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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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\date{\today} |
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\maketitle |
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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 alkylthiolate 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 relatively 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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as: |
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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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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 withour 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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\begin{figure} |
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\includegraphics[width=\linewidth]{demoPic} |
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\caption{A sample showing how a metal slab has its (111) surface |
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covered by capping agent molecules and solvated by hexane.} |
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\label{demoPic} |
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\end{figure} |
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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{The 1st and 2nd derivatives of temperature profile can be |
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obtained with finite difference approximation.} |
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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 $35 \sim 60$\AA. |
291 |
|
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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 |
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derivatives of $T$ with respect to slab number $n$ can be directly |
309 |
used for $G^\prime$ calculations: |
310 |
\begin{equation} |
311 |
G^\prime = |J_z|\Big|\frac{\partial^2 T}{\partial z^2}\Big| |
312 |
\Big/\left(\frac{\partial T}{\partial z}\right)^2 |
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= |J_z|\Big|\frac{1}{d^2}\frac{\partial^2 T}{\partial n^2}\Big| |
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\Big/\left(\frac{1}{d}\frac{\partial T}{\partial n}\right)^2 |
315 |
= |J_z|\Big|\frac{\partial^2 T}{\partial n^2}\Big| |
316 |
\Big/\left(\frac{\partial T}{\partial n}\right)^2 |
317 |
\label{derivativeG2} |
318 |
\end{equation} |
319 |
|
320 |
\subsection{Force Field Parameters} |
321 |
Our simulations include various components. Therefore, force field |
322 |
parameter descriptions are needed for interactions both between the |
323 |
same type of particles and between particles of different species. |
324 |
|
325 |
The Au-Au interactions in metal lattice slab is described by the |
326 |
quantum Sutton-Chen (QSC) formulation\cite{PhysRevB.59.3527}. The QSC |
327 |
potentials include zero-point quantum corrections and are |
328 |
reparametrized for accurate surface energies compared to the |
329 |
Sutton-Chen potentials\cite{Chen90}. |
330 |
|
331 |
Figure \ref{demoMol} demonstrates how we name our pseudo-atoms of the |
332 |
organic solvent molecules in our simulations. |
333 |
|
334 |
\begin{figure} |
335 |
\includegraphics[width=\linewidth]{demoMol} |
336 |
\caption{Denomination of atoms or pseudo-atoms in our simulations: a) |
337 |
UA-hexane; b) AA-hexane; c) UA-toluene; d) AA-toluene.} |
338 |
\label{demoMol} |
339 |
\end{figure} |
340 |
|
341 |
For both solvent molecules, straight chain {\it n}-hexane and aromatic |
342 |
toluene, United-Atom (UA) and All-Atom (AA) models are used |
343 |
respectively. The TraPPE-UA |
344 |
parameters\cite{TraPPE-UA.alkanes,TraPPE-UA.alkylbenzenes} are used |
345 |
for our UA solvent molecules. In these models, pseudo-atoms are |
346 |
located at the carbon centers for alkyl groups. By eliminating |
347 |
explicit hydrogen atoms, these models are simple and computationally |
348 |
