## Optimal packing of a finite collection of deformable objects

### 9th November 2022:

## Simon Cox

### Aberystwyth University

## Abstract

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.

**Measures of order for imperfect two-dimensional patterns**

### 26th October 2022

## Patrick Shipman

### Colorado State University

## Abstract

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.

**Mathematical Modelling of Programmable Polymorphism of Protein Cages**

### 4th May 2022

## Farzad Fatehi Chenar

### University of York

## Abstract

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.

**Cumulative Geometric Frustration and the Intrinsic Approach to Physical Assemblies**

### 20th April 2022

## Efi Efrati

### Weizmann Institute of Science

## Abstract

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.

**The topological origin of the Peierls-Nabarro barrier**

### 6th April 2022

## Randall Kamien

### University of Pennsylvania

## Abstract

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.

**Geometry, Topology and Defects in the Programmable Assembly of Nanoparticles**

### 23rd March 2022

## Alex Travesset

### Iowa State University and Ames Lab

## Abstract

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.

**Fractional defect charges for liquid crystals on cones**

### 8th December 2021

## David Nelson

### Harvard University

## Abstract

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.

**Slimming down through frustration**

### 24th November 2021

## Martin Lenz

### Laboratory of Theoretical Physics and Statistical Models, Université Paris-Saclay, CNRS

## Abstract

In many diseases, proteins aggregate into fibers. Why? One could think of molecular reasons, but here we try something more general. We propose that when particles with complex shapes aggregate, geometrical frustration builds up and fibers generically appear. Such a rule could be very useful in designing artificial self-assembling systems, *e.g.*, out of colloids or DNA origami.

**Colloidal clusters from confined self-assembly: Structure – Thermodynamics – Formation kinetics**

### 10th November 2021

## Nicolas Vogel

### Friedrich-Alexander University Erlangen-Nürnberg

## Abstract

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.

**From cylinder packings to auxetic periodic tensegrity structures**

### 27th October 2021

## Myfanwy Evans

### Universität Potsdam

## Abstract

will present a chiral, triply-periodic tensegrity structure, which displays local reentrant geometry at its vertices. Our tensegrity structure is based on the β−Mn cylinder packing from structural chemistry, and contains all symmetries of the cylinder packing itself. The tensegrity structure is auxetic, as demonstrated by the modelling of a quasi-static extension and compression deformation. This simple three-dimensional analogue to the two-dimensional reentrant honeycomb is an interesting design target for soft metamaterials.

**Symmetry Making and Breaking in Seeded Syntheses of Metal Nanocrystals**

### 13th October 2021

## Sara Skrabalak

### Indiana University

## Abstract:

Crystal growth theory predicts that heterogeneous nucleation will occur preferentially at defect sites, such as the vertices rather than the faces of shape-controlled seeds. Platonic metal solids are generally assumed to have vertices with nearly identical chemical potentials, and also nearly identical faces, leading to the useful generality that heterogeneous nucleation preserves the symmetry of the original seeds in the final product. This presentation will discuss how this generality can be used to access stellated metal nanocrystals with high and tunable symmetries for applications in plasmonics. This presentation will also discuss the limits of this generality in the extreme of low supersaturation. A strategy for favoring localized deposition that differentiates between both different vertices and different edges or faces, i.e., regioselective deposition, will be demonstrated. Such regioselective heterogeneous nucleation was achieved at low supersaturation by a kinetic preference for high-energy defect-rich sites over lower-energy sites. This outcome was enhanced by using capping agents to passivate facet sites where deposition was not desired. Collectively, the results presented provide a model for breaking the symmetry of seeded growth and for achieving regioselective deposition.

## The Wrinkle-to-Crumple Transition in Thin Elastic Solids

### 29th September 2021

## Joey Paulson

### Syracuse University

## Abstract:

The last decade has seen a renaissance in the buckling of thin elastic solids, in part because of its impact on the mechanics of synthetic skins, biological tissues, textiles, and 2D materials like graphene. Significant effort has been devoted to understanding and exploiting buckled morphologies that decorate otherwise planar surfaces. Yet, we show that one of the most-studied morphologies—sinusoidal wrinkling—does not persist in a variety of curved geometries. Instead, smooth wrinkles generically give way to sharp “crumples”. We characterize this transition in a suite of experiments across geometries and scales, using ultrathin polymer films on liquid droplets and macroscopic membranes that we inflate with gas. These setups reveal robust morphological features of the crumpled phase, and they allow us to disentangle the effects of curvature and compression. Our work highlights the need for a theoretical understanding of this ubiquitous elastic building block, and we lay out concrete directions for such studies.

