Soft sphere potential

The soft sphere potential is defined as

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

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 \(n \le 3\) 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 \(k_BT/\epsilon=1.0\) 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 \(k_BT/\epsilon=1.0\) (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	\(\rho_{\mathrm {melting}}\)	\(\rho_{\mathrm {freezing}}\)	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	\(\rho_{\mathrm {melting}}\)	\(\rho_{\mathrm {freezing}}\)	Reference
36.36(10)	1.4406(12)	1.4053(14)	Table 3

For \(n=6\)

pressure	\(\rho_{\mathrm {melting}}\)	\(\rho_{\mathrm {freezing}}\)	Reference
100.1(3)	2.320(2)	2.295(2)	Table 4