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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 chuckv 3226 $\mathrm{Ag}_6\mathrm{Cu}_4$. 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 gezelter 3221 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 chuckv 3226 Maxwell-Boltzmann distribution at each temperature.
16 gezelter 3221
17     To mimic the effects of the heating due to laser irradiation, the
18     particles were allowed to melt by sampling velocities from the Maxwell
19     Boltzmann distribution at a temperature of 900 K. The particles were
20     run under microcanonical simulation conditions for 1 ns of simualtion
21     time prior to studying the effects of heat transfer to the solvent.
22     In all cases, center of mass translational and rotational motion of
23     the particles were set to zero before any data collection was
24     undertaken. Structural features (pair distribution functions) were
25     used to verify that the particles were indeed liquid droplets before
26     cooling simulations took place.
27    
28     \subsection{Modeling random alloy and core shell particles in solution
29     phase environments}
30    
31     To approximate the effects of rapid heat transfer to the solvent
32     following a heating at the plasmon resonance, we utilized a
33     methodology in which atoms contained in the outer $4$ {\AA} radius of
34     the nanoparticle evolved under Langevin Dynamics with a solvent
35     friction approximating the contribution from the solvent and capping
36     agent. Atoms located in the interior of the nanoparticle evolved
37     under Newtonian dynamics. The set-up of our simulations is nearly
38     identical with the ``stochastic boundary molecular dynamics'' ({\sc
39     sbmd}) method that has seen wide use in the protein simulation
40     community.\cite{BROOKS:1985kx,BROOKS:1983uq,BRUNGER:1984fj} A sketch
41     of this setup can be found in Fig. \ref{fig:langevinSketch}. For a
42     spherical atom of radius $a$, the Langevin frictional forces can be
43     determined by Stokes' law
44     \begin{equation}
45     \mathbf{F}_{\mathrm{frictional}}=6\pi a \eta \mathbf{v}
46 chuckv 3208 \end{equation}
47 gezelter 3221 where $\eta$ is the effective viscosity of the solvent in which the
48     particle is embedded. Due to the presence of the capping agent and
49     the lack of details about the atomic-scale interactions between the
50     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.
53 chuckv 3222 \begin{figure}[htbp]
54     \centering
55     \includegraphics[width=\linewidth]{images/stochbound.pdf}
56     \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.}
57     \label{fig:langevinSketch}
58     \end{figure}
59 gezelter 3221 The viscosity ($\eta$) can be tuned by comparing the cooling rate that
60     a set of nanoparticles experience with the known cooling rates for
61     those particles obtained via the laser heating experiments.
62     Essentially, we tune the solvent viscosity until the thermal decay
63     profile matches a heat-transfer model using reasonable values for the
64     interfacial conductance and the thermal conductivity of the solvent.
65    
66     Cooling rates for the experimentally-observed nanoparticles were
67     calculated from the heat transfer equations for a spherical particle
68     embedded in a ambient medium that allows for diffusive heat
69     transport. The heat transfer model is a set of two coupled
70     differential equations in the Laplace domain,
71 chuckv 3208 \begin{eqnarray}
72 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\\
73     \left(\frac{\partial}{\partial r} T_{f}(r,s)\right)_{r=R} +
74     \frac{G}{K}(T_{p}(s)-T_{f}(r,s) = 0
75     \label{eq:heateqn}
76 chuckv 3208 \end{eqnarray}
77 gezelter 3221 where $s$ is the time-conjugate variable in Laplace space. The
78     variables in these equations describe a nanoparticle of radius $R$,
79     mass $M$, and specific heat $c_{p}$ at an initial temperature
80     $T_0$. The surrounding solvent has a thermal profile $T_f(r,t)$,
81     thermal conductivity $K$, density $\rho$, and specific heat $c$. $G$
82     is the interfacial conductance between the nanoparticle and the
83     surrounding solvent, and contains information about heat transfer to
84     the capping agent as well as the direct metal-to-solvent heat loss.
85     The temperature of the nanoparticle as a function of time can then
86     obtained by the inverse Laplace transform,
87 chuckv 3208 \begin{equation}
88 gezelter 3221 T_{p}(t)=\frac{2 k R^2 g^2
89     T_0}{\pi}\int_{0}^{\infty}\frac{\exp(-\kappa u^2
90     t/R^2)u^2}{(u^2(1 + R g) - k R g)^2+(u^3 - k R g u)^2}\mathrm{d}u.
91     \label{eq:laplacetransform}
92 chuckv 3208 \end{equation}
93 gezelter 3221 For simplicity, we have introduced the thermal diffusivity $\kappa =
94     K/(\rho c)$, and defined $k=4\pi R^3 \rho c /(M c_p)$ and $g = G/K$ in
95     Eq. \ref{eq:laplacetransform}.
96 chuckv 3208
97 gezelter 3221 Eq. \ref{eq:laplacetransform} was solved numerically for the Ag-Cu
98     system using mole-fraction weighted values for $c_p$ and $\rho_p$ of
99     0.295 $(\mathrm{J g^{-1} K^{-1}})$ and $9.826\times 10^6$ $(\mathrm{g
100     m^{-3}})$ respectively. Since most of the laser excitation experiments
101     have been done in aqueous solutions, parameters used for the fluid are
102     $K$ of $0.6$ $(\mathrm{Wm^{-1}K^{-1}})$, $\rho$ of $1.0\times10^6$ $(\mathrm{g
103     m^{-3}})$ and $c$ of $4.184$ $(\mathrm{J g^{-1} K^{-1}})$.
104 chuckv 3208
105 gezelter 3221 Values for the interfacial conductance have been determined by a
106     number of groups for similar nanoparticles and range from a low
107     $87.5\times 10^{6}$ $(\mathrm{Wm^{-2}K^{-1}})$ to $120\times 10^{6}$
108 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
109     $G=130\pm 15$ $(\mathrm{Wm^{-2}K^{-1}})$ for Pt nanoparticles.\cite{plech:195423,PhysRevB.66.224301}
110 gezelter 3221
111     We conducted our simulations at both ends of the range of
112     experimentally-determined values for the interfacial conductance.
113     This allows us to observe both the slowest and fastest heat transfers
114     from the nanoparticle to the solvent that are consistent with
115     experimental observations. For the slowest heat transfer, a value for
116     G of $87.5\times 10^{6}$ $(\mathrm{Wm^{-2}K^{-1}})$ was used and for
117     the fastest heat transfer, a value of $117\times 10^{6}$
118     $(\mathrm{Wm^{-2}K^{-1}})$ was used. Based on calculations we have
119     done using raw data from the Hartland group's thermal half-time
120     experiments on Au nanospheres, we believe that the true G values are
121 chuckv 3222 closer to the faster regime: $117\times 10^{6}$ $(\mathrm{Wm^{-2}K^{-1}})$.
122 gezelter 3221
123 chuckv 3222
124 gezelter 3221 The rate of cooling for the nanoparticles in a molecular dynamics
125     simulation can then be tuned by changing the effective solvent
126     viscosity ($\eta$) until the nanoparticle cooling rate matches the
127     cooling rate described by the heat-transfer equations
128     (\ref{eq:heateqn}). The effective solvent viscosity (in poise) for a G
129     of $87.5\times 10^{6}$ $(\mathrm{Wm^{-2}K^{-1}})$ is 0.17, 0.20, and
130     0.22 for 20 {\AA}, 30 {\AA}, and 40 {\AA} particles, respectively. The
131     effective solvent viscosity (again in poise) for an interfacial
132     conductance of $117\times 10^{6}$ $(\mathrm{Wm^{-2}K^{-1}})$ is 0.23,
133     0.29, and 0.30 for 20 {\AA}, 30 {\AA} and 40 {\AA} particles. Cooling
134     traces for each particle size are presented in
135     Fig. \ref{fig:images_cooling_plot}. It should be noted that the
136     Langevin thermostat produces cooling curves that are consistent with
137     Newtonian (single-exponential) cooling, which cannot match the cooling
138     profiles from Eq. \ref{eq:laplacetransform} exactly.
