[ViewVC] Diff of: group/interfacial/interfacial.tex

Comparing interfacial/interfacial.tex (file contents):
Revision 3726 by skuang, Mon May 9 19:08:08 2011 UTC vs.
Revision 3727 by skuang, Fri Jun 24 16:59:37 2011 UTC

+\setlength{\abovecaptionskip}{20 pt}
+\setlength{\belowcaptionskip}{30 pt}
-<
+%\renewcommand\citemid{\ } % no comma in optional referenc note
->
+%\renewcommand\citemid{\ } % no comma in optional reference note
+\bibpunct{[}{]}{,}{s}{}{;}
+\bibliographystyle{aip}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\section{Introduction}
-<
->
+[BACKGROUND FOR INTERFACIAL THERMAL CONDUCTANCE PROBLEM]
+Interfacial thermal conductance is extensively studied both
+experimentally and computationally, and systems with interfaces
+present are generally heterogeneous. Although interfaces are commonly
+\section{Methodology}
+\subsection{Algorithm}
-+
+[BACKGROUND FOR MD METHODS]
+There have been many algorithms for computing thermal conductivity
+using molecular dynamics simulations. However, interfacial conductance
+is at least an order of magnitude smaller. This would make the
+algorithm conserves momenta and energy and does not depend on an
+external thermostat.
-<
+(wondering how much detail of algorithm should be put here...)
->
+\subsection{Defining Interfacial Thermal Conductivity $G$}
->
+For interfaces with a relatively low interfacial conductance, the bulk
->
+regions on either side of an interface rapidly come to a state in
->
+which the two phases have relatively homogeneous (but distinct)
->
+temperatures. The interfacial thermal conductivity $G$ can therefore
->
+be approximated as:
->
+\begin{equation}
->
+G = \frac{E_{total}}{2 t L_x L_y \left( \langle T_\mathrm{hot}\rangle -
->
+    \langle T_\mathrm{cold}\rangle \right)}
->
+\label{lowG}
->
+\end{equation}
->
+where ${E_{total}}$ is the imposed non-physical kinetic energy
->
+transfer and ${\langle T_\mathrm{hot}\rangle}$ and ${\langle
->
+  T_\mathrm{cold}\rangle}$ are the average observed temperature of the
->
+two separated phases.
-+
+When the interfacial conductance is {\it not} small, two ways can be
-+
+used to define $G$.
-+
-+
+One way is to assume the temperature is discretely different on two
-+
+sides of the interface, $G$ can be calculated with the thermal flux
-+
+applied $J$ and the maximum temperature difference measured along the
-+
+thermal gradient max($\Delta T$), which occurs at the interface, as:
-+
+\begin{equation}
-+
+G=\frac{J}{\Delta T}
-+
+\label{discreteG}
-+
+\end{equation}
-+
-+
+The other approach is to assume a continuous temperature profile along
-+
+the thermal gradient axis (e.g. $z$) and define $G$ at the point where
-+
+the magnitude of thermal conductivity $\lambda$ change reach its
-+
+maximum, given that $\lambda$ is well-defined throughout the space:
-+
+\begin{equation}
-+
+G^\prime = \Big|\frac{\partial\lambda}{\partial z}\Big|
-+
+         = \Big|\frac{\partial}{\partial z}\left(-J_z\Big/
-+
+           \left(\frac{\partial T}{\partial z}\right)\right)\Big|
-+
+         = |J_z|\Big|\frac{\partial^2 T}{\partial z^2}\Big|
-+
+         \Big/\left(\frac{\partial T}{\partial z}\right)^2
-+
+\label{derivativeG}
-+
+\end{equation}
-+
-+
+With the temperature profile obtained from simulations, one is able to
-+
+approximate the first and second derivatives of $T$ with finite
-+
+difference method and thus calculate $G^\prime$.
-+
-+
+In what follows, both definitions are used for calculation and comparison.
-+
-+
+[IMPOSE G DEFINITION INTO OUR SYSTEMS]
-+
+To facilitate the use of the above definitions in calculating $G$ and
-+
+$G^\prime$, we have a metal slab with its (111) surfaces perpendicular
-+
+to the $z$-axis of our simulation cells. With or withour capping
-+
+agents on the surfaces, the metal slab is solvated with organic
-+
+solvents, as illustrated in Figure \ref{demoPic}.
-+
-+
+\begin{figure}
-+
+\includegraphics[width=\linewidth]{demoPic}
-+
