Study on the Melting Temperature, the Jumps of Volume, Enthalpy and Entropy at Melting Point, and the Debye Temperature for the BCC Defective and Perfect Interstitial Alloy WSi under Pressure

Study on the Melting Temperature, the Jumps of Volume, Enthalpy and Entropy at Melting Point, and the Debye Temperature for the BCC Defective and Perfect Interstitial Abstract: The objective of this study is to determine the analytic expressions of the Helmholtz free energy, the equilibrium vacancy concentration, the melting temperature, the jumps of volume, enthalpy the mean nearest neighbor distance and entropy at melting point, the Debye temperature for the BCC defective, the limiting temperature of absolute stability for the crystalline state, and for the perfect binary interstitial alloy. The results obtained from the expressions are combined with the statistical moment method, the limiting condition of the absolute stability at the crystalline state, the Clausius–Clapeyron equation, the Debye model and the Gruneisen equation. Our numerical calculations of obtained theoretical results were carried out for alloy WSi under high temperature and pressure. Our calculated melting curve and relation between the melting temperature and the silicon concentration for WSi are in good agreement with other calculations. Our calculations for the jumps of volume, enthalpy and entropy, and the Debye temperature for WSi predict and orient experimental results in the future.

Different researchers have studied the dependence of the mechanical and thermodynamic properties of materials on the temperature (T), pressure (P), and concentration of components for these materials. It is known that point defects such as vacancies have important contributions to the properties of materials [18][19][20][21][22][23][24]. At the melting point, the equilibrium vacancy concentration of metals changes from 10 −4 to 10 −2 [20]; therefore, it has a significant influence on the thermodynamic quantities of crystals at high temperatures. W has very high melting temperatures and can provide fundamental information on the equilibrium vacancy concentrations in metals and alloy metals with structured Body-Centered-Cubic (BCC) and their temperature dependences [20][21][22]. At P = 0.1 MPa, W has a BCC structure with lattice constant (a), a = 3.1649 × 10 −10 m at T = 300 K and a melting point at T = 3690 K. The melting curve of W was studied by the optical method at P = 5 GPa and T = (4050 ± 200) K with dT/dP = 75 K/GPa [25] and up to P = 90 GPa and T~(4000 ± 100) K using the Laser-Heated Diamond Anvil Cell (LHDAC) for optical measurements [26].
In various studies, researchers determined the melting temperature (T m ) of a crystal only from the solid phase and applied the statistical moment method (SMM) [27][28][29][30][31][32][33]. From the SMM method, the stability temperature (T S ) can be determined at different pressures, and the corresponding calculations to find T m from T S , and then T S at crystalline state is defined by ∂P ∂V T s = 0. Then, the isothermal compressibility of the crystal is equal to infinite. Several SMM calculations are more consistent with experiments than those obtained from other calculations.
In this research, the melting temperature, the jumps of volume, enthalpy and entropy at the melting point, and the Debye temperature for the BCC defective and perfect binary interstitial alloy by combining the SMM, the limiting condition of the absolute stability of the crystalline state, the Clapeyron-Clausius equation, the Debye model and the Gruneisen equation are studied. The theoretical results are numerically performed for alloy WSi, and some of our calculated results are compared with experiments and other calculations. In our model, the interstitial alloy AB with the concentration c B << c A , was applied. In it, the A metal has atoms in the center and peaks of cubic, while the atoms of B are in the center of cubic. With the parameters k, γ 1 , γ 2 , γ, the cohesive energy u 0 with the sphere center at the position of B, radii r 1B and r 2B are determined as follows [1][2][3][4]29,33]:

