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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 3230 $\mathrm{Ag}_6\mathrm{Cu}_4$. All three compositions 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
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.
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     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
43     community.\cite{BROOKS:1985kx,BROOKS:1983uq,BRUNGER:1984fj} A sketch
44     of this setup can be found in Fig. \ref{fig:langevinSketch}. For a
45     spherical atom of radius $a$, the Langevin frictional forces can be
46     determined by Stokes' law
47     \begin{equation}
48     \mathbf{F}_{\mathrm{frictional}}=6\pi a \eta \mathbf{v}
49 chuckv 3208 \end{equation}
50 gezelter 3221 where $\eta$ is the effective viscosity of the solvent in which the
51     particle is embedded. Due to the presence of the capping agent and
52     the lack of details about the atomic-scale interactions between the
53     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.
56 chuckv 3222 \begin{figure}[htbp]
57     \centering
58     \includegraphics[width=\linewidth]{images/stochbound.pdf}
59 gezelter 3230 \caption{Methodology used to mimic the experimental cooling conditions
60     of a hot nanoparticle surrounded by a solvent. Atoms in the core of
61     the particle evolved under Newtonian dynamics, while atoms that were
62     in the outer skin of the particle evolved under Langevin dynamics.
63     The radial cutoff between the two dynamical regions was set to 4 {\AA}
64     smaller than the original radius of the liquid droplet.}
65 chuckv 3222 \label{fig:langevinSketch}
66     \end{figure}
67 gezelter 3230
68 gezelter 3221 The viscosity ($\eta$) can be tuned by comparing the cooling rate that
69     a set of nanoparticles experience with the known cooling rates for
70 gezelter 3230 similar particles obtained via the laser heating experiments.
71 gezelter 3221 Essentially, we tune the solvent viscosity until the thermal decay
72     profile matches a heat-transfer model using reasonable values for the
73     interfacial conductance and the thermal conductivity of the solvent.
74    
75     Cooling rates for the experimentally-observed nanoparticles were
76     calculated from the heat transfer equations for a spherical particle
77 gezelter 3230 embedded in a ambient medium that allows for diffusive heat transport.
78     Following Plech {\it et al.},\cite{plech:195423} we use a heat
79     transfer model that consists of two coupled differential equations
80     in the Laplace domain,
81 chuckv 3208 \begin{eqnarray}
82 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\\
83     \left(\frac{\partial}{\partial r} T_{f}(r,s)\right)_{r=R} +
84     \frac{G}{K}(T_{p}(s)-T_{f}(r,s) = 0
85     \label{eq:heateqn}
86 chuckv 3208 \end{eqnarray}
87 gezelter 3221 where $s$ is the time-conjugate variable in Laplace space. The
88     variables in these equations describe a nanoparticle of radius $R$,
89     mass $M$, and specific heat $c_{p}$ at an initial temperature
90     $T_0$. The surrounding solvent has a thermal profile $T_f(r,t)$,
91     thermal conductivity $K$, density $\rho$, and specific heat $c$. $G$
92     is the interfacial conductance between the nanoparticle and the
93     surrounding solvent, and contains information about heat transfer to
94     the capping agent as well as the direct metal-to-solvent heat loss.
95     The temperature of the nanoparticle as a function of time can then
96     obtained by the inverse Laplace transform,
97 chuckv 3208 \begin{equation}
98 gezelter 3221 T_{p}(t)=\frac{2 k R^2 g^2
99     T_0}{\pi}\int_{0}^{\infty}\frac{\exp(-\kappa u^2
100     t/R^2)u^2}{(u^2(1 + R g) - k R g)^2+(u^3 - k R g u)^2}\mathrm{d}u.
101     \label{eq:laplacetransform}
102 chuckv 3208 \end{equation}
103 gezelter 3221 For simplicity, we have introduced the thermal diffusivity $\kappa =
104     K/(\rho c)$, and defined $k=4\pi R^3 \rho c /(M c_p)$ and $g = G/K$ in
105     Eq. \ref{eq:laplacetransform}.
106 chuckv 3208
107 gezelter 3221 Eq. \ref{eq:laplacetransform} was solved numerically for the Ag-Cu
108     system using mole-fraction weighted values for $c_p$ and $\rho_p$ of
109     0.295 $(\mathrm{J g^{-1} K^{-1}})$ and $9.826\times 10^6$ $(\mathrm{g
110     m^{-3}})$ respectively. Since most of the laser excitation experiments
111     have been done in aqueous solutions, parameters used for the fluid are
112 gezelter 3230 $K$ of $0.6$ $(\mathrm{Wm^{-1}K^{-1}})$, $\rho$ of $1.0\times10^6$
113     $(\mathrm{g m^{-3}})$ and $c$ of $4.184$ $(\mathrm{J g^{-1} K^{-1}})$.
