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\section{Introduction} |
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Interfacial thermal conductance is extensively studied both |
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experimentally and computationally\cite{cahill:793}, due to its |
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importance in nanoscale science and technology. Reliability of |
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nanoscale devices depends on their thermal transport |
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properties. Unlike bulk homogeneous materials, nanoscale materials |
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features significant presence of interfaces, and these interfaces |
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could dominate the heat transfer behavior of these |
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materials. Furthermore, these materials are generally heterogeneous, |
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which challenges traditional research methods for homogeneous |
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systems. |
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Due to the importance of heat flow in nanotechnology, interfacial |
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thermal conductance has been studied extensively both experimentally |
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and computationally.\cite{cahill:793} Unlike bulk materials, nanoscale |
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materials have a significant fraction of their atoms at interfaces, |
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and the chemical details of these interfaces govern the heat transfer |
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behavior. Furthermore, the interfaces are |
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heterogeneous (e.g. solid - liquid), which provides a challenge to |
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traditional methods developed for homogeneous systems. |
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|
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Heat conductance of molecular and nano-scale interfaces will be |
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affected by the chemical details of the surface. Experimentally, |
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various interfaces have been investigated for their thermal |
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conductance properties. Wang {\it et al.} studied heat transport |
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through long-chain hydrocarbon monolayers on gold substrate at |
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individual molecular level\cite{Wang10082007}; Schmidt {\it et al.} |
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studied the role of CTAB on thermal transport between gold nanorods |
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and solvent\cite{doi:10.1021/jp8051888}; Juv\'e {\it et al.} studied |
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Experimentally, various interfaces have been investigated for their |
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thermal conductance. Wang {\it et al.} studied heat transport through |
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long-chain hydrocarbon monolayers on gold substrate at individual |
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molecular level,\cite{Wang10082007} Schmidt {\it et al.} studied the |
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role of CTAB on thermal transport between gold nanorods and |
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solvent,\cite{doi:10.1021/jp8051888} and Juv\'e {\it et al.} studied |
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the cooling dynamics, which is controlled by thermal interface |
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resistence of glass-embedded metal |
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nanoparticles\cite{PhysRevB.80.195406}. Although interfaces are |
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commonly barriers for heat transport, Alper {\it et al.} suggested |
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that specific ligands (capping agents) could completely eliminate this |
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barrier ($G\rightarrow\infty$)\cite{doi:10.1021/la904855s}. |
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nanoparticles.\cite{PhysRevB.80.195406} Although interfaces are |
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normally considered barriers for heat transport, Alper {\it et al.} |
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suggested that specific ligands (capping agents) could completely |
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eliminate this barrier |
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($G\rightarrow\infty$).\cite{doi:10.1021/la904855s} |
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|
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Theoretical and computational models have also been used to study the |
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interfacial thermal transport in order to gain an understanding of |
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employed Non-Equilibrium Molecular Dynamics (NEMD) simulations to |
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study thermal transport from hot Au(111) substrate to a self-assembled |
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monolayer of alkylthiol with relatively long chain (8-20 carbon |
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atoms)\cite{hase:2010,hase:2011}. However, ensemble averaged |
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atoms).\cite{hase:2010,hase:2011} However, ensemble averaged |
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measurements for heat conductance of interfaces between the capping |
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monolayer on Au and a solvent phase has yet to be studied. |
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The comparatively low thermal flux through interfaces is |
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monolayer on Au and a solvent phase have yet to be studied with their |
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approach. The comparatively low thermal flux through interfaces is |
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difficult to measure with Equilibrium MD or forward NEMD simulation |
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methods. Therefore, the Reverse NEMD (RNEMD) methods would have the |
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advantage of having this difficult to measure flux known when studying |
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the thermal transport across interfaces, given that the simulation |
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methods being able to effectively apply an unphysical flux in |
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non-homogeneous systems. |
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methods. Therefore, the Reverse NEMD (RNEMD) |
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methods\cite{MullerPlathe:1997xw,kuang:164101} would have the |
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advantage of applying this difficult to measure flux (while measuring |
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the resulting gradient), given that the simulation methods being able |
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to effectively apply an unphysical flux in non-homogeneous systems. |
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Garde and coworkers\cite{garde:nl2005,garde:PhysRevLett2009} applied |
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this approach to various liquid interfaces and studied how thermal |
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conductance (or resistance) is dependent on chemistry details of |
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interfaces, e.g. hydrophobic and hydrophilic aqueous interfaces. |
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|
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Recently, we have developed the Non-Isotropic Velocity Scaling (NIVS) |
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Recently, we have developed a Non-Isotropic Velocity Scaling (NIVS) |
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algorithm for RNEMD simulations\cite{kuang:164101}. This algorithm |
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retains the desirable features of RNEMD (conservation of linear |
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momentum and total energy, compatibility with periodic boundary |
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properties. Different models were used for both the capping agent and |
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the solvent force field parameters. Using the NIVS algorithm, the |
