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[[Image:sphere_green.png|thumb|right]] | [[Image:sphere_green.png|thumb|right]] | ||
The '''hard sphere model''' (sometimes known as the ''rigid sphere model'') is defined as | The '''hard sphere model''' (sometimes known as the ''rigid sphere model'') is defined as | ||
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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 | ==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, 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 | to understanding the thermodynamics of the fluid 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/10.1063/1.1740207 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/10.1063/1.1743956 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> | <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> | ||
== | |||
The following are a series of plots of the hard sphere [[radial distribution function]] <ref>The [[total correlation function]] data was produced using the [ | ==Fluid 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 [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]</ref>. 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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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>[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> | :<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]]. | ||
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<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/10.1063/1.1979488 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> | <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> | ||
== | ==Fluid-solid transition== | ||
The hard sphere system undergoes a [[Solid-liquid phase transitions | | The hard sphere system undergoes a [[Solid-liquid phase transitions |fluid-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>, now referred 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 | 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 { | | <math>\rho^*_{\mathrm {solid}}</math> || <math>\rho^*_{\mathrm {fluid}}</math> || Reference | ||
|- | |- | ||
| 1.041 | | 1.041|| 0.945 || <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> | | 1.0376|| 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> | ||
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| 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>[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. | | 1.035(3) || 0.936(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> | ||
|} | |} | ||
The coexistence [[pressure]] | 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 | ||
|- | |- | ||
| 11.567|| <ref name="FrenkelSmitBook"> </ref> | |||
| 11.567|| <ref name="FrenkelSmitBook"></ref> | |||
|- | |- | ||
| 11. | | 11.57(10) || <ref name="Fortini"> </ref> | ||
|- | |- | ||
| 11.54(4) || <ref name="Noya"></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> | | 11.50(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. | | 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. | | 11.48(11) || <ref name="Miguel"> </ref> | ||
|} | |} | ||
The coexistence [[chemical potential]] | 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 | | 15.980(11) || <ref name="Miguel"> </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 { | | <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 name="Miguel"> </ref> | ||
|} | |} | ||
==Helmholtz energy function== | ==Helmholtz energy function== | ||
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| <math>\rho^*</math> || <math>A/(Nk_BT)</math>|| Reference | | <math>\rho^*</math> || <math>A/(Nk_BT)</math>|| Reference | ||
|- | |- | ||
| 0.25 || | | 0.25 || 0.620 <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 || | | 0.50 || 1.541 <math>\pm</math> 0.002 || Table I <ref name="Schilling"> </ref> | ||
|- | |- | ||
| 0.75 || | | 0.75 || 3.009 <math>\pm</math> 0.002 || Table I <ref name="Schilling"> </ref> | ||
|- | |- | ||
| 1.04086 || 4.959 || Table VI <ref name="VegaNoya"></ref> | | 1.04086 || 4.959 || Table VI <ref name="VegaNoya"> </ref> | ||
|- | |- | ||
| 1.099975 || 5.631 || 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> | | 1.150000 || 6.274 || Table VI <ref name="VegaNoya"> </ref> | ||
|} | |} | ||
==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 | 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> | ||
<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/10.1103/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> | <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> | ||
*See also: [[Equations of state for crystals of hard spheres]] | *See also: [[Equations of state for crystals of hard spheres]] | ||
==Direct correlation function== | ==Direct correlation function== | ||
For the [[direct correlation function]] see: | For the [[direct correlation function]] see: | ||
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'''Related reading''' | '''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/10.1007/978-3-540-78767-9 "Theory and Simulation of Hard-Sphere Fluids and Related Systems", Lecture Notes in Physics '''753/2008''' Springer (2008)] | ||
==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]] |