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\begin{document} |
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|
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\title{Simulating interfacial thermal conductance at metal-solvent |
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interfaces: the role of chemical capping agents} |
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\title{Simulating Interfacial Thermal Conductance at Metal-Solvent |
32 |
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Interfaces: the Role of Chemical Capping Agents} |
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|
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\author{Shenyu Kuang and J. Daniel |
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Gezelter\footnote{Corresponding author. \ Electronic mail: gezelter@nd.edu} \\ |
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\begin{doublespace} |
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|
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\begin{abstract} |
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With the Non-Isotropic Velocity Scaling (NIVS) approach to Reverse |
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Non-Equilibrium Molecular Dynamics (RNEMD) it is possible to impose |
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an unphysical thermal flux between different regions of |
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inhomogeneous systems such as solid / liquid interfaces. We have |
51 |
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applied NIVS to compute the interfacial thermal conductance at a |
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metal / organic solvent interface that has been chemically capped by |
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butanethiol molecules. Our calculations suggest that vibrational |
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coupling between the metal and liquid phases is enhanced by the |
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capping agents, leading to a greatly enhanced conductivity at the |
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interface. Specifically, the chemical bond between the metal and |
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the capping agent introduces a vibrational overlap that is not |
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present without the capping agent, and the overlap between the |
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vibrational spectra (metal to cap, cap to solvent) provides a |
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mechanism for rapid thermal transport across the interface. Our |
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calculations also suggest that this is a non-monotonic function of |
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the fractional coverage of the surface, as moderate coverages allow |
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diffusive heat transport of solvent molecules that have been in |
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close contact with the capping agent. |
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|
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With the Non-Isotropic Velocity Scaling algorithm (NIVS) we have |
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developed, an unphysical thermal flux can be effectively set up even |
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for non-homogeneous systems like interfaces in non-equilibrium |
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molecular dynamics simulations. In this work, this algorithm is |
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applied for simulating thermal conductance at metal / organic solvent |
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interfaces with various coverages of butanethiol capping |
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agents. Different solvents and force field models were tested. Our |
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results suggest that the United-Atom models are able to provide an |
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estimate of the interfacial thermal conductivity comparable to |
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experiments in our simulations with satisfactory computational |
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efficiency. From our results, the acoustic impedance mismatch between |
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metal and liquid phase is effectively reduced by the capping |
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agents, and thus leads to interfacial thermal conductance |
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enhancement. Furthermore, this effect is closely related to the |
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capping agent coverage on the metal surfaces and the type of solvent |
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molecules, and is affected by the models used in the simulations. |
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|
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Keywords: non-equilibrium, molecular dynamics, vibrational overlap, |
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coverage dependent. |
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\end{abstract} |
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\newpage |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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\section{Introduction} |
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Interfacial thermal conductance is extensively studied both |
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experimentally and computationally\cite{cahill:793}, due to its |
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importance in nanoscale science and technology. Reliability of |
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nanoscale devices depends on their thermal transport |
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properties. Unlike bulk homogeneous materials, nanoscale materials |
84 |
< |
features significant presence of interfaces, and these interfaces |
85 |
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could dominate the heat transfer behavior of these |
86 |
< |
materials. Furthermore, these materials are generally heterogeneous, |
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which challenges traditional research methods for homogeneous |
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systems. |
79 |
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Due to the importance of heat flow (and heat removal) in |
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nanotechnology, interfacial thermal conductance has been studied |
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extensively both experimentally and computationally.\cite{cahill:793} |