+\caption{A sample showing how a metal slab has its (111) surface
-+
+  covered by capping agent molecules and solvated by hexane.}
-+
+\label{demoPic}
-+
+\end{figure}
-+
-+
+With a simulation cell setup following the above manner, one is able
-+
+to equilibrate the system and impose an unphysical thermal flux
-+
+between the liquid and the metal phase with the NIVS algorithm. Under
-+
+a stablized thermal gradient induced by periodically applying the
-+
+unphysical flux, one is able to obtain a temperature profile and the
-+
+physical thermal flux corresponding to it, which equals to the
-+
+unphysical flux applied by NIVS. These data enables the evaluation of
-+
+the interfacial thermal conductance of a surface. Figure \ref{gradT}
-+
+is an example how those stablized thermal gradient can be used to
-+
+obtain the 1st and 2nd derivatives of the temperature profile.
-+
-+
+\begin{figure}
-+
+\includegraphics[width=\linewidth]{gradT}
-+
+\caption{The 1st and 2nd derivatives of temperature profile can be
-+
+  obtained with finite difference approximation.}
-+
+\label{gradT}
-+
+\end{figure}
-+
-+
+\section{Computational Details}
-+
+\subsection{System Geometry}
-+
+In our simulations, Au is used to construct a metal slab with bare
-+
+(111) surface perpendicular to the $z$-axis. Different slab thickness
-+
+(layer numbers of Au) are simulated. This metal slab is first
-+
+equilibrated under normal pressure (1 atm) and a desired
-+
+temperature. After equilibration, butanethiol is used as the capping
-+
+agent molecule to cover the bare Au (111) surfaces evenly. The sulfur
-+
+atoms in the butanethiol molecules would occupy the three-fold sites
-+
+of the surfaces, and the maximal butanethiol capacity on Au surface is
-+
+$1/3$ of the total number of surface Au atoms[CITATION]. A series of
-+
+different coverage surfaces is investigated in order to study the
-+
+relation between coverage and conductance.
-+
-+
+[COVERAGE DISCRIPTION] However, since the interactions between surface
-+
+Au and butanethiol is non-bonded, the capping agent molecules are
-+
+allowed to migrate to an empty neighbor three-fold site during a
-+
+simulation. Therefore, the initial configuration would not severely
-+
+affect the sampling of a variety of configurations of the same
-+
+coverage, and the final conductance measurement would be an average
-+
+effect of these configurations explored in the simulations. [MAY NEED FIGURES]
-+
-+
+After the modified Au-butanethiol surface systems are equilibrated
-+
+under canonical ensemble, Packmol\cite{packmol} is used to pack
-+
+organic solvent molecules in the previously vacuum part of the
-+
+simulation cells, which guarantees that short range repulsive
-+
+interactions do not disrupt the simulations. Two solvents are
-+
+investigated, one which has little vibrational overlap with the
-+
+alkanethiol and plane-like shape (toluene), and one which has similar
-+
+vibrational frequencies and chain-like shape ({\it n}-hexane). The
-+
+initial configurations generated by Packmol are further equilibrated
-+
+with the $x$ and $y$ dimensions fixed, only allowing length scale
-+
+change in $z$ dimension. This is to ensure that the equilibration of
-+
+liquid phase does not affect the metal crystal structure in $x$ and
-+
+$y$ dimensions. Further equilibration are run under NVT and then NVE ensembles.
-+
-+
+After the systems reach equilibrium, NIVS is implemented to impose a
-+
+periodic unphysical thermal flux between the metal and the liquid
-+
+phase. Most of our simulations have this flux from the metal to the
-+
+liquid so that the liquid has a higher temperature and would not
-+
+freeze due to excessively low temperature. This induced temperature
-+
+gradient is stablized and the simulation cell is devided evenly into
-+
+N slabs along the $z$-axis and the temperatures of each slab are
-+
+recorded. When the slab width $d$ of each slab is the same, the
-+
