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\title{A method for creating thermal and angular momentum fluxes in |
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non-periodic simulations} |
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\author{Kelsey M. Stocker} |
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\author{J. Daniel Gezelter} |
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\email{gezelter@nd.edu} |
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\affiliation[University of Notre Dame]{251 Nieuwland Science Hall\\ Department of Chemistry and Biochemistry\\ University of Notre Dame\\ Notre Dame, Indiana 46556} |
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\begin{document} |
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\begin{tocentry} |
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\includegraphics[width=3.6cm]{figures/NP20} |
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\end{tocentry} |
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% \author{Kelsey M. Stocker and J. Daniel |
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% Gezelter\footnote{Corresponding author. \ Electronic mail: |
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% gezelter@nd.edu} \\ |
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% 251 Nieuwland Science Hall, \\ |
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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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We present a new reverse non-equilibrium molecular dynamics (RNEMD) |
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method that can be used with non-periodic simulation cells. This |
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method applies thermal and/or angular momentum fluxes between two |
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arbitrary regions of the simulation, and is capable of creating |
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stable temperature and angular velocity gradients while conserving |
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total energy and angular momentum. One particularly useful |
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application is the exchange of kinetic energy between two concentric |
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spherical regions, which can be used to generate thermal transport |
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between nanoparticles and the solvent that surrounds them. The |
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rotational couple to the solvent (a measure of interfacial friction) |
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is also available via this method. As demonstrations and tests of |
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the new method, we have computed the thermal conductivities of gold |
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nanoparticles and water clusters, the shear viscosity of a water |
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cluster, the interfacial thermal conductivity ($G$) of a solvated |
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gold nanoparticle and the interfacial friction of a variety of |
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solvated gold nanostructures. |
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\end{abstract} |
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\newpage |
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%\narrowtext |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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% **INTRODUCTION** |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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\section{Introduction} |
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|
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Non-equilibrium molecular dynamics (NEMD) methods impose a temperature |
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or velocity {\it gradient} on a |
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system,\cite{Ashurst:1975eu,Evans:1982oq,Erpenbeck:1984qe,Evans:1986nx,Vogelsang:1988qv,Maginn:1993kl,Hess:2002nr,Schelling:2002dp,Berthier:2002ai,Evans:2002tg,Vasquez:2004ty,Backer:2005sf,Jiang:2008hc,Picalek:2009rz} |
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and use linear response theory to connect the resulting thermal or |
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momentum {\it flux} to transport coefficients of bulk materials, |
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\begin{equation} |
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j_z(p_x) = -\eta \frac{\partial v_x}{\partial z}, \hspace{0.5in} |
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J_z = \lambda \frac{\partial T}{\partial z}. |
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\end{equation} |
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Here, $\frac{\partial T}{\partial z}$ and $\frac{\partial |
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v_x}{\partial z}$ are the imposed thermal and momentum gradients, |
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and as long as the imposed gradients are relatively small, the |
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corresponding fluxes, $J_z$ and $j_z(p_x)$, have a linear relationship |
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to the gradients. The coefficients that provide this relationship |
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correspond to physical properties of the bulk material, either the |
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shear viscosity $(\eta)$ or thermal conductivity $(\lambda)$. For |
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systems which include phase boundaries or interfaces, it is often |
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unclear what gradient (or discontinuity) should be imposed at the |
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boundary between materials. |
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In contrast, reverse Non-Equilibrium Molecular Dynamics (RNEMD) |
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methods impose an unphysical {\it flux} between different regions or |
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``slabs'' of the simulation |
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box.\cite{Muller-Plathe:1997wq,Muller-Plathe:1999ao,Patel:2005zm,Shenogina:2009ix,Tenney:2010rp,Kuang:2010if,Kuang:2011ef,Kuang:2012fe,Stocker:2013cl} |
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The system responds by developing a temperature or velocity {\it |
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gradient} between the two regions. The gradients which develop in |
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response to the applied flux have the same linear response |
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relationships to the transport coefficient of interest. Since the |
