group/trunk/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}

\begin{document}

\newcolumntype{A}{p{1.5in}}
\newcolumntype{B}{p{0.75in}}

% \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}

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

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. 

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.

Hexane molecules are described by the TraPPE united atom model. This model provides good computational efficiency and reasonable accuracy for bulk thermal conductivity values.

Metal-nonmetal interactions are governed by parameters derived from Luedtke and Landman.


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

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. 

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

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.

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.

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

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

Gold Nanoparticle:\\
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.\\

\begin{longtable}{p{2.7cm} p{2.5cm} p{2.5cm}}
        \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
                \centering {$J_r$} & \centering\arraybackslash {$\langle dT / dr \rangle$} & \centering\arraybackslash {$\boldsymbol \lambda$}\\ 
        \centering {\small(kcal fs$^{-1}$ \AA$^{-2}$)} & \centering\arraybackslash {\small(K \AA$^{-1}$)} & \centering\arraybackslash {\small(W m$^{-1}$ K$^{-1}$)}\\ \hline
\endhead
\hline
% \endfoot
\centering3.25$\times 10^{-6}$ & \centering\arraybackslash 0.11435 & \centering\arraybackslash 1.9753 \\
\centering6.50$\times 10^{-6}$ & \centering\arraybackslash 0.2324 & \centering\arraybackslash 1.9438 \\
\centering1.30$\times 10^{-5}$ & \centering\arraybackslash 0.44922 & \centering\arraybackslash 2.0113 \\
\centering3.25$\times 10^{-5}$ & \centering\arraybackslash 1.1802 & \centering\arraybackslash 1.9139 \\
\centering6.50$\times 10^{-5}$ & \centering\arraybackslash 2.339 & \centering\arraybackslash 1.9314
\\ \hline \hline
\label{table:goldconductivity}
\end{longtable}
        
SPC/E Water Cluster:

\begin{longtable}{p{2.7cm} p{2.5cm} p{2.5cm}}
        \caption{Calculated thermal conductivity of a cluster of 6912 SPC/E water molecules. Calculations were performed at 300 K and ambient density.}
        \\ \hline \hline
                \centering {$J_r$} & \centering\arraybackslash {$\langle dT / dr \rangle$} & \centering\arraybackslash {$\boldsymbol \lambda$}\\ 
        \centering {\small(kcal fs$^{-1}$ \AA$^{-2}$)} & \centering\arraybackslash {\small(K \AA$^{-1}$)} & \centering\arraybackslash {\small(W m$^{-1}$ K$^{-1}$)}\\ \hline
\endhead
\hline
% \endfoot

\\ \hline \hline
\label{table:waterconductivity}
\end{longtable}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% SHEAR VISCOSITY
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Shear viscosity}

SPC/E Water Cluster:

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

Gold Nanoparticle in Hexane:

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

Gold Nanoparticle and Ellipsoid in Hexane:

\begin{longtable}{p{2.5cm} p{3cm} p{2.5cm} p{2.5cm}}
        \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.}
        \\ \hline \hline
                {Structure} & \centering{Axis of rotation} & \centering\arraybackslash {$\kappa$} & \centering\arraybackslash {$\delta$}\\ 
        \centering {} & {} & \centering\arraybackslash {\small($10^4$ Pa s m$^{-1}$)} & \centering\arraybackslash {\small(nm)}\\ \hline
\endhead
\hline
% \endfoot
Nanoparticle & \centering$x = y = z$ & & \\
Ellipsoid & \centering$x = y$ & & \\
Ellipsoid & \centering$z$ & &
\\ \hline \hline
\label{table:interfacialfriction}
\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

