Combining rules: Difference between revisions
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The '''combining rules''' (also known as ''mixing rules'') for binary [[mixtures]] are variously given by | The '''combining rules''' (also known as ''mixing rules'') for binary [[mixtures]] are variously given by | ||
==Admur and Mason== | |||
For the [[second virial coefficient]] of a mixture | For the [[second virial coefficient]] of a mixture | ||
<ref>[http://dx.doi.org/10.1063/1.1724353 I. Amdur and E. A. Mason "Properties of Gases at Very High Temperatures", Physics of Fluids '''1''' pp. 370-383 (1958)]</ref> | <ref>[http://dx.doi.org/10.1063/1.1724353 I. Amdur and E. A. Mason "Properties of Gases at Very High Temperatures", Physics of Fluids '''1''' pp. 370-383 (1958)]</ref> | ||
:<math>B_{ij} = \frac{\left(B_{ii}^{1/3}+B_{jj}^{1/3}\right)^3}{8}</math> | :<math>B_{ij} = \frac{\left(B_{ii}^{1/3}+B_{jj}^{1/3}\right)^3}{8}</math> | ||
==== | ==Böhm-Ahlrichs== | ||
:< | <ref>[http://dx.doi.org/10.1063/1.444057 Hans‐Joachim Böhm and Reinhart Ahlrichs "A study of short‐range repulsions", Journal of Chemical Physics '''77''' pp. 2028- (1982)]</ref> | ||
==Diaz Peña-Pando-Renuncio== | |||
<ref>[http://dx.doi.org/10.1063/1.442726 M. Diaz Peña, C. Pando, and J. A. R. Renuncio "Combination rules for intermolecular potential parameters. I. Rules based on approximations for the long-range dispersion energy", Journal of Chemical Physics '''76''' pp. 325- (1982)]</ref> | |||
<ref>[http://dx.doi.org/10.1063/1.442727 M. Diaz Peña, C. Pando, and J. A. R. Renuncio "Combination rules for intermolecular potential parameters. II. Rules based on approximations for the long-range dispersion energy and an atomic distortion model for the repulsive interactions", Journal of Chemical Physics '''76''' pp. 333- (1982)]</ref> | |||
==Fender-Halsey== | |||
The Fender-Halsey combining rule for the [[Lennard-Jones model]] is given by <ref>[http://dx.doi.org/10.1063/1.1701284 B. E. F. Fender and G. D. Halsey, Jr. "Second Virial Coefficients of Argon, Krypton, and Argon-Krypton Mixtures at Low Temperatures", Journal of Chemical Physics '''36''' pp. 1881-1888 (1962)]</ref> | The Fender-Halsey combining rule for the [[Lennard-Jones model]] is given by <ref>[http://dx.doi.org/10.1063/1.1701284 B. E. F. Fender and G. D. Halsey, Jr. "Second Virial Coefficients of Argon, Krypton, and Argon-Krypton Mixtures at Low Temperatures", Journal of Chemical Physics '''36''' pp. 1881-1888 (1962)]</ref> | ||
:<math>\epsilon_{ij} = \frac{2 \epsilon_i \epsilon_j}{\epsilon_i + \epsilon_j}</math> | :<math>\epsilon_{ij} = \frac{2 \epsilon_i \epsilon_j}{\epsilon_i + \epsilon_j}</math> | ||
====Hudson and McCoubrey | ==Gilbert-Smith== | ||
The Gilbert-Smith rules for the [[Born-Huggins-Meyer potential]]<ref>[http://dx.doi.org/10.1063/1.1670463 T. L. Gilbert "Soft‐Sphere Model for Closed‐Shell Atoms and Ions", Journal of Chemical Physics '''49''' pp. 2640- (1968)]</ref><ref>[http://dx.doi.org/10.1063/1.431848 T. L. Gilbert, O. C. Simpson, and M. A. Williamson "Relation between charge and force parameters of closed‐shell atoms and ions", Journal of Chemical Physics '''63''' pp. 4061- (1975)]</ref><ref>[http://dx.doi.org/10.1103/PhysRevA.5.1708 Felix T. Smith "Atomic Distortion and the Combining Rule for Repulsive Potentials", Physical Review A '''5''' pp. 1708-1713 (1972)]</ref>. | |||
==Good-Hope rule== | |||
The Good-Hope rule for [[Mie potential |Mie]]–[[Lennard-Jones model |Lennard‐Jones]] or [[Buckingham potential]]s <ref>[http://dx.doi.org/10.1063/1.1674022 Robert J. Good and Christopher J. Hope "New Combining Rule for Intermolecular Distances in Intermolecular Potential Functions", Journal of Chemical Physics '''53''' pp. 540- (1970)]</ref> is given by (Eq. 2): | |||
:<math>\sigma_{ij} = \sqrt{\sigma_{ii} \sigma_{jj}}</math> | |||
==Hudson and McCoubrey== | |||