efficient, while maintains good accuracy. However, the TraPPE-UA for |
349 |
alkanes is known to predict a lower boiling point than experimental |
350 |
values. Considering that after an unphysical thermal flux is applied |
351 |
to a system, the temperature of ``hot'' area in the liquid phase would be |
352 |
significantly higher than the average, to prevent over heating and |
353 |
boiling of the liquid phase, the average temperature in our |
354 |
simulations should be much lower than the liquid boiling point. [MORE DISCUSSION] |
355 |
For UA-toluene model, rigid body constraints are applied, so that the |
356 |
benzene ring and the methyl-CRar bond are kept rigid. This would save |
357 |
computational time.[MORE DETAILS] |
358 |
|
359 |
Besides the TraPPE-UA models, AA models for both organic solvents are |
360 |
included in our studies as well. For hexane, the OPLS-AA\cite{OPLSAA} |
361 |
force field is used. [MORE DETAILS] |
362 |
For toluene, the United Force Field developed by Rapp\'{e} {\it et |
363 |
al.}\cite{doi:10.1021/ja00051a040} is adopted.[MORE DETAILS] |
364 |
|
365 |
The capping agent in our simulations, the butanethiol molecules can |
366 |
either use UA or AA model. The TraPPE-UA force fields includes |
367 |
parameters for thiol molecules\cite{TraPPE-UA.thiols} and are used for |
368 |
UA butanethiol model in our simulations. The OPLS-AA also provides |
369 |
parameters for alkyl thiols. However, alkyl thiols adsorbed on Au(111) |
370 |
surfaces do not have the hydrogen atom bonded to sulfur. To adapt this |
371 |
change and derive suitable parameters for butanethiol adsorbed on |
372 |
Au(111) surfaces, we adopt the S parameters from Luedtke and |
373 |
Landman\cite{landman:1998} and modify parameters for its neighbor C |
374 |
atom for charge balance in the molecule. Note that the model choice |
375 |
(UA or AA) of capping agent can be different from the |
376 |
solvent. Regardless of model choice, the force field parameters for |
377 |
interactions between capping agent and solvent can be derived using |
378 |
Lorentz-Berthelot Mixing Rule:[EQN'S] |
379 |
|
380 |
|
381 |
To describe the interactions between metal Au and non-metal capping |
382 |
agent and solvent particles, we refer to an adsorption study of alkyl |
383 |
thiols on gold surfaces by Vlugt {\it et |
384 |
al.}\cite{vlugt:cpc2007154} They fitted an effective Lennard-Jones |
385 |
form of potential parameters for the interaction between Au and |
386 |
pseudo-atoms CH$_x$ and S based on a well-established and widely-used |
387 |
effective potential of Hautman and Klein\cite{hautman:4994} for the |
388 |
Au(111) surface. As our simulations require the gold lattice slab to |
389 |
be non-rigid so that it could accommodate kinetic energy for thermal |
390 |
transport study purpose, the pair-wise form of potentials is |
391 |
preferred. |
392 |
|
393 |
Besides, the potentials developed from {\it ab initio} calculations by |
394 |
Leng {\it et al.}\cite{doi:10.1021/jp034405s} are adopted for the |
395 |
interactions between Au and aromatic C/H atoms in toluene.[MORE DETAILS] |
396 |
|
397 |
However, the Lennard-Jones parameters between Au and other types of |
398 |
particles in our simulations are not yet well-established. For these |
399 |
interactions, we attempt to derive their parameters using the Mixing |
400 |
Rule. To do this, the ``Metal-non-Metal'' (MnM) interaction parameters |
401 |
for Au is first extracted from the Au-CH$_x$ parameters by applying |
402 |
the Mixing Rule reversely. Table \ref{MnM} summarizes these ``MnM'' |
403 |
parameters in our simulations. |
404 |
|
405 |
\begin{table*} |
406 |
\begin{minipage}{\linewidth} |
407 |
\begin{center} |
408 |
\caption{Lennard-Jones parameters for Au-non-Metal |
409 |
interactions in our simulations.} |
410 |
|
411 |
\begin{tabular}{ccc} |
412 |
\hline\hline |
413 |
Non-metal atom & $\sigma$ & $\epsilon$ \\ |
414 |
(or pseudo-atom) & \AA & kcal/mol \\ |
415 |
\hline |
416 |
S & 2.40 & 8.465 \\ |
417 |
CH3 & 3.54 & 0.2146 \\ |
418 |
CH2 & 3.54 & 0.1749 \\ |
419 |
CT3 & 3.365 & 0.1373 \\ |
