Thermodynamic Properties and Equation of State for Tungsten

: A high-temperature equation of state for tungsten was constructed in this study using experimental data on its thermodynamic properties, thermal expansion, compressibility, and bulk compression modulus. The totality of experimental data were optimized by the temperature-dependent Tait equation over a pressure range from 0 up to 1000 kbar and over a temperature range from 20 K to the melting point. An extended Einstein model was used to describe the temperature dependence of thermodynamic and thermophysical parameters. A linear temperature dependence was embraced for the derivative of the isothermal bulk modulus. The resultant equation of state provides a good ﬁt to the whole set of experimental data within measurement uncertainties associated with individual quantities.


Introduction
Tungsten has normally a body-centered cubic (bcc) lattice and retains this structure at least up to 3640 kbar [1].Tungsten was theoretically predicted to transit into a hexagonal close-packed (hcp) modification at 12.5 Mbar [2] and 9.2 Mbar [3] at room temperature, and further into a face-centered cubic (fcc) phase at 14.4 Mbar [2,3].
Tungsten and its alloys are widely used in various science and technology fields.One needs to know the equation of state (EoS) in order to correctly describe the behavior of tungsten and its alloys in a wide range of pressures and temperatures.The experimental data from measurements of different properties of tungsten include measured isothermal compression, thermal expansion, and elastic constants at normal pressure.
A number of papers have been published in which equations of state for tungsten suitable for the estimation of its properties at temperatures ranging from room temperature to the melting point under pressures up to several Mbar were formulated using different approximations [4][5][6][7].Those works share common shortcomings as follows: the lack of a quantitative evaluation of the estimation error relative to the experimental data used; in most instances, only graphical comparisons are given; all the works only examine the temperature range above room temperature (298.15K) and do not cover the lowtemperature region.It should also be noted that those studies are based on a limited set of the existing experimental data and disregard more recent measurement results.
Therefore, the aim of the present study was to examine and analyze the currently available experimental data, bring the thermodynamic and thermophysical parameters into agreement, and formulate an equation of state for the bcc phase of tungsten.

Physicochemical Model
The optimization of the thermophysical and thermodynamic parameters of tungsten was performed taking account of the suggestions reported in [8] by using a model that approximates well and reproduces experimental data for lead [9], aluminum [10], and copper [11] in a wide temperature and pressure range.

Thermodynamic Functions
The thermodynamic properties of tungsten in the standard state were described in this study by the three-term Einstein equation having an additional correction power term to account for anharmonic effects.The following forms of thermodynamic functions at zero pressure were used: where T is the absolute temperature; H is the enthalpy; C P is the isobaric heat capacity; S is the entropy; Y i , θ i , h, and m are constants (fit parameters); and ∆S 0 is the integration constant.The molar Gibbs energy is defined by the common relation:

Molar Volume
The Tait equation was used for the description of the pressure-dependent molar volume of tungsten [12,13].Unlike the studies [9][10][11], and accounting for the very high melting temperature of tungsten (T m = 3687 ± 7 K [14]), a temperature dependence of the pressure derivative of the isothermal bulk modulus was incorporated into the equation.The final form of the equation is written as: where P is the pressure; V is the volume; B T and V T are the bulk modulus and the molar volume at temperature T and zero pressure; and n T is the pressure derivative of the bulk modulus.This study adopted a linear dependence of n T on temperature: To describe the thermal expansion of tungsten in a wide temperature interval, the following relationship similar to that for enthalpy (1) was used: where V T S and V 0 S are the molar volume at zero pressure and temperatures T and T = 0, respectively; X i , Θ i , g, and k are constants.It should be noted that an analogous three-term Einstein function but having another anharmonic term was employed in [15] to describe the thermal expansion of refractory metals, including tungsten.
Thus, the temperature-dependent molar volume in EoS (5) at the specified pressure is defined by the temperature dependences of V T , B T , and n T .

Isothermal Bulk Modulus
The function chosen in the studies [9][10][11] to describe lead, aluminum, and copper was employed in this study for the description of the isothermal bulk modulus of tungsten.This function is based on the equation that calculates the isothermal compressibility, as suggested in [16].To ensure that the adiabatic modulus behavior at high pressures is Crystals 2023, 13, 1470 3 of 14 realistic, three summands should be taken into account in summation.The final form of the expression is written as: where B T and B 0 are the bulk compression moduli at temperatures T and 0 K, respectively; w i and s i are constants.This equation has an advantage of providing an adequate description of the measured data at low temperatures.Moreover, this equation guarantees the non-negativity of the modulus at very high temperatures.