<ref>[http://dx.doi.org/10.1039/TF9605600761 G. H. Hudson and J. C. McCoubrey "Intermolecular forces between unlike molecules. A more complete form of the combining rules", Transactions of the Faraday Society '''56''' pp. 761-766 (1960)]</ref> | <ref>[http://dx.doi.org/10.1039/TF9605600761 G. H. Hudson and J. C. McCoubrey "Intermolecular forces between unlike molecules. A more complete form of the combining rules", Transactions of the Faraday Society '''56''' pp. 761-766 (1960)]</ref> | ||
==Kong rules== | |||
<ref>[http://dx.doi.org/10.1063/1.1680358 Chang Lyoul Kong "Combining rules for intermolecular potential parameters. II. Rules for the Lennard-Jones (12–6) potential and the Morse potential", Journal of Chemical Physics '''59''' pp. 2464-2467 (1973)]</ref> | <ref>[http://dx.doi.org/10.1063/1.1680358 Chang Lyoul Kong "Combining rules for intermolecular potential parameters. II. Rules for the Lennard-Jones (12–6) potential and the Morse potential", Journal of Chemical Physics '''59''' pp. 2464-2467 (1973)]</ref> | ||
====Lorentz rule | ==Lorentz-Berthelot rules== | ||
The Lorentz rule is given by <ref>[h | |||
:<math>\sigma_{ij} = \frac{\sigma_{ii} + \sigma_{jj}}{2}</math> | :<math>\sigma_{ij} = \frac{\sigma_{ii} + \sigma_{jj}}{2}</math> | ||
= | which is only really valid for the [[hard sphere model]]. | ||
The Berthelot rule is given by | |||
:<math>\epsilon_{ij} = \sqrt{\epsilon_{ii} \epsilon_{jj}}</math> | |||
These rules are simple and widely used, but are not without their failings <ref>[http://dx.doi.org/10.1080/00268970010020041 Jérôme Delhommelle; Philippe Millié "Inadequacy of the Lorentz-Berthelot combining rules for accurate predictions of equilibrium properties by molecular simulation", Molecular Physics '''99''' pp. 619-625 (2001)]</ref> | |||
<ref>[http://dx.doi.org/10.1080/00268970802471137 Dezso Boda and Douglas Henderson "The effects of deviations from Lorentz-Berthelot rules on the properties of a simple mixture", Molecular Physics '''106''' pp. 2367-2370 (2008)]</ref> | |||
<ref>[http://dx.doi.org/10.1063/1.1610435 W. Song, P. J. Rossky, and M. Maroncelli "Modeling alkane+perfluoroalkane interactions using all-atom potentials: Failure of the usual combining rules", Journal of Chemical Physics '''119''' pp. 9145- (2003)]</ref>. | |||
==Tang and Toennies== | |||
<ref>[http://dx.doi.org/10.1007/BF01384663 K. T. Tang and J. Peter Toennies "New combining rules for well parameters and shapes of the van der Waals potential of mixed rare gas systems", Zeitschrift für Physik D Atoms, Molecules and Clusters '''1''' pp. 91-101 (1986)]</ref> | <ref>[http://dx.doi.org/10.1007/BF01384663 K. T. Tang and J. Peter Toennies "New combining rules for well parameters and shapes of the van der Waals potential of mixed rare gas systems", Zeitschrift für Physik D Atoms, Molecules and Clusters '''1''' pp. 91-101 (1986)]</ref> | ||
==Waldman-Hagler rules== | |||
<ref>[http://dx.doi.org/10.1002/jcc.540140909 M. Waldman and A. T. Hagler "New combining rules for rare-gas Van der-Waals parameters", Journal of Computational Chemistry '''14''' pp. 1077-1084 (1993)]</ref> | <ref>[http://dx.doi.org/10.1002/jcc.540140909 M. Waldman and A. T. Hagler "New combining rules for rare-gas Van der-Waals parameters", Journal of Computational Chemistry '''14''' pp. 1077-1084 (1993)]</ref> | ||
==References== | ==References== | ||
<references/> | <references/> | ||
'''Related reading''' | '''Related reading''' | ||
*[http://dx.doi.org/10. | *[http://dx.doi.org/10.1021/ja00046a032 Thomas A. Halgren "The representation of van der Waals (vdW) interactions in molecular mechanics force fields: potential form, combination rules, and vdW parameters", Journal of the American Chemical Society '''114''' pp. 7827-7843 (1992)] | ||
[[category: mixtures]] | [[category: mixtures]] | ||
Revision as of 17:18, 7 July 2011
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The combining rules (also known as mixing rules) for binary mixtures are variously given by
Admur and Mason
For the second virial coefficient of a mixture [1]
Böhm-Ahlrichs
Diaz Peña-Pando-Renuncio
Fender-Halsey
The Fender-Halsey combining rule for the Lennard-Jones model is given by [5]
Gilbert-Smith
The Gilbert-Smith rules for the Born-Huggins-Meyer potential[6][7][8].
