| 7 |
|
|
| 8 |
|
Cu-core / Ag-shell and random alloy structures were constructed on an |
| 9 |
|
underlying FCC lattice (4.09 {\AA}) at the bulk eutectic composition |
| 10 |
< |
$\mathrm{Ag}_6\mathrm{Cu}_4$. All three compositions were considered |
| 10 |
> |
$\mathrm{Ag}_6\mathrm{Cu}_4$. Both initial geometries were considered |
| 11 |
|
although experimental results suggest that the random structure is the |
| 12 |
< |
most likely composition after |
| 13 |
< |
synthesis.\cite{Jiang:2005lr,gonzalo:5163} Three different sizes of |
| 12 |
> |
most likely structure to be found following |
| 13 |
> |
synthesis.\cite{Jiang:2005lr,gonzalo:5163} Three different sizes of |
| 14 |
|
nanoparticles corresponding to a 20 \AA radius (1961 atoms), 30 {\AA} |
| 15 |
|
radius (6603 atoms) and 40 {\AA} radius (15683 atoms) were |
| 16 |
|
constructed. These initial structures were relaxed to their |
| 17 |
|
equilibrium structures at 20 K for 20 ps and again at 300 K for 100 ps |
| 18 |
< |
sampling from a Maxwell-Boltzmann distribution at each temperature. |
| 18 |
> |
sampling from a Maxwell-Boltzmann distribution at each temperature. All simulations were conducted using the {\sc OOPSE} molecular dynamics package.\cite{Meineke:2004uq} |
| 19 |
|
|
| 20 |
|
To mimic the effects of the heating due to laser irradiation, the |
| 21 |
|
particles were allowed to melt by sampling velocities from the Maxwell |
| 34 |
|
To approximate the effects of rapid heat transfer to the solvent |
| 35 |
|
following a heating at the plasmon resonance, we utilized a |
| 36 |
|
methodology in which atoms contained in the outer $4$ {\AA} radius of |
| 37 |
< |
the nanoparticle evolved under Langevin Dynamics with a solvent |
| 38 |
< |
friction approximating the contribution from the solvent and capping |
| 39 |
< |
agent. Atoms located in the interior of the nanoparticle evolved |
| 40 |
< |
under Newtonian dynamics. The set-up of our simulations is nearly |
| 41 |
< |
identical with the ``stochastic boundary molecular dynamics'' ({\sc |
| 42 |
< |
sbmd}) method that has seen wide use in the protein simulation |
| 37 |
> |
the nanoparticle evolved under Langevin Dynamics, |
| 38 |
> |
\begin{equation} |
| 39 |
> |
m \frac{\partial^2 \vec{x}}{\partial t^2} = F_\textrm{sys}(\vec{x}(t)) |
| 40 |
> |
- 6 \pi a \eta \vec{v}(t) + F_\textrm{ran} |
| 41 |
> |
\label{eq:langevin} |
| 42 |
> |
\end{equation} |
| 43 |
> |
with a solvent friction ($\eta$) approximating the contribution from |
| 44 |
> |
the solvent and capping agent. Atoms located in the interior of the |
| 45 |
> |
nanoparticle evolved under Newtonian dynamics. The set-up of our |
| 46 |
> |
simulations is nearly identical with the ``stochastic boundary |
| 47 |
> |
molecular dynamics'' ({\sc sbmd}) method that has seen wide use in the |
| 48 |
> |
protein simulation |
| 49 |
|
community.\cite{BROOKS:1985kx,BROOKS:1983uq,BRUNGER:1984fj} A sketch |
| 50 |
< |
of this setup can be found in Fig. \ref{fig:langevinSketch}. For a |
| 51 |
< |
spherical atom of radius $a$, the Langevin frictional forces can be |
| 52 |
< |
determined by Stokes' law |
| 53 |
< |
\begin{equation} |
| 54 |
< |
\mathbf{F}_{\mathrm{frictional}}=6\pi a \eta \mathbf{v} |
| 50 |
> |
of this setup can be found in Fig. \ref{fig:langevinSketch}. In |
| 51 |
> |
equation \ref{eq:langevin} the frictional forces of a spherical atom |
| 52 |
> |
of radius $a$ depend on the solvent viscosity. The random forces are |
