--- trunk/langevinHull/langevinHull.tex	2010/11/04 16:01:58	3668
+++ trunk/langevinHull/langevinHull.tex	2011/01/12 22:03:36	3713
@@ -40,18 +40,18 @@ Notre Dame, Indiana 46556}
   We have developed a new isobaric-isothermal (NPT) algorithm which
   applies an external pressure to the facets comprising the convex
   hull surrounding the system.  A Langevin thermostat is also applied
-  to facets of the hull to mimic contact with an external heat
-  bath. This new method, the ``Langevin Hull'', performs better than
-  traditional affine transform methods for systems containing
-  heterogeneous mixtures of materials with different
-  compressibilities. It does not suffer from the edge effects of
-  boundary potential methods, and allows realistic treatment of both
-  external pressure and thermal conductivity to an implicit solvent.
-  We apply this method to several different systems including bare
-  metal nanoparticles, nanoparticles in an explicit solvent, as well
-  as clusters of liquid water. The predicted mechanical properties of
-  these systems are in good agreement with experimental data and
-  previous simulation work.
+  to the facets to mimic contact with an external heat bath. This new
+  method, the ``Langevin Hull'', can handle heterogeneous mixtures of
+  materials with different compressibilities.  These are systems that
+  are problematic for traditional affine transform methods.  The
+  Langevin Hull does not suffer from the edge effects of boundary
+  potential methods, and allows realistic treatment of both external
+  pressure and thermal conductivity due to the presence of an implicit
+  solvent.  We apply this method to several different systems
+  including bare metal nanoparticles, nanoparticles in an explicit
+  solvent, as well as clusters of liquid water. The predicted
+  mechanical properties of these systems are in good agreement with
+  experimental data and previous simulation work.
 \end{abstract}
 
 \newpage
@@ -115,11 +115,19 @@ effect.  For example, calculations using typical hydra
 pressure conditions. The use of periodic boxes to enforce a system
 volume requires either effective solute concentrations that are much
 higher than desirable, or unreasonable system sizes to avoid this
-effect.  For example, calculations using typical hydration shells
+effect.  For example, calculations using typical hydration boxes
 solvating a protein under periodic boundary conditions are quite
-expensive. [CALCULATE EFFECTIVE PROTEIN CONCENTRATIONS IN TYPICAL
-SIMULATIONS]
+expensive.  A 62 \AA$^3$ box of water solvating a moderately small
+protein like hen egg white lysozyme (PDB code: 1LYZ) yields an
+effective protein concentration of 100 mg/mL.\cite{Asthagiri20053300}
 
+{\it Yotal} protein concentrations in the cell are typically on the
+order of 160-310 mg/ml,\cite{Brown1991195} and individual proteins
+have concentrations orders of magnitude lower than this in the
+cellular environment. The effective concentrations of single proteins
+in simulations may have significant effects on the structure and
+dynamics of simulated structures.
+
 \subsection*{Boundary Methods}
 There have been a number of approaches to handle simulations of
 explicitly non-periodic systems that focus on constant or
@@ -131,7 +139,7 @@ used widely for protein simulations. [CITATIONS NEEDED
 fixed regions were defined relative to a central atom.  Brooks and
 Karplus extended this method to include deformable stochastic
 boundaries.\cite{iii:6312} The stochastic boundary approach has been
-used widely for protein simulations. [CITATIONS NEEDED]
+used widely for protein simulations.
 
 The electrostatic and dispersive behavior near the boundary has long
 been a cause for concern when performing simulations of explicitly
@@ -142,23 +150,23 @@ affecting most of molecules in the simulation.  This r
 simulated clusters of TIPS2 water surrounded by a hydrophobic bounding
 potential. The spherical hydrophobic boundary induced dangling
 hydrogen bonds at the surface that propagated deep into the cluster,
-affecting most of molecules in the simulation.  This result echoes an
-earlier study which showed that an extended planar hydrophobic surface
-caused orientational preference at the surface which extended
-relatively deep (7 \r{A}) into the liquid simulation
-cell.\cite{Lee1984} The surface constrained all-atom solvent (SCAAS)
-model \cite{King1989} improved upon its SCSSD predecessor. The SCAAS
-model utilizes a polarization constraint which is applied to the
-surface molecules to maintain bulk-like structure at the cluster
-surface. A radial constraint is used to maintain the desired bulk
-density of the liquid. Both constraint forces are applied only to a
-pre-determined number of the outermost molecules.
+affecting most of the molecules in the simulation.  This result echoes
+an earlier study which showed that an extended planar hydrophobic
+surface caused orientational preferences at the surface which extended
+relatively deep (7 \AA) into the liquid simulation cell.\cite{Lee1984}
+The surface constrained all-atom solvent (SCAAS) model \cite{King1989}
+improved upon its SCSSD predecessor. The SCAAS model utilizes a
+polarization constraint which is applied to the surface molecules to
+maintain bulk-like structure at the cluster surface. A radial
+constraint is used to maintain the desired bulk density of the
+liquid. Both constraint forces are applied only to a pre-determined
+number of the outermost molecules.
 
