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%!TEX root = /Users/charles/Desktop/nanoglass/nanoglass.tex |
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\section{Analysis} |
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Frank first proposed local icosahedral ordering of atoms as an |
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explanation for supercooled atomic (specifically metallic) liquids, |
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and further showed that a 13-atom icosahedral cluster has a 8.4\% |
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higher binding energy the either a face centered cubic ({\sc fcc}) or |
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hexagonal close-packed ({\sc hcp}) crystal structures.\cite{19521106} |
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Icosahedra also have six five-fold symmetry axes that cannot be |
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extended indefinitely in three dimensions, which makes them long-range |
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translational order incommensurate with local icosahedral ordering. |
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This does not preclude icosahedral clusters from possessing long-range |
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{\it orientational} order. The ``frustrated'' packing of these |
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icosahedral structures into dense clusters has been proposed as a |
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model for glass formation.\cite{19871127} The size of the icosahedral |
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clusters is thought to increase until frustration prevents any further |
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growth.\cite{HOARE:1976fk} Molecular dynamics simulations of a |
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two-component Lennard-Jones glass showed that clusters of face-sharing |
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icosahedra are distributed throughout the |
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material.\cite{PhysRevLett.60.2295} Simulations of freezing of single |
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component metalic nanoclusters have shown a tendency for icosohedral |
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structure formation particularly at the surfaces of these |
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clusters.\cite{Gafner:2004bg,PhysRevLett.89.275502,Ascencio:2000qy,Chen:2004ec} |
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Experimentally, the splitting (or shoulder) on the second peak of the |
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X-ray structure factor in binary metallic glasses has been attributed |
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to the formation of tetrahedra that share faces of adjoining |
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icosahedra.\cite{Waal:1995lr} |
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|
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Various structural probes have been used to characterize structural |
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order in molecular systems including: common neighbor analysis, |
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Voronoi tesselations, and orientational bond-order |
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parameters.\cite{HoneycuttJ.Dana_j100303a014,Iwamatsu:2007lr,hsu:4974,nose:1803} |
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The method that has been used most extensively for determining local |
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and extended orientational symmetry in condensed phases is the |
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bond-orientational analysis formulated by Steinhart {\it et |
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al.}\cite{Steinhardt:1983mo} In this model, a set of spherical |
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harmonics is associated with each of the near neighbors of a central |
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atom. Neighbors (or ``bonds'') are defined as having a distance from |
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the central atom that is within the first peak in the radial |
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distribution function. The spherical harmonic between a central atom |
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$i$ and a neighboring atom $j$ is |
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\begin{equation} |
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Y_{lm}\left(\vec{r}_{ij}\right)=Y_{lm}\left(\theta(\vec{r}_{ij}),\phi(\vec{r}_{ij})\right) |
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\label{eq:spharm} |
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\end{equation} |
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where, $\{ Y_{lm}(\theta,\phi)\}$ are spherical harmonics, and |
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$\theta(\vec{r})$ and $\phi(\vec{r})$ are the polar and azimuthal |
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angles made by the bond vector $\vec{r}$ with respect to a reference |
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coordinate system. We chose for simplicity the origin as defined by |
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the coordinates for our nanoparticle. (Only even-$l$ spherical |
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harmonics are considered since permutation of a pair of identical |
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particles should not affect the bond-order parameter.) The local |
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environment surrounding atom $i$ can be defined by |
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the average over all neighbors, $N_b(i)$, surrounding that atom, |
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\begin{equation} |
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\bar{q}_{lm}(i) = \frac{1}{N_{b}(i)}\sum_{j=1}^{N_b(i)} Y_{lm}(\vec{r}_{ij}). |
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\label{eq:local_avg_bo} |
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\end{equation} |
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We can further define a global average orientational-bond order over |
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all $\bar{q}_{lm}$ by calculating the average of $\bar{q}_{lm}(i)$ |
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over all $N$ particles |
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\begin{equation} |
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\bar{Q}_{lm} = \frac{\sum^{N}_{i=1}N_{b}(i)\bar{q}_{lm}(i)}{\sum^{N}_{i=1}N_{b}(i)} |
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\label{eq:sys_avg_bo} |
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\end{equation} |
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The $\bar{Q}_{lm}$ contained in equation \ref{eq:sys_avg_bo} is not |
