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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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|
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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 earlier |
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Reverse Non-Equilibrium Molecular Dynamics (RNEMD) methods which use |
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momentum exchange swapping moves that can create non-thermal |
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velocity distributions and are difficult to use for interfacial |
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calculations. By using non-isotropic velocity scaling (NIVS) on the |
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molecules in specific regions of a system, it is possible to impose |
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momentum or thermal flux between regions of a simulation and stable |
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thermal and momentum gradients can then be established. The scaling |
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method we have developed conserves the total linear momentum and |
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total energy of the system. To test the methods, we have computed |
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the thermal conductivity of model liquid and solid systems as well |
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as the interfacial thermal conductivity of a metal-water interface. |
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We find that the NIVS-RNEMD improves the problematic velocity |
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distributions that develop in other RNEMD methods. |
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\end{abstract} |
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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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kinetic energy from the cold slab to the hot slab. If the hot and |
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cold slabs are separated along the z-axis, the energy flux is given |
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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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\sum_{\alpha = x,y,z} K_c^\alpha \alpha^2 = \sum_{\alpha = x,y,z} |
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K_c^\alpha -J_z \Delta t |
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\label{eq:fluxEllipsoid} |
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\end{equation} |
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The spatial extent of the {\it thermal flux ellipsoid} 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,\cite{Hoffman:2001sf,384119} but numerically finding |
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the roots of high-degree polynomials is generally an ill-conditioned |
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problem.\cite{Hoffman:2001sf}[P157] One simplification is to maintain velocity |
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scalings that are {\it as isotropic as possible}. To do this, we |
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impose $x=y$, and to treat both the constraint and flux ellipsoids as |
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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 three |
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scaling 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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It is also possible to use this method to magnify the random |
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fluctuations of the average momentum in each of the bins to induce a |
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momentum flux. Doing this repeatedly will create a shear stress on |
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the system which will respond with an easily-measured strain. The |
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momentum flux (say along the $x$-direction) may be 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} |
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\end{equation} |