Thermodynamic Properties
The thermodynamic functions of tungsten are outlined in the reference literature and review papers [4,[17][18][19][20][21][22].Some studies examined only separate properties such as heat capacity and enthalpy [23] or only heat capacity [24,25].The comparative values of heat capacity from the cited works are outlined in Table 1.The isobaric heat capacity (C P ) values reported in the different literature sources are in good agreement with each other (the maximum difference not exceeding 0.3%) in the lowtemperature region (up to 300 K).The difference increases at higher temperatures, exceeding 20% near the melting point.The difference in the enthalpy values is considerably lower and is 0.03% in the low-temperature region, less than 0.4% in the mid-temperature region (up to 2600 K) and 1.5-1.8%near the melting temperature (except the earlier work [17] in which the difference attained 3.8%).
A more thorough review and analysis of the literature data on the thermodynamic properties of tungsten was carried out in the study [22] which was taken as the basis in this study.

Thermophysical Properties
The measured data on thermal expansion, adiabatic bulk modulus, and isothermal compressibility were used in this study to formulate an equation of state for solid tungsten.The isothermal bulk modulus involved in the equation of state is defined by the well-known relation [26]: where α is the bulk thermal expansion coefficient and B S is the adiabatic bulk modulus.

Molar Volume
In this study, the molar volume estimated by Ablaster [27], who carried out a critical analysis of 38 experimental measurements from the different literature sources, was used as the benchmark value: 9.548 ± 0.002 cm 3 •mol −1 (density 19.254 g•cm −3 ) at 293.15 K.The recalculation using the thermal expansion coefficient from the same study gave almost the same molar volume of 9.549 ± 0.002 cm 3 •mol −1 at standard temperature.

Thermal Expansion
The data on the thermal expansion of tungsten are presented in handbooks and review papers [4,5,15,24,25,[28][29][30][31][32][33][34].The details of the listed studies are given in Table 2.The discrepancy in the molar volume values reported in the above-listed works does not exceed 0.08% over a temperature range from 50 K to the melting point, except the study [4] in which the discrepancy was slightly higher and was 0.17% at the melting temperature (the data in study [5] were obtained from study [4]).At the same time, the difference in the thermal expansion coefficients reaches 11% up to 1000 K and exceeds 20% near the melting point.Therefore, this study employed the molar volumes from the paper [27], which values exhibit a good alignment with the other reference data.
In this study, the optimization was conducted within the pressure range from 0 up to 1000 kbar.To find the EoS parameters for tungsten, we used all the original measured data, except the data from [34,37].The measurement results reported in [34] were recalculated in the study [40] by using a more precise pressure calibration scale.The data from the study [37] were not utilized because of the overestimated (exaggerated) pressures compared to all the other experimental data available.This is probably associated with an inaccuracy of the equation of state for gold [44], which was used for pressure calibration.

Calculation Procedure
As an optimization criterion, the error function was used that represents a weighted root-mean-squared deviation: where N is the total number of experimental points; D i is the value of different parameters (molar volume, enthalpy, heat capacity, etc.); and w i is the weighting coefficients of these parameters.Indices c and m are the calculated and measured properties, respectively.The weighting coefficients were evaluated using relative measurement errors of different parameters.
The function was minimized by the Nelder-Mead simplex method for multidimensional minimization [52].

Results and Discussion
The EoS parameters obtained by the optimization are presented in Table 4.The comparison with experimental data is displayed in Figures 1-6.

