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\title{The Thermal Conductance of Alkanethiolate-Protected Gold |
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Nanospheres: Effects of Curvature and Chain Length} |
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|
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\author{Kelsey M. Stocker} |
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\author{J. Daniel Gezelter} |
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\email{gezelter@nd.edu} |
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\affiliation[University of Notre Dame]{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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\keywords{Nanoparticles, interfaces, thermal conductance} |
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\begin{document} |
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\begin{tocentry} |
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\center\includegraphics[width=3.9cm]{figures/TOC} |
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\begin{abstract} |
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\end{abstract} |
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\newpage |
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|
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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 thermal properties of various nanostructured interfaces have been |
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the subject of intense experimental |
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interest,\cite{Wilson:2002uq,PhysRevB.67.054302,doi:10.1021/jp048375k,PhysRevLett.96.186101,Wang10082007,doi:10.1021/jp8051888,PhysRevB.80.195406,doi:10.1021/la904855s} |
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and the interfacial thermal conductance is the principal quantity of |
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interest for understanding interfacial heat |
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transport.\cite{cahill:793} Because nanoparticles have a significant |
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fraction of their atoms at the particle / solvent interface, the |
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chemical details of these interfaces govern the thermal transport |
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properties. |
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|
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Previously, reverse non-equilibrium molecular dynamics (RNEMD) methods |
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have been applied to calculate the interfacial thermal conductance at |
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flat (111) metal / organic solvent interfaces that had been chemically |
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protected by mixed-chain alkanethiolate groups.\cite{kuang:AuThl} |
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These simulations suggested an explanation for the increase in thermal |
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conductivity at alkanethiol-capped metal surfaces compared with bare |
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metal interfaces. Specifically, the chemical bond between the metal |
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and the ligand introduces a vibrational overlap that is not present |
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without the protecting group, and the overlap between the vibrational |
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spectra (metal to ligand, ligand to solvent) provides a mechanism for |
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rapid thermal transport across the interface. The simulations also |
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suggested that this phenomenon is a non-monotonic function of the |
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fractional coverage of the surface, as moderate coverages allow |
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diffusive heat transport of solvent molecules that have been in close |
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contact with the ligands. |
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|
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Simulations of {\it mixed-chain} alkylthiolate surfaces showed that |
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solvent trapped close to the interface can be very efficient at moving |
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thermal energy away from the surface.\cite{Stocker:2013cl} Trapped |
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solvent molecules that were orientationally ordered with nearby |
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ligands (but which were less able to diffuse into the bulk) were able |
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to double the thermal conductance of the interface. This indicates |
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that the ligand-to-solvent vibrational energy transfer is the key |
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feature for increasing particle-to-solvent thermal conductance. |
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|
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Recently, we extended RNEMD methods for use in non-periodic geometries |
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by creating scaling/shearing moves between concentric regions of the |
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simulation.\cite{Stocker:2014qq} In this work, we apply this |
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non-periodic variant of RNEMD to investigate the role that {\it |
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curved} nanoparticle surfaces play in heat and mass transport. On |
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planar surfaces, we discovered that orientational ordering of surface |
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protecting ligands had a large effect on the heat conduction from the |
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metal to the solvent. Smaller nanoparticles have high surface |
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curvature that creates gaps in well-ordered self-assembled monolayers, |
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and the effects those gaps have on the thermal conductance are unknown. |
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|
