Fabrication-Directed Entanglement for Designing Chiral and Anisotropic Metamaterial Foams

Entangled networks are fundamental in various systems, from biological structures to engineered materials. Current techniques for programming entanglement often rely on intricate chemistry or result in statistically homogeneous networks, limiting the ability to create spatially patterned structures with precisely engineered functions. Thus, a key challenge remains in developing approaches to program complex mechanical behaviors, such as anisotropy and chirality, within monolithic entangled structures. This work introduces Fabrication-Directed Entanglement (FDE), a methodology integrating viscous thread printing (VTP) and topology optimization (TO) to program the entanglement of a single homogeneous filament. By spatially adjusting VTP parameters (deposition height, speed), we control local coiling density, creating quasi-two-phase (dense/sparse) regions within a monolithic entangled foam. Topology optimization guides the placement of these regions to achieve target macroscopic mechanical properties. Here we show foam-like mechanical metamaterials with tunable compliance, rigidity, and chirality. Experimental testing and simulation confirm that FDE expands the achievable material property space compared to homogeneous VTP foams, enabling properties like tunable directional stiffness, Poisson's ratios from $\nu\approx0.06~\text{to}~0.56$, and novel significant normal-shear coupling ($\eta_{212}\approx0.72$) from a single base material. This approach offers a viable new pathway for designing complex, functional entangled foam structures with tailored mechanical behaviors.