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\documentclass[aps,jcp,preprint,showpacs,superscriptaddress,groupedaddress]{revtex4-1} % for double-spaced preprint |
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\usepackage[version=3]{mhchem} |
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\begin{document} |
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|
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\title{Interfacial Thermal Conductance of Thiolate-Protected |
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Gold Nanospheres} |
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\author{Kelsey M. Stocker} |
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\author{Suzanne M. Neidhart} |
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\author{J. Daniel Gezelter} |
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\email{gezelter@nd.edu} |
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\affiliation{Department of Chemistry and Biochemistry, University of |
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Notre Dame, Notre Dame, IN 46556} |
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|
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\begin{abstract} |
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Molecular dynamics simulations of thiolate-protected and solvated |
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gold nanoparticles were carried out in the presence of a |
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non-equilibrium heat flux between the solvent and the core of the |
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particle. The interfacial thermal conductance ($G$) was computed for |
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these interfaces, and the behavior of the thermal conductance was |
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studied as a function of particle size, ligand flexibility, and |
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ligand chain length. In all cases, thermal conductance of the |
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ligand-protected particles was higher than the bare metal--solvent |
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interface. A number of mechanisms for the enhanced conductance were |
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investigated, including thiolate-driven corrugation of the metal |
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surface, solvent ordering at the interface, solvent-ligand |
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interpenetration, and ligand ordering relative to the particle |
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surface. Only the smallest particles exhibited significant |
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corrugation. All ligands permitted substantial solvent-ligand |
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interpenetration, and ligand chain length has a significant |
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influence on the orientational ordering of interfacial solvent. |
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Solvent -- ligand vibrational overlap, particularly in the low |
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frequency range ($< 80 \mathrm{cm}^{-1}$) was significantly altered |
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by ligand rigidity, and had direct influence on the interfacial |
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thermal conductance. |
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\end{abstract} |
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|
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\pacs{} |
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\keywords{} |
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\maketitle |
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|
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\section{Introduction} |
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|
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Heat transport across various nanostructured interfaces has been the |
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subject of intense experimental |
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interest,\cite{Wilson:2002uq,Ge:2004yg,Shenogina:2009ix,Wang10082007,Schmidt:2008ad,Juve:2009pt,Alper:2010pd,Harikrishna:2013ys} |
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and the interfacial thermal conductance, $G$, is the principal |
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quantity of interest for understanding interfacial heat |
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transport.\cite{Cahill:2003fk} Because nanoparticles have a |
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significant fraction of their atoms at the particle / solvent |
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interface, the chemical details of these interfaces govern the thermal |
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transport properties. For ligand-protected nanoparticles, there may |
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be three distinct heat transfer processes: (1) from the particles to |
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the ligands, (2) vibrational energy tranfer along the length of the |
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ligand, followed by (3) heat transport from the ligand to the |
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surrounding solvent.\cite{Ge:2006kx} |
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|
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Heat transport at the gold-alkylthiolate-solvent interface has been |
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previously explored both through molecular dynamics simulations and |
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via time domain |
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thermoreflectance.\cite{Kikugawa:2009vn,Kuang:2011ef,Stocker:2013cl,Tian:2015uq} |
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Most of these studies have found that alkylthiolates enhance the |
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thermal conductance to the solvent, and that the vibrational overlap |