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|
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|
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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% INTERFACIAL THERMAL CONDUCTANCE OF METALLIC NANOPARTICLES |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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%\section{Interfacial Thermal Conductance of Metallic Nanoparticles} |
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|
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For a solvated nanoparticle, it is possible to define a critical value |
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for the 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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\end{equation} |
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which depends on the solvent heat capacity, $C_s$, solvent thermal |
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conductivity, $\Lambda_s$, particle radius, $R$, and nanoparticle heat |
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capacity, $C_p$.\cite{Wilson:2002uq} In the limit of infinite |
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interfacial thermal conductance, $G \gg G_c$, cooling of the |
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nanoparticle is limited by the solvent properties, $C_s$ and |
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$\Lambda_s$. In the opposite limit, $G \ll G_c$, the heat dissipation |
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is controlled by the thermal conductance of the particle / fluid |
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interface. It is this regime with which we are concerned, where |
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properties of ligands and the particle surface may be tuned to |
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manipulate the rate of cooling for solvated nanoparticles. Based on |
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estimates of $G$ from previous simulations as well as experimental |
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results for solvated nanostructures, gold nanoparticles solvated in |
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hexane are in the $G \ll G_c$ regime for radii smaller than 40 nm. The |
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particles included in this study are more than an order of magnitude |
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smaller than this critical radius, so the heat dissipation should be |
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controlled entirely by the surface features of the particle / ligand / |
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solvent interface. |
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|
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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% STRUCTURE OF SELF-ASSEMBLED MONOLAYERS ON NANOPARTICLES |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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\subsection{Structures of Self-Assembled Monolayers on Nanoparticles} |
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|
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Though the ligand packing on planar surfaces is characterized for many |
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different ligands and surface facets, it is not obvious \emph{a |
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priori} how the same ligands will behave on the highly curved |
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surfaces of spherical nanoparticles. Thus, as more applications of |
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ligand-stabilized nanostructures have become apparent, the structure |
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and dynamics of ligands on metallic nanoparticles have been studied |
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extensively.\cite{Badia1996:2,Badia1996,Henz2007,Badia1997:2,Badia1997,Badia2000} |
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Badia, \textit{et al.} used transmission electron microscopy to |
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determine that alkanethiol ligands on gold nanoparticles pack |
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approximately 30\% more densely than on planar Au(111) |
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surfaces.\cite{Badia1996:2} Subsequent experiments demonstrated that |
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even at full coverages, surface curvature creates voids between linear |
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ligand chains that can be filled via interdigitation of ligands on |
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neighboring particles.\cite{Badia1996} The molecular dynamics |
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simulations of Henz, \textit{et al.} indicate that at low coverages, |
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the thiolate alkane chains will lie flat on the nanoparticle |
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surface\cite{Henz2007} Above 90\% coverage, the ligands stand upright |
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and recover the rigidity and tilt angle displayed on planar |
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facets. Their simulations also indicate a high degree of mixing |
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between the thiolate sulfur atoms and surface gold atoms at high |
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coverages. |
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|
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To model thiolated gold nanospheres in this work, gold nanoparticles |
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with radii ranging from 10 - 25 \AA\ were created from a bulk fcc |
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lattice. To match surface coverages previously reported by Badia, |
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\textit{et al.}\cite{Badia1996:2}, these particles were passivated |
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with a 50\% coverage of a selection of alkyl thiolates of varying |
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chain lengths. The passivated particles were then solvated in hexane. |
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Details of the models and simulation protocol follow in the next |
