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

The already century-old Murray’s law, originally formulated to describe optimal transport in biological vascular systems, continues to be an inspiration in the design of hierarchical nanomaterials. However, at the nanoscale, its fundamental assumptions of fluid homogeneity and interfacial slip-negligibility no longer hold, limiting its validity. In this work, we re-derived Murray’s law to incorporate nanoscale effects, including slip boundary conditions and confinement-induced variations in fluid density and viscosity. Our theoretical analysis reveals a transition from traditional viscous flow dominance at larger scales to interfacial slip-driven transport in microporous channels, resulting in significant deviations from the original predictions of Murray’s law. We experimentally validated these theoretical insights using atomic force microscopy-based hydrodynamic measurements. Remarkably, our nanoscale-adapted Murray's law uncovers a generalized resistance matching principle for the design of hierarchical structures: optimal transport is achieved when the hydraulic resistances of parent and children channels match precisely. This principle is experimentally confirmed in two structurally diverse nanosystems, biological-skeleton carbon and zeolite molecular sieves, demonstrating its broad applicability. These findings establish both a generalizable theoretical foundation and a practical benchmark for the rational engineering of advanced hierarchical nanomaterials. Bridging the century-old biological principle with modern nanofluidics, this resistance matching principle promises a significant impact on fields such as heterogeneous catalysis, membrane technology, and energy storage.