--- trunk/nonperiodicVSS/nonperiodicVSS.tex	2014/01/14 19:50:22	3995
+++ trunk/nonperiodicVSS/nonperiodicVSS.tex	2014/01/17 19:45:57	4003
@@ -101,7 +101,7 @@ box.\cite{MullerPlathe:1997xw,ISI:000080382700030,Kuan
 Reverse Non-Equilibrium Molecular Dynamics (RNEMD) methods impose an
 unphysical {\it flux} between different regions or ``slabs'' of the
 simulation
-box.\cite{MullerPlathe:1997xw,ISI:000080382700030,Kuang:2010uq} The
+box.\cite{MullerPlathe:1997xw,ISI:000080382700030,Kuang2010} The
 system responds by developing a temperature or velocity {\it gradient}
 between the two regions.  The gradients which develop in response to
 the applied flux are then related (via linear response theory) to the
@@ -122,8 +122,9 @@ The VSS-RNEMD approach uses a series of simultaneous v
 % **METHODOLOGY**
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \section{Velocity Shearing and Scaling (VSS) for non-periodic systems}
+
 The VSS-RNEMD approach uses a series of simultaneous velocity shearing
-and scaling exchanges between the two slabs.\cite{2012MolPh.110..691K}
+and scaling exchanges between the two slabs.\cite{Kuang2012}
 This method imposes energy and momentum conservation constraints while
 simultaneously creating a desired flux between the two slabs.  These
 constraints ensure that all configurations are sampled from the same
@@ -132,7 +133,7 @@ concentric spheres (as in figure \ref{fig:npVSS}), or 
 We have extended the VSS method for use in {\it non-periodic}
 simulations, in which the ``slabs'' have been generalized to two
 separated regions of space.  These regions could be defined as
-concentric spheres (as in figure \ref{fig:npVSS}), or one of the regions
+concentric spheres (as in figure \ref{fig:VSS}), or one of the regions
 can be defined in terms of a dynamically changing ``hull'' comprising
 the surface atoms of the cluster.  This latter definition is identical
 to the hull used in the Langevin Hull algorithm.
@@ -193,7 +194,7 @@ $j_r(\mathbf{L})$, and thermal flux, $J_r$ values.  Th
 is the instantaneous translational kinetic energy of each shell.  At
 each time interval, we solve for $a$, $b$, $\mathbf{c}_a$, and
 $\mathbf{c}_b$, subject to the imposed angular momentum flux,
-$j_r(\mathbf{L})$, and thermal flux, $J_r$ values.  The new particle
+$j_r(\mathbf{L})$, and thermal flux, $J_r$, values.  The new particle
 velocities are computed, and the simulation continues. System
 configurations after the transformations have exactly the same energy
 ({\it and} angular momentum) as before the moves.
@@ -203,13 +204,12 @@ gradients, it is quite simple to obtain both the therm
 temperature and/or angular velocity gradient in response to the
 applied flux.  Using the slope of the radial temperature or velocity
 gradients, it is quite simple to obtain both the thermal conductivity
-($\lambda$) and shear viscosity ($\eta$),
-\begin{equation}
-  J_r = -\lambda \frac{\partial T}{\partial
-    r} \hspace{2in} j_r(\mathbf{L}_z) = -\eta \frac{\partial
-    \omega_z}{\partial r}
-\end{equation}
-of a liquid cluster.
+($\lambda$), interfacial thermal conductance ($G$), or rotational friction coefficients ($\Xi^{rr}$) of any nonperiodic system.
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+% **COMPUTATIONAL DETAILS**
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\section{Computational Details}
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 % NON-PERIODIC DYNAMICS
@@ -217,18 +217,14 @@ several different systems including bare metal nanopar
 \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 flux. Systems containing liquids were run under moderate pressure ($\sim$ 5 atm) to avoid the formation of a substantial vapor phase. 
+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.
 
