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

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

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

ORDER PARAMETER FIELD Tensorial nematic order parameter Q (director n, degree of order S, biaxiality P) : eigen frame: 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 metastable nematic structures are determined via minimization of F that leads to the solving of the corresponding differential equations.

DEFECTS discontinues director fields & variation in nematic order 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

SPHERICAL HOMEOTROPIC PARTICLES CONFINED TO A HOMOGENOUS NEMATIC 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

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

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

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

RESTRUCTURING OF DIMERS

DISCLINATION LINE AS A RIBBON

RIBBONS in form of LOOPS LINKING NUMBER, WRITHE, AND TWIST Linking number (L) of a closed ribbon is equal to a number of times it twists around 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 zero for all dimer loop structures ! 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.

WRITHE IN TANGENT SPACE Writhe change due to a terahedron rotation for 120o: DWr = 2/3

FIGURE OF EIGHT 3D loop 2D

NEMATIC COLLOIDAL DIMERS Disjoint Saturns Wr = 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

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)