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\title{The Thermal Conductance of Alkanethiolate-Protected Gold
  Nanospheres: Effects of Curvature and Chain Length}

\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}


\keywords{Nanoparticles, interfaces, thermal conductance}

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\begin{abstract}
        
\end{abstract}

\newpage

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

The thermal properties of various nanostructured interfaces have been
the subject of intense experimental
interest,\cite{Wilson:2002uq,PhysRevB.67.054302,doi:10.1021/jp048375k,PhysRevLett.96.186101,Wang10082007,doi:10.1021/jp8051888,PhysRevB.80.195406,doi:10.1021/la904855s}
and the interfacial thermal conductance is the principal quantity of
interest for understanding interfacial heat
transport.\cite{cahill:793} Because nanoparticles have a significant
fraction of their atoms at the particle / solvent interface, the
chemical details of these interfaces govern the thermal transport
properties.

Previously, reverse non-equilibrium molecular dynamics (RNEMD) methods
have been applied to calculate the interfacial thermal conductance at
flat (111) metal / organic solvent interfaces that had been chemically
protected by mixed-chain alkanethiolate groups.\cite{kuang:AuThl}
These simulations suggested an explanation for the increase in thermal
conductivity at alkanethiol-capped metal surfaces compared with bare
metal interfaces.  Specifically, the chemical bond between the metal
and the ligand introduces a vibrational overlap that is not present
without the protecting group, and the overlap between the vibrational
spectra (metal to ligand, ligand to solvent) provides a mechanism for
rapid thermal transport across the interface. The simulations also
suggested that this phenomenon is a non-monotonic function of the
fractional coverage of the surface, as moderate coverages allow
diffusive heat transport of solvent molecules that have been in close
contact with the ligands.

Simulations of {\it mixed-chain} alkylthiolate surfaces showed that
solvent trapped close to the interface can be very efficient at moving
thermal energy away from the surface.\cite{Stocker:2013cl} Trapped
solvent molecules that were orientationally ordered with nearby
ligands (but which were less able to diffuse into the bulk) were able
to double the thermal conductance of the interface.  This indicates
that the ligand-to-solvent vibrational energy transfer is the key
feature for increasing particle-to-solvent thermal conductance.

Recently, we extended RNEMD methods for use in non-periodic geometries
by creating scaling/shearing moves between concentric regions of the
simulation.\cite{Stocker:2014qq} In this work, we apply this
non-periodic variant of RNEMD to investigate the role that {\it
  curved} nanoparticle surfaces play in heat and mass transport.  On
planar surfaces, we discovered that orientational ordering of surface
protecting ligands had a large effect on the heat conduction from the
metal to the solvent.  Smaller nanoparticles have high surface
curvature that creates gaps in well-ordered self-assembled monolayers,
and the effects those gaps have on the thermal conductance are unknown.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%               INTERFACIAL THERMAL CONDUCTANCE OF METALLIC NANOPARTICLES
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\section{Interfacial Thermal Conductance of Metallic Nanoparticles}

For a solvated nanoparticle, it is possible to define a critical value
for the interfacial thermal conductance,
\begin{equation}
G_c = \frac{3 C_s \Lambda_s}{R C_p}
\end{equation}
which depends on the solvent heat capacity, $C_s$, solvent thermal
conductivity, $\Lambda_s$, particle radius, $R$, and nanoparticle heat
capacity, $C_p$.\cite{Wilson:2002uq} In the limit of infinite
interfacial thermal conductance, $G \gg G_c$, cooling of the
nanoparticle is limited by the solvent properties, $C_s$ and
$\Lambda_s$.  In the opposite limit, $G \ll G_c$, the heat dissipation
is controlled by the thermal conductance of the particle / fluid
interface. It is this regime with which we are concerned, where
properties of ligands and the particle surface may be tuned to
manipulate the rate of cooling for solvated nanoparticles.  Based on
estimates of $G$ from previous simulations as well as experimental
results for solvated nanostructures, gold nanoparticles solvated in
hexane are in the $G \ll G_c$ regime for radii smaller than 40 nm. The
particles included in this study are more than an order of magnitude
smaller than this critical radius, so the heat dissipation should be
controlled entirely by the surface features of the particle / ligand /
solvent interface.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%               STRUCTURE OF SELF-ASSEMBLED MONOLAYERS ON NANOPARTICLES
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Structures of Self-Assembled Monolayers on Nanoparticles}

