trunk/nonperiodicVSS/nonperiodicVSS.tex

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\title{A method for creating thermal and angular momentum fluxes in
  non-periodic simulations}

\author{Kelsey M. Stocker}
\author{J. Daniel Gezelter}
\email{gezelter@nd.edu}
\affiliation[University of Notre Dame]{251 Nieuwland Science Hall\\ Department of Chemistry and Biochemistry\\ University of Notre Dame\\ Notre Dame, Indiana 46556}

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% \author{Kelsey M. Stocker and J. Daniel
%   Gezelter\footnote{Corresponding author. \ Electronic mail:
%     gezelter@nd.edu} \\
%   251 Nieuwland Science Hall, \\
%       Department of Chemistry and Biochemistry,\\
%       University of Notre Dame\\
%       Notre Dame, Indiana 46556}

%\date{\today}

%\maketitle 

%\begin{doublespace}

\begin{abstract}

  We present a new reverse non-equilibrium molecular dynamics (RNEMD)
  method that can be used with non-periodic simulation cells. This
  method applies thermal and/or angular momentum fluxes between two
  arbitrary regions of the simulation, and is capable of creating
  stable temperature and angular velocity gradients while conserving
  total energy and angular momentum.  One particularly useful
  application is the exchange of kinetic energy between two concentric
  spherical regions, which can be used to generate thermal transport
  between nanoparticles and the solvent that surrounds them.  The
  rotational couple to the solvent (a measure of interfacial friction)
  is also available via this method.  As demonstrations and tests of
  the new method, we have computed the thermal conductivities of gold
  nanoparticles and water clusters, the shear viscosity of a water
  cluster, the interfacial thermal conductivity ($G$) of a solvated
  gold nanoparticle and the interfacial friction of a variety of
  solvated gold nanostructures.

\end{abstract}

\newpage

%\narrowtext

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% **INTRODUCTION**
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Introduction}

Non-equilibrium molecular dynamics (NEMD) methods impose a temperature
or velocity {\it gradient} on a
system,\cite{Ashurst:1975eu,Evans:1982oq,Erpenbeck:1984qe,Evans:1986nx,Vogelsang:1988qv,Maginn:1993kl,Hess:2002nr,Schelling:2002dp,Berthier:2002ai,Evans:2002tg,Vasquez:2004ty,Backer:2005sf,Jiang:2008hc,Picalek:2009rz}
and use linear response theory to connect the resulting thermal or
momentum {\it flux} to transport coefficients of bulk materials,
\begin{equation}
j_z(p_x) = -\eta \frac{\partial v_x}{\partial z},  \hspace{0.5in}
J_z = \lambda \frac{\partial T}{\partial z}.
\end{equation}
Here, $\frac{\partial T}{\partial z}$ and $\frac{\partial
  v_x}{\partial z}$ are the imposed thermal and momentum gradients,
and as long as the imposed gradients are relatively small, the
corresponding fluxes, $J_z$ and $j_z(p_x)$, have a linear relationship
to the gradients.  The coefficients that provide this relationship
correspond to physical properties of the bulk material, either the
shear viscosity $(\eta)$ or thermal conductivity $(\lambda)$.  For
systems which include phase boundaries or interfaces, it is often
unclear what gradient (or discontinuity) should be imposed at the
boundary between materials.

In contrast, reverse Non-Equilibrium Molecular Dynamics (RNEMD)
methods impose an unphysical {\it flux} between different regions or
``slabs'' of the simulation
box.\cite{Muller-Plathe:1997wq,Muller-Plathe:1999ao,Patel:2005zm,Shenogina:2009ix,Tenney:2010rp,Kuang:2010if,Kuang:2011ef,Kuang:2012fe,Stocker:2013cl}
The system responds by developing a temperature or velocity {\it
  gradient} between the two regions. The gradients which develop in
response to the applied flux have the same linear response
relationships to the transport coefficient of interest. Since the
amount of the applied flux is known exactly, and measurement of a
gradient is generally less complicated, imposed-flux methods typically
take shorter simulation times to obtain converged results. At
interfaces, the observed gradients often exhibit near-discontinuities
at the boundaries between dissimilar materials. RNEMD methods do not
need many trajectories to provide information about transport
properties, and they have become widely used to compute thermal and
mechanical transport in both homogeneous liquids and
solids~\cite{Muller-Plathe:1997wq,Muller-Plathe:1999ao,Tenney:2010rp}
as well as heterogeneous
interfaces.\cite{Patel:2005zm,Shenogina:2009ix,Kuang:2010if,Kuang:2011ef,Kuang:2012fe,Stocker:2013cl}

