trunk/NPthiols/NPthiols.tex

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\begin{document}

\title{Interfacial Thermal Conductance of Thiolate-Protected
  Gold Nanospheres}
\author{Kelsey M. Stocker}
\author{Suzanne Kucera}
\author{J. Daniel Gezelter}
\email{gezelter@nd.edu}
\affiliation{Department of Chemistry and Biochemistry, University of
  Notre Dame, Notre Dame, IN 46556}

\begin{abstract}
  Molecular dynamics simulations of thiolate-protected and solvated
  gold nanoparticles were carried out in the presence of a
  non-equilibrium heat flux between the solvent and the core of the
  particle.  The interfacial thermal conductance ($G$) was computed
  for these interfaces, and the behavior of the thermal conductance
  was studied as a function of particle size, ligand flexibility, and
  ligand chain length. In all cases, thermal conductance of the
  ligand-protected particles was higher than the bare metal--solvent
  interface.  A number of mechanisms for the enhanced conductance were
  investigated, including thiolate-driven corrugation of the metal
  surface, solvent ordering at the interface, solvent-ligand
  interpenetration, and ligand ordering relative to the particle
  surface.  MORE HERE.
\end{abstract}

\pacs{}
\keywords{}
\maketitle

\section{Introduction}

Heat transport across various nanostructured interfaces has 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, $G$, 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 nonequilibrium 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 varying coverages of alkanethiolate groups.\cite{kuang:AuThl}
These simulations suggested an explanation for the increased 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 come into close
contact with the ligands.

Simulations of {\it mixed-chain} alkylthiolate surfaces showed that
solvent trapped close to the interface can be efficient at moving
thermal energy away from the surface.\cite{Stocker:2013cl} Trapped
solvent molecules that were aligned with nearby
ligands (but which were less able to diffuse into the bulk) were able
to increase the thermal conductance of the interface.  This indicates
that the ligand-to-solvent vibrational energy transfer is a 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 a
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 effect of those gaps on the thermal conductance is 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 has been 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 new applications of
ligand-stabilized nanostructures have been proposed, the structure and
dynamics of ligands on metallic nanoparticles have been studied using
molecular simulation,\cite{Henz2007,Henz:2008qf} NMR, XPS, FTIR,
calorimetry, and surface
microscopies.\cite{Badia1996:2,Badia1996,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,Henz:2008qf} 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.

In this work, thiolated gold nanospheres were modeled using a united
atom force field and non-equilibrium molecular dynamics. Gold
nanoparticles with radii ranging from 10 - 25 \AA\ were created from a
bulk fcc lattice.  These particles were passivated with a 50\%
coverage -- based on coverage densities reported by Badia \textit{et
  al.} -- of a selection of thiolates of varying chain lengths and
flexibilities. The passivated particles were then solvated in hexane.
Details of the models and simulation protocol follow in the next
section.

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

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%               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
intermediate 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.

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

Throughout this work, gold -- gold interactions are described by the
quantum Sutton-Chen (QSC) model.\cite{PhysRevB.59.3527} Previous
work\cite{kuang:AuThl} has demonstrated that the electronic
contributions to heat conduction (which are missing from the QSC
model) across heterogeneous metal / non-metal interfaces are
negligible compared to phonon excitation, which is captured by the
classical model. 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}

Additional terms to represent thiolated alkenes and conjugated ligand
moieties were parameterized as part of this work and are available in
the supporting information.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%               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}. We have chosen three lengths for the
straight-chain ligands, $C_4$, $C_8$, and $C_{12}$, differentiated by
the number of carbons in the chains.  Additionally, to explore the
effects of ligand flexibility, we have used three levels of ligand
``stiffness''.  The most flexible chain is a fully saturated
alkanethiolate, while moderate rigidity is introduced using an alkene
thiolate with one double bond in the penultimate (solvent-facing)
carbon-carbon location.  The most rigid ligands are fully-conjugated
chains where all of the carbons are represented with conjugated (aryl)
united-atom carbon atoms (CHar or terminal CH2ar).

The nanoparticle / ligand complexes were thermally equilibrated to
allow for ligand conformational flexibility. Packmol was then 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 $C_{12}$ alkane thiolate 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.

\begin{figure}
        \includegraphics[width=\linewidth]{figures/temp_profile}
        \caption{Radial temperature profile for a 25 \AA\ radius
          particle protected with a 50\% coverage of TraPPE-UA
          butanethiolate (C$_4$) ligands and solvated in TraPPE-UA
          hexane. A kinetic energy flux is applied between RNEMD
          region A and RNEMD region B. The size of the temperature
          discontinuity at the interface is governed by the
          interfacial thermal conductance.}
        \label{fig:temp_profile}
\end{figure}

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 statistical independence, five separate configurations were
simulated for each particle radius and ligand. The structures 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 systematic 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.

\begin{figure}
  \includegraphics[width=\linewidth]{figures/G3}
  \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$_{12}$ thiolate
      ligands. Ligands of different flexibility are shown in separate
      panels.  The middle panel indicates ligands which have a single
      carbon-carbon double bond in the penultimate position.}
  \label{fig:NPthiols_G}
\end{figure}

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

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 identity. A half-monolayer coverage of ligands yields
interfacial conductance that is strongly dependent on both ligand
length and flexibility.

