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\begin{document} |
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\title{A gentler approach to RNEMD: Non-isotropic Velocity Scaling for computing Thermal Conductivity and Shear Viscosity} |
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|
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\author{Shenyu Kuang and J. Daniel |
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Gezelter\footnote{Corresponding author. \ Electronic mail: gezelter@nd.edu} \\ |
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Department of Chemistry and Biochemistry,\\ |
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University of Notre Dame\\ |
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Notre Dame, Indiana 46556} |
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\date{\today} |
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\maketitle |
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\begin{doublespace} |
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\begin{abstract} |
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A molecular simulation method is developed based on the previous RNEMD |
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algorithm. With scaling the velocities of molecules in specific |
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regions of a system, unphysical thermal transfer can be achieved and |
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thermal / momentum gradient can be established, as well as |
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conservation of linear momentum and translational kinetic |
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energy. Thermal conductivity calculations of Lennard-Jones fluid water |
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(SPC/E model) and crystal gold (QSC and EAM model) are performed and |
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demostrate the validity of the method. Furthermore, NIVS-RNEMD |
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improves the non-Maxwell-Boltzmann velocity distributions in previous |
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RNEMD methods, where unphysical momentum transfer occurs. NIVS-RNEMD |
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also extends its application to interfacial thermal conductivity |
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calculations, thanks to its novel means in kinetic energy transfer. |
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\end{abstract} |
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\newpage |
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% BODY OF TEXT |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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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'' either |
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the entire momentum vector $\vec{p}$ or single components of this |
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vector ($p_x$) between molecules in each of the two slabs. If the two |
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slabs are separated along the $z$ coordinate, the imposed flux is either |
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directional ($j_z(p_x)$) or isotropic ($J_z$), and the response of a |
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simulated system to the imposed momentum flux will typically be a |
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velocity or thermal gradient (Fig. \ref{thermalDemo}). The transport |
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coefficients (shear viscosity and thermal conductivity) are easily |
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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 thermal |
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conductivities and shear viscosities in a wide range of materials, |
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from liquid copper to monatomic liquids to molecular fluids |
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(e.g. ionic liquids).\cite{ISI:000246190100032} |
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|
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\begin{figure} |
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\includegraphics[width=\linewidth]{thermalDemo} |
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\caption{Demostration of thermal gradient estalished by RNEMD |
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method. Physical thermal flow directs from high temperature region |
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to low temperature region. Unphysical thermal transfer counteracts |
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it and maintains a steady thermal gradient.} |
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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 methods because |
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it imposes what is typically difficult to measure (a flux or stress) |
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and it is typically much easier to compute momentum gradients or |
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strains (the response). For similar reasons, RNEMD is also preferable |
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to slowly-converging equilibrium methods for measuring thermal |
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conductivity and shear viscosity (using Green-Kubo relations or the |
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Helfand moment approach of Viscardy {\it et |
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al.}\cite{Viscardy:2007bh,Viscardy:2007lq}) because these rely on |
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computing difficult to measure quantities. |
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|
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Another attractive feature of RNEMD is that it conserves both total |
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linear momentum and total energy during the swaps (as long as the two |
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molecules have the same identity), so the swapped configurations are |
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typically samples from the same manifold of states in the |
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microcanonical ensemble. |
