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\begin{document} |
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\newcolumntype{A}{p{1.5in}} |
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\title{Simulations of heat conduction at thiolate-capped gold |
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surfaces: The role of chain length and solvent penetration} |
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\author{Kelsey M. Stocker and J. Daniel |
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Gezelter\footnote{Corresponding author. \ Electronic mail: |
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gezelter@nd.edu} \\ |
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251 Nieuwland Science Hall, \\ |
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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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We report on simulations of heat conduction from Au(111) / hexane interfaces in which the surface has been protected by a mix of short and long chain alkanethiolate ligands. A variant of reverse non-equilibrium molecular dynamics (RNEMD) was used to create a thermal flux between the metal and solvent, and thermal conductance was computed using the resulting thermal profiles of the interface. We find a non-monotonic dependence of the interfacial thermal conductance on the fraction of long chains present at the interface, and correlate this behavior to solvent ordering and escape from the thiolate layer immediately in contact with the metal surface. |
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Our results suggest that a mixed vibrational transfer / convection model is |
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necessary to explain the features of heat transfer at this interface. The |
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alignment of the solvent chains with the ordered ligand allows rapid transfer |
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of energy to the trapped solvent (and this is the dominant feature for ordered |
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ligand layers). Diffusion of the vibrationally excited solvent into the bulk |
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also plays a significant role when the ligands are less tightly packed. |
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\end{abstract} |
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\newpage |
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%\narrowtext |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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% **INTRODUCTION** |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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\section{Introduction} |
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|
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The structural details of the interfaces of metal nanoparticles |
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determines how energy flows between these particles and their |
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surroundings. Understanding this energy flow is essential in designing |
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and functionalizing metallic nanoparticles for plasmonic photothermal |
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therapies,\cite{Jain:2007ux,Petrova:2007ad,Gnyawali:2008lp,Mazzaglia:2008to,Huff:2007ye,Larson:2007hw} |
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which rely on the ability of metallic nanoparticles to absorb light in |
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the near-IR, a portion of the spectrum in which living tissue is very |
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nearly transparent. The relevant physical property controlling the |
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transfer of this energy as heat into the surrounding tissue is the |
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interfacial thermal conductance, $G$, which can be somewhat difficult |
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to determine experimentally.\cite{Wilson:2002uq,Plech:2005kx} |
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|
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Metallic particles have also been proposed for use in highly efficient |
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thermal-transfer fluids, although careful experiments by Eapen {\it et |
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al.} have shown that metal-particle-based nanofluids have thermal |
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conductivities that match Maxwell predictions.\cite{Eapen:2007th} The |
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likely cause of previously reported non-Maxwell |
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behavior\cite{Eastman:2001wb,Keblinski:2002bx,Lee:1999ct,Xue:2003ya,Xue:2004oa} |
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is percolation networks of nanoparticles exchanging energy via the |
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solvent,\cite{Eapen:2007mw} so it is important to get a detailed |
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molecular picture of particle-ligand and ligand-solvent interactions |
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in order to understand the thermal behavior of complex fluids. To |
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date, there have been few reported values (either from theory or |
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experiment) for $G$ for ligand-protected nanoparticles embedded in |
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liquids, and there is a significant gap in knowledge about how |
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chemically distinct ligands or protecting groups will affect heat |
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transport from the particles. |
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|
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Experimentally, the thermal properties of various nanostructured |
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interfaces have been investigated by a number of groups. Cahill and |
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coworkers studied thermal transport from metal nanoparticle/fluid |
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interfaces, epitaxial TiN/single crystal oxide interfaces, and |
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hydrophilic and hydrophobic interfaces between water and solids with |
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different self-assembled |
