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\begin{document} |
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\title{ENTER TITLE HERE} |
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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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\date{\today} |
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\maketitle |
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\begin{doublespace} |
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\begin{abstract} |
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REPLACE ABSTRACT HERE |
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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 |
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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 coupling |
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between the metal and liquid phases is enhanced by the capping |
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agents, leading to a greatly enhanced conductivity at the interface. |
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Specifically, the chemical bond between the metal and the capping |
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agent introduces a vibrational overlap that is not present without |
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the capping agent, and the overlap between the vibrational spectra |
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(metal to cap, cap to solvent) provides a mechanism for rapid |
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thermal transport across the interface. Our calculations also |
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suggest that this is a non-monotonic function of the fractional |
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coverage of the surface, as moderate coverages allow diffusive heat |
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transport of solvent molecules that have been in close contact with |
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the capping agent. |
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|
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\end{abstract} |
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\newpage |
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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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[REFINE LATER, ADD MORE REF.S] |
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Imposed-flux methods in Molecular Dynamics (MD) |
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simulations\cite{MullerPlathe:1997xw} can establish steady state |
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systems with a set applied flux vs a corresponding gradient that can |
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be measured. These methods does not need many trajectories to provide |
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information of transport properties of a given system. Thus, they are |
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utilized in computing thermal and mechanical transfer of homogeneous |
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or bulk systems as well as heterogeneous systems such as liquid-solid |
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interfaces.\cite{kuang:AuThl} |
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|
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The Reverse Non-Equilibrium MD (RNEMD) methods adopt constraints that |
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satisfy linear momentum and total energy conservation of a system when |
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imposing fluxes in a simulation. Thus they are compatible with various |
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ensembles, including the micro-canonical (NVE) ensemble, without the |
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need of an external thermostat. The original approaches by |
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M\"{u}ller-Plathe {\it et |
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al.}\cite{MullerPlathe:1997xw,ISI:000080382700030} utilize simple |
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momentum swapping for generating energy/momentum fluxes, which is also |
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compatible with particles of different identities. Although simple to |
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implement in a simulation, this approach can create nonthermal |
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velocity distributions, as discovered by Tenney and |
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Maginn\cite{Maginn:2010}. Furthermore, this approach to kinetic energy |
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transfer between particles of different identities is less efficient |
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when the mass difference between the particles becomes significant, |
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which also limits its application on heterogeneous interfacial |
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systems. |
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|
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Recently, we developed a different approach, using Non-Isotropic |
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Velocity Scaling (NIVS) \cite{kuang:164101} algorithm to impose |
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fluxes. Compared to the momentum swapping move, it scales the velocity |
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vectors in two separate regions of a simulated system with respective |
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diagonal scaling matrices. These matrices are determined by solving a |
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set of equations including linear momentum and kinetic energy |
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conservation constraints and target flux satisfaction. This method is |
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able to effectively impose a wide range of kinetic energy fluxes |
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without obvious perturbation to the velocity distributions of the |
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simulated systems, regardless of the presence of heterogeneous |