Though the ligand packing on planar surfaces is characterized for many
different ligands and surface facets, it is not obvious \emph{a
  priori} how the same ligands will behave on the highly curved
surfaces of spherical nanoparticles. Thus, as more applications of
ligand-stabilized nanostructures have become apparent, the structure
and dynamics of ligands on metallic nanoparticles have been studied
extensively.\cite{Badia1996:2,Badia1996,Henz2007,Badia1997:2,Badia1997,Badia2000}
Badia, \textit{et al.} used transmission electron microscopy to
determine that alkanethiol ligands on gold nanoparticles pack
approximately 30\% more densely than on planar Au(111)
surfaces.\cite{Badia1996:2} Subsequent experiments demonstrated that
even at full coverages, surface curvature creates voids between linear
ligand chains that can be filled via interdigitation of ligands on
neighboring particles.\cite{Badia1996} The molecular dynamics
simulations of Henz, \textit{et al.} indicate that at low coverages,
the thiolate alkane chains will lie flat on the nanoparticle
surface\cite{Henz2007} Above 90\% coverage, the ligands stand upright
and recover the rigidity and tilt angle displayed on planar
facets. Their simulations also indicate a high degree of mixing
between the thiolate sulfur atoms and surface gold atoms at high
coverages.

To model thiolated gold nanospheres in this work, gold nanoparticles
with radii ranging from 10 - 25 \AA\ were created from a bulk fcc
lattice.  To match surface coverages previously reported by Badia,
\textit{et al.}\cite{Badia1996:2}, these particles were passivated
with a 50\% coverage of a selection of alkyl thiolates of varying
chain lengths. The passivated particles were then solvated in hexane.
Details of the models and simulation protocol follow in the next
section.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%               NON-PERIODIC VSS-RNEMD METHODOLOGY
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Creating a thermal flux between particles and solvent}

The non-periodic variant of VSS-RNEMD\cite{Stocker:2014qq} applies a
series of velocity scaling and shearing moves at regular intervals to
impose a flux between two concentric spherical regions. To impose a
thermal flux between the shells (without an accompanying angular
shear), we solve for scaling coefficients $a$ and $b$,
\begin{eqnarray}
        a = \sqrt{1 - \frac{q_r \Delta t}{K_a - K_a^\mathrm{rot}}}\\ \nonumber\\
        b = \sqrt{1 + \frac{q_r \Delta t}{K_b - K_b^\mathrm{rot}}}
\end{eqnarray}
at each time interval.  These scaling coefficients conserve total
kinetic energy and angular momentum subject to an imposed heat rate,
$q_r$.  The coefficients also depend on the instantaneous kinetic
energy, $K_{\{a,b\}}$, and the total rotational kinetic energy of each
shell, $K_{\{a,b\}}^\mathrm{rot} = \sum_i m_i \left( \mathbf{v}_i
  \times \mathbf{r}_i \right)^2 / 2$.

The scaling coefficients are determined and the velocity changes are
applied at regular intervals, 
\begin{eqnarray}
        \mathbf{v}_i \leftarrow a \left ( \mathbf{v}_i - \left < \omega_a \right > \times \mathbf{r}_i \right ) + \left < \omega_a \right > \times \mathbf{r}_i~~\:\\
        \mathbf{v}_j \leftarrow b \left ( \mathbf{v}_j - \left < \omega_b \right > \times \mathbf{r}_j \right ) + \left < \omega_b \right > \times \mathbf{r}_j.
\end{eqnarray}
Here $\left < \omega_a \right > \times \mathbf{r}_i$ is the
contribution to the velocity of particle $i$ due to the overall
angular velocity of the $a$ shell. In the absence of an angular
momentum flux, the angular velocity $\left < \omega_a \right >$ of the
shell is nearly 0 and the resultant particle velocity is a nearly
linear scaling of the initial velocity by the coefficient $a$ or $b$.

