| 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} |
| 208 |
> |
Frenkel.\cite{duijneveldt:4655} |
| 209 |
|
|
| 210 |
+ |
|
| 211 |
|
\begin{figure}[htbp] |
| 212 |
|
\centering |
| 213 |
|
\includegraphics[width=5in]{images/freeEnergyVsW6.pdf} |
| 339 |
|
the surface.} |
| 340 |
|
\label{fig:Surface} |
| 341 |
|
\end{figure} |
| 342 |
+ |
|
| 343 |
+ |
The methods used by Sheng, He, and Ma to estimate the glass transition |
| 344 |
+ |
temperature, $T_g$, in bulk Ag-Cu alloys involve finding |
| 345 |
+ |
discontinuities in the slope of the average atomic volume, $\langle V |
| 346 |
+ |
\rangle / N$, or enthalpy when plotted against the temperature of the |
| 347 |
+ |
alloy. They obtained a bulk glass transition temperature, $T_g$ = 510 |
| 348 |
+ |
K for a quenching rate of $2.5 \times 10^{13}$ K/s. |
| 349 |
+ |
|
| 350 |
+ |
For simulations of nanoparticles, there is no periodic box, and |
| 351 |
+ |
therefore, no easy way to compute the volume exactly. Instead, we |
| 352 |
+ |
estimate the volume of our nanoparticles using Barber {\it et al.}'s |
| 353 |
+ |
very fast quickhull algorithm to obtain the convex hull for the |
| 354 |
+ |
collection of 3-d coordinates of all of atoms at each point in |
| 355 |
+ |
time.~\cite{Barber96,qhull} The convex hull is the smallest convex |
| 356 |
+ |
polyhedron which includes all of the atoms, so the volume of this |
| 357 |
+ |
polyhedron is an excellent estimate of the volume of the nanoparticle. |
| 358 |
+ |
This method of estimating the volume will be problematic if the |
| 359 |
+ |
nanoparticle breaks into pieces (i.e. if the bounding surface becomes |
| 360 |
+ |
concave), but for the relatively short trajectories used in this |
| 361 |
+ |
study, it provides an excellent measure of particle volume as a |
| 362 |
+ |
function of time (and temperature). |
| 363 |
+ |
|
| 364 |
+ |
Using the discontinuity in the slope of the average atomic volume |
| 365 |
+ |
vs. temperature, we arrive at an estimate of $T_g$ that is |
| 366 |
+ |
approximately 490 K. We note that this temperature is somewhat below |
| 367 |
+ |
the onset of icosahedral ordering exhibited in the bond orientational |
| 368 |
+ |
order parameters. It appears that icosahedral ordering sets in while |
| 369 |
+ |
the system is still somewhat fluid, and is locked in place once the |
| 370 |
+ |
temperature falls below $T_g$. We did not observe any dependence of |
| 371 |
+ |
our estimates for $T_g$ on either the nanoparticle size or the value |
| 372 |
+ |
of the interfacial conductance. However, the cooling rates and size |
| 373 |
+ |
ranges we utilized are all sampled from a relatively narrow range, and |
| 374 |
+ |
it is possible that much larger particles would have substantially |
| 375 |
+ |
different values for $T_g$. Our estimates for the glass transition |
| 376 |
+ |
temperatures for all three particle sizes and both interfacial |
| 377 |
+ |
conductance values are shown in table \ref{table:Tg}. |
| 378 |
+ |
|
| 379 |
+ |
\begin{table} |
| 380 |
+ |
\caption{Estimates of the glass transition temperatures $T_g$ for |
| 381 |
+ |
three different sizes of bimetallic Ag$_6$Cu$_4$ nanoparticles cooled |
| 382 |
+ |
under two different values of the interfacial conductance, $G$.} |
| 383 |
+ |
\begin{center} |
| 384 |
+ |
\begin{tabular}{ccccc} |
| 385 |
+ |
\hline |
| 386 |
+ |
\hline |
| 387 |
+ |
Radius (\AA\ ) & Interfacial conductance & Effective cooling rate |
| 388 |
+ |
(K/s) & & $T_g$ (K) \\ |
| 389 |
+ |
20 & 87.5 & & 477 \\ |
| 390 |
+ |
20 & 117 & & 502 \\ |
| 391 |
+ |
30 & 87.5 & & 512 \\ |
| 392 |
+ |
30 & 117 & & 493 \\ |
| 393 |
+ |
40 & 87.5 & & 476 \\ |
| 394 |
+ |
40 & 117 & & 487 \\ |
| 395 |
+ |
\hline |
| 396 |
+ |
\end{tabular} |
| 397 |
+ |
\end{center} |
| 398 |
+ |
\label{table:Tg} |
| 399 |
+ |
\end{table} |