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thermal transport across these interfaces was studied and the |
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underlying mechanism for this phenomena was investigated. |
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underlying mechanism for the phenomena was investigated. |
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|
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[MAY ADD WHY STUDY AU-THIOL SURFACE; CITE SHAOYI JIANG] |
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|
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\section{Methodology} |
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\subsection{Imposd-Flux Methods in MD Simulations} |
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For systems with low interfacial conductivity one must have a method |
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capable of generating relatively small fluxes, compared to those |
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required for bulk conductivity. This requirement makes the calculation |
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even more difficult for those slowly-converging equilibrium |
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methods\cite{Viscardy:2007lq}. |
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Forward methods impose gradient, but in interfacail conditions it is |
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not clear what behavior to impose at the boundary... |
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Imposed-flux reverse non-equilibrium |
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methods\cite{MullerPlathe:1997xw} have the flux set {\it a priori} and |
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the thermal response becomes easier to |
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measure than the flux. Although M\"{u}ller-Plathe's original momentum |
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swapping approach can be used for exchanging energy between particles |
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of different identity, the kinetic energy transfer efficiency is |
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affected by the mass difference between the particles, which limits |
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its application on heterogeneous interfacial systems. |
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Steady state MD simulations have an advantage in that not many |
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trajectories are needed to study the relationship between thermal flux |
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and thermal gradients. For systems with low interfacial conductance, |
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one must have a method capable of generating or measuring relatively |
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small fluxes, compared to those required for bulk conductivity. This |
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requirement makes the calculation even more difficult for |
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slowly-converging equilibrium methods.\cite{Viscardy:2007lq} Forward |
146 |
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NEMD methods impose a gradient (and measure a flux), but at interfaces |
147 |
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it is not clear what behavior should be imposed at the boundaries |
148 |
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between materials. Imposed-flux reverse non-equilibrium |
149 |
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methods\cite{MullerPlathe:1997xw} set the flux {\it a priori} and |
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the thermal response becomes an easy-to-measure quantity. Although |
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M\"{u}ller-Plathe's original momentum swapping approach can be used |
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for exchanging energy between particles of different identity, the |
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kinetic energy transfer efficiency is affected by the mass difference |
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between the particles, which limits its application on heterogeneous |
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interfacial systems. |
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|
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The non-isotropic velocity scaling (NIVS)\cite{kuang:164101} approach to |
158 |
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non-equilibrium MD simulations is able to impose a wide range of |
157 |
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The non-isotropic velocity scaling (NIVS) \cite{kuang:164101} approach |
158 |
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to non-equilibrium MD simulations is able to impose a wide range of |
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kinetic energy fluxes without obvious perturbation to the velocity |
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distributions of the simulated systems. Furthermore, this approach has |
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the advantage in heterogeneous interfaces in that kinetic energy flux |
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for computing thermal conductivities. The NIVS algorithm conserves |
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momenta and energy and does not depend on an external thermostat. |
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|
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\subsection{Defining Interfacial Thermal Conductivity $G$} |
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For interfaces with a relatively low interfacial conductance, the bulk |
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regions on either side of an interface rapidly come to a state in |
178 |
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which the two phases have relatively homogeneous (but distinct) |
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temperatures. The interfacial thermal conductivity $G$ can therefore |
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be approximated as: |
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\subsection{Defining Interfacial Thermal Conductivity ($G$)} |
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|
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For an interface with relatively low interfacial conductance, and a |
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thermal flux between two distinct bulk regions, the regions on either |
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> |
side of the interface rapidly come to a state in which the two phases |
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have relatively homogeneous (but distinct) temperatures. The |
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interfacial thermal conductivity $G$ can therefore be approximated as: |
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\begin{equation} |
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G = \frac{E_{total}}{2 t L_x L_y \left( \langle T_\mathrm{hot}\rangle - |
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G = \frac{E_{total}}{2 t L_x L_y \left( \langle T_\mathrm{hot}\rangle - |
184 |
|
\langle T_\mathrm{cold}\rangle \right)} |
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\label{lowG} |
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\end{equation} |
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where ${E_{total}}$ is the imposed non-physical kinetic energy |
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transfer and ${\langle T_\mathrm{hot}\rangle}$ and ${\langle |
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T_\mathrm{cold}\rangle}$ are the average observed temperature of the |
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two separated phases. |
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where ${E_{total}}$ is the total imposed non-physical kinetic energy |
188 |
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transfer during the simulation and ${\langle T_\mathrm{hot}\rangle}$ |
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> |
and ${\langle T_\mathrm{cold}\rangle}$ are the average observed |
190 |
> |
temperature of the two separated phases. |
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|
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When the interfacial conductance is {\it not} small, there are two |
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ways to define $G$. |
194 |
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|
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One way is to assume the temperature is discrete on the two sides of |
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the interface. $G$ can be calculated using the applied thermal flux |
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$J$ and the maximum temperature difference measured along the thermal |
196 |
< |
