trunk/chainLength/GoldThiolsPaper.tex

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\begin{document}

\title{Interfacial Thermal Conductance of Thiolate-Capped Gold}

\author{Kelsey M. Stocker and J. Daniel Gezelter\footnote{Corresponding author. \ Electronic mail: gezelter@nd.edu} \\
        Department of Chemistry and Biochemistry,\\
        University of Notre Dame\\
        Notre Dame, Indiana 46556}

\date{\today}

\maketitle 

\begin{doublespace}

\begin{abstract}
        ABSTRACT
\end{abstract}

\newpage

%\narrowtext

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%                          **INTRODUCTION**
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Introduction}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%                          **METHODOLOGY**
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Methodology}

        %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
        %                           VSS-RNEMD
        %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
        \subsection{VSS-RNEMD}
        
        \begin{figure}
                \includegraphics[width=\linewidth]{figures/rnemd}
                \caption{VSS-RNEMD}
                \label{fig:rnemd}
        \end{figure}
        
        \begin{equation}
                \boldsymbol{\nu}_{i} \leftarrow c \cdot \left( \boldsymbol{\nu}_{i} - \langle \boldsymbol{\nu}_{c} \rangle \right) + \left( \langle \boldsymbol{\nu}_{c} \rangle + \bold{a}_{c} \right),
        \end{equation}
        
        \begin{equation}
                \boldsymbol{\nu}_{j} \leftarrow h \cdot \left( \boldsymbol{\nu}_{j} - \langle \boldsymbol{\nu}_{h} \rangle \right) + \left( \langle \boldsymbol{\nu}_{h} \rangle + \bold{a}_{h} \right),
        \end{equation}
        
        \begin{equation}
                K_{c} - J_{z} \Delta t = c^{2} \left( K_{c} - \frac{1}{2} M_{c} \langle \boldsymbol{\nu}_{c} \rangle ^{2}  \right) + \frac{1}{2} M_{c} \left( \langle \boldsymbol{\nu}_{c} \rangle + \bold{a}_{c} \right) ^{2},
        \end{equation}
        
        \begin{equation}
                K_{h} + J_{z} \Delta t = h^{2} \left( K_{h} - \frac{1}{2} M_{h} \langle \boldsymbol{\nu}_{h} \rangle ^{2}  \right) + \frac{1}{2} M_{h} \left( \langle \boldsymbol{\nu}_{h} \rangle + \bold{a}_{h} \right) ^{2},
        \end{equation}
                
        
        %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
        %                          INTERFACIAL CONDUCTANCE
        %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
        \subsection{Defining Interfacial Conductance (G)}
        
        \begin{figure}
                \includegraphics[width=\linewidth]{figures/resistor_series}
                \caption{RESISTOR SERIES}
                \label{fig:resistor_series}
        \end{figure}
        
        
There are two interfaces involved in the transfer of heat from the gold slab to the solvent: the gold/thiolate interface and the thiolate/solvent interface. We can treat the temperature on each side of an interface as discrete, making the interfacial conductance the inverse of the Kaptiza resistance, or $G = \frac{J}{\Delta T}$. To model the total conductance across multiple interfaces, we treat the interfaces as resistors in series. Resistors in series are additive, ($R_{total} = R_{1} + R_{2} + R_{3} + ...$) and the total conductance is the inverse of the total resistance, or $G = \frac{1}{(R_{1} + R_{2} + R_{3} + ...}$). We treat each bin in the VSS-RNEMD temperature profile as a resistor with resistance $\frac{T_{2}-T_{1}}{J}$, $\frac{T_{3}-T_{2}}{J}$, etc. The sum of this resistor series which spans the gold/thiolate interface, thiolate chains, and thiolate/solvent interface simplifies to 

\begin{equation}
        \frac{T_{n}-T_{1}}{J},
        \label{eq:finalG}
        \end{equation} 

or the temperature difference between the gold side of the gold/thiolate interface and the solvent side of the thiolate/solvent interface over the applied flux. 
        
