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\bibpunct{[}{]}{,}{s}{}{;} |
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\bibliographystyle{achemso} |
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\begin{document} |
| 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 |
| 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 |
| 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 |
| 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. |
| 497 |
> |
of 175.53 GPa.\cite{QSC2} Using the same force field, we have performed |
| 498 |
> |
a series of 1 ns simulations on gold nanoparticles of three different radii under the Langevin Hull at a variety of applied pressures ranging from 0 -- 10 GPa. For the 40 \AA~ radius nanoparticle we obtain a value of 177.55 GPa for the bulk modulus of gold, in close agreement with both previous simulations and the experimental bulk modulus reported for gold single crystals.\cite{Collard1991} Polycrystalline gold has a reported bulk modulus of 220 GPa. The smaller gold nanoparticles (30 and 20 \AA~ radii) have calculated bulk moduli of 215.58 and 208.86 GPa, respectively, indicating that smaller nanoparticles approach the polycrystalline bulk modulus value while larger nanoparticles approach the single crystal value. As nanoparticle size decreases, the bulk modulus becomes larger and the nanoparticle is less compressible. This stiffening of the small nanoparticles may be related to their high degree of surface curvature, resulting in a lower coordination number of surface atoms relative to the the surface atoms in the 40 \AA~ radius particle. |
| 499 |
> |
|
| 500 |
> |
We measure a gold lattice constant of 4.051 \AA~ using the Langevin Hull at 1 atm, close to the experimentally-determined value for bulk gold and the value for gold simulated using the QSC potential and periodic boundary conditions (4.079 \AA~ and 4.088\AA~, respectively).\cite{QSC2} The slightly smaller calculated lattice constant is most likely due to the presence of surface tension in the non-periodic Langevin Hull cluster, an effect absent from a bulk simulation. The specific heat of a 40 \AA~ gold nanoparticle under the Langevin Hull at 1 atm is 24.914 $\mathrm {\frac{J}{mol \, K}}$, which compares very well with the experimental value of 25.42 $\mathrm {\frac{J}{mol \, K}}$. |
| 501 |
|
|
| 502 |
|
\begin{figure} |
| 503 |
|
\includegraphics[width=\linewidth]{stacked} |
| 515 |
|
temperature respond to the Langevin Hull for nanoparticles that were |
| 516 |
|
initialized far from the target pressure and temperature. As |
| 517 |
|
expected, the rate at which thermal equilibrium is achieved depends on |
| 518 |
< |
the total surface area of the cluter exposed to the bath as well as |
| 518 |
> |
the total surface area of the cluster exposed to the bath as well as |
| 519 |
|
the bath viscosity. Pressure that is applied suddenly to a cluster |
| 520 |
|
can excite breathing vibrations, but these rapidly damp out (on time |
| 521 |
< |
scales of 30-50 ps). |
| 521 |
> |
scales of 30 -- 50 ps). |
| 522 |
|
|
| 523 |
|
\subsection{Compressibility of SPC/E water clusters} |
| 524 |
|
|
| 527 |
|
ensembles) have yielded values for the isothermal compressibility that |
| 528 |
|
agree well with experiment.\cite{Fine1973} The results of two |
| 529 |
|
different approaches for computing the isothermal compressibility from |
| 530 |
< |
Langevin Hull simulations for pressures between 1 and 6500 atm are |
| 530 |
> |
Langevin Hull simulations for pressures between 1 and 3000 atm are |
| 531 |
|
shown in Fig. \ref{fig:compWater} along with compressibility values |
| 532 |
|
obtained from both other SPC/E simulations and experiment. |
| 533 |
|
|
| 542 |
|
and previous simulation work throughout the 1 -- 1000 atm pressure |
| 543 |
|
regime. Compressibilities computed using the Hull volume, however, |
| 544 |
|
deviate dramatically from the experimental values at low applied |
| 545 |
< |
pressures. The reason for this deviation is quite simple; at low |
| 545 |
> |
pressures. The reason for this deviation is quite simple: at low |
| 546 |
|
applied pressures, the liquid is in equilibrium with a vapor phase, |
| 547 |
|
and it is entirely possible for one (or a few) molecules to drift away |
| 548 |
|
from the liquid cluster (see Fig. \ref{fig:coneOfShame}). At low |
| 572 |
|
different pressures must be done to compute the first derivatives. It |
| 573 |
|
is also possible to compute the compressibility using the fluctuation |
| 574 |