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provided by the chemically-bound ligand species plays a role in this |
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enhancement. |
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|
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Reverse nonequilibrium molecular dynamics (RNEMD) methods have been |
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previously applied to calculate the thermal conductance at flat (111) |
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metal / organic solvent interfaces that had been chemically protected |
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by varying coverages of alkanethiolate groups.\cite{Kuang:2011ef} |
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These simulations suggested an explanation for the increased 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 energy |
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transfer to solvent molecules that come into close contact with the |
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ligands. |
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|
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Similarly, simulations of {\it mixed-chain} alkylthiolate surfaces |
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showed that solvent trapped close to the interface can be efficient at |
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moving thermal energy away from the surface.\cite{Stocker:2013cl} |
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Trapped solvent molecules that were orientationally aligned with |
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nearby ligands were able to increase the thermal conductance of the |
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interface. This indicates that the ligand-to-solvent vibrational |
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energy transfer is a key feature for increasing particle-to-solvent |
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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 a |
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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 effect of those gaps on the thermal conductance is unknown. |
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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 has been characterized |
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for many different ligands and surface facets, it is not obvious |
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\emph{a priori} how the same ligands will behave on the highly curved |
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surfaces of spherical nanoparticles. Thus, as new applications of |
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ligand-stabilized nanostructures have been proposed, the structure and |
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dynamics of ligands on metallic nanoparticles have been studied using |
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molecular simulation,\cite{Henz:2008qf} NMR, XPS, FTIR, |
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calorimetry, and surface |
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microscopies.\cite{Badia1996:2,Badia1996,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{Henz:2008qf} Above 90\% coverage, the ligands |
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stand upright and recover the rigidity and tilt angle displayed on |
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planar 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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In this work, thiolated gold nanospheres were modeled using a united |
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atom force field and non-equilibrium molecular dynamics. Gold |
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nanoparticles with radii ranging from 10 - 25 \AA\ were created from a |
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bulk fcc lattice. These particles were passivated with a 50\% |
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coverage -- based on (compared with the coverage densities reported by |
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Badia \textit{et al.}) of a selection of thiolates. Three |
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straight-chain thiolates of varying chain lengths and rigidities were |
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utilized. These are summarized in Fig. \ref{fig:structures}. The |
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passivated particles were then solvated in hexane. Details on the |
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united atom force field are given below and in the supporting |
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information. |
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|
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\begin{figure} |
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\includegraphics[width=\linewidth]{figures/structures} |
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\caption{Topologies of the thiolate capping agents and solvent |
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utilized in the simulations. The chemically-distinct sites (S, \ce{CH} |
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\ce{CH2}, \ce{CH3}, and \ce{CHa}) are treated as united atoms. Most |
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parameters are taken from references \bibpunct{}{}{,}{n}{}{,} |
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\protect\cite{TraPPE-UA.alkanes} and |
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\protect\cite{TraPPE-UA.thiols}. Cross-interactions with the Au |
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atoms were adapted from references |
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\protect\cite{landman:1998},~\protect\cite{vlugt:cpc2007154},~and |
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\protect\cite{hautman:4994}.} |
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\label{fig:structures} |
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\bibpunct{[}{]}{,}{n}{}{,} |
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\end{figure} |
