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Revision 3767 by gezelter, Fri Sep 30 19:37:13 2011 UTC

#	Line 28 | Line 28
28
29		\begin{document}
30
31	<	\title{Simulating interfacial thermal conductance at metal-solvent
32	<	interfaces: the role of chemical capping agents}
31	>	\title{Simulating Interfacial Thermal Conductance at Metal-Solvent
32	>	Interfaces: the Role of Chemical Capping Agents}
33
34		\author{Shenyu Kuang and J. Daniel
35		Gezelter\footnote{Corresponding author. \ Electronic mail: gezelter@nd.edu} \\
#	Line 44 | Line 44 | Notre Dame, Indiana 46556}
44		\begin{doublespace}
45
46		\begin{abstract}
47	+	With the Non-Isotropic Velocity Scaling (NIVS) approach to Reverse
48	+	Non-Equilibrium Molecular Dynamics (RNEMD) it is possible to impose
49	+	an unphysical thermal flux between different regions of
50	+	inhomogeneous systems such as solid / liquid interfaces.  We have
51	+	applied NIVS to compute the interfacial thermal conductance at a
52	+	metal / organic solvent interface that has been chemically capped by
53	+	butanethiol molecules.  Our calculations suggest that vibrational
54	+	coupling between the metal and liquid phases is enhanced by the
55	+	capping agents, leading to a greatly enhanced conductivity at the
56	+	interface.  Specifically, the chemical bond between the metal and
57	+	the capping agent introduces a vibrational overlap that is not
58	+	present without the capping agent, and the overlap between the
59	+	vibrational spectra (metal to cap, cap to solvent) provides a
60	+	mechanism for rapid thermal transport across the interface. Our
61	+	calculations also suggest that this is a non-monotonic function of
62	+	the fractional coverage of the surface, as moderate coverages allow
63	+	diffusive heat transport of solvent molecules that have been in
64	+	close contact with the capping agent.
65
66	<	With the Non-Isotropic Velocity Scaling algorithm (NIVS) we have
67	<	developed, an unphysical thermal flux can be effectively set up even
50	<	for non-homogeneous systems like interfaces in non-equilibrium
51	<	molecular dynamics simulations. In this work, this algorithm is
52	<	applied for simulating thermal conductance at metal / organic solvent
53	<	interfaces with various coverages of butanethiol capping
54	<	agents. Different solvents and force field models were tested. Our
55	<	results suggest that the United-Atom models are able to provide an
56	<	estimate of the interfacial thermal conductivity comparable to
57	<	experiments in our simulations with satisfactory computational
58	<	efficiency. From our results, the acoustic impedance mismatch between
59	<	metal and liquid phase is effectively reduced by the capping
60	<	agents, and thus leads to interfacial thermal conductance
61	<	enhancement. Furthermore, this effect is closely related to the
62	<	capping agent coverage on the metal surfaces and the type of solvent
63	<	molecules, and is affected by the models used in the simulations.
64	<
66	>	Keywords: non-equilibrium, molecular dynamics, vibrational overlap,
67	>	coverage dependent.
68		\end{abstract}
69
70		\newpage
#	Line 73 | Line 76 | Interfacial thermal conductance is extensively studied
76		%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
77
78		\section{Introduction}
79	<	Interfacial thermal conductance is extensively studied both
80	<	experimentally and computationally\cite{cahill:793}, due to its
81	<	importance in nanoscale science and technology. Reliability of
82	<	nanoscale devices depends on their thermal transport
83	<	properties. Unlike bulk homogeneous materials, nanoscale materials
84	<	features significant presence of interfaces, and these interfaces
85	<	could dominate the heat transfer behavior of these
86	<	materials. Furthermore, these materials are generally heterogeneous,
87	<	which challenges traditional research methods for homogeneous
85	<	systems.
79	>	Due to the importance of heat flow (and heat removal) in
80	>	nanotechnology, interfacial thermal conductance has been studied
81	>	extensively both experimentally and computationally.\cite{cahill:793}
82	>	Nanoscale materials have a significant fraction of their atoms at
83	>	interfaces, and the chemical details of these interfaces govern the
84	>	thermal transport properties.  Furthermore, the interfaces are often
85	>	heterogeneous (e.g. solid - liquid), which provides a challenge to
86	>	computational methods which have been developed for homogeneous or
87	>	bulk systems.
88
89	<	Heat conductance of molecular and nano-scale interfaces will be
90	<	affected by the chemical details of the surface. Experimentally,
91	<	various interfaces have been investigated for their thermal
92	<	conductance properties. Wang {\it et al.} studied heat transport
93	<	through long-chain hydrocarbon monolayers on gold substrate at
94	<	individual molecular level\cite{Wang10082007}; Schmidt {\it et al.}
95	<	studied the role of CTAB on thermal transport between gold nanorods
96	<	and solvent\cite{doi:10.1021/jp8051888}; Juv\'e {\it et al.} studied
97	<	the cooling dynamics, which is controlled by thermal interface
98	<	resistence of glass-embedded metal
99	<	nanoparticles\cite{PhysRevB.80.195406}. Although interfaces are
100	<	commonly barriers for heat transport, Alper {\it et al.} suggested
101	<	that specific ligands (capping agents) could completely eliminate this
102	<	barrier ($G\rightarrow\infty$)\cite{doi:10.1021/la904855s}.
89	>	Experimentally, the thermal properties of a number of interfaces have
90	>	been investigated.  Cahill and coworkers studied nanoscale thermal
91	>	transport from metal nanoparticle/fluid interfaces, to epitaxial
92	>	TiN/single crystal oxides interfaces, and hydrophilic and hydrophobic
93	>	interfaces between water and solids with different self-assembled
94	>	monolayers.\cite{Wilson:2002uq,PhysRevB.67.054302,doi:10.1021/jp048375k,PhysRevLett.96.186101}
95	>	Wang {\it et al.} studied heat transport through long-chain
96	>	hydrocarbon monolayers on gold substrate at individual molecular
97	>	level,\cite{Wang10082007} Schmidt {\it et al.} studied the role of
98	>	cetyltrimethylammonium bromide (CTAB) on the thermal transport between
99	>	gold nanorods and solvent,\cite{doi:10.1021/jp8051888} and Juv\'e {\it
100	>	et al.} studied the cooling dynamics, which is controlled by thermal
101	>	interface resistance of glass-embedded metal
102	>	nanoparticles.\cite{PhysRevB.80.195406} Although interfaces are
103	>	normally considered barriers for heat transport, Alper {\it et al.}
104	>	suggested that specific ligands (capping agents) could completely
105	>	eliminate this barrier
106	>	($G\rightarrow\infty$).\cite{doi:10.1021/la904855s}
107
108	<	Theoretical and computational models have also been used to study the
108	>	The acoustic mismatch model for interfacial conductance utilizes the
109	>	acoustic impedance ($Z_a = \rho_a v^s_a$) on both sides of the
110	>	interface.\cite{schwartz} Here, $\rho_a$ and $v^s_a$ are the density
111	>	and speed of sound in material $a$.  The phonon transmission
112	>	probability at the $a-b$ interface is
113	>	\begin{equation}
114	>	t_{ab} = \frac{4 Z_a Z_b}{\left(Z_a + Z_b \right)^2},
115	>	\end{equation}
116	>	and the interfacial conductance can then be approximated as
117	>	\begin{equation}
118	>	G_{ab} \approx \frac{1}{4} C_D v_D t_{ab}
119	>	\end{equation}
120	>	where $C_D$ is the Debye heat capacity of the hot side, and $v_D$ is
121	>	the Debye phonon velocity ($1/v_D^3 = 1/3v_L^3 + 2/3 v_T^3$) where
122	>	$v_L$ and $v_T$ are the longitudinal and transverse speeds of sound,
123	>	respectively.  For the Au/hexane and Au/toluene interfaces, the
124	>	acoustic mismatch model predicts room-temperature $G \approx 87 \mbox{
125	>	and } 129$ MW m$^{-2}$ K$^{-1}$, respectively.  However, it is not
126	>	clear how one might apply the acoustic mismatch model to a
127	>	chemically-modified surface, particularly when the acoustic properties
128	>	of a monolayer film may not be well characterized.
129	>
130	>	More precise computational models have also been used to study the
131		interfacial thermal transport in order to gain an understanding of
132		this phenomena at the molecular level. Recently, Hase and coworkers
133		employed Non-Equilibrium Molecular Dynamics (NEMD) simulations to
134		study thermal transport from hot Au(111) substrate to a self-assembled
135		monolayer of alkylthiol with relatively long chain (8-20 carbon
136	<	atoms)\cite{hase:2010,hase:2011}. However, ensemble averaged
136	>	atoms).\cite{hase:2010,hase:2011} However, ensemble averaged
137		measurements for heat conductance of interfaces between the capping
138	<	monolayer on Au and a solvent phase has yet to be studied.
139	<	The comparatively low thermal flux through interfaces is
140	<	difficult to measure with Equilibrium MD or forward NEMD simulation
141	<	methods. Therefore, the Reverse NEMD (RNEMD) methods would have the
142	<	advantage of having this difficult to measure flux known when studying
143	<	the thermal transport across interfaces, given that the simulation
144	<	methods being able to effectively apply an unphysical flux in
145	<	non-homogeneous systems.
138	>	monolayer on Au and a solvent phase have yet to be studied with their
139	>	approach. The comparatively low thermal flux through interfaces is
140	>	difficult to measure with Equilibrium
141	>	MD\cite{doi:10.1080/0026897031000068578} or forward NEMD simulation
142	>	methods. Therefore, the Reverse NEMD (RNEMD)
143	>	methods\cite{MullerPlathe:1997xw,kuang:164101} would be advantageous
144	>	in that they {\it apply} the difficult to measure quantity (flux),
145	>	while {\it measuring} the easily-computed quantity (the thermal
146	>	gradient).  This is particularly true for inhomogeneous interfaces
147	>	where it would not be clear how to apply a gradient {\it a priori}.
148	>	Garde and coworkers\cite{garde:nl2005,garde:PhysRevLett2009} applied
149	>	this approach to various liquid interfaces and studied how thermal
150	>	conductance (or resistance) is dependent on chemical details of a
151	>	number of hydrophobic and hydrophilic aqueous interfaces. And
152	>	recently, Luo {\it et al.} studied the thermal conductance of
153	>	Au-SAM-Au junctions using the same approach, comparing to a constant
154	>	temperature difference method.\cite{Luo20101} While this latter
155	>	approach establishes more ideal Maxwell-Boltzmann distributions than
156	>	previous RNEMD methods, it does not guarantee momentum or kinetic
157	>	energy conservation.
158
159	<	Recently, we have developed the Non-Isotropic Velocity Scaling (NIVS)
159	>	Recently, we have developed a Non-Isotropic Velocity Scaling (NIVS)
160		algorithm for RNEMD simulations\cite{kuang:164101}. This algorithm
161		retains the desirable features of RNEMD (conservation of linear
162		momentum and total energy, compatibility with periodic boundary
#	Line 133 | Line 173 | underlying mechanism for the phenomena was investigate
173		thermal transport across these interfaces was studied and the
174		underlying mechanism for the phenomena was investigated.
175
136	–	[MAY ADD WHY STUDY AU-THIOL SURFACE; CITE SHAOYI JIANG]
137	–
176		\section{Methodology}
177	<	\subsection{Imposd-Flux Methods in MD Simulations}
178	<	[CF. CAHILL]
179	<	For systems with low interfacial conductivity one must have a method
180	<	capable of generating relatively small fluxes, compared to those
181	<	required for bulk conductivity. This requirement makes the calculation
182	<	even more difficult for those slowly-converging equilibrium
183	<	methods\cite{Viscardy:2007lq}.
184	<	Forward methods impose gradient, but in interfacial conditions it is
185	<	not clear what behavior to impose at the boundary...