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interfaces. We have successfully applied this approach in studying the |
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interfacial thermal conductance at metal-solvent |
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interfaces.\cite{kuang:AuThl} |
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|
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However, the NIVS approach limits its application in imposing momentum |
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fluxes. Temperature anisotropy can happen under high momentum fluxes, |
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due to the nature of the algorithm. Thus, combining thermal and |
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momentum flux is also difficult to implement with this |
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approach. However, such combination may provide a means to simulate |
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thermal/momentum gradient coupled processes such as freeze |
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desalination. Therefore, developing novel approaches to extend the |
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application of imposed-flux method is desired. |
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|
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In this paper, we improve the NIVS method and propose a novel approach |
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to impose fluxes. This approach separate the means of applying |
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momentum and thermal flux with operations in one time step and thus is |
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able to simutaneously impose thermal and momentum flux. Furthermore, |
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the approach retains desirable features of previous RNEMD approaches |
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and is simpler to implement compared to the NIVS method. In what |
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follows, we first present the method to implement the method in a |
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simulation. Then we compare the method on bulk fluids to previous |
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methods. Also, interfacial frictions are computed for a series of |
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interfaces. |
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|
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\section{Methodology} |
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Similar to the NIVS methodology,\cite{kuang:164101} we consider a |
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periodic system divided into a series of slabs along a certain axis |
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(e.g. $z$). The unphysical thermal and/or momentum flux is designated |
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from the center slab to one of the end slabs, and thus the center slab |
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would have a lower temperature than the end slab (unless the thermal |
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flux is negative). Therefore, the center slab is denoted as ``$c$'' |
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while the end slab as ``$h$''. |
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|
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To impose these fluxes, we periodically apply separate operations to |
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velocities of particles {$i$} within the center slab and of particles |
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{$j$} within the end slab: |
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\begin{eqnarray} |
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\vec{v}_i & \leftarrow & c\cdot\left(\vec{v}_i - \langle\vec{v}_c |
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\rangle\right) + \left(\langle\vec{v}_c\rangle + \vec{a}_c\right) \\ |
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\vec{v}_j & \leftarrow & h\cdot\left(\vec{v}_j - \langle\vec{v}_h |
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\rangle\right) + \left(\langle\vec{v}_h\rangle + \vec{a}_h\right) |
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\end{eqnarray} |
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where $\langle\vec{v}_c\rangle$ and $\langle\vec{v}_h\rangle$ denotes |
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the instantaneous bulk velocity of slabs $c$ and $h$ respectively |
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before an operation occurs. When a momentum flux $\vec{j}_z(\vec{p})$ |
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presents, these bulk velocities would have a corresponding change |
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($\vec{a}_c$ and $\vec{a}_h$ respectively) according to Newton's |
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second law: |
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\begin{eqnarray} |
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M_c \vec{a}_c & = & -\vec{j}_z(\vec{p}) \Delta t \\ |
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M_h \vec{a}_h & = & \vec{j}_z(\vec{p}) \Delta t |
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\end{eqnarray} |
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where |
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\begin{eqnarray} |
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M_c & = & \sum_{i = 1}^{N_c} m_i \\ |
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M_h & = & \sum_{j = 1}^{N_h} m_j |
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\end{eqnarray} |
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and $\Delta t$ is the interval between two operations. |
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|
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The above operations conserve the linear momentum of a periodic |
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system. To satisfy total energy conservation as well as to impose a |
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thermal flux $J_z$, one would have |
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[SUPPORT INFO MIGHT BE NECESSARY TO PUT EXTRA MATH IN] |
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\begin{eqnarray} |