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This {\it momentum flux plane} is perpendicular to the $x$-axis, with |
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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 |
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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 other |
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words, 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 code, |
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OpenMD,\cite{Meineke:2005gd,openmd} performing the NIVS scaling moves |
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after an MD step with a variable frequency. We have tested the method |
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in a variety of different systems, including homogeneous fluids |
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(Lennard-Jones and SPC/E water), crystalline solids ({\sc |
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eam}~\cite{PhysRevB.33.7983} and quantum Sutton-Chen ({\sc |
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q-sc})~\cite{PhysRevB.59.3527} models for Gold), and heterogeneous |
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interfaces (QSC gold - SPC/E water). The last of these systems would |
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have been difficult to study using previous RNEMD methods, but using |
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velocity scaling moves, we can even obtain estimates of the |
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interfacial thermal conductivities ($G$). |
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|
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\subsection{Simulation Cells} |
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|
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In each of the systems studied, the dynamics was carried out in a |
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rectangular simulation cell using periodic boundary conditions in all |
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three dimensions. The cells were longer along the $z$ axis and the |
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space was divided into $N$ slabs along this axis (typically $N=20$). |
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The top slab ($n=1$) was designated the ``hot'' slab, while the |
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central slab ($n= N/2 + 1$) was designated the ``cold'' slab. In all |
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cases, simulations were first thermalized in canonical ensemble (NVT) |
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using a Nos\'{e}-Hoover thermostat.\cite{Hoover85} then equilibrated in |
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microcanonical ensemble (NVE) before introducing any non-equilibrium |
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method. |
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|
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\subsection{RNEMD with M\"{u}ller-Plathe swaps} |
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|
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In order to compare our new methodology with the original |
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M\"{u}ller-Plathe swapping algorithm,\cite{ISI:000080382700030} we |
356 |
first performed simulations using the original technique. |
357 |
|
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\subsection{RNEMD with NIVS scaling} |
359 |
|
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For each simulation utilizing the swapping method, a corresponding |
361 |
NIVS-RNEMD simulation was carried out using a target momentum flux set |
362 |
to produce a the same momentum or energy flux exhibited in the |
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swapping simulation. |
364 |
|
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To test the temperature homogeneity (and to compute transport |
366 |
coefficients), directional momentum and temperature distributions were |
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accumulated for molecules in each of the slabs. |
368 |
|
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\subsection{Shear viscosities} |
370 |
|