gradient max($\Delta T$), which occurs at the Gibbs deviding surface, |
197 |
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as: |
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> |
ways to define $G$. One common way is to assume the temperature is |
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> |
discrete on the two sides of the interface. $G$ can be calculated |
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> |
using the applied thermal flux $J$ and the maximum temperature |
196 |
> |
difference measured along the thermal gradient max($\Delta T$), which |
197 |
> |
occurs at the Gibbs deviding surface (Figure \ref{demoPic}): |
198 |
|
\begin{equation} |
199 |
< |
G=\frac{J}{\Delta T} |
199 |
> |
G=\frac{J}{\Delta T} |
200 |
|
\label{discreteG} |
201 |
|
\end{equation} |
202 |
|
|
203 |
+ |
\begin{figure} |
204 |
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\includegraphics[width=\linewidth]{method} |
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\caption{Interfacial conductance can be calculated by applying an |
206 |
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(unphysical) kinetic energy flux between two slabs, one located |
207 |
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within the metal and another on the edge of the periodic box. The |
208 |
+ |
system responds by forming a thermal response or a gradient. In |
209 |
+ |
bulk liquids, this gradient typically has a single slope, but in |
210 |
+ |
interfacial systems, there are distinct thermal conductivity |
211 |
+ |
domains. The interfacial conductance, $G$ is found by measuring the |
212 |
+ |
temperature gap at the Gibbs dividing surface, or by using second |
213 |
+ |
derivatives of the thermal profile.} |
214 |
+ |
\label{demoPic} |
215 |
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\end{figure} |
216 |
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|
217 |
|
The other approach is to assume a continuous temperature profile along |
218 |
|
the thermal gradient axis (e.g. $z$) and define $G$ at the point where |
219 |
< |
the magnitude of thermal conductivity $\lambda$ change reach its |
219 |
> |
the magnitude of thermal conductivity ($\lambda$) change reaches its |
220 |
|
maximum, given that $\lambda$ is well-defined throughout the space: |
221 |
|
\begin{equation} |
222 |
|
G^\prime = \Big|\frac{\partial\lambda}{\partial z}\Big| |
227 |
|
\label{derivativeG} |
228 |
|
\end{equation} |
229 |
|
|
230 |
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With the temperature profile obtained from simulations, one is able to |
230 |
> |
With temperature profiles obtained from simulation, one is able to |
231 |
|
approximate the first and second derivatives of $T$ with finite |
232 |
< |
difference methods and thus calculate $G^\prime$. |
232 |
> |
difference methods and calculate $G^\prime$. In what follows, both |
233 |
> |
definitions have been used, and are compared in the results. |
234 |
|
|
235 |
< |
In what follows, both definitions have been used for calculation and |
236 |
< |
are compared in the results. |
235 |
> |
To investigate the interfacial conductivity at metal / solvent |
236 |
> |
interfaces, we have modeled a metal slab with its (111) surfaces |
237 |
> |
perpendicular to the $z$-axis of our simulation cells. The metal slab |
238 |
> |
has been prepared both with and without capping agents on the exposed |
239 |
> |
surface, and has been solvated with simple organic solvents, as |
240 |
> |
illustrated in Figure \ref{gradT}. |
241 |
|
|
223 |
– |
To compare the above definitions ($G$ and $G^\prime$), we have modeled |
224 |
– |
a metal slab with its (111) surfaces perpendicular to the $z$-axis of |
225 |
– |
our simulation cells. Both with and withour capping agents on the |
226 |
– |
surfaces, the metal slab is solvated with simple organic solvents, as |
227 |
– |
illustrated in Figure \ref{demoPic}. |
228 |
– |
|
229 |
– |
\begin{figure} |
230 |
– |
\includegraphics[width=\linewidth]{method} |
231 |
– |
\caption{Interfacial conductance can be calculated by applying an |
232 |
– |
(unphysical) kinetic energy flux between two slabs, one located |
233 |
– |
within the metal and another on the edge of the periodic box. The |
234 |
– |
system responds by forming a thermal response or a gradient. In |
235 |
– |
bulk liquids, this gradient typically has a single slope, but in |
236 |
– |
interfacial systems, there are distinct thermal conductivity |
237 |
– |
domains. The interfacial conductance, $G$ is found by measuring the |
238 |
– |
temperature gap at the Gibbs dividing surface, or by using second |
239 |
– |
derivatives of the thermal profile.} |
240 |
– |
\label{demoPic} |
241 |
– |
\end{figure} |
242 |
– |
|
242 |
|
With the simulation cell described above, we are able to equilibrate |
243 |
|
the system and impose an unphysical thermal flux between the liquid |
244 |
|
and the metal phase using the NIVS algorithm. By periodically applying |
245 |
< |
the unphysical flux, we are able to obtain a temperature profile and |
246 |
< |
its spatial derivatives. These quantities enable the evaluation of the |
247 |
< |
interfacial thermal conductance of a surface. Figure \ref{gradT} is an |
248 |
< |
example how those applied thermal fluxes can be used to obtain the 1st |
250 |
< |
and 2nd derivatives of the temperature profile. |
245 |
> |
the unphysical flux, we obtained a temperature profile and its spatial |
246 |
> |
derivatives. Figure \ref{gradT} shows how an applied thermal flux can |
247 |
> |
be used to obtain the 1st and 2nd derivatives of the temperature |
248 |
> |
profile. |
249 |
|
|
250 |
|
\begin{figure} |
251 |
|
\includegraphics[width=\linewidth]{gradT} |
252 |
< |
\caption{The 1st and 2nd derivatives of temperature profile can be |
253 |
< |
obtained with finite difference approximation.} |
252 |
> |
\caption{A sample of Au-butanethiol/hexane interfacial system and the |
253 |
> |
temperature profile after a kinetic energy flux is imposed to |
254 |
> |
it. The 1st and 2nd derivatives of the temperature profile can be |
255 |
> |
obtained with finite difference approximation (lower panel).} |
256 |
|
\label{gradT} |
257 |
|
\end{figure} |
258 |
|
|
259 |
|
\section{Computational Details} |
260 |
|
\subsection{Simulation Protocol} |
261 |
|
The NIVS algorithm has been implemented in our MD simulation code, |
262 |
< |
OpenMD\cite{Meineke:2005gd,openmd}, and was used for our |
263 |
< |
simulations. Different slab thickness (layer numbers of Au) were |
264 |
< |
simulated. Metal slabs were first equilibrated under atmospheric |
265 |
< |
pressure (1 atm) and a desired temperature (e.g. 200K). After |
266 |
< |
equilibration, butanethiol capping agents were placed at three-fold |
267 |
< |
sites on the Au(111) surfaces. The maximum butanethiol capacity on Au |
268 |
< |
surface is $1/3$ of the total number of surface Au |
269 |
< |
atoms\cite{vlugt:cpc2007154}. A series of different coverages was |
270 |
< |
investigated in order to study the relation between coverage and |
271 |
< |
interfacial conductance. |
262 |
> |
OpenMD\cite{Meineke:2005gd,openmd}, and was used for our simulations. |
263 |
> |
Metal slabs of 6 or 11 layers of Au atoms were first equilibrated |
264 |
> |
under atmospheric pressure (1 atm) and 200K. After equilibration, |
265 |
> |
butanethiol capping agents were placed at three-fold hollow sites on |
266 |
> |
the Au(111) surfaces. These sites are either {\it fcc} or {\it |
267 |
> |
hcp} sites, although Hase {\it et al.} found that they are |
268 |
> |
equivalent in a heat transfer process,\cite{hase:2010} so we did not |
269 |
> |
distinguish between these sites in our study. The maximum butanethiol |
270 |
> |
capacity on Au surface is $1/3$ of the total number of surface Au |
271 |
> |
atoms, and the packing forms a $(\sqrt{3}\times\sqrt{3})R30^\circ$ |
272 |
> |
structure\cite{doi:10.1021/ja00008a001,doi:10.1021/cr9801317}. A |
273 |
> |
series of lower coverages was also prepared by eliminating |
274 |
> |
butanethiols from the higher coverage surface in a regular manner. The |
275 |
> |
lower coverages were prepared in order to study the relation between |
276 |
> |
coverage and interfacial conductance. |
277 |
|
|
278 |
|
The capping agent molecules were allowed to migrate during the |
279 |
|
simulations. They distributed themselves uniformly and sampled a |
280 |
|
number of three-fold sites throughout out study. Therefore, the |
281 |
< |
initial configuration would not noticeably affect the sampling of a |
281 |
> |
initial configuration does not noticeably affect the sampling of a |
282 |
|
variety of configurations of the same coverage, and the final |
283 |
|
conductance measurement would be an average effect of these |
284 |
< |
configurations explored in the simulations. [MAY NEED FIGURES] |
284 |
> |
configurations explored in the simulations. |
285 |
|
|
286 |
< |
After the modified Au-butanethiol surface systems were equilibrated |
287 |
< |
under canonical ensemble, organic solvent molecules were packed in the |
288 |
< |
previously empty part of the simulation cells\cite{packmol}. Two |
286 |
> |
After the modified Au-butanethiol surface systems were equilibrated in |
287 |
> |