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Nanoscale materials have a significant fraction of their atoms at |
83 |
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interfaces, and the chemical details of these interfaces govern the |
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thermal transport properties. Furthermore, the interfaces are often |
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heterogeneous (e.g. solid - liquid), which provides a challenge to |
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computational methods which have been developed for homogeneous or |
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bulk systems. |
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|
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Heat conductance of molecular and nano-scale interfaces will be |
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affected by the chemical details of the surface. Experimentally, |
91 |
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various interfaces have been investigated for their thermal |
92 |
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conductance properties. Wang {\it et al.} studied heat transport |
93 |
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through long-chain hydrocarbon monolayers on gold substrate at |
94 |
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individual molecular level\cite{Wang10082007}; Schmidt {\it et al.} |
95 |
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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 |
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the cooling dynamics, which is controlled by thermal interface |
98 |
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resistence of glass-embedded metal |
99 |
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nanoparticles\cite{PhysRevB.80.195406}. Although interfaces are |
100 |
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commonly barriers for heat transport, Alper {\it et al.} suggested |
101 |
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that specific ligands (capping agents) could completely eliminate this |
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barrier ($G\rightarrow\infty$)\cite{doi:10.1021/la904855s}. |
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Experimentally, the thermal properties of a number of interfaces have |
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been investigated. Cahill and coworkers studied nanoscale thermal |
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transport from metal nanoparticle/fluid interfaces, to epitaxial |
92 |
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TiN/single crystal oxides interfaces, and hydrophilic and hydrophobic |
93 |
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interfaces between water and solids with different self-assembled |
94 |
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monolayers.\cite{Wilson:2002uq,PhysRevB.67.054302,doi:10.1021/jp048375k,PhysRevLett.96.186101} |
95 |
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Wang {\it et al.} studied heat transport through long-chain |
96 |
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hydrocarbon monolayers on gold substrate at individual molecular |
97 |
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level,\cite{Wang10082007} Schmidt {\it et al.} studied the role of |
98 |
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cetyltrimethylammonium bromide (CTAB) on the thermal transport between |
99 |
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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 |
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interface resistance of glass-embedded metal |
102 |
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nanoparticles.\cite{PhysRevB.80.195406} Although interfaces are |
103 |
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normally considered barriers for heat transport, Alper {\it et al.} |
104 |
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suggested that specific ligands (capping agents) could completely |
105 |
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eliminate this barrier |
106 |
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($G\rightarrow\infty$).\cite{doi:10.1021/la904855s} |
107 |
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|
108 |
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Theoretical and computational models have also been used to study the |
108 |
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The acoustic mismatch model for interfacial conductance utilizes the |
109 |
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acoustic impedance ($Z_a = \rho_a v^s_a$) on both sides of the |
110 |
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interface.\cite{schwartz} Here, $\rho_a$ and $v^s_a$ are the density |
111 |
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and speed of sound in material $a$. The phonon transmission |
112 |
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probability at the $a-b$ interface is |
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\begin{equation} |
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t_{ab} = \frac{4 Z_a Z_b}{\left(Z_a + Z_b \right)^2}, |
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\end{equation} |
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and the interfacial conductance can then be approximated as |
117 |
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\begin{equation} |
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G_{ab} \approx \frac{1}{4} C_D v_D t_{ab} |
119 |
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\end{equation} |
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where $C_D$ is the Debye heat capacity of the hot side, and $v_D$ is |
121 |
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the Debye phonon velocity ($1/v_D^3 = 1/3v_L^3 + 2/3 v_T^3$) where |
122 |
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$v_L$ and $v_T$ are the longitudinal and transverse speeds of sound, |
123 |
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respectively. For the Au/hexane and Au/toluene interfaces, the |
124 |
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acoustic mismatch model predicts room-temperature $G \approx 87 \mbox{ |
125 |
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and } 129$ MW m$^{-2}$ K$^{-1}$, respectively. However, it is not |
126 |
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clear how one might apply the acoustic mismatch model to a |
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chemically-modified surface, particularly when the acoustic properties |
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of a monolayer film may not be well characterized. |
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|
