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Understanding the stability mechanism of surface micro/nanobubbles adhered to gasevolving electrodes is essential for improving the efficiency of water electrolysis, which is known to be hindered by the bubble coverage on electrodes. Using molecular simulations, the diffusion-controlled evolution of single electrolytic nanobubbles on wettability-patterned nanoelectrodes is investigated. These nanoelectrodes feature hydrophobic islands as preferential nucleation sites and allow the growth of nanobubbles in the pinning mode. In these simulations, a threshold current density distinguishing stable nanobubbles from unstable nanobubbles is found. When the current density remains below the threshold value, nucleated nanobubbles grow to their equilibrium states, maintaining their nanoscopic size. However, for the current density above the threshold value, nanobubbles undergo unlimited growth and can eventually detach due to buoyancy. Increasing the pinning length of nanobubbles increases the degree of nanobubble instability. By connecting the current density with the local gas oversaturation, an extension of the stability theory for surface nanobubbles [Lohse and Zhang, Phys. Rev. E 91, 031003(R) (2015)] accurately predicts the nanobubble behavior found in molecular simulations, including equilibrium contact angles and the threshold current density. For larger systems that are not accessible to molecular simulations, continuum numerical simulations with the finite difference method combined with the immersed boundary method are performed, again demonstrating good agreement between numerics and theories. [ABSTRACT FROM AUTHOR]