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|
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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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% 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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intermediate 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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% 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{Qi:1999ph} Previous |
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work\cite{Kuang:2011ef} has demonstrated that the electronic |
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contributions to heat conduction (which are missing from the QSC |
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model) across heterogeneous metal / non-metal interfaces are |
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negligible compared to phonon excitation, which is captured by the |
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classical model. The hexane solvent is described by the TraPPE united |
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atom model,\cite{TraPPE-UA.alkanes} where sites are located at the |
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carbon centers for alkyl groups. The TraPPE-UA model for hexane |
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provides both computational efficiency and reasonable accuracy for |
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bulk thermal 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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The TraPPE-UA force field includes parameters for thiol |
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molecules\cite{TraPPE-UA.thiols} as well as unsaturated and aromatic |
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carbon sites.\cite{TraPPE-UA.alkylbenzenes} These were used for the |
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thiolate molecules in our simulations, and missing parameters for the |
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ligands were supplemented using fits described in the supporting |
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information. Bonds are typically rigid in TraPPE-UA, so although |
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equilibrium bond distances were taken from TraPPE-UA, flexible bonds |
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were allowed bond stretching spring constants from the OPLS-AA force |
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field.\cite{Jorgensen:1996sf} |
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|
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To derive suitable parameters for the thiolates adsorbed on Au(111) |
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surfaces, we adopted the S parameters from Luedtke and |
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Landman\cite{landman:1998} and modified the parameters for the CTS |
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atom to maintain charge neutrality in the molecule. |
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|
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Other interactions between metal (Au) and non-metal atoms were adapted |
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from an adsorption study of alkyl thiols on gold surfaces by Vlugt, |
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\textit{et 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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All additional terms to represent thiolated alkenes and conjugated |
334 |
ligand moieties were parameterized as part of this work and are |
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available in the supporting information. |
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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 |
343 |
from a bulk fcc lattice and were thermally equilibrated prior to the |
344 |
addition of ligands. A 50\% coverage of ligands (based on coverages |
345 |
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}. We have chosen three lengths for the |
348 |
straight-chain ligands, $C_4$, $C_8$, and $C_{12}$, differentiated by |
349 |
the number of carbons in the chains. Additionally, to explore the |
350 |
effects of ligand flexibility, we have used three levels of ligand |
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``stiffness''. The most flexible chain is a fully saturated |
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alkanethiolate, while moderate rigidity is introduced using an alkene |
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thiolate with one double bond in the penultimate (solvent-facing) |
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carbon-carbon location. The most rigid ligands are fully-conjugated |
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chains where all of the carbons are represented with conjugated (aryl) |
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united-atom carbon atoms (CHar or terminal CH2ar). |
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|
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The nanoparticle / ligand complexes were thermally equilibrated to |
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allow for ligand conformational flexibility. Packmol was then used to |
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solvate the structures inside a spherical droplet of hexane. The |
361 |
thickness of the solvent layer was chosen to be at least 1.5$\times$ |
362 |
the combined radius of the nanoparticle / ligand structure. The fully |
363 |
solvated system was equilibrated for at least 1 ns using the Langevin |
364 |
Hull to apply 50 atm of pressure and a target temperature of 250 |
365 |
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 |
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sites for the 25 \AA\ particles with $C_{12}$ ligands. Figure |
368 |
\ref{fig:NP25_C12h1} shows one of the solvated 25 \AA\ nanoparticles |
369 |
passivated with the $C_{12}$ alkane thiolate ligands. |
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|
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\begin{figure} |
372 |