186	<	Imposed-flux reverse non-equilibrium
187	<	methods\cite{MullerPlathe:1997xw} have the flux set {\it a priori} and
188	<	the thermal response becomes easier to
189	<	measure than the flux. Although M\"{u}ller-Plathe's original momentum
190	<	swapping approach can be used for exchanging energy between particles
191	<	of different identity, the kinetic energy transfer efficiency is
192	<	affected by the mass difference between the particles, which limits
193	<	its application on heterogeneous interfacial systems.
177	>	\subsection{Imposed-Flux Methods in MD Simulations}
178	>	Steady state MD simulations have an advantage in that not many
179	>	trajectories are needed to study the relationship between thermal flux
180	>	and thermal gradients. For systems with low interfacial conductance,
181	>	one must have a method capable of generating or measuring relatively
182	>	small fluxes, compared to those required for bulk conductivity. This
183	>	requirement makes the calculation even more difficult for
184	>	slowly-converging equilibrium methods.\cite{Viscardy:2007lq} Forward
185	>	NEMD methods impose a gradient (and measure a flux), but at interfaces
186	>	it is not clear what behavior should be imposed at the boundaries
187	>	between materials.  Imposed-flux reverse non-equilibrium
188	>	methods\cite{MullerPlathe:1997xw} set the flux {\it a priori} and
189	>	the thermal response becomes an easy-to-measure quantity.  Although
190	>	M\"{u}ller-Plathe's original momentum swapping approach can be used
191	>	for exchanging energy between particles of different identity, the
192	>	kinetic energy transfer efficiency is affected by the mass difference
193	>	between the particles, which limits its application on heterogeneous
194	>	interfacial systems.
195
196	<	The non-isotropic velocity scaling (NIVS)\cite{kuang:164101} approach to
197	<	non-equilibrium MD simulations is able to impose a wide range of
196	>	The non-isotropic velocity scaling (NIVS) \cite{kuang:164101} approach
197	>	to non-equilibrium MD simulations is able to impose a wide range of
198		kinetic energy fluxes without obvious perturbation to the velocity
199		distributions of the simulated systems. Furthermore, this approach has
200		the advantage in heterogeneous interfaces in that kinetic energy flux
201	<	can be applied between regions of particles of arbitary identity, and
201	>	can be applied between regions of particles of arbitrary identity, and
202		the flux will not be restricted by difference in particle mass.
203
204		The NIVS algorithm scales the velocity vectors in two separate regions
205	<	of a simulation system with respective diagonal scaling matricies. To
206	<	determine these scaling factors in the matricies, a set of equations
205	>	of a simulation system with respective diagonal scaling matrices. To
206	>	determine these scaling factors in the matrices, a set of equations
207		including linear momentum conservation and kinetic energy conservation
208		constraints and target energy flux satisfaction is solved. With the
209		scaling operation applied to the system in a set frequency, bulk
#	Line 172 | Line 211 | momenta and energy and does not depend on an external
211		for computing thermal conductivities. The NIVS algorithm conserves
212		momenta and energy and does not depend on an external thermostat.
213
214	<	\subsection{Defining Interfacial Thermal Conductivity $G$}
215	<	Given a system with thermal gradients and the corresponding thermal
216	<	flux, for interfaces with a relatively low interfacial conductance,
217	<	the bulk regions on either side of an interface rapidly come to a
218	<	state in which the two phases have relatively homogeneous (but
219	<	distinct) temperatures. The interfacial thermal conductivity $G$ can
220	<	therefore be approximated as:
214	>	\subsection{Defining Interfacial Thermal Conductivity ($G$)}
215	>
216	>	For an interface with relatively low interfacial conductance, and a
217	>	thermal flux between two distinct bulk regions, the regions on either
218	>	side of the interface rapidly come to a state in which the two phases
219	>	have relatively homogeneous (but distinct) temperatures. The
220	>	interfacial thermal conductivity $G$ can therefore be approximated as:
221		\begin{equation}
222	<	G = \frac{E_{total}}{2 t L_x L_y \left( \langle T_\mathrm{hot}\rangle -
222	>	G = \frac{E_{total}}{2 t L_x L_y \left( \langle T_\mathrm{hot}\rangle -
223		\langle T_\mathrm{cold}\rangle \right)}
224		\label{lowG}
225		\end{equation}
226	<	where ${E_{total}}$ is the imposed non-physical kinetic energy
227	<	transfer and ${\langle T_\mathrm{hot}\rangle}$ and ${\langle
228	<	T_\mathrm{cold}\rangle}$ are the average observed temperature of the
229	<	two separated phases.
226	>	where ${E_{total}}$ is the total imposed non-physical kinetic energy
227	>	transfer during the simulation and ${\langle T_\mathrm{hot}\rangle}$
228	>	and ${\langle T_\mathrm{cold}\rangle}$ are the average observed
229	>	temperature of the two separated phases.  For an applied flux $J_z$
230	>	operating over a simulation time $t$ on a periodically-replicated slab
231	>	of dimensions $L_x \times L_y$, $E_{total} = J_z *(t)*(2 L_x L_y)$.
232
233		When the interfacial conductance is {\it not} small, there are two
234	<	ways to define $G$.
235	<
236	<	One way is to assume the temperature is discrete on the two sides of
237	<	the interface. $G$ can be calculated using the applied thermal flux
238	<	$J$ and the maximum temperature difference measured along the thermal
239	<	gradient max($\Delta T$), which occurs at the Gibbs deviding surface
240	<	(Figure \ref{demoPic}):
234	>	ways to define $G$. One common way is to assume the temperature is
235	>	discrete on the two sides of the interface. $G$ can be calculated
236	>	using the applied thermal flux $J$ and the maximum temperature
237	>	difference measured along the thermal gradient max($\Delta T$), which
238	>	occurs at the Gibbs dividing surface (Figure \ref{demoPic}). This is
239	>	known as the Kapitza conductance, which is the inverse of the Kapitza
240	>	resistance.
241		\begin{equation}
242	<	G=\frac{J}{\Delta T}
242	>	G=\frac{J}{\Delta T}
243		\label{discreteG}
244		\end{equation}
245
#	Line 207 | Line 248 | G=\frac{J}{\Delta T}
248		\caption{Interfacial conductance can be calculated by applying an
249		(unphysical) kinetic energy flux between two slabs, one located
250		within the metal and another on the edge of the periodic box.  The
251	<	system responds by forming a thermal response or a gradient.  In
252	<	bulk liquids, this gradient typically has a single slope, but in
253	<	interfacial systems, there are distinct thermal conductivity
254	<	domains.  The interfacial conductance, $G$ is found by measuring the
255	<	temperature gap at the Gibbs dividing surface, or by using second
256	<	derivatives of the thermal profile.}
251	>	system responds by forming a thermal gradient.  In bulk liquids,
252	>	this gradient typically has a single slope, but in interfacial
253	>	systems, there are distinct thermal conductivity domains.  The
254	>	interfacial conductance, $G$ is found by measuring the temperature
255	>	gap at the Gibbs dividing surface, or by using second derivatives of
256	>	the thermal profile.}
257		\label{demoPic}
258		\end{figure}
259
260	<	The other approach is to assume a continuous temperature profile along
261	<	the thermal gradient axis (e.g. $z$) and define $G$ at the point where
262	<	the magnitude of thermal conductivity $\lambda$ change reach its
263	<	maximum, given that $\lambda$ is well-defined throughout the space:
264	<	\begin{equation}
265	<	G^\prime = \Big|\frac{\partial\lambda}{\partial z}\Big|
266	<	= \Big|\frac{\partial}{\partial z}\left(-J_z\Big/
267	<	\left(\frac{\partial T}{\partial z}\right)\right)\Big|
268	<	= |J_z|\Big|\frac{\partial^2 T}{\partial z^2}\Big|
269	<	\Big/\left(\frac{\partial T}{\partial z}\right)^2
270	<	\label{derivativeG}
271	<	\end{equation}
260	>	Another approach is to assume that the temperature is continuous and
261	>	differentiable throughout the space. Given that $\lambda$ is also
262	>	differentiable, $G$ can be defined as its gradient ($\nabla\lambda$)
263	>	projected along a vector normal to the interface ($\mathbf{\hat{n}}$)
264	>	and evaluated at the interface location ($z_0$). This quantity,
265	>	\begin{align}
266	>	G^\prime &= \left(\nabla\lambda \cdot \mathbf{\hat{n}}\right)_{z_0} \\
267	>	&= \frac{\partial}{\partial z}\left(-\frac{J_z}{
268	>	\left(\frac{\partial T}{\partial z}\right)}\right)_{z_0} \\
269	>	&= J_z\left(\frac{\partial^2 T}{\partial z^2}\right)_{z_0} \Big/
270	>	\left(\frac{\partial T}{\partial z}\right)_{z_0}^2 \label{derivativeG}
271	>	\end{align}
272	>	has the same units as the common definition for $G$, and the maximum
273	>	of its magnitude denotes where thermal conductivity has the largest
274	>	change, i.e. the interface.  In the geometry used in this study, the
275	>	vector normal to the interface points along the $z$ axis, as do
276	>	$\vec{J}$ and the thermal gradient.  This yields the simplified
277	>	expressions in Eq. \ref{derivativeG}.
278
279	<	With the temperature profile obtained from simulations, one is able to
279	>	With temperature profiles obtained from simulation, one is able to
280		approximate the first and second derivatives of $T$ with finite
281	<	difference methods and thus calculate $G^\prime$.
281	>	difference methods and calculate $G^\prime$. In what follows, both
282	>	definitions have been used, and are compared in the results.
283
284	<	In what follows, both definitions have been used for calculation and
285	<	are compared in the results.
286	<
287	<	To compare the above definitions ($G$ and $G^\prime$), we have modeled
288	<	a metal slab with its (111) surfaces perpendicular to the $z$-axis of
241	<	our simulation cells. Both with and without capping agents on the
242	<	surfaces, the metal slab is solvated with simple organic solvents, as
284	>	To investigate the interfacial conductivity at metal / solvent
285	>	interfaces, we have modeled a metal slab with its (111) surfaces
286	>	perpendicular to the $z$-axis of our simulation cells. The metal slab
287	>	has been prepared both with and without capping agents on the exposed
288	>	surface, and has been solvated with simple organic solvents, as
289		illustrated in Figure \ref{gradT}.
290
291		With the simulation cell described above, we are able to equilibrate
292		the system and impose an unphysical thermal flux between the liquid
293		and the metal phase using the NIVS algorithm. By periodically applying
294	<	the unphysical flux, we are able to obtain a temperature profile and
295	<	its spatial derivatives. These quantities enable the evaluation of the
296	<	interfacial thermal conductance of a surface. Figure \ref{gradT} is an
297	<	example of how an applied thermal flux can be used to obtain the 1st
252	<	and 2nd derivatives of the temperature profile.
294	>	the unphysical flux, we obtained a temperature profile and its spatial
295	>	derivatives. Figure \ref{gradT} shows how an applied thermal flux can
296	>	be used to obtain the 1st and 2nd derivatives of the temperature
297	>	profile.
