Quantitative analysis of the grain morphology in self-assembled hexagonal lattices

We present a methodology for the analysis of the grain morphology of self-ordered hexagonal lattices and for the quantitative comparison of the quality of their grain ordering based on the distances between nearest neighbors and their angular order. Two approaches to grain identification and evaluation are introduced: (i) color coding the relative angular orientation of hexagons containing a central entity and its six nearest neighbors, and (ii) incorporating triangles comprising three nearest neighbors into grains or repelling them from grains based on deviations of the side lengths and the internal angles of the triangles from those of an ideal equilateral triangle. A spreading algorithm with tolerance parameters allows single grains to be identified, which can thus be ranked according to their size. Hence, grain size distributions are accessible. For the practical evaluation of micrographs displaying self-ordered structures, we suggest using the size of the largest identified grain as a quality measure. Quantitative analyses of grain morphologies are key to the systematic and rational optimization of the fabrication of self-assembled materials.