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K_c - J_z\Delta t & = & c^2 (K_c - \frac{1}{2}M_c \langle\vec{v}_c |
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\rangle^2) + \frac{1}{2}M_c (\langle \vec{v}_c \rangle + \vec{a}_c)^2 \\ |
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K_h + J_z\Delta t & = & h^2 (K_h - \frac{1}{2}M_h \langle\vec{v}_h |
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\rangle^2) + \frac{1}{2}M_h (\langle \vec{v}_h \rangle + \vec{a}_h)^2 |
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\end{eqnarray} |
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where $K_c$ and $K_h$ denotes translational kinetic energy of slabs |
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$c$ and $h$ respectively before an operation occurs. These |
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translational kinetic energy conservation equations are sufficient to |
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ensure total energy conservation, as the operations applied do not |
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change the potential energy of a system, given that the potential |
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energy does not depend on particle velocity. |
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|
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The above sets of equations are sufficient to determine the velocity |
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scaling coefficients ($c$ and $h$) as well as $\vec{a}_c$ and |
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$\vec{a}_h$. Note that two roots of $c$ and $h$ exist |
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respectively. However, to avoid dramatic perturbations to a system, |
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the positive roots (which are closer to 1) are chosen. Figure |
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\ref{method} illustrates the implementation of this algorithm in an |
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individual step. |
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|
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\begin{figure} |
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\includegraphics[width=\linewidth]{method} |
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\caption{Illustration of the implementation of the algorithm in a |
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single step. Starting from an ideal velocity distribution, the |
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transformation is used to apply both thermal and momentum flux from |
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the ``c'' slab to the ``h'' slab. As the figure shows, the thermal |
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distributions preserve after this operation.} |
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\label{method} |
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\end{figure} |
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|
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By implementing these operations at a certain frequency, a steady |
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thermal and/or momentum flux can be applied and the corresponding |
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temperature and/or momentum gradients can be established. |
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|
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This approach is more computationaly efficient compared to the |
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previous NIVS method, in that only quadratic equations are involved, |
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while the NIVS method needs to solve a quartic equations. Furthermore, |
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the method implements isotropic scaling of velocities in respective |
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slabs, unlike the NIVS, where an extra criteria function is necessary |
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to choose a set of coefficients that performs the most isotropic |
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scaling. More importantly, separating the momentum flux imposing from |
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velocity scaling avoids the underlying cause that NIVS produced |
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thermal anisotropy when applying a momentum flux. |
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|
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The advantages of the approach over the original momentum swapping |
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approach lies in its nature to preserve a Gaussian |
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distribution. Because the momentum swapping tends to render a |
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nonthermal distribution, when the imposed flux is relatively large, |
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diffusion of the neighboring slabs could no longer remedy this effect, |
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and nonthermal distributions would be observed. Results in later |
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section will illustrate this effect. |
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|
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\section{Computational Details} |
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The algorithm has been implemented in our MD simulation code, |
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OpenMD\cite{Meineke:2005gd,openmd}. We compare the method with |
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previous RNEMD methods or equilibrium MD methods in homogeneous fluids |
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(Lennard-Jones and SPC/E water). And taking advantage of the method, |
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we simulate the interfacial friction of different heterogeneous |
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interfaces (gold-organic solvent and gold-SPC/E water and ice-liquid |
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water). |
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|
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\subsection{Simulation Protocols} |