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|
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Recently, Tenney and Maginn\cite{ISI:000273472300004} have discovered |
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some problems with the original RNEMD swap technique. Notably, large |
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momentum fluxes (equivalent to frequent momentum swaps between the |
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slabs) can result in ``notched'', ``peaked'' and generally non-thermal |
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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-RNEMD) which retains the desirable features of RNEMD |
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(conservation of linear momentum and total energy, compatibility with |
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periodic boundary conditions) while establishing true thermal |
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distributions in each of the two slabs. In the next section, we |
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develop the method for determining the scaling constraints. We then |
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test the method on both single component, multi-component, and |
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non-isotropic mixtures and show that it is capable of providing |
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reasonable estimates of the thermal conductivity and shear viscosity |
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in these cases. |
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|
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\section{Methodology} |
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We retain the basic idea of Muller-Plathe's RNEMD method; the periodic |
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system is partitioned into a series of thin slabs along a particular |
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axis ($z$). One of the slabs at the end of the periodic box is |
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designated the ``hot'' slab, while the slab in the center of the box |
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is designated the ``cold'' slab. The artificial momentum flux will be |
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established by transferring momentum from the cold slab and into the |
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hot slab. |
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|
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Rather than using momentum swaps, we use a series of velocity scaling |
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moves. For molecules $\{i\}$ located within the cold slab, |
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\begin{equation} |
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\vec{v}_i \leftarrow \left( \begin{array}{ccc} |
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x & 0 & 0 \\ |
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0 & y & 0 \\ |
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0 & 0 & z \\ |
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\end{array} \right) \cdot \vec{v}_i |
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\end{equation} |
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where ${x, y, z}$ are a set of 3 scaling variables for each of the |
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three directions in the system. Likewise, the molecules $\{j\}$ |
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located in the hot slab will see a concomitant scaling of velocities, |
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\begin{equation} |
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\vec{v}_j \leftarrow \left( \begin{array}{ccc} |
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x^\prime & 0 & 0 \\ |
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0 & y^\prime & 0 \\ |
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0 & 0 & z^\prime \\ |
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\end{array} \right) \cdot \vec{v}_j |
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\end{equation} |
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|
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Conservation of linear momentum in each of the three directions |
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($\alpha = x,y,z$) ties the values of the hot and cold bin scaling |
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parameters together: |
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\begin{equation} |
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P_h^\alpha + P_c^\alpha = \alpha^\prime P_h^\alpha + \alpha P_c^\alpha |
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\end{equation} |
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where |
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\begin{eqnarray} |
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P_c^\alpha & = & \sum_{i = 1}^{N_c} m_i \left[\vec{v}_i\right]_\alpha \\ |
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P_h^\alpha & = & \sum_{j = 1}^{N_h} m_j \left[\vec{v}_j\right]_\alpha |
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\label{eq:momentumdef} |
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\end{eqnarray} |
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Therefore, for each of the three directions, the hot scaling |
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parameters are a simple function of the cold scaling parameters and |
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the instantaneous linear momentum in each of the two slabs. |
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\begin{equation} |
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\alpha^\prime = 1 + (1 - \alpha) p_\alpha |
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\label{eq:hotcoldscaling} |
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\end{equation} |
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where |
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\begin{equation} |
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p_\alpha = \frac{P_c^\alpha}{P_h^\alpha} |
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\end{equation} |
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for convenience. |
