| 64 |
|
\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 |
|
\end{equation} |
| 67 |
< |
The $\bar{Q}_{lm}$ contained in equation \ref{eq:sys_avg_bo} is not |
| 67 |
> |
The $\bar{Q}_{lm}$ contained in Eq. (\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 |
| 83 |
|
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 |
|
\label{eq:third_inv} |
| 85 |
|
\end{equation} |
| 86 |
< |
The factor in parentheses in Eq. \ref{eq:third_inv} is the Wigner-3$j$ |
| 86 |
> |
The factor in parentheses in Eq. (\ref{eq:third_inv}) is the Wigner-3$j$ |
| 87 |
|
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 |
|
|
| 146 |
|
structures. |
| 147 |
|
|
| 148 |
|
The distributions of atomic $Q_6$ and $\hat{W}_6$ values are plotted |
| 149 |
< |
as a function of temperature for our nanoparticles in figures |
| 149 |
> |
as a function of temperature for our nanoparticles in Fig. |
| 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 = |
| 155 |
> |
sharp peak appearing in Fig. \ref{fig:w6} at $\hat{W}_6=-0.17$ and |
| 156 |
> |
to the broad shoulder appearing in Fig. \ref{fig:q6} at $Q_6 = |
| 157 |
|
0.663$.) |
| 158 |
|
|
| 159 |
|
\begin{figure}[htbp] |
| 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 |
| 199 |
> |
of temperatures are shown in Fig. \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 |
| 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 |
| 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, |
| 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, |
| 235 |
> |
particles is given in Fig. \ref{fig:ficos}. As the particles cool, |
| 236 |
|
the fraction of local icosahedral ordering rises smoothly to a plateau |
| 237 |
|
value. The smaller particles (particularly the ones that were cooled |
| 238 |
|
in a higher viscosity solvent) show a slightly larger tendency towards |
| 270 |
|
\end{figure} |
| 271 |
|
|
| 272 |
|
The locations of these icosahedral centers are not uniformly |
| 273 |
< |
distrubted throughout the particles. In figure \ref{fig:icoscluster} |
| 273 |
> |
distrubted throughout the particles. In Fig. \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 |
| 303 |
|
|
| 304 |
|
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 |
| 306 |
> |
the nanoparticles. Fig. \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}. |
| 313 |
> |
can be seen visually in Fig. \ref{fig:cross_sections} as well as in |
| 314 |
> |
the density plots in the bottom panel of Fig. \ref{fig:Surface}. |
| 315 |
|
This observation is consistent with previous experimental and |
| 316 |
|
theoretical studies on bimetallic alloys composed of noble |
| 317 |
|
metals.\cite{MainardiD.S._la0014306,HuangS.-P._jp0204206,Ramirez-Caballero:2006lr} |