the canonical (NVT) ensemble, organic solvent molecules were packed in |
288 |
> |
the previously empty part of the simulation cells.\cite{packmol} Two |
289 |
|
solvents were investigated, one which has little vibrational overlap |
290 |
< |
with the alkanethiol and a planar shape (toluene), and one which has |
291 |
< |
similar vibrational frequencies and chain-like shape ({\it n}-hexane). |
290 |
> |
with the alkanethiol and which has a planar shape (toluene), and one |
291 |
> |
which has similar vibrational frequencies to the capping agent and |
292 |
> |
chain-like shape ({\it n}-hexane). |
293 |
|
|
294 |
< |
The space filled by solvent molecules, i.e. the gap between |
295 |
< |
periodically repeated Au-butanethiol surfaces should be carefully |
296 |
< |
chosen. A very long length scale for the thermal gradient axis ($z$) |
291 |
< |
may cause excessively hot or cold temperatures in the middle of the |
294 |
> |
The simulation cells were not particularly extensive along the |
295 |
> |
$z$-axis, as a very long length scale for the thermal gradient may |
296 |
> |
cause excessively hot or cold temperatures in the middle of the |
297 |
|
solvent region and lead to undesired phenomena such as solvent boiling |
298 |
|
or freezing when a thermal flux is applied. Conversely, too few |
299 |
|
solvent molecules would change the normal behavior of the liquid |
300 |
|
phase. Therefore, our $N_{solvent}$ values were chosen to ensure that |
301 |
< |
these extreme cases did not happen to our simulations. And the |
302 |
< |
corresponding spacing is usually $35 \sim 60$\AA. |
301 |
> |
these extreme cases did not happen to our simulations. The spacing |
302 |
> |
between periodic images of the gold interfaces is $45 \sim 75$\AA. |
303 |
|
|
304 |
< |
The initial configurations generated by Packmol are further |
305 |
< |
equilibrated with the $x$ and $y$ dimensions fixed, only allowing |
306 |
< |
length scale change in $z$ dimension. This is to ensure that the |
307 |
< |
equilibration of liquid phase does not affect the metal crystal |
308 |
< |
structure in $x$ and $y$ dimensions. Further equilibration are run |
309 |
< |
under NVT and then NVE ensembles. |
304 |
> |
The initial configurations generated are further equilibrated with the |
305 |
> |
$x$ and $y$ dimensions fixed, only allowing the $z$-length scale to |
306 |
> |
change. This is to ensure that the equilibration of liquid phase does |
307 |
> |
not affect the metal's crystalline structure. Comparisons were made |
308 |
> |
with simulations that allowed changes of $L_x$ and $L_y$ during NPT |
309 |
> |
equilibration. No substantial changes in the box geometry were noticed |
310 |
> |
in these simulations. After ensuring the liquid phase reaches |
311 |
> |
equilibrium at atmospheric pressure (1 atm), further equilibration was |
312 |
> |
carried out under canonical (NVT) and microcanonical (NVE) ensembles. |
313 |
|
|
314 |
< |
After the systems reach equilibrium, NIVS is implemented to impose a |
315 |
< |
periodic unphysical thermal flux between the metal and the liquid |
316 |
< |
phase. Most of our simulations are under an average temperature of |
317 |
< |
$\sim$200K. Therefore, this flux usually comes from the metal to the |
314 |
> |
After the systems reach equilibrium, NIVS was used to impose an |
315 |
> |
unphysical thermal flux between the metal and the liquid phases. Most |
316 |
> |
of our simulations were done under an average temperature of |
317 |
> |
$\sim$200K. Therefore, thermal flux usually came from the metal to the |
318 |
|
liquid so that the liquid has a higher temperature and would not |
319 |
< |
freeze due to excessively low temperature. This induced temperature |
320 |
< |
gradient is stablized and the simulation cell is devided evenly into |
321 |
< |
N slabs along the $z$-axis and the temperatures of each slab are |
322 |
< |
recorded. When the slab width $d$ of each slab is the same, the |
323 |
< |
derivatives of $T$ with respect to slab number $n$ can be directly |
324 |
< |
used for $G^\prime$ calculations: |
325 |
< |
\begin{equation} |
326 |
< |
G^\prime = |J_z|\Big|\frac{\partial^2 T}{\partial z^2}\Big| |
319 |
> |
freeze due to lowered temperatures. After this induced temperature |
320 |
> |
gradient had stablized, the temperature profile of the simulation cell |
321 |
> |
was recorded. To do this, the simulation cell is devided evenly into |
322 |
> |
$N$ slabs along the $z$-axis. The average temperatures of each slab |
323 |
> |
are recorded for 1$\sim$2 ns. When the slab width $d$ of each slab is |
324 |
> |
the same, the derivatives of $T$ with respect to slab number $n$ can |
325 |
> |
be directly used for $G^\prime$ calculations: \begin{equation} |
326 |
> |
G^\prime = |J_z|\Big|\frac{\partial^2 T}{\partial z^2}\Big| |
327 |
|
\Big/\left(\frac{\partial T}{\partial z}\right)^2 |
328 |
|
= |J_z|\Big|\frac{1}{d^2}\frac{\partial^2 T}{\partial n^2}\Big| |
329 |
|
\Big/\left(\frac{1}{d}\frac{\partial T}{\partial n}\right)^2 |
332 |
|
\label{derivativeG2} |
333 |
|
\end{equation} |
334 |
|
|
335 |
+ |
All of the above simulation procedures use a time step of 1 fs. Each |
336 |
+ |
equilibration stage took a minimum of 100 ps, although in some cases, |
337 |
+ |
longer equilibration stages were utilized. |
338 |
+ |
|
339 |
|
\subsection{Force Field Parameters} |
340 |
< |
Our simulations include various components. Therefore, force field |
341 |
< |
parameter descriptions are needed for interactions both between the |
342 |
< |
same type of particles and between particles of different species. |
340 |
> |
Our simulations include a number of chemically distinct components. |
341 |
> |
Figure \ref{demoMol} demonstrates the sites defined for both |
342 |
> |
United-Atom and All-Atom models of the organic solvent and capping |
343 |
> |
agents in our simulations. Force field parameters are needed for |
344 |
> |
interactions both between the same type of particles and between |
345 |
> |
particles of different species. |
346 |
|
|
332 |
– |
The Au-Au interactions in metal lattice slab is described by the |
333 |
– |
quantum Sutton-Chen (QSC) formulation\cite{PhysRevB.59.3527}. The QSC |
334 |
– |
potentials include zero-point quantum corrections and are |
335 |
– |
reparametrized for accurate surface energies compared to the |
336 |
– |
Sutton-Chen potentials\cite{Chen90}. |
337 |
– |
|
338 |
– |
Figure \ref{demoMol} demonstrates how we name our pseudo-atoms of the |
339 |
– |
organic solvent molecules in our simulations. |
340 |
– |
|
347 |
|
\begin{figure} |
348 |
|
\includegraphics[width=\linewidth]{structures} |
349 |
|
\caption{Structures of the capping agent and solvents utilized in |
350 |
|
these simulations. The chemically-distinct sites (a-e) are expanded |
351 |
|
in terms of constituent atoms for both United Atom (UA) and All Atom |
352 |
|
(AA) force fields. Most parameters are from |
353 |
< |
Refs. \protect\cite{TraPPE-UA.alkanes,TraPPE-UA.alkylbenzenes} (UA) and |
348 |
< |
\protect\cite{OPLSAA} (AA). Cross-interactions with the Au atoms are given |
349 |
< |
in Table \ref{MnM}.} |
353 |
> |
Refs. \protect\cite{TraPPE-UA.alkanes,TraPPE-UA.alkylbenzenes,TraPPE-UA.thiols} (UA) and \protect\cite{OPLSAA} (AA). Cross-interactions with the Au atoms are given in Table \ref{MnM}.} |
354 |
|
\label{demoMol} |
355 |
|
\end{figure} |
356 |
|
|
357 |
< |
For both solvent molecules, straight chain {\it n}-hexane and aromatic |
358 |
< |
toluene, United-Atom (UA) and All-Atom (AA) models are used |
359 |
< |
respectively. The TraPPE-UA |
357 |
> |
The Au-Au interactions in metal lattice slab is described by the |
358 |
> |
quantum Sutton-Chen (QSC) formulation.\cite{PhysRevB.59.3527} The QSC |
359 |
> |
potentials include zero-point quantum corrections and are |
360 |
> |
reparametrized for accurate surface energies compared to the |
361 |
> |
Sutton-Chen potentials.\cite{Chen90} |
362 |
> |
|
363 |
> |
For the two solvent molecules, {\it n}-hexane and toluene, two |
364 |
> |
different atomistic models were utilized. Both solvents were modeled |
365 |
> |
using United-Atom (UA) and All-Atom (AA) models. The TraPPE-UA |
366 |
|
parameters\cite{TraPPE-UA.alkanes,TraPPE-UA.alkylbenzenes} are used |
367 |
< |
for our UA solvent molecules. In these models, pseudo-atoms are |
368 |
< |
located at the carbon centers for alkyl groups. By eliminating |
369 |
< |
explicit hydrogen atoms, these models are simple and computationally |
370 |
< |
efficient, while maintains good accuracy. However, the TraPPE-UA for |
371 |
< |
alkanes is known to predict a lower boiling point than experimental |
362 |
< |
values. Considering that after an unphysical thermal flux is applied |
363 |
< |
to a system, the temperature of ``hot'' area in the liquid phase would be |
364 |
< |
significantly higher than the average, to prevent over heating and |
365 |
< |
boiling of the liquid phase, the average temperature in our |
366 |
< |
simulations should be much lower than the liquid boiling point. [MORE DISCUSSION] |
367 |
< |
For UA-toluene model, rigid body constraints are applied, so that the |
368 |
< |
benzene ring and the methyl-CRar bond are kept rigid. This would save |
369 |
< |
computational time.[MORE DETAILS] |
367 |
> |
for our UA solvent molecules. In these models, sites are located at |
