trunk/nanoglass/analysis.tex

%!TEX root = /Users/charles/Desktop/nanoglass/nanoglass.tex

\section{Analysis}

Frank first proposed local icosahedral ordering of atoms as an
explanation for supercooled atomic (specifically metallic) liquids,
and further showed that a 13-atom icosahedral cluster has a 8.4\%
higher binding energy the either a face centered cubic ({\sc fcc}) or
hexagonal close-packed ({\sc hcp}) crystal structures.\cite{19521106}
Icosahedra also have six five-fold symmetry axes that cannot be
extended indefinitely in three dimensions; long-range translational
order is therefore incommensurate with local icosahedral ordering.
This does not preclude icosahedral clusters from possessing long-range
{\it orientational} order. The ``frustrated'' packing of these
icosahedral structures into dense clusters has been proposed as a
model for glass formation.\cite{19871127} The size of the icosahedral
clusters is thought to increase until frustration prevents any further
growth.\cite{HOARE:1976fk} Molecular dynamics simulations of a
two-component Lennard-Jones glass showed that clusters of face-sharing
icosahedra are distributed throughout the
material.\cite{PhysRevLett.60.2295} Simulations of freezing of single
component metalic nanoclusters have shown a tendency for icosohedral
structure formation particularly at the surfaces of these
clusters.\cite{Gafner:2004bg,PhysRevLett.89.275502,Chen:2004ec}
Experimentally, the splitting (or shoulder) on the second peak of the
X-ray structure factor in binary metallic glasses has been attributed
to the formation of tetrahedra that share faces of adjoining
icosahedra.\cite{Waal:1995lr}

Various structural probes have been used to characterize structural
order in molecular systems including: common neighbor analysis,
Voronoi tesselations, and orientational bond-order
parameters.\cite{HoneycuttJ.Dana_j100303a014,Iwamatsu:2007lr,hsu:4974,nose:1803}
The method that has been used most extensively for determining local
and extended orientational symmetry in condensed phases is the
bond-orientational analysis formulated by Steinhart {\it et
al.}\cite{Steinhardt:1983mo} In this model, a set of spherical
harmonics is associated with each of the near neighbors of a central
atom.  Neighbors (or ``bonds'') are defined as having a distance from
the central atom that is within the first peak in the radial
distribution function. The spherical harmonic between a central atom
$i$ and a neighboring atom $j$ is
\begin{equation}
 Y_{lm}\left(\vec{r}_{ij}\right)=Y_{lm}\left(\theta(\vec{r}_{ij}),\phi(\vec{r}_{ij})\right)
\label{eq:spharm}
\end{equation}
where, $\{ Y_{lm}(\theta,\phi)\}$ are spherical harmonics, and
$\theta(\vec{r})$ and $\phi(\vec{r})$ are the polar and azimuthal
angles made by the bond vector $\vec{r}$ with respect to a reference
coordinate system. We chose for simplicity the origin as defined by
the coordinates for our nanoparticle. (Only even-$l$ spherical
harmonics are considered since permutation of a pair of identical
particles should not affect the bond-order parameter.) The local
environment surrounding atom $i$ can be defined by
the average over all neighbors, $N_b(i)$, surrounding that atom,
\begin{equation}
\bar{q}_{lm}(i) = \frac{1}{N_{b}(i)}\sum_{j=1}^{N_b(i)} Y_{lm}(\vec{r}_{ij}).
\label{eq:local_avg_bo}
\end{equation}
We can further define a global average orientational-bond order over
all $\bar{q}_{lm}$ by calculating the average of $\bar{q}_{lm}(i)$
over all $N$ particles 
\begin{equation}
\bar{Q}_{lm} = \frac{\sum^{N}_{i=1}N_{b}(i)\bar{q}_{lm}(i)}{\sum^{N}_{i=1}N_{b}(i)}
\label{eq:sys_avg_bo}
\end{equation}
The $\bar{Q}_{lm}$ contained in equation \ref{eq:sys_avg_bo} is not
necessarily invariant under rotations of the arbitrary reference
coordinate system.  Second- and third-order rotationally invariant
combinations, $Q_l$ and $W_l$, can be taken by summing over $m$ values
of $\bar{Q}_{lm}$,
\begin{equation}
Q_{l} = \left( \frac{4\pi}{2 l+1} \sum^{l}_{m=-l} \left| \bar{Q}_{lm} \right|^2\right)^{1/2}
\label{eq:sec_ord_inv}
\end{equation}
and
\begin{equation} 
\hat{W}_{l} = \frac{W_{l}}{\left( \sum^{l}_{m=-l} \left| \bar{Q}_{lm} \right|^2\right)^{3/2}}
\label{eq:third_ord_inv}
\end{equation}
where
\begin{equation} 
W_{l} = \sum_{\substack{m_1,m_2,m_3 \\m_1 + m_2 + m_3 = 0}} \left( \begin{array}{ccc} l & l & l \\ m_1 & m_2 & m_3 \end{array}\right) \\ \times \bar{Q}_{lm_1}\bar{Q}_{lm_2}\bar{Q}_{lm_3}.
\label{eq:third_inv}
\end{equation}
The factor in parentheses in Eq. \ref{eq:third_inv} is the Wigner-3$j$
symbol, and the sum is over all valid ($|m| \leq l$) values of $m_1$,
$m_2$, and $m_3$ which sum to zero.

