The SPOCK equation of state for condensed phases under arbitrary compression

SUMMARY

This short paper presents a new equation of state for condensed phases. The equation of state is built on the premise that $K^{'}$ ⁠, the first derivative of the bulk modulus, monotonically increases with volume according to a power law. The input parameters are the zero-pressure volume $V_{0}$ ⁠, bulk modulus $K_{0}$ ⁠, and first and second derivatives of the bulk modulus, $K_{0}^{'}$ and $K_{0}^{''}$ and also $K_{\infty}^{'}$ ⁠, the value of $K^{'}$ at infinite compression. Expressions are provided for the internal energy, pressure, and bulk modulus. The equation of state is robust for all compressions as long as $K_{0}^{''} < 0$ and $K_{\infty}^{'} < K_{0}^{'}$ ⁠. Heuristic values are suggested for situations in which available data is not sufficient to independently constrain $K_{0}^{''}$ and $K_{\infty}^{'}$ ⁠. The equation of state compares favourably with other equations of state using recently published experimental data on Au and Pt.

Elasticity and anelasticity, Equations of state, High-pressure behaviour

Subject

Rock and Mineral Physics, Rheology

Issue Section:

Rock and mineral physics, rheology

1 INTRODUCTION

There are many ways to construct equations of state. Many of the early equations of state were derived from simple models for atomic potentials as a function of interatomic separations. Such equations include the Morse (1929) equation of state used by Slater (1939), and the Rydberg (1932) equation of state rederived by Vinet et al. (1987) and modified by Holzapfel (1998). Other equations of state were derived using Taylor expansions of the internal energy in terms of finite strain. These include the popular second, third and fourth order Birch–Murnaghan equations of state (Murnaghan 1937; Birch 1947), expanded as a Taylor series in Eulerian strain (⁠ $((V / V_{0})^{- 2 / 3} - 1) / 2$ ⁠; see a simple derivation in Katsura & Tange 2019), and the Logarithmic equations of state (Poirier & Tarantola 1998) expanded as a function of the logarithmic (Hencky) strain (⁠ $\ln (V / V_{0}) / 3$ ⁠). A third class of equation of state functions were chosen purely because they provided good approximations to experimental P(V) data. These include the Modified Tait Equation of State (Huang & Chow 1974) used by Holland & Powell (2011). Some of these equations of state have forms that can incorporate constraints from theoretical physics (Holzapfel 1998; Stacey 2000; Lozano & Aslam 2022).

A particular challenge when building equations of state is that it is difficult to obtain a simple expression that simultaneously satisfies all the physical and thermodynamic constraints and has the flexibility to fit experimental or computational data. Good experimental data over experimentally feasible ranges of compression (⁠ $\sim 0.4 \leq V / V_{0} \leq 1.0$ ⁠) can provide estimates of the standard-state (zero-pressure) volume $V_{0}$ ⁠, Reuss bulk modulus $K_{0}$ ⁠, and its first and second derivatives with respect to pressure $K_{0}^{'}$ ⁠, and $K_{0}^{''}$ ⁠, although the last may be associated with significant uncertainty (Freund & Ingalls 1989). There is no theoretical basis suggesting that any of these parameters are dependent on the others; therefore, a comprehensive high-pressure equation of state should include at least four parameters. The current study describes a five-parameter equation of state, where the additional free parameter, the first derivative of the bulk modulus at infinite compression (⁠ $K_{\infty}^{'}$ ⁠), is required to ensure that compressibility remains thermodynamically consistent at extreme compressions (Holzapfel 1996; Stacey 2000).

2 THE SPOCK EQUATION OF STATE

2.1 Desired properties

Any equation of state should ideally satisfy a number of physical, empirical and pragmatic requirements. Physical constraints include the requirement that bulk modulus K should be positive for all volumes (e.g. Lozano & Aslam 2022). In the absence of phase, spin or structural transitions, K should decrease monotonically as a function of volume, and approach zero at infinite volume. The bulk modulus should also be differentiable with respect to volume. Finally, $K^{'}$ should approach a constant value at infinite compressions that is greater than the Fermi–Thomas limit (⁠ $K^{'} \geq 5 / 3$ ⁠, Holzapfel 1998; Stacey 2000; Lozano & Aslam 2022). These physical requirements are not met by many popular equations of state, for which the bulk modulus can become negative at large compressions or tensions (Table 1). There are two important cases in which the tensional part of an equation of state (where $V > V_{0}$ ⁠) can be important. First, there are physical situations where negative pressures can be achieved, such as during the passage of a shock wave. These negative pressures represent metastable states that will relax over time by processes such as cavitation or spallation (Rajendran et al. 1989; Addessio & Johnson 1993; Menikoff 2007). Simulating these states and processes requires equations of state in which the bulk modulus remains positive under large tensions. Secondly, thermal equations of state often use the tensional part of the reference equation of state to model properties at high temperatures and low but positive pressures. The failure of the third-order Birch–Murnaghan equation of state (Birch 1947) in tension can cause the breakdown of some high-temperature equations of state (e.g. in data sets using the thermal equation of state of Stixrude & Lithgow-Bertelloni 2005).

