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[[Image:sphere_green.png|thumb|right]]
[[Image:sphere_green.png|thumb|right]]
[[Image:Hard-sphere phase diagram pressure vs packing fraction.png|thumb|right|Phase diagram (pressure vs packing fraction) of hard sphere system (Solid line - stable branch, dashed line - metastable branch)]]
== Interaction Potential ==
The '''hard sphere model''' (sometimes known as the  ''rigid sphere model'') is defined as
The hard sphere interaction potential is given by


: <math>
: <math>
\Phi_{12}\left( r \right) = \left\{ \begin{array}{lll}
V \left( r \right) = \left\{ \begin{array}{lll}
\infty & ; & r <  \sigma \\
\infty & ; & r <  \sigma \\
0      & ; & r \ge \sigma \end{array} \right.
0      & ; & r \ge \sigma \end{array} \right.
</math>
</math>


where <math> \Phi_{12}\left(r \right) </math> is the [[intermolecular pair potential]] between two spheres at a distance <math>r := |\mathbf{r}_1 - \mathbf{r}_2|</math>, and <math> \sigma </math> 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, <math>a=b=c</math>.
==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 transitions | phase transition]]
<ref>[http://dx.doi.org/10.1063/1.1740207 Marshall N. Rosenbluth and Arianna W. Rosenbluth "Further Results on Monte Carlo Equations of State", Journal of Chemical Physics '''22''' pp. 881-884  (1954)]</ref>
<ref>[http://dx.doi.org/10.1063/1.1743956    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)]</ref>
<ref>[http://dx.doi.org/10.1063/1.1743957    B. J. Alder and T. E. Wainwright "Phase Transition for a Hard Sphere System", Journal of Chemical Physics '''27''' pp. 1208-1209 (1957)]</ref>, much of this work undertaken at the Los Alamos Scientific Laboratory on the world's first electronic digital computer  ENIAC <ref>[http://ftp.arl.army.mil/~mike/comphist/eniac-story.html The ENIAC Story]</ref>.
==Liquid phase radial distribution function==
The following are a series of plots of the hard sphere [[radial distribution function]] <ref>The [[total correlation function]] data was produced using the [https://old.vscht.cz/fch/software/hsmd/hspline-8-2004.zip computer code] written by [https://web.vscht.cz/~kolafaj/ Jiří Kolafa]</ref> shown for different values of the number density <math>\rho</math>.  The horizontal axis is in units of <math>\sigma</math> where <math>\sigma</math> is set to be 1. Click on image of interest to see a larger view.
:{| border="1"
|-
|<math>\rho=0.2</math>  [[Image:HS_0.2_rdf.png|center|220px]] ||<math>\rho=0.3</math>  [[Image:HS_0.3_rdf.png|center|220px]]  || <math>\rho=0.4</math>  [[Image:HS_0.4_rdf.png|center|220px]]
|-
|<math>\rho=0.5</math>  [[Image:HS_0.5_rdf.png|center|220px]] ||<math>\rho=0.6</math>  [[Image:HS_0.6_rdf.png|center|220px]]  || <math>\rho=0.7</math>  [[Image:HS_0.7_rdf.png|center|220px]]
|-
|<math>\rho=0.8</math>  [[Image:HS_0.8_rdf.png|center|220px]] ||<math>\rho=0.85</math>  [[Image:HS_0.85_rdf.png|center|220px]]  ||  <math>\rho=0.9</math> [[Image:HS_0.9_rdf.png|center|220px]]
|}
The value of the radial distribution at contact, <math>{\mathrm g}(\sigma^+)</math>, can be used to calculate the [[pressure]] via the [[equations of state |equation of state]] (Eq. 1 in <ref name="Tao1"> [http://dx.doi.org/10.1103/PhysRevA.46.8007 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)]</ref>)
:<math>\frac{p}{\rho k_BT}= 1 + B_2 \rho {\mathrm g}(\sigma^+)</math>
where the [[second virial coefficient]], <math>B_2</math>, is given by
:<math>B_2 = \frac{2\pi}{3}\sigma^3</math>.
Carnahan and Starling <ref>[http://dx.doi.org/10.1063/1.1672048 N. F.Carnahan and K. E.Starling,"Equation of State for Nonattracting Rigid Spheres"  Journal of Chemical Physics '''51''' pp. 635-636 (1969)]</ref> provided the following expression for <math>{\mathrm g}(\sigma^+)</math> (Eq. 3 in <ref name="Tao1" ></ref>)
:<math>{\mathrm g}(\sigma^+)= \frac{1-\eta/2}{(1-\eta)^3}</math>
where <math>\eta</math> is the [[packing fraction]].


