--- trunk/nanoglass/experimental.tex	2007/09/11 15:23:24	3222
+++ trunk/nanoglass/experimental.tex	2007/10/02 22:01:48	3242
@@ -1,3 +1,5 @@
+%!TEX root = /Users/charles/Desktop/nanoglass/nanoglass.tex
+
 \section{Computational Methodology} 
 \label{sec:details} 
 
@@ -5,12 +7,15 @@ $\mathrm{Ag}_6\mathrm{Cu}_4$.  Three different sizes o
 
 Cu-core / Ag-shell and random alloy structures were constructed on an
 underlying FCC lattice (4.09 {\AA}) at the bulk eutectic composition
-$\mathrm{Ag}_6\mathrm{Cu}_4$.  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.
+$\mathrm{Ag}_6\mathrm{Cu}_4$.  Both initial geometries were considered
+although experimental results suggest that the random structure is the
+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.
 
 To mimic the effects of the heating due to laser irradiation, the
 particles were allowed to melt by sampling velocities from the Maxwell
@@ -29,43 +34,59 @@ 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}
-\caption{Methodology for nanoparticle cooling. Equations of motion for metal atoms contained in the outer 4 {\AA} were determined by Langevins' Equations of motion. Metal atoms outside this region were allowed to evolve under Newtonian dynamics.}
+\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 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}
+
 The viscosity ($\eta$) can be tuned by comparing the cooling rate that
 a set of nanoparticles experience with the known cooling rates for
-those particles obtained via the laser heating experiments.
+similar particles obtained via the laser heating experiments.
 Essentially, we tune the solvent viscosity until the thermal decay
 profile matches a heat-transfer model using reasonable values for the
 interfacial conductance and the thermal conductivity of the solvent.
 
 Cooling rates for the experimentally-observed nanoparticles were
 calculated from the heat transfer equations for a spherical particle
-embedded in a ambient medium that allows for diffusive heat
-transport. The heat transfer model is a set of two coupled
-differential equations in the Laplace domain,
+embedded in a ambient medium that allows for diffusive heat transport.
+Following Plech {\it et al.},\cite{plech:195423} we use a heat
+transfer model that consists of two coupled differential equations
+in the Laplace domain,
 \begin{eqnarray}
 Mc_{P}\cdot(s\cdot T_{p}(s)-T_{0})+4\pi R^{2} G\cdot(T_{p}(s)-T_{f}(r=R,s)=0\\
 \left(\frac{\partial}{\partial r} T_{f}(r,s)\right)_{r=R} +
@@ -97,14 +118,17 @@ $K$ of $0.6$  $(\mathrm{Wm^{-1}K^{-1}})$, $\rho$ of $1
 0.295 $(\mathrm{J g^{-1} K^{-1}})$ and $9.826\times 10^6$ $(\mathrm{g
 m^{-3}})$ respectively. Since most of the laser excitation experiments
 have been done in aqueous solutions, parameters used for the fluid are
-$K$ of $0.6$  $(\mathrm{Wm^{-1}K^{-1}})$, $\rho$ of $1.0\times10^6$ $(\mathrm{g
-m^{-3}})$ and $c$ of $4.184$ $(\mathrm{J g^{-1} K^{-1}})$. 
+$K$ of $0.6$ $(\mathrm{Wm^{-1}K^{-1}})$, $\rho$ of $1.0\times10^6$
+$(\mathrm{g m^{-3}})$ and $c$ of $4.184$ $(\mathrm{J g^{-1} K^{-1}})$.
 
 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} 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 nanoparticles.\cite{plech:195423,PhysRevB.66.224301}
+$(\mathrm{Wm^{-2}K^{-1}})$.\cite{hartlandPrv2007} 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
+nanoparticles.\cite{plech:195423,PhysRevB.66.224301}
 
 We conducted our simulations at both ends of the range of
 experimentally-determined values for the interfacial conductance.
@@ -115,10 +139,9 @@ experiments on Au nanospheres, we believe that the tru
 the fastest heat transfer, a value of $117\times 10^{6}$
 $(\mathrm{Wm^{-2}K^{-1}})$ was used.  Based on calculations we have
 done using raw data from the Hartland group's thermal half-time
-experiments on Au nanospheres, we believe that the true G values are
-closer to the faster regime: $117\times 10^{6}$ $(\mathrm{Wm^{-2}K^{-1}})$.
+experiments on Au nanospheres, the true G values are probably in the
+faster regime: $117\times 10^{6}$ $(\mathrm{Wm^{-2}K^{-1}})$.
 
-
 The rate of cooling for the nanoparticles in a molecular dynamics
 simulation can then be tuned by changing the effective solvent
 viscosity ($\eta$) until the nanoparticle cooling rate matches the
@@ -133,19 +156,29 @@ profiles from Eq. \ref{eq:laplacetransform} exactly.
 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
-profiles from Eq. \ref{eq:laplacetransform} exactly.
+profiles from Eq. \ref{eq:laplacetransform} exactly. Fitting the
+Langevin cooling profiles to a single-exponential produces
+$\tau=25.576$ ps, $\tau=43.786$ ps, and $\tau=56.621$ ps for the 20,
+30 and 40 {\AA} nanoparticles and a G of $87.5\times 10^{6}$
+$(\mathrm{Wm^{-2}K^{-1}})$.  For comparison's sake, similar
+single-exponential fits with an interfacial conductance of G of
+$117\times 10^{6}$ $(\mathrm{Wm^{-2}K^{-1}})$ produced a $\tau=13.391$
+ps, $\tau=30.426$ ps, $\tau=43.857$ ps for the 20, 30 and 40 {\AA}
+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.}
+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.
+}
 \label{fig:images_cooling_plot}
 \end{figure}
 
@@ -197,7 +230,37 @@ metals.
 $2\alpha_{ij}$ and include up to the sixth coordination shell in FCC
 metals.
 
-\subsection{Sampling single-temperature configurations from a cooling
-trajectory} 
+%\subsection{Sampling single-temperature configurations from a cooling
+%trajectory} 
 
-ffdsafjdksalfdsa
+To better understand the structural changes occurring in the
+nanoparticles throughout the cooling trajectory, configurations were
+sampled at regular intervals during the cooling trajectory. These
+configurations were then allowed to evolve under NVE dynamics to
+sample from the proper distribution in phase space. Figure
+\ref{fig:images_cooling_time_traces} illustrates this sampling.
+
+
+\begin{figure}[htbp]
+	\centering
+		\includegraphics[height=3in]{images/cooling_time_traces.pdf}
+	\caption{Illustrative cooling profile for the 40 {\AA}
+nanoparticle evolving under stochastic boundary conditions
+corresponding to $G=$$87.5\times 10^{6}$
+$(\mathrm{Wm^{-2}K^{-1}})$. At temperatures along the cooling
+trajectory, configurations were sampled and allowed to evolve in the
+NVE ensemble. These subsequent trajectories were analyzed for
+structural features associated with bulk glass formation.}
+	\label{fig:images_cooling_time_traces}
+\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}