Polygonal tessellations as predictive models of molecular monolayers

Molecular self-assembly plays a very important role in various aspects of technology as well as in biological systems. Governed by 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.

分子自组装（Molecular self-assembly）在技术的各个领域以及生物系统中都发挥着至关重要的作用。在共价键（covalent bond）、氢键（hydrogen bond）或范德华相互作用（van der Waals interactions）的调控下，相似分子的自组装即使在二维（2D）空间中也能形成多种多样的复杂图案。二维分子网络的图案形成预测极为重要，但也极具挑战性；迄今为止，该预测依赖于计算密集型方法，例如密度泛函理论（density functional theory）、经典分子动力学（classical molecular dynamics）、蒙特卡洛（Monte Carlo）或机器学习（machine learning）。然而，这些方法无法保证涵盖所有可能的图案，且往往依赖于直觉。本文提出一种更简洁但严谨的分层几何模型，该模型基于二维多边形镶嵌（2D polygonal tessellations）的平均场理论（mean-field theory），可根据分子水平的信息预测扩展网络图案。基于图论（graph theory），该方法可在明确界定的范围内实现图案分类与预测。将该模型应用于现有实验数据时，可为自组装分子图案提供全新视角，进而对可容许图案及潜在的额外相态做出有趣的预测。尽管该模型最初是为氢键系统开发的，但可扩展至共价键合的石墨烯衍生材料或富勒烯（fullerenes）等三维结构，从而显著拓宽未来潜在应用的范围。