\includegraphics[width=\linewidth]{figures/NP25_C12h1} |
373 |
\caption{A 25 \AA\ radius gold nanoparticle protected with a |
374 |
half-monolayer of TraPPE-UA dodecanethiolate (C$_{12}$) ligands |
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and solvated in TraPPE-UA hexane. The interfacial thermal |
376 |
conductance is computed by applying a kinetic energy flux between |
377 |
the nanoparticle and an outer shell of solvent.} |
378 |
\label{fig:NP25_C12h1} |
379 |
\end{figure} |
380 |
|
381 |
Once equilibrated, thermal fluxes were applied for 1 ns, until stable |
382 |
temperature gradients had developed. Systems were run under moderate |
383 |
pressure (50 atm) with an average temperature (250K) that maintained a |
384 |
compact solvent cluster and avoided formation of a vapor layer near |
385 |
the heated metal surface. Pressure was applied to the system via the |
386 |
non-periodic Langevin Hull.\cite{Vardeman2011} However, thermal |
387 |
coupling to the external temperature bath was removed to avoid |
388 |
interference with the imposed RNEMD flux. |
389 |
|
390 |
\begin{figure} |
391 |
\includegraphics[width=\linewidth]{figures/temp_profile} |
392 |
\caption{Radial temperature profile for a 25 \AA\ radius |
393 |
particle protected with a 50\% coverage of TraPPE-UA |
394 |
butanethiolate (C$_4$) ligands and solvated in TraPPE-UA |
395 |
hexane. A kinetic energy flux is applied between RNEMD |
396 |
region A and RNEMD region B. The size of the temperature |
397 |
discontinuity at the interface is governed by the |
398 |
interfacial thermal conductance.} |
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\label{fig:temp_profile} |
400 |
\end{figure} |
401 |
|
402 |
Because the method conserves \emph{total} angular momentum and energy, |
403 |
systems which contain a metal nanoparticle embedded in a significant |
404 |
volume of solvent will still experience nanoparticle diffusion inside |
405 |
the solvent droplet. To aid in measuring an accurate temperature |
406 |
profile for these systems, a single gold atom at the origin of the |
407 |
coordinate system was assigned a mass $10,000 \times$ its original |
408 |
mass. The bonded and nonbonded interactions for this atom remain |
409 |
unchanged and the heavy atom is excluded from the RNEMD velocity |
410 |
scaling. The only effect of this gold atom is to effectively pin the |
411 |
nanoparticle at the origin of the coordinate system, thereby |
412 |
preventing translational diffusion of the nanoparticle due to Brownian |
413 |
motion. |
414 |
|
415 |
To provide statistical independence, five separate configurations were |
416 |
simulated for each particle radius and ligand. The structures were |
417 |
unique, starting at the point of ligand placement, in order to sample |
418 |
multiple surface-ligand configurations. |
419 |
|
420 |
|
421 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
422 |
% EFFECT OF PARTICLE SIZE |
423 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
424 |
\section{Results} |
425 |
|
426 |
We modeled four sizes of nanoparticles ($R =$ 10, 15, 20, and 25 |
427 |
\AA). The smallest particle size produces the lowest interfacial |
428 |
thermal conductance values for most of the of protecting groups |
429 |
(Fig. \ref{fig:NPthiols_G}). Between the other three sizes of |
430 |
nanoparticles, there is no systematic dependence of the interfacial |
431 |
thermal conductance on the nanoparticle size. It is likely that the |
432 |
differences in local curvature of the nanoparticle sizes studied here |
433 |
do not disrupt the ligand packing and behavior in drastically |
434 |
different ways. |
435 |
|
436 |
\begin{figure} |
437 |
\includegraphics[width=\linewidth]{figures/G3} |
438 |
\caption{Interfacial thermal conductance ($G$) values for 4 |
439 |
sizes of solvated nanoparticles that are bare or protected with |
440 |
a 50\% coverage of C$_{4}$, C$_{8}$, or C$_{12}$ thiolate |
441 |
ligands. Ligands of different flexibility are shown in separate |
442 |
panels. The middle panel indicates ligands which have a single |
443 |
carbon-carbon double bond in the penultimate position.} |
444 |
\label{fig:NPthiols_G} |
445 |
\end{figure} |
446 |
|
447 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
448 |
% EFFECT OF LIGAND CHAIN LENGTH |
449 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
450 |
|
451 |
Unlike our previous study of varying thiolate ligand chain lengths on |
452 |
planar Au(111) surfaces, the interfacial thermal conductance of |
453 |
ligand-protected nanospheres exhibits a distinct dependence on the |
454 |
ligand identity. A half-monolayer coverage of ligands yields |
455 |
interfacial conductance that is strongly dependent on both ligand |
456 |
length and flexibility. |
457 |
|
458 |
There are many factors that could be playing a role in the |
459 |
ligand-dependent conductuance. The sulfur-gold interaction is |
460 |
particularly strong, and the presence of the ligands can easily |
461 |
disrupt the crystalline structure of the gold at the surface of the |
462 |
particles, providing more efficient scattering of phonons into the |
463 |
ligand / solvent layer. This effect would be particularly important at |
464 |
small particle sizes. |
465 |
|
466 |
In previous studies of mixed-length ligand layers with full coverage, |
467 |
we observed that ligand-solvent alignment was an important factor for |
468 |
heat transfer into the solvent. With high surface curvature and lower |
469 |
effective coverages, ligand behavior also becomes more complex. Some |
470 |
chains may be lying down on the surface, and solvent may not be |
471 |
penetrating the ligand layer to the same degree as in the planar |
472 |
surfaces. |
473 |
|
474 |
Additionally, the ligand flexibility directly alters the vibrational |
475 |
density of states for the layer that mediates the transfer of phonons |
476 |
between the metal and the solvent. This could be a partial explanation |
477 |
for the observed differences between the fully conjugated and more |
478 |
flexible ligands. |
479 |
|
480 |
In the following sections we provide details on how we |
481 |
measure surface corrugation, solvent-ligand interpenetration, and |
482 |
ordering of the solvent and ligand at the surfaces of the |
483 |
nanospheres. We also investigate the overlap between vibrational |
484 |
densities of states for the various ligands. |
