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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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With the Non-Isotropic Velocity Scaling (NIVS) approach to Reverse |
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Non-Equilibrium Molecular Dynamics (RNEMD) it is possible to impose |
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an unphysical thermal flux between different regions of |
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inhomogeneous systems such as solid / liquid interfaces. We have |
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applied NIVS to compute the interfacial thermal conductance at a |
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metal / organic solvent interface that has been chemically capped by |
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butanethiol molecules. Our calculations suggest that the acoustic |
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impedance mismatch between the metal and liquid phases is |
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effectively reduced by the capping agents, leading to a greatly |
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enhanced conductivity at the interface. Specifically, the chemical |
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bond between the metal and the capping agent introduces a |
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vibrational overlap that is not present without the capping agent, |
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and the overlap between the vibrational spectra (metal to cap, cap |
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to solvent) provides a mechanism for rapid thermal transport across |
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the interface. Our calculations also suggest that this is a |
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non-monotonic function of the fractional coverage of the surface, as |
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moderate coverages allow convective heat transport of solvent |
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molecules that have been in close contact with the capping agent. |
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\end{abstract} |
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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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Due to the importance of heat flow (and heat removal) in |
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nanotechnology, interfacial thermal conductance has been studied |
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extensively both experimentally and computationally.\cite{cahill:793} |
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Nanoscale materials have a significant fraction of their atoms at |
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interfaces, and the chemical details of these interfaces govern the |
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thermal transport properties. Furthermore, the interfaces are often |
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heterogeneous (e.g. solid - liquid), which provides a challenge to |
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computational methods which have been developed for homogeneous or |
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bulk systems. |
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|
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Experimentally, the thermal properties of a number of interfaces have |
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been investigated. Cahill and coworkers studied nanoscale thermal |
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transport from metal nanoparticle/fluid interfaces, to epitaxial |
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TiN/single crystal oxides interfaces, and hydrophilic and hydrophobic |
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interfaces between water and solids with different self-assembled |
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monolayers.\cite{Wilson:2002uq,PhysRevB.67.054302,doi:10.1021/jp048375k,PhysRevLett.96.186101} |
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Wang {\it et al.} studied heat transport through long-chain |
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hydrocarbon monolayers on gold substrate at individual molecular |
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level,\cite{Wang10082007} Schmidt {\it et al.} studied the role of |
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cetyltrimethylammonium bromide (CTAB) on the thermal transport between |
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gold nanorods and solvent,\cite{doi:10.1021/jp8051888} and Juv\'e {\it |
97 |
et al.} studied the cooling dynamics, which is controlled by thermal |
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interface resistance of glass-embedded metal |
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nanoparticles.\cite{PhysRevB.80.195406} Although interfaces are |
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normally considered barriers for heat transport, Alper {\it et al.} |
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suggested that specific ligands (capping agents) could completely |
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eliminate this barrier |
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($G\rightarrow\infty$).\cite{doi:10.1021/la904855s} |
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|
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Theoretical and computational models have also been used to study the |
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interfacial thermal transport in order to gain an understanding of |
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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 have yet to be studied with their |
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approach. The comparatively low thermal flux through interfaces is |
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difficult to measure with Equilibrium |
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MD\cite{doi:10.1080/0026897031000068578} or forward NEMD simulation |
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methods. Therefore, the Reverse NEMD (RNEMD) |
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methods\cite{MullerPlathe:1997xw,kuang:164101} would be advantageous |
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in that they {\it apply} the difficult to measure quantity (flux), |
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while {\it measuring} the easily-computed quantity (the thermal |
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gradient). This is particularly true for inhomogeneous interfaces |
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where it would not be clear how to apply a gradient {\it a priori}. |
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Garde and coworkers\cite{garde:nl2005,garde:PhysRevLett2009} applied |
124 |
this approach to various liquid interfaces and studied how thermal |
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conductance (or resistance) is dependent on chemical details of a |
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number of hydrophobic and hydrophilic aqueous interfaces. {\bf And |
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Luo {\it et al.} studied the thermal conductance of Au-SAM-Au |
