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== | ||
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): | 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): | |||
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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== | ||
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): | 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): | ||
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==Glass transition== | ==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> | <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><ref>[http://dx.doi.org/10.1063/1.3554378 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)]</ref> | ||
==Transport coefficients== | ==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> | <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== | ||
<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}} | {{numeric}} | ||
[[category: models]] | [[category: models]] | ||
Latest revision as of 14:49, 5 April 2016
The soft sphere potential is defined as
- 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) = \left\{ \begin{array}{lll} \epsilon \left( \frac{\sigma}{r}\right) ^n & ; & r \le \sigma \\ 0 & ; & r > \sigma \end{array} \right. }
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 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} 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[edit]
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 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 k_BT/\epsilon=1.0} 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 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 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=9}
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[edit]
Tan, Schultz and Kofke[3] have calculated the virial coefficients at 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 k_BT/\epsilon=1.0} (Table 1):
n=12 n=9 n=6 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 B_3} 3.79106644 4.27563423 5.55199919 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 B_4} 3.52761(6) 3.43029(7) 1.44261(4) 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 B_5} 2.1149(2) 1.08341(7) -1.68834(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 B_6} 0.7695(2) -0.21449(11) 1.8935(5) 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 B_7} 0.0908(5) -0.0895(7) -1.700(3) 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 B_8} -0.074(2) 0.071(4) 0.44(2)
Melting point[edit]
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=12}
pressure 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 \rho_{\mathrm {melting}}} 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 \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 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=9}
pressure 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 \rho_{\mathrm {melting}}} 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 \rho_{\mathrm {freezing}}} Reference 36.36(10) 1.4406(12) 1.4053(14) Table 3 [3]
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}
pressure 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 \rho_{\mathrm {melting}}} 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 \rho_{\mathrm {freezing}}} Reference 100.1(3) 2.320(2) 2.295(2) Table 4 [3]
Glass transition[edit]
Transport coefficients[edit]
Radial distribution function[edit]
References[edit]
- ↑ 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 3.2 3.3 3.4 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)
- ↑ 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)
- ↑ 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)
- ↑ 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)
- ↑ 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)
- ↑ A. C. Brańka and D. M. Heyes "Pair correlation function of soft-sphere fluids", Journal of Chemical Physics 134 064115 (2011)
- Related reading
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