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monolayers.\cite{cahill:793,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 the 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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have 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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In previous simulations, we applied a variant of reverse |
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non-equilibrium molecular dynamics (RNEMD) to calculate the |
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interfacial thermal conductance at a metal / organic solvent interface |
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that had been chemically protected by butanethiolate groups. Our |
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calculations suggested an explanation for the very large thermal |
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conductivity at alkanethiol-capped metal surfaces when compared with |
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bare metal/solvent interfaces. Specifically, the chemical bond |
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between the metal and the ligand introduces a vibrational overlap that |
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is not present without the protecting group, and the overlap between |
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the vibrational spectra (metal to ligand, ligand to solvent) provides |
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a mechanism for rapid thermal transport across the interface. |
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A notable result of the previous simulations was the non-monotonic |
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dependence of $G$ on the fractional coverage of the metal surface by |
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the chemical protecting group. Gaps in surface coverage allow the |
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solvent molecules come into direct contact with ligands that had been |
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heated by the metal surface, absorb thermal energy from the ligands, |
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and then diffuse away. Quantifying the role of surface coverage is |
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difficult as the ligands have lateral mobility on the surface and can |
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aggregate to form domains on the timescale of the simulation. |
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To isolate this effect without worrying about lateral mobility of the |
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surface ligands, the current work involves mixed-chain-length |
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monolayers in which the length mismatch between long and short chains |
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is sufficient to accommodate solvent molecules. These completely |
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covered (but mixed-chain) surfaces are significantly less prone to |
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surface domain formation, and allow us to further investigate the |
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mechanism of heat transport to the solvent. A thermal flux is created |
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using velocity shearing and scaling reverse non-equilibrium molecular |
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dynamics (VSS-RNEMD), and the resulting temperature profiles are |
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analyzed to yield information about the interfacial thermal |
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conductance. |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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% **METHODOLOGY** |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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\section{Methodology} |
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There are many ways to compute bulk transport properties from |
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classical molecular dynamics simulations. Equilibrium Molecular |
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Dynamics (EMD) simulations can be used by computing relevant time |
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correlation functions and assuming that linear response theory |
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holds.\cite{PhysRevB.37.5677,MASSOBRIO:1984bl,PhysRev.119.1,Viscardy:2007rp,che:6888,kinaci:014106} |
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For some transport properties (notably thermal conductivity), EMD |
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approaches are subject to noise and poor convergence of the relevant |
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correlation functions. Traditional Non-equilibrium Molecular Dynamics |
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(NEMD) methods impose a gradient (e.g. thermal or momentum) on a |
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simulation.\cite{ASHURST:1975tg,Evans:1982zk,ERPENBECK:1984sp,MAGINN:1993hc,Berthier:2002ij,Evans:2002ai,Schelling:2002dp,PhysRevA.34.1449,JiangHao_jp802942v} |
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However, the resulting flux is often difficult to |
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measure. Furthermore, problems arise for NEMD simulations of |
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heterogeneous systems, such as phase-phase boundaries or interfaces, |
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where the type of gradient to enforce at the boundary between |
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materials is unclear. |
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{\it Reverse} Non-Equilibrium Molecular Dynamics (RNEMD) methods adopt |
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a different approach in that an unphysical {\it flux} is imposed |
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between different regions or ``slabs'' of the simulation |
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box.\cite{MullerPlathe:1997xw,ISI:000080382700030,Kuang2010} The |
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system responds by developing a temperature or momentum {\it gradient} |
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between the two regions. Since the amount of the applied flux is known |
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exactly, and the measurement of a gradient is generally less |
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complicated, imposed-flux methods typically take shorter simulation |
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times to obtain converged results for transport properties. The |
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corresponding temperature or velocity gradients which develop in |