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necessarily invariant under rotations of the arbitrary reference |
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coordinate system. Second- and third-order rotationally invariant |
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combinations, $Q_l$ and $W_l$, can be taken by summing over $m$ values |
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of $\bar{Q}_{lm}$, |
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\begin{equation} |
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Q_{l} = \left( \frac{4\pi}{2 l+1} \sum^{l}_{m=-l} \left| \bar{Q}_{lm} \right|^2\right)^{1/2} |
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\label{eq:sec_ord_inv} |
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\end{equation} |
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and |
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\begin{equation} |
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\hat{W}_{l} = \frac{W_{l}}{\left( \sum^{l}_{m=-l} \left| \bar{Q}_{lm} \right|^2\right)^{3/2}} |
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\label{eq:third_ord_inv} |
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\end{equation} |
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where |
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\begin{equation} |
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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}. |
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\label{eq:third_inv} |
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\end{equation} |
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The factor in parentheses in Eq. \ref{eq:third_inv} is the Wigner-3$j$ |
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symbol, and the sum is over all valid ($|m| \leq l$) values of $m_1$, |
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$m_2$, and $m_3$ which sum to zero. |
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\begin{table} |
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\caption{Values of bond orientational order parameters for |
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simple structures cooresponding to $l=4$ and $l=6$ spherical harmonic |
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functions.\cite{wolde:9932} $Q_4$ and $\hat{W}_4$ have values for {\it |
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individual} icosahedral clusters, but these values are not invariant |
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under rotations of the reference coordinate systems. Similar behavior |
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is observed in the bond-orientational order parameters for individual |
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liquid-like structures.} |
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\begin{center} |
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\begin{tabular}{ccccc} |
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\hline |
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\hline |
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& $Q_4$ & $Q_6$ & $\hat{W}_4$ & $\hat{W}_6$\\ |
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fcc & 0.191 & 0.575 & -0.159 & -0.013\\ |
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hcp & 0.097 & 0.485 & 0.134 & -0.012\\ |
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bcc & 0.036 & 0.511 & 0.159 & 0.013\\ |
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sc & 0.764 & 0.354 & 0.159 & 0.013\\ |
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Icosahedral & - & 0.663 & - & -0.170\\ |
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|
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(liquid) & - & - & - & -\\ |
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\hline |
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\end{tabular} |
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\end{center} |
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\label{table:bopval} |
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\end{table} |
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For ideal face-centered cubic ({\sc fcc}), body-centered cubic ({\sc |
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bcc}) and simple cubic ({\sc sc}) as well as hexagonally close-packed |
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({\sc hcp}) structures, these rotationally invariant bond order |
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parameters have fixed values independent of the choice of coordinate |
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reference frames. For ideal icosahedral structures, the $l=6$ |
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invariants, $Q_6$ and $\hat{W}_6$ are also independent of the |
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coordinate system. $Q_4$ and $\hat{W}_4$ will have non-vanishing |
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values for {\it individual} icosahedral clusters, but these values are |
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not invariant under rotations of the reference coordinate systems. |
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Similar behavior is observed in the bond-orientational order |
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parameters for individual liquid-like structures. |
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Additionally, both $Q_6$ and $\hat{W}_6$ are thought to have extreme |
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values for the icosahedral clusters.\cite{Steinhardt:1983mo} This |
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makes the $l=6$ bond-orientational order parameters particularly |
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useful in identifying the extent of local icosahedral ordering in |
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condensed phases. For example, a local structure which exhibits |
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$\hat{W}_6$ values near -0.17 is easily identified as an icosahedral |
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cluster and cannot be mistaken for distorted cubic or liquid-like |
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structures. |
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One may use these bond orientational order parameters as an averaged |
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property to obtain the extent of icosahedral ordering in a supercooled |
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liquid or cluster. It is also possible to accumulate information |
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about the {\it distributions} of local bond orientational order |
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parameters, $p(\hat{W}_6)$ and $p(Q_6)$, which provide information |
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about individual atomic sites that are central to local icosahedral |
