# Lebwohl-Lasher model

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

\[\Phi_{ij} = -\epsilon P_2 (\cos \beta_{ij}) \]

where \(\epsilon > 0\), \(\beta_{ij}\) is the angle between the axes of nearest neighbour particles \(i\) and \(j\), and \(P_2\) is a second order Legendre polynomial.

## Contents

## Isotropic-nematic transition[edit]

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

\[T^*_{NI}= \frac{k_BT_{NI}}{\epsilon}=1.1232 \pm 0.0006\]

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

\[T^*_{NI}= \frac{k_BT_{NI}}{\epsilon}=1.1225 \pm 0.0001 \]

See also the paper by Zhang *et al.* ^{[5]} and that of Shekhar et al. ^{[6]}.

## Confined systems[edit]

The Lebwohl-Lasher model has been used to study the effect of confinement in the phase
behavior of nematogens ^{[7]}

## Planar Lebwohl–Lasher model[edit]

The planar Lebwohl-Lasher appears when the lattice considered is two-dimensional. The square lattice is the usual choice for most of the simulation studies.
This system exhibits a continuous transition. The ascription of such a transition to the
Kosterlitz-Touless type is still under discussion
^{[8]}
^{[9]}
^{[10]}
^{[11]}.

## Lattice Gas Lebwohl-Lasher model[edit]

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
^{[12]}.

## References[edit]

- ↑ P. A. Lebwohl and G. Lasher "Nematic-Liquid-Crystal Order—A Monte Carlo Calculation", Physical Review A
**6**pp. 426 - 429 (1972) - ↑ Erratum, Physical Review A
**7**p. 2222 (1973) - ↑ 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) - ↑ N. V. Priezjev and Robert A. Pelcovits
*Cluster Monte Carlo simulations of the nematic-isotropic transition*Phys. Rev. E 63, 062702 (2001) [4 pages] - ↑ 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) - ↑ Raj Shekhar, Jonathan K. Whitmer, Rohit Malshe, J. A. Moreno-Razo, Tyler F. Roberts, and Juan J. de Pablo "Isotropic–nematic phase transition in the Lebwohl–Lasher model from density of states simulations", Journal of Chemical Physics
**136**234503 (2012) - ↑ Douglas J. Cleaver and Michael P. Allen, " Computer simulation of liquid crystal films", Molecular Physics
**80**pp 253-276 (1993) - ↑ Enakshi Mondal and Soumen Kumar Roy "Finite size scaling in the planar Lebwohl–Lasher model", Physics Letters A
**312**pp. 397-410 (2003) - ↑ C. Chiccoli, P. Pasini, and C. Zannoni "A Monte Carlo investigation of the planar Lebwohl-Lasher lattice model", Physica A
**148**pp. 298-311 (1988) - ↑ H. Kunz, and G. Zumbach "Topological phase transition in a two-dimensional nematic n-vector model: A numerical study" Physical Review B
**46**, 662-673 (1992) - ↑ Ricardo Paredes V., Ana Isabel Fariñas-Sánchez, and Robert Botet "No quasi-long-range order in a two-dimensional liquid crystal", Physical Review E 78, 051706 (2008)
- ↑ Martin A. Bates "Computer simulation study of the phase behavior of a nematogenic lattice-gas model", Physical Review E
**64**051702 (2001)