Enumerating Stable Nanopores in Graphene and Their Geometrical Properties Using the Combinatorics of Hexagonal Lattices

Abstract

Nanopores in two-dimensional (2D) materials, including graphene, can be used for a variety of applications, such as gas separations, water desalination, and DNA sequencing. So far, however, all plausible isomeric shapes of graphene nanopores have not been enumerated. Instead, a probabilistic approach has been followed to predict nanopore shapes in 2D materials, due to the exponential increase in the number of nanopores as the size of the vacancy increases. For example, there are 12 possible isomers when N = 6 atoms are removed, a number that theoretically increases to 11.7 million when N = 20 atoms are removed from the graphene lattice. In this regard, the development of a smaller, exhaustive data set of stable nanopore shapes can help future experimental and theoretical studies focused on using nanoporous 2D materials in various applications. In this work, we use the theory of 2D triangular "lattice animals" to create a library of all stable graphene nanopore shapes based on a modification of a well-known algorithm in the mathematical combinatorics of polyforms known as Redelmeier's algorithm. We show that there exists a correspondence between graphene nanopores and triangular polyforms (called polyiamonds) as well as hexagonal polyforms (called polyhexes). We develop the concept of a polyiamond ID to identify unique nanopore isomers. We also use concepts from polyiamond and polyhex geometries to eliminate unstable nanopores containing dangling atoms, bonds, and moieties. We verify using density functional theory calculations that such pores are indeed unstable. The exclusion of these unstable nanopores leads to a remarkable reduction in the possible nanopores from 11.7 million for N = 20 to only 0.184 million nanopores, thereby indicating that the number of stable nanopores is almost 2 orders of magnitude lower and is much more tractable. Not only that, by extracting the polyhex outline, our algorithm allows searching for nanopores with dimensions and shape factors in a specified range, thus aiding the design of the geometrical properties of nanopores for specific applications. We also provide the coordinate files of the stable nanopores as a library to facilitate future theoretical studies of these nanopores.

Machine Learning in Membrane Design: From Property Prediction to AI-Guided Optimization

“…Nanopore optimization on membranes is another field where ML excels besides membrane material screening. On 2D membranes such as graphene, there is a myriad of variations in nanopore geometry and the geometries determine the membrane performance in applications like water desalination. , Optimizing the nanopore geometry for desired performance can be a Herculean task because of the sheer size of the search space. To tackle such a problem, Wang et al designed a deep reinforcement learning (DRL) framework to optimize the graphene nanopore geometry for RO water desalination (Figure b).…”

“…10 However, the number of possible geometry of nanopores is astronomical (i.e., theoretically 11.7 million when pore size is 20 atoms on a graphene lattice). 12,26 In the work of Wang et al, 27 a convolutional neural network (ResNet) 28 was used to predict the water flux and ion rejection rate of graphene nanopores.…”

“…Besides the material of the membrane, the geometry of nanopores was also shown to influence the membrane performance in RO water desalination . However, the number of possible geometry of nanopores is astronomical (i.e., theoretically 11.7 million when pore size is 20 atoms on a graphene lattice). , In the work of Wang et al, a convolutional neural network (ResNet) was used to predict the water flux and ion rejection rate of graphene nanopores. Since the desalination conditions and the membrane material were fixed, the model takes only an image of the nanoporous graphene membrane as input and automatically extracts geometrical features to make accurate predictions.…”

“…7 For 2D material membranes, their performances depend not only on the type of materials 8,9 but also on the size and geometry of the artificial nanopores. 10,11 Due to the higher cost of synthesis and the sheer amount of possible nanopore geometries on 2D materials, 12 exhaustive experimental methods fall short again in efficiently discovering or searching the optimal membrane for desired properties. Furthermore, physics-based computational approaches, such as molecular dynamics simulation, face challenges in conducting exhaustive searches due to their computational intensity, which limits the assessment of a wide range of candidate membranes.…”