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\begin{document} |
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\title{A minimal perturbation approach to RNEMD able to simultaneously |
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impose thermal and momentum gradients} |
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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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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 while maintaining |
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thermal velocity distributions. It also avoid thermal anisotropy |
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occured in previous NIVS simulations by using isotropic velocity |
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scaling on the molecules in specific regions of a system. To test |
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the method, we have computed the thermal conductivity and shear |
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viscosity of model liquid systems as well as the interfacial |
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frictions of a series of metal/liquid interfaces. Its ability to |
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combine the thermal and momentum gradients allows us to obtain shear |
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viscosity data for a range of temperatures in only one trajectory. |
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|
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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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|
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\section{Introduction} |
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Imposed-flux methods in Molecular Dynamics (MD) |
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simulations\cite{MullerPlathe:1997xw,ISI:000080382700030,kuang:164101} |
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can establish steady state systems with an applied flux set vs a |
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corresponding gradient that can be measured. These methods does not |
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need many trajectories to provide information of transport properties |
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of a given system. Thus, they are utilized in computing thermal and |
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mechanical transfer of homogeneous bulk systems as well as |
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heterogeneous systems such as solid-liquid |
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interfaces.\cite{garde:nl2005,garde:PhysRevLett2009,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 proposed 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 can |
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also be compatible with particles of different identities. Although |
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simple to implement in a simulation, this approach can create |
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nonthermal velocity distributions, as discovered by Tenney and |
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Maginn\cite{Maginn:2010}. Furthermore, this approach is less efficient |
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for kinetic energy transfer between particles of different identities, |
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especially when the mass difference between the particles becomes |
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significant. This also limits its applications on heterogeneous |
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interfacial 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 has limited applications in imposing |
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momentum fluxes. Temperature anisotropy could happen under high |
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momentum fluxes due to the implementation of this algorithm. Thus, |
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combining thermal and momentum flux is also difficult to obtain with |
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this approach. However, such combination may provide a means to |
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simulate thermal/momentum gradient coupled processes such as Soret |
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effect in liquid flows. Therefore, developing improved approaches to |
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extend the applications of the imposed-flux method is desirable. |
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|
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In this paper, we improve the RNEMD methods by proposing a novel |
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approach 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 and its implementation 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 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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|
