--- interfacial/interfacial.tex 2011/01/27 16:29:20 3717 +++ interfacial/interfacial.tex 2011/06/29 18:14:23 3729 @@ -22,7 +22,7 @@ \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} @@ -44,7 +44,22 @@ The abstract \begin{doublespace} \begin{abstract} -The abstract + +We have developed a Non-Isotropic Velocity Scaling algorithm for +setting up and maintaining stable thermal gradients in non-equilibrium +molecular dynamics simulations. This approach effectively imposes +unphysical thermal flux even between particles of different +identities, conserves linear momentum and kinetic energy, and +minimally perturbs the velocity profile of a system when compared with +previous RNEMD methods. We have used this method to simulate thermal +conductance at metal / organic solvent interfaces both with and +without the presence of thiol-based capping agents. We obtained +values comparable with experimental values, and observed significant +conductance enhancement with the presence of capping agents. Computed +power spectra indicate the acoustic impedance mismatch between metal +and liquid phase is greatly reduced by the capping agents and thus +leads to higher interfacial thermal transfer efficiency. + \end{abstract} \newpage @@ -56,9 +71,458 @@ The abstract %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \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 +barriers to heat transfer, it has been +reported\cite{doi:10.1021/la904855s} that under specific circustances, +e.g. with certain capping agents present on the surface, interfacial +conductance can be significantly enhanced. However, heat conductance +of molecular and nano-scale interfaces will be affected by the +chemical details of the surface and is challenging to +experimentalist. The lower thermal flux through interfaces is even +more difficult to measure with EMD and forward NEMD simulation +methods. Therefore, developing good simulation methods will be +desirable in order to investigate thermal transport across interfaces. -The intro. +Recently, we have developed the Non-Isotropic Velocity Scaling (NIVS) +algorithm for RNEMD simulations\cite{kuang:164101}. This algorithm +retains the desirable features of RNEMD (conservation of linear +momentum and total energy, compatibility with periodic boundary +conditions) while establishing true thermal distributions in each of +the two slabs. Furthermore, it allows more effective thermal exchange +between particles of different identities, and thus enables extensive +study of interfacial conductance. + +\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 +calculation even more difficult for those slowly-converging +equilibrium methods. Imposed-flux non-equilibrium +methods\cite{MullerPlathe:1997xw} have the flux set {\it a priori} and +the response of temperature or momentum gradients are easier to +measure than the flux, if unknown, and thus, is a preferable way to +the forward NEMD methods. Although the momentum swapping approach for +flux-imposing can be used for exchanging energy between particles of +different identity, the kinetic energy transfer efficiency is affected +by the mass difference between the particles, which limits its +application on heterogeneous interfacial systems. + +The non-isotropic velocity scaling (NIVS)\cite{kuang:164101} approach in +non-equilibrium MD simulations is able to impose relatively large +kinetic energy flux without obvious perturbation to the velocity +distribution of the simulated systems. Furthermore, this approach has +the advantage in heterogeneous interfaces in that kinetic energy flux +can be applied between regions of particles of arbitary identity, and +the flux quantity is not restricted by particle mass difference. + +The NIVS algorithm scales the velocity vectors in two separate regions +of a simulation system with respective diagonal scaling matricies. To +determine these scaling factors in the matricies, a set of equations +including linear momentum conservation and kinetic energy conservation +constraints and target momentum/energy flux satisfaction is +solved. With the scaling operation applied to the system in a set +frequency, corresponding momentum/temperature gradients can be built, +which can be used for computing transportation properties and other +applications related to momentum/temperature gradients. The NIVS +algorithm conserves momenta and energy and does not depend on an +external thermostat. + +\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 +spacing filled by solvent molecules, i.e. the gap between periodically +repeated Au-butanethiol surfaces should be carefully chosen so that it +would not be too short to affect the liquid phase structure, nor too +long, leading to over cooling (freezing) or heating (boiling) when a +thermal flux is applied. In our simulations, this spacing is usually +$35 \sim 60$\AA. + +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 are under an average temperature of +$\sim$200K. Therefore, this flux usually comes 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 simulations include various components. Therefore, force field +parameter descriptions are needed for interactions both between the +same type of particles and between particles of different species. + +The Au-Au interactions in metal lattice slab is described by the +quantum Sutton-Chen (QSC) formulation.