There are many factors that could be playing a role in the
ligand-dependent conductuance.  The sulfur-gold interaction is
particularly strong, and the presence of the ligands can easily
disrupt the crystalline structure of the gold at the surface of the
particles, providing more efficient scattering of phonons into the
ligand / solvent layer. This effect would be particularly important at
small particle sizes.

In previous studies of mixed-length ligand layers with full coverage,
we observed that ligand-solvent alignment was an important factor for
heat transfer into the solvent.  With high surface curvature and lower
effective coverages, ligand behavior also becomes more complex. Some
chains may be lying down on the surface, and solvent may not be
penetrating the ligand layer to the same degree as in the planar
surfaces.  

Additionally, the ligand flexibility directly alters the vibrational
density of states for the layer that mediates the transfer of phonons
between the metal and the solvent. This could be a partial explanation
for the observed differences between the fully conjugated and more
flexible ligands.

In the following sections we provide details on how we
measure surface corrugation, solvent-ligand interpenetration, and
ordering of the solvent and ligand at the surfaces of the
nanospheres.  We also investigate the overlap between vibrational
densities of states for the various ligands.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%               CORRUGATION OF PARTICLE SURFACE
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Corrugation of the 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 sulfur
atoms 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,Henz:2008qf} 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} The local bonding
environment, $\bar{q}_{\ell m}$, for each atom in the system is
determined by averaging over the spherical harmonics between that 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, it is 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{{\bf 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 steep 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_corrugation} 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.

\begin{figure}
  \includegraphics[width=\linewidth]{figures/C3.pdf}
  \caption{Computed corrugation values for 4 sizes of solvated
    nanoparticles that are bare or protected with a 50\% coverage of
    C$_{4}$, C$_{8}$, or C$_{12}$ thiolate ligands.  The smallest (10
    \AA ) particles show significant disruption to their crystal
    structures, and the length and stiffness of the ligands is a
    contributing factor to the surface disruption.}
  \label{fig:NPthiols_corrugation}
\end{figure}

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 when ligands are added.

% \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.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%               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 saturated 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}

An order parameter describing 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 have
$P_2$ 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_P2} 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 $\cos{(\theta)} \sim \pm 1$ 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 molecules near 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.

\begin{figure}
  \includegraphics[width=\linewidth]{figures/P2_3.pdf}
  \caption{Computed ligand 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. Increasing stiffness of the ligand orients
    these molecules normal to the particle surface, while the length
    of the ligand chains works to prevent solvent from lying flat on
    the surface.}
  \label{fig:NPthiols_P2}
\end{figure}

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.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%               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.

\subsection{Ligand-mediated Vibrational Overlap}

In phonon scattering models for interfacial thermal conductance, the
frequency-dependent transmission probability
($t_{a \rightarrow b}(\omega)$) predicts phonon transfer between
materials $a$ and $b$.  Many of the models for phonon transmission
through an interface estimate this quantity using the phonon density
of states and group velocity, and make use of a Debye model for the
density of states in the solid.

A consensus picture is that in order to transfer the heat carried by
an incoming phonon of frequency $\omega$ on the $a$ side, the phonon
density of states on the $b$ side must have a matching phonon.  The
overlap of the densities of states, particularly at relatively low
frequencies will therefore govern the transfer of thermal energy.

To compute the relevant densities of states, we have first projected
the velocity of each atom $i$ in the region of the interface onto a
direction normal to the interface.  For the nanosphere geometries
studied here, the normal direction depends on the instantaneous
positon of the atom relative to the center of mass of the particle.
\begin{equation}
v_i^\perp(t) = \mathbf{v}_i(t) \cdot \frac{\mathbf{r}_i(t)}{\left|\mathbf{r}_i(t)\right|}
\end{equation}
The perpendicular velocities of the atoms in the interfacial region
can then be correlated,
\begin{equation}
C_\perp(t) = \left< v_i^\perp(t) \cdot v_i^\perp(0) \right>,
\end{equation}
and the time-symmetrized autocorrelation function is Fourier
transformed to provide an estimate of the vibrational density of
states,
\begin{equation}
\rho(\omega) = \frac{1}{\tau} \int_{-\tau/2}^{\tau/2} C_\perp(t) e^{-i
  \omega t} dt.
\end{equation}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%               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 shortest and least
flexible ligand ($C_4$), which exhibits the highest interfacial
thermal conductance value, has a smaller range of available angles relative to
the surface normal. 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. 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% **ACKNOWLEDGMENTS**
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{acknowledgments}
  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{acknowledgments}

\newpage
\bibliographystyle{aip}
\bibliography{NPthiols}

\end{document}
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