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-% **COMPUTATIONAL DETAILS**
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\section{Computational Details}
-
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 % SIMULATION PROTOCOL
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
 \subsection{Simulation protocol}
 
+Systems containing liquids were run under moderate pressure ($\sim$ 5 atm) to avoid the formation of a substantial vapor phase. Thermal coupling to the Langevin Hull external bath was turned off to avoid interference with any imposed flux.
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 % FORCE FIELD PARAMETERS
@@ -257,32 +253,27 @@ Fourier's Law of heat conduction in radial coordinates
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \subsection{Thermal conductivities}
 
-Fourier's Law of heat conduction in radial coordinates is
-
-\begin{equation}
-	q_r = -\lambda A \frac{dT}{dr}
-	\label{eq:fourier}
-\end{equation}
+Fourier's Law of heat conduction in radial coordinates yields an expression for the heat flow between the concentric spherical RNEMD shells:
 
-Substituting the area of a sphere and integrating between $r = r_1$ and $r_2$ and $T = T_1$ and $T_2$, we arrive at an expression for the heat flow between the concentric spherical RNEMD shells:
-
 \begin{equation}
-	q_r = - \frac{4 \pi \lambda (T_2 - T_1)}{\frac{1}{r_1} - \frac{1}{r_2}}
-	\label{eq:Q}
+	q_r = - \frac{4 \pi \lambda (T_b - T_a)}{\frac{1}{r_a} - \frac{1}{r_b}}
+\label{eq:Q}
 \end{equation}
 
-Once a stable thermal gradient has been established between the two regions, the thermal conductivity, $\lambda$, can be calculated using the the temperature difference between the selected RNEMD regions, the radii of the two shells, and the heat, $q_r$, transferred between the regions.
+where $\lambda$ is the thermal conductivity, and $T_{a,b}$ and $r_{a,b}$ are the temperatures and radii of the two RNEMD regions, respectively.
 
+Once a stable thermal gradient has been established between the two regions, the thermal conductivity, $\lambda$, can be calculated using a linear regression of the thermal gradient, $\langle \frac{dT}{dr} \rangle$:
+
 \begin{equation}
-	\lambda = \frac{q_r (\frac{1}{r_2} - \frac{1}{r_1})}{4 \pi (T_2 - T_1)}
-	\label{eq:lambda}
+	\lambda = \frac{r_a}{r_b} \frac{q_r}{4 \pi \langle \frac{dT}{dr} \rangle}
+\label{eq:lambda}
 \end{equation}
 
-The heat transferred between the two RNEMD regions is the amount of transferred kinetic energy over the length of the simulation, t
+The rate of heat transfer between the two RNEMD regions is the amount of transferred kinetic energy over the length of the simulation, t
 
 \begin{equation}
 	q_r = \frac{KE}{t}
-	\label{eq:heat}
+\label{eq:heat}
 \end{equation}
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@@ -290,17 +281,24 @@ A thermal flux is created using VSS-RNEMD moves, and t
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \subsection{Interfacial thermal conductance}
 
-A thermal flux is created using VSS-RNEMD moves, and the resulting temperature
+A thermal flux is created using VSS-RNEMD moves, and the temperature in each of the radial shells is recorded. 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 $r_0$, 
+flux. We can treat the temperature in each radial shell as discrete, making the thermal conductance, $G$, of each shell the inverse of its Kapitza resistance, $R_K$. To model the thermal conductance across an interface (or multiple interfaces) it is useful to consider the shells as resistors wired in series. The total resistance of the shells is then additive, and the interfacial thermal conductance is the inverse of the total Kapitza resistance. The thermal resistance of each shell is
+
 \begin{equation}
-	G = \frac{J_r}{\Delta T_{r_0}}, \label{eq:G}
+	R_K = \frac{1}{q_r} \Delta T 4 \pi r^2
+\label{eq:RK}
 \end{equation}
-is useful once the RNEMD approach has generated a
-stable temperature gap across the interface.
 