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response to the applied flux are then related (via linear response |
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theory) to the transport coefficient of interest. These methods are |
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quite efficient, in that they do not need many trajectories to provide |
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information about transport properties. To date, they have been |
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utilized in computing thermal and mechanical transfer of both |
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homogeneous |
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liquids~\cite{MullerPlathe:1997xw,ISI:000080382700030,Maginn:2010} as |
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well as heterogeneous |
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systems.\cite{garde:nl2005,garde:PhysRevLett2009,kuang:AuThl} |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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% VSS-RNEMD |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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\subsection{VSS-RNEMD} |
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The original ``swapping'' approaches by M\"{u}ller-Plathe {\it et |
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al.}\cite{ISI:000080382700030,MullerPlathe:1997xw} can be understood |
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as a sequence of imaginary elastic collisions between particles in |
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regions separated by half of the simulation cell. In each collision, |
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the entire momentum vectors of both particles may be exchanged to |
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generate a thermal flux. Alternatively, a single component of the |
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momentum vectors may be exchanged to generate a shear flux. This |
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algorithm turns out to be quite useful in many simulations of bulk |
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liquids. However, the M\"{u}ller-Plathe swapping approach perturbs the |
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system away from ideal Maxwell-Boltzmann distributions, and this has |
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undesirable side-effects when the applied flux becomes |
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large.\cite{Maginn:2010} |
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The most useful alternative RNEMD approach developed so far utilizes a |
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series of simultaneous velocity shearing and scaling exchanges between |
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the two slabs.\cite{Kuang2012} This method provides a set of |
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conservation constraints while simultaneously creating a desired flux |
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between the two slabs. Satisfying the constraint equations ensures |
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that the new configurations are sampled from the same NVE ensemble. |
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The VSS moves are applied periodically to scale and shift the particle |
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velocities ($\mathbf{v}_i$ and $\mathbf{v}_j$) in two slabs ($H$ and |
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$C$) which are separated by half of the simulation box, |
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\begin{displaymath} |
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\begin{array}{rclcl} |
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& \underline{\mathrm{shearing}} & & |
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\underline{~~~~~~~~~~~~\mathrm{scaling}~~~~~~~~~~~~} \\ \\ |
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\mathbf{v}_i \leftarrow & |
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\mathbf{a}_c\ & + & c\cdot\left(\mathbf{v}_i - \langle\mathbf{v}_c |
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\rangle\right) + \langle\mathbf{v}_c\rangle \\ |
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\mathbf{v}_j \leftarrow & |
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\mathbf{a}_h & + & h\cdot\left(\mathbf{v}_j - \langle\mathbf{v}_h |
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\rangle\right) + \langle\mathbf{v}_h\rangle . |
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\end{array} |
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\end{displaymath} |
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Here $\langle\mathbf{v}_c\rangle$ and $\langle\mathbf{v}_h\rangle$ are |
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the center of mass velocities in the $C$ and $H$ slabs, respectively. |
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Within the two slabs, particles receive incremental changes or a |
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``shear'' to their velocities. The amount of shear is governed by the |
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imposed momentum flux, $j_z(\mathbf{p})$ |
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\begin{eqnarray} |
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\mathbf{a}_c & = & - \mathbf{j}_z(\mathbf{p}) \Delta t / M_c \label{vss1}\\ |
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\mathbf{a}_h & = & + \mathbf{j}_z(\mathbf{p}) \Delta t / M_h \label{vss2} |
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\end{eqnarray} |
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where $M_{\{c,h\}}$ is total mass of particles within each slab and $\Delta t$ |
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is the interval between two separate operations. |
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To simultaneously impose a thermal flux ($J_z$) between the slabs we |
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use energy conservation constraints, |
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\begin{eqnarray} |
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K_c - J_z\Delta t & = & c^2 (K_c - \frac{1}{2}M_c \langle\mathbf{v}_c |
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\rangle^2) + \frac{1}{2}M_c (\langle \mathbf{v}_c \rangle + \mathbf{a}_c)^2 \label{vss3}\\ |
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K_h + J_z\Delta t & = & h^2 (K_h - \frac{1}{2}M_h \langle\mathbf{v}_h |
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\rangle^2) + \frac{1}{2}M_h (\langle \mathbf{v}_h \rangle + |
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\mathbf{a}_h)^2 \label{vss4}. |
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\label{constraint} |
