Nematic colloids for photonic systems
(with schemes for complex structures)
Iztok Bajc
Adviser: Prof. dr. Slobodan Žumer
Fakulteta za matematiko in fiziko
Univerza v Ljubljani
Slovenija
Outline
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Motivations, classical and new applications
Nematic liquid crystals
Colloidal particles in nematic
Modeling requirements for large 3D systems
Test calculations (3D)
Future work: external fields for photonic systems
Motivations,
classical and new applications
Motivations
Why to approach this thematic?
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Interesting and fast evolving field.
Liquid crystals well represented field in Slovenia.
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New potential applications:
• Metamaterials.
• Microcavities - microresonators.
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M. Ravnik, S. Žumer, Soft Matter, 2009.
(Hot topics!)
One of the priorities of the EU project (Hierarchy)
in which I’m involved.
Requirement of very effective modeling codes.
Challenge to find the right approaches.
M. Humar, M. Ravnik, S. Pajk, I. Muševič,
Nature Photonics, 2009.
Classical applications of liquid crystals
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LCD (Liquid Crystal Displays).
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Eye protecting filters for
welding helmets (Balder)
Liquid crystals
have unique
optical properties.
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Polarizing glasses for 3D vision
New potential applications:
metamaterials, microresonators
• Photonic crystals:
Nematic droplet.
Whispering Gallery Modes (WGM ) in a microresonator.
• Solid state metamaterials:
• Soft metamaterials?
Figures: I. Muševič, CLC Ljubljana Conference, 2010.
Nematic Liquid Crystals
Nematic liquid crystals
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Liquid crystals are a liquid, oily material.
They flow like a liquid...
... but can be partially ordered - like
crystals.

E
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Molecules are rodlike.
Tend to align in a preferred direction.
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Electric or magnetic field can change
their phase form isotropic liquid to
partially ordered mesophase.
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(The same happens, if temperature is lowered)
Description of nematic liquid crystals
• Basic quantities
Director

n (r )

n 1
Points in preferenced orientation.
Scalar order
parameter

S (r )
1
  S 1
2
Quantifies the degree of order of the
orientation:
-1/2  ideal biaxial liquid
0  isotropic liquid
1  ideally aligned liquid
(all molecules parallel)
Alternative description with
Q-tensor field

New quantity: tensor order parameter Q (r ) :
S  
P    
Q  3n  n  I   e1  e1  e2  e2 
2
2

S its largest eigenvector and n its corrispondent eigenvalue.
• Q traceless:
• Q symmetric:
Q11  Q22  Q33  0
Qij  Q ji
Only 5 independent
components of Q are required.
Q33  Q11  Q22
 Q11 Q12

