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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 icosahedral arrangement of atoms as a model for |
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structure supercooled atomic liquids.\cite{19521106} The ability to |
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cool simple liquid metals well below their equilibrium melting |
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temperatures was attributed to this local icosahedral ordering. Frank |
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further showed that a 13-atom icosahedral cluster has a 8.4\% higher |
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binding energy the either a face centered cubic ({\sc fcc}) or |
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hexagonal close-packed ({\sc hcp}) crystal structure. Icosahedra also |
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have six fivefold symmetry axes that cannot be extended indefinitely |
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in three dimensions making them incommensurate with long-range |
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translational order. This does not preclude icosahedral clusters from |
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possessing long-range {\it orientational} order. The ``frustrated'' |
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packing of these icosahedral structures into dense clusters has been |
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proposed as a model for glass formation.\cite{19871127} The size of |
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the icosahedral clusters is thought to increase until frustration |
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prevents any further growth.\cite{HOARE:1976fk} Molecular dynamics |
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simulations of a two-component Lennard-Jones glass showed that |
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clusters of face-sharing 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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gezelter |
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Various structural probes have been used to characterize structural |
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order in systems including: common neighbor analysis, voronoi-analysis |
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and orientational bond-order |
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parameters.\cite{HoneycuttJ.Dana_j100303a014,Iwamatsu:2007lr,hsu:4974,nose:1803} |
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One method that has been used extensively for determining local and |
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extended orientational symmetry in condensed phases is the |
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bond-orientational analysis formulated by Steinhart |
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{\it et 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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(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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gezelter |
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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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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=\linewidth]{images/w6fig.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.} |
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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=\linewidth]{images/q6fig.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 looked at the fraction of atomic centers which have local |
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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 lower tendency towards icosahedral |
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ordering. |
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\begin{figure}[htbp] |
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\centering |
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%\includegraphics[width=\linewidth]{images/ficos.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:q6} |
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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 local icosahedral |
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ordering. This is probably due to local packing competition of the |
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larger silver atoms around the copper, which would tend to favor |
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icosahedral structures over the more densely packed cubic structures. |
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\begin{figure}[htbp] |
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\centering |
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%\includegraphics[width=\linewidth]{images/AgVsCu.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:q6} |
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\end{figure} |
