Gay-Berne model

The Gay-Berne model (Ref. 1) is used extensively in simulations of liquid crystalline systems. The Gay-Berne model is an anistropic form of the Lennard-Jones 12:6 potential.

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],

where, in the limit of one of the particles being spherical, gives:

\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}

and

\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}]}

with

{\frac {\chi }{\alpha ^{2}}}={\frac {l_{j}^{2}-d_{j}^{2}}{l_{j}^{2}+d_{i}^{2}}}

and

{\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