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\begin{document} |
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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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\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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The structural and dynamical details of interfaces between metal |
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nanoparticles and solvents determines how energy flows between these |
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particles and their surroundings. Understanding this energy flow is |
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essential in designing and functionalizing metallic nanoparticles for |
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plasmonic photothermal |
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therapies,\cite{Jain:2007ux,Petrova:2007ad,Gnyawali:2008lp,Mazzaglia:2008to,Huff:2007ye,Larson:2007hw} which rely on the ability of metallic |
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nanoparticles to absorb light in the near-IR, a portion of the |
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spectrum in which living tissue is very nearly transparent. The |
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principle of this therapy is to pump the particles at high power at |
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the plasmon resonance and to allow heat dissipation to kill targeted |
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(e.g. cancerous) cells. The relevant physical property controlling |
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this transfer of energy is the interfacial thermal conductance, $G$, |
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which can be somewhat difficult to determine |
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experimentally.\cite{Wilson:2002uq,Plech:2005kx} |
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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 al.} |
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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 vital to get a detailed molecular |
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picture of particle-solvent interactions in order to understand the |
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thermal behavior of complex fluids. To date, there have been few |
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reported values (either from theory or experiment) for $G$ for |
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ligand-protected nanoparticles embedded in liquids, and there is a |
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significant gap in knowledge about how chemically distinct ligands or |
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protecting groups will affect heat transport from the particles. |
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The thermal properties of various nanostructured interfaces have been |
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investigated experimentally by a number of groups: Cahill and |
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coworkers studied nanoscale thermal transport from metal |
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nanoparticle/fluid interfaces, to epitaxial TiN/single crystal oxides |
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interfaces, and hydrophilic and hydrophobic interfaces between water |
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and solids with 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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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 suggest 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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One interesting result of our previous work was the observation of |
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{\it non-monotonic dependence} of the thermal conductance on the |
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coverage of a metal surface by a chemical protecting group. Our |
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explanation for this behavior was that gaps in surface coverage |
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allowed solvent to penetrate close to the capping molecules that had |
