain pressures (0, 50, and 100 GPa). Due to the anharmonic approximations of the Debye model used, the CV values of Co2Zr and Co2Ti rely on temperature and pressure when T < 1300 K and a small amount of temperature difference will cause a great change in heat capacity. At a high temperature, the curve tends to be horizontal, which means that the heat capacity is close to constant, and the effects of temperature and pressure can be ignored. The heat capacity increases with the increasing temperature, but decreases with the increasing pressure, which indicates that the decrease in temperature and the increase in pressure have the same effect on the heat capacity of CV. In addition, under the same conditions, the heat capacity of Co2Zr is always greater than Co2Ti. In the Debye model, the vibration of the elastic wave of the crystal is the main factor contributing to the heat capacity that reflects the ability to absorb heat from the surrounding environment. Figure 4a-c, respectively, show how the C V of Co 2 Zr and Co 2 Ti vary with temperature under certain pressures (0, 50, and 100 GPa). Due to the anharmonic approximations of the Debye model used, the C V values of Co 2 Zr and Co 2 Ti rely on temperature and pressure when T < 1300 K and a small amount of temperature difference will cause a great change in heat capacity. At a high temperature, the curve tends to be horizontal, which means that the heat capacity is close to constant, and the effects of temperature and pressure can be ignored. The heat capacity increases with the increasing temperature, but decreases with the increasing pressure, which indicates that the decrease in temperature and the increase in pressure have the same effect on the heat capacity of C V . In addition, under the same conditions, the heat capacity of Co 2 Zr is always greater than Co 2 Ti. In the Debye model, the Debye temperature Θ often reflects the vibrational frequency of crystal atoms [38]. Figure 5 illustrates the change curve of Debye temperature Θ that varies with pressure at the given temperature (0, 1500 K). As the pressure goes up, the Debye temperatures of two crystals increase monotonously in the similar trend. Yet the effect of temperature is much less from 0 to 1500 K as the amount of variation of Debye temperature is very small, and the amount of variation decreases continuously with increasing pressure. If the pressure is stable, no matter how temperature changes, the Debye temperature of Co2Ti is always greater than that of Co2Zr, as shown in Table 3. As a vital physical property of the materials, the thermal expansion can be used to investigate the various issues of related solid state. The thermal expansion coefficient α indicates the ability of an object to expand and shrink due to temperature changes. Figure 6a,b show the thermal expansion α as a function of temperature T at different pressures (10, 30, and 90 GPa) and as a function of In the Debye model, the Debye temperature Θ often reflects the vibrational frequency of crystal atoms [38]. Figure 5 illustrates the change curve of Debye temperature Θ that varies with pressure at the given temperature (0, 1500 K). As the pressure goes up, the Debye temperatures of two crystals increase monotonously in the similar trend. Yet the effect of temperature is much less from 0 to 1500 K as the amount of variation of Debye temperature is very small, and the amount of variation decreases continuously with increasing pressure. If the pressure is stable, no matter how temperature changes, the Debye temperature of Co 2 Ti is always greater than that of Co 2 Zr, as shown in Table 3.  In the Debye model, the Debye temperature Θ often reflects the vibrational frequency of crystal atoms [38]. Figure 5 illustrates the change curve of Debye temperature Θ that varies with pressure at the given temperature (0, 1500 K). As the pressure goes up, the Debye temperatures of two crystals increase monotonously in the similar trend. Yet the effect of temperature is much less from 0 to 1500 K as the amount of variation of Debye temperature is very small, and the amount of variation decreases continuously with increasing pressure. If the pressure is stable, no matter how temperature changes, the Debye temperature of Co2Ti is always greater than that of Co2Zr, as shown in Table 3. As a vital physical property of the materials, the thermal expansion can be used to investigate the various issues of related solid state. The thermal expansion coefficient α indicates the ability of an object to expand and shrink due to temperature changes. Figure 6a,b show the thermal expansion α as a function of temperature T at different pressures (10, 30, and 90 GPa) and as a function of As a vital physical property of the materials, the thermal expansion can be used to investigate the various issues of related solid state. The thermal expansion coefficient α indicates the ability of an object to expand and shrink due to temperature changes. Figure 6a,b show the thermal expansion α as a function of temperature T at different pressures (10, 30, and 90 GPa) and as a function of pressure P at different temperatures (500, 1000, and 1500 K), respectively. From Figure 6a, the thermal expansion coefficient α of Co 2 Zr increases fast with increasing temperature when T < 700 K and the variation of it with temperature T is like that of C V . When T > 700 K, the thermal expansion coefficient α approaches a linear increase but the amount of increase is small, indicating that α is close to a constant. The thermal expansion coefficient α of Co 2 Zr is larger than that of Co 2 Ti at low temperatures while it is opposite at high temperatures under the same pressure. From Figure 6b, it is seen that the thermal expansion coefficient α of Co 2 Zr and Co 2 Ti decreases exponentially with increasing pressure and the variation of it with pressure P is similar with that of V/V 0 , which means that there is a large thermal expansion at low pressure. Moreover, the influence of pressure on thermal expansion coefficient is almost equal at high and low temperatures.
