Hard sphere model

2008-12-31T01:06:37Z

59.92.86.24: /* Fluid phase radial distribution function */ the plots are of C(r1,r2) not g(r1,r2)

[[Image:sphere_green.png|thumb|right]]
The '''hard sphere''' [[intermolecular pair potential]] is defined as

: <math>
\Phi_{12}\left( r \right) = \left\{ \begin{array}{lll}
\infty & ; & r < \sigma \\
0 & ; & r \ge \sigma \end{array} \right.
</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==
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 fluid and solid phases and their corresponding [[Phase transitions | phase transition]]. The following are a sample of some of the very first works:
*[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)]
*[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)]
*[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)]

==Fluid phase radial distribution function==
The following are a series of plots of the hard sphere [[total correlation function]], 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"
|-
|<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]] (Ref 5 Eq. 1)
:<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. 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>
where <math>\eta</math> is the [[packing fraction]].

Over the years many groups have studied the radial distribution function of the hard sphere model:
*[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)]
*[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)]
*[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)]
*[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)]
*[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)]
*[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)]
*[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)]
*[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)]
*[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)]
*[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)]
*[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)]

==Direct correlation function==
For the [[direct correlation function]] see:
#[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)]

==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"
|-
| <math>\rho^*_{\mathrm {solid}}</math> || <math>\rho^*_{\mathrm {fluid}}</math> || Reference
|-
| 1.041|| 0.945 ||Ref. 1
|-
| 1.0376|| 0.9391 ||Ref. 2
|-
| 1.0367(10) || 0.9387(10) ||Ref. 3
|-
| 1.0372 || 0.9387 || Ref. 4
|-
| 1.0369(33) || 0.9375(14) || Ref. 5
|-
| 1.037 || 0.938 || Ref. 6
|-
| 1.035(3) || 0.936(2) || Ref. 9
|}
The coexistence [[pressure]] is given by
:{| border="1"
|-
| <math>p (k_BT/\sigma^3) </math> || Reference
|-
| 11.567|| Ref. 2
|-
| 11.57(10) || Ref. 3
|-
| 11.54(4) || Ref. 5
|-
| 11.50(9) || Ref. 7
|-
| 11.55(11) || Ref. 8
|-
| 11.48(11) || Ref. 9
|}
The coexistence [[chemical potential]] is given by
:{| border="1"
|-
| <math>\mu (k_BT) </math> || Reference
|-
| 15.980(11) || Ref. 9
|}
The [[Helmholtz energy function]] (in units of <math>Nk_BT</math>) is given by
:{| border="1"
|-
| <math>A_{\mathrm {solid}}</math> || <math>A_{\mathrm {fluid}}</math> || Reference
|-
| 4.887(3) || 3.719(8) ||Ref. 9
|}
#[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==
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).
#[http://dx.doi.org/10.1038/26609 Neil J. A. Sloane "Kepler's conjecture confirmed", Nature '''395''' pp. 435-436 (1998)]
#[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)]
#[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]]

== Equations of state ==
:''Main article: [[Equations of state for hard spheres]]''
==Virial coefficients==
:''Main article: ''[[Hard sphere: virial coefficients]]''
==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]]
== 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==
#[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/10.1088/0953-8984/10/20/006 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)]
#[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==
*[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: hard sphere]]

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Hard sphere model