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\begin{document} |
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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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|
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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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Department of Chemistry and Biochemistry,\\ |
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University of Notre Dame\\ |
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Notre Dame, Indiana 46556} |
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|
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\date{\today} |
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|
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\maketitle |
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|
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\begin{doublespace} |
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|
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\begin{abstract} |
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|
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We have developed a Non-Isotropic Velocity Scaling algorithm for |
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setting up and maintaining stable thermal gradients in non-equilibrium |
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molecular dynamics simulations. This approach effectively imposes |
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unphysical thermal flux even between particles of different |
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identities, conserves linear momentum and kinetic energy, and |
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minimally perturbs the velocity profile of a system when compared with |
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previous RNEMD methods. We have used this method to simulate thermal |
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conductance at metal / organic solvent interfaces both with and |
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without the presence of thiol-based capping agents. We obtained |
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values comparable with experimental values, and observed significant |
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conductance enhancement with the presence of capping agents. Computed |
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power spectra indicate the acoustic impedance mismatch between metal |
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and liquid phase is greatly reduced by the capping agents and thus |
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leads to higher interfacial thermal transfer efficiency. |
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|
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\end{abstract} |
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|
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\newpage |
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|
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%\narrowtext |
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|
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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% BODY OF TEXT |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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|
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\section{Introduction} |
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|
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Interfacial thermal conductance is extensively studied both |
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experimentally and computationally, and systems with interfaces |
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present are generally heterogeneous. Although interfaces are commonly |
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barriers to heat transfer, it has been |
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reported\cite{doi:10.1021/la904855s} that under specific circustances, |
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e.g. with certain capping agents present on the surface, interfacial |
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conductance can be significantly enhanced. However, heat conductance |
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of molecular and nano-scale interfaces will be affected by the |
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chemical details of the surface and is challenging to |
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experimentalist. The lower thermal flux through interfaces is even |
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more difficult to measure with EMD and forward NEMD simulation |
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methods. Therefore, developing good simulation methods will be |
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desirable in order to investigate thermal transport across interfaces. |
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|
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Recently, we have developed the Non-Isotropic Velocity Scaling (NIVS) |
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algorithm for RNEMD simulations\cite{kuang:164101}. This algorithm |
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retains the desirable features of RNEMD (conservation of linear |
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momentum and total energy, compatibility with periodic boundary |
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conditions) while establishing true thermal distributions in each of |
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the two slabs. Furthermore, it allows more effective thermal exchange |
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between particles of different identities, and thus enables extensive |
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study of interfacial conductance. |
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|
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\section{Methodology} |
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\subsection{Algorithm} |
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There have been many algorithms for computing thermal conductivity |
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using molecular dynamics simulations. However, interfacial conductance |
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is at least an order of magnitude smaller. This would make the |
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calculation even more difficult for those slowly-converging |
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equilibrium methods. Imposed-flux non-equilibrium |
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methods\cite{MullerPlathe:1997xw} have the flux set {\it a priori} and |
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the response of temperature or momentum gradients are easier to |
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measure than the flux, if unknown, and thus, is a preferable way to |
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the forward NEMD methods. Although the momentum swapping approach for |
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flux-imposing can be used for exchanging energy between particles of |
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different identity, the kinetic energy transfer efficiency is affected |
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by the mass difference between the particles, which limits its |
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application on heterogeneous interfacial systems. |
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|
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The non-isotropic velocity scaling (NIVS)\cite{kuang:164101} approach in |
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non-equilibrium MD simulations is able to impose relatively large |
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kinetic energy flux without obvious perturbation to the velocity |
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distribution of the simulated systems. Furthermore, this approach has |
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the advantage in heterogeneous interfaces in that kinetic energy flux |
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can be applied between regions of particles of arbitary identity, and |
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the flux quantity is not restricted by particle mass difference. |
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|
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The NIVS algorithm scales the velocity vectors in two separate regions |
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of a simulation system with respective diagonal scaling matricies. To |
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determine these scaling factors in the matricies, a set of equations |
