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\title{{\sc OpenMD}: Molecular Dynamics in the Open} |
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\author{Shenyu Kuang, Chunlei Li, Charles F. Vardeman II, \\ |
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Teng Lin, Christopher J. Fennell, Xiuquan Sun, \\ |
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Kyle Daily, Yang Zheng, Matthew A. Meineke, and J. Daniel Gezelter\\ |
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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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\author{Kelsey M. Stocker, Shenyu Kuang, Charles F. Vardeman II, \\ |
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Teng Lin, Christopher J. Fennell, Xiuquan Sun, \\ |
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Chunlei Li, Kyle Daily, Yang Zheng, Matthew A. Meineke, and \\ |
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J. Daniel Gezelter \\ |
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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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\maketitle |
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that is easy to learn. |
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\tableofcontents |
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%\listoffigures |
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%\listoftables |
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\listoffigures |
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\listoftables |
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\mainmatter |
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|
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{\tt minimizer} & string & Chooses a minimizer & Possible minimizers |
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are SD and CG. Either {\tt ensemble} or {\tt minimizer} must be specified. \\ |
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{\tt ensemble} & string & Sets the ensemble. & Possible ensembles are |
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NVE, NVT, NPTi, NPAT, NPTf, NPTxyz, and LD. Either {\tt ensemble} |
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NVE, NVT, NPTi, NPAT, NPTf, NPTxyz, LD and LangevinHull. Either {\tt ensemble} |
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or {\tt minimizer} must be specified. \\ |
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{\tt dt} & fs & Sets the time step. & Selection of {\tt dt} should be |
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small enough to sample the fastest motion of the simulation. ({\tt |
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<MetaData> |
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molecule{ |
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name = "I2"; |
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atom[0]{ |
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type = "I"; |
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} |
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atom[1]{ |
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type = "I"; |
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} |
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bond{ |
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members( 0, 1); |
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} |
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atom[0]{ type = "I"; } |
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atom[1]{ type = "I"; } |
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bond{ members( 0, 1); } |
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} |
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molecule{ |
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name = "HCl" |
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atom[0]{ |
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type = "H"; |
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} |
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atom[1]{ |
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type = "Cl"; |
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} |
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bond{ |
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members( 0, 1); |
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} |
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atom[0]{ type = "H";} |
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atom[1]{ type = "Cl";} |
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bond{ members( 0, 1); } |
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} |
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component{ |
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type = "HCl"; |
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& (with separate barostats on each box dimension) & \\ |
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LD & Langevin Dynamics & {\tt ensemble = LD;} \\ |
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& (approximates the effects of an implicit solvent) & \\ |
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LangevinHull & Non-periodic Langevin Dynamics & {\tt ensemble = LangevinHull;} \\ |
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& (Langevin Dynamics for molecules on convex hull;\\ |
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& Newtonian for interior molecules) & \\ |
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\end{tabular} |
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\end{center} |
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|
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in the body-fixed frame) and ${\bf V}$ is a generalized velocity, |
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${\bf V} = |
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\left\{{\bf v},{\bf \omega}\right\}$. The right side of |
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Eq.~\ref{LDGeneralizedForm} consists of three generalized forces: a |
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Eq. \ref{LDGeneralizedForm} consists of three generalized forces: a |
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system force (${\bf F}_{s}$), a frictional or dissipative force (${\bf |
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F}_{f}$) and a stochastic force (${\bf F}_{r}$). While the evolution |
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of the system in Newtonian mechanics is typically done in the lab |
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\endhead |
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\hline |
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\endfoot |
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{\tt viscosity} & centipoise & Sets the value of viscosity of the implicit |
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{\tt viscosity} & poise & Sets the value of viscosity of the implicit |
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solvent \\ |
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{\tt targetTemp} & K & Sets the target temperature of the system. |
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This parameter must be specified to use Langevin dynamics. \\ |
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{\tt HydroPropFile} & string & Specifies the name of the resistance |
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tensor (usually a {\tt .diff} file) which is precalculated by {\tt |
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Hydro}. This keyworkd is not necessary if the simulation contains only |
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Hydro}. This keyword is not necessary if the simulation contains only |
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simple bodies (spheres and ellipsoids). \\ |
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{\tt beadSize} & $\mbox{\AA}$ & Sets the diameter of the beads to use |
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when the {\tt RoughShell} model is used to approximate the resistance |
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tensor. |
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\label{table:ldParameters} |
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\end{longtable} |
