| 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.} |
| 254 |
> |
between the interior (Newtonian) region and the Langevin Hull.} |
| 255 |
|
\label{fig:hullSample} |
| 256 |
|
\end{figure} |
| 257 |
|
|
| 266 |
|
potential energy. For atoms on the exterior of the cluster |
| 267 |
|
(i.e. those that occupy one of the vertices of the convex hull), the |
| 268 |
|
equation of motion is modified with an external force, ${\mathbf |
| 269 |
< |
F}_i^{\mathrm ext}$, |
| 269 |
> |
F}_i^{\mathrm ext}$: |
| 270 |
|
\begin{equation} |
| 271 |
|
m_i \dot{\mathbf v}_i(t)=-{\mathbf \nabla}_i U + {\mathbf F}_i^{\mathrm ext}. |
| 272 |
|
\end{equation} |
| 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, |
| 415 |
> |
heterogeneous mixture (gold nanoparticles in an SPC/E water droplet). In each case, we have computed properties that depend on the external applied pressure. Of particular interest for the single-phase systems is the isothermal compressibility, |
| 416 |
|
\begin{equation} |
| 417 |
|
\kappa_{T} = -\frac{1}{V} \left ( \frac{\partial V}{\partial P} \right |
| 418 |
|
)_{T}. |
| 421 |
|
|
| 422 |
|
One problem with eliminating periodic boundary conditions and |
| 423 |
|
simulation boxes is that the volume of a three-dimensional point cloud |
| 424 |
< |
is not well-defined. In order to compute the compressibility of a |
| 424 |
> |
is not well-defined. In order to compute the compressibility of a |
| 425 |
|
bulk material, we make an assumption that the number density, $\rho = |
| 426 |
< |
\frac{N}{V}$, is uniform within some region of the point cloud. The |
| 426 |
> |
\frac{N}{V}$, is uniform within some region of the point cloud. The |
| 427 |
|
compressibility can then be expressed in terms of the average number |
| 428 |
|
of particles in that region, |
| 429 |
|
\begin{equation} |
| 430 |
|
\kappa_{T} = -\frac{1}{N} \left ( \frac{\partial N}{\partial P} \right |
| 431 |
< |
)_{T} |
| 431 |
> |
)_{T}. |
| 432 |
|
\label{eq:BMN} |
| 433 |
|
\end{equation} |
| 434 |
|
The region we used is a spherical volume of 10 \AA\ radius centered in |
| 521 |
|
the total surface area of the cluter exposed to the bath as well as |
| 522 |
|
the bath viscosity. Pressure that is applied suddenly to a cluster |
| 523 |
|
can excite breathing vibrations, but these rapidly damp out (on time |
| 524 |
< |
scales of 30-50 ps). |
| 524 |
> |
scales of 30 -- 50 ps). |
| 525 |
|
|
| 526 |
|
\subsection{Compressibility of SPC/E water clusters} |
| 527 |
|
|
| 585 |
|
fixed region, |
| 586 |
|
\begin{equation} |
| 587 |
|
\kappa_{T} = \frac{\left \langle N^{2} \right \rangle - \left \langle |
| 588 |
< |
N \right \rangle ^{2}}{N \, k_{B} \, T}, |
| 588 |
> |
N \right \rangle ^{2}}{N \, k_{B} \, T}. |
| 589 |
|
\label{eq:BMNfluct} |
| 590 |
|
\end{equation} |
| 591 |
|
Thus, the compressibility of each simulation can be calculated |
| 643 |
|
\includegraphics[width=\linewidth]{pAngle} |
| 644 |
|
\caption{Distribution of $\cos{\theta}$ values for molecules on the |
| 645 |
|
interior of the cluster (squares) and for those participating in the |
| 646 |
< |
convex hull (circles) at a variety of pressures. The Langevin hull |
| 646 |
> |
convex hull (circles) at a variety of pressures. The Langevin Hull |
| 647 |
|
exhibits minor dewetting behavior with exposed oxygen sites on the |
| 648 |
|
hull water molecules. The orientational preference for exposed |
| 649 |
|
oxygen appears to be independent of applied pressure. } |
| 657 |
|