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|
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Conservation of total energy also places constraints on the scaling: |
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\begin{equation} |
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\sum_{\alpha = x,y,z} K_h^\alpha + K_c^\alpha = \sum_{\alpha = x,y,z} |
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\left(\alpha^\prime\right)^2 K_h^\alpha + \alpha^2 K_c^\alpha |
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\end{equation} |
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where the 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 equation}: |
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\begin{equation} |
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\sum_{\alpha = x,y,z} a_\alpha \alpha^2 + b_\alpha \alpha + c_\alpha = 0 |
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\label{eq:constraintEllipsoid} |
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\end{equation} |
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where the constants are obtained from the instantaneous values of the |
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linear momenta and kinetic energies for the hot and cold slabs, |
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\begin{eqnarray} |
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a_\alpha & = &\left(K_c^\alpha + K_h^\alpha |
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\left(p_\alpha\right)^2\right) \\ |
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b_\alpha & = & -2 K_h^\alpha p_\alpha \left( 1 + p_\alpha\right) \\ |
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c_\alpha & = & K_h^\alpha p_\alpha^2 + 2 K_h^\alpha p_\alpha - K_c^\alpha |
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\label{eq:constraintEllipsoidConsts} |
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\end{eqnarray} |
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This ellipsoid equation defines the set of cold slab scaling |
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parameters which can be applied while preserving both linear momentum |
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in all three directions as well as kinetic energy. |
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|
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The goal of using velocity scaling variables is to transfer linear |
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momentum or kinetic energy from the cold slab to the hot slab. If the |
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hot and cold slabs are separated along the z-axis, the energy flux is |
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given simply by the decrease in kinetic energy of the cold bin: |
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\begin{equation} |
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(1-x^2) K_c^x + (1-y^2) K_c^y + (1-z^2) K_c^z = J_z\Delta t |
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\end{equation} |
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The expression for the energy flux can be re-written as another |
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ellipsoid centered on $(x,y,z) = 0$: |
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\begin{equation} |
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x^2 K_c^x + y^2 K_c^y + z^2 K_c^z = K_c^x + K_c^y + K_c^z - J_z\Delta t |
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\label{eq:fluxEllipsoid} |
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\end{equation} |
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The spatial extent of the {\it thermal flux ellipsoid equation} is |
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governed both by a targetted value, $J_z$ as well as the instantaneous |
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values of 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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it is sufficient to determine the points ${x,y,z}$ which lie on both |
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the constraint ellipsoid (Eq. \ref{eq:constraintEllipsoid}) and the |
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flux ellipsoid (Eq. \ref{eq:fluxEllipsoid}), i.e. the intersection of |
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the two ellipsoids in 3-dimensional space. |
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|
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\begin{figure} |
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\includegraphics[width=\linewidth]{ellipsoids} |
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\caption{Scaling points which maintain both constant energy and |
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constant linear momentum of the system lie on the surface of the |
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{\it constraint ellipsoid} while points which generate the target |
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momentum flux lie on the surface of the {\it flux ellipsoid}. The |
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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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One may also define momentum flux (say along the $x$-direction) as: |
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\begin{equation} |
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(1-x) P_c^x = j_z(p_x)\Delta t |
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\label{eq:fluxPlane} |
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\end{equation} |
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The above {\it momentum flux equation} is essentially a plane which is |
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perpendicular to the $x$-axis, with its position governed both by a |
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target value, $j_z(p_x)$ as well as the instantaneous value of the |
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momentum along the $x$-direction. |
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|
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Similarly, to satisfy a momentum flux as well as the conservation |
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constraints, it is sufficient to determine the points ${x,y,z}$ which |
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lie on both the constraint ellipsoid (Eq. \ref{eq:constraintEllipsoid}) |
