3-dimensional hard rods

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3-dimensional hard rods are line segments that are unable to intersect. The can also be considered to be a limiting case of the hard spherocylinder model whose diameter is zero.

Minimum distance[edit]

The minimum distance between two hard rods in three dimensions can be calculated using an algorithm published by Vega and Lago (Ref. 1). The source code can be found here. Such an algorithm is essential in, for example, a Monte Carlo simulation, in order to check for overlaps between two sites.

  1. Carlos Vega and Santiago Lago "A fast algorithm to evaluate the shortest distance between rods", Computers & Chemistry 18 pp. 55-59 (1994)

Density-functional theory[edit]

Density-functional theory

  1. A. Poniewierski and R. Holyst "Density-functional theory for systems of hard rods", Physical Review A 41 pp. 6871-6880 (1990)

Infinitely long hard rods[edit]

Main article: Onsager theory
  1. Lars Onsager "The effects of shape on the interaction of colloidal particles", Annals of the New York Academy of Sciences 51 pp. 627-659 (1949)


  1. Szabolcs Varga, András Gábor, Enrique Velasco, Luis Mederos and Franz J. Vesely "Demixed and ordered phases in hard-rod mixtures", Molecular Physics 106 pp. 1939-1947 (2008)

Isotropic-nematic transition[edit]

Hard rods of sufficient length are able to display liquid crystalline behaviour, specifically, an isotropic to nematic transition, as well as the smectic phase.

  1. Ping Sheng "Hard rod model of the nematic-isotropic phase transition", RCA Review 35 pp. 132-143 (1974)
  2. J. D. Parsons "Nematic ordering in a system of rods" Physical Review A 19 1225-1230 (1979)
  3. H. H. Wensink and G. J. Vroege "Isotropic–nematic phase behavior of length-polydisperse hard rods", Journal of Chemical Physics 119 6868 (2003)
  4. D. Frenkel, H. N. W. Lekkerkerker and A. Stroobants "Thermodynamic stability of a smectic phase in a system of hard rods", Nature 332 pp. 822-823 (1988)

Related models[edit]