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amount of the applied flux is known exactly, and measurement of a |
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gradient is generally less complicated, imposed-flux methods typically |
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take shorter simulation times to obtain converged results. At |
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interfaces, the observed gradients often exhibit near-discontinuities |
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at the boundaries between dissimilar materials. RNEMD methods do not |
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need many trajectories to provide information about transport |
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properties, and they have become widely used to compute thermal and |
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mechanical transport in both homogeneous liquids and |
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solids~\cite{Muller-Plathe:1997wq,Muller-Plathe:1999ao,Tenney:2010rp} |
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as well as heterogeneous |
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interfaces.\cite{Patel:2005zm,Shenogina:2009ix,Kuang:2010if,Kuang:2011ef,Kuang:2012fe,Stocker:2013cl} |
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The strengths of specific algorithms for imposing the flux between two |
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different slabs of the simulation cell has been the subject of some |
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renewed interest. The original RNEMD approach used kinetic energy or |
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momentum exchange between particles in the two slabs, either through |
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direct swapping of momentum vectors or via virtual elastic collisions |
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between atoms in the two regions. There have been recent |
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methodological advances which involve scaling all particle velocities |
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in both slabs.\cite{Kuang:2010if,Kuang:2012fe} Constraint equations |
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are simultaneously imposed to require the simulation to conserve both |
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total energy and total linear momentum. The most recent and simplest |
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of the velocity scaling approaches allows for simultaneous shearing |
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(to provide viscosity estimates) as well as scaling (to provide |
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information about thermal conductivity).\cite{Kuang:2012fe} |
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To date, however, the RNEMD methods have only been used in periodic |
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simulation cells where the exchange regions are physically separated |
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along one of the axes of the simulation cell. This limits the |
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applicability to infinite planar interfaces which are perpendicular to |
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the applied flux. In order to model steady-state non-equilibrium |
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distributions for curved surfaces (e.g. hot nanoparticles in contact |
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with colder solvent), or for regions that are not planar slabs, the |
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method requires some generalization for non-parallel exchange regions. |
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In the following sections, we present a new velocity shearing and |
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scaling (VSS) RNEMD algorithm which has been explicitly designed for |
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non-periodic simulations, and use the method to compute some thermal |
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transport and solid-liquid friction at the surfaces of spherical and |
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ellipsoidal nanoparticles. |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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% **METHODOLOGY** |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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\section{Velocity shearing and scaling (VSS) for non-periodic systems} |
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The original periodic VSS-RNEMD approach uses a series of simultaneous |
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velocity shearing and scaling exchanges between the two |
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slabs.\cite{Kuang:2012fe} This method imposes energy and linear |
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momentum conservation constraints while simultaneously creating a |
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desired flux between the two slabs. These constraints ensure that all |
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configurations are sampled from the same microcanonical (NVE) |
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ensemble. |
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\begin{figure} |
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\includegraphics[width=\linewidth]{figures/npVSS} |
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\caption{Schematics of periodic (left) and non-periodic (right) |
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Velocity Shearing and Scaling RNEMD. A kinetic energy or momentum |
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flux is applied from region B to region A. Thermal gradients are |
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depicted by a color gradient. Linear or angular velocity gradients |
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are shown as arrows.} |
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\label{fig:VSS} |
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\end{figure} |
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We have extended the VSS method for use in {\it non-periodic} |
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simulations, in which the ``slabs'' have been generalized to two |
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separated regions of space. These regions could be defined as |
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concentric spheres (as in figure \ref{fig:VSS}), or one of the regions |
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can be defined in terms of a dynamically changing ``hull'' comprising |
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the surface atoms of the cluster. This latter definition is identical |
