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\begin{document} |
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\title{Simulating interfacial thermal conductance at metal-solvent |
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interfaces: the role of chemical capping agents} |
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|
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\author{Shenyu Kuang and J. Daniel |
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Gezelter\footnote{Corresponding author. \ Electronic mail: gezelter@nd.edu} \\ |
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Department of Chemistry and Biochemistry,\\ |
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University of Notre Dame\\ |
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Notre Dame, Indiana 46556} |
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|
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\date{\today} |
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\maketitle |
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|
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\begin{doublespace} |
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|
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\begin{abstract} |
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|
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With the Non-Isotropic Velocity Scaling algorithm (NIVS) we have |
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developed, an unphysical thermal flux can be effectively set up even |
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for non-homogeneous systems like interfaces in non-equilibrium |
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molecular dynamics simulations. In this work, this algorithm is |
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applied for simulating thermal conductance at metal / organic solvent |
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interfaces with various coverages of butanethiol capping |
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agents. Different solvents and force field models were tested. Our |
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results suggest that the United-Atom models are able to provide an |
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estimate of the interfacial thermal conductivity comparable to |
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experiments in our simulations with satisfactory computational |
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efficiency. From our results, the acoustic impedance mismatch between |
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metal and liquid phase is effectively reduced by the capping |
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agents, and thus leads to interfacial thermal conductance |
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enhancement. Furthermore, this effect is closely related to the |
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capping agent coverage on the metal surfaces and the type of solvent |
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molecules, and is affected by the models used in the simulations. |
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|
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\end{abstract} |
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|
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\newpage |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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% BODY OF TEXT |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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|
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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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|
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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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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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|
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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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this phenomena at the molecular level. Recently, Hase and coworkers |
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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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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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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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|
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Recently, we have developed the 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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conditions) while establishing true thermal distributions in each of |
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the two slabs. Furthermore, it allows effective thermal exchange |
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between particles of different identities, and thus makes the study of |
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interfacial conductance much simpler. |
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|
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The work presented here deals with the Au(111) surface covered to |
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varying degrees by butanethiol, a capping agent with short carbon |
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chain, and solvated with organic solvents of different molecular |
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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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|
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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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|
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The non-isotropic velocity scaling (NIVS)\cite{kuang:164101} approach to |
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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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can be applied between regions of particles of arbitary identity, and |
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the flux will not be restricted by difference in particle mass. |
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|
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The NIVS algorithm scales the velocity vectors in two separate regions |
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of a simulation system with respective diagonal scaling matricies. To |
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determine these scaling factors in the matricies, a set of equations |