\begin{table}
\caption{Values of bond orientational order parameters for
simple structures cooresponding to $l=4$ and $l=6$ spherical harmonic
functions.\cite{wolde:9932} $Q_4$ and $\hat{W}_4$ have values for {\it
individual} icosahedral clusters, but these values are not invariant
under rotations of the reference coordinate systems.  Similar behavior
is observed in the bond-orientational order parameters for individual
liquid-like structures.}
\begin{center}
\begin{tabular}{ccccc}
\hline
\hline
   & $Q_4$ & $Q_6$ & $\hat{W}_4$ & $\hat{W}_6$\\

fcc & 0.191 & 0.575 & -0.159 & -0.013\\

hcp & 0.097 & 0.485 & 0.134 & -0.012\\

bcc & 0.036 & 0.511 & 0.159 & 0.013\\

sc & 0.764 & 0.354 & 0.159 & 0.013\\

Icosahedral & - & 0.663 & - & -0.170\\

(liquid) & - & - & - &  -\\
\hline
\end{tabular}
\end{center}
\label{table:bopval}
\end{table} 
 
For ideal face-centered cubic ({\sc fcc}), body-centered cubic ({\sc
bcc}) and simple cubic ({\sc sc}) as well as hexagonally close-packed
({\sc hcp}) structures, these rotationally invariant bond order
parameters have fixed values independent of the choice of coordinate
reference frames.  For ideal icosahedral structures, the $l=6$
invariants, $Q_6$ and $\hat{W}_6$ are also independent of the
coordinate system.  $Q_4$ and $\hat{W}_4$ will have non-vanishing
values for {\it individual} icosahedral clusters, but these values are
not invariant under rotations of the reference coordinate systems.
Similar behavior is observed in the bond-orientational order
parameters for individual liquid-like structures.  Additionally, both
$Q_6$ and $\hat{W}_6$ are thought to have extreme values for the
icosahedral clusters.\cite{Steinhardt:1983mo} This makes the $l=6$
bond-orientational order parameters particularly useful in identifying
the extent of local icosahedral ordering in condensed phases.  For
example, a local structure which exhibits $\hat{W}_6$ values near
-0.17 is easily identified as an icosahedral cluster and cannot be
mistaken for distorted cubic or liquid-like structures.

One may use these bond orientational order parameters as an averaged
property to obtain the extent of icosahedral ordering in a supercooled
liquid or cluster.  It is also possible to accumulate information
about the {\it distributions} of local bond orientational order
parameters, $p(\hat{W}_6)$ and $p(Q_6)$, which provide information
about individual atomic sites that are central to local icosahedral
structures.

The distributions of atomic $Q_6$ and $\hat{W}_6$ values are plotted
as a function of temperature for our nanoparticles in figures
\ref{fig:q6} and \ref{fig:w6}.   At high temperatures, the
distributions are unstructured and are broadly distributed across the
entire range of values.  As the particles are cooled, however, there
is a dramatic increase in the fraction of atomic sites which have
local icosahedral ordering around them.  (This corresponds to the
sharp peak appearing in figure \ref{fig:w6} at $\hat{W}_6=-0.17$ and
to the broad shoulder appearing in figure \ref{fig:q6} at $Q_6 =
0.663$.)

\begin{figure}[htbp]
\centering
\includegraphics[width=5in]{images/w6_stacked_plot.pdf}
\caption{Distributions of the bond orientational order parameter
($\hat{W}_6$) at different temperatures.  The upper, middle, and lower
panels are for 20, 30, and 40 \AA\ particles, respectively.  The
left-hand column used cooling rates commensurate with a low
interfacial conductance ($87.5 \times 10^{6}$
$\mathrm{Wm^{-2}K^{-1}}$), while the right-hand column used a more
physically reasonable value of $117 \times 10^{6}$
$\mathrm{Wm^{-2}K^{-1}}$.  The peak at $\hat{W}_6 \approx -0.17$ is
due to local icosahedral structures.  The different curves in each of
the panels indicate the distribution of $\hat{W}_6$ values for samples
taken at different times along the cooling trajectory.  The initial
and final temperatures (in K) are indicated on the plots adjacent to
their respective distributions.}
\label{fig:w6}
\end{figure}

\begin{figure}[htbp]
\centering
\includegraphics[width=5in]{images/q6_stacked_plot.pdf}
\caption{Distributions of the bond orientational order parameter
($Q_6$) at different temperatures.  The curves in the six panels in
this figure were computed at identical conditions to the same panels in
figure \ref{fig:w6}.}
\label{fig:q6}
\end{figure}

