Mechanism and Prediction of Gas Permeation through Sub-Nanometer Graphene Pores: Comparison of Theory and Simulation

Due to its atomic thickness, porous graphene with sub-nanometer pore sizes constitutes a promising candidate for gas separation membranes that exhibit ultrahigh permeances. While graphene pores can greatly facilitate gas mixture separation, there is currently no validated analytical framework with which one can predict gas permeation through a given graphene pore. In this work, we simulate the permeation of adsorptive gases, such as CO₂ and CH₄, through sub-nanometer graphene pores using molecular dynamics simulations. We show that gas permeation can typically be decoupled into two steps: (1) adsorption of gas molecules to the pore mouth and (2) translocation of gas molecules from the pore mouth on one side of the graphene membrane to the pore mouth on the other side. We find that the translocation rate coefficient can be expressed using an Arrhenius-type equation, where the energy barrier and the pre-exponential factor can be theoretically predicted using the transition state theory for classical barrier crossing events. We propose a relation between the pre-exponential factor and the entropy penalty of a gas molecule crossing the pore. Furthermore, on the basis of the theory, we propose an efficient algorithm to calculate CO₂ and CH₄ permeances per pore for sub-nanometer graphene pores of any shape. For the CO₂/CH₄ mixture, the graphene nanopores exhibit a trade-off between the CO₂ permeance and the CO₂/CH₄ separation factor. This upper bound on a Robeson plot of selectivity versus permeance for a given pore density is predicted and described by the theory. Pores with CO₂/CH₄ separation factors higher than 10² have CO₂ permeances per pore lower than 10^–22 mol s^–1 Pa^–1, and pores with separation factors of ∼10 have CO₂ permeances per pore between 10^–22 and 10^–21 mol s^–1 Pa^–1. Finally, we show that a pore density of 10¹⁴ m^–2 is required for a porous graphene membrane to exceed the permeance-selectivity upper bound of polymeric materials. Moreover, we show that a higher pore density can potentially further boost the permeation performance of a porous graphene membrane above all existing membranes. Our findings provide insights into the potential and the limitations of porous graphene membranes for gas separation and provide an efficient methodology for screening nanopore configurations and sizes for the efficient separation of desired gas mixtures.