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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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%\narrowtext |
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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,Muller-Plathe:1999ek} 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 have discovered some problems with the |
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original RNEMD swap technique. Notably, large momentum fluxes |
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(equivalent to frequent momentum swaps between the slabs) can result |
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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}{c} |
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x \\ |
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y \\ |
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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}{c} |
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x^\prime \\ |
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y^\prime \\ |
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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{equation} |
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\begin{array}{rcl} |
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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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\end{array} |
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\label{eq:momentumdef} |
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\end{equation} |
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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{equation} |
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\begin{array}{rcl} |
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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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\end{array} |
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\label{eq:constraintEllipsoidConsts} |
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\end{equation} |
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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 |
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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) |
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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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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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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) |
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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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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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|
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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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|
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\section{Computational Details} |
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Our simulation consists of a series of systems. |
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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 0.849, which enabled direct |
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comparison between our results and others. |
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|
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For shear viscosity calculation, the reduced temperature was ${T^* = |
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k_B T / \epsilon = 0.72}$. Simulations were run in microcanonical |
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ensemble (NVE). 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 20 slabs. In each swap, |
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the top slab ${(n = 0)}$ exchange the most negative $x$ momentum with the |
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most positive $x$ momentum in the center slab ${(n = N/2)}$. Referring |
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to Tenney {\it et al.}\cite{tenneyANDmaginn}, a series of swapping |
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frequency were chosen. Corresponding to each result from swapping |
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RNEMD, scaling RNEMD simulations were run with the target momentum |
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flux parameter set to produce a similar momentum flux and shear |
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rate. Furthermore, various scaling frequencies and corresponding flux |
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can be tested for one swapping rate. |
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|
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After each simulation, the shear viscosities were calculated in |
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reduced unit. |
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|
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\section{Acknowledgments} |
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Support for this project was provided by the National Science |
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Foundation under grant CHE-0848243. Computational time was provided by |
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the Center for Research Computing (CRC) at the University of Notre |
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Dame. \newpage |
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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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