Over the years many groups have studied the radial distribution function of the hard sphere model:
<ref>[http://dx.doi.org/10.1063/1.1747854 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)]</ref>
<ref>[http://dx.doi.org/10.1103/PhysRev.85.777  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)]</ref>
<ref>[http://dx.doi.org/10.1063/1.1742004 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)]</ref>
<ref>[http://dx.doi.org/10.1063/1.1727245 Francis H. Ree, R. Norris Keeler, and Shaun L. McCarthy "Radial Distribution Function of Hard Spheres", Journal of Chemical Physics '''44''' pp. 3407- (1966)]</ref>
<ref>[http://dx.doi.org/10.1080/00268977000101421 W. R. Smith and D. Henderson "Analytical representation of the Percus-Yevick hard-sphere radial distribution function", Molecular Physics '''19''' pp. 411-415 (1970)]</ref>
<ref>[http://dx.doi.org/10.1080/00268977100101331 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)]</ref>
<ref>[http://dx.doi.org/10.1080/00268977700102241 J. M. Kincaid and J. J. Weis "Radial distribution function of a hard-sphere solid", Molecular Physics '''34''' pp. 931-938 (1977)]</ref>
<ref>[http://dx.doi.org/10.1103/PhysRevA.43.5418      S. Bravo Yuste and A. Santos "Radial distribution function for hard spheres", Physical Review A '''43''' pp. 5418-5423 (1991)]</ref>
<ref>[http://dx.doi.org/10.1080/00268979400100491 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)]</ref>
<ref>[http://dx.doi.org/10.1063/1.1979488 Andrij Trokhymchuk, Ivo Nezbeda and Jan Jirsák "Hard-sphere radial distribution function again",  Journal of Chemical Physics '''123''' 024501 (2005)]</ref>
<ref>[http://dx.doi.org/10.1063/1.2201699 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)]</ref>
==Liquid-solid transition==
The hard sphere system undergoes a [[Solid-liquid phase transitions |liquid-solid]] [[First-order transitions |first order transition]] <ref name="HooverRee">[http://dx.doi.org/10.1063/1.1670641    William G. Hoover and Francis H. Ree "Melting Transition and Communal Entropy for Hard Spheres", Journal of Chemical Physics '''49''' pp. 3609-3617  (1968)]</ref>
<ref>[http://dx.doi.org/10.1063/1.4870524 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)]</ref>, sometimes referred to as the Kirkwood-Alder transition <ref name="GastRussel">[http://dx.doi.org/10.1063/1.882495 Alice P. Gast and William B. Russel "Simple Ordering in Complex Fluids", Physics Today '''51''' (12) pp. 24-30  (1998)]</ref>.
The liquid-solid coexistence densities (<math>\rho^* = \rho \sigma^3=6\eta/\pi</math>) has been calculated to be
:{| border="1"
|-
| <math>\rho^*_{\mathrm {solid}}</math> || <math>\rho^*_{\mathrm {liquid}}</math> || Reference
|-
| 1.041(4)|| 0.943(4) || <ref name="HooverRee"></ref>
|-
| 1.0376|| 0.9391 || <ref name="FrenkelSmitBook">Daan Frenkel and Berend Smit "Understanding Molecular Simulation: From Algorithms to Applications", Second Edition (2002) (ISBN 0-12-267351-4) p. 261.</ref>