 Beglov and Roux have developed a boundary model in which the hard
 sphere boundary has a radius that varies with the instantaneous
 configuration of the solute (and solvent) molecules.\cite{beglov:9050}
 This model contains a clear pressure and surface tension contribution
-to the free energy which XXX.
+to the free energy.
 
 \subsection*{Restraining Potentials}
 Restraining {\it potentials} introduce repulsive potentials at the
@@ -173,7 +181,7 @@ position of the nearest solute atom.\cite{LiY._jp04685
 Recently, Krilov {\it et al.} introduced a {\it flexible} boundary
 model that uses a Lennard-Jones potential between the solvent
 molecules and a boundary which is determined dynamically from the
-position of the nearest solute atom.\cite{LiY._jp046852t,Zhu:xw} This
+position of the nearest solute atom.\cite{LiY._jp046852t,Zhu:2008fk} This
 approach allows the confining potential to prevent solvent molecules
 from migrating too far from the solute surface, while providing a weak
 attractive force pulling the solvent molecules towards a fictitious
@@ -189,8 +197,8 @@ force in a direction that is inward-facing relative to
 into non-periodic simulations.\cite{Kohanoff:2005qm,Baltazar:2006ru}
 This method is based on standard Langevin dynamics, but the Brownian
 or random forces are allowed to act only on peripheral atoms and exert
-force in a direction that is inward-facing relative to the facets of a
-closed bounding surface.  The statistical distribution of the random
+forces in a direction that is inward-facing relative to the facets of
+a closed bounding surface.  The statistical distribution of the random
 forces are uniquely tied to the pressure in the external reservoir, so
 the method can be shown to sample the isobaric-isothermal ensemble.
 Kohanoff {\it et al.} used a Delaunay tessellation to generate a
@@ -214,17 +222,18 @@ The Langevin Hull uses an external bath at a fixed con
 \label{sec:meth}
 
 The Langevin Hull uses an external bath at a fixed constant pressure
-($P$) and temperature ($T$).  This bath interacts only with the
-objects on the exterior hull of the system.  Defining the hull of the
-simulation is done in a manner similar to the approach of Kohanoff,
-Caro and Finnis.\cite{Kohanoff:2005qm} That is, any instantaneous
-configuration of the atoms in the system is considered as a point
-cloud in three dimensional space.  Delaunay triangulation is used to
-find all facets between coplanar
-neighbors.\cite{delaunay,springerlink:10.1007/BF00977785}  In highly
+($P$) and temperature ($T$) with an effective solvent viscosity
+($\eta$).  This bath interacts only with the objects on the exterior
+hull of the system.  Defining the hull of the atoms in a simulation is
+done in a manner similar to the approach of Kohanoff, Caro and
+Finnis.\cite{Kohanoff:2005qm} That is, any instantaneous configuration
+of the atoms in the system is considered as a point cloud in three
+dimensional space.  Delaunay triangulation is used to find all facets
+between coplanar
+neighbors.\cite{delaunay,springerlink:10.1007/BF00977785} In highly
 symmetric point clouds, facets can contain many atoms, but in all but
-the most symmetric of cases the facets are simple triangles in 3-space
-that contain exactly three atoms.
+the most symmetric of cases, the facets are simple triangles in
+3-space which contain exactly three atoms.
 
