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NEMATIC COLLOID AS A TOPOLOGICAL PLAYGROUND

NEMATIC COLLOID AS A TOPOLOGICAL PLAYGROUND. S. ŽUMER University of Ljubljana & Jozef Stefan Inst itute , Ljubljana, Slovenia. Confined Liquid Crystals: Perspectives and Landmarks June 19-20, 2010     Ljubljana. COWORKER : S. Čopar

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NEMATIC COLLOID AS A TOPOLOGICAL PLAYGROUND

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  1. NEMATIC COLLOID AS A TOPOLOGICAL PLAYGROUND S. ŽUMER University of Ljubljana & Jozef Stefan Institute, Ljubljana, Slovenia Confined Liquid Crystals: Perspectives and Landmarks June 19-20, 2010     Ljubljana COWORKER:S. Čopar COLLABORATIONS: B. Črnko, T. Lubensky, I. Muševič, M. Ravnik,… Supports of Slovenian Research Agency, Center of Excellence NAMASTE, EU ITN HIERARCHY are acknowledged

  2. MOTIVATION • Nematic braids & nematic colloids • structures entangled by disclinations • modeling • experiments • spontaneous & mediated formation • quench laser tweezers d = 1mm, h = 2mm director S=0.5 surface of -1/2 defect line (Sbulk=0.533)

  3. OUTLINE • order parameter field • defects & colloidal particles • colloidal dimer in a homogenous nematic field • local restructuring of a disclination crossings • writhe & twist (geometry and topology of entangled dimers) • conclusions

  4. ORDER PARAMETER FIELD Tensorialnematic order parameter Q (directorn, degree of orderS, biaxialityP): eigenframe: n, e(1), e(2) Landau - de Gennes free energy with elastic (gradient) term and standard phase term is complemented by a surface term introducing homeotropic anchoring on colloidal surfaces. Geometry of confinement yields together with anchoring boundary conditions. Equilibrium and metastablenematic structures are determined via minimization of F that leads to the solving of the corresponding differential equations.

  5. DEFECTS • discontinues director fields & variation innematicorder • defects are formed after fast cooling, or by other external perturbations, • topological picture (director fields, equivalence, and conservation laws): - point defects: topological charge - line defects (disclinations) : • winding number, • topological charge of a loop • core structure (topology & energy): • singular (half- integer) disclination lines biaxiality & decrease of order • nonsigular(integer) disclination lines

  6. SPHERICAL HOMEOTROPIC PARTICLES CONFINED TO A HOMOGENOUSNEMATIC FIELD zero topological charge 2.5 mm cell 2 mm particle Strong anchoring S=0.5 surface of defect(Sbulk=0.533) Saturn ring (quadrupolar symmetry) dipole (dipolar symmetry) <= Stark et al., NATO Science Series Kluwer 02

  7. COLLOIDAL DIMER IN A HOMOGENOUS NEMATIC FIELD zero topological charge cell thickness: h = 2 mm , colloid diameter: d = 1 mm director figure of eight figure of omega entangled hyperbolic defect director In homogenous cells these structures are obtained only via melting & quenching

  8. LOCAL RESTRUCTURING OF DISLINATIONS Orthogonal crossing of disclinations in a tetrahedron Restructuring via tetrahedron reorientation

  9. LOCAL RESTRUCTURING via tetrahedron reorientation Director field on the surface of a tetrahedron

  10. RESTRUCTURING OF DIMERS

  11. DISCLINATION LINE AS A RIBBON

  12. RIBBONS in form of LOOPS LINKING NUMBER, WRITHE, AND TWIST Linking number (L) of a closed ribbon is equal to a number of times ittwistsaround itself before closing a loop. Calugareanu theorem (1959): writhe and twist are given by well known expressions L = Wr + Tw Symmetric planar loops (like Saturn)Tw = 0 and Wr = 0 Our tetrahedron transformation does not add twist. Twist is zerofor all dimerloopstructures ! L = Wr Following Fuller (1978) writhe is calculated in tangent representation on a unit sphere Wr = A/(2p) -1 mod 2 A - surface on a unit sphere encircled by the tangent.

  13. WRITHE IN TANGENT SPACE Writhechange due to a terahedron rotationfor 120o: DWr = 2/3

  14. FIGURE OF EIGHT 3D loop 2D

  15. NEMATIC COLLOIDAL DIMERS Disjoint SaturnsWr = 0 Entangled hyperbolic defect Wr = 0 Figure of eight Wr = + 2/3 Figure of omega Wr = + 2/3 twist: Tw =0 linking number L = Tw + Wr

  16. CONCLUSIONS • Desription: restructuring of an orthogonal line crossing via a tetrahedron rotation. • Clasification of colloidal dimers via linking number, writhe, and twist. • Further chalanges: • complex nematic (also chiral and biaxial) braids • chiral nematic offers further line crossings that for: • colloids easily lead to the formation of links • and knots in the disclination network • (Tkalec, Ravnik, Muševič,…) • confined blue phases enables restructuring • among numerous structures (Fukuda & Žumer)

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