The probability distributions of local order can be used to generate
free energy surfaces using the local orientational ordering as a
reaction coordinate.  By making the simple statistical equivalence
between the free energy and the probabilities of occupying certain
states,
\begin{equation}
g(\hat{W}_6) = - k_B T \ln p(\hat{W}_6),
\end{equation}
we can obtain a sequence of free energy surfaces (as a function of
temperature) for the local ordering around central atoms within our
particles.  Free energy surfaces for the 40 \AA\ particle at a range
of temperatures are shown in figure \ref{fig:freeEnergy}.  Note that
at all temperatures, the liquid-like structures are global minima on
the free energy surface, while the local icosahedra appear as local
minima once the temperature has fallen below 528 K.  As the
temperature falls, it is possible for substructures to become trapped
in the local icosahedral well, and if the cooling is rapid enough,
this trapping leads to vitrification.  A similar analysis of the free
energy surface for orientational order in bulk glass formers can be
found in the work of van~Duijneveldt and
Frenkel.\cite{duijneveldt:4655} 

\begin{figure}[htbp]
\centering
\includegraphics[width=5in]{images/freeEnergyVsW6.pdf}
\caption{Free energy as a function of the orientational order
parameter ($\hat{W}_6$) for 40 \AA bimetallic nanoparticles as they
are cooled from 902 K to 310 K.  As the particles cool below 528 K, a
local minimum in the free energy surface appears near the perfect
icosahedral ordering ($\hat{W}_6 = -0.17$).  At all temperatures,
liquid-like structures are a global minimum on the free energy
surface, but if the cooling rate is fast enough, substructures
may become trapped with local icosahedral order, leading to the
formation of a glass.}
\label{fig:freeEnergy}
\end{figure}

We have also calculated the fraction of atomic centers which have
strong local icosahedral order:
\begin{equation}
f_\textrm{icos} = \int_{-\infty}^{w_i} p(\hat{W}_6) d \hat{W}_6
\label{eq:ficos}
\end{equation}
where $w_i$ is a cutoff value in $\hat{W}_6$ for atomic centers that
are displaying icosahedral environments.  We have chosen a (somewhat
arbitrary) value of $w_i= -0.15$ for the purposes of this work.  A
plot of $f_\textrm{icos}(T)$ as a function of temperature of the
particles is given in figure \ref{fig:ficos}.  As the particles cool,
the fraction of local icosahedral ordering rises smoothly to a plateau
value.  The smaller particles (particularly the ones that were cooled
in a higher viscosity solvent) show a slightly larger tendency towards
icosahedral ordering.

\begin{figure}[htbp]
\centering
\includegraphics[width=5in]{images/fraction_icos.pdf}
\caption{Temperautre dependence of the fraction of atoms with local
icosahedral ordering, $f_\textrm{icos}(T)$ for 20, 30, and 40 \AA\
particles cooled at two different values of the interfacial
conductance.} 
\label{fig:ficos}
\end{figure}

Since we have atomic-level resolution of the local bond-orientational
ordering information, we can also look at the local ordering as a
function of the identities of the central atoms.  In figure
\ref{fig:AgVsCu} we display the distributions of $\hat{W}_6$ values
for both the silver and copper atoms, and we note a strong
predilection for the copper atoms to be central to icosahedra.  This
is probably due to local packing competition of the larger silver
atoms around the copper, which would tend to favor icosahedral
structures over the more densely packed cubic structures.

\begin{figure}[htbp]
\centering
\includegraphics[width=5in]{images/w6_stacked_bytype_plot.pdf}
\caption{Distributions of the bond orientational order parameter
($\hat{W}_6$) for the two different elements present in the
nanoparticles.  This distribution was taken from the fully-cooled 40
\AA\ nanoparticle.  Local icosahedral ordering around copper atoms is
much more prevalent than around silver atoms.}
\label{fig:AgVsCu}
\end{figure}

The locations of these icosahedral centers are not uniformly
distrubted throughout the particles.  In figure \ref{fig:icoscluster}
we show snapshots of the centers of the local icosahedra (i.e. any
atom which exhibits a local bond orientational order parameter
$\hat{W}_6 < -0.15$).  At high temperatures, the icosahedral centers
are transitory, existing only for a few fs before being reabsorbed
into the liquid droplet.  As the particle cools, these centers become
fixed at certain locations, and additional icosahedra develop
throughout the particle, clustering around the sites where the
structures originated.  There is a strong preference for icosahedral
ordering near the surface of the particles.  Identification of these
structures by the type of atom shows that the silver-centered
icosahedra are evident only at the surface of the particles.

\begin{figure}[htbp]
\centering
\begin{tabular}{c c c}
\includegraphics[width=2.1in]{images/Cu_Ag_random_30A_liq_icosonly.pdf}
\includegraphics[width=2.1in]{images/Cu_Ag_random_30A__0007_icosonly.pdf}
\includegraphics[width=2.1in]{images/Cu_Ag_random_30A_glass_icosonly.pdf}
\end{tabular}
\caption{Centers of local icosahedral order ($\hat{W}_6<0.15$) at 900 
K, 471 K and 315 K for the 30 \AA\ nanoparticle cooled with an
interfacial conductance $G = 87.5 \times 10^{6}$
$\mathrm{Wm^{-2}K^{-1}}$. Silver atoms (blue) exhibit icosahedral
order at the surface of the nanoparticle while copper icosahedral
centers (green) are distributed throughout the nanoparticle.  The
icosahedral centers appear to cluster together and these clusters
increase in size with decreasing temperature.}
\label{fig:icoscluster}
\end{figure}

In contrast with the silver ordering behavior, the copper atoms which
have local icosahedral ordering are distributed more evenly throughout
the nanoparticles.  Figure \ref{fig:Surface} shows this tendency as a
function of distance from the center of the nanoparticle.  Silver,
since it has a lower surface free energy than copper, tends to coat
the skins of the mixed particles.\cite{Zhu:1997lr} This is true even
for bimetallic particles that have been prepared in the Ag (core) / Cu
(shell) configuration.  Upon forming a liquid droplet, approximately 1
monolayer of Ag atoms will rise to the surface of the particles.  This
can be seen visually in figure \ref{fig:cross_sections} as well as in
the density plots in the bottom panel of figure \ref{fig:Surface}.
This observation is consistent with previous experimental and
theoretical studies on bimetallic alloys composed of noble
metals.\cite{MainardiD.S._la0014306,HuangS.-P._jp0204206,Ramirez-Caballero:2006lr}
Bond order parameters for surface atoms are averaged only over the
neighboring atoms, so packing constraints that may prevent icosahedral
ordering around silver in the bulk are removed near the surface.  It
would certainly be interesting to see if the relative tendency of
silver and copper to form local icosahedral structures in a bulk glass
differs from our observations on nanoparticles.

\begin{figure}[htbp]
\centering
\includegraphics[width=5in]{images/dens_fracr_stacked_plot.pdf}
\caption{Appearance of icosahedral clusters around central silver atoms
is largely due to the presence of these silver atoms at or near the
surface of the nanoparticle. The upper panel shows the fraction of
icosahedral atoms ($f_\textrm{icos}(r)$ for each of the two metallic
atoms as a function of distance from the center of the nanoparticle
($r$).  The lower panel shows the radial density of the two
constituent metals (relative to the overall density of the
nanoparticle).  Icosahedral clustering around copper atoms are more
evenly distributed throughout the particle, while icosahedral
clustering around silver is largely confined to the silver atoms at
the surface.}
\label{fig:Surface}
\end{figure}
Revision:	3259
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File size:	16758 byte(s)
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#	User	Rev	Content
1	chuckv	3226	%!TEX root = /Users/charles/Desktop/nanoglass/nanoglass.tex
2
3	chuckv	3213	\section{Analysis}
4
5	gezelter	3242	Frank first proposed local icosahedral ordering of atoms as an
6			explanation for supercooled atomic (specifically metallic) liquids,
7			and further showed that a 13-atom icosahedral cluster has a 8.4\%
8			higher binding energy the either a face centered cubic ({\sc fcc}) or
9			hexagonal close-packed ({\sc hcp}) crystal structures.\cite{19521106}
10			Icosahedra also have six five-fold symmetry axes that cannot be
11	gezelter	3259	extended indefinitely in three dimensions; long-range translational
12			order is therefore incommensurate with local icosahedral ordering.
13	gezelter	3242	This does not preclude icosahedral clusters from possessing long-range
14			{\it orientational} order. The ``frustrated'' packing of these
15			icosahedral structures into dense clusters has been proposed as a
16			model for glass formation.\cite{19871127} The size of the icosahedral
17			clusters is thought to increase until frustration prevents any further
18			growth.\cite{HOARE:1976fk} Molecular dynamics simulations of a
19			two-component Lennard-Jones glass showed that clusters of face-sharing
20			icosahedra are distributed throughout the
21	gezelter	3230	material.\cite{PhysRevLett.60.2295} Simulations of freezing of single
22			component metalic nanoclusters have shown a tendency for icosohedral
23			structure formation particularly at the surfaces of these
24	chuckv	3254	clusters.\cite{Gafner:2004bg,PhysRevLett.89.275502,Chen:2004ec}
25	gezelter	3230	Experimentally, the splitting (or shoulder) on the second peak of the
26			X-ray structure factor in binary metallic glasses has been attributed
27			to the formation of tetrahedra that share faces of adjoining
28			icosahedra.\cite{Waal:1995lr}
29	chuckv	3226
30	gezelter	3230	Various structural probes have been used to characterize structural
31	gezelter	3242	order in molecular systems including: common neighbor analysis,
32			Voronoi tesselations, and orientational bond-order
33	gezelter	3230	parameters.\cite{HoneycuttJ.Dana_j100303a014,Iwamatsu:2007lr,hsu:4974,nose:1803}
34	gezelter	3242	The method that has been used most extensively for determining local
35			and extended orientational symmetry in condensed phases is the
36			bond-orientational analysis formulated by Steinhart {\it et
37			al.}\cite{Steinhardt:1983mo} In this model, a set of spherical
38	gezelter	3230	harmonics is associated with each of the near neighbors of a central
39			atom. Neighbors (or ``bonds'') are defined as having a distance from
40			the central atom that is within the first peak in the radial
41			distribution function. The spherical harmonic between a central atom
42			$i$ and a neighboring atom $j$ is
43	chuckv	3213	\begin{equation}
44	gezelter	3230	Y_{lm}\left(\vec{r}_{ij}\right)=Y_{lm}\left(\theta(\vec{r}_{ij}),\phi(\vec{r}_{ij})\right)
45			\label{eq:spharm}
46	chuckv	3213	\end{equation}
47	gezelter	3230	where, $\{ Y_{lm}(\theta,\phi)\}$ are spherical harmonics, and
48			$\theta(\vec{r})$ and $\phi(\vec{r})$ are the polar and azimuthal
49			angles made by the bond vector $\vec{r}$ with respect to a reference
50			coordinate system. We chose for simplicity the origin as defined by
51			the coordinates for our nanoparticle. (Only even-$l$ spherical
52			harmonics are considered since permutation of a pair of identical
53			particles should not affect the bond-order parameter.) The local
54			environment surrounding atom $i$ can be defined by
55			the average over all neighbors, $N_b(i)$, surrounding that atom,
56	chuckv	3213	\begin{equation}
57	gezelter	3230	\bar{q}_{lm}(i) = \frac{1}{N_{b}(i)}\sum_{j=1}^{N_b(i)} Y_{lm}(\vec{r}_{ij}).
58			\label{eq:local_avg_bo}
59	chuckv	3213	\end{equation}
60	gezelter	3230	We can further define a global average orientational-bond order over
61			all $\bar{q}_{lm}$ by calculating the average of $\bar{q}_{lm}(i)$
62			over all $N$ particles
63	chuckv	3213	\begin{equation}
64	gezelter	3230	\bar{Q}_{lm} = \frac{\sum^{N}_{i=1}N_{b}(i)\bar{q}_{lm}(i)}{\sum^{N}_{i=1}N_{b}(i)}
