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\begin{document} |
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\title{ENTER TITLE HERE} |
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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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We present a new method for introducing stable nonequilibrium |
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velocity and temperature gradients in molecular dynamics simulations |
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of heterogeneous systems. This method conserves the linear momentum |
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and total energy of the system and improves previous Reverse |
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Non-Equilibrium Molecular Dynamics (RNEMD) methods and maintains |
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thermal velocity distributions. It also avoid thermal anisotropy |
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occured in NIVS simulations by using isotropic velocity scaling on |
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the molecules in specific regions of a system. To test the method, |
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we have computed the thermal conductivity and shear viscosity of |
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model liquid systems as well as the interfacial frictions of a |
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series of metal/liquid interfaces. |
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\end{abstract} |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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% BODY OF TEXT |
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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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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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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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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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\section{Methodology} |
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Similar to the NIVS method,\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 thermal |
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flux results in a lower temperature of the center slab than the end |
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slab, and the momentum flux results in negative center slab momentum |
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with positive end slab momentum (unless these fluxes are set |
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negative). Therefore, the center slab is denoted as ``$c$'', while the |
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end slab as ``$h$''. |
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To impose these fluxes, we periodically apply different set of |
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operations on velocities of particles {$i$} within the center slab and |
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those of particles {$j$} 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 is applied. 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 $M$ denotes total mass of particles within a slab: |
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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 separate operations. |
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The above operations already conserve the linear momentum of a |
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periodic system. To further satisfy total energy conservation as well |
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as to impose the thermal flux $J_z$, the following equations are |
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included as well: |
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[SUPPORT INFO OR APPENDIX 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 is applied. 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 in our |
