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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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|
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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'', performs |
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better than traditional affine transform methods for systems |
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containing heterogeneous mixtures of materials with different |
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compressibilities. It does not suffer from the edge effects of |
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boundary potential methods, and allows realistic treatment of both |
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external pressure and thermal conductivity to an implicit solvent. |
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We apply this method to several different systems including bare |
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nanoparticles, nanoparticles in an explicit solvent, as well as |
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clusters of liquid water and ice. The predicted mechanical and |
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thermal properties of these systems are in good agreement with |
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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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\section{Introduction} |
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|
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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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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 |
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barostat}, which is an extra degree of freedom propagated along with |
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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 |
> |
using {\it affine transforms} of the system geometry. An affine |
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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 coordinate |
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transformation to adjust the box volume. As long as the material in |
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the simulation box is essentially a bulk-like liquid which has a |
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relatively uniform compressibility, the standard affine transform |
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approach provides an excellent way of adjusting the volume of the |
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system and applying pressure directly via the interactions between |
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atomic sites. |
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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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The problem with this approach becomes apparent when the material |
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being simulated is an inhomogeneous mixture in which portions of the |
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simulation box are incompressible relative to other portions. |
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Examples include simulations of metallic nanoparticles in liquid |
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environments, proteins at interfaces, as well as other multi-phase or |
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One problem with this approach appears when the system being simulated |
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is an inhomogeneous mixture in which portions of the simulation box |
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are incompressible relative to other portions. Examples include |
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simulations of metallic nanoparticles in liquid environments, proteins |
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at ice / water interfaces, as well as other heterogeneous or |
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interfacial environments. In these cases, the affine transform of |
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atomic coordinates will either cause numerical instability when the |
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sites in the incompressible medium collide with each other, or lead to |
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inefficient sampling of system volumes if the barostat is set slow |
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enough to avoid the instabilities in the incompressible region. |
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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. |
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|
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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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\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 |
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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 |
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compressibility regions to collide.} |
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\label{affineScale} |
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\end{figure} |
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|
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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 either requires effective solute concentrations that are much |
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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 shells |
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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 |
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expensive. [CALCULATE EFFECTIVE PROTEIN CONCENTRATIONS IN TYPICAL |
| 121 |
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SIMULATIONS] |
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expensive. A 62 \AA$^3$ box of water solvating a moderately small |
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protein like hen egg white lysozyme (PDB code: 1LYZ) yields an |
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effective protein concentration of 100 mg/mL.\cite{Asthagiri20053300} |
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|
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There have been a number of other approaches to explicit |
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non-periodicity that focus on constant or nearly-constant {\it volume} |
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conditions while maintaining bulk-like behavior. Berkowitz and |
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McCammon introduced a stochastic (Langevin) boundary layer inside a |
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region of fixed molecules which effectively enforces constant |
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temperature and volume (NVT) conditions.\cite{Berkowitz1982} In this |
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approach, the stochastic and fixed regions were defined relative to a |
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central atom. Brooks and Karplus extended this method to include |
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deformable stochastic boundaries.\cite{iii:6312} The stochastic |
| 132 |
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boundary approach has been used widely for protein |
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simulations. [CITATIONS NEEDED] |
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{\it Yotal} protein concentrations in the cell are typically on the |
| 125 |
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order of 160-310 mg/ml,\cite{Brown1991195} and individual proteins |
| 126 |
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have concentrations orders of magnitude lower than this in the |
| 127 |
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cellular environment. The effective concentrations of single proteins |
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in simulations may have significant effects on the structure and |
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dynamics of simulated structures. |
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|
