git-svn-id: svn://svn.icms.temple.edu/lammps-ro/trunk@1085 f3b2605a-c512-4ea7-a41b-209d697bcdaa

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

doc/Eqs/pair_resquared.tex

@@ -0,0 +1,7 @@
+\documentstyle[12pt]{article}
+
+\begin{document}
+
+$$ A_{12} = 4\pi^2\epsilon_{\mathrm{LJ}}(\rho\sigma^3)^2 $$
+
+\end{document}

doc/Eqs/pair_resquared2.tex

@@ -0,0 +1,7 @@
+\documentstyle[12pt]{article}
+
+\begin{document}
+
+$$ A_{12} = 4\pi^2\epsilon_{\mathrm{LJ}}(\rho\sigma^3) $$
+
+\end{document}

doc/Eqs/pair_resquared3.tex

@@ -0,0 +1,7 @@
+\documentstyle[12pt]{article}
+
+\begin{document}
+
+$$ A_{12} = \epsilon_{\mathrm{LJ}} $$
+
+\end{document}

doc/Eqs/pair_resquared4.tex

@@ -0,0 +1,9 @@
+\documentstyle[12pt]{article}
+
+\begin{document}
+
+$$ \epsilon_a = \sigma \cdot { \frac{a}{ b \cdot c } }; \epsilon_b =
+\sigma \cdot { \frac{b}{ a \cdot c } }; \epsilon_c = \sigma \cdot {
+\frac{c}{ a \cdot b } } $$
+
+\end{document}

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}