298
299		\begin{figure}
300		\includegraphics[width=\linewidth]{gradT}
301	<	\caption{A sample of Au-butanethiol/hexane interfacial system and the
302	<	temperature profile after a kinetic energy flux is imposed to
303	<	it. The 1st and 2nd derivatives of the temperature profile can be
304	<	obtained with finite difference approximation (lower panel).}
301	>	\caption{A sample of Au (111) / butanethiol / hexane interfacial
302	>	system with the temperature profile after a kinetic energy flux has
303	>	been imposed.  Note that the largest temperature jump in the thermal
304	>	profile (corresponding to the lowest interfacial conductance) is at
305	>	the interface between the butanethiol molecules (blue) and the
306	>	solvent (grey).  First and second derivatives of the temperature
307	>	profile are obtained using a finite difference approximation (lower
308	>	panel).}
309		\label{gradT}
310		\end{figure}
311
312		\section{Computational Details}
313		\subsection{Simulation Protocol}
314		The NIVS algorithm has been implemented in our MD simulation code,
315	<	OpenMD\cite{Meineke:2005gd,openmd}, and was used for our
316	<	simulations. Different metal slab thickness (layer numbers of Au) was
317	<	simulated. Metal slabs were first equilibrated under atmospheric
318	<	pressure (1 atm) and a desired temperature (e.g. 200K). After
319	<	equilibration, butanethiol capping agents were placed at three-fold
320	<	hollow sites on the Au(111) surfaces. These sites could be either a
321	<	{\it fcc} or {\it hcp} site. However, Hase {\it et al.} found that
322	<	they are equivalent in a heat transfer process\cite{hase:2010}, so
274	<	they are not distinguished in our study. The maximum butanethiol
315	>	OpenMD\cite{Meineke:2005gd,openmd}, and was used for our simulations.
316	>	Metal slabs of 6 or 11 layers of Au atoms were first equilibrated
317	>	under atmospheric pressure (1 atm) and 200K. After equilibration,
318	>	butanethiol capping agents were placed at three-fold hollow sites on
319	>	the Au(111) surfaces. These sites are either {\it fcc} or {\it
320	>	hcp} sites, although Hase {\it et al.} found that they are
321	>	equivalent in a heat transfer process,\cite{hase:2010} so we did not
322	>	distinguish between these sites in our study. The maximum butanethiol
323		capacity on Au surface is $1/3$ of the total number of surface Au
324		atoms, and the packing forms a $(\sqrt{3}\times\sqrt{3})R30^\circ$
325	<	structure[CITE PORTER].
326	<	A series of different coverages was derived by evenly eliminating
327	<	butanethiols on the surfaces, and was investigated in order to study
328	<	the relation between coverage and interfacial conductance.
325	>	structure\cite{doi:10.1021/ja00008a001,doi:10.1021/cr9801317}. A
326	>	series of lower coverages was also prepared by eliminating
327	>	butanethiols from the higher coverage surface in a regular manner. The
328	>	lower coverages were prepared in order to study the relation between
329	>	coverage and interfacial conductance.
330
331		The capping agent molecules were allowed to migrate during the
332		simulations. They distributed themselves uniformly and sampled a
333		number of three-fold sites throughout out study. Therefore, the
334	<	initial configuration would not noticeably affect the sampling of a
334	>	initial configuration does not noticeably affect the sampling of a
335		variety of configurations of the same coverage, and the final
336		conductance measurement would be an average effect of these
337	<	configurations explored in the simulations. [MAY NEED SNAPSHOTS]
337	>	configurations explored in the simulations.
338
339	<	After the modified Au-butanethiol surface systems were equilibrated
340	<	under canonical ensemble, organic solvent molecules were packed in the
341	<	previously empty part of the simulation cells\cite{packmol}. Two
339	>	After the modified Au-butanethiol surface systems were equilibrated in
340	>	the canonical (NVT) ensemble, organic solvent molecules were packed in
341	>	the previously empty part of the simulation cells.\cite{packmol} Two
342		solvents were investigated, one which has little vibrational overlap
343	<	with the alkanethiol and a planar shape (toluene), and one which has
344	<	similar vibrational frequencies and chain-like shape ({\it n}-hexane).
343	>	with the alkanethiol and which has a planar shape (toluene), and one
344	>	which has similar vibrational frequencies to the capping agent and
345	>	chain-like shape ({\it n}-hexane).
346
347	<	The space filled by solvent molecules, i.e. the gap between
348	<	periodically repeated Au-butanethiol surfaces should be carefully
349	<	chosen. A very long length scale for the thermal gradient axis ($z$)
300	<	may cause excessively hot or cold temperatures in the middle of the
347	>	The simulation cells were not particularly extensive along the
348	>	$z$-axis, as a very long length scale for the thermal gradient may
349	>	cause excessively hot or cold temperatures in the middle of the
350		solvent region and lead to undesired phenomena such as solvent boiling
351		or freezing when a thermal flux is applied. Conversely, too few
352		solvent molecules would change the normal behavior of the liquid
353		phase. Therefore, our $N_{solvent}$ values were chosen to ensure that
354	<	these extreme cases did not happen to our simulations. And the
355	<	corresponding spacing is usually $35[DOUBLE CHECK] \sim 75$\AA.
354	>	these extreme cases did not happen to our simulations. The spacing
355	>	between periodic images of the gold interfaces is $45 \sim 75$\AA in
356	>	our simulations.
357
358		The initial configurations generated are further equilibrated with the
359	<	$x$ and $y$ dimensions fixed, only allowing length scale change in $z$
360	<	dimension. This is to ensure that the equilibration of liquid phase
361	<	does not affect the metal crystal structure in $x$ and $y$ dimensions.
362	<	To investigate this effect, comparisons were made with simulations
363	<	that allow changes of $L_x$ and $L_y$ during NPT equilibration, and
364	<	the results are shown in later sections. After ensuring the liquid
365	<	phase reaches equilibrium at atmospheric pressure (1 atm), further
366	<	equilibration are followed under NVT and then NVE ensembles.
359	>	$x$ and $y$ dimensions fixed, only allowing the $z$-length scale to
360	>	change. This is to ensure that the equilibration of liquid phase does
361	>	not affect the metal's crystalline structure. Comparisons were made
362	>	with simulations that allowed changes of $L_x$ and $L_y$ during NPT
363	>	equilibration. No substantial changes in the box geometry were noticed
364	>	in these simulations. After ensuring the liquid phase reaches
365	>	equilibrium at atmospheric pressure (1 atm), further equilibration was
366	>	carried out under canonical (NVT) and microcanonical (NVE) ensembles.
367
368	<	After the systems reach equilibrium, NIVS is implemented to impose a
369	<	periodic unphysical thermal flux between the metal and the liquid
370	<	phase. Most of our simulations are under an average temperature of
371	<	$\sim$200K. Therefore, this flux usually comes from the metal to the
368	>	After the systems reach equilibrium, NIVS was used to impose an
369	>	unphysical thermal flux between the metal and the liquid phases. Most
370	>	of our simulations were done under an average temperature of
371	>	$\sim$200K. Therefore, thermal flux usually came from the metal to the
372		liquid so that the liquid has a higher temperature and would not
373	<	freeze due to excessively low temperature. After this induced
374	<	temperature gradient is stablized, the temperature profile of the
375	<	simulation cell is recorded. To do this, the simulation cell is
376	<	devided evenly into $N$ slabs along the $z$-axis and $N$ is maximized
327	<	for highest possible spatial resolution but not too many to have some
328	<	slabs empty most of the time. The average temperatures of each slab
373	>	freeze due to lowered temperatures. After this induced temperature
374	>	gradient had stabilized, the temperature profile of the simulation cell
375	>	was recorded. To do this, the simulation cell is divided evenly into
376	>	$N$ slabs along the $z$-axis. The average temperatures of each slab
377		are recorded for 1$\sim$2 ns. When the slab width $d$ of each slab is
378		the same, the derivatives of $T$ with respect to slab number $n$ can
379	<	be directly used for $G^\prime$ calculations:
380	<	\begin{equation}
333	<	G^\prime = |J_z|\Big|\frac{\partial^2 T}{\partial z^2}\Big|
379	>	be directly used for $G^\prime$ calculations: \begin{equation}
380	>	G^\prime = |J_z|\Big|\frac{\partial^2 T}{\partial z^2}\Big|
381		\Big/\left(\frac{\partial T}{\partial z}\right)^2
382		= |J_z|\Big|\frac{1}{d^2}\frac{\partial^2 T}{\partial n^2}\Big|
383		\Big/\left(\frac{1}{d}\frac{\partial T}{\partial n}\right)^2
#	Line 338 | Line 385 | G^\prime = |J_z|\Big|\frac{\partial^2 T}{\partial z^2}
385		\Big/\left(\frac{\partial T}{\partial n}\right)^2
386		\label{derivativeG2}
387		\end{equation}
388	+	The absolute values in Eq. \ref{derivativeG2} appear because the
389	+	direction of the flux $\vec{J}$ is in an opposing direction on either
390	+	side of the metal slab.
391
392	<	All of the above simulation procedures use a time step of 1 fs. And
393	<	each equilibration / stabilization step usually takes 100 ps, or
394	<	longer, if necessary.
392	>	All of the above simulation procedures use a time step of 1 fs. Each
393	>	equilibration stage took a minimum of 100 ps, although in some cases,
394	>	longer equilibration stages were utilized.
395
396		\subsection{Force Field Parameters}
397	<	Our simulations include various components. Figure \ref{demoMol}
398	<	demonstrates the sites defined for both United-Atom and All-Atom
399	<	models of the organic solvent and capping agent molecules in our
400	<	simulations. Force field parameter descriptions are needed for
397	>	Our simulations include a number of chemically distinct components.
398	>	Figure \ref{demoMol} demonstrates the sites defined for both
399	>	United-Atom and All-Atom models of the organic solvent and capping
400	>	agents in our simulations. Force field parameters are needed for
401		interactions both between the same type of particles and between
402		particles of different species.
403
#	Line 356 | Line 406 | particles of different species.
406		\caption{Structures of the capping agent and solvents utilized in
407		these simulations. The chemically-distinct sites (a-e) are expanded
408		in terms of constituent atoms for both United Atom (UA) and All Atom
409	<	(AA) force fields.  Most parameters are from
410	<	Refs. \protect\cite{TraPPE-UA.alkanes,TraPPE-UA.alkylbenzenes} (UA) and
411	<	\protect\cite{OPLSAA} (AA).  Cross-interactions with the Au atoms are given
412	<	in Table \ref{MnM}.}
409	>	(AA) force fields.  Most parameters are from References
410	>	\protect\cite{TraPPE-UA.alkanes,TraPPE-UA.alkylbenzenes,TraPPE-UA.thiols}
411	>	(UA) and \protect\cite{OPLSAA} (AA). Cross-interactions with the Au
412	>	atoms are given in Table \ref{MnM}.}
413		\label{demoMol}
414		\end{figure}
415
416		The Au-Au interactions in metal lattice slab is described by the
417	<	quantum Sutton-Chen (QSC) formulation\cite{PhysRevB.59.3527}. The QSC
417	>	quantum Sutton-Chen (QSC) formulation.\cite{PhysRevB.59.3527} The QSC
418		potentials include zero-point quantum corrections and are
419		reparametrized for accurate surface energies compared to the
420	<	Sutton-Chen potentials\cite{Chen90}.
420	>	Sutton-Chen potentials.\cite{Chen90}
421
422	<	For both solvent molecules, straight chain {\it n}-hexane and aromatic
423	<	toluene, United-Atom (UA) and All-Atom (AA) models are used
424	<	respectively. The TraPPE-UA
422	>	For the two solvent molecules, {\it n}-hexane and toluene, two
423	>	different atomistic models were utilized. Both solvents were modeled
424	>	using United-Atom (UA) and All-Atom (AA) models. The TraPPE-UA
425		parameters\cite{TraPPE-UA.alkanes,TraPPE-UA.alkylbenzenes} are used
426		for our UA solvent molecules. In these models, sites are located at
427		the carbon centers for alkyl groups. Bonding interactions, including
428		bond stretches and bends and torsions, were used for intra-molecular
429	<	sites not separated by more than 3 bonds. Otherwise, for non-bonded
430	<	interactions, Lennard-Jones potentials are used. [CHECK CITATION]
429	>	sites closer than 3 bonds. For non-bonded interactions, Lennard-Jones
430	>	potentials are used.