Equation Parameter Value
Thermodynamic functions (1)-( 3)  The estimated heat capacity of tungsten is compared with the other literature data in Figures 1 and 2. The relationship obtained in the present study is a good approximation of the data reported in [22].The root-mean-square deviation (RMS) of the calculation from the experiment in the temperature range from 50 K to the melting temperature was 0.58%.This deviation of enthalpy was significantly lower: RMS = 0.12% for the whole temperature interval.
It should be noted that the calculated data in the low-temperature region are well consistent with those from all the literature sources outlined in Table 1.The calculation in the high-temperature region through to the melting point almost coincides with the data reported in [24,25] (the error less than 1%) and aligns with the works reported in [4,23] (the error not higher than 3.7%).For the studies [18][19][20][21], the calculation discrepancy from the presented values is considerably higher and reaches 10-18%.The highest discrepancy exceeding 20% is observed when compared to the study [17].Such differences between the calculated and reference data are explained by the different number of the literature sources considered and by the averaging method of the experimental values.The greatest number of primary experimental data were examined and critically reviewed in the studies [22,24,25] which agree excellently with each other and with the calculated data obtained in the present study.
Figures 3 and 4 depict the bulk thermal expansion coefficient of tungsten plotted versus temperature.The calculated relationship gives a good fit to the data reported in [27].The root-mean-square deviation (RMS) between the calculated data and the experiment at temperatures ranging from 20 K to the melting point was 0.66%.The molar volume calculation showed a significantly higher accuracy.The average absolute deviation was 0.001 cm 3 •mol -1 (RMS = 0.009%).The calculated molar volume at 298.15 K was 9.548 cm 3 •mol -1 , being almost no different from the benchmark value.At the temperature of 0 K, the estimated molar volume of 9.522 cm 3 •mol -1 differs from that reported in [53] (9.524 cm 3 •mol -1 ) by 0.02% and from that reported in [54] (9.5175 cm 3 •mol -1 ) by 0.05%.
The low-temperature coefficient of thermal expansion (Figure 3) is in agreement with all the data reviewed, except the study [29].This is associated with the fact that a small number of experimental data were averaged in the study [29] and exhibited a significant spread in addition.
Figures 3 and 4 depict the bulk thermal expansion coefficient of tungsten plotted versus temperature.The calculated relationship gives a good fit to the data reported in [27].The root-mean-square deviation (RMS) between the calculated data and the experiment at temperatures ranging from 20 K to the melting point was 0.66%.The molar volume calculation showed a significantly higher accuracy.The average absolute deviation was 0.001 cm 3 •mol −1 (RMS = 0.009%).The calculated molar volume at 298.15 K was 9.548 cm 3 •mol −1 , being almost no different from the benchmark value.At the temperature of 0 K, the estimated molar volume of 9.522 cm 3 •mol −1 differs from that reported in [53]  (9.524 cm 3 •mol −1 ) by 0.02% and from that reported in [54] (9.5175 cm 3 •mol −1 ) by 0.05%.
The low-temperature coefficient of thermal expansion (Figure 3) is in agreement with all the data reviewed, except the study [29].This is associated with the fact that a small number of experimental data were averaged in the study [29] and exhibited a significant spread in addition.
Figure 5 plots the molar volume of tungsten versus pressure at 298.15 K.The total number of experimental points is 177.The average absolute deviation of the calculation from the experiment was 0.005 cm 3 •mol −1 , RMS = 0.08%.The isothermal bulk modulus at 298.15 K was calculated to be 3072.8kbar, with a pressure derivative of 4.009, in agreement with the literature data set forth in Table 3, considering the different EoS types used.
Figure 6 plots the molar volume of tungsten versus pressure up to 350 kbar at different temperatures compared to the experimental data from [36].Since the experimental points at different temperatures are in very close proximity to each other, the figure displays only data at 300, 673, 1073, and 1473 K for visual clarity (the study [36] also performed measurements at 473, 873, 1273, and 1673 K).Moreover, the data at 298.15 K from the other authors are also included.The calculation and experiment are seen to match fairly well with each other.The least root-mean-square deviation of 0.04% was observed at 473 K and increased to 0.09% at 1673 K.
Figure 7 illustrates the bulk modulus plotted versus temperature.The temperature dependence of the adiabatic bulk modulus obtained in this study approximates well the measured data from [51] and agrees with the data reported in [48] within an error.Moreover, this temperature dependence almost coincides with the calculation results presented in [6].This indicates the adequacy of the model used in the present study, as compared to the studies [4,36] in which the calculated relationships differ noticeably from the experimental data.
It should be noted that the calculated data obtained in the present study and in the paper [6] differ noticeably from the low-temperature ones reported in [49,50].This might be due to an inaccuracy of the measurements performed in these works.This is evidenced by the incorrect behavior of the adiabatic bulk modulus.For instance, the adiabatic modulus reported in the work [49] begins to increase abruptly as the temperature tends to zero, whereas this parameter rises and then begins to diminish abruptly as the temperature increases from 0 K to 160 K in the study [50].(Vaidya (1970) [32], Ming (1978) [33], Litasov (2013) [36], Qi (2018) [39], Dewaele (2019) [40], Anzellini (2022) [41]).
It should be noted that the calculated data obtained in the present study and in the paper [6] differ noticeably from the low-temperature ones reported in [49,50].This might be due to an inaccuracy of the measurements performed in these works.This is evidenced by the incorrect behavior of the adiabatic bulk modulus.For instance, the adiabatic modulus reported in the work [49] begins to increase abruptly as the temperature tends to zero, whereas this parameter rises and then begins to diminish abruptly as the temperature increases from 0 K to 160 K in the study [50].

Conclusions
The obtained results show that the proposed model allows the description of the experimental data for tungsten in a wide range of pressures and temperatures within experimental measurement uncertainties.The high prediction accuracy is achieved for the pressure-dependent molar volume, the absolute deviation between the calculation and experiment being 0.005 cm 3 •mol −1 (RMS = 0.08%).An even higher accuracy is reached for the temperature-dependent molar volume (0.001 cm 3 •mol −1 , RMS = 0.009%) in the temperature range from 20 K to the melting point.The estimated isothermal bulk compression modulus at 298.15 K was 3072.8 kbar, with a derivative pressure of 4.009, which is consistent with the literature data.It worth noting here that the thermophysical and thermodynamic parameters of tungsten have been reconciled with each other by using the well-known thermodynamic relations in a pressure range from 0 up to 1000 kbar and in a temperature range from 20 K to the melting point.

Table 1 .
Results taken from basic handbooks and review papers on thermodynamic functions of tungsten (C P , J•mol −1 •K −1 ).

Table 2 .
Basic handbooks and review papers on thermal expansion data for tungsten.
Note: a-lattice parameter.Form of data presentation: T-Table, P-Plot, A-Approximation.

Table 3 .
Isothermal compressibility measurement data for tungsten.

Table 4 .
Summary of optimized EoS parameters for tungsten.