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29		\begin{document}
30
#	Line 73 | Line 73 | Interfacial thermal conductance is extensively studied
73		%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
74
75		\section{Introduction}
76	<	Interfacial thermal conductance is extensively studied both
77	<	experimentally and computationally\cite{cahill:793}, due to its
78	<	importance in nanoscale science and technology. Reliability of
79	<	nanoscale devices depends on their thermal transport
80	<	properties. Unlike bulk homogeneous materials, nanoscale materials
81	<	features significant presence of interfaces, and these interfaces
82	<	could dominate the heat transfer behavior of these
83	<	materials. Furthermore, these materials are generally heterogeneous,
84	<	which challenges traditional research methods for homogeneous
85	<	systems.
76	>	Due to the importance of heat flow in nanotechnology, interfacial
77	>	thermal conductance has been studied extensively both experimentally
78	>	and computationally.\cite{cahill:793} Unlike bulk materials, nanoscale
79	>	materials have a significant fraction of their atoms at interfaces,
80	>	and the chemical details of these interfaces govern the heat transfer
81	>	behavior. Furthermore, the interfaces are
82	>	heterogeneous (e.g. solid - liquid), which provides a challenge to
83	>	traditional methods developed for homogeneous systems.
84
85	<	Heat conductance of molecular and nano-scale interfaces will be
86	<	affected by the chemical details of the surface. Experimentally,
87	<	various interfaces have been investigated for their thermal
88	<	conductance properties. Wang {\it et al.} studied heat transport
89	<	through long-chain hydrocarbon monolayers on gold substrate at
90	<	individual molecular level\cite{Wang10082007}; Schmidt {\it et al.}
91	<	studied the role of CTAB on thermal transport between gold nanorods
92	<	and solvent\cite{doi:10.1021/jp8051888}; Juv\'e {\it et al.} studied
85	>	Experimentally, various interfaces have been investigated for their
86	>	thermal conductance. Cahill and coworkers studied nanoscale thermal
87	>	transport from metal nanoparticle/fluid interfaces, to epitaxial
88	>	TiN/single crystal oxides interfaces, to hydrophilic and hydrophobic
89	>	interfaces between water and solids with different self-assembled
90	>	monolayers.\cite{Wilson:2002uq,PhysRevB.67.054302,doi:10.1021/jp048375k,PhysRevLett.96.186101}
91	>	Wang {\it et al.} studied heat transport through
92	>	long-chain hydrocarbon monolayers on gold substrate at individual
93	>	molecular level,\cite{Wang10082007} Schmidt {\it et al.} studied the
94	>	role of CTAB on thermal transport between gold nanorods and
95	>	solvent,\cite{doi:10.1021/jp8051888} and Juv\'e {\it et al.} studied
96		the cooling dynamics, which is controlled by thermal interface
97		resistence of glass-embedded metal
98	<	nanoparticles\cite{PhysRevB.80.195406}. Although interfaces are
99	<	commonly barriers for heat transport, Alper {\it et al.} suggested
100	<	that specific ligands (capping agents) could completely eliminate this
101	<	barrier ($G\rightarrow\infty$)\cite{doi:10.1021/la904855s}.
98	>	nanoparticles.\cite{PhysRevB.80.195406} Although interfaces are
99	>	normally considered barriers for heat transport, Alper {\it et al.}
100	>	suggested that specific ligands (capping agents) could completely
101	>	eliminate this barrier
102	>	($G\rightarrow\infty$).\cite{doi:10.1021/la904855s}
103
104		Theoretical and computational models have also been used to study the
105		interfacial thermal transport in order to gain an understanding of
106		this phenomena at the molecular level. Recently, Hase and coworkers
107		employed Non-Equilibrium Molecular Dynamics (NEMD) simulations to
108		study thermal transport from hot Au(111) substrate to a self-assembled
109	<	monolayer of alkylthiolate with relatively long chain (8-20 carbon
110	<	atoms)\cite{hase:2010,hase:2011}. However, ensemble averaged
109	>	monolayer of alkylthiol with relatively long chain (8-20 carbon
110	>	atoms).\cite{hase:2010,hase:2011} However, ensemble averaged
111		measurements for heat conductance of interfaces between the capping
112	<	monolayer on Au and a solvent phase has yet to be studied.
113	<	The relatively low thermal flux through interfaces is
114	<	difficult to measure with Equilibrium MD or forward NEMD simulation
115	<	methods. Therefore, the Reverse NEMD (RNEMD) methods would have the
116	<	advantage of having this difficult to measure flux known when studying
117	<	the thermal transport across interfaces, given that the simulation
118	<	methods being able to effectively apply an unphysical flux in
119	<	non-homogeneous systems.
112	>	monolayer on Au and a solvent phase have yet to be studied with their
113	>	approach. The comparatively low thermal flux through interfaces is
114	>	difficult to measure with Equilibrium
115	>	MD\cite{doi:10.1080/0026897031000068578} or forward NEMD simulation
116	>	methods. Therefore, the Reverse NEMD (RNEMD)
117	>	methods\cite{MullerPlathe:1997xw,kuang:164101} would have the
118	>	advantage of applying this difficult to measure flux (while measuring
119	>	the resulting gradient), given that the simulation methods being able
120	>	to effectively apply an unphysical flux in non-homogeneous systems.
121	>	Garde and coworkers\cite{garde:nl2005,garde:PhysRevLett2009} applied
122	>	this approach to various liquid interfaces and studied how thermal
123	>	conductance (or resistance) is dependent on chemistry details of
124	>	interfaces, e.g. hydrophobic and hydrophilic aqueous interfaces.
125
126	<	Recently, we have developed the Non-Isotropic Velocity Scaling (NIVS)
126	>	Recently, we have developed a Non-Isotropic Velocity Scaling (NIVS)
127		algorithm for RNEMD simulations\cite{kuang:164101}. This algorithm
128		retains the desirable features of RNEMD (conservation of linear
129		momentum and total energy, compatibility with periodic boundary
#	Line 131 | Line 138 | underlying mechanism for this phenomena was investigat
138		properties. Different models were used for both the capping agent and
139		the solvent force field parameters. Using the NIVS algorithm, the
140		thermal transport across these interfaces was studied and the
141	<	underlying mechanism for this phenomena was investigated.
141	>	underlying mechanism for the phenomena was investigated.
142
136	–	[MAY ADD WHY STUDY AU-THIOL SURFACE; CITE SHAOYI JIANG]
137	–
143		\section{Methodology}
144		\subsection{Imposd-Flux Methods in MD Simulations}
145	<	For systems with low interfacial conductivity one must have a method
146	<	capable of generating relatively small fluxes, compared to those
147	<	required for bulk conductivity. This requirement makes the calculation
148	<	even more difficult for those slowly-converging equilibrium
149	<	methods\cite{Viscardy:2007lq}.
150	<	Forward methods impose gradient, but in interfacail conditions it is
151	<	not clear what behavior to impose at the boundary...
152	<	Imposed-flux reverse non-equilibrium
153	<	methods\cite{MullerPlathe:1997xw} have the flux set {\it a priori} and
154	<	the thermal response becomes easier to
155	<	measure than the flux. Although M\"{u}ller-Plathe's original momentum
156	<	swapping approach can be used for exchanging energy between particles
157	<	of different identity, the kinetic energy transfer efficiency is
158	<	affected by the mass difference between the particles, which limits
159	<	its application on heterogeneous interfacial systems.
145	>	Steady state MD simulations have an advantage in that not many
146	>	trajectories are needed to study the relationship between thermal flux
147	>	and thermal gradients. For systems with low interfacial conductance,
148	>	one must have a method capable of generating or measuring relatively
149	>	small fluxes, compared to those required for bulk conductivity. This
150	>	requirement makes the calculation even more difficult for
151	>	slowly-converging equilibrium methods.\cite{Viscardy:2007lq} Forward
152	>	NEMD methods impose a gradient (and measure a flux), but at interfaces
153	>	it is not clear what behavior should be imposed at the boundaries
154	>	between materials.  Imposed-flux reverse non-equilibrium
155	>	methods\cite{MullerPlathe:1997xw} set the flux {\it a priori} and
156	>	the thermal response becomes an easy-to-measure quantity.  Although
157	>	M\"{u}ller-Plathe's original momentum swapping approach can be used
158	>	for exchanging energy between particles of different identity, the
159	>	kinetic energy transfer efficiency is affected by the mass difference
160	>	between the particles, which limits its application on heterogeneous
161	>	interfacial systems.
162
163	<	The non-isotropic velocity scaling (NIVS)\cite{kuang:164101} approach to
164	<	non-equilibrium MD simulations is able to impose a wide range of
163	>	The non-isotropic velocity scaling (NIVS) \cite{kuang:164101} approach
164	>	to non-equilibrium MD simulations is able to impose a wide range of
165		kinetic energy fluxes without obvious perturbation to the velocity
166		distributions of the simulated systems. Furthermore, this approach has
167		the advantage in heterogeneous interfaces in that kinetic energy flux
#	Line 171 | Line 178 | momenta and energy and does not depend on an external
178		for computing thermal conductivities. The NIVS algorithm conserves
179		momenta and energy and does not depend on an external thermostat.
180
181	<	\subsection{Defining Interfacial Thermal Conductivity $G$}
182	<	For interfaces with a relatively low interfacial conductance, the bulk
183	<	regions on either side of an interface rapidly come to a state in
184	<	which the two phases have relatively homogeneous (but distinct)
185	<	temperatures. The interfacial thermal conductivity $G$ can therefore
186	<	be approximated as:
181	>	\subsection{Defining Interfacial Thermal Conductivity ($G$)}
182	>
183	>	For an interface with relatively low interfacial conductance, and a
184	>	thermal flux between two distinct bulk regions, the regions on either
185	>	side of the interface rapidly come to a state in which the two phases
186	>	have relatively homogeneous (but distinct) temperatures. The
187	>	interfacial thermal conductivity $G$ can therefore be approximated as:
188		\begin{equation}
189	<	G = \frac{E_{total}}{2 t L_x L_y \left( \langle T_\mathrm{hot}\rangle -
189	>	G = \frac{E_{total}}{2 t L_x L_y \left( \langle T_\mathrm{hot}\rangle -
190		\langle T_\mathrm{cold}\rangle \right)}
191		\label{lowG}
192		\end{equation}
193	<	where ${E_{total}}$ is the imposed non-physical kinetic energy
194	<	transfer and ${\langle T_\mathrm{hot}\rangle}$ and ${\langle
195	<	T_\mathrm{cold}\rangle}$ are the average observed temperature of the
196	<	two separated phases.
193	>	where ${E_{total}}$ is the total imposed non-physical kinetic energy
194	>	transfer during the simulation and ${\langle T_\mathrm{hot}\rangle}$
195	>	and ${\langle T_\mathrm{cold}\rangle}$ are the average observed
196	>	temperature of the two separated phases.  For an applied flux $J_z$
197	>	operating over a simulation time $t$ on a periodically-replicated slab
198	>	of dimensions $L_x \times L_y$, $E_{total} = J_z *(t)*(2 L_x L_y)$.
