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1 gezelter 3221 \section{Computational Methodology}
2     \label{sec:details}
3    
4     \subsection{Initial Geometries and Heating}
5    
6     Cu-core / Ag-shell and random alloy structures were constructed on an
7     underlying FCC lattice (4.09 {\AA}) at the bulk eutectic composition
8     $\mathrm{Ag}_6\mathrm{Cu}_4$. Three different sizes of nanoparticles
9     corresponding to a 20 \AA radius (1961 atoms), 30 {\AA} radius (6603
10     atoms) and 40 {\AA} radius (15683 atoms) were constructed. These
11     initial structures were relaxed to their equilibrium structures at 20
12     K for 20 ps and again at 300 K for 100 ps sampling from a
13     Maxwell-Boltzmann distribution at each temperature.
14    
15     To mimic the effects of the heating due to laser irradiation, the
16     particles were allowed to melt by sampling velocities from the Maxwell
17     Boltzmann distribution at a temperature of 900 K. The particles were
18     run under microcanonical simulation conditions for 1 ns of simualtion
19     time prior to studying the effects of heat transfer to the solvent.
20     In all cases, center of mass translational and rotational motion of
21     the particles were set to zero before any data collection was
22     undertaken. Structural features (pair distribution functions) were
23     used to verify that the particles were indeed liquid droplets before
24     cooling simulations took place.
25    
26     \subsection{Modeling random alloy and core shell particles in solution
27     phase environments}
28    
29     To approximate the effects of rapid heat transfer to the solvent
30     following a heating at the plasmon resonance, we utilized a
31     methodology in which atoms contained in the outer $4$ {\AA} radius of
32     the nanoparticle evolved under Langevin Dynamics with a solvent
33     friction approximating the contribution from the solvent and capping
34     agent. Atoms located in the interior of the nanoparticle evolved
35     under Newtonian dynamics. The set-up of our simulations is nearly
36     identical with the ``stochastic boundary molecular dynamics'' ({\sc
37     sbmd}) method that has seen wide use in the protein simulation
38     community.\cite{BROOKS:1985kx,BROOKS:1983uq,BRUNGER:1984fj} A sketch
39     of this setup can be found in Fig. \ref{fig:langevinSketch}. For a
40     spherical atom of radius $a$, the Langevin frictional forces can be
41     determined by Stokes' law
42     \begin{equation}
43     \mathbf{F}_{\mathrm{frictional}}=6\pi a \eta \mathbf{v}
44 chuckv 3208 \end{equation}
45 gezelter 3221 where $\eta$ is the effective viscosity of the solvent in which the
46     particle is embedded. Due to the presence of the capping agent and
47     the lack of details about the atomic-scale interactions between the
48     metallic atoms and the solvent, the effective viscosity is a
49     essentially a free parameter that must be tuned to give experimentally
50     relevant simulations.
51 chuckv 3222 \begin{figure}[htbp]
52     \centering
53     \includegraphics[width=\linewidth]{images/stochbound.pdf}
54     \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.}
55     \label{fig:langevinSketch}
56     \end{figure}
57 gezelter 3221 The viscosity ($\eta$) can be tuned by comparing the cooling rate that
58     a set of nanoparticles experience with the known cooling rates for
59     those particles obtained via the laser heating experiments.
60     Essentially, we tune the solvent viscosity until the thermal decay
61     profile matches a heat-transfer model using reasonable values for the
62     interfacial conductance and the thermal conductivity of the solvent.
63    
64     Cooling rates for the experimentally-observed nanoparticles were
65     calculated from the heat transfer equations for a spherical particle
66     embedded in a ambient medium that allows for diffusive heat
67     transport. The heat transfer model is a set of two coupled
68     differential equations in the Laplace domain,
69 chuckv 3208 \begin{eqnarray}
70 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\\
71     \left(\frac{\partial}{\partial r} T_{f}(r,s)\right)_{r=R} +
72     \frac{G}{K}(T_{p}(s)-T_{f}(r,s) = 0
73     \label{eq:heateqn}
74 chuckv 3208 \end{eqnarray}
75 gezelter 3221 where $s$ is the time-conjugate variable in Laplace space. The
76     variables in these equations describe a nanoparticle of radius $R$,
77     mass $M$, and specific heat $c_{p}$ at an initial temperature
78     $T_0$. The surrounding solvent has a thermal profile $T_f(r,t)$,
79     thermal conductivity $K$, density $\rho$, and specific heat $c$. $G$
80     is the interfacial conductance between the nanoparticle and the
81     surrounding solvent, and contains information about heat transfer to
82     the capping agent as well as the direct metal-to-solvent heat loss.