Repeated application of this thermal energy exchange yields a radial
temperature profile for the solvated nanoparticles that depends
linearly on the applied heat rate, $q_r$. Similar to the behavior in
the slab geometries, the temperature profiles have discontinuities at
the interfaces between dissimilar materials.  The size of the
discontinuity depends on the interfacial thermal conductance, which is
the primary quantity of interest.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%               CALCULATING TRANSPORT PROPERTIES
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%               INTERFACIAL THERMAL CONDUCTANCE
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Interfacial Thermal Conductance}

As described in earlier work,\cite{Stocker:2014qq} the thermal
conductance of each spherical shell may be defined as the inverse
Kapitza resistance of the shell. To describe the thermal conductance
of an interface of considerable thickness -- such as the ligand layers
shown here -- we can sum the individual thermal resistances of each
concentric spherical shell to arrive at the inverse of the total
interfacial thermal conductance. In slab geometries, the intermediate
temperatures cancel, but for concentric spherical shells, the
intermeidate temperatures and surface areas remain in the final sum,
requiring the use of a series of individual resistance terms:

\begin{equation}
  \frac{1}{G} = R_\mathrm{total} = \frac{1}{q_r} \sum_i \left(T_{i+i} -
    T_i\right) 4 \pi r_i^2.
\end{equation}

The longest ligand considered here is in excess of 15 \AA\ in length,
and we use 10 concentric spherical shells to describe the total
interfacial thermal conductance of the ligand layer.

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

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%               FORCE FIELDS
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Force Fields}

Throughout this work, gold -- gold interactions are described by the
quantum Sutton-Chen (QSC) model.\cite{PhysRevB.59.3527} The hexane
solvent is described by the TraPPE united atom
model,\cite{TraPPE-UA.alkanes} where sites are located at the carbon
centers for alkyl groups. The TraPPE-UA model for hexane provides both
computational efficiency and reasonable accuracy for bulk thermal
conductivity values. Bonding interactions were used for
intra-molecular sites closer than 3 bonds. Effective Lennard-Jones
potentials were used for non-bonded interactions.

To describe the interactions between metal (Au) and non-metal atoms,
potential energy terms were adapted from an adsorption study of alkyl
thiols on gold surfaces by Vlugt, \textit{et
  al.}\cite{vlugt:cpc2007154} They fit an effective pair-wise
Lennard-Jones form of potential parameters for the interaction between
Au and pseudo-atoms CH$_x$ and S based on a well-established and
widely-used effective potential of Hautman and Klein for the Au(111)
surface.\cite{hautman:4994}


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

Gold nanospheres with radii ranging from 10 - 25 \AA\ were created
from a bulk fcc lattice and were thermally equilibrated prior to the
addition of ligands. A 50\% coverage of ligands (based on coverages
reported by Badia, \textit{et al.}\cite{Badia1996:2}) were placed on
the surface of the equilibrated nanoparticles using
Packmol\cite{packmol}. The nanoparticle / ligand complexes were
thermally equilibrated before Packmol was used to solvate the
structures inside a spherical droplet of hexane. The thickness of the
solvent layer was chosen to be at least 1.5$\times$ the combined
radius of the nanoparticle / ligand structure. The fully solvated
system was equilibrated for at least 1 ns using the Langevin Hull to
apply 50 atm of pressure and a target temperature of 250
K.\cite{Vardeman2011} Typical system sizes ranged from 18,310 united
atom sites for the 10 \AA particles with $C_4$ ligands to 89,490 sites
for the 25 \AA particles with $C_{12}$ ligands.  Figure
\ref{fig:NP25_C12h1} shows one of the solvated 25 \AA nanoparticles
passivated with the $C_{12}$ ligands.  

