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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)]]
The '''hard sphere'''  [[intermolecular pair potential]] is defined as
The '''hard sphere model''' (sometimes known as the ''rigid sphere model'') is defined as


: <math>
: <math>
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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.
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>.
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)==
==First simulations  of hard spheres==
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
The hard sphere model 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]]
to understanding the thermodynamics of the fluid and solid phases and their corresponding [[Phase transitions | phase transition]]. The following are a sample of some of the very first works:
<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>
*[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>[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>
*[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>[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>.
*[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)]
==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.
==Fluid phase radial distribution function==
The following are a series of plots of the hard sphere [[radial distribution function]] (the [[total correlation function]] data was produced using the [http://www.vscht.cz/fch/software/hsmd/hspline-8-2004.zip computer code] written by [http://www.vscht.cz/fch/en/people/Jiri.Kolafa.html Jiří Kolafa]). 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"
:{| border="1"
|-  
|-  
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|<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]]
|<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>)
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]] (Ref 5 Eq. 1)
:<math>\frac{p}{\rho k_BT}= 1 + B_2 \rho {\mathrm g}(\sigma^+)</math>
:<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  
where the [[second virial coefficient]], <math>B_2</math>, is given by  
:<math>B_2 = \frac{2\pi}{3}\sigma^3</math>.
:<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>)
Carnahan and Starling (Ref. 6) provided the following expression for <math>{\mathrm g}(\sigma^+)</math> (Ref. 5 Eq. 3)
:<math>{\mathrm g}(\sigma^+)= \frac{1-\eta/2}{(1-\eta)^3}</math>
:<math>{\mathrm g}(\sigma^+)= \frac{1-\eta/2}{(1-\eta)^3}</math>
where <math>\eta</math> is the [[packing fraction]].
where <math>\eta</math> is the [[packing fraction]].


Over the years many groups have studied the radial distribution function of the hard sphere model:
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>
*[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>[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>
*[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>[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>
*[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>[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>
*[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>[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>
*[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>[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>
*[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>[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>
*[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>[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>
*[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>[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>
*[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>[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>
*[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>[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>
*[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)]
==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>
==Direct correlation function==
<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>.
For the [[direct correlation function]] see:
The liquid-solid coexistence densities (<math>\rho^* = \rho \sigma^3=6\eta/\pi</math>) has been calculated to be
#[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)]
==Bridge function==
Details of the [[bridge function]] for hard sphere can be found in the following publication
*[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)]
==Fluid-solid transition==
The hard sphere system undergoes a [[Solid-liquid phase transitions |fluid-solid]] [[First-order transitions |first order transition]] (Ref. 1).
The fluid-solid coexistence densities (<math>\rho^* = \rho \sigma^3</math>) are given by
:{| border="1"
:{| border="1"
|-  
|-  
| <math>\rho^*_{\mathrm {solid}}</math> || <math>\rho^*_{\mathrm {liquid}}</math> || Reference
| <math>\rho^*_{\mathrm {solid}}</math> || <math>\rho^*_{\mathrm {fluid}}</math> || Reference
|-  
|-  
| 1.041(4)|| 0.943(4) || <ref name="HooverRee"></ref>
| 1.041|| 0.945 ||Ref. 1
|-  
|-  
| 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>
| 1.0376|| 0.9391 ||Ref. 2
|-  
|-  
| 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>
| 1.0367(10) || 0.9387(10) ||Ref. 3
|-  
|-  
| 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>
| 1.0372 || 0.9387  || Ref. 4
|-  
|-  
| 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>
| 1.0369(33) || 0.9375(14) || Ref. 5
|-  
|-  
| 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>
| 1.037 || 0.938 || Ref. 6
|-  
|-  
| 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>
| 1.035(3) || 0.936(2) || Ref. 9
|-
| 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
The coexistence [[pressure]] is given by
:{| border="1"
:{| border="1"
|-  
|-  
| <math>p (k_BT/\sigma^3) </math> || Reference
| <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>
| 11.567|| Ref. 2
|-
| 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>
| 11.57(10) || Ref. 3
|-  
|-  
| 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>
| 11.54(4) || Ref. 5
|-  
|-  
| 11.48(11) || <ref name="Miguel"></ref>
| 11.50(9) || Ref. 7
|-  
|-  
| 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>
| 11.55(11) || Ref. 8
|-  
|-  
| 11.550(4) || <ref name="MoirEtAl2021"></ref>
| 11.48(11) || Ref. 9
|}
|}
The coexistence [[chemical potential]] has been calculated to be
The coexistence [[chemical potential]] is given by
:{| border="1"
:{| border="1"
|-  
|-  
| <math>\mu (k_BT) </math> || Reference
| <math>\mu (k_BT) </math> || Reference
|-  
|-  
| 15.980(11) || <ref name="Miguel"></ref>
| 15.980(11) || Ref. 9
|-
| 16.053(4) || <ref name="MoirEtAl2021"></ref>
|}
|}
The [[Helmholtz energy function]] (in units of <math>Nk_BT</math>) is given by  
The [[Helmholtz energy function]] (in units of <math>Nk_BT</math>) is given by  
:{| border="1"
:{| border="1"
|-  
|-  
| <math>A_{\mathrm {solid}}</math> || <math>A_{\mathrm {liquid}}</math> || Reference
| <math>A_{\mathrm {solid}}</math> || <math>A_{\mathrm {fluid}}</math> || Reference
|-  
|-  
| 4.887(3) || 3.719(8) || <ref name="Miguel"></ref>
| 4.887(3) || 3.719(8) ||Ref. 9
|}
 