Table 1.

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Favourable characteristics of different equations of state for physically reasonable parameter values. Equations of state are Rydberg–Vinet (Rydberg 1932; Vinet et al. 1987), Birch–Murnaghan of orders 3 and 4 (BM3, BM4; Birch 1947), Modified Tait (MT; Huang & Chow 1974), Reciprocal K-prime (RK; Stacey 2000), MACAW (Lozano & Aslam 2022) and SPOCK (this study).

	$K > 0$ as $P \to \infty$	$K > 0$ as $V \to \infty$	$K_{\infty}^{'}$ control	$K_{0}^{''}$ control	$K^{''} < 0$
Rydberg–Vinet	Yes	–	–	–	Yes
BM3	If $K_{0}^{'} > 4$	–	–	–	–
BM4	Not guaranteed	–	–	Yes	–
Modified Tait	Not guaranteed	Yes	–	Yes	–
RK	Yes	–	Yes	–	Yes
MACAW	Yes	Yes	Yes	–	If $K_{\infty}^{'} > 2 + 16 / 29$
SPOCK	Yes	Yes	Yes	Yes	Yes

	$K > 0$ as $P \to \infty$	$K > 0$ as $V \to \infty$	$K_{\infty}^{'}$ control	$K_{0}^{''}$ control	$K^{''} < 0$
Rydberg–Vinet	Yes	–	–	–	Yes
BM3	If $K_{0}^{'} > 4$	–	–	–	–
BM4	Not guaranteed	–	–	Yes	–
Modified Tait	Not guaranteed	Yes	–	Yes	–
RK	Yes	–	Yes	–	Yes
MACAW	Yes	Yes	Yes	–	If $K_{\infty}^{'} > 2 + 16 / 29$
SPOCK	Yes	Yes	Yes	Yes	Yes

Table 1.

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	$K > 0$ as $P \to \infty$	$K > 0$ as $V \to \infty$	$K_{\infty}^{'}$ control	$K_{0}^{''}$ control	$K^{''} < 0$
Rydberg–Vinet	Yes	–	–	–	Yes
BM3	If $K_{0}^{'} > 4$	–	–	–	–
BM4	Not guaranteed	–	–	Yes	–
Modified Tait	Not guaranteed	Yes	–	Yes	–
RK	Yes	–	Yes	–	Yes
MACAW	Yes	Yes	Yes	–	If $K_{\infty}^{'} > 2 + 16 / 29$
SPOCK	Yes	Yes	Yes	Yes	Yes

	$K > 0$ as $P \to \infty$	$K > 0$ as $V \to \infty$	$K_{\infty}^{'}$ control	$K_{0}^{''}$ control	$K^{''} < 0$
Rydberg–Vinet	Yes	–	–	–	Yes
BM3	If $K_{0}^{'} > 4$	–	–	–	–
BM4	Not guaranteed	–	–	Yes	–
Modified Tait	Not guaranteed	Yes	–	Yes	–
RK	Yes	–	Yes	–	Yes
MACAW	Yes	Yes	Yes	–	If $K_{\infty}^{'} > 2 + 16 / 29$
SPOCK	Yes	Yes	Yes	Yes	Yes

To place all of these physical requirements on a mathematical basis, we define the following thermodynamic identities:

\begin{array}{r} P \equiv - \frac{\partial E}{\partial V}, \end{array}

(1)

\begin{array}{r} K \equiv - \frac{\partial P}{\partial f}, \end{array}

(2)

\begin{array}{r} K^{'} \equiv \frac{\partial K}{\partial P} \equiv - \frac{\partial \ln K}{\partial f}, \end{array}

(3)

\begin{array}{r} K^{''} K \equiv - \frac{\partial K^{'}}{\partial f}, \end{array}

(4)

\begin{array}{r} f \equiv \ln (\frac{V}{V_{0}}), \end{array}

(5)

where $K^{'}$ and $K^{''}$ are the first and derivatives of K with respect to P. The physical requirements of an equation of state can be summed up by two constraints on $K^{'}$ ⁠:

\begin{array}{r} lim_{f \to - \infty} K^{'} (f) = c \geq 5 / 3 . \end{array}

(6)

\begin{array}{r} \frac{\partial K^{'}}{\partial f} > 0 for all f \end{array}

(7)

The empirical requirement for equations of state is that they should be able to accurately represent available experimental data with a minimal number of physically meaningful parameters. Most importantly, it should be possible to fit the observed standard state volume $V_{0}$ ⁠, bulk modulus $K_{0}$ and first derivative with respect to pressure $K_{0}^{'}$ ⁠. In some cases, data are sufficient to also estimate the second derivative $K_{0}^{''}$ ⁠. High-pressure extensions of the curves should be able to pass through all data points without breaking the physical constraints mentioned above. Finally, a pragmatic characteristic of successful equations of state is that they should provide simple expressions for the internal energy $E$ and pressure P as a function of volume V, or Gibbs energy $G$ and volume V as a function of pressure P.

