| 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 although experimental results suggest that the random structure is the most likely composition after synthesis.\cite{Jiang:2005lr,gonzalo:5163} Three different sizes of nanoparticles |
| 11 |
< |
corresponding to a 20 \AA radius (1961 atoms), 30 {\AA} radius (6603 |
| 12 |
< |
atoms) and 40 {\AA} radius (15683 atoms) were constructed. These |
| 13 |
< |
initial structures were relaxed to their equilibrium structures at 20 |
| 14 |
< |
K for 20 ps and again at 300 K for 100 ps sampling from a |
| 15 |
< |
Maxwell-Boltzmann distribution at each temperature. |
| 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 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 |
| 19 |
> |
temperature. All simulations were conducted using the {\sc oopse} |
| 20 |
> |
molecular dynamics package.\cite{Meineke:2004uq} |
| 21 |
|
|
| 22 |
|
To mimic the effects of the heating due to laser irradiation, the |
| 23 |
|
particles were allowed to melt by sampling velocities from the Maxwell |
| 36 |
|
To approximate the effects of rapid heat transfer to the solvent |
| 37 |
|
following a heating at the plasmon resonance, we utilized a |
| 38 |
|
methodology in which atoms contained in the outer $4$ {\AA} radius of |
| 39 |
< |
the nanoparticle evolved under Langevin Dynamics with a solvent |
| 40 |
< |
friction approximating the contribution from the solvent and capping |
| 41 |
< |
agent. Atoms located in the interior of the nanoparticle evolved |
| 42 |
< |
under Newtonian dynamics. The set-up of our simulations is nearly |
| 43 |
< |
identical with the ``stochastic boundary molecular dynamics'' ({\sc |
| 44 |
< |
sbmd}) method that has seen wide use in the protein simulation |
| 39 |
> |
the nanoparticle evolved under Langevin Dynamics, |
| 40 |
> |
\begin{equation} |
| 41 |
> |
m \frac{\partial^2 \vec{x}}{\partial t^2} = F_\textrm{sys}(\vec{x}(t)) |
| 42 |
> |
- 6 \pi a \eta \vec{v}(t) + F_\textrm{ran} |
| 43 |
> |
\label{eq:langevin} |
| 44 |
> |
\end{equation} |
| 45 |
> |
with a solvent friction ($\eta$) approximating the contribution from |
| 46 |
> |
the solvent and capping agent. Atoms located in the interior of the |
| 47 |
> |
nanoparticle evolved under Newtonian dynamics. The set-up of our |
| 48 |
> |
simulations is nearly identical with the ``stochastic boundary |
| 49 |
> |
molecular dynamics'' ({\sc sbmd}) method that has seen wide use in the |
| 50 |
> |
protein simulation |
| 51 |
|
community.\cite{BROOKS:1985kx,BROOKS:1983uq,BRUNGER:1984fj} A sketch |
| 52 |
< |
of this setup can be found in Fig. \ref{fig:langevinSketch}. For a |
| 53 |
< |
spherical atom of radius $a$, the Langevin frictional forces can be |
| 54 |
< |
determined by Stokes' law |
| 55 |
< |
\begin{equation} |
| 56 |
< |
\mathbf{F}_{\mathrm{frictional}}=6\pi a \eta \mathbf{v} |
| 52 |
> |
of this setup can be found in Fig. \ref{fig:langevinSketch}. In |
| 53 |
> |
equation \ref{eq:langevin} the frictional forces of a spherical atom |
| 54 |
> |
of radius $a$ depend on the solvent viscosity. The random forces are |
| 55 |
> |
usually taken as gaussian random variables with zero mean and a |
| 56 |
> |
variance tied to the solvent viscosity and temperature, |
| 57 |
> |
\begin{equation} |
| 58 |
> |
\langle F_\textrm{ran}(t) \cdot F_\textrm{ran} (t') |
| 59 |
> |
\rangle = 2 k_B T (6 \pi \eta a) \delta(t - t') |
| 60 |
> |
\label{eq:stochastic} |
| 61 |
|
\end{equation} |
| 62 |
< |
where $\eta$ is the effective viscosity of the solvent in which the |
| 63 |
< |
particle is embedded. Due to the presence of the capping agent and |
