trunk/langevinHull/langevinHull.tex

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\begin{document}

\title{The Langevin Hull: Constant pressure and temperature dynamics for non-periodic systems}

\author{Charles F. Vardeman II, Kelsey M. Stocker, and J. Daniel
Gezelter\footnote{Corresponding author. \ Electronic mail: gezelter@nd.edu} \\
Department of Chemistry and Biochemistry,\\
University of Notre Dame\\
Notre Dame, Indiana 46556}

\date{\today}

\maketitle 

\begin{doublespace}

\begin{abstract}
  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.
\end{abstract}

\newpage

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\section{Introduction}

The most common molecular dynamics methods for sampling configurations
from an isobaric-isothermal (NPT) ensemble maintain a target pressure
in a simulation by coupling the volume of the system to a {\it
  barostat}, which is an extra degree of freedom propagated along with
the particle coordinates.  These methods require periodic boundary
conditions, because when the instantaneous pressure in the system
differs from the target pressure, the volume is reduced or expanded
using {\it affine transforms} of the system geometry. An affine
transform scales the size and shape of the periodic box as well as the
particle positions within the box (but not the sizes of the
particles). The most common constant pressure methods, including the
Melchionna modification\cite{Melchionna1993} to the
Nos\'e-Hoover-Andersen equations of
motion,\cite{Hoover85,ANDERSEN:1980vn,Sturgeon:2000kx} the Berendsen
pressure bath,\cite{ISI:A1984TQ73500045} and the Langevin
Piston,\cite{FELLER:1995fk,Jakobsen:2005uq} all utilize scaled
coordinate transformation to adjust the box volume.  As long as the
material in the simulation box has a relatively uniform
compressibility, the standard affine transform approach provides an
excellent way of adjusting the volume of the system and applying
pressure directly via the interactions between atomic sites.

One problem with this approach appears when the system being simulated
is an inhomogeneous mixture in which portions of the simulation box
are incompressible relative to other portions.  Examples include
simulations of metallic nanoparticles in liquid environments, proteins
at ice / water interfaces, as well as other heterogeneous or
interfacial environments.  In these cases, the affine transform of
atomic coordinates will either cause numerical instability when the
sites in the incompressible medium collide with each other, or will
lead to inefficient sampling of system volumes if the barostat is set
slow enough to avoid the instabilities in the incompressible region.

\begin{figure}
\includegraphics[width=\linewidth]{AffineScale2}
\caption{Affine scaling methods use box-length scaling to adjust the
  volume to adjust to under- or over-pressure conditions. In a system
  with a uniform compressibility (e.g. bulk fluids) these methods can
  work well.  In systems containing heterogeneous mixtures, the affine
  scaling moves required to adjust the pressure in the
  high-compressibility regions can cause molecules in low
  compressibility regions to collide.}
\label{affineScale}
\end{figure}

One may also wish to avoid affine transform periodic boundary methods
to simulate {\it explicitly non-periodic systems} under constant
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
solvating a protein under periodic boundary conditions are quite
expensive. [CALCULATE EFFECTIVE PROTEIN CONCENTRATIONS IN TYPICAL
SIMULATIONS]

\subsection*{Boundary Methods}
There have been a number of approaches to handle simulations of
explicitly non-periodic systems that focus on constant or
nearly-constant {\it volume} conditions while maintaining bulk-like
behavior.  Berkowitz and McCammon introduced a stochastic (Langevin)
boundary layer inside a region of fixed molecules which effectively
enforces constant temperature and volume (NVT)
conditions.\cite{Berkowitz1982} In this approach, the stochastic and
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]

The electrostatic and dispersive behavior near the boundary has long
been a cause for concern when performing simulations of explicitly
non-periodic systems.  Early work led to the surface constrained soft
sphere dipole model (SCSSD)\cite{Warshel1978} in which the surface
molecules are fixed in a random orientation representative of the bulk
solvent structural properties. Belch {\it et al.}\cite{Belch1985}
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.

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.

\subsection*{Restraining Potentials}
Restraining {\it potentials} introduce repulsive potentials at the
surface of a sphere or other geometry.  The solute and any explicit
solvent are therefore restrained inside the range defined by the
external potential.  Often the potentials include a weak short-range
attraction to maintain the correct density at the boundary.  Beglov
and Roux have also introduced a restraining boundary potential which
relaxes dynamically depending on the solute geometry and the force the
explicit system exerts on the shell.\cite{Beglov:1995fk}

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
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
bulk solvent.  Although this approach is appealing and has physical
motivation, nanoparticles do not deform far from their original
geometries even at temperatures which vaporize the nearby solvent. For
the systems like this, the flexible boundary model will be nearly
identical to a fixed-volume restraining potential.

