Biomimetic diffusion metamaterials enabled by physical-geometric duality

Structural design typically faces a fundamental trade-off between advanced functionality and practical manufacturability, a challenge that is especially acute in freeform metamaterials, where omnidirectional performance often demands intricate fabrication. Inspired by the angle-preserving growth of leaves, we address this long-standing challenge by establishing a physical-geometric duality for metamaterial design. This duality enables geometric transformations to be derived directly from material parameters and fields obtained via physical equivalence. The resulting diffusion metamaterials exhibit a distinctive behavior: despite strong anisotropy and heterogeneity in thermal conductivity, isotherms and heat streamlines remain mutually orthogonal. This property effectively reduces the realization of complex thermal conductivity tensors to the direct alignment of laminated composites along local field lines without optimization. We experimentally demonstrate a freeform thermal cloak that achieves omnidirectional invisibility. Our work establishes a biomimetic design paradigm that reconciles functionality with manufacturability, with broad implications for thermal management and diffusive transport. Leaf-inspired, this work links physical equivalence and geometric transformation into a physical-geometric duality, reconciling structural functionality and manufacturability to directly fabricate complex, omnidirectional diffusion metamaterials.