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% \bibliographystyle{achemso} |
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\title{A Method for Creating Thermal and Angular Momentum Fluxes in Non-Periodic Systems} |
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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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% \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 have adapted the Velocity Shearing and Scaling Reverse Non-Equilibium Molecular Dynamics (VSS-RNEMD) method for use with aperiodic system geometries. This new method is capable of creating stable temperature and angular velocity gradients in heterogeneous non-periodic systems while conserving total energy and angular momentum. To demonstrate the method, we have computed the thermal conductivities of a gold nanoparticle and water cluster, the shear viscosity of a water cluster, the interfacial thermal conductivity of a solvated gold nanoparticle and the interfacial friction of 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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% **METHODOLOGY** |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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\section{Methodology} |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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% FORCE FIELD PARAMETERS |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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\subsection{Force field parameters} |
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We have chosen the SPC/E water model for these simulations. There are many values for physical properties from previous simulations available for direct comparison. |
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Gold-gold interactions are described by the quantum Sutton-Chen (QSC) model. The QSC parameters are tuned to experimental properties such as density, cohesive energy, and elastic moduli and include zero-point quantum corrections. |
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Hexane molecules are described by the TraPPE united atom model. This model provides good computational efficiency and reasonable accuracy for bulk thermal conductivity values. |
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Metal-nonmetal interactions are governed by parameters derived from Luedtke and Landman. |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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% NON-PERIODIC DYNAMICS |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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\subsection{Dynamics for non-periodic systems} |
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We have run all tests using the Langevin Hull nonperiodic simulation methodology. The Langevin Hull uses a Delaunay tesselation to create a dynamic convex hull composed of triangular facets with vertices at atomic sites. Atomic sites included in the hull are coupled to an external bath defined by a temperature, pressure and viscosity. Atoms not included in the hull are subject to standard Newtonian dynamics. Thermal coupling to the bath was turned off to avoid interference with any imposed kinetic flux. Systems containing liquids were run under moderate pressure ($\sim$ 5 atm) to avoid the formation of a substantial vapor phase, which would have a hull created out of a relatively small number of molecules. |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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% NON-PERIODIC RNEMD |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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\subsection{VSS-RNEMD for non-periodic systems} |
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The adaptation of VSS-RNEMD for non-periodic systems is relatively |
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straightforward. The major modifications to the method are the addition of a rotational shearing term and the use of more versatile hot / cold regions instead |
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of rectangular slabs. A temperature profile along the $r$ coordinate is created by recording the average temperature of concentric spherical shells. |
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At each time interval, the particle velocities ($\mathbf{v}_i$ and |
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$\mathbf{v}_j$) in the cold and hot shells ($C$ and $H$) are modified by a |
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velocity scaling coefficient ($c$ and $h$), an additive linear velocity shearing |
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term ($\mathbf{a}_c$ and $\mathbf{a}_h$), and an additive angular velocity |
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shearing term ($\mathbf{b}_c$ and $\mathbf{b}_h$). Here, $\langle |
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\mathbf{v}_{i,j} \rangle$ and $\langle \mathbf{\omega}_{i,j} \rangle$ are the |
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average linear and angular velocities for each shell. |
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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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c \, \left(\mathbf{v}_i - \langle \omega_c |
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\rangle \times r_i\right) & + & \mathbf{b}_c \times r_i \\ |
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\mathbf{v}_j $~~~$\leftarrow & |
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h \, \left(\mathbf{v}_j - \langle \omega_h |
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\rangle \times r_j\right) & + & \mathbf{b}_h \times r_j |
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\end{array} |
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\end{displaymath} |
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\begin{eqnarray} |
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\mathbf{b}_c & = & - \mathbf{j}_r(\mathbf{L}) \; \Delta \, t \; I_c^{-1} + \langle \omega_c \rangle \label{eq:bc}\\ |
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\mathbf{b}_h & = & + \mathbf{j}_r(\mathbf{L}) \; \Delta \, t \; I_h^{-1} + \langle \omega_h \rangle \label{eq:bh} |
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\end{eqnarray} |
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The total energy is constrained via two quadratic formulae, |
