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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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|
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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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% BODY OF TEXT |
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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. The transport coefficients (shear |
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viscosity and thermal conductivity) are easily obtained by assuming |
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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 & = & \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 method.} |
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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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|
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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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|
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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 momentum |
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distributions in the two slabs, as well as non-linear thermal and |
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velocity distributions along the direction of the imposed flux ($z$). |
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Tenney and Maginn obtained reasonable limits on imposed flux and |
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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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|
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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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|
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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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|
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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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|
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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 kinetic energies, $K_h^\alpha$ and $K_c^\alpha$, are computed |
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for each of the three directions in a similar manner to the linear momenta |
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(Eq. \ref{eq:momentumdef}). Substituting in the expressions for the |
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hot scaling parameters ($\alpha^\prime$) from Eq. (\ref{eq:hotcoldscaling}), |
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we obtain the {\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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|
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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 flux ellipsoid equation} is governed |
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both by a targetted value, $J_z$ as well as the instantaneous values of the |
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kinetic energy components in the cold bin. |
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|
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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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|
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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 hot 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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|
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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 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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targetted 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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|
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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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|
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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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|
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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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|
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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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|
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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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|
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\section{Computational Details} |
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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 methods. |
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|
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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 |
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to Tenney {\it et al.}\cite{ISI:000273472300004}, a series of swapping |
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frequency were chosen. According to each result from swapping |
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RNEMD, scaling RNEMD simulations were run with the target momentum |
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flux set to produce a similar momentum flux and shear |
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rate. Furthermore, various scaling frequencies can be tested for one |
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single swapping rate. To compare the performance between swapping and |
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scaling method, temperatures of different dimensions in all the slabs |
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were observed. Most of the simulations include $10^5$ steps of |
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equilibration without imposing momentum flux, $10^5$ steps of |
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stablization with imposing momentum transfer, and $10^6$ steps of data |
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collection under RNEMD. For relatively high momentum flux simulations, |
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${5\times10^5}$ step data collection is sufficient. For some low momentum |
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flux simulations, ${2\times10^6}$ steps were necessary. |
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|
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After each simulation, the shear viscosity was calculated in reduced |
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unit. The momentum flux was calculated with total unphysical |
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transferred momentum ${P_x}$ and data collection time $t$: |
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\begin{equation} |
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j_z(p_x) = \frac{P_x}{2 t L_x L_y} |
331 |
\end{equation} |
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And the velocity gradient ${\langle \partial v_x /\partial z \rangle}$ |
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can be obtained by a linear regression of the velocity profile. From |
334 |
the shear viscosity $\eta$ calculated with the above parameters, one |
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can further convert it into reduced unit ${\eta^* = \eta \sigma^2 |
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(\varepsilon m)^{-1/2}}$. |
337 |
|
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For thermal conductivity calculation, simulations were first run under |
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reduced temperature ${T^* = 0.72}$ in NVE ensemble. Muller-Plathe's |
340 |
algorithm was adopted in the swapping method. Under identical |
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simulation box parameters, in each swap, the top slab exchange the |
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molecule with least kinetic energy with the molecule in the center |
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slab with most kinetic energy, unless this ``coldest'' molecule in the |
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``hot'' slab is still ``hotter'' than the ``hottest'' molecule in the ``cold'' |
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slab. According to swapping RNEMD results, target energy flux for |
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scaling RNEMD simulations can be set. Also, various scaling |
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frequencies can be tested for one target energy flux. To compare the |
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performance between swapping and scaling method, distributions of |
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velocity and speed in different slabs were observed. |
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|
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For each swapping rate, thermal conductivity was calculated in reduced |
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unit. The energy flux was calculated similarly to the momentum flux, |
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with total unphysical transferred energy ${E_{total}}$ and data collection |
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time $t$: |
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\begin{equation} |
356 |
J_z = \frac{E_{total}}{2 t L_x L_y} |
357 |
\end{equation} |
358 |
And the temperature gradient ${\langle\partial T/\partial z\rangle}$ |
359 |
can be obtained by a linear regression of the temperature |
360 |
profile. From the thermal conductivity $\lambda$ calculated, one can |
361 |
further convert it into reduced unit ${\lambda^*=\lambda \sigma^2 |
362 |
m^{1/2} k_B^{-1}\varepsilon^{-1/2}}$. |
363 |
|
