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Order Parameters and Defects in Liquid Crystals
Ferroelectric phenomena in liquid crystalsJune, 2007
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Liquid Crystal-substance with a degree of crystalline order that remains in a liquid state
• Simple picture- long thin molecules
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Energy to describe static stable configurations
Made up of competing energies• Molecular packing (molecular alignment)• Thermal effects (randomize alignment)• External stresses (mechanical, electrical..)
Temperature dependent-different terms become dominant at different temperatureselectr
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Isotropic
Nematic
SmecticT
• Nematic phase-energy seeks local alignment of long axes.
• n(x) is the “local average”of long axes.• |n(x)|=1
n(x)
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n(x) is a unit vector field
• Molecules do not have a head and tail as n(x) does.• In problems where this does not lead to inconsistencies n(x) is the
simplist way to describe a nematic liquid crystal.• n(x)-director field
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Oseen Frank Energy
• Energy records the cost of distortions away from n(x)=const.
• Written in terms of pure splay, twist, and bend. These have K1 ,K2, K3 as coefficients respectively
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Pure Distortions
Splay Bend Twist
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ionapproximatconstant one 0,K, case, Special 4321 ==== KKKK
vector identity
=⇒ )(nFFelectronic-
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Boundary Conditions
Ω∂= on 0nn1)
2) 2))(( xencFb ⋅= ∫ Ω∂
1) Strong anchoring problem2) F=FF+Fb weak anchoring
If c<0 then n(x) tends to be parallel to e(x). e(x) is the easy axis.
If n(x) or e(x) are parallel to the boundary normal
we are promoting “homeotropic” boundary values.
1, 0 =n
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If c>0 then n(x) tends to be e(x) on the boundary. If e(x) is parallel to the boundary normal then n(x) tends
towards tangential boundary values.
⊥
ThmAssume is a smooth surface, n0 is smooth, K1 ,K2, K3 >0.Then there is a minimizer for F(n) in the class
}||,|,:|{ 20
2 ∞<∇=→Ω= ∫ ΩΩ∂ unuSuuA
Ω∂
.nonemptyisAprovided
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n(x) may have singularities, “defects”
• ex . n(x)=x/|x|“Hedgehog”. n(x) has homeotropic b.v. and
a defect at one point. Minimizers from the theorem can not be singular on a curve. Line singularities in liquid crystals do exist, “disclination lines”.
Try n(r, ,z)=er and get
RB=Ω
θ
∞=∇∫ 2|| nelectr
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Cladis and Kleman consider :
withzr ererrn )(sin)(cos)( ϕϕ += 0)( =Rϕ
R
rz
r
zn(r)
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rdrr
dxdynF RBR
)cos)'((2|| 2
22
02 ϕϕπ +=∇= ∫∫
They prove the minimizer is )(tan22
)( 1
Rrr −−=
πϕ
R
Solution is not singular. It escapes to the third dimension.
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f( ) probability density that the director of molecules near x are within an angle
Of n(x).f0( ) =s(x) is the the isotropic order parameter .Measures how concentrated f is near =0.
π41
ϕ
))((cos23)(
20
2∫∫ −=S
ffxs ϕ
ϕ
ϕ
Introduce an order parameter that allows n to “melt’ near r=0
s=1 strongly nematic, s=0 isotropicelectronic-
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Ericksen considers a nematic described by the pair (s,n)
2s
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}0)(,0)(,)(sin)(cos),({ 0 >==+ sRsRererrs zr ϕϕϕ
Mizel, Roccato, and Virga prove that this has a minimizer
( ) in the family
of the form 0)0(~,00)(~,0~ =>>≡ sforrsϕ
•
ns ~,~
The solution is allowed to melt into the isotropicn phase at r=0,
has finite energy, and has a disclination line along the z axis.
1≤kIf
.2
)0(~0)0(~1 πϕ =>> andshassolutionthethenkIf
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uniaxial
biaxial
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Smectics
∫=π
ρ2
)(x
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Molecules locally align and in addition form layers
ν
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• Layer normal • Density modulation
• , = layer thickness• If then layers exist - smectic phase• If then no layers - nematic phase
01 >ρ
)2cos()( 10 xd
x ⋅+= νπρρρ
dq π2=
ν
d
01 =ρelectr
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• If - smectic A• If not - smectic C• Complex order parameter
• nematic, smectic
n||ν
)()( xiex ϕρψ =
0=ψ
*1 cos2 ψψϕρ +=
0≠ψ
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Energy
III II I
A>0
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• I prefers smectic phase• II locally the level sets of are
the layers,
• III penalizes phase transitions
11 =ρ
,|| nϕ∇ q≈∇ || ϕ
ϕ
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0
r
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Rffrdr surfR +∫ 0
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reasy
iiii
eekSK
== ,2
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Cholesteric Phases
τπ
=
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Chirality introduces a stress onto the liquid crystal that affects its phase transition.
τ lowers the transition temperature
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Smectic C
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Set )(xie ωψ =
Then
Now let xk ⋅=ω and let n= const , then
k| k
n⋅
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This minimizes for =2|| k
and =⋅
=||
cosknkθ
θ
||2kπ
||2kπ
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dxFF FSm )(F +∑
= ∫
dxFF FSm )(F +∑
= ∫electr
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Boundary value problem for F.M. Calderer and J.Park
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