Quasicrystals: how and why do they form?

Crystals are formed from ordered arrangements of atoms with rotation and translation symmetries, i.e. rotating or shifting the crystal by certain specific values leaves the crystal looking unchanged. However, some rather striking materials, named quasicrystals (QCs), were discovered in 1982, later attracting the Nobel Prize for Chemistry in 2011. These lack the translation symmetries of crystals and yet they have rotation symmetry on average. Quasicrystals are usually formed from metallic alloys made from at least two types of atoms and hundreds of examples have been discovered.

Crystallisation is not limited to atoms, and quasicrystals formed from micellar copolymers, dendrimers or other particles have been discovered recently. Polymers are string-like molecules and copolymers are polymers that are made of two or more chemically different types of polymers that are bonded together. The micelles are formed from (for example) dendrimers which comprise a hydrophobic polymer core surrounded by a corona of hydrophilic polymer. Dendrimers are polymeric molecules made by joining branched polymers in successive layers, in a tree-like structure. The main theoretical approach to investigating the formation and stability of soft-matter quasicrystals involves minimising an appropriate free energy, but the principle(s) underlying their stability are only beginning to be understood.

One central idea of this proposal is to bring ideas and insights from the mathematics of pattern formation and nonlinear dynamics to bear on this physical problem. Patterns with quasicrystalline structure, or quasipatterns, were discovered in Faraday wave experiments in the 1990s. In these experiments, a tray of liquid is subjected to vertical vibrations. If the forcing is strong enough, the flat surface of the liquid becomes unstable and a pattern or quasipattern of standing waves is formed. Recent progress in understanding the formation mechanism for quasipatterns has confirmed the key ingredient is the nonlinear interaction of waves with two different wavelengths.

Starting from the effective interaction potential between dendrimers, and the statistical physics of many interacting particles, we will link to pattern formation via two intermediate theoretical frameworks, each representing an increase in degree of level of detail - i.e. in coarse graining. These are Dynamical Density Functional Theory and Phase Field Crystal Partial Differential Equations. The first of these is a theory for the average density of particles, and can be derived from the interaction potential between the dendrimers; the second is a simplification of the first, and is directly amenable to techniques from pattern formation theory. Each step in this process involves approximations and simplifications, but the approximations can be controlled and the simplifications can be tested.

Having worked out from pattern formation theory the ingredients for forming and stabilising QCs, we can go back through the simplifications and bring the new insights into the design principles for dendrimers that are likely to produce QCs.

This proposal lies at the boundary between the EPSRC research areas 'non-linear systems' and 'biophysics and soft matter physics', and will contribute substantially to the understanding of the basic mechanisms of self-assembly in soft-matter colloids that have complex structure. The immediate impact of the research will be on mathematicians and physicists working on basic nonlinear problems in pattern formation and phase field crystals, and on particle simulation and crystallisation phenomena in soft-matter science. There are several research groups around the world making and simulating three-dimensional quasicrystals, but most of the theory giving insight is in two dimensions, so our three-dimensional work has the potential to generate significant new ideas. The work will also be of interest to the community working on aperiodic order and tilings: the structures we will generate will in turn produce three-dimensional aperiodic tilings, which are much more challenging (and less well understood) than two-dimensional aperiodic tilings such as Penrose tilings.

We will reach these communities not only through publications and presentations at conferences, but also through visits in either direction during the course of the project, important for communicating our results and for developing a wider perspective on our work.

Quasicrystals are beautiful structures and in two dimensions, they have captured the public imagination for decades. The three-dimensional work at the core of this proposal will likewise be of great general interest, so there is potential for impact on the general public. An important aspect of popular communication is visual impact, and we will ensure that our three dimensional images are presented in an as appealing way as possible. We will give popular lectures on quasicrystals, write popular articles and participate in national outreach events.

This project is a key part of our long-term ambition to transform how those working on soft matter understand and think about quasicrystal formation, and our work will clearly be relevant to the EPSRC Physics Grand Challenges of 'Nanoscale Design of Functional Materials' and 'Emergence and Physics Far From Equilibrium'.

In this project, we will concentrate on developing basic mathematical ideas and using these to understand the behaviour of systems that model particle self-assembly. The next stage will be to link more strongly to soft-matter physicists and chemists with an eventual goal of contributing to the fabrication of novel materials with quasicrystalline structure. Such materials could have application as ingredients in coatings or in photonics: recent experiments on mesoscale quasicrystals have highlighted the potential for their use in photonics, arising from their unusual optical bandgap properties.