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\setlength{\belowcaptionskip}{30 pt} |
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\bibpunct{[}{]}{,}{s}{}{;} |
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\bibliographystyle{aip} |
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\bibliographystyle{achemso} |
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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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|
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\author{Charles F. Varedeman II, Kelsey Stocker, and J. Daniel |
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\author{Charles F. Vardeman II, Kelsey M. 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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\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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hull surrounding the system. A Langevin thermostat is also applied |
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to the facets to mimic contact with an external heat bath. This new |
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method, the ``Langevin Hull'', can handle heterogeneous mixtures of |
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materials with different compressibilities. These are systems that |
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are problematic for traditional affine transform methods. The |
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Langevin Hull does not suffer from the edge effects of boundary |
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potential methods, and allows realistic treatment of both external |
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pressure and thermal conductivity due to the presence of an implicit |
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solvent. We apply this method to several different systems |
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including bare metal nanoparticles, nanoparticles in an explicit |
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solvent, as well as clusters of liquid water. The predicted |
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mechanical properties of these systems are in good agreement with |
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experimental data and previous simulation work. |
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\end{abstract} |
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|
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\newpage |
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|
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\section{Introduction} |
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|
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Affine transform methods |
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The most common molecular dynamics methods for sampling configurations |
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from an isobaric-isothermal (NPT) ensemble maintain a target pressure |
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in a simulation by coupling the volume of the system to a {\it |
| 71 |
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barostat}, which is an extra degree of freedom propagated along with |
| 72 |
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the particle coordinates. These methods require periodic boundary |
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conditions, because when the instantaneous pressure in the system |
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> |
differs from the target pressure, the volume is reduced or expanded |
| 75 |
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using {\it affine transforms} of the system geometry. An affine |
| 76 |
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transform scales the size and shape of the periodic box as well as the |
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particle positions within the box (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{Melchionna1993} to the |
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Nos\'e-Hoover-Andersen equations of |
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motion,\cite{Hoover85,ANDERSEN:1980vn,Sturgeon:2000kx} the Berendsen |
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pressure bath,\cite{ISI:A1984TQ73500045} and the Langevin |
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Piston,\cite{FELLER:1995fk,Jakobsen:2005uq} all utilize scaled |
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coordinate transformation to adjust the box volume. As long as the |
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material in the simulation box has a relatively uniform |
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compressibility, the standard affine transform approach provides an |
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excellent way of adjusting the volume of the system and applying |
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pressure directly via the interactions between atomic sites. |
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|
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\begin{figure} |
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\includegraphics[width=\linewidth]{AffineScale} |
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\caption{Affine Scale} |
| 93 |
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\label{affineScale} |
| 94 |
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\end{figure} |
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One problem with this approach appears when the system being simulated |
| 91 |
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is an inhomogeneous mixture in which portions of the simulation box |
| 92 |
> |
are incompressible relative to other portions. Examples include |
| 93 |
> |
simulations of metallic nanoparticles in liquid environments, proteins |
| 94 |
> |
at ice / water interfaces, as well as other heterogeneous or |
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interfacial environments. In these cases, the affine transform of |
| 96 |
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atomic coordinates will either cause numerical instability when the |
| 97 |
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sites in the incompressible medium collide with each other, or will |
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lead to inefficient sampling of system volumes if the barostat is set |
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slow enough to avoid the instabilities in the incompressible region. |
| 100 |
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|
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– |
|
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\begin{figure} |
| 102 |
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\includegraphics[width=\linewidth]{AffineScale2} |
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\caption{Affine Scale2} |
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\label{affineScale2} |