| 53 |
> |
usually taken as gaussian random variables with zero mean and a |
| 54 |
> |
variance tied to the solvent viscosity and temperature, |
| 55 |
> |
\begin{equation} |
| 56 |
> |
\langle F_\textrm{ran}(t) \cdot F_\textrm{ran} (t') |
| 57 |
> |
\rangle = 2 k_B T (6 \pi \eta a) \delta(t - t') |
| 58 |
> |
\label{eq:stochastic} |
| 59 |
|
\end{equation} |
| 60 |
< |
where $\eta$ is the effective viscosity of the solvent in which the |
| 61 |
< |
particle is embedded. Due to the presence of the capping agent and |
| 62 |
< |
the lack of details about the atomic-scale interactions between the |
| 63 |
< |
metallic atoms and the solvent, the effective viscosity is a |
| 54 |
< |
essentially a free parameter that must be tuned to give experimentally |
| 55 |
< |
relevant simulations. |
| 60 |
> |
Due to the presence of the capping agent and the lack of details about |
| 61 |
> |
the atomic-scale interactions between the metallic atoms and the |
| 62 |
> |
solvent, the effective viscosity is a essentially a free parameter |
| 63 |
> |
that must be tuned to give experimentally relevant simulations. |
| 64 |
|
\begin{figure}[htbp] |
| 65 |
|
\centering |
| 66 |
< |
\includegraphics[width=\linewidth]{images/stochbound.pdf} |
| 66 |
> |
\includegraphics[width=5in]{images/stochbound.pdf} |
| 67 |
|
\caption{Methodology used to mimic the experimental cooling conditions |
| 68 |
|
of a hot nanoparticle surrounded by a solvent. Atoms in the core of |
| 69 |
|
the particle evolved under Newtonian dynamics, while atoms that were |
| 70 |
|
in the outer skin of the particle evolved under Langevin dynamics. |
| 71 |
< |
The radial cutoff between the two dynamical regions was set to 4 {\AA} |
| 72 |
< |
smaller than the original radius of the liquid droplet.} |
| 71 |
> |
The radius of the spherical region operating under Newtonian dynamics, |
| 72 |
> |
$r_\textrm{Newton}$ was set to be 4 {\AA} smaller than the original |
| 73 |
> |
radius ($R$) of the liquid droplet.} |
| 74 |
|
\label{fig:langevinSketch} |
| 75 |
|
\end{figure} |
| 76 |
|
|
| 124 |
|
Values for the interfacial conductance have been determined by a |
| 125 |
|
number of groups for similar nanoparticles and range from a low |
| 126 |
|
$87.5\times 10^{6}$ $(\mathrm{Wm^{-2}K^{-1}})$ to $120\times 10^{6}$ |
| 127 |
< |
$(\mathrm{Wm^{-2}K^{-1}})$.\cite{hartlandPrv2007} Similarly, Plech |
| 127 |
> |
$(\mathrm{Wm^{-2}K^{-1}})$.\cite{Wilson:2002uq} Similarly, Plech |
| 128 |
|
{\it et al.} reported a value for the interfacial conductance of |
| 129 |
|
$G=105\pm 15$ $(\mathrm{Wm^{-2}K^{-1}})$ and $G=130\pm 15$ |
| 130 |
|
$(\mathrm{Wm^{-2}K^{-1}})$ for Pt |
| 146 |
|
simulation can then be tuned by changing the effective solvent |
| 147 |
|
viscosity ($\eta$) until the nanoparticle cooling rate matches the |
| 148 |
|
cooling rate described by the heat-transfer equations |
| 149 |
< |
(\ref{eq:heateqn}). The effective solvent viscosity (in poise) for a G |
| 150 |
< |
of $87.5\times 10^{6}$ $(\mathrm{Wm^{-2}K^{-1}})$ is 0.17, 0.20, and |
| 151 |
< |
0.22 for 20 {\AA}, 30 {\AA}, and 40 {\AA} particles, respectively. The |
| 152 |
< |
effective solvent viscosity (again in poise) for an interfacial |
| 153 |
< |
conductance of $117\times 10^{6}$ $(\mathrm{Wm^{-2}K^{-1}})$ is 0.23, |
| 154 |
< |
0.29, and 0.30 for 20 {\AA}, 30 {\AA} and 40 {\AA} particles. Cooling |
| 155 |
< |
traces for each particle size are presented in |
| 149 |
> |