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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 |
| 135 |
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behavior. Berkowitz and McCammon introduced a stochastic (Langevin) |
| 136 |
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boundary layer inside a region of fixed molecules which effectively |
| 137 |
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enforces constant temperature and volume (NVT) |
| 138 |
+ |
conditions.\cite{Berkowitz1982} In this approach, the stochastic and |
| 139 |
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fixed regions were defined relative to a central atom. Brooks and |
| 140 |
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Karplus extended this method to include deformable stochastic |
| 141 |
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boundaries.\cite{iii:6312} The stochastic boundary approach has been |
| 142 |
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used widely for protein simulations. |
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|
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The electrostatic and dispersive behavior near the boundary has long |
| 145 |
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been a cause for concern. King and Warshel introduced a surface |
| 146 |
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constrained all-atom solvent (SCAAS) which included polarization |
| 147 |
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effects of a fixed spherical boundary to mimic bulk-like behavior |
| 148 |
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without periodic boundaries.\cite{king:3647} In the SCAAS model, a |
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layer of fixed solvent molecules surrounds the solute and any explicit |
| 150 |
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solvent, and this in turn is surrounded by a continuum dielectric. |
| 151 |
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MORE HERE. WHAT DID THEY FIND? |
| 145 |
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been a cause for concern when performing simulations of explicitly |
| 146 |
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non-periodic systems. Early work led to the surface constrained soft |
| 147 |
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sphere dipole model (SCSSD)\cite{Warshel1978} in which the surface |
| 148 |
> |
molecules are fixed in a random orientation representative of the bulk |
| 149 |
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solvent structural properties. Belch {\it et al.}\cite{Belch1985} |
| 150 |
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simulated clusters of TIPS2 water surrounded by a hydrophobic bounding |
| 151 |
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potential. The spherical hydrophobic boundary induced dangling |
| 152 |
> |
hydrogen bonds at the surface that propagated deep into the cluster, |
| 153 |
> |
affecting most of the molecules in the simulation. This result echoes |
| 154 |
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an earlier study which showed that an extended planar hydrophobic |
| 155 |
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surface caused orientational preferences at the surface which extended |
| 156 |
> |
relatively deep (7 \AA) into the liquid simulation cell.\cite{Lee1984} |
| 157 |
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The surface constrained all-atom solvent (SCAAS) model \cite{King1989} |
| 158 |
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improved upon its SCSSD predecessor. The SCAAS model utilizes a |
| 159 |
> |
polarization constraint which is applied to the surface molecules to |
| 160 |
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maintain bulk-like structure at the cluster surface. A radial |
| 161 |
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constraint is used to maintain the desired bulk density of the |
| 162 |
> |
liquid. Both constraint forces are applied only to a pre-determined |
| 163 |
> |
number of the outermost molecules. |
| 164 |
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|
| 165 |
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Beglov and Roux developed a boundary model in which the hard sphere |
| 166 |
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boundary has a radius that varies with the instantaneous configuration |
| 167 |
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of the solute (and solvent) molecules.\cite{beglov:9050} This model |
| 168 |
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contains a clear pressure and surface tension contribution to the free |
| 169 |
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energy which XXX. |
| 165 |
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Beglov and Roux have developed a boundary model in which the hard |
| 166 |
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sphere boundary has a radius that varies with the instantaneous |
| 167 |
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configuration of the solute (and solvent) molecules.\cite{beglov:9050} |
| 168 |
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This model contains a clear pressure and surface tension contribution |
| 169 |
> |
to the free energy. |
| 170 |
|
|
| 171 |
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\subsection*{Restraining Potentials} |
| 172 |
|
Restraining {\it potentials} introduce repulsive potentials at the |
| 173 |
|
surface of a sphere or other geometry. The solute and any explicit |
| 174 |
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solvent are therefore restrained inside this potential. Often the |
| 175 |
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potentials include a weak short-range attraction to maintain the |
| 176 |
< |
correct density at the boundary. Beglov and Roux have also introduced |
| 177 |
< |
a restraining boundary potential which relaxes dynamically depending |
| 178 |
< |
on the solute geometry and the force the explicit system exerts on the |
| 179 |
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shell.\cite{Beglov:1995fk} |
| 174 |
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solvent are therefore restrained inside the range defined by the |
| 175 |
> |
external potential. Often the potentials include a weak short-range |
| 176 |
> |
attraction to maintain the correct density at the boundary. Beglov |
| 177 |
> |
and Roux have also introduced a restraining boundary potential which |
| 178 |
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relaxes dynamically depending on the solute geometry and the force the |
| 179 |
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explicit system exerts on the shell.\cite{Beglov:1995fk} |
| 180 |
|
|
| 181 |
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Recently, Krilov {\it et al.} introduced a flexible boundary model |
| 182 |
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that uses a Lennard-Jones potential between the solvent molecules and |
| 183 |
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a boundary which is determined dynamically from the position of the |
| 184 |
< |
nearest solute atom.\cite{LiY._jp046852t,Zhu:xw} This approach allows |
| 185 |
< |
the confining potential to prevent solvent molecules from migrating |
| 186 |
< |
too far from the solute surface, while providing a weak attractive |
| 187 |
< |
force pulling the solvent molecules towards a fictitious bulk solvent. |
| 188 |
< |
Although this approach is appealing and has physical motivation, |
| 189 |
< |
nanoparticles do not deform far from their original geometries even at |
| 190 |
< |
temperatures which vaporize the nearby solvent. For the systems like |
| 191 |
< |
the one described, the flexible boundary model will be nearly |
| 181 |
> |
Recently, Krilov {\it et al.} introduced a {\it flexible} boundary |
| 182 |
> |
model that uses a Lennard-Jones potential between the solvent |
| 183 |
> |
molecules and a boundary which is determined dynamically from the |
| 184 |
> |
position of the nearest solute atom.\cite{LiY._jp046852t,Zhu:2008fk} This |
| 185 |
> |
approach allows the confining potential to prevent solvent molecules |
| 186 |
> |
from migrating too far from the solute surface, while providing a weak |
| 187 |
> |
attractive force pulling the solvent molecules towards a fictitious |
| 188 |
> |
bulk solvent. Although this approach is appealing and has physical |
| 189 |
> |
motivation, nanoparticles do not deform far from their original |
| 190 |
> |
