Hard sphere model

The hard sphere intermolecular pair potential is defined as

\Phi _{12}\left(r\right)=\left\{{\begin{array}{lll}\infty &;&r<\sigma \\0&;&r\geq \sigma \end{array}}\right.

where $\Phi _{12}\left(r\right)$ is the intermolecular pair potential between two spheres at a distance $r:=|\mathbf {r} _{1}-\mathbf {r} _{2}|$ , and $\sigma$ is the diameter of the sphere. The hard sphere model can be considered to be a special case of the hard ellipsoid model, where each of the semi-axes has the same length, $a=b=c$ .

First simulations of hard spheres

The hard sphere model was one of the first ever systems studied using computer simulation techniques with a view to understanding the thermodynamics of the fluid and solid phases and their corresponding phase transition. The following are a sample of some of the very first works:

Fluid phase radial distribution function

The following are a series of plots of the hard sphere radial distribution function (the total correlation function data was produced using the computer code written by Jiří Kolafa). The horizontal axis is in units of $\sigma$ where $\sigma$ is set to be 1. Click on image of interest to see a larger view.

$\rho =0.2$	$\rho =0.3$	$\rho =0.4$
$\rho =0.5$	$\rho =0.6$	$\rho =0.7$
$\rho =0.8$	$\rho =0.85$	$\rho =0.9$

The value of the radial distribution at contact, ${\mathrm {g} }(\sigma ^{+})$ , can be used to calculate the pressure via the equation of state (Ref 5 Eq. 1)

{\frac {p}{\rho k_{B}T}}=1+B_{2}\rho {\mathrm {g} }(\sigma ^{+})

where the second virial coefficient, $B_{2}$ , is given by

B_{2}={\frac {2\pi }{3}}\sigma ^{3}

.

Carnahan and Starling (Ref. 6) provided the following expression for ${\mathrm {g} }(\sigma ^{+})$ (Ref. 5 Eq. 3)

{\mathrm {g} }(\sigma ^{+})={\frac {1-\eta /2}{(1-\eta )^{3}}}

where $\eta$ is the packing fraction.

Over the years many groups have studied the radial distribution function of the hard sphere model:

Direct correlation function

For the direct correlation function see:

Bridge function

Details of the bridge function for hard sphere can be found in the following publication

Jiri Kolafa, Stanislav Labik and Anatol Malijevsky "The bridge function of hard spheres by direct inversion of computer simulation data", Molecular Physics 100 pp. 2629-2640 (2002)

Fluid-solid transition

The hard sphere system undergoes a fluid-solid first order transition (Ref. 1). The fluid-solid coexistence densities ( $\rho ^{*}=\rho \sigma ^{3}$ ) are given by

$\rho _{\mathrm {solid} }^{*}$	$\rho _{\mathrm {fluid} }^{*}$	Reference
1.041	0.945	Ref. 1
1.0376	0.9391	Ref. 2
1.0367(10)	0.9387(10)	Ref. 3
1.0372	0.9387	Ref. 4
1.0369(33)	0.9375(14)	Ref. 5
1.037	0.938	Ref. 6
1.035(3)	0.936(2)	Ref. 9

The coexistence pressure is given by

$p(k_{B}T/\sigma ^{3})$	Reference
11.567	Ref. 2
11.57(10)	Ref. 3
11.54(4)	Ref. 5
11.50(9)	Ref. 7
11.55(11)	Ref. 8
11.48(11)	Ref. 9

The coexistence chemical potential is given by

$\mu (k_{B}T)$	Reference
15.980(11)	Ref. 9

The Helmholtz energy function (in units of $Nk_{B}T$ ) is given by

$A_{\mathrm {solid} }$	$A_{\mathrm {fluid} }$	Reference
4.887(3)	3.719(8)	Ref. 9

Solid structure

The Kepler conjecture states that the optimal packing for three dimensional spheres is either cubic or hexagonal close packing, both of which have maximum densities of $\pi /(3{\sqrt {2}})\approx 74.048\%$ . However, for hard spheres at close packing the face centred cubic phase is the more stable (Ref. 3).

See also: Equations of state for crystals of hard spheres

Equations of state

Main article: Equations of state for hard spheres

Virial coefficients

Main article: Hard sphere: virial coefficients

Mixtures

Related systems

Hard spheres in other dimensions:

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

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.

Interesting reading

"Theory and Simulation of Hard-Sphere Fluids and Related Systems", Lecture Notes in Physics 753/2008 Springer (2008)

References

External links

Hard disks and spheres computer code on SMAC-wiki.

Hard sphere model

Contents

First simulations of hard spheres

Fluid phase radial distribution function

Direct correlation function

Bridge function

Fluid-solid transition

Solid structure

Equations of state

Virial coefficients

Mixtures

Related systems

Experimental results

Interesting reading

References

External links

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Hard sphere model

First simulations of hard spheres

Fluid phase radial distribution function

Direct correlation function

Bridge function

Fluid-solid transition

Solid structure

Equations of state

Virial coefficients

Mixtures

Related systems

Experimental results

Interesting reading

References

External links

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