\begin{figure}
        \includegraphics[width=\linewidth]{figures/NP25_C12h1}
        \caption{A 25 \AA\ radius gold nanoparticle protected with a
          half-monolayer of TraPPE-UA dodecanethiolate (C$_{12}$)
          ligands and solvated in TraPPE-UA hexane. The interfacial
          thermal conductance is computed by applying a kinetic energy
          flux between the nanoparticle and an outer shell of
          solvent.}
        \label{fig:NP25_C12h1}
\end{figure}

Once equilibrated, thermal fluxes were applied for 1 ns, until stable
temperature gradients had developed. Systems were run under moderate
pressure (50 atm) with an average temperature (250K) that maintained a
compact solvent cluster and avoided formation of a vapor layer near
the heated metal surface.  Pressure was applied to the system via the
non-periodic Langevin Hull.\cite{Vardeman2011} However, thermal
coupling to the external temperature bath was removed to avoid
interference with the imposed RNEMD flux.

Because the method conserves \emph{total} angular momentum and energy,
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 measuring an accurate temperature
profile for 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 velocity
scaling.  The only effect of this gold atom is to effectively pin the
nanoparticle at the origin of the coordinate system, thereby
preventing translational diffusion of the nanoparticle due to Brownian
motion.

To provide statisical independence, five separate configurations were
simulated for each particle radius and ligand length. The
configurations were unique starting at the point of ligand placement
in order to sample multiple surface-ligand configurations.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%               EFFECT OF PARTICLE SIZE
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Results}

We modeled four sizes of nanoparticles ($R =$ 10, 15, 20, and 25
\AA). The smallest particle size produces the lowest interfacial
thermal conductance values for most of the of protecting groups
(Fig. \ref{fig:NPthiols_G}).  Between the other three sizes of
nanoparticles, there is no discernible dependence of the interfacial
thermal conductance on the nanoparticle size. It is likely that the
differences in local curvature of the nanoparticle sizes studied here
do not disrupt the ligand packing and behavior in drastically
different ways.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%               EFFECT OF LIGAND CHAIN LENGTH
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

We have also utilized half-monolayers of three lengths of
alkanethiolate ligands -- S(CH$_2$)$_3$CH$_3$, S(CH$_2$)$_7$CH$_3$,
and S(CH$_2$)$_{11}$CH$_3$ -- referred to as C$_4$, C$_8$, and
C$_{12}$ respectively, in this study.

Unlike our previous study of varying thiolate ligand chain lengths on
planar Au(111) surfaces, the interfacial thermal conductance of
ligand-protected nanospheres exhibits a distinct dependence on the
ligand length. For the three largest particle sizes, a half-monolayer
coverage of $C_4$ yields the highest interfacial thermal conductance
and the next-longest ligand, $C_8$, provides a similar boost. The
longest ligand, $C_{12}$, offers only a nominal ($\sim$ 10 \%)
increase in the interfacial thermal conductance over the bare
nanoparticles.

\begin{figure}
        \includegraphics[width=\linewidth]{figures/NPthiols_G}
        \caption{Interfacial thermal conductance ($G$) values for 4
          sizes of solvated nanoparticles that are bare or protected
          with a 50\% coverage of C$_{4}$, C$_{8}$, or C$_{12}$
          alkanethiolate ligands.}
        \label{fig:NPthiols_G}
\end{figure}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%               HEAT TRANSFER MECHANISMS
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Mechanisms for Ligand-Enhanced Heat Transfer}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%               CORRUGATION OF PARTICLE SURFACE
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Corrugation of Particle Surface}

The bonding sites for thiols on gold surfaces have been studied
extensively and include configurations beyond the traditional atop,
bridge, and hollow sites found on planar surfaces. In particular, the
deep potential well between the gold atoms and the thiolate sulfurs
leads to insertion of the sulfur into the gold lattice and
displacement of interfacial gold atoms. The degree of ligand-induced
surface restructuring may have an impact on the interfacial thermal
conductance and is an important phenomenon to quantify.

