# Hard sphere model

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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

$\Phi_{12}\left( r \right) = \left\{ \begin{array}{lll} \infty & ; & r < \sigma \\ 0 & ; & r \ge \sigma \end{array} \right.$

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 [1] [2] [3], much of this work undertaken at the Los Alamos Scientific Laboratory on the world's first electronic digital computer ENIAC [4].

## Liquid phase radial distribution function

The following are a series of plots of the hard sphere radial distribution function [5] 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 [6])

$\frac{p}{\rho k_BT}= 1 + B_2 \rho {\mathrm g}(\sigma^+)$

where the second virial coefficient, $B_2$, is given by

$B_2 = \frac{2\pi}{3}\sigma^3$.

Carnahan and Starling [7] provided the following expression for ${\mathrm g}(\sigma^+)$ (Eq. 3 in [6])

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

where $\eta$ is the packing fraction.

Over the years many groups have studied the radial distribution function of the hard sphere model: [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18]

## Liquid-solid transition

The hard sphere system undergoes a liquid-solid first order transition [19] [20], sometimes referred to as the Kirkwood-Alder transition [21]. 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 0.945 [19] 1.0376 0.9391 [22] 1.0367(10) 0.9387(10) [23] 1.0372 0.9387 [24] 1.0369(33) 0.9375(14) [25] 1.037 0.938 [26] 1.035(3) 0.936(2) [27]

The coexistence pressure has been calculated to be

 $p (k_BT/\sigma^3)$ Reference 11.5727(10) [28] 11.57(10) [23] 11.567 [22] 11.55(11) [29] 11.54(4) [25] 11.50(9) [30] 11.48(11) [27] 11.43(17) [31]

The coexistence chemical potential has been calculated to be

 $\mu (k_BT)$ Reference 15.980(11) [27]

The Helmholtz energy function (in units of $Nk_BT$) is given by

 $A_{\mathrm {solid}}$ $A_{\mathrm {liquid}}$ Reference 4.887(3) 3.719(8) [27]

The melting and crystallization process has been studied by Isobe and Krauth [32].

## Helmholtz energy function

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

 $\rho^*$ $A/(Nk_BT)$ Reference 0.25 −1.766 $\pm$ 0.002 Table I [33] 0.50 −0.152 $\pm$ 0.002 Table I [33] 0.75 1.721 $\pm$ 0.002 Table I [33] 1.04086 4.959 Table VI [24] 1.099975 5.631 Table VI [24] 1.150000 6.274 Table VI [24]

In [33] 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. [34] Table I):

 work per unit area/$(k_BT/\sigma^2)$ $\gamma_{\{100\}}$ 0.5820(19) $\gamma_{\{100\}}$ 0.636(11) [28] $\gamma_{\{110\}}$ 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 $\pi/(3 \sqrt{2}) \approx 0.74048%$[35] [36] [37]. However, for hard spheres at close packing the face centred cubic phase is the more stable [38], 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) $Nk_BT$[39]. Recently evidence has been found for a metastable cI16 phase [40] 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".

## Direct correlation function

For the direct correlation function see: [41] [42]

## Bridge function

Details of the bridge function for hard sphere can be found in the following publication [43]

## 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 [44] For results obtained from the Colloidal Disorder - Order Transition (CDOT) experiments performed on-board the Space Shuttles Columbia and Discovery see Ref. [45]

## Related systems

Hard spheres in other dimensions: