trunk/nonperiodicVSS/nonperiodicVSS.tex

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\title{A Method for Creating Thermal and Angular Momentum Fluxes in Non-Periodic Systems}

\author{Kelsey M. Stocker}
\author{J. Daniel Gezelter}
\email{gezelter@nd.edu}
\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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% \author{Kelsey M. Stocker and J. Daniel
%   Gezelter\footnote{Corresponding author. \ Electronic mail:
%     gezelter@nd.edu} \\
%   251 Nieuwland Science Hall, \\
%       Department of Chemistry and Biochemistry,\\
%       University of Notre Dame\\
%       Notre Dame, Indiana 46556}

\date{\today}

\maketitle 

\begin{doublespace}

\begin{abstract}

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.

\end{abstract}

\newpage

%\narrowtext

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% **INTRODUCTION**
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Introduction}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% **METHODOLOGY**
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Methodology}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% NON-PERIODIC DYNAMICS
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Dynamics for non-periodic systems}

We have run all tests using the Langevin Hull nonperiodic simulation methodology.\cite{Vardeman2011} The Langevin Hull is especially useful for simulating heterogeneous mixtures of materials with different compressibilities, which are typically problematic for traditional affine transform methods. We have had success applying this method to
several different systems including bare metal nanoparticles, liquid water clusters, and explicitly solvated nanoparticles. Calculated physical properties such as the isothermal compressibility of water and the bulk modulus of gold nanoparticles are in good agreement with previous theoretical and experimental results. 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. For these tests, 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. 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% NON-PERIODIC RNEMD
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{VSS-RNEMD for non-periodic systems}

The most useful RNEMD approach developed so far utilizes a series of
simultaneous velocity shearing and scaling (VSS) exchanges between the two
regions.\cite{Kuang2012} This method provides a set of conservation constraints
while simultaneously creating a desired flux between the two regions. Satisfying
the constraint equations ensures that the new configurations are sampled from the
same NVE ensemble.

We have implemented this new method in OpenMD, our group molecular dynamics code.\cite{openmd} The adaptation of VSS-RNEMD for non-periodic systems is relatively
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
of rectangular slabs. A temperature profile along the $r$ coordinate is created by recording the average temperature of concentric spherical shells.

\begin{figure}
        \center{\includegraphics[width=7in]{figures/VSS}}
        \caption{Schematics of periodic (left) and nonperiodic (right) Velocity Shearing and Scaling RNEMD. A kinetic energy or momentum flux is applied from region B to region A. Thermal gradients are depicted by a color gradient. Linear or angular velocity gradients are shown as arrows.}
        \label{fig:VSS}
\end{figure}

At each time interval, the particle velocities ($\mathbf{v}_i$ and
$\mathbf{v}_j$) in the cold and hot shells ($C$ and $H$) are modified by a
velocity scaling coefficient ($c$ and $h$), an additive linear velocity shearing
term ($\mathbf{a}_c$ and $\mathbf{a}_h$), and an additive angular velocity
shearing term ($\mathbf{b}_c$ and $\mathbf{b}_h$). Here, $\langle
\mathbf{v}_{i,j} \rangle$ and $\langle \mathbf{\omega}_{i,j} \rangle$ are the
average linear and angular velocities for each shell.
\begin{displaymath}
\begin{array}{rclcl}
 & \underline{~~~~~~~~~~\mathrm{scaling}~~~~~~~~~~} & &
 \underline{\mathrm{rotational \; shearing}} \\  \\
\mathbf{v}_i $~~~$\leftarrow & 
  c \, \left(\mathbf{v}_i - \langle \omega_c
  \rangle \times r_i\right) & + & \mathbf{b}_c \times r_i \\
\mathbf{v}_j $~~~$\leftarrow & 
  h \, \left(\mathbf{v}_j - \langle \omega_h
  \rangle \times r_j\right) & + & \mathbf{b}_h \times r_j
\end{array}
\end{displaymath}

\begin{eqnarray}
\mathbf{b}_c & = & - \mathbf{j}_r(\mathbf{L}) \; \Delta \, t \; I_c^{-1} + \langle \omega_c \rangle \label{eq:bc}\\
\mathbf{b}_h & = & + \mathbf{j}_r(\mathbf{L}) \; \Delta \, t \; I_h^{-1} + \langle \omega_h \rangle \label{eq:bh}
\end{eqnarray}

The total energy is constrained via two quadratic formulae,
\begin{eqnarray}
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}\\
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}
\end{eqnarray}

the simultaneous
solution of which provide the velocity scaling coefficients $c$ and $h$. Given an
imposed angular momentum flux, $\mathbf{j}_{r} \left( \mathbf{L} \right)$, and/or
thermal flux, $J_r$, equations \ref{eq:bc} - \ref{eq:Kh} are sufficient to obtain
the velocity scaling ($c$ and $h$) and shearing ($\mathbf{b}_c,\,$ and $\mathbf{b}_h$) at each time interval.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% **COMPUTATIONAL DETAILS**
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Computational Details}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% SIMULATION PROTOCOL
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Simulation protocol}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% FORCE FIELD PARAMETERS
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Force field parameters}