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\caption{Affine scaling methods use box-length scaling to adjust the |
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volume to adjust to under- or over-pressure conditions. In a system |
| 105 |
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with a uniform compressibility (e.g. bulk fluids) these methods can |
| 106 |
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work well. In systems containing heterogeneous mixtures, the affine |
| 107 |
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scaling moves required to adjust the pressure in the |
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high-compressibility regions can cause molecules in low |
| 109 |
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compressibility regions to collide.} |
| 110 |
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\label{affineScale} |
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\end{figure} |
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|
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Heterogeneous mixtures of materials with different compressibilities? |
| 113 |
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One may also wish to avoid affine transform periodic boundary methods |
| 114 |
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to simulate {\it explicitly non-periodic systems} under constant |
| 115 |
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pressure conditions. The use of periodic boxes to enforce a system |
| 116 |
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volume requires either effective solute concentrations that are much |
| 117 |
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higher than desirable, or unreasonable system sizes to avoid this |
| 118 |
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effect. For example, calculations using typical hydration boxes |
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solvating a protein under periodic boundary conditions are quite |
| 120 |
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expensive. A 62 $\AA^3$ box of water solvating a moderately small |
| 121 |
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protein like hen egg white lysozyme (PDB code: 1LYZ) yields an |
| 122 |
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effective protein concentration of 100 mg/mL.\cite{Asthagiri20053300} |
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|
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Explicitly non-periodic systems |
| 124 |
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Typically protein concentrations in the cell are on the order of |
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160-310 mg/ml,\cite{Brown1991195} and the factor of 20 difference |
| 126 |
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between simulations and the cellular environment may have significant |
| 127 |
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effects on the structure and dynamics of simulated protein structures. |
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|
| 87 |
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Elastic Bag |
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|
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Spherical Boundary approaches |
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\subsection*{Boundary Methods} |
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There have been a number of approaches to handle simulations of |
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explicitly non-periodic systems that focus on constant or |
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nearly-constant {\it volume} conditions while maintaining bulk-like |
| 134 |
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behavior. Berkowitz and McCammon introduced a stochastic (Langevin) |
| 135 |
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boundary layer inside a region of fixed molecules which effectively |
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enforces constant temperature and volume (NVT) |
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conditions.\cite{Berkowitz1982} In this approach, the stochastic and |
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fixed regions were defined relative to a central atom. Brooks and |
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Karplus extended this method to include deformable stochastic |
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boundaries.\cite{iii:6312} The stochastic boundary approach has been |
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used widely for protein simulations. [CITATIONS NEEDED] |
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|
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\section{Methodology} |
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The electrostatic and dispersive behavior near the boundary has long |
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been a cause for concern when performing simulations of explicitly |
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non-periodic systems. Early work led to the surface constrained soft |
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sphere dipole model (SCSSD)\cite{Warshel1978} in which the surface |
| 147 |
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molecules are fixed in a random orientation representative of the bulk |
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solvent structural properties. Belch {\it et al.}\cite{Belch1985} |
| 149 |
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simulated clusters of TIPS2 water surrounded by a hydrophobic bounding |
| 150 |
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potential. The spherical hydrophobic boundary induced dangling |
| 151 |
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hydrogen bonds at the surface that propagated deep into the cluster, |
| 152 |
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affecting most of molecules in the simulation. This result echoes an |
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earlier study which showed that an extended planar hydrophobic surface |
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caused orientational preference at the surface which extended |
| 155 |
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relatively deep (7 \r{A}) into the liquid simulation |
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cell.\cite{Lee1984} The surface constrained all-atom solvent (SCAAS) |
| 157 |
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model \cite{King1989} improved upon its SCSSD predecessor. The SCAAS |
| 158 |
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model utilizes a polarization constraint which is applied to the |
| 159 |
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surface molecules to maintain bulk-like structure at the cluster |
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surface. A radial constraint is used to maintain the desired bulk |
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density of the liquid. Both constraint forces are applied only to a |
