Universal Murray's law for optimised fluid transport in synthetic structures

The study addresses how to design synthetic hierarchically porous materials that realize the optimal mass transport predicted by Murray's law. While Murray's law, derived from biological circular tubular networks, has guided design of hierarchical materials, synthetic systems typically feature non-cylindrical, shape-varying pores across scales due to fabrication constraints. This mismatch undermines theoretical optimality and has prevented convincing experimental verification of superior mass transport in synthetic Murray materials. The authors aim to generalize Murray's law beyond circular tubes and conserved mass to arbitrary channel geometries, hierarchical structures (including planar forms), and diverse transport processes, enabling practical optimization in real materials.

Murray's law, originally derived for vascular networks to minimize the sum of hydraulic power and metabolic costs, predicts that the sum of cubes of radii is conserved across branching for laminar flow, and a square law applies for diffusion, ionic migration, and electron transport. Numerous biological transport networks obey these relationships. In materials science, biomimetic efforts have attempted to implement Murray-inspired hierarchies to enhance catalysis, sensing, and energy storage. However, these efforts typically deviate from ideal circular cylindrical pores and controlled branching geometries available in biology, limiting the applicability of the classic theory and compromising mass transfer benefits. Prior generalizations introduced mass loss ratios to account for reactions/adsorption but still focused on tubular geometry. No rigorous and experimentally validated framework had existed for arbitrary pore shapes or planar hierarchical structures common in synthetic materials.

Theory: The authors re-derive Murray's law by minimizing total transport resistance R rather than power, enabling extension to various transport types where energy accounting is difficult. For an arbitrary channel size variable x with cross-sectional area A = k1 x^α and generalized mass flow rate Q = k2 x^β / l (or more generally Q ∝ x^γ), minimizing R yields a universal optimality condition for an i-level hierarchy: sum x^{(α+β)/2} is equal across branching levels (or sum x^γ for processes where Q ∝ x^γ). This Universal Murray's law applies to networks with similar or dissimilar shapes between levels, including non-circular channels and planar structures. It recovers classic results for laminar flow in circular tubes (∑r^3), diffusion/ionic/electronic transport (∑r^2), and extends to turbulent flow, non-Newtonian laminar flow, Knudsen diffusion (∑r^3), and planar parallel-plate hierarchies (e.g., laminar flow: ∑h^2).

Extension to materials: For diffusion and ionic/electronic migration, Q ∝ A independent of shape. Choosing A as the size variable (α=1, β=1) gives ∑A conserved across levels, enabling optimization independent of pore shape. For planar laminar flow between parallel plates with fixed width, selecting channel height h as x yields α=1, β=3 and optimal ∑h^2 conservation across levels.

Experiments: Graphene oxide aerogels (GOA) were fabricated via freeze-casting to create hierarchical planar (bidirectional freezing) and tubular (unidirectional freezing) structures. Unidirectional freeze-casting created vertically aligned pores; bidirectional freeze-casting using a 30° PDMS wedge imposed horizontal and vertical gradients to form lamellae. Pore sizes and lamellar spacings were tuned by freezing temperature (e.g., vertically porous average diameters from ~7.85 to 39.8 µm; lamellar spacings ~30.5 to 97.1 µm). Porosity ~98.2%. Image-processing codes quantified pore sizes, layer spacing, and orientation.

Hierarchical test structures with three levels were built by shaping bulk GOA to control channel numbers/heights/diameters under fixed total volumes and lengths. For planar hierarchies (lamellar GOA), Murray-compliant designs satisfied n1 h1 = n2 h2 = n3 h3 (equivalently H1 h1 = H2 h2 = H3 h3), corresponding to ∑h^2 conservation. Comparison structures deviated by enforcing ∑h^x with x ≠ 2 (e.g., x=1, 1.5, 2.5, 3). For tubular hierarchies (vertically porous GOA), designs enforced ∑r^x with x = 1–5, including the Murray case x=3. Smooth transitions between sections were included to avoid flow concentration.

Measurements: Flow resistance was measured for water, air, and several organic solvents (2-butanol, hexane, ethanol, toluene) using a syringe pump to control volumetric flow and a differential pressure gauge across the sample; laminar conditions were confirmed via low Reynolds numbers (<50). CFD simulations (ANSYS Fluent) on scaled-down geometrical models corroborated experiments, assuming laminar flow and smooth walls. Diffusion scenarios were also simulated.

Application: A GOA-based gas sensor was fabricated by decorating GO sheets with SnO2 quantum dots via hydrothermal synthesis and subsequent unidirectional freeze-casting to form hierarchical porous sensing elements. Two sensor geometries (a straight hierarchical three-section cylinder and a Murray-optimized variant with adjusted section diameters to satisfy ∑r^3) were compared. Gas sensing to 1 ppm NO2, NH3, and CH2O was measured at room temperature; airflow resistance through scaled models was simulated.