        
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%                          **COMPUTATIONAL DETAILS**
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
\section{Computational Details}

        %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
        %                          SIMULATION PROTOCOL
        %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
        \subsection{Simulation Protocol}
        
We have implemented the VSS-RNEMD algorithm in OpenMD, our group molecular dynamics code. A gold slab 11 layers thick was equilibrated at 1 atm and 200 K. The periodic box was expanded in the z direction to expose two Au(111) faces. 

A full monolayer of thiolates (1/3 the number of surface gold atoms) was placed on three-fold hollow sites on the gold interfaces. To efficiently test the effect of thiolate binding sites on the thermal conductance, all systems had one gold interface with thiolates placed only on fcc hollow sites and the other interface with thiolates only on hcp hollow sites. To test the effect of thiolate chain length on interfacial thermal conductance, full coverages of five chain lengths were tested: butanethiolate, hexanethiolate, octanethiolate, decanethiolate, and dodecanethiolate. To test the effect of mixed chain lengths, full coverages of butanethiolate/decanethiolate and butanethiolate/dodecanethiolate mixtures were created in short/long chain ratios of 25/75, 50/50, and 75/25. The short and long chains were placed on the surface in a random configuration.

The simulation box z dimension was set to roughly double the length of the gold/thiolate block. Solvent molecules were placed in the vacant portion of the box using the packmol algorithm. Two solvent molecules were examined: hexane and toluene. Hexane, a straight chain flexible alkane, is very structurally similar to the thiolate alkane tails while toluene is a rigid planar molecule.

The system was equilibrated to 220 K in the NVT ensemble, allowing the thiolates and solvent to warm gradually. Pressure correction to 1 atm was done in an NPT ensemble (NPAT) that allowed expansion or contraction only in the z direction, so as not to disrupt the crystalline structure of the gold. The diagonal elements of the pressure tensor were monitored during the pressure correction step. The zz element was successfully equilibrated during the NPAT simulation. If the xx and/or yy elements had a mean above zero throughout the simulation -- indicating residual pressure in the plane of the gold slab -- an additional short NPT equilibration step was performed allowing all box dimensions to change. Once the pressure was stable at 1 atm, a final NVT simulation was performed.

A kinetic energy flux was imposed using VSS-RNEMD in the microcanonical (NVE) ensemble. The total simulation time was 3 nanoseconds, with velocity scaling/swapping occurring every 10 femtoseconds. The hot slab was centered in the gold and the cold slab was placed in the center of the solvent region. The average temperature was 220 K, with a temperature difference between the hot and cold slabs of approximately 30 K. The average temperature and kinetic energy flux were carefully selected with two considerations in mind: 1) if the cold bin gets too cold (below ~180 K) the solvent may freeze or undergo a glassy transition, and 2) the deep sulfur-gold potential well makes the sulfur prone to insertion into the gold slab, particularly at temperatures above 250 K. Simulation conditions were chosen to avoid both of these undesirable situations. A reversed flux direction resulted in frozen long chain thiolates and solvent too near its boiling point.

SOMETHING ABOUT VSS

        %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
        %                          FORCE-FIELD PARAMETERS
        %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
        \subsection{Force-Field Parameters}
        
        
        \begin{figure}
                \includegraphics[width=\linewidth]{figures/structures}
                \caption{STRUCTURES}
                \label{fig:structures}
        \end{figure}

        The gold-gold interactions are modeled using the quantum Sutton-Chen (QSC) force field. 
                
        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.
                
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%                          **RESULTS**
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%                         
\section{Results}


        %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
        %                          CHAIN LENGTH
        %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
        \subsection{Effect of Chain Length}
        
        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 Angstrom gap between the solvent region and the thiolates. Consequently, all chain lengths had low thermal coupling between the solvent and thiolate molecules. The trend of interfacial conductance is 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. Additionally, it suggests that end-to-end or side-to-end alignment of the solvent and thiolate molecules is an inefficient mode of thermal energy transfer.
        
        %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
        %                          MIXED CHAINS
        %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
        \subsection{Effect of Mixed Chain Lengths}
        
        Previous work showed 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 mixture 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.
        
        Mixtures of butanethiolate and decanethiolate (n = 4, 10) have a peak interfacial condutance for equal amounts of short and long chains. The difference in the lengths of the short and long chains is equivalent to the length of the solvent molecule, meaning that the entire hexane molecule cannot fit into the gap between the long chains without getting unfavorably close to the butanethiolate below.
        
        For mixtures of butanethiolate and dodecanethiolate (n = 4, 12) the interfacial conductance reaches a maximum value when there is a 50/50 blend of short and long chains. In this case, $n_{long} - n_{short} > n_{solvent}$, enabling entire hexane solvent molecules to fit into the gaps between the dodecanethiolate chains without hitting the butanethiolates below. This configuration allows for efficient thermal energy exchange between the thiolate tail and the solvent molecules. Once the solvent molecules have picked up thermal energy from the thiolates, when they diffuse back into the bulk solvent they carry heat away from the gold. When the proportion of long chains increases, there are fewer gaps to be filled by solvent, decreasing the number of solvent molecules that can pick up thermal energy from the thiolates and carry it into the bulk solvent. When the short/long chain ratio is below 50/50, the tails of the long chains are much more disordered and do not provide channels for the solvent to efficiently pack into. 
        
        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, which should correspond to a high interfacial conductance. 


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%                          **DISCUSSION**
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Discussion}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%                          **ACKNOWLEDGMENTS**
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section*{Acknowledgments}
Support for this project was provided by the
National Science Foundation under grant CHE-0848243. Computational
time was provided by the Center for Research Computing (CRC) at the
University of Notre Dame.  

\newpage

\bibliography{thiolsRNEMD}

\end{doublespace}
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#	User	Rev	Content
1	kstocke1	3801	\documentclass[11pt]{article}
2			\usepackage{amsmath}
3			\usepackage{amssymb}
4			\usepackage{setspace}
5			\usepackage{endfloat}
6			\usepackage{caption}
7			\usepackage{graphicx}
8			\usepackage{multirow}
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12			\evensidemargin 0.0cm \topmargin -21pt \headsep 10pt \textheight
13			9.0in \textwidth 6.5in \brokenpenalty=10000
14
15			% double space list of tables and figures
16			%\AtBeginDelayedFloats{\renewcomand{\baselinestretch}{1.66}}
17			\setlength{\abovecaptionskip}{20 pt}
18			\setlength{\belowcaptionskip}{30 pt}
19
20			\bibpunct{}{}{,}{s}{}{;}
21			\bibliographystyle{achemso}
22
23			\begin{document}
24
25			\title{Interfacial Thermal Conductance of Thiolate-Capped Gold}
26
27			\author{Kelsey M. Stocker and J. Daniel Gezelter\footnote{Corresponding author. \ Electronic mail: gezelter@nd.edu} \\
28			Department of Chemistry and Biochemistry,\\
29			University of Notre Dame\\
30			Notre Dame, Indiana 46556}
31
32			\date{\today}
33
34			\maketitle
35
36			\begin{doublespace}
37
38			\begin{abstract}
39			ABSTRACT
40			\end{abstract}
41
42			\newpage
43
44			%\narrowtext
45
46			%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
47			% INTRODUCTION
48			%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
49			\section{Introduction}
50
51			%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
52			% METHODOLOGY
53			%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
54			\section{Methodology}
55
56			%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
57			% VSS-RNEMD
58			%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
59			\subsection{VSS-RNEMD}
60
61			\begin{figure}
62			\includegraphics[width=\linewidth]{figures/rnemd}
63			\caption{VSS-RNEMD}
64			\label{fig:rnemd}
65			\end{figure}
66
67			\begin{equation}
68			\boldsymbol{\nu}_{i} \leftarrow c \cdot \left( \boldsymbol{\nu}_{i} - \langle \boldsymbol{\nu}_{c} \rangle \right) + \left( \langle \boldsymbol{\nu}_{c} \rangle + \bold{a}_{c} \right),
69			\end{equation}
70
71			\begin{equation}
72			\boldsymbol{\nu}_{j} \leftarrow h \cdot \left( \boldsymbol{\nu}_{j} - \langle \boldsymbol{\nu}_{h} \rangle \right) + \left( \langle \boldsymbol{\nu}_{h} \rangle + \bold{a}_{h} \right),
73			\end{equation}
74
75			\begin{equation}
76			K_{c} - J_{z} \Delta t = c^{2} \left( K_{c} - \frac{1}{2} M_{c} \langle \boldsymbol{\nu}_{c} \rangle ^{2} \right) + \frac{1}{2} M_{c} \left( \langle \boldsymbol{\nu}_{c} \rangle + \bold{a}_{c} \right) ^{2},
77			\end{equation}
78
79			\begin{equation}
80			K_{h} + J_{z} \Delta t = h^{2} \left( K_{h} - \frac{1}{2} M_{h} \langle \boldsymbol{\nu}_{h} \rangle ^{2} \right) + \frac{1}{2} M_{h} \left( \langle \boldsymbol{\nu}_{h} \rangle + \bold{a}_{h} \right) ^{2},
81			\end{equation}
82
83
84