431
432	<	By eliminating explicit hydrogen atoms, these models are simple and
433	<	computationally efficient, while maintains good accuracy. However, the
434	<	TraPPE-UA for alkanes is known to predict a lower boiling point than
435	<	experimental values. Considering that after an unphysical thermal flux
436	<	is applied to a system, the temperature of ``hot'' area in the liquid
437	<	phase would be significantly higher than the average of the system, to
438	<	prevent over heating and boiling of the liquid phase, the average
439	<	temperature in our simulations should be much lower than the liquid
440	<	boiling point.
432	>	By eliminating explicit hydrogen atoms, the TraPPE-UA models are
433	>	simple and computationally efficient, while maintaining good accuracy.
434	>	However, the TraPPE-UA model for alkanes is known to predict a slightly
435	>	lower boiling point than experimental values. This is one of the
436	>	reasons we used a lower average temperature (200K) for our
437	>	simulations. If heat is transferred to the liquid phase during the
438	>	NIVS simulation, the liquid in the hot slab can actually be
439	>	substantially warmer than the mean temperature in the simulation. The
440	>	lower mean temperatures therefore prevent solvent boiling.
441
442	<	For UA-toluene model, the non-bonded potentials between
443	<	inter-molecular sites have a similar Lennard-Jones formulation. For
444	<	intra-molecular interactions, considering the stiffness of the benzene
445	<	ring, rigid body constraints are applied for further computational
446	<	efficiency. All bonds in the benzene ring and between the ring and the
397	<	methyl group remain rigid during the progress of simulations.
442	>	For UA-toluene, the non-bonded potentials between intermolecular sites
443	>	have a similar Lennard-Jones formulation. The toluene molecules were
444	>	treated as a single rigid body, so there was no need for
445	>	intramolecular interactions (including bonds, bends, or torsions) in
446	>	this solvent model.
447
448		Besides the TraPPE-UA models, AA models for both organic solvents are
449	<	included in our studies as well. For hexane, the OPLS-AA\cite{OPLSAA}
450	<	force field is used. Additional explicit hydrogen sites were
449	>	included in our studies as well. The OPLS-AA\cite{OPLSAA} force fields
450	>	were used. For hexane, additional explicit hydrogen sites were
451		included. Besides bonding and non-bonded site-site interactions,
452		partial charges and the electrostatic interactions were added to each
453	<	CT and HC site. For toluene, the United Force Field developed by
454	<	Rapp\'{e} {\it et al.}\cite{doi:10.1021/ja00051a040} is
455	<	adopted. Without the rigid body constraints, bonding interactions were
407	<	included. For the aromatic ring, improper torsions (inversions) were
408	<	added as an extra potential for maintaining the planar shape.
409	<	[CHECK CITATION]
453	>	CT and HC site. For toluene, a flexible model for the toluene molecule
454	>	was utilized which included bond, bend, torsion, and inversion
455	>	potentials to enforce ring planarity.
456
457	<	The capping agent in our simulations, the butanethiol molecules can
458	<	either use UA or AA model. The TraPPE-UA force fields includes
457	>	The butanethiol capping agent in our simulations, were also modeled
458	>	with both UA and AA model. The TraPPE-UA force field includes
459		parameters for thiol molecules\cite{TraPPE-UA.thiols} and are used for
460		UA butanethiol model in our simulations. The OPLS-AA also provides
461		parameters for alkyl thiols. However, alkyl thiols adsorbed on Au(111)
462	<	surfaces do not have the hydrogen atom bonded to sulfur. To adapt this
463	<	change and derive suitable parameters for butanethiol adsorbed on
464	<	Au(111) surfaces, we adopt the S parameters from Luedtke and
465	<	Landman\cite{landman:1998}[CHECK CITATION]
466	<	and modify parameters for its neighbor C
467	<	atom for charge balance in the molecule. Note that the model choice
468	<	(UA or AA) of capping agent can be different from the
469	<	solvent. Regardless of model choice, the force field parameters for
424	<	interactions between capping agent and solvent can be derived using
425	<	Lorentz-Berthelot Mixing Rule:
462	>	surfaces do not have the hydrogen atom bonded to sulfur. To derive
463	>	suitable parameters for butanethiol adsorbed on Au(111) surfaces, we
464	>	adopt the S parameters from Luedtke and Landman\cite{landman:1998} and
465	>	modify the parameters for the CTS atom to maintain charge neutrality
466	>	in the molecule.  Note that the model choice (UA or AA) for the capping
467	>	agent can be different from the solvent. Regardless of model choice,
468	>	the force field parameters for interactions between capping agent and
469	>	solvent can be derived using Lorentz-Berthelot Mixing Rule:
470		\begin{eqnarray}
471	<	\sigma_{ij} & = & \frac{1}{2} \left(\sigma_{ii} + \sigma_{jj}\right) \\
472	<	\epsilon_{ij} & = & \sqrt{\epsilon_{ii}\epsilon_{jj}}
471	>	\sigma_{ij} & = & \frac{1}{2} \left(\sigma_{ii} + \sigma_{jj}\right) \\
472	>	\epsilon_{ij} & = & \sqrt{\epsilon_{ii}\epsilon_{jj}}
473		\end{eqnarray}
474
475	<	To describe the interactions between metal Au and non-metal capping
476	<	agent and solvent particles, we refer to an adsorption study of alkyl
477	<	thiols on gold surfaces by Vlugt {\it et
478	<	al.}\cite{vlugt:cpc2007154} They fitted an effective Lennard-Jones
479	<	form of potential parameters for the interaction between Au and
480	<	pseudo-atoms CH$_x$ and S based on a well-established and widely-used
481	<	effective potential of Hautman and Klein\cite{hautman:4994} for the
482	<	Au(111) surface. As our simulations require the gold lattice slab to
483	<	be non-rigid so that it could accommodate kinetic energy for thermal
440	<	transport study purpose, the pair-wise form of potentials is
441	<	preferred.
442	<
443	<	Besides, the potentials developed from {\it ab initio} calculations by
444	<	Leng {\it et al.}\cite{doi:10.1021/jp034405s} are adopted for the
445	<	interactions between Au and aromatic C/H atoms in toluene. A set of
446	<	pseudo Lennard-Jones parameters were provided for Au in their force
447	<	fields. By using the Mixing Rule, this can be used to derive pair-wise
448	<	potentials for non-bonded interactions between Au and non-metal sites.
475	>	To describe the interactions between metal (Au) and non-metal atoms,
476	>	we refer to an adsorption study of alkyl thiols on gold surfaces by
477	>	Vlugt {\it et al.}\cite{vlugt:cpc2007154} They fitted an effective
478	>	Lennard-Jones form of potential parameters for the interaction between
479	>	Au and pseudo-atoms CH$_x$ and S based on a well-established and
480	>	widely-used effective potential of Hautman and Klein for the Au(111)
481	>	surface.\cite{hautman:4994} As our simulations require the gold slab
482	>	to be flexible to accommodate thermal excitation, the pair-wise form
483	>	of potentials they developed was used for our study.
484
485	<	However, the Lennard-Jones parameters between Au and other types of
486	<	particles, such as All-Atom normal alkanes in our simulations are not
487	<	yet well-established. For these interactions, we attempt to derive
488	<	their parameters using the Mixing Rule. To do this, Au pseudo
489	<	Lennard-Jones parameters for ``Metal-non-Metal'' (MnM) interactions
490	<	were first extracted from the Au-CH$_x$ parameters by applying the
491	<	Mixing Rule reversely. Table \ref{MnM} summarizes these ``MnM''
492	<	parameters in our simulations.
485	>	The potentials developed from {\it ab initio} calculations by Leng
486	>	{\it et al.}\cite{doi:10.1021/jp034405s} are adopted for the
487	>	interactions between Au and aromatic C/H atoms in toluene. However,
488	>	the Lennard-Jones parameters between Au and other types of particles,
489	>	(e.g. AA alkanes) have not yet been established. For these
490	>	interactions, the Lorentz-Berthelot mixing rule can be used to derive
491	>	effective single-atom LJ parameters for the metal using the fit values
492	>	for toluene. These are then used to construct reasonable mixing
493	>	parameters for the interactions between the gold and other atoms.
494	>	Table \ref{MnM} summarizes the ``metal/non-metal'' parameters used in
495	>	our simulations.
496
497		\begin{table*}
498		\begin{minipage}{\linewidth}
#	Line 491 | Line 529 | parameters in our simulations.
529		\end{minipage}
530		\end{table*}
531
494	–	\subsection{Vibrational Spectrum}
495	–	To investigate the mechanism of interfacial thermal conductance, the
496	–	vibrational spectrum is utilized as a complementary tool. Vibrational
497	–	spectra were taken for individual components in different
498	–	simulations. To obtain these spectra, simulations were run after
499	–	equilibration, in the NVE ensemble. Snapshots of configurations were
500	–	collected at a frequency that is higher than that of the fastest
501	–	vibrations occuring in the simulations. With these configurations, the
502	–	velocity auto-correlation functions can be computed:
503	–	\begin{equation}
504	–	C_A (t) = \langle\vec{v}_A (t)\cdot\vec{v}_A (0)\rangle
505	–	\label{vCorr}
506	–	\end{equation}
532
533	<	Followed by Fourier transforms, the power spectrum can be constructed:
534	<	\begin{equation}
535	<	\hat{f}(\omega) = \int_{-\infty}^{\infty} C_A (t) e^{-2\pi it\omega}\,dt
536	<	\label{fourier}
537	<	\end{equation}
533	>	\section{Results}
534	>	There are many factors contributing to the measured interfacial
535	>	conductance; some of these factors are physically motivated
536	>	(e.g. coverage of the surface by the capping agent coverage and
537	>	solvent identity), while some are governed by parameters of the
538	>	methodology (e.g. applied flux and the formulas used to obtain the
539	>	conductance). In this section we discuss the major physical and
540	>	calculational effects on the computed conductivity.
541
542	<	\section{Results and Discussions}
515	<	In what follows, how the parameters and protocol of simulations would
516	<	affect the measurement of $G$'s is first discussed. With a reliable
517	<	protocol and set of parameters, the influence of capping agent
518	<	coverage on thermal conductance is investigated. Besides, different
519	<	force field models for both solvents and selected deuterated models
520	<	were tested and compared. Finally, a summary of the role of capping
521	<	agent in the interfacial thermal transport process is given.
542	>	\subsection{Effects due to capping agent coverage}
543
544	<	\subsection{How Simulation Parameters Affects $G$}
545	<	We have varied our protocol or other parameters of the simulations in
546	<	order to investigate how these factors would affect the measurement of
547	<	$G$'s. It turned out that while some of these parameters would not
548	<	affect the results substantially, some other changes to the
549	<	simulations would have a significant impact on the measurement
529	<	results.
530	<
531	<	In some of our simulations, we allowed $L_x$ and $L_y$ to change
532	<	during equilibrating the liquid phase. Due to the stiffness of the
533	<	crystalline Au structure, $L_x$ and $L_y$ would not change noticeably
534	<	after equilibration. Although $L_z$ could fluctuate $\sim$1\% after a
535	<	system is fully equilibrated in the NPT ensemble, this fluctuation, as
536	<	well as those of $L_x$ and $L_y$ (which is significantly smaller),
537	<	would not be magnified on the calculated $G$'s, as shown in Table
538	<	\ref{AuThiolHexaneUA}. This insensivity to $L_i$ fluctuations allows
539	<	reliable measurement of $G$'s without the necessity of extremely
540	<	cautious equilibration process.
541	<
542	<	As stated in our computational details, the spacing filled with
543	<	solvent molecules can be chosen within a range. This allows some
544	<	change of solvent molecule numbers for the same Au-butanethiol
545	<	surfaces. We did this study on our Au-butanethiol/hexane
546	<	simulations. Nevertheless, the results obtained from systems of
547	<	different $N_{hexane}$ did not indicate that the measurement of $G$ is
548	<	susceptible to this parameter. For computational efficiency concern,
549	<	smaller system size would be preferable, given that the liquid phase
550	<	structure is not affected.