199
200		When the interfacial conductance is {\it not} small, there are two
201	<	ways to define $G$.
202	<
203	<	One way is to assume the temperature is discrete on the two sides of
204	<	the interface. $G$ can be calculated using the applied thermal flux
205	<	$J$ and the maximum temperature difference measured along the thermal
206	<	gradient max($\Delta T$), which occurs at the Gibbs deviding surface,
207	<	as:
201	>	ways to define $G$. One common way is to assume the temperature is
202	>	discrete on the two sides of the interface. $G$ can be calculated
203	>	using the applied thermal flux $J$ and the maximum temperature
204	>	difference measured along the thermal gradient max($\Delta T$), which
205	>	occurs at the Gibbs deviding surface (Figure \ref{demoPic}). This is
206	>	known as the Kapitza conductance, which is the inverse of the Kapitza
207	>	resistance.
208		\begin{equation}
209	<	G=\frac{J}{\Delta T}
209	>	G=\frac{J}{\Delta T}
210		\label{discreteG}
211		\end{equation}
212
213	+	\begin{figure}
214	+	\includegraphics[width=\linewidth]{method}
215	+	\caption{Interfacial conductance can be calculated by applying an
216	+	(unphysical) kinetic energy flux between two slabs, one located
217	+	within the metal and another on the edge of the periodic box.  The
218	+	system responds by forming a thermal response or a gradient.  In
219	+	bulk liquids, this gradient typically has a single slope, but in
220	+	interfacial systems, there are distinct thermal conductivity
221	+	domains.  The interfacial conductance, $G$ is found by measuring the
222	+	temperature gap at the Gibbs dividing surface, or by using second
223	+	derivatives of the thermal profile.}
224	+	\label{demoPic}
225	+	\end{figure}
226	+
227		The other approach is to assume a continuous temperature profile along
228		the thermal gradient axis (e.g. $z$) and define $G$ at the point where
229	<	the magnitude of thermal conductivity $\lambda$ change reach its
229	>	the magnitude of thermal conductivity ($\lambda$) change reaches its
230		maximum, given that $\lambda$ is well-defined throughout the space:
231		\begin{equation}
232		G^\prime = \Big|\frac{\partial\lambda}{\partial z}\Big|
#	Line 213 | Line 237 | With the temperature profile obtained from simulations
237		\label{derivativeG}
238		\end{equation}
239
240	<	With the temperature profile obtained from simulations, one is able to
240	>	With temperature profiles obtained from simulation, one is able to
241		approximate the first and second derivatives of $T$ with finite
242	<	difference methods and thus calculate $G^\prime$.
242	>	difference methods and calculate $G^\prime$. In what follows, both
243	>	definitions have been used, and are compared in the results.
244
245	<	In what follows, both definitions have been used for calculation and
246	<	are compared in the results.
247	<
248	<	To compare the above definitions ($G$ and $G^\prime$), we have modeled
249	<	a metal slab with its (111) surfaces perpendicular to the $z$-axis of
250	<	our simulation cells. Both with and withour capping agents on the
226	<	surfaces, the metal slab is solvated with simple organic solvents, as
227	<	illustrated in Figure \ref{demoPic}.
245	>	To investigate the interfacial conductivity at metal / solvent
246	>	interfaces, we have modeled a metal slab with its (111) surfaces
247	>	perpendicular to the $z$-axis of our simulation cells. The metal slab
248	>	has been prepared both with and without capping agents on the exposed
249	>	surface, and has been solvated with simple organic solvents, as
250	>	illustrated in Figure \ref{gradT}.
251
229	–	\begin{figure}
230	–	\includegraphics[width=\linewidth]{demoPic}
231	–	\caption{A sample showing how a metal slab has its (111) surface
232	–	covered by capping agent molecules and solvated by hexane.}
233	–	\label{demoPic}
234	–	\end{figure}
235	–
252		With the simulation cell described above, we are able to equilibrate
253		the system and impose an unphysical thermal flux between the liquid
254		and the metal phase using the NIVS algorithm. By periodically applying
255	<	the unphysical flux, we are able to obtain a temperature profile and
256	<	its spatial derivatives. These quantities enable the evaluation of the
257	<	interfacial thermal conductance of a surface. Figure \ref{gradT} is an
258	<	example how those applied thermal fluxes can be used to obtain the 1st
243	<	and 2nd derivatives of the temperature profile.
255	>	the unphysical flux, we obtained a temperature profile and its spatial
256	>	derivatives. Figure \ref{gradT} shows how an applied thermal flux can
257	>	be used to obtain the 1st and 2nd derivatives of the temperature
258	>	profile.
259
260		\begin{figure}
261		\includegraphics[width=\linewidth]{gradT}
262	<	\caption{The 1st and 2nd derivatives of temperature profile can be
263	<	obtained with finite difference approximation.}
262	>	\caption{A sample of Au-butanethiol/hexane interfacial system and the
263	>	temperature profile after a kinetic energy flux is imposed to
264	>	it. The 1st and 2nd derivatives of the temperature profile can be
265	>	obtained with finite difference approximation (lower panel).}
266		\label{gradT}
267		\end{figure}
268
269		\section{Computational Details}
270		\subsection{Simulation Protocol}
271		The NIVS algorithm has been implemented in our MD simulation code,
272	<	OpenMD\cite{Meineke:2005gd,openmd}, and was used for our
273	<	simulations. Different slab thickness (layer numbers of Au) were
274	<	simulated. Metal slabs were first equilibrated under atmospheric
275	<	pressure (1 atm) and a desired temperature (e.g. 200K). After
276	<	equilibration, butanethiol capping agents were placed at three-fold
277	<	sites on the Au(111) surfaces. The maximum butanethiol capacity on Au
278	<	surface is $1/3$ of the total number of surface Au
279	<	atoms\cite{vlugt:cpc2007154}. A series of different coverages was
280	<	investigated in order to study the relation between coverage and
281	<	interfacial conductance.
272	>	OpenMD\cite{Meineke:2005gd,openmd}, and was used for our simulations.
273	>	Metal slabs of 6 or 11 layers of Au atoms were first equilibrated
274	>	under atmospheric pressure (1 atm) and 200K. After equilibration,
275	>	butanethiol capping agents were placed at three-fold hollow sites on
276	>	the Au(111) surfaces. These sites are either {\it fcc} or {\it
277	>	hcp} sites, although Hase {\it et al.} found that they are
278	>	equivalent in a heat transfer process,\cite{hase:2010} so we did not
279	>	distinguish between these sites in our study. The maximum butanethiol
280	>	capacity on Au surface is $1/3$ of the total number of surface Au
281	>	atoms, and the packing forms a $(\sqrt{3}\times\sqrt{3})R30^\circ$
282	>	structure\cite{doi:10.1021/ja00008a001,doi:10.1021/cr9801317}. A
283	>	series of lower coverages was also prepared by eliminating
284	>	butanethiols from the higher coverage surface in a regular manner. The
285	>	lower coverages were prepared in order to study the relation between
286	>	coverage and interfacial conductance.
287
288		The capping agent molecules were allowed to migrate during the
289		simulations. They distributed themselves uniformly and sampled a
290		number of three-fold sites throughout out study. Therefore, the
291	<	initial configuration would not noticeably affect the sampling of a
291	>	initial configuration does not noticeably affect the sampling of a
292		variety of configurations of the same coverage, and the final
293		conductance measurement would be an average effect of these
294	<	configurations explored in the simulations. [MAY NEED FIGURES]
294	>	configurations explored in the simulations.
295
296	<	After the modified Au-butanethiol surface systems were equilibrated
297	<	under canonical ensemble, organic solvent molecules were packed in the
298	<	previously empty part of the simulation cells\cite{packmol}. Two
296	>	After the modified Au-butanethiol surface systems were equilibrated in
297	>	the canonical (NVT) ensemble, organic solvent molecules were packed in
298	>	the previously empty part of the simulation cells.\cite{packmol} Two
299		solvents were investigated, one which has little vibrational overlap
300	<	with the alkanethiol and a planar shape (toluene), and one which has
301	<	similar vibrational frequencies and chain-like shape ({\it n}-hexane).
300	>	with the alkanethiol and which has a planar shape (toluene), and one
301	>	which has similar vibrational frequencies to the capping agent and
302	>	chain-like shape ({\it n}-hexane).
303
304	<	The space filled by solvent molecules, i.e. the gap between
305	<	periodically repeated Au-butanethiol surfaces should be carefully
306	<	chosen. A very long length scale for the thermal gradient axis ($z$)
284	<	may cause excessively hot or cold temperatures in the middle of the
304	>	The simulation cells were not particularly extensive along the
305	>	$z$-axis, as a very long length scale for the thermal gradient may
306	>	cause excessively hot or cold temperatures in the middle of the
307		solvent region and lead to undesired phenomena such as solvent boiling
308		or freezing when a thermal flux is applied. Conversely, too few
309		solvent molecules would change the normal behavior of the liquid
310		phase. Therefore, our $N_{solvent}$ values were chosen to ensure that
311	<	these extreme cases did not happen to our simulations. And the
312	<	corresponding spacing is usually $35 \sim 60$\AA.
311	>	these extreme cases did not happen to our simulations. The spacing
312	>	between periodic images of the gold interfaces is $45 \sim 75$\AA.
313
314	<	The initial configurations generated by Packmol are further
315	<	equilibrated with the $x$ and $y$ dimensions fixed, only allowing
316	<	length scale change in $z$ dimension. This is to ensure that the
317	<	equilibration of liquid phase does not affect the metal crystal
318	<	structure in $x$ and $y$ dimensions. Further equilibration are run
319	<	under NVT and then NVE ensembles.
314	>	The initial configurations generated are further equilibrated with the
315	>	$x$ and $y$ dimensions fixed, only allowing the $z$-length scale to
316	>	change. This is to ensure that the equilibration of liquid phase does
317	>	not affect the metal's crystalline structure. Comparisons were made
318	>	with simulations that allowed changes of $L_x$ and $L_y$ during NPT
319	>	equilibration. No substantial changes in the box geometry were noticed
320	>	in these simulations. After ensuring the liquid phase reaches
321	>	equilibrium at atmospheric pressure (1 atm), further equilibration was
322	>	carried out under canonical (NVT) and microcanonical (NVE) ensembles.
