Polar flexoelectric deformations and second order elasticity in nematic liquid crystals

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Polar flexoelectric deformations and second order elasticity in nematic liquid crystals

A.I. Derzhanski, H.P. Hinov

To cite this version:

A.I. Derzhanski, H.P. Hinov. Polar flexoelectric deformations and second order elasticity in nematic liquid crystals. Journal de Physique, 1977, 38 (8), pp.1013-1023. �10.1051/jphys:019770038080101300�.

�jpa-00208654�

POLAR FLEXOELECTRIC DEFORMATIONS

AND SECOND ORDER ELASTICITY IN NEMATIC LIQUID ^CRYSTALS

A. I. DERZHANSKI and H. P. HINOV

Institut de

Physique

^des

Solides,

Académie des Sciences de

Bulgarie,

^Sofia

1113, Bulgarie (Reçu

^le

29 juin ^1976,

^{revise le 25 avril}

1977, accepte

^{le 2 mai}

1977)

Résumé. ²⁰¹⁴ On a étudié théoriquement l’influence de l’élasticité de second ordre

(K13~

^{0) sur les}

déformations flexoélectriques

polaires

^{dans une couche} homéotrope

nématique

(cas A) ^{et dans une}

couche planaire nématique (cas B). ^Le champ électrique ^est

perpendiculaire

^{à la couche,}

l’anisotropie

diélectrique peut ^être

positive

^ou négative. ^Les conditions de seuil exactes sont obtenues numérique-

ment à

partir

^des équations pour différentes valeurs des énergies de surface

anisotropes WS1

^et

WS2,

du coefficient flexoélectrique total e1z ⁺ e3x ^et de K13. ^Pour

K13~

^{0 et} quelques ^valeurs

critiques

de

WS1 et WS2

les déformations

apparaissent

soudainement; l’énergie nécessaire à la création des déformations

flexoélectriques

(cas ^{A et} B) ^est considérablement modifiée.

Abstract. ²⁰¹⁴ The influence of second order elasticity

(K13 ~

0) ^{on the} flexoelectric polar ^defor-

mations in a

homeotropic

(case A) ^{and a} homogeneous (case B) ^nematic layer ^with

negative

^and positive dielectric anisotropy ^{and an} electric field normal to the layer is considered theoretically ⁱⁿ

this paper. The ^exact threshold condition is obtained, transcendental

equations

^{are solved} numerically

for different values of the anisotropic interfacial energies

Ws1

^and

Ws2,

the total flexoelectric coeffi- cient e1z + e3x ^and K13. ^For K13 ~ ^{0 and some values of}

Ws1

^or

Ws2

(critical values), the deformations start with a

jump.

^The required electric energy for the appearance of the flexoelectric deformations

(case ^{A and} B) is corrected considerably.

Classification

Physics ^Abstracts

7.130

1. Introduction. ^- The fundamentals of flexo- electric

theory

^of

liquid crystals

^{were laid}

by

^R.

Meyer [1].

^He

pointed

^out that nematic and cholesteric

liquid crystals

^with

asymmetric

^and

polar

^molecules

may deform under the influence of external electric fields ^- a linear flexoelectric effect ^- and vice versa

in the case of deformation :

they

^are

polarized.

The

subtlety

^and

diversity

^{of this}

phenomenon

^was

shown once

again by

^the concept ^{of a}

polar

^flexo-

electric effect in the case of interaction of a homeotro-

pic

^nematic

layer

^{with As 0 and}

asymmetric

^ani-

sotropic

interfacial _energy

(zero

^and

infinity)

^{and a}

constant electric field

parallel

^{to the} initial orientation of the

liquid crystal proposed by

W. Helfrich

[2].

In these conditions he

pointed

^out that the flexo- electric deformation due to the interaction between the

applied

electric field and the flexoelectric

polari-

zation

parallel

^to

it,

may exists for

only

^one

sign

^{of the}

applied

^electric field. The critical threshold

voltage

obtained

by

Helfrich has the form :

where

K33

is the Frank elastic coefficient of bend and _{e 1z}

+ e3x

is the total flexoelectric coefficient.

W. Helfrich’s concept ^{for a}

polar

flexoelectric effect

(case A)

is elaborated _upon

theoretically

ⁱⁿ

this _paper for different values of the

anisotropic

interfacial _energy

WS1

^and

WS2

^{and for different}

^signs

of the dielectric

anisotropy.

The elastic surface terms due to the second derivative of the director n

(unit

vector

following

the orientation of the

liquid crystal molecules)

^with respect ^{to the} coordinates

(elastic

coefficient

K13 = 0)

^are included in the surface _energy.