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More precise computational models have also been used to study the |
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|
interfacial thermal transport in order to gain an understanding of |
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this phenomena at the molecular level. Recently, Hase and coworkers |
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employed Non-Equilibrium Molecular Dynamics (NEMD) simulations to |
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study thermal transport from hot Au(111) substrate to a self-assembled |
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monolayer of alkylthiol with relatively long chain (8-20 carbon |
136 |
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atoms)\cite{hase:2010,hase:2011}. However, ensemble averaged |
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atoms).\cite{hase:2010,hase:2011} However, ensemble averaged |
137 |
|
measurements for heat conductance of interfaces between the capping |
138 |
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monolayer on Au and a solvent phase has yet to be studied. |
139 |
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The comparatively low thermal flux through interfaces is |
140 |
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difficult to measure with Equilibrium MD or forward NEMD simulation |
141 |
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methods. Therefore, the Reverse NEMD (RNEMD) methods would have the |
142 |
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advantage of having this difficult to measure flux known when studying |
143 |
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the thermal transport across interfaces, given that the simulation |
144 |
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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 |
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MD\cite{doi:10.1080/0026897031000068578} or forward NEMD simulation |
142 |
> |
methods. Therefore, the Reverse NEMD (RNEMD) |
143 |
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methods\cite{MullerPlathe:1997xw,kuang:164101} would be advantageous |
144 |
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in that they {\it apply} the difficult to measure quantity (flux), |
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while {\it measuring} the easily-computed quantity (the thermal |
146 |
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gradient). This is particularly true for inhomogeneous interfaces |
147 |
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where it would not be clear how to apply a gradient {\it a priori}. |
148 |
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Garde and coworkers\cite{garde:nl2005,garde:PhysRevLett2009} applied |
149 |
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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 |
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recently, Luo {\it et al.} studied the thermal conductance of |
153 |
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Au-SAM-Au junctions using the same approach, comparing to a constant |
154 |
> |
temperature difference method.\cite{Luo20101} While this latter |
155 |
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approach establishes more ideal Maxwell-Boltzmann distributions than |
156 |
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previous RNEMD methods, it does not guarantee momentum or kinetic |
157 |
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energy conservation. |
158 |
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|
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Recently, we have developed the Non-Isotropic Velocity Scaling (NIVS) |
159 |
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Recently, we have developed a Non-Isotropic Velocity Scaling (NIVS) |
160 |
|
algorithm for RNEMD simulations\cite{kuang:164101}. This algorithm |
161 |
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retains the desirable features of RNEMD (conservation of linear |
162 |
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momentum and total energy, compatibility with periodic boundary |
171 |
|
properties. Different models were used for both the capping agent and |
172 |
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the solvent force field parameters. Using the NIVS algorithm, the |
173 |
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thermal transport across these interfaces was studied and the |
174 |
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underlying mechanism for this phenomena was investigated. |
174 |
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underlying mechanism for the phenomena was investigated. |
175 |
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|
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[MAY ADD WHY STUDY AU-THIOL SURFACE; CITE SHAOYI JIANG] |
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|
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|
\section{Methodology} |
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\subsection{Imposd-Flux Methods in MD Simulations} |
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For systems with low interfacial conductivity one must have a method |
179 |
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capable of generating relatively small fluxes, compared to those |
180 |
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required for bulk conductivity. This requirement makes the calculation |
181 |
< |
even more difficult for those slowly-converging equilibrium |
182 |
< |
methods\cite{Viscardy:2007lq}. |
183 |
< |
Forward methods impose gradient, but in interfacail conditions it is |
184 |
< |
not clear what behavior to impose at the boundary... |
185 |
< |
Imposed-flux reverse non-equilibrium |
186 |
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methods\cite{MullerPlathe:1997xw} have the flux set {\it a priori} and |
187 |
< |
the thermal response becomes easier to |
188 |
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measure than the flux. Although M\"{u}ller-Plathe's original momentum |
189 |
< |
swapping approach can be used for exchanging energy between particles |
190 |
< |
of different identity, the kinetic energy transfer efficiency is |
191 |
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affected by the mass difference between the particles, which limits |
192 |
< |
its application on heterogeneous interfacial systems. |
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> |
\subsection{Imposed-Flux Methods in MD Simulations} |
178 |
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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 |
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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. |
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|
|
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|
