Kirigami, the traditional art of paper cutting, has recently emerged as a promising paradigm for mechanical metamaterials. While many prior works have studied the geometry and mechanics of certain periodic kirigami tessellations, the computational design of more complex structures is less understood. In this talk, I will present several mathematical design frameworks for modulating the geometry of kirigami tessellations. In particular, by identifying the geometric constraints controlling the compatibility, compact reconfigurability and rigid-deployability of the kirigami structures, we can achieve a wide range of patterns that can be deployed into pre-specified shapes in two or three dimensions.

]]>What do wrinkled shells and shape-morphing kirigami sheets have in common? The answer is fine scale buckling — a patterned response driven by mechanical instabilities enabling macroscopic shape change beyond bulk elasticity. This talk will present two recent developments on (i) a hidden duality between the wrinkles of positively and negatively curved confined shells [1], and (ii) the aggregate deformations of kirigami sheets made by removing a lattice of holes [2]. In both cases, a homogenization-style argument leads to a coarse-grained description of the pattern which is simpler to study, and is even explicitly solvable sometimes. Wrinkle patterns become the subdifferentials of pairs of convex functions; kirigami is shown to deform along a family of non-Euclidean isometries set by the holes. The resulting predictions match the results of simulations and experiments across scales.

[1] I. Tobasco, Y. Timounay, D. Todorova, G. C. Leggat, J. D. Paulsen, and E. Katifori, “Exact Solutions for the Wrinkle Patterns of Confined Elastic Shells,” Nat. Phys. 18 (2022) 1099-1104

[2] Y. Zheng, I. Niloy, P. Celli, I. Tobasco, and P. Plucinsky, “Continuum Field Theory for the Deformations of Planar Kirigami,” Phys. Rev. Lett. 128 (2022) 208003

How should space-filling, deformable objects be packed so as to minimize the area of the interfaces between them? In 3D the search for a packing of equal-volume shapes with minimum surface area is known as the Kelvin problem. In 2D the hexagonal honeycomb is the least perimeter way to divide the plane into regions of equal area.

I will describe contributions to finite clusters of 2D objects, that is, the arrangement of N deformable “bubbles” that minimizes the total perimeter. I will describe conjectured optimal candidates (without proving anything!) under the assumption that each bubble is connected. For equal-area free clusters the candidates consist mostly of hexagons with defects confined to the perimeter. When the bubbles are confined within a fixed boundary the location of the defects becomes, in certain cases, easier to predict. For bidisperse clusters the mixing of the large and small bubbles in the optimal candidates is rather sensitive to the area ratio. And in the case of very large clusters, of more than about 5000 bubbles, extra peripheral defects are introduced to reduce the perimeter.

]]>Motivated by patterns with defects in natural and laboratory systems, we develop two quantitative measures of order for imperfect Bravais lattices in the plane. A tool from topological data analysis called persistent homology combined with the sliced Wasserstein distance, a metric on point distributions, are the key components for defining these measures. We also study imperfect hexagonal, square, and rhombic arrangements of nanodots produced by numerical simulations of pattern-forming partial differential equations and apply the measures to study packing on surfaces.

]]>Protein cages, convex polyhedral protein containers self-assembled from multiple copies of identical protein subunits, are pillars of nanotechnology. They include naturally occurring virus capsids (virus-like particles, VLPs) which are spherical as well as a plethora of particles with diverse symmetries such as tubes. VLPs have an innate ability to encapsulate nucleic acids, which makes them attractive as DNA/RNA delivery vehicles and in vaccine development. In this talk I will discuss a mathematical framework for the classification of the structures of such VLPs. This has been used, in collaboration with experimentalists, to programmably control VLPs size and symmetry for a model system, achieving larger particle sizes with higher carrying capacity than wild-type virus. I will also present a model of VLP assembly, that incorporates different assembly pathways leading to distinct particle geometries. The model reveals a mechanism by which particle size can be controlled and can thus serve as a guide for the design of desired particle morphologies for applications in virus nanotechnology. At the end, I will present a mathematical analysis of tubular particles arising from the self-assembly of variants of the Lumazine Synthase (LS) enzyme and explain the mathematical principles of tubular architectures that have recently been observed experimentally.

]]>Geometric frustration arises whenever the constituents of a physical assembly locally favor an arrangement that is incompatible with the geometry or topology of the space in which it resides. It naturally arises in a variety of fields ranging from macromolecular assemblies to liquid-crystals to spin models, yet in distinct systems, geometric frustration may be associated with different phenomena. For example, in liquid crystals frustration may lead to spontaneous size limitation and to a unique ground state whose energy grows super-extensively, while for the Ising antiferromagnet on triangular lattice frustration leads to a highly degenerate ground state of extensive energy.

