Piezoelectricity in cholesteric liquid crystalline structures

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Submitted on 1 Jan 1993

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structures

Harald Pleiner, Helmut Brand

To cite this version:

Harald Pleiner, Helmut Brand. Piezoelectricity in cholesteric liquid crystalline structures. Journal de

Physique II, EDP Sciences, 1993, 3 (9), pp.1397-1409. �10.1051/jp2:1993200�. �jpa-00247915�

Classification Physics Abstracts

61.30 61.40K 77.60 64.70M

Piezoelectricity in cholesteric liquid crystalline structures

Harald Pleiner

(~)

and Helmut R. Brand (2,~)

(~) FB7, Physik, Universit£t Essen, D 45117 Essen, Germany

(2)

Theoretische Physik III, Universitit Bayreuth, Postfach 10 12 51, D 95440 Bayreuth, Germany

(Received

25 January 1993, revised 28 April 1993, accepted 2 June

1993)

Abstract. Using symmetry arguments _we discuss the various possible helical structures that _can show _a longitudinal piezoelectric effect. Due to the recent experimental verification of such _an elsect in _some elastomeric cholesteric liquid crystals, _{we propose} that what is called

cholesterics could actually represent dilserent helical structures, which _can be _more complicated

than the conventional ii-e- simple

screw)

model.

1 Introduction.

Conventionally

the _structure of cholesteric

liquid crystals

is described _{as a} twisted nematic _one

Ill,

I-e- there is orientational order in _a

given layer,

where the

preferred

^direction

(the

director

fi)

rotates

helically

when

going

from _one

layer

to the next. This

picture

is obtained from _a

Ginzburg-Landau

functional

using

_a uniaxal nematic order parameter and Dm symmetry for the molecules

(I,e.

fi to -i1

symmetry).

Due to the lack of inversion symmetry _a contribution to the _energy linear in the twist becomes

possible,

which leads _to the

simple

helical structure with

D2

symmetry

(like

_an infinite

screw).

The three 2-fold rotation _{axes are} the helical axis

fi,

the director fi and

fix

i1, ^{where the latter} two _are not constant in space, but rotate

helically.

Thus, there is D2 symmetry ^for every

layer locally,

but with two of the

rotationally

_symmetry

axes

changing

^directions when

going

from _one

layer

to the next. In _{a coarse}

grained ("global"

description, averaged

_{over many}

pitch lengths,

the symmetry ^is

Dm,

with the helical axis _as

preferred

direction. This

picture

is not

changed qualitatively,

if _a small local

biaxiality

is introduced

perturbatively

[2].

Such _a

simple

_screw like structure e-g- is

responsible

for the

strong optical activity (strong optical

_rotatory

power)

and the

spatially periodic

textures well-known in cholesteric

liquid crystals [1,3].

^Due to the existence of _a two-fold rotation axis

locally

in every

layer,

which is

perpendicular

to the helical axis, the latter cannot be _a

polar

axis. This

implies

that there is _no

ferroelectricity along

the helix axis, nor can be there _a

longitudinal piezoelectric

effect.

According

to the Curie

principle

_{[4] a}

nonpolar

medium

subject

to _a

nonpolar

external force

(like

_an unidirectional _stress

along

the helical

axis)

cannot exhibit _a

polar

result

la polarization along

the helical

axis),

which excludes the

possibility

of _a

longitudinal piezoelectric

effect.

In

addition,

also the

shear-piezo

effect

(the

shear

plane

contains the helical

axis,

but the

resulting polarization

is normal to that

plane)

is _zero for Dm symmetry in the electrostatic

approximation,

where curl E _must be _zero.

Recently,

in _a series of _very

thorough investigations

[5-7] ^{the existence of} a

longitudinal piezoelectric

effect has been demonstrated

experimentally

in _some elastomeric cholesteric

liq-

uid

crystals.

These

experiments

have been

performed

_on

samples

of

cylindrical shape

for which

height

and diameter _were

typically

of the order of1 _cm. The

samples

_were obtained

by

_cross-

linking

_a nematic

liquid crystalline polymer

and

by swelling

it with _a low molecular

weight

cholesteric

liquid crystal.

^{A sufficient} amount of static

compression gives

rise to _a cholesteric monodomain [5-7] ^{with the helical axis}

parallel

to the axis of the

cylinders. Increasing

_com-

pression gives

rise to _a static

voltage

parallel to the helical axis

(longitudinal piezoelectric effect) [5,7].