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and the flux plane (Eq. \ref{eq:fluxPlane}), i.e. the intersection of |
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an ellipsoid and a plane in 3-dimensional space. |
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|
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To summarize, by solving respective equation sets, one can determine |
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possible sets of scaling variables for cold slab. And corresponding |
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sets of scaling variables for hot slab can be determine as well. |
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|
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The following problem will be choosing an optimal set of scaling |
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variables among the possible sets. Although this method is inherently |
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non-isotropic, the goal is still to maintain the system as isotropic |
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as possible. Under this consideration, one would like the kinetic |
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energies in different directions could become as close as each other |
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after each scaling. Simultaneously, one would also like each scaling |
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as gentle as possible, i.e. ${x,y,z \rightarrow 1}$, in order to avoid |
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large perturbation to the system. Therefore, one approach to obtain the |
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scaling variables would be constructing an criteria function, with |
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constraints as above equation sets, and solving the function's minimum |
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by method like Lagrange multipliers. |
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|
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In order to save computation time, we have a different approach to a |
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relatively good set of scaling variables with much less calculation |
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than above. Here is the detail of our simplification of the problem. |
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|
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In the case of kinetic energy transfer, we impose another constraint |
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${x = y}$, into the equation sets. Consequently, there are two |
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variables left. And now one only needs to solve a set of two {\it |
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ellipses equations}. This problem would be transformed into solving |
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one quartic equation for one of the two variables. There are known |
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generic methods that solve real roots of quartic equations. Then one |
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can determine the other variable and obtain sets of scaling |
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variables. Among these sets, one can apply the above criteria to |
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choose the best set, while much faster with only a few sets to choose. |
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|
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In the case of momentum flux transfer, we impose another constraint to |
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set the kinetic energy transfer as zero. In another word, we apply |
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Eq. \ref{eq:fluxEllipsoid} and let ${J_z = 0}$. After that, with one |
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variable fixed by Eq. \ref{eq:fluxPlane}, one now have a similar set |
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of equations on the above kinetic energy transfer problem. Therefore, |
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an approach similar to the above would be sufficient for this as well. |
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|
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\section{Computational Details} |
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\subsection{Lennard-Jones Fluid} |
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Our simulation consists of a series of systems. All of these |
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simulations were run with the OpenMD simulation software |
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package\cite{Meineke:2005gd} integrated with RNEMD codes. |
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|
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A Lennard-Jones fluid system was built and tested first. In order to |
310 |
compare our method with swapping RNEMD, a series of simulations were |
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performed to calculate the shear viscosity and thermal conductivity of |
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argon. 2592 atoms were in a orthorhombic cell, which was ${10.06\sigma |
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\times 10.06\sigma \times 30.18\sigma}$ by size. The reduced density |
314 |
${\rho^* = \rho\sigma^3}$ was thus 0.85, which enabled direct |
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comparison between our results and others. These simulations used |
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velocity Verlet algorithm with reduced timestep ${\tau^* = |
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4.6\times10^{-4}}$. |
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|
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For shear viscosity calculation, the reduced temperature was ${T^* = |
320 |
k_B T/\varepsilon = 0.72}$. Simulations were first equilibrated in canonical |
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ensemble (NVT), then equilibrated in microcanonical ensemble |
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(NVE). Establishing and stablizing momentum gradient were followed |
