Transcript Computer Modeling of Nematic Liquid Crystals
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
• • • • • •
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?
•
Interesting
and
fast evolving
field.
• Liquid crystals
well represented
field in Slovenia.
• New potential applications: • •
Metamaterials .
Microcavities microresonators .
( Hot topics!
) M. Ravnik, S. Žumer,
Soft Matter
, 2009. • 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 •
LCD
(Liquid Crystal Displays).
•
Eye protecting filters
for welding helmets (Balder) Liquid crystals have unique
optical
properties.
•
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 • • •
Liquid crystals
are a liquid, oily material.
They flow like a
liquid
...
... but can be
partially ordered
- like
crystals
.
E
• • • Molecules are
rodlike.
Tend to align in a
preferred direction
.
Electric
or
magnetic field
can
change
their phase form isotropic liquid to
partially ordered mesophase
.
• (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 2
S
1 Quantifies the
degree of order
of the orientation:
-1/2
ideal biaxial
liquid
0 1
isotropic
liquid
ideally aligned
liquid (all molecules parallel)
Alternative description with Q-tensor field New quantity:
tensor order parameter
Q
(
r
) :
Q
S
3
n
n
I
P
e
1
e
1
e
2
e
2
S
2 2
n
its largest eigenvector and its corrispondent eigenvalue.
•
Q
traceless
: •
Q
symmetric
:
Q
11
Q
22
Q
33 0
Q ij
Q ji Q
33
Q
11
Q
22 Only
5 independent components
of
Q
are required.
Q
Q
11
Q
12
Q
22
Q
13
Q Q
23 11
Q
22
Free-energy functional
•
Director
and
order
nematic structure follow from the
Landau-de Gennes functional
:
minimizing
F
(
Q
)
bulk f bulk
(
Q
,
Q
)
dV
border f surf
(
Q
,
Q
)
dV
f bulk 1 2 L Q ij x k Q ij x k 1 2 AQ ij Q ij 1 3 BQ ij Q jk Q ki 1 4 C(Q ij Q ij ) 2 Elastic energy Thermodynamic energy
L
– elastic constants
A, B, C
– material constants
W
– surface energy f surf 1 2 W (Q ij Q ij ( 0 ) ) 2 Surface energy
Colloidal particles
in nematic
Inclusion of colloidal particles • Inclusion of
colloidal particles
in a thin sheet of nematic LC.
• 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
' (
x
) 0 Newton iteration: First variation of
functional
:
F
(
Q
)
F
' (
Q
) 0 Newton iteration:
x k
1
x k
f f
' (
x k
'' (
x k
) )
F
' ' (
Q k
)
Q k
F
' (
Q k
)
Q k
1
Q k
Q k test functions
)
Test calculations in 3D:
One
colloidal particle
• • • Central section of 3D simulation box mesh Mesh points:
17 000
; Tetrahedra:
100 000
Mesh generation’s time:
5 sec
(TetGen)
2 microns
• • Central section:
director
field
n
(in green).
Newton’s method took
19
iterations (total time:
54 min
).
2 microns
• • Central section of the order parameter field S. In green: sections of
Saturn ring defect.
Topological defect 2 microns
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 A large field
E
change
Q
.
(
Q
)
Iteration needed
we
switch
between
optical modes
.
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.
• Aim:
band structure
of
photonic crystals
.
(
r
) 1
H
H
0
c
2
H
Eigenequation
(+ condition) •
Periodic
boundary conditions.
• Reduces to a matrix eigenproblem :
Ax
2
Bx Pictures from site of Steve Johnoson (MIT).
2) Frequency domain responses • Seeking for
stationary state
.
• Aims:
absorption
&
transmittivity.
• At
fixed
?
E
H
1
c
(
r
)
t
1
c
H
t
E i
c J
H
i
c
E
+
Absorbing Boundary Conditions
(ABC).
• Reduces to a matrix linear system :
Ax
b
3) Time-domain propagation
Time evolution
of electromagnetic waves.
?
?
?
?
Micro-waveguides?
Micro-optical elements?
Start with
FDTD
(Finite Difference Time Domain) numerical method: • •
1.
Ready code
freely available.
Easily supports
nonlinear optical
effects.
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