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\bibliographystyle{achemso} |
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\begin{document} |
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configurations, so this appears to be a reasonably good approximation. |
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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 |
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integrator in our code, OpenMD.\cite{Meineke2005,open_md} At each |
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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 |
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using calls to the qhull library.\cite{Q_hull} There is a minimal |
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penalty for computing the convex hull and resistance tensors at each |
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step in the molecular dynamics simulation (roughly 0.02 $\times$ cost |
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of a single force evaluation), and the convex hull is remarkably easy |
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\label{eq:BMN} |
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\end{equation} |
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The region we used is a spherical volume of 20 \AA\ radius centered in |
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the middle of the cluster. $N$ is the average number of molecules |
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the middle of the cluster with a roughly 25 \AA\ radius. $N$ is the average number of molecules |
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found within this region throughout a given simulation. The geometry |
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and size of the region is arbitrary, and any bulk-like portion of the |
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cluster can be used to compute the compressibility. |
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of the region is arbitrary, and any bulk-like portion of the |
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cluster can be used to compute the compressibility. |
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One might assume that the volume of the convex hull could simply be |
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taken as the system volume $V$ in the compressibility expression |
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energy, and elastic moduli for FCC transition metals. The quantum |
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Sutton-Chen (QSC) formulation matches these properties while including |
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zero-point quantum corrections for different transition |
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metals.\cite{PhysRevB.59.3527,QSC} |
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metals.\cite{PhysRevB.59.3527,QSC2} |
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In bulk gold, the experimentally-measured value for the bulk modulus |
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is 180.32 GPa, while previous calculations on the QSC potential in |
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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 |
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a series of 1 ns simulations on 40 \AA~ radius |
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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 performed |
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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. |
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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 $\mathrm {\frac{J}{mol \, K}}$. |
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\begin{figure} |
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\includegraphics[width=\linewidth]{stacked} |
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\caption{The response of the internal pressure and temperature of gold |
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ensembles) have yielded values for the isothermal compressibility that |
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agree well with experiment.\cite{Fine1973} The results of two |
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different approaches for computing the isothermal compressibility from |
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Langevin Hull simulations for pressures between 1 and 6500 atm are |
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Langevin Hull simulations for pressures between 1 and 3000 atm are |
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shown in Fig. \ref{fig:compWater} along with compressibility values |
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obtained from both other SPC/E simulations and experiment. |
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|
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effects of the empty space due to the vapor phase; for this reason, we |
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recommend using the number density (Eq. \ref{eq:BMN}) or number |
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density fluctuations (Eq. \ref{eq:BMNfluct}) for computing |
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compressibilities. |
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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. |
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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. |
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\subsection{Molecular orientation distribution at cluster boundary} |
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In order for a non-periodic boundary method to be widely applicable, |
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methods (e.g. hydrophobic boundary potentials) induced spurious |
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orientational correlations deep within the simulated |
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system.\cite{Lee1984,Belch1985} This behavior spawned many methods for |
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fixing and characterizing the effects of artifical boundaries |
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fixing and characterizing the effects of artificial boundaries |
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including methods which fix the orientations of a set of edge |
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molecules.\cite{Warshel1978,King1989} |
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\subsection{Heterogeneous nanoparticle / water mixtures} |
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To further test the method, we simulated gold nanopartices ($r = 18$ |
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To further test the method, we simulated gold nanoparticles ($r = 18$ |
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\AA) solvated by explicit SPC/E water clusters using a model for the |
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gold / water interactions that has been used by Dou {\it et. al.} for |
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investigating the separation of water films near hot metal |