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also in NVE ensemble. For the swapping part, Muller-Plathe's algorithm was |
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adopted.\cite{ISI:000080382700030} The simulation box was under |
325 |
periodic boundary condition, and devided into ${N = 20}$ slabs. In each swap, |
326 |
the top slab ${(n = 1)}$ exchange the most negative $x$ momentum with the |
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most positive $x$ momentum in the center slab ${(n = N/2 + 1)}$. Referred |
328 |
to Tenney {\it et al.}\cite{ISI:000273472300004}, a series of swapping |
329 |
frequency were chosen. According to each result from swapping |
330 |
RNEMD, scaling RNEMD simulations were run with the target momentum |
331 |
flux set to produce a similar momentum flux, and consequently shear |
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rate. Furthermore, various scaling frequencies can be tested for one |
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single swapping rate. To test the temperature homogeneity in our |
334 |
system of swapping and scaling methods, temperatures of different |
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dimensions in all the slabs were observed. Most of the simulations |
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include $10^5$ steps of equilibration without imposing momentum flux, |
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$10^5$ steps of stablization with imposing unphysical momentum |
338 |
transfer, and $10^6$ steps of data collection under RNEMD. For |
339 |
relatively high momentum flux simulations, ${5\times10^5}$ step data |
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collection is sufficient. For some low momentum flux simulations, |
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${2\times10^6}$ steps were necessary. |
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|
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After each simulation, the shear viscosity was calculated in reduced |
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unit. The momentum flux was calculated with total unphysical |
345 |
transferred momentum ${P_x}$ and data collection time $t$: |
346 |
\begin{equation} |
347 |
j_z(p_x) = \frac{P_x}{2 t L_x L_y} |
348 |
\end{equation} |
349 |
where $L_x$ and $L_y$ denotes $x$ and $y$ lengths of the simulation |
350 |
box, and physical momentum transfer occurs in two ways due to our |
351 |
periodic boundary condition settings. And the velocity gradient |
352 |
${\langle \partial v_x /\partial z \rangle}$ can be obtained by a |
353 |
linear regression of the velocity profile. From the shear viscosity |
354 |
$\eta$ calculated with the above parameters, one can further convert |
355 |
it into reduced unit ${\eta^* = \eta \sigma^2 (\varepsilon m)^{-1/2}}$. |
356 |
|
357 |
For thermal conductivity calculations, simulations were first run under |
358 |
reduced temperature ${\langle T^*\rangle = 0.72}$ in NVE |
359 |
ensemble. Muller-Plathe's algorithm was adopted in the swapping |
360 |
method. Under identical simulation box parameters with our shear |
361 |
viscosity calculations, in each swap, the top slab exchanges all three |
362 |
translational momentum components of the molecule with least kinetic |
363 |
energy with the same components of the molecule in the center slab |
364 |
with most kinetic energy, unless this ``coldest'' molecule in the |
365 |
``hot'' slab is still ``hotter'' than the ``hottest'' molecule in the |
366 |
``cold'' slab. According to swapping RNEMD results, target energy flux |
367 |
for scaling RNEMD simulations can be set. Also, various scaling |
368 |
frequencies can be tested for one target energy flux. To compare the |
369 |
performance between swapping and scaling method, distributions of |
370 |
velocity and speed in different slabs were observed. |
371 |
|
372 |
For each swapping rate, thermal conductivity was calculated in reduced |
373 |
unit. The energy flux was calculated similarly to the momentum flux, |
374 |
with total unphysical transferred energy ${E_{total}}$ and data collection |
375 |
time $t$: |
376 |
\begin{equation} |
377 |
J_z = \frac{E_{total}}{2 t L_x L_y} |
378 |
\end{equation} |
379 |
And the temperature gradient ${\langle\partial T/\partial z\rangle}$ |
380 |
can be obtained by a linear regression of the temperature |
381 |
profile. From the thermal conductivity $\lambda$ calculated, one can |
382 |
further convert it into reduced unit ${\lambda^*=\lambda \sigma^2 |
383 |
m^{1/2} k_B^{-1}\varepsilon^{-1/2}}$. |
384 |
|
385 |
\subsection{ Water / Metal Thermal Conductivity} |
386 |
Another series of our simulation is the calculation of interfacial |
387 |
thermal conductivity of a Au/H$_2$O system. Respective calculations of |
388 |
liquid water (Extended Simple Point Charge model) and crystal gold |
389 |
thermal conductivity were performed and compared with current results |
390 |
to ensure the validity of NIVS-RNEMD. After that, a mixture system was |
391 |
simulated. |
392 |
|
393 |
For thermal conductivity calculation of bulk water, a simulation box |
394 |
consisting of 1000 molecules were first equilibrated under ambient |
395 |
pressure and temperature conditions using NPT ensemble, followed by |
396 |
equilibration in fixed volume (NVT). The system was then equilibrated in |
397 |
microcanonical ensemble (NVE). Also in NVE ensemble, establishing a |
398 |
stable thermal gradient was followed. The simulation box was under |
399 |
periodic boundary condition and devided into 10 slabs. Data collection |
400 |
process was similar to Lennard-Jones fluid system. |
401 |
|
402 |
Thermal conductivity calculation of bulk crystal gold used a similar |
403 |
protocol. Two types of force field parameters, Embedded Atom Method |
404 |
(EAM) and Quantum Sutten-Chen (QSC) force field were used |
405 |
respectively. The face-centered cubic crystal simulation box consists of |
406 |
2880 Au atoms. The lattice was first allowed volume change to relax |
407 |
under ambient temperature and pressure. Equilibrations in canonical and |
408 |
microcanonical ensemble were followed in order. With the simulation |
409 |
lattice devided evenly into 10 slabs, different thermal gradients were |
410 |
established by applying a set of target thermal transfer flux. Data of |
411 |
the series of thermal gradients was collected for calculation. |
412 |
|
413 |
After simulations of bulk water and crystal gold, a mixture system was |
414 |
constructed, consisting of 1188 Au atoms and 1862 H$_2$O |
415 |
