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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 gezelter 3242 \includegraphics[width=5in]{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 gezelter 3242 The radius of the spherical region operating under Newtonian dynamics,
72     $r_\textrm{Newton}$ was set to be 4 {\AA} smaller than the original
73     radius ($R$) of the liquid droplet.}
74 chuckv 3222 \label{fig:langevinSketch}
75     \end{figure}
76 gezelter 3230
77 gezelter 3221 The viscosity ($\eta$) can be tuned by comparing the cooling rate that
78     a set of nanoparticles experience with the known cooling rates for
79 gezelter 3230 similar particles obtained via the laser heating experiments.
80 gezelter 3221 Essentially, we tune the solvent viscosity until the thermal decay
81     profile matches a heat-transfer model using reasonable values for the
82     interfacial conductance and the thermal conductivity of the solvent.
83    
84     Cooling rates for the experimentally-observed nanoparticles were
85     calculated from the heat transfer equations for a spherical particle
86 gezelter 3230 embedded in a ambient medium that allows for diffusive heat transport.
87     Following Plech {\it et al.},\cite{plech:195423} we use a heat
88     transfer model that consists of two coupled differential equations
89     in the Laplace domain,
90 chuckv 3208 \begin{eqnarray}
91 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\\
92     \left(\frac{\partial}{\partial r} T_{f}(r,s)\right)_{r=R} +
93     \frac{G}{K}(T_{p}(s)-T_{f}(r,s) = 0
94     \label{eq:heateqn}
95 chuckv 3208 \end{eqnarray}
96 gezelter 3221 where $s$ is the time-conjugate variable in Laplace space. The
97     variables in these equations describe a nanoparticle of radius $R$,
98     mass $M$, and specific heat $c_{p}$ at an initial temperature
99     $T_0$. The surrounding solvent has a thermal profile $T_f(r,t)$,
100     thermal conductivity $K$, density $\rho$, and specific heat $c$. $G$
101     is the interfacial conductance between the nanoparticle and the
102     surrounding solvent, and contains information about heat transfer to
103     the capping agent as well as the direct metal-to-solvent heat loss.
104     The temperature of the nanoparticle as a function of time can then
105     obtained by the inverse Laplace transform,
106 chuckv 3208 \begin{equation}
107 gezelter 3221 T_{p}(t)=\frac{2 k R^2 g^2
108     T_0}{\pi}\int_{0}^{\infty}\frac{\exp(-\kappa u^2
109     t/R^2)u^2}{(u^2(1 + R g) - k R g)^2+(u^3 - k R g u)^2}\mathrm{d}u.
110     \label{eq:laplacetransform}
111 chuckv 3208 \end{equation}
112 gezelter 3221 For simplicity, we have introduced the thermal diffusivity $\kappa =
113     K/(\rho c)$, and defined $k=4\pi R^3 \rho c /(M c_p)$ and $g = G/K$ in
114     Eq. \ref{eq:laplacetransform}.
115 chuckv 3208
116 gezelter 3221 Eq. \ref{eq:laplacetransform} was solved numerically for the Ag-Cu
117     system using mole-fraction weighted values for $c_p$ and $\rho_p$ of
118     0.295 $(\mathrm{J g^{-1} K^{-1}})$ and $9.826\times 10^6$ $(\mathrm{g
119     m^{-3}})$ respectively. Since most of the laser excitation experiments
120     have been done in aqueous solutions, parameters used for the fluid are
121 gezelter 3230 $K$ of $0.6$ $(\mathrm{Wm^{-1}K^{-1}})$, $\rho$ of $1.0\times10^6$
122     $(\mathrm{g m^{-3}})$ and $c$ of $4.184$ $(\mathrm{J g^{-1} K^{-1}})$.
123 chuckv 3208
124 gezelter 3221 Values for the interfacial conductance have been determined by a
125     number of groups for similar nanoparticles and range from a low
126     $87.5\times 10^{6}$ $(\mathrm{Wm^{-2}K^{-1}})$ to $120\times 10^{6}$
127 gezelter 3230 $(\mathrm{Wm^{-2}K^{-1}})$.\cite{hartlandPrv2007} Similarly, Plech
128     {\it et al.} reported a value for the interfacial conductance of
129     $G=105\pm 15$ $(\mathrm{Wm^{-2}K^{-1}})$ and $G=130\pm 15$
130     $(\mathrm{Wm^{-2}K^{-1}})$ for Pt
131     nanoparticles.\cite{plech:195423,PhysRevB.66.224301}
132 gezelter 3221
133     We conducted our simulations at both ends of the range of
134     experimentally-determined values for the interfacial conductance.
