Analytical Virial Coefficients and New Equations of State of Hard Ellipsoid Fluids
Chemical and Biomolecular Engineering
Volume 2, Issue 1, March 2017, Pages: 27-40
Received: Jan. 2, 2017; Accepted: Jan. 19, 2017; Published: Feb. 22, 2017
Views 2621      Downloads 94
Maryam Hashemi, Department of Physics, Yasouj University, Yasouj, Iran
Abolghasem Avazpour, Department of Physics, Yasouj University, Yasouj, Iran
Shaker Hajati, Department of Physics, Yasouj University, Yasouj, Iran
Article Tools
Follow on us
The complexity of calculations for high order virial coefficients of ellipsoids makes it difficult to obtain accurate analytical high order coefficients and equation of state for such systems. Using analytical method, the virial coefficients up to third order are calculated. For higher ones, the numerical values were taken from publications of other researchers, based on Monte Carlo integration method. By fitting the available numerical virial coefficients, sixth to eighth order, the two shape parameter analytical expressions of the hard convex molecules are obtained. Using these available data, up to eighth order, we have obtained the approximate one shape parameter analytical expressions of the hard prolate and oblate ellipsoid molecules. The fitted virial coefficients are in agreement with the simulation results. Moreover, the approximate analytical expressions for the equation of state of isotropic hard ellipsoid fluids are proposed. The proposed equations of state are in good agreement with the simulations up to medium elongations. In addition, our equations show a better agreement comparing to other works. Also, the newest equation is used for both prolate and oblate ellipsoid fluids and is convenient for elongations, k<10.0.
Virial Coefficients, Equation of State, Hard Ellipsoid, Isotropic Fluid, Prolate Molecule, Oblate Molecule
To cite this article
Maryam Hashemi, Abolghasem Avazpour, Shaker Hajati, Analytical Virial Coefficients and New Equations of State of Hard Ellipsoid Fluids, Chemical and Biomolecular Engineering. Vol. 2, No. 1, 2017, pp. 27-40. doi: 10.11648/j.cbe.20170201.15
Copyright © 2017 Authors retain the copyright of this article.
This article is an open access article distributed under the Creative Commons Attribution License ( which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
J. P. Hansen, I. R. McDonald, Theory of Simple Liquids, (Academic Press, London, 1976).
P. Hohenberg and W. Kohn, Inhomogeneous Electron Gas, Phys. Rev. B 136, 864 (1964).
A. Mulero, Theory and simulation of hard- sphere fluids and related systems, (Springer, Berlin, Heidelbrg, 2008).
J. S D. van der Waals, The equation of state for gases and liquids, (Nobel Lecture, December 12, 1910).
M. Thiesen, "Untersuchungenüber die Zustandsgleichung", Annalen der Physik 24, 467 (1885).
M. Luban and A. Baram, Third and fourth virial coefficients of hard hyperspheres of arbitrary dimensionality, J. Chem. Phys. 76, 3233 (1982).
H. Reiss, H. L. Frisch, J. L. Lebowitze, Statistical Mechanics of Rigid Spheres, J. Chem. Phys. 31, 369 (1959).
E. Thiele, Equation of State for Hard Spheres, J. Chem. Phys. 39, 474 (1963).
F. H. Ree, W. G. Hoover, Fifth and Sixth Virial Coefficients for Hard Spheres and Hard Disks, J. Chem. Phys. 40, 939 (1964).
E. A. Guggenheim, Variations on van der Waals' equation of state for high densities, Mol. Phys. 9, 199 (1965).
N. F. Carnahan, K. E. Starling, Equation of State for Non attracting Rigid Spheres, J. Chem. Phys. 51, 635 (1969).
K. R. Hall, Another Hard‐Sphere Equation of State, J. Chem. Phys. 57, 2252 (1972.
F. C. Andrews, Simple approach to the equilibrium statistical mechanics of the hard sphere fluid, J. Chem. Phys. 62, 272 (1975).
J. J. Erpenbeck, W. W. Wood, Molecular dynamics calculations of the hard-sphere equation of state, J. Stat. Phys. 35, 321 (1984).
M. Baus, J. L. Colot, Thermodynamics and structure of a fluid of hard rods, disks, spheres, or hyperspheres from rescaled virial expansions, Phys. Rev. A 36, 3912 (1987).
I. C. Sanchez, Virial coefficients and close‐packing of hard spheres and disks, J. Chem. Phys. 101, 7003 (1994).
G. S. Singh, B. Kumar, Molecular fluids and liquid crystals in convex-body coordinate systems, J. Ann. Phys. 294, 24 (2001).
T. Boublik, Molecular-Based studies of Fluids, Adv. Chem. Ser. 204, 173 (1983).
T. Booublik, Equation of state of hard convex body fluids, Mol. Phys. 42, 209 (1981).
I. Nezbeda, Virial expansion and an improved equation of state for the hard convex molecule system, J. Chem. Phys. Lett. 41, 55 (1976).
Y. Song, E. A. Mason, Equation of state for a fluid of hard convex bodies in any number of dimensions, Phys. Rev. A 41, 3121 (1990).
K. H. Naumann, P. Y. Chen, T. W. Leland, Ber. Bunsenges. Conformal-Solution-Theorie fuer Mischungen konvexer Molekuele, J. Phys. Chem. 85, 1029 (1981).
B. Barboy, W. M. Gelbart, Series representation of the equation of state for hard particle fluids, J. Chem. Phys. 71, 3053 (1979).
