Soft sphere potential

The soft sphere potential is defined as

$\Phi _{12}\left(r\right)=\left\{{\begin{array}{lll}\epsilon \left({\frac {\sigma }{r}}\right)^{n}&;&r\leq \sigma \\0&;&r>\sigma \end{array}}\right.$

where $\Phi _{12}\left(r\right)$ is the intermolecular pair potential between two soft spheres separated by a distance $r:=|\mathbf {r} _{1}-\mathbf {r} _{2}|$, $\epsilon$ 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 $n\rightarrow \infty$ one has the hard sphere model. For $n\leq 3$ 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):

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