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The momentum flux was calculated using the total non-physical momentum |
372 |
transferred (${P_x}$) and the data collection time ($t$): |
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\begin{equation} |
374 |
j_z(p_x) = \frac{P_x}{2 t L_x L_y} |
375 |
\end{equation} |
376 |
where $L_x$ and $L_y$ denote the $x$ and $y$ lengths of the simulation |
377 |
box. The factor of two in the denominator is present because physical |
378 |
momentum transfer occurs in two directions due to our periodic |
379 |
boundary conditions. The velocity gradient ${\langle \partial v_x |
380 |
/\partial z \rangle}$ was obtained using linear regression of the |
381 |
velocity profiles in the bins. For Lennard-Jones simulations, shear |
382 |
viscosities are reporte in reduced units (${\eta^* = \eta \sigma^2 |
383 |
(\varepsilon m)^{-1/2}}$). |
384 |
|
385 |
\subsection{Thermal Conductivities} |
386 |
|
387 |
The energy flux was calculated similarly to the momentum flux, using |
388 |
the total non-physical energy transferred (${E_{total}}$) and the data |
389 |
collection time $t$: |
390 |
\begin{equation} |
391 |
J_z = \frac{E_{total}}{2 t L_x L_y} |
392 |
\end{equation} |
393 |
The temperature gradient ${\langle\partial T/\partial z\rangle}$ was |
394 |
obtained by a linear regression of the temperature profile. For |
395 |
Lennard-Jones simulations, thermal conductivities are reported in |
396 |
reduced units (${\lambda^*=\lambda \sigma^2 m^{1/2} |
397 |
k_B^{-1}\varepsilon^{-1/2}}$). |
398 |
|
399 |
\subsection{Interfacial Thermal Conductivities} |
400 |
|
401 |
For materials with a relatively low interfacial conductance, and in |
402 |
cases where the flux between the materials is small, the bulk regions |
403 |
on either side of an interface rapidly come to a state in which the |
404 |
two phases have relatively homogeneous (but distinct) temperatures. |
405 |
In calculating the interfacial thermal conductivity $G$, this |
406 |
assumption was made, and the conductance can be approximated as: |
407 |
|
408 |
\begin{equation} |
409 |
G = \frac{E_{total}}{2 t L_x L_y \left( \langle T_{gold}\rangle - |
410 |
\langle T_{water}\rangle \right)} |
411 |
\label{interfaceCalc} |
412 |
\end{equation} |
413 |
where ${E_{total}}$ is the imposed non-physical kinetic energy |
414 |
transfer and ${\langle T_{gold}\rangle}$ and ${\langle |
415 |
T_{water}\rangle}$ are the average observed temperature of gold and |
416 |
water phases respectively. |
417 |
|
418 |
\section{Results} |
419 |
|
420 |
\subsection{Lennard-Jones Fluid} |
421 |
2592 Lennard-Jones atoms were placed in an orthorhombic cell |
422 |
${10.06\sigma \times 10.06\sigma \times 30.18\sigma}$ on a side. The |
423 |
reduced density ${\rho^* = \rho\sigma^3}$ was thus 0.85, which enabled |
424 |
direct comparison between our results and previous methods. These |
425 |
simulations were carried out with a reduced timestep ${\tau^* = |
426 |
4.6\times10^{-4}}$. For the shear viscosity calculations, the mean |
427 |
temperature was ${T^* = k_B T/\varepsilon = 0.72}$. For thermal |
428 |
conductivity calculations, simulations were first run under reduced |
429 |
temperature ${\langle T^*\rangle = 0.72}$ in the microcanonical |
430 |
ensemble, but other temperatures ([XXX, YYY, and ZZZ]) were also |
431 |
sampled. The simulations included $10^5$ steps of equilibration |
432 |
without any momentum flux, $10^5$ steps of stablization with an |
433 |
imposed momentum transfer to create a gradient, and $10^6$ steps of |
434 |
data collection under RNEMD. |
435 |
|
436 |
\subsubsection*{Thermal Conductivity} |
437 |
|
438 |
Our thermal conductivity calculations show that the NIVS method agrees |
439 |
well with the swapping method. Four different swap intervals were |
440 |
tested (Table \ref{LJ}). With a fixed scaling interval of 10 time steps, |
441 |
the target exchange kinetic energy produced equivalent kinetic energy |
442 |
flux as in the swapping method. Similar thermal gradients were |
443 |
observed with similar thermal flux under the two different methods |
444 |
(Figure \ref{thermalGrad}). |
445 |
|
446 |
\begin{table*} |
447 |
\begin{minipage}{\linewidth} |
448 |
\begin{center} |
449 |
|
450 |
\caption{Thermal conductivity ($\lambda^*$) and shear viscosity |
451 |
($\eta^*$) (in reduced units) of a Lennard-Jones fluid at |
452 |
${\langle T^* \rangle = 0.72}$ and ${\rho^* = 0.85}$ computed |
