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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 gradients in molecular dynamics simulations |
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of heterogeneous systems. This method extends earlier Reverse |
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Non-Equilibrium Molecular Dynamics (RNEMD) methods which use |
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momentum exchange swapping moves. The standard swapping moves can |
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create non-thermal velocity distributions and 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 while conserving the linear momentum and total energy of |
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the system. To test the method, we have computed the thermal |
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conductivity of model liquid and solid systems as well as the |
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interfacial thermal conductivity of a metal-water interface. We |
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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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|
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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% BODY OF TEXT |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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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 and integrating long-time correlation functions that are |
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subject to noise issues. |
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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. Furthermore, the method is applicable with |
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different ensembles, unlike the heat-exchange algorithm proposed by |
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Hafskjold {\it et al.} \cite{HeX:1994,HeX:1993}, which is incompatible |
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with non-microcanonical ensemble. |
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|
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Recently, Tenney and Maginn\cite{Maginn:2010} have discovered some |
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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 proposed self-adjusting metrics for retaining the usability of the |
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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 v_{i\alpha} \\ |
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P_h^\alpha & = & \sum_{j = 1}^{N_h} m_j v_{j\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 momenta 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} \left\{ K_h^\alpha + K_c^\alpha \right\} = \sum_{\alpha = x,y,z} |
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\left\{ \left(\alpha^\prime\right)^2 K_h^\alpha + \alpha^2 K_c^\alpha \right\} |
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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} One simplification is 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 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 to be as gentle as possible, i.e. |
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${x,y,z \rightarrow 1}$, in order to avoid large perturbations to the |
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system. To do this, we pick the intersection point which maintains |
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the three scaling variables ${x, y, z}$ as well as the ratio of |
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kinetic energies ${K_c^z/K_c^x, K_c^z/K_c^y}$ as close as possible to |
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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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with a variable frequency after the molecular dynamics (MD) steps. We |
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have tested the method in a variety of different systems, including: |
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homogeneous fluids (Lennard-Jones and SPC/E water), crystalline |
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solids, using both the embedded atom method |