135     This allows us to observe both the slowest and fastest heat transfers
136     from the nanoparticle to the solvent that are consistent with
137     experimental observations. For the slowest heat transfer, a value for
138     G of $87.5\times 10^{6}$ $(\mathrm{Wm^{-2}K^{-1}})$ was used and for
139     the fastest heat transfer, a value of $117\times 10^{6}$
140     $(\mathrm{Wm^{-2}K^{-1}})$ was used. Based on calculations we have
141     done using raw data from the Hartland group's thermal half-time
142 gezelter 3230 experiments on Au nanospheres, the true G values are probably in the
143     faster regime: $117\times 10^{6}$ $(\mathrm{Wm^{-2}K^{-1}})$.
144 gezelter 3221
145     The rate of cooling for the nanoparticles in a molecular dynamics
146     simulation can then be tuned by changing the effective solvent
147     viscosity ($\eta$) until the nanoparticle cooling rate matches the
148     cooling rate described by the heat-transfer equations
149 gezelter 3247 (\ref{eq:heateqn}). The effective solvent viscosity (in Pa s) for a G
150     of $87.5\times 10^{6}$ $(\mathrm{Wm^{-2}K^{-1}})$ is $4.2 \times
151     10^{-6}$, $5.0 \times 10^{-6}$, and
152     $5.5 \times 10^{-6}$ for 20 {\AA}, 30 {\AA}, and 40 {\AA} particles, respectively. The
153     effective solvent viscosity (again in Pa s) for an interfacial
154     conductance of $117\times 10^{6}$ $(\mathrm{Wm^{-2}K^{-1}})$ is $5.7
155     \times 10^{-6}$, $7.2 \times 10^{-6}$, and $7.5 \times 10^{-6}$
156     for 20 {\AA}, 30 {\AA} and 40 {\AA} particles. Cooling traces for
157     each particle size are presented in
158 gezelter 3221 Fig. \ref{fig:images_cooling_plot}. It should be noted that the
159     Langevin thermostat produces cooling curves that are consistent with
160     Newtonian (single-exponential) cooling, which cannot match the cooling
161 gezelter 3230 profiles from Eq. \ref{eq:laplacetransform} exactly. Fitting the
162     Langevin cooling profiles to a single-exponential produces
163     $\tau=25.576$ ps, $\tau=43.786$ ps, and $\tau=56.621$ ps for the 20,
164     30 and 40 {\AA} nanoparticles and a G of $87.5\times 10^{6}$
165     $(\mathrm{Wm^{-2}K^{-1}})$. For comparison's sake, similar
166     single-exponential fits with an interfacial conductance of G of
167     $117\times 10^{6}$ $(\mathrm{Wm^{-2}K^{-1}})$ produced a $\tau=13.391$
168     ps, $\tau=30.426$ ps, $\tau=43.857$ ps for the 20, 30 and 40 {\AA}
169     nanoparticles.
170 gezelter 3221
171 chuckv 3213 \begin{figure}[htbp]
172 gezelter 3221 \centering
173 gezelter 3242 \includegraphics[width=5in]{images/cooling_plot.pdf}
174 gezelter 3221 \caption{Thermal cooling curves obtained from the inverse Laplace
175     transform heat model in Eq. \ref{eq:laplacetransform} (solid line) as
176     well as from molecular dynamics simulations (circles). Effective
177 gezelter 3247 solvent viscosities of 4.2-7.5 $\times 10^{-6}$ Pa s (depending on the
178     radius of the particle) give the best fit to the experimental cooling
179     curves. This viscosity suggests that the nanoparticles in these
180     experiments are surrounded by a vapor layer (which is a reasonable
181     assumptions given the initial temperatures of the particles). }
182 gezelter 3221 \label{fig:images_cooling_plot}
183 chuckv 3213 \end{figure}
184 chuckv 3208
185 gezelter 3221 \subsection{Potentials for classical simulations of bimetallic
186     nanoparticles}
187 chuckv 3208
188 gezelter 3221 Several different potential models have been developed that reasonably
189     describe interactions in transition metals. In particular, the
190     Embedded Atom Model ({\sc eam})~\cite{PhysRevB.33.7983} and
191     Sutton-Chen ({\sc sc})~\cite{Chen90} potential have been used to study
192     a wide range of phenomena in both bulk materials and
193     nanoparticles.\cite{Vardeman-II:2001jn,ShibataT._ja026764r} Both
194     potentials are based on a model of a metal which treats the nuclei and
195     core electrons as pseudo-atoms embedded in the electron density due to
196     the valence electrons on all of the other atoms in the system. The
197     {\sc sc} potential has a simple form that closely resembles that of
198     the ubiquitous Lennard Jones potential,
199     \begin{equation}
200     \label{eq:SCP1}
201     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] ,
202     \end{equation}
203     where $V^{pair}_{ij}$ and $\rho_{i}$ are given by
204     \begin{equation}
205     \label{eq:SCP2}
206     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}}.
