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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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|
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\begin{doublespace} |
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|
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\begin{abstract} |
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We present a new method for introducing stable non-equilibrium |
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velocity and temperature distributions in molecular dynamics |
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simulations of heterogeneous systems. This method extends some |
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earlier Reverse Non-Equilibrium Molecular Dynamics (RNEMD) methods |
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which use momentum exchange swapping moves that can create |
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non-thermal velocity distributions (and which are difficult to use |
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for interfacial calculations). By using non-isotropic velocity |
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scaling (NIVS) on the molecules in specific regions of a system, it |
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is possible to impose momentum or thermal flux between regions of a |
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simulation and stable thermal and momentum gradients can then be |
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established. The scaling method we have developed conserves the |
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total linear momentum and total energy of the system. To test the |
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methods, we have computed the thermal conductivity of model liquid |
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and solid systems as well as the interfacial thermal conductivity of |
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a metal-water interface. We find that the NIVS-RNEMD improves the |
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problematic velocity distributions that develop in other RNEMD |
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methods. |
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\end{abstract} |
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|
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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% BODY OF TEXT |
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|
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\section{Introduction} |
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The original formulation of Reverse Non-equilibrium Molecular Dynamics |
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(RNEMD) obtains transport coefficients (thermal conductivity and shear |
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viscosity) in a fluid by imposing an artificial momentum flux between |
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two thin parallel slabs of material that are spatially separated in |
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the simulation cell.\cite{MullerPlathe:1997xw,ISI:000080382700030} The |
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artificial flux is typically created by periodically ``swapping'' |
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either the entire momentum vector $\vec{p}$ or single components of |
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this vector ($p_x$) between molecules in each of the two slabs. If |
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the two slabs are separated along the $z$ coordinate, the imposed flux |
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is either directional ($j_z(p_x)$) or isotropic ($J_z$), and the |
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response of a simulated system to the imposed momentum flux will |
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typically be a velocity or thermal gradient (Fig. \ref{thermalDemo}). |
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The shear viscosity ($\eta$) and thermal conductivity ($\lambda$) are |
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easily obtained by assuming linear response of the system, |
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\begin{eqnarray} |
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j_z(p_x) & = & -\eta \frac{\partial v_x}{\partial z}\\ |
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J_z & = & \lambda \frac{\partial T}{\partial z} |
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\end{eqnarray} |
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RNEMD has been widely used to provide computational estimates of |
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thermal conductivities and shear viscosities in a wide range of |
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materials, from liquid copper to both monatomic and molecular fluids |
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(e.g. ionic |
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liquids).\cite{Bedrov:2000-1,Bedrov:2000,Muller-Plathe:2002,ISI:000184808400018,ISI:000231042800044,Maginn:2007,Muller-Plathe:2008,ISI:000258460400020,ISI:000258840700015,ISI:000261835100054} |
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|
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\begin{figure} |
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\includegraphics[width=\linewidth]{thermalDemo} |
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\caption{RNEMD methods impose an unphysical transfer of momentum or |
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kinetic energy between a ``hot'' slab and a ``cold'' slab in the |
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simulation box. The molecular system responds to this imposed flux |
