trunk/nonperiodicVSS/nonperiodicVSS.tex

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\title{A Method for Creating Thermal and Angular Momentum Fluxes in Non-Periodic Systems}

\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 have adapted the Velocity Shearing and Scaling Reverse Non-Equilibium Molecular Dynamics (VSS-RNEMD) method
for use with non-periodic system geometries. This new method is capable of creating stable temperature and
angular velocity gradients in heterogeneous non-periodic systems while conserving total energy and angular
momentum. To demonstrate the method, we have computed the thermal conductivities of a gold nanoparticle and
water cluster, the shear viscosity of a water cluster, the interfacial thermal conductivity of a solvated gold
nanoparticle and the interfacial friction 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:1975tg,Evans:1982zk,ERPENBECK:1984sp,MAGINN:1993hc,Berthier:2002ij,Evans:2002ai,Schelling:2
002dp,PhysRevA.34.1449,JiangHao_jp802942v} and use linear response theory to connect the resulting thermal or
momentum flux to transport coefficients of bulk materials. However, for heterogeneous systems, such as phase
boundaries or interfaces, it is often unclear what shape of gradient should be imposed at the boundary between
materials.

Reverse Non-Equilibrium Molecular Dynamics (RNEMD) methods impose an unphysical {\it flux} between different
regions or ``slabs'' of the simulation box.\cite{MullerPlathe:1997xw,ISI:000080382700030,Kuang2010} 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 are then related (via linear response theory) 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{MullerPlathe:1997xw,ISI:000080382700030,Maginn:2010} as well as heterogeneous
interfaces.\cite{garde:nl2005,garde:PhysRevLett2009,kuang:AuThl}


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

The VSS-RNEMD approach uses a series of simultaneous velocity shearing and scaling exchanges between the two
slabs.\cite{Kuang2012} This method imposes energy and 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.

We present here a new set of constraints that are more general than the VSS constraints. 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 each $\Delta t$ time interval, 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 $\mathbf{c}_b$).

\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 quite simple 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{openmd} We have used 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 parameters}

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.

We have chosen the SPC/E water model for these simulations, which is particularly useful for method validation
as there are many values for physical properties from previous simulations available for direct
comparison.\cite{Bedrov:2000, Kuang2010}

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:AuThl,Kuang2012} 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
conductivity.

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 fully 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 ellipsoid were first created from a bulk lattice and
thermally equilibrated before being solvated in hexane. Packmol\cite{packmol} was used to solvate previously
equilibrated gold nanostructures within a droplet of hexane.

Once fully equilibrated, a thermal or angular momentum flux was applied for 1 - 2
ns, until a stable temperature or angular velocity gradient had developed. Systems containing liquids were run
under moderate pressure (5 atm) and temperatures (230 K) to avoid the formation of a substantial vapor phase. Pressure was applied to the system via the non-periodic convex Langevin Hull. Thermal coupling to the external temperature and pressure bath was removed to avoid interference with any
imposed flux.

To stabilize the gold nanoparticle under the imposed angular momentum flux we altered the gold atom at the
designated coordinate origin to have $10,000$ times its original mass. The nonbonded interactions remain
unchanged. The heavy atom is excluded from the angular momentum exchange. For rotation of the ellipsoid about
its long axis we have added two heavy atoms along the axis of rotation, one at each end of the rod. We collected angular velocity data for the heterogeneous systems after a brief VSS-RNEMD simulation 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}

Fourier's Law of heat conduction in radial coordinates yields an expression for the heat flow between the
concentric spherical RNEMD 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 significant 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. 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}

The effective rotational friction coefficient 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.

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{Kuang2012} 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.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% **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
{\footnotesize W / m $\cdot$ K}\cite{Kuang2010}, though still significantly lower than the experimental value
of 320 {\footnotesize 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, $0.8841$ {\footnotesize 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 explicit TraPPE-UA hexane. The nanoparticle $G$ values are compared to previous 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.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 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 conditions. 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 may at first seem 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.

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% **DISCUSSION**
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\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.

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% **ACKNOWLEDGMENTS**
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\section*{Acknowledgments} 

We gratefully acknowledge conversations with Dr. Shenyu Kuang. 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

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