Theoretical Predictions for the Equation of State of Metal Nickel at Extreme Conditions

Abstract

A recently developed approach to the partition function with very high efficiency was applied to study the equation of state (EOS) of metal nickel (Ni) up to 3000 K and concurrently 500 GPa. The theoretical results agree very well with previous hydrostatic experiments at room temperature, and at high temperatures, the deviation of our calculated pressures from the latest hydrostatic experiments up to 109 GPa is less than 4.16%, 4.95%, and 5.53% at 1000, 2000, and 3000 K, respectively. Furthermore, an analytical EOS model with only two parameters was developed for common metals at high temperatures, and the analytical EOS of metal Ni was obtained to produce the map of pressure over the temperature–volume plane, which should be helpful to understand the thermodynamic properties of Ni-based alloys.

1. Introduction

In the design of new materials, theoretical predictions of the equation of state (EOS) without experimental data support is crucial to understand the thermodynamic properties. In fact, even for existing materials, the predictions are also necessary since experimental measurement of EOS still confronts some problems, especially at high pressures and high temperatures.

Taking nickel (Ni) as an example, its content is as large as 5% in the Earth’s core [1], where the temperature and pressure reaches 6000 K and 350 GPa, and Ni-based alloys have been widely applied in industries. However, rare experiments concern the EOS of Ni-based alloys, and even for the metal of pure Ni, hydrostatic measurements at room temperature are limited to 156 GPa [2,3,4,5] so far, and at high temperatures, there are only two sets of experiments with pulsed laser to heat the specimen, resulting in the uncertainty of the temperature measurements being as large as ~100 K above 1000 K [3,4], and the results of the two experiments are distinctly different.

According to the ensemble theory, the EOS and all the other thermodynamic properties of materials can be theoretically calculated from the partition function (PF) without any experimental measurement. However, this fundamental approach faces formidable computational challenges: for an N-atom system, a 3N-dimensional configuration integral is required to be solved, rendering it computationally intractable for realistic solids containing ~10²³ atoms. In spite of a century of research, this fundamental limitation persists in contemporary materials science.

Quasi-harmonic approximation (QHA) [6,7,8] has emerged as an effective compromise, achieving wide applications by simplifying lattice dynamics through harmonic vibrational modes. However, its inherent assumption of small atomic displacements becomes inadequate at elevated temperatures where an-harmonic effects dominate. In fact, a recent study [9] has quantitatively demonstrated that the accuracy of QHA deteriorates progressively with both increasing temperature and atomic volume. Alternative approaches have also been developed to circumvent this limitation. The particle-in-a-cell model (PCM) [10,11,12] reduces the 3N-dimensional integral into a 3-fold one. Nevertheless, achieving the 3-dimensional integral at the ab initio level remains a technical challenge. Moreover, PCM’s validity is restricted to the conditions above the Debye temperature [13]. A more radical simplification appears in the classical mean field (CMF) [14,15,16,17] method, which reduces the 3-fold integral in PCM into a one-fold spherical approximation. This approach fundamentally assumes isotropic atomic interactions, a significant oversimplification that introduces substantial errors for most crystalline metals. The anisotropic nature of interatomic forces in crystalline systems, particularly along different crystallographic directions, renders CMF’s spherical symmetry assumption physically inappropriate for accurate thermodynamic predictions.

Although the methods mentioned above could be valid in some cases [10,11,12,13,14,15,16,17], they are not universal, and it is unsure which system they are well applicable to. Distinctly different from these methods, a recently developed direct integral approach (DIA) [18] introduces a novel perspective by transforming the 3N-fold integration in PF into a 3N-dimensional effective volume, enabling the PF to be solved with ultra-high efficiency at the ab initio level with high precision. Up to now, DIA has been successfully employed to calculate the EOS and phase transitions of several metals. DIA without any artificial or empirical parameters is applicable to study the thermodynamic properties of any solids in principle. However, considering DIA is a newly developed approach, it needs to be examined by comparing with as many realistic systems as possible.

In the present work, we take Ni as a prototype, and apply DIA to calculate the EOS up to 3000 K and concurrently 500 GPa to further check the reliability of DIA, and simultaneously give theoretical predictions for the EOS at extreme conditions. In order to obtain an analytical EOS of the metal Ni, we applied the widely used Mie–Grüneisen–Debye (MGD) formalism [19,20,21] to fit our calculated results, showing that one of the three parameters in the MGD model, the Debye temperature, could be removed without losing any accuracy of the fitting. Furthermore, a much more simplified and universal model is developed to produce analytical EOS for common metals.