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been heated by the metal surface, to absorb thermal energy from these |
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molecules, and then diffuse away. The effect of surface coverage is |
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relatively difficult to study as the individual protecting groups have |
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lateral mobility on the surface and can aggregate to form domains on |
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the timescale of the simulation. |
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The work reported here involves the use of velocity shearing and |
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scaling reverse non-equilibrium molecular dynamics (VSS-RNEMD) to |
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study surfaces composed of mixed-length chains which collectively form |
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a complete monolayer on the surfaces. These complete (but |
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mixed-chain) surfaces are significantly less prone to surface domain |
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formation, and would allow us to further investigate the mechanism of |
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heat transport to the solvent. |
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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,Kuang:2010uq} 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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Instead of having momentum exchange between {\it individual particles} |
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in each slab, the NIVS algorithm applies velocity scaling to all of |
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the selected particles in both slabs.\cite{Kuang:2010uq} A combination |
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of linear momentum, kinetic energy, and flux constraint equations |
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governs the amount of velocity scaling performed at each step. NIVS |
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has been shown to be very effective at producing thermal gradients and |
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for computing thermal conductivities, particularly for heterogeneous |
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interfaces. Although the NIVS algorithm can also be applied to impose |
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a directional momentum flux, thermal anisotropy was observed in |
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relatively high flux simulations, and the method is not suitable for |
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imposing a shear flux or for computing shear viscosities. |
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The most useful RNEMD |
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approach developed so far utilizes a series of simultaneous velocity |
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shearing and scaling exchanges between the two |
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slabs.\cite{2012MolPh.110..691K} 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 {\it single 1 ns simulation}.\cite{2012MolPh.110..691K} |
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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 |
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change in the thermal conductivity ($\lambda$) across the boundary |
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between materials: |
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\begin{align} |
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G &= \left(\nabla\lambda \cdot \mathbf{\hat{n}}\right)_{z_0} \\ |
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|
&= J_z\left(\frac{\partial^2 T}{\partial z^2}\right)_{z_0} \Big/ |
327 |
|
|
\left(\frac{\partial T}{\partial z}\right)_{z_0}^2. |
328 |
|
|
\label{derivativeG} |
329 |
|
|
\end{align} |
330 |
|
|
where $z_0$ is the location of the interface between two materials and |
331 |
|
|
$\mathbf{\hat{n}}$ is a unit vector normal to the interface (assumed |
332 |
|
|
to be the $z$ direction from here on). RNEMD simulations, and |
333 |
|
|
particularly the VSS-RNEMD approach, function by applying a momentum |
334 |
|
|
or thermal flux and watching the gradient response of the |
335 |