\subsection*{Hull methods}
The approach of Kohanoff, Caro, and Finnis is the most promising of
the methods for introducing both constant pressure and temperature
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 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
bounding surface surrounding the outermost atoms in the simulated
system.  This is not the only possible triangulated outer surface, but
guarantees that all of the random forces point inward towards the
cluster.

In the following sections, we extend and generalize the approach of
Kohanoff, Caro, and Finnis. The new method, which we are calling the
``Langevin Hull'' applies the external pressure, Langevin drag, and
random forces on the {\it facets of the hull} instead of the atomic
sites comprising the vertices of the hull.  This allows us to decouple
the external pressure contribution from the drag and random force.
The methodology is introduced in section \ref{sec:meth}, tests on
crystalline nanoparticles, liquid clusters, and heterogeneous mixtures
are detailed in section \ref{sec:tests}.  Section \ref{sec:discussion}
summarizes our findings.

\section{Methodology}
\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
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 convex hull is the set of facets that have {\it no concave
  corners} at an atomic site.\cite{Barber96,EDELSBRUNNER:1994oq} This
eliminates all facets on the interior of the point cloud, leaving only
those exposed to the bath. Sites on the convex hull are dynamic; as
molecules re-enter the cluster, all interactions between atoms on that
molecule and the external bath are removed.  Since the edge is
determined dynamically as the simulation progresses, no {\it a priori}
geometry is defined. The pressure and temperature bath interacts only
with the atoms on the edge and not with atoms interior to the
simulation.

\begin{figure}
\includegraphics[width=\linewidth]{hullSample}
\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.}
\label{fig:hullSample}
\end{figure}

Atomic sites in the interior of the simulation move under standard
Newtonian dynamics,
\begin{equation}
m_i \dot{\mathbf v}_i(t)=-{\mathbf \nabla}_i U,
\label{eq:Newton}
\end{equation}
where $m_i$ is the mass of site $i$, ${\mathbf v}_i(t)$ is the
instantaneous velocity of site $i$ at time $t$, and $U$ is the total
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, ${\mathbf
  F}_i^{\mathrm ext}$,
\begin{equation}
m_i \dot{\mathbf v}_i(t)=-{\mathbf \nabla}_i U + {\mathbf F}_i^{\mathrm ext}.
\end{equation}

The external bath interacts indirectly with the atomic sites through
the intermediary of the hull facets.  Since each vertex (or atom)
provides one corner of a triangular facet, the force on the facets are
divided equally to each vertex.  However, each vertex can participate
in multiple facets, so the resultant force is a sum over all facets
$f$ containing vertex $i$:
\begin{equation}
{\mathbf F}_{i}^{\mathrm ext} = \sum_{\begin{array}{c}\mathrm{facets\
    } f \\ \mathrm{containing\ } i\end{array}} \frac{1}{3}\  {\mathbf
  F}_f^{\mathrm ext}
\end{equation}

The external pressure bath applies a force to the facets of the convex
hull in direct proportion to the area of the facet, while the thermal
coupling depends on the solvent temperature, viscosity and the size
and shape of each facet. The thermal interactions are expressed as a
standard Langevin description of the forces,
\begin{equation}
\begin{array}{rclclcl}
{\mathbf F}_f^{\text{ext}} & = &  \text{external pressure} & + & \text{drag force} & + & \text{random force} \\
& = &  -\hat{n}_f P A_f  & - & \Xi_f(t) {\mathbf v}_f(t)  & + & {\mathbf R}_f(t)
\end{array}
\end{equation}
Here, $A_f$ and $\hat{n}_f$ are the area and (outward-facing) normal
vectors for facet $f$, respectively.  ${\mathbf v}_f(t)$ is the
velocity of the facet centroid,
\begin{equation}
{\mathbf v}_f(t) =  \frac{1}{3} \sum_{i=1}^{3} {\mathbf v}_i,
\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
fluctuations of the random force, $\mathbf{R}(t)$, by the
fluctuation-dissipation theorem,
\begin{eqnarray}
\left< {\mathbf R}_f(t) \right> & = & 0 \\
\left<{\mathbf R}_f(t) {\mathbf R}_f^T(t^\prime)\right> & = & 2 k_B T\
\Xi_f(t)\delta(t-t^\prime).
\label{eq:randomForce}
\end{eqnarray}