2. Calculation Method

From the ensemble theory, it is known that for a system containing N atoms occupying the spatial position r^N = {r₁, r₂, …, r_N}, the PF reads

$$Z={\displaystyle \frac{1}{N!}}{\left({\displaystyle \frac{2\pi m{k}_{\mathrm{B}}T}{h^2}}\right)}^{{\displaystyle \frac{3}{2}}N}{\displaystyle \int \mathrm{d}{r}^{\mathrm{N}}{\mathrm{e}}^{-U\left(r^{\mathrm{N}}\right)/k_{\mathrm{B}}T}},$$

(1)

in which U(r^N) is the interaction energy of the system, and m, h, k_B is the atomic mass, Planck constant, and Boltzmann constant, respectively. If the configurational integral

$$Q={\displaystyle \int \mathrm{d}{r}^{\mathrm{N}}{\mathrm{e}}^{-U\left(r^{\mathrm{N}}\right)/k_{\mathrm{B}}T}}$$

(2)

is solved, the pressure (P) as a function of V and T, i.e., the isothermal EOS could be obtained by

$$P=k_{\mathrm{B}}T{\displaystyle \frac{\partial \ln Z}{\partial V}},$$

(3)

and all the other thermodynamic quantities can also be achieved without any experimental support.

To solve the 3N-dimensional integral in Equation (2), an approach was developed recently by Ning et al. [18], and its essential framework is as follows.

For a crystal in an equilibrium state with the N atoms located at their ideal lattice sites R^N = {R₁, R₂, …, R_N} and the total energy of U₀(R^N), a transformation is firstly introduced,

$${r^{\prime }}^{\mathrm{N}}=q^{\mathrm{N}}-R^{\mathrm{N}},U^{\prime }\left({r^{\prime }}^{\mathrm{N}}\right)=U\left(r^{\mathrm{N}}\right)-U_0\left(R^{\mathrm{N}}\right),$$

(4)

so that r′^N represents the displacements of atoms away from their equilibrium positions. Then, the Q in Equation (2) is expressed as

$$Q={\mathrm{e}}^{-U_0/k_{\mathrm{B}}T}{\displaystyle \int \mathrm{d}{r^{\prime }}^{\mathrm{N}}{\mathrm{e}}^{-U^{\prime }\left({r^{\prime }}^{\mathrm{N}}\right)/k_{\mathrm{B}}T}}.$$

(5)

Here, we define a function

$$F\left({r^{\prime }}^{\mathrm{N}}\right)=U^{\prime }\left({r^{\prime }}^{\mathrm{N}}\right)-{\displaystyle \sum _{i=1}^N{\displaystyle \sum _{j=x,y,z}{U}^{\prime }\left(0\dots {r^{\prime }}_{ij}\dots 0\right)}},$$

(6)

and then the 3N-fold integral in Equation (5) turns into

$$Q={\mathrm{e}}^{-U_0/k_{\mathrm{B}}T}{\displaystyle \prod _{i=1}^N\left(L_{ix}{L}_{iy}{L}_{iz}\right)}{\mathrm{e}}^{-F\left({r^{\prime }}^{\mathrm{N}}\right)/k_{\mathrm{B}}T},$$

(7)

in which

$$\begin{array}{l}{L}_{ix}={\displaystyle \int {\mathrm{e}}^{-U^{\prime }\left(0\dots {r^{\prime }}_{ix}\dots 0\right)/k_{\mathrm{B}}T}\mathrm{d}{r^{\prime }}_{ix}},\\ L_{iy}={\displaystyle \int {\mathrm{e}}^{-U^{\prime }\left(0\dots {r^{\prime }}_{iy}\dots 0\right)/k_{\mathrm{B}}T}\mathrm{d}{r^{\prime }}_{iy}},\\ L_{iz}={\displaystyle \int {\mathrm{e}}^{-U^{\prime }\left(0\dots {r^{\prime }}_{iz}\dots 0\right)/k_{\mathrm{B}}T}\mathrm{d}{r^{\prime }}_{iz}},\end{array}$$

(8)

is defined as the effective length, and r′_ix(r′_iy or r′_iz) is the x (y or z) coordinate of the ith atom away from its ideal lattice site with simultaneously the other two degrees of freedom and all the other atoms fixed at the ideal lattice sites.