139    
140 chuckv 3213 \begin{figure}[htbp]
141 gezelter 3221 \centering
142     \includegraphics[width=\linewidth]{images/cooling_plot.pdf}
143     \caption{Thermal cooling curves obtained from the inverse Laplace
144     transform heat model in Eq. \ref{eq:laplacetransform} (solid line) as
145     well as from molecular dynamics simulations (circles). Effective
146     solvent viscosities of 0.23-0.30 poise (depending on the radius of the
147     particle) give the best fit to the experimental cooling curves. Since
148     this viscosity is substantially in excess of the viscosity of liquid
149     water, much of the thermal transfer to the surroundings is probably
150     due to the capping agent.}
151     \label{fig:images_cooling_plot}
152 chuckv 3213 \end{figure}
153 chuckv 3208
154 gezelter 3221 \subsection{Potentials for classical simulations of bimetallic
155     nanoparticles}
156 chuckv 3208
157 gezelter 3221 Several different potential models have been developed that reasonably
158     describe interactions in transition metals. In particular, the
159     Embedded Atom Model ({\sc eam})~\cite{PhysRevB.33.7983} and
160     Sutton-Chen ({\sc sc})~\cite{Chen90} potential have been used to study
161     a wide range of phenomena in both bulk materials and
162     nanoparticles.\cite{Vardeman-II:2001jn,ShibataT._ja026764r} Both
163     potentials are based on a model of a metal which treats the nuclei and
164     core electrons as pseudo-atoms embedded in the electron density due to
165     the valence electrons on all of the other atoms in the system. The
166     {\sc sc} potential has a simple form that closely resembles that of
167     the ubiquitous Lennard Jones potential,
168     \begin{equation}
169     \label{eq:SCP1}
170     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] ,
171     \end{equation}
172     where $V^{pair}_{ij}$ and $\rho_{i}$ are given by
173     \begin{equation}
174     \label{eq:SCP2}
175     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}}.
176     \end{equation}
177     $V^{pair}_{ij}$ is a repulsive pairwise potential that accounts for
178     interactions between the pseudoatom cores. The $\sqrt{\rho_i}$ term in
179     Eq. (\ref{eq:SCP1}) is an attractive many-body potential that models
180     the interactions between the valence electrons and the cores of the
181     pseudo-atoms. $D_{ij}$, $D_{ii}$ set the appropriate overall energy
182     scale, $c_i$ scales the attractive portion of the potential relative
183     to the repulsive interaction and $\alpha_{ij}$ is a length parameter
184     that assures a dimensionless form for $\rho$. These parameters are
185     tuned to various experimental properties such as the density, cohesive
186     energy, and elastic moduli for FCC transition metals. The quantum
187     Sutton-Chen ({\sc q-sc}) formulation matches these properties while
188     including zero-point quantum corrections for different transition
189     metals.\cite{PhysRevB.59.3527} This particular parametarization has
190     been shown to reproduce the experimentally available heat of mixing
191     data for both FCC solid solutions of Ag-Cu and the high-temperature
192     liquid.\cite{sheng:184203} In contrast, the {\sc eam} potential does
193     not reproduce the experimentally observed heat of mixing for the
194     liquid alloy.\cite{MURRAY:1984lr} Combination rules for the alloy were
195     taken to be the arithmatic average of the atomic parameters with the
196     exception of $c_i$ since its values is only dependent on the identity
197     of the atom where the density is evaluated. For the {\sc q-sc}
198     potential, cutoff distances are traditionally taken to be
199     $2\alpha_{ij}$ and include up to the sixth coordination shell in FCC
200     metals.
201 chuckv 3213
202 chuckv 3226 %\subsection{Sampling single-temperature configurations from a cooling
203     %trajectory}
204 chuckv 3213
205 chuckv 3226 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.
206    
207    
208     \begin{figure}[htbp]
209     \centering
210     \includegraphics[height=3in]{images/cooling_time_traces.pdf}
211     \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.}
212     \label{fig:images_cooling_time_traces}
213     \end{figure}
214    
215