| 217 |
|
defined. The pressure and temperature bath interacts {\it directly} |
| 218 |
|
with the atoms on the edge and not with atoms interior to the |
| 219 |
|
simulation. |
| 220 |
+ |
|
| 221 |
+ |
|
| 222 |
+ |
\begin{figure} |
| 223 |
+ |
\includegraphics[width=\linewidth]{hullSample} |
| 224 |
+ |
\caption{The external temperature and pressure bath interacts only |
| 225 |
+ |
with those atoms on the convex hull (grey surface). The hull is |
| 226 |
+ |
computed dynamically at each time step, and molecules dynamically |
| 227 |
+ |
move between the interior (Newtonian) region and the Langevin hull.} |
| 228 |
+ |
\label{fig:hullSample} |
| 229 |
+ |
\end{figure} |
| 230 |
|
|
| 231 |
+ |
|
| 232 |
|
Atomic sites in the interior of the point cloud move under standard |
| 233 |
|
Newtonian dynamics, |
| 234 |
|
\begin{equation} |
| 310 |
|
tensor that includes translational and rotational drag as well as |
| 311 |
|
translational-rotational coupling. The computation of resistance |
| 312 |
|
tensors for rigid bodies has been detailed |
| 313 |
< |
elsewhere,\cite{JoseGarciadelaTorre02012000,Garcia-de-la-Torre:2001wd,GarciadelaTorreJ2002,Sun2008} |
| 313 |
> |
elsewhere,\cite{JoseGarciadelaTorre02012000,Garcia-de-la-Torre:2001wd,GarciadelaTorreJ2002,Sun:2008fk} |
| 314 |
|
but the standard approach involving bead approximations would be |
| 315 |
|
prohibitively expensive if it were recomputed at each step in a |
| 316 |
|
molecular dynamics simulation. |
| 353 |
|
We have implemented this method by extending the Langevin dynamics |
| 354 |
|
integrator in our code, OpenMD.\cite{Meineke2005,openmd} The Delaunay |
| 355 |
|
triangulation and computation of the convex hull are done using calls |
| 356 |
< |
to the qhull library.\cite{qhull} There is a moderate penalty for |
| 356 |
> |
to the qhull library.\cite{Qhull} There is a moderate penalty for |
| 357 |
|
computing the convex hull at each step in the molecular dynamics |
| 358 |
|
simulation (HOW MUCH?), but the convex hull is remarkably easy to |
| 359 |
|
parallelize on distributed memory machines (see Appendix A). |
| 361 |
|
\section{Tests \& Applications} |
| 362 |
|
\label{sec:tests} |
| 363 |
|
|
| 364 |
< |
In order to test this method, we have carried out simulations using |
| 365 |
< |
the Langevin Hull on a crystalline system (gold nanoparticles), a |
| 366 |
< |
liquid droplet (SPC/E water), and a heterogeneous mixture (gold |
| 367 |
< |
nanoparticles in a water droplet). In each case, we have computed |
| 368 |
< |
bulk properties that depend on the external applied pressure. Of |
| 369 |
< |
particular interest is the bulk modulus, |
| 364 |
> |
To test the new method, we have carried out simulations using the |
| 365 |
> |
Langevin Hull on: 1) a crystalline system (gold nanoparticles), 2) a |
| 366 |
> |
liquid droplet (SPC/E water),\cite{SPCE} and 3) a heterogeneous |
| 367 |
> |
mixture (gold nanoparticles in a water droplet). In each case, we have |
| 368 |
> |
computed properties that depend on the external applied pressure. Of |
| 369 |
> |
particular interest for the single-phase systems is the bulk modulus, |
| 370 |
|
\begin{equation} |
| 371 |
|
\kappa_{T} = -\frac{1}{V} \left ( \frac{\partial V}{\partial P} \right |
| 372 |
|
)_{T}. |
| 385 |
|
)_{T} |
| 386 |
|
\label{eq:BMN} |
| 387 |
|
\end{equation} |
| 388 |
< |
The region we pick is a spherical volume of 10 \AA radius centered in |
| 389 |
< |
the middle of the cluster. This radius is arbitrary, and any |
| 390 |
< |
bulk-like portion of the cluster can be used to compute the bulk |
| 391 |
< |
modulus. |