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section. |
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|
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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% NON-PERIODIC VSS-RNEMD METHODOLOGY |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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\subsection{Creating a thermal flux between particles and solvent} |
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|
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The non-periodic variant of VSS-RNEMD\cite{Stocker:2014qq} applies a |
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series of velocity scaling and shearing moves at regular intervals to |
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impose a flux between two concentric spherical regions. To impose a |
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thermal flux between the shells (without an accompanying angular |
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shear), we solve for scaling coefficients $a$ and $b$, |
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\begin{eqnarray} |
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a = \sqrt{1 - \frac{q_r \Delta t}{K_a - K_a^\mathrm{rot}}}\\ \nonumber\\ |
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b = \sqrt{1 + \frac{q_r \Delta t}{K_b - K_b^\mathrm{rot}}} |
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\end{eqnarray} |
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at each time interval. These scaling coefficients conserve total |
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kinetic energy and angular momentum subject to an imposed heat rate, |
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$q_r$. The coefficients also depend on the instantaneous kinetic |
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energy, $K_{\{a,b\}}$, and the total rotational kinetic energy of each |
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shell, $K_{\{a,b\}}^\mathrm{rot} = \sum_i m_i \left( \mathbf{v}_i |
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\times \mathbf{r}_i \right)^2 / 2$. |
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|
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The scaling coefficients are determined and the velocity changes are |
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applied at regular intervals, |
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\begin{eqnarray} |
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\mathbf{v}_i \leftarrow a \left ( \mathbf{v}_i - \left < \omega_a \right > \times \mathbf{r}_i \right ) + \left < \omega_a \right > \times \mathbf{r}_i~~\:\\ |
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\mathbf{v}_j \leftarrow b \left ( \mathbf{v}_j - \left < \omega_b \right > \times \mathbf{r}_j \right ) + \left < \omega_b \right > \times \mathbf{r}_j. |
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\end{eqnarray} |
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Here $\left < \omega_a \right > \times \mathbf{r}_i$ is the |
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contribution to the velocity of particle $i$ due to the overall |
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angular velocity of the $a$ shell. In the absence of an angular |
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momentum flux, the angular velocity $\left < \omega_a \right >$ of the |
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shell is nearly 0 and the resultant particle velocity is a nearly |
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linear scaling of the initial velocity by the coefficient $a$ or $b$. |
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|
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Repeated application of this thermal energy exchange yields a radial |
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temperature profile for the solvated nanoparticles that depends |
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linearly on the applied heat rate, $q_r$. Similar to the behavior in |
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the slab geometries, the temperature profiles have discontinuities at |
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the interfaces between dissimilar materials. The size of the |
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discontinuity depends on the interfacial thermal conductance, which is |
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the primary quantity of interest. |
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|
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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% CALCULATING TRANSPORT PROPERTIES |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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% INTERFACIAL THERMAL CONDUCTANCE |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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\subsection{Interfacial Thermal Conductance} |
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|
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As described in earlier work,\cite{Stocker:2014qq} the thermal |
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conductance of each spherical shell may be defined as the inverse |
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Kapitza resistance of the shell. To describe the thermal conductance |
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of an interface of considerable thickness -- such as the ligand layers |
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shown here -- we can sum the individual thermal resistances of each |
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concentric spherical shell to arrive at the inverse of the total |
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interfacial thermal conductance. In slab geometries, the intermediate |
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temperatures cancel, but for concentric spherical shells, the |
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intermeidate temperatures and surface areas remain in the final sum, |
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requiring the use of a series of individual resistance terms: |
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|
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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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|
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The longest ligand considered here is in excess of 15 \AA\ in length, |
