1 |
chuckv |
3226 |
%!TEX root = /Users/charles/Desktop/nanoglass/nanoglass.tex |
2 |
|
|
|
3 |
chuckv |
3213 |
\section{Analysis} |
4 |
|
|
|
5 |
gezelter |
3242 |
Frank first proposed local icosahedral ordering of atoms as an |
6 |
|
|
explanation for supercooled atomic (specifically metallic) liquids, |
7 |
|
|
and further showed that a 13-atom icosahedral cluster has a 8.4\% |
8 |
|
|
higher binding energy the either a face centered cubic ({\sc fcc}) or |
9 |
|
|
hexagonal close-packed ({\sc hcp}) crystal structures.\cite{19521106} |
10 |
|
|
Icosahedra also have six five-fold symmetry axes that cannot be |
11 |
gezelter |
3259 |
extended indefinitely in three dimensions; long-range translational |
12 |
|
|
order is therefore incommensurate with local icosahedral ordering. |
13 |
gezelter |
3242 |
This does not preclude icosahedral clusters from possessing long-range |
14 |
|
|
{\it orientational} order. The ``frustrated'' packing of these |
15 |
|
|
icosahedral structures into dense clusters has been proposed as a |
16 |
|
|
model for glass formation.\cite{19871127} The size of the icosahedral |
17 |
|
|
clusters is thought to increase until frustration prevents any further |
18 |
|
|
growth.\cite{HOARE:1976fk} Molecular dynamics simulations of a |
19 |
|
|
two-component Lennard-Jones glass showed that clusters of face-sharing |
20 |
|
|
icosahedra are distributed throughout the |
21 |
gezelter |
3230 |
material.\cite{PhysRevLett.60.2295} Simulations of freezing of single |
22 |
|
|
component metalic nanoclusters have shown a tendency for icosohedral |
23 |
|
|
structure formation particularly at the surfaces of these |
24 |
chuckv |
3254 |
clusters.\cite{Gafner:2004bg,PhysRevLett.89.275502,Chen:2004ec} |
25 |
gezelter |
3230 |
Experimentally, the splitting (or shoulder) on the second peak of the |
26 |
|
|
X-ray structure factor in binary metallic glasses has been attributed |
27 |
|
|
to the formation of tetrahedra that share faces of adjoining |
28 |
|
|
icosahedra.\cite{Waal:1995lr} |
29 |
chuckv |
3226 |
|
30 |
gezelter |
3230 |
Various structural probes have been used to characterize structural |
31 |
gezelter |
3242 |
order in molecular systems including: common neighbor analysis, |
32 |
|
|
Voronoi tesselations, and orientational bond-order |
33 |
gezelter |
3230 |
parameters.\cite{HoneycuttJ.Dana_j100303a014,Iwamatsu:2007lr,hsu:4974,nose:1803} |
34 |
gezelter |
3242 |
The method that has been used most extensively for determining local |
35 |
|
|
and extended orientational symmetry in condensed phases is the |
36 |
|
|
bond-orientational analysis formulated by Steinhart {\it et |
37 |
|
|
al.}\cite{Steinhardt:1983mo} In this model, a set of spherical |
38 |
gezelter |
3230 |
harmonics is associated with each of the near neighbors of a central |
39 |
|
|
atom. Neighbors (or ``bonds'') are defined as having a distance from |
40 |
|
|
the central atom that is within the first peak in the radial |
41 |
|
|
distribution function. The spherical harmonic between a central atom |
42 |
|
|
$i$ and a neighboring atom $j$ is |
43 |
chuckv |
3213 |
\begin{equation} |
44 |
gezelter |
3230 |
Y_{lm}\left(\vec{r}_{ij}\right)=Y_{lm}\left(\theta(\vec{r}_{ij}),\phi(\vec{r}_{ij})\right) |
45 |
|
|
\label{eq:spharm} |
46 |
chuckv |
3213 |
\end{equation} |
47 |
gezelter |
3230 |
where, $\{ Y_{lm}(\theta,\phi)\}$ are spherical harmonics, and |
48 |
|
|
$\theta(\vec{r})$ and $\phi(\vec{r})$ are the polar and azimuthal |
49 |
|
|
angles made by the bond vector $\vec{r}$ with respect to a reference |
50 |
|
|
coordinate system. We chose for simplicity the origin as defined by |
51 |
|
|
the coordinates for our nanoparticle. (Only even-$l$ spherical |
52 |
|
|
harmonics are considered since permutation of a pair of identical |
53 |
|
|
particles should not affect the bond-order parameter.) The local |
54 |
|
|
environment surrounding atom $i$ can be defined by |
55 |
|
|
the average over all neighbors, $N_b(i)$, surrounding that atom, |
56 |
chuckv |
3213 |
\begin{equation} |
57 |
gezelter |
3230 |
\bar{q}_{lm}(i) = \frac{1}{N_{b}(i)}\sum_{j=1}^{N_b(i)} Y_{lm}(\vec{r}_{ij}). |
58 |
|
|
\label{eq:local_avg_bo} |
59 |
chuckv |
3213 |
\end{equation} |
60 |
gezelter |
3230 |
We can further define a global average orientational-bond order over |
61 |
|
|
all $\bar{q}_{lm}$ by calculating the average of $\bar{q}_{lm}(i)$ |
62 |
|
|
over all $N$ particles |
63 |
chuckv |
3213 |
\begin{equation} |
64 |
gezelter |
3230 |
\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 |
chuckv |
3213 |
\end{equation} |
67 |
chuckv |
3261 |
The $\bar{Q}_{lm}$ contained in Eq. (\ref{eq:sys_avg_bo}) is not |
68 |
gezelter |
3233 |
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 |
71 |
|
|
of $\bar{Q}_{lm}$, |
72 |
chuckv |
3222 |
\begin{equation} |
73 |
gezelter |
3230 |
Q_{l} = \left( \frac{4\pi}{2 l+1} \sum^{l}_{m=-l} \left| \bar{Q}_{lm} \right|^2\right)^{1/2} |
74 |
|
|
\label{eq:sec_ord_inv} |
75 |
chuckv |
3222 |
\end{equation} |
76 |
|
|
and |
77 |
|
|
\begin{equation} |
78 |
|
|
\hat{W}_{l} = \frac{W_{l}}{\left( \sum^{l}_{m=-l} \left| \bar{Q}_{lm} \right|^2\right)^{3/2}} |
79 |
|
|
\label{eq:third_ord_inv} |
80 |
|
|
\end{equation} |
81 |
gezelter |
3230 |
where |
82 |
chuckv |
3222 |
\begin{equation} |
83 |
gezelter |
3230 |
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 |
chuckv |
3222 |
\label{eq:third_inv} |
85 |
|
|
\end{equation} |
86 |
chuckv |
3261 |
The factor in parentheses in Eq. (\ref{eq:third_inv}) is the Wigner-3$j$ |
87 |
gezelter |
3233 |
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 |
chuckv |
3226 |
|
90 |
|
|
\begin{table} |
91 |
gezelter |
3230 |
\caption{Values of bond orientational order parameters for |
92 |
|
|
simple structures cooresponding to $l=4$ and $l=6$ spherical harmonic |
93 |
|
|
functions.\cite{wolde:9932} $Q_4$ and $\hat{W}_4$ have values for {\it |
94 |
|
|
individual} icosahedral clusters, but these values are not invariant |
95 |
|
|
under rotations of the reference coordinate systems. Similar behavior |
96 |
|
|
is observed in the bond-orientational order parameters for individual |
97 |
|
|
liquid-like structures.} |
98 |
chuckv |
3226 |
\begin{center} |
99 |
|
|
\begin{tabular}{ccccc} |
100 |
|
|
\hline |
101 |
|
|
\hline |
102 |
|
|
& $Q_4$ & $Q_6$ & $\hat{W}_4$ & $\hat{W}_6$\\ |
103 |
|
|
|
104 |
|
|
fcc & 0.191 & 0.575 & -0.159 & -0.013\\ |
105 |
|
|
|
106 |
|
|
hcp & 0.097 & 0.485 & 0.134 & -0.012\\ |
107 |
|
|
|
108 |
|
|
bcc & 0.036 & 0.511 & 0.159 & 0.013\\ |
109 |
|
|
|
110 |
|
|
sc & 0.764 & 0.354 & 0.159 & 0.013\\ |
111 |
|
|
|
112 |
gezelter |
3230 |
Icosahedral & - & 0.663 & - & -0.170\\ |
113 |
chuckv |
3226 |
|
114 |
gezelter |
3230 |
(liquid) & - & - & - & -\\ |
115 |
chuckv |
3226 |
\hline |
116 |
|
|
\end{tabular} |
117 |
|
|
\end{center} |
118 |
|
|
\label{table:bopval} |
119 |
|
|
\end{table} |
120 |
|
|
|
121 |
gezelter |
3230 |
For ideal face-centered cubic ({\sc fcc}), body-centered cubic ({\sc |
122 |
|
|
bcc}) and simple cubic ({\sc sc}) as well as hexagonally close-packed |