Henz, \textit{et al.}\cite{Henz2007} used the metal density as a
function of radius to measure the degree of mixing between the thiol
sulfurs and surface gold atoms at the edge of a nanoparticle. Although
metal density is important, disruption of the local crystalline
ordering would also have a large effect on the phonon spectrum in the
particles. To measure this effect, we use the fraction of gold atoms
exhibiting local fcc ordering as a function of radius to describe the
ligand-induced disruption of the nanoparticle surface.

The local bond orientational order can be described using the method
of Steinhardt \textit{et al.},\cite{Steinhardt1983} where spherical
harmonics are associated with a central atom and its nearest
neighbors. The local bonding environment, $\bar{q}_{\ell m}$, for each
atom in the system can be determined by averaging over the spherical
harmonics between the central atom and each of its neighbors,
\begin{equation}
\bar{q}_{\ell m} = \sum_i Y_\ell^m(\theta_i, \phi_i)
\end{equation}
where $\theta_i$ and $\phi_i$ are the relative angular coordinates of
neighbor $i$ in the laboratory frame.  A global average orientational
bond order parameter, $\bar{Q}_{\ell m}$, is the average over each
$\bar{q}_{\ell m}$ for all atoms in the system. To remove the
dependence on the laboratory coordinate frame, the third order
rotationally invariant combination of $\bar{Q}_{\ell m}$,
$\hat{w}_\ell$, is utilized here.\cite{Steinhardt1983,Vardeman:2008fk}

For $\ell=4$, the ideal face-centered cubic (fcc), body-centered cubic
(bcc), hexagonally close-packed (hcp), and simple cubic (sc) local
structures exhibit $\hat{w}_4$ values of -0.159, 0.134, 0.159, and
0.159, respectively. Because $\hat{w}_4$ exhibits an extreme value for
fcc structures, this makes it ideal for measuring local fcc
ordering. The spatial distribution of $\hat{w}_4$ local bond
orientational order parameters, $p(\hat{w}_4 , r)$, can provide
information about the location of individual atoms that are central to
local fcc ordering.

The fraction of fcc-ordered gold atoms at a given radius in the
nanoparticle,
\begin{equation}
        f_\mathrm{fcc}(r) = \int_{-\infty}^{w_c} p(\hat{w}_4, r) d \hat{w}_4
\end{equation}
is described by the distribution of the local bond orientational order
parameters, $p(\hat{w}_4, r)$, and $w_c$, a cutoff for the peak
$\hat{w}_4$ value displayed by fcc structures. A $w_c$ value of -0.12
was chosen to isolate the fcc peak in $\hat{w}_4$.

As illustrated in Figure \ref{fig:Corrugation}, the presence of
ligands decreases the fcc ordering of the gold atoms at the
nanoparticle surface. For the smaller nanoparticles, this disruption
extends into the core of the nanoparticle, indicating widespread
disruption of the lattice.

\begin{figure}
        \includegraphics[width=\linewidth]{figures/NP10_fcc}
        \caption{Fraction of gold atoms with fcc ordering as a
          function of radius for a 10 \AA\ radius nanoparticle. The
          decreased fraction of fcc-ordered atoms in ligand-protected
          nanoparticles relative to bare particles indicates
          restructuring of the nanoparticle surface by the thiolate
          sulfur atoms.}
        \label{fig:Corrugation}
\end{figure}

We may describe the thickness of the disrupted nanoparticle surface by
defining a corrugation factor, $c$, as the ratio of the radius at
which the fraction of gold atoms with fcc ordering is 0.9 and the
radius at which the fraction is 0.5.

\begin{equation}
        c = 1 - \frac{r(f_\mathrm{fcc} = 0.9)}{r(f_\mathrm{fcc} = 0.5)}
\end{equation}

A sharp interface will have a sharp drop in $f_\mathrm{fcc}$ at the
edge of the particle ($c \rightarrow$ 0). In the opposite limit where
the entire nanoparticle surface is restructured by ligands, the radius
at which there is a high probability of fcc ordering moves
dramatically inward ($c \rightarrow$ 1).