323
324	<	After the systems reach equilibrium, NIVS is implemented to impose a
325	<	periodic unphysical thermal flux between the metal and the liquid
326	<	phase. Most of our simulations are under an average temperature of
327	<	$\sim$200K. Therefore, this flux usually comes from the metal to the
324	>	After the systems reach equilibrium, NIVS was used to impose an
325	>	unphysical thermal flux between the metal and the liquid phases. Most
326	>	of our simulations were done under an average temperature of
327	>	$\sim$200K. Therefore, thermal flux usually came from the metal to the
328		liquid so that the liquid has a higher temperature and would not
329	<	freeze due to excessively low temperature. This induced temperature
330	<	gradient is stablized and the simulation cell is devided evenly into
331	<	N slabs along the $z$-axis and the temperatures of each slab are
332	<	recorded. When the slab width $d$ of each slab is the same, the
333	<	derivatives of $T$ with respect to slab number $n$ can be directly
334	<	used for $G^\prime$ calculations:
335	<	\begin{equation}
336	<	G^\prime = |J_z|\Big|\frac{\partial^2 T}{\partial z^2}\Big|
329	>	freeze due to lowered temperatures. After this induced temperature
330	>	gradient had stablized, the temperature profile of the simulation cell
331	>	was recorded. To do this, the simulation cell is devided evenly into
332	>	$N$ slabs along the $z$-axis. The average temperatures of each slab
333	>	are recorded for 1$\sim$2 ns. When the slab width $d$ of each slab is
334	>	the same, the derivatives of $T$ with respect to slab number $n$ can
335	>	be directly used for $G^\prime$ calculations: \begin{equation}
336	>	G^\prime = |J_z|\Big|\frac{\partial^2 T}{\partial z^2}\Big|
337		\Big/\left(\frac{\partial T}{\partial z}\right)^2
338		= |J_z|\Big|\frac{1}{d^2}\frac{\partial^2 T}{\partial n^2}\Big|
339		\Big/\left(\frac{1}{d}\frac{\partial T}{\partial n}\right)^2
#	Line 317 | Line 342 | G^\prime = |J_z|\Big|\frac{\partial^2 T}{\partial z^2}
342		\label{derivativeG2}
343		\end{equation}
344
345	+	All of the above simulation procedures use a time step of 1 fs. Each
346	+	equilibration stage took a minimum of 100 ps, although in some cases,
347	+	longer equilibration stages were utilized.
348	+
349		\subsection{Force Field Parameters}
350	<	Our simulations include various components. Therefore, force field
351	<	parameter descriptions are needed for interactions both between the
352	<	same type of particles and between particles of different species.
350	>	Our simulations include a number of chemically distinct components.
351	>	Figure \ref{demoMol} demonstrates the sites defined for both
352	>	United-Atom and All-Atom models of the organic solvent and capping
353	>	agents in our simulations. Force field parameters are needed for
354	>	interactions both between the same type of particles and between
355	>	particles of different species.
356
357	+	\begin{figure}
358	+	\includegraphics[width=\linewidth]{structures}
359	+	\caption{Structures of the capping agent and solvents utilized in
360	+	these simulations. The chemically-distinct sites (a-e) are expanded
361	+	in terms of constituent atoms for both United Atom (UA) and All Atom
362	+	(AA) force fields.  Most parameters are from
363	+	Refs. \protect\cite{TraPPE-UA.alkanes,TraPPE-UA.alkylbenzenes,TraPPE-UA.thiols}
364	+	(UA) and \protect\cite{OPLSAA} (AA). Cross-interactions with the Au
365	+	atoms are given in Table \ref{MnM}.}
366	+	\label{demoMol}
367	+	\end{figure}
368	+
369		The Au-Au interactions in metal lattice slab is described by the
370	<	quantum Sutton-Chen (QSC) formulation\cite{PhysRevB.59.3527}. The QSC
370	>	quantum Sutton-Chen (QSC) formulation.\cite{PhysRevB.59.3527} The QSC
371		potentials include zero-point quantum corrections and are
372		reparametrized for accurate surface energies compared to the
373	<	Sutton-Chen potentials\cite{Chen90}.
373	>	Sutton-Chen potentials.\cite{Chen90}
374
375	<	Figure \ref{demoMol} demonstrates how we name our pseudo-atoms of the
376	<	organic solvent molecules in our simulations.
375	>	For the two solvent molecules, {\it n}-hexane and toluene, two
376	>	different atomistic models were utilized. Both solvents were modeled
377	>	using United-Atom (UA) and All-Atom (AA) models. The TraPPE-UA
378	>	parameters\cite{TraPPE-UA.alkanes,TraPPE-UA.alkylbenzenes} are used
379	>	for our UA solvent molecules. In these models, sites are located at
380	>	the carbon centers for alkyl groups. Bonding interactions, including
381	>	bond stretches and bends and torsions, were used for intra-molecular
382	>	sites closer than 3 bonds. For non-bonded interactions, Lennard-Jones
383	>	potentials are used.
384
385	<	\begin{figure}
386	<	\includegraphics[width=\linewidth]{demoMol}
387	<	\caption{Denomination of atoms or pseudo-atoms in our simulations: a)
388	<	UA-hexane; b) AA-hexane; c) UA-toluene; d) AA-toluene.}
389	<	\label{demoMol}
390	<	\end{figure}
385	>	By eliminating explicit hydrogen atoms, the TraPPE-UA models are
386	>	simple and computationally efficient, while maintaining good accuracy.
387	>	However, the TraPPE-UA model for alkanes is known to predict a slighly
388	>	lower boiling point than experimental values. This is one of the
389	>	reasons we used a lower average temperature (200K) for our
390	>	simulations. If heat is transferred to the liquid phase during the
391	>	NIVS simulation, the liquid in the hot slab can actually be
392	>	substantially warmer than the mean temperature in the simulation. The
393	>	lower mean temperatures therefore prevent solvent boiling.
394
395	<	For both solvent molecules, straight chain {\it n}-hexane and aromatic
396	<	toluene, United-Atom (UA) and All-Atom (AA) models are used
397	<	respectively. The TraPPE-UA
398	<	parameters\cite{TraPPE-UA.alkanes,TraPPE-UA.alkylbenzenes} are used
399	<	for our UA solvent molecules. In these models, pseudo-atoms are
346	<	located at the carbon centers for alkyl groups. By eliminating
347	<	explicit hydrogen atoms, these models are simple and computationally
348	<	efficient, while maintains good accuracy. However, the TraPPE-UA for
349	<	alkanes is known to predict a lower boiling point than experimental
350	<	values. Considering that after an unphysical thermal flux is applied
351	<	to a system, the temperature of ``hot'' area in the liquid phase would be
352	<	significantly higher than the average, to prevent over heating and
353	<	boiling of the liquid phase, the average temperature in our
354	<	simulations should be much lower than the liquid boiling point. [MORE DISCUSSION]
355	<	For UA-toluene model, rigid body constraints are applied, so that the
356	<	benzene ring and the methyl-CRar bond are kept rigid. This would save
357	<	computational time.[MORE DETAILS]
395	>	For UA-toluene, the non-bonded potentials between intermolecular sites
396	>	have a similar Lennard-Jones formulation. The toluene molecules were
397	>	treated as a single rigid body, so there was no need for
398	>	intramolecular interactions (including bonds, bends, or torsions) in
399	>	this solvent model.
400
401		Besides the TraPPE-UA models, AA models for both organic solvents are
402	<	included in our studies as well. For hexane, the OPLS-AA\cite{OPLSAA}
403	<	force field is used. [MORE DETAILS]
404	<	For toluene, the United Force Field developed by Rapp\'{e} {\it et
405	<	al.}\cite{doi:10.1021/ja00051a040} is adopted.[MORE DETAILS]
402	>	included in our studies as well. The OPLS-AA\cite{OPLSAA} force fields
403	>	were used. For hexane, additional explicit hydrogen sites were
404	>	included. Besides bonding and non-bonded site-site interactions,
405	>	partial charges and the electrostatic interactions were added to each
406	>	CT and HC site. For toluene, a flexible model for the toluene molecule
407	>	was utilized which included bond, bend, torsion, and inversion
408	>	potentials to enforce ring planarity.
409
410	<	The capping agent in our simulations, the butanethiol molecules can
411	<	either use UA or AA model. The TraPPE-UA force fields includes
410	>	The butanethiol capping agent in our simulations, were also modeled
411	>	with both UA and AA model. The TraPPE-UA force field includes
412		parameters for thiol molecules\cite{TraPPE-UA.thiols} and are used for
413		UA butanethiol model in our simulations. The OPLS-AA also provides
414		parameters for alkyl thiols. However, alkyl thiols adsorbed on Au(111)
415	<	surfaces do not have the hydrogen atom bonded to sulfur. To adapt this
416	<	change and derive suitable parameters for butanethiol adsorbed on
417	<	Au(111) surfaces, we adopt the S parameters from Luedtke and
418	<	Landman\cite{landman:1998} and modify parameters for its neighbor C
419	<	atom for charge balance in the molecule. Note that the model choice
420	<	(UA or AA) of capping agent can be different from the
421	<	solvent. Regardless of model choice, the force field parameters for
422	<	interactions between capping agent and solvent can be derived using
423	<	Lorentz-Berthelot Mixing Rule:[EQN'S]
415	>	surfaces do not have the hydrogen atom bonded to sulfur. To derive
416	>	suitable parameters for butanethiol adsorbed on Au(111) surfaces, we
417	>	adopt the S parameters from Luedtke and Landman\cite{landman:1998} and
418	>	modify the parameters for the CTS atom to maintain charge neutrality
419	>	in the molecule.  Note that the model choice (UA or AA) for the capping
420	>	agent can be different from the solvent. Regardless of model choice,
421	>	the force field parameters for interactions between capping agent and
422	>	solvent can be derived using Lorentz-Berthelot Mixing Rule:
423	>	\begin{eqnarray}
424	>	\sigma_{ij} & = & \frac{1}{2} \left(\sigma_{ii} + \sigma_{jj}\right) \\
425	>	\epsilon_{ij} & = & \sqrt{\epsilon_{ii}\epsilon_{jj}}
426	>	\end{eqnarray}
427
428	+	To describe the interactions between metal (Au) and non-metal atoms,
429	+	we refer to an adsorption study of alkyl thiols on gold surfaces by
430	+	Vlugt {\it et al.}\cite{vlugt:cpc2007154} They fitted an effective
431	+	Lennard-Jones form of potential parameters for the interaction between
432	+	Au and pseudo-atoms CH$_x$ and S based on a well-established and
433	+	widely-used effective potential of Hautman and Klein for the Au(111)
434	+	surface.\cite{hautman:4994} As our simulations require the gold slab
435	+	to be flexible to accommodate thermal excitation, the pair-wise form
436	+	of potentials they developed was used for our study.
437
438	<	To describe the interactions between metal Au and non-metal capping
439	<	agent and solvent particles, we refer to an adsorption study of alkyl
440	<	thiols on gold surfaces by Vlugt {\it et
441	<	al.}\cite{vlugt:cpc2007154} They fitted an effective Lennard-Jones
442	<	form of potential parameters for the interaction between Au and
443	<	pseudo-atoms CH$_x$ and S based on a well-established and widely-used
444	<	effective potential of Hautman and Klein\cite{hautman:4994} for the
445	<	Au(111) surface. As our simulations require the gold lattice slab to
446	<	be non-rigid so that it could accommodate kinetic energy for thermal
447	<	transport study purpose, the pair-wise form of potentials is
448	<	preferred.