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by generating a momentum or temperature gradient. The slope of the |
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gradient can then be used to compute transport properties (e.g. |
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shear viscosity and thermal conductivity).} |
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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 |
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methods\cite{ISI:A1988Q205300014,hess:209,Vasquez:2004fk,backer:154503,ISI:000266247600008} |
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because it imposes what is typically difficult to measure (a flux or |
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stress) and it is typically much easier to compute the response |
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(momentum gradients or strains. For similar reasons, RNEMD is also |
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preferable to slowly-converging equilibrium methods for measuring |
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thermal conductivity and shear viscosity (using Green-Kubo |
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relations\cite{daivis:541,mondello:9327} or the Helfand moment |
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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{Maginn:2010} have discovered |
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some problems with the original RNEMD swap technique. Notably, large |
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momentum fluxes (equivalent to frequent momentum swaps between the |
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slabs) can result in ``notched'', ``peaked'' and generally non-thermal |
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momentum distributions in the two slabs, as well as non-linear thermal |
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and velocity distributions along the direction of the imposed flux |
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($z$). Tenney and Maginn obtained reasonable limits on imposed flux |
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and self-adjusting metrics for retaining the usability of the method. |
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|
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In this paper, we develop and test a method for non-isotropic velocity |
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scaling (NIVS) 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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present the method for determining the scaling constraints. We then |
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test the method on both liquids and solids as well as a non-isotropic |
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liquid-solid interface 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 all of these cases. |
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|
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\section{Methodology} |
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We retain the basic idea of M\"{u}ller-Plathe's RNEMD method; the |
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periodic system is partitioned into a series of thin slabs along one |
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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 velocity-scaling variables for each |
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of the three directions in the system. Likewise, the molecules |
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$\{j\}$ located in the hot slab will see a concomitant scaling of |
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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 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 translational kinetic energies, $K_h^\alpha$ and |
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$K_c^\alpha$, are computed for each of the three directions in a |
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similar manner to the linear momenta (Eq. \ref{eq:momentumdef}). |
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Substituting in the expressions for the hot scaling parameters |
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($\alpha^\prime$) from Eq. (\ref{eq:hotcoldscaling}), we obtain the |
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{\it constraint ellipsoid}: |
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\begin{equation} |
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\sum_{\alpha = x,y,z} \left( a_\alpha \alpha^2 + b_\alpha \alpha + |
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c_\alpha \right) = 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 (Eq. \ref{eq:constraintEllipsoid}) defines the set of |
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cold slab scaling parameters which, when applied, preserve the linear |
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momentum of the system in all three directions as well as total |
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kinetic energy. |
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|
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The goal of using these velocity scaling variables is to transfer |
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linear momentum or kinetic energy from the cold slab to the hot slab. |
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If the hot and cold slabs are separated along the z-axis, the energy |
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flux is given simply by the decrease in kinetic energy of the cold |
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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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\sum_{\alpha = x,y,z} K_c^\alpha \alpha^2 = -J_z \Delta t + |