This is based on the theoretical conclusions drawn

by

^J.

Nehring

^{and A.}

Saupe [3]

^{that the} second-order elastic deformation _energy is comensurate with the first-order elastic deformation _energy under the condition that all elastic coefficients are of the same

order

[4].

^On the other hand the

polar

flexoelectric effects

require

small surface _energy of interaction of the

liquid crystal

molecules with one of the surface walls which can be corrected to a

great

^extent

by including

^the surface elastic deformation _energy due to the second derivatives as well.

A new case B is also considered where the

polar

flexoelectric effect ^- the interaction between a homo- geneous nematic

layer

^with

negative

^and

positive

dielectric

anisotropy

^{and a constant electric field}

normal to the

layer

is demonstrated as well.

(A

^unified

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:019770038080101300

description

of flexoelectric effects for various _expe- rimental

geometries

^{- 16 different cases - for}

K13

^{= 0 is}

given by

^A. Derzhanski and A. Petrov

[5].)

The differential

equations

^{and the}

boundary

^condi-

tions are obtained after minimization of the func- tional of the

liquid crystal

^free energy with respect ^to the deformation

angle 8(z).

^The solutions demon- strated a

deforming

action of the linear flexoelectric surface torques ^{as well as a}

deforming (case A,

As

0 ;

^case B, ^As >

0)

^{and an}

orienting (case

^A,

As >

0 ;

^case B, ^As

0)

action of the bulk dielectric torque.

The character of the

boundary

conditions

brings

about a threshold

requirement

for these deformations.

The threshold transcendental

equations,

^obtained

from the

vanishing

^condition

O(z)

--> 0 are solved

numerically

^{for the most}

interesting

^cases.

Taking

the second-order

elasticity

^{for some values}

of the

anisotropic

interfacial _energy,

depending strong- ly

^on the value of the total flexoelectric coefficient e ,- ⁺ e3x

(Critical

Interfacial

Energy)

^{into account}

brings

^{about an}

abrupt

threshold with considerable deformations. Two

examples

where the flexoelectric interfacial _energy is

exactly equal

^{to the}

anisotropic

interfacial _energy

(infinite anisotropic

interfacial energy ^at the other

wall)

for ±

K13

^{were solved. The}

sign

of the elastic coefficient

K13

^{is of} great

importance

for the type of deformations in the

layer.

^For

example

for case

A,

^As

0,

^the

negative sign

of this coefficient for

large

^or small values of the

anisotropic

interfacial energy

(differing

from the critical interfacial

energy)

ensures deformations

changing smoothly

^{with the}

electric field. The

positive sign of K13

for small values of the

anisotropic

interfacial _energy

leads,

^{above a} certain value of the electric

field,

^to smooth deforma- tions up ^to the maximum

possible.

The influence of

the

sign

of the elastic coefficient

K13

may be

replaced by

^a

change

^of

sign

^{of the} dielectric

anisotropy

^{of the}

liquid crystal.

A characteristic feature of the case A, ^As > 0 and case B, ^{As 0} is the absence of flexoelectric deformations for

relatively

small values of the total flexoelectric coefficient _{e lz.} + e3x.

The exact solution of the

problem

allows for the effective

preliminary assignment

^of

appropriate

^sur-

face

energies

of interaction of the

liquid crystal

molecules with the walls attainable in the

experiment.

The inclusion of the dielectric

anisotropy,

^{as noted}

by

^Helfrich

[2],

^{on the one hand}

changes

^the

polarity

with mixed-flexoelectric and dielectric deformations

(case A,

^{As 0 and case}

B,

^As >

0)

and the existence of a threshold for both directions of the electric

field,

but on the other hand it lowers the

voltage required

for the effect. From here it follows that the

operating voltage

of this effect can

change

^{with the}

change

^{of As.}

In the cases where

only

flexoelectric deformations exist

(case A,

^As >

0,

^case

B,

^As

0), important

relations between all constants of the

liquid crystal (with

^a

preliminary

measurement of the values of the surface

energies)

^may be obtained.

2. Case

A, theory

^As

0,

^{As =} _sll ^- 8ol8 ^- The interaction between a thin

homeotropic

^nematic

monocrystal

^{with As 0} and flexoelectric

properties (e lz :f:.

^0,

e3x = 0)

^{and finite}

anisotropic

^elastic energy of interaction of the interface molecules with the cell walls

WS1

^and

WS2,

^{and a constant} electric field

parallel

to the initial orientation is considered

theoretically

in this section.

(The conductivity

^{effects are}

neglected.)

