Duality and Sheared Analytic Response in Mechanism-Based Metamaterials

Transformable Superisostatic Crystals Self-Assembled from Segment Colloidal Rods

“…We find that in the range of 60–120°, v is positive, while in the range of 0–60° and 120–180°, v is negative (auxetic) . Similar results were also reported in ref . It should be emphasized that structures with ϑ and 180° – ϑ are in fact the same structure connected by 90° rotation due to the symmetry of this structure.…”

“…69 Similar results were also reported in ref. 48 It should be emphasized that structures with ϑ and 180°− ϑ are in fact the same structure connected by 90°rotation due to the symmetry of this structure.…”

“…Recently, a theoretical framework based on complex analyticity has been developed from CEM theory. 48 In order to verify the above CEM theory, we calculate the phononic spectrum for pgg crystals with different open angles ϑ under periodic boundary conditions (PBCs). We confirm that the soft bulk modes only exist in the shear dominant regime 60°≤ ϑ ≤ 120°and the soft mode direction is along a 1 and a 2 , fully agreeing with the theoretical predictions.…”

“…Notably, ϑ can be smoothly adjusted without any energy cost, making the pgg structures transformable superisostatic structures. 46,48,72 Examining the self-assembled pgg structures in Figure 2f, we observe that when α is approximately 260°, the permissible range of ϑ spans from 40°to 80°, encompassing the transition point at ϑ = 60°. Consequently, the self-assembled structures are capable of transforming between two different soft states through stretching or compression.…”

“…For example, it was found that spherical two-end-patchy colloidal particle can self-assemble into deformable kagome lattice stabilized by vibrational entropy. , This structure is at the verge of mechanical stability since the degrees of freedom in the structure are merely balanced by the mechanical constraint, leaving only a few global soft modes. Such deformable structures are called isostatic (Maxwell) lattices, which generally satisfy z = 2 d , with z being the coordination number of the structure and d being the spatial dimension. − Recently, it has been found that half-cylinder colloidal rods can self-assemble into self-dual isostatic crystals with hidden-symmetry, which exhibit the classical analog of Kramers degeneracy and critically tilted Dirac cone in quantum systems. − On the other side, theoretical studies have shown that other isostatic structures can exhibit topological phase transitions and robust topologically protected soft edge states. ,,− Considering the fact that current phononic and mechanical metamaterials can only be fabricated through top-down methods, e.g, 3D printing techniques, the self-assembly of deformable crystals provides us another option to obtain these materials at microscale with potential applications in acoustic cloaking, focusing, shock/noise reduction, and nanomechanically engineering. − …”