Thermal equation of state of solids

• Comment: The article obviously has merit, but the writing style such as "Based on the discussion above" or "for detail, please refer to the original paper" is not the correct tone for Wikipedia. ✠ SunDawn ✠ (contact) 03:27, 7 November 2023 (UTC)

• Comment: This is an essay, it needs to be rewritten as an encyclopedoa article Stuartyeates (talk) 00:29, 3 September 2023 (UTC)

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Thermal pressure/thermal expansion in the thermal equation of the states of solids

Thermal equation of state of solids

The thermal equation of state (TEOS) is a mathematical expression of pressure, temperature, and, volume. For the solids, the TEOS, P(V, T) = P(V, T₀) + P_th(V, T), is the pressure as a function of temperature and volume, and abbreviated as PVT TEOS. It consists of an isothermal compression P(V, T₀) at room temperature, and an isochoric heating (P_th(V, T) at high temperature. The latter is called thermal pressure, which is the pressure change in isochoric heating. For the isothermal compression term, both the Mao ruby scale of Ag, Cu, Mo, Pd up to 1 Mba , and third-order Birch–Murnaghan equation of state of Ag, Cu, and MgO are consistent in a wide pressure range. However, at high P-T conditions, there have been large discrepancies in pressure determination using different pressure standards or different thermal equations of state for the same standard , and Fig.1 is one schematic plot showing the discrepancy in paper .

As the pressures in room temperature isothermal compression P(V, T₀) are consistent, the discrepancy of the total pressure should be from the thermal pressure P_th(V, T) at high temperature.

Anderson thermal pressure model

The first and also the most common thermal pressure model is the Anderson model. It was first developed for the thermal gradient (∂T/∂P)_v=(α•K)^-1 in 1968 , and then used to correlate the pressure and temperature in an isochoric heating process by (∂P/∂T)_v=α•K in 1992 . The pressure change in this process is called thermal pressure, which is the integration of α•K. In other words, thermal pressure is the pressure to keep the volume constant during an isochoric heating process.

Experimentally determine the Anderson thermal pressure

As the thermal pressure is the pressure change in an isochoric heating, to experimentally measure the thermal pressure, it should meet two requirements, (1) reliable pressure determination at high P-T conditions, (2) the heating process is isochoric. As mentioned above, there are large discrepancies in total pressure determination using different pressure standards or different thermal equations of state for the same standard, therefore, it is challenging to accurate thermal pressure determination at high P-T conditions.

An isochoric heating can be easily achieved for a gas or fluid. When a gas/fluid is filled in a solid firm container, the thermal expansions of gas/fluid are constrained inside the container, and the heating is isochoric. On the other hand, the heating for a solid sample in a diamond anvil cell (DAC) or large volume press (LVP) is not the case. When a solid sample is loaded into a soft pressure medium in a DAC, the samples can thermally expand inside its surrounding soft pressure medium, therefore, the heating is not isochoric heating for the sample in a DAC at high P-T conditions. Therefore, thermal pressure from a DAC or LVP can't be accurately determined directly at high P-T conditions.

Model calculated Anderson thermal pressure

According to the most common and the first thermal pressure model, Anderson model, thermal pressure is the integration of the product α_p and K_T,i.e. P_th=∫(α•K)dt. Here both α_p and K_T are pressure dependent and temperature dependent, so integrating the α_p and K_T from Anderson model is not straight forward. To bypass this issue, an approximation to reduce the P-T dependent α_p and K_T to a constant α₀ and K₀ was adopted . But in publication , authors demonstrated that the reduced Anderson model prediction of Au and MgO at ambient pressure from its experimental values, and the higher temperature, the higher deviate. A cartoon plot for the pressures predicted from PVT TEOS in paper is shown in Fig. 2, and for detail, please refer to the original paper.

An alternate way to make the integration of (α_p·K_T) possible, authors assumes a pressure independent thermal expansion to reduce the P-T dependent α_p and K_T to only temperature dependent α and K_T, but in the preprint paper preprint, author proved that pressure independent thermal expansion leads to the bulk modulus to be temperature independent, which again reduce the P-T dependent α_p and K_T to constant α₀ and K₀.

There are various thermal pressure models, but to prove these models, accurately experimentally determined thermal pressures are required.

Thermal expansion model

Based on the discussion above, the thermal pressure can neither be accurately determined experimentally, nor accurately calculated from the most common model, a thermal expansion model was recommended to replace it, which is expressed the volume as a function of pressure and temperature, called VPT TEOS. In this paper, the pressure in an isothermal compression from V₀ to V₁ at room temperature is expressed as

P=f^-1(T₀,V₁,V₀,K₀,K₀') ....................... (1)

After integrating the isothermal compression at room temperature with the thermal expansion at high pressure, the general form of VPT TEOS is expressed as

P=f^-1(T₀,V_M,V₀,K₀,K₀') ........................(2)

While V_M= V·exp(-∫α_p·dx) in an isobaric heating process, while V is the volume after the isobaric heating. For the detail of how to derive Eq (2), please refer to paper. Here the mathematical expression of f^-1 could be any of the isothermal compression models, like Murnaghan, modified Tait, natural strain, Vinet, Birch-Murnagha, and others.

To partially prove the VPT TEOS, a set of x-ray diffraction data at various temperatures at ambient pressure was collected. In this case, the ambient pressure P=0 GPA, so, the volume, pressure, and temperature are all known, then, the model predicted pressure value at (V, T) could be compared with the known experimental value of 0 GPa, and it matches well. In the future, the validation of the VPT TEOS at high P-T conditions is required.

The pressure dependent α_p has to be determined from an isobaric heating process. It has been reported that the DAC heatings at high P-T with membrane were isobaric. Authors in the paper propose a reversible isobaric heating concept, in which the plotted heating data and cooling data stay on the same curve, authors thought this heating and cooling process is very close to the ideal isobaric. A cartoon plot of reversible heating/cooling proposed in paper is shown as Fig. 3

, and for detail, please refer to the original paper. In a reversible heating process, no pressure determination at high P-T is required, and accurate pressure determination at high P-T is not easy at the moment.