207     \end{equation}
208     $V^{pair}_{ij}$ is a repulsive pairwise potential that accounts for
209     interactions between the pseudoatom cores. The $\sqrt{\rho_i}$ term in
210     Eq. (\ref{eq:SCP1}) is an attractive many-body potential that models
211     the interactions between the valence electrons and the cores of the
212     pseudo-atoms. $D_{ij}$, $D_{ii}$ set the appropriate overall energy
213     scale, $c_i$ scales the attractive portion of the potential relative
214     to the repulsive interaction and $\alpha_{ij}$ is a length parameter
215     that assures a dimensionless form for $\rho$. These parameters are
216     tuned to various experimental properties such as the density, cohesive
217     energy, and elastic moduli for FCC transition metals. The quantum
218     Sutton-Chen ({\sc q-sc}) formulation matches these properties while
219     including zero-point quantum corrections for different transition
220     metals.\cite{PhysRevB.59.3527} This particular parametarization has
221     been shown to reproduce the experimentally available heat of mixing
222     data for both FCC solid solutions of Ag-Cu and the high-temperature
223     liquid.\cite{sheng:184203} In contrast, the {\sc eam} potential does
224     not reproduce the experimentally observed heat of mixing for the
225     liquid alloy.\cite{MURRAY:1984lr} Combination rules for the alloy were
226     taken to be the arithmatic average of the atomic parameters with the
227     exception of $c_i$ since its values is only dependent on the identity
228     of the atom where the density is evaluated. For the {\sc q-sc}
229     potential, cutoff distances are traditionally taken to be
230     $2\alpha_{ij}$ and include up to the sixth coordination shell in FCC
231     metals.
232 chuckv 3213
233 chuckv 3226 %\subsection{Sampling single-temperature configurations from a cooling
234     %trajectory}
235 chuckv 3213
236 gezelter 3230 To better understand the structural changes occurring in the
237     nanoparticles throughout the cooling trajectory, configurations were
238     sampled at regular intervals during the cooling trajectory. These
239     configurations were then allowed to evolve under NVE dynamics to
240     sample from the proper distribution in phase space. Figure
241     \ref{fig:images_cooling_time_traces} illustrates this sampling.
242 chuckv 3226
243    
244     \begin{figure}[htbp]
245     \centering
246     \includegraphics[height=3in]{images/cooling_time_traces.pdf}
247 gezelter 3230 \caption{Illustrative cooling profile for the 40 {\AA}
248     nanoparticle evolving under stochastic boundary conditions
249     corresponding to $G=$$87.5\times 10^{6}$
250     $(\mathrm{Wm^{-2}K^{-1}})$. At temperatures along the cooling
251     trajectory, configurations were sampled and allowed to evolve in the
252     NVE ensemble. These subsequent trajectories were analyzed for
253     structural features associated with bulk glass formation.}
254 chuckv 3226 \label{fig:images_cooling_time_traces}
255     \end{figure}
256    
257    
258 gezelter 3233 \begin{figure}[htbp]
259     \centering
260 gezelter 3242 \includegraphics[width=5in]{images/cross_section_array.jpg}
261 gezelter 3233 \caption{Cutaway views of 30 \AA\ Ag-Cu nanoparticle structures for
262     random alloy (top) and Cu (core) / Ag (shell) initial conditions
263     (bottom). Shown from left to right are the crystalline, liquid
264     droplet, and final glassy bead configurations.}
265 gezelter 3242 \label{fig:cross_sections}
266 gezelter 3233 \end{figure}