The strengths of specific algorithms for imposing the flux between two
different slabs of the simulation cell has been the subject of some
renewed interest.  The original RNEMD approach used kinetic energy or
momentum exchange between particles in the two slabs, either through
direct swapping of momentum vectors or via virtual elastic collisions
between atoms in the two regions.  There have been recent
methodological advances which involve scaling all particle velocities
in both slabs.\cite{Kuang:2010if,Kuang:2012fe} Constraint equations
are simultaneously imposed to require the simulation to conserve both
total energy and total linear momentum.  The most recent and simplest
of the velocity scaling approaches allows for simultaneous shearing
(to provide viscosity estimates) as well as scaling (to provide
information about thermal conductivity).\cite{Kuang:2012fe}

To date, however, the RNEMD methods have only been used in periodic
simulation cells where the exchange regions are physically separated
along one of the axes of the simulation cell. This limits the
applicability to infinite planar interfaces which are perpendicular to
the applied flux.  In order to model steady-state non-equilibrium
distributions for curved surfaces (e.g. hot nanoparticles in contact
with colder solvent), or for regions that are not planar slabs, the
method requires some generalization for non-parallel exchange regions.
In the following sections, we present a new velocity shearing and
scaling (VSS) RNEMD algorithm which has been explicitly designed for
non-periodic simulations, and use the method to compute some thermal
transport and solid-liquid friction at the surfaces of spherical and
ellipsoidal nanoparticles.  

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% **METHODOLOGY**
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Velocity shearing and scaling (VSS) for non-periodic systems}

The original periodic VSS-RNEMD approach uses a series of simultaneous
velocity shearing and scaling exchanges between the two
slabs.\cite{Kuang:2012fe} This method imposes energy and linear
momentum conservation constraints while simultaneously creating a
desired flux between the two slabs. These constraints ensure that all
configurations are sampled from the same microcanonical (NVE)
ensemble.

\begin{figure}
\includegraphics[width=\linewidth]{figures/npVSS}
\caption{Schematics of periodic (left) and non-periodic (right)
  Velocity Shearing and Scaling RNEMD. A kinetic energy or momentum
  flux is applied from region B to region A. Thermal gradients are
  depicted by a color gradient. Linear or angular velocity gradients
  are shown as arrows.}
\label{fig:VSS}
\end{figure}

We have extended the VSS method for use in {\it non-periodic}
simulations, in which the ``slabs'' have been generalized to two
separated regions of space. These regions could be defined as
concentric spheres (as in figure \ref{fig:VSS}), or one of the regions
can be defined in terms of a dynamically changing ``hull'' comprising
the surface atoms of the cluster. This latter definition is identical
to the hull used in the Langevin Hull algorithm.\cite{Vardeman2011}
For the non-periodic variant, the constraints fix both the total
energy and total {\it angular} momentum of the system while
simultaneously imposing a thermal and angular momentum flux between
the two regions.

After a time interval of $\Delta t$, the particle velocities
($\mathbf{v}_i$ and $\mathbf{v}_j$) in the two shells ($A$ and $B$)
are modified by a velocity scaling coefficient ($a$ and $b$) and by a
rotational shearing term ($\mathbf{c}_a$ and $\mathbf{c}_b$).  The
scalars $a$ and $b$ collectively provide a thermal exchange between
the two regions.  One of the values is larger than 1, and the other
smaller. To conserve total energy and angular momentum, the values of
these two scalars are coupled.  The vectors ($\mathbf{c}_a$ and
$\mathbf{c}_b$) provide a relative rotational shear to the velocities
of the particles within the two regions, and these vectors must also
be coupled to constrain the total angular momentum.