 The convex hull is the set of facets that have {\it no concave
   corners} at an atomic site.\cite{Barber96,EDELSBRUNNER:1994oq} This
@@ -238,11 +247,11 @@ simulation.
 simulation.
 
 \begin{figure}
-\includegraphics[width=\linewidth]{hullSample}
+\includegraphics[width=\linewidth]{solvatedNano}
 \caption{The external temperature and pressure bath interacts only
   with those atoms on the convex hull (grey surface).  The hull is
   computed dynamically at each time step, and molecules can move 
-  between the interior (Newtonian) region and the Langevin hull.}
+  between the interior (Newtonian) region and the Langevin Hull.}
 \label{fig:hullSample}
 \end{figure}
 
@@ -257,7 +266,7 @@ equation of motion is modified with an external force,
 potential energy.  For atoms on the exterior of the cluster
 (i.e. those that occupy one of the vertices of the convex hull), the
 equation of motion is modified with an external force, ${\mathbf
-  F}_i^{\mathrm ext}$,
+  F}_i^{\mathrm ext}$:
 \begin{equation}
 m_i \dot{\mathbf v}_i(t)=-{\mathbf \nabla}_i U + {\mathbf F}_i^{\mathrm ext}.
 \end{equation}
@@ -293,7 +302,7 @@ viscosity of the fluid.  The resistance tensor is rela
 \end{equation}
 and $\Xi_f(t)$ is an approximate ($3 \times 3$) resistance tensor that
 depends on the geometry and surface area of facet $f$ and the
-viscosity of the fluid.  The resistance tensor is related to the
+viscosity of the bath.  The resistance tensor is related to the
 fluctuations of the random force, $\mathbf{R}(t)$, by the
 fluctuation-dissipation theorem,
 \begin{eqnarray}
@@ -323,7 +332,7 @@ Our treatment of the resistance tensor is approximate.
 random force, ${\bf R}_{f} = {\bf S} {\bf Z}$, can be shown to
 have the correct properties required by Eq. (\ref{eq:randomForce}).
 
-Our treatment of the resistance tensor is approximate.  $\Xi$ for a
+Our treatment of the resistance tensor is approximate.  $\Xi_f$ for a
 rigid triangular plate would normally be treated as a $6 \times 6$
 tensor that includes translational and rotational drag as well as
 translational-rotational coupling. The computation of resistance
@@ -378,7 +387,7 @@ molecular dynamics time step, the following process is
 \item The convex hull is computed and facets are identified.
 \item For each facet:
 \begin{itemize}
-\item[a.] The force from the pressure bath ($-PA_f\hat{n}_f$) is
+\item[a.] The force from the pressure bath ($-\hat{n}_fPA_f$) is
   computed.
 \item[b.] The resistance tensor ($\Xi_f(t)$) is computed using the
   viscosity ($\eta$) of the bath.
@@ -403,10 +412,7 @@ heterogeneous mixture (gold nanoparticles in a water d
 To test the new method, we have carried out simulations using the
 Langevin Hull on: 1) a crystalline system (gold nanoparticles), 2) a
 liquid droplet (SPC/E water),\cite{Berendsen1987} and 3) a
-heterogeneous mixture (gold nanoparticles in a 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,
+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,
 \begin{equation}
 \kappa_{T} = -\frac{1}{V} \left ( \frac{\partial V}{\partial P} \right
 )_{T}.
@@ -415,17 +421,17 @@ is not well-defined.  In order to compute the compress
 
 One problem with eliminating periodic boundary conditions and
 simulation boxes is that the volume of a three-dimensional point cloud
-is not well-defined.  In order to compute the compressibility of a
+is not well-defined. In order to compute the compressibility of a
 bulk material, we make an assumption that the number density, $\rho =
-\frac{N}{V}$, is uniform within some region of the point cloud.  The
+\frac{N}{V}$, is uniform within some region of the point cloud. The
 compressibility can then be expressed in terms of the average number
 of particles in that region,
 \begin{equation}
 \kappa_{T} = -\frac{1}{N} \left ( \frac{\partial N}{\partial P} \right
-)_{T}
+)_{T}.
 \label{eq:BMN}
 \end{equation}
-The region we used is a spherical volume of 10 \AA\ radius centered in
+The region we used is a spherical volume of 20 \AA\ radius centered in
 the middle of the cluster. $N$ is the average number of molecules
 found within this region throughout a given simulation. The geometry
 and size of the region is arbitrary, and any bulk-like portion of the
@@ -447,34 +453,76 @@ atoms and the SPC/E water molecules.\cite{ISI:00016776
 Spohr potential was adopted in depicting the interaction between metal
 atoms and the SPC/E water molecules.\cite{ISI:000167766600035}
 
-\subsection{Compressibility of gold nanoparticles}
+\subsection{Bulk Modulus of gold nanoparticles}
 
-The compressibility is well-known for gold, and it provides a good first
-test of how the method compares to other similar methods.   
+The compressibility (and its inverse, the bulk modulus) is well-known
+for gold, and is captured well by the embedded atom method
+(EAM)~\cite{PhysRevB.33.7983} potential and related multi-body force
+fields.  In particular, the quantum Sutton-Chen potential gets nearly
+quantitative agreement with the experimental bulk modulus values, and
+makes a good first test of how the Langevin Hull will perform at large
+applied pressures.
 
-\begin{figure}
-\includegraphics[width=\linewidth]{P_T_combined}
-\caption{Pressure and temperature response of an 18 \AA\ gold
-  nanoparticle initially when first placed in the Langevin Hull
-  ($T_\mathrm{bath}$ = 300K, $P_\mathrm{bath}$ = 4 GPa) and starting
-  from initial conditions that were far from the bath pressure and
-  temperature.  The pressure response is rapid, and the thermal
-  equilibration depends on both total surface area and the viscosity
-  of the bath.}
-\label{pressureResponse}
-\end{figure}
-
+The Sutton-Chen (SC) potentials are based on a model of a metal which
+treats the nuclei and core electrons as pseudo-atoms embedded in the
+electron density due to the valence electrons on all of the other
+atoms in the system.\cite{Chen90} The SC potential has a simple form
+that closely resembles the Lennard Jones potential,
 \begin{equation}
-\kappa_T=-\frac{1}{V_{\mathrm{eq}}}\left(\frac{\partial V}{\partial
-    P}\right)
+\label{eq:SCP1} 
+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] , 
 \end{equation}
+where $V^{pair}_{ij}$ and $\rho_{i}$ are given by 
+\begin{equation}
+\label{eq:SCP2} 
+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}}. 
+\end{equation}
+$V^{pair}_{ij}$ is a repulsive pairwise potential that accounts for
+interactions between the pseudoatom cores. The $\sqrt{\rho_i}$ term in
+Eq. (\ref{eq:SCP1}) is an attractive many-body potential that models
+the interactions between the valence electrons and the cores of the
+pseudo-atoms. $D_{ij}$, $D_{ii}$ set the appropriate overall energy
+scale, $c_i$ scales the attractive portion of the potential relative
+to the repulsive interaction and $\alpha_{ij}$ is a length parameter
+that assures a dimensionless form for $\rho$. These parameters are
+tuned to various experimental properties such as the density, cohesive
+energy, and elastic moduli for FCC transition metals. The quantum
+Sutton-Chen (QSC) formulation matches these properties while including
+zero-point quantum corrections for different transition
+metals.\cite{PhysRevB.59.3527,QSC}
 
+In bulk gold, the experimentally-measured value for the bulk modulus
+is 180.32 GPa, while previous calculations on the QSC potential in
+periodic-boundary simulations of the bulk crystal have yielded values
+of 175.53 GPa.\cite{QSC} Using the same force field, we have performed
+a series of 1 ns simulations on 40 \AA~ radius
+nanoparticles under the Langevin Hull at a variety of applied
+pressures ranging from 0 -- 10 GPa.  We obtain a value of 177.55 GPa
+for the bulk modulus of gold using this technique, in close agreement
+with both previous simulations and the experimental bulk modulus of
+gold.
+
 \begin{figure}
-\includegraphics[width=\linewidth]{compress_tb}
-\caption{Isothermal Compressibility (18 \AA gold nanoparticle)}
-\label{temperatureResponse}
+\includegraphics[width=\linewidth]{stacked}
+\caption{The response of the internal pressure and temperature of gold
+  nanoparticles when first placed in the Langevin Hull
+  ($T_\mathrm{bath}$ = 300K, $P_\mathrm{bath}$ = 4 GPa), starting
+  from initial conditions that were far from the bath pressure and
+  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).}
+\label{fig:pressureResponse}
 \end{figure}
 
+We note that the Langevin Hull produces rapidly-converging behavior
+for structures that are started far from equilibrium.  In
+Fig. \ref{fig:pressureResponse} we show how the pressure and
+temperature respond to the Langevin Hull for nanoparticles that were
+initialized far from the target pressure and temperature.  As
+expected, the rate at which thermal equilibrium is achieved depends on
+the total surface area of the cluster exposed to the bath as well as
+the bath viscosity.  Pressure that is applied suddenly to a cluster
+can excite breathing vibrations, but these rapidly damp out (on time
+scales of 30 -- 50 ps).
+
 \subsection{Compressibility of SPC/E water clusters}
 
 Prior molecular dynamics simulations on SPC/E water (both in
@@ -485,8 +533,6 @@ Compressibility values from all references are for app
 Langevin Hull simulations for pressures between 1 and 6500 atm are
 shown in Fig. \ref{fig:compWater} along with compressibility values
 obtained from both other SPC/E simulations and experiment.
-Compressibility values from all references are for applied pressures
-within the range 1 - 1000 atm.
 
 \begin{figure}
 \includegraphics[width=\linewidth]{new_isothermalN}
@@ -496,7 +542,7 @@ and previous simulation work throughout the 1 - 1000 a
 
 Isothermal compressibility values calculated using the number density
 (Eq. \ref{eq:BMN}) expression are in good agreement with experimental
-and previous simulation work throughout the 1 - 1000 atm pressure
+and previous simulation work throughout the 1 -- 1000 atm pressure
 regime.  Compressibilities computed using the Hull volume, however,
 deviate dramatically from the experimental values at low applied
 pressures.  The reason for this deviation is quite simple; at low
@@ -508,11 +554,11 @@ geometries which include large volumes of empty space.
 geometries which include large volumes of empty space.
 
 \begin{figure}
-\includegraphics[width=\linewidth]{flytest2}
+\includegraphics[width=\linewidth]{coneOfShame}
 \caption{At low pressures, the liquid is in equilibrium with the vapor
   phase, and isolated molecules can detach from the liquid droplet.
   This is expected behavior, but the volume of the convex hull
-  includes large regions of empty space.  For this reason,
+  includes large regions of empty space. For this reason,
   compressibilities are computed using local number densities rather
   than hull volumes.}
 \label{fig:coneOfShame}
@@ -533,27 +579,30 @@ volume,\cite{Debenedetti1986},
 \begin{equation}
 \kappa_{T} = \frac{\left \langle V^{2} \right \rangle - \left \langle
     V \right \rangle ^{2}}{V \, k_{B} \, T},
+\label{eq:BMVfluct}
 \end{equation}
 or, equivalently, fluctuations in the number of molecules within the
 fixed region,
 \begin{equation}
 \kappa_{T} = \frac{\left \langle N^{2} \right \rangle - \left \langle
-    N \right \rangle ^{2}}{N \, k_{B} \, T},
+    N \right \rangle ^{2}}{N \, k_{B} \, T}.
+\label{eq:BMNfluct}
 \end{equation}
 Thus, the compressibility of each simulation can be calculated
-entirely independently from all other trajectories. However, the
-resulting compressibilities were still as much as an order of
-magnitude larger than the reference values. However, compressibility
-calculation that relies on the hull volume will suffer these effects.
-WE NEED MORE HERE.
+entirely independently from other trajectories.  Compressibility
+calculations that rely on the hull volume will still suffer the
+effects of the empty space due to the vapor phase; for this reason, we
+recommend using the number density (Eq. \ref{eq:BMN}) or number
+density fluctuations (Eq. \ref{eq:BMNfluct}) for computing
+compressibilities.
 
