Icosahedral Orientational Order in Glasses

 

    Characterization of glassy behavior by molecular dynamics simulations is typically done using dynamic measurements such as the mean squared displacement, <r2(t)>. Liquids exhibit a mean squared displacement that is linear in time. Glassy materials deviate significantly from this linear behavior at intermediate times, entering a sub-linear regime with a return to linear behavior in the infinite time limit. Diffusion in nanoparticles differs significantly from the bulk in that atoms are confined to a roughly spherical volume and cannot explore any region larger than the particle radius. In these confined geometries,

<r2(t)> in the radial direction approaches a limiting value of 6R2/40.


    However, glassy materials exhibit strong icosahedral ordering among nearest-neghbors in contrast to crystalline or liquid structures. Steinhart, et al., defined an orientational bond order parameter that is sensitive to the nearest-neighbor environment by using invariant combinations of spherical harmonics Yl,m(θ,ϕ).[12] Spherical harmonics involving
the Y6,m(θ,ϕ) are particularly sensitive to icosohedral order among nearest neighbors as can be seen in the cartoon to the left. The second and third-order invariants, Q6 and Ŵ6 are used to determine the level of icosahedral order present in a quenched nanoparticle. Perfect icosahedral structures have a maximal value of 0.663 for Q6 and -0.170 for Ŵ6. A plot of the distributions of Q6 and Ŵ6 with cooling temperature indicates increasing icosahedral order with decreasing temperature.  This is a clear indication that glassy structures are forming as the nanoparticles are quenched.
 
Distributions for Q6 at different temperatures corresponding to the same conditions of the Ŵ6 profile. Glass formation is indicated by the shoulder growth at Q6 = 0.663.
 
Distributions of the bond orientational order parameter Ŵ6 at different temperatures. The upper, middle, and lower panels are for 20, 30, and 40 Å particles, respectively. The left-hand column used cooling rates assuming low interfacial conductance (87.5 × 106 Wm−2 K−1 ), while the right-hand column used a more physically reasonable value of 117×106 Wm−2 K−1 . The peak at Ŵ6 ≈ −0.17 is due to local icosahedral structures. The different curves in each of the panels indicate the distribution of Ŵ6 values for samples taken at different times along the cooling trajectory. The initial and final temperatures (in K) are indicated on the plots adjacent to their respective distributions.
 
Distributions for Ŵ6 from a fully-cooled 40 Å by atomic identity of the central atom. Local icosahedral ordering around the copper atoms is much more prevalent than around silver atoms.
 

We can visualize the growth of icosahedral order (Ŵ6<-0.15) at 900 K, 471 K and 315 K for the 30 Å cooled at the slower cooling rate corresponding to 87.5 × 106 Wm−2 K−1. Silver atoms (blue) exhibit icosahedral order at the surface while copper centers are distributed throughout the nanoparticle. Growth of the copper centered clusters appears more prominent at lower temperature and the clusters increase in size with decreasing temperature. Click the images for larger versions.