Once the resistance tensor is known for a given facet, a stochastic
vector that has the properties in Eq. (\ref{eq:randomForce}) can be
calculated efficiently by carrying out a Cholesky decomposition to
obtain the square root matrix of the resistance tensor,
\begin{equation} 
\Xi_f = {\bf S} {\bf S}^{T},
\label{eq:Cholesky}
\end{equation} 
where ${\bf S}$ is a lower triangular matrix.\cite{Schlick2002} A
vector with the statistics required for the random force can then be
obtained by multiplying ${\bf S}$ onto a random 3-vector ${\bf Z}$ which
has elements chosen from a Gaussian distribution, such that:
\begin{equation}
\langle {\bf Z}_i \rangle = 0, \hspace{1in} \langle {\bf Z}_i \cdot
{\bf Z}_j \rangle = \frac{2 k_B T}{\delta t} \delta_{ij},
\end{equation}
where $\delta t$ is the timestep in use during the simulation. The
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
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
tensors for rigid bodies has been detailed
elsewhere,\cite{JoseGarciadelaTorre02012000,Garcia-de-la-Torre:2001wd,GarciadelaTorreJ2002,Sun:2008fk}
but the standard approach involving bead approximations would be
prohibitively expensive if it were recomputed at each step in a
molecular dynamics simulation.

Instead, we are utilizing an approximate resistance tensor obtained by
first constructing the Oseen tensor for the interaction of the
centroid of the facet ($f$) with each of the subfacets $\ell=1,2,3$,
\begin{equation}
T_{\ell f}=\frac{A_\ell}{8\pi\eta R_{\ell f}}\left(I +
  \frac{\mathbf{R}_{\ell f}\mathbf{R}_{\ell f}^T}{R_{\ell f}^2}\right)
\end{equation}
Here, $A_\ell$ is the area of subfacet $\ell$ which is a triangle
containing two of the vertices of the facet along with the centroid.
$\mathbf{R}_{\ell f}$ is the vector between the centroid of facet $f$
and the centroid of sub-facet $\ell$, and $I$ is the ($3 \times 3$)
identity matrix.  $\eta$ is the viscosity of the external bath.

\begin{figure}
\includegraphics[width=\linewidth]{hydro}
\caption{The resistance tensor $\Xi$ for a facet comprising sites $i$,
  $j$, and $k$ is constructed using Oseen tensor contributions between
  the centoid of the facet $f$ and each of the sub-facets ($i,f,j$),
  ($j,f,k$), and ($k,f,i$). The centroids of the sub-facets are
  located at $1$, $2$, and $3$, and the area of each sub-facet is
  easily computed using half the cross product of two of the edges.}
\label{hydro}
\end{figure}

The tensors for each of the sub-facets are added together, and the
resulting matrix is inverted to give a $3 \times 3$ resistance tensor
for translations of the triangular facet,
\begin{equation}
\Xi_f(t) =\left[\sum_{i=1}^3 T_{if}\right]^{-1}.
\end{equation}
Note that this treatment ignores rotations (and
translational-rotational coupling) of the facet.  In compact systems,
the facets stay relatively fixed in orientation between
configurations, so this appears to be a reasonably good approximation.

We have implemented this method by extending the Langevin dynamics
integrator in our code, OpenMD.\cite{Meineke2005,openmd}  At each
molecular dynamics time step, the following process is carried out:
\begin{enumerate}
\item The standard inter-atomic forces ($\nabla_iU$) are computed.
\item Delaunay triangulation is carried out using the current atomic
  configuration.
\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
  computed.
\item[b.] The resistance tensor ($\Xi_f(t)$) is computed using the
  viscosity ($\eta$) of the bath.
\item[c.] Facet drag ($-\Xi_f(t) \mathbf{v}_f(t)$) forces are
  computed.
\item[d.] Random forces ($\mathbf{R}_f(t)$) are computed using the
  resistance tensor and the temperature ($T$) of the bath.
\end{itemize}
\item The facet forces are divided equally among the vertex atoms.
\item Atomic positions and velocities are propagated.
\end{enumerate}
The Delaunay triangulation and computation of the convex hull are done
using calls to the qhull library.\cite{Qhull} There is a minimal
penalty for computing the convex hull and resistance tensors at each
step in the molecular dynamics simulation (roughly 0.02 $\times$ cost
of a single force evaluation), and the convex hull is remarkably easy
to parallelize on distributed memory machines (see Appendix A).

\section{Tests \& Applications}
\label{sec:tests}

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,
\begin{equation}
\kappa_{T} = -\frac{1}{V} \left ( \frac{\partial V}{\partial P} \right
)_{T}.
\label{eq:BM}
\end{equation}

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
bulk material, we make an assumption that the number density, $\rho =
\frac{N}{V}$, 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}
\label{eq:BMN}
\end{equation}
The region we used is a spherical volume of 10 \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
cluster can be used to compute the compressibility.

