Hard sphere model

The hard sphere model (sometimes known as the rigid sphere model) is defined as

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

where ${\displaystyle \Phi _{12}\left(r\right)}$ is the intermolecular pair potential between two spheres at a distance ${\displaystyle r:=|\mathbf {r} _{1}-\mathbf {r} _{2}|}$, and ${\displaystyle \sigma }$ 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, ${\displaystyle a=b=c}$.

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].

The following are a series of plots of the hard sphere radial distribution function [5] shown for different values of the number density ${\displaystyle \rho }$. The horizontal axis is in units of ${\displaystyle \sigma }$ where ${\displaystyle \sigma }$ is set to be 1. Click on image of interest to see a larger view.

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

The value of the radial distribution at contact, ${\displaystyle {\mathrm {g} }(\sigma ^{+})}$, can be used to calculate the pressure via the equation of state (Eq. 1 in [6])

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

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

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

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

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

where ${\displaystyle \eta }$ 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 (${\displaystyle \rho ^{*}=\rho \sigma ^{3}=6\eta /\pi }$) has been calculated to be

 ${\displaystyle \rho _{\mathrm {solid} }^{*}}$ ${\displaystyle \rho _{\mathrm {liquid} }^{*}}$ Reference 1.041(4) 0.943(4) [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] 1.033(3) 0.935(2) [27] 1.03715(9) 0.93890(7) [28]

The coexistence pressure has been calculated to be

 ${\displaystyle p(k_{B}T/\sigma ^{3})}$ Reference 11.5727(10) [29] 11.57(10) [23] 11.567 [22] 11.55(11) [30] 11.54(4) [25] 11.50(9) [31] 11.48(11) [27] 11.43(17) [32] 11.550(4) [28]

The coexistence chemical potential has been calculated to be

 ${\displaystyle \mu (k_{B}T)}$ Reference 15.980(11) [27] 16.053(4) [28]

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

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

The melting and crystallization process has been studied by Isobe and Krauth [33].

Helmholtz energy function

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

 $\displaystyle \rho^*$ ${\displaystyle A/(Nk_{B}T)}$ Reference 0.25 −1.766 ${\displaystyle \pm }$ 0.002 Table I [34] 0.50 −0.152 ${\displaystyle \pm }$ 0.002 Table I [34] 0.75 1.721 ${\displaystyle \pm }$ 0.002 Table I [34] 1.04086 4.959 Table VI [24] 1.099975 5.631 Table VI [24] 1.150000 6.274 Table VI [24]

In [34] the free energies are given without the ideal gas contribution ${\displaystyle \ln(\rho ^{*})-1}$ . 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. [35] Table I):

 work per unit area/${\displaystyle (k_{B}T/\sigma ^{2})}$ ${\displaystyle \gamma _{\{100\}}}$ 0.5820(19) $\displaystyle \gamma_{\{100\}}$ 0.636(11) [29] ${\displaystyle \gamma _{\{110\}}}$ 0.5590(20) ${\displaystyle \gamma _{\{111\}}}$ 0.5416(31) ${\displaystyle \gamma _{\{120\}}}$ 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 ${\displaystyle \pi /(3{\sqrt {2}})\approx 74.048\%}$[36] [37] [38]. However, for hard spheres at close packing the face centred cubic phase is the more stable [39], 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) ${\displaystyle Nk_{B}T}$[40]. Recently evidence has been found for a metastable cI16 phase [41] 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: [42] [43]

Bridge function

Details of the bridge function for hard sphere can be found in the following publication [44]

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 ${\displaystyle \pm }$10 nm, suspended in poly-12-hydroxystearic acid [45] For results obtained from the Colloidal Disorder - Order Transition (CDOT) experiments performed on-board the Space Shuttles Columbia and Discovery see Ref. [46]

Related systems

Hard spheres in other dimensions: