Electric field induced static modulated structures in nematics

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Electric field induced static modulated structures in

nematics

U.D. Kini

To cite this version:

Electric field induced

static

modulated

structures

in nematics

U. D. Kini

Raman Research _Institute,

_{Bangalore -}

560 080, India

(Reçu

le 17 août

1989,

révisé le 24 novembre

_{1989, accepté}

le 30 novembre

₁₉₈₉₎

Résumé. 2014 On étudie la

possibilité

d’existence de structures

_statiques

modulées

_(MS)

sous

l’action d’un

_{champ électrique}

_{(E) appliqué parallèlement}

aux

_plans

de _{l’échantillon,} avec

l’orientation initiale du directeur uniforme tournée par rapport aux limites de l’échantillon, dans

un

_plan

_{perpendiculaire}

à E. Avec

l’hypothèse

d’un ancrage

rigide,

on montre _{que la formation de}

MS avec une

_{périodicité}

dans la direction de E est favorisée

_quand

un

_champ

_magnétique

(H~)

stabilisateur d’intensité suffisante est

_appliqué

dans la direction initiale du directeur. Quand

l’angle

de rotation du directeur initial est suffisamment loin de

_{l’homéotropie,}

MS _peut ne _pas

apparaître ;

en

_particulier,

MS ne _peut exister dans une

_géométrie

de twist. Ces résultats sont en

accord

_qualitatif

avec

_{quelques expériences}

récentes. Il semble

_possible

de freiner la formation de MS en utilisant des

_{champs électriques}

de haute

fréquence

sur des matériaux ayant tendance à

présenter

une forte relaxation

_{diélectrique.}

L’action favorable

_{de H~}

en faveur de MS est assez

analogue

à celle observée en

_{hydrodynamique.}

Les effets

_d’ancrage

faible du directeur et la flexoélectricité sont

_plus

forts seulement

_quand

_H~

est suffisamment

_petit.

Il est _surprenant de

constater

_qu’un

mode d’instabilité de

_{périodicité}

_parallèle

aux

_plans

de l’échantillon mais

perpendiculaire

à E peut amener à une situation

_comparable

à celle de MS, ce

_qui

montre le besoin de

plus

d’études à la fois

_théoriques

et

_{expérimentales}

_qui

_prennent en _compte la cellule de l’échantillon.

Abstract. 2014

Linearized continuum

_theory

is used to

_investigate

the

_possible

occurrence of static

modulated

structures

_(MS)

under the action of an electric field

_{(E) applied}

_parallel

to the

_sample

planes

with the initial uniform director orientation tilted with respect to the

_sample

boundaries in

a

_plane

normal to E. Under the

_rigid

_anchoring

_hypothesis

it is shown that the formation of MS with

_periodicity

_along

E is favoured when a

_{stabilizing magnetic}

field

_(H~)

of sufficient

_strength

is

impressed along

the initial director orientation. When the initial director tilt is

_sufficiently

_away from the

_homeotropic,

MS may not occur ; in

_particular,

MS cannot occur in the twist _geometry. These results are in

_qualitative

accord with some recent

_{experimental investigations.}

_{It appears}

possible

to deter the formation of MS

_{by using high frequency}

electric fields in materials which have a

_propensity

towards

_exhibiting

_strong dielectric relaxation. The

_facilitating

action of

H~

towards MS is rather reminiscent of a

_{hydrodynamical}

_analogue.

The effects of weak director

anchoring

and

_{flexoelectricity}

are found to be

_{pronounced only}

_{when H~}

is small

_enough.

Intriguingly,

it appears that an

_instability

mode

_having

_{periodicity parallel}

to the

_{sample planes}

but normal to _{E may} set in at a threshold

_comparable

to that of MS thus

_indicating

a need for

more detailed

_experimental

and theoretical

_{investigations}

_taking

_proper account of the _aspect ratio of the

_sample

cell.

Classification

Physics

Abstracts 61.30 - 62.20D -

41.40D - 68.45

1. Introduction.

The varied responses of nematic

liquid crystals

to

_extemally

_impressed

electric

_(E)

and

magnetic

(H)

fields have

_provided

an

_interesting

field of

_experimental

and theoretical

_study.

The

_homogeneous

deformation

_(HD)

which occurs above a Freedericksz threshold is well understood on the basis of the Oseen-Frank continuum

_theory

of curvature

_elasticity

_[1-7].

The detection of the Freedericksz transition is an

_important

tool in the determination of the

three curvature elastic constants of a nematic.

When a nematic possesses

_large

elastic

_{anisotropies,}

HD may not be the

_only

deformation

possible.

For

_instance,

a deformed nematic

_layer

cooled towards the nematic-smectic A transition

_{(TNA) or a}

nematic

_subjected

to H in the bend

_geometry

close to

_TNA

exhibits the

stripe phase

(SP) [8, 9] ;

the

_stripes

_appear

_parallel

to the

_{plane containing}

H and the initial

director orientation _no. Allender et al.

_[10]

have used the continuum

_theory

to

_explain

the

occurrence of SP.

It is also known that HD _may not occur at all in the

_splay

Freedericksz

_geometry

in certain

polymer

nematics

_having

a

_{large splay}

constant

_K,

_compared

to the twist elastic constant

K2.

