Difference between revisions of "Hard spherocylinders"

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The '''hard spherocylinder''' model consists of an  impenetrable cylinder, capped at both ends  
 
The '''hard spherocylinder''' model consists of an  impenetrable cylinder, capped at both ends  
 
by hemispheres whose diameters are the same as the diameter of the cylinder. The hard spherocylinder model
 
by hemispheres whose diameters are the same as the diameter of the cylinder. The hard spherocylinder model
has been studied extensively because of its propensity to form both [[Nematic phase | nematic]] and [[Smectic phases | smectic]] [[Liquid crystals | liquid crystalline]] phases. One of the first studies of  hard spherocylinders was undertaken by Cotter and Martire (Ref. 1) using [[scaled-particle theory]], and one of the first simulations was in the classic work of Jacques Vieillard-Baron  (Ref. 2). In the limit of zero diameter the hard spherocylinder becomes a line segment, often known as the [[3-dimensional hard rods |hard rod model]].
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has been studied extensively because of its propensity to form both [[Nematic phase | nematic]] and [[Smectic phases | smectic]] [[Liquid crystals | liquid crystalline]] phases. One of the first studies of  hard spherocylinders was undertaken by Cotter and Martire <ref>[http://dx.doi.org/10.1063/1.1673232  Martha A. Cotter and Daniel E. Martire "Statistical Mechanics of Rodlike Particles. II. A Scaled Particle Investigation of the Aligned to Isotropic Transition in a Fluid of Rigid Spherocylinders", Journal of Chemical Physics  '''52''' pp. 1909-1919 (1970)]</ref> using [[scaled-particle theory]], and one of the first simulations was in the classic work of Jacques Vieillard-Baron <ref>[http://dx.doi.org/10.1080/00268977400102161 Jacques Vieillard-Baron  "The equation of state of a system of hard spherocylinders", Molecular Physics '''28''' pp. 809-818 (1974)]</ref>. In the limit of zero diameter the hard spherocylinder becomes a line segment, often known as the [[3-dimensional hard rods |hard rod model]].
 
==Volume==
 
==Volume==
 
The molecular volume of the spherocylinder  is given by  
 
The molecular volume of the spherocylinder  is given by  
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where <math>L</math> is the length of the cylindrical part of the spherocylinder and <math>D</math> is the diameter.
 
where <math>L</math> is the length of the cylindrical part of the spherocylinder and <math>D</math> is the diameter.
 
==Minimum distance==
 
==Minimum distance==
The minimum distance between two spherocylinders can be calculated using an algorithm published by Vega and Lago (Ref. 1). The [[Source code for the minimum distance between two rods | 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.
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The minimum distance between two spherocylinders can be calculated using an algorithm published by Vega and Lago <ref>[http://dx.doi.org/10.1016/0097-8485(94)80023-5  Carlos Vega and Santiago Lago "A fast algorithm to evaluate the shortest distance between rods", Computers & Chemistry  '''18''' pp. 55-59 (1994)]</ref>. The [[Source code for the minimum distance between two rods | 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.
#[http://dx.doi.org/10.1016/0097-8485(94)80023-5  Carlos Vega and Santiago Lago "A fast algorithm to evaluate the shortest distance between rods", Computers & Chemistry  '''18''' pp. 55-59 (1994)]
 
 
==Virial coefficients==
 
==Virial coefficients==
 
:''Main article: [[Hard spherocylinders: virial coefficients]]''
 
:''Main article: [[Hard spherocylinders: virial coefficients]]''
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*[[Charged hard spherocylinders]]
 
*[[Charged hard spherocylinders]]
 
==References==
 
==References==
#[http://dx.doi.org/10.1063/1.1673232  Martha A. Cotter and Daniel E. Martire "Statistical Mechanics of Rodlike Particles. II. A Scaled Particle Investigation of the Aligned to Isotropic Transition in a Fluid of Rigid Spherocylinders", Journal of Chemical Physics  '''52''' pp. 1909-1919 (1970)]
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<references/>
#[http://dx.doi.org/10.1080/00268977400102161 Jacques Vieillard-Baron  "The equation of state of a system of hard spherocylinders", Molecular Physics '''28''' pp. 809-818 (1974)]
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'''Related reading'''
#[http://dx.doi.org/10.1021/j100303a008 Daan Frenkel "Onsager's spherocylinders revisited", Journal of Physical Chemistry '''91''' pp. 4912-4916 (1987)]
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*[http://dx.doi.org/10.1021/j100303a008 Daan Frenkel "Onsager's spherocylinders revisited", Journal of Physical Chemistry '''91''' pp. 4912-4916 (1987)]
#[http://dx.doi.org/10.1038/332822a0 D. Frenkel, H. N. W. Lekkerkerker and A. Stroobants "Thermodynamic stability of a smectic phase in a system of hard rods", Nature '''332''' p. 822 (1988)]
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*[http://dx.doi.org/10.1038/332822a0 D. Frenkel, H. N. W. Lekkerkerker and A. Stroobants "Thermodynamic stability of a smectic phase in a system of hard rods", Nature '''332''' p. 822 (1988)]
 
[[Category: Models]]
 
[[Category: Models]]

Revision as of 18:37, 30 November 2010

Spherocylinder purple.png

The hard spherocylinder model consists of an impenetrable cylinder, capped at both ends by hemispheres whose diameters are the same as the diameter of the cylinder. The hard spherocylinder model has been studied extensively because of its propensity to form both nematic and smectic liquid crystalline phases. One of the first studies of hard spherocylinders was undertaken by Cotter and Martire [1] using scaled-particle theory, and one of the first simulations was in the classic work of Jacques Vieillard-Baron [2]. In the limit of zero diameter the hard spherocylinder becomes a line segment, often known as the hard rod model.

Volume

The molecular volume of the spherocylinder is given by

v_0 = \pi \left( \frac{LD^2}{4} + \frac{D^3}{6} \right)

where L is the length of the cylindrical part of the spherocylinder and D is the diameter.

Minimum distance

The minimum distance between two spherocylinders can be calculated using an algorithm published by Vega and Lago [3]. 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.

Virial coefficients

Main article: Hard spherocylinders: virial coefficients

Phase diagram

Main aritcle: Phase diagram of the hard spherocylinder model

See also

References

Related reading