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\bibpunct{[}{]}{,}{s}{}{;} |
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\bibliographystyle{achemso} |
| 22 |
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|
| 23 |
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\begin{document} |
| 42 |
|
hull surrounding the system. A Langevin thermostat is also applied |
| 43 |
|
to the facets to mimic contact with an external heat bath. This new |
| 44 |
|
method, the ``Langevin Hull'', can handle heterogeneous mixtures of |
| 45 |
< |
materials with different compressibilities. These are systems that |
| 46 |
< |
are problematic for traditional affine transform methods. The |
| 47 |
< |
Langevin Hull does not suffer from the edge effects of boundary |
| 48 |
< |
potential methods, and allows realistic treatment of both external |
| 49 |
< |
pressure and thermal conductivity due to the presence of an implicit |
| 50 |
< |
solvent. We apply this method to several different systems |
| 51 |
< |
including bare metal nanoparticles, nanoparticles in an explicit |
| 52 |
< |
solvent, as well as clusters of liquid water. The predicted |
| 53 |
< |
mechanical properties of these systems are in good agreement with |
| 54 |
< |
experimental data and previous simulation work. |
| 45 |
> |
materials with different compressibilities. These systems are |
| 46 |
> |
problematic for traditional affine transform methods. The Langevin |
| 47 |
> |
Hull does not suffer from the edge effects of boundary potential |
| 48 |
> |
methods, and allows realistic treatment of both external pressure |
| 49 |
> |
and thermal conductivity due to the presence of an implicit solvent. |
| 50 |
> |
We apply this method to several different systems including bare |
| 51 |
> |
metal nanoparticles, nanoparticles in an explicit solvent, as well |
| 52 |
> |
as clusters of liquid water. The predicted mechanical properties of |
| 53 |
> |
these systems are in good agreement with experimental data and |
| 54 |
> |
previous simulation work. |
| 55 |
|
\end{abstract} |
| 56 |
|
|
| 57 |
|
\newpage |
| 121 |
|
protein like hen egg white lysozyme (PDB code: 1LYZ) yields an |
| 122 |
|
effective protein concentration of 100 mg/mL.\cite{Asthagiri20053300} |
| 123 |
|
|
| 124 |
< |
Typically {\it total} protein concentrations in the cell are on the |
| 124 |
> |
{\it Total} protein concentrations in the cell are typically on the |
| 125 |
|
order of 160-310 mg/ml,\cite{Brown1991195} and individual proteins |
| 126 |
|
have concentrations orders of magnitude lower than this in the |
| 127 |
|
cellular environment. The effective concentrations of single proteins |
| 128 |
|
in simulations may have significant effects on the structure and |
| 129 |
< |
dynamics of simulated structures. |
| 129 |
> |
dynamics of simulated systems. |
| 130 |
|
|
| 131 |
|
\subsection*{Boundary Methods} |
| 132 |
|
There have been a number of approaches to handle simulations of |
| 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}$: |
| 270 |
|
\begin{equation} |
| 271 |
|
m_i \dot{\mathbf v}_i(t)=-{\mathbf \nabla}_i U + {\mathbf F}_i^{\mathrm ext}. |
| 272 |
|
\end{equation} |
| 378 |
|
configurations, so this appears to be a reasonably good approximation. |
| 379 |
|
|
| 380 |
|
We have implemented this method by extending the Langevin dynamics |
| 381 |
< |
integrator in our code, OpenMD.\cite{Meineke2005,openmd} At each |
| 381 |
> |
integrator in our code, OpenMD.\cite{Meineke2005,open_md} At each |
| 382 |
|
molecular dynamics time step, the following process is carried out: |
| 383 |
|
\begin{enumerate} |
| 384 |
|