As demonstrated

_{experimentally by Lonberg}

and

_{Meyer [11]}

in this case the distortion

that occurs above a threshold is not HD but a

_periodic

deformation

_(PD)

_having

the direction

of

_periodicity

normal to the

_{(no, H)}

_plane.

It has been

_{suggested theoretically}

_[12-16]

that PD may be

suppressed by

the proper choice of field

direction,

director

anchoring strengths,

initial director tilt relative to the

_sample

boundaries etc.

Electrically

and

_magnetically

induced HD in most _{nematics with average elastic}

_anisotropies

have,

so

far,

been understood to occur above a second order Freedericksz transition.

Recently,

however,

it was

_{experimentally}

shown

_by

Frisken and

_{Palffy-Muhoray}

_[17]

that

electrically

driven twist and

_splay

Freedericksz transitions in nematic 5CB are first order and

associated with

_hysterisis.

These authors have made another

_{interesting discovery.}

In the bend

_geometry

when the initial

_homeotropic

_alignment

is stabilized

_by

a

sufficiently

_strong

magnetic

field

_{Hp , they}

find that the deformation that occurs above a threshold is modulated

with wavevector

_along

E and

_wavelength

of the order of the

_sample

thickness.

_They

do not

observe such modulated structures

_(MS)

in the twist

_geometry

and claim not to have seen evidence of

_{electrohydrodynamic}

flow.

_{Interestingly,}

the transition width

_{experimentally}

determined

_by

the authors is about an order of

_magnitude

smaller than their theoretical

estimate. In a recent _paper Allender et al.

_[18]

have

_given

a

_qualitative

_argument

to show that MS

_might

occur above a first order transition.

At this

_stage

_{it may be necessary} to make a few observations. Is it

_quite

_possible

that as in

the case of

_magnetically

induced SP

_{[9, 10],}

in the

_present

case

_also,

HD occurs

_initially

and

subsequently

becomes unstable

_{against periodic}

fluctuations

_leading

to the formation of MS. If this were the case _{it would be necessary} to first solve for HD

_(as

done in

_[17])

and then

consider

_stability

of HD as E is increased above HD threshold

_(analogous

to what has been done in

_[10]

for the case of

_SP).

However,

as shown in

_[10],

this

_problem

is

_{quite complicated}

and can be dealt

with

_{facility only approximately.}

In the case of SP there is no alternative as

there exists no linear threshold for a direct transformation of the initial

_homeotropic

alignment

into

SP ;

the intermediate HD’ is

_absolutely

_{necessary. This has been substantiated}

by

an accurate determination of the SP threshold

_[19].

A

_possible

_explanation

for this could be that the director distortions do not

_{substantially}

alter a

_magnetic

field inside the

_sample.

If,

on the other

hand,

we want to treat the case of MS

_occurring

via a first order transition

directly

from the

_aligned

nematic state it becomes _necessary to consider terms in the free energy

density

of order

higher

than the second in the

_{perturbations}

and director

_{gradients ;}

the

_{perturbations}

will

_depend

on more than one

_spatial

coordinate. In addition modifications

equations.

This will mean

_solving

a set of

_coupled

nonlinear

_partial

differential

_equations

-

a

formidable task. On the other

hand,

linear threshold calculations are

_{simpler especially}

because of the ease with which

_periodicity

can be included as a closed solution. This

_provides

motivation for

_{investigating}

a linear threshold.

There seems to be an

_analogy

between MS and HD on the one hand and roll

_instability

_(RI)

and

_{homogeneous instability}

_(HI)

in shear flow of

_aligning

nematics on the other

_[20-23].

It is found that HI threshold is

_generally

lower _{than the RI threshold. But in the presence of} a

sufficiently

strong

HII,

the RI threshold _can become lower than the HI threshold. The theoretical model

_developed

in

_[22]

for

_{understanding}

RI and HI is based on linear

perturbations.

Experiments

suggest

[17]

that MS does not manifest in twist

_geometry

_(homogeneous

alignment).

This

_suggests

that there must exist a cut off director

_tilt,

_{away from the}

homeotropic, beyond

which MS is unfavourable - rather reminiscent of similar

suggestions

made in connection with the

_suppression

of PD

_[12-16]

in

_polymer

nematics. This

_provides

incentive to

_study

the effect of

_tilting

the initial director orientation in a

_plane

normal to

E and also consider the variation of other

_controlling

_parameters.

The

_importance

of

_{flexoelectricity}

_[24-26]

in electric field effects can

_hardly

be

_exaggerated.

It appears [25] that this is a

_property

possessed by

_{every nematic}

_system

whereby

deformation

in the director field

_produces

an associated electric

polarization ;

it is difficult to exclude

flexoelectricity

from a

_complete

treatment of electric field effects in nematics.

Thus

motivated,

the

_general

differential

_{equations governing}

linear

_{perturbations}

are set _up

and relevant

_boundary

conditions described in section 2. Sections 3 and 4 describe the behaviour of MS for

_{rigid anchoring}

and weak

_{anchoring, respectively,}

in the absence of

flexoelectricity.

Section 5 deals

_briefly

with the effects of

_{flexoelectricity}

for the case of

_rigid

anchoring.