|
|
material. This means that the {\it interfacial} conductance is a |
336 |
|
|
second derivative property which is subject to significant noise and |
337 |
|
|
therefore requires extensive sampling. We have been able to |
338 |
|
|
demonstrate the use of the second derivative approach to compute |
339 |
|
|
interfacial conductance at chemically-modified metal / solvent |
340 |
|
|
interfaces. However, a definition of the interfacial conductance in |
341 |
|
|
terms of a temperature difference ($\Delta T$) measured at $z_0$, |
342 |
|
|
\begin{displaymath} |
343 |
|
|
G = \frac{J_z}{\Delta T_{z_0}}, |
344 |
|
|
\end{displaymath} |
345 |
|
|
is useful once the RNEMD approach has generated a stable temperature |
346 |
|
|
gap across the interface. |
347 |
|
|
|
348 |
|
|
\begin{figure} |
349 |
|
|
\includegraphics[width=\linewidth]{figures/resistor_series} |
350 |
gezelter |
3822 |
\caption{The inverse of the interfacial thermal conductance, $G$, is |
351 |
|
|
the Kapitza resistance, $R_K$. Because the gold / thiolate/ |
352 |
|
|
solvent interface extends a significant distance from the metal |
353 |
|
|
surface, the interfacial resistance $R_K$ can be computed by |
354 |
|
|
summing a series of temperature drops between adjacent temperature |
355 |
|
|
bins along the $z$ axis.} |
356 |
gezelter |
3803 |
\label{fig:resistor_series} |
357 |
|
|
\end{figure} |
358 |
|
|
|
359 |
|
|
In the particular case we are studying here, there are two interfaces |
360 |
|
|
involved in the transfer of heat from the gold slab to the solvent: |
361 |
|
|
the gold/thiolate interface and the thiolate/solvent interface. We |
362 |
|
|
could treat the temperature on each side of an interface as discrete, |
363 |
|
|
making the interfacial conductance the inverse of the Kaptiza |
364 |
|
|
resistance, or $G = \frac{1}{R_k}$. To model the total conductance |
365 |
|
|
across multiple interfaces, it is useful to think of the interfaces as |
366 |
|
|
a set of resistors wired in series. The total resistance is then |
367 |
|
|
additive, $R_{total} = \sum_i R_{i}$ and the interfacial conductance |
368 |
|
|
is the inverse of the total resistance, or $G = \frac{1}{\sum_i |
369 |
|
|
R_i}$). In the interfacial region, we treat each bin in the |
370 |
|
|
VSS-RNEMD temperature profile as a resistor with resistance |
371 |
|
|
$\frac{T_{2}-T_{1}}{J_z}$, $\frac{T_{3}-T_{2}}{J_z}$, etc. The sum of |
372 |
|
|
the set of resistors which spans the gold/thiolate interface, thiolate |
373 |
|
|
chains, and thiolate/solvent interface simplifies to |
374 |
kstocke1 |
3801 |
\begin{equation} |
375 |
gezelter |
3825 |
R_{K} = \frac{T_{n}-T_{1}}{J_z}, |
376 |
gezelter |
3803 |
\label{eq:finalG} |
377 |
|
|
\end{equation} |
378 |
|
|
or the temperature difference between the gold side of the |
379 |
|
|
gold/thiolate interface and the solvent side of the thiolate/solvent |
380 |
|
|
interface over the applied flux. |
381 |
kstocke1 |
3801 |
|
382 |
|
|
|
383 |
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
384 |
|
|
% **COMPUTATIONAL DETAILS** |
385 |
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
386 |
|
|
\section{Computational Details} |
387 |
|
|
|
388 |
gezelter |
3803 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
389 |
gezelter |
3819 |
% FORCE-FIELD PARAMETERS |
390 |
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
391 |
|
|
\subsection{Force-Field Parameters} |
392 |
kstocke1 |
3801 |
|
393 |
gezelter |
3819 |
Our simulations include a number of chemically distinct components. |
394 |
|
|
Figure \ref{fig:structures} demonstrates the sites defined for both |
395 |
|
|
the {\it n}-hexane and alkanethiolate ligands present in our |
396 |
|
|
simulations. Force field parameters are needed for interactions both |
397 |
|
|
between the same type of particles and between particles of different |
398 |
|
|
species. |
399 |
|
|
|
400 |
|
|
\begin{figure} |
401 |
|
|
\includegraphics[width=\linewidth]{figures/structures} |
402 |
gezelter |
3822 |
\caption{Topologies of the thiolate capping agents and solvent |
403 |
|
|
utilized in the simulations. The chemically-distinct sites (S, |
404 |
|
|
\ce{CH2}, and \ce{CH3}) are treated as united atoms. Most |
405 |
|
|
parameters are taken from references \bibpunct{}{}{,}{n}{}{,} |
406 |
|
|
\protect\cite{TraPPE-UA.alkanes} and |
407 |
|
|
\protect\cite{TraPPE-UA.thiols}. Cross-interactions with the Au |
408 |
|
|
atoms were adapted from references |
409 |
|
|
\protect\cite{landman:1998},~\protect\cite{vlugt:cpc2007154},~and |