65			\label{eq:sys_avg_bo}
66	chuckv	3213	\end{equation}
67	gezelter	3233	The $\bar{Q}_{lm}$ contained in equation \ref{eq:sys_avg_bo} is not
68			necessarily invariant under rotations of the arbitrary reference
69			coordinate system. Second- and third-order rotationally invariant
70			combinations, $Q_l$ and $W_l$, can be taken by summing over $m$ values
71			of $\bar{Q}_{lm}$,
72	chuckv	3222	\begin{equation}
73	gezelter	3230	Q_{l} = \left( \frac{4\pi}{2 l+1} \sum^{l}_{m=-l} \left\| \bar{Q}_{lm} \right\|^2\right)^{1/2}
74			\label{eq:sec_ord_inv}
75	chuckv	3222	\end{equation}
76			and
77			\begin{equation}
78			\hat{W}_{l} = \frac{W_{l}}{\left( \sum^{l}_{m=-l} \left\| \bar{Q}_{lm} \right\|^2\right)^{3/2}}
79			\label{eq:third_ord_inv}
80			\end{equation}
81	gezelter	3230	where
82	chuckv	3222	\begin{equation}
83	gezelter	3230	W_{l} = \sum_{\substack{m_1,m_2,m_3 \\m_1 + m_2 + m_3 = 0}} \left( \begin{array}{ccc} l & l & l \\ m_1 & m_2 & m_3 \end{array}\right) \\ \times \bar{Q}_{lm_1}\bar{Q}_{lm_2}\bar{Q}_{lm_3}.
84	chuckv	3222	\label{eq:third_inv}
85			\end{equation}
86	gezelter	3230	The factor in parentheses in Eq. \ref{eq:third_inv} is the Wigner-3$j$
87	gezelter	3233	symbol, and the sum is over all valid ($\|m\| \leq l$) values of $m_1$,
88			$m_2$, and $m_3$ which sum to zero.
89	chuckv	3226
90			\begin{table}
91	gezelter	3230	\caption{Values of bond orientational order parameters for
92			simple structures cooresponding to $l=4$ and $l=6$ spherical harmonic
93			functions.\cite{wolde:9932} $Q_4$ and $\hat{W}_4$ have values for {\it
94			individual} icosahedral clusters, but these values are not invariant
95			under rotations of the reference coordinate systems. Similar behavior
96			is observed in the bond-orientational order parameters for individual
97			liquid-like structures.}
98	chuckv	3226	\begin{center}
99			\begin{tabular}{ccccc}
100			\hline
101			\hline
102			& $Q_4$ & $Q_6$ & $\hat{W}_4$ & $\hat{W}_6$\\
103
104			fcc & 0.191 & 0.575 & -0.159 & -0.013\\
105
106			hcp & 0.097 & 0.485 & 0.134 & -0.012\\
107
108			bcc & 0.036 & 0.511 & 0.159 & 0.013\\
109
110			sc & 0.764 & 0.354 & 0.159 & 0.013\\
111
112	gezelter	3230	Icosahedral & - & 0.663 & - & -0.170\\
113	chuckv	3226
114	gezelter	3230	(liquid) & - & - & - & -\\
115	chuckv	3226	\hline
116			\end{tabular}
117			\end{center}
118			\label{table:bopval}
119			\end{table}
120
121	gezelter	3230	For ideal face-centered cubic ({\sc fcc}), body-centered cubic ({\sc
122			bcc}) and simple cubic ({\sc sc}) as well as hexagonally close-packed
123			({\sc hcp}) structures, these rotationally invariant bond order
124			parameters have fixed values independent of the choice of coordinate
125			reference frames. For ideal icosahedral structures, the $l=6$
126			invariants, $Q_6$ and $\hat{W}_6$ are also independent of the
127			coordinate system. $Q_4$ and $\hat{W}_4$ will have non-vanishing
128			values for {\it individual} icosahedral clusters, but these values are
129			not invariant under rotations of the reference coordinate systems.
130			Similar behavior is observed in the bond-orientational order
131	gezelter	3259	parameters for individual liquid-like structures. Additionally, both
132			$Q_6$ and $\hat{W}_6$ are thought to have extreme values for the
133			icosahedral clusters.\cite{Steinhardt:1983mo} This makes the $l=6$
134			bond-orientational order parameters particularly useful in identifying
135			the extent of local icosahedral ordering in condensed phases. For
136			example, a local structure which exhibits $\hat{W}_6$ values near
137			-0.17 is easily identified as an icosahedral cluster and cannot be
138			mistaken for distorted cubic or liquid-like structures.
139	gezelter	3230
140	gezelter	3233	One may use these bond orientational order parameters as an averaged
141			property to obtain the extent of icosahedral ordering in a supercooled
142			liquid or cluster. It is also possible to accumulate information
143			about the {\it distributions} of local bond orientational order
144			parameters, $p(\hat{W}_6)$ and $p(Q_6)$, which provide information
145			about individual atomic sites that are central to local icosahedral
146			structures.
147	gezelter	3230
148	gezelter	3233	The distributions of atomic $Q_6$ and $\hat{W}_6$ values are plotted
149			as a function of temperature for our nanoparticles in figures
150			\ref{fig:q6} and \ref{fig:w6}. At high temperatures, the
151			distributions are unstructured and are broadly distributed across the
152			entire range of values. As the particles are cooled, however, there
153			is a dramatic increase in the fraction of atomic sites which have
154			local icosahedral ordering around them. (This corresponds to the
155			sharp peak appearing in figure \ref{fig:w6} at $\hat{W}_6=-0.17$ and
156			to the broad shoulder appearing in figure \ref{fig:q6} at $Q_6 =
157			0.663$.)
158
159			\begin{figure}[htbp]
160			\centering
161	gezelter	3242	\includegraphics[width=5in]{images/w6_stacked_plot.pdf}
162	gezelter	3233	\caption{Distributions of the bond orientational order parameter
