Hard sphere model

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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 using computer simulation techniques with a view to understanding the thermodynamics of the fluid and solid phases and their corresponding phase transition. The following are a sample of some of the very first works:

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 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
1.035(3) 0.936(2) Ref. 9

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
11.48(11) Ref. 9

The coexistence chemical potential is given by

Reference
15.980(11) Ref. 9

The Helmholtz energy function (in units of ) is given by

Reference
4.887(3) 3.719(8) Ref. 9
  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)
  9. 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 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