| 166 |
|
|
| 167 |
|
Calculated values for the thermal conductivity of a 40 \AA$~$ radius gold nanoparticle (15707 atoms) at different kinetic energy flux values are shown in Table \ref{table:goldconductivity}. For these calculations, the hot and cold slabs were excluded from the linear regression of the thermal gradient. The measured linear slope $\langle dT / dr \rangle$ is linearly dependent on the applied kinetic energy flux $J_r$. Calculated thermal conductivity values compare well with previous bulk QSC values of 1.08 - 1.26 W m$^{-1}$ K$^{-1}$ [cite NIVS paper], though still significantly lower than the experimental value of 320 W m$^{-1}$ K$^{-1}$, as the QSC force field neglects significant electronic contributions to heat conduction. The small increase relative to previous simulated bulk values is due to a slight increase in gold density -- as expected, an increase in density results in higher thermal conductivity values. The increased density is a result of nanoparticle curvature relative to an infinite bulk slab, which introduces surface tension that increases ambient density. |
| 168 |
|
|
| 169 |
< |
\begin{longtable}{p{2.7cm} p{2.5cm} p{2.5cm}} |
| 170 |
< |
\caption{Calculated thermal conductivity of a crystalline gold nanoparticle of radius 40 \AA. Calculations were performed at 300 K and ambient density. Gold-gold interactions are described by the Quantum Sutton-Chen potential.} |
| 171 |
< |
\\ \hline \hline |
| 172 |
< |
\centering {$J_r$} & \centering\arraybackslash {$\langle dT / dr \rangle$} & \centering\arraybackslash {$\boldsymbol \lambda$}\\ |
| 173 |
< |
\centering {\small(kcal fs$^{-1}$ \AA$^{-2}$)} & \centering\arraybackslash {\small(K \AA$^{-1}$)} & \centering\arraybackslash {\small(W m$^{-1}$ K$^{-1}$)}\\ \hline |
| 174 |
< |
\endhead |
| 175 |
< |
\hline |
| 176 |
< |
% \endfoot |
| 177 |
< |
\centering3.25$\times 10^{-6}$ & \centering\arraybackslash 0.11435 & \centering\arraybackslash 1.9753 \\ |
| 178 |
< |
\centering6.50$\times 10^{-6}$ & \centering\arraybackslash 0.2324 & \centering\arraybackslash 1.9438 \\ |
| 179 |
< |
\centering1.30$\times 10^{-5}$ & \centering\arraybackslash 0.44922 & \centering\arraybackslash 2.0113 \\ |
| 180 |
< |
\centering3.25$\times 10^{-5}$ & \centering\arraybackslash 1.1802 & \centering\arraybackslash 1.9139 \\ |
| 181 |
< |
\centering6.50$\times 10^{-5}$ & \centering\arraybackslash 2.339 & \centering\arraybackslash 1.9314 |
| 169 |
> |
\begin{longtable}{ccc} |
| 170 |
> |
\caption{Calculated thermal conductivity of a crystalline gold nanoparticle of radius 40 \AA. Calculations were performed at 300 K and ambient density. Gold-gold interactions are described by the Quantum Sutton-Chen potential.} |
| 171 |
|
\\ \hline \hline |
| 172 |
+ |
{$J_r$} & {$\langle dT / dr \rangle$} & {$\boldsymbol \lambda$}\\ |
| 173 |
+ |
{\small(kcal fs$^{-1}$ \AA$^{-2}$)} & {\small(K \AA$^{-1}$)} & {\small(W m$^{-1}$ K$^{-1}$)}\\ \hline |
| 174 |
+ |
3.25$\times 10^{-6}$ & 0.11435 & 1.9753 \\ |
| 175 |
+ |
6.50$\times 10^{-6}$ & 0.2324 & 1.9438 \\ |
| 176 |
+ |
1.30$\times 10^{-5}$ & 0.44922 & 2.0113 \\ |
