From 28b1e0de2bad7eca92eed140143cbdb418184a6f Mon Sep 17 00:00:00 2001 From: sjplimp Date: Tue, 13 Dec 2011 21:38:11 +0000 Subject: [PATCH] git-svn-id: svn://svn.icms.temple.edu/lammps-ro/trunk@7359 f3b2605a-c512-4ea7-a41b-209d697bcdaa --- doc/PDF/pair_gayberne_extra.tex | 167 +++++++++++++++++++++++++++++++ doc/PDF/pair_resquared_extra.tex | 113 +++++++++++++++++++++ 2 files changed, 280 insertions(+) create mode 100644 doc/PDF/pair_gayberne_extra.tex create mode 100644 doc/PDF/pair_resquared_extra.tex diff --git a/doc/PDF/pair_gayberne_extra.tex b/doc/PDF/pair_gayberne_extra.tex new file mode 100644 index 0000000000..5f0d97d7ab --- /dev/null +++ b/doc/PDF/pair_gayberne_extra.tex @@ -0,0 +1,167 @@ +\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} diff --git a/doc/PDF/pair_resquared_extra.tex b/doc/PDF/pair_resquared_extra.tex new file mode 100644 index 0000000000..945ee562d7 --- /dev/null +++ b/doc/PDF/pair_resquared_extra.tex @@ -0,0 +1,113 @@ +\documentstyle[12pt]{article} + +\begin{document} + +\begin{center} + +\large{Additional documentation for the RE-squared ellipsoidal potential \\ + as implemented in LAMMPS} + +\end{center} + +\centerline{Mike Brown, Sandia National Labs, October 2007} + +\vspace{0.3in} + +Let the shape matrices $\mathbf{S}_i=\mbox{diag}(a_i, b_i, c_i)$ be +given by the ellipsoid radii. 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). +Let $\mathbf{A}_1$ and $\mathbf{A}_2$ be the transformation matrices +from the simulation box frame to the body frame and $\mathbf{r}$ +be the center to center vector between the particles. Let $A_{12}$ be +the Hamaker constant for the interaction given in LJ units by +$A_{12}=4\pi^2\epsilon_{\mathrm{LJ}}(\rho\sigma^3)^2$. + +\vspace{0.3in} + +The RE-squared anisotropic interaction between pairs of +ellipsoidal particles is given by + +$$ U=U_A+U_R, $$ + +$$ U_\alpha=\frac{A_{12}}{m_\alpha}(\frac\sigma{h})^{n_\alpha} +(1+o_\alpha\eta\chi\frac\sigma{h}) \times \prod_i{ +\frac{a_ib_ic_i}{(a_i+h/p_\alpha)(b_i+h/p_\alpha)(c_i+h/p_\alpha)}}, $$ + +$$ m_A=-36, n_A=0, o_A=3, p_A=2, $$ + +$$ m_R=2025, n_R=6, o_R=45/56, p_R=60^{1/3}, $$ + +$$ \chi = 2 \hat{\mathbf{r}}^T \mathbf{B}^{-1} +\hat{\mathbf{r}}, $$ + +$$ \hat{\mathbf{r}} = { \mathbf{r} } / |\mathbf{r}|, $$ + +$$ \mathbf{B} = \mathbf{A}_1^T \mathbf{E}_1 \mathbf{A}_1 + +\mathbf{A}_2^T \mathbf{E}_2 \mathbf{A}_2 $$ + +$$ \eta = \frac{ \det[\mathbf{S}_1]/\sigma_1^2+ +det[\mathbf{S}_2]/\sigma_2^2}{[\det[\mathbf{H}]/ +(\sigma_1+\sigma_2)]^{1/2}}, $$ + +$$ \sigma_i = (\hat{\mathbf{r}}^T\mathbf{A}_i^T\mathbf{S}_i^{-2} +\mathbf{A}_i\hat{\mathbf{r}})^{-1/2}, $$ + +$$ \mathbf{H} = \frac{1}{\sigma_1}\mathbf{A}_1^T \mathbf{S}_1^2 \mathbf{A}_1 + +\frac{1}{\sigma_2}\mathbf{A}_2^T \mathbf{S}_2^2 \mathbf{A}_2 $$ + + +Here, we use the distance of closest approach approximation given by the +Perram reference, namely + +$$ h = |r| - \sigma_{12}, $$ + +$$ \sigma_{12} = [ \frac{1}{2} \hat{\mathbf{r}}^T +\mathbf{G}^{-1} \hat{\mathbf{r}}]^{ -1/2 }, $$ + +and + +$$ \mathbf{G} = \mathbf{A}_1^T \mathbf{S}_1^2 \mathbf{A}_1 + +\mathbf{A}_2^T \mathbf{S}_2^2 \mathbf{A}_2 $$ + +\vspace{0.3in} + +The RE-squared anisotropic interaction between a +ellipsoidal particle and a Lennard-Jones sphere is defined +as the $\lim_{a_2->0}U$ under the constraints that +$a_2=b_2=c_2$ and $\frac{4}{3}\pi a_2^3\rho=1$: + +$$ U_{\mathrm{elj}}=U_{A_{\mathrm{elj}}}+U_{R_{\mathrm{elj}}}, $$ + +$$ U_{\alpha_{\mathrm{elj}}}=(\frac{3\sigma^3c_\alpha^3} +{4\pi h_{\mathrm{elj}}^3})\frac{A_{12_{\mathrm{elj}}}} +{m_\alpha}(\frac\sigma{h_{\mathrm{elj}}})^{n_\alpha} +(1+o_\alpha\chi_{\mathrm{elj}}\frac\sigma{h_{\mathrm{elj}}}) \times +\frac{a_1b_1c_1}{(a_1+h_{\mathrm{elj}}/p_\alpha) +(b_1+h_{\mathrm{elj}}/p_\alpha)(c_1+h_{\mathrm{elj}}/p_\alpha)}, $$ + +$$ A_{12_{\mathrm{elj}}}=4\pi^2\epsilon_{\mathrm{LJ}}(\rho\sigma^3), $$ + +with $h_{\mathrm{elj}}$ and $\chi_{\mathrm{elj}}$ calculated as above +by replacing $B$ with $B_{\mathrm{elj}}$ and $G$ with $G_{\mathrm{elj}}$: + +$$ \mathbf{B}_{\mathrm{elj}} = \mathbf{A}_1^T \mathbf{E}_1 \mathbf{A}_1 + I, $$ + +$$ \mathbf{G}_{\mathrm{elj}} = \mathbf{A}_1^T \mathbf{S}_1^2 \mathbf{A}_1.$$ + +\vspace{0.3in} + +The interaction between two LJ spheres is calculated as: + +$$ + U_{\mathrm{lj}} = 4 \epsilon \left[ \left(\frac{\sigma}{|\mathbf{r}|}\right)^{12} - + \left(\frac{\sigma}{|\mathbf{r}|}\right)^6 \right] +$$ + +\vspace{0.3in} + +The analytic derivatives are used for all force and torque calculation. + +\end{document}