# Hard sphere model

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.

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

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

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

Reference 15.980(11) ^{[27]}16.053(4) ^{[28]}

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

Reference 4.887(3) 3.719(8) ^{[27]}

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

## Helmholtz energy function[edit]

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

**Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \rho^*}**Reference 0.25 −1.766 0.002 Table I ^{[34]}0.50 −0.152 0.002 Table I ^{[34]}0.75 1.721 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 . Hence, it was added to the free energies in the table.

## Interfacial Helmholtz energy function[edit]

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/ 0.5820(19) **Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \gamma_{\{100\}}}**0.636(11) ^{[29]}0.5590(20) 0.5416(31) 0.5669(20)

## 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 ^{[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) ^{[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[edit]

For the direct correlation function see:
^{[42]}
^{[43]}

## Bridge function[edit]

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

## 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 ^{[45]}
For results obtained from the Colloidal Disorder - Order Transition (CDOT) experiments performed on-board the Space Shuttles *Columbia* and *Discovery* see Ref. ^{[46]}

## Mixtures[edit]

## Related systems[edit]

Hard spheres in other dimensions:

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

## References[edit]

- ↑ Marshall N. Rosenbluth and Arianna W. Rosenbluth "Further Results on Monte Carlo Equations of State", Journal of Chemical Physics
**22**pp. 881-884 (1954) - ↑ W. W. Wood and J. D. Jacobson "Preliminary Results from a Recalculation of the Monte Carlo Equation of State of Hard Spheres", Journal of Chemical Physics
**27**pp. 1207-1208 (1957) - ↑ B. J. Alder and T. E. Wainwright "Phase Transition for a Hard Sphere System", Journal of Chemical Physics
**27**pp. 1208-1209 (1957) - ↑ The ENIAC Story
- ↑ The total correlation function data was produced using the computer code written by Jiří Kolafa
- ↑
^{6.0}^{6.1}Fu-Ming Tao, Yuhua Song, and E. A. Mason "Derivative of the hard-sphere radial distribution function at contact", Physical Review A**46**pp. 8007-8008 (1992) - ↑ N. F.Carnahan and K. E.Starling,"Equation of State for Nonattracting Rigid Spheres" Journal of Chemical Physics
**51**pp. 635-636 (1969) - ↑ John G. Kirkwood, Eugene K. Maun, and Berni J. Alder "Radial Distribution Functions and the Equation of State of a Fluid Composed of Rigid Spherical Molecules", Journal of Chemical Physics
**18**pp. 1040- (1950) - ↑ B. R. A. Nijboer and L. Van Hove "Radial Distribution Function of a Gas of Hard Spheres and the Superposition Approximation", Physical Review
**85**pp. 777 - 783 (1952) - ↑ B. J. Alder, S. P. Frankel, and V. A. Lewinson "Radial Distribution Function Calculated by the Monte-Carlo Method for a Hard Sphere Fluid", Journal of Chemical Physics
**23**pp. 417- (1955) - ↑ Francis H. Ree, R. Norris Keeler, and Shaun L. McCarthy "Radial Distribution Function of Hard Spheres", Journal of Chemical Physics
**44**pp. 3407- (1966) - ↑ W. R. Smith and D. Henderson "Analytical representation of the Percus-Yevick hard-sphere radial distribution function", Molecular Physics
**19**pp. 411-415 (1970) - ↑ J. A. Barker and D. Henderson "Monte Carlo values for the radial distribution function of a system of fluid hard spheres", Molecular Physics
**21**pp. 187-191 (1971) - ↑ J. M. Kincaid and J. J. Weis "Radial distribution function of a hard-sphere solid", Molecular Physics
**34**pp. 931-938 (1977) - ↑ S. Bravo Yuste and A. Santos "Radial distribution function for hard spheres", Physical Review A
**43**pp. 5418-5423 (1991) - ↑ Jaeeon Chang and Stanley I. Sandler "A real function representation for the structure of the hard-sphere fluid", Molecular Physics
**81**pp. 735-744 (1994) - ↑ Andrij Trokhymchuk, Ivo Nezbeda and Jan Jirsák "Hard-sphere radial distribution function again", Journal of Chemical Physics
**123**024501 (2005) - ↑ M. López de Haro, A. Santos and S. B. Yuste "On the radial distribution function of a hard-sphere fluid", Journal of Chemical Physics
**124**236102 (2006) - ↑
^{19.0}^{19.1}William G. Hoover and Francis H. Ree "Melting Transition and Communal Entropy for Hard Spheres", Journal of Chemical Physics**49**pp. 3609-3617 (1968) - ↑ Miguel Robles, Mariano López de Haro and Andrés Santos "Note: Equation of state and the freezing point in the hard-sphere model", Journal of Chemical Physics
