Reconfigurable knotting and joining in chiral nematic colloids
(2011 Science article)
Knotting or constructing microscale rings of high molecular, macromolecular or complex materials is a challenging task in materials science. By using a laser cymbal as a micromanipulation tool, the microscopic topological defect lines in the chiral nematic liquid crystal colloid are knotted and joined at any degree of complexity. All knots and connections shown include the Hopf connection, the Star of David, and the Borromean ring, which has up to six crossovers and stabilizes the colloidal particles in unusual soft materials. The knots in the chiral nematic colloid are classified according to the number of self-joins, geometric direct measurements or Berry phases. The construction of arbitrary microscale knots and connections in chiral nematic colloids demonstrates the importance of topology in soft material engineering.
Fig. 1. Knotting and joining of topological defect lines in chiral nematic colloids (A) A twisted defect ring is topologically equivalent to untangling and spontaneously surrounding a single small sphere. The molecular orientation of the top and bottom of the unit is the same as the direction of the crossed polarizers. (B to E) Colloidal dimers, trimers, and tetramer-deficient rings are equivalently unfolded. (F) Weave the first unusual Hopf connection using two inner joint defect rings. In (A) to (F), the construction of the corresponding ring is quantified by the Landau-Derna model. (13). (G to J) A series of alternating ring-shaped junctions and connections on the pxq particle array are induced by laser The defects are fused to weave. The defect line (33) is again rendered using a knotting procedure to show the image projected from the initial plane to the final knotting plot, which is achieved by a series of Meister movements. The nomenclature uses the standard method Ci, N, where C indicates the minimum number of intersections, i is used to distinguish the types of different junctions, and N indicates the number of rings in a multi-component connection. Ruler, 5um.
Fig. 3. Topological classification and custom assembly of the nematic tweezers of the joints and junctions (A) The defect ring has a three-fold rotational symmetry of a local hyperbolic cross section and corresponds to a 3-band surface, similar to a Mobis ring. They can be clearly identified by the fractional number of rings N and the number of self-joining SLs. (B) All possible junctions and connections on the 3x4 particle array are classified by SL and N. The hierarchical order of knots and connections, shown in different colors and standard symbols. (C) Custom assembly of the Borromean ring on the Pxq particle array. The feasible kink combination is tested using a numerical algorithm based on the Jones polynomial and the Kaufman scaffold method (24, 28). The selected configuration is directly compared by polynomial, at T a The ble of Knot Invariants (29) is enumerated and assembled with a laser. (D) The distribution of primary junctions and connections on a 4x4 particle array demonstrates the diversity of a large topology with a minimum number of crossings of up to 10. The likelihood of a particular knot or connection occurring decreases with complexity, and the complexity is determined by the minimum number of crossings measured.
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2011 science - Reconfigurable Knots and Links in Chiral NematicColloids
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