Hard spheres

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.

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.[2]

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.

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

• Classical fluid

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