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radius, so the heat dissipation should be controlled entirely by the |
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surface features of the particle / ligand / solvent interface. |
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% 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} |
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% |
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% 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. |
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% STRUCTURE OF SELF-ASSEMBLED MONOLAYERS ON NANOPARTICLES |
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\section{Non-Periodic Velocity Shearing and Scaling RNEMD Methodology} |
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< |
Non-periodic VSS-RNEMD,\cite{Stocker:2014qq} periodically applies a |
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> |
Non-periodic VSS-RNEMD\cite{Stocker:2014qq} periodically applies a |
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series of velocity scaling and shearing moves at regular intervals to |
| 167 |
< |
impose a flux between two concentric spherical regions. |
| 167 |
> |
impose a flux between two concentric spherical regions. To impose a thermal flux between the shells while conserving total kinetic energy and angular momentum we solve for scaling coefficients $a$ and $b$ |
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|
| 173 |
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To simultaneously impose a thermal flux ($J_r$) between the shells we |
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use energy conservation constraints, |
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\begin{eqnarray} |
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< |
K_a - J_r \Delta t & = & a^2 \left(K_a - \frac{1}{2}\langle |
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< |
\omega_a \rangle \cdot \overleftrightarrow{I_a} \cdot \langle \omega_a |
| 178 |
< |
\rangle \right) + \frac{1}{2} \mathbf{c}_a \cdot \overleftrightarrow{I_a} |
| 179 |
< |
\cdot \mathbf{c}_a \label{eq:Kc}\\ |
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< |
K_b + J_r \Delta t & = & b^2 \left(K_b - \frac{1}{2}\langle |
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< |
\omega_b \rangle \cdot \overleftrightarrow{I_b} \cdot \langle \omega_b |
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< |
\rangle \right) + \frac{1}{2} \mathbf{c}_b \cdot \overleftrightarrow{I_b} \cdot \mathbf{c}_b \label{eq:Kh} |
| 170 |
> |
a = \sqrt{1 - \frac{q_r \Delta t}{K_a - K_a^r}}\\ \nonumber\\ |
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> |
b = \sqrt{1 + \frac{q_r \Delta t}{K_b - K_b^r}} |
| 172 |
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\end{eqnarray} |
| 173 |
< |
Simultaneous solution of these quadratic formulae for the scaling coefficients, $a$ and $b$, will ensure that |
| 174 |
< |
the simulation samples from the original microcanonical (NVE) ensemble. Here $K_{\{a,b\}}$ is the instantaneous |
| 186 |
< |
translational kinetic energy of each shell. At each time interval, we solve for $a$, $b$, $\mathbf{c}_a$, and |
| 187 |
< |
$\mathbf{c}_b$, subject to the imposed angular momentum flux, $j_r(\mathbf{L})$, and thermal flux, $J_r$, |
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< |
values. |
| 173 |
> |
|
| 174 |
> |
at each time interval subject to the imposed thermal flux, $q_r$, the instantaneous translational kinetic energy of each shell, $K_{\{a,b\}}$, and the instatanteous rotational kinetic energy of each shell, $K_{\{a,b\}}^r$. |
| 175 |
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|
| 176 |
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The scaling coefficients are determined and the velocity changes are applied |
| 177 |
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\begin{eqnarray} |
| 178 |
+ |
\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~~\:\\ |
| 179 |
+ |
\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. |
| 180 |
+ |
\end{eqnarray} |
| 181 |
+ |
|
| 182 |
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Here $\left < \omega_a \right > \times \mathbf{r}_i$ is the velocity for particle $i$ due to the overal 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 near 0 and the resultant particle velocity is a nearly linear scaling of the initial velocity by the coefficient $a$ or $b$. |
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% CALCULATING TRANSPORT PROPERTIES |
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\subsection{Force Fields} |
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| 218 |
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Gold -- gold interactions are described by the quantum Sutton-Chen (QSC) model,\cite{PhysRevB.59.3527} described in detail in Chapter 1. |
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> |
Gold -- gold interactions are described by the quantum Sutton-Chen (QSC) model.\cite{PhysRevB.59.3527} Hexane molecules are described by the TraPPE united atom model,\cite{TraPPE-UA.alkanes} which provides good computational efficiency and reasonable accuracy for bulk thermal conductivity values. In this model, sites are located at the carbon centers for alkyl groups. Bonding interactions, including bond stretches, bends and torsions, were used for intra-molecular sites closer than 3 bonds. For non-bonded interactions, Lennard-Jones potentials were used. |
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| 226 |
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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. |
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| 220 |
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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} |