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method do not change the kinetic energy related to other degrees of |
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freedom or the potential energy of a system, given that its potential |
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energy does not depend on particle velocity. |
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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 there are two roots respectively for $c$ and |
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$h$. However, the positive roots (which are closer to 1) are chosen so |
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that the perturbations to a system can be reduced to a minimum. Figure |
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\ref{method} illustrates the implementation sketch of this algorithm |
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in an individual step. |
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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 the effect of both a thermal and a |
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momentum flux from the ``c'' slab to the ``h'' slab. As the figure |
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shows, thermal distributions can preserve after this operation.} |
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\label{method} |
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\end{figure} |
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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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Compared to the previous NIVS method, this approach is computationally |
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more efficient in that only quadratic equations are involved to |
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determine a set of scaling coefficients, while the NIVS method needs |
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to solve quartic equations. Furthermore, this method implements |
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isotropic scaling of velocities in respective slabs, unlike the NIVS, |
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where an extra criteria function is necessary to choose a set of |
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coefficients that performs a scaling as isotropic as possible. More |
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importantly, separating the means of momentum flux imposing from |
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velocity scaling avoids the underlying cause to thermal anisotropy in |
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NIVS when applying a momentum flux. And later sections will |
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demonstrate that this can improve the performance in shear viscosity |
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simulations. |
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The advantages of this approach over the original momentum swapping |
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one lies in its nature of preserve a Maxwell-Boltzmann distribution |
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(mathimatically a Gaussian function). However, because the momentum |
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swapping steps tend to result in a nonthermal distribution, when an |
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imposed flux is relatively large, diffusions from the neighboring |
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slabs could no longer remedy this effect, problematic distributions |
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would be observed. Results in later section will illustrate this |
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effect in more detail. |
