Lebwohl-Lasher model: Difference between revisions

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m (→‎Isotropic-nematic transition: Added an internal link.)
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where <math>\epsilon_{ij} > 0</math>, <math>\beta_{ij}</math> is the angle between the axes of nearest neighbour particles <math>i</math> and <math>j</math>, and <math>P_2</math> is a second order [[Legendre polynomials |Legendre polynomial]].
where <math>\epsilon_{ij} > 0</math>, <math>\beta_{ij}</math> is the angle between the axes of nearest neighbour particles <math>i</math> and <math>j</math>, and <math>P_2</math> is a second order [[Legendre polynomials |Legendre polynomial]].
==Isotropic-nematic transition==
==Isotropic-nematic transition==
Fabbri and Zannoni estimated the transition temperature <ref>[http://dx.doi.org/10.1080/00268978600101561 U. Fabbri and C. Zannoni "A Monte Carlo investigation of the Lebwohl-Lasher lattice model in the vicinity of its orientational phase transition", Molecular Physics pp. 763-788 '''58''' (1986)]</ref> using Monte Carlo simulation:
Fabbri and Zannoni estimated the transition temperature <ref>[http://dx.doi.org/10.1080/00268978600101561 U. Fabbri and C. Zannoni "A Monte Carlo investigation of the Lebwohl-Lasher lattice model in the vicinity of its orientational phase transition", Molecular Physics pp. 763-788 '''58''' (1986)]</ref> via a [[Monte Carlo]] simulation:


:<math>T^*_{NI}= \frac{k_BT_{NI}}{\epsilon}=1.1232 \pm 0.0006</math>
:<math>T^*_{NI}= \frac{k_BT_{NI}}{\epsilon}=1.1232 \pm 0.0006</math>
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:<math>T^*_{NI}= \frac{k_BT_{NI}}{\epsilon}=1.1225 \pm 0.0001 </math>
:<math>T^*_{NI}= \frac{k_BT_{NI}}{\epsilon}=1.1225 \pm 0.0001 </math>


See also the paper by Zhang ''et al.'' <ref>[http://dx.doi.org/10.1103/PhysRevLett.69.2803  Zhengping Zhang, Ole G. Mouritsen, and Martin J. Zuckermann, "Weak first-order orientational transition in the Lebwohl-Lasher model for liquid crystals", Physical Review Letters '''69''' pp. 2803-2806 (1992)]</ref>
See also the paper by Zhang ''et al.'' <ref>[http://dx.doi.org/10.1103/PhysRevLett.69.2803  Zhengping Zhang, Ole G. Mouritsen, and Martin J. Zuckermann, "Weak first-order orientational transition in the Lebwohl-Lasher model for liquid crystals", Physical Review Letters '''69''' pp. 2803-2806 (1992)]</ref>.


==Planar Lebwohl–Lasher model ==
==Planar Lebwohl–Lasher model ==

Revision as of 12:05, 15 April 2009

The Lebwohl-Lasher model is a lattice version of the Maier-Saupe mean field model of a nematic liquid crystal [1][2]. The Lebwohl-Lasher model consists of a cubic lattice occupied by uniaxial nematogenic particles with the pair potential

where , is the angle between the axes of nearest neighbour particles and , and is a second order Legendre polynomial.

Isotropic-nematic transition

Fabbri and Zannoni estimated the transition temperature [3] via a Monte Carlo simulation:

More recently N. V. Priezjev and Robert A. Pelcovits [4] used a Monte Carlo cluster algorithm and obtained:

See also the paper by Zhang et al. [5].

Planar Lebwohl–Lasher model

The planar Lebwohl-Lasher appears when the lattice considered is two-dimensional. This system exhibits a continuous transition. The ascription of such a transition to the Kosterlitz-Touless type is still under discussion [6] [7] [8] [9].

Lattice Gas Lebwohl-Lasher model

This model is the lattice gas version of the Lebwohl-Lasher model. In this case the sites of the lattice can be occupied by particles or empty. The interaction between nearest-neighbour particles is that of the Lebwohl-Lasher model. This model has been studied in [10].

References