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.

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U_{ij}^{\mathrm LJ/GB} = 4 \epsilon_0^{\mathrm LJ/GB} [\epsilon^{\mathrm LJ/GB}]^{\mu} ( {\mathbf {\hat u}}_j , {\mathbf {\hat r}}_{ij} ) \times \left[ \left( \frac{\sigma_0^{\mathrm LJ/GB} } { r_{ij} - \sigma^{\mathrm LJ/GB} ({\mathbf {\hat{u}}}_j, {\mathbf {\hat{r}}}_{ij} ) + {\sigma_0}^{\mathrm LJ/GB} } \right)^{12} - \left( \frac { \sigma_0^{\mathrm LJ/GB} } { r_{ij} - \sigma^{\mathrm LJ/GB} ({\mathbf {\hat{u}}}_j, {\mathbf {\hat{r}}}_{ij} ) + {\sigma_0}^{\mathrm LJ/GB} } \right)^{6} \right], }

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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma^{\mathrm LJ/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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \epsilon^{\mathrm LJ/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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\chi}{\alpha^{2}}=\frac{l_{j}^{2}-d_{j}^{2}}{l_{j}^{2}+d_{i}^{2}}}

and

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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

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)