The computed corrugation factors are shown in Figure
\ref{fig:NPthiols_combo} for bare nanoparticles and for
ligand-protected particles as a function of ligand chain length. The
largest nanoparticles are only slightly restructured by the presence
of ligands on the surface, while the smallest particle ($r$ = 10 \AA)
exhibits significant disruption of the original fcc ordering when
covered with a half-monolayer of thiol ligands.

Because the thiolate ligands do not significantly alter the larger
particle crystallinity, the surface corrugation does not seem to be a
likely candidate to explain the large increase in thermal conductance
at the interface.

% \begin{equation}
%       C = \frac{r_{bare}(\rho_{\scriptscriptstyle{0.85}}) - r_{capped}(\rho_{\scriptscriptstyle{0.85}})}{r_{bare}(\rho_{\scriptscriptstyle{0.85}})}.
% \end{equation}
% 
% Here, $r_{bare}(\rho_{\scriptscriptstyle{0.85}})$ is the radius of a bare nanoparticle at which the density is $85\%$ the bulk value and $r_{capped}(\rho_{\scriptscriptstyle{0.85}})$ is the corresponding radius for a particle of the same size with a layer of ligands. $C$ has a value of 0 for a bare particle and approaches $1$ as the degree of surface atom mixing increases.


\begin{figure}
        \includegraphics[width=\linewidth]{figures/NPthiols_combo}
        \caption{Computed corrugation values, solvent escape rates,
          ligand orientational $P_2$ values, and interfacial solvent
          orientational $P_2$ values for 4 sizes of solvated
          nanoparticles that are bare or protected with a 50\%
          coverage of C$_{4}$, C$_{8}$, or C$_{12}$ alkanethiolate
          ligands.}
        \label{fig:NPthiols_combo}
\end{figure}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%               MOBILITY OF INTERFACIAL SOLVENT
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Mobility of Interfacial Solvent}

Another possible mechanism for increasing interfacial conductance is
the mobility of the interfacial solvent.  We used a survival
correlation function, $C(t)$, to measure the residence time of a
solvent molecule in the nanoparticle thiolate
layer.\cite{Stocker:2013cl} This function correlates the identity of
all hexane molecules within the radial range of the thiolate layer at
two separate times. If the solvent molecule is present at both times,
the configuration contributes a $1$, while the absence of the molecule
at the later time indicates that the solvent molecule has migrated
into the bulk, and this configuration contributes a $0$. A steep decay
in $C(t)$ indicates a high turnover rate of solvent molecules from the
chain region to the bulk. We may define the escape rate for trapped
solvent molecules at the interface as
\begin{equation}
 k_\mathrm{escape} = \left( \int_0^T C(t) dt \right)^{-1}
  \label{eq:mobility}
\end{equation}
where T is the length of the simulation. This is a direct measure of
the rate at which solvent molecules initially entangled in the
thiolate layer can escape into the bulk. When $k_\mathrm{escape}
\rightarrow 0$, the solvent becomes permanently trapped in the
interfacial region.