2.2 Derivation

Given the important role played by $K^{'}$ in eqs (6) and (7), it seems reasonable to use an expression for $K^{'}$ to derive an equation of state. This was the starting point chosen for the equation of state of Murnaghan (1937), who assumed that $K^{'}$ was constant. Unfortunately, the Murnaghan equation of state is too incompressible to match experimental data at high pressure. Here, let us instead express $K^{'}$ as a power law function of the volume to create the ‘Scaled Power Of Compression for K-prime’ (SPOCK) Equation of State:

\begin{array}{r} K^{'} (f) = a b {(\frac{V}{V_{0}})}^{a} + c = a b \exp (a f) + c . \end{array}

(8)

The parameters a, b and c can be related to standard state properties:

\begin{array}{r} a = - \frac{K_{0}^{''} K_{0}}{(K_{0}^{'} - K_{\infty}^{'})}, \end{array}

(9)

\begin{array}{r} b = \frac{(K_{0}^{'} - K_{\infty}^{'})}{a} = - \frac{(K_{0}^{'} - K_{\infty}^{'})^{2}}{K_{0}^{''} K_{0}}, \end{array}

(10)

\begin{array}{r} c = K_{\infty}^{'}, \end{array}

(11)

where we have used eq. (4) in the expression for a. Combining eq. (4) with eq. (8) implies that $K^{''} K$ is a linear function of $K^{'}$ ⁠:

\begin{array}{r} K^{''} K = - a^{2} b {(\frac{V}{V_{0}})}^{a}, \end{array}

(12)

\begin{array}{r} = - a (K^{'} - K_{\infty}^{'}), \end{array}

(13)

\begin{array}{r} = K_{0}^{''} K_{0} (\frac{K^{'} - K_{\infty}^{'}}{K_{0}^{'} - K_{\infty}^{'}}) . \end{array}

(14)

The bulk modulus (eq. 3) can be found by integrating and exponentiating eq. (8):

\begin{array}{r} K (f) = K_{0} \exp (b) \exp (- b (\exp (a f)) - c f) . \end{array}

(15)

To find an expression for the pressure (eq. 2), let $u = \exp (a f)$ ⁠, such that $d f = (a u)^{- 1} d u$ ⁠:

\begin{array}{r} P (f) - P_{0} = - \int_{0}^{f} K (f^{'}) d f^{'}, \end{array}

(16)

\begin{array}{r} = - K_{0} \exp (b) \int_{1}^{\exp (a f)} \exp (- b u - \frac{c}{a} \ln (u)) (a u)^{- 1} d u, \end{array}

(17)

\begin{array}{r} = - K_{0} \frac{\exp (b)}{a} \int_{1}^{u} u^{- 1 - \frac{c}{a}} \exp (- b u) d u . \end{array}

(18)

For positive a, b, c and u:

\begin{array}{r} \int_{1}^{u} u^{- 1 - \frac{c}{a}} \exp (- b u) d u = - b^{c / a} Γ (- \frac{c}{a}, b u, b), \end{array}

(19)

where $Γ (α, β_{1}, β_{2})$ is the incomplete gamma function:

\begin{array}{r} Γ (α, β_{1}, β_{2}) = \int_{β_{1}}^{β_{2}} t^{α - 1} e^{- t} d t . \end{array}

(20)

The incomplete gamma function has been implemented as part of many commonly used computational libraries (e.g. BOOST, scipy, mpmath), and can also be implemented in Excel (see Data Availability statement). Using the incomplete gamma function, the change in pressure (eq. 18) can be expressed as:

\begin{array}{r} Δ P = P (f) - P_{0} = K_{0} \frac{\exp (b)}{a} b^{c / a} Γ (- \frac{c}{a}, b \exp (a f), b), \end{array}

(21)

\begin{array}{r} = P (V) - P_{0} = K_{0} \frac{\exp (b)}{a} b^{c / a} Γ (- \frac{c}{a}, b {(\frac{V}{V_{0}})}^{a}, b) . \end{array}

(22)

Finally the change in internal energy is given by (eq. 1):

\begin{array}{r} Δ E (f) = E (f) - E_{0} = - \int_{V_{0}}^{V} P (V^{'}) d V^{'}, \end{array}

(23)

\begin{array}{r} = E (f) - E_{0} = - \int_{V_{0}}^{V} Δ P (V^{'}) d V^{'} - P_{0} (V - V_{0}) . \end{array}

