A general topological network criterion for exploring the structure of icy nanoribbons and monolayers

We develop intuitive metrics for quantifying complex nucleating systems under confinement. These are shown to arise naturally from the analysis of the topological ring network, and are amenable for use as order parameters for such systems. Drawing inspiration from qualitative visual inspection, we introduce a general topological criterion for elucidating the ordered structures of confined water, using a graph theoretic approach. Our criterion is based on primitive rings, and reinterprets the hydrogen-bond-network in terms of these primitives. This approach has no a priori assumptions, except the hydrogen bond definition, and may be used as an exploratory tool for the automated discovery of new ordered phases. We demonstrate the versatility of our criterion by applying it to analyse well-known monolayer ices. Our methodology is then extended to identify the building blocks of one-dimensional n-sided prismatic nanoribbon ices.