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Anisotropic and isotropic electroconvectionCollaborators: L.Kramer, W.Pesch (Univ. Bayreuth/Germany and N.Eber (Inst. Solid State Phys./Hungary)
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OR(low f)
NR
(high f)
I. (anis.) III. (isotr.)II. (interm.)
+ +
x
y
H
planar homeotropic homeotropic
ELECTROHYDRODYNAMICS OF NEMATICS
€
F = FK + FE
n × S = 0
ρd
dtv =∇T + f
∇v = 0
∇(ε E) = ρ e,∇ × E = 0
∇(σ E + ρ e v) + ∂tρ e = 0
- free energy density
- balance of torques
- equation of motion
- incompressibility
- equation of electrostatics
-charge conservation
€
FK = 12 K11 ∇n( )
2+ K22 n ∇ × n( )[ ]
2+ K33 n × ∇ × n( )[ ]
2
{ }
( ) ;221 EnF aoE εε−=
kikjkjkiijijjijikllkij AnnAnnANnNnnnAnnt 654321 αααααα +++++=
STANDARD
MODEL
(SM)
K11... K33, α1....α6 , εa, σa, ρ
Material parameters:
Boundary conditions: planar or homeotropic
Relevant: alignment + sign of εa and σa 8 combinations
I. planar, εa < 0, σa > 0 anisotropic
II. homeotropic, εa < 0, σa > 0 intermediate
III. homeotropic, εa > 0, σa < 0 isotropic
IV. planar, εa < 0, σa < 0 non-standard
SM
H =0 H ≠0
NR
OR
H drives between semi-isotropic and anisotropic
- soft <-> patterning mode
- direct transition to STC
- AR-s
- chevron formation
- defect glide
- 2 LP-s
Homeotropic alignment (standard, semi-isotropic)
τ∂t A = [1+ ∂x2 + (∂y − iϕ )2 − A
2+ iβ yϕ ,y ]A
∂tϕ = ∂y2ϕ + K3∂x
2ϕ − ε −1H 2ϕ + Γ[−iA*(∂y − iϕ )A + c.c]
(A.Rossberg, L.Kramer)
Voltage
Frequency
conductive dielectric
LPOR
NR
TW
AR
qc ~d-1
qc ≈ (μ )m -1
fc
UC ( )= .d const U
C~d
defects
dRchevrons
Freedericksz
theor. exp.
OR
NR
nonlinear regime: hard squares
:
exp. theo.
f
At onset:
Swift-Hohenberg eq. (W.Pesch, L.Kramer, B.Dressel)
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soft squares
not reproduced
- PR or oblique
- nz= 0, no shadowgraph
- ny (?) oscillates
- Uc~ d, f
- qc is d indep.
Experimental:
Dielectric mode! (LK)
1. Dielectric mode for MBBA (planar, εa < 0, σa > 0)
2. Dielectric mode for MBBA (planar, εa < 0, σa < 0) - no pattern
Flexoelectricity
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Flexoelectricity
Pfl =e1n(divn)
Pfl =e3[n×(curl n)]
Ffl =−E Pfl
Effect on the roll angle, only for d.c. (only in conductive)
3. Dielectric mode for MBBA (planar, εa < 0, σa < 0) + flexoelectricity
finite threshold!
obliqueness!
e1- e3= 1.34e1+ e3= -7.84
4. Dielectric mode for MBBA (planar, εa < 0, σa < 0) + flexoelectricity
e1- e3= 2.68e1+ e3= -7.84
(A.Krekhov, W.Pesch)