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including linear momentum conservation and kinetic energy conservation |
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constraints and target energy flux satisfaction is solved. With the |
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scaling operation applied to the system in a set frequency, bulk |
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temperature gradients can be easily established, and these can be used |
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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 |
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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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\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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\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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|
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When the interfacial conductance is {\it not} small, there are two |
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ways to define $G$. |
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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 |
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gradient max($\Delta T$), which occurs at the Gibbs deviding surface |
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(Figure \ref{demoPic}): |
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\begin{equation} |
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G=\frac{J}{\Delta T} |
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\label{discreteG} |
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\end{equation} |
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|
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\begin{figure} |
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\includegraphics[width=\linewidth]{method} |
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\caption{Interfacial conductance can be calculated by applying an |
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(unphysical) kinetic energy flux between two slabs, one located |
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within the metal and another on the edge of the periodic box. The |
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system responds by forming a thermal response or a gradient. In |
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bulk liquids, this gradient typically has a single slope, but in |
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interfacial systems, there are distinct thermal conductivity |
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domains. The interfacial conductance, $G$ is found by measuring the |
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temperature gap at the Gibbs dividing surface, or by using second |
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derivatives of the thermal profile.} |
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\label{demoPic} |
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\end{figure} |
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|
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The other approach is to assume a continuous temperature profile along |
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the thermal gradient axis (e.g. $z$) and define $G$ at the point where |
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the magnitude of thermal conductivity $\lambda$ change reach its |
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maximum, given that $\lambda$ is well-defined throughout the space: |
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\begin{equation} |
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G^\prime = \Big|\frac{\partial\lambda}{\partial z}\Big| |
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= \Big|\frac{\partial}{\partial z}\left(-J_z\Big/ |
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\left(\frac{\partial T}{\partial z}\right)\right)\Big| |
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= |J_z|\Big|\frac{\partial^2 T}{\partial z^2}\Big| |
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\Big/\left(\frac{\partial T}{\partial z}\right)^2 |
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\label{derivativeG} |
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\end{equation} |
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|
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With the temperature profile obtained from simulations, one is able to |
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approximate the first and second derivatives of $T$ with finite |
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difference methods and thus calculate $G^\prime$. |
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|
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In what follows, both definitions have been used for calculation and |
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are compared in the results. |
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|
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To compare the above definitions ($G$ and $G^\prime$), we have modeled |
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a metal slab with its (111) surfaces perpendicular to the $z$-axis of |
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our simulation cells. Both with and without capping agents on the |
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surfaces, the metal slab is solvated with simple organic solvents, as |
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illustrated in Figure \ref{demoPic}. |
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|
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With the simulation cell described above, we are able to equilibrate |
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the system and impose an unphysical thermal flux between the liquid |
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and the metal phase using the NIVS algorithm. By periodically applying |
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the unphysical flux, we are able to obtain a temperature profile and |
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its spatial derivatives. These quantities enable the evaluation of the |
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interfacial thermal conductance of a surface. Figure \ref{gradT} is an |
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example how those applied thermal fluxes can be used to obtain the 1st |
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and 2nd derivatives of the temperature profile. |
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|
