From bb5e1cb67f74efd92f6c0098858187b636d5f300 Mon Sep 17 00:00:00 2001 From: sjplimp Date: Mon, 22 Oct 2007 21:43:27 +0000 Subject: [PATCH] git-svn-id: svn://svn.icms.temple.edu/lammps-ro/trunk@1085 f3b2605a-c512-4ea7-a41b-209d697bcdaa --- doc/Eqs/Script.create | 4 ++ doc/Eqs/pair_resquared.jpg | Bin 0 -> 1660 bytes doc/Eqs/pair_resquared.tex | 7 ++ doc/Eqs/pair_resquared2.jpg | Bin 0 -> 1504 bytes doc/Eqs/pair_resquared2.tex | 7 ++ doc/Eqs/pair_resquared3.jpg | Bin 0 -> 679 bytes doc/Eqs/pair_resquared3.tex | 7 ++ doc/Eqs/pair_resquared4.jpg | Bin 0 -> 2959 bytes doc/Eqs/pair_resquared4.tex | 9 +++ doc/Eqs/pair_resquared_extra.pdf | Bin 0 -> 57660 bytes doc/Eqs/pair_resquared_extra.tex | 113 +++++++++++++++++++++++++++++++ 11 files changed, 147 insertions(+) create mode 100644 doc/Eqs/pair_resquared.jpg create mode 100755 doc/Eqs/pair_resquared.tex create mode 100644 doc/Eqs/pair_resquared2.jpg create mode 100755 doc/Eqs/pair_resquared2.tex create mode 100644 doc/Eqs/pair_resquared3.jpg create mode 100755 doc/Eqs/pair_resquared3.tex create mode 100644 doc/Eqs/pair_resquared4.jpg create mode 100755 doc/Eqs/pair_resquared4.tex create mode 100644 doc/Eqs/pair_resquared_extra.pdf create mode 100755 doc/Eqs/pair_resquared_extra.tex diff --git a/doc/Eqs/Script.create b/doc/Eqs/Script.create index 7061f6fe66..8f473006ab 100755 --- a/doc/Eqs/Script.create +++ b/doc/Eqs/Script.create @@ -46,6 +46,10 @@ latex pair_lj_smooth latex pair_lubricate latex pair_meam latex pair_morse +latex pair_resquared +latex pair_resquared2 +latex pair_resquared3 +latex pair_resquared4 latex pair_soft latex pair_sw latex pair_tersoff diff --git a/doc/Eqs/pair_resquared.jpg b/doc/Eqs/pair_resquared.jpg new file mode 100644 index 0000000000000000000000000000000000000000..de3e0fa887518900b0a8b772059f91a161c57091 GIT binary patch literal 1660 zcmV-?27~$k*#F=F5K2Z#MgRc;0RTtG1Oo*G1Ox>I1qKHU2nPoT z2M-Gi2@DYr5)u&)5fKv>92XN58Wj-{7$F!M9UUJZ9}^cMBqAOp93CGY|G)qX2mmVp 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z2K^T0$Er^m6=DmYJFP zrBcx^G#1b}fqMHF8Z$FJ2omjA8WTG^Bd9}u@cyj4yy0ztUJ(UsltvG)8t1P~C4dP=kMa7DhIv-*vD6=|RaFzsl0HFudS$ z{Gx-NK=bc! z^NR65j3pyGkpA~~V*(SI9XmS5MZlY_pw zm9YaH7Z)5r(cHuMWnTj**xEV~{;_3)He_)d6I;SR_SBcZyl{?A`VLNi?17*(9u{^u LGBROV5xD;akWl#Q literal 0 HcmV?d00001 diff --git a/doc/Eqs/pair_resquared_extra.tex b/doc/Eqs/pair_resquared_extra.tex new file mode 100755 index 0000000000..02958906b8 --- /dev/null +++ b/doc/Eqs/pair_resquared_extra.tex @@ -0,0 +1,113 @@ +\documentstyle[12pt]{article} + +\begin{document} + +\begin{center} + +\large{Additional documention 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}