420 |
CT2 & 3.365 & 0.1373 \\ |
421 |
CTT & 3.365 & 0.1373 \\ |
422 |
HC & 2.865 & 0.09256 \\ |
423 |
CHar & 3.4625 & 0.1680 \\ |
424 |
CRar & 3.555 & 0.1604 \\ |
425 |
CA & 3.173 & 0.0640 \\ |
426 |
HA & 2.746 & 0.0414 \\ |
427 |
\hline\hline |
428 |
\end{tabular} |
429 |
\label{MnM} |
430 |
\end{center} |
431 |
\end{minipage} |
432 |
\end{table*} |
433 |
|
434 |
|
435 |
\section{Results and Discussions} |
436 |
[MAY HAVE A BRIEF SUMMARY] |
437 |
\subsection{How Simulation Parameters Affects $G$} |
438 |
[MAY NOT PUT AT FIRST] |
439 |
We have varied our protocol or other parameters of the simulations in |
440 |
order to investigate how these factors would affect the measurement of |
441 |
$G$'s. It turned out that while some of these parameters would not |
442 |
affect the results substantially, some other changes to the |
443 |
simulations would have a significant impact on the measurement |
444 |
results. |
445 |
|
446 |
In some of our simulations, we allowed $L_x$ and $L_y$ to change |
447 |
during equilibrating the liquid phase. Due to the stiffness of the Au |
448 |
slab, $L_x$ and $L_y$ would not change noticeably after |
449 |
equilibration. Although $L_z$ could fluctuate $\sim$1\% after a system |
450 |
is fully equilibrated in the NPT ensemble, this fluctuation, as well |
451 |
as those comparably smaller to $L_x$ and $L_y$, would not be magnified |
452 |
on the calculated $G$'s, as shown in Table \ref{AuThiolHexaneUA}. This |
453 |
insensivity to $L_i$ fluctuations allows reliable measurement of $G$'s |
454 |
without the necessity of extremely cautious equilibration process. |
455 |
|
456 |
As stated in our computational details, the spacing filled with |
457 |
solvent molecules can be chosen within a range. This allows some |
458 |
change of solvent molecule numbers for the same Au-butanethiol |
459 |
surfaces. We did this study on our Au-butanethiol/hexane |
460 |
simulations. Nevertheless, the results obtained from systems of |
461 |
different $N_{hexane}$ did not indicate that the measurement of $G$ is |
462 |
susceptible to this parameter. For computational efficiency concern, |
463 |
smaller system size would be preferable, given that the liquid phase |
464 |
structure is not affected. |
465 |
|
466 |
Our NIVS algorithm allows change of unphysical thermal flux both in |
467 |
direction and in quantity. This feature extends our investigation of |
468 |
interfacial thermal conductance. However, the magnitude of this |
469 |
thermal flux is not arbitary if one aims to obtain a stable and |
470 |
reliable thermal gradient. A temperature profile would be |
471 |
substantially affected by noise when $|J_z|$ has a much too low |
472 |
magnitude; while an excessively large $|J_z|$ that overwhelms the |
473 |
conductance capacity of the interface would prevent a thermal gradient |
474 |
to reach a stablized steady state. NIVS has the advantage of allowing |
475 |
$J$ to vary in a wide range such that the optimal flux range for $G$ |
476 |
measurement can generally be simulated by the algorithm. Within the |
477 |
optimal range, we were able to study how $G$ would change according to |
478 |
the thermal flux across the interface. For our simulations, we denote |
479 |
$J_z$ to be positive when the physical thermal flux is from the liquid |
480 |
to metal, and negative vice versa. The $G$'s measured under different |
481 |
$J_z$ is listed in Table \ref{AuThiolHexaneUA} and [REF]. These |
482 |
results do not suggest that $G$ is dependent on $J_z$ within this flux |
483 |
range. The linear response of flux to thermal gradient simplifies our |
484 |
investigations in that we can rely on $G$ measurement with only a |
485 |
couple $J_z$'s and do not need to test a large series of fluxes. |
486 |
|
487 |
%ADD MORE TO TABLE |
488 |
\begin{table*} |
489 |
\begin{minipage}{\linewidth} |
490 |
\begin{center} |
491 |
\caption{Computed interfacial thermal conductivity ($G$ and |
492 |
$G^\prime$) values for the 100\% covered Au-butanethiol/hexane |
493 |
interfaces with UA model and different hexane molecule numbers |
494 |