Equation (6) could be expanded by a Taylor series as

$$\begin{array}{c}F\left({r^{\prime }}^{\mathrm{N}}\right)=F\left(0^{\mathrm{N}}\right)+{\displaystyle \sum _{i=1}^N{\displaystyle \sum _{j=x,y,z}{\left(\frac{\partial F}{\partial {r^{\prime }}_{ij}}\right)}_0}}\Delta {r^{\prime }}_{ij}+{\displaystyle \sum _{i=1}^N{\displaystyle \sum _{j,k=x,y,z}\frac{1}{2}{\left(\frac{{\partial }^{2}F}{\partial {r^{\prime }}_{ij}\partial {r^{\prime }}_{ik}}\right)}_0}\Delta {r^{\prime }}_{ij}\Delta {r^{\prime }}_{ik}}+\dots \\ ={\displaystyle \sum _{i=1}^N{\displaystyle \sum _{j,k=x,y,z}^{j\ne k}\frac{1}{2}{\left(\frac{{\partial }^2{U}^{\prime }\left({r^{\prime }}^{\mathrm{N}}\right)}{\partial {r^{\prime }}_{ij}\partial {r^{\prime }}_{ik}}\right)}_0\Delta {r^{\prime }}_{ij}\Delta {r^{\prime }}_{ik}}}+\dots \end{array}$$

(9)

Clearly, if U′(r′^N) changes slow enough with respect to r′^N, ∂²U′(r′^N)/∂r′_ij∂r′_ik and further F(r′^N) would approach zero, so that the Q of Equation (7) is close to

$$Q={\mathrm{e}}^{-U_0/k_{\mathrm{B}}T}{\displaystyle \prod _{i=1}^N\left(L_{ix}{L}_{iy}{L}_{iz}\right)}.$$

(10)

To ensure Equation (10) is a good approximation, the axes of a Cartesian coordinate system should be set as the direction along which U′(r′^N) changes slowest.

For face-centered cubic metal Ni, if the lattice orientations [100], [010], and [001] are selected as the axes of a Cartesian system, the variations of U′(r′^N) along all the three axes are the slowest. Thus, L_x is equal to L_y and L_z, so that

$$Q={\mathrm{e}}^{-U_0/k_{\mathrm{B}}T}{\left(L_x\right)}^{3N}.$$

(11)

As shown in Figure 1, a perfect 3 × 3 × 3 supercell containing 27 atoms was constructed for calculating the PF of metal Ni. For a given atomic volume, an arbitrary atom was moved along the [100] direction step by step until the U′(x′) in Equation (4) is larger than 3.5 eV, ensuring convergence of the effective length up to 3000 K. The cubic spline interpolation algorithm [22] is used to smooth U′(x′), then L_x was calculated via Equation (8), following which the corresponding PF could be gained. As shown in Figure 1b, by increasing the lattice constant from 0.81 a₀ to 1.03 a₀ (a₀ = 3. 5236 Å), the PF at 15 volumes can be obtained. For a given temperature, the cubic spline algorithm was again used to fit the PF versus volume data, and from the first-order derivative of the PF as a function of the volume we can obtain the pressure at arbitrary volume.

First principles calculations were performed using the Vienna Ab Initio Simulation Package (VASP) [23,24] to obtain U′(x′), and the computational parameters were set as follows: (1) the projector augmented wave (PAW) pseudopotential [25,26] is used for describing the electron–ion interactions with 3d⁹4s¹ selected as the valence orbitals; (2) the Perdew–Burke–Ernzerhof (PBE) [27] exchange–correlation functional within the generalized gradient approximation (GGA) framework is employed to describe the electron–electron interactions; (3) the electronic self-consistency criterion is set as 10⁻⁶ eV; (4) a Γ-centered 15 × 15 × 15 k-mesh generated via the Monkhorst–Pack method [28] is adopted to sample the Brillouin zone; (5) the tetrahedron method with Blöchl corrections [25] is used to determine the electron orbital partial occupancy with the plane-wave cutoff energy set as 400 eV; (6) although the magnetic moments of metal Ni were experimentally measured as 0.61 μ_B at room temperature [29], tests in our calculations showed that 0.61, or 1 μ_B has negligible effects on the calculated isothermal P-V relationship. For convenience, the tolerant 1 μ_B was adopted in the present ab initio calculations. The convergence of all these parameters was verified across all the considered volumes with effective length fluctuating below 10⁻⁵ Å.