^The variation of this

piezoeffect

with the cholesteric

pitch

and with temperature

has been measured and it has been demonstrated that in the racemate _no

piezoelectric

effect exists [5]. ^The

vanishing

effect in the racemate

clearly

shows that

potential

flexoelectric effects due to the _presence of deformations of the director field _are absent. We also note, that the measured

piezoelectric voltage

was

strictly

linear in the

applied compression

_{once a} mondo- main had been formed. Thus electrostrictive

effects,

which would be

quadratic

in the electric

field,

do not contribute [5]. ^{This has also been found} to be

applicable

for the measurement of the inverse

piezoelectric

effect [6]. ^{Also in} this _{case one} does not observe _a static linear

electromechanical effect in the racemate.

Obviously,

these cholesteric systems cannot be of the conventional

D2

symmetry. ^{The follow-}

ing questions immediately

arise: what is the actual structure and symmetry ^{of these} cholesteric

phases?

Is the unconventional structure

typical

for elastomeric _or

polymeric

systems _{or can} it

occur in low molecular

weight

systems as well? In the latter _{case a} direct

experimental proof

for _a

longitudinal

piezo ^effect seems

hardly manageable,

because of the

fluidity

of such systems.

There

is, however,

_an additional hint for non-standard cholesteric _structures due _to the report of _an electroclinic effect [8,9] ^{in certain low molecular}

weight

cholesteric

liquid crystals, which, again,

is not

compatible

^with the D2 symmetry. ^The electroclinic _{response was} detected either

by switching

unwound,

large pitch

systems, _[8] or

by

dielectric measurements _on

wound-up,

small

pitch

systems [9].

In the

following

_we will discuss

using

symmetry and

Ginzburg-Landau

type arguments which unconventional structures are

possible

and under which conditions

(which

_may not

always

be fulfilled in _a real

situation) they

_{can occur.} A discussion of the

hydrodynamic description

of the conventional _as well _as unconventional _structures is

given

ⁱⁿ

appendix

B.

2. Conic structures in low molecular

weight

systems.

Recently

_we have shown

theoretically

_[10] that _a chiralized biaxial nematic _can show either _a conic helical structure _or the

simple

_screw structure

depending

_{on some} material parameters

(Ginzburg-Landau coefficients).

In the former _case,

generally

the two

orthogonal

nematic

directors,

i1 and

rh, spiral together

around the helix axis

fi,

but _are tilted out of the

plane perpendicular

to

fi.

Such _a conic helical _state

generally

lacks any two-fold rotational symmetry axis and is of monoclinic

(Ci)

symmetry

(Fig.1).

Coarse

grained

_{over many}

pitch lengths

it is

uniaxial,

but

polar (Cm symmetry)

with the helix axis _as

polar

axis.

Thus,

it shows

ferroelectricity (or

^ferri- or

antiferroelectricity)

and _a

longitudinal piezoelectric

effect.

~

A

m ^~

Fig. I. Two orthogonal directions fi, _1i1, both spiraling about the helical axis fi and both tilted out of the plane perpendicular to fi by the angle en and em, respectively. The structure has locally Ci

symmetry

(except

for fi, _1i1, and fi being

coplanar).

We will show _now that _even for uniaxial nematics, ^if

chiralized,

such _a conic helical struc- ture is

possible.

The conventional

Ginzburg-Landau description Ill

^of an

isotropic

to cholesteric

phase

transition makes _use of the uniaxial nematic order parameter

Qzj

"

S(ninj )bij) by expanding

the free _energy into powers of

Qij

and its

gradients. Usually, only

_one chiral term,

QijeikiT7kQij,

is added to the non-chiral nematic terms. Then minimization leads

straightfor- wardly

to the

simple

_screw structure

ni(r)

= i~

cos#

+

ii

_sin

_#,

_with

_#

=

qofi

_r, where £ and

fi

are any

pair

of

orthogonal

unit vectors in the

plane perpendicular

to

fi

and _qo is the helical

wavelength,

related to the

pitch

do ₌

7r/qo. However,

since the

isotropic

to cholesteric

phase

transition is

generally

^first

order,

there is _{no reason} to stop ^the

Ginzburg-Landau expansion

after the

quadratic order,

I-e- the cubic order is necessary, ^too. In that order additional _non- chiral terms occur, e-g- of the form

QQT7Q,

_{T7QT7QT7Q or T7T7QT7Q}

(for

details cf.

Appendix A),

where the latter _are not

negligible compared

to the

former,

since _{we are}

looking

for _a

spatially non-homogeneous

structure and

gradients

need not be small.

