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Liquid Crystals: Lecture 1
Basic properties
Support: NSF
Oleg D. Lavrentovich
Liquid Crystal Institute and
Chemical Physics Interdisciplinary Program,
Kent State University, Kent, OH 44242
Boulder School for Condensed Matter and Materials Physics,
Soft Matter In and Out of Equilibrium,
6-31 July, 2015
Acknowledgements
Volodymyr Borshch, Mingxia Gu, Tomohiro Ishikawa, Israel Lazo, Young-Ki Kim, Heung-Shik Park, Chenhui Peng, Oleg Pishnyak, Ivan Smalyukh, Bohdan Senyuk, Jie Xiang, Shuang Zhou (LCI, Kent State University)
Drs. Sergii Shiyanovskii, Antal Jakli (LCI, Kent State University)
Maurice Kleman (Paris), Grigory Volovik (Moscow), Vasyl Nazarenko (Kyiv), Igor Aranson (Argonne NL) Georg Mehl (Hull), Corrie Imrie (Aberdeen)
The lectures are based on the book by
Maurice Kleman and O. D. Lavrentovich
“Soft Matter Physics: An Introduction” (Springer, 2003)
3
Content: Lecture 1
Types of liquid crystalline order Nematic
Cholesteric and blue phases
Twist bend nematic
Smectic A
Lyotropic and Chromonic LCs
Basic Physics Dielectric anisotropy
Surface anchoring
Elasticity
Frederiks effect and modern LCDs
Optics Birefringence
Polarizing microscopy: 2D imaging
Fluorescence Confocal Polarizing microscopy: 3D imaging
4
Content: Lecture 2
Topological defects and droplets Disclinations in uniaxial nematics
Singular
Nonsingular
Homotopy classification
Drops and Conservation laws of topological defects
Cholesteric droplets, Dirac monopole
Chromonic tactoids
Broken chiral symmetry
Wulff construction for liquid crystals
Lamellar phases Free energy density for weak and strong perturbations
Long-range character of deformations; undulations
Focal conic domains
5
Content: Lecture 3
Dynamics of director realignment
Anisotropy of viscosity
Coupling of director reorientation and flow
Statics of colloids in nematic LC
Levitation
Dynamics of colloids in nematic LC
Brownian motion
LC-enabled electrophoresis
LC with patterned orientation as an active medium
Living LC
Swimming bacteria in LC; individual and collective effects
6
Liquid crystal:
a state of matter with long-range orientational order and
complete (nematic)
partial (smectics, columnar phases)
absence of long-range positional order of “building units”
(molecules, viruses, aggregates, etc.)
Our goal: Develop an
intuitive understanding
of what kind of new
physics the orientational
order brings to soft
matter
Crystals, liquid crystals, liquids
pentyl-cyanobiphenyl (5CB):
room temperature nematic
1.2 nm
Temperature
orientational and positional order: Molecular crystal
orientational order; no positional order:
Nematic LC
no orientational and no positional order
Isotropic liquid
Director n = optic axis
7
8
Nematic LC: nema=thread; aka “disclination”
1922, G. Friedel:
Named “nematics”, the simplest
LC, after observing linear defects,
nema=thread, under a polarized
light microscope
Nematic droplet-pancake sheared between two
glass plates; polarizing optical microscope
50 microns
9
Nematic LC: Calamitic and Discotic
10
Nematic LC: Calamitic and Discotic. Biaxial?
Uniaxial
calamitic
Uniaxial
discotic
Existence established
Existence debated, Uniaxial N often mimics biaxiality
Biaxial
Chiral molecule (does not overlap with its chiral
image)
Carbon atom with 4 different attachments
Cholesterol benzoate: Rod-like molecule
with a chiral C atom;
A similar cholesterol derivative was the
subject of the first known publication on
LCs, reporting double melting point and
selective reflection of light, J. Planer, Ann
Chem Pharm 118, 25 (1861)
11
Add chiral molecules to nematic and obtain a
cholesteric:
12
Where do the colors come from? pitch~0.5 mm
Bragg reflection at the periodic structure with period close to the
wavelength of visible light
Why P>>molecular scale? Weakness of chiral contribution to the
intermolecular potential, see Harris et al, Rev Mod Phys 71, 1745
(1999)
Comparative chemistry
CN
Cholesteric CB15 molecule
Nematic 5CB molecule
What would happen when two CB molecules are connected by a flexible
aliphatic chain and form a “bimesogen” or “dimer”?