3. Results and Discussion

3.1. The Room Temperature Isotherm

Based on DIA, the room temperature P-V relationship of Ni up to 500 GPa is obtained and shown as the red solid line in Figure 2, which includes four sets of experimental data [2,3,4,5] and the results are calculated by QHA [30]. It can be roughly seen that the calculations of both QHA and DIA agree well with the experiments. To specifically inspect the feasibility of DIA, under the same volume, differences between the pressure calculated by DIA (P_DIA) or QHA (P_QHA) and the experimental measurement (P_EXP) are conducted and presented in Figure 3.

In the experiments of Ref. [2], two groups of pressure measurements were provided under two calibrations of the ruby gauge, one of which was established by Mao et al. in 1986 [31], and the other is from Dewaele et al. in 2004 [32]. However, several works [33,34,35,36] have demonstrated that the ruby standard by Mao et al. [31] always underestimates pressures, so we chose the data calibrated by the one of Dewaele et al. to make comparisons with our calculations [32]. As shown in Figure 3a, P_DIA is larger than P_EXP in the whole pressure range of 0–156.6 GPa, and the difference increases progressively with increasing pressure to ~11 GPa, corresponding to a relative deviation (|P_DIA − P_EXP|/P_EXP) of 7.3%. On the other hand, P_QHA seems much better than P_DIA, and the largest deviation of P_QHA from P_EXP is about 2.15 GPa, whereas the ruby standard by Dewaele et al. [32] was still proven to underestimate the pressure by subsequent studies of Refs. [36,37]. Under this situation, employing the ruby scale built in Ref. [37] we re-calibrated the pressures in the experiments of Ref. [2]. As shown by the olive stars and magenta diamonds in Figure 3a, the largest deviation of P_DIA from the modified P_EXP decreases to 6.88 GPa. Meanwhile, the difference between P_QHA and the re-calibrated P_EXP increases to 3.38 GPa. Evidently, the experimental pressures are directly determined by the accuracy of the ruby scale. In fact, relatively comprehensive parameters of the different ruby gauges developed before 2020 were summarized in Ref. [38], and the authors suggested that a 2.5% uncertainty of the ruby scales should be accepted by the practitioners. Actually, the ruby gauge, as an international practical pressure scale, has been constantly studied, and a new developed one may be built in the near future [39].

As shown in Figure 3b, the differences between P_DIA and the P_EXP of Ref. [3] distribute evenly around 0, and all the values of |P_EXP − P_QHA| are less than 3 GPa, except for the one at 22.1 GPa, which should result from the accidental error in the experiments since the P-V curve from DIA is continuously progressive. Comparatively, nearly all the P_QHA are smaller than the P_EXP, and present an average shift of 2.85 GPa.

A larger range of diamond anvil cell (DAC) compressions of Ni up to 98 GPa were reported in Ref. [4]. As shown in Figure 3c, although the deviations of P_DIA from the P_EXP of Ref. [4] distribute around zero in the whole experimental condition, the differences above 37 GPa are larger than that between P_DIA and P_EXP of Ref. [3], which may be attributed to the fact that the NaCl pressure transmitting medium (PTM) used in Ref. [3] could supply better hydrostatic conditions than the MgO PTM in Ref. [4]. On the other hand, nearly all the P_QHA are larger than the P_EXP with an average deviation of 2.59 GPa.

Recently, a set of volume measurements of Ni up to 368 GPa was reported in Ref. [5], but their DAC compressions above >21 GPa were performed without any PTM. As shown in Figure 3d, most of the non-hydrostatic P_EXP are smaller than P_DIA, which qualitatively indicates the reliability of DIA since the pressures under non-hydrostatic conditions are recognized to be smaller than that under hydrostatic conditions. On the other hand, it is noticed that although P_EXP below 21 GPa in Ref. [5] was obtained within the helium PTM, all these hydrostatic P_EXP are still smaller than P_DIA. As we know that the pressures determined in DAC experiments depend not only on the experimental conditions but also on the accuracy of the adopted pressure standards, it is difficult for us to judge which factor should be attributed for the current difference between P_DIA and the hydrostatic P_EXP of Ref. [5] since some P_EXP < 21 GPa in both Refs. [3,4] are larger than the corresponding P_DIA. As the pressure calculated by QHA in Ref. [30] is just up to 180 GPa, comparisons between P_QHA and the experiment of Ref. [5] are only within 0–180 GPa. Obviously, nearly all the P_EXP are larger than P_QHA.

