Quasicrystals: how and why do they form?

晶体是由具有旋转和平移对称性的原子有序排列形成的，即通过某些特定值旋转或移动晶体使晶体看起来保持不变。然而，1982年发现了一些相当引人注目的材料，称为准晶体（QCs），后来在2011年获得了诺贝尔化学奖。它们缺乏晶体的平移对称性，但它们平均具有旋转对称性。准晶通常由至少两种原子组成的金属合金形成，已经发现了数百个例子。结晶不限于原子，最近已经发现了由胶束共聚物，树枝状聚合物或其他颗粒形成的准晶。聚合物是线状分子，共聚物是由两种或更多种化学上不同类型的聚合物键合在一起制成的聚合物。胶束由（例如）树枝状聚合物形成，所述树枝状聚合物包含被亲水性聚合物的冠包围的疏水性聚合物核。树枝状聚合物是通过将支链聚合物以树状结构连接在连续层中而制成的聚合物分子。研究软物质准晶的形成和稳定性的主要理论方法涉及最小化适当的自由能，但其稳定性背后的原理才刚刚开始被理解。这一提议的一个中心思想是从图案形成和非线性动力学的数学中引入思想和见解来解决这个物理问题。具有准晶结构的图案，或准图案，是在20世纪90年代的法拉第波实验中发现的。在这些实验中，一盘液体受到垂直振动。如果强迫足够强，液体的平坦表面变得不稳定，并且形成驻波的图案或准图案。近年来，对准斑图形成机制的研究已经证实，其关键因素是两种不同波长的波的非线性相互作用。我们将从树枝状聚合物之间的有效相互作用势和许多相互作用粒子的统计物理出发，通过两个中间理论框架与斑图形成联系起来，每一个都表示细节级别的程度（即粗粒化）的增加。这些是动力学密度泛函理论和相场晶体偏微分方程。其中第一个是颗粒平均密度的理论，可以从树枝状聚合物之间的相互作用势导出;第二个是第一个的简化，并且直接服从于图案形成理论的技术。在这个过程中的每一步都涉及到近似和简化，但是近似是可以控制的，简化是可以检验的。从模式形成理论中得出了形成和稳定QC的成分之后，我们可以回顾简化过程，并将新的见解带入可能产生QC的树枝状聚合物的设计原理中。

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.