Answer: depends on odd-even character of aliphatic chain
even number m of CH2 groups
How can we pack these molecules in space?
odd number m of CH2 groups
Packing bimesogens
Even m, rod-like molecules:
Easy! Nematic!
Odd m, bent shape:
Difficult…cannot
sustain uniform bend in
2D…
Predictions:
R.B. Meyer (1973, Les Houches)
R. Kamien, J. Phys II 6, 461 (1996)
I. Dozov, EPL 56, 247 (2001)
J. Selinger et al, PRE 87, 052503
(2013)
E. Virga, PRE 89 052502 (2014)
0 0 0ˆ sin cos , sin sin , costz tz =n
Go to 3D:
Uniform bent
is achieved
through twist!
Freeze Fracture TEM, “planar” fractures
0 . 1 µ m0 . 1 µ m
FFT
t̂
Period 8.05 nm (molecular scale!!! Frozen rotations?
Not visible under regular optical microscope; need an electron microscope)
Borshch, Gao et al, Nature Comm 4, 2635 (2013)
Chen, Walba, Clark et al, PNAS 110, 15931 (2013)
CB7CB quenched from the “X” temperature range and viewed under TEM
Freeze-Fractures in TEM: most likely to occur parallel to the long axes of molecules, thanks to the lowest molecular density
8 nm period
Freeze Fracture TEM, “planar” fractures
The 8 nm period is not visible in X-ray studies; thus there is no
modulation of density
How do we know that the structure is indeed twist-bend as opposed
to a simple cholesteric, or, say, splay-bend, also predicted to exist?
Tilted fractures: Bouligand arches are
different in Ch and Ntb
Cholesteric:
Bouligand
arches are
symmetric
0
0
z'
0
0
0 0
0
ln 2cos sin sin 'sin tan cot
sin cos
tb
tb
t yx x
t
=
Ntb: Bouligand arches
are either asymmetric or
incomplete
Bouligand et al, Chromosoma 24, 251 (1968)
FF TEM, Asymmetric Bouligand arches
Both types of Ntb asymmetric Bouligand arches are observed
Borshch et al, Nature Comm 4, 2635 (2013)
Nematic and Cholesteric are merely two point ends
of The Twist-Bend Nematic World…
Cholesteric
: molecular tilt angle
Twist-bend Nematic
0 0 =
0 0 0ˆ sin cos , sin sin , cos =n
0 / 2 =00 / 2
2 /t P=0
Nematic
tz =
Add a 1D positional order to obtain a smectic…
1-4 nm
Smectic A (SmA)
Periodic modulation of density (verified by X-
ray experiments), unlike in Ntb phase
1910, G. Friedel, F. Grandjean:
Deciphered SmA structure
from observation of
focal conic domains;
X-ray was not available
Smectics and focal conic domains
mm40
21
22
Smectics: Optical Microscopy at its best
Smectic A
1 nm
1910, G. Friedel, F. Grandjean:
Deciphered SmA structure
from observation of
focal conic domains;
X-ray was not available,
but the mere fact of existence of
ellipses and hyperbolae led to a
correct conclusion: SmA is a
system of equidistant flexible
fluid layers, i.e, a 1D periodic
structure
mm40
Smectics A, C, C*, etc
23
SmC SmC*
Smectic A: Molecules normal to the layers; Smectic C; molecules are tilted
Chiral smectic C; molecules are tilted and follow an oblique helicoid
Twist Grain Boundary Phases
24
Combination of smectics and cholesterics; formed by chiral molecules;
sophisticated analog of Abrikosov phase in superconductors; smectic is
penetrated by a lattice of screw dislocations that allows the smectic to twist
