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Soft materials are attractive to engineers due to their low density, ability to sustain large deformations and customization of properties. These custom properties are realized by tailoring the microstructural architecture and optimizing the constituent materials. However, achieving complex behavior at the macroscale is difficult through conventional engineering top-down approaches. Instead, we draw inspiration from a more biological bottom-up approach based on the concept of emergence. Slime molds and fungi are fascinating models for emergence; formed by aggregation of almost identical cells, their internal networks optimize transport better than engineers, solve mazes, detect masses at a distance, or memorize periodic events. We establish here a theoretical and computational framework to explain, quantify and reproduce how this feedback between macroscopic features and local tuning of mechanical properties determines an emergent coordinated response of the network morphology. These networks have the ability to grow, survive, or die in the presence of different concentrations of nutrients and to redistribute them, thus optimizing their proliferation. We propose a phase-field scalar variable to represent the network matrix evolution and a diffusive-advective process for nutrient distribution. This framework enables high-fidelity simulations of slime molds in three-dimensional space, which are challenging due to the coupled physics involved, high-order partial differential equations, and the existence of a highly complex evolving geometry. The emergent properties of these organisms are similar in nature to tissues with transport networks, and we can exploit them to design self-healing materials, fungal sustainable building elements, or even coordinate drone swarms. This investigation is a road to the microstructural design of active soft matter, an object of interest in many fields of engineering and science.
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