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The ''' | The '''Lebwohl-Lasher model''' is a lattice version of the [[Maier-Saupe mean field model]] of a [[Nematic phase | nematic liquid crystal]] | ||
<ref>[http://dx.doi.org/10.1103/PhysRevA.6.426 P. A. Lebwohl and G. Lasher "Nematic-Liquid-Crystal Order—A Monte Carlo Calculation", Physical Review A '''6''' pp. 426 - 429 (1972)]</ref><ref>[http://dx.doi.org/10.1103/PhysRevA.7.2222.3 Erratum, Physical Review A '''7''' p. 2222 (1973)]</ref>. | <ref>[http://dx.doi.org/10.1103/PhysRevA.6.426 P. A. Lebwohl and G. Lasher "Nematic-Liquid-Crystal Order—A Monte Carlo Calculation", Physical Review A '''6''' pp. 426 - 429 (1972)]</ref><ref>[http://dx.doi.org/10.1103/PhysRevA.7.2222.3 Erratum, Physical Review A '''7''' p. 2222 (1973)]</ref>. | ||
The | The Lebwohl-Lasher model consists of a cubic lattice occupied by uniaxial [[Nematic phase|nematogenic]] particles with the [[Intermolecular pair potential | pair potential]] | ||
:<math>\Phi_{ij} = -\ | :<math>\Phi_{ij} = -\epsilon_{ij} P_2 (\cos \beta_{ij}) </math> | ||
where <math>\ | 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 | 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: | ||
:<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> | ||
More recently N. V. Priezjev and Robert A. Pelcovits <ref>[http://dx.doi.org/10.1103/PhysRevE.63.062702 | More recently N. V. Priezjev and Robert A. Pelcovits <ref>[http://dx.doi.org/10.1103/PhysRevE.63.062702 | ||
''Cluster Monte Carlo simulations of the nematic-isotropic transition'' Phys. Rev. E 63, 062702 (2001) [4 pages] | |||
]</ref> used a Monte Carlo [[cluster algorithms|cluster algorithm]] and got: | |||
:<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> | ||
==Planar Lebwohl–Lasher model == | ==Planar Lebwohl–Lasher model == | ||
The planar | The planar Lebwohl-Lasher appears when the lattice considered is two-dimensional. | ||
This system exhibits a | This system exhibits a [[Kosterlitz-Thouless transition|Kosterlitz-Touless]] continuous transition | ||
[[Kosterlitz-Thouless transition|Kosterlitz-Touless]] | |||
<ref>[http://dx.doi.org/10.1016/S0375-9601(03)00576-0 Enakshi Mondal and Soumen Kumar Roy "Finite size scaling in the planar Lebwohl–Lasher model", Physics Letters A '''312''' pp. 397-410 (2003)]</ref> | <ref>[http://dx.doi.org/10.1016/S0375-9601(03)00576-0 Enakshi Mondal and Soumen Kumar Roy "Finite size scaling in the planar Lebwohl–Lasher model", Physics Letters A '''312''' pp. 397-410 (2003)]</ref> | ||
<ref>[http://dx.doi.org/10.1016/0378-4371(88)90148-3 C. Chiccoli, P. Pasini, and C. Zannoni "A Monte Carlo investigation of the planar | <ref>[http://dx.doi.org/10.1016/0378-4371(88)90148-3 C. Chiccoli, P. Pasini, and C. Zannoni "A Monte Carlo investigation of the planar Lebwohl-Lasher lattice model", ĥysica A '''148''' pp. 298-311 (1988)]</ref>. | ||
==Lattice Gas Lebwohl-Lasher model== | |||
This model is the [[lattice gas]] version of the Lebwohl-Lasher model. In this case | |||
==Lattice Gas | |||
This model is the [[lattice gas]] version of the | |||
the sites of the lattice can be occupied by particles or empty. The interaction | the sites of the lattice can be occupied by particles or empty. The interaction | ||
between nearest-neighbour particles is that of the | between nearest-neighbour particles is that of the Lebwohl-Lasher model. | ||
This model has been studied in | This model has been studied in | ||
<ref>[http://dx.doi.org/10.1103/PhysRevE.64.051702 Martin A. Bates "Computer simulation study of the phase behavior of a nematogenic lattice-gas model", Physical Review E '''64''' 051702 (2001)]</ref>. | <ref>[http://dx.doi.org/10.1103/PhysRevE.64.051702 Martin A. Bates "Computer simulation study of the phase behavior of a nematogenic lattice-gas model", Physical Review E '''64''' 051702 (2001)]</ref>. |