at different temperatures using a range of energy fluxes.} |
495 |
|
496 |
\begin{tabular}{cccccccc} |
497 |
\hline\hline |
498 |
$\langle T\rangle$ & & $L_x$ & $L_y$ & $L_z$ & $J_z$ & |
499 |
$G$ & $G^\prime$ \\ |
500 |
(K) & $N_{hexane}$ & \multicolumn{3}{c}{(\AA)} & (GW/m$^2$) & |
501 |
\multicolumn{2}{c}{(MW/m$^2$/K)} \\ |
502 |
\hline |
503 |
200 & 266 & 29.86 & 25.80 & 113.1 & -0.96 & |
504 |
102() & 80.0() \\ |
505 |
& 200 & 29.84 & 25.81 & 93.9 & 1.92 & |
506 |
129() & 87.3() \\ |
507 |
& & 29.84 & 25.81 & 95.3 & 1.93 & |
508 |
131() & 77.5() \\ |
509 |
& 166 & 29.84 & 25.81 & 85.7 & 0.97 & |
510 |
115() & 69.3() \\ |
511 |
& & & & & 1.94 & |
512 |
125() & 87.1() \\ |
513 |
250 & 200 & 29.84 & 25.87 & 106.8 & 0.96 & |
514 |
81.8() & 67.0() \\ |
515 |
& 166 & 29.87 & 25.84 & 94.8 & 0.98 & |
516 |
79.0() & 62.9() \\ |
517 |
& & 29.84 & 25.85 & 95.0 & 1.44 & |
518 |
76.2() & 64.8() \\ |
519 |
\hline\hline |
520 |
\end{tabular} |
521 |
\label{AuThiolHexaneUA} |
522 |
\end{center} |
523 |
\end{minipage} |
524 |
\end{table*} |
525 |
|
526 |
Furthermore, we also attempted to increase system average temperatures |
527 |
to above 200K. These simulations are first equilibrated in the NPT |
528 |
ensemble under normal pressure. As stated above, the TraPPE-UA model |
529 |
for hexane tends to predict a lower boiling point. In our simulations, |
530 |
hexane had diffculty to remain in liquid phase when NPT equilibration |
531 |
temperature is higher than 250K. Additionally, the equilibrated liquid |
532 |
hexane density under 250K becomes lower than experimental value. This |
533 |
expanded liquid phase leads to lower contact between hexane and |
534 |
butanethiol as well.[MAY NEED FIGURE] And this reduced contact would |
535 |
probably be accountable for a lower interfacial thermal conductance, |
536 |
as shown in Table \ref{AuThiolHexaneUA}. |
537 |
|
538 |
A similar study for TraPPE-UA toluene agrees with the above result as |
539 |
well. Having a higher boiling point, toluene tends to remain liquid in |
540 |
our simulations even equilibrated under 300K in NPT |
541 |
ensembles. Furthermore, the expansion of the toluene liquid phase is |
542 |
not as significant as that of the hexane. This prevents severe |
543 |
decrease of liquid-capping agent contact and the results (Table |
544 |
\ref{AuThiolToluene}) show only a slightly decreased interface |
545 |
conductance. Therefore, solvent-capping agent contact should play an |
546 |
important role in the thermal transport process across the interface |
547 |
in that higher degree of contact could yield increased conductance. |
548 |
|
549 |
[ADD Lxyz AND ERROR ESTIMATE TO TABLE] |
550 |
\begin{table*} |
551 |
\begin{minipage}{\linewidth} |
552 |
\begin{center} |
553 |
\caption{Computed interfacial thermal conductivity ($G$ and |
554 |
$G^\prime$) values for a 90\% coverage Au-butanethiol/toluene |
555 |
interface at different temperatures using a range of energy |
556 |
fluxes.} |
557 |
|
558 |
\begin{tabular}{cccc} |
559 |
\hline\hline |
560 |
$\langle T\rangle$ & $J_z$ & $G$ & $G^\prime$ \\ |
561 |
(K) & (GW/m$^2$) & \multicolumn{2}{c}{(MW/m$^2$/K)} \\ |
562 |
\hline |
563 |
200 & -1.86 & 180() & 135() \\ |
564 |
& 2.15 & 204() & 113() \\ |
565 |
& -3.93 & 175() & 114() \\ |
566 |
300 & -1.91 & 143() & 125() \\ |
567 |
& -4.19 & 134() & 113() \\ |
568 |
\hline\hline |
569 |
\end{tabular} |
570 |
\label{AuThiolToluene} |
571 |
\end{center} |
572 |
\end{minipage} |
573 |
\end{table*} |
574 |
|
575 |
Besides lower interfacial thermal conductance, surfaces in relatively |
576 |
high temperatures are susceptible to reconstructions, when |
577 |
butanethiols have a full coverage on the Au(111) surface. These |
578 |
reconstructions include surface Au atoms migrated outward to the S |
579 |
atom layer, and butanethiol molecules embedded into the original |
580 |
surface Au layer. The driving force for this behavior is the strong |
581 |
Au-S interactions in our simulations. And these reconstructions lead |
582 |
to higher ratio of Au-S attraction and thus is energetically |
583 |