(24)

By letting $u = b (V / V_{0})^{a}$ ⁠, such that $d V = V_{0} / (a b^{1 / a}) u^{(1 / a - 1)} d u$ ⁠, we have:

\begin{array}{rcl} \int_{V_{0}}^{V} Δ P (V^{'}) d V^{'} \\ = K_{0} \frac{\exp (b)}{a} b^{c / a} \int_{V_{0}}^{V} Γ (- \frac{c}{a}, b {(\frac{V}{V_{0}})}^{a}, b) d V^{'}, \end{array}

(25)

\begin{array}{r} = K_{0} \frac{\exp (b)}{a} b^{c / a} \int_{u_{0}}^{u} \frac{V_{0}}{a b^{1 / a}} u^{' (\frac{1}{a} - 1)} Γ (- \frac{c}{a}, u^{'}, b) d u^{'}, \end{array}

(26)

\begin{array}{r} = V_{0} K_{0} \frac{\exp (b)}{a} b^{(c - 1) / a} I, \end{array}

(27)

where

\begin{array}{r} I = \int_{u_{0}}^{u} \frac{u^{' (\frac{1}{a} - 1)}}{a} Γ (- \frac{c}{a}, u^{'}, b) d u^{'}, \end{array}

(28)

\begin{array}{r} = u^{\frac{1}{a}} Γ (- \frac{c}{a}, u, u_{0}) - Γ (\frac{1 - c}{a}, u, u_{0}), \end{array}

(29)

\begin{array}{r} = b^{\frac{1}{a}} \frac{V}{V_{0}} Γ (- \frac{c}{a}, b {(\frac{V}{V_{0}})}^{a}, b) - Γ (\frac{1 - c}{a}, b {(\frac{V}{V_{0}})}^{a}, b) \\ . \end{array}

(30)

2.3 Constraints and heuristics

The SPOCK equation of state has one or two more free parameters than most equations of state, and limited data may not be able to independently constrain all five parameters. For this reason, it is useful to provide heuristics that can be used as weak priors during model fitting.

2.3.1 $K_{\infty}^{'}$

The SPOCK equation of state requires that $K_{\infty}^{'} < K_{0}^{'}$ ⁠, and $K_{\infty}^{'}$ must be higher than the Fermi–Thomas limit for a free-electron gas (⁠ $K_{\infty, TF}^{'} = 5 / 3$ ⁠) (Stacey 2000; Stacey & Davis 2004; Stacey 2005). Optimized values for gold (⁠ $K_{\infty}^{'} \sim 2.4$ ⁠) and platinum (⁠ $K_{\infty}^{'} \sim 3.0$ ⁠) given in Section 3 suggest that $K_{\infty}^{'} / K_{0}^{'} \sim 0.43$ ⁠, but it is unlikely that this value is appropriate for all materials, given that compression mechanisms can change as a function of pressure. If the value of $K_{\infty}^{'}$ cannot be constrained by other means, any value $2 ≲ K_{\infty}^{'} ≲ 3$ will return a physically reasonable equation of state.

2.3.2 $K_{0}^{''}$

A negative value for $K_{0}^{''}$ is a requirement for physically justified equations of state (Stacey 2005). $K_{0} K_{0}^{''}$ has traditionally been subject to uncertainties greater than or similar to 100 per cent (Freund & Ingalls 1989). Based on published periclase data, Holland & Powell (2011) proposed the heuristic

\begin{array}{r} - K_{0}^{''} K_{0} \sim K_{0}^{'} \end{array}

(31)

A more flexible constraint is to assume that $K^{''} K$ has a finite derivative with respect to V at infinite compression, which for the SPOCK EoS would imply that the parameter a in eq. (12) is $\geq 1$ ⁠, and therefore, from eq. (9):

\begin{array}{r} - K_{0}^{''} K_{0} \geq K_{0}^{'} - K_{\infty}^{'} \end{array}

(32)

The Holland & Powell (2011) heuristic satisfies this inequality. Enforcing eq. (32) limits the magnitude of the first argument (⁠ $α$ ⁠) passed to the incomplete gamma function (eqs 22 and 30). When $α = - c / a$ ⁠:

\begin{array}{r} - K_{\infty}^{'} \leq α < 0 \end{array}

(33)

and when $α = (1 - c) / a$ ⁠:

\begin{array}{r} - (K_{\infty}^{'} + 1) \leq α < 0 \end{array}

(34)

These limits are computationally beneficial because evaluating the incomplete gamma function with negative $α$ requires use of the recurrence relation:

\begin{array}{r} Γ (α, β_{1}, β_{2}) = \frac{1}{α} (Γ (α + 1, β_{1}, β_{2}) - β_{1}^{α} e^{- β_{1}} + β_{2}^{α} e^{- β_{2}}) . \end{array}

(35)

By precluding large negative values of $α$ ⁠, this recurrence relation only needs to be used at most a few times, ensuring precision and relatively short calculation times.

