Soft sphere potential: Difference between revisions

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(Added two EOS (for n=6 and 9))
(Added table of virial coefficients)
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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.
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==
==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> and experssed in terms of [[Padé approximants]]. For <math>n=6</math> one has (Eq. 8):
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> and expressed in terms of [[Padé approximants]]. For <math>k_BT/\epsilon=1.0</math> and <math>n=6</math> one has (Eq. 8):




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:<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>
:<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==
==Virial coefficients==
<ref name="Tan">[ </ref>
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)
|}
==Solid phase==
==Solid phase==
<ref>[http://dx.doi.org/10.1080/00268970802603507 Nigel B. Wilding "Freezing parameters of soft spheres", Molecular Physics '''107''' pp. 295-299 (2009)]</ref>
<ref>[http://dx.doi.org/10.1080/00268970802603507 Nigel B. Wilding "Freezing parameters of soft spheres", Molecular Physics '''107''' pp. 295-299 (2009)]</ref>
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==References==
==References==
<references/>
<references/>
{{numeric}}
[[category: models]]
[[category: models]]

Revision as of 17:08, 17 January 2011

The soft sphere potential is defined as

where is the intermolecular pair potential between two soft spheres separated by a distance , is the interaction strength and is the diameter of the sphere. Frequently the value of is taken to be 12, thus the model effectively becomes the high temperature limit of the Lennard-Jones model [1]. If one has the hard sphere model. For 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] and expressed in terms of Padé approximants. For and one has (Eq. 8):



and for one has (Eq. 9):


Virial coefficients

Tan, Schultz and Kofke[3] have calculated the virial coefficients at (Table 1):

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

Solid phase

[4]

Glass transition

[5]

Transport coefficients

[6]

References

This page contains numerical values and/or equations. If you intend to use ANY of the numbers or equations found in SklogWiki in any way, you MUST take them from the original published article or book, and cite the relevant source accordingly.