Equation of State (EoS) fit

Overview

The equation of state (EoS) of a material describe how its volume changes with pressure. Here, the EoS formulations coded into Quantas are briefly presented.

Last update

Jul 25, 2024

Author

Gianfranco Ulian

Note

Currently, Quantas provides isothermal equation of states. Thermal expansion models and P-T-V EoS will be provided in future releases.

Equation of State formulations

In both experimental and theoretical settings, the volumetric behaviour of a solid phase with pressure can be described in a functional form called equation of state (EoS).

The volume of the solid is related to its unit cell, which is obtained from high-pressure diffraction experiments. The equation of state is a parametrized function, containing from two to four parameters that are adjusted to fit the experimental data. There are several works in literature that provide a detailed description of the theory behind the EoS formulations, 1, 2 and here only the relevant information is discussed.

In Quantas, there are five isothermal equation of state formulation coded:

  1. Murnaghan: 3

\[P_{VT} = \frac{K_{0T}}{K^{\prime}_{0T}} \Bigg[ \Big( \frac{V_{0T}}{V} \Big)^{K^{\prime}_{0T}} - 1 \Bigg]\]

  1. Birch-Murnaghan: 4

\[P_{VT} = 3K_{0T}f_E \big(1 + 2f_E \big)^{5/2} \Bigg[ 1 + \frac{3}{2}\big(K^{\prime}_{0T} - 4\big)f_E +\]
\[\frac{3}{2} \Big(K_{0T}K^{\prime \prime}_{0T}+ \big(K^{\prime}_{0T} - 4\big)\big(K^{\prime}_{0T} - 3\big) + \frac{35}{9} \Big) f^2_E \Bigg]\]
\[f_E = \frac{1}{2}\Bigg[\bigg( \frac{V_{0T}}{V} \bigg)^{\frac{2}{3}} -1 \Bigg]\]

  1. Natural Strain: 5

\[P_{VT} = 3K_{0T}\Big(\frac{V_{0T}}{V_{PT}}\Big)f_N \Bigg[ 1 + af_N + bf^2_N \Bigg]\]
\[f_N = \frac{1}{3} ln\Big(\frac{V_0}{V}\Big)\]
\[a = \frac{3}{2} \big(K^{\prime}_{0T} - 2 \big)\]
\[b = \frac{3}{2} \Big[ 1 + K_{0T} K^{\prime \prime}_{0T} + \big(K^{\prime}_{0T} -2\big) + \big(K^{\prime}_{0T} -2\big)^2 \Big]\]

  1. Vinet: 6

\[P_{VT} = K_{0T} \frac{3f_V}{\big(1-f_V\big)^2} exp\big(\eta f_V\big)\]
\[f_V = 1- \Big( \frac{V_{PT}}{V_{0T}} \Big)^{1/3}\]
\[\eta = \frac{3}{2} \big(K^{\prime}_{0T} - 1 \big)\]

  1. Tait: 7

\[P_{VT} = \frac{1}{b} \Bigg\{ \Bigg[\frac{\big(V_{PT}/V_{0T}\big) + a - 1 }{a} \Bigg]^{-1/c} - 1 \Bigg\}\]
\[a = \frac{1 + K^{\prime}_{0T}}{1 + K^{\prime}_{0T} + K_{0T} K^{\prime \prime}_{0T}}\]
\[b = \frac{K^{\prime}_{0T}}{K_{0T}} - \frac{K^{\prime \prime}_{0T}}{1 + K^{\prime}_{0T}}\]
\[c = \frac{1+K^{\prime}_{0T}+K_{0T} K^{\prime \prime}_{0T}}{\big(K^{\prime}_{0T} \big)^2 + K^{\prime}_{0T} - K_{0T} K^{\prime \prime}_{0T}}\]

where \(V_{0T}\) is the unit cell volume, \(K_{0T}\) is the bulk modulus, \(K^{\prime}_{0T}\) is the first derivative of the bulk modulus with respect to pressure and \(K^{\prime \prime}_{0T}\) is the second derivative of \(K_{0T}\). Worth remembering that the subscripts 0 and T mean that each parameter is obtained at reference pressure zero and reference temperature T.