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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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|
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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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[MAY PUT EXTRA MATH IN SUPPORT INFO OR APPENDIX] |
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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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|
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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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|
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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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|
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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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|
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This approach is advantageous over the original momentum swapping in |
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many aspects. In one swapping, the velocity vectors involved are |
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usually very different (or the generated flux is trivial to obtain |
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gradients), thus the swapping tends to incur perturbations to the |
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neighbors of the particles involved. Comparatively, our approach |
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disperse the flux to every selected particle in a slab so that |
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perturbations in the flux generating region could be |
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minimized. Additionally, because the momentum swapping steps tend to |
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result in a nonthermal distribution, when an imposed flux is |
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relatively large and diffusions from the neighboring slabs could no |
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longer remedy this effect, problematic distributions would be |
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observed. In comparison, the operations of our approach has the nature |
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of preserving the equilibrium velocity distributions (commonly |
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Maxwell-Boltzmann), and results in later section will illustrate that |
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this is helpful to retain thermal distributions in a simulation. |
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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 (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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|
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\subsection{Simulation Protocols} |
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The systems to be investigated are set up in orthorhombic simulation |
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cells with periodic boundary conditions in all three dimensions. The |
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$z$ axis of these cells were longer and set as the temperature and/or |
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momentum gradient axis. And the cells were evenly 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 of the same length in homogeneous |
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systems or had length scale close to each other where heterogeneous |
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interfaces presents. In all cases, before introducing a nonequilibrium |
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method to establish steady thermal and/or momentum gradients for |
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further measurements and calculations, canonical ensemble with a |
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Nos\'e-Hoover thermostat\cite{hoover85} and microcanonical ensemble |
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equilibrations were used before data collections. For SPC/E water |
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simulations, isobaric-isothermal equilibrations\cite{melchionna93} are |
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performed before the above to reach normal pressure (1 bar); for |
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interfacial systems, similar equilibrations are used to relax the |
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surface tensions of the $xy$ plane. |
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|
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While homogeneous fluid systems can be set up with rather random |
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configurations, our interfacial systems needs a series of steps to |