These cubic terms result not

only

in _{a more}

complicated

temperature

dependence

of the

pitch,

but also in the

possibility

^of _a conic helical structure _ni

jr)

=

iii

_cos

#

+

it

_sin

_#)

cos

en

₊

(£

_x

fi)I

sin

en

with

en

= const.

(cf. Appendix A).

^This state _can have the lower free _energy

compared

to the untilted state, ^{if the}

Ginzburg-Landau

coefficients

satisfy

certain conditions.

There is _{even a} first order

phase

transition

possible

between these _{two states}

(which

also have different

qo). However,

for

large pitch lengths (small qo) always

the untilted _state

(the simple screw)

_occurs.

As in the untilted _case [2], ^{also in the conic helical state the helix induces} a small

biaxiality,

characterized

by

_a second

preferred direction, rh, spiraling

about

fi,

too.

Again

this direction rf1

can be untilted

(rh

I

fi,

I-e- em ₌

0)

_or tilted

jam # 0)

with respect to the

plane perpendicular

to the helix axis

(Fig.1) depending

_{on some}

Ginzburg-Landau

parameters

(Appendix A).

In the former _case

(em

=

0)

_or if

fi, rh,

and

fi

_are

coplanar,

the structure is of C2 symmetry

locally,

with the two-fold rotational axis

(symmetry axis) perpendicular

_to

fi. However, only

the rotational symmetry axis itself _can be _a

polar

direction since all _axes

perpendicular

to the symmetry ^axis are

subject

to the two-fold rotational symmetry.

Thus,

there is _no

longitudinal

piezoelectric

effect

(I.e. parallel

to the helix

axis) possible.

The

(spiraling) polar

axis and the

accompanying piezoelectric

^effect

along

this axis

(which

_can be detected in

large pitch

_or

unwound

systems)

_are

averaged

out _on

length

scales

large compared

to the

pitch leading

to

Dm

symmetry

globally (as

in the

simple

_{screw case,} ^where fi is

untilted).

The other

possible

state, where also rh is tilted

(rh. fi # 0,1)

and not

coplanar

with i1 and

fi,

is

locally

of the

Ci (globally

Cm symmetry ^discussed at the

beginning

of this section,

showing

a

longitudinal piezoelectric

effect.

Nevertheless,

due to the many conditions to be

fulfilled,

it

seems rather

unlikely

that

uniaxial,

low molecular

weight

chiralized nematics will be

frequently

of

Ci

_symmetry

locally.

3.

Oblique

biaxial nematic order.

Recently

it has been

proposed

_{[8] to} describe certain chiral nematogens not

only by

the usual uniaxial order parameter

Qzj

=

S(fi~fij (1/3)bij _),

but in addition

by

_a second

oblique

biaxial one, tip =

T(fi~fij~~

⁺fijfi)~~

(2/3)bijfikfif~ ),

thus

introducing

_a second director i1(2J, which is

oblique compared

to

fi,

I-e- fi

fi(2)=

cos 8

#

o,1. It is assumed [8] ^{that the nematic}

ordering

is still

given by

i1

(representing

the

long

molecular

axis),

while fi(21 represents a less

anisotropic

molecular axis and is

averaged

out in _a

perfectly

ordered nematic state, ^but becomes manifest in the presence of _an electric field

(as

in Ref. [8]) or a helix.

Adopting

this scenario _we will

show that the cholesteric state _can then support _a

longitudinal piezoelectric

effect.

The

Ginzburg-Landau

free _energy

density

for

tjj

can now be written _as

(note

that the tensors b to

k(")

_{in equation}

(3.1)

_are not

only

functions of b~j and e~jk ^as

usual,

but also of the conventional order parameter

Qij)

f

₌

aoijt~~

_{+ b~~ki}t~~tki +

c~jkim(timv~ojk

+

Qimv~t~k)

+ d~~kim

t~~vktim

~~~Uk~~"?~~~~~

_~~2/

~~

t

)t

T7 T7

Q

(31)