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\sum_{\alpha = x,y,z} K_c^\alpha |
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\label{eq:fluxEllipsoid} |
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\end{equation} |
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The spatial extent of the {\it thermal flux ellipsoid} is governed |
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both by the target flux, $J_z$ as well as the instantaneous values of |
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the 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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we must determine the points ${x,y,z}$ that lie on both the constraint |
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ellipsoid (Eq. \ref{eq:constraintEllipsoid}) and the flux ellipsoid |
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(Eq. \ref{eq:fluxEllipsoid}), i.e. the intersection of the two |
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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{Velocity scaling coefficients which maintain both constant |
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energy and constant linear momentum of the system lie on the surface |
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of the {\it constraint ellipsoid} while points which generate the |
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target momentum flux lie on the surface of the {\it flux ellipsoid}. |
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The velocity distributions in the cold bin are scaled by only those |
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points which lie on both ellipsoids.} |
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\label{ellipsoids} |
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\end{figure} |
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|
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Since ellipsoids can be expressed as polynomials up to second order in |
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each of the three coordinates, finding the the intersection points of |
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two ellipsoids is isomorphic to finding the roots a polynomial of |
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degree 16. There are a number of polynomial root-finding methods in |
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the literature, [CITATIONS NEEDED] but numerically finding the roots |
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of high-degree polynomials is generally an ill-conditioned |
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problem.[CITATION NEEDED] One way around this is to try to maintain |
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velocity scalings that are {\it as isotropic as possible}. To do |
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this, we impose $x=y$, and to treat both the constraint and flux |
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ellipsoids as 2-dimensional ellipses. In reduced dimensionality, the |
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intersecting-ellipse problem reduces to finding the roots of |
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polynomials of degree 4. |
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|
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Depending on the target flux and current velocity distributions, the |
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ellipsoids can have between 0 and 4 intersection points. If there are |
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no intersection points, it is not possible to satisfy the constraints |
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while performing a non-equilibrium scaling move, and no change is made |
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to the dynamics. |
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|
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With multiple intersection points, any of the scaling points will |
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conserve the linear momentum and kinetic energy of the system and will |
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generate the correct target flux. Although this method is inherently |
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non-isotropic, the goal is still to maintain the system as close to an |
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isotropic fluid as possible. With this in mind, we would like the |
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kinetic energies in the three different directions could become as |
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close as each other as possible after each scaling. Simultaneously, |
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one would also like each scaling as gentle as possible, i.e. ${x,y,z |
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\rightarrow 1}$, in order to avoid large perturbation to the system. |
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To do this, we pick the intersection point which maintains the scaling |
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variables ${x=y, z}$ as well as the ratio of kinetic energies |
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${K_c^z/K_c^x, K_c^z/K_c^y}$ as close as possible to 1. |
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|
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After the valid scaling parameters are arrived at by solving geometric |
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intersection problems in $x, y, z$ space in order to obtain cold slab |
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scaling parameters, Eq. (\ref{eq:hotcoldscaling}) is used to |
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determine the conjugate hot slab scaling variables. |