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to the hull used in the Langevin Hull algorithm.\cite{Vardeman2011} |
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For the non-periodic variant, the constraints fix both the total |
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energy and total {\it angular} momentum of the system while |
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simultaneously imposing a thermal and angular momentum flux between |
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the two regions. |
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After a time interval of $\Delta t$, the particle velocities |
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($\mathbf{v}_i$ and $\mathbf{v}_j$) in the two shells ($A$ and $B$) |
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are modified by a velocity scaling coefficient ($a$ and $b$) and by a |
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rotational shearing term ($\mathbf{c}_a$ and $\mathbf{c}_b$). The |
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scalars $a$ and $b$ collectively provide a thermal exchange between |
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the two regions. One of the values is larger than 1, and the other |
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smaller. To conserve total energy and angular momentum, the values of |
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these two scalars are coupled. The vectors ($\mathbf{c}_a$ and |
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$\mathbf{c}_b$) provide a relative rotational shear to the velocities |
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of the particles within the two regions, and these vectors must also |
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be coupled to constrain the total angular momentum. |
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Once the values of the scaling and shearing factors are known, the |
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velocity changes are applied, |
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\begin{displaymath} |
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\begin{array}{rclcl} |
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& \underline{~~~~~~~~\mathrm{scaling}~~~~~~~~} & & |
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\underline{\mathrm{rotational~shearing}} \\ \\ |
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\mathbf{v}_i $~~~$\leftarrow & |
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a \left(\mathbf{v}_i - \langle \omega_a |
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\rangle \times \mathbf{r}_i\right) & + & \mathbf{c}_a \times \mathbf{r}_i \\ |
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\mathbf{v}_j $~~~$\leftarrow & |
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b \left(\mathbf{v}_j - \langle \omega_b |
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\rangle \times \mathbf{r}_j\right) & + & \mathbf{c}_b \times \mathbf{r}_j |
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\end{array} |
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\end{displaymath} |
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Here $\langle\mathbf{\omega}_a\rangle$ and $\langle\mathbf{\omega}_b\rangle$ are the instantaneous angular |
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velocities of each shell, and $\mathbf{r}_i$ is the position of particle $i$ relative to a fixed point in space |
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(usually the center of mass of the cluster). Particles in the shells also receive an additive ``angular shear'' |
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to their velocities. The amount of shear is governed by the imposed angular momentum flux, |
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$\mathbf{j}_r(\mathbf{L})$, |
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\begin{eqnarray} |
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\mathbf{c}_a & = & - \mathbf{j}_r(\mathbf{L}) \cdot |
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\overleftrightarrow{I_a}^{-1} \Delta t + \langle \omega_a \rangle \label{eq:bc}\\ |
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\mathbf{c}_b & = & + \mathbf{j}_r(\mathbf{L}) \cdot |
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\overleftrightarrow{I_b}^{-1} \Delta t + \langle \omega_b \rangle \label{eq:bh} |
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\end{eqnarray} |
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where $\overleftrightarrow{I}_{\{a,b\}}$ is the moment of inertia |
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tensor for each of the two shells. |
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To simultaneously impose a thermal flux ($J_r$) between the shells we |
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use energy conservation constraints, |
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\begin{eqnarray} |
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K_a - J_r \Delta t & = & a^2 \left(K_a - \frac{1}{2}\langle |
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\omega_a \rangle \cdot \overleftrightarrow{I_a} \cdot \langle \omega_a |
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\rangle \right) + \frac{1}{2} \mathbf{c}_a \cdot \overleftrightarrow{I_a} |
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\cdot \mathbf{c}_a \label{eq:Kc}\\ |
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K_b + J_r \Delta t & = & b^2 \left(K_b - \frac{1}{2}\langle |
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\omega_b \rangle \cdot \overleftrightarrow{I_b} \cdot \langle \omega_b |
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\rangle \right) + \frac{1}{2} \mathbf{c}_b \cdot \overleftrightarrow{I_b} \cdot \mathbf{c}_b \label{eq:Kh} |
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\end{eqnarray} |
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Simultaneous solution of these quadratic formulae for the scaling coefficients, $a$ and $b$, will ensure that |
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the simulation samples from the original microcanonical (NVE) ensemble. Here $K_{\{a,b\}}$ is the instantaneous |
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translational kinetic energy of each shell. At each time interval, we solve for $a$, $b$, $\mathbf{c}_a$, and |
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$\mathbf{c}_b$, subject to the imposed angular momentum flux, $j_r(\mathbf{L})$, and thermal flux, $J_r$, |
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values. The new particle velocities are computed, and the simulation continues. System configurations after the |
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transformations have exactly the same energy ({\it and} angular momentum) as before the moves. |
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As the simulation progresses, the velocity transformations can be |