123 |
|
|
({\sc hcp}) structures, these rotationally invariant bond order |
124 |
|
|
parameters have fixed values independent of the choice of coordinate |
125 |
|
|
reference frames. For ideal icosahedral structures, the $l=6$ |
126 |
|
|
invariants, $Q_6$ and $\hat{W}_6$ are also independent of the |
127 |
|
|
coordinate system. $Q_4$ and $\hat{W}_4$ will have non-vanishing |
128 |
|
|
values for {\it individual} icosahedral clusters, but these values are |
129 |
|
|
not invariant under rotations of the reference coordinate systems. |
130 |
|
|
Similar behavior is observed in the bond-orientational order |
131 |
gezelter |
3259 |
parameters for individual liquid-like structures. Additionally, both |
132 |
|
|
$Q_6$ and $\hat{W}_6$ are thought to have extreme values for the |
133 |
|
|
icosahedral clusters.\cite{Steinhardt:1983mo} This makes the $l=6$ |
134 |
|
|
bond-orientational order parameters particularly useful in identifying |
135 |
|
|
the extent of local icosahedral ordering in condensed phases. For |
136 |
|
|
example, a local structure which exhibits $\hat{W}_6$ values near |
137 |
|
|
-0.17 is easily identified as an icosahedral cluster and cannot be |
138 |
|
|
mistaken for distorted cubic or liquid-like structures. |
139 |
gezelter |
3230 |
|
140 |
gezelter |
3233 |
One may use these bond orientational order parameters as an averaged |
141 |
|
|
property to obtain the extent of icosahedral ordering in a supercooled |
142 |
|
|
liquid or cluster. It is also possible to accumulate information |
143 |
|
|
about the {\it distributions} of local bond orientational order |
144 |
|
|
parameters, $p(\hat{W}_6)$ and $p(Q_6)$, which provide information |
145 |
|
|
about individual atomic sites that are central to local icosahedral |
146 |
|
|
structures. |
147 |
gezelter |
3230 |
|
148 |
gezelter |
3233 |
The distributions of atomic $Q_6$ and $\hat{W}_6$ values are plotted |
149 |
chuckv |
3261 |
as a function of temperature for our nanoparticles in Fig. |
150 |
gezelter |
3233 |
\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 |
chuckv |
3261 |
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 $Q_6 = |
157 |
gezelter |
3233 |
0.663$.) |
158 |
|
|
|
159 |
|
|
\begin{figure}[htbp] |
160 |
|
|
\centering |
161 |
gezelter |
3242 |
\includegraphics[width=5in]{images/w6_stacked_plot.pdf} |
162 |
gezelter |
3233 |
\caption{Distributions of the bond orientational order parameter |
163 |
|
|
($\hat{W}_6$) at different temperatures. The upper, middle, and lower |
164 |
|
|
panels are for 20, 30, and 40 \AA\ particles, respectively. The |
165 |
|
|
left-hand column used cooling rates commensurate with a low |
166 |
|
|
interfacial conductance ($87.5 \times 10^{6}$ |
167 |
|
|
$\mathrm{Wm^{-2}K^{-1}}$), while the right-hand column used a more |
168 |
|
|
physically reasonable value of $117 \times 10^{6}$ |
169 |
|
|
$\mathrm{Wm^{-2}K^{-1}}$. The peak at $\hat{W}_6 \approx -0.17$ is |
170 |
gezelter |
3239 |
due to local icosahedral structures. The different curves in each of |
171 |
|
|
the panels indicate the distribution of $\hat{W}_6$ values for samples |
172 |
|
|
taken at different times along the cooling trajectory. The initial |
173 |
|
|
and final temperatures (in K) are indicated on the plots adjacent to |
174 |
|
|
their respective distributions.} |
175 |
gezelter |
3233 |
\label{fig:w6} |
176 |
|
|
\end{figure} |
177 |
|
|
|
178 |
|
|
\begin{figure}[htbp] |
179 |
|
|
\centering |
180 |
gezelter |
3242 |
\includegraphics[width=5in]{images/q6_stacked_plot.pdf} |