The free _energy of this nematic

layer

written in an

invariant vector form

following Nehring

^and

Saupe [3]

and

Meyer [ 1]

^{reads :}

It is a sum of the elastic _energy

(with

^an accuracy to the second derivatives of the director n with

respect

to the

coordinates),

^the flexoelectric _energy and

approximate

dielectric _energy, accurate to a

large

extent for nematic

liquid crystals

with small dielectric

anisotropy ((Asls_L) 1 ).

^The

following physical

quan- tities in CGSE units are introduced in

[1] : K11 K22

and

K33

^are elastic coefficients of

splay,

^{twist and bend}

respectively;

e1z, and e3x ^are flexoelectric coefficients of

splay

^and

bend; sll

^{|| and El are the}

permittivities

^of

the

liquid crystal along

^the

liquid crystal optical

^axis

(the long

axis of the

liquid crystal molecules)

^and

along

^the

normal ; Kl3

is the second-order elastic

coefficient ; WS1

^and

Ws2

^{are the}

anisotropic

interfacial

energies

^{of the}

liquid crystal (theoretically

^between

zero and

infinity);

V1.2 ^are the normals to the nematic

layer

^surfaces.

The introduction of a

rectangular

coordinate system OXYZ is most convenient for the

problem

^{thus set}

where the X axis coincides with the lower interface

surface,

^{Z is} the normal and Y

brings

^the system ^to the

right

orientation.

The constant electric field E

applied along

^{+ Z}

is denoted

by TE

^and

along -

^Z

by JE.

^The

anisotropic

interfacial

energies

^{ensure that the}

preferred

^orien-

tation of the

liquid crystal

^is

homeotropic (WS1

^{of the}

lower surface and

Ws2

^{of the upper}

^surface, Fig. 1).

We suppose them to be

substantially depending

^{on z.}

FIG. 1. ^- Polar flexoelectric deformations in nematic layers ^with

a negative dielectric anisotropy : upper cells (case A), lower cells

(case B).

Substituting

^the components ^{of the vector field} nx ⁼ sin

0,

_nZ ^{= cos 0}

and Ez

^{= E in the}

expression

for the free _energy

[1]

transforms the latter into a

functional of the deformation

angle O(z). Varying

^this

functional with respect ^to

8(z) (HEIO(z)

is transformed into a functional of the inverse function

z(O)

^{and is}

varied as a functional with movable

boundaries) gives

^the differential

equation describing

^{the defor-}

mations in the

layer :

where

and the rather

complicated boundary

^conditions

for the values of

O(z)

^{at the two} interfacial surfaces :

where

dO/dz =

^0, d is the thickness of the

liquid crystal layer.

The _presence of the second derivative in the

boundary

conditions is a result of the elastic surface _energy, due to second-order

elasticity (Kl 3 =A 0).

^{It is} clear that this _energy introduces

essentially

^{nonlinear terms in}

the

boundary

conditions and

complicates

^the

problem

very much. Another characteristic feature of these boun-

dary

conditions is their manifest

dependence

^on the electric field E.

(Similar boundary

conditions for

K13

^{= 0}

were first obtained

by

^{A. G. Petrov}

[6]

^{in a} theoretical

investigation

^{of low}

frequency

flexoelectric oscillations and

by

Derzhanski and Petrov

[7]

^when

investigating

^low

frequency

flexoelectric oscillations and deformations for

asymmetric

interfacial _energy without

taking

^{into account} the second-order

elasticity.)

A

quadratic equation

^for

d8/dz

is obtained for the two surfaces

(see

Derzhanski and Hinov

[8])

^from

(2)

and

(3)

where

The

following important

conclusions can be drawn from

(4) :

the value of the derivative of the deformation

angle O(z)

^at the surfaces

depends

^not

only

^on the flexoelectric _energy

(elz

⁺

e3,)

^{E but on} the dielectric _energy

as well

(I

^As

1/4 n) ^E2. (This

^{case was touched upon}

^by

Barratt and Jenkins

[9].)

^The

sign

of the coefficient b is of great

importance.

^It is evident that it is determined from the relation between the flexoelectric and the

anisotropic

interfacial _energy. The

sign

^{of b at the two} surfaces dictates the

sign

^of

d8/dz

^with respect ^{to that} of

O(z)

^at the interfaces which determines the type of deformation in the

liquid crystal layer

^under

investigation.

Two types of deformation _may be realized in the

problem

^under consideration with a

special

^selection

of the interfacial

energies.

The monotone deformation is determined

by

^the

expressions :

where

where

The interfacial

angles

^are smaller than

Tr/2

because of the unidirectional action of the volume and surface torques

(see

^Barratt and Jenkins

[9]).

