| 40 |
|
We have developed a new isobaric-isothermal (NPT) algorithm which |
| 41 |
|
applies an external pressure to the facets comprising the convex |
| 42 |
|
hull surrounding the system. A Langevin thermostat is also applied |
| 43 |
< |
to facets of the hull to mimic contact with an external heat |
| 44 |
< |
bath. This new method, the ``Langevin Hull'', performs better than |
| 45 |
< |
traditional affine transform methods for systems containing |
| 46 |
< |
heterogeneous mixtures of materials with different |
| 47 |
< |
compressibilities. It does not suffer from the edge effects of |
| 48 |
< |
boundary potential methods, and allows realistic treatment of both |
| 49 |
< |
external pressure and thermal conductivity to 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. |
| 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. |
| 55 |
|
\end{abstract} |
| 56 |
|
|
| 57 |
|
\newpage |
| 115 |
|
pressure conditions. The use of periodic boxes to enforce a system |
| 116 |
|
volume requires either effective solute concentrations that are much |
| 117 |
|
higher than desirable, or unreasonable system sizes to avoid this |
| 118 |
< |
effect. For example, calculations using typical hydration shells |
| 118 |
> |
effect. For example, calculations using typical hydration boxes |
| 119 |
|
solvating a protein under periodic boundary conditions are quite |
| 120 |
< |
expensive. [CALCULATE EFFECTIVE PROTEIN CONCENTRATIONS IN TYPICAL |
| 121 |
< |
SIMULATIONS] |
| 120 |
> |
expensive. A 62 $\AA^3$ box of water solvating a moderately small |
| 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 protein concentrations in the cell are on the order of |
| 125 |
> |
160-310 mg/ml,\cite{Brown1991195} and the factor of 20 difference |
| 126 |
> |
between simulations and the cellular environment may have significant |
| 127 |
> |
effects on the structure and dynamics of simulated protein structures. |
| 128 |
|
|
| 129 |
+ |
|
| 130 |
|
\subsection*{Boundary Methods} |
| 131 |
|
There have been a number of approaches to handle simulations of |
| 132 |
|
explicitly non-periodic systems that focus on constant or |
| 456 |
|
|
| 457 |
|
\subsection{Compressibility of gold nanoparticles} |
| 458 |
|
|
| 459 |
< |
The compressibility is well-known for gold, and it provides a good first |
| 460 |
< |
test of how the method compares to other similar methods. |
| 459 |
> |
The compressibility (and its inverse, the bulk modulus) is well-known |
| 460 |
> |
for gold, and is captured well by the embedded atom method |
| 461 |
> |
(EAM)~\cite{PhysRevB.33.7983} potential |
| 462 |
> |
and related multi-body force fields. In particular, the quantum |
| 463 |
> |
Sutton-Chen potential gets nearly quantitative agreement with the |
| 464 |
> |
experimental bulk modulus values, and makes a good first test of how |
| 465 |
> |
the Langevin Hull will perform at large applied pressures. |
| 466 |
|
|
| 467 |
< |
\begin{figure} |
| 468 |
< |
\includegraphics[width=\linewidth]{P_T_combined} |
| 469 |
< |
\caption{Pressure and temperature response of an 18 \AA\ gold |
| 470 |
< |
nanoparticle initially when first placed in the Langevin Hull |
| 471 |
< |
($T_\mathrm{bath}$ = 300K, $P_\mathrm{bath}$ = 4 GPa) and starting |
| 467 |
> |
The Sutton-Chen (SC) potentials are based on a model of a metal which |
| 468 |
> |
treats the nuclei and core electrons as pseudo-atoms embedded in the |
| 469 |
> |