3 EXAMPLES AND COMPARISON WITH OTHER EQUATIONS OF STATE

In the following sections, the SPOCK equation of state is directly compared with the Rydberg–Vinet (Rydberg 1932; Vinet et al. 1987), Birch–Murnaghan (BM3, BM4; Birch 1947), Modified Tait (MT; Huang & Chow 1974), Reciprocal K-prime (RK; Stacey 2000), and Murnaghan combined with the Walsh mirror-image (MACAW; Lozano & Aslam 2022) equations of state. None of these equations have as many adjustable parameters as the SPOCK equation of state, and so for these the other parameters are implicitly defined. For example, $K_{0}^{''}$ in the MACAW equation of state (Lozano & Aslam 2022) is implicitly defined by:

\begin{array}{r} K_{0}^{''} K_{0} = \frac{3 C (n + m)}{8 (B + \frac{C}{2} + (B + C)^{2})^{2} - 16 C (B + (B + C)^{2})}, \end{array}

(36)

\begin{array}{r} n = 4 B^{4} + B^{3} (16 C - 4) + B^{2} (24 C^{2} - 12 C - 5), \end{array}

(37)

\begin{array}{r} m = B (16 C^{3} - 12 C^{2} - 28 C + 3) + 4 C^{2} (C^{2} - C + 1), \end{array}

(38)

where functions for A, B and C in terms of $K_{0}$ ⁠, $K_{0}^{'}$ and $K_{\infty}^{'}$ are given in their Appendix B.

3.1 Platinum

Equation of state parameter values for Pt have been optimized using the high pressure experimental data from Fratanduono et al. (2021) (Table 2). A graphical comparison between each equation of state and the experimental data is provided (Fig. 1).

Figure 1.

(a) Volumes and (b) bulk moduli for Pt using different equations of state (abbreviations as in Table 1). Differences between model and observed values are given in panels (c) and (d). Parameters for each equation of state are fitted to the isothermal Pt data of Fratanduono et al. (2021). The bulk moduli data have been estimated from the gradient of the pressure with respect to the volume. Weighted sum of squared residuals (WSS) for each model are shown in the legend of panel (a).

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Table 2.

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Optimized Pt equation of state parameters using the data of Fratanduono et al. (2021). Equation of state abbreviations are given in Table 1.

	$V_{0}$ (cm $^{3}$ /mol)	$K_{0}$ (GPa)	$K_{0}^{'}$	$K_{\infty}^{'}$	$K_{0}^{''}$ (10 $^{- 11}$ Pa $^{- 1}$ ⁠)	$- K_{0}^{''} K_{0}$	WSS
Rydberg-Vinet	9.0943(5)	256.9(3)	5.893(5)	–	–	–	0.63
BM3	9.0880(18)	266.9(11)	5.38(2)	–	–	–	9.20
BM4	9.0958(3)	254.5(3)	6.023(13)	–	−4.85(5)	12.3	0.26
Modified Tait	9.0936(5)	259.8(4)	5.525(15)	–	−1.98(3)	5.1	0.50
RK	9.0976(6)	249.4(6)	6.64(4)	2.77(3)	–	–	0.84
MACAW	9.0948(3)	257.2(3)	5.731(10)	2.578(11)	–	–	0.26
SPOCK	9.0951(4)	256.5(5)	5.79(3)	2.42(15)	−3.12(15)	8.0	0.25

	$V_{0}$ (cm $^{3}$ /mol)	$K_{0}$ (GPa)	$K_{0}^{'}$	$K_{\infty}^{'}$	$K_{0}^{''}$ (10 $^{- 11}$ Pa $^{- 1}$ ⁠)	$- K_{0}^{''} K_{0}$	WSS
Rydberg-Vinet	9.0943(5)	256.9(3)	5.893(5)	–	–	–	0.63
BM3	9.0880(18)	266.9(11)	5.38(2)	–	–	–	9.20
BM4	9.0958(3)	254.5(3)	6.023(13)	–	−4.85(5)	12.3	0.26
Modified Tait	9.0936(5)	259.8(4)	5.525(15)	–	−1.98(3)	5.1	0.50
RK	9.0976(6)	249.4(6)	6.64(4)	2.77(3)	–	–	0.84
MACAW	9.0948(3)	257.2(3)	5.731(10)	2.578(11)	–	–	0.26
SPOCK	9.0951(4)	256.5(5)	5.79(3)	2.42(15)	−3.12(15)	8.0	0.25

Table 2.

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Optimized Pt equation of state parameters using the data of Fratanduono et al. (2021). Equation of state abbreviations are given in Table 1.