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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 (EMD) 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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\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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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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\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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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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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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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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\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 |
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assume linear response of the temperature gradient with respect to the |
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thermal flux in general case. And the thermal conductivity ($\lambda$) |
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can be obtained with the imposed kinetic energy flux and the measured |
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thermal gradient: |
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\begin{equation} |
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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 |
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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 |
324 |
|
|
the time for data collection $t$: |
325 |
|
|
\begin{equation} |
326 |
|
|
J_z = \frac{E_{total}}{2 t L_x L_y} |
327 |
|
|
\end{equation} |
328 |
|
|
where $L_x$ and $L_y$ denotes the dimensions of the plane in a |
329 |
|
|
simulation cell perpendicular to the thermal gradient, and a factor of |
330 |
|
|
two in the denominator is present for the heat transport occurs in |
331 |
|
|
both $+z$ and $-z$ directions. The temperature gradient |
332 |
|
|
${\langle\partial T/\partial z\rangle}$ can be obtained by a linear |
333 |
|
|
regression of the temperature profile, which is recorded during a |
334 |
|
|
simulation for each slab in a cell. For Lennard-Jones simulations, |
335 |
|
|
thermal conductivities are reported in reduced units |
336 |
|
|
(${\lambda^*=\lambda \sigma^2 m^{1/2} k_B^{-1}\varepsilon^{-1/2}}$). |
337 |
|
|
|
338 |
skuang |
3774 |
\subsection{Shear viscosities} |
339 |
skuang |
3775 |
Alternatively, the method can carry out shear viscosity calculations |
340 |
|
|
by switching off $J_z$. One can specify the vector |
341 |
|
|
$\vec{j}_z(\vec{p})$ by choosing the three components |
342 |
|
|
respectively. For shear viscosity simulations, $j_z(p_z)$ is usually |
343 |
|
|
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 |
345 |
|
|
of arbitarily specifying the vector direction in our method provides |
346 |
|
|
convenience in anisotropic simulations. |
347 |
|
|
|
348 |
|
|
Similar to thermal conductivity computations, linear response of the |
349 |
|
|
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: |
352 |
|
|
\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 |
|
|
|
367 |
skuang |
3774 |
\subsection{Interfacial friction and Slip length} |
368 |
skuang |
3775 |
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 |
skuang |
3776 |
solid-liquid interface. [MAY INCLUDE A SNAPSHOT IN FIGURE] |
390 |
gezelter |
3769 |
|
391 |
skuang |
3775 |
\begin{figure} |
392 |
|
|
\includegraphics[width=\linewidth]{defDelta} |
393 |
|
|
\caption{The slip length $\delta$ can be obtained from a velocity |
394 |
skuang |
3780 |
profile of a solid-liquid interface simulation. An example of |
395 |
|
|
Au/hexane interfaces is shown. Calculation for the left side is |
396 |
|
|
illustrated. The right side is similar to the left side.} |
397 |
skuang |
3775 |
\label{slipLength} |
398 |
|
|
\end{figure} |
399 |
gezelter |
3769 |
|
400 |
skuang |
3775 |
In our method, a shear stress can be applied similar to shear |
401 |
|
|
viscosity computations by applying an unphysical momentum flux |
402 |
|
|
(e.g. $j_z(p_x)$). A corresponding velocity profile can be obtained as |
403 |
|
|
shown in Figure \ref{slipLength}, in which the velocity gradients |
404 |
|
|
within liquid phase and velocity difference at the liquid-solid |
405 |
|
|
interface can be measured respectively. Further calculations and |
406 |
|
|
