Hydration structures at crystal surfaces play important roles in crystal growth or dissolution processes in liquid environments. Recently developed two-dimensional (2D) and three-dimensional (3D) force mapping techniques using frequency-modulation atomic force microscopy (FM-AFM) allow us to visualize the hydration structures at the solid–liquid interfaces at angstrom-scale resolution in real space. Up to now, the experimental and theoretical studies on local hydration structures have mainly focused on those on the terrace, but little work has looked at step edges, usually the key areas in dissolution and growth. In this study, we measured local hydration structures on water-soluble alkali halide crystal surfaces by 2D force mapping FM-AFM. The atomic-scale hydration structures observed on the terraces agree well with molecular-dynamics (MD) simulations. We also measured the hydration structures at the step edge of the NaCl(001) surface, which was constantly dissolving and growing, leading to the clear observation of atomic fluctuations. We found, with the support of MD simulations, that the hydration structures measured by FM-AFM at a time scale of a minute can be interpreted as the time-average of the hydration structures on the upper terrace and those on the lower terrace.

Representing various networked data as multiplex networks, networks of networks and other multilayer networks can reveal completely new types of structures in these systems. We introduce a general and principled graphlet framework for multilayer networks which allows one to break any multilayer network into small multilayered building blocks. These multilayer graphlets can be either analysed themselves or used to do tasks such as comparing different systems. The method is flexible in terms of multilayer isomorphism, automorphism orbit definition and the type of multilayer network. We illustrate our method for multiplex networks and show how it can be used to distinguish networks produced with multiple models from each other in an unsupervised way. In addition, we include an automatic way of generating the hundreds of dependency equations between the orbit counts needed to remove redundant orbit counts. The framework introduced here allows one to analyse multilayer networks with versatile semantics, and these methods can thus be used to analyse the structural building blocks of myriad multilayer networks.