Thermodynamic origin of root-like architectures in topology-optimized porous electrodes

Recent studies have shown that topology optimization can generate high-performing porous architectures, often exhibiting graded, branched, or root-like morphologies. However, the physical origin of these recurring structures remains unclear. In this work, we develop a theoretical framework linking reaction–diffusion transport, topology optimization, and nonequilibrium thermodynamics to explain the emergence of root-like porous electrodes. Starting from a simplified first-order reaction–diffusion model with porosity-dependent effective properties, classical relationships for effectiveness factor and characteristic penetration depth are revisited and extended to graded media. Scaling arguments are then proposed for branch spacing, root thickness, and root length by balancing axial reactant delivery with lateral consumption in the surrounding reactive matrix. A thermodynamic interpretation is further introduced in which the penetration depth represents a crossover distance beyond which transport-related irreversibility becomes increasingly dominant. Density-based topology optimization is then employed to maximize overall reaction rate under finite porosity constraints. The optimized structures evolve from uniform to hierarchical root-like networks, showing substantially improved utilization compared with conventional uniform porous layers. The predicted geometric trends are consistent with the optimized morphologies. These results suggest that root-like porous architectures are not arbitrary numerical artifacts, but physically meaningful responses to coupled transport, reaction, and thermodynamic constraints. The present framework provides mechanistic design principles for reaction-diffusion-limited porous electrodes.