molecules. Spohr potential was adopted in depicting the interaction |
416 |
between metal atom and water molecule.\cite{ISI:000167766600035} A |
417 |
similar protocol of equilibration was followed. Several thermal |
418 |
gradients was built under different target thermal flux. It was found |
419 |
out that compared to our previous simulation systems, the two phases |
420 |
could have large temperature difference even under a relatively low |
421 |
thermal flux. Therefore, under our low flux conditions, it is assumed |
422 |
that the metal and water phases have respectively homogeneous |
423 |
temperature, excluding the surface regions. In calculating the |
424 |
interfacial thermal conductivity $G$, this assumptioin was applied and |
425 |
thus our formula becomes: |
426 |
|
427 |
\begin{equation} |
428 |
G = \frac{E_{total}}{2 t L_x L_y \left( \langle T_{gold}\rangle - |
429 |
\langle T_{water}\rangle \right)} |
430 |
\label{interfaceCalc} |
431 |
\end{equation} |
432 |
where ${E_{total}}$ is the imposed unphysical kinetic energy transfer |
433 |
and ${\langle T_{gold}\rangle}$ and ${\langle T_{water}\rangle}$ are the |
434 |
average observed temperature of gold and water phases respectively. |
435 |
|
436 |
\section{Results And Discussions} |
437 |
\subsection{Thermal Conductivity} |
438 |
\subsubsection{Lennard-Jones Fluid} |
439 |
Our thermal conductivity calculations show that scaling method results |
440 |
agree with swapping method. Four different exchange intervals were |
441 |
tested (Table \ref{thermalLJRes}) using swapping method. With a fixed |
442 |
10fs exchange interval, target exchange kinetic energy was set to |
443 |
produce equivalent kinetic energy flux as in swapping method. And |
444 |
similar thermal gradients were observed with similar thermal flux in |
445 |
two simulation methods (Figure \ref{thermalGrad}). |
446 |
|
447 |
\begin{table*} |
448 |
\begin{minipage}{\linewidth} |
449 |
\begin{center} |
450 |
|
451 |
\caption{Calculation results for thermal conductivity of Lennard-Jones |
452 |
fluid at ${\langle T^* \rangle = 0.72}$ and ${\rho^* = 0.85}$, with |
453 |
swap and scale methods at various kinetic energy exchange rates. Results |
454 |
in reduced unit. Errors of calculations in parentheses.} |
455 |
|
456 |
\begin{tabular}{ccc} |
457 |
\hline |
458 |
(Equilvalent) Exchange Interval (fs) & $\lambda^*_{swap}$ & |
459 |
$\lambda^*_{scale}$\\ |
460 |
\hline |
461 |
250 & 7.03(0.34) & 7.30(0.10)\\ |
462 |
500 & 7.03(0.14) & 6.95(0.09)\\ |
463 |
1000 & 6.91(0.42) & 7.19(0.07)\\ |
464 |
2000 & 7.52(0.15) & 7.19(0.28)\\ |
465 |
\hline |
466 |
\end{tabular} |
467 |
\label{thermalLJRes} |
468 |
\end{center} |
469 |
\end{minipage} |
470 |
\end{table*} |
471 |
|
472 |
\begin{figure} |
473 |
\includegraphics[width=\linewidth]{thermalGrad} |
474 |
\caption{Temperature gradients under various kinetic energy flux of |
475 |
thermal conductivity simulations} |
476 |
\label{thermalGrad} |
477 |
\end{figure} |
478 |
|
479 |
During these simulations, molecule velocities were recorded in 1000 of |
480 |
all the snapshots of one single data collection process. These |
481 |
velocity data were used to produce histograms of velocity and speed |
482 |
distribution in different slabs. From these histograms, it is observed |
483 |
that under relatively high unphysical kinetic energy flux, speed and |
484 |
velocity distribution of molecules in slabs where swapping occured |
485 |
could deviate from Maxwell-Boltzmann distribution. Figure |
486 |
\ref{histSwap} illustrates how these distributions deviate from an |
487 |
ideal distribution. In high temperature slab, probability density in |
488 |
low speed is confidently smaller than ideal curve fit; in low |
489 |
temperature slab, probability density in high speed is smaller than |
490 |
ideal, while larger than ideal in low speed. This phenomenon is more |
491 |
obvious in our high swapping rate simulations. And this deviation |
492 |
could also leads to deviation of distribution of velocity in various |
493 |
dimensions. One feature of these deviated distribution is that in high |
494 |
temperature slab, the ideal Gaussian peak was changed into a |
495 |
relatively flat plateau; while in low temperature slab, that peak |
496 |
appears sharper. This problem is rooted in the mechanism of the |
497 |
swapping method. Continually depleting low (high) speed particles in |
498 |
the high (low) temperature slab could not be complemented by |
499 |
diffusions of low (high) speed particles from neighbor slabs, unless |
500 |
in suffciently low swapping rate. Simutaneously, surplus low speed |
501 |
particles in the low temperature slab do not have sufficient time to |
502 |
diffuse to neighbor slabs. However, thermal exchange rate should reach |
503 |
a minimum level to produce an observable thermal gradient under noise |
504 |
interference. Consequently, swapping RNEMD has a relatively narrow |
505 |
choice of swapping rate to satisfy these above restrictions. |
506 |
|
507 |
\begin{figure} |
508 |
\includegraphics[width=\linewidth]{histSwap} |
509 |
\caption{Speed distribution for thermal conductivity using swapping |
510 |
RNEMD. Shown is from the simulation with 250 fs exchange interval.} |
511 |
\label{histSwap} |
512 |
\end{figure} |
513 |
|
514 |
Comparatively, NIVS-RNEMD has a speed distribution closer to the ideal |
515 |
curve fit (Figure \ref{histScale}). Essentially, after scaling, a |
516 |
Gaussian distribution function would remain Gaussian. Although a |
517 |
single scaling is non-isotropic in all three dimensions, our scaling |
518 |
coefficient criteria could help maintian the scaling region as |
519 |
isotropic as possible. On the other hand, scaling coefficients are |
520 |
preferred to be as close to 1 as possible, which also helps minimize |
521 |
the difference among different dimensions. This is possible if scaling |
522 |
interval and one-time thermal transfer energy are well |
523 |
chosen. Consequently, NIVS-RNEMD is able to impose an unphysical |
524 |
thermal flux as the previous RNEMD method without large perturbation |
525 |
to the distribution of velocity and speed in the exchange regions. |
526 |
|
527 |
\begin{figure} |
528 |
\includegraphics[width=\linewidth]{histScale} |
529 |
\caption{Speed distribution for thermal conductivity using scaling |
530 |
RNEMD. Shown is from the simulation with an equilvalent thermal flux |
531 |
as an 250 fs exchange interval swapping simulation.} |
532 |
\label{histScale} |
533 |
\end{figure} |
534 |
|
535 |
\subsubsection{SPC/E Water} |
536 |