368 |
> |
the carbon centers for alkyl groups. Bonding interactions, including |
369 |
> |
bond stretches and bends and torsions, were used for intra-molecular |
370 |
> |
sites closer than 3 bonds. For non-bonded interactions, Lennard-Jones |
371 |
> |
potentials are used. |
372 |
|
|
373 |
+ |
By eliminating explicit hydrogen atoms, the TraPPE-UA models are |
374 |
+ |
simple and computationally efficient, while maintaining good accuracy. |
375 |
+ |
However, the TraPPE-UA model for alkanes is known to predict a slighly |
376 |
+ |
lower boiling point than experimental values. This is one of the |
377 |
+ |
reasons we used a lower average temperature (200K) for our |
378 |
+ |
simulations. If heat is transferred to the liquid phase during the |
379 |
+ |
NIVS simulation, the liquid in the hot slab can actually be |
380 |
+ |
substantially warmer than the mean temperature in the simulation. The |
381 |
+ |
lower mean temperatures therefore prevent solvent boiling. |
382 |
+ |
|
383 |
+ |
For UA-toluene, the non-bonded potentials between intermolecular sites |
384 |
+ |
have a similar Lennard-Jones formulation. The toluene molecules were |
385 |
+ |
treated as a single rigid body, so there was no need for |
386 |
+ |
intramolecular interactions (including bonds, bends, or torsions) in |
387 |
+ |
this solvent model. |
388 |
+ |
|
389 |
|
Besides the TraPPE-UA models, AA models for both organic solvents are |
390 |
< |
included in our studies as well. For hexane, the OPLS-AA\cite{OPLSAA} |
391 |
< |
force field is used. [MORE DETAILS] |
392 |
< |
For toluene, the United Force Field developed by Rapp\'{e} {\it et |
393 |
< |
al.}\cite{doi:10.1021/ja00051a040} is adopted.[MORE DETAILS] |
390 |
> |
included in our studies as well. The OPLS-AA\cite{OPLSAA} force fields |
391 |
> |
were used. For hexane, additional explicit hydrogen sites were |
392 |
> |
included. Besides bonding and non-bonded site-site interactions, |
393 |
> |
partial charges and the electrostatic interactions were added to each |
394 |
> |
CT and HC site. For toluene, a flexible model for the toluene molecule |
395 |
> |
was utilized which included bond, bend, torsion, and inversion |
396 |
> |
potentials to enforce ring planarity. |
397 |
|
|
398 |
< |
The capping agent in our simulations, the butanethiol molecules can |
399 |
< |
either use UA or AA model. The TraPPE-UA force fields includes |
398 |
> |
The butanethiol capping agent in our simulations, were also modeled |
399 |
> |
with both UA and AA model. The TraPPE-UA force field includes |
400 |
|
parameters for thiol molecules\cite{TraPPE-UA.thiols} and are used for |
401 |
|
UA butanethiol model in our simulations. The OPLS-AA also provides |
402 |
|
parameters for alkyl thiols. However, alkyl thiols adsorbed on Au(111) |
403 |
< |
surfaces do not have the hydrogen atom bonded to sulfur. To adapt this |
404 |
< |
change and derive suitable parameters for butanethiol adsorbed on |
405 |
< |
Au(111) surfaces, we adopt the S parameters from Luedtke and |
406 |
< |
Landman\cite{landman:1998} and modify parameters for its neighbor C |
407 |
< |
atom for charge balance in the molecule. Note that the model choice |
408 |
< |
(UA or AA) of capping agent can be different from the |
409 |
< |
solvent. Regardless of model choice, the force field parameters for |
410 |
< |
interactions between capping agent and solvent can be derived using |
390 |
< |
Lorentz-Berthelot Mixing Rule: |
403 |
> |
surfaces do not have the hydrogen atom bonded to sulfur. To derive |
404 |
> |
suitable parameters for butanethiol adsorbed on Au(111) surfaces, we |
405 |
> |
adopt the S parameters from Luedtke and Landman\cite{landman:1998} and |
406 |
> |
modify the parameters for the CTS atom to maintain charge neutrality |
407 |
> |
in the molecule. Note that the model choice (UA or AA) for the capping |
408 |
> |
agent can be different from the solvent. Regardless of model choice, |
409 |
> |
the force field parameters for interactions between capping agent and |
410 |
> |
solvent can be derived using Lorentz-Berthelot Mixing Rule: |
411 |
|
\begin{eqnarray} |
412 |
< |
\sigma_{ij} & = & \frac{1}{2} \left(\sigma_{ii} + \sigma_{jj}\right) \\ |
413 |
< |
\epsilon_{ij} & = & \sqrt{\epsilon_{ii}\epsilon_{jj}} |
412 |
> |
\sigma_{ij} & = & \frac{1}{2} \left(\sigma_{ii} + \sigma_{jj}\right) \\ |
413 |
> |
\epsilon_{ij} & = & \sqrt{\epsilon_{ii}\epsilon_{jj}} |
414 |
|
\end{eqnarray} |
415 |
|
|
416 |
< |
To describe the interactions between metal Au and non-metal capping |
417 |
< |
agent and solvent particles, we refer to an adsorption study of alkyl |
418 |
< |
thiols on gold surfaces by Vlugt {\it et |
419 |
< |
al.}\cite{vlugt:cpc2007154} They fitted an effective Lennard-Jones |
420 |
< |
form of potential parameters for the interaction between Au and |
421 |
< |
pseudo-atoms CH$_x$ and S based on a well-established and widely-used |
422 |
< |
effective potential of Hautman and Klein\cite{hautman:4994} for the |
423 |
< |
Au(111) surface. As our simulations require the gold lattice slab to |
424 |
< |
be non-rigid so that it could accommodate kinetic energy for thermal |
405 |
< |
transport study purpose, the pair-wise form of potentials is |
406 |
< |
preferred. |
416 |
> |
To describe the interactions between metal (Au) and non-metal atoms, |
417 |
> |
we refer to an adsorption study of alkyl thiols on gold surfaces by |
418 |
> |
Vlugt {\it et al.}\cite{vlugt:cpc2007154} They fitted an effective |
419 |
> |
Lennard-Jones form of potential parameters for the interaction between |
420 |
> |
Au and pseudo-atoms CH$_x$ and S based on a well-established and |
421 |
> |
widely-used effective potential of Hautman and Klein for the Au(111) |
422 |
> |
surface.\cite{hautman:4994} As our simulations require the gold slab |
423 |
> |
to be flexible to accommodate thermal excitation, the pair-wise form |
424 |
> |
of potentials they developed was used for our study. |
425 |
|
|
426 |
< |
Besides, the potentials developed from {\it ab initio} calculations by |
427 |
< |
Leng {\it et al.}\cite{doi:10.1021/jp034405s} are adopted for the |
428 |
< |
interactions between Au and aromatic C/H atoms in toluene.[MORE DETAILS] |
426 |
> |
The potentials developed from {\it ab initio} calculations by Leng |
427 |
> |
{\it et al.}\cite{doi:10.1021/jp034405s} are adopted for the |
428 |
> |
interactions between Au and aromatic C/H atoms in toluene. However, |
429 |
> |
the Lennard-Jones parameters between Au and other types of particles, |
430 |
> |
(e.g. AA alkanes) have not yet been established. For these |
431 |
> |
interactions, the Lorentz-Berthelot mixing rule can be used to derive |
432 |
> |
effective single-atom LJ parameters for the metal using the fit values |
433 |
> |
for toluene. These are then used to construct reasonable mixing |
434 |
> |
parameters for the interactions between the gold and other atoms. |
435 |
> |
Table \ref{MnM} summarizes the ``metal/non-metal'' parameters used in |
436 |
> |
our simulations. |
437 |
|
|
412 |
– |
However, the Lennard-Jones parameters between Au and other types of |
413 |
– |
particles in our simulations are not yet well-established. For these |
414 |
– |
interactions, we attempt to derive their parameters using the Mixing |
415 |
– |
Rule. To do this, the ``Metal-non-Metal'' (MnM) interaction parameters |
416 |
– |
for Au is first extracted from the Au-CH$_x$ parameters by applying |
417 |
– |
the Mixing Rule reversely. Table \ref{MnM} summarizes these ``MnM'' |
418 |
– |
parameters in our simulations. |
419 |
– |
|
438 |
|
\begin{table*} |
439 |
|
\begin{minipage}{\linewidth} |
440 |
|
\begin{center} |
461 |
|
&CA & 3.55 & 0.070 & -0.115 & 3.173 & 0.0640 \\ |
462 |
|
&HA & 2.42 & 0.030 & 0.115 & 2.746 & 0.0414 \\ |
463 |
|
\hline |
464 |
< |
Both UA and AA & S & 4.45 & 0.25 & - & 2.40 & 8.465 \\ |
464 |
> |
Both UA and AA |
465 |
> |
& S & 4.45 & 0.25 & - & 2.40 & 8.465 \\ |
466 |
|
\hline\hline |
467 |
|
\end{tabular} |
468 |
|
\label{MnM} |
470 |
|
\end{minipage} |
471 |
|
\end{table*} |
472 |
|
|
473 |
+ |
\subsection{Vibrational Power Spectrum} |
474 |
|
|
475 |
+ |
To investigate the mechanism of interfacial thermal conductance, the |
476 |
+ |
vibrational power spectrum was computed. Power spectra were taken for |
477 |
+ |
individual components in different simulations. To obtain these |
478 |
+ |
spectra, simulations were run after equilibration, in the NVE |
479 |
+ |
ensemble, and without a thermal gradient. Snapshots of configurations |
480 |
+ |
were collected at a frequency that is higher than that of the fastest |
481 |
+ |
vibrations occuring in the simulations. With these configurations, the |
482 |
+ |
velocity auto-correlation functions can be computed: |
483 |
+ |
\begin{equation} |
484 |
+ |
C_A (t) = \langle\vec{v}_A (t)\cdot\vec{v}_A (0)\rangle |
485 |
+ |
\label{vCorr} |
486 |
+ |
\end{equation} |
487 |
+ |
The power spectrum is constructed via a Fourier transform of the |
488 |
+ |
symmetrized velocity autocorrelation function, |
489 |
+ |
\begin{equation} |
490 |
+ |
\hat{f}(\omega) = |
491 |
+ |
\int_{-\infty}^{\infty} C_A (t) e^{-2\pi it\omega}\,dt |
492 |
+ |
\label{fourier} |
493 |
+ |
\end{equation} |
494 |
+ |
|
495 |
|
\section{Results and Discussions} |
496 |
