From 62e304c0d325cb222e5c625c8b3bbab3e1e99e4a Mon Sep 17 00:00:00 2001 From: sjplimp Date: Tue, 13 Dec 2011 21:36:06 +0000 Subject: [PATCH] git-svn-id: svn://svn.icms.temple.edu/lammps-ro/trunk@7357 f3b2605a-c512-4ea7-a41b-209d697bcdaa --- doc/Eqs/angle_charmm.jpg | Bin 2185 -> 3932 bytes doc/Eqs/angle_class2.jpg | Bin 8873 -> 16202 bytes doc/Eqs/angle_cosine.jpg | Bin 1443 -> 2380 bytes doc/Eqs/angle_cosine_delta.jpg | Bin 1630 -> 3276 bytes doc/Eqs/angle_cosine_periodic.jpg | Bin 1949 -> 3362 bytes doc/Eqs/angle_cosine_shift.jpg | Bin 2435 -> 3933 bytes doc/Eqs/angle_cosine_shift_exp.jpg | Bin 4705 -> 7884 bytes doc/Eqs/angle_cosine_squared.jpg | Bin 1961 -> 3265 bytes doc/Eqs/angle_harmonic.jpg | Bin 1266 -> 2005 bytes doc/Eqs/bond_class2.jpg | Bin 2690 -> 5142 bytes doc/Eqs/bond_fene.jpg | Bin 4017 -> 8322 bytes doc/Eqs/bond_fene_expand.jpg | Bin 6844 -> 11506 bytes doc/Eqs/bond_harmonic.jpg | Bin 1211 -> 1861 bytes doc/Eqs/bond_harmonic_shift.jpg | Bin 3087 -> 5078 bytes 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--git a/doc/Eqs/pair_resquared_extra.tex b/doc/Eqs/pair_resquared_extra.tex deleted file mode 100755 index 945ee562d7..0000000000 --- a/doc/Eqs/pair_resquared_extra.tex +++ /dev/null @@ -1,113 +0,0 @@ -\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} diff --git a/doc/Eqs/pair_soft.jpg 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