410 |
|
|
\protect\cite{hautman:4994}.} |
411 |
gezelter |
3819 |
\label{fig:structures} |
412 |
|
|
\end{figure} |
413 |
|
|
|
414 |
kstocke1 |
3829 |
The Au-Au interactions in the metal lattice slab are described by the |
415 |
gezelter |
3819 |
quantum Sutton-Chen (QSC) formulation.\cite{PhysRevB.59.3527} The QSC |
416 |
|
|
potentials include zero-point quantum corrections and are |
417 |
|
|
reparametrized for accurate surface energies compared to the |
418 |
|
|
Sutton-Chen potentials.\cite{Chen90} |
419 |
|
|
|
420 |
|
|
For the {\it n}-hexane solvent molecules, the TraPPE-UA |
421 |
|
|
parameters\cite{TraPPE-UA.alkanes} were utilized. In this model, |
422 |
|
|
sites are located at the carbon centers for alkyl groups. Bonding |
423 |
|
|
interactions, including bond stretches and bends and torsions, were |
424 |
|
|
used for intra-molecular sites closer than 3 bonds. For non-bonded |
425 |
|
|
interactions, Lennard-Jones potentials are used. We have previously |
426 |
|
|
utilized both united atom (UA) and all-atom (AA) force fields for |
427 |
|
|
thermal conductivity in previous work,\cite{} and since the united |
428 |
|
|
atom force fields cannot populate the high-frequency modes that are |
429 |
|
|
present in AA force fields, they appear to work better for modeling |
430 |
|
|
thermal conductivity. The TraPPE-UA model for alkanes is known to |
431 |
|
|
predict a slightly lower boiling point than experimental values. This |
432 |
|
|
is one of the reasons we used a lower average temperature (200K) for |
433 |
|
|
our simulations. |
434 |
|
|
|
435 |
|
|
The TraPPE-UA force field includes parameters for thiol |
436 |
|
|
molecules\cite{TraPPE-UA.thiols} which were used for the |
437 |
|
|
alkanethiolate molecules in our simulations. To derive suitable |
438 |
|
|
parameters for butanethiol adsorbed on Au(111) surfaces, we adopted |
439 |
|
|
the S parameters from Luedtke and Landman\cite{landman:1998} and |
440 |
|
|
modified the parameters for the CTS atom to maintain charge neutrality |
441 |
|
|
in the molecule. |
442 |
|
|
|
443 |
|
|
To describe the interactions between metal (Au) and non-metal atoms, |
444 |
|
|
we refer to an adsorption study of alkyl thiols on gold surfaces by |
445 |
|
|
Vlugt {\it et al.}\cite{vlugt:cpc2007154} They fitted an effective |
446 |
|
|
Lennard-Jones form of potential parameters for the interaction between |
447 |
|
|
Au and pseudo-atoms CH$_x$ and S based on a well-established and |
448 |
|
|
widely-used effective potential of Hautman and Klein for the Au(111) |
449 |
|
|
surface.\cite{hautman:4994} As our simulations require the gold slab |
450 |
|
|
to be flexible to accommodate thermal excitation, the pair-wise form |
451 |
kstocke1 |
3832 |
of potentials they developed was used for our study. |
452 |
kstocke1 |
3821 |
|
453 |
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
454 |
|
|
% SIMULATION PROTOCOL |
455 |
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
456 |
|
|
\subsection{Simulation Protocol} |
457 |
|
|
|
458 |
|
|
We have implemented the VSS-RNEMD algorithm in OpenMD, our |
459 |
|
|
group molecular dynamics code. A gold slab 11 atoms thick was |
460 |
|
|
equilibrated at 1 atm and 200 K. The periodic box was expanded |
461 |
|
|
in the z direction to expose two Au(111) faces. |
462 |
|
|
|
463 |
|
|
A full monolayer of thiolates (1/3 the number of surface gold atoms) was placed on three-fold hollow sites on the gold interfaces. To efficiently test the effect of thiolate binding sites on the thermal conductance, all systems had one gold interface with thiolates placed only on fcc hollow sites and the other interface with thiolates only on hcp hollow sites. To test the effect of thiolate chain length on interfacial thermal conductance, full coverages of five chain lengths were tested: butanethiolate, hexanethiolate, octanethiolate, decanethiolate, and dodecanethiolate. To test the effect of mixed chain lengths, full coverages of butanethiolate/decanethiolate and butanethiolate/dodecanethiolate mixtures were created in short/long chain percentages of 25/75, 50/50, 62.5/37.5, 75/25, and 87.5/12.5. The short and long chains were placed on the surface hollow sites in a random configuration. |
464 |
|
|
|
465 |
|
|
