trunk/nanoglass/experimental.tex

\section{Computational Methodology} 
\label{sec:details} 

\subsection{Initial Geometries and Heating} 

Cu-core / Ag-shell and random alloy structures were constructed on an
underlying FCC lattice (4.09 {\AA}) at the bulk eutectic composition
$\mathrm{Ag}_6\mathrm{Cu}_4$.  Three different sizes of nanoparticles
corresponding to a 20 \AA radius (1961 atoms), 30 {\AA} radius (6603
atoms) and 40 {\AA} radius (15683 atoms) were constructed.  These
initial structures were relaxed to their equilibrium structures at 20
K for 20 ps and again at 300 K for 100 ps sampling from a
Maxwell-Boltzmann distribution at each temperature.

To mimic the effects of the heating due to laser irradiation, the
particles were allowed to melt by sampling velocities from the Maxwell
Boltzmann distribution at a temperature of 900 K.  The particles were
run under microcanonical simulation conditions for 1 ns of simualtion
time prior to studying the effects of heat transfer to the solvent.
In all cases, center of mass translational and rotational motion of
the particles were set to zero before any data collection was
undertaken.  Structural features (pair distribution functions) were
used to verify that the particles were indeed liquid droplets before
cooling simulations took place.

\subsection{Modeling random alloy and core shell particles in solution
phase environments}

To approximate the effects of rapid heat transfer to the solvent
following a heating at the plasmon resonance, we utilized a
methodology in which atoms contained in the outer $4$ {\AA} radius of
the nanoparticle evolved under Langevin Dynamics with a solvent
friction approximating the contribution from the solvent and capping
agent.  Atoms located in the interior of the nanoparticle evolved
under Newtonian dynamics.  The set-up of our simulations is nearly
identical with the ``stochastic boundary molecular dynamics'' ({\sc
sbmd}) method that has seen wide use in the protein simulation
community.\cite{BROOKS:1985kx,BROOKS:1983uq,BRUNGER:1984fj} A sketch
of this setup can be found in Fig. \ref{fig:langevinSketch}.  For a
spherical atom of radius $a$, the Langevin frictional forces can be
determined by Stokes' law
\begin{equation} 
\mathbf{F}_{\mathrm{frictional}}=6\pi a \eta \mathbf{v}
\end{equation}
where $\eta$ is the effective viscosity of the solvent in which the
particle is embedded.  Due to the presence of the capping agent and
the lack of details about the atomic-scale interactions between the
metallic atoms and the solvent, the effective viscosity is a
essentially a free parameter that must be tuned to give experimentally
relevant simulations.

The viscosity ($\eta$) can be tuned by comparing the cooling rate that
a set of nanoparticles experience with the known cooling rates for
those particles obtained via the laser heating experiments.
Essentially, we tune the solvent viscosity until the thermal decay
profile matches a heat-transfer model using reasonable values for the
interfacial conductance and the thermal conductivity of the solvent.

Cooling rates for the experimentally-observed nanoparticles were
calculated from the heat transfer equations for a spherical particle
embedded in a ambient medium that allows for diffusive heat
transport. The heat transfer model is a set of two coupled
differential equations in the Laplace domain,
\begin{eqnarray}
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\\
\left(\frac{\partial}{\partial r} T_{f}(r,s)\right)_{r=R} +
\frac{G}{K}(T_{p}(s)-T_{f}(r,s) = 0 
\label{eq:heateqn} 
\end{eqnarray}
where $s$ is the time-conjugate variable in Laplace space. The
variables in these equations describe a nanoparticle of radius $R$,
mass $M$, and specific heat $c_{p}$ at an initial temperature
$T_0$. The surrounding solvent has a thermal profile $T_f(r,t)$,
thermal conductivity $K$, density $\rho$, and specific heat $c$. $G$
is the interfacial conductance between the nanoparticle and the
surrounding solvent, and contains information about heat transfer to
the capping agent as well as the direct metal-to-solvent heat loss.
The temperature of the nanoparticle as a function of time can then
obtained by the inverse Laplace transform,
\begin{equation}
T_{p}(t)=\frac{2 k R^2 g^2
T_0}{\pi}\int_{0}^{\infty}\frac{\exp(-\kappa u^2
t/R^2)u^2}{(u^2(1 + R g) - k R g)^2+(u^3 - k R g u)^2}\mathrm{d}u. 
\label{eq:laplacetransform}
\end{equation}
For simplicity, we have introduced the thermal diffusivity $\kappa =
K/(\rho c)$,  and defined $k=4\pi R^3 \rho c /(M c_p)$ and $g = G/K$ in
Eq. \ref{eq:laplacetransform}.