\bibliography{nonperiodicVSS}

\end{doublespace}
\end{document}
Revision:	3926
Committed:	Fri Aug 2 19:28:10 2013 UTC (10 years, 10 months ago) by kstocke1
Content type:	application/x-tex
File size:	12009 byte(s)
Log Message:	Initial Import
#	Content
1	\documentclass[journal = jctcce, manuscript = article]{achemso}
2	\setkeys{acs}{usetitle = true}
3
4	\usepackage{caption}
5	\usepackage{float}
6	\usepackage{geometry}
7	\usepackage{natbib}
8	\usepackage{setspace}
9	\usepackage{xkeyval}
10	%%%%%%%%%%%%%%%%%%%%%%%
11	\usepackage{amsmath}
12	\usepackage{amssymb}
13	\usepackage{times}
14	\usepackage{mathptm}
15	\usepackage{setspace}
16	\usepackage{endfloat}
17	\usepackage{caption}
18	\usepackage{tabularx}
19	\usepackage{longtable}
20	\usepackage{graphicx}
21	\usepackage{multirow}
22	\usepackage{multicol}
23	\usepackage{achemso}
24	\usepackage[version=3]{mhchem} % this is a great package for formatting chemical reactions
25	% \usepackage[square, comma, sort&compress]{natbib}
26	\usepackage{url}
27	\pagestyle{plain} \pagenumbering{arabic} \oddsidemargin 0.0cm
28	\evensidemargin 0.0cm \topmargin -21pt \headsep 10pt \textheight
29	9.0in \textwidth 6.5in \brokenpenalty=10000
30
31	% double space list of tables and figures
32	% \AtBeginDelayedFloats{\renewcomand{\baselinestretch}{1.66}}
33	\setlength{\abovecaptionskip}{20 pt}
34	\setlength{\belowcaptionskip}{30 pt}
35
36	% \bibpunct{}{}{,}{s}{}{;}
37
38	% \citestyle{nature}
39	% \bibliographystyle{achemso}
40
41	\title{A Method for Creating Thermal and Angular Momentum Fluxes in Non-Periodic Systems}
42
43	\author{Kelsey M. Stocker}
44	\author{J. Daniel Gezelter}
45	\email{gezelter@nd.edu}
46	\affiliation[University of Notre Dame]{251 Nieuwland Science Hall\\ Department of Chemistry and Biochemistry\\ University of Notre Dame\\ Notre Dame, Indiana 46556}
47
48	\begin{document}
49
50	\newcolumntype{A}{p{1.5in}}
51	\newcolumntype{B}{p{0.75in}}
52
53	% \author{Kelsey M. Stocker and J. Daniel
54	% Gezelter\footnote{Corresponding author. \ Electronic mail:
55	% gezelter@nd.edu} \\
56	% 251 Nieuwland Science Hall, \\
57	% Department of Chemistry and Biochemistry,\\
58	% University of Notre Dame\\
59	% Notre Dame, Indiana 46556}
60
61	\date{\today}
62
63	\maketitle
64
65	\begin{doublespace}
66
67	\begin{abstract}
68
69	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.
70
71	\end{abstract}
72
73	\newpage
74
75	%\narrowtext
76
77	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
78	% INTRODUCTION
79	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
80	\section{Introduction}
81
82
83	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
84	% METHODOLOGY
85	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
86	\section{Methodology}
87
88	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
89	% FORCE FIELD PARAMETERS
90	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
91	\subsection{Force field parameters}
92
93	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.
94
95	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.
96
97	Hexane molecules are described by the TraPPE united atom model. This model provides good computational efficiency and reasonable accuracy for bulk thermal conductivity values.
98
99	Metal-nonmetal interactions are governed by parameters derived from Luedtke and Landman.
100
101
102	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
103	% NON-PERIODIC DYNAMICS
104	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
105	\subsection{Dynamics for non-periodic systems}
106
107	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.
108
109	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
110	% NON-PERIODIC RNEMD
111	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
112	\subsection{VSS-RNEMD for non-periodic systems}
113
114	The adaptation of VSS-RNEMD for non-periodic systems is relatively
115	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
116	of rectangular slabs. A temperature profile along the $r$ coordinate is created by recording the average temperature of concentric spherical shells.
117
118	At each time interval, the particle velocities ($\mathbf{v}_i$ and
119	$\mathbf{v}_j$) in the cold and hot shells ($C$ and $H$) are modified by a
120	velocity scaling coefficient ($c$ and $h$), an additive linear velocity shearing
121	term ($\mathbf{a}_c$ and $\mathbf{a}_h$), and an additive angular velocity
122	shearing term ($\mathbf{b}_c$ and $\mathbf{b}_h$). Here, $\langle
123	\mathbf{v}_{i,j} \rangle$ and $\langle \mathbf{\omega}_{i,j} \rangle$ are the
124	average linear and angular velocities for each shell.
125
126	\begin{displaymath}
127	\begin{array}{rclcl}
128	& \underline{~~~~~~~~~~\mathrm{scaling}~~~~~~~~~~} & &
129	\underline{\mathrm{rotational \; shearing}} \\ \\
130	\mathbf{v}_i $~~~$\leftarrow &
131	c \, \left(\mathbf{v}_i - \langle \omega_c
132	\rangle \times r_i\right) & + & \mathbf{b}_c \times r_i \\
133	\mathbf{v}_j $~~~$\leftarrow &
134	h \, \left(\mathbf{v}_j - \langle \omega_h
135	\rangle \times r_j\right) & + & \mathbf{b}_h \times r_j
136	\end{array}
137	\end{displaymath}
138
139	\begin{eqnarray}
140	\mathbf{b}_c & = & - \mathbf{j}_r(\mathbf{L}) \; \Delta \, t \; I_c^{-1} + \langle \omega_c \rangle \label{eq:bc}\\
141	\mathbf{b}_h & = & + \mathbf{j}_r(\mathbf{L}) \; \Delta \, t \; I_h^{-1} + \langle \omega_h \rangle \label{eq:bh}
142	\end{eqnarray}
143
