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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 gezelter 3259 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 gezelter 3221
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
24     Boltzmann distribution at a temperature of 900 K. The particles were
25     run under microcanonical simulation conditions for 1 ns of simualtion
26     time prior to studying the effects of heat transfer to the solvent.
27     In all cases, center of mass translational and rotational motion of
28     the particles were set to zero before any data collection was
29     undertaken. Structural features (pair distribution functions) were
30     used to verify that the particles were indeed liquid droplets before
31     cooling simulations took place.
32    
33     \subsection{Modeling random alloy and core shell particles in solution
34     phase environments}
35    
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 gezelter 3233 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 gezelter 3221 community.\cite{BROOKS:1985kx,BROOKS:1983uq,BRUNGER:1984fj} A sketch
52 gezelter 3233 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 chuckv 3208 \end{equation}
62 gezelter 3233 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 chuckv 3222 \begin{figure}[htbp]
67     \centering
68 gezelter 3242 \includegraphics[width=5in]{images/stochbound.pdf}
69 gezelter 3230 \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 gezelter 3242 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 chuckv 3222 \label{fig:langevinSketch}
77     \end{figure}
78 gezelter 3230
79 gezelter 3221 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 gezelter 3230 similar particles obtained via the laser heating experiments.
82 gezelter 3221 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 gezelter 3230 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 chuckv 3208 \begin{eqnarray}
93 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\\
94     \left(\frac{\partial}{\partial r} T_{f}(r,s)\right)_{r=R} +
95     \frac{G}{K}(T_{p}(s)-T_{f}(r,s) = 0
96     \label{eq:heateqn}
97 chuckv 3208 \end{eqnarray}
98 gezelter 3221 where $s$ is the time-conjugate variable in Laplace space. The
99     variables in these equations describe a nanoparticle of radius $R$,
100     mass $M$, and specific heat $c_{p}$ at an initial temperature
101     $T_0$. The surrounding solvent has a thermal profile $T_f(r,t)$,
102     thermal conductivity $K$, density $\rho$, and specific heat $c$. $G$
103     is the interfacial conductance between the nanoparticle and the
104     surrounding solvent, and contains information about heat transfer to
105     the capping agent as well as the direct metal-to-solvent heat loss.
106     The temperature of the nanoparticle as a function of time can then
107     obtained by the inverse Laplace transform,
108 chuckv 3208 \begin{equation}
109 gezelter 3221 T_{p}(t)=\frac{2 k R^2 g^2
110     T_0}{\pi}\int_{0}^{\infty}\frac{\exp(-\kappa u^2
111     t/R^2)u^2}{(u^2(1 + R g) - k R g)^2+(u^3 - k R g u)^2}\mathrm{d}u.
112     \label{eq:laplacetransform}
113 chuckv 3208 \end{equation}
114 gezelter 3221 For simplicity, we have introduced the thermal diffusivity $\kappa =
115     K/(\rho c)$, and defined $k=4\pi R^3 \rho c /(M c_p)$ and $g = G/K$ in
116     Eq. \ref{eq:laplacetransform}.
117 chuckv 3208
118 gezelter 3221 Eq. \ref{eq:laplacetransform} was solved numerically for the Ag-Cu
119     system using mole-fraction weighted values for $c_p$ and $\rho_p$ of
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 gezelter 3230 $K$ 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 chuckv 3208
126 gezelter 3221 Values for the interfacial conductance have been determined by a
127     number of groups for similar nanoparticles and range from a low
128 gezelter 3259 $87.5\times 10^{6} (\mathrm{Wm^{-2}K^{-1}})$ 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 gezelter 3221
136     We conducted our simulations at both ends of the range of
137     experimentally-determined values for the interfacial conductance.
138     This allows us to observe both the slowest and fastest heat transfers
139     from the nanoparticle to the solvent that are consistent with
140     experimental observations. For the slowest heat transfer, a value for
141     G of $87.5\times 10^{6}$ $(\mathrm{Wm^{-2}K^{-1}})$ was used and for
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 gezelter 3259 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 gezelter 3221
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 gezelter 3247 (\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 gezelter 3259 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 gezelter 3221 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 gezelter 3230 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 gezelter 3221
185 chuckv 3213 \begin{figure}[htbp]
186 gezelter 3221 \centering
187 gezelter 3242 \includegraphics[width=5in]{images/cooling_plot.pdf}
188 gezelter 3221 \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 gezelter 3247 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 gezelter 3221 \label{fig:images_cooling_plot}
197 chuckv 3213 \end{figure}
198 chuckv 3208
199 gezelter 3221 \subsection{Potentials for classical simulations of bimetallic
200     nanoparticles}
201 chuckv 3208
202 gezelter 3221 Several different potential models have been developed that reasonably
203     describe interactions in transition metals. In particular, the
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 chuckv 3256 nanoparticles.\cite{Vardeman-II:2001jn,ShibataT._ja026764r,Sankaranarayanan:2005lr,Chui:2003fk,Wang:2005qy,Medasani:2007uq} Both
208 gezelter 3221 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
211     {\sc sc} potential has a simple form that closely resembles that of
212     the ubiquitous Lennard Jones potential,
213     \begin{equation}
214     \label{eq:SCP1}
215     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] ,
216     \end{equation}
217     where $V^{pair}_{ij}$ and $\rho_{i}$ are given by
218     \begin{equation}
219     \label{eq:SCP2}
220     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}}.
221     \end{equation}
222     $V^{pair}_{ij}$ is a repulsive pairwise potential that accounts for
223     interactions between the pseudoatom cores. The $\sqrt{\rho_i}$ term in
224     Eq. (\ref{eq:SCP1}) is an attractive many-body potential that models
225     the interactions between the valence electrons and the cores of the
226     pseudo-atoms. $D_{ij}$, $D_{ii}$ set the appropriate overall energy
227     scale, $c_i$ scales the attractive portion of the potential relative
228     to the repulsive interaction and $\alpha_{ij}$ is a length parameter
229     that assures a dimensionless form for $\rho$. These parameters are
230     tuned to various experimental properties such as the density, cohesive
231     energy, and elastic moduli for FCC transition metals. The quantum
232     Sutton-Chen ({\sc q-sc}) formulation matches these properties while
233     including zero-point quantum corrections for different transition
234     metals.\cite{PhysRevB.59.3527} This particular parametarization has
235     been shown to reproduce the experimentally available heat of mixing
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 gezelter 3259 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 chuckv 3213
248 chuckv 3226 %\subsection{Sampling single-temperature configurations from a cooling
249     %trajectory}
250 chuckv 3213
251 gezelter 3230 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 chuckv 3226
258    
259     \begin{figure}[htbp]
260     \centering
261     \includegraphics[height=3in]{images/cooling_time_traces.pdf}
262 gezelter 3230 \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 chuckv 3226 \label{fig:images_cooling_time_traces}
270     \end{figure}
271    
272    
273 gezelter 3233 \begin{figure}[htbp]
274     \centering
275 gezelter 3242 \includegraphics[width=5in]{images/cross_section_array.jpg}
276 gezelter 3233 \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 gezelter 3242 \label{fig:cross_sections}
281 gezelter 3233 \end{figure}