We have chosen the SPC/E water model for these simulations, which is particularly useful for method validation as there are many values for physical properties from previous simulations available for direct comparison.\cite{Bedrov:2000, Kuang2010}

Gold -- gold interactions are described by the quantum Sutton-Chen (QSC) model.\cite{PhysRevB.59.3527} The QSC parameters are tuned to experimental properties such as density, cohesive energy, and elastic moduli and include zero-point quantum corrections.

Hexane molecules are described by the TraPPE united atom model,\cite{TraPPE-UA.alkanes} which provides good computational efficiency and reasonable accuracy for bulk thermal conductivity values. In this model,
sites are located at the carbon centers for alkyl groups. Bonding
interactions, including bond stretches and bends and torsions, were
used for intra-molecular sites closer than 3 bonds. For non-bonded
interactions, Lennard-Jones potentials were used.  We have previously
utilized both united atom (UA) and all-atom (AA) force fields for
thermal conductivity,\cite{kuang:AuThl,Kuang2012} and since the united
atom force fields cannot populate the high-frequency modes that are
present in AA force fields, they appear to work better for modeling
thermal conductivity.

Gold -- hexane nonbonded interactions are governed by pairwise Lennard-Jones parameters derived from Vlugt \emph{et al}.\cite{vlugt:cpc2007154} They fitted parameters for the interaction between Au and CH$_{\emph x}$ pseudo-atoms based on the effective potential of Hautman and Klein for the Au(111) surface.\cite{hautman:4994}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% THERMAL CONDUCTIVITIES
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Thermal conductivities}

The thermal conductivity, $\lambda$, can be calculated using the imposed kinetic energy flux, $J_r$, and thermal gradient, $\frac{\partial T}{\partial r}$ measured from the resulting temperature profile

\begin{equation}
        J_r = -\lambda \frac{\partial T}{\partial r}
\end{equation}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% INTERFACIAL THERMAL CONDUCTANCE
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Interfacial thermal conductance}

A thermal flux is created using VSS-RNEMD moves, and the resulting temperature
profiles are analyzed to yield information about the interfacial thermal
conductance. As the simulation progresses, the VSS moves are performed on a regular basis, and
the system develops a thermal or velocity gradient in response to the applied
flux. A definition of the interfacial conductance in terms of a temperature difference ($\Delta T$) measured at $r_0$, 
\begin{equation}
        G = \frac{J_r}{\Delta T_{r_0}}, \label{eq:G}
\end{equation}
is useful once the RNEMD approach has generated a
stable temperature gap across the interface.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% INTERFACIAL FRICTION
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Interfacial friction}

The slip length, $\delta$, is defined as the ratio of the interfacial friction coefficient, $\kappa$, and dynamic solvent viscosity, $\eta$

\begin{equation}
        \delta = \frac{\eta}{\kappa}
\end{equation}

and is measured from a radial angular momentum profile generated from a nonperiodic VSS-RNEMD simulation.

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

\begin{equation}
        \Xi = 8 \pi \eta r^3 \label{eq:Xi}.
\end{equation}

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.

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, $S$,

\begin{equation}
        S = \frac{2}{\sqrt{a^2 - b^2}} ln \left[ \frac{a + \sqrt{a^2 - b^2}}{b} \right]. \label{eq:S}
\end{equation}

For a prolate ellipsoidal rod, demonstrated here, the rotational resistance tensor $\Xi$ is a $3 \times 3$ diagonal matrix with elements
\begin{equation}
        \Xi^{rr}_a = \frac{32 \pi}{2} \eta \frac{ \left( a^2 - b^2 \right) b^2}{2a - b^2 S}
\label{eq:Xia}
\end{equation}
\begin{equation}
        \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}
\end{equation}

% However, the friction between hexane solvent and gold more likely falls within ``slip'' boundary conditions. Hu and Zwanzig\cite{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.

