Difference between revisions of "Hard disk model"

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m (Phase transitions: Added a recent publication)
m (Phase transitions: Added a recent publication)
 
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where <math> \Phi_{12}\left(r \right) </math> is the [[intermolecular pair potential]] between two disks at a distance <math>r := |\mathbf{r}_1 - \mathbf{r}_2|</math>, and <math> \sigma </math> is the diameter of the disk. This page treats hard disks in a two-dimensional space, for three dimensions see the page [[hard disks in a three dimensional space]].
 
where <math> \Phi_{12}\left(r \right) </math> is the [[intermolecular pair potential]] between two disks at a distance <math>r := |\mathbf{r}_1 - \mathbf{r}_2|</math>, and <math> \sigma </math> is the diameter of the disk. This page treats hard disks in a two-dimensional space, for three dimensions see the page [[hard disks in a three dimensional space]].
 
==Phase transitions==
 
==Phase transitions==
Despite the apparent simplicity of this model/system, the phase behaviour and the nature of the phase transitions remains an area of active study ever since the early work of Alder and Wainwright <ref>[http://dx.doi.org/10.1103/PhysRev.127.359 B. J. Alder and T. E. Wainwright "Phase Transition in Elastic Disks", Physical Review '''127''' pp. 359-361 (1962)]</ref>. In a recent publication by Mak <ref>[http://dx.doi.org/10.1103/PhysRevE.73.065104 C. H. Mak "Large-scale simulations of the two-dimensional melting of hard disks", Physical Review E '''73''' 065104(R) (2006)]</ref>  using over 4 million particles <math>(2048^2)</math> one appears to have the phase diagram isotropic <math>(\eta < 0.699)</math>, a  hexatic phase, and a solid phase <math>(\eta > 0.723)</math> (the maximum possible packing fraction is given by <math>\eta = \pi / \sqrt{12} \approx 0.906899...</math> <ref>[http://dx.doi.org/10.1007/BF01181430 L. Fejes Tóth "Über einen geometrischen Satz." Mathematische Zeitschrift '''46''' pp. 83-85 (1940)]</ref>) . Similar results have been found using the [[BBGKY hierarchy]] <ref>[http://dx.doi.org/10.1063/1.3491039  Jarosław Piasecki, Piotr Szymczak, and John J. Kozak "Prediction of a structural transition in the hard disk fluid", Journal of Chemical Physics '''133''' 164507 (2010)]</ref> and by studying tessellations (the hexatic region: <math>0.680 < \eta < 0.729</math>) <ref>[http://dx.doi.org/10.1021/jp806287e John J. Kozak, Jack Brzezinski and Stuart A. Rice "A Conjecture Concerning the Symmetries of Planar Nets and the Hard disk Freezing Transition", Journal of Physical Chemistry B '''112''' pp. 16059-16069 (2008)]</ref>.
+
Despite the apparent simplicity of this model/system, the phase behaviour and the nature of the phase transitions remains an area of active study ever since the early work of Alder and Wainwright <ref>[http://dx.doi.org/10.1103/PhysRev.127.359 B. J. Alder and T. E. Wainwright "Phase Transition in Elastic Disks", Physical Review '''127''' pp. 359-361 (1962)]</ref>. In a recent publication by Mak <ref>[http://dx.doi.org/10.1103/PhysRevE.73.065104 C. H. Mak "Large-scale simulations of the two-dimensional melting of hard disks", Physical Review E '''73''' 065104(R) (2006)]</ref>  using over 4 million particles <math>(2048^2)</math> one appears to have the phase diagram isotropic <math>(\eta < 0.699)</math>, a  hexatic phase, and a solid phase <math>(\eta > 0.723)</math> (the maximum possible packing fraction is given by <math>\eta = \pi / \sqrt{12} \approx 0.906899...</math> <ref>[http://dx.doi.org/10.1007/BF01181430 L. Fejes Tóth "Über einen geometrischen Satz." Mathematische Zeitschrift '''46''' pp. 83-85 (1940)]</ref>) . Similar results have been found using the [[BBGKY hierarchy]] <ref>[http://dx.doi.org/10.1063/1.3491039  Jarosław Piasecki, Piotr Szymczak, and John J. Kozak "Prediction of a structural transition in the hard disk fluid", Journal of Chemical Physics '''133''' 164507 (2010)]</ref> and by studying tessellations (the hexatic region: <math>0.680 < \eta < 0.729</math>) <ref>[http://dx.doi.org/10.1021/jp806287e John J. Kozak, Jack Brzezinski and Stuart A. Rice "A Conjecture Concerning the Symmetries of Planar Nets and the Hard disk Freezing Transition", Journal of Physical Chemistry B '''112''' pp. 16059-16069 (2008)]</ref>. Also studied via [[integral equations]] <ref>[https://doi.org/10.1063/1.5026496  Luis Mier-y-Terán, Brian Ignacio Machorro-Martínez, Gustavo A. Chapela, and Fernando del Río "Study of the hard-disk system at high densities: the fluid-hexatic phase transition", Journal of Chemical Physics '''148''' 234502 (2018)]</ref>.
 
