The Voronoi entropy is a mathematical tool for quantitative characterization of the orderliness of points distributed on a surface. The tool is useful to study various surface self-assembly processes. We provide the historical background, from Kepler and Descartes to our days, and discuss topological properties of the Voronoi tessellation, upon which the entropy concept is based, and its scaling properties, known as the Lewis and Aboav-Weaire laws. The Voronoi entropy has been successfully applied to recently discovered self-assembled structures, such as patterned microporous polymer surfaces obtained by the breath figure method and levitating ordered water microdroplet clusters. Entropy 2018, 20, 956 11 of 13 functionals and Fourier analysis remains an open problem. The relation between the Voronoi entropy and thermodynamic entropy remains unclear and calls for future investigations.

We have compared the results of dynamical modeling of the fission process with predictions of the Kramers formulas. For the case of large dissipation, there are two of them: the integral rate R I and its approximation R O . As the ratio of the fission barrier height B f to the temperature T reaches 4, any analytical rate is expected to agree with the dynamical quasistationary rate R D within 2%. The latter has been obtained using numerical modeling with six different potentials. It has been found that the difference between R O and R D sometimes exceeds 20%. The features of the potentials used that are responsible for this disagreement are identified and studied. It is demonstrated that it is R I , not R O , that meets this expectation regardless of the potential used.

The condensational growth of spherical water microdroplets is studied in a laboratory setup and with a mathematical model. In the experiment, droplet clusters are kept in a freely levitated state within an upward-oriented flow of water vapor. In the presence of an electrostatic field of 1.5 • 10 5 V m -1 , droplet growth is accelerated by factors 1.5 to 2.0 as compared to conditions without any external electric field. Presumably water molecules in the ambient air are accelerated through the presence of the electric field. A kinetic model to predict the acceleration of condensational growth confirms this hypothesis to be feasible. The droplets themselves are polarized so that the deposition of steam molecules is facilitated in the electric field. The simplifications and limitations of the model are discussed.

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A self-assembled cluster of microdroplets levitating over a heated water surface is a fascinating phenomenon with potential applications for microreactors and for chemical and biological analysis of small volumes of liquids. Recently, we suggested a method to synthesize a cluster with an arbitrary number of small monodisperse droplets. However, the interactions, which control the structure of the cluster, are still not well understood. Here we propose a Langevin computational model considering the aerodynamic forces between the droplets and random diffusion-like fluctuations. Characteristic length and time scales and scaling relationships of interactions are discussed. The model shows excellent agreement with experimental observations for a small number of droplets.