# 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

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

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]

## Helmholtz energy function

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

 $$\rho^*$$ $$A/(Nk_BT)$$ Reference 0.25 0.620 $$\pm$$ 0.002 Table I [32] 0.50 1.541 $$\pm$$ 0.002 Table I [32] 0.75 3.009 $$\pm$$ 0.002 Table I [32] 1.04086 4.959 Table VI [24] 1.099975 5.631 Table VI [24] 1.150000 6.274 Table VI [24]

## Interfacial Helmholtz energy function

The Helmholtz energy function of the solid–liquid interface has been calculated using the cleaving method giving (Ref. [33] 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 $$\pi/(3 \sqrt{2}) \approx 0.74048%$$ [34] [35]. However, for hard spheres at close packing the face centred cubic phase is the more stable [36]

## Direct correlation function

For the direct correlation function see: [37] [38]

## Bridge function

Details of the bridge function for hard sphere can be found in the following publication [39]

## 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 [40] For results obtained from the Colloidal Disorder - Order Transition (CDOT) experiments performed on-board the Space Shuttles Columbia and Discovery see Ref. [41]

## Related systems

Hard spheres in other dimensions: