Two-dimensional (2D) crystalline materials possess unique structural, mechanical, and electronic properties1,2, which have rendered them highly attractive in many applications3-5. Although there have been advances in preparing 2D materials that consist of one or few atomic/molecular layers6,7, bottom-up assembly of 2D crystalline materials remains a considerable challenge and an active area of development8-10. Even more challenging is the design of dynamic 2D lattices that can undergo large-scale motions without loss of crystallinity. Dynamicity in porous 3D crystalline solids has been exploited for stimuli-responsive functions and adaptive behavior11-13. As in the case of such 3D materials, integrating flexibility/adaptiveness into crystalline 2D lattices would greatly broaden the functional scope of 2D materials. Here we report the self-assembly of unsupported, 2D protein lattices with precise spatial arrangements and patterns through a readily accessible design strategy. Three single- or double-point mutants of the C4 symmetric protein RhuA were designed to assemble via different modes of intermolecular interactions (single disulfide, double disulfide and metal coordination) into crystalline 2D arrays. Owing to the flexibility of the single disulfide interactions, the lattices of one of the variants (C98RhuA) are essentially defect-free and undergo substantial but fully correlated changes in molecular arrangement, giving coherently dynamic 2D molecular lattices. Notably, C98RhuA lattices possess a Poisson's ratio of −1, the lowest thermodynamically possible value for an isotropic material.

Biological materials found in Nature such as nacre and bone are well recognized as light-weight, strong, and tough structural materials. The remarkable toughness and damage tolerance of such biological materials are conferred through hierarchical assembly of their multiscale (i.e., atomicto macroscale) architectures and components. Herein, the toughening mechanisms of different organisms at multilength scales are identified and summarized: macromolecular deformation, chemical bond breakage, and biomineral crystal imperfections at the atomic scale; biopolymer fibril reconfiguration/deformation and biomineral nanoparticle/nanoplatelet/ nanorod translation, and crack reorientation at the nanoscale; crack deflection and twisting by characteristic features such as tubules and lamellae at the microscale; and structure and morphology optimization at the macroscale. In addition, the actual loading conditions of the natural organisms are different, leading to energy dissipation occurring at different time scales. These toughening mechanisms are further illustrated by comparing the experimental results with computational modeling. Modeling methods at different length and time scales are reviewed. Examples of biomimetic designs that realize the multiscale toughening mechanisms in engineering materials are introduced. Indeed, there is still plenty of room mimicking the strong and tough biological designs at the multilength and time scale in Nature.

A phase transition is the transformation of a thermodynamic system from one phase to another. This transformation occurs in a wide range of structure and systems by the deformation of the crystal into a new structure changing the packing arrangement of the atoms in the unit cell. The different phases exhibit distinct properties and this change in the properties is exploited by technological innovations. Shape memory alloys are materials that present phase transformation alloys where martensitic phase transformation occurs by the deformation of the crystal into a new structure. A characteristic feature of the phase transformation in memory alloys is a periodic saw tooth pattern in the stress plateau of the stress-strain curve. Many biological components present behaviors that resemble the phase transformation of shape memory alloys; e.g., certain structural proteins exhibits saw tooth patterns when switching from a folded to unfolded confi guration. In addition, researchers have shown that nacre achieves its remarkable toughness, without sacrifi cing its strength and stiffness, from its wavy brick-and-mortar-like microstructure. The wavy bricks, in conjunction with the mineral bridges and the organic glue, activate a very unique compression-tension behavior that leads to irreversible bistable mechanisms resulting in an effi cient process of energy dissipation and spreading of damage. In this article, we propose to extend this notion of diffusionless solid-state phase transformations to cellular solids, where we defi ne a phase transformation to represent a change in the geometry of the unit cell. This is achieved by utilizing either bistable or metastable mechanisms as base for the unit cells of the cellular material. The phase transformation is due to a progressive change of confi gurations from cell to cell leading to a serrated force displacement behavior. The cells are design in such way that the deformations remain in the elastic regime making the process reversible. Analytical and computational micromechanics based models will be presented and used to estimate the effective properties of the phase transforming cellular material in each of its phases. In addition, initial studies on the characterization of the material moduli, energy absorption, volume change, self-excited dynamic response associated with the phase transformation and effect of phase change on the wave guiding properties of the material among other properties will be presented.

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