Anisotropic reaction–transport dynamics in Ti-substituted tunnel sodium cathodes: A three-dimensional critical-rate framework

High-rate performance in anisotropic sodium-ion cathodes emerges from coupled particle-scale transport and reaction processes that cannot be represented adequately by a single effective diffusion coefficient. This study establishes a three-dimensional finite-element framework for rod-like Ti-substituted Na₄TiᵧMn₉₋ᵧO₁₈ tunnel cathodes by coupling tunnel-informed anisotropic Na transport, state-dependent diffusivity, concentration-dependent Butler–Volmer kinetics, finite electronic transport, and composition-dependent material properties. A model-derived critical rate, C crit , is introduced to identify the transition from quasi-uniform Na redistribution to transport-limited operation using the coupled evolution of concentration heterogeneity, reaction localization, and Na utilization. For the reference low-Ti particle, C crit was 2.35 ± 0.18C2.35\pm0.18 C. Increasing transport anisotropy raised C crit from 1.92 C at AD= 2–2.61 C at A D = 10, followed by a decline to 2.48 C at AD= 20, revealing diminishing benefit once transverse redistribution becomes limiting. Ti substitution produced a similarly non-monotonic response, with C crit increasing from 1.95 C for the Ti-free composition to 2.55 C at x Ti = 0.29, before decreasing to 1.88 C at x Ti = 0.56. Increasing particle aspect ratio from 3 to 8 reduced the threshold from 2.72 C to 1.86 C. Independent validation produced a GITT deviation of 9.7%, a rate-response NRMSE of 5.64%, and a MAPE of 6.38%. Notably, reaction localization emerged before pronounced loss of average Na utilization, identifying spatial reaction heterogeneity as an early marker of transport instability. The framework defines a composition–anisotropy–morphology design space for particle-scale control of high-rate sodium storage rather than a single optimum material configuration.