438	>	The potentials developed from {\it ab initio} calculations by Leng
439	>	{\it et al.}\cite{doi:10.1021/jp034405s} are adopted for the
440	>	interactions between Au and aromatic C/H atoms in toluene. However,
441	>	the Lennard-Jones parameters between Au and other types of particles,
442	>	(e.g. AA alkanes) have not yet been established. For these
443	>	interactions, the Lorentz-Berthelot mixing rule can be used to derive
444	>	effective single-atom LJ parameters for the metal using the fit values
445	>	for toluene. These are then used to construct reasonable mixing
446	>	parameters for the interactions between the gold and other atoms.
447	>	Table \ref{MnM} summarizes the ``metal/non-metal'' parameters used in
448	>	our simulations.
449
393	–	Besides, the potentials developed from {\it ab initio} calculations by
394	–	Leng {\it et al.}\cite{doi:10.1021/jp034405s} are adopted for the
395	–	interactions between Au and aromatic C/H atoms in toluene.[MORE DETAILS]
396	–
397	–	However, the Lennard-Jones parameters between Au and other types of
398	–	particles in our simulations are not yet well-established. For these
399	–	interactions, we attempt to derive their parameters using the Mixing
400	–	Rule. To do this, the ``Metal-non-Metal'' (MnM) interaction parameters
401	–	for Au is first extracted from the Au-CH$_x$ parameters by applying
402	–	the Mixing Rule reversely. Table \ref{MnM} summarizes these ``MnM''
403	–	parameters in our simulations.
404	–
450		\begin{table*}
451		\begin{minipage}{\linewidth}
452		\begin{center}
453	<	\caption{Lennard-Jones parameters for Au-non-Metal
454	<	interactions in our simulations.}
455	<
456	<	\begin{tabular}{ccc}
453	>	\caption{Non-bonded interaction parameters (including cross
454	>	interactions with Au atoms) for both force fields used in this
455	>	work.}
456	>	\begin{tabular}{lllllll}
457		\hline\hline
458	<	Non-metal atom   & $\sigma$ & $\epsilon$ \\
459	<	(or pseudo-atom) & \AA      & kcal/mol  \\
458	>	& Site  & $\sigma_{ii}$ & $\epsilon_{ii}$ & $q_i$ &
459	>	$\sigma_{Au-i}$ & $\epsilon_{Au-i}$ \\
460	>	& & (\AA) & (kcal/mol) & ($e$) & (\AA) & (kcal/mol) \\
461		\hline
462	<	S    & 2.40   & 8.465   \\
463	<	CH3  & 3.54   & 0.2146  \\
464	<	CH2  & 3.54   & 0.1749  \\
465	<	CT3  & 3.365  & 0.1373  \\
466	<	CT2  & 3.365  & 0.1373  \\
467	<	CTT  & 3.365  & 0.1373  \\
468	<	HC   & 2.865  & 0.09256 \\
469	<	CHar & 3.4625 & 0.1680  \\
470	<	CRar & 3.555  & 0.1604  \\
471	<	CA   & 3.173  & 0.0640  \\
472	<	HA   & 2.746  & 0.0414  \\
462	>	United Atom (UA)
463	>	&CH3  & 3.75  & 0.1947  & -      & 3.54   & 0.2146  \\
464	>	&CH2  & 3.95  & 0.0914  & -      & 3.54   & 0.1749  \\
465	>	&CHar & 3.695 & 0.1003  & -      & 3.4625 & 0.1680  \\
466	>	&CRar & 3.88  & 0.04173 & -      & 3.555  & 0.1604  \\
467	>	\hline
468	>	All Atom (AA)
469	>	&CT3  & 3.50  & 0.066   & -0.18  & 3.365  & 0.1373  \\
470	>	&CT2  & 3.50  & 0.066   & -0.12  & 3.365  & 0.1373  \\
471	>	&CTT  & 3.50  & 0.066   & -0.065 & 3.365  & 0.1373  \\
472	>	&HC   & 2.50  & 0.030   &  0.06  & 2.865  & 0.09256 \\
473	>	&CA   & 3.55  & 0.070   & -0.115 & 3.173  & 0.0640  \\
474	>	&HA   & 2.42  & 0.030   &  0.115 & 2.746  & 0.0414  \\
475	>	\hline
476	>	Both UA and AA
477	>	& S   & 4.45  & 0.25    & -      & 2.40   & 8.465   \\
478		\hline\hline
479		\end{tabular}
480		\label{MnM}
#	Line 432 | Line 483 | parameters in our simulations.
483		\end{table*}
484
485
486	<	\section{Results and Discussions}
487	<	[MAY HAVE A BRIEF SUMMARY]
488	<	\subsection{How Simulation Parameters Affects $G$}
489	<	[MAY NOT PUT AT FIRST]
490	<	We have varied our protocol or other parameters of the simulations in
491	<	order to investigate how these factors would affect the measurement of
492	<	$G$'s. It turned out that while some of these parameters would not
493	<	affect the results substantially, some other changes to the
443	<	simulations would have a significant impact on the measurement
444	<	results.
486	>	\section{Results}
487	>	There are many factors contributing to the measured interfacial
488	>	conductance; some of these factors are physically motivated
489	>	(e.g. coverage of the surface by the capping agent coverage and
490	>	solvent identity), while some are governed by parameters of the
491	>	methodology (e.g. applied flux and the formulas used to obtain the
492	>	conductance). In this section we discuss the major physical and
493	>	calculational effects on the computed conductivity.
494
495	<	In some of our simulations, we allowed $L_x$ and $L_y$ to change
447	<	during equilibrating the liquid phase. Due to the stiffness of the Au
448	<	slab, $L_x$ and $L_y$ would not change noticeably after
449	<	equilibration. Although $L_z$ could fluctuate $\sim$1\% after a system
450	<	is fully equilibrated in the NPT ensemble, this fluctuation, as well
451	<	as those comparably smaller to $L_x$ and $L_y$, would not be magnified
452	<	on the calculated $G$'s, as shown in Table \ref{AuThiolHexaneUA}. This
453	<	insensivity to $L_i$ fluctuations allows reliable measurement of $G$'s
454	<	without the necessity of extremely cautious equilibration process.
495	>	\subsection{Effects due to capping agent coverage}
496
497	<	As stated in our computational details, the spacing filled with
498	<	solvent molecules can be chosen within a range. This allows some
499	<	change of solvent molecule numbers for the same Au-butanethiol
500	<	surfaces. We did this study on our Au-butanethiol/hexane
501	<	simulations. Nevertheless, the results obtained from systems of
502	<	different $N_{hexane}$ did not indicate that the measurement of $G$ is
462	<	susceptible to this parameter. For computational efficiency concern,
463	<	smaller system size would be preferable, given that the liquid phase
464	<	structure is not affected.
497	>	A series of different initial conditions with a range of surface
498	>	coverages was prepared and solvated with various with both of the
499	>	solvent molecules. These systems were then equilibrated and their
500	>	interfacial thermal conductivity was measured with the NIVS
501	>	algorithm. Figure \ref{coverage} demonstrates the trend of conductance
502	>	with respect to surface coverage.
503
504	<	Our NIVS algorithm allows change of unphysical thermal flux both in
505	<	direction and in quantity. This feature extends our investigation of
506	<	interfacial thermal conductance. However, the magnitude of this
507	<	thermal flux is not arbitary if one aims to obtain a stable and
508	<	reliable thermal gradient. A temperature profile would be
509	<	substantially affected by noise when $|J_z|$ has a much too low
510	<	magnitude; while an excessively large $|J_z|$ that overwhelms the
473	<	conductance capacity of the interface would prevent a thermal gradient
474	<	to reach a stablized steady state. NIVS has the advantage of allowing
475	<	$J$ to vary in a wide range such that the optimal flux range for $G$
476	<	measurement can generally be simulated by the algorithm. Within the
477	<	optimal range, we were able to study how $G$ would change according to
478	<	the thermal flux across the interface. For our simulations, we denote
479	<	$J_z$ to be positive when the physical thermal flux is from the liquid
480	<	to metal, and negative vice versa. The $G$'s measured under different
481	<	$J_z$ is listed in Table \ref{AuThiolHexaneUA} and [REF]. These
482	<	results do not suggest that $G$ is dependent on $J_z$ within this flux
483	<	range. The linear response of flux to thermal gradient simplifies our
484	<	investigations in that we can rely on $G$ measurement with only a
485	<	couple $J_z$'s and do not need to test a large series of fluxes.
504	>	\begin{figure}
505	>	\includegraphics[width=\linewidth]{coverage}
506	>	\caption{Comparison of interfacial thermal conductivity ($G$) values
507	>	for the Au-butanethiol/solvent interface with various UA models and
508	>	different capping agent coverages at $\langle T\rangle\sim$200K.}
509	>	\label{coverage}
510	>	\end{figure}
511
512	<	%ADD MORE TO TABLE
512	>	In partially covered surfaces, the derivative definition for
513	>	$G^\prime$ (Eq. \ref{derivativeG}) becomes difficult to apply, as the
514	>	location of maximum change of $\lambda$ becomes washed out.  The
515	>	discrete definition (Eq. \ref{discreteG}) is easier to apply, as the
516	>	Gibbs dividing surface is still well-defined. Therefore, $G$ (not
517	>	$G^\prime$) was used in this section.
518	>
519	>	From Figure \ref{coverage}, one can see the significance of the
520	>	presence of capping agents. When even a small fraction of the Au(111)
521	>	surface sites are covered with butanethiols, the conductivity exhibits
522	>	an enhancement by at least a factor of 3.  Cappping agents are clearly
523	>	playing a major role in thermal transport at metal / organic solvent
524	>	surfaces.
525	>
526	>	We note a non-monotonic behavior in the interfacial conductance as a
527	>	function of surface coverage. The maximum conductance (largest $G$)
528	>	happens when the surfaces are about 75\% covered with butanethiol
529	>	caps.  The reason for this behavior is not entirely clear.  One
530	>	explanation is that incomplete butanethiol coverage allows small gaps
531	>	between butanethiols to form. These gaps can be filled by transient
532	>	solvent molecules.  These solvent molecules couple very strongly with
533	>	the hot capping agent molecules near the surface, and can then carry
534	>	away (diffusively) the excess thermal energy from the surface.
535	>
536	>	There appears to be a competition between the conduction of the
537	>	thermal energy away from the surface by the capping agents (enhanced
538	>	by greater coverage) and the coupling of the capping agents with the
539	>	solvent (enhanced by interdigitation at lower coverages).  This
540	>	competition would lead to the non-monotonic coverage behavior observed
541	>	here.
542	>
543	>	Results for rigid body toluene solvent, as well as the UA hexane, are
544	>	within the ranges expected from prior experimental
545	>	work.\cite{Wilson:2002uq,cahill:793,PhysRevB.80.195406} This suggests
546	>	that explicit hydrogen atoms might not be required for modeling
547	>	thermal transport in these systems.  C-H vibrational modes do not see
548	>	significant excited state population at low temperatures, and are not
549	>	likely to carry lower frequency excitations from the solid layer into
550	>	the bulk liquid.