181 |
gezelter |
3233 |
\caption{Distributions of the bond orientational order parameter |
182 |
|
|
($Q_6$) at different temperatures. The curves in the six panels in |
183 |
|
|
this figure were computed at identical conditions to the same panels in |
184 |
|
|
figure \ref{fig:w6}.} |
185 |
|
|
\label{fig:q6} |
186 |
|
|
\end{figure} |
187 |
|
|
|
188 |
gezelter |
3252 |
The probability distributions of local order can be used to generate |
189 |
|
|
free energy surfaces using the local orientational ordering as a |
190 |
|
|
reaction coordinate. By making the simple statistical equivalence |
191 |
|
|
between the free energy and the probabilities of occupying certain |
192 |
|
|
states, |
193 |
|
|
\begin{equation} |
194 |
|
|
g(\hat{W}_6) = - k_B T \ln p(\hat{W}_6), |
195 |
|
|
\end{equation} |
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 |
chuckv |
3261 |
of temperatures are shown in Fig. \ref{fig:freeEnergy}. Note that |
200 |
gezelter |
3252 |
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 |
203 |
|
|
temperature falls, it is possible for substructures to become trapped |
204 |
|
|
in the local icosahedral well, and if the cooling is rapid enough, |
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 |
gezelter |
3279 |
Frenkel.\cite{duijneveldt:4655} |
209 |
gezelter |
3252 |
|
210 |
gezelter |
3279 |
|
211 |
gezelter |
3252 |
\begin{figure}[htbp] |
212 |
|
|
\centering |
213 |
|
|
\includegraphics[width=5in]{images/freeEnergyVsW6.pdf} |
214 |
|
|
\caption{Free energy as a function of the orientational order |
215 |
chuckv |
3261 |
parameter ($\hat{W}_6$) for 40 {\AA} bimetallic nanoparticles as they |
216 |
gezelter |
3252 |
are cooled from 902 K to 310 K. As the particles cool below 528 K, a |
217 |
|
|
local minimum in the free energy surface appears near the perfect |
218 |
|
|
icosahedral ordering ($\hat{W}_6 = -0.17$). At all temperatures, |
219 |
|
|
liquid-like structures are a global minimum on the free energy |
220 |
|
|
surface, but if the cooling rate is fast enough, substructures |
221 |
|
|
may become trapped with local icosahedral order, leading to the |
222 |
|
|
formation of a glass.} |
223 |
|
|
\label{fig:freeEnergy} |
224 |
|
|
\end{figure} |
225 |
|
|
|
226 |
gezelter |
3242 |
We have also calculated the fraction of atomic centers which have |
227 |
|
|
strong local icosahedral order: |
228 |
gezelter |
3233 |
\begin{equation} |
229 |
|
|
f_\textrm{icos} = \int_{-\infty}^{w_i} p(\hat{W}_6) d \hat{W}_6 |
230 |
|
|
\label{eq:ficos} |
231 |
|
|
\end{equation} |
232 |
|
|
where $w_i$ is a cutoff value in $\hat{W}_6$ for atomic centers that |
233 |
|
|
are displaying icosahedral environments. We have chosen a (somewhat |
234 |
|
|
arbitrary) value of $w_i= -0.15$ for the purposes of this work. A |
235 |
|
|
plot of $f_\textrm{icos}(T)$ as a function of temperature of the |
236 |
chuckv |
3261 |
particles is given in Fig. \ref{fig:ficos}. As the particles cool, |
237 |
gezelter |
3233 |
the fraction of local icosahedral ordering rises smoothly to a plateau |
238 |
gezelter |
3259 |
value. The smaller particles (particularly the ones that were cooled |
239 |
|
|
in a higher viscosity solvent) show a slightly larger tendency towards |
240 |
gezelter |
3242 |
icosahedral ordering. |
241 |
gezelter |
3233 |
|
242 |
|
|
\begin{figure}[htbp] |
243 |
|
|
\centering |
244 |
gezelter |
3242 |
\includegraphics[width=5in]{images/fraction_icos.pdf} |
245 |
gezelter |
3233 |
\caption{Temperautre dependence of the fraction of atoms with local |