electron density due to the valence electrons on all of the other |
| 470 |
> |
atoms in the system.\cite{Chen90} The SC potential has a simple form that closely |
| 471 |
> |
resembles the Lennard Jones potential, |
| 472 |
> |
\begin{equation} |
| 473 |
> |
\label{eq:SCP1} |
| 474 |
> |
U_{tot}=\sum _{i}\left[ \frac{1}{2}\sum _{j\neq i}D_{ij}V^{pair}_{ij}(r_{ij})-c_{i}D_{ii}\sqrt{\rho_{i}}\right] , |
| 475 |
> |
\end{equation} |
| 476 |
> |
where $V^{pair}_{ij}$ and $\rho_{i}$ are given by |
| 477 |
> |
\begin{equation} |
| 478 |
> |
\label{eq:SCP2} |
| 479 |
> |
V^{pair}_{ij}(r)=\left( \frac{\alpha_{ij}}{r_{ij}}\right)^{n_{ij}}, \rho_{i}=\sum_{j\neq i}\left( \frac{\alpha_{ij}}{r_{ij}}\right) ^{m_{ij}}. |
| 480 |
> |
\end{equation} |
| 481 |
> |
$V^{pair}_{ij}$ is a repulsive pairwise potential that accounts for |
| 482 |
> |
interactions between the pseudoatom cores. The $\sqrt{\rho_i}$ term in |
| 483 |
> |
Eq. (\ref{eq:SCP1}) is an attractive many-body potential that models |
| 484 |
> |
the interactions between the valence electrons and the cores of the |
| 485 |
> |
pseudo-atoms. $D_{ij}$, $D_{ii}$ set the appropriate overall energy |
| 486 |
> |
scale, $c_i$ scales the attractive portion of the potential relative |
| 487 |
> |
to the repulsive interaction and $\alpha_{ij}$ is a length parameter |
| 488 |
> |
that assures a dimensionless form for $\rho$. These parameters are |
| 489 |
> |
tuned to various experimental properties such as the density, cohesive |
| 490 |
> |
energy, and elastic moduli for FCC transition metals. The quantum |
| 491 |
> |
Sutton-Chen (QSC) formulation matches these properties while including |
| 492 |
> |
zero-point quantum corrections for different transition |
| 493 |
> |
metals.\cite{PhysRevB.59.3527} |
| 494 |
> |
|
| 495 |
> |
In bulk gold, the experimentally-measured value for the bulk modulus |
| 496 |
> |
is 180.32 GPa, while previous calculations on the QSC potential in |
| 497 |
> |
periodic-boundary simulations of the bulk have yielded values of |
| 498 |
> |
175.53 GPa.\cite{XXX} Using the same force field, we have performed a |
| 499 |
> |
series of relatively short (200 ps) simulations on 40 \r{A} radius |
| 500 |
> |
nanoparticles under the Langevin Hull at a variety of applied |
| 501 |
> |
pressures ranging from 0 GPa to XXX. We obtain a value of 177.547 GPa |
| 502 |
> |
for the bulk modulus for gold using this echnique. |
| 503 |
> |
|
| 504 |
> |
\begin{figure} |
| 505 |
> |
\includegraphics[width=\linewidth]{stacked} |
| 506 |
> |
\caption{The response of the internal pressure and temperature of gold |
| 507 |
> |
nanoparticles when first placed in the Langevin Hull |
| 508 |
> |
($T_\mathrm{bath}$ = 300K, $P_\mathrm{bath}$ = 4 GPa), starting |
| 509 |
|
from initial conditions that were far from the bath pressure and |
| 510 |
< |
temperature. The pressure response is rapid, and the thermal |
| 462 |
< |
equilibration depends on both total surface area and the viscosity |
| 463 |
< |
of the bath.} |
| 510 |
> |
temperature. The pressure response is rapid (after the breathing mode oscillations in the nanoparticle die out), and the rate of thermal equilibration depends on both exposed surface area (top panel) and the viscosity of the bath (middle panel).} |
| 511 |
|
\label{pressureResponse} |
| 512 |
|
\end{figure} |
| 513 |
|
|
| 516 |
|
P}\right) |
| 517 |
|
\end{equation} |
| 518 |
|
|
| 472 |
– |
\begin{figure} |
| 473 |
– |
\includegraphics[width=\linewidth]{compress_tb} |
