The Gay-Berne model is used extensively in simulations of liquid crystalline systems. The Gay-Berne model
is an anistropic form of the Lennard-Jones 12:6 potential.
![{\displaystyle U_{ij}^{\mathrm {L} J/GB}=4\epsilon _{0}^{\mathrm {L} J/GB}[\epsilon ^{\mathrm {L} J/GB}]^{\mu }({\mathbf {\hat {u}} }_{j},{\mathbf {\hat {r}} }_{ij})\times \left[\left({\frac {\sigma _{0}^{\mathrm {L} J/GB}}{r_{ij}-\sigma ^{\mathrm {L} J/GB}({\mathbf {\hat {u}} }_{j},{\mathbf {\hat {r}} }_{ij})+{\sigma _{0}}^{\mathrm {L} J/GB}}}\right)^{12}-\left({\frac {\sigma _{0}^{\mathrm {L} J/GB}}{r_{ij}-\sigma ^{\mathrm {L} J/GB}({\mathbf {\hat {u}} }_{j},{\mathbf {\hat {r}} }_{ij})+{\sigma _{0}}^{\mathrm {L} J/GB}}}\right)^{6}\right],}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2772fe11a58f7457bf6d9b2d0c7647032a2019b8)
where, in the limit of one of the particles being spherical, gives:
![{\displaystyle \sigma ^{\mathrm {L} J/GB}({\mathbf {\hat {u}} }_{j},{\mathbf {\hat {r}} }_{ij})={\sigma _{0}}{[1-\chi \alpha ^{-2}{({\mathbf {\hat {r}} }_{ij}\cdot {\mathbf {\hat {u}} }_{j})}^{2}]}^{-1/2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0f559304db8b0a3db5a0c82414e2d4289ea52e01)
and
![{\displaystyle \epsilon ^{\mathrm {L} J/GB}({\mathbf {\hat {u}} }_{j},{\mathbf {\hat {r}} }_{ij})={\epsilon _{0}}{[1-\chi \prime \alpha \prime ^{-2}{({\mathbf {\hat {r}} }_{ij}\cdot {\mathbf {\hat {u}} }_{j})}^{2}]}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/278dbbf31474c1792aa835e8e9e9f9a3da8ab095)
with
- Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\frac {\chi }{\alpha ^{2}}}={\frac {l_{j}^{2}-d_{j}^{2}}{l_{j}^{2}+d_{i}^{2}}}}
and
- Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\frac {\chi \prime }{\alpha \prime ^{2}}}=1-{\left({\frac {\epsilon _{ee}}{\epsilon _{ss}}}\right)}^{\frac {1}{\mu }}.}
Phase diagram
- Main article: Phase diagram of the Gay-Berne model
References
- J. G. Gay and B. J. Berne "Modification of the overlap potential to mimic a linear site–site potential", Journal of Chemical Physics 74 pp. 3316-3319 (1981)
- Douglas J. Cleaver, Christopher M. Care, Michael P. Allen, and Maureen P. Neal "Extension and generalization of the Gay-Berne potential" Physical Review E 54 pp. 559 - 567 (1996)