| 449 |
|
|
| 450 |
|
\subsection{Compressibility of gold nanoparticles} |
| 451 |
|
|
| 452 |
< |
The compressibility is well-known for gold, and it provides a good first |
| 453 |
< |
test of how the method compares to other similar methods. |
| 452 |
> |
The compressibility (and its inverse, the bulk modulus) is well-known |
| 453 |
> |
for gold, and is captured well by the embedded atom method |
| 454 |
> |
(EAM)~\cite{PhysRevB.33.7983} potential |
| 455 |
> |
and related multi-body force fields. In particular, the quantum |
| 456 |
> |
Sutton-Chen potential gets nearly quantitative agreement with the |
| 457 |
> |
experimental bulk modulus values, and makes a good first test of how |
| 458 |
> |
the Langevin Hull will perform at large applied pressures. |
| 459 |
|
|
| 460 |
< |
\begin{figure} |
| 461 |
< |
\includegraphics[width=\linewidth]{P_T_combined} |
| 462 |
< |
\caption{Pressure and temperature response of an 18 \AA\ gold |
| 463 |
< |
nanoparticle initially when first placed in the Langevin Hull |
| 464 |
< |
($T_\mathrm{bath}$ = 300K, $P_\mathrm{bath}$ = 4 GPa) and starting |
| 460 |
> |
The Sutton-Chen (SC) potentials are based on a model of a metal which |
| 461 |
> |
treats the nuclei and core electrons as pseudo-atoms embedded in the |
| 462 |
> |
electron density due to the valence electrons on all of the other |
| 463 |
> |
atoms in the system.\cite{Chen90} The SC potential has a simple form that closely |
| 464 |
> |
resembles the Lennard Jones potential, |
| 465 |
> |
\begin{equation} |
| 466 |
> |
\label{eq:SCP1} |
| 467 |
> |
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] , |
| 468 |
> |
\end{equation} |
| 469 |
> |
where $V^{pair}_{ij}$ and $\rho_{i}$ are given by |
| 470 |
> |
\begin{equation} |
| 471 |
> |
\label{eq:SCP2} |
| 472 |
> |
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}}. |
| 473 |
> |
\end{equation} |
| 474 |
> |
$V^{pair}_{ij}$ is a repulsive pairwise potential that accounts for |
| 475 |
> |
interactions between the pseudoatom cores. The $\sqrt{\rho_i}$ term in |
| 476 |
> |
Eq. (\ref{eq:SCP1}) is an attractive many-body potential that models |
| 477 |
> |
the interactions between the valence electrons and the cores of the |
| 478 |
> |
pseudo-atoms. $D_{ij}$, $D_{ii}$ set the appropriate overall energy |
| 479 |
> |
scale, $c_i$ scales the attractive portion of the potential relative |
| 480 |
> |
to the repulsive interaction and $\alpha_{ij}$ is a length parameter |
| 481 |
> |
that assures a dimensionless form for $\rho$. These parameters are |
| 482 |
> |
tuned to various experimental properties such as the density, cohesive |
| 483 |
> |
energy, and elastic moduli for FCC transition metals. The quantum |
| 484 |
> |
Sutton-Chen (QSC) formulation matches these properties while including |
| 485 |
> |
zero-point quantum corrections for different transition |
| 486 |
> |
metals.\cite{PhysRevB.59.3527} |
| 487 |
> |
|
| 488 |
> |
In bulk gold, the experimentally-measured value for the bulk modulus |
| 489 |
> |
is 180.32 GPa, while previous calculations on the QSC potential in |
| 490 |
> |
periodic-boundary simulations of the bulk have yielded values of |
| 491 |
> |
175.53 GPa.\cite{XXX} Using the same force field, we have performed a |
| 492 |
> |
series of relatively short (200 ps) simulations on 40 \r{A} radius |
| 493 |
> |
nanoparticles under the Langevin Hull at a variety of applied |
| 494 |
> |
pressures ranging from 0 GPa to XXX. We obtain a value of 177.547 GPa |
| 495 |
> |
for the bulk modulus for gold using this echnique. |
| 496 |
> |
|
| 497 |
> |
\begin{figure} |
| 498 |
> |
\includegraphics[width=\linewidth]{stacked} |
| 499 |
> |
\caption{The response of the internal pressure and temperature of gold |
| 500 |
> |
nanoparticles when first placed in the Langevin Hull |
| 501 |
> |
($T_\mathrm{bath}$ = 300K, $P_\mathrm{bath}$ = 4 GPa), starting |
| 502 |
|
from initial conditions that were far from the bath pressure and |
| 503 |
< |
temperature. The pressure response is rapid, and the thermal |
| 462 |
< |
equilibration depends on both total surface area and the viscosity |
| 463 |
< |
of the bath.} |
| 503 |
> |
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).} |
| 504 |
|
\label{pressureResponse} |
| 505 |
|
\end{figure} |
| 506 |
|
|
| 509 |
|
P}\right) |
| 510 |
|
\end{equation} |
| 511 |
|
|
| 472 |
– |
\begin{figure} |
| 473 |
– |
\includegraphics[width=\linewidth]{compress_tb} |
| 474 |
– |
\caption{Isothermal Compressibility (18 \AA gold nanoparticle)} |
| 475 |
– |
\label{temperatureResponse} |
| 476 |
– |
\end{figure} |
| 477 |
– |
|
| 512 |
|
\subsection{Compressibility of SPC/E water clusters} |
| 513 |
|
|
| 514 |
|
Prior molecular dynamics simulations on SPC/E water (both in |
| 614 |
|
\end{equation} |
| 615 |
|
where $\vec{r}_{i}$ is the vector between molecule {\it i}'s center of |
| 616 |
|
mass and the cluster center of mass and $\vec{\mu}_{i}$ is the vector |
| 617 |
< |
bisecting the H-O-H angle of molecule {\it i} (See |
| 618 |
< |
Fig. \ref{fig:coords}). |
| 619 |
< |
\begin{figure} |
| 620 |
< |
\includegraphics[width=\linewidth]{g_r_theta} |
| 621 |
< |
\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} |
| 617 |
> |
bisecting the H-O-H angle of molecule {\it i} Bulk-like distributions |
| 618 |
> |
will result in $\langle \cos \theta \rangle$ values close to zero. If |
| 619 |
> |
the hull exhibits an overabundance of externally-oriented oxygen sites |
| 620 |
> |
the average orientation will be negative, while dangling hydrogen |
| 621 |
> |
sites will result in positive average orientations. |
| 622 |
|
|
| 623 |
|
Fig. \ref{fig:pAngle} shows the distribution of $\cos{\theta}$ values |
| 624 |
|
for molecules in the interior of the cluster (squares) and for |
