User:Xyzheng/distance of closest approach of two hard ellipses
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The distance of closest approach of hard particles is a key parameter of their interaction and plays an important role in the resulting phase behavior. For non-spherical particles, the distance of closest approach depends on orientation, and its calculation can be surprisingly difficult. Although overlap criteria have been developed for use in computer simulations [1][2], analytic solutions for the distance of closest approach and the location of the point of contact have only recently become available [3][4]. The details of the calculations are provided in Ref. [5]. The Fortran90 subroutine are provided in Ref.[6]
The Method
[edit]The procedure constists of three steps:
- Transformation of the two tangent ellipses and , whose centers are joined by the vector , into a circle and an ellipse , whose centers are joined by the vector . The circle and the ellipse remain tangent after the transformation.
- Determination of the distance of closest approach of and analytically. It requires the appropriate solution of a quartic equation. The normal is calculated.
- Determination of the distance of closest approach and the location of the point of contact of and by the inverse transformations of the vectors and .
Input:
- lengths of the semiaxes ,
- unit vectors , along major axes of both ellipses, and
- unit vector joining the centers of the two ellipses.
Output:
- distance between the centers when the ellipses and are externally tangent, and
- location of point of contact in terms of ,.
References
[edit]- ^ J. Vieillard-Baron, "Phase transition of the classical hard ellipse system" J. Chem. Phys., 56(10), 4729 (1972).
- ^ J. W. Perram and M. S. Wertheim, "Statistical mechanics of hard ellipsoids. I. overlap algorithm and the contact function", J. Comput. Phys., 58, 409 (1985).
- ^ X. Zheng and P. Palffy-Muhoray, Distance of closest approach of two arbitrary hard ellipses in two dimensions, electronic Liquid Crystal Communications, 2007
- ^ X. Zheng and P. Palffy-Muhoray, Distance of closest approach of two arbitrary hard ellipses in two dimensions, Phys. Rev. E, 75, 061709 (2007).
- ^ X. Zheng and P. Palffy-Muhoray, Complete version containing contact point algorithm, May 4, 2009.
- ^ Fortran90 subroutine by X. Zheng and P. Palffy-Muhoray