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ensure 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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in details 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 all possible proton order configurations. We refer to |
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their results and choose the configuration of the lowest energy after |
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geometry optimization as the unit cell for our ice lattices. Although |
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experimental solid/liquid coexistant temperature under normal pressure |
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should be close to 273K, Bryk and Haymet's simulations of ice/liquid |
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water interfaces with different models suggest that for SPC/E, the |
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most stable interface is observed at 225$\pm$5K.\cite{bryk:10258} |
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Therefore, our ice/liquid water simulations were carried out at |
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225K. To have extra protection of the ice lattice during initial |
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equilibration (when the randomly generated liquid phase configuration |
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could release large amount of energy in relaxation), restraints were |
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applied to the ice lattice to avoid inadvertent melting by the heat |
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dissipated from the high enery configurations. |
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[MAY ADD A SNAPSHOT FOR BASAL PLANE] |
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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 |
294 |
force field parameters consistent with previous simulations. Argon is |
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the Lennard-Jones fluid used here, and its results are reported in |
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reduced unit 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 |
304 |
the quantum Sutton-Chen (QSC) formulation.\cite{PhysRevB.59.3527} The |
305 |
QSC potentials include zero-point quantum corrections and are |
306 |
reparametrized for accurate surface energies compared to the |
307 |
Sutton-Chen potentials.\cite{Chen90} For gold/water interfaces, the |
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Spohr potential was adopted\cite{ISI:000167766600035} to depict |
309 |
Au-H$_2$O interactions. |
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|
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For our gold/organic liquid interfaces, the small organic molecules |
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included in our simulations are the Au surface capping agent |
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butanethiol and liquid hexane and toluene. The United-Atom |
314 |
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 |
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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 ($\lambda$) |
327 |
can be obtained with the imposed kinetic energy flux and the measured |
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thermal gradient: |
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\begin{equation} |
330 |
J_z = -\lambda \frac{\partial T}{\partial z} |
331 |
\end{equation} |
332 |
Like other imposed-flux methods, the energy flux was calculated using |
333 |
the total non-physical energy transferred (${E_{total}}$) from slab |
334 |
``c'' to slab ``h'', which is recorded throughout a simulation, and |
335 |
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} |
338 |
\end{equation} |
339 |
where $L_x$ and $L_y$ denotes the dimensions of the plane in a |
340 |
simulation cell perpendicular to the thermal gradient, and a factor of |
341 |
two in the denominator is necessary for the heat transport occurs in |
342 |
both $+z$ and $-z$ directions. The average temperature gradient |
343 |
${\langle\partial T/\partial z\rangle}$ can be obtained by a linear |
344 |
regression of the temperature profile, which is recorded during a |
345 |
simulation for each slab in a cell. For Lennard-Jones simulations, |
346 |
thermal conductivities are reported in reduced units |
347 |
(${\lambda^*=\lambda \sigma^2 m^{1/2} k_B^{-1}\varepsilon^{-1/2}}$). |
348 |
|
349 |
\subsection{Shear viscosities} |
350 |
Alternatively, the method can carry out shear viscosity calculations |
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by specify a momentum flux. In our algorithm, one can specify the |
352 |
three components of the flux vector $\vec{j}_z(\vec{p})$ |
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respectively. For shear viscosity simulations, $j_z(p_z)$ is usually |
354 |
set to zero. For isotropic systems, the direction of |
355 |
$\vec{j}_z(\vec{p})$ on the $xy$ plane should not matter, but the |
356 |
ability of arbitarily specifying the vector direction in our method |