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\subsection{Effect of Particle Size} |
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| 258 |
< |
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. |
| 258 |
> |
We 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. |
| 259 |
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\begin{figure} |
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< |
\includegraphics[width=\linewidth]{figures/NPthiols_Gcombo} |
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< |
\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.} |
| 263 |
< |
\label{fig:NPthiols_Gcombo} |
| 261 |
> |
\includegraphics[width=\linewidth]{figures/NP25_C12h1} |
| 262 |
> |
\caption{25 \AA\ radius gold nanoparticle protected with a half-monolayer of TraPPE-UA dodecanethiolate (C$_{12}$) ligands and solvated in TraPPE-UA hexane. The kinetic energy flux is imposed between the nanoparticle and an outer shell of solvent to compute the interfacial thermal conductance.} |
| 263 |
> |
\label{fig:NP25_C12h1} |
| 264 |
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\end{figure} |
| 265 |
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| 266 |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
| 268 |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
| 269 |
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\subsection{Effect of Ligand Chain Length} |
| 270 |
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| 271 |
< |
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. |
| 271 |
> |
We have utilized a half-monolayer 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. |
| 272 |
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|
| 273 |
< |
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. |
| 273 |
> |
Unlike our 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 nominal ($\sim$ 10 \%) increase in the interfacial thermal conductance over a bare nanoparticle. |
| 274 |
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|
| 275 |
+ |
\begin{figure} |
| 276 |
+ |
\includegraphics[width=\linewidth]{figures/NPthiols_Gcombo} |
| 277 |
+ |
\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.} |
| 278 |
+ |
\label{fig:NPthiols_Gcombo} |
| 279 |
+ |
\end{figure} |
| 280 |
+ |
|
| 281 |
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
| 282 |
|
% HEAT TRANSFER MECHANISMS |
| 283 |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
| 285 |
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|
| 286 |
|
\begin{figure} |
| 287 |
|
\includegraphics[width=\linewidth]{figures/NPthiols_combo} |
| 288 |
< |
\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.} |
| 288 |
> |
\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.} |
| 289 |
|
\label{fig:NPthiols_combo} |
| 290 |
|
\end{figure} |
| 291 |
|
|
| 296 |
|
|
| 297 |
|
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. |
| 298 |
|
|
| 299 |
< |
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. |
| 299 |
> |
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. |
| 300 |
|
|
| 301 |
|
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. |
| 302 |
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|
| 363 |
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
| 364 |
|
\subsection{Orientation of Ligand Chains} |
| 365 |
|
|
| 366 |
< |
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 $\cos{(\theta)}$, |
| 366 |
> |
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 we examine the probability of each $\cos{(\theta)}$, |
| 367 |
|
|
| 368 |
|
\begin{equation} |
| 369 |
|
\cos{(\theta)}=\frac{\vec{r}_i\cdot\hat{u}_i}{|\vec{r}_i||\hat{u}_i|} |
| 370 |
|
\end{equation} |
| 371 |
|
|
| 372 |
< |
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 $\cos{(\theta)} = 0$. |
| 372 |
> |
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 likely to lie down on the nanoparticle surface and there will be increased population at $\cos{(\theta)} = 0$. |
| 373 |
|
|
| 374 |
|
\begin{figure} |
| 375 |
|
\includegraphics[width=\linewidth]{figures/NP_pAngle} |
| 389 |
|
P_2(\cos(\theta)) = \left < \frac{1}{2} \left (3\cos^2(\theta) - 1 \right ) \right > |
| 390 |
|
\end{equation} |
| 391 |
|
|
| 392 |
< |
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. |
| 392 |
> |
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 $-0.25$. 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. |
| 393 |
|
|
| 394 |
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
| 395 |
|
% ORIENTATION OF INTERFACIAL SOLVENT |
| 396 |
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
| 397 |
|
\subsection{Orientation of Interfacial Solvent} |
| 398 |
|
|
| 399 |
< |
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. |
| 399 |
> |
Similarly, we 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, 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 $\cos{(\theta)} \cong -1$ 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. |
| 400 |
|
|
| 401 |
< |
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. |
| 401 |
> |
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 randomly oriented as the longer ligand lengths. |
| 402 |
|
|
| 403 |
< |
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. |
| 403 |
> |
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 very disordered ligand layer that lacks well-formed channels for the solvent molecules to occupy. |
| 404 |
|
|
| 405 |
|
% \begin{figure} |
| 406 |
|
% \includegraphics[width=\linewidth]{figures/hex_pAngle} |
| 430 |
|
|
| 431 |
|
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. |
| 432 |
|
|
| 433 |
< |
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. |
| 433 |
> |
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. |
| 434 |
|
|
| 435 |
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
| 436 |
|
% **ACKNOWLEDGMENTS** |