\item The standard inter-atomic forces ($\nabla_iU$) are computed. |
| 387 |
|
\item The convex hull is computed and facets are identified. |
| 388 |
|
\item For each facet: |
| 389 |
|
\begin{itemize} |
| 390 |
< |
\item[a.] The force from the pressure bath ($-PA_f\hat{n}_f$) is |
| 390 |
> |
\item[a.] The force from the pressure bath ($-\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. |
| 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 |
| 403 |
> |
using calls to the qhull library.\cite{Q_hull} 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 |
| 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 |
| 435 |
< |
the middle of the cluster. $N$ is the average number of molecules |
| 434 |
> |
The region we used is a spherical volume of 20 \AA\ radius centered in |
| 435 |
> |
the middle of the cluster with a roughly 25 \AA\ radius. $N$ is the average number of molecules |
| 436 |
|
found within this region throughout a given simulation. The geometry |
| 437 |
< |
and size of the region is arbitrary, and any bulk-like portion of the |
| 438 |
< |
cluster can be used to compute the compressibility. |
| 437 |
> |
of the region is arbitrary, and any bulk-like portion of the |
| 438 |
> |
cluster can be used to compute the compressibility. |
| 439 |
|
|
| 440 |
|
One might assume that the volume of the convex hull could simply be |
| 441 |
|
taken as the system volume $V$ in the compressibility expression |
| 481 |
|
interactions between the pseudoatom cores. The $\sqrt{\rho_i}$ term in |
| 482 |
|
Eq. (\ref{eq:SCP1}) is an attractive many-body potential that models |
| 483 |
|
the interactions between the valence electrons and the cores of the |
| 484 |
< |
pseudo-atoms. $D_{ij}$, $D_{ii}$ set the appropriate overall energy |
| 484 |
> |
pseudo-atoms. $D_{ij}$ and $D_{ii}$ set the appropriate overall energy |
| 485 |
|
scale, $c_i$ scales the attractive portion of the potential relative |
| 486 |
|
to the repulsive interaction and $\alpha_{ij}$ is a length parameter |
| 487 |
|
that assures a dimensionless form for $\rho$. These parameters are |
| 489 |
|
energy, and elastic moduli for FCC transition metals. The quantum |
| 490 |
|
Sutton-Chen (QSC) formulation matches these properties while including |
| 491 |
|
zero-point quantum corrections for different transition |
| 492 |
< |
metals.\cite{PhysRevB.59.3527,QSC} |
| 492 |
> |
metals.\cite{PhysRevB.59.3527,QSC2} |
| 493 |
|
|
| 494 |
|
In bulk gold, the experimentally-measured value for the bulk modulus |
| 495 |
|
is 180.32 GPa, while previous calculations on the QSC potential in |
| 496 |
|
periodic-boundary simulations of the bulk crystal have yielded values |
| 497 |
< |
of 175.53 GPa.\cite{QSC} Using the same force field, we have performed |
| 498 |
< |
a series of relatively short (200 ps) simulations on 40 \AA~ radius |
| 499 |
< |
nanoparticles under the Langevin Hull at a variety of applied |
| 500 |
< |
pressures ranging from 0 -- 10 GPa. We obtain a value of 177.55 GPa |
| 501 |
< |
for the bulk modulus of gold using this technique, in close agreement |
| 502 |
< |
with both previous simulations and the experimental bulk modulus of |
| 503 |
< |
gold. |
| 497 |
> |
of 175.53 GPa.\cite{QSC2} Using the same force field, we have |
| 498 |
> |
performed a series of 1 ns simulations on gold nanoparticles of three |
| 499 |
> |
different radii: 20 \AA~ (1985 atoms), 30 \AA~ (6699 atoms), and 40 |
| 500 |
> |
\AA~ (15707 atoms) utilizing the Langevin Hull at a variety of applied |
| 501 |
> |
pressures ranging from 0 -- 10 GPa. For the 40 \AA~ radius |
| 502 |
> |