357 |
could provide convenience in anisotropic simulations. |
358 |
|
359 |
Similar to thermal conductivity computations, for a homogeneous |
360 |
system, linear response of the momentum gradient with respect to the |
361 |
shear stress is assumed, and the shear viscosity ($\eta$) can be |
362 |
obtained with the imposed momentum flux (e.g. in $x$ direction) and |
363 |
the measured gradient: |
364 |
\begin{equation} |
365 |
j_z(p_x) = -\eta \frac{\partial v_x}{\partial z} |
366 |
\end{equation} |
367 |
where the flux is similarly defined: |
368 |
\begin{equation} |
369 |
j_z(p_x) = \frac{P_x}{2 t L_x L_y} |
370 |
\end{equation} |
371 |
with $P_x$ being the total non-physical momentum transferred within |
372 |
the data collection time. Also, the averaged velocity gradient |
373 |
${\langle\partial v_x/\partial z\rangle}$ can be obtained using linear |
374 |
regression of the $x$ component of the mean velocity ($\langle |
375 |
v_x\rangle$) in each of the bins. For Lennard-Jones simulations, shear |
376 |
viscosities are also reported in reduced units |
377 |
(${\eta^*=\eta\sigma^2(\varepsilon m)^{-1/2}}$). |
378 |
|
379 |
Although $J_z$ may be switched off for shear viscosity simulations at |
380 |
a certain temperature, our method's ability to impose both a thermal |
381 |
and a momentum flux in one simulation allows the combination of a |
382 |
temperature and a velocity gradient. In this case, since viscosity is |
383 |
generally a function of temperature, the local viscosity also depends |
384 |
on the local temperature. Therefore, in one such simulation, viscosity |
385 |
at $z$ (corresponding to a certain $T$) can be computed with the |
386 |
applied shear flux and the local velocity gradient (which can be |
387 |
obtained by finite difference approximation). As a whole, the |
388 |
viscosity can be mapped out as the function of temperature in one |
389 |
single trajectory of simulation. Results for shear viscosity |
390 |
computations of SPC/E water will demonstrate its effectiveness in |
391 |
detail. |
392 |
|
393 |
\subsection{Interfacial friction and Slip length} |
394 |
While the shear stress results in a velocity gradient within bulk |
395 |
fluid phase, its effect at a solid-liquid interface could vary due to |
396 |
the interaction strength between the two phases. The interfacial |
397 |
friction coefficient $\kappa$ is defined to relate the shear stress |
398 |
(e.g. along $x$-axis) with the relative fluid velocity tangent to the |
399 |
interface: |
400 |
\begin{equation} |
401 |
j_z(p_x)|_{interface} = \kappa\Delta v_x|_{interface} |
402 |
\end{equation} |
403 |
Under ``stick'' boundary condition, $\Delta v_x|_{interface} |
404 |
\rightarrow 0$, which leads to $\kappa\rightarrow\infty$. However, for |
405 |
``slip'' boundary conditions at the solid-liquid interfaces, $\kappa$ |
406 |
becomes finite. To characterize the interfacial boundary conditions, |
407 |
slip length ($\delta$) is defined using $\kappa$ and the shear |
408 |
viscocity of liquid phase ($\eta$): |
409 |
\begin{equation} |
410 |
\delta = \frac{\eta}{\kappa} |
411 |
\end{equation} |
412 |
so that $\delta\rightarrow 0$ in the ``no-slip'' boundary condition, |
413 |
and depicts how ``slippery'' an interface is. Figure \ref{slipLength} |
414 |
illustrates how this quantity is defined and computed for a |
415 |
solid-liquid interface. [MAY INCLUDE SNAPSHOT IN FIGURE] |
416 |
|
417 |
\begin{figure} |
418 |
\includegraphics[width=\linewidth]{defDelta} |
419 |
\caption{The slip length $\delta$ can be obtained from a velocity |
420 |
profile of a solid-liquid interface simulation, when a momentum flux |
421 |
is applied. An example of Au/hexane interfaces is shown, and the |
422 |
calculation for the left side is illustrated. The calculation for |
423 |
the right side is similar to the left.} |
424 |
\label{slipLength} |
425 |
\end{figure} |
426 |
|
427 |
In our method, a shear stress can be applied similar to shear |
428 |
viscosity computations by applying an unphysical momentum flux |
429 |
(e.g. $j_z(p_x)$). A corresponding velocity profile can be obtained as |
430 |
shown in Figure \ref{slipLength}, in which the velocity gradients |
431 |
within liquid phase and velocity difference at the liquid-solid |
432 |
interface can be measured respectively. Further calculations and |
433 |
characterizations of the interface can be carried out using these |
434 |
data. |
435 |
|
436 |
\section{Results and Discussions} |
437 |
\subsection{Lennard-Jones fluid} |
438 |
Our orthorhombic simulation cell of Lennard-Jones fluid has identical |
439 |
parameters to our previous work\cite{kuang:164101} to facilitate |
440 |
comparison. Thermal conductivitis and shear viscosities were computed |
441 |
with the algorithm applied to the simulations. The results of thermal |
442 |
conductivity are compared with our previous NIVS algorithm. However, |
443 |
since the NIVS algorithm could produce temperature anisotropy for |
444 |
shear viscocity computations, these results are instead compared to |
445 |
the momentum swapping approaches. Table \ref{LJ} lists these |
446 |
calculations with various fluxes in reduced units. |
447 |
|