+making the total resistance of two neighboring shells
+
+\begin{equation}
+	R_{total} = \frac{1}{q_r} \left [ (T_2 - T_1) 4 \pi r^2_1 + (T_3 - T_2) 4 \pi r^2_2 \right ]
+\label{eq:Rtotal}
+\end{equation}
+
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 % INTERFACIAL FRICTION
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@@ -310,47 +308,46 @@ Analytical solutions for the rotational friction coeff
 
 \begin{equation}
 	\Xi^{rr} = 8 \pi \eta r^3
-	\label{eq:Xistick}.
+\label{eq:Xistick}.
 \end{equation}
 
-where $\eta$ is the dynamic viscosity of the surrounding solvent and was determined for TraPPE-UA hexane by applying a traditional VSS-RNEMD linear momentum flux to a periodic box of solvent under the same temperature and pressure conditions as the nonperiodic systems.
+where $\eta$ is the dynamic viscosity of the surrounding solvent. An $\eta$ value for UA hexane under these particular temperature and pressure conditions was determined 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}
+	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}
+\label{eq:Xia}
 \end{equation}\vspace{-0.45in}\\
 \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}
+\label{eq:Xibc}
 \end{equation}
 
 The effective rotational friction coefficient at the interface can be extracted from nonperiodic VSS-RNEMD simulations quite easily using the applied torque ($\tau$) and the observed angular velocity of the gold structure ($\omega_{Au}$)
 
 \begin{equation}
 	\Xi^{rr}_{\mathit{eff}} = \frac{\tau}{\omega_{Au}}
-	\label{eq:Xieff}
+\label{eq:Xieff}
 \end{equation}
 
 The applied torque required to overcome the interfacial friction and maintain constant rotation of the gold is 
 
 \begin{equation}
 	\tau = \frac{L}{2 t}
-	\label{eq:tau}	
+\label{eq:tau}	
 \end{equation}
 
 where $L$ is the total angular momentum exchanged between the two RNEMD regions and $t$ is the length of the simulation.
 
-% 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.
+Previous VSS-RNEMD simulations of the interfacial friction of the planar Au(111) / hexane interface have shown that the interface exists within ``slip'' boundary conditions.\cite{Kuang2012} Hu and Zwanzig\cite{Zwanzig} investigated the rotational friction coefficients for spheroids under slip boundary conditions and obtained numerial results for a scaling factor to be applied to $\Xi^{rr}_{\mathit{stick}}$ 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, $\Xi^{rr}_{\mathit{slip}}$ for any sphere approaches $0$, while the ellipsoidal $\Xi^{rr}_{\mathit{slip}}$ is the analytical $\Xi^{rr}_{\mathit{stick}}$ result scaled by a factor of $0.359$ 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**
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@@ -361,15 +358,15 @@ Calculated values for the thermal conductivity of a 40
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \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:goldTC}. 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. 
+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:goldTC}. For these calculations, the hot and cold slabs were excluded from the linear regression of the thermal gradient. The measured linear slope $\langle \frac{dT}{dr} \rangle$ 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 {\footnotesize W / m $\cdot$ K}\cite{Kuang2010}, though still significantly lower than the experimental value of 320 {\footnotesize W / m $\cdot$ K}, 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
+{$J_r$} & {$\langle \frac{dT}{dr} \rangle$} & {$\boldsymbol \lambda$}\\ 
+{\footnotesize(kcal / fs $\cdot$ \AA$^{2}$)} & {\footnotesize(K / \AA)} & {\footnotesize(W / m $\cdot$ K)}\\ \hline
 3.25$\times 10^{-6}$ & 0.11435 & 1.0143 \\
 6.50$\times 10^{-6}$ & 0.2324 & 0.9982 \\
 1.30$\times 10^{-5}$ & 0.44922 & 1.0328 \\
@@ -381,13 +378,13 @@ Calculated values for the thermal conductivity of a cl
 \label{table:goldTC}
 \end{longtable}
 
