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\begin{document} |
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\title{A gentler approach to RNEMD: Non-isotropic Velocity Scaling for computing Thermal Conductivity and Shear Viscosity} |
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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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\end{abstract} |
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\newpage |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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% BODY OF TEXT |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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\section{Introduction} |
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The original formulation of Reverse Non-equilibrium Molecular Dynamics |
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(RNEMD) obtains transport coefficients (thermal conductivity and shear |
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viscosity) in a fluid by imposing an artificial momentum flux between |
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two thin parallel slabs of material that are spatially separated in |
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the simulation cell.\cite{MullerPlathe:1997xw,ISI:000080382700030} The |
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artificial flux is typically created by periodically ``swapping'' either |
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the entire momentum vector $\vec{p}$ or single components of this |
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vector ($p_x$) between molecules in each of the two slabs. If the two |
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slabs are separated along the $z$ coordinate, the imposed flux is either |
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directional ($j_z(p_x)$) or isotropic ($J_z$), and the response of a |
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simulated system to the imposed momentum flux will typically be a |
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velocity or thermal gradient (Fig. \ref{thermalDemo}). The transport |
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coefficients (shear viscosity and thermal conductivity) are easily |
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obtained by assuming linear response of the system, |
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\begin{eqnarray} |
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j_z(p_x) & = & -\eta \frac{\partial v_x}{\partial z}\\ |
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J_z & = & \lambda \frac{\partial T}{\partial z} |
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\end{eqnarray} |
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RNEMD has been widely used to provide computational estimates of thermal |
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conductivities and shear viscosities in a wide range of materials, |
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from liquid copper to monatomic liquids to molecular fluids |
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(e.g. ionic liquids).\cite{ISI:000246190100032} |
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|
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\begin{figure} |
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\includegraphics[width=\linewidth]{thermalDemo} |
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\caption{Demostration of thermal gradient estalished by RNEMD |
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method. Physical thermal flow directs from high temperature region |
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to low temperature region. Unphysical thermal transfer counteracts |
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it and maintains a steady thermal gradient.} |
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\label{thermalDemo} |
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\end{figure} |
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|
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RNEMD is preferable in many ways to the forward NEMD methods because |
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it imposes what is typically difficult to measure (a flux or stress) |
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and it is typically much easier to compute momentum gradients or |
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strains (the response). For similar reasons, RNEMD is also preferable |
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to slowly-converging equilibrium methods for measuring thermal |
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conductivity and shear viscosity (using Green-Kubo relations or the |
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Helfand moment approach of Viscardy {\it et |
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al.}\cite{Viscardy:2007bh,Viscardy:2007lq}) because these rely on |
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computing difficult to measure quantities. |
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Another attractive feature of RNEMD is that it conserves both total |
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linear momentum and total energy during the swaps (as long as the two |
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molecules have the same identity), so the swapped configurations are |
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typically samples from the same manifold of states in the |
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microcanonical ensemble. |
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Recently, Tenney and Maginn\cite{ISI:000273472300004} have discovered |
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some problems with the original RNEMD swap technique. Notably, large |
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momentum fluxes (equivalent to frequent momentum swaps between the |