246 |
|
|
icosahedral ordering, $f_\textrm{icos}(T)$ for 20, 30, and 40 \AA\ |
247 |
|
|
particles cooled at two different values of the interfacial |
248 |
|
|
conductance.} |
249 |
gezelter |
3242 |
\label{fig:ficos} |
250 |
gezelter |
3233 |
\end{figure} |
251 |
|
|
|
252 |
|
|
Since we have atomic-level resolution of the local bond-orientational |
253 |
|
|
ordering information, we can also look at the local ordering as a |
254 |
|
|
function of the identities of the central atoms. In figure |
255 |
|
|
\ref{fig:AgVsCu} we display the distributions of $\hat{W}_6$ values |
256 |
|
|
for both the silver and copper atoms, and we note a strong |
257 |
gezelter |
3242 |
predilection for the copper atoms to be central to icosahedra. This |
258 |
|
|
is probably due to local packing competition of the larger silver |
259 |
|
|
atoms around the copper, which would tend to favor icosahedral |
260 |
|
|
structures over the more densely packed cubic structures. |
261 |
gezelter |
3233 |
|
262 |
|
|
\begin{figure}[htbp] |
263 |
|
|
\centering |
264 |
gezelter |
3242 |
\includegraphics[width=5in]{images/w6_stacked_bytype_plot.pdf} |
265 |
gezelter |
3233 |
\caption{Distributions of the bond orientational order parameter |
266 |
|
|
($\hat{W}_6$) for the two different elements present in the |
267 |
|
|
nanoparticles. This distribution was taken from the fully-cooled 40 |
268 |
|
|
\AA\ nanoparticle. Local icosahedral ordering around copper atoms is |
269 |
|
|
much more prevalent than around silver atoms.} |
270 |
gezelter |
3242 |
\label{fig:AgVsCu} |
271 |
gezelter |
3233 |
\end{figure} |
272 |
gezelter |
3242 |
|
273 |
gezelter |
3252 |
The locations of these icosahedral centers are not uniformly |
274 |
chuckv |
3261 |
distrubted throughout the particles. In Fig. \ref{fig:icoscluster} |
275 |
gezelter |
3252 |
we show snapshots of the centers of the local icosahedra (i.e. any |
276 |
|
|
atom which exhibits a local bond orientational order parameter |
277 |
|
|
$\hat{W}_6 < -0.15$). At high temperatures, the icosahedral centers |
278 |
|
|
are transitory, existing only for a few fs before being reabsorbed |
279 |
|
|
into the liquid droplet. As the particle cools, these centers become |
280 |
|
|
fixed at certain locations, and additional icosahedra develop |
281 |
|
|
throughout the particle, clustering around the sites where the |
282 |
|
|
structures originated. There is a strong preference for icosahedral |
283 |
|
|
ordering near the surface of the particles. Identification of these |
284 |
|
|
structures by the type of atom shows that the silver-centered |
285 |
|
|
icosahedra are evident only at the surface of the particles. |
286 |
|
|
|
287 |
|
|
\begin{figure}[htbp] |
288 |
|
|
\centering |
289 |
|
|
\begin{tabular}{c c c} |
290 |
|
|
\includegraphics[width=2.1in]{images/Cu_Ag_random_30A_liq_icosonly.pdf} |
291 |
|
|
\includegraphics[width=2.1in]{images/Cu_Ag_random_30A__0007_icosonly.pdf} |
292 |
|
|
\includegraphics[width=2.1in]{images/Cu_Ag_random_30A_glass_icosonly.pdf} |
293 |
|
|
\end{tabular} |
294 |
|
|
\caption{Centers of local icosahedral order ($\hat{W}_6<0.15$) at 900 |
295 |
|
|
K, 471 K and 315 K for the 30 \AA\ nanoparticle cooled with an |
296 |
|
|
interfacial conductance $G = 87.5 \times 10^{6}$ |
297 |
|
|
$\mathrm{Wm^{-2}K^{-1}}$. Silver atoms (blue) exhibit icosahedral |
298 |
|
|
order at the surface of the nanoparticle while copper icosahedral |
299 |
|
|
centers (green) are distributed throughout the nanoparticle. The |
300 |
|
|
