Editing Lennard-Jones equation of state

{{Numeric}}
The equation of state of the [[Lennard-Jones model]].
==Equation of state==
Johnson et al [1] propose an equation of state based on 33 parameters, which accurately reproduces the [[Density-temperature | vapor liquid equilibrium]] curve.
==Melting line==
The solid and liquid densities along the melting line are given by the equations of Mastny and  de Pablo (Ref. 5 Eqs. 20 and 21):

:<math>\rho_{\mathrm {solid}} = \beta^{-1/4} \left[ 0.908629 - 0.041510 \beta + 0.514632 \beta^2 -0.708590\beta^3 + 0.428351 \beta^4 -0.095229 \beta^5\right]</math> 

and

:<math>\rho_{\mathrm {liquid}} = \beta^{-1/4} \left[ 0.90735 - 0.27120 \beta + 0.91784 \beta^2 -1.16270\beta^3 + 0.68012 \beta^4 -0.15284 \beta^5\right]</math>

==References==
#[http://dx.doi.org/10.1080/00268979300100411 J. Karl Johnson, John A. Zollweg and Keith E. Gubbins "The Lennard-Jones equation of state revisited", Molecular Physics '''78''' pp. 591-618 (1993)]
#[http://dx.doi.org/10.1016/0378-3812(93)87002-I   Karel Aim, Jirí Kolafa, Ivo Nezbeda and Horst L. Vörtler "The Lennard-Jones fluid revisited: new thermodynamic data and new equation of state", Fluid Phase Equilibria  '''83''' pp. 15-22 (1993)]
#[http://dx.doi.org/10.1023/A:1023394122000 G. Sh. Boltachev and V. G. Baidakov "Equation of State for Lennard-Jones Fluid", High Temperature '''41''' pp. 270-272 (2003)]
#[http://dx.doi.org/10.1021/ie0495628 Hertanto Adidharma and Maciej Radosz "The LJ-Solid Equation of State Extended to Thermal Properties, Chain Molecules, and Mixtures", Industrial and Engineering Chemistry Research '''43''' pp. 6890 - 6897 (2004)]
#[http://dx.doi.org/10.1063/1.1823371     David M. Eike, Joan F. Brennecke, and Edward J. Maginn "Toward a robust and general molecular simulation method for computing solid-liquid coexistence", Journal of Chemical Physics '''122''' 014115 (2005)]
#[http://dx.doi.org/10.1063/1.2753149     Ethan A. Mastny and Juan J. de Pablo "Melting line of the Lennard-Jones system, infinite size, and full potential", Journal of Chemical Physics '''127''' 104504 (2007)]
[[category: equations of state]]
@@ Line 1: / Line 1: @@
-The [[equations of state |equation of state]] (EOS) of the [[Lennard-Jones model]]. Lennard-Jones EOS are widely used &ndash; especially in soft matter physics. Lennard-Jones EOS are also often used as a point of departure for the development of models of complex fluids. A large number of Lennard-Jones EOS have been developed in the past. Several popular Lennard-Jones EOS for the fluid phases were systematically compared and evaluated to simulation data <ref name="Stephan">[https://doi.org/10.1016/j.fluid.2020.112772  Simon Stephan, Jens Staubach, Hans Hasse "Review and comparison of equations of state for the Lennard-Jones fluid", Fluid Phase Equilibria '''523''' pp. 112772 (2020)]</ref><ref>[https://doi.org/10.1007/s10765-020-02721-9  Simon Stephan, Ulrich K. Deiters "Characteristic Curves of the Lennard‑Jones Fluid" International Journal of Thermophysics '''41''' 147 pp. 112772 (2020)]</ref>. The one of Kolafa and Nezbeda was therein found to be the most robust and accurate Lennard-Jones EOS. Ref. <ref name="Stephan"></ref> gives a comprehensive review of EOS of the Lennard-Jones fluid. Overall, it was found that none of the presently available EOS gives a satisfactory description of the Lennard-Jones fluid, which makes the development of LJ EOS still an active field.
+{{Numeric}}
-==Johnson, Zollweg and Gubbins==
+The equation of state of the [[Lennard-Jones model]].