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slabs) can result in ``notched'', ``peaked'' and generally non-thermal |
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momentum distributions in the two slabs, as well as non-linear thermal |
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and velocity distributions along the direction of the imposed flux |
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($z$). Tenney and Maginn obtained reasonable limits on imposed flux |
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and self-adjusting metrics for retaining the usability of the method. |
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|
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In this paper, we develop and test a method for non-isotropic velocity |
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scaling (NIVS-RNEMD) which retains the desirable features of RNEMD |
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(conservation of linear momentum and total energy, compatibility with |
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periodic boundary conditions) while establishing true thermal |
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distributions in each of the two slabs. In the next section, we |
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develop the method for determining the scaling constraints. We then |
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test the method on both single component, multi-component, and |
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non-isotropic mixtures and show that it is capable of providing |
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reasonable estimates of the thermal conductivity and shear viscosity |
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in these cases. |
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\section{Methodology} |
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We retain the basic idea of Muller-Plathe's RNEMD method; the periodic |
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system is partitioned into a series of thin slabs along a particular |
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axis ($z$). One of the slabs at the end of the periodic box is |
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designated the ``hot'' slab, while the slab in the center of the box |
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is designated the ``cold'' slab. The artificial momentum flux will be |
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established by transferring momentum from the cold slab and into the |
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hot slab. |
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Rather than using momentum swaps, we use a series of velocity scaling |
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moves. For molecules $\{i\}$ located within the cold slab, |
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\begin{equation} |
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\vec{v}_i \leftarrow \left( \begin{array}{ccc} |
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x & 0 & 0 \\ |
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0 & y & 0 \\ |
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0 & 0 & z \\ |
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\end{array} \right) \cdot \vec{v}_i |
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\end{equation} |
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where ${x, y, z}$ are a set of 3 scaling variables for each of the |
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three directions in the system. Likewise, the molecules $\{j\}$ |
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located in the hot slab will see a concomitant scaling of velocities, |
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\begin{equation} |
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\vec{v}_j \leftarrow \left( \begin{array}{ccc} |
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x^\prime & 0 & 0 \\ |
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0 & y^\prime & 0 \\ |
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0 & 0 & z^\prime \\ |
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\end{array} \right) \cdot \vec{v}_j |
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\end{equation} |
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Conservation of linear momentum in each of the three directions |
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($\alpha = x,y,z$) ties the values of the hot and cold bin scaling |
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parameters together: |
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\begin{equation} |
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P_h^\alpha + P_c^\alpha = \alpha^\prime P_h^\alpha + \alpha P_c^\alpha |
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\end{equation} |
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where |
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\begin{eqnarray} |
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P_c^\alpha & = & \sum_{i = 1}^{N_c} m_i \left[\vec{v}_i\right]_\alpha \\ |
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P_h^\alpha & = & \sum_{j = 1}^{N_h} m_j \left[\vec{v}_j\right]_\alpha |
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\label{eq:momentumdef} |
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\end{eqnarray} |
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Therefore, for each of the three directions, the hot scaling |
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parameters are a simple function of the cold scaling parameters and |
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the instantaneous linear momentum in each of the two slabs. |
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\begin{equation} |
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\alpha^\prime = 1 + (1 - \alpha) p_\alpha |
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\label{eq:hotcoldscaling} |
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\end{equation} |
