Closed virial equations for hard parallel cubes andsquares

Leslie V. Woodcock

.

DOI: 10.4236/ns.2011.37085                

Abstract

A correlation between maxima in virial coefficients (B_n), and “kissing” numbers for hard hyper-spheres up to dimension D = 5, indicates a virial equation and close-packing relationship. Known virial coefficients up to B₇, both for hard parallel cubes and squares, indicate that the limiting differences B_n – B_n-1 behave similar to spheres and disks, in the respective expan-sions relative to maximum close packing. In all cases, the difference B_n – B_n-1 is approaching a negative constant with similar functional form in each dimen-sion. This observation enables closed-virial equa-tions-of-state for cubes and squares to be obtained. In both the 3D and 2D cases, the virial pressures begin to deviate from MD thermodynamic pressures at densities well below crystallization. These results consolidate the general conclusion, from previous papers on spheres and disks, that the Mayer cluster expansion cannot represent the thermodynamic fluid phases up to freezing as commonly assumed in statistical theories.

Keywords

Virial Equations Liquid Theory

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	Woodcock, L.V. (2008) Virial equation-of-state for hard spheres. ArXiv condensed matter. http://arxiv.org/abs/0801.4846
[2]	Bannerman M., Lue L. and Woodcock L.V. (2010) Thermodynamic pressures for hard spheres and closed virial equation-of-state. Journal of Chemical Physics, 132, 084507-084513.
[3]	Woodcock, L.V. (2011) Percolation transitions in the hard-sphere fluid. AICHE Journal (accepted and published online). doi:10.1002/aic.12666
[4]	Woodcock, L.V. (2008) Virial equation-of-state for the hard disk fluid. ArXiv condensed matter.
[5]	Beris, A. and Woodcock L.V. (2010) Closed virial equation-of- state for the hard-disk fluid. ArXiv condensed matter.
[6]	Hoover, W.G., Hoover, N.E. and Hanson, Exact, K. (1979) hard-disk free volumes. Journal of Chemical Physics, 70, 1837-1844.
[7]	Clisby, N. and McCoy, B.N. (2006) Ninth and tenth virial coefficients for hard hyper-spheres in D-dimensions. Physics and Astronomy, 122, 15-57. doi:10.1007/s10955-005-8080-0
[8]	Zwanzig, R.W. (1956) Virial coefficients of parallel square and parallel cube gases. Journal of Chemical Physics, 24, 855-856. doi:10.1063/1.1742621
[9]	Hoover W.G. and De Rocco, A.G. (1961) Sixth virial coefficients for gases of parallel hard lines, squares, and cubes. Journal of Chemical Physics, 34, 1059-1060. doi:10.1063/1.1731634
[10]	Hoover W.G. and De Rocco, A.G. (1962) Sixth and seventh virial coefficients for the parallel hard cube model. Journal of Chemical Physics, 36, 3141-3162. doi:10.1063/1.1732443
[11]	van Swol F. and Woodcock L.V. (1987) Percolation transition in the parallel hard cube model fluid, Molecular Simulation, 1, 95-108. doi:10.1080/08927028708080934
[12]	Kratky, K.W. (1988) Is the percolation transition of hard spheres a thermodynamic phase transition? Journal of Statistical Physics, 52, 1413-1421. doi:10.1007/BF01011656
[13]	Geometry: Multidimensional Geometry: n-Dimensional Geometry http://mathworld.wolfram.com/KissingNumber.html
[14]	Hoover, W.G., Hoover, C. and Bannerman, M. (2009) Single speed molecular dynamics of hard parallel squares and cubes. Journal of Statistical Physics, 136, 715-732. doi:10.1007/s10955-009-9795-0
[15]	Mayer J.E. and Mayer, M.G. (1940) Statistical Mechanics John Wiley, New York.
[16]	Hansen J.-P. and McDonald I.R. (2006) The Theory of Simple Liquids, 3rd Edition, Academic Press, Oxford.