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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
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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
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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
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2) thin nematic shells• new 2D Q-tensor mesoscopic approach• character of the I-N transition• defect structures on ellipsoidal shells
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Complex behavior :
1)Boojum2)Frustration3)Finite size effects4)External field
I) THIN NEMATIC HYBRID FILMS
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Half of hedgehog ?
Well known HEDGEHOG • biaxial structure• includes order reconstruction
Expected boojum structure ?
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h > hc h < hc
Order reconstruction: h<hc
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Mesoscopic modelling
F
Uniaxial states :
Degree of biaxiality :
biaxialitymaximal,1
states uniaxial,0
1,0
2
2
2
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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).
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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
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I) RESULS
A
A
B
C
C
?
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Half uniaxial hedgehog
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Half biaxial hedgehog
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Observed boojum structure
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“Finger” boojum structure, 2(r,z)
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Naively expected biaxial boojum structure
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Interaction boojum : order reconstruction structure in thin films
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A
B C
D
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boojum can lift the order reconstruction structure
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Boojum pushed to the top, the order reconstruction structure locally follows it
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SS=0
),(2 z
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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
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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
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Width of the elongated boojum ?
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Planar radial with a negative uniaxial core
ER = escaped radialPhys.Rev.E 60, 1858 (1999).
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Phys.Rev.E 66, 021703 (2002).
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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
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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
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b
def
2
2
18 e
b
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II) THIN NEMATIC SHELLS
eigenframe
general frame
ei : chosen along the lines of principal curvatures of the surface
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The surface gradients:
curvaturetotalH
curvatureGaussianK
curvaturesprincipal
curvaturesgeodesic
i
gi
:
:
:
:
21
21
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Symmetry invariant terms entering the free energy density
Condensation term
Elastic term, K: Gaussian curvature
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Uniaxial ellipsoidal surfaces obtained by rotating the ellipse
v : meridiansu : parallels
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In agreement with :
Director field representation
semi-microscopic simulations
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Oblate surfaces (sphere : =1) 20/ bR ab /
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20/ bR
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50/ bR
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Prolate surfaces (sphere : =1) 20/ bR ba /
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Prolate surfaces (sphere : =1) 50/ bR ba /
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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
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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)