< |
[MAY HAVE A BRIEF SUMMARY] |
496 |
> |
In what follows, how the parameters and protocol of simulations would |
497 |
> |
affect the measurement of $G$'s is first discussed. With a reliable |
498 |
> |
protocol and set of parameters, the influence of capping agent |
499 |
> |
coverage on thermal conductance is investigated. Besides, different |
500 |
> |
force field models for both solvents and selected deuterated models |
501 |
> |
were tested and compared. Finally, a summary of the role of capping |
502 |
> |
agent in the interfacial thermal transport process is given. |
503 |
> |
|
504 |
|
\subsection{How Simulation Parameters Affects $G$} |
458 |
– |
[MAY NOT PUT AT FIRST] |
505 |
|
We have varied our protocol or other parameters of the simulations in |
506 |
|
order to investigate how these factors would affect the measurement of |
507 |
|
$G$'s. It turned out that while some of these parameters would not |
510 |
|
results. |
511 |
|
|
512 |
|
In some of our simulations, we allowed $L_x$ and $L_y$ to change |
513 |
< |
during equilibrating the liquid phase. Due to the stiffness of the Au |
514 |
< |
slab, $L_x$ and $L_y$ would not change noticeably after |
515 |
< |
equilibration. Although $L_z$ could fluctuate $\sim$1\% after a system |
516 |
< |
is fully equilibrated in the NPT ensemble, this fluctuation, as well |
517 |
< |
as those comparably smaller to $L_x$ and $L_y$, would not be magnified |
518 |
< |
on the calculated $G$'s, as shown in Table \ref{AuThiolHexaneUA}. This |
519 |
< |
insensivity to $L_i$ fluctuations allows reliable measurement of $G$'s |
520 |
< |
without the necessity of extremely cautious equilibration process. |
513 |
> |
during equilibrating the liquid phase. Due to the stiffness of the |
514 |
> |
crystalline Au structure, $L_x$ and $L_y$ would not change noticeably |
515 |
> |
after equilibration. Although $L_z$ could fluctuate $\sim$1\% after a |
516 |
> |
system is fully equilibrated in the NPT ensemble, this fluctuation, as |
517 |
> |
well as those of $L_x$ and $L_y$ (which is significantly smaller), |
518 |
> |
would not be magnified on the calculated $G$'s, as shown in Table |
519 |
> |
\ref{AuThiolHexaneUA}. This insensivity to $L_i$ fluctuations allows |
520 |
> |
reliable measurement of $G$'s without the necessity of extremely |
521 |
> |
cautious equilibration process. |
522 |
|
|
523 |
|
As stated in our computational details, the spacing filled with |
524 |
|
solvent molecules can be chosen within a range. This allows some |
545 |
|
the thermal flux across the interface. For our simulations, we denote |
546 |
|
$J_z$ to be positive when the physical thermal flux is from the liquid |
547 |
|
to metal, and negative vice versa. The $G$'s measured under different |
548 |
< |
$J_z$ is listed in Table \ref{AuThiolHexaneUA} and [REF]. These |
549 |
< |
results do not suggest that $G$ is dependent on $J_z$ within this flux |
550 |
< |
range. The linear response of flux to thermal gradient simplifies our |
551 |
< |
investigations in that we can rely on $G$ measurement with only a |
552 |
< |
couple $J_z$'s and do not need to test a large series of fluxes. |
548 |
> |
$J_z$ is listed in Table \ref{AuThiolHexaneUA} and |
549 |
> |
\ref{AuThiolToluene}. These results do not suggest that $G$ is |
550 |
> |
dependent on $J_z$ within this flux range. The linear response of flux |
551 |
> |
to thermal gradient simplifies our investigations in that we can rely |
552 |
> |
on $G$ measurement with only a couple $J_z$'s and do not need to test |
553 |
> |
a large series of fluxes. |
554 |
|
|
507 |
– |
%ADD MORE TO TABLE |
555 |
|
\begin{table*} |
556 |
|
\begin{minipage}{\linewidth} |
557 |
|
\begin{center} |
558 |
|
\caption{Computed interfacial thermal conductivity ($G$ and |
559 |
|
$G^\prime$) values for the 100\% covered Au-butanethiol/hexane |
560 |
|
interfaces with UA model and different hexane molecule numbers |
561 |
< |
at different temperatures using a range of energy fluxes.} |
561 |
> |
at different temperatures using a range of energy |
562 |
> |
fluxes. Error estimates indicated in parenthesis.} |
563 |
|
|
564 |
|
\begin{tabular}{ccccccc} |
565 |
|
\hline\hline |
568 |
|
(K) & & $L_x$ \& $L_y$? & (g/cm$^3$) & (GW/m$^2$) & |
569 |
|
\multicolumn{2}{c}{(MW/m$^2$/K)} \\ |
570 |
|
\hline |
571 |
< |
200 & 266 & No & 0.672 & -0.96 & 102() & 80.0() \\ |
572 |
< |
& 200 & Yes & 0.694 & 1.92 & 129() & 87.3() \\ |
573 |
< |
& & Yes & 0.672 & 1.93 & 131() & 77.5() \\ |
574 |
< |
& & No & 0.688 & 0.96 & 125() & 90.2() \\ |
575 |
< |
& & & & 1.91 & 139() & 101() \\ |
576 |
< |
& & & & 2.83 & 141() & 89.9() \\ |
577 |
< |
& 166 & Yes & 0.679 & 0.97 & 115() & 69.3() \\ |
578 |
< |
& & & & 1.94 & 125() & 87.1() \\ |
579 |
< |
& & No & 0.681 & 0.97 & 141() & 77.7() \\ |
580 |
< |
& & & & 1.92 & 138() & 98.9() \\ |
571 |
> |
200 & 266 & No & 0.672 & -0.96 & 102(3) & 80.0(0.8) \\ |
572 |
> |
& 200 & Yes & 0.694 & 1.92 & 129(11) & 87.3(0.3) \\ |
573 |
> |
& & Yes & 0.672 & 1.93 & 131(16) & 78(13) \\ |
574 |
> |
& & No & 0.688 & 0.96 & 125(16) & 90.2(15) \\ |
575 |
> |
& & & & 1.91 & 139(10) & 101(10) \\ |
576 |
> |
& & & & 2.83 & 141(6) & 89.9(9.8) \\ |
577 |
> |
& 166 & Yes & 0.679 & 0.97 & 115(19) & 69(18) \\ |
578 |
> |
& & & & 1.94 & 125(9) & 87.1(0.2) \\ |
579 |
> |
& & No & 0.681 & 0.97 & 141(30) & 78(22) \\ |
580 |
> |
& & & & 1.92 & 138(4) & 98.9(9.5) \\ |
581 |
|
\hline |
582 |
< |
250 & 200 & No & 0.560 & 0.96 & 74.8() & 61.8() \\ |
583 |
< |
& & & & -0.95 & 49.4() & 45.7() \\ |
584 |
< |
& 166 & Yes & 0.570 & 0.98 & 79.0() & 62.9() \\ |
585 |
< |
& & No & 0.569 & 0.97 & 80.3() & 67.1() \\ |
586 |
< |
& & & & 1.44 & 76.2() & 64.8() \\ |
587 |
< |
& & & & -0.95 & 56.4() & 54.4() \\ |
588 |
< |
& & & & -1.85 & 47.8() & 53.5() \\ |
582 |
> |
250 & 200 & No & 0.560 & 0.96 & 75(10) & 61.8(7.3) \\ |
583 |
> |
& & & & -0.95 & 49.4(0.3) & 45.7(2.1) \\ |
584 |
> |
& 166 & Yes & 0.570 & 0.98 & 79.0(3.5) & 62.9(3.0) \\ |
585 |
> |
& & No & 0.569 & 0.97 & 80.3(0.6) & 67(11) \\ |
586 |
> |
& & & & 1.44 & 76.2(5.0) & 64.8(3.8) \\ |
587 |
> |
& & & & -0.95 & 56.4(2.5) & 54.4(1.1) \\ |
588 |
> |
& & & & -1.85 & 47.8(1.1) & 53.5(1.5) \\ |
589 |
|
\hline\hline |
590 |
|
\end{tabular} |
591 |
|
\label{AuThiolHexaneUA} |
601 |
|
temperature is higher than 250K. Additionally, the equilibrated liquid |
602 |
|
hexane density under 250K becomes lower than experimental value. This |
603 |
|
expanded liquid phase leads to lower contact between hexane and |
604 |
< |
butanethiol as well.[MAY NEED FIGURE] And this reduced contact would |
604 |
> |
butanethiol as well.[MAY NEED SLAB DENSITY FIGURE] |
605 |
> |
And this reduced contact would |
606 |
|
probably be accountable for a lower interfacial thermal conductance, |
607 |
|
as shown in Table \ref{AuThiolHexaneUA}. |
608 |
|
|
617 |
|
important role in the thermal transport process across the interface |
618 |
|
in that higher degree of contact could yield increased conductance. |
619 |
|
|
571 |
– |
[ADD ERROR ESTIMATE TO TABLE] |
620 |
|
\begin{table*} |
621 |
|
\begin{minipage}{\linewidth} |
622 |
|
\begin{center} |
623 |
|
\caption{Computed interfacial thermal conductivity ($G$ and |
624 |
|
$G^\prime$) values for a 90\% coverage Au-butanethiol/toluene |
625 |
|
interface at different temperatures using a range of energy |
626 |
< |
fluxes.} |
626 |
> |
fluxes. Error estimates indicated in parenthesis.} |
627 |
|
|
628 |
|
\begin{tabular}{ccccc} |
629 |
|
\hline\hline |
630 |
|
$\langle T\rangle$ & $\rho_{toluene}$ & $J_z$ & $G$ & $G^\prime$ \\ |
631 |
|
(K) & (g/cm$^3$) & (GW/m$^2$) & \multicolumn{2}{c}{(MW/m$^2$/K)} \\ |
632 |
|
\hline |
633 |
< |
200 & 0.933 & -1.86 & 180() & 135() \\ |
634 |
< |
& & 2.15 & 204() & 113() \\ |
635 |
< |
& & -3.93 & 175() & 114() \\ |
633 |
> |
200 & 0.933 & 2.15 & 204(12) & 113(12) \\ |
634 |
> |
& & -1.86 & 180(3) & 135(21) \\ |
635 |
> |
& & -3.93 & 176(5) & 113(12) \\ |
636 |
|
\hline |
637 |
< |
300 & 0.855 & -1.91 & 143() & 125() \\ |
638 |
< |
& & -4.19 & 134() & 113() \\ |
637 |
> |
300 & 0.855 & -1.91 & 143(5) & 125(2) \\ |
638 |
> |
& & -4.19 & 135(9) & 113(12) \\ |
639 |
|
\hline\hline |
640 |
|
\end{tabular} |
641 |
|
\label{AuThiolToluene} |
667 |
|
|
668 |
|
However, when the surface is not completely covered by butanethiols, |
669 |
|
the simulated system is more resistent to the reconstruction |
670 |
< |
above. Our Au-butanethiol/toluene system did not see this phenomena |
671 |
< |
even at $\langle T\rangle\sim$300K. The Au(111) surfaces have a 90\% |
672 |
< |
coverage of butanethiols and have empty three-fold sites. These empty |
673 |
< |
sites could help prevent surface reconstruction in that they provide |
674 |
< |
other means of capping agent relaxation. It is observed that |
670 |
> |
above. Our Au-butanethiol/toluene system had the Au(111) surfaces 90\% |
671 |
> |
covered by butanethiols, but did not see this above phenomena even at |
672 |
> |
$\langle T\rangle\sim$300K. The empty three-fold sites not occupied by |
673 |
> |
capping agents could help prevent surface reconstruction in that they |
674 |
> |
provide other means of capping agent relaxation. It is observed that |
675 |
|
butanethiols can migrate to their neighbor empty sites during a |
676 |
|
simulation. Therefore, we were able to obtain $G$'s for these |
