Hard sphere model

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Sphere green.png
Phase diagram (pressure vs packing fraction) of hard sphere system (Solid line - stable branch, dashed line - metastable branch)

The hard sphere model (sometimes known as the rigid sphere model) is defined as

\[ \Phi_{12}\left( r \right) = \left\{ \begin{array}{lll} \infty & ; & r < \sigma \\ 0 & ; & r \ge \sigma \end{array} \right. \]

where \( \Phi_{12}\left(r \right) \) is the intermolecular pair potential between two spheres at a distance \(r := |\mathbf{r}_1 - \mathbf{r}_2|\), and \( \sigma \) 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, \(a=b=c\).

First simulations of hard spheres (1954-1957)[edit]

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 liquid and solid phases and their corresponding phase transition [1] [2] [3], much of this work undertaken at the Los Alamos Scientific Laboratory on the world's first electronic digital computer ENIAC [4].

Liquid phase radial distribution function[edit]

The following are a series of plots of the hard sphere radial distribution function [5] shown for different values of the number density \(\rho\). The horizontal axis is in units of \(\sigma\) where \(\sigma\) is set to be 1. Click on image of interest to see a larger view.

\(\rho=0.2\)
HS 0.2 rdf.png
\(\rho=0.3\)
HS 0.3 rdf.png
\(\rho=0.4\)
HS 0.4 rdf.png
\(\rho=0.5\)
HS 0.5 rdf.png
\(\rho=0.6\)
HS 0.6 rdf.png
\(\rho=0.7\)
HS 0.7 rdf.png
\(\rho=0.8\)
HS 0.8 rdf.png
\(\rho=0.85\)
HS 0.85 rdf.png
\(\rho=0.9\)
HS 0.9 rdf.png

The value of the radial distribution at contact, \({\mathrm g}(\sigma^+)\), can be used to calculate the pressure via the equation of state (Eq. 1 in [6]) \[\frac{p}{\rho k_BT}= 1 + B_2 \rho {\mathrm g}(\sigma^+)\] where the second virial coefficient, \(B_2\), is given by \[B_2 = \frac{2\pi}{3}\sigma^3\]. Carnahan and Starling [7] provided the following expression for \({\mathrm g}(\sigma^+)\) (Eq. 3 in [6]) \[{\mathrm g}(\sigma^+)= \frac{1-\eta/2}{(1-\eta)^3}\] where \(\eta\) is the packing fraction.

Over the years many groups have studied the radial distribution function of the hard sphere model: [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18]

Liquid-solid transition[edit]

The hard sphere system undergoes a liquid-solid first order transition [19] [20], sometimes referred to as the Kirkwood-Alder transition [21]. The liquid-solid coexistence densities (\(\rho^* = \rho \sigma^3=6\eta/\pi\)) has been calculated to be

\(\rho^*_{\mathrm {solid}}\) \(\rho^*_{\mathrm {liquid}}\) Reference
1.041 0.945 [19]
1.0376 0.9391 [22]
1.0367(10) 0.9387(10) [23]
1.0372 0.9387 [24]
1.0369(33) 0.9375(14) [25]
1.037 0.938 [26]
1.035(3) 0.936(2) [27]

The coexistence pressure has been calculated to be

\(p (k_BT/\sigma^3) \) Reference
11.5727(10) [28]
11.57(10) [23]
11.567 [22]
11.55(11) [29]
11.54(4) [25]
11.50(9) [30]
11.48(11) [27]
11.43(17) [31]

The coexistence chemical potential has been calculated to be

\(\mu (k_BT) \) Reference
15.980(11) [27]

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

\(A_{\mathrm {solid}}\) \(A_{\mathrm {liquid}}\) Reference
4.887(3) 3.719(8) [27]

Helmholtz energy function[edit]

Values for the Helmholtz energy function (\(A\)) are given in the following Table:

\(\rho^*\) \(A/(Nk_BT)\) Reference
0.25 0.620 \(\pm\) 0.002 Table I [32]
0.50 1.541 \(\pm\) 0.002 Table I [32]
0.75 3.009 \(\pm\) 0.002 Table I [32]
1.04086 4.959 Table VI [24]
1.099975 5.631 Table VI [24]
1.150000 6.274 Table VI [24]

Interfacial Helmholtz energy function[edit]

The Helmholtz energy function of the solid–liquid interface has been calculated using the cleaving method giving (Ref. [33] Table I):

work per unit area/\((k_BT/\sigma^2)\)
\(\gamma_{\{100\}}\) 0.5820(19)
\(\gamma_{\{100\}}\) 0.636(11) [28]
\(\gamma_{\{110\}}\) 0.5590(20)
\(\gamma_{\{111\}}\) 0.5416(31)
\(\gamma_{\{120\}}\) 0.5669(20)

Solid structure[edit]

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 \(\pi/(3 \sqrt{2}) \approx 0.74048%\) [34] [35]. However, for hard spheres at close packing the face centred cubic phase is the more stable [36]

Direct correlation function[edit]

For the direct correlation function see: [37] [38]

Bridge function[edit]

Details of the bridge function for hard sphere can be found in the following publication [39]

Equations of state[edit]

Main article: Equations of state for hard spheres

Virial coefficients[edit]

Main article: Hard sphere: virial coefficients

Experimental results[edit]

Pusey and van Megen used a suspension of PMMA particles of radius 305 \(\pm\)10 nm, suspended in poly-12-hydroxystearic acid [40] For results obtained from the Colloidal Disorder - Order Transition (CDOT) experiments performed on-board the Space Shuttles Columbia and Discovery see Ref. [41]