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junctions using the same approach, with comparison to a constant |
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temperature difference method\cite{Luo20101}. While this latter |
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approach establishes more thermal distributions compared to the |
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former RNEMD methods, it does not guarantee momentum or kinetic |
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energy conservations.} |
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|
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Recently, we have developed a Non-Isotropic Velocity Scaling (NIVS) |
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algorithm for RNEMD simulations\cite{kuang:164101}. This algorithm |
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retains the desirable features of RNEMD (conservation of linear |
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momentum and total energy, compatibility with periodic boundary |
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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. {\bf To our knowledge, few previous MD inverstigations |
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have been found to address to these systems yet.} Different models |
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were used for both the capping agent and the solvent force field |
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parameters. Using the NIVS algorithm, the thermal transport across |
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these interfaces was studied and the underlying mechanism for the |
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phenomena was investigated. |
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|
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\section{Methodology} |
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\subsection{Imposed-Flux Methods in MD Simulations} |
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Steady state MD simulations have an advantage in that not many |
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trajectories are needed to study the relationship between thermal flux |
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and thermal gradients. For systems with low interfacial conductance, |
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one must have a method capable of generating or measuring relatively |
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small fluxes, compared to those required for bulk conductivity. This |
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requirement makes the calculation even more difficult for |
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slowly-converging equilibrium methods.\cite{Viscardy:2007lq} Forward |
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NEMD methods impose a gradient (and measure a flux), but at interfaces |
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it is not clear what behavior should be imposed at the boundaries |
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between materials. Imposed-flux reverse non-equilibrium |
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methods\cite{MullerPlathe:1997xw} set the flux {\it a priori} and |
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the thermal response becomes an easy-to-measure quantity. Although |
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M\"{u}ller-Plathe's original momentum swapping approach can be used |
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for exchanging energy between particles of different identity, the |
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kinetic energy transfer efficiency is affected by the mass difference |
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between the particles, which limits its application on heterogeneous |
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interfacial systems. |
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|
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The non-isotropic velocity scaling (NIVS) \cite{kuang:164101} approach |
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to non-equilibrium MD simulations is able to impose a wide range of |
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kinetic energy fluxes without obvious perturbation to the velocity |
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distributions of the simulated systems. Furthermore, this approach has |
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the advantage in heterogeneous interfaces in that kinetic energy flux |
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can be applied between regions of particles of arbitrary 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 matrices. To |
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determine these scaling factors in the matrices, 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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|
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For an interface with relatively low interfacial conductance, and a |
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thermal flux between two distinct bulk regions, the regions on either |
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side of the interface rapidly come to a state in which the two phases |
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have relatively homogeneous (but distinct) temperatures. The |
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interfacial thermal conductivity $G$ can therefore be approximated as: |
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\begin{equation} |
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G = \frac{E_{total}}{2 t L_x L_y \left( \langle T_\mathrm{hot}\rangle - |
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\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 total imposed non-physical kinetic energy |
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transfer during the simulation and ${\langle T_\mathrm{hot}\rangle}$ |
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and ${\langle T_\mathrm{cold}\rangle}$ are the average observed |
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temperature of the two separated phases. For an applied flux $J_z$ |
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operating over a simulation time $t$ on a periodically-replicated slab |
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of dimensions $L_x \times L_y$, $E_{total} = J_z *(t)*(2 L_x L_y)$. |
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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 $G$. One common way is to assume the temperature is |
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discrete on the two sides of the interface. $G$ can be calculated |
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using the applied thermal flux $J$ and the maximum temperature |
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difference measured along the thermal gradient max($\Delta T$), which |
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occurs at the Gibbs dividing surface (Figure \ref{demoPic}). This is |
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known as the Kapitza conductance, which is the inverse of the Kapitza |
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resistance. |
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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 gradient. In bulk liquids, |