characterizations of the interface can be carried out using these |
407 |
|
|
data. |
408 |
|
|
|
409 |
skuang |
3776 |
\section{Results and Discussions} |
410 |
|
|
\subsection{Lennard-Jones fluid} |
411 |
|
|
Our orthorhombic simulation cell of Lennard-Jones fluid has identical |
412 |
|
|
parameters to our previous work\cite{kuang:164101} to facilitate |
413 |
|
|
comparison. Thermal conductivitis and shear viscosities were computed |
414 |
|
|
with the algorithm applied to the simulations. The results of thermal |
415 |
|
|
conductivity are compared with our previous NIVS algorithm. However, |
416 |
|
|
since the NIVS algorithm could produce temperature anisotropy for |
417 |
|
|
shear viscocity computations, these results are instead compared to |
418 |
|
|
the momentum swapping approaches. Table \ref{LJ} lists these |
419 |
|
|
calculations with various fluxes in reduced units. |
420 |
skuang |
3770 |
|
421 |
skuang |
3776 |
\begin{table*} |
422 |
|
|
\begin{minipage}{\linewidth} |
423 |
|
|
\begin{center} |
424 |
gezelter |
3769 |
|
425 |
skuang |
3776 |
\caption{Thermal conductivity ($\lambda^*$) and shear viscosity |
426 |
|
|
($\eta^*$) (in reduced units) of a Lennard-Jones fluid at |
427 |
|
|
${\langle T^* \rangle = 0.72}$ and ${\rho^* = 0.85}$ computed |
428 |
|
|
at various momentum fluxes. The new method yields similar |
429 |
|
|
results to previous RNEMD methods. All results are reported in |
430 |
|
|
reduced unit. Uncertainties are indicated in parentheses.} |
431 |
|
|
|
432 |
|
|
\begin{tabular}{cccccc} |
433 |
|
|
\hline\hline |
434 |
|
|
\multicolumn{2}{c}{Momentum Exchange} & |
435 |
|
|
\multicolumn{2}{c}{$\lambda^*$} & |
436 |
|
|
\multicolumn{2}{c}{$\eta^*$} \\ |
437 |
|
|
\hline |
438 |
skuang |
3779 |
Swap Interval $t^*$ & Equivalent $J_z^*$ or $j_z^*(p_x)$ & |
439 |
skuang |
3776 |
NIVS & This work & Swapping & This work \\ |
440 |
|
|
\hline |
441 |
|
|
0.116 & 0.16 & 7.30(0.10) & 6.25(0.23) & 3.57(0.06) & 3.53(0.16)\\ |
442 |
|
|
0.232 & 0.09 & 6.95(0.09) & 8.02(0.56) & 3.64(0.05) & 3.43(0.17)\\ |
443 |
|
|
0.463 & 0.047 & 7.19(0.07) & 6.48(0.15) & 3.52(0.16) & 3.51(0.08)\\ |
444 |
|
|
0.926 & 0.024 & 7.19(0.28) & 7.02(0.13) & 3.72(0.05) & 3.79(0.11)\\ |
445 |
|
|
1.158 & 0.019 & 7.98(0.33) & 7.37(0.23) & 3.42(0.06) & 3.53(0.06)\\ |
446 |
|
|
\hline\hline |
447 |
|
|
\end{tabular} |
448 |
|
|
\label{LJ} |
449 |
|
|
\end{center} |
450 |
|
|
\end{minipage} |
451 |
|
|
\end{table*} |
452 |
gezelter |
3769 |
|
453 |
skuang |
3776 |
\subsubsection{Thermal conductivity} |
454 |
|
|
Our thermal conductivity calculations with this method yields |
455 |
|
|
comparable results to the previous NIVS algorithm. This indicates that |
456 |
|
|
the thermal gradients rendered using this method are also close to |
457 |
|
|
previous RNEMD methods. Simulations with moderately higher thermal |
458 |
|
|
fluxes tend to yield more reliable thermal gradients and thus avoid |
459 |
|
|
large errors, while overly high thermal fluxes could introduce side |
460 |
|
|
effects such as non-linear temperature gradient response or |
461 |
|
|
inadvertent phase transitions. |
462 |
gezelter |
3769 |
|
463 |
skuang |
3776 |
Since the scaling operation is isotropic in this method, one does not |
464 |
|
|
need extra care to ensure temperature isotropy between the $x$, $y$ |
465 |
|
|
and $z$ axes, while thermal anisotropy might happen if the criteria |
466 |
|
|
function for choosing scaling coefficients does not perform as |
467 |
|
|
expected. Furthermore, this method avoids inadvertent concomitant |
468 |
|
|
momentum flux when only thermal flux is imposed, which could not be |
469 |
|
|
achieved with swapping or NIVS approaches. The thermal energy exchange |
470 |
skuang |
3778 |
in swapping ($\vec{p}_i$ in slab ``c'' with $\vec{p}_j$ in slab ``h'') |
471 |
skuang |
3776 |
or NIVS (total slab momentum conponemts $P^\alpha$ scaled to $\alpha |
472 |
|
|
P^\alpha$) would not obtain this result unless thermal flux vanishes |
473 |
|
|
(i.e. $\vec{p}_i=\vec{p}_j$ or $\alpha=1$ which are trivial to apply a |
474 |
|
|
thermal flux). In this sense, this method contributes to having |
475 |
|
|
minimal perturbation to a simulation while imposing thermal flux. |
476 |