nanoparticle we obtain a value of 177.55 GPa for the bulk modulus of |
| 503 |
> |
gold, in close agreement with both previous simulations and the |
| 504 |
> |
experimental bulk modulus reported for gold single |
| 505 |
> |
crystals.\cite{Collard1991} The smaller gold nanoparticles (30 and 20 |
| 506 |
> |
\AA~ radii) have calculated bulk moduli of 215.58 and 208.86 GPa, |
| 507 |
> |
respectively, indicating that smaller nanoparticles are somewhat |
| 508 |
> |
stiffer (less compressible) than the larger nanoparticles. This |
| 509 |
> |
stiffening of the small nanoparticles may be related to their high |
| 510 |
> |
degree of surface curvature, resulting in a lower coordination number |
| 511 |
> |
of surface atoms relative to the the surface atoms in the 40 \AA~ |
| 512 |
> |
radius particle. |
| 513 |
|
|
| 514 |
+ |
We obtain a gold lattice constant of 4.051 \AA~ using the Langevin |
| 515 |
+ |
Hull at 1 atm, close to the experimentally-determined value for bulk |
| 516 |
+ |
gold and the value for gold simulated using the QSC potential and |
| 517 |
+ |
periodic boundary conditions (4.079 \AA~ and 4.088\AA~, |
| 518 |
+ |
respectively).\cite{QSC2} The slightly smaller calculated lattice |
| 519 |
+ |
constant is most likely due to the presence of surface tension in the |
| 520 |
+ |
non-periodic Langevin Hull cluster, an effect absent from a bulk |
| 521 |
+ |
simulation. The specific heat of a 40 \AA~ gold nanoparticle under the |
| 522 |
+ |
Langevin Hull at 1 atm is 24.914 $\mathrm {\frac{J}{mol \, K}}$, which |
| 523 |
+ |
compares very well with the experimental value of 25.42 $\mathrm |
| 524 |
+ |
{\frac{J}{mol \, K}}$. |
| 525 |
+ |
|
| 526 |
|
\begin{figure} |
| 527 |
|
\includegraphics[width=\linewidth]{stacked} |
| 528 |
|
\caption{The response of the internal pressure and temperature of gold |
| 539 |
|
temperature respond to the Langevin Hull for nanoparticles that were |
| 540 |
|
initialized far from the target pressure and temperature. As |
| 541 |
|
expected, the rate at which thermal equilibrium is achieved depends on |
| 542 |
< |
the total surface area of the cluter exposed to the bath as well as |
| 542 |
> |
the total surface area of the cluster exposed to the bath as well as |
| 543 |
|
the bath viscosity. Pressure that is applied suddenly to a cluster |
| 544 |
|
can excite breathing vibrations, but these rapidly damp out (on time |
| 545 |
< |
scales of 30-50 ps). |
| 545 |
> |
scales of 30 -- 50 ps). |
| 546 |
|
|
| 547 |
|
\subsection{Compressibility of SPC/E water clusters} |
| 548 |
|
|
| 551 |
|
ensembles) have yielded values for the isothermal compressibility that |
| 552 |
|
agree well with experiment.\cite{Fine1973} The results of two |
| 553 |
|
different approaches for computing the isothermal compressibility from |
| 554 |
< |
Langevin Hull simulations for pressures between 1 and 6500 atm are |
| 554 |
> |
Langevin Hull simulations for pressures between 1 and 3000 atm are |
| 555 |
|
shown in Fig. \ref{fig:compWater} along with compressibility values |
| 556 |
|
obtained from both other SPC/E simulations and experiment. |
| 557 |
|
|
| 566 |
|
and previous simulation work throughout the 1 -- 1000 atm pressure |
| 567 |
|
regime. Compressibilities computed using the Hull volume, however, |
| 568 |
|
deviate dramatically from the experimental values at low applied |
| 569 |
< |
pressures. The reason for this deviation is quite simple; at low |
| 569 |
> |
pressures. The reason for this deviation is quite simple: at low |
| 570 |
|
applied pressures, the liquid is in equilibrium with a vapor phase, |
| 571 |
|
and it is entirely possible for one (or a few) molecules to drift away |