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|
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\subsection{Introducing shear stress via velocity scaling} |
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Rather than using this method to induce a thermal flux, it is possible |
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to use the random fluctuations of the average momentum in each of the |
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bins to induce a momentum flux. Doing this repeatedly will create a |
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shear stress on the system which will respond with an easily-measured |
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strain. The momentum flux (say along the $x$-direction) may be |
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defined 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} |
310 |
\end{equation} |
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This {\it momentum flux plane} is perpendicular to the $x$-axis, with |
312 |
its position governed both by a target value, $j_z(p_x)$ as well as |
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the instantaneous value of the momentum along the $x$-direction. |
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|
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In order to satisfy a momentum flux as well as the conservation |
316 |
constraints, we must 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 plane (Eq. \ref{eq:fluxPlane}), i.e. the intersection of an |
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ellipsoid and a plane in 3-dimensional space. |
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|
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In the case of momentum flux transfer, we also impose another |
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constraint to set the kinetic energy transfer as zero. In another |
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word, we apply Eq. \ref{eq:fluxEllipsoid} and let ${J_z = 0}$. With |
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one variable fixed by Eq. \ref{eq:fluxPlane}, one now have a similar |
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set of quartic equations to the above kinetic energy transfer problem. |
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|
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\section{Computational Details} |
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|
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We have implemented this methodology in our molecular dynamics |
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code,\cite{Meineke:2005gd} by performing the NIVS scaling moves after |
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each MD step. We have tested it for a variety of different |
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situations, including homogeneous fluids (Lennard-Jones and SPC/E |
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water), crystalline solids (EAM and Sutton-Chen models for Gold), and |
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heterogeneous interfaces (EAM gold - SPC/E water). The last of these |
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systems would have been very difficult to study using previous RNEMD |
336 |
methods, but using velocity scaling moves, we can even obtain |
337 |
estimates of the interfacial thermal conductivity. |
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|
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\subsection{Lennard-Jones Fluid} |
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|
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2592 Lennard-Jones atoms were placed in an orthorhombic cell |
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${10.06\sigma \times 10.06\sigma \times 30.18\sigma}$ on a side. The |
343 |
reduced density ${\rho^* = \rho\sigma^3}$ was thus 0.85, which enabled |
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direct comparison between our results and previous methods. These |
345 |
simulations were carried out with a reduced timestep ${\tau^* = |
346 |
4.6\times10^{-4}}$. For the shear viscosity calculation, the mean |
347 |
temperature was ${T^* = k_B T/\varepsilon = 0.72}$. Simulations were |
348 |
first thermalized in canonical ensemble (NVT), then equilibrated in |
349 |
microcanonical ensemble (NVE) before introducing any non-equilibrium |
350 |
method. |
351 |
|
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We have compared the momentum gradients established using our method |
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to those obtained using the original M\"{u}ller-Plathe swapping |
354 |
algorithm.\cite{ISI:000080382700030} In both cases, the simulation box |
355 |
was divided into ${N = 20}$ slabs. In the swapping algorithm, the top |
356 |
slab $(n = 1)$ exchanges the most negative $x$ momentum with the most |
357 |
positive $x$ momentum in the center slab $(n = N/2 + 1)$. The rate at |
358 |
which the swapping moves are carried out defines the momentum or |
359 |
thermal flux between the two slabs. In their work, Tenney {\it et |
360 |
al.}\cite{Maginn:2010} found problematic behavior with large swap |
361 |
frequencies. |
362 |
|
363 |
According to each result from swapping RNEMD, scaling RNEMD |
364 |
simulations were run with the target momentum flux set to produce a |
365 |
similar momentum flux, and consequently shear rate. Furthermore, |
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various scaling frequencies can be tested for one single swapping |
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rate. To test the temperature homogeneity in our system of swapping |
368 |
and scaling methods, temperatures of different dimensions in all the |
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slabs were observed. Most of the simulations include $10^5$ steps of |
