Triangular well model: Difference between revisions

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{{stub-general}}
The '''triangular well model''', proposed by T. Nagayimain one dimension <ref>[http://www.journalarchive.jst.go.jp/english//jnlabstract_en.php?cdjournal=ppmsj1919&cdvol=22&noissue=8-9&startpage=705 T. Nagayima "Statistical Mechanics of One-dimensional Substances I", Proceedings of the Physico-Mathematical Society of Japan '''22''' pp.  705-720 (1940)]</ref><ref>[http://www.journalarchive.jst.go.jp/english//jnlabstract_en.php?cdjournal=ppmsj1919&cdvol=22&noissue=12&startpage=1034 T. Nagayima "Statistical Mechanics of One-dimensional Substances. II", Proceedings of the Physico-Mathematical Society of Japan '''22''' pp.  1034-1047 (1940)]</ref>, is given by
The '''triangular well model''' is given by


:<math>
:<math>
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where <math>\Phi_{12}(r)</math> is the [[intermolecular pair potential]], <math>r</math> is the distance <math>r := |\mathbf{r}_1 - \mathbf{r}_2|</math>, <math>\sigma</math> is the hard diameter, <math>\epsilon</math> is the well depth
where <math>\Phi_{12}(r)</math> is the [[intermolecular pair potential]], <math>r</math> is the distance <math>r := |\mathbf{r}_1 - \mathbf{r}_2|</math>, <math>\sigma</math> is the hard diameter, <math>\epsilon</math> is the well depth
and &lambda; &gt; 1.
and &lambda; &gt; 1.
This model was first proposed by T. Nagayima in 1940 (Refs. 1-3) in one dimension.
==Equation of state==
==Equation of state==
:''Main article: [[Equations of state for the triangular well model]]''
:''Main article: [[Equations of state for the triangular well model]]''
==Virial coefficients==
==Virial coefficients==
<math>B_2</math> and <math>B_3</math>:
<math>B_2</math>, <math>B_3</math> <ref>[http://dx.doi.org/10.1063/1.1725746  M. J. Feinberg and Andrew G. De Rocco "Intermolecular Forces: The Triangle Well and Some Comparisons with the Square Well and Lennard-Jones", Journal of Chemical Physics '''41''' pp. 3439-3450 (1964)]</ref>
*[http://dx.doi.org/10.1063/1.1725746  M. J. Feinberg and Andrew G. De Rocco "Intermolecular Forces: The Triangle Well and Some Comparisons with the Square Well and Lennard-Jones", Journal of Chemical Physics '''41''' pp. 3439-3450 (1964)]
<ref>[http://dx.doi.org/10.1063/1.1696837 R. H. Fowler, H. W. Graben, Andrew G. De Rocco and M. J. Feinberg "Some Additional Results for the Triangle-Well Potential Model", Journal of Chemical Physics '''43''' pp. 1083-1084 (1965)]</ref>
*[http://dx.doi.org/10.1063/1.1696837 R. H. Fowler, H. W. Graben, Andrew G. De Rocco and M. J. Feinberg "Some Additional Results for the Triangle-Well Potential Model", Journal of Chemical Physics '''43''' pp. 1083-1084 (1965)]
and <math>B_4</math> <ref>[http://dx.doi.org/10.1063/1.1675133 W. C. Farrar and Andrew G. De Rocco "Perturbation Theory for a High-Temperature Triangle-Well Fluid", Journal of Chemical Physics '''54''' pp. 2024-2025 (1971)]</ref>
<math>B_4</math>:
*[http://dx.doi.org/10.1063/1.1675133 W. C. Farrar and Andrew G. De Rocco "Perturbation Theory for a High-Temperature Triangle-Well Fluid", Journal of Chemical Physics '''54''' pp. 2024-2025 (1971)]
==Critical point==
==Critical point==
==Solid phase==
==Solid phase==
#[http://dx.doi.org/10.1080/00268970110120300 Jhumpa Adhikari and David A. Kofke "Monte Carlo and cell model calculations for the solid-fluid phase behaviour of the triangle-well model", Molecular Physics '''100''' pp. 1543-1550  (2002)]
<ref>[http://dx.doi.org/10.1080/00268970110120300 Jhumpa Adhikari and David A. Kofke "Monte Carlo and cell model calculations for the solid-fluid phase behaviour of the triangle-well model", Molecular Physics '''100''' pp. 1543-1550  (2002)]</ref>
==References==
==References==
(Note: the following three Nagayima references have not yet been checked. Numbers 1 and 3 were found in the [http://dx.doi.org/10.1063/1.1725746 Feinberg and De Rocco] paper, and Ref 2 in the [http://dx.doi.org/10.1063/1.3049399 Shiqi Zhou] paper).
<references/>
#T. Nagayima "", Proceedings of the Physico-Mathematical Society of Japan '''22''' pp.  705- (1940)
;Related reading
#T. Nagayima "", Proceedings of the Physico-Mathematical Society of Japan '''22''' pp.  855- (1940)
*[http://dx.doi.org/10.1139/p72-195 Damon N. Card and John Walkley "Perturbation Calculations for a Triangular Well Potential at Low Densities", Canadian Journal of Physics '''50''' pp. 1419–1426 (1972)]
#T. Nagayima "", Proceedings of the Physico-Mathematical Society of Japan '''22''' pp.  1034- (1940)
*[http://dx.doi.org/10.1139/p74-010 Damon N. Card and John Walkley "Monte Carlo and Perturbation Calculations for a Triangular Well Fluid", Canadian Journal of Physics '''52''' pp. 80-88 (1974)]
#[http://dx.doi.org/10.1139/p72-195 Damon N. Card and John Walkley "Perturbation Calculations for a Triangular Well Potential at Low Densities", Canadian Journal of Physics '''50''' pp. 1419–1426 (1972)]
*[http://dx.doi.org/10.1016/S0378-4371(00)00232-6  J. Largo and J. R. Solana "A simplified perturbation theory for equilibrium properties of triangular-well fluids", Physica A '''284''' pp. 68-78 (2000)]
#[http://dx.doi.org/10.1139/p74-010 Damon N. Card and John Walkley "Monte Carlo and Perturbation Calculations for a Triangular Well Fluid", Canadian Journal of Physics '''52''' pp. 80-88 (1974)]
*[http://dx.doi.org/10.1080/00268970701725013 F. F. Betancourt-Cárdenas, L. A. Galicia-Luna and S. I. Sandler "Thermodynamic properties for the triangular-well fluid", Molecular Physics '''105''' pp. 2987-2998 (2007)]
#[http://dx.doi.org/10.1016/S0378-4371(00)00232-6  J. Largo and J. R. Solana "A simplified perturbation theory for equilibrium properties of triangular-well fluids", Physica A '''284''' pp. 68-78 (2000)]
*[http://dx.doi.org/10.1080/00268970701832397 F. F. Betancourt-Cárdenas,  L. A. Galicia-Luna,  A. L. Benavides,  J. A. Ramírez and E. Schöll-Paschinger "Thermodynamics of a long-range triangle-well fluid", Molecular Physics '''106''' pp. 113-126 (2008)]
#[http://dx.doi.org/10.1080/00268970701725013 F. F. Betancourt-Cárdenas, L. A. Galicia-Luna and S. I. Sandler "Thermodynamic properties for the triangular-well fluid", Molecular Physics '''105''' pp. 2987-2998 (2007)]
*[http://dx.doi.org/10.1063/1.3049399 Shiqi Zhou "Thermodynamics and phase behavior of a triangle-well model and density-dependent variety", Journal of Chemical Physics '''130''' 014502 (2009)]
#[http://dx.doi.org/10.1080/00268970701832397 F. F. Betancourt-Cárdenas,  L. A. Galicia-Luna,  A. L. Benavides,  J. A. Ramírez and E. Schöll-Paschinger "Thermodynamics of a long-range triangle-well fluid", Molecular Physics '''106''' pp. 113-126 (2008)]
#[http://dx.doi.org/10.1063/1.3049399 Shiqi Zhou "Thermodynamics and phase behavior of a triangle-well model and density-dependent variety", Journal of Chemical Physics '''130''' 014502 (2009)]
[[category: models]]
[[category: models]]

