distance of closest approach

{{main|Excluded volume}}

The excluded volume of particles (the volume excluded to the centers of other particles due to the presence of one) is a key parameter in such descriptions,;T.L. Hill, An Introduction to Statistical Thermodynamics (Addison Wesley, London, 1960)T.A. Witten, and P.A. Pincus, Structured Fluids (Oxford University Press, Oxford, 2004) the distance of closest approach is required to calculate the excluded volume. The excluded volume for identical spheres is just four times the volume of one sphere. For other anisotropic objects, the excluded volume depends on orientation, and its calculation can be surprising difficult.Forces, Growth and Form in Soft Condensed Matter: At the Interface between Physics and Biology, ed. A.T. Skjeltrop and A.V. Belushkin, (NATO Science Series II: Mathematics, Physics and Chemistry, 2009), The simplest shapes after spheres are ellipses and ellipsoids; these have received [http://www.cnn.com/2004/TECH/science/02/16/science.candy.reut/ considerable attention],{{cite journal | last1=Donev | first1=Aleksandar | last2=Stillinger | first2=Frank H. | last3=Chaikin | first3=P. M. | last4=Torquato | first4=Salvatore | title=Unusually Dense Crystal Packings of Ellipsoids | journal=Physical Review Letters | publisher=American Physical Society (APS) | volume=92 | issue=25 | date=2004-06-23 | issn=0031-9007 | doi=10.1103/physrevlett.92.255506 | page=255506| pmid=15245027 | arxiv=cond-mat/0403286 | s2cid=7982407 }} yet their excluded volume is not known. Vieillard Baron was able to provide an overlap criterion for two ellipses. His results were useful for computer simulations of hard particle systems and for packing problems using Monte Carlo simulations.

File:Ellipses.png

The one anisotropic shape whose excluded volume can be expressed analytically is the spherocylinder; the solution of this problem is a classic work by Onsager.{{cite journal | last=Onsager | first=Lars | title=The Effects of Shape on the Interaction of Colloidal Particles | journal=Annals of the New York Academy of Sciences | publisher=Wiley | volume=51 | issue=4 | year=1949 | issn=0077-8923 | doi=10.1111/j.1749-6632.1949.tb27296.x | pages=627–659| s2cid=84562683 }} The problem was tackled by considering the distance between two line segments, which are the center lines of the capped cylinders. Results for other shapes are not readily available.

The orientation dependence of the distance of closest approach has surprising consequences. Systems of hard particles, whose interactions are only entropic, can become ordered. Hard spherocylinders form not only orientationally ordered nematic, but also positionally ordered smectic phases.{{cite journal | last=Frenkel | first=Daan. | title=Onsager's spherocylinders revisited | journal=The Journal of Physical Chemistry | publisher=American Chemical Society (ACS) | volume=91 | issue=19 | date=1987-09-10 | issn=0022-3654 | doi=10.1021/j100303a008 | pages=4912–4916| hdl=1874/8823 | hdl-access=free }} Here, the system gives up some (orientational and even positional) disorder to gain disorder and entropy elsewhere.

Vieillard Baron first investigated this problem, and although he did not obtain a result for the distance of closest approaches, he derived the overlap criterion for two ellipses. His final results were useful for the study of the phase behavior of hard particles and for the packing problem using Monte Carlo simulations. Although overlap criteria have been developed,{{cite journal | last=Vieillard‐Baron | first=Jacques | title=Phase Transitions of the Classical Hard‐Ellipse System | journal=The Journal of Chemical Physics | publisher=AIP Publishing | volume=56 | issue=10 | date=1972-05-15 | issn=0021-9606 | doi=10.1063/1.1676946 | pages=4729–4744}}{{cite journal | last1=Perram | first1=John W. | last2=Wertheim | first2=M.S. | title=Statistical mechanics of hard ellipsoids. I. Overlap algorithm and the contact function | journal=Journal of Computational Physics | publisher=Elsevier BV | volume=58 | issue=3 | year=1985 | issn=0021-9991 | doi=10.1016/0021-9991(85)90171-8 | pages=409–416}} analytic solutions for the distance of closest approach and the location of the point of contact have only recently become available.X. Zheng and P. Palffy-Muhoray, "Distance of closest approach of two arbitrary hard ellipses in two dimensions", [http://www.e-lc.org/docs/2007_01_17_00_46_52 electronic Liquid Crystal Communications] {{Webarchive|url=https://web.archive.org/web/20160305073734/http://www.e-lc.org/docs/2007_01_17_00_46_52 |date=2016-03-05 }}, 2007{{cite journal | last1=Zheng | first1=Xiaoyu | last2=Palffy-Muhoray | first2=Peter | title=Distance of closest approach of two arbitrary hard ellipses in two dimensions | journal=Physical Review E | volume=75 | issue=6 | date=2007-06-26 | issn=1539-3755 | doi=10.1103/physreve.75.061709 | page=061709| pmid=17677285 |arxiv=0911.3420| s2cid=7576313 }} The details of the calculations are provided in Ref.X. Zheng and P. Palffy-Muhoray, [http://www.math.kent.edu/~zheng/ellipse.pdf Complete version containing contact point algorithm, May 4, 2009.] The Fortran 90 subroutine is provided in Ref.[http://www.math.kent.edu/~zheng/ellipses.f90 Fortran90 subroutine for contact distance and contact point for 2D ellipses] by X. Zheng and P. Palffy-Muhoray, May 2009.