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this gradient typically has a single slope, but in interfacial |
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systems, there are distinct thermal conductivity domains. The |
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interfacial conductance, $G$ is found by measuring the temperature |
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gap at the Gibbs dividing surface, or by using second derivatives of |
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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 reaches 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 temperature profiles obtained from simulation, 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 calculate $G^\prime$. In what follows, both |
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definitions have been used, and are compared in the results. |
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|
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To investigate the interfacial conductivity at metal / solvent |
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interfaces, we have modeled a metal slab with its (111) surfaces |
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perpendicular to the $z$-axis of our simulation cells. The metal slab |
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has been prepared both with and without capping agents on the exposed |
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surface, and has been solvated with simple organic solvents, as |
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illustrated in Figure \ref{gradT}. |
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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 obtained a temperature profile and its spatial |
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derivatives. Figure \ref{gradT} shows how an applied thermal flux can |
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be used to obtain the 1st and 2nd derivatives of the temperature |
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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 (111) / butanethiol / hexane interfacial |
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system with the temperature profile after a kinetic energy flux has |
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been imposed. Note that the largest temperature jump in the thermal |
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profile (corresponding to the lowest interfacial conductance) is at |
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the interface between the butanethiol molecules (blue) and the |
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solvent (grey). First and second derivatives of the temperature |
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profile are obtained using a finite difference approximation (lower |
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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 simulations. |
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Metal slabs of 6 or 11 layers of Au atoms were first equilibrated |
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under atmospheric pressure (1 atm) and 200K. After equilibration, |
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butanethiol capping agents were placed at three-fold hollow sites on |
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the Au(111) surfaces. These sites are either {\it fcc} or {\it |
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hcp} sites, although Hase {\it et al.} found that they are |
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equivalent in a heat transfer process,\cite{hase:2010} so we did not |
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distinguish between these sites in our study. The maximum butanethiol |
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capacity on Au surface is $1/3$ of the total number of surface Au |
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atoms, and the packing forms a $(\sqrt{3}\times\sqrt{3})R30^\circ$ |
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structure\cite{doi:10.1021/ja00008a001,doi:10.1021/cr9801317}. A |
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series of lower coverages was also prepared by eliminating |
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butanethiols from the higher coverage surface in a regular manner. The |
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lower coverages were prepared in order to study the relation between |
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coverage and 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 does 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. |
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|
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After the modified Au-butanethiol surface systems were equilibrated in |
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the canonical (NVT) ensemble, organic solvent molecules were packed in |
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the 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 which has a planar shape (toluene), and one |
315 |
which has similar vibrational frequencies to the capping agent and |
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chain-like shape ({\it n}-hexane). |
317 |
|
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The simulation cells were not particularly extensive along the |
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$z$-axis, as a very long length scale for the thermal gradient may |
320 |
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. The spacing |
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between periodic images of the gold interfaces is $45 \sim 75$\AA in |
327 |
our simulations. |
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|
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The initial configurations generated are further equilibrated with the |
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$x$ and $y$ dimensions fixed, only allowing the $z$-length scale to |
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change. This is to ensure that the equilibration of liquid phase does |
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not affect the metal's crystalline structure. Comparisons were made |
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with simulations that allowed changes of $L_x$ and $L_y$ during NPT |
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equilibration. No substantial changes in the box geometry were noticed |
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in these simulations. After ensuring the liquid phase reaches |
336 |
equilibrium at atmospheric pressure (1 atm), further equilibration was |
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carried out under canonical (NVT) and microcanonical (NVE) ensembles. |
338 |
|
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After the systems reach equilibrium, NIVS was used to impose an |
340 |
unphysical thermal flux between the metal and the liquid phases. Most |
341 |
of our simulations were done under an average temperature of |
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$\sim$200K. Therefore, thermal flux usually came from the metal to the |
343 |
liquid so that the liquid has a higher temperature and would not |
344 |
freeze due to lowered temperatures. After this induced temperature |
345 |
gradient had stabilized, the temperature profile of the simulation cell |
346 |