M. Wojcik, K. E. Gubbins, Thermodynamics and structure of hard oblate spherocylinder fluids, Mol. Phys. 53, 397 (1984).
J. D. Parsons, Nematic ordering in a system of rods, Phys. Rev. A 19, 1225 (1979).
N. Clisby, B. M. McCoy, Ninth and Tenth Order Virial Coefficients for Hard Spheres in D Dimensions, J. Stat. Phys. 114, 747 (2005).
I. Lyberg, The Fourth Virial Coefficient of a Fluid of Hard Spheres in Odd Dimensions, J. Stat. Phys. 119, 747 (2005).
W. G. Hoover, A. G. De Rocco, Second Virial Coefficient for the Spherical Shell Potential, J. Chem. Phys. 36, 3141 (1962).
A. Isihara, Determination of Molecular Shape by Osmotic Measurement, J. Chem. Phys. 18, 1446 (1950).
M. Rigby, Hard ellipsoids of revolution Virial coefficients for prolate and oblate molecules, Mol. Phys. 66, 1261 (1989).
C. Vega, Virial coefficients and equation of state of hard ellipsoids, Mol. Phys. 92, 651 (1997).
A. Yu. Vlasov, X. M. You, A. J. Masters, Monte-Carlo integration for virial coefficients re-visited: hard convex bodies, spheres with a square-well potential and mixtures of hard spheres, Mol. Phys. 100, 3313 (2002).
X. M. You, A. Yu. Vlasov, A. J. Masters, The equation of state of isotropic fluids of hard convex bodies from a high-level virial expansion, J. Chem. Phys. 123, 034510 (2005).
D. Frenkel, Structure of hard-core models for liquid crystals, J. Phys. Chem. 92, 3280 (1988).
J. A. C. Veerman, D. Frenkel, Phase behavior of disklike hard-core mesogens, Phys. Rev. A 45, 5632 (1992).
M. Rigby, Hard Gaussian overlap fluids, Mol. Phys. 68, 687 (1989).
W. R. Cooney, S. M. Thompson, K. E. Gubbins, Virial coefficients for the hard oblate spherocylinder fluid, Mol. Phys. 66, 1269 (1989).
I. Nezbeda, S. Labik, Fluids of general hard triatomic molecules, Mol. Phys. 47, 1087 (1982).
A. Yu. Vlasov, A. J. Masters, Binary mixtures of hard spheres: how far can one go with the virial equation of state, Fluid Phase Equilib. 212, 183 (2003).
J. S. Rowlinson, Virial Expansions in an Inhomogeneous System, Proc. R. Soc. A. 402, 67 (1985).
D. A. McQuarrie, Statistical Mechanics, (Harper and Row, New York, 1979).
B. J. Berne, P. Pechukas, Gaussian Model Potentials for Molecular Interactions, J. Chem. Phys. 64, 4213 (1972).
G. Rickayzen, A model for the study of the structure of hard molecular fluids, Mol. Phys. 95, 393 (1998).
P. J. Camp, C. P. Mason, M. P. Allen, A. A. Khare and D. A. Kofke, The isotropic–nematic phase transition in uniaxial hard ellipsoid fluids: Coexistence data and the approach to the Onsager limit, J. Chem. Phys. 105, 2837 (1996).
A. Avazpour and M. Moradi, The direct correlation functions of hard Gaussian overlap and hard ellipsoidal fluids, Physica B 392, 242 (2007).
A. Isihara and T. Hayashida, Theory of High Polymer Solutions. I. Second Virial Coefficient for Rigid Ovaloids Model, J. Phys. Soc. Japan. 6, 40 (1950).
I. S. Gradshteyn, I. M. Ryzhik, Tables of Integrals, Series and Products, 4th ed., (Academic, London, 1980).
H. Reiss, Scaled Particle Methods in the Statistical Thermodynamics of Fluids Adv. Chem. Phys. 9, 1 (1956).
R. M. Gibbons, The scaled particle theory for particles of arbitrary shape, Mol. Phys. 17, 81 (1969).
J. Vieillard- Baron, Phase Transitions of the Classical Hard‐Ellipse System, J. Chem. Phys. 56, 4729 (1972).
D. Frenkel, B. Mulder, The hard ellipsoid-of-revolution fluid: I. Monte Carlo simulations, Mol. Phys. 55, 1171 (1985).
M. P. Allen, Computer simulation of a biaxial liquid crystal, Liq. Cryst. 8, 499 (1990).
M. Dennison, A. J. Masters, High-level virial theory of hard spheroids, Phys. Rev. E 84, 021709 (2011).
G. R. Luckhurst, T. J. Sluckin, Biaxial Nematic Liquid Crystals: Theory, Simulation and Experiment, 1th ed.,(John wiley, UK, 2015).
S. Ho. Ryu and D. Ki. Yoon, Liquid crystal phases in confined geometries, Liq. Cryst. 43, 1951 (2016).
H. Hadwiger, Altes und NeuesuberkonvexeKorper, (Birkhauser, Basel, 1955).
K. H. Naumann, T. W. Leland, Conformal Solution Methods Based on the Hard Convex Body Expansion Theory, Fluid Phase Equilibria, 18, 1 (1984).
A. J. Masters, Virial expansions, J. Phys: Condens. Matter 20, 283102 (2008).
Science Publishing Group
1 Rockefeller Plaza,
10th and 11th Floors,
New York, NY 10020
Tel: (001)347-983-5186