677 |
|
interfaces even at a relatively high temperature without being |
682 |
|
thermal conductance, a series of different coverage Au-butanethiol |
683 |
|
surfaces is prepared and solvated with various organic |
684 |
|
molecules. These systems are then equilibrated and their interfacial |
685 |
< |
thermal conductivity are measured with our NIVS algorithm. Table |
686 |
< |
\ref{tlnUhxnUhxnD} lists these results for direct comparison between |
687 |
< |
different coverages of butanethiol. To study the isotope effect in |
688 |
< |
interfacial thermal conductance, deuterated UA-hexane is included as |
689 |
< |
well. |
685 |
> |
thermal conductivity are measured with our NIVS algorithm. Figure |
686 |
> |
\ref{coverage} demonstrates the trend of conductance change with |
687 |
> |
respect to different coverages of butanethiol. To study the isotope |
688 |
> |
effect in interfacial thermal conductance, deuterated UA-hexane is |
689 |
> |
included as well. |
690 |
|
|
691 |
< |
It turned out that with partial covered butanethiol on the Au(111) |
692 |
< |
surface, the derivative definition for $G$ (Eq. \ref{derivativeG}) has |
693 |
< |
difficulty to apply, due to the difficulty in locating the maximum of |
694 |
< |
change of $\lambda$. Instead, the discrete definition |
695 |
< |
(Eq. \ref{discreteG}) is easier to apply, as max($\Delta T$) can still |
696 |
< |
be well-defined. Therefore, $G$'s (not $G^\prime$) are used for this |
697 |
< |
section. |
691 |
> |
\begin{figure} |
692 |
> |
\includegraphics[width=\linewidth]{coverage} |
693 |
> |
\caption{Comparison of interfacial thermal conductivity ($G$) values |
694 |
> |
for the Au-butanethiol/solvent interface with various UA models and |
695 |
> |
different capping agent coverages at $\langle T\rangle\sim$200K |
696 |
> |
using certain energy flux respectively.} |
697 |
> |
\label{coverage} |
698 |
> |
\end{figure} |
699 |
|
|
700 |
< |
From Table \ref{tlnUhxnUhxnD}, one can see the significance of the |
700 |
> |
It turned out that with partial covered butanethiol on the Au(111) |
701 |
> |
surface, the derivative definition for $G^\prime$ |
702 |
> |
(Eq. \ref{derivativeG}) was difficult to apply, due to the difficulty |
703 |
> |
in locating the maximum of change of $\lambda$. Instead, the discrete |
704 |
> |
definition (Eq. \ref{discreteG}) is easier to apply, as the Gibbs |
705 |
> |
deviding surface can still be well-defined. Therefore, $G$ (not |
706 |
> |
$G^\prime$) was used for this section. |
707 |
> |
|
708 |
> |
From Figure \ref{coverage}, one can see the significance of the |
709 |
|
presence of capping agents. Even when a fraction of the Au(111) |
710 |
|
surface sites are covered with butanethiols, the conductivity would |
711 |
|
see an enhancement by at least a factor of 3. This indicates the |
712 |
|
important role cappping agent is playing for thermal transport |
713 |
< |
phenomena on metal/organic solvent surfaces. |
713 |
> |
phenomena on metal / organic solvent surfaces. |
714 |
|
|
715 |
|
Interestingly, as one could observe from our results, the maximum |
716 |
|
conductance enhancement (largest $G$) happens while the surfaces are |
729 |
|
would not offset this effect. Eventually, when butanethiol coverage |
730 |
|
continues to decrease, solvent-capping agent contact actually |
731 |
|
decreases with the disappearing of butanethiol molecules. In this |
732 |
< |
case, $G$ decrease could not be offset but instead accelerated. |
732 |
> |
case, $G$ decrease could not be offset but instead accelerated. [MAY NEED |
733 |
> |
SNAPSHOT SHOWING THE PHENOMENA / SLAB DENSITY ANALYSIS] |
734 |
|
|
735 |
|
A comparison of the results obtained from differenet organic solvents |
736 |
|
can also provide useful information of the interfacial thermal |
740 |
|
studies, even though eliminating C-H vibration samplings, still have |
741 |
|
C-C vibrational frequencies different from each other. However, these |
742 |
|
differences in the infrared range do not seem to produce an observable |
743 |
< |
difference for the results of $G$. [MAY NEED FIGURE] |
743 |
> |
difference for the results of $G$ (Figure \ref{uahxnua}). |
744 |
|
|
745 |
+ |
\begin{figure} |
746 |
+ |
\includegraphics[width=\linewidth]{uahxnua} |
747 |
+ |
\caption{Vibrational spectra obtained for normal (upper) and |
748 |
+ |
deuterated (lower) hexane in Au-butanethiol/hexane |
749 |
+ |
systems. Butanethiol spectra are shown as reference. Both hexane and |
750 |
+ |
butanethiol were using United-Atom models.} |
751 |
+ |
\label{uahxnua} |
752 |
+ |
\end{figure} |
753 |
+ |
|
754 |
|
Furthermore, results for rigid body toluene solvent, as well as other |
755 |
|
UA-hexane solvents, are reasonable within the general experimental |
756 |
< |
ranges[CITATIONS]. This suggests that explicit hydrogen might not be a |
757 |
< |
required factor for modeling thermal transport phenomena of systems |
758 |
< |
such as Au-thiol/organic solvent. |
756 |
> |
ranges\cite{Wilson:2002uq,cahill:793,PhysRevB.80.195406}. This |
757 |
> |
suggests that explicit hydrogen might not be a required factor for |
758 |
> |
modeling thermal transport phenomena of systems such as |
759 |
> |
Au-thiol/organic solvent. |
760 |
|
|
761 |
|
However, results for Au-butanethiol/toluene do not show an identical |
762 |
< |
trend with those for Au-butanethiol/hexane in that $G$'s remain at |
762 |
> |
trend with those for Au-butanethiol/hexane in that $G$ remains at |
763 |
|
approximately the same magnitue when butanethiol coverage differs from |
764 |
|
25\% to 75\%. This might be rooted in the molecule shape difference |
765 |
< |
for plane-like toluene and chain-like {\it n}-hexane. Due to this |
765 |
> |
for planar toluene and chain-like {\it n}-hexane. Due to this |
766 |
|
difference, toluene molecules have more difficulty in occupying |
767 |
|
relatively small gaps among capping agents when their coverage is not |
768 |
|
too low. Therefore, the solvent-capping agent contact may keep |
771 |
|
its effect to the process of interfacial thermal transport. Thus, one |
772 |
|
can see a plateau of $G$ vs. butanethiol coverage in our results. |
773 |
|
|
706 |
– |
\begin{figure} |
707 |
– |
\includegraphics[width=\linewidth]{coverage} |
708 |
– |
\caption{Comparison of interfacial thermal conductivity ($G$) values |
709 |
– |
for the Au-butanethiol/solvent interface with various UA models and |
710 |
– |
different capping agent coverages at $\langle T\rangle\sim$200K |
711 |
– |
using certain energy flux respectively.} |
712 |
– |
\label{coverage} |
713 |
– |
\end{figure} |
714 |
– |
|
774 |
|
\subsection{Influence of Chosen Molecule Model on $G$} |
716 |
– |
[MAY COMBINE W MECHANISM STUDY] |
717 |
– |
|
775 |
|
In addition to UA solvent/capping agent models, AA models are included |
776 |
|
in our simulations as well. Besides simulations of the same (UA or AA) |
777 |
|
model for solvent and capping agent, different models can be applied |
840 |
|
interfaces, using AA model for both butanethiol and hexane yields |
841 |
|
substantially higher conductivity values than using UA model for at |
842 |
|
least one component of the solvent and capping agent, which exceeds |
843 |
< |
the upper bond of experimental value range. This is probably due to |
844 |
< |
the classically treated C-H vibrations in the AA model, which should |
845 |
< |
not be appreciably populated at normal temperatures. In comparison, |
846 |
< |
once either the hexanes or the butanethiols are deuterated, one can |
847 |
< |
see a significantly lower $G$ and $G^\prime$. In either of these |
848 |
< |
cases, the C-H(D) vibrational overlap between the solvent and the |
849 |
< |
capping agent is removed. [MAY NEED FIGURE] Conclusively, the |
850 |
< |
improperly treated C-H vibration in the AA model produced |
851 |
< |
over-predicted results accordingly. Compared to the AA model, the UA |
852 |
< |
model yields more reasonable results with higher computational |
853 |
< |
efficiency. |
843 |
> |
the general range of experimental measurement results. This is |
844 |
> |
probably due to the classically treated C-H vibrations in the AA |
845 |
> |
model, which should not be appreciably populated at normal |
846 |
> |
temperatures. In comparison, once either the hexanes or the |
847 |
> |
butanethiols are deuterated, one can see a significantly lower $G$ and |
848 |
> |
$G^\prime$. In either of these cases, the C-H(D) vibrational overlap |
849 |
> |
between the solvent and the capping agent is removed (Figure |
850 |
> |
\ref{aahxntln}). Conclusively, the improperly treated C-H vibration in |
851 |
> |
the AA model produced over-predicted results accordingly. Compared to |
852 |
> |
the AA model, the UA model yields more reasonable results with higher |
853 |
> |
computational efficiency. |
854 |
|
|
855 |
+ |
\begin{figure} |
856 |
+ |
\includegraphics[width=\linewidth]{aahxntln} |
857 |
+ |
\caption{Spectra obtained for All-Atom model Au-butanethil/solvent |
858 |
+ |
systems. When butanethiol is deuterated (lower left), its |
859 |
+ |
vibrational overlap with hexane would decrease significantly, |
860 |
+ |
compared with normal butanethiol (upper left). However, this |
861 |
+ |
dramatic change does not apply to toluene as much (right).} |
