| 42 |
|
hull surrounding the system. A Langevin thermostat is also applied |
| 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. |
| 45 |
> |
materials with different compressibilities. These systems are |
| 46 |
> |
problematic for traditional affine transform methods. The Langevin |
| 47 |
> |
Hull does not suffer from the edge effects of boundary potential |
| 48 |
> |
methods, and allows realistic treatment of both external pressure |
| 49 |
> |
and thermal conductivity due to the presence of 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. |
| 55 |
|
\end{abstract} |
| 56 |
|
|
| 57 |
|
\newpage |
| 126 |
|
have concentrations orders of magnitude lower than this in the |
| 127 |
|
cellular environment. The effective concentrations of single proteins |
| 128 |
|
in simulations may have significant effects on the structure and |
| 129 |
< |
dynamics of simulated structures. |
| 129 |
> |
dynamics of simulated systems. |
| 130 |
|
|
| 131 |
|
\subsection*{Boundary Methods} |
| 132 |
|
There have been a number of approaches to handle simulations of |
| 481 |
|
interactions between the pseudoatom cores. The $\sqrt{\rho_i}$ term in |
| 482 |
|
Eq. (\ref{eq:SCP1}) is an attractive many-body potential that models |
| 483 |
|
the interactions between the valence electrons and the cores of the |
| 484 |
< |
pseudo-atoms. $D_{ij}$, $D_{ii}$ set the appropriate overall energy |
| 484 |
> |
pseudo-atoms. $D_{ij}$ and $D_{ii}$ set the appropriate overall energy |
| 485 |
|
scale, $c_i$ scales the attractive portion of the potential relative |
| 486 |
|
to the repulsive interaction and $\alpha_{ij}$ is a length parameter |
| 487 |
|
that assures a dimensionless form for $\rho$. These parameters are |
| 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{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. |
| 497 |
> |
of 175.53 GPa.\cite{QSC2} Using the same force field, we have |
| 498 |
> |
performed a series of 1 ns simulations on gold nanoparticles of three |
| 499 |
> |
different radii: 20 \AA~ (1985 atoms), 30 \AA~ (6699 atoms), and 40 |
| 500 |
> |
\AA~ (15707 atoms) utilizing the Langevin Hull at a variety of applied |
| 501 |
> |
pressures ranging from 0 -- 10 GPa. For the 40 \AA~ radius |
| 502 |
> |
nanoparticle we obtain a value of 177.55 GPa for the bulk modulus of |
| 503 |
> |
gold, in close agreement with both previous simulations and the |
| 504 |
> |
experimental bulk modulus reported for gold single |
| 505 |
> |
crystals.\cite{Collard1991} The smaller gold nanoparticles (30 and 20 |
| 506 |
> |
\AA~ radii) have calculated bulk moduli of 215.58 and 208.86 GPa, |
| 507 |
> |
respectively, indicating that smaller nanoparticles are somewhat |
| 508 |
> |
stiffer (less compressible) than the larger nanoparticles. This |
| 509 |
> |
stiffening of the small nanoparticles may be related to their high |
| 510 |
> |
degree of surface curvature, resulting in a lower coordination number |
| 511 |
> |
of surface atoms relative to the the surface atoms in the 40 \AA~ |
| 512 |
> |
radius particle. |
| 513 |
|
|
| 514 |
< |
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 $\mathrm {\frac{J}{mol \, K}}$. |
| 514 |
> |
We obtain a gold lattice constant of 4.051 \AA~ using the Langevin |
| 515 |
> |
Hull at 1 atm, close to the experimentally-determined value for bulk |
| 516 |
> |
gold and the value for gold simulated using the QSC potential and |
| 517 |
> |
periodic boundary conditions (4.079 \AA~ and 4.088\AA~, |
| 518 |
> |
respectively).\cite{QSC2} The slightly smaller calculated lattice |
| 519 |
> |
constant is most likely due to the presence of surface tension in the |
| 520 |
> |
non-periodic Langevin Hull cluster, an effect absent from a bulk |
| 521 |
> |
simulation. The specific heat of a 40 \AA~ gold nanoparticle under the |
| 522 |
> |
Langevin Hull at 1 atm is 24.914 $\mathrm {\frac{J}{mol \, K}}$, which |
| 523 |
> |
compares very well with the experimental value of 25.42 $\mathrm |
| 524 |
> |
{\frac{J}{mol \, K}}$. |
| 525 |
|
|
| 526 |
|
\begin{figure} |
| 527 |
|
\includegraphics[width=\linewidth]{stacked} |
| 615 |
|
effects of the empty space due to the vapor phase; for this reason, we |
| 616 |
|
recommend using the number density (Eq. \ref{eq:BMN}) or number |
| 617 |
|
density fluctuations (Eq. \ref{eq:BMNfluct}) for computing |