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(EAM)~\cite{PhysRevB.33.7983} and quantum Sutton-Chen |
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(QSC)~\cite{PhysRevB.59.3527} models for Gold, and heterogeneous |
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interfaces (QSC gold - SPC/E water). Even though previous RNEMD |
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methods might remain usable for the last of these systems, energy |
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transfer from imaginary elastic collisions would be less effective |
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when the two particles involved have larger mass difference, and thus |
348 |
affect the actuall implementation of these methods. However, our |
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current method can easily provide estimates of the interfacial thermal |
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conductivity ($G$). |
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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 |
359 |
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 |
363 |
method. |
364 |
|
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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 |
368 |
M\"{u}ller-Plathe swapping algorithm,\cite{ISI:000080382700030} we |
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first performed simulations using the original technique. At fixed |
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intervals, kinetic energy or momentum exchange moves were performed |
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between the hot and the cold slabs. The interval between exchange |
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moves governs the effective momentum flux ($j_z(p_x)$) or energy flux |
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($J_z$) between the two slabs so to vary this quantity, we performed |
374 |
simulations with a variety of delay intervals between the swapping moves. |
375 |
|
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For thermal conductivity measurements, the particle with smallest |
377 |
speed in the hot slab and the one with largest speed in the cold slab |
378 |
had their entire momentum vectors swapped. In the test cases run |
379 |
here, all particles had the same chemical identity and mass, so this |
380 |
move preserves both total linear momentum and total energy. It is |
381 |
also possible to exchange energy by assuming an elastic collision |
382 |
between the two particles which are exchanging energy. |
383 |
|
384 |
For shear stress simulations, the particle with the most negative |
385 |
$p_x$ in the hot slab and the one with the most positive $p_x$ in the |
386 |
cold slab exchanged only this component of their momentum vectors. |
387 |
|
388 |
\subsection{RNEMD with NIVS scaling} |
389 |
|
390 |
For each simulation utilizing the swapping method, a corresponding |
391 |
NIVS-RNEMD simulation was carried out using a target momentum flux set |
392 |
to produce the same flux experienced in the swapping simulation. |
393 |
|
394 |
To test the temperature homogeneity, momentum and temperature |
395 |
distributions (for all three dimensions) were accumulated for |
396 |
molecules in each of the slabs. Transport coefficients were computed |
397 |
using the temperature (and momentum) gradients across the $z$-axis as |
398 |
well as the total momentum flux the system experienced during data |
399 |
collection portion of the simulation. |
400 |
|
401 |
\subsection{Shear viscosities} |
402 |
|
403 |
The momentum flux was calculated using the total non-physical momentum |
404 |
transferred (${P_x}$) and the data collection time ($t$): |
405 |
\begin{equation} |
406 |
j_z(p_x) = \frac{P_x}{2 t L_x L_y} |
407 |
\end{equation} |
408 |
where $L_x$ and $L_y$ denote the $x$ and $y$ lengths of the simulation |
409 |
box. The factor of two in the denominator is present because physical |
410 |
momentum transfer between the slabs occurs in two directions ($+z$ and |
411 |
$-z$). The velocity gradient ${\langle \partial v_x /\partial z |
412 |
\rangle}$ was obtained using linear regression of the mean $x$ |
413 |
component of the velocity, $\langle v_x \rangle$, in each of the bins. |
