# Yair Shokef's

Research Group

#### Our research is all about geometric frustration in mesoscopic lattice-based soft-matter systems…

Frustration is all around us, from personal debates between conflicting considerations in our everyday lives to trying to tile our kitchen floor with regular pentagons, not to mention the complex folding of proteins in the cells of every living creature. In condensed-matter physics, frustration is considered in terms of competing interactions or incommensurability between different symmetries, and leads to highly-degenerate, disordered ground-states, to jamming and glassiness, and to peculiar excitations and response functions. Recent experiments allow to generate soft-matter mesoscopic systems, which are built of macroscopic building blocks. The latter have dimensions that are significantly larger than the atomic length scale and thus may be individually visualized and even manipulated. Artificial frustrated matter is formed when multiple such units are spatially ordered in a geometrically-frustrated manner. Our research employs theoretical tools from condensed-matter and statistical physics in order to analyze frustration in such systems. The building blocks comprising the systems that we study range from micrometer-sized colloidal spheres to centimeter-sized 3D-printed elastic structures, or mechanical metamaterials. Our research on soft-matter mesoscopic systems contributes to the understanding of atomic systems like ice, glasses and magnets, but also leads to applications at the macro-scale such as designable rigidity and personalized mechanics.

Geometric frustration has been traditionally considered separately in soft vs. hard condensed-matter physics; Soft-matter systems such as granular materials and colloidal suspensions are governed by disordered (off-lattice) packings. There, the tendency to close-pack leads locally to tetrahedral motifs, which are inconsistent with the globally optimal crystalline packing. On the other hand, in hard-condensed-matter antiferromagnets, the magnetic moments of neighboring particles on a periodic lattice tend to be oriented in opposite directions. If the lattice these particles reside on contains loops with an odd number of sites, such as the two-dimensional triangular lattice, all pairs of nearest-neighbor spins cannot be simultaneously antiparallel.

We identified a colloidal system that connects these two seemingly different types of frustration; for a buckled monolayer of spheres, packing considerations induce frustration that is similar to that appearing in the triangular-lattice antiferromagnet. Interestingly, small out-of-plane displacements of the spheres translate into in-plane lattice deformations. We found that when lattice deformability is added to the theoretical model, we could explain the experimental results concerning the glassy relaxation dynamics and the partial spatial ordering in the form of zigzagging stripes, shown in the image. In this lattice system, disorder emerges naturally from the inherent frustration of the interactions, and the underlying lattice enables many results to be obtained for it theoretically, therefore many insights may be learned from it on the mechanisms governing disordered glassy systems.

In conjunction with our efforts to understand this colloidal antiferromagnet we have been studying a related colloidal ice. Here, particles reside in a two-dimensional array of double-well traps such that at each vertex of the lattice, four traps meet, and the magnetic repulsion between colloidal particles imposes a 2-in-2-out ice rule. The two-dimensional nature of this system significantly reduces the ground-state degeneracy, which appears in three-dimensional ice. We suggested to shear the square lattice, as shown in the image in order to partially restore this degeneracy. This enabled us to identify and study topological excitations that restrict the restoration of degeneracy.

We have also introduced the concepts of disorder and frustration into the realm of mechanical metamaterials. These are built of repeating, macroscopic unit cells, and are designed so that their cooperative local deformations will lead to unusual mechanical behavior at the system level. We introduced three-dimensional structures with anisotropic unit cells. When the orientation of each one is set at random, they typically form a non-periodic structure, in which adjacent unit cells may not all deform self-consistently, thus constituting a frustrated mechanical spin glass. By mapping to a discrete model, we presented a combinatorial strategy for the design of a multitude of non-periodic, yet frustration-free metamaterials that exhibit spatially-textured responses. We demonstrated these by designing three-dimensional metacubes, which when compressed can deform to give any pre-defined texture on their faces. Moreover, we quantitatively explained how pressing on a metacube with the wrong texture increases its overall rigidity. Taken together, these functionalities, which are embedded within the metamaterial constitute it as a “machine material”.

In two dimensions we reached a much deeper theoretical understanding, and by controlling the degree of frustration we could steer the mechanical response of a textured metamaterial. Using anisotropic triangular building blocks, we constructed non-periodic networks whereby the mutual orientations of the triangles determine their mechanical compatibility. Specifically, we can design an extensive set of frustration-free systems. More interestingly, we can introduce into these systems topological defects that induce frustration at an arbitrary distance from the defect. This allows, for instance to steer the mechanical stress to one side of the system and displacements to the other side. We subsequently translated the insight gained in two dimensions to a generalized framework for describing mechanical frustration and topological defects also in three-dimensional metamaterials.

These concentrated efforts promote our group’s major research track on geometric frustration in mesoscopic lattice-based soft-matter systems. They integrate with our broader activities on non-equilibrium statistical mechanics of soft-matter systems, which in recent years have included on one hand stochastic dynamics in lattice gas models, and in particular kinetically-constrained models, and on the other hand the physics of living systems and active matter, and specifically cell mechanics as well as synthetic active metamaterials.