The above discussions show that when compared to the latest two hydrostatic experiments of Refs. [3,4], DIA manifests better than QHA.

3.2. The High-Temperature Isotherms

On the P-V relationship of Ni at high temperatures (>300 K), Campbell et al. [3] using a multi-anvil press and laser-heated diamond anvil cells (LHDACs) completed a group of measurements with the covered P, T reaching 2457 K and 66 GPa. Pigott et al. [4] using LHDAC achieved the volume measurements of Ni up to 109 GPa and 2941 K. Unfortunately, a large uncertainty of the manipulated temperature is noted in both the experiments of Refs. [3,4], so that the measured P-V-T data are isolated points instead of isotherms. To inspect the feasibility of DIA, we calculated the pressure at each pair of the experimentally measured (V, T) of the Ni sample in Refs. [3,4], and then performed specific comparisons with the experiments. In order to show the differences clearly, we arranged the pressures in Refs. [3,4] in an ascending order mapped by the Arabic numerals shown as the blue symbols in Figure 4a,c, where the red symbols are the corresponding results by DIA (Supplementary Materials, Tables S1 and S2).

From Figure 4b,d, we can see that the deviations of our calculated pressures from the ones in both the experiments of Refs. [3,4] distribute at the two sides of the gray dashed line, but as denoted by the red dashed line in the two figures, P_DIA presents an increasing deviation from the P_EXP of Ref. [3], while it exhibits an average offset from the P_EXP of Ref. [4]. In contrast to the room temperature cases in Figure 3b,c, it is obvious that the deviations of P_DIA from the P_EXP of both Refs. [3,4] become larger at high temperatures, and the largest |P_EXP − P_QHA| reaches ~8 GPa. It is known that controlling the temperature becomes harder and harder with increasing temperatures; the uncertainty reaches more than 300 K at ~1200 K, and up to 500 K at ~1900 K for the pressure gauges in the two experiments, which may be one of the predominant factors in giving rise to large errors for the experimental pressure determinations.

For realistic applications, having a good command of the isotherms of Ni at high temperatures is demanded. In Refs. [3,4], the measured P-V-T data were fitted by the Mie–Grüneisen–Debye formalism (MGD) [19,20,21], and from the fitted parameters, the isotherms of Ni could be obtained. However, the results in Refs. [3,4] are effective just to 66 GPa, 2457 K, and 109 GPa, 2941 K, respectively. In the present work, we calculated the isotherms of Ni up to 3000 K and simultaneously 500 GPa, which are listed in

Table 1. Based on DIA, the calculated volume (ų/atom) of Ni as a function of the temperature and pressure.