	$V_{0}$ (cm $^{3}$ /mol)	$K_{0}$ (GPa)	$K_{0}^{'}$	$K_{\infty}^{'}$	$K_{0}^{''}$ (10 $^{- 11}$ Pa $^{- 1}$ ⁠)	$- K_{0}^{''} K_{0}$	WSS
Rydberg-Vinet	9.0943(5)	256.9(3)	5.893(5)	–	–	–	0.63
BM3	9.0880(18)	266.9(11)	5.38(2)	–	–	–	9.20
BM4	9.0958(3)	254.5(3)	6.023(13)	–	−4.85(5)	12.3	0.26
Modified Tait	9.0936(5)	259.8(4)	5.525(15)	–	−1.98(3)	5.1	0.50
RK	9.0976(6)	249.4(6)	6.64(4)	2.77(3)	–	–	0.84
MACAW	9.0948(3)	257.2(3)	5.731(10)	2.578(11)	–	–	0.26
SPOCK	9.0951(4)	256.5(5)	5.79(3)	2.42(15)	−3.12(15)	8.0	0.25

	$V_{0}$ (cm $^{3}$ /mol)	$K_{0}$ (GPa)	$K_{0}^{'}$	$K_{\infty}^{'}$	$K_{0}^{''}$ (10 $^{- 11}$ Pa $^{- 1}$ ⁠)	$- K_{0}^{''} K_{0}$	WSS
Rydberg-Vinet	9.0943(5)	256.9(3)	5.893(5)	–	–	–	0.63
BM3	9.0880(18)	266.9(11)	5.38(2)	–	–	–	9.20
BM4	9.0958(3)	254.5(3)	6.023(13)	–	−4.85(5)	12.3	0.26
Modified Tait	9.0936(5)	259.8(4)	5.525(15)	–	−1.98(3)	5.1	0.50
RK	9.0976(6)	249.4(6)	6.64(4)	2.77(3)	–	–	0.84
MACAW	9.0948(3)	257.2(3)	5.731(10)	2.578(11)	–	–	0.26
SPOCK	9.0951(4)	256.5(5)	5.79(3)	2.42(15)	−3.12(15)	8.0	0.25

For this equation of state, all evaluated equations of state perform acceptably, apart from the BM3 equation of state, which has a WSS residuals of 9.20. As expected, the three-parameter Rydberg–Vinet equation of state has a higher WSS residual than most of the four parameter equations (BM4, MT, MACAW). The four parameter RK equation of state performs slightly less well than any of these. The five parameter SPOCK equation of state slightly outperforms all other equations of state. For this data set, there is little to differentiate between MACAW and SPOCK; BM4 and MT have slightly larger residuals at low pressure (Fig. 1c).

The variation in parameter values between models is about ten times larger than the formal uncertainties. Ignoring the BM3 fit, the range of estimated $V_{0}$ is 9.0936–9.0976 cm $^{3}$ mol⁻¹, $K_{0}$ is 249.4–259.8 GPa, $K_{0}^{'}$ is 5.53–6.64 and $K_{0}^{''}$ is −2.0 to −4.8. These ranges are in good qualitative agreement with the suggestion of Freund & Ingalls (1989) that model-free uncertainties for $K_{0}$ ⁠, $K_{0}^{'}$ and $K_{0}^{''}$ are on the order of 1, 10 and 100 per cent. The smaller formal uncertainties arise from the restrictive nature of each of the equations of state. If the model approximates the true functional form of the equation of state, this reduction in error is good. The differences between the models indicate that, despite their ability to fit the data, they cannot all be good representations of the true functional form over the pressure range of the data.

3.2 Gold

Equation of state parameter values for Au have been optimized using the high-pressure experimental data from Fratanduono et al. (2021) (Table 3). A graphical comparison between each equation of state and the experimental data is provided (Fig. 2).

Figure 2.

(a) Volumes and (b) bulk moduli for Au using different equations of state (abbreviations as in Table 1). Differences between model and observed values are given in panels (c) and (d). Parameters for each equation of state are fitted to the isothermal Au data of Fratanduono et al. (2021). The bulk moduli data have been estimated from the gradient of the pressure with respect to the volume. Weighted sum of squared residuals (WSS) for each model are shown in the legend of panel (a).

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Table 3.

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Optimized Au equation of state parameters using the data of Fratanduono et al. (2021). Equation of state abbreviations are given in Table 1.

