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\begin{document} |
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\title{The Langevin Hull: Constant pressure and temperature dynamics for non-periodic systems} |
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\author{Charles F. Varedeman II, Kelsey Stocker, and J. Daniel |
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Gezelter\footnote{Corresponding author. \ Electronic mail: gezelter@nd.edu} \\ |
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Department of Chemistry and Biochemistry,\\ |
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University of Notre Dame\\ |
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Notre Dame, Indiana 46556} |
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\date{\today} |
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\maketitle |
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\begin{doublespace} |
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\begin{abstract} |
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We have developed a new isobaric-isothermal (NPT) algorithm which |
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applies an external pressure to the facets comprising the convex |
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hull surrounding the objects in the system. Additionally, a Langevin |
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thermostat is applied to facets of the hull to mimic contact with an |
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external heat bath. This new method, the ``Langevin Hull'', |
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performs better than traditional affine transform methods for |
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systems containing heterogeneous mixtures of materials with |
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different compressibilities. It does not suffer from the edge |
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effects of boundary potential methods, and allows realistic |
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treatment of both external pressure and thermal conductivity to an |
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implicit solvents. We apply this method to several different |
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systems including bare nano-particles, nano-particles in explicit |
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solvent, as well as clusters of liquid water and ice. The predicted |
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mechanical and thermal properties of these systems are in good |
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agreement with experimental data. |
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\end{abstract} |
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\newpage |
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%\narrowtext |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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% BODY OF TEXT |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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\section{Introduction} |
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The most common molecular dynamics methods for sampling configurations |
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of an isobaric-isothermal (NPT) ensemble attempt to maintain a target |
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pressure in a simulation by coupling the volume of the system to an |
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extra degree of freedom, the {\it barostat}. These methods require |
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periodic boundary conditions, because when the instantaneous pressure |
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in the system differs from the target pressure, the volume is |
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typically reduced or expanded using {\it affine transforms} of the |
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system geometry. An affine transform scales both the box lengths as |
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well as the scaled particle positions (but not the sizes of the |
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particles). The most common constant pressure methods, including the |
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Melchionna modification\cite{melchionna93} to the |
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Nos\'e-Hoover-Andersen equations of motion, the Berendsen pressure |
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bath, and the Langevin Piston, all utilize coordinate transformation |
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to adjust the box volume. |
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\begin{figure} |
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\includegraphics[width=\linewidth]{AffineScale2} |
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\caption{Affine Scaling constant pressure methods use box-length |
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scaling to adjust the volume to adjust to under- or over-pressure |
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conditions. In a system with a uniform compressibility (e.g. bulk |
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fluids) these methods can work well. In systems containing |
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heterogeneous mixtures, the affine scaling moves required to adjust |
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the pressure in the high-compressibility regions can cause molecules |
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in low compressibility regions to collide.} |
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\label{affineScale} |
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\end{figure} |
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Heterogeneous mixtures of materials with different compressibilities? |
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Explicitly non-periodic systems |
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Elastic Bag |
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Spherical Boundary approaches |
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\section{Methodology} |
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A new method which uses a constant pressure and temperature bath that |
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interacts with the objects that are currently at the edge of the |
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system. |
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Novel features: No a priori geometry is defined, No affine transforms, |
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No fictitious particles, No bounding potentials. |
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Simulation starts as a collection of atomic locations in 3D (a point |
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cloud). |
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Delaunay triangulation finds all facets between coplanar neighbors. |
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The Convex Hull is the set of facets that have no concave corners at a |
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vertex. |
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Molecules on the convex hull are dynamic. As they re-enter the |
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cluster, all interactions with the external bath are removed.The |
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external bath applies pressure to the facets of the convex hull in |
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direct proportion to the area of the facet.Thermal coupling depends on |
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the solvent temperature, friction and the size and shape of each |
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facet. |
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\begin{equation} |
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m_i \dot{\mathbf v}_i(t)=-{\mathbf \nabla}_i U |
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\end{equation} |
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\begin{equation} |
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m_i \dot{\mathbf v}_i(t)=-{\mathbf \nabla}_i U + {\mathbf F}_i^{\mathrm ext} |