| 64 |
< |
the lack of details about the atomic-scale interactions between the |
| 65 |
< |
metallic atoms and the solvent, the effective viscosity is a |
| 51 |
< |
essentially a free parameter that must be tuned to give experimentally |
| 52 |
< |
relevant simulations. |
| 62 |
> |
Due to the presence of the capping agent and the lack of details about |
| 63 |
> |
the atomic-scale interactions between the metallic atoms and the |
| 64 |
> |
solvent, the effective viscosity is a essentially a free parameter |
| 65 |
> |
that must be tuned to give experimentally relevant simulations. |
| 66 |
|
\begin{figure}[htbp] |
| 67 |
|
\centering |
| 68 |
< |
\includegraphics[width=\linewidth]{images/stochbound.pdf} |
| 69 |
< |
\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.} |
| 68 |
> |
\includegraphics[width=5in]{images/stochbound.pdf} |
| 69 |
> |
\caption{Methodology used to mimic the experimental cooling conditions |
| 70 |
> |
of a hot nanoparticle surrounded by a solvent. Atoms in the core of |
| 71 |
> |
the particle evolved under Newtonian dynamics, while atoms that were |
| 72 |
> |
in the outer skin of the particle evolved under Langevin dynamics. |
| 73 |
> |
The radius of the spherical region operating under Newtonian dynamics, |
| 74 |
> |
$r_\textrm{Newton}$ was set to be 4 {\AA} smaller than the original |
| 75 |
> |
radius ($R$) of the liquid droplet.} |
| 76 |
|
\label{fig:langevinSketch} |
| 77 |
|
\end{figure} |
| 78 |
+ |
|
| 79 |
|
The viscosity ($\eta$) can be tuned by comparing the cooling rate that |
| 80 |
|
a set of nanoparticles experience with the known cooling rates for |
| 81 |
< |
those particles obtained via the laser heating experiments. |
| 81 |
> |
similar particles obtained via the laser heating experiments. |
| 82 |
|
Essentially, we tune the solvent viscosity until the thermal decay |
| 83 |
|
profile matches a heat-transfer model using reasonable values for the |
| 84 |
|
interfacial conductance and the thermal conductivity of the solvent. |
| 85 |
|
|
| 86 |
|
Cooling rates for the experimentally-observed nanoparticles were |
| 87 |
|
calculated from the heat transfer equations for a spherical particle |
| 88 |
< |
embedded in a ambient medium that allows for diffusive heat |
| 89 |
< |
transport. The heat transfer model is a set of two coupled |
| 90 |
< |
differential equations in the Laplace domain, |
| 88 |
> |
embedded in a ambient medium that allows for diffusive heat transport. |
| 89 |
> |
Following Plech {\it et al.},\cite{plech:195423} we use a heat |
| 90 |
> |
transfer model that consists of two coupled differential equations |
| 91 |
> |
in the Laplace domain, |
| 92 |
|
\begin{eqnarray} |
| 93 |
|
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\\ |
| 94 |
|
\left(\frac{\partial}{\partial r} T_{f}(r,s)\right)_{r=R} + |
| 120 |
|
0.295 $(\mathrm{J g^{-1} K^{-1}})$ and $9.826\times 10^6$ $(\mathrm{g |
| 121 |
|
m^{-3}})$ respectively. Since most of the laser excitation experiments |
| 122 |
|
have been done in aqueous solutions, parameters used for the fluid are |
| 123 |
< |
$K$ of $0.6$ $(\mathrm{Wm^{-1}K^{-1}})$, $\rho$ of $1.0\times10^6$ $(\mathrm{g |
| 124 |
< |
m^{-3}})$ and $c$ of $4.184$ $(\mathrm{J g^{-1} K^{-1}})$. |
| 123 |
> |
$K$ of $0.6$ $(\mathrm{Wm^{-1}K^{-1}})$, $\rho$ of $1.0\times10^6$ |
| 124 |
> |
$(\mathrm{g m^{-3}})$ and $c$ of $4.184$ $(\mathrm{J g^{-1} K^{-1}})$. |
| 125 |
|
|
| 126 |
|
Values for the interfacial conductance have been determined by a |
| 127 |
|
number of groups for similar nanoparticles and range from a low |
| 128 |
< |