In this talk, I will discuss how the intrinsic approach, in which matter is described only through local properties available to an observer residing within the material, overcomes the lack of a stress-free rest state and leads to a general framework in which the geometric compatibility conditions assume a central role. This framework, in particular, allows predicting the super-extensive energy exponent for sufficiently small systems and explains the origin of the large variety of phenomena attributed to geometric frustration. I will discuss its application to several specific systems exhibiting geometric frustration including growing elastic bodies, frustrated liquid crystals, and frustrated spin systems. Time permitting, I will discuss how the discretization of the degrees of freedom affects a system’s response to geometric frustration.

]]>Crystals and other condensed matter systems described by density waves often exhibit dislocations. Here we show, by considering the topology of the ground state manifolds (GSMs) of such systems, that dislocations in the density phase field always split into disclinations, and that the disclinations themselves are constrained to sit at particular points in the GSM. Consequently, the topology of the GSM forbids zero-energy dislocation glide, giving rise to a Peierls-Nabarro barrier.

]]>Materials whose fundamental units are nanocrystals (NC)s, instead of atoms or molecules, are emerging as major candidates to solve many of the technological challenges of our century. Those materials display unique structural, dynamical and thermodynamical properties, often reflecting deep underlying geometric, packing and topological constraints. In this talk, I will briefly introduce different NC systems studied in our group, such as those whose assembly is driven by DNA, electrostatic phase separation of neutral polymers, attachment of irreversible dithiol linkers, interpolymer complexation, Nanocomposite Tectons and also, via solvent evaporation. I will elaborate on this latter strategy and present the Orbifold Topological Model (OTM), which accurately describes the structure of crystal or quasicrystal arrangements of NCs (superlattices). The OTM represents capping ligands as Skyrmion textures, with “atomic orbitals” consisting of vortices, which enable the generation of a spontaneous valence and reveal the universal tendency of these systems towards icosahedral order. I will elaborate on the success of the OTM in describing existing experimental structural data on single component and binary superlattices with spherical NCs. Finally, I will present the diverse phase diagram of mixed Perovskite (cubic) and spherical NCs and emphasize the optoelectronic properties of these superlattices, thus demonstrating how a precise control of the structure determines the function.

Alex Travesset got his PhD from the Universitat de Barcelona in 1997. After Postdoc positions in Syracuse University and University of Illinois at Urbana Champaign, he joined the faculty at the Department of Physics and Astronomy at Iowa State University, where he is now full professor. He also holds an appointment in Materials Science and Engineering and is an associated scientist at the Ames lab.

]]>We study two-dimensional liquid crystals with p-fold rotational symmetry (p-atics) on the surfaces of cones, and find both the ground state(s) and a ladder of quantized metastable states as a function of both the cone angle and the liquid crystal symmetry p. We find that these states are characterized by a fractional defect charge at the apex and that the ground states are in general frustrated due to effects of parallel transport along the azimuthal direction of the cone. We check our predictions numerically for a set of commensurate cone angles, whose surfaces can be polygonized as a perfect triangular or square mesh, and find good agreement.

]]>The spontaneous organization of individual building blocks into ordered structures is extensively used in nature and found at all length scales, from crystallization processes, via composite materials, to living cells constituting complex tissue. Understanding the relationship between building blocks, environmental conditions, and resulting structure is of fundamental importance for controlling materials properties. Confining elements imposed upon the self-organizing particles can significantly alter the assembly process and may lead to entirely different colloidal crystals. Especially interesting confinements are emulsion droplets that prevent the formation of periodic structures by introducing boundaries and curvature.

Here, we explore the surprising diversity of crystal structures and symmetries that can form in this confining element. We create a phase diagram of observed crystal phases in dependence of the number of colloidal particles within the confinement and support our model by event-driven molecular dynamics simulations of hard-spheres in a spherical confinement. A closer look at the thermodynamics in such systems shows that certain configurations exist as minimum energy structures, a signature associated with magic number clusters which are well known in the atomic world, but have not been observed in the colloidal realm. Importantly, and differing from their atomic analogues, the occurrence of such magic number states is not driven by the mutual attraction of the individual building blocks. Instead, the thermodynamics in our colloidal system is entirely governed by entropy maximisation. In this presentation, I introduce synthetic requirements that are necessary for the self-assembly of magic colloidal clusters and present a detailed study on the structures, thermodynamics and formation kinetics of this confined self-assembly process.

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