-Johnson, Zollweg and Gubbins <ref>[http://dx.doi.org/10.1080/00268979300100411 J. Karl Johnson, John A. Zollweg and Keith E. Gubbins "The Lennard-Jones equation of state revisited", Molecular Physics '''78''' pp. 591-618 (1993)]</ref> proposed an equation of state based on 33 parameters within a modified [[Benedict, Webb and Rubin equation of state]], which accurately reproduces the [[Density-temperature | vapour-liquid equilibrium]] curve.
+==Equation of state==
+Johnson et al [1] propose an equation of state based on 33 parameters, which accurately reproduces the [[Density-temperature | vapor liquid equilibrium]] curve.
-==Kolafa and Nezbeda==
+==Melting line==
-The Kolafa and Nezbeda equation of state <ref>[http://dx.doi.org/10.1016/0378-3812(94)80001-4  Jirí Kolafa, Ivo Nezbeda "The Lennard-Jones fluid: an accurate analytic and theoretically-based equation of state", Fluid Phase Equilibria '''100''' pp. 1-34 (1994)]</ref>
+The solid and liquid densities along the melting line are given by the equations of Mastny and  de Pablo (Ref. 5 Eqs. 20 and 21):
-provides us with the [[Helmholtz energy function]]: (Eq. 30):
-:<math>A=A_{\mathrm{HS}} + \exp (-\gamma \rho^2) \rho T \Delta B_{2,{\mathrm{hBH}}} + \sum_{ij} C_{ij} T^{i/2} \rho^j</math>
-the [[compressibility factor]] (Eq. 31)
+:<math>\rho_{\mathrm {solid}} = \beta^{-1/4} \left[ 0.908629 - 0.041510 \beta + 0.514632 \beta^2 -0.708590\beta^3 + 0.428351 \beta^4 -0.095229 \beta^5\right]</math>
-:<math>z \equiv \frac{P}{\rho T}= z_{\mathrm{HS}} +  \rho(1-2\gamma\rho^2) \exp (-\gamma \rho^2) \Delta B_{2,{\mathrm{hBH}}} + \sum_{ij} jC_{ij} T^{i/2-1} \rho^j</math>
+and
-and the [[internal energy]] (Eq. 32)
+:<math>\rho_{\mathrm {liquid}} = \beta^{-1/4} \left[ 0.90735 - 0.27120 \beta + 0.91784 \beta^2 -1.16270\beta^3 + 0.68012 \beta^4 -0.15284 \beta^5\right]</math>
-:<math>U=
- {3(z_{\rm HS}-1)\over d_{\rm hBH}}\,
- {\partial d_{\rm hBH}\over \partial (1/T)}
- + \rho \exp(-\gamma\rho^2)\,{\partial \Delta B_{\rm2,hBH}\over\partial (1/T)}
- - \sum_{ij} \left({i\over2}-1\right) C_{ij}\, T^{i/2} \rho^j
-</math>
-On the following page is the [[FORTRAN code for the Kolafa and Nezbeda equation of state]].
-==Ree==
-The Ree equation of state <ref>[http://dx.doi.org/10.1063/1.439940 Francis H. Ree "Analytic representation of thermodynamic data for the Lennard‐Jones fluid", Journal of Chemical Physics '''73''' pp. 5401-5403 (1980)]</ref> is an extension of the earlier work of Hansen <ref>[http://dx.doi.org/10.1103/PhysRevA.2.221 Jean-Pierre Hansen "Phase Transition of the Lennard-Jones System. II. High-Temperature Limit", Physical Review A '''2''' pp. 221-230 (1970)]</ref> in the high temperature region.
-==Boltachev and Baidakov==
-Boltachev and Baidakov have paid particular attention to including data from the metastable region <ref>[http://dx.doi.org/10.1023/A:1023394122000 G. Sh. Boltachev and V. G. Baidakov "Equation of State for Lennard-Jones Fluid", High Temperature '''41''' pp. 270-272 (2003)]</ref>.
-==Pieprzyk-Brańka-Maćkowiak and Heyes==
-The Pieprzyk-Brańka-Maćkowiak and Heyes equation of state <ref>[https://doi.org/10.1063/1.5021560 S. Pieprzyk, A. C. Brańka, Sz. Maćkowiak and D. M. Heyes "Comprehensive representation of the Lennard-Jones equation of state based on molecular dynamics simulation data", Journal of Chemical Physics '''148''' 114505 (2018)]</ref>
-consists of a parameterisation of the  modified [[Benedict, Webb and Rubin equation of state]].
-==PeTS==
