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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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\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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\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 coupling |
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between the metal and liquid phases is enhanced by the capping |
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agents, leading to a greatly enhanced conductivity at the interface. |
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Specifically, the chemical bond between the metal and the capping |
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agent introduces a vibrational overlap that is not present without |
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the capping agent, and the overlap between the vibrational spectra |
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(metal to cap, cap to solvent) provides a mechanism for rapid |
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thermal transport across the interface. Our calculations also |
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suggest that this is a non-monotonic function of the fractional |
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coverage of the surface, as moderate coverages allow diffusive heat |
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transport of solvent molecules that have been in close contact with |
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the capping agent. |
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{\bf Keywords: non-equilibrium, molecular dynamics, vibrational |
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overlap, coverage dependent.} |
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\end{abstract} |
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\newpage |
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%\narrowtext |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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% BODY OF TEXT |
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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 |
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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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The acoustic mismatch model for interfacial conductance utilizes the |
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acoustic impedance ($Z_a = \rho_a v^s_a$) on both sides of the |
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interface.\cite{swartz1989} Here, $\rho_a$ and $v^s_a$ are the density |
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and speed of sound in material $a$. The phonon transmission |
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probability at the $a-b$ interface is |
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\begin{equation} |
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t_{ab} = \frac{4 Z_a Z_b}{\left(Z_a + Z_b \right)^2}, |
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\end{equation} |
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and the interfacial conductance can then be approximated as |
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\begin{equation} |
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G_{ab} \approx \frac{1}{4} C_D v_D t_{ab} |
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\end{equation} |
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where $C_D$ is the Debye heat capacity of the hot side, and $v_D$ is |
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the Debye phonon velocity ($1/v_D^3 = 1/3v_L^3 + 2/3 v_T^3$) where |
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$v_L$ and $v_T$ are the longitudinal and transverse speeds of sound, |
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respectively. For the Au/hexane and Au/toluene interfaces, the |
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acoustic mismatch model predicts room-temperature $G \approx 87 \mbox{ |
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and } 129$ MW m$^{-2}$ K$^{-1}$, respectively. However, it is not |
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clear how to apply the acoustic mismatch model to a |
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chemically-modified surface, particularly when the acoustic properties |
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of a monolayer film may not be well characterized. |
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More precise 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 |
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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. And |
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recently, Luo {\it et al.} studied the thermal conductance of |
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Au-SAM-Au junctions using the same approach, comparing to a constant |
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temperature difference method.\cite{Luo20101} While this latter |
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approach establishes more ideal Maxwell-Boltzmann distributions than |
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previous RNEMD methods, it does not guarantee momentum or kinetic |
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energy conservation. |
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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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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 the phenomena was investigated. |
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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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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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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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\subsection{Defining Interfacial Thermal Conductivity ($G$)} |