The NIVS algorithm scales the velocity vectors in two separate regions |
205 |
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of a simulation system with respective diagonal scaling matricies. To |
206 |
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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 |
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|
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 |
211 |
|
for computing thermal conductivities. The NIVS algorithm conserves |
212 |
|
momenta and energy and does not depend on an external thermostat. |
213 |
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|
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\subsection{Defining Interfacial Thermal Conductivity $G$} |
215 |
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For interfaces with a relatively low interfacial conductance, the bulk |
216 |
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regions on either side of an interface rapidly come to a state in |
217 |
< |
which the two phases have relatively homogeneous (but distinct) |
218 |
< |
temperatures. The interfacial thermal conductivity $G$ can therefore |
219 |
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be approximated as: |
214 |
> |
\subsection{Defining Interfacial Thermal Conductivity ($G$)} |
215 |
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|
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 |
< |
as: |
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 |
|
|
203 |
– |
The other approach is to assume a continuous temperature profile along |
204 |
– |
the thermal gradient axis (e.g. $z$) and define $G$ at the point where |
205 |
– |
the magnitude of thermal conductivity $\lambda$ change reach its |
206 |
– |
maximum, given that $\lambda$ is well-defined throughout the space: |
207 |
– |
\begin{equation} |
208 |
– |
G^\prime = \Big|\frac{\partial\lambda}{\partial z}\Big| |
209 |
– |
= \Big|\frac{\partial}{\partial z}\left(-J_z\Big/ |
210 |
– |
\left(\frac{\partial T}{\partial z}\right)\right)\Big| |
211 |
– |
= |J_z|\Big|\frac{\partial^2 T}{\partial z^2}\Big| |
212 |
– |
\Big/\left(\frac{\partial T}{\partial z}\right)^2 |
213 |
– |
\label{derivativeG} |
214 |
– |
\end{equation} |
215 |
– |
|
216 |
– |
With the temperature profile obtained from simulations, one is able to |
217 |
– |
approximate the first and second derivatives of $T$ with finite |
218 |
– |
difference methods and thus calculate $G^\prime$. |
219 |
– |
|
220 |
– |
In what follows, both definitions have been used for calculation and |
221 |
– |
are compared in the results. |
222 |
– |
|
223 |
– |
To compare the above definitions ($G$ and $G^\prime$), we have modeled |
224 |
– |
a metal slab with its (111) surfaces perpendicular to the $z$-axis of |
225 |
– |
our simulation cells. Both with and withour capping agents on the |
226 |
– |
surfaces, the metal slab is solvated with simple organic solvents, as |
227 |
– |
illustrated in Figure \ref{demoPic}. |
228 |
– |
|
246 |
|
\begin{figure} |
247 |
|
\includegraphics[width=\linewidth]{method} |
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 |
+ |
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 temperature profiles obtained from simulation, one is able to |
280 |
+ |
approximate the first and second derivatives of $T$ with finite |
281 |
+ |
difference methods and calculate $G^\prime$. In what follows, both |
282 |
+ |
definitions have been used, and are compared in the results. |
283 |
+ |
|
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 how those applied thermal fluxes can be used to obtain the 1st |
250 |
< |
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{The 1st and 2nd derivatives of temperature profile can be |
302 |
< |
obtained with finite difference approximation.} |
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 slab thickness (layer numbers of Au) were |
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 |
< |
sites on the Au(111) surfaces. The maximum butanethiol capacity on Au |
321 |
< |
surface is $1/3$ of the total number of surface Au |
322 |
< |
atoms\cite{vlugt:cpc2007154}. A series of different coverages was |
323 |
< |
investigated in order to study the relation between coverage and |
324 |
< |
interfacial conductance. |
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{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 FIGURES] |
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 ($z$) |
291 |
< |
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 $35 \sim 60$\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 by Packmol are further |
359 |
< |
equilibrated with the $x$ and $y$ dimensions fixed, only allowing |
360 |
< |
length scale change in $z$ dimension. This is to ensure that the |
361 |
< |
equilibration of liquid phase does not affect the metal crystal |
362 |
< |
structure in $x$ and $y$ dimensions. Further equilibration are run |
363 |
< |
under NVT and then NVE ensembles. |
358 |
> |
The initial configurations generated are further equilibrated with the |
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. This induced temperature |
374 |
< |
gradient is stablized and the simulation cell is devided evenly into |
375 |
< |
N slabs along the $z$-axis and the temperatures of each slab are |
376 |
< |
recorded. When the slab width $d$ of each slab is the same, the |
377 |
< |
derivatives of $T$ with respect to slab number $n$ can be directly |
378 |
< |
used for $G^\prime$ calculations: |
379 |
< |
\begin{equation} |
380 |
< |
G^\prime = |J_z|\Big|\frac{\partial^2 T}{\partial z^2}\Big| |
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: \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 |
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. 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. Therefore, force field |
398 |
< |
parameter descriptions are needed for interactions both between the |
399 |
< |
same type of particles and between particles of different species. |
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 |
|
|
332 |
– |
The Au-Au interactions in metal lattice slab is described by the |
333 |
– |
quantum Sutton-Chen (QSC) formulation\cite{PhysRevB.59.3527}. The QSC |
334 |
– |
potentials include zero-point quantum corrections and are |
335 |
– |
reparametrized for accurate surface energies compared to the |
336 |
– |
Sutton-Chen potentials\cite{Chen90}. |
337 |
– |
|
338 |
– |
Figure \ref{demoMol} demonstrates how we name our pseudo-atoms of the |
339 |
– |
organic solvent molecules in our simulations. |
340 |
– |
|
404 |
|
\begin{figure} |
405 |
|
\includegraphics[width=\linewidth]{structures} |
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 |
< |
For both solvent molecules, straight chain {\it n}-hexane and aromatic |
417 |
< |
toluene, United-Atom (UA) and All-Atom (AA) models are used |
418 |
< |
respectively. The TraPPE-UA |
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 |
418 |
> |
potentials include zero-point quantum corrections and are |
419 |
> |
