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1 chuckv 3226 %!TEX root = /Users/charles/Desktop/nanoglass/nanoglass.tex
2    
3 gezelter 3221 \section{Computational Methodology}
4     \label{sec:details}
5    
6     \subsection{Initial Geometries and Heating}
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 gezelter 3233 $\mathrm{Ag}_6\mathrm{Cu}_4$. Both initial geometries were considered
11 gezelter 3230 although experimental results suggest that the random structure is the
12 gezelter 3233 most likely structure to be found following
13     synthesis.\cite{Jiang:2005lr,gonzalo:5163} Three different sizes of
14 gezelter 3230 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.
19 gezelter 3221
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
22     Boltzmann distribution at a temperature of 900 K. The particles were
23     run under microcanonical simulation conditions for 1 ns of simualtion
24     time prior to studying the effects of heat transfer to the solvent.
25     In all cases, center of mass translational and rotational motion of
26     the particles were set to zero before any data collection was
27     undertaken. Structural features (pair distribution functions) were
28     used to verify that the particles were indeed liquid droplets before
29     cooling simulations took place.
30    
31     \subsection{Modeling random alloy and core shell particles in solution
32     phase environments}
33    
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 gezelter 3233 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 gezelter 3221 community.\cite{BROOKS:1985kx,BROOKS:1983uq,BRUNGER:1984fj} A sketch
50 gezelter 3233 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 chuckv 3208 \end{equation}
60 gezelter 3233 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 chuckv 3222 \begin{figure}[htbp]
65     \centering
66     \includegraphics[width=\linewidth]{images/stochbound.pdf}
67 gezelter 3230 \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.}
73 chuckv 3222 \label{fig:langevinSketch}
74     \end{figure}
75 gezelter 3230
76 gezelter 3221 The viscosity ($\eta$) can be tuned by comparing the cooling rate that
77     a set of nanoparticles experience with the known cooling rates for
78 gezelter 3230 similar particles obtained via the laser heating experiments.
79 gezelter 3221 Essentially, we tune the solvent viscosity until the thermal decay
80     profile matches a heat-transfer model using reasonable values for the
81     interfacial conductance and the thermal conductivity of the solvent.
82    
83     Cooling rates for the experimentally-observed nanoparticles were
84     calculated from the heat transfer equations for a spherical particle
85 gezelter 3230 embedded in a ambient medium that allows for diffusive heat transport.
86     Following Plech {\it et al.},\cite{plech:195423} we use a heat
87     transfer model that consists of two coupled differential equations
88     in the Laplace domain,
89 chuckv 3208 \begin{eqnarray}
90 gezelter 3221 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\\
91     \left(\frac{\partial}{\partial r} T_{f}(r,s)\right)_{r=R} +
92     \frac{G}{K}(T_{p}(s)-T_{f}(r,s) = 0
93     \label{eq:heateqn}
94 chuckv 3208 \end{eqnarray}
95 gezelter 3221 where $s$ is the time-conjugate variable in Laplace space. The
96     variables in these equations describe a nanoparticle of radius $R$,
97     mass $M$, and specific heat $c_{p}$ at an initial temperature
98     $T_0$. The surrounding solvent has a thermal profile $T_f(r,t)$,
99     thermal conductivity $K$, density $\rho$, and specific heat $c$. $G$
100     is the interfacial conductance between the nanoparticle and the
101     surrounding solvent, and contains information about heat transfer to
102     the capping agent as well as the direct metal-to-solvent heat loss.
103     The temperature of the nanoparticle as a function of time can then
104     obtained by the inverse Laplace transform,
105 chuckv 3208 \begin{equation}
106 gezelter 3221 T_{p}(t)=\frac{2 k R^2 g^2
107     T_0}{\pi}\int_{0}^{\infty}\frac{\exp(-\kappa u^2
108     t/R^2)u^2}{(u^2(1 + R g) - k R g)^2+(u^3 - k R g u)^2}\mathrm{d}u.
109     \label{eq:laplacetransform}
110 chuckv 3208 \end{equation}
111 gezelter 3221 For simplicity, we have introduced the thermal diffusivity $\kappa =
112     K/(\rho c)$, and defined $k=4\pi R^3 \rho c /(M c_p)$ and $g = G/K$ in
113     Eq. \ref{eq:laplacetransform}.
114 chuckv 3208
115 gezelter 3221 Eq. \ref{eq:laplacetransform} was solved numerically for the Ag-Cu
116     system using mole-fraction weighted values for $c_p$ and $\rho_p$ of
117     0.295 $(\mathrm{J g^{-1} K^{-1}})$ and $9.826\times 10^6$ $(\mathrm{g
118     m^{-3}})$ respectively. Since most of the laser excitation experiments
119     have been done in aqueous solutions, parameters used for the fluid are
120 gezelter 3230 $K$ of $0.6$ $(\mathrm{Wm^{-1}K^{-1}})$, $\rho$ of $1.0\times10^6$
121     $(\mathrm{g m^{-3}})$ and $c$ of $4.184$ $(\mathrm{J g^{-1} K^{-1}})$.