Eq. \ref{eq:laplacetransform} was solved numerically for the Ag-Cu
system using mole-fraction weighted values for $c_p$ and $\rho_p$ of
0.295 $(\mathrm{J g^{-1} K^{-1}})$ and $9.826\times 10^6$ $(\mathrm{g
m^{-3}})$ respectively. Since most of the laser excitation experiments
have been done in aqueous solutions, parameters used for the fluid are
$K$ of $0.6$  $(\mathrm{Wm^{-1}K^{-1}})$, $\rho$ of $1.0\times10^6$ $(\mathrm{g
m^{-3}})$ and $c$ of $4.184$ $(\mathrm{J g^{-1} K^{-1}})$. 

Values for the interfacial conductance have been determined by a
number of groups for similar nanoparticles and range from a low
$87.5\times 10^{6}$ $(\mathrm{Wm^{-2}K^{-1}})$ to $120\times 10^{6}$
$(\mathrm{Wm^{-2}K^{-1}})$.\cite{XXX}

We conducted our simulations at both ends of the range of
experimentally-determined values for the interfacial conductance.
This allows us to observe both the slowest and fastest heat transfers
from the nanoparticle to the solvent that are consistent with
experimental observations.  For the slowest heat transfer, a value for
G of $87.5\times 10^{6}$ $(\mathrm{Wm^{-2}K^{-1}})$ was used and for
the fastest heat transfer, a value of $117\times 10^{6}$
$(\mathrm{Wm^{-2}K^{-1}})$ was used.  Based on calculations we have
done using raw data from the Hartland group's thermal half-time
experiments on Au nanospheres, we believe that the true G values are
closer to the faster regime: $117\times 10^{6}$
$(\mathrm{Wm^{-2}K^{-1}})$.

The rate of cooling for the nanoparticles in a molecular dynamics
simulation can then be tuned by changing the effective solvent
viscosity ($\eta$) until the nanoparticle cooling rate matches the
cooling rate described by the heat-transfer equations
(\ref{eq:heateqn}). The effective solvent viscosity (in poise) for a G
of $87.5\times 10^{6}$ $(\mathrm{Wm^{-2}K^{-1}})$ is 0.17, 0.20, and
0.22 for 20 {\AA}, 30 {\AA}, and 40 {\AA} particles, respectively. The
effective solvent viscosity (again in poise) for an interfacial
conductance of $117\times 10^{6}$ $(\mathrm{Wm^{-2}K^{-1}})$ is 0.23,
0.29, and 0.30 for 20 {\AA}, 30 {\AA} and 40 {\AA} particles.  Cooling
traces for each particle size are presented in
Fig. \ref{fig:images_cooling_plot}. It should be noted that the
Langevin thermostat produces cooling curves that are consistent with
Newtonian (single-exponential) cooling, which cannot match the cooling
profiles from Eq. \ref{eq:laplacetransform} exactly.

\begin{figure}[htbp]
\centering
\includegraphics[width=\linewidth]{images/cooling_plot.pdf}
\caption{Thermal cooling curves obtained from the inverse Laplace
transform heat model in Eq. \ref{eq:laplacetransform} (solid line) as
well as from molecular dynamics simulations (circles).  Effective
solvent viscosities of 0.23-0.30 poise (depending on the radius of the
particle) give the best fit to the experimental cooling curves.  Since
this viscosity is substantially in excess of the viscosity of liquid
water, much of the thermal transfer to the surroundings is probably
due to the capping agent.}
\label{fig:images_cooling_plot}
\end{figure}

