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Surface nanobubbles are nanoscopic spherical-cap shaped gaseous domains onimmersed substrates which are stable, even for days. After the stability of a{\it single} surface nanobubble has been theoretically explained, i.e. contactline pinning and gas oversaturation are required to stabilize them againstdiffusive dissolution [Lohse and Zhang, Phys.\ Rev.\ E 91, 031003 (R) (2015)],here we focus on the {\it collective} diffusive interaction of {\it multiple}nanobubbles. For that purpose we develop a finite difference scheme for thediffusion equation with the appropriate boundary conditions and with theimmersed boundary method used to represent the growing or shrinking bubbles.After validation of the scheme against the exact results of Epstein and Plessetfor a bulk bubble [J. Chem. Phys. 18, 1505 (1950)] and of Lohse and Zhang for asurface bubble, the framework of these simulations is used to describe thecoarsening process of competitively growing nanobubbles. The coarsening processfor such diffusively interacting nanobubbles slows down with advancing time andthus increasing bubble distance. The present results for surface nanobubblesare also applicable for immersed surface nanodroplets, for which bettercontrolled experimental results of the coarsening process exist.

Diffusive interaction of multiple surface nanobubbles and nanodroplets: shrinkage, growth, and coarsening