$87.5\times 10^{6}$ $(\mathrm{Wm^{-2}K^{-1}})$ to $120\times 10^{6}$ |
| 129 |
< |
$(\mathrm{Wm^{-2}K^{-1}})$.\cite{hartlandPrv2007} Plech {\it et al.} reported a value for the interfacial conductance of $G=105\pm 15$ $(\mathrm{Wm^{-2}K^{-1}})$ and |
| 130 |
< |
$G=130\pm 15$ $(\mathrm{Wm^{-2}K^{-1}})$ for Pt nanoparticles.\cite{plech:195423,PhysRevB.66.224301} |
| 128 |
> |
$87.5\times 10^{6} (\mathrm{Wm^{-2}K^{-1}})$ to $130\times 10^{6} |
| 129 |
> |
(\mathrm{Wm^{-2}K^{-1}})$.\cite{XXXHartland,Wilson:2002uq} Wilson {\it |
| 130 |
> |
et al.} worked with Au, Pt, and AuPd nanoparticles and obtained an |
| 131 |
> |
estimate for the interfacial conductance of $G=130 |
| 132 |
> |
(\mathrm{Wm^{-2}K^{-1}})$.\cite{Wilson:2002uq} Similarly, Plech {\it |
| 133 |
> |
et al.} reported a value for the interfacial conductance of $G=105\pm |
| 134 |
> |
15 (\mathrm{Wm^{-2}K^{-1}})$ for Au nanoparticles.\cite{plech:195423} |
| 135 |
|
|
| 136 |
|
We conducted our simulations at both ends of the range of |
| 137 |
|
experimentally-determined values for the interfacial conductance. |
| 142 |
|
the fastest heat transfer, a value of $117\times 10^{6}$ |
| 143 |
|
$(\mathrm{Wm^{-2}K^{-1}})$ was used. Based on calculations we have |
| 144 |
|
done using raw data from the Hartland group's thermal half-time |
| 145 |
< |
experiments on Au nanospheres, we believe that the true G values are |
| 146 |
< |
closer to the faster regime: $117\times 10^{6}$ $(\mathrm{Wm^{-2}K^{-1}})$. |
| 145 |
> |
experiments on Au nanospheres,\cite{HuM._jp020581+} the true G values |
| 146 |
> |
are probably in the faster regime: $117\times 10^{6}$ |
| 147 |
> |
$(\mathrm{Wm^{-2}K^{-1}})$. |
| 148 |
|
|
| 123 |
– |
|
| 149 |
|
The rate of cooling for the nanoparticles in a molecular dynamics |
| 150 |
|
simulation can then be tuned by changing the effective solvent |
| 151 |
|
viscosity ($\eta$) until the nanoparticle cooling rate matches the |
| 152 |
|
cooling rate described by the heat-transfer equations |
| 153 |
< |
(\ref{eq:heateqn}). The effective solvent viscosity (in poise) for a G |
| 154 |
< |
of $87.5\times 10^{6}$ $(\mathrm{Wm^{-2}K^{-1}})$ is 0.17, 0.20, and |
| 155 |
< |
0.22 for 20 {\AA}, 30 {\AA}, and 40 {\AA} particles, respectively. The |
| 156 |
< |
effective solvent viscosity (again in poise) for an interfacial |
| 157 |
< |
conductance of $117\times 10^{6}$ $(\mathrm{Wm^{-2}K^{-1}})$ is 0.23, |
| 158 |
< |
0.29, and 0.30 for 20 {\AA}, 30 {\AA} and 40 {\AA} particles. Cooling |
| 159 |
< |
traces for each particle size are presented in |
| 153 |
> |
(\ref{eq:heateqn}). The effective solvent viscosity (in Pa s) for a G |
| 154 |
> |
of $87.5\times 10^{6}$ $(\mathrm{Wm^{-2}K^{-1}})$ is $4.2 \times |
| 155 |
> |
10^{-6}$, $5.0 \times 10^{-6}$, and |
| 156 |
> |
$5.5 \times 10^{-6}$ for 20 {\AA}, 30 {\AA}, and 40 {\AA} particles, respectively. The |
| 157 |
> |
effective solvent viscosity (again in Pa s) for an interfacial |
| 158 |
> |
conductance of $117\times 10^{6}$ $(\mathrm{Wm^{-2}K^{-1}})$ is $5.7 |
| 159 |
> |
\times 10^{-6}$, $7.2 \times 10^{-6}$, and $7.5 \times 10^{-6}$ |
| 160 |
> |
for 20 {\AA}, 30 {\AA} and 40 {\AA} particles. These viscosities are |
| 161 |
> |
essentially gas-phase values, a fact which is consistent with the |
| 162 |
> |
initial temperatures of the particles being well into the |
| 163 |
> |
super-critical region for the aqueous environment. Gas bubble |
| 164 |
> |
generation has also been seen experimentally around gold nanoparticles |
| 165 |
> |
in water.\cite{kotaidis:184702} Instead of a single value for the |
| 166 |
> |