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where |
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\begin{equation} |
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p_\alpha = \frac{P_c^\alpha}{P_h^\alpha} |
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\end{equation} |
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for convenience. |
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Conservation of total energy also places constraints on the scaling: |
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\begin{equation} |
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\sum_{\alpha = x,y,z} K_h^\alpha + K_c^\alpha = \sum_{\alpha = x,y,z} |
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\left(\alpha^\prime\right)^2 K_h^\alpha + \alpha^2 K_c^\alpha |
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\end{equation} |
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where the translational kinetic energies, $K_h^\alpha$ and |
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$K_c^\alpha$, are computed for each of the three directions in a |
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similar manner to the linear momenta (Eq. \ref{eq:momentumdef}). |
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Substituting in the expressions for the hot scaling parameters |
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($\alpha^\prime$) from Eq. (\ref{eq:hotcoldscaling}), we obtain the |
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{\it constraint ellipsoid equation}: |
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\begin{equation} |
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\sum_{\alpha = x,y,z} a_\alpha \alpha^2 + b_\alpha \alpha + c_\alpha = 0 |
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\label{eq:constraintEllipsoid} |
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\end{equation} |
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where the constants are obtained from the instantaneous values of the |
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linear momenta and kinetic energies for the hot and cold slabs, |
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\begin{eqnarray} |
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a_\alpha & = &\left(K_c^\alpha + K_h^\alpha |
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\left(p_\alpha\right)^2\right) \\ |
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b_\alpha & = & -2 K_h^\alpha p_\alpha \left( 1 + p_\alpha\right) \\ |
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c_\alpha & = & K_h^\alpha p_\alpha^2 + 2 K_h^\alpha p_\alpha - K_c^\alpha |
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\label{eq:constraintEllipsoidConsts} |
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\end{eqnarray} |
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This ellipsoid equation defines the set of cold slab scaling |
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parameters which can be applied while preserving both linear momentum |
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in all three directions as well as kinetic energy. |
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The goal of using velocity scaling variables is to transfer linear |
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momentum or kinetic energy from the cold slab to the hot slab. If the |
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hot and cold slabs are separated along the z-axis, the energy flux is |
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given simply by the decrease in kinetic energy of the cold bin: |
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\begin{equation} |
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(1-x^2) K_c^x + (1-y^2) K_c^y + (1-z^2) K_c^z = J_z\Delta t |
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\end{equation} |
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The expression for the energy flux can be re-written as another |
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ellipsoid centered on $(x,y,z) = 0$: |
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\begin{equation} |
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x^2 K_c^x + y^2 K_c^y + z^2 K_c^z = K_c^x + K_c^y + K_c^z - J_z\Delta t |
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\label{eq:fluxEllipsoid} |
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\end{equation} |
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The spatial extent of the {\it thermal flux ellipsoid equation} is |
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governed both by a targetted value, $J_z$ as well as the instantaneous |
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values of the kinetic energy components in the cold bin. |
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To satisfy an energetic flux as well as the conservation constraints, |
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it is sufficient to determine the points ${x,y,z}$ which lie on both |
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the constraint ellipsoid (Eq. \ref{eq:constraintEllipsoid}) and the |
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flux ellipsoid (Eq. \ref{eq:fluxEllipsoid}), i.e. the intersection of |
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the two ellipsoids in 3-dimensional space. |
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\begin{figure} |
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\includegraphics[width=\linewidth]{ellipsoids} |
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\caption{Scaling points which maintain both constant energy and |
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constant linear momentum of the system lie on the surface of the |
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{\it constraint ellipsoid} while points which generate the target |
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momentum flux lie on the surface of the {\it flux ellipsoid}. The |
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velocity distributions in the cold bin are scaled by only those |
