NEMATIC COLLOID AS A TOPOLOGICAL PLAYGROUND 1 S. ŽUMER University of Ljubljana & Jozef Stefan...
Transcript of NEMATIC COLLOID AS A TOPOLOGICAL PLAYGROUND 1 S. ŽUMER University of Ljubljana & Jozef Stefan...
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)