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equilibration without imposing momentum flux, $10^5$ steps of |
371 |
stablization with imposing unphysical momentum transfer, and $10^6$ |
372 |
steps of data collection under RNEMD. For relatively high momentum |
373 |
flux simulations, ${5\times10^5}$ step data collection is sufficient. |
374 |
For some low momentum flux simulations, ${2\times10^6}$ steps were |
375 |
necessary. |
376 |
|
377 |
After each simulation, the shear viscosity was calculated in reduced |
378 |
unit. The momentum flux was calculated with total unphysical |
379 |
transferred momentum ${P_x}$ and data collection time $t$: |
380 |
\begin{equation} |
381 |
j_z(p_x) = \frac{P_x}{2 t L_x L_y} |
382 |
\end{equation} |
383 |
where $L_x$ and $L_y$ denotes $x$ and $y$ lengths of the simulation |
384 |
box, and physical momentum transfer occurs in two ways due to our |
385 |
periodic boundary condition settings. And the velocity gradient |
386 |
${\langle \partial v_x /\partial z \rangle}$ can be obtained by a |
387 |
linear regression of the velocity profile. From the shear viscosity |
388 |
$\eta$ calculated with the above parameters, one can further convert |
389 |
it into reduced unit ${\eta^* = \eta \sigma^2 (\varepsilon m)^{-1/2}}$. |
390 |
|
391 |
For thermal conductivity calculations, simulations were first run under |
392 |
reduced temperature ${\langle T^*\rangle = 0.72}$ in NVE |
393 |
ensemble. Muller-Plathe's algorithm was adopted in the swapping |
394 |
method. Under identical simulation box parameters with our shear |
395 |
viscosity calculations, in each swap, the top slab exchanges all three |
396 |
translational momentum components of the molecule with least kinetic |
397 |
energy with the same components of the molecule in the center slab |
398 |
with most kinetic energy, unless this ``coldest'' molecule in the |
399 |
``hot'' slab is still ``hotter'' than the ``hottest'' molecule in the |
400 |
``cold'' slab. According to swapping RNEMD results, target energy flux |
401 |
for scaling RNEMD simulations can be set. Also, various scaling |
402 |
frequencies can be tested for one target energy flux. To compare the |
403 |
performance between swapping and scaling method, distributions of |
404 |
velocity and speed in different slabs were observed. |
405 |
|
406 |
For each swapping rate, thermal conductivity was calculated in reduced |
407 |
unit. The energy flux was calculated similarly to the momentum flux, |
408 |
with total unphysical transferred energy ${E_{total}}$ and data collection |
409 |
time $t$: |
410 |
\begin{equation} |
411 |
J_z = \frac{E_{total}}{2 t L_x L_y} |
412 |
\end{equation} |
413 |
And the temperature gradient ${\langle\partial T/\partial z\rangle}$ |
414 |
can be obtained by a linear regression of the temperature |
415 |
profile. From the thermal conductivity $\lambda$ calculated, one can |
416 |
further convert it into reduced unit ${\lambda^*=\lambda \sigma^2 |
417 |
m^{1/2} k_B^{-1}\varepsilon^{-1/2}}$. |
418 |
|
419 |
\subsection{ Water / Metal Thermal Conductivity} |
420 |
Another series of our simulation is the calculation of interfacial |
421 |
thermal conductivity of a Au/H$_2$O system. Respective calculations of |
422 |
liquid water (Extended Simple Point Charge model) and crystal gold |
423 |
thermal conductivity were performed and compared with current results |
424 |
to ensure the validity of NIVS-RNEMD. After that, a mixture system was |
425 |
simulated. |
426 |
|
427 |
For thermal conductivity calculation of bulk water, a simulation box |
428 |
consisting of 1000 molecules were first equilibrated under ambient |
429 |
pressure and temperature conditions using NPT ensemble, followed by |
430 |
equilibration in fixed volume (NVT). The system was then equilibrated in |
431 |
microcanonical ensemble (NVE). Also in NVE ensemble, establishing a |
432 |
stable thermal gradient was followed. The simulation box was under |
433 |
periodic boundary condition and devided into 10 slabs. Data collection |
434 |
process was similar to Lennard-Jones fluid system. |
435 |
|
436 |
Thermal conductivity calculation of bulk crystal gold used a similar |
437 |
protocol. Two types of force field parameters, Embedded Atom Method |
438 |
(EAM) and Quantum Sutten-Chen (QSC) force field were used |
439 |
respectively. The face-centered cubic crystal simulation box consists of |
440 |
2880 Au atoms. The lattice was first allowed volume change to relax |
441 |
under ambient temperature and pressure. Equilibrations in canonical and |
442 |
microcanonical ensemble were followed in order. With the simulation |
443 |
lattice devided evenly into 10 slabs, different thermal gradients were |
444 |
established by applying a set of target thermal transfer flux. Data of |
445 |
the series of thermal gradients was collected for calculation. |
446 |
|
447 |
After simulations of bulk water and crystal gold, a mixture system was |
448 |
constructed, consisting of 1188 Au atoms and 1862 H$_2$O |
449 |
molecules. Spohr potential was adopted in depicting the interaction |
450 |
between metal atom and water molecule.\cite{ISI:000167766600035} A |
451 |
similar protocol of equilibration was followed. Several thermal |
452 |
gradients was built under different target thermal flux. It was found |
453 |
out that compared to our previous simulation systems, the two phases |
454 |
could have large temperature difference even under a relatively low |
455 |
thermal flux. Therefore, under our low flux conditions, it is assumed |
456 |
that the metal and water phases have respectively homogeneous |
457 |
temperature, excluding the surface regions. In calculating the |
458 |
interfacial thermal conductivity $G$, this assumptioin was applied and |
459 |
thus our formula becomes: |
460 |
|
461 |
\begin{equation} |
462 |
G = \frac{E_{total}}{2 t L_x L_y \left( \langle T_{gold}\rangle - |
463 |
\langle T_{water}\rangle \right)} |
464 |
\label{interfaceCalc} |