453 |
at various momentum fluxes. The original swapping method and |
454 |
the velocity scaling method give similar results. |
455 |
Uncertainties are indicated in parentheses.} |
456 |
|
457 |
\begin{tabular}{|cc|cc|cc|} |
458 |
\hline |
459 |
\multicolumn{2}{|c}{Momentum Exchange} & |
460 |
\multicolumn{2}{|c}{Swapping RNEMD} & |
461 |
\multicolumn{2}{|c|}{NIVS-RNEMD} \\ |
462 |
\hline |
463 |
\multirow{2}{*}{Swap Interval (fs)} & Equivalent $J_z^*$ or & |
464 |
|
465 |
\multirow{2}{*}{$\lambda^*_{swap}$} & |
466 |
\multirow{2}{*}{$\eta^*_{swap}$} & |
467 |
\multirow{2}{*}{$\lambda^*_{scale}$} & |
468 |
\multirow{2}{*}{$\eta^*_{scale}$} \\ |
469 |
& $j_p^*(v_x)$ (reduced units) & & & & \\ |
470 |
\hline |
471 |
250 & 0.16 & 7.03(0.34) & & 7.30(0.10) & \\ |
472 |
500 & 0.09 & 7.03(0.14) & 3.64(0.05) & 6.95(0.09) & 3.76(0.09)\\ |
473 |
1000 & 0.047 & 6.91(0.42) & 3.52(0.16) & 7.19(0.07) & 3.66(0.06)\\ |
474 |
2000 & 0.024 & 7.52(0.15) & 3.72(0.05) & 7.19(0.28) & 3.32(0.18)\\ |
475 |
2500 & 0.019 & & 3.42(0.06) & & 3.43(0.08)\\ |
476 |
\hline |
477 |
\end{tabular} |
478 |
\label{LJ} |
479 |
\end{center} |
480 |
\end{minipage} |
481 |
\end{table*} |
482 |
|
483 |
\begin{figure} |
484 |
\includegraphics[width=\linewidth]{thermalGrad} |
485 |
\caption{NIVS-RNEMD method creates similar temperature gradients |
486 |
compared with the swapping method under a variety of imposed kinetic |
487 |
energy flux values.} |
488 |
\label{thermalGrad} |
489 |
\end{figure} |
490 |
|
491 |
\subsubsection*{Velocity Distributions} |
492 |
|
493 |
During these simulations, velocities were recorded every 1000 steps |
494 |
and was used to produce distributions of both velocity and speed in |
495 |
each of the slabs. From these distributions, we observed that under |
496 |
relatively high non-physical kinetic energy flux, the speed of |
497 |
molecules in slabs where swapping occured could deviate from the |
498 |
Maxwell-Boltzmann distribution. This behavior was also noted by Tenney |
499 |
and Maginn.\cite{Maginn:2010} Figure \ref{thermalHist} shows how these |
500 |
distributions deviate from an ideal distribution. In the ``hot'' slab, |
501 |
the probability density is notched at low speeds and has a substantial |
502 |
shoulder at higher speeds relative to the ideal MB distribution. In |
503 |
the cold slab, the opposite notching and shouldering occurs. This |
504 |
phenomenon is more obvious at higher swapping rates. |
505 |
|
506 |
In the velocity distributions, the ideal Gaussian peak is |
507 |
substantially flattened in the hot slab, and is overly sharp (with |
508 |
truncated wings) in the cold slab. This problem is rooted in the |
509 |
mechanism of the swapping method. Continually depleting low (high) |
510 |
speed particles in the high (low) temperature slab is not complemented |
511 |
by diffusions of low (high) speed particles from neighboring slabs, |
512 |
unless the swapping rate is sufficiently small. Simutaneously, surplus |
513 |
low speed particles in the low temperature slab do not have sufficient |
514 |
time to diffuse to neighboring slabs. Since the thermal exchange rate |
515 |
must reach a minimum level to produce an observable thermal gradient, |
516 |
the swapping-method RNEMD has a relatively narrow choice of exchange |
517 |
times that can be utilized. |
518 |
|
519 |
For comparison, NIVS-RNEMD produces a speed distribution closer to the |
520 |
Maxwell-Boltzmann curve (Figure \ref{thermalHist}). The reason for |
521 |
this is simple; upon velocity scaling, a Gaussian distribution remains |
522 |
Gaussian. Although a single scaling move is non-isotropic in three |
523 |
dimensions, our criteria for choosing a set of scaling coefficients |
524 |
helps maintain the distributions as close to isotropic as possible. |
525 |
Consequently, NIVS-RNEMD is able to impose an unphysical thermal flux |
526 |
as the previous RNEMD methods but without large perturbations to the |
527 |
velocity distributions in the two slabs. |
528 |
|
529 |
\begin{figure} |
530 |
\includegraphics[width=\linewidth]{thermalHist} |
531 |
\caption{Speed distribution for thermal conductivity using a) |
532 |
``swapping'' and b) NIVS- RNEMD methods. Shown is from the |
533 |
simulations with an exchange or equilvalent exchange interval of 250 |
534 |
fs. In circled areas, distributions from ``swapping'' RNEMD |
535 |
