Multipolar Models for Anisotropic Vacancy Interactions in MXenes

Atomic-scale defects form during the synthesis of two-dimensional (2D) MXenes, and their interactions and surface migration lead to spatial distributions that affect their electronic, catalytic, or sensing properties. We investigate here the interaction energy of vacancies on Ti_m+1C_m MXenes, in particular their anisotropic dependence on the intervacancy separation, r. Using molecular statics calculations based on bond-order interatomic potentials, we extract the strain-mediated interaction energy of vacancies in MXenes and uncover their spatial dependence. At short range, the interactions display attraction-repulsion oscillations. At separations larger than several lattice constants, we show that the vacancies can be modeled as superposed in-plane force tripoles and out-of-plane force dipoles, resulting in a combination of two power laws (1/r² and 1/r⁴) for their interaction. This is similar to the interaction of proteins on biological membranes and in contrast to surface defects on an elastic semi-infinite medium─whose long-range interactions vary as 1/r³. The origin of the spatial dependence of vacancy interactions is robust, as we illustrate it for 2D MXene systems with different atomic layer stackings, surface layers (Ti or C), and for vacancies along different crystallographic directions. Depending on the specific atomic-scale details, the two-multipole model presented here may be applicable for defects or adatoms on other 2D materials, or it can be adapted to capture more complex interactions. While our results on the power-law combination can be verified in future experiments by performing statistical analysis of vacancy-vacancy separations, the attractive interaction we have found for nearest-neighbor Ti vacancies already provides an explanation for the previously observed clustering of Ti vacancies on MXenes.