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\end{eqnarray} |
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Simultaneous solution of these quadratic formulae for the scaling |
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coefficients, $c$ and $h$, will ensure that the simulation samples from |
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the original microcanonical (NVE) ensemble. Here $K_{\{c,h\}}$ is the |
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instantaneous translational kinetic energy of each slab. At each time |
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interval, it is a simple matter to solve for $c$, $h$, $\mathbf{a}_c$, |
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and $\mathbf{a}_h$, subject to the imposed momentum flux, |
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$j_z(\mathbf{p})$, and thermal flux, $J_z$ values. Since the VSS |
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operations do not change the kinetic energy due to orientational |
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degrees of freedom or the potential energy of a system, configurations |
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after the VSS move have exactly the same energy ({\it and} linear |
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momentum) as before the move. |
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As the simulation progresses, the VSS moves are performed on a regular |
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basis, and the system develops a thermal or velocity gradient in |
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response to the applied flux. Using the slope of the temperature or |
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velocity gradient, it is quite simple to obtain of thermal |
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conductivity ($\lambda$), |
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\begin{equation} |
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J_z = -\lambda \frac{\partial T}{\partial z}, |
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\end{equation} |
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and shear viscosity ($\eta$), |
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\begin{equation} |
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j_z(p_x) = -\eta \frac{\partial \langle v_x \rangle}{\partial z}. |
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\end{equation} |
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Here, the quantities on the left hand side are the imposed flux |
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values, while the slopes are obtained from linear fits to the |
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gradients that develop in the simulated system. |
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The VSS-RNEMD approach is versatile in that it may be used to |
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implement both thermal and shear transport either separately or |
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simultaneously. Perturbations of velocities away from the ideal |
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Maxwell-Boltzmann distributions are minimal, and thermal anisotropy is |
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kept to a minimum. This ability to generate simultaneous thermal and |
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shear fluxes has been previously utilized to map out the shear |
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viscosity of SPC/E water over a wide range of temperatures (90~K) with |
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a single 1 ns simulation.\cite{Kuang2012} |
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|
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\begin{figure} |
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\includegraphics[width=\linewidth]{figures/rnemd} |
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\caption{The VSS-RNEMD approach imposes unphysical transfer of |
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linear momentum or kinetic energy between a ``hot'' slab and a |
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``cold'' slab in the simulation box. The system responds to this |
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imposed flux by generating velocity or temperature gradients. The |
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slope of the gradients can then be used to compute transport |
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properties (e.g. shear viscosity or thermal conductivity).} |
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\label{fig:rnemd} |
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\end{figure} |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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% INTERFACIAL CONDUCTANCE |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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\subsection{Reverse Non-Equilibrium Molecular Dynamics approaches |
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to interfacial transport} |
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Interfaces between dissimilar materials have transport properties |
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which can be defined as derivatives of the standard transport |
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coefficients in a direction normal to the interface. For example, the |
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|
{\it interfacial} thermal conductance ($G$) can be thought of as the |
318 |
|
|
change in the thermal conductivity ($\lambda$) across the boundary |
319 |
|
|
between materials: |
320 |
|
|
\begin{align} |
321 |
|
|
G &= \left(\nabla\lambda \cdot \mathbf{\hat{n}}\right)_{z_0} \\ |
322 |
|
|
&= J_z\left(\frac{\partial^2 T}{\partial z^2}\right)_{z_0} \Big/ |
323 |
|
|
\left(\frac{\partial T}{\partial z}\right)_{z_0}^2. |
324 |
|
|
\label{derivativeG} |
325 |
|
|
\end{align} |
326 |
|
|
where $z_0$ is the location of the interface between two materials and |
327 |
|
|
$\mathbf{\hat{n}}$ is a unit vector normal to the interface (assumed |
328 |
|
|
to be the $z$ direction from here on). RNEMD simulations, and |
329 |
|
|
particularly the VSS-RNEMD approach, function by applying a momentum |
330 |
|
|
or thermal flux and watching the gradient response of the |
331 |
|
|
material. This means that the {\it interfacial} conductance is a |
332 |
|
|
second derivative property which is subject to significant noise and |
333 |
|
|
therefore requires extensive sampling. We have been able to |
334 |
|
|
demonstrate the use of the second derivative approach to compute |
335 |
|
|
interfacial conductance at chemically-modified metal / solvent |
336 |
|
|
interfaces. However, a definition of the interfacial conductance in |
337 |
|
|
