Thermodynamic properties of tin: Part I Experimental investigation, ab-initio modelling of α-, β-phase and a thermodynamic description for pure metal in solid and liquid state from 0 K

Thermodynamic modelling of the Ca–O system including 3rd generation description of CaO and CaO2

“…The model was successfully used by various authors [5][6][7] to describe the thermodynamic properties of Mn, Co and Pb. However, a satisfying fit could not be obtained using a single characteristic Einstein temperature for C [4] and Sn [3]. Using multiple Einstein temperatures, Eq.…”

“…It was recommended as a compromise to describe the heat capacity of compounds either using an Einstein or a Debye model with a constant Debye or Einstein temperature and additional terms given in terms of polynomial functions of temperature to account for the rise of anharmonic contributions at higher temperatures. Although the Debye model gives better results in describing the heat capacity at low temperature, the Einstein model was preferred by Chen and Sundman [9] and following modelers [3][4][5][6][7] as it is easier to implement. The heat capacity of a crystalline phases can then be expressed, excluding any potential magnetic transformations, as [9]:…”

“…3, therefore it was removed from the expression to keep it as simple as possible. Then, a T/(B+CT) 3 term was used, where the B parameter allows to shift the heat capacity dampening along the temperature axis, and the C parameter enable to fit the required curvature at the junction. The reason the denominator is up to the power 3 is to avoid having any logarithmic terms appearing when integrating the heat capacity expression to obtain S, H and G, as negative values would then result in errors in the calculations.…”

“…Then, the heat capacity of crystalline CaO was dampened above the selected melting point of 3222 K in order to avoid unrealistic extrapolation at very high temperature. In order to obtain continuous functions at the junction for the heat capacity and its derivative, the T/(B+CT) 3 term proposed in Eq. 9 had to be used instead of the traditional negative powers of T as discussed in the corresponding section.…”

“…The motivation behind this change is that using the traditional polynomial description of the temperature dependence, extrapolations beyond the stability range of the phases may lead to spurious predictions, such as artificial phase stability, or the so-called Kauzmann and inverse Kauzmann paradoxes [2]. Following those guidelines, various authors [3][4][5][6][7][8][9] are currently working toward the development of the so-called 3 rd generation databases.…”