-Calculated values for the thermal conductivity of a cluster of 6912 SPC/E water molecules are shown in Table \ref{table:waterTC}. As with the gold nanoparticle thermal conductivity calculations, the RNEMD regions were excluded from the $\langle dT / dr \rangle$ fit. Again, the measured slope is linearly dependent on the applied kinetic energy flux $J_r$. The average calculated thermal conductivity from this work, $0.8841$ W m$^{-1}$ K$^{-1}$, compares very well to previous nonequilibrium molecular dynamics results (0.81 and 0.87 W m$^{-1}$ K$^{-1}$\cite{Romer2012, Zhang2005}) and experimental values (0.607 W m$^{-1}$ K$^{-1}$\cite{WagnerKruse})
+Calculated values for the thermal conductivity of a cluster of 6912 SPC/E water molecules are shown in Table \ref{table:waterTC}. As with the gold nanoparticle thermal conductivity calculations, the RNEMD regions were excluded from the $\langle \frac{dT}{dr} \rangle$ fit. Again, the measured slope is linearly dependent on the applied kinetic energy flux $J_r$. The average calculated thermal conductivity from this work, $0.8841$ {\footnotesize W / m $\cdot$ K}, compares very well to previous nonequilibrium molecular dynamics results\cite{Romer2012, Zhang2005} and experimental values.\cite{WagnerKruse}
 
 \begin{longtable}{ccc}
 \caption{Calculated thermal conductivity of a cluster of 6912 SPC/E water molecules. Calculations were performed at 300 K and 5 atm.}
 \\ \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
+{$J_r$} & {$\langle \frac{dT}{dr} \rangle$} & {$\boldsymbol \lambda$}\\ 
+{\footnotesize(kcal / fs $\cdot$ \AA$^{2}$)} & {\footnotesize(K / \AA)} & {\footnotesize(W / m $\cdot$ K)}\\ \hline
 1$\times 10^{-5}$ & 0.38683 & 0.8698 \\
 3$\times 10^{-5}$ & 1.1643 & 0.9098 \\
 6$\times 10^{-5}$ & 2.2262 & 0.8727 \\
@@ -405,16 +402,18 @@ Experiment\cite{WagnerKruse} & & 0.61
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \subsection{Interfacial thermal conductance}
 
+Calculated interfacial thermal conductance ($G$) 
+
 \begin{longtable}{ccc}
-\caption{Calculated interfacial thermal conductance (G) values for gold nanoparticles of varying radii solvated in explicit TraPPE-UA hexane. The nanoparticle G values are compared to previous results for a gold slab in TraPPE-UA hexane, revealing increased interfacial thermal conductance for non-planar interfaces.}
+\caption{Calculated interfacial thermal conductance ($G$) values for gold nanoparticles of varying radii solvated in explicit TraPPE-UA hexane. The nanoparticle $G$ values are compared to previous results for a gold slab in TraPPE-UA hexane, revealing increased interfacial thermal conductance for non-planar interfaces.}
 \\ \hline \hline
-{Nanoparticle Radius} & {G}\\ 
-{\small(\AA)} & {\small(MW m$^{-2}$ K$^{-1}$)}\\ \hline
-20 & {47.10} \\
-30 & {45.41} \\
-40 & {46.51} \\
-slab & {30.2} \\
-\hline \hline
+{Nanoparticle Radius} & {$G$}\\ 
+{\footnotesize(\AA)} & {\footnotesize(MW / m$^{2}$ $\cdot$ K)}\\ \hline
+20 & {47.1} \\
+30 & {45.4} \\
+40 & {46.5} \\
+slab & {30.2} 
+\\ \hline \hline
 \label{table:interfacialconductance}
 \end{longtable}
 
@@ -425,22 +424,22 @@ Table \ref{table:couple} shows the calculated rotation
 
 Table \ref{table:couple} shows the calculated rotational friction coefficients $\Xi^{rr}$ 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 causes the gold structure to rotate about the prescribed axis.
 	