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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 $L_x \times L_y$, $E_{total} = 2 J_z t L_x L_y$. |
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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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\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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Another approach is to assume that the temperature is continuous and |
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differentiable throughout the space. Given that $\lambda$ is also |
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differentiable, $G$ can be defined as its gradient ($\nabla\lambda$) |
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projected along a vector normal to the interface ($\mathbf{\hat{n}}$) |
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and evaluated at the interface location ($z_0$). This quantity, |
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\begin{align} |
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G^\prime &= \left(\nabla\lambda \cdot \mathbf{\hat{n}}\right)_{z_0} \\ |
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&= \frac{\partial}{\partial z}\left(-\frac{J_z}{ |
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\left(\frac{\partial T}{\partial z}\right)}\right)_{z_0} \\ |
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&= J_z\left(\frac{\partial^2 T}{\partial z^2}\right)_{z_0} \Big/ |
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\left(\frac{\partial T}{\partial z}\right)_{z_0}^2 \label{derivativeG} |
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\end{align} |
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has the same units as the common definition for $G$, and the maximum |
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of its magnitude denotes where thermal conductivity has the largest |
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change, i.e. the interface. In the geometry used in this study, the |
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vector normal to the interface points along the $z$ axis, as do |
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$\vec{J}$ and the thermal gradient. This yields the simplified |
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expressions in Eq. \ref{derivativeG}. |
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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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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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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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\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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\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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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 |
336 |
|
|
conductance measurement would be an average effect of these |
337 |
|
|
configurations explored in the simulations. |
338 |
|
|
|
339 |
|
|
After the modified Au-butanethiol surface systems were equilibrated in |
340 |
|
|
the canonical (NVT) ensemble, organic solvent molecules were packed in |
341 |
|
|
the previously empty part of the simulation cells.\cite{packmol} Two |
342 |
|
|
solvents were investigated, one which has little vibrational overlap |
343 |
|
|
with the alkanethiol and which has a planar shape (toluene), and one |
344 |
|
|
which has similar vibrational frequencies to the capping agent and |
345 |
|
|
chain-like shape ({\it n}-hexane). |
346 |
|
|
|
347 |
|
|
The simulation cells were not particularly extensive along the |
348 |
|
|
$z$-axis, as a very long length scale for the thermal gradient may |
349 |
|
|
cause excessively hot or cold temperatures in the middle of the |
350 |
|
|
solvent region and lead to undesired phenomena such as solvent boiling |
351 |
|
|
or freezing when a thermal flux is applied. Conversely, too few |
352 |
|
|
solvent molecules would change the normal behavior of the liquid |
353 |
|
|
phase. Therefore, our $N_{solvent}$ values were chosen to ensure that |
354 |
|
|
these extreme cases did not happen to our simulations. The spacing |
355 |
|
|
between periodic images of the gold interfaces is $45 \sim 75$\AA in |
356 |
|
|
our simulations. |
357 |
|
|
|
358 |
|
|
The initial configurations generated are further equilibrated with the |
359 |
|
|
$x$ and $y$ dimensions fixed, only allowing the $z$-length scale to |
360 |
|
|
change. This is to ensure that the equilibration of liquid phase does |
361 |
|
|
not affect the metal's crystalline structure. Comparisons were made |
362 |
|
|
with simulations that allowed changes of $L_x$ and $L_y$ during NPT |
363 |
|
|
equilibration. No substantial changes in the box geometry were noticed |
364 |
|
|
in these simulations. After ensuring the liquid phase reaches |
365 |
|
|
equilibrium at atmospheric pressure (1 atm), further equilibration was |
366 |
|
|
carried out under canonical (NVT) and microcanonical (NVE) ensembles. |
367 |
|
|
|
368 |
|
|
After the systems reach equilibrium, NIVS was used to impose an |
369 |
|
|
unphysical thermal flux between the metal and the liquid phases. Most |
370 |
|
|
of our simulations were done under an average temperature of |
371 |
|
|
$\sim$200K. Therefore, thermal flux usually came from the metal to the |
372 |
|
|
liquid so that the liquid has a higher temperature and would not |
373 |
|
|
freeze due to lowered temperatures. After this induced temperature |
374 |
|
|
gradient had stabilized, the temperature profile of the simulation cell |
375 |
|
|
was recorded. To do this, the simulation cell is divided evenly into |
376 |
|
|
$N$ slabs along the $z$-axis. The average temperatures of each slab |
377 |
|
|
are recorded for 1$\sim$2 ns. When the slab width $d$ of each slab is |
