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We investigate three-dimensional Turing patterns in two-component reaction diffusion systems. The FitzHugh-Nagumo equation, the Brusselator, and the Gray-Scott model are solved numerically in three dimensions. Several interconnected structures of domains as well as lamellar, hexagonal, and spherical domains are obtained as stable motionless equilibrium patterns. The relative stability of these structures is studied analytically based on the reduction approximation. The relation with the microphase-separated structures in block copolymers is also discussed.

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This study demonstrates that the Marangoni flow in a channel network can solve maze problems such as exploring and visualizing the shortest path and finding all possible solutions in a parallel fashion. The Marangoni flow is generated by the pH gradient in a maze filled with an alkaline solution of a fatty acid by introducing a hydrogel block soaked with an acid at the exit. The pH gradient changes the protonation rate of fatty acid molecules, which translates into the surface tension gradient at the liquid-air interface through the maze. Fluid flow maintained by the surface tension gradient (Marangoni flow) can drag water-soluble dye particles toward low pH (exit) at the liquid-air interface. Dye particles placed at the entrance of the maze dissolve during this motion, thus exhibiting and finding the shortest path and all possible paths in a maze.