Content of Research
where u iβ , n i , and r 1B = r 01B + y 0A 1 (T) are the displacements of the ith atom in the direction of β, α, β = x, y, z, α = β, the number of atoms on the ith coordination sphere with radius r i (i = 1, 2), and the nearest neighbor distance between the interstitial atom B and the main metal A in the alloy at temperature T; y 0A 1 (T) and r 01B are the displacements of atom A 1 (atom A in the body center of the cubic unit cell) from the equilibrium position at temperature T, determined from the minimum condition of the cohesive energy u 0B , and the nearest neighbor distance between the interstitial atom B and the main metal A in the alloy at temperature 0 K; ϕ AB is the interaction potential between atom A and atom B. The cohesive energy u 0 and the alloy parameters k, γ 1 , γ 2 , γ for atom A 1 (atom A in the body center of the cubic unit cell) in the approximation of three coordination spheres with the sphere center at the position of A 1 and the radius r 1A 1 of the third coordination sphere are determined as follows [1][2][3][4]29,33]: where r 1A 1 ≈ r 1B is the nearest neighbor distance between atom A 1 and the other atoms in the alloy. The u 0 , k, γ 1 , γ 2 , γ are the cohesive energy and the alloy parameters for atom A 2 (atom A in the peaks of the cubic unit cell) in the approximation of three coordination spheres, with A 2 and radius r 1A 2 of the third coordination sphere as follows [1][2][3][4]29,33]: where r 1A 2 = r 01A 2 + y 0B (T), r 01A 2 and u 0A 2 , y 0B (T) are the nearest neighbor distances between atom A 2 and the other atoms in the alloy, and is determined from the minimum condition of the cohesive energy in the displacement of atom B from the equilibrium position at temperature T. In Equations (6)- (15), u 0A , k A , γ 1A , γ 2A are the cohesive energy and the metal parameters for atom A in the clean metal A in the approximation of two coordination spheres with the sphere center at the position of A and radii r 1A and r 2A and have the following forms [29,33]: The equations of state for the BCC alloy AB at P and T, and at P and T = 0 K, respectively, are determined by the following relations [29,33]: where r 1, v, x = ω 2θ , θ = k Bo T, ω = k m , k Bo are the nearest neighbor distance between two atoms in the alloy, the volume of the cubic unit cell per atom, and the Boltzmann constant. If the form of the interaction potential between two atoms X (X = A, A 1 , A 2 , B) is known, from Equation (22) we can find the nearest neighbor distance between two r 01X (P,0) and the alloy parameters k X (P, 0), γ 1 (P, 0), γ 2 (P, 0), γ(P, 0) for atom X at T = 0 K and P. From that, we can determine the displacement y X (P, T) of atom X from the equilibrium position at T and P as follows [29,33]: The nearest neighbor distance r 1X (P, T) is given by the following relations [1][2][3][4]: The mean nearest neighbor distance r 1A (P, T) between two atoms A in the alloy is derived from the following expressions [1][2][3][4]: where r 01A (P, 0), r 01A (P, 0), r 1A (P, T), r 01A (P, 0) and c B are the mean nearest neighbor distances between two atoms A in the clean metal A at pressure P and temperature 0 K; the region containing the interstitial atom C of the alloy at P and T = 0 K, and the concentration of the interstitial atoms B and nearest neighbor distance between two atoms A in the alloy at P and T and at P and T = 0 K. The Helmholtz free energy of the BCC perfect interstitial alloy AB with the condition c B << c A (c A , c B , respectively, are the concentrations of atoms A and B) can be calculated with the following relations [1][2][3][4]: where N is the number of atoms in the alloy, ψ X is the Helmholtz free energy of atom X in the clean material X, and S C is the configurational entropy of the interstitial alloy AB.
The concentrations of atoms A, A 1 , A 2 and B are determined by the following relations: From the condition of absolute stability limit expressed as: and from the equation of state for the interstitial alloy AB expressed as: the absolute stability temperature for the crystalline state can be derived in the following expression [3,4,29,32,33]: where are the Gruneisen parameters of the alloy. The right side of Equation (30) must be determined at T S . By solving Equation (30) the value of T S can be obtained.
The melting temperature T m is derived from the absolute stability temperature T S by the following relation [3,4,29,32,33]: where where a AB , ∂u 0X ∂r 1X , γ T G are determined at T S (0). Temperature T S at pressure P can be calculated with the following relation [3,4]: Here (33) can be solved by the approximate iteration method applied at low pressure.
In the case of high pressure, temperature T m at pressure P is calculated by the following relation [3,4]: where T m (P), T m (0), G and B T are the melting T at P and at zero P, respectively; the bulk modulus and the isothermal elastic modulus are calculated by the following relations [1][2][3][4]29,33]: where ε is the strain of the alloy, and ν AB , ν A are the Poisson ratios of alloy AB and the main metal A, respectively. The equilibrium vacancy concentration n v of the alloy is determined from the minimum condition of the real Gibbs thermodynamic potential G R AB of the defective alloy AB, and has the following relation form: where n and g f vAB are the numbers of vacancies in the alloy and the change in the Gibbs thermodynamic potential when a vacancy is formulated, and is determined from the distribution of atomic concentrations c X as follows: where B X ≈ 1 + u 0X ψ X . Therefore, g f vX ≈ − u 0X 4 and according to ref. [34]: At constant P and constant interstitial atom concentration, the melting temperature T R m of a defective crystal is the function of the equilibrium vacancy concentration n v . Approximately, the following relations can be applied [2,34]: where α TX , α T , respectively, are the thermal expansion coefficients of atom X in alloy AB.
The jumping of volume at melting point for the alloy can be found from the following expression [35]: where ε is a constant depending on the nature of the alloy and we take the value 0.01 as for metal [36], and u = y = ∑ X c X y X is the mean displacement of the main metal atom A from the equilibrium position as in Equation (25). In order to determine the jumping of volume ∆v m at pressure P and temperature T, it is necessary to determine a AB , u at pressure P and temperature T. The alloy parameters k, γ are determined with respect to a AB at pressure P and temperature T.
After finding the melting temperature T m and the melting T m (P), we can calculate the slope of this curve, and the derivative ∂T m ∂P . If T m , ∂T m ∂P and ∆v m , are known, the jumping of enthalpy at melting point from the Clausius-Clapeyron equation can be derived according to the following relation: and the jumping of entropy at melting point: Firstly, we find the isothermal compressibility χ T of the BCC interstitial alloy AB according to Equation (35) and the thermal expansion coefficient α T of the alloy according to Equation (40). The specific capacity at a constant volume of the alloy is given by the following relations [2,29,33]: The Gruneisen parameter of the alloy has the following relation [29,33]: Then, from the mean nearest neighbor distance a AB the quantities V, χ T , α T , C V and γ G at pressure P and temperature T can be determined. For the BCC lattice, the following relation is known: Graf et al. [37] proposed the following expression for the Gruneisen parameter: where γ G = γ G (P, T), γ G0 = γ G (0, T) and q are material constants and q > 0. Therefore, by using the SMM, γ G , γ G0 , V V 0 can be calculated from a AB , and by using Equation (49) the value of q can be calculated.
According to the Debye model, the Gruneisen parameter is defined as follows: where ω D is the Debye frequency and T D is the Debye temperature. By substituting the Gruneisen parameter from Equation (47) into Equation (48) and taking the integration, we derived the dependence of the Debye temperature T D (P) at P on the Debye temperature T D0 , and the Gruneisen parameter γ G0 at zero P, as well as the volume ratio V V 0 [7]: The Debye temperature T D0 of the alloy at zero pressure is given by the following relation: The Debye frequency ω D (0, T) at zero pressure and temperature T is related to the Einstein frequency ω E (0, T) at zero pressure and temperature T by the following relation [38]: where k(0,T) is the harmonic parameter of the alloy at zero pressure and temperature T. Therefore, it can be obtained through the following expression: Equations (1)- (40) are used in our previous papers [1-9] on elastic, thermodynamic and melting properties of metals and interstitial alloys. Equations (41)-(52) only are used to study the jumps of volume, enthalpy and entropy, and the Debye temperature of metals. In this study, for the first time, we apply Equations (41)-(52) to study the jumps of volume, enthalpy and entropy, and the Debye temperature of the BCC interstitial alloy AB.

