  Molecular self-assembly plays a very important role in various aspects of
technology as well as in biological systems. Governed by the covalent, hydrogen
or van der Waals interactions - self-assembly of alike molecules results in a
large variety of complex patterns even in two dimensions (2D). Prediction of
pattern formation for 2D molecular networks is extremely important, though very
challenging, and so far, relied on computationally involved approaches such as
density functional theory, classical molecular dynamics, Monte Carlo, or
machine learning. Such methods, however, do not guarantee that all possible
patterns will be considered and often rely on intuition. Here we introduce a
much simpler, though rigorous, hierarchical geometric model founded on the
mean-field theory of 2D polygonal tessellations to predict extended network
patterns based on molecular-level information. Based on graph theory, this
approach yields pattern classification and pattern prediction within
well-defined ranges. When applied to existing experimental data, our model
provides an entirely new view of self-assembled molecular patterns, leading to
interesting predictions on admissible patterns and potential additional phases.
While developed for hydrogen-bonded systems, an extension to covalently bonded
graphene-derived materials or 3D structures such as fullerenes is possible,
significantly opening the range of potential future applications.