114 chuckv 3208
115 gezelter 3221 Values for the interfacial conductance have been determined by a
116     number of groups for similar nanoparticles and range from a low
117     $87.5\times 10^{6}$ $(\mathrm{Wm^{-2}K^{-1}})$ to $120\times 10^{6}$
118 gezelter 3230 $(\mathrm{Wm^{-2}K^{-1}})$.\cite{hartlandPrv2007} Similarly, Plech
119     {\it et al.} reported a value for the interfacial conductance of
120     $G=105\pm 15$ $(\mathrm{Wm^{-2}K^{-1}})$ and $G=130\pm 15$
121     $(\mathrm{Wm^{-2}K^{-1}})$ for Pt
122     nanoparticles.\cite{plech:195423,PhysRevB.66.224301}
123 gezelter 3221
124     We conducted our simulations at both ends of the range of
125     experimentally-determined values for the interfacial conductance.
126     This allows us to observe both the slowest and fastest heat transfers
127     from the nanoparticle to the solvent that are consistent with
128     experimental observations. For the slowest heat transfer, a value for
129     G of $87.5\times 10^{6}$ $(\mathrm{Wm^{-2}K^{-1}})$ was used and for
130     the fastest heat transfer, a value of $117\times 10^{6}$
131     $(\mathrm{Wm^{-2}K^{-1}})$ was used. Based on calculations we have
132     done using raw data from the Hartland group's thermal half-time
133 gezelter 3230 experiments on Au nanospheres, the true G values are probably in the
134     faster regime: $117\times 10^{6}$ $(\mathrm{Wm^{-2}K^{-1}})$.
135 gezelter 3221
136     The rate of cooling for the nanoparticles in a molecular dynamics
137     simulation can then be tuned by changing the effective solvent
138     viscosity ($\eta$) until the nanoparticle cooling rate matches the
139     cooling rate described by the heat-transfer equations
140     (\ref{eq:heateqn}). The effective solvent viscosity (in poise) for a G
141     of $87.5\times 10^{6}$ $(\mathrm{Wm^{-2}K^{-1}})$ is 0.17, 0.20, and
142     0.22 for 20 {\AA}, 30 {\AA}, and 40 {\AA} particles, respectively. The
143     effective solvent viscosity (again in poise) for an interfacial
144     conductance of $117\times 10^{6}$ $(\mathrm{Wm^{-2}K^{-1}})$ is 0.23,
145     0.29, and 0.30 for 20 {\AA}, 30 {\AA} and 40 {\AA} particles. Cooling
146     traces for each particle size are presented in
147     Fig. \ref{fig:images_cooling_plot}. It should be noted that the
148     Langevin thermostat produces cooling curves that are consistent with
149     Newtonian (single-exponential) cooling, which cannot match the cooling
150 gezelter 3230 profiles from Eq. \ref{eq:laplacetransform} exactly. Fitting the
151     Langevin cooling profiles to a single-exponential produces
152     $\tau=25.576$ ps, $\tau=43.786$ ps, and $\tau=56.621$ ps for the 20,
153     30 and 40 {\AA} nanoparticles and a G of $87.5\times 10^{6}$
154     $(\mathrm{Wm^{-2}K^{-1}})$. For comparison's sake, similar
155     single-exponential fits with an interfacial conductance of G of
156     $117\times 10^{6}$ $(\mathrm{Wm^{-2}K^{-1}})$ produced a $\tau=13.391$
157     ps, $\tau=30.426$ ps, $\tau=43.857$ ps for the 20, 30 and 40 {\AA}
158     nanoparticles.
159 gezelter 3221
160 chuckv 3213 \begin{figure}[htbp]
161 gezelter 3221 \centering
162     \includegraphics[width=\linewidth]{images/cooling_plot.pdf}
163     \caption{Thermal cooling curves obtained from the inverse Laplace
164     transform heat model in Eq. \ref{eq:laplacetransform} (solid line) as
165     well as from molecular dynamics simulations (circles). Effective
166     solvent viscosities of 0.23-0.30 poise (depending on the radius of the
167 gezelter 3230 particle) give the best fit to the experimental cooling curves.