| 177 |
+ |
3.25$\times 10^{-5}$ & 1.1802 & 1.9139 \\ |
| 178 |
+ |
6.50$\times 10^{-5}$ & 2.339 & 1.9314 |
| 179 |
+ |
\\ \hline \hline |
| 180 |
|
\label{table:goldconductivity} |
| 181 |
|
\end{longtable} |
| 182 |
|
|
| 183 |
|
SPC/E Water Cluster: |
| 184 |
|
|
| 185 |
< |
\begin{longtable}{p{2.7cm} p{2.5cm} p{2.5cm}} |
| 186 |
< |
\caption{Calculated thermal conductivity of a cluster of 6912 SPC/E water molecules. Calculations were performed at 300 K and ambient density.} |
| 190 |
< |
\\ \hline \hline |
| 191 |
< |
\centering {$J_r$} & \centering\arraybackslash {$\langle dT / dr \rangle$} & \centering\arraybackslash {$\boldsymbol \lambda$}\\ |
| 192 |
< |
\centering {\small(kcal fs$^{-1}$ \AA$^{-2}$)} & \centering\arraybackslash {\small(K \AA$^{-1}$)} & \centering\arraybackslash {\small(W m$^{-1}$ K$^{-1}$)}\\ \hline |
| 193 |
< |
\endhead |
| 194 |
< |
\hline |
| 195 |
< |
% \endfoot |
| 196 |
< |
|
| 185 |
> |
\begin{longtable}{ccc} |
| 186 |
> |
\caption{Calculated thermal conductivity of a cluster of 6912 SPC/E water molecules. Calculations were performed at 300 K and ambient density.} |
| 187 |
|
\\ \hline \hline |
| 188 |
+ |
{$J_r$} & {$\langle dT / dr \rangle$} & {$\boldsymbol \lambda$}\\ |
| 189 |
+ |
{\small(kcal fs$^{-1}$ \AA$^{-2}$)} & {\small(K \AA$^{-1}$)} & {\small(W m$^{-1}$ K$^{-1}$)}\\ \hline |
| 190 |
+ |
\\ \hline \hline |
| 191 |
|
\label{table:waterconductivity} |
| 192 |
|
\end{longtable} |
| 193 |
|
|
| 203 |
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
| 204 |
|
\subsection{Interfacial thermal conductance} |
| 205 |
|
|
| 206 |
< |
Gold Nanoparticle in Hexane: |
| 206 |
> |
The interfacial thermal conductance, $G$, is calculated by defining a temperature difference $\Delta T$ across a given interface. |
| 207 |
|
|
| 208 |
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
| 209 |
|
% INTERFACIAL FRICTION |
| 210 |
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
| 211 |
|
\subsection{Interfacial friction} |
| 212 |
|
|
| 213 |
< |
Table \ref{table:interfacialfriction} gives the calculated interfacial friction coefficients $\kappa$ for a spherical gold nanoparticle and prolate ellipsoidal gold nanorod in TraPPE-UA hexane. An angular momentum flux was applied between the A and B regions defined as the gold structure and hexane molecules included in the convex hull, respectively. The resulting angular velocity gradient resulted in the gold structure rotating about the prescribed axis within the solvent. |
| 213 |
> |
The interfacial friction coefficient, $\kappa$, can be calculated from the solvent dynamic viscosity, $\eta$, and the slip length, $\delta$. The slip length is measured from a radial angular momentum profile generated from a nonperiodic VSS-RNEMD simulation, as shown in Figure X. |
| 214 |
|
|
| 215 |
+ |
Table \ref{table:interfacialfriction} shows the calculated interfacial friction coefficients $\kappa$ for a spherical gold nanoparticle and prolate ellipsoidal gold nanorod in TraPPE-UA hexane. An angular momentum flux was applied between the A and B regions defined as the gold structure and hexane molecules included in the convex hull, respectively. The resulting angular velocity gradient resulted in the gold structure rotating about the prescribed axis within the solvent. |