|
|
|
477 |
|
|
\subsubsection{Shear viscosity} |
478 |
|
|
Table \ref{LJ} also compares our shear viscosity results with momentum |
479 |
|
|
swapping approach. Our calculations show that our method predicted |
480 |
|
|
similar values for shear viscosities to the momentum swapping |
481 |
|
|
approach, as well as the velocity gradient profiles. Moderately larger |
482 |
|
|
momentum fluxes are helpful to reduce the errors of measured velocity |
483 |
|
|
gradients and thus the final result. However, it is pointed out that |
484 |
|
|
the momentum swapping approach tends to produce nonthermal velocity |
485 |
|
|
distributions.\cite{Maginn:2010} |
486 |
|
|
|
487 |
|
|
To examine that temperature isotropy holds in simulations using our |
488 |
|
|
method, we measured the three one-dimensional temperatures in each of |
489 |
|
|
the slabs (Figure \ref{tempXyz}). Note that the $x$-dimensional |
490 |
|
|
temperatures were calculated after subtracting the effects from bulk |
491 |
|
|
velocities of the slabs. The one-dimensional temperature profiles |
492 |
|
|
showed no observable difference between the three dimensions. This |
493 |
|
|
ensures that isotropic scaling automatically preserves temperature |
494 |
|
|
isotropy and that our method is useful in shear viscosity |
495 |
|
|
computations. |
496 |
|
|
|
497 |
gezelter |
3769 |
\begin{figure} |
498 |
skuang |
3776 |
\includegraphics[width=\linewidth]{tempXyz} |
499 |
skuang |
3777 |
\caption{Unlike the previous NIVS algorithm, the new method does not |
500 |
|
|
produce a thermal anisotropy. No temperature difference between |
501 |
|
|
different dimensions were observed beyond the magnitude of the error |
502 |
|
|
bars. Note that the two ``hotter'' regions are caused by the shear |
503 |
|
|
stress, as reported by Tenney and Maginn\cite{Maginn:2010}, but not |
504 |
|
|
an effect that only observed in our methods.} |
505 |
skuang |
3776 |
\label{tempXyz} |
506 |
gezelter |
3769 |
\end{figure} |
507 |
|
|
|
508 |
skuang |
3776 |
Furthermore, the velocity distribution profiles are tested by imposing |
509 |
|
|
a large shear stress into the simulations. Figure \ref{vDist} |
510 |
|
|
demonstrates how our method is able to maintain thermal velocity |
511 |
|
|
distributions against the momentum swapping approach even under large |
512 |
|
|
imposed fluxes. Previous swapping methods tend to deplete particles of |
513 |
|
|
positive velocities in the negative velocity slab (``c'') and vice |
514 |
|
|
versa in slab ``h'', where the distributions leave a notch. This |
515 |
|
|
problematic profiles become significant when the imposed-flux becomes |
516 |
|
|
larger and diffusions from neighboring slabs could not offset the |
517 |
|
|
depletion. Simutaneously, abnormal peaks appear corresponding to |
518 |
|
|
excessive velocity swapped from the other slab. This nonthermal |
519 |
|
|
distributions limit applications of the swapping approach in shear |
520 |
|
|
stress simulations. Our method avoids the above problematic |
521 |
|
|
distributions by altering the means of applying momentum |
522 |
|
|
fluxes. Comparatively, velocity distributions recorded from |
523 |
|
|
simulations with our method is so close to the ideal thermal |
524 |
|
|
prediction that no observable difference is shown in Figure |
525 |
|
|
\ref{vDist}. Conclusively, our method avoids problems happened in |
526 |
|
|
previous RNEMD methods and provides a useful means for shear viscosity |
527 |
|
|
computations. |
528 |
gezelter |
3769 |
|
529 |
skuang |
3776 |
\begin{figure} |
530 |
|
|
\includegraphics[width=\linewidth]{velDist} |
531 |
skuang |
3777 |
\caption{Velocity distributions that develop under the swapping and |
532 |
|
|
our methods at high flux. These distributions were obtained from |
533 |
skuang |
3779 |
Lennard-Jones simulations with $j_z^*(p_x)\sim 0.4$ (equivalent to a |
534 |
skuang |
3777 |
swapping interval of 20 time steps). This is a relatively large flux |
535 |
|
|
to demonstrate the nonthermal distributions that develop under the |
536 |
|
|
swapping method. Distributions produced by our method are very close |
537 |
|
|
to the ideal thermal situations.} |
538 |
skuang |
3776 |
\label{vDist} |
539 |
|