terms of a temperature difference ($\Delta T$) measured at $z_0$, |
338 |
|
|
\begin{displaymath} |
339 |
|
|
G = \frac{J_z}{\Delta T_{z_0}}, |
340 |
|
|
\end{displaymath} |
341 |
|
|
is useful once the RNEMD approach has generated a stable temperature |
342 |
|
|
gap across the interface. |
343 |
|
|
|
344 |
|
|
\begin{figure} |
345 |
|
|
\includegraphics[width=\linewidth]{figures/resistor_series} |
346 |
gezelter |
3822 |
\caption{The inverse of the interfacial thermal conductance, $G$, is |
347 |
|
|
the Kapitza resistance, $R_K$. Because the gold / thiolate/ |
348 |
|
|
solvent interface extends a significant distance from the metal |
349 |
|
|
surface, the interfacial resistance $R_K$ can be computed by |
350 |
|
|
summing a series of temperature drops between adjacent temperature |
351 |
|
|
bins along the $z$ axis.} |
352 |
gezelter |
3803 |
\label{fig:resistor_series} |
353 |
|
|
\end{figure} |
354 |
|
|
|
355 |
|
|
In the particular case we are studying here, there are two interfaces |
356 |
|
|
involved in the transfer of heat from the gold slab to the solvent: |
357 |
gezelter |
3850 |
the metal/thiolate interface and the thiolate/solvent interface. We |
358 |
gezelter |
3803 |
could treat the temperature on each side of an interface as discrete, |
359 |
|
|
making the interfacial conductance the inverse of the Kaptiza |
360 |
|
|
resistance, or $G = \frac{1}{R_k}$. To model the total conductance |
361 |
|
|
across multiple interfaces, it is useful to think of the interfaces as |
362 |
|
|
a set of resistors wired in series. The total resistance is then |
363 |
|
|
additive, $R_{total} = \sum_i R_{i}$ and the interfacial conductance |
364 |
|
|
is the inverse of the total resistance, or $G = \frac{1}{\sum_i |
365 |
|
|
R_i}$). In the interfacial region, we treat each bin in the |
366 |
|
|
VSS-RNEMD temperature profile as a resistor with resistance |
367 |
|
|
$\frac{T_{2}-T_{1}}{J_z}$, $\frac{T_{3}-T_{2}}{J_z}$, etc. The sum of |
368 |
|
|
the set of resistors which spans the gold/thiolate interface, thiolate |
369 |
|
|
chains, and thiolate/solvent interface simplifies to |
370 |
kstocke1 |
3801 |
\begin{equation} |
371 |
gezelter |
3825 |
R_{K} = \frac{T_{n}-T_{1}}{J_z}, |
372 |
gezelter |
3803 |
\label{eq:finalG} |
373 |
|
|
\end{equation} |
374 |
|
|
or the temperature difference between the gold side of the |
375 |
|
|
gold/thiolate interface and the solvent side of the thiolate/solvent |
376 |
|
|
interface over the applied flux. |
377 |
kstocke1 |
3801 |
|
378 |
|
|
|
379 |
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
380 |
|
|
% **COMPUTATIONAL DETAILS** |
381 |
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
382 |
|
|
\section{Computational Details} |
383 |
|
|
|
384 |
gezelter |
3803 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
385 |
gezelter |
3819 |
% FORCE-FIELD PARAMETERS |
386 |
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
387 |
|
|
\subsection{Force-Field Parameters} |
388 |
kstocke1 |
3801 |
|
389 |
gezelter |
3819 |
Our simulations include a number of chemically distinct components. |
390 |
|
|
Figure \ref{fig:structures} demonstrates the sites defined for both |
391 |
|
|
the {\it n}-hexane and alkanethiolate ligands present in our |
392 |
gezelter |
3850 |
simulations. |
393 |
gezelter |
3819 |
|
394 |
|
|
\begin{figure} |
395 |
|
|
\includegraphics[width=\linewidth]{figures/structures} |
396 |
gezelter |
3822 |
\caption{Topologies of the thiolate capping agents and solvent |
397 |
|
|
utilized in the simulations. The chemically-distinct sites (S, |
398 |
|
|
\ce{CH2}, and \ce{CH3}) are treated as united atoms. Most |
399 |
|
|
parameters are taken from references \bibpunct{}{}{,}{n}{}{,} |
400 |
|
|
\protect\cite{TraPPE-UA.alkanes} and |
401 |
|
|
\protect\cite{TraPPE-UA.thiols}. Cross-interactions with the Au |
402 |
|
|
atoms were adapted from references |
403 |
|
|
\protect\cite{landman:1998},~\protect\cite{vlugt:cpc2007154},~and |
404 |
|
|
\protect\cite{hautman:4994}.} |
405 |
gezelter |
3819 |
\label{fig:structures} |
406 |
|
|
\end{figure} |
407 |
|
|
|
408 |
kstocke1 |
3829 |
The Au-Au interactions in the metal lattice slab are described by the |
409 |
gezelter |
3819 |
quantum Sutton-Chen (QSC) formulation.\cite{PhysRevB.59.3527} The QSC |
410 |
|
|
potentials include zero-point quantum corrections and are |
411 |
|
|
reparametrized for accurate surface energies compared to the |
412 |
|
|
Sutton-Chen potentials.\cite{Chen90} |
413 |
|
|
|
414 |
|
|
For the {\it n}-hexane solvent molecules, the TraPPE-UA |
415 |
|
|
parameters\cite{TraPPE-UA.alkanes} were utilized. In this model, |
416 |
|
|
sites are located at the carbon centers for alkyl groups. Bonding |
417 |
|
|
interactions, including bond stretches and bends and torsions, were |
418 |
|
|
used for intra-molecular sites closer than 3 bonds. For non-bonded |
419 |
gezelter |
3850 |
interactions, Lennard-Jones potentials were used. We have previously |
420 |
gezelter |
3819 |
utilized both united atom (UA) and all-atom (AA) force fields for |
421 |
gezelter |
3850 |
thermal conductivity,\cite{kuang:AuThl,Kuang2012} and since the united |
422 |
gezelter |
3819 |
atom force fields cannot populate the high-frequency modes that are |
423 |
|
|
present in AA force fields, they appear to work better for modeling |
424 |
|
|
thermal conductivity. The TraPPE-UA model for alkanes is known to |
425 |
|
|
predict a slightly lower boiling point than experimental values. This |
426 |
|
|
is one of the reasons we used a lower average temperature (200K) for |
427 |
gezelter |
3850 |
our simulations. |
428 |
gezelter |
3819 |
|
429 |
|
|
The TraPPE-UA force field includes parameters for thiol |
430 |
|
|
molecules\cite{TraPPE-UA.thiols} which were used for the |
431 |
|
|
alkanethiolate molecules in our simulations. To derive suitable |
432 |
|
|
parameters for butanethiol adsorbed on Au(111) surfaces, we adopted |
433 |
|
|
the S parameters from Luedtke and Landman\cite{landman:1998} and |
434 |
|
|
modified the parameters for the CTS atom to maintain charge neutrality |
435 |
|
|
in the molecule. |
436 |
|
|
|
437 |
|
|
To describe the interactions between metal (Au) and non-metal atoms, |
438 |
|
|
we refer to an adsorption study of alkyl thiols on gold surfaces by |
439 |
|
|
Vlugt {\it et al.}\cite{vlugt:cpc2007154} They fitted an effective |
440 |
|
|
Lennard-Jones form of potential parameters for the interaction between |
441 |
|
|
Au and pseudo-atoms CH$_x$ and S based on a well-established and |
442 |
|
|
widely-used effective potential of Hautman and Klein for the Au(111) |
443 |
|
|
surface.\cite{hautman:4994} As our simulations require the gold slab |
444 |
|
|
to be flexible to accommodate thermal excitation, the pair-wise form |
445 |
kstocke1 |
3832 |
of potentials they developed was used for our study. |
446 |
kstocke1 |
3821 |
|
447 |
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
448 |
|
|
% SIMULATION PROTOCOL |
449 |
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
450 |