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pre-determined number of the outermost molecules. |
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|
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A new method which uses a constant pressure and temperature bath that |
| 165 |
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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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Beglov and Roux have developed a boundary model in which the hard |
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sphere boundary has a radius that varies with the instantaneous |
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configuration of the solute (and solvent) molecules.\cite{beglov:9050} |
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This model contains a clear pressure and surface tension contribution |
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to the free energy which XXX. |
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|
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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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\subsection*{Restraining Potentials} |
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Restraining {\it potentials} introduce repulsive potentials at the |
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surface of a sphere or other geometry. The solute and any explicit |
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solvent are therefore restrained inside the range defined by the |
| 174 |
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external potential. Often the potentials include a weak short-range |
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attraction to maintain the correct density at the boundary. Beglov |
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and Roux have also introduced a restraining boundary potential which |
| 177 |
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relaxes dynamically depending on the solute geometry and the force the |
| 178 |
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explicit system exerts on the shell.\cite{Beglov:1995fk} |
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|
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Simulation starts as a collection of atomic locations in 3D (a point |
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cloud). |
| 180 |
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Recently, Krilov {\it et al.} introduced a {\it flexible} boundary |
| 181 |
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model that uses a Lennard-Jones potential between the solvent |
| 182 |
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molecules and a boundary which is determined dynamically from the |
| 183 |
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position of the nearest solute atom.\cite{LiY._jp046852t,Zhu:xw} This |
| 184 |
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approach allows the confining potential to prevent solvent molecules |
| 185 |
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from migrating too far from the solute surface, while providing a weak |
| 186 |
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attractive force pulling the solvent molecules towards a fictitious |
| 187 |
> |
bulk solvent. Although this approach is appealing and has physical |
| 188 |
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motivation, nanoparticles do not deform far from their original |
| 189 |
> |
geometries even at temperatures which vaporize the nearby solvent. For |
| 190 |
> |
the systems like this, the flexible boundary model will be nearly |
| 191 |
> |
identical to a fixed-volume restraining potential. |
| 192 |
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|
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< |
Delaunay triangulation finds all facets between coplanar neighbors. |
| 193 |
> |
\subsection*{Hull methods} |
| 194 |
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The approach of Kohanoff, Caro, and Finnis is the most promising of |
| 195 |
> |
the methods for introducing both constant pressure and temperature |
| 196 |
> |
into non-periodic simulations.\cite{Kohanoff:2005qm,Baltazar:2006ru} |
| 197 |
> |
This method is based on standard Langevin dynamics, but the Brownian |
| 198 |
> |
or random forces are allowed to act only on peripheral atoms and exert |
| 199 |
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force in a direction that is inward-facing relative to the facets of a |
| 200 |
> |
closed bounding surface. The statistical distribution of the random |
| 201 |
> |
forces are uniquely tied to the pressure in the external reservoir, so |
| 202 |
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the method can be shown to sample the isobaric-isothermal ensemble. |
| 203 |
> |
Kohanoff {\it et al.} used a Delaunay tessellation to generate a |
| 204 |
> |
bounding surface surrounding the outermost atoms in the simulated |
| 205 |
> |
system. This is not the only possible triangulated outer surface, but |
| 206 |
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guarantees that all of the random forces point inward towards the |
| 207 |
> |
cluster. |
| 208 |
|
|
| 209 |
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The Convex Hull is the set of facets that have no concave corners at a |
| 210 |
< |
vertex. |
| 209 |
> |
In the following sections, we extend and generalize the approach of |
| 210 |
> |
Kohanoff, Caro, and Finnis. The new method, which we are calling the |
| 211 |
> |
``Langevin Hull'' applies the external pressure, Langevin drag, and |
| 212 |
> |
random forces on the {\it facets of the hull} instead of the atomic |
| 213 |
> |
sites comprising the vertices of the hull. This allows us to decouple |
| 214 |
> |
the external pressure contribution from the drag and random force. |
| 215 |
> |
The methodology is introduced in section \ref{sec:meth}, tests on |
| 216 |
> |
crystalline nanoparticles, liquid clusters, and heterogeneous mixtures |
| 217 |
> |
are detailed in section \ref{sec:tests}. Section \ref{sec:discussion} |
| 218 |
> |
summarizes our findings. |
| 219 |
|
|
| 220 |
< |
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 |
| 110 |
< |
external bath applies pressure to the facets of the convex hull in |
| 111 |
< |
direct proportion to the area of the facet.Thermal coupling depends on |
| 112 |
< |
the solvent temperature, friction and the size and shape of each |
| 113 |
< |
facet. |
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> |
\section{Methodology} |
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\label{sec:meth} |
| 222 |
|
|
| 223 |
+ |
The Langevin Hull uses an external bath at a fixed constant pressure |
| 224 |
+ |
($P$) and temperature ($T$). This bath interacts only with the |
| 225 |
+ |
objects on the exterior hull of the system. Defining the hull of the |
| 226 |
+ |
simulation is done in a manner similar to the approach of Kohanoff, |
| 227 |
+ |
Caro and Finnis.\cite{Kohanoff:2005qm} That is, any instantaneous |
| 228 |
