| 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} |
| 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 |
| 521 |
|
the total surface area of the cluter exposed to the bath as well as |
| 522 |
|
the bath viscosity. Pressure that is applied suddenly to a cluster |
| 523 |
|
can excite breathing vibrations, but these rapidly damp out (on time |
| 524 |
< |
scales of 30-50 ps). |
| 524 |
> |
scales of 30 -- 50 ps). |
| 525 |
|
|
| 526 |
|
\subsection{Compressibility of SPC/E water clusters} |
| 527 |
|
|
| 585 |
|
fixed region, |
| 586 |
|
\begin{equation} |
| 587 |
|
\kappa_{T} = \frac{\left \langle N^{2} \right \rangle - \left \langle |
| 588 |
< |
N \right \rangle ^{2}}{N \, k_{B} \, T}, |
| 588 |
> |
N \right \rangle ^{2}}{N \, k_{B} \, T}. |
| 589 |
|
\label{eq:BMNfluct} |
| 590 |
|
\end{equation} |
| 591 |
|
Thus, the compressibility of each simulation can be calculated |
| 657 |
|
to molecules with oxygen directed toward the exterior of the cluster, |
| 658 |
|
forming a dangling hydrogen bond acceptor site. |
| 659 |
|
|
| 660 |
< |
Previous molecular dynamics simulations |
| 664 |
< |
of SPC/E water using periodic boundary conditions have shown that molecules on the liquid side of the liquid/vapor interface favor a similar orientation where oxygen is directed away from the bulk.\cite{Taylor1996} These simulations had both a liquid phase and a well-defined vapor phase in equilibrium and showed that vapor molecules generally had one hydrogen protruding from the surface, forming a dangling hydrogen bond donor. Our water cluster simulations do not have a true lasting vapor phase, but rather a few transient molecules that leave the liquid droplet. Thus we are unable to comment on the orientational preference of vapor phase molecules in a Langevin Hull simulation. |
| 660 |
> |
The orientational preference exhibited by liquid phase hull molecules in the Langevin Hull 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. |
| 661 |
|
|
| 662 |
< |
However, the orientational preference exhibited by liquid phase hull molecules in the Langevin Hull 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. |
| 662 |
> |
Previous molecular dynamics simulations |
| 663 |
> |
of SPC/E water using periodic boundary conditions have shown that molecules on the liquid side of the liquid/vapor interface favor a similar orientation where oxygen is directed away from the bulk.\cite{Taylor1996} These simulations had both a liquid phase and a well-defined vapor phase in equilibrium and showed that vapor molecules generally had one hydrogen protruding from the surface, forming a dangling hydrogen bond donor. Our water cluster simulations do not have a true lasting vapor phase, but rather a few transient molecules that leave the liquid droplet. Thus while we are unable to comment on the orientational preference of vapor phase molecules in a Langevin Hull simulation, we achieve good agreement for the orientation of liquid phase molecules at the interface. |
| 664 |
|
|
| 665 |
|
\subsection{Heterogeneous nanoparticle / water mixtures} |
| 666 |
|
|
| 674 |
|
nanoparticle. |
| 675 |
|
|
| 676 |
|
\begin{figure} |
| 677 |
< |
|
| 678 |
< |
\caption{interesting plot showing cluster behavior} |
| 677 |
> |
\includegraphics[width=\linewidth]{RhoR} |
| 678 |
> |
\caption{Densities of gold and water at the nanoparticle surface. Higher applied pressures de-structure both the gold nanoparticle and water at the metal/water interface.} |
| 679 |
|
\label{fig:RhoR} |
| 680 |
|
\end{figure} |
| 681 |
|
|