was recorded. To do this, the simulation cell is divided evenly into |
347 |
$N$ slabs along the $z$-axis. The average temperatures of each slab |
348 |
are recorded for 1$\sim$2 ns. When the slab width $d$ of each slab is |
349 |
the same, the derivatives of $T$ with respect to slab number $n$ can |
350 |
be directly used for $G^\prime$ calculations: \begin{equation} |
351 |
G^\prime = |J_z|\Big|\frac{\partial^2 T}{\partial z^2}\Big| |
352 |
\Big/\left(\frac{\partial T}{\partial z}\right)^2 |
353 |
= |J_z|\Big|\frac{1}{d^2}\frac{\partial^2 T}{\partial n^2}\Big| |
354 |
\Big/\left(\frac{1}{d}\frac{\partial T}{\partial n}\right)^2 |
355 |
= |J_z|\Big|\frac{\partial^2 T}{\partial n^2}\Big| |
356 |
\Big/\left(\frac{\partial T}{\partial n}\right)^2 |
357 |
\label{derivativeG2} |
358 |
\end{equation} |
359 |
|
360 |
All of the above simulation procedures use a time step of 1 fs. Each |
361 |
equilibration stage took a minimum of 100 ps, although in some cases, |
362 |
longer equilibration stages were utilized. |
363 |
|
364 |
\subsection{Force Field Parameters} |
365 |
Our simulations include a number of chemically distinct components. |
366 |
Figure \ref{demoMol} demonstrates the sites defined for both |
367 |
United-Atom and All-Atom models of the organic solvent and capping |
368 |
agents in our simulations. Force field parameters are needed for |
369 |
interactions both between the same type of particles and between |
370 |
particles of different species. |
371 |
|
372 |
\begin{figure} |
373 |
\includegraphics[width=\linewidth]{structures} |
374 |
\caption{Structures of the capping agent and solvents utilized in |
375 |
these simulations. The chemically-distinct sites (a-e) are expanded |
376 |
in terms of constituent atoms for both United Atom (UA) and All Atom |
377 |
(AA) force fields. Most parameters are from References |
378 |
\protect\cite{TraPPE-UA.alkanes,TraPPE-UA.alkylbenzenes,TraPPE-UA.thiols} |
379 |
(UA) and \protect\cite{OPLSAA} (AA). Cross-interactions with the Au |
380 |
atoms are given in Table \ref{MnM}.} |
381 |
\label{demoMol} |
382 |
\end{figure} |
383 |
|
384 |
The Au-Au interactions in metal lattice slab is described by the |
385 |
quantum Sutton-Chen (QSC) formulation.\cite{PhysRevB.59.3527} The QSC |
386 |
potentials include zero-point quantum corrections and are |
387 |
reparametrized for accurate surface energies compared to the |
388 |
Sutton-Chen potentials.\cite{Chen90} |
389 |
|
390 |
For the two solvent molecules, {\it n}-hexane and toluene, two |
391 |
different atomistic models were utilized. Both solvents were modeled |
392 |
using United-Atom (UA) and All-Atom (AA) models. The TraPPE-UA |
393 |
parameters\cite{TraPPE-UA.alkanes,TraPPE-UA.alkylbenzenes} are used |
394 |
for our UA solvent molecules. In these models, sites are located at |
395 |
the carbon centers for alkyl groups. Bonding interactions, including |
396 |
bond stretches and bends and torsions, were used for intra-molecular |
397 |
sites closer than 3 bonds. For non-bonded interactions, Lennard-Jones |
398 |
potentials are used. |
399 |
|
400 |
By eliminating explicit hydrogen atoms, the TraPPE-UA models are |
401 |
simple and computationally efficient, while maintaining good accuracy. |
402 |
However, the TraPPE-UA model for alkanes is known to predict a slightly |
403 |
lower boiling point than experimental values. This is one of the |
404 |
reasons we used a lower average temperature (200K) for our |
405 |
simulations. If heat is transferred to the liquid phase during the |
406 |
NIVS simulation, the liquid in the hot slab can actually be |
407 |
substantially warmer than the mean temperature in the simulation. The |
408 |
lower mean temperatures therefore prevent solvent boiling. |
409 |
|
410 |
For UA-toluene, the non-bonded potentials between intermolecular sites |
411 |
have a similar Lennard-Jones formulation. The toluene molecules were |
412 |
treated as a single rigid body, so there was no need for |
413 |
intramolecular interactions (including bonds, bends, or torsions) in |
414 |
this solvent model. |
415 |
|
416 |
Besides the TraPPE-UA models, AA models for both organic solvents are |
417 |
included in our studies as well. The OPLS-AA\cite{OPLSAA} force fields |
418 |
were used. For hexane, additional explicit hydrogen sites were |
419 |
included. Besides bonding and non-bonded site-site interactions, |
420 |
partial charges and the electrostatic interactions were added to each |
421 |
CT and HC site. For toluene, a flexible model for the toluene molecule |
422 |
was utilized which included bond, bend, torsion, and inversion |
423 |
potentials to enforce ring planarity. |
424 |
|
425 |
The butanethiol capping agent in our simulations, were also modeled |
426 |
with both UA and AA model. The TraPPE-UA force field includes |
427 |
parameters for thiol molecules\cite{TraPPE-UA.thiols} and are used for |
428 |
UA butanethiol model in our simulations. The OPLS-AA also provides |
429 |
parameters for alkyl thiols. However, alkyl thiols adsorbed on Au(111) |
430 |
surfaces do not have the hydrogen atom bonded to sulfur. To derive |
431 |
suitable parameters for butanethiol adsorbed on Au(111) surfaces, we |
432 |
adopt the S parameters from Luedtke and Landman\cite{landman:1998} and |
433 |
modify the parameters for the CTS atom to maintain charge neutrality |
434 |
in the molecule. Note that the model choice (UA or AA) for the capping |
435 |
agent can be different from the solvent. Regardless of model choice, |
436 |
the force field parameters for interactions between capping agent and |
437 |
solvent can be derived using Lorentz-Berthelot Mixing Rule: |
438 |
\begin{eqnarray} |
439 |
\sigma_{ij} & = & \frac{1}{2} \left(\sigma_{ii} + \sigma_{jj}\right) \\ |
440 |
\epsilon_{ij} & = & \sqrt{\epsilon_{ii}\epsilon_{jj}} |
441 |
\end{eqnarray} |
442 |
|
443 |
To describe the interactions between metal (Au) and non-metal atoms, |
444 |
we refer to an adsorption study of alkyl thiols on gold surfaces by |
445 |
Vlugt {\it et al.}\cite{vlugt:cpc2007154} They fitted an effective |
446 |
Lennard-Jones form of potential parameters for the interaction between |
447 |
Au and pseudo-atoms CH$_x$ and S based on a well-established and |
448 |
widely-used effective potential of Hautman and Klein for the Au(111) |
449 |
surface.\cite{hautman:4994} As our simulations require the gold slab |
450 |
to be flexible to accommodate thermal excitation, the pair-wise form |
451 |
of potentials they developed was used for our study. |
452 |
|
453 |
The potentials developed from {\it ab initio} calculations by Leng |
454 |
{\it et al.}\cite{doi:10.1021/jp034405s} are adopted for the |
455 |
interactions between Au and aromatic C/H atoms in toluene. However, |
456 |
the Lennard-Jones parameters between Au and other types of particles, |
457 |
(e.g. AA alkanes) have not yet been established. For these |
458 |
interactions, the Lorentz-Berthelot mixing rule can be used to derive |
459 |
effective single-atom LJ parameters for the metal using the fit values |
460 |
for toluene. These are then used to construct reasonable mixing |
461 |
parameters for the interactions between the gold and other atoms. |
462 |
Table \ref{MnM} summarizes the ``metal/non-metal'' parameters used in |
463 |
our simulations. |
464 |
|
465 |
\begin{table*} |
466 |
\begin{minipage}{\linewidth} |
467 |
\begin{center} |
468 |
\caption{Non-bonded interaction parameters (including cross |
469 |
interactions with Au atoms) for both force fields used in this |
470 |
work.} |
471 |
\begin{tabular}{lllllll} |
472 |
\hline\hline |
473 |
& Site & $\sigma_{ii}$ & $\epsilon_{ii}$ & $q_i$ & |
474 |
$\sigma_{Au-i}$ & $\epsilon_{Au-i}$ \\ |
