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\documentclass[11pt]{article} |
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\usepackage{amsmath} |
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\usepackage{amssymb} |
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\usepackage{times} |
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\usepackage{mathptm} |
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\usepackage{setspace} |
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\usepackage{endfloat} |
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\usepackage{caption} |
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\begin{document} |
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\title{The role of chain length and solvent penetration in the |
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interfacial thermal conductance of thiolate-capped gold surfaces} |
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\title{Simulations of heat conduction at thiolate-capped gold |
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surfaces: The role of chain length and solvent penetration} |
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\author{Kelsey M. Stocker and J. Daniel |
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Gezelter\footnote{Corresponding author. \ Electronic mail: |
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particles and their surroundings. Understanding this energy flow is |
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essential in designing and functionalizing metallic nanoparticles for |
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plasmonic photothermal |
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therapies,\cite{Jain:2007ux,Petrova:2007ad,Gnyawali:2008lp,Mazzaglia:2008to,Huff |
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:2007ye,Larson:2007hw} which rely on the ability of metallic |
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therapies,\cite{Jain:2007ux,Petrova:2007ad,Gnyawali:2008lp,Mazzaglia:2008to,Huff:2007ye,Larson:2007hw} which rely on the ability of metallic |
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nanoparticles to absorb light in the near-IR, a portion of the |
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spectrum in which living tissue is very nearly transparent. The |
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principle of this therapy is to pump the particles at high power at |
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nanoparticle/fluid interfaces, to epitaxial TiN/single crystal oxides |
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interfaces, and hydrophilic and hydrophobic interfaces between water |
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and solids with different self-assembled |
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monolayers.\cite{cahill:793,Wilson:2002uq,PhysRevB.67.054302,doi:10.1021/jp048375k,PhysRevL |
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ett.96.186101} |
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monolayers.\cite{cahill:793,Wilson:2002uq,PhysRevB.67.054302,doi:10.1021/jp048375k,PhysRevLett.96.186101} |
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Wang {\it et al.} studied heat transport through long-chain |
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hydrocarbon monolayers on gold substrate at the individual molecular |
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level,\cite{Wang10082007} Schmidt {\it et al.} studied the role of |
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A temperature profile of the system was created by dividing the box into $\sim$ 3 \AA \, bins along the z axis and recording the average temperature of each bin. |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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% FORCE-FIELD PARAMETERS |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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\subsection{Force-Field Parameters} |
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\begin{figure} |
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\includegraphics[width=\linewidth]{figures/structures} |
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\caption{STRUCTURES} |
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\label{fig:structures} |
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\end{figure} |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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% FORCE-FIELD PARAMETERS |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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\subsection{Force-Field Parameters} |
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The gold-gold interactions are modeled using the quantum Sutton-Chen (QSC) force field.\cite{Goddard1998} |
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The TraPPE-UA parameters are used for the united atom n-hexane solvent molecules. Intramolecular bends, torsions, and bond stretching are applied to intramolecular sites that are within three bonds. Intermolecular interactions are modeled by a Lennard-Jones potential. |
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Our simulations include a number of chemically distinct components. |
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Figure \ref{fig:structures} demonstrates the sites defined for both |
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the {\it n}-hexane and alkanethiolate ligands present in our |
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simulations. Force field parameters are needed for interactions both |
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between the same type of particles and between particles of different |
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species. |
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\begin{figure} |
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\includegraphics[width=\linewidth]{figures/structures} |
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\caption{STRUCTURES} |
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\label{fig:structures} |
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\end{figure} |
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The Au-Au interactions in metal lattice slab is described by the |
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quantum Sutton-Chen (QSC) formulation.\cite{PhysRevB.59.3527} The QSC |
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potentials include zero-point quantum corrections and are |
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reparametrized for accurate surface energies compared to the |
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Sutton-Chen potentials.\cite{Chen90} |
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|
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For the {\it n}-hexane solvent molecules, the TraPPE-UA |
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parameters\cite{TraPPE-UA.alkanes} were utilized. In this model, |
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sites are located at the carbon centers for alkyl groups. Bonding |
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interactions, including bond stretches and bends and torsions, were |
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used for intra-molecular sites closer than 3 bonds. For non-bonded |
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interactions, Lennard-Jones potentials are used. We have previously |
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utilized both united atom (UA) and all-atom (AA) force fields for |
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thermal conductivity in previous work,\cite{} and since the united |