Numerical Results and Discussions for Alloy WSi
In order to study alloy WSi, we applied the Mie-Lennard-Jones (MLJ) pair interaction potential as follows [39]: where D, r 0 , m and n are the depths of potential well corresponding to the equilibrium distance-they are determined empirically. Then, the potential parameters for the interaction W-Si are determined by the following relations [28]: We find n W-Si, m W-Si by fitting the experimental data and the theoretical result for the Young modulus of the interstitial alloy WSi at room temperature. The Mie-Lennard-Jones potential's parameters for the interactions of W-W and Si-Si are given in Table 1. Our computed results are summarised from Tables 2-9 and are illustrated in Figures 1-9. We calculated the silicon concentration and pressure of the volume, the isothermal compressibility, the Gruneisen parameter and the Debye temperature, the thermal expansion coefficient, the specific heat at constant volume in Tables 2-7 and in Figures 1-6. According to our obtained results, for WSi at the same temperature and silicon concentration when pressure increases, the volume, the isothermal compressibility, the thermal expansion coefficient, the Gruneisen parameter decreases, the Debye temperature increases, and the specific heat at constant volume. For WSi at the same temperature and pressure when the silicon concentration increases, the volume, the specific heat at constant volume, the Gruneisen parameter increases, the thermal expansion coefficient, the Debye temperature decreases and the isothermal compressibility.                    Figure 8 shows the melting curve of WSi at cSi = 1% obtained from the perfect model and the SMM for the defective model [3]. In the range of press 0 GPa to P = 80 GPa, the m T of the perfect W0.99Si0.01 increases from T = 3 6088 K and the m T of the defective W0.99Si0.01 increases from T = 3383 K to T . For WSi at P = 0 obtained from the SMM for the perfect model [3], the SMM fective model [3] and CALPHAD [40]. . For WSi at P = 0 obtained from the SMM for the perfect model [3], the SMM for the defective model [3] and CALPHAD [40].

