Soft sphere potential: Difference between revisions
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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== | ||
<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> | 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): | ||
:<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== | ==Virial coefficients== | ||
<ref>[ | <ref name="Tan">[ </ref> | ||
==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> | ||
Revision as of 16:57, 17 January 2011
The soft sphere potential is defined as
where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Phi_{12}\left(r \right) } is the intermolecular pair potential between two soft spheres separated by a distance Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r := |\mathbf{r}_1 - \mathbf{r}_2|} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \epsilon } is the interaction strength and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma } 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n\rightarrow \infty} one has the hard sphere model. For Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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] and experssed in terms of Padé approximants. For Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n=6} one has (Eq. 8):
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 one has (Eq. 9):
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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
Solid phase
Glass transition
Transport coefficients
References
- ↑ Jean-Pierre Hansen "Phase Transition of the Lennard-Jones System. II. High-Temperature Limit", Physical Review A 2 pp. 221-230 (1970)
- ↑ 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)
- ↑ 3.0 3.1 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 - ↑ Nigel B. Wilding "Freezing parameters of soft spheres", Molecular Physics 107 pp. 295-299 (2009)
- ↑ D. M. Heyes, S. M. Clarke, and A. C. Brańka "Soft-sphere soft glasses", Journal of Chemical Physics 131 204506 (2009)
- ↑ 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)