| 1 |
chuckv |
3226 |
%!TEX root = /Users/charles/Desktop/nanoglass/nanoglass.tex |
| 2 |
|
|
|
| 3 |
chuckv |
3213 |
\section{Analysis} |
| 4 |
|
|
|
| 5 |
gezelter |
3230 |
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 |
chuckv |
3226 |
|
| 31 |
gezelter |
3230 |
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 |
gezelter |
3233 |
{\it et al.}\cite{Steinhardt:1983mo} In this model, a set of spherical |
| 39 |
gezelter |
3230 |
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 |
chuckv |
3213 |
\begin{equation} |
| 45 |
gezelter |
3230 |
Y_{lm}\left(\vec{r}_{ij}\right)=Y_{lm}\left(\theta(\vec{r}_{ij}),\phi(\vec{r}_{ij})\right) |
| 46 |
|
|
\label{eq:spharm} |
| 47 |
chuckv |
3213 |
\end{equation} |
| 48 |
gezelter |
3230 |
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 |
chuckv |
3213 |
\begin{equation} |
| 58 |
gezelter |
3230 |
\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 |
chuckv |
3213 |
\end{equation} |
| 61 |
gezelter |
3230 |
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 |
chuckv |
3213 |
\begin{equation} |
| 65 |
gezelter |
3230 |
\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 |
chuckv |
3213 |
\end{equation} |
| 68 |
gezelter |
3233 |
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 |
chuckv |
3222 |
\begin{equation} |
| 74 |
gezelter |
3230 |
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 |
chuckv |
3222 |
\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 |
gezelter |
3230 |
where |
| 83 |
chuckv |
3222 |
\begin{equation} |
| 84 |
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}. |
| 85 |
chuckv |
3222 |
\label{eq:third_inv} |
| 86 |
|
|
\end{equation} |
| 87 |
gezelter |
3230 |
The factor in parentheses in Eq. \ref{eq:third_inv} is the Wigner-3$j$ |
| 88 |
gezelter |
3233 |
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 |
chuckv |
3226 |
|
| 91 |
|
|
\begin{table} |
| 92 |
gezelter |
3230 |
\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 |
chuckv |
3226 |
\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 |
gezelter |
3230 |
Icosahedral & - & 0.663 & - & -0.170\\ |
| 114 |
chuckv |
3226 |
|
| 115 |
gezelter |
3230 |
(liquid) & - & - & - & -\\ |
| 116 |
chuckv |
3226 |
\hline |
| 117 |
|
|
\end{tabular} |
| 118 |
|
|
\end{center} |
| 119 |
|
|
\label{table:bopval} |
| 120 |
|
|
\end{table} |
| 121 |
|
|
|
| 122 |
gezelter |
3230 |
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 |
gezelter |
3233 |
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 |
gezelter |
3230 |
|
| 143 |
gezelter |
3233 |
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 |
gezelter |
3230 |
|
| 151 |
gezelter |
3233 |
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 |
gezelter |
3239 |
\includegraphics[width=\linewidth]{images/w6_stacked_plot.pdf} |
| 165 |
gezelter |
3233 |
\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 |
gezelter |
3239 |
due to local icosahedral structures. The different curves in each of |
| 174 |
|
|
the panels indicate the distribution of $\hat{W}_6$ values for samples |
| 175 |
|
|
taken at different times along the cooling trajectory. The initial |
| 176 |
|
|
and final temperatures (in K) are indicated on the plots adjacent to |
| 177 |
|
|
their respective distributions.} |
| 178 |
gezelter |
3233 |
\label{fig:w6} |
| 179 |
|
|
\end{figure} |
| 180 |
|
|