475 |
& & (\AA) & (kcal/mol) & ($e$) & (\AA) & (kcal/mol) \\ |
476 |
\hline |
477 |
United Atom (UA) |
478 |
&CH3 & 3.75 & 0.1947 & - & 3.54 & 0.2146 \\ |
479 |
&CH2 & 3.95 & 0.0914 & - & 3.54 & 0.1749 \\ |
480 |
&CHar & 3.695 & 0.1003 & - & 3.4625 & 0.1680 \\ |
481 |
&CRar & 3.88 & 0.04173 & - & 3.555 & 0.1604 \\ |
482 |
\hline |
483 |
All Atom (AA) |
484 |
&CT3 & 3.50 & 0.066 & -0.18 & 3.365 & 0.1373 \\ |
485 |
&CT2 & 3.50 & 0.066 & -0.12 & 3.365 & 0.1373 \\ |
486 |
&CTT & 3.50 & 0.066 & -0.065 & 3.365 & 0.1373 \\ |
487 |
&HC & 2.50 & 0.030 & 0.06 & 2.865 & 0.09256 \\ |
488 |
&CA & 3.55 & 0.070 & -0.115 & 3.173 & 0.0640 \\ |
489 |
&HA & 2.42 & 0.030 & 0.115 & 2.746 & 0.0414 \\ |
490 |
\hline |
491 |
Both UA and AA |
492 |
& S & 4.45 & 0.25 & - & 2.40 & 8.465 \\ |
493 |
\hline\hline |
494 |
\end{tabular} |
495 |
\label{MnM} |
496 |
\end{center} |
497 |
\end{minipage} |
498 |
\end{table*} |
499 |
|
500 |
|
501 |
\section{Results} |
502 |
There are many factors contributing to the measured interfacial |
503 |
conductance; some of these factors are physically motivated |
504 |
(e.g. coverage of the surface by the capping agent coverage and |
505 |
solvent identity), while some are governed by parameters of the |
506 |
methodology (e.g. applied flux and the formulas used to obtain the |
507 |
conductance). In this section we discuss the major physical and |
508 |
calculational effects on the computed conductivity. |
509 |
|
510 |
\subsection{Effects due to capping agent coverage} |
511 |
|
512 |
A series of different initial conditions with a range of surface |
513 |
coverages was prepared and solvated with various with both of the |
514 |
solvent molecules. These systems were then equilibrated and their |
515 |
interfacial thermal conductivity was measured with the NIVS |
516 |
algorithm. Figure \ref{coverage} demonstrates the trend of conductance |
517 |
with respect to surface coverage. |
518 |
|
519 |
\begin{figure} |
520 |
\includegraphics[width=\linewidth]{coverage} |
521 |
\caption{The interfacial thermal conductivity ($G$) has a |
522 |
non-monotonic dependence on the degree of surface capping. This |
523 |
data is for the Au(111) / butanethiol / solvent interface with |
524 |
various UA force fields at $\langle T\rangle \sim $200K.} |
525 |
\label{coverage} |
526 |
\end{figure} |
527 |
|
528 |
In partially covered surfaces, the derivative definition for |
529 |
$G^\prime$ (Eq. \ref{derivativeG}) becomes difficult to apply, as the |
530 |
location of maximum change of $\lambda$ becomes washed out. The |
531 |
discrete definition (Eq. \ref{discreteG}) is easier to apply, as the |
532 |
Gibbs dividing surface is still well-defined. Therefore, $G$ (not |
533 |
$G^\prime$) was used in this section. |
534 |
|
535 |
From Figure \ref{coverage}, one can see the significance of the |
536 |
presence of capping agents. When even a small fraction of the Au(111) |
537 |
surface sites are covered with butanethiols, the conductivity exhibits |
538 |
an enhancement by at least a factor of 3. Capping agents are clearly |
539 |
playing a major role in thermal transport at metal / organic solvent |
540 |
surfaces. |
541 |
|
542 |
We note a non-monotonic behavior in the interfacial conductance as a |
543 |
function of surface coverage. The maximum conductance (largest $G$) |
544 |
happens when the surfaces are about 75\% covered with butanethiol |
545 |
caps. The reason for this behavior is not entirely clear. One |
546 |
explanation is that incomplete butanethiol coverage allows small gaps |
547 |
between butanethiols to form. These gaps can be filled by transient |
548 |
solvent molecules. These solvent molecules couple very strongly with |
549 |
the hot capping agent molecules near the surface, and can then carry |
550 |
away (diffusively) the excess thermal energy from the surface. |
551 |
|
552 |
There appears to be a competition between the conduction of the |
553 |
thermal energy away from the surface by the capping agents (enhanced |
554 |
by greater coverage) and the coupling of the capping agents with the |
555 |
solvent (enhanced by interdigitation at lower coverages). This |
556 |
competition would lead to the non-monotonic coverage behavior observed |
557 |
here. |
558 |
|
559 |
Results for rigid body toluene solvent, as well as the UA hexane, are |
560 |
within the ranges expected from prior experimental |
561 |
work.\cite{Wilson:2002uq,cahill:793,PhysRevB.80.195406} This suggests |
562 |
that explicit hydrogen atoms might not be required for modeling |
563 |
thermal transport in these systems. C-H vibrational modes do not see |
564 |
significant excited state population at low temperatures, and are not |
565 |
likely to carry lower frequency excitations from the solid layer into |
566 |
the bulk liquid. |
567 |
|
568 |
The toluene solvent does not exhibit the same behavior as hexane in |
569 |
that $G$ remains at approximately the same magnitude when the capping |
570 |
coverage increases from 25\% to 75\%. Toluene, as a rigid planar |
571 |
molecule, cannot occupy the relatively small gaps between the capping |
572 |
agents as easily as the chain-like {\it n}-hexane. The effect of |
573 |
solvent coupling to the capping agent is therefore weaker in toluene |
574 |
except at the very lowest coverage levels. This effect counters the |
575 |
coverage-dependent conduction of heat away from the metal surface, |
576 |
leading to a much flatter $G$ vs. coverage trend than is observed in |
577 |
{\it n}-hexane. |
578 |
|
579 |
\subsection{Effects due to Solvent \& Solvent Models} |
580 |
In addition to UA solvent and capping agent models, AA models have |
581 |
also been included in our simulations. In most of this work, the same |
582 |
(UA or AA) model for solvent and capping agent was used, but it is |
583 |
also possible to utilize different models for different components. |
584 |
We have also included isotopic substitutions (Hydrogen to Deuterium) |
585 |
to decrease the explicit vibrational overlap between solvent and |
586 |
capping agent. Table \ref{modelTest} summarizes the results of these |
587 |
studies. |
588 |
|
589 |
\begin{table*} |
590 |
\begin{minipage}{\linewidth} |
591 |
\begin{center} |
592 |
|
593 |
\caption{Computed interfacial thermal conductance ($G$ and |
594 |
$G^\prime$) values for interfaces using various models for |
595 |
solvent and capping agent (or without capping agent) at |
596 |
$\langle T\rangle\sim$200K. Here ``D'' stands for deuterated |
597 |
solvent or capping agent molecules; ``Avg.'' denotes results |
598 |
that are averages of simulations under different applied |
599 |
thermal flux $(J_z)$ values. Error estimates are indicated in |
600 |
parentheses.} |
601 |
|
602 |
\begin{tabular}{llccc} |
603 |
\hline\hline |
604 |
Butanethiol model & Solvent & $J_z$ & $G$ & $G^\prime$ \\ |
605 |
(or bare surface) & model & (GW/m$^2$) & |
606 |
\multicolumn{2}{c}{(MW/m$^2$/K)} \\ |
607 |
\hline |
608 |
UA & UA hexane & Avg. & 131(9) & 87(10) \\ |
609 |
& UA hexane(D) & 1.95 & 153(5) & 136(13) \\ |
610 |
& AA hexane & Avg. & 131(6) & 122(10) \\ |
611 |
& UA toluene & 1.96 & 187(16) & 151(11) \\ |
612 |
& AA toluene & 1.89 & 200(36) & 149(53) \\ |
613 |
\hline |
614 |
AA & UA hexane & 1.94 & 116(9) & 129(8) \\ |
615 |
& AA hexane & Avg. & 442(14) & 356(31) \\ |
616 |
& AA hexane(D) & 1.93 & 222(12) & 234(54) \\ |
617 |
& UA toluene & 1.98 & 125(25) & 97(60) \\ |
618 |
& AA toluene & 3.79 & 487(56) & 290(42) \\ |
619 |
\hline |
620 |
AA(D) & UA hexane & 1.94 & 158(25) & 172(4) \\ |
621 |
& AA hexane & 1.92 & 243(29) & 191(11) \\ |
622 |
& AA toluene & 1.93 & 364(36) & 322(67) \\ |
623 |
\hline |
624 |
