Topological colloids

Abstract: 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 Poincare-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.

• The paper demonstrates that colloid topology directly influences defect formation in nematic liquid crystals using advanced optical imaging methods.
• The study employs precise fabrication and holographic optical tweezers to control silica colloids and reveal complex director field configurations.
• Results confirm topological charge conservation, highlighting implications for reconfigurable materials and topological memory device applications.

An Examination of Topological Colloids and Their Interactions with Liquid Crystals

The paper "Topological Colloids" advances the understanding of how colloidal particles with varying topology influence nematic liquid crystal environments. Unlike traditional colloidal particles shaped by minimizing interfacial tension, such as spherical or prismatic structures, this study focuses on colloids with non-trivial topologies, categorized by genus $g$, which extends from 1 to 5 in this research. The investigation provides compelling insights into the interactions between these uniquely shaped colloids and nematic liquid crystals, highlighting the significance of their topology on induced defect structures.

Experimental Design and Methodology

The study employs advanced fabrication methods to produce silica colloids with handlebody topologies, characterized by their genus and Euler characteristic $\chi = 2 - 2g$. These particles are then embedded in a nematic liquid crystal medium, where their presence instigates complex director field configurations and topological defects. The research leverages optical microscopy techniques, including polarizing microscopy (PM) and three-photon excitation fluorescence polarizing microscopy (3PEF-PM), to meticulously elucidate the resulting director fields and defect structures. Holographic optical tweezers facilitate non-contact manipulation, allowing for controlled experiments on how these colloids influence, and are influenced by, their liquid crystal environment.

Observations and Findings

Key findings from the study include the observation that the topology of colloidal particles dictates the formation and type of defects within nematic liquid crystals. Colloids with a higher genus induce a corresponding increase in the number of internal defects, either as singular disclination loops or hyperbolic point defects. Moreover, the experiments confirm that topological charge is conserved in these interactions, following the Gauss–Bonnet and Poincaré–Hopf index theorems. Such conservation is consistently observed regardless of colloid orientation relative to the far-field director.

Colloidal alignment within the liquid crystal is predominantly driven by the minimization of distortion in the director field, leading to configurations that align perpendicularly to the director field in homeotropic settings. The interplay of elastic and gravitational forces also influences the positioning of the colloids within the cell medium, leading to nuanced understandings of colloidal behavior in complex liquid crystal environments.

Theoretical and Practical Implications

The theoretical implications of this research are substantial, offering new paradigms for the understanding of topological aspects in colloidal systems, particularly in how particle topology can influence defect dynamics in liquid crystals. Practically, the findings have potential applications in the development of topological memory devices, novel self-assembly structures, and in the experimental study of low-dimensional topology. The work lays a foundation for innovations in liquid crystal technology, particularly in the fields of data storage and materials science where the control over defect structures can lead to reconfigurable materials.

Future Directions

Looking forward, the study suggests several avenues for continued research. One promising direction is the exploration of how external fields, such as electric and magnetic fields, can further manipulate and stabilize the director fields and induced defects around topological colloids. Additionally, investigations into other liquid crystal phases and their interactions with topologically complex colloids could yield further insights into the fundamental science and technological applications of these systems.

In conclusion, this paper contributes significantly to our understanding of the role of particle topology in colloidal systems and sets the stage for further exploration into the vast possibilities presented by the intersection of topology and soft matter physics.