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

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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

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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

I. planar, εa < 0, σa > 0

anisotropic

MBBA: - 0.13

Ginzburg-Landau description works

At threshold, increasing f (planar, εa > 0, σ a < 0):

OR NR

TW (non-stand.) DR

n

II. homeotropic, εa < 0, σa > 0

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

III. Homeotropic, εa > 0, σa < 0 ( truly isotropic)

Direct transition to isotropic EC

Direct transition to EC -> SM

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

IV. planar, εa < 0, σa < 0: no standard pattern (conductive)

- PR or oblique

- nz= 0, no shadowgraph

- ny (?) oscillates

- Uc~ d, f

- qc is d indep.

Experimental:

Dielectric mode! (LK)

I and II- conductiveIII and IV - dielectric

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)

- dielectric mode at low f- SM + flexoelectricity- why is DM more sensitive to flexo, than CM?

planar, εa < 0, σa < 0: no standard pattern (conductive)