+ |
configuration of the atoms in the system is considered as a point |
| 229 |
+ |
cloud in three dimensional space. Delaunay triangulation is used to |
| 230 |
+ |
find all facets between coplanar |
| 231 |
+ |
neighbors.\cite{delaunay,springerlink:10.1007/BF00977785} In highly |
| 232 |
+ |
symmetric point clouds, facets can contain many atoms, but in all but |
| 233 |
+ |
the most symmetric of cases the facets are simple triangles in 3-space |
| 234 |
+ |
that contain exactly three atoms. |
| 235 |
+ |
|
| 236 |
+ |
The convex hull is the set of facets that have {\it no concave |
| 237 |
+ |
corners} at an atomic site.\cite{Barber96,EDELSBRUNNER:1994oq} This |
| 238 |
+ |
eliminates all facets on the interior of the point cloud, leaving only |
| 239 |
+ |
those exposed to the bath. Sites on the convex hull are dynamic; as |
| 240 |
+ |
molecules re-enter the cluster, all interactions between atoms on that |
| 241 |
+ |
molecule and the external bath are removed. Since the edge is |
| 242 |
+ |
determined dynamically as the simulation progresses, no {\it a priori} |
| 243 |
+ |
geometry is defined. The pressure and temperature bath interacts only |
| 244 |
+ |
with the atoms on the edge and not with atoms interior to the |
| 245 |
+ |
simulation. |
| 246 |
+ |
|
| 247 |
+ |
\begin{figure} |
| 248 |
+ |
\includegraphics[width=\linewidth]{hullSample} |
| 249 |
+ |
\caption{The external temperature and pressure bath interacts only |
| 250 |
+ |
with those atoms on the convex hull (grey surface). The hull is |
| 251 |
+ |
computed dynamically at each time step, and molecules can move |
| 252 |
+ |
between the interior (Newtonian) region and the Langevin hull.} |
| 253 |
+ |
\label{fig:hullSample} |
| 254 |
+ |
\end{figure} |
| 255 |
+ |
|
| 256 |
+ |
Atomic sites in the interior of the simulation move under standard |
| 257 |
+ |
Newtonian dynamics, |
| 258 |
|
\begin{equation} |
| 259 |
< |
m_i \dot{\mathbf v}_i(t)=-{\mathbf \nabla}_i U |
| 259 |
> |
m_i \dot{\mathbf v}_i(t)=-{\mathbf \nabla}_i U, |
| 260 |
> |
\label{eq:Newton} |
| 261 |
|
\end{equation} |
| 262 |
< |
|
| 262 |
> |
where $m_i$ is the mass of site $i$, ${\mathbf v}_i(t)$ is the |
| 263 |
> |
instantaneous velocity of site $i$ at time $t$, and $U$ is the total |
| 264 |
> |
potential energy. For atoms on the exterior of the cluster |
| 265 |
> |
(i.e. those that occupy one of the vertices of the convex hull), the |
| 266 |
> |
equation of motion is modified with an external force, ${\mathbf |
| 267 |
> |
F}_i^{\mathrm ext}$, |
| 268 |
|
\begin{equation} |
| 269 |
< |
m_i \dot{\mathbf v}_i(t)=-{\mathbf \nabla}_i U + {\mathbf F}_i^{\mathrm ext} |
| 269 |
> |
m_i \dot{\mathbf v}_i(t)=-{\mathbf \nabla}_i U + {\mathbf F}_i^{\mathrm ext}. |
| 270 |
|
\end{equation} |
| 271 |
|
|
| 272 |
+ |
The external bath interacts indirectly with the atomic sites through |
| 273 |
+ |
the intermediary of the hull facets. Since each vertex (or atom) |
| 274 |
+ |
provides one corner of a triangular facet, the force on the facets are |
| 275 |
+ |
divided equally to each vertex. However, each vertex can participate |
| 276 |
+ |
in multiple facets, so the resultant force is a sum over all facets |
| 277 |
+ |
$f$ containing vertex $i$: |
| 278 |
|
\begin{equation} |
| 279 |
|
{\mathbf F}_{i}^{\mathrm ext} = \sum_{\begin{array}{c}\mathrm{facets\ |
| 280 |
|
} f \\ \mathrm{containing\ } i\end{array}} \frac{1}{3}\ {\mathbf |
| 281 |
|
F}_f^{\mathrm ext} |
| 282 |
|
\end{equation} |
| 283 |
|
|
| 284 |
+ |
The external pressure bath applies a force to the facets of the convex |
| 285 |
+ |
hull in direct proportion to the area of the facet, while the thermal |
| 286 |
+ |
coupling depends on the solvent temperature, viscosity and the size |
| 287 |
+ |
and shape of each facet. The thermal interactions are expressed as a |
| 288 |
+ |
standard Langevin description of the forces, |
| 289 |
|
\begin{equation} |
| 290 |
|
\begin{array}{rclclcl} |
| 291 |
|
{\mathbf F}_f^{\text{ext}} & = & \text{external pressure} & + & \text{drag force} & + & \text{random force} \\ |
| 292 |
|
& = & -\hat{n}_f P A_f & - & \Xi_f(t) {\mathbf v}_f(t) & + & {\mathbf R}_f(t) |
| 293 |
|
\end{array} |
| 294 |
|
\end{equation} |
| 295 |
< |
|
| 295 |
> |
Here, $A_f$ and $\hat{n}_f$ are the area and (outward-facing) normal |
| 296 |
> |
vectors for facet $f$, respectively. ${\mathbf v}_f(t)$ is the |
| 297 |
> |
velocity of the facet centroid, |
| 298 |
> |
\begin{equation} |
| 299 |
> |
{\mathbf v}_f(t) = \frac{1}{3} \sum_{i=1}^{3} {\mathbf v}_i, |
| 300 |
> |
\end{equation} |
| 301 |
> |
and $\Xi_f(t)$ is an approximate ($3 \times 3$) resistance tensor that |
| 302 |
> |
depends on the geometry and surface area of facet $f$ and the |
| 303 |
> |
viscosity of the fluid. The resistance tensor is related to the |
| 304 |
> |
fluctuations of the random force, $\mathbf{R}(t)$, by the |
| 305 |
> |
fluctuation-dissipation theorem, |
| 306 |
|
\begin{eqnarray} |
| 137 |
– |
A_f & = & \text{area of facet}\ f \\ |
| 138 |
– |
\hat{n}_f & = & \text{facet normal} \\ |
| 139 |
– |
P & = & \text{external pressure} |
| 140 |
– |
\end{eqnarray} |
| 141 |
– |
|
| 142 |
– |
\begin{eqnarray} |
| 143 |
– |
{\mathbf v}_f(t) & = & \text{velocity of facet} \\ |
| 144 |
– |
& = & \frac{1}{3} \sum_{i=1}^{3} {\mathbf v}_i \\ |
| 145 |
– |
\Xi_f(t) & = & \text{is a hydrodynamic tensor that depends} \\ |
| 146 |
– |
& & \text{on the geometry and surface area of} \\ |
| 147 |
– |
& & \text{facet} \ f\ \text{and the viscosity of the fluid.} |
| 148 |
– |
\end{eqnarray} |
| 149 |
– |
|
| 150 |
– |
\begin{eqnarray} |
| 307 |
|
\left< {\mathbf R}_f(t) \right> & = & 0 \\ |
| 308 |
|
\left<{\mathbf R}_f(t) {\mathbf R}_f^T(t^\prime)\right> & = & 2 k_B T\ |
| 309 |
< |
\Xi_f(t)\delta(t-t^\prime) |
| 309 |
> |
\Xi_f(t)\delta(t-t^\prime). |
| 310 |
> |
\label{eq:randomForce} |
| 311 |
|
\end{eqnarray} |
| 312 |
|
|
| 313 |
< |
Implemented in OpenMD.\cite{Meineke:2005gd,openmd} |
| 313 |
> |
Once the resistance tensor is known for a given facet, a stochastic |
| 314 |
> |
vector that has the properties in Eq. (\ref{eq:randomForce}) can be |
| 315 |
> |
calculated efficiently by carrying out a Cholesky decomposition to |
| 316 |
> |
obtain the square root matrix of the resistance tensor, |
| 317 |
> |
\begin{equation} |
| 318 |
> |
\Xi_f = {\bf S} {\bf S}^{T}, |
| 319 |
> |
\label{eq:Cholesky} |
| 320 |
> |
\end{equation} |
| 321 |
> |
where ${\bf S}$ is a lower triangular matrix.\cite{Schlick2002} A |
| 322 |
> |
vector with the statistics required for the random force can then be |
| 323 |
> |
obtained by multiplying ${\bf S}$ onto a random 3-vector ${\bf Z}$ which |
| 324 |
> |
has elements chosen from a Gaussian distribution, such that: |
| 325 |
> |
\begin{equation} |
| 326 |
> |
\langle {\bf Z}_i \rangle = 0, \hspace{1in} \langle {\bf Z}_i \cdot |
| 327 |
> |
{\bf Z}_j \rangle = \frac{2 k_B T}{\delta t} \delta_{ij}, |
| 328 |
> |
\end{equation} |
| 329 |
> |
where $\delta t$ is the timestep in use during the simulation. The |
| 330 |
> |
random force, ${\bf R}_{f} = {\bf S} {\bf Z}$, can be shown to |
| 331 |
> |
have the correct properties required by Eq. (\ref{eq:randomForce}). |
| 332 |
|
|
| 333 |
< |
\section{Tests \& Applications} |