This solution

requires

^little

anisotropic

interfacial _energy at the upper wall ^- weak

anchoring

^{of the}

liquid crystal

molecules with this wall

((ei

⁺

e3x) Er > WS2’ ^{(el +} e3x) > 0,

Fig. 1).

The deformation with a maximum in the

liquid crystal layer

is determined

by

^the

expressions :

where

In the cases

WS2

>

(ei

⁺

e3x) Et

^and

WSl ^(el. + e3x) ^Et, dOr ,,,Idz change signs

which leads to the monotone deformation

changing

^{into a} deformation with ^a maximum in the

liquid crystal layer

^{and the defor-}

mation with a maximum in the cell turns into a monotone one.

The differential _eq.

(2)

^{and the}

boundary

conditions

(3)

^{allow, for}

WS1 :f:. WS2 =1=

^{00 the}

elementary

^solu-

tions 0=0 and 0 ⁼

n/2

^{as well. In order to} find which is the real

physical

solution in this case, a

dynamic stability investigation

^is

required.

^{This is}

impossible

^{at the moment. For this reason we} shall follow Dafermos

[10],

Leslie

[11]

^and Barratt and Jenkins

[9]

^{and shall assume that the}

only

^{solution over} the threshold is the one with lower energy.

(At

^{the same time we shall}

keep

in mind that other solutions with lower _energy than the solutions

considered _may

exist.)

The criterion leads to the

following

conclusions : the deformations

(5)

^and

(6)

^{are of}

lower _energy than the

homeotropic

orientation if the

following inequalities

^{are valid}

(see

^Barratt and Jenkins

[9]) :

where

For a deformation with a maximum inside the

liquid crystal layer :

The

homogeneous

orientation

(8 = n/2)

will be of lower _energy than the

homeotropic (0

⁼

0)

^{when the}

following inequality

^{is valid :}

The solution

(5)

^and

(6)

^{are exact} solutions and

fully

exhaust the

problem

^of

polar

flexoelectric deformation in the

problem

^thus

posed

^{for case} A. We shall however in this _paper touch _upon the

problem

of threshold characteristics

alone, giving

^{the cases most}

advantageous

^for

practical applications

^when

realizing polar

^flexo-

electric deformations. The

complete

tabulation of

(5)

^and

(6)

^{will be}

carried

out in another _paper. The

following

very

important

threshold transcendental

equations

for the non-dimensional parameter

U / Uo (Uo

is the threshold for fixed

vanishing boundary

conditions for the

homeotropic

^nematic

layer)

^{were obtained}

following

^Barratt

and Jenkins

[9]

and Leslie

[11]

^from

(5)

^and

(6)

with Z = d after

changing

the variables

respectively

^to

sin A ⁼ sin

0/k

^{and sin A = sin}

0/sin (Jm

^and

vanishing

conditions 0

(z

⁼

0, d)

^and

Om ->

^{0 :}

Monotone deformation :

where

Deformation with a maximum in the

liquid crysta 1 1 ’yer :

where

The threshold conditions

(7)

^and

(8)

^{are solved}

numerically

^{for the cases :}

1. Infinite

anisotropic

interfacial _energy

(practically Ws1 d

⁼

^{10-4 erg/cm)}

and finite interfacial _energy

WS2

for MBBA at room

temperature (T

^{= 21}

OC)

^{for the}

following

^values

of K13

⁰

(see ^Nehring

^and

^Saupe [4]) :

0; 1.6; 3.2;

^{and 4.8}

dyn

^{and the}

following

values of the total flexoelectric coefficient

(elz

⁺

e3x)

⁼

10- 4;

3 ^x

10 - 4 ;

^{and 6 x}

^{10 - 4}

in cgs units

(see

^{W. Helfrich}

[12]).

2.

For K13 ^{= 0,} WS2 d = 10- 8 ; 10-7; 10- 6 ; ^{10- 5}

^and

^{10- 4} erg/cm,

^finite

anisotropic

interfacial energy

WS1

^{and the same} values of the total flexoelectric coefficient

(e1z

⁺

e3x)’

3. Case

B, theory

^{As 0. - The} interaction between a thin

homogeneously

^{oriented nematic}

monocrystal

with a

negative

dielectric

anisotropy

^and flexoelectric

properties (e lz :f:. ^0, e 3x :f:. 0)

^{and finite}

anisotropic

^elastic

energy of interaction of the

liquid crystal

molecules with the cell walls

WS1

^and

WS2

^{and a constant} electric field E normal to the

layer

is considered

theoretically

in this section.

