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1 %!TEX root = /Users/charles/Desktop/nanoglass/nanoglass.tex
2
3 \section{Analysis}
4
5 Frank first proposed icosahedral arrangement of atoms as a model for
6 structure supercooled atomic liquids.\cite{19521106} The ability to
7 cool simple liquid metals well below their equilibrium melting
8 temperatures was attributed to this local icosahedral ordering. Frank
9 further showed that a 13-atom icosahedral cluster has a 8.4\% higher
10 binding energy the either a face centered cubic ({\sc fcc}) or
11 hexagonal close-packed ({\sc hcp}) crystal structure. Icosahedra also
12 have six fivefold symmetry axes that cannot be extended indefinitely
13 in three dimensions making them incommensurate with long-range
14 translational order. This does not preclude icosahedral clusters from
15 possessing long-range {\it orientational} order. The ``frustrated''
16 packing of these icosahedral structures into dense clusters has been
17 proposed as a model for glass formation.\cite{19871127} The size of
18 the icosahedral clusters is thought to increase until frustration
19 prevents any further growth.\cite{HOARE:1976fk} Molecular dynamics
20 simulations of a two-component Lennard-Jones glass showed that
21 clusters of face-sharing icosahedra are distributed throughout the
22 material.\cite{PhysRevLett.60.2295} Simulations of freezing of single
23 component metalic nanoclusters have shown a tendency for icosohedral
24 structure formation particularly at the surfaces of these
25 clusters.\cite{Gafner:2004bg,PhysRevLett.89.275502,Ascencio:2000qy,Chen:2004ec}
26 Experimentally, the splitting (or shoulder) on the second peak of the
27 X-ray structure factor in binary metallic glasses has been attributed
28 to the formation of tetrahedra that share faces of adjoining
29 icosahedra.\cite{Waal:1995lr}
30
31 Various structural probes have been used to characterize structural
32 order in systems including: common neighbor analysis, voronoi-analysis
33 and orientational bond-order
34 parameters.\cite{HoneycuttJ.Dana_j100303a014,Iwamatsu:2007lr,hsu:4974,nose:1803}
35 One method that has been used extensively for determining local and
36 extended orientational symmetry in condensed phases is the
37 bond-orientational analysis formulated by Steinhart
38 {\it et al.}\cite{Steinhardt:1983mo} In this model, a set of spherical
39 harmonics is associated with each of the near neighbors of a central
40 atom. Neighbors (or ``bonds'') are defined as having a distance from
41 the central atom that is within the first peak in the radial
42 distribution function. The spherical harmonic between a central atom
43 $i$ and a neighboring atom $j$ is
44 \begin{equation}
45 Y_{lm}\left(\vec{r}_{ij}\right)=Y_{lm}\left(\theta(\vec{r}_{ij}),\phi(\vec{r}_{ij})\right)
46 \label{eq:spharm}
47 \end{equation}
48 where, $\{ Y_{lm}(\theta,\phi)\}$ are spherical harmonics, and
49 $\theta(\vec{r})$ and $\phi(\vec{r})$ are the polar and azimuthal
50 angles made by the bond vector $\vec{r}$ with respect to a reference
51 coordinate system. We chose for simplicity the origin as defined by
52 the coordinates for our nanoparticle. (Only even-$l$ spherical
53 harmonics are considered since permutation of a pair of identical
54 particles should not affect the bond-order parameter.) The local
55 environment surrounding atom $i$ can be defined by
56 the average over all neighbors, $N_b(i)$, surrounding that atom,
57 \begin{equation}
58 \bar{q}_{lm}(i) = \frac{1}{N_{b}(i)}\sum_{j=1}^{N_b(i)} Y_{lm}(\vec{r}_{ij}).