~~~j~j~~~~~~~~~~~~~~~~~~~

~

~~~~~~~~~~~~~

~~

jj[)))jjj)[([)~)()j[(~[((~[li _Qmpvqtrs

+

°iT~)

where _we have written down

only

terms up ^to order

q(

^and

T2,

_since

we expect T to be small. Terms linear in

T7itjk

are

possible

due _to the existence of

Qij.

When

evaluating

their

precise

structure, the tensors b to

k("I

are

expanded

into the

isotropic

tensors

bjj

^and _e~jk,

as well _{as into}

Qjj,

^whose

equilibrium

^form is

given by

^the usual

Ginzburg-Landau expansion

in

Qij. Explicitly

_we have _e.g. b~jki =

bib~kbji

+

b2(Qikbji

+

Qjibik)

+

b3QijQki

₊

b4QzkQji,

Cijklm " Cl

flij~km

+

C2Qpkfzlp~jm

^{+ and}

dijklm

"

dlfikl~jm

+

d2fiklQjm

+

AltlloUgll

_a precise

physical meaning

cannot be

given

for all the parameters involved in equation

(3.1),

the

quantities bv

la

_are related _to the biaxial order due to the _presence of

fi)~l,

^while the others govern the

spatial

structure

(cf. below).

Assuming

_a simple screw structure for fi

(a

conic structure _as discussed in

chapter

2 would not alter the conclusions of this

chapter)

_we make the ansatz

fi~ = bm cos qoz + bj~ ^sin qoz

fi)~~ =

bi~(cos

_qoz cos 8 Sin _{qoz cos ~b}Sin

8) (3.2)

+b~~

(sin

_{qoz cos} 8 + _cos qoz

coslb

sin

8)

_{+ b~z sin ~b} sin 8

where cos8

= fi i1(21 is related _to the molecular geometry. ^Since intramolecular

energies

are much

higher

than the orientational

energies

_{we are}

dealing

with in the

Ginzburg-Landau expansions,

_{we can}

keep

fixed the molecular

angle 8,

when

determining

the orientation of i1 and

fi(~l.

The

angle1b (yet

to be

determined)

_governs ^the direction of

fi(~l

on the _cone around

fi,

which is accessible for fixed

8,

I-e- _1b

= 0 _means

i1(21i fi (fi

is the helix

axis,

here taken

as the

z-direction), while1b

=

7r/2

makes i1,

rh,

and

fi coplanar.

With the

help

of equation

(3.2)

and

assuming

that 8

and1b

_are

spatially homogeneous,

the free _energy

density (Eq.(3.1))

reduces to

f

₌ aT +

(P

_{+ ~t}

cos~lb)T~

₊

O(T~ (3.3)

from which the

equilibrium

structure is obtained

by

minimization with respect to T and _1b.

The coefficients _o,

P

and _{~t are}

complicated

functions of the

angle 8,

the coefficients contained in

equation (3.1)

and the helical _wave

length.

There _are three different helical states (qo

# o)

possible,

with:

I)

_1b =

0;

_{11) 1b =}

7r/2

and

iii)

_1b

=lbo # 0,7r/2 (Fig.2).

The first structure

I)

A A

P

P

,(i)

8

A

~

8

j(2)

A(2) n

8

iii)

Fig. 2. Two oblique directions fi, fi(~)

(e

# 0,

x/2)

spiraling about the helix axis fi, where: I) both

are untilted

it)

they _are coplanar with fi and iii) fi(~) is tilted and not coplanar

(general case).

The symmetry ^of the structures is locally C2 with the polar axis parallel

(case

I) _or perpendicular

(case

it)

to fi while _case

iii)

corresponds to Ci symmetry locally.

is of

C2

symmetry

locally

^with the two-fold rotation axis

being

^the helix

axis,

I-e- _one obtains Cm symmetry

globally

when _coarse

grained

_{over many}

pitch lengths. Thus,

this is _an

untilted, polar phase,

which supports a

longitudinal piezoelectric

effect. The second _one also has C2

symmetry, ^{but the} rotation axis is normal to the helix axis

(reminiscent

of conventional smectic C*

structures)

and does not show _a

longitudinal piezoelectric

effect

(it

is of Dm symmetry

globally).

The latter structure has Cl symmetry

lacking

_any rotational symmetry axis.

Thus,

this structure also supports a

longitudinal piezoelectric effect,

but it is _a

possible equilibrium

state

only,

if

higher

order terms

(at

least cubic in

T)

in

equation (3.3)

are

important (I.e.

T is not

small),

while the structures I) and

it)

are

possible

also for small T. Within the truncated

expansion (3.3)

the

polar

state I) ^{is stable}

compared

to state

it),

if either

fl

and _{~t are}

negative,

or ~t ^< 0 and

P

> -~t, or

fl

< 0 and _{~t >}