551	<
552	<	Our NIVS algorithm allows change of unphysical thermal flux both in
553	<	direction and in quantity. This feature extends our investigation of
554	<	interfacial thermal conductance. However, the magnitude of this
555	<	thermal flux is not arbitary if one aims to obtain a stable and
556	<	reliable thermal gradient. A temperature profile would be
557	<	substantially affected by noise when $|J_z|$ has a much too low
558	<	magnitude; while an excessively large $|J_z|$ that overwhelms the
559	<	conductance capacity of the interface would prevent a thermal gradient
560	<	to reach a stablized steady state. NIVS has the advantage of allowing
561	<	$J$ to vary in a wide range such that the optimal flux range for $G$
562	<	measurement can generally be simulated by the algorithm. Within the
563	<	optimal range, we were able to study how $G$ would change according to
564	<	the thermal flux across the interface. For our simulations, we denote
565	<	$J_z$ to be positive when the physical thermal flux is from the liquid
566	<	to metal, and negative vice versa. The $G$'s measured under different
567	<	$J_z$ is listed in Table \ref{AuThiolHexaneUA} and
568	<	\ref{AuThiolToluene}. These results do not suggest that $G$ is
569	<	dependent on $J_z$ within this flux range. The linear response of flux
570	<	to thermal gradient simplifies our investigations in that we can rely
571	<	on $G$ measurement with only a couple $J_z$'s and do not need to test
572	<	a large series of fluxes.
573	<
574	<	\begin{table*}
575	<	\begin{minipage}{\linewidth}
576	<	\begin{center}
577	<	\caption{Computed interfacial thermal conductivity ($G$ and
578	<	$G^\prime$) values for the 100\% covered Au-butanethiol/hexane
579	<	interfaces with UA model and different hexane molecule numbers
580	<	at different temperatures using a range of energy
581	<	fluxes. Error estimates indicated in parenthesis.}
582	<
583	<	\begin{tabular}{ccccccc}
584	<	\hline\hline
585	<	$\langle T\rangle$ & $N_{hexane}$ & Fixed & $\rho_{hexane}$ &
586	<	$J_z$ & $G$ & $G^\prime$ \\
587	<	(K) & & $L_x$ \& $L_y$? & (g/cm$^3$) & (GW/m$^2$) &
588	<	\multicolumn{2}{c}{(MW/m$^2$/K)} \\
589	<	\hline
590	<	200 & 266 & No  & 0.672 & -0.96 & 102(3)    & 80.0(0.8) \\
591	<	& 200 & Yes & 0.694 &  1.92 & 129(11)   & 87.3(0.3) \\
592	<	&     & Yes & 0.672 &  1.93 & 131(16)   & 78(13)    \\
593	<	&     & No  & 0.688 &  0.96 & 125(16)   & 90.2(15)  \\
594	<	&     &     &       &  1.91 & 139(10)   & 101(10)   \\
595	<	&     &     &       &  2.83 & 141(6)    & 89.9(9.8) \\
596	<	& 166 & Yes & 0.679 &  0.97 & 115(19)   & 69(18)    \\
597	<	&     &     &       &  1.94 & 125(9)    & 87.1(0.2) \\
598	<	&     & No  & 0.681 &  0.97 & 141(30)   & 78(22)    \\
599	<	&     &     &       &  1.92 & 138(4)    & 98.9(9.5) \\
600	<	\hline
601	<	250 & 200 & No  & 0.560 &  0.96 & 75(10)    & 61.8(7.3) \\
602	<	&     &     &       & -0.95 & 49.4(0.3) & 45.7(2.1) \\
603	<	& 166 & Yes & 0.570 &  0.98 & 79.0(3.5) & 62.9(3.0) \\
604	<	&     & No  & 0.569 &  0.97 & 80.3(0.6) & 67(11)    \\
605	<	&     &     &       &  1.44 & 76.2(5.0) & 64.8(3.8) \\
606	<	&     &     &       & -0.95 & 56.4(2.5) & 54.4(1.1) \\
607	<	&     &     &       & -1.85 & 47.8(1.1) & 53.5(1.5) \\
608	<	\hline\hline
609	<	\end{tabular}
610	<	\label{AuThiolHexaneUA}
611	<	\end{center}
612	<	\end{minipage}
613	<	\end{table*}
614	<
615	<	Furthermore, we also attempted to increase system average temperatures
616	<	to above 200K. These simulations are first equilibrated in the NPT
617	<	ensemble under normal pressure. As stated above, the TraPPE-UA model
618	<	for hexane tends to predict a lower boiling point. In our simulations,
619	<	hexane had diffculty to remain in liquid phase when NPT equilibration
620	<	temperature is higher than 250K. Additionally, the equilibrated liquid
621	<	hexane density under 250K becomes lower than experimental value. This
622	<	expanded liquid phase leads to lower contact between hexane and
623	<	butanethiol as well.[MAY NEED SLAB DENSITY FIGURE]
624	<	And this reduced contact would
625	<	probably be accountable for a lower interfacial thermal conductance,
626	<	as shown in Table \ref{AuThiolHexaneUA}.
627	<
628	<	A similar study for TraPPE-UA toluene agrees with the above result as
629	<	well. Having a higher boiling point, toluene tends to remain liquid in
630	<	our simulations even equilibrated under 300K in NPT
631	<	ensembles. Furthermore, the expansion of the toluene liquid phase is
632	<	not as significant as that of the hexane. This prevents severe
633	<	decrease of liquid-capping agent contact and the results (Table
634	<	\ref{AuThiolToluene}) show only a slightly decreased interface
635	<	conductance. Therefore, solvent-capping agent contact should play an
636	<	important role in the thermal transport process across the interface
637	<	in that higher degree of contact could yield increased conductance.
544	>	A series of different initial conditions with a range of surface
545	>	coverages was prepared and solvated with various with both of the
546	>	solvent molecules. These systems were then equilibrated and their
547	>	interfacial thermal conductivity was measured with the NIVS
548	>	algorithm. Figure \ref{coverage} demonstrates the trend of conductance
549	>	with respect to surface coverage.
550
551	<	\begin{table*}
552	<	\begin{minipage}{\linewidth}
553	<	\begin{center}
554	<	\caption{Computed interfacial thermal conductivity ($G$ and
555	<	$G^\prime$) values for a 90\% coverage Au-butanethiol/toluene
556	<	interface at different temperatures using a range of energy
557	<	fluxes. Error estimates indicated in parenthesis.}
558	<
647	<	\begin{tabular}{ccccc}
648	<	\hline\hline
649	<	$\langle T\rangle$ & $\rho_{toluene}$ & $J_z$ & $G$ & $G^\prime$ \\
650	<	(K) & (g/cm$^3$) & (GW/m$^2$) & \multicolumn{2}{c}{(MW/m$^2$/K)} \\
651	<	\hline
652	<	200 & 0.933 &  2.15 & 204(12) & 113(12) \\
653	<	&       & -1.86 & 180(3)  & 135(21) \\
654	<	&       & -3.93 & 176(5)  & 113(12) \\
655	<	\hline
656	<	300 & 0.855 & -1.91 & 143(5)  & 125(2)  \\
657	<	&       & -4.19 & 135(9)  & 113(12) \\
658	<	\hline\hline
659	<	\end{tabular}
660	<	\label{AuThiolToluene}
661	<	\end{center}
662	<	\end{minipage}
663	<	\end{table*}
551	>	\begin{figure}
552	>	\includegraphics[width=\linewidth]{coverage}
553	>	\caption{The interfacial thermal conductivity ($G$) has a
554	>	non-monotonic dependence on the degree of surface capping.  This
555	>	data is for the Au(111) / butanethiol / solvent interface with
556	>	various UA force fields at $\langle T\rangle \sim $200K.}
557	>	\label{coverage}
558	>	\end{figure}
559
560	<	Besides lower interfacial thermal conductance, surfaces in relatively
561	<	high temperatures are susceptible to reconstructions, when
562	<	butanethiols have a full coverage on the Au(111) surface. These
563	<	reconstructions include surface Au atoms migrated outward to the S
564	<	atom layer, and butanethiol molecules embedded into the original
565	<	surface Au layer. The driving force for this behavior is the strong
671	<	Au-S interactions in our simulations. And these reconstructions lead
672	<	to higher ratio of Au-S attraction and thus is energetically
673	<	favorable. Furthermore, this phenomenon agrees with experimental
674	<	results\cite{doi:10.1021/j100035a033,doi:10.1021/la026493y}. Vlugt
675	<	{\it et al.} had kept their Au(111) slab rigid so that their
676	<	simulations can reach 300K without surface reconstructions. Without
677	<	this practice, simulating 100\% thiol covered interfaces under higher
678	<	temperatures could hardly avoid surface reconstructions. However, our
679	<	measurement is based on assuming homogeneity on $x$ and $y$ dimensions
680	<	so that measurement of $T$ at particular $z$ would be an effective
681	<	average of the particles of the same type. Since surface
682	<	reconstructions could eliminate the original $x$ and $y$ dimensional
683	<	homogeneity, measurement of $G$ is more difficult to conduct under
684	<	higher temperatures. Therefore, most of our measurements are
685	<	undertaken at $\langle T\rangle\sim$200K.
560	>	In partially covered surfaces, the derivative definition for
561	>	$G^\prime$ (Eq. \ref{derivativeG}) becomes difficult to apply, as the
562	>	location of maximum change of $\lambda$ becomes washed out.  The
563	>	discrete definition (Eq. \ref{discreteG}) is easier to apply, as the
564	>	Gibbs dividing surface is still well-defined. Therefore, $G$ (not
565	>	$G^\prime$) was used in this section.
566
687	–	However, when the surface is not completely covered by butanethiols,
688	–	the simulated system is more resistent to the reconstruction
689	–	above. Our Au-butanethiol/toluene system had the Au(111) surfaces 90\%
690	–	covered by butanethiols, but did not see this above phenomena even at
691	–	$\langle T\rangle\sim$300K. The empty three-fold sites not occupied by
692	–	capping agents could help prevent surface reconstruction in that they
693	–	provide other means of capping agent relaxation. It is observed that
694	–	butanethiols can migrate to their neighbor empty sites during a
695	–	simulation. Therefore, we were able to obtain $G$'s for these
696	–	interfaces even at a relatively high temperature without being
697	–	affected by surface reconstructions.
698	–
699	–	\subsection{Influence of Capping Agent Coverage on $G$}
700	–	To investigate the influence of butanethiol coverage on interfacial
701	–	thermal conductance, a series of different coverage Au-butanethiol
702	–	surfaces is prepared and solvated with various organic
703	–	molecules. These systems are then equilibrated and their interfacial
704	–	thermal conductivity are measured with our NIVS algorithm. Figure
705	–	\ref{coverage} demonstrates the trend of conductance change with
706	–	respect to different coverages of butanethiol. To study the isotope
707	–	effect in interfacial thermal conductance, deuterated UA-hexane is
708	–	included as well.
709	–
710	–	It turned out that with partial covered butanethiol on the Au(111)
711	–	surface, the derivative definition for $G^\prime$
712	–	(Eq. \ref{derivativeG}) was difficult to apply, due to the difficulty
713	–	in locating the maximum of change of $\lambda$. Instead, the discrete
714	–	definition (Eq. \ref{discreteG}) is easier to apply, as the Gibbs
715	–	deviding surface can still be well-defined. Therefore, $G$ (not
716	–	$G^\prime$) was used for this section.
717	–
567		From Figure \ref{coverage}, one can see the significance of the
568	<	presence of capping agents. Even when a fraction of the Au(111)
569	<	surface sites are covered with butanethiols, the conductivity would
570	<	see an enhancement by at least a factor of 3. This indicates the
571	<	important role cappping agent is playing for thermal transport
572	<	phenomena on metal / organic solvent surfaces.