| 216 |
+ |
|
| 217 |
|
Analytical solutions for the rotational friction coefficients for a solvated spherical body of radius $r$ under ``stick'' boundary conditions can be estimated using Stokes' law |
| 218 |
|
|
| 219 |
|
\begin{equation} |
| 220 |
< |
f_r = 8 \pi \eta r^3 \label{eq:fr}. |
| 220 |
> |
\Xi = 8 \pi \eta r^3 \label{eq:Xi}. |
| 221 |
|
\end{equation} |
| 222 |
|
|
| 223 |
< |
For general ellipsoids with semiaxes $a$, $b$, and $c$, Perrin's extension of Stokes' law provides exact solutions for prolate $(a \geq b = c)$ and oblate $(a < b = c)$ ellipsoids. For a prolate ellipsoidal rod, demonstrated here, |
| 223 |
> |
where $\eta$ is the dynamic viscosity of the surrounding solvent and was determined for TraPPE-UA hexane under these condition by applying a traditional VSS-RNEMD linear momentum flux to a periodic box of solvent. |
| 224 |
|
|
| 225 |
+ |
For general ellipsoids with semiaxes $a$, $b$, and $c$, Perrin's extension of Stokes' law provides exact solutions for symmetric prolate $(a \geq b = c)$ and oblate $(a < b = c)$ ellipsoids. For simplicity, we define a Perrin Factor, $S$, |
| 226 |
+ |
|
| 227 |
+ |
\begin{equation} |
| 228 |
+ |
S = \frac{2}{\sqrt{a^2 - b^2}} ln \left[ \frac{a + \sqrt{a^2 - b^2}}{b} \right]. \label{eq:S} |
| 229 |
+ |
\end{equation} |
| 230 |
+ |
|
| 231 |
+ |
For a prolate ellipsoidal rod, demonstrated here, the rotational resistance tensor $\Xi$ is a $3 \times 3$ diagonal matrix with elements |
| 232 |
+ |
|
| 233 |
|
\begin{eqnarray} |
| 234 |
< |
f_a = \label{eq:fa}\\ |
| 235 |
< |
f_b = f_c = \label{eq:fb} |
| 234 |
> |
\Xi^{rr}_a = \frac{32 \pi}{2} \eta \frac{ \left( a^2 - b^2 \right) b^2}{2a - b^2 S} \label{eq:Xia}\\ |
| 235 |
> |
\Xi^{rr}_{b,c} = \frac{32 \pi}{2} \eta \frac{ \left( a^4 - b^4 \right)}{ \left( 2a^2 - b^2 \right)S - 2a}. \label{eq:Xibc} |
| 236 |
|
\end{eqnarray} |
| 237 |
|
|
| 238 |
< |
The dynamic viscosity of the solvent, $\eta$, was calculated by applying a linear momentum flux to a periodic box of TraPPE-UA hexane. |
| 238 |
> |
However, the friction between hexane solvent and gold more likely falls within ``slip'' boundary conditions. Hu and Zwanzig investigated the rotational friction coefficients for spheroids under slip boundary conditions and obtained numerial results for a scaling factor as a function of $\tau$, the ratio of the shorter semiaxes and the longer semiaxis of the spheroid. For the sphere and prolate ellipsoid shown here, the values of $\tau$ are $1$ and $0.3939$, respectively. According to the values tabulated by Hu and Zwanzig, the sphere friction coefficient approaches $0$, while the ellipsoidal friction coefficient must be scaled by a factor of $0.880$ to account for the reduced interfacial friction under ``slip'' boundary conditions. |
| 239 |
|
|
| 240 |
< |
\begin{longtable}{p{3.8cm} p{3cm} p{2.8cm} p{2.5cm} p{2.5cm}} |