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\begin{figure} |
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\includegraphics[width=\linewidth]{gradT} |
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\caption{A sample of Au-butanethiol/hexane interfacial system and the |
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temperature profile after a kinetic energy flux is imposed to |
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it. The 1st and 2nd derivatives of the temperature profile can be |
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obtained with finite difference approximation (lower panel).} |
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\label{gradT} |
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\end{figure} |
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|
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\section{Computational Details} |
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\subsection{Simulation Protocol} |
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The NIVS algorithm has been implemented in our MD simulation code, |
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OpenMD\cite{Meineke:2005gd,openmd}, and was used for our |
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simulations. Different slab thickness (layer numbers of Au) were |
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simulated. Metal slabs were first equilibrated under atmospheric |
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pressure (1 atm) and a desired temperature (e.g. 200K). After |
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equilibration, butanethiol capping agents were placed at three-fold |
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sites on the Au(111) surfaces. The maximum butanethiol capacity on Au |
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surface is $1/3$ of the total number of surface Au |
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atoms\cite{vlugt:cpc2007154}. A series of different coverages was |
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investigated in order to study the relation between coverage and |
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interfacial conductance. |
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|
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The capping agent molecules were allowed to migrate during the |
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simulations. They distributed themselves uniformly and sampled a |
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number of three-fold sites throughout out study. Therefore, the |
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initial configuration would not noticeably affect the sampling of a |
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variety of configurations of the same coverage, and the final |
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conductance measurement would be an average effect of these |
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configurations explored in the simulations. [MAY NEED FIGURES] |
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|
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After the modified Au-butanethiol surface systems were equilibrated |
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under canonical ensemble, organic solvent molecules were packed in the |
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previously empty part of the simulation cells\cite{packmol}. Two |
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solvents were investigated, one which has little vibrational overlap |
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with the alkanethiol and a planar shape (toluene), and one which has |
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similar vibrational frequencies and chain-like shape ({\it n}-hexane). |
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|
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The space filled by solvent molecules, i.e. the gap between |
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periodically repeated Au-butanethiol surfaces should be carefully |
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chosen. A very long length scale for the thermal gradient axis ($z$) |
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may cause excessively hot or cold temperatures in the middle of the |
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solvent region and lead to undesired phenomena such as solvent boiling |
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or freezing when a thermal flux is applied. Conversely, too few |
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solvent molecules would change the normal behavior of the liquid |
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phase. Therefore, our $N_{solvent}$ values were chosen to ensure that |
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these extreme cases did not happen to our simulations. And the |
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corresponding spacing is usually $35 \sim 75$\AA. |
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|
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The initial configurations generated by Packmol are further |
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equilibrated with the $x$ and $y$ dimensions fixed, only allowing |
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length scale change in $z$ dimension. This is to ensure that the |
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equilibration of liquid phase does not affect the metal crystal |
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structure in $x$ and $y$ dimensions. Further equilibration are run |
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under NVT and then NVE ensembles. |
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|
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After the systems reach equilibrium, NIVS is implemented to impose a |
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periodic unphysical thermal flux between the metal and the liquid |
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phase. Most of our simulations are under an average temperature of |
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$\sim$200K. Therefore, this flux usually comes from the metal to the |
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liquid so that the liquid has a higher temperature and would not |
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freeze due to excessively low temperature. This induced temperature |
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gradient is stablized and the simulation cell is devided evenly into |
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N slabs along the $z$-axis and the temperatures of each slab are |