378 |
|
|
the same, the derivatives of $T$ with respect to slab number $n$ can |
379 |
|
|
be directly used for $G^\prime$ calculations: \begin{equation} |
380 |
|
|
G^\prime = |J_z|\Big|\frac{\partial^2 T}{\partial z^2}\Big| |
381 |
|
|
\Big/\left(\frac{\partial T}{\partial z}\right)^2 |
382 |
|
|
= |J_z|\Big|\frac{1}{d^2}\frac{\partial^2 T}{\partial n^2}\Big| |
383 |
|
|
\Big/\left(\frac{1}{d}\frac{\partial T}{\partial n}\right)^2 |
384 |
|
|
= |J_z|\Big|\frac{\partial^2 T}{\partial n^2}\Big| |
385 |
|
|
\Big/\left(\frac{\partial T}{\partial n}\right)^2 |
386 |
|
|
\label{derivativeG2} |
387 |
|
|
\end{equation} |
388 |
|
|
The absolute values in Eq. \ref{derivativeG2} appear because the |
389 |
|
|
direction of the flux $\vec{J}$ is in an opposing direction on either |
390 |
|
|
side of the metal slab. |
391 |
|
|
|
392 |
|
|
All of the above simulation procedures use a time step of 1 fs. Each |
393 |
|
|
equilibration stage took a minimum of 100 ps, although in some cases, |
394 |
|
|
longer equilibration stages were utilized. |
395 |
|
|
|
396 |
|
|
\subsection{Force Field Parameters} |
397 |
|
|
Our simulations include a number of chemically distinct components. |
398 |
|
|
Figure \ref{demoMol} demonstrates the sites defined for both |
399 |
|
|
United-Atom and All-Atom models of the organic solvent and capping |
400 |
|
|
agents in our simulations. Force field parameters are needed for |
401 |
|
|
interactions both between the same type of particles and between |
402 |
|
|
particles of different species. |
403 |
|
|
|
404 |
|
|
\begin{figure} |
405 |
|
|
\includegraphics[width=\linewidth]{structures} |
406 |
|
|
\caption{Structures of the capping agent and solvents utilized in |
407 |
|
|
these simulations. The chemically-distinct sites (a-e) are expanded |
408 |
|
|
in terms of constituent atoms for both United Atom (UA) and All Atom |
409 |
|
|
(AA) force fields. Most parameters are from References |
410 |
|
|
\protect\cite{TraPPE-UA.alkanes,TraPPE-UA.alkylbenzenes,TraPPE-UA.thiols} |
411 |
|
|
(UA) and \protect\cite{OPLSAA} (AA). Cross-interactions with the Au |
412 |
|
|
atoms are given in Table 1 in the supporting information.} |
413 |
|
|
\label{demoMol} |
414 |
|
|
\end{figure} |
415 |
|
|
|
416 |
|
|
The Au-Au interactions in metal lattice slab is described by the |
417 |
|
|
quantum Sutton-Chen (QSC) formulation.\cite{PhysRevB.59.3527} The QSC |
418 |
|
|
potentials include zero-point quantum corrections and are |
419 |
|
|
reparametrized for accurate surface energies compared to the |
420 |
|
|
Sutton-Chen potentials.\cite{Chen90} |
421 |
|
|
|
422 |
|
|
For the two solvent molecules, {\it n}-hexane and toluene, two |
423 |
|
|
different atomistic models were utilized. Both solvents were modeled |
424 |
|
|
using United-Atom (UA) and All-Atom (AA) models. The TraPPE-UA |
425 |
|
|
parameters\cite{TraPPE-UA.alkanes,TraPPE-UA.alkylbenzenes} are used |
426 |
|
|
for our UA solvent molecules. In these models, sites are located at |
427 |
|
|
the carbon centers for alkyl groups. Bonding interactions, including |
428 |
|
|
bond stretches and bends and torsions, were used for intra-molecular |
429 |
|
|
sites closer than 3 bonds. For non-bonded interactions, Lennard-Jones |
430 |
|
|
potentials are used. |
431 |
|
|
|
432 |
|
|
By eliminating explicit hydrogen atoms, the TraPPE-UA models are |
433 |
|
|
simple and computationally efficient, while maintaining good accuracy. |
434 |
|
|
However, the TraPPE-UA model for alkanes is known to predict a slightly |
435 |
|
|
lower boiling point than experimental values. This is one of the |
436 |
|
|
reasons we used a lower average temperature (200K) for our |
437 |
|
|
simulations. If heat is transferred to the liquid phase during the |
438 |
|
|
NIVS simulation, the liquid in the hot slab can actually be |
439 |
|
|
substantially warmer than the mean temperature in the simulation. The |
440 |
|
|
lower mean temperatures therefore prevent solvent boiling. |
441 |
|
|
|
442 |
|
|
For UA-toluene, the non-bonded potentials between intermolecular sites |
443 |
|
|
have a similar Lennard-Jones formulation. The toluene molecules were |
444 |
|
|
treated as a single rigid body, so there was no need for |
445 |
|
|
intramolecular interactions (including bonds, bends, or torsions) in |
446 |
|
|
this solvent model. |
447 |
|
|
|
448 |
|
|
Besides the TraPPE-UA models, AA models for both organic solvents are |
449 |
|
|
included in our studies as well. The OPLS-AA\cite{OPLSAA} force fields |
450 |
|
|
were used. For hexane, additional explicit hydrogen sites were |
451 |
|
|
included. Besides bonding and non-bonded site-site interactions, |
452 |
|
|
partial charges and the electrostatic interactions were added to each |
453 |
|
|
CT and HC site. For toluene, a flexible model for the toluene molecule |
454 |
|
|
was utilized which included bond, bend, torsion, and inversion |
455 |
|
|
potentials to enforce ring planarity. |
456 |
|
|
|
457 |
|
|
The butanethiol capping agent in our simulations, were also modeled |
458 |
|
|
with both UA and AA model. The TraPPE-UA force field includes |
459 |
|
|
parameters for thiol molecules\cite{TraPPE-UA.thiols} and are used for |
460 |
|
|
UA butanethiol model in our simulations. The OPLS-AA also provides |
461 |
|
|
parameters for alkyl thiols. However, alkyl thiols adsorbed on Au(111) |
462 |
|
|
surfaces do not have the hydrogen atom bonded to sulfur. To derive |
463 |
|
|
suitable parameters for butanethiol adsorbed on Au(111) surfaces, we |
464 |
|
|
adopt the S parameters from Luedtke and Landman\cite{landman:1998} and |
465 |
|
|
modify the parameters for the CTS atom to maintain charge neutrality |
466 |
|
|
in the molecule. Note that the model choice (UA or AA) for the capping |
467 |
|
|
agent can be different from the solvent. Regardless of model choice, |
468 |
|
|
the force field parameters for interactions between capping agent and |
469 |
|
|
solvent can be derived using Lorentz-Berthelot Mixing Rule: |
470 |
|
|
\begin{eqnarray} |
471 |
|
|
\sigma_{ij} & = & \frac{1}{2} \left(\sigma_{ii} + \sigma_{jj}\right) \\ |