After

substituting

^the components ^{of the vector field} nx ^{= cos}

0,

nZ ⁼ sin 0 and

E,

^{= E in the}

expression

for the free _energy

(1)

^and

minimizing

the latter with respect ^to

0(z)

^we obtain the

following

differential

equation, describing

the deformations in the

liquid crystal layer :

where

and the

following boundary

conditions for

d0/dz

^{at the two surfaces :}

where

Only

^{a monotone} deformation, defined with the

expressions

^from

[9],

is realized in the

problem

considered :

where

The differential _eq.

(9)

^{and the}

boundary

^condition

(10)

^{allow for}

WS1 = WS2 the elementary

solution 0=0

as well.

Similarly

^{to case A} this solution will be with an energy lower than

(11)

^{if the}

inequalities :

are valid.

The solution

(11)

^{is the exact} solution and

fully

^{covers the}

problem

of flexoelectric deformation thus

posed

for case B. We shall however

only

consider the threshold characteristics

showing

^{the most} favourable cases

from a

practical point

of view for

realizing polar

flexoelectric deformations for this

configuration.

^The

complete

tabulation of

(11)

^{will be}

given

^{in a} separate paper.

In

analogy

^{to case}

A,

^after

changing

the variables

sh A,,2

^{= sin}

O/hsl,2

^{and the}

^vanishing

conditions

OS1

and

as2

^-

^0,

^we obtain the

following important

threshold transcendental

equation

for the non-dimensional parameter

Ut/ Uo ( Uo

is the threshold for the fixed

vanishing boundary

condition for the

homogeneous layer) :

where

The threshold condition

(12)

^{is solved}

numerically

^{for the}

following

^{cases : infinite}

anisotropic

interfacial elastic _energy of the lower

limiting

^surface

(in ^practice WS1 d

⁼

^{10 - 4 erg/cm}

^{and finite}

anisotropic

interfacial elastic _energy

WS2

^{for MBBA at room} temperature

(T

^{= 21}

°C)

^{and the}

following

values for

K13

^{0 :}

^{0 ;} 1.6 ; 3.2 ;

^4.8

dyn

^{and the}

following

^{values of} the total flexoelectric coefficient

(el

⁺

e3x) 10-4;

^{3 x}

10-4 ;

6 x

10 - 4

in _cgs

units).

Case A and B, ^As > 0

(Fig. 2).

^{- A constant} electric field

applied

in the direction of initial orientation

(along Z)

^{of a}

homeotropic

^nematic

layer

^with

positive

dielectric

anisotropy

stabilizes the

liquid crystal

^with respect ^{to As. The same} field however destabilizes in a dielectric as well as in a flexoelectric sense the nematic

layer homogeneously

^oriented

along

^{the X axis with a}

positive

dielectric

anisotropy.

The threshold conditions for these two cases are

simply

obtained from

( 12), (7)

^and

(8)

^{after a} substitution of

!E, TE, K33, K11

^and

K13

for

iE, J,E, K11 , K33

^{and -}

K13 respectively.

The

following

relative difference in the thresholds is obtained from

(7)

^and

(8) (case A,

^Ae >

0)

^{after this}

substitution for the case

WS1

^{= oo and}

WS2

^{= 0}

^(for

^two

^possible

directions of the electric field

E) :

A numerical tabulation is carried out for formula

(13). Here,

in the relations between the dielectric and flexoelectric constants

given by

^Helfrich

[12]

FIG. 2. ^- Polar flexoelectric deformations in nematic layers ^with

a positive dielectric anisotropy : upper cells (case A), lower cells

(case B).

Frank’s elastic constants

K 11

^and

K33

^are

replaced by

^{the elastic constants}

of Nehring

^and

Saupe K’l 1

^and

K33

and the

sign

^for

inequality

^is

replaced by

^an

equality.

Since in nematic

liquid crystals

^the

positive

^dielectric

anisotropy changes

^{over a wide} range, ^we have used the

following

values for

s and 8.L ^covering

^{in our}

^opinion

the dielectric and flexoelectric

properties

^{of a}

large

number of nematic

liquid crystals :

8.L ⁼

5, 8

|| ⁼

20,15,10.

The

following

data for the elastic coefficients were used for the tabulation :

K

⁼ 10 -6

dyne, K13 =

⁺

^{4.8 ;}

+

3.2;

± 1.6 ^x

10 -’ dyne.

The numerical values are

given

^on

figure

^9.

In a similar manner for case

B,

^Ae > 0 from

(12)

^{after a} substitution

of Ki 1, K13

^and

TE

^for

K33, - K13

and

JE respectively,

the threshold condition is obtained

showing

^the electric field for which flexoelectric defor- mations will arise in this

configuration.