| 572 |
|
from the liquid cluster (see Fig. \ref{fig:coneOfShame}). At low |
| 596 |
|
different pressures must be done to compute the first derivatives. It |
| 597 |
|
is also possible to compute the compressibility using the fluctuation |
| 598 |
|
dissipation theorem using either fluctuations in the |
| 599 |
< |
volume,\cite{Debenedetti1986}, |
| 599 |
> |
volume,\cite{Debenedetti1986} |
| 600 |
|
\begin{equation} |
| 601 |
|
\kappa_{T} = \frac{\left \langle V^{2} \right \rangle - \left \langle |
| 602 |
|
V \right \rangle ^{2}}{V \, k_{B} \, T}, |
| 606 |
|
fixed region, |
| 607 |
|
\begin{equation} |
| 608 |
|
\kappa_{T} = \frac{\left \langle N^{2} \right \rangle - \left \langle |
| 609 |
< |
N \right \rangle ^{2}}{N \, k_{B} \, T}, |
| 609 |
> |
N \right \rangle ^{2}}{N \, k_{B} \, T}. |
| 610 |
|
\label{eq:BMNfluct} |
| 611 |
|
\end{equation} |
| 612 |
|
Thus, the compressibility of each simulation can be calculated |
| 615 |
|
effects of the empty space due to the vapor phase; for this reason, we |
| 616 |
|
recommend using the number density (Eq. \ref{eq:BMN}) or number |
| 617 |
|
density fluctuations (Eq. \ref{eq:BMNfluct}) for computing |
| 618 |
< |
compressibilities. |
| 618 |
> |
compressibilities. We obtained the results in |
| 619 |
> |
Fig. \ref{fig:compWater} using a sampling radius that was |
| 620 |
> |
approximately 80\% of the mean distance between the center of mass of |
| 621 |
> |
the cluster and the hull atoms. This ratio of sampling radius to |
| 622 |
> |
average hull radius excludes the problematic vapor phase on the |
| 623 |
> |
outside of the cluster while including enough of the liquid phase to |
| 624 |
> |
avoid poor statistics due to fluctuating local densities. |
| 625 |
> |
|
| 626 |
> |
A comparison of the oxygen-oxygen radial distribution functions for |
| 627 |
> |
SPC/E water simulated using both the Langevin Hull and more |
| 628 |
> |
traditional periodic boundary methods -- both at 1 atm and 300K -- |
| 629 |
> |
reveals an understructuring of water in the Langevin Hull that |
| 630 |
> |
manifests as a slight broadening of the solvation shells. This effect |
| 631 |
> |
may be due to the introduction of a liquid-vapor interface in the |
| 632 |
> |
Langevin Hull simulations (an interface which is missing in most |
| 633 |
> |
periodic simulations of bulk water). Vapor-phase molecules contribute |
| 634 |
> |
a small but nearly flat portion of the radial distribution function. |
| 635 |
|
|
| 636 |
|
\subsection{Molecular orientation distribution at cluster boundary} |
| 637 |
|
|
| 638 |
|
In order for a non-periodic boundary method to be widely applicable, |
| 639 |
< |
they must be constructed in such a way that they allow a finite system |
| 639 |
> |
it must be constructed in such a way that they allow a finite system |
| 640 |
|
to replicate the properties of the bulk. Early non-periodic simulation |
| 641 |
|
methods (e.g. hydrophobic boundary potentials) induced spurious |
| 642 |
|
orientational correlations deep within the simulated |
| 643 |
|
system.\cite{Lee1984,Belch1985} This behavior spawned many methods for |
| 644 |
< |
fixing and characterizing the effects of artifical boundaries |
| 644 |
> |
fixing and characterizing the effects of artificial boundaries |
| 645 |
|
including methods which fix the orientations of a set of edge |
| 646 |
|
molecules.\cite{Warshel1978,King1989} |
| 647 |
|
|
| 650 |
|
hydrophobic boundary, or orientational or radial constraints. |
| 651 |
|
Therefore, the orientational correlations of the molecules in water |
| 652 |
|