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and we use 10 concentric spherical shells to describe the total |
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interfacial thermal conductance of the ligand layer. |
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|
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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% COMPUTATIONAL DETAILS |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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\section{Computational Details} |
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|
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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% FORCE FIELDS |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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\subsection{Force Fields} |
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|
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Throughout this work, gold -- gold interactions are described by the |
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quantum Sutton-Chen (QSC) model.\cite{PhysRevB.59.3527} The hexane |
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solvent is described by the TraPPE united atom |
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model,\cite{TraPPE-UA.alkanes} where sites are located at the carbon |
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centers for alkyl groups. The TraPPE-UA model for hexane provides both |
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computational efficiency and reasonable accuracy for bulk thermal |
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conductivity values. Bonding interactions were used for |
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intra-molecular sites closer than 3 bonds. Effective Lennard-Jones |
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potentials were used for non-bonded interactions. |
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|
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To describe the interactions between metal (Au) and non-metal atoms, |
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potential energy terms were adapted from an adsorption study of alkyl |
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thiols on gold surfaces by Vlugt, \textit{et |
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al.}\cite{vlugt:cpc2007154} They fit an effective pair-wise |
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Lennard-Jones form of potential parameters for the interaction between |
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Au and pseudo-atoms CH$_x$ and S based on a well-established and |
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widely-used effective potential of Hautman and Klein for the Au(111) |
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surface.\cite{hautman:4994} |
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|
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|
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|
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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% SIMULATION PROTOCOL |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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\subsection{Simulation Protocol} |
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|
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Gold nanospheres with radii ranging from 10 - 25 \AA\ were created |
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from a bulk fcc lattice and were thermally equilibrated prior to the |
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addition of ligands. A 50\% coverage of ligands (based on coverages |
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reported by Badia, \textit{et al.}\cite{Badia1996:2}) were placed on |
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the surface of the equilibrated nanoparticles using |
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Packmol\cite{packmol}. The nanoparticle / ligand complexes were |
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thermally equilibrated before Packmol was used to solvate the |
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structures inside a spherical droplet of hexane. The thickness of the |
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solvent layer was chosen to be at least 1.5$\times$ the combined |
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radius of the nanoparticle / ligand structure. The fully solvated |
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system was equilibrated for at least 1 ns using the Langevin Hull to |
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apply 50 atm of pressure and a target temperature of 250 |
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K.\cite{Vardeman2011} Typical system sizes ranged from 18,310 united |
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atom sites for the 10 \AA particles with $C_4$ ligands to 89,490 sites |
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for the 25 \AA particles with $C_{12}$ ligands. Figure |
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\ref{fig:NP25_C12h1} shows one of the solvated 25 \AA nanoparticles |
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passivated with the $C_{12}$ ligands. |
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|
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\begin{figure} |
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\includegraphics[width=\linewidth]{figures/NP25_C12h1} |
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\caption{A 25 \AA\ radius gold nanoparticle protected with a |
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half-monolayer of TraPPE-UA dodecanethiolate (C$_{12}$) |
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ligands and solvated in TraPPE-UA hexane. The interfacial |
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thermal conductance is computed by applying a kinetic energy |
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flux between the nanoparticle and an outer shell of |
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solvent.} |
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\label{fig:NP25_C12h1} |
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\end{figure} |
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|
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Once equilibrated, thermal fluxes were applied for 1 ns, until stable |