**140**136101 (2014) - ↑ Alice P. Gast and William B. Russel "Simple Ordering in Complex Fluids", Physics Today
**51**(12) pp. 24-30 (1998) - ↑
^{22.0}^{22.1}Daan Frenkel and Berend Smit "Understanding Molecular Simulation: From Algorithms to Applications", Second Edition (2002) (ISBN 0-12-267351-4) p. 261. - ↑
^{23.0}^{23.1}Andrea Fortini and Marjolein Dijkstra "Phase behaviour of hard spheres confined between parallel hard plates: manipulation of colloidal crystal structures by confinement", Journal of Physics: Condensed Matter**18**pp. L371-L378 (2006) - ↑
^{24.0}^{24.1}^{24.2}^{24.3}Carlos Vega and Eva G. Noya "Revisiting the Frenkel-Ladd method to compute the free energy of solids: The Einstein molecule approach", Journal of Chemical Physics**127**154113 (2007) - ↑
^{25.0}^{25.1}Eva G. Noya, Carlos Vega, and Enrique de Miguel "Determination of the melting point of hard spheres from direct coexistence simulation methods", Journal of Chemical Physics**128**154507 (2008) - ↑ Ruslan L. Davidchack and Brian B. Laird "Simulation of the hard-sphere crystal–melt interface", Journal of Chemical Physics
**108**pp. 9452-9462 (1998) - ↑
^{27.0}^{27.1}^{27.2}^{27.3}Enrique de Miguel "Estimating errors in free energy calculations from thermodynamic integration using fitted data", Journal of Chemical Physics**129**214112 (2008) - ↑
^{28.0}^{28.1}^{28.2}Craig Moir, Leo Lue, and Marcus N. Bannerman "Tethered-particle model: The calculation of free energies for hard-sphere systems", Journal of Chemical Physics**155**064504 (2021) - ↑
^{29.0}^{29.1}L. A. Fernández, V. Martín-Mayor, B. Seoane, and P. Verrocchio "Equilibrium Fluid-Solid Coexistence of Hard Spheres", Physical Review Letters**108**165701 (2012) - ↑ Robin J. Speedy "Pressure of the metastable hard-sphere fluid", Journal of Physics: Condensed Matter
**9**pp. 8591-8599 (1997) - ↑ N. B. Wilding and A. D. Bruce "Freezing by Monte Carlo Phase Switch", Physical Review Letters
**85**pp. 5138-5141 (2000) - ↑ G. Odriozola "Replica exchange Monte Carlo applied to hard spheres", Journal of Chemical Physics
**131**144107 (2009) - ↑ Masaharu Isobe and Werner Krauth "Hard-sphere melting and crystallization with event-chain Monte Carlo", Journal of Chemical Physics
**143**084509 (2015) - ↑
^{34.0}^{34.1}^{34.2}^{34.3}T. Schilling and F. Schmid "Computing absolute free energies of disordered structures by molecular simulation", Journal of Chemical Physics**131**231102 (2009) - ↑ Ruslan L. Davidchack "Hard spheres revisited: Accurate calculation of the solid–liquid interfacial free energy", Journal of Chemical Physics
**133**234701 (2010) - ↑ Neil J. A. Sloane "Kepler's conjecture confirmed", Nature
**395**pp. 435-436 (1998) - ↑ Jacob Aron "Proof confirmed of 400-year-old fruit-stacking problem", New Scientist daily news 12 August (2014)
- ↑ C. F. Tejero, M. S. Ripoll, and A. Pérez "Pressure of the hard-sphere solid", Physical Review E
**52**pp. 3632-3636 (1995) - ↑ Leslie V. Woodcock "Computation of the free energy for alternative crystal structures of hard spheres", Faraday Discussions
**106**pp. 325-338 (1997) - ↑ Eva G. Noya and Noé G. Almarza "Entropy of hard spheres in the close-packing limit", Molecular Physics
**113**pp. 1061-1068 (2015) - ↑ Vadim B. Warshavsky, David M. Ford, and Peter A. Monson "On the mechanical stability of the body-centered cubic phase and the emergence of a metastable cI16 phase in classical hard sphere solids", Journal of Chemical Physics
**148**024502 (2018) - ↑ C. F. Tejero and M. López De Haro "Direct correlation function of the hard-sphere fluid", Molecular Physics
**105**pp. 2999-3004 (2007) - ↑ Matthew Dennison, Andrew J. Masters, David L. Cheung, and Michael P. Allen "Calculation of direct correlation function for hard particles using a virial expansion", Molecular Physics pp. 375-382 (2009)
- ↑ Jiri Kolafa, Stanislav Labik and Anatol Malijevsky "The bridge function of hard spheres by direct inversion of computer simulation data", Molecular Physics
**100**pp. 2629-2640 (2002) - ↑ P. N. Pusey and W. van Megen "Phase behaviour of concentrated suspensions of nearly hard colloidal spheres", Nature
**320**pp. 340-342 (1986) - ↑ Z. Chenga, P. M. Chaikina, W. B. Russelb, W. V. Meyerc, J. Zhub, R. B. Rogersc and R. H. Ottewilld, "Phase diagram of hard spheres", Materials & Design
**22**pp. 529-534 (2001)

**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[edit]

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