**From viral capsids to self-assembling nanoparticles**

### 2nd June 2021

## Giuliana Indelicato

### University of York

## Abstract:

Viruses are examples of biological nanoparticles. They are made of a highly symmetric protein container, the capsid, that contains the nucleic acid and self-assembles from several copies of a single protein. The relative position of the proteins in the capsids is conventionally described by the Caspar-Klug scheme, that was inspired by techniques developed by Fuller to construct geodetic domes.

Much interest has recently arisen in the engineering of self-assembling protein nanoparticles. In general, these nanoparticles do not fit into the Caspar-Klug framework, and conventional experimental methods only yield partial information on their structure: therefore, an original approach is needed to provide blueprints for the determination of their structure.

In this talk I will discuss a mathematical framework that enables the investigation of a specific class of self-assembling protein nanoparticles (SAPNs) which have been designed to act as antigen display systems for vaccines.

I will show that graph-theoretical tools, combined with symmetry methods based on the Goldberg construction for symmetric polyhedra, allow to study the topology of the protein networks. This approach unveils a hidden relation with fullerene geometries and enables the classification of the high and low symmetry particles seen in experiments.

**Braiding wires using capillary forces**

### 19th May 2021

## Vinothan N. Manoharan

### Harvard University

## Abstract:

Electrical conductors that can carry frequencies of tens of GHz are needed for next-generation telecommunications networks. In principle, such conductors can be made from braided conducting filaments. However, maximizing the current-carrying capacity and minimizing loss requires each filament to have a diameter approximately equal to the skin depth, which is on the order of 1 micrometer at 10 GHz. Because such small filaments break easily, current manufacturing techniques cannot braid them. We have developed a technique to braid such small filaments using repulsive capillary forces. We attach microscale filaments to polymer “floats” that sit at a water-air interface such that the capillary force between the floats and the container walls is repulsive. As a result, the floats – and therefore the filaments – can be translated or rotated by moving the walls of the container. I will explain how the containers can be designed using principles of braid theory, and I will show how they can be used to braid wires into arbitrary topologies.

This research is in collaboration with Cheng Zeng, John Miles Faaborg, Ahmed Sherif, Ming Xiao, Martin Falk. Rozhin Hajian, Yohai Bar Sinai, and Michael P. Brenner

**Biological tissues as mechanical metamaterials**

### 5th May 2021

## M. Lisa Manning

### Syracuse University

## Abstract:

In multicellular organisms, properly programmed collective motion is required to form tissues and organs, and this programming breaks down in diseases like cancer. Recent experimental work highlights that some organisms tune the global mechanical properties of a tissue across a fluid-solid transition to allow or prohibit cell motion and control processes such as body axis elongation. What is the physical origin of such rigidity transitions? Is it similar to zero-temperature jamming transitions in particulate matter, or glass transitions in molecular or colloidal materials? Over the past decade, our group and others have shown that models for confluent tissues, where there are no gaps or overlaps between cells, exhibit a rigidity transition that depends on cell shape. A similar transition is also seen in models for biopolymer networks. I will use the framework of “higher-order rigidity”, recently implicated in origami rigidification, to discuss similarities and differences between rigidity in particulate matter and rigidity in confluent tissues and fiber networks. This suggests that many biological tissues may tune their rigidity using the same mechanisms as mechanical metamaterials. I will also discuss recent work to test which mechanisms are operating in real biological systems.

**Placement and Symmetry of Singularities on Curved Surfaces**

### 21st April 2021

## Sascha Hilgenfeldt

### University of Illinois

## Abstract:

In various contexts, biological structures resemble regular two-dimensional lattices made from smaller units like macromolecules (on the nanoscale) or cells (on the tissue scale). This regularity carries an inherent elastic energy penalty when the 2D manifold is also required to have intrinsic curvature: Virus capsids have to close, and arthropod eyes need to reconcile the periodicity of ommatidia with eye curvature, both of which are crucial for proper function. We derive a new theoretical criterion directly from the shape properties of the manifold that predicts whether in its ground state of mechanical energy a curved structure can be regular (defect-free) or whether defects are energetically favorable. In contrast to previous approaches, the criterion does not prescribe local or globally integrated curvature, but sets a universal value for a specific weighted integral Gaussian curvature. Verified against direct numerical computation, this formalism is simple, easy to employ, and accurate for a vast variety of curved surfaces.

