Soft sphere potential

The soft sphere potential is defined as

${\mathrm{\Phi }}_{12}(r)=\{\begin{array}{ccc}ϵ{(\frac{\sigma }{r})}^n& ;& r\le \sigma \\ 0& ;& r>\sigma \end{array}$

where ${\mathrm{\Phi }}_{12}(r)$ is the intermolecular pair potential between two soft spheres separated by a distance $r:=|{\mathbf{r}}_1-{\mathbf{r}}_2|$, $ϵ$ is the interaction strength and $\sigma$ is the diameter of the sphere. Frequently the value of $n$ is taken to be 12, thus the model effectively becomes the high temperature limit of the Lennard-Jones model . If $n\to \mathrm{\infty }$ one has the hard sphere model. For $n\le 3$ no thermodynamically stable phases are found.

Equation of state

The soft-sphere equation of state has recently been studied by Tan, Schultz and Kofke and experssed in terms of Padé approximants. For $n=6$ one has (Eq. 8):

$Z_{n=6}=\frac{1+7.432255\rho +23.854807{\rho }^2+40.330195{\rho }^3+34.393896{\rho }^4+10.723480{\rho }^5}{1+3.720037\rho +4.493218{\rho }^2+1.554135{\rho }^3}$

and for $n=9$ one has (Eq. 9):

$Z_{n=9}=\frac{1+3.098829\rho +5.188915{\rho }^2+5.019851{\rho }^3+2.673385{\rho }^4+0.601529{\rho }^5}{1+0.262771\rho +0.168052{\rho }^2-0.010554{\rho }^3}$