448 |
\begin{table*} |
449 |
\begin{minipage}{\linewidth} |
450 |
\begin{center} |
451 |
|
452 |
\caption{Thermal conductivity ($\lambda^*$) and shear viscosity |
453 |
($\eta^*$) (in reduced units) of a Lennard-Jones fluid at |
454 |
${\langle T^* \rangle = 0.72}$ and ${\rho^* = 0.85}$ computed |
455 |
at various momentum fluxes. The new method yields similar |
456 |
results to previous RNEMD methods. All results are reported in |
457 |
reduced unit. Uncertainties are indicated in parentheses.} |
458 |
|
459 |
\begin{tabular}{cccccc} |
460 |
\hline\hline |
461 |
\multicolumn{2}{c}{Momentum Exchange} & |
462 |
\multicolumn{2}{c}{$\lambda^*$} & |
463 |
\multicolumn{2}{c}{$\eta^*$} \\ |
464 |
\hline |
465 |
Swap Interval $t^*$ & Equivalent $J_z^*$ or $j_z^*(p_x)$ & |
466 |
NIVS\cite{kuang:164101} & This work & Swapping & This work \\ |
467 |
\hline |
468 |
0.116 & 0.16 & 7.30(0.10) & 6.25(0.23) & 3.57(0.06) & 3.53(0.16)\\ |
469 |
0.232 & 0.09 & 6.95(0.09) & 8.02(0.56) & 3.64(0.05) & 3.43(0.17)\\ |
470 |
0.463 & 0.047 & 7.19(0.07) & 6.48(0.15) & 3.52(0.16) & 3.51(0.08)\\ |
471 |
0.926 & 0.024 & 7.19(0.28) & 7.02(0.13) & 3.72(0.05) & 3.79(0.11)\\ |
472 |
1.158 & 0.019 & 7.98(0.33) & 7.37(0.23) & 3.42(0.06) & 3.53(0.06)\\ |
473 |
\hline\hline |
474 |
\end{tabular} |
475 |
\label{LJ} |
476 |
\end{center} |
477 |
\end{minipage} |
478 |
\end{table*} |
479 |
|
480 |
\subsubsection{Thermal conductivity} |
481 |
Our thermal conductivity calculations with this method yields |
482 |
comparable results to the previous NIVS algorithm. This indicates that |
483 |
the thermal gradients introduced using this method are also close to |
484 |
previous RNEMD methods. Simulations with moderately higher thermal |
485 |
fluxes tend to yield more reliable thermal gradients and thus avoid |
486 |
large errors, while overly high thermal fluxes could introduce side |
487 |
effects such as non-linear temperature gradient response or |
488 |
inadvertent phase transitions. |
489 |
|
490 |
Since the scaling operation is isotropic in this method, one does not |
491 |
need extra care to ensure temperature isotropy between the $x$, $y$ |
492 |
and $z$ axes, while for NIVS, thermal anisotropy might happen if the |
493 |
criteria function for choosing scaling coefficients does not perform |
494 |
as expected. Furthermore, this method avoids inadvertent concomitant |
495 |
momentum flux when only thermal flux is imposed, which could not be |
496 |
achieved with swapping or NIVS approaches. The thermal energy exchange |
497 |
in swapping ($\vec{p}_i$ in slab ``c'' with $\vec{p}_j$ in slab ``h'') |
498 |
or NIVS (total slab momentum conponemts $P^\alpha$ scaled to $\alpha |
499 |
P^\alpha$) would not achieve this effect unless thermal flux vanishes |
500 |
(i.e. $\vec{p}_i=\vec{p}_j$ or $\alpha=1$ which do not contribute to |
501 |
applying a thermal flux). In this sense, this method aids to achieve |
502 |
minimal perturbation to a simulation while imposing a thermal flux. |
503 |
|
504 |
\subsubsection{Shear viscosity} |
505 |
Table \ref{LJ} also compares our shear viscosity results with the |
506 |
momentum swapping approach. Our calculations show that our method |
507 |
predicted similar values of shear viscosities to the momentum swapping |
508 |
approach, as well as the velocity gradient profiles. Moderately larger |
509 |
momentum fluxes are helpful to reduce the errors of measured velocity |
510 |
gradients and thus the final result. However, it is pointed out that |
511 |
the momentum swapping approach tends to produce nonthermal velocity |
512 |
distributions.\cite{Maginn:2010} |
513 |
|
514 |
To examine that temperature isotropy holds in simulations using our |
515 |
method, we measured the three one-dimensional temperatures in each of |
516 |
the slabs (Figure \ref{tempXyz}). Note that the $x$-dimensional |
517 |
temperatures were calculated after subtracting the contribution from |
518 |
bulk velocities of the slabs. The one-dimensional temperature profiles |
519 |
showed no observable difference between the three dimensions. This |
520 |
ensures that isotropic scaling automatically preserves temperature |
521 |
isotropy and that our method is useful in shear viscosity |
522 |
computations. |
523 |
|
524 |
\begin{figure} |
525 |
\includegraphics[width=\linewidth]{tempXyz} |
526 |
\caption{Unlike the previous NIVS algorithm, the new method does not |
527 |
produce a thermal anisotropy. No temperature difference between |
528 |
different dimensions were observed beyond the magnitude of the error |
529 |
bars. Note that the two ``hotter'' regions are caused by the shear |
530 |
stress, as reported by Tenney and Maginn\cite{Maginn:2010}, but not |
531 |
an effect that only observed in our methods.} |
532 |
\label{tempXyz} |
533 |
\end{figure} |
534 |
|
535 |
Furthermore, the velocity distribution profiles are tested by imposing |
536 |
a large shear stress into the simulations. Figure \ref{vDist} |
537 |
demonstrates how our method is able to maintain thermal velocity |
538 |
distributions against the momentum swapping approach even under large |
539 |
imposed fluxes. Previous swapping methods tend to deplete particles of |
540 |