414 |
For Lennard-Jones simulations, shear viscosities are reported in |
415 |
reduced units (${\eta^* = \eta \sigma^2 (\varepsilon m)^{-1/2}}$). |
416 |
|
417 |
\subsection{Thermal Conductivities} |
418 |
|
419 |
The energy flux was calculated in a similar manner to the momentum |
420 |
flux, using the total non-physical energy transferred (${E_{total}}$) |
421 |
and the data collection time $t$: |
422 |
\begin{equation} |
423 |
J_z = \frac{E_{total}}{2 t L_x L_y} |
424 |
\end{equation} |
425 |
The temperature gradient ${\langle\partial T/\partial z\rangle}$ was |
426 |
obtained by a linear regression of the temperature profile. For |
427 |
Lennard-Jones simulations, thermal conductivities are reported in |
428 |
reduced units (${\lambda^*=\lambda \sigma^2 m^{1/2} |
429 |
k_B^{-1}\varepsilon^{-1/2}}$). |
430 |
|
431 |
\subsection{Interfacial Thermal Conductivities} |
432 |
|
433 |
For interfaces with a relatively low interfacial conductance, the bulk |
434 |
regions on either side of an interface rapidly come to a state in |
435 |
which the two phases have relatively homogeneous (but distinct) |
436 |
temperatures. The interfacial thermal conductivity $G$ can therefore |
437 |
be approximated as: |
438 |
|
439 |
\begin{equation} |
440 |
G = \frac{E_{total}}{2 t L_x L_y \left( \langle T_\mathrm{hot}\rangle - |
441 |
\langle T_\mathrm{cold}\rangle \right)} |
442 |
\label{interfaceCalc} |
443 |
\end{equation} |
444 |
where ${E_{total}}$ is the imposed non-physical kinetic energy |
445 |
transfer and ${\langle T_\mathrm{hot}\rangle}$ and ${\langle |
446 |
T_\mathrm{cold}\rangle}$ are the average observed temperature of the |
447 |
two separated phases. If the interfacial conductance is {\it not} |
448 |
small, it would also be possible to compute the interfacial thermal |
449 |
conductivity using this method by computing the change in the slope of |
450 |
the thermal gradient ($\partial^2 \langle T \rangle / |
451 |
\partial z^2$) at the interface. |
452 |
|
453 |
\section{Results} |
454 |
|
455 |
\subsection{Lennard-Jones Fluid} |
456 |
2592 Lennard-Jones atoms were placed in an orthorhombic cell |
457 |
${10.06\sigma \times 10.06\sigma \times 30.18\sigma}$ on a side. The |
458 |
reduced density ${\rho^* = \rho\sigma^3}$ was thus 0.85, which enabled |
459 |
direct comparison between our results and previous methods. These |
460 |
simulations were carried out with a reduced timestep ${\tau^* = |
461 |
4.6\times10^{-4}}$. For the shear viscosity calculations, the mean |
462 |
temperature was ${T^* = k_B T/\varepsilon = 0.72}$. For thermal |
463 |
conductivity calculations, simulations were run under reduced |
464 |
temperature ${\langle T^*\rangle = 0.72}$ in the microcanonical |
465 |
ensemble. The simulations included $10^5$ steps of equilibration |
466 |
without any momentum flux, $10^5$ steps of stablization with an |
467 |
imposed momentum transfer to create a gradient, and $10^6$ steps of |
468 |
data collection under RNEMD. |
469 |
|
470 |
\subsubsection*{Thermal Conductivity} |
471 |
|
472 |
Our thermal conductivity calculations show that the NIVS method agrees |
473 |
well with the swapping method. Five different swap intervals were |
474 |
tested (Table \ref{LJ}). Similar thermal gradients were observed with |
475 |
similar thermal flux under the two different methods (Fig. |
476 |
\ref{thermalGrad}). Furthermore, the 1-d temperature profiles showed |
477 |
no observable differences between the $x$, $y$ and $z$ axes (lower |
478 |
panel of Fig. \ref{thermalGrad}), so even though we are using a |
479 |
non-isotropic scaling method, none of the three directions are |
480 |
experience disproportionate heating due to the velocity scaling. |
481 |
|
482 |
\begin{table*} |
483 |
\begin{minipage}{\linewidth} |
484 |
\begin{center} |
485 |
|
486 |
\caption{Thermal conductivity ($\lambda^*$) and shear viscosity |
487 |
($\eta^*$) (in reduced units) of a Lennard-Jones fluid at |
488 |
${\langle T^* \rangle = 0.72}$ and ${\rho^* = 0.85}$ computed |
489 |
at various momentum fluxes. The original swapping method and |
490 |
the velocity scaling method give similar results. |
491 |
Uncertainties are indicated in parentheses.} |
492 |
|
493 |
\begin{tabular}{|cc|cc|cc|} |
494 |
\hline |
495 |
\multicolumn{2}{|c}{Momentum Exchange} & |
496 |
