Department of Physics, University of Colorado, Boulder, Colorado 80309, USA.
Nature. 2013 Jan 10;493(7431):200-5. doi: 10.1038/nature11710. Epub 2012 Dec 23.
Smoke, fog, jelly, paints, milk and shaving cream are common everyday examples of colloids, a type of soft matter consisting of tiny particles dispersed in chemically distinct host media. Being abundant in nature, colloids also find increasingly important applications in science and technology, ranging from direct probing of kinetics in crystals and glasses to fabrication of third-generation quantum-dot solar cells. Because naturally occurring colloids have a shape that is typically determined by minimization of interfacial tension (for example, during phase separation) or faceted crystal growth, their surfaces tend to have minimum-area spherical or topologically equivalent shapes such as prisms and irregular grains (all continuously deformable--homeomorphic--to spheres). Although toroidal DNA condensates and vesicles with different numbers of handles can exist and soft matter defects can be shaped as rings and knots, the role of particle topology in colloidal systems remains unexplored. Here we fabricate and study colloidal particles with different numbers of handles and genus g ranging from 1 to 5. When introduced into a nematic liquid crystal--a fluid made of rod-like molecules that spontaneously align along the so-called 'director'--these particles induce three-dimensional director fields and topological defects dictated by colloidal topology. Whereas electric fields, photothermal melting and laser tweezing cause transformations between configurations of particle-induced structures, three-dimensional nonlinear optical imaging reveals that topological charge is conserved and that the total charge of particle-induced defects always obeys predictions of the Gauss-Bonnet and Poincaré-Hopf index theorems. This allows us to establish and experimentally test the procedure for assignment and summation of topological charges in three-dimensional director fields. Our findings lay the groundwork for new applications of colloids and liquid crystals that range from topological memory devices, through new types of self-assembly, to the experimental study of low-dimensional topology.
烟雾、雾、果冻、油漆、牛奶和剃须膏是胶体的常见日常示例，胶体是一种由微小颗粒分散在化学上不同的主体介质中的软物质。胶体在自然界中大量存在，也在科学技术中找到了越来越重要的应用，从晶体和玻璃中动力学的直接探测到第三代量子点太阳能电池的制造。由于天然胶体的形状通常是由界面张力最小化（例如，在相分离期间）或有面晶体生长决定的，因此它们的表面往往具有最小面积的球形或拓扑等效形状，如棱镜和不规则颗粒（都可以连续变形--同胚--到球体）。尽管环形 DNA 凝聚体和具有不同手柄数的囊泡可以存在，并且软物质缺陷可以形成环和结，但胶体系统中颗粒拓扑的作用仍然没有得到探索。在这里，我们制造并研究了具有不同手柄数和 genus g 的胶体颗粒，其范围从 1 到 5。当它们被引入向列液晶--一种由沿所谓的 'director' 自发排列的棒状分子组成的流体时，这些颗粒会诱导出由胶体拓扑决定的三维 director 场和拓扑缺陷。虽然电场、光热熔化和激光镊子会导致颗粒诱导结构的构型之间的转变，但三维非线性光学成像表明拓扑电荷是守恒的，并且颗粒诱导缺陷的总电荷始终遵循高斯-博内特和庞加莱-霍普夫指标定理的预测。这使我们能够建立并通过实验测试三维 director 场中拓扑电荷的分配和求和的程序。我们的发现为胶体和液晶的新应用奠定了基础，这些应用从拓扑存储设备到新型自组装，再到低维拓扑的实验研究。