1989, Renn, Lubensky, Pindak, Goodby et al.:
Crystals, thermotropic liquid crystals, liquids
pentyl-cyanobiphenyl (5CB):
room temperature nematic
1.2 nm
Temperature
orientational and positional order: Molecular crystal
orientational order; no positional order:
Nematic LC
no orientational and no positional order
Isotropic liquid
Director n = optic axis
25
Lyotropic Liquid Crystals: Power of Entropy
Concentration, c
Onsager (1949): Nematic order in solution
of long thin rods, thanks to translational vs
orientational entropy trade-off
Lyotropic Liquid Crystals: Power of Entropy
Onsager (1949): Nematic order in solution
of long thin rods, thanks to translational vs
orientational entropy trade-off
Concentration, c
Tobacco mosaic virus (TMV)
J. Bernal and I. Fankuchen, J. Gen. Physiol. (1941): tactoids as N nuclei in tobacco mosaic virus dispersions (1939: First images of TMVs)
Phase diagram does not depend on
temperature
Molecular structure of phospholipids
Molecular structure of phospholipids. The
number of carbon atoms in the aliphatic
chains varies, usually between 16 and 20.
Two chains might be of different length.
phosphatidylcholine (lecithin) phosphatidylserine
phosphatidylethanolamine
N
O
N
O
N
N
SO3NH4NH4O3S
+ +
2 nm
Chromonic molecules:
1. Rigid plank-like polyaromatic core
2. Ionic groups at periphery
Lyotropic Chromonic LCs: special case
Violet 20
Mechanism of LC formation: through
aggregation: balance of entropy and
association energy d; average length
of aggregates
n
d
exp2 B
L ck T
d
29
LCLCs=Mesomorphic phases (N and Col) resulting from 1D non-covalent molecular self-assembly
Common occurrence in dyes, drugs, proteins, nucleic acids
Similar to living polymers, wormlike micelles of surfactants
Promising applications as sensing material, in optical components, organic semiconductors, alignment of nanotubes, assembly of nanorods, etc.
Nakata, Clark et al. Science 318, 1276
(2007)
Kuriabova, Betterton, Glaser,
J. Mat. Chem. 20,10366 (2010)
Lyotropic Chromonic LCs: special case
Chromonics :
controlled by both c
and T
Thermotropic LC (in your
displays): same molecules,
c=const, aligned at low T
Onsager systems:
same rods,
athermal,
aligned at high c
T
c
T
c
HS Park, ODL in Liquid crystals beyond displays, Editor Q. Li (2012)
Lyotropic Chromonic LCs: special case
HS Park, ODL in Liquid crystals beyond displays, Editor Q. Li (2012)
Columnar phase: 2D positional order of
columns (aggregates)
Nuclei of columnar
phase in isotropic
fluid: Bend
deformations that
preserve the
equidistance of 2D
lattices
Toroids rather than
spheres
Disodium
cromoglycate + water
+PEG
Content
Types of liquid crystalline order Nematic
Cholesteric and blue phases
Twist bend nematic
Smectic A
Lyotropic and Chromonic LCs
Basic Physics Dielectric anisotropy
Surface anchoring
Elasticity
Frederiks effect and modern LCDs
pentyl-cyanobiphenyl (5CB)
Anisotropy of uniaxial nematic
Director n=optic axis
Anisotropy of all properties: Optical, Dielectric, Diamagnetic,
Electric conductivity, Elasticity, Viscosity, etc.