551	>
552	>	The toluene solvent does not exhibit the same behavior as hexane in
553	>	that $G$ remains at approximately the same magnitude when the capping
554	>	coverage increases from 25\% to 75\%.  Toluene, as a rigid planar
555	>	molecule, cannot occupy the relatively small gaps between the capping
556	>	agents as easily as the chain-like {\it n}-hexane.  The effect of
557	>	solvent coupling to the capping agent is therefore weaker in toluene
558	>	except at the very lowest coverage levels.  This effect counters the
559	>	coverage-dependent conduction of heat away from the metal surface,
560	>	leading to a much flatter $G$ vs. coverage trend than is observed in
561	>	{\it n}-hexane.
562	>
563	>	\subsection{Effects due to Solvent \& Solvent Models}
564	>	In addition to UA solvent and capping agent models, AA models have
565	>	also been included in our simulations.  In most of this work, the same
566	>	(UA or AA) model for solvent and capping agent was used, but it is
567	>	also possible to utilize different models for different components.
568	>	We have also included isotopic substitutions (Hydrogen to Deuterium)
569	>	to decrease the explicit vibrational overlap between solvent and
570	>	capping agent. Table \ref{modelTest} summarizes the results of these
571	>	studies.
572	>
573		\begin{table*}
574		\begin{minipage}{\linewidth}
575		\begin{center}
491	–	\caption{Computed interfacial thermal conductivity ($G$ and
492	–	$G^\prime$) values for the 100\% covered Au-butanethiol/hexane
493	–	interfaces with UA model and different hexane molecule numbers
494	–	at different temperatures using a range of energy fluxes.}
576
577	<	\begin{tabular}{cccccccc}
577	>	\caption{Computed interfacial thermal conductance ($G$ and
578	>	$G^\prime$) values for interfaces using various models for
579	>	solvent and capping agent (or without capping agent) at
580	>	$\langle T\rangle\sim$200K. (D stands for deuterated solvent
581	>	or capping agent molecules; ``Avg.'' denotes results that are
582	>	averages of simulations under different applied thermal flux
583	>	values $(J_z)$. Error estimates are indicated in
584	>	parentheses.)}
585	>
586	>	\begin{tabular}{llccc}
587		\hline\hline
588	<	$\langle T\rangle$ & & $L_x$ & $L_y$ & $L_z$ & $J_z$ &
589	<	$G$ & $G^\prime$ \\
500	<	(K) & $N_{hexane}$ & \multicolumn{3}{c}{(\AA)} & (GW/m$^2$) &
588	>	Butanethiol model & Solvent & $J_z$ & $G$ & $G^\prime$ \\
589	>	(or bare surface) & model & (GW/m$^2$) &
590		\multicolumn{2}{c}{(MW/m$^2$/K)} \\
591		\hline
592	<	200 & 266 & 29.86 & 25.80 & 113.1 & -0.96 &
593	<	102()  & 80.0() \\
594	<	& 200 & 29.84 & 25.81 &  93.9 &  1.92 &
595	<	129()  & 87.3() \\
596	<	&     & 29.84 & 25.81 &  95.3 &  1.93 &
597	<	131()  & 77.5() \\
598	<	& 166 & 29.84 & 25.81 &  85.7 &  0.97 &
599	<	115()  & 69.3() \\
600	<	&     &       &       &       &  1.94 &
601	<	125()  & 87.1() \\
602	<	250 & 200 & 29.84 & 25.87 & 106.8 &  0.96 &
603	<	81.8() & 67.0() \\
604	<	& 166 & 29.87 & 25.84 &  94.8 &  0.98 &
605	<	79.0() & 62.9() \\
606	<	&     & 29.84 & 25.85 &  95.0 &  1.44 &
518	<	76.2() & 64.8() \\
519	<	\hline\hline
520	<	\end{tabular}
521	<	\label{AuThiolHexaneUA}
522	<	\end{center}
523	<	\end{minipage}
524	<	\end{table*}
525	<
526	<	Furthermore, we also attempted to increase system average temperatures
527	<	to above 200K. These simulations are first equilibrated in the NPT
528	<	ensemble under normal pressure. As stated above, the TraPPE-UA model
529	<	for hexane tends to predict a lower boiling point. In our simulations,
530	<	hexane had diffculty to remain in liquid phase when NPT equilibration
531	<	temperature is higher than 250K. Additionally, the equilibrated liquid
532	<	hexane density under 250K becomes lower than experimental value. This
533	<	expanded liquid phase leads to lower contact between hexane and
534	<	butanethiol as well.[MAY NEED FIGURE] And this reduced contact would
535	<	probably be accountable for a lower interfacial thermal conductance,
536	<	as shown in Table \ref{AuThiolHexaneUA}.
537	<
538	<	A similar study for TraPPE-UA toluene agrees with the above result as
539	<	well. Having a higher boiling point, toluene tends to remain liquid in
540	<	our simulations even equilibrated under 300K in NPT
541	<	ensembles. Furthermore, the expansion of the toluene liquid phase is
542	<	not as significant as that of the hexane. This prevents severe
543	<	decrease of liquid-capping agent contact and the results (Table
544	<	\ref{AuThiolToluene}) show only a slightly decreased interface
545	<	conductance. Therefore, solvent-capping agent contact should play an
546	<	important role in the thermal transport process across the interface
547	<	in that higher degree of contact could yield increased conductance.
548	<
549	<	[ADD Lxyz AND ERROR ESTIMATE TO TABLE]
550	<	\begin{table*}
551	<	\begin{minipage}{\linewidth}
552	<	\begin{center}
553	<	\caption{Computed interfacial thermal conductivity ($G$ and
554	<	$G^\prime$) values for a 90\% coverage Au-butanethiol/toluene
555	<	interface at different temperatures using a range of energy
556	<	fluxes.}
557	<
558	<	\begin{tabular}{cccc}
559	<	\hline\hline
560	<	$\langle T\rangle$ & $J_z$ & $G$ & $G^\prime$ \\
561	<	(K) & (GW/m$^2$) & \multicolumn{2}{c}{(MW/m$^2$/K)} \\
592	>	UA    & UA hexane    & Avg. & 131(9)    & 87(10)    \\
593	>	& UA hexane(D) & 1.95 & 153(5)    & 136(13)   \\
594	>	& AA hexane    & Avg. & 131(6)    & 122(10)   \\
595	>	& UA toluene   & 1.96 & 187(16)   & 151(11)   \\
596	>	& AA toluene   & 1.89 & 200(36)   & 149(53)   \\
597	>	\hline
598	>	AA    & UA hexane    & 1.94 & 116(9)    & 129(8)    \\
599	>	& AA hexane    & Avg. & 442(14)   & 356(31)   \\
600	>	& AA hexane(D) & 1.93 & 222(12)   & 234(54)   \\
601	>	& UA toluene   & 1.98 & 125(25)   & 97(60)    \\
602	>	& AA toluene   & 3.79 & 487(56)   & 290(42)   \\
603	>	\hline
604	>	AA(D) & UA hexane    & 1.94 & 158(25)   & 172(4)    \\
605	>	& AA hexane    & 1.92 & 243(29)   & 191(11)   \\
606	>	& AA toluene   & 1.93 & 364(36)   & 322(67)   \\
607		\hline
608	<	200 & -1.86 & 180() & 135() \\
609	<	&  2.15 & 204() & 113() \\
610	<	& -3.93 & 175() & 114() \\
611	<	300 & -1.91 & 143() & 125() \\
567	<	& -4.19 & 134() & 113() \\
608	>	bare  & UA hexane    & Avg. & 46.5(3.2) & 49.4(4.5) \\
609	>	& UA hexane(D) & 0.98 & 43.9(4.6) & 43.0(2.0) \\
610	>	& AA hexane    & 0.96 & 31.0(1.4) & 29.4(1.3) \\
611	>	& UA toluene   & 1.99 & 70.1(1.3) & 65.8(0.5) \\
612		\hline\hline
613		\end{tabular}
614	<	\label{AuThiolToluene}
614	>	\label{modelTest}
615		\end{center}
616		\end{minipage}
617		\end{table*}
618
619	<	Besides lower interfacial thermal conductance, surfaces in relatively
620	<	high temperatures are susceptible to reconstructions, when
621	<	butanethiols have a full coverage on the Au(111) surface. These
578	<	reconstructions include surface Au atoms migrated outward to the S
579	<	atom layer, and butanethiol molecules embedded into the original
580	<	surface Au layer. The driving force for this behavior is the strong
581	<	Au-S interactions in our simulations. And these reconstructions lead
582	<	to higher ratio of Au-S attraction and thus is energetically
583	<	favorable. Furthermore, this phenomenon agrees with experimental
584	<	results\cite{doi:10.1021/j100035a033,doi:10.1021/la026493y}. Vlugt
585	<	{\it et al.} had kept their Au(111) slab rigid so that their
586	<	simulations can reach 300K without surface reconstructions. Without
587	<	this practice, simulating 100\% thiol covered interfaces under higher
588	<	temperatures could hardly avoid surface reconstructions. However, our
589	<	measurement is based on assuming homogeneity on $x$ and $y$ dimensions
590	<	so that measurement of $T$ at particular $z$ would be an effective
591	<	average of the particles of the same type. Since surface
592	<	reconstructions could eliminate the original $x$ and $y$ dimensional
593	<	homogeneity, measurement of $G$ is more difficult to conduct under
594	<	higher temperatures. Therefore, most of our measurements are
595	<	undertaken at $\langle T\rangle\sim$200K.
619	>	To facilitate direct comparison between force fields, systems with the
620	>	same capping agent and solvent were prepared with the same length
621	>	scales for the simulation cells.
622
623	<	However, when the surface is not completely covered by butanethiols,
624	<	the simulated system is more resistent to the reconstruction
625	<	above. Our Au-butanethiol/toluene system did not see this phenomena
626	<	even at $\langle T\rangle\sim$300K. The Au(111) surfaces have a 90\% coverage of
627	<	butanethiols and have empty three-fold sites. These empty sites could
628	<	help prevent surface reconstruction in that they provide other means
603	<	of capping agent relaxation. It is observed that butanethiols can
604	<	migrate to their neighbor empty sites during a simulation. Therefore,
605	<	we were able to obtain $G$'s for these interfaces even at a relatively
606	<	high temperature without being affected by surface reconstructions.
623	>	On bare metal / solvent surfaces, different force field models for
624	>	hexane yield similar results for both $G$ and $G^\prime$, and these
625	>	two definitions agree with each other very well. This is primarily an
626	>	indicator of weak interactions between the metal and the solvent, and
627	>	is a typical case for acoustic impedance mismatch between these two
628	>	phases.