465 |
\end{equation} |
466 |
where ${E_{total}}$ is the imposed unphysical kinetic energy transfer |
467 |
and ${\langle T_{gold}\rangle}$ and ${\langle T_{water}\rangle}$ are the |
468 |
average observed temperature of gold and water phases respectively. |
469 |
|
470 |
\section{Results And Discussions} |
471 |
\subsection{Thermal Conductivity} |
472 |
\subsubsection{Lennard-Jones Fluid} |
473 |
Our thermal conductivity calculations show that scaling method results |
474 |
agree with swapping method. Four different exchange intervals were |
475 |
tested (Table \ref{thermalLJRes}) using swapping method. With a fixed |
476 |
10fs exchange interval, target exchange kinetic energy was set to |
477 |
produce equivalent kinetic energy flux as in swapping method. And |
478 |
similar thermal gradients were observed with similar thermal flux in |
479 |
two simulation methods (Figure \ref{thermalGrad}). |
480 |
|
481 |
\begin{table*} |
482 |
\begin{minipage}{\linewidth} |
483 |
\begin{center} |
484 |
|
485 |
\caption{Calculation results for thermal conductivity of Lennard-Jones |
486 |
fluid at ${\langle T^* \rangle = 0.72}$ and ${\rho^* = 0.85}$, with |
487 |
swap and scale methods at various kinetic energy exchange rates. Results |
488 |
in reduced unit. Errors of calculations in parentheses.} |
489 |
|
490 |
\begin{tabular}{ccc} |
491 |
\hline |
492 |
(Equilvalent) Exchange Interval (fs) & $\lambda^*_{swap}$ & |
493 |
$\lambda^*_{scale}$\\ |
494 |
\hline |
495 |
250 & 7.03(0.34) & 7.30(0.10)\\ |
496 |
500 & 7.03(0.14) & 6.95(0.09)\\ |
497 |
1000 & 6.91(0.42) & 7.19(0.07)\\ |
498 |
2000 & 7.52(0.15) & 7.19(0.28)\\ |
499 |
\hline |
500 |
\end{tabular} |
501 |
\label{thermalLJRes} |
502 |
\end{center} |
503 |
\end{minipage} |
504 |
\end{table*} |
505 |
|
506 |
\begin{figure} |
507 |
\includegraphics[width=\linewidth]{thermalGrad} |
508 |
\caption{NIVS-RNEMD method introduced similar temperature gradients |
509 |
compared to ``swapping'' method under various kinetic energy flux in |
510 |
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 of one single data collection process. These |
516 |
velocity data were used to produce histograms of velocity and speed |
517 |
distribution in different slabs. From these histograms, it is observed |
518 |
that under relatively high unphysical kinetic energy flux, speed and |
519 |
velocity distribution of molecules in slabs where swapping occured |
520 |
could deviate from Maxwell-Boltzmann distribution. Figure |
521 |
\ref{thermalHist} a) illustrates how these distributions deviate from an |
522 |
ideal distribution. In high temperature slab, probability density in |
523 |
low speed is confidently smaller than ideal curve fit; in low |
524 |
temperature slab, probability density in high speed is smaller than |
525 |
ideal, while larger than ideal in low speed. This phenomenon is more |
526 |
obvious in our high swapping rate simulations. And this deviation |
527 |
could also leads to deviation of distribution of velocity in various |
528 |
dimensions. One feature of these deviated distribution is that in high |
529 |
temperature slab, the ideal Gaussian peak was changed into a |
530 |
relatively flat plateau; while in low temperature slab, that peak |
531 |
appears sharper. This problem is rooted in the mechanism of the |
532 |
swapping method. Continually depleting low (high) speed particles in |
533 |
the high (low) temperature slab could not be complemented by |
534 |
diffusions of low (high) speed particles from neighbor slabs, unless |
535 |
in suffciently low swapping rate. Simutaneously, surplus low speed |
536 |
particles in the low temperature slab do not have sufficient time to |
537 |
diffuse to neighbor slabs. However, thermal exchange rate should reach |
538 |
a minimum level to produce an observable thermal gradient under noise |
539 |
interference. Consequently, swapping RNEMD has a relatively narrow |
540 |
choice of swapping rate to satisfy these above restrictions. |
541 |
|
542 |
Comparatively, NIVS-RNEMD has a speed distribution closer to the ideal |
543 |
curve fit (Figure \ref{thermalHist} b). Essentially, after scaling, a |
544 |
Gaussian distribution function would remain Gaussian. Although a |
545 |
single scaling is non-isotropic in all three dimensions, our scaling |
546 |
coefficient criteria could help maintian the scaling region as |
547 |
isotropic as possible. On the other hand, scaling coefficients are |
548 |
preferred to be as close to 1 as possible, which also helps minimize |
549 |
the difference among different dimensions. This is possible if scaling |
550 |
interval and one-time thermal transfer energy are well |
551 |
chosen. Consequently, NIVS-RNEMD is able to impose an unphysical |
552 |
thermal flux as the previous RNEMD method without large perturbation |
553 |
to the distribution of velocity and speed in the exchange regions. |
554 |
|
555 |
\begin{figure} |
556 |
\includegraphics[width=\linewidth]{thermalHist} |
557 |
\caption{Speed distribution for thermal conductivity using a) |
558 |
``swapping'' and b) NIVS- RNEMD methods. Shown is from the |
559 |
simulations with an exchange or equilvalent exchange interval of 250 |
560 |
fs. In circled areas, distributions from ``swapping'' RNEMD |
561 |
simulation have deviation from ideal Maxwell-Boltzmann distribution |
562 |
(curves fit for each distribution).} |
563 |
\label{thermalHist} |
564 |
\end{figure} |
565 |
|
566 |
\subsubsection{SPC/E Water} |
567 |
Our results of SPC/E water thermal conductivity are comparable to |
568 |
Bedrov {\it et al.}\cite{Bedrov:2000}, which employed the |
569 |
previous swapping RNEMD method for their calculation. Bedrov {\it et |
570 |
al.}\cite{Bedrov:2000} argued that exchange of the molecule |
571 |
center-of-mass velocities instead of single atom velocities in a |
572 |
molecule conserves the total kinetic energy and linear momentum. This |
573 |
principle is adopted in our simulations. Scaling is applied to the |
574 |
