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\bibliographystyle{achemso} |
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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 under the Langevin Hull at a variety of applied |
500 |
> |
pressures ranging from 0 -- 10 GPa. For the 40 \AA~ radius |
501 |
> |
nanoparticle we obtain a value of 177.55 GPa for the bulk modulus of |
502 |
> |
gold, in close agreement with both previous simulations and the |
503 |
> |
experimental bulk modulus reported for gold single |
504 |
> |
crystals.\cite{Collard1991} The smaller gold nanoparticles (30 and 20 |
505 |
> |
\AA~ radii) have calculated bulk moduli of 215.58 and 208.86 GPa, |
506 |
> |
respectively, indicating that smaller nanoparticles are somewhat |
507 |
> |
stiffer (less compressible) than the larger nanoparticles. This |
508 |
> |
stiffening of the small nanoparticles may be related to their high |
509 |
> |
degree of surface curvature, resulting in a lower coordination number |
510 |
> |
of surface atoms relative to the the surface atoms in the 40 \AA~ |
511 |
> |
radius particle. |
512 |
|
|
513 |
+ |
We obtain a gold lattice constant of 4.051 \AA~ using the Langevin |
514 |
+ |
Hull at 1 atm, close to the experimentally-determined value for bulk |
515 |
+ |
gold and the value for gold simulated using the QSC potential and |
516 |
+ |
periodic boundary conditions (4.079 \AA~ and 4.088\AA~, |
517 |
+ |
respectively).\cite{QSC2} The slightly smaller calculated lattice |
518 |
+ |
constant is most likely due to the presence of surface tension in the |
519 |
+ |
non-periodic Langevin Hull cluster, an effect absent from a bulk |
520 |
+ |
simulation. The specific heat of a 40 \AA~ gold nanoparticle under the |
521 |
+ |
Langevin Hull at 1 atm is 24.914 $\mathrm {\frac{J}{mol \, K}}$, which |
522 |
+ |
compares very well with the experimental value of 25.42 $\mathrm |
523 |
+ |
{\frac{J}{mol \, K}}$. |
524 |
+ |
|
525 |
|
\begin{figure} |
526 |
|
\includegraphics[width=\linewidth]{stacked} |
527 |
|
\caption{The response of the internal pressure and temperature of gold |
538 |
|
temperature respond to the Langevin Hull for nanoparticles that were |
539 |
|
initialized far from the target pressure and temperature. As |
540 |
|
expected, the rate at which thermal equilibrium is achieved depends on |
541 |
< |
the total surface area of the cluter exposed to the bath as well as |
541 |
> |
the total surface area of the cluster exposed to the bath as well as |
542 |
|
the bath viscosity. Pressure that is applied suddenly to a cluster |
543 |
|
can excite breathing vibrations, but these rapidly damp out (on time |
544 |
< |
scales of 30-50 ps). |
544 |
> |
scales of 30 -- 50 ps). |
545 |
|
|
546 |
|
\subsection{Compressibility of SPC/E water clusters} |
547 |
|
|
550 |
|
ensembles) have yielded values for the isothermal compressibility that |
551 |
|
agree well with experiment.\cite{Fine1973} The results of two |
552 |
|
different approaches for computing the isothermal compressibility from |
553 |
< |
Langevin Hull simulations for pressures between 1 and 6500 atm are |
553 |
> |
Langevin Hull simulations for pressures between 1 and 3000 atm are |
554 |
|
shown in Fig. \ref{fig:compWater} along with compressibility values |
555 |
|
obtained from both other SPC/E simulations and experiment. |
556 |
|
|
565 |
|
and previous simulation work throughout the 1 -- 1000 atm pressure |
566 |
|
regime. Compressibilities computed using the Hull volume, however, |
567 |
|
deviate dramatically from the experimental values at low applied |
568 |
< |
pressures. The reason for this deviation is quite simple; at low |
568 |
> |
pressures. The reason for this deviation is quite simple: at low |
569 |
|
applied pressures, the liquid is in equilibrium with a vapor phase, |
570 |
|
and it is entirely possible for one (or a few) molecules to drift away |
571 |
|
from the liquid cluster (see Fig. \ref{fig:coneOfShame}). At low |
595 |
|
different pressures must be done to compute the first derivatives. It |
596 |
|
is also possible to compute the compressibility using the fluctuation |
597 |
|
dissipation theorem using either fluctuations in the |
598 |
< |
volume,\cite{Debenedetti1986}, |
598 |
> |
volume,\cite{Debenedetti1986} |
599 |
|
\begin{equation} |
600 |
|
\kappa_{T} = \frac{\left \langle V^{2} \right \rangle - \left \langle |
601 |
|
V \right \rangle ^{2}}{V \, k_{B} \, T}, |
605 |
|