\subsection{Potentials for classical simulations of bimetallic
nanoparticles}

Several different potential models have been developed that reasonably
describe interactions in transition metals. In particular, the
Embedded Atom Model ({\sc eam})~\cite{PhysRevB.33.7983} and
Sutton-Chen ({\sc sc})~\cite{Chen90} potential have been used to study
a wide range of phenomena in both bulk materials and
nanoparticles.\cite{Vardeman-II:2001jn,ShibataT._ja026764r} Both
potentials are based on a model of a metal which treats the nuclei and
core electrons as pseudo-atoms embedded in the electron density due to
the valence electrons on all of the other atoms in the system. The
{\sc sc} potential has a simple form that closely resembles that of
the ubiquitous Lennard Jones potential,
\begin{equation}
\label{eq:SCP1} 
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] , 
\end{equation}
where $V^{pair}_{ij}$ and $\rho_{i}$ are given by 
\begin{equation}
\label{eq:SCP2} 
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}}. 
\end{equation}
$V^{pair}_{ij}$ is a repulsive pairwise potential that accounts for
interactions between the pseudoatom cores. The $\sqrt{\rho_i}$ term in
Eq. (\ref{eq:SCP1}) is an attractive many-body potential that models
the interactions between the valence electrons and the cores of the
pseudo-atoms. $D_{ij}$, $D_{ii}$ set the appropriate overall energy
scale, $c_i$ scales the attractive portion of the potential relative
to the repulsive interaction and $\alpha_{ij}$ is a length parameter
that assures a dimensionless form for $\rho$. These parameters are
tuned to various experimental properties such as the density, cohesive
energy, and elastic moduli for FCC transition metals. The quantum
Sutton-Chen ({\sc q-sc}) formulation matches these properties while
including zero-point quantum corrections for different transition
metals.\cite{PhysRevB.59.3527} This particular parametarization has
been shown to reproduce the experimentally available heat of mixing
data for both FCC solid solutions of Ag-Cu and the high-temperature
liquid.\cite{sheng:184203} In contrast, the {\sc eam} potential does
not reproduce the experimentally observed heat of mixing for the
liquid alloy.\cite{MURRAY:1984lr} Combination rules for the alloy were
taken to be the arithmatic average of the atomic parameters with the
exception of $c_i$ since its values is only dependent on the identity
of the atom where the density is evaluated.  For the {\sc q-sc}
potential, cutoff distances are traditionally taken to be
$2\alpha_{ij}$ and include up to the sixth coordination shell in FCC
metals.

\subsection{Sampling single-temperature configurations from a cooling
trajectory} 