In section

_6,

a different

_instability

mode

(Yi)

is described whose behaviour

requires comparison

with that of MS. Section 7 concludes the discussion.

2.

Governing équations, boundary

conditions.

We

try

to

_essentially

simulate the

_experimental

_geometry

used in

_[17].

Consider a nematic

uniformly aligned

between dielectric

_plates

z = ± h

_(sample

thickness 2

_h)

such that the initial director orientation is

_{given by}

lying

in the yz

plane parallel

to the two electrodes situated at x = ±

_{w /2}

_(distance

between

electrodes =

w).

A

_{stabilizing Ifi =}

_{(0, HI}

_S,

_{Hp C )}

and a

_{destabilizing}

E =

(Ex0,

0,

0)

_are

applied

with

_Exo

=

Vo/w ;

Vo

is the

potential

difference between the two electrodes. Under

perturbation

the director and electric fields inside the

_sample

become

where

_{X a (> 0 )}

is the

_diamagnetic

_{susceptibility}

_{anisotropy ;}

_{e , e 1.} are the dielectric constants

along

and normal to the nematic

_{director, respectively ;}

_{el, e3} are two flexoelectric

summed over. The

_equations

of

_equilibrium

are written down

_{by extremizing}

the total free energy (keeping

Vo

constant) :

These are

_{supplemented by}

the Maxwell

equations

[27]

corresponding

to the additional

_assumption

that free

_{charge density}

is absent.

_Linearizing

₍₄₎

and

₍₅₎

with

_respect

to the

_{perturbations 0,}

4>,

Ex,

_Ey,

E,,

one

_gets

where the different

_{f ’s}

are defined in the

_appendix.

The

_above,

rather

general

derivation has

been

_presented

so that it is convenient to take subsets of terms from the above

_equations

in

subsequent

sections for

_studying

different situations. The

_boundary

conditions for

_rigid

anchoring

become

_{[27, 28]}

In

_(6),

the Maxwell’s curl

_{equation (5b)}

_permits

us to write

_{Ex, y,}

_{Ex,z, Ez, y}

for

Ey, X,

EZ, x,

Ey, z,

respectively.

For the case of weak director

_anchoring

_[29-31]

we

shall

write down the

_boundary

conditions

_by

and

_by.

3. x, z

_{dependence ; rigid anchoring ;}

no

_{flexoelectricity.}

Keeping

in mind the

_{experimental findings,}

we assume that all

_{perturbations depend}

on x and z. For the sake of

simplicity,

we also

_{ignore flexoelectricity.}

The cases of

_homeotropic

and

_general

tilted orientations will be considered

_separately.

3.1 HOMEOTROPIC ORIENTATION ;

çb 0

= 0. - The

homogeneous

deformation

_(HD)

involves

only 9

and

_Ez.

From

_(6a)

and

₍₇₎

the HD threshold value of

_Exo

is found to be

_[28]

Assuming

next

_dependence

on both x and z, it is found that

(6a)

and

(6c)

result in two

coupled equations

while 0

damps

out. The

_equations

_support

two

_independent

modes-mode

1 with 0 even and

_Ex

odd with

_respect

to the

_sample

centre and mode 2

_having

the

_opposite

solutions of the form

_{0 - 0 0 cos qz cos q., x,}

_{Ex - Eo sin qz sin q x x}

_{where qx}

is real and

90,

Eo

are constants.

_Then,

₍₇₎

leads to the condition of

_{compatibility}

which determines the

critical

values

_Ex,(qx)

of

_Exo

as a _{function of qx :}

Obviously, Exc ~ EH

when

_{Qx ~}

0. When

_Qx

is

increased,

Exc

decreases and attains a

minimum

_Exc(QM)

=

EM

when Qx

=

QM

which is

given by

Using

the data for 5CB

_{[17, 32]}

as also for other relevant

_parameters

, , , -- - - _

-one gets,

where

_VH

and

_VM

are the

_voltages

_{corresponding}

to the HD and MS

_thresholds,

_{respectively.}

Thus,

for

_sufficiently

_strong

_HI,

the MS threshold is found to be lower than the HD threshold.

The domain

_wavelength

= 2

7Th/QM

corresponding

to

QM

= 4.8 is

roughly

0.65 times the

sample

thickness,

in fair

_agreement

with

_[17].

Another check is to count the number of MS

domains

_(NM

=

WQM/7Th)

sandwiched between the two

electrodes ;

for the

_present

_case,

NM

=

21 ;

figure

5 of

[17]

does _{appear to} show

roughly

the _same number of MS domains

between the two electrodes.

Equation

(10)

shows that

_{when HI}

_Hc

with

MS cannot

_{exist ;}

as

_{Hj -}

_Hc,

_QM

-

0,

EM

-

EH.

For 5CB

parameters

(11),

Hc -

200

oersteds

_(h

= 0.025

cm).

Equation

(13)

gives

_{a cut} off field for _a

given sample

thickness _as

well as a cut off

_sample

thickness for a

_given

field

_strength

_HI.

For fixed

_HI,

it should be

possible

to

_quench

MS

_{by using}

a

_sufficiently

thin

_sample.