\multicolumn{2}{|c}{Swapping RNEMD} & |
497 |
\multicolumn{2}{|c|}{NIVS-RNEMD} \\ |
498 |
\hline |
499 |
\multirow{2}{*}{Swap Interval (timesteps)} & Equivalent $J_z^*$ or & |
500 |
\multirow{2}{*}{$\lambda^*_{swap}$} & |
501 |
\multirow{2}{*}{$\eta^*_{swap}$} & |
502 |
\multirow{2}{*}{$\lambda^*_{scale}$} & |
503 |
\multirow{2}{*}{$\eta^*_{scale}$} \\ |
504 |
& $j_z^*(p_x)$ (reduced units) & & & & \\ |
505 |
\hline |
506 |
250 & 0.16 & 7.03(0.34) & 3.57(0.06) & 7.30(0.10) & 3.54(0.04)\\ |
507 |
500 & 0.09 & 7.03(0.14) & 3.64(0.05) & 6.95(0.09) & 3.76(0.09)\\ |
508 |
1000 & 0.047 & 6.91(0.42) & 3.52(0.16) & 7.19(0.07) & 3.66(0.06)\\ |
509 |
2000 & 0.024 & 7.52(0.15) & 3.72(0.05) & 7.19(0.28) & 3.32(0.18)\\ |
510 |
2500 & 0.019 & 7.41(0.29) & 3.42(0.06) & 7.98(0.33) & 3.43(0.08)\\ |
511 |
\hline |
512 |
\end{tabular} |
513 |
\label{LJ} |
514 |
\end{center} |
515 |
\end{minipage} |
516 |
\end{table*} |
517 |
|
518 |
\begin{figure} |
519 |
\includegraphics[width=\linewidth]{thermalGrad} |
520 |
\caption{The NIVS-RNEMD method creates similar temperature gradients |
521 |
compared with the swapping method under a variety of imposed |
522 |
kinetic energy flux values. Furthermore, the implementation of |
523 |
Non-Isotropic Velocity Scaling does not cause temperature |
524 |
anisotropy to develop in thermal conductivity calculations.} |
525 |
\label{thermalGrad} |
526 |
\end{figure} |
527 |
|
528 |
\subsubsection*{Velocity Distributions} |
529 |
|
530 |
To test the effects on the velocity distributions, we accumulated |
531 |
velocities every 100 steps and produced distributions of both velocity |
532 |
and speed in each of the slabs. From these distributions, we observed |
533 |
that under high non-physical kinetic energy flux, the speed of |
534 |
molecules in slabs where {\it swapping} occured could deviate from the |
535 |
Maxwell-Boltzmann distribution. This behavior was also noted by Tenney |
536 |
and Maginn.\cite{Maginn:2010} Figure \ref{thermalHist} shows how these |
537 |
distributions deviate from an ideal distribution. In the ``hot'' slab, |
538 |
the probability density is notched at low speeds and has a substantial |
539 |
shoulder at higher speeds relative to the ideal distribution. In the |
540 |
cold slab, the opposite notching and shouldering occurs. This |
541 |
phenomenon is more obvious at high swapping rates. |
542 |
|
543 |
The peak of the velocity distribution is substantially flattened in |
544 |
the hot slab, and is overly sharp (with truncated wings) in the cold |
545 |
slab. This problem is rooted in the mechanism of the swapping method. |
546 |
Continually depleting low (high) speed particles in the high (low) |
547 |
temperature slab is not complemented by diffusion of low (high) speed |
548 |
particles from neighboring slabs unless the swapping rate is |
549 |
sufficiently small. Simutaneously, surplus low speed particles in the |
550 |
cold slab do not have sufficient time to diffuse to neighboring slabs. |
551 |
Since the thermal exchange rate must reach a minimum level to produce |
552 |
an observable thermal gradient, the swapping-method RNEMD has a |
553 |
relatively narrow choice of exchange times that can be utilized. |
554 |
|
555 |
For comparison, NIVS-RNEMD produces a speed distribution closer to the |
556 |
Maxwell-Boltzmann curve (Fig. \ref{thermalHist}). The reason for this |
557 |
is simple; upon velocity scaling, a Gaussian distribution remains |
558 |
Gaussian. Although a single scaling move is non-isotropic in three |
559 |
dimensions, our criteria for choosing a set of scaling coefficients |
560 |
helps maintain the distributions as close to isotropic as possible. |
561 |
Consequently, NIVS-RNEMD is able to impose a non-physical thermal flux |
562 |
without large perturbations to the velocity distributions in the two |
563 |
slabs. |
564 |
|
565 |
\begin{figure} |
566 |
\includegraphics[width=\linewidth]{thermalHist} |
567 |
\caption{Velocity and speed distributions that develop under the |
568 |
swapping and NIVS-RNEMD methods at high flux. The distributions for |
569 |
the hot bins (upper panels) and cold bins (lower panels) were |
570 |
obtained from Lennard-Jones simulations with $\langle T^* \rangle = |
571 |
4.5$ with a flux of $J_z^* \sim 5$ (equivalent to a swapping interval |
572 |
of 10 time steps). This is a relatively large flux which shows the |