geometries even at temperatures which vaporize the nearby solvent. For |
| 191 |
> |
the systems like this, the flexible boundary model will be nearly |
| 192 |
|
identical to a fixed-volume restraining potential. |
| 193 |
|
|
| 194 |
+ |
\subsection*{Hull methods} |
| 195 |
|
The approach of Kohanoff, Caro, and Finnis is the most promising of |
| 196 |
|
the methods for introducing both constant pressure and temperature |
| 197 |
|
into non-periodic simulations.\cite{Kohanoff:2005qm,Baltazar:2006ru} |
| 198 |
|
This method is based on standard Langevin dynamics, but the Brownian |
| 199 |
|
or random forces are allowed to act only on peripheral atoms and exert |
| 200 |
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force in a direction that is inward-facing relative to the facets of a |
| 201 |
< |
closed bounding surface. The statistical distribution of the random |
| 200 |
> |
forces in a direction that is inward-facing relative to the facets of |
| 201 |
> |
a closed bounding surface. The statistical distribution of the random |
| 202 |
|
forces are uniquely tied to the pressure in the external reservoir, so |
| 203 |
|
the method can be shown to sample the isobaric-isothermal ensemble. |
| 204 |
|
Kohanoff {\it et al.} used a Delaunay tessellation to generate a |
| 210 |
|
In the following sections, we extend and generalize the approach of |
| 211 |
|
Kohanoff, Caro, and Finnis. The new method, which we are calling the |
| 212 |
|
``Langevin Hull'' applies the external pressure, Langevin drag, and |
| 213 |
< |
random forces on the facets of the {\it hull itself} instead of the |
| 214 |
< |
atomic sites comprising the vertices of the hull. This allows us to |
| 215 |
< |
decouple the external pressure contribution from the drag and random |
| 216 |
< |
force. Section \ref{sec:meth} |
| 213 |
> |
random forces on the {\it facets of the hull} instead of the atomic |
| 214 |
> |
sites comprising the vertices of the hull. This allows us to decouple |
| 215 |
> |
the external pressure contribution from the drag and random force. |
| 216 |
> |
The methodology is introduced in section \ref{sec:meth}, tests on |
| 217 |
> |
crystalline nanoparticles, liquid clusters, and heterogeneous mixtures |
| 218 |
> |
are detailed in section \ref{sec:tests}. Section \ref{sec:discussion} |
| 219 |
> |
summarizes our findings. |
| 220 |
|
|
| 221 |
|
\section{Methodology} |
| 222 |
|
\label{sec:meth} |
| 223 |
|
|
| 224 |
< |
We have developed a new method which uses a constant pressure and |
| 225 |
< |
temperature bath. This bath interacts only with the objects that are |
| 226 |
< |
currently at the edge of the system. Since the edge is determined |
| 227 |
< |
dynamically as the simulation progresses, no {\it a priori} geometry |
| 228 |
< |
is defined. The pressure and temperature bath interacts {\it |
| 229 |
< |
directly} with the atoms on the edge and not with atoms interior to |
| 230 |
< |
the simulation. This means that there are no affine transforms |
| 231 |
< |
required. There are also no fictitious particles or bounding |
| 232 |
< |
potentials used in this approach. |
| 224 |
> |
The Langevin Hull uses an external bath at a fixed constant pressure |
| 225 |
> |
($P$) and temperature ($T$) with an effective solvent viscosity |
| 226 |
> |
($\eta$). This bath interacts only with the objects on the exterior |
| 227 |
> |
hull of the system. Defining the hull of the atoms in a simulation is |
| 228 |
> |
done in a manner similar to the approach of Kohanoff, Caro and |
| 229 |
> |
Finnis.\cite{Kohanoff:2005qm} That is, any instantaneous configuration |
| 230 |
> |
of the atoms in the system is considered as a point cloud in three |
| 231 |
> |
dimensional space. Delaunay triangulation is used to find all facets |
| 232 |
> |
between coplanar |
| 233 |
> |
neighbors.\cite{delaunay,springerlink:10.1007/BF00977785} In highly |
| 234 |
> |
symmetric point clouds, facets can contain many atoms, but in all but |
| 235 |
> |
the most symmetric of cases, the facets are simple triangles in |
| 236 |
> |
3-space which contain exactly three atoms. |
| 237 |
|
|
| 208 |
– |
The basics of the method are as follows. The simulation starts as a |
| 209 |
– |
collection of atomic locations in three dimensions (a point cloud). |
| 210 |
– |
Delaunay triangulation is used to find all facets between coplanar |
| 211 |
– |
neighbors. In highly symmetric point clouds, facets can contain many |
| 212 |
– |
atoms, but in all but the most symmetric of cases one might experience |
| 213 |
– |
in a molecular dynamics simulation, the facets are simple triangles in |
| 214 |
– |
3-space that contain exactly three atoms. |
| 215 |
– |
|
| 238 |
|
The convex hull is the set of facets that have {\it no concave |
| 239 |
< |
corners} at an atomic site. This eliminates all facets on the |
| 240 |
< |
interior of the point cloud, leaving only those exposed to the |
| 241 |
< |
bath. Sites on the convex hull are dynamic. As molecules re-enter the |
| 242 |
< |
cluster, all interactions between atoms on that molecule and the |
| 243 |
< |
external bath are removed. |
| 239 |
> |
corners} at an atomic site.\cite{Barber96,EDELSBRUNNER:1994oq} This |
| 240 |
> |
eliminates all facets on the interior of the point cloud, leaving only |
| 241 |
> |
those exposed to the bath. Sites on the convex hull are dynamic; as |
| 242 |
> |
molecules re-enter the cluster, all interactions between atoms on that |
| 243 |
> |
molecule and the external bath are removed. Since the edge is |
| 244 |
> |
determined dynamically as the simulation progresses, no {\it a priori} |
| 245 |
> |
geometry is defined. The pressure and temperature bath interacts only |
| 246 |
> |
with the atoms on the edge and not with atoms interior to the |
| 247 |
> |
simulation. |
| 248 |
|
|
| 249 |
< |
For atomic sites in the interior of the point cloud, the equations of |
| 250 |
< |
motion are simple Newtonian dynamics, |
| 249 |
> |
\begin{figure} |
| 250 |
> |
\includegraphics[width=\linewidth]{solvatedNano} |
| 251 |
> |
\caption{The external temperature and pressure bath interacts only |
| 252 |
> |
with those atoms on the convex hull (grey surface). The hull is |
| 253 |
> |
computed dynamically at each time step, and molecules can move |
| 254 |
> |
between the interior (Newtonian) region and the Langevin hull.} |
| 255 |
> |
\label{fig:hullSample} |
| 256 |
> |
\end{figure} |
| 257 |
> |
|
| 258 |
> |
Atomic sites in the interior of the simulation move under standard |
| 259 |
> |
Newtonian dynamics, |
| 260 |
|
\begin{equation} |
| 261 |
|
m_i \dot{\mathbf v}_i(t)=-{\mathbf \nabla}_i U, |
| 262 |
|
\label{eq:Newton} |
| 271 |
|
m_i \dot{\mathbf v}_i(t)=-{\mathbf \nabla}_i U + {\mathbf F}_i^{\mathrm ext}. |
| 272 |
|
\end{equation} |
| 273 |
|
|
| 274 |
< |
The external bath interacts directly with the facets of the convex |
| 275 |
< |
hull. Since each vertex (or atom) provides one corner of a triangular |
| 276 |
< |
facet, the force on the facets are divided equally to each vertex. |
| 277 |
< |
However, each vertex can participate in multiple facets, so the resultant |
| 278 |
< |
force is a sum over all facets $f$ containing vertex $i$: |
| 274 |
> |
The external bath interacts indirectly with the atomic sites through |
| 275 |
> |
the intermediary of the hull facets. Since each vertex (or atom) |
| 276 |
> |
provides one corner of a triangular facet, the force on the facets are |
| 277 |
> |
divided equally to each vertex. However, each vertex can participate |
| 278 |
> |
in multiple facets, so the resultant force is a sum over all facets |
| 279 |
> |
$f$ containing vertex $i$: |
| 280 |
|
\begin{equation} |
| 281 |
|
{\mathbf F}_{i}^{\mathrm ext} = \sum_{\begin{array}{c}\mathrm{facets\ |
| 282 |
|
} f \\ \mathrm{containing\ } i\end{array}} \frac{1}{3}\ {\mathbf |
| 285 |
|
|
| 286 |
|
The external pressure bath applies a force to the facets of the convex |
| 287 |
|
hull in direct proportion to the area of the facet, while the thermal |
| 288 |
< |
coupling depends on the solvent temperature, friction and the size and |
| 289 |
< |
shape of each facet. The thermal interactions are expressed as a |
| 290 |
< |
typical Langevin description of the forces, |
| 288 |
> |
coupling depends on the solvent temperature, viscosity and the size |
| 289 |
> |
and shape of each facet. The thermal interactions are expressed as a |
| 290 |
> |
standard Langevin description of the forces, |
| 291 |
|
\begin{equation} |
| 292 |
|
\begin{array}{rclclcl} |
| 293 |
|
{\mathbf F}_f^{\text{ext}} & = & \text{external pressure} & + & \text{drag force} & + & \text{random force} \\ |
| 294 |
|
& = & -\hat{n}_f P A_f & - & \Xi_f(t) {\mathbf v}_f(t) & + & {\mathbf R}_f(t) |