485 |
|
486 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
487 |
% CORRUGATION OF PARTICLE SURFACE |
488 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
489 |
\subsection{Corrugation of the Particle Surface} |
490 |
|
491 |
The bonding sites for thiols on gold surfaces have been studied |
492 |
extensively and include configurations beyond the traditional atop, |
493 |
bridge, and hollow sites found on planar surfaces. In particular, the |
494 |
deep potential well between the gold atoms and the thiolate sulfur |
495 |
atoms leads to insertion of the sulfur into the gold lattice and |
496 |
displacement of interfacial gold atoms. The degree of ligand-induced |
497 |
surface restructuring may have an impact on the interfacial thermal |
498 |
conductance and is an important phenomenon to quantify. |
499 |
|
500 |
Henz, \textit{et al.}\cite{Henz:2008qf} used the metal |
501 |
density as a function of radius to measure the degree of mixing |
502 |
between the thiol sulfurs and surface gold atoms at the edge of a |
503 |
nanoparticle. Although metal density is important, disruption of the |
504 |
local crystalline ordering would also have a large effect on the |
505 |
phonon spectrum in the particles. To measure this effect, we use the |
506 |
fraction of gold atoms exhibiting local fcc ordering as a function of |
507 |
radius to describe the ligand-induced disruption of the nanoparticle |
508 |
surface. |
509 |
|
510 |
The local bond orientational order can be described using the method |
511 |
of Steinhardt \textit{et al.}\cite{Steinhardt1983} The local bonding |
512 |
environment, $\bar{q}_{\ell m}$, for each atom in the system is |
513 |
determined by averaging over the spherical harmonics between that atom |
514 |
and each of its neighbors, |
515 |
\begin{equation} |
516 |
\bar{q}_{\ell m} = \sum_i Y_\ell^m(\theta_i, \phi_i) |
517 |
\end{equation} |
518 |
where $\theta_i$ and $\phi_i$ are the relative angular coordinates of |
519 |
neighbor $i$ in the laboratory frame. A global average orientational |
520 |
bond order parameter, $\bar{Q}_{\ell m}$, is the average over each |
521 |
$\bar{q}_{\ell m}$ for all atoms in the system. To remove the |
522 |
dependence on the laboratory coordinate frame, the third order |
523 |
rotationally invariant combination of $\bar{Q}_{\ell m}$, |
524 |
$\hat{w}_\ell$, is utilized here.\cite{Steinhardt1983,Vardeman:2008fk} |
525 |
|
526 |
For $\ell=4$, the ideal face-centered cubic (fcc), body-centered cubic |
527 |
(bcc), hexagonally close-packed (hcp), and simple cubic (sc) local |
528 |
structures exhibit $\hat{w}_4$ values of -0.159, 0.134, 0.159, and |
529 |
0.159, respectively. Because $\hat{w}_4$ exhibits an extreme value for |
530 |
fcc structures, it is ideal for measuring local fcc |
531 |
ordering. The spatial distribution of $\hat{w}_4$ local bond |
532 |
orientational order parameters, $p(\hat{w}_4 , r)$, can provide |
533 |
information about the location of individual atoms that are central to |
534 |
local fcc ordering. |
535 |
|
536 |
The fraction of fcc-ordered gold atoms at a given radius in the |
537 |
nanoparticle, |
538 |
\begin{equation} |
539 |
f_\mathrm{fcc}(r) = \int_{-\infty}^{w_c} p(\hat{w}_4, r) d \hat{w}_4 |
540 |
\end{equation} |
541 |
is described by the distribution of the local bond orientational order |
542 |
parameters, $p(\hat{w}_4, r)$, and $w_c$, a cutoff for the peak |
543 |
$\hat{w}_4$ value displayed by fcc structures. A $w_c$ value of -0.12 |
544 |
was chosen to isolate the fcc peak in $\hat{w}_4$. |
545 |
|
546 |
As illustrated in Figure \ref{fig:Corrugation}, the presence of |
547 |
ligands decreases the fcc ordering of the gold atoms at the |
548 |
nanoparticle surface. For the smaller nanoparticles, this disruption |
549 |
extends into the core of the nanoparticle, indicating widespread |
550 |
disruption of the lattice. |
551 |
|
552 |
\begin{figure} |
553 |
\includegraphics[width=\linewidth]{figures/fcc} |
554 |
\caption{Fraction of gold atoms with fcc ordering as a function of |
555 |
radius for a 10 \AA\ radius nanoparticle. The decreased fraction |
556 |
of fcc-ordered atoms in ligand-protected nanoparticles relative to |
557 |
bare particles indicates restructuring of the nanoparticle surface |
558 |
by the thiolate sulfur atoms.} |
559 |
\label{fig:Corrugation} |
560 |
\end{figure} |
561 |
|
562 |
We may describe the thickness of the disrupted nanoparticle surface by |
563 |
defining a corrugation factor, $c$, as the ratio of the radius at |
564 |
which the fraction of gold atoms with fcc ordering is 0.9 and the |
565 |
radius at which the fraction is 0.5. |
566 |
|
567 |
\begin{equation} |
568 |
c = 1 - \frac{r(f_\mathrm{fcc} = 0.9)}{r(f_\mathrm{fcc} = 0.5)} |
569 |
\end{equation} |
570 |
|
571 |
A sharp interface will have a steep drop in $f_\mathrm{fcc}$ at the |
572 |
edge of the particle ($c \rightarrow$ 0). In the opposite limit where |
573 |
the entire nanoparticle surface is restructured by ligands, the radius |
574 |
at which there is a high probability of fcc ordering moves |
575 |
dramatically inward ($c \rightarrow$ 1). |
576 |
|
577 |
The computed corrugation factors are shown in Figure |
578 |
\ref{fig:NPthiols_corrugation} for bare nanoparticles and for |
579 |
ligand-protected particles as a function of ligand chain length. The |
580 |
largest nanoparticles are only slightly restructured by the presence |
581 |
of ligands on the surface, while the smallest particle ($r$ = 10 \AA) |
582 |
exhibits significant disruption of the original fcc ordering when |
583 |
covered with a half-monolayer of thiol ligands. |
584 |
|
585 |
\begin{figure} |
586 |
\includegraphics[width=\linewidth]{figures/C3.pdf} |
587 |
\caption{Computed corrugation values for 4 sizes of solvated |
588 |
nanoparticles that are bare or protected with a 50\% coverage of |
589 |
C$_{4}$, C$_{8}$, or C$_{12}$ thiolate ligands. The smallest (10 |
590 |
\AA ) particles show significant disruption to their crystal |
591 |
structures, and the length and stiffness of the ligands is a |
592 |
contributing factor to the surface disruption.} |
593 |
\label{fig:NPthiols_corrugation} |
594 |
\end{figure} |
595 |
|
596 |
Because the thiolate ligands do not significantly alter the larger |
597 |
particle crystallinity, the surface corrugation does not seem to be a |
598 |