|-
| 1.0367(10) || 0.9387(10) || <ref name="Fortini">[http://dx.doi.org/10.1088/0953-8984/18/28/L02  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)]</ref>
|-
| 1.0372 || 0.9387  || <ref name="VegaNoya"> [http://dx.doi.org/10.1063/1.2790426 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)]</ref>
|-
| 1.0369(33) || 0.9375(14) || <ref name="Noya"> [http://dx.doi.org/10.1063/1.2901172 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)]</ref>
|-
| 1.037 || 0.938 || <ref>[http://dx.doi.org/10.1063/1.476396 Ruslan L. Davidchack and Brian B. Laird "Simulation of the hard-sphere crystal–melt interface",  Journal of Chemical Physics '''108''' pp.  9452-9462 (1998)]</ref>
|-
| 1.033(3) || 0.935(2) || <ref name="Miguel"> [http://dx.doi.org/10.1063/1.3023062 Enrique de Miguel "Estimating errors in free energy calculations from thermodynamic integration using fitted data", Journal of Chemical Physics '''129''' 214112 (2008)]</ref>
|-
| 1.03715(9) || 0.93890(7) || <ref name="MoirEtAl2021"> [https://doi.org/10.1063/5.0058892 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)]</ref>
|}
The coexistence [[pressure]] has been calculated to be
:{| border="1"
|-
| <math>p (k_BT/\sigma^3) </math> || Reference
|-
| 11.5727(10)|| <ref name="FernandezUCM">[http://dx.doi.org/10.1103/PhysRevLett.108.165701 L. A. Fernández, V. Martín-Mayor, B. Seoane, and P. Verrocchio "Equilibrium Fluid-Solid Coexistence of Hard Spheres", Physical Review Letters '''108''' 165701 (2012)]</ref>
|-
| 11.57(10) || <ref name="Fortini"></ref>
|-
| 11.567|| <ref name="FrenkelSmitBook"></ref>
|-
| 11.55(11) || <ref>[http://dx.doi.org/10.1088/0953-8984/9/41/006 Robin J. Speedy "Pressure of the metastable hard-sphere fluid", Journal of  Physics: Condensed Matter '''9''' pp. 8591-8599 (1997)]</ref>
|-
| 11.54(4) || <ref name="Noya"></ref>
|-
| 11.50(9) || <ref>[http://dx.doi.org/10.1103/PhysRevLett.85.5138 N. B. Wilding and A. D. Bruce "Freezing by Monte Carlo Phase Switch", Physical Review Letters '''85''' pp. 5138-5141 (2000)]</ref>
|-
| 11.48(11) || <ref name="Miguel"></ref>
|-
| 11.43(17) || <ref>[http://dx.doi.org/10.1063/1.3244562 G. Odriozola "Replica exchange Monte Carlo applied to hard spheres", Journal of Chemical Physics '''131''' 144107 (2009)]</ref>
|-
| 11.550(4) || <ref name="MoirEtAl2021"></ref>
|}
The coexistence [[chemical potential]] has been calculated to be
:{| border="1"
|-
| <math>\mu (k_BT) </math> || Reference
|-
| 15.980(11) || <ref name="Miguel"></ref>
|-
| 16.053(4) || <ref name="MoirEtAl2021"></ref>
|}
The [[Helmholtz energy function]] (in units of <math>Nk_BT</math>) is given by
:{| border="1"
|-
| <math>A_{\mathrm {solid}}</math> || <math>A_{\mathrm {liquid}}</math> || Reference
|-
| 4.887(3) || 3.719(8) || <ref name="Miguel"></ref>
|}