Model calculations are

presented

for three sets of elastic values of 5CB

_[32]

which are

shown

_along

with other relevant data :

Figures

1 and 2 which

_{depict plots}

of R =

EM/EH

and

QM

_as functions of

HI

and the

. . ii

Fig.

1. - Plots of R =

EM/EH

and

QM

as functions of

_HI.

_{Homeotropic alignment. Rigid anchoring.}

H~

is the

_stabilizing

field

_{strength. EM}

and

_EH

_are,

_{respectively,}

the electric thresholds for MS and HD.

QM

is the scaled MS domain wave vector. Curves 1, 2, 3 are _drawn,

_{respectively,}

for the elastic sets

_(i),

(ii), (iii),

of

₍₁₄₎

from which the rest of the _parameters are also chosen. For a

_{given kl}

=

KI/ K3,

MS

cannot exist for

_Hp

:

Hc.

kl

increases from curve 1 to 3. Thus,

Hc

decreases as

_kl

increases.

Il 1

Fig.

2. - Variations of Rand

QM

with the dielectric constants at

_H1

= 600 oersteds.

Homeotropic

alignment. Rigid

anchoring.

Curves 1, 2, 3 are for

_{(i), (ii), (iii)}

of

_{(14). a), b)}

Plots of Rand

_QM

_{versus Et for e1} = 8.2.

c), d)

Plots of R and

QM

_versus

El for El = 18.8. The

positive

definiteness

_{of ea}

is necessary for MS to occur

_(e.

=

increase

_{when HII}

is

diminished ;

(ii)

MS cannot exist when

_HII

the cut off field

Hc ;

(iii)

a decrease in

_kl

increases

_H,

thus

_shrinking

the

_Hl

_{range of existence of MS.}

According

to

_figure

_2,

MS cannot exist in materials

_{having negative}

dielectric

_anisotropy

(Ea

0).

Hence,

if a nematic

_exhibiting

MS under the action of

_HI

has also the

_property

of

sufficiently

strong

dielectric

_relaxation,

_{it may be}

_possible

to deter the formation of MS in such a material

_{by using}

time

_varying

electric fields of

_{high enough frequency.}

In

_particular,

figure

2 indicates that as Ea -

0,

the MS domain width must increase at constant

HII ;

this may be

capable

of

_being

checked

_{experimentally by changing}

the

_frequency

of the

applied

electric field.

3.2 TILTED

ORIENTATION; cf> =F

0. - In this

case

_again,

HD is determined

by 0

and

Ez

and has the threshold value of

_{EA given by}

The

_tilting

of the director field away from the

homeotropic brings

in the tilt

_angle

and

K2

as additional

_parameters.

When we consider x, z

dependence,

however,

the

picture changes

as

_compared

to the

homeotropic

case

_{(Sect. 3.1).}

_Now,

₀

also

gets

coupled

to 0 and

_E,,.

_Seeking

solutions

- exp (i qx x )

for x

_{dependence, equations (6)}

result in three

_{coupled, ordinary}

differential

equations

in

_0,

₀

and

_Ex

to be solved with

_(7).

The

_equations

_support

two

_independent

modes

out of which we

_study

the mode with 0 even and

_¢,

_Ex

odd relative to the

_sample

centre

_(the

other mode with

_opposite

_symmetry

is

_ignored

as it will

_generally

have a

_higher

_threshold).

The solution is most

_conveniently

effected

_using

the series method

_(see,

for

instance,

[12]).

The

_{compatibility}

condition

_yields

the critical value of

_Exo,

_Exc(Qx,

_cl>o),

as a function of

Qx

for a

_given

set of

_parameters.

The minimum of the neutral

_stability

curve occurs at

QM(OO)

and

_corresponds

to the MS threshold

_Em(oo)

at the tilt

_{angle 0().}

Figures

3a,

3b illustrate the variation of

_{R (0 0)}

=

Em (0 0)/EH (0 0)

and

Qm (0 (»

_as

functions of

₀₍₎

at

_Hp

= 600 oersteds for the three elastic _sets

(14) (as

given

in Sect.

2,

the

direction of

_{HII changes}

with

_{00 ;}

the

_magnitude

_Hp

is held

_constant).

For a

_given

elastic set,

R(oo)

increases and

_Qm(oo)

decreases as

_{ci> 0}

is enhanced from its

_homeotropic

value

cl>o

=

0 ;

when

cl>o -+

_{an upper limit}

0 c

(

«

w /2

corresponding

to

_homogeneous

_orientation)

R (0 0) 1, Qm (0 0) - 0

showing

that

_{for cl>o::> cl>c,}

MS cannot exist. This is rather reminiscent of similar results obtained earlier

_{[12, 14]}

in connection with the

_possible

suppression

of PD

_{by resorting}

to

_changes

_{in initial tilt of the director field. It may be noted}

that

_cl>c

increases from set

_(i)

to set

_(iii).

Thus the

_cl>o

_{range of existence of MS} broadens when

kl, k2

increase.

Figures

3c,

3d show the

_dependence

of

_R(cI>o)

and

_{QM(cI>O)}

on

_HI

at three different

00

for the elastic set

_(i)

of

_(14).