P (GPa)	300 K	500 K	1000 K	1500 K	2000 K	2500 K	3000 K
0.0001	11.0328	11.1324	11.4000	11.7070
10	10.4944	10.5697	10.7763	10.9992	11.2423
20	10.0904	10.1468	10.2977	10.4650	10.6542
30	9.7656	9.8131	9.9364	10.0679	10.2091	10.3625
40	9.4884	9.5299	9.6365	9.7479	9.8651	9.9890
50	9.2444	9.2809	9.3750	9.4735	9.5758	9.6822
60	9.0317	9.0640	9.1464	9.2321	9.3219	9.4164	9.5146
70	8.8413	8.8707	8.9454	9.0221	9.1013	9.1835	9.2693
80	8.6688	8.6956	8.7641	8.8343	8.9063	8.9803	9.0565
90	8.5127	8.5373	8.5999	8.6641	8.7300	8.7978	8.8674
100	8.3693	8.3919	8.4498	8.5091	8.5698	8.6321	8.6958
110	8.2358	8.2567	8.3106	8.3659	8.4225	8.4804	8.5394
120	8.1127	8.1316	8.1809	8.2323	8.2854	8.3396	8.3949
130	7.9993	8.0169	8.0622	8.1092	8.1578	8.2080	8.2599
140	7.8938	7.9103	7.9526	7.9962	8.0411	8.0872	8.1346
150	7.7946	7.8104	7.8503	7.8912	7.9331	7.9760	8.0200
160	7.7008	7.7160	7.7540	7.7927	7.8322	7.8725	7.9137
170	7.6114	7.6261	7.6627	7.6996	7.7371	7.7752	7.8142
180	7.5258	7.5401	7.5755	7.6110	7.6469	7.6832	7.7202
190	7.4437	7.4575	7.4920	7.5264	7.5609	7.5958	7.6311
200	7.3660	7.3790	7.4120	7.4452	7.4786	7.5123	7.5463
210	7.2928	7.3051	7.3364	7.3680	7.4000	7.4323	7.4651
220	7.2235	7.2352	7.2648	7.2949	7.3254	7.3563	7.3876
230	7.1574	7.1685	7.1966	7.2252	7.2544	7.2839	7.3138
240	7.0936	7.1043	7.1312	7.1586	7.1864	7.2147	7.2435
250	7.0319	7.0422	7.0681	7.0944	7.1212	7.1484	7.1760
260	6.9720	6.9820	7.0070	7.0325	7.0584	7.0846	7.1112
270	6.9141	6.9236	6.9479	6.9726	6.9976	7.0230	7.0487
280	6.8585	6.8677	6.8910	6.9147	6.9389	6.9635	6.9884
290	6.8050	6.8138	6.8363	6.8592	6.8825	6.9061	6.9302
300	6.7534	6.7619	6.7837	6.8058	6.8283	6.8511	6.8742
310	6.7034	6.7117	6.7329	6.7544	6.7761	6.7981	6.8204
320	6.6551	6.6632	6.6838	6.7046	6.7257	6.7470	6.7686
330	6.6085	6.6164	6.6363	6.6565	6.6769	6.6975	6.7185
340	6.5633	6.5710	6.5904	6.6100	6.6298	6.6498	6.6700
350	6.5194	6.5269	6.5459	6.5650	6.5842	6.6036	6.6232
360	6.4767	6.4841	6.5026	6.5212	6.5399	6.5588	6.5779
370	6.4352	6.4424	6.4605	6.4786	6.4969	6.5154	6.5339
380	6.3947	6.4017	6.4194	6.4372	6.4551	6.4731	6.4912
390	6.3551	6.3620	6.3794	6.3968	6.4143	6.4319	6.4496
400	6.3165	6.3233	6.3403	6.3573	6.3745	6.3917	6.4090
410	6.2787	6.2853	6.3020	6.3187	6.3356	6.3525	6.3695
420	6.2416	6.2482	6.2646	6.2810	6.2975	6.3141	6.3308
430	6.2056	6.2120	6.2279	6.2440	6.2603	6.2766	6.2930
440	6.1706	6.1768	6.1923	6.2080	6.2239	6.2399	6.2560
450	6.1364	6.1424	6.1576	6.1730	6.1884	6.2041	6.2198
460	6.1031	6.1090	6.1238	6.1388	6.1539	6.1692	6.1846
470	6.0705	6.0763	6.0907	6.1054	6.1202	6.1351	6.1502
480	6.0387	6.0443	6.0585	6.0727	6.0872	6.1018	6.1165
490	6.0076	6.0130	6.0269	6.0408	6.0550	6.0692	6.0837
500	5.9770	5.9824	5.9959	6.0096	6.0234	6.0374	6.0515

