NBIA Workshop: A Copenhagen afternoon on geometry and topology in soft materials

Monday 19 Feb 2024, 12:00 → 18:00 Europe/Copenhagen

Aud. A (Niels Bohr Institute)

 

The growth and form of natural materials, at the micro- and macro-scale, and the relationship between function and structure continues to be a fascinating topic of research for both materials science, biomimetics and biology alike. Modern imaging techniques produce ever more intricate and detailed observations, and enable swift progress in our understanding. This afternoon workshop is dedicated to theoretical, computational and conceptual ideas that help us understand how geometry emerges, such as in self-assembly or growth, and how it determines the functional properties of a material or tissue. Where there is material, there is geometry.

1 Martin Cramer Pedersen (University of Copenhagen): Welcome

2 Gerd Schröder-Türk (Murdoch University): The Gyroid - Nature's best attempt at embedding the Hyperbolic Plane in Euclidean space

By Hilbert's famous theorem, all infinite symmetric saddle surface with zero mean curvature (ie minimal surfaces) must have variations of the Gauss curvature. For bicontinuous minimal surfaces which divide space into two domains and define network-like domains, the Euler-Poincare number and hence the average Gauss curvature is negative. However, these surfaces also always have flat points with Gauss curvature zero. In Euclidean space, the perfect solution of a minimal surface with no variations in Gauss curvature does not exist. But is the Gyroid the minimal surface with the smallest degree of the unavoidable variations of Gauss curvature?

3 Amin Doostmohammadi (University of Copenhagen): Topological defects in biological matter

I will discuss various examples of how the emergence of orientational order and its breakdown are generic themes in fundamental processes in cell biology and how they provide a universal mechanism for choreographing the directional passage of mechanical and biochemical information in time and across different scales.

14:00 Coffee break

4 Myfanwy Evans (University of Potsdam): Geometry-based simulation of self assembly

The morphometric approach to solvation free energy is a geometry-based theory that incorporates a weighted combination of geometric measures over the solvent accessible surface for solute configurations in a solvent. In this talk, I will demonstrate that employing this geometric technique in simulating the self assembly of sphere clusters and small loops results in an assortment of interesting geometric configurations. This gives insight into the role of shape in the physical process of self assembly, potentially relevant to proteins, viruses and other complex systems.

5 Jacob Kirkensgaard (University of Copenhagen): Exploring pattern formation on negatively curved surfaces via the hyperbolic plane

We investigate the self-assembly behaviour of block copolymers constrained to thin hyperbolic films. Specifically, we study the pattern formation on the three-periodic cubic minimal surfaces P, D and G found ubiquitously in soft matter material science. We use a new method for visualisation and analysis of the patterns by mapping to two-dimensional hyperbolic space analogous to stereographic projections in cartography thus effectively creating a more accessible ‘‘hyperbolic map’’ of the pattern. This allows us to pinpoint in detail the role of intrinsic geometry and to probe the role of negative curvature on the resulting assemblies as contrasted with flat and positively curved films. We present results from AB diblock copolymers and ABC mikto-arm star terpolymers of varying composition. In the case of compositionally balanced diblocks, the resulting patterns are related to ``free’’ tilings of the hyperbolic plane while unbalanced molecules form a plethora of disc packings. Star polymers form three-colored cellular patterns related to curved graphene-like schwarzites constrained to only form even polygons.

15:30 Coffee break

6 Adil Mughal (Aberystwyth University): Peierls-Nabarro potential for a confined chain of hard spheres under compression

Examples of bifurcation diagrams are presented for a chain of hard spheres under compression, which are confined by a transverse potential. The diagrams are modified in the presence of an axial force and show interesting topological rearrangements due to the reconnection of loci of different equilibrium solutions. The corresponding Peierls-Nabarro potential is defined and calculated for various values of compression. This relatively tractable and transparent system can illustrate in detail the general features of such a potential and its relation to experiment.

7 Matthias Himmelmann (University of Potsdam): An Empirical Treatment of Disordered Minimal Surfaces

Triply periodic minimal surfaces -- especially the Primitive, Diamond and Gyroid -- appear throughout nature. While traditionally, the focus has been on highly symmetric phases, recently attention has shifted towards disordered structures. For example, these surfaces manifest as intermediate phases, sponge phases, and materials such as amorphous silicon. In this work, we employ two distinct approaches -- one geometric and one of topological nature -- to computationally generate disordered minimal surfaces. By empirically comparing the resulting structures and examining key invariants such as curvature fluctuation and isotropy, we aim to enhance our understanding of these surfaces' quality in relation to ordered phases.