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performed on a regular basis, and the system will develop a |
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temperature and/or angular velocity gradient in response to the |
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applied flux. By fitting the radial temperature gradient, it is |
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straightforward to obtain the bulk thermal conductivity, |
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\begin{equation} |
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J_r = -\lambda \left( \frac{\partial T}{\partial r}\right) |
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\end{equation} |
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from the radial kinetic energy flux $(J_r)$ that was applied during |
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the simulation. In practice, it is significantly easier to use the |
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integrated form of Fourier's law for spherical geometries. In |
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sections \ref{sec:thermal} -- \ref{sec:rotation} we outline ways of |
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obtaining interfacial transport coefficients from these RNEMD |
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simulations. |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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% **COMPUTATIONAL DETAILS** |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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\section{Computational Details} |
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The new VSS-RNEMD methodology for non-periodic system geometries has |
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been implemented in our group molecular dynamics code, |
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OpenMD.\cite{Meineke:2005gd,openmd} We have tested the new method to |
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calculate the thermal conductance of a gold nanoparticle and SPC/E |
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water cluster, and compared the results with previous bulk RNEMD |
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values, as well as experiment. We have also investigated the |
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interfacial thermal conductance and interfacial rotational friction |
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for gold nanostructures solvated in hexane as a function of |
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nanoparticle size and shape. |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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% FORCE FIELD PARAMETERS |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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\subsection{Force field} |
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Gold -- gold interactions are described by the quantum Sutton-Chen |
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(QSC) model.\cite{PhysRevB.59.3527} The QSC parameters are tuned to |
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experimental properties such as density, cohesive energy, and elastic |
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moduli and include zero-point quantum corrections. |
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The SPC/E water model~\cite{Berendsen87} is particularly useful for |
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validation of conductivities and shear viscosities. This model has |
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been used to previously test other RNEMD and NEMD approaches, and |
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there are reported values for thermal conductivies and shear |
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viscosities at a wide range of thermodynamic conditions that are |
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available for direct comparison.\cite{Bedrov:2000,Kuang:2010if} |
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Hexane molecules are described by the TraPPE united atom |
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model,\cite{TraPPE-UA.alkanes} which provides good computational |
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efficiency and reasonable accuracy for bulk thermal conductivity |
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values. In this model, sites are located at the carbon centers for |
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alkyl groups. Bonding interactions, including bond stretches and bends |
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and torsions, were used for intra-molecular sites closer than 3 |
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bonds. For non-bonded interactions, Lennard-Jones potentials were |
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used. We have previously utilized both united atom (UA) and all-atom |
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(AA) force fields for thermal |
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conductivity,\cite{Kuang:2011ef,Kuang:2012fe,Stocker:2013cl} and since |
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the united atom force fields cannot populate the high-frequency modes |
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that are present in AA force fields, they appear to work better for |
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modeling thermal conductance at metal/ligand interfaces. |
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Gold -- hexane nonbonded interactions are governed by pairwise |
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Lennard-Jones parameters derived from Vlugt \emph{et |
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al}.\cite{vlugt:cpc2007154} They fitted parameters for the |
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interaction between Au and CH$_{\emph x}$ pseudo-atoms based on the |
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effective potential of Hautman and Klein for the Au(111) |
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surface.\cite{hautman:4994} |
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|
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kstocke1 |
3947 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
315 |
kstocke1 |
4009 |
% NON-PERIODIC DYNAMICS |
316 |
kstocke1 |
3947 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
317 |
kstocke1 |
4009 |
% \subsection{Dynamics for non-periodic systems} |
318 |
|
|
% |
319 |
|
|
% We have run all tests using the Langevin Hull non-periodic simulation methodology.\cite{Vardeman2011} The |
320 |
|
|
% Langevin Hull is especially useful for simulating heterogeneous mixtures of materials with different |
321 |
|
|
% compressibilities, which are typically problematic for traditional affine transform methods. We have had |
322 |
|
|