clusters are of particular interest in testing this method. Ideally, |
| 653 |
< |
the water molecules on the surfaces of the clusterss will have enough |
| 653 |
> |
the water molecules on the surfaces of the clusters will have enough |
| 654 |
|
mobility into and out of the center of the cluster to maintain |
| 655 |
|
bulk-like orientational distribution in the absence of orientational |
| 656 |
|
and radial constraints. However, since the number of hydrogen bonding |
| 658 |
|
likely that there will be an effective hydrophobicity of the hull. |
| 659 |
|
|
| 660 |
|
To determine the extent of these effects, we examined the |
| 661 |
< |
orientationations exhibited by SPC/E water in a cluster of 1372 |
| 661 |
> |
orientations exhibited by SPC/E water in a cluster of 1372 |
| 662 |
|
molecules at 300 K and at pressures ranging from 1 -- 1000 atm. The |
| 663 |
< |
orientational angle of a water molecule is described |
| 663 |
> |
orientational angle of a water molecule is described by |
| 664 |
|
\begin{equation} |
| 665 |
|
\cos{\theta}=\frac{\vec{r}_i\cdot\vec{\mu}_i}{|\vec{r}_i||\vec{\mu}_i|} |
| 666 |
|
\end{equation} |
| 680 |
|
\includegraphics[width=\linewidth]{pAngle} |
| 681 |
|
\caption{Distribution of $\cos{\theta}$ values for molecules on the |
| 682 |
|
interior of the cluster (squares) and for those participating in the |
| 683 |
< |
convex hull (circles) at a variety of pressures. The Langevin hull |
| 683 |
> |
convex hull (circles) at a variety of pressures. The Langevin Hull |
| 684 |
|
exhibits minor dewetting behavior with exposed oxygen sites on the |
| 685 |
|
hull water molecules. The orientational preference for exposed |
| 686 |
|
oxygen appears to be independent of applied pressure. } |
| 692 |
|
orientations. Molecules included in the convex hull show a slight |
| 693 |
|
preference for values of $\cos{\theta} < 0.$ These values correspond |
| 694 |
|
to molecules with oxygen directed toward the exterior of the cluster, |
| 695 |
< |
forming a dangling hydrogen bond acceptor site. |
| 695 |
> |
forming dangling hydrogen bond acceptor sites. |
| 696 |
|
|
| 697 |
< |
In the absence of an electrostatic contribution from the exterior |
| 698 |
< |
bath, the orientational distribution of water molecules included in |
| 699 |
< |
the Langevin Hull will slightly resemble the distribution at a neat |
| 700 |
< |
water liquid/vapor interface. Previous molecular dynamics simulations |
| 701 |
< |
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. |
| 697 |
> |
The orientational preference exhibited by water molecules on the hull |
| 698 |
> |
is significantly weaker than the preference caused by an explicit |
| 699 |
> |
hydrophobic bounding potential. Additionally, the Langevin Hull does |
| 700 |
> |
not require that the orientation of any molecules be fixed in order to |
| 701 |
> |
maintain bulk-like structure, even near the cluster surface. |
| 702 |
|
|
| 703 |
< |
The orientational preference exhibited by hull molecules in the |
| 704 |
< |
Langevin hull is significantly weaker than the preference caused by an |
| 705 |
< |
explicit hydrophobic bounding potential. Additionally, the Langevin |
| 706 |
< |
Hull does not require that the orientation of any molecules be fixed |
| 707 |
< |
in order to maintain bulk-like structure, even at the cluster surface. |
| 703 |
> |
Previous molecular dynamics simulations of SPC/E liquid / vapor |
| 704 |
> |
interfaces using periodic boundary conditions have shown that |
| 705 |
> |
molecules on the liquid side of interface favor a similar orientation |
| 706 |
> |
where oxygen is directed away from the bulk.\cite{Taylor1996} These |