(\ref{eq:heateqn}). The effective solvent viscosity (in Pa s) for a G |
| 150 |
> |
of $87.5\times 10^{6}$ $(\mathrm{Wm^{-2}K^{-1}})$ is $4.2 \times |
| 151 |
> |
10^{-6}$, $5.0 \times 10^{-6}$, and |
| 152 |
> |
$5.5 \times 10^{-6}$ for 20 {\AA}, 30 {\AA}, and 40 {\AA} particles, respectively. The |
| 153 |
> |
effective solvent viscosity (again in Pa s) for an interfacial |
| 154 |
> |
conductance of $117\times 10^{6}$ $(\mathrm{Wm^{-2}K^{-1}})$ is $5.7 |
| 155 |
> |
\times 10^{-6}$, $7.2 \times 10^{-6}$, and $7.5 \times 10^{-6}$ |
| 156 |
> |
for 20 {\AA}, 30 {\AA} and 40 {\AA} particles. Cooling traces for |
| 157 |
> |
each particle size are presented in |
| 158 |
|
Fig. \ref{fig:images_cooling_plot}. It should be noted that the |
| 159 |
|
Langevin thermostat produces cooling curves that are consistent with |
| 160 |
|
Newtonian (single-exponential) cooling, which cannot match the cooling |
| 170 |
|
|
| 171 |
|
\begin{figure}[htbp] |
| 172 |
|
\centering |
| 173 |
< |
\includegraphics[width=\linewidth]{images/cooling_plot.pdf} |
| 173 |
> |
\includegraphics[width=5in]{images/cooling_plot.pdf} |
| 174 |
|
\caption{Thermal cooling curves obtained from the inverse Laplace |
| 175 |
|
transform heat model in Eq. \ref{eq:laplacetransform} (solid line) as |
| 176 |
|
well as from molecular dynamics simulations (circles). Effective |
| 177 |
< |
solvent viscosities of 0.23-0.30 poise (depending on the radius of the |
| 178 |
< |
particle) give the best fit to the experimental cooling curves. |
| 179 |
< |
%Since |
| 180 |
< |
%this viscosity is substantially in excess of the viscosity of liquid |
| 181 |
< |
%water, much of the thermal transfer to the surroundings is probably |
| 171 |
< |
%due to the capping agent. |
| 172 |
< |
} |
| 177 |
> |
solvent viscosities of 4.2-7.5 $\times 10^{-6}$ Pa s (depending on the |
| 178 |
> |
radius of the particle) give the best fit to the experimental cooling |
| 179 |
> |
curves. This viscosity suggests that the nanoparticles in these |
| 180 |
> |
experiments are surrounded by a vapor layer (which is a reasonable |
| 181 |
> |
assumptions given the initial temperatures of the particles). } |
| 182 |
|
\label{fig:images_cooling_plot} |
| 183 |
|
\end{figure} |
| 184 |
|
|
| 190 |
|
Embedded Atom Model ({\sc eam})~\cite{PhysRevB.33.7983} and |
| 191 |
|
Sutton-Chen ({\sc sc})~\cite{Chen90} potential have been used to study |
| 192 |
|
a wide range of phenomena in both bulk materials and |
| 193 |
< |
nanoparticles.\cite{Vardeman-II:2001jn,ShibataT._ja026764r} Both |
| 193 |
> |
nanoparticles.\cite{Vardeman-II:2001jn,ShibataT._ja026764r,Sankaranarayanan:2005lr,Chui:2003fk,Wang:2005qy,Medasani:2007uq} Both |
| 194 |
|
potentials are based on a model of a metal which treats the nuclei and |
| 195 |
|
core electrons as pseudo-atoms embedded in the electron density due to |
| 196 |
|
the valence electrons on all of the other atoms in the system. The |
| 255 |
|
\end{figure} |
| 256 |
|
|
| 257 |
|
|
| 258 |
+ |
\begin{figure}[htbp] |
| 259 |
+ |
\centering |
| 260 |
+ |
\includegraphics[width=5in]{images/cross_section_array.jpg} |
| 261 |
+ |
\caption{Cutaway views of 30 \AA\ Ag-Cu nanoparticle structures for |
| 262 |
+ |
random alloy (top) and Cu (core) / Ag (shell) initial conditions |
| 263 |
+ |
(bottom). Shown from left to right are the crystalline, liquid |
| 264 |
+ |
droplet, and final glassy bead configurations.} |
| 265 |
+ |
\label{fig:cross_sections} |
| 266 |
+ |
\end{figure} |