| 241 |
< |
\caption{Calculated interfacial friction coefficients ($\kappa$) and slip length ($\delta$) of gold nanostructures solvated in TraPPE-UA hexane. The ellipsoid is oriented with the long axis along the $z$ direction.} |
| 242 |
< |
\\ \hline \hline |
| 243 |
< |
{Structure} & \centering{Axis of rotation} & \centering\arraybackslash {$\kappa$} & \centering\arraybackslash {$\delta$} & \centering\arraybackslash Stokes' Law $F$\\ |
| 241 |
< |
\centering {} & {} & \centering\arraybackslash {\small($10^4$ Pa s m$^{-1}$)} & \centering\arraybackslash {\small(nm)} & \centering\arraybackslash{\small()}\\ \hline |
| 242 |
< |
\endhead |
| 243 |
< |
\hline |
| 244 |
< |
% \endfoot |
| 245 |
< |
Nanoparticle & \centering$x = y = z$ & & & \\ |
| 246 |
< |
Prolate Ellipsoidal rod & \centering$x = y$ & & & \\ |
| 247 |
< |
Prolate Ellipsoidal rod & \centering$z$ & & & |
| 240 |
> |
Another useful quantity is the friction factor $f$, or the friction coefficient of a non-spherical shape divided by the friction coefficient of a sphere of equivalent volume. The nanoparticle and ellipsoidal nanorod dimensions used here were specifically chosen to create structures with equal volumes. |
| 241 |
> |
|
| 242 |
> |
\begin{longtable}{lccccc} |
| 243 |
> |
\caption{Calculated ``stick'' interfacial friction coefficients ($\kappa$) and friction factors ($f$) of gold nanostructures solvated in TraPPE-UA hexane. The ellipsoid is oriented with the long axis along the $z$ direction.} |
| 244 |
|
\\ \hline \hline |
| 245 |
+ |
{Structure} & {Axis of rotation} & {$\kappa_{VSS}$} & {$\kappa_{calc}$} & {$f_{VSS}$} & {$f_{calc}$}\\ |
| 246 |
+ |
{} & {} & {\small($10^{-29}$ Pa s m$^{3}$)} & {\small($10^{-29}$ Pa s m$^{3}$)} & {} & {}\\ \hline |
| 247 |
+ |
{Sphere} & {$x = y = z$} & {} & {5.37237} & {1} & {1}\\ |
| 248 |
+ |
{Prolate Ellipsoid} & {$x = y$} & {} & {3.59881} & {} & {0.768726}\\ |
| 249 |
+ |
{Prolate Ellipsoid} & {$z$} & {} & {9.01084} & {} & {1.92477}\\ \hline \hline |
| 250 |
|
\label{table:interfacialfriction} |
| 251 |
|
\end{longtable} |
| 252 |
|
|
| 253 |
+ |
\begin{longtable}{lccc} |
| 254 |
+ |
\caption{Calculated ``slip'' interfacial friction coefficients ($\kappa$) and friction factors ($f$) of gold nanostructures solvated in TraPPE-UA hexane. The ellipsoid is oriented with the long axis along the $z$ direction.} |
| 255 |
+ |
\\ \hline \hline |
| 256 |
+ |
{Structure} & {Axis of rotation} & {$\kappa_{VSS}$} & {$\kappa_{calc}$}\\ |
| 257 |
+ |
{} & {} & {\small($10^{-29}$ Pa s m$^{3}$)} & {\small($10^{-29}$ Pa s m$^{3}$)}\\ \hline |
| 258 |
+ |
{Sphere} & {$x = y = z$} & {} & {0}\\ |
| 259 |
+ |
{Prolate Ellipsoid} & {$x = y$} & {} & {0}\\ |
| 260 |
+ |
{Prolate Ellipsoid} & {$z$} & {} & {7.9295392}\\ \hline \hline |
| 261 |
+ |
\label{table:interfacialfriction} |
| 262 |
+ |
\end{longtable} |
| 263 |
|
|
| 264 |
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
| 265 |
|
% **DISCUSSION** |
| 282 |
|
\bibliography{nonperiodicVSS} |
| 283 |
|
|
| 284 |
|
\end{doublespace} |
| 285 |
< |
\end{document} |
| 285 |
> |
\end{document} |