| 295 |
|
\end{array} |
| 296 |
|
\end{equation} |
| 297 |
< |
Here, $P$ is the external pressure, $A_f$ and $\hat{n}_f$ are the area |
| 298 |
< |
and normal vectors for facet $f$, respectively. ${\mathbf v}_f(t)$ is |
| 299 |
< |
the velocity of the facet, |
| 297 |
> |
Here, $A_f$ and $\hat{n}_f$ are the area and (outward-facing) normal |
| 298 |
> |
vectors for facet $f$, respectively. ${\mathbf v}_f(t)$ is the |
| 299 |
> |
velocity of the facet centroid, |
| 300 |
|
\begin{equation} |
| 301 |
|
{\mathbf v}_f(t) = \frac{1}{3} \sum_{i=1}^{3} {\mathbf v}_i, |
| 302 |
|
\end{equation} |
| 303 |
< |
and $\Xi_f(t)$ is an approximate ($3 \times 3$) hydrodynamic tensor |
| 304 |
< |
that depends on the geometry and surface area of facet $f$ and the |
| 305 |
< |
viscosity of the fluid (See Appendix A). The hydrodynamic tensor is |
| 306 |
< |
related to the fluctuations of the random force, $\mathbf{R}(t)$, by |
| 307 |
< |
the fluctuation-dissipation theorem, |
| 303 |
> |
and $\Xi_f(t)$ is an approximate ($3 \times 3$) resistance tensor that |
| 304 |
> |
depends on the geometry and surface area of facet $f$ and the |
| 305 |
> |
viscosity of the bath. The resistance tensor is related to the |
| 306 |
> |
fluctuations of the random force, $\mathbf{R}(t)$, by the |
| 307 |
> |
fluctuation-dissipation theorem, |
| 308 |
|
\begin{eqnarray} |
| 309 |
|
\left< {\mathbf R}_f(t) \right> & = & 0 \\ |
| 310 |
|
\left<{\mathbf R}_f(t) {\mathbf R}_f^T(t^\prime)\right> & = & 2 k_B T\ |
| 312 |
|
\label{eq:randomForce} |
| 313 |
|
\end{eqnarray} |
| 314 |
|
|
| 315 |
< |
Once the hydrodynamic tensor is known for a given facet (see Appendix |
| 316 |
< |
A) obtaining a stochastic vector that has the properties in |
| 317 |
< |
Eq. (\ref{eq:randomForce}) can be done efficiently by carrying out a |
| 318 |
< |
one-time Cholesky decomposition to obtain the square root matrix of |
| 283 |
< |
the resistance tensor, |
| 315 |
> |
Once the resistance tensor is known for a given facet, a stochastic |
| 316 |
> |
vector that has the properties in Eq. (\ref{eq:randomForce}) can be |
| 317 |
> |
calculated efficiently by carrying out a Cholesky decomposition to |
| 318 |
> |
obtain the square root matrix of the resistance tensor, |
| 319 |
|
\begin{equation} |
| 320 |
|
\Xi_f = {\bf S} {\bf S}^{T}, |
| 321 |
|
\label{eq:Cholesky} |
| 332 |
|
random force, ${\bf R}_{f} = {\bf S} {\bf Z}$, can be shown to |
| 333 |
|
have the correct properties required by Eq. (\ref{eq:randomForce}). |
| 334 |
|
|
| 335 |
< |
Our treatment of the hydrodynamic tensor must be approximate. $\Xi$ |
| 336 |
< |
for a triangular plate would normally be treated as a $6 \times 6$ |
| 335 |
> |
Our treatment of the resistance tensor is approximate. $\Xi_f$ for a |
| 336 |
> |
rigid triangular plate would normally be treated as a $6 \times 6$ |
| 337 |
|
tensor that includes translational and rotational drag as well as |
| 338 |
< |
translational-rotational coupling. The computation of hydrodynamic |
| 338 |
> |
translational-rotational coupling. The computation of resistance |
| 339 |
|
tensors for rigid bodies has been detailed |
| 340 |
< |
elsewhere,\cite{JoseGarciadelaTorre02012000,Garcia-de-la-Torre:2001wd,GarciadelaTorreJ2002,Sun2008} |
| 340 |
> |
elsewhere,\cite{JoseGarciadelaTorre02012000,Garcia-de-la-Torre:2001wd,GarciadelaTorreJ2002,Sun:2008fk} |
| 341 |
|
but the standard approach involving bead approximations would be |
| 342 |
|
prohibitively expensive if it were recomputed at each step in a |
| 343 |
|
molecular dynamics simulation. |
| 344 |
|
|
| 345 |
< |
We are utilizing an approximate hydrodynamic tensor obtained by first |
| 346 |
< |
constructing the Oseen tensor for the interaction of the centroid of |
| 347 |
< |
the facet ($f$) with each of the subfacets $j$, |
| 345 |
> |
Instead, we are utilizing an approximate resistance tensor obtained by |
| 346 |
> |
first constructing the Oseen tensor for the interaction of the |
| 347 |
> |
centroid of the facet ($f$) with each of the subfacets $\ell=1,2,3$, |
| 348 |
|
\begin{equation} |
| 349 |
< |
T_{jf}=\frac{A_j}{8\pi\eta R_{jf}}\left(I + |
| 350 |
< |
\frac{\mathbf{R}_{jf}\mathbf{R}_{jf}^T}{R_{jf}^2}\right) |
| 349 |
> |
T_{\ell f}=\frac{A_\ell}{8\pi\eta R_{\ell f}}\left(I + |
| 350 |
> |
\frac{\mathbf{R}_{\ell f}\mathbf{R}_{\ell f}^T}{R_{\ell f}^2}\right) |
| 351 |
|
\end{equation} |
| 352 |
< |
Here, $A_j$ is the area of subfacet $j$ which is a triangle containing |
| 353 |
< |
two of the vertices of the facet along with the centroid. |
| 354 |
< |
$\mathbf{R}_{jf}$ is the vector between the centroid of facet $f$ and |
| 355 |
< |
the centroid of sub-facet $j$, and $I$ is the ($3 \times 3$) identity |
| 356 |
< |
matrix. $\eta$ is the viscosity of the external bath. |
| 352 |
> |
Here, $A_\ell$ is the area of subfacet $\ell$ which is a triangle |
| 353 |
> |
containing two of the vertices of the facet along with the centroid. |
| 354 |
> |
$\mathbf{R}_{\ell f}$ is the vector between the centroid of facet $f$ |
| 355 |
> |
and the centroid of sub-facet $\ell$, and $I$ is the ($3 \times 3$) |
| 356 |
> |
identity matrix. $\eta$ is the viscosity of the external bath. |
| 357 |
|
|
| 358 |
|
\begin{figure} |
| 359 |
|
\includegraphics[width=\linewidth]{hydro} |
| 360 |
< |
\caption{The hydrodynamic tensor $\Xi$ for a facet comprising sites $i$, |
| 361 |
< |
$j$, and $k$ is constructed using Oseen tensor contributions |
| 362 |
< |
between the centoid of the facet $f$ and each of the sub-facets |
| 363 |
< |
($i,f,j$), ($j,f,k$), and ($k,f,i$). The centroids of the sub-facets |
| 364 |
< |
are located at $1$, $2$, and $3$, and the area of each sub-facet is |
| 360 |
> |
\caption{The resistance tensor $\Xi$ for a facet comprising sites $i$, |
| 361 |
> |
$j$, and $k$ is constructed using Oseen tensor contributions between |
| 362 |
> |
the centoid of the facet $f$ and each of the sub-facets ($i,f,j$), |
| 363 |
> |
($j,f,k$), and ($k,f,i$). The centroids of the sub-facets are |
| 364 |
> |
located at $1$, $2$, and $3$, and the area of each sub-facet is |
| 365 |
|
easily computed using half the cross product of two of the edges.} |
| 366 |
|
\label{hydro} |
| 367 |
|
\end{figure} |
| 368 |
|
|
| 369 |
< |
The Oseen tensors for each of the sub-facets are summed, and the |
| 370 |
< |
resulting matrix is inverted to give a $3 \times 3$ hydrodynamic |
| 371 |
< |
tensor for translations of the triangular plate, |
| 369 |
> |
The tensors for each of the sub-facets are added together, and the |
| 370 |
> |
resulting matrix is inverted to give a $3 \times 3$ resistance tensor |
| 371 |
> |
for translations of the triangular facet, |
| 372 |
|
\begin{equation} |
| 373 |
|
\Xi_f(t) =\left[\sum_{i=1}^3 T_{if}\right]^{-1}. |
| 374 |
|
\end{equation} |
| 375 |
+ |
Note that this treatment ignores rotations (and |
| 376 |
+ |
translational-rotational coupling) of the facet. In compact systems, |
| 377 |
+ |
the facets stay relatively fixed in orientation between |
| 378 |
+ |
configurations, so this appears to be a reasonably good approximation. |
| 379 |
+ |
|
| 380 |
|
We have implemented this method by extending the Langevin dynamics |
| 381 |
< |
integrator in our group code, OpenMD.\cite{Meineke2005,openmd} There |
| 382 |
< |
is a moderate penalty for computing the convex hull at each step in |
| 383 |
< |
the molecular dynamics simulation (HOW MUCH?), but the convex hull is |
| 384 |
< |
remarkably easy to parallelize on distributed memory machines (see |
| 385 |
< |
Appendix B). |
| 381 |
> |
integrator in our code, OpenMD.\cite{Meineke2005,openmd} At each |
| 382 |
> |
molecular dynamics time step, the following process is carried out: |
| 383 |
> |
\begin{enumerate} |
| 384 |
> |
\item The standard inter-atomic forces ($\nabla_iU$) are computed. |
| 385 |
> |
\item Delaunay triangulation is carried out using the current atomic |
| 386 |
> |
configuration. |
| 387 |
> |
\item The convex hull is computed and facets are identified. |
| 388 |
> |
\item For each facet: |
| 389 |
> |
\begin{itemize} |
| 390 |
> |
\item[a.] The force from the pressure bath ($-\hat{n}_fPA_f$) is |
| 391 |
> |
computed. |
| 392 |
> |
\item[b.] The resistance tensor ($\Xi_f(t)$) is computed using the |
| 393 |
> |
viscosity ($\eta$) of the bath. |
| 394 |
> |
\item[c.] Facet drag ($-\Xi_f(t) \mathbf{v}_f(t)$) forces are |
| 395 |
> |
computed. |
| 396 |
> |
\item[d.] Random forces ($\mathbf{R}_f(t)$) are computed using the |
| 397 |
> |
resistance tensor and the temperature ($T$) of the bath. |
| 398 |
> |
\end{itemize} |
| 399 |
> |
\item The facet forces are divided equally among the vertex atoms. |
| 400 |
> |