| 333 |
> |
Our treatment of the resistance tensor is approximate. $\Xi$ for a |
| 334 |
> |
rigid triangular plate would normally be treated as a $6 \times 6$ |
| 335 |
> |
tensor that includes translational and rotational drag as well as |
| 336 |
> |
translational-rotational coupling. The computation of resistance |
| 337 |
> |
tensors for rigid bodies has been detailed |
| 338 |
> |
elsewhere,\cite{JoseGarciadelaTorre02012000,Garcia-de-la-Torre:2001wd,GarciadelaTorreJ2002,Sun:2008fk} |
| 339 |
> |
but the standard approach involving bead approximations would be |
| 340 |
> |
prohibitively expensive if it were recomputed at each step in a |
| 341 |
> |
molecular dynamics simulation. |
| 342 |
|
|
| 343 |
< |
\subsection{Bulk modulus of gold nanoparticles} |
| 343 |
> |
Instead, we are utilizing an approximate resistance tensor obtained by |
| 344 |
> |
first constructing the Oseen tensor for the interaction of the |
| 345 |
> |
centroid of the facet ($f$) with each of the subfacets $\ell=1,2,3$, |
| 346 |
> |
\begin{equation} |
| 347 |
> |
T_{\ell f}=\frac{A_\ell}{8\pi\eta R_{\ell f}}\left(I + |
| 348 |
> |
\frac{\mathbf{R}_{\ell f}\mathbf{R}_{\ell f}^T}{R_{\ell f}^2}\right) |
| 349 |
> |
\end{equation} |
| 350 |
> |
Here, $A_\ell$ is the area of subfacet $\ell$ which is a triangle |
| 351 |
> |
containing two of the vertices of the facet along with the centroid. |
| 352 |
> |
$\mathbf{R}_{\ell f}$ is the vector between the centroid of facet $f$ |
| 353 |
> |
and the centroid of sub-facet $\ell$, and $I$ is the ($3 \times 3$) |
| 354 |
> |
identity matrix. $\eta$ is the viscosity of the external bath. |
| 355 |
|
|
| 356 |
|
\begin{figure} |
| 357 |
< |
\includegraphics[width=\linewidth]{pressure_tb} |
| 358 |
< |
\caption{Pressure response is rapid (18 \AA gold nanoparticle), target |
| 359 |
< |
pressure = 4 GPa} |
| 360 |
< |
\label{pressureResponse} |
| 357 |
> |
\includegraphics[width=\linewidth]{hydro} |
| 358 |
> |
\caption{The resistance tensor $\Xi$ for a facet comprising sites $i$, |
| 359 |
> |
$j$, and $k$ is constructed using Oseen tensor contributions between |
| 360 |
> |
the centoid of the facet $f$ and each of the sub-facets ($i,f,j$), |
| 361 |
> |
($j,f,k$), and ($k,f,i$). The centroids of the sub-facets are |
| 362 |
> |
located at $1$, $2$, and $3$, and the area of each sub-facet is |
| 363 |
> |
easily computed using half the cross product of two of the edges.} |
| 364 |
> |
\label{hydro} |
| 365 |
|
\end{figure} |
| 366 |
+ |
|
| 367 |
+ |
The tensors for each of the sub-facets are added together, and the |
| 368 |
+ |
resulting matrix is inverted to give a $3 \times 3$ resistance tensor |
| 369 |
+ |
for translations of the triangular facet, |
| 370 |
+ |
\begin{equation} |
| 371 |
+ |
\Xi_f(t) =\left[\sum_{i=1}^3 T_{if}\right]^{-1}. |
| 372 |
+ |
\end{equation} |
| 373 |
+ |
Note that this treatment ignores rotations (and |
| 374 |
+ |
translational-rotational coupling) of the facet. In compact systems, |
| 375 |
+ |
the facets stay relatively fixed in orientation between |
| 376 |
+ |
configurations, so this appears to be a reasonably good approximation. |
| 377 |
+ |
|
| 378 |
+ |
We have implemented this method by extending the Langevin dynamics |
| 379 |
+ |
integrator in our code, OpenMD.\cite{Meineke2005,openmd} At each |
| 380 |
+ |
molecular dynamics time step, the following process is carried out: |
| 381 |
+ |
\begin{enumerate} |
| 382 |
+ |
\item The standard inter-atomic forces ($\nabla_iU$) are computed. |
| 383 |
+ |
\item Delaunay triangulation is carried out using the current atomic |
| 384 |
+ |
configuration. |
| 385 |
+ |
\item The convex hull is computed and facets are identified. |
| 386 |
+ |
\item For each facet: |
| 387 |
+ |
\begin{itemize} |
| 388 |
+ |
\item[a.] The force from the pressure bath ($-PA_f\hat{n}_f$) is |
| 389 |
+ |
computed. |
| 390 |
+ |
\item[b.] The resistance tensor ($\Xi_f(t)$) is computed using the |
| 391 |
+ |
viscosity ($\eta$) of the bath. |
| 392 |
+ |
\item[c.] Facet drag ($-\Xi_f(t) \mathbf{v}_f(t)$) forces are |
| 393 |
+ |
computed. |
| 394 |
+ |
\item[d.] Random forces ($\mathbf{R}_f(t)$) are computed using the |
| 395 |
+ |
resistance tensor and the temperature ($T$) of the bath. |
| 396 |
+ |
\end{itemize} |
| 397 |
+ |
\item The facet forces are divided equally among the vertex atoms. |
| 398 |
+ |
\item Atomic positions and velocities are propagated. |
| 399 |
+ |
\end{enumerate} |
| 400 |
+ |
The Delaunay triangulation and computation of the convex hull are done |
| 401 |
+ |
using calls to the qhull library.\cite{Qhull} There is a minimal |
| 402 |
+ |
penalty for computing the convex hull and resistance tensors at each |
| 403 |
+ |
step in the molecular dynamics simulation (roughly 0.02 $\times$ cost |
| 404 |
+ |
of a single force evaluation), and the convex hull is remarkably easy |
| 405 |
+ |
to parallelize on distributed memory machines (see Appendix A). |
| 406 |
|
|
| 407 |
+ |
\section{Tests \& Applications} |
| 408 |
+ |
\label{sec:tests} |
| 409 |
+ |
|
| 410 |
+ |
To test the new method, we have carried out simulations using the |
| 411 |
+ |
Langevin Hull on: 1) a crystalline system (gold nanoparticles), 2) a |
| 412 |
+ |
liquid droplet (SPC/E water),\cite{Berendsen1987} and 3) a |
| 413 |
+ |
heterogeneous mixture (gold nanoparticles in a water droplet). In each |
| 414 |
+ |
case, we have computed properties that depend on the external applied |
| 415 |
+ |
pressure. Of particular interest for the single-phase systems is the |
| 416 |
+ |
isothermal compressibility, |
| 417 |
+ |
\begin{equation} |
| 418 |
+ |
\kappa_{T} = -\frac{1}{V} \left ( \frac{\partial V}{\partial P} \right |
| 419 |
+ |
)_{T}. |
| 420 |
+ |
\label{eq:BM} |
| 421 |
+ |
\end{equation} |
| 422 |
+ |
|
| 423 |
+ |
One problem with eliminating periodic boundary conditions and |
| 424 |
+ |
simulation boxes is that the volume of a three-dimensional point cloud |
| 425 |
+ |
is not well-defined. In order to compute the compressibility of a |
| 426 |
+ |
bulk material, we make an assumption that the number density, $\rho = |
| 427 |
+ |
\frac{N}{V}$, is uniform within some region of the point cloud. The |
| 428 |
+ |
compressibility can then be expressed in terms of the average number |
| 429 |
+ |
of particles in that region, |
| 430 |
+ |
\begin{equation} |
| 431 |
+ |
\kappa_{T} = -\frac{1}{N} \left ( \frac{\partial N}{\partial P} \right |
| 432 |
+ |
)_{T} |
| 433 |
+ |
\label{eq:BMN} |
| 434 |
+ |
\end{equation} |
| 435 |
+ |
The region we used is a spherical volume of 10 \AA\ radius centered in |
| 436 |
+ |
the middle of the cluster. $N$ is the average number of molecules |
| 437 |
+ |
found within this region throughout a given simulation. The geometry |
| 438 |
+ |
and size of the region is arbitrary, and any bulk-like portion of the |
| 439 |
+ |
cluster can be used to compute the compressibility. |
| 440 |
+ |
|
| 441 |
+ |
One might assume that the volume of the convex hull could simply be |
| 442 |
+ |
taken as the system volume $V$ in the compressibility expression |
| 443 |
+ |
(Eq. \ref{eq:BM}), but this has implications at lower pressures (which |
| 444 |
+ |
are explored in detail in the section on water droplets). |