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The systems to be investigated are set up in a orthorhombic simulation |
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cell with periodic boundary conditions in all three dimensions. The |
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$z$ axis of these cells were longer and was set as the gradient axis |
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of temperature and/or momentum. Thus the cells were divided into $N$ |
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slabs along this axis, with various $N$ depending on individual |
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system. The $x$ and $y$ axis were usually of the same length in |
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homogeneous systems or close to each other where interfaces |
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presents. In all cases, before introducing a nonequilibrium method to |
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establish steady thermal and/or momentum gradients for further |
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measurements and calculations, canonical ensemble with a Nos\'e-Hoover |
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thermostat\cite{hoover85} and microcanonical ensemble equilibrations |
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were used to prepare systems ready for data |
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collections. Isobaric-isothermal equilibrations are performed before |
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this for SPC/E water systems to reach normal pressure (1 bar), while |
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similar equilibrations are used for interfacial systems to relax the |
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surface tensions. |
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|
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While homogeneous fluid systems can be set up with random |
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configurations, our interfacial systems needs extra steps to ensure |
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the interfaces be established properly for computations. The |
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preparation and equilibration of butanethiol covered gold (111) |
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surface and further solvation and equilibration process is described |
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as in reference \cite{kuang:AuThl}. |
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|
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As for the ice/liquid water interfaces, the basal surface of ice |
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lattice was first constructed. Hirsch {\it et |
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al.}\cite{doi:10.1021/jp048434u} explored the energetics of ice |
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lattices with different proton orders. We refer to their results and |
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choose the configuration of the lowest energy after geometry |
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optimization as the unit cells of our ice lattices. Although |
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experimental solid/liquid coexistant temperature near normal pressure |
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is 273K, Bryk and Haymet's simulations of ice/liquid water interfaces |
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with different models suggest that for SPC/E, the most stable |
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interface is observed at 225$\pm$5K. Therefore, all our ice/liquid |
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water simulations were carried out under 225K. To have extra |
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protection of the ice lattice during initial equilibration (when the |
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randomly generated liquid phase configuration could release large |
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amount of energy in relaxation), a constraint method (REF?) was |
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adopted until the high energy configuration was relaxed. |
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[MAY ADD A FIGURE HERE FOR BASAL PLANE, MAY INCLUDE PRISM IF POSSIBLE] |
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|
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\subsection{Force Field Parameters} |
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For comparison of our new method with previous work, we retain our |
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force field parameters consistent with the results we will compare |
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with. The Lennard-Jones fluid used here for argon , and reduced unit |
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results are reported for direct comparison purpose. |
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|
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As for our water simulations, SPC/E model is used throughout this work |
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for consistency. Previous work for transport properties of SPC/E water |
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model is available\cite{Bedrov:2000,10.1063/1.3330544,Medina2011} so |
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that unnecessary repetition of previous methods can be avoided. |
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|
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The Au-Au interaction parameters in all simulations are described by |
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the quantum Sutton-Chen (QSC) formulation.\cite{PhysRevB.59.3527} The |
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QSC 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} For gold/water interfaces, the |
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Spohr potential was adopted\cite{ISI:000167766600035} to depict |
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Au-H$_2$O interactions. |
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|
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The small organic molecules included in our simulations are the Au |
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surface capping agent butanethiol and liquid hexane and toluene. The |