. To discuss the differences among the high-temperature isotherms of Ni reported by different experiments and theoretical calculations, the P-V curves of Ni under 1000, 2000, and 3000 K from Refs. [3,4], DIA and QHA are displayed in Figure 5a–c. Since the largest calculated pressure in Ref. [30] is limited to 180 GPa, and according to Ref. [40], solid Ni exists above 7.5 GPa at 2000 K and 50.6 GPa at 3000 K, the comparisons and discussions in this section are just within these conditions. Under the same volume, the differences between P_QHA, P_EXP of Ref. [3], or Ref. [4] and P_DIA at 1000, 2000, and 3000 K are correspondingly displayed in Figure 5d–f, from which we can see that P_QHA [30] is relatively close to the P_EXP of Ref. [4] in the whole pressure and temperature range with the largest difference of less than 3.2 GPa, whereas both P_QHA and P_EXP of Ref. [4] are consistently smaller than P_DIA. The deviation of P_EXP in Ref. [4] (P_QHA) from P_DIA increases gradually with increasing pressure with the largest relative error ((P_EXP − P_DIA)/P_DIA) of 5.25%, 5.78%, and 6.19% (6.58%, 6.64% and 6.65%) at 1000, 2000, and 3000 K, respectively. It is noted that the pressures in the experiment of Ref. [3] were also determined based on the MGD model, but the derived isotherms are far away from the ones reported in Ref. [4]. As shown in Figure 5d–f, the largest relative difference between P_EXP in Ref. [3] and P_DIA reaches 15.58%, 14.08%, and 12.46% at 1000, 2000, and 3000 K, respectively.

The above discussions show that our high-temperature P-V relationship agrees better with the fitted EOS under the experiment of Ref. [4] than Ref. [3]. In addition, it is noted that QHA is rarely applied to such high temperatures up to 3000 K due to its effectiveness only with respect to small atomic vibrations. Based on the mathematic foundation, the precision of DIA may decrease with increasing temperature, but the accuracy indeed increases with decreasing volume [9], so the predictions for the isotherms of Ni up to 3000 K and simultaneously 500 GPa are expected to be proven by future experiments.

3.3. The Analytical EOS

For practical application of metal Ni, it is necessary to obtain an analytical isothermal EOS based on the calculated results of DIA to determine the pressure for arbitrary volume V and temperature T. For this purpose, the MGD model [19,20,21] should be a good choice since it has been applied successfully to fit experimental data to obtain analytical EOS at high temperatures [3,4]. In this model, the total pressure P is considered to be the summation of thermal pressure Pth and the room temperature one P_300K. Pth as a function of V and T is determined by three parameters via the following equations,

$$P_{\mathrm{th}}={\displaystyle \frac{\gamma \left(U_{\mathrm{T}}-U_{300\mathrm{K}}\right)}{V}}$$

(12)

and

$$U_{\mathrm{T}}=9nR\left(\theta /8+T{\left(T/\theta \right)}^3{\displaystyle {\int }_0^{\theta /T}\frac{x^3}{{\mathrm{e}}^x-1}\mathrm{dx}}\right),$$

(13)

where R is the gas constant and n is the mole number of atoms confined within volume V. The Grüneisen parameter γ and Debye temperature θ are regarded to be only volume dependent via

$$\gamma ={\gamma }_0{\left(V/V_0\right)}^q$$

(14)

and

$$\theta ={\theta }_0\exp \left({\displaystyle \frac{{\gamma }_0-\gamma }{q}}\right),$$

(15)

in which the quantities with subscript 0 represent the corresponding values at 300 K and one atmospheric pressure. γ₀, q, and θ₀ are the three parameters to be decided. P_300K can be described by the third-order Birch–Murnaghan (BM) [41] EOS

$$P_{300\mathrm{K}}={\displaystyle \frac{3}{2}}{K}_0\left[{\left(V_0/V\right)}^{{\displaystyle \frac{7}{3}}}-{\left(V_0/V\right)}^{{\displaystyle \frac{5}{3}}}\right]\left[1+{\displaystyle \frac{3}{4}}\left({K^{\prime }}_0-4\right)\left({\left(V_0/V\right)}^{{\displaystyle \frac{2}{3}}}-1\right)\right]$$

(16)

with K₀ and K₀′ being the bulk modulus and its pressure derivative at one atmospheric pressure.

Although K₀ and K₀′ could be easily obtained by fitting experimental data via Equation (16), determining γ₀, q, and θ₀ will cost too many computer hours because an integral equation (Equation (13)) is involved in the MGD model. Actually, in previous applications of the model [3,4], the parameter θ₀ measured by other experiments was pre-introduced. For example, Ref. [4] employed an MGD model fitting the experimental P-V-T data of metal Ni with θ₀ of 415 K [42], and γ₀ and q were then determined to be 1.98 and 1.3. In fact, the fitted results are insensitive to θ₀. When we varied the value of θ₀ from 1 to 500 K without changing the value of γ₀ and q, the obtained pressures deviate from the original ones with θ₀ of 415 K by no more than 0.45 GPa (Figure 6), which is significantly smaller than the uncertainty of the experimental measurements.