reparametrized for accurate surface energies compared to the |
420 |
> |
Sutton-Chen potentials.\cite{Chen90} |
421 |
> |
|
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, pseudo-atoms are |
427 |
< |
located at the carbon centers for alkyl groups. By eliminating |
428 |
< |
explicit hydrogen atoms, these models are simple and computationally |
429 |
< |
efficient, while maintains good accuracy. However, the TraPPE-UA for |
430 |
< |
alkanes is known to predict a lower boiling point than experimental |
362 |
< |
values. Considering that after an unphysical thermal flux is applied |
363 |
< |
to a system, the temperature of ``hot'' area in the liquid phase would be |
364 |
< |
significantly higher than the average, to prevent over heating and |
365 |
< |
boiling of the liquid phase, the average temperature in our |
366 |
< |
simulations should be much lower than the liquid boiling point. [MORE DISCUSSION] |
367 |
< |
For UA-toluene model, rigid body constraints are applied, so that the |
368 |
< |
benzene ring and the methyl-CRar bond are kept rigid. This would save |
369 |
< |
computational time.[MORE DETAILS] |
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 closer than 3 bonds. For non-bonded interactions, Lennard-Jones |
430 |
> |
potentials are used. |
431 |
|
|
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, 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. [MORE DETAILS] |
451 |
< |
For toluene, the United Force Field developed by Rapp\'{e} {\it et |
452 |
< |
al.}\cite{doi:10.1021/ja00051a040} is adopted.[MORE DETAILS] |
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, 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} and modify parameters for its neighbor C |
466 |
< |
atom for charge balance in the molecule. Note that the model choice |
467 |
< |
(UA or AA) of capping agent can be different from the |
468 |
< |
solvent. Regardless of model choice, the force field parameters for |
469 |
< |
interactions between capping agent and solvent can be derived using |
390 |
< |
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 |
405 |
< |
transport study purpose, the pair-wise form of potentials is |
406 |
< |
preferred. |
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 |
< |
Besides, the potentials developed from {\it ab initio} calculations by |
486 |
< |
Leng {\it et al.}\cite{doi:10.1021/jp034405s} are adopted for the |
487 |
< |
interactions between Au and aromatic C/H atoms in toluene.[MORE DETAILS] |
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 |
|
|
412 |
– |
However, the Lennard-Jones parameters between Au and other types of |
413 |
– |
particles in our simulations are not yet well-established. For these |
414 |
– |
interactions, we attempt to derive their parameters using the Mixing |
415 |
– |
Rule. To do this, the ``Metal-non-Metal'' (MnM) interaction parameters |
416 |
– |
for Au is first extracted from the Au-CH$_x$ parameters by applying |
417 |
– |
the Mixing Rule reversely. Table \ref{MnM} summarizes these ``MnM'' |
418 |
– |
parameters in our simulations. |
419 |
– |
|
497 |
|
\begin{table*} |
498 |
|
\begin{minipage}{\linewidth} |
499 |
|
\begin{center} |
520 |
|
&CA & 3.55 & 0.070 & -0.115 & 3.173 & 0.0640 \\ |
521 |
|
&HA & 2.42 & 0.030 & 0.115 & 2.746 & 0.0414 \\ |
522 |
|
\hline |
523 |
< |
Both UA and AA & S & 4.45 & 0.25 & - & 2.40 & 8.465 \\ |
523 |
> |
Both UA and AA |
524 |
> |
& S & 4.45 & 0.25 & - & 2.40 & 8.465 \\ |
525 |
|
\hline\hline |
526 |
|
\end{tabular} |
527 |
|
\label{MnM} |
530 |
|
\end{table*} |
531 |
|
|
532 |
|
|
533 |
< |
\section{Results and Discussions} |
534 |
< |
[MAY HAVE A BRIEF SUMMARY] |
535 |
< |
\subsection{How Simulation Parameters Affects $G$} |
536 |
< |
[MAY NOT PUT AT FIRST] |
537 |
< |
We have varied our protocol or other parameters of the simulations in |
538 |
< |
order to investigate how these factors would affect the measurement of |
539 |
< |
$G$'s. It turned out that while some of these parameters would not |
540 |
< |
affect the results substantially, some other changes to the |
463 |
< |
simulations would have a significant impact on the measurement |
464 |
< |
results. |
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 |
< |
In some of our simulations, we allowed $L_x$ and $L_y$ to change |
467 |
< |
during equilibrating the liquid phase. Due to the stiffness of the Au |
468 |
< |
slab, $L_x$ and $L_y$ would not change noticeably after |
469 |
< |
equilibration. Although $L_z$ could fluctuate $\sim$1\% after a system |
470 |
< |
is fully equilibrated in the NPT ensemble, this fluctuation, as well |
471 |
< |
as those comparably smaller to $L_x$ and $L_y$, would not be magnified |
472 |
< |
on the calculated $G$'s, as shown in Table \ref{AuThiolHexaneUA}. This |
473 |
< |
insensivity to $L_i$ fluctuations allows reliable measurement of $G$'s |
474 |
< |
without the necessity of extremely cautious equilibration process. |
542 |
> |
\subsection{Effects due to capping agent coverage} |
543 |
|
|
544 |
< |
As stated in our computational details, the spacing filled with |
545 |
< |
solvent molecules can be chosen within a range. This allows some |
546 |
< |
change of solvent molecule numbers for the same Au-butanethiol |
547 |
< |
surfaces. We did this study on our Au-butanethiol/hexane |
548 |
< |
simulations. Nevertheless, the results obtained from systems of |
549 |
< |
different $N_{hexane}$ did not indicate that the measurement of $G$ is |
482 |
< |
susceptible to this parameter. For computational efficiency concern, |
483 |
< |
smaller system size would be preferable, given that the liquid phase |
484 |
< |
structure is not affected. |
485 |
< |
|
486 |
< |
Our NIVS algorithm allows change of unphysical thermal flux both in |
487 |
< |
direction and in quantity. This feature extends our investigation of |
488 |
< |
interfacial thermal conductance. However, the magnitude of this |
489 |
< |
thermal flux is not arbitary if one aims to obtain a stable and |
490 |
< |
reliable thermal gradient. A temperature profile would be |
491 |
< |
substantially affected by noise when $|J_z|$ has a much too low |
492 |
< |
magnitude; while an excessively large $|J_z|$ that overwhelms the |
493 |
< |
conductance capacity of the interface would prevent a thermal gradient |
494 |
< |
to reach a stablized steady state. NIVS has the advantage of allowing |
495 |
< |
$J$ to vary in a wide range such that the optimal flux range for $G$ |
496 |
< |
measurement can generally be simulated by the algorithm. Within the |
497 |
< |
optimal range, we were able to study how $G$ would change according to |
498 |
< |
the thermal flux across the interface. For our simulations, we denote |
499 |
< |
$J_z$ to be positive when the physical thermal flux is from the liquid |
500 |
< |
to metal, and negative vice versa. The $G$'s measured under different |
501 |
< |
$J_z$ is listed in Table \ref{AuThiolHexaneUA} and [REF]. These |
502 |
< |
results do not suggest that $G$ is dependent on $J_z$ within this flux |
503 |
< |