122 chuckv 3208
123 gezelter 3221 Values for the interfacial conductance have been determined by a
124     number of groups for similar nanoparticles and range from a low
125     $87.5\times 10^{6}$ $(\mathrm{Wm^{-2}K^{-1}})$ to $120\times 10^{6}$
126 gezelter 3230 $(\mathrm{Wm^{-2}K^{-1}})$.\cite{hartlandPrv2007} Similarly, Plech
127     {\it et al.} reported a value for the interfacial conductance of
128     $G=105\pm 15$ $(\mathrm{Wm^{-2}K^{-1}})$ and $G=130\pm 15$
129     $(\mathrm{Wm^{-2}K^{-1}})$ for Pt
130     nanoparticles.\cite{plech:195423,PhysRevB.66.224301}
131 gezelter 3221
132     We conducted our simulations at both ends of the range of
133     experimentally-determined values for the interfacial conductance.
134     This allows us to observe both the slowest and fastest heat transfers
135     from the nanoparticle to the solvent that are consistent with
136     experimental observations. For the slowest heat transfer, a value for
137     G of $87.5\times 10^{6}$ $(\mathrm{Wm^{-2}K^{-1}})$ was used and for
138     the fastest heat transfer, a value of $117\times 10^{6}$
139     $(\mathrm{Wm^{-2}K^{-1}})$ was used. Based on calculations we have
140     done using raw data from the Hartland group's thermal half-time
141 gezelter 3230 experiments on Au nanospheres, the true G values are probably in the
142     faster regime: $117\times 10^{6}$ $(\mathrm{Wm^{-2}K^{-1}})$.
143 gezelter 3221
144     The rate of cooling for the nanoparticles in a molecular dynamics
145     simulation can then be tuned by changing the effective solvent
146     viscosity ($\eta$) until the nanoparticle cooling rate matches the
147     cooling rate described by the heat-transfer equations
148     (\ref{eq:heateqn}). The effective solvent viscosity (in poise) for a G
149     of $87.5\times 10^{6}$ $(\mathrm{Wm^{-2}K^{-1}})$ is 0.17, 0.20, and
150     0.22 for 20 {\AA}, 30 {\AA}, and 40 {\AA} particles, respectively. The
151     effective solvent viscosity (again in poise) for an interfacial
152     conductance of $117\times 10^{6}$ $(\mathrm{Wm^{-2}K^{-1}})$ is 0.23,
153     0.29, and 0.30 for 20 {\AA}, 30 {\AA} and 40 {\AA} particles. Cooling
154     traces for each particle size are presented in
155     Fig. \ref{fig:images_cooling_plot}. It should be noted that the
156     Langevin thermostat produces cooling curves that are consistent with
157     Newtonian (single-exponential) cooling, which cannot match the cooling
158 gezelter 3230 profiles from Eq. \ref{eq:laplacetransform} exactly. Fitting the
159     Langevin cooling profiles to a single-exponential produces
160     $\tau=25.576$ ps, $\tau=43.786$ ps, and $\tau=56.621$ ps for the 20,
161     30 and 40 {\AA} nanoparticles and a G of $87.5\times 10^{6}$
162     $(\mathrm{Wm^{-2}K^{-1}})$. For comparison's sake, similar
163     single-exponential fits with an interfacial conductance of G of
164     $117\times 10^{6}$ $(\mathrm{Wm^{-2}K^{-1}})$ produced a $\tau=13.391$
165     ps, $\tau=30.426$ ps, $\tau=43.857$ ps for the 20, 30 and 40 {\AA}
166     nanoparticles.
167 gezelter 3221
168 chuckv 3213 \begin{figure}[htbp]
169 gezelter 3221 \centering
170     \includegraphics[width=\linewidth]{images/cooling_plot.pdf}
171     \caption{Thermal cooling curves obtained from the inverse Laplace
172     transform heat model in Eq. \ref{eq:laplacetransform} (solid line) as
173     well as from molecular dynamics simulations (circles). Effective
174     solvent viscosities of 0.23-0.30 poise (depending on the radius of the
175 gezelter 3230 particle) give the best fit to the experimental cooling curves.
176     %Since
177     %this viscosity is substantially in excess of the viscosity of liquid
178     %water, much of the thermal transfer to the surroundings is probably
179     %due to the capping agent.