-fl, respectively,

where

however, taking

into account the cubic terms in

equation (3.3),

the

stability

relation _among the three

possible

states is much _more

complicated.

A

polar

axis

parallel

to the helix axis _can be defined

by Qkivtki

_or fib

Vfif~,

which reads

(cf.(3.2))

fikT7jfif~

=

-qobjz coslb

sing

(3.4)

showing again

that _a

polar

axis exists

only

for _case

I)

and

iii).

4. Biaxial

polymeric

systems.

The scenarios outlined in the

previous chapters

_are not

only applicable

for low molecular

weight liquid crystals,

but _are valid for side-chain

liquid crystalline polymers

_as well.

However,

due to the existence of the

polymeric

^{backbone besides the}

mesogenic side-chains,

there _are additional

features and scenarios

possible.

We will not deal here with the relative

macroscopic

variables

Ill,12]

characteristic for the _presence of

both, polymeric

and

liquid crystalline subsystems,

but rather concentrate on additional

equilibrium

structures and their

symmetries possible

in such

hybrid

systems.

In the

isotropic phase

of side-chain

polymers generally

the orientations of the backbone chain

as well _as those of the side-chains _are random and

isotropic. However,

in the nematic

phase,

e-g- the orientational order of the side-chain

usually

also has _an orientational

(albeit weak)

effect _on the backbone [13]. ^The

averaged

conformation of the

polymeric

chain is _no

longer spherical,

but

slightly uniaxial,

where the

preferred

direction does not

necessarily

coincide with the nematic direction of the side-chains _nor is it related _to the

angle

between individual side- chains and the local backbone segment. We will _now discuss what _{structures can} be obtained

by chiralizing

^such _a

weakly

biaxial nematic system.

We will concentrate _on two

possibilities:

either the

averaged angle

between the individual side-chain and the local backbone segment is fixed and

preserved

_upon

chiralizing,

_or the

conformational

anisotropy

of the backbone is _{more or} less unaffected

by

the formation of the helix in the side-chain system. ^{In the latter} case

(conformational

biaxial cholesterics _or

CBC)

the

preferred

direction of the backbone conformation and the helical axis

generally

do not

coincide,

while in the former _case

(fixed

_cone

cholesterics, FCC) they

_can be

expected

to be

equal.

4.I CONFORMATIONAL BIAXIALITY. In the CBC _case it is assumed that the

mesogenic

side-chains form _a helix _upon chiralization _as

they

would do without the backbone

(I.e.

as a

low molecular

weight system). Thus,

the

resulting

helix is most

probably

of the

simple

_screw type,

although

conic structures are

possible

in

principle,

too

(cf.

Sect.

2). However,

there is in

addition the

(weak)

conformational

anisotropy

of the

backbone,

whose

preferred direction, I,

is

independent from,

and

generically

not

parallel with,

the helix axis

fi.

Since this

biaxiality

_may

be weak

(as

it is in

appropriate

nematic

phases),

it may be difficult _to detect the

biaxiality directly

in

experiments.

In the cholesteric

phase (even

if fi is

untilted)

the symmetry is either

C2 locally (for

I

fi)

with the axis of rotational symmetry

parallel

to the helix

axis,

_or

Cl (for I _fi # 0,1) everywhere,

except in the

single planes,

where fi is either

perpendicular

to

both, I

and

fi,

_or

coplanar

with these two

globally preferred

directions

(Fig.3). Thus,

there is _a

polar preferred

direction

along

the helical axis almost

everywhere (in both,

the

locally

Cl and C2

symmetric

_case, which

give Cm

symmetry

globally)

and _a

longitudinal piezoelectric

effect _can be

expected.

With the

simple

_screw ansatz for

i1including

_an

arbitrary

helical

phase lo,

i1

=

£cos(qoz

+

lo

+

fi sin(qoz

+

lo ),

^{and the} conformational

anisotropy

axis tilted _away from the helix

axis,

=

ficos8

+

fi

_sin

_8,

one can construct the