positive velocities in the negative velocity slab (``c'') and vice |
541 |
versa in slab ``h'', where the distributions leave notchs. This |
542 |
problematic profiles become significant when the imposed-flux becomes |
543 |
larger and diffusions from neighboring slabs could not offset the |
544 |
depletions. Simutaneously, abnormal peaks appear corresponding to |
545 |
excessive particles having velocity swapped from the other slab. These |
546 |
nonthermal distributions limit applications of the swapping approach |
547 |
in shear stress simulations. Our method avoids the above problematic |
548 |
distributions by altering the means of applying momentum |
549 |
fluxes. Comparatively, velocity distributions recorded from |
550 |
simulations with our method is so close to the ideal thermal |
551 |
prediction that no obvious difference is shown in Figure |
552 |
\ref{vDist}. Conclusively, our method avoids problems that occurs in |
553 |
previous RNEMD methods and provides a useful means for shear viscosity |
554 |
computations. |
555 |
|
556 |
\begin{figure} |
557 |
\includegraphics[width=\linewidth]{velDist} |
558 |
\caption{Velocity distributions that develop under the swapping and |
559 |
our methods at a large flux. These distributions were obtained from |
560 |
Lennard-Jones simulations with $j_z^*(p_x)\sim 0.4$ (equivalent to a |
561 |
swapping interval of 20 time steps). This is a relatively large flux |
562 |
to demonstrate the nonthermal distributions that develop under the |
563 |
swapping method. In comparison, distributions produced by our method |
564 |
are very close to the ideal thermal situations.} |
565 |
\label{vDist} |
566 |
\end{figure} |
567 |
|
568 |
\subsection{Bulk SPC/E water} |
569 |
We extend our applications of thermal conductivity and shear viscosity |
570 |
computations to a complex fluid model of SPC/E water. A simulation |
571 |
cell with 1000 molecules was set up in the similar manner as in |
572 |
\cite{kuang:164101}. For thermal conductivity simulations, |
573 |
measurements were taken to compare with previous RNEMD methods; for |
574 |
shear viscosity computations, simulations were run under a series of |
575 |
temperatures (with corresponding pressure relaxation using the |
576 |
isobaric-isothermal ensemble\cite{melchionna93}), and results were |
577 |
compared to available data from EMD |
578 |
methods\cite{10.1063/1.3330544,Medina2011}. Besides, a simulation with |
579 |
both thermal and momentum gradient was carried out to map out shear |
580 |
viscosity as a function of temperature to see the effectiveness and |
581 |
accuracy our method could reach. |
582 |
|
583 |
\subsubsection{Thermal conductivity} |
584 |
Table \ref{spceThermal} summarizes our thermal conductivity |
585 |
computations under different temperatures and thermal gradients, in |
586 |
comparison to the previous NIVS results\cite{kuang:164101} and |
587 |
experimental measurements\cite{WagnerKruse}. Note that no appreciable |
588 |
drift of total system energy or temperature was observed when our |
589 |
method is applied, which indicates that our algorithm conserves total |
590 |
energy well for systems involving electrostatic interactions. |
591 |
|
592 |
Measurements using our method established similar temperature |
593 |
gradients to the previous NIVS method. Our simulation results are in |
594 |
good agreement with those from previous simulations. And both methods |
595 |
yield values in reasonable agreement with experimental |
596 |
values. Simulations using moderately higher thermal gradient or those |
597 |
with longer gradient axis ($z$) for measurement seem to have better |
598 |
accuracy, from our results. |
599 |
|
600 |
\begin{table*} |
601 |
\begin{minipage}{\linewidth} |
602 |
\begin{center} |
603 |
|
604 |
\caption{Thermal conductivity of SPC/E water under various |
605 |
imposed thermal gradients. Uncertainties are indicated in |
606 |
parentheses.} |
607 |
|
608 |
\begin{tabular}{ccccc} |
609 |
\hline\hline |
610 |
$\langle T\rangle$ & $\langle dT/dz\rangle$ & \multicolumn{3}{c} |
611 |
{$\lambda (\mathrm{W m}^{-1} \mathrm{K}^{-1})$} \\ |
612 |
(K) & (K/\AA) & This work & Previous NIVS\cite{kuang:164101} & |
613 |
Experiment\cite{WagnerKruse} \\ |
614 |
\hline |
615 |
300 & 0.8 & 0.815(0.027) & 0.770(0.008) & 0.61 \\ |
616 |
318 & 0.8 & 0.801(0.024) & 0.750(0.032) & 0.64 \\ |
617 |
& 1.6 & 0.766(0.007) & 0.778(0.019) & \\ |
618 |
& 0.8 & 0.786(0.009)\footnote{Simulation with $L_z$ |
619 |
twice as long.} & & \\ |
620 |
\hline\hline |
621 |
\end{tabular} |
622 |
\label{spceThermal} |
623 |
\end{center} |
624 |
\end{minipage} |
625 |
\end{table*} |
626 |
|
627 |
\subsubsection{Shear viscosity} |
628 |
The improvement our method achieves for shear viscosity computations |
629 |
enables us to apply it on SPC/E water models. The series of |
630 |
temperatures under which our shear viscosity calculations were carried |
631 |
out covers the liquid range under normal pressure. Our simulations |
632 |
predict a similar trend of $\eta$ vs. $T$ to EMD results we refer to |
633 |
(Table \ref{spceShear}). Considering subtlties such as temperature or |
634 |
pressure/density errors in these two series of measurements, our |
635 |