An extension of this approach furthermore elucidates the continuous or discontinuous character of this transition to disorder: The shape of the energy landscape determines if a transition preserves the symmetry of defect placement on surfaces of revolution because an energy barrier must be overcome, or breaks the symmetry due to stable intermediate positions. Our analytical theory predicts these transition characteristics a priori for general surface shapes, demonstrating a universal scale of energy barrier that is modified, and sometimes eliminated, by local shape properties.

Our results give practical insight into the bounds of perfect order in the growth of lattices on manifolds in many biological systems that also inspire technological applications such as microlens arrays. They inform material design considerations through geometric control, while also improving our fundamental understanding.

## Link

**Packing, geometry & entropy: Crystallization of spheres in a sphere**

### 7th April 2021

## Marjolein Dijkstra

### Universiteit Utrecht

## Abstract:

When twelve equally sized spheres are packed around a central sphere, the smallest volume of the resulting cluster and thus the highest packing is obtained for a regular icosahedron, a shape with twenty equilateral triangles as faces and a five-fold symmetry. This five-fold symmetry makes this packing interesting, because it can be proven that clusters with such a symmetry cannot fully pack 3D space.

However, it has been proven only very recently that a close-packed arrangement of equally sized spheres, composed of stackings of hexagonal layers of spheres, is the closest-packed arrangement in 3D of any single-sized sphere packing. However, if the twelve spheres are arranged in this close-packed arrangement, the local packing is less dense than the one with the icosahedral symmetry. A high local density is incompatible with a high global density.

In this talk, I will show that if spheres crystallize in a spherical confinement in experiments and simulations, clusters with icosahedral symmetry will be formed up to about 100.000 spheres [1]. Using simulations, I will show that the free energy of these icosahedral clusters is lower than that of the close-packed crystal for as many as 100.000 spheres. With larger numbers of spheres, the system crystallizes in the close-packed bulk crystal.

In the case of a mixture of two sizes of hard spheres, with a size ratio around 0.8, I will show that such a mixture of spheres would not form the bulk equilibrium MgZn_{2 }phase, but instead a so-called binary icosahedral cluster consisting of MgCu_{2 }crystalline domains [2]. The resulting clusters, a.k.a. supraparticles, can be used for a next self-assembly step in order to structure matter over multiple length scales.

References:

[1] Entropy-driven formation of large icosahedral colloidal clusters by spherical confinement

B. de Nijs, S. Dussi, F. Smallenburg, J.D. Meeldijk, D.J. Groenendijk, L. Filion, A. Imhof, A. van Blaaderen, and M. Dijkstra, *Nature Materials* **14**, 56-60 (2015).

[2] Binary icosahedral clusters of hard spheres in spherical confinement

D. Wang, T. Dasgupta, E.B. van der Wee, D. Zanaga, T. Altantzis, Y. Wu, G.M. Coli, C.B. Murray, S. Bals, M. Dijkstra and A. van Blaaderen, *Nature Physics 17, 128–134 (*2021).

## Link

**Twisted topological tangles or: the knot theory of knitting**

### 24th March 2021

## Elisabetta Matsumoto

### Georgia Tech

## Abstract:

Imagine a 1D curve, then use it to fill a 2D manifold that covers an arbitrary 3D object – this computationally intensive materials challenge has been realized in the ancient technology known as knitting. This process for making functional materials 2D materials from 1D portable cloth dates back to prehistory, with the oldest known examples dating from the 11th century CE. Knitted textiles are ubiquitous as they are easy and cheap to create, lightweight, portable, flexible and stretchy. As with many functional materials, the key to knitting’s extraordinary properties lies in its microstructure.

At the 1D level, knits are composed of an interlocking series of slip knots. At the most basic level there is only one manipulation that creates a knitted stitch – pulling a loop of yarn through another loop. However, there exist hundreds of books with thousands of patterns of stitches with seemingly unbounded complexity.

The topology of knitted stitches has a profound impact on the geometry and elasticity of the resulting fabric. This puts a new spin on additive manufacturing – not only can stitch pattern control the local and global geometry of a textile, but the creation process encodes mechanical properties within the material itself. Unlike standard additive manufacturing techniques, the innate properties of the yarn and the stitch microstructure has a direct effect on the global geometric and mechanical outcome of knitted fabrics.