862 |
+ |
\label{aahxntln} |
863 |
+ |
\end{figure} |
864 |
+ |
|
865 |
|
However, for Au-butanethiol/toluene interfaces, having the AA |
866 |
|
butanethiol deuterated did not yield a significant change in the |
867 |
|
measurement results. Compared to the C-H vibrational overlap between |
868 |
|
hexane and butanethiol, both of which have alkyl chains, that overlap |
869 |
|
between toluene and butanethiol is not so significant and thus does |
870 |
< |
not have as much contribution to the ``Intramolecular Vibration |
871 |
< |
Redistribution''[CITE HASE]. Conversely, extra degrees of freedom such |
872 |
< |
as the C-H vibrations could yield higher heat exchange rate between |
873 |
< |
these two phases and result in a much higher conductivity. |
870 |
> |
not have as much contribution to the heat exchange |
871 |
> |
process. Conversely, extra degrees of freedom such as the C-H |
872 |
> |
vibrations could yield higher heat exchange rate between these two |
873 |
> |
phases and result in a much higher conductivity. |
874 |
|
|
875 |
|
Although the QSC model for Au is known to predict an overly low value |
876 |
|
for bulk metal gold conductivity\cite{kuang:164101}, our computational |
880 |
|
the accuracy of the interaction descriptions between components |
881 |
|
occupying the interfaces. |
882 |
|
|
883 |
< |
\subsection{Mechanism of Interfacial Thermal Conductance Enhancement |
884 |
< |
by Capping Agent} |
885 |
< |
%OR\subsection{Vibrational spectrum study on conductance mechanism} |
883 |
> |
\subsection{Role of Capping Agent in Interfacial Thermal Conductance} |
884 |
> |
The vibrational spectra for gold slabs in different environments are |
885 |
> |
shown as in Figure \ref{specAu}. Regardless of the presence of |
886 |
> |
solvent, the gold surfaces covered by butanethiol molecules, compared |
887 |
> |
to bare gold surfaces, exhibit an additional peak observed at the |
888 |
> |
frequency of $\sim$170cm$^{-1}$, which is attributed to the S-Au |
889 |
> |
bonding vibration. This vibration enables efficient thermal transport |
890 |
> |
from surface Au layer to the capping agents. Therefore, in our |
891 |
> |
simulations, the Au/S interfaces do not appear major heat barriers |
892 |
> |
compared to the butanethiol / solvent interfaces. |
893 |
|
|
894 |
< |
[MAY INTRODUCE PROTOCOL IN METHODOLOGY/COMPUTATIONAL DETAIL, EQN'S] |
894 |
> |
Simultaneously, the vibrational overlap between butanethiol and |
895 |
> |
organic solvents suggests higher thermal exchange efficiency between |
896 |
> |
these two components. Even exessively high heat transport was observed |
897 |
> |
when All-Atom models were used and C-H vibrations were treated |
898 |
> |
classically. Compared to metal and organic liquid phase, the heat |
899 |
> |
transfer efficiency between butanethiol and organic solvents is closer |
900 |
> |
to that within bulk liquid phase. |
901 |
|
|
902 |
< |
To investigate the mechanism of this interfacial thermal conductance, |
903 |
< |
the vibrational spectra of various gold systems were obtained and are |
904 |
< |
shown as in the upper panel of Fig. \ref{vibration}. To obtain these |
905 |
< |
spectra, one first runs a simulation in the NVE ensemble and collects |
906 |
< |
snapshots of configurations; these configurations are used to compute |
907 |
< |
the velocity auto-correlation functions, which is used to construct a |
908 |
< |
power spectrum via a Fourier transform. |
902 |
> |
Furthermore, our observation validated previous |
903 |
> |
results\cite{hase:2010} that the intramolecular heat transport of |
904 |
> |
alkylthiols is highly effecient. As a combinational effects of these |
905 |
> |
phenomena, butanethiol acts as a channel to expedite thermal transport |
906 |
> |
process. The acoustic impedance mismatch between the metal and the |
907 |
> |
liquid phase can be effectively reduced with the presence of suitable |
908 |
> |
capping agents. |
909 |
|
|
830 |
– |
[MAY RELATE TO HASE'S] |
831 |
– |
The gold surfaces covered by |
832 |
– |
butanethiol molecules, compared to bare gold surfaces, exhibit an |
833 |
– |
additional peak observed at a frequency of $\sim$170cm$^{-1}$, which |
834 |
– |
is attributed to the vibration of the S-Au bond. This vibration |
835 |
– |
enables efficient thermal transport from surface Au atoms to the |
836 |
– |
capping agents. Simultaneously, as shown in the lower panel of |
837 |
– |
Fig. \ref{vibration}, the large overlap of the vibration spectra of |
838 |
– |
butanethiol and hexane in the all-atom model, including the C-H |
839 |
– |
vibration, also suggests high thermal exchange efficiency. The |
840 |
– |
combination of these two effects produces the drastic interfacial |
841 |
– |
thermal conductance enhancement in the all-atom model. |
842 |
– |
|
843 |
– |
[REDO. MAY NEED TO CONVERT TO JPEG] |
910 |
|
\begin{figure} |
911 |
|
\includegraphics[width=\linewidth]{vibration} |
912 |
|
\caption{Vibrational spectra obtained for gold in different |
913 |
< |
environments (upper panel) and for Au/thiol/hexane simulation in |
914 |
< |
all-atom model (lower panel).} |
849 |
< |
\label{vibration} |
913 |
> |
environments.} |
914 |
> |
\label{specAu} |
915 |
|
\end{figure} |
916 |
|
|
917 |
< |
[COMPARISON OF TWO G'S; AU SLAB WIDTHS; ETC] |
853 |
< |
% The results show that the two definitions used for $G$ yield |
854 |
< |
% comparable values, though $G^\prime$ tends to be smaller. |
917 |
> |
[MAY ADD COMPARISON OF AU SLAB WIDTHS] |
918 |
|
|
919 |
|
\section{Conclusions} |
920 |
|
The NIVS algorithm we developed has been applied to simulations of |
922 |
|
effective unphysical thermal flux transferred between the metal and |
923 |
|
the liquid phase. With the flux applied, we were able to measure the |
924 |
|
corresponding thermal gradient and to obtain interfacial thermal |
925 |
< |
conductivities. Our simulations have seen significant conductance |
926 |
< |
enhancement with the presence of capping agent, compared to the bare |
927 |
< |
gold/liquid interfaces. The acoustic impedance mismatch between the |
928 |
< |
metal and the liquid phase is effectively eliminated by proper capping |
925 |
> |
conductivities. Under steady states, single trajectory simulation |
926 |
> |
would be enough for accurate measurement. This would be advantageous |
927 |
> |
compared to transient state simulations, which need multiple |
928 |
> |
trajectories to produce reliable average results. |
929 |
> |
|
930 |
> |
Our simulations have seen significant conductance enhancement with the |
931 |
> |
presence of capping agent, compared to the bare gold / liquid |
932 |
> |
interfaces. The acoustic impedance mismatch between the metal and the |
933 |
> |
liquid phase is effectively eliminated by proper capping |
934 |
|
agent. Furthermore, the coverage precentage of the capping agent plays |
935 |
< |
an important role in the interfacial thermal transport process. |
935 |
> |
an important role in the interfacial thermal transport |
936 |
> |
process. Moderately lower coverages allow higher contact between |
937 |
> |
capping agent and solvent, and thus could further enhance the heat |
938 |
> |
transfer process. |
939 |
|
|
940 |
|
Our measurement results, particularly of the UA models, agree with |
941 |
|
available experimental data. This indicates that our force field |
945 |
|
vibration would be overly sampled. Compared to the AA models, the UA |
946 |
|
models have higher computational efficiency with satisfactory |
947 |
|
accuracy, and thus are preferable in interfacial thermal transport |
948 |
< |
modelings. |
948 |
> |
modelings. Of the two definitions for $G$, the discrete form |
949 |
> |
(Eq. \ref{discreteG}) was easier to use and gives out relatively |
950 |
> |
consistent results, while the derivative form (Eq. \ref{derivativeG}) |
951 |
> |
is not as versatile. Although $G^\prime$ gives out comparable results |
952 |
> |
and follows similar trend with $G$ when measuring close to fully |
953 |
> |
covered or bare surfaces, the spatial resolution of $T$ profile is |
954 |
> |
limited for accurate computation of derivatives data. |
955 |
|
|
956 |
|
Vlugt {\it et al.} has investigated the surface thiol structures for |
957 |
|
nanocrystal gold and pointed out that they differs from those of the |
958 |
< |
Au(111) surface\cite{vlugt:cpc2007154}. This difference might lead to |
959 |
< |
change of interfacial thermal transport behavior as well. To |
960 |
< |
investigate this problem, an effective means to introduce thermal flux |
961 |
< |
and measure the corresponding thermal gradient is desirable for |
962 |
< |
simulating structures with spherical symmetry. |
958 |
> |
Au(111) surface\cite{landman:1998,vlugt:cpc2007154}. This difference |
959 |
> |
might lead to change of interfacial thermal transport behavior as |
960 |
> |
well. To investigate this problem, an effective means to introduce |
961 |
> |
thermal flux and measure the corresponding thermal gradient is |
962 |
> |
desirable for simulating structures with spherical symmetry. |
963 |
|
|
887 |
– |
|
964 |
|
\section{Acknowledgments} |
965 |
|
Support for this project was provided by the National Science |
966 |
|
Foundation under grant CHE-0848243. Computational time was provided by |