range. The linear response of flux to thermal gradient simplifies our |
504 |
< |
investigations in that we can rely on $G$ measurement with only a |
505 |
< |
couple $J_z$'s and do not need to test a large series of fluxes. |
506 |
< |
|
507 |
< |
%ADD MORE TO TABLE |
508 |
< |
\begin{table*} |
509 |
< |
\begin{minipage}{\linewidth} |
510 |
< |
\begin{center} |
511 |
< |
\caption{Computed interfacial thermal conductivity ($G$ and |
512 |
< |
$G^\prime$) values for the 100\% covered Au-butanethiol/hexane |
513 |
< |
interfaces with UA model and different hexane molecule numbers |
514 |
< |
at different temperatures using a range of energy fluxes.} |
515 |
< |
|
516 |
< |
\begin{tabular}{ccccccc} |
517 |
< |
\hline\hline |
518 |
< |
$\langle T\rangle$ & $N_{hexane}$ & Fixed & $\rho_{hexane}$ & |
519 |
< |
$J_z$ & $G$ & $G^\prime$ \\ |
520 |
< |
(K) & & $L_x$ \& $L_y$? & (g/cm$^3$) & (GW/m$^2$) & |
521 |
< |
\multicolumn{2}{c}{(MW/m$^2$/K)} \\ |
522 |
< |
\hline |
523 |
< |
200 & 266 & No & 0.672 & -0.96 & 102() & 80.0() \\ |
524 |
< |
& 200 & Yes & 0.694 & 1.92 & 129() & 87.3() \\ |
525 |
< |
& & Yes & 0.672 & 1.93 & 131() & 77.5() \\ |
526 |
< |
& & No & 0.688 & 0.96 & 125() & 90.2() \\ |
527 |
< |
& & & & 1.91 & 139() & 101() \\ |
528 |
< |
& & & & 2.83 & 141() & 89.9() \\ |
529 |
< |
& 166 & Yes & 0.679 & 0.97 & 115() & 69.3() \\ |
530 |
< |
& & & & 1.94 & 125() & 87.1() \\ |
531 |
< |
& & No & 0.681 & 0.97 & 141() & 77.7() \\ |
532 |
< |
& & & & 1.92 & 138() & 98.9() \\ |
533 |
< |
\hline |
534 |
< |
250 & 200 & No & 0.560 & 0.96 & 74.8() & 61.8() \\ |
535 |
< |
& & & & -0.95 & 49.4() & 45.7() \\ |
536 |
< |
& 166 & Yes & 0.570 & 0.98 & 79.0() & 62.9() \\ |
537 |
< |
& & No & 0.569 & 0.97 & 80.3() & 67.1() \\ |
538 |
< |
& & & & 1.44 & 76.2() & 64.8() \\ |
539 |
< |
& & & & -0.95 & 56.4() & 54.4() \\ |
540 |
< |
& & & & -1.85 & 47.8() & 53.5() \\ |
541 |
< |
\hline\hline |
542 |
< |
\end{tabular} |
543 |
< |
\label{AuThiolHexaneUA} |
544 |
< |
\end{center} |
545 |
< |
\end{minipage} |
546 |
< |
\end{table*} |
547 |
< |
|
548 |
< |
Furthermore, we also attempted to increase system average temperatures |
549 |
< |
to above 200K. These simulations are first equilibrated in the NPT |
550 |
< |
ensemble under normal pressure. As stated above, the TraPPE-UA model |
551 |
< |
for hexane tends to predict a lower boiling point. In our simulations, |
552 |
< |
hexane had diffculty to remain in liquid phase when NPT equilibration |
553 |
< |
temperature is higher than 250K. Additionally, the equilibrated liquid |
554 |
< |
hexane density under 250K becomes lower than experimental value. This |
555 |
< |
expanded liquid phase leads to lower contact between hexane and |
556 |
< |
butanethiol as well.[MAY NEED FIGURE] And this reduced contact would |
557 |
< |
probably be accountable for a lower interfacial thermal conductance, |
558 |
< |
as shown in Table \ref{AuThiolHexaneUA}. |
559 |
< |
|
560 |
< |
A similar study for TraPPE-UA toluene agrees with the above result as |
561 |
< |
well. Having a higher boiling point, toluene tends to remain liquid in |
562 |
< |
our simulations even equilibrated under 300K in NPT |
563 |
< |
ensembles. Furthermore, the expansion of the toluene liquid phase is |
564 |
< |
not as significant as that of the hexane. This prevents severe |
565 |
< |
decrease of liquid-capping agent contact and the results (Table |
566 |
< |
\ref{AuThiolToluene}) show only a slightly decreased interface |
567 |
< |
conductance. Therefore, solvent-capping agent contact should play an |
568 |
< |
important role in the thermal transport process across the interface |
569 |
< |
in that higher degree of contact could yield increased conductance. |
570 |
< |
|
571 |
< |
[ADD ERROR ESTIMATE TO TABLE] |
572 |
< |
\begin{table*} |
573 |
< |
\begin{minipage}{\linewidth} |
574 |
< |
\begin{center} |
575 |
< |
\caption{Computed interfacial thermal conductivity ($G$ and |
576 |
< |
$G^\prime$) values for a 90\% coverage Au-butanethiol/toluene |
577 |
< |
interface at different temperatures using a range of energy |
578 |
< |
fluxes.} |
579 |
< |
|
580 |
< |
\begin{tabular}{ccccc} |
581 |
< |
\hline\hline |
582 |
< |
$\langle T\rangle$ & $\rho_{toluene}$ & $J_z$ & $G$ & $G^\prime$ \\ |
583 |
< |
(K) & (g/cm$^3$) & (GW/m$^2$) & \multicolumn{2}{c}{(MW/m$^2$/K)} \\ |
584 |
< |
\hline |
585 |
< |
200 & 0.933 & -1.86 & 180() & 135() \\ |
586 |
< |
& & 2.15 & 204() & 113() \\ |
587 |
< |
& & -3.93 & 175() & 114() \\ |
588 |
< |
\hline |
589 |
< |
300 & 0.855 & -1.91 & 143() & 125() \\ |
590 |
< |
& & -4.19 & 134() & 113() \\ |
591 |
< |
\hline\hline |
592 |
< |
\end{tabular} |
593 |
< |
\label{AuThiolToluene} |
594 |
< |
\end{center} |
595 |
< |
\end{minipage} |
596 |
< |
\end{table*} |
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 |
< |
Besides lower interfacial thermal conductance, surfaces in relatively |
552 |
< |
high temperatures are susceptible to reconstructions, when |
553 |
< |
butanethiols have a full coverage on the Au(111) surface. These |
554 |
< |
reconstructions include surface Au atoms migrated outward to the S |
555 |
< |
atom layer, and butanethiol molecules embedded into the original |
556 |
< |
surface Au layer. The driving force for this behavior is the strong |
557 |
< |
Au-S interactions in our simulations. And these reconstructions lead |
558 |
< |
to higher ratio of Au-S attraction and thus is energetically |
606 |
< |
favorable. Furthermore, this phenomenon agrees with experimental |
607 |
< |
results\cite{doi:10.1021/j100035a033,doi:10.1021/la026493y}. Vlugt |
608 |
< |
{\it et al.} had kept their Au(111) slab rigid so that their |
609 |
< |
simulations can reach 300K without surface reconstructions. Without |
610 |
< |
this practice, simulating 100\% thiol covered interfaces under higher |
611 |
< |
temperatures could hardly avoid surface reconstructions. However, our |
612 |
< |
measurement is based on assuming homogeneity on $x$ and $y$ dimensions |
613 |
< |
so that measurement of $T$ at particular $z$ would be an effective |
614 |
< |
average of the particles of the same type. Since surface |
615 |
< |
reconstructions could eliminate the original $x$ and $y$ dimensional |
616 |
< |
homogeneity, measurement of $G$ is more difficult to conduct under |
617 |
< |
higher temperatures. Therefore, most of our measurements are |
618 |
< |
undertaken at $\langle T\rangle\sim$200K. |
551 |
> |
\begin{figure} |
552 |
> |
\includegraphics[width=\linewidth]{coverage} |
553 |
> |
\caption{The interfacial thermal conductivity ($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 |
< |
However, when the surface is not completely covered by butanethiols, |
561 |
< |
the simulated system is more resistent to the reconstruction |
562 |
< |
above. Our Au-butanethiol/toluene system did not see this phenomena |
563 |
< |
even at $\langle T\rangle\sim$300K. The Au(111) surfaces have a 90\% |
564 |
< |
coverage of butanethiols and have empty three-fold sites. These empty |
565 |
< |
sites could help prevent surface reconstruction in that they provide |