Once the values of the scaling and shearing factors are known, the
velocity changes are applied,
\begin{displaymath}
\begin{array}{rclcl}
 & \underline{~~~~~~~~\mathrm{scaling}~~~~~~~~} & &
 \underline{\mathrm{rotational~shearing}} \\  \\
\mathbf{v}_i $~~~$\leftarrow & 
  a \left(\mathbf{v}_i - \langle \omega_a
  \rangle \times \mathbf{r}_i\right) & + & \mathbf{c}_a \times \mathbf{r}_i \\
\mathbf{v}_j $~~~$\leftarrow & 
  b  \left(\mathbf{v}_j - \langle \omega_b
  \rangle \times \mathbf{r}_j\right) & + & \mathbf{c}_b \times \mathbf{r}_j
\end{array}
\end{displaymath}
Here $\langle\mathbf{\omega}_a\rangle$ and $\langle\mathbf{\omega}_b\rangle$ are the instantaneous angular
velocities of each shell, and $\mathbf{r}_i$ is the position of particle $i$ relative to a fixed point in space
(usually the center of mass of the cluster). Particles in the shells also receive an additive ``angular shear''
to their velocities. The amount of shear is governed by the imposed angular momentum flux,
$\mathbf{j}_r(\mathbf{L})$,
\begin{eqnarray}
\mathbf{c}_a & = & - \mathbf{j}_r(\mathbf{L}) \cdot
\overleftrightarrow{I_a}^{-1} \Delta t   + \langle \omega_a \rangle \label{eq:bc}\\
\mathbf{c}_b & = & + \mathbf{j}_r(\mathbf{L}) \cdot
\overleftrightarrow{I_b}^{-1}  \Delta t  + \langle \omega_b \rangle \label{eq:bh}
\end{eqnarray}
where $\overleftrightarrow{I}_{\{a,b\}}$ is the moment of inertia
tensor for each of the two shells. 

To simultaneously impose a thermal flux ($J_r$) between the shells we
use energy conservation constraints, 
\begin{eqnarray}
K_a - J_r \Delta t & = & a^2 \left(K_a - \frac{1}{2}\langle
\omega_a \rangle \cdot \overleftrightarrow{I_a} \cdot \langle \omega_a
\rangle \right) + \frac{1}{2} \mathbf{c}_a \cdot \overleftrightarrow{I_a}
\cdot \mathbf{c}_a \label{eq:Kc}\\
K_b + J_r \Delta t & = & b^2 \left(K_b - \frac{1}{2}\langle
\omega_b \rangle \cdot \overleftrightarrow{I_b} \cdot \langle \omega_b
\rangle \right) + \frac{1}{2} \mathbf{c}_b \cdot \overleftrightarrow{I_b} \cdot \mathbf{c}_b \label{eq:Kh}
\end{eqnarray}
Simultaneous solution of these quadratic formulae for the scaling coefficients, $a$ and $b$, will ensure that
the simulation samples from the original microcanonical (NVE) ensemble. Here $K_{\{a,b\}}$ is the instantaneous
translational kinetic energy of each shell. At each time interval, we solve for $a$, $b$, $\mathbf{c}_a$, and
$\mathbf{c}_b$, subject to the imposed angular momentum flux, $j_r(\mathbf{L})$, and thermal flux, $J_r$,
values. The new particle velocities are computed, and the simulation continues. System configurations after the
transformations have exactly the same energy ({\it and} angular momentum) as before the moves.

As the simulation progresses, the velocity transformations can be
performed on a regular basis, and the system will develop a
temperature and/or angular velocity gradient in response to the
applied flux. Using the slope of the radial temperature or velocity
gradients, it is straightforward to obtain both the thermal
conductivity ($\lambda$), interfacial thermal conductance ($G$), or
rotational friction coefficients ($\Xi^{rr}$) of any non-periodic
system.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% **COMPUTATIONAL DETAILS**
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Computational Details}

The new VSS-RNEMD methodology for non-periodic system geometries has
been implemented in our group molecular dynamics code,
OpenMD.\cite{Meineke:2005gd,openmd} We have tested the new method to
calculate the thermal conductance of a gold nanoparticle and SPC/E
water cluster, and compared the results with previous bulk RNEMD
values, as well as experiment. We have also investigated the
interfacial thermal conductance and interfacial rotational friction
for gold nanostructures solvated in hexane as a function of
nanoparticle size and shape.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% FORCE FIELD PARAMETERS
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Force field}

Gold -- gold interactions are described by the quantum Sutton-Chen
(QSC) model.\cite{PhysRevB.59.3527} The QSC parameters are tuned to
experimental properties such as density, cohesive energy, and elastic
moduli and include zero-point quantum corrections.