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United-Atom |
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models\cite{TraPPE-UA.thiols,TraPPE-UA.alkanes,TraPPE-UA.alkylbenzenes} |
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for these components were used in this work for better computational |
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efficiency, while maintaining good accuracy. We refer readers to our |
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previous work\cite{kuang:AuThl} for further details of these models, |
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as well as the interactions between Au and the above organic molecule |
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components. |
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|
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\subsection{Thermal conductivities} |
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When $\vec{j}_z(\vec{p})$ is set to zero and a target $J_z$ is set to |
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impose kinetic energy transfer, the method can be used for thermal |
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conductivity computations. Similar to previous RNEMD methods, we |
314 |
assume linear response of the temperature gradient with respect to the |
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thermal flux in general case. And the thermal conductivity ($\lambda$) |
316 |
can be obtained with the imposed kinetic energy flux and the measured |
317 |
thermal gradient: |
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\begin{equation} |
319 |
J_z = -\lambda \frac{\partial T}{\partial z} |
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\end{equation} |
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Like other imposed-flux methods, the energy flux was calculated using |
322 |
the total non-physical energy transferred (${E_{total}}$) from slab |
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``c'' to slab ``h'', which is recorded throughout a simulation, and |
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the time for data collection $t$: |
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\begin{equation} |
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J_z = \frac{E_{total}}{2 t L_x L_y} |
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\end{equation} |
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where $L_x$ and $L_y$ denotes the dimensions of the plane in a |
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simulation cell perpendicular to the thermal gradient, and a factor of |
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two in the denominator is present for the heat transport occurs in |
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both $+z$ and $-z$ directions. The temperature gradient |
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${\langle\partial T/\partial z\rangle}$ can be obtained by a linear |
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regression of the temperature profile, which is recorded during a |
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simulation for each slab in a cell. For Lennard-Jones simulations, |
335 |
thermal conductivities are reported in reduced units |
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(${\lambda^*=\lambda \sigma^2 m^{1/2} k_B^{-1}\varepsilon^{-1/2}}$). |
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|
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\subsection{Shear viscosities} |
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Alternatively, the method can carry out shear viscosity calculations |
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by switching off $J_z$. One can specify the vector |
341 |
$\vec{j}_z(\vec{p})$ by choosing the three components |
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respectively. For shear viscosity simulations, $j_z(p_z)$ is usually |
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set to zero. Although for isotropic systems, the direction of |
344 |
$\vec{j}_z(\vec{p})$ on the $xy$ plane should not matter, the ability |
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of arbitarily specifying the vector direction in our method provides |
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convenience in anisotropic simulations. |
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|
348 |
Similar to thermal conductivity computations, linear response of the |
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momentum gradient with respect to the shear stress is assumed, and the |
350 |
shear viscosity ($\eta$) can be obtained with the imposed momentum |
351 |
flux (e.g. in $x$ direction) and the measured gradient: |
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\begin{equation} |
353 |
j_z(p_x) = -\eta \frac{\partial v_x}{\partial z} |
354 |
\end{equation} |
355 |
where the flux is similarly defined: |
356 |
\begin{equation} |
357 |
j_z(p_x) = \frac{P_x}{2 t L_x L_y} |
358 |
\end{equation} |
359 |
with $P_x$ being the total non-physical momentum transferred within |
360 |
the data collection time. Also, the velocity gradient |
361 |
${\langle\partial v_x/\partial z\rangle}$ can be obtained using linear |
362 |
regression of the $x$ component of the mean velocity, $\langle |
363 |
v_x\rangle$, in each of the bins. For Lennard-Jones simulations, shear |
364 |
viscosities are reported in reduced units |
365 |
(${\eta^*=\eta\sigma^2(\varepsilon m)^{-1/2}}$). |
366 |
|
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\subsection{Interfacial friction and Slip length} |
368 |
While the shear stress results in a velocity gradient within bulk |
369 |
fluid phase, its effect at a solid-liquid interface could vary due to |
370 |
the interaction strength between the two phases. The interfacial |
371 |
friction coefficient $\kappa$ is defined to relate the shear stress |
372 |
(e.g. along $x$-axis) and the relative fluid velocity tangent to the |
373 |
interface: |
374 |
\begin{equation} |
375 |
j_z(p_x)|_{interface} = \kappa\Delta v_x|_{interface} |
376 |
\end{equation} |
377 |
Under ``stick'' boundary condition, $\Delta v_x|_{interface} |
378 |
\rightarrow 0$, which leads to $\kappa\rightarrow\infty$. However, for |
379 |
``slip'' boundary condition at the solid-liquid interface, $\kappa$ |
380 |
becomes finite. To characterize the interfacial boundary conditions, |
381 |
slip length ($\delta$) is defined using $\kappa$ and the shear |