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recorded. When the slab width $d$ of each slab is the same, the |
317 |
derivatives of $T$ with respect to slab number $n$ can be directly |
318 |
used for $G^\prime$ calculations: |
319 |
\begin{equation} |
320 |
G^\prime = |J_z|\Big|\frac{\partial^2 T}{\partial z^2}\Big| |
321 |
\Big/\left(\frac{\partial T}{\partial z}\right)^2 |
322 |
= |J_z|\Big|\frac{1}{d^2}\frac{\partial^2 T}{\partial n^2}\Big| |
323 |
\Big/\left(\frac{1}{d}\frac{\partial T}{\partial n}\right)^2 |
324 |
= |J_z|\Big|\frac{\partial^2 T}{\partial n^2}\Big| |
325 |
\Big/\left(\frac{\partial T}{\partial n}\right)^2 |
326 |
\label{derivativeG2} |
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\end{equation} |
328 |
|
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\subsection{Force Field Parameters} |
330 |
Our simulations include various components. Figure \ref{demoMol} |
331 |
demonstrates the sites defined for both United-Atom and All-Atom |
332 |
models of the organic solvent and capping agent molecules in our |
333 |
simulations. Force field parameter descriptions are needed for |
334 |
interactions both between the same type of particles and between |
335 |
particles of different species. |
336 |
|
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\begin{figure} |
338 |
\includegraphics[width=\linewidth]{structures} |
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\caption{Structures of the capping agent and solvents utilized in |
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these simulations. The chemically-distinct sites (a-e) are expanded |
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in terms of constituent atoms for both United Atom (UA) and All Atom |
342 |
(AA) force fields. Most parameters are from |
343 |
Refs. \protect\cite{TraPPE-UA.alkanes,TraPPE-UA.alkylbenzenes} (UA) and |
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\protect\cite{OPLSAA} (AA). Cross-interactions with the Au atoms are given |
345 |
in Table \ref{MnM}.} |
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\label{demoMol} |
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\end{figure} |
348 |
|
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The Au-Au interactions in metal lattice slab is described by the |
350 |
quantum Sutton-Chen (QSC) formulation\cite{PhysRevB.59.3527}. The QSC |
351 |
potentials include zero-point quantum corrections and are |
352 |
reparametrized for accurate surface energies compared to the |
353 |
Sutton-Chen potentials\cite{Chen90}. |
354 |
|
355 |
For both solvent molecules, straight chain {\it n}-hexane and aromatic |
356 |
toluene, United-Atom (UA) and All-Atom (AA) models are used |
357 |
respectively. The TraPPE-UA |
358 |
parameters\cite{TraPPE-UA.alkanes,TraPPE-UA.alkylbenzenes} are used |
359 |
for our UA solvent molecules. In these models, sites are located at |
360 |
the carbon centers for alkyl groups. Bonding interactions, including |
361 |
bond stretches and bends and torsions, were used for intra-molecular |
362 |
sites not separated by more than 3 bonds. Otherwise, for non-bonded |
363 |
interactions, Lennard-Jones potentials are used. [MORE CITATION?] |
364 |
|
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By eliminating explicit hydrogen atoms, these models are simple and |
366 |
computationally efficient, while maintains good accuracy. However, the |
367 |
TraPPE-UA for alkanes is known to predict a lower boiling point than |
368 |
experimental values. Considering that after an unphysical thermal flux |
369 |
is applied to a system, the temperature of ``hot'' area in the liquid |
370 |
phase would be significantly higher than the average, to prevent over |
371 |
heating and boiling of the liquid phase, the average temperature in |
372 |
our simulations should be much lower than the liquid boiling point. |
373 |
|
374 |
For UA-toluene model, the non-bonded potentials between |
375 |
inter-molecular sites have a similar Lennard-Jones formulation. For |
376 |
intra-molecular interactions, considering the stiffness of the benzene |
377 |
ring, rigid body constraints are applied for further computational |
378 |
efficiency. All bonds in the benzene ring and between the ring and the |
379 |
methyl group remain rigid during the progress of simulations. |
380 |
|
381 |
Besides the TraPPE-UA models, AA models for both organic solvents are |
382 |
included in our studies as well. For hexane, the OPLS-AA\cite{OPLSAA} |
383 |
force field is used. Additional explicit hydrogen sites were |
384 |
included. Besides bonding and non-bonded site-site interactions, |
385 |
partial charges and the electrostatic interactions were added to each |
386 |
CT and HC site. For toluene, the United Force Field developed by |
387 |
Rapp\'{e} {\it et al.}\cite{doi:10.1021/ja00051a040} is |
388 |
adopted. Without the rigid body constraints, bonding interactions were |
389 |
included. For the aromatic ring, improper torsions (inversions) were |
390 |
added as an extra potential for maintaining the planar shape. |
391 |
[MORE CITATION?] |
392 |
|
393 |
The capping agent in our simulations, the butanethiol molecules can |
394 |
either use UA or AA model. The TraPPE-UA force fields includes |
395 |
parameters for thiol molecules\cite{TraPPE-UA.thiols} and are used for |
396 |
UA butanethiol model in our simulations. The OPLS-AA also provides |
397 |
parameters for alkyl thiols. However, alkyl thiols adsorbed on Au(111) |
398 |
surfaces do not have the hydrogen atom bonded to sulfur. To adapt this |
399 |
change and derive suitable parameters for butanethiol adsorbed on |
400 |
Au(111) surfaces, we adopt the S parameters from Luedtke and |
401 |
Landman\cite{landman:1998} and modify parameters for its neighbor C |
402 |
atom for charge balance in the molecule. Note that the model choice |
403 |
(UA or AA) of capping agent can be different from the |
404 |
solvent. Regardless of model choice, the force field parameters for |
405 |
interactions between capping agent and solvent can be derived using |
406 |
Lorentz-Berthelot Mixing Rule: |
407 |
\begin{eqnarray} |
408 |
\sigma_{ij} & = & \frac{1}{2} \left(\sigma_{ii} + \sigma_{jj}\right) \\ |
409 |
\epsilon_{ij} & = & \sqrt{\epsilon_{ii}\epsilon_{jj}} |
410 |
\end{eqnarray} |
411 |
|
412 |
To describe the interactions between metal Au and non-metal capping |
413 |
agent and solvent particles, we refer to an adsorption study of alkyl |
414 |
thiols on gold surfaces by Vlugt {\it et |
415 |
al.}\cite{vlugt:cpc2007154} They fitted an effective Lennard-Jones |
416 |
form of potential parameters for the interaction between Au and |
417 |
pseudo-atoms CH$_x$ and S based on a well-established and widely-used |
418 |
effective potential of Hautman and Klein\cite{hautman:4994} for the |
419 |
Au(111) surface. As our simulations require the gold lattice slab to |
420 |
be non-rigid so that it could accommodate kinetic energy for thermal |