\cite{PhysRevB.59.3527} The QSC +potentials include zero-point quantum corrections and are +reparametrized for accurate surface energies compared to the +Sutton-Chen potentials\cite{Chen90}. + +For both solvent molecules, straight chain {\it n}-hexane and aromatic +toluene, United-Atom (UA) and All-Atom (AA) models are used +respectively. The TraPPE-UA +parameters\cite{TraPPE-UA.alkanes,TraPPE-UA.alkylbenzenes} are used +for our UA solvent molecules. In these models, pseudo-atoms are +located at the carbon centers for alkyl groups. By eliminating +explicit hydrogen atoms, these models are simple and computationally +efficient, while maintains good accuracy. However, the TraPPE-UA for +alkanes is known to predict a lower boiling point than experimental +values. Considering that after an unphysical thermal flux is applied +to a system, the temperature of ``hot'' area in the liquid phase would be +significantly higher than the average, to prevent over heating and +boiling of the liquid phase, the average temperature in our +simulations should be much lower than the liquid boiling point. [NEED MORE DISCUSSION] +For UA-toluene model, rigid body constraints are applied, so that the +benzene ring and the methyl-C(aromatic) bond are kept rigid. This +would save computational time.[MORE DETAILS NEEDED] +Besides the TraPPE-UA models, AA models for both organic solvents are +included in our studies as well. For hexane, the OPLS +all-atom\cite{OPLSAA} force field is used. [MORE DETAILS] +For toluene, the United Force Field developed by Rapp\'{e} {\it et + al.}\cite{doi:10.1021/ja00051a040} is adopted.[MORE DETAILS] + +The capping agent in our simulations, the butanethiol molecules can +either use UA or AA model. The TraPPE-UA force fields includes +parameters for thiol molecules\cite{TraPPE-UA.thiols} and are used in +our simulations corresponding to our TraPPE-UA models for solvent. + and All-Atom models [NEED CITATIONS] +However, the model choice (UA or AA) of capping agent can be different +from the solvent. Regardless of model choice, the force field +parameters for interactions between capping agent and solvent can be +derived using Lorentz-Berthelot Mixing Rule. + +To describe the interactions between metal Au and non-metal capping +agent and solvent, we refer to Vlugt\cite{vlugt:cpc2007154} and derive +other interactions which are not yet finely parametrized. [can add +hautman and klein's paper here and more discussion; need to put +aromatic-metal interaction approximation here]\cite{doi:10.1021/jp034405s} + +[TABULATED FORCE FIELD PARAMETERS NEEDED] + + +[SURFACE RECONSTRUCTION PREVENTS SIMULATION TEMP TO GO HIGHER] + + +\section{Results} +[REARRANGEMENT NEEDED] +\subsection{Toluene Solvent} + +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 +predicting an accurate bulk metal thermal conductivity, does not +greatly interfere with the simulation of the thermal conductance +behavior across a non-metal interface. The solvent model is not +particularly volatile, so the simulation cell does not expand +significantly under higher temperature. We did not observe a +significant conductance decrease when the temperature was increased to +300K. The results show that the two definitions used for $G$ yield +comparable values, though $G^\prime$ tends to be smaller. + +\begin{table*} + \begin{minipage}{\linewidth} + \begin{center} + \caption{Computed interfacial thermal conductivity ($G$ and + $G^\prime$) values for the Au/butanethiol/toluene interface at + different temperatures using a range of energy fluxes.} + + \begin{tabular}{cccc} + \hline\hline + $\langle T\rangle$ & $J_z$ & $G$ & $G^\prime$ \\ + (K) & (GW/m$^2$) & \multicolumn{2}{c}{(MW/m$^2$/K)} \\ + \hline + 200 & 1.86 & 180 & 135 \\ + & 2.15 & 204 & 113 \\ + & 3.93 & 175 & 114 \\ + 300 & 1.91 & 143 & 125 \\ + & 4.19 & 134 & 113 \\ + \hline\hline + \end{tabular} + \label{AuThiolToluene} + \end{center} + \end{minipage} +\end{table*} + +\subsection{Hexane Solvent} + +Using the united-atom model, different coverages of capping agent, +temperatures of simulations and numbers of solvent molecules were all +investigated and Table \ref{AuThiolHexaneUA} shows the results of +these computations. The number of hexane molecules in our simulations +does not affect the calculations significantly. However, a very long +length scale for the thermal gradient axis ($z$) may cause excessively +hot or cold temperatures in the middle of the solvent region and lead +to undesired phenomena such as solvent boiling or freezing, while too +few solvent molecules would change the normal behavior of the liquid +phase. Our $N_{hexane}$ values were chosen to ensure that these +extreme cases did not happen to our simulations. + +Table \ref{AuThiolHexaneUA} enables direct comparison between +different coverages of capping agent, when other system parameters are +held constant. With high coverage of butanethiol on the gold surface, +the interfacial thermal conductance is enhanced +significantly. Interestingly, a slightly lower butanethiol coverage +leads to a moderately higher conductivity. This is probably due to +more solvent/capping agent contact when butanethiol molecules are +not densely packed, which enhances the interactions between the two +phases and lowers the thermal transfer barrier of this interface. +% [COMPARE TO AU/WATER IN PAPER] + +It is also noted that the overall simulation temperature is another +factor that affects the interfacial thermal conductance. One +possibility of this effect