+derivatives of $T$ with respect to slab number $n$ can be directly
-+
+used for $G^\prime$ calculations:
-+
+\begin{equation}
-+
+G^\prime = |J_z|\Big|\frac{\partial^2 T}{\partial z^2}\Big|
-+
+         \Big/\left(\frac{\partial T}{\partial z}\right)^2
-+
+         = |J_z|\Big|\frac{1}{d^2}\frac{\partial^2 T}{\partial n^2}\Big|
-+
+         \Big/\left(\frac{1}{d}\frac{\partial T}{\partial n}\right)^2
-+
+         = |J_z|\Big|\frac{\partial^2 T}{\partial n^2}\Big|
-+
+         \Big/\left(\frac{\partial T}{\partial n}\right)^2
-+
+\label{derivativeG2}
-+
+\end{equation}
-+
-+
+\subsection{Force Field Parameters}
-–
+Our simulation systems consists of metal gold lattice slab solvated by
-–
+organic solvents. In order to study the role of capping agents in
-–
+interfacial thermal conductance, butanethiol is chosen to cover gold
-–
+surfaces in comparison to no capping agent present.
+The Au-Au interactions in metal lattice slab is described by the
+quantum Sutton-Chen (QSC) formulation.\cite{PhysRevB.59.3527} The QSC
+[TABULATED FORCE FIELD PARAMETERS NEEDED]
-–
+\section{Computational Details}
-–
+\subsection{System Geometry}
-–
+Our simulation systems consists of a lattice Au slab with the (111)
-–
+surface perpendicular to the $z$-axis, and a solvent layer between the
-–
+periodic Au slabs along the $z$-axis. To set up the interfacial
-–
+system, the Au slab is first equilibrated without solvent under room
-–
+pressure and a desired temperature. After the metal slab is
-–
+equilibrated, United-Atom or All-Atom butanethiols are replicated on
-–
+the Au surface, each occupying the (??) among three Au atoms, and is
-–
+equilibrated under NVT ensemble. According to (CITATION), the maximal
-–
+thiol capacity on Au surface is $1/3$ of the total number of surface
-–
+Au atoms.
-–
-–
+\cite{packmol}
-–
-–
+\subsection{Simulation Parameters}
-–
-–
+When the interfacial conductance is {\it not} small, there are two
-–
+ways to define $G$. If we assume the temperature is discretely
-–
+different on two sides of the interface, $G$ can be calculated with
-–
+the thermal flux applied $J$ and the temperature difference measured
-–
+$\Delta T$ as:
-–
+\begin{equation}
-–
+G=\frac{J}{\Delta T}
-–
+\label{discreteG}
-–
+\end{equation}
-–
+We can as well assume a continuous temperature profile along the
-–
+thermal gradient axis $z$ and define $G$ as the change of bulk thermal
-–
+conductivity $\lambda$ at a defined interfacial point:
-–
+\begin{equation}
-–
+G^\prime = \Big|\frac{\partial\lambda}{\partial z}\Big|
-–
+         = \Big|\frac{\partial}{\partial z}\left(-J_z\Big/
-–
+           \left(\frac{\partial T}{\partial z}\right)\right)\Big|
-–
+         = J_z\Big|\frac{\partial^2 T}{\partial z^2}\Big|
-–
+         \Big/\left(\frac{\partial T}{\partial z}\right)^2
-–
+\label{derivativeG}
-–
+\end{equation}
-–
+With the temperature profile obtained from simulations, one is able to
-–
+approximate the first and second derivatives of $T$ with finite
-–
+difference method and thus calculate $G^\prime$.
-–
-–
+In what follows, both definitions are used for calculation and comparison.
-–
+\section{Results}
+\subsection{Toluene Solvent}
-<
+The simulations follow a protocol similar to the previous gold/water
-<
+interfacial systems. The results (Table \ref{AuThiolToluene}) show a
->
+The results (Table \ref{AuThiolToluene}) show a
+significant conductance enhancement compared to the gold/water
+interface without capping agent and agree with available experimental
+data. This indicates that the metal-metal potential, though not

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Comparing interfacial/interfacial.tex (file contents): Revision 3726 by skuang, Mon May 9 19:08:08 2011 UTC vs. Revision 3727 by skuang, Fri Jun 24 16:59:37 2011 UTC

Diff Legend

Comparing interfacial/interfacial.tex (file contents):
Revision 3726 by skuang, Mon May 9 19:08:08 2011 UTC vs.
Revision 3727 by skuang, Fri Jun 24 16:59:37 2011 UTC