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

Abstract

Materials following Murray's law are of significant interest due to their unique porous structure and optimal mass transfer ability. However, it is challenging to construct such biomimetic hierarchical channels with perfectly cylindrical pores in synthetic systems following the existing theory. Achieving superior mass transport capacity revealed by Murray's law in nanostructured materials has thus far remained out of reach. We propose a Universal Murray's law applicable to a wide range of hierarchical structures, shapes and generalised transfer processes. We experimentally demonstrate optimal flow of various fluids in hierarchically planar and tubular graphene aerogel structures to validate the proposed law. By adjusting the macroscopic pores in such aerogel-based gas sensors, we also show a significantly improved sensor response dynamics. In this work, we provide a solid framework for designing synthetic Murray materials with arbitrarily shaped channels for superior mass transfer capabilities, with future implications in catalysis, sensing and energy applications.

Fig. 1. Murray’s law in hierarchical structures.

a Schematic illustration of branching columnar tubes and corresponding initial expressions of Murray’s law. Here r₁, r₂, and r₃ represent the radii of tubes at different levels. b Schematic illustration of demonstrative hierarchical structure in materials with comprehensive pore shapes and corresponding expression of Murray’s law. Here A₁, A₂, and A₃ represent the cross-sectional area of pores at different levels. c Schematic illustration of hierarchically tubular network with arbitrary shape and corresponding expressions of Murray’s law. Here x₁, x₂, and x₃ represent the selected variables of channels at different levels. d Schematic illustration of hierarchical lamellar structure and corresponding expressions of Murray’s law. Here h₁, h₂, and h₃ represent the heights of planar channels at different levels, d denotes the width of the plates.

Fig. 2. The structure of unidirectional and bidirectional freeze-cast graphene oxide aerogel (GOA).

a Schematic illustration of unidirectional freeze-casting method. Here ΔT_V denotes the temperature difference in the vertical direction. (b-d) Top-view SEM images of unidirectional freeze-cast GOA frozen at (b) −20 ^∘C, (c) −40 ^∘C, and (d) −70 ^∘C. Insets: Fourier transform images. e The average pore size of unidirectionally freeze-cast GOA frozen at different temperatures and GOA frozen by liquid nitrogen. f Schematic illustration of bidirectional freeze-casting method. Here ΔT_V denotes the temperature difference in the vertical direction and ΔT_H represents the temperature difference in the horizontal direction. g–i Top-view SEM images of bidirectionally freeze-cast GOA frozen at (g) −10 ^∘C, (h) −30 ^∘C, and (i) −50 ^∘C. Insets: Fourier transform images. j Average layer height of bidirectional freeze-cast GOA at different temperatures. k Orientation degree of unidirectionally and bidirectionally freeze-cast GOA. Scale bars: 100 μm. All error bars represent standard deviations of pore size calculated by the corresponding image processing programs (see Methods).

Fig. 3. The experimental and simulation validation of Universal Murray’s law.

a, b The experimental flow resistance of (a) water and (b) air in hierarchical lamellar GOA with a fixed total volume and corresponding simulated resistance in scaled-down models. c Changes in the estimated laminar flow resistance and section volume with exponent x in the conservation. Here R₁, R₂, R₃, and R represent the estimated resistances of different sections and the total resistance. V₁, V₂, V₃, and V represent the volumes of different sections and the total volume. d, e The experimental flow resistance of (d) water and (e) air in hierarchically tubular GOA and corresponding simulated results in scaled-down models. f–i The experimental flow resistance of (f) 2-butanol, (g) hexane, (h) ethanol, and (i) toluene in hierarchical lamellar GOA. j–m The experimental flow resistance of (j) 2-butanol, (k) hexane, (l) ethanol, and (m) toluene in hierarchical tubular GOA. The error bars arise from the linear fit of pressure drop to the flow rate as discussed in Methods.

Fig. 4. Optimising tubular GOA-based gas sensor by Murray’s law.

a Schematic illustration of the synthesis of SnO₂ quantum dot (QD)-decorated graphene oxide aerogel (GOA) and the assembly of hierarchical gas sensor. b–d TEM images of SnO₂ QD-decorated GO ink. Scale bars: (b) 100 nm, (c) 10 nm and (d) 2 nm. e–g Top-view SEM images of unidirectional freeze-cast GOA frozen at (e) −20 ^∘C, (f) −40 ^∘C, and (g) −70 ^∘C. Scale bars: 100 μm. h Response time (τ_res) and recovery time (τ_rec) of SnO₂ QD-doped GOA in the hierarchical straight pipe and the pipe optimised by Murray’s law to 1 ppm nitrogen dioxide, ammonia, and formaldehyde. i Air flow simulation in the scaled-down models of hierarchical and Murray GOA. The standard deviations of pore size are calculated by the corresponding image processing programs (see Methods).