Editing Soft sphere potential

The '''soft sphere potential''' is defined as

: <math>
\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.
</math>

where <math> \Phi_{12}\left(r \right) </math> is the [[intermolecular pair potential]] between two soft spheres separated by  a distance <math>r := |\mathbf{r}_1 - \mathbf{r}_2|</math>, <math>\epsilon </math> is the interaction strength and <math> \sigma </math> is the diameter of the sphere. Frequently the value of <math>n</math> is taken to be 12, thus the model effectively becomes the high temperature limit of the [[Lennard-Jones model]] <ref>[http://dx.doi.org/10.1103/PhysRevA.2.221 Jean-Pierre Hansen "Phase Transition of the Lennard-Jones System. II. High-Temperature Limit", Physical Review A  '''2''' pp. 221-230 (1970)]</ref>. If <math>n\rightarrow \infty</math> one has the [[hard sphere model]]. For <math>n \le 3</math> no thermodynamically stable phases are found.
==Equation of state==
The soft-sphere [[Equations of state | equation of state]]<ref>[http://dx.doi.org/10.1063/1.1672728 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)]</ref> has recently been studied by Tan, Schultz and Kofke<ref name="Tan">[http://dx.doi.org/10.1080/00268976.2010.520041 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)]</ref> 
<ref>[http://dx.doi.org/10.1063/1.4767065  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)]</ref>
and expressed in terms of [[Padé approximants]]. For <math>k_BT/\epsilon=1.0</math> and <math>n=6</math> one has (Eq. 8):


:<math>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}</math>


and for <math>n=9</math> one has (Eq. 9):


:<math>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}</math>

==Virial coefficients==
Tan, Schultz and Kofke<ref name="Tan"> </ref> have calculated the [[Virial equation of state | virial coefficients]] at <math>k_BT/\epsilon=1.0</math> (Table 1):

:{| border="1"
|- 
| || n=12 || n=9 || n=6
|-  
| <math>B_3</math>  || 3.79106644 || 4.27563423  || 5.55199919
|-  
| <math>B_4</math>  || 3.52761(6)  || 3.43029(7) || 1.44261(4)
|-  
| <math>B_5</math>  || 2.1149(2)  || 1.08341(7) || -1.68834(9)
|-  
| <math>B_6</math>  || 0.7695(2)  || -0.21449(11) || 1.8935(5)
|-  
| <math>B_7</math>  || 0.0908(5)  || -0.0895(7)  || -1.700(3)
|-  
| <math>B_8</math>  || -0.074(2)  || 0.071(4) || 0.44(2)
|}
==Melting point==
For <math>n=12</math>
:{| border="1"
|- 
| pressure || <math>\rho_{\mathrm {melting}}</math> || <math>\rho_{\mathrm {freezing}}</math>  || Reference
|-  
| 22.66(1) || 1.195(6) || 1.152(6) || Table 1 <ref>[http://dx.doi.org/10.1080/00268970802603507 Nigel B. Wilding "Freezing parameters of soft spheres", Molecular Physics '''107''' pp. 295-299 (2009)]</ref>
|-  
| 23.24(4) || 1.2035(6) || 1.1602(7) || Table 2 <ref name="Tan"> </ref>
|}
For <math>n=9</math>
:{| border="1"
|- 
| pressure || <math>\rho_{\mathrm {melting}}</math> || <math>\rho_{\mathrm {freezing}}</math>  || Reference
|-  
| 36.36(10) || 1.4406(12) || 1.4053(14) || Table 3 <ref name="Tan"> </ref>
|}
For <math>n=6</math>
:{| border="1"
|- 
| pressure || <math>\rho_{\mathrm {melting}}</math> || <math>\rho_{\mathrm {freezing}}</math>  || Reference
|-  
| 100.1(3) || 2.320(2) || 2.295(2) || Table 4 <ref name="Tan"> </ref>
|}
==Glass transition==
<ref>[http://dx.doi.org/10.1063/1.3266845 D. M. Heyes, S. M. Clarke, and A. C. Brańka "Soft-sphere soft glasses", Journal of Chemical Physics '''131''' 204506 (2009)]</ref>
==Transport coefficients==
<ref>[http://dx.doi.org/10.1080/00268970802712563 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)]</ref>
==Radial distribution function==
<ref>[http://dx.doi.org/10.1063/1.3554363 A. C. Brańka and D. M. Heyes "Pair correlation function of soft-sphere fluids", Journal of Chemical Physics '''134''' 064115 (2011)]</ref>
==References==
<references/>
;Related reading
*[http://dx.doi.org/10.1063/1.4944824  Sergey A. Khrapak "Note: Sound velocity of a soft sphere model near the fluid-solid phase transition", Journal of Chemical Physics '''144''' 126101 (2016)]

{{numeric}}
[[category: models]]