The simulation box z dimension was set to roughly double the length of the gold/thiolate block. Hexane solvent molecules were placed in the vacant portion of the box using the packmol algorithm. Hexane, a straight chain flexible alkane, is very structurally similar to the thiolate alkane tails; previous work has shown that UA models of hexane and butanethiolate have a high degree of vibrational overlap.\cite{Kuang2011} This overlap should provide a mechanism for thermal energy transfer from the thiolates to the solvent. |
466 |
|
|
|
467 |
|
|
The system was equilibrated to 220 K in the NVT ensemble, allowing the thiolates and solvent to warm gradually. Pressure correction to 1 atm was done in an NPT ensemble that allowed expansion or contraction only in the z direction, so as not to disrupt the crystalline structure of the gold. The diagonal elements of the pressure tensor were monitored during the pressure correction step. If the xx and/or yy elements had a mean above zero throughout the simulation -- indicating residual pressure in the plane of the gold slab -- an additional short NPT equilibration step was performed allowing all box dimensions to change. Once the pressure was stable at 1 atm, a final NVT simulation was performed. All systems were equilibrated in the microcanonical (NVE) ensemble before proceeding with the VSS-RNEMD step. |
468 |
|
|
|
469 |
|
|
A kinetic energy flux was applied using VSS-RNEMD in the NVE ensemble. The total simulation time was 3 nanoseconds, with velocity scaling occurring every 10 femtoseconds. The hot slab was centered in the gold and the cold slab was placed in the center of the solvent region. The average temperature was 220 K, with a temperature difference between the hot and cold slabs of approximately 30 K. The average temperature and kinetic energy flux were carefully selected with two considerations in mind: 1) if the cold bin gets too cold (below ~180 K) the solvent may freeze or undergo a glassy transition, and 2) due to the deep sulfur-gold potential well, sulfur atoms routinely embed into the gold slab, particularly at temperatures above 250 K. Simulation conditions were chosen to avoid both of these undesirable situations. A reversed flux direction resulted in frozen long chain thiolates and solvent too near its boiling point. |
470 |
|
|
|
471 |
|
|
A temperature profile of the system was created by dividing the box into $\sim$ 3 \AA \, bins along the z axis and recording the average temperature of each bin. |
472 |
|
|
|
473 |
|
|
|
474 |
kstocke1 |
3801 |
|
475 |
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
476 |
|
|
% **RESULTS** |
477 |
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
478 |
|
|
\section{Results} |
479 |
|
|
|
480 |
|
|
|
481 |
gezelter |
3819 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
482 |
|
|
% CHAIN LENGTH |
483 |
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
484 |
|
|
\subsection{Effect of Chain Length} |
485 |
|
|
|
486 |
kstocke1 |
3830 |
We examined full coverages of five chain lengths, n = 4, 6, 8, 10, 12. As shown in table \ref{table:chainlengthG}, the trend of interfacial conductance is mostly flat 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. 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. |
487 |
|
|
\begin{longtable}{p{4cm} p{3cm}} |
488 |
|
|
\caption{Computed interfacial thermal conductance (G) values for bare gold and 100\% coverages of various thiolate alkyl chain lengths.} |
489 |
|
|
\\ |
490 |
kstocke1 |
3831 |
\centering {\bf Chain Length (n)} & \centering\arraybackslash {\bf G (MW/m$^2$/K)} \\ \hline |
491 |
kstocke1 |
3830 |
\endhead |
492 |
|
|
\hline |
493 |
|
|
\endfoot |
494 |
kstocke1 |
3832 |
\centering bare gold & \centering\arraybackslash 30.2 \\ |
495 |
|
|
\centering 4 & \centering\arraybackslash 59.4 \\ |
496 |
|
|
\centering 6 & \centering\arraybackslash 60.2 \\ |
497 |
|
|
\centering 8 & \centering\arraybackslash 61.0 \\ |
498 |
|
|
\centering 10 & \centering\arraybackslash 58.2 \\ |
499 |
|
|
\centering 12 & \centering\arraybackslash 58.8 |
500 |
kstocke1 |
3830 |
\label{table:chainlengthG} |
501 |
|
|
\end{longtable} |
502 |
kstocke1 |
3801 |
|
503 |
gezelter |
3819 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
504 |
|
|
% MIXED CHAINS |
505 |
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
506 |
|
|
\subsection{Effect of Mixed Chain Lengths} |
507 |
|
|
|
508 |
kstocke1 |
3815 |