629
630	<	\subsection{Influence of Capping Agent Coverage on $G$}
631	<	To investigate the influence of butanethiol coverage on interfacial
632	<	thermal conductance, a series of different coverage Au-butanethiol
633	<	surfaces is prepared and solvated with various organic
634	<	molecules. These systems are then equilibrated and their interfacial
635	<	thermal conductivity are measured with our NIVS algorithm. Table
636	<	\ref{tlnUhxnUhxnD} lists these results for direct comparison between
637	<	different coverages of butanethiol. To study the isotope effect in
638	<	interfacial thermal conductance, deuterated UA-hexane is included as
639	<	well.
630	>	For the fully-covered surfaces, the choice of force field for the
631	>	capping agent and solvent has a large impact on the calulated values
632	>	of $G$ and $G^\prime$.  The AA thiol to AA solvent conductivities are
633	>	much larger than their UA to UA counterparts, and these values exceed
634	>	the experimental estimates by a large measure.  The AA force field
635	>	allows significant energy to go into C-H (or C-D) stretching modes,
636	>	and since these modes are high frequency, this non-quantum behavior is
637	>	likely responsible for the overestimate of the conductivity.  Compared
638	>	to the AA model, the UA model yields more reasonable conductivity
639	>	values with much higher computational efficiency.
640
641	<	It turned out that with partial covered butanethiol on the Au(111)
642	<	surface, the derivative definition for $G$ (Eq. \ref{derivativeG}) has
643	<	difficulty to apply, due to the difficulty in locating the maximum of
644	<	change of $\lambda$. Instead, the discrete definition
645	<	(Eq. \ref{discreteG}) is easier to apply, as max($\Delta T$) can still
646	<	be well-defined. Therefore, $G$'s (not $G^\prime$) are used for this
647	<	section.
641	>	\subsubsection{Are electronic excitations in the metal important?}
642	>	Because they lack electronic excitations, the QSC and related embedded
643	>	atom method (EAM) models for gold are known to predict unreasonably
644	>	low values for bulk conductivity
645	>	($\lambda$).\cite{kuang:164101,ISI:000207079300006,Clancy:1992} If the
646	>	conductance between the phases ($G$) is governed primarily by phonon
647	>	excitation (and not electronic degrees of freedom), one would expect a
648	>	classical model to capture most of the interfacial thermal
649	>	conductance.  Our results for $G$ and $G^\prime$ indicate that this is
650	>	indeed the case, and suggest that the modeling of interfacial thermal
651	>	transport depends primarily on the description of the interactions
652	>	between the various components at the interface.  When the metal is
653	>	chemically capped, the primary barrier to thermal conductivity appears
654	>	to be the interface between the capping agent and the surrounding
655	>	solvent, so the excitations in the metal have little impact on the
656	>	value of $G$.
657
658	<	From Table \ref{tlnUhxnUhxnD}, one can see the significance of the
628	<	presence of capping agents. Even when a fraction of the Au(111)
629	<	surface sites are covered with butanethiols, the conductivity would
630	<	see an enhancement by at least a factor of 3. This indicates the
631	<	important role cappping agent is playing for thermal transport
632	<	phenomena on metal/organic solvent surfaces.
658	>	\subsection{Effects due to methodology and simulation parameters}
659
660	<	Interestingly, as one could observe from our results, the maximum
661	<	conductance enhancement (largest $G$) happens while the surfaces are
662	<	about 75\% covered with butanethiols. This again indicates that
663	<	solvent-capping agent contact has an important role of the thermal
664	<	transport process. Slightly lower butanethiol coverage allows small
665	<	gaps between butanethiols to form. And these gaps could be filled with
666	<	solvent molecules, which acts like ``heat conductors'' on the
667	<	surface. The higher degree of interaction between these solvent
668	<	molecules and capping agents increases the enhancement effect and thus
643	<	produces a higher $G$ than densely packed butanethiol arrays. However,
644	<	once this maximum conductance enhancement is reached, $G$ decreases
645	<	when butanethiol coverage continues to decrease. Each capping agent
646	<	molecule reaches its maximum capacity for thermal
647	<	conductance. Therefore, even higher solvent-capping agent contact
648	<	would not offset this effect. Eventually, when butanethiol coverage
649	<	continues to decrease, solvent-capping agent contact actually
650	<	decreases with the disappearing of butanethiol molecules. In this
651	<	case, $G$ decrease could not be offset but instead accelerated.
660	>	We have varied the parameters of the simulations in order to
661	>	investigate how these factors would affect the computation of $G$.  Of
662	>	particular interest are: 1) the length scale for the applied thermal
663	>	gradient (modified by increasing the amount of solvent in the system),
664	>	2) the sign and magnitude of the applied thermal flux, 3) the average
665	>	temperature of the simulation (which alters the solvent density during
666	>	equilibration), and 4) the definition of the interfacial conductance
667	>	(Eqs. (\ref{discreteG}) or (\ref{derivativeG})) used in the
668	>	calculation.
669
670	<	A comparison of the results obtained from differenet organic solvents
671	<	can also provide useful information of the interfacial thermal
672	<	transport process. The deuterated hexane (UA) results do not appear to
673	<	be much different from those of normal hexane (UA), given that
674	<	butanethiol (UA) is non-deuterated for both solvents. These UA model
675	<	studies, even though eliminating C-H vibration samplings, still have
676	<	C-C vibrational frequencies different from each other. However, these
677	<	differences in the infrared range do not seem to produce an observable
678	<	difference for the results of $G$. [MAY NEED FIGURE]
670	>	Systems of different lengths were prepared by altering the number of
671	>	solvent molecules and extending the length of the box along the $z$
672	>	axis to accomodate the extra solvent.  Equilibration at the same
673	>	temperature and pressure conditions led to nearly identical surface
674	>	areas ($L_x$ and $L_y$) available to the metal and capping agent,
675	>	while the extra solvent served mainly to lengthen the axis that was
676	>	used to apply the thermal flux.  For a given value of the applied
677	>	flux, the different $z$ length scale has only a weak effect on the
678	>	computed conductivities (Table \ref{AuThiolHexaneUA}).
679
680	<	Furthermore, results for rigid body toluene solvent, as well as other
681	<	UA-hexane solvents, are reasonable within the general experimental
682	<	ranges[CITATIONS]. This suggests that explicit hydrogen might not be a
683	<	required factor for modeling thermal transport phenomena of systems
684	<	such as Au-thiol/organic solvent.
680	>	\subsubsection{Effects of applied flux}
681	>	The NIVS algorithm allows changes in both the sign and magnitude of
682	>	the applied flux.  It is possible to reverse the direction of heat
683	>	flow simply by changing the sign of the flux, and thermal gradients
684	>	which would be difficult to obtain experimentally ($5$ K/\AA) can be
685	>	easily simulated.  However, the magnitude of the applied flux is not
686	>	arbitary if one aims to obtain a stable and reliable thermal gradient.
687	>	A temperature gradient can be lost in the noise if $|J_z|$ is too
688	>	small, and excessive $|J_z|$ values can cause phase transitions if the
689	>	extremes of the simulation cell become widely separated in
690	>	temperature.  Also, if $|J_z|$ is too large for the bulk conductivity
691	>	of the materials, the thermal gradient will never reach a stable
692	>	state.
693
694	<	However, results for Au-butanethiol/toluene do not show an identical
695	<	trend with those for Au-butanethiol/hexane in that $G$'s remain at
696	<	approximately the same magnitue when butanethiol coverage differs from
697	<	25\% to 75\%. This might be rooted in the molecule shape difference
698	<	for plane-like toluene and chain-like {\it n}-hexane. Due to this
699	<	difference, toluene molecules have more difficulty in occupying
700	<	relatively small gaps among capping agents when their coverage is not
701	<	too low. Therefore, the solvent-capping agent contact may keep
702	<	increasing until the capping agent coverage reaches a relatively low
703	<	level. This becomes an offset for decreasing butanethiol molecules on
704	<	its effect to the process of interfacial thermal transport. Thus, one
705	<	can see a plateau of $G$ vs. butanethiol coverage in our results.
694	>	Within a reasonable range of $J_z$ values, we were able to study how
695	>	$G$ changes as a function of this flux.  In what follows, we use
696	>	positive $J_z$ values to denote the case where energy is being
697	>	transferred by the method from the metal phase and into the liquid.
698	>	The resulting gradient therefore has a higher temperature in the
699	>	liquid phase.  Negative flux values reverse this transfer, and result
700	>	in higher temperature metal phases.  The conductance measured under
701	>	different applied $J_z$ values is listed in Tables
702	>	\ref{AuThiolHexaneUA} and \ref{AuThiolToluene}. These results do not
703	>	indicate that $G$ depends strongly on $J_z$ within this flux
704	>	range. The linear response of flux to thermal gradient simplifies our
705	>	investigations in that we can rely on $G$ measurement with only a
706	>	small number $J_z$ values.
707
682	–	[NEED ERROR ESTIMATE, MAY ALSO PUT J HERE]
708		\begin{table*}
709		\begin{minipage}{\linewidth}
710		\begin{center}
711	<	\caption{Computed interfacial thermal conductivity ($G$) values
712	<	for the Au-butanethiol/solvent interface with various UA
713	<	models and different capping agent coverages at $\langle
714	<	T\rangle\sim$200K using certain energy flux respectively.}
711	>	\caption{Computed interfacial thermal conductivity ($G$ and
712	>	$G^\prime$) values for the 100\% covered Au-butanethiol/hexane
713	>	interfaces with UA model and different hexane molecule numbers
714	>	at different temperatures using a range of energy
715	>	fluxes. Error estimates indicated in parenthesis.}
716
717	<	\begin{tabular}{cccc}
717	>	\begin{tabular}{ccccccc}
718		\hline\hline
719	<	Thiol & \multicolumn{3}{c}{$G$(MW/m$^2$/K)} \\
720	<	coverage (\%) & hexane & hexane(D) & toluene \\
719	>	$\langle T\rangle$ & $N_{hexane}$  & $\rho_{hexane}$ &
720	>	$J_z$ & $G$ & $G^\prime$ \\
721	>	(K) &  & (g/cm$^3$) & (GW/m$^2$) &
722	>	\multicolumn{2}{c}{(MW/m$^2$/K)} \\
723		\hline
724	<	0.0   & 46.5() & 43.9() & 70.1() \\
725	<	25.0  & 151()  & 153()  & 249()  \\
726	<	50.0  & 172()  & 182()  & 214()  \\
727	<	75.0  & 242()  & 229()  & 244()  \\
728	<	88.9  & 178()  & -      & -      \\
729	<	100.0 & 137()  & 153()  & 187()  \\
724	>	200 & 266 &  0.672 & -0.96 & 102(3)    & 80.0(0.8) \\
725	>	& 200 &  0.688 &  0.96 & 125(16)   & 90.2(15)  \\
726	>	&     &        &  1.91 & 139(10)   & 101(10)   \\
727	>	&     &        &  2.83 & 141(6)    & 89.9(9.8) \\
728	>	& 166 &  0.681 &  0.97 & 141(30)   & 78(22)    \\
729	>	&     &        &  1.92 & 138(4)    & 98.9(9.5) \\
730	>	\hline
731	>	250 & 200 &  0.560 &  0.96 & 75(10)    & 61.8(7.3) \\
732	>	&     &        & -0.95 & 49.4(0.3) & 45.7(2.1) \\
733	>	& 166 &  0.569 &  0.97 & 80.3(0.6) & 67(11)    \\
734	>	&     &        &  1.44 & 76.2(5.0) & 64.8(3.8) \\
735	>	&     &        & -0.95 & 56.4(2.5) & 54.4(1.1) \\
736	>	&     &        & -1.85 & 47.8(1.1) & 53.5(1.5) \\
737		\hline\hline
738		\end{tabular}
739	<	\label{tlnUhxnUhxnD}
739	>	\label{AuThiolHexaneUA}
740		\end{center}
741		\end{minipage}
742		\end{table*}
743
744	<	\subsection{Influence of Chosen Molecule Model on $G$}
745	<	[MAY COMBINE W MECHANISM STUDY]
744	>	The sign of $J_z$ is a different matter, however, as this can alter
745	>	the temperature on the two sides of the interface. The average
746	>	temperature values reported are for the entire system, and not for the
747	>	liquid phase, so at a given $\langle T \rangle$, the system with
748	>	positive $J_z$ has a warmer liquid phase.  This means that if the
749	>	liquid carries thermal energy via convective transport, {\it positive}
750	>	$J_z$ values will result in increased molecular motion on the liquid
751	>	side of the interface, and this will increase the measured
752	>	conductivity.