Appl. Sci. 2020, 10, x FOR PEER REVIEW 10 of 15 pressure P at different temperatures (500, 1000, and 1500 K), respectively. From Figure 6a, the thermal expansion coefficient α of Co2Zr increases fast with increasing temperature when T < 700 K and the variation of it with temperature T is like that of CV. When T > 700 K, the thermal expansion coefficient α approaches a linear increase but the amount of increase is small, indicating that α is close to a constant. The thermal expansion coefficient α of Co2Zr is larger than that of Co2Ti at low temperatures while it is opposite at high temperatures under the same pressure. From Figure 6b, it is seen that the thermal expansion coefficient α of Co2Zr and Co2Ti decreases exponentially with increasing pressure and the variation of it with pressure P is similar with that of V/V0, which means that there is a large thermal expansion at low pressure. Moreover, the influence of pressure on thermal expansion coefficient is almost equal at high and low temperatures.

Electronic Properties
To find origins of band structures, the total density of states and partial density of states are calculated, as shown in Figure 7. From Figure 7a,c, the main bonding peak between −2.5eV and −0.5eV are dominated by Co-d and Zr-p, Zr-d for Co2Zr (see Figure 7a), and between −3.5 eV and −0.5eV are contributed by Co-d and Ti-p, Ti-d for Co2Ti (see Figure 7d) at 0 GPa. It is illustrated that pressure has little effect on the density of states of Co2Zr and Co2Ti in Figure 7b,c,e,f. However, the energy dispersion spectrum crosses the Fermi level (Ef = 0 eV), indicating Co2Zr and Co2Ti compounds are both electrical conductors. In addition, there are two spikes on both sides of the Fermi level where the value of DOS (density of states) is not zero, and the width between two spikes represents the pseudo gap. The wider the pseudo gap is, the stronger covalence is. Therefore, Co2Zr has greater covalence than Co2Ti.
Electron density difference can be used to study transfer and accumulation of electrons when atoms form compounds. Positive values indicate an increase in electron density while negative ones show a loss of electron density. To see the electron transfer and bonding property of Co2Zr and Co2Ti, we calculated the electron density difference in the (110) plane of Co2Zr and Co2Ti, as shown in Figure  8 at 0, 10 and 30 GPa. It is seen that the atoms at the four vertices of the (110) plane of two crystals form metal ions with losing electrons while electrons accumulate at the center of the plane to form electron clouds. There are no electrons around the rest of the atoms, and they exist in crystalline atomic form. Therefore, there are both metal ions and metal atoms in the two crystals, indicating that both Co2Zr and Co2Ti are metal crystals. With the increase in pressure, the degree of electron accumulation deepens. Comparing the electron density difference of two substances at the same pressure, the gain and loss of electron capacity of Co2Ti is stronger than that of Co2Zr, which proves that the covalent property of Co2Zr is stronger than that of Co2Ti.

Electronic Properties
To find origins of band structures, the total density of states and partial density of states are calculated, as shown in Figure 7. From Figure 7a,c, the main bonding peak between −2.5 eV and −0.5 eV are dominated by Co-d and Zr-p, Zr-d for Co 2 Zr (see Figure 7a), and between −3.5 eV and −0.5 eV are contributed by Co-d and Ti-p, Ti-d for Co 2 Ti (see Figure 7d) at 0 GPa. It is illustrated that pressure has little effect on the density of states of Co 2 Zr and Co 2 Ti in Figure 7b,c,e,f. However, the energy dispersion spectrum crosses the Fermi level (E f = 0 eV), indicating Co 2 Zr and Co 2 Ti compounds are both electrical conductors. In addition, there are two spikes on both sides of the Fermi level where the value of DOS (density of states) is not zero, and the width between two spikes represents the pseudo gap. The wider the pseudo gap is, the stronger covalence is. Therefore, Co 2 Zr has greater covalence than Co 2 Ti.