|
dissipation theorem using either fluctuations in the |
| 575 |
< |
volume,\cite{Debenedetti1986}, |
| 575 |
> |
volume,\cite{Debenedetti1986} |
| 576 |
|
\begin{equation} |
| 577 |
|
\kappa_{T} = \frac{\left \langle V^{2} \right \rangle - \left \langle |
| 578 |
|
V \right \rangle ^{2}}{V \, k_{B} \, T}, |
| 582 |
|
fixed region, |
| 583 |
|
\begin{equation} |
| 584 |
|
\kappa_{T} = \frac{\left \langle N^{2} \right \rangle - \left \langle |
| 585 |
< |
N \right \rangle ^{2}}{N \, k_{B} \, T}, |
| 585 |
> |
N \right \rangle ^{2}}{N \, k_{B} \, T}. |
| 586 |
|
\label{eq:BMNfluct} |
| 587 |
|
\end{equation} |
| 588 |
|
Thus, the compressibility of each simulation can be calculated |
| 591 |
|
effects of the empty space due to the vapor phase; for this reason, we |
| 592 |
|
recommend using the number density (Eq. \ref{eq:BMN}) or number |
| 593 |
|
density fluctuations (Eq. \ref{eq:BMNfluct}) for computing |
| 594 |
< |
compressibilities. |
| 594 |
> |
compressibilities. We achieved the best results using a sampling radius approximately 80\% of the cluster radius. This ratio of sampling radius to cluster radius excludes the problematic vapor phase on the outside of the cluster while including enough of the liquid phase to avoid poor statistics due to fluctuating local densities. |
| 595 |
|
|
| 596 |
+ |
A comparison of the oxygen-oxygen radial distribution functions for SPC/E water simulated using the Langevin Hull and bulk SPC/E using periodic boundary conditions -- both at 1 atm and 300K -- reveals a slight understructuring of water in the Langevin Hull that manifests as a minor broadening of the solvation shells. This effect may be related to the introduction of surface tension around the entire cluster, an effect absent in bulk systems. As a result, molecules on the hull may experience an increased inward force, slightly compressing the solvation shell structure. |
| 597 |
+ |
|
| 598 |
|
\subsection{Molecular orientation distribution at cluster boundary} |
| 599 |
|
|
| 600 |
|
In order for a non-periodic boundary method to be widely applicable, |
| 601 |
< |
they must be constructed in such a way that they allow a finite system |
| 601 |
> |
it must be constructed in such a way that they allow a finite system |
| 602 |
|
to replicate the properties of the bulk. Early non-periodic simulation |
| 603 |
|
methods (e.g. hydrophobic boundary potentials) induced spurious |
| 604 |
|
orientational correlations deep within the simulated |
| 605 |
|
system.\cite{Lee1984,Belch1985} This behavior spawned many methods for |
| 606 |
< |
fixing and characterizing the effects of artifical boundaries |
| 606 |
> |
fixing and characterizing the effects of artificial boundaries |
| 607 |
|
including methods which fix the orientations of a set of edge |
| 608 |
|
molecules.\cite{Warshel1978,King1989} |
| 609 |
|
|
| 612 |
|
hydrophobic boundary, or orientational or radial constraints. |
| 613 |
|
Therefore, the orientational correlations of the molecules in water |
| 614 |
|
clusters are of particular interest in testing this method. Ideally, |
| 615 |
< |
the water molecules on the surfaces of the clusterss will have enough |
| 615 |
> |
the water molecules on the surfaces of the clusters will have enough |
| 616 |
|
mobility into and out of the center of the cluster to maintain |
| 617 |
|
bulk-like orientational distribution in the absence of orientational |
| 618 |
|
and radial constraints. However, since the number of hydrogen bonding |
| 620 |
|
likely that there will be an effective hydrophobicity of the hull. |
| 621 |
|
|
| 622 |
|
To determine the extent of these effects, we examined the |
| 623 |
< |
orientationations exhibited by SPC/E water in a cluster of 1372 |
| 623 |
> |
orientations exhibited by SPC/E water in a cluster of 1372 |
| 624 |
|
molecules at 300 K and at pressures ranging from 1 -- 1000 atm. The |
| 625 |
< |
orientational angle of a water molecule is described |
| 625 |
> |
orientational angle of a water molecule is described by |
| 626 |
|
\begin{equation} |
| 627 |
|
\cos{\theta}=\frac{\vec{r}_i\cdot\vec{\mu}_i}{|\vec{r}_i||\vec{\mu}_i|} |
| 628 |
|
\end{equation} |
| 642 |
|
\includegraphics[width=\linewidth]{pAngle} |
| 643 |
|
\caption{Distribution of $\cos{\theta}$ values for molecules on the |
| 644 |
|
interior of the cluster (squares) and for those participating in the |
| 645 |
< |
convex hull (circles) at a variety of pressures. The Langevin hull |
| 645 |
> |
convex hull (circles) at a variety of pressures. The Langevin Hull |