364 |
Another series of our simulation is to calculate the interfacial |
365 |
thermal conductivity of a Au/H$_2$O system. Respective calculations of |
366 |
water (SPC/E) and gold (QSC) thermal conductivity were performed and |
367 |
compared with current results to ensure the validity of |
368 |
NIVS-RNEMD. After that, a mixture system was simulated. |
369 |
|
370 |
For thermal conductivity calculation of bulk water, a simulation box |
371 |
consisting of 1000 molecules were first equilibrated under ambient |
372 |
pressure and temperature conditions (NPT), followed by equilibration |
373 |
in fixed volume (NVT). The system was then equilibrated in |
374 |
microcanonical ensemble (NVE). Also in NVE ensemble, establishing |
375 |
stable thermal gradient was followed. The simulation box was under |
376 |
periodic boundary condition and devided into 10 slabs. Data collection |
377 |
process was similar to Lennard-Jones fluid system. Thermal |
378 |
conductivity calculation of bulk crystal gold used a similar |
379 |
protocol. And the face centered cubic crystal simulation box consists |
380 |
of 2880 Au atoms. |
381 |
|
382 |
After simulations of bulk water and crystal gold, a mixture system was |
383 |
constructed, consisting of 1188 Au atoms and 1862 H$_2$O |
384 |
molecules. Spohr potential was adopted in depicting the interaction |
385 |
between metal atom and water molecule.\cite{ISI:000167766600035} A |
386 |
similar protocol of equilibration was followed. A thermal gradient was |
387 |
built. It was found out that compared to homogeneous systems, the two |
388 |
phases could have large temperature difference under a relatively low |
389 |
thermal flux. Therefore, under our low flux condition, it is assumed |
390 |
that the metal and water phases have respectively homogeneous |
391 |
temperature. In calculating the interfacial thermal conductivity $G$, |
392 |
this assumptioin was applied and thus our formula becomes: |
393 |
|
394 |
\begin{equation} |
395 |
G = \frac{E_{total}}{2 t L_x L_y \left( \langle T_{gold}\rangle - |
396 |
\langle T_{water}\rangle \right)} |
397 |
\label{interfaceCalc} |
398 |
\end{equation} |
399 |
where ${E_{total}}$ is the imposed unphysical kinetic energy transfer |
400 |
and ${\langle T_{gold}\rangle}$ and ${\langle T_{water}\rangle}$ are the |
401 |
average observed temperature of gold and water phases respectively. |
402 |
|
403 |
\section{Results And Discussion} |
404 |
\subsection{Shear Viscosity} |
405 |
Our calculations (Table \ref{shearRate}) shows that scale RNEMD method |
406 |
produced comparable shear viscosity to swap RNEMD method. In Table |
407 |
\ref{shearRate}, the names of the calculated samples are devided into |
408 |
two parts. The first number refers to total slabs in one simulation |
409 |
box. The second number refers to the swapping interval in swap method, or |
410 |
in scale method the equilvalent swapping interval that the same |
411 |
momentum flux would theoretically result in swap method. All the scale |
412 |
method results were from simulations that had a scaling interval of 10 |
413 |
time steps. The average molecular momentum gradients of these samples |
414 |
are shown in Figure \ref{shearGrad}. |
415 |
|
416 |
\begin{table*} |
417 |
\begin{minipage}{\linewidth} |
418 |
\begin{center} |
419 |
|
420 |
\caption{Calculation results for shear viscosity of Lennard-Jones |
421 |
fluid at ${T^* = 0.72}$ and ${\rho^* = 0.85}$, with swap and scale |
422 |
methods at various momentum exchange rates. Results in reduced |
423 |
unit. Errors of calculations in parentheses. } |
424 |
|
425 |
\begin{tabular}{ccc} |
426 |
\hline |
427 |
Series & $\eta^*_{swap}$ & $\eta^*_{scale}$\\ |
428 |
\hline |
429 |
20-500 & 3.64(0.05) & 3.76(0.09)\\ |
430 |
20-1000 & 3.52(0.16) & 3.66(0.06)\\ |
431 |
20-2000 & 3.72(0.05) & 3.32(0.18)\\ |
432 |
20-2500 & 3.42(0.06) & 3.43(0.08)\\ |
433 |
\hline |
434 |
\end{tabular} |
435 |
\label{shearRate} |
436 |
\end{center} |
437 |
\end{minipage} |
438 |
\end{table*} |
439 |
|
440 |
\begin{figure} |
441 |
\includegraphics[width=\linewidth]{shearGrad} |
442 |
\caption{Average momentum gradients of shear viscosity simulations} |
443 |
\label{shearGrad} |
444 |
\end{figure} |
445 |
|
446 |
\begin{figure} |
447 |
\includegraphics[width=\linewidth]{shearTempScale} |
448 |
\caption{Temperature profile for scaling RNEMD simulation.} |
449 |
\label{shearTempScale} |
450 |
\end{figure} |
451 |
However, observations of temperatures along three dimensions show that |
452 |
inhomogeneity occurs in scaling RNEMD simulations, particularly in the |
453 |
two slabs which were scaled. Figure \ref{shearTempScale} indicate that with |
454 |
relatively large imposed momentum flux, the temperature difference among $x$ |
455 |
and the other two dimensions was significant. This would result from the |
456 |
algorithm of scaling method. From Eq. \ref{eq:fluxPlane}, after |
457 |
momentum gradient is set up, $P_c^x$ would be roughly stable |
458 |
($<0$). Consequently, scaling factor $x$ would most probably larger |
459 |
than 1. Therefore, the kinetic energy in $x$-dimension $K_c^x$ would |
460 |
keep increase after most scaling steps. And if there is not enough time |
