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 expressed in terms of Padé approximants. For $k_{B}T/ϵ=1.0$ and $n=6$ 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):

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

Virial coefficients

Tan, Schultz and Kofke have calculated the virial coefficients at $k_{B}T/ϵ=1.0$ (Table 1):

	n=12	n=9	n=6
$B_3$	3.79106644	4.27563423	5.55199919
$B_4$	3.52761(6)	3.43029(7)	1.44261(4)
$B_5$	2.1149(2)	1.08341(7)	-1.68834(9)
$B_6$	0.7695(2)	-0.21449(11)	1.8935(5)
$B_7$	0.0908(5)	-0.0895(7)	-1.700(3)
$B_8$	-0.074(2)	0.071(4)	0.44(2)

Melting point

For $n=12$

pressure	${\rho }_{\mathrm{m}\mathrm{e}\mathrm{l}\mathrm{t}\mathrm{i}\mathrm{n}\mathrm{g}}$	${\rho }_{\mathrm{f}\mathrm{r}\mathrm{e}\mathrm{e}\mathrm{z}\mathrm{i}\mathrm{n}\mathrm{g}}$	Reference
22.66(1)	1.195(6)	1.152(6)	Table 1
23.24(4)	1.2035(6)	1.1602(7)	Table 2

For $n=9$

pressure	${\rho }_{\mathrm{m}\mathrm{e}\mathrm{l}\mathrm{t}\mathrm{i}\mathrm{n}\mathrm{g}}$	${\rho }_{\mathrm{f}\mathrm{r}\mathrm{e}\mathrm{e}\mathrm{z}\mathrm{i}\mathrm{n}\mathrm{g}}$	Reference
36.36(10)	1.4406(12)	1.4053(14)	Table 3

For $n=6$

pressure	${\rho }_{\mathrm{m}\mathrm{e}\mathrm{l}\mathrm{t}\mathrm{i}\mathrm{n}\mathrm{g}}$	${\rho }_{\mathrm{f}\mathrm{r}\mathrm{e}\mathrm{e}\mathrm{z}\mathrm{i}\mathrm{n}\mathrm{g}}$	Reference
100.1(3)	2.320(2)	2.295(2)	Table 4