|
|
\subsection{Simulation Protocol} |
451 |
|
|
|
452 |
gezelter |
3850 |
We have implemented the VSS-RNEMD algorithm in OpenMD, our group |
453 |
|
|
molecular dynamics code.\cite{openmd} An 1188 atom gold slab was |
454 |
|
|
equilibrated at 1 atm and 200 K. The periodic box was then expanded |
455 |
|
|
in the $z$-direction to expose two Au(111) faces on either side of the |
456 |
|
|
11-atom thick slab. |
457 |
kstocke1 |
3821 |
|
458 |
gezelter |
3850 |
A full monolayer of thiolates (1/3 the number of surface gold atoms) |
459 |
|
|
was placed on three-fold hollow sites on the gold interfaces. The |
460 |
|
|
effect of thiolate binding sites on the thermal conductance was tested |
461 |
|
|
by placing thiolates at both fcc and hcp hollow sites. No appreciable |
462 |
|
|
difference in the temperature profile was noted due to the location of |
463 |
|
|
thiolate binding. |
464 |
kstocke1 |
3821 |
|
465 |
gezelter |
3850 |
To test the role of thiolate chain length on interfacial thermal |
466 |
|
|
conductance, full coverages of each of five chain lengths were tested: |
467 |
|
|
butanethiolate (C$_4$), hexanethiolate (C$_6$), octanethiolate |
468 |
|
|
(C$_8$), decanethiolate (C$_{10}$), and dodecanethiolate |
469 |
|
|
(C$_{12}$). To test the effect of mixed chain lengths, full coverages |
470 |
|
|
of C$_4$ / C$_{10}$ and C$_4$ / C$_{12}$ mixtures were created in |
471 |
|
|
short/long chain percentages of 25/75, 50/50, 62.5/37.5, 75/25, and |
472 |
|
|
87.5/12.5. The short and long chains were placed on the surface hollow |
473 |
|
|
sites in a random configuration. |
474 |
kstocke1 |
3821 |
|
475 |
gezelter |
3850 |
The simulation box $z$-dimension was set to roughly double the length |
476 |
|
|
of the gold/thiolate block. Hexane solvent molecules were placed in |
477 |
kstocke1 |
3851 |
the vacant portion of the box using the packmol algorithm. Figure \ref{fig:timelapse} shows two of the mixed chain length systems both before and after the RNEMD simulation. |
478 |
kstocke1 |
3821 |
|
479 |
kstocke1 |
3851 |
\begin{figure} |
480 |
|
|
\includegraphics[width=\linewidth]{figures/timelapse} |
481 |
|
|
\caption{Images of 25\%/75\% C$_4$/C$_{12}$ (top panel) and 75\%/25\% C$_4$/C$_{12}$ (bottom panel) systems at the beginning and end of a 3 ns simulation. Initial interfacial solvent molecules are colored light blue. The solvent-thiolate disorder and interfacial solvent turnover of the 25\%/75\% short/long system stand in stark contrast to the highly ordered but entrapped interfacial solvent of the 75\%/25\% system.} |
482 |
|
|
\label{fig:timelapse} |
483 |
|
|
\end{figure} |
484 |
|
|
|
485 |
gezelter |
3850 |
The system was equilibrated to 220 K in the canonical (NVT) ensemble, |
486 |
|
|
allowing the thiolates and solvent to warm gradually. Pressure |
487 |
|
|
correction to 1 atm was done using an isobaric-isothermal (NPT) |
488 |
|
|
integrator that allowed expansion or contraction only in the $z$ |
489 |
|
|
direction, maintaining the crystalline structure of the gold as close |
490 |
|
|
to the bulk result as possible. The diagonal elements of the pressure |
491 |
|
|
tensor were monitored during the pressure equilibration stage. If the |
492 |
|
|
$xx$ and/or $yy$ elements had a mean above zero throughout the |
493 |
|
|
simulation -- indicating residual surface tension in the plane of the |
494 |
|
|
gold slab -- an additional short NPT equilibration step was performed |
495 |
|
|
allowing all box dimensions to change. Once the pressure was stable |
496 |
|
|
at 1 atm, a final equilibration stage was performed at constant |
497 |
|
|
temperature. All systems were equilibrated in the microcanonical (NVE) |
498 |
|
|
ensemble before proceeding with the VSS-RNEMD and data collection |
499 |
|
|
stages. |
500 |
kstocke1 |
3821 |
|
501 |
gezelter |
3850 |
A kinetic energy flux was applied using VSS-RNEMD. The total |
502 |
|
|
simulation time was 3 nanoseconds, with velocity scaling occurring |
503 |
|
|
every 10 femtoseconds. The hot slab was centered in the gold and the |
504 |
|
|
cold slab was placed in the center of the solvent region. The entire |
505 |
|
|
system has a (time-averaged) temperature of 220 K, with a temperature |
506 |
|
|
difference between the hot and cold slabs of approximately 30 K. The |
507 |
|
|
average temperature and kinetic energy flux were selected to prevent |
508 |
|
|
solvent freezing (or glass formation) and to not allow the thiolates |
509 |
|
|
to bury in the gold slab. The Au-S interaction has a deep potential |
510 |
|
|
energy well, which allows sulfur atoms to embed into the gold slab at |
511 |
|
|
temperatures above 250 K. Simulation conditions were chosen to avoid |
512 |
|
|
both of these undesirable situations. |
513 |
kstocke1 |
3821 |
|
514 |
gezelter |
3850 |
Temperature profiles of the system were created by dividing the box |
515 |
|
|
into $\sim$ 3 \AA \, bins along the $z$ axis and recording the average |
516 |
|
|
temperature of each bin. |
517 |
kstocke1 |
3821 |
|
518 |
gezelter |
3850 |
|
519 |
kstocke1 |
3801 |
|
520 |
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
521 |
|
|
% **RESULTS** |
522 |
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
523 |
|
|
\section{Results} |
524 |
|
|
|
525 |
kstocke1 |
3851 |
The solvent, hexane, is a straight chain flexible alkane that is structurally |
526 |
|
|
similar to the thiolate alkane tails. Previous work has shown that UA models |
527 |
|
|
of hexane and butanethiolate have a high degree of vibrational |
528 |
|
|
overlap.\cite{kuang:AuThl} This overlap provides a mechanism for thermal |
529 |
|
|
energy conduction from the thiolates to the solvent. Indeed, we observe that |
530 |
|
|
the interfacial conductance is twice as large with the thiolate monolayers (of |
531 |
|
|
all chain lengths) than with the bare metal surface. |
532 |
kstocke1 |
3801 |
|
533 |
gezelter |
3819 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
534 |
|
|
% CHAIN LENGTH |
535 |
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
536 |
|
|
\subsection{Effect of Chain Length} |
537 |
|
|
|
538 |
kstocke1 |
3851 |
We examined full coverages of five chain lengths, n = 4, 6, 8, 10, and 12. As shown in table \ref{table:chainlengthG}, the interfacial conductance is roughly constant as a function of chain length, indicating that the length of the thiolate alkyl chains does not play a significant role in the transport of heat across the gold/thiolate and thiolate/solvent interfaces. In all cases, the hexane solvent was unable to penetrate into the thiolate layer, leading to a persistent 2-4 \AA \, gap between the solvent region and the thiolates. However, while the identity of the alkyl thiolate capping agent has little effect on the interfacial thermal conductance, the presence of a full monolayer of capping agent provides a two-fold increase in the G value relative to a bare gold surface. |
539 |
kstocke1 |
3830 |
\begin{longtable}{p{4cm} p{3cm}} |
540 |
kstocke1 |
3851 |
