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\bibpunct{[}{]}{,}{s}{}{;} |
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\bibliographystyle{achemso} |
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\begin{document} |
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hull surrounding the system. A Langevin thermostat is also applied |
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to the facets to mimic contact with an external heat bath. This new |
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method, the ``Langevin Hull'', can handle heterogeneous mixtures of |
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materials with different compressibilities. These are systems that |
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are problematic for traditional affine transform methods. The |
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Langevin Hull does not suffer from the edge effects of boundary |
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potential methods, and allows realistic treatment of both external |
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pressure and thermal conductivity due to the presence of an implicit |
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solvent. We apply this method to several different systems |
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including bare metal nanoparticles, nanoparticles in an explicit |
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solvent, as well as clusters of liquid water. The predicted |
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mechanical properties of these systems are in good agreement with |
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experimental data and previous simulation work. |
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materials with different compressibilities. These systems are |
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problematic for traditional affine transform methods. The Langevin |
| 47 |
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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 |
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metal nanoparticles, nanoparticles in an explicit solvent, as well |
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as clusters of liquid water. The predicted mechanical properties of |
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these systems are in good agreement with experimental data and |
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previous simulation work. |
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\end{abstract} |
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|
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\newpage |
| 121 |
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protein like hen egg white lysozyme (PDB code: 1LYZ) yields an |
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effective protein concentration of 100 mg/mL.\cite{Asthagiri20053300} |
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|
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{\it Yotal} protein concentrations in the cell are typically on the |
| 124 |
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{\it Total} protein concentrations in the cell are typically on the |
| 125 |
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order of 160-310 mg/ml,\cite{Brown1991195} and individual proteins |
| 126 |
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have concentrations orders of magnitude lower than this in the |
| 127 |
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cellular environment. The effective concentrations of single proteins |
| 128 |
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in simulations may have significant effects on the structure and |
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dynamics of simulated structures. |
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dynamics of simulated systems. |
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|
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\subsection*{Boundary Methods} |
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There have been a number of approaches to handle simulations of |
| 378 |
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configurations, so this appears to be a reasonably good approximation. |
| 379 |
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|
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We have implemented this method by extending the Langevin dynamics |
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integrator in our code, OpenMD.\cite{Meineke2005,openmd} At each |
| 381 |
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integrator in our code, OpenMD.\cite{Meineke2005,open_md} At each |
| 382 |
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molecular dynamics time step, the following process is carried out: |
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\begin{enumerate} |
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\item The standard inter-atomic forces ($\nabla_iU$) are computed. |
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\item Atomic positions and velocities are propagated. |
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\end{enumerate} |
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The Delaunay triangulation and computation of the convex hull are done |
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using calls to the qhull library.\cite{Qhull} There is a minimal |
| 403 |
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using calls to the qhull library.\cite{Q_hull} There is a minimal |
| 404 |
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penalty for computing the convex hull and resistance tensors at each |
| 405 |
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step in the molecular dynamics simulation (roughly 0.02 $\times$ cost |
| 406 |
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of a single force evaluation), and the convex hull is remarkably easy |
| 431 |
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)_{T}. |
| 432 |
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\label{eq:BMN} |
| 433 |
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\end{equation} |
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The region we used is a spherical volume of 10 \AA\ radius centered in |
| 435 |
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the middle of the cluster. $N$ is the average number of molecules |
| 434 |
> |
The region we used is a spherical volume of 20 \AA\ radius centered in |
| 435 |
> |
the middle of the cluster with a roughly 25 \AA\ radius. $N$ is the average number of molecules |
| 436 |
|
found within this region throughout a given simulation. The geometry |
| 437 |
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and size of the region is arbitrary, and any bulk-like portion of the |
| 438 |
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cluster can be used to compute the compressibility. |
| 437 |
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of the region is arbitrary, and any bulk-like portion of the |
| 438 |
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cluster can be used to compute the compressibility. |
| 439 |
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|
| 440 |
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One might assume that the volume of the convex hull could simply be |
| 441 |
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taken as the system volume $V$ in the compressibility expression |
| 481 |
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interactions between the pseudoatom cores. The $\sqrt{\rho_i}$ term in |
| 482 |
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Eq. (\ref{eq:SCP1}) is an attractive many-body potential that models |
| 483 |