bare & UA hexane & Avg. & 46.5(3.2) & 49.4(4.5) \\ |
625 |
& UA hexane(D) & 0.98 & 43.9(4.6) & 43.0(2.0) \\ |
626 |
& AA hexane & 0.96 & 31.0(1.4) & 29.4(1.3) \\ |
627 |
& UA toluene & 1.99 & 70.1(1.3) & 65.8(0.5) \\ |
628 |
\hline\hline |
629 |
\end{tabular} |
630 |
\label{modelTest} |
631 |
\end{center} |
632 |
\end{minipage} |
633 |
\end{table*} |
634 |
|
635 |
To facilitate direct comparison between force fields, systems with the |
636 |
same capping agent and solvent were prepared with the same length |
637 |
scales for the simulation cells. |
638 |
|
639 |
On bare metal / solvent surfaces, different force field models for |
640 |
hexane yield similar results for both $G$ and $G^\prime$, and these |
641 |
two definitions agree with each other very well. This is primarily an |
642 |
indicator of weak interactions between the metal and the solvent, and |
643 |
is a typical case for acoustic impedance mismatch between these two |
644 |
phases. |
645 |
|
646 |
For the fully-covered surfaces, the choice of force field for the |
647 |
capping agent and solvent has a large impact on the calculated values |
648 |
of $G$ and $G^\prime$. The AA thiol to AA solvent conductivities are |
649 |
much larger than their UA to UA counterparts, and these values exceed |
650 |
the experimental estimates by a large measure. The AA force field |
651 |
allows significant energy to go into C-H (or C-D) stretching modes, |
652 |
and since these modes are high frequency, this non-quantum behavior is |
653 |
likely responsible for the overestimate of the conductivity. Compared |
654 |
to the AA model, the UA model yields more reasonable conductivity |
655 |
values with much higher computational efficiency. |
656 |
|
657 |
\subsubsection{Are electronic excitations in the metal important?} |
658 |
Because they lack electronic excitations, the QSC and related embedded |
659 |
atom method (EAM) models for gold are known to predict unreasonably |
660 |
low values for bulk conductivity |
661 |
($\lambda$).\cite{kuang:164101,ISI:000207079300006,Clancy:1992} If the |
662 |
conductance between the phases ($G$) is governed primarily by phonon |
663 |
excitation (and not electronic degrees of freedom), one would expect a |
664 |
classical model to capture most of the interfacial thermal |
665 |
conductance. Our results for $G$ and $G^\prime$ indicate that this is |
666 |
indeed the case, and suggest that the modeling of interfacial thermal |
667 |
transport depends primarily on the description of the interactions |
668 |
between the various components at the interface. When the metal is |
669 |
chemically capped, the primary barrier to thermal conductivity appears |
670 |
to be the interface between the capping agent and the surrounding |
671 |
solvent, so the excitations in the metal have little impact on the |
672 |
value of $G$. |
673 |
|
674 |
\subsection{Effects due to methodology and simulation parameters} |
675 |
|
676 |
We have varied the parameters of the simulations in order to |
677 |
investigate how these factors would affect the computation of $G$. Of |
678 |
particular interest are: 1) the length scale for the applied thermal |
679 |
gradient (modified by increasing the amount of solvent in the system), |
680 |
2) the sign and magnitude of the applied thermal flux, 3) the average |
681 |
temperature of the simulation (which alters the solvent density during |
682 |
equilibration), and 4) the definition of the interfacial conductance |
683 |
(Eqs. (\ref{discreteG}) or (\ref{derivativeG})) used in the |
684 |
calculation. |
685 |
|
686 |
Systems of different lengths were prepared by altering the number of |
687 |
solvent molecules and extending the length of the box along the $z$ |
688 |
axis to accomodate the extra solvent. Equilibration at the same |
689 |
temperature and pressure conditions led to nearly identical surface |
690 |
areas ($L_x$ and $L_y$) available to the metal and capping agent, |
691 |
while the extra solvent served mainly to lengthen the axis that was |
692 |
used to apply the thermal flux. For a given value of the applied |
693 |
flux, the different $z$ length scale has only a weak effect on the |
694 |
computed conductivities (Table \ref{AuThiolHexaneUA}). |
695 |
|
696 |
\subsubsection{Effects of applied flux} |
697 |
The NIVS algorithm allows changes in both the sign and magnitude of |
698 |
the applied flux. It is possible to reverse the direction of heat |
699 |
flow simply by changing the sign of the flux, and thermal gradients |
700 |
which would be difficult to obtain experimentally ($5$ K/\AA) can be |
701 |
easily simulated. However, the magnitude of the applied flux is not |
702 |
arbitrary if one aims to obtain a stable and reliable thermal gradient. |
703 |
A temperature gradient can be lost in the noise if $|J_z|$ is too |
704 |
small, and excessive $|J_z|$ values can cause phase transitions if the |
705 |
extremes of the simulation cell become widely separated in |
706 |
temperature. Also, if $|J_z|$ is too large for the bulk conductivity |
707 |
of the materials, the thermal gradient will never reach a stable |
708 |
state. |
709 |
|
710 |
Within a reasonable range of $J_z$ values, we were able to study how |
711 |
$G$ changes as a function of this flux. In what follows, we use |
712 |
positive $J_z$ values to denote the case where energy is being |
713 |
transferred by the method from the metal phase and into the liquid. |
714 |
The resulting gradient therefore has a higher temperature in the |
715 |
liquid phase. Negative flux values reverse this transfer, and result |
716 |
in higher temperature metal phases. The conductance measured under |
717 |
different applied $J_z$ values is listed in Tables |
718 |
\ref{AuThiolHexaneUA} and \ref{AuThiolToluene}. These results do not |
719 |
indicate that $G$ depends strongly on $J_z$ within this flux |
720 |
range. The linear response of flux to thermal gradient simplifies our |
721 |
investigations in that we can rely on $G$ measurement with only a |
722 |
small number $J_z$ values. |
723 |
|
724 |
\begin{table*} |
725 |
\begin{minipage}{\linewidth} |
726 |
\begin{center} |
727 |
\caption{In the hexane-solvated interfaces, the system size has |
728 |
little effect on the calculated values for interfacial |
729 |
conductance ($G$ and $G^\prime$), but the direction of heat |
730 |
flow (i.e. the sign of $J_z$) can alter the average |
731 |
temperature of the liquid phase and this can alter the |
732 |
computed conductivity.} |
733 |
|
734 |
\begin{tabular}{ccccccc} |
735 |
\hline\hline |
736 |
$\langle T\rangle$ & $N_{hexane}$ & $\rho_{hexane}$ & |
737 |
$J_z$ & $G$ & $G^\prime$ \\ |
738 |
(K) & & (g/cm$^3$) & (GW/m$^2$) & |
739 |
\multicolumn{2}{c}{(MW/m$^2$/K)} \\ |
740 |
\hline |
741 |
200 & 266 & 0.672 & -0.96 & 102(3) & 80.0(0.8) \\ |
742 |
& 200 & 0.688 & 0.96 & 125(16) & 90.2(15) \\ |
743 |
& & & 1.91 & 139(10) & 101(10) \\ |
744 |
& & & 2.83 & 141(6) & 89.9(9.8) \\ |
745 |
& 166 & 0.681 & 0.97 & 141(30) & 78(22) \\ |
746 |
& & & 1.92 & 138(4) & 98.9(9.5) \\ |
747 |
\hline |
748 |
250 & 200 & 0.560 & 0.96 & 75(10) & 61.8(7.3) \\ |
749 |
& & & -0.95 & 49.4(0.3) & 45.7(2.1) \\ |
750 |
& 166 & 0.569 & 0.97 & 80.3(0.6) & 67(11) \\ |
751 |
& & & 1.44 & 76.2(5.0) & 64.8(3.8) \\ |
752 |
& & & -0.95 & 56.4(2.5) & 54.4(1.1) \\ |
753 |
& & & -1.85 & 47.8(1.1) & 53.5(1.5) \\ |
754 |
\hline\hline |
755 |
\end{tabular} |
756 |
\label{AuThiolHexaneUA} |
757 |
\end{center} |
758 |
\end{minipage} |
759 |
\end{table*} |
760 |
|
761 |
The sign of $J_z$ is a different matter, however, as this can alter |
762 |
the temperature on the two sides of the interface. The average |
763 |