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\end{equation} |
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\begin{equation} |
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{\mathbf F}_{i}^{\mathrm ext} = \sum_{\begin{array}{c}\mathrm{facets\ |
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} f \\ \mathrm{containing\ } i\end{array}} \frac{1}{3}\ {\mathbf |
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F}_f^{\mathrm ext} |
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\end{equation} |
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\begin{equation} |
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\begin{array}{rclclcl} |
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{\mathbf F}_f^{\text{ext}} & = & \text{external pressure} & + & \text{drag force} & + & \text{random force} \\ |
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& = & -\hat{n}_f P A_f & - & \Xi_f(t) {\mathbf v}_f(t) & + & {\mathbf R}_f(t) |
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\end{array} |
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\end{equation} |
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\begin{eqnarray} |
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A_f & = & \text{area of facet}\ f \\ |
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\hat{n}_f & = & \text{facet normal} \\ |
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P & = & \text{external pressure} |
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\end{eqnarray} |
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\begin{eqnarray} |
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{\mathbf v}_f(t) & = & \text{velocity of facet} \\ |
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& = & \frac{1}{3} \sum_{i=1}^{3} {\mathbf v}_i \\ |
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\Xi_f(t) & = & \text{is a hydrodynamic tensor that depends} \\ |
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& & \text{on the geometry and surface area of} \\ |
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& & \text{facet} \ f\ \text{and the viscosity of the fluid.} |
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\end{eqnarray} |
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\begin{eqnarray} |
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\left< {\mathbf R}_f(t) \right> & = & 0 \\ |
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\left<{\mathbf R}_f(t) {\mathbf R}_f^T(t^\prime)\right> & = & 2 k_B T\ |
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\Xi_f(t)\delta(t-t^\prime) |
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\end{eqnarray} |
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Implemented in OpenMD.\cite{Meineke:2005gd,openmd} |
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\section{Tests \& Applications} |
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\subsection{Bulk modulus of gold nanoparticles} |
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\begin{figure} |
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\includegraphics[width=\linewidth]{pressure_tb} |
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\caption{Pressure response is rapid (18 \AA gold nanoparticle), target |
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pressure = 4 GPa} |
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\label{pressureResponse} |
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\end{figure} |
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\begin{figure} |
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\includegraphics[width=\linewidth]{temperature_tb} |
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\caption{Temperature equilibration depends on surface area and bath |
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viscosity. Target Temperature = 300K} |
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\label{temperatureResponse} |
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\end{figure} |
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\begin{equation} |
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\kappa_T=-\frac{1}{V_{\mathrm{eq}}}\left(\frac{\partial V}{\partial |
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P}\right) |
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\end{equation} |
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\begin{figure} |
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\includegraphics[width=\linewidth]{compress_tb} |
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\caption{Isothermal Compressibility (18 \AA gold nanoparticle)} |
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\label{temperatureResponse} |
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\end{figure} |
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\subsection{Compressibility of SPC/E water clusters} |
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\begin{figure} |
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\includegraphics[width=\linewidth]{g_r_theta} |
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\caption{Definition of coordinates} |
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\label{coords} |
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\end{figure} |
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\begin{equation} |
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\cos{\theta}=\frac{\vec{r}_i\cdot\vec{\mu}_i}{|\vec{r}_i||\vec{\mu}_i|} |
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\end{equation} |
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\begin{figure} |
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\includegraphics[width=\linewidth]{pAngle} |
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\caption{SPC/E water clusters: only minor dewetting at the boundary} |
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\label{pAngle} |
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\end{figure} |
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\begin{figure} |
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\includegraphics[width=\linewidth]{isothermal} |
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\caption{Compressibility of SPC/E water} |
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\label{compWater} |
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\end{figure} |
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\subsection{Heterogeneous nanoparticle / water mixtures} |
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\section{Appendix A: Hydrodynamic tensor for triangular facets} |
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\begin{figure} |
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\includegraphics[width=\linewidth]{hydro} |
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\caption{Hydro} |
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\label{hydro} |
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\end{figure} |
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\begin{equation} |
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\Xi_f(t) =\left[\sum_{i=1}^3 T_{if}\right]^{-1} |
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\end{equation} |
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\begin{equation} |
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T_{if}=\frac{A_i}{8\pi\eta R_{if}}\left(I + |
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\frac{\mathbf{R}_{if}\mathbf{R}_{if}^T}{R_{if}^2}\right) |
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\end{equation} |
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\section{Appendix B: Computing Convex Hulls on Parallel Computers} |
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\section{Acknowledgments} |
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Support for this project was provided by the |
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National Science Foundation under grant CHE-0848243. Computational |
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time was provided by the Center for Research Computing (CRC) at the |
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University of Notre Dame. |
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\newpage |
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\bibliography{langevinHull} |
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\end{doublespace} |
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\end{document} |