168     %Since
169     %this viscosity is substantially in excess of the viscosity of liquid
170     %water, much of the thermal transfer to the surroundings is probably
171     %due to the capping agent.
172     }
173 gezelter 3221 \label{fig:images_cooling_plot}
174 chuckv 3213 \end{figure}
175 chuckv 3208
176 gezelter 3221 \subsection{Potentials for classical simulations of bimetallic
177     nanoparticles}
178 chuckv 3208
179 gezelter 3221 Several different potential models have been developed that reasonably
180     describe interactions in transition metals. In particular, the
181     Embedded Atom Model ({\sc eam})~\cite{PhysRevB.33.7983} and
182     Sutton-Chen ({\sc sc})~\cite{Chen90} potential have been used to study
183     a wide range of phenomena in both bulk materials and
184     nanoparticles.\cite{Vardeman-II:2001jn,ShibataT._ja026764r} Both
185     potentials are based on a model of a metal which treats the nuclei and
186     core electrons as pseudo-atoms embedded in the electron density due to
187     the valence electrons on all of the other atoms in the system. The
188     {\sc sc} potential has a simple form that closely resembles that of
189     the ubiquitous Lennard Jones potential,
190     \begin{equation}
191     \label{eq:SCP1}
192     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] ,
193     \end{equation}
194     where $V^{pair}_{ij}$ and $\rho_{i}$ are given by
195     \begin{equation}
196     \label{eq:SCP2}
197     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}}.
198     \end{equation}
199     $V^{pair}_{ij}$ is a repulsive pairwise potential that accounts for
200     interactions between the pseudoatom cores. The $\sqrt{\rho_i}$ term in
201     Eq. (\ref{eq:SCP1}) is an attractive many-body potential that models
202     the interactions between the valence electrons and the cores of the
203     pseudo-atoms. $D_{ij}$, $D_{ii}$ set the appropriate overall energy
204     scale, $c_i$ scales the attractive portion of the potential relative
205     to the repulsive interaction and $\alpha_{ij}$ is a length parameter
206     that assures a dimensionless form for $\rho$. These parameters are
207     tuned to various experimental properties such as the density, cohesive
208     energy, and elastic moduli for FCC transition metals. The quantum
209     Sutton-Chen ({\sc q-sc}) formulation matches these properties while
210     including zero-point quantum corrections for different transition
211     metals.\cite{PhysRevB.59.3527} This particular parametarization has
212     been shown to reproduce the experimentally available heat of mixing
213     data for both FCC solid solutions of Ag-Cu and the high-temperature
214     liquid.\cite{sheng:184203} In contrast, the {\sc eam} potential does
215     not reproduce the experimentally observed heat of mixing for the
216     liquid alloy.\cite{MURRAY:1984lr} Combination rules for the alloy were
217     taken to be the arithmatic average of the atomic parameters with the
218     exception of $c_i$ since its values is only dependent on the identity
219     of the atom where the density is evaluated. For the {\sc q-sc}
220     potential, cutoff distances are traditionally taken to be
221     $2\alpha_{ij}$ and include up to the sixth coordination shell in FCC
222     metals.
223 chuckv 3213
224 chuckv 3226 %\subsection{Sampling single-temperature configurations from a cooling
225     %trajectory}
226 chuckv 3213
227 gezelter 3230 To better understand the structural changes occurring in the
228     nanoparticles throughout the cooling trajectory, configurations were
229     sampled at regular intervals during the cooling trajectory. These
230     configurations were then allowed to evolve under NVE dynamics to
231     sample from the proper distribution in phase space. Figure
232     \ref{fig:images_cooling_time_traces} illustrates this sampling.
233 chuckv 3226
234    
235     \begin{figure}[htbp]
236     \centering
237     \includegraphics[height=3in]{images/cooling_time_traces.pdf}
238 gezelter 3230 \caption{Illustrative cooling profile for the 40 {\AA}
239     nanoparticle evolving under stochastic boundary conditions
240     corresponding to $G=$$87.5\times 10^{6}$
241     $(\mathrm{Wm^{-2}K^{-1}})$. At temperatures along the cooling
242     trajectory, configurations were sampled and allowed to evolve in the
243     NVE ensemble. These subsequent trajectories were analyzed for
244     structural features associated with bulk glass formation.}
245 chuckv 3226 \label{fig:images_cooling_time_traces}
246     \end{figure}
247    
248