753
754	<	In addition to UA solvent/capping agent models, AA models are included
713	<	in our simulations as well. Besides simulations of the same (UA or AA)
714	<	model for solvent and capping agent, different models can be applied
715	<	to different components. Furthermore, regardless of models chosen,
716	<	either the solvent or the capping agent can be deuterated, similar to
717	<	the previous section. Table \ref{modelTest} summarizes the results of
718	<	these studies.
754	>	\subsubsection{Effects due to average temperature}
755
756	<	[MORE DATA; ERROR ESTIMATE]
756	>	We also studied the effect of average system temperature on the
757	>	interfacial conductance.  The simulations are first equilibrated in
758	>	the NPT ensemble at 1 atm.  The TraPPE-UA model for hexane tends to
759	>	predict a lower boiling point (and liquid state density) than
760	>	experiments.  This lower-density liquid phase leads to reduced contact
761	>	between the hexane and butanethiol, and this accounts for our
762	>	observation of lower conductance at higher temperatures as shown in
763	>	Table \ref{AuThiolHexaneUA}.  In raising the average temperature from
764	>	200K to 250K, the density drop of ~20\% in the solvent phase leads to
765	>	a ~65\% drop in the conductance.
766	>
767	>	Similar behavior is observed in the TraPPE-UA model for toluene,
768	>	although this model has better agreement with the experimental
769	>	densities of toluene.  The expansion of the toluene liquid phase is
770	>	not as significant as that of the hexane (8.3\% over 100K), and this
771	>	limits the effect to ~20\% drop in thermal conductivity  (Table
772	>	\ref{AuThiolToluene}).
773	>
774	>	Although we have not mapped out the behavior at a large number of
775	>	temperatures, is clear that there will be a strong temperature
776	>	dependence in the interfacial conductance when the physical properties
777	>	of one side of the interface (notably the density) change rapidly as a
778	>	function of temperature.
779	>
780		\begin{table*}
781		\begin{minipage}{\linewidth}
782		\begin{center}
724	–
783		\caption{Computed interfacial thermal conductivity ($G$ and
784	<	$G^\prime$) values for interfaces using various models for
785	<	solvent and capping agent (or without capping agent) at
786	<	$\langle T\rangle\sim$200K.}
784	>	$G^\prime$) values for a 90\% coverage Au-butanethiol/toluene
785	>	interface at different temperatures using a range of energy
786	>	fluxes. Error estimates indicated in parenthesis.}
787
788		\begin{tabular}{ccccc}
789		\hline\hline
790	<	Butanethiol model & Solvent & $J_z$ & $G$ & $G^\prime$ \\
791	<	(or bare surface) & model & (GW/m$^2$) &
734	<	\multicolumn{2}{c}{(MW/m$^2$/K)} \\
790	>	$\langle T\rangle$ & $\rho_{toluene}$ & $J_z$ & $G$ & $G^\prime$ \\
791	>	(K) & (g/cm$^3$) & (GW/m$^2$) & \multicolumn{2}{c}{(MW/m$^2$/K)} \\
792		\hline
793	<	UA    & AA hexane  & 1.94 & 135()  & 129()  \\
794	<	&            & 2.86 & 126()  & 115()  \\
795	<	& AA toluene & 1.89 & 200()  & 149()  \\
796	<	AA    & UA hexane  & 1.94 & 116()  & 129()  \\
797	<	& AA hexane  & 3.76 & 451()  & 378()  \\
798	<	&            & 4.71 & 432()  & 334()  \\
742	<	& AA toluene & 3.79 & 487()  & 290()  \\
743	<	AA(D) & UA hexane  & 1.94 & 158()  & 172()  \\
744	<	bare  & AA hexane  & 0.96 & 31.0() & 29.4() \\
793	>	200 & 0.933 &  2.15 & 204(12) & 113(12) \\
794	>	&       & -1.86 & 180(3)  & 135(21) \\
795	>	&       & -3.93 & 176(5)  & 113(12) \\
796	>	\hline
797	>	300 & 0.855 & -1.91 & 143(5)  & 125(2)  \\
798	>	&       & -4.19 & 135(9)  & 113(12) \\
799		\hline\hline
800		\end{tabular}
801	<	\label{modelTest}
801	>	\label{AuThiolToluene}
802		\end{center}
803		\end{minipage}
804		\end{table*}
805
806	<	To facilitate direct comparison, the same system with differnt models
807	<	for different components uses the same length scale for their
808	<	simulation cells. Without the presence of capping agent, using
809	<	different models for hexane yields similar results for both $G$ and
810	<	$G^\prime$, and these two definitions agree with eath other very
811	<	well. This indicates very weak interaction between the metal and the
812	<	solvent, and is a typical case for acoustic impedance mismatch between
813	<	these two phases.
806	>	Besides the lower interfacial thermal conductance, surfaces at
807	>	relatively high temperatures are susceptible to reconstructions,
808	>	particularly when butanethiols fully cover the Au(111) surface. These
809	>	reconstructions include surface Au atoms which migrate outward to the
810	>	S atom layer, and butanethiol molecules which embed into the surface
811	>	Au layer. The driving force for this behavior is the strong Au-S
812	>	interactions which are modeled here with a deep Lennard-Jones
813	>	potential. This phenomenon agrees with reconstructions that have beeen
814	>	experimentally
815	>	observed.\cite{doi:10.1021/j100035a033,doi:10.1021/la026493y}.  Vlugt
816	>	{\it et al.} kept their Au(111) slab rigid so that their simulations
817	>	could reach 300K without surface
818	>	reconstructions.\cite{vlugt:cpc2007154} Since surface reconstructions
819	>	blur the interface, the measurement of $G$ becomes more difficult to
820	>	conduct at higher temperatures.  For this reason, most of our
821	>	measurements are undertaken at $\langle T\rangle\sim$200K where
822	>	reconstruction is minimized.
823
824	<	As for Au(111) surfaces completely covered by butanethiols, the choice
825	<	of models for capping agent and solvent could impact the measurement
826	<	of $G$ and $G^\prime$ quite significantly. For Au-butanethiol/hexane
827	<	interfaces, using AA model for both butanethiol and hexane yields
828	<	substantially higher conductivity values than using UA model for at
829	<	least one component of the solvent and capping agent, which exceeds
830	<	the upper bond of experimental value range. This is probably due to
831	<	the classically treated C-H vibrations in the AA model, which should
832	<	not be appreciably populated at normal temperatures. In comparison,
770	<	once either the hexanes or the butanethiols are deuterated, one can
771	<	see a significantly lower $G$ and $G^\prime$. In either of these
772	<	cases, the C-H(D) vibrational overlap between the solvent and the
773	<	capping agent is removed. [MAY NEED FIGURE] Conclusively, the
774	<	improperly treated C-H vibration in the AA model produced
775	<	over-predicted results accordingly. Compared to the AA model, the UA
776	<	model yields more reasonable results with higher computational
777	<	efficiency.
824	>	However, when the surface is not completely covered by butanethiols,
825	>	the simulated system appears to be more resistent to the
826	>	reconstruction. O ur Au / butanethiol / toluene system had the Au(111)
827	>	surfaces 90\% covered by butanethiols, but did not see this above
828	>	phenomena even at $\langle T\rangle\sim$300K.  That said, we did
829	>	observe butanethiols migrating to neighboring three-fold sites during
830	>	a simulation.  Since the interface persisted in these simulations,
831	>	were able to obtain $G$'s for these interfaces even at a relatively
832	>	high temperature without being affected by surface reconstructions.
833
834	<	However, for Au-butanethiol/toluene interfaces, having the AA
780	<	butanethiol deuterated did not yield a significant change in the
781	<	measurement results.
782	<	. , so extra degrees of freedom
783	<	such as the C-H vibrations could enhance heat exchange between these
784	<	two phases and result in a much higher conductivity.
834	>	\section{Discussion}
835
836	+	The primary result of this work is that the capping agent acts as an
837	+	efficient thermal coupler between solid and solvent phases.  One of
838	+	the ways the capping agent can carry out this role is to down-shift
839	+	between the phonon vibrations in the solid (which carry the heat from
840	+	the gold) and the molecular vibrations in the liquid (which carry some
841	+	of the heat in the solvent).
842
843	<	Although the QSC model for Au is known to predict an overly low value
844	<	for bulk metal gold conductivity[CITE NIVSRNEMD], our computational
845	<	results for $G$ and $G^\prime$ do not seem to be affected by this
846	<	drawback of the model for metal. Instead, the modeling of interfacial
847	<	thermal transport behavior relies mainly on an accurate description of
848	<	the interactions between components occupying the interfaces.
843	>	To investigate the mechanism of interfacial thermal conductance, the
844	>	vibrational power spectrum was computed. Power spectra were taken for
845	>	individual components in different simulations. To obtain these
846	>	spectra, simulations were run after equilibration in the
847	>	microcanonical (NVE) ensemble and without a thermal
848	>	gradient. Snapshots of configurations were collected at a frequency
849	>	that is higher than that of the fastest vibrations occuring in the
850	>	simulations. With these configurations, the velocity auto-correlation
851	>	functions can be computed:
852	>	\begin{equation}
853	>	C_A (t) = \langle\vec{v}_A (t)\cdot\vec{v}_A (0)\rangle
854	>	\label{vCorr}
855	>	\end{equation}
856	>	The power spectrum is constructed via a Fourier transform of the
857	>	symmetrized velocity autocorrelation function,
858	>	\begin{equation}
859	>	\hat{f}(\omega) =
860	>	\int_{-\infty}^{\infty} C_A (t) e^{-2\pi it\omega}\,dt
861	>	\label{fourier}
862	>	\end{equation}
863
864	<	\subsection{Mechanism of Interfacial Thermal Conductance Enhancement
865	<	by Capping Agent}
866	<	%OR\subsection{Vibrational spectrum study on conductance mechanism}
864	>	\subsection{The role of specific vibrations}
865	>	The vibrational spectra for gold slabs in different environments are
866	>	shown as in Figure \ref{specAu}. Regardless of the presence of
867	>	solvent, the gold surfaces which are covered by butanethiol molecules
868	>	exhibit an additional peak observed at a frequency of
869	>	$\sim$165cm$^{-1}$.  We attribute this peak to the S-Au bonding
870	>	vibration. This vibration enables efficient thermal coupling of the
871	>	surface Au layer to the capping agents. Therefore, in our simulations,
872	>	the Au / S interfaces do not appear to be the primary barrier to
873	>	thermal transport when compared with the butanethiol / solvent
874	>	interfaces.
