Hard sphere model

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Sphere green.png
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

where is the intermolecular pair potential between two spheres at a distance , and 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, .

First simulations of hard spheres (1954-1957)[edit]

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[edit]

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

HS 0.2 rdf.png
HS 0.3 rdf.png
HS 0.4 rdf.png
HS 0.5 rdf.png
HS 0.6 rdf.png
HS 0.7 rdf.png
HS 0.8 rdf.png
HS 0.85 rdf.png
HS 0.9 rdf.png

The value of the radial distribution at contact, , can be used to calculate the pressure via the equation of state (Eq. 1 in [6])

where the second virial coefficient, , is given by


Carnahan and Starling [7] provided the following expression for (Eq. 3 in [6])

where 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[edit]

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 () has been calculated to be

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

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

15.980(11) [27]

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

4.887(3) 3.719(8) [27]

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

Helmholtz energy function[edit]

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

0.25 0.620 0.002 Table I [33]
0.50 1.541 0.002 Table I [33]
0.75 3.009 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]

Interfacial Helmholtz energy function[edit]

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/
0.636(11) [28]

Solid structure[edit]

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 [35] [36]. However, for hard spheres at close packing the face centred cubic phase is the more stable [37], 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) [38].

Direct correlation function[edit]

For the direct correlation function see: [39] [40]

Bridge function[edit]

Details of the bridge function for hard sphere can be found in the following publication [41]

Equations of state[edit]

Main article: Equations of state for hard spheres

Virial coefficients[edit]

Main article: Hard sphere: virial coefficients

Experimental results[edit]

Pusey and van Megen used a suspension of PMMA particles of radius 305 10 nm, suspended in poly-12-hydroxystearic acid [42] For results obtained from the Colloidal Disorder - Order Transition (CDOT) experiments performed on-board the Space Shuttles Columbia and Discovery see Ref. [43]


Related systems[edit]

Hard spheres in other dimensions:


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  3. B. J. Alder and T. E. Wainwright "Phase Transition for a Hard Sphere System", Journal of Chemical Physics 27 pp. 1208-1209 (1957)
  4. The ENIAC Story
  5. The total correlation function data was produced using the computer code written by Jiří Kolafa
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Related reading

External links[edit]