144	The total energy is constrained via two quadratic formulae,
145
146	\begin{eqnarray}
147	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}\\
148	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}
149	\end{eqnarray}
150
151	the simultaneous
152	solution of which provide the velocity scaling coefficients $c$ and $h$. Given an
153	imposed angular momentum flux, $\mathbf{j}_{r} \left( \mathbf{L} \right)$, and/or
154	thermal flux, $J_r$, equations \ref{eq:bc} - \ref{eq:Kh} are sufficient to obtain
155	the velocity scaling ($c$ and $h$) and shearing ($\mathbf{b}_c,\,$ and $\mathbf{b}_h$) at each time interval.
156
157	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
158	% TESTS AND APPLICATIONS
159	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
160	\section{Tests and Applications}
161
162	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
163	% THERMAL CONDUCTIVITIES
164	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
165	\subsection{Thermal conductivities}
166
167	Gold Nanoparticle:\\
168	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$100 times lower than experiment (limitations of potential -- neglects electronic contributions to heat conduction). Increase relative to bulk may be due to slight increase in gold density. Curvature of nanoparticle introduces higher surface tension and increases density of gold.\\
169
170	\begin{longtable}{p{2.7cm} p{2.5cm} p{2.5cm}}
171	\caption{Calculated thermal conductivity of a crystalline gold nanoparticle of radius 40 \AA. Calculations were performed at 300 K and ambient density. Gold-gold interactions are described by the Quantum Sutton-Chen potential.}
172	\\ \hline \hline
173	\centering {$J_r$} & \centering\arraybackslash {$\langle dT / dr \rangle$} & \centering\arraybackslash {$\boldsymbol \lambda$}\\
174	\centering {\small(kcal fs$^{-1}$ \AA$^{-2}$)} & \centering\arraybackslash {\small(K \AA$^{-1}$)} & \centering\arraybackslash {\small(W m$^{-1}$ K$^{-1}$)}\\ \hline
175	\endhead
176	\hline
177	% \endfoot
178	\centering3.25$\times 10^{-6}$ & \centering\arraybackslash 0.11435 & \centering\arraybackslash 1.9753 \\
179	\centering6.50$\times 10^{-6}$ & \centering\arraybackslash 0.2324 & \centering\arraybackslash 1.9438 \\
180	\centering1.30$\times 10^{-5}$ & \centering\arraybackslash 0.44922 & \centering\arraybackslash 2.0113 \\
181	\centering3.25$\times 10^{-5}$ & \centering\arraybackslash 1.1802 & \centering\arraybackslash 1.9139 \\
182	\centering6.50$\times 10^{-5}$ & \centering\arraybackslash 2.339 & \centering\arraybackslash 1.9314
183	\\ \hline \hline
184	\label{table:goldconductivity}
185	\end{longtable}
186
187	SPC/E Water Cluster:
188
189	\begin{longtable}{p{2.7cm} p{2.5cm} p{2.5cm}}
190	\caption{Calculated thermal conductivity of a cluster of 6912 SPC/E water molecules. Calculations were performed at 300 K and ambient density.}
191	\\ \hline \hline
192	\centering {$J_r$} & \centering\arraybackslash {$\langle dT / dr \rangle$} & \centering\arraybackslash {$\boldsymbol \lambda$}\\
193	\centering {\small(kcal fs$^{-1}$ \AA$^{-2}$)} & \centering\arraybackslash {\small(K \AA$^{-1}$)} & \centering\arraybackslash {\small(W m$^{-1}$ K$^{-1}$)}\\ \hline
194	\endhead
195	\hline
196	% \endfoot
197
198	\\ \hline \hline
199	\label{table:waterconductivity}
200	\end{longtable}
201
202	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
203	% SHEAR VISCOSITY
204	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
205	\subsection{Shear viscosity}
206
207	SPC/E Water Cluster:
208
209	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
210	% INTERFACIAL THERMAL CONDUCTANCE
211	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
212	\subsection{Interfacial thermal conductance}
213
214	Gold Nanoparticle in Hexane:
215
216	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
217	% INTERFACIAL FRICTION
218	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
219	\subsection{Interfacial friction}
220
221	Gold Nanoparticle and Ellipsoid in Hexane:
222
223	\begin{longtable}{p{2.5cm} p{3cm} p{2.5cm} p{2.5cm}}
224	\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.}
225	\\ \hline \hline
226	{Structure} & \centering{Axis of rotation} & \centering\arraybackslash {$\kappa$} & \centering\arraybackslash {$\delta$}\\
227	\centering {} & {} & \centering\arraybackslash {\small($10^4$ Pa s m$^{-1}$)} & \centering\arraybackslash {\small(nm)}\\ \hline
228	\endhead
229	\hline
230	% \endfoot
231	Nanoparticle & \centering$x = y = z$ & & \\
232	Ellipsoid & \centering$x = y$ & & \\
233	Ellipsoid & \centering$z$ & &
234	\\ \hline \hline
235	\label{table:interfacialfriction}
236	\end{longtable}
237
238
239	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
240	% DISCUSSION
241	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
242	\section{Discussion}
243
244
245	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
246	% ACKNOWLEDGMENTS
247	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
248	\section*{Acknowledgments}
249
250	We gratefully acknowledge conversations with Dr. Shenyu Kuang. Support for
251	this project was provided by the National Science Foundation under grant
252	CHE-0848243. Computational time was provided by the Center for Research
253	Computing (CRC) at the University of Notre Dame.
254
255	\newpage
256
257	\bibliography{nonperiodicVSS}
258
259	\end{doublespace}
260	\end{document}