The calculations

analogous

^to

(13)

^from

for

WS1

^{= oo,}

WS2

^{= 0 are} also shown on

figure

^9.

4. A discussion of the influence of the

physical quantities

^on the obtained theoretical threshold condi- tions. ^- 4. .1 THE INFLUENCE OF ASYMMETRY IN THE SURFACE ANISOTROPIC ENERGY OF INTERACTION OF LIQUID ^{CRYSTAL MOLECULES} WITH THE BOUNDARY WALLS. - In the

problem

under consideration sur-

face _energy

plays

^a considerable part. ^{In the consi-} deration of most of the

liquid crystal problems

^{it is}

usually

assumed that the surface _energy is of consi- derable value

(1-10 erg/cm’)

^{which in}

practice assigns

a fixed surface deformation

angle

which remains

unchanged

^{under the} influence of

applied

^external

forces, deforming

^the

liquid crystal

^{even when the}

cells are very thin

(with

^thickness of the order of 1

um).

In the case under consideration considerable surface energy of interaction of the

liquid crystal

^molecules

with the walls should be

assigned

^{to one} of the surfaces

only.

^The

sign

of the total flexoelectric coefficient e1z + e3x ^shows which of the surface

energies

^should

be

larger (e.g. Ws1)

^{under the}

^given

direction of the

applied

electric field

(JE

^or

TE).

^{Small surface} energy

(WS2 WS1)

^{should be}

^assigned

^{to the other surface,}

which

together

^{with the} flexoelectric

properties

^{of the}

liquid crystal

determines the

polarity

of the effect.

The

investigation

of the theoretical curves obtained

(the

^most

interesting

^{are shown in}

figures

^{3-6 and}

figure 8)

^{for MBBA leads to the}

following important

conclusions

regarding

^the

practical application

^of

this effect

(for

^{LC’s with a} small dielectric

anisotropy) :

1. The value of the

important parameter

may be assumed for the lower limit of a rather

weakly

fixed nematic

liquid crystal.

2. For the _upper limit of a rather

strongly

^fixed

liquid crystal,

^one may ^assume the value

3. If the difference

(in

^the

pointed interval)

^between

the two

anisotropic

^surface

energies

^{is at least one}

FIG. 3. ^- Polar theoretical curves for a homeotropic ^MBBA layer.

FIG. 4. ^- Polar theoretical curves for a homeotropic ^MBBA layer.

order,

^{one obtains} very clear distinctions in the thresholds for the

polar

flexoelectric effect.

FIG. 5. ^- Polar theoretical curves for a homeotropic ^MBBA layer.

FIG. 6. ^- Polar theoretical curves for a homeotropic ^MBBA layer.

4. The threshold conditions obtained for a homeo-

tropic

^nematic

layer

^{with a}

negative

dielectric aniso- tropy ^{and an}

applied

electric field

perpendicular

^to

the

layer

^{make it}

possible

^{to draw the}

following

more

important generalizations regarding

^{the values}

of the

anisotropic energies

^which

might

^{ensure a}

good polar

flexoelectric effect

(for WS1

>

WSZ) :

where

U|UE

^and

UltE

^are the threshold

voltages

^for

WS1

^{= oo and}

WS2

^{= 0.}

In a similar _way, the

following inequalities (with WS1

>

Ws2)

^are

^given

^{for the}

respective

limitations

on the values of the surface

energies

^{in the case of a}

homogeneous

^nematic

layer

^with

positive

^dielectric

anisotropy

^{and an}

applied

electric field

perpendicular

to the

layer :

where

again UtTE

^and

UIIE

^are the threshold

voltages

for the case

WS1 = 00

^and

WS2

^{= 0.}

4.2 INFLUENCE OF THE FLEXOELECTRIC PROPERTIES OF THE LIQUID CRYSTALS UNDER CONSIDERATION. ^-

The

deforming

^{moments due to} the flexoelectric

properties

^{of a nematic}

liquid crystal

may be in the bulk as well as the surface

depending

^{on the formu-}

lation of the

problem,

^the type and the direction of the

applied

electric field

interacting

^{with the}

crystal.

In the

problem

^under

consideration,

the flexoelectric energy is

simply

^{added to or} subtracted from the

anisotropic

^surface energy. The balance between these two

energies

determines the type of deformation

- monotone or with a maximum in the

liquid crystal layer

^{- as well as the}

polarity

^{of the}

effect,

^{i.e. the}

deformation and the threshold value of the

applied voltages

^{for a}

given

direction of the electric field.

