Lebwohl-Lasher model: Difference between revisions
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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> | 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
- ↑ 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)
- ↑ 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)