% success applying this method to several different systems including bare metal nanoparticles, liquid water |
323 |
|
|
% clusters, and explicitly solvated nanoparticles. Calculated physical properties such as the isothermal |
324 |
|
|
% compressibility of water and the bulk modulus of gold nanoparticles are in good agreement with previous |
325 |
|
|
% theoretical and experimental results. The Langevin Hull uses a Delaunay tesselation to create a dynamic convex |
326 |
|
|
% hull composed of triangular facets with vertices at atomic sites. Atomic sites included in the hull are coupled |
327 |
|
|
% to an external bath defined by a temperature, pressure and viscosity. Atoms not included in the hull are |
328 |
|
|
% subject to standard Newtonian dynamics. |
329 |
kstocke1 |
3947 |
|
330 |
kstocke1 |
4009 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
331 |
|
|
% SIMULATION PROTOCOL |
332 |
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
333 |
|
|
\subsection{Simulation protocol} |
334 |
kstocke1 |
3947 |
|
335 |
gezelter |
4063 |
In all cases, systems were equilibrated under non-periodic |
336 |
|
|
isobaric-isothermal (NPT) conditions -- using the Langevin Hull |
337 |
|
|
methodology\cite{Vardeman2011} -- before any non-equilibrium methods |
338 |
|
|
were introduced. For heterogeneous systems, the gold nanoparticles and |
339 |
|
|
ellipsoids were created from a bulk fcc lattice and were thermally |
340 |
|
|
equilibrated before being solvated in hexane. Packmol\cite{packmol} |
341 |
|
|
was used to solvate previously equilibrated gold nanostructures within |
342 |
|
|
a spherical droplet of hexane. |
343 |
kstocke1 |
3947 |
|
344 |
gezelter |
4063 |
Once equilibrated, thermal or angular momentum fluxes were applied for |
345 |
|
|
1 - 2 ns, until stable temperature or angular velocity gradients had |
346 |
|
|
developed. Systems containing liquids were run under moderate pressure |
347 |
|
|
(5 atm) and temperatures (230 K) to avoid the formation of a vapor |
348 |
|
|
layer at the boundary of the cluster. Pressure was applied to the |
349 |
|
|
system via the non-periodic Langevin Hull.\cite{Vardeman2011} However, |
350 |
|
|
thermal coupling to the external temperature and pressure bath was |
351 |
|
|
removed to avoid interference with the imposed RNEMD flux. |
352 |
kstocke1 |
3947 |
|
353 |
gezelter |
4063 |
Because the method conserves \emph{total} angular momentum, systems |
354 |
|
|
which contain a metal nanoparticle embedded in a significant volume of |
355 |
|
|
solvent will still experience nanoparticle diffusion inside the |
356 |
|
|
solvent droplet. To aid in computing the rotational friction in these |
357 |
|
|
systems, a single gold atom at the origin of the coordinate system was |
358 |
|
|
assigned a mass $10,000 \times$ its original mass. The bonded and |
359 |
|
|
nonbonded interactions for this atom remain unchanged and the heavy |
360 |
|
|
atom is excluded from the RNEMD exchanges. The only effect of this |
361 |
|
|
gold atom is to effectively pin the nanoparticle at the origin of the |
362 |
|
|
coordinate system, while still allowing for rotation. For rotation of |
363 |
|
|
the gold ellipsoids we added two of these heavy atoms along the axis |
364 |
|
|
of rotation, separated by an equal distance from the origin of the |
365 |
|
|
coordinate system. These heavy atoms prevent off-axis tumbling of the |
366 |
|
|
nanoparticle and allow for measurement of rotational friction relative |
367 |
|
|
to a particular axis of the ellipsoid. |
368 |
kstocke1 |
3947 |
|
369 |
gezelter |
4063 |
Angular velocity data was collected for the heterogeneous systems |
370 |
|
|
after a brief period of imposed flux to initialize rotation of the |
371 |
|
|
solvated nanostructure. Doing so ensures that we overcome the initial |
372 |
|
|
static friction and calculate only the \emph{dynamic} interfacial |
373 |
|
|
rotational friction. |
374 |
|
|
|
375 |
kstocke1 |
3947 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
376 |
|
|
% THERMAL CONDUCTIVITIES |
377 |
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
378 |
|
|
\subsection{Thermal conductivities} |
379 |
gezelter |
4064 |
\label{sec:thermal} |
380 |
kstocke1 |
3947 |
|
381 |
gezelter |
4064 |
To compute the thermal conductivities of bulk materials, the |
382 |
|
|
integrated form of Fourier's Law of heat conduction in radial |
383 |
|
|
coordinates can be used to obtain an expression for the heat transfer |
384 |
|
|
rate between concentric spherical shells: |
385 |
kstocke1 |
3947 |
\begin{equation} |
386 |
gezelter |
4064 |
q_r = - \frac{4 \pi \lambda (T_b - T_a)}{\frac{1}{r_a} - \frac{1}{r_b}} |
387 |
kstocke1 |
4003 |
\label{eq:Q} |
388 |
kstocke1 |
3947 |
\end{equation} |
389 |
gezelter |
4064 |
The heat transfer rate, $q_r$, is constant in spherical geometries, |
390 |
|
|
while the heat flux, $J_r$ depends on the surface area of the two |
391 |
|
|
shells. $\lambda$ is the thermal conductivity, and $T_{a,b}$ and |
392 |
|
|
$r_{a,b}$ are the temperatures and radii of the two concentric RNEMD |
393 |
|
|
regions, respectively. |
394 |
kstocke1 |
3947 |
|
395 |
gezelter |
4064 |
A kinetic energy flux is created using VSS-RNEMD moves, and the |
396 |
|
|
temperature in each of the radial shells is recorded. The resulting |
397 |
|
|
temperature profiles are analyzed to yield information about the |
398 |
|
|
interfacial thermal conductance. As the simulation progresses, the |
399 |
|
|
VSS moves are performed on a regular basis, and the system develops a |
400 |
|
|
thermal or velocity gradient in response to the applied flux. Once a |
401 |
|
|
stable thermal gradient has been established between the two regions, |
402 |
|
|
the thermal conductivity, $\lambda$, can be calculated using a linear |
403 |
gezelter |
4063 |
regression of the thermal gradient, $\langle \frac{dT}{dr} \rangle$: |
404 |
kstocke1 |
3991 |
|
405 |
|
|
\begin{equation} |
406 |
gezelter |
4064 |
\lambda = \frac{r_a}{r_b} \frac{q_r}{4 \pi \langle \frac{dT}{dr} \rangle} |
407 |
kstocke1 |
4003 |
\label{eq:lambda} |
408 |
kstocke1 |
3991 |
\end{equation} |
409 |
|
|
|
410 |
gezelter |
4064 |
The rate of heat transfer, $q_r$, between the two RNEMD regions is |
411 |
|
|
easily obtained from either the applied kinetic energy flux and the |
412 |
|
|
area of the smaller of the two regions, or from the total amount of |
413 |
|
|
transferred kinetic energy and the run time of the simulation. |
414 |
kstocke1 |
3991 |
\begin{equation} |
415 |
gezelter |
4064 |
q_r = J_r A = \frac{KE}{t} |
416 |
kstocke1 |
4003 |
\label{eq:heat} |
417 |
kstocke1 |
3991 |
\end{equation} |
418 |
|
|
|
419 |
gezelter |
4065 |
\subsubsection{Thermal conductivity of nanocrystalline gold} |
420 |
|
|
Calculated values for the thermal conductivity of a 40 \AA$~$ radius |
421 |
|
|
gold nanoparticle (15707 atoms) at a range of kinetic energy flux |
422 |
|
|
values are shown in Table \ref{table:goldTC}. For these calculations, |
423 |
|
|
the hot and cold slabs were excluded from the linear regression of the |
424 |
|
|
thermal gradient. |
425 |
|
|
|
426 |
|
|
\begin{longtable}{ccc} |
427 |
|
|