=
||e
e
e
e
00
00
00
10414 == eee ||a
1.7 1.5 0.2e on n n =
Field-induced reorientation of LC optic axis caused by diamagnetic or
dielectric anisotropy: Frederiks effect (Leningrad, USSR, 1920-1930ies);
Optic axis: Easily deformable by E-field: Frederiks effect
Optical response to the electric field puts liquid crystals
at the heart of modern informational displays technologies
n̂
E
Guide to eye, director line
1927, Vsevolod Frederiks and Antonina Repiova
Magnetic field reorients LC
Frederiks effect: Field reorients LC
1934, V. Frederiks and V. Tsvetkov: Electric field reorients LC
(1885-1943)
cB d const=
1969, Kent, Ohio: James Fergason
patents Twisted nematic cell, the first field-operated LCD
(M. Schadt and W. Helfrich filed a similar patent in Europe and
also published an article)
Field effects = LC displays
First product,
Fergason’s ILIXCO (Kent, 1970)
1934-2008
2006: Lemelson-MIT $500,000 award
LC displays: Control of polarized light
Dielectric anisotropy and birefringence of LC enabled
revolution in informational portable displays
3.5 billion LCD panels sold in
2014 (all sizes)
39
Elasticity vs. Anchoring
Elasticity: Director gradients cost energy
2
2
1~
2
i
j
n Kf K
x L
=
~F f dV KL=
Elasticity vs. Anchoring
Elasticity: Director gradients cost energy
Surface anchoring
2
2
1~
2
i
j
n Kf K
x L
=
~F f dV KL=
Elasticity vs. Anchoring
Elasticity: Director gradients cost energy
Surface anchoring
2 212
; ~s sf W F WL
External Torque
2
2
1~
2
i
j
n Kf K
x L
=
~F f dV KL=
Elasticity vs. Anchoring
Elasticity: Director gradients cost energy
Surface anchoring
2 212
; ~s sf W F WL
2
2
1~
2
i
j
n Kf K
x L
=
~F f dV KL=
Elasticity vs. Anchoring
Elasticity: Director gradients cost energy
Surface anchoring
Anchoring extrapolation length
W/K=
2 212
; ~s sf W F WL
2
2
1~
2
i
j
n Kf K
x L
=
~F f dV KL=
Material parameters: orders of magnitude
263 J/m1010 ~W
Anisotropy of surface tension: Intuition fails; experiments say
(weak as compared to surface tension)
? ? ?K W 21
9
5 10 J~ ~ ~ 5 pN
10 m
B NIk TK
a
asizemolecular~W
K
==
μm10nm10J/m1010
N10263
11
21-2 2
2 18
5 10~ ~ ~10 J/m
10
Bk T
a
Surface tension
Elastic constants
Elasticity of N: Oseen, 1933, Frank, 1958
F. C. Frank, Disc. Faraday Soc. 25, 19 (1958)
ˆ ˆ x, y,z=n n
Elastic free energy density-?
2
0 ....i iij ijkl
j j
n nf f
x xa
=
1
molecular size
i
j
n
x
Requirements:
-n and n are invariant;
-central inversion about any point;
-invariance under any rotation around n.
ˆdivn ˆ ˆcurln nTwo scalar invariants linear in derivatives:
2
curln n…thus can also be 2 2 2
ˆ ˆ ˆ ˆ ˆcurl curl curl= n n n n n
Invariant and Quadratic in derivatives: 2
ˆ ˆcurln n 2
ˆcurln 2
ˆdivn
2 2 2
1 2 3
1 1 1ˆ ˆ ˆ ˆ ˆdiv curl curl
2 2 2FOf K K K= n n n n n
splay twist bend
Frank-Oseen elastic free
energy density:
divn=1
r
nx = cos, ny = sin, nz = 0
divn=2
r
nx =x
r, ny =
y
r, nz =
z
r
2D and 3D splay
q = n curl n
nx = cosqz, ny = sinqz, nz = 0
q =a
d=
2
p
Twist (one-directional)
n curln
nx = sin, ny = cos, nz = 0
Bend
Elasticity vs Anchoring: Hybrid-Aligned Nematic Film
1. Infinitely strong anchoring
Fixed boundary conditions
Problem: Find the director dependence on z in equilibrium
Assumption #1:
2
3
2
1 cos2
1sin
2
1
=
dz
dK
dz
dKfFO
ˆ , , sin , 0, cosx y zn n n z z = =nn̂
K1 = K3 = KAssumption #2:
2
12FO
df K
dz
=
00z = = dz d = =
The problem reduces to finding (z) that minimizes the integral
2
12
0
z d
FO
z
dF K dz
dz
=
=
=
Euler-Lagrange eq. (next slide gives the outline, home assignment if you do not know it yet):
2
20 0
'
FO FOf fd d
dz dz
= =
0
0
d zz
d
=
2
012
d
FOF Kd
=
eq z
za=
z = eq z a z
z= 0 = z = d = 0
where is such that
z
F z = f eq z a z , 'eq z a' z , z dz0
d
F a
a
a=0
=0 Condition of the extremum:
Euler-Lagrange equation for 1D problem, fixed boundary conditions (leisure time reading)
FFO = f FO ,' ,z dzz =0
z=d
FFO = 12 K '
2dz
z=0
z=d
?
