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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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Gold Nanoparticle:\\ |
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Measured linear slope $\left( \langle dT / dr \rangle \right)$ is linearly dependent on applied kinetic energy flux. Calculated thermal conductivity compares well with previous bulk QSC values. Still $\sim$100 times lower than experiment (limitations of potential -- neglects electronic contributions to heat conduction). Increase relative to bulk may be due to slight increase in gold density. Curvature of nanoparticle introduces higher surface tension and increases density of gold.\\ |
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\begin{longtable}{p{2.7cm} p{2.5cm} p{2.5cm}} |
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\caption{Calculated thermal conductivity of a crystalline gold nanoparticle of radius 40 \AA. Calculations were performed at 300 K and ambient density. Gold-gold interactions are described by the Quantum Sutton-Chen potential.} |
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\\ \hline \hline |
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\centering {$J_r$} & \centering\arraybackslash {$\langle dT / dr \rangle$} & \centering\arraybackslash {$\boldsymbol \lambda$}\\ |
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\centering {\small(kcal fs$^{-1}$ \AA$^{-2}$)} & \centering\arraybackslash {\small(K \AA$^{-1}$)} & \centering\arraybackslash {\small(W m$^{-1}$ K$^{-1}$)}\\ \hline |
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\endhead |
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\hline |
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% \endfoot |
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\centering3.25$\times 10^{-6}$ & \centering\arraybackslash 0.11435 & \centering\arraybackslash 1.9753 \\ |
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\centering6.50$\times 10^{-6}$ & \centering\arraybackslash 0.2324 & \centering\arraybackslash 1.9438 \\ |
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\centering1.30$\times 10^{-5}$ & \centering\arraybackslash 0.44922 & \centering\arraybackslash 2.0113 \\ |
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\centering3.25$\times 10^{-5}$ & \centering\arraybackslash 1.1802 & \centering\arraybackslash 1.9139 \\ |
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\centering6.50$\times 10^{-5}$ & \centering\arraybackslash 2.339 & \centering\arraybackslash 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}{p{2.7cm} p{2.5cm} p{2.5cm}} |
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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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\centering {$J_r$} & \centering\arraybackslash {$\langle dT / dr \rangle$} & \centering\arraybackslash {$\boldsymbol \lambda$}\\ |
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\centering {\small(kcal fs$^{-1}$ \AA$^{-2}$)} & \centering\arraybackslash {\small(K \AA$^{-1}$)} & \centering\arraybackslash {\small(W m$^{-1}$ K$^{-1}$)}\\ \hline |
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\endhead |
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\hline |
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% \endfoot |
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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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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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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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Gold Nanoparticle in Hexane: |
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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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Gold Nanoparticle and Ellipsoid in Hexane: |
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\begin{longtable}{p{2.5cm} p{3cm} p{2.5cm} p{2.5cm}} |
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\caption{Calculated interfacial friction coefficients ($\kappa$) and slip length ($\delta$) of gold nanostructures solvated in TraPPE-UA hexane. The ellipsoid is aligned with the long axis along the $z$ direction.} |
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\\ \hline \hline |
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{Structure} & \centering{Axis of rotation} & \centering\arraybackslash {$\kappa$} & \centering\arraybackslash {$\delta$}\\ |
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\centering {} & {} & \centering\arraybackslash {\small($10^4$ Pa s m$^{-1}$)} & \centering\arraybackslash {\small(nm)}\\ \hline |
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\endhead |
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\hline |
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% \endfoot |
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Nanoparticle & \centering$x = y = z$ & & \\ |
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Ellipsoid & \centering$x = y$ & & \\ |
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Ellipsoid & \centering$z$ & & |
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\\ \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} |