\caption{Calculated thermal conductivity of a crystalline gold |
428 |
|
|
nanoparticle of radius 40 \AA. Calculations were performed at 300 |
429 |
|
|
K and ambient density. Gold-gold interactions are described by the |
430 |
|
|
Quantum Sutton-Chen potential.} |
431 |
|
|
\\ \hline \hline |
432 |
|
|
{$J_r$} & {$\langle \frac{dT}{dr} \rangle$} & {$\boldsymbol \lambda$}\\ |
433 |
|
|
{\footnotesize(kcal / fs $\cdot$ \AA$^{2}$)} & {\footnotesize(K / \AA)} & {\footnotesize(W / m $\cdot$ K)}\\ \hline |
434 |
|
|
3.25$\times 10^{-6}$ & 0.11435 & 1.0143 \\ |
435 |
|
|
6.50$\times 10^{-6}$ & 0.2324 & 0.9982 \\ |
436 |
|
|
1.30$\times 10^{-5}$ & 0.44922 & 1.0328 \\ |
437 |
|
|
3.25$\times 10^{-5}$ & 1.1802 & 0.9828 \\ |
438 |
|
|
6.50$\times 10^{-5}$ & 2.339 & 0.9918 \\ |
439 |
|
|
\hline |
440 |
|
|
This work & & 1.0040 |
441 |
|
|
\\ \hline \hline |
442 |
|
|
\label{table:goldTC} |
443 |
|
|
\end{longtable} |
444 |
|
|
|
445 |
|
|
The measured linear slope $\langle \frac{dT}{dr} \rangle$ is linearly |
446 |
|
|
dependent on the applied kinetic energy flux $J_r$. Calculated thermal |
447 |
|
|
conductivity values compare well with previous bulk QSC values of 1.08 |
448 |
|
|
-- 1.26 W / m $\cdot$ K\cite{Kuang:2010if}, though still significantly |
449 |
|
|
lower than the experimental value of 320 W / m $\cdot$ K, as the QSC |
450 |
|
|
force field neglects significant electronic contributions to heat |
451 |
|
|
conduction. |
452 |
|
|
|
453 |
|
|
\subsubsection{Thermal conductivity of a droplet of SPC/E water} |
454 |
|
|
|
455 |
|
|
Calculated values for the thermal conductivity of a cluster of 6912 |
456 |
|
|
SPC/E water molecules are shown in Table \ref{table:waterTC}. As with |
457 |
|
|
the gold nanoparticle thermal conductivity calculations, the RNEMD |
458 |
|
|
regions were excluded from the $\langle \frac{dT}{dr} \rangle$ fit. |
459 |
|
|
|
460 |
|
|
\begin{longtable}{ccc} |
461 |
|
|
\caption{Calculated thermal conductivity of a cluster of 6912 SPC/E |
462 |
|
|
water molecules. Calculations were performed at 300 K and 5 atm.} |
463 |
|
|
\\ \hline \hline |
464 |
|
|
{$J_r$} & {$\langle \frac{dT}{dr} \rangle$} & {$\boldsymbol \lambda$}\\ |
465 |
|
|
{\footnotesize(kcal / fs $\cdot$ \AA$^{2}$)} & {\footnotesize(K / \AA)} & {\footnotesize(W / m $\cdot$ K)}\\ \hline |
466 |
|
|
1$\times 10^{-5}$ & 0.38683 & 0.8698 \\ |
467 |
|
|
3$\times 10^{-5}$ & 1.1643 & 0.9098 \\ |
468 |
|
|
6$\times 10^{-5}$ & 2.2262 & 0.8727 \\ |
469 |
|
|
\hline |
470 |
|
|
This work & & 0.8841 \\ |
471 |
|
|
Zhang, et al\cite{Zhang2005} & & 0.81 \\ |
472 |
|
|
R$\ddot{\mathrm{o}}$mer, et al\cite{Romer2012} & & 0.87 \\ |
473 |
|
|
Experiment\cite{WagnerKruse} & & 0.61 |
474 |
|
|
\\ \hline \hline |
475 |
|
|
\label{table:waterTC} |
476 |
|
|
\end{longtable} |
477 |
|
|
|
478 |
|
|
Again, the measured slope is linearly dependent on the applied kinetic |
479 |
|
|
energy flux $J_r$. The average calculated thermal conductivity from |
480 |
|
|
this work, $0.8841$ W / m $\cdot$ K, compares very well to previous |
481 |
|
|
non-equilibrium molecular dynamics results\cite{Romer2012, Zhang2005} |
482 |
|
|
and experimental values.\cite{WagnerKruse} |
483 |
|
|
|
484 |
|
|
|
485 |
kstocke1 |
3947 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
486 |
|
|
% INTERFACIAL THERMAL CONDUCTANCE |
487 |
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
488 |
|
|
\subsection{Interfacial thermal conductance} |
489 |
gezelter |
4064 |
\label{sec:interfacial} |
490 |
kstocke1 |
3947 |
|
491 |
gezelter |
4064 |
The interfacial thermal conductance, $G$, of a heterogeneous interface |
492 |
|
|
located at $r_0$ can be understood as the change in thermal |
493 |
|
|
conductivity in a direction normal $(\mathbf{\hat{n}})$ to the |
494 |
|
|
interface, |
495 |
|
|
\begin{equation} |
496 |
|
|
G = \left(\nabla\lambda \cdot \mathbf{\hat{n}}\right)_{r_0} |
497 |
|
|
\end{equation} |
498 |
|
|
For heterogeneous systems such as the solvated nanoparticle shown in |
499 |
|
|
Figure \ref{fig:NP20}, the interfacial thermal conductance at the |
500 |
|
|
surface of the nanoparticle can be determined using a kinetic energy |
501 |
|
|
flux applied using the RNEMD method developed above. It is most |
502 |
|
|
convenient to compute the Kaptiza or interfacial thermal resistance |
503 |
|
|
for a thin spherical shell, |
504 |
|
|
\begin{equation} |
505 |
|
|
R_K = \frac{1}{G} = \frac{\Delta |
506 |
|
|
T}{J_r} |
507 |
|
|
\end{equation} |
508 |
|
|
where $\Delta T$ is the temperature drop from the interior to the |
509 |
|
|
exterior of the shell. For two neighboring shells, the kinetic energy |
510 |
|
|
flux ($J_r$) is not the same (as the surface areas are not the same), |
511 |
|
|
but the heat transfer rate, $q_r = J_r A$ is constant. The thermal |
512 |
|
|
resistance of a shell with interior radius $r$ is most conveniently |
513 |
|
|
written as |
514 |
|
|
\begin{equation} |
515 |
|
|
R_K = \frac{1}{q_r} \Delta T 4 \pi r^2. |
516 |
|
|
\label{eq:RK} |
517 |
|
|
\end{equation} |
518 |
|
|
|
519 |
|
|
|
520 |
kstocke1 |
4009 |
\begin{figure} |
521 |
|
|
\includegraphics[width=\linewidth]{figures/NP20} |
522 |
gezelter |
4064 |
\caption{A gold nanoparticle with a radius of 20 \AA$\,$ solvated in |
523 |
|
|
TraPPE-UA hexane. A kinetic energy flux is applied between the |
524 |
|
|
nanoparticle and an outer shell of solvent to obtain the interfacial |
525 |
|
|
thermal conductance, $G$, and the interfacial rotational resistance |
526 |
|
|
$\Xi^{rr}$ is determined using an angular momentum flux. } |
527 |
kstocke1 |
4009 |
\label{fig:NP20} |
528 |
|
|
\end{figure} |
529 |
kstocke1 |
4003 |
|
530 |
gezelter |
4064 |
To model the thermal conductance across an interface (or multiple |
531 |
|
|
interfaces) it is useful to consider the shells as resistors wired in |
532 |
|
|
series. The resistance of the shells is then additive, and the |
533 |
|
|
interfacial thermal conductance is the inverse of the total Kapitza |
534 |
|
|
resistance: |
535 |
kstocke1 |
3947 |
\begin{equation} |
536 |
gezelter |
4064 |
\frac{1}{G} = R_\mathrm{total} = \frac{1}{q_r} \sum_i \left(T_{i+i} - |
537 |
|
|
T_i\right) 4 \pi r_i^2 |
538 |
kstocke1 |
3947 |
\end{equation} |
539 |
gezelter |
4064 |
This series can be expanded for any number of adjacent shells, |
540 |
|
|
allowing for the calculation of the interfacial thermal conductance |
541 |
|
|
for interfaces of considerable thickness, such as self-assembled |
542 |
|
|
ligand monolayers on a metal surface. |
543 |
kstocke1 |
3947 |
|
544 |
gezelter |
4065 |
\subsubsection{Interfacial thermal conductance of solvated gold |
545 |
|
|
nanoparticles} |
546 |
|
|
Calculated interfacial thermal conductance ($G$) values for three |
547 |
|
|
sizes of gold nanoparticles and a flat Au(111) surface solvated in |
548 |
|
|
TraPPE-UA hexane are shown in Table \ref{table:G}. |
549 |
kstocke1 |
4003 |
|
550 |
gezelter |
4065 |
\begin{longtable}{ccc} |
551 |
|
|
\caption{Calculated interfacial thermal conductance ($G$) values for |
552 |
|
|
gold nanoparticles of varying radii solvated in TraPPE-UA |
553 |
|
|
hexane. The nanoparticle $G$ values are compared to previous |