-\begin{longtable}{lcccc}
-\caption{Comparison of rotational friction coefficients under ideal ``stick'' conditions ($\Xi^{rr}_{\mathit{stick}}$) calculated via Stokes' and Perrin's laws and effective rotational friction coefficients ($\Xi^{rr}_{\mathit{eff}}$) of gold nanostructures solvated in TraPPE-UA hexane at 230 K. The ellipsoid is oriented with the long axis along the $z$ direction.}
+\begin{longtable}{lccccc}
+\caption{Comparison of rotational friction coefficients under ideal ``slip'' ($\Xi^{rr}_{\mathit{slip}}$) and ``stick'' conditions ($\Xi^{rr}_{\mathit{stick}}$) and effective rotational friction coefficients ($\Xi^{rr}_{\mathit{eff}}$) 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} & {$\Xi^{rr}_{\mathit{stick}}$} & {$\Xi^{rr}_{\mathit{eff}}$} & {$\Xi^{rr}_{\mathit{eff}}$ / $\Xi^{rr}_{\mathit{stick}}$}\\ 
-{} & {} & {\small(amu A$^2$ fs$^{-1}$)} & {\small(amu A$^2$ fs$^{-1}$)} & \\  \hline
-Sphere (r = 20 \AA) & {$x = y = z$} & {3314} & {2386} & {0.720}\\
-Sphere (r = 30 \AA) & {$x = y = z$} & {11749} & {8415} & {0.716}\\
-Sphere (r = 40 \AA) & {$x = y = z$} & {34464} & {47544} & {1.380}\\ 
-Prolate Ellipsoid & {$x = y$} & {4991} & {3128} & {0.627}\\
-Prolate Ellipsoid & {$z$} & {1993} & {1590} & {0.798}\\ 
-  \hline \hline
+{Structure} & {Axis of Rotation} & {$\Xi^{rr}_{\mathit{slip}}$} & {$\Xi^{rr}_{\mathit{eff}}$} & {$\Xi^{rr}_{\mathit{stick}}$} & {$\Xi^{rr}_{\mathit{eff}}$ / $\Xi^{rr}_{\mathit{stick}}$}\\ 
+{} & {} & {\footnotesize(amu A$^2$ / fs)} & {\footnotesize(amu A$^2$ / fs)} & {\footnotesize(amu A$^2$ / fs)} & \\  \hline
+Sphere (r = 20 \AA) & {$x = y = z$} & 0 & {2386} & {3314} & {0.720}\\
+Sphere (r = 30 \AA) & {$x = y = z$} & 0 & {8415} & {11749} & {0.716}\\
+Sphere (r = 40 \AA) & {$x = y = z$} & 0 & {47544} & {34464} & {1.380}\\ 
+Prolate Ellipsoid & {$x = y$} & {1792} & {3128} & {4991} & {0.627}\\
+Prolate Ellipsoid & {$z$} & {716} & {1590} & {1993} & {0.798}
+\\ \hline \hline
 \label{table:couple}
 \end{longtable}
 
+The results for $\Xi^{rr}_{\mathit{eff}}$ show that, contrary to the flat Au(111) / hexane interface, gold structures solvated by hexane do not exist in the ``slip'' boundary conditions. At this length scale, the nanostructures are not perfect spheroids due to atomic `roughening' of the surface and therefore experience increased interfacial friction which deviates from the ideal ``slip'' case. The 20 and 30 \AA$\,$ radius nanoparticles experience approximately 70\% of the ideal ``stick'' boundary interfacial friction. Rotation of the ellipsoid about its long axis more closely approaches the ``stick'' limit than rotation about the short axis, which may at first seem counterintuitive. However, the `propellor' motion caused by rotation about short axis may exclude solvent from the rotation cavity or move a sufficient amount of solvent along with the gold that a smaller interfacial friction is actually experienced. The largest nanoparticle (40 \AA$\,$ radius) appears to experience more than the ``stick'' limit of interfacial friction, which may be a consequence of surface features or anomalous solvent behaviors that are not fully understood at this time.
 
-
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 % **DISCUSSION**
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
