Nanoparticle shapes: Quantification by elongation, convexity and circularity measures

The goal of the nanoparticle synthesis is, first of all, the production of nanoparticles that will be more similar in size and shape. This is very important for the possibility of studying and applying nanomaterials because of their characteristics that are very sensitive to size and shape such as, for example, magnetic properties. In this paper, we propose the shape analysis of the nanoparticles using three shape descriptors – elongation, convexity and circularity. Experimental results were obtained by using TEM images of hematite nanoparticles that were, first of all, subjected to segmentation in order to obtain isolated nanoparticles, and then the values of elongation, convexity and circularity were measured. Convexity Cx(S) is regarded as the ratio between shape’s area and area of the its convex hull. The convexity measure defines the degree to which a shape differs from a convex shape while the circularity measure defines the degree to which a shape differs from an ideal circle. The range of convexity and circularity values is (0, 1], while the range of elongation values is [1, ∞). The circle has lowest elongation (ε = 1), while it has biggest convexity and circularity values (Cx = 1; C = 1). The measures ε(S), Cx(S), C(S) proposed and used in the experiment have the few desirable properties and give intuitively expected results. None of the measures is good enough to describe all the shapes, and therefore it is suggested to use a variety of measures so that the shapes can be described better and then classify and control during the synthesis process.

A Bound on a Convexity Measure for Point Sets

A planar point set is in convex position precisely when it has a convex polygonization, that is, a polygonization with maximum interior angle measure at most [Formula: see text]. We can thus talk about the convexity of a set of points in terms of its min-max interior angle measure. The main result presented here is a nontrivial upper bound of the min-max value in terms of the number of points in the set. Motivated by a particular construction, we also pose a natural conjecture for the best upper bound.