472 |
|
|
\epsilon_{ij} & = & \sqrt{\epsilon_{ii}\epsilon_{jj}} |
473 |
|
|
\end{eqnarray} |
474 |
|
|
|
475 |
|
|
To describe the interactions between metal (Au) and non-metal atoms, |
476 |
|
|
we refer to an adsorption study of alkyl thiols on gold surfaces by |
477 |
|
|
Vlugt {\it et al.}\cite{vlugt:cpc2007154} They fitted an effective |
478 |
|
|
Lennard-Jones form of potential parameters for the interaction between |
479 |
|
|
Au and pseudo-atoms CH$_x$ and S based on a well-established and |
480 |
|
|
widely-used effective potential of Hautman and Klein for the Au(111) |
481 |
|
|
surface.\cite{hautman:4994} As our simulations require the gold slab |
482 |
|
|
to be flexible to accommodate thermal excitation, the pair-wise form |
483 |
|
|
of potentials they developed was used for our study. |
484 |
|
|
|
485 |
|
|
The potentials developed from {\it ab initio} calculations by Leng |
486 |
|
|
{\it et al.}\cite{doi:10.1021/jp034405s} are adopted for the |
487 |
|
|
interactions between Au and aromatic C/H atoms in toluene. However, |
488 |
|
|
the Lennard-Jones parameters between Au and other types of particles, |
489 |
|
|
(e.g. AA alkanes) have not yet been established. For these |
490 |
|
|
interactions, the Lorentz-Berthelot mixing rule can be used to derive |
491 |
|
|
effective single-atom LJ parameters for the metal using the fit values |
492 |
|
|
for toluene. These are then used to construct reasonable mixing |
493 |
|
|
parameters for the interactions between the gold and other atoms. |
494 |
|
|
Table 1 in the supporting information summarizes the |
495 |
|
|
``metal/non-metal'' parameters utilized in our simulations. |
496 |
|
|
|
497 |
|
|
\section{Results} |
498 |
|
|
There are many factors contributing to the measured interfacial |
499 |
|
|
conductance; some of these factors are physically motivated |
500 |
|
|
(e.g. coverage of the surface by the capping agent coverage and |
501 |
|
|
solvent identity), while some are governed by parameters of the |
502 |
|
|
methodology (e.g. applied flux and the formulas used to obtain the |
503 |
|
|
conductance). In this section we discuss the major physical and |
504 |
|
|
calculational effects on the computed conductivity. |
505 |
|
|
|
506 |
|
|
\subsection{Effects due to capping agent coverage} |
507 |
|
|
|
508 |
|
|
A series of different initial conditions with a range of surface |
509 |
|
|
coverages was prepared and solvated with various with both of the |
510 |
|
|
solvent molecules. These systems were then equilibrated and their |
511 |
|
|
interfacial thermal conductivity was measured with the NIVS |
512 |
|
|
algorithm. Figure \ref{coverage} demonstrates the trend of conductance |
513 |
|
|
with respect to surface coverage. |
514 |
|
|
|
515 |
|
|
\begin{figure} |
516 |
|
|
\includegraphics[width=\linewidth]{coverage} |
517 |
|
|
\caption{The interfacial thermal conductivity ($G$) has a |
518 |
|
|
non-monotonic dependence on the degree of surface capping. This |
519 |
|
|
data is for the Au(111) / butanethiol / solvent interface with |
520 |
|
|
various UA force fields at $\langle T\rangle \sim $200K.} |
521 |
|
|
\label{coverage} |
522 |
|
|
\end{figure} |
523 |
|
|
|
524 |
|
|
In partially covered surfaces, the derivative definition for |
525 |
|
|
$G^\prime$ (Eq. \ref{derivativeG}) becomes difficult to apply, as the |
526 |
|
|
location of maximum change of $\lambda$ becomes washed out. The |
527 |
|
|
discrete definition (Eq. \ref{discreteG}) is easier to apply, as the |
528 |
|
|
Gibbs dividing surface is still well-defined. Therefore, $G$ (not |
529 |
|
|
$G^\prime$) was used in this section. |
530 |
|
|
|
531 |
|
|
From Figure \ref{coverage}, one can see the significance of the |
532 |
|
|
presence of capping agents. When even a small fraction of the Au(111) |
533 |
|
|
surface sites are covered with butanethiols, the conductivity exhibits |
534 |
|
|
an enhancement by at least a factor of 3. Capping agents are clearly |
535 |
|
|
playing a major role in thermal transport at metal / organic solvent |
536 |
|
|
surfaces. |
537 |
|
|
|
538 |
|
|
We note a non-monotonic behavior in the interfacial conductance as a |
539 |
|
|
function of surface coverage. The maximum conductance (largest $G$) |
540 |
|
|
happens when the surfaces are about 75\% covered with butanethiol |
541 |
|
|
caps. The reason for this behavior is not entirely clear. One |
542 |
|
|
explanation is that incomplete butanethiol coverage allows small gaps |
543 |
|
|
between butanethiols to form. These gaps can be filled by transient |
544 |
|
|
solvent molecules. These solvent molecules couple very strongly with |
545 |
|
|
the hot capping agent molecules near the surface, and can then carry |
546 |
|
|
away (diffusively) the excess thermal energy from the surface. |
547 |
|
|
|
548 |
|
|
There appears to be a competition between the conduction of the |
549 |
|
|
thermal energy away from the surface by the capping agents (enhanced |
550 |
|
|
by greater coverage) and the coupling of the capping agents with the |
551 |
|
|
solvent (enhanced by interdigitation at lower coverages). This |
552 |
|
|
competition would lead to the non-monotonic coverage behavior observed |
553 |
|
|
here. |
554 |
|
|
|
555 |
|
|
Results for rigid body toluene solvent, as well as the UA hexane, are |
556 |
|
|
within the ranges expected from prior experimental |
557 |
|
|
work.\cite{Wilson:2002uq,cahill:793,PhysRevB.80.195406} This suggests |
558 |
|
|
that explicit hydrogen atoms might not be required for modeling |
559 |
|
|
thermal transport in these systems. C-H vibrational modes do not see |
560 |
|
|
significant excited state population at low temperatures, and are not |
561 |
|
|
likely to carry lower frequency excitations from the solid layer into |
562 |
|
|
the bulk liquid. |
563 |
|
|
|
564 |
|
|
The toluene solvent does not exhibit the same behavior as hexane in |
565 |
|
|
that $G$ remains at approximately the same magnitude when the capping |
566 |
|
|
coverage increases from 25\% to 75\%. Toluene, as a rigid planar |
567 |
|
|
