Hard sphere model: Difference between revisions

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m (→‎Direct correlation function: Added internal link.)
(→‎Fluid-solid transition: Added value for coexistence pressure.)
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==Fluid-solid transition==
==Fluid-solid transition==
The hard sphere system undergoes a [[Solid-liquid phase transitions |fluid-solid]] [[First-order transitions |first order transition]] at <math>\rho d^3 = 0.94</math>,  
The hard sphere system undergoes a [[Solid-liquid phase transitions |fluid-solid]] [[First-order transitions |first order transition]] at <math>\rho d^3 = 0.94</math>,  
<math>\eta_A = \frac{\pi \rho d^3}{6} = 0.49218</math>.
<math>\eta_A = \frac{\pi \rho d^3}{6} = 0.49218</math> (Ref. 1). The coexistence [[pressure]] is given by <math>p= 11.54(4) k_BT/\sigma^3</math> (Ref. 2).
*[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)]
#[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)]
#[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)]
 
==Solid structure==
==Solid structure==
*[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)]
*[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)]

Revision as of 13:25, 17 April 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.

First simulations of hard spheres

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

Radial distribution function

The following is a plot of the hard sphere radial distribution function at (produced using the computer code written by Jiří Kolafa)

Typical radial distribution function for hard spheres.
Typical radial distribution function for hard spheres.

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

  1. 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)
  2. 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)
  3. 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)
  4. Francis H. Ree, R. Norris Keeler, and Shaun L. McCarthy "Radial Distribution Function of Hard Spheres", Journal of Chemical Physics 44 pp. 3407- (1966)
  5. W. R. Smith and D. Henderson "Analytical representation of the Percus-Yevick hard-sphere radial distribution function", Molecular Physics 19 pp. 411-415 (1970)
  6. 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)
  7. J. M. Kincaid and J. J. Weis "Radial distribution function of a hard-sphere solid", Molecular Physics 34 pp. 931-938 (1977)
  8. S. Bravo Yuste and A. Santos "Radial distribution function for hard spheres", Physical Review A 43 pp. 5418-5423 (1991)
  9. 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)
  10. Andrij Trokhymchuk, Ivo Nezbeda and Jan Jirsák "Hard-sphere radial distribution function again", Journal of Chemical Physics 123 024501 (2005)
  11. 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:

  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 at , (Ref. 1). The coexistence pressure is given by (Ref. 2).

  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. 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)

Solid structure

Equations of state

Main article: Equations of state for hard spheres

Virial coefficients

Main article: Hard sphere: virial coefficients

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.

External links

Related systems

Hard spheres in other dimensions

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)