| 474 |
– |
\caption{Isothermal Compressibility (18 \AA gold nanoparticle)} |
| 475 |
– |
\label{temperatureResponse} |
| 476 |
– |
\end{figure} |
| 477 |
– |
|
| 519 |
|
\subsection{Compressibility of SPC/E water clusters} |
| 520 |
|
|
| 521 |
|
Prior molecular dynamics simulations on SPC/E water (both in |
| 621 |
|
\end{equation} |
| 622 |
|
where $\vec{r}_{i}$ is the vector between molecule {\it i}'s center of |
| 623 |
|
mass and the cluster center of mass and $\vec{\mu}_{i}$ is the vector |
| 624 |
< |
bisecting the H-O-H angle of molecule {\it i} (See |
| 625 |
< |
Fig. \ref{fig:coords}). |
| 626 |
< |
\begin{figure} |
| 627 |
< |
\includegraphics[width=\linewidth]{g_r_theta} |
| 628 |
< |
\caption{Orientation angle of the water molecules relative to the |
| 588 |
< |
center of the cluster. Bulk-like distributions will result in |
| 589 |
< |
$\langle \cos \theta \rangle$ values close to zero. If the hull |
| 590 |
< |
exhibits an overabundance of externally-oriented oxygen sites the |
| 591 |
< |
average orientation will be negative, while dangling hydrogen sites |
| 592 |
< |
will result in positive average orientations.} |
| 593 |
< |
\label{fig:coords} |
| 594 |
< |
\end{figure} |
| 624 |
> |
bisecting the H-O-H angle of molecule {\it i} Bulk-like distributions |
| 625 |
> |
will result in $\langle \cos \theta \rangle$ values close to zero. If |
| 626 |
> |
the hull exhibits an overabundance of externally-oriented oxygen sites |
| 627 |
> |
the average orientation will be negative, while dangling hydrogen |
| 628 |
> |
sites will result in positive average orientations. |
| 629 |
|
|
| 630 |
|
Fig. \ref{fig:pAngle} shows the distribution of $\cos{\theta}$ values |
| 631 |
|
for molecules in the interior of the cluster (squares) and for |
| 709 |
|
\begin{enumerate} |
| 710 |
|
\item Each processor computes the convex hull for its own atomic sites |
| 711 |
|
(left panel in Fig. \ref{fig:parallel}). |
| 712 |
< |
\item The Hull vertices from each processor are passed out to all of |
| 712 |
> |
\item The Hull vertices from each processor are communicated to all of |
| 713 |
|
the processors, and each processor assembles a complete list of hull |
| 714 |
|
sites (this is much smaller than the original number of points in |
| 715 |
|
the point cloud). |
| 725 |
|
computation, the processors first compute the convex hulls for their |
| 726 |
|
own sites (dashed lines in left panel). The positions of the sites |
| 727 |
|
that make up the subset hulls are then communicated to all |
| 728 |
< |
processors (middle panel). The convex hull of the system (solid line in right panel) is the convex hull of the points on the union of the subset hulls.} |
| 728 |
> |
processors (middle panel). The convex hull of the system (solid line in |
| 729 |
> |
right panel) is the convex hull of the points on the union of the subset |
| 730 |
> |
hulls.} |
| 731 |
|
\label{fig:parallel} |
| 732 |
|
\end{figure} |
| 733 |
|
|
| 752 |
|
time was provided by the Center for Research Computing (CRC) at the |
| 753 |
|
University of Notre Dame. |
| 754 |
|
|
| 755 |
+ |
Molecular graphics images were produced using the UCSF Chimera package from |
| 756 |
+ |
the Resource for Biocomputing, Visualization, and Informatics at the |
| 757 |
+ |
University of California, San Francisco (supported by NIH P41 RR001081). |
| 758 |
|
\newpage |
| 759 |
|
|
| 760 |
|
\bibliography{langevinHull} |
