Coupled theory for transient responses of conductive hydrogels with multi-stimuli

Hydrogels experience complex environment stimuli (evaporation, temperature, electric fields, external loads, and constraints) when used in stretchable electronic devices, soft robots, and many other applications. In addition, it always needs time for hydrogels to respond when these stimuli change. Therefore, the transient responses of hydrogels to multi-stimuli should be studied carefully. Existing studies have usually treated the physical processes independently and have ignored the correlations among them; thus, these studies have been unable to describe certain experimental and natural phenomena. Herein, we propose coupled non-linear governing equations for the transient responses of conductive hydrogels to multi-stimuli, based on nonequilibrium thermodynamics. The coupling effects among the solvent diffusion, heat conduction, and electric conduction in hydrogels are investigated. Moreover, a new formulation of free energy for conductive hydrogels is introduced, which combines swelling the polymer and stretching the swollen hydrogel. The new free energy is easier to use and has clearer physical meaning. Basing on our coupled theory, the drying process (evaporation) of hydrogel circuits attached to a substrate with various manufacturing designs is studied. Remarkably, this theory can reflect the thermal diffusion effect and the cooling phenomenon due to the evaporation of the solvent, which is not possible when neglecting the coupling effect between diffusion and heat conduction. This work can serve as a guide for application of hydrogels in stretchable electronic devices, soft robots, and other devices.