Lebwohl-Lasher model

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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.

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]

  1. P. A. Lebwohl and G. Lasher "Nematic-Liquid-Crystal Order—A Monte Carlo Calculation", Physical Review A 6 pp. 426 - 429 (1972)
  2. Erratum, Physical Review A 7 p. 2222 (1973)
  3. 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)
  4. N. V. Priezjev and Robert A. Pelcovits Cluster Monte Carlo simulations of the nematic-isotropic transition Phys. Rev. E 63, 062702 (2001) [4 pages]
  5. 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)
  6. 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)
  7. Douglas J. Cleaver and Michael P. Allen, " Computer simulation of liquid crystal films", Molecular Physics 80 pp 253-276 (1993)
  8. Enakshi Mondal and Soumen Kumar Roy "Finite size scaling in the planar Lebwohl–Lasher model", Physics Letters A 312 pp. 397-410 (2003)
  9. 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)
  10. 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)
  11. 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)
  12. Martin A. Bates "Computer simulation study of the phase behavior of a nematogenic lattice-gas model", Physical Review E 64 051702 (2001)