85			%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
86			% INTERFACIAL CONDUCTANCE
87			%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
88			\subsection{Defining Interfacial Conductance (G)}
89
90			\begin{figure}
91			\includegraphics[width=\linewidth]{figures/resistor_series}
92			\caption{RESISTOR SERIES}
93			\label{fig:resistor_series}
94			\end{figure}
95
96
97			There are two interfaces involved in the transfer of heat from the gold slab to the solvent: the gold/thiolate interface and the thiolate/solvent interface. We can treat the temperature on each side of an interface as discrete, making the interfacial conductance the inverse of the Kaptiza resistance, or $G = \frac{J}{\Delta T}$. To model the total conductance across multiple interfaces, we treat the interfaces as resistors in series. Resistors in series are additive, ($R_{total} = R_{1} + R_{2} + R_{3} + ...$) and the total conductance is the inverse of the total resistance, or $G = \frac{1}{(R_{1} + R_{2} + R_{3} + ...}$). We treat each bin in the VSS-RNEMD temperature profile as a resistor with resistance $\frac{T_{2}-T_{1}}{J}$, $\frac{T_{3}-T_{2}}{J}$, etc. The sum of this resistor series which spans the gold/thiolate interface, thiolate chains, and thiolate/solvent interface simplifies to
98
99			\begin{equation}
100			\frac{T_{n}-T_{1}}{J},
101			\label{eq:finalG}
102			\end{equation}
103
104			or the temperature difference between the gold side of the gold/thiolate interface and the solvent side of the thiolate/solvent interface over the applied flux.
105
106
107			%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
108			% COMPUTATIONAL DETAILS
109			%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
110			\section{Computational Details}
111
112			%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
113			% SIMULATION PROTOCOL
114			%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
115			\subsection{Simulation Protocol}
116
117			We have implemented the VSS-RNEMD algorithm in OpenMD, our group molecular dynamics code. A gold slab 11 layers thick was equilibrated at 1 atm and 200 K. The periodic box was expanded in the z direction to expose two Au(111) faces.
118
119			A full monolayer of thiolates (1/3 the number of surface gold atoms) was placed on three-fold hollow sites on the gold interfaces. To efficiently test the effect of thiolate binding sites on the thermal conductance, all systems had one gold interface with thiolates placed only on fcc hollow sites and the other interface with thiolates only on hcp hollow sites. To test the effect of thiolate chain length on interfacial thermal conductance, full coverages of five chain lengths were tested: butanethiolate, hexanethiolate, octanethiolate, decanethiolate, and dodecanethiolate. To test the effect of mixed chain lengths, full coverages of butanethiolate/decanethiolate and butanethiolate/dodecanethiolate mixtures were created in short/long chain ratios of 25/75, 50/50, and 75/25. The short and long chains were placed on the surface in a random configuration.
120
121			The simulation box z dimension was set to roughly double the length of the gold/thiolate block. Solvent molecules were placed in the vacant portion of the box using the packmol algorithm. Two solvent molecules were examined: hexane and toluene. Hexane, a straight chain flexible alkane, is very structurally similar to the thiolate alkane tails while toluene is a rigid planar molecule.
122
123			The system was equilibrated to 220 K in the NVT ensemble, allowing the thiolates and solvent to warm gradually. Pressure correction to 1 atm was done in an NPT ensemble (NPAT) that allowed expansion or contraction only in the z direction, so as not to disrupt the crystalline structure of the gold. The diagonal elements of the pressure tensor were monitored during the pressure correction step. The zz element was successfully equilibrated during the NPAT simulation. If the xx and/or yy elements had a mean above zero throughout the simulation -- indicating residual pressure in the plane of the gold slab -- an additional short NPT equilibration step was performed allowing all box dimensions to change. Once the pressure was stable at 1 atm, a final NVT simulation was performed.
124