F a
a=
f
af
'
'
a
dz
0
d
=f
z
f
'
d z
dz
dz
0
d
d z
dz
f
'dz
0
d
= z f
' z=0
z=d
0
z d
dz
f
'dz
0
d
f
d
dz
f
'
z dz = 0
0
d
f
d
dz
f
'
ddz
0
d
= aF a
a
a =0
= dF = 0or
f
d
dz
f
'= 0
Euler-Lagrange equation for 1D problem; its solution
has 2 constants of integration defined
from the boundary conditions, for example, = z,c1,c2
z = 0 = 0 z = d = dand
F = f ,' ,z dz0
d
f s0 0 0 f sd d d
zzz eq ad ==
z is not necessarily 0 at the boundaries!
dfs 0 0 da
=d
dafs 0 0,eq a z = 0 = 0
dfs0
d0
F a
a=
f
af
'
'
a
dz
0
d
dfs 0
d0
0 dfsd
dd
d
d z
dz
f
'dz
0
d
= z f
' z=0
z=d
0
z d
dz
f
'dz
0
d
f
d
dz
f
'
z dz
f
'
dfs0
d
z=0
0 f
'
dfsd
d
0
d
z=d
d = 0
f
d
dz
f
'= 0
Euler-Lagrange equation for 1D problem; its solution
has 2 constants of integration defined
from the boundary conditions = z,c1,c2
f
'
dfs0
d
z = 0
= 0f
'
dfsd
d
z= d
= 0and
Euler-Lagrange equation for 1D problem, soft boundary conditions (leisure time reading)
Elasticity vs Anchoring: Hybrid-Aligned Nematic Film
2. Finite (weak) anchoring
Problem: Find the director vs z in equilibrium with new
boundary conditions:
0
0
0'
FO s
z
f df
d =
=
n̂
2
12FO
df K
dz
=
21
0 0 0 02sf W = 2
12sd d d df W =
n̂
0'
FO sd
z d
f df
d =
=
0
0
d zz
d
=
0 0
0 0
0
d
d
L
d L L
=
2
012
0
d
FO
d
F Kd L L
=
0 0 0 0 0dK W d = 0 0d d d dK W d =
0
0
d d
d d
d
L
d L L
=
Finite anchoring makes the director gradients weaker
and the energy smaller, effectively increasing the cell
thickness
0 0/L K W=
/d dL K W=
0
0
d zz
d
=
Summary of hybrid aligned N film
Ld = K /Wd
0
0
d zz
d
=
2
012
d
FOF Kd
=
0
0
d zz
d
=
0 0
0 0
0
d
d
L
d L L
=
0
0
d d
d d
d
L
d L L
=
2
012
0
d
FO
d
F Kd L L
=
0 dd d L L
L: anchoring extrapolation length
Infinitely strong Finite anchoring
NB: at large scales, surface anchoring takes over, enslaving the director field, the director follows the “easy axis” prescribed by surface anchoring potential
~FOF Kl2~anchoringF Wl
00z = =
dz d= =
End of story for hybrid aligned films?