likely candidate to explain the large increase in thermal conductance |
599 |
at the interface when ligands are added. |
600 |
|
601 |
% \begin{equation} |
602 |
% C = \frac{r_{bare}(\rho_{\scriptscriptstyle{0.85}}) - r_{capped}(\rho_{\scriptscriptstyle{0.85}})}{r_{bare}(\rho_{\scriptscriptstyle{0.85}})}. |
603 |
% \end{equation} |
604 |
% |
605 |
% 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. |
606 |
|
607 |
|
608 |
|
609 |
|
610 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
611 |
% MOBILITY OF INTERFACIAL SOLVENT |
612 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
613 |
% \subsection{Mobility of Interfacial Solvent} |
614 |
|
615 |
% Another possible mechanism for increasing interfacial conductance is |
616 |
% the mobility of the interfacial solvent. We used a survival |
617 |
% correlation function, $C(t)$, to measure the residence time of a |
618 |
% solvent molecule in the nanoparticle thiolate |
619 |
% layer.\cite{Stocker:2013cl} This function correlates the identity of |
620 |
% all hexane molecules within the radial range of the thiolate layer at |
621 |
% two separate times. If the solvent molecule is present at both times, |
622 |
% the configuration contributes a $1$, while the absence of the molecule |
623 |
% at the later time indicates that the solvent molecule has migrated |
624 |
% into the bulk, and this configuration contributes a $0$. A steep decay |
625 |
% in $C(t)$ indicates a high turnover rate of solvent molecules from the |
626 |
% chain region to the bulk. We may define the escape rate for trapped |
627 |
% solvent molecules at the interface as |
628 |
% \begin{equation} |
629 |
% k_\mathrm{escape} = \left( \int_0^T C(t) dt \right)^{-1} |
630 |
% \label{eq:mobility} |
631 |
% \end{equation} |
632 |
% where T is the length of the simulation. This is a direct measure of |
633 |
% the rate at which solvent molecules initially entangled in the |
634 |
% thiolate layer can escape into the bulk. When $k_\mathrm{escape} |
635 |
% \rightarrow 0$, the solvent becomes permanently trapped in the |
636 |
% interfacial region. |
637 |
|
638 |
% The solvent escape rates for bare and ligand-protected nanoparticles |
639 |
% are shown in Figure \ref{fig:NPthiols_combo}. As the ligand chain |
640 |
% becomes longer and more flexible, interfacial solvent molecules become |
641 |
% trapped in the ligand layer and the solvent escape rate decreases. |
642 |
% This mechanism contributes a partial explanation as to why the longer |
643 |
% ligands have significantly lower thermal conductance. |
644 |
|
645 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
646 |
% ORIENTATION OF LIGAND CHAINS |
647 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
648 |
\subsection{Orientation of Ligand Chains} |
649 |
|
650 |
As the saturated ligand chain length increases in length, it exhibits |
651 |
significantly more conformational flexibility. Thus, different lengths |
652 |
of ligands should favor different chain orientations on the surface of |
653 |
the nanoparticle. To determine the distribution of ligand orientations |
654 |
relative to the particle surface we examine the probability of finding |
655 |
a ligand with a particular orientation relative to the surface normal |
656 |
of the nanoparticle, |
657 |
\begin{equation} |
658 |
\cos{(\theta)}=\frac{\vec{r}_i\cdot\hat{u}_i}{|\vec{r}_i||\hat{u}_i|} |
659 |
\end{equation} |
660 |
where $\vec{r}_{i}$ is the vector between the cluster center of mass |
661 |
and the sulfur atom on ligand molecule {\it i}, and $\hat{u}_{i}$ is |
662 |
the vector between the sulfur atom and \ce{CH3} pseudo-atom on ligand |
663 |
molecule {\it i}. As depicted in Figure \ref{fig:NP_pAngle}, $\theta |
664 |
\rightarrow 180^{\circ}$ for a ligand chain standing upright on the |
665 |
particle ($\cos{(\theta)} \rightarrow -1$) and $\theta \rightarrow |
666 |
90^{\circ}$ for a ligand chain lying down on the surface |
667 |
($\cos{(\theta)} \rightarrow 0$). As the thiolate alkane chain |
668 |
increases in length and becomes more flexible, the ligands are more |
669 |
willing to lie down on the nanoparticle surface and exhibit increased |
670 |
population at $\cos{(\theta)} = 0$. |
671 |
|
672 |
\begin{figure} |
673 |
\includegraphics[width=\linewidth]{figures/NP_pAngle} |
674 |
\caption{The two extreme cases of ligand orientation relative to the |
675 |
nanoparticle surface: the ligand completely outstretched |
676 |
($\cos{(\theta)} = -1$) and the ligand fully lying down on the |
677 |
particle surface ($\cos{(\theta)} = 0$).} |
678 |
\label{fig:NP_pAngle} |
679 |
\end{figure} |
680 |
|
681 |
An order parameter describing the average ligand chain orientation relative to |
682 |
the nanoparticle surface is available using the second order Legendre |
683 |
parameter, |
684 |
\begin{equation} |
685 |
P_2 = \left< \frac{1}{2} \left(3\cos^2(\theta) - 1 \right) \right> |
686 |
\end{equation} |
687 |
|
688 |
Ligand populations that are perpendicular to the particle surface have |
689 |
$P_2$ values of 1, while ligand populations lying flat on the |
690 |
nanoparticle surface have $P_2$ values of $-0.5$. Disordered ligand |
691 |
layers will exhibit mean $P_2$ values of 0. As shown in Figure |
692 |
\ref{fig:NPthiols_P2} the ligand $P_2$ values approaches 0 as |
693 |
ligand chain length -- and ligand flexibility -- increases. |
694 |
|
695 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
696 |
% ORIENTATION OF INTERFACIAL SOLVENT |
697 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
698 |
\subsection{Orientation of Interfacial Solvent} |
699 |
|
700 |
Similarly, we examined the distribution of \emph{hexane} molecule |
701 |
orientations relative to the particle surface using the same angular |
702 |
analysis utilized for the ligand chain orientations. In this case, |
703 |
$\vec{r}_i$ is the vector between the particle center of mass and one |
704 |
of the \ce{CH2} pseudo-atoms in the middle of hexane molecule $i$ and |
705 |
$\hat{u}_i$ is the vector between the two \ce{CH3} pseudo-atoms on |
706 |
molecule $i$. Since we are only interested in the orientation of |
707 |
solvent molecules near the ligand layer, we select only the hexane |
708 |
molecules within a specific $r$-range, between the edge of the |
709 |
particle and the end of the ligand chains. A large population of |
710 |
hexane molecules with $\cos{(\theta)} \sim \pm 1$ would indicate |