results show no significant difference from those with EMD |
636 |
methods. Since each value reported using our method takes only one |
637 |
single trajectory of simulation, instead of average from many |
638 |
trajectories when using EMD, our method provides an effective means |
639 |
for shear viscosity computations. |
640 |
|
641 |
\begin{table*} |
642 |
\begin{minipage}{\linewidth} |
643 |
\begin{center} |
644 |
|
645 |
\caption{Computed shear viscosity of SPC/E water under different |
646 |
temperatures. Results are compared to those obtained with EMD |
647 |
method[CITATION]. Uncertainties are indicated in parentheses.} |
648 |
|
649 |
\begin{tabular}{cccc} |
650 |
\hline\hline |
651 |
$T$ & $\partial v_x/\partial z$ & \multicolumn{2}{c} |
652 |
{$\eta (\mathrm{mPa}\cdot\mathrm{s})$} \\ |
653 |
(K) & (10$^{10}$s$^{-1}$) & This work & Previous |
654 |
simulations\cite{Medina2011} \\ |
655 |
\hline |
656 |
273 & 1.12 & 1.218(0.004) & 1.282(0.048) \\ |
657 |
& 1.79 & 1.140(0.012) & \\ |
658 |
303 & 2.09 & 0.646(0.008) & 0.643(0.019) \\ |
659 |
318 & 2.50 & 0.536(0.007) & \\ |
660 |
& 5.25 & 0.510(0.007) & \\ |
661 |
& 2.82 & 0.474(0.003)\footnote{Simulation with $L_z$ twice |
662 |
as long.} & \\ |
663 |
333 & 3.10 & 0.428(0.002) & 0.421(0.008) \\ |
664 |
363 & 2.34 & 0.279(0.014) & 0.291(0.005) \\ |
665 |
& 4.26 & 0.306(0.001) & \\ |
666 |
\hline\hline |
667 |
\end{tabular} |
668 |
\label{spceShear} |
669 |
\end{center} |
670 |
\end{minipage} |
671 |
\end{table*} |
672 |
|
673 |
A more effective way to map out $\eta$ vs $T$ is to combine a momentum |
674 |
flux with a thermal flux. Figure \ref{Tvxdvdz} shows the thermal and |
675 |
velocity gradient in one such simulation. At different positions with |
676 |
different temperatures, the velocity gradient is not a constant but |
677 |
can be computed locally. With the data provided in Figure |
678 |
\ref{Tvxdvdz}, a series of $\eta$ is calculated as in Figure |
679 |
\ref{etaT} and a linear fit was performed to $\partial v_x/\partial z$ |
680 |
vs. $z$ so that the resulted $\eta$ can be present as a curve as |
681 |
well. For comparison, other results are also mapped in the figure. |
682 |
|
683 |
\begin{figure} |
684 |
\includegraphics[width=\linewidth]{tvxdvdz} |
685 |
\caption{With a combination of a thermal and a momentum flux, a |
686 |
simulation can have both a temperature (top) and a velocity (middle) |
687 |
gradient. Due to the thermal gradient, $\partial v_x/\partial z$ is |
688 |
not constant but can be computed using finite difference |
689 |
approximations (lower). These data can be used further to calculate |
690 |
$\eta$ vs $T$ (Figure \ref{etaT}).} |
691 |
\label{Tvxdvdz} |
692 |
\end{figure} |
693 |
|
694 |
From Figure \ref{etaT}, one can see that the generated curve agrees |
695 |
well with the above RNEMD simulations at different temperatures, as |
696 |
well as results reported using EMD |
697 |
methods\cite{10.1063/1.3330544,Medina2011} in much of the temperature |
698 |
range simulated. However, this curve has relatively large error in |
699 |
lower temperature regions and has some difference in predicting $\eta$ |
700 |
near 273K. Provided that this curve only takes one trajectory to |
701 |
generate, these results are of satisfactory efficiency and |
702 |
accuracy. Since previous work already pointed out that the SPC/E model |
703 |
tends to predict lower viscosity compared to experimental |
704 |
data,\cite{Medina2011} experimental comparison are not given here. |
705 |
|
706 |
\begin{figure} |
707 |
\includegraphics[width=\linewidth]{etaT} |
708 |
\caption{The curve generated by single simulation with thermal and |
709 |
momentum gradient predicts satisfatory values in much of the |
710 |
temperature range under test.} |
711 |
\label{etaT} |
712 |
\end{figure} |
713 |
|
714 |
\subsection{Interfacial frictions and slip lengths} |
715 |
Another attractive aspect of our method is the ability to apply |
716 |
momentum and/or thermal flux in nonhomogeneous systems, where |
717 |
molecules of different identities (or phases) are segregated in |
718 |
different regions. We have previously studied the interfacial thermal |
719 |
transport of a series of metal gold-liquid |
720 |
surfaces\cite{kuang:164101,kuang:AuThl}, and would like to further |
721 |
investigate the relationship between this phenomenon and the |
722 |
interfacial frictions. |
723 |
|
724 |
Table \ref{etaKappaDelta} includes these computations and previous |
725 |
calculations of corresponding interfacial thermal conductance. For |
726 |
bare Au(111) surfaces, slip boundary conditions were observed for both |
727 |
organic and aqueous liquid phases, corresponding to previously |
728 |
computed low interfacial thermal conductance. In comparison, the |
729 |
butanethiol covered Au(111) surface appeared to be sticky to the |
730 |
organic liquid layers in our simulations. We have reported conductance |
731 |
enhancement effect for this surface capping agent,\cite{kuang:AuThl} |
732 |
and these observations have a qualitative agreement with the thermal |
733 |
conductance results. This agreement also supports discussions on the |
734 |
relationship between surface wetting and slip effect and thermal |
