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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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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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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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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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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} |
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\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 |
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the shear viscosity $\eta$ calculated with the above parameters, one |
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can further convert it into reduced unit ${\eta^* = \eta \sigma^2 |
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(\varepsilon m)^{-1/2}}$. |
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|
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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 |
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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} |
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J_z = \frac{E_{total}}{2 t L_x L_y} |
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\end{equation} |
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And the temperature gradient ${\langle\partial T/\partial z\rangle}$ |
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can be obtained by a linear regression of the temperature |
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profile. From the thermal conductivity $\lambda$ calculated, one can |
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further convert it into reduced unit ${\lambda^*=\lambda \sigma^2 |
345 |
m^{1/2} k_B^{-1}\varepsilon^{-1/2}}$. |
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|
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Another series of our simulation is to calculate the interfacial |
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thermal conductivity of a Au/H${_2}$O system. Respective calculations of |
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water (SPC/E) and gold (QSC) thermal conductivity were performed and |
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compared with current results to ensure the validity of |
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NIVS-RNEMD. After that, the mixture system was simulated. |
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|
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\section{Results And Discussion} |
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\subsection{Shear Viscosity} |
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Our calculations (Table \ref{shearRate}) shows that scale RNEMD method |
356 |
produced comparable shear viscosity to swap RNEMD method. In Table |
357 |
\ref{shearRate}, the names of the calculated samples are devided into |
358 |
two parts. The first number refers to total slabs in one simulation |
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box. The second number refers to the swapping interval in swap method, or |
360 |
in scale method the equilvalent swapping interval that the same |
361 |
momentum flux would theoretically result in swap method. All the scale |
362 |
method results were from simulations that had a scaling interval of 10 |
363 |
time steps. The average molecular momentum gradients of these samples |
364 |
are shown in Figure \ref{shearGrad}. |
365 |
|
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\begin{table*} |
367 |
\begin{minipage}{\linewidth} |
368 |
\begin{center} |
369 |
|
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\caption{Calculation results for shear viscosity of Lennard-Jones |
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fluid at ${T^* = 0.72}$ and ${\rho^* = 0.85}$, with swap and scale |
372 |
methods at various momentum exchange rates. Results in reduced |
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unit. Errors of calculations in parentheses. } |
374 |
|
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\begin{tabular}{ccc} |
376 |
\hline |
377 |
Series & $\eta^*_{swap}$ & $\eta^*_{scale}$\\ |
378 |
\hline |
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20-500 & 3.64(0.05) & 3.76(0.09)\\ |
380 |
20-1000 & 3.52(0.16) & 3.66(0.06)\\ |
381 |
20-2000 & 3.72(0.05) & 3.32(0.18)\\ |
382 |
20-2500 & 3.42(0.06) & 3.43(0.08)\\ |
383 |
\hline |
384 |
\end{tabular} |
385 |
\label{shearRate} |
386 |
\end{center} |
387 |
\end{minipage} |
388 |
\end{table*} |
389 |
|
390 |
\begin{figure} |
391 |
\includegraphics[width=\linewidth]{shearGrad} |
392 |
\caption{Average momentum gradients of shear viscosity simulations} |
393 |
\label{shearGrad} |
394 |
\end{figure} |
395 |
|
396 |
\begin{figure} |
397 |
\includegraphics[width=\linewidth]{shearTempScale} |
398 |
\caption{Temperature profile for scaling RNEMD simulation.} |
399 |
\label{shearTempScale} |
400 |
\end{figure} |
401 |
However, observations of temperatures along three dimensions show that |
402 |
inhomogeneity occurs in scaling RNEMD simulations, particularly in the |
403 |
two slabs which were scaled. Figure \ref{shearTempScale} indicate that with |