711 |
interdigitation of the solvent molecules between the upright ligand |
712 |
chains. A more random distribution of $\cos{(\theta)}$ values |
713 |
indicates a disordered arrangement of solvent molecules near the particle |
714 |
surface. Again, $P_2$ order parameter values provide a population |
715 |
analysis for the solvent that is close to the particle surface. |
716 |
|
717 |
The average orientation of the interfacial solvent molecules is |
718 |
notably flat on the \emph{bare} nanoparticle surfaces. This blanket of |
719 |
hexane molecules on the particle surface may act as an insulating |
720 |
layer, increasing the interfacial thermal resistance. As the length |
721 |
(and flexibility) of the ligand increases, the average interfacial |
722 |
solvent P$_2$ value approaches 0, indicating a more random orientation |
723 |
of the ligand chains. The average orientation of solvent within the |
724 |
$C_8$ and $C_{12}$ ligand layers is essentially random. Solvent |
725 |
molecules in the interfacial region of $C_4$ ligand-protected |
726 |
nanoparticles do not lie as flat on the surface as in the case of the |
727 |
bare particles, but are not as randomly oriented as the longer ligand |
728 |
lengths. |
729 |
|
730 |
\begin{figure} |
731 |
\includegraphics[width=\linewidth]{figures/P2_3.pdf} |
732 |
\caption{Computed ligand and interfacial solvent orientational $P_2$ |
733 |
values for 4 sizes of solvated nanoparticles that are bare or |
734 |
protected with a 50\% coverage of C$_{4}$, C$_{8}$, or C$_{12}$ |
735 |
alkanethiolate ligands. Increasing stiffness of the ligand orients |
736 |
these molecules normal to the particle surface, while the length |
737 |
of the ligand chains works to prevent solvent from lying flat on |
738 |
the surface.} |
739 |
\label{fig:NPthiols_P2} |
740 |
\end{figure} |
741 |
|
742 |
These results are particularly interesting in light of our previous |
743 |
results\cite{Stocker:2013cl}, where solvent molecules readily filled |
744 |
the vertical gaps between neighboring ligand chains and there was a |
745 |
strong correlation between ligand and solvent molecular |
746 |
orientations. It appears that the introduction of surface curvature |
747 |
and a lower ligand packing density creates a disordered ligand layer |
748 |
that lacks well-formed channels for the solvent molecules to occupy. |
749 |
|
750 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
751 |
% SOLVENT PENETRATION OF LIGAND LAYER |
752 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
753 |
\subsection{Solvent Penetration of Ligand Layer} |
754 |
|
755 |
The extent of ligand -- solvent interaction is also determined by the |
756 |
degree to which these components occupy the same region of space |
757 |
adjacent to the nanoparticle. The radial density profiles of these |
758 |
components help determine this degree of interaction. Figure |
759 |
\ref{fig:density} shows representative density profiles for solvated |
760 |
25 \AA\ radius nanoparticles with no ligands, and with a 50\% coverage |
761 |
of C$_{4}$, C$_{8}$, and C$_{12}$ thiolates. |
762 |
|
763 |
\begin{figure} |
764 |
\includegraphics[width=\linewidth]{figures/density} |
765 |
\caption{Radial density profiles for 25 \AA\ radius nanoparticles |
766 |
with no ligands (circles), C$_{4}$ ligands (squares), C$_{8}$ |
767 |
ligands (diamonds), and C$_{12}$ ligands (triangles). Ligand |
768 |
density is indicated with filled symbols, solvent (hexane) density |
769 |
is indicated with open symbols. As ligand chain length increases, |
770 |
the nearby solvent is excluded from the ligand layer. The |
771 |
conjugated ligands (upper panel) can create a separated solvent |
772 |
shell within the ligand layer and also allow significantly more |
773 |
solvent to penetrate close to the particle.} |
774 |
\label{fig:density} |
775 |
\end{figure} |
776 |
|
777 |
The differences between the radii at which the hexane surrounding the |
778 |
ligand-covered particles reaches bulk density correspond nearly |
779 |
exactly to the differences between the lengths of the ligand |
780 |
chains. Beyond the edge of the ligand layer, the solvent reaches its |
781 |
bulk density within a few angstroms. The differing shapes of the |
782 |
density curves indicate that the solvent is increasingly excluded from |
783 |
the ligand layer as the chain length increases. |
784 |
|
785 |
The conjugated ligands create a distinct solvent shell within the |
786 |
ligand layer and also allow significantly more solvent to penetrate |
787 |
close to the particle. We define a density overlap parameter, |
788 |
\begin{equation} |
789 |
O_{l-s} = \frac{1}{V} \int_0^{r_\mathrm{max}} 4 \pi r^2 \frac{4 \rho_l(r) \rho_s(r)}{\left(\rho_l(r) + |
790 |
\rho_s(r)\right)^2} dr |
791 |
\end{equation} |
792 |
where $\rho_l(r)$ and $\rho_s(r)$ are the ligand and solvent densities |
793 |
at a radius $r$, and $V$ is the total integration volume |
794 |
($V = 4\pi r_\mathrm{max}^3 / 3$). The fraction in the integrand is a |
795 |
dimensionless quantity that is unity when ligand and solvent densities |
796 |
are identical at radius $r$, but falls to zero when either of the two |
797 |
components are excluded from that region. |
798 |
|
799 |
\begin{figure} |
800 |
\includegraphics[width=\linewidth]{figures/rho3} |
801 |
\caption{Density overlap parameters ($O_{l-s}$) for solvated |
802 |
nanoparticles protected by thiolate ligands. In general, the |
803 |
rigidity of the fully-conjugated ligands provides the easiest |
804 |
route for solvent to enter the interfacial region. Additionally, |
805 |
shorter chains allow a greater degree of solvent penetration of |
806 |
the ligand layer.} |
807 |
\label{fig:rho3} |
808 |
\end{figure} |
809 |
|
810 |
The density overlap parameters are shown in Fig. \ref{fig:rho3}. The |
811 |
calculated overlap parameters indicate that the conjugated ligand |
812 |
allows for the most solvent penetration close to the particle, and |
813 |
that shorter chains generally permit greater solvent penetration in |
814 |
the interfacial region. Increasing overlap can certainly allow for |
815 |
enhanced thermal transport, but this is clearly not the only |
816 |
contributing factor. Even when the solvent and ligand are in close |
817 |
physical contact, there must also be good vibrational overlap between |
818 |