421 |
transport study purpose, the pair-wise form of potentials is |
422 |
preferred. |
423 |
|
424 |
Besides, the potentials developed from {\it ab initio} calculations by |
425 |
Leng {\it et al.}\cite{doi:10.1021/jp034405s} are adopted for the |
426 |
interactions between Au and aromatic C/H atoms in toluene. A set of |
427 |
pseudo Lennard-Jones parameters were provided for Au in their force |
428 |
fields. By using the Mixing Rule, this can be used to derive pair-wise |
429 |
potentials for non-bonded interactions between Au and non-metal sites. |
430 |
|
431 |
However, the Lennard-Jones parameters between Au and other types of |
432 |
particles, such as All-Atom normal alkanes in our simulations are not |
433 |
yet well-established. For these interactions, we attempt to derive |
434 |
their parameters using the Mixing Rule. To do this, Au pseudo |
435 |
Lennard-Jones parameters for ``Metal-non-Metal'' (MnM) interactions |
436 |
were first extracted from the Au-CH$_x$ parameters by applying the |
437 |
Mixing Rule reversely. Table \ref{MnM} summarizes these ``MnM'' |
438 |
parameters in our simulations. |
439 |
|
440 |
\begin{table*} |
441 |
\begin{minipage}{\linewidth} |
442 |
\begin{center} |
443 |
\caption{Non-bonded interaction parameters (including cross |
444 |
interactions with Au atoms) for both force fields used in this |
445 |
work.} |
446 |
\begin{tabular}{lllllll} |
447 |
\hline\hline |
448 |
& Site & $\sigma_{ii}$ & $\epsilon_{ii}$ & $q_i$ & |
449 |
$\sigma_{Au-i}$ & $\epsilon_{Au-i}$ \\ |
450 |
& & (\AA) & (kcal/mol) & ($e$) & (\AA) & (kcal/mol) \\ |
451 |
\hline |
452 |
United Atom (UA) |
453 |
&CH3 & 3.75 & 0.1947 & - & 3.54 & 0.2146 \\ |
454 |
&CH2 & 3.95 & 0.0914 & - & 3.54 & 0.1749 \\ |
455 |
&CHar & 3.695 & 0.1003 & - & 3.4625 & 0.1680 \\ |
456 |
&CRar & 3.88 & 0.04173 & - & 3.555 & 0.1604 \\ |
457 |
\hline |
458 |
All Atom (AA) |
459 |
&CT3 & 3.50 & 0.066 & -0.18 & 3.365 & 0.1373 \\ |
460 |
&CT2 & 3.50 & 0.066 & -0.12 & 3.365 & 0.1373 \\ |
461 |
&CTT & 3.50 & 0.066 & -0.065 & 3.365 & 0.1373 \\ |
462 |
&HC & 2.50 & 0.030 & 0.06 & 2.865 & 0.09256 \\ |
463 |
&CA & 3.55 & 0.070 & -0.115 & 3.173 & 0.0640 \\ |
464 |
&HA & 2.42 & 0.030 & 0.115 & 2.746 & 0.0414 \\ |
465 |
\hline |
466 |
Both UA and AA |
467 |
& S & 4.45 & 0.25 & - & 2.40 & 8.465 \\ |
468 |
\hline\hline |
469 |
\end{tabular} |
470 |
\label{MnM} |
471 |
\end{center} |
472 |
\end{minipage} |
473 |
\end{table*} |
474 |
|
475 |
|
476 |
\section{Results and Discussions} |
477 |
[MAY HAVE A BRIEF SUMMARY] |
478 |
\subsection{How Simulation Parameters Affects $G$} |
479 |
[MAY NOT PUT AT FIRST] |
480 |
We have varied our protocol or other parameters of the simulations in |
481 |
order to investigate how these factors would affect the measurement of |
482 |
$G$'s. It turned out that while some of these parameters would not |
483 |
affect the results substantially, some other changes to the |
484 |
simulations would have a significant impact on the measurement |
485 |
results. |
486 |
|
487 |
In some of our simulations, we allowed $L_x$ and $L_y$ to change |
488 |
during equilibrating the liquid phase. Due to the stiffness of the |
489 |
crystalline Au structure, $L_x$ and $L_y$ would not change noticeably |
490 |
after equilibration. Although $L_z$ could fluctuate $\sim$1\% after a |
491 |
system is fully equilibrated in the NPT ensemble, this fluctuation, as |
492 |
well as those of $L_x$ and $L_y$ (which is significantly smaller), |
493 |
would not be magnified on the calculated $G$'s, as shown in Table |
494 |
\ref{AuThiolHexaneUA}. This insensivity to $L_i$ fluctuations allows |
495 |
reliable measurement of $G$'s without the necessity of extremely |
496 |
cautious equilibration process. |
497 |
|
498 |
As stated in our computational details, the spacing filled with |
499 |
solvent molecules can be chosen within a range. This allows some |
500 |
change of solvent molecule numbers for the same Au-butanethiol |
501 |
surfaces. We did this study on our Au-butanethiol/hexane |
502 |
simulations. Nevertheless, the results obtained from systems of |
503 |
different $N_{hexane}$ did not indicate that the measurement of $G$ is |
504 |
susceptible to this parameter. For computational efficiency concern, |
505 |
smaller system size would be preferable, given that the liquid phase |
506 |
structure is not affected. |
507 |
|
508 |
Our NIVS algorithm allows change of unphysical thermal flux both in |
509 |
direction and in quantity. This feature extends our investigation of |
510 |
interfacial thermal conductance. However, the magnitude of this |
511 |
thermal flux is not arbitary if one aims to obtain a stable and |
512 |
reliable thermal gradient. A temperature profile would be |
513 |
substantially affected by noise when $|J_z|$ has a much too low |
514 |
magnitude; while an excessively large $|J_z|$ that overwhelms the |
515 |
conductance capacity of the interface would prevent a thermal gradient |
516 |
to reach a stablized steady state. NIVS has the advantage of allowing |
517 |
$J$ to vary in a wide range such that the optimal flux range for $G$ |
518 |
measurement can generally be simulated by the algorithm. Within the |
519 |
optimal range, we were able to study how $G$ would change according to |
520 |
the thermal flux across the interface. For our simulations, we denote |
521 |
$J_z$ to be positive when the physical thermal flux is from the liquid |
522 |
to metal, and negative vice versa. The $G$'s measured under different |
523 |
$J_z$ is listed in Table \ref{AuThiolHexaneUA} and |
524 |
\ref{AuThiolToluene}. These results do not suggest that $G$ is |
525 |
dependent on $J_z$ within this flux range. The linear response of flux |
526 |
to thermal gradient simplifies our investigations in that we can rely |
527 |
on $G$ measurement with only a couple $J_z$'s and do not need to test |
528 |
a large series of fluxes. |
529 |
|
530 |
\begin{table*} |
531 |
\begin{minipage}{\linewidth} |
532 |
\begin{center} |
533 |
\caption{Computed interfacial thermal conductivity ($G$ and |
534 |
$G^\prime$) values for the 100\% covered Au-butanethiol/hexane |
535 |
interfaces with UA model and different hexane molecule numbers |
536 |
at different temperatures using a range of energy |
537 |
fluxes. Error estimates indicated in parenthesis.} |
538 |
|
539 |
\begin{tabular}{ccccccc} |
540 |
\hline\hline |
541 |
$\langle T\rangle$ & $N_{hexane}$ & Fixed & $\rho_{hexane}$ & |
542 |
$J_z$ & $G$ & $G^\prime$ \\ |
543 |
(K) & & $L_x$ \& $L_y$? & (g/cm$^3$) & (GW/m$^2$) & |
544 |
\multicolumn{2}{c}{(MW/m$^2$/K)} \\ |
545 |
\hline |
546 |
200 & 266 & No & 0.672 & -0.96 & 102(3) & 80.0(0.8) \\ |
547 |
& 200 & Yes & 0.694 & 1.92 & 129(11) & 87.3(0.3) \\ |
548 |
& & Yes & 0.672 & 1.93 & 131(16) & 78(13) \\ |
549 |
& & No & 0.688 & 0.96 & 125(16) & 90.2(15) \\ |
550 |
& & & & 1.91 & 139(10) & 101(10) \\ |
551 |
& & & & 2.83 & 141(6) & 89.9(9.8) \\ |
552 |
