Beyond Murray’s Law: Resistance Matching Principle for Optimal Fluid Transport in Hierarchical Nanomaterials

The century-old Murray's law, originally formulated to describe optimal transport in biological vascular systems, continues to inspire the design of hierarchical nanomaterials. However, at the nanoscale, its fundamental assumptions of fluid homogeneity and negligible interfacial slip no longer hold, limiting its validity. In this work, Murray's law is extended to incorporate nanoscale effects, including slip boundary conditions and confinement-induced variations in fluid density and viscosity. Quantitative calculations reveal a transition from traditional viscous flow dominance at larger scales to interfacial slip-driven transport in microporous channels, leading to significant deviations from the original predictions of Murray's law. Furthermore, the physical foundation of the nanoscale-adapted Murray's law, namely minimum energy dissipation in nonequilibrium thermodynamics, is restated as a generalized resistance matching principle, offering a practical framework for designing hierarchical structures. This principle is experimentally validated in two structurally diverse nanosystems-biological-skeleton carbon and zeolite molecular sieves-demonstrating its broad applicability. The work provides a generalizable theoretical foundation and a practical benchmark for the rational engineering of advanced hierarchical nanomaterials. By bridging a century-old biological principle with modern nanofluidics, the proposed resistance-matching principle is expected to influence fields such as heterogeneous catalysis, membrane technology, and energy storage.