\item Atomic positions and velocities are propagated. |
| 401 |
> |
\end{enumerate} |
| 402 |
> |
The Delaunay triangulation and computation of the convex hull are done |
| 403 |
> |
using calls to the qhull library.\cite{Qhull} There is a minimal |
| 404 |
> |
penalty for computing the convex hull and resistance tensors at each |
| 405 |
> |
step in the molecular dynamics simulation (roughly 0.02 $\times$ cost |
| 406 |
> |
of a single force evaluation), and the convex hull is remarkably easy |
| 407 |
> |
to parallelize on distributed memory machines (see Appendix A). |
| 408 |
|
|
| 409 |
|
\section{Tests \& Applications} |
| 410 |
|
\label{sec:tests} |
| 411 |
|
|
| 412 |
< |
\subsection{Bulk modulus of gold nanoparticles} |
| 413 |
< |
|
| 414 |
< |
\begin{figure} |
| 415 |
< |
\includegraphics[width=\linewidth]{pressure_tb} |
| 416 |
< |
\caption{Pressure response is rapid (18 \AA gold nanoparticle), target |
| 417 |
< |
pressure = 4 GPa} |
| 418 |
< |
\label{pressureResponse} |
| 419 |
< |
\end{figure} |
| 412 |
> |
To test the new method, we have carried out simulations using the |
| 413 |
> |
Langevin Hull on: 1) a crystalline system (gold nanoparticles), 2) a |
| 414 |
> |
liquid droplet (SPC/E water),\cite{Berendsen1987} and 3) a |
| 415 |
> |
heterogeneous mixture (gold nanoparticles in a water droplet). In each |
| 416 |
> |
case, we have computed properties that depend on the external applied |
| 417 |
> |
pressure. Of particular interest for the single-phase systems is the |
| 418 |
> |
isothermal compressibility, |
| 419 |
> |
\begin{equation} |
| 420 |
> |
\kappa_{T} = -\frac{1}{V} \left ( \frac{\partial V}{\partial P} \right |
| 421 |
> |
)_{T}. |
| 422 |
> |
\label{eq:BM} |
| 423 |
> |
\end{equation} |
| 424 |
|
|
| 425 |
< |
\begin{figure} |
| 426 |
< |
\includegraphics[width=\linewidth]{temperature_tb} |
| 427 |
< |
\caption{Temperature equilibration depends on surface area and bath |
| 428 |
< |
viscosity. Target Temperature = 300K} |
| 429 |
< |
\label{temperatureResponse} |
| 430 |
< |
\end{figure} |
| 425 |
> |
One problem with eliminating periodic boundary conditions and |
| 426 |
> |
simulation boxes is that the volume of a three-dimensional point cloud |
| 427 |
> |
is not well-defined. In order to compute the compressibility of a |
| 428 |
> |
bulk material, we make an assumption that the number density, $\rho = |
| 429 |
> |
\frac{N}{V}$, is uniform within some region of the point cloud. The |
| 430 |
> |
compressibility can then be expressed in terms of the average number |
| 431 |
> |
of particles in that region, |
| 432 |
> |
\begin{equation} |
| 433 |
> |
\kappa_{T} = -\frac{1}{N} \left ( \frac{\partial N}{\partial P} \right |
| 434 |
> |
)_{T} |
| 435 |
> |
\label{eq:BMN} |
| 436 |
> |
\end{equation} |
| 437 |
> |
The region we used is a spherical volume of 10 \AA\ radius centered in |
| 438 |
> |
the middle of the cluster. $N$ is the average number of molecules |
| 439 |
> |
found within this region throughout a given simulation. The geometry |
| 440 |
> |
and size of the region is arbitrary, and any bulk-like portion of the |
| 441 |
> |
cluster can be used to compute the compressibility. |
| 442 |
|
|
| 443 |
+ |
One might assume that the volume of the convex hull could simply be |
| 444 |
+ |
taken as the system volume $V$ in the compressibility expression |
| 445 |
+ |
(Eq. \ref{eq:BM}), but this has implications at lower pressures (which |
| 446 |
+ |
are explored in detail in the section on water droplets). |
| 447 |
+ |
|
| 448 |
+ |
The metallic force field in use for the gold nanoparticles is the |
| 449 |
+ |
quantum Sutton-Chen (QSC) model.\cite{PhysRevB.59.3527} In all |
| 450 |
+ |
simulations involving point charges, we utilized damped shifted-force |
| 451 |
+ |
(DSF) electrostatics\cite{Fennell06} which is a variant of the Wolf |
| 452 |
+ |
summation\cite{wolf:8254} that has been shown to provide good forces |
| 453 |
+ |
and torques on molecular models for water in a computationally |
| 454 |
+ |
efficient manner.\cite{Fennell06} The damping parameter ($\alpha$) was |
| 455 |
+ |
set to 0.18 \AA$^{-1}$, and the cutoff radius was set to 12 \AA. The |
| 456 |
+ |
Spohr potential was adopted in depicting the interaction between metal |
| 457 |
+ |
atoms and the SPC/E water molecules.\cite{ISI:000167766600035} |
| 458 |
+ |
|
| 459 |
+ |
\subsection{Bulk Modulus of gold nanoparticles} |
| 460 |
+ |
|
| 461 |
+ |
The compressibility (and its inverse, the bulk modulus) is well-known |
| 462 |
+ |
for gold, and is captured well by the embedded atom method |
| 463 |
+ |
(EAM)~\cite{PhysRevB.33.7983} potential and related multi-body force |
| 464 |
+ |
fields. In particular, the quantum Sutton-Chen potential gets nearly |
| 465 |
+ |
quantitative agreement with the experimental bulk modulus values, and |
| 466 |
+ |
makes a good first test of how the Langevin Hull will perform at large |
| 467 |
+ |
applied pressures. |
| 468 |
+ |
|
| 469 |
+ |
The Sutton-Chen (SC) potentials are based on a model of a metal which |
| 470 |
+ |
treats the nuclei and core electrons as pseudo-atoms embedded in the |
| 471 |
+ |
electron density due to the valence electrons on all of the other |
| 472 |
+ |
atoms in the system.\cite{Chen90} The SC potential has a simple form |
| 473 |
+ |
that closely resembles the Lennard Jones potential, |
| 474 |
|
\begin{equation} |
| 475 |
< |
\kappa_T=-\frac{1}{V_{\mathrm{eq}}}\left(\frac{\partial V}{\partial |
| 476 |
< |
P}\right) |
| 475 |
> |
\label{eq:SCP1} |
| 476 |
> |
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] , |
| 477 |
|
\end{equation} |
| 478 |
+ |
where $V^{pair}_{ij}$ and $\rho_{i}$ are given by |
| 479 |
+ |
\begin{equation} |
| 480 |
+ |
\label{eq:SCP2} |
| 481 |
+ |
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}}. |
| 482 |
+ |
\end{equation} |
| 483 |
+ |
$V^{pair}_{ij}$ is a repulsive pairwise potential that accounts for |
| 484 |
+ |
interactions between the pseudoatom cores. The $\sqrt{\rho_i}$ term in |
| 485 |
+ |
Eq. (\ref{eq:SCP1}) is an attractive many-body potential that models |
| 486 |
+ |
the interactions between the valence electrons and the cores of the |
| 487 |
+ |
pseudo-atoms. $D_{ij}$, $D_{ii}$ set the appropriate overall energy |
| 488 |
+ |
scale, $c_i$ scales the attractive portion of the potential relative |
| 489 |
+ |
to the repulsive interaction and $\alpha_{ij}$ is a length parameter |
| 490 |
+ |
that assures a dimensionless form for $\rho$. These parameters are |
| 491 |
+ |
tuned to various experimental properties such as the density, cohesive |
| 492 |
+ |
energy, and elastic moduli for FCC transition metals. The quantum |
| 493 |
+ |
Sutton-Chen (QSC) formulation matches these properties while including |
| 494 |
+ |
zero-point quantum corrections for different transition |
| 495 |
+ |
metals.\cite{PhysRevB.59.3527,QSC} |
| 496 |
|
|
| 497 |
+ |
In bulk gold, the experimentally-measured value for the bulk modulus |
| 498 |
+ |
is 180.32 GPa, while previous calculations on the QSC potential in |
| 499 |
+ |
periodic-boundary simulations of the bulk crystal have yielded values |
| 500 |
+ |
of 175.53 GPa.\cite{QSC} Using the same force field, we have performed |
| 501 |
+ |
a series of 1 ns simulations on 40 \AA~ radius |
| 502 |
+ |
nanoparticles under the Langevin Hull at a variety of applied |
| 503 |
+ |
pressures ranging from 0 -- 10 GPa. We obtain a value of 177.55 GPa |
| 504 |
+ |
for the bulk modulus of gold using this technique, in close agreement |
| 505 |
+ |
with both previous simulations and the experimental bulk modulus of |
| 506 |
+ |
gold. |
| 507 |
+ |
|
| 508 |
|
\begin{figure} |
| 509 |
< |
\includegraphics[width=\linewidth]{compress_tb} |
| 510 |
< |
\caption{Isothermal Compressibility (18 \AA gold nanoparticle)} |
| 511 |
< |
\label{temperatureResponse} |
| 509 |
> |
\includegraphics[width=\linewidth]{stacked} |
| 510 |
> |
\caption{The response of the internal pressure and temperature of gold |
| 511 |
> |
nanoparticles when first placed in the Langevin Hull |
| 512 |
> |
($T_\mathrm{bath}$ = 300K, $P_\mathrm{bath}$ = 4 GPa), starting |
| 513 |
> |
from initial conditions that were far from the bath pressure and |
| 514 |
> |
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).} |
| 515 |
> |
\label{fig:pressureResponse} |
| 516 |
|
\end{figure} |
| 517 |
|
|
| 518 |
+ |
We note that the Langevin Hull produces rapidly-converging behavior |
| 519 |
+ |
for structures that are started far from equilibrium. In |
| 520 |
+ |
Fig. \ref{fig:pressureResponse} we show how the pressure and |
| 521 |
+ |
temperature respond to the Langevin Hull for nanoparticles that were |
| 522 |
+ |
initialized far from the target pressure and temperature. As |
