New Method for Describing Graphene Simplifies Analysis of Nanomaterials
An international team, including scientists from HSE University, has proposed a new mathematical method to analyse the structure of graphene. The scientists demonstrated that the characteristics of a graphene lattice can be represented using a three-step random walk model of a particle. This approach allows the lattice to be described more quickly and without cumbersome calculations. The study has been published in Journal of Physics A: Mathematical and Theoretical.
Graphene is one of the most recent and widely discussed materials of the 21st century. It consists of a single layer of carbon atoms arranged in a honeycomb, hexagonal lattice structure. Graphene is exceptionally strong, an excellent conductor of electricity, nearly transparent, and highly flexible. It has already been used in the production of conductive films, sensors, and miniature transistors. Similar structures occur in other carbon forms, such as fullerenes—closed spherical molecules composed of pentagons and hexagons—used in drug delivery and solar cell technology. These materials have numerous structural variants that directly influence their properties, including molecular stability. Experimentally testing each variant is costly and difficult, so scientists are seeking simpler methods to predict their characteristics.
An international team of scientists from Germany, Russia, France, and Japan, including researchers from HSE University, has proposed such a method. They simplified the description of the key parameters that determine the lattice’s behaviour into a three-step random walk model.
In this model, an imaginary particle starts at the origin of a plane and takes three equal-length steps in random directions. The target lattice parameter is then defined by the final position of the particle along the x-axis. Mathematically, this is expressed as the sum of the cosines of three random numbers corresponding to the step directions. For the calculations, it is enough to repeatedly generate random numbers, substitute them into the formulas, and add the results. Repeating this procedure many times yields values that capture the key properties of the lattice. This method describes the material without complex computations and simplifies the analysis.
This simplified calculation method is useful not only for graphene. The authors suggest that their approach could also be applied to other carbon structures, such as fullerenes.
Victor Buchstaber
'We hypothesised that as the size of a molecule increases, random fullerenes become locally more similar in structure to an infinite graphene lattice. If this can be rigorously proven, the spectral properties of fullerenes could be derived from those of graphene, greatly simplifying their analysis,' explains Viktor Buchstaber from the International Laboratory of Algebraic Topology and Its Applications at the HSE Faculty of Computer Science.
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