velocities of the rigid bodies of SPC/E model water molecules, instead |
575 |
of each hydrogen and oxygen atoms in relevant water molecules. As |
576 |
shown in Figure \ref{spceGrad}, temperature gradients were established |
577 |
similar to their system. However, the average temperature of our |
578 |
system is 300K, while theirs is 318K, which would be attributed for |
579 |
part of the difference between the final calculation results (Table |
580 |
\ref{spceThermal}). Both methods yields values in agreement with |
581 |
experiment. And this shows the applicability of our method to |
582 |
multi-atom molecular system. |
583 |
|
584 |
\begin{figure} |
585 |
\includegraphics[width=\linewidth]{spceGrad} |
586 |
\caption{Temperature gradients in SPC/E water thermal conductivity |
587 |
simulations.} |
588 |
\label{spceGrad} |
589 |
\end{figure} |
590 |
|
591 |
\begin{table*} |
592 |
\begin{minipage}{\linewidth} |
593 |
\begin{center} |
594 |
|
595 |
\caption{Calculation results for thermal conductivity of SPC/E water |
596 |
at ${\langle T\rangle}$ = 300K at various thermal exchange rates. Errors of |
597 |
calculations in parentheses. } |
598 |
|
599 |
\begin{tabular}{cccc} |
600 |
\hline |
601 |
$\langle dT/dz\rangle$(K/\AA) & & $\lambda$(W/m/K) & \\ |
602 |
& This work & Previous simulations\cite{Bedrov:2000} & |
603 |
Experiment$^a$\\ |
604 |
\hline |
605 |
0.38 & 0.816(0.044) & & 0.64\\ |
606 |
0.81 & 0.770(0.008) & 0.784\\ |
607 |
1.54 & 0.813(0.007) & 0.730\\ |
608 |
\hline |
609 |
\end{tabular} |
610 |
\label{spceThermal} |
611 |
\end{center} |
612 |
\end{minipage} |
613 |
\end{table*} |
614 |
|
615 |
\subsubsection{Crystal Gold} |
616 |
Our results of gold thermal conductivity using two force fields are |
617 |
shown in Table \ref{AuThermal}. In these calculations,the end and |
618 |
middle slabs were excluded in thermal gradient regession and only used |
619 |
as heat source and drain in the systems. Our yielded values using EAM |
620 |
force field are slightly larger than those using QSC force |
621 |
field. However, both series are significantly smaller than |
622 |
experimental value by a factor of more than 200. It has been verified |
623 |
that this difference is mainly attributed to the lack of electron |
624 |
interaction representation in these force field parameters. Richardson |
625 |
{\it et al.}\cite{Clancy:1992} used EAM force field parameters in |
626 |
their metal thermal conductivity calculations. The Non-Equilibrium MD |
627 |
method they employed in their simulations produced comparable results |
628 |
to ours. As Zhang {\it et al.}\cite{ISI:000231042800044} stated, |
629 |
thermal conductivity values are influenced mainly by force |
630 |
field. Another factor that affects the calculation results could be |
631 |
the density of the metal. After equilibration under |
632 |
isobaric-isothermal conditions, our crystall simulation cell expanded |
633 |
by the order of 1\%. Under longer lattice constant than default, |
634 |
lower thermal conductance would be expected. Furthermore, the result |
635 |
of Richardson {\it et al.} were obtained between 300K and 850K, which |
636 |
are significantly higher than in our simulations. Therefore, it is |
637 |
still confident to conclude that NIVS-RNEMD is applicable to metal |
638 |
force field system. |
639 |
|
640 |
\begin{table*} |
641 |
\begin{minipage}{\linewidth} |
642 |
\begin{center} |
643 |
|
644 |
\caption{Calculation results for thermal conductivity of crystal gold |
645 |
using different force fields at ${\langle T\rangle}$ = 300K at |
646 |
various thermal exchange rates. Errors of calculations in parentheses.} |
647 |
|
648 |
\begin{tabular}{ccc} |
649 |
\hline |
650 |
Force Field Used & $\langle dT/dz\rangle$(K/\AA) & $\lambda$(W/m/K)\\ |
651 |
\hline |
652 |
& 1.44 & 1.10(0.01)\\ |
653 |
QSC & 2.86 & 1.08(0.02)\\ |
654 |
& 5.14 & 1.15(0.01)\\ |
655 |
\hline |
656 |
& 1.24 & 1.24(0.06)\\ |
657 |
EAM & 2.06 & 1.37(0.04)\\ |
658 |
& 2.55 & 1.41(0.03)\\ |
659 |
\hline |
660 |
\end{tabular} |
661 |
\label{AuThermal} |
662 |
\end{center} |
663 |
\end{minipage} |
664 |
\end{table*} |
665 |
|
666 |
|
667 |
\subsection{Interfaciel Thermal Conductivity} |
668 |
After simulations of homogeneous water and gold systems using |
669 |
NIVS-RNEMD method were proved valid, calculation of gold/water |
670 |
interfacial thermal conductivity was followed. It is found out that |
671 |
the low interfacial conductance is probably due to the hydrophobic |
672 |
surface in our system. Figure \ref{interface} (a) demonstrates mass |
673 |
density change along $z$-axis, which is perpendicular to the |
674 |
gold/water interface. It is observed that water density significantly |
675 |
decreases when approaching the surface. Under this low thermal |
676 |
conductance, both gold and water phase have sufficient time to |
677 |
eliminate temperature difference inside respectively (Figure |
678 |
\ref{interface} b). With indistinguishable temperature difference |
679 |
within respective phase, it is valid to assume that the temperature |
680 |
difference between gold and water on surface would be approximately |
681 |
the same as the difference between the gold and water phase. This |
682 |
assumption enables convenient calculation of $G$ using |
683 |
Eq. \ref{interfaceCalc} instead of measuring temperatures of thin |
684 |
layer of water and gold close enough to surface, which would have |
685 |
greater fluctuation and lower accuracy. Reported results (Table |
686 |
\ref{interfaceRes}) are of two orders of magnitude smaller than our |
687 |
calculations on homogeneous systems, and thus have larger relative |
688 |
errors than our calculation results on homogeneous systems. |
689 |
|
690 |
\begin{figure} |
691 |
\includegraphics[width=\linewidth]{interface} |
692 |
\caption{Simulation results for Gold/Water interfacial thermal |
693 |
conductivity: (a) Significant water density decrease is observed on |
694 |
crystalline gold surface, which indicates low surface contact and |