573 |
non-thermal distributions that develop under the swapping method. |
574 |
NIVS does a better job of producing near-thermal distributions in |
575 |
the bins.} |
576 |
\label{thermalHist} |
577 |
\end{figure} |
578 |
|
579 |
|
580 |
\subsubsection*{Shear Viscosity} |
581 |
Our calculations (Table \ref{LJ}) show that velocity-scaling RNEMD |
582 |
predicted similar values for shear viscosities to the swapping RNEMD |
583 |
method. The average molecular momentum gradients of these samples are |
584 |
shown in the upper two panels of Fig. \ref{shear}. |
585 |
|
586 |
\begin{figure} |
587 |
\includegraphics[width=\linewidth]{shear} |
588 |
\caption{Average momentum gradients in shear viscosity simulations, |
589 |
using ``swapping'' method (top panel) and NIVS-RNEMD method |
590 |
(middle panel). NIVS-RNEMD produces a thermal anisotropy artifact |
591 |
in the hot and cold bins when used for shear viscosity. This |
592 |
artifact does not appear in thermal conductivity calculations.} |
593 |
\label{shear} |
594 |
\end{figure} |
595 |
|
596 |
Observations of the three one-dimensional temperatures in each of the |
597 |
slabs shows that NIVS-RNEMD does produce some thermal anisotropy, |
598 |
particularly in the hot and cold slabs. Note that these temperature |
599 |
measurements have been taken into account of the kinetic energy |
600 |
contributed by the slab field velocity. However, this contribution has |
601 |
only a comparable order of magnitude to the errors of data, and does |
602 |
not significantly affect our observation. The lower panel of Fig. |
603 |
\ref{shear} indicates that with a relatively large imposed momentum |
604 |
flux, $j_z(p_x)$, the $x$ direction approaches a different temperature |
605 |
from the $y$ and $z$ directions in both the hot and cold bins. This |
606 |
is an artifact of the scaling constraints. After a momentum gradient |
607 |
has been established, $P_c^x$ is always less than zero. Consequently, |
608 |
the scaling factor $x$ is always greater than one in order to satisfy |
609 |
the constraints. This will continually increase the kinetic energy in |
610 |
$x$-dimension, $K_c^x$. If there is not enough time for the kinetic |
611 |
energy to exchange among different directions and different slabs, the |
612 |
system will exhibit the observed thermal anisotropy in the hot and |
613 |
cold bins. |
614 |
|
615 |
Although results between scaling and swapping methods are comparable, |
616 |
the inherent temperature anisotropy does make NIVS-RNEMD method less |
617 |
attractive than swapping RNEMD for shear viscosity calculations. We |
618 |
note that this problem appears only when a large {\it linear} momentum |
619 |
flux is applied, and does not appear in {\it thermal} flux |
620 |
calculations. |
621 |
|
622 |
\subsection{Bulk SPC/E water} |
623 |
|
624 |
We compared the thermal conductivity of SPC/E water using NIVS-RNEMD |
625 |
to the work of Bedrov {\it et al.}\cite{Bedrov:2000}, which employed |
626 |
the original swapping RNEMD method. Bedrov {\it et |
627 |
al.}\cite{Bedrov:2000} argued that exchange of the molecular |
628 |
center-of-mass velocities instead of single atom velocities conserves |
629 |
the total kinetic energy and linear momentum. This principle is also |
630 |
adopted in our simulations. Scaling was applied to the center-of-mass |
631 |
velocities of SPC/E water molecules. |
632 |
|
633 |
To construct the simulations, a simulation box consisting of 1000 |
634 |
molecules were first equilibrated under ambient pressure and |
635 |
temperature conditions using the isobaric-isothermal (NPT) |
636 |
ensemble.\cite{melchionna93} A fixed volume was chosen to match the |
637 |
average volume observed in the NPT simulations, and this was followed |
638 |
by equilibration, first in the canonical (NVT) ensemble, followed by a |
639 |
100~ps period under constant-NVE conditions without any momentum flux. |
640 |
Another 100~ps was allowed to stabilize the system with an imposed |
641 |
momentum transfer to create a gradient, and 1~ns was allotted for data |
642 |
collection under RNEMD. Total system energy is recorded to ensure that |
643 |
it is not drifted noticeably without a thermostat although |
644 |
electrostatic interactions are involved. |
645 |
|
646 |