|
\end{figure} |
540 |
gezelter |
3769 |
|
541 |
skuang |
3776 |
\subsection{Bulk SPC/E water} |
542 |
skuang |
3778 |
Since our method was in good performance of thermal conductivity and |
543 |
|
|
shear viscosity computations for simple Lennard-Jones fluid, we extend |
544 |
|
|
our applications of these simulations to complex fluid like SPC/E |
545 |
|
|
water model. A simulation cell with 1000 molecules was set up in the |
546 |
|
|
same manner as in \cite{kuang:164101}. For thermal conductivity |
547 |
|
|
simulations, measurements were taken to compare with previous RNEMD |
548 |
|
|
methods; for shear viscosity computations, simulations were run under |
549 |
|
|
a series of temperatures (with corresponding pressure relaxation using |
550 |
|
|
the isobaric-isothermal ensemble[CITE NIVS REF 32]), and results were |
551 |
|
|
compared to available data from Equilibrium MD methods[CITATIONS]. |
552 |
skuang |
3777 |
|
553 |
skuang |
3776 |
\subsubsection{Thermal conductivity} |
554 |
skuang |
3778 |
Table \ref{spceThermal} summarizes our thermal conductivity |
555 |
|
|
computations under different temperatures and thermal gradients, in |
556 |
|
|
comparison to the previous NIVS results\cite{kuang:164101} and |
557 |
|
|
experimental measurements\cite{WagnerKruse}. Note that no appreciable |
558 |
|
|
drift of total system energy or temperature was observed when our |
559 |
|
|
method is applied, which indicates that our algorithm conserves total |
560 |
|
|
energy even for systems involving electrostatic interactions. |
561 |
gezelter |
3769 |
|
562 |
skuang |
3778 |
Measurements using our method established similar temperature |
563 |
|
|
gradients to the previous NIVS method. Our simulation results are in |
564 |
|
|
good agreement with those from previous simulations. And both methods |
565 |
|
|
yield values in reasonable agreement with experimental |
566 |
|
|
values. Simulations using moderately higher thermal gradient or those |
567 |
|
|
with longer gradient axis ($z$) for measurement seem to have better |
568 |
|
|
accuracy, from our results. |
569 |
|
|
|
570 |
|
|
\begin{table*} |
571 |
|
|
\begin{minipage}{\linewidth} |
572 |
|
|
\begin{center} |
573 |
|
|
|
574 |
|
|
\caption{Thermal conductivity of SPC/E water under various |
575 |
|
|
imposed thermal gradients. Uncertainties are indicated in |
576 |
|
|
parentheses.} |
577 |
|
|
|
578 |
|
|
\begin{tabular}{ccccc} |
579 |
|
|
\hline\hline |
580 |
|
|
$\langle T\rangle$ & $\langle dT/dz\rangle$ & \multicolumn{3}{c} |
581 |
|
|
{$\lambda (\mathrm{W m}^{-1} \mathrm{K}^{-1})$} \\ |
582 |
|
|
(K) & (K/\AA) & This work & Previous NIVS\cite{kuang:164101} & |
583 |
|
|
Experiment\cite{WagnerKruse} \\ |
584 |
|
|
\hline |
585 |
|
|
300 & 0.8 & 0.815(0.027) & 0.770(0.008) & 0.61 \\ |
586 |
|
|
318 & 0.8 & 0.801(0.024) & 0.750(0.032) & 0.64 \\ |
587 |
|
|
& 1.6 & 0.766(0.007) & 0.778(0.019) & \\ |
588 |
skuang |
3779 |
& 0.8 & 0.786(0.009)\footnote{Simulation with $L_z$ |
589 |
|
|
twice as long.} & & \\ |
590 |
skuang |
3778 |
\hline\hline |
591 |
|
|
\end{tabular} |
592 |
|
|
\label{spceThermal} |
593 |
|
|
\end{center} |
594 |
|
|
\end{minipage} |
595 |
|
|
\end{table*} |
596 |
|
|
|
597 |
skuang |
3776 |
\subsubsection{Shear viscosity} |
598 |
skuang |
3778 |
The improvement our method achieves for shear viscosity computations |
599 |
|
|
enables us to apply it on SPC/E water models. The series of |
600 |
|
|
temperatures under which our shear viscosity calculations were carried |
601 |
|
|
out covers the liquid range under normal pressure. Our simulations |
602 |
|
|
predict a similar trend of $\eta$ vs. $T$ to EMD results we refer to |
603 |
skuang |
3779 |
(Table \ref{spceShear}). Considering subtlties such as temperature or |
604 |
|
|
pressure/density errors in these two series of measurements, our |
605 |
|
|
results show no significant difference from those with EMD |
606 |
|
|
methods. Since each value reported using our method takes only one |
607 |
|
|
single trajectory of simulation, instead of average from many |
608 |
|
|
trajectories when using EMD, our method provides an effective means |
609 |
|
|
for shear viscosity computations. |
610 |
gezelter |
3769 |
|
611 |
skuang |