polar

_vector

(along

the helix axis

fi (cf. Fig. 3),

here taken _as the

z-axis) by

fi~iiijT7kfij

=

~qobkzsin(2qoz)sin~8 (4.1)

2

A

n ~

n

~~ ii)

Fig. 3. The direction fi spirals about the helical axis fi and is perpendicular to fi

(simple

_screw

structure);

the direction I _{of the backbone} conformational anisotropy is: I) oblique _or

it)

perpendicular

to fi. The local symmetry ^is Ci

(case

I) _or C2

(case it)

with the symmetry axis parallel to fi

(except

for the planes, where fi, f1, and I

are coplanar).

Obviously

^this

polar

direction does not exist for 8

= 0

(I

ii

fi),

since then the symmetry is

D2 locally (Dm globally).

The

polar

vector

changes sign periodically

when

going along

^the

helix axis

(the z-axis),

thus

representing

_a

polarization

_wave with the

polarization changing sign

_every ^half

wavelength.

Of _course,

simple

_coarse

graining

_{over many}

pitch lengths

would eliminate this

polar axis,

but

(naive)

_coarse

graining

is _a

procedure

that _never makes _sense in

polarization

_wave states _{or more}

classically

in _an antiferroelectric state.

4. 2 FIXED CONE CHOLESTERICS. In the FCC _case it is assumed that fbo, ^the

angle

^between

the direction of the side-chains and the orientation of the local backbone segment, is fixed in the _mean

by

some local interactions and does not

change

_very much when the side-chains

order in _a nematic _or cholesteric structure

giving

rise to the

macroscopic

director fi. To form _a helical arrangement ^of the side-chains

keeping

the local

angle1bo,

_a

large

portion of the backbone has to be oriented

parallel

to the helical axis.

Thus,

_a

global anisotropy

of the backbone conformation _is

induced,

where the

preferred

^direction due to the

backbones, I,

coincide with the helical axis. If the local

angle1bo

is

accidentally equal

to

7r/2,

then the system is

isomorphic

to the low molecular

weight

systems discussed in

chapter

2.

However,

in the

general case1bo # 7r/2,

_a conic helical structure arises

(Fig.4),

_as is obtained

by minimizing

the

gradient

free _energy

density

f

=

~

(coslbo

i1)~ +

Cifi

curlfi + C2

Ii

_x i1)~ijT7~fij + C3

Ii

_x i1)~ijT7jfi~ ⁺ ~~

(divfi)~

2 2

+~~ (fi curlfi)~

+

~~ IA

x

curlfi)~

₊

~ _{(i curlfi)~}

+

i~ ^Ii

x

cur1il)~

2 2

(4.2)

fi

~

Fig. 4. The direction f1spirals about the helical axis fi including _a fixed angle ~bo(lbo # 0,

x/2)

with the helical axis fi The _average preferred direction of the backbone is parallel to the helical axis fi

where

f

is written _as the _sum of _an elastic energy, where the coefficient B denotes the

strength

of the interaction to

keep

the local

angle

lbo, and _a

gradient

part ^{obtained from}

general

_symmetry

considerations,

where

C~

and

K~

characterize the various chiral and achiral curvature elastic

energies, respectively. Ki, K2, K3

and

Cl

_are the elastic coefficients familiar from

ordinary

cholesterics. With the _ansatz fi

=

ii

_cos

#+I

_sin

_{#) sin1b+I coslb}

_where

_#

=

qoi.r

and

£, fi

_and

I

_form

an

orthogonal triad,

the

equilibrium

structure is found _to be

always

_a tilted

(conic helical)

_state, where the _cone

angle1b

is somewhat different from lbo, due to the presence of the helix. For small _qo

(large pitch)

_{or strong} B the solution _can be

simplified

into

~°~'~

"

~°S1l0 Ii

⁺

^~~ j ^~5

qi)

~~

~~

~ ~5

~~ ^~K~

Sin~ffi~

~~ ~~

Equation (4.3)

represents a

locally

C2

symmetric

structure

(with

the symmetry axis perpen- dicular _to the helix

axis) quite

^similar to conventional smectic C* structures.

However,

there is _an additional local

biaxility

due to the different orientations of the side- chains and the backbone. In contrast to the helix induced

(small) biaxiality

in low molecular

weight

systems, ^this

biaxiality

does not vanish with

q(

and does not have to be small. The

biaxiality

induces _a

preferred direction,

^rf1(°) _w IA x

I)/ _ifi