404 |
relatively large imposed momentum flux, the temperature difference among $x$ |
405 |
and the other two dimensions was significant. This would result from the |
406 |
algorithm of scaling method. From Eq. \ref{eq:fluxPlane}, after |
407 |
momentum gradient is set up, $P_c^x$ would be roughly stable |
408 |
($<0$). Consequently, scaling factor $x$ would most probably larger |
409 |
than 1. Therefore, the kinetic energy in $x$-dimension $K_c^x$ would |
410 |
keep increase after most scaling steps. And if there is not enough time |
411 |
for the kinetic energy to exchange among different dimensions and |
412 |
different slabs, the system would finally build up temperature |
413 |
(kinetic energy) difference among the three dimensions. Also, between |
414 |
$y$ and $z$ dimensions in the scaled slabs, temperatures of $z$-axis |
415 |
are closer to neighbor slabs. This is due to momentum transfer along |
416 |
$z$ dimension between slabs. |
417 |
|
418 |
Although results between scaling and swapping methods are comparable, |
419 |
the inherent temperature inhomogeneity even in relatively low imposed |
420 |
exchange momentum flux simulations makes scaling RNEMD method less |
421 |
attractive than swapping RNEMD in shear viscosity calculation. |
422 |
|
423 |
\subsection{Thermal Conductivity} |
424 |
|
425 |
Our thermal conductivity calculations also show that scaling method results |
426 |
agree with swapping method. Table \ref{thermal} lists our simulation |
427 |
results with similar manner we used in shear viscosity |
428 |
calculation. All the data reported from scaling method were obtained |
429 |
by simulations of 10-step exchange frequency, and the target exchange |
430 |
kinetic energy were set to produce equivalent kinetic energy flux as |
431 |
in swapping method. Figure \ref{thermalGrad} exhibits similar thermal |
432 |
gradients of respective similar kinetic energy flux. |
433 |
|
434 |
\begin{table*} |
435 |
\begin{minipage}{\linewidth} |
436 |
\begin{center} |
437 |
|
438 |
\caption{Calculation results for thermal conductivity of Lennard-Jones |
439 |
fluid at ${\langle T^* \rangle = 0.72}$ and ${\rho^* = 0.85}$, with |
440 |
swap and scale methods at various kinetic energy exchange rates. Results |
441 |
in reduced unit. Errors of calculations in parentheses.} |
442 |
|
443 |
\begin{tabular}{ccc} |
444 |
\hline |
445 |
Series & $\lambda^*_{swap}$ & $\lambda^*_{scale}$\\ |
446 |
\hline |
447 |
20-250 & 7.03(0.34) & 7.30(0.10)\\ |
448 |
20-500 & 7.03(0.14) & 6.95(0.09)\\ |
449 |
20-1000 & 6.91(0.42) & 7.19(0.07)\\ |
450 |
20-2000 & 7.52(0.15) & 7.19(0.28)\\ |
451 |
\hline |
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\end{tabular} |
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\label{thermal} |
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\end{center} |
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\end{minipage} |
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\end{table*} |
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|
458 |
\begin{figure} |
459 |
\includegraphics[width=\linewidth]{thermalGrad} |
460 |
\caption{Temperature gradients of thermal conductivity simulations} |
461 |
\label{thermalGrad} |
462 |
\end{figure} |
463 |
|
464 |
During these simulations, molecule velocities were recorded in 1000 of |
465 |
all the snapshots. These velocity data were used to produce histograms |
466 |
of velocity and speed distribution in different slabs. From these |
467 |
histograms, it is observed that with increasing unphysical kinetic |
468 |
energy flux, speed and velocity distribution of molecules in slabs |
469 |
where swapping occured could deviate from Maxwell-Boltzmann |
470 |
distribution. Figure \ref{histSwap} indicates how these distributions |
471 |
deviate from ideal condition. In high temperature slabs, probability |
472 |
density in low speed is confidently smaller than ideal distribution; |
473 |
in low temperature slabs, probability density in high speed is smaller |
474 |
than ideal. This phenomenon is observable even in our relatively low |
475 |
swpping rate simulations. And this deviation could also leads to |
476 |
deviation of distribution of velocity in various dimensions. One |
477 |
feature of these deviated distribution is that in high temperature |
478 |
slab, the ideal Gaussian peak was changed into a relatively flat |
479 |
plateau; while in low temperature slab, that peak appears sharper. |
480 |
|
481 |
\begin{figure} |
482 |
\includegraphics[width=\linewidth]{histSwap} |
483 |
\caption{Speed distribution for thermal conductivity using swapping RNEMD.} |
484 |
\label{histSwap} |
485 |
\end{figure} |
486 |
|
487 |
\subsection{Interfaciel Thermal Conductivity} |
488 |
|
489 |
\section{Acknowledgments} |
490 |
Support for this project was provided by the National Science |
491 |
Foundation under grant CHE-0848243. Computational time was provided by |
492 |
the Center for Research Computing (CRC) at the University of Notre |
493 |
Dame. \newpage |
494 |
|
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\bibliographystyle{jcp2} |
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\bibliography{nivsRnemd} |
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\end{doublespace} |
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\end{document} |
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|