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>.
 
==Helmholtz energy function==
Values for the [[Helmholtz energy function]] (<math>A</math>) are given in the following Table:
:{| border="1"
|-
| <math>\rho^*</math> || <math>A/(Nk_BT)</math>|| Reference
|-
| 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>
|-
| 0.50 || −0.152 <math>\pm</math> 0.002  || Table I <ref name="Schilling"></ref>
|-
| 0.75 || 1.721 <math>\pm</math> 0.002  || Table I <ref name="Schilling"></ref>
|-
| 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.
 
==Interfacial Helmholtz energy function==
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)
|}
|}
#[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)]
#Daan Frenkel and Berend Smit "Understanding Molecular Simulation: From Algorithms to Applications", Second Edition (2002) (ISBN 0-12-267351-4) p. 261.
#[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)]
#[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)]
#[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)]
#[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)]
#[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)]
#[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://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)]


==Solid structure==
==Solid structure==
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>
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>. However, for hard spheres at close packing the [[Building up a face centered cubic lattice |face centred cubic]] phase is the more stable (Ref. 3).
<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>
#[http://dx.doi.org/10.1038/26609 Neil J. A. Sloane "Kepler's conjecture confirmed", Nature '''395''' pp. 435-436 (1998)]
<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
#[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>[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".''
#[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)]
*See also: [[Equations of state for crystals of hard spheres]]
*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 ==  
== Equations of state ==  
:''Main article: [[Equations of state for hard spheres]]''
:''Main article: [[Equations of state for hard spheres]]''
==Virial coefficients==
==Virial coefficients==
:''Main article: [[Hard sphere: virial coefficients]]''
:''Main article: ''[[Hard sphere: virial coefficients]]''
== 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>
==Mixtures==
==Mixtures==
*[[Binary hard-sphere mixtures]]
*[[Binary hard-sphere mixtures]]
Line 183: Line 138:
* 2-dimensional case: [[Hard disks | hard disks]].
* 2-dimensional case: [[Hard disks | hard disks]].
* [[Hard hyperspheres]]
* [[Hard hyperspheres]]
== 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:
*[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)]
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.
==References==
==References==
<references/>
#[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)]
'''Related reading'''
#[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/10.1007/978-3-540-78767-9 "Theory and Simulation of Hard-Sphere Fluids and Related Systems", Lecture Notes in Physics '''753/2008''' Springer (2008)]
#[http://dx.doi.org/10.1016/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)]
*[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)]
#[http://dx.doi.org/10.1080/00268970701628423 W. R. Smith,  D. J. Henderson, P. J. Leonard, J. A. Barker and E. W. Grundke "Fortran codes for the correlation functions of hard sphere fluids", Molecular Physics '''106''' pp. 3-7 (2008)]
 
#[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)]
#[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)]
==External links==
==External links==
*[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://www.smac.lps.ens.fr/index.php/Programs_Chapter_2:_Hard_disks_and_spheres Hard disks and spheres] computer code on SMAC-wiki.
[[Category:Models]]
[[Category:Models]]
[[category: hard sphere]]
[[category: hard sphere]]
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