695 |
leads to low thermal conductance. (b) Temperature profiles for a |
696 |
series of simulations. Temperatures of different slabs in the same |
697 |
phase show no significant differences.} |
698 |
\label{interface} |
699 |
\end{figure} |
700 |
|
701 |
\begin{table*} |
702 |
\begin{minipage}{\linewidth} |
703 |
\begin{center} |
704 |
|
705 |
\caption{Calculation results for interfacial thermal conductivity |
706 |
at ${\langle T\rangle \sim}$ 300K at various thermal exchange |
707 |
rates. Errors of calculations in parentheses. } |
708 |
|
709 |
\begin{tabular}{cccc} |
710 |
\hline |
711 |
$J_z$(MW/m$^2$) & $T_{gold}$ & $T_{water}$ & $G$(MW/m$^2$/K)\\ |
712 |
\hline |
713 |
98.0 & 355.2 & 295.8 & 1.65(0.21) \\ |
714 |
78.8 & 343.8 & 298.0 & 1.72(0.32) \\ |
715 |
73.6 & 344.3 & 298.0 & 1.59(0.24) \\ |
716 |
49.2 & 330.1 & 300.4 & 1.65(0.35) \\ |
717 |
\hline |
718 |
\end{tabular} |
719 |
\label{interfaceRes} |
720 |
\end{center} |
721 |
\end{minipage} |
722 |
\end{table*} |
723 |
|
724 |
\subsection{Shear Viscosity} |
725 |
Our calculations (Table \ref{shearRate}) shows that scale RNEMD method |
726 |
produced comparable shear viscosity to swap RNEMD method. In Table |
727 |
\ref{shearRate}, the names of the calculated samples are devided into |
728 |
two parts. The first number refers to total slabs in one simulation |
729 |
box. The second number refers to the swapping interval in swap method, or |
730 |
in scale method the equilvalent swapping interval that the same |
731 |
momentum flux would theoretically result in swap method. All the scale |
732 |
method results were from simulations that had a scaling interval of 10 |
733 |
time steps. The average molecular momentum gradients of these samples |
734 |
are shown in Figure \ref{shear} (a) and (b). |
735 |
|
736 |
\begin{table*} |
737 |
\begin{minipage}{\linewidth} |
738 |
\begin{center} |
739 |
|
740 |
\caption{Calculation results for shear viscosity of Lennard-Jones |
741 |
fluid at ${T^* = 0.72}$ and ${\rho^* = 0.85}$, with swap and scale |
742 |
methods at various momentum exchange rates. Results in reduced |
743 |
unit. Errors of calculations in parentheses. } |
744 |
|
745 |
\begin{tabular}{ccc} |
746 |
\hline |
747 |
(Equilvalent) Exchange Interval (fs) & $\eta^*_{swap}$ & |
748 |
$\eta^*_{scale}$\\ |
749 |
\hline |
750 |
500 & 3.64(0.05) & 3.76(0.09)\\ |
751 |
1000 & 3.52(0.16) & 3.66(0.06)\\ |
752 |
2000 & 3.72(0.05) & 3.32(0.18)\\ |
753 |
2500 & 3.42(0.06) & 3.43(0.08)\\ |
754 |
\hline |
755 |
\end{tabular} |
756 |
\label{shearRate} |
757 |
\end{center} |
758 |
\end{minipage} |
759 |
\end{table*} |
760 |
|
761 |
\begin{figure} |
762 |
\includegraphics[width=\linewidth]{shear} |
763 |
\caption{Average momentum gradients in shear viscosity simulations, |
764 |
using (a) ``swapping'' method and (b) NIVS-RNEMD method |
765 |
respectively. (c) Temperature difference among x and y, z dimensions |
766 |
observed when using NIVS-RNEMD with equivalent exchange interval of |
767 |
500 fs.} |
768 |
\label{shear} |
769 |
\end{figure} |
770 |
|
771 |
However, observations of temperatures along three dimensions show that |
772 |
inhomogeneity occurs in scaling RNEMD simulations, particularly in the |
773 |
two slabs which were scaled. Figure \ref{shear} (c) indicate that with |
774 |
relatively large imposed momentum flux, the temperature difference among $x$ |
775 |
and the other two dimensions was significant. This would result from the |
776 |
algorithm of scaling method. From Eq. \ref{eq:fluxPlane}, after |
777 |
momentum gradient is set up, $P_c^x$ would be roughly stable |
778 |
($<0$). Consequently, scaling factor $x$ would most probably larger |
779 |
than 1. Therefore, the kinetic energy in $x$-dimension $K_c^x$ would |
780 |
keep increase after most scaling steps. And if there is not enough time |
781 |
for the kinetic energy to exchange among different dimensions and |
782 |
different slabs, the system would finally build up temperature |
783 |
(kinetic energy) difference among the three dimensions. Also, between |
784 |
$y$ and $z$ dimensions in the scaled slabs, temperatures of $z$-axis |
785 |
are closer to neighbor slabs. This is due to momentum transfer along |
786 |
$z$ dimension between slabs. |
787 |
|
788 |
Although results between scaling and swapping methods are comparable, |
789 |
the inherent temperature inhomogeneity even in relatively low imposed |
790 |
exchange momentum flux simulations makes scaling RNEMD method less |
791 |
attractive than swapping RNEMD in shear viscosity calculation. |
792 |
|
793 |
\section{Conclusions} |
794 |
NIVS-RNEMD simulation method is developed and tested on various |
795 |
systems. Simulation results demonstrate its validity in thermal |
796 |
conductivity calculations, from Lennard-Jones fluid to multi-atom |
797 |
molecule like water and metal crystals. NIVS-RNEMD improves |
798 |
non-Boltzmann-Maxwell distributions, which exist in previous RNEMD |
799 |
methods. Furthermore, it develops a valid means for unphysical thermal |
800 |
transfer between different species of molecules, and thus extends its |
801 |
applicability to interfacial systems. Our calculation of gold/water |
802 |
interfacial thermal conductivity demonstrates this advantage over |
803 |
previous RNEMD methods. NIVS-RNEMD has also limited application on |
804 |
shear viscosity calculations, but could cause temperature difference |
805 |
among different dimensions under high momentum flux. Modification is |
806 |
necessary to extend the applicability of NIVS-RNEMD in shear viscosity |
807 |
calculations. |
808 |
|
809 |
\section{Acknowledgments} |
810 |
Support for this project was provided by the National Science |
811 |
Foundation under grant CHE-0848243. Computational time was provided by |
812 |
the Center for Research Computing (CRC) at the University of Notre |
813 |
Dame. \newpage |
814 |
|
815 |
\bibliographystyle{aip} |
816 |
\bibliography{nivsRnemd} |
817 |
|
818 |
\end{doublespace} |
819 |
\end{document} |
820 |
|