& 166 & Yes & 0.679 & 0.97 & 115(19) & 69(18) \\ |
553 |
& & & & 1.94 & 125(9) & 87.1(0.2) \\ |
554 |
& & No & 0.681 & 0.97 & 141(30) & 78(22) \\ |
555 |
& & & & 1.92 & 138(4) & 98.9(9.5) \\ |
556 |
\hline |
557 |
250 & 200 & No & 0.560 & 0.96 & 75(10) & 61.8(7.3) \\ |
558 |
& & & & -0.95 & 49.4(0.3) & 45.7(2.1) \\ |
559 |
& 166 & Yes & 0.570 & 0.98 & 79.0(3.5) & 62.9(3.0) \\ |
560 |
& & No & 0.569 & 0.97 & 80.3(0.6) & 67(11) \\ |
561 |
& & & & 1.44 & 76.2(5.0) & 64.8(3.8) \\ |
562 |
& & & & -0.95 & 56.4(2.5) & 54.4(1.1) \\ |
563 |
& & & & -1.85 & 47.8(1.1) & 53.5(1.5) \\ |
564 |
\hline\hline |
565 |
\end{tabular} |
566 |
\label{AuThiolHexaneUA} |
567 |
\end{center} |
568 |
\end{minipage} |
569 |
\end{table*} |
570 |
|
571 |
Furthermore, we also attempted to increase system average temperatures |
572 |
to above 200K. These simulations are first equilibrated in the NPT |
573 |
ensemble under normal pressure. As stated above, the TraPPE-UA model |
574 |
for hexane tends to predict a lower boiling point. In our simulations, |
575 |
hexane had diffculty to remain in liquid phase when NPT equilibration |
576 |
temperature is higher than 250K. Additionally, the equilibrated liquid |
577 |
hexane density under 250K becomes lower than experimental value. This |
578 |
expanded liquid phase leads to lower contact between hexane and |
579 |
butanethiol as well.[MAY NEED SLAB DENSITY FIGURE] |
580 |
And this reduced contact would |
581 |
probably be accountable for a lower interfacial thermal conductance, |
582 |
as shown in Table \ref{AuThiolHexaneUA}. |
583 |
|
584 |
A similar study for TraPPE-UA toluene agrees with the above result as |
585 |
well. Having a higher boiling point, toluene tends to remain liquid in |
586 |
our simulations even equilibrated under 300K in NPT |
587 |
ensembles. Furthermore, the expansion of the toluene liquid phase is |
588 |
not as significant as that of the hexane. This prevents severe |
589 |
decrease of liquid-capping agent contact and the results (Table |
590 |
\ref{AuThiolToluene}) show only a slightly decreased interface |
591 |
conductance. Therefore, solvent-capping agent contact should play an |
592 |
important role in the thermal transport process across the interface |
593 |
in that higher degree of contact could yield increased conductance. |
594 |
|
595 |
\begin{table*} |
596 |
\begin{minipage}{\linewidth} |
597 |
\begin{center} |
598 |
\caption{Computed interfacial thermal conductivity ($G$ and |
599 |
$G^\prime$) values for a 90\% coverage Au-butanethiol/toluene |
600 |
interface at different temperatures using a range of energy |
601 |
fluxes. Error estimates indicated in parenthesis.} |
602 |
|
603 |
\begin{tabular}{ccccc} |
604 |
\hline\hline |
605 |
$\langle T\rangle$ & $\rho_{toluene}$ & $J_z$ & $G$ & $G^\prime$ \\ |
606 |
(K) & (g/cm$^3$) & (GW/m$^2$) & \multicolumn{2}{c}{(MW/m$^2$/K)} \\ |
607 |
\hline |
608 |
200 & 0.933 & 2.15 & 204(12) & 113(12) \\ |
609 |
& & -1.86 & 180(3) & 135(21) \\ |
610 |
& & -3.93 & 176(5) & 113(12) \\ |
611 |
\hline |
612 |
300 & 0.855 & -1.91 & 143(5) & 125(2) \\ |
613 |
& & -4.19 & 135(9) & 113(12) \\ |
614 |
\hline\hline |
615 |
\end{tabular} |
616 |
\label{AuThiolToluene} |
617 |
\end{center} |
618 |
\end{minipage} |
619 |
\end{table*} |
620 |
|
621 |
Besides lower interfacial thermal conductance, surfaces in relatively |
622 |
high temperatures are susceptible to reconstructions, when |
623 |
butanethiols have a full coverage on the Au(111) surface. These |
624 |
reconstructions include surface Au atoms migrated outward to the S |
625 |
atom layer, and butanethiol molecules embedded into the original |
626 |
surface Au layer. The driving force for this behavior is the strong |
627 |
Au-S interactions in our simulations. And these reconstructions lead |
628 |
to higher ratio of Au-S attraction and thus is energetically |
629 |
favorable. Furthermore, this phenomenon agrees with experimental |
630 |
results\cite{doi:10.1021/j100035a033,doi:10.1021/la026493y}. Vlugt |
631 |
{\it et al.} had kept their Au(111) slab rigid so that their |
632 |
simulations can reach 300K without surface reconstructions. Without |
633 |
this practice, simulating 100\% thiol covered interfaces under higher |
634 |
temperatures could hardly avoid surface reconstructions. However, our |
635 |
measurement is based on assuming homogeneity on $x$ and $y$ dimensions |
636 |
so that measurement of $T$ at particular $z$ would be an effective |
637 |
average of the particles of the same type. Since surface |
638 |
reconstructions could eliminate the original $x$ and $y$ dimensional |
639 |
homogeneity, measurement of $G$ is more difficult to conduct under |
640 |
higher temperatures. Therefore, most of our measurements are |
641 |
undertaken at $\langle T\rangle\sim$200K. |
642 |
|
643 |
However, when the surface is not completely covered by butanethiols, |
644 |
the simulated system is more resistent to the reconstruction |
645 |
above. Our Au-butanethiol/toluene system had the Au(111) surfaces 90\% |
646 |
covered by butanethiols, but did not see this above phenomena even at |
647 |
$\langle T\rangle\sim$300K. The empty three-fold sites not occupied by |
648 |
capping agents could help prevent surface reconstruction in that they |
649 |
provide other means of capping agent relaxation. It is observed that |
650 |
butanethiols can migrate to their neighbor empty sites during a |
651 |
simulation. Therefore, we were able to obtain $G$'s for these |
652 |
interfaces even at a relatively high temperature without being |
653 |
affected by surface reconstructions. |
654 |
|
655 |
\subsection{Influence of Capping Agent Coverage on $G$} |
656 |
To investigate the influence of butanethiol coverage on interfacial |
657 |
thermal conductance, a series of different coverage Au-butanethiol |
658 |
surfaces is prepared and solvated with various organic |
659 |
molecules. These systems are then equilibrated and their interfacial |
660 |
thermal conductivity are measured with our NIVS algorithm. Figure |
661 |
\ref{coverage} demonstrates the trend of conductance change with |
662 |
respect to different coverages of butanethiol. To study the isotope |
663 |
effect in interfacial thermal conductance, deuterated UA-hexane is |
664 |
included as well. |
665 |
|
666 |
It turned out that with partial covered butanethiol on the Au(111) |
667 |
surface, the derivative definition for $G^\prime$ |
668 |
(Eq. \ref{derivativeG}) was difficult to apply, due to the difficulty |
669 |
in locating the maximum of change of $\lambda$. Instead, the discrete |
670 |
definition (Eq. \ref{discreteG}) is easier to apply, as the Gibbs |
671 |
deviding surface can still be well-defined. Therefore, $G$ (not |
672 |
$G^\prime$) was used for this section. |
673 |
|
674 |
From Figure \ref{coverage}, one can see the significance of the |
675 |
presence of capping agents. Even when a fraction of the Au(111) |
676 |
surface sites are covered with butanethiols, the conductivity would |
