git-svn-id: svn://svn.icms.temple.edu/lammps-ro/trunk@7357 f3b2605a-c512-4ea7-a41b-209d697bcdaa

doc/Eqs/pair_gayberne_extra.tex

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-\documentstyle[12pt]{article}
-
-\begin{document}
-
-\begin{center}
-
-\large{Additional documentation for the Gay-Berne ellipsoidal potential \\
-  as implemented in LAMMPS}
-
-\end{center}
-
-\centerline{Mike Brown, Sandia National Labs, April 2007}
-
-\vspace{0.3in}
-
-The Gay-Berne anisotropic LJ interaction between pairs of dissimilar
-ellipsoidal particles is given by
-
-$$ U ( \mathbf{A}_1, \mathbf{A}_2, \mathbf{r}_{12} ) = U_r (
-\mathbf{A}_1, \mathbf{A}_2, \mathbf{r}_{12}, \gamma ) \cdot \eta_{12} (
-\mathbf{A}_1, \mathbf{A}_2, \upsilon ) \cdot \chi_{12} ( \mathbf{A}_1,
-\mathbf{A}_2, \mathbf{r}_{12}, \mu ) $$
-
-where $\mathbf{A}_1$ and $\mathbf{A}_2$ are the transformation
-matrices from the simulation box frame to the body frame and
-$\mathbf{r}_{12}$ is the center to center vector between the
-particles.  $U_r$ controls the shifted distance dependent interaction
-based on the distance of closest approach of the two particles
-($h_{12}$) and the user-specified shift parameter gamma:
-
-$$ U_r = 4 \epsilon ( \varrho^{12} - \varrho^6) $$
-
-$$ \varrho = \frac{\sigma}{ h_{12} + \gamma \sigma} $$
-
-Let the shape matrices $\mathbf{S}_i=\mbox{diag}(a_i, b_i, c_i)$ be 
-given by the ellipsoid radii. The $\eta$ orientation-dependent energy 
-based on the user-specified exponent $\upsilon$ is given by
-
-$$ \eta_{12} = [ \frac{ 2 s_1 s_2 }{\det ( \mathbf{G}_{12} )}]^{
-\upsilon / 2 } , $$
-
-$$ s_i = [a_i b_i + c_i c_i][a_i b_i]^{ 1 / 2 }, $$
-
-and
-
-$$ \mathbf{G}_{12} = \mathbf{A}_1^T \mathbf{S}_1^2 \mathbf{A}_1 +
-\mathbf{A}_2^T \mathbf{S}_2^2 \mathbf{A}_2 = \mathbf{G}_1 +
-\mathbf{G}_2. $$
-
-Let the relative energy matrices $\mathbf{E}_i = \mbox{diag}
-(\epsilon_{ia}, \epsilon_{ib}, \epsilon_{ic})$ be given by 
-the relative well depths (dimensionless energy scales 
-inversely proportional to the well-depths of the respective 
-orthogonal configurations of the interacting molecules). The 
-$\chi$ orientation-dependent energy based on the user-specified 
-exponent $\mu$ is given by
-
-$$ \chi_{12} = [2 \hat{\mathbf{r}}_{12}^T \mathbf{B}_{12}^{-1}
-\hat{\mathbf{r}}_{12}]^\mu, $$
-
-$$ \hat{\mathbf{r}}_{12} = { \mathbf{r}_{12} } / |\mathbf{r}_{12}|, $$
-
-and
-
-$$ \mathbf{B}_{12} = \mathbf{A}_1^T \mathbf{E}_1^2 \mathbf{A}_1 +
-\mathbf{A}_2^T \mathbf{E}_2^2 \mathbf{A}_2 = \mathbf{B}_1 +
-\mathbf{B}_2. $$
-
-Here, we use the distance of closest approach approximation given by the
-Perram reference, namely
-
-$$ h_{12} = r - \sigma_{12} ( \mathbf{A}_1, \mathbf{A}_2,
-\mathbf{r}_{12} ), $$
-
-$$ r = |\mathbf{r}_{12}|, $$
-
-and
-
-$$ \sigma_{12} = [ \frac{1}{2} \hat{\mathbf{r}}_{12}^T
-\mathbf{G}_{12}^{-1} \hat{\mathbf{r}}_{12}.]^{ -1/2 } $$
-
-Forces and Torques: Because the analytic forces and torques have not
-been published for this potential, we list them here:
-