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temperature gradients had developed. Systems were run under moderate |
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pressure (50 atm) with an average temperature (250K) that maintained a |
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compact solvent cluster and avoided formation of a vapor layer near |
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the heated metal surface. Pressure was applied to the system via the |
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non-periodic Langevin Hull.\cite{Vardeman2011} However, thermal |
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coupling to the external temperature bath was removed to avoid |
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interference with the imposed RNEMD flux. |
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|
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Because the method conserves \emph{total} angular momentum and energy, |
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systems which contain a metal nanoparticle embedded in a significant |
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volume of solvent will still experience nanoparticle diffusion inside |
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the solvent droplet. To aid in measuring an accurate temperature |
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profile for these systems, a single gold atom at the origin of the |
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coordinate system was assigned a mass $10,000 \times$ its original |
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mass. The bonded and nonbonded interactions for this atom remain |
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unchanged and the heavy atom is excluded from the RNEMD velocity |
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scaling. The only effect of this gold atom is to effectively pin the |
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nanoparticle at the origin of the coordinate system, thereby |
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preventing translational diffusion of the nanoparticle due to Brownian |
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motion. |
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|
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To provide statisical independence, five separate configurations were |
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simulated for each particle radius and ligand length. The |
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configurations were unique starting at the point of ligand placement |
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in order to sample multiple surface-ligand configurations. |
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|
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|
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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% EFFECT OF PARTICLE SIZE |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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\section{Results} |
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|
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We modeled four sizes of nanoparticles ($R =$ 10, 15, 20, and 25 |
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\AA). The smallest particle size produces the lowest interfacial |
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thermal conductance values for most of the of protecting groups |
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(Fig. \ref{fig:NPthiols_G}). Between the other three sizes of |
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nanoparticles, there is no discernible dependence of the interfacial |
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thermal conductance on the nanoparticle size. It is likely that the |
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differences in local curvature of the nanoparticle sizes studied here |
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do not disrupt the ligand packing and behavior in drastically |
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different ways. |
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|
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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% EFFECT OF LIGAND CHAIN LENGTH |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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|
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We have also utilized half-monolayers of three lengths of |
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alkanethiolate ligands -- S(CH$_2$)$_3$CH$_3$, S(CH$_2$)$_7$CH$_3$, |
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and S(CH$_2$)$_{11}$CH$_3$ -- referred to as C$_4$, C$_8$, and |
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C$_{12}$ respectively, in this study. |
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|
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Unlike our previous study of varying thiolate ligand chain lengths on |
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planar Au(111) surfaces, the interfacial thermal conductance of |
359 |
ligand-protected nanospheres exhibits a distinct dependence on the |
360 |
ligand length. For the three largest particle sizes, a half-monolayer |
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coverage of $C_4$ yields the highest interfacial thermal conductance |
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and the next-longest ligand, $C_8$, provides a similar boost. The |
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longest ligand, $C_{12}$, offers only a nominal ($\sim$ 10 \%) |
364 |
increase in the interfacial thermal conductance over the bare |
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nanoparticles. |
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|
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\begin{figure} |
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\includegraphics[width=\linewidth]{figures/NPthiols_G} |
369 |
\caption{Interfacial thermal conductance ($G$) values for 4 |
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sizes of solvated nanoparticles that are bare or protected |
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with a 50\% coverage of C$_{4}$, C$_{8}$, or C$_{12}$ |
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alkanethiolate ligands.} |
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\label{fig:NPthiols_G} |
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\end{figure} |