	$V_{0}$ (cm $^{3}$ mol⁻¹)	$K_{0}$ (GPa)	$K_{0}^{'}$	$K_{\infty}^{'}$	$K_{0}^{''}$ (10 $^{- 11}$ Pa $^{- 1}$ ⁠)	$- K_{0}^{''} K_{0}$	WSS
Rydberg–Vinet	10.183(2)	167.3(6)	5.999(14)	–	–	–	12.29
BM3	10.171(4)	177.7(13)	5.33(3)	–	–	–	59.06
BM4	10.1926(10)	159.8(4)	6.52(3)	–	−9.97(17)	15.9	2.40
Modified Tait	10.1887(15)	165.1(7)	5.80(3)	–	−3.51(11)	5.8	6.24
RK	10.1984(6)	151.9(4)	7.84(4)	2.15(2)	–	–	1.07
MACAW	10.1912(10)	162.3(4)	6.092(19)	2.420(15)	–	–	2.64
SPOCK	10.1959(5)	156.6(4)	6.76(4)	2.97(3)	−10.0(3)	15.6	0.59

	$V_{0}$ (cm $^{3}$ mol⁻¹)	$K_{0}$ (GPa)	$K_{0}^{'}$	$K_{\infty}^{'}$	$K_{0}^{''}$ (10 $^{- 11}$ Pa $^{- 1}$ ⁠)	$- K_{0}^{''} K_{0}$	WSS
Rydberg–Vinet	10.183(2)	167.3(6)	5.999(14)	–	–	–	12.29
BM3	10.171(4)	177.7(13)	5.33(3)	–	–	–	59.06
BM4	10.1926(10)	159.8(4)	6.52(3)	–	−9.97(17)	15.9	2.40
Modified Tait	10.1887(15)	165.1(7)	5.80(3)	–	−3.51(11)	5.8	6.24
RK	10.1984(6)	151.9(4)	7.84(4)	2.15(2)	–	–	1.07
MACAW	10.1912(10)	162.3(4)	6.092(19)	2.420(15)	–	–	2.64
SPOCK	10.1959(5)	156.6(4)	6.76(4)	2.97(3)	−10.0(3)	15.6	0.59

Table 3.

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Optimized Au equation of state parameters using the data of Fratanduono et al. (2021). Equation of state abbreviations are given in Table 1.

	$V_{0}$ (cm $^{3}$ mol⁻¹)	$K_{0}$ (GPa)	$K_{0}^{'}$	$K_{\infty}^{'}$	$K_{0}^{''}$ (10 $^{- 11}$ Pa $^{- 1}$ ⁠)	$- K_{0}^{''} K_{0}$	WSS
Rydberg–Vinet	10.183(2)	167.3(6)	5.999(14)	–	–	–	12.29
BM3	10.171(4)	177.7(13)	5.33(3)	–	–	–	59.06
BM4	10.1926(10)	159.8(4)	6.52(3)	–	−9.97(17)	15.9	2.40
Modified Tait	10.1887(15)	165.1(7)	5.80(3)	–	−3.51(11)	5.8	6.24
RK	10.1984(6)	151.9(4)	7.84(4)	2.15(2)	–	–	1.07
MACAW	10.1912(10)	162.3(4)	6.092(19)	2.420(15)	–	–	2.64
SPOCK	10.1959(5)	156.6(4)	6.76(4)	2.97(3)	−10.0(3)	15.6	0.59

	$V_{0}$ (cm $^{3}$ mol⁻¹)	$K_{0}$ (GPa)	$K_{0}^{'}$	$K_{\infty}^{'}$	$K_{0}^{''}$ (10 $^{- 11}$ Pa $^{- 1}$ ⁠)	$- K_{0}^{''} K_{0}$	WSS
Rydberg–Vinet	10.183(2)	167.3(6)	5.999(14)	–	–	–	12.29
BM3	10.171(4)	177.7(13)	5.33(3)	–	–	–	59.06
BM4	10.1926(10)	159.8(4)	6.52(3)	–	−9.97(17)	15.9	2.40
Modified Tait	10.1887(15)	165.1(7)	5.80(3)	–	−3.51(11)	5.8	6.24
RK	10.1984(6)	151.9(4)	7.84(4)	2.15(2)	–	–	1.07
MACAW	10.1912(10)	162.3(4)	6.092(19)	2.420(15)	–	–	2.64
SPOCK	10.1959(5)	156.6(4)	6.76(4)	2.97(3)	−10.0(3)	15.6	0.59

The WSS residuals for the Au models are significantly higher than they were for Pt. The three-parameter models (Vinet, BM3) perform poorly, the four-parameter models (BM4, Modified Tait, RK, MACAW) have WSS residuals of 1–10, and the SPOCK equation of state has a WSS residual of 0.6. Similar to the Pt models, parameter variation between models is about ten times greater than formal uncertainties. The volume residuals between the data and the RK and SPOCK equations of state at $<$ 100 GPa are distributed quasirandomly about zero, while the other equations of state exhibit systematic positive and negative deviations about zero (Fig. 2c).

At first glance, the good performance of the Reciprocal K-prime (RK) equation of state relative to the BM4 and MT equations is surprising. How can an equation of state that constrains $K_{\infty}^{''}$ rather than $K_{0}^{''}$ outperform two equations of state that directly constrain $K_{0}^{''}$ ? At least part of the answer is that both BM4 and MT become unstable at high pressure when $| K_{0}^{''} |$ is large, so that model fits to data that span a wide range of compressions with large $| K_{0}^{''} |$ will tend to skew $K_{0}^{''}$ to smaller values.