| 618 |
< |
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. |
| 618 |
> |
compressibilities. We obtained the results in |
| 619 |
> |
Fig. \ref{fig:compWater} using a sampling radius that was |
| 620 |
> |
approximately 80\% of the mean distance between the center of mass of |
| 621 |
> |
the cluster and the hull atoms. This ratio of sampling radius to |
| 622 |
> |
average hull radius excludes the problematic vapor phase on the |
| 623 |
> |
outside of the cluster while including enough of the liquid phase to |
| 624 |
> |
avoid poor statistics due to fluctuating local densities. |
| 625 |
|
|
| 626 |
< |
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. |
| 626 |
> |
A comparison of the oxygen-oxygen radial distribution functions for |
| 627 |
> |
SPC/E water simulated using both the Langevin Hull and more |
| 628 |
> |
traditional periodic boundary methods -- both at 1 atm and 300K -- |
| 629 |
> |
reveals an understructuring of water in the Langevin Hull that |
| 630 |
> |
manifests as a slight broadening of the solvation shells. This effect |
| 631 |
> |
may be due to the introduction of a liquid-vapor interface in the |
| 632 |
> |
Langevin Hull simulations (an interface which is missing in most |
| 633 |
> |
periodic simulations of bulk water). Vapor-phase molecules contribute |
| 634 |
> |
a small but nearly flat portion of the radial distribution function. |
| 635 |
|
|
| 636 |
|
\subsection{Molecular orientation distribution at cluster boundary} |
| 637 |
|
|
| 718 |
|
\subsection{Heterogeneous nanoparticle / water mixtures} |
| 719 |
|
|
| 720 |
|
To further test the method, we simulated gold nanoparticles ($r = 18$ |
| 721 |
< |
\AA) solvated by explicit SPC/E water clusters using a model for the |
| 722 |
< |
gold / water interactions that has been used by Dou {\it et. al.} for |
| 723 |
< |
investigating the separation of water films near hot metal |
| 724 |
< |
surfaces.\cite{ISI:000167766600035} The Langevin Hull was used to |
| 725 |
< |
sample pressures of 1, 2, 5, 10, 20, 50, 100 and 200 atm, while all |
| 726 |
< |
simulations were done at a temperature of 300 K. At these |
| 727 |
< |
temperatures and pressures, there is no observed separation of the |
| 728 |
< |
water film from the surface. |
| 721 |
> |
\AA~, 1433 atoms) solvated by explicit SPC/E water clusters (5000 |
| 722 |
> |
molecules) using a model for the gold / water interactions that has |
| 723 |
> |
been used by Dou {\it et. al.} for investigating the separation of |
| 724 |
> |
water films near hot metal surfaces.\cite{ISI:000167766600035} The |
| 725 |
> |
Langevin Hull was used to sample pressures of 1, 2, 5, 10, 20, 50, 100 |
| 726 |
> |
and 200 atm, while all simulations were done at a temperature of 300 |
| 727 |
> |
K. At these temperatures and pressures, there is no observed |
| 728 |
> |
separation of the water film from the surface. |
| 729 |
|
|
| 730 |
|
In Fig. \ref{fig:RhoR} we show the density of water and gold as a |
| 731 |
|
function of the distance from the center of the nanoparticle. Higher |
| 739 |
|
\includegraphics[width=\linewidth]{RhoR} |
| 740 |
|
\caption{Density profiles of gold and water at the nanoparticle |
| 741 |
|
surface. Each curve has been normalized by the average density in |
| 742 |
< |
the bulk-like region available to the corresponding material. Higher applied pressures |
| 743 |
< |
de-structure both the gold nanoparticle surface and water at the |
| 744 |
< |
metal/water interface.} |
| 742 |
> |
the bulk-like region available to the corresponding material. |
| 743 |
> |
Higher applied pressures de-structure both the gold nanoparticle |
| 744 |
> |
surface and water at the metal/water interface.} |
| 745 |
|
\label{fig:RhoR} |
| 746 |
|
\end{figure} |
| 747 |
|
|
| 754 |
|
to prevent water from diffusing into the center of the gold |
| 755 |
|
nanoparticles. This behavior is likely not a realistic description of |
| 756 |
|
the real physics of the situation. A better model of the gold-water |
| 757 |
< |
adsorption behavior appears to require harder repulsive walls to |
| 758 |
< |
prevent this behavior. |
| 757 |
> |
adsorption behavior would require harder repulsive walls to prevent |
| 758 |
> |
this behavior. |
| 759 |
|
|
| 760 |
|
\section{Discussion} |
| 761 |
|
\label{sec:discussion} |