382 |
viscocity of liquid phase ($\eta$): |
383 |
\begin{equation} |
384 |
\delta = \frac{\eta}{\kappa} |
385 |
\end{equation} |
386 |
so that $\delta\rightarrow 0$ in the ``no-slip'' boundary condition, |
387 |
and depicts how ``slippery'' an interface is. Figure \ref{slipLength} |
388 |
illustrates how this quantity is defined and computed for a |
389 |
solid-liquid interface. [MAY INCLUDE A SNAPSHOT IN FIGURE] |
390 |
|
391 |
\begin{figure} |
392 |
\includegraphics[width=\linewidth]{defDelta} |
393 |
\caption{The slip length $\delta$ can be obtained from a velocity |
394 |
profile of a solid-liquid interface. An example of Au/hexane |
395 |
interfaces is shown.} |
396 |
\label{slipLength} |
397 |
\end{figure} |
398 |
|
399 |
In our method, a shear stress can be applied similar to shear |
400 |
viscosity computations by applying an unphysical momentum flux |
401 |
(e.g. $j_z(p_x)$). A corresponding velocity profile can be obtained as |
402 |
shown in Figure \ref{slipLength}, in which the velocity gradients |
403 |
within liquid phase and velocity difference at the liquid-solid |
404 |
interface can be measured respectively. Further calculations and |
405 |
characterizations of the interface can be carried out using these |
406 |
data. |
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|
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\section{Results and Discussions} |
409 |
\subsection{Lennard-Jones fluid} |
410 |
Our orthorhombic simulation cell of Lennard-Jones fluid has identical |
411 |
parameters to our previous work\cite{kuang:164101} to facilitate |
412 |
comparison. Thermal conductivitis and shear viscosities were computed |
413 |
with the algorithm applied to the simulations. The results of thermal |
414 |
conductivity are compared with our previous NIVS algorithm. However, |
415 |
since the NIVS algorithm could produce temperature anisotropy for |
416 |
shear viscocity computations, these results are instead compared to |
417 |
the momentum swapping approaches. Table \ref{LJ} lists these |
418 |
calculations with various fluxes in reduced units. |
419 |
|
420 |
\begin{table*} |
421 |
\begin{minipage}{\linewidth} |
422 |
\begin{center} |
423 |
|
424 |
\caption{Thermal conductivity ($\lambda^*$) and shear viscosity |
425 |
($\eta^*$) (in reduced units) of a Lennard-Jones fluid at |
426 |
${\langle T^* \rangle = 0.72}$ and ${\rho^* = 0.85}$ computed |
427 |
at various momentum fluxes. The new method yields similar |
428 |
results to previous RNEMD methods. All results are reported in |
429 |
reduced unit. Uncertainties are indicated in parentheses.} |
430 |
|
431 |
\begin{tabular}{cccccc} |
432 |
\hline\hline |
433 |
\multicolumn{2}{c}{Momentum Exchange} & |
434 |
\multicolumn{2}{c}{$\lambda^*$} & |
435 |
\multicolumn{2}{c}{$\eta^*$} \\ |
436 |
\hline |
437 |
Swap Interval & Equivalent $J_z^*$ or $j_z^*(p_x)$ & |
438 |
NIVS & This work & Swapping & This work \\ |
439 |
\hline |
440 |
0.116 & 0.16 & 7.30(0.10) & 6.25(0.23) & 3.57(0.06) & 3.53(0.16)\\ |
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0.232 & 0.09 & 6.95(0.09) & 8.02(0.56) & 3.64(0.05) & 3.43(0.17)\\ |
442 |
0.463 & 0.047 & 7.19(0.07) & 6.48(0.15) & 3.52(0.16) & 3.51(0.08)\\ |
443 |
0.926 & 0.024 & 7.19(0.28) & 7.02(0.13) & 3.72(0.05) & 3.79(0.11)\\ |
444 |
1.158 & 0.019 & 7.98(0.33) & 7.37(0.23) & 3.42(0.06) & 3.53(0.06)\\ |
445 |
\hline\hline |
446 |
\end{tabular} |
447 |
\label{LJ} |
448 |
\end{center} |
449 |
\end{minipage} |
450 |
\end{table*} |
451 |
|
452 |
\subsubsection{Thermal conductivity} |
453 |
Our thermal conductivity calculations with this method yields |
454 |
comparable results to the previous NIVS algorithm. This indicates that |
455 |
the thermal gradients rendered using this method are also close to |
456 |
previous RNEMD methods. Simulations with moderately higher thermal |
457 |
fluxes tend to yield more reliable thermal gradients and thus avoid |
458 |
large errors, while overly high thermal fluxes could introduce side |
459 |
effects such as non-linear temperature gradient response or |
460 |
inadvertent phase transitions. |
461 |
|
462 |
Since the scaling operation is isotropic in this method, one does not |
463 |
need extra care to ensure temperature isotropy between the $x$, $y$ |
464 |
and $z$ axes, while thermal anisotropy might happen if the criteria |
465 |
function for choosing scaling coefficients does not perform as |
466 |
expected. Furthermore, this method avoids inadvertent concomitant |
467 |
momentum flux when only thermal flux is imposed, which could not be |
468 |
achieved with swapping or NIVS approaches. The thermal energy exchange |
469 |
in swapping ($\vec{p}_i$ in slab ``c'' with $\vec{p}_j$ in slab ``h'') |
470 |
or NIVS (total slab momentum conponemts $P^\alpha$ scaled to $\alpha |
471 |
P^\alpha$) would not obtain this result unless thermal flux vanishes |
472 |
(i.e. $\vec{p}_i=\vec{p}_j$ or $\alpha=1$ which are trivial to apply a |
473 |
thermal flux). In this sense, this method contributes to having |
474 |
minimal perturbation to a simulation while imposing thermal flux. |
475 |
|
476 |
\subsubsection{Shear viscosity} |
477 |
Table \ref{LJ} also compares our shear viscosity results with momentum |
478 |
swapping approach. Our calculations show that our method predicted |
479 |
similar values for shear viscosities to the momentum swapping |
480 |
approach, as well as the velocity gradient profiles. Moderately larger |
481 |
momentum fluxes are helpful to reduce the errors of measured velocity |
482 |
gradients and thus the final result. However, it is pointed out that |
483 |
the momentum swapping approach tends to produce nonthermal velocity |
484 |
distributions.\cite{Maginn:2010} |
485 |
|
486 |
To examine that temperature isotropy holds in simulations using our |
487 |
method, we measured the three one-dimensional temperatures in each of |
488 |
the slabs (Figure \ref{tempXyz}). Note that the $x$-dimensional |
489 |
temperatures were calculated after subtracting the effects from bulk |
490 |
velocities of the slabs. The one-dimensional temperature profiles |
491 |
showed no observable difference between the three dimensions. This |
492 |
ensures that isotropic scaling automatically preserves temperature |
493 |
isotropy and that our method is useful in shear viscosity |
494 |
computations. |
495 |
|
496 |
\begin{figure} |
497 |
\includegraphics[width=\linewidth]{tempXyz} |
498 |
\caption{Unlike the previous NIVS algorithm, the new method does not |
499 |
produce a thermal anisotropy. No temperature difference between |
