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profiles are analyzed to yield information about the interfacial thermal |
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conductance. As the simulation progresses, the VSS moves are performed on a regular basis, and |
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the system develops a thermal or velocity gradient in response to the applied |
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< |
flux. We can treat the temperature in each radial shell as discrete, making the thermal conductance, $G$, of each shell the inverse of its Kapitza resistance, $R_K$. To model the thermal conductance across an interface (or multiple interfaces) it is useful to consider the shells as resistors wired in series. The total resistance of the shells is then additive, and the interfacial thermal conductance is the inverse of the total Kapitza resistance. The thermal resistance of each shell is |
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> |
flux. We can treat the temperature in each radial shell as discrete, making the thermal conductance, $G$, of each shell the inverse of its Kapitza resistance, $R_K$. To model the thermal conductance across an interface (or multiple interfaces) it is useful to consider the shells as resistors wired in series. The resistance of the shells is then additive, and the interfacial thermal conductance is the inverse of the total Kapitza resistance. The thermal resistance of each shell is |
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|
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\begin{equation} |
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R_K = \frac{1}{q_r} \Delta T 4 \pi r^2 |
| 295 |
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making the total resistance of two neighboring shells |
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|
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\begin{equation} |
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< |
R_{total} = \frac{1}{q_r} \left [ (T_2 - T_1) 4 \pi r^2_1 + (T_3 - T_2) 4 \pi r^2_2 \right ] |
| 298 |
> |
R_{total} = \frac{1}{q_r} \left [ (T_2 - T_1) 4 \pi r^2_1 + (T_3 - T_2) 4 \pi r^2_2 \right ] = \frac{1}{G} |
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\label{eq:Rtotal} |
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\end{equation} |
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+ |
|
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+ |
This series can be expanded for any number of adjacent shells, allowing for the calculation of the interfacial thermal conductance for interfaces of significant thickness, such as self-assembled ligand monolayers on a metal surface. |
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|
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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% INTERFACIAL FRICTION |
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|
| 349 |
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where $L$ is the total angular momentum exchanged between the two RNEMD regions and $t$ is the length of the simulation. |
| 350 |
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|
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< |
Previous VSS-RNEMD simulations of the interfacial friction of the planar Au(111) / hexane interface have shown that the interface exists within ``slip'' boundary conditions.\cite{Kuang2012} Hu and Zwanzig\cite{Zwanzig} investigated the rotational friction coefficients for spheroids under slip boundary conditions and obtained numerial results for a scaling factor to be applied to $\Xi^{rr}_{\mathit{stick}}$ 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, $\Xi^{rr}_{\mathit{slip}}$ for any sphere approaches $0$, while the ellipsoidal $\Xi^{rr}_{\mathit{slip}}$ is the analytical $\Xi^{rr}_{\mathit{stick}}$ result scaled by a factor of $0.359$ to account for the reduced interfacial friction under ``slip'' boundary conditions. |
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> |
Previous VSS-RNEMD simulations of the interfacial friction of the planar Au(111) / hexane interface have shown that the interface exists within ``slip'' boundary conditions.\cite{Kuang2012} Hu and Zwanzig\cite{Zwanzig} investigated the rotational friction coefficients for spheroids under slip boundary conditions and obtained numerial results for a scaling factor to be applied to $\Xi^{rr}_{\mathit{stick}}$ 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. Under ``slip'' conditions, $\Xi^{rr}_{\mathit{slip}}$ for any sphere and rotation of the prolate ellipsoid about its long axis approaches $0$, as no solvent is displaced by either of these rotations. $\Xi^{rr}_{\mathit{slip}}$ for rotation of the prolate ellipsoid about its short axis is $35.9\%$ of the analytical $\Xi^{rr}_{\mathit{stick}}$ result, accounting for the reduced interfacial friction under ``slip'' boundary conditions. |
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|
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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% **TESTS AND APPLICATIONS** |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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\subsection{Interfacial thermal conductance} |
| 406 |
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|
| 407 |
< |
Calculated interfacial thermal conductance ($G$) |
| 407 |
> |
Calculated interfacial thermal conductance ($G$) values for three sizes of gold nanoparticles and Au(111) surface solvated in TraPPE-UA hexane. The introduction of surface curvature increases the interfacial thermal conductance by a factor of approximately $1.5$ relative to the flat interface. There are no significant differences in the $G$ values for the varying nanoparticle sizes. It seems likely that for the range of nanoparticle sizes represented here, size effects are not evident. |
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|
| 409 |
|
\begin{longtable}{ccc} |
| 410 |
< |
\caption{Calculated interfacial thermal conductance ($G$) values for gold nanoparticles of varying radii solvated in explicit TraPPE-UA hexane. The nanoparticle $G$ values are compared to previous results for a gold slab in TraPPE-UA hexane, revealing increased interfacial thermal conductance for non-planar interfaces.} |
| 410 |
> |
\caption{Calculated interfacial thermal conductance ($G$) values for gold nanoparticles of varying radii solvated in explicit TraPPE-UA hexane. The nanoparticle $G$ values are compared to previous results for a Au(111) interface in TraPPE-UA hexane, revealing increased interfacial thermal conductance for non-planar interfaces.} |
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\\ \hline \hline |
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{Nanoparticle Radius} & {$G$}\\ |
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{\footnotesize(\AA)} & {\footnotesize(MW / m$^{2}$ $\cdot$ K)}\\ \hline |
| 414 |
|
20 & {47.1} \\ |
| 415 |
|
30 & {45.4} \\ |
| 416 |
|
40 & {46.5} \\ |
| 417 |
< |
slab & {30.2} |
| 417 |
> |
\hline |
| 418 |
> |
Au(111) & {30.2} |
| 419 |
|
\\ \hline \hline |
| 420 |
|
\label{table:interfacialconductance} |
| 421 |
|
\end{longtable} |
| 426 |
|
\subsection{Interfacial friction} |
| 427 |
|
|
| 428 |
|
Table \ref{table:couple} shows the calculated rotational friction coefficients $\Xi^{rr}$ for spherical gold nanoparticles and a 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 beyond a certain radius, respectively. The resulting angular velocity gradient causes the gold structure to rotate about the prescribed axis. |
| 429 |
< |
|
| 429 |
> |
|
| 430 |
|
\begin{longtable}{lccccc} |
| 431 |
|
\caption{Comparison of rotational friction coefficients under ideal ``slip'' ($\Xi^{rr}_{\mathit{slip}}$) and ``stick'' conditions ($\Xi^{rr}_{\mathit{stick}}$) and effective rotational friction coefficients ($\Xi^{rr}_{\mathit{eff}}$) of gold nanostructures solvated in TraPPE-UA hexane at 230 K. The ellipsoid is oriented with the long axis along the $z$ direction.} |
| 432 |
|
\\ \hline \hline |
| 436 |
|
Sphere (r = 30 \AA) & {$x = y = z$} & 0 & {8415} & {11749} & {0.716}\\ |
| 437 |
|
Sphere (r = 40 \AA) & {$x = y = z$} & 0 & {47544} & {34464} & {1.380}\\ |
| 438 |
|
Prolate Ellipsoid & {$x = y$} & {1792} & {3128} & {4991} & {0.627}\\ |
| 439 |
< |
Prolate Ellipsoid & {$z$} & {716} & {1590} & {1993} & {0.798} |
| 439 |
> |
Prolate Ellipsoid & {$z$} & 0 & {1590} & {1993} & {0.798} |
| 440 |
|
\\ \hline \hline |
| 441 |
|
\label{table:couple} |
| 442 |
|
\end{longtable} |