The modified Tait (T) and Murnaghan (M) EoS are invertible formulations, as it is possible to express the unit cell volume as a function of pressure by inverting the equation. Also, the Tait equation of state can be reduced to the Murnaghan one by imposing \(K^{\prime \prime}_{0T} = 0\).

Birch-Murnaghan (BM), Natural strain (NS) equations of state are finite strain EoS, which were formulated considering that the energy of the compressed solid can be expressed as a Taylor series in the linear strain \(f\) (Eulerian strain, \(f_E\), for the BM EoS and Natural strain, \(f_N\), for the NS EoS). Both of them are 4th-order expansions, but they can be truncated to 3th- and 2nd-order expressions by using implied values for the \(K^{\prime}_{0T}\) and \(K^{\prime \prime}_{0T}\) parameters, respectively.

The Vinet (V) equation of state was derived from molecular mechanics models for very high compression regimes and is a third-order EoS.

Data fitting approach

According to literature and to the reported formulations, the fitting strategy considers the volume as the independent variable and the pressure as the dependent one, as the experimental uncertainties on \(V\) are generally much lower those on \(P\). Then, a least-squares method is used to fit the data, employing the errors on the variables as weights during the procedure. For example, this is the approach adopted by the well-known EOSFit software. 8 The goodness of fit is given by the residual variance (weighted chi-squared, \(\chi^2\)), which is equal to unity (1) if the EoS model perfectly match the weighted experimental data. On the contrary, if \(\chi^2 > 1\) it means that the equation of state correctly represent only a portion of the data, for several possible reasons. Some compression states were not adequately obtained (wrong data), the errors of the values were underestimated or the model is not accurate enough to describe all the data set (for example, it is discouraged to use the Murnaghan EoS for unit cell compressions higher that 10%). A value of \(\chi^2 < 1\) does not represent a better fit and may be also an overfitting of the data.

In Quantas, a different fitting approach in employed, namely the software employs Orthogonal Distance Regression (ODR) to calculate the parameters of the different EoS formulations. 9 With this approach, the experimental uncertainties on pressure can also be included as a weight in the fitting procedure.

Workflow of EoS fitting procedure

The following scheme reports how the Equation of State fitting procedure is performed in Quantas. It is worth noting that the procedure is interactive, meaning that the user has complete control on the choices reported in the graph.

Workflow of the Equation of State fitting procedure followed by Quantas

References

1

Anderson, O.L., 1995. Equation of state of solids for geophysics and ceramic science. Oxford University Press, New York, US.

2

Angel, R.J., Gonzalez-Platas, J., Alvaro, M., 2014. EosFit7c and a Fortran module (library) for equation of state calculations. Z. Kristallogr. 229, 405-419.

3

Murnaghan, F.D., 1937. Finite deformations of an elastic solid. American Journal of Mathematics 49, 235-260.

4

Birch, F., 1947. Finite elastic strain of cubic crystal. Physical Review 71, 809-824.

5

Poirier, J.P., Tarantola, A., 1998. A logarithmic equation of state. Phys. Earth Planet. Inter. 109, 1-8.

6

Vinet, P., Ferrante, J., Rose, J.H., Smith, J.R., 1987. Compressibility of Solids. J Geophys Res-Solid 92, 9319-9325.

7

Freund, J., Ingalls, R., 1989. Inverted Isothermal Equations of State and Determination of B0’ and B0’’. J. Phys. Chem. Solids 50, 263-268.

8

Angel, R.J., 2001. EOS-FIT6.0. Computer Program (http://www.rossangel.com).

9

P. T. Boggs and J. E. Rogers, Orthogonal Distance Regression, in Statistical analysis of measurement error models and applications: proceedings of the AMS-IMS-SIAM joint summer research conference held June 10-16, 1989, Contemporary Mathematics, vol. 112, pg. 186, 1990.