| 523 |
+ |
expected, the rate at which thermal equilibrium is achieved depends on |
| 524 |
+ |
the total surface area of the cluter exposed to the bath as well as |
| 525 |
+ |
the bath viscosity. Pressure that is applied suddenly to a cluster |
| 526 |
+ |
can excite breathing vibrations, but these rapidly damp out (on time |
| 527 |
+ |
scales of 30-50 ps). |
| 528 |
+ |
|
| 529 |
|
\subsection{Compressibility of SPC/E water clusters} |
| 530 |
|
|
| 531 |
< |
Both NVT \cite{Glattli2002} and NPT \cite{Motakabbir1990, Pi2009} molecular dynamics simulations of SPC/E water have yielded values for the isothermal compressibility of water that agree well with experiment \cite{Fine1973}. The results of three different methods for computing the isothermal compressibility from Langevin Hull simulations for pressures between 1 and 6500 atm are shown in Fig. 5 along with compressibility values obtained from both other SPC/E simulations and experiment. Compressibility values from all references are for applied pressures within the range 1 - 1000 atm. |
| 531 |
> |
Prior molecular dynamics simulations on SPC/E water (both in |
| 532 |
> |
NVT~\cite{Glattli2002} and NPT~\cite{Motakabbir1990, Pi2009} |
| 533 |
> |
ensembles) have yielded values for the isothermal compressibility that |
| 534 |
> |
agree well with experiment.\cite{Fine1973} The results of two |
| 535 |
> |
different approaches for computing the isothermal compressibility from |
| 536 |
> |
Langevin Hull simulations for pressures between 1 and 6500 atm are |
| 537 |
> |
shown in Fig. \ref{fig:compWater} along with compressibility values |
| 538 |
> |
obtained from both other SPC/E simulations and experiment. |
| 539 |
|
|
| 540 |
|
\begin{figure} |
| 541 |
|
\includegraphics[width=\linewidth]{new_isothermalN} |
| 542 |
|
\caption{Compressibility of SPC/E water} |
| 543 |
< |
\label{compWater} |
| 543 |
> |
\label{fig:compWater} |
| 544 |
|
\end{figure} |
| 545 |
|
|
| 546 |
< |
The volume of a three-dimensional point cloud is not an obvious property to calculate. In order to calculate the isothermal compressibility we adapted the classic compressibility formula so that the compressibility could be calculated using information about the local density instead of the total volume of the convex hull. |
| 546 |
> |
Isothermal compressibility values calculated using the number density |
| 547 |
> |
(Eq. \ref{eq:BMN}) expression are in good agreement with experimental |
| 548 |
> |
and previous simulation work throughout the 1 -- 1000 atm pressure |
| 549 |
> |
regime. Compressibilities computed using the Hull volume, however, |
| 550 |
> |
deviate dramatically from the experimental values at low applied |
| 551 |
> |
pressures. The reason for this deviation is quite simple; at low |
| 552 |
> |
applied pressures, the liquid is in equilibrium with a vapor phase, |
| 553 |
> |
and it is entirely possible for one (or a few) molecules to drift away |
| 554 |
> |
from the liquid cluster (see Fig. \ref{fig:coneOfShame}). At low |
| 555 |
> |
pressures, the restoring forces on the facets are very gentle, and |
| 556 |
> |
this means that the hulls often take on relatively distorted |
| 557 |
> |
geometries which include large volumes of empty space. |
| 558 |
|
|
| 559 |
< |
\begin{equation} |
| 560 |
< |
\kappa_{T} = -\frac{1}{V} \left ( \frac{\partial V}{\partial P} \right )_{T} |
| 561 |
< |
\end{equation} |
| 559 |
> |
\begin{figure} |
| 560 |
> |
\includegraphics[width=\linewidth]{coneOfShame} |
| 561 |
> |
\caption{At low pressures, the liquid is in equilibrium with the vapor |
| 562 |
> |
phase, and isolated molecules can detach from the liquid droplet. |
| 563 |
> |
This is expected behavior, but the volume of the convex hull |
| 564 |
> |
includes large regions of empty space. For this reason, |
| 565 |
> |
compressibilities are computed using local number densities rather |
| 566 |
> |
than hull volumes.} |
| 567 |
> |
\label{fig:coneOfShame} |
| 568 |
> |
\end{figure} |
| 569 |
|
|
| 570 |
+ |
At higher pressures, the equilibrium strongly favors the liquid phase, |
| 571 |
+ |
and the hull geometries are much more compact. Because of the |
| 572 |
+ |
liquid-vapor effect on the convex hull, the regional number density |
| 573 |
+ |
approach (Eq. \ref{eq:BMN}) provides more reliable estimates of the |
| 574 |
+ |
compressibility. |
| 575 |
|
|
| 576 |
< |
Assuming a uniform density, we can use the relationship $\rho = \frac{N}{V}$ to rewrite the isothermal compressibility formula as |
| 577 |
< |
|
| 576 |
> |
In both the traditional compressibility formula (Eq. \ref{eq:BM}) and |
| 577 |
> |
the number density version (Eq. \ref{eq:BMN}), multiple simulations at |
| 578 |
> |
different pressures must be done to compute the first derivatives. It |
| 579 |
> |
is also possible to compute the compressibility using the fluctuation |
| 580 |
> |
dissipation theorem using either fluctuations in the |
| 581 |
> |
volume,\cite{Debenedetti1986}, |
| 582 |
|
\begin{equation} |
| 583 |
< |
\kappa_{T} = \frac{1}{N} \left ( \frac{\partial N}{\partial P} \right )_{T} |
| 583 |
> |
\kappa_{T} = \frac{\left \langle V^{2} \right \rangle - \left \langle |
| 584 |
> |
V \right \rangle ^{2}}{V \, k_{B} \, T}, |
| 585 |
> |
\label{eq:BMVfluct} |
| 586 |
|
\end{equation} |
| 587 |
< |
|
| 588 |
< |
Isothermal compressibility values calculated using this modified expression are in good agreement with the reference values throughout the 1 - 1000 atm pressure regime. Regardless of the difficulty in obtaining accurate hull volumes at low temperature and pressures, the Langevin Hull NPT method provides reasonable isothermal compressibility values for water through a large range of pressures. |
| 401 |
< |
|
| 402 |
< |
We initially used the classic compressibility formula to calculate the the isothermal compressibility at each target pressure. These calculations yielded compressibility values that were dramatically higher than both previous simulations and experiment. The particular compressibility expression used requires the calculation of both a volume and pressure differential, thereby stipulating that the data from at least two simulations at different pressures must be used to calculate the isothermal compressibility at one pressure. |
| 403 |
< |
|
| 404 |
< |
Per the fluctuation dissipation theorem \cite{Debenedetti1986}, the hull volume fluctuation in any given simulation can be used to calculated the isothermal compressibility at that particular pressure |
| 405 |
< |
|
| 587 |
> |
or, equivalently, fluctuations in the number of molecules within the |
| 588 |
> |
fixed region, |
| 589 |
|
\begin{equation} |
| 590 |
< |
\kappa_{T} = \frac{\left \langle V^{2} \right \rangle - \left \langle V \right \rangle ^{2}}{V \, k_{B} \, T} |
| 590 |
> |
\kappa_{T} = \frac{\left \langle N^{2} \right \rangle - \left \langle |
| 591 |
> |
N \right \rangle ^{2}}{N \, k_{B} \, T}, |
| 592 |
> |
\label{eq:BMNfluct} |
| 593 |
|
\end{equation} |
| 594 |
+ |
Thus, the compressibility of each simulation can be calculated |
| 595 |
+ |
entirely independently from other trajectories. Compressibility |
| 596 |
+ |
calculations that rely on the hull volume will still suffer the |
| 597 |
+ |
effects of the empty space due to the vapor phase; for this reason, we |
| 598 |
+ |
recommend using the number density (Eq. \ref{eq:BMN}) or number |
| 599 |
+ |
density fluctuations (Eq. \ref{eq:BMNfluct}) for computing |
| 600 |
+ |
compressibilities. |
| 601 |
|
|
| 410 |
– |
Thus, the compressibility of each simulation run can be calculated entirely independently from all other trajectories. However, the resulting compressibilities were still as much as an order of magnitude larger than the reference values. The effect was particularly pronounced at the low end of the pressure range. At ambient temperature and low pressures, there exists an equilibrium between vapor and liquid phases. Vapor molecules are naturally more diffuse around the exterior of the cluster, causing artificially large cluster volumes. Any compressibility calculation that relies on the hull volume will suffer these effects. |
| 411 |
– |
|
| 412 |
– |
|
| 602 |
|
\subsection{Molecular orientation distribution at cluster boundary} |
| 603 |
|
|
| 604 |
< |
In order for non-periodic boundary conditions to be widely applicable, they must be constructed in such a way that they allow a finite, usually small, simulated system to replicate the properties of an infinite bulk system. Naturally, this requirement has spawned many methods for inserting boundaries into simulated systems [REF... ?]. Of particular interest to our characterization of the Langevin Hull is the orientation of water molecules included in the geometric hull. Ideally, all molecules in the cluster will have the same orientational distribution as bulk water. |