• Universal Murray's law: A generalized optimality condition minimizing transport resistance in arbitrary hierarchical networks and transfer types was derived: for Q ∝ x^β / l with A ∝ x^α, the optimum satisfies equal sums of x^{(α+β)/2} across levels; for Q ∝ x^γ, the optimum satisfies equal sums of x^γ. It recovers classic forms (∑r^3 for laminar flow in tubes; ∑r^2 for diffusion/ionic/electronic transport) and extends to non-circular channels, networks with level-dependent shapes, planar hierarchies, turbulent and non-Newtonian flows, and Knudsen diffusion.
• Planar laminar flow validation: In three-level lamellar GOA planar channels with fixed total volume and length, experimental flow resistance for water and air exhibited a U-shaped dependence versus conservation exponent x, with the minimum at x=2 (i.e., ∑h^2), matching Universal Murray's law. CFD on scaled-down models reproduced the minima and overall trends. The optimal design balances sectionwise resistances by appropriately allocating volume to higher-level sections.
• Tubular laminar flow validation: In three-level tubular GOA channels, experiments and CFD confirmed minimal resistance at x=3 (∑r^3), with resistance increasing as x deviated from 3.
• Universality across fluids: The U-shaped resistance curves with minima at the Murray exponents were observed for multiple fluids under laminar flow, including 2-butanol (high viscosity), hexane (low viscosity), ethanol (polar), and toluene (non-polar), in both planar and tubular structures.
• Diffusion: Simulations showed that planar and tubular structures following the corresponding Universal Murray's law for diffusion achieve optimal diffusion efficiency.
• Structural characterization: Freeze-cast GOA had ~98.2% porosity; vertically porous pore diameters ranged from ~7.85–39.8 µm depending on freezing temperature; liquid nitrogen-frozen GOA ~6.23 µm; lamellar spacings ~30.5–97.1 µm. Orientation degree for lamellar GOA ranged ~0.62–0.88, higher than vertically porous/isotropic GOA (~0.18–0.30).
• Sensor application: A simple macroscopic shape adjustment to satisfy ∑r^3 in a three-section tubular GOA sensor reduced simulated air-flow resistance by ~12.3% and improved dynamic performance: response and recovery times shortened by ~8.6% to 18.2% for 1 ppm NO2, NH3, and CH2O compared to a non-optimized hierarchical straight cylinder of equal volume and length.

The findings confirm that optimizing hierarchical porous networks by the Universal Murray's law minimizes transport resistance under a fixed volume constraint, directly addressing the challenge of applying classic Murray's law to synthetic materials with non-ideal geometries. By formulating the optimality condition in terms of general size exponents and accommodating arbitrary shapes (including planar hierarchies), the work establishes a geometry-agnostic design rule. Experimental validations across planar and tubular GOA structures, with multiple fluids, demonstrate that adherence to the prescribed exponent (x=2 for planar laminar flow; x=3 for tubular laminar flow) yields minimal resistance, while deviations increase resistance, producing a characteristic U-shaped behavior. Simulations corroborate the even flow distribution assumption and the resistance balance across sections. The demonstrated improvements in gas sensor response dynamics from simple geometric adjustments underscore the practical relevance, suggesting broad applicability to catalysis, sensing, energy storage, and other mass-transfer-limited systems.

The study introduces a Universal Murray's law that generalizes optimal transport in hierarchical networks to arbitrary channel shapes and a broad set of transfer processes by minimizing resistance. It recovers known results, extends to previously unexplored structures (planar) and regimes (e.g., Knudsen diffusion), and provides a direct, shape-independent formulation for diffusion and ionic/electronic transport. Experimental and CFD evidence using freeze-cast graphene oxide aerogels validate the theory for planar and tubular laminar flows, across various fluids. A practical demonstration in a gas sensor shows that simple, Murray-guided geometric optimization enhances mass transport and sensor dynamics without altering material chemistry. Future work could experimentally probe additional transport regimes (e.g., turbulent and non-Newtonian flows), apply the framework to other materials and multiscale architectures, and integrate reaction kinetics and adsorption into coupled optimization for catalytic and electrochemical systems.

• Experiments focused on laminar flow; although the theory covers other regimes (turbulent, non-Newtonian), these were not experimentally validated.
• CFD used scaled-down geometrical models and assumed smooth walls and laminar flow; real materials may have surface roughness and defects affecting resistance.
• The even flow distribution within sections is an assumption; while simulations suggest approximate uniformity, deviations in real samples could occur.
• Freeze-casting affords limited precise control of pore sizes; the study tuned channel numbers and macroscopic geometry rather than finely controlling pore dimensions at each level.
• Pore wall thickness was considered negligible; this approximation may affect exact resistance estimates in denser materials.
• Electrical double layer effects were neglected based on micron-scale pores; results may not directly translate to nanofluidic regimes where EDL is significant.
• Knudsen diffusion and diffusion optimizations were discussed and simulated but not experimentally tested in this work.