Hard sphere model

Interaction Potential

The hard sphere intermolecular pair potential is given by

$\Phi \left(r\right)=\left\{{\begin{array}{lll}\infty &;&r<\sigma \\0&;&r\geq \sigma \end{array}}\right.$

where $\Phi \left(r\right)$ is the intermolecular pair potential between two spheres at a distance $r$, and $\sigma$ is the diameter of the sphere.

Equations of state

See: Equations of state for hard spheres (for example, the celebrated Carnahan-Starling equation of state). For the virial equation of state associated with the hard sphere model see: Hard sphere: virial coefficients

Fluid-solid transition

The hard sphere system undergoes a fluid-solid first order transition at $\rho d^{3}=0.94$, $\eta _{A}={\frac {\pi \rho d^{3}}{6}}=0.49218$.

• William G. Hoover and Francis H. Ree "Melting Transition and Communal Entropy for Hard Spheres", Journal of Chemical Physics 49 pp. 3609-3617 (1968)

Solid structure

• Leslie V. Woodcock "Computation of the free energy for alternative crystal structures of hard spheres", Faraday Discussions 106 pp. 325 - 338 (1997)

First 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)

Experimental results

Pusey and van Megen used a suspension of PMMA particles of radius 305 $\pm$10 nm, suspended in poly-12-hydroxystearic acid:

• P. N. Pusey and W. van Megen "Phase behaviour of concentrated suspensions of nearly hard colloidal spheres", Nature 320 pp. 340 - 342 (1986)

For results obtained from the Colloidal Disorder - Order Transition (CDOT) experiments performed on-board the Space Shuttles Columbia and Discovery see Ref. 3.

External links

• Hard disks and spheres computer code on SMAC-wiki.

Related systems

• Polydisperse hard spheres
• Quantum hard spheres
• Dipolar hard spheres

Hard spheres in other dimensions

• 1-dimensional case: hard rods.
• 2-dimensional case: hard disks.
• Hard hyperspheres

References

1. Robin J. Speedy "Pressure of the metastable hard-sphere fluid", Journal of Physics: Condensed Matter 9 pp. 8591-8599 (1997)
2. Robin J. Speedy "Pressure and entropy of hard-sphere crystals", Journal of Physics: Condensed Matter 10 pp. 4387-4391 (1998)
3. 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)
4. W. R. Smith, D. J. Henderson, P. J. Leonard, J. A. Barker and E. W. Grundke "Fortran codes for the correlation functions of hard sphere fluids", Molecular Physics (article ahead of the print edition)