 \subsection{Molecular orientation distribution at cluster boundary}
 
-In order for non-periodic boundary conditions to be widely applicable,
-they must be constructed in such a way that they allow a finite system
-to replicate the properties of the bulk.  Early non-periodic
-simulation methods (e.g. hydrophobic boundary potentials) induced
-spurious orientational correlations deep within the simulated
+In order for a non-periodic boundary method to be widely applicable,
+it must be constructed in such a way that they allow a finite system
+to replicate the properties of the bulk. Early non-periodic simulation
+methods (e.g. hydrophobic boundary potentials) induced spurious
+orientational correlations deep within the simulated
 system.\cite{Lee1984,Belch1985} This behavior spawned many methods for
 fixing and characterizing the effects of artifical boundaries
 including methods which fix the orientations of a set of edge
@@ -561,37 +610,31 @@ hydrophobic boundary, orientational constraint or radi
 
 As described above, the Langevin Hull does not require that the
 orientation of molecules be fixed, nor does it utilize an explicitly
-hydrophobic boundary, orientational constraint or radial constraint.
-Therefore, the orientational correlations of the molecules in a water
-cluster are of particular interest in testing this method.  Ideally,
-the water molecules on the surface of the cluster will have enough
-mobility into and out of the center of the cluster to maintain a
+hydrophobic boundary, or orientational or radial constraints.
+Therefore, the orientational correlations of the molecules in water
+clusters are of particular interest in testing this method.  Ideally,
+the water molecules on the surfaces of the clusters will have enough
+mobility into and out of the center of the cluster to maintain
 bulk-like orientational distribution in the absence of orientational
 and radial constraints.  However, since the number of hydrogen bonding
 partners available to molecules on the exterior are limited, it is
-likely that there will be some effective hydrophobicity of the hull.
+likely that there will be an effective hydrophobicity of the hull.
 
-To determine the extent of these effects demonstrated by the Langevin
-Hull, we examined the orientationations exhibited by SPC/E water in a
-cluster of 1372 molecules at 300 K and at pressures ranging from 1 -
-1000 atm.  The orientational angle of a water molecule is described
+To determine the extent of these effects, we examined the
+orientations exhibited by SPC/E water in a cluster of 1372
+molecules at 300 K and at pressures ranging from 1 -- 1000 atm.  The
+orientational angle of a water molecule is described by
 \begin{equation} 
 \cos{\theta}=\frac{\vec{r}_i\cdot\vec{\mu}_i}{|\vec{r}_i||\vec{\mu}_i|}
 \end{equation}
 where $\vec{r}_{i}$ is the vector between molecule {\it i}'s center of
-mass and the cluster center of mass and $\vec{\mu}_{i}$ is the vector
-bisecting the H-O-H angle of molecule {\it i} (See
-Fig. \ref{fig:coords}).
-\begin{figure}
-\includegraphics[width=\linewidth]{g_r_theta}
-\caption{Orientation angle of the water molecules relative to the
-  center of the cluster.  Bulk-like distributions will result in
-  $\langle \cos \theta \rangle$ values close to zero.  If the hull
-  exhibits an overabundance of externally-oriented oxygen sites the
-  average orientation will be negative, while dangling hydrogen sites
-  will result in positive average orientations.}
-\label{fig:coords}
-\end{figure}
+mass and the cluster center of mass, and $\vec{\mu}_{i}$ is the vector
+bisecting the H-O-H angle of molecule {\it i}.  Bulk-like
+distributions will result in $\langle \cos \theta \rangle$ values
+close to zero.  If the hull exhibits an overabundance of
+externally-oriented oxygen sites, the average orientation will be
+negative, while dangling hydrogen sites will result in positive
+average orientations.
 
 Fig. \ref{fig:pAngle} shows the distribution of $\cos{\theta}$ values
 for molecules in the interior of the cluster (squares) and for
@@ -600,7 +643,7 @@ molecules included in the convex hull (circles).
 \includegraphics[width=\linewidth]{pAngle}
 \caption{Distribution of $\cos{\theta}$ values for molecules on the
   interior of the cluster (squares) and for those participating in the
-  convex hull (circles) at a variety of pressures.  The Langevin hull
+  convex hull (circles) at a variety of pressures.  The Langevin Hull
   exhibits minor dewetting behavior with exposed oxygen sites on the
   hull water molecules.  The orientational preference for exposed
   oxygen appears to be independent of applied pressure. }
@@ -612,45 +655,86 @@ forming a dangling hydrogen bond acceptor site.
 orientations. Molecules included in the convex hull show a slight
 preference for values of $\cos{\theta} < 0.$ These values correspond
 to molecules with oxygen directed toward the exterior of the cluster,
-forming a dangling hydrogen bond acceptor site.
+forming dangling hydrogen bond acceptor sites.
 