125			A kinetic energy flux was imposed using VSS-RNEMD in the microcanonical (NVE) ensemble. The total simulation time was 3 nanoseconds, with velocity scaling/swapping occurring every 10 femtoseconds. The hot slab was centered in the gold and the cold slab was placed in the center of the solvent region. The average temperature was 220 K, with a temperature difference between the hot and cold slabs of approximately 30 K. The average temperature and kinetic energy flux were carefully selected with two considerations in mind: 1) if the cold bin gets too cold (below ~180 K) the solvent may freeze or undergo a glassy transition, and 2) the deep sulfur-gold potential well makes the sulfur prone to insertion into the gold slab, particularly at temperatures above 250 K. Simulation conditions were chosen to avoid both of these undesirable situations. A reversed flux direction resulted in frozen long chain thiolates and solvent too near its boiling point.
126
127			SOMETHING ABOUT VSS
128
129			%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
130			% FORCE-FIELD PARAMETERS
131			%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
132			\subsection{Force-Field Parameters}
133
134
135			\begin{figure}
136			\includegraphics[width=\linewidth]{figures/structures}
137			\caption{STRUCTURES}
138			\label{fig:structures}
139			\end{figure}
140
141			The gold-gold interactions are modeled using the quantum Sutton-Chen (QSC) force field.
142
143			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.
144
145			%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
146			% RESULTS
147			%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
148			\section{Results}
149
150
151			%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
152			% CHAIN LENGTH
153			%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
154			\subsection{Effect of Chain Length}
155
156			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 Angstrom gap between the solvent region and the thiolates. Consequently, all chain lengths had low thermal coupling between the solvent and thiolate molecules. The trend of interfacial conductance is 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. Additionally, it suggests that end-to-end or side-to-end alignment of the solvent and thiolate molecules is an inefficient mode of thermal energy transfer.
157
158			%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
159			% MIXED CHAINS
160			%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
161			\subsection{Effect of Mixed Chain Lengths}
162
163			Previous work showed 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 mixture of short (butanethiolate, n = 4) and long (decanethiolate, n = 10 or dodecanethiolate, n = 12).
164
165			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.
166
167			Mixtures of butanethiolate and decanethiolate (n = 4, 10) have a peak interfacial condutance for equal amounts of short and long chains. The difference in the lengths of the short and long chains is equivalent to the length of the solvent molecule, meaning that the entire hexane molecule cannot fit into the gap between the long chains without getting unfavorably close to the butanethiolate below.
168
169			For mixtures of butanethiolate and dodecanethiolate (n = 4, 12) the interfacial conductance reaches a maximum value when there is a 50/50 blend of short and long chains. In this case, $n_{long} - n_{short} > n_{solvent}$, enabling entire hexane solvent molecules to fit into the gaps between the dodecanethiolate chains without hitting the butanethiolates below. This configuration allows for efficient thermal energy exchange between the thiolate tail and the solvent molecules. Once the solvent molecules have picked up thermal energy from the thiolates, when they diffuse back into the bulk solvent they carry heat away from the gold. When the proportion of long chains increases, there are fewer gaps to be filled by solvent, decreasing the number of solvent molecules that can pick up thermal energy from the thiolates and carry it into the bulk solvent. When the short/long chain ratio is below 50/50, the tails of the long chains are much more disordered and do not provide channels for the solvent to efficiently pack into.
170
171			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, which should correspond to a high interfacial conductance.
172
173
174			%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
175			% DISCUSSION
176			%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
177			\section{Discussion}
178
179			%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
180			% ACKNOWLEDGMENTS
181			%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
182			\section*{Acknowledgments}
183			Support for this project was provided by the
184			National Science Foundation under grant CHE-0848243. Computational
185			time was provided by the Center for Research Computing (CRC) at the
186			University of Notre Dame.
187
188			\newpage
189
190			\bibliography{thiolsRNEMD}
191
192			\end{doublespace}
193			\end{document}
194
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201