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|
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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% HEAT TRANSFER MECHANISMS |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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\section{Mechanisms for Ligand-Enhanced Heat Transfer} |
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|
381 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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% CORRUGATION OF PARTICLE SURFACE |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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\subsection{Corrugation of Particle Surface} |
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|
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The bonding sites for thiols on gold surfaces have been studied |
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extensively and include configurations beyond the traditional atop, |
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bridge, and hollow sites found on planar surfaces. In particular, the |
389 |
deep potential well between the gold atoms and the thiolate sulfurs |
390 |
leads to insertion of the sulfur into the gold lattice and |
391 |
displacement of interfacial gold atoms. The degree of ligand-induced |
392 |
surface restructuring may have an impact on the interfacial thermal |
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conductance and is an important phenomenon to quantify. |
394 |
|
395 |
Henz, \textit{et al.}\cite{Henz2007} used the metal density as a |
396 |
function of radius to measure the degree of mixing between the thiol |
397 |
sulfurs and surface gold atoms at the edge of a nanoparticle. Although |
398 |
metal density is important, disruption of the local crystalline |
399 |
ordering would also have a large effect on the phonon spectrum in the |
400 |
particles. To measure this effect, we use the fraction of gold atoms |
401 |
exhibiting local fcc ordering as a function of radius to describe the |
402 |
ligand-induced disruption of the nanoparticle surface. |
403 |
|
404 |
The local bond orientational order can be described using the method |
405 |
of Steinhardt \textit{et al.},\cite{Steinhardt1983} where spherical |
406 |
harmonics are associated with a central atom and its nearest |
407 |
neighbors. The local bonding environment, $\bar{q}_{\ell m}$, for each |
408 |
atom in the system can be determined by averaging over the spherical |
409 |
harmonics between the central atom and each of its neighbors, |
410 |
\begin{equation} |
411 |
\bar{q}_{\ell m} = \sum_i Y_\ell^m(\theta_i, \phi_i) |
412 |
\end{equation} |
413 |
where $\theta_i$ and $\phi_i$ are the relative angular coordinates of |
414 |
neighbor $i$ in the laboratory frame. A global average orientational |
415 |
bond order parameter, $\bar{Q}_{\ell m}$, is the average over each |
416 |
$\bar{q}_{\ell m}$ for all atoms in the system. To remove the |
417 |
dependence on the laboratory coordinate frame, the third order |
418 |
rotationally invariant combination of $\bar{Q}_{\ell m}$, |
419 |
$\hat{w}_\ell$, is utilized here.\cite{Steinhardt1983,Vardeman:2008fk} |
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|
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For $\ell=4$, the ideal face-centered cubic (fcc), body-centered cubic |
422 |
(bcc), hexagonally close-packed (hcp), and simple cubic (sc) local |
423 |
structures exhibit $\hat{w}_4$ values of -0.159, 0.134, 0.159, and |
424 |
0.159, respectively. Because $\hat{w}_4$ exhibits an extreme value for |
425 |
fcc structures, this makes it ideal for measuring local fcc |
426 |
ordering. The spatial distribution of $\hat{w}_4$ local bond |
427 |
orientational order parameters, $p(\hat{w}_4 , r)$, can provide |
428 |
information about the location of individual atoms that are central to |
429 |
local fcc ordering. |
430 |
|
431 |
The fraction of fcc-ordered gold atoms at a given radius in the |
432 |
nanoparticle, |
433 |
\begin{equation} |
434 |
f_\mathrm{fcc}(r) = \int_{-\infty}^{w_c} p(\hat{w}_4, r) d \hat{w}_4 |
435 |
\end{equation} |
436 |
is described by the distribution of the local bond orientational order |
437 |
parameters, $p(\hat{w}_4, r)$, and $w_c$, a cutoff for the peak |
438 |
$\hat{w}_4$ value displayed by fcc structures. A $w_c$ value of -0.12 |
439 |
was chosen to isolate the fcc peak in $\hat{w}_4$. |
440 |
|
441 |
As illustrated in Figure \ref{fig:Corrugation}, the presence of |
442 |
ligands decreases the fcc ordering of the gold atoms at the |
443 |
nanoparticle surface. For the smaller nanoparticles, this disruption |
444 |
extends into the core of the nanoparticle, indicating widespread |
445 |
disruption of the lattice. |
446 |
|
447 |
\begin{figure} |
448 |
\includegraphics[width=\linewidth]{figures/NP10_fcc} |
449 |
\caption{Fraction of gold atoms with fcc ordering as a |
450 |
function of radius for a 10 \AA\ radius nanoparticle. The |
451 |
decreased fraction of fcc-ordered atoms in ligand-protected |
452 |
nanoparticles relative to bare particles indicates |
453 |
restructuring of the nanoparticle surface by the thiolate |
454 |
sulfur atoms.} |
455 |
\label{fig:Corrugation} |
456 |
\end{figure} |
457 |
|
458 |
We may describe the thickness of the disrupted nanoparticle surface by |
459 |
defining a corrugation factor, $c$, as the ratio of the radius at |
460 |
which the fraction of gold atoms with fcc ordering is 0.9 and the |
461 |
radius at which the fraction is 0.5. |
462 |
|
463 |
\begin{equation} |
464 |
c = 1 - \frac{r(f_\mathrm{fcc} = 0.9)}{r(f_\mathrm{fcc} = 0.5)} |
465 |
\end{equation} |
466 |
|
467 |
A sharp interface will have a sharp drop in $f_\mathrm{fcc}$ at the |
468 |
edge of the particle ($c \rightarrow$ 0). In the opposite limit where |
469 |
the entire nanoparticle surface is restructured by ligands, the radius |
470 |
at which there is a high probability of fcc ordering moves |
471 |
dramatically inward ($c \rightarrow$ 1). |
472 |
|
473 |
The computed corrugation factors are shown in Figure |
474 |
\ref{fig:NPthiols_combo} for bare nanoparticles and for |
475 |
ligand-protected particles as a function of ligand chain length. The |
476 |
largest nanoparticles are only slightly restructured by the presence |
477 |
of ligands on the surface, while the smallest particle ($r$ = 10 \AA) |
478 |
exhibits significant disruption of the original fcc ordering when |
479 |
covered with a half-monolayer of thiol ligands. |
480 |
|
481 |
Because the thiolate ligands do not significantly alter the larger |
482 |
particle crystallinity, the surface corrugation does not seem to be a |