| 707 |
> |
simulations had well-defined liquid and vapor phase regions |
| 708 |
> |
equilibrium and it was observed that {\it vapor} molecules generally |
| 709 |
> |
had one hydrogen protruding from the surface, forming a dangling |
| 710 |
> |
hydrogen bond donor. Our water clusters do not have a true vapor |
| 711 |
> |
region, but rather a few transient molecules that leave the liquid |
| 712 |
> |
droplet (and which return to the droplet relatively quickly). |
| 713 |
> |
Although we cannot obtain an orientational preference of vapor phase |
| 714 |
> |
molecules in a Langevin Hull simulation, but we do agree with previous |
| 715 |
> |
estimates of the orientation of {\it liquid phase} molecules at the |
| 716 |
> |
interface. |
| 717 |
|
|
| 718 |
|
\subsection{Heterogeneous nanoparticle / water mixtures} |
| 719 |
|
|
| 720 |
< |
To further test the method, we simulated gold nanopartices ($r = 18$ |
| 721 |
< |
\AA) solvated by explicit SPC/E water clusters using the Langevin |
| 722 |
< |
hull. This was done at pressures of 1, 2, 5, 10, 20, 50 and 100 atm |
| 723 |
< |
in order to observe the effects of pressure on the ordering of water |
| 724 |
< |
ordering at the surface. In Fig. \ref{fig:RhoR} we show the density |
| 725 |
< |
of water adjacent to the surface as a function of pressure, as well as |
| 726 |
< |
the orientational ordering of water at the surface of the |
| 727 |
< |
nanoparticle. |
| 720 |
> |
To further test the method, we simulated gold nanoparticles ($r = 18$ |
| 721 |
> |
\AA~, 1433 atoms) solvated by explicit SPC/E water clusters (5000 |
| 722 |
> |
molecules) using a model for the gold / water interactions that has |
| 723 |
> |
been used by Dou {\it et. al.} for investigating the separation of |
| 724 |
> |
water films near hot metal surfaces.\cite{ISI:000167766600035} The |
| 725 |
> |
Langevin Hull was used to sample pressures of 1, 2, 5, 10, 20, 50, 100 |
| 726 |
> |
and 200 atm, while all simulations were done at a temperature of 300 |
| 727 |
> |
K. At these temperatures and pressures, there is no observed |
| 728 |
> |
separation of the water film from the surface. |
| 729 |
|
|
| 730 |
< |
\begin{figure} |
| 730 |
> |
In Fig. \ref{fig:RhoR} we show the density of water and gold as a |
| 731 |
> |
function of the distance from the center of the nanoparticle. Higher |
| 732 |
> |
applied pressures appear to destroy structural correlations in the |
| 733 |
> |
outermost monolayer of the gold nanoparticle as well as in the water |
| 734 |
> |
at the near the metal / water interface. Simulations at increased |
| 735 |
> |
pressures exhibit significant overlap of the gold and water densities, |
| 736 |
> |
indicating a less well-defined interfacial surface. |
| 737 |
|
|
| 738 |
< |
\caption{interesting plot showing cluster behavior} |
| 738 |
> |
\begin{figure} |
| 739 |
> |
\includegraphics[width=\linewidth]{RhoR} |
| 740 |
> |
\caption{Density profiles of gold and water at the nanoparticle |
| 741 |
> |
surface. Each curve has been normalized by the average density in |
| 742 |
> |
the bulk-like region available to the corresponding material. |
| 743 |
> |
Higher applied pressures de-structure both the gold nanoparticle |
| 744 |
> |
surface and water at the metal/water interface.} |
| 745 |
|
\label{fig:RhoR} |
| 746 |
|
\end{figure} |
| 747 |
|
|
| 748 |
< |
At higher pressures, problems with the gold - water interaction |
| 749 |
< |
potential became apparent. The model we are using (due to Spohr) was |
| 750 |
< |
intended for relatively low pressures; it utilizes both shifted Morse |
| 751 |
< |