59 \label{eq:local_avg_bo}
60 \end{equation}
61 We can further define a global average orientational-bond order over
62 all $\bar{q}_{lm}$ by calculating the average of $\bar{q}_{lm}(i)$
63 over all $N$ particles
64 \begin{equation}
65 \bar{Q}_{lm} = \frac{\sum^{N}_{i=1}N_{b}(i)\bar{q}_{lm}(i)}{\sum^{N}_{i=1}N_{b}(i)}
66 \label{eq:sys_avg_bo}
67 \end{equation}
68 The $\bar{Q}_{lm}$ contained in equation \ref{eq:sys_avg_bo} is not
69 necessarily invariant under rotations of the arbitrary reference
70 coordinate system. Second- and third-order rotationally invariant
71 combinations, $Q_l$ and $W_l$, can be taken by summing over $m$ values
72 of $\bar{Q}_{lm}$,
73 \begin{equation}
74 Q_{l} = \left( \frac{4\pi}{2 l+1} \sum^{l}_{m=-l} \left| \bar{Q}_{lm} \right|^2\right)^{1/2}
75 \label{eq:sec_ord_inv}
76 \end{equation}
77 and
78 \begin{equation}
79 \hat{W}_{l} = \frac{W_{l}}{\left( \sum^{l}_{m=-l} \left| \bar{Q}_{lm} \right|^2\right)^{3/2}}
80 \label{eq:third_ord_inv}
81 \end{equation}
82 where
83 \begin{equation}
84 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}.
85 \label{eq:third_inv}
86 \end{equation}
87 The factor in parentheses in Eq. \ref{eq:third_inv} is the Wigner-3$j$
88 symbol, and the sum is over all valid ($|m| \leq l$) values of $m_1$,
89 $m_2$, and $m_3$ which sum to zero.
90
91 \begin{table}
92 \caption{Values of bond orientational order parameters for
93 simple structures cooresponding to $l=4$ and $l=6$ spherical harmonic
94 functions.\cite{wolde:9932} $Q_4$ and $\hat{W}_4$ have values for {\it
95 individual} icosahedral clusters, but these values are not invariant
96 under rotations of the reference coordinate systems. Similar behavior
97 is observed in the bond-orientational order parameters for individual
98 liquid-like structures.}
99 \begin{center}
100 \begin{tabular}{ccccc}
101 \hline
102 \hline
103 & $Q_4$ & $Q_6$ & $\hat{W}_4$ & $\hat{W}_6$\\
104
105 fcc & 0.191 & 0.575 & -0.159 & -0.013\\
106
107 hcp & 0.097 & 0.485 & 0.134 & -0.012\\
108
109 bcc & 0.036 & 0.511 & 0.159 & 0.013\\
110
111 sc & 0.764 & 0.354 & 0.159 & 0.013\\
112
113 Icosahedral & - & 0.663 & - & -0.170\\
114
115 (liquid) & - & - & - & -\\
116 \hline
117 \end{tabular}
118 \end{center}
119 \label{table:bopval}
120 \end{table}
121
122 For ideal face-centered cubic ({\sc fcc}), body-centered cubic ({\sc
123 bcc}) and simple cubic ({\sc sc}) as well as hexagonally close-packed
124 ({\sc hcp}) structures, these rotationally invariant bond order
125 parameters have fixed values independent of the choice of coordinate
126 reference frames. For ideal icosahedral structures, the $l=6$
127 invariants, $Q_6$ and $\hat{W}_6$ are also independent of the
128 coordinate system. $Q_4$ and $\hat{W}_4$ will have non-vanishing
129 values for {\it individual} icosahedral clusters, but these values are
130 not invariant under rotations of the reference coordinate systems.
131 Similar behavior is observed in the bond-orientational order
132 parameters for individual liquid-like structures.
133
134 Additionally, both $Q_6$ and $\hat{W}_6$ are thought to have extreme
135 values for the icosahedral clusters.\cite{Steinhardt:1983mo} This
136 makes the $l=6$ bond-orientational order parameters particularly
137 useful in identifying the extent of local icosahedral ordering in
138 condensed phases. For example, a local structure which exhibits
139 $\hat{W}_6$ values near -0.17 is easily identified as an icosahedral
140 cluster and cannot be mistaken for distorted cubic or liquid-like
141 structures.
142
143 One may use these bond orientational order parameters as an averaged
144 property to obtain the extent of icosahedral ordering in a supercooled
145 liquid or cluster. It is also possible to accumulate information
146 about the {\it distributions} of local bond orientational order
147 parameters, $p(\hat{W}_6)$ and $p(Q_6)$, which provide information
148 about individual atomic sites that are central to local icosahedral
149 structures.