163			($\hat{W}_6$) at different temperatures. The upper, middle, and lower
164			panels are for 20, 30, and 40 \AA\ particles, respectively. The
165			left-hand column used cooling rates commensurate with a low
166			interfacial conductance ($87.5 \times 10^{6}$
167			$\mathrm{Wm^{-2}K^{-1}}$), while the right-hand column used a more
168			physically reasonable value of $117 \times 10^{6}$
169			$\mathrm{Wm^{-2}K^{-1}}$. The peak at $\hat{W}_6 \approx -0.17$ is
170	gezelter	3239	due to local icosahedral structures. The different curves in each of
171			the panels indicate the distribution of $\hat{W}_6$ values for samples
172			taken at different times along the cooling trajectory. The initial
173			and final temperatures (in K) are indicated on the plots adjacent to
174			their respective distributions.}
175	gezelter	3233	\label{fig:w6}
176			\end{figure}
177
178			\begin{figure}[htbp]
179			\centering
180	gezelter	3242	\includegraphics[width=5in]{images/q6_stacked_plot.pdf}
181	gezelter	3233	\caption{Distributions of the bond orientational order parameter
182			($Q_6$) at different temperatures. The curves in the six panels in
183			this figure were computed at identical conditions to the same panels in
184			figure \ref{fig:w6}.}
185			\label{fig:q6}
186			\end{figure}
187
188	gezelter	3252	The probability distributions of local order can be used to generate
189			free energy surfaces using the local orientational ordering as a
190			reaction coordinate. By making the simple statistical equivalence
191			between the free energy and the probabilities of occupying certain
192			states,
193			\begin{equation}
194			g(\hat{W}_6) = - k_B T \ln p(\hat{W}_6),
195			\end{equation}
196			we can obtain a sequence of free energy surfaces (as a function of
197			temperature) for the local ordering around central atoms within our
198			particles. Free energy surfaces for the 40 \AA\ particle at a range
199			of temperatures are shown in figure \ref{fig:freeEnergy}. Note that
200			at all temperatures, the liquid-like structures are global minima on
201			the free energy surface, while the local icosahedra appear as local
202			minima once the temperature has fallen below 528 K. As the
203			temperature falls, it is possible for substructures to become trapped
204			in the local icosahedral well, and if the cooling is rapid enough,
205			this trapping leads to vitrification. A similar analysis of the free
206			energy surface for orientational order in bulk glass formers can be
207			found in the work of van~Duijneveldt and
208			Frenkel.\cite{duijneveldt:4655}
209
210			\begin{figure}[htbp]
211			\centering
212			\includegraphics[width=5in]{images/freeEnergyVsW6.pdf}
213			\caption{Free energy as a function of the orientational order
214			parameter ($\hat{W}_6$) for 40 \AA bimetallic nanoparticles as they
215			are cooled from 902 K to 310 K. As the particles cool below 528 K, a
216			local minimum in the free energy surface appears near the perfect
217			icosahedral ordering ($\hat{W}_6 = -0.17$). At all temperatures,
218			liquid-like structures are a global minimum on the free energy
219			surface, but if the cooling rate is fast enough, substructures
220			may become trapped with local icosahedral order, leading to the
221			formation of a glass.}
222			\label{fig:freeEnergy}
223			\end{figure}
224
225	gezelter	3242	We have also calculated the fraction of atomic centers which have
226			strong local icosahedral order:
227	gezelter	3233	\begin{equation}
228			f_\textrm{icos} = \int_{-\infty}^{w_i} p(\hat{W}_6) d \hat{W}_6
229			\label{eq:ficos}
230			\end{equation}
231			where $w_i$ is a cutoff value in $\hat{W}_6$ for atomic centers that
232			are displaying icosahedral environments. We have chosen a (somewhat
233			arbitrary) value of $w_i= -0.15$ for the purposes of this work. A
234			plot of $f_\textrm{icos}(T)$ as a function of temperature of the
235			particles is given in figure \ref{fig:ficos}. As the particles cool,
236			the fraction of local icosahedral ordering rises smoothly to a plateau
237	gezelter	3259	value. The smaller particles (particularly the ones that were cooled
238			in a higher viscosity solvent) show a slightly larger tendency towards
239	gezelter	3242	icosahedral ordering.
240	gezelter	3233
241			\begin{figure}[htbp]
242			\centering
243	gezelter	3242	\includegraphics[width=5in]{images/fraction_icos.pdf}
244	gezelter	3233	\caption{Temperautre dependence of the fraction of atoms with local
245			icosahedral ordering, $f_\textrm{icos}(T)$ for 20, 30, and 40 \AA\
246			particles cooled at two different values of the interfacial
247			conductance.}
248	gezelter	3242	\label{fig:ficos}
249	gezelter	3233	\end{figure}
250
251			Since we have atomic-level resolution of the local bond-orientational
252			ordering information, we can also look at the local ordering as a
253			function of the identities of the central atoms. In figure
254			\ref{fig:AgVsCu} we display the distributions of $\hat{W}_6$ values
255			for both the silver and copper atoms, and we note a strong