The theoretical ^curves obtained demonstrate

(case A, negative

dielectric

anisotropy,

^case

B, positive

dielectric

anisotropy)

^{that the}

larger

^{value total}

flexoelectric coefficient assures better

polarity

^{of the}

effect

(a larger

^relative

change

^{in the}

thresholds)

^for

fixed values of the

remaining physical quantities, lowering

^the

requirements

for the values of the surface

energies

^and

increasing

^the upper

voltage

^threshold.

The

sign

of this coefficient

shows,

^{for a}

given

asym- metry ^{in the surface} energy

(e.g. WS1

>

Ws2),

^for

which direction of the electric field the smaller defor- mation will appear and for which the

larger

^{one will}

(see Fig.

^{3-6 and}

8).

For the case

A,

^As > 0 and

B,

^{As 0} flexoelectric deformations are obtained

only

when the total flexoelectric coefficient has a considerable value.

In this case

only

^the surface flexoelectric moments

are

deforming

^{and a} relation exists between the

dielectric,

flexoelectric and elastic coefficients of the

liquid crystal

which forbids the appearance of the flexoelectric deformation

(see

^Fan

[13]).

For case A

For case B

These are the

only

known relations between the elastic constants of

Frank,

^of

Nehring

^and

Saupe

and the flexoelectric coefficients of R.

Meyer

^at present. ^The

larger

value total flexoelectric coefficient e1z ⁺ e3x determines a flexoelectric deformation even

for a

larger anisotropic

^surface energy and at the same

FIG. 7. ^- The angle ^{0 at} the interface for K13 0, Ag 0

(case A) ^and K13 > 0, ^As > 0 (case B) and various values of

WS2 ^{d =} U*(elz ⁺ e3x) (WS1 ⁼ (0).

FIG. 8. ^- Polar threshold theoeretical curves for a homogeneous

MBBA layer.

time it

gives

^the

larger

threshold above which these deformations would take

place.

4.3 INFLUENCE OF THE DIELECTRIC ANISOTROPY. ^-

In all cases

investigated

^the

dielectric anisotropy

worsens the

polarity

of the effect

(the

^dielectric

deforming

^or

stabilizing

^{moments are}

quadratic

^with

respect

^{to the}

applied

^electric

field)

^and

changes

^the

voltage

thresholds

by lowering

^them

(see

^{W. Hel-}

frich

[2]).

The dielectric

anisotropy

^however

plays

^a

positive

role as well since the

lowering

^{of the} upper threshold lowers the

operating voltage

of this effect. Variation of this

voltage

^{in a wide} range may therefore be

accomplished

^{with the}

change

^{of Ag.}

The influence of the dielectric

anisotropy

^{on the}

participation

of the flexoelectric

properties

^{in the}

polar

effects is

considerably

^more

complicated.

^{It is}

difficult to evaluate the influence of the various

parameters

^{in case the}

larger

dielectric

anisotropy

leads to a

larger

difference in the flexoelectric coef- ficients as well. The conclusion can be drawn from the

case considered

( WS1

^{= oo,}

WS2

⁼

^0),

^{shown on}

figure

⁹

(the inequalities (14)

^{show that} the theoretical

curves on

figure

⁹ for real values of _{e lz}

+ e3x

^must

be translated in the direction of smaller

As)

^{that for}

nematic

liquid crystals

^{with As} >

0,

^{the relative}

change

ⁱⁿ

polarity

^{is reduced on a} small scale with the increase in

anisotropy (case B,

^As >

0).

^{In case}

^A,

As > 0, ^{the ratio}

UI,IUO

is also reduced on a consi- derable

larger

^{scale, but with} decrease in

anisotropy.

FIG. 9. ^- Relative change ⁱⁿ polarity (case B, As -> 0) ^and U, I E / UO (case A, ^As > 0). ^{For case} A, AE 0 and case B, ^{As 0 the mirror} images around the Z axis of the curves are valid - E 11 -+ e,, 8, -+ 8 11 - (The theoretical curves for real values of el, + e3x must be translated

in the direction of smaller As.)

8jj

^||

and 81

^remain

unchanged

^{in a wide} range for the

widely investigated

^nematic

liquid crystals

^with

negative

dielectric

anisotropy

and therefore the com-

parison

between Helfrich’s threshold

(3 V)

^{and our}

thresholds

(calculated

^for

MBBA)

demonstrates to a

large

^extent the influence of the dielectric

anisotropy

in these

crystals

^{on the}

polar

effects under conside- ration.

For

liquid crystals

of nematic type ^with

large negative

dielectric

anisotropy

the mirror

images

around the Z axis of the curves

given

^on

figure

^{9 are}

valid

(81

^-

E || E

II ->

El).