No. At submicron thicknesses, the director is unstable
w.r.t. in-plane deformations (similar to buckling)
2
2
1 1
1 2
1 1 1 1ˆdiv ~
2 2K K
R R
n
Int. J. Mod. Phys 9 2389 (1995)
1 2
1 1ˆ ˆ ˆ ˆdiv div curl
2R R= n n n n 24
24
1 2
2ˆ ˆ ˆ ˆdiv div curl
KK
R R =n n n n
Free Energy of a Nematic in an External Field
Dielectric case (no ions, no flexoelectric/surface polarization)
21
02ˆ
FO af f e e= n E
211
02ˆ
FO af f m = n B m0 = 4 107
Henry / m
Energy density should depend on two vectors, the field and the director
ˆ ˆ= n nE
E D= e0eE2 e0ea n E
2
a = e e e
0i ij jD E= e e
||
0 0
0 0
0 0
e
e e
e
=
Electric displacement:
Energy density (x 2):
FFO = f FO dV
U=const
FE = f E dV =12 E D dV
dFE = 12 E dD dV
dDz
The energy of the electric field
and its change:
Which leads to a change in the
surface charge density by
When n reorients, surface charge density changes;
to keep the voltage constant at the electrodes, one
needs to supply energy from the electric source:
dFG = dDz
A
dA= div dD dVElectric potential
div dD =divdDdD
E =
dFG = E dD dV = 2dFE
The minimum of the total bulk free energy is achieved when dFFO dFE dFG = d FFO FE = 0
D = e0eE e0e| |E| | = e0eE e0ea E n n E D= e0eE2 e0ea n E
2
21
02ˆ
FO af f e e= n E
211
02ˆ
FO af f m = n B m0 = 4 107
Henry / m
Free Energy of a Nematic in an External Field
Dielectric case (no ions, no flexoelectric/surface polarization)
Splay Frederiks Transitions
2 21
1 0
1 1ˆ ˆdiv
2 2af K = m n B n
Assuming deviations are small:
m
221
0
2
2
1 sin2
1cos
2
1B
dz
dKf a
=
221
0
2
12
1
2
1m
B
dz
dKf a
=
=0 at z=0, z=d
E-L equation: 02
22 =
dz
d
za
za sincos 21 =General solution: Boundary conditions yield and 01 =a nd =/
, the critical field for the Frederiks tra nsition Non-trivial solution at a
c
K
dBB
m
1
0
1
=
a
K
B m
1
0
11
=
, , cos ,0, sinx y zn n n z z=
Three basic geometries of Frederiks effect
a
c
K
dB
m
1
0
1
=
a
c
K
dB
m
1
0
2
=
a
c
K
dB
m
1
0
3
=
Electric field case can be treated similarly
Splay deformation
Twist deformation
Bend deformation
Heliconical director in electric field
0cE
p PE
=
Bimesogens: Ideal for electrically induced twist bend, since the bend constant is
very small
A cholesteric with a small bend-twist ratio K3 /K2 =1 and e>0 adopts an
oblique helicoidal shape in an electric field with the field-dependent pitch:
EE
0e
Color tuning: Vertical field
E
J. Xiang et al, Advanced Materials, 3014 (2015)
E
Liquid Crystals. Lecture 1.2
Optics
Support: NSF
Oleg D. Lavrentovich Liquid Crystal Institute and
Chemical Physics Interdisciplinary Program,
Kent State University, Kent, OH 44242
Boulder School for Condensed Matter and Materials Physics,
Soft Matter In and Out of Equilibrium,
6-31 July, 2015
LCs: Birefringent materials
1.7
extraordinary wave
en
Director n
=Optic axis
1.5
ordinary wave
on
Polarization E of light
Birefringence: Double refraction of light in an ordered material,
manifested through dependence of refractive indices on
polarization of light
0.2e on n n =
Birefringence revealed through pair of
polarizers: textures and LCDs
LCs: Ordinary and Extraordinary waves
/ t = E B / t = H D 0 =B 0 =D
0i ij jD Ee e= 0i ij ij jB Hm d =
||
0 0
0 0