-$$ \mathbf{f} = - \eta_{12} ( U_r \cdot { \frac{\partial \chi_{12}
-}{\partial r} } + \chi_{12} \cdot { \frac{\partial U_r }{\partial r} }
-) $$
-
-where the derivative of $U_r$ is given by (see Allen reference)
-
-$$ \frac{\partial U_r }{\partial r} = \frac{ \partial U_{SLJ} }{
-\partial r } \hat{\mathbf{r}}_{12} + r^{-2} \frac{ \partial U_{SLJ} }{
-\partial \varphi } [ \mathbf{\kappa} - ( \mathbf{\kappa}^T \cdot
-\hat{\mathbf{r}}_{12}) \hat{\mathbf{r}}_{12} ], $$
-
-$$ \frac{ \partial U_{SLJ} }{ \partial \varphi } = 24 \epsilon ( 2
-\varrho^{13} - \varrho^7 ) \sigma_{12}^3 / 2 \sigma, $$
-
-$$ \frac{ \partial U_{SLJ} }{ \partial r } = 24 \epsilon ( 2
-\varrho^{13} - \varrho^7 ) / \sigma, $$
-
-and
-
-$$ \mathbf{\kappa} = \mathbf{G}_{12}^{-1} \cdot \mathbf{r}_{12}. $$
-
-The derivate of the $\chi$ term is given by
-
-$$ \frac{\partial \chi_{12} }{\partial r} = - r^{-2} \cdot 4.0 \cdot [
-\mathbf{\iota} - ( \mathbf{\iota}^T \cdot \hat{\mathbf{r}}_{12} )
-\hat{\mathbf{r}}_{12} ] \cdot \mu \cdot \chi_{12}^{ ( \mu -1 ) / \mu
-}, $$
-
-and
-
-$$ \mathbf{\iota} = \mathbf{B}_{12}^{-1} \cdot \mathbf{r}_{12}. $$
-
-The torque is given by:
-
-$$ \mathbf{\tau}_i = U_r \eta_{12} \frac{ \partial \chi_{12} }{
-\partial \mathbf{q}_i } + \chi_{12} ( U_r \frac{ \partial \eta_{12} }{
-\partial \mathbf{q}_i } + \eta_{12} \frac{ \partial U_r }{ \partial
-\mathbf{q}_i } ), $$
-
-$$ \frac{ \partial U_r }{ \partial \mathbf{q}_i } = \mathbf{A}_i \cdot
-(- \mathbf{\kappa}^T \cdot \mathbf{G}_i \times \mathbf{f}_k ), $$
-
-$$ \mathbf{f}_k = - r^{-2} \frac{ \delta U_{SLJ} }{ \delta \varphi }
-\mathbf{\kappa}, $$
-
-and
-
-$$ \frac{ \partial \chi_{12} }{ \partial \mathbf{q}_i } = 4.0 \cdot
-r^{-2} \cdot \mathbf{A}_i (- \mathbf{\iota}^T \cdot \mathbf{B}_i
-\times \mathbf{\iota} ). $$
-
-For the derivative of the $\eta$ term, we were unable to find a matrix
-expression due to the determinant. Let $a_{mi}$ be the mth row of the
-rotation matrix $A_i$. Then,
-
-$$ \frac{ \partial \eta_{12} }{ \partial \mathbf{q}_i } = \mathbf{A}_i
-\cdot \sum_m \mathbf{a}_{mi} \times \frac{ \partial \eta_{12} }{
-\partial \mathbf{a}_{mi} } = \mathbf{A}_i \cdot \sum_m \mathbf{a}_{mi}
-\times \mathbf{d}_{mi}, $$
-
-where $d_mi$ represents the mth row of a derivative matrix $D_i$,
-
-$$ \mathbf{D}_i = - \frac{1}{2} \cdot ( \frac{2s1s2}{\det (
-\mathbf{G}_{12} ) } )^{ \upsilon / 2 } \cdot {\frac{\upsilon}{\det (
-\mathbf{G}_{12} ) }} \cdot \mathbf{E}, $$
-
-where the matrix $E$ gives the derivate with respect to the rotation
-matrix,
-
-$$ \mathbf{E} = [ e_{my} ] = \frac{ \partial \eta_{12} }{ \partial
-\mathbf{A}_i }, $$
-
-and
-
-$$ e_{my} = \det ( \mathbf{G}_{12} ) \cdot \mbox{trace} [
-\mathbf{G}_{12}^{-1} \cdot ( \hat{\mathbf{p}}_y \otimes \mathbf{a}_m +
-\mathbf{a}_m \otimes \hat{\mathbf{p}}_y ) \cdot s_{mm}^2 ]. $$
-
-Here, $p_v$ is the unit vector for the axes in the lab frame $(p1=[1, 0,
-0], p2=[0, 1, 0], and p3=[0, 0, 1])$ and $s_{mm}$ gives the mth radius of
-the ellipsoid $i$.
-
-\end{document}