--- trunk/nanoglass/experimental.tex	2007/09/25 19:23:21	3230
+++ trunk/nanoglass/experimental.tex	2007/10/11 18:53:11	3256
@@ -7,15 +7,15 @@ $\mathrm{Ag}_6\mathrm{Cu}_4$.  All three compositions 
 
 Cu-core / Ag-shell and random alloy structures were constructed on an
 underlying FCC lattice (4.09 {\AA}) at the bulk eutectic composition
-$\mathrm{Ag}_6\mathrm{Cu}_4$.  All three compositions were considered
+$\mathrm{Ag}_6\mathrm{Cu}_4$.  Both initial geometries were considered
 although experimental results suggest that the random structure is the
-most likely composition after
-synthesis.\cite{Jiang:2005lr,gonzalo:5163}  Three different sizes of
+most likely structure to be found following
+synthesis.\cite{Jiang:2005lr,gonzalo:5163} Three different sizes of
 nanoparticles corresponding to a 20 \AA radius (1961 atoms), 30 {\AA}
 radius (6603 atoms) and 40 {\AA} radius (15683 atoms) were
 constructed.  These initial structures were relaxed to their
 equilibrium structures at 20 K for 20 ps and again at 300 K for 100 ps
-sampling from a Maxwell-Boltzmann distribution at each temperature.
+sampling from a Maxwell-Boltzmann distribution at each temperature. All simulations were conducted using the {\sc OOPSE} molecular dynamics package.\cite{Meineke:2004uq}
 
 To mimic the effects of the heating due to laser irradiation, the
 particles were allowed to melt by sampling velocities from the Maxwell
@@ -34,34 +34,43 @@ the nanoparticle evolved under Langevin Dynamics with 
 To approximate the effects of rapid heat transfer to the solvent
 following a heating at the plasmon resonance, we utilized a
 methodology in which atoms contained in the outer $4$ {\AA} radius of
-the nanoparticle evolved under Langevin Dynamics with a solvent
-friction approximating the contribution from the solvent and capping
-agent.  Atoms located in the interior of the nanoparticle evolved
-under Newtonian dynamics.  The set-up of our simulations is nearly
-identical with the ``stochastic boundary molecular dynamics'' ({\sc
-sbmd}) method that has seen wide use in the protein simulation
+the nanoparticle evolved under Langevin Dynamics,
+\begin{equation}
+m \frac{\partial^2 \vec{x}}{\partial t^2} = F_\textrm{sys}(\vec{x}(t))
+- 6 \pi a \eta \vec{v}(t)  + F_\textrm{ran}
+\label{eq:langevin}
+\end{equation}
+with a solvent friction ($\eta$) approximating the contribution from
+the solvent and capping agent.  Atoms located in the interior of the
+nanoparticle evolved under Newtonian dynamics.  The set-up of our
+simulations is nearly identical with the ``stochastic boundary
+molecular dynamics'' ({\sc sbmd}) method that has seen wide use in the
+protein simulation
 community.\cite{BROOKS:1985kx,BROOKS:1983uq,BRUNGER:1984fj} A sketch
-of this setup can be found in Fig. \ref{fig:langevinSketch}.  For a
-spherical atom of radius $a$, the Langevin frictional forces can be
-determined by Stokes' law
-\begin{equation} 
-\mathbf{F}_{\mathrm{frictional}}=6\pi a \eta \mathbf{v}
+of this setup can be found in Fig. \ref{fig:langevinSketch}.  In
+equation \ref{eq:langevin} the frictional forces of a spherical atom
+of radius $a$ depend on the solvent viscosity.  The random forces are
+usually taken as gaussian random variables with zero mean and a
+variance tied to the solvent viscosity and temperature,
+\begin{equation}
+\langle F_\textrm{ran}(t) \cdot F_\textrm{ran} (t')
+\rangle = 2 k_B T (6 \pi \eta a) \delta(t - t')
+\label{eq:stochastic}
 \end{equation}
-where $\eta$ is the effective viscosity of the solvent in which the
-particle is embedded.  Due to the presence of the capping agent and
-the lack of details about the atomic-scale interactions between the
-metallic atoms and the solvent, the effective viscosity is a
-essentially a free parameter that must be tuned to give experimentally
-relevant simulations.
+Due to the presence of the capping agent and the lack of details about
+the atomic-scale interactions between the metallic atoms and the
+solvent, the effective viscosity is a essentially a free parameter
+that must be tuned to give experimentally relevant simulations.
 \begin{figure}[htbp]
 \centering
-\includegraphics[width=\linewidth]{images/stochbound.pdf}
+\includegraphics[width=5in]{images/stochbound.pdf}
 \caption{Methodology used to mimic the experimental cooling conditions
 of a hot nanoparticle surrounded by a solvent.  Atoms in the core of
 the particle evolved under Newtonian dynamics, while atoms that were
 in the outer skin of the particle evolved under Langevin dynamics.
-The radial cutoff between the two dynamical regions was set to 4 {\AA}
-smaller than the original radius of the liquid droplet.}
+The radius of the spherical region operating under Newtonian dynamics,
+$r_\textrm{Newton}$ was set to be 4 {\AA} smaller than the original
+radius ($R$) of the liquid droplet.}
 \label{fig:langevinSketch}
 \end{figure}
 