875
876	<	[MAY INTRODUCE PROTOCOL IN METHODOLOGY/COMPUTATIONAL DETAIL, EQN'S]
876	>	\begin{figure}
877	>	\includegraphics[width=\linewidth]{vibration}
878	>	\caption{Vibrational power spectra for gold in different solvent
879	>	environments.  The presence of the butanethiol capping molecules
880	>	adds a vibrational peak at $\sim$165cm$^{-1}$. The butanethiol
881	>	spectra exhibit a corresponding peak.}
882	>	\label{specAu}
883	>	\end{figure}
884
885	<	To investigate the mechanism of this interfacial thermal conductance,
886	<	the vibrational spectra of various gold systems were obtained and are
887	<	shown as in the upper panel of Fig. \ref{vibration}. To obtain these
803	<	spectra, one first runs a simulation in the NVE ensemble and collects
804	<	snapshots of configurations; these configurations are used to compute
805	<	the velocity auto-correlation functions, which is used to construct a
806	<	power spectrum via a Fourier transform.
885	>	Also in this figure, we show the vibrational power spectrum for the
886	>	bound butanethiol molecules, which also exhibits the same
887	>	$\sim$165cm$^{-1}$ peak.
888
889	<	The gold surfaces covered by
890	<	butanethiol molecules, compared to bare gold surfaces, exhibit an
891	<	additional peak observed at a frequency of $\sim$170cm$^{-1}$, which
892	<	is attributed to the vibration of the S-Au bond. This vibration
893	<	enables efficient thermal transport from surface Au atoms to the
894	<	capping agents. Simultaneously, as shown in the lower panel of
895	<	Fig. \ref{vibration}, the large overlap of the vibration spectra of
896	<	butanethiol and hexane in the all-atom model, including the C-H
897	<	vibration, also suggests high thermal exchange efficiency. The
898	<	combination of these two effects produces the drastic interfacial
899	<	thermal conductance enhancement in the all-atom model.
889	>	\subsection{Overlap of power spectra}
890	>	A comparison of the results obtained from the two different organic
891	>	solvents can also provide useful information of the interfacial
892	>	thermal transport process.  In particular, the vibrational overlap
893	>	between the butanethiol and the organic solvents suggests a highly
894	>	efficient thermal exchange between these components.  Very high
895	>	thermal conductivity was observed when AA models were used and C-H
896	>	vibrations were treated classically.  The presence of extra degrees of
897	>	freedom in the AA force field yields higher heat exchange rates
898	>	between the two phases and results in a much higher conductivity than
899	>	in the UA force field.
900
901	<	[MAY NEED TO CONVERT TO JPEG]
901	>	The similarity in the vibrational modes available to solvent and
902	>	capping agent can be reduced by deuterating one of the two components
903	>	(Fig. \ref{aahxntln}).  Once either the hexanes or the butanethiols
904	>	are deuterated, one can observe a significantly lower $G$ and
905	>	$G^\prime$ values (Table \ref{modelTest}).
906	>
907		\begin{figure}
908	<	\includegraphics[width=\linewidth]{vibration}
909	<	\caption{Vibrational spectra obtained for gold in different
910	<	environments (upper panel) and for Au/thiol/hexane simulation in
911	<	all-atom model (lower panel).}
912	<	\label{vibration}
908	>	\includegraphics[width=\linewidth]{aahxntln}
909	>	\caption{Spectra obtained for all-atom (AA) Au / butanethiol / solvent
910	>	systems. When butanethiol is deuterated (lower left), its
911	>	vibrational overlap with hexane decreases significantly.  Since
912	>	aromatic molecules and the butanethiol are vibrationally dissimilar,
913	>	the change is not as dramatic when toluene is the solvent (right).}
914	>	\label{aahxntln}
915		\end{figure}
916
917	<	[COMPARISON OF TWO G'S; AU SLAB WIDTHS; ETC]
918	<	% The results show that the two definitions used for $G$ yield
919	<	% comparable values, though $G^\prime$ tends to be smaller.
917	>	For the Au / butanethiol / toluene interfaces, having the AA
918	>	butanethiol deuterated did not yield a significant change in the
919	>	measured conductance. Compared to the C-H vibrational overlap between
920	>	hexane and butanethiol, both of which have alkyl chains, the overlap
921	>	between toluene and butanethiol is not as significant and thus does
922	>	not contribute as much to the heat exchange process.
923
924	+	Previous observations of Zhang {\it et al.}\cite{hase:2010} indicate
925	+	that the {\it intra}molecular heat transport due to alkylthiols is
926	+	highly efficient.  Combining our observations with those of Zhang {\it
927	+	et al.}, it appears that butanethiol acts as a channel to expedite
928	+	heat flow from the gold surface and into the alkyl chain.  The
929	+	acoustic impedance mismatch between the metal and the liquid phase can
930	+	therefore be effectively reduced with the presence of suitable capping
931	+	agents.
932	+
933	+	Deuterated models in the UA force field did not decouple the thermal
934	+	transport as well as in the AA force field.  The UA models, even
935	+	though they have eliminated the high frequency C-H vibrational
936	+	overlap, still have significant overlap in the lower-frequency
937	+	portions of the infrared spectra (Figure \ref{uahxnua}).  Deuterating
938	+	the UA models did not decouple the low frequency region enough to
939	+	produce an observable difference for the results of $G$ (Table
940	+	\ref{modelTest}).
941	+
942	+	\begin{figure}
943	+	\includegraphics[width=\linewidth]{uahxnua}
944	+	\caption{Vibrational spectra obtained for normal (upper) and
945	+	deuterated (lower) hexane in Au-butanethiol/hexane
946	+	systems. Butanethiol spectra are shown as reference. Both hexane and
947	+	butanethiol were using United-Atom models.}
948	+	\label{uahxnua}
949	+	\end{figure}
950	+
951		\section{Conclusions}
952	<	The NIVS algorithm we developed has been applied to simulations of
953	<	Au-butanethiol surfaces with organic solvents. This algorithm allows
954	<	effective unphysical thermal flux transferred between the metal and
955	<	the liquid phase. With the flux applied, we were able to measure the
956	<	corresponding thermal gradient and to obtain interfacial thermal
957	<	conductivities. Our simulations have seen significant conductance
958	<	enhancement with the presence of capping agent, compared to the bare
959	<	gold/liquid interfaces. The acoustic impedance mismatch between the
960	<	metal and the liquid phase is effectively eliminated by proper capping
952	>	The NIVS algorithm has been applied to simulations of
953	>	butanethiol-capped Au(111) surfaces in the presence of organic
954	>	solvents. This algorithm allows the application of unphysical thermal
955	>	flux to transfer heat between the metal and the liquid phase. With the
956	>	flux applied, we were able to measure the corresponding thermal
957	>	gradients and to obtain interfacial thermal conductivities. Under
958	>	steady states, 2-3 ns trajectory simulations are sufficient for
959	>	computation of this quantity.
960	>
961	>	Our simulations have seen significant conductance enhancement in the
962	>	presence of capping agent, compared with the bare gold / liquid
963	>	interfaces. The acoustic impedance mismatch between the metal and the
964	>	liquid phase is effectively eliminated by a chemically-bonded capping
965		agent. Furthermore, the coverage precentage of the capping agent plays
966	<	an important role in the interfacial thermal transport process.
966	>	an important role in the interfacial thermal transport
967	>	process. Moderately low coverages allow higher contact between capping
968	>	agent and solvent, and thus could further enhance the heat transfer
969	>	process, giving a non-monotonic behavior of conductance with
970	>	increasing coverage.
971
972	<	Our measurement results, particularly of the UA models, agree with
973	<	available experimental data. This indicates that our force field
848	<	parameters have a nice description of the interactions between the
849	<	particles at the interfaces. AA models tend to overestimate the
972	>	Our results, particularly using the UA models, agree well with
973	>	available experimental data.  The AA models tend to overestimate the
974		interfacial thermal conductance in that the classically treated C-H
975	<	vibration would be overly sampled. Compared to the AA models, the UA
976	<	models have higher computational efficiency with satisfactory
977	<	accuracy, and thus are preferable in interfacial thermal transport
978	<	modelings.
975	>	vibrations become too easily populated. Compared to the AA models, the
976	>	UA models have higher computational efficiency with satisfactory
977	>	accuracy, and thus are preferable in modeling interfacial thermal
978	>	transport.
979
980	<	Vlugt {\it et al.} has investigated the surface thiol structures for
981	<	nanocrystal gold and pointed out that they differs from those of the
982	<	Au(111) surface\cite{vlugt:cpc2007154}. This difference might lead to
983	<	change of interfacial thermal transport behavior as well. To
984	<	investigate this problem, an effective means to introduce thermal flux
985	<	and measure the corresponding thermal gradient is desirable for
986	<	simulating structures with spherical symmetry.
980	>	Of the two definitions for $G$, the discrete form
981	>	(Eq. \ref{discreteG}) was easier to use and gives out relatively
982	>	consistent results, while the derivative form (Eq. \ref{derivativeG})
983	>	is not as versatile. Although $G^\prime$ gives out comparable results
984	>	and follows similar trend with $G$ when measuring close to fully
985	>	covered or bare surfaces, the spatial resolution of $T$ profile
986	>	required for the use of a derivative form is limited by the number of
987	>	bins and the sampling required to obtain thermal gradient information.
988
989	+	Vlugt {\it et al.} have investigated the surface thiol structures for
990	+	nanocrystalline gold and pointed out that they differ from those of
991	+	the Au(111) surface.\cite{landman:1998,vlugt:cpc2007154} This
992	+	difference could also cause differences in the interfacial thermal
993	+	transport behavior. To investigate this problem, one would need an
994	+	effective method for applying thermal gradients in non-planar
995	+	(i.e. spherical) geometries.
996
997		\section{Acknowledgments}
998		Support for this project was provided by the National Science
999		Foundation under grant CHE-0848243. Computational time was provided by
1000		the Center for Research Computing (CRC) at the University of Notre
1001	<	Dame. \newpage
1001	>	Dame.
1002	>	\newpage
1003
1004		\bibliography{interfacial}
1005

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