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\begin{eqnarray} |
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K_c - J_r \; \Delta \, t & = & c^2 \, (K_c - \frac{1}{2}\langle \omega_c \rangle \cdot I_c\langle \omega_c \rangle) + \frac{1}{2} \mathbf{b}_c \cdot I_c \, \mathbf{b}_c \label{eq:Kc}\\ |
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K_h + J_r \; \Delta \, t & = & h^2 \, (K_h - \frac{1}{2}\langle \omega_h \rangle \cdot I_h\langle \omega_h \rangle) + \frac{1}{2} \mathbf{b}_h \cdot I_h \, \mathbf{b}_h \label{eq:Kh} |
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\end{eqnarray} |
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the simultaneous |
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solution of which provide the velocity scaling coefficients $c$ and $h$. Given an |
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imposed angular momentum flux, $\mathbf{j}_{r} \left( \mathbf{L} \right)$, and/or |
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thermal flux, $J_r$, equations \ref{eq:bc} - \ref{eq:Kh} are sufficient to obtain |
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the velocity scaling ($c$ and $h$) and shearing ($\mathbf{b}_c,\,$ and $\mathbf{b}_h$) at each time interval. |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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% **TESTS AND APPLICATIONS** |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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\section{Tests and Applications} |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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% THERMAL CONDUCTIVITIES |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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\subsection{Thermal conductivities} |
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Calculated values for the thermal conductivity of a 40 \AA$~$ radius gold nanoparticle (15707 atoms) at different kinetic energy flux values are shown in Table \ref{table:goldconductivity}. For these calculations, the hot and cold slabs were excluded from the linear regression of the thermal gradient. The measured linear slope $\langle dT / dr \rangle$ is linearly dependent on the applied kinetic energy flux $J_r$. Calculated thermal conductivity values compare well with previous bulk QSC values of 1.08 - 1.26 W m$^{-1}$ K$^{-1}$ [cite NIVS paper], though still significantly lower than the experimental value of 320 W m$^{-1}$ K$^{-1}$, as the QSC force field neglects significant electronic contributions to heat conduction. The small increase relative to previous simulated bulk values is due to a slight increase in gold density -- as expected, an increase in density results in higher thermal conductivity values. The increased density is a result of nanoparticle curvature relative to an infinite bulk slab, which introduces surface tension that increases ambient density. |
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\begin{longtable}{ccc} |
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\caption{Calculated thermal conductivity of a crystalline gold nanoparticle of radius 40 \AA. Calculations were performed at 300 K and ambient density. Gold-gold interactions are described by the Quantum Sutton-Chen potential.} |
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\\ \hline \hline |
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{$J_r$} & {$\langle dT / dr \rangle$} & {$\boldsymbol \lambda$}\\ |
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{\small(kcal fs$^{-1}$ \AA$^{-2}$)} & {\small(K \AA$^{-1}$)} & {\small(W m$^{-1}$ K$^{-1}$)}\\ \hline |
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3.25$\times 10^{-6}$ & 0.11435 & 1.9753 \\ |
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6.50$\times 10^{-6}$ & 0.2324 & 1.9438 \\ |
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1.30$\times 10^{-5}$ & 0.44922 & 2.0113 \\ |
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3.25$\times 10^{-5}$ & 1.1802 & 1.9139 \\ |
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6.50$\times 10^{-5}$ & 2.339 & 1.9314 |
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\\ \hline \hline |
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\label{table:goldconductivity} |
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\end{longtable} |
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SPC/E Water Cluster: |
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\begin{longtable}{ccc} |
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\caption{Calculated thermal conductivity of a cluster of 6912 SPC/E water molecules. Calculations were performed at 300 K and ambient density.} |
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\\ \hline \hline |
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{$J_r$} & {$\langle dT / dr \rangle$} & {$\boldsymbol \lambda$}\\ |
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{\small(kcal fs$^{-1}$ \AA$^{-2}$)} & {\small(K \AA$^{-1}$)} & {\small(W m$^{-1}$ K$^{-1}$)}\\ \hline |
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\\ \hline \hline |
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\label{table:waterconductivity} |
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\end{longtable} |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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% SHEAR VISCOSITY |
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\subsection{Shear viscosity} |
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SPC/E Water Cluster: |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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% INTERFACIAL THERMAL CONDUCTANCE |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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\subsection{Interfacial thermal conductance} |
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The interfacial thermal conductance, $G$, is calculated by defining a temperature difference $\Delta T$ across a given interface. |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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% INTERFACIAL FRICTION |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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\subsection{Interfacial friction} |
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kstocke1 |