@@ -115,7 +124,7 @@ $(\mathrm{Wm^{-2}K^{-1}})$.\cite{hartlandPrv2007} Simi
 Values for the interfacial conductance have been determined by a
 number of groups for similar nanoparticles and range from a low
 $87.5\times 10^{6}$ $(\mathrm{Wm^{-2}K^{-1}})$ to $120\times 10^{6}$
-$(\mathrm{Wm^{-2}K^{-1}})$.\cite{hartlandPrv2007} Similarly, Plech
+$(\mathrm{Wm^{-2}K^{-1}})$.\cite{Wilson:2002uq} Similarly, Plech
 {\it et al.}  reported a value for the interfacial conductance of
 $G=105\pm 15$ $(\mathrm{Wm^{-2}K^{-1}})$ and $G=130\pm 15$
 $(\mathrm{Wm^{-2}K^{-1}})$ for Pt
@@ -137,13 +146,15 @@ cooling rate described by the heat-transfer equations
 simulation can then be tuned by changing the effective solvent
 viscosity ($\eta$) until the nanoparticle cooling rate matches the
 cooling rate described by the heat-transfer equations
-(\ref{eq:heateqn}). The effective solvent viscosity (in poise) for a G
-of $87.5\times 10^{6}$ $(\mathrm{Wm^{-2}K^{-1}})$ is 0.17, 0.20, and
-0.22 for 20 {\AA}, 30 {\AA}, and 40 {\AA} particles, respectively. The
-effective solvent viscosity (again in poise) for an interfacial
-conductance of $117\times 10^{6}$ $(\mathrm{Wm^{-2}K^{-1}})$ is 0.23,
-0.29, and 0.30 for 20 {\AA}, 30 {\AA} and 40 {\AA} particles.  Cooling
-traces for each particle size are presented in
+(\ref{eq:heateqn}). The effective solvent viscosity (in Pa s) for a G
+of $87.5\times 10^{6}$ $(\mathrm{Wm^{-2}K^{-1}})$ is $4.2 \times
+10^{-6}$, $5.0 \times 10^{-6}$, and
+$5.5 \times 10^{-6}$ for 20 {\AA}, 30 {\AA}, and 40 {\AA} particles, respectively. The
+effective solvent viscosity (again in Pa s) for an interfacial
+conductance of $117\times 10^{6}$ $(\mathrm{Wm^{-2}K^{-1}})$ is $5.7
+\times 10^{-6}$, $7.2 \times 10^{-6}$, and $7.5 \times 10^{-6}$
+for 20 {\AA}, 30 {\AA} and 40 {\AA} particles.  Cooling traces for
+each particle size are presented in
 Fig. \ref{fig:images_cooling_plot}. It should be noted that the
 Langevin thermostat produces cooling curves that are consistent with
 Newtonian (single-exponential) cooling, which cannot match the cooling
@@ -159,17 +170,15 @@ nanoparticles.
 
 \begin{figure}[htbp]
 \centering
-\includegraphics[width=\linewidth]{images/cooling_plot.pdf}
+\includegraphics[width=5in]{images/cooling_plot.pdf}
 \caption{Thermal cooling curves obtained from the inverse Laplace
 transform heat model in Eq. \ref{eq:laplacetransform} (solid line) as
 well as from molecular dynamics simulations (circles).  Effective
-solvent viscosities of 0.23-0.30 poise (depending on the radius of the
-particle) give the best fit to the experimental cooling curves. 
-%Since
-%this viscosity is substantially in excess of the viscosity of liquid
-%water, much of the thermal transfer to the surroundings is probably
-%due to the capping agent.
-}
+solvent viscosities of 4.2-7.5 $\times 10^{-6}$ Pa s (depending on the
+radius of the particle) give the best fit to the experimental cooling
+curves.  This viscosity suggests that the nanoparticles in these
+experiments are surrounded by a vapor layer (which is a reasonable
+assumptions given the initial temperatures of the particles).  }
 \label{fig:images_cooling_plot}
 \end{figure}
 
@@ -181,7 +190,7 @@ nanoparticles.\cite{Vardeman-II:2001jn,ShibataT._ja026
 Embedded Atom Model ({\sc eam})~\cite{PhysRevB.33.7983} and
 Sutton-Chen ({\sc sc})~\cite{Chen90} potential have been used to study
 a wide range of phenomena in both bulk materials and
-nanoparticles.\cite{Vardeman-II:2001jn,ShibataT._ja026764r} Both
+nanoparticles.\cite{Vardeman-II:2001jn,ShibataT._ja026764r,Sankaranarayanan:2005lr,Chui:2003fk,Wang:2005qy,Medasani:2007uq} Both
 potentials are based on a model of a metal which treats the nuclei and
 core electrons as pseudo-atoms embedded in the electron density due to
 the valence electrons on all of the other atoms in the system. The
@@ -246,3 +255,12 @@ structural features associated with bulk glass formati
 \end{figure}
 
 
+\begin{figure}[htbp]
+\centering
+\includegraphics[width=5in]{images/cross_section_array.jpg}
+\caption{Cutaway views of 30 \AA\ Ag-Cu nanoparticle structures for
+random alloy (top) and Cu (core) / Ag (shell) initial conditions
+(bottom).  Shown from left to right are the crystalline, liquid
+droplet, and final glassy bead configurations.}
+\label{fig:cross_sections}
+\end{figure}