The solvent escape rates for bare and ligand-protected nanoparticles
are shown in Figure \ref{fig:NPthiols_combo}. As the ligand chain
becomes longer and more flexible, interfacial solvent molecules become
trapped in the ligand layer and the solvent escape rate decreases.
This mechanism contributes a partial explanation as to why the longer
ligands have significantly lower thermal conductance.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%               ORIENTATION OF LIGAND CHAINS
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Orientation of Ligand Chains}

As the ligand chain length increases in length, it exhibits
significantly more conformational flexibility. Thus, different lengths
of ligands should favor different chain orientations on the surface of
the nanoparticle. To determine the distribution of ligand orientations
relative to the particle surface we examine the probability of
finding a ligand with a particular orientation relative to the surface
normal of the nanoparticle,
\begin{equation} 
\cos{(\theta)}=\frac{\vec{r}_i\cdot\hat{u}_i}{|\vec{r}_i||\hat{u}_i|}
\end{equation}
where $\vec{r}_{i}$ is the vector between the cluster center of mass
and the sulfur atom on ligand molecule {\it i}, and $\hat{u}_{i}$ is
the vector between the sulfur atom and \ce{CH3} pseudo-atom on ligand
molecule {\it i}. As depicted in Figure \ref{fig:NP_pAngle}, $\theta
\rightarrow 180^{\circ}$ for a ligand chain standing upright on the
particle ($\cos{(\theta)} \rightarrow -1$) and $\theta \rightarrow
90^{\circ}$ for a ligand chain lying down on the surface
($\cos{(\theta)} \rightarrow 0$). As the thiolate alkane chain
increases in length and becomes more flexible, the ligands are more
willing to lie down on the nanoparticle surface and exhibit increased
population at $\cos{(\theta)} = 0$.

\begin{figure}
        \includegraphics[width=\linewidth]{figures/NP_pAngle}
        \caption{The two extreme cases of ligand orientation relative
          to the nanoparticle surface: the ligand completely
          outstretched ($\cos{(\theta)} = -1$) and the ligand fully
          lying down on the particle surface ($\cos{(\theta)} = 0$).}
        \label{fig:NP_pAngle}
\end{figure}

% \begin{figure}
%       \includegraphics[width=\linewidth]{figures/thiol_pAngle}
%       \caption{}
%       \label{fig:thiol_pAngle}
% \end{figure}

An order parameter the average ligand chain orientation relative to
the nanoparticle surface is available using the second order Legendre
parameter,
\begin{equation}
        P_2 = \left< \frac{1}{2} \left(3\cos^2(\theta) - 1 \right) \right>
\end{equation}

Ligand populations that are perpendicular to the particle surface hav
P$_2$ values of 1, while ligand populations lying flat on the
nanoparticle surface have P$_2$ values of $-0.5$. Disordered ligand
layers will exhibit mean P$_2$ values of 0. As shown in Figure
\ref{fig:NPthiols_combo} the ligand P$_2$ values approaches 0 as
ligand chain length -- and ligand flexibility -- increases.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%               ORIENTATION OF INTERFACIAL SOLVENT
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Orientation of Interfacial Solvent}

Similarly, we examined the distribution of \emph{hexane} molecule
orientations relative to the particle surface using the same angular
analysis utilized for the ligand chain orientations. In this case,
$\vec{r}_i$ is the vector between the particle center of mass and one
of the \ce{CH2} pseudo-atoms in the middle of hexane molecule $i$ and
$\hat{u}_i$ is the vector between the two \ce{CH3} pseudo-atoms on
molecule $i$. Since we are only interested in the orientation of
solvent molecules near the ligand layer, we select only the hexane
molecules within a specific $r$-range, between the edge of the
particle and the end of the ligand chains. A large population of
hexane molecules with $\cos{(\theta)} \cong \pm 1$ would indicate
interdigitation of the solvent molecules between the upright ligand
chains. A more random distribution of $\cos{(\theta)}$ values
indicates a disordered arrangement of solvent chains on the particle
surface. Again, P$_2$ order parameter values provide a population
analysis for the solvent that is close to the particle surface.