0 0
e
e e
e
=
Light propagation in a homogeneous medium
El. displacement Mag. field induction El. field Mag.field strength
0, exp , , ...t i i t t= =E r E k r H r
=k E B =k H D 0 =k H0 =k D
Consider a plane monochromatic wave:
Eliminating mag. field H: 2 2
0 k m = D E k Ek
2 0ij i j ij jN N N Ed e =0 0
1 c
v e m= N kRefractive index “vector”
and using constitutive eq. for D:
2 2
0 0 ij j i i j jE k E k E k= m e e
Fresnel equation; Propagation of Light in LC
The three homogeneous equations have a nontrivial solution only if the
determinant of coefficients vanishes (Fresnel equation):
2 2 2 2
|| ||Det , 0z x yN N N N e e e e e = =
k
Two waves: ordinary, with
and extraordinary, with refractive index that
depends on the angle between the wave-vector
and the optic axis:
oN n e= =
,2 2 2 2cos sin
o ee eff
e o
n nN n
n n = =
on e=
en e=
optic axis
2 0ij i j ij jN N N Ed e =
Polarizing microscopy: Principle
sin
cos:Entry
A
A
Phase difference =2d
0
ne no
dntA
dntA
e
o
0
0
2cossin
2coscos
:Exit
nematic slab
n X
Y d
polarizer
aryextraordinc
dn
ordinaryc
dn
e
o
,
,:propagate toTime
Projected onto analyzer’ direction:
=
=
dntAA
dntAA
e
o
0
2
0
1
2coscossin
2coscossin
Acos t Resulting vibration:
with amplitude 2
2 2
1 2 1 22 cosA A A A A =
= oe nn
dII
0
22
0 sin2sin
nematic slab
n X
Y d
polarizer
Y
X
analyzer
Polarizing microscopy: Principle
= oe nn
dII
0
22
0 sin2sin
Dark Brushes:
regions where n is || or to polarizers
0; / 2; ... =
P A
Polarizing microscopy: Principle
Polarizing microscopy:
Degeneracy of two director fields
= oe nn
dII
0
22
0 sin2sin
P A
Polarizing microscopy: Degeneracy of two
director fields: Radial or Circular?
= oe nn
dII
0
22
0 sin2sin
P A
PM problem: Two orthogonal n fields
Solution: optical compensator (quartz wedge, Red plate, etc.)
en
on
en
on
en
on
en
on
en
on
Lower retardation,
yellow of the
first order of
interference
Higher retardation,
blue of the
second order of
interference
Radial and circular patterns produce different pattern
of interference colors when the compensator is added
Ultimate Compensator: LC PolScope
R. Oldenburg, G. Mei, US Patent 5,521,705
YK Kim et al, J Phys Cond Phys 25, 404202 (2013)
LC PolScope image of chromonic: shows both
the in-plane director and retardance
(a) T = 32.9 oC (c) T = 30.6 oC, t = 0 s (d) T = 30.6 oC, t = 156 s (b) T = 30.7 oC
0 nm
136 nm
50 μm
1
2
1 2
1
2
1
2
1 2 1 2 Retardance:
; e o
eff
d n planar n n n
d n titled director
= =
=
PolScope creates a map of the orientation of the optic axis in the sample and of the local value of the
optical phase retardation;
Nematic
Isotropic
Polarizing microscopy
= oe nn
dII
0
22
0 sin2sin
Interference colors: 0I I =
Visible when 0~ 1 3e od n n
Limitation: 2D image
Why the interference colors change?
= oe nn
dII
0
22
0 sin2sin
Limitation: 2D image
= oe nn
dII
0
22
0 sin2sin
Derived in approximation of planar director,
and constant thickness d ˆ , ,0x yn n=n
BTW! When , how does the texture look like? , , 0, 0,1x y zn n n =
Limitation: 2D image
2 2
0
0
sin 2 sin , , , , , , ,
d
effI I f x y z n x y z d x y dz
= n
When and , one deals with: , ,x y z=n n ,d d x y=
Limitation: 2D image
??????????????