It is notable that the internal energy expressed by Equation (13) is based on the harmonic approximation, and tends to be U_T = 3nRT when the temperature T is larger than the Debye temperature. It is a fact that the Debye temperature for most solids is in the range of 200–400 K, and therefore the internal energy at temperature above 300 K should be independent of volume and can be simplified as 3nRT. Accordingly, the expression of the thermal pressure Pth can be simplified as

$$P_{\mathrm{th}}=3nR\left(T-300\right){\displaystyle \frac{{\gamma }_0}{V}}{\left({\displaystyle \frac{V}{V_0}}\right)}^q.$$

(17)

Thus, the MGD model is reduced to a simplified one without parameter θ₀ and the integral equation Equation (13). We apply this simplified model to calculate the pressure of metal Ni with parameter γ₀ and q taken as 1.98 and 1.3 determined in Ref. [4]; the results deviate from the predictions under the MGD model by no more than 0.42 GPa (Figure 6), which is significantly smaller than the uncertainty of the experimental measurements. Furthermore, we obtained an analytical EOS of metal Ni by fitting the calculated results (Table 1) of DIA via the simplified model, showing that the fitted pressure deviates from the original one by no more than 2 GPa (Figure 7). As shown in

Table 2. The parameters in the MGD model (or the simplified one) determined by experimental data (or DIA calculations).

	DIA	Campbell et al. [3]	Pigott et al. [4]
V0 (Å3/atom)	10.972 (0)	10.939	10.926
K0 (GPa)	197.256 (0.126)	179 (3)	201 (6)
K0′	4.626 (0.001)	4.3 (0.2)	4.4 (0.3)
θ0 (K)	/	415	415
γ0	2.014 (0.002)	2.50 (0.06)	1.98 (0.08)
q	0.745 (0.002)	1	1.3 (0.2)

, the parameters of the EOS determined by DIA present some difference from the ones determined by fitting experimental data under the MGD model reported in Refs. [3,4]. Relatively, our theoretical parameters are closer to the latter ones.

Applying the analytical EOS of metal Ni, we obtained a map of pressure distribution over the T-V plane shown in Figure 8a, which is similar to the empirical predictions (Figure 8b) based on the MGD fitting experimental data [4]. The isobars in Figure 8a,b indicate clearly that the pressure depends mainly upon the volume instead of the temperature varying from 300 to 3000 K. For a given pressure, the volume expands linearly with the temperature and the expansion rate becomes slower and slower with increasing pressure. When the pressure is larger than 400 GPa, the volume remains nearly unchanged even if the temperature increases from 300 to 3000 K. For arbitrary volume (or pressure) and temperature, the pressure (or the volume) can be easily read from this pressure map.

The difference between our theoretical predictions and the empirical ones take place mainly in the high-pressure region. As shown in Figure 8c, there exists a T-V curve along which our theoretical prediction is the same as the ones of the empirical prediction, and the theoretical pressure is larger than the empirical one by 5 GPa when the pressure is ~200 GPa. As the pressure increases to 500 GPa, the pressure difference reaches 31.1 GPa, which can be understood in consideration of the fact that the empirical predictions using the MGD model are based on the experimental data below 109 GPa, resulting in the corresponding extrapolations perhaps being unreliable, and on the other hand, it is really not sure if the pseudopotential and the exchange–correlation function in the ab initio calculations is still efficient up to 500 GPa.

4. Conclusions

As a recently developed approach to the classical PF, DIA should be applied to more realistic materials to check its reliability. In this work, we applied DIA at the ab initio level to study the EOS of metal Ni up to 3000 K and simultaneously 500 GPa, and obtained an analytical expression of the EOS. The theoretical results are in good agreement with available hydrostatic experiments, suggesting that DIA is a good approximation method and that the predictions under extreme conditions are expected to be confirmed by future experiments. The map of pressure distribution over the T-V plane should be helpful to understand the thermodynamic properties of Ni-based alloys. The simplified EOS model would find its vast applications for obtaining the EOS of various solids by fitting limited experimental or theoretical data, which should be valuable in the field of high-temperature and high-pressure research. It should be pointed out that DIA is based on the classical statistical mechanics, which does not include the quantum effects at low temperatures, so DIA fails to work when the temperature is far below the Debye temperature.