molecule, cannot occupy the relatively small gaps between the capping |
568 |
|
|
agents as easily as the chain-like {\it n}-hexane. The effect of |
569 |
|
|
solvent coupling to the capping agent is therefore weaker in toluene |
570 |
|
|
except at the very lowest coverage levels. This effect counters the |
571 |
|
|
coverage-dependent conduction of heat away from the metal surface, |
572 |
|
|
leading to a much flatter $G$ vs. coverage trend than is observed in |
573 |
|
|
{\it n}-hexane. |
574 |
|
|
|
575 |
|
|
\subsection{Effects due to Solvent \& Solvent Models} |
576 |
|
|
In addition to UA solvent and capping agent models, AA models have |
577 |
|
|
also been included in our simulations. In most of this work, the same |
578 |
|
|
(UA or AA) model for solvent and capping agent was used, but it is |
579 |
|
|
also possible to utilize different models for different components. |
580 |
|
|
We have also included isotopic substitutions (Hydrogen to Deuterium) |
581 |
|
|
to decrease the explicit vibrational overlap between solvent and |
582 |
|
|
capping agent. Table \ref{modelTest} summarizes the results of these |
583 |
|
|
studies. |
584 |
|
|
|
585 |
|
|
\begin{table*} |
586 |
|
|
\begin{minipage}{\linewidth} |
587 |
|
|
\begin{center} |
588 |
|
|
|
589 |
|
|
\caption{Computed interfacial thermal conductance ($G$ and |
590 |
|
|
$G^\prime$) values for interfaces using various models for |
591 |
|
|
solvent and capping agent (or without capping agent) at |
592 |
|
|
$\langle T\rangle\sim$200K. Here ``D'' stands for deuterated |
593 |
|
|
solvent or capping agent molecules. Error estimates are |
594 |
|
|
indicated in parentheses.} |
595 |
|
|
|
596 |
|
|
\begin{tabular}{llccc} |
597 |
|
|
\hline\hline |
598 |
|
|
Butanethiol model & Solvent & $G$ & $G^\prime$ \\ |
599 |
|
|
(or bare surface) & model & \multicolumn{2}{c}{(MW/m$^2$/K)} \\ |
600 |
|
|
\hline |
601 |
|
|
UA & UA hexane & 131(9) & 87(10) \\ |
602 |
|
|
& UA hexane(D) & 153(5) & 136(13) \\ |
603 |
|
|
& AA hexane & 131(6) & 122(10) \\ |
604 |
|
|
& UA toluene & 187(16) & 151(11) \\ |
605 |
|
|
& AA toluene & 200(36) & 149(53) \\ |
606 |
|
|
\hline |
607 |
|
|
AA & UA hexane & 116(9) & 129(8) \\ |
608 |
|
|
& AA hexane & 442(14) & 356(31) \\ |
609 |
|
|
& AA hexane(D) & 222(12) & 234(54) \\ |
610 |
|
|
& UA toluene & 125(25) & 97(60) \\ |
611 |
|
|
& AA toluene & 487(56) & 290(42) \\ |
612 |
|
|
\hline |
613 |
|
|
AA(D) & UA hexane & 158(25) & 172(4) \\ |
614 |
|
|
& AA hexane & 243(29) & 191(11) \\ |
615 |
|
|
& AA toluene & 364(36) & 322(67) \\ |
616 |
|
|
\hline |
617 |
|
|
bare & UA hexane & 46.5(3.2) & 49.4(4.5) \\ |
618 |
|
|
& UA hexane(D) & 43.9(4.6) & 43.0(2.0) \\ |
619 |
|
|
& AA hexane & 31.0(1.4) & 29.4(1.3) \\ |
620 |
|
|
& UA toluene & 70.1(1.3) & 65.8(0.5) \\ |
621 |
|
|
\hline\hline |
622 |
|
|
\end{tabular} |
623 |
|
|
\label{modelTest} |
624 |
|
|
\end{center} |
625 |
|
|
\end{minipage} |
626 |
|
|
\end{table*} |
627 |
|
|
|
628 |
|
|
To facilitate direct comparison between force fields, systems with the |
629 |
|
|
same capping agent and solvent were prepared with the same length |
630 |
|
|
scales for the simulation cells. |
631 |
|
|
|
632 |
|
|
On bare metal / solvent surfaces, different force field models for |
633 |
|
|
hexane yield similar results for both $G$ and $G^\prime$, and these |
634 |
|
|
two definitions agree with each other very well. This is primarily an |
635 |
|
|
indicator of weak interactions between the metal and the solvent. |
636 |
|
|
|
637 |
|
|
For the fully-covered surfaces, the choice of force field for the |
638 |
|
|
capping agent and solvent has a large impact on the calculated values |
639 |
|
|
of $G$ and $G^\prime$. The AA thiol to AA solvent conductivities are |
640 |
|
|
much larger than their UA to UA counterparts, and these values exceed |
641 |
|
|
the experimental estimates by a large measure. The AA force field |
642 |
|
|
allows significant energy to go into C-H (or C-D) stretching modes, |
643 |
|
|
and since these modes are high frequency, this non-quantum behavior is |
644 |
|
|
likely responsible for the overestimate of the conductivity. Compared |
645 |
|
|
to the AA model, the UA model yields more reasonable conductivity |
646 |
|
|
values with much higher computational efficiency. |
647 |
|
|
|
648 |
|
|
\subsubsection{Are electronic excitations in the metal important?} |
649 |
|
|
Because they lack electronic excitations, the QSC and related embedded |
650 |
|
|
atom method (EAM) models for gold are known to predict unreasonably |
651 |
|
|
low values for bulk conductivity |
652 |
|
|
($\lambda$).\cite{kuang:164101,ISI:000207079300006,Clancy:1992} If the |
653 |
|
|
conductance between the phases ($G$) is governed primarily by phonon |
654 |
|
|
excitation (and not electronic degrees of freedom), one would expect a |
655 |
|
|
classical model to capture most of the interfacial thermal |
656 |
|
|
conductance. Our results for $G$ and $G^\prime$ indicate that this is |
657 |
|
|
indeed the case, and suggest that the modeling of interfacial thermal |
658 |
|
|
transport depends primarily on the description of the interactions |
659 |
|
|
between the various components at the interface. When the metal is |
660 |
|
|
chemically capped, the primary barrier to thermal conductivity appears |
661 |
|
|
to be the interface between the capping agent and the surrounding |
662 |
|
|
solvent, so the excitations in the metal have little impact on the |
663 |
|
|
value of $G$. |
664 |
|
|
|
665 |
|
|
\subsection{Effects due to methodology and simulation parameters} |
666 |
|
|
|
667 |
|
|
We have varied the parameters of the simulations in order to |
668 |
|
|
investigate how these factors would affect the computation of $G$. Of |
669 |
|
|
particular interest are: 1) the length scale for the applied thermal |
670 |
|
|
gradient (modified by increasing the amount of solvent in the system), |
671 |
|
|
2) the sign and magnitude of the applied thermal flux, 3) the average |
672 |
|
|
temperature of the simulation (which alters the solvent density during |
673 |
|
|
equilibration), and 4) the definition of the interfacial conductance |