| 445 |
+ |
|
| 446 |
+ |
The metallic force field in use for the gold nanoparticles is the |
| 447 |
+ |
quantum Sutton-Chen (QSC) model.\cite{PhysRevB.59.3527} In all |
| 448 |
+ |
simulations involving point charges, we utilized damped shifted-force |
| 449 |
+ |
(DSF) electrostatics\cite{Fennell06} which is a variant of the Wolf |
| 450 |
+ |
summation\cite{wolf:8254} that has been shown to provide good forces |
| 451 |
+ |
and torques on molecular models for water in a computationally |
| 452 |
+ |
efficient manner.\cite{Fennell06} The damping parameter ($\alpha$) was |
| 453 |
+ |
set to 0.18 \AA$^{-1}$, and the cutoff radius was set to 12 \AA. The |
| 454 |
+ |
Spohr potential was adopted in depicting the interaction between metal |
| 455 |
+ |
atoms and the SPC/E water molecules.\cite{ISI:000167766600035} |
| 456 |
+ |
|
| 457 |
+ |
\subsection{Compressibility of gold nanoparticles} |
| 458 |
+ |
|
| 459 |
+ |
The compressibility (and its inverse, the bulk modulus) is well-known |
| 460 |
+ |
for gold, and is captured well by the embedded atom method |
| 461 |
+ |
(EAM)~\cite{PhysRevB.33.7983} potential |
| 462 |
+ |
and related multi-body force fields. In particular, the quantum |
| 463 |
+ |
Sutton-Chen potential gets nearly quantitative agreement with the |
| 464 |
+ |
experimental bulk modulus values, and makes a good first test of how |
| 465 |
+ |
the Langevin Hull will perform at large applied pressures. |
| 466 |
+ |
|
| 467 |
+ |
The Sutton-Chen (SC) potentials are based on a model of a metal which |
| 468 |
+ |
treats the nuclei and core electrons as pseudo-atoms embedded in the |
| 469 |
+ |
electron density due to the valence electrons on all of the other |
| 470 |
+ |
atoms in the system.\cite{Chen90} The SC potential has a simple form that closely |
| 471 |
+ |
resembles the Lennard Jones potential, |
| 472 |
+ |
\begin{equation} |
| 473 |
+ |
\label{eq:SCP1} |
| 474 |
+ |
U_{tot}=\sum _{i}\left[ \frac{1}{2}\sum _{j\neq i}D_{ij}V^{pair}_{ij}(r_{ij})-c_{i}D_{ii}\sqrt{\rho_{i}}\right] , |
| 475 |
+ |
\end{equation} |
| 476 |
+ |
where $V^{pair}_{ij}$ and $\rho_{i}$ are given by |
| 477 |
+ |
\begin{equation} |
| 478 |
+ |
\label{eq:SCP2} |
| 479 |
+ |
V^{pair}_{ij}(r)=\left( \frac{\alpha_{ij}}{r_{ij}}\right)^{n_{ij}}, \rho_{i}=\sum_{j\neq i}\left( \frac{\alpha_{ij}}{r_{ij}}\right) ^{m_{ij}}. |
| 480 |
+ |
\end{equation} |
| 481 |
+ |
$V^{pair}_{ij}$ is a repulsive pairwise potential that accounts for |
| 482 |
+ |
interactions between the pseudoatom cores. The $\sqrt{\rho_i}$ term in |
| 483 |
+ |
Eq. (\ref{eq:SCP1}) is an attractive many-body potential that models |
| 484 |
+ |
the interactions between the valence electrons and the cores of the |
| 485 |
+ |
pseudo-atoms. $D_{ij}$, $D_{ii}$ set the appropriate overall energy |
| 486 |
+ |
scale, $c_i$ scales the attractive portion of the potential relative |
| 487 |
+ |
to the repulsive interaction and $\alpha_{ij}$ is a length parameter |
| 488 |
+ |
that assures a dimensionless form for $\rho$. These parameters are |
| 489 |
+ |
tuned to various experimental properties such as the density, cohesive |
| 490 |
+ |
energy, and elastic moduli for FCC transition metals. The quantum |
| 491 |
+ |
Sutton-Chen (QSC) formulation matches these properties while including |
| 492 |
+ |
zero-point quantum corrections for different transition |
| 493 |
+ |
metals.\cite{PhysRevB.59.3527} |
| 494 |
+ |
|
| 495 |
+ |
In bulk gold, the experimentally-measured value for the bulk modulus |
| 496 |
+ |
is 180.32 GPa, while previous calculations on the QSC potential in |
| 497 |
+ |
periodic-boundary simulations of the bulk have yielded values of |
| 498 |
+ |
175.53 GPa.\cite{XXX} Using the same force field, we have performed a |
| 499 |
+ |
series of relatively short (200 ps) simulations on 40 \r{A} radius |
| 500 |
+ |
nanoparticles under the Langevin Hull at a variety of applied |
| 501 |
+ |
pressures ranging from 0 GPa to XXX. We obtain a value of 177.547 GPa |
| 502 |
+ |
for the bulk modulus for gold using this echnique. |
| 503 |
+ |
|
| 504 |
|
\begin{figure} |
| 505 |
< |
\includegraphics[width=\linewidth]{temperature_tb} |
| 506 |
< |
\caption{Temperature equilibration depends on surface area and bath |
| 507 |
< |
viscosity. Target Temperature = 300K} |
| 508 |
< |
\label{temperatureResponse} |
| 505 |
> |
\includegraphics[width=\linewidth]{stacked} |
| 506 |
> |
\caption{The response of the internal pressure and temperature of gold |
| 507 |
> |
nanoparticles when first placed in the Langevin Hull |
| 508 |
> |
($T_\mathrm{bath}$ = 300K, $P_\mathrm{bath}$ = 4 GPa), starting |
| 509 |
> |
from initial conditions that were far from the bath pressure and |
| 510 |
> |
temperature. The pressure response is rapid (after the breathing mode oscillations in the nanoparticle die out), and the rate of thermal equilibration depends on both exposed surface area (top panel) and the viscosity of the bath (middle panel).} |
| 511 |
> |
\label{pressureResponse} |
| 512 |
|
\end{figure} |
| 513 |
|
|
| 514 |
|
\begin{equation} |
| 516 |
|
P}\right) |
| 517 |
|
\end{equation} |
| 518 |
|
|
| 519 |
+ |
\subsection{Compressibility of SPC/E water clusters} |
| 520 |
+ |
|
| 521 |
+ |
Prior molecular dynamics simulations on SPC/E water (both in |
| 522 |
+ |
NVT~\cite{Glattli2002} and NPT~\cite{Motakabbir1990, Pi2009} |
| 523 |
+ |
ensembles) have yielded values for the isothermal compressibility that |
| 524 |
+ |
agree well with experiment.\cite{Fine1973} The results of two |
| 525 |
+ |
different approaches for computing the isothermal compressibility from |
| 526 |
+ |
Langevin Hull simulations for pressures between 1 and 6500 atm are |
| 527 |
+ |
shown in Fig. \ref{fig:compWater} along with compressibility values |
| 528 |
+ |
obtained from both other SPC/E simulations and experiment. |
| 529 |
+ |
Compressibility values from all references are for applied pressures |
| 530 |
+ |
within the range 1 - 1000 atm. |
| 531 |
+ |
|
| 532 |
|
\begin{figure} |
| 533 |
< |
\includegraphics[width=\linewidth]{compress_tb} |
| 534 |
< |
\caption{Isothermal Compressibility (18 \AA gold nanoparticle)} |
| 535 |
< |
\label{temperatureResponse} |
| 533 |
> |
\includegraphics[width=\linewidth]{new_isothermalN} |
| 534 |
> |
\caption{Compressibility of SPC/E water} |
| 535 |
> |
\label{fig:compWater} |
| 536 |
|
\end{figure} |
| 537 |
|
|
| 538 |
< |
\subsection{Compressibility of SPC/E water clusters} |
| 538 |
> |
Isothermal compressibility values calculated using the number density |
| 539 |
> |
(Eq. \ref{eq:BMN}) expression are in good agreement with experimental |
| 540 |
> |
and previous simulation work throughout the 1 - 1000 atm pressure |
| 541 |
> |
regime. Compressibilities computed using the Hull volume, however, |
| 542 |
> |
deviate dramatically from the experimental values at low applied |
| 543 |
> |
pressures. The reason for this deviation is quite simple; at low |
| 544 |
> |
applied pressures, the liquid is in equilibrium with a vapor phase, |
| 545 |
> |
and it is entirely possible for one (or a few) molecules to drift away |
| 546 |
> |
from the liquid cluster (see Fig. \ref{fig:coneOfShame}). At low |
| 547 |
> |
pressures, the restoring forces on the facets are very gentle, and |
