Fig. 3: Multi-physics Modeling of CO₂R in Model II.

a Scheme of CO₂ and H₂O transport in the polymer modified catalyst (left) and the graphical illustration of the modeling domains (right). The gas and liquid transport are decoupled, leading to a three-phase unsaturated layer for enhanced electron participation in desired electrochemical reaction and reduced the CO₂ transport length to the catalyst surface. The source of protons for CO₂R is the water taken up by the polymer coating. b Modeled CO₂ availability for the desired CO₂R. c Cathodic potential. d C₂₊ products current density and e C₂₊ selectivity with the variation of the polymer porosity and local H₂O/CO₂ ratio at the same applied cathodic potential (−1.426 V vs. SHE). The inserts in panels b and d are the enlarged areas indicated by the red dashed line. CO₂ flux boundary conditions are set at the upper boundaries of both the liquid electrolyte ___domain (\({{Flux}}_{{{CO}}_{2}({EL})}\)) and the polymer ___domain (\({{Flux}}_{{{CO}}_{2}({PL})}\)). A symmetrical condition is imposed at the right boundary to model a confined pore geometry in the catalyst layer (CL). The electrolyte potential \({\psi }_{l}\) at this boundary is set to 0 V to provide a reference for solving the electric field. Due to the continuous flow of fresh 2 M KOH electrolyte into the cathode chamber, an open boundary condition is imposed at the bottom boundary, and the equilibrium concentration values of K⁺ (\({c}_{{K}^{+}}^{{eq}}\)) and OH⁻ (\({c}_{{{OH}}^{-}}^{{eq}}\)) are set at this boundary (Supplementary Table 1-3). A fixed cathodic potential \({\psi }_{s}{=V}_{{app}}\) and electrolyte current density \({J}_{l}=0\) are imposed at the left boundary. Relevant source data are provided as a Source Data file.