and repulsive Morse potentials to model the Au/O and Au/H |
| 752 |
< |
interactions, respectively. The repulsive wall of the Morse potential |
| 753 |
< |
does not diverge quickly enough at short distances to prevent water |
| 754 |
< |
from diffusing into the center of the gold nanoparticles. This |
| 755 |
< |
behavior is likely not a realistic description of the real physics of |
| 756 |
< |
the situation. A better model of the gold-water adsorption behavior |
| 757 |
< |
appears to require harder repulsive walls to prevent this behavior. |
| 748 |
> |
At even higher pressures (500 atm and above), problems with the metal |
| 749 |
> |
- water interaction potential became quite clear. The model we are |
| 750 |
> |
using appears to have been parameterized for relatively low pressures; |
| 751 |
> |
it utilizes both shifted Morse and repulsive Morse potentials to model |
| 752 |
> |
the Au/O and Au/H interactions, respectively. The repulsive wall of |
| 753 |
> |
the Morse potential does not diverge quickly enough at short distances |
| 754 |
> |
to prevent water from diffusing into the center of the gold |
| 755 |
> |
nanoparticles. This behavior is likely not a realistic description of |
| 756 |
> |
the real physics of the situation. A better model of the gold-water |
| 757 |
> |
adsorption behavior would require harder repulsive walls to prevent |
| 758 |
> |
this behavior. |
| 759 |
|
|
| 760 |
|
\section{Discussion} |
| 761 |
|
\label{sec:discussion} |
| 763 |
|
The Langevin Hull samples the isobaric-isothermal ensemble for |
| 764 |
|
non-periodic systems by coupling the system to a bath characterized by |
| 765 |
|
pressure, temperature, and solvent viscosity. This enables the |
| 766 |
< |
simulation of heterogeneous systems composed of materials of |
| 766 |
> |
simulation of heterogeneous systems composed of materials with |
| 767 |
|
significantly different compressibilities. Because the boundary is |
| 768 |
|
dynamically determined during the simulation and the molecules |
| 769 |
< |
interacting with the boundary can change, the method and has minimal |
| 769 |
> |
interacting with the boundary can change, the method inflicts minimal |
| 770 |
|
perturbations on the behavior of molecules at the edges of the |
| 771 |
|
simulation. Further work on this method will involve implicit |
| 772 |
|
electrostatics at the boundary (which is missing in the current |
| 821 |
|
The individual hull operations scale with |
| 822 |
|
$\mathcal{O}(\frac{n}{p}\log\frac{n}{p})$ where $n$ is the total |
| 823 |
|
number of sites, and $p$ is the number of processors. These local |
| 824 |
< |
hull operations create a set of $p$ hulls each with approximately |
| 825 |
< |
$\frac{n}{3pr}$ sites (for a cluster of radius $r$). The worst-case |
| 824 |
> |
hull operations create a set of $p$ hulls, each with approximately |
| 825 |
> |
$\frac{n}{3pr}$ sites for a cluster of radius $r$. The worst-case |
| 826 |
|
communication cost for using a ``gather'' operation to distribute this |
| 827 |
|
information to all processors is $\mathcal{O}( \alpha (p-1) + \frac{n |
| 828 |
|
\beta (p-1)}{3 r p^2})$, while the final computation of the system |
| 830 |
|
|
| 831 |
|
For a large number of atoms on a moderately parallel machine, the |
| 832 |
|
total costs are dominated by the computations of the individual hulls, |
| 833 |
< |
and communication of these hulls to so the Langevin hull sees roughly |
| 833 |
> |
and communication of these hulls to create the Langevin Hull sees roughly |
| 834 |
|
linear speed-up with increasing processor counts. |
| 835 |
|
|
| 836 |
|
\section*{Acknowledgments} |