temperature values reported are for the entire system, and not for the |
764 |
liquid phase, so at a given $\langle T \rangle$, the system with |
765 |
positive $J_z$ has a warmer liquid phase. This means that if the |
766 |
liquid carries thermal energy via convective transport, {\it positive} |
767 |
$J_z$ values will result in increased molecular motion on the liquid |
768 |
side of the interface, and this will increase the measured |
769 |
conductivity. |
770 |
|
771 |
\subsubsection{Effects due to average temperature} |
772 |
|
773 |
We also studied the effect of average system temperature on the |
774 |
interfacial conductance. The simulations are first equilibrated in |
775 |
the NPT ensemble at 1 atm. The TraPPE-UA model for hexane tends to |
776 |
predict a lower boiling point (and liquid state density) than |
777 |
experiments. This lower-density liquid phase leads to reduced contact |
778 |
between the hexane and butanethiol, and this accounts for our |
779 |
observation of lower conductance at higher temperatures as shown in |
780 |
Table \ref{AuThiolHexaneUA}. In raising the average temperature from |
781 |
200K to 250K, the density drop of $\sim$20\% in the solvent phase |
782 |
leads to a $\sim$40\% drop in the conductance. |
783 |
|
784 |
Similar behavior is observed in the TraPPE-UA model for toluene, |
785 |
although this model has better agreement with the experimental |
786 |
densities of toluene. The expansion of the toluene liquid phase is |
787 |
not as significant as that of the hexane (8.3\% over 100K), and this |
788 |
limits the effect to $\sim$20\% drop in thermal conductivity (Table |
789 |
\ref{AuThiolToluene}). |
790 |
|
791 |
Although we have not mapped out the behavior at a large number of |
792 |
temperatures, is clear that there will be a strong temperature |
793 |
dependence in the interfacial conductance when the physical properties |
794 |
of one side of the interface (notably the density) change rapidly as a |
795 |
function of temperature. |
796 |
|
797 |
\begin{table*} |
798 |
\begin{minipage}{\linewidth} |
799 |
\begin{center} |
800 |
\caption{When toluene is the solvent, the interfacial thermal |
801 |
conductivity is less sensitive to temperature, but again, the |
802 |
direction of the heat flow can alter the solvent temperature |
803 |
and can change the computed conductance values.} |
804 |
|
805 |
\begin{tabular}{ccccc} |
806 |
\hline\hline |
807 |
$\langle T\rangle$ & $\rho_{toluene}$ & $J_z$ & $G$ & $G^\prime$ \\ |
808 |
(K) & (g/cm$^3$) & (GW/m$^2$) & \multicolumn{2}{c}{(MW/m$^2$/K)} \\ |
809 |
\hline |
810 |
200 & 0.933 & 2.15 & 204(12) & 113(12) \\ |
811 |
& & -1.86 & 180(3) & 135(21) \\ |
812 |
& & -3.93 & 176(5) & 113(12) \\ |
813 |
\hline |
814 |
300 & 0.855 & -1.91 & 143(5) & 125(2) \\ |
815 |
& & -4.19 & 135(9) & 113(12) \\ |
816 |
\hline\hline |
817 |
\end{tabular} |
818 |
\label{AuThiolToluene} |
819 |
\end{center} |
820 |
\end{minipage} |
821 |
\end{table*} |
822 |
|
823 |
Besides the lower interfacial thermal conductance, surfaces at |
824 |
relatively high temperatures are susceptible to reconstructions, |
825 |
particularly when butanethiols fully cover the Au(111) surface. These |
826 |
reconstructions include surface Au atoms which migrate outward to the |
827 |
S atom layer, and butanethiol molecules which embed into the surface |
828 |
Au layer. The driving force for this behavior is the strong Au-S |
829 |
interactions which are modeled here with a deep Lennard-Jones |
830 |
potential. This phenomenon agrees with reconstructions that have been |
831 |
experimentally |
832 |
observed.\cite{doi:10.1021/j100035a033,doi:10.1021/la026493y}. Vlugt |
833 |
{\it et al.} kept their Au(111) slab rigid so that their simulations |
834 |
could reach 300K without surface |
835 |
reconstructions.\cite{vlugt:cpc2007154} Since surface reconstructions |
836 |
blur the interface, the measurement of $G$ becomes more difficult to |
837 |
conduct at higher temperatures. For this reason, most of our |
838 |
measurements are undertaken at $\langle T\rangle\sim$200K where |
839 |
reconstruction is minimized. |
840 |
|
841 |
However, when the surface is not completely covered by butanethiols, |
842 |
the simulated system appears to be more resistent to the |
843 |
reconstruction. Our Au / butanethiol / toluene system had the Au(111) |
844 |
surfaces 90\% covered by butanethiols, but did not see this above |
845 |
phenomena even at $\langle T\rangle\sim$300K. That said, we did |
846 |
observe butanethiols migrating to neighboring three-fold sites during |
847 |
a simulation. Since the interface persisted in these simulations, we |
848 |
were able to obtain $G$'s for these interfaces even at a relatively |
849 |
high temperature without being affected by surface reconstructions. |
850 |
|
851 |
\section{Discussion} |
852 |
|
853 |
The primary result of this work is that the capping agent acts as an |
854 |
efficient thermal coupler between solid and solvent phases. One of |
855 |
the ways the capping agent can carry out this role is to down-shift |
856 |
between the phonon vibrations in the solid (which carry the heat from |
857 |
the gold) and the molecular vibrations in the liquid (which carry some |
858 |
of the heat in the solvent). |
859 |
|
860 |
To investigate the mechanism of interfacial thermal conductance, the |
861 |
vibrational power spectrum was computed. Power spectra were taken for |
862 |
individual components in different simulations. To obtain these |
863 |
spectra, simulations were run after equilibration in the |
864 |
microcanonical (NVE) ensemble and without a thermal |
865 |
gradient. Snapshots of configurations were collected at a frequency |
866 |
that is higher than that of the fastest vibrations occurring in the |
867 |
simulations. With these configurations, the velocity auto-correlation |
868 |
functions can be computed: |
869 |
\begin{equation} |
870 |
C_A (t) = \langle\vec{v}_A (t)\cdot\vec{v}_A (0)\rangle |
871 |
\label{vCorr} |
872 |
\end{equation} |
873 |
The power spectrum is constructed via a Fourier transform of the |
874 |
symmetrized velocity autocorrelation function, |
875 |
\begin{equation} |
876 |
\hat{f}(\omega) = |
877 |
\int_{-\infty}^{\infty} C_A (t) e^{-2\pi it\omega}\,dt |
878 |
\label{fourier} |
879 |
\end{equation} |
880 |
|
881 |
\subsection{The role of specific vibrations} |
882 |
The vibrational spectra for gold slabs in different environments are |
883 |
shown as in Figure \ref{specAu}. Regardless of the presence of |
884 |
solvent, the gold surfaces which are covered by butanethiol molecules |
885 |
exhibit an additional peak observed at a frequency of |
886 |
$\sim$165cm$^{-1}$. We attribute this peak to the S-Au bonding |
887 |
vibration. This vibration enables efficient thermal coupling of the |
888 |
surface Au layer to the capping agents. Therefore, in our simulations, |
889 |
the Au / S interfaces do not appear to be the primary barrier to |
890 |
thermal transport when compared with the butanethiol / solvent |
891 |
interfaces. {\bf This confirms the results from Luo {\it et |
892 |
al.}\cite{Luo20101}, which reported $G$ for Au-SAM junctions |
893 |
generally twice larger than what we have computed for the |
894 |
thiol-liquid interfaces.} |
895 |
|
896 |
\begin{figure} |
897 |
\includegraphics[width=\linewidth]{vibration} |
898 |
\caption{The vibrational power spectrum for thiol-capped gold has an |
899 |
additional vibrational peak at $\sim $165cm$^{-1}$. Bare gold |
900 |
surfaces (both with and without a solvent over-layer) are missing |
901 |
this peak. A similar peak at $\sim $165cm$^{-1}$ also appears in |
902 |
the vibrational power spectrum for the butanethiol capping agents.} |