Mixtures[edit]

Related systems[edit]

Hard spheres in other dimensions:

References[edit]

  1. Marshall N. Rosenbluth and Arianna W. Rosenbluth "Further Results on Monte Carlo Equations of State", Journal of Chemical Physics 22 pp. 881-884 (1954)
  2. 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)
  3. B. J. Alder and T. E. Wainwright "Phase Transition for a Hard Sphere System", Journal of Chemical Physics 27 pp. 1208-1209 (1957)
  4. The ENIAC Story
  5. The total correlation function data was produced using the computer code written by Jiří Kolafa
  6. 6.0 6.1 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)
  7. N. F.Carnahan and K. E.Starling,"Equation of State for Nonattracting Rigid Spheres" Journal of Chemical Physics 51 pp. 635-636 (1969)
  8. 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)
  9. 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)
  10. 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)
  11. Francis H. Ree, R. Norris Keeler, and Shaun L. McCarthy "Radial Distribution Function of Hard Spheres", Journal of Chemical Physics 44 pp. 3407- (1966)
  12. W. R. Smith and D. Henderson "Analytical representation of the Percus-Yevick hard-sphere radial distribution function", Molecular Physics 19 pp. 411-415 (1970)
  13. 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)
  14. J. M. Kincaid and J. J. Weis "Radial distribution function of a hard-sphere solid", Molecular Physics 34 pp. 931-938 (1977)
  15. S. Bravo Yuste and A. Santos "Radial distribution function for hard spheres", Physical Review A 43 pp. 5418-5423 (1991)
  16. 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)
  17. Andrij Trokhymchuk, Ivo Nezbeda and Jan Jirsák "Hard-sphere radial distribution function again", Journal of Chemical Physics 123 024501 (2005)
  18. 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)
  19. 19.0 19.1 William G. Hoover and Francis H. Ree "Melting Transition and Communal Entropy for Hard Spheres", Journal of Chemical Physics 49 pp. 3609-3617 (1968)
  20. Miguel Robles, Mariano López de Haro and Andrés Santos "Note: Equation of state and the freezing point in the hard-sphere model", Journal of Chemical Physics 140 136101 (2014)
  21. Alice P. Gast and William B. Russel "Simple Ordering in Complex Fluids", Physics Today 51 (12) pp. 24-30 (1998)
  22. 22.0 22.1 Daan Frenkel and Berend Smit "Understanding Molecular Simulation: From Algorithms to Applications", Second Edition (2002) (ISBN 0-12-267351-4) p. 261.
  23. 23.0 23.1 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)
  24. 24.0 24.1 24.2 24.3 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)
  25. 25.0 25.1 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)
  26. Ruslan L. Davidchack and Brian B. Laird "Simulation of the hard-sphere crystal–melt interface", Journal of Chemical Physics 108 pp. 9452-9462 (1998)
  27. 27.0 27.1 27.2 27.3 Enrique de Miguel "Estimating errors in free energy calculations from thermodynamic integration using fitted data", Journal of Chemical Physics 129 214112 (2008)
  28. 28.0 28.1 L. A. Fernández, V. Martín-Mayor, B. Seoane, and P. Verrocchio "Equilibrium Fluid-Solid Coexistence of Hard Spheres", Physical Review Letters 108 165701 (2012)
  29. Robin J. Speedy "Pressure of the metastable hard-sphere fluid", Journal of Physics: Condensed Matter 9 pp. 8591-8599 (1997)
  30. N. B. Wilding and A. D. Bruce "Freezing by Monte Carlo Phase Switch", Physical Review Letters 85 pp. 5138-5141 (2000)
  31. G. Odriozola "Replica exchange Monte Carlo applied to hard spheres", Journal of Chemical Physics 131 144107 (2009)
  32. 32.0 32.1 32.2 T. Schilling and F. Schmid "Computing absolute free energies of disordered structures by molecular simulation", Journal of Chemical Physics 131 231102 (2009)
  33. Ruslan L. Davidchack "Hard spheres revisited: Accurate calculation of the solid–liquid interfacial free energy", Journal of Chemical Physics 133 234701 (2010)
  34. Neil J. A. Sloane "Kepler's conjecture confirmed", Nature 395 pp. 435-436 (1998)
  35. C. F. Tejero, M. S. Ripoll, and A. Pérez "Pressure of the hard-sphere solid", Physical Review E 52 pp. 3632-3636 (1995)
  36. Leslie V. Woodcock "Computation of the free energy for alternative crystal structures of hard spheres", Faraday Discussions 106 pp. 325 - 338 (1997)
  37. C. F. Tejero and M. López De Haro "Direct correlation function of the hard-sphere fluid", Molecular Physics 105 pp. 2999-3004 (2007)
  38. Matthew Dennison, Andrew J. Masters, David L. Cheung, and Michael P. Allen "Calculation of direct correlation function for hard particles using a virial expansion", Molecular Physics pp. 375-382 (2009)
  39. Jiri Kolafa, Stanislav Labik and Anatol Malijevsky "The bridge function of hard spheres by direct inversion of computer simulation data", Molecular Physics 100 pp. 2629-2640 (2002)
  40. P. N. Pusey and W. van Megen "Phase behaviour of concentrated suspensions of nearly hard colloidal spheres", Nature 320 pp. 340-342 (1986)
  41. 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)

Related reading

External links[edit]