Previous work demonstrated that for butanethiolate monolayers on a Au(111) surface, the interfacial conductance was a non-monotonic function of the percent coverage. This is believed to be due to enhanced solvent-thiolate coupling through greater penetration of solvent molecules into the thiolate layer. At lower coverages, hexane solvent can more easily line up lengthwise with the thiolate tails by fitting into gaps between the thiolates. However, a side effect of low coverages is surface aggregation of the thiolates. To simulate the effect of low coverages while preventing aggregation we maintain 100\% thiolate coverage while varying the proportions of short (butanethiolate, n = 4) and long (decanethiolate, n = 10 or dodecanethiolate, n = 12). In systems where there is a mix of short and long chain thiolates, interfacial conductance is a non-monotonic function of the percent of long chains. The depth of the gaps between the long chains is $n_{long} - n_{short}$, which has implications for the ability of the hexane solvent to fill in the gaps between the long chains. |
509 |
kstocke1 |
3829 |
|
510 |
|
|
\begin{figure} |
511 |
|
|
\includegraphics[width=\linewidth]{figures/Gstacks} |
512 |
|
|
\caption{} |
513 |
|
|
\label{fig:Gstacks} |
514 |
|
|
\end{figure} |
515 |
kstocke1 |
3801 |
|
516 |
kstocke1 |
3815 |
\subsubsection{Butanethiolate/Decanethiolate} |
517 |
|
|
Mixtures of butanethiolate/decanethiolate (n = 4, 10) have a peak interfacial condutance for 25\%/75\% short/long chains. |
518 |
kstocke1 |
3801 |
|
519 |
kstocke1 |
3815 |
\subsubsection{Butanethiolate/Dodecanethiolate} |
520 |
kstocke1 |
3821 |
Mixtures of butanethiolate/dodecanethiolate (n = 4, 12) have a peak interfacial conductance for 12.5\%/87.5\% short/long chains. |
521 |
kstocke1 |
3815 |
|
522 |
|
|
|
523 |
kstocke1 |
3801 |
|
524 |
|
|
|
525 |
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
526 |
|
|
% **DISCUSSION** |
527 |
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
528 |
|
|
\section{Discussion} |
529 |
|
|
|
530 |
gezelter |
3819 |
In the mixed chain-length simulations, solvent molecules can become |
531 |
|
|
temporarily trapped or entangled with the thiolate chains. Their |
532 |
|
|
residence in close proximity to the higher temperature environment |
533 |
|
|
close to the surface allows them to carry heat away from the surface |
534 |
|
|
quite efficiently. There are two aspects of this behavior that are |
535 |
|
|
relevant to thermal conductance of the interface: the residence time |
536 |
|
|
of solvent molecules in the thiolate layer, and the orientational |
537 |
|
|
ordering of the C-C chains as a mechanism for transferring vibrational |
538 |
|
|
energy to these entrapped solvent molecules. |
539 |
|
|
|
540 |
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
541 |
|
|
% RESIDENCE TIME |
542 |
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
543 |
kstocke1 |
3821 |
\subsection{Mobility for solvent in the interfacial layer} |
544 |
gezelter |
3819 |
|
545 |
|
|
We use a simple survival correlation function, $C(t)$, to quantify the |
546 |
|
|
residence time of a solvent molecule in the long thiolate chain |
547 |
|
|
layer. This function correlates the identity of all hexane molecules |
548 |
|
|
within the $z$-coordinate range of the thiolate layer at two separate |
549 |
|
|
times. If the solvent molecule is present at both times, the |
550 |
|
|
configuration contributes a $1$, while the absence of the molecule at |
551 |
|
|
the later time indicates that the solvent molecule has migrated into |
552 |
|
|
the bulk, and this configuration contributes a $0$. A steep decay in |
553 |
|
|
$C(t)$ indicates a high turnover rate of solvent molecules from the |
554 |
kstocke1 |
3837 |
chain region to the bulk. We define the escape rate for trapped solvent molecules at the interface as |
555 |
kstocke1 |
3815 |
\begin{equation} |
556 |
gezelter |
3825 |
k_{escape} = \left( \int_0^T C(t) dt \right)^{-1} |
557 |
kstocke1 |
3821 |
\label{eq:mobility} |
558 |
kstocke1 |
3815 |
\end{equation} |
559 |
gezelter |
3825 |
where T is the length of the simulation. This is a direct measure of |
560 |
|
|
the rate at which solvent molecules entangled in the thiolate layer |
561 |
|
|
can escape into the bulk. As $k_{escape} \rightarrow \infty$, the |
562 |
|
|
solvent has become permanently trapped in the thiolate layer. In |
563 |
kstocke1 |
3829 |
figure \ref{figure:Gstack} we show that interfacial solvent mobility |
564 |
gezelter |
3825 |
decreases as the percentage of long thiolate chains increases. |
565 |
gezelter |
3819 |
|
566 |
kstocke1 |
3832 |
\begin{figure} |
567 |
kstocke1 |
3837 |
\includegraphics[width=\linewidth]{figures/timelapse} |
568 |
kstocke1 |
3832 |
\caption{} |
569 |
|
|
\label{fig:timelapse} |
