Hard sphere model

Phase diagram (pressure vs packing fraction) of hard sphere system (Solid line - stable branch, dashed line - metastable branch)

The hard sphere model (sometimes known as the rigid sphere model) 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 (1954-1957)

The hard sphere model, along with its two-dimensional manifestation hard disks, was one of the first ever systems studied using computer simulation techniques with a view to understanding the thermodynamics of the liquid and solid phases and their corresponding phase transition , much of this work undertaken at the Los Alamos Scientific Laboratory on the world's first electronic digital computer ENIAC .

Liquid phase radial distribution function

The following are a series of plots of the hard sphere radial distribution function shown for different values of the number density $\rho$. 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 (Eq. 1 in )

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

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

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

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

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

where $\eta$ is the packing fraction.

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

Liquid-solid transition

The hard sphere system undergoes a liquid-solid first order transition , sometimes referred to as the Kirkwood-Alder transition . The liquid-solid coexistence densities ($\rho ^{*}=\rho \sigma ^{3}=6\eta /\pi$) has been calculated to be

$\rho _{\mathrm {solid} }^{*}$	$\rho _{\mathrm {liquid} }^{*}$	Reference
1.041(4)	0.943(4)
1.0376	0.9391
1.0367(10)	0.9387(10)
1.0372	0.9387
1.0369(33)	0.9375(14)
1.037	0.938
1.033(3)	0.935(2)
1.03715(9)	0.93890(7)

The coexistence pressure has been calculated to be

$p(k_{B}T/\sigma ^{3})$	Reference
11.5727(10)
11.57(10)
11.567
11.55(11)
11.54(4)
11.50(9)
11.48(11)
11.43(17)
11.550(4)

The coexistence chemical potential has been calculated to be

$\mu (k_{B}T)$	Reference
15.980(11)
16.053(4)

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

$A_{\mathrm {solid} }$	$A_{\mathrm {liquid} }$	Reference
4.887(3)	3.719(8)

The melting and crystallization process has been studied by Isobe and Krauth .

Helmholtz energy function

Values for the Helmholtz energy function ($A$) are given in the following Table:

$\rho ^{*}$	$A/(Nk_{B}T)$	Reference
0.25	−1.766 $\pm$ 0.002	Table I
0.50	−0.152 $\pm$ 0.002	Table I
0.75	1.721 $\pm$ 0.002	Table I
1.04086	4.959	Table VI
1.099975	5.631	Table VI
1.150000	6.274	Table VI

In the free energies are given without the ideal gas contribution $\ln(\rho ^{*})-1$ . Hence, it was added to the free energies in the table.

Interfacial Helmholtz energy function

The Helmholtz energy function of the solid–liquid interface has been calculated using the cleaving method giving (Ref. Table I):

	work per unit area/$(k_{B}T/\sigma ^{2})$
$\gamma _{\{100\}}$	0.5820(19)
$\gamma _{\{100\}}$	0.636(11)
$\gamma _{\{110\}}$	0.5590(20)
$\gamma _{\{111\}}$	0.5416(31)
$\gamma _{\{120\}}$	0.5669(20)

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 , with a Helmholtz energy function difference in the thermodynamic limit between the hexagonal close packed and face centered cubic crystals at close packing of 0.001164(8) $Nk_{B}T$. Recently evidence has been found for a metastable cI16 phase indicating the "cI16 is a mechanically stable structure that can spontaneously emerge from a bcc starting point but it is thermodynamically metastable relative to fcc or hcp".

• See also: Equations of state for crystals of hard spheres

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

Equations of state

Main article: Equations of state for hard spheres

Virial coefficients

Main article: Hard sphere: virial coefficients

Experimental results

Pusey and van Megen used a suspension of PMMA particles of radius 305 $\pm$10 nm, suspended in poly-12-hydroxystearic acid For results obtained from the Colloidal Disorder - Order Transition (CDOT) experiments performed on-board the Space Shuttles Columbia and Discovery see Ref.

Mixtures

• Binary hard-sphere mixtures
• Multicomponent hard-sphere mixtures

Related systems

• Quantum hard spheres
• Dipolar hard spheres
• Lattice hard spheres

Hard spheres in other dimensions:

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

References

Related reading

• "Theory and Simulation of Hard-Sphere Fluids and Related Systems", Lecture Notes in Physics 753/2008 Springer (2008)
• Laura Filion, Michiel Hermes, Ran Ni and Marjolein Dijkstra "Crystal nucleation of hard spheres using molecular dynamics, umbrella sampling, and forward flux sampling: A comparison of simulation techniques", Journal of Chemical Physics 133 244115 (2010)

External links

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