Hard sphere model: Difference between revisions
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[[Image:sphere_green.png|thumb|right]] | [[Image:sphere_green.png|thumb|right]] | ||
== Interaction Potential == | == Interaction Potential == | ||
The hard sphere | The hard sphere [[intermolecular pair potential]] is given by | ||
: <math> | : <math> | ||
\Phi\left( r \right) = \left\{ \begin{array}{lll} | |||
\infty & ; & r < \sigma \\ | \infty & ; & r < \sigma \\ | ||
0 & ; & r \ge \sigma \end{array} \right. | 0 & ; & r \ge \sigma \end{array} \right. | ||
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where <math> | where <math> \Phi\left(r \right) </math> is the potential energy between two spheres at a distance <math> r </math>, and <math> \sigma </math> is the diameter of the sphere. | ||
== Equations of state == | == Equations of state == | ||
Revision as of 13:36, 21 June 2007
Interaction Potential
The hard sphere intermolecular pair potential is given by
- 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\left( r \right) = \left\{ \begin{array}{lll} \infty & ; & r < \sigma \\ 0 & ; & r \ge \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\left(r \right) } is the potential energy between two spheres at 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 } , 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.
Equations of state
Hard sphere fluid:
- See Carnahan-Starling (three dimensions)
- See Ref.1
Hard sphere solid:
- See Ref. 2
Fluid-solid transition
The hard sphere fluid undergoes a fluid-solid first order transition 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 \rho d^3 = 0.94} , .
First ever simulations of hard spheres
- Marshall N. Rosenbluth and Arianna W. Rosenbluth "Further Results on Monte Carlo Equations of State", Journal of Chemical Physics 22 pp. 881-884 (1954)
- W. W. Wood and J. D. Jacobson "Preliminary Results from a Recalculation of the Monte Carlo Equation of State of Hard Spheres", Journal of Chemical Physics 27 pp. 1207-1208 (1957)
- B. J. Alder and T. E. Wainwright "Phase Transition for a Hard Sphere System", Journal of Chemical Physics 27 pp. 1208-1209 (1957)
Related systems
Other dimensions
- 1-dimensional case: hard rods.
- 2-dimensional case: hard disks.
- Hard hyperspheres
Data
Virial coefficients of hard spheres and hard disks are to be found in the table.
Experimental results
For results obtained from the Colloidal Disorder - Order Transition (CDOT) experiments performed on-board the Space Shuttles Columbia and Discovery see Ref. 3.
References
- Robin J. Speedy "Pressure of the metastable hard-sphere fluid", Journal of Physics: Condensed Matter 9 pp. 8591-8599 (1997)
- Robin J. Speedy "Pressure and entropy of hard-sphere crystals", Journal of Physics: Condensed Matter 10 pp. 4387-4391 (1998)
- Z. Chenga, P. M. Chaikina, W. B. Russelb, W. V. Meyerc, J. Zhub, R. B. Rogersc and R. H. Ottewilld, "Phase diagram of hard spheres", Materials & Design 22 pp. 529-534 (2001)