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|
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\section{Constant Pressure without periodic boundary conditions (The LangevinHull)} |
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|
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The Langevin Hull uses an external bath at a fixed constant pressure |
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($P$) and temperature ($T$) with an effective solvent viscosity |
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($\eta$). This bath interacts only with the objects on the exterior |
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hull of the system. Defining the hull of the atoms in a simulation is |
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done in a manner similar to the approach of Kohanoff, Caro and |
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Finnis.\cite{Kohanoff:2005qm} That is, any instantaneous configuration |
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of the atoms in the system is considered as a point cloud in three |
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dimensional space. Delaunay triangulation is used to find all facets |
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between coplanar |
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neighbors.\cite{delaunay,springerlink:10.1007/BF00977785} In highly |
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symmetric point clouds, facets can contain many atoms, but in all but |
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the most symmetric of cases, the facets are simple triangles in |
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3-space which contain exactly three atoms. |
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|
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The convex hull is the set of facets that have {\it no concave |
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corners} at an atomic site.\cite{Barber96,EDELSBRUNNER:1994oq} This |
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eliminates all facets on the interior of the point cloud, leaving only |
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those exposed to the bath. Sites on the convex hull are dynamic; as |
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molecules re-enter the cluster, all interactions between atoms on that |
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molecule and the external bath are removed. Since the edge is |
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determined dynamically as the simulation progresses, no {\it a priori} |
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geometry is defined. The pressure and temperature bath interacts only |
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with the atoms on the edge and not with atoms interior to the |
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simulation. |
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|
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Atomic sites in the interior of the simulation move under standard |
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Newtonian dynamics, |
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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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\label{eq:Newton} |
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\end{equation} |
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where $m_i$ is the mass of site $i$, ${\mathbf v}_i(t)$ is the |
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instantaneous velocity of site $i$ at time $t$, and $U$ is the total |
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potential energy. For atoms on the exterior of the cluster |
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(i.e. those that occupy one of the vertices of the convex hull), the |
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equation of motion is modified with an external force, ${\mathbf |
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F}_i^{\mathrm ext}$: |
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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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|
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The external bath interacts indirectly with the atomic sites through |
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the intermediary of the hull facets. Since each vertex (or atom) |
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provides one corner of a triangular facet, the force on the facets are |
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divided equally to each vertex. However, each vertex can participate |
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in multiple facets, so the resultant force is a sum over all facets |
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$f$ containing vertex $i$: |
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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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|
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The external pressure bath applies a force to the facets of the convex |
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hull in direct proportion to the area of the facet, while the thermal |
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coupling depends on the solvent temperature, viscosity and the size |
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and shape of each facet. The thermal interactions are expressed as a |
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standard Langevin description of the forces, |
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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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Here, $A_f$ and $\hat{n}_f$ are the area and (outward-facing) normal |
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vectors for facet $f$, respectively. ${\mathbf v}_f(t)$ is the |
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velocity of the facet centroid, |
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\begin{equation} |
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{\mathbf v}_f(t) = \frac{1}{3} \sum_{i=1}^{3} {\mathbf v}_i, |
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\end{equation} |
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and $\Xi_f(t)$ is an approximate ($3 \times 3$) resistance tensor that |
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depends on the geometry and surface area of facet $f$ and the |
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viscosity of the bath. The resistance tensor is related to the |
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fluctuations of the random force, $\mathbf{R}(t)$, by the |
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fluctuation-dissipation theorem (see Eq. \ref{eq:randomForce}). |
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|
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Once the resistance tensor is known for a given facet, a stochastic |
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vector that has the properties in Eq. (\ref{eq:randomForce}) can be |
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calculated efficiently by carrying out a Cholesky decomposition to |
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obtain the square root matrix of the resistance tensor (see |
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Eq. \ref{eq:Cholesky}). |
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|
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Our treatment of the resistance tensor for the Langevin Hull facets is |
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approximate. $\Xi_f$ for a rigid triangular plate would normally be |
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treated as a $6 \times 6$ tensor that includes translational and |
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rotational drag as well as translational-rotational coupling. The |
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computation of resistance tensors for rigid bodies has been detailed |
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elsewhere,\cite{JoseGarciadelaTorre02012000,Garcia-de-la-Torre:2001wd,GarciadelaTorreJ2002,Sun:2008fk} |