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The interfacial friction coefficient, $\kappa$, can be calculated from the solvent dynamic viscosity, $\eta$, and the slip length, $\delta$. The slip length is measured from a radial angular momentum profile generated from a nonperiodic VSS-RNEMD simulation, as shown in Figure X. |
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Table \ref{table:interfacialfriction} shows the calculated interfacial friction coefficients $\kappa$ for a spherical gold nanoparticle and prolate ellipsoidal gold nanorod in TraPPE-UA hexane. An angular momentum flux was applied between the A and B regions defined as the gold structure and hexane molecules included in the convex hull, respectively. The resulting angular velocity gradient resulted in the gold structure rotating about the prescribed axis within the solvent. |
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Analytical solutions for the rotational friction coefficients for a solvated spherical body of radius $r$ under ``stick'' boundary conditions can be estimated using Stokes' law |
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\begin{equation} |
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\Xi = 8 \pi \eta r^3 \label{eq:Xi}. |
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\end{equation} |
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where $\eta$ is the dynamic viscosity of the surrounding solvent and was determined for TraPPE-UA hexane under these condition by applying a traditional VSS-RNEMD linear momentum flux to a periodic box of solvent. |
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For general ellipsoids with semiaxes $a$, $b$, and $c$, Perrin's extension of Stokes' law provides exact solutions for symmetric prolate $(a \geq b = c)$ and oblate $(a < b = c)$ ellipsoids. For simplicity, we define a Perrin Factor, $S$, |
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\begin{equation} |
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S = \frac{2}{\sqrt{a^2 - b^2}} ln \left[ \frac{a + \sqrt{a^2 - b^2}}{b} \right]. \label{eq:S} |
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\end{equation} |
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For a prolate ellipsoidal rod, demonstrated here, the rotational resistance tensor $\Xi$ is a $3 \times 3$ diagonal matrix with elements |
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\begin{eqnarray} |
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\Xi^{rr}_a = \frac{32 \pi}{2} \eta \frac{ \left( a^2 - b^2 \right) b^2}{2a - b^2 S} \label{eq:Xia}\\ |
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\Xi^{rr}_{b,c} = \frac{32 \pi}{2} \eta \frac{ \left( a^4 - b^4 \right)}{ \left( 2a^2 - b^2 \right)S - 2a}. \label{eq:Xibc} |
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\end{eqnarray} |
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However, the friction between hexane solvent and gold more likely falls within ``slip'' boundary conditions. Hu and Zwanzig investigated the rotational friction coefficients for spheroids under slip boundary conditions and obtained numerial results for a scaling factor as a function of $\tau$, the ratio of the shorter semiaxes and the longer semiaxis of the spheroid. For the sphere and prolate ellipsoid shown here, the values of $\tau$ are $1$ and $0.3939$, respectively. According to the values tabulated by Hu and Zwanzig, the sphere friction coefficient approaches $0$, while the ellipsoidal friction coefficient must be scaled by a factor of $0.880$ to account for the reduced interfacial friction under ``slip'' boundary conditions. |
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Another useful quantity is the friction factor $f$, or the friction coefficient of a non-spherical shape divided by the friction coefficient of a sphere of equivalent volume. The nanoparticle and ellipsoidal nanorod dimensions used here were specifically chosen to create structures with equal volumes. |
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\begin{longtable}{lccccc} |
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\caption{Calculated ``slip'' interfacial friction coefficients ($\kappa$) and friction factors ($f$) of gold nanostructures solvated in TraPPE-UA hexane. The ellipsoid is oriented with the long axis along the $z$ direction.} |
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\\ \hline \hline |
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{Structure} & {Axis of rotation} & {$\kappa_{VSS}$} & {$\kappa_{calc}$} & {$f_{VSS}$} & {$f_{calc}$}\\ |
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{} & {} & {\small($10^4$ Pa s m$^{-1}$)} & {\small($10^4$ Pa s m$^{-1}$)} & {} & {}\\ \hline |
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{Sphere} & {$x = y = z$} & {} & {} & {1} & {1}\\ |
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{Prolate Ellipsoid} & {$x = y$} & {} & {} & {} & {}\\ |
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{Prolate Ellipsoid} & {$z$} & {} & {} & {} & {}\\ \hline \hline |
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\label{table:interfacialfriction} |
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\end{longtable} |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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% **DISCUSSION** |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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\section{Discussion} |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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% **ACKNOWLEDGMENTS** |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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\section*{Acknowledgments} |
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We gratefully acknowledge conversations with Dr. Shenyu Kuang. Support for |
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this project was provided by the National Science Foundation under grant |
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CHE-0848243. Computational time was provided by the Center for Research |
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Computing (CRC) at the University of Notre Dame. |
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\newpage |
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\bibliography{nonperiodicVSS} |
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\end{doublespace} |
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\end{document} |