677 |
see an enhancement by at least a factor of 3. This indicates the |
678 |
important role cappping agent is playing for thermal transport |
679 |
phenomena on metal / organic solvent surfaces. |
680 |
|
681 |
Interestingly, as one could observe from our results, the maximum |
682 |
conductance enhancement (largest $G$) happens while the surfaces are |
683 |
about 75\% covered with butanethiols. This again indicates that |
684 |
solvent-capping agent contact has an important role of the thermal |
685 |
transport process. Slightly lower butanethiol coverage allows small |
686 |
gaps between butanethiols to form. And these gaps could be filled with |
687 |
solvent molecules, which acts like ``heat conductors'' on the |
688 |
surface. The higher degree of interaction between these solvent |
689 |
molecules and capping agents increases the enhancement effect and thus |
690 |
produces a higher $G$ than densely packed butanethiol arrays. However, |
691 |
once this maximum conductance enhancement is reached, $G$ decreases |
692 |
when butanethiol coverage continues to decrease. Each capping agent |
693 |
molecule reaches its maximum capacity for thermal |
694 |
conductance. Therefore, even higher solvent-capping agent contact |
695 |
would not offset this effect. Eventually, when butanethiol coverage |
696 |
continues to decrease, solvent-capping agent contact actually |
697 |
decreases with the disappearing of butanethiol molecules. In this |
698 |
case, $G$ decrease could not be offset but instead accelerated. [NEED |
699 |
SNAPSHOT SHOWING THE PHENOMENA] |
700 |
|
701 |
A comparison of the results obtained from differenet organic solvents |
702 |
can also provide useful information of the interfacial thermal |
703 |
transport process. The deuterated hexane (UA) results do not appear to |
704 |
be much different from those of normal hexane (UA), given that |
705 |
butanethiol (UA) is non-deuterated for both solvents. These UA model |
706 |
studies, even though eliminating C-H vibration samplings, still have |
707 |
C-C vibrational frequencies different from each other. However, these |
708 |
differences in the infrared range do not seem to produce an observable |
709 |
difference for the results of $G$. [MAY NEED SPECTRA FIGURE] |
710 |
|
711 |
Furthermore, results for rigid body toluene solvent, as well as other |
712 |
UA-hexane solvents, are reasonable within the general experimental |
713 |
ranges[CITATIONS]. This suggests that explicit hydrogen might not be a |
714 |
required factor for modeling thermal transport phenomena of systems |
715 |
such as Au-thiol/organic solvent. |
716 |
|
717 |
However, results for Au-butanethiol/toluene do not show an identical |
718 |
trend with those for Au-butanethiol/hexane in that $G$ remains at |
719 |
approximately the same magnitue when butanethiol coverage differs from |
720 |
25\% to 75\%. This might be rooted in the molecule shape difference |
721 |
for planar toluene and chain-like {\it n}-hexane. Due to this |
722 |
difference, toluene molecules have more difficulty in occupying |
723 |
relatively small gaps among capping agents when their coverage is not |
724 |
too low. Therefore, the solvent-capping agent contact may keep |
725 |
increasing until the capping agent coverage reaches a relatively low |
726 |
level. This becomes an offset for decreasing butanethiol molecules on |
727 |
its effect to the process of interfacial thermal transport. Thus, one |
728 |
can see a plateau of $G$ vs. butanethiol coverage in our results. |
729 |
|
730 |
\begin{figure} |
731 |
\includegraphics[width=\linewidth]{coverage} |
732 |
\caption{Comparison of interfacial thermal conductivity ($G$) values |
733 |
for the Au-butanethiol/solvent interface with various UA models and |
734 |
different capping agent coverages at $\langle T\rangle\sim$200K |
735 |
using certain energy flux respectively.} |
736 |
\label{coverage} |
737 |
\end{figure} |
738 |
|
739 |
\subsection{Influence of Chosen Molecule Model on $G$} |
740 |
[MAY COMBINE W MECHANISM STUDY] |
741 |
|
742 |
In addition to UA solvent/capping agent models, AA models are included |
743 |
in our simulations as well. Besides simulations of the same (UA or AA) |
744 |
model for solvent and capping agent, different models can be applied |
745 |
to different components. Furthermore, regardless of models chosen, |
746 |
either the solvent or the capping agent can be deuterated, similar to |
747 |
the previous section. Table \ref{modelTest} summarizes the results of |
748 |
these studies. |
749 |
|
750 |
\begin{table*} |
751 |
\begin{minipage}{\linewidth} |
752 |
\begin{center} |
753 |
|
754 |
\caption{Computed interfacial thermal conductivity ($G$ and |
755 |
$G^\prime$) values for interfaces using various models for |
756 |
solvent and capping agent (or without capping agent) at |
757 |
$\langle T\rangle\sim$200K. (D stands for deuterated solvent |
758 |
or capping agent molecules; ``Avg.'' denotes results that are |
759 |
averages of simulations under different $J_z$'s. Error |
760 |
estimates indicated in parenthesis.)} |
761 |
|
762 |
\begin{tabular}{llccc} |
763 |
\hline\hline |
764 |
Butanethiol model & Solvent & $J_z$ & $G$ & $G^\prime$ \\ |
765 |
(or bare surface) & model & (GW/m$^2$) & |
766 |
\multicolumn{2}{c}{(MW/m$^2$/K)} \\ |
767 |
\hline |
768 |
UA & UA hexane & Avg. & 131(9) & 87(10) \\ |
769 |
& UA hexane(D) & 1.95 & 153(5) & 136(13) \\ |
770 |
& AA hexane & Avg. & 131(6) & 122(10) \\ |
771 |
& UA toluene & 1.96 & 187(16) & 151(11) \\ |
772 |
& AA toluene & 1.89 & 200(36) & 149(53) \\ |
773 |
\hline |
774 |
AA & UA hexane & 1.94 & 116(9) & 129(8) \\ |
775 |
& AA hexane & Avg. & 442(14) & 356(31) \\ |
776 |
& AA hexane(D) & 1.93 & 222(12) & 234(54) \\ |
777 |
& UA toluene & 1.98 & 125(25) & 97(60) \\ |
778 |
& AA toluene & 3.79 & 487(56) & 290(42) \\ |
779 |
\hline |
780 |
AA(D) & UA hexane & 1.94 & 158(25) & 172(4) \\ |
781 |
& AA hexane & 1.92 & 243(29) & 191(11) \\ |
782 |
& AA toluene & 1.93 & 364(36) & 322(67) \\ |
783 |
\hline |
784 |
bare & UA hexane & Avg. & 46.5(3.2) & 49.4(4.5) \\ |
785 |
& UA hexane(D) & 0.98 & 43.9(4.6) & 43.0(2.0) \\ |
786 |
& AA hexane & 0.96 & 31.0(1.4) & 29.4(1.3) \\ |
787 |
& UA toluene & 1.99 & 70.1(1.3) & 65.8(0.5) \\ |
788 |
\hline\hline |
789 |
\end{tabular} |
790 |
\label{modelTest} |
791 |
\end{center} |
792 |
\end{minipage} |
793 |
\end{table*} |
794 |
|
795 |
To facilitate direct comparison, the same system with differnt models |
796 |
for different components uses the same length scale for their |
797 |
simulation cells. Without the presence of capping agent, using |
798 |
different models for hexane yields similar results for both $G$ and |
799 |
$G^\prime$, and these two definitions agree with eath other very |
800 |
well. This indicates very weak interaction between the metal and the |
801 |
solvent, and is a typical case for acoustic impedance mismatch between |