| 548 |
> |
this means that the hulls often take on relatively distorted |
| 549 |
> |
geometries which include large volumes of empty space. |
| 550 |
|
|
| 551 |
|
\begin{figure} |
| 552 |
< |
\includegraphics[width=\linewidth]{g_r_theta} |
| 553 |
< |
\caption{Definition of coordinates} |
| 554 |
< |
\label{coords} |
| 552 |
> |
\includegraphics[width=\linewidth]{flytest2} |
| 553 |
> |
\caption{At low pressures, the liquid is in equilibrium with the vapor |
| 554 |
> |
phase, and isolated molecules can detach from the liquid droplet. |
| 555 |
> |
This is expected behavior, but the volume of the convex hull |
| 556 |
> |
includes large regions of empty space. For this reason, |
| 557 |
> |
compressibilities are computed using local number densities rather |
| 558 |
> |
than hull volumes.} |
| 559 |
> |
\label{fig:coneOfShame} |
| 560 |
|
\end{figure} |
| 561 |
|
|
| 562 |
+ |
At higher pressures, the equilibrium strongly favors the liquid phase, |
| 563 |
+ |
and the hull geometries are much more compact. Because of the |
| 564 |
+ |
liquid-vapor effect on the convex hull, the regional number density |
| 565 |
+ |
approach (Eq. \ref{eq:BMN}) provides more reliable estimates of the |
| 566 |
+ |
compressibility. |
| 567 |
+ |
|
| 568 |
+ |
In both the traditional compressibility formula (Eq. \ref{eq:BM}) and |
| 569 |
+ |
the number density version (Eq. \ref{eq:BMN}), multiple simulations at |
| 570 |
+ |
different pressures must be done to compute the first derivatives. It |
| 571 |
+ |
is also possible to compute the compressibility using the fluctuation |
| 572 |
+ |
dissipation theorem using either fluctuations in the |
| 573 |
+ |
volume,\cite{Debenedetti1986}, |
| 574 |
+ |
\begin{equation} |
| 575 |
+ |
\kappa_{T} = \frac{\left \langle V^{2} \right \rangle - \left \langle |
| 576 |
+ |
V \right \rangle ^{2}}{V \, k_{B} \, T}, |
| 577 |
+ |
\end{equation} |
| 578 |
+ |
or, equivalently, fluctuations in the number of molecules within the |
| 579 |
+ |
fixed region, |
| 580 |
+ |
\begin{equation} |
| 581 |
+ |
\kappa_{T} = \frac{\left \langle N^{2} \right \rangle - \left \langle |
| 582 |
+ |
N \right \rangle ^{2}}{N \, k_{B} \, T}, |
| 583 |
+ |
\end{equation} |
| 584 |
+ |
Thus, the compressibility of each simulation can be calculated |
| 585 |
+ |
entirely independently from all other trajectories. However, the |
| 586 |
+ |
resulting compressibilities were still as much as an order of |
| 587 |
+ |
magnitude larger than the reference values. However, compressibility |
| 588 |
+ |
calculation that relies on the hull volume will suffer these effects. |
| 589 |
+ |
WE NEED MORE HERE. |
| 590 |
+ |
|
| 591 |
+ |
\subsection{Molecular orientation distribution at cluster boundary} |
| 592 |
+ |
|
| 593 |
+ |
In order for non-periodic boundary conditions to be widely applicable, |
| 594 |
+ |
they must be constructed in such a way that they allow a finite system |
| 595 |
+ |
to replicate the properties of the bulk. Early non-periodic |
| 596 |
+ |
simulation methods (e.g. hydrophobic boundary potentials) induced |
| 597 |
+ |
spurious orientational correlations deep within the simulated |
| 598 |
+ |
system.\cite{Lee1984,Belch1985} This behavior spawned many methods for |
| 599 |
+ |
fixing and characterizing the effects of artifical boundaries |
| 600 |
+ |
including methods which fix the orientations of a set of edge |
| 601 |
+ |
molecules.\cite{Warshel1978,King1989} |
| 602 |
+ |
|
| 603 |
+ |
As described above, the Langevin Hull does not require that the |
| 604 |
+ |
orientation of molecules be fixed, nor does it utilize an explicitly |
| 605 |
+ |
hydrophobic boundary, orientational constraint or radial constraint. |
| 606 |
+ |
Therefore, the orientational correlations of the molecules in a water |
| 607 |
+ |
cluster are of particular interest in testing this method. Ideally, |
| 608 |
+ |
the water molecules on the surface of the cluster will have enough |
| 609 |
+ |
mobility into and out of the center of the cluster to maintain a |
| 610 |
+ |
bulk-like orientational distribution in the absence of orientational |
| 611 |
+ |
and radial constraints. However, since the number of hydrogen bonding |
| 612 |
+ |
partners available to molecules on the exterior are limited, it is |
| 613 |
+ |
likely that there will be some effective hydrophobicity of the hull. |
| 614 |
+ |
|
| 615 |
+ |
To determine the extent of these effects demonstrated by the Langevin |
| 616 |
+ |
Hull, we examined the orientationations exhibited by SPC/E water in a |
| 617 |
+ |
cluster of 1372 molecules at 300 K and at pressures ranging from 1 - |
| 618 |
+ |
1000 atm. The orientational angle of a water molecule is described |
| 619 |
|
\begin{equation} |
| 620 |
|
\cos{\theta}=\frac{\vec{r}_i\cdot\vec{\mu}_i}{|\vec{r}_i||\vec{\mu}_i|} |
| 621 |
|
\end{equation} |
| 622 |
+ |
where $\vec{r}_{i}$ is the vector between molecule {\it i}'s center of |
| 623 |
+ |
mass and the cluster center of mass and $\vec{\mu}_{i}$ is the vector |
| 624 |
+ |
bisecting the H-O-H angle of molecule {\it i} Bulk-like distributions |
| 625 |
+ |
will result in $\langle \cos \theta \rangle$ values close to zero. If |
| 626 |
+ |
the hull exhibits an overabundance of externally-oriented oxygen sites |
| 627 |
+ |
the average orientation will be negative, while dangling hydrogen |
| 628 |
+ |
sites will result in positive average orientations. |
| 629 |
|
|
| 630 |
+ |
Fig. \ref{fig:pAngle} shows the distribution of $\cos{\theta}$ values |
| 631 |
+ |
for molecules in the interior of the cluster (squares) and for |
| 632 |
+ |
molecules included in the convex hull (circles). |
| 633 |
|
\begin{figure} |
| 634 |
|
\includegraphics[width=\linewidth]{pAngle} |
| 635 |
< |
\caption{SPC/E water clusters: only minor dewetting at the boundary} |
| 636 |
< |
\label{pAngle} |
| 635 |
> |
\caption{Distribution of $\cos{\theta}$ values for molecules on the |
| 636 |
> |
interior of the cluster (squares) and for those participating in the |
| 637 |
> |
convex hull (circles) at a variety of pressures. The Langevin hull |
| 638 |
> |
exhibits minor dewetting behavior with exposed oxygen sites on the |
| 639 |
> |
hull water molecules. The orientational preference for exposed |
| 640 |
> |
oxygen appears to be independent of applied pressure. } |
| 641 |
> |
\label{fig:pAngle} |
| 642 |
|
\end{figure} |
| 643 |
|
|
| 644 |
< |
\begin{figure} |
| 645 |
< |
\includegraphics[width=\linewidth]{isothermal} |
| 646 |
< |
\caption{Compressibility of SPC/E water} |
| 647 |
< |
\label{compWater} |
| 648 |
< |
\end{figure} |
| 644 |
> |
As expected, interior molecules (those not included in the convex |
| 645 |
> |
hull) maintain a bulk-like structure with a uniform distribution of |
| 646 |
> |
orientations. Molecules included in the convex hull show a slight |
| 647 |
> |
preference for values of $\cos{\theta} < 0.$ These values correspond |
| 648 |
> |
to molecules with oxygen directed toward the exterior of the cluster, |
| 649 |
> |
forming a dangling hydrogen bond acceptor site. |
| 650 |
|
|
| 651 |
+ |
In the absence of an electrostatic contribution from the exterior |
| 652 |
+ |
bath, the orientational distribution of water molecules included in |
| 653 |