icosahedral centers appear to cluster together and these clusters |
301 |
|
|
increase in size with decreasing temperature.} |
302 |
|
|
\label{fig:icoscluster} |
303 |
|
|
\end{figure} |
304 |
|
|
|
305 |
gezelter |
3259 |
In contrast with the silver ordering behavior, the copper atoms which |
306 |
|
|
have local icosahedral ordering are distributed more evenly throughout |
307 |
chuckv |
3261 |
the nanoparticles. Fig. \ref{fig:Surface} shows this tendency as a |
308 |
gezelter |
3259 |
function of distance from the center of the nanoparticle. Silver, |
309 |
|
|
since it has a lower surface free energy than copper, tends to coat |
310 |
|
|
the skins of the mixed particles.\cite{Zhu:1997lr} This is true even |
311 |
|
|
for bimetallic particles that have been prepared in the Ag (core) / Cu |
312 |
|
|
(shell) configuration. Upon forming a liquid droplet, approximately 1 |
313 |
|
|
monolayer of Ag atoms will rise to the surface of the particles. This |
314 |
chuckv |
3261 |
can be seen visually in Fig. \ref{fig:cross_sections} as well as in |
315 |
|
|
the density plots in the bottom panel of Fig. \ref{fig:Surface}. |
316 |
gezelter |
3259 |
This observation is consistent with previous experimental and |
317 |
|
|
theoretical studies on bimetallic alloys composed of noble |
318 |
gezelter |
3247 |
metals.\cite{MainardiD.S._la0014306,HuangS.-P._jp0204206,Ramirez-Caballero:2006lr} |
319 |
|
|
Bond order parameters for surface atoms are averaged only over the |
320 |
|
|
neighboring atoms, so packing constraints that may prevent icosahedral |
321 |
|
|
ordering around silver in the bulk are removed near the surface. It |
322 |
|
|
would certainly be interesting to see if the relative tendency of |
323 |
|
|
silver and copper to form local icosahedral structures in a bulk glass |
324 |
|
|
differs from our observations on nanoparticles. |
325 |
|
|
|
326 |
|
|
\begin{figure}[htbp] |
327 |
|
|
\centering |
328 |
|
|
\includegraphics[width=5in]{images/dens_fracr_stacked_plot.pdf} |
329 |
|
|
\caption{Appearance of icosahedral clusters around central silver atoms |
330 |
|
|
is largely due to the presence of these silver atoms at or near the |
331 |
|
|
surface of the nanoparticle. The upper panel shows the fraction of |
332 |
|
|
icosahedral atoms ($f_\textrm{icos}(r)$ for each of the two metallic |
333 |
|
|
atoms as a function of distance from the center of the nanoparticle |
334 |
|
|
($r$). The lower panel shows the radial density of the two |
335 |
|
|
constituent metals (relative to the overall density of the |
336 |
|
|
nanoparticle). Icosahedral clustering around copper atoms are more |
337 |
|
|
evenly distributed throughout the particle, while icosahedral |
338 |
|
|
clustering around silver is largely confined to the silver atoms at |
339 |
|
|
the surface.} |
340 |
|
|
\label{fig:Surface} |
341 |
|
|
\end{figure} |
342 |
gezelter |
3279 |
|
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 |
chuckv |
3280 |
approximately 488 K. We note that this temperature is somewhat below |
367 |
gezelter |
3279 |
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 |
chuckv |
3280 |
(K/s $\times 10^{13}$) & & $T_g$ (K) \\ |
389 |
|
|
20 & 87.5 & 2.4 & 477 \\ |
390 |
|
|
20 & 117 & 4.5 & 502 \\ |
391 |
|
|
30 & 87.5 & 1.3 & 491 \\ |
392 |
|
|
30 & 117 & 1.9 & 493 \\ |
393 |
|
|
40 & 87.5 & 1.0 & 476 \\ |
394 |
|
|
40 & 117 & 1.3 & 487 \\ |
395 |
gezelter |
3279 |
\hline |
396 |
|
|
\end{tabular} |
397 |
|
|
\end{center} |
398 |
|
|
\label{table:Tg} |
399 |
|
|
\end{table} |