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\section{Analysis} |
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|
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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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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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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 systems including: common neighbor analysis, voronoi-analysis |
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and orientational bond-order |
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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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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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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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|
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\begin{figure}[htbp] |
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\centering |
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\includegraphics[width=\linewidth]{images/w6_stacked_plot.pdf} |
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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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|
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\begin{figure}[htbp] |
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\centering |
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\includegraphics[width=\linewidth]{images/q6_stacked_plot.pdf} |
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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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\label{fig:q6} |
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\end{figure} |
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|
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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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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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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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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=\linewidth]{images/fraction_icos.pdf} |
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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:q6} |
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\label{fig:ficos} |
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\end{figure} |
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|
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Since we have atomic-level resolution of the local bond-orientational |
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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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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=\linewidth]{images/w6_stacked_bytype_plot.pdf} |
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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:q6} |
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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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Silver, since it has a lower surface free energy than copper, tends to |
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coat the skins of the mixed particles. This is true even for |
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bimetallic particles that have been prepared in the Ag (core) / Cu |
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(shell) configuration. Upon forming a liquid droplet, approximately 1 |
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monolayer of Ag atoms will rise to the surface of the particles. This |
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can be seen visually in figure \ref{fig:cross_sections}. Bond order |
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parameters for surface atoms are averaged only over the neighboring |
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atoms, so packing constraints that may prevent icosahedral ordering |
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around silver in the bulk are removed near the surface. It would |
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certainly be interesting to see if the relative tendency of silver and |
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copper to form local icosahedral structures in a bulk glass differs |
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from our observations on nanoparticles. |