\caption{Computed interfacial thermal conductance ($G$) values for bare gold and 100\% coverages of various thiolate alkyl chain lengths.} |
541 |
kstocke1 |
3830 |
\\ |
542 |
kstocke1 |
3831 |
\centering {\bf Chain Length (n)} & \centering\arraybackslash {\bf G (MW/m$^2$/K)} \\ \hline |
543 |
kstocke1 |
3830 |
\endhead |
544 |
|
|
\hline |
545 |
|
|
\endfoot |
546 |
kstocke1 |
3851 |
\centering bare metal & \centering\arraybackslash 30.2 \\ |
547 |
kstocke1 |
3832 |
\centering 4 & \centering\arraybackslash 59.4 \\ |
548 |
|
|
\centering 6 & \centering\arraybackslash 60.2 \\ |
549 |
|
|
\centering 8 & \centering\arraybackslash 61.0 \\ |
550 |
|
|
\centering 10 & \centering\arraybackslash 58.2 \\ |
551 |
|
|
\centering 12 & \centering\arraybackslash 58.8 |
552 |
kstocke1 |
3830 |
\label{table:chainlengthG} |
553 |
|
|
\end{longtable} |
554 |
kstocke1 |
3801 |
|
555 |
gezelter |
3819 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
556 |
|
|
% MIXED CHAINS |
557 |
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
558 |
|
|
\subsection{Effect of Mixed Chain Lengths} |
559 |
|
|
|
560 |
kstocke1 |
3851 |
Previous simulations have demonstrated a non-monotonic behavior for $G$ as a function of the surface coverage. One difficulty with the previous study was the ability of butanethiolate ligands to migrate on the Au(111) surface and to form segregated domains. To simulate the effect of low coverages while preventing thiolate domain formation, we maintain 100\% thiolate coverage while varying the proportions of short (butanethiolate, C$_4$ and long (decanethiolate, C$_{10}$, or dodecanethiolate, C$_{12}$) alkyl chains. Data on the conductance as the fraction of long chains was varied is shown in figure \ref{fig:Gstacks}. Note that as in the previous study, $G$ is depends on solvent accessibility to thermally excited ligands. Our simulations indicate a similar (but not as dramatic) non-monotonic dependence on the fraction of long chains. |
561 |
kstocke1 |
3829 |
\begin{figure} |
562 |
|
|
\includegraphics[width=\linewidth]{figures/Gstacks} |
563 |
kstocke1 |
3851 |
\caption{Interfacial thermal conductivity of mixed-chains has a non-monotonic dependence on the fraction of long chains (lower panels). At low fractions of long chains, the solvent escape rate ($k_{escape}$) dominates the heat transfer process, while the solvent-thiolate orientational ordering dominates in systems with higher fractions of long chains (upper panels).} |
564 |
kstocke1 |
3829 |
\label{fig:Gstacks} |
565 |
|
|
\end{figure} |
566 |
kstocke1 |
3815 |
|
567 |
kstocke1 |
3801 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
568 |
|
|
% **DISCUSSION** |
569 |
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
570 |
|
|
\section{Discussion} |
571 |
gezelter |
3819 |
In the mixed chain-length simulations, solvent molecules can become |
572 |
|
|
temporarily trapped or entangled with the thiolate chains. Their |
573 |
|
|
residence in close proximity to the higher temperature environment |
574 |
|
|
close to the surface allows them to carry heat away from the surface |
575 |
|
|
quite efficiently. There are two aspects of this behavior that are |
576 |
|
|
relevant to thermal conductance of the interface: the residence time |
577 |
kstocke1 |
3851 |
of solvent molecules in the thiolate layer, and the alignment |
578 |
|
|
of the C-C chains as a mechanism for transferring vibrational |
579 |
|
|
energy to these entrapped solvent molecules. To quantify these competing effects, we have computed solvent escape rates from the ligand layer as well as a joint orientational order parameter between the trapped solvent and the thiolate ligands. |
580 |
gezelter |
3819 |
|
581 |
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
582 |
|
|
% RESIDENCE TIME |
583 |
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
584 |
kstocke1 |
3841 |
\subsection{Mobility of solvent in the interfacial layer} |
585 |
gezelter |
3819 |
|
586 |
kstocke1 |
3851 |
We use a simple survival correlation function, $C(t)$, to measure the |
587 |
|
|
residence time of a solvent molecule in the thiolate |
588 |
gezelter |
3819 |
layer. This function correlates the identity of all hexane molecules |
589 |
|
|
within the $z$-coordinate range of the thiolate layer at two separate |
590 |
|
|
times. If the solvent molecule is present at both times, the |
591 |
|
|
configuration contributes a $1$, while the absence of the molecule at |
592 |
|
|
the later time indicates that the solvent molecule has migrated into |
593 |
|
|
the bulk, and this configuration contributes a $0$. A steep decay in |
594 |
|
|
$C(t)$ indicates a high turnover rate of solvent molecules from the |
595 |
kstocke1 |
3837 |
chain region to the bulk. We define the escape rate for trapped solvent molecules at the interface as |
596 |
kstocke1 |
3815 |
\begin{equation} |
597 |
gezelter |
3825 |
k_{escape} = \left( \int_0^T C(t) dt \right)^{-1} |
598 |
kstocke1 |
3821 |
\label{eq:mobility} |
599 |
kstocke1 |
3815 |
\end{equation} |
600 |
gezelter |
3825 |
where T is the length of the simulation. This is a direct measure of |
601 |
kstocke1 |
3851 |
the rate at which solvent molecules initially entangled in the thiolate layer |
602 |
kstocke1 |
3843 |
can escape into the bulk. As $k_{escape} \rightarrow 0$, the |
603 |
gezelter |
3825 |
solvent has become permanently trapped in the thiolate layer. In |
604 |
kstocke1 |
3841 |
figure \ref{fig:Gstacks} we show that interfacial solvent mobility |
605 |
gezelter |
3825 |
decreases as the percentage of long thiolate chains increases. |
606 |
gezelter |
3819 |
|
607 |
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
608 |
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% ORDER PARAMETER |
609 |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
610 |
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611 |
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\subsection{Vibrational coupling via orientational ordering} |
612 |
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|
613 |
kstocke1 |
3851 |
As the fraction of long-chain thiolates increases, the entrapped |
614 |
gezelter |
3819 |
solvent molecules must find specific orientations relative to the mean |
615 |
kstocke1 |
3851 |
orientation of the thiolate chains. This alignment allows for |
616 |
gezelter |
3819 |
efficient thermal energy exchange between the thiolate alkyl chain and |
617 |
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|
the solvent molecules. |
618 |
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|
619 |
kstocke1 |
3851 |
To measure this cooperative ordering, we computed the orientational order |
620 |
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parameters and director axes for both the thiolate chains and for the |
621 |
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entrapped solvent. The director axis can be easily obtained by diagonalization |
622 |
|
|
of the order parameter tensor, |
623 |
gezelter |
3819 |
\begin{equation} |
624 |
|
|