674 |
|
|
(Eqs. (\ref{discreteG}) or (\ref{derivativeG})) used in the |
675 |
|
|
calculation. |
676 |
|
|
|
677 |
|
|
Systems of different lengths were prepared by altering the number of |
678 |
|
|
solvent molecules and extending the length of the box along the $z$ |
679 |
|
|
axis to accomodate the extra solvent. Equilibration at the same |
680 |
|
|
temperature and pressure conditions led to nearly identical surface |
681 |
|
|
areas ($L_x$ and $L_y$) available to the metal and capping agent, |
682 |
|
|
while the extra solvent served mainly to lengthen the axis that was |
683 |
|
|
used to apply the thermal flux. For a given value of the applied |
684 |
|
|
flux, the different $z$ length scale has only a weak effect on the |
685 |
|
|
computed conductivities. |
686 |
|
|
|
687 |
|
|
\subsubsection{Effects of applied flux} |
688 |
|
|
The NIVS algorithm allows changes in both the sign and magnitude of |
689 |
|
|
the applied flux. It is possible to reverse the direction of heat |
690 |
|
|
flow simply by changing the sign of the flux, and thermal gradients |
691 |
|
|
which would be difficult to obtain experimentally ($5$ K/\AA) can be |
692 |
|
|
easily simulated. However, the magnitude of the applied flux is not |
693 |
|
|
arbitrary if one aims to obtain a stable and reliable thermal gradient. |
694 |
|
|
A temperature gradient can be lost in the noise if $|J_z|$ is too |
695 |
|
|
small, and excessive $|J_z|$ values can cause phase transitions if the |
696 |
|
|
extremes of the simulation cell become widely separated in |
697 |
|
|
temperature. Also, if $|J_z|$ is too large for the bulk conductivity |
698 |
|
|
of the materials, the thermal gradient will never reach a stable |
699 |
|
|
state. |
700 |
|
|
|
701 |
|
|
Within a reasonable range of $J_z$ values, we were able to study how |
702 |
|
|
$G$ changes as a function of this flux. In what follows, we use |
703 |
|
|
positive $J_z$ values to denote the case where energy is being |
704 |
|
|
transferred by the method from the metal phase and into the liquid. |
705 |
|
|
The resulting gradient therefore has a higher temperature in the |
706 |
|
|
liquid phase. Negative flux values reverse this transfer, and result |
707 |
|
|
in higher temperature metal phases. The conductance measured under |
708 |
|
|
different applied $J_z$ values is listed in Tables 2 and 3 in the |
709 |
|
|
supporting information. These results do not indicate that $G$ depends |
710 |
|
|
strongly on $J_z$ within this flux range. The linear response of flux |
711 |
|
|
to thermal gradient simplifies our investigations in that we can rely |
712 |
|
|
on $G$ measurement with only a small number $J_z$ values. |
713 |
|
|
|
714 |
|
|
The sign of $J_z$ is a different matter, however, as this can alter |
715 |
|
|
the temperature on the two sides of the interface. The average |
716 |
|
|
temperature values reported are for the entire system, and not for the |
717 |
|
|
liquid phase, so at a given $\langle T \rangle$, the system with |
718 |
|
|
positive $J_z$ has a warmer liquid phase. This means that if the |
719 |
|
|
liquid carries thermal energy via diffusive transport, {\it positive} |
720 |
|
|
$J_z$ values will result in increased molecular motion on the liquid |
721 |
|
|
side of the interface, and this will increase the measured |
722 |
|
|
conductivity. |
723 |
|
|
|
724 |
|
|
\subsubsection{Effects due to average temperature} |
725 |
|
|
|
726 |
|
|
We also studied the effect of average system temperature on the |
727 |
|
|
interfacial conductance. The simulations are first equilibrated in |
728 |
|
|
the NPT ensemble at 1 atm. The TraPPE-UA model for hexane tends to |
729 |
|
|
predict a lower boiling point (and liquid state density) than |
730 |
|
|
experiments. This lower-density liquid phase leads to reduced contact |
731 |
|
|
between the hexane and butanethiol, and this accounts for our |
732 |
|
|
observation of lower conductance at higher temperatures. In raising |
733 |
|
|
the average temperature from 200K to 250K, the density drop of |
734 |
|
|
$\sim$20\% in the solvent phase leads to a $\sim$40\% drop in the |
735 |
|
|
conductance. |
736 |
|
|
|
737 |
|
|
Similar behavior is observed in the TraPPE-UA model for toluene, |
738 |
|
|
although this model has better agreement with the experimental |
739 |
|
|
densities of toluene. The expansion of the toluene liquid phase is |
740 |
|
|
not as significant as that of the hexane (8.3\% over 100K), and this |
741 |
|
|
limits the effect to $\sim$20\% drop in thermal conductivity. |
742 |
|
|
|
743 |
|
|
Although we have not mapped out the behavior at a large number of |
744 |
|
|
temperatures, is clear that there will be a strong temperature |
745 |
|
|
dependence in the interfacial conductance when the physical properties |
746 |
|
|
of one side of the interface (notably the density) change rapidly as a |
747 |
|
|
function of temperature. |
748 |
|
|
|
749 |
|
|
Besides the lower interfacial thermal conductance, surfaces at |
750 |
|
|
relatively high temperatures are susceptible to reconstructions, |
751 |
|
|
particularly when butanethiols fully cover the Au(111) surface. These |
752 |
|
|
reconstructions include surface Au atoms which migrate outward to the |
753 |
|
|
S atom layer, and butanethiol molecules which embed into the surface |
754 |
|
|
Au layer. The driving force for this behavior is the strong Au-S |
755 |
|
|
interactions which are modeled here with a deep Lennard-Jones |
756 |
|
|
potential. This phenomenon agrees with reconstructions that have been |
757 |
|
|
experimentally |
758 |
|
|
observed.\cite{doi:10.1021/j100035a033,doi:10.1021/la026493y}. Vlugt |
759 |
|
|
{\it et al.} kept their Au(111) slab rigid so that their simulations |
760 |
|
|
could reach 300K without surface |
761 |
|
|
reconstructions.\cite{vlugt:cpc2007154} Since surface reconstructions |
762 |
|
|
blur the interface, the measurement of $G$ becomes more difficult to |
763 |
|
|
conduct at higher temperatures. For this reason, most of our |
764 |
|
|
measurements are undertaken at $\langle T\rangle\sim$200K where |