Electron density difference can be used to study transfer and accumulation of electrons when atoms form compounds. Positive values indicate an increase in electron density while negative ones show a loss of electron density. To see the electron transfer and bonding property of Co 2 Zr and Co 2 Ti, we calculated the electron density difference in the (110) plane of Co 2 Zr and Co 2 Ti, as shown in Figure 8 at 0, 10 and 30 GPa. It is seen that the atoms at the four vertices of the (110) plane of two crystals form metal ions with losing electrons while electrons accumulate at the center of the plane to form electron clouds. There are no electrons around the rest of the atoms, and they exist in crystalline atomic form. Therefore, there are both metal ions and metal atoms in the two crystals, indicating that both Co 2 Zr and Co 2 Ti are metal crystals. With the increase in pressure, the degree of electron accumulation deepens. Comparing the electron density difference of two substances at the same pressure, the gain and loss of electron capacity of Co 2 Ti is stronger than that of Co 2 Zr, which proves that the covalent property of Co 2 Zr is stronger than that of Co 2 Ti. Appl. Sci. 2020, 10,

Conclusions
In this work, the elastic, thermodynamic and electronic properties of intermetallic compounds Co2Zr and Co2Ti in cubic structure are investigated and compared by first principles calculations. Detailed structural information of Co2Zr and Co2Ti are presented, and the optimized lattice constant, crystal volume, is in accordance with the available experimental value. The elastic constants, bulk modulus B, shear modulus G, anisotropy, ductility, heat capacity CV, Debye temperature, thermal expansion coefficient α are functions of pressure and temperature. Co2Zr is always stable while Co2Ti is unstable when the pressure is more than 80 GPa, and it is possible that a transition phase exists under the pressure of 85 GPa. Co2Zr crystals are brittle at 10 and 30 GPa, whereas the toughness of Co2Zr increases with the increment of pressure except at 10 and 30 GPa. The toughness of Co2Ti is weaker than that of Co2Zr and reaches the highest point at 60 and 70 GPa. It is also found that the anisotropy of Co2Zr and Co2Ti reaches its highest degree at 30 and 70 GPa. Moreover, the anisotropy of Co2Zr is larger than that of Co2Ti only at 10 and 30 GPa. The thermodynamic properties are also

Conclusions
In this work, the elastic, thermodynamic and electronic properties of intermetallic compounds Co 2 Zr and Co 2 Ti in cubic structure are investigated and compared by first principles calculations. Detailed structural information of Co 2 Zr and Co 2 Ti are presented, and the optimized lattice constant, crystal volume, is in accordance with the available experimental value. The elastic constants, bulk modulus B, shear modulus G, anisotropy, ductility, heat capacity C V , Debye temperature, thermal expansion coefficient α are functions of pressure and temperature. Co 2 Zr is always stable while Co 2 Ti is unstable when the pressure is more than 80 GPa, and it is possible that a transition phase exists under the pressure of 85 GPa. Co 2 Zr crystals are brittle at 10 and 30 GPa, whereas the toughness of Co 2 Zr increases with the increment of pressure except at 10 and 30 GPa. The toughness of Co 2 Ti is weaker than that of Co 2 Zr and reaches the highest point at 60 and 70 GPa. It is also found that the anisotropy of Co 2 Zr and Co 2 Ti reaches its highest degree at 30 and 70 GPa. Moreover, the anisotropy of Co 2 Zr is larger than that of Co 2 Ti only at 10 and 30 GPa. The thermodynamic properties are also predicted successfully. Debye temperature is a function of pressure and increases monotonically with increasing pressure. The Debye temperature of Co 2 Zr is always larger than that of Co 2 Ti at various temperatures under the same pressure. Both the heat capacity C V and thermal expansion coefficient α of Co 2 Zr and Co 2 Ti increase non-linearly with increasing temperature and decrease monotonically with increasing pressure, showing that the effect of increasing temperature is just the same as that of decreasing pressure. In addition, Co 2 Zr always has larger heat capacity C V than Co 2 Ti under same condition. The thermal expansion coefficient α of Co 2 Zr is larger than that of Co 2 Ti at low temperature, while it is the opposite at high temperature under same pressure. Finally, the electron density difference and density of states of Co 2 Zr and Co 2 Ti are investigated. What is more, the total density of states and partial density of states are calculated to find origins of band structures. Co 2 Zr and Co 2 Ti are all metal crystals, but Co 2 Zr has greater covalence than Co 2 Ti.