fixed region, |
606 |
|
\begin{equation} |
607 |
|
\kappa_{T} = \frac{\left \langle N^{2} \right \rangle - \left \langle |
608 |
< |
N \right \rangle ^{2}}{N \, k_{B} \, T}, |
608 |
> |
N \right \rangle ^{2}}{N \, k_{B} \, T}. |
609 |
|
\label{eq:BMNfluct} |
610 |
|
\end{equation} |
611 |
|
Thus, the compressibility of each simulation can be calculated |
614 |
|
effects of the empty space due to the vapor phase; for this reason, we |
615 |
|
recommend using the number density (Eq. \ref{eq:BMN}) or number |
616 |
|
density fluctuations (Eq. \ref{eq:BMNfluct}) for computing |
617 |
< |
compressibilities. |
617 |
> |
compressibilities. We achieved the best results using a sampling |
618 |
> |
radius approximately 80\% of the cluster radius. This ratio of |
619 |
> |
sampling radius to cluster radius excludes the problematic vapor phase |
620 |
> |
on the outside of the cluster while including enough of the liquid |
621 |
> |
phase to avoid poor statistics due to fluctuating local densities. |
622 |
|
|
623 |
+ |
A comparison of the oxygen-oxygen radial distribution functions for |
624 |
+ |
SPC/E water simulated using the Langevin Hull and bulk SPC/E using |
625 |
+ |
periodic boundary conditions -- both at 1 atm and 300K -- reveals an |
626 |
+ |
understructuring of water in the Langevin Hull that manifests as a |
627 |
+ |
slight broadening of the solvation shells. This effect may be related |
628 |
+ |
to the introduction of surface tension around the entire cluster, an |
629 |
+ |
effect absent in bulk systems. As a result, molecules on the hull may |
630 |
+ |
experience an increased inward force, slightly compressing the |
631 |
+ |
solvation shell for these molecules. |
632 |
+ |
|
633 |
|
\subsection{Molecular orientation distribution at cluster boundary} |
634 |
|
|
635 |
|
In order for a non-periodic boundary method to be widely applicable, |
636 |
< |
they must be constructed in such a way that they allow a finite system |
636 |
> |
it must be constructed in such a way that they allow a finite system |
637 |
|
to replicate the properties of the bulk. Early non-periodic simulation |
638 |
|
methods (e.g. hydrophobic boundary potentials) induced spurious |
639 |
|
orientational correlations deep within the simulated |
640 |
|
system.\cite{Lee1984,Belch1985} This behavior spawned many methods for |
641 |
< |
fixing and characterizing the effects of artifical boundaries |
641 |
> |
fixing and characterizing the effects of artificial boundaries |
642 |
|
including methods which fix the orientations of a set of edge |
643 |
|
molecules.\cite{Warshel1978,King1989} |
644 |
|
|
647 |
|
hydrophobic boundary, or orientational or radial constraints. |
648 |
|
Therefore, the orientational correlations of the molecules in water |
649 |
|
clusters are of particular interest in testing this method. Ideally, |
650 |
< |
the water molecules on the surfaces of the clusterss will have enough |
650 |
> |
the water molecules on the surfaces of the clusters will have enough |
651 |
|
mobility into and out of the center of the cluster to maintain |
652 |
|
bulk-like orientational distribution in the absence of orientational |
653 |
|
and radial constraints. However, since the number of hydrogen bonding |
655 |
|
likely that there will be an effective hydrophobicity of the hull. |
656 |
|
|
657 |
|
To determine the extent of these effects, we examined the |
658 |
< |
orientationations exhibited by SPC/E water in a cluster of 1372 |
658 |
> |
orientations exhibited by SPC/E water in a cluster of 1372 |
659 |
|
molecules at 300 K and at pressures ranging from 1 -- 1000 atm. The |
660 |
< |
orientational angle of a water molecule is described |
660 |
> |
orientational angle of a water molecule is described by |
661 |
|
\begin{equation} |
662 |
|
\cos{\theta}=\frac{\vec{r}_i\cdot\vec{\mu}_i}{|\vec{r}_i||\vec{\mu}_i|} |
663 |
|
\end{equation} |
677 |
|
\includegraphics[width=\linewidth]{pAngle} |
678 |
|
\caption{Distribution of $\cos{\theta}$ values for molecules on the |
679 |
|
interior of the cluster (squares) and for those participating in the |
680 |
< |
convex hull (circles) at a variety of pressures. The Langevin hull |
680 |
> |
convex hull (circles) at a variety of pressures. The Langevin Hull |
681 |
|
exhibits minor dewetting behavior with exposed oxygen sites on the |
682 |
|
hull water molecules. The orientational preference for exposed |
683 |
|
oxygen appears to be independent of applied pressure. } |