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structures. |
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|
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The distributions of atomic $Q_6$ and $\hat{W}_6$ values are plotted |
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as a function of temperature for our nanoparticles in figures |
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\ref{fig:q6} and \ref{fig:w6}. At high temperatures, the |
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distributions are unstructured and are broadly distributed across the |
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entire range of values. As the particles are cooled, however, there |
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is a dramatic increase in the fraction of atomic sites which have |
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local icosahedral ordering around them. (This corresponds to the |
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sharp peak appearing in figure \ref{fig:w6} at $\hat{W}_6=-0.17$ and |
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to the broad shoulder appearing in figure \ref{fig:q6} at $Q_6 = |
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0.663$.) |
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\begin{figure}[htbp] |
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\centering |
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\includegraphics[width=5in]{images/w6_stacked_plot.pdf} |
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\caption{Distributions of the bond orientational order parameter |
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($\hat{W}_6$) at different temperatures. The upper, middle, and lower |
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panels are for 20, 30, and 40 \AA\ particles, respectively. The |
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left-hand column used cooling rates commensurate with a low |
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interfacial conductance ($87.5 \times 10^{6}$ |
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$\mathrm{Wm^{-2}K^{-1}}$), while the right-hand column used a more |
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physically reasonable value of $117 \times 10^{6}$ |
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$\mathrm{Wm^{-2}K^{-1}}$. The peak at $\hat{W}_6 \approx -0.17$ is |
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due to local icosahedral structures. The different curves in each of |
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the panels indicate the distribution of $\hat{W}_6$ values for samples |
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taken at different times along the cooling trajectory. The initial |
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and final temperatures (in K) are indicated on the plots adjacent to |
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their respective distributions.} |
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\label{fig:w6} |
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\end{figure} |
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\begin{figure}[htbp] |
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\centering |
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\includegraphics[width=5in]{images/q6_stacked_plot.pdf} |
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\caption{Distributions of the bond orientational order parameter |
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($Q_6$) at different temperatures. The curves in the six panels in |
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this figure were computed at identical conditions to the same panels in |
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figure \ref{fig:w6}.} |
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\label{fig:q6} |
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\end{figure} |
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We have also calculated the fraction of atomic centers which have |
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strong local icosahedral order: |
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\begin{equation} |
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f_\textrm{icos} = \int_{-\infty}^{w_i} p(\hat{W}_6) d \hat{W}_6 |
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\label{eq:ficos} |
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\end{equation} |
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where $w_i$ is a cutoff value in $\hat{W}_6$ for atomic centers that |
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are displaying icosahedral environments. We have chosen a (somewhat |
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arbitrary) value of $w_i= -0.15$ for the purposes of this work. A |
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plot of $f_\textrm{icos}(T)$ as a function of temperature of the |
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particles is given in figure \ref{fig:ficos}. As the particles cool, |
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the fraction of local icosahedral ordering rises smoothly to a plateau |
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value. The larger particles (particularly the ones that were cooled |
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in a lower viscosity solvent) show a slightly smaller tendency towards |
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icosahedral ordering. |
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|
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\begin{figure}[htbp] |
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\centering |
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\includegraphics[width=5in]{images/fraction_icos.pdf} |
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\caption{Temperautre dependence of the fraction of atoms with local |
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icosahedral ordering, $f_\textrm{icos}(T)$ for 20, 30, and 40 \AA\ |
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particles cooled at two different values of the interfacial |
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conductance.} |
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\label{fig:ficos} |
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\end{figure} |
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Since we have atomic-level resolution of the local bond-orientational |
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ordering information, we can also look at the local ordering as a |
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function of the identities of the central atoms. In figure |
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\ref{fig:AgVsCu} we display the distributions of $\hat{W}_6$ values |
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for both the silver and copper atoms, and we note a strong |
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predilection for the copper atoms to be central to icosahedra. This |