554 |
|
|
simulation results for a Au(111) interface in TraPPE-UA hexane.} |
555 |
|
|
\\ \hline \hline |
556 |
|
|
{Nanoparticle Radius} & {$G$}\\ |
557 |
|
|
{\footnotesize(\AA)} & {\footnotesize(MW / m$^{2}$ $\cdot$ K)}\\ \hline |
558 |
|
|
20 & {47.1} \\ |
559 |
|
|
30 & {45.4} \\ |
560 |
|
|
40 & {46.5} \\ |
561 |
|
|
\hline |
562 |
|
|
Au(111) & {30.2} |
563 |
|
|
\\ \hline \hline |
564 |
|
|
\label{table:G} |
565 |
|
|
\end{longtable} |
566 |
|
|
|
567 |
|
|
The introduction of surface curvature increases the interfacial |
568 |
|
|
thermal conductance by a factor of approximately $1.5$ relative to the |
569 |
|
|
flat interface. There are no significant differences in the $G$ values |
570 |
|
|
for the varying nanoparticle sizes. It seems likely that for the range |
571 |
|
|
of nanoparticle sizes represented here, any particle size effects are |
572 |
|
|
not evident. Simulations of larger nanoparticles may yet demonstrate |
573 |
|
|
a limiting $G$ value close tothe flat Au(111) slab but these would |
574 |
|
|
require prohibitively costly numbers of atoms. |
575 |
|
|
|
576 |
kstocke1 |
3947 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
577 |
kstocke1 |
4009 |
% INTERFACIAL ROTATIONAL FRICTION |
578 |
kstocke1 |
3947 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
579 |
kstocke1 |
4009 |
\subsection{Interfacial rotational friction} |
580 |
gezelter |
4064 |
\label{sec:rotation} |
581 |
|
|
The interfacial rotational resistance tensor, $\Xi^{rr}$, can be |
582 |
|
|
calculated for heterogeneous nanostructure/solvent systems by applying |
583 |
|
|
an angular momentum flux between the solvated nanostructure and a |
584 |
|
|
spherical shell of solvent at the outer edge of the cluster. An |
585 |
|
|
angular velocity gradient develops in response to the applied flux, |
586 |
|
|
causing the nanostructure and solvent shell to rotate in opposite |
587 |
|
|
directions about a given axis. |
588 |
kstocke1 |
3947 |
|
589 |
kstocke1 |
4009 |
\begin{figure} |
590 |
|
|
\includegraphics[width=\linewidth]{figures/E25-75} |
591 |
gezelter |
4064 |
\caption{A gold prolate ellipsoid of length 65 \AA$\,$ and width 25 |
592 |
|
|
\AA$\,$ solvated by TraPPE-UA hexane. An angular momentum flux is |
593 |
|
|
applied between the ellipsoid and an outer shell of solvent.} |
594 |
kstocke1 |
4009 |
\label{fig:E25-75} |
595 |
|
|
\end{figure} |
596 |
|
|
|
597 |
gezelter |
4064 |
Analytical solutions for the diagonal elements of the rotational |
598 |
|
|
resistance tensor for solvated spherical body of radius $r$ under |
599 |
|
|
ideal stick boundary conditions can be estimated using Stokes' law |
600 |
kstocke1 |
3947 |
\begin{equation} |
601 |
gezelter |
4064 |
\Xi^{rr}_{stick} = 8 \pi \eta r^3, |
602 |
|
|
\label{eq:Xisphere} |
603 |
kstocke1 |
3947 |
\end{equation} |
604 |
gezelter |
4064 |
where $\eta$ is the dynamic viscosity of the surrounding solvent. |
605 |
kstocke1 |
3947 |
|
606 |
gezelter |
4064 |
For general ellipsoids with semiaxes $\alpha$, $\beta$, and $\gamma$, |
607 |
|
|
Perrin's extension of Stokes' law provides exact solutions for |
608 |
|
|
symmetric prolate $(\alpha \geq \beta = \gamma)$ and oblate $(\alpha < |
609 |
|
|
\beta = \gamma)$ ellipsoids under ideal stick conditions. For |
610 |
gezelter |
4066 |
simplicity, we define the Perrin factor, $S$, |
611 |
kstocke1 |
3947 |
|
612 |
|
|
\begin{equation} |
613 |
gezelter |
4064 |
S = \frac{2}{\sqrt{\alpha^2 - \beta^2}} \ln \left[ \frac{\alpha + \sqrt{\alpha^2 - \beta^2}}{\beta} \right]. |
614 |
kstocke1 |
4003 |
\label{eq:S} |
615 |
kstocke1 |
3947 |
\end{equation} |
616 |
|
|
|
617 |
gezelter |
4064 |
For a prolate ellipsoidal nanoparticle (see Fig. \ref{fig:E25-75}), |
618 |
|
|
the rotational resistance tensor $\Xi^{rr}$ is a $3 \times 3$ diagonal |
619 |
|
|
matrix with elements |
620 |
gezelter |
4066 |
\begin{align} |
621 |
|
|
\Xi^{rr}_\alpha =& \frac{32 \pi}{3} \eta \frac{ \left( |
622 |
|
|
\alpha^2 - \beta^2 \right) \beta^2}{2\alpha - \beta^2 S} |
623 |
|
|
\\ \nonumber \\ |
624 |
|
|
\Xi^{rr}_{\beta,\gamma} =& \frac{32 \pi}{3} \eta \frac{ \left( \alpha^4 - \beta^4 \right)}{ \left( 2\alpha^2 - \beta^2 \right)S - 2\alpha}, |
625 |
|
|
\label{eq:Xirr} |
626 |
|
|
\end{align} |
627 |
gezelter |
4064 |
corresponding to rotation about the long axis ($\alpha$), and each of |
628 |
|
|
the equivalent short axes ($\beta$ and $\gamma$), respectively. |
629 |
kstocke1 |
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|
630 |
gezelter |
4064 |
Previous VSS-RNEMD simulations of the interfacial friction of the |
631 |
|
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planar Au(111) / hexane interface have shown that the interface exists |
632 |
|
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within slip boundary conditions.\cite{Kuang:2012fe} Hu and |
633 |
|
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Zwanzig\cite{Zwanzig} investigated the rotational friction |
634 |
|
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coefficients for spheroids under slip boundary conditions and obtained |
635 |
|
|
numerial results for a scaling factor to be applied to |
636 |
|
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$\Xi^{rr}_{\mathit{stick}}$ as a function of $\tau$, the ratio of the |
637 |
|
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shorter semiaxes and the longer semiaxis of the spheroid. For the |
638 |
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sphere and prolate ellipsoid shown here, the values of $\tau$ are $1$ |
639 |
|
|
and $0.3939$, respectively. Under ``slip'' conditions, |
640 |
|
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$\Xi^{rr}_{\mathit{slip}}$ for any sphere and rotation of the prolate |
641 |
|
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ellipsoid about its long axis approaches $0$, as no solvent is |
642 |
|
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displaced by either of these rotations. $\Xi^{rr}_{\mathit{slip}}$ for |
643 |
|
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rotation of the prolate ellipsoid about its short axis is $35.9\%$ of |
644 |
|
|
the analytical $\Xi^{rr}_{\mathit{stick}}$ result, accounting for the |
645 |
|
|
reduced interfacial friction under ``slip'' boundary conditions. |
646 |
kstocke1 |
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|
647 |
kstocke1 |
4058 |
|
648 |
gezelter |
4064 |
An |
649 |
|
|
$\eta$ value for TraPPE-UA hexane under these particular temperature |
650 |
|
|
and pressure conditions was determined by applying a traditional |
651 |
|
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VSS-RNEMD linear momentum flux to a periodic box of solvent. |
652 |
|
|
|
653 |
kstocke1 |
4058 |
The effective rotational friction coefficient, $\Xi^{rr}_{\mathit{eff}}$ at the interface can be extracted from non-periodic VSS-RNEMD simulations quite easily using the applied torque ($\tau$) and the observed angular velocity of the gold structure ($\omega_{Au}$) |
654 |
|
|
|
655 |
kstocke1 |
3947 |
\begin{equation} |
656 |
kstocke1 |
3991 |
\Xi^{rr}_{\mathit{eff}} = \frac{\tau}{\omega_{Au}} |