|
| 181 |
|
|
\begin{figure}[htbp] |
| 182 |
|
|
\centering |
| 183 |
gezelter |
3239 |
\includegraphics[width=\linewidth]{images/q6_stacked_plot.pdf} |
| 184 |
gezelter |
3233 |
\caption{Distributions of the bond orientational order parameter |
| 185 |
|
|
($Q_6$) at different temperatures. The curves in the six panels in |
| 186 |
|
|
this figure were computed at identical conditions to the same panels in |
| 187 |
|
|
figure \ref{fig:w6}.} |
| 188 |
|
|
\label{fig:q6} |
| 189 |
|
|
\end{figure} |
| 190 |
|
|
|
| 191 |
|
|
We have also looked at the fraction of atomic centers which have local |
| 192 |
|
|
icosahedral order: |
| 193 |
|
|
\begin{equation} |
| 194 |
|
|
f_\textrm{icos} = \int_{-\infty}^{w_i} p(\hat{W}_6) d \hat{W}_6 |
| 195 |
|
|
\label{eq:ficos} |
| 196 |
|
|
\end{equation} |
| 197 |
|
|
where $w_i$ is a cutoff value in $\hat{W}_6$ for atomic centers that |
| 198 |
|
|
are displaying icosahedral environments. We have chosen a (somewhat |
| 199 |
|
|
arbitrary) value of $w_i= -0.15$ for the purposes of this work. A |
| 200 |
|
|
plot of $f_\textrm{icos}(T)$ as a function of temperature of the |
| 201 |
|
|
particles is given in figure \ref{fig:ficos}. As the particles cool, |
| 202 |
|
|
the fraction of local icosahedral ordering rises smoothly to a plateau |
| 203 |
|
|
value. The larger particles (particularly the ones that were cooled |
| 204 |
|
|
in a lower viscosity solvent) show a lower tendency towards icosahedral |
| 205 |
|
|
ordering. |
| 206 |
|
|
|
| 207 |
|
|
\begin{figure}[htbp] |
| 208 |
|
|
\centering |
| 209 |
gezelter |
3239 |
\includegraphics[width=\linewidth]{images/fraction_icos.pdf} |
| 210 |
gezelter |
3233 |
\caption{Temperautre dependence of the fraction of atoms with local |
| 211 |
|
|
icosahedral ordering, $f_\textrm{icos}(T)$ for 20, 30, and 40 \AA\ |
| 212 |
|
|
particles cooled at two different values of the interfacial |
| 213 |
|
|
conductance.} |
| 214 |
|
|
\label{fig:q6} |
| 215 |
|
|
\end{figure} |
| 216 |
|
|
|
| 217 |
|
|
Since we have atomic-level resolution of the local bond-orientational |
| 218 |
|
|
ordering information, we can also look at the local ordering as a |
| 219 |
|
|
function of the identities of the central atoms. In figure |
| 220 |
|
|
\ref{fig:AgVsCu} we display the distributions of $\hat{W}_6$ values |
| 221 |
|
|
for both the silver and copper atoms, and we note a strong |
| 222 |
|
|
predilection for the copper atoms to be central to local icosahedral |
| 223 |
|
|
ordering. This is probably due to local packing competition of the |
| 224 |
|
|
larger silver atoms around the copper, which would tend to favor |
| 225 |
|
|
icosahedral structures over the more densely packed cubic structures. |
| 226 |
|
|
|
| 227 |
|
|
\begin{figure}[htbp] |
| 228 |
|
|
\centering |
| 229 |
gezelter |
3239 |
\includegraphics[width=\linewidth]{images/w6_stacked_bytype_plot.pdf} |
| 230 |
gezelter |
3233 |
\caption{Distributions of the bond orientational order parameter |
| 231 |
|
|
($\hat{W}_6$) for the two different elements present in the |
| 232 |
|
|
nanoparticles. This distribution was taken from the fully-cooled 40 |
| 233 |
|
|
\AA\ nanoparticle. Local icosahedral ordering around copper atoms is |
| 234 |
|
|
much more prevalent than around silver atoms.} |
| 235 |
|
|
\label{fig:q6} |
| 236 |
|
|
\end{figure} |
