Combinatorial design of textured mechanical metamaterials.

The structural complexity of metamaterials is limitless, but, in practice, most designs comprise periodic architectures that lead to materials with spatially homogeneous features. More advanced applications in soft robotics, prosthetics and wearable technology involve spatially textured mechanical functionality, which requires aperiodic architectures. However, a naive implementation of such structural complexity invariably leads to geometrical frustration (whereby local constraints cannot be satisfied everywhere), which prevents coherent operation and impedes functionality. Here we introduce a combinatorial strategy for the design of aperiodic, yet frustration-free, mechanical metamaterials that exhibit spatially textured functionalities. We implement this strategy using cubic building blocks-voxels-that deform anisotropically, a local stacking rule that allows cooperative shape changes by guaranteeing that deformed building blocks fit together as in a three-dimensional jigsaw puzzle, and three-dimensional printing. These aperiodic metamaterials exhibit long-range holographic order, whereby the two-dimensional pixelated surface texture dictates the three-dimensional interior voxel arrangement. They also act as programmable shape-shifters, morphing into spatially complex, but predictable and designable, shapes when uniaxially compressed. Finally, their mechanical response to compression by a textured surface reveals their ability to perform sensing and pattern analysis. Combinatorial design thus opens up a new avenue towards mechanical metamaterials with unusual order and machine-like functionalities.

Ordered hexagonal patterns via notch-delta signaling.

Many developmental processes in biology utilize notch-delta signaling to construct an ordered pattern of cellular differentiation. This signaling modality is based on nearest-neighbor contact, as opposed to the more familiar mechanism driven by the release of diffusible ligands. Here, exploiting this 'juxtacrine' property, we present an exact treatment of the pattern formation problem via a system of nine coupled ordinary differential equations. The possible patterns that are realized for realistic parameters can be analyzed by considering a co-dimension 2 pitchfork bifurcation of this system. This analysis explains the observed prevalence of hexagonal patterns with high delta at their center, as opposed to those with central high notch levels (referred to as anti-hexagons). We show that outside this range of parameters, in particular for lowcis-coupling, a novel kind of pattern is produced, where high delta cells have high notch as well. It also suggests that the biological system is only weakly first order, so that an additional mechanism is required to generate the observed defect-free patterns. We construct a simple strategy for producing such defect-free patterns.

Finite-density effects in the Fredrickson-Andersen and Kob-Andersen kinetically-constrained models.

We calculate the corrections to the thermodynamic limit of the critical density for jamming in the Kob-Andersen and Fredrickson-Andersen kinetically-constrained models, and find them to be finite-density corrections, and not finite-size corrections. We do this by introducing a new numerical algorithm, which requires negligible computer memory since contrary to alternative approaches, it generates at each point only the necessary data. The algorithm starts from a single unfrozen site and at each step randomly generates the neighbors of the unfrozen region and checks whether they are frozen or not. Our results correspond to systems of size greater than 10(7) × 10(7), much larger than any simulated before, and are consistent with the rigorous bounds on the asymptotic corrections. We also find that the average number of sites that seed a critical droplet is greater than 1.

Constraint relaxation leads to jamming.

Adding transitions to an equilibrium system increases the activity. Naively, one would expect this to hold also in out-of-equilibrium systems. We demonstrate, using relatively simple models, how adding transitions to an out of equilibrium system may in fact reduce the activity and even cause it to vanish. This surprising effect is caused by adding heretofore forbidden transitions into less and less active states. We investigate six related kinetically constrained lattice gas models, some of which behave as naively expected while others exhibit this nonintuitive behavior. These models exhibit an absorbing state phase transition, which is also affected by the added transitions. We introduce a semi-mean-field approximation describing the models, which agrees qualitatively with our numerical simulation.

Hydrodynamics in kinetically constrained lattice-gas models.

Kinetically constrained models are lattice-gas models that are used for describing glassy systems. By construction, their equilibrium state is trivial and there are no equal-time correlations between the occupancy of different sites. We drive such models out of equilibrium by connecting them to two reservoirs of different densities, and we measure the response of the system to this perturbation. We find that under the proper coarse-graining, the behavior of these models may be expressed by a nonlinear diffusion equation, with a model- and density-dependent diffusion coefficient. We find a simple approximation for the diffusion coefficient, and we show that the relatively mild discrepancy between the approximation and our numerical results arises due to non-negligible correlations that appear as the system is driven out of equilibrium, even when the density gradient is infinitesimally small. Similar correlations appear when such kinetically constrained models are driven out of equilibrium by applying a uniform external force. We suggest that these correlations are the reason for the same discrepancy between the approximate diffusion coefficient and the numerical results for a broader group of models-nongradient lattice-gas models-for which kinetically constrained models are arguably the simplest example thereof.

Relation between structure of blocked clusters and relaxation dynamics in kinetically constrained models.

We investigate the relation between the cooperative length and relaxation time, represented, respectively, by the culling time and the persistence time, in the Fredrickson-Andersen, Kob-Andersen, and spiral kinetically constrained models. By mapping the dynamics to diffusion of defects, we find a relation between the persistence time, τ_{p}, which is the time until a particle moves for the first time, and the culling time, τ_{c}, which is the minimal number of particles that need to move before a specific particle can move, τ_{p}=τ_{c}^{γ}, where γ is model- and dimension-dependent. We also show that the persistence function in the Kob-Andersen and Fredrickson-Andersen models decays subexponentially in time, P(t)=exp[-(t/τ)^{ß}], but unlike previous works, we find that the exponent ß appears to decay to 0 as the particle density approaches 1.

Jamming by shape in kinetically constrained models.

We derive expressions for the critical density for jamming in a hyper-rhomboid system of arbitrary shape in any dimension for the Kob-Andersen and Fredrickson-Andersen kinetically constrained models. We find that changing the system's shape without altering its total volume or particle density may induce jamming. We also find a transition between shapes in which the correlation length between jammed particles is infinite and shapes that have a finite correlation length.

Jamming transition of kinetically constrained models in rectangular systems.

We theoretically calculate the average fraction of frozen particles in rectangular systems of arbitrary dimensions for the Kob-Andersen and Fredrickson-Andersen kinetically constrained models. We find the aspect ratio of the rectangle's length to width, which distinguishes short, square-like rectangles from long, tunnel-like rectangles, and show how changing it can effect the jamming transition. We find how the critical vacancy density converges to zero in infinite systems for different aspect ratios: For long and wide channels it decreases algebraically v(c) ~ W(-1/2) with the system's width W, while in square systems it decreases logarithmically vc) ~ 1/lnL with length L. Although derived for asymptotically wide rectangles, our analytical results agree with numerical data for systems as small as W ≈ 10.