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points which lie on both ellipsoids.} |
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\label{ellipsoids} |
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\end{figure} |
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One may also define momentum flux (say along the $x$-direction) as: |
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\begin{equation} |
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(1-x) P_c^x = j_z(p_x)\Delta t |
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\label{eq:fluxPlane} |
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\end{equation} |
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The above {\it momentum flux equation} is essentially a plane which is |
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perpendicular to the $x$-axis, with its position governed both by a |
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target value, $j_z(p_x)$ as well as the instantaneous value of the |
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momentum along the $x$-direction. |
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Similarly, to satisfy a momentum flux as well as the conservation |
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constraints, it is sufficient to determine the points ${x,y,z}$ which |
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lie on both the constraint ellipsoid (Eq. \ref{eq:constraintEllipsoid}) |
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and the flux plane (Eq. \ref{eq:fluxPlane}), i.e. the intersection of |
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an ellipsoid and a plane in 3-dimensional space. |
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To summarize, by solving respective equation sets, one can determine |
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possible sets of scaling variables for cold slab. And corresponding |
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sets of scaling variables for hot slab can be determine as well. |
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The following problem will be choosing an optimal set of scaling |
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variables among the possible sets. Although this method is inherently |
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non-isotropic, the goal is still to maintain the system as isotropic |
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as possible. Under this consideration, one would like the kinetic |
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energies in different directions could become as close as each other |
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after each scaling. Simultaneously, one would also like each scaling |
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as gentle as possible, i.e. ${x,y,z \rightarrow 1}$, in order to avoid |
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large perturbation to the system. Therefore, one approach to obtain the |
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scaling variables would be constructing an criteria function, with |
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constraints as above equation sets, and solving the function's minimum |
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by method like Lagrange multipliers. |
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|
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In order to save computation time, we have a different approach to a |
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relatively good set of scaling variables with much less calculation |
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than above. Here is the detail of our simplification of the problem. |
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In the case of kinetic energy transfer, we impose another constraint |
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${x = y}$, into the equation sets. Consequently, there are two |
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variables left. And now one only needs to solve a set of two {\it |
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ellipses equations}. This problem would be transformed into solving |
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one quartic equation for one of the two variables. There are known |
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generic methods that solve real roots of quartic equations. Then one |
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can determine the other variable and obtain sets of scaling |
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variables. Among these sets, one can apply the above criteria to |
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choose the best set, while much faster with only a few sets to choose. |
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In the case of momentum flux transfer, we impose another constraint to |
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set the kinetic energy transfer as zero. In another word, we apply |
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Eq. \ref{eq:fluxEllipsoid} and let ${J_z = 0}$. After that, with one |
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variable fixed by Eq. \ref{eq:fluxPlane}, one now have a similar set |
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of equations on the above kinetic energy transfer problem. Therefore, |
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an approach similar to the above would be sufficient for this as well. |
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\section{Computational Details} |
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\subsection{Lennard-Jones Fluid} |
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Our simulation consists of a series of systems. All of these |
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simulations were run with the OpenMD simulation software |