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atom force fields cannot populate the high-frequency modes that are |
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present in AA force fields, they appear to work better for modeling |
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thermal conductivity. The TraPPE-UA model for alkanes is known to |
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predict a slightly lower boiling point than experimental values. This |
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is one of the reasons we used a lower average temperature (200K) for |
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our simulations. |
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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} which were used for the |
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alkanethiolate molecules in our simulations. To derive suitable |
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parameters for butanethiol adsorbed on Au(111) surfaces, we adopted |
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the S parameters from Luedtke and Landman\cite{landman:1998} and |
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modified the parameters for the CTS atom to maintain charge neutrality |
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in the molecule. |
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|
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To describe the interactions between metal (Au) and non-metal atoms, |
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we refer to an adsorption study of alkyl thiols on gold surfaces by |
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Vlugt {\it et al.}\cite{vlugt:cpc2007154} They fitted an effective |
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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} As our simulations require the gold slab |
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to be flexible to accommodate thermal excitation, the pair-wise form |
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of potentials they developed was used for our study. Table |
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\ref{table:pars} in the supporting information summarizes the |
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``metal/non-metal'' parameters utilized in our simulations. |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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% **RESULTS** |
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\section{Results} |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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% CHAIN LENGTH |
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\subsection{Effect of Chain Length} |
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% CHAIN LENGTH |
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\subsection{Effect of Chain Length} |
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We examined full coverages of five chain lengths, n = 4, 6, 8, 10, 12. In all cases, the hexane solvent was unable to penetrate into the thiolate layer, leading to a persistent 2-4 \AA \, gap between the solvent region and the thiolates. The trend of interfacial conductance is mostly flat as a function of chain length, indicating that the length of the thiolate alkyl chains does not play a significant role in the transport of heat across the gold/thiolate and thiolate/solvent interfaces. There is, however, a peak in conductance for a chain length of 6 (hexanethiolate). This may be due to the equivalent chain lengths of the hexane solvent and the alkyl chain of the capping agent, leading to an especially high degree of vibrational overlap between the thiolate and solvent. Strong vibrational overlap would allow for efficient thermal energy transfer across the thiolate/solvent interface. |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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% MIXED CHAINS |
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\subsection{Effect of Mixed Chain Lengths} |
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% MIXED CHAINS |
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\subsection{Effect of Mixed Chain Lengths} |
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Previous work demonstrated that for butanethiolate monolayers on a Au(111) surface, the interfacial conductance was a non-monotonic function of the percent coverage. This is believed to be due to enhanced solvent-thiolate coupling through greater penetration of solvent molecules into the thiolate layer. At lower coverages, hexane solvent can more easily line up lengthwise with the thiolate tails by fitting into gaps between the thiolates. However, a side effect of low coverages is surface aggregation of the thiolates. To simulate the effect of low coverages while preventing aggregation we maintain 100\% thiolate coverage while varying the proportions of short (butanethiolate, n = 4) and long (decanethiolate, n = 10 or dodecanethiolate, n = 12). In systems where there is a mix of short and long chain thiolates, interfacial conductance is a non-monotonic function of the percent of long chains. The depth of the gaps between the long chains is $n_{long} - n_{short}$, which has implications for the ability of the hexane solvent to fill in the gaps between the long chains. |
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\subsubsection{Butanethiolate/Decanethiolate} |
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\section{Discussion} |
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% RESIDENCE TIME |
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\subsection{Solvent Molecule Residence Time} |
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We use a selection correlation function to quantify the residence time of a solvent molecule in the long thiolate chain layer. This function compares the identity of all hexane molecules within the z-coordinate range of the thiolate layer at each timestep to the identities of solvent molecules in that range at time zero. A steep decay in the correlation function indicates a high turnover rate of solvent molecules within the thiolate chains. We use a biexponential fit |
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In the mixed chain-length simulations, solvent molecules can become |
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temporarily trapped or entangled with the thiolate chains. Their |
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residence in close proximity to the higher temperature environment |
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close to the surface allows them to carry heat away from the surface |
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quite efficiently. There are two aspects of this behavior that are |
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relevant to thermal conductance of the interface: the residence time |
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of solvent molecules in the thiolate layer, and the orientational |
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ordering of the C-C chains as a mechanism for transferring vibrational |
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energy to these entrapped solvent molecules. |
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|
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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% RESIDENCE TIME |
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\subsection{Residence times for solvent in the interfacial layer} |
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We use a simple survival correlation function, $C(t)$, to quantify the |
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residence time of a solvent molecule in the long thiolate chain |