Conclusions
The analytic expressions for structural and thermodynamic quantities alloy parameters, the mean nearest neighbor distance, the melting temp Helmholtz free energy, the equilibrium vacancy concentration, the cohesive thalpy and entropy, the isothermal compressibility, the limiting temperatur stability for the crystalline state, the thermal expansion coefficient, the jump the heat capacity at constant volume, the Gruneisen parameter and the Deb ture for the defective and perfect binary interstitial alloy with a BCC stru rived by combining the limiting condition of the absolute stability of the cry with the statistical moment method, the Clapeyron-Clausius equation, the D and the Gruneisen equation. Our numerical calculations of the melting cu relation are carried out for alloy WSi under a pressure of up to P = 80 GPa lated melting curve and relation between the melting temperature and the centration for WSi are in good agreement with other calculations. Our cal the melting curve, the jumping of volume, enthalpy and entropy, and the perature for WSi predict and orient experimental results in the future.

Data Availability Statement:
The data that support the findings of this study are a the corresponding author upon reasonable request. According to Figure 7 [3], when c Si increases from 0 to 5.5%, the T m of WSi decreases (from 3810 to T = 2459 K) from the SMM for the perfect model, (from T = 3609 K to T = 2366 K) from the SMM for the defective model and from T = 3695 K to T = 2460 K from CALPHAD [40]. The melting slope dT m dc Si for WSi at zero pressure is 245.6 K/% from the SMM for the perfect alloy, 226 K/% from the SMM for the defective alloy and 224.5K/% from CALPHAD [40]. The SMM calculations for the melting slope of the defective alloy are in good agreement with the CALPHAD calculations. Figure 8 shows the melting curve of WSi at c Si = 1% obtained from the SMM for the perfect model and the SMM for the defective model [3]. In the range of pressure from P = 0 GPa to P = 80 GPa, the T m of the perfect W 0.99 Si 0.01 increases from T = 3564 K to T = 6088 K and the T m of the defective W 0.99 Si 0.01 increases from T = 3383 K to T = 5534 K.

Conclusions
The analytic expressions for structural and thermodynamic quantities such as the alloy parameters, the mean nearest neighbor distance, the melting temperature, the Helmholtz free energy, the equilibrium vacancy concentration, the cohesive energy, enthalpy and entropy, the isothermal compressibility, the limiting temperature of absolute stability for the crystalline state, the thermal expansion coefficient, the jumps of volume, the heat capacity at constant volume, the Gruneisen parameter and the Debye temperature for the defective and perfect binary interstitial alloy with a BCC structure are derived by combining the limiting condition of the absolute stability of the crystalline state with the statistical moment method, the Clapeyron-Clausius equation, the Debye model and the Gruneisen equation. Our numerical calculations of the melting curve and the relation are carried out for alloy WSi under a pressure of up to P = 80 GPa. Our calculated melting curve and relation between the melting temperature and the silicon concentration for WSi are in good agreement with other calculations. Our calculations for the melting curve, the jumping of volume, enthalpy and entropy, and the Debye temperature for WSi predict and orient experimental results in the future. Funding: This research received no external funding.

Data Availability Statement:
The data that support the findings of this study are available from the corresponding author upon reasonable request.

Conflicts of Interest:
The authors declare no conflict of interest.