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package\cite{Meineke:2005gd} integrated with RNEMD codes. |
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A Lennard-Jones fluid system was built and tested first. In order to |
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compare our method with swapping RNEMD, a series of simulations were |
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performed to calculate the shear viscosity and thermal conductivity of |
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argon. 2592 atoms were in a orthorhombic cell, which was ${10.06\sigma |
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\times 10.06\sigma \times 30.18\sigma}$ by size. The reduced density |
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${\rho^* = \rho\sigma^3}$ was thus 0.85, which enabled direct |
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comparison between our results and others. These simulations used |
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velocity Verlet algorithm with reduced timestep ${\tau^* = |
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4.6\times10^{-4}}$. |
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|
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For shear viscosity calculation, the reduced temperature was ${T^* = |
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k_B T/\varepsilon = 0.72}$. Simulations were first equilibrated in canonical |
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ensemble (NVT), then equilibrated in microcanonical ensemble |
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(NVE). Establishing and stablizing momentum gradient were followed |
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also in NVE ensemble. For the swapping part, Muller-Plathe's algorithm was |
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adopted.\cite{ISI:000080382700030} The simulation box was under |
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periodic boundary condition, and devided into ${N = 20}$ slabs. In each swap, |
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the top slab ${(n = 1)}$ exchange the most negative $x$ momentum with the |
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|
|
most positive $x$ momentum in the center slab ${(n = N/2 + 1)}$. Referred |
317 |
skuang |
3565 |
to Tenney {\it et al.}\cite{ISI:000273472300004}, a series of swapping |
318 |
skuang |
3534 |
frequency were chosen. According to each result from swapping |
319 |
skuang |
3532 |
RNEMD, scaling RNEMD simulations were run with the target momentum |
320 |
skuang |
3576 |
flux set to produce a similar momentum flux, and consequently shear |
321 |
skuang |
3534 |
rate. Furthermore, various scaling frequencies can be tested for one |
322 |
skuang |
3576 |
single swapping rate. To test the temperature homogeneity in our |
323 |
|
|
system of swapping and scaling methods, temperatures of different |
324 |
|
|
dimensions in all the slabs were observed. Most of the simulations |
325 |
|
|
include $10^5$ steps of equilibration without imposing momentum flux, |
326 |
|
|
$10^5$ steps of stablization with imposing unphysical momentum |
327 |
|
|
transfer, and $10^6$ steps of data collection under RNEMD. For |
328 |
|
|
relatively high momentum flux simulations, ${5\times10^5}$ step data |
329 |
|
|
collection is sufficient. For some low momentum flux simulations, |
330 |
|
|
${2\times10^6}$ steps were necessary. |
331 |
skuang |
3532 |
|
332 |
skuang |
3534 |
After each simulation, the shear viscosity was calculated in reduced |
333 |
|
|
unit. The momentum flux was calculated with total unphysical |
334 |
skuang |
3565 |
transferred momentum ${P_x}$ and data collection time $t$: |
335 |
skuang |
3534 |
\begin{equation} |
336 |
|
|
j_z(p_x) = \frac{P_x}{2 t L_x L_y} |
337 |
|
|
\end{equation} |
338 |
skuang |
3576 |
where $L_x$ and $L_y$ denotes $x$ and $y$ lengths of the simulation |
339 |
|
|
box, and physical momentum transfer occurs in two ways due to our |
340 |
|
|
periodic boundary condition settings. And the velocity gradient |
341 |
|
|
${\langle \partial v_x /\partial z \rangle}$ can be obtained by a |
342 |
|
|
linear regression of the velocity profile. From the shear viscosity |
343 |
|
|
$\eta$ calculated with the above parameters, one can further convert |
344 |
|
|
it into reduced unit ${\eta^* = \eta \sigma^2 (\varepsilon m)^{-1/2}}$. |
345 |
skuang |
3532 |
|
346 |
skuang |
3576 |
For thermal conductivity calculations, simulations were first run under |
347 |
|
|
reduced temperature ${\langle T^*\rangle = 0.72}$ in NVE |
348 |
|
|
ensemble. Muller-Plathe's algorithm was adopted in the swapping |
349 |
|
|
method. Under identical simulation box parameters with our shear |
350 |
|
|
viscosity calculations, in each swap, the top slab exchanges all three |
351 |
|
|
translational momentum components of the molecule with least kinetic |
352 |
|
|
energy with the same components of the molecule in the center slab |
353 |
|
|
with most kinetic energy, unless this ``coldest'' molecule in the |
354 |
|
|
``hot'' slab is still ``hotter'' than the ``hottest'' molecule in the |
355 |
|
|
``cold'' slab. According to swapping RNEMD results, target energy flux |
356 |
|
|
for scaling RNEMD simulations can be set. Also, various scaling |
357 |
skuang |
3534 |
frequencies can be tested for one target energy flux. To compare the |
358 |
|
|
performance between swapping and scaling method, distributions of |
359 |
|
|
velocity and speed in different slabs were observed. |
360 |
|
|
|
361 |
|
|
For each swapping rate, thermal conductivity was calculated in reduced |
362 |
|
|
unit. The energy flux was calculated similarly to the momentum flux, |
363 |
skuang |
3565 |
with total unphysical transferred energy ${E_{total}}$ and data collection |
364 |
skuang |
3534 |
time $t$: |
365 |
|
|
\begin{equation} |
366 |
|
|
J_z = \frac{E_{total}}{2 t L_x L_y} |
367 |
|
|
\end{equation} |
368 |
|
|
And the temperature gradient ${\langle\partial T/\partial z\rangle}$ |
369 |
|
|
can be obtained by a linear regression of the temperature |
370 |
|
|
profile. From the thermal conductivity $\lambda$ calculated, one can |