simulation have deviation from ideal Maxwell-Boltzmann distribution |
536 |
(curves fit for each distribution).} |
537 |
\label{thermalHist} |
538 |
\end{figure} |
539 |
|
540 |
|
541 |
\subsubsection*{Shear Viscosity} |
542 |
Our calculations (Table \ref{LJ}) show that velocity-scaling |
543 |
RNEMD predicted comparable shear viscosities to swap RNEMD method. All |
544 |
the scale method results were from simulations that had a scaling |
545 |
interval of 10 time steps. The average molecular momentum gradients of |
546 |
these samples are shown in Figure \ref{shear} (a) and (b). |
547 |
|
548 |
\begin{figure} |
549 |
\includegraphics[width=\linewidth]{shear} |
550 |
\caption{Average momentum gradients in shear viscosity simulations, |
551 |
using (a) ``swapping'' method and (b) NIVS-RNEMD method |
552 |
respectively. (c) Temperature difference among x and y, z dimensions |
553 |
observed when using NIVS-RNEMD with equivalent exchange interval of |
554 |
500 fs.} |
555 |
\label{shear} |
556 |
\end{figure} |
557 |
|
558 |
However, observations of temperatures along three dimensions show that |
559 |
inhomogeneity occurs in scaling RNEMD simulations, particularly in the |
560 |
two slabs which were scaled. Figure \ref{shear} (c) indicate that with |
561 |
relatively large imposed momentum flux, the temperature difference among $x$ |
562 |
and the other two dimensions was significant. This would result from the |
563 |
algorithm of scaling method. From Eq. \ref{eq:fluxPlane}, after |
564 |
momentum gradient is set up, $P_c^x$ would be roughly stable |
565 |
($<0$). Consequently, scaling factor $x$ would most probably larger |
566 |
than 1. Therefore, the kinetic energy in $x$-dimension $K_c^x$ would |
567 |
keep increase after most scaling steps. And if there is not enough time |
568 |
for the kinetic energy to exchange among different dimensions and |
569 |
different slabs, the system would finally build up temperature |
570 |
(kinetic energy) difference among the three dimensions. Also, between |
571 |
$y$ and $z$ dimensions in the scaled slabs, temperatures of $z$-axis |
572 |
are closer to neighbor slabs. This is due to momentum transfer along |
573 |
$z$ dimension between slabs. |
574 |
|
575 |
Although results between scaling and swapping methods are comparable, |
576 |
the inherent temperature inhomogeneity even in relatively low imposed |
577 |
exchange momentum flux simulations makes scaling RNEMD method less |
578 |
attractive than swapping RNEMD in shear viscosity calculation. |
579 |
|
580 |
|
581 |
\subsection{Bulk SPC/E water} |
582 |
|
583 |
We compared the thermal conductivity of SPC/E water using NIVS-RNEMD |
584 |
to the work of Bedrov {\it et al.}\cite{Bedrov:2000}, which employed |
585 |
the original swapping RNEMD method. Bedrov {\it et |
586 |
al.}\cite{Bedrov:2000} argued that exchange of the molecule |
587 |
center-of-mass velocities instead of single atom velocities in a |
588 |
molecule conserves the total kinetic energy and linear momentum. This |
589 |
principle is also adopted in our simulations. Scaling was applied to |
590 |
the center-of-mass velocities of the rigid bodies of SPC/E model water |
591 |
molecules. |
592 |
|
593 |
To construct the simulations, a simulation box consisting of 1000 |
594 |
molecules were first equilibrated under ambient pressure and |
595 |
temperature conditions using the isobaric-isothermal (NPT) |
596 |
ensemble.\cite{melchionna93} A fixed volume was chosen to match the |
597 |
average volume observed in the NPT simulations, and this was followed |
598 |
by equilibration, first in the canonical (NVT) ensemble, followed by a |
599 |
100ps period under constant-NVE conditions without any momentum |
600 |
flux. 100ps was allowed to stabilize the system with an imposed |
601 |
momentum transfer to create a gradient, and 1ns was alotted for |
602 |
data collection under RNEMD. |
603 |
|
604 |
As shown in Figure \ref{spceGrad}, temperature gradients were |
605 |
established similar to the previous work. However, the average |
606 |
temperature of our system is 300K, while that in Bedrov {\it et al.} |
607 |
is 318K, which would be attributed for part of the difference between |
608 |
the final calculation results (Table \ref{spceThermal}). [WHY DIDN'T |
609 |
WE DO 318 K?] Both methods yield values in reasonable agreement with |
610 |
experiment [DONE AT WHAT TEMPERATURE?] |
611 |
|
612 |
\begin{figure} |