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the interactions between the valence electrons and the cores of the |
| 484 |
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pseudo-atoms. $D_{ij}$, $D_{ii}$ set the appropriate overall energy |
| 484 |
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pseudo-atoms. $D_{ij}$ and $D_{ii}$ set the appropriate overall energy |
| 485 |
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scale, $c_i$ scales the attractive portion of the potential relative |
| 486 |
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to the repulsive interaction and $\alpha_{ij}$ is a length parameter |
| 487 |
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that assures a dimensionless form for $\rho$. These parameters are |
| 489 |
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energy, and elastic moduli for FCC transition metals. The quantum |
| 490 |
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Sutton-Chen (QSC) formulation matches these properties while including |
| 491 |
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zero-point quantum corrections for different transition |
| 492 |
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metals.\cite{PhysRevB.59.3527,QSC} |
| 492 |
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metals.\cite{PhysRevB.59.3527,QSC2} |
| 493 |
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|
| 494 |
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In bulk gold, the experimentally-measured value for the bulk modulus |
| 495 |
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is 180.32 GPa, while previous calculations on the QSC potential in |
| 496 |
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periodic-boundary simulations of the bulk crystal have yielded values |
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of 175.53 GPa.\cite{QSC} Using the same force field, we have performed |
| 498 |
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a series of 1 ns simulations on 40 \AA~ radius |
| 499 |
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nanoparticles under the Langevin Hull at a variety of applied |
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pressures ranging from 0 -- 10 GPa. We obtain a value of 177.55 GPa |
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for the bulk modulus of gold using this technique, in close agreement |
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with both previous simulations and the experimental bulk modulus of |
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gold. |
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of 175.53 GPa.\cite{QSC2} Using the same force field, we have |
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performed a series of 1 ns simulations on gold nanoparticles of three |
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different radii under the Langevin Hull at a variety of applied |
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pressures ranging from 0 -- 10 GPa. For the 40 \AA~ radius |
| 501 |
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nanoparticle we obtain a value of 177.55 GPa for the bulk modulus of |
| 502 |
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gold, in close agreement with both previous simulations and the |
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experimental bulk modulus reported for gold single |
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crystals.\cite{Collard1991} The smaller gold nanoparticles (30 and 20 |
| 505 |
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\AA~ radii) have calculated bulk moduli of 215.58 and 208.86 GPa, |
| 506 |
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respectively, indicating that smaller nanoparticles are somewhat |
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stiffer (less compressible) than the larger nanoparticles. This |
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stiffening of the small nanoparticles may be related to their high |
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degree of surface curvature, resulting in a lower coordination number |
| 510 |
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of surface atoms relative to the the surface atoms in the 40 \AA~ |
| 511 |
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radius particle. |
| 512 |
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|
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We obtain a gold lattice constant of 4.051 \AA~ using the Langevin |
| 514 |
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Hull at 1 atm, close to the experimentally-determined value for bulk |
| 515 |
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gold and the value for gold simulated using the QSC potential and |
| 516 |
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periodic boundary conditions (4.079 \AA~ and 4.088\AA~, |
| 517 |
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respectively).\cite{QSC2} The slightly smaller calculated lattice |
| 518 |
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constant is most likely due to the presence of surface tension in the |
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non-periodic Langevin Hull cluster, an effect absent from a bulk |
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simulation. The specific heat of a 40 \AA~ gold nanoparticle under the |
| 521 |
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Langevin Hull at 1 atm is 24.914 $\mathrm {\frac{J}{mol \, K}}$, which |
| 522 |
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compares very well with the experimental value of 25.42 $\mathrm |
| 523 |
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{\frac{J}{mol \, K}}$. |
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|
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\begin{figure} |
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\includegraphics[width=\linewidth]{stacked} |
| 527 |
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\caption{The response of the internal pressure and temperature of gold |
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temperature respond to the Langevin Hull for nanoparticles that were |
| 539 |
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initialized far from the target pressure and temperature. As |
| 540 |
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expected, the rate at which thermal equilibrium is achieved depends on |
| 541 |
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the total surface area of the cluter exposed to the bath as well as |
| 541 |
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the total surface area of the cluster exposed to the bath as well as |
| 542 |
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the bath viscosity. Pressure that is applied suddenly to a cluster |
| 543 |
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can excite breathing vibrations, but these rapidly damp out (on time |
| 544 |
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scales of 30 -- 50 ps). |
| 550 |
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ensembles) have yielded values for the isothermal compressibility that |
| 551 |
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agree well with experiment.\cite{Fine1973} The results of two |
| 552 |
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different approaches for computing the isothermal compressibility from |
| 553 |
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Langevin Hull simulations for pressures between 1 and 6500 atm are |
| 553 |
> |
Langevin Hull simulations for pressures between 1 and 3000 atm are |
| 554 |
|
shown in Fig. \ref{fig:compWater} along with compressibility values |
| 555 |
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obtained from both other SPC/E simulations and experiment. |
| 556 |
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|
