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%\section{Interfacial Thermal Conductance of Metallic Nanoparticles} |
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For a solvated nanoparticle, we can define a critical value for the |
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For a solvated nanoparticle, there is a critical value for the |
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interfacial thermal conductance, |
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\begin{equation} |
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G_c = \frac{3 C_s \Lambda_s}{R C_p} |
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\subsection{Interfacial Thermal Conductance} |
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We can describe the thermal conductance of each spherical shell as the |
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inverse Kapitza resistance. To describe the thermal conductance for an |
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interface of considerable thickness, such as the ligand layers shown |
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here, we can sum the individual thermal resistances of each concentric |
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As described in our earlier work, the thermal conductance of each spherical shell may be defined as the |
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inverse Kapitza resistance of the shell. To describe the thermal conductance of an |
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interface of considerable thickness -- such as the ligand layers shown |
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here -- we can sum the individual thermal resistances of each concentric |
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spherical shell to arrive at the total thermal resistance, or the |
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inverse of the total interfacial thermal conductance: |
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inverse of the total interfacial thermal conductance. Unlike the periodic case, the intermediate temperature terms remain in the final sum, requiring the use of a series of individual resistance terms: |
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\begin{equation} |
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\frac{1}{G} = R_\mathrm{total} = \frac{1}{q_r} \sum_i \left(T_{i+i} - |
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T_i\right) 4 \pi r_i^2. |
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\end{equation} |
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The longest ligand considered here is in excess of 15 \AA\ in length, requiring the use of at least 10 spherical shells to describe the total interfacial thermal conductance. |
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The longest ligand considered here is in excess of 15 \AA\ in length, and we use 10 concentric spherical shells to describe the total interfacial thermal conductance of the ligand layer. |
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% COMPUTATIONAL DETAILS |
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\subsection{Force Fields} |
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Gold -- gold interactions are described by the quantum Sutton-Chen (QSC) model.\cite{PhysRevB.59.3527} Hexane molecules are described by the TraPPE united atom model,\cite{TraPPE-UA.alkanes} which provides good computational efficiency and reasonable accuracy for bulk thermal conductivity values. In this model, sites are located at the carbon centers for alkyl groups. Bonding interactions, including bond stretches, bends and torsions, were used for intra-molecular sites closer than 3 bonds. For non-bonded interactions, Lennard-Jones potentials were used. |
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Gold -- gold interactions are described by the quantum Sutton-Chen (QSC) model.\cite{PhysRevB.59.3527} Hexane solvent is described by the TraPPE united atom model,\cite{TraPPE-UA.alkanes} where sites are located at the carbon centers for alkyl groups. The TraPPE-UA model for hexane provides both computational efficiency and reasonable accuracy for bulk thermal conductivity values. Bonding interactions were used for intra-molecular sites closer than 3 bonds. Effective Lennard-Jones potentials were used for non-bonded interactions. |
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To describe the interactions between metal (Au) and non-metal atoms, potential energy terms were adapted from an adsorption study of alkyl thiols on gold surfaces by Vlugt, \textit{et al.}\cite{vlugt:cpc2007154} They fit an effective pair-wise Lennard-Jones form of potential parameters for the interaction between Au and pseudo-atoms CH$_x$ and S based on a well-established and widely-used effective potential of Hautman and Klein for the Au(111) surface.\cite{hautman:4994} |
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\subsection{Simulation Protocol} |
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The various sized gold nanoparticles were created from a bulk fcc lattice and were thermally equilibrated prior to the addition of ligands. A 50\% coverage of ligands (based on coverages reported by Badia, \textit{et al.}\cite{Badia1996:2}) were placed on the surface of the equilibrated nanoparticles using Packmol\cite{packmol}. The nanoparticle / ligand complexes were briefly thermally equilibrated before Packmol was used to solvate the structures within a spherical droplet of hexane. The thickness of the solvent layer was chosen to be at least 1.5$\times$ the radius of the nanoparticle / ligand structure. The fully solvated system was equilibrated in the Langevin Hull under 50 atm of pressure with a target temperature of 250 K for at least 1 nanosecond. |
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The various sized gold nanoparticles were created from a bulk fcc lattice and were thermally equilibrated prior to the addition of ligands. A 50\% coverage of ligands (based on coverages reported by Badia, \textit{et al.}\cite{Badia1996:2}) were placed on the surface of the equilibrated nanoparticles using Packmol\cite{packmol}. The nanoparticle / ligand complexes were briefly thermally equilibrated before Packmol was used to solvate the structures within a spherical droplet of hexane. The thickness of the solvent layer was chosen to be at least 1.5$\times$ the radius of the nanoparticle / ligand structure. The fully solvated system was equilibrated for at least 1 nanoseconds using the Langevin Hull to apply 50 atm of pressure and a target temperature of 250 K. |
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Once equilibrated, thermal fluxes were applied for |
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1 nanosecond, until stable temperature gradients had |
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\begin{figure} |
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\includegraphics[width=\linewidth]{figures/NP25_C12h1} |
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\caption{25 \AA\ radius gold nanoparticle protected with a half-monolayer of TraPPE-UA dodecanethiolate (C$_{12}$) ligands and solvated in TraPPE-UA hexane. The kinetic energy flux is imposed between the nanoparticle and an outer shell of solvent to compute the interfacial thermal conductance.} |
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\caption{A 25 \AA\ radius gold nanoparticle protected with a half-monolayer of TraPPE-UA dodecanethiolate (C$_{12}$) ligands and solvated in TraPPE-UA hexane. The interfacial thermal conductance is computed by applying a kinetic energy flux between the nanoparticle and an outer shell of solvent.} |
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\label{fig:NP25_C12h1} |
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\end{figure} |
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We have utilized a half-monolayer of three lengths of alkanethiolate ligands -- S(CH$_2$)$_3$CH$_3$, S(CH$_2$)$_7$CH$_3$, and S(CH$_2$)$_{11}$CH$_3$ -- referred to as C$_4$, C$_8$, and C$_{12}$ respectively, in this study. |
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Unlike our previous study of varying thiolate ligand chain lengths on Au(111) surfaces, the interfacial thermal conductance of ligand-protected nanoparticles exhibits a distinct non-monotonic dependence on the ligand length. For the three largest particle sizes, a half-monolayer coverage of $C_4$ yields the highest interfacial thermal conductance and the next-longest ligand $C_8$ provides a nearly equivalent boost. The longest ligand $C_{12}$ offers only a nominal ($\sim$ 10 \%) increase in the interfacial thermal conductance over a bare nanoparticle. |
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Unlike our previous study of varying thiolate ligand chain lengths on Au(111) surfaces, the interfacial thermal conductance of ligand-protected nanoparticles exhibits a distinct dependence on the ligand length. For the three largest particle sizes, a half-monolayer coverage of $C_4$ yields the highest interfacial thermal conductance and the next-longest ligand $C_8$ provides a nearly equivalent boost. The longest ligand $C_{12}$ offers only a nominal ($\sim$ 10 \%) increase in the interfacial thermal conductance over a bare nanoparticle. |
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\begin{figure} |
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\includegraphics[width=\linewidth]{figures/NPthiols_Gcombo} |
