Equations of State & Elasticity
This page is a very brief introduction to the concept of Equations of State of solids. If you want more information read the following paper and the references given in it:

Angel RJ, Gonzalez-Platas J, Alvaro M (2014) EosFit-7c and a Fortran module (library) for equation of state calculations. Zeitschrift für Kristallographie, 229, 405-419. A reprint is available as a pdf file, Copyright © De Gruyter.

Equations of state (EoS) describe how the volume or density of a material varies with changes in pressure and temperature. They also define how some of the elastic properties of the material change in response to compression and expansion. Equations of state therefore provide fundamental thermodynamic data that is required for the calculation of equilibrium phase diagrams, and they provide insights in to the details of interatomic interactions within the solid state, as it is these that resist the externally-applied compressive stresses and control the dynamics that lead to thermal expansion.

Isothermal equations of state

The variation of the volume of a solid with hydrostatic pressure at fixed temperature is termed its ‘isothermal equation of state’. It is characterised by the bulk modulus of the material, K = -V(dP/dV) , which is a function of both temperature and pressure. For infinitesimal changes in pressure which give rise to infinitesimal changes in volume, the bulk modulus can also be defined in terms of the elastic tensor of the material by applying linear elasticity theory, or ‘Hooke’s law’. Hydrostatic pressure is a special stress state in which the normal stresses are all equal and there are no shear stresses. Thus, at any pressure P, the strains caused by an infinitesimal increase in pressure can be calculated from linear elasticity theory. Such an algebraic exercise shows that the bulk modulus for hydrostatic compression of a solid, whether a powder or single crystal, is equal to the Reuss bound on the bulk modulus of a polycrystal made of the same material where it represents the volume response when every constituent grain is subject to the same stress.

While linear elasticity thus defines the bulk modulus of a material under hydrostatic compression at any pressure, it cannot define an equation of state which describes the large (finite) changes in volume due to large (finite) changes in pressure. In this sense an equation of state is an extension of linear elasticity; although normally defined in terms of the volume variation with pressure, it can also be seen as a definition of the variation of bulk modulus with pressure. Because there is no absolute thermodynamic basis for specifying how the bulk modulus K varies with pressure, all EoS that have been developed and that are in widespread use are based upon a number of assumptions. The validity of such assumptions can only be judged in terms of whether the derived EoS reproduces experimental data for volume or elasticity. For materials that do not exhibit phase transitions, isothermal equations of state are usually parameterized in terms of the values of the bulk modulus and its pressure derivatives, K' = dK/dP and K''=dK'/dP, evaluated at a reference pressure, normally taken as zero pressure.

Thermal expansion

The volume thermal expansion of a material is defined as alpha = 1/V(dV/dT). The only thermodynamic constraints on the form of the function for alpha are that alpha = d(alpha)/dT = 0 at absolute zero. Consequently many different forms have been proposed in the literature. Some simple formulations that describe thermal expansion at high temperatures very well do not include the low-T saturation, but are widely used and are perfectly adequate for thermodynamic databases. On the other hand, some equations explicitly handle the saturation in thermal expansion as the temperature drops towards absolute zero, but these often result in unphysical values of thermal expansion at high temperatures in excess of 1000 K, where experiments indicate that alpha increases approximately linearly with temperature.

P-V-T equations of state

Equations to describe the variation of volume with both pressure and temperature can be developed by combining any thermal expansion model with any isothermal equation of state, and a model of the variation of bulk modulus with temperature at room pressure, dK/dT. The simplest approach of assuming a linear variation of with temperature, so dK/dT is constant, is certainly justified at high temperatures by direct measurements of the bulk moduli of many materials by elasticity measurements.

However, as Hellfrich & Connolly (2009) pointed out, this formulation with a constant dK/dT often leads to the prediction of non-physical negative thermal expansion coefficients at reasonably modest pressures for a large number of materials. We have recently introduced a new way of handling the temperature variation of the bulk modulus and it's pressure derivative (K'=dK/dP) that is thermodynamically-correct at low temperatures and avoids negative thermal expansion at high pressures. This new 'isothermal' EoS has been added to the EosFit program and is described in detail in a paper by Angel et al. (2017).

Another approach that yields indistinguishable P-V-T relationships and also avoids negative values of thermal expansion, is the concept of thermal pressure. The idea of thermal pressure is that the total pressure at a given V and T can be expressed as the sum of two terms:
P(V,T) = P(V,Tref) + Pth(T)
The function P(V,Tref) is the isothermal equation of state for the material at the reference temperature Tref, but using the ‘observed’ volume from P and T. The thermal-pressure function Pth(T) is the pressure that would be created by increasing the temperature from Tref to T at constant volume at room pressure. The thermal pressure at Tref is thus zero, so at Tref the thermal-pressure EoS reduces to the isothermal EoS.

Determining EoS

The majority of equation of state studies measure the volume or unit-cell parameter variations with pressure (and/or temperature), with the aim of deriving elastic parameters that are derivatives of the data. The bulk moduli of most inorganic solids range from ~40 to 400 GPa; organic molecular solids and frameworks are softer. Therefore the volume changes induced by compression over the relatively-easily accessible experimental pressure range (0-10 GPa) are only larger than the experimental uncertainties by 1 or 2 orders of magnitude. The need to obtain parameters that are derivatives of the original data, in combination with the small data range, makes the reliable determination of the parameters difficult to achieve. Strong correlations between parameters, which include the pressure derivatives of K, exacerbate the problems. These problems are overcome in the EosFit program which provides the capability of fitting P-V (and unit-cell parameter data) with various EoS, with the correct algebra and with options to fully weight the data with the measurement uncertainties. The Eosfit program includes utilities to do further EoS calculations beyond just fitting EoS parameters to P-V data.

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