Fig. 3 illustrates the differences between the different equations of state by assuming the same parameter values as those optimized for the SPOCK equation of state (last line of Table 3). Particularly prominent in all subfigures are the failures of the BM4 and MT equations of state at extreme compression noted in Table 1. Also notable are the failures of the Vinet, BM3, BM4 and RK equations of state in tension (Fig. 3b). The RK equation of state performs particularly poorly in tension, despite showing promising behaviour at positive compressions (Fig. 2).

$A comparison of different equations of state using the optimized parameters from the Au inversion for the SPOCK equation of state (last line of Table 3). Equation of state abbreviations are given in Table 1. Subplots are the pressure (a), bulk modulus (b), internal energy (c) and $K^{\prime }$ (d).$

Figure 3.

A comparison of different equations of state using the optimized parameters from the Au inversion for the SPOCK equation of state (last line of Table 3). Equation of state abbreviations are given in Table 1. Subplots are the pressure (a), bulk modulus (b), internal energy (c) and $K^{'}$ (d).

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Both MACAW and SPOCK are robust equations of state at all volumes, but their responses are quite different. Notably, the magnitude of $K_{0}^{''}$ implicitly defined by the MACAW equation of state (eq. 36) is much smaller than the optimized value for the SPOCK equation of state (Fig. 3d). This difference leads to the optimized MACAW Au EoS having a larger WSS residual than the SPOCK Au EoS (Table 3).

4 SUMMARY AND CONCLUSIONS

The SPOCK equation of state can accurately reproduce experimental data to extreme pressure. It exhibits robust behaviour both in compression and in tension, properties required by the underlying physics of compression in compact media (Stacey 2000; Lozano & Aslam 2022). It allows the user to fit $K_{0}$ ⁠, $K_{0}^{'}$ and $K_{0}^{''}$ independently of $K_{\infty}^{'}$ ⁠. This last point separates the SPOCK equation of state from the otherwise excellent MACAW equation of state (Lozano & Aslam 2022) which restricts $K_{0}^{''}$ to small magnitudes.

One limitation of the new equation of state is its reliance on the incomplete gamma function, which is expensive to evaluate relative to (for example) the exponential function. Even in applications where the calculation cost precludes direct use of the SPOCK equation of state, it could remain useful for initial calibration to experimental data. A secondary fitting step could then convert the SPOCK calibration to a cheaper approximation using splines, Chebyshev polynomials, or machine-learning models. This two-stage calibration would enable generation of realistic equations of state with many parameters while avoiding overfitting.

FUNDING

This work was supported by the Natural Environment Research Council (Large Grant MC-squared; Award No. NE/T012633/1) and the Science and Technology Facilities Council (Grant No. ST/R001332/1).

ACKNOWLEDGEMENTS

I would like to thank Eduardo Lozano and Jesse Walters for their careful reviews of this paper, and Tobias Keller for his editorial handling. Any mistakes or oversights are my own.

DATA AVAILABILITY

Equation of state fitting and all the figures in this study were produced using the BurnMan Python package (Myhill et al. 2023, 2024), and are provided as contributions to that package at https://github.com/geodynamics/burnman. An excel spreadsheet implementing the SPOCK EoS can be found in the contrib directory of that repository.

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Article Contents

The SPOCK equation of state for condensed phases under arbitrary compression

SUMMARY

1 INTRODUCTION

2 THE SPOCK EQUATION OF STATE

2.1 Desired properties

2.2 Derivation

2.3 Constraints and heuristics

2.3.1 $K_{\infty}^{'}$

2.3.2 $K_{0}^{''}$

3 EXAMPLES AND COMPARISON WITH OTHER EQUATIONS OF STATE

3.1 Platinum

3.2 Gold

4 SUMMARY AND CONCLUSIONS

FUNDING

ACKNOWLEDGEMENTS

DATA AVAILABILITY

REFERENCES

Citations

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Article Contents

The SPOCK equation of state for condensed phases under arbitrary compression

SUMMARY

1 INTRODUCTION

2 THE SPOCK EQUATION OF STATE

2.1 Desired properties

2.2 Derivation

2.3 Constraints and heuristics

2.3.1 K∞′

2.3.2 K0′′

3 EXAMPLES AND COMPARISON WITH OTHER EQUATIONS OF STATE

3.1 Platinum

3.2 Gold

4 SUMMARY AND CONCLUSIONS

FUNDING

ACKNOWLEDGEMENTS

DATA AVAILABILITY

REFERENCES

Citations

Views

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2.3.1 $K_{\infty}^{'}$

2.3.2 $K_{0}^{''}$