461 |
for the kinetic energy to exchange among different dimensions and |
462 |
different slabs, the system would finally build up temperature |
463 |
(kinetic energy) difference among the three dimensions. Also, between |
464 |
$y$ and $z$ dimensions in the scaled slabs, temperatures of $z$-axis |
465 |
are closer to neighbor slabs. This is due to momentum transfer along |
466 |
$z$ dimension between slabs. |
467 |
|
468 |
Although results between scaling and swapping methods are comparable, |
469 |
the inherent temperature inhomogeneity even in relatively low imposed |
470 |
exchange momentum flux simulations makes scaling RNEMD method less |
471 |
attractive than swapping RNEMD in shear viscosity calculation. |
472 |
|
473 |
\subsection{Thermal Conductivity} |
474 |
\subsubsection{Lennard-Jones Fluid} |
475 |
Our thermal conductivity calculations also show that scaling method results |
476 |
agree with swapping method. Table \ref{thermal} lists our simulation |
477 |
results with similar manner we used in shear viscosity |
478 |
calculation. All the data reported from scaling method were obtained |
479 |
by simulations of 10-step exchange frequency, and the target exchange |
480 |
kinetic energy were set to produce equivalent kinetic energy flux as |
481 |
in swapping method. Figure \ref{thermalGrad} exhibits similar thermal |
482 |
gradients of respective similar kinetic energy flux. |
483 |
|
484 |
\begin{table*} |
485 |
\begin{minipage}{\linewidth} |
486 |
\begin{center} |
487 |
|
488 |
\caption{Calculation results for thermal conductivity of Lennard-Jones |
489 |
fluid at ${\langle T^* \rangle = 0.72}$ and ${\rho^* = 0.85}$, with |
490 |
swap and scale methods at various kinetic energy exchange rates. Results |
491 |
in reduced unit. Errors of calculations in parentheses.} |
492 |
|
493 |
\begin{tabular}{ccc} |
494 |
\hline |
495 |
Series & $\lambda^*_{swap}$ & $\lambda^*_{scale}$\\ |
496 |
\hline |
497 |
20-250 & 7.03(0.34) & 7.30(0.10)\\ |
498 |
20-500 & 7.03(0.14) & 6.95(0.09)\\ |
499 |
20-1000 & 6.91(0.42) & 7.19(0.07)\\ |
500 |
20-2000 & 7.52(0.15) & 7.19(0.28)\\ |
501 |
\hline |
502 |
\end{tabular} |
503 |
\label{thermal} |
504 |
\end{center} |
505 |
\end{minipage} |
506 |
\end{table*} |
507 |
|
508 |
\begin{figure} |
509 |
\includegraphics[width=\linewidth]{thermalGrad} |
510 |
\caption{Temperature gradients of thermal conductivity simulations} |
511 |
\label{thermalGrad} |
512 |
\end{figure} |
513 |
|
514 |
During these simulations, molecule velocities were recorded in 1000 of |
515 |
all the snapshots. These velocity data were used to produce histograms |
516 |
of velocity and speed distribution in different slabs. From these |
517 |
histograms, it is observed that with increasing unphysical kinetic |
518 |
energy flux, speed and velocity distribution of molecules in slabs |
519 |
where swapping occured could deviate from Maxwell-Boltzmann |
520 |
distribution. Figure \ref{histSwap} indicates how these distributions |
521 |
deviate from ideal condition. In high temperature slabs, probability |
522 |
density in low speed is confidently smaller than ideal distribution; |
523 |
in low temperature slabs, probability density in high speed is smaller |
524 |
than ideal. This phenomenon is observable even in our relatively low |
525 |
swapping rate simulations. And this deviation could also leads to |
526 |
deviation of distribution of velocity in various dimensions. One |
527 |
feature of these deviated distribution is that in high temperature |
528 |
slab, the ideal Gaussian peak was changed into a relatively flat |
529 |
plateau; while in low temperature slab, that peak appears sharper. |
530 |
|
531 |
\begin{figure} |
532 |
\includegraphics[width=\linewidth]{histSwap} |
533 |
\caption{Speed distribution for thermal conductivity using swapping RNEMD.} |
534 |
\label{histSwap} |
535 |
\end{figure} |
536 |
|
537 |
\begin{figure} |
538 |
\includegraphics[width=\linewidth]{histScale} |
539 |
\caption{Speed distribution for thermal conductivity using scaling RNEMD.} |
540 |
\label{histScale} |
541 |
\end{figure} |
542 |
|
543 |
\subsubsection{SPC/E Water} |
544 |
Our results of SPC/E water thermal conductivity are comparable to |
545 |
Bedrov {\it et al.}\cite{ISI:000090151400044}, which employed the |
546 |
previous swapping RNEMD method for their calculation. Our simulations |
547 |
were able to produce a similar temperature gradient to their |
548 |
system. However, the average temperature of our system is 300K, while |
549 |
theirs is 318K, which would be attributed for part of the difference |
550 |
between the two series of results. Both methods yields values in |
551 |
agreement with experiment. And this shows the applicability of our |
552 |
method to multi-atom molecular system. |
553 |
|
554 |
\begin{figure} |
555 |
\includegraphics[width=\linewidth]{spceGrad} |
556 |
\caption{Temperature gradients for SPC/E water thermal conductivity.} |
557 |
\label{spceGrad} |
558 |
\end{figure} |
559 |
|
560 |
\begin{table*} |
561 |
\begin{minipage}{\linewidth} |
562 |
\begin{center} |
563 |
|
564 |
\caption{Calculation results for thermal conductivity of SPC/E water |