the phonon densities of states in the ligand and solvent to transmit |
819 |
vibrational energy between the two materials. |
820 |
|
821 |
\subsection{Ligand-mediated Vibrational Overlap} |
822 |
|
823 |
In phonon scattering models for interfacial thermal |
824 |
conductance,\cite{Swartz:1989uq,Young:1989xy,Cahill:2003fk,Reddy:2005fk,Schmidt:2010nr} |
825 |
the frequency-dependent transmission probability |
826 |
($t_{a \rightarrow b}(\omega)$) predicts phonon transfer between |
827 |
materials $a$ and $b$. Many of the models for interfacial phonon |
828 |
transmission estimate this quantity using the phonon density of states |
829 |
and group velocity, and make use of a Debye model for the density of |
830 |
states in the solid. |
831 |
|
832 |
A consensus picture is that in order to transfer the energy carried by |
833 |
an incoming phonon of frequency $\omega$ on the $a$ side, the phonon |
834 |
density of states on the $b$ side must have a phonon of the same |
835 |
frequency. The overlap of the phonon densities of states, particularly |
836 |
at low frequencies, therefore contributes to the transfer of heat. |
837 |
Phonon scattering must also be done in a direction perpendicular to |
838 |
the interface. In the geometries described here, there are two |
839 |
interfaces (particle $\rightarrow$ ligand, and ligand $\rightarrow$ |
840 |
solvent), and the vibrational overlap between the ligand and the other |
841 |
two components is going to be relevant to heat transfer. |
842 |
|
843 |
To estimate the relevant densities of states, we have projected the |
844 |
velocity of each atom $i$ in the region of the interface onto a |
845 |
direction normal to the interface. For the nanosphere geometries |
846 |
studied here, the normal direction depends on the instantaneous |
847 |
positon of the atom relative to the center of mass of the particle. |
848 |
\begin{equation} |
849 |
v_\perp(t) = \mathbf{v}(t) \cdot \frac{\mathbf{r}(t)}{\left|\mathbf{r}(t)\right|} |
850 |
\end{equation} |
851 |
The quantity $v_\perp(t)$ measures the instantaneous velocity of an |
852 |
atom in a direction perpendicular to the nanoparticle interface. In |
853 |
the interfacial region, the autocorrelation function of these |
854 |
velocities, |
855 |
\begin{equation} |
856 |
C_\perp(t) = \left< v_\perp(t) \cdot v_\perp(0) \right>, |
857 |
\end{equation} |
858 |
will include contributions from all of the phonon modes present at the |
859 |
interface. The Fourier transform of the time-symmetrized |
860 |
autocorrelation function provides an estimate of the vibrational |
861 |
density of states,\cite{Shin:2010sf} |
862 |
\begin{equation} |
863 |
\rho(\omega) = \frac{1}{\tau} \int_{-\tau/2}^{\tau/2} C_\perp(t) e^{-i |
864 |
\omega t} dt. |
865 |
\end{equation} |
866 |
Here $\tau$ is the total observation time for the autocorrelation |
867 |
function. In Fig.~\ref{fig:vdos} we show the low-frequency region of |
868 |
the normalized vibrational densities of states for the three chemical |
869 |
components (gold nanoparticle, C$_{12}$ ligands, and interfacial |
870 |
solvent). The double bond in the penultimate location is a small |
871 |
perturbation on ligands of this size, and that is reflected in |
872 |
relatively similar spectra in the lower panels. The fully conjugated |
873 |
ligand, however, pushes the peak in the lowest frequency band from |
874 |
$\sim 29 \mathrm{cm}^{-1}$ to $\sim 55 \mathrm{cm}^{-1}$, yielding |
875 |
significant overlap with the density of states in the nanoparticle. |
876 |
This ligand also increases the overlap with the solvent density of |
877 |
states in a band between 280 and 380 $\mathrm{cm}^{-1}$. This |
878 |
provides some physical basis for the high interfacial conductance |
879 |
observed for the fully conjugated $C_8$ and $C_{12}$ ligands. |
880 |
|
881 |
\begin{figure} |
882 |
\includegraphics[width=\linewidth]{figures/rho_omega_12} |
883 |
\caption{The low frequency portion of the vibrational density of |
884 |
states for three chemical components (gold nanoparticles, C$_{12}$ |
885 |
ligands, and hexane solvent). These densities of states were |
886 |
computed using the velocity autocorrelation functions for atoms in |
887 |
the interfacial region, constructed with velocities projected onto |
888 |
a direction normal to the interface.} |
889 |
\label{fig:vdos} |
890 |
\end{figure} |
891 |
|
892 |
The similarity between the density of states for the alkanethiolate |
893 |
and penultimate ligands also helps explain why the interfacial |
894 |
conductance is nearly the same for these two ligands, particularly at |
895 |
longer chain lengths. |
896 |
|
897 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
898 |
% DISCUSSION |
899 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
900 |
\section{Discussion} |
901 |
|
902 |
The chemical bond between the metal and the ligand introduces |
903 |
vibrational overlap that is not present between the bare metal surface |
904 |
and solvent. Thus, regardless of ligand identity or chain length, the |
905 |
presence of a half-monolayer ligand coverage yields a higher |
906 |
interfacial thermal conductance value than the bare nanoparticle. The |
907 |
mechanism for the varying conductance for the different ligands is |
908 |
somewhat less clear. Ligand-based alterations to vibrational density |
909 |
of states is a major contributor, but some of the ligands can disrupt |
910 |
the crystalline structure of the smaller nanospheres, while others can |
911 |
re-order the interfacial solvent and alter the interpenetration |
912 |
profile between ligand and solvent chains. Further work to separate |
913 |
the effects of ligand-solvent interpenetration and surface |
914 |
reconstruction is clearly needed for a complete picture of the heat |
915 |
transport in these systems. |
916 |
|
917 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
918 |
% **ACKNOWLEDGMENTS** |
919 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
920 |
\begin{acknowledgments} |
921 |
Support for this project was provided by the National Science Foundation |
922 |
under grant CHE-1362211. Computational time was provided by the |
923 |
Center for Research Computing (CRC) at the University of Notre Dame. |
924 |
\end{acknowledgments} |
925 |
|
926 |
\newpage |
927 |
\bibliographystyle{aip} |
928 |
\bibliography{NPthiols} |
929 |
|
930 |
\end{document} |