Our results of SPC/E water thermal conductivity are comparable to |
537 |
Bedrov {\it et al.}\cite{ISI:000090151400044}, which employed the |
538 |
previous swapping RNEMD method for their calculation. Bedrov {\it et |
539 |
al.}\cite{ISI:000090151400044} argued that exchange of the molecule |
540 |
center-of-mass velocities instead of single atom velocities in a |
541 |
molecule conserves the total kinetic energy and linear momentum. This |
542 |
principle is adopted in our simulations. Scaling is applied to the |
543 |
velocities of the rigid bodies of SPC/E model water molecules, instead |
544 |
of each hydrogen and oxygen atoms in relevant water molecules. As |
545 |
shown in Figure \ref{spceGrad}, temperature gradients were established |
546 |
similar to their system. However, the average temperature of our |
547 |
system is 300K, while theirs is 318K, which would be attributed for |
548 |
part of the difference between the final calculation results (Table |
549 |
\ref{spceThermal}). Both methods yields values in agreement with |
550 |
experiment. And this shows the applicability of our method to |
551 |
multi-atom molecular system. |
552 |
|
553 |
\begin{figure} |
554 |
\includegraphics[width=\linewidth]{spceGrad} |
555 |
\caption{Temperature gradients for SPC/E water thermal conductivity.} |
556 |
\label{spceGrad} |
557 |
\end{figure} |
558 |
|
559 |
\begin{table*} |
560 |
\begin{minipage}{\linewidth} |
561 |
\begin{center} |
562 |
|
563 |
\caption{Calculation results for thermal conductivity of SPC/E water |
564 |
at ${\langle T\rangle}$ = 300K at various thermal exchange rates. Errors of |
565 |
calculations in parentheses. } |
566 |
|
567 |
\begin{tabular}{cccc} |
568 |
\hline |
569 |
$\langle dT/dz\rangle$(K/\AA) & & $\lambda$(W/m/K) & \\ |
570 |
& This work & Previous simulations\cite{ISI:000090151400044} & |
571 |
Experiment$^a$\\ |
572 |
\hline |
573 |
0.38 & 0.816(0.044) & & 0.64\\ |
574 |
0.81 & 0.770(0.008) & 0.784\\ |
575 |
1.54 & 0.813(0.007) & 0.730\\ |
576 |
\hline |
577 |
\end{tabular} |
578 |
\label{spceThermal} |
579 |
\end{center} |
580 |
\end{minipage} |
581 |
\end{table*} |
582 |
|
583 |
\subsubsection{Crystal Gold} |
584 |
Our results of gold thermal conductivity using two force fields are |
585 |
shown separately in Table \ref{qscThermal} and \ref{eamThermal}. In |
586 |
these calculations,the end and middle slabs were excluded in thermal |
587 |
gradient regession and only used as heat source and drain in the |
588 |
systems. Our yielded values using EAM force field are slightly larger |
589 |
than those using QSC force field. However, both series are |
590 |
significantly smaller than experimental value by an order of more than |
591 |
100. It has been verified that this difference is mainly attributed to |
592 |
the lack of electron interaction representation in these force field |
593 |
parameters. Richardson {\it et al.}\cite{Clancy:1992} used EAM |
594 |
force field parameters in their metal thermal conductivity |
595 |
calculations. The Non-Equilibrium MD method they employed in their |
596 |
simulations produced comparable results to ours. As Zhang {\it et |
597 |
al.}\cite{ISI:000231042800044} stated, thermal conductivity values |
598 |
are influenced mainly by force field. Therefore, it is confident to |
599 |
conclude that NIVS-RNEMD is applicable to metal force field system. |
600 |
|
601 |
\begin{figure} |
602 |
\includegraphics[width=\linewidth]{AuGrad} |
603 |
\caption{Temperature gradients for thermal conductivity calculation of |
604 |
crystal gold using QSC force field.} |
605 |
\label{AuGrad} |
606 |
\end{figure} |
607 |
|
608 |
\begin{table*} |
609 |
\begin{minipage}{\linewidth} |
610 |
\begin{center} |
611 |
|
612 |
\caption{Calculation results for thermal conductivity of crystal gold |
613 |
using QSC force field at ${\langle T\rangle}$ = 300K at various |
614 |
thermal exchange rates. Errors of calculations in parentheses. } |
615 |
|
616 |
\begin{tabular}{cc} |
617 |
\hline |
618 |
$\langle dT/dz\rangle$(K/\AA) & $\lambda$(W/m/K)\\ |
619 |
\hline |
620 |
1.44 & 1.10(0.01)\\ |
621 |
2.86 & 1.08(0.02)\\ |
622 |
5.14 & 1.15(0.01)\\ |
623 |
\hline |
624 |
\end{tabular} |
625 |
\label{qscThermal} |
626 |
\end{center} |
627 |
\end{minipage} |
628 |
\end{table*} |
629 |
|
630 |
\begin{figure} |
631 |
\includegraphics[width=\linewidth]{eamGrad} |
632 |
\caption{Temperature gradients for thermal conductivity calculation of |
633 |
crystal gold using EAM force field.} |
634 |
\label{eamGrad} |
635 |
\end{figure} |
636 |
|
637 |
\begin{table*} |
638 |
\begin{minipage}{\linewidth} |
639 |
\begin{center} |
640 |
|
641 |
\caption{Calculation results for thermal conductivity of crystal gold |
642 |
using EAM force field at ${\langle T\rangle}$ = 300K at various |
643 |
thermal exchange rates. Errors of calculations in parentheses. } |
644 |
|
645 |
\begin{tabular}{cc} |
646 |
\hline |
647 |
$\langle dT/dz\rangle$(K/\AA) & $\lambda$(W/m/K)\\ |
648 |
\hline |
649 |
1.24 & 1.24(0.06)\\ |
650 |
2.06 & 1.37(0.04)\\ |
651 |
2.55 & 1.41(0.03)\\ |
652 |
\hline |
653 |
\end{tabular} |
654 |
\label{eamThermal} |
655 |
\end{center} |
656 |
\end{minipage} |
657 |
\end{table*} |
658 |
|
659 |
|
660 |
\subsection{Interfaciel Thermal Conductivity} |
661 |
After simulations of homogeneous water and gold systems using |
662 |
NIVS-RNEMD method were proved valid, calculation of gold/water |
663 |
interfacial thermal conductivity was followed. It is found out that |
664 |
the low interfacial conductance is probably due to the hydrophobic |
665 |
surface in our system. Figure \ref{interfaceDensity} demonstrates mass |
666 |
density change along $z$-axis, which is perpendicular to the |
667 |
gold/water interface. It is observed that water density significantly |
668 |
decreases when approaching the surface. Under this low thermal |
669 |
conductance, both gold and water phase have sufficient time to |
670 |
eliminate temperature difference inside respectively (Figure |
671 |
\ref{interfaceGrad}). With indistinguishable temperature difference |
672 |
within respective phase, it is valid to assume that the temperature |
673 |
difference between gold and water on surface would be approximately |
674 |
the same as the difference between the gold and water phase. This |
675 |
assumption enables convenient calculation of $G$ using |
676 |
Eq. \ref{interfaceCalc} instead of measuring temperatures of thin |
677 |
layer of water and gold close enough to surface, which would have |
678 |
greater fluctuation and lower accuracy. Reported results (Table |
