Transport-Regime-Optimal Branching Ratios for Hierarchical Electrode Porosity: A Generalized Murray's Law Framework for Electrochemical Systems

This preprint derives transport-regime-optimal branching ratios for hierarchical battery electrode porosity using generalized Murray's Law applied to mixed diffusive-migrative ion transport governed by the Nernst-Planck equation. The optimal parent-to-child radius ratio r₀/r₁ = N^(1/α) depends on branching geometry N and transport exponent α, which lies between 2.0 (pure diffusion) and 2.5 (mixed electrochemical conditions) for practical lithium-ion battery electrodes. For hexagonal branching (N=6), the physics-derived optimum is 1.82–2.45 — consistently above the golden ratio φ ≈ 1.618 across all electrochemical regimes. A geometry-dependent finding emerges: φ over-shoots the optimum for triangular branching under migration-dominated conditions, demonstrating that φ is not a universal heuristic but a geometry-specific approximation. The paper further proposes aperiodic (quasicrystalline-inspired) hierarchical electrode architectures for superior mechanical resilience against resonant crack propagation during cyclic volume expansion — the failure mode that destroys ordered periodic electrode structures in high-expansion chemistries. This is the first paper in a planned series advancing geometric optimization for next-generation energy storage. Presignal Research Initiative Preprint #16.