83     The temperature of the nanoparticle as a function of time can then
84     obtained by the inverse Laplace transform,
85 chuckv 3208 \begin{equation}
86 gezelter 3221 T_{p}(t)=\frac{2 k R^2 g^2
87     T_0}{\pi}\int_{0}^{\infty}\frac{\exp(-\kappa u^2
88     t/R^2)u^2}{(u^2(1 + R g) - k R g)^2+(u^3 - k R g u)^2}\mathrm{d}u.
89     \label{eq:laplacetransform}
90 chuckv 3208 \end{equation}
91 gezelter 3221 For simplicity, we have introduced the thermal diffusivity $\kappa =
92     K/(\rho c)$, and defined $k=4\pi R^3 \rho c /(M c_p)$ and $g = G/K$ in
93     Eq. \ref{eq:laplacetransform}.
94 chuckv 3208
95 gezelter 3221 Eq. \ref{eq:laplacetransform} was solved numerically for the Ag-Cu
96     system using mole-fraction weighted values for $c_p$ and $\rho_p$ of
97     0.295 $(\mathrm{J g^{-1} K^{-1}})$ and $9.826\times 10^6$ $(\mathrm{g
98     m^{-3}})$ respectively. Since most of the laser excitation experiments
99     have been done in aqueous solutions, parameters used for the fluid are
100     $K$ of $0.6$ $(\mathrm{Wm^{-1}K^{-1}})$, $\rho$ of $1.0\times10^6$ $(\mathrm{g
101     m^{-3}})$ and $c$ of $4.184$ $(\mathrm{J g^{-1} K^{-1}})$.
102 chuckv 3208
103 gezelter 3221 Values for the interfacial conductance have been determined by a
104     number of groups for similar nanoparticles and range from a low
105     $87.5\times 10^{6}$ $(\mathrm{Wm^{-2}K^{-1}})$ to $120\times 10^{6}$
106 chuckv 3222 $(\mathrm{Wm^{-2}K^{-1}})$.\cite{hartlandPrv2007} Plech {\it et al.} reported a value for the interfacial conductance of $G=105\pm 15$ $(\mathrm{Wm^{-2}K^{-1}})$ and
107     $G=130\pm 15$ $(\mathrm{Wm^{-2}K^{-1}})$ for Pt nanoparticles.\cite{plech:195423,PhysRevB.66.224301}
108 gezelter 3221
109     We conducted our simulations at both ends of the range of
110     experimentally-determined values for the interfacial conductance.
111     This allows us to observe both the slowest and fastest heat transfers
112     from the nanoparticle to the solvent that are consistent with
113     experimental observations. For the slowest heat transfer, a value for
114     G of $87.5\times 10^{6}$ $(\mathrm{Wm^{-2}K^{-1}})$ was used and for
115     the fastest heat transfer, a value of $117\times 10^{6}$
116     $(\mathrm{Wm^{-2}K^{-1}})$ was used. Based on calculations we have
117     done using raw data from the Hartland group's thermal half-time
118     experiments on Au nanospheres, we believe that the true G values are
119 chuckv 3222 closer to the faster regime: $117\times 10^{6}$ $(\mathrm{Wm^{-2}K^{-1}})$.
120 gezelter 3221
121 chuckv 3222
122 gezelter 3221 The rate of cooling for the nanoparticles in a molecular dynamics
123     simulation can then be tuned by changing the effective solvent
124     viscosity ($\eta$) until the nanoparticle cooling rate matches the
125     cooling rate described by the heat-transfer equations
126     (\ref{eq:heateqn}). The effective solvent viscosity (in poise) for a G
127     of $87.5\times 10^{6}$ $(\mathrm{Wm^{-2}K^{-1}})$ is 0.17, 0.20, and
128     0.22 for 20 {\AA}, 30 {\AA}, and 40 {\AA} particles, respectively. The
129     effective solvent viscosity (again in poise) for an interfacial
130     conductance of $117\times 10^{6}$ $(\mathrm{Wm^{-2}K^{-1}})$ is 0.23,
131     0.29, and 0.30 for 20 {\AA}, 30 {\AA} and 40 {\AA} particles. Cooling
132     traces for each particle size are presented in
133     Fig. \ref{fig:images_cooling_plot}. It should be noted that the
134     Langevin thermostat produces cooling curves that are consistent with
135     Newtonian (single-exponential) cooling, which cannot match the cooling
136     profiles from Eq. \ref{eq:laplacetransform} exactly.