483 |
likely candidate to explain the large increase in thermal conductance |
484 |
at the interface. |
485 |
|
486 |
% \begin{equation} |
487 |
% C = \frac{r_{bare}(\rho_{\scriptscriptstyle{0.85}}) - r_{capped}(\rho_{\scriptscriptstyle{0.85}})}{r_{bare}(\rho_{\scriptscriptstyle{0.85}})}. |
488 |
% \end{equation} |
489 |
% |
490 |
% Here, $r_{bare}(\rho_{\scriptscriptstyle{0.85}})$ is the radius of a bare nanoparticle at which the density is $85\%$ the bulk value and $r_{capped}(\rho_{\scriptscriptstyle{0.85}})$ is the corresponding radius for a particle of the same size with a layer of ligands. $C$ has a value of 0 for a bare particle and approaches $1$ as the degree of surface atom mixing increases. |
491 |
|
492 |
|
493 |
|
494 |
\begin{figure} |
495 |
\includegraphics[width=\linewidth]{figures/NPthiols_combo} |
496 |
\caption{Computed corrugation values, solvent escape rates, |
497 |
ligand orientational $P_2$ values, and interfacial solvent |
498 |
orientational $P_2$ values for 4 sizes of solvated |
499 |
nanoparticles that are bare or protected with a 50\% |
500 |
coverage of C$_{4}$, C$_{8}$, or C$_{12}$ alkanethiolate |
501 |
ligands.} |
502 |
\label{fig:NPthiols_combo} |
503 |
\end{figure} |
504 |
|
505 |
|
506 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
507 |
% MOBILITY OF INTERFACIAL SOLVENT |
508 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
509 |
\subsection{Mobility of Interfacial Solvent} |
510 |
|
511 |
Another possible mechanism for increasing interfacial conductance is |
512 |
the mobility of the interfacial solvent. We used a survival |
513 |
correlation function, $C(t)$, to measure the residence time of a |
514 |
solvent molecule in the nanoparticle thiolate |
515 |
layer.\cite{Stocker:2013cl} This function correlates the identity of |
516 |
all hexane molecules within the radial range of the thiolate layer at |
517 |
two separate times. If the solvent molecule is present at both times, |
518 |
the configuration contributes a $1$, while the absence of the molecule |
519 |
at the later time indicates that the solvent molecule has migrated |
520 |
into the bulk, and this configuration contributes a $0$. A steep decay |
521 |
in $C(t)$ indicates a high turnover rate of solvent molecules from the |
522 |
chain region to the bulk. We may define the escape rate for trapped |
523 |
solvent molecules at the interface as |
524 |
\begin{equation} |
525 |
k_\mathrm{escape} = \left( \int_0^T C(t) dt \right)^{-1} |
526 |
\label{eq:mobility} |
527 |
\end{equation} |
528 |
where T is the length of the simulation. This is a direct measure of |
529 |
the rate at which solvent molecules initially entangled in the |
530 |
thiolate layer can escape into the bulk. When $k_\mathrm{escape} |
531 |
\rightarrow 0$, the solvent becomes permanently trapped in the |
532 |
interfacial region. |
533 |
|
534 |
The solvent escape rates for bare and ligand-protected nanoparticles |
535 |
are shown in Figure \ref{fig:NPthiols_combo}. As the ligand chain |
536 |
becomes longer and more flexible, interfacial solvent molecules become |
537 |
trapped in the ligand layer and the solvent escape rate decreases. |
538 |
This mechanism contributes a partial explanation as to why the longer |
539 |
ligands have significantly lower thermal conductance. |
540 |
|
541 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
542 |
% ORIENTATION OF LIGAND CHAINS |
543 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
544 |
\subsection{Orientation of Ligand Chains} |
545 |
|
546 |
As the ligand chain length increases in length, it exhibits |
547 |
significantly more conformational flexibility. Thus, different lengths |
548 |
of ligands should favor different chain orientations on the surface of |
549 |
the nanoparticle. To determine the distribution of ligand orientations |
550 |
relative to the particle surface we examine the probability of |
551 |
finding a ligand with a particular orientation relative to the surface |
552 |
normal of the nanoparticle, |
553 |
\begin{equation} |
554 |
\cos{(\theta)}=\frac{\vec{r}_i\cdot\hat{u}_i}{|\vec{r}_i||\hat{u}_i|} |
555 |
\end{equation} |
556 |
where $\vec{r}_{i}$ is the vector between the cluster center of mass |
557 |
and the sulfur atom on ligand molecule {\it i}, and $\hat{u}_{i}$ is |
558 |
the vector between the sulfur atom and \ce{CH3} pseudo-atom on ligand |
559 |
molecule {\it i}. As depicted in Figure \ref{fig:NP_pAngle}, $\theta |
560 |
\rightarrow 180^{\circ}$ for a ligand chain standing upright on the |
561 |
particle ($\cos{(\theta)} \rightarrow -1$) and $\theta \rightarrow |
562 |
90^{\circ}$ for a ligand chain lying down on the surface |
563 |
($\cos{(\theta)} \rightarrow 0$). As the thiolate alkane chain |
564 |
increases in length and becomes more flexible, the ligands are more |
565 |
willing to lie down on the nanoparticle surface and exhibit increased |
566 |
population at $\cos{(\theta)} = 0$. |
567 |
|
568 |
\begin{figure} |
569 |
\includegraphics[width=\linewidth]{figures/NP_pAngle} |
570 |
\caption{The two extreme cases of ligand orientation relative |
571 |
to the nanoparticle surface: the ligand completely |
572 |
outstretched ($\cos{(\theta)} = -1$) and the ligand fully |
573 |
lying down on the particle surface ($\cos{(\theta)} = 0$).} |
574 |
\label{fig:NP_pAngle} |
575 |
\end{figure} |
576 |
|
577 |
% \begin{figure} |
578 |
% \includegraphics[width=\linewidth]{figures/thiol_pAngle} |
579 |
% \caption{} |
580 |
% \label{fig:thiol_pAngle} |
581 |
% \end{figure} |
582 |
|
583 |
An order parameter the average ligand chain orientation relative to |
584 |
the nanoparticle surface is available using the second order Legendre |
585 |
parameter, |
586 |
\begin{equation} |
587 |
P_2 = \left< \frac{1}{2} \left(3\cos^2(\theta) - 1 \right) \right> |
588 |
\end{equation} |
589 |
|
590 |
Ligand populations that are perpendicular to the particle surface hav |
591 |
P$_2$ values of 1, while ligand populations lying flat on the |
592 |
nanoparticle surface have P$_2$ values of $-0.5$. Disordered ligand |
593 |
layers will exhibit mean P$_2$ values of 0. As shown in Figure |
594 |
\ref{fig:NPthiols_combo} the ligand P$_2$ values approaches 0 as |
595 |
ligand chain length -- and ligand flexibility -- increases. |
596 |
|
597 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
598 |
% ORIENTATION OF INTERFACIAL SOLVENT |
599 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
600 |
\subsection{Orientation of Interfacial Solvent} |
601 |
|
602 |
Similarly, we examined the distribution of \emph{hexane} molecule |
603 |
orientations relative to the particle surface using the same angular |
604 |