effective viscosity, a time-dependent parameter might be a better |
| 167 |
> |
mimic of the cooling vapor layer that surrounds the hot particles. |
| 168 |
> |
This may also be a contributing factor to the size-dependence of the |
| 169 |
> |
effective viscosities in our simulations. |
| 170 |
> |
|
| 171 |
> |
Cooling traces for each particle size are presented in |
| 172 |
|
Fig. \ref{fig:images_cooling_plot}. It should be noted that the |
| 173 |
|
Langevin thermostat produces cooling curves that are consistent with |
| 174 |
|
Newtonian (single-exponential) cooling, which cannot match the cooling |
| 175 |
< |
profiles from Eq. \ref{eq:laplacetransform} exactly. |
| 175 |
> |
profiles from Eq. \ref{eq:laplacetransform} exactly. Fitting the |
| 176 |
> |
Langevin cooling profiles to a single-exponential produces |
| 177 |
> |
$\tau=25.576$ ps, $\tau=43.786$ ps, and $\tau=56.621$ ps for the 20, |
| 178 |
> |
30 and 40 {\AA} nanoparticles and a G of $87.5\times 10^{6}$ |
| 179 |
> |
$(\mathrm{Wm^{-2}K^{-1}})$. For comparison's sake, similar |
| 180 |
> |
single-exponential fits with an interfacial conductance of G of |
| 181 |
> |
$117\times 10^{6}$ $(\mathrm{Wm^{-2}K^{-1}})$ produced a $\tau=13.391$ |
| 182 |
> |
ps, $\tau=30.426$ ps, $\tau=43.857$ ps for the 20, 30 and 40 {\AA} |
| 183 |
> |
nanoparticles. |
| 184 |
|
|
| 185 |
|
\begin{figure}[htbp] |
| 186 |
|
\centering |
| 187 |
< |
\includegraphics[width=\linewidth]{images/cooling_plot.pdf} |
| 187 |
> |
\includegraphics[width=5in]{images/cooling_plot.pdf} |
| 188 |
|
\caption{Thermal cooling curves obtained from the inverse Laplace |
| 189 |
|
transform heat model in Eq. \ref{eq:laplacetransform} (solid line) as |
| 190 |
|
well as from molecular dynamics simulations (circles). Effective |
| 191 |
< |
solvent viscosities of 0.23-0.30 poise (depending on the radius of the |
| 192 |
< |
particle) give the best fit to the experimental cooling curves. Since |
| 193 |
< |
this viscosity is substantially in excess of the viscosity of liquid |
| 194 |
< |
water, much of the thermal transfer to the surroundings is probably |
| 195 |
< |
due to the capping agent.} |
| 191 |
> |
solvent viscosities of 4.2-7.5 $\times 10^{-6}$ Pa s (depending on the |
| 192 |
> |
radius of the particle) give the best fit to the experimental cooling |
| 193 |
> |
curves. This viscosity suggests that the nanoparticles in these |
| 194 |
> |
experiments are surrounded by a vapor layer (which is a reasonable |
| 195 |
> |
assumptions given the initial temperatures of the particles). } |
| 196 |
|
\label{fig:images_cooling_plot} |
| 197 |
|
\end{figure} |
| 198 |
|
|
| 204 |
|
Embedded Atom Model ({\sc eam})~\cite{PhysRevB.33.7983} and |
| 205 |
|
Sutton-Chen ({\sc sc})~\cite{Chen90} potential have been used to study |
| 206 |
|
a wide range of phenomena in both bulk materials and |
| 207 |
< |
nanoparticles.\cite{Vardeman-II:2001jn,ShibataT._ja026764r} Both |
| 207 |
> |
nanoparticles.\cite{Vardeman-II:2001jn,ShibataT._ja026764r,Sankaranarayanan:2005lr,Chui:2003fk,Wang:2005qy,Medasani:2007uq} Both |
| 208 |
|
potentials are based on a model of a metal which treats the nuclei and |
| 209 |
|
core electrons as pseudo-atoms embedded in the electron density due to |
| 210 |
|
the valence electrons on all of the other atoms in the system. The |
| 236 |
|
data for both FCC solid solutions of Ag-Cu and the high-temperature |
| 237 |
|
liquid.\cite{sheng:184203} In contrast, the {\sc eam} potential does |
| 238 |
|
not reproduce the experimentally observed heat of mixing for the |
| 239 |
< |