-In the absence of an electrostatic contribution from the exterior
-bath, the orientational distribution of water molecules included in
-the Langevin Hull will slightly resemble the distribution at a neat
-water liquid/vapor interface.  Previous molecular dynamics simulations
-of SPC/E water \cite{Taylor1996} have shown that molecules at the
-liquid/vapor interface favor an orientation where one hydrogen
-protrudes from the liquid phase. This behavior is demonstrated by
-experiments \cite{Du1994} \cite{Scatena2001} showing that
-approximately one-quarter of water molecules at the liquid/vapor
-interface form dangling hydrogen bonds. The negligible preference
-shown in these cluster simulations could be removed through the
-introduction of an implicit solvent model, which would provide the
-missing electrostatic interactions between the cluster molecules and
-the surrounding temperature/pressure bath.
+The orientational preference exhibited by water molecules on the 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 near the cluster surface.
 
-The orientational preference exhibited by 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.
+Previous molecular dynamics simulations of SPC/E liquid / vapor
+interfaces using periodic boundary conditions have shown that
+molecules on the liquid side of interface favor a similar orientation
+where oxygen is directed away from the bulk.\cite{Taylor1996} These
+simulations had well-defined liquid and vapor phase regions
+equilibrium and it was observed that {\it vapor} molecules generally
+had one hydrogen protruding from the surface, forming a dangling
+hydrogen bond donor. Our water clusters do not have a true vapor
+region, but rather a few transient molecules that leave the liquid
+droplet (and which return to the droplet relatively quickly).
+Although we cannot obtain an orientational preference of vapor phase
+molecules in a Langevin Hull simulation, but we do agree with previous
+estimates of the orientation of {\it liquid phase} molecules at the
+interface.
 
 \subsection{Heterogeneous nanoparticle / water mixtures}
 
+To further test the method, we simulated gold nanopartices ($r = 18$
+\AA) solvated by explicit SPC/E water clusters using a model for the
+gold / water interactions that has been used by Dou {\it et. al.} for
+investigating the separation of water films near hot metal
+surfaces.\cite{ISI:000167766600035} The Langevin Hull was used to
+sample pressures of 1, 2, 5, 10, 20, 50, 100 and 200 atm, while all
+simulations were done at a temperature of 300 K.   At these
+temperatures and pressures, there is no observed separation of the
+water film from the surface.  
+
+In Fig. \ref{fig:RhoR} we show the density of water and gold as a
+function of the distance from the center of the nanoparticle.  Higher
+applied pressures appear to destroy structural correlations in the
+outermost monolayer of the gold nanoparticle as well as in the water
+at the near the metal / water interface.  Simulations at increased
+pressures exhibit significant overlap of the gold and water densities,
+indicating a less well-defined interfacial surface.
+
+\begin{figure}
+\includegraphics[width=\linewidth]{RhoR}
+\caption{Density profiles of gold and water at the nanoparticle
+  surface. Each curve has been normalized by the average density in
+  the bulk-like region available to the corresponding material.  Higher applied pressures
+  de-structure both the gold nanoparticle surface and water at the
+  metal/water interface.}
+\label{fig:RhoR}
+\end{figure}
+
+At even higher pressures (500 atm and above), problems with the metal
+- water interaction potential became quite clear.  The model we are
+using appears to have been parameterized for relatively low pressures;
+it utilizes both shifted Morse and repulsive Morse potentials to model
+the Au/O and Au/H interactions, respectively.  The repulsive wall of
+the Morse potential does not diverge quickly enough at short distances
+to prevent water from diffusing into the center of the gold
+nanoparticles.  This behavior is likely not a realistic description of
+the real physics of the situation.  A better model of the gold-water
+adsorption behavior appears to require harder repulsive walls to
+prevent this behavior.
+
 \section{Discussion}
 \label{sec:discussion}
 