150
151 The distributions of atomic $Q_6$ and $\hat{W}_6$ values are plotted
152 as a function of temperature for our nanoparticles in figures
153 \ref{fig:q6} and \ref{fig:w6}. At high temperatures, the
154 distributions are unstructured and are broadly distributed across the
155 entire range of values. As the particles are cooled, however, there
156 is a dramatic increase in the fraction of atomic sites which have
157 local icosahedral ordering around them. (This corresponds to the
158 sharp peak appearing in figure \ref{fig:w6} at $\hat{W}_6=-0.17$ and
159 to the broad shoulder appearing in figure \ref{fig:q6} at $Q_6 =
160 0.663$.)
161
162 \begin{figure}[htbp]
163 \centering
164 %\includegraphics[width=\linewidth]{images/w6fig.pdf}
165 \caption{Distributions of the bond orientational order parameter
166 ($\hat{W}_6$) at different temperatures. The upper, middle, and lower
167 panels are for 20, 30, and 40 \AA\ particles, respectively. The
168 left-hand column used cooling rates commensurate with a low
169 interfacial conductance ($87.5 \times 10^{6}$
170 $\mathrm{Wm^{-2}K^{-1}}$), while the right-hand column used a more
171 physically reasonable value of $117 \times 10^{6}$
172 $\mathrm{Wm^{-2}K^{-1}}$. The peak at $\hat{W}_6 \approx -0.17$ is
173 due to local icosahedral structures.}
174 \label{fig:w6}
175 \end{figure}
176
177 \begin{figure}[htbp]
178 \centering
179 %\includegraphics[width=\linewidth]{images/q6fig.pdf}
180 \caption{Distributions of the bond orientational order parameter
181 ($Q_6$) at different temperatures. The curves in the six panels in
182 this figure were computed at identical conditions to the same panels in
183 figure \ref{fig:w6}.}
184 \label{fig:q6}
185 \end{figure}
186
187 We have also looked at the fraction of atomic centers which have local
188 icosahedral order:
189 \begin{equation}
190 f_\textrm{icos} = \int_{-\infty}^{w_i} p(\hat{W}_6) d \hat{W}_6
191 \label{eq:ficos}
192 \end{equation}
193 where $w_i$ is a cutoff value in $\hat{W}_6$ for atomic centers that
194 are displaying icosahedral environments. We have chosen a (somewhat
195 arbitrary) value of $w_i= -0.15$ for the purposes of this work. A
196 plot of $f_\textrm{icos}(T)$ as a function of temperature of the
197 particles is given in figure \ref{fig:ficos}. As the particles cool,
198 the fraction of local icosahedral ordering rises smoothly to a plateau
199 value. The larger particles (particularly the ones that were cooled
200 in a lower viscosity solvent) show a lower tendency towards icosahedral
201 ordering.
202
203 \begin{figure}[htbp]
204 \centering
205 %\includegraphics[width=\linewidth]{images/ficos.pdf}
206 \caption{Temperautre dependence of the fraction of atoms with local
207 icosahedral ordering, $f_\textrm{icos}(T)$ for 20, 30, and 40 \AA\
208 particles cooled at two different values of the interfacial
209 conductance.}
210 \label{fig:q6}
211 \end{figure}
212
213 Since we have atomic-level resolution of the local bond-orientational
214 ordering information, we can also look at the local ordering as a
215 function of the identities of the central atoms. In figure
216 \ref{fig:AgVsCu} we display the distributions of $\hat{W}_6$ values
217 for both the silver and copper atoms, and we note a strong
218 predilection for the copper atoms to be central to local icosahedral
219 ordering. This is probably due to local packing competition of the
220 larger silver atoms around the copper, which would tend to favor
221 icosahedral structures over the more densely packed cubic structures.
222
223 \begin{figure}[htbp]
224 \centering
225 %\includegraphics[width=\linewidth]{images/AgVsCu.pdf}
226 \caption{Distributions of the bond orientational order parameter
227 ($\hat{W}_6$) for the two different elements present in the
228 nanoparticles. This distribution was taken from the fully-cooled 40
229 \AA\ nanoparticle. Local icosahedral ordering around copper atoms is
230 much more prevalent than around silver atoms.}
231 \label{fig:q6}
232 \end{figure}