In our simulations, the established temperature gradients were similar |
647 |
to the previous work. Our simulation results at 318K are in good |
648 |
agreement with those from Bedrov {\it et al.} (Table |
649 |
\ref{spceThermal}). And both methods yield values in reasonable |
650 |
agreement with experimental values. |
651 |
|
652 |
\begin{table*} |
653 |
\begin{minipage}{\linewidth} |
654 |
\begin{center} |
655 |
|
656 |
\caption{Thermal conductivity of SPC/E water under various |
657 |
imposed thermal gradients. Uncertainties are indicated in |
658 |
parentheses.} |
659 |
|
660 |
\begin{tabular}{|c|c|ccc|} |
661 |
\hline |
662 |
\multirow{2}{*}{$\langle T\rangle$(K)} & |
663 |
\multirow{2}{*}{$\langle dT/dz\rangle$(K/\AA)} & |
664 |
\multicolumn{3}{|c|}{$\lambda (\mathrm{W m}^{-1} |
665 |
\mathrm{K}^{-1})$} \\ |
666 |
& & This work & Previous simulations\cite{Bedrov:2000} & |
667 |
Experiment\cite{WagnerKruse}\\ |
668 |
\hline |
669 |
\multirow{3}{*}{300} & 0.38 & 0.816(0.044) & & |
670 |
\multirow{3}{*}{0.61}\\ |
671 |
& 0.81 & 0.770(0.008) & & \\ |
672 |
& 1.54 & 0.813(0.007) & & \\ |
673 |
\hline |
674 |
\multirow{2}{*}{318} & 0.75 & 0.750(0.032) & 0.784 & |
675 |
\multirow{2}{*}{0.64}\\ |
676 |
& 1.59 & 0.778(0.019) & 0.730 & \\ |
677 |
\hline |
678 |
\end{tabular} |
679 |
\label{spceThermal} |
680 |
\end{center} |
681 |
\end{minipage} |
682 |
\end{table*} |
683 |
|
684 |
\subsection{Crystalline Gold} |
685 |
|
686 |
To see how the method performed in a solid, we calculated thermal |
687 |
conductivities using two atomistic models for gold. Several different |
688 |
potential models have been developed that reasonably describe |
689 |
interactions in transition metals. In particular, the Embedded Atom |
690 |
Model (EAM)~\cite{PhysRevB.33.7983} and Sutton-Chen (SC)~\cite{Chen90} |
691 |
potential have been used to study a wide range of phenomena in both |
692 |
bulk materials and |
693 |
nanoparticles.\cite{ISI:000207079300006,Clancy:1992,Vardeman:2008fk,Vardeman-II:2001jn,ShibataT._ja026764r,Sankaranarayanan:2005lr,Chui:2003fk,Wang:2005qy,Medasani:2007uq} |
694 |
Both potentials are based on a model of a metal which treats the |
695 |
nuclei and core electrons as pseudo-atoms embedded in the electron |
696 |
density due to the valence electrons on all of the other atoms in the |
697 |
system. The SC potential has a simple form that closely resembles the |
698 |
Lennard Jones potential, |
699 |
\begin{equation} |
700 |
\label{eq:SCP1} |
701 |
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] , |
702 |
\end{equation} |
703 |
where $V^{pair}_{ij}$ and $\rho_{i}$ are given by |
704 |
\begin{equation} |
705 |
\label{eq:SCP2} |
706 |
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}}. |
707 |
\end{equation} |
708 |
$V^{pair}_{ij}$ is a repulsive pairwise potential that accounts for |
709 |
interactions between the pseudoatom cores. The $\sqrt{\rho_i}$ term in |
710 |
Eq. (\ref{eq:SCP1}) is an attractive many-body potential that models |
711 |
the interactions between the valence electrons and the cores of the |
712 |
pseudo-atoms. $D_{ij}$, $D_{ii}$ set the appropriate overall energy |
713 |
scale, $c_i$ scales the attractive portion of the potential relative |
714 |
to the repulsive interaction and $\alpha_{ij}$ is a length parameter |
715 |
that assures a dimensionless form for $\rho$. These parameters are |
716 |
tuned to various experimental properties such as the density, cohesive |
717 |
energy, and elastic moduli for FCC transition metals. The quantum |
718 |
Sutton-Chen (QSC) formulation matches these properties while including |
719 |
zero-point quantum corrections for different transition |
720 |
metals.\cite{PhysRevB.59.3527} The EAM functional forms differ |
721 |
slightly from SC but the overall method is very similar. |
722 |
|
723 |
In this work, we have utilized both the EAM and the QSC potentials to |
724 |
test the behavior of scaling RNEMD. |
725 |
|
726 |
A face-centered-cubic (FCC) lattice was prepared containing 2880 Au |
727 |
atoms (i.e. ${6\times 6\times 20}$ unit cells). Simulations were run |
728 |
both with and without isobaric-isothermal (NPT)~\cite{melchionna93} |
729 |
pre-equilibration at a target pressure of 1 atm. When equilibrated |
730 |
under NPT conditions, our simulation box expanded by approximately 1\% |
731 |