The procedure consists of three steps:

1. Transformation of the two tangent ellipses $E\_1$ and $E\_2$, whose centers are joined by the vector $d$, into a circle $C\_1\text{'}$ and an ellipse {{nowrap|$E\_2\text{'}$,}} whose centers are joined by the vector $d\text{'}$. The circle $C\_1\text{'}$ and the ellipse $E\_2\text{'}$ remain tangent after the transformation.
2. Determination of the distance $d\text{'}$ of closest approach of $C\_1\text{'}$ and $E\_2\text{'}$ analytically. It requires the appropriate solution of a quartic equation. The normal $n\text{'}$ is calculated.
3. Determination of the distance $d$ of closest approach and the location of the point of contact of $E\_1$ and $E\_2$ by the inverse transformations of the vectors $d\text{'}$ and {{nowrap|$n\text{'}$.}}

Input:

• lengths of the semiaxes $a\_1,b\_1,a\_2,\; b\_2$,
• unit vectors $k\_1$,$k\_2$ along major axes of both ellipses, and
• unit vector $d$ joining the centers of the two ellipses.

Output:

• distance $d$ between the centers when the ellipses $E\_1$ and $E\_2$ are externally tangent, and
• location of point of contact in terms of $k\_1$,$k\_2$.

Consider two ellipsoids, each with a given shape and orientation, whose centers are on a line with given direction. We wish to determine the distance between centers when the ellipsoids are in point contact externally. This distance of closest approach is a function of the shapes of the ellipsoids and their orientation. There is no analytic solution for this problem, since solving for the distance requires the solution of a sixth order polynomial equation. Here an algorithm is developed to determine this distance, based on the analytic results for the distance of closest approach of ellipses in 2D, which can be implemented numerically. Details are given in publications.{{cite journal | last1=Zheng | first1=Xiaoyu | last2=Iglesias | first2=Wilder | last3=Palffy-Muhoray | first3=Peter | title=Distance of closest approach of two arbitrary hard ellipsoids | journal=Physical Review E | publisher=American Physical Society (APS) | volume=79 | issue=5 | date=2009-05-20 | issn=1539-3755 | doi=10.1103/physreve.79.057702 | page=057702| pmid=19518604 }}X. Zheng, W. Iglesias, P. Palffy-Muhoray, "Distance of closest approach of two arbitrary hard ellipsoids", [http://www.e-lc.org/docs/2008_10_12_23_11_56 electronic Liquid Crystal Communications], 2008 Subroutines are provided in two formats: Fortran90 [http://www.math.kent.edu/~zheng/ellipsoid.f90 Fortran90 subroutine for distance of closest approach of ellipsoids] and C.[http://www.math.kent.edu/~zheng/ellipsoid.c C subroutine for distance of closest approach of ellipsoids]

The algorithm consists of three steps.

1. Constructing a plane containing the line joining the centers of the two ellipsoids, and finding the equations of the ellipses formed by the intersection of this plane and the ellipsoids.
2. Determining the distance of closest approach of the ellipses; that is the distance between the centers of the ellipses when they are in point contact externally.
3. Rotating the plane until the distance of closest approach of the ellipses is a maximum. The distance of closest approach of the ellipsoids is this maximum distance.

• Apsis
• Impact parameter

{{DEFAULTSORT:Distance Of Closest Approach Of Ellipses And Ellipsoids}}

Category:Conic sections

Category:Distance