371 |
|
|
further convert it into reduced unit ${\lambda^*=\lambda \sigma^2 |
372 |
|
|
m^{1/2} k_B^{-1}\varepsilon^{-1/2}}$. |
373 |
|
|
|
374 |
skuang |
3576 |
\subsection{ Water / Metal Thermal Conductivity} |
375 |
|
|
Another series of our simulation is the calculation of interfacial |
376 |
skuang |
3573 |
thermal conductivity of a Au/H$_2$O system. Respective calculations of |
377 |
skuang |
3576 |
liquid water (SPC/E) and crystal gold (QSC) thermal conductivity were |
378 |
|
|
performed and compared with current results to ensure the validity of |
379 |
skuang |
3573 |
NIVS-RNEMD. After that, a mixture system was simulated. |
380 |
skuang |
3563 |
|
381 |
skuang |
3573 |
For thermal conductivity calculation of bulk water, a simulation box |
382 |
|
|
consisting of 1000 molecules were first equilibrated under ambient |
383 |
skuang |
3576 |
pressure and temperature conditions using NPT ensemble, followed by |
384 |
|
|
equilibration in fixed volume (NVT). The system was then equilibrated in |
385 |
|
|
microcanonical ensemble (NVE). Also in NVE ensemble, establishing a |
386 |
skuang |
3573 |
stable thermal gradient was followed. The simulation box was under |
387 |
|
|
periodic boundary condition and devided into 10 slabs. Data collection |
388 |
skuang |
3576 |
process was similar to Lennard-Jones fluid system. |
389 |
skuang |
3573 |
|
390 |
skuang |
3576 |
Thermal conductivity calculation of bulk crystal gold used a similar |
391 |
|
|
protocol. The face centered cubic crystal simulation box consists of |
392 |
|
|
2880 Au atoms. The lattice was first allowed volume change to relax |
393 |
|
|
under ambient temperature and pressure. Equilibrations in canonical and |
394 |
|
|
microcanonical ensemble were followed in order. With the simulation |
395 |
|
|
lattice devided evenly into 10 slabs, different thermal gradients were |
396 |
|
|
established by applying a set of target thermal transfer flux. Data of |
397 |
|
|
the series of thermal gradients was collected for calculation. |
398 |
|
|
|
399 |
skuang |
3573 |
After simulations of bulk water and crystal gold, a mixture system was |
400 |
|
|
constructed, consisting of 1188 Au atoms and 1862 H$_2$O |
401 |
|
|
molecules. Spohr potential was adopted in depicting the interaction |
402 |
|
|
between metal atom and water molecule.\cite{ISI:000167766600035} A |
403 |
skuang |
3576 |
similar protocol of equilibration was followed. Several thermal |
404 |
|
|
gradients was built under different target thermal flux. It was found |
405 |
|
|
out that compared to our previous simulation systems, the two phases |
406 |
|
|
could have large temperature difference even under a relatively low |
407 |
|
|
thermal flux. Therefore, under our low flux conditions, it is assumed |
408 |
skuang |
3573 |
that the metal and water phases have respectively homogeneous |
409 |
skuang |
3576 |
temperature, excluding the surface regions. In calculating the |
410 |
|
|
interfacial thermal conductivity $G$, this assumptioin was applied and |
411 |
|
|
thus our formula becomes: |
412 |
skuang |
3573 |
|
413 |
|
|
\begin{equation} |
414 |
|
|
G = \frac{E_{total}}{2 t L_x L_y \left( \langle T_{gold}\rangle - |
415 |
|
|
\langle T_{water}\rangle \right)} |
416 |
|
|
\label{interfaceCalc} |
417 |
|
|
\end{equation} |
418 |
|
|
where ${E_{total}}$ is the imposed unphysical kinetic energy transfer |
419 |
|
|
and ${\langle T_{gold}\rangle}$ and ${\langle T_{water}\rangle}$ are the |
420 |
|
|
average observed temperature of gold and water phases respectively. |
421 |
|
|
|
422 |
skuang |
3577 |
\section{Results And Discussions} |
423 |
skuang |
3538 |
\subsection{Thermal Conductivity} |
424 |
skuang |
3573 |
\subsubsection{Lennard-Jones Fluid} |
425 |
skuang |
3577 |
Our thermal conductivity calculations show that scaling method results |
426 |
|
|
agree with swapping method. Four different exchange intervals were |
427 |
|
|
tested (Table \ref{thermalLJRes}) using swapping method. With a fixed |
428 |
|
|
10fs exchange interval, target exchange kinetic energy was set to |
429 |
|
|
produce equivalent kinetic energy flux as in swapping method. And |
430 |
|
|
similar thermal gradients were observed with similar thermal flux in |
431 |
|
|
two simulation methods (Figure \ref{thermalGrad}). |
432 |
skuang |
3538 |
|
433 |
skuang |
3563 |
\begin{table*} |
434 |
|
|
\begin{minipage}{\linewidth} |
435 |
|
|
\begin{center} |
436 |
skuang |
3538 |
|
437 |
skuang |
3563 |
\caption{Calculation results for thermal conductivity of Lennard-Jones |
438 |
skuang |
3565 |
fluid at ${\langle T^* \rangle = 0.72}$ and ${\rho^* = 0.85}$, with |
439 |
skuang |
3563 |
swap and scale methods at various kinetic energy exchange rates. Results |
440 |
|
|
in reduced unit. Errors of calculations in parentheses.} |
441 |
|
|
|
442 |
skuang |
3565 |
\begin{tabular}{ccc} |
443 |
skuang |
3563 |
\hline |
444 |
skuang |
3577 |
(Equilvalent) Exchange Interval (fs) & $\lambda^*_{swap}$ & |
445 |
|
|
$\lambda^*_{scale}$\\ |
446 |
skuang |
3565 |
\hline |
447 |
skuang |
3577 |
250 & 7.03(0.34) & 7.30(0.10)\\ |
448 |
|
|
500 & 7.03(0.14) & 6.95(0.09)\\ |
449 |
|
|
1000 & 6.91(0.42) & 7.19(0.07)\\ |
450 |
|
|
2000 & 7.52(0.15) & 7.19(0.28)\\ |
451 |
skuang |
3566 |
\hline |
452 |
skuang |
3563 |
\end{tabular} |
453 |
skuang |
3577 |
\label{thermalLJRes} |
454 |
skuang |
3563 |
\end{center} |
455 |
|
|
\end{minipage} |
456 |
|
|
\end{table*} |
457 |
|
|
|
458 |
|
|
\begin{figure} |
459 |
skuang |
3567 |
\includegraphics[width=\linewidth]{thermalGrad} |
460 |
skuang |
3577 |
\caption{Temperature gradients under various kinetic energy flux of |
461 |
|
|
thermal conductivity simulations} |
462 |
skuang |
3567 |
\label{thermalGrad} |
463 |
skuang |
3563 |
\end{figure} |
464 |
|
|
|
465 |
|
|
During these simulations, molecule velocities were recorded in 1000 of |
466 |
|
|
all the snapshots. These velocity data were used to produce histograms |
467 |
|
|
of velocity and speed distribution in different slabs. From these |
468 |
|
|
histograms, it is observed that with increasing unphysical kinetic |
469 |
|
|
energy flux, speed and velocity distribution of molecules in slabs |
470 |
