Sufficient Conditions for the Global Rigidity of Periodic Graphs

Tanigawa (2016) showed that vertex-redundant rigidity of a graph implies its global rigidity in arbitrary dimension. We extend this result to periodic frameworks under fixed lattice representations. That is, we show that if a generic periodic framework is vertex-redundantly rigid, in the sense that the deletion of a single vertex orbit under the periodicity results in a periodically rigid framework, then it is also periodically globally rigid. Our proof is similar to the one of Tanigawa, but there are some added difficulties. First, it is not known whether periodic global rigidity is a generic property in dimension $$d>2$$ d > 2 . We work around this issue by using slight modifications of recent results of Kaszanitzky et al. (2021). Secondly, while the rigidity of finite frameworks in $${\mathbb {R}}^d$$ R d on at most d vertices obviously implies their global rigidity, it is non-trivial to prove a similar result for periodic frameworks. This is accomplished by extending a result of Bezdek and Connelly (2002) on the existence of a continuous motion between two equivalent d-dimensional realisations of a single graph in $${\mathbb {R}}^{2d}$$ R 2 d to periodic frameworks. As an application of our result, we give a necessary and sufficient condition for the global rigidity of generic periodic body-bar frameworks in arbitrary dimension. This provides a periodic counterpart to a result of Connelly et al. (2013) regarding the global rigidity of generic finite body-bar frameworks.

PEG-stabilized coaxial stacking of two-dimensional covalent organic frameworks for enhanced photocatalytic hydrogen evolution

Two-dimensional covalent organic frameworks (2D COFs) featuring periodic frameworks, extended π-conjugation and layered stacking structures, have emerged as a promising class of materials for photocatalytic hydrogen evolution. Nevertheless, the layer-by-layer assembly in 2D COFs is not stable during the photocatalytic cycling in water, causing disordered stacking and declined activity. Here, we report an innovative strategy to stabilize the ordered arrangement of layered structures in 2D COFs for hydrogen evolution. Polyethylene glycol is filled up in the mesopore channels of a β-ketoenamine-linked COF containing benzothiadiazole moiety. This unique feature suppresses the dislocation of neighbouring layers and retains the columnar π-orbital arrays to facilitate free charge transport. The hydrogen evolution rate is therefore remarkably promoted under visible irradiation compared with that of the pristine COF. This study provides a general post-functionalization strategy for 2D COFs to enhance photocatalytic performances.

Evaluation of Cellular Solids Derived From Triply Periodic Minimal Surfaces

Cellular solids are a class of materials that have many interesting engineering applications, including ultralight structural materials [1]. The traditional method for analyzing these solids uses convex uniform polyhedral honeycombs to represent the geometry of the material [2], and this approach has carried over into the design of digital cellular solids [3]. However, the use of such honeycomb-derived lattices makes the problem of decomposing a three-dimensional lattice into a library of two-dimensional parts non-trivial. We introduce a method for generating periodic frameworks from Triply Periodic Minimal Surfaces (TPMS), which result in geometries that are easier to decompose into digital parts. Additionally, we perform multi-scale analysis of two cellular solids generated from two TPMS, the P- and D-Schwarz, and two cellular solids, the Kelvin and Octet honeycombs. We show that the simulated behavior of these TMPS-derived structures shows the expected modulus of the cellular solid scaling linearly with relative density, and matches the behavior of the octet truss.

Geometric auxetics

We formulate a mathematical theory of auxetic behaviour based on one-parameter deformations of periodic frameworks. Our approach is purely geome- tric, relies on the evolution of the periodicity lattice and works in any dimension. We demonstrate its usefulness by predicting or recognizing, without experiment, computer simulations or numerical approximations, the auxetic capabilities of several well-known structures available in the literature. We propose new principles of auxetic design and rely on the stronger notion of expansive behaviour to provide an infinite supply of planar auxetic mechanisms and several new three-dimensional structures.