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including linear momentum conservation and kinetic energy conservation |
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constraints and target momentum/energy flux satisfaction is |
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solved. With the scaling operation applied to the system in a set |
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frequency, corresponding momentum/temperature gradients can be built, |
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which can be used for computing transportation properties and other |
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applications related to momentum/temperature gradients. The NIVS |
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algorithm conserves momenta and energy and does not depend on an |
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external thermostat. |
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|
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(wondering how much detail of algorithm should be put here...) |
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|
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\subsection{Force Field Parameters} |
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Our simulation systems consists of metal gold lattice slab solvated by |
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organic solvents. In order to study the role of capping agents in |
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interfacial thermal conductance, butanethiol is chosen to cover gold |
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surfaces in comparison to no capping agent present. |
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|
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The Au-Au interactions in metal lattice slab is described by the |
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quantum Sutton-Chen (QSC) formulation.\cite{PhysRevB.59.3527} The QSC |
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potentials include zero-point quantum corrections and are |
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reparametrized for accurate surface energies compared to the |
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Sutton-Chen potentials\cite{Chen90}. |
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|
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Straight chain {\it n}-hexane and aromatic toluene are respectively |
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used as solvents. For hexane, both United-Atom\cite{TraPPE-UA.alkanes} |
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and All-Atom\cite{OPLSAA} force fields are used for comparison; for |
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toluene, United-Atom\cite{TraPPE-UA.alkylbenzenes} force fields are |
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used with rigid body constraints applied. (maybe needs more details |
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about rigid body) |
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|
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Buatnethiol molecules are used as capping agent for some of our |
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simulations. United-Atom\cite{TraPPE-UA.thiols} and All-Atom models |
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are respectively used corresponding to the force field type of |
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solvent. |
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|
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To describe the interactions between metal Au and non-metal capping |
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agent and solvent, we refer to Vlugt\cite{vlugt:cpc2007154} and derive |
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other interactions which are not parametrized in their work. (can add |
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hautman and klein's paper here and more discussion; need to put |
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aromatic-metal interaction approximation here) |
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|
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[TABULATED FORCE FIELD PARAMETERS NEEDED] |
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|
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\section{Computational Details} |
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\subsection{System Geometry} |
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Our simulation systems consists of a lattice Au slab with the (111) |
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surface perpendicular to the $z$-axis, and a solvent layer between the |
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periodic Au slabs along the $z$-axis. To set up the interfacial |
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system, the Au slab is first equilibrated without solvent under room |
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pressure and a desired temperature. After the metal slab is |
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equilibrated, United-Atom or All-Atom butanethiols are replicated on |
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the Au surface, each occupying the (??) among three Au atoms, and is |
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equilibrated under NVT ensemble. According to (CITATION), the maximal |
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thiol capacity on Au surface is $1/3$ of the total number of surface |
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Au atoms. |
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|
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\cite{packmol} |
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|
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\subsection{Simulation Parameters} |
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|
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When the interfacial conductance is {\it not} small, there are two |
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ways to define $G$. If we assume the temperature is discretely |
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different on two sides of the interface, $G$ can be calculated with |
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the thermal flux applied $J$ and the temperature difference measured |
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$\Delta T$ as: |
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\begin{equation} |
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G=\frac{J}{\Delta T} |
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\label{discreteG} |
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\end{equation} |
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We can as well assume a continuous temperature profile along the |
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thermal gradient axis $z$ and define $G$ as the change of bulk thermal |
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conductivity $\lambda$ at a defined interfacial point: |
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\begin{equation} |
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G^\prime = \Big|\frac{\partial\lambda}{\partial z}\Big| |
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= \Big|\frac{\partial}{\partial z}\left(-J_z\Big/ |
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\left(\frac{\partial T}{\partial z}\right)\right)\Big| |
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= J_z\Big|\frac{\partial^2 T}{\partial z^2}\Big| |
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\Big/\left(\frac{\partial T}{\partial z}\right)^2 |
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\label{derivativeG} |
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\end{equation} |
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With the temperature profile obtained from simulations, one is able to |
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approximate the first and second derivatives of $T$ with finite |
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difference method and thus calculate $G^\prime$. |
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|
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In what follows, both definitions are used for calculation and comparison. |
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|
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\section{Results} |
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\subsection{Toluene Solvent} |
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|
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The simulations follow a protocol similar to the previous gold/water |
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interfacial systems. The results (Table \ref{AuThiolToluene}) show a |
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significant conductance enhancement compared to the gold/water |
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interface without capping agent and agree with available experimental |
