# Hard spheres

Hard spheres are widely used as model particles in the statistical mechanical theory of fluids and solids. They are defined simply as impenetrable spheres that cannot overlap in space. They mimic the extremely strong ("infinitely elastic bouncing") repulsion that atoms and spherical molecules experience at very close distances. Hard spheres systems are studied by analytical means, by molecular dynamics simulations, and by the experimental study of certain colloidal model systems. The hard-sphere system provides a generic model that explains the quasiuniversal structure and dynamics of simple liquids.[1]

## Formal definition

Hard spheres of diameter ${\displaystyle \sigma }$  are particles with the following pairwise interaction potential:

${\displaystyle V(\mathbf {r} _{1},\mathbf {r} _{2})=\left\{{\begin{matrix}0&{\mbox{if}}\quad |\mathbf {r} _{1}-\mathbf {r} _{2}|\geq \sigma \\\infty &{\mbox{if}}\quad |\mathbf {r} _{1}-\mathbf {r} _{2}|<\sigma \end{matrix}}\right.}$

where ${\displaystyle \mathbf {r} _{1}}$  and ${\displaystyle \mathbf {r} _{2}}$  are the positions of the two particles.

## Hard-spheres gas

The first three virial coefficients for hard spheres can be determined analytically

 ${\displaystyle {\frac {B_{2}}{v_{0}}}}$ = ${\displaystyle 4{\frac {}{}}}$ ${\displaystyle {\frac {B_{3}}{{v_{0}}^{2}}}}$ = ${\displaystyle 10{\frac {}{}}}$ ${\displaystyle {\frac {B_{4}}{{v_{0}}^{3}}}}$ = ${\displaystyle -{\frac {712}{35}}+{\frac {219{\sqrt {2}}}{35\pi }}+{\frac {4131}{35\pi }}\arccos {\frac {1}{\sqrt {3}}}\approx 18.365}$

Higher-order ones can be determined numerically using Monte Carlo integration. We list

 ${\displaystyle {\frac {B_{5}}{{v_{0}}^{4}}}}$ = ${\displaystyle 28.24\pm 0.08}$ ${\displaystyle {\frac {B_{6}}{{v_{0}}^{5}}}}$ = ${\displaystyle 39.5\pm 0.4}$ ${\displaystyle {\frac {B_{7}}{{v_{0}}^{6}}}}$ = ${\displaystyle 56.5\pm 1.6}$

A table of virial coefficients for up to eight dimensions can be found on the page Hard sphere: virial coefficients.

Phase diagram of hard sphere system (Solid line - stable branch, dashed line - metastable branch): Pressure ${\displaystyle P}$  as a function of the volume fraction (or packing fraction) ${\displaystyle \eta }$

The hard sphere system exhibits a fluid-solid phase transition between the volume fractions of freezing ${\displaystyle \eta _{\mathrm {f} }\approx 0.494}$  and melting ${\displaystyle \eta _{\mathrm {m} }\approx 0.545}$ . The pressure diverges at random close packing ${\displaystyle \eta _{\mathrm {rcp} }\approx 0.644}$  for the metastable liquid branch and at close packing ${\displaystyle \eta _{\mathrm {cp} }={\sqrt {2}}\pi /6\approx 0.74048}$  for the stable solid branch.

## Hard-spheres liquid

The static structure factor of the hard-spheres liquid can be calculated using the Percus–Yevick approximation.

## Literature

• J. P. Hansen and I. R. McDonald Theory of Simple Liquids Academic Press, London (1986)
• Hard sphere model page on SklogWiki.

## References

1. ^ Dyre, Jeppe C (2016). "Simple liquids' quasiuniversality and the hard-sphere paradigm". Journal of Physics: Condensed Matter. 28 (32): 323001. doi:10.1088/0953-8984/28/32/323001. ISSN 0953-8984.