Confocal Microscopy: 3D image of 3D
Usual
microscope
objective Confocal
microscope
Voxel=3D pixel
~1 mm 3
objective
Pinhole #1 Pinhole #2
(Minsky, 1957)
Computer 3D image Scanning
PMT PMT
Light
source
Light
source
Confocal Microscopy: Principle
Pinzhole #1 Pinhole #2
Fluorescence Confocal Microscopy
3D image of concentration (positional distribution) of
fluorescent dopant….
Fluorescent tag increases contrast of tissues
Living cell
PMT PMT
Light
source
Light
source
1980 M. Petran and A. Boyde
Two distinctive features:
1. Anisometric fluorescent dye aligned by LC
2. Polarized light
….but we are interested in orientations
rather than in concentrations…
Fluorescence Confocal Polarizing Microscopy
FCPM: Fluorescent anisometric dyes
N,N'-Bis(2,5-di-tert-butylphenyl)-3,4,9,10-
perylenedicarboximide)
BTBP
t
BTBP
350 400 450 500 550 600 650
Wavelength, nm
Ab
sorp
tion
, arb
.u
Ar Laser Excitation at
=488nm
Flu
ore
scen
ce E
mis
sion
, arb
.u.
A | |
A |
F | |
F |
0.01 % of BTBP in ZLI-3412
FCPM: Anisotropic Fluorescence
Fluorescence signal = f (orientation of dye)
n
n
P
minimum signal
n
n
P P
a
I ~ cos4a
n
n
P P
maximum signal
I. Smalyukh et al, Chem Phys Lett 336, 88 (2001)
FCPM: Anisotropic Fluorescence
Objective:
40X NA0.6;
60x NA1.4
Sample
Focal plane
PMT
Light source
Illuminating aperture
Confocal aperture
Beam splitter
filter
488 nm,
100 nW
510nm-550 nm
z
Polarizer/Analyzer
mznnn oe m10;1.0/||
mz m1~
/e oz n n z n
I. Smalyukh et al Chem Phys Lett 336, 88 (2001)
FCPM: Frederiks Effect
Relative Intensity scale:
Imin Imax
I ~ cos4a
P 0V
a=0
d=15mm
3V
a= f(z)
P
d=15mm V
I. Smalyukh et al Chem Phys Lett 336, 88 (2001)
FCPM: Planar Cholesteric
P
8 mm
Pn/2 <1 (to avoid the Mauguin regime)
I. Smalyukh et al, Phys. Rev. Lett. 90, 085503 (2003)
Summary: What have you learned Liquid crystals: Orientationally ordered media
Thermotropic (t-driven), lyotropic (c-driven), chromonic (t,c-driven)
Uniaxial nematic, twist bend, cholesteric, smectic, columnar, dramatic dependence on molecular structure…
Orientational elasticity vs surface anchoring Frank elastic constants ~ 5 pN
Equilibrium director defined by boundary conditions (anchoring), elasticity, and external field
As the system become larger, anchoring imposes stronger restrictions on the director
Frederiks transitions @ heart of modern LCDs
Optics LCs are birefringent; ordinary and extraordinary waves
Polarizing microscope: 2D image of 3D sample
Fluorescence confocal polarizing microscopy: 3D image of 3D orientational order
Cholesteric structure of DNA in chromosomes:
Bouligand arches
Chromosoma (Berl.) 24, 251--287 (1968)
La structure fibrillaire et l'orientation des chromosomes
chez les Dinoflagellés
Y. BOULIGAND, M.-O. SOYER et S. PUISEUX-DAO
89
Double twisted cylinders stabilized by a 3D network of topological defects –
disclinations.
BPII
BPI
100 nm
Main problem for applications:
Temperature range of BPs is narrow,
~1oC
One approach: Polymerization
H. Kikuchi et al, Nature Mat. 1, 64
(2002)
0ˆ || Zn
n̂
X
Y
Clolesteric: 1D twist; Blue phases: 3D twist
90
Polymer mesh formed in BPII
Can be refilled with N for
electro-optic applications
J. Xiang et al, APL 103, 051112
(2013)
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