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data. This indicates that the metal-metal potential, though not |
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predicting an accurate bulk metal thermal conductivity, does not |
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greatly interfere with the simulation of the thermal conductance |
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behavior across a non-metal interface. The solvent model is not |
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particularly volatile, so the simulation cell does not expand |
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significantly under higher temperature. We did not observe a |
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significant conductance decrease when the temperature was increased to |
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300K. The results show that the two definitions used for $G$ yield |
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comparable values, though $G^\prime$ tends to be smaller. |
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|
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\begin{table*} |
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\begin{minipage}{\linewidth} |
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\begin{center} |
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\caption{Computed interfacial thermal conductivity ($G$ and |
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$G^\prime$) values for the Au/butanethiol/toluene interface at |
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different temperatures using a range of energy fluxes.} |
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|
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\begin{tabular}{cccc} |
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\hline\hline |
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$\langle T\rangle$ & $J_z$ & $G$ & $G^\prime$ \\ |
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(K) & (GW/m$^2$) & \multicolumn{2}{c}{(MW/m$^2$/K)} \\ |
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\hline |
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200 & 1.86 & 180 & 135 \\ |
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& 2.15 & 204 & 113 \\ |
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& 3.93 & 175 & 114 \\ |
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300 & 1.91 & 143 & 125 \\ |
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& 4.19 & 134 & 113 \\ |
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\hline\hline |
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\end{tabular} |
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\label{AuThiolToluene} |
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\end{center} |
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\end{minipage} |
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\end{table*} |
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|
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\subsection{Hexane Solvent} |
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|
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Using the united-atom model, different coverages of capping agent, |
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temperatures of simulations and numbers of solvent molecules were all |
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investigated and Table \ref{AuThiolHexaneUA} shows the results of |
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these computations. The number of hexane molecules in our simulations |
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does not affect the calculations significantly. However, a very long |
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length scale for the thermal gradient axis ($z$) may cause excessively |
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hot or cold temperatures in the middle of the solvent region and lead |
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to undesired phenomena such as solvent boiling or freezing, while too |
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few solvent molecules would change the normal behavior of the liquid |
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phase. Our $N_{hexane}$ values were chosen to ensure that these |
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extreme cases did not happen to our simulations. |
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|
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Table \ref{AuThiolHexaneUA} enables direct comparison between |
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different coverages of capping agent, when other system parameters are |
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held constant. With high coverage of butanethiol on the gold surface, |
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the interfacial thermal conductance is enhanced |
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significantly. Interestingly, a slightly lower butanethiol coverage |
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leads to a moderately higher conductivity. This is probably due to |
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more solvent/capping agent contact when butanethiol molecules are |
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not densely packed, which enhances the interactions between the two |
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phases and lowers the thermal transfer barrier of this interface. |
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% [COMPARE TO AU/WATER IN PAPER] |
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|
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It is also noted that the overall simulation temperature is another |
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factor that affects the interfacial thermal conductance. One |
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possibility of this effect may be rooted in the decrease in density of |
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the liquid phase. We observed that when the average temperature |
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increases from 200K to 250K, the bulk hexane density becomes lower |
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than experimental value, as the system is equilibrated under NPT |
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ensemble. This leads to lower contact between solvent and capping |
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agent, and thus lower conductivity. |
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|
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Conductivity values are more difficult to obtain under higher |
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temperatures. This is because the Au surface tends to undergo |
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reconstructions in relatively high temperatures. Surface Au atoms can |
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migrate outward to reach higher Au-S contact; and capping agent |
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molecules can be embedded into the surface Au layer due to the same |
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driving force. This phenomenon agrees with experimental |
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results\cite{doi:10.1021/j100035a033,doi:10.1021/la026493y}. A surface |
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fully covered in capping agent is more susceptible to reconstruction, |
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possibly because fully coverage prevents other means of capping agent |
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relaxation, such as migration to an empty neighbor three-fold site. |
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|
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%MAY ADD MORE DATA TO TABLE |
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\begin{table*} |
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\begin{minipage}{\linewidth} |
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\begin{center} |
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\caption{Computed interfacial thermal conductivity ($G$ and |
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$G^\prime$) values for the Au/butanethiol/hexane interface |
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with united-atom model and different capping agent coverage |
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and solvent molecule numbers at different temperatures using a |
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range of energy fluxes.} |
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|
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\begin{tabular}{cccccc} |
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\hline\hline |
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Thiol & $\langle T\rangle$ & & $J_z$ & $G$ & $G^\prime$ \\ |
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coverage (\%) & (K) & $N_{hexane}$ & (GW/m$^2$) & |
