Quasicrystals: how and why do they form?

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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.