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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. |
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\begin{longtable}{p{2.7cm} p{2.5cm} p{2.5cm}} |
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\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.} |
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\\ \hline \hline |
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\centering {$J_r$} & \centering\arraybackslash {$\langle dT / dr \rangle$} & \centering\arraybackslash {$\boldsymbol \lambda$}\\ |
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\centering {\small(kcal fs$^{-1}$ \AA$^{-2}$)} & \centering\arraybackslash {\small(K \AA$^{-1}$)} & \centering\arraybackslash {\small(W m$^{-1}$ K$^{-1}$)}\\ \hline |
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\endhead |
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\hline |
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% \endfoot |
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\centering3.25$\times 10^{-6}$ & \centering\arraybackslash 0.11435 & \centering\arraybackslash 1.9753 \\ |
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\centering6.50$\times 10^{-6}$ & \centering\arraybackslash 0.2324 & \centering\arraybackslash 1.9438 \\ |
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\centering1.30$\times 10^{-5}$ & \centering\arraybackslash 0.44922 & \centering\arraybackslash 2.0113 \\ |
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\centering3.25$\times 10^{-5}$ & \centering\arraybackslash 1.1802 & \centering\arraybackslash 1.9139 \\ |
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\centering6.50$\times 10^{-5}$ & \centering\arraybackslash 2.339 & \centering\arraybackslash 1.9314 |
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\begin{longtable}{ccc} |
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\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.} |
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\\ \hline \hline |
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{$J_r$} & {$\langle dT / dr \rangle$} & {$\boldsymbol \lambda$}\\ |
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{\small(kcal fs$^{-1}$ \AA$^{-2}$)} & {\small(K \AA$^{-1}$)} & {\small(W m$^{-1}$ K$^{-1}$)}\\ \hline |
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3.25$\times 10^{-6}$ & 0.11435 & 1.9753 \\ |
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6.50$\times 10^{-6}$ & 0.2324 & 1.9438 \\ |
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1.30$\times 10^{-5}$ & 0.44922 & 2.0113 \\ |
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3.25$\times 10^{-5}$ & 1.1802 & 1.9139 \\ |
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6.50$\times 10^{-5}$ & 2.339 & 1.9314 |
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\\ \hline \hline |
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\label{table:goldconductivity} |
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\end{longtable} |
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SPC/E Water Cluster: |
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|
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\begin{longtable}{p{2.7cm} p{2.5cm} p{2.5cm}} |
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\caption{Calculated thermal conductivity of a cluster of 6912 SPC/E water molecules. Calculations were performed at 300 K and ambient density.} |
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\\ \hline \hline |
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\centering {$J_r$} & \centering\arraybackslash {$\langle dT / dr \rangle$} & \centering\arraybackslash {$\boldsymbol \lambda$}\\ |
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\centering {\small(kcal fs$^{-1}$ \AA$^{-2}$)} & \centering\arraybackslash {\small(K \AA$^{-1}$)} & \centering\arraybackslash {\small(W m$^{-1}$ K$^{-1}$)}\\ \hline |
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\endhead |
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\hline |
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% \endfoot |
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|
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\begin{longtable}{ccc} |
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\caption{Calculated thermal conductivity of a cluster of 6912 SPC/E water molecules. Calculations were performed at 300 K and ambient density.} |
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\\ \hline \hline |
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{$J_r$} & {$\langle dT / dr \rangle$} & {$\boldsymbol \lambda$}\\ |
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{\small(kcal fs$^{-1}$ \AA$^{-2}$)} & {\small(K \AA$^{-1}$)} & {\small(W m$^{-1}$ K$^{-1}$)}\\ \hline |
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\\ \hline \hline |
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\label{table:waterconductivity} |
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\end{longtable} |
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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 |
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|
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\begin{equation} |
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f_r = 8 \pi \eta r^3 \label{eq:fr}. |
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\Xi = 8 \pi \eta r^3 \label{eq:Xi}. |
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\end{equation} |
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|
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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, |
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where $\eta$ is the dynamic viscosity of the surrounding solvent. |
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|
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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, $S$, |
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|
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\begin{equation} |
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S = \frac{2}{\sqrt{a^2 - b^2}} ln \left[ \frac{a + \sqrt{a^2 - b^2}}{b} \right]. \label{eq:S} |
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\end{equation} |
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|
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For a prolate ellipsoidal rod, demonstrated here, the rotational resistance tensor $\Xi$ is a $3 \times 3$ diagonal matrix with elements |
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|
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\begin{eqnarray} |
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f_a = \label{eq:fa}\\ |
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f_b = f_c = \label{eq:fb} |
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\Xi^{rr}_a = \frac{32 \Pi}{2} \eta \frac{ \left( a^2 - b^2 \right) b^2}{2a - b^2 S} \label{eq:Xia}\\ |
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\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} |
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\end{eqnarray} |
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|
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The dynamic viscosity of the solvent, $\eta$, was calculated by applying a linear momentum flux to a periodic box of TraPPE-UA hexane. |
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The dynamic viscosity of the solvent, $\eta$, was determined by applying a traditional VSS-RNEMD linear momentum flux to a periodic box of TraPPE-UA hexane. |
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|
| 238 |
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\begin{longtable}{p{3.8cm} p{3cm} p{2.8cm} p{2.5cm} p{2.5cm}} |
| 239 |
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\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.} |
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\\ \hline \hline |
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{Structure} & \centering{Axis of rotation} & \centering\arraybackslash {$\kappa$} & \centering\arraybackslash {$\delta$} & \centering\arraybackslash Stokes' Law $F$\\ |
| 241 |
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\centering {} & {} & \centering\arraybackslash {\small($10^4$ Pa s m$^{-1}$)} & \centering\arraybackslash {\small(nm)} & \centering\arraybackslash{\small()}\\ \hline |
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\endhead |
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\hline |
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% \endfoot |
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Nanoparticle & \centering$x = y = z$ & & & \\ |
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Prolate Ellipsoidal rod & \centering$x = y$ & & & \\ |
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Prolate Ellipsoidal rod & \centering$z$ & & & |
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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. |
| 239 |
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|
| 240 |
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\begin{longtable}{lccccc} |
| 241 |
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\caption{Calculated 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.} |
| 242 |
|
\\ \hline \hline |
| 243 |
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{Structure} & {Axis of rotation} & {$\kappa_{VSS}$} & {$\kappa_{Stokes-Perrin}$} & {$f_{VSS}$} & {$f_{Stokes-Perrin}$}\\ |
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{} & {} & {\small($10^4$ Pa s m$^{-1}$)} & {\small($10^4$ Pa s m$^{-1}$)} & {} & {}\\ \hline |
| 245 |
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{Sphere} & {$x = y = z$} & {} & {} & {1} & {1}\\ |
| 246 |
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{Prolate Ellipsoid} & {$x = y$} & {} & {} & {} & {}\\ |
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{Prolate Ellipsoid} & {$z$} & {} & {} & {} & {}\\ \hline \hline |
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\label{table:interfacialfriction} |
| 249 |
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\end{longtable} |
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|
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|
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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% **DISCUSSION** |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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\bibliography{nonperiodicVSS} |
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|
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\end{doublespace} |
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\end{document} |
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\end{document} |