568	>	presence of capping agents. When even a small fraction of the Au(111)
569	>	surface sites are covered with butanethiols, the conductivity exhibits
570	>	an enhancement by at least a factor of 3.  Capping agents are clearly
571	>	playing a major role in thermal transport at metal / organic solvent
572	>	surfaces.
573
574	<	Interestingly, as one could observe from our results, the maximum
575	<	conductance enhancement (largest $G$) happens while the surfaces are
576	<	about 75\% covered with butanethiols. This again indicates that
577	<	solvent-capping agent contact has an important role of the thermal
578	<	transport process. Slightly lower butanethiol coverage allows small
579	<	gaps between butanethiols to form. And these gaps could be filled with
580	<	solvent molecules, which acts like ``heat conductors'' on the
581	<	surface. The higher degree of interaction between these solvent
582	<	molecules and capping agents increases the enhancement effect and thus
734	<	produces a higher $G$ than densely packed butanethiol arrays. However,
735	<	once this maximum conductance enhancement is reached, $G$ decreases
736	<	when butanethiol coverage continues to decrease. Each capping agent
737	<	molecule reaches its maximum capacity for thermal
738	<	conductance. Therefore, even higher solvent-capping agent contact
739	<	would not offset this effect. Eventually, when butanethiol coverage
740	<	continues to decrease, solvent-capping agent contact actually
741	<	decreases with the disappearing of butanethiol molecules. In this
742	<	case, $G$ decrease could not be offset but instead accelerated. [NEED
743	<	SNAPSHOT SHOWING THE PHENOMENA / SLAB DENSITY ANALYSIS]
574	>	We note a non-monotonic behavior in the interfacial conductance as a
575	>	function of surface coverage. The maximum conductance (largest $G$)
576	>	happens when the surfaces are about 75\% covered with butanethiol
577	>	caps.  The reason for this behavior is not entirely clear.  One
578	>	explanation is that incomplete butanethiol coverage allows small gaps
579	>	between butanethiols to form. These gaps can be filled by transient
580	>	solvent molecules.  These solvent molecules couple very strongly with
581	>	the hot capping agent molecules near the surface, and can then carry
582	>	away (diffusively) the excess thermal energy from the surface.
583
584	<	A comparison of the results obtained from differenet organic solvents
585	<	can also provide useful information of the interfacial thermal
586	<	transport process. The deuterated hexane (UA) results do not appear to
587	<	be much different from those of normal hexane (UA), given that
588	<	butanethiol (UA) is non-deuterated for both solvents. These UA model
589	<	studies, even though eliminating C-H vibration samplings, still have
751	<	C-C vibrational frequencies different from each other. However, these
752	<	differences in the infrared range do not seem to produce an observable
753	<	difference for the results of $G$. [MAY NEED SPECTRA FIGURE]
584	>	There appears to be a competition between the conduction of the
585	>	thermal energy away from the surface by the capping agents (enhanced
586	>	by greater coverage) and the coupling of the capping agents with the
587	>	solvent (enhanced by interdigitation at lower coverages).  This
588	>	competition would lead to the non-monotonic coverage behavior observed
589	>	here.
590
591	<	Furthermore, results for rigid body toluene solvent, as well as other
592	<	UA-hexane solvents, are reasonable within the general experimental
593	<	ranges[CITATIONS]. This suggests that explicit hydrogen might not be a
594	<	required factor for modeling thermal transport phenomena of systems
595	<	such as Au-thiol/organic solvent.
591	>	Results for rigid body toluene solvent, as well as the UA hexane, are
592	>	within the ranges expected from prior experimental
593	>	work.\cite{Wilson:2002uq,cahill:793,PhysRevB.80.195406} This suggests
594	>	that explicit hydrogen atoms might not be required for modeling
595	>	thermal transport in these systems.  C-H vibrational modes do not see
596	>	significant excited state population at low temperatures, and are not
597	>	likely to carry lower frequency excitations from the solid layer into
598	>	the bulk liquid.
599
600	<	However, results for Au-butanethiol/toluene do not show an identical
601	<	trend with those for Au-butanethiol/hexane in that $G$ remains at
602	<	approximately the same magnitue when butanethiol coverage differs from
603	<	25\% to 75\%. This might be rooted in the molecule shape difference
604	<	for planar toluene and chain-like {\it n}-hexane. Due to this
605	<	difference, toluene molecules have more difficulty in occupying
606	<	relatively small gaps among capping agents when their coverage is not
607	<	too low. Therefore, the solvent-capping agent contact may keep
608	<	increasing until the capping agent coverage reaches a relatively low
609	<	level. This becomes an offset for decreasing butanethiol molecules on
771	<	its effect to the process of interfacial thermal transport. Thus, one
772	<	can see a plateau of $G$ vs. butanethiol coverage in our results.
600	>	The toluene solvent does not exhibit the same behavior as hexane in
601	>	that $G$ remains at approximately the same magnitude when the capping
602	>	coverage increases from 25\% to 75\%.  Toluene, as a rigid planar
603	>	molecule, cannot occupy the relatively small gaps between the capping
604	>	agents as easily as the chain-like {\it n}-hexane.  The effect of
605	>	solvent coupling to the capping agent is therefore weaker in toluene
606	>	except at the very lowest coverage levels.  This effect counters the
607	>	coverage-dependent conduction of heat away from the metal surface,
608	>	leading to a much flatter $G$ vs. coverage trend than is observed in
609	>	{\it n}-hexane.
610
611	<	\begin{figure}
612	<	\includegraphics[width=\linewidth]{coverage}
613	<	\caption{Comparison of interfacial thermal conductivity ($G$) values
614	<	for the Au-butanethiol/solvent interface with various UA models and
615	<	different capping agent coverages at $\langle T\rangle\sim$200K
616	<	using certain energy flux respectively.}
617	<	\label{coverage}
618	<	\end{figure}
611	>	\subsection{Effects due to Solvent \& Solvent Models}
612	>	In addition to UA solvent and capping agent models, AA models have
613	>	also been included in our simulations.  In most of this work, the same
614	>	(UA or AA) model for solvent and capping agent was used, but it is
615	>	also possible to utilize different models for different components.
616	>	We have also included isotopic substitutions (Hydrogen to Deuterium)
617	>	to decrease the explicit vibrational overlap between solvent and
618	>	capping agent. Table \ref{modelTest} summarizes the results of these
619	>	studies.
620
783	–	\subsection{Influence of Chosen Molecule Model on $G$}
784	–	In addition to UA solvent/capping agent models, AA models are included
785	–	in our simulations as well. Besides simulations of the same (UA or AA)
786	–	model for solvent and capping agent, different models can be applied
787	–	to different components. Furthermore, regardless of models chosen,
788	–	either the solvent or the capping agent can be deuterated, similar to
789	–	the previous section. Table \ref{modelTest} summarizes the results of
790	–	these studies.
791	–
621		\begin{table*}
622		\begin{minipage}{\linewidth}
623		\begin{center}
624
625	<	\caption{Computed interfacial thermal conductivity ($G$ and
625	>	\caption{Computed interfacial thermal conductance ($G$ and
626		$G^\prime$) values for interfaces using various models for
627		solvent and capping agent (or without capping agent) at
628	<	$\langle T\rangle\sim$200K. (D stands for deuterated solvent
629	<	or capping agent molecules; ``Avg.'' denotes results that are
630	<	averages of simulations under different $J_z$'s. Error
802	<	estimates indicated in parenthesis.)}
628	>	$\langle T\rangle\sim$200K.  Here ``D'' stands for deuterated
629	>	solvent or capping agent molecules. Error estimates are
630	>	indicated in parentheses.}
631
632		\begin{tabular}{llccc}
633		\hline\hline
634	<	Butanethiol model & Solvent & $J_z$ & $G$ & $G^\prime$ \\
635	<	(or bare surface) & model & (GW/m$^2$) &
634	>	Butanethiol model & Solvent & $G$ & $G^\prime$ \\
635	>	(or bare surface) & model &
636		\multicolumn{2}{c}{(MW/m$^2$/K)} \\
637		\hline
638	<	UA    & UA hexane    & Avg. & 131(9)    & 87(10)    \\
639	<	& UA hexane(D) & 1.95 & 153(5)    & 136(13)   \\
640	<	& AA hexane    & Avg. & 131(6)    & 122(10)   \\
641	<	& UA toluene   & 1.96 & 187(16)   & 151(11)   \\
642	<	& AA toluene   & 1.89 & 200(36)   & 149(53)   \\
638	>	UA    & UA hexane    & 131(9)    & 87(10)    \\
639	>	& UA hexane(D) & 153(5)    & 136(13)   \\
640	>	& AA hexane    & 131(6)    & 122(10)   \\
641	>	& UA toluene   & 187(16)   & 151(11)   \\
642	>	& AA toluene   & 200(36)   & 149(53)   \\
643		\hline
644	<	AA    & UA hexane    & 1.94 & 116(9)    & 129(8)    \\
645	<	& AA hexane    & Avg. & 442(14)   & 356(31)   \\
646	<	& AA hexane(D) & 1.93 & 222(12)   & 234(54)   \\
647	<	& UA toluene   & 1.98 & 125(25)   & 97(60)    \\
648	<	& AA toluene   & 3.79 & 487(56)   & 290(42)   \\
644	>	AA    & UA hexane    & 116(9)    & 129(8)    \\
645	>	& AA hexane    & 442(14)   & 356(31)   \\
646	>	& AA hexane(D) & 222(12)   & 234(54)   \\
647	>	& UA toluene   & 125(25)   & 97(60)    \\
648	>	& AA toluene   & 487(56)   & 290(42)   \\
649		\hline
650	<	AA(D) & UA hexane    & 1.94 & 158(25)   & 172(4)    \\
651	<	& AA hexane    & 1.92 & 243(29)   & 191(11)   \\
652	<	& AA toluene   & 1.93 & 364(36)   & 322(67)   \\
650	>	AA(D) & UA hexane    & 158(25)   & 172(4)    \\
651	>	& AA hexane    & 243(29)   & 191(11)   \\
652	>	& AA toluene   & 364(36)   & 322(67)   \\
653		\hline
654	<	bare  & UA hexane    & Avg. & 46.5(3.2) & 49.4(4.5) \\
655	<	& UA hexane(D) & 0.98 & 43.9(4.6) & 43.0(2.0) \\
656	<	& AA hexane    & 0.96 & 31.0(1.4) & 29.4(1.3) \\
657	<	& UA toluene   & 1.99 & 70.1(1.3) & 65.8(0.5) \\
654	>	bare  & UA hexane    & 46.5(3.2) & 49.4(4.5) \\
655	>	& UA hexane(D) & 43.9(4.6) & 43.0(2.0) \\
656	>	& AA hexane    & 31.0(1.4) & 29.4(1.3) \\
657	>	& UA toluene   & 70.1(1.3) & 65.8(0.5) \\
658		\hline\hline
659		\end{tabular}
660		\label{modelTest}
#	Line 834 | Line 662 | To facilitate direct comparison, the same system with
662		\end{minipage}
663		\end{table*}
664
665	<	To facilitate direct comparison, the same system with differnt models
666	<	for different components uses the same length scale for their
667	<	simulation cells. Without the presence of capping agent, using
840	<	different models for hexane yields similar results for both $G$ and
841	<	$G^\prime$, and these two definitions agree with eath other very
842	<	well. This indicates very weak interaction between the metal and the
843	<	solvent, and is a typical case for acoustic impedance mismatch between
844	<	these two phases.
665	>	To facilitate direct comparison between force fields, systems with the
666	>	same capping agent and solvent were prepared with the same length
667	>	scales for the simulation cells.