570 |
|
|
\end{figure} |
571 |
gezelter |
3819 |
|
572 |
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
573 |
|
|
% ORDER PARAMETER |
574 |
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
575 |
|
|
|
576 |
|
|
\subsection{Vibrational coupling via orientational ordering} |
577 |
|
|
|
578 |
|
|
As the fraction of long-chain thiolates becomes large, the entrapped |
579 |
|
|
solvent molecules must find specific orientations relative to the mean |
580 |
|
|
orientation of the thiolate chains. This configuration allows for |
581 |
|
|
efficient thermal energy exchange between the thiolate alkyl chain and |
582 |
|
|
the solvent molecules. |
583 |
|
|
|
584 |
kstocke1 |
3821 |
To quantify this cooperative |
585 |
gezelter |
3819 |
ordering, we computed the orientational order parameters and director |
586 |
|
|
axes for both the thiolate chains and for the entrapped solvent. The |
587 |
|
|
director axis can be easily obtained by diagonalization of the order |
588 |
|
|
parameter tensor, |
589 |
|
|
\begin{equation} |
590 |
|
|
\mathsf{Q}_{\alpha \beta} = \frac{1}{2 N} \sum_{i=1}^{N} \left( 3 \mathbf{e}_{i |
591 |
|
|
\alpha} \mathbf{e}_{i \beta} - \delta_{\alpha \beta} \right) |
592 |
kstocke1 |
3815 |
\end{equation} |
593 |
gezelter |
3819 |
where $\mathbf{e}_{i \alpha}$ was the $\alpha = x,y,z$ component of |
594 |
|
|
the unit vector $\mathbf{e}_{i}$ along the long axis of molecule $i$. |
595 |
|
|
For both kinds of molecules, the $\mathbf{e}$ vector is defined using |
596 |
|
|
the terminal atoms of the chains. |
597 |
|
|
|
598 |
|
|
The largest eigenvalue of $\overleftrightarrow{\mathsf{Q}}$ is |
599 |
|
|
traditionally used to obtain orientational order parameter, while the |
600 |
|
|
eigenvector corresponding to the order parameter yields the director |
601 |
|
|
axis ($\mathbf{d}(t)$) which defines the average direction of |
602 |
|
|
molecular alignment at any time $t$. The overlap between the director |
603 |
|
|
axes of the thiolates and the entrapped solvent is time-averaged, |
604 |
kstocke1 |
3815 |
\begin{equation} |
605 |
gezelter |
3825 |
\bar{d} = \langle \mathbf{d}_{thiolates} \left( t \right) \cdot |
606 |
|
|
\mathbf{d}_{solvent} \left( t \right) \rangle_t |
607 |
kstocke1 |
3815 |
\label{eq:orientation} |
608 |
|
|
\end{equation} |
609 |
gezelter |
3825 |
and reported in figure \ref{fig:Gstack}. |
610 |
kstocke1 |
3815 |
|
611 |
gezelter |
3819 |
Once the solvent molecules have picked up thermal energy from the |
612 |
|
|
thiolates, they carry heat away from the gold as they diffuse back |
613 |
|
|
into the bulk solvent. When the percentage of long chains decreases, |
614 |
|
|
the tails of the long chains are much more disordered and do not |
615 |
|
|
provide structured channels for the solvent to fill. |
616 |
kstocke1 |
3815 |
|
617 |
gezelter |
3819 |
Although the alignment of the chains with the entrapped solvent is one |
618 |
|
|
possible mechanism for the non-monotonic increase in the conductance |
619 |
|
|
as a function |
620 |
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|
621 |
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622 |
kstocke1 |
3801 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
623 |
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% **ACKNOWLEDGMENTS** |
624 |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
625 |
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\section*{Acknowledgments} |
626 |
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Support for this project was provided by the |
627 |
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National Science Foundation under grant CHE-0848243. Computational |
628 |
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time was provided by the Center for Research Computing (CRC) at the |
629 |
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University of Notre Dame. |
630 |
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631 |
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\newpage |
632 |
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633 |
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\bibliography{thiolsRNEMD} |
634 |
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635 |
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\end{doublespace} |
636 |
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\end{document} |
637 |
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640 |
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