| 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 |
< |
et.al.\cite{Steinhardt:1983mo} In this model, a set of spherical |
| 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 |
| 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 with respect to rotation of the arbitrary reference |
| 70 |
< |
coordinate system. |
| 71 |
< |
Second- and third-order rotationally invariant combinations, $Q_l$ and |
| 72 |
< |
$W_l$, can be taken by summing over $m$ values of $\bar{Q}_{lm}$, |
| 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} |
| 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. |
| 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 |
| 131 |
|
Similar behavior is observed in the bond-orientational order |
| 132 |
|
parameters for individual liquid-like structures. |
| 133 |
|
|
| 134 |
< |
Additionally, both $Q_6$ and $\hat{W}_6$ have extreme values for the |
| 135 |
< |
icosahedral clusters. This makes the $l=6$ bond-orientational order |
| 136 |
< |
parameters particularly useful in identifying the extent of local |
| 137 |
< |
icosahedral ordering in condensed phases. For example, a local |
| 138 |
< |
structure which exhibits $\hat{W}_6$ values near -0.17 is easily |
| 139 |
< |
identified as an icosahedral cluster and cannot be mistaken for |
| 140 |
< |
distorted cubic or liquid-like structures. |
| 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} |
