How heterogeneous wettability enhances boiling

For super-heated water on a substrate with hydrophobic patches immersed in a hydrophilic matrix, one can choose the temperature so that micro-bubbles will form, grow and merge on the hydrophobic patches and not on the hydrophilic matrix. Until covering a patch, making a pinned macro-bubble, a bubble has a contact angle $\pi-\theta_2$, where $\theta_2$ is the receding contact angle of water on the patch material. This pinned macro-bubble serves as the initial condition of a quasi-static growth process, \`a la Landau, leading to detachment through the formation of a neck, so long as depinning and dewetting of the hydrophilic matrix was avoided during the growth of the pinned bubble: the bubble contact angle should not exceed $\pi-\theta_1$, where $\theta_1$ is the receding contact angle of water on the matrix material. The boiling process may then enter a cycle of macro-bubbles forming and detaching on the patches; the radii of these patches can be optimized for maximizing the heat transfer for a given substrate area. For this analysis to become quantitative, we revisit the Young-Laplace quasi-static evolution of key physical quantities, such as bubble energy, as functions of bubble growing volume, when gravity is either significative or negligible: this concerns both pinned bubbles on a fixed circular footprint (Dirichlet boundary conditions) and free un-pinned bubbles with a fixed contact angle (Neumann boundary conditions).