The Simon–Glatzel equation^[1] is an empirical correlation describing the pressure dependence of the melting temperature of a solid. The pressure dependence of the melting temperature is small for small pressure changes because the volume change during fusion or melting is rather small. However, at very high pressures higher melting temperatures are generally observed as the liquid usually occupies a larger volume than the solid making melting more thermodynamically unfavorable at elevated pressure. If the liquid has a smaller volume than the solid (as for ice and liquid water) a higher pressure leads to a lower melting point.

The equation and its variations

${\displaystyle T_{m}=T_{\text{ref}}\left({\frac {P-P_{\text{ref}}}{a}}+1\right)^{b}}$

${\displaystyle T_{\text{ref}}}$ and ${\displaystyle P_{\text{ref}}}$ are normally the temperature and the pressure of the triple point, but the normal melting temperature at atmospheric pressure are also commonly used as reference point because the normal melting point is much more easily accessible. Typically ${\displaystyle P_{\text{ref}}}$ is then set to 0. ${\displaystyle a}$ and ${\displaystyle b}$ are component-specific parameters.

The Simon–Glatzel equation can be viewed as a combination of the Murnaghan equation of state and the Lindemann law,^[2] and an alternative form was proposed by J. J. Gilvarry (1956):^[3]

${\displaystyle T_{m}=T_{\text{ref}}\left(K_{0}^{'}{\frac {P-P_{\text{ref}}}{K_{0}}}+1\right)^{\frac {2(\gamma -1)}{3+f}}}$

where ${\displaystyle K_{0}}$ is general ${\displaystyle K}$ at ${\displaystyle P=0}$, ${\displaystyle K_{0}^{'}}$ is pressure derivative ${\displaystyle K}$ at ${\displaystyle P=0}$, ${\displaystyle \gamma }$ is Grüneisen ratio, and ${\displaystyle f}$ is the coefficient in Morse potential.

Example parameters

Methanol melting temperature versus pressure

For methanol the following parameters^[4] can be obtained:

The reference temperature has been T_ref = 174.61 K and the reference pressure P_ref has been set to 0 kPa.

Methanol is a component where the Simon–Glatzel works well in the given validity range.

Extensions and generalizations

The Simon–Glatzel equation is a monotonically increasing function. It can only describe the melting curves that rise indefinitely with increasing pressure. It may fail to describe the melting curves with a negative pressure dependence or local maximums. A damping term that asymptotically slopes down under pressure, ${\displaystyle D(P)=\exp[-c(P-P_{\text{ref}})]}$ (c is another component-specific parameter), is introduced by Vladimir V. Kechin to extend the Simon–Glatzel equation^[5] so that all melting curves, rising, falling, and flattening, as well as curves with a maximum, can be described by a unified equation:

${\displaystyle T_{m}=F(P)\cdot D(P)}$

where ${\displaystyle F(P)}$ is the Simon–Glatzel equation (rising) and ${\displaystyle D(P)}$ is the damping term (falling or flattening).

The unified equation may be rewritten as:

${\displaystyle T_{m}=T_{\text{ref}}\left({\frac {P-P_{\text{ref}}}{a}}+1\right)^{b}\cdot \exp[-c(P-P_{\text{ref}})]}$

This form predicts that all solids have a maximum melting temperature at a positive or (fictitious) negative pressure.

References