-The PeTS (perturbed truncated and shifted) equation of state for pure components <ref>[https://doi.org/10.1080/00268976.2018.1447153 Michaela Heier, Simon Stephan, Jinlu Liu, Walter G. Chapman, Hans Hasse and Kai Langenbach "Equation of state for the Lennard-Jones truncated and shifted fluid with a cut-off radius of 2.5σ based on perturbation theory and its applications to interfacial thermodynamics", Molecular Physics '''116''' pp. 2083-2094 (2018)]</ref> and mixtures <ref>[https://doi.org/10.1063/1.5093603 Simon Stephan, Kai Langenbach, Hans Hasse "interfacial properties of binary Lennard-Jones mixtures by molecular simulation and density gradient theory", Journal of Chemical Physics '''150''' pp. 174704 (2019)]</ref> (only for the Lennard-Jones truncated and shifted fluid).
 ==References==
-<references/>
+#[http://dx.doi.org/10.1080/00268979300100411 J. Karl Johnson, John A. Zollweg and Keith E. Gubbins "The Lennard-Jones equation of state revisited", Molecular Physics '''78''' pp. 591-618 (1993)]
-'''Related reading'''
+#[http://dx.doi.org/10.1016/0378-3812(93)87002-I   Karel Aim, Jirí Kolafa, Ivo Nezbeda and Horst L. Vörtler "The Lennard-Jones fluid revisited: new thermodynamic data and new equation of state", Fluid Phase Equilibria  '''83''' pp. 15-22 (1993)]
-*[http://dx.doi.org/10.1080/00268977900101051 J. J. Nicolas, K. E. Gubbins,  W. B. Streett and D. J. Tildesley "Equation of state for the Lennard-Jones fluid", Molecular Physics '''37''' pp. 1429-1454 (1979)]
+#[http://dx.doi.org/10.1023/A:1023394122000 G. Sh. Boltachev and V. G. Baidakov "Equation of State for Lennard-Jones Fluid", High Temperature '''41''' pp. 270-272 (2003)]
-*[http://dx.doi.org/10.1016/0378-3812(93)87002-I   Karel Aim, Jirí Kolafa, Ivo Nezbeda and Horst L. Vörtler "The Lennard-Jones fluid revisited: new thermodynamic data and new equation of state", Fluid Phase Equilibria  '''83''' pp. 15-22 (1993)]
+#[http://dx.doi.org/10.1021/ie0495628 Hertanto Adidharma and Maciej Radosz "The LJ-Solid Equation of State Extended to Thermal Properties, Chain Molecules, and Mixtures", Industrial and Engineering Chemistry Research '''43''' pp. 6890 - 6897 (2004)]
-*[http://dx.doi.org/10.1021/ie0495628 Hertanto Adidharma and Maciej Radosz "The LJ-Solid Equation of State Extended to Thermal Properties, Chain Molecules, and Mixtures", Industrial and Engineering Chemistry Research '''43''' pp. 6890 - 6897 (2004)]
+#[http://dx.doi.org/10.1063/1.1823371     David M. Eike, Joan F. Brennecke, and Edward J. Maginn "Toward a robust and general molecular simulation method for computing solid-liquid coexistence", Journal of Chemical Physics '''122''' 014115 (2005)]
-*[http://dx.doi.org/10.1063/1.1823371     David M. Eike, Joan F. Brennecke, and Edward J. Maginn "Toward a robust and general molecular simulation method for computing solid-liquid coexistence", Journal of Chemical Physics '''122''' 014115 (2005)]
+#[http://dx.doi.org/10.1063/1.2753149     Ethan A. Mastny and Juan J. de Pablo "Melting line of the Lennard-Jones system, infinite size, and full potential", Journal of Chemical Physics '''127''' 104504 (2007)]
-*[http://dx.doi.org/10.1063/1.3561698 Sergey A. Khrapak and Gregor E. Morfill "Accurate freezing and melting equations for the Lennard-Jones system", Journal of Chemical Physics '''134''' 094108 (2011)]
-*[https://doi.org/10.1063/1.4945000 Monika Thol, Gabor Rutkai, Andreas Köster, Rolf Lustig, Roland Span, and Jadran Vrabec "Equation of State for the Lennard-Jones Fluid", Journal of Physical and Chemical Reference Data '''45''' 023101 (2016)]
-{{Numeric}}
 [[category: equations of state]]