Magical Mathematical Formulas for Nanoboxes.

Hollow nanostructures are at the forefront of many scientific endeavors. These consist of nanoboxes, nanocages, nanoframes, and nanotubes. We examine the mathematics of atomic coordination in nanoboxes. Such structures consist of a hollow box with n shells and t outer layers. The magical formulas we derive depend on both n and t. We find that nanoboxes with t = 2  or  3, or walls with only a few layers generally have bulk coordinated atoms. The benefits of low-coordination in nanostructures is shown to only occur when the wall thickness is much thinner than normally synthesized. The case where t = 1 is unique, and has distinct magic formulas. Such low-coordinated nanoboxes are of interest for a myriad variety of applications, including batteries, fuel cells, plasmonic, catalytic and biomedical uses. Given these formulas, it is possible to determine the surface dispersion of the nanoboxes. We expect these formulas to be useful in understanding how the atomic coordination varies with n and t within a nanobox.

Magic Mathematical Relationships for Nanoclusters-Errata and Addendum.

We correct magic formulas for body centered cubic (bcc) structures. The logical rational for this is further corroborated by calculations of the radial distribution function (RDF) for several crystal structures. We add results for truncated cubes which may be found in nature.

Magic Mathematical Relationships for Nanoclusters.

Size and surface properties such as catalysis, optical quantum dot photoluminescense, and surface plasmon resonances depend on the coordination and chemistry of metal and semiconducting nanoclusters. Such coordination-dependent properties are quantified herein via "magic formulas" for the number of shells, n, in the cluster. We investigate face-centered cubic, body-centered cubic, simple cubic clusters, hexagonal close-packed clusters, and the diamond cubic structure as a function of the number of cluster shells, n. In addition, we examine the Platonic solids in the form of multi-shell clusters, for a total of 19 cluster types. The number of bonds and atoms and coordination numbers exhibit magic number characteristics versus n, as the size of the clusters increases. Starting with only the spatial coordinates, we create an adjacency and distance matrix that facilitates the calculation of topological indices, including the Wiener, hyper-Wiener, reverse Wiener, and Szeged indices. Some known topological formulas for some Platonic solids when n=1 are computationally verified. These indices have magic formulas for many of the clusters. The simple cubic structure is the least complex of our clusters as measured by the topological complexity derived from the information content of the vertex-degree distribution. The dispersion, or relative percentage of surface atoms, is measured quantitatively with respect to size and shape dependence for some types of clusters with catalytic applications.

Size, shape, and compositional effects on the order-disorder phase transitions in Au-Cu and Pt-M (M = Fe, Co, and Ni) nanocluster alloys.

Au-Cu and Pt-M (M = Fe, Co, and Ni) nanocluster alloys are currently being investigated world-wide by many researchers for their interesting catalytic and nanophase properties. The low temperature behavior of the phase diagrams is not well understood for alloys with nanometer sizes and shapes. We consider two models for low temperature ordering in the phase diagrams of Au-Cu and Pt-M nanocluster alloys. These models are valid for sizes ∼5 nm and approach bulk values for sizes ∼20 nm. We study the phase transitions in nanoclusters with cubic, octahedral, and cuboctahedral shapes, covering the compositions of interest. These models are based on studying the melting temperatures in nanoclusters using the regular solution, mixing model for alloys. From our data, experiments on nanocubes about 5 nm in size, of stoichiometric AuCu and PtM composition, could help differentiate between the models. Dispersion data shows that for the three shapes considered, octahedra have the highest percentage of surface atoms for the same relative diameter. We summarize the effects of structural ordering on the catalytic activity and suggest a method to avoid sintering during annealing of Pt-M alloys.

Order parameters from image analysis: a honeycomb example.

Honeybee combs have aroused interest in the ability of honeybees to form regular hexagonal geometric constructs since ancient times. Here we use a real space technique based on the pair distribution function (PDF) and radial distribution function (RDF), and a reciprocal space method utilizing the Debye-Waller Factor (DWF) to quantify the order for a range of honeycombs made by Apis mellifera ligustica. The PDFs and RDFs are fit with a series of Gaussian curves. We characterize the order in the honeycomb using a real space order parameter, OP ( 3 ), to describe the order in the combs and a two-dimensional Fourier transform from which a Debye-Waller order parameter, u, is derived. Both OP ( 3 ) and u take values from [0, 1] where the value one represents perfect order. The analyzed combs have values of OP ( 3 ) from 0.33 to 0.60 and values of u from 0.59 to 0.69. RDF fits of honeycomb histograms show that naturally made comb can be crystalline in a 2D ordered structural sense, yet is more 'liquid-like' than cells made on 'foundation' wax. We show that with the assistance of man-made foundation wax, honeybees can manufacture highly ordered arrays of hexagonal cells. This is the first description of honeycomb utilizing the Debye-Waller Factor, and provides a complete analysis of the order in comb from a real-space order parameter and a reciprocal space order parameter. It is noted that the techniques used are general in nature and could be applied to any digital photograph of an ordered array.