689 |
|
orientations. Molecules included in the convex hull show a slight |
690 |
|
preference for values of $\cos{\theta} < 0.$ These values correspond |
691 |
|
to molecules with oxygen directed toward the exterior of the cluster, |
692 |
< |
forming a dangling hydrogen bond acceptor site. |
692 |
> |
forming dangling hydrogen bond acceptor sites. |
693 |
|
|
694 |
< |
In the absence of an electrostatic contribution from the exterior |
695 |
< |
bath, the orientational distribution of water molecules included in |
696 |
< |
the Langevin Hull will slightly resemble the distribution at a neat |
697 |
< |
water liquid/vapor interface. Previous molecular dynamics simulations |
698 |
< |
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. |
694 |
> |
The orientational preference exhibited by water molecules on the hull |
695 |
> |
is significantly weaker than the preference caused by an explicit |
696 |
> |
hydrophobic bounding potential. Additionally, the Langevin Hull does |
697 |
> |
not require that the orientation of any molecules be fixed in order to |
698 |
> |
maintain bulk-like structure, even near the cluster surface. |
699 |
|
|
700 |
< |
The orientational preference exhibited by hull molecules in the |
701 |
< |
Langevin hull is significantly weaker than the preference caused by an |
702 |
< |
explicit hydrophobic bounding potential. Additionally, the Langevin |
703 |
< |
Hull does not require that the orientation of any molecules be fixed |
704 |
< |
in order to maintain bulk-like structure, even at the cluster surface. |
700 |
> |
Previous molecular dynamics simulations of SPC/E liquid / vapor |
701 |
> |
interfaces using periodic boundary conditions have shown that |
702 |
> |
molecules on the liquid side of interface favor a similar orientation |
703 |
> |
where oxygen is directed away from the bulk.\cite{Taylor1996} These |
704 |
> |
simulations had well-defined liquid and vapor phase regions |
705 |
> |
equilibrium and it was observed that {\it vapor} molecules generally |
706 |
> |
had one hydrogen protruding from the surface, forming a dangling |
707 |
> |
hydrogen bond donor. Our water clusters do not have a true vapor |
708 |
> |
region, but rather a few transient molecules that leave the liquid |
709 |
> |
droplet (and which return to the droplet relatively quickly). |
710 |
> |
Although we cannot obtain an orientational preference of vapor phase |
711 |
> |
molecules in a Langevin Hull simulation, but we do agree with previous |
712 |
> |
estimates of the orientation of {\it liquid phase} molecules at the |
713 |
> |
interface. |
714 |
|
|
715 |
|
\subsection{Heterogeneous nanoparticle / water mixtures} |
716 |
|
|
717 |
< |
To further test the method, we simulated gold nanopartices ($r = 18$ |
718 |
< |
\AA) solvated by explicit SPC/E water clusters using the Langevin |
719 |
< |
hull. This was done at pressures of 1, 2, 5, 10, 20, 50 and 100 atm |
720 |
< |
in order to observe the effects of pressure on the ordering of water |
721 |
< |
ordering at the surface. In Fig. \ref{fig:RhoR} we show the density |
722 |
< |
of water adjacent to the surface as a function of pressure, as well as |
723 |
< |
the orientational ordering of water at the surface of the |
724 |
< |
nanoparticle. |
717 |
> |
To further test the method, we simulated gold nanoparticles ($r = 18$ |
718 |
> |
\AA) solvated by explicit SPC/E water clusters using a model for the |
719 |
> |
gold / water interactions that has been used by Dou {\it et. al.} for |
720 |
> |
investigating the separation of water films near hot metal |
721 |
> |
surfaces.\cite{ISI:000167766600035} The Langevin Hull was used to |
722 |
> |
sample pressures of 1, 2, 5, 10, 20, 50, 100 and 200 atm, while all |
723 |
> |
simulations were done at a temperature of 300 K. At these |
724 |
> |
temperatures and pressures, there is no observed separation of the |
725 |
> |
water film from the surface. |
726 |
|
|
727 |
< |
\begin{figure} |
727 |
> |
In Fig. \ref{fig:RhoR} we show the density of water and gold as a |
728 |
> |
function of the distance from the center of the nanoparticle. Higher |
729 |
> |
applied pressures appear to destroy structural correlations in the |
730 |
> |
outermost monolayer of the gold nanoparticle as well as in the water |
731 |
> |
at the near the metal / water interface. Simulations at increased |
732 |
> |
pressures exhibit significant overlap of the gold and water densities, |