735 |
conductance of the |
736 |
interface.\cite{PhysRevLett.82.4671,doi:10.1080/0026897031000068578,garde:PhysRevLett2009} |
737 |
|
738 |
\begin{table*} |
739 |
\begin{minipage}{\linewidth} |
740 |
\begin{center} |
741 |
|
742 |
\caption{Computed interfacial friction coefficient values for |
743 |
interfaces with various components for liquid and solid |
744 |
phase. Error estimates are indicated in parentheses.} |
745 |
|
746 |
\begin{tabular}{llcccccc} |
747 |
\hline\hline |
748 |
Solid & Liquid & $T$ & $j_z(p_x)$ & $\eta_{liquid}$ & $\kappa$ |
749 |
& $\delta$ & $G$\footnote{References \cite{kuang:AuThl} and |
750 |
\cite{kuang:164101}.} \\ |
751 |
surface & molecules & K & MPa & mPa$\cdot$s & |
752 |
10$^4$Pa$\cdot$s/m & nm & MW/m$^2$/K \\ |
753 |
\hline |
754 |
Au(111) & hexane & 200 & 1.08 & 0.197(0.009) & 5.30(0.36) & |
755 |
3.72 & 46.5 \\ |
756 |
& & & 2.15 & 0.141(0.002) & 5.31(0.26) & |
757 |
2.76 & \\ |
758 |
Au-SC$_4$H$_9$ & hexane & 200 & 2.16 & 0.286(0.019) & $\infty$ |
759 |
& 0 & 131 \\ |
760 |
& & & 5.39 & 0.320(0.006) & $\infty$ |
761 |
& 0 & \\ |
762 |
\hline |
763 |
Au(111) & toluene & 200 & 1.08 & 0.722(0.035) & 15.7(0.7) & |
764 |
4.60 & 70.1 \\ |
765 |
& & & 2.16 & 0.544(0.030) & 11.2(0.5) & |
766 |
4.86 & \\ |
767 |
Au-SC$_4$H$_9$ & toluene & 200 & 5.39 & 0.980(0.057) & |
768 |
$\infty$ & 0 & 187 \\ |
769 |
& & & 10.8 & 0.995(0.005) & |
770 |
$\infty$ & 0 & \\ |
771 |
\hline |
772 |
Au(111) & water & 300 & 1.08 & 0.399(0.050) & 1.928(0.022) & |
773 |
20.7 & 1.65 \\ |
774 |
& & & 2.16 & 0.794(0.255) & 1.895(0.003) & |
775 |
41.9 & \\ |
776 |
\hline |
777 |
ice(basal) & water & 225 & 19.4 & 15.8(0.2) & $\infty$ & 0 & \\ |
778 |
\hline\hline |
779 |
\end{tabular} |
780 |
\label{etaKappaDelta} |
781 |
\end{center} |
782 |
\end{minipage} |
783 |
\end{table*} |
784 |
|
785 |
An interesting effect alongside the surface friction change is |
786 |
observed on the shear viscosity of liquids in the regions close to the |
787 |
solid surface. In our results, $\eta$ measured near a ``slip'' surface |
788 |
tends to be smaller than that near a ``stick'' surface. This may |
789 |
suggest the influence from an interface on the dynamic properties of |
790 |
liquid within its neighbor regions. It is known that diffusions of |
791 |
solid particles in liquid phase is affected by their surface |
792 |
conditions (stick or slip boundary).\cite{10.1063/1.1610442} Our |
793 |
observations could provide a support to this phenomenon. |
794 |
|
795 |
In addition to these previously studied interfaces, we attempt to |
796 |
construct ice-water interfaces and the basal plane of ice lattice was |
797 |
studied here. In contrast to the Au(111)/water interface, where the |
798 |
friction coefficient is substantially small and large slip effect |
799 |
presents, the ice/liquid water interface demonstrates strong |
800 |
solid-liquid interactions and appears to be sticky. The supercooled |
801 |
liquid phase is an order of magnitude more viscous than measurements |
802 |
in previous section. It would be of interst to investigate the effect |
803 |
of different ice lattice planes (such as prism and other surfaces) on |
804 |
interfacial friction and the corresponding liquid viscosity. |
805 |
|
806 |
\section{Conclusions} |
807 |
Our simulations demonstrate the validity of our method in RNEMD |
808 |
computations of thermal conductivity and shear viscosity in atomic and |
809 |
molecular liquids. Our method maintains thermal velocity distributions |
810 |
and avoids thermal anisotropy in previous NIVS shear stress |
811 |
simulations, as well as retains attractive features of previous RNEMD |
812 |
methods. There is no {\it a priori} restrictions to the method to be |
813 |
applied in various ensembles, so prospective applications to |
814 |
extended-system methods are possible. |
815 |
|
816 |
Our method is capable of effectively imposing thermal and/or momentum |
817 |
flux accross an interface. This facilitates studies that relates |
818 |
dynamic property measurements to the chemical details of an |
819 |
interface. Therefore, investigations can be carried out to |
820 |
characterize interfacial interactions using the method. |
821 |
|
822 |
Another attractive feature of our method is the ability of |
823 |
simultaneously introducing thermal and momentum gradients in a |
824 |
system. This facilitates us to effectively map out the shear viscosity |
825 |
with respect to a range of temperature in single trajectory of |
826 |
simulation with satisafactory accuracy. Complex systems that involve |
827 |
thermal and momentum gradients might potentially benefit from |
828 |
this. For example, the Soret effects under a velocity gradient might |
829 |
be models of interest to purification and separation researches. |
830 |
|
831 |
\section{Acknowledgments} |
832 |
Support for this project was provided by the National Science |
833 |
Foundation under grant CHE-0848243. Computational time was provided by |
834 |
the Center for Research Computing (CRC) at the University of Notre |
835 |
Dame. |
836 |
|
837 |
\newpage |
838 |
|
839 |
\bibliography{stokes} |
840 |
|
841 |
\end{doublespace} |
842 |
\end{document} |