137    
138 chuckv 3213 \begin{figure}[htbp]
139 gezelter 3221 \centering
140     \includegraphics[width=\linewidth]{images/cooling_plot.pdf}
141     \caption{Thermal cooling curves obtained from the inverse Laplace
142     transform heat model in Eq. \ref{eq:laplacetransform} (solid line) as
143     well as from molecular dynamics simulations (circles). Effective
144     solvent viscosities of 0.23-0.30 poise (depending on the radius of the
145     particle) give the best fit to the experimental cooling curves. Since
146     this viscosity is substantially in excess of the viscosity of liquid
147     water, much of the thermal transfer to the surroundings is probably
148     due to the capping agent.}
149     \label{fig:images_cooling_plot}
150 chuckv 3213 \end{figure}
151 chuckv 3208
152 gezelter 3221 \subsection{Potentials for classical simulations of bimetallic
153     nanoparticles}
154 chuckv 3208
155 gezelter 3221 Several different potential models have been developed that reasonably
156     describe interactions in transition metals. In particular, the
157     Embedded Atom Model ({\sc eam})~\cite{PhysRevB.33.7983} and
158     Sutton-Chen ({\sc sc})~\cite{Chen90} potential have been used to study
159     a wide range of phenomena in both bulk materials and
160     nanoparticles.\cite{Vardeman-II:2001jn,ShibataT._ja026764r} Both
161     potentials are based on a model of a metal which treats the nuclei and
162     core electrons as pseudo-atoms embedded in the electron density due to
163     the valence electrons on all of the other atoms in the system. The
164     {\sc sc} potential has a simple form that closely resembles that of
165     the ubiquitous Lennard Jones potential,
166     \begin{equation}
167     \label{eq:SCP1}
168     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] ,
169     \end{equation}
170     where $V^{pair}_{ij}$ and $\rho_{i}$ are given by
171     \begin{equation}
172     \label{eq:SCP2}
173     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}}.
174     \end{equation}
175     $V^{pair}_{ij}$ is a repulsive pairwise potential that accounts for
176     interactions between the pseudoatom cores. The $\sqrt{\rho_i}$ term in
177     Eq. (\ref{eq:SCP1}) is an attractive many-body potential that models
178     the interactions between the valence electrons and the cores of the
179     pseudo-atoms. $D_{ij}$, $D_{ii}$ set the appropriate overall energy
180     scale, $c_i$ scales the attractive portion of the potential relative
181     to the repulsive interaction and $\alpha_{ij}$ is a length parameter
182     that assures a dimensionless form for $\rho$. These parameters are
183     tuned to various experimental properties such as the density, cohesive
184     energy, and elastic moduli for FCC transition metals. The quantum
185     Sutton-Chen ({\sc q-sc}) formulation matches these properties while
186     including zero-point quantum corrections for different transition
187     metals.\cite{PhysRevB.59.3527} This particular parametarization has
188     been shown to reproduce the experimentally available heat of mixing
189     data for both FCC solid solutions of Ag-Cu and the high-temperature
190     liquid.\cite{sheng:184203} In contrast, the {\sc eam} potential does
191     not reproduce the experimentally observed heat of mixing for the
192     liquid alloy.\cite{MURRAY:1984lr} Combination rules for the alloy were
193     taken to be the arithmatic average of the atomic parameters with the
194     exception of $c_i$ since its values is only dependent on the identity
195     of the atom where the density is evaluated. For the {\sc q-sc}
196     potential, cutoff distances are traditionally taken to be
197     $2\alpha_{ij}$ and include up to the sixth coordination shell in FCC
198     metals.
199 chuckv 3213
200 gezelter 3221 \subsection{Sampling single-temperature configurations from a cooling
201     trajectory}
202 chuckv 3213
203 gezelter 3221 ffdsafjdksalfdsa