analysis utilized for the ligand chain orientations. In this case, |
605 |
$\vec{r}_i$ is the vector between the particle center of mass and one |
606 |
of the \ce{CH2} pseudo-atoms in the middle of hexane molecule $i$ and |
607 |
$\hat{u}_i$ is the vector between the two \ce{CH3} pseudo-atoms on |
608 |
molecule $i$. Since we are only interested in the orientation of |
609 |
solvent molecules near the ligand layer, we select only the hexane |
610 |
molecules within a specific $r$-range, between the edge of the |
611 |
particle and the end of the ligand chains. A large population of |
612 |
hexane molecules with $\cos{(\theta)} \cong \pm 1$ would indicate |
613 |
interdigitation of the solvent molecules between the upright ligand |
614 |
chains. A more random distribution of $\cos{(\theta)}$ values |
615 |
indicates a disordered arrangement of solvent chains on the particle |
616 |
surface. Again, P$_2$ order parameter values provide a population |
617 |
analysis for the solvent that is close to the particle surface. |
618 |
|
619 |
The average orientation of the interfacial solvent molecules is |
620 |
notably flat on the \emph{bare} nanoparticle surfaces. This blanket of |
621 |
hexane molecules on the particle surface may act as an insulating |
622 |
layer, increasing the interfacial thermal resistance. As the length |
623 |
(and flexibility) of the ligand increases, the average interfacial |
624 |
solvent P$_2$ value approaches 0, indicating a more random orientation |
625 |
of the ligand chains. The average orientation of solvent within the |
626 |
$C_8$ and $C_{12}$ ligand layers is essentially random. Solvent |
627 |
molecules in the interfacial region of $C_4$ ligand-protected |
628 |
nanoparticles do not lie as flat on the surface as in the case of the |
629 |
bare particles, but are not as randomly oriented as the longer ligand |
630 |
lengths. |
631 |
|
632 |
These results are particularly interesting in light of our previous |
633 |
results\cite{Stocker:2013cl}, where solvent molecules readily filled |
634 |
the vertical gaps between neighboring ligand chains and there was a |
635 |
strong correlation between ligand and solvent molecular |
636 |
orientations. It appears that the introduction of surface curvature |
637 |
and a lower ligand packing density creates a disordered ligand layer |
638 |
that lacks well-formed channels for the solvent molecules to occupy. |
639 |
|
640 |
% \begin{figure} |
641 |
% \includegraphics[width=\linewidth]{figures/hex_pAngle} |
642 |
% \caption{} |
643 |
% \label{fig:hex_pAngle} |
644 |
% \end{figure} |
645 |
|
646 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
647 |
% SOLVENT PENETRATION OF LIGAND LAYER |
648 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
649 |
\subsection{Solvent Penetration of Ligand Layer} |
650 |
|
651 |
We may also determine the extent of ligand -- solvent interaction by |
652 |
calculating the hexane density as a function of radius. Figure |
653 |
\ref{fig:hex_density} shows representative radial hexane density |
654 |
profiles for a solvated 25 \AA\ radius nanoparticle with no ligands, |
655 |
and 50\% coverage of C$_{4}$, C$_{8}$, and C$_{12}$ thiolates. |
656 |
|
657 |
\begin{figure} |
658 |
\includegraphics[width=\linewidth]{figures/hex_density} |
659 |
\caption{Radial hexane density profiles for 25 \AA\ radius |
660 |
nanoparticles with no ligands (circles), C$_{4}$ ligands |
661 |
(squares), C$_{8}$ ligands (triangles), and C$_{12}$ ligands |
662 |
(diamonds). As ligand chain length increases, the nearby |
663 |
solvent is excluded from the ligand layer. Some solvent is |
664 |
present inside the particle $r_{max}$ location due to |
665 |
faceting of the nanoparticle surface.} |
666 |
\label{fig:hex_density} |
667 |
\end{figure} |
668 |
|
669 |
The differences between the radii at which the hexane surrounding the |
670 |
ligand-covered particles reaches bulk density correspond nearly |
671 |
exactly to the differences between the lengths of the ligand |
672 |
chains. Beyond the edge of the ligand layer, the solvent reaches its |
673 |
bulk density within a few angstroms. The differing shapes of the |
674 |
density curves indicate that the solvent is increasingly excluded from |
675 |
the ligand layer as the chain length increases. |
676 |
|
677 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
678 |
% DISCUSSION |
679 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
680 |
\section{Discussion} |
681 |
|
682 |
The chemical bond between the metal and the ligand introduces |
683 |
vibrational overlap that is not present between the bare metal surface |
684 |
and solvent. Thus, regardless of ligand chain length, the presence of |
685 |
a half-monolayer ligand coverage yields a higher interfacial thermal |
686 |
conductance value than the bare nanoparticle. The dependence of the |
687 |
interfacial thermal conductance on ligand chain length is primarily |
688 |
explained by increased ligand flexibility and a corresponding decrease |
689 |
in solvent mobility away from the particles. The shortest and least |
690 |
flexible ligand ($C_4$), which exhibits the highest interfacial |
691 |
thermal conductance value, has a smaller range of angles relative to |
692 |
the surface normal and is least likely to trap solvent molecules |
693 |
within the ligand layer. The longer $C_8$ and $C_{12}$ ligands have |
694 |
increasingly disordered orientations and correspondingly lower solvent |
695 |
escape rates. |
696 |
|
697 |
When the ligands are less tightly packed, the cooperative |
698 |
orientational ordering between the ligand and solvent decreases |
699 |
dramatically and the conductive heat transfer model plays a much |
700 |
smaller role in determining the total interfacial thermal |
701 |
conductance. Thus, heat transfer into the solvent relies primarily on |
702 |
the convective model, where solvent molecules pick up thermal energy |
703 |
from the ligands and diffuse into the bulk solvent. This mode of heat |
704 |
transfer is hampered by a slow solvent escape rate, which is clearly |
705 |
present in the randomly ordered long ligand layers. |
706 |
|
707 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
708 |
% **ACKNOWLEDGMENTS** |
709 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
710 |
\begin{acknowledgement} |
711 |
Support for this project was provided by the National Science Foundation |
712 |
under grant CHE-1362211. Computational time was provided by the |
713 |
Center for Research Computing (CRC) at the University of Notre Dame. |
714 |
\end{acknowledgement} |
715 |
|
716 |
|
717 |
\newpage |
718 |
|
719 |
\bibliography{NPthiols} |
720 |
|
721 |
\end{document} |