liquid alloy.\cite{MURRAY:1984lr} Combination rules for the alloy were |
| 240 |
< |
taken to be the arithmatic average of the atomic parameters with the |
| 241 |
< |
exception of $c_i$ since its values is only dependent on the identity |
| 242 |
< |
of the atom where the density is evaluated. For the {\sc q-sc} |
| 243 |
< |
potential, cutoff distances are traditionally taken to be |
| 244 |
< |
$2\alpha_{ij}$ and include up to the sixth coordination shell in FCC |
| 245 |
< |
metals. |
| 239 |
> |
liquid alloy.\cite{MURRAY:1984lr} In this work, we have utilized the |
| 240 |
> |
{\sc q-sc} formulation for our potential energies and forces. |
| 241 |
> |
Combination rules for the alloy were taken to be the arithmetic |
| 242 |
> |
average of the atomic parameters with the exception of $c_i$ since its |
| 243 |
> |
values is only dependent on the identity of the atom where the density |
| 244 |
> |
is evaluated. For the {\sc q-sc} potential, cutoff distances are |
| 245 |
> |
traditionally taken to be $2\alpha_{ij}$ and include up to the sixth |
| 246 |
> |
coordination shell in FCC metals. |
| 247 |
|
|
| 248 |
|
%\subsection{Sampling single-temperature configurations from a cooling |
| 249 |
|
%trajectory} |
| 250 |
|
|
| 251 |
< |
To better understand the structural changes occurring in the nanoparticles throughout the cooling trajectory, configurations were sampled at temperatures throughout 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. |
| 251 |
> |
To better understand the structural changes occurring in the |
| 252 |
> |
nanoparticles throughout the cooling trajectory, configurations were |
| 253 |
> |
sampled at regular intervals during the cooling trajectory. These |
| 254 |
> |
configurations were then allowed to evolve under NVE dynamics to |
| 255 |
> |
sample from the proper distribution in phase space. Figure |
| 256 |
> |
\ref{fig:images_cooling_time_traces} illustrates this sampling. |
| 257 |
|
|
| 258 |
|
|
| 259 |
|
\begin{figure}[htbp] |
| 260 |
|
\centering |
| 261 |
|
\includegraphics[height=3in]{images/cooling_time_traces.pdf} |
| 262 |
< |
\caption{Illustrative cooling profile for the 40 {\AA} nanoparticle evolving under stochastic boundary conditions corresponding to $G=$$87.5\times 10^{6}$ $(\mathrm{Wm^{-2}K^{-1}})$. At temperatures along the cooling trajectory, configurations were sampled and allowed to evolve in the NVE ensemble. These subsequent trajectories were analyzed for structural features associated with bulk glass formation.} |
| 262 |
> |
\caption{Illustrative cooling profile for the 40 {\AA} |
| 263 |
> |
nanoparticle evolving under stochastic boundary conditions |
| 264 |
> |
corresponding to $G=$$87.5\times 10^{6}$ |
| 265 |
> |
$(\mathrm{Wm^{-2}K^{-1}})$. At temperatures along the cooling |
| 266 |
> |
trajectory, configurations were sampled and allowed to evolve in the |
| 267 |
> |
NVE ensemble. These subsequent trajectories were analyzed for |
| 268 |
> |
structural features associated with bulk glass formation.} |
| 269 |
|
\label{fig:images_cooling_time_traces} |
| 270 |
|
\end{figure} |
| 271 |
|
|
| 272 |
|
|
| 273 |
+ |
\begin{figure}[htbp] |
| 274 |
+ |
\centering |
| 275 |
+ |
\includegraphics[width=5in]{images/cross_section_array.jpg} |
| 276 |
+ |
\caption{Cutaway views of 30 \AA\ Ag-Cu nanoparticle structures for |
| 277 |
+ |
random alloy (top) and Cu (core) / Ag (shell) initial conditions |
| 278 |
+ |
(bottom). Shown from left to right are the crystalline, liquid |
| 279 |
+ |
droplet, and final glassy bead configurations.} |
| 280 |
+ |
\label{fig:cross_sections} |
| 281 |
+ |
\end{figure} |