Q
Q22




Q23

 Q11  Q22 
Q13
Free-energy functional
• Director and order nematic structure follow from minimizing
the Landau-de Gennes functional:
F (Q ) 
f
bulk
(Q, Q )dV 
bulk
f bulk
f
surf
(Q, Q )dV
border
1 Qij Qij 1
1
1
 L
 AQijQij  BQijQ jkQki  C(QijQij )2
2 x k x k 2
3
4
Elastic energy
Thermodynamic energy
L – elastic constants
A, B, C – material constants
W – surface energy
f surf
1
( 0)
 W (Q ij - Qij ) 2 Surface energy
2
Colloidal particles
in nematic
Inclusion of colloidal particles
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Inclusion of colloidal particles in a thin sheet of nematic LC.
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We get disclination lines (topological defects) around the particles:
Strong attractive forces
between particles.
Colloidal structures
-crystals in nematic.
Structures of colloidal particles
in nematic
1D structures
3D structures
2D structures - crystals
Large 3D structures:
12- and 10- cluster in 90° twisted nematic cell.
Experiments by U. Tkalec, 2010 (to be published).
3×3×3 dipolar crystal in
homeotropically oriented nematic.
Experiment by Andriy Nych, 2010
(to be published).
Modeling Requirements
Computations until now
Actual finite difference code in C is:
• Robust and effective for smaller or periodic systems.
• But uses uniform grid (uniform resolution).
Example:
A job needs 2h to converge.
You double the resolution
Then it will run for 2 days.
New modeling requirements
Mesh adaptivity in 3D, preferably
with anisotropic metric.
Moving objects (due to
nematic elastic forces).
Parallel processing
(computer clusters).
Meshes by Cécile Dobrzynski, Institut de Mathématiques de Bordeaux.
Finite Element Method (FEM)
Advantages:
– Mesh can be locally refined
less mesh point needed.
– Around each point we have an interpolating function (spline).
Newton iteration of tensor fields
If function (of one variable):
F (Q)  F ' (Q)  0
f ' ( x)  0
Newton iteration:
Newton iteration:
xk 1  xk 
First variation of functional:
f '( x k )
f '' ( x k )
F ' ' (Qk )Qk   F ' (Qk )
Qk 1  Qk  Qk
( - test functions)
Test calculations in 3D:
One colloidal particle
2 microns
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Central section of 3D simulation box mesh
Mesh points: 17 000; Tetrahedra: 100 000
Mesh generation’s time: 5 sec (TetGen)
2 microns
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Central section: director field n (in green).
Newton’s method took 19 iterations (total time: 54 min).
Topological
defect
2 microns
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Central section of the order parameter field S.
In green: sections of Saturn ring defect.
Test calculations in 3D :
More particles
Future work: external fields
for photonic systems
Electric field on a nematic droplet
E 0
By tuning electric field
we switch between optical modes.
A large field E change Q.
   (Q)
Also  changes.
Iteration needed
Figures: I. Muševič, CLC Ljubljana Conference, 2010.
Electromagnetic waves –
linear/nonlinear optics
• Detail dimensions comparable with wavelength.
2 microns
Ray optics not adequate.
• Full system description needed (diffraction,...).
• Nematic is a lossy medium.
• Also nonhomegeneously anisotropic.
Birefringence
Computational electromagnetics
Basis:
Numerical solution of Maxwell equations
Computational photonics
Mature field for homogeneous
medium and periodic structures
(e.g. photonic crystals).
But young for nonhomegenously
anysotropic media !
Computational soft photonics
Computational approaches
Book Joannopoulos et alt., Photonic Crystals, points out
three cathegories of problems:
1) Frequency-domain eigenproblems
2) Frequency-domain response
3) Time-domain propagation
[1] Joannopoulos et alt., Photonic Crystals, Molding the flow of Light, 2nd ed, Princeton University Press, 2008.
1) Frequency domain eigenproblems
• Seeking for eigenmodes.
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• Aim: band structure (k ) of photonic crystals.
2

 1      
   (r )   H    H
Eigenequation
c
 
(+ condition)
 H  0
• Periodic boundary conditions.
• Reduces to a matrix eigenproblem:
Ax   Bx
2
Pictures from site of
Steve Johnoson (MIT).
2)
Frequency domain responses
• Seeking for stationary state.
• Aims: absorption & transmittivity.
• At fixed frequency
?
.
 
1 
 
E  
H i H
c t
c
 
 1  
 
  H   (r )
E  J   i E
c t
c
+ Absorbing Boundary Conditions (ABC).
• Reduces to a matrix linear system:
Ax  b
3)
Time-domain propagation
Time evolution of electromagnetic waves.
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Micro-waveguides?
Micro-optical elements?
Start with FDTD (Finite Difference Time Domain) numerical method:
1.
Ready code freely available.
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Easily supports nonlinear optical effects.
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Gain feeling and experience for smaller systems.
Next: possibility of passing to FEM will be considered.
Acknowledgments:
• Slobodan Žumer (adviser)
• Miha Ravnik, Rudolf Peierls Centre for Theoretical Physics, Univerza v Oxfordu, in FMF-UL.
• Frédéric Hecht, Laboratoire Jacques-Louis Lyon, UPMC, Paris 6.
• Daniel Svenšek
• Igor Muševič
• Miha Škarabot
• Martin Čopič
• Uroš Tkalec
Work has been finansed by EU:
Hierarchy Project, Marie-Curie Actions