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Revision:	3221
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#	User	Rev	Content
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	3208
52	gezelter	3221	The viscosity ($\eta$) can be tuned by comparing the cooling rate that
53			a set of nanoparticles experience with the known cooling rates for
54			those particles obtained via the laser heating experiments.
55			Essentially, we tune the solvent viscosity until the thermal decay
56			profile matches a heat-transfer model using reasonable values for the
57			interfacial conductance and the thermal conductivity of the solvent.
58
59			Cooling rates for the experimentally-observed nanoparticles were
60			calculated from the heat transfer equations for a spherical particle
61			embedded in a ambient medium that allows for diffusive heat
62			transport. The heat transfer model is a set of two coupled
63			differential equations in the Laplace domain,
64	chuckv	3208	\begin{eqnarray}
65	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\\
66			\left(\frac{\partial}{\partial r} T_{f}(r,s)\right)_{r=R} +
67			\frac{G}{K}(T_{p}(s)-T_{f}(r,s) = 0
68			\label{eq:heateqn}
69	chuckv	3208	\end{eqnarray}
70	gezelter	3221	where $s$ is the time-conjugate variable in Laplace space. The
71			variables in these equations describe a nanoparticle of radius $R$,
72			mass $M$, and specific heat $c_{p}$ at an initial temperature
73			$T_0$. The surrounding solvent has a thermal profile $T_f(r,t)$,
74			thermal conductivity $K$, density $\rho$, and specific heat $c$. $G$
75			is the interfacial conductance between the nanoparticle and the
76			surrounding solvent, and contains information about heat transfer to
77			the capping agent as well as the direct metal-to-solvent heat loss.
78			The temperature of the nanoparticle as a function of time can then
79			obtained by the inverse Laplace transform,
80	chuckv	3208	\begin{equation}
81	gezelter	3221	T_{p}(t)=\frac{2 k R^2 g^2
82			T_0}{\pi}\int_{0}^{\infty}\frac{\exp(-\kappa u^2
83			t/R^2)u^2}{(u^2(1 + R g) - k R g)^2+(u^3 - k R g u)^2}\mathrm{d}u.
84			\label{eq:laplacetransform}
85	chuckv	3208	\end{equation}
86	gezelter	3221	For simplicity, we have introduced the thermal diffusivity $\kappa =
87			K/(\rho c)$, and defined $k=4\pi R^3 \rho c /(M c_p)$ and $g = G/K$ in
88			Eq. \ref{eq:laplacetransform}.
89	chuckv	3208
90	gezelter	3221	Eq. \ref{eq:laplacetransform} was solved numerically for the Ag-Cu
91			system using mole-fraction weighted values for $c_p$ and $\rho_p$ of
92			0.295 $(\mathrm{J g^{-1} K^{-1}})$ and $9.826\times 10^6$ $(\mathrm{g
93			m^{-3}})$ respectively. Since most of the laser excitation experiments
94			have been done in aqueous solutions, parameters used for the fluid are
95			$K$ of $0.6$ $(\mathrm{Wm^{-1}K^{-1}})$, $\rho$ of $1.0\times10^6$ $(\mathrm{g
96			m^{-3}})$ and $c$ of $4.184$ $(\mathrm{J g^{-1} K^{-1}})$.
97	chuckv	3208
98	gezelter	3221	Values for the interfacial conductance have been determined by a
99			number of groups for similar nanoparticles and range from a low
100			$87.5\times 10^{6}$ $(\mathrm{Wm^{-2}K^{-1}})$ to $120\times 10^{6}$
101			$(\mathrm{Wm^{-2}K^{-1}})$.\cite{XXX}
102
103			We conducted our simulations at both ends of the range of
104			experimentally-determined values for the interfacial conductance.
105			This allows us to observe both the slowest and fastest heat transfers
106			from the nanoparticle to the solvent that are consistent with
107			experimental observations. For the slowest heat transfer, a value for
108			G of $87.5\times 10^{6}$ $(\mathrm{Wm^{-2}K^{-1}})$ was used and for
109			the fastest heat transfer, a value of $117\times 10^{6}$
110			$(\mathrm{Wm^{-2}K^{-1}})$ was used. Based on calculations we have
111			done using raw data from the Hartland group's thermal half-time
112			experiments on Au nanospheres, we believe that the true G values are
113			closer to the faster regime: $117\times 10^{6}$
114			$(\mathrm{Wm^{-2}K^{-1}})$.
115
116			The rate of cooling for the nanoparticles in a molecular dynamics
117			simulation can then be tuned by changing the effective solvent
118			viscosity ($\eta$) until the nanoparticle cooling rate matches the
119			cooling rate described by the heat-transfer equations
120			(\ref{eq:heateqn}). The effective solvent viscosity (in poise) for a G
121			of $87.5\times 10^{6}$ $(\mathrm{Wm^{-2}K^{-1}})$ is 0.17, 0.20, and