The variations of R and

_QM

are similar to those of

_figure

1. It is

_found,

rather

_obviously,

that the

_{HI range of existence of MS shrinks for}

a

_given

material

when the initial director tilt is increased away from the

homeotropic.

4. x, z

_{dependence ;}

weak

_{anchoring ;}

no

_{flexoelectricity.}

It is

_again

convenient to

_separately

treat the cases of

_homeotropic

and tilted

_alignments.

4.1 HOMEOTROPIC ALIGNMENT ;

fl#o

= 0. -

. 11

Fig.

3.

- Plots of

R(~o)

=

E,(çbo)IE,(00)

and

_QM(~0)

as functions of

₀₀

_{(Figs. 3a, 3b)}

and

Hl (Figs.

3c,

3d).

Initial director orientation is tilted at

_{angle çbo}

to the

_homeotropic.

_EM(OO)

and

EH (0 0)

are the MS and HD thresholds,

respectively.

Qm (0 0) is

the scaled MS wavevector at threshold. In a and b, curves 1, 2, 3 are drawn for

_{(i), (ii), (iii)}

of

_{(14). Hl}

the

_magnitude

of the

_{stabilizing magnetic}

field is fixed at 600 oersteds. MS is

_extinguished

before

_~0

reaches

_ir/2

radian

_(homogeneous

alignment) showing

thereby

that MS cannot exists in the twist _geometry. In c and d,

only

the _parameter

set

(i)

of

₍₁₄₎

has been used.

₀₀

takes the values

(1)

0.05

₍₂₎

0.5

(3)

1.0 radian. For a

_given

00,

the results are

_analogous

to those of

_figure

1. The

₀₀

range of existence of MS broadens when

k, and k2

increase for a fixed

_HI.

_Similarly,

the

_Hp

_{range of MS shrinks for} a

_given

material when

cfJo

is enhanced.

where B is the

_{anchoring strength}

_(as

the initial orientation has uniaxial

_symmetry

about

z, we take B to be the

_anchoring

_strength

for both

_{perturbations).}

In the absence of

flexoelectricity,

the

_boundary

conditions

_(7a)

and

_(7b)

on 6

_{and çb,}

_{respectively,}

take the

form

In

_general,

_therefore,

the

_{perturbations 0}

_{and 0}

do not vanish at the boundaries. When

03C3 >

_1,

we recover the

_rigid

_anchoring

_boundary

conditions

_{(7a), (7b) ;}

this limit is realized for

B -

_{10- 2 dyne}

_cm-1(cgs).

The HD threshold value of

_{Exo (for}

_{perturbations 0}

and

_{E, ; 0}

_damps

out

_again)

is

For a

_given

set of

_parameters,

_{qH is first} determined.

_Using

_{this value of qH, the HD} threshold

EH (B )

is

_computed

for the

_given

B.

_Obviously,

when 03C3>

1,

qH:5

ir/2 ;

when a decreases

(when

the

_anchoring

_weakens),

so does qH.

For ’xz

_dependence,

the MS threshold for mode 1 is calculated

_{by solving}

_{(6a), (6c)}

with the

boundary

conditions

_(17a)

and

_{(7c) (it}

must be noted

_{that 0}

_again

_gets

_decoupled).

The

Figures

4a,

4b show the

_dependence

of

_{R (B ) = EM (B )/EH (B )}

and

_{QM (B )}

on the

anchoring strength B

at a constant

_stabilizing

field

_HII

= 300 oersteds. The three elastic _sets

(14)

have been chosen. When B is

_large,

_{R (B )}

and

_{QM (B )}

take values close to those in the

rigid anchoring

case

_{(Fig. 1).}

_Initially,

when B is decreased

_{R (B )}

increases and

_{QM (B )}

diminishes ;

when B attains still lower

_values,

_{R (B )}

and

_{QM (B ) appear}

to saturate. Thus a

weakening

of the

_{anchoring strength}

broadens the MS domains. It is found that at

_higher

Hg (1200

oersteds),

variation of B has little or no effect on

_{R (B )}

and

_{Qm (B).}

Thus,

the

effects of

_changing

director

_{anchoring strength}

should be discernible

_only

at

_sufficiently

low

stabilizing

fields.

This is reflected in the variations of

_{R (B )}

and

_{QM (B )}

with the

_stabilizing

field

HII

(Figs.

4c,

4d).

Two values of B

₍

_{=10- 2,}

_{10- 6 cgs)}

have been chosen. Both curves have been drawn for the elastic set

_(i)

of

_(14).

The curves are similar to those of

_figure

1.

Now,

however,

the cut off

_{strength, H,,}

is a function of

B ;

when B is diminished

_Hc

increases

showing

that the

_HII

_{range of existence of MS shrinks when the director}

_anchoring

is weakened.

Fig.

4. -

Dependence

of

_{R (B )}

and

_{QM (B )}

on B and

_{Hr .}

_Homeotropic

orientation. B is the director

anchoring strength

at the

_{sample planes.}

_{R (B )}

=

EM (B )/EH (B )

where

_{EM (B )}

and

_{EH (B )}

are the MS and HD electric thresholds for a

_{given anchoring strength}

B.

_{QM (B )}

is the

_{corresponding}

scaled MS domain wavevector.