One might assume that the volume of the convex hull could simply be
taken as the system volume $V$ in the compressibility expression
(Eq. \ref{eq:BM}), but this has implications at lower pressures (which
are explored in detail in the section on water droplets).

The metallic force field in use for the gold nanoparticles is the
quantum Sutton-Chen (QSC) model.\cite{PhysRevB.59.3527} In all
simulations involving point charges, we utilized damped shifted-force
(DSF) electrostatics\cite{Fennell06} which is a variant of the Wolf
summation\cite{wolf:8254} that has been shown to provide good forces
and torques on molecular models for water in a computationally
efficient manner.\cite{Fennell06} The damping parameter ($\alpha$) was
set to 0.18 \AA$^{-1}$, and the cutoff radius was set to 12 \AA.  The
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}

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.

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}
\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}

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 have yielded values of
175.53 GPa.\cite{XXX} Using the same force field, we have performed a
series of relatively short (200 ps) simulations on 40 \r{A} radius
nanoparticles under the Langevin Hull at a variety of applied
pressures ranging from 0 GPa to XXX.  We obtain a value of 177.547 GPa
for the bulk modulus for gold using this echnique.

\begin{figure}
\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{pressureResponse}
\end{figure}

\begin{equation}
\kappa_T=-\frac{1}{V_{\mathrm{eq}}}\left(\frac{\partial V}{\partial
    P}\right)
\end{equation}

\subsection{Compressibility of SPC/E water clusters}

Prior molecular dynamics simulations on SPC/E water (both in
NVT~\cite{Glattli2002} and NPT~\cite{Motakabbir1990, Pi2009}
ensembles) have yielded values for the isothermal compressibility that
agree well with experiment.\cite{Fine1973} The results of two
different approaches for computing the isothermal compressibility from
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}
\caption{Compressibility of SPC/E water}
\label{fig:compWater}
\end{figure}

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
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
applied pressures, the liquid is in equilibrium with a vapor phase,
and it is entirely possible for one (or a few) molecules to drift away
from the liquid cluster (see Fig. \ref{fig:coneOfShame}).  At low
pressures, the restoring forces on the facets are very gentle, and
this means that the hulls often take on relatively distorted
geometries which include large volumes of empty space.

\begin{figure}
\includegraphics[width=\linewidth]{flytest2}
\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,
  compressibilities are computed using local number densities rather
  than hull volumes.}
\label{fig:coneOfShame}
\end{figure}

At higher pressures, the equilibrium strongly favors the liquid phase,
and the hull geometries are much more compact.  Because of the
liquid-vapor effect on the convex hull, the regional number density
approach (Eq. \ref{eq:BMN}) provides more reliable estimates of the
compressibility.

In both the traditional compressibility formula (Eq. \ref{eq:BM}) and
the number density version (Eq. \ref{eq:BMN}), multiple simulations at
different pressures must be done to compute the first derivatives.  It
is also possible to compute the compressibility using the fluctuation
dissipation theorem using either fluctuations in the
volume,\cite{Debenedetti1986},
\begin{equation}
\kappa_{T} = \frac{\left \langle V^{2} \right \rangle - \left \langle
    V \right \rangle ^{2}}{V \, k_{B} \, T},
\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},
\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.

\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
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
molecules.\cite{Warshel1978,King1989}

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
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.

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
\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} 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
molecules included in the convex hull (circles).
\begin{figure}
\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
  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. }
\label{fig:pAngle}
\end{figure}

As expected, interior molecules (those not included in the convex
hull) maintain a bulk-like structure with a uniform distribution of
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.

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 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.

\subsection{Heterogeneous nanoparticle / water mixtures}

\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
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
($\mathcal{O}(N \log N)$), but would enable the use of hull volumes
directly in computing the compressibility of the sample.

\section*{Appendix A: Computing Convex Hulls on Parallel Computers} 

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
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.

The basic idea involves splitting the point cloud into
spatially-overlapping subsets and computing the convex hulls for each
of the subsets.  The points on the convex hull of the entire system
are all present on at least one of the subset hulls. The algorithm
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
  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).
\item Each processor computes the global convex hull (right panel in
  Fig. \ref{fig:parallel}) using only those points that are the union
  of sites gathered from all of the subset hulls.  Delaunay
  triangulation is then done to obtain the facets of the global hull.
\end{enumerate}

\begin{figure}
\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 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{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
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
hull scales as $\mathcal{O}(\frac{n}{3r}\log\frac{n}{3r})$.

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
linear speed-up with increasing processor counts.

\section*{Acknowledgments}
Support for this project was provided by the
National Science Foundation under grant CHE-0848243. Computational
time was provided by the Center for Research Computing (CRC) at the
University of Notre Dame.  

\newpage

\bibliography{langevinHull}

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