 The Langevin Hull samples the isobaric-isothermal ensemble for
-non-periodic systems by coupling the system to an bath characterized
-by pressure, temperature, and solvent viscosity.  This enables the
-study of heterogeneous systems composed of materials of significantly
-different compressibilities.  Because the boundary is dynamically
-determined during the simulation and the molecules interacting with
-the boundary can change, the method and has minimal perturbations on
-the behavior of molecules at the edges of the simulation.  Further
-work on this method will involve implicit electrostatics at the
-boundary (which is missing in the current implementation) as well as
-more sophisticated treatments of the surface geometry (alpha
+non-periodic systems by coupling the system to a bath characterized by
+pressure, temperature, and solvent viscosity.  This enables the
+simulation of heterogeneous systems composed of materials with
+significantly different compressibilities.  Because the boundary is
+dynamically determined during the simulation and the molecules
+interacting with the boundary can change, the method inflicts minimal
+perturbations on the behavior of molecules at the edges of the
+simulation.  Further work on this method will involve implicit
+electrostatics at the boundary (which is missing in the current
+implementation) as well as more sophisticated treatments of the
+surface geometry (alpha
 shapes\cite{EDELSBRUNNER:1994oq,EDELSBRUNNER:1995cj} and Tight
 Cocone\cite{Dey:2003ts}). The non-convex hull geometries are
 significantly more expensive ($\mathcal{O}(N^2)$) than the convex hull
@@ -661,8 +745,8 @@ surface, facets, and resistance tensors when the proce
 
 In order to use the Langevin Hull for simulations on parallel
 computers, one of the more difficult tasks is to compute the bounding
-surface, facets, and resistance tensors when the processors have
-incomplete information about the entire system's topology.  Most
+surface, facets, and resistance tensors when the individual processors
+have incomplete information about the entire system's topology.  Most
 parallel decomposition methods assign primary responsibility for the
 motion of an atomic site to a single processor, and we can exploit
 this to efficiently compute the convex hull for the entire system.
@@ -675,7 +759,7 @@ works as follows:
 \begin{enumerate}
 \item Each processor computes the convex hull for its own atomic sites
   (left panel in Fig. \ref{fig:parallel}).
-\item The Hull vertices from each processor are passed out to all of
+\item The Hull vertices from each processor are communicated to all of
   the processors, and each processor assembles a complete list of hull
   sites (this is much smaller than the original number of points in
   the point cloud).
@@ -686,24 +770,22 @@ works as follows:
 \end{enumerate}
 
 \begin{figure}
-\begin{centering}
 \includegraphics[width=\linewidth]{parallel}
 \caption{When the sites are distributed among many nodes for parallel
   computation, the processors first compute the convex hulls for their
   own sites (dashed lines in left panel). The positions of the sites
-  that make up the convex hulls are then communicated to all
-  processors (middle panel).  The convex hull of the system (solid line in right panel) is the convex hull of the points on the hulls for all
-  processors.}
+  that make up the subset hulls are then communicated to all
+  processors (middle panel).  The convex hull of the system (solid line in 
+  right panel) is the convex hull of the points on the union of the subset 
+  hulls.}
 \label{fig:parallel}
-\end{centering}
-\label{fig:parallel}
 \end{figure}
 
 The individual hull operations scale with
 $\mathcal{O}(\frac{n}{p}\log\frac{n}{p})$ where $n$ is the total
 number of sites, and $p$ is the number of processors.  These local
-hull operations create a set of $p$ hulls each with approximately
-$\frac{n}{3pr}$ sites (for a cluster of radius $r$). The worst-case
+hull operations create a set of $p$ hulls, each with approximately
+$\frac{n}{3pr}$ sites for a cluster of radius $r$. The worst-case
 communication cost for using a ``gather'' operation to distribute this
 information to all processors is $\mathcal{O}( \alpha (p-1) + \frac{n
   \beta (p-1)}{3 r p^2})$, while the final computation of the system
@@ -711,7 +793,7 @@ and communication of these hulls to so the Langevin hu
 
 For a large number of atoms on a moderately parallel machine, the
 total costs are dominated by the computations of the individual hulls,
-and communication of these hulls to so the Langevin hull sees roughly
+and communication of these hulls to create the Langevin Hull sees roughly
 linear speed-up with increasing processor counts.
 
 \section*{Acknowledgments}
@@ -720,6 +802,9 @@ University of Notre Dame.  
 time was provided by the Center for Research Computing (CRC) at the
 University of Notre Dame.  
 
+Molecular graphics images were produced using the UCSF Chimera package from 
+the Resource for Biocomputing, Visualization, and Informatics at the 
+University of California, San Francisco (supported by NIH P41 RR001081).
 \newpage
 
 \bibliography{langevinHull}