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.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% **TESTS AND APPLICATIONS**
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Tests and Applications} 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% THERMAL CONDUCTIVITIES
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Thermal conductivities}

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{Kuang2010}, 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.

\begin{longtable}{ccc}
\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.}
\\ \hline \hline
{$J_r$} & {$\langle dT / dr \rangle$} & {$\boldsymbol \lambda$}\\ 
{\small(kcal fs$^{-1}$ \AA$^{-2}$)} & {\small(K \AA$^{-1}$)} & {\small(W m$^{-1}$ K$^{-1}$)}\\ \hline
3.25$\times 10^{-6}$ & 0.11435 & 1.9753 \\
6.50$\times 10^{-6}$ & 0.2324 & 1.9438 \\
1.30$\times 10^{-5}$ & 0.44922 & 2.0113 \\
3.25$\times 10^{-5}$ & 1.1802 & 1.9139 \\
6.50$\times 10^{-5}$ & 2.339 & 1.9314
\\ \hline \hline
\label{table:goldconductivity}
\end{longtable}

Calculated values for the thermal conductivity of a cluster of 6912 SPC/E water molecules are shown in Table \ref{table:waterconductivity}. 

\begin{longtable}{ccc}
\caption{Calculated thermal conductivity of a cluster of 6912 SPC/E water molecules. Calculations were performed at 300 K and ambient density.}
\\ \hline \hline
{$J_r$} & {$\langle dT / dr \rangle$} & {$\boldsymbol \lambda$}\\ 
{\small(kcal fs$^{-1}$ \AA$^{-2}$)} & {\small(K \AA$^{-1}$)} & {\small(W m$^{-1}$ K$^{-1}$)}\\ \hline
1.00$\times 10^{-5}$ & 0.38683 & 1.79665486\\
3.00$\times 10^{-5}$ & 1.1643 & 1.79077557\\
6.00$\times 10^{-5}$ & 2.2262 & 1.87314707\\
\hline \hline
\label{table:waterconductivity}
\end{longtable}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% INTERFACIAL THERMAL CONDUCTANCE
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Interfacial thermal conductance}

\begin{longtable}{ccc}
\caption{Caption.}
\\ \hline \hline
{Nanoparticle Radius} & {$\boldsymbol \lambda$}\\ 
{\small(\AA)} & {\small(W m$^{-1}$ K$^{-1}$)}\\ \hline
20 & 59.66\\
30 & 57.88\\
40 & \\
$\infty$ & \\
\hline \hline
\label{table:waterconductivity}
\end{longtable}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% INTERFACIAL FRICTION
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Interfacial friction}

Table \ref{table:interfacialfrictionstick} shows the calculated interfacial friction coefficients $\kappa$ for spherical gold nanoparticles and a 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 beyond a certain radius, respectively. The resulting angular velocity gradient resulted in the gold structure rotating about the prescribed axis within the solvent.
        
\begin{longtable}{lccccc}
\caption{Calculated ``stick'' interfacial friction coefficients ($\kappa$) and friction factors ($f$) of gold nanostructures solvated in TraPPE-UA hexane at 230 K. The ellipsoid is oriented with the long axis along the $z$ direction.}
\\ \hline \hline
{Structure} & {Axis of Rotation} & {$\kappa_{VSS}$} & {$\kappa_{calc}$} & {$f_{VSS}$} & {$f_{calc}$}\\ 
{} & {} & {\small($10^{-29}$ Pa s m$^{3}$)} & {\small($10^{-29}$ Pa s m$^{3}$)} & {} & {}\\  \hline
{Sphere (r = 20 \AA)} & {$x = y = z$} & {} & {6.13239} & {1} & {1}\\
{Sphere (r = 30 \AA)} & {$x = y = z$} & {} & {20.6968} & {1} & {1}\\
{Sphere (r = 40 \AA)} & {$x = y = z$} & {} & {49.0591} & {1} & {1}\\ 
{Prolate Ellipsoid} & {$x = y$} & {} & {8.22846} & {} & {1.92477}\\
{Prolate Ellipsoid} & {$z$} & {} & {3.28634} & {} & {0.768726}\\ 
  \hline \hline
\label{table:interfacialfrictionstick}
\end{longtable}

% \begin{longtable}{lccc}
% \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.}
% \\ \hline \hline
% {Structure} & {Axis of rotation} & {$\kappa_{VSS}$} & {$\kappa_{calc}$}\\ 
% {} & {} & {\small($10^{-29}$ Pa s m$^{3}$)} & {\small($10^{-29}$ Pa s m$^{3}$)}\\  \hline
% {Sphere} & {$x = y = z$} & {} & {0}\\ 
% {Prolate Ellipsoid} & {$x = y$} & {} & {0}\\ 
% {Prolate Ellipsoid} & {$z$} & {} & {7.9295392}\\  \hline \hline
% \label{table:interfacialfrictionslip}
% \end{longtable}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% **DISCUSSION**
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Discussion} 


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% **ACKNOWLEDGMENTS**
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section*{Acknowledgments} 

We gratefully acknowledge conversations with Dr. Shenyu Kuang. Support for
this project was provided by the National Science Foundation under grant
CHE-0848243. Computational time was provided by the Center for Research
Computing (CRC) at the University of Notre Dame.

\newpage

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