657 |
kstocke1 |
4003 |
\label{eq:Xieff} |
658 |
kstocke1 |
3947 |
\end{equation} |
659 |
|
|
|
660 |
kstocke1 |
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The applied torque required to overcome the interfacial friction and maintain constant rotation of the gold is |
661 |
|
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|
662 |
|
|
\begin{equation} |
663 |
|
|
\tau = \frac{L}{2 t} |
664 |
kstocke1 |
4003 |
\label{eq:tau} |
665 |
kstocke1 |
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\end{equation} |
666 |
|
|
|
667 |
|
|
where $L$ is the total angular momentum exchanged between the two RNEMD regions and $t$ is the length of the simulation. |
668 |
|
|
|
669 |
kstocke1 |
3927 |
|
670 |
kstocke1 |
4009 |
Table \ref{table:couple} shows the calculated rotational friction coefficients $\Xi^{rr}$ for spherical gold |
671 |
|
|
nanoparticles and a prolate ellipsoidal gold nanorod in TraPPE-UA hexane. An angular momentum flux was applied |
672 |
|
|
between the A and B regions defined as the gold structure and hexane molecules beyond a certain radius, |
673 |
|
|
respectively. |
674 |
kstocke1 |
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|
675 |
kstocke1 |
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\begin{longtable}{lccccc} |
676 |
kstocke1 |
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\caption{Comparison of rotational friction coefficients under ideal ``slip'' ($\Xi^{rr}_{\mathit{slip}}$) and ``stick'' ($\Xi^{rr}_{\mathit{stick}}$) conditions and effective ($\Xi^{rr}_{\mathit{eff}}$) rotational friction coefficients of gold nanostructures solvated in TraPPE-UA hexane at 230 K. The ellipsoid is oriented with the long axis along the $z$ direction.} |
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kstocke1 |
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\\ \hline \hline |
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kstocke1 |
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{Structure} & {Axis of Rotation} & {$\Xi^{rr}_{\mathit{slip}}$} & {$\Xi^{rr}_{\mathit{eff}}$} & {$\Xi^{rr}_{\mathit{stick}}$} & {$\Xi^{rr}_{\mathit{eff}}$ / $\Xi^{rr}_{\mathit{stick}}$}\\ |
679 |
|
|
{} & {} & {\footnotesize(amu A$^2$ / fs)} & {\footnotesize(amu A$^2$ / fs)} & {\footnotesize(amu A$^2$ / fs)} & \\ \hline |
680 |
kstocke1 |
4067 |
Sphere (r = 20 \AA) & {$x = y = z$} & 0 & {2386 $\pm$ 14} & {3314} & {0.720}\\ |
681 |
|
|
Sphere (r = 30 \AA) & {$x = y = z$} & 0 & {8415 $\pm$ 274} & {11749} & {0.716}\\ |
682 |
|
|
Sphere (r = 40 \AA) & {$x = y = z$} & 0 & {47544 $\pm$ 3051} & {34464} & {1.380}\\ |
683 |
|
|
Prolate Ellipsoid & {$x = y$} & {1792} & {3128 $\pm$ 166} & {4991} & {0.627}\\ |
684 |
|
|
Prolate Ellipsoid & {$z$} & 0 & {1590 $\pm$ 30} & {1993} & {0.798} |
685 |
kstocke1 |
4003 |
\\ \hline \hline |
686 |
kstocke1 |
3991 |
\label{table:couple} |
687 |
kstocke1 |
3927 |
\end{longtable} |
688 |
|
|
|
689 |
kstocke1 |
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The results for $\Xi^{rr}_{\mathit{eff}}$ show that, contrary to the flat Au(111) / hexane interface, gold |
690 |
kstocke1 |
4058 |
structures solvated by hexane do not exist in the ``slip'' boundary condition. At this length scale, the |
691 |
kstocke1 |
4009 |
nanostructures are not perfect spheroids due to atomic `roughening' of the surface and therefore experience |
692 |
|
|
increased interfacial friction which deviates from the ideal ``slip'' case. The 20 and 30 \AA$\,$ radius |
693 |
|
|
nanoparticles experience approximately 70\% of the ideal ``stick'' boundary interfacial friction. Rotation of |
694 |
|
|
the ellipsoid about its long axis more closely approaches the ``stick'' limit than rotation about the short |
695 |
kstocke1 |
4058 |
axis, which at first seems counterintuitive. However, the `propellor' motion caused by rotation about the |
696 |
kstocke1 |
4009 |
short axis may exclude solvent from the rotation cavity or move a sufficient amount of solvent along with the |
697 |
|
|
gold that a smaller interfacial friction is actually experienced. The largest nanoparticle (40 \AA$\,$ radius) |
698 |
|
|
appears to experience more than the ``stick'' limit of interfacial friction, which may be a consequence of |
699 |
|
|
surface features or anomalous solvent behaviors that are not fully understood at this time. |
700 |
kstocke1 |
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|
701 |
kstocke1 |
3927 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
702 |
|
|
% **DISCUSSION** |
703 |
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
704 |
|
|
\section{Discussion} |
705 |
|
|
|
706 |
kstocke1 |
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We have demonstrated a novel adaptation of the VSS-RNEMD methodology for the application of thermal and angular momentum fluxes in explicitly non-periodic system geometries. Non-periodic VSS-RNEMD preserves the virtues of traditional VSS-RNEMD, namely Boltzmann thermal velocity distributions and minimal thermal anisotropy, while extending the constraints to conserve total energy and total \emph{angular} momentum. We also still have the ability to impose the thermal and angular momentum fluxes simultaneously or individually. |
707 |
kstocke1 |
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|
708 |
kstocke1 |
4010 |
Most strikingly, this method enables calculation of thermal conductivity in homogeneous non-periodic geometries, as well as interfacial thermal conductance and interfacial rotational friction in heterogeneous clusters. The ability to interrogate explicitly non-periodic effects -- such as surface curvature -- on interfacial transport properties is an exciting prospect that will be explored in the future. |
709 |
kstocke1 |
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|
710 |
kstocke1 |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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|
|
% **ACKNOWLEDGMENTS** |
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|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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gezelter |
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\begin{acknowledgement} |
714 |
|
|
The authors thank Dr. Shenyu Kuang for helpful discussions. Support |
715 |
|
|
for this project was provided by the National Science Foundation |
716 |
|
|
under grant CHE-0848243. Computational time was provided by the |
717 |
|
|
Center for Research Computing (CRC) at the University of Notre Dame. |
718 |
|
|
\end{acknowledgement} |
719 |
kstocke1 |
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|
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|
|
|
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|
|
\newpage |
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|
|
|
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|
|
\bibliography{nonperiodicVSS} |
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|
|
|
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gezelter |
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%\end{doublespace} |
726 |
kstocke1 |
3934 |
\end{document} |