Good-Hope rule
The Good-Hope rule for Mie–Lennard‐Jones or Buckingham potentials [9] is given by (Eq. 2):
Hudson and McCoubrey
Kong rules
Lorentz-Berthelot rules
The Lorentz rule is given by Cite error: Closing </ref> missing for <ref> tag
[12]
[13].
Tang and Toennies
Waldman-Hagler rules
References
- ↑ I. Amdur and E. A. Mason "Properties of Gases at Very High Temperatures", Physics of Fluids 1 pp. 370-383 (1958)
- ↑ Hans‐Joachim Böhm and Reinhart Ahlrichs "A study of short‐range repulsions", Journal of Chemical Physics 77 pp. 2028- (1982)
- ↑ M. Diaz Peña, C. Pando, and J. A. R. Renuncio "Combination rules for intermolecular potential parameters. I. Rules based on approximations for the long-range dispersion energy", Journal of Chemical Physics 76 pp. 325- (1982)
- ↑ M. Diaz Peña, C. Pando, and J. A. R. Renuncio "Combination rules for intermolecular potential parameters. II. Rules based on approximations for the long-range dispersion energy and an atomic distortion model for the repulsive interactions", Journal of Chemical Physics 76 pp. 333- (1982)
- ↑ B. E. F. Fender and G. D. Halsey, Jr. "Second Virial Coefficients of Argon, Krypton, and Argon-Krypton Mixtures at Low Temperatures", Journal of Chemical Physics 36 pp. 1881-1888 (1962)
- ↑ T. L. Gilbert "Soft‐Sphere Model for Closed‐Shell Atoms and Ions", Journal of Chemical Physics 49 pp. 2640- (1968)
- ↑ T. L. Gilbert, O. C. Simpson, and M. A. Williamson "Relation between charge and force parameters of closed‐shell atoms and ions", Journal of Chemical Physics 63 pp. 4061- (1975)
- ↑ Felix T. Smith "Atomic Distortion and the Combining Rule for Repulsive Potentials", Physical Review A 5 pp. 1708-1713 (1972)
- ↑ Robert J. Good and Christopher J. Hope "New Combining Rule for Intermolecular Distances in Intermolecular Potential Functions", Journal of Chemical Physics 53 pp. 540- (1970)
- ↑ G. H. Hudson and J. C. McCoubrey "Intermolecular forces between unlike molecules. A more complete form of the combining rules", Transactions of the Faraday Society 56 pp. 761-766 (1960)
- ↑ Chang Lyoul Kong "Combining rules for intermolecular potential parameters. II. Rules for the Lennard-Jones (12–6) potential and the Morse potential", Journal of Chemical Physics 59 pp. 2464-2467 (1973)
- ↑ Dezso Boda and Douglas Henderson "The effects of deviations from Lorentz-Berthelot rules on the properties of a simple mixture", Molecular Physics 106 pp. 2367-2370 (2008)
- ↑ W. Song, P. J. Rossky, and M. Maroncelli "Modeling alkane+perfluoroalkane interactions using all-atom potentials: Failure of the usual combining rules", Journal of Chemical Physics 119 pp. 9145- (2003)
- ↑ K. T. Tang and J. Peter Toennies "New combining rules for well parameters and shapes of the van der Waals potential of mixed rare gas systems", Zeitschrift für Physik D Atoms, Molecules and Clusters 1 pp. 91-101 (1986)
- ↑ M. Waldman and A. T. Hagler "New combining rules for rare-gas Van der-Waals parameters", Journal of Computational Chemistry 14 pp. 1077-1084 (1993)
Related reading