733 |
> |
indicating a less well-defined interfacial surface. |
734 |
|
|
735 |
< |
\caption{interesting plot showing cluster behavior} |
735 |
> |
\begin{figure} |
736 |
> |
\includegraphics[width=\linewidth]{RhoR} |
737 |
> |
\caption{Density profiles of gold and water at the nanoparticle |
738 |
> |
surface. Each curve has been normalized by the average density in |
739 |
> |
the bulk-like region available to the corresponding material. |
740 |
> |
Higher applied pressures de-structure both the gold nanoparticle |
741 |
> |
surface and water at the metal/water interface.} |
742 |
|
\label{fig:RhoR} |
743 |
|
\end{figure} |
744 |
|
|
745 |
< |
At higher pressures, problems with the gold - water interaction |
746 |
< |
potential became apparent. The model we are using (due to Spohr) was |
747 |
< |
intended for relatively low pressures; it utilizes both shifted Morse |
748 |
< |
and repulsive Morse potentials to model the Au/O and Au/H |
749 |
< |
interactions, respectively. The repulsive wall of the Morse potential |
750 |
< |
does not diverge quickly enough at short distances to prevent water |
751 |
< |
from diffusing into the center of the gold nanoparticles. This |
752 |
< |
behavior is likely not a realistic description of the real physics of |
753 |
< |
the situation. A better model of the gold-water adsorption behavior |
754 |
< |
appears to require harder repulsive walls to prevent this behavior. |
745 |
> |
At even higher pressures (500 atm and above), problems with the metal |
746 |
> |
- water interaction potential became quite clear. The model we are |
747 |
> |
using appears to have been parameterized for relatively low pressures; |
748 |
> |
it utilizes both shifted Morse and repulsive Morse potentials to model |
749 |
> |
the Au/O and Au/H interactions, respectively. The repulsive wall of |
750 |
> |
the Morse potential does not diverge quickly enough at short distances |
751 |
> |
to prevent water from diffusing into the center of the gold |
752 |
> |
nanoparticles. This behavior is likely not a realistic description of |
753 |
> |
the real physics of the situation. A better model of the gold-water |
754 |
> |
adsorption behavior would require harder repulsive walls to prevent |
755 |
> |
this behavior. |
756 |
|
|
757 |
|
\section{Discussion} |
758 |
|
\label{sec:discussion} |
760 |
|
The Langevin Hull samples the isobaric-isothermal ensemble for |
761 |
|
non-periodic systems by coupling the system to a bath characterized by |
762 |
|
pressure, temperature, and solvent viscosity. This enables the |
763 |
< |
simulation of heterogeneous systems composed of materials of |
763 |
> |
simulation of heterogeneous systems composed of materials with |
764 |
|
significantly different compressibilities. Because the boundary is |
765 |
|
dynamically determined during the simulation and the molecules |
766 |
< |
interacting with the boundary can change, the method and has minimal |
766 |
> |
interacting with the boundary can change, the method inflicts minimal |
767 |
|
perturbations on the behavior of molecules at the edges of the |
768 |
|
simulation. Further work on this method will involve implicit |
769 |
|
electrostatics at the boundary (which is missing in the current |
818 |
|
The individual hull operations scale with |
819 |
|
$\mathcal{O}(\frac{n}{p}\log\frac{n}{p})$ where $n$ is the total |
820 |
|
number of sites, and $p$ is the number of processors. These local |
821 |
< |
hull operations create a set of $p$ hulls each with approximately |
822 |
< |
$\frac{n}{3pr}$ sites (for a cluster of radius $r$). The worst-case |
821 |
> |
hull operations create a set of $p$ hulls, each with approximately |
822 |
> |
$\frac{n}{3pr}$ sites for a cluster of radius $r$. The worst-case |
823 |
|
communication cost for using a ``gather'' operation to distribute this |
824 |
|
information to all processors is $\mathcal{O}( \alpha (p-1) + \frac{n |
825 |
|
\beta (p-1)}{3 r p^2})$, while the final computation of the system |
827 |
|
|
828 |
|
For a large number of atoms on a moderately parallel machine, the |
829 |
|
total costs are dominated by the computations of the individual hulls, |
830 |
< |
and communication of these hulls to so the Langevin hull sees roughly |
830 |
> |
and communication of these hulls to create the Langevin Hull sees roughly |
831 |
|
linear speed-up with increasing processor counts. |
832 |
|
|
833 |
|
\section*{Acknowledgments} |