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is probably due to local packing competition of the larger silver |
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atoms around the copper, which would tend to favor icosahedral |
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structures over the more densely packed cubic structures. |
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|
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\begin{figure}[htbp] |
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\centering |
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\includegraphics[width=5in]{images/w6_stacked_bytype_plot.pdf} |
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\caption{Distributions of the bond orientational order parameter |
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($\hat{W}_6$) for the two different elements present in the |
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nanoparticles. This distribution was taken from the fully-cooled 40 |
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\AA\ nanoparticle. Local icosahedral ordering around copper atoms is |
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much more prevalent than around silver atoms.} |
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\label{fig:AgVsCu} |
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\end{figure} |
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|
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Additionally, we have observed that those silver atoms that {\it do} |
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form local icosahedral structures are usually on the surface of the |
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nanoparticle, while the copper atoms which have local icosahedral |
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ordering are distributed more evenly throughout the nanoparticles. |
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Figure \ref{fig:Surface} shows this tendency as a function of distance |
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from the center of the nanoparticle. Silver, since it has a lower |
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surface free energy than copper, tends to coat the skins of the mixed |
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particles.\cite{Zhu:1997lr} This is true even for bimetallic particles |
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that have been prepared in the Ag (core) / Cu (shell) configuration. |
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Upon forming a liquid droplet, approximately 1 monolayer of Ag atoms |
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will rise to the surface of the particles. This can be seen visually |
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in figure \ref{fig:cross_sections} as well as in the density plots in |
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the bottom panel of figure \ref{fig:Surface}. This observation is |
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consistent with previous experimental and theoretical studies on |
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bimetallic alloys composed of noble |
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metals.\cite{MainardiD.S._la0014306,HuangS.-P._jp0204206,Ramirez-Caballero:2006lr} |
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Bond order parameters for surface atoms are averaged only over the |
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neighboring atoms, so packing constraints that may prevent icosahedral |
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ordering around silver in the bulk are removed near the surface. It |
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would certainly be interesting to see if the relative tendency of |
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silver and copper to form local icosahedral structures in a bulk glass |
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differs from our observations on nanoparticles. |
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|
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|
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\begin{figure}[htbp] |
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\centering |
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\begin{tabular}{c c c} |
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\includegraphics[width=2.1in]{images/Cu_Ag_random_30A_liq_icosonly.pdf} |
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\includegraphics[width=2.1in]{images/Cu_Ag_random_30A__0007_icosonly.pdf} |
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\includegraphics[width=2.1in]{images/Cu_Ag_random_30A_glass_icosonly.pdf} |
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\end{tabular} |
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\caption{Appearance of icosahedral clusters ($\hat{W}_6<0.15$) at 900 K, 471 K and 315 K for the 30 \AA\ nanoparticle cooled at the slower cooling rate. Silver atoms (blue) mostly exhibit icosahedral order at the surface whereas clusters of Copper atoms (green) with icosahedral order are distributed throughout the nanoparticle and increase in size with decreasing temperature.} |
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\label{fig:icoscluster} |
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\end{figure} |
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\begin{figure}[htbp] |
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\centering |
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\includegraphics[width=5in]{images/dens_fracr_stacked_plot.pdf} |
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\caption{Appearance of icosahedral clusters around central silver atoms |
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is largely due to the presence of these silver atoms at or near the |
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surface of the nanoparticle. The upper panel shows the fraction of |
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icosahedral atoms ($f_\textrm{icos}(r)$ for each of the two metallic |
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atoms as a function of distance from the center of the nanoparticle |
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($r$). The lower panel shows the radial density of the two |
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constituent metals (relative to the overall density of the |
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nanoparticle). Icosahedral clustering around copper atoms are more |
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evenly distributed throughout the particle, while icosahedral |
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clustering around silver is largely confined to the silver atoms at |
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the surface.} |
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\label{fig:Surface} |
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\end{figure} |