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\multicolumn{2}{c}{(MW/m$^2$/K)} \\ |
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\hline |
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0.0 & 200 & 200 & 0.96 & 43.3 & 42.7 \\ |
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& & & 1.91 & 45.7 & 42.9 \\ |
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& & 166 & 0.96 & 43.1 & 53.4 \\ |
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88.9 & 200 & 166 & 1.94 & 172 & 108 \\ |
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100.0 & 250 & 200 & 0.96 & 81.8 & 67.0 \\ |
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& & 166 & 0.98 & 79.0 & 62.9 \\ |
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& & & 1.44 & 76.2 & 64.8 \\ |
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& 200 & 200 & 1.92 & 129 & 87.3 \\ |
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& & & 1.93 & 131 & 77.5 \\ |
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& & 166 & 0.97 & 115 & 69.3 \\ |
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& & & 1.94 & 125 & 87.1 \\ |
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\hline\hline |
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\end{tabular} |
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\label{AuThiolHexaneUA} |
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\end{center} |
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\end{minipage} |
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\end{table*} |
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|
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For the all-atom model, the liquid hexane phase was not stable under NPT |
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conditions. Therefore, the simulation length scale parameters are |
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adopted from previous equilibration results of the united-atom model |
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at 200K. Table \ref{AuThiolHexaneAA} shows the results of these |
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simulations. The conductivity values calculated with full capping |
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agent coverage are substantially larger than observed in the |
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united-atom model, and is even higher than predicted by |
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experiments. It is possible that our parameters for metal-non-metal |
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particle interactions lead to an overestimate of the interfacial |
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thermal conductivity, although the active C-H vibrations in the |
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all-atom model (which should not be appreciably populated at normal |
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temperatures) could also account for this high conductivity. The major |
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thermal transfer barrier of Au/butanethiol/hexane interface is between |
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the liquid phase and the capping agent, so extra degrees of freedom |
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such as the C-H vibrations could enhance heat exchange between these |
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two phases and result in a much higher conductivity. |
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|
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\begin{table*} |
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\begin{minipage}{\linewidth} |
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\begin{center} |
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|
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\caption{Computed interfacial thermal conductivity ($G$ and |
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$G^\prime$) values for the Au/butanethiol/hexane interface |
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with all-atom model and different capping agent coverage at |
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200K using a range of energy fluxes.} |
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|
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\begin{tabular}{cccc} |
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\hline\hline |
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Thiol & $J_z$ & $G$ & $G^\prime$ \\ |
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coverage (\%) & (GW/m$^2$) & \multicolumn{2}{c}{(MW/m$^2$/K)} \\ |
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\hline |
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0.0 & 0.95 & 28.5 & 27.2 \\ |
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& 1.88 & 30.3 & 28.9 \\ |
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100.0 & 2.87 & 551 & 294 \\ |
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& 3.81 & 494 & 193 \\ |
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\hline\hline |
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\end{tabular} |
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\label{AuThiolHexaneAA} |
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\end{center} |
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\end{minipage} |
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\end{table*} |
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|
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%subsubsection{Vibrational spectrum study on conductance mechanism} |
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To investigate the mechanism of this interfacial thermal conductance, |
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the vibrational spectra of various gold systems were obtained and are |
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shown as in the upper panel of Fig. \ref{vibration}. To obtain these |
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spectra, one first runs a simulation in the NVE ensemble and collects |
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snapshots of configurations; these configurations are used to compute |
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the velocity auto-correlation functions, which is used to construct a |
| 380 |
power spectrum via a Fourier transform. The gold surfaces covered by |
| 381 |
butanethiol molecules exhibit an additional peak observed at a |
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frequency of $\sim$170cm$^{-1}$, which is attributed to the vibration |
| 383 |
of the S-Au bond. This vibration enables efficient thermal transport |
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from surface Au atoms to the capping agents. Simultaneously, as shown |
| 385 |
in the lower panel of Fig. \ref{vibration}, the large overlap of the |
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vibration spectra of butanethiol and hexane in the all-atom model, |
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including the C-H vibration, also suggests high thermal exchange |
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efficiency. The combination of these two effects produces the drastic |
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interfacial thermal conductance enhancement in the all-atom model. |
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|
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\begin{figure} |
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\includegraphics[width=\linewidth]{vibration} |
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\caption{Vibrational spectra obtained for gold in different |
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environments (upper panel) and for Au/thiol/hexane simulation in |
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all-atom model (lower panel).} |
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\label{vibration} |
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\end{figure} |
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% 600dpi, letter size. too large? |
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|
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|
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\section{Acknowledgments} |
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Support for this project was provided by the National Science |
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Foundation under grant CHE-0848243. Computational time was provided by |
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the Center for Research Computing (CRC) at the University of Notre |
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Dame. \newpage |
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|
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\bibliography{interfacial} |
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|
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\end{doublespace} |
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\end{document} |
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|
