Download 3. Liquid crystals

3. Liquid crystals
3.1.
Thermotropic liquid crystals
3.1.1
3.1.2
Calamitic liquid crystals
Discotic liquid crystals
3.2.
Phase behavior
3.2.1
3.2.2
3.2.3
3.2.4
3.2.5
Positional order
Anisotropic properties
Defects in LCs
Orientational order
Theory for orientational order
3.3.
Applications
3.3.1
3.3.2
3.3.3
3.3.4
The Fréedericksz transition
The twisted nematic cell
Thin-film transistor displays
Polymer-dispersed liquid crystals
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3.2 Phase behavior
3.2.1 Positional order
radial distribution functions of positional order g(r):
g (r )r 2 dr : probability that molecule is located
in the range dr at distance r
from another molecule
crystal
Figures from Hamley
smectic liquid crystal
isotropic liquid or
nematic liquid crystal
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Positional distribution function
crystal:
• peaks due to periodic spacing of molecules
• broadened by thermal fluctuations
liquids:
• no long-range positional order, only local packing of molecules
→ weak oscillations in g(r)
→ decay as exp(-r/ξ), ξ: positional correlation length
nematic LCs:
• same g(r) as liquid
smectic LCs:
• periodic in 1D (normal to layers)
• but only quasi-long-range ordering
• slow decay as r-η, η: temperature-dependent exponent, 0.1-0.4
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3.2.2 Anisotropic properties of liquid crystals
anisometry of mesogens
→ induction of dipole moment by electric field
→ orientation of dipole in direction of field
→ response to external fields (electric and magnetic)
polarization induced by electric field:
r
r
P = ε 0 χE
P:
ε0:
χ:
E:
polarization
permittivity of vacuum
electric susceptibility
electric field
polarization different if E parallel or perpendicular to director:
 χ⊥
 Px 

 
 Py  = ε 0  0
 0
P 
 z

0
χ⊥
0
0   Ex 
  
0  ⋅  Ey 
χ||   E z 
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Anisotropic polarization
r
t r
P = ε0χ ⋅ E
r t r
P =ε ⋅E
electric susceptibility tensor (second rank)
anisotropic permittivity
ε ⊥ , ε ||
nematic LC having
permanent electric dipole
parallel to long axis
of molecule
left: E = 0:
no net dipole moment
top right: E || n
→ large induced dipole
bottom right: E ⊥ n
→ smaller degree
of polarization
→ ε|| > ε⊥
Figure from Jones
alignment along electric field lowers energy
→ electric fields can align nematic liquid crystals
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Optical anisotropy
nematic, smectic, columnar LCs: optically anisotropic
refractive indices along director, n||, and
perpendicular to director, n⊥, different
→ birefringence:
∆n = n|| − n⊥
→ use in LC displays
→ milky appearance of nematics
n||
n⊥
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Anisotropy near solid substrate
planar or homogeneous orientation:
director parallel to surface (usual case),
induced e.g. by crystalline structure of surface
or by polyimide rubbed by velvet cloth
(aligned polymers)
polyimide
homeotropic:
director perpendicular to surface,
e.g. when substrate coated by surfactant
→ dark field in polarizing optical microscopy
sensitivity to alignment:
broken symmetry of nematic phase
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Alignment by flow
if applied stress leads to deformation
that pertubs long-ranged order of system
→ increase in elastic energy opposing deformation
if not: flow → viscous response
shear stress
→ long-ranged orientational order unchanged
→ liquid-like flow
three fundamental
types of deformation
splay deformation
→ increase in elastic energy
3.2.3 Defects in liquid crystals
identification in polarizing optical microscopy
by means of defects
disclinations:
line defects unique to LCs
discontinuity of orientation of director field
disclinations in nematic phase
→ Schlieren texture
director orientation:
rˆ
n = [cos(θ (r ) ), sin (θ (r ) )]
r
 y
r = ( x, y ) → θ = s ⋅ arctan  + θ 0
x
disclination
→ increase of elastic energy
because of distortion
of director field
s = +1/2
s = -1/2
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3.2.4 Orientational order
z
definition of orientational order parameter
β
n̂
orientational distribution function f(β):
probability for molecules to be oriented
at angle β with respect to the average
 U (β ) 
f ( β ) = Z exp −