The SPC/E water model~\cite{Berendsen87} is particularly useful for
validation of conductivities and shear viscosities.  This model has
been used to previously test other RNEMD and NEMD approaches, and
there are reported values for thermal conductivies and shear
viscosities at a wide range of thermodynamic conditions that are
available for direct comparison.\cite{Bedrov:2000,Kuang:2010if}

Hexane molecules are described by the TraPPE united atom
model,\cite{TraPPE-UA.alkanes} which provides good computational
efficiency and reasonable accuracy for bulk thermal conductivity
values. In this model, sites are located at the carbon centers for
alkyl groups. Bonding interactions, including bond stretches and bends
and torsions, were used for intra-molecular sites closer than 3
bonds. For non-bonded interactions, Lennard-Jones potentials were
used. We have previously utilized both united atom (UA) and all-atom
(AA) force fields for thermal
conductivity,\cite{Kuang:2011ef,Kuang:2012fe,Stocker:2013cl} and since
the united atom force fields cannot populate the high-frequency modes
that are present in AA force fields, they appear to work better for
modeling thermal conductance at metal/ligand interfaces.

Gold -- hexane nonbonded interactions are governed by pairwise
Lennard-Jones parameters derived from Vlugt \emph{et
  al}.\cite{vlugt:cpc2007154} They fitted parameters for the
interaction between Au and CH$_{\emph x}$ pseudo-atoms based on the
effective potential of Hautman and Klein for the Au(111)
surface.\cite{hautman:4994}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% NON-PERIODIC DYNAMICS
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \subsection{Dynamics for non-periodic systems}
% 
% We have run all tests using the Langevin Hull non-periodic simulation methodology.\cite{Vardeman2011} The
% Langevin Hull is especially useful for simulating heterogeneous mixtures of materials with different
% compressibilities, which are typically problematic for traditional affine transform methods. We have had
% success applying this method to several different systems including bare metal nanoparticles, liquid water
% clusters, and explicitly solvated nanoparticles. Calculated physical properties such as the isothermal
% compressibility of water and the bulk modulus of gold nanoparticles are in good agreement with previous
% theoretical and experimental results. The Langevin Hull uses a Delaunay tesselation to create a dynamic convex
% hull composed of triangular facets with vertices at atomic sites. Atomic sites included in the hull are coupled
% to an external bath defined by a temperature, pressure and viscosity. Atoms not included in the hull are
% subject to standard Newtonian dynamics.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% SIMULATION PROTOCOL
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
\subsection{Simulation protocol}

In all cases, systems were equilibrated under non-periodic
isobaric-isothermal (NPT) conditions -- using the Langevin Hull
methodology\cite{Vardeman2011} -- before any non-equilibrium methods
were introduced. For heterogeneous systems, the gold nanoparticles and
ellipsoids were created from a bulk fcc lattice and were thermally
equilibrated before being solvated in hexane.  Packmol\cite{packmol}
was used to solvate previously equilibrated gold nanostructures within
a spherical droplet of hexane.

Once equilibrated, thermal or angular momentum fluxes were applied for
1 - 2 ns, until stable temperature or angular velocity gradients had
developed. Systems containing liquids were run under moderate pressure
(5 atm) and temperatures (230 K) to avoid the formation of a vapor
layer at the boundary of the cluster.  Pressure was applied to the
system via the non-periodic Langevin Hull.\cite{Vardeman2011} However,
thermal coupling to the external temperature and pressure bath was
removed to avoid interference with the imposed RNEMD flux.

Because the method conserves \emph{total} angular momentum, systems
which contain a metal nanoparticle embedded in a significant volume of
solvent will still experience nanoparticle diffusion inside the
solvent droplet.  To aid in computing the rotational friction in these
systems, a single gold atom at the origin of the coordinate system was
assigned a mass $10,000 \times$ its original mass. The bonded and
nonbonded interactions for this atom remain unchanged and the heavy
atom is excluded from the RNEMD exchanges.  The only effect of this
gold atom is to effectively pin the nanoparticle at the origin of the
coordinate system, while still allowing for rotation. For rotation of
the gold ellipsoids we added two of these heavy atoms along the axis
of rotation, separated by an equal distance from the origin of the
coordinate system.  These heavy atoms prevent off-axis tumbling of the
nanoparticle and allow for measurement of rotational friction relative
to a particular axis of the ellipsoid.