| 646 |
|
exhibits minor dewetting behavior with exposed oxygen sites on the |
| 647 |
|
hull water molecules. The orientational preference for exposed |
| 648 |
|
oxygen appears to be independent of applied pressure. } |
| 654 |
|
orientations. Molecules included in the convex hull show a slight |
| 655 |
|
preference for values of $\cos{\theta} < 0.$ These values correspond |
| 656 |
|
to molecules with oxygen directed toward the exterior of the cluster, |
| 657 |
< |
forming a dangling hydrogen bond acceptor site. |
| 662 |
< |
|
| 663 |
< |
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. |
| 657 |
> |
forming dangling hydrogen bond acceptor sites. |
| 658 |
|
|
| 659 |
< |
The orientational preference exhibited by hull molecules in the |
| 660 |
< |
Langevin hull is significantly weaker than the preference caused by an |
| 661 |
< |
explicit hydrophobic bounding potential. Additionally, the Langevin |
| 662 |
< |
Hull does not require that the orientation of any molecules be fixed |
| 663 |
< |
in order to maintain bulk-like structure, even at the cluster surface. |
| 659 |
> |
The orientational preference exhibited by water molecules on the hull |
| 660 |
> |
is significantly weaker than the preference caused by an explicit |
| 661 |
> |
hydrophobic bounding potential. Additionally, the Langevin Hull does |
| 662 |
> |
not require that the orientation of any molecules be fixed in order to |
| 663 |
> |
maintain bulk-like structure, even near the cluster surface. |
| 664 |
|
|
| 665 |
+ |
Previous molecular dynamics simulations of SPC/E liquid / vapor |
| 666 |
+ |
interfaces using periodic boundary conditions have shown that |
| 667 |
+ |
molecules on the liquid side of interface favor a similar orientation |
| 668 |
+ |
where oxygen is directed away from the bulk.\cite{Taylor1996} These |
| 669 |
+ |
simulations had well-defined liquid and vapor phase regions |
| 670 |
+ |
equilibrium and it was observed that {\it vapor} molecules generally |
| 671 |
+ |
had one hydrogen protruding from the surface, forming a dangling |
| 672 |
+ |
hydrogen bond donor. Our water clusters do not have a true vapor |
| 673 |
+ |
region, but rather a few transient molecules that leave the liquid |
| 674 |
+ |
droplet (and which return to the droplet relatively quickly). |
| 675 |
+ |
Although we cannot obtain an orientational preference of vapor phase |
| 676 |
+ |
molecules in a Langevin Hull simulation, but we do agree with previous |
| 677 |
+ |
estimates of the orientation of {\it liquid phase} molecules at the |
| 678 |
+ |
interface. |
| 679 |
+ |
|
| 680 |
|
\subsection{Heterogeneous nanoparticle / water mixtures} |
| 681 |
|
|
| 682 |
< |
To further test the method, we simulated gold nanopartices ($r = 18$ |
| 683 |
< |
\AA) solvated by explicit SPC/E water clusters using the Langevin |
| 684 |
< |
hull. This was done at pressures of 1, 2, 5, 10, 20, 50 and 100 atm |
| 685 |
< |
in order to observe the effects of pressure on the ordering of water |
| 686 |
< |
ordering at the surface. In Fig. \ref{fig:RhoR} we show the density |
| 687 |
< |
of water adjacent to the surface as a function of pressure, as well as |
| 688 |
< |
the orientational ordering of water at the surface of the |
| 689 |
< |
nanoparticle. |
| 682 |
> |
To further test the method, we simulated gold nanoparticles ($r = 18$ |
| 683 |
> |
\AA) solvated by explicit SPC/E water clusters using a model for the |
| 684 |
> |
gold / water interactions that has been used by Dou {\it et. al.} for |
| 685 |
> |
investigating the separation of water films near hot metal |
| 686 |
> |
surfaces.\cite{ISI:000167766600035} The Langevin Hull was used to |
| 687 |
> |
sample pressures of 1, 2, 5, 10, 20, 50, 100 and 200 atm, while all |
| 688 |
> |
simulations were done at a temperature of 300 K. At these |
| 689 |
> |
temperatures and pressures, there is no observed separation of the |
| 690 |
> |
water film from the surface. |
| 691 |
|
|
| 692 |
< |
\begin{figure} |
| 692 |
> |
In Fig. \ref{fig:RhoR} we show the density of water and gold as a |
| 693 |
> |
function of the distance from the center of the nanoparticle. Higher |
| 694 |
> |
applied pressures appear to destroy structural correlations in the |
| 695 |
> |
outermost monolayer of the gold nanoparticle as well as in the water |
| 696 |
> |
at the near the metal / water interface. Simulations at increased |
| 697 |
> |
pressures exhibit significant overlap of the gold and water densities, |
| 698 |
> |
indicating a less well-defined interfacial surface. |