_{a), b) R (B )}

and

_{QM (B )}

versus B for

_Hl

= 300 oersteds. Curves 1, 2, 3 _are drawn

for the _parameter sets

_{(i), (ii),}

_(iii)

of

_(14).

The increase of

_{R (B )}

with

_{diminishing B}

for a

_given

k,

shows that

_weakening

of

_anchoring

_strength

enhances the MS threshold with _respect to the HD threshold. This trend is reflected in

_figures

4c, 4d which

_depict

variations of

_{R (B )}

and

_{QM (B )}

with

Hp

for B =

_{(1 )}

10- 2

_{(2 )}

_{10- 6 dyne cm-l.}

Both curves are drawn for the elastic set

_(i)

of

_(14).

The eut off field

_H,,

which is now a function of B, increases when B is

_{diminished ;}

thus a

_weakening

of the director

anchoring

shrinks the

_HI

_{range of MS.}

4.2 TILTED ALIGNMENT ;

cf>o =F

0. -

The easy axis is assumed to be

_{given by}

_(1).

We start

with the director field

_{aligned along}

_{the easy direction. For small fluctuations}

03B8,

where

Be,

Bo

are the twist and

_{splay anchoring strengths, respectively. Ignoring}

flexoelectrici-ty,

the

_boundary

conditions

_{(7a), (7b)}

now take the form

This

_equation

has been written for the

_general

case ; in this section we

_ignore

terms

_containing

y derivatives as we consider

_only

x, z

_dependence.

The HD threshold

_EH

_(~0),

_BO),

for

_{perturbations}

0 and

_Ez,

is

_{given by}

₍₁₅₎

_except

that we

replace

(03C02/4) by qT

where _{qT is determined from}

For x, z

dependence,

the MS threshold is

computed by solving

(6)

along

with the

boundary

conditions

₍₇₎

and

_(20).

Once more we consider the mode

_having

the

_symmetry

described in section 3.2. It should be

_kept

in mind that while the HD threshold is influenced

_by

Bo

alone,

the MS threshold

_Em(0(),

_BB,

_{B~ )}

is

_affected,

in

_{principle, by}

variations of both the

anchoring strengths.

Figures

5a-5d

_{depict, briefly,}

_{the way in}

which

the

_HII

_{range of existence of MS} is affected

by changes

in

_anchoring

conditions for a

_given

set of elastic

_{parameters ;}

three different orientations have been chosen. A

_comparison

between

_figures

5a,

5b and

_figures

_5c,

5d shows that for a

_given

orientation the

_HII

_{range of existence of MS broadens when the twist}

anchoring strength

is increased and the

_{splay anchoring strength}

diminished. Once more, the effect of

_changing

the

_anchoring

_{energy is}

_apparent

_only

in the

_region

of small

HII .

Fig.

5. - Plots of

R (~0)

and

_{Qm (~0)}

as functions of

_HI.

Tilted orientation. Weak director

_anchoring.

Curves are all drawn for set

_(i)

of

_(14).

The orientation

_angles

are

₀₀

=

(1 )

0.05

(2)

0.5

(3)

1.0 radian.

In

_figures

_{5a, 5b,}

_B,

_(twist

_anchoring

_strength)

=10- 2

and

_B.

_(splay

_anchoring

_strength)

=10- 6 dyne cm-’.

In

_figures

5c, 5d,

Bo =10- 6

and

_Bo

_{=10- 2 dyne cm-1.}

A diminution of twist

anchoring

strength

and an increase of

_{splay anchoring strength}

curtails the

_HI

_{range of MS. The effect of}

5. Effect of

_{flexoelectricity ;}

_homeotropic

initial

alignment ;

rigid

anchoring ; x,

z

_dependence.

As mentioned

_already,

the effects of

_{flexoelectricity}

must be included while

_studying

electric field effects in nematics. A detailed

_{exposition being}

rather difficult in the

_present

context we shall choose the case of

_homeotropic

_alignment

(~0

=

0)

and

rigid anchoring.

It may be noted

that the HD threshold

₍₈₎

is not affected

_by

the inclusion of

_{flexoelectricity.}

For

_MS,

we are

left to solve a

_pair

of

_coupled

_equations

in 0 and

_Ex

with

_boundary

conditions

(7).

_Seeking

solutions of the form exp

(i q,, x )

we obtain from

(6)

a

_pair

of

_coupled

second order differential

equations

with

_complex

coefficients

_(the

coefficients become

_{complex owing}

to the

_spatial

dependence

of the flexoelectric

_{polarization).}

_Again,

we consider mode 1. The solution is

effected as in section 3.2

_except

that the

compatibility

condition now involves the

_vanishing

of the modulus of a

_complex

number.

_Using

the

_{compatibility}

condition the neutral

_stability

curve is

_computed

from which the MS threshold and wavevector are determined.

As the flexoelectric

_parameters

are not

_readily

available for

5CB,

the estimates of el

and e3

are made as follows. It has been shown

_by

Helfrich

[33]

that the

_magnitudes

of

el and e3 cannot _{exceed el m and e3 m,}

_{respectively,}

such that

_[34]

Using

the elastic set

_(i)

of

₍₁₄₎

we find

_that

_{el m 1}

= 9.3 x

10- 4,

_{e3 m}

₁

= 6.9 _x

10- 4

esu. Determination _{of el and e3 for MBBA}

_{[35, 36]}

have shown that both these coefficients are

negative.