500 |
different dimensions were observed beyond the magnitude of the error |
501 |
bars. Note that the two ``hotter'' regions are caused by the shear |
502 |
stress, as reported by Tenney and Maginn\cite{Maginn:2010}, but not |
503 |
an effect that only observed in our methods.} |
504 |
\label{tempXyz} |
505 |
\end{figure} |
506 |
|
507 |
Furthermore, the velocity distribution profiles are tested by imposing |
508 |
a large shear stress into the simulations. Figure \ref{vDist} |
509 |
demonstrates how our method is able to maintain thermal velocity |
510 |
distributions against the momentum swapping approach even under large |
511 |
imposed fluxes. Previous swapping methods tend to deplete particles of |
512 |
positive velocities in the negative velocity slab (``c'') and vice |
513 |
versa in slab ``h'', where the distributions leave a notch. This |
514 |
problematic profiles become significant when the imposed-flux becomes |
515 |
larger and diffusions from neighboring slabs could not offset the |
516 |
depletion. Simutaneously, abnormal peaks appear corresponding to |
517 |
excessive velocity swapped from the other slab. This nonthermal |
518 |
distributions limit applications of the swapping approach in shear |
519 |
stress simulations. Our method avoids the above problematic |
520 |
distributions by altering the means of applying momentum |
521 |
fluxes. Comparatively, velocity distributions recorded from |
522 |
simulations with our method is so close to the ideal thermal |
523 |
prediction that no observable difference is shown in Figure |
524 |
\ref{vDist}. Conclusively, our method avoids problems happened in |
525 |
previous RNEMD methods and provides a useful means for shear viscosity |
526 |
computations. |
527 |
|
528 |
\begin{figure} |
529 |
\includegraphics[width=\linewidth]{velDist} |
530 |
\caption{Velocity distributions that develop under the swapping and |
531 |
our methods at high flux. These distributions were obtained from |
532 |
Lennard-Jones simulations with $j_z(p_x)\sim 0.4$ (equivalent to a |
533 |
swapping interval of 20 time steps). This is a relatively large flux |
534 |
to demonstrate the nonthermal distributions that develop under the |
535 |
swapping method. Distributions produced by our method are very close |
536 |
to the ideal thermal situations.} |
537 |
\label{vDist} |
538 |
\end{figure} |
539 |
|
540 |
\subsection{Bulk SPC/E water} |
541 |
Since our method was in good performance of thermal conductivity and |
542 |
shear viscosity computations for simple Lennard-Jones fluid, we extend |
543 |
our applications of these simulations to complex fluid like SPC/E |
544 |
water model. A simulation cell with 1000 molecules was set up in the |
545 |
same manner as in \cite{kuang:164101}. For thermal conductivity |
546 |
simulations, measurements were taken to compare with previous RNEMD |
547 |
methods; for shear viscosity computations, simulations were run under |
548 |
a series of temperatures (with corresponding pressure relaxation using |
549 |
the isobaric-isothermal ensemble[CITE NIVS REF 32]), and results were |
550 |
compared to available data from Equilibrium MD methods[CITATIONS]. |
551 |
|
552 |
\subsubsection{Thermal conductivity} |
553 |
Table \ref{spceThermal} summarizes our thermal conductivity |
554 |
computations under different temperatures and thermal gradients, in |
555 |
comparison to the previous NIVS results\cite{kuang:164101} and |
556 |
experimental measurements\cite{WagnerKruse}. Note that no appreciable |
557 |
drift of total system energy or temperature was observed when our |
558 |
method is applied, which indicates that our algorithm conserves total |
559 |
energy even for systems involving electrostatic interactions. |
560 |
|
561 |
Measurements using our method established similar temperature |
562 |
gradients to the previous NIVS method. Our simulation results are in |
563 |
good agreement with those from previous simulations. And both methods |
564 |
yield values in reasonable agreement with experimental |
565 |
values. Simulations using moderately higher thermal gradient or those |
566 |
with longer gradient axis ($z$) for measurement seem to have better |
567 |
accuracy, from our results. |
568 |
|
569 |
\begin{table*} |
570 |
\begin{minipage}{\linewidth} |
571 |
\begin{center} |
572 |
|
573 |
\caption{Thermal conductivity of SPC/E water under various |
574 |
imposed thermal gradients. Uncertainties are indicated in |
575 |
parentheses.} |
576 |
|
577 |
\begin{tabular}{ccccc} |
578 |
\hline\hline |
579 |
$\langle T\rangle$ & $\langle dT/dz\rangle$ & \multicolumn{3}{c} |
580 |
{$\lambda (\mathrm{W m}^{-1} \mathrm{K}^{-1})$} \\ |
581 |
(K) & (K/\AA) & This work & Previous NIVS\cite{kuang:164101} & |
582 |
Experiment\cite{WagnerKruse} \\ |
583 |
\hline |
584 |
300 & 0.8 & 0.815(0.027) & 0.770(0.008) & 0.61 \\ |
585 |
318 & 0.8 & 0.801(0.024) & 0.750(0.032) & 0.64 \\ |
586 |
& 1.6 & 0.766(0.007) & 0.778(0.019) & \\ |
587 |
& 0.8 & 0.786(0.009)$^a$ & & \\ |
588 |
\hline\hline |
589 |
\end{tabular} |
590 |
$^a$Simulation with $L_z$ twice as long. |
591 |
\label{spceThermal} |
592 |
\end{center} |
593 |
\end{minipage} |
594 |
\end{table*} |
595 |
|
596 |
\subsubsection{Shear viscosity} |
597 |
The improvement our method achieves for shear viscosity computations |
598 |
enables us to apply it on SPC/E water models. The series of |
599 |
temperatures under which our shear viscosity calculations were carried |
600 |
out covers the liquid range under normal pressure. Our simulations |
601 |
predict a similar trend of $\eta$ vs. $T$ to EMD results we refer to |
602 |
(Table \ref{spceShear}). |
603 |
|
604 |
\begin{table*} |
605 |
\begin{minipage}{\linewidth} |
606 |
\begin{center} |
607 |
|
608 |
\caption{Computed shear viscosity of SPC/E water under different |
609 |
temperatures. Results are compared to those obtained with EMD |
610 |
method[CITATION]. Uncertainties are indicated in parentheses.} |
611 |
|
612 |
\begin{tabular}{cccc} |
613 |
\hline\hline |
614 |
$T$ & $\partial v_x/\partial z$ & \multicolumn{2}{c} |
615 |
{$\eta (\mathrm{mPa}\cdot\mathrm{s})$} \\ |
616 |
(K) & (10$^{10}$s$^{-1}$) & This work & Previous simulations[CITATION]\\ |
617 |
\hline |
618 |
273 & & 1.218(0.004) & \\ |
619 |
& & 1.140(0.012) & \\ |
620 |
303 & & 0.646(0.008) & \\ |
621 |
318 & & 0.536(0.007) & \\ |
622 |
& & 0.510(0.007) & \\ |
623 |
& & & \\ |
624 |
333 & & 0.428(0.002) & \\ |
625 |
363 & & 0.279(0.014) & \\ |
626 |
& & 0.306(0.001) & \\ |
627 |
\hline\hline |
628 |
\end{tabular} |
629 |
\label{spceShear} |
630 |
\end{center} |
