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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}

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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%               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
suggest 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.

Additionally, simulations of {\it mixed-chain} alkylthiolate surfaces
showed that entrapped solvent can be very efficient at moving thermal
energy away from the surface.\cite{Stocker:2013cl} Trapped solvent that
is orientationally coupled to the ordered ligands (and is less able to
diffuse into the bulk) were able to double the thermal conductance of
the interface.

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} The primary reason for developing a
non-periodic variant of RNEMD is 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 those gaps will have on the thermal conductance are
unknown.


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

For a solvated nanoparticle, we can 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, $G >> G_c$, cooling of the
nanoparticle is limited by the solvent properties, $C_s$ and
$\Lambda_s$.  In the opposite limit ($G << G_c$), 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 the interface may be tuned to manipulate the rate of
cooling for a solvated nanoparticle.  Based on estimates of $G$ from
previous simulations of gold nanoparticles solvated in hexane and
experimental results for solvated nanostructures, it appears that we
are in the $G << G_c$ regime for gold nanoparticles with radii smaller
than 40 nm when solvated in hexane. 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.

% Understanding how the structural details of the interfaces affect the energy flow between the particle and its surroundings is essential in designing and functionalizing metallic nanoparticles for use in plasmonic photothermal therapies,\cite{Jain:2007ux,Petrova:2007ad,Gnyawali:2008lp,Mazzaglia:2008to,Huff:2007ye,Larson:2007hw} which rely on the ability of metallic nanoparticles to absorb light in the near-IR, a portion of the spectrum in which living tissue is very nearly transparent. The relevant physical property controlling the transfer of this energy as heat into the surrounding tissue is the interfacial thermal conductance, $G$, which can be somewhat difficult to determine experimentally.\cite{Wilson:2002uq,Plech:2005kx}
% 
% Metallic particles have also been proposed for use in efficient thermal-transfer fluids, although careful experiments by Eapen \textit{et al.} have shown that metal-particle-based nanofluids have thermal conductivities that match Maxwell predictions.\cite{Eapen:2007th} The likely cause of previously reported non-Maxwell behavior\cite{Eastman:2001wb,Keblinski:2002bx,Lee:1999ct,Xue:2003ya,Xue:2004oa} is percolation networks of nanoparticles exchanging energy via the solvent,\cite{Eapen:2007mw} so it is important to get a detailed molecular picture of particle-ligand and ligand-solvent interactions in order to understand the thermal behavior of complex fluids. To date, there have been some reported values from experiment\cite{Wilson:2002uq,doi:10.1021jp8051888,doi:10.1021jp048375k,Ge2005,Park2012}) of $G$ for ligand-protected nanoparticles embedded in liquids, but there is still a significant gap in knowledge about how chemically distinct ligands or protecting groups will affect heat transport from the particles. In particular, the dearth of atomistic, dynamic information available from molecular dynamics simulations means that the heat transfer mechanisms at these nanoparticle surfaces remain largely unclear.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%               STRUCTURE OF SELF-ASSEMBLED MONOLAYERS ON NANOPARTICLES
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Structure 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 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.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%               NON-PERIODIC VSS-RNEMD METHODOLOGY
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Non-Periodic Velocity Shearing and Scaling RNEMD Methodology}

Non-periodic VSS-RNEMD,\cite{Stocker:2014qq} periodically applies a
series of velocity scaling and shearing moves at regular intervals to
impose a flux between two concentric spherical regions.

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.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%               CALCULATING TRANSPORT PROPERTIES
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Calculating Transport Properties from Non-Periodic VSS-RNEMD}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%               INTERFACIAL THERMAL CONDUCTANCE
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Interfacial Thermal Conductance}

We can describe the thermal conductance of each spherical shell as the
inverse Kapitza resistance. To describe the thermal conductance for 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 total thermal resistance, or the
inverse of the total interfacial thermal conductance:

\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, requiring the use of at least 10 spherical shells to describe the total interfacial thermal conductance.

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

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

Gold -- gold interactions are described by the quantum Sutton-Chen (QSC) model,\cite{PhysRevB.59.3527} described in detail in Chapter 1.

Hexane molecules are described by the TraPPE united atom model,\cite{TraPPE-UA.alkanes} detailed in Chapter 3, 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, bends and torsions, were used for intra-molecular sites closer than 3 bonds. For non-bonded interactions, Lennard-Jones potentials were used.

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}

The various sized gold nanoparticles 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 briefly thermally equilibrated before Packmol was used to solvate the structures within a spherical droplet of hexane. The thickness of the solvent layer was chosen to be at least 1.5$\times$ the radius of the nanoparticle / ligand structure. The fully solvated system was equilibrated in the Langevin Hull under 50 atm of pressure with a target temperature of 250 K for at least 1 nanosecond.