| 604 |
> |
In order for a non-periodic boundary method to be widely applicable, |
| 605 |
> |
it must be constructed in such a way that they allow a finite system |
| 606 |
> |
to replicate the properties of the bulk. Early non-periodic simulation |
| 607 |
> |
methods (e.g. hydrophobic boundary potentials) induced spurious |
| 608 |
> |
orientational correlations deep within the simulated |
| 609 |
> |
system.\cite{Lee1984,Belch1985} This behavior spawned many methods for |
| 610 |
> |
fixing and characterizing the effects of artifical boundaries |
| 611 |
> |
including methods which fix the orientations of a set of edge |
| 612 |
> |
molecules.\cite{Warshel1978,King1989} |
| 613 |
|
|
| 614 |
< |
The orientation of molecules at the edges of a simulated cluster has long been a concern when performing simulations of explicitly non-periodic systems. Early work led to the surface constrained soft sphere dipole model (SCSSD) \cite{Warshel1978} in which the surface molecules are fixed in a random orientation representative of the bulk solvent structural properties. Belch, et al \cite{Belch1985} simulated clusters of TIPS2 water surrounded by a hydrophobic bounding potential. The spherical hydrophobic boundary induced dangling hydrogen bonds at the surface that propagated deep into the cluster, affecting 70\% of the 100 molecules in the simulation. This result echoes an earlier study which showed that an extended planar hydrophobic surface caused orientational preference at the surface which extended 7 \r{A} into the liquid simulation cell \cite{Lee1984}. The surface constrained all-atom solvent (SCAAS) model \cite{King1989} improved upon its SCSSD predecessor. The SCAAS model utilizes a polarization constraint which is applied to the surface molecules to maintain bulk-like structure at the cluster surface. A radial constraint is used to maintain the desired bulk density of the liquid. Both constraint forces are applied only to a pre-determined number of the outermost molecules. |
| 614 |
> |
As described above, the Langevin Hull does not require that the |
| 615 |
> |
orientation of molecules be fixed, nor does it utilize an explicitly |
| 616 |
> |
hydrophobic boundary, or orientational or radial constraints. |
| 617 |
> |
Therefore, the orientational correlations of the molecules in water |
| 618 |
> |
clusters are of particular interest in testing this method. Ideally, |
| 619 |
> |
the water molecules on the surfaces of the clusterss will have enough |
| 620 |
> |
mobility into and out of the center of the cluster to maintain |
| 621 |
> |
bulk-like orientational distribution in the absence of orientational |
| 622 |
> |
and radial constraints. However, since the number of hydrogen bonding |
| 623 |
> |
partners available to molecules on the exterior are limited, it is |
| 624 |
> |
likely that there will be an effective hydrophobicity of the hull. |
| 625 |
|
|
| 626 |
< |
In contrast, the Langevin Hull does not require that the orientation of molecules be fixed, nor does it utilize an explicitly hydrophobic boundary, orientational constraint or radial constraint. The number and identity of the molecules included on the convex hull are dynamic properties, thus avoiding the formation of an artificial solvent boundary layer. The hope is that the water molecules on the surface of the cluster, if left to their own devices in the absence of orientational and radial constraints, will maintain a bulk-like orientational distribution. |
| 627 |
< |
|
| 628 |
< |
To determine the extent of these effects demonstrated by the Langevin Hull, we examined the orientations exhibited by SPC/E water in a cluster of 1372 molecules at 300 K and at pressures ranging from 1 - 1000 atm. |
| 629 |
< |
|
| 423 |
< |
The orientation of a water molecule is described by |
| 424 |
< |
|
| 626 |
> |
To determine the extent of these effects, we examined the |
| 627 |
> |
orientations exhibited by SPC/E water in a cluster of 1372 |
| 628 |
> |
molecules at 300 K and at pressures ranging from 1 -- 1000 atm. The |
| 629 |
> |
orientational angle of a water molecule is described by |
| 630 |
|
\begin{equation} |
| 631 |
|
\cos{\theta}=\frac{\vec{r}_i\cdot\vec{\mu}_i}{|\vec{r}_i||\vec{\mu}_i|} |
| 632 |
|
\end{equation} |
| 633 |
+ |
where $\vec{r}_{i}$ is the vector between molecule {\it i}'s center of |
| 634 |
+ |
mass and the cluster center of mass, and $\vec{\mu}_{i}$ is the vector |
| 635 |
+ |
bisecting the H-O-H angle of molecule {\it i}. Bulk-like |
| 636 |
+ |
distributions will result in $\langle \cos \theta \rangle$ values |
| 637 |
+ |
close to zero. If the hull exhibits an overabundance of |
| 638 |
+ |
externally-oriented oxygen sites, the average orientation will be |
| 639 |
+ |
negative, while dangling hydrogen sites will result in positive |
| 640 |
+ |
average orientations. |
| 641 |
|
|
| 642 |
< |
where $\vec{r}_{i}$ is the vector between molecule {\it i}'s center of mass and the cluster center of mass and $\vec{\mu}_{i}$ is the vector bisecting the H-O-H angle of molecule {\it i}. |
| 643 |
< |
|
| 642 |
> |
Fig. \ref{fig:pAngle} shows the distribution of $\cos{\theta}$ values |
| 643 |
> |
for molecules in the interior of the cluster (squares) and for |
| 644 |
> |
molecules included in the convex hull (circles). |
| 645 |
|
\begin{figure} |
| 646 |
< |
\includegraphics[width=\linewidth]{g_r_theta} |
| 647 |
< |
\caption{Definition of coordinates} |
| 648 |
< |
\label{coords} |
| 646 |
> |
\includegraphics[width=\linewidth]{pAngle} |
| 647 |
> |
\caption{Distribution of $\cos{\theta}$ values for molecules on the |
| 648 |
> |
interior of the cluster (squares) and for those participating in the |
| 649 |
> |
convex hull (circles) at a variety of pressures. The Langevin hull |
| 650 |
> |
exhibits minor dewetting behavior with exposed oxygen sites on the |
| 651 |
> |
hull water molecules. The orientational preference for exposed |
| 652 |
> |
oxygen appears to be independent of applied pressure. } |
| 653 |
> |
\label{fig:pAngle} |
| 654 |
|
\end{figure} |
| 655 |
|
|
| 656 |
< |
Fig. 7 shows the probability of each value of $\cos{\theta}$ for molecules in the interior of the cluster (squares) and for molecules included in the convex hull (circles). |
| 656 |
> |
As expected, interior molecules (those not included in the convex |
| 657 |
> |
hull) maintain a bulk-like structure with a uniform distribution of |
| 658 |
> |
orientations. Molecules included in the convex hull show a slight |
| 659 |
> |
preference for values of $\cos{\theta} < 0.$ These values correspond |
| 660 |
> |
to molecules with oxygen directed toward the exterior of the cluster, |
| 661 |
> |
forming a dangling hydrogen bond acceptor site. |
| 662 |
|
|
| 663 |
+ |
Previous molecular dynamics simulations |
| 664 |
+ |
of SPC/E water that use periodic boundary conditions have shown that molecules on the liquid side of the liquid/vapor interface favor a similar orientation where oxygen is directed away from the bulk.\cite{Taylor1996} These simulations had both a liquid phase and a well-defined vapor phase in equilibrium and showed that vapor molecules generally had one hydrogen protruding from the surface, forming a dangling hydrogen bond donor. Our water cluster simulations do not have a true lasting vapor phase, but rather a few transient molecules that leave the liquid droplet. Thus we are unable to comment on the orientational preference of vapor phase molecules in a Langevin Hull simulation. |
| 665 |
+ |
|
| 666 |
+ |
However, the orientational preference exhibited by liquid phase hull molecules in the Langevin hull is significantly weaker than the preference caused by an explicit hydrophobic bounding potential. Additionally, the Langevin Hull does not require that the orientation of any molecules be fixed in order to maintain bulk-like structure, even at the cluster surface. |
| 667 |
+ |
|
| 668 |
+ |
\subsection{Heterogeneous nanoparticle / water mixtures} |
| 669 |
+ |
|
| 670 |
+ |
To further test the method, we simulated gold nanopartices ($r = 18$ |
| 671 |
+ |
\AA) solvated by explicit SPC/E water clusters using the Langevin |
| 672 |
+ |
hull. This was done at pressures of 1, 2, 5, 10, 20, 50, 100 and 200 atm |
| 673 |
+ |
in order to observe the effects of pressure on the ordering of water |
| 674 |
+ |
ordering at the surface. In Fig. \ref{fig:RhoR} we show the density |
| 675 |
+ |
of water adjacent to the surface as a function of pressure, as well as |
| 676 |
+ |
the orientational ordering of water at the surface of the |