626 |
< |
other means of capping agent relaxation. It is observed that |
627 |
< |
butanethiols can migrate to their neighbor empty sites during a |
628 |
< |
simulation. Therefore, we were able to obtain $G$'s for these |
629 |
< |
interfaces even at a relatively high temperature without being |
630 |
< |
affected by surface reconstructions. |
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 |
|
|
567 |
< |
\subsection{Influence of Capping Agent Coverage on $G$} |
568 |
< |
To investigate the influence of butanethiol coverage on interfacial |
569 |
< |
thermal conductance, a series of different coverage Au-butanethiol |
570 |
< |
surfaces is prepared and solvated with various organic |
571 |
< |
molecules. These systems are then equilibrated and their interfacial |
572 |
< |
thermal conductivity are measured with our NIVS algorithm. Table |
638 |
< |
\ref{tlnUhxnUhxnD} lists these results for direct comparison between |
639 |
< |
different coverages of butanethiol. To study the isotope effect in |
640 |
< |
interfacial thermal conductance, deuterated UA-hexane is included as |
641 |
< |
well. |
567 |
> |
From Figure \ref{coverage}, one can see the significance of the |
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 |
< |
It turned out that with partial covered butanethiol on the Au(111) |
575 |
< |
surface, the derivative definition for $G$ (Eq. \ref{derivativeG}) has |
576 |
< |
difficulty to apply, due to the difficulty in locating the maximum of |
577 |
< |
change of $\lambda$. Instead, the discrete definition |
578 |
< |
(Eq. \ref{discreteG}) is easier to apply, as max($\Delta T$) can still |
579 |
< |
be well-defined. Therefore, $G$'s (not $G^\prime$) are used for this |
580 |
< |
section. |
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 |
< |
From Table \ref{tlnUhxnUhxnD}, one can see the significance of the |
585 |
< |
presence of capping agents. Even when a fraction of the Au(111) |
586 |
< |
surface sites are covered with butanethiols, the conductivity would |
587 |
< |
see an enhancement by at least a factor of 3. This indicates the |
588 |
< |
important role cappping agent is playing for thermal transport |
589 |
< |
phenomena on metal/organic solvent surfaces. |
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 |
< |
Interestingly, as one could observe from our results, the maximum |
592 |
< |
conductance enhancement (largest $G$) happens while the surfaces are |
593 |
< |
about 75\% covered with butanethiols. This again indicates that |
594 |
< |
solvent-capping agent contact has an important role of the thermal |
595 |
< |
transport process. Slightly lower butanethiol coverage allows small |
596 |
< |
gaps between butanethiols to form. And these gaps could be filled with |
597 |
< |
solvent molecules, which acts like ``heat conductors'' on the |
598 |
< |
surface. The higher degree of interaction between these solvent |
666 |
< |
molecules and capping agents increases the enhancement effect and thus |
667 |
< |
produces a higher $G$ than densely packed butanethiol arrays. However, |
668 |
< |
once this maximum conductance enhancement is reached, $G$ decreases |
669 |
< |
when butanethiol coverage continues to decrease. Each capping agent |
670 |
< |
molecule reaches its maximum capacity for thermal |
671 |
< |
conductance. Therefore, even higher solvent-capping agent contact |
672 |
< |
would not offset this effect. Eventually, when butanethiol coverage |
673 |
< |
continues to decrease, solvent-capping agent contact actually |
674 |
< |
decreases with the disappearing of butanethiol molecules. In this |
675 |
< |
case, $G$ decrease could not be offset but instead accelerated. |
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 |
< |
A comparison of the results obtained from differenet organic solvents |
601 |
< |
can also provide useful information of the interfacial thermal |
602 |
< |
transport process. The deuterated hexane (UA) results do not appear to |
603 |
< |
be much different from those of normal hexane (UA), given that |
604 |
< |
butanethiol (UA) is non-deuterated for both solvents. These UA model |
605 |
< |
studies, even though eliminating C-H vibration samplings, still have |
606 |
< |
C-C vibrational frequencies different from each other. However, these |
607 |
< |
differences in the infrared range do not seem to produce an observable |
608 |
< |
difference for the results of $G$. [MAY NEED FIGURE] |
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 |
< |
Furthermore, results for rigid body toluene solvent, as well as other |
612 |
< |
UA-hexane solvents, are reasonable within the general experimental |
613 |
< |
ranges[CITATIONS]. This suggests that explicit hydrogen might not be a |
614 |
< |
required factor for modeling thermal transport phenomena of systems |
615 |
< |
such as Au-thiol/organic solvent. |
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 |
|
|
693 |
– |
However, results for Au-butanethiol/toluene do not show an identical |
694 |
– |
trend with those for Au-butanethiol/hexane in that $G$'s remain at |
695 |
– |
approximately the same magnitue when butanethiol coverage differs from |
696 |
– |
25\% to 75\%. This might be rooted in the molecule shape difference |
697 |
– |
for plane-like toluene and chain-like {\it n}-hexane. Due to this |
698 |
– |
difference, toluene molecules have more difficulty in occupying |
699 |
– |
relatively small gaps among capping agents when their coverage is not |
700 |
– |
too low. Therefore, the solvent-capping agent contact may keep |
701 |
– |
increasing until the capping agent coverage reaches a relatively low |
702 |
– |
level. This becomes an offset for decreasing butanethiol molecules on |
703 |
– |
its effect to the process of interfacial thermal transport. Thus, one |
704 |
– |
can see a plateau of $G$ vs. butanethiol coverage in our results. |
705 |
– |
|
706 |
– |
[NEED ERROR ESTIMATE] |
707 |
– |
\begin{figure} |
708 |
– |
\includegraphics[width=\linewidth]{coverage} |
709 |
– |
\caption{Comparison of interfacial thermal conductivity ($G$) values |
710 |
– |
for the Au-butanethiol/solvent interface with various UA models and |
711 |
– |
different capping agent coverages at $\langle T\rangle\sim$200K |
712 |
– |
using certain energy flux respectively.} |
713 |
– |
\label{coverage} |
714 |
– |
\end{figure} |
715 |
– |
|
716 |
– |
\subsection{Influence of Chosen Molecule Model on $G$} |
717 |
– |
[MAY COMBINE W MECHANISM STUDY] |
718 |
– |
|
719 |
– |
In addition to UA solvent/capping agent models, AA models are included |
720 |
– |
in our simulations as well. Besides simulations of the same (UA or AA) |
721 |
– |
model for solvent and capping agent, different models can be applied |
722 |
– |
to different components. Furthermore, regardless of models chosen, |
723 |
– |
either the solvent or the capping agent can be deuterated, similar to |
724 |
– |
the previous section. Table \ref{modelTest} summarizes the results of |
725 |
– |
these studies. |
726 |
– |
|
727 |
– |