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layer. This function correlates the identity of all hexane molecules |
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within the $z$-coordinate range of the thiolate layer at two separate |
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times. If the solvent molecule is present at both times, the |
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configuration contributes a $1$, while the absence of the molecule at |
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the later time indicates that the solvent molecule has migrated into |
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the bulk, and this configuration contributes a $0$. A steep decay in |
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$C(t)$ indicates a high turnover rate of solvent molecules from the |
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chain region to the bulk. The correlation function is easily fit |
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using a biexponential, |
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\begin{equation} |
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N_{short} \, e^{-t/\tau_{short}} + N_{long} \, e^{-t/\tau_{long}} |
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C(t) = A \, e^{-t/\tau_{short}} + (1-A) e^{-t/\tau_{long}} |
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\label{eq:biexponential} |
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\end{equation} |
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\begin{equation} |
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N_{short} + N_{long} = 1 |
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\label{eq:biexponential2} |
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\end{equation} |
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to determine a short and long residence timescale and their relative populations for solvent molecules within the thiolate alkyl chain region. |
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to determine short and long residence timescales and the relative |
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populations of solvent molecules that can escape rapidly. In table |
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\ref{table:res} we show that the timescales... |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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% ORDER PARAMETER |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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\subsection{Orientational Order Parameter} |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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% ORDER PARAMETER |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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\subsection{Vibrational coupling via orientational ordering} |
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As the fraction of long-chain thiolates becomes large, the entrapped |
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solvent molecules must find specific orientations relative to the mean |
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orientation of the thiolate chains. This configuration allows for |
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efficient thermal energy exchange between the thiolate alkyl chain and |
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the solvent molecules. |
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To quantify this cooperative |
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ordering, we computed the orientational order parameters and director |
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axes for both the thiolate chains and for the entrapped solvent. The |
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director axis can be easily obtained by diagonalization of the order |
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parameter tensor, |
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\begin{equation} |
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\mathsf{Q}_{\alpha \beta} = \frac{1}{2 N} \sum_{i=1}^{N} \left( 3 \mathbf{e}_{i |
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\alpha} \mathbf{e}_{i \beta} - \delta_{\alpha \beta} \right) |
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\end{equation} |
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where $\mathbf{e}_{i \alpha}$ was the $\alpha = x,y,z$ component of |
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the unit vector $\mathbf{e}_{i}$ along the long axis of molecule $i$. |
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For both kinds of molecules, the $\mathbf{e}$ vector is defined using |
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the terminal atoms of the chains. |
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The largest eigenvalue of $\overleftrightarrow{\mathsf{Q}}$ is |
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traditionally used to obtain orientational order parameter, while the |
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eigenvector corresponding to the order parameter yields the director |
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axis ($\mathbf{d}(t)$) which defines the average direction of |
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molecular alignment at any time $t$. The overlap between the director |
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axes of the thiolates and the entrapped solvent is time-averaged, |
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\begin{equation} |
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\left \langle \vec{d}_{thiolates} \left( t \right) \cdot \vec{d}_{solvent} \left( t \right) \right \rangle |
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\left \langle \mathbf{d}_{thiolates} \left( t \right) \cdot |
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\mathbf{d}_{solvent} \left( t \right) \right \rangle, |
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\label{eq:orientation} |
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\end{equation} |
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This configuration allows for efficient thermal energy exchange between the thiolate alkyl chain and the solvent molecules. Once the solvent molecules have picked up thermal energy from the thiolates, they carry heat away from the gold as they diffuse back into the bulk solvent. When the percentage of long chains decreases, the tails of the long chains are much more disordered and do not provide structured channels for the solvent to fill. |
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and reported in table \ref{table:ordering}. |
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Once the solvent molecules have picked up thermal energy from the |
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thiolates, they carry heat away from the gold as they diffuse back |
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into the bulk solvent. When the percentage of long chains decreases, |
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the tails of the long chains are much more disordered and do not |
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provide structured channels for the solvent to fill. |
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Although the alignment of the chains with the entrapped solvent is one |
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possible mechanism for the non-monotonic increase in the conductance |
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as a function |
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|
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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% **ACKNOWLEDGMENTS** |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
