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 0000000000..de3e0fa887 Binary files /dev/null and b/doc/Eqs/pair_resquared.jpg differ diff --git a/doc/Eqs/pair_resquared.tex b/doc/Eqs/pair_resquared.tex new file mode 100755 index 0000000000..e5c758aec0 --- /dev/null +++ b/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} diff --git a/doc/Eqs/pair_resquared2.jpg b/doc/Eqs/pair_resquared2.jpg new file mode 100644 index 0000000000..deec547829 Binary files /dev/null and b/doc/Eqs/pair_resquared2.jpg differ diff --git a/doc/Eqs/pair_resquared2.tex b/doc/Eqs/pair_resquared2.tex new file mode 100755 index 0000000000..ff2c3ca749 --- /dev/null +++ b/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} diff --git a/doc/Eqs/pair_resquared3.jpg b/doc/Eqs/pair_resquared3.jpg new file mode 100644 index 0000000000..8e58c660ed Binary files /dev/null and b/doc/Eqs/pair_resquared3.jpg differ diff --git a/doc/Eqs/pair_resquared3.tex b/doc/Eqs/pair_resquared3.tex new file mode 100755 index 0000000000..136bcf73dc --- /dev/null +++ b/doc/Eqs/pair_resquared3.tex @@ -0,0 +1,7 @@ +\documentstyle[12pt]{article} + +\begin{document} + +$$ A_{12} = \epsilon_{\mathrm{LJ}} $$ + +\end{document} diff --git a/doc/Eqs/pair_resquared4.jpg b/doc/Eqs/pair_resquared4.jpg new file mode 100644 index 0000000000..a654952d2f Binary files /dev/null and b/doc/Eqs/pair_resquared4.jpg differ diff --git a/doc/Eqs/pair_resquared4.tex b/doc/Eqs/pair_resquared4.tex new file mode 100755 index 0000000000..6efb755774 --- /dev/null +++ b/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} diff --git a/doc/Eqs/pair_resquared_extra.pdf b/doc/Eqs/pair_resquared_extra.pdf new file mode 100644 index 0000000000..28be0fd6f4 Binary files /dev/null and b/doc/Eqs/pair_resquared_extra.pdf differ 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}