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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} = -\epsilon P_2 (\cos \beta_{ij}) </math> | :<math>\Phi_{ij} = -\epsilon P_2 (\cos \beta_{ij}) </math> | ||
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where <math>\epsilon > 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 > 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> 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 | 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> and that of Shekhar et al. <ref>[http://dx.doi.org/10.1063/1.4722209 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)]</ref>. | ||
==Confined systems== | ==Confined systems== | ||
The | The Lebwohl-Lasher model has been used to study the effect of [[Confined systems |confinement]] in the phase | ||
behavior of nematogens <ref>[http://dx.doi.org/10.1080/00268979300102251 Douglas J. Cleaver and Michael P. Allen, " Computer simulation of liquid crystal films", Molecular Physics '''80''' pp 253-276 (1993) ]</ref> | behavior of nematogens <ref>[http://dx.doi.org/10.1080/00268979300102251 Douglas J. Cleaver and Michael P. Allen, " Computer simulation of liquid crystal films", Molecular Physics '''80''' pp 253-276 (1993) ]</ref> | ||
==Planar Lebwohl–Lasher model == | ==Planar Lebwohl–Lasher model == | ||
The planar | 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 | This system exhibits a continuous transition. The ascription of such a transition to the | ||
[[Kosterlitz-Thouless transition|Kosterlitz-Touless]] type is still under discussion | [[Kosterlitz-Thouless transition|Kosterlitz-Touless]] type is still under discussion | ||
<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", Physica A '''148''' pp. 298-311 (1988)]</ref> | ||
<ref> [http://link.aps.org/doi/10.1103/PhysRevB.46.662 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) ]</ref> | <ref> [http://link.aps.org/doi/10.1103/PhysRevB.46.662 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) ]</ref> | ||
<ref>[http://link.aps.org/doi/10.1103/PhysRevE.78.051706 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)]</ref>. | <ref>[http://link.aps.org/doi/10.1103/PhysRevE.78.051706 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)]</ref>. | ||
==Lattice Gas | ==Lattice Gas Lebwohl-Lasher model== | ||
This model is the [[lattice gas]] version of the | 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 | 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>. | ||