The average orientation of the interfacial solvent molecules is
notably flat on the \emph{bare} nanoparticle surfaces. This blanket of
hexane molecules on the particle surface may act as an insulating
layer, increasing the interfacial thermal resistance. As the length
(and flexibility) of the ligand increases, the average interfacial
solvent P$_2$ value approaches 0, indicating a more random orientation
of the ligand chains. The average orientation of solvent within the
$C_8$ and $C_{12}$ ligand layers is essentially random. Solvent
molecules in the interfacial region of $C_4$ ligand-protected
nanoparticles do not lie as flat on the surface as in the case of the
bare particles, but are not as randomly oriented as the longer ligand
lengths.

These results are particularly interesting in light of our previous
results\cite{Stocker:2013cl}, where solvent molecules readily filled
the vertical gaps between neighboring ligand chains and there was a
strong correlation between ligand and solvent molecular
orientations. It appears that the introduction of surface curvature
and a lower ligand packing density creates a disordered ligand layer
that lacks well-formed channels for the solvent molecules to occupy.

% \begin{figure}
%       \includegraphics[width=\linewidth]{figures/hex_pAngle}
%       \caption{}
%       \label{fig:hex_pAngle}
% \end{figure}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%               SOLVENT PENETRATION OF LIGAND LAYER
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Solvent Penetration of Ligand Layer}

We may also determine the extent of ligand -- solvent interaction by
calculating the hexane density as a function of radius. Figure
\ref{fig:hex_density} shows representative radial hexane density
profiles for a solvated 25 \AA\ radius nanoparticle with no ligands,
and 50\% coverage of C$_{4}$, C$_{8}$, and C$_{12}$ thiolates.

\begin{figure}
        \includegraphics[width=\linewidth]{figures/hex_density}
        \caption{Radial hexane density profiles for 25 \AA\ radius
          nanoparticles with no ligands (circles), C$_{4}$ ligands
          (squares), C$_{8}$ ligands (triangles), and C$_{12}$ ligands
          (diamonds). As ligand chain length increases, the nearby
          solvent is excluded from the ligand layer. Some solvent is
          present inside the particle $r_{max}$ location due to
          faceting of the nanoparticle surface.}
        \label{fig:hex_density}
\end{figure}

The differences between the radii at which the hexane surrounding the
ligand-covered particles reaches bulk density correspond nearly
exactly to the differences between the lengths of the ligand
chains. Beyond the edge of the ligand layer, the solvent reaches its
bulk density within a few angstroms. The differing shapes of the
density curves indicate that the solvent is increasingly excluded from
the ligand layer as the chain length increases.

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

The chemical bond between the metal and the ligand introduces
vibrational overlap that is not present between the bare metal surface
and solvent. Thus, regardless of ligand chain length, the presence of
a half-monolayer ligand coverage yields a higher interfacial thermal
conductance value than the bare nanoparticle. The dependence of the
interfacial thermal conductance on ligand chain length is primarily
explained by increased ligand flexibility and a corresponding decrease
in solvent mobility away from the particles.  The shortest and least
flexible ligand ($C_4$), which exhibits the highest interfacial
thermal conductance value, has a smaller range of angles relative to
the surface normal and is least likely to trap solvent molecules
within the ligand layer. The longer $C_8$ and $C_{12}$ ligands have
increasingly disordered orientations and correspondingly lower solvent
escape rates.

When the ligands are less tightly packed, the cooperative
orientational ordering between the ligand and solvent decreases
dramatically and the conductive heat transfer model plays a much
smaller role in determining the total interfacial thermal
conductance. Thus, heat transfer into the solvent relies primarily on
the convective model, where solvent molecules pick up thermal energy
from the ligands and diffuse into the bulk solvent. This mode of heat
transfer is hampered by a slow solvent escape rate, which is clearly
present in the randomly ordered long ligand layers.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% **ACKNOWLEDGMENTS**
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{acknowledgement}
  Support for this project was provided by the National Science Foundation
  under grant CHE-1362211. Computational time was provided by the
  Center for Research Computing (CRC) at the University of Notre Dame.
\end{acknowledgement}


\newpage

\bibliography{NPthiols}

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