Revision as of 13:27, 17 February 2011

The triangular well model, proposed by T. Nagayimain one dimension [1][2], is given by

where is the intermolecular pair potential, is the distance , is the hard diameter, is the well depth and λ > 1.

Equation of state

Main article: Equations of state for the triangular well model

Virial coefficients

, [3] [4] and [5]

Critical point

Solid phase

[6]

References

  1. ↑ T. Nagayima "Statistical Mechanics of One-dimensional Substances I", Proceedings of the Physico-Mathematical Society of Japan 22 pp. 705-720 (1940)
  2. ↑ T. Nagayima "Statistical Mechanics of One-dimensional Substances. II", Proceedings of the Physico-Mathematical Society of Japan 22 pp. 1034-1047 (1940)
  3. ↑ M. J. Feinberg and Andrew G. De Rocco "Intermolecular Forces: The Triangle Well and Some Comparisons with the Square Well and Lennard-Jones", Journal of Chemical Physics 41 pp. 3439-3450 (1964)
  4. ↑ R. H. Fowler, H. W. Graben, Andrew G. De Rocco and M. J. Feinberg "Some Additional Results for the Triangle-Well Potential Model", Journal of Chemical Physics 43 pp. 1083-1084 (1965)
  5. ↑ W. C. Farrar and Andrew G. De Rocco "Perturbation Theory for a High-Temperature Triangle-Well Fluid", Journal of Chemical Physics 54 pp. 2024-2025 (1971)
  6. ↑ Jhumpa Adhikari and David A. Kofke "Monte Carlo and cell model calculations for the solid-fluid phase behaviour of the triangle-well model", Molecular Physics 100 pp. 1543-1550 (2002)
Related reading
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