Once equilibrated, thermal fluxes were applied for
1 nanosecond, until stable temperature gradients had
developed. Systems were run under moderate pressure
(50 atm) and average temperature (250K) to maintain a compact solvent cluster and avoid formation of a vapor phase 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 and pressure 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.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%               INTERFACIAL THERMAL CONDUCTANCE
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Interfacial Thermal Conductance}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%               EFFECT OF PARTICLE SIZE
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Effect of Particle Size}

I have modeled four sizes of nanoparticles ($r =$ 10, 15, 20, and 25 \AA). The smallest particle size produces the lowest interfacial thermal conductance value regardless of protecting group. 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.

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

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%               EFFECT OF LIGAND CHAIN LENGTH
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Effect of Ligand Chain Length}

I have studied 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, on each of the four nanoparticle sizes.

Unlike my previous study of varying thiolate ligand chain lengths on Au(111) surfaces, the interfacial thermal conductance of ligand-protected nanoparticles exhibits a distinct non-monotonic 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 nearly equivalent boost. The longest ligand $C_{12}$ offers only a marginal ($\sim$ 10 \%) increase in the interfacial thermal conductance over a bare nanoparticle.

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

\begin{figure}
        \includegraphics[width=\linewidth]{figures/NPthiols_combo}
        \caption{Computed 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}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%               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 have a large effect on the phonon spectrum in the particles. To measure this effect, I used 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 model proposed by 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. 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. The third-order rotationally invariant combination of $\bar{Q}_{\ell m}$, $\hat{W}_4$, is utilized here. Ideal face-centered cubic (fcc), body-centered cubic (bcc), hexagonally close-packed (hcp), and simple cubic (sc), have values in the $\ell$ = 4 $\hat{W}$ invariant of -0.159, 0.134, 0.159, and 0.159, respectively. $\hat{W}_4$ has an extreme value for fcc structures, making it ideal for measuring local fcc order. The distribution of $\hat{W}_4$ local bond orientational order parameters, $p(\hat{W}_4)$, can provide information about individual atoms that are central to local fcc ordering.

The fraction of fcc ordered gold atoms at a given radius

\begin{equation}
        f_{fcc} = \int_{-\infty}^{w_i} p(\hat{W}_4) d \hat{W}_4
\end{equation}

 is described by the distribution of the local bond orientational order parameter, $p(\hat{W}_4)$, and $w_i$, a cutoff for the peak $\hat{W}_4$ value displayed by fcc structures. A $w_i$ 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_{fcc} = 0.9)}{r(f_{fcc} = 0.5)}
\end{equation}

A clean, unstructured interface will have a sharp drop in $f_{fcc}$ at the edge of the particle ($c \rightarrow$ 0). In the opposite limit where the entire nanoparticle surface is restructured, 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_Gcombo} 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{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}

We use 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_{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. As $k_{escape} \rightarrow 0$, 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 becomes trapped in the ligand layer and the solvent escape rate decreases.

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

I have previously observed that as the ligand chain length increases in length, it becomes significantly more flexible. 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 I examine the probability of each $\cos{(\theta)}$,

\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 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 likely to lie down on the nanoparticle surface and there will be 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}

A single number describing the average ligand chain orientation relative to the nanoparticle surface may be achieved by calculating a P$_2$ order parameter from the distribution of $\cos(\theta)$ values.

\begin{equation}
        P_2(\cos(\theta)) = \left < \frac{1}{2} \left (3\cos^2(\theta) - 1 \right ) \right >
\end{equation}

A ligand chain that is perpendicular to the particle surface has a P$_2$ value of 1, while a ligand chain lying flat on the nanoparticle surface has a P$_2$ value of $-\frac{1}{2}$. Disordered ligand layers will exhibit a mean P$_2$ value of 0. As shown in Figure \ref{fig:NPthiols_combo} the ligand P$_2$ value approaches 0 as ligand chain length -- and ligand flexibility -- increases.

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

I also examined the distribution of \emph{hexane} molecule orientations relative to the particle surface using the same $\cos{(\theta)}$ 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, I selected 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 $\cos{(\theta)} \cong -1$ would indicate interdigitation of the solvent molecules between the upright ligand chains. A more random distribution of $\cos{(\theta)}$ values indicates either little penetration of the ligand layer by the solvent, or a very disordered arrangement of ligand chains on the particle surface. Again, P$_2$ order parameter values may be obtained from the distribution of $\cos(\theta)$ values.

The average orientation of the interfacial solvent molecules is notably flat on the \emph{bare} nanoparticle surface. 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 random orientation of the ligand chains. The average orientation of solvent within the $C_8$ and $C_{12}$ ligand layers is essentially totally 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 random as the longer ligand lengths.

These results are particularly interesting in light of the results described in Chapter 3, 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 very 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 $r$. 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. The shortest and least flexible ligand ($C_4$), which exhibits the highest interfacial thermal conductance value, is oriented more normal to the particle surface than the longer ligands and is least likely to trap solvent molecules within the ligand layer. The longer $C_8$ and $C_{12}$ ligands have increasingly disordered average orientations and correspondingly lower solvent escape rates.

The heat transfer mechanisms proposed in Chapter 3 can also be applied to the non-periodic case. 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.

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


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