Samo Kralj1,2, Riccardo Rosso3, Epifanio G. Virga3

1Faculty of Natural Sciences and Mathematics, University of Maribor, Slovenia

2Jozef Stefan Institute, Ljubljana, Slovenia3Department of Mathematics, University of Pavia, Italy

LIQUID CRYSTAL NEMATIC CONFIGURATIONS ON THIN

FILMS

Liquid crystal phases :

Important role in several natural systems.

Main advantages: • softness (= susceptibility)• optical transparency + anisotropy• richness of phases & structures

Confinement :• surface local interactions (affecting translational&orientational ordering)• symmetry breaking• finite size effects

LIQUID CRZSTALSFocus

1) complex behavior in thin nematic hybrid films (frustrations + topological defects)• new boojum structure• interaction of boojum with elastic distortions• boojum dragging towards cell interior• defect core enhancement

2) thin nematic shells• new 2D Q-tensor mesoscopic approach• character of the I-N transition• defect structures on ellipsoidal shells

Complex behavior :

1)Boojum2)Frustration3)Finite size effects4)External field

I) THIN NEMATIC HYBRID FILMS

Half of hedgehog ?

Well known HEDGEHOG • biaxial structure• includes order reconstruction

Expected boojum structure ?

h > hc h < hc

Order reconstruction: h<hc

Mesoscopic modelling

F

Uniaxial states :

Degree of biaxiality :

biaxialitymaximal,1

states uniaxial,0

1,0

2

2

2

qqq

q

qqq

zrQ

m

m

0

0

0

0

020

0

,

Cylindrical coordinate system, parametrization (cylindrical symmetry, no twist)

Phys. Rev. E 78, 031701 (2008); 81, 021702 (2010).

Adequate parametrization for visualization of biaxial states

TrQ=0

Phys. Rev. E 81, 021702 (2010).

states with negative uniaxiality

states with positive uniaxiality

states with maximal biaxiality

I) RESULS

A

A

B

C

C

?

Half uniaxial hedgehog

Half biaxial hedgehog

Observed boojum structure

“Finger” boojum structure, 2(r,z)

Naively expected biaxial boojum structure

Interaction boojum : order reconstruction structure in thin films

A

B C

D

boojum can lift the order reconstruction structure

Boojum pushed to the top, the order reconstruction structure locally follows it

SS=0

),(2 z

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1B=400, h=10

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1B=0, h=10

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1B=200, h=10

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1B=400, h=10

Width of the elongated boojum ?

Planar radial with a negative uniaxial core

ER = escaped radialPhys.Rev.E 60, 1858 (1999).

Phys.Rev.E 66, 021703 (2002).

Dimensionless excess free energy :

Phys.Rev.E 60, 1858 (1999).

3/cos4

1)3cos(1

18 2

22

2

2

eb

RRf

External field contribution

Bulk nematic ordering: =-/3

= negative uniaxiality

fieldexternalcriticale

b )(

10

0 1 2 3 4 5 6 7 8 9 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1beta, E2=0

0 1 2 3 4 5 6 7 8 9 100.4

0.5

0.6

0.7

0.8

0.9

1OP, E2=0

bss

2

2

3TrQs

2

br / br /

0 1 2 3 4 5 6 7 8 9 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1beta, E2=-20

0 1 2 3 4 5 6 7 8 9 100.98

1

1.02

1.04

1.06

1.08

1.1

1.12

1.14

1.16

1.18OP, E2=-20

2bss

br / br /

def

b

def

2

2

18 e

b

II) THIN NEMATIC SHELLS

eigenframe

general frame

ei : chosen along the lines of principal curvatures of the surface

The surface gradients:

curvaturetotalH

curvatureGaussianK

curvaturesprincipal

curvaturesgeodesic

i

gi

:

:

:

:

21

21

Symmetry invariant terms entering the free energy density

Condensation term

Elastic term, K: Gaussian curvature

Uniaxial ellipsoidal surfaces obtained by rotating the ellipse

v : meridiansu : parallels

In agreement with :

Director field representation

semi-microscopic simulations

Oblate surfaces (sphere : =1) 20/ bR ab /

20/ bR

50/ bR

Prolate surfaces (sphere : =1) 20/ bR ba /

Prolate surfaces (sphere : =1) 50/ bR ba /

Conclusions

• Complex nematic structures in thin films

• Rich variety of structures -> interplay among geometrical constraints, elastic forces and finite size effects

• Of interest for future nanobased electrooptic devices

Nelson, Nano. Lett. 2, 1125 (2002)

Nematic shells immersed in a solution of an isotropic liquid and flexible linkers

= SCALED ATOMA (defect sites > determine valence)