k
T


B
−1
U(β): anisotropic
intermolecular potential
y
x
Z: orientational partition function
 U (β ) 
Z = ∫ exp −
d (cos β )
 k BT 
θ=β
for nematic phase:
f(β) = f(π- β)
Figure from Jones
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Definition of orientational order parameter
should be one in ordered phase
and zero in isotropic phase
only polar angle β relevant
→ use a function of cos(β)
nematic phase:
β and π-β are equally likely
→ use a function of cos 2(β)
average of cos 2(β) for isotropic distribution is 1/3
→ use order parameter
P2 =
3
1
cos 2 ( β ) −
2
2
K : taken over orientational distribution function
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Calculation of order parameter
0
P2 = 
1
for isotropic phase
for completely oriented phase
average 〈…〉 means:
P2 = ∫ P2 (cos β ) f ( β )d (cosβ )
complete specification of orientational ordering:
knowledge of
P2 , P4 , P6 , K required
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Non-cylindrically symmetric mesogens
second-rank tensor needed to describe orientation
of phases with inversion plane of symmetry
e.g. N, SmA,…
orientational order tensor (Saupe matrix):
relates orientation of vector in molecular frame (x,y,z)
to that in reference frame of director
 S xx
t 
S =  S yx
S
 zx
S xy
S yy
S zy
S xz 

S yx 
S zz 
for cylindrically symmetric phases
of cylindrically symmetric molecules:
S zz = P2
other elements:
3 r ˆ r ˆ 1
Sij = n ⋅ i n ⋅ j − δ ij
2
2
( )( )
i, j = x, y, z
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3.2.5 Theory for orientational order
two types of theories:
Jones: Soft Condensed Matter
Maier-Saupe theory:
• orientational ordering in nematic phases
• long-range attractive interactions
Onsager model:
• short-range steric interactions
• excluded volume for rod-like particles
→ nematic phase at high volume fractions
Maier-Saupe theory:
long-range attractive contributions to intermolecular potential
• maximized when mesogens are aligned
• important for small molecules
short-range repulsive contributions to intermolecular potential
• better packing of aligned mesogens
• most important for long, rigid molecules
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Calculation of free energy
entropy loss when molecules become oriented
S orient = −k B ∫ f ( β ) ln f ( β )dΩ
in isotropic state: f ( β ) =
1
4π
→ ∆S = − k B ∫ f ( β ) ln[4πf ( β )]dΩ
Maier-Saupe theory:
phenomenological assumption that
energetic interaction between molecules
is quadratic function of order parameter
S2
→ ∆F = −u
+ k BT ∫ f ( β ) ln[4πf ( β )]dΩ
2
find f(β) which minimizes ∆F
in self-consistent way!
Jones: Soft Condensed Matter
u: parameter related to strength
of favorable interaction
between molecules
S: order parameter,
defined in terms of f(β)
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Free energy as function of order parameter
free energy as function
of order parameter S
for various values of u/kBT
small values of u/kBT:
minimum free energy at S = 0
→ orientational entropy term dominates
→ equilibrium state is isotropic
at higher u/kBT:
minimum of free energy for non-zero value of S
→ equilibrium state is nematic
critical value: u/kBT = 4.55
Figure from Jones
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Nematic-isotropic phase transition
calculation of S as a function of u/kBT
→ character of transition
→ at u/kBT = 4.55 discontinuous change of S from 0 to 0.44
→ first-order phase transition
but: minimum of free energy shallow
→ fluctuations at transition important
→ weakly first oder transition
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Figure from Jones
Comparison with experiment
assumption: u independent of temperature
e.g. because entirely due to van der Waals forces
order parameter S as a function
of temperature for p-azoxyanisole
as measured using refractometry and
diamagnetic anisotropy (symbols)
and from Maier-Saupe theory
(phase transition temperature adapted)
Figure from Jones
good agreement:
• small degree of order at transition
• good description of shape
discrepancies:
• temperature-dependence of u due to excluded volume interactions
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• neglect of fluctations – important near phase transition
McMillan theory – ordering in the SmA phase
describes nematic-smectic A phase transition
coupling orientational – positional order


σ = cos 2π
z  3
1
2
 cos ( β ) − 
d  2
2
d: period of density modulation
dependence of anisotropic intermolecular potential
on intermolecular separation: Gaussian function
 πr0  2 
α = 2 exp  −  
 d  
Hamley
r0: range of attractive interaction
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Predictions of McMillan theory
P2, σ: orientational and translational order parameters
SmA-I 1st order
SmA-N 1st order
N-I 1st order
SmA-N 2nd order
N-I 1st order
large α:
• d large, smectic ordering favored
→ first-oder transition Sm-I
lower α:
• nematic phase between smectic and isotropic
• transition smectic-nematic first- or second-order
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Twisted nematic (TN) cell
TN cell:
light is rotated by 90°
super twisted nematic
(STN) LCD:
light can rotate up to 270°
http://dictionary.zdnet.com/definition/LCD.html
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