903 |
\label{specAu} |
904 |
\end{figure} |
905 |
|
906 |
Also in this figure, we show the vibrational power spectrum for the |
907 |
bound butanethiol molecules, which also exhibits the same |
908 |
$\sim$165cm$^{-1}$ peak. |
909 |
|
910 |
\subsection{Overlap of power spectra} |
911 |
A comparison of the results obtained from the two different organic |
912 |
solvents can also provide useful information of the interfacial |
913 |
thermal transport process. In particular, the vibrational overlap |
914 |
between the butanethiol and the organic solvents suggests a highly |
915 |
efficient thermal exchange between these components. Very high |
916 |
thermal conductivity was observed when AA models were used and C-H |
917 |
vibrations were treated classically. The presence of extra degrees of |
918 |
freedom in the AA force field yields higher heat exchange rates |
919 |
between the two phases and results in a much higher conductivity than |
920 |
in the UA force field. {\bf Due to the classical models used, this |
921 |
even includes those high frequency modes which should be unpopulated |
922 |
at our relatively low temperatures. This artifact causes high |
923 |
frequency vibrations accountable for thermal transport in classical |
924 |
MD simulations.} |
925 |
|
926 |
The similarity in the vibrational modes available to solvent and |
927 |
capping agent can be reduced by deuterating one of the two components |
928 |
(Fig. \ref{aahxntln}). Once either the hexanes or the butanethiols |
929 |
are deuterated, one can observe a significantly lower $G$ and |
930 |
$G^\prime$ values (Table \ref{modelTest}). |
931 |
|
932 |
\begin{figure} |
933 |
\includegraphics[width=\linewidth]{aahxntln} |
934 |
\caption{Spectra obtained for all-atom (AA) Au / butanethiol / solvent |
935 |
systems. When butanethiol is deuterated (lower left), its |
936 |
vibrational overlap with hexane decreases significantly. Since |
937 |
aromatic molecules and the butanethiol are vibrationally dissimilar, |
938 |
the change is not as dramatic when toluene is the solvent (right).} |
939 |
\label{aahxntln} |
940 |
\end{figure} |
941 |
|
942 |
For the Au / butanethiol / toluene interfaces, having the AA |
943 |
butanethiol deuterated did not yield a significant change in the |
944 |
measured conductance. Compared to the C-H vibrational overlap between |
945 |
hexane and butanethiol, both of which have alkyl chains, the overlap |
946 |
between toluene and butanethiol is not as significant and thus does |
947 |
not contribute as much to the heat exchange process. |
948 |
|
949 |
Previous observations of Zhang {\it et al.}\cite{hase:2010} indicate |
950 |
that the {\it intra}molecular heat transport due to alkylthiols is |
951 |
highly efficient. Combining our observations with those of Zhang {\it |
952 |
et al.}, it appears that butanethiol acts as a channel to expedite |
953 |
heat flow from the gold surface and into the alkyl chain. The |
954 |
acoustic impedance mismatch between the metal and the liquid phase can |
955 |
therefore be effectively reduced with the presence of suitable capping |
956 |
agents. |
957 |
|
958 |
Deuterated models in the UA force field did not decouple the thermal |
959 |
transport as well as in the AA force field. The UA models, even |
960 |
though they have eliminated the high frequency C-H vibrational |
961 |
overlap, still have significant overlap in the lower-frequency |
962 |
portions of the infrared spectra (Figure \ref{uahxnua}). Deuterating |
963 |
the UA models did not decouple the low frequency region enough to |
964 |
produce an observable difference for the results of $G$ (Table |
965 |
\ref{modelTest}). |
966 |
|
967 |
\begin{figure} |
968 |
\includegraphics[width=\linewidth]{uahxnua} |
969 |
\caption{Vibrational power spectra for UA models for the butanethiol |
970 |
and hexane solvent (upper panel) show the high degree of overlap |
971 |
between these two molecules, particularly at lower frequencies. |
972 |
Deuterating a UA model for the solvent (lower panel) does not |
973 |
decouple the two spectra to the same degree as in the AA force |
974 |
field (see Fig \ref{aahxntln}).} |
975 |
\label{uahxnua} |
976 |
\end{figure} |
977 |
|
978 |
\section{Conclusions} |
979 |
The NIVS algorithm has been applied to simulations of |
980 |
butanethiol-capped Au(111) surfaces in the presence of organic |
981 |
solvents. This algorithm allows the application of unphysical thermal |
982 |
flux to transfer heat between the metal and the liquid phase. With the |
983 |
flux applied, we were able to measure the corresponding thermal |
984 |
gradients and to obtain interfacial thermal conductivities. Under |
985 |
steady states, 2-3 ns trajectory simulations are sufficient for |
986 |
computation of this quantity. |
987 |
|
988 |
Our simulations have seen significant conductance enhancement in the |
989 |
presence of capping agent, compared with the bare gold / liquid |
990 |
interfaces. The acoustic impedance mismatch between the metal and the |
991 |
liquid phase is effectively eliminated by a chemically-bonded capping |
992 |
agent. Furthermore, the coverage percentage of the capping agent plays |
993 |
an important role in the interfacial thermal transport |
994 |
process. Moderately low coverages allow higher contact between capping |
995 |
agent and solvent, and thus could further enhance the heat transfer |
996 |
process, giving a non-monotonic behavior of conductance with |
997 |
increasing coverage. |
998 |
|
999 |
Our results, particularly using the UA models, agree well with |
1000 |
available experimental data. The AA models tend to overestimate the |
1001 |
interfacial thermal conductance in that the classically treated C-H |
1002 |
vibrations become too easily populated. Compared to the AA models, the |
1003 |
UA models have higher computational efficiency with satisfactory |
1004 |
accuracy, and thus are preferable in modeling interfacial thermal |
1005 |
transport. |
1006 |
|
1007 |
Of the two definitions for $G$, the discrete form |
1008 |
(Eq. \ref{discreteG}) was easier to use and gives out relatively |
1009 |
consistent results, while the derivative form (Eq. \ref{derivativeG}) |
1010 |
is not as versatile. Although $G^\prime$ gives out comparable results |
1011 |
and follows similar trend with $G$ when measuring close to fully |
1012 |
covered or bare surfaces, the spatial resolution of $T$ profile |
1013 |
required for the use of a derivative form is limited by the number of |
1014 |
bins and the sampling required to obtain thermal gradient information. |
1015 |
|
1016 |
Vlugt {\it et al.} have investigated the surface thiol structures for |
1017 |
nanocrystalline gold and pointed out that they differ from those of |
1018 |
the Au(111) surface.\cite{landman:1998,vlugt:cpc2007154} This |
1019 |
difference could also cause differences in the interfacial thermal |
1020 |
transport behavior. To investigate this problem, one would need an |
1021 |
effective method for applying thermal gradients in non-planar |
1022 |
(i.e. spherical) geometries. |
1023 |
|
1024 |
\section{Acknowledgments} |
1025 |
Support for this project was provided by the National Science |
1026 |
Foundation under grant CHE-0848243. Computational time was provided by |
1027 |
the Center for Research Computing (CRC) at the University of Notre |
1028 |
Dame. |
1029 |
|
1030 |
\section{Supporting Information} |
1031 |
This information is available free of charge via the Internet at |
1032 |
http://pubs.acs.org. |
1033 |
|
1034 |
\newpage |
1035 |
|
1036 |
\bibliography{interfacial} |
1037 |
|
1038 |
\end{doublespace} |
1039 |
\end{document} |
1040 |
|