We shall assume that this is valid for 5CB also and hence choose values of the flexoelectric coefficients _{such that el m} > _el > - _{el m ; e3 m} > _{e3 > - e3}

m-Figure

6

_presents

the variation of R =

EM/EH

and

QM

with

HI’ ;

EH,

the HD threshold is

given by

(8)

and is unaffected

_by

the introduction of

_{flexoelectricity}

but

_EM,

the MS

threshold,

Fig.

6. - Variations of R =

EM/EH

and

_QM

as functions of

_{HI. Homeotropic}

initial

alignment. Rigid

director

_anchoring

at the boundaries.

_{Flexoelectricity}

is taken into account. Elastic set

_(i)

of

₍₁₄₎

is

employed.

Curves are drawn for the flexoelectric _parameters

_(el,

_{e3) = (1) (0.0, 0.0) ; (2) ( -}

6 x

10-4,

- 4 _x

10- 4) ;

may be influenced. Three different sets of values

_{of el,}

_e3 are chosen for the same elastic set

_(i)

of

_(14).

It is seen that the effect of

_{flexoelectricity}

is

_negligible

_{when HII}

is

_large.

In the

_region

of small

_{Hll ,}

however,

changes

in el, e3 can determine at what

_HI

MS is

_{quenched. It appears}

that the

_Hll

_{range of existence of MS}

_gets

curtailed when the

_magnitudes

_{of el,} e3 are enhanced. More detailed calculations

_(on

the effects of variation of the individual flexoelectric

_{coefficients,}

_change

of initial director

tilt,

etc.),

through

desirable,

have not been

attempted

here

_keeping

in mind the rather

_preliminary

nature of the

_present

_{investigations.}

6.

_Possibility

of

_{PD ;}

the _y, z

_instability

mode.

So far the results have been

_presented

for an

_instability

mode whose direction of

_periodicity

is

along

E ;

this has been done

_keeping

in mind

_explicity

the

_experimental

_findings.

_Physical

reasoning compels

one to

_study

the other

_possibility

_{- the y,} z mode in which

_{perturbations}

vary

periodically

with wavevector

_along

_y.

_Bearing

in mind that

_K2

is smaller than

Kl

and

_K3,

the

_question

does arise as to whether a twist out of

the x,

z

_plane

can also diminish the total _{free energy of the}

_{sample by}

the formation of PD.

For

_simplicity

we consider

_{only homeotropic}

initial

_alignment

_{( ~0 = 0 )}

and

_neglect

flexoelectricity ;

the

_anchoring

is also taken to be

_rigid.

When

_{perturbations}

are assumed to

depend

on _y, z, it is found that

_Ey

couples

with 03B8 We solve

(6)

with

(7)

for

y

dependence

of the form

exp (i qy y).

The mode

Y,,

has 03B8 even and

_Ey

odd relative to the

sample

centre. The mode

_YI

threshold

_EM,

scaled wavevector at threshold

_QM

and cutoff field

R

Fig.

7. -

Instability

mode

_Yi

_having

direction

_{of periodicity}

_along

y, normal to both the

_stabilizing

magnetic

field and

_{destabilizing}

electric field.

_{Homeotropic alignment.}

_Rigid

_anchoring.

Plots of

R’ = EM/ EH

and

_QM

versus

_stabilizing

field

_strength

_HI.

_EM

is the mode

_Y1

1 threshold and

0M

the dimensionless wavevector. Curves 1, 2, 3 are for the sets

_{(i), (ii), (iii)}

of

_(14).

This

_figure

is to be

compared

with

_figure

1 drawn for MS

_(periodicity

_along

_x).

Even

_{when Hj}

= 0, mode

Y,

threshold

exists and is

_slightly

lower than the HD threshold. For

_Hp

= 0, the mode

Hc

(if

one

_exists)

are described

_by

_{(9), (10), (13)}

with

_{(Qx, Kl)}

_{replaced by}

_(Qy,

_{K2) ;}

Qy

= qy

h. The HD threshold for

perturbations 0

and

E,

is still

(8).

Using

the

parameters

for

33.4 °C

_(11),

one

_gets

for the mode

_YI

threshold :

Thus the mode

_YI

threshold tums out to be lower than the MS threshold

_(12).

Figure

7 illustrates the variations of R’ =

EM/EH

and

QM

as functions of

_HII.

The three

elastic sets of

₍₁₄₎

have been

_{chosen ;}

_{figure 7}

is to be

_compared

with

_figure

1.

_Though

R’ increases and

_QM

diminishes as

_HN

is

decreased,

neither does R’ tend to 1 nor does

QM

approach

0

_{when Hj -}

0.

Thus,

even in the absence of a

_stabilizing

field mode

Y,

may set in at a threshold lower than the HD

threshold ;

the mode

_YI

_{domain size may be} rather

_large

_(roughly

thrice the

_sample

_thickness).

The absence of a cut off field

_implies

that

Hc2

is

_{negative ;}

this is indeed the case for all three elastic sets of

_(14).

