Quantitative Characterization of Hexagonal Packings in Nanoporous Alumina Arrays: A Case Study

Herein is presented a methodology to quantify the degree of hexagonal order in nanoporous alumina arrays (NAA). The approach is inspired by the theory of two-dimensional melting, developed to describe phase transitions in two-dimensional systems that present liquid-crystal-like structures. A local order parameter (LOP) is defined to quantify the degree of hexagonal order of each pore without any arbitrary parameters. Using this LOP, three main qualitative and quantitative analytical tools were developed: (i) a color code to create a map of the LOP, which is a visual tool to identify the degree of order; (ii) quantitative measurements of the average hexagonal order of the sample by measuring the distribution of the LOP and the distribution of the number of neighbors of each pore, and (iii) a quantification of the spatial correlation of the LOP, which indicates how far the hexagonal order is spread in a sample. Because this approach has a strong support on tools developed in statistical mechanics, one can go beyond a simple characterization and interpret the results in terms of phases, as in other physical systems. This may help to unveil the mechanisms behind the self-organization process and long-range order observed in NAA. Moreover, this approach can be trivially extended to characterize other physical systems that form hexagonal packings.