in volume. Following adjustment of the box volume, equilibrations in |
732 |
both the canonical and microcanonical ensembles were carried out. With |
733 |
the simulation cell divided evenly into 10 slabs, different thermal |
734 |
gradients were established by applying a set of target thermal fluxes. |
735 |
|
736 |
The results for the thermal conductivity of gold are shown in Table |
737 |
\ref{AuThermal}. In these calculations, the end and middle slabs were |
738 |
excluded from the thermal gradient linear regession. EAM predicts |
739 |
slightly larger thermal conductivities than QSC. However, both values |
740 |
are smaller than experimental value by a factor of more than 200. This |
741 |
behavior has been observed previously by Richardson and Clancy, and |
742 |
has been attributed to the lack of electronic contribution in these |
743 |
force fields.\cite{Clancy:1992} It should be noted that the density of |
744 |
the metal being simulated has an effect on thermal conductance. With |
745 |
an expanded lattice, lower thermal conductance is expected (and |
746 |
observed). We also observed a decrease in thermal conductance at |
747 |
higher temperatures, a trend that agrees with experimental |
748 |
measurements.\cite{AshcroftMermin} |
749 |
|
750 |
\begin{table*} |
751 |
\begin{minipage}{\linewidth} |
752 |
\begin{center} |
753 |
|
754 |
\caption{Calculated thermal conductivity of crystalline gold |
755 |
using two related force fields. Calculations were done at both |
756 |
experimental and equilibrated densities and at a range of |
757 |
temperatures and thermal flux rates. Uncertainties are |
758 |
indicated in parentheses. Richardson {\it et |
759 |
al.}\cite{Clancy:1992} give an estimate of 1.74 $\mathrm{W |
760 |
m}^{-1}\mathrm{K}^{-1}$ for EAM gold |
761 |
at a density of 19.263 g / cm$^3$.} |
762 |
|
763 |
\begin{tabular}{|c|c|c|cc|} |
764 |
\hline |
765 |
Force Field & $\rho$ (g/cm$^3$) & ${\langle T\rangle}$ (K) & |
766 |
$\langle dT/dz\rangle$ (K/\AA) & $\lambda (\mathrm{W m}^{-1} \mathrm{K}^{-1})$\\ |
767 |
\hline |
768 |
\multirow{7}{*}{QSC} & \multirow{3}{*}{19.188} & \multirow{3}{*}{300} & 1.44 & 1.10(0.06)\\ |
769 |
& & & 2.86 & 1.08(0.05)\\ |
770 |
& & & 5.14 & 1.15(0.07)\\\cline{2-5} |
771 |
& \multirow{4}{*}{19.263} & \multirow{2}{*}{300} & 2.31 & 1.25(0.06)\\ |
772 |
& & & 3.02 & 1.26(0.05)\\\cline{3-5} |
773 |
& & \multirow{2}{*}{575} & 3.02 & 1.02(0.07)\\ |
774 |
& & & 4.84 & 0.92(0.05)\\ |
775 |
\hline |
776 |
\multirow{8}{*}{EAM} & \multirow{3}{*}{19.045} & \multirow{3}{*}{300} & 1.24 & 1.24(0.16)\\ |
777 |
& & & 2.06 & 1.37(0.04)\\ |
778 |
& & & 2.55 & 1.41(0.07)\\\cline{2-5} |
779 |
& \multirow{5}{*}{19.263} & \multirow{3}{*}{300} & 1.06 & 1.45(0.13)\\ |
780 |
& & & 2.04 & 1.41(0.07)\\ |
781 |
& & & 2.41 & 1.53(0.10)\\\cline{3-5} |
782 |
& & \multirow{2}{*}{575} & 2.82 & 1.08(0.03)\\ |
783 |
& & & 4.14 & 1.08(0.05)\\ |
784 |
\hline |
785 |
\end{tabular} |
786 |
\label{AuThermal} |
787 |
\end{center} |
788 |
\end{minipage} |
789 |
\end{table*} |
790 |
|
791 |
\subsection{Thermal Conductance at the Au/H$_2$O interface} |
792 |
The most attractive aspect of the scaling approach for RNEMD is the |
793 |
ability to use the method in non-homogeneous systems, where molecules |
794 |
of different identities are segregated in different slabs. To test |
795 |
this application, we simulated a Gold (111) / water interface. To |
796 |
construct the interface, a box containing a lattice of 1188 Au atoms |
797 |
(with the 111 surface in the $+z$ and $-z$ directions) was allowed to |
798 |
relax under ambient temperature and pressure. A separate (but |
799 |
identically sized) box of SPC/E water was also equilibrated at ambient |
800 |
conditions. The two boxes were combined by removing all water |
801 |
molecules within 3 \AA~ radius of any gold atom. The final |
802 |
configuration contained 1862 SPC/E water molecules. |
803 |
|
804 |
The Spohr potential was adopted in depicting the interaction between |
805 |
metal atoms and water molecules.\cite{ISI:000167766600035} A similar |
806 |
protocol of equilibration to our water simulations was followed. We |
807 |
observed that the two phases developed large temperature differences |
808 |
even under a relatively low thermal flux. |
809 |
|
810 |
The low interfacial conductance is probably due to an acoustic |
811 |
