Hard sphere model: Difference between revisions

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


: <math>
: <math>
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==Virial coefficients==
==Virial coefficients==
:''Main article: ''[[Hard sphere: virial coefficients]]''
:''Main article: ''[[Hard sphere: virial coefficients]]''
== Experimental results ==
==Mixtures==
Pusey and  van Megen used a suspension of PMMA particles of radius 305 <math>\pm</math>10 nm,  suspended in poly-12-hydroxystearic acid:
*[[Binary hard-sphere mixtures]]
*[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)]
*[[Multicomponent hard-sphere mixtures]]
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.
==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.
== Related systems ==  
== Related systems ==  
*[[Polydisperse hard spheres]]
*[[Quantum hard spheres]]
*[[Quantum hard spheres]]
*[[Dipolar hard spheres]]
*[[Dipolar hard spheres]]
*[[Lattice hard spheres]]
*[[Lattice hard spheres]]
====Hard spheres in other dimensions====
Hard spheres in other dimensions:
* 1-dimensional case: [[1-dimensional hard rods | hard rods]].
* 1-dimensional case: [[1-dimensional hard rods | hard rods]].
* 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==
#[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/10.1088/0953-8984/9/41/006 Robin J. Speedy "Pressure of the metastable hard-sphere fluid", Journal of  Physics: Condensed Matter '''9''' pp. 8591-8599    (1997)]
Line 126: Line 125:
#[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/10.1103/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)]
#[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:Models]]
[[category: hard sphere]]
[[category: hard sphere]]

Revision as of 14:49, 27 November 2008

The hard sphere intermolecular pair potential 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

The hard sphere model was one of the first ever systems studied:

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 computer code written by Jiří Kolafa). 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 (Ref 5 Eq. 1)

where the second virial coefficient, , is given by

.

Carnahan and Starling (Ref. 6) provided the following expression for (Ref. 5 Eq. 3)

where is the packing fraction.

Over the years many groups have studied the radial distribution function of the hard sphere model:

Direct correlation function

For the direct correlation function see:

  1. 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 fluid-solid first order transition (Ref. 1). The fluid-solid coexistence densities () are given by

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

The coexistence pressure is given by

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
  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)
  2. Daan Frenkel and Berend Smit "Understanding Molecular Simulation: From Algorithms to Applications", Second Edition (2002) (ISBN 0-12-267351-4) p. 261.
  3. 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)
  4. 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)
  5. 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)
  6. Ruslan L. Davidchack and Brian B. Laird "Simulation of the hard-sphere crystal–melt interface", Journal of Chemical Physics 108 pp. 9452-9462 (1998)
  7. N. B. Wilding and A. D. Bruce "Freezing by Monte Carlo Phase Switch", Physical Review Letters 85 pp. 5138-5141 (2000)
  8. Robin J. Speedy "Pressure of the metastable hard-sphere fluid", Journal of Physics: Condensed Matter 9 pp. 8591-8599 (1997)

Solid structure

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 . However, for hard spheres at close packing the face centred cubic phase is the more stable (Ref. 3).

  1. Neil J. A. Sloane "Kepler's conjecture confirmed", Nature 395 pp. 435-436 (1998)
  2. C. F. Tejero, M. S. Ripoll, and A. Pérez "Pressure of the hard-sphere solid", Physical Review E 52 pp. 3632-3636 (1995)
  3. Leslie V. Woodcock "Computation of the free energy for alternative crystal structures of hard spheres", Faraday Discussions 106 pp. 325 - 338 (1997)

Equations of state

Main article: Equations of state for hard spheres

Virial coefficients

Main article: Hard sphere: virial coefficients

Mixtures

Related systems

Hard spheres in other dimensions:

Experimental results

Pusey and van Megen used a suspension of PMMA particles of radius 305 10 nm, suspended in poly-12-hydroxystearic acid:

For results obtained from the Colloidal Disorder - Order Transition (CDOT) experiments performed on-board the Space Shuttles Columbia and Discovery see Ref. 3.

References

  1. Robin J. Speedy "Pressure of the metastable hard-sphere fluid", Journal of Physics: Condensed Matter 9 pp. 8591-8599 (1997)
  2. Robin J. Speedy "Pressure and entropy of hard-sphere crystals", Journal of Physics: Condensed Matter 10 pp. 4387-4391 (1998)
  3. 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)
  4. 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)
  5. 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)
  6. 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