765 |
|
|
reconstruction is minimized. |
766 |
|
|
|
767 |
|
|
However, when the surface is not completely covered by butanethiols, |
768 |
|
|
the simulated system appears to be more resistent to the |
769 |
|
|
reconstruction. Our Au / butanethiol / toluene system had the Au(111) |
770 |
|
|
surfaces 90\% covered by butanethiols, but did not see this above |
771 |
|
|
phenomena even at $\langle T\rangle\sim$300K. That said, we did |
772 |
|
|
observe butanethiols migrating to neighboring three-fold sites during |
773 |
|
|
a simulation. Since the interface persisted in these simulations, we |
774 |
|
|
were able to obtain $G$'s for these interfaces even at a relatively |
775 |
|
|
high temperature without being affected by surface reconstructions. |
776 |
|
|
|
777 |
|
|
\section{Discussion} |
778 |
|
|
|
779 |
|
|
The primary result of this work is that the capping agent acts as an |
780 |
|
|
efficient thermal coupler between solid and solvent phases. One of |
781 |
|
|
the ways the capping agent can carry out this role is to down-shift |
782 |
|
|
between the phonon vibrations in the solid (which carry the heat from |
783 |
|
|
the gold) and the molecular vibrations in the liquid (which carry some |
784 |
|
|
of the heat in the solvent). |
785 |
|
|
|
786 |
|
|
To investigate the mechanism of interfacial thermal conductance, the |
787 |
|
|
vibrational power spectrum was computed. Power spectra were taken for |
788 |
|
|
individual components in different simulations. To obtain these |
789 |
|
|
spectra, simulations were run after equilibration in the |
790 |
|
|
microcanonical (NVE) ensemble and without a thermal |
791 |
|
|
gradient. Snapshots of configurations were collected at a frequency |
792 |
|
|
that is higher than that of the fastest vibrations occurring in the |
793 |
|
|
simulations. With these configurations, the velocity auto-correlation |
794 |
|
|
functions can be computed: |
795 |
|
|
\begin{equation} |
796 |
|
|
C_A (t) = \langle\vec{v}_A (t)\cdot\vec{v}_A (0)\rangle |
797 |
|
|
\label{vCorr} |
798 |
|
|
\end{equation} |
799 |
|
|
The power spectrum is constructed via a Fourier transform of the |
800 |
|
|
symmetrized velocity autocorrelation function, |
801 |
|
|
\begin{equation} |
802 |
|
|
\hat{f}(\omega) = |
803 |
|
|
\int_{-\infty}^{\infty} C_A (t) e^{-2\pi it\omega}\,dt |
804 |
|
|
\label{fourier} |
805 |
|
|
\end{equation} |
806 |
|
|
|
807 |
|
|
\subsection{The role of specific vibrations} |
808 |
|
|
The vibrational spectra for gold slabs in different environments are |
809 |
|
|
shown as in Figure \ref{specAu}. Regardless of the presence of |
810 |
|
|
solvent, the gold surfaces which are covered by butanethiol molecules |
811 |
|
|
exhibit an additional peak observed at a frequency of |
812 |
|
|
$\sim$165cm$^{-1}$. We attribute this peak to the S-Au bonding |
813 |
|
|
vibration. This vibration enables efficient thermal coupling of the |
814 |
|
|
surface Au layer to the capping agents. Therefore, in our simulations, |
815 |
|
|
the Au / S interfaces do not appear to be the primary barrier to |
816 |
|
|
thermal transport when compared with the butanethiol / solvent |
817 |
|
|
interfaces. This supports the results of Luo {\it et |
818 |
|
|
al.}\cite{Luo20101}, who reported $G$ for Au-SAM junctions roughly |
819 |
|
|
twice as large as what we have computed for the thiol-liquid |
820 |
|
|
interfaces. |
821 |
|
|
|
822 |
|
|
\begin{figure} |
823 |
|
|
\includegraphics[width=\linewidth]{vibration} |
824 |
|
|
\caption{The vibrational power spectrum for thiol-capped gold has an |
825 |
|
|
additional vibrational peak at $\sim $165cm$^{-1}$. Bare gold |
826 |
|
|
surfaces (both with and without a solvent over-layer) are missing |
827 |
|
|
this peak. A similar peak at $\sim $165cm$^{-1}$ also appears in |
828 |
|
|
the vibrational power spectrum for the butanethiol capping agents.} |
829 |
|
|
\label{specAu} |
830 |
|
|
\end{figure} |
831 |
|
|
|
832 |
|
|
Also in this figure, we show the vibrational power spectrum for the |
833 |
|
|
bound butanethiol molecules, which also exhibits the same |
834 |
|
|
$\sim$165cm$^{-1}$ peak. |
835 |
|
|
|
836 |
|
|
\subsection{Overlap of power spectra} |
837 |
|
|
A comparison of the results obtained from the two different organic |
838 |
|
|
solvents can also provide useful information of the interfacial |
839 |
|
|
thermal transport process. In particular, the vibrational overlap |
840 |
|
|
between the butanethiol and the organic solvents suggests a highly |
841 |
|
|
efficient thermal exchange between these components. Very high |
842 |
|
|
thermal conductivity was observed when AA models were used and C-H |
843 |
|
|
vibrations were treated classically. The presence of extra degrees of |
844 |
|
|
freedom in the AA force field yields higher heat exchange rates |
845 |
|
|
between the two phases and results in a much higher conductivity than |
846 |
|
|
in the UA force field. The all-atom classical models include high |
847 |
|
|
frequency modes which should be unpopulated at our relatively low |
848 |
|
|
temperatures. This artifact is likely the cause of the high thermal |
849 |
|
|
conductance in all-atom MD simulations. |
850 |
|
|
|
851 |
|
|
The similarity in the vibrational modes available to solvent and |
852 |
|
|
capping agent can be reduced by deuterating one of the two components |
853 |
|
|
(Fig. \ref{aahxntln}). Once either the hexanes or the butanethiols |
854 |
|
|
are deuterated, one can observe a significantly lower $G$ and |
855 |
|
|
$G^\prime$ values (Table \ref{modelTest}). |
856 |
|
|
|
857 |
|
|
\begin{figure} |
858 |
|
|
\includegraphics[width=\linewidth]{aahxntln} |
859 |
|
|
\caption{Spectra obtained for all-atom (AA) Au / butanethiol / solvent |
860 |
|
|
systems. When butanethiol is deuterated (lower left), its |
861 |
|
|
vibrational overlap with hexane decreases significantly. Since |
862 |
|
|
aromatic molecules and the butanethiol are vibrationally dissimilar, |
863 |
|
|
the change is not as dramatic when toluene is the solvent (right).} |
864 |
|
|
\label{aahxntln} |
