Quasicrystals: how and why do they form?

晶体是由具有旋转和平移对称性的原子的有序排列形成的，即旋转或移动晶体的某些特定值使晶体看起来不变。然而，1982年发现了一些相当惊人的材料，称为准晶体（qc），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。准晶体通常是由至少两种类型的原子组成的金属合金形成的，已经发现了数百个这样的例子。结晶不局限于原子，最近发现了由胶束共聚物、树状聚合物或其他颗粒形成的准晶体。聚合物是线状分子，共聚物是由两种或两种以上化学性质不同的聚合物结合而成的聚合物。胶束由（例如）树状大分子形成，树状大分子包括由亲水聚合物的电晕包围的疏水聚合物核。树状大分子是一种聚合分子，由连续的分支聚合物层连接而成，呈树形结构。研究软物质准晶体的形成和稳定性的主要理论方法包括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。这个建议的一个中心思想是将模式形成和非线性动力学的数学思想和见解带到这个物理问题上。具有准晶体结构的图案，或称准图案，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，一个装有液体的托盘受到垂直振动。如果作用力足够强，液体的平坦表面就会变得不稳定，形成驻波模式或准驻波模式。在对准图案形成机制的理解方面的最新进展已经证实，关键因素是两种不同波长的波的非线性相互作用。从树状大分子之间的有效相互作用势和许多相互作用粒子的统计物理开始，我们将通过两个中间理论框架将模式形成联系起来，每个理论框架都代表了细节程度的增加-即粗粒化。这些是动态密度泛函理论和相场晶体偏微分方程。第一个是关于粒子平均密度的理论，可以从树状大分子之间的相互作用势推导出来；第二种方法是第一种方法的简化，并且直接适用于模式形成理论的技术。这个过程中的每一步都涉及近似和简化，但近似是可以控制的，简化是可以测试的。从模式形成理论中得出了形成和稳定qc的成分，我们可以通过简化并将新的见解带入可能产生qc的树状大分子的设计原则中。

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.