| 677 |
+ |
nanoparticle. |
| 678 |
+ |
|
| 679 |
|
\begin{figure} |
| 680 |
< |
\includegraphics[width=\linewidth]{pAngle} |
| 681 |
< |
\caption{SPC/E water clusters: only minor dewetting at the boundary} |
| 682 |
< |
\label{pAngle} |
| 680 |
> |
|
| 681 |
> |
\caption{interesting plot showing cluster behavior} |
| 682 |
> |
\label{fig:RhoR} |
| 683 |
|
\end{figure} |
| 684 |
|
|
| 685 |
< |
As expected, interior molecules (those not included in the convex hull) maintain a bulk-like structure with a uniform distribution of orientations. Molecules included in the convex hull show a slight preference for values of $\cos{\theta} < 0.$ These values correspond to molecules with a hydrogen directed toward the exterior of the cluster, forming a dangling hydrogen bond. |
| 685 |
> |
At higher pressures, problems with the gold - water interaction |
| 686 |
> |
potential became apparent. The model we are using (due to Spohr) was |
| 687 |
> |
intended for relatively low pressures; it utilizes both shifted Morse |
| 688 |
> |
and repulsive Morse potentials to model the Au/O and Au/H |
| 689 |
> |
interactions, respectively. The repulsive wall of the Morse potential |
| 690 |
> |
does not diverge quickly enough at short distances to prevent water |
| 691 |
> |
from diffusing into the center of the gold nanoparticles. This |
| 692 |
> |
behavior is likely not a realistic description of the real physics of |
| 693 |
> |
the situation. A better model of the gold-water adsorption behavior |
| 694 |
> |
appears to require harder repulsive walls to prevent this behavior. |
| 695 |
|
|
| 696 |
< |
In the absence of an electrostatic contribution from the exterior bath, the orientational distribution of water molecules included in the Langevin Hull will slightly resemble the distribution at a neat water liquid/vapor interface. Previous molecular dynamics simulations of SPC/E water \cite{Taylor1996} have shown that molecules at the liquid/vapor interface favor an orientation where one hydrogen protrudes from the liquid phase. This behavior is demonstrated by experiments \cite{Du1994} \cite{Scatena2001} showing that approximately one-quarter of water molecules at the liquid/vapor interface form dangling hydrogen bonds. The negligible preference shown in these cluster simulations could be removed through the introduction of an implicit solvent model, which would provide the missing electrostatic interactions between the cluster molecules and the surrounding temperature/pressure bath. |
| 696 |
> |
\section{Discussion} |
| 697 |
> |
\label{sec:discussion} |
| 698 |
|
|
| 699 |
< |
The orientational preference exhibited by hull molecules is significantly weaker than the preference caused by an explicit hydrophobic bounding potential. Additionally, the Langevin Hull does not require that the orientation of any molecules be fixed in order to maintain bulk-like structure, even at the cluster surface. |
| 699 |
> |
The Langevin Hull samples the isobaric-isothermal ensemble for |
| 700 |
> |
non-periodic systems by coupling the system to a bath characterized by |
| 701 |
> |
pressure, temperature, and solvent viscosity. This enables the |
| 702 |
> |
simulation of heterogeneous systems composed of materials with |
| 703 |
> |
significantly different compressibilities. Because the boundary is |
| 704 |
> |
dynamically determined during the simulation and the molecules |
| 705 |
> |
interacting with the boundary can change, the method inflicts minimal |
| 706 |
> |
perturbations on the behavior of molecules at the edges of the |
| 707 |
> |
simulation. Further work on this method will involve implicit |
| 708 |
> |
electrostatics at the boundary (which is missing in the current |
| 709 |
> |
implementation) as well as more sophisticated treatments of the |
| 710 |
> |
surface geometry (alpha |
| 711 |
> |
shapes\cite{EDELSBRUNNER:1994oq,EDELSBRUNNER:1995cj} and Tight |
| 712 |
> |
Cocone\cite{Dey:2003ts}). The non-convex hull geometries are |
| 713 |
> |
significantly more expensive ($\mathcal{O}(N^2)$) than the convex hull |
| 714 |
> |
($\mathcal{O}(N \log N)$), but would enable the use of hull volumes |
| 715 |
> |
directly in computing the compressibility of the sample. |
| 716 |
|
|
| 717 |
+ |
\section*{Appendix A: Computing Convex Hulls on Parallel Computers} |
| 718 |
|
|
| 719 |
< |
\subsection{Heterogeneous nanoparticle / water mixtures} |
| 719 |
> |
In order to use the Langevin Hull for simulations on parallel |
| 720 |
> |
computers, one of the more difficult tasks is to compute the bounding |
| 721 |
> |
surface, facets, and resistance tensors when the individual processors |
| 722 |
> |
have incomplete information about the entire system's topology. Most |
| 723 |
> |
parallel decomposition methods assign primary responsibility for the |
| 724 |
> |
motion of an atomic site to a single processor, and we can exploit |
| 725 |
> |
this to efficiently compute the convex hull for the entire system. |
| 726 |
|
|
| 727 |
+ |
The basic idea involves splitting the point cloud into |
| 728 |
+ |
spatially-overlapping subsets and computing the convex hulls for each |
| 729 |
+ |
of the subsets. The points on the convex hull of the entire system |
| 730 |
+ |
are all present on at least one of the subset hulls. The algorithm |
| 731 |
+ |
works as follows: |
| 732 |
+ |
\begin{enumerate} |
| 733 |
+ |
\item Each processor computes the convex hull for its own atomic sites |
| 734 |
+ |
(left panel in Fig. \ref{fig:parallel}). |
| 735 |
+ |
\item The Hull vertices from each processor are communicated to all of |
| 736 |
+ |
the processors, and each processor assembles a complete list of hull |
| 737 |
+ |
sites (this is much smaller than the original number of points in |
| 738 |
+ |
the point cloud). |
| 739 |
+ |
\item Each processor computes the global convex hull (right panel in |
| 740 |
+ |
Fig. \ref{fig:parallel}) using only those points that are the union |
| 741 |
+ |
of sites gathered from all of the subset hulls. Delaunay |
| 742 |
+ |
triangulation is then done to obtain the facets of the global hull. |
| 743 |
+ |
\end{enumerate} |
| 744 |
|
|
| 745 |
< |
\section{Appendix A: Hydrodynamic tensor for triangular facets} |
| 745 |
> |
\begin{figure} |
| 746 |
> |
\includegraphics[width=\linewidth]{parallel} |
| 747 |
> |
\caption{When the sites are distributed among many nodes for parallel |
| 748 |
> |
computation, the processors first compute the convex hulls for their |
| 749 |
> |
own sites (dashed lines in left panel). The positions of the sites |
| 750 |
> |
that make up the subset hulls are then communicated to all |
| 751 |
> |
processors (middle panel). The convex hull of the system (solid line in |
| 752 |
> |
right panel) is the convex hull of the points on the union of the subset |
| 753 |
> |
hulls.} |
| 754 |
> |
\label{fig:parallel} |
| 755 |
> |
\end{figure} |
| 756 |
|
|
| 757 |
< |
\section{Appendix B: Computing Convex Hulls on Parallel Computers} |
| 757 |
> |
The individual hull operations scale with |
| 758 |
> |
$\mathcal{O}(\frac{n}{p}\log\frac{n}{p})$ where $n$ is the total |
| 759 |
> |
number of sites, and $p$ is the number of processors. These local |
| 760 |
> |
hull operations create a set of $p$ hulls, each with approximately |
| 761 |
> |
$\frac{n}{3pr}$ sites for a cluster of radius $r$. The worst-case |
| 762 |
> |
communication cost for using a ``gather'' operation to distribute this |
| 763 |
> |
information to all processors is $\mathcal{O}( \alpha (p-1) + \frac{n |
| 764 |
> |
\beta (p-1)}{3 r p^2})$, while the final computation of the system |
| 765 |
> |
hull scales as $\mathcal{O}(\frac{n}{3r}\log\frac{n}{3r})$. |
| 766 |
|
|
| 767 |
< |
\section{Acknowledgments} |
| 767 |
> |
For a large number of atoms on a moderately parallel machine, the |
| 768 |
> |
total costs are dominated by the computations of the individual hulls, |
| 769 |
> |
and communication of these hulls to create the Langevin hull sees roughly |
| 770 |
> |
linear speed-up with increasing processor counts. |
| 771 |
> |
|
| 772 |
> |
\section*{Acknowledgments} |
| 773 |
|
Support for this project was provided by the |
| 774 |
|
National Science Foundation under grant CHE-0848243. Computational |
| 775 |
|
time was provided by the Center for Research Computing (CRC) at the |
| 776 |
|
University of Notre Dame. |
| 777 |
|
|
| 778 |
+ |
Molecular graphics images were produced using the UCSF Chimera package from |
| 779 |
+ |
the Resource for Biocomputing, Visualization, and Informatics at the |
| 780 |
+ |
University of California, San Francisco (supported by NIH P41 RR001081). |
| 781 |
|
\newpage |
| 782 |
|
|
| 783 |
|
\bibliography{langevinHull} |