565 |
at ${\langle T\rangle}$ = 300K at various thermal exchange rates. Errors of |
566 |
calculations in parentheses. } |
567 |
|
568 |
\begin{tabular}{cccc} |
569 |
\hline |
570 |
$\langle dT/dz\rangle$(K/\AA) & & $\lambda$(W/m/K) & \\ |
571 |
& This work & Previous simulations\cite{ISI:000090151400044} & |
572 |
Experiment$^a$\\ |
573 |
\hline |
574 |
0.38 & 0.816(0.044) & & 0.64\\ |
575 |
0.81 & 0.770(0.008) & 0.784\\ |
576 |
1.54 & 0.813(0.007) & 0.730\\ |
577 |
\hline |
578 |
\end{tabular} |
579 |
\label{spceThermal} |
580 |
\end{center} |
581 |
\end{minipage} |
582 |
\end{table*} |
583 |
|
584 |
\subsubsection{Crystal Gold} |
585 |
Our results of gold thermal conductivity used QSC force field are |
586 |
shown in Table \ref{AuThermal}. Although our calculation is smaller |
587 |
than experimental value by an order of more than 100, this difference |
588 |
is mainly attributed to the lack of electron interaction |
589 |
representation in our force field parameters. Richardson {\it et |
590 |
al.}\cite{ISI:A1992HX37800010} used similar force field parameters |
591 |
in their metal thermal conductivity calculations. The EMD method they |
592 |
employed in their simulations produced comparable results to |
593 |
ours. Therefore, it is confident to conclude that NIVS-RNEMD is |
594 |
applicable to metal force field system. |
595 |
|
596 |
\begin{figure} |
597 |
\includegraphics[width=\linewidth]{AuGrad} |
598 |
\caption{Temperature gradients for crystal gold thermal conductivity.} |
599 |
\label{AuGrad} |
600 |
\end{figure} |
601 |
|
602 |
\begin{table*} |
603 |
\begin{minipage}{\linewidth} |
604 |
\begin{center} |
605 |
|
606 |
\caption{Calculation results for thermal conductivity of crystal gold |
607 |
at ${\langle T\rangle}$ = 300K at various thermal exchange rates. Errors of |
608 |
calculations in parentheses. } |
609 |
|
610 |
\begin{tabular}{ccc} |
611 |
\hline |
612 |
$\langle dT/dz\rangle$(K/\AA) & $\lambda$(W/m/K)\\ |
613 |
& This work & Previous simulations\cite{ISI:A1992HX37800010} \\ |
614 |
\hline |
615 |
1.44 & 1.10(0.01) & \\ |
616 |
2.86 & 1.08(0.02) & \\ |
617 |
5.14 & 1.15(0.01) & \\ |
618 |
\hline |
619 |
\end{tabular} |
620 |
\label{AuThermal} |
621 |
\end{center} |
622 |
\end{minipage} |
623 |
\end{table*} |
624 |
|
625 |
\subsection{Interfaciel Thermal Conductivity} |
626 |
After valid simulations of homogeneous water and gold systems using |
627 |
NIVS-RNEMD method, calculation of gold/water interfacial thermal |
628 |
conductivity was followed. It is found out that the interfacial |
629 |
conductance is low due to a hydrophobic surface in our system. Figure |
630 |
\ref{interfaceDensity} demonstrates this observance. Consequently, our |
631 |
reported results (Table \ref{interfaceRes}) are of two orders of |
632 |
magnitude smaller than our calculations on homogeneous systems. |
633 |
|
634 |
\begin{figure} |
635 |
\includegraphics[width=\linewidth]{interfaceDensity} |
636 |
\caption{Density profile for interfacial thermal conductivity |
637 |
simulation box.} |
638 |
\label{interfaceDensity} |
639 |
\end{figure} |
640 |
|
641 |
\begin{figure} |
642 |
\includegraphics[width=\linewidth]{interfaceGrad} |
643 |
\caption{Temperature profiles for interfacial thermal conductivity |
644 |
simulation box.} |
645 |
\label{interfaceGrad} |
646 |
\end{figure} |
647 |
|
648 |
\begin{table*} |
649 |
\begin{minipage}{\linewidth} |
650 |
\begin{center} |
651 |
|
652 |
\caption{Calculation results for interfacial thermal conductivity |
653 |
at ${\langle T\rangle \sim}$ 300K at various thermal exchange |
654 |
rates. Errors of calculations in parentheses. } |
655 |
|
656 |
\begin{tabular}{cccc} |
657 |
\hline |
658 |
$J_z$(MW/m$^2$) & $T_{gold}$ & $T_{water}$ & $G$(MW/m$^2$/K)\\ |
659 |
\hline |
660 |
98.0 & 355.2 & 295.8 & 1.65(0.21) \\ |
661 |
78.8 & 343.8 & 298.0 & 1.72(0.32) \\ |
662 |
73.6 & 344.3 & 298.0 & 1.59(0.24) \\ |
663 |
49.2 & 330.1 & 300.4 & 1.65(0.35) \\ |
664 |
\hline |
665 |
\end{tabular} |
666 |
\label{interfaceRes} |
667 |
\end{center} |
668 |
\end{minipage} |
669 |
\end{table*} |
670 |
|
671 |
\section{Conclusions} |
672 |
NIVS-RNEMD simulation method is developed and tested on various |
673 |
systems. Simulation results demonstrate its validity of thermal |
674 |
conductivity calculations. NIVS-RNEMD improves non-Boltzmann-Maxwell |
675 |
distributions existing in previous RNEMD methods, and extends its |
676 |
applicability to interfacial systems. NIVS-RNEMD has also limited |
677 |
application on shear viscosity calculations, but under high momentum |
678 |
flux, it could cause temperature difference among different |
679 |
dimensions. Modification is necessary to extend the applicability of |
680 |
NIVS-RNEMD in shear viscosity calculations. |
681 |
|
682 |
\section{Acknowledgments} |
683 |
Support for this project was provided by the National Science |
684 |
Foundation under grant CHE-0848243. Computational time was provided by |
685 |
the Center for Research Computing (CRC) at the University of Notre |
686 |
Dame. \newpage |
687 |
|
688 |
\bibliographystyle{jcp2} |
689 |
\bibliography{nivsRnemd} |
690 |
\end{doublespace} |
691 |
\end{document} |
692 |
|