favorable. Furthermore, this phenomenon agrees with experimental |
584 |
results\cite{doi:10.1021/j100035a033,doi:10.1021/la026493y}. Vlugt |
585 |
{\it et al.} had kept their Au(111) slab rigid so that their |
586 |
simulations can reach 300K without surface reconstructions. Without |
587 |
this practice, simulating 100\% thiol covered interfaces under higher |
588 |
temperatures could hardly avoid surface reconstructions. However, our |
589 |
measurement is based on assuming homogeneity on $x$ and $y$ dimensions |
590 |
so that measurement of $T$ at particular $z$ would be an effective |
591 |
average of the particles of the same type. Since surface |
592 |
reconstructions could eliminate the original $x$ and $y$ dimensional |
593 |
homogeneity, measurement of $G$ is more difficult to conduct under |
594 |
higher temperatures. Therefore, most of our measurements are |
595 |
undertaken at $\langle T\rangle\sim$200K. |
596 |
|
597 |
However, when the surface is not completely covered by butanethiols, |
598 |
the simulated system is more resistent to the reconstruction |
599 |
above. Our Au-butanethiol/toluene system did not see this phenomena |
600 |
even at $\langle T\rangle\sim$300K. The Au(111) surfaces have a 90\% coverage of |
601 |
butanethiols and have empty three-fold sites. These empty sites could |
602 |
help prevent surface reconstruction in that they provide other means |
603 |
of capping agent relaxation. It is observed that butanethiols can |
604 |
migrate to their neighbor empty sites during a simulation. Therefore, |
605 |
we were able to obtain $G$'s for these interfaces even at a relatively |
606 |
high temperature without being affected by surface reconstructions. |
607 |
|
608 |
\subsection{Influence of Capping Agent Coverage on $G$} |
609 |
To investigate the influence of butanethiol coverage on interfacial |
610 |
thermal conductance, a series of different coverage Au-butanethiol |
611 |
surfaces is prepared and solvated with various organic |
612 |
molecules. These systems are then equilibrated and their interfacial |
613 |
thermal conductivity are measured with our NIVS algorithm. Table |
614 |
\ref{tlnUhxnUhxnD} lists these results for direct comparison between |
615 |
different coverages of butanethiol. To study the isotope effect in |
616 |
interfacial thermal conductance, deuterated UA-hexane is included as |
617 |
well. |
618 |
|
619 |
It turned out that with partial covered butanethiol on the Au(111) |
620 |
surface, the derivative definition for $G$ (Eq. \ref{derivativeG}) has |
621 |
difficulty to apply, due to the difficulty in locating the maximum of |
622 |
change of $\lambda$. Instead, the discrete definition |
623 |
(Eq. \ref{discreteG}) is easier to apply, as max($\Delta T$) can still |
624 |
be well-defined. Therefore, $G$'s (not $G^\prime$) are used for this |
625 |
section. |
626 |
|
627 |
From Table \ref{tlnUhxnUhxnD}, one can see the significance of the |
628 |
presence of capping agents. Even when a fraction of the Au(111) |
629 |
surface sites are covered with butanethiols, the conductivity would |
630 |
see an enhancement by at least a factor of 3. This indicates the |
631 |
important role cappping agent is playing for thermal transport |
632 |
phenomena on metal/organic solvent surfaces. |
633 |
|
634 |
Interestingly, as one could observe from our results, the maximum |
635 |
conductance enhancement (largest $G$) happens while the surfaces are |
636 |
about 75\% covered with butanethiols. This again indicates that |
637 |
solvent-capping agent contact has an important role of the thermal |
638 |
transport process. Slightly lower butanethiol coverage allows small |
639 |
gaps between butanethiols to form. And these gaps could be filled with |
640 |
solvent molecules, which acts like ``heat conductors'' on the |
641 |
surface. The higher degree of interaction between these solvent |
642 |
molecules and capping agents increases the enhancement effect and thus |
643 |
produces a higher $G$ than densely packed butanethiol arrays. However, |
644 |
once this maximum conductance enhancement is reached, $G$ decreases |