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and previous simulation work throughout the 1 -- 1000 atm pressure |
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regime. Compressibilities computed using the Hull volume, however, |
| 567 |
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deviate dramatically from the experimental values at low applied |
| 568 |
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pressures. The reason for this deviation is quite simple; at low |
| 568 |
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pressures. The reason for this deviation is quite simple: at low |
| 569 |
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applied pressures, the liquid is in equilibrium with a vapor phase, |
| 570 |
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and it is entirely possible for one (or a few) molecules to drift away |
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from the liquid cluster (see Fig. \ref{fig:coneOfShame}). At low |
| 595 |
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different pressures must be done to compute the first derivatives. It |
| 596 |
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is also possible to compute the compressibility using the fluctuation |
| 597 |
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dissipation theorem using either fluctuations in the |
| 598 |
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volume,\cite{Debenedetti1986}, |
| 598 |
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volume,\cite{Debenedetti1986} |
| 599 |
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\begin{equation} |
| 600 |
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\kappa_{T} = \frac{\left \langle V^{2} \right \rangle - \left \langle |
| 601 |
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V \right \rangle ^{2}}{V \, k_{B} \, T}, |
| 614 |
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effects of the empty space due to the vapor phase; for this reason, we |
| 615 |
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recommend using the number density (Eq. \ref{eq:BMN}) or number |
| 616 |
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density fluctuations (Eq. \ref{eq:BMNfluct}) for computing |
| 617 |
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compressibilities. |
| 617 |
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compressibilities. We achieved the best results using a sampling |
| 618 |
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radius approximately 80\% of the cluster radius. This ratio of |
| 619 |
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sampling radius to cluster radius excludes the problematic vapor phase |
| 620 |
> |
on the outside of the cluster while including enough of the liquid |
| 621 |
> |
phase to avoid poor statistics due to fluctuating local densities. |
| 622 |
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|
| 623 |
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A comparison of the oxygen-oxygen radial distribution functions for |
| 624 |
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SPC/E water simulated using the Langevin Hull and bulk SPC/E using |
| 625 |
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periodic boundary conditions -- both at 1 atm and 300K -- reveals an |
| 626 |
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understructuring of water in the Langevin Hull that manifests as a |
| 627 |
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slight broadening of the solvation shells. This effect may be related |
| 628 |
+ |
to the introduction of surface tension around the entire cluster, an |
| 629 |
+ |
effect absent in bulk systems. As a result, molecules on the hull may |
| 630 |
+ |
experience an increased inward force, slightly compressing the |
| 631 |
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solvation shell for these molecules. |
| 632 |
+ |
|
| 633 |
|
\subsection{Molecular orientation distribution at cluster boundary} |
| 634 |
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|
| 635 |
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In order for a non-periodic boundary method to be widely applicable, |
| 638 |
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methods (e.g. hydrophobic boundary potentials) induced spurious |
| 639 |
|
orientational correlations deep within the simulated |
| 640 |
|
system.\cite{Lee1984,Belch1985} This behavior spawned many methods for |
| 641 |
< |
fixing and characterizing the effects of artifical boundaries |
| 641 |
> |
fixing and characterizing the effects of artificial boundaries |
| 642 |
|
including methods which fix the orientations of a set of edge |
| 643 |
|
molecules.\cite{Warshel1978,King1989} |
| 644 |
|
|
| 647 |
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hydrophobic boundary, or orientational or radial constraints. |
| 648 |
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Therefore, the orientational correlations of the molecules in water |
| 649 |
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clusters are of particular interest in testing this method. Ideally, |
| 650 |
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the water molecules on the surfaces of the clusterss will have enough |
| 650 |
> |
the water molecules on the surfaces of the clusters will have enough |
| 651 |
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mobility into and out of the center of the cluster to maintain |
| 652 |
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bulk-like orientational distribution in the absence of orientational |
| 653 |
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and radial constraints. However, since the number of hydrogen bonding |
| 714 |
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|
| 715 |
|
\subsection{Heterogeneous nanoparticle / water mixtures} |
| 716 |
|
|
| 717 |
< |
To further test the method, we simulated gold nanopartices ($r = 18$ |
| 717 |
> |
To further test the method, we simulated gold nanoparticles ($r = 18$ |
| 718 |
|
\AA) solvated by explicit SPC/E water clusters using a model for the |
| 719 |
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gold / water interactions that has been used by Dou {\it et. al.} for |
| 720 |
|
investigating the separation of water films near hot metal |
| 736 |
|
\includegraphics[width=\linewidth]{RhoR} |
| 737 |
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\caption{Density profiles of gold and water at the nanoparticle |
| 738 |
|
surface. Each curve has been normalized by the average density in |
| 739 |
< |
the bulk-like region available to the corresponding material. Higher applied pressures |
| 740 |
< |
de-structure both the gold nanoparticle surface and water at the |
| 741 |
< |
metal/water interface.} |
| 739 |
> |
the bulk-like region available to the corresponding material. |
| 740 |
> |
Higher applied pressures de-structure both the gold nanoparticle |
| 741 |
> |
surface and water at the metal/water interface.} |
| 742 |
|
\label{fig:RhoR} |
| 743 |
|
\end{figure} |
| 744 |
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|
| 751 |
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to prevent water from diffusing into the center of the gold |
| 752 |
|
nanoparticles. This behavior is likely not a realistic description of |
| 753 |
|
the real physics of the situation. A better model of the gold-water |
| 754 |
< |
adsorption behavior appears to require harder repulsive walls to |
| 755 |
< |
prevent this behavior. |
| 754 |
> |
adsorption behavior would require harder repulsive walls to prevent |
| 755 |
> |
this behavior. |
| 756 |
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|
| 757 |
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\section{Discussion} |
| 758 |
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\label{sec:discussion} |
