Modification of Lu's (2005) high pressure model for improved high pressure/high temperature extrapolations. Part I: Modeling of platinum at high pressure/high temperature
Introduction
Modeling of both thermodynamic properties and phase diagrams at high pressure is very important not only for geological applications. It is also needed for the description of materials under thermomechanical stress, impact or explosion, for the prediction of molar volume... High pressure properties are obviously also strongly related to the basic physics of the materials. The models used must combine the possibility to reproduce well the data, the ease of use and accurate extrapolations. There exist different high pressure models in the literature that can be used in the frame of the Calphad methodology: Mie-Grüneisen (1912), Murnaghan (1944), Birch-Murnaghan (1947), Refs. [[1], [2], [3], [4], [5], [6], [7], [8]]. However, there is no consensus about which is preferable.
Among them, the model by Lu et al. [2] has had indisputable achievements. It is particularly interesting because it has been implemented in the Thermo-Calc software, it is relatively easy to handle, it can treat any phase, solid or liquid, stable or metastable, and it follows the Calphad formalism in its extrapolation possibilities to multi-component systems. However, it has been criticized for leading to sometimes erroneous extrapolations at high pressure and/or temperature (negative entropy and specific heat). We will describe the limitations of this model in more detail and their origin will be investigated. Solutions will be proposed in order to avoid the problems of extrapolation. It will be finally demonstrated that it may be used safely with the proposed modifications and that, in most cases, it should be preferred as being simpler, easier to handle and because it is better adapted to the frame of the Calphad method, to the description of complete systems and to the extrapolation to multicomponent systems.
As an application, the example of platinum at extreme pressure and temperature will be detailed in the present paper. A thermodynamic assessment of the available experimental data together with calculated data obtained in this work will be described. In the second part of this study [9], this model will be applied to the complete description of Os–Pt binary system .
Section snippets
Description of Lu's model
In high pressure models, the Gibbs energy is described with an additional contribution taking into account the temperature and pressure dependence of the volume:
In Lu's model [2], the volume is described as a function of pressure using the Grover equation, itself derived from the Murnaghan model (see the original paper for the complete description and derivation of the equations). Finally, the volume is calculated as a function of temperature and pressure in Equation (9)
Reported problems
Soon criticisms about this model appeared in the literature. The main criticism is that the model yields nonphysical extrapolated values of Cp and entropy. As an example, in Fig. 1, the Cp of bcc Fe is calculated from the assessment of Lu et al. [2]. The figure shows that negative Cp are obtained at high temperature at pressures above 100 GPa.
Brosh et al. [5] analyzed the problem as being due to an ‘incompatibility’ between the SGTE description of the Cp and the Mie-Grüneisen equation. By
Possible solution
It is always possible to fit the volume as a linear function of temperature (no curvature in the volume curve or constant thermal expansion). As can be inferred from Eq. (9), it would allow to maintain a constant Cp as a function of pressure. This would however be a crude approximation resulting in a poor description of the data and therefore of the thermodynamic properties. Another option is to keep a good description of the low pressure thermal expansion data but make the thermal expansion
Application
We present here an application of the modified model to the description of platinum. This system has been chosen because a lot of data is available up to very high pressure. It is also important because platinum may be used as a pressure standard in high pressure diffraction experiments. Both fcc and liquid phases will be described allowing a calculation of the pressure–temperature phase diagram.
Other models
As observed in the introduction, many models are available in the literature. We will limit our comparison to the most recent ones [[5], [6], [7],51]. These models are based on the quasi-harmonic approximation, the Debye-Grüneisen model, or the modeling of phonon frequencies, respectively. They offer very sound and complete approaches encompassing a closer relation to the physics and a superior use of DFT data and phonon calculations.
However, as a consequence of their link to the DFT or phonon
Conclusion
A modification has been proposed to Lu's (2005) model to suppress the anomalous extrapolation of specific heat and entropy at high temperatures and pressures. With this modification, we have shown that such anomalies are not present in a typical system like platinum which has been described completely including the liquid phase. A model is always a tradeoff between accuracy, predictability and usability. Modified Lu's model is indeed so easy to handle compared to many other models that it
Declaration of competing interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgements
DFT calculations were performed using HPC resources from GENCI CINES (Grant A0080906175).
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