may be rooted in the decrease in density of +the liquid phase. We observed that when the average temperature +increases from 200K to 250K, the bulk hexane density becomes lower +than experimental value, as the system is equilibrated under NPT +ensemble. This leads to lower contact between solvent and capping +agent, and thus lower conductivity. + +Conductivity values are more difficult to obtain under higher +temperatures. This is because the Au surface tends to undergo +reconstructions in relatively high temperatures. Surface Au atoms can +migrate outward to reach higher Au-S contact; and capping agent +molecules can be embedded into the surface Au layer due to the same +driving force. This phenomenon agrees with experimental +results\cite{doi:10.1021/j100035a033,doi:10.1021/la026493y}. A surface +fully covered in capping agent is more susceptible to reconstruction, +possibly because fully coverage prevents other means of capping agent +relaxation, such as migration to an empty neighbor three-fold site. + +%MAY ADD MORE DATA TO TABLE +\begin{table*} + \begin{minipage}{\linewidth} + \begin{center} + \caption{Computed interfacial thermal conductivity ($G$ and + $G^\prime$) values for the Au/butanethiol/hexane interface + with united-atom model and different capping agent coverage + and solvent molecule numbers at different temperatures using a + range of energy fluxes.} + + \begin{tabular}{cccccc} + \hline\hline + Thiol & $\langle T\rangle$ & & $J_z$ & $G$ & $G^\prime$ \\ + coverage (\%) & (K) & $N_{hexane}$ & (GW/m$^2$) & + \multicolumn{2}{c}{(MW/m$^2$/K)} \\ + \hline + 0.0 & 200 & 200 & 0.96 & 43.3 & 42.7 \\ + & & & 1.91 & 45.7 & 42.9 \\ + & & 166 & 0.96 & 43.1 & 53.4 \\ + 88.9 & 200 & 166 & 1.94 & 172 & 108 \\ + 100.0 & 250 & 200 & 0.96 & 81.8 & 67.0 \\ + & & 166 & 0.98 & 79.0 & 62.9 \\ + & & & 1.44 & 76.2 & 64.8 \\ + & 200 & 200 & 1.92 & 129 & 87.3 \\ + & & & 1.93 & 131 & 77.5 \\ + & & 166 & 0.97 & 115 & 69.3 \\ + & & & 1.94 & 125 & 87.1 \\ + \hline\hline + \end{tabular} + \label{AuThiolHexaneUA} + \end{center} + \end{minipage} +\end{table*} + +For the all-atom model, the liquid hexane phase was not stable under NPT +conditions. Therefore, the simulation length scale parameters are +adopted from previous equilibration results of the united-atom model +at 200K. Table \ref{AuThiolHexaneAA} shows the results of these +simulations. The conductivity values calculated with full capping +agent coverage are substantially larger than observed in the +united-atom model, and is even higher than predicted by +experiments. It is possible that our parameters for metal-non-metal +particle interactions lead to an overestimate of the interfacial +thermal conductivity, although the active C-H vibrations in the +all-atom model (which should not be appreciably populated at normal +temperatures) could also account for this high conductivity. The major +thermal transfer barrier of Au/butanethiol/hexane interface is between +the liquid phase and the capping agent, so extra degrees of freedom +such as the C-H vibrations could enhance heat exchange between these +two phases and result in a much higher conductivity. + +\begin{table*} + \begin{minipage}{\linewidth} + \begin{center} + + \caption{Computed interfacial thermal conductivity ($G$ and + $G^\prime$) values for the Au/butanethiol/hexane interface + with all-atom model and different capping agent coverage at + 200K using a range of energy fluxes.} + + \begin{tabular}{cccc} + \hline\hline + Thiol & $J_z$ & $G$ & $G^\prime$ \\ + coverage (\%) & (GW/m$^2$) & \multicolumn{2}{c}{(MW/m$^2$/K)} \\ + \hline + 0.0 & 0.95 & 28.5 & 27.2 \\ + & 1.88 & 30.3 & 28.9 \\ + 100.0 & 2.87 & 551 & 294 \\ + & 3.81 & 494 & 193 \\ + \hline\hline + \end{tabular} + \label{AuThiolHexaneAA} + \end{center} + \end{minipage} +\end{table*} + +%subsubsection{Vibrational spectrum study on conductance mechanism} +To investigate the mechanism of this interfacial thermal conductance, +the vibrational spectra of various gold systems were obtained and are +shown as in the upper panel of Fig. \ref{vibration}. To obtain these +spectra, one first runs a simulation in the NVE ensemble and collects +snapshots of configurations; these configurations are used to compute +the velocity auto-correlation functions, which is used to construct a +power spectrum via a Fourier transform. The gold surfaces covered by +butanethiol molecules exhibit an additional peak observed at a +frequency of $\sim$170cm$^{-1}$, which is attributed to the vibration +of the S-Au bond. This vibration enables efficient thermal transport +from surface Au atoms to the capping agents. Simultaneously, as shown +in the lower panel of Fig. \ref{vibration}, the large overlap of the +vibration spectra of butanethiol and hexane in the all-atom model, +including the C-H vibration, also suggests high thermal exchange +efficiency. The combination of these two effects produces the drastic +interfacial thermal conductance enhancement in the all-atom model. + +\begin{figure} +\includegraphics[width=\linewidth]{vibration} +\caption{Vibrational spectra obtained for gold in different + environments (upper panel) and for Au/thiol/hexane simulation in + all-atom model (lower panel).} +\label{vibration} +\end{figure} +% 600dpi, letter size. too large? + + \section{Acknowledgments} Support for this project was provided by the National Science Foundation under grant CHE-0848243. Computational time was provided by