Editing Gay-Berne model

The '''Gay-Berne model''' <ref>[http://dx.doi.org/10.1063/1.441483  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)]</ref> is used extensively in simulations of [[liquid crystals | liquid crystalline]] systems. The Gay-Berne model
is an anisotropic form of the [[Lennard-Jones model | Lennard-Jones 12:6 potential]].
:<math>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],
</math>

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

:<math>\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}</math>

and

:<math>\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}]}</math>

with

:<math>\frac{\chi}{\alpha^{2}}=\frac{l_{j}^{2}-d_{j}^{2}}{l_{j}^{2}+d_{i}^{2}}</math>

and

:<math>\frac{\chi \prime }{\alpha \prime^{2}}=1- {\left(\frac{\epsilon_{ee}}{\epsilon_{ss}}\right)} ^{\frac{1}{\mu}}.</math>
==Phase diagram==
:''Main article: [[Phase diagram of the Gay-Berne model]]''
==References==
<references/>
'''Related reading'''
*[http://dx.doi.org/10.1016/0009-2614(95)00212-M R. Berardi, C. Fava and C. Zannoni "A generalized Gay-Berne intermolecular potential for biaxial particles", Chemical Physics Letters '''236''' pp. 462-468 (1995)]
*[http://dx.doi.org/10.1103/PhysRevE.54.559  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)]
*[http://dx.doi.org/10.1016/S0009-2614(98)01090-2 Roberto Berardi, Carlo Fava, Claudio Zannoni "A Gay–Berne potential for dissimilar biaxial particles",  Chemical Physics Letters '''297''' pp. 8-14 (1998)]
[[category:liquid crystals]]
[[category:models]]