+ |
the Langevin Hull will slightly resemble the distribution at a neat |
| 654 |
+ |
water liquid/vapor interface. Previous molecular dynamics simulations |
| 655 |
+ |
of SPC/E water \cite{Taylor1996} have shown that molecules at the |
| 656 |
+ |
liquid/vapor interface favor an orientation where one hydrogen |
| 657 |
+ |
protrudes from the liquid phase. This behavior is demonstrated by |
| 658 |
+ |
experiments \cite{Du1994} \cite{Scatena2001} showing that |
| 659 |
+ |
approximately one-quarter of water molecules at the liquid/vapor |
| 660 |
+ |
interface form dangling hydrogen bonds. The negligible preference |
| 661 |
+ |
shown in these cluster simulations could be removed through the |
| 662 |
+ |
introduction of an implicit solvent model, which would provide the |
| 663 |
+ |
missing electrostatic interactions between the cluster molecules and |
| 664 |
+ |
the surrounding temperature/pressure bath. |
| 665 |
+ |
|
| 666 |
+ |
The orientational preference exhibited by hull molecules in the |
| 667 |
+ |
Langevin hull is significantly weaker than the preference caused by an |
| 668 |
+ |
explicit hydrophobic bounding potential. Additionally, the Langevin |
| 669 |
+ |
Hull does not require that the orientation of any molecules be fixed |
| 670 |
+ |
in order to maintain bulk-like structure, even at the cluster surface. |
| 671 |
+ |
|
| 672 |
|
\subsection{Heterogeneous nanoparticle / water mixtures} |
| 673 |
|
|
| 674 |
+ |
\section{Discussion} |
| 675 |
+ |
\label{sec:discussion} |
| 676 |
|
|
| 677 |
< |
\section{Appendix A: Hydrodynamic tensor for triangular facets} |
| 677 |
> |
The Langevin Hull samples the isobaric-isothermal ensemble for |
| 678 |
> |
non-periodic systems by coupling the system to an bath characterized |
| 679 |
> |
by pressure, temperature, and solvent viscosity. This enables the |
| 680 |
> |
study of heterogeneous systems composed of materials of significantly |
| 681 |
> |
different compressibilities. Because the boundary is dynamically |
| 682 |
> |
determined during the simulation and the molecules interacting with |
| 683 |
> |
the boundary can change, the method and has minimal perturbations on |
| 684 |
> |
the behavior of molecules at the edges of the simulation. Further |
| 685 |
> |
work on this method will involve implicit electrostatics at the |
| 686 |
> |
boundary (which is missing in the current implementation) as well as |
| 687 |
> |
more sophisticated treatments of the surface geometry (alpha |
| 688 |
> |
shapes\cite{EDELSBRUNNER:1994oq,EDELSBRUNNER:1995cj} and Tight |
| 689 |
> |
Cocone\cite{Dey:2003ts}). The non-convex hull geometries are |
| 690 |
> |
significantly more expensive ($\mathcal{O}(N^2)$) than the convex hull |
| 691 |
> |
($\mathcal{O}(N \log N)$), but would enable the use of hull volumes |
| 692 |
> |
directly in computing the compressibility of the sample. |
| 693 |
|
|
| 694 |
+ |
\section*{Appendix A: Computing Convex Hulls on Parallel Computers} |
| 695 |
+ |
|
| 696 |
+ |
In order to use the Langevin Hull for simulations on parallel |
| 697 |
+ |
computers, one of the more difficult tasks is to compute the bounding |
| 698 |
+ |
surface, facets, and resistance tensors when the processors have |
| 699 |
+ |
incomplete information about the entire system's topology. Most |
| 700 |
+ |
parallel decomposition methods assign primary responsibility for the |
| 701 |
+ |
motion of an atomic site to a single processor, and we can exploit |
| 702 |
+ |
this to efficiently compute the convex hull for the entire system. |
| 703 |
+ |
|
| 704 |
+ |
The basic idea involves splitting the point cloud into |
| 705 |
+ |
spatially-overlapping subsets and computing the convex hulls for each |
| 706 |
+ |
of the subsets. The points on the convex hull of the entire system |
| 707 |
+ |
are all present on at least one of the subset hulls. The algorithm |
| 708 |
+ |
works as follows: |
| 709 |
+ |
\begin{enumerate} |
| 710 |
+ |
\item Each processor computes the convex hull for its own atomic sites |
| 711 |
+ |
(left panel in Fig. \ref{fig:parallel}). |
| 712 |
+ |
\item The Hull vertices from each processor are communicated to all of |
| 713 |
+ |
the processors, and each processor assembles a complete list of hull |
| 714 |
+ |
sites (this is much smaller than the original number of points in |
| 715 |
+ |
the point cloud). |
| 716 |
+ |
\item Each processor computes the global convex hull (right panel in |
| 717 |
+ |
Fig. \ref{fig:parallel}) using only those points that are the union |
| 718 |
+ |
of sites gathered from all of the subset hulls. Delaunay |
| 719 |
+ |
triangulation is then done to obtain the facets of the global hull. |
| 720 |
+ |
\end{enumerate} |
| 721 |
+ |
|
| 722 |
|
\begin{figure} |
| 723 |
< |
\includegraphics[width=\linewidth]{hydro} |
| 724 |
< |
\caption{Hydro} |
| 725 |
< |
\label{hydro} |
| 723 |
> |
\includegraphics[width=\linewidth]{parallel} |
| 724 |
> |
\caption{When the sites are distributed among many nodes for parallel |
| 725 |
> |
computation, the processors first compute the convex hulls for their |
| 726 |
> |
own sites (dashed lines in left panel). The positions of the sites |
| 727 |
> |
that make up the subset hulls are then communicated to all |
| 728 |
> |
processors (middle panel). The convex hull of the system (solid line in |
| 729 |
> |
right panel) is the convex hull of the points on the union of the subset |
| 730 |
> |
hulls.} |
| 731 |
> |
\label{fig:parallel} |
| 732 |
|
\end{figure} |
| 733 |
|
|
| 734 |
< |
\begin{equation} |
| 735 |
< |
\Xi_f(t) =\left[\sum_{i=1}^3 T_{if}\right]^{-1} |
| 736 |
< |
\end{equation} |
| 734 |
> |
The individual hull operations scale with |
| 735 |
> |
$\mathcal{O}(\frac{n}{p}\log\frac{n}{p})$ where $n$ is the total |
| 736 |
> |
number of sites, and $p$ is the number of processors. These local |
| 737 |
> |
hull operations create a set of $p$ hulls each with approximately |
| 738 |
> |
$\frac{n}{3pr}$ sites (for a cluster of radius $r$). The worst-case |
| 739 |
> |
communication cost for using a ``gather'' operation to distribute this |
| 740 |
> |
information to all processors is $\mathcal{O}( \alpha (p-1) + \frac{n |
| 741 |
> |
\beta (p-1)}{3 r p^2})$, while the final computation of the system |
| 742 |
> |
hull scales as $\mathcal{O}(\frac{n}{3r}\log\frac{n}{3r})$. |
| 743 |
|
|
| 744 |
< |
\begin{equation} |
| 745 |
< |
T_{if}=\frac{A_i}{8\pi\eta R_{if}}\left(I + |
| 746 |
< |
\frac{\mathbf{R}_{if}\mathbf{R}_{if}^T}{R_{if}^2}\right) |
| 747 |
< |
\end{equation} |
| 744 |
> |
For a large number of atoms on a moderately parallel machine, the |
| 745 |
> |
total costs are dominated by the computations of the individual hulls, |
| 746 |
> |
and communication of these hulls to so the Langevin hull sees roughly |
| 747 |
> |
linear speed-up with increasing processor counts. |
| 748 |
|
|
| 749 |
< |
\section{Appendix B: Computing Convex Hulls on Parallel Computers} |
| 232 |
< |
|
| 233 |
< |
\section{Acknowledgments} |
| 749 |
> |
\section*{Acknowledgments} |
| 750 |
|
Support for this project was provided by the |
| 751 |
|
National Science Foundation under grant CHE-0848243. Computational |
| 752 |
|
time was provided by the Center for Research Computing (CRC) at the |