Angular velocity data was collected for the heterogeneous systems
after a brief period of imposed flux to initialize rotation of the
solvated nanostructure. Doing so ensures that we overcome the initial
static friction and calculate only the \emph{dynamic} interfacial
rotational friction.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% THERMAL CONDUCTIVITIES
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Thermal conductivities}

To compute the thermal conductivities of bulk materials, Fourier's Law
of heat conduction in radial coordinates yields an expression for the
heat flow between the concentric spherical shells:
\begin{equation}
        q_r = - \frac{4 \pi \lambda (T_b - T_a)}{\frac{1}{r_a} - \frac{1}{r_b}}
\label{eq:Q}
\end{equation}
where $\lambda$ is the thermal conductivity, and $T_{a,b}$ and
$r_{a,b}$ are the temperatures and radii of the two RNEMD regions,
respectively.

A thermal flux is created using VSS-RNEMD moves, and the temperature
in each of the radial shells is recorded.  The resulting temperature
profiles are analyzed to yield information about the interfacial
thermal conductance.  As the simulation progresses, the VSS moves are
performed on a regular basis, and the system develops a thermal or
velocity gradient in response to the applied flux. Once a stable
thermal gradient has been established between the two regions, the
thermal conductivity, $\lambda$, can be calculated using a linear
regression of the thermal gradient, $\langle \frac{dT}{dr} \rangle$:

\begin{equation}
        \lambda = \frac{r_a}{r_b} \frac{q_r}{4 \pi \langle \frac{dT}{dr} \rangle}
\label{eq:lambda}
\end{equation}

The rate of heat transfer between the two RNEMD regions is the amount of transferred kinetic energy over the
length of the simulation, t

\begin{equation}
        q_r = \frac{KE}{t}
\label{eq:heat}
\end{equation}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% INTERFACIAL THERMAL CONDUCTANCE
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Interfacial thermal conductance}

\begin{figure}
\includegraphics[width=\linewidth]{figures/NP20}
\caption{A gold nanoparticle with a radius of 20 \AA$\,$ solvated in TraPPE-UA hexane. A thermal flux is applied        between the nanoparticle and an outer shell of solvent.}
\label{fig:NP20}
\end{figure}

For heterogeneous systems such as a solvated nanoparticle, shown in Figure \ref{fig:NP20}, the interfacial
thermal conductance at the surface of the nanoparticle can be determined using an applied kinetic energy flux.
We can treat the temperature in each radial shell as discrete, making the thermal conductance, $G$, of each
shell the inverse of its Kapitza resistance, $R_K$. To model the thermal conductance across an interface (or
multiple interfaces) it is useful to consider the shells as resistors wired in series. The resistance of the
shells is then additive, and the interfacial thermal conductance is the inverse of the total Kapitza
resistance. The thermal resistance of each shell is

\begin{equation}
        R_K = \frac{1}{q_r} \Delta T 4 \pi r^2
\label{eq:RK}
\end{equation}

making the total resistance of two neighboring shells

\begin{equation}
        R_{total} = \frac{1}{q_r} \left [ (T_2 - T_1) 4 \pi r^2_1 + (T_3 - T_2) 4 \pi r^2_2 \right ] = \frac{1}{G}
\label{eq:Rtotal}
\end{equation}

This series can be expanded for any number of adjacent shells, allowing for the calculation of the interfacial
thermal conductance for interfaces of considerable thickness, such as self-assembled ligand monolayers on a
metal surface.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% INTERFACIAL ROTATIONAL FRICTION
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Interfacial rotational friction}

The interfacial rotational friction, $\Xi^{rr}$, can be calculated for heterogeneous nanostructure/solvent
systems by applying an angular momentum flux between the solvated nanostructure and a spherical shell of
solvent at the boundary of the cluster. An angular velocity gradient develops in response to the applied flux,
causing the nanostructure and solvent shell to rotate in opposite directions about a given axis.

\begin{figure}
\includegraphics[width=\linewidth]{figures/E25-75}
\caption{A gold prolate ellipsoid of length 65 \AA$\,$ and width 25 \AA$\,$ solvated by TraPPE-UA hexane. An angular momentum flux is applied between the ellipsoid and an outer shell of solvent.}
\label{fig:E25-75}
\end{figure}

Analytical solutions for the rotational friction coefficients for a solvated spherical body of radius $r$ under ideal ``stick'' boundary conditions can be estimated using Stokes' law

\begin{equation}
        \Xi^{rr}_{stick} = 8 \pi \eta r^3
\label{eq:Xisphere}.
\end{equation}

where $\eta$ is the dynamic viscosity of the surrounding solvent. An $\eta$ value for TraPPE-UA hexane under
these particular temperature and pressure conditions was determined by applying a traditional VSS-RNEMD linear
momentum flux to a periodic box of solvent.