The melting and crystallization process has been studied by Isobe and Krauth <ref>[http://dx.doi.org/10.1063/1.4929529  Masaharu Isobe and Werner Krauth "Hard-sphere melting and crystallization with event-chain Monte Carlo", Journal of Chemical Physics '''143''' 084509 (2015)]</ref>.
where <math> V\left(r \right) </math> is the potential energy between two spheres at a distance <math> r </math>, and <math> \sigma </math> is the sphere diameter.


==Helmholtz energy function==
== Equations of state ==  
Values for the [[Helmholtz energy function]] (<math>A</math>) are given in the following Table:
Hard sphere fluid:
:{| border="1"
# See [[Carnahan-Starling]] (three dimensions)
|-
# See Ref.1
| <math>\rho^*</math> || <math>A/(Nk_BT)</math>|| Reference
Hard sphere solid:
|-
# See Ref. 2
| 0.25 || −1.766 <math>\pm</math> 0.002 || Table I <ref name="Schilling"> [http://dx.doi.org/10.1063/1.3274951 T. Schilling and F. Schmid "Computing absolute free energies of disordered structures by molecular simulation", Journal of Chemical Physics '''131''' 231102 (2009)]</ref>
 
|-
== Related systems ==  
| 0.50 || −0.152 <math>\pm</math> 0.002  || Table I <ref name="Schilling"></ref>
 
|-
[[Hard disk|Hard disks]] in a 2-dimensional space
| 0.75 || 1.721 <math>\pm</math> 0.002  || Table I <ref name="Schilling"></ref>
 
|-
[[Hard rods]] in one dimension
| 1.04086 || 4.959 || Table VI <ref name="VegaNoya"></ref>
|-
| 1.099975 || 5.631 || Table VI <ref name="VegaNoya"></ref>
|-  
| 1.150000 || 6.274 || Table VI <ref name="VegaNoya"></ref>
|}


In <ref name="Schilling"></ref> the free energies are given without the ideal gas contribution <math>\ln(\rho^*)-1</math> . Hence, it was added to the free energies in the table.
[[Widom-Rowlinson model]]


==Interfacial Helmholtz energy function==
[[Polydisperse hard spheres]]
The [[Helmholtz energy function]] of the solid–liquid [[interface]] has been calculated using the [[cleaving method]] giving (Ref. <ref>[http://dx.doi.org/10.1063/1.3514144 Ruslan L. Davidchack "Hard spheres revisited: Accurate calculation of the solid–liquid interfacial free energy", Journal of Chemical Physics '''133''' 234701 (2010)]</ref> Table I):
:{| border="1"
|-
|  || [[work]] per unit area/<math>(k_BT/\sigma^2)</math>
|-
| <math>\gamma_{\{100\}}</math> || 0.5820(19)
|-
| <math>\gamma_{\{100\}}</math> || 0.636(11) <ref name="FernandezUCM"></ref>
|-
| <math>\gamma_{\{110\}}</math> || 0.5590(20)
|-
| <math>\gamma_{\{111\}}</math> || 0.5416(31)
|-
| <math>\gamma_{\{120\}}</math> || 0.5669(20)
|}


==Solid structure==
== Data ==
The [http://mathworld.wolfram.com/KeplerConjecture.html Kepler conjecture] states that the optimal packing for three dimensional spheres is either cubic or hexagonal close [[Lattice Structures | packing]], both of which have maximum densities of <math>\pi/(3 \sqrt{2}) \approx 74.048%</math><ref>[http://dx.doi.org/10.1038/26609 Neil J. A. Sloane "Kepler's conjecture confirmed", Nature '''395''' pp. 435-436 (1998)]</ref>
[[Hard sphere: virial equation of state|Virial coefficients]] of hard spheres and hard disks
<ref>[https://www.newscientist.com/article/dn26041-proof-confirmed-of-400-year-old-fruit-stacking-problem/ Jacob Aron "Proof confirmed of 400-year-old fruit-stacking problem", New Scientist daily news 12 August (2014)]</ref>
<ref>[http://dx.doi.org/10.1103/PhysRevE.52.3632 C. F. Tejero, M. S. Ripoll, and A. Pérez "Pressure of the hard-sphere solid", Physical Review E  '''52''' pp. 3632-3636 (1995)]</ref>. However, for hard spheres at close packing the [[Building up a face centered cubic lattice |face centred cubic]] phase is the more stable
<ref>[http://dx.doi.org/10.1039/a701761h Leslie V. Woodcock "Computation of the free energy for alternative crystal structures of hard spheres", Faraday Discussions '''106''' pp. 325-338 (1997)]</ref>, 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) <math>Nk_BT</math><ref>[http://dx.doi.org/10.1080/00268976.2014.982736 Eva G. Noya and Noé G. Almarza "Entropy of hard spheres in the close-packing limit", Molecular Physics '''113''' pp. 1061-1068 (2015)]</ref>. Recently evidence has been found for a metastable cI16 phase <ref>[https://doi.org/10.1063/1.5009099 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)]</ref> 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".''
*See also: [[Equations of state for crystals of hard spheres]]