668
669	<	As for Au(111) surfaces completely covered by butanethiols, the choice
670	<	of models for capping agent and solvent could impact the measurement
671	<	of $G$ and $G^\prime$ quite significantly. For Au-butanethiol/hexane
672	<	interfaces, using AA model for both butanethiol and hexane yields
850	<	substantially higher conductivity values than using UA model for at
851	<	least one component of the solvent and capping agent, which exceeds
852	<	the general range of experimental measurement results. This is
853	<	probably due to the classically treated C-H vibrations in the AA
854	<	model, which should not be appreciably populated at normal
855	<	temperatures. In comparison, once either the hexanes or the
856	<	butanethiols are deuterated, one can see a significantly lower $G$ and
857	<	$G^\prime$. In either of these cases, the C-H(D) vibrational overlap
858	<	between the solvent and the capping agent is removed.
859	<	[ NEED SPECTRA FIGURE] Conclusively, the
860	<	improperly treated C-H vibration in the AA model produced
861	<	over-predicted results accordingly. Compared to the AA model, the UA
862	<	model yields more reasonable results with higher computational
863	<	efficiency.
669	>	On bare metal / solvent surfaces, different force field models for
670	>	hexane yield similar results for both $G$ and $G^\prime$, and these
671	>	two definitions agree with each other very well. This is primarily an
672	>	indicator of weak interactions between the metal and the solvent.
673
674	<	However, for Au-butanethiol/toluene interfaces, having the AA
675	<	butanethiol deuterated did not yield a significant change in the
676	<	measurement results. Compared to the C-H vibrational overlap between
677	<	hexane and butanethiol, both of which have alkyl chains, that overlap
678	<	between toluene and butanethiol is not so significant and thus does
679	<	not have as much contribution to the ``Intramolecular Vibration
680	<	Redistribution''[CITE HASE]. Conversely, extra degrees of freedom such
681	<	as the C-H vibrations could yield higher heat exchange rate between
682	<	these two phases and result in a much higher conductivity.
674	>	For the fully-covered surfaces, the choice of force field for the
675	>	capping agent and solvent has a large impact on the calculated values
676	>	of $G$ and $G^\prime$.  The AA thiol to AA solvent conductivities are
677	>	much larger than their UA to UA counterparts, and these values exceed
678	>	the experimental estimates by a large measure.  The AA force field
679	>	allows significant energy to go into C-H (or C-D) stretching modes,
680	>	and since these modes are high frequency, this non-quantum behavior is
681	>	likely responsible for the overestimate of the conductivity.  Compared
682	>	to the AA model, the UA model yields more reasonable conductivity
683	>	values with much higher computational efficiency.
684
685	<	Although the QSC model for Au is known to predict an overly low value
686	<	for bulk metal gold conductivity\cite{kuang:164101}, our computational
687	<	results for $G$ and $G^\prime$ do not seem to be affected by this
688	<	drawback of the model for metal. Instead, our results suggest that the
689	<	modeling of interfacial thermal transport behavior relies mainly on
690	<	the accuracy of the interaction descriptions between components
691	<	occupying the interfaces.
685	>	\subsubsection{Are electronic excitations in the metal important?}
686	>	Because they lack electronic excitations, the QSC and related embedded
687	>	atom method (EAM) models for gold are known to predict unreasonably
688	>	low values for bulk conductivity
689	>	($\lambda$).\cite{kuang:164101,ISI:000207079300006,Clancy:1992} If the
690	>	conductance between the phases ($G$) is governed primarily by phonon
691	>	excitation (and not electronic degrees of freedom), one would expect a
692	>	classical model to capture most of the interfacial thermal
693	>	conductance.  Our results for $G$ and $G^\prime$ indicate that this is
694	>	indeed the case, and suggest that the modeling of interfacial thermal
695	>	transport depends primarily on the description of the interactions
696	>	between the various components at the interface.  When the metal is
697	>	chemically capped, the primary barrier to thermal conductivity appears
698	>	to be the interface between the capping agent and the surrounding
699	>	solvent, so the excitations in the metal have little impact on the
700	>	value of $G$.
701
702	<	\subsection{Role of Capping Agent in Interfacial Thermal Conductance}
703	<	The vibrational spectra for gold slabs in different environments are
704	<	shown as in Figure \ref{specAu}. Regardless of the presence of
705	<	solvent, the gold surfaces covered by butanethiol molecules, compared
706	<	to bare gold surfaces, exhibit an additional peak observed at the
707	<	frequency of $\sim$170cm$^{-1}$, which is attributed to the S-Au
708	<	bonding vibration. This vibration enables efficient thermal transport
709	<	from surface Au layer to the capping agents. Therefore, in our
710	<	simulations, the Au/S interfaces do not appear major heat barriers
711	<	compared to the butanethiol / solvent interfaces.
702	>	\subsection{Effects due to methodology and simulation parameters}
703	>
704	>	We have varied the parameters of the simulations in order to
705	>	investigate how these factors would affect the computation of $G$.  Of
706	>	particular interest are: 1) the length scale for the applied thermal
707	>	gradient (modified by increasing the amount of solvent in the system),
708	>	2) the sign and magnitude of the applied thermal flux, 3) the average
709	>	temperature of the simulation (which alters the solvent density during
710	>	equilibration), and 4) the definition of the interfacial conductance
711	>	(Eqs. (\ref{discreteG}) or (\ref{derivativeG})) used in the
712	>	calculation.
713
714	<	Simultaneously, the vibrational overlap between butanethiol and
715	<	organic solvents suggests higher thermal exchange efficiency between
716	<	these two components. Even exessively high heat transport was observed
717	<	when All-Atom models were used and C-H vibrations were treated
718	<	classically. Compared to metal and organic liquid phase, the heat
719	<	transfer efficiency between butanethiol and organic solvents is closer
720	<	to that within bulk liquid phase.
714	>	Systems of different lengths were prepared by altering the number of
715	>	solvent molecules and extending the length of the box along the $z$
716	>	axis to accomodate the extra solvent.  Equilibration at the same
717	>	temperature and pressure conditions led to nearly identical surface
718	>	areas ($L_x$ and $L_y$) available to the metal and capping agent,
719	>	while the extra solvent served mainly to lengthen the axis that was
720	>	used to apply the thermal flux.  For a given value of the applied
721	>	flux, the different $z$ length scale has only a weak effect on the
722	>	computed conductivities (Table \ref{AuThiolHexaneUA}).
723
724	<	As a combinational effects of the above two, butanethiol acts as a
725	<	channel to expedite thermal transport process. The acoustic impedance
726	<	mismatch between the metal and the liquid phase can be effectively
727	<	reduced with the presence of suitable capping agents.
724	>	\subsubsection{Effects of applied flux}
725	>	The NIVS algorithm allows changes in both the sign and magnitude of
726	>	the applied flux.  It is possible to reverse the direction of heat
727	>	flow simply by changing the sign of the flux, and thermal gradients
728	>	which would be difficult to obtain experimentally ($5$ K/\AA) can be
729	>	easily simulated.  However, the magnitude of the applied flux is not
730	>	arbitrary if one aims to obtain a stable and reliable thermal gradient.
731	>	A temperature gradient can be lost in the noise if $|J_z|$ is too
732	>	small, and excessive $|J_z|$ values can cause phase transitions if the
733	>	extremes of the simulation cell become widely separated in
734	>	temperature.  Also, if $|J_z|$ is too large for the bulk conductivity
735	>	of the materials, the thermal gradient will never reach a stable
736	>	state.
737
738	+	Within a reasonable range of $J_z$ values, we were able to study how
739	+	$G$ changes as a function of this flux.  In what follows, we use
740	+	positive $J_z$ values to denote the case where energy is being
741	+	transferred by the method from the metal phase and into the liquid.
742	+	The resulting gradient therefore has a higher temperature in the
743	+	liquid phase.  Negative flux values reverse this transfer, and result
744	+	in higher temperature metal phases.  The conductance measured under
745	+	different applied $J_z$ values is listed in Tables 1 and 2 in the
746	+	supporting information. These results do not indicate that $G$ depends
747	+	strongly on $J_z$ within this flux range. The linear response of flux
748	+	to thermal gradient simplifies our investigations in that we can rely
749	+	on $G$ measurement with only a small number $J_z$ values.
750	+
751	+	The sign of $J_z$ is a different matter, however, as this can alter
752	+	the temperature on the two sides of the interface. The average
753	+	temperature values reported are for the entire system, and not for the
754	+	liquid phase, so at a given $\langle T \rangle$, the system with
755	+	positive $J_z$ has a warmer liquid phase.  This means that if the
756	+	liquid carries thermal energy via diffusive transport, {\it positive}
757	+	$J_z$ values will result in increased molecular motion on the liquid
758	+	side of the interface, and this will increase the measured
759	+	conductivity.
760	+
761	+	\subsubsection{Effects due to average temperature}
762	+
763	+	We also studied the effect of average system temperature on the
764	+	interfacial conductance.  The simulations are first equilibrated in
765	+	the NPT ensemble at 1 atm.  The TraPPE-UA model for hexane tends to
766	+	predict a lower boiling point (and liquid state density) than
767	+	experiments.  This lower-density liquid phase leads to reduced contact
768	+	between the hexane and butanethiol, and this accounts for our
769	+	observation of lower conductance at higher temperatures.  In raising
770	+	the average temperature from 200K to 250K, the density drop of
771	+	$\sim$20\% in the solvent phase leads to a $\sim$40\% drop in the
772	+	conductance.
773	+
774	+	Similar behavior is observed in the TraPPE-UA model for toluene,
775	+	although this model has better agreement with the experimental
776	+	densities of toluene.  The expansion of the toluene liquid phase is
777	+	not as significant as that of the hexane (8.3\% over 100K), and this
778	+	limits the effect to $\sim$20\% drop in thermal conductivity.
779	+
780	+	Although we have not mapped out the behavior at a large number of
781	+	temperatures, is clear that there will be a strong temperature
782	+	dependence in the interfacial conductance when the physical properties
783	+	of one side of the interface (notably the density) change rapidly as a
784	+	function of temperature.
785	+
786	+	Besides the lower interfacial thermal conductance, surfaces at
787	+	relatively high temperatures are susceptible to reconstructions,
788	+	particularly when butanethiols fully cover the Au(111) surface. These
789	+	reconstructions include surface Au atoms which migrate outward to the
790	+	S atom layer, and butanethiol molecules which embed into the surface
791	+	Au layer. The driving force for this behavior is the strong Au-S
792	+	interactions which are modeled here with a deep Lennard-Jones
793	+	potential. This phenomenon agrees with reconstructions that have been
794	+	experimentally
795	+	observed.\cite{doi:10.1021/j100035a033,doi:10.1021/la026493y}.  Vlugt
796	+	{\it et al.} kept their Au(111) slab rigid so that their simulations
797	+	could reach 300K without surface
798	+	reconstructions.\cite{vlugt:cpc2007154} Since surface reconstructions
799	+	blur the interface, the measurement of $G$ becomes more difficult to
800	+	conduct at higher temperatures.  For this reason, most of our
801	+	measurements are undertaken at $\langle T\rangle\sim$200K where
802	+	reconstruction is minimized.
803	+
804	+	However, when the surface is not completely covered by butanethiols,
805	+	the simulated system appears to be more resistent to the
806	+	reconstruction. Our Au / butanethiol / toluene system had the Au(111)
807	+	surfaces 90\% covered by butanethiols, but did not see this above
808	+	phenomena even at $\langle T\rangle\sim$300K.  That said, we did
809	+	observe butanethiols migrating to neighboring three-fold sites during
810	+	a simulation.  Since the interface persisted in these simulations, we
811	+	were able to obtain $G$'s for these interfaces even at a relatively
812	+	high temperature without being affected by surface reconstructions.