For general ellipsoids with semiaxes $a$, $b$, and $c$, Perrin's extension of Stokes' law provides exact
solutions for symmetric prolate $(a \geq b = c)$ and oblate $(a < b = c)$ ellipsoids under ideal ``stick'' conditions. For simplicity, we define
a Perrin Factor, $S$,

\begin{equation}
        S = \frac{2}{\sqrt{a^2 - b^2}} ln \left[ \frac{a + \sqrt{a^2 - b^2}}{b} \right].
\label{eq:S}
\end{equation}

For a prolate ellipsoidal rod, demonstrated here, the rotational resistance tensor $\Xi$ is a $3 \times 3$ diagonal matrix with elements
\begin{equation}
        \Xi^{rr}_a = \frac{32 \pi}{2} \eta \frac{ \left( a^2 - b^2 \right) b^2}{2a - b^2 S}
\label{eq:Xia}
\end{equation}\vspace{-0.45in}\\
\begin{equation}
        \Xi^{rr}_{b,c} = \frac{32 \pi}{2} \eta \frac{ \left( a^4 - b^4 \right)}{ \left( 2a^2 - b^2 \right)S - 2a}.
\label{eq:Xibc}
\end{equation}

corresponding to rotation about the long axis ($a$), and each of the equivalent short axes ($b$ and $c$), respectively.

Previous VSS-RNEMD simulations of the interfacial friction of the planar Au(111) / hexane interface have shown
that the interface exists within ``slip'' boundary conditions.\cite{Kuang:2012fe} Hu and Zwanzig\cite{Zwanzig}
investigated the rotational friction coefficients for spheroids under slip boundary conditions and obtained
numerial results for a scaling factor to be applied to $\Xi^{rr}_{\mathit{stick}}$ as a function of $\tau$, the
ratio of the shorter semiaxes and the longer semiaxis of the spheroid. For the sphere and prolate ellipsoid
shown here, the values of $\tau$ are $1$ and $0.3939$, respectively. Under ``slip'' conditions,
$\Xi^{rr}_{\mathit{slip}}$ for any sphere and rotation of the prolate ellipsoid about its long axis approaches
$0$, as no solvent is displaced by either of these rotations. $\Xi^{rr}_{\mathit{slip}}$ for rotation of the
prolate ellipsoid about its short axis is $35.9\%$ of the analytical $\Xi^{rr}_{\mathit{stick}}$ result,
accounting for the reduced interfacial friction under ``slip'' boundary conditions.

The effective rotational friction coefficient, $\Xi^{rr}_{\mathit{eff}}$ at the interface can be extracted from non-periodic VSS-RNEMD simulations quite easily using the applied torque ($\tau$) and the observed angular velocity of the gold structure ($\omega_{Au}$)

\begin{equation}
        \Xi^{rr}_{\mathit{eff}} = \frac{\tau}{\omega_{Au}}
\label{eq:Xieff}
\end{equation}

The applied torque required to overcome the interfacial friction and maintain constant rotation of the gold is 

\begin{equation}
        \tau = \frac{L}{2 t}
\label{eq:tau}  
\end{equation}

where $L$ is the total angular momentum exchanged between the two RNEMD regions and $t$ is the length of the simulation.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% **TESTS AND APPLICATIONS**
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Tests and Applications} 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% THERMAL CONDUCTIVITIES
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Thermal conductivities}

Calculated values for the thermal conductivity of a 40 \AA$~$ radius gold nanoparticle (15707 atoms) at
different kinetic energy flux values are shown in Table \ref{table:goldTC}. For these calculations, the hot and
cold slabs were excluded from the linear regression of the thermal gradient.

