Difference between revisions of "Hard spherocylinders"

From SklogWiki
Jump to: navigation, search
m (Changed references to Cite format)
m (Included related stub pages.)
 
(8 intermediate revisions by 4 users not shown)
Line 2: Line 2:
 
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>[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]].
+
has been studied extensively because of its propensity to form both [[Nematic phase | nematic]] and [[Smectic phases | smectic]] [[Liquid crystals | liquid crystalline]] phases <ref>[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)]</ref> as well as forming a [[Plastic crystals | plastic crystal]] phase for short <math>L</math> <ref>[http://dx.doi.org/10.1063/1.474626 C. Vega and P. A. Monson "Plastic crystal phases of hard dumbbells and hard spherocylinders", Journal of Chemical Physics '''107''' pp. 2696-2697 (1997)]</ref> . 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]], and in the limit <math>L=0</math> one has the [[hard sphere model]]. A closely related model is that of the [[Oblate hard spherocylinders | oblate hard spherocylinder]].
 
==Volume==
 
==Volume==
 
The molecular volume of the spherocylinder  is given by  
 
The molecular volume of the spherocylinder  is given by  
Line 11: Line 11:
 
==Minimum distance==
 
==Minimum distance==
 
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.
 
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.
 +
 +
The original code did not give the symmetry property of the distance for almost parallel rods. The revised algorithm in C for systems with periodic boundary conditions can be found [[Rev. source code for the minimum distance between two rods in C | here]].
 +
 
==Virial coefficients==
 
==Virial coefficients==
:''Main article: [[Hard spherocylinders: virial coefficients]]''
+
[[Virial equation of state | Virial coefficients]] of the hard spherocylinder model
 +
<ref>[http://dx.doi.org/10.1080/00268977800100991 Peter A. Monson and Maurice Rigby "Virial equation of state for rigid spherocylinders", Molecular Physics '''35''' pp.  1337-1342 (1978)]</ref>
 +
<ref>[http://dx.doi.org/10.1080/00268978900100861 W. R. Cooney, S. M. Thompson and K. E. Gubbins " Virial coefficients for the hard oblate spherocylinder fluid", Molecular Physics '''66''' pp. 1269-1272  (1989)]</ref>
 +
<ref>[http://dx.doi.org/10.1021/jp049502f Tomás Boublík "Third and Fourth Virial Coefficients and the Equation of State of Hard Prolate Spherocylinders", Journal of Physical Chemistry B '''108''' pp. 7424-7429  (2004)]</ref>.
 
==Phase diagram==
 
==Phase diagram==
:''Main aritcle: [[Phase diagram of the hard spherocylinder model]]''
+
[[Phase diagrams |Phase diagram]] of the hard spherocylinder model
 +
<ref>[http://dx.doi.org/10.1063/1.471343  S. C. McGrother and D. C. Williamson and G. Jackson "A re-examination of the phase diagram of hard spherocylinders", Journal of Chemical Physics '''104''' pp.  6755-6771  (1996)]</ref>
 +
<ref>[http://dx.doi.org/10.1063/1.473404  P. Bolhuis and D. Frenkel "Tracing the phase boundaries of hard spherocylinders", Journal of Chemical Physics '''106''' pp. 666-687  (1997)]</ref>
 
==See also==
 
==See also==
 
*[[Charged hard spherocylinders]]
 
*[[Charged hard spherocylinders]]
 +
*[[Oblate hard spherocylinders]]
 +
 
==References==
 
==References==
 
<references/>
 
<references/>
 
'''Related reading'''
 
'''Related reading'''
 
*[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/10.1021/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)]
+
*[http://dx.doi.org/10.1103/PhysRevLett.98.09570 Alejandro Cuetos and Marjolein Dijkstra "Kinetic Pathways for the Isotropic-Nematic Phase Transition in a System of Colloidal Hard Rods: A Simulation Study", Physical Review Letters '''98''' 095701 (2007)]
 +
*[http://dx.doi.org/10.1103/PhysRevLett.105.088302 Ran Ni, Simone Belli, René van Roij, and Marjolein Dijkstra "Glassy Dynamics, Spinodal Fluctuations, and the Kinetic Limit of Nucleation in Suspensions of Colloidal Hard Rods", Physical Review Letters '''105''' 088302 (2010)]
 
[[Category: Models]]
 
[[Category: Models]]

Latest revision as of 11:48, 28 April 2011

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 [1] as well as forming a plastic crystal phase for short L [2] . One of the first studies of hard spherocylinders was undertaken by Cotter and Martire [3] using scaled-particle theory, and one of the first simulations was in the classic work of Jacques Vieillard-Baron [4]. In the limit of zero diameter the hard spherocylinder becomes a line segment, often known as the hard rod model, and in the limit L=0 one has the hard sphere model. A closely related model is that of the oblate hard spherocylinder.

Volume[edit]

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[edit]

The minimum distance between two spherocylinders can be calculated using an algorithm published by Vega and Lago [5]. 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.

The original code did not give the symmetry property of the distance for almost parallel rods. The revised algorithm in C for systems with periodic boundary conditions can be found here.

Virial coefficients[edit]

Virial coefficients of the hard spherocylinder model [6] [7] [8].

Phase diagram[edit]

Phase diagram of the hard spherocylinder model [9] [10]

See also[edit]

References[edit]

  1. 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)
  2. C. Vega and P. A. Monson "Plastic crystal phases of hard dumbbells and hard spherocylinders", Journal of Chemical Physics 107 pp. 2696-2697 (1997)
  3. 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)
  4. Jacques Vieillard-Baron "The equation of state of a system of hard spherocylinders", Molecular Physics 28 pp. 809-818 (1974)
  5. Carlos Vega and Santiago Lago "A fast algorithm to evaluate the shortest distance between rods", Computers & Chemistry 18 pp. 55-59 (1994)
  6. Peter A. Monson and Maurice Rigby "Virial equation of state for rigid spherocylinders", Molecular Physics 35 pp. 1337-1342 (1978)
  7. W. R. Cooney, S. M. Thompson and K. E. Gubbins " Virial coefficients for the hard oblate spherocylinder fluid", Molecular Physics 66 pp. 1269-1272 (1989)
  8. Tomás Boublík "Third and Fourth Virial Coefficients and the Equation of State of Hard Prolate Spherocylinders", Journal of Physical Chemistry B 108 pp. 7424-7429 (2004)
  9. S. C. McGrother and D. C. Williamson and G. Jackson "A re-examination of the phase diagram of hard spherocylinders", Journal of Chemical Physics 104 pp. 6755-6771 (1996)
  10. P. Bolhuis and D. Frenkel "Tracing the phase boundaries of hard spherocylinders", Journal of Chemical Physics 106 pp. 666-687 (1997)

Related reading