Gay-Berne model

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. $U_{ij}^{\mathrm {L} J/GB}=4\epsilon _{0}^{\mathrm {L} J/GB}[\epsilon ^{\mathrm {L} J/GB}]^{\mu }({\mathbf {\hat {u}} }_{j},{\mathbf {\hat {r}} }_{ij})\times \left[\left({\frac {\sigma _{0}^{\mathrm {L} J/GB}}{r_{ij}-\sigma ^{\mathrm {L} J/GB}({\mathbf {\hat {u}} }_{j},{\mathbf {\hat {r}} }_{ij})+{\sigma _{0}}^{\mathrm {L} J/GB}}}\right)^{12}-\left({\frac {\sigma _{0}^{\mathrm {L} J/GB}}{r_{ij}-\sigma ^{\mathrm {L} J/GB}({\mathbf {\hat {u}} }_{j},{\mathbf {\hat {r}} }_{ij})+{\sigma _{0}}^{\mathrm {L} J/GB}}}\right)^{6}\right],$ 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

${\frac {\chi \prime }{\alpha \prime ^{2}}}=1-{\left({\frac {\epsilon _{ee}}{\epsilon _{ss}}}\right)}^{\frac {1}{\mu }}.$

References

1. 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)
2. 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)
3. Enrique De Miguel "Reexamining the phase diagram of the Gay-Berne fluid", Molecular Physics 100 pp. 2449-2459 (2002)
4. Julian T. Brown, Michael P. Allen, Elvira Martín del Río and Enrique de Miguel "Effects of elongation on the phase behavior of the Gay-Berne fluid", Physical Review E 57 pp. 6685 - 6699 (1998)