256	gezelter	3242	predilection for the copper atoms to be central to icosahedra. This
257			is probably due to local packing competition of the larger silver
258			atoms around the copper, which would tend to favor icosahedral
259			structures over the more densely packed cubic structures.
260	gezelter	3233
261			\begin{figure}[htbp]
262			\centering
263	gezelter	3242	\includegraphics[width=5in]{images/w6_stacked_bytype_plot.pdf}
264	gezelter	3233	\caption{Distributions of the bond orientational order parameter
265			($\hat{W}_6$) for the two different elements present in the
266			nanoparticles. This distribution was taken from the fully-cooled 40
267			\AA\ nanoparticle. Local icosahedral ordering around copper atoms is
268			much more prevalent than around silver atoms.}
269	gezelter	3242	\label{fig:AgVsCu}
270	gezelter	3233	\end{figure}
271	gezelter	3242
272	gezelter	3252	The locations of these icosahedral centers are not uniformly
273			distrubted throughout the particles. In figure \ref{fig:icoscluster}
274			we show snapshots of the centers of the local icosahedra (i.e. any
275			atom which exhibits a local bond orientational order parameter
276			$\hat{W}_6 < -0.15$). At high temperatures, the icosahedral centers
277			are transitory, existing only for a few fs before being reabsorbed
278			into the liquid droplet. As the particle cools, these centers become
279			fixed at certain locations, and additional icosahedra develop
280			throughout the particle, clustering around the sites where the
281			structures originated. There is a strong preference for icosahedral
282			ordering near the surface of the particles. Identification of these
283			structures by the type of atom shows that the silver-centered
284			icosahedra are evident only at the surface of the particles.
285
286			\begin{figure}[htbp]
287			\centering
288			\begin{tabular}{c c c}
289			\includegraphics[width=2.1in]{images/Cu_Ag_random_30A_liq_icosonly.pdf}
290			\includegraphics[width=2.1in]{images/Cu_Ag_random_30A__0007_icosonly.pdf}
291			\includegraphics[width=2.1in]{images/Cu_Ag_random_30A_glass_icosonly.pdf}
292			\end{tabular}
293			\caption{Centers of local icosahedral order ($\hat{W}_6<0.15$) at 900
294			K, 471 K and 315 K for the 30 \AA\ nanoparticle cooled with an
295			interfacial conductance $G = 87.5 \times 10^{6}$
296			$\mathrm{Wm^{-2}K^{-1}}$. Silver atoms (blue) exhibit icosahedral
297			order at the surface of the nanoparticle while copper icosahedral
298			centers (green) are distributed throughout the nanoparticle. The
299			icosahedral centers appear to cluster together and these clusters
300			increase in size with decreasing temperature.}
301			\label{fig:icoscluster}
302			\end{figure}
303
304	gezelter	3259	In contrast with the silver ordering behavior, the copper atoms which
305			have local icosahedral ordering are distributed more evenly throughout
306			the nanoparticles. Figure \ref{fig:Surface} shows this tendency as a
307			function of distance from the center of the nanoparticle. Silver,
308			since it has a lower surface free energy than copper, tends to coat
309			the skins of the mixed particles.\cite{Zhu:1997lr} This is true even
310			for bimetallic particles that have been prepared in the Ag (core) / Cu
311			(shell) configuration. Upon forming a liquid droplet, approximately 1
312			monolayer of Ag atoms will rise to the surface of the particles. This
313			can be seen visually in figure \ref{fig:cross_sections} as well as in
314			the density plots in the bottom panel of figure \ref{fig:Surface}.
315			This observation is consistent with previous experimental and
316			theoretical studies on bimetallic alloys composed of noble
317	gezelter	3247	metals.\cite{MainardiD.S._la0014306,HuangS.-P._jp0204206,Ramirez-Caballero:2006lr}
318			Bond order parameters for surface atoms are averaged only over the
319			neighboring atoms, so packing constraints that may prevent icosahedral
320			ordering around silver in the bulk are removed near the surface. It
321			would certainly be interesting to see if the relative tendency of
322			silver and copper to form local icosahedral structures in a bulk glass
323			differs from our observations on nanoparticles.
324
325			\begin{figure}[htbp]
326			\centering
327			\includegraphics[width=5in]{images/dens_fracr_stacked_plot.pdf}
328			\caption{Appearance of icosahedral clusters around central silver atoms
329			is largely due to the presence of these silver atoms at or near the
330			surface of the nanoparticle. The upper panel shows the fraction of
331			icosahedral atoms ($f_\textrm{icos}(r)$ for each of the two metallic
332			atoms as a function of distance from the center of the nanoparticle
333			($r$). The lower panel shows the radial density of the two
334			constituent metals (relative to the overall density of the
335			nanoparticle). Icosahedral clustering around copper atoms are more
336			evenly distributed throughout the particle, while icosahedral
337			clustering around silver is largely confined to the silver atoms at
338			the surface.}
339			\label{fig:Surface}
340			\end{figure}