| 699 |
|
|
| 700 |
< |
\caption{interesting plot showing cluster behavior} |
| 700 |
> |
\begin{figure} |
| 701 |
> |
\includegraphics[width=\linewidth]{RhoR} |
| 702 |
> |
\caption{Density profiles of gold and water at the nanoparticle |
| 703 |
> |
surface. Each curve has been normalized by the average density in |
| 704 |
> |
the bulk-like region available to the corresponding material. Higher applied pressures |
| 705 |
> |
de-structure both the gold nanoparticle surface and water at the |
| 706 |
> |
metal/water interface.} |
| 707 |
|
\label{fig:RhoR} |
| 708 |
|
\end{figure} |
| 709 |
|
|
| 710 |
< |
At higher pressures, problems with the gold - water interaction |
| 711 |
< |
potential became apparent. The model we are using (due to Spohr) was |
| 712 |
< |
intended for relatively low pressures; it utilizes both shifted Morse |
| 713 |
< |
and repulsive Morse potentials to model the Au/O and Au/H |
| 714 |
< |
interactions, respectively. The repulsive wall of the Morse potential |
| 715 |
< |
does not diverge quickly enough at short distances to prevent water |
| 716 |
< |
from diffusing into the center of the gold nanoparticles. This |
| 717 |
< |
behavior is likely not a realistic description of the real physics of |
| 718 |
< |
the situation. A better model of the gold-water adsorption behavior |
| 719 |
< |
appears to require harder repulsive walls to prevent this behavior. |
| 710 |
> |
At even higher pressures (500 atm and above), problems with the metal |
| 711 |
> |
- water interaction potential became quite clear. The model we are |
| 712 |
> |
using appears to have been parameterized for relatively low pressures; |
| 713 |
> |
it utilizes both shifted Morse and repulsive Morse potentials to model |
| 714 |
> |
the Au/O and Au/H interactions, respectively. The repulsive wall of |
| 715 |
> |
the Morse potential does not diverge quickly enough at short distances |
| 716 |
> |
to prevent water from diffusing into the center of the gold |
| 717 |
> |
nanoparticles. This behavior is likely not a realistic description of |
| 718 |
> |
the real physics of the situation. A better model of the gold-water |
| 719 |
> |
adsorption behavior appears to require harder repulsive walls to |
| 720 |
> |
prevent this behavior. |
| 721 |
|
|
| 722 |
|
\section{Discussion} |
| 723 |
|
\label{sec:discussion} |
| 725 |
|
The Langevin Hull samples the isobaric-isothermal ensemble for |
| 726 |
|
non-periodic systems by coupling the system to a bath characterized by |
| 727 |
|
pressure, temperature, and solvent viscosity. This enables the |
| 728 |
< |
simulation of heterogeneous systems composed of materials of |
| 728 |
> |
simulation of heterogeneous systems composed of materials with |
| 729 |
|
significantly different compressibilities. Because the boundary is |
| 730 |
|
dynamically determined during the simulation and the molecules |
| 731 |
< |
interacting with the boundary can change, the method and has minimal |
| 731 |
> |
interacting with the boundary can change, the method inflicts minimal |
| 732 |
|
perturbations on the behavior of molecules at the edges of the |
| 733 |
|
simulation. Further work on this method will involve implicit |
| 734 |
|
electrostatics at the boundary (which is missing in the current |
| 783 |
|
The individual hull operations scale with |
| 784 |
|
$\mathcal{O}(\frac{n}{p}\log\frac{n}{p})$ where $n$ is the total |
| 785 |
|
number of sites, and $p$ is the number of processors. These local |
| 786 |
< |
hull operations create a set of $p$ hulls each with approximately |
| 787 |
< |
$\frac{n}{3pr}$ sites (for a cluster of radius $r$). The worst-case |
| 786 |
> |
hull operations create a set of $p$ hulls, each with approximately |
| 787 |
> |
$\frac{n}{3pr}$ sites for a cluster of radius $r$. The worst-case |
| 788 |
|
communication cost for using a ``gather'' operation to distribute this |
| 789 |
|
information to all processors is $\mathcal{O}( \alpha (p-1) + \frac{n |
| 790 |
|
\beta (p-1)}{3 r p^2})$, while the final computation of the system |
| 792 |
|
|
| 793 |
|
For a large number of atoms on a moderately parallel machine, the |
| 794 |
|
total costs are dominated by the computations of the individual hulls, |
| 795 |
< |
and communication of these hulls to so the Langevin hull sees roughly |
| 795 |
> |
and communication of these hulls to create the Langevin Hull sees roughly |
| 796 |
|
linear speed-up with increasing processor counts. |
| 797 |
|
|
| 798 |
|
\section*{Acknowledgments} |