[MORE DATA; ERROR ESTIMATE] |
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 several simulations.)} |
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}{ccccc} |
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() & 86.5() \\ |
639 |
< |
& UA hexane(D) & 1.95 & 153() & 136() \\ |
640 |
< |
& AA hexane & 1.94 & 135() & 129() \\ |
641 |
< |
& & 2.86 & 126() & 115() \\ |
642 |
< |
& UA toluene & 1.96 & 187() & 151() \\ |
750 |
< |
& AA toluene & 1.89 & 200() & 149() \\ |
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() & 129() \\ |
645 |
< |
& AA hexane & Avg. & 442() & 356() \\ |
646 |
< |
& AA hexane(D) & 1.93 & 222() & 234() \\ |
647 |
< |
& UA toluene & 1.98 & 125() & 96.5() \\ |
648 |
< |
& AA toluene & 3.79 & 487() & 290() \\ |
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() & 172() \\ |
651 |
< |
& AA hexane & 1.92 & 243() & 191() \\ |
652 |
< |
& AA toluene & 1.93 & 364() & 322() \\ |
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() & 49.4() \\ |
655 |
< |
& UA hexane(D) & 0.98 & 43.9() & 43.0() \\ |
656 |
< |
& AA hexane & 0.96 & 31.0() & 29.4() \\ |
657 |
< |
& UA toluene & 1.99 & 70.1() & 65.8() \\ |
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} |
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 |
776 |
< |
different models for hexane yields similar results for both $G$ and |
777 |
< |
$G^\prime$, and these two definitions agree with eath other very |
778 |
< |
well. This indicates very weak interaction between the metal and the |
779 |
< |
solvent, and is a typical case for acoustic impedance mismatch between |
780 |
< |
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 |
786 |
< |
substantially higher conductivity values than using UA model for at |
787 |
< |
least one component of the solvent and capping agent, which exceeds |
788 |
< |
the upper bond of experimental value range. This is probably due to |
789 |
< |
the classically treated C-H vibrations in the AA model, which should |
790 |
< |
not be appreciably populated at normal temperatures. In comparison, |
791 |
< |
once either the hexanes or the butanethiols are deuterated, one can |
792 |
< |
see a significantly lower $G$ and $G^\prime$. In either of these |
793 |
< |
cases, the C-H(D) vibrational overlap between the solvent and the |
794 |
< |
capping agent is removed. [MAY NEED FIGURE] Conclusively, the |
795 |
< |
improperly treated C-H vibration in the AA model produced |
796 |
< |
over-predicted results accordingly. Compared to the AA model, the UA |
797 |
< |
model yields more reasonable results with higher computational |
798 |
< |
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. |
692 |
< |
|
693 |
< |
\subsection{Mechanism of Interfacial Thermal Conductance Enhancement |
694 |
< |
by Capping Agent} |
695 |
< |
%OR\subsection{Vibrational spectrum study on conductance mechanism} |
696 |
< |
|
697 |
< |
[MAY INTRODUCE PROTOCOL IN METHODOLOGY/COMPUTATIONAL DETAIL, EQN'S] |
698 |
< |
|
699 |
< |
To investigate the mechanism of this interfacial thermal conductance, |
700 |
< |
the vibrational spectra of various gold systems were obtained and are |
701 |
< |
shown as in the upper panel of Fig. \ref{vibration}. To obtain these |
702 |
< |
spectra, one first runs a simulation in the NVE ensemble and collects |
703 |
< |
snapshots of configurations; these configurations are used to compute |
704 |
< |
the velocity auto-correlation functions, which is used to construct a |
705 |
< |
power spectrum via a Fourier transform. |
706 |
< |
|
707 |
< |
[MAY RELATE TO HASE'S] |
708 |
< |
The gold surfaces covered by |
709 |
< |
butanethiol molecules, compared to bare gold surfaces, exhibit an |
710 |
< |
additional peak observed at a frequency of $\sim$170cm$^{-1}$, which |
711 |
< |
is attributed to the vibration of the S-Au bond. This vibration |
712 |
< |
enables efficient thermal transport from surface Au atoms to the |
713 |
< |
capping agents. Simultaneously, as shown in the lower panel of |
714 |
< |
Fig. \ref{vibration}, the large overlap of the vibration spectra of |
715 |
< |
butanethiol and hexane in the all-atom model, including the C-H |
716 |
< |
vibration, also suggests high thermal exchange efficiency. The |
717 |
< |
combination of these two effects produces the drastic interfacial |
718 |
< |
thermal conductance enhancement in the all-atom model. |
719 |
< |
|
720 |
< |
[REDO. MAY NEED TO CONVERT TO JPEG] |
685 |
> |
\subsubsection{Are electronic excitations in the metal important?} |
686 |
> |
Because they lack electronic excitations, the QSC and related embedded |
687 |
> |
atom method (EAM) models for gold are known to predict unreasonably |
688 |
> |
low values for bulk conductivity |
689 |
> |
($\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{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 |
> |
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 |
> |
\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 (upper panel) and for Au/thiol/hexane simulation in |
863 |
< |
all-atom model (lower panel).} |
864 |
< |
\label{vibration} |
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 |
< |
[COMPARISON OF TWO G'S; AU SLAB WIDTHS; ETC] |
870 |
< |
% The results show that the two definitions used for $G$ yield |
871 |
< |
% comparable values, though $G^\prime$ tends to be smaller. |
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. Our simulations have seen significant conductance |
946 |
< |
enhancement with the presence of capping agent, compared to the bare |
947 |
< |
gold/liquid interfaces. The acoustic impedance mismatch between the |
867 |
< |
metal and the liquid phase is effectively eliminated by proper capping |
868 |
< |
agent. Furthermore, the coverage precentage of the capping agent plays |
869 |
< |
an important role in the interfacial thermal transport process. |
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 measurement results, particularly of the UA models, agree with |
950 |
< |
available experimental data. This indicates that our force field |
951 |
< |
parameters have a nice description of the interactions between the |
952 |
< |
particles at the interfaces. AA models tend to overestimate the |
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 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. |
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 |
< |
Vlugt {\it et al.} has investigated the surface thiol structures for |
968 |
< |
nanocrystal gold and pointed out that they differs from those of the |
969 |
< |
Au(111) surface\cite{vlugt:cpc2007154}. This difference might lead to |
970 |
< |
change of interfacial thermal transport behavior as well. To |
971 |
< |
investigate this problem, an effective means to introduce thermal flux |
972 |
< |
and measure the corresponding thermal gradient is desirable for |
973 |
< |
simulating structures with spherical symmetry. |
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 |
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.} 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} |