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 = 1 mm, h = 2 mm
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 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.

5. 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

6. 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

7. 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

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 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) mod 2
A - surface on a unit sphere encircled by the tangent.

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

14. FIGURE OF EIGHT
3D loop D

15. 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

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)