613 |
\includegraphics[width=\linewidth]{spceGrad} |
614 |
\caption{Temperature gradients in SPC/E water thermal conductivity |
615 |
simulations.} |
616 |
\label{spceGrad} |
617 |
\end{figure} |
618 |
|
619 |
\begin{table*} |
620 |
\begin{minipage}{\linewidth} |
621 |
\begin{center} |
622 |
|
623 |
\caption{Thermal conductivity of SPC/E water under various |
624 |
imposed thermal gradients. Uncertainties are indicated in |
625 |
parentheses.} |
626 |
|
627 |
\begin{tabular}{|c|ccc|} |
628 |
\hline |
629 |
\multirow{2}{*}{$\langle dT/dz\rangle$(K/\AA)} & \multicolumn{3}{|c|}{$\lambda |
630 |
(\mathrm{W m}^{-1} \mathrm{K}^{-1})$} \\ |
631 |
& This work (300K) & Previous simulations (318K)\cite{Bedrov:2000} & |
632 |
Experiment\cite{WagnerKruse}\\ |
633 |
\hline |
634 |
0.38 & 0.816(0.044) & & 0.64\\ |
635 |
0.81 & 0.770(0.008) & 0.784 & \\ |
636 |
1.54 & 0.813(0.007) & 0.730 & \\ |
637 |
\hline |
638 |
\end{tabular} |
639 |
\label{spceThermal} |
640 |
\end{center} |
641 |
\end{minipage} |
642 |
\end{table*} |
643 |
|
644 |
\subsection{Crystalline Gold} |
645 |
|
646 |
To see how the method performed in a solid, we calculated thermal |
647 |
conductivities using two atomistic models for gold. Several different |
648 |
potential models have been developed that reasonably describe |
649 |
interactions in transition metals. In particular, the Embedded Atom |
650 |
Model ({\sc eam})~\cite{PhysRevB.33.7983} and Sutton-Chen ({\sc |
651 |
sc})~\cite{Chen90} potential have been used to study a wide range of |
652 |
phenomena in both bulk materials and |
653 |
nanoparticles.\cite{ISI:000207079300006,Clancy:1992,Vardeman:2008fk,Vardeman-II:2001jn,ShibataT._ja026764r,Sankaranarayanan:2005lr,Chui:2003fk,Wang:2005qy,Medasani:2007uq} |
654 |
Both potentials are based on a model of a metal which treats the |
655 |
nuclei and core electrons as pseudo-atoms embedded in the electron |
656 |
density due to the valence electrons on all of the other atoms in the |
657 |
system. The {\sc sc} potential has a simple form that closely |
658 |
resembles the Lennard Jones potential, |
659 |
\begin{equation} |
660 |
\label{eq:SCP1} |
661 |
U_{tot}=\sum _{i}\left[ \frac{1}{2}\sum _{j\neq i}D_{ij}V^{pair}_{ij}(r_{ij})-c_{i}D_{ii}\sqrt{\rho_{i}}\right] , |
662 |
\end{equation} |
663 |
where $V^{pair}_{ij}$ and $\rho_{i}$ are given by |
664 |
\begin{equation} |
665 |
\label{eq:SCP2} |
666 |
V^{pair}_{ij}(r)=\left( \frac{\alpha_{ij}}{r_{ij}}\right)^{n_{ij}}, \rho_{i}=\sum_{j\neq i}\left( \frac{\alpha_{ij}}{r_{ij}}\right) ^{m_{ij}}. |
667 |
\end{equation} |
668 |
$V^{pair}_{ij}$ is a repulsive pairwise potential that accounts for |
669 |
interactions between the pseudoatom cores. The $\sqrt{\rho_i}$ term in |
670 |
Eq. (\ref{eq:SCP1}) is an attractive many-body potential that models |
671 |
the interactions between the valence electrons and the cores of the |
672 |
pseudo-atoms. $D_{ij}$, $D_{ii}$ set the appropriate overall energy |
673 |
scale, $c_i$ scales the attractive portion of the potential relative |
674 |
to the repulsive interaction and $\alpha_{ij}$ is a length parameter |
675 |
that assures a dimensionless form for $\rho$. These parameters are |
676 |
tuned to various experimental properties such as the density, cohesive |
677 |
energy, and elastic moduli for FCC transition metals. The quantum |
678 |
Sutton-Chen ({\sc q-sc}) formulation matches these properties while |
679 |
including zero-point quantum corrections for different transition |
680 |
metals.\cite{PhysRevB.59.3527} The {\sc eam} functional forms differ |
681 |
slightly from {\sc sc} but the overall method is very similar. |
682 |
|
683 |
In this work, we have utilized both the {\sc eam} and the {\sc q-sc} |
684 |
potentials to test the behavior of scaling RNEMD. |
685 |
|
686 |
A face-centered-cubic (FCC) lattice was prepared containing 2880 Au |
687 |
atoms (i.e. ${6\times 6\times 20}$ unit cells). Simulations were run |
688 |
both with and without isobaric-isothermal (NPT)~\cite{melchionna93} |
689 |
pre-equilibration at a target pressure of 1 atm. When equilibrated |
690 |
under NPT conditions, our simulation box expanded by approximately 1\% |
691 |
in volume. Following adjustment of the box volume, equilibrations in |
692 |
both the canonical and microcanonical ensembles were carried out. With |
693 |
the simulation cell divided evenly into 10 slabs, different thermal |