122			0.22 for 20 {\AA}, 30 {\AA}, and 40 {\AA} particles, respectively. The
123			effective solvent viscosity (again in poise) for an interfacial
124			conductance of $117\times 10^{6}$ $(\mathrm{Wm^{-2}K^{-1}})$ is 0.23,
125			0.29, and 0.30 for 20 {\AA}, 30 {\AA} and 40 {\AA} particles. Cooling
126			traces for each particle size are presented in
127			Fig. \ref{fig:images_cooling_plot}. It should be noted that the
128			Langevin thermostat produces cooling curves that are consistent with
129			Newtonian (single-exponential) cooling, which cannot match the cooling
130			profiles from Eq. \ref{eq:laplacetransform} exactly.
131
132	chuckv	3213	\begin{figure}[htbp]
133	gezelter	3221	\centering
134			\includegraphics[width=\linewidth]{images/cooling_plot.pdf}
135			\caption{Thermal cooling curves obtained from the inverse Laplace
136			transform heat model in Eq. \ref{eq:laplacetransform} (solid line) as
137			well as from molecular dynamics simulations (circles). Effective
138			solvent viscosities of 0.23-0.30 poise (depending on the radius of the
139			particle) give the best fit to the experimental cooling curves. Since
140			this viscosity is substantially in excess of the viscosity of liquid
141			water, much of the thermal transfer to the surroundings is probably
142			due to the capping agent.}
143			\label{fig:images_cooling_plot}
144	chuckv	3213	\end{figure}
145	chuckv	3208
146	gezelter	3221	\subsection{Potentials for classical simulations of bimetallic
147			nanoparticles}
148	chuckv	3208
149	gezelter	3221	Several different potential models have been developed that reasonably
150			describe interactions in transition metals. In particular, the
151			Embedded Atom Model ({\sc eam})~\cite{PhysRevB.33.7983} and
152			Sutton-Chen ({\sc sc})~\cite{Chen90} potential have been used to study
153			a wide range of phenomena in both bulk materials and
154			nanoparticles.\cite{Vardeman-II:2001jn,ShibataT._ja026764r} Both
155			potentials are based on a model of a metal which treats the nuclei and
156			core electrons as pseudo-atoms embedded in the electron density due to
157			the valence electrons on all of the other atoms in the system. The
158			{\sc sc} potential has a simple form that closely resembles that of
159			the ubiquitous Lennard Jones potential,
160			\begin{equation}
161			\label{eq:SCP1}
162			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] ,
163			\end{equation}
164			where $V^{pair}_{ij}$ and $\rho_{i}$ are given by
165			\begin{equation}
166			\label{eq:SCP2}
167			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}}.
168			\end{equation}
169			$V^{pair}_{ij}$ is a repulsive pairwise potential that accounts for
170			interactions between the pseudoatom cores. The $\sqrt{\rho_i}$ term in
171			Eq. (\ref{eq:SCP1}) is an attractive many-body potential that models
172			the interactions between the valence electrons and the cores of the
173			pseudo-atoms. $D_{ij}$, $D_{ii}$ set the appropriate overall energy
174			scale, $c_i$ scales the attractive portion of the potential relative
175			to the repulsive interaction and $\alpha_{ij}$ is a length parameter
176			that assures a dimensionless form for $\rho$. These parameters are
177			tuned to various experimental properties such as the density, cohesive
178			energy, and elastic moduli for FCC transition metals. The quantum
179			Sutton-Chen ({\sc q-sc}) formulation matches these properties while
180			including zero-point quantum corrections for different transition
181			metals.\cite{PhysRevB.59.3527} This particular parametarization has
182			been shown to reproduce the experimentally available heat of mixing
183			data for both FCC solid solutions of Ag-Cu and the high-temperature
184			liquid.\cite{sheng:184203} In contrast, the {\sc eam} potential does
185			not reproduce the experimentally observed heat of mixing for the
186			liquid alloy.\cite{MURRAY:1984lr} Combination rules for the alloy were
187			taken to be the arithmatic average of the atomic parameters with the
188			exception of $c_i$ since its values is only dependent on the identity
189			of the atom where the density is evaluated. For the {\sc q-sc}
190			potential, cutoff distances are traditionally taken to be
191			$2\alpha_{ij}$ and include up to the sixth coordination shell in FCC
192			metals.
193	chuckv	3213
194	gezelter	3221	\subsection{Sampling single-temperature configurations from a cooling
195			trajectory}
196	chuckv	3213
197	gezelter	3221	ffdsafjdksalfdsa