|
|
where swapping occured could deviate from Maxwell-Boltzmann |
471 |
|
|
distribution. Figure \ref{histSwap} indicates how these distributions |
472 |
|
|
deviate from ideal condition. In high temperature slabs, probability |
473 |
|
|
density in low speed is confidently smaller than ideal distribution; |
474 |
|
|
in low temperature slabs, probability density in high speed is smaller |
475 |
|
|
than ideal. This phenomenon is observable even in our relatively low |
476 |
skuang |
3568 |
swapping rate simulations. And this deviation could also leads to |
477 |
skuang |
3563 |
deviation of distribution of velocity in various dimensions. One |
478 |
|
|
feature of these deviated distribution is that in high temperature |
479 |
|
|
slab, the ideal Gaussian peak was changed into a relatively flat |
480 |
|
|
plateau; while in low temperature slab, that peak appears sharper. |
481 |
|
|
|
482 |
|
|
\begin{figure} |
483 |
skuang |
3565 |
\includegraphics[width=\linewidth]{histSwap} |
484 |
skuang |
3563 |
\caption{Speed distribution for thermal conductivity using swapping RNEMD.} |
485 |
|
|
\label{histSwap} |
486 |
|
|
\end{figure} |
487 |
|
|
|
488 |
skuang |
3568 |
\begin{figure} |
489 |
|
|
\includegraphics[width=\linewidth]{histScale} |
490 |
|
|
\caption{Speed distribution for thermal conductivity using scaling RNEMD.} |
491 |
|
|
\label{histScale} |
492 |
|
|
\end{figure} |
493 |
|
|
|
494 |
skuang |
3573 |
\subsubsection{SPC/E Water} |
495 |
|
|
Our results of SPC/E water thermal conductivity are comparable to |
496 |
|
|
Bedrov {\it et al.}\cite{ISI:000090151400044}, which employed the |
497 |
|
|
previous swapping RNEMD method for their calculation. Our simulations |
498 |
|
|
were able to produce a similar temperature gradient to their |
499 |
|
|
system. However, the average temperature of our system is 300K, while |
500 |
|
|
theirs is 318K, which would be attributed for part of the difference |
501 |
|
|
between the two series of results. Both methods yields values in |
502 |
|
|
agreement with experiment. And this shows the applicability of our |
503 |
|
|
method to multi-atom molecular system. |
504 |
skuang |
3563 |
|
505 |
skuang |
3570 |
\begin{figure} |
506 |
|
|
\includegraphics[width=\linewidth]{spceGrad} |
507 |
|
|
\caption{Temperature gradients for SPC/E water thermal conductivity.} |
508 |
|
|
\label{spceGrad} |
509 |
|
|
\end{figure} |
510 |
|
|
|
511 |
|
|
\begin{table*} |
512 |
|
|
\begin{minipage}{\linewidth} |
513 |
|
|
\begin{center} |
514 |
|
|
|
515 |
|
|
\caption{Calculation results for thermal conductivity of SPC/E water |
516 |
|
|
at ${\langle T\rangle}$ = 300K at various thermal exchange rates. Errors of |
517 |
|
|
calculations in parentheses. } |
518 |
|
|
|
519 |
|
|
\begin{tabular}{cccc} |
520 |
|
|
\hline |
521 |
|
|
$\langle dT/dz\rangle$(K/\AA) & & $\lambda$(W/m/K) & \\ |
522 |
skuang |
3573 |
& This work & Previous simulations\cite{ISI:000090151400044} & |
523 |
|
|
Experiment$^a$\\ |
524 |
skuang |
3570 |
\hline |
525 |
skuang |
3573 |
0.38 & 0.816(0.044) & & 0.64\\ |
526 |
|
|
0.81 & 0.770(0.008) & 0.784\\ |
527 |
|
|
1.54 & 0.813(0.007) & 0.730\\ |
528 |
skuang |
3570 |
\hline |
529 |
|
|
\end{tabular} |
530 |
|
|
\label{spceThermal} |
531 |
|
|
\end{center} |
532 |
|
|
\end{minipage} |
533 |
|
|
\end{table*} |
534 |
|
|
|
535 |
skuang |
3573 |
\subsubsection{Crystal Gold} |
536 |
skuang |
3574 |
Our results of gold thermal conductivity used QSC force field are |
537 |
|
|
shown in Table \ref{AuThermal}. Although our calculation is smaller |
538 |
|
|
than experimental value by an order of more than 100, this difference |
539 |
|
|
is mainly attributed to the lack of electron interaction |
540 |
|
|
representation in our force field parameters. Richardson {\it et |
541 |
|
|
al.}\cite{ISI:A1992HX37800010} used similar force field parameters |
542 |
|
|
in their metal thermal conductivity calculations. The EMD method they |
543 |
|
|
employed in their simulations produced comparable results to |
544 |
|
|
ours. Therefore, it is confident to conclude that NIVS-RNEMD is |
545 |
|
|
applicable to metal force field system. |
546 |
skuang |
3570 |
|
547 |
|
|
\begin{figure} |
548 |
|
|
\includegraphics[width=\linewidth]{AuGrad} |
549 |
|
|
\caption{Temperature gradients for crystal gold thermal conductivity.} |
550 |
|
|
\label{AuGrad} |
551 |
|
|
\end{figure} |
552 |
|
|
|
553 |
|
|
\begin{table*} |
554 |
|
|
\begin{minipage}{\linewidth} |
555 |
|
|
\begin{center} |
556 |
|
|
|
557 |
|
|
\caption{Calculation results for thermal conductivity of crystal gold |
558 |
|
|
at ${\langle T\rangle}$ = 300K at various thermal exchange rates. Errors of |
559 |
|
|
calculations in parentheses. } |
560 |
|
|
|
561 |
|
|
\begin{tabular}{ccc} |
562 |
|
|
\hline |
563 |
|
|
$\langle dT/dz\rangle$(K/\AA) & $\lambda$(W/m/K)\\ |
564 |
skuang |
3573 |
& This work & Previous simulations\cite{ISI:A1992HX37800010} \\ |
565 |
skuang |
3570 |
\hline |
566 |
skuang |
3571 |
1.44 & 1.10(0.01) & \\ |
567 |
|
|
2.86 & 1.08(0.02) & \\ |
568 |
|
|
5.14 & 1.15(0.01) & \\ |
569 |
skuang |
3570 |
\hline |
570 |
|
|
\end{tabular} |
571 |
|
|
\label{AuThermal} |
572 |
|
|
\end{center} |
573 |
|
|
\end{minipage} |
574 |
|
|
\end{table*} |
575 |
|
|
|
576 |
skuang |
3573 |
\subsection{Interfaciel Thermal Conductivity} |
577 |
skuang |
3574 |
After valid simulations of homogeneous water and gold systems using |
578 |
|
|
NIVS-RNEMD method, calculation of gold/water interfacial thermal |
579 |
|
|
conductivity was followed. It is found out that the interfacial |
580 |
|
|
conductance is low due to a hydrophobic surface in our system. Figure |
581 |
|
|
\ref{interfaceDensity} demonstrates this observance. Consequently, our |
582 |
|
|
reported results (Table \ref{interfaceRes}) are of two orders of |
583 |
|
|
magnitude smaller than our calculations on homogeneous systems. |
584 |
skuang |
3573 |
|
585 |
skuang |
3571 |
\begin{figure} |
586 |
|
|
\includegraphics[width=\linewidth]{interfaceDensity} |
587 |
|
|
\caption{Density profile for interfacial thermal conductivity |
588 |
|
|
simulation box.} |
589 |
|
|
\label{interfaceDensity} |
590 |
|
|
\end{figure} |
591 |
|