impedance mismatch between the solid and the liquid |
812 |
phase.\cite{Cahill:793,RevModPhys.61.605} Experiments on the thermal |
813 |
conductivity of gold nanoparticles and nanorods in solvent and in |
814 |
glass cages have predicted values for $G$ between 100 and 350 |
815 |
(MW/m$^2$/K), two orders of magnitude larger than the value reported |
816 |
here. The experiments typically have multiple surfaces that have been |
817 |
protected by ionic surfactants, either |
818 |
citrate\cite{Wilson:2002uq,plech:195423} or cetyltrimethylammonium |
819 |
bromide (CTAB), or which are in direct contact with various glassy |
820 |
solids. In these cases, the acoustic impedance mismatch would be |
821 |
substantially reduced, leading to much higher interfacial |
822 |
conductances. It is also possible, however, that the lack of |
823 |
electronic effects that gives rise to the low thermal conductivity of |
824 |
EAM gold is also causing a low reading for this particular interface. |
825 |
|
826 |
Under this low thermal conductance, both gold and water phases have |
827 |
sufficient time to eliminate local temperature differences (Fig. |
828 |
\ref{interface}). With flat thermal profiles within each phase, it is |
829 |
valid to assume that the temperature difference between gold and water |
830 |
surfaces would be approximately the same as the difference between the |
831 |
gold and water bulk regions. This assumption enables convenient |
832 |
calculation of $G$ using Eq. \ref{interfaceCalc}. |
833 |
|
834 |
\begin{figure} |
835 |
\includegraphics[width=\linewidth]{interface} |
836 |
\caption{Temperature profiles of the Gold / Water interface at four |
837 |
different values for the thermal flux. Temperatures for slabs |
838 |
either in the gold or in the water show no significant differences, |
839 |
although there is a large discontinuity between the materials |
840 |
because of the relatively low interfacial thermal conductivity.} |
841 |
\label{interface} |
842 |
\end{figure} |
843 |
|
844 |
\begin{table*} |
845 |
\begin{minipage}{\linewidth} |
846 |
\begin{center} |
847 |
|
848 |
\caption{Computed interfacial thermal conductivity ($G$) values |
849 |
for the Au(111) / water interface at ${\langle T\rangle \sim}$ |
850 |
300K using a range of energy fluxes. Uncertainties are |
851 |
indicated in parentheses. } |
852 |
|
853 |
\begin{tabular}{|cccc| } |
854 |
\hline |
855 |
$J_z$ (MW/m$^2$) & $\langle T_{gold} \rangle$ (K) & $\langle |
856 |
T_{water} \rangle$ (K) & $G$ |
857 |
(MW/m$^2$/K)\\ |
858 |
\hline |
859 |
98.0 & 355.2 & 295.8 & 1.65(0.21) \\ |
860 |
78.8 & 343.8 & 298.0 & 1.72(0.32) \\ |
861 |
73.6 & 344.3 & 298.0 & 1.59(0.24) \\ |
862 |
49.2 & 330.1 & 300.4 & 1.65(0.35) \\ |
863 |
\hline |
864 |
\end{tabular} |
865 |
\label{interfaceRes} |
866 |
\end{center} |
867 |
\end{minipage} |
868 |
\end{table*} |
869 |
|
870 |
|
871 |
\section{Conclusions} |
872 |
|
873 |
Our simulations demonstrate that validity of non-isotropic velocity |
874 |
scaling (NIVS) in RNEMD calculations of thermal conductivity in atomic |
875 |
and molecular liquids and solids. NIVS-RNEMD improves the problematic |
876 |
velocity distributions which can develop in other RNEMD methods. |
877 |
Furthermore, it provides a means for carrying out non-physical thermal |
878 |
transfer between different species of molecules, and thus extends its |
879 |
applicability to interfacial systems. Our calculation of the gold / |
880 |
water interfacial thermal conductivity demonstrates this advantage |
881 |
over previous RNEMD methods. NIVS-RNEMD also has limited applications |
882 |
for shear viscosity calculations, but has the potential to cause |
883 |
temperature anisotropy under high momentum fluxes. Further work will |
884 |
be necessary to eliminate the one-dimensional heating if shear |
885 |
viscosities are required. |
886 |
|
887 |
\section{Acknowledgments} |
888 |
The authors would like to thank Craig Tenney and Ed Maginn for many |
889 |
helpful discussions. Support for this project was provided by the |
890 |
National Science Foundation under grant CHE-0848243. Computational |
891 |
time was provided by the Center for Research Computing (CRC) at the |
892 |
University of Notre Dame. |
893 |
\newpage |
894 |
|
895 |
\bibliography{nivsRnemd} |
896 |
|
897 |
\end{doublespace} |
898 |
\end{document} |
899 |
|