645 |
when butanethiol coverage continues to decrease. Each capping agent |
646 |
molecule reaches its maximum capacity for thermal |
647 |
conductance. Therefore, even higher solvent-capping agent contact |
648 |
would not offset this effect. Eventually, when butanethiol coverage |
649 |
continues to decrease, solvent-capping agent contact actually |
650 |
decreases with the disappearing of butanethiol molecules. In this |
651 |
case, $G$ decrease could not be offset but instead accelerated. |
652 |
|
653 |
A comparison of the results obtained from differenet organic solvents |
654 |
can also provide useful information of the interfacial thermal |
655 |
transport process. The deuterated hexane (UA) results do not appear to |
656 |
be much different from those of normal hexane (UA), given that |
657 |
butanethiol (UA) is non-deuterated for both solvents. These UA model |
658 |
studies, even though eliminating C-H vibration samplings, still have |
659 |
C-C vibrational frequencies different from each other. However, these |
660 |
differences in the infrared range do not seem to produce an observable |
661 |
difference for the results of $G$. [MAY NEED FIGURE] |
662 |
|
663 |
Furthermore, results for rigid body toluene solvent, as well as other |
664 |
UA-hexane solvents, are reasonable within the general experimental |
665 |
ranges[CITATIONS]. This suggests that explicit hydrogen might not be a |
666 |
required factor for modeling thermal transport phenomena of systems |
667 |
such as Au-thiol/organic solvent. |
668 |
|
669 |
However, results for Au-butanethiol/toluene do not show an identical |
670 |
trend with those for Au-butanethiol/hexane in that $G$'s remain at |
671 |
approximately the same magnitue when butanethiol coverage differs from |
672 |
25\% to 75\%. This might be rooted in the molecule shape difference |
673 |
for plane-like toluene and chain-like {\it n}-hexane. Due to this |
674 |
difference, toluene molecules have more difficulty in occupying |
675 |
relatively small gaps among capping agents when their coverage is not |
676 |
too low. Therefore, the solvent-capping agent contact may keep |
677 |
increasing until the capping agent coverage reaches a relatively low |
678 |
level. This becomes an offset for decreasing butanethiol molecules on |
679 |
its effect to the process of interfacial thermal transport. Thus, one |
680 |
can see a plateau of $G$ vs. butanethiol coverage in our results. |
681 |
|
682 |
[NEED ERROR ESTIMATE, MAY ALSO PUT J HERE] |
683 |
\begin{table*} |
684 |
\begin{minipage}{\linewidth} |
685 |
\begin{center} |
686 |
\caption{Computed interfacial thermal conductivity ($G$) values |
687 |
for the Au-butanethiol/solvent interface with various UA |
688 |
models and different capping agent coverages at $\langle |
689 |
T\rangle\sim$200K using certain energy flux respectively.} |
690 |
|
691 |
\begin{tabular}{cccc} |
692 |
\hline\hline |
693 |
Thiol & \multicolumn{3}{c}{$G$(MW/m$^2$/K)} \\ |
694 |
coverage (\%) & hexane & hexane(D) & toluene \\ |
695 |
\hline |
696 |
0.0 & 46.5() & 43.9() & 70.1() \\ |
697 |
25.0 & 151() & 153() & 249() \\ |
698 |
50.0 & 172() & 182() & 214() \\ |
699 |
75.0 & 242() & 229() & 244() \\ |
700 |
88.9 & 178() & - & - \\ |
701 |
100.0 & 137() & 153() & 187() \\ |
702 |
\hline\hline |
703 |
\end{tabular} |
704 |
\label{tlnUhxnUhxnD} |
705 |
\end{center} |
706 |
\end{minipage} |
707 |
\end{table*} |
708 |
|
709 |
\subsection{Influence of Chosen Molecule Model on $G$} |
710 |
[MAY COMBINE W MECHANISM STUDY] |
711 |
|
712 |
In addition to UA solvent/capping agent models, AA models are included |
713 |
in our simulations as well. Besides simulations of the same (UA or AA) |
714 |
model for solvent and capping agent, different models can be applied |
715 |
to different components. Furthermore, regardless of models chosen, |
716 |
either the solvent or the capping agent can be deuterated, similar to |
717 |
the previous section. Table \ref{modelTest} summarizes the results of |
718 |
these studies. |
719 |
|
720 |
[MORE DATA; ERROR ESTIMATE] |
721 |
\begin{table*} |
722 |