_x

ii,

in the

plane perpendicular

to i1. In _a

homogeneous

_Ii-e-

nematic)

state this second

preferred

^direction is also in the

plane perpendicular

to

I,

while in _a helical

phase

it _can be tilted _out of this

plane

_(rf1(°1

#

rf1 and

rh.I

a

coslbm #

0, ^cf.

Fig.1

^{with instead of}

fi)

_{as we} will show in the

following. Describing

the tilted

biaxiality perturbatively by

_an order parameter

M~j

e

M(flijflij fli)°~flij°I

a

Ginzburg-

Landau free _energy

(density)

of the form of

equation (3.1)

is

obtained,

where tip ^is

replaced by Mjj

^and

Qij

^{stands for}

either,

the uniaxial nematic order parameter Q~j ~

(fi~fij (1/3)b~j) (with

fi

tilted)

_or the backbone order parameter L~j ~ (i~ij

(1/3)b~j).

In terms of M and rh

this _{energy can} be written

fm =M(Dl Ii rh)~

₊ El ^{rh curlrh} +

E2fi~

_{(i1 x}

rh)

Vfli~ + E3flii

(rh

_{x i1)} Vfi~

+

E4ii ii

_x

rh)

Vii ₊

jfi

_IA

_curlrh)~

₊

_jF2 Ii curlrh)~

+ F3

IA curlfi)~

2

+

F4 IA

_x

curlfi)~

₊

F5 Ii

x

curlrh)~

₊

jF6

^{IA x}

^curlrh)~

⁺

Glldivrh)~)

^{~~ ~~}

2 2 2

+

M~ (D2

+

G2 IA curlrh)~

+

G3(rh

_x

curlrh)~

+. +

O(M~, q()

2 2

Using

_an

explicit representation for1ii, fi,

and

I (e.g.

_i~

= b~z, i1and rh

given by Eqs. (A.2, A.5), respectively)

the free _energy is written _as

fm =M(Dl sin~ em

_{+ qo}

(El

+

El ^sin~ 8m]

+

q( [F[

+

F[ sin~ Bm]

+M2 (D2

+

q( [G[

+

G[ sin~ em

+

G[ ^sin~ _Bm]

^~~'~~

Minimizing

equation

(4.5)

with respect to the tilt order parameter M leads to

fm

₌

^sin~

em

(-&

+fisin~

_{em +}

isin~ em). Minimizing

with respect to the tilt

angle

of

rh,

em

(Fig.1)

then

generally gives

a non-zero tilt

angle.

The condition for this _to be the _case reduces for small _qo to

qoEi

> 0, which is

always

fulfilled under the usual

assumption

_[2]

Ei/G2

_~

Ci/K2 (cf. Eq.

(4.2))

~ qo. In that _case the tilt order parameter ^{is small of order}

q(.

This tilted

(conic)

structure, where

both,

fi and rh _are

tilted,

has _an overall

Cl

symmetry

(except

in the _very

special

_case that

rh, fi,

and

I

are

accidentally coplanar).

The

polar

axis

(along

the helix

axis)

_can be defined _as in equation

(A.7).

There is also the

possibility

that the local

forces,

which fix the

angle

between

I

_and

_fi,

also lead to _a

preferred angle,

lbmo, ^{between and}

rh,

which is then

generically

different from

7r/2.

In that _case equation

(4.4)

carries the term

MBm(coslbmo

&

I)~

instead of the term

proportional

to Di

Thus _{we can} expect under the FCC scenario to have conic helical structures with Ci _sym- metry

locally, although

local C2 symmetry

(rh ^I

₌

0)

^{with the}

polar

axis

perpendicular

to the helix axis cannot be ruled out

completely.

The former _case supports

ferroelectricity

and

longitudinal piezoelectricity,

while the latter does _not.

5. Discussion and conclusion.

From the

analysis

in the last three sections _we extract as the

major

conclusion that there

are several

possibilities,

both for low molecular

weight

and for

polymeric

cholesteric

liquid crystals,

to obtain _a

globally polar

structure with

Cm

symmetry, where the

polar

axis coincides with the helical axis. For such

polar

cholesteric _{structures one} obtains [14] ^{for the relation} between the

macroscopic polarization

P~ and the

compression

of the helix T7~R~

(where

Ri is the

displacement vector)

Pi

=

(~~lqoPiPjPkv~R~ is-1)

where p~ is the

polar

direction

parallel

to the helical axis. For

polar

cholesteric

liquid crystalline

elastomers and sidechain

polymers

above their Maxwell

frequency

_one has in addition [15]

l~ ₌

()~lqoejk ^15.2)

where ejk ^is the strain tensor associated with the permanent

(elastomeric)

_or transient

(poly- meric)

network. The associated

piezoelectric

tensor

()$~

^{takes for}

^Cm

^{symmetry the form}

(~$~ ^"

(~~~PiPJPk

+ (~~~Pz~j~ ⁺

(~~~(PJ~(~ ~Pk~()) ^(5.3)

with the transverse Kronecker

