The spreading of droplets of liquid healing agents on both horizontal and tilted intact surfaces is considered and compared with that on porous nanofiber (NF) mats in Sects. 3.1 and 3.2, respectively. The intact surfaces and NF mats serve as macroscopic models of self-healing engineering materials with vascular networks, where the healing agents have been released from the NFs in a damaged domain. The spreading of droplets on NF mats show significant deviations from that on the intact surfaces because of the imbibition of liquid into the inter-fiber pores. The model macroscopic experiments with a single crack tip in Sect. 3.3 elucidate the self-healing mechanism, namely, that the epoxy resin and hardener released into the tip react with each other, yielding a cured and hardened epoxy that heals the crack tip. Then, in Sect. 3.4, a microfluidic chip-like setup comprising a vascular system of microchannels alternatingly filled with either a resin monomer or a curing agent is used to study the additional intrinsic aspects of the physical healing mechanism in self-healing engineering materials. The model demonstrates that, as a pre-notched crack propagates across the chip, the resin and curing agent are released from the damaged channels, wet the surrounding matrix, spread over the banks of the crack, mix, and finally polymerize. Moreover, the polymerized domains form a system of pillars, which stitch the crack banks on opposite sides, thus preventing further propagation of the crack.