4.4 INFLUENCE OF SECOND-ORDER ELASTICITY. ^-

The presence of elastic _energy of the second order

brings

^{about a}

change

in the bulk deformation _energy and a deformation surface _energy

strongly dependent

on the type of deformation of the bulk

liquid crystal layer.

The theoretical results demonstrated that for ani-

sotropic

^surface

energies Ws1 = oo

^and

WS2

^{= 0}

(Fig. 9) :

^.

a)

^For

positive

^and

negative

^{values of}

K13

^{in case B,}

Ag > 0, the relative

change

ⁱⁿ

polarity

is increased or

lowered

respectively. (For

^{case A, A8 0 the reverse}

is

true.)

b)

^The

negative

^{values of}

K13

^{for case}

^A,

^A8 > 0

(dielectrically

^stable

liquid crystal) considerably change

^{the ratio}

U, IEI Uo.

^{For case}

^B,

^{A8 0 this is}

valid for the

positive

^{values of}

K13.

The theoretical results also

unexpectedly

^demon-

strated

(case A,

^A8

0,

^case

B,

^A8 >

0)

^{that for some}

Critical Values of the

Anisotropic

Elastic Surface

Energy

of interaction of the

liquid crystal

^molecules

with the walls

(strongly dependent

^on the values of the total flexoelectric coefficient e1Z ⁺

e3x)

the deforma- tions start with a

jump.

^{The case}

WS d

⁼

(e 1 _,

⁺

e3x)

^U*

is considered for a

simple example (infinite anisotropic

interfacial _energy at the other

wall).

^For these values of

WS

^{a monotone} solution with a

jump

is obtained

(K13

>

0,

^case

B,

^A8 >

0)

^{or a} solution with a maxi-

mum in the

liquid crystal layer

also with a

jump (K13 ^0,

^case

^A,

^A8

0).

These results are shown

on

figure

^7.

^They

^are tabulated for

isotropic elasticity

- K ⁼ 7.5 x

10-’ dyne, K13

^{= 3.2 x}

^10-’ dyn.

^It

is evident from the

figure

^{that for some values of}

WS

considerable deformations _may be established in the

cell for one direction of the

applied

electric field. For the reverse direction of the field the nematic

layer

^is

stable thus

assuring

^a much better

polar

flexoelectric effect.

Similar deformations _may be obtained for values of the electric field

considerably

^over the threshold for As

0, Kl 3

> 0 in case A where

or when

K13 ^0,As

> 0 for case B as well.

These results are

mathematically explained by

^the

constraint on the derivative

d8/dz

^{at the walls}

(z

⁼

d, 0) coming

^{from the}

boundary

conditions and the pre-

sence of surface flexoelectric _energy.

Physically

this constraint means that for a smooth

change

ⁱⁿ the electric field for Critical values of the surface

energies,

^a smooth transition from a monotone deformation to a deformation with a maximum in the

liquid crystal layer

^{cannot take}

place.

References [1] MEYER, R. B., Phys. ^{Rev. Lett. 22} (1969) ^918.

[2] HELFRICH, W., Appl. Phys. ^{Lett. 24} (1974) ^451.

[3] NEHRING, J., SAUPE, A., ^{J. Chem.} Phys. ⁵⁴ (1971) ^337.

[4] NEHRING, J., SAUPE, A., ^{J. Chem.} Phys. ⁵⁶ (1972) ^5527.

[5] DERZHANSKI, ^A. I., PETROV, ^A. G., ^VI Internat. LC’s conf., Kent, U.S.A. (1976) ^{Abstracts A-2.}

[6] PETROV, A. G., VI Internat. Conf. Spectroscopy, Sunny Beach, Bulgaria (1974) Abstracts 248.

[7] DERZHANSKI, ^A. I., PETROV, ^A. G., I Internat. LC’s conf. of soc.

countries, Halle, ^DDR (1976) Abstracts 78.

[8] DERZHANSKI, ^A. I., HINOV, ^H. P., Phys. ^{Lett. 56A} (1976) ^465.

[9] BARRATT, ^P. J., JENKINS, J., ^J. Phys. ^{A :} Math. Nucl. Gen. 6

(1973) 756.

[10] DAFERMOS, ^C. M., ^SIAM J. appl. ^{Math. 16} (1968) ^1305.

[11] LESLIE, F. M., J. Phys. ^{D :} Appl. Phys. ³ (1970) ^889.

[12] HELFRICH, W., Mol. Cryst. Liq. Cryst. ²⁶ (1974) ^1.

[13] FAN, C., ^Mol. Cryst. Liq. Cryst. ¹³ (1971) ^9.