679 |
\ref{interfaceRes}) are of two orders of magnitude smaller than our |
680 |
calculations on homogeneous systems, and thus have larger relative |
681 |
errors than our calculation results on homogeneous systems. |
682 |
|
683 |
\begin{figure} |
684 |
\includegraphics[width=\linewidth]{interfaceDensity} |
685 |
\caption{Density profile for interfacial thermal conductivity |
686 |
simulation box. Significant water density decrease is observed on |
687 |
gold surface.} |
688 |
\label{interfaceDensity} |
689 |
\end{figure} |
690 |
|
691 |
\begin{figure} |
692 |
\includegraphics[width=\linewidth]{interfaceGrad} |
693 |
\caption{Temperature profiles for interfacial thermal conductivity |
694 |
simulation box. Temperatures of different slabs in the same phase |
695 |
show no significant difference.} |
696 |
\label{interfaceGrad} |
697 |
\end{figure} |
698 |
|
699 |
\begin{table*} |
700 |
\begin{minipage}{\linewidth} |
701 |
\begin{center} |
702 |
|
703 |
\caption{Calculation results for interfacial thermal conductivity |
704 |
at ${\langle T\rangle \sim}$ 300K at various thermal exchange |
705 |
rates. Errors of calculations in parentheses. } |
706 |
|
707 |
\begin{tabular}{cccc} |
708 |
\hline |
709 |
$J_z$(MW/m$^2$) & $T_{gold}$ & $T_{water}$ & $G$(MW/m$^2$/K)\\ |
710 |
\hline |
711 |
98.0 & 355.2 & 295.8 & 1.65(0.21) \\ |
712 |
78.8 & 343.8 & 298.0 & 1.72(0.32) \\ |
713 |
73.6 & 344.3 & 298.0 & 1.59(0.24) \\ |
714 |
49.2 & 330.1 & 300.4 & 1.65(0.35) \\ |
715 |
\hline |
716 |
\end{tabular} |
717 |
\label{interfaceRes} |
718 |
\end{center} |
719 |
\end{minipage} |
720 |
\end{table*} |
721 |
|
722 |
\subsection{Shear Viscosity} |
723 |
Our calculations (Table \ref{shearRate}) shows that scale RNEMD method |
724 |
produced comparable shear viscosity to swap RNEMD method. In Table |
725 |
\ref{shearRate}, the names of the calculated samples are devided into |
726 |
two parts. The first number refers to total slabs in one simulation |
727 |
box. The second number refers to the swapping interval in swap method, or |
728 |
in scale method the equilvalent swapping interval that the same |
729 |
momentum flux would theoretically result in swap method. All the scale |
730 |
method results were from simulations that had a scaling interval of 10 |
731 |
time steps. The average molecular momentum gradients of these samples |
732 |
are shown in Figure \ref{shearGrad}. |
733 |
|
734 |
\begin{table*} |
735 |
\begin{minipage}{\linewidth} |
736 |
\begin{center} |
737 |
|
738 |
\caption{Calculation results for shear viscosity of Lennard-Jones |
739 |
fluid at ${T^* = 0.72}$ and ${\rho^* = 0.85}$, with swap and scale |
740 |
methods at various momentum exchange rates. Results in reduced |
741 |
unit. Errors of calculations in parentheses. } |
742 |
|
743 |
\begin{tabular}{ccc} |
744 |
\hline |
745 |
Series & $\eta^*_{swap}$ & $\eta^*_{scale}$\\ |
746 |
\hline |
747 |
20-500 & 3.64(0.05) & 3.76(0.09)\\ |
748 |
20-1000 & 3.52(0.16) & 3.66(0.06)\\ |
749 |
20-2000 & 3.72(0.05) & 3.32(0.18)\\ |
750 |
20-2500 & 3.42(0.06) & 3.43(0.08)\\ |
751 |
\hline |
752 |
\end{tabular} |
753 |
\label{shearRate} |
754 |
\end{center} |
755 |
\end{minipage} |
756 |
\end{table*} |
757 |
|
758 |
\begin{figure} |
759 |
\includegraphics[width=\linewidth]{shearGrad} |
760 |
\caption{Average momentum gradients of shear viscosity simulations} |
761 |
\label{shearGrad} |
762 |
\end{figure} |
763 |
|
764 |
\begin{figure} |
765 |
\includegraphics[width=\linewidth]{shearTempScale} |
766 |
\caption{Temperature profile for scaling RNEMD simulation.} |
767 |
\label{shearTempScale} |
768 |
\end{figure} |
769 |
However, observations of temperatures along three dimensions show that |
770 |
inhomogeneity occurs in scaling RNEMD simulations, particularly in the |
771 |
two slabs which were scaled. Figure \ref{shearTempScale} indicate that with |
772 |
relatively large imposed momentum flux, the temperature difference among $x$ |
773 |
and the other two dimensions was significant. This would result from the |
774 |
algorithm of scaling method. From Eq. \ref{eq:fluxPlane}, after |
775 |
momentum gradient is set up, $P_c^x$ would be roughly stable |
776 |
($<0$). Consequently, scaling factor $x$ would most probably larger |
777 |
than 1. Therefore, the kinetic energy in $x$-dimension $K_c^x$ would |
778 |
keep increase after most scaling steps. And if there is not enough time |
779 |
for the kinetic energy to exchange among different dimensions and |
780 |
different slabs, the system would finally build up temperature |
781 |
(kinetic energy) difference among the three dimensions. Also, between |
782 |
$y$ and $z$ dimensions in the scaled slabs, temperatures of $z$-axis |
783 |
are closer to neighbor slabs. This is due to momentum transfer along |
784 |
$z$ dimension between slabs. |
785 |
|
786 |
Although results between scaling and swapping methods are comparable, |
787 |
the inherent temperature inhomogeneity even in relatively low imposed |
788 |
exchange momentum flux simulations makes scaling RNEMD method less |
789 |
attractive than swapping RNEMD in shear viscosity calculation. |
790 |
|
791 |
\section{Conclusions} |
792 |
NIVS-RNEMD simulation method is developed and tested on various |
793 |
systems. Simulation results demonstrate its validity in thermal |
794 |
conductivity calculations, from Lennard-Jones fluid to multi-atom |
795 |
molecule like water and metal crystals. NIVS-RNEMD improves |
796 |
non-Boltzmann-Maxwell distributions, which exist in previous RNEMD |
797 |
methods. Furthermore, it develops a valid means for unphysical thermal |
798 |
transfer between different species of molecules, and thus extends its |
799 |
applicability to interfacial systems. Our calculation of gold/water |
800 |
interfacial thermal conductivity demonstrates this advantage over |
801 |
previous RNEMD methods. NIVS-RNEMD has also limited application on |
802 |
shear viscosity calculations, but could cause temperature difference |
803 |
among different dimensions under high momentum flux. Modification is |
804 |
necessary to extend the applicability of NIVS-RNEMD in shear viscosity |
805 |
calculations. |
806 |
|
807 |
\section{Acknowledgments} |
808 |
Support for this project was provided by the National Science |
809 |
Foundation under grant CHE-0848243. Computational time was provided by |
810 |
the Center for Research Computing (CRC) at the University of Notre |
811 |
Dame. \newpage |
812 |
|
813 |
\bibliographystyle{jcp2} |
814 |
\bibliography{nivsRnemd} |
815 |
\end{doublespace} |
816 |
\end{document} |
817 |
|