Theoretical model of dynamics and stability of nanobubbles on heterogeneous surfaces

[Display omitted] •Describing the properties of heterogeneous surfaces via a surface energy function.•The evolution of surface nanobubbles follows the proposed theoretical model.•The contact line slips on the homogeneous regions of the substrate, while pinning always occurs at the boundary sites.•The phase diagram shows the critical nucleation, evolution trends and stable states of the surface nanobubbles.•The nanoscale motion of the contact line can be explained in terms of the forces applied. Surface nanobubbles have revealed a new mechanism of gas–liquid–solid interaction at the nanoscale; however, the nanobubble evolution on real substrates is still veiled, because the experimental observation of contact line motions at the nanoscale is too difficult. This study proposes a theoretical model to describe the dynamics and stability of nanobubbles on heterogeneous substrates. It simultaneously considers the diffusive equilibrium of the liquid–gas interface and the mechanical equilibrium at the contact line, and introduces a surface energy function to express the substrate’s heterogeneity. The present model unifies the nanoscale stability and the microscale instability of surface bubbles. The theoretical predictions are highly consistent to the nanobubble morphology on heterogeneous surfaces observed in experiments. As the nanobubbles grow, a lower Laplace pressure leads to weaker gas adsorption, and the mechanical equilibrium can eventually revert to the classical Young-Laplace equation above microscale. The analysis results indicate that both the decrease in substrate surface energy and the increase in gas oversaturation are more conducive to the nucleation and growth of surface nanobubbles, leading to larger stable sizes. The larger surface energy barriers result in the stronger pinning, which is beneficial for achieving stability of the pinned bubbles. The present model is able to reproduce the continual behaviors of the three-phase contact line during the nanobubble evolution, e.g., “pinning, depinning, slipping and jumping” induced by the nanoscale defects.