865 |
|
|
\end{figure} |
866 |
|
|
|
867 |
|
|
For the Au / butanethiol / toluene interfaces, having the AA |
868 |
|
|
butanethiol deuterated did not yield a significant change in the |
869 |
|
|
measured conductance. Compared to the C-H vibrational overlap between |
870 |
|
|
hexane and butanethiol, both of which have alkyl chains, the overlap |
871 |
|
|
between toluene and butanethiol is not as significant and thus does |
872 |
|
|
not contribute as much to the heat exchange process. |
873 |
|
|
|
874 |
|
|
Previous observations of Zhang {\it et al.}\cite{hase:2010} indicate |
875 |
|
|
that the {\it intra}molecular heat transport due to alkylthiols is |
876 |
|
|
highly efficient. Combining our observations with those of Zhang {\it |
877 |
|
|
et al.}, it appears that butanethiol acts as a channel to expedite |
878 |
|
|
heat flow from the gold surface and into the alkyl chain. The |
879 |
|
|
vibrational coupling between the metal and the liquid phase can |
880 |
|
|
therefore be enhanced with the presence of suitable capping agents. |
881 |
|
|
|
882 |
|
|
Deuterated models in the UA force field did not decouple the thermal |
883 |
|
|
transport as well as in the AA force field. The UA models, even |
884 |
|
|
though they have eliminated the high frequency C-H vibrational |
885 |
|
|
overlap, still have significant overlap in the lower-frequency |
886 |
|
|
portions of the infrared spectra (Figure \ref{uahxnua}). Deuterating |
887 |
|
|
the UA models did not decouple the low frequency region enough to |
888 |
|
|
produce an observable difference for the results of $G$ (Table |
889 |
|
|
\ref{modelTest}). |
890 |
|
|
|
891 |
|
|
\begin{figure} |
892 |
|
|
\includegraphics[width=\linewidth]{uahxnua} |
893 |
|
|
\caption{Vibrational power spectra for UA models for the butanethiol |
894 |
|
|
and hexane solvent (upper panel) show the high degree of overlap |
895 |
|
|
between these two molecules, particularly at lower frequencies. |
896 |
|
|
Deuterating a UA model for the solvent (lower panel) does not |
897 |
|
|
decouple the two spectra to the same degree as in the AA force |
898 |
|
|
field (see Fig \ref{aahxntln}).} |
899 |
|
|
\label{uahxnua} |
900 |
|
|
\end{figure} |
901 |
|
|
|
902 |
|
|
\section{Conclusions} |
903 |
|
|
The NIVS algorithm has been applied to simulations of |
904 |
|
|
butanethiol-capped Au(111) surfaces in the presence of organic |
905 |
|
|
solvents. This algorithm allows the application of unphysical thermal |
906 |
|
|
flux to transfer heat between the metal and the liquid phase. With the |
907 |
|
|
flux applied, we were able to measure the corresponding thermal |
908 |
|
|
gradients and to obtain interfacial thermal conductivities. Under |
909 |
|
|
steady states, 2-3 ns trajectory simulations are sufficient for |
910 |
|
|
computation of this quantity. |
911 |
|
|
|
912 |
|
|
Our simulations have seen significant conductance enhancement in the |
913 |
|
|
presence of capping agent, compared with the bare gold / liquid |
914 |
|
|
interfaces. The vibrational coupling between the metal and the liquid |
915 |
|
|
phase is enhanced by a chemically-bonded capping agent. Furthermore, |
916 |
|
|
the coverage percentage of the capping agent plays an important role |
917 |
|
|
in the interfacial thermal transport process. Moderately low coverages |
918 |
|
|
allow higher contact between capping agent and solvent, and thus could |
919 |
|
|
further enhance the heat transfer process, giving a non-monotonic |
920 |
|
|
behavior of conductance with increasing coverage. |
921 |
|
|
|
922 |
|
|
Our results, particularly using the UA models, agree well with |
923 |
|
|
available experimental data. The AA models tend to overestimate the |
924 |
|
|
interfacial thermal conductance in that the classically treated C-H |
925 |
|
|
vibrations become too easily populated. Compared to the AA models, the |
926 |
|
|
UA models have higher computational efficiency with satisfactory |
927 |
|
|
accuracy, and thus are preferable in modeling interfacial thermal |
928 |
|
|
transport. |
929 |
|
|
|
930 |
|
|
Of the two definitions for $G$, the discrete form |
931 |
|
|
(Eq. \ref{discreteG}) was easier to use and gives out relatively |
932 |
|
|
consistent results, while the derivative form (Eq. \ref{derivativeG}) |
933 |
|
|
is not as versatile. Although $G^\prime$ gives out comparable results |
934 |
|
|
and follows similar trend with $G$ when measuring close to fully |
935 |
|
|
covered or bare surfaces, the spatial resolution of $T$ profile |
936 |
|
|
required for the use of a derivative form is limited by the number of |
937 |
|
|
bins and the sampling required to obtain thermal gradient information. |
938 |
|
|
|
939 |
|
|
Vlugt {\it et al.} have investigated the surface thiol structures for |
940 |
|
|
nanocrystalline gold and pointed out that they differ from those of |
941 |
|
|
the Au(111) surface.\cite{landman:1998,vlugt:cpc2007154} This |
942 |
|
|
difference could also cause differences in the interfacial thermal |
943 |
|
|
transport behavior. To investigate this problem, one would need an |
944 |
|
|
effective method for applying thermal gradients in non-planar |
945 |
|
|
(i.e. spherical) geometries. |
946 |
|
|
|
947 |
|
|
\section{Acknowledgments} |
948 |
|
|
Support for this project was provided by the National Science |
949 |
|
|
Foundation under grant CHE-0848243. Computational time was provided by |
950 |
|
|
the Center for Research Computing (CRC) at the University of Notre |
951 |
|
|
Dame. |
952 |
|
|
|
953 |
|
|
{\bf Supporting Information Available:} Force field parameters |
954 |
|
|
utilized in the simulation are available in the supporting |
955 |
|
|
information, as are thermal conductivity data for a range of applied |
956 |
|
|
thermal fluxes, simulation temperatures, and solvent densities. This |
957 |
|
|
information is available free of charge via the Internet at |
958 |
|
|
http://pubs.acs.org. |
959 |
|
|
|
960 |
|
|
\newpage |
961 |
|
|
|
962 |
|
|
\bibliography{interfacial} |
963 |
|
|
|
964 |
|
|
\end{doublespace} |
965 |
|
|
\end{document} |
966 |
|
|
|