\begin{minipage}{\linewidth} |
723 |
\begin{center} |
724 |
|
725 |
\caption{Computed interfacial thermal conductivity ($G$ and |
726 |
$G^\prime$) values for interfaces using various models for |
727 |
solvent and capping agent (or without capping agent) at |
728 |
$\langle T\rangle\sim$200K.} |
729 |
|
730 |
\begin{tabular}{ccccc} |
731 |
\hline\hline |
732 |
Butanethiol model & Solvent & $J_z$ & $G$ & $G^\prime$ \\ |
733 |
(or bare surface) & model & (GW/m$^2$) & |
734 |
\multicolumn{2}{c}{(MW/m$^2$/K)} \\ |
735 |
\hline |
736 |
UA & AA hexane & 1.94 & 135() & 129() \\ |
737 |
& & 2.86 & 126() & 115() \\ |
738 |
& AA toluene & 1.89 & 200() & 149() \\ |
739 |
AA & UA hexane & 1.94 & 116() & 129() \\ |
740 |
& AA hexane & 3.76 & 451() & 378() \\ |
741 |
& & 4.71 & 432() & 334() \\ |
742 |
& AA toluene & 3.79 & 487() & 290() \\ |
743 |
AA(D) & UA hexane & 1.94 & 158() & 172() \\ |
744 |
bare & AA hexane & 0.96 & 31.0() & 29.4() \\ |
745 |
\hline\hline |
746 |
\end{tabular} |
747 |
\label{modelTest} |
748 |
\end{center} |
749 |
\end{minipage} |
750 |
\end{table*} |
751 |
|
752 |
To facilitate direct comparison, the same system with differnt models |
753 |
for different components uses the same length scale for their |
754 |
simulation cells. Without the presence of capping agent, using |
755 |
different models for hexane yields similar results for both $G$ and |
756 |
$G^\prime$, and these two definitions agree with eath other very |
757 |
well. This indicates very weak interaction between the metal and the |
758 |
solvent, and is a typical case for acoustic impedance mismatch between |
759 |
these two phases. |
760 |
|
761 |
As for Au(111) surfaces completely covered by butanethiols, the choice |
762 |
of models for capping agent and solvent could impact the measurement |
763 |
of $G$ and $G^\prime$ quite significantly. For Au-butanethiol/hexane |
764 |
interfaces, using AA model for both butanethiol and hexane yields |
765 |
substantially higher conductivity values than using UA model for at |
766 |
least one component of the solvent and capping agent, which exceeds |
767 |
the upper bond of experimental value range. This is probably due to |
768 |
the classically treated C-H vibrations in the AA model, which should |
769 |
not be appreciably populated at normal temperatures. In comparison, |
770 |
once either the hexanes or the butanethiols are deuterated, one can |
771 |
see a significantly lower $G$ and $G^\prime$. In either of these |
772 |
cases, the C-H(D) vibrational overlap between the solvent and the |
773 |
capping agent is removed. [MAY NEED FIGURE] Conclusively, the |
774 |
improperly treated C-H vibration in the AA model produced |
775 |
over-predicted results accordingly. Compared to the AA model, the UA |
776 |
model yields more reasonable results with higher computational |
777 |
efficiency. |
778 |
|
779 |
However, for Au-butanethiol/toluene interfaces, having the AA |
780 |
butanethiol deuterated did not yield a significant change in the |
781 |
measurement results. |
782 |
. , so extra degrees of freedom |
783 |
such as the C-H vibrations could enhance heat exchange between these |
784 |
two phases and result in a much higher conductivity. |
785 |
|
786 |
|
787 |
Although the QSC model for Au is known to predict an overly low value |
788 |
for bulk metal gold conductivity[CITE NIVSRNEMD], our computational |
789 |
results for $G$ and $G^\prime$ do not seem to be affected by this |
790 |
drawback of the model for metal. Instead, the modeling of interfacial |
791 |
thermal transport behavior relies mainly on an accurate description of |
792 |
the interactions between components occupying the interfaces. |
793 |
|
794 |
\subsection{Mechanism of Interfacial Thermal Conductance Enhancement |
795 |
by Capping Agent} |
796 |
%OR\subsection{Vibrational spectrum study on conductance mechanism} |
797 |
|
798 |
[MAY INTRODUCE PROTOCOL IN METHODOLOGY/COMPUTATIONAL DETAIL, EQN'S] |
799 |
|
800 |
To investigate the mechanism of this interfacial thermal conductance, |
801 |
the vibrational spectra of various gold systems were obtained and are |