813	+
814	+	\section{Discussion}
815	+
816	+	The primary result of this work is that the capping agent acts as an
817	+	efficient thermal coupler between solid and solvent phases.  One of
818	+	the ways the capping agent can carry out this role is to down-shift
819	+	between the phonon vibrations in the solid (which carry the heat from
820	+	the gold) and the molecular vibrations in the liquid (which carry some
821	+	of the heat in the solvent).
822	+
823	+	To investigate the mechanism of interfacial thermal conductance, the
824	+	vibrational power spectrum was computed. Power spectra were taken for
825	+	individual components in different simulations. To obtain these
826	+	spectra, simulations were run after equilibration in the
827	+	microcanonical (NVE) ensemble and without a thermal
828	+	gradient. Snapshots of configurations were collected at a frequency
829	+	that is higher than that of the fastest vibrations occurring in the
830	+	simulations. With these configurations, the velocity auto-correlation
831	+	functions can be computed:
832	+	\begin{equation}
833	+	C_A (t) = \langle\vec{v}_A (t)\cdot\vec{v}_A (0)\rangle
834	+	\label{vCorr}
835	+	\end{equation}
836	+	The power spectrum is constructed via a Fourier transform of the
837	+	symmetrized velocity autocorrelation function,
838	+	\begin{equation}
839	+	\hat{f}(\omega) =
840	+	\int_{-\infty}^{\infty} C_A (t) e^{-2\pi it\omega}\,dt
841	+	\label{fourier}
842	+	\end{equation}
843	+
844	+	\subsection{The role of specific vibrations}
845	+	The vibrational spectra for gold slabs in different environments are
846	+	shown as in Figure \ref{specAu}. Regardless of the presence of
847	+	solvent, the gold surfaces which are covered by butanethiol molecules
848	+	exhibit an additional peak observed at a frequency of
849	+	$\sim$165cm$^{-1}$.  We attribute this peak to the S-Au bonding
850	+	vibration. This vibration enables efficient thermal coupling of the
851	+	surface Au layer to the capping agents. Therefore, in our simulations,
852	+	the Au / S interfaces do not appear to be the primary barrier to
853	+	thermal transport when compared with the butanethiol / solvent
854	+	interfaces.  This supports the results of Luo {\it et
855	+	al.}\cite{Luo20101}, who reported $G$ for Au-SAM junctions roughly
856	+	twice as large as what we have computed for the thiol-liquid
857	+	interfaces.
858	+
859		\begin{figure}
860		\includegraphics[width=\linewidth]{vibration}
861	<	\caption{Vibrational spectra obtained for gold in different
862	<	environments.}
861	>	\caption{The vibrational power spectrum for thiol-capped gold has an
862	>	additional vibrational peak at $\sim $165cm$^{-1}$.  Bare gold
863	>	surfaces (both with and without a solvent over-layer) are missing
864	>	this peak.   A similar peak at  $\sim $165cm$^{-1}$ also appears in
865	>	the vibrational power spectrum for the butanethiol capping agents.}
866		\label{specAu}
867		\end{figure}
868
869	<	[MAY ADD COMPARISON OF AU SLAB WIDTHS]
869	>	Also in this figure, we show the vibrational power spectrum for the
870	>	bound butanethiol molecules, which also exhibits the same
871	>	$\sim$165cm$^{-1}$ peak.
872
873	+	\subsection{Overlap of power spectra}
874	+	A comparison of the results obtained from the two different organic
875	+	solvents can also provide useful information of the interfacial
876	+	thermal transport process.  In particular, the vibrational overlap
877	+	between the butanethiol and the organic solvents suggests a highly
878	+	efficient thermal exchange between these components.  Very high
879	+	thermal conductivity was observed when AA models were used and C-H
880	+	vibrations were treated classically. The presence of extra degrees of
881	+	freedom in the AA force field yields higher heat exchange rates
882	+	between the two phases and results in a much higher conductivity than
883	+	in the UA force field. The all-atom classical models include high
884	+	frequency modes which should be unpopulated at our relatively low
885	+	temperatures.  This artifact is likely the cause of the high thermal
886	+	conductance in all-atom MD simulations.
887	+
888	+	The similarity in the vibrational modes available to solvent and
889	+	capping agent can be reduced by deuterating one of the two components
890	+	(Fig. \ref{aahxntln}).  Once either the hexanes or the butanethiols
891	+	are deuterated, one can observe a significantly lower $G$ and
892	+	$G^\prime$ values (Table \ref{modelTest}).
893	+
894	+	\begin{figure}
895	+	\includegraphics[width=\linewidth]{aahxntln}
896	+	\caption{Spectra obtained for all-atom (AA) Au / butanethiol / solvent
897	+	systems. When butanethiol is deuterated (lower left), its
898	+	vibrational overlap with hexane decreases significantly.  Since
899	+	aromatic molecules and the butanethiol are vibrationally dissimilar,
900	+	the change is not as dramatic when toluene is the solvent (right).}
901	+	\label{aahxntln}
902	+	\end{figure}
903	+
904	+	For the Au / butanethiol / toluene interfaces, having the AA
905	+	butanethiol deuterated did not yield a significant change in the
906	+	measured conductance. Compared to the C-H vibrational overlap between
907	+	hexane and butanethiol, both of which have alkyl chains, the overlap
908	+	between toluene and butanethiol is not as significant and thus does
909	+	not contribute as much to the heat exchange process.
910	+
911	+	Previous observations of Zhang {\it et al.}\cite{hase:2010} indicate
912	+	that the {\it intra}molecular heat transport due to alkylthiols is
913	+	highly efficient.  Combining our observations with those of Zhang {\it
914	+	et al.}, it appears that butanethiol acts as a channel to expedite
915	+	heat flow from the gold surface and into the alkyl chain.  The
916	+	vibrational coupling between the metal and the liquid phase can
917	+	therefore be enhanced with the presence of suitable capping agents.
918	+
919	+	Deuterated models in the UA force field did not decouple the thermal
920	+	transport as well as in the AA force field.  The UA models, even
921	+	though they have eliminated the high frequency C-H vibrational
922	+	overlap, still have significant overlap in the lower-frequency
923	+	portions of the infrared spectra (Figure \ref{uahxnua}).  Deuterating
924	+	the UA models did not decouple the low frequency region enough to
925	+	produce an observable difference for the results of $G$ (Table
926	+	\ref{modelTest}).
927	+
928	+	\begin{figure}
929	+	\includegraphics[width=\linewidth]{uahxnua}
930	+	\caption{Vibrational power spectra for UA models for the butanethiol
931	+	and hexane solvent (upper panel) show the high degree of overlap
932	+	between these two molecules, particularly at lower frequencies.
933	+	Deuterating a UA model for the solvent (lower panel) does not
934	+	decouple the two spectra to the same degree as in the AA force
935	+	field (see Fig \ref{aahxntln}).}
936	+	\label{uahxnua}
937	+	\end{figure}
938	+
939		\section{Conclusions}
940	<	The NIVS algorithm we developed has been applied to simulations of
941	<	Au-butanethiol surfaces with organic solvents. This algorithm allows
942	<	effective unphysical thermal flux transferred between the metal and
943	<	the liquid phase. With the flux applied, we were able to measure the
944	<	corresponding thermal gradient and to obtain interfacial thermal
945	<	conductivities. Under steady states, single trajectory simulation
946	<	would be enough for accurate measurement. This would be advantageous
947	<	compared to transient state simulations, which need multiple
925	<	trajectories to produce reliable average results.
940	>	The NIVS algorithm has been applied to simulations of
941	>	butanethiol-capped Au(111) surfaces in the presence of organic
942	>	solvents. This algorithm allows the application of unphysical thermal
943	>	flux to transfer heat between the metal and the liquid phase. With the
944	>	flux applied, we were able to measure the corresponding thermal
945	>	gradients and to obtain interfacial thermal conductivities. Under
946	>	steady states, 2-3 ns trajectory simulations are sufficient for
947	>	computation of this quantity.
948
949	<	Our simulations have seen significant conductance enhancement with the
950	<	presence of capping agent, compared to the bare gold / liquid
951	<	interfaces. The acoustic impedance mismatch between the metal and the
952	<	liquid phase is effectively eliminated by proper capping
953	<	agent. Furthermore, the coverage precentage of the capping agent plays
954	<	an important role in the interfacial thermal transport
955	<	process. Moderately lower coverages allow higher contact between
956	<	capping agent and solvent, and thus could further enhance the heat
957	<	transfer process.
949	>	Our simulations have seen significant conductance enhancement in the
950	>	presence of capping agent, compared with the bare gold / liquid
951	>	interfaces. The vibrational coupling between the metal and the liquid
952	>	phase is enhanced by a chemically-bonded capping agent. Furthermore,
953	>	the coverage percentage of the capping agent plays an important role
954	>	in the interfacial thermal transport process. Moderately low coverages
955	>	allow higher contact between capping agent and solvent, and thus could
956	>	further enhance the heat transfer process, giving a non-monotonic
957	>	behavior of conductance with increasing coverage.
958
959	<	Our measurement results, particularly of the UA models, agree with
960	<	available experimental data. This indicates that our force field
939	<	parameters have a nice description of the interactions between the
940	<	particles at the interfaces. AA models tend to overestimate the
959	>	Our results, particularly using the UA models, agree well with
960	>	available experimental data.  The AA models tend to overestimate the
961		interfacial thermal conductance in that the classically treated C-H
962	<	vibration would be overly sampled. Compared to the AA models, the UA
963	<	models have higher computational efficiency with satisfactory
964	<	accuracy, and thus are preferable in interfacial thermal transport
965	<	modelings. Of the two definitions for $G$, the discrete form
962	>	vibrations become too easily populated. Compared to the AA models, the
963	>	UA models have higher computational efficiency with satisfactory
964	>	accuracy, and thus are preferable in modeling interfacial thermal
965	>	transport.
966	>
967	>	Of the two definitions for $G$, the discrete form
968		(Eq. \ref{discreteG}) was easier to use and gives out relatively
969		consistent results, while the derivative form (Eq. \ref{derivativeG})
970		is not as versatile. Although $G^\prime$ gives out comparable results
971		and follows similar trend with $G$ when measuring close to fully
972	<	covered or bare surfaces, the spatial resolution of $T$ profile is
973	<	limited for accurate computation of derivatives data.
972	>	covered or bare surfaces, the spatial resolution of $T$ profile
973	>	required for the use of a derivative form is limited by the number of
974	>	bins and the sampling required to obtain thermal gradient information.
975
976	<	Vlugt {\it et al.} has investigated the surface thiol structures for
977	<	nanocrystal gold and pointed out that they differs from those of the
978	<	Au(111) surface\cite{vlugt:cpc2007154}. This difference might lead to
979	<	change of interfacial thermal transport behavior as well. To
980	<	investigate this problem, an effective means to introduce thermal flux
981	<	and measure the corresponding thermal gradient is desirable for
982	<	simulating structures with spherical symmetry.
976	>	Vlugt {\it et al.} have investigated the surface thiol structures for
977	>	nanocrystalline gold and pointed out that they differ from those of
978	>	the Au(111) surface.\cite{landman:1998,vlugt:cpc2007154} This
979	>	difference could also cause differences in the interfacial thermal
980	>	transport behavior. To investigate this problem, one would need an
981	>	effective method for applying thermal gradients in non-planar
982	>	(i.e. spherical) geometries.
983
984		\section{Acknowledgments}
985		Support for this project was provided by the National Science
986		Foundation under grant CHE-0848243. Computational time was provided by
987		the Center for Research Computing (CRC) at the University of Notre
988	<	Dame. \newpage
988	>	Dame.
989
990	+	\section{Supporting Information}
991	+	This information is available free of charge via the Internet at
992	+	http://pubs.acs.org.
993	+
994	+	\newpage
995	+
996		\bibliography{interfacial}
997
998		\end{doublespace}

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