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but the standard approach involving bead approximations would be |
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prohibitively expensive if it were recomputed at each step in a |
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molecular dynamics simulation. |
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|
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Instead, we are utilizing an approximate resistance tensor obtained by |
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first constructing the Oseen tensor for the interaction of the |
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centroid of the facet ($f$) with each of the subfacets $\ell=1,2,3$, |
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\begin{equation} |
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T_{\ell f}=\frac{A_\ell}{8\pi\eta R_{\ell f}}\left(I + |
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\frac{\mathbf{R}_{\ell f}\mathbf{R}_{\ell f}^T}{R_{\ell f}^2}\right) |
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\end{equation} |
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Here, $A_\ell$ is the area of subfacet $\ell$ which is a triangle |
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containing two of the vertices of the facet along with the centroid. |
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$\mathbf{R}_{\ell f}$ is the vector between the centroid of facet $f$ |
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and the centroid of sub-facet $\ell$, and $I$ is the ($3 \times 3$) |
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identity matrix. $\eta$ is the viscosity of the external bath. |
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|
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The tensors for each of the sub-facets are added together, and the |
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resulting matrix is inverted to give a $3 \times 3$ resistance tensor |
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for translations of the triangular facet, |
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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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Note that this treatment ignores rotations (and |
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translational-rotational coupling) of the facet. In compact systems, |
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the facets stay relatively fixed in orientation between |
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configurations, so this appears to be a reasonably good approximation. |
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|
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At each |
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molecular dynamics time step, the following process is carried out: |
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\begin{enumerate} |
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\item The standard inter-atomic forces ($\nabla_iU$) are computed. |
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\item Delaunay triangulation is carried out using the current atomic |
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configuration. |
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\item The convex hull is computed and facets are identified. |
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\item For each facet: |
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\begin{itemize} |
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\item[a.] The force from the pressure bath ($-\hat{n}_fPA_f$) is |
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computed. |
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\item[b.] The resistance tensor ($\Xi_f(t)$) is computed using the |
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viscosity ($\eta$) of the bath. |
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\item[c.] Facet drag ($-\Xi_f(t) \mathbf{v}_f(t)$) forces are |
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computed. |
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\item[d.] Random forces ($\mathbf{R}_f(t)$) are computed using the |
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resistance tensor and the temperature ($T$) of the bath. |
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\end{itemize} |
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\item The facet forces are divided equally among the vertex atoms. |
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\item Atomic positions and velocities are propagated. |
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\end{enumerate} |
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The Delaunay triangulation and computation of the convex hull are done |
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using calls to the qhull library,\cite{Qhull} and for this reason, if |
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qhull is not detected during the build, the Langevin Hull integrator |
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will not be available. There is a minimal penalty for computing the |
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convex hull and resistance tensors at each step in the molecular |
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dynamics simulation (roughly 0.02 $\times$ cost of a single force |
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evaluation). |
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|
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\begin{longtable}[c]{GBF} |
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\caption{Meta-data Keywords: Required parameters for the Langevin Hull integrator} |
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\\ |
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{\bf keyword} & {\bf units} & {\bf use} \\ \hline |
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\endhead |
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\hline |
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\endfoot |
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{\tt viscosity} & poise & Sets the value of viscosity of the implicit |
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solven . \\ |
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{\tt targetTemp} & K & Sets the target temperature of the system. |
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This parameter must be specified to use Langevin Hull dynamics. \\ |
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{\tt targetPressure} & atm & Sets the target pressure of the system. |
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This parameter must be specified to use Langevin Hull dynamics. \\ |
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{\tt usePeriodicBoundaryConditions} & logical & Turns off periodic boundary conditions. |
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This parameter must be set to \tt false \\ |
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\label{table:lhullParameters} |
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\end{longtable} |
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|
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|
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\section{\label{sec:constraints}Constraint Methods} |
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|
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\subsection{\label{section:rattle}The {\sc rattle} Method for Bond |
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\end{center} |
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|
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For example, the phrase {\tt select mass > 16.0 and charge < -2} |
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wouldselect StuntDoubles which have mass greater than 16.0 and charges |
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would select StuntDoubles which have mass greater than 16.0 and charges |
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less than -2. |
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|
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\subsection{\label{section:within}Within expressions} |