| 392 |
< |
|
| 382 |
< |
One might also assume that the volume of the convex hull could be |
| 383 |
< |
taken as the system volume in the compressibility expression |
| 384 |
< |
(Eq. \ref{eq:BM}), but this has implications at lower pressures (which |
| 385 |
< |
are explored in detail in the section on water droplets). |
| 388 |
> |
The region we used is a spherical volume of 10 \AA\ radius centered in |
| 389 |
> |
the middle of the cluster. $N$ is the average number of molecules |
| 390 |
> |
found within this region throughout a given simulation. The geometry |
| 391 |
> |
and size of the region is arbitrary, and any bulk-like portion of the |
| 392 |
> |
cluster can be used to compute the bulk modulus. |
| 393 |
|
|
| 394 |
+ |
One might assume that the volume of the convex hull could be taken as |
| 395 |
+ |
the system volume in the compressibility expression (Eq. \ref{eq:BM}), |
| 396 |
+ |
but this has implications at lower pressures (which are explored in |
| 397 |
+ |
detail in the section on water droplets). |
| 398 |
+ |
|
| 399 |
+ |
The metallic force field in use for the gold nanoparticles is the |
| 400 |
+ |
quantum Sutton-Chen (QSC) model.\cite{PhysRevB.59.3527} In all |
| 401 |
+ |
simulations involving point charges, we utilized damped shifted-force |
| 402 |
+ |
(DSF) electrostatics\cite{Fennell06} which is a variant of the Wolf |
| 403 |
+ |
summation\cite{wolf:8254} that has been shown to provide good forces |
| 404 |
+ |
and torques on molecular models for water in a computationally |
| 405 |
+ |
efficient manner.\cite{Fennell06} The damping parameter ($\alpha$) was |
| 406 |
+ |
set to 0.18 \AA$^{-1}$, and the cutoff radius was set to 12 \AA. The |
| 407 |
+ |
Spohr potential was adopted in depicting the interaction between metal |
| 408 |
+ |
atoms and the SPC/E water molecules.\cite{ISI:000167766600035} |
| 409 |
+ |
|
| 410 |
|
\subsection{Bulk modulus of gold nanoparticles} |
| 411 |
|
|
| 412 |
+ |
The bulk modulus is well-known for gold, and it provides a good first |
| 413 |
+ |
test of how the method compares to other similar methods. |
| 414 |
+ |
|
| 415 |
+ |
|
| 416 |
|
\begin{figure} |
| 417 |
|
\includegraphics[width=\linewidth]{pressure_tb} |
| 418 |
|
\caption{Pressure response is rapid (18 \AA gold nanoparticle), target |
| 476 |
|
phase, and isolated molecules can detach from the liquid droplet. |
| 477 |
|
This is expected behavior, but the reported volume of the convex |
| 478 |
|
hull includes large regions of empty space. For this reason, |
| 479 |
< |
compressibilities should be computed using local number densities |
| 480 |
< |
rather than hull volumes.} |
| 479 |
> |
compressibilities are computed using local number densities rather |
| 480 |
> |
than hull volumes.} |
| 481 |
|
\label{fig:coneOfShame} |
| 482 |
|
\end{figure} |
| 483 |
|
|
| 541 |
|
|
| 542 |
|
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. |
| 543 |
|
|
| 517 |
– |
|
| 544 |
|
\subsection{Heterogeneous nanoparticle / water mixtures} |
| 545 |
|
|
| 546 |
|
|
| 547 |
< |
\section{Appendix A: Hydrodynamic tensor for triangular facets} |
| 547 |
> |
\section*{Appendix A: Computing Convex Hulls on Parallel Computers} |
| 548 |
|
|
| 549 |
< |
\section{Appendix B: Computing Convex Hulls on Parallel Computers} |
| 524 |
< |
|
| 525 |
< |
\section{Acknowledgments} |
| 549 |
> |
\section*{Acknowledgments} |
| 550 |
|
Support for this project was provided by the |
| 551 |
|
National Science Foundation under grant CHE-0848243. Computational |
| 552 |
|
time was provided by the Center for Research Computing (CRC) at the |