802 |
these two phases. |
803 |
|
804 |
As for Au(111) surfaces completely covered by butanethiols, the choice |
805 |
of models for capping agent and solvent could impact the measurement |
806 |
of $G$ and $G^\prime$ quite significantly. For Au-butanethiol/hexane |
807 |
interfaces, using AA model for both butanethiol and hexane yields |
808 |
substantially higher conductivity values than using UA model for at |
809 |
least one component of the solvent and capping agent, which exceeds |
810 |
the general range of experimental measurement results. This is |
811 |
probably due to the classically treated C-H vibrations in the AA |
812 |
model, which should not be appreciably populated at normal |
813 |
temperatures. In comparison, once either the hexanes or the |
814 |
butanethiols are deuterated, one can see a significantly lower $G$ and |
815 |
$G^\prime$. In either of these cases, the C-H(D) vibrational overlap |
816 |
between the solvent and the capping agent is removed. |
817 |
[MAY NEED SPECTRA FIGURE] Conclusively, the |
818 |
improperly treated C-H vibration in the AA model produced |
819 |
over-predicted results accordingly. Compared to the AA model, the UA |
820 |
model yields more reasonable results with higher computational |
821 |
efficiency. |
822 |
|
823 |
However, for Au-butanethiol/toluene interfaces, having the AA |
824 |
butanethiol deuterated did not yield a significant change in the |
825 |
measurement results. Compared to the C-H vibrational overlap between |
826 |
hexane and butanethiol, both of which have alkyl chains, that overlap |
827 |
between toluene and butanethiol is not so significant and thus does |
828 |
not have as much contribution to the ``Intramolecular Vibration |
829 |
Redistribution''[CITE HASE]. Conversely, extra degrees of freedom such |
830 |
as the C-H vibrations could yield higher heat exchange rate between |
831 |
these two phases and result in a much higher conductivity. |
832 |
|
833 |
Although the QSC model for Au is known to predict an overly low value |
834 |
for bulk metal gold conductivity\cite{kuang:164101}, our computational |
835 |
results for $G$ and $G^\prime$ do not seem to be affected by this |
836 |
drawback of the model for metal. Instead, our results suggest that the |
837 |
modeling of interfacial thermal transport behavior relies mainly on |
838 |
the accuracy of the interaction descriptions between components |
839 |
occupying the interfaces. |
840 |
|
841 |
\subsection{Mechanism of Interfacial Thermal Conductance Enhancement |
842 |
by Capping Agent} |
843 |
[OR: Vibrational Spectrum Study on Conductance Mechanism] |
844 |
|
845 |
[MAY INTRODUCE PROTOCOL IN METHODOLOGY/COMPUTATIONAL DETAIL, EQN'S] |
846 |
|
847 |
To investigate the mechanism of this interfacial thermal conductance, |
848 |
the vibrational spectra of various gold systems were obtained and are |
849 |
shown as in the upper panel of Fig. \ref{vibration}. To obtain these |
850 |
spectra, one first runs a simulation in the NVE ensemble and collects |
851 |
snapshots of configurations; these configurations are used to compute |
852 |
the velocity auto-correlation functions, which is used to construct a |
853 |
power spectrum via a Fourier transform. |
854 |
|
855 |
[MAY RELATE TO HASE'S] |
856 |
The gold surfaces covered by butanethiol molecules, compared to bare |
857 |
gold surfaces, exhibit an additional peak observed at the frequency of |
858 |
$\sim$170cm$^{-1}$, which is attributed to the S-Au bonding |
859 |
vibration. This vibration enables efficient thermal transport from |
860 |
surface Au layer to the capping agents. |
861 |
[MAY PUT IN OTHER SECTION] Simultaneously, as shown in |
862 |
the lower panel of Fig. \ref{vibration}, the large overlap of the |
863 |
vibration spectra of butanethiol and hexane in the All-Atom model, |
864 |
including the C-H vibration, also suggests high thermal exchange |
865 |
efficiency. The combination of these two effects produces the drastic |
866 |
interfacial thermal conductance enhancement in the All-Atom model. |
867 |
|
868 |
[NEED SEPARATE FIGURE. MAY NEED TO CONVERT TO JPEG] |
869 |
\begin{figure} |
870 |
\includegraphics[width=\linewidth]{vibration} |
871 |
\caption{Vibrational spectra obtained for gold in different |
872 |
environments.} |
873 |
\label{vibration} |
874 |
\end{figure} |
875 |
|
876 |
[MAY ADD COMPARISON OF G AND G', AU SLAB WIDTHS, ETC] |
877 |
% The results show that the two definitions used for $G$ yield |
878 |
% comparable values, though $G^\prime$ tends to be smaller. |
879 |
|
880 |
\section{Conclusions} |
881 |
The NIVS algorithm we developed has been applied to simulations of |
882 |
Au-butanethiol surfaces with organic solvents. This algorithm allows |
883 |
effective unphysical thermal flux transferred between the metal and |
884 |
the liquid phase. With the flux applied, we were able to measure the |
885 |
corresponding thermal gradient and to obtain interfacial thermal |
886 |
conductivities. Our simulations have seen significant conductance |
887 |
enhancement with the presence of capping agent, compared to the bare |
888 |
gold / liquid interfaces. The acoustic impedance mismatch between the |
889 |
metal and the liquid phase is effectively eliminated by proper capping |
890 |
agent. Furthermore, the coverage precentage of the capping agent plays |
891 |
an important role in the interfacial thermal transport process. |
892 |
|
893 |
Our measurement results, particularly of the UA models, agree with |
894 |
available experimental data. This indicates that our force field |
895 |
parameters have a nice description of the interactions between the |
896 |
particles at the interfaces. AA models tend to overestimate the |
897 |
interfacial thermal conductance in that the classically treated C-H |
898 |
vibration would be overly sampled. Compared to the AA models, the UA |
899 |
models have higher computational efficiency with satisfactory |
900 |
accuracy, and thus are preferable in interfacial thermal transport |
901 |
modelings. |
902 |
|
903 |
Vlugt {\it et al.} has investigated the surface thiol structures for |
904 |
nanocrystal gold and pointed out that they differs from those of the |
905 |
Au(111) surface\cite{vlugt:cpc2007154}. This difference might lead to |
906 |
change of interfacial thermal transport behavior as well. To |
907 |
investigate this problem, an effective means to introduce thermal flux |
908 |
and measure the corresponding thermal gradient is desirable for |
909 |
simulating structures with spherical symmetry. |
910 |
|
911 |
|
912 |
\section{Acknowledgments} |
913 |
Support for this project was provided by the National Science |
914 |
Foundation under grant CHE-0848243. Computational time was provided by |
915 |
the Center for Research Computing (CRC) at the University of Notre |
916 |
Dame. \newpage |
917 |
|
918 |
\bibliography{interfacial} |
919 |
|
920 |
\end{doublespace} |
921 |
\end{document} |
922 |
|