|
|
592 |
skuang |
3572 |
\begin{figure} |
593 |
|
|
\includegraphics[width=\linewidth]{interfaceGrad} |
594 |
|
|
\caption{Temperature profiles for interfacial thermal conductivity |
595 |
|
|
simulation box.} |
596 |
|
|
\label{interfaceGrad} |
597 |
|
|
\end{figure} |
598 |
|
|
|
599 |
|
|
\begin{table*} |
600 |
|
|
\begin{minipage}{\linewidth} |
601 |
|
|
\begin{center} |
602 |
|
|
|
603 |
|
|
\caption{Calculation results for interfacial thermal conductivity |
604 |
|
|
at ${\langle T\rangle \sim}$ 300K at various thermal exchange |
605 |
|
|
rates. Errors of calculations in parentheses. } |
606 |
|
|
|
607 |
|
|
\begin{tabular}{cccc} |
608 |
|
|
\hline |
609 |
|
|
$J_z$(MW/m$^2$) & $T_{gold}$ & $T_{water}$ & $G$(MW/m$^2$/K)\\ |
610 |
|
|
\hline |
611 |
skuang |
3573 |
98.0 & 355.2 & 295.8 & 1.65(0.21) \\ |
612 |
|
|
78.8 & 343.8 & 298.0 & 1.72(0.32) \\ |
613 |
|
|
73.6 & 344.3 & 298.0 & 1.59(0.24) \\ |
614 |
|
|
49.2 & 330.1 & 300.4 & 1.65(0.35) \\ |
615 |
skuang |
3572 |
\hline |
616 |
|
|
\end{tabular} |
617 |
skuang |
3574 |
\label{interfaceRes} |
618 |
skuang |
3572 |
\end{center} |
619 |
|
|
\end{minipage} |
620 |
|
|
\end{table*} |
621 |
|
|
|
622 |
skuang |
3576 |
\subsection{Shear Viscosity} |
623 |
|
|
Our calculations (Table \ref{shearRate}) shows that scale RNEMD method |
624 |
|
|
produced comparable shear viscosity to swap RNEMD method. In Table |
625 |
|
|
\ref{shearRate}, the names of the calculated samples are devided into |
626 |
|
|
two parts. The first number refers to total slabs in one simulation |
627 |
|
|
box. The second number refers to the swapping interval in swap method, or |
628 |
|
|
in scale method the equilvalent swapping interval that the same |
629 |
|
|
momentum flux would theoretically result in swap method. All the scale |
630 |
|
|
method results were from simulations that had a scaling interval of 10 |
631 |
|
|
time steps. The average molecular momentum gradients of these samples |
632 |
|
|
are shown in Figure \ref{shearGrad}. |
633 |
|
|
|
634 |
|
|
\begin{table*} |
635 |
|
|
\begin{minipage}{\linewidth} |
636 |
|
|
\begin{center} |
637 |
|
|
|
638 |
|
|
\caption{Calculation results for shear viscosity of Lennard-Jones |
639 |
|
|
fluid at ${T^* = 0.72}$ and ${\rho^* = 0.85}$, with swap and scale |
640 |
|
|
methods at various momentum exchange rates. Results in reduced |
641 |
|
|
unit. Errors of calculations in parentheses. } |
642 |
|
|
|
643 |
|
|
\begin{tabular}{ccc} |
644 |
|
|
\hline |
645 |
|
|
Series & $\eta^*_{swap}$ & $\eta^*_{scale}$\\ |
646 |
|
|
\hline |
647 |
|
|
20-500 & 3.64(0.05) & 3.76(0.09)\\ |
648 |
|
|
20-1000 & 3.52(0.16) & 3.66(0.06)\\ |
649 |
|
|
20-2000 & 3.72(0.05) & 3.32(0.18)\\ |
650 |
|
|
20-2500 & 3.42(0.06) & 3.43(0.08)\\ |
651 |
|
|
\hline |
652 |
|
|
\end{tabular} |
653 |
|
|
\label{shearRate} |
654 |
|
|
\end{center} |
655 |
|
|
\end{minipage} |
656 |
|
|
\end{table*} |
657 |
|
|
|
658 |
|
|
\begin{figure} |
659 |
|
|
\includegraphics[width=\linewidth]{shearGrad} |
660 |
|
|
\caption{Average momentum gradients of shear viscosity simulations} |
661 |
|
|
\label{shearGrad} |
662 |
|
|
\end{figure} |
663 |
|
|
|
664 |
|
|
\begin{figure} |
665 |
|
|
\includegraphics[width=\linewidth]{shearTempScale} |
666 |
|
|
\caption{Temperature profile for scaling RNEMD simulation.} |
667 |
|
|
\label{shearTempScale} |
668 |
|
|
\end{figure} |
669 |
|
|
However, observations of temperatures along three dimensions show that |
670 |
|
|
inhomogeneity occurs in scaling RNEMD simulations, particularly in the |
671 |
|
|
two slabs which were scaled. Figure \ref{shearTempScale} indicate that with |
672 |
|
|
relatively large imposed momentum flux, the temperature difference among $x$ |
673 |
|
|
and the other two dimensions was significant. This would result from the |
674 |
|
|
algorithm of scaling method. From Eq. \ref{eq:fluxPlane}, after |
675 |
|
|
momentum gradient is set up, $P_c^x$ would be roughly stable |
676 |
|
|
($<0$). Consequently, scaling factor $x$ would most probably larger |
677 |
|
|
than 1. Therefore, the kinetic energy in $x$-dimension $K_c^x$ would |
678 |
|
|
keep increase after most scaling steps. And if there is not enough time |
679 |
|
|
for the kinetic energy to exchange among different dimensions and |
680 |
|
|
different slabs, the system would finally build up temperature |
681 |
|
|
(kinetic energy) difference among the three dimensions. Also, between |
682 |
|
|
$y$ and $z$ dimensions in the scaled slabs, temperatures of $z$-axis |
683 |
|
|
are closer to neighbor slabs. This is due to momentum transfer along |
684 |
|
|
$z$ dimension between slabs. |
685 |
|
|
|
686 |
|
|
Although results between scaling and swapping methods are comparable, |
687 |
|
|
the inherent temperature inhomogeneity even in relatively low imposed |
688 |
|
|
exchange momentum flux simulations makes scaling RNEMD method less |
689 |
|
|
attractive than swapping RNEMD in shear viscosity calculation. |
690 |
|
|
|
691 |
skuang |
3574 |
\section{Conclusions} |
692 |
|
|
NIVS-RNEMD simulation method is developed and tested on various |
693 |
|
|
systems. Simulation results demonstrate its validity of thermal |
694 |
|
|
conductivity calculations. NIVS-RNEMD improves non-Boltzmann-Maxwell |
695 |
|
|
distributions existing in previous RNEMD methods, and extends its |
696 |
|
|
applicability to interfacial systems. NIVS-RNEMD has also limited |
697 |
|
|
application on shear viscosity calculations, but under high momentum |
698 |
|
|
flux, it could cause temperature difference among different |
699 |
|
|
dimensions. Modification is necessary to extend the applicability of |
700 |
|
|
NIVS-RNEMD in shear viscosity calculations. |
701 |
skuang |
3572 |
|
702 |
gezelter |
3524 |
\section{Acknowledgments} |
703 |
|
|
Support for this project was provided by the National Science |
704 |
|
|
Foundation under grant CHE-0848243. Computational time was provided by |
705 |
|
|
the Center for Research Computing (CRC) at the University of Notre |
706 |
|
|
Dame. \newpage |
707 |
|
|
|
708 |
|
|
\bibliographystyle{jcp2} |
709 |
|
|
\bibliography{nivsRnemd} |
710 |
|
|
\end{doublespace} |
711 |
|
|
\end{document} |
712 |
|
|
|