to molecules with oxygen directed toward the exterior of the cluster, |
| 658 |
|
forming a dangling hydrogen bond acceptor site. |
| 659 |
|
|
| 660 |
< |
In the absence of an electrostatic contribution from the exterior |
| 664 |
< |
bath, the orientational distribution of water molecules included in |
| 665 |
< |
the Langevin Hull will slightly resemble the distribution at a neat |
| 666 |
< |
water liquid/vapor interface. Previous molecular dynamics simulations |
| 667 |
< |
of SPC/E water \cite{Taylor1996} have shown that molecules at the |
| 668 |
< |
liquid/vapor interface favor an orientation where one hydrogen |
| 669 |
< |
protrudes from the liquid phase. This behavior is demonstrated by |
| 670 |
< |
experiments \cite{Du1994} \cite{Scatena2001} showing that |
| 671 |
< |
approximately one-quarter of water molecules at the liquid/vapor |
| 672 |
< |
interface form dangling hydrogen bonds. The negligible preference |
| 673 |
< |
shown in these cluster simulations could be removed through the |
| 674 |
< |
introduction of an implicit solvent model, which would provide the |
| 675 |
< |
missing electrostatic interactions between the cluster molecules and |
| 676 |
< |
the surrounding temperature/pressure bath. |
| 660 |
> |
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. |
| 661 |
|
|
| 662 |
< |
The orientational preference exhibited by hull molecules in the |
| 663 |
< |
Langevin hull is significantly weaker than the preference caused by an |
| 680 |
< |
explicit hydrophobic bounding potential. Additionally, the Langevin |
| 681 |
< |
Hull does not require that the orientation of any molecules be fixed |
| 682 |
< |
in order to maintain bulk-like structure, even at the cluster surface. |
| 662 |
> |
Previous molecular dynamics simulations |
| 663 |
> |
of SPC/E water using 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 while we are unable to comment on the orientational preference of vapor phase molecules in a Langevin Hull simulation, we achieve good agreement for the orientation of liquid phase molecules at the interface. |
| 664 |
|
|
| 665 |
|
\subsection{Heterogeneous nanoparticle / water mixtures} |
| 666 |
|
|
| 667 |
|
To further test the method, we simulated gold nanopartices ($r = 18$ |
| 668 |
|
\AA) solvated by explicit SPC/E water clusters using the Langevin |
| 669 |
< |
hull. This was done at pressures of 1, 2, 5, 10, 20, 50, 100 and 200 atm |
| 669 |
> |
Hull. This was done at pressures of 1, 2, 5, 10, 20, 50, 100 and 200 atm |
| 670 |
|
in order to observe the effects of pressure on the ordering of water |
| 671 |
|
ordering at the surface. In Fig. \ref{fig:RhoR} we show the density |
| 672 |
|
of water adjacent to the surface as a function of pressure, as well as |
| 675 |
|
|
| 676 |
|
\begin{figure} |
| 677 |
|
|
| 678 |
< |
\caption{interesting plot showing cluster behavior} |
| 678 |
> |
\caption{Higher applied pressures de-structure both the gold nanoparticle and water at the metal/water interface.} |
| 679 |
|
\label{fig:RhoR} |
| 680 |
|
\end{figure} |
| 681 |
|
|
| 763 |
|
|
| 764 |
|
For a large number of atoms on a moderately parallel machine, the |
| 765 |
|
total costs are dominated by the computations of the individual hulls, |
| 766 |
< |
and communication of these hulls to create the Langevin hull sees roughly |
| 766 |
> |
and communication of these hulls to create the Langevin Hull sees roughly |
| 767 |
|
linear speed-up with increasing processor counts. |
| 768 |
|
|
| 769 |
|
\section*{Acknowledgments} |