Experimental results <ref>[http://dx.doi.org/10.1103/PhysRevLett.118.158001 Alice L. Thorneywork, Joshua L. Abbott, Dirk G. A. L. Aarts, and Roel P. A. Dullens "Two-Dimensional Melting of Colloidal Hard Spheres", Physical Review Letters '''118''' 158001 (2017)]</ref>.
 
Experimental results <ref>[http://dx.doi.org/10.1103/PhysRevLett.118.158001 Alice L. Thorneywork, Joshua L. Abbott, Dirk G. A. L. Aarts, and Roel P. A. Dullens "Two-Dimensional Melting of Colloidal Hard Spheres", Physical Review Letters '''118''' 158001 (2017)]</ref>.
  

Latest revision as of 11:45, 25 June 2018

Hard disks are hard spheres in two dimensions. The hard disk intermolecular pair potential is given by[1] [2]


\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 disks at a distance r := |\mathbf{r}_1 - \mathbf{r}_2|, and  \sigma is the diameter of the disk. This page treats hard disks in a two-dimensional space, for three dimensions see the page hard disks in a three dimensional space.

Phase transitions[edit]

Despite the apparent simplicity of this model/system, the phase behaviour and the nature of the phase transitions remains an area of active study ever since the early work of Alder and Wainwright [3]. In a recent publication by Mak [4] using over 4 million particles (2048^2) one appears to have the phase diagram isotropic (\eta < 0.699), a hexatic phase, and a solid phase (\eta > 0.723) (the maximum possible packing fraction is given by \eta = \pi / \sqrt{12} \approx 0.906899... [5]) . Similar results have been found using the BBGKY hierarchy [6] and by studying tessellations (the hexatic region: 0.680 < \eta < 0.729) [7]. Also studied via integral equations [8]. Experimental results [9].

Equations of state[edit]

Main article: Equations of state for hard disks

Virial coefficients[edit]

Main article: Hard sphere: virial coefficients

See also[edit]

References[edit]

  1. Nicholas Metropolis, Arianna W. Rosenbluth, Marshall N. Rosenbluth, Augusta H. Teller and Edward Teller, "Equation of State Calculations by Fast Computing Machines", Journal of Chemical Physics 21 pp.1087-1092 (1953)
  2. W. W. Wood "Monte Carlo calculations of the equation of state of systems of 12 and 48 hard circles", Los Alamos Scientific Laboratory Report LA-2827 (1963)
  3. B. J. Alder and T. E. Wainwright "Phase Transition in Elastic Disks", Physical Review 127 pp. 359-361 (1962)
  4. C. H. Mak "Large-scale simulations of the two-dimensional melting of hard disks", Physical Review E 73 065104(R) (2006)
  5. L. Fejes Tóth "Über einen geometrischen Satz." Mathematische Zeitschrift 46 pp. 83-85 (1940)
  6. Jarosław Piasecki, Piotr Szymczak, and John J. Kozak "Prediction of a structural transition in the hard disk fluid", Journal of Chemical Physics 133 164507 (2010)
  7. John J. Kozak, Jack Brzezinski and Stuart A. Rice "A Conjecture Concerning the Symmetries of Planar Nets and the Hard disk Freezing Transition", Journal of Physical Chemistry B 112 pp. 16059-16069 (2008)
  8. Luis Mier-y-Terán, Brian Ignacio Machorro-Martínez, Gustavo A. Chapela, and Fernando del Río "Study of the hard-disk system at high densities: the fluid-hexatic phase transition", Journal of Chemical Physics 148 234502 (2018)
  9. Alice L. Thorneywork, Joshua L. Abbott, Dirk G. A. L. Aarts, and Roel P. A. Dullens "Two-Dimensional Melting of Colloidal Hard Spheres", Physical Review Letters 118 158001 (2017)

Related reading

External links[edit]