3778 |
\begin{table*} |
612 |
|
|
\begin{minipage}{\linewidth} |
613 |
|
|
\begin{center} |
614 |
|
|
|
615 |
|
|
\caption{Computed shear viscosity of SPC/E water under different |
616 |
|
|
temperatures. Results are compared to those obtained with EMD |
617 |
|
|
method[CITATION]. Uncertainties are indicated in parentheses.} |
618 |
|
|
|
619 |
|
|
\begin{tabular}{cccc} |
620 |
|
|
\hline\hline |
621 |
|
|
$T$ & $\partial v_x/\partial z$ & \multicolumn{2}{c} |
622 |
|
|
{$\eta (\mathrm{mPa}\cdot\mathrm{s})$} \\ |
623 |
|
|
(K) & (10$^{10}$s$^{-1}$) & This work & Previous simulations[CITATION]\\ |
624 |
|
|
\hline |
625 |
|
|
273 & & 1.218(0.004) & \\ |
626 |
|
|
& & 1.140(0.012) & \\ |
627 |
|
|
303 & & 0.646(0.008) & \\ |
628 |
|
|
318 & & 0.536(0.007) & \\ |
629 |
|
|
& & 0.510(0.007) & \\ |
630 |
|
|
& & & \\ |
631 |
|
|
333 & & 0.428(0.002) & \\ |
632 |
|
|
363 & & 0.279(0.014) & \\ |
633 |
|
|
& & 0.306(0.001) & \\ |
634 |
|
|
\hline\hline |
635 |
|
|
\end{tabular} |
636 |
|
|
\label{spceShear} |
637 |
|
|
\end{center} |
638 |
|
|
\end{minipage} |
639 |
|
|
\end{table*} |
640 |
gezelter |
3769 |
|
641 |
skuang |
3776 |
[MAY COMBINE JZPX AND JZKE TO RUN ONE JOB BUT GET ETA=F(T)] |
642 |
skuang |
3777 |
[PUT RESULTS AND FIGURE HERE IF IT WORKS] |
643 |
skuang |
3779 |
\subsection{Interfacial frictions and slip lengths} |
644 |
|
|
An attractive aspect of our method is the ability to apply momentum |
645 |
|
|
and/or thermal flux in nonhomogeneous systems, where molecules of |
646 |
|
|
different identities (or phases) are segregated in different |
647 |
|
|
regions. We have previously studied the interfacial thermal transport |
648 |
|
|
of a series of metal gold-liquid |
649 |
|
|
surfaces\cite{kuang:164101,kuang:AuThl}, and attemptions have been |
650 |
|
|
made to investigate the relationship between this phenomenon and the |
651 |
|
|
interfacial frictions. |
652 |
gezelter |
3769 |
|
653 |
skuang |
3779 |
Table \ref{etaKappaDelta} includes these computations and previous |
654 |
|
|
calculations of corresponding interfacial thermal conductance. For |
655 |
|
|
bare Au(111) surfaces, slip boundary conditions were observed for both |
656 |
|
|
organic and aqueous liquid phases, corresponding to previously |
657 |
|
|
computed low interfacial thermal conductance. Instead, the butanethiol |
658 |
|
|
covered Au(111) surface appeared to be sticky to the organic liquid |
659 |
|
|
molecules in our simulations. We have reported conductance enhancement |
660 |
|
|
effect for this surface capping agent,\cite{kuang:AuThl} and these |
661 |
|
|
observations have a qualitative agreement with the thermal conductance |
662 |
|
|
results. This agreement also supports discussions on the relationship |
663 |
|
|
between surface wetting and slip effect and thermal conductance of the |
664 |
|
|
interface.[CITE BARRAT, GARDE] |
665 |
skuang |
3776 |
|
666 |
gezelter |
3769 |
\begin{table*} |
667 |
|
|
\begin{minipage}{\linewidth} |
668 |
|
|
\begin{center} |
669 |
|
|
|
670 |
skuang |
3779 |
\caption{Computed interfacial friction coefficient values for |
671 |
|
|
interfaces with various components for liquid and solid |
672 |
|
|
phase. Error estimates are indicated in parentheses.} |
673 |
gezelter |
3769 |
|
674 |
skuang |
3779 |
\begin{tabular}{llcccccc} |
675 |
gezelter |
3769 |
\hline\hline |
676 |
skuang |
3779 |
Solid & Liquid & $T$ & $j_z(p_x)$ & $\eta_{liquid}$ & $\kappa$ |
677 |
|
|
& $\delta$ & $G$\footnote{References \cite{kuang:AuThl} and |
678 |
|
|
\cite{kuang:164101}.} \\ |
679 |
skuang |
3780 |
surface & molecules & K & MPa & mPa$\cdot$s & Pa$\cdot$s/m & nm |
680 |
|
|
& MW/m$^2$/K \\ |
681 |
gezelter |
3769 |
\hline |
682 |
skuang |
3779 |
Au(111) & hexane & 200 & 1.08 & 0.20() & 5.3$\times$10$^4$() & |
683 |
|
|
3.7 & 46.5 \\ |
684 |
|
|
& & & 2.15 & 0.14() & 5.3$\times$10$^4$() & |
685 |
|
|
2.7 & \\ |
686 |
|
|
Au-SC$_4$H$_9$ & hexane & 200 & 2.16 & 0.29() & $\infty$ & 0 & |
687 |
|
|
131 \\ |
688 |
|
|
& & & 5.39 & 0.32() & $\infty$ & 0 & |
689 |
|
|
\\ |
690 |
gezelter |
3769 |
\hline |
691 |
skuang |
3779 |
Au(111) & toluene & 200 & 1.08 & 0.72() & 1.?$\times$10$^5$() & |
692 |
|
|
4.6 & 70.1 \\ |
693 |
|
|
& & & 2.16 & 0.54() & 1.?$\times$10$^5$() & |