The

_experimental

evidence

_[17]

states that the wavevector of the distortion is

_{primarily along}

E. It is

_possible

that the formation of the Y mode is deterred due to some other

_physical

mechanism. This

requires

to be

_investigated

because in all earlier sections we have

_compared

MS threshold

with HD threshold. We now face a situation where not

_only

can mode

_YI

threshold be less

than the MS threshold with

_H);,

mode

_YI

threshold may even be lower than HD threshold in

the absence of

_Hll.

7.

_Concluding

remark.

Using

linearized continuum

_theory

an

_attempt

has been made to account for MS discovered

recently

[17]

under the

_joint

action of a

_{stabilizing Ifi}

and a

_{destabilizing}

E

_applied

to a

homeotropic sample.

The theoretical view

_point,

which

_essentially

treats the

_instability

as

arising

out of a second order

transition,

appears to

_{yield good}

orders of

_magnitudes

for the MS domain size. It appears that with a

_strong

_enough

_Hll,

the director field can minimize its total _{free energy}

_{by forming}

MS with

_{periodicity along}

E. MS is described in terms of

_splay

and bend distortions. MS can be

_{quenched when HII}

is small

_enough.

It must be borne in mind that in contrast to PD

_(in

_polymer

_nematics)

or SP

_(which

occurs near

TNA)

whose formation

is

_strongly

controlled

_by

elastic

_anisotropy,

MS sets in

_possibly

due to the modification of the electric field inside the

_sample.

The nonoccurrence of MS in twist

_geometry

is also

_{qualitatively}

accounted for. It is shown that MS domain width can be controlled

_{by changing}

the initial director tilt in a

_plane

normal

to

E ;

when the director tilt crosses a critical

_value,

MS will not form. The formation of MS

can be affected

_{by varying}

_{Hp ,}

_sample

thickness and the dielectric constants. _{MS may be}

suppressed

in nematics

_exhibiting

dielectric relaxation with

_sign

reversal

_{Of ea ; this may be}

possible by

the use of time

_varying

electric fields of

_{high enough frequency. Flexoelectricity}

and weak director

_anchoring

affect MS

_only

for low

_HII.

Physical

considerations necessitate a

_study

of the

_instability

mode

_{(YI) having}

the direction

of

_periodicity

normal to both

_HH

and E. In the

_homeotropic

_geometry

for

_{rigid anchoring}

the mode

_YI

threshold tums out to _{be less than the MS threshold in the presence of}

Hll

and also lower than HD threshold in the field free case.

_Figure

5 of

_{[17] does appear}

to

indicate the

_possibility

of a modulation

_along

_{y also.}

_Still,

_{it may be rather}

_premature

from the

point

of view of a

_preliminary

_{communication,}

to read too much detail into a

_solitary

observation. It may be necessary,

however,

to

_explore

the

_possibility

of

_oblique

domains.

The

_present

work does

_help,

to a certain extent, in

_appreciating

the

_discrepancy

between the

experiment

and

_{theory given}

in

_[17].

In

_[17]

the authors have shown

_{experimentally}

as also

transition for

_{Hp -}

_{0. In the presence of}

_HII

the authors

_again

observe a first order

_{transition ;}

however when

they again

extend the HD

_{theory they}

_get

a theoretical width of transition about an order

_larger

than the

_experimental

value. This could be due to the fact that with

HII ,

the authors observe MS and not HD. In this case _{it would be necessary} to

_study

nonlinear

perturbations

above MS threshold and then deduce the width of the

transition ;

needless to

say, this is a difficult task.

Another

_aspect

of the

_{theory developed}

in the

_present

work must be borne in mind. While

treating

MS,

the

_{periodic dependence}

of the

_perturbation

on x has been taken to be

sinusoidal. This is

_strictly

valid

_provided

that the

_sample

can be assumed to be have lateral dimensions

_(along

x and

y)

which are

_{large compared}

to the

_sample

thickness. In the

experiment

[17]

the ratio of width

_along

x

(distance

between electrodes = w = 0.33

cm)

to the

sample

thickness

₍₂

h = 0.05

cm )

is not _very

large ;

the

aspect

ratio of the

sample

cell may

have some influence not

_only

over the value of the MS threshold but also over the nature of the transition itself.

Finally

it must be remembered that while

_experiment

describes MS as

_occurring

above a

first order transition the

_present

work has taken the

_{mathematically}

easier view

_point

of

employing

small

_{perturbation analysis}

to account for

MS ;

this

_inevitably

views the transition

as a second order one. It is

_satisfying,

however,

that this

_simple

mathematical model does

yield

results some of which are in

_qualitative

_agreement

with

_experiment.

Efforts must be made to

_develop

a

_rigorous

_theory

of a first order transition to account for MS more

completely.

It will be all the more

_satisfying

if such an effort

_yields

a first order MS threshold lower than the

_{corresponding}

second order threshold

_presented

in this work.

Appendix.

Acknowledgement.

The author

_acknowledges

useful comments from a referee towards

_improving

a

previous

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Helfrich differentiates between the « free » and «

_clamped

» dielectric constants and enunciates the relations

₍₂₂₎

in terms of the « free » _dielectric constants. As we are interested in orders of

magnitudes, mainly

from the

_point

of view of a model calculation, we shall

ignore

this difference.