694 |
gradients were established by applying a set of target thermal |
695 |
transfer fluxes. |
696 |
|
697 |
The results for the thermal conductivity of gold are shown in Table |
698 |
\ref{AuThermal}. In these calculations, the end and middle slabs were |
699 |
excluded in thermal gradient linear regession. {\sc eam} predicts |
700 |
slightly larger thermal conductivities than {\sc q-sc}. However, both |
701 |
values are smaller than experimental value by a factor of more than |
702 |
200. This behavior has been observed previously by Richardson and |
703 |
Clancy, and has been attributed to the lack of electronic effects in |
704 |
these force fields.\cite{Clancy:1992} The non-equilibrium MD method |
705 |
employed in their simulations gave an thermal conductance estimation |
706 |
of [FORCE FIELD] gold as [RESULT IN REF], which is comparable to ours. It |
707 |
should be noted that the density of the metal being simulated also |
708 |
greatly affects the thermal conductivity. With an expanded lattice, |
709 |
lower thermal conductance is expected (and observed). We also observed |
710 |
a decrease in thermal conductance at higher temperatures, a trend that |
711 |
agrees with experimental measurements [PAGE |
712 |
NUMBERS?].\cite{AshcroftMermin} |
713 |
|
714 |
\begin{table*} |
715 |
\begin{minipage}{\linewidth} |
716 |
\begin{center} |
717 |
|
718 |
\caption{Calculated thermal conductivity of crystalline gold |
719 |
using two related force fields. Calculations were done at both |
720 |
experimental and equilibrated densities and at a range of |
721 |
temperatures and thermal flux rates. Uncertainties are |
722 |
indicated in parentheses. [SWAPPING COMPARISON?]} |
723 |
|
724 |
\begin{tabular}{|c|c|c|cc|} |
725 |
\hline |
726 |
Force Field & $\rho$ (g/cm$^3$) & ${\langle T\rangle}$ (K) & |
727 |
$\langle dT/dz\rangle$ (K/\AA) & $\lambda (\mathrm{W m}^{-1} \mathrm{K}^{-1})$\\ |
728 |
\hline |
729 |
\multirow{7}{*}{\sc q-sc} & \multirow{3}{*}{19.188} & \multirow{3}{*}{300} & 1.44 & 1.10(0.06)\\ |
730 |
& & & 2.86 & 1.08(0.05)\\ |
731 |
& & & 5.14 & 1.15(0.07)\\\cline{2-5} |
732 |
& \multirow{4}{*}{19.263} & \multirow{2}{*}{300} & 2.31 & 1.25(0.06)\\ |
733 |
& & & 3.02 & 1.26(0.05)\\\cline{3-5} |
734 |
& & \multirow{2}{*}{575} & 3.02 & 1.02(0.07)\\ |
735 |
& & & 4.84 & 0.92(0.05)\\ |
736 |
\hline |
737 |
\multirow{8}{*}{\sc eam} & \multirow{3}{*}{19.045} & \multirow{3}{*}{300} & 1.24 & 1.24(0.16)\\ |
738 |
& & & 2.06 & 1.37(0.04)\\ |
739 |
& & & 2.55 & 1.41(0.07)\\\cline{2-5} |
740 |
& \multirow{5}{*}{19.263} & \multirow{3}{*}{300} & 1.06 & 1.45(0.13)\\ |
741 |
& & & 2.04 & 1.41(0.07)\\ |
742 |
& & & 2.41 & 1.53(0.10)\\\cline{3-5} |
743 |
& & \multirow{2}{*}{575} & 2.82 & 1.08(0.03)\\ |
744 |
& & & 4.14 & 1.08(0.05)\\ |
745 |
\hline |
746 |
\end{tabular} |
747 |
\label{AuThermal} |
748 |
\end{center} |
749 |
\end{minipage} |
750 |
\end{table*} |
751 |
|
752 |
\subsection{Thermal Conductance at the Au/H$_2$O interface} |
753 |
The most attractive aspect of the scaling approach for RNEMD is the |
754 |
ability to use the method in non-homogeneous systems, where molecules |
755 |
of different identities are segregated in different slabs. To test |
756 |
this application, we simulated a Gold (111) / water interface. To |
757 |
construct the interface, a box containing a lattice of 1188 Au atoms |
758 |
(with the 111 surface in the +z and -z directions) was allowed to |
759 |
relax under ambient temperature and pressure. A separate (but |
760 |
identically sized) box of SPC/E water was also equilibrated at ambient |
761 |
conditions. The two boxes were combined by removing all water |
762 |
molecules within 3 \AA radius of any gold atom. The final |
763 |
configuration contained 1862 SPC/E water molecules. |
764 |
|
765 |
After simulations of bulk water and crystal gold, a mixture system was |
766 |
constructed, consisting of 1188 Au atoms and 1862 H$_2$O |
767 |
molecules. Spohr potential was adopted in depicting the interaction |
768 |
between metal atom and water molecule.\cite{ISI:000167766600035} A |
769 |
similar protocol of equilibration was followed. Several thermal |
770 |
gradients was built under different target thermal flux. It was found |
771 |
out that compared to our previous simulation systems, the two phases |
772 |
could have large temperature difference even under a relatively low |
773 |
thermal flux. |
774 |