631 |
\end{minipage} |
632 |
\end{table*} |
633 |
|
634 |
[MAY COMBINE JZPX AND JZKE TO RUN ONE JOB BUT GET ETA=F(T)] |
635 |
[PUT RESULTS AND FIGURE HERE IF IT WORKS] |
636 |
\subsection{Interfacial frictions} |
637 |
[SLIP BOUNDARY VS STICK BOUNDARY] |
638 |
|
639 |
qualitative agreement w interfacial thermal conductance |
640 |
|
641 |
[ETA OBTAINED HERE DOES NOT NECESSARILY EQUAL TO BULK VALUES] |
642 |
|
643 |
|
644 |
[ATTEMPT TO CONSTRUCT BASAL PLANE ICE-WATER INTERFACE] |
645 |
|
646 |
[FUTURE WORK HERE OR IN CONCLUSIONS] |
647 |
|
648 |
|
649 |
\begin{table*} |
650 |
\begin{minipage}{\linewidth} |
651 |
\begin{center} |
652 |
|
653 |
\caption{Computed interfacial thermal conductance ($G$ and |
654 |
$G^\prime$) values for interfaces using various models for |
655 |
solvent and capping agent (or without capping agent) at |
656 |
$\langle T\rangle\sim$200K. Here ``D'' stands for deuterated |
657 |
solvent or capping agent molecules. Error estimates are |
658 |
indicated in parentheses.} |
659 |
|
660 |
\begin{tabular}{llccc} |
661 |
\hline\hline |
662 |
Butanethiol model & Solvent & $G$ & $G^\prime$ \\ |
663 |
(or bare surface) & model & \multicolumn{2}{c}{(MW/m$^2$/K)} \\ |
664 |
\hline |
665 |
UA & UA hexane & 131(9) & 87(10) \\ |
666 |
& UA hexane(D) & 153(5) & 136(13) \\ |
667 |
& AA hexane & 131(6) & 122(10) \\ |
668 |
& UA toluene & 187(16) & 151(11) \\ |
669 |
& AA toluene & 200(36) & 149(53) \\ |
670 |
\hline |
671 |
bare & UA hexane & 46.5(3.2) & 49.4(4.5) \\ |
672 |
& UA hexane(D) & 43.9(4.6) & 43.0(2.0) \\ |
673 |
& AA hexane & 31.0(1.4) & 29.4(1.3) \\ |
674 |
& UA toluene & 70.1(1.3) & 65.8(0.5) \\ |
675 |
\hline\hline |
676 |
\end{tabular} |
677 |
\label{modelTest} |
678 |
\end{center} |
679 |
\end{minipage} |
680 |
\end{table*} |
681 |
|
682 |
On bare metal / solvent surfaces, different force field models for |
683 |
hexane yield similar results for both $G$ and $G^\prime$, and these |
684 |
two definitions agree with each other very well. This is primarily an |
685 |
indicator of weak interactions between the metal and the solvent. |
686 |
|
687 |
For the fully-covered surfaces, the choice of force field for the |
688 |
capping agent and solvent has a large impact on the calculated values |
689 |
of $G$ and $G^\prime$. The AA thiol to AA solvent conductivities are |
690 |
much larger than their UA to UA counterparts, and these values exceed |
691 |
the experimental estimates by a large measure. The AA force field |
692 |
allows significant energy to go into C-H (or C-D) stretching modes, |
693 |
and since these modes are high frequency, this non-quantum behavior is |
694 |
likely responsible for the overestimate of the conductivity. Compared |
695 |
to the AA model, the UA model yields more reasonable conductivity |
696 |
values with much higher computational efficiency. |
697 |
|
698 |
\subsubsection{Effects due to average temperature} |
699 |
|
700 |
We also studied the effect of average system temperature on the |
701 |
interfacial conductance. The simulations are first equilibrated in |
702 |
the NPT ensemble at 1 atm. The TraPPE-UA model for hexane tends to |
703 |
predict a lower boiling point (and liquid state density) than |
704 |
experiments. This lower-density liquid phase leads to reduced contact |
705 |
between the hexane and butanethiol, and this accounts for our |
706 |
observation of lower conductance at higher temperatures. In raising |
707 |
the average temperature from 200K to 250K, the density drop of |
708 |
$\sim$20\% in the solvent phase leads to a $\sim$40\% drop in the |
709 |
conductance. |
710 |
|
711 |
Similar behavior is observed in the TraPPE-UA model for toluene, |
712 |
although this model has better agreement with the experimental |
713 |
densities of toluene. The expansion of the toluene liquid phase is |
714 |
not as significant as that of the hexane (8.3\% over 100K), and this |
715 |
limits the effect to $\sim$20\% drop in thermal conductivity. |
716 |
|
717 |
Although we have not mapped out the behavior at a large number of |
718 |
temperatures, is clear that there will be a strong temperature |
719 |
dependence in the interfacial conductance when the physical properties |
720 |
of one side of the interface (notably the density) change rapidly as a |
721 |
function of temperature. |
722 |
|
723 |
\section{Conclusions} |
724 |
[VSIS WORKS! COMBINES NICE FEATURES OF PREVIOUS RNEMD METHODS AND |
725 |
IMPROVEMENTS TO THEIR PROBLEMS! ROBUST AND VERSATILE!] |
726 |
|
727 |
The NIVS algorithm has been applied to simulations of |
728 |
butanethiol-capped Au(111) surfaces in the presence of organic |
729 |
solvents. This algorithm allows the application of unphysical thermal |
730 |
flux to transfer heat between the metal and the liquid phase. With the |
731 |
flux applied, we were able to measure the corresponding thermal |
732 |
gradients and to obtain interfacial thermal conductivities. Under |
733 |
steady states, 2-3 ns trajectory simulations are sufficient for |
734 |
computation of this quantity. |
735 |
|
736 |
Our simulations have seen significant conductance enhancement in the |
737 |
presence of capping agent, compared with the bare gold / liquid |
738 |
interfaces. The vibrational coupling between the metal and the liquid |
739 |
phase is enhanced by a chemically-bonded capping agent. Furthermore, |
740 |
the coverage percentage of the capping agent plays an important role |
741 |
in the interfacial thermal transport process. Moderately low coverages |
742 |
allow higher contact between capping agent and solvent, and thus could |
743 |
further enhance the heat transfer process, giving a non-monotonic |
744 |
behavior of conductance with increasing coverage. |
745 |
|
746 |
Our results, particularly using the UA models, agree well with |
747 |
available experimental data. The AA models tend to overestimate the |
748 |
interfacial thermal conductance in that the classically treated C-H |
749 |
vibrations become too easily populated. Compared to the AA models, the |
750 |
UA models have higher computational efficiency with satisfactory |
751 |
accuracy, and thus are preferable in modeling interfacial thermal |
752 |
transport. |
753 |
|
754 |
\section{Acknowledgments} |
755 |
Support for this project was provided by the National Science |
756 |
Foundation under grant CHE-0848243. Computational time was provided by |
757 |
the Center for Research Computing (CRC) at the University of Notre |
758 |
Dame. |
759 |
|
760 |
\newpage |
761 |
|
762 |
\bibliography{stokes} |
763 |
|
764 |
\end{doublespace} |
765 |
\end{document} |