\mathsf{Q}_{\alpha \beta} = \frac{1}{2 N} \sum_{i=1}^{N} \left( 3 \mathbf{e}_{i |
625 |
|
|
\alpha} \mathbf{e}_{i \beta} - \delta_{\alpha \beta} \right) |
626 |
kstocke1 |
3815 |
\end{equation} |
627 |
gezelter |
3819 |
where $\mathbf{e}_{i \alpha}$ was the $\alpha = x,y,z$ component of |
628 |
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|
the unit vector $\mathbf{e}_{i}$ along the long axis of molecule $i$. |
629 |
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For both kinds of molecules, the $\mathbf{e}$ vector is defined using |
630 |
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the terminal atoms of the chains. |
631 |
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|
632 |
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|
The largest eigenvalue of $\overleftrightarrow{\mathsf{Q}}$ is |
633 |
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|
traditionally used to obtain orientational order parameter, while the |
634 |
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|
eigenvector corresponding to the order parameter yields the director |
635 |
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|
axis ($\mathbf{d}(t)$) which defines the average direction of |
636 |
|
|
molecular alignment at any time $t$. The overlap between the director |
637 |
|
|
axes of the thiolates and the entrapped solvent is time-averaged, |
638 |
kstocke1 |
3815 |
\begin{equation} |
639 |
kstocke1 |
3851 |
\langle d \rangle = \langle \mathbf{d}_{thiolates} \left( t \right) \cdot |
640 |
gezelter |
3825 |
\mathbf{d}_{solvent} \left( t \right) \rangle_t |
641 |
kstocke1 |
3815 |
\label{eq:orientation} |
642 |
|
|
\end{equation} |
643 |
kstocke1 |
3840 |
and reported in figure \ref{fig:Gstacks}. |
644 |
kstocke1 |
3815 |
|
645 |
gezelter |
3819 |
Once the solvent molecules have picked up thermal energy from the |
646 |
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thiolates, they carry heat away from the gold as they diffuse back |
647 |
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into the bulk solvent. When the percentage of long chains decreases, |
648 |
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the tails of the long chains are much more disordered and do not |
649 |
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provide structured channels for the solvent to fill. |
650 |
kstocke1 |
3815 |
|
651 |
kstocke1 |
3851 |
C$_4$ / C$_{10}$ mixed monolayers have a peak interfacial conductance with 75\% long chains. At this fraction of long chains, the cooperative orientational ordering of the solvent molecules and chains becomes the dominant effect while the solvent escape rate sharply decreases. C$_4$ / C$_{12}$ mixtures have a peak interfacial conductance for 87.5\% long chains. The solvent-thiolate orientational ordering reaches its maximum value at this long chain fraction. Long chain fractions of over $0.5$ for the n = 4, 12 system are well ordered, but this effect is tempered by the exceptionally slow solvent escape rate. |
652 |
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|
653 |
kstocke1 |
3843 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
654 |
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% **CONCLUSIONS** |
655 |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
656 |
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\section{Conclusions} |
657 |
kstocke1 |
3851 |
Our results suggest that a mixed vibrational transfer / convection model is necessary to explain the features of heat transfer at this interface. The alignment of the solvent chains with the ordered ligand allows rapid transfer of energy to the trapped solvent (and this is the dominant feature for ordered ligand layers). Diffusion of the vibrationally excited solvent into the bulk also plays a significant role when the ligands are less tightly packed. |
658 |
kstocke1 |
3843 |
|
659 |
kstocke1 |
3851 |
In the language of earlier continuum approaches to interfacial |
660 |
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|
conductance,\cite{RevModPhys.61.605} the alignment of the chains is an |
661 |
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|
important factor in the transfer of phonons from the thiolate layer to the |
662 |
|
|
trapped solvent. The aligned solvent and thiolate chains have nearly identical |
663 |
|
|
acoustic impedances and the phonons can scatter directly into a solvent |
664 |
|
|
molecule that has been forced into alignment. When the entrapped solvent has |
665 |
|
|
more configurations available, the likelihood of an impedance mismatch is |
666 |
|
|
higher, and the phonon scatters into the solvent with lower |
667 |
|
|
efficiency. The fractional coverage of the long chains is therefore a simple |
668 |
|
|
way of tuning the acoustic mismatch between the thiolate layer and the hexane |
669 |
|
|
solvent. |
670 |
kstocke1 |
3843 |
|
671 |
kstocke1 |
3851 |
Efficient heat transfer also can be accomplished via convective or diffusive motion of vibrationally excited solvent back into the bulk. Once the entrapped solvent becomes too tightly aligned with the ligands, however, the convective avenue of heat transfer is cut off. |
672 |
kstocke1 |
3843 |
|
673 |
kstocke1 |
3851 |
Our simulations suggest a number of routes to make interfaces with high thermal conductance. If it is possible to create an interface which forces the solvent into alignment with a ligand that shares many of the solvent's vibrational modes, while simultaneously preserving solvent mobility back to the bulk, we would expect a significant jump in the interfacial conductance. One possible way to do this is to use polyene ligands with alternating unsaturated bonds. These are significantly more rigid than long alkanes, and could force solvent alignment (even at low relative coverages) while preserving mobility into the bulk. |
674 |
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|
675 |
kstocke1 |
3801 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
676 |
kstocke1 |
3851 |
% **ACKNOWLEDGMENTS** |
677 |
kstocke1 |
3801 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
678 |
kstocke1 |
3851 |
\section*{Acknowledgments} |
679 |
kstocke1 |
3801 |
|
680 |
kstocke1 |
3851 |
We gratefully acknowledge conversations with Dr. Shenyu Kuang. Support for |
681 |
|
|
this project was provided by the National Science Foundation under grant |
682 |
|
|
CHE-0848243. Computational time was provided by the Center for Research |
683 |
|
|
Computing (CRC) at the University of Notre Dame. |
684 |
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|
685 |
kstocke1 |
3801 |
\newpage |
686 |
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687 |
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\bibliography{thiolsRNEMD} |
688 |
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689 |
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\end{doublespace} |
690 |
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\end{document} |
691 |
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