802 |
shown as in the upper panel of Fig. \ref{vibration}. To obtain these |
803 |
spectra, one first runs a simulation in the NVE ensemble and collects |
804 |
snapshots of configurations; these configurations are used to compute |
805 |
the velocity auto-correlation functions, which is used to construct a |
806 |
power spectrum via a Fourier transform. |
807 |
|
808 |
The gold surfaces covered by |
809 |
butanethiol molecules, compared to bare gold surfaces, exhibit an |
810 |
additional peak observed at a frequency of $\sim$170cm$^{-1}$, which |
811 |
is attributed to the vibration of the S-Au bond. This vibration |
812 |
enables efficient thermal transport from surface Au atoms to the |
813 |
capping agents. Simultaneously, as shown in the lower panel of |
814 |
Fig. \ref{vibration}, the large overlap of the vibration spectra of |
815 |
butanethiol and hexane in the all-atom model, including the C-H |
816 |
vibration, also suggests high thermal exchange efficiency. The |
817 |
combination of these two effects produces the drastic interfacial |
818 |
thermal conductance enhancement in the all-atom model. |
819 |
|
820 |
[MAY NEED TO CONVERT TO JPEG] |
821 |
\begin{figure} |
822 |
\includegraphics[width=\linewidth]{vibration} |
823 |
\caption{Vibrational spectra obtained for gold in different |
824 |
environments (upper panel) and for Au/thiol/hexane simulation in |
825 |
all-atom model (lower panel).} |
826 |
\label{vibration} |
827 |
\end{figure} |
828 |
|
829 |
[COMPARISON OF TWO G'S; AU SLAB WIDTHS; ETC] |
830 |
% The results show that the two definitions used for $G$ yield |
831 |
% comparable values, though $G^\prime$ tends to be smaller. |
832 |
|
833 |
\section{Conclusions} |
834 |
The NIVS algorithm we developed has been applied to simulations of |
835 |
Au-butanethiol surfaces with organic solvents. This algorithm allows |
836 |
effective unphysical thermal flux transferred between the metal and |
837 |
the liquid phase. With the flux applied, we were able to measure the |
838 |
corresponding thermal gradient and to obtain interfacial thermal |
839 |
conductivities. Our simulations have seen significant conductance |
840 |
enhancement with the presence of capping agent, compared to the bare |
841 |
gold/liquid interfaces. The acoustic impedance mismatch between the |
842 |
metal and the liquid phase is effectively eliminated by proper capping |
843 |
agent. Furthermore, the coverage precentage of the capping agent plays |
844 |
an important role in the interfacial thermal transport process. |
845 |
|
846 |
Our measurement results, particularly of the UA models, agree with |
847 |
available experimental data. This indicates that our force field |
848 |
parameters have a nice description of the interactions between the |
849 |
particles at the interfaces. AA models tend to overestimate the |
850 |
interfacial thermal conductance in that the classically treated C-H |
851 |
vibration would be overly sampled. Compared to the AA models, the UA |
852 |
models have higher computational efficiency with satisfactory |
853 |
accuracy, and thus are preferable in interfacial thermal transport |
854 |
modelings. |
855 |
|
856 |
Vlugt {\it et al.} has investigated the surface thiol structures for |
857 |
nanocrystal gold and pointed out that they differs from those of the |
858 |
Au(111) surface\cite{vlugt:cpc2007154}. This difference might lead to |
859 |
change of interfacial thermal transport behavior as well. To |
860 |
investigate this problem, an effective means to introduce thermal flux |
861 |
and measure the corresponding thermal gradient is desirable for |
862 |
simulating structures with spherical symmetry. |
863 |
|
864 |
|
865 |
\section{Acknowledgments} |
866 |
Support for this project was provided by the National Science |
867 |
Foundation under grant CHE-0848243. Computational time was provided by |
868 |
the Center for Research Computing (CRC) at the University of Notre |
869 |
Dame. \newpage |
870 |
|
871 |
\bibliography{interfacial} |
872 |
|
873 |
\end{doublespace} |
874 |
\end{document} |
875 |
|