# 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 [1]. 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[2] has recently been studied by Tan, Schultz and Kofke[3] [4] 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[3] 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 [5] 23.24(4) 1.2035(6) 1.1602(7) Table 2 [3]

For $n=9$

 pressure $\rho_{\mathrm {melting}}$ $\rho_{\mathrm {freezing}}$ Reference 36.36(10) 1.4406(12) 1.4053(14) Table 3 [3]

For $n=6$

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

## References

1. Jean-Pierre Hansen "Phase Transition of the Lennard-Jones System. II. High-Temperature Limit", Physical Review A 2 pp. 221-230 (1970)
2. William G. Hoover, Marvin Ross, Keith W. Johnson, Douglas Henderson, John A. Barker and Bryan C. Brown "Soft-Sphere Equation of State", Journal of Chemical Physics 52 pp. 4931-4941 (1970)
3. Tai Boon Tan, Andrew J. Schultz and David A. Kofke "Virial coefficients, equation of state, and solid-fluid coexistence for the soft sphere model", Molecular Physics 109 pp. 123-132 (2011) Cite error: Invalid <ref> tag; name "Tan" defined multiple times with different content Cite error: Invalid <ref> tag; name "Tan" defined multiple times with different content Cite error: Invalid <ref> tag; name "Tan" defined multiple times with different content Cite error: Invalid <ref> tag; name "Tan" defined multiple times with different content
4. N. S. Barlow, A. J. Schultz, S. J. Weinstein, and D. A. Kofke "An asymptotically consistent approximant method with application to soft- and hard-sphere fluids", Journal of Chemical Physics 137 204102 (2012)
5. Nigel B. Wilding "Freezing parameters of soft spheres", Molecular Physics 107 pp. 295-299 (2009)
6. D. M. Heyes, S. M. Clarke, and A. C. Brańka "Soft-sphere soft glasses", Journal of Chemical Physics 131 204506 (2009)
7. Junko Habasaki and Akira Ueda "Several routes to the glassy states in the one component soft core system: Revisited by molecular dynamics", Journal of Chemical Physics 134 084505 (2011)
8. D. M. Heyes and A. C. Branka "Density and pressure dependence of the equation of state and transport coefficients of soft-sphere fluids", Molecular Physics 107 pp. 309-319 (2009)
9. A. C. Brańka and D. M. Heyes "Pair correlation function of soft-sphere fluids", Journal of Chemical Physics 134 064115 (2011)