694 |
|
|
4.9 & \\ |
695 |
|
|
Au-SC$_4$H$_9$ & toluene & 200 & 5.39 & 0.98() & $\infty$ & 0 |
696 |
|
|
& 187 \\ |
697 |
|
|
& & & 10.8 & 0.99() & $\infty$ & 0 |
698 |
|
|
& \\ |
699 |
|
|
\hline |
700 |
|
|
Au(111) & water & 300 & 1.08 & 0.40() & 1.9$\times$10$^4$() & |
701 |
|
|
20.7 & 1.65 \\ |
702 |
|
|
& & & 2.16 & 0.79() & 1.9$\times$10$^4$() & |
703 |
|
|
41.9 & \\ |
704 |
|
|
\hline |
705 |
|
|
ice(basal) & water & 225 & 19.4 & 15.8() & $\infty$ & 0 & \\ |
706 |
gezelter |
3769 |
\hline\hline |
707 |
|
|
\end{tabular} |
708 |
skuang |
3779 |
\label{etaKappaDelta} |
709 |
gezelter |
3769 |
\end{center} |
710 |
|
|
\end{minipage} |
711 |
|
|
\end{table*} |
712 |
|
|
|
713 |
skuang |
3779 |
An interesting effect alongside the surface friction change is |
714 |
|
|
observed on the shear viscosity of liquids in the regions close to the |
715 |
|
|
solid surface. Note that $\eta$ measured near a ``slip'' surface tends |
716 |
|
|
to be smaller than that near a ``stick'' surface. This suggests that |
717 |
|
|
an interface could affect the dynamic properties on its neighbor |
718 |
|
|
regions. It is known that diffusions of solid particles in liquid |
719 |
|
|
phase is affected by their surface conditions (stick or slip |
720 |
|
|
boundary).[CITE SCHMIDT AND SKINNER] Our observations could provide |
721 |
|
|
support to this phenomenon. |
722 |
gezelter |
3769 |
|
723 |
skuang |
3779 |
In addition to these previously studied interfaces, we attempt to |
724 |
|
|
construct ice-water interfaces and the basal plane of ice lattice was |
725 |
|
|
first studied. In contrast to the Au(111)/water interface, where the |
726 |
|
|
friction coefficient is relatively small and large slip effect |
727 |
|
|
presents, the ice/liquid water interface demonstrates strong |
728 |
|
|
interactions and appears to be sticky. The supercooled liquid phase is |
729 |
|
|
an order of magnitude viscous than measurements in previous |
730 |
|
|
section. It would be of interst to investigate the effect of different |
731 |
|
|
ice lattice planes (such as prism surface) on interfacial friction and |
732 |
|
|
corresponding liquid viscosity. |
733 |
gezelter |
3769 |
|
734 |
skuang |
3776 |
\section{Conclusions} |
735 |
skuang |
3779 |
Our simulations demonstrate the validity of our method in RNEMD |
736 |
|
|
computations of thermal conductivity and shear viscosity in atomic and |
737 |
|
|
molecular liquids. Our method maintains thermal velocity distributions |
738 |
|
|
and avoids thermal anisotropy in previous NIVS shear stress |
739 |
|
|
simulations, as well as retains attractive features of previous RNEMD |
740 |
|
|
methods. There is no {\it a priori} restrictions to the method to be |
741 |
|
|
applied in various ensembles, so prospective applications to |
742 |
|
|
extended-system methods are possible. |
743 |
gezelter |
3769 |
|
744 |
skuang |
3779 |
Furthermore, using this method, investigations can be carried out to |
745 |
|
|
characterize interfacial interactions. Our method is capable of |
746 |
|
|
effectively imposing both thermal and momentum flux accross an |
747 |
|
|
interface and thus facilitates studies that relates dynamic property |
748 |
|
|
measurements to the chemical details of an interface. |
749 |
gezelter |
3769 |
|
750 |
skuang |
3779 |
Another attractive feature of our method is the ability of |
751 |
|
|
simultaneously imposing thermal and momentum flux in a |
752 |
|
|
system. potential researches that might be benefit include complex |
753 |
|
|
systems that involve thermal and momentum gradients. For example, the |
754 |
|
|
Soret effects under a velocity gradient would be of interest to |
755 |
|
|
purification and separation researches. |
756 |
gezelter |
3769 |
|
757 |
|
|
\section{Acknowledgments} |
758 |
|
|
Support for this project was provided by the National Science |
759 |
|
|
Foundation under grant CHE-0848243. Computational time was provided by |
760 |
|
|
the Center for Research Computing (CRC) at the University of Notre |
761 |
|
|
Dame. |
762 |
|
|
|
763 |
|
|
\newpage |
764 |
|
|
|
765 |
skuang |
3770 |
\bibliography{stokes} |
766 |
gezelter |
3769 |
|
767 |
|
|
\end{doublespace} |
768 |
|
|
\end{document} |