\begin{longtable}{ccc}
\caption{Calculated thermal conductivity of a crystalline gold nanoparticle of radius 40 \AA. Calculations were performed at 300 K and ambient density. Gold-gold interactions are described by the Quantum Sutton-Chen potential.}
\\ \hline \hline
{$J_r$} & {$\langle \frac{dT}{dr} \rangle$} & {$\boldsymbol \lambda$}\\ 
{\footnotesize(kcal / fs $\cdot$ \AA$^{2}$)} & {\footnotesize(K / \AA)} & {\footnotesize(W / m $\cdot$ K)}\\ \hline
3.25$\times 10^{-6}$ & 0.11435 & 1.0143 \\
6.50$\times 10^{-6}$ & 0.2324 & 0.9982 \\
1.30$\times 10^{-5}$ & 0.44922 & 1.0328 \\
3.25$\times 10^{-5}$ & 1.1802 & 0.9828 \\
6.50$\times 10^{-5}$ & 2.339 & 0.9918 \\
\hline
This work & & 1.0040
\\ \hline \hline
\label{table:goldTC}
\end{longtable}

The measured linear slope $\langle \frac{dT}{dr} \rangle$ is linearly dependent on the applied kinetic energy
flux $J_r$. Calculated thermal conductivity values compare well with previous bulk QSC values of 1.08 -- 1.26 W / m $\cdot$ K\cite{Kuang:2010if}, though still significantly lower than the experimental value
of 320 W / m $\cdot$ K, as the QSC force field neglects significant electronic contributions to
heat conduction.

Calculated values for the thermal conductivity of a cluster of 6912 SPC/E water molecules are shown in Table
\ref{table:waterTC}. As with the gold nanoparticle thermal conductivity calculations, the RNEMD regions were
excluded from the $\langle \frac{dT}{dr} \rangle$ fit.

\begin{longtable}{ccc}
\caption{Calculated thermal conductivity of a cluster of 6912 SPC/E water molecules. Calculations were performed at 300 K and 5 atm.}
\\ \hline \hline
{$J_r$} & {$\langle \frac{dT}{dr} \rangle$} & {$\boldsymbol \lambda$}\\ 
{\footnotesize(kcal / fs $\cdot$ \AA$^{2}$)} & {\footnotesize(K / \AA)} & {\footnotesize(W / m $\cdot$ K)}\\ \hline
1$\times 10^{-5}$ & 0.38683 & 0.8698 \\
3$\times 10^{-5}$ & 1.1643 & 0.9098 \\
6$\times 10^{-5}$ & 2.2262 & 0.8727 \\
\hline
This work & & 0.8841 \\
Zhang, et al\cite{Zhang2005} & & 0.81 \\
R$\ddot{\mathrm{o}}$mer, et al\cite{Romer2012} & & 0.87 \\
Experiment\cite{WagnerKruse} & & 0.61
\\ \hline \hline
\label{table:waterTC}
\end{longtable}

Again, the measured slope is linearly dependent on the applied kinetic energy flux $J_r$. The average
calculated thermal conductivity from this work, $0.8841$ W / m $\cdot$ K, compares very well to
previous non-equilibrium molecular dynamics results\cite{Romer2012, Zhang2005} and experimental
values.\cite{WagnerKruse}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% INTERFACIAL THERMAL CONDUCTANCE
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Interfacial thermal conductance}

Calculated interfacial thermal conductance ($G$) values for three sizes of gold nanoparticles and Au(111)
surface solvated in TraPPE-UA hexane are shown in Table \ref{table:G}.

\begin{longtable}{ccc}
\caption{Calculated interfacial thermal conductance ($G$) values for gold nanoparticles of varying radii solvated in TraPPE-UA hexane. The nanoparticle $G$ values are compared to previous simulation results for a Au(111) interface in TraPPE-UA hexane.}
\\ \hline \hline
{Nanoparticle Radius} & {$G$}\\ 
{\footnotesize(\AA)} & {\footnotesize(MW / m$^{2}$ $\cdot$ K)}\\ \hline
20 & {47.1} \\
30 & {45.4} \\
40 & {46.5} \\
\hline
Au(111) & {30.2} 
\\ \hline \hline
\label{table:G}
\end{longtable}

The introduction of surface curvature increases the interfacial thermal conductance by a factor of
approximately $1.5$ relative to the flat interface. There are no significant differences in the $G$ values for
the varying nanoparticle sizes. It seems likely that for the range of nanoparticle sizes represented here, any
particle size effects are not evident. The simulation of larger nanoparticles may demonstrate an approach to the $G$ value of a flat Au(111) slab but would require prohibitively costly numbers of atoms.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% INTERFACIAL FRICTION
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Interfacial friction}