180     }
181 gezelter 3221 \label{fig:images_cooling_plot}
182 chuckv 3213 \end{figure}
183 chuckv 3208
184 gezelter 3221 \subsection{Potentials for classical simulations of bimetallic
185     nanoparticles}
186 chuckv 3208
187 gezelter 3221 Several different potential models have been developed that reasonably
188     describe interactions in transition metals. In particular, the
189     Embedded Atom Model ({\sc eam})~\cite{PhysRevB.33.7983} and
190     Sutton-Chen ({\sc sc})~\cite{Chen90} potential have been used to study
191     a wide range of phenomena in both bulk materials and
192     nanoparticles.\cite{Vardeman-II:2001jn,ShibataT._ja026764r} Both
193     potentials are based on a model of a metal which treats the nuclei and
194     core electrons as pseudo-atoms embedded in the electron density due to
195     the valence electrons on all of the other atoms in the system. The
196     {\sc sc} potential has a simple form that closely resembles that of
197     the ubiquitous Lennard Jones potential,
198     \begin{equation}
199     \label{eq:SCP1}
200     U_{tot}=\sum _{i}\left[ \frac{1}{2}\sum _{j\neq i}D_{ij}V^{pair}_{ij}(r_{ij})-c_{i}D_{ii}\sqrt{\rho_{i}}\right] ,
201     \end{equation}
202     where $V^{pair}_{ij}$ and $\rho_{i}$ are given by
203     \begin{equation}
204     \label{eq:SCP2}
205     V^{pair}_{ij}(r)=\left( \frac{\alpha_{ij}}{r_{ij}}\right)^{n_{ij}}, \rho_{i}=\sum_{j\neq i}\left( \frac{\alpha_{ij}}{r_{ij}}\right) ^{m_{ij}}.
206     \end{equation}
207     $V^{pair}_{ij}$ is a repulsive pairwise potential that accounts for
208     interactions between the pseudoatom cores. The $\sqrt{\rho_i}$ term in
209     Eq. (\ref{eq:SCP1}) is an attractive many-body potential that models
210     the interactions between the valence electrons and the cores of the
211     pseudo-atoms. $D_{ij}$, $D_{ii}$ set the appropriate overall energy
212     scale, $c_i$ scales the attractive portion of the potential relative
213     to the repulsive interaction and $\alpha_{ij}$ is a length parameter
214     that assures a dimensionless form for $\rho$. These parameters are
215     tuned to various experimental properties such as the density, cohesive
216     energy, and elastic moduli for FCC transition metals. The quantum
217     Sutton-Chen ({\sc q-sc}) formulation matches these properties while
218     including zero-point quantum corrections for different transition
219     metals.\cite{PhysRevB.59.3527} This particular parametarization has
220     been shown to reproduce the experimentally available heat of mixing
221     data for both FCC solid solutions of Ag-Cu and the high-temperature
222     liquid.\cite{sheng:184203} In contrast, the {\sc eam} potential does
223     not reproduce the experimentally observed heat of mixing for the
224     liquid alloy.\cite{MURRAY:1984lr} Combination rules for the alloy were
225     taken to be the arithmatic average of the atomic parameters with the
226     exception of $c_i$ since its values is only dependent on the identity
227     of the atom where the density is evaluated. For the {\sc q-sc}
228     potential, cutoff distances are traditionally taken to be
229     $2\alpha_{ij}$ and include up to the sixth coordination shell in FCC
230     metals.
231 chuckv 3213
232 chuckv 3226 %\subsection{Sampling single-temperature configurations from a cooling
233     %trajectory}
234 chuckv 3213
235 gezelter 3230 To better understand the structural changes occurring in the
236     nanoparticles throughout the cooling trajectory, configurations were
237     sampled at regular intervals during the cooling trajectory. These
238     configurations were then allowed to evolve under NVE dynamics to
239     sample from the proper distribution in phase space. Figure
240     \ref{fig:images_cooling_time_traces} illustrates this sampling.
241 chuckv 3226
242    
243     \begin{figure}[htbp]
244     \centering
245     \includegraphics[height=3in]{images/cooling_time_traces.pdf}
246 gezelter 3230 \caption{Illustrative cooling profile for the 40 {\AA}
247     nanoparticle evolving under stochastic boundary conditions
248     corresponding to $G=$$87.5\times 10^{6}$
249     $(\mathrm{Wm^{-2}K^{-1}})$. At temperatures along the cooling
250     trajectory, configurations were sampled and allowed to evolve in the
251     NVE ensemble. These subsequent trajectories were analyzed for
252     structural features associated with bulk glass formation.}
253 chuckv 3226 \label{fig:images_cooling_time_traces}
254     \end{figure}
255    
256    
257 gezelter 3233 \begin{figure}[htbp]
258     \centering
259     \includegraphics[width=\linewidth]{images/cross_section_array.jpg}
260     \caption{Cutaway views of 30 \AA\ Ag-Cu nanoparticle structures for
261     random alloy (top) and Cu (core) / Ag (shell) initial conditions
262     (bottom). Shown from left to right are the crystalline, liquid
263     droplet, and final glassy bead configurations.}
264     \label{fig:q6}
265     \end{figure}