The egalitarianism of dual bubbles ripening on heterogeneous substrate with constant contact radius

In this work, we study the reaction-controlled dual bubbles ripening on a heterogeneous substrate with high surface wettability hysteresis, where the bubbles evolve with constant contact radius but varied contact angle. We first theoretically derived the governing kinetic equation of bubble curvature radius R_B, based on which we surprisingly found three possible ripening processes under six different conditions, i.e. the classical Ostwald ripening (the bubble with the larger curvature radius R_B exhibits an increase in R_B, while the bubble with the smaller curvature radius R_B experiences a decrease in R_B), the reversed ripening (converse to Ostwald ripening), and the consistent ripening (R_B of both bubbles increases or reduces consistently). Further analyses from the aspects of chemical potential and free energy lead to an interesting finding that the RB of two bubbles finally reach egalitarianism, independently of different ripening processes. Numerical results obtained from two-phase lattice Boltzmann modelling demonstrate excellent agreement with theoretical predictions, specifically concerning the kinetic equation, the various ripening processes, and the egalitarianism of bubble radii R_B after ripening completion.