NEMATIC COLLOID AS A TOPOLOGICAL PLAYGROUND

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S. ŽUMER

University of Ljubljana & Jozef Stefan Institute, Ljubljana, Slovenia

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

Confined Liquid Crystals: Perspectives

and Landmarks

June 19-20, 2010    Ljubljana

MOTIVATION

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S=0.5 surface of -1/2 defect line (Sbulk=0.533)

d = 1 m, h = 2 m director

OUTLINE

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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

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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.

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):

o singular (half- integer) disclination lines biaxiality & decrease of order

o nonsigular (integer) disclination lines

DEFECTS

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SPHERICAL HOMEOTROPIC PARTICLES

6 Stark et al., NATO Science Series Kluwer 02

CONFINED TO A HOMOGENOUS NEMATIC FIELD zero topological charge

Saturn ring(quadrupolar symmetry)

dipole(dipolar symmetry)

2.5 m cell2 m particle

Strong anchoringS=0.5 surface of defect (Sbulk=0.533)

<=

COLLOIDAL DIMER IN A HOMOGENOUS NEMATIC FIELD

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cell thickness: h = 2 m , colloid diameter: d = 1 m

In homogenous cells these structures are obtained only via melting & quenching

director

figure of eight figure of omega entangled hyperbolic defect

director

zero topological charge

LOCAL RESTRUCTURING OF DISLINATIONS

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Orthogonal crossing of disclinations in a tetrahedron

Restructuring via tetrahedron reorientation

LOCAL RESTRUCTURING

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Director field on the surface of a tetrahedron

via tetrahedron reorientation

RESTRUCTURING OF DIMERS

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DISCLINATION LINE AS A RIBBON

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RIBBONS in form of LOOPS LINKING NUMBER, WRITHE, AND TWIST

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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/(2) - 1 mod 2

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

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

WRITHE IN TANGENT SPACE

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Writhe change due to a terahedron rotation for 120o:

Wr = 2/3

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FIGURE OF EIGHT

3D loop 2D

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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

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• 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)