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Hard spheres

Beside being a model of theoretical significance, the hard-sphere system is used as a basis in the formulation of several modern, predictive Equations of State for real fluids through the SAFT approach, and models for transport properties in gases through Chapman-Enskog Theory.

Formal definition[edit]

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[edit]

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.^[1]

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[edit]

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

The Carnahan-Starling Equation of State[edit]

A simple, yet popular equation of state describing systems of pure hard spheres was developed in 1969 by N. F. Carnahan and K. E. Starling.^[2] By expressing the compressibility of a hard-sphere system as a geometric series, the expression

${\displaystyle Z={\frac {pV}{nRT}}={\frac {1+\eta +\eta ^{2}-\eta ^{3}}{(1-\eta )^{3}}}}$

is obtained, where ${\displaystyle \eta }$ is the packing fraction, given by

${\displaystyle \eta ={\frac {N_{A}\pi n\sigma ^{3}}{6V}}}$

where ${\displaystyle N_{A}}$isAvogadros number, ${\displaystyle n/V}$ is the molar density of the fluid, and ${\displaystyle \sigma }$ is the diameter of the hard-spheres. From this Equation of State, one can obtain the residual Helmholtz energy,^[3]

${\displaystyle {\frac {A_{res}}{nRT}}={\frac {4\eta -3\eta ^{2}}{(1-\eta )^{2}}}}$,

which yields the residual chemical potential

${\displaystyle {\frac {\mu _{res}}{RT}}={\frac {8\eta -9\eta ^{2}+3\eta ^{3}}{(1-\eta )^{3}}}}$.

One can also obtain the value of the radial distribution function, ${\displaystyle g($r)} , evaluated at the surface of a sphere,^[3]

${\displaystyle g(\sigma )={\frac {1-{\frac {1}{2}}\eta }{(1-\eta )^{3}}}}$.

The latter is of significant importance to accurate descriptions of more advanced intermolecular potentials based on perturbation theory, such as SAFT, where a system of hard spheres is taken as a reference system, and the complete pair-potential is described by perturbations to the underlying hard-sphere system. Computation of the transport properties of hard-sphere gases at moderate densities using Revised Enskog Theory also relies on an accurate value for ${\displaystyle g(\sigma )}$, and the Carnahan-Starling Equation of State has been used for this purpose to large success.^[4]

See also[edit]

• Classical fluid

Literature[edit]

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

References[edit]