|
775 |
|
776 |
After simulations of homogeneous water and gold systems using |
777 |
NIVS-RNEMD method were proved valid, calculation of gold/water |
778 |
interfacial thermal conductivity was followed. It is found out that |
779 |
the low interfacial conductance is probably due to the hydrophobic |
780 |
surface in our system. Figure \ref{interface} (a) demonstrates mass |
781 |
density change along $z$-axis, which is perpendicular to the |
782 |
gold/water interface. It is observed that water density significantly |
783 |
decreases when approaching the surface. Under this low thermal |
784 |
conductance, both gold and water phase have sufficient time to |
785 |
eliminate temperature difference inside respectively (Figure |
786 |
\ref{interface} b). With indistinguishable temperature difference |
787 |
within respective phase, it is valid to assume that the temperature |
788 |
difference between gold and water on surface would be approximately |
789 |
the same as the difference between the gold and water phase. This |
790 |
assumption enables convenient calculation of $G$ using |
791 |
Eq. \ref{interfaceCalc} instead of measuring temperatures of thin |
792 |
layer of water and gold close enough to surface, which would have |
793 |
greater fluctuation and lower accuracy. Reported results (Table |
794 |
\ref{interfaceRes}) are of two orders of magnitude smaller than our |
795 |
calculations on homogeneous systems, and thus have larger relative |
796 |
errors than our calculation results on homogeneous systems. |
797 |
|
798 |
\begin{figure} |
799 |
\includegraphics[width=\linewidth]{interface} |
800 |
\caption{Simulation results for Gold/Water interfacial thermal |
801 |
conductivity: (a) Significant water density decrease is observed on |
802 |
crystalline gold surface, which indicates low surface contact and |
803 |
leads to low thermal conductance. (b) Temperature profiles for a |
804 |
series of simulations. Temperatures of different slabs in the same |
805 |
phase show no significant differences.} |
806 |
\label{interface} |
807 |
\end{figure} |
808 |
|
809 |
\begin{table*} |
810 |
\begin{minipage}{\linewidth} |
811 |
\begin{center} |
812 |
|
813 |
\caption{Computed interfacial thermal conductivity ($G$) values |
814 |
for the Au(111) / water interface at ${\langle T\rangle \sim}$ |
815 |
300K using a range of energy fluxes. Uncertainties are |
816 |
indicated in parentheses. } |
817 |
|
818 |
\begin{tabular}{|cccc|} |
819 |
\hline |
820 |
$J_z$ (MW/m$^2$) & $\langle T_{gold} \rangle$ (K) & $\langle |
821 |
T_{water} \rangle$ (K) & $G$ |
822 |
(MW/m$^2$/K)\\ |
823 |
\hline |
824 |
98.0 & 355.2 & 295.8 & 1.65(0.21) \\ |
825 |
78.8 & 343.8 & 298.0 & 1.72(0.32) \\ |
826 |
73.6 & 344.3 & 298.0 & 1.59(0.24) \\ |
827 |
49.2 & 330.1 & 300.4 & 1.65(0.35) \\ |
828 |
\hline |
829 |
\end{tabular} |
830 |
\label{interfaceRes} |
831 |
\end{center} |
832 |
\end{minipage} |
833 |
\end{table*} |
834 |
|
835 |
|
836 |
\section{Conclusions} |
837 |
NIVS-RNEMD simulation method is developed and tested on various |
838 |
systems. Simulation results demonstrate its validity in thermal |
839 |
conductivity calculations, from Lennard-Jones fluid to multi-atom |
840 |
molecule like water and metal crystals. NIVS-RNEMD improves |
841 |
non-Boltzmann-Maxwell distributions, which exist in previous RNEMD |
842 |
methods. Furthermore, it develops a valid means for unphysical thermal |
843 |
transfer between different species of molecules, and thus extends its |
844 |
applicability to interfacial systems. Our calculation of gold/water |
845 |
interfacial thermal conductivity demonstrates this advantage over |
846 |
previous RNEMD methods. NIVS-RNEMD has also limited application on |
847 |
shear viscosity calculations, but could cause temperature difference |
848 |
among different dimensions under high momentum flux. Modification is |
849 |
necessary to extend the applicability of NIVS-RNEMD in shear viscosity |
850 |
calculations. |
851 |
|
852 |
\section{Acknowledgments} |
853 |
Support for this project was provided by the National Science |
854 |
Foundation under grant CHE-0848243. Computational time was provided by |
855 |
the Center for Research Computing (CRC) at the University of Notre |
856 |
Dame. \newpage |
857 |
|
858 |
\bibliographystyle{aip} |
859 |
\bibliography{nivsRnemd} |
860 |
|
861 |
\end{doublespace} |
862 |
\end{document} |
863 |
|