Research project is (co) funded by the Slovenian Research Agency.

**UL Member:** Faculty of Mathematics and Physics**Code:** J1-9149**Project:** Orientational Interactions in a Generalized Thomson Problem: Dipole-Stabilized Spherical Nanocontainers**Period:** 1.7.2018 - 30.6.2022 **Range per year:** 1,50 FTE, category C**Head:** Simon Čopar**Research activity:** Natural sciences and mathematics**Research Organisations:** link on SICRIS**Researchers:** link on SICRIS**Citations for bibliographic records:** link on SICRIS

**Project description:**

The question of particle arrangements on curved surfaces is, at a first glance, a simple question, which has nonetheless yielded an astounding amount of interesting results, both addressing some of the fundamental questions in physics as well as finding application in numerous systems in soft matter, biophysics, and materials science. In this project proposal, we address an open problem of finding the ground states of particle arrangements on spherical lattices, where the particles interact through anisotropic electrostatic interactions.

Assemblies that take on a (near-)spherical shape are ubiquitous in biophysics and soft matter and take on the form of viral capsids, proteinaceous nanocages, micelles, lipid vesicles, cells, and colloidal structures. In addition to the hard-core repulsion, the interactions between their molecular-to-micron-sized building blocks are largely based on electrostatic interactions of different types. These assemblies are often highly symmetric, and their self-assembly, biological function and biophysical properties are all underpinned by symmetry-breaking transitions.

For the most part, the state-of-the-art studies of assembly on spherical surfaces assume that the interactions between the building blocks are isotropic, resulting in symmetric patterns wherein the building blocks lie on regular lattices. These are most often modelled as Caspar-Klug lattices, solutions of either classical or modified Thomson problem, or maxima of symmetrized spherical harmonics. And while the role of various anisotropic interactions -- and most prominently, dipolar interaction -- has been thoroughly investigated in planar assemblies, this remains an unanswered question in the case of assembly on spheres and other curved surfaces.

We will consider the possibility of stabilizing or destabilizing spherical structures through dipolar interactions, classify the equilibrium states by their symmetry properties, and investigate phase transitions between them. Using numerical minimization with symmetry constraints, we will tackle the problem first for dipoles on fixed lattices with variable orientation order. Later we will extend the possible interactions to include screened, multipolar, van der Waals, and nearest-neighbor interactions, and we will use full positional and orientational minimization to obtain equilibrium states. In this way, we will be able to explore the various possibilities that may arise in different biological systems and under different biochemical conditions in the solvent. We will also construct a theoretical framework to study the effects of thermal fluctuations and external symmetry-breaking effects that may lead to novel ways of controlling assembly and disassembly of spherical nanostructures in biophysics or, on the micron-scale, in colloidal and material science.

The results of this project will benefit various scientific fields. Understanding the ground states of finite, ordered systems in non-trivial topologies is of fundamental importance in the context of hard proteinaceous shells, such as viral capsids, wiffle balls, ferritin cages, carboxysomes or synthetic virus-like particles. The use of specific interactions for the design of self-assembling nanocontainers is an invaluable concept, particularly important for targeted drug delivery. More broadly, the main objectives of the project are also of interest in mathematics, as the fundamental geometric questions related to the Thomson problem and its generalizations remain largely unanswered even on the basic theoretical level. The project will also have an impact beyond its scientific results, establishing new collaborations between two of the leading physics institutions in Slovenia. Lastly, the project will also serve an educational purpose, promoting the modern development of science among the general public and in particular among physics students at various levels, who will be able to join in the activities of the project.