==Direct correlation function==
For the [[direct correlation function]] see:
<ref>[http://dx.doi.org/10.1080/00268970701725021 C. F. Tejero and M. López De Haro "Direct correlation function of the hard-sphere fluid", Molecular Physics '''105''' pp. 2999-3004 (2007)]</ref>
<ref>[http://dx.doi.org/10.1080/00268970902784934 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)]</ref>
==Bridge function==
Details of the [[bridge function]] for hard sphere can be found in the following publication
<ref>[http://dx.doi.org/10.1080/00268970210136357 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)]</ref>
== Equations of state ==
:''Main article: [[Equations of state for hard spheres]]''
==Virial coefficients==
:''Main article: [[Hard sphere: virial coefficients]]''
== Experimental results ==
== Experimental results ==
Pusey and  van Megen used a suspension of PMMA particles of radius 305 <math>\pm</math>10 nm,  suspended in poly-12-hydroxystearic acid <ref>[http://dx.doi.org/10.1038/320340a0 P. N. Pusey and W. van Megen "Phase behaviour of concentrated suspensions of nearly hard colloidal spheres", Nature '''320''' pp. 340-342 (1986)]</ref>
 
For results obtained from the [http://exploration.grc.nasa.gov/expr2/cdot.html Colloidal Disorder - Order Transition] (CDOT) experiments performed on-board the Space Shuttles ''Columbia'' and ''Discovery'' see Ref. <ref>[http://dx.doi.org/10.1016/S0261-3069(01)00015-2 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)]</ref>
For results obtained from the [http://exploration.grc.nasa.gov/expr2/cdot.html Colloidal Disorder - Order Transition] (CDOT) experiments performed on-board the Space Shuttles ''Columbia'' and ''Discovery'' see Ref. 3.
==Mixtures==
 
*[[Binary hard-sphere mixtures]]
*[[Multicomponent hard-sphere mixtures]]
== Related systems ==
*[[Quantum hard spheres]]
*[[Dipolar hard spheres]]
*[[Lattice hard spheres]]
Hard spheres in other dimensions:
* 1-dimensional case: [[1-dimensional hard rods | hard rods]].
* 2-dimensional case: [[Hard disks | hard disks]].
* [[Hard hyperspheres]]
==References==
==References==
<references/>
'''Related reading'''
*[http://dx.doi.org/10.1007/978-3-540-78767-9 "Theory and Simulation of Hard-Sphere Fluids and Related Systems", Lecture Notes in Physics  '''753/2008'''  Springer  (2008)]
*[http://dx.doi.org/10.1063/1.3506838 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==
#[http://dx.doi.org/10.1088/0953-8984/9/41/006 Robin J. Speedy "Pressure of the metastable hard-sphere fluid", Journal of  Physics: Condensed Matter '''9''' pp. 8591-8599    (1997)]
*[http://www.smac.lps.ens.fr/index.php/Programs_Chapter_2:_Hard_disks_and_spheres Hard disks and spheres] computer code on SMAC-wiki.
#[http://dx.doi.org/10.1088/0953-8984/10/20/006 Robin J. Speedy "Pressure and entropy of hard-sphere crystals", Journal of  Physics: Condensed Matter '''10''' pp.   4387-4391  (1998)]
#[http://dx.doi.org/10.1016/S0261-3069(01)00015-2 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)]
 
[[Category:Models]]
[[Category:Models]]
[[Category:Equations of state]]
[[category: hard sphere]]
[[category: hard sphere]]
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