Table \ref{table:couple} shows the calculated rotational friction coefficients $\Xi^{rr}$ for spherical gold
nanoparticles and a prolate ellipsoidal gold nanorod in TraPPE-UA hexane. An angular momentum flux was applied
between the A and B regions defined as the gold structure and hexane molecules beyond a certain radius,
respectively.

\begin{longtable}{lccccc}
\caption{Comparison of rotational friction coefficients under ideal ``slip'' ($\Xi^{rr}_{\mathit{slip}}$) and ``stick'' ($\Xi^{rr}_{\mathit{stick}}$) conditions and effective ($\Xi^{rr}_{\mathit{eff}}$) rotational friction coefficients of gold nanostructures solvated in TraPPE-UA hexane at 230 K. The ellipsoid is oriented with the long axis along the $z$ direction.}
\\ \hline \hline
{Structure} & {Axis of Rotation} & {$\Xi^{rr}_{\mathit{slip}}$} & {$\Xi^{rr}_{\mathit{eff}}$} & {$\Xi^{rr}_{\mathit{stick}}$} & {$\Xi^{rr}_{\mathit{eff}}$ / $\Xi^{rr}_{\mathit{stick}}$}\\ 
{} & {} & {\footnotesize(amu A$^2$ / fs)} & {\footnotesize(amu A$^2$ / fs)} & {\footnotesize(amu A$^2$ / fs)} & \\  \hline
Sphere (r = 20 \AA) & {$x = y = z$} & 0 & {2386} & {3314} & {0.720}\\
Sphere (r = 30 \AA) & {$x = y = z$} & 0 & {8415} & {11749} & {0.716}\\
Sphere (r = 40 \AA) & {$x = y = z$} & 0 & {47544} & {34464} & {1.380}\\ 
Prolate Ellipsoid & {$x = y$} & {1792} & {3128} & {4991} & {0.627}\\
Prolate Ellipsoid & {$z$} & 0 & {1590} & {1993} & {0.798}
\\ \hline \hline
\label{table:couple}
\end{longtable}

The results for $\Xi^{rr}_{\mathit{eff}}$ show that, contrary to the flat Au(111) / hexane interface, gold
structures solvated by hexane do not exist in the ``slip'' boundary condition. At this length scale, the
nanostructures are not perfect spheroids due to atomic `roughening' of the surface and therefore experience
increased interfacial friction which deviates from the ideal ``slip'' case. The 20 and 30 \AA$\,$ radius
nanoparticles experience approximately 70\% of the ideal ``stick'' boundary interfacial friction. Rotation of
the ellipsoid about its long axis more closely approaches the ``stick'' limit than rotation about the short
axis, which at first seems counterintuitive. However, the `propellor' motion caused by rotation about the
short axis may exclude solvent from the rotation cavity or move a sufficient amount of solvent along with the
gold that a smaller interfacial friction is actually experienced. The largest nanoparticle (40 \AA$\,$ radius)
appears to experience more than the ``stick'' limit of interfacial friction, which may be a consequence of
surface features or anomalous solvent behaviors that are not fully understood at this time.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% **DISCUSSION**
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Discussion} 

We have demonstrated a novel adaptation of the VSS-RNEMD methodology for the application of thermal and angular momentum fluxes in explicitly non-periodic system geometries. Non-periodic VSS-RNEMD preserves the virtues of traditional VSS-RNEMD, namely Boltzmann thermal velocity distributions and minimal thermal anisotropy, while extending the constraints to conserve total energy and total \emph{angular} momentum. We also still have the ability to impose the thermal and angular momentum fluxes simultaneously or individually.

Most strikingly, this method enables calculation of thermal conductivity in homogeneous non-periodic geometries, as well as interfacial thermal conductance and interfacial rotational friction in heterogeneous clusters. The ability to interrogate explicitly non-periodic effects -- such as surface curvature -- on interfacial transport properties is an exciting prospect that will be explored in the future.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% **ACKNOWLEDGMENTS**
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{acknowledgement}
  The authors thank Dr. Shenyu Kuang for helpful discussions. 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.
\end{acknowledgement}


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