trunk/nanoglass/analysis.tex

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\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 Eq. (\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 Fig.
\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 Fig. \ref{fig:w6} at $\hat{W}_6=-0.17$ and
to the broad shoulder appearing in Fig. \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 Fig. \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 Fig. \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 Fig. \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.  Fig. \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 Fig. \ref{fig:cross_sections} as well as in
the density plots in the bottom panel of Fig. \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}

The methods used by Sheng, He, and Ma to estimate the glass transition
temperature, $T_g$, in bulk Ag-Cu alloys involve finding
discontinuities in the slope of the average atomic volume, $\langle V
\rangle / N$, or enthalpy when plotted against the temperature of the
alloy.  They obtained a bulk glass transition temperature, $T_g$ = 510
K for a quenching rate of $2.5 \times 10^{13}$ K/s.

For simulations of nanoparticles, there is no periodic box, and
therefore, no easy way to compute the volume exactly.  Instead, we
estimate the volume of our nanoparticles using Barber {\it et al.}'s
very fast quickhull algorithm to obtain the convex hull for the
collection of 3-d coordinates of all of atoms at each point in
time.~\cite{Barber96,qhull} The convex hull is the smallest convex
polyhedron which includes all of the atoms, so the volume of this
polyhedron is an excellent estimate of the volume of the nanoparticle.
This method of estimating the volume will be problematic if the
nanoparticle breaks into pieces (i.e. if the bounding surface becomes
concave), but for the relatively short trajectories used in this
study, it provides an excellent measure of particle volume as a
function of time (and temperature).

Using the discontinuity in the slope of the average atomic volume
vs. temperature, we arrive at an estimate of $T_g$ that is
approximately 490 K.  We note that this temperature is somewhat below
the onset of icosahedral ordering exhibited in the bond orientational
order parameters. It appears that icosahedral ordering sets in while
the system is still somewhat fluid, and is locked in place once the
temperature falls below $T_g$.  We did not observe any dependence of
our estimates for $T_g$ on either the nanoparticle size or the value
of the interfacial conductance.  However, the cooling rates and size
ranges we utilized are all sampled from a relatively narrow range, and
it is possible that much larger particles would have substantially
different values for $T_g$.  Our estimates for the glass transition
temperatures for all three particle sizes and both interfacial
conductance values are shown in table \ref{table:Tg}.

\begin{table}
\caption{Estimates of the glass transition temperatures $T_g$ for
three different sizes of bimetallic Ag$_6$Cu$_4$ nanoparticles cooled
under two different values of the interfacial conductance, $G$.} 
\begin{center}
\begin{tabular}{ccccc}
\hline
\hline
Radius (\AA\ ) & Interfacial conductance & Effective cooling rate
(K/s) &  & $T_g$ (K) \\
20 & 87.5 &  & 477 \\
20 & 117  &  & 502 \\
30 & 87.5 &  & 512 \\
30 & 117  &  & 493 \\
40 & 87.5 &  & 476 \\
40 & 117  &  & 487 \\
\hline
\end{tabular}
\end{center}
\label{table:Tg}
\end{table} 
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