symbol b)j

= b~j pipj

projecting

into the

plane perpendicular

to the

polar

^helical axis _pi. In _a local

description

of cholesteric structures

having

Ci symmetry

locally (or locally C2 symmetric

_ones with the symmetry axis

parallel

to the helical

axis)

_one

finds

again

_a linear relation between the

macroscopic polarization

P~ and deformations of the helix and the network

(the

latter for

polymeric

and elastomeric materials

only)

similar to

equations (5.1, 5.2)

except that _{now one} has due to the lower symmetry _a

larger

number of

piezoelectric

moduli.

In conclusion _{we can} state that

locally

Cl

symmetric

cholesteric _structures

(or locally

C2

symmetric

_ones with the symmetry axis

parallel

to the helical

axis) supporting

_a

longitudinal piezoelectric effect,

_{can occur} in low molecular

weight

systems as well _as in

polymeric

and elastomeric systems, ^{but it is much} more

likely

to find them _in the latter _cases. For the _sys- tems, where

experimentally

_a

longitudinal piezoelectric

effect has been

demonstrated,

further

investigations

_{are necessary} to reveal the detailed structure of their cholesteric state.

Acknowledgements.

Support by

the Deutsche

Forscllungsgemeinschaft

is

gratefully acknowledged.

Appendix

A.

In this

appendix

_we

give

a

Ginzburg-Landau description

of the helix formation at a first order

isotropic

to cholesteric

phase

transition in low molecular

weight

systems. ^{The uniaxial}

orientational order S is not

infinitesimally

small and is assumed to be

given by

the

homogeneous

part ^{of the}

Ginzburg-Landau

free _energy. The

gradient

part ^{of this free} _energy

(density),

which _governs the

spatial

structure of the state, is

expanded

_up to cubic order in

Qij

₌

S(fiifij (1/3)bjj) reading

f

_{" a}

Qijfzkl$7kolj

+ _Cl

($7joij)($7lozl)

~ _C2

($7joik)($7joik)

+

(bl

_~jl~pq + b2

~jp~lq)fmik($7m$7jail)($7pokq)

+ d

Qklojlfpik$7poij

+

elokl (i7jQzk )(i7jQil

₊

e2Qzl(i7jQij)(i7kQlk) (A.I)

+

fjljq» ez~miv~o~j )ivioip)iv~o»)

+

fjljq» ei~m(v~oji )iv~o~q)iv~o~~)

+

gzjklmpqrsoij(vkolm) (vpvqors

+

o(s~, _ql

With the conic helical _ansatz

ni(r)

=

(i~ cos[qofi r]

+ _$~

sin[qofi r]

_cos en +

(£

_x

fi)I

sin

en (A.2) equation (A.1)

reduces _to

f

=

cos~ en lqoc

⁺ ₂

^{q( [Di}

⁺

^{D2 cos~ en]}

⁺ ₃

^{q( [Bi}

^{+ 82}

^{cos~ en}

⁺

^{83 cos~ en] (A.3)}

where the coefficients

C,

B~ and D~ follow from the parameters and tensors a to g of equation

(A.1). Minimizing

with respect to the helical

wavelength

_qo and the tilt

angle en

then leads to two

possible equilibrium

states with either

en

=

0,

the untilted

simple

_screw structure, _or

en #

0, the tilted

(conic)

structure. The latter _can exist

only,

if the coefficients in

equation (A.3)

^fulfil certain

complicated

conditions. In

particular,

for small _qo, the conic structure does

not exist.

In the untilted _case [2] as well _as in the conic _case the helix introduces _a small

biaxiality,

I-e- _a

preferred direction, rh,

in the

plane perpendicular

_to fi. With _a biaxial order parameter

Bij

_{e Q~j}

+Mij

with

Mij

_a

M(flijflij (1/3)b~j ),

where the biaxial order M vanishes _as

q(

and

Q~j

is assumed _to be

given by

the

procedure

^outlined

above,

the

Ginzburg-Landau expansion

leads _{to a} free energy

(density)

^for _M~j and its

gradients

of the form of

equation (3.1),

when t~j ^is

replaced by Mij

there. In terms of rh and fi this

equation

has the form

fm

₌ Firh curl& + F2fi~ (i1 x

rh) Vfli~

+

F3flij (rh

_x

fi) Vfi~

+

)Gi ^(divrh)~

+

)G2 IA curlrh)~

+

)G3Irh

^x

^curlrh)~

⁺

)G4Ifi ^{curlrh)~ (A.4)}

+

)G5

^{IA x}

^curlrh)~

⁺

)G6 ^{IA curlfi)~}

⁺

^G7(rh

^x

^cur1il)~

2 With the

general

ansatz

m~(r) =it(sin[qofi r] sin1b cos[qofi r]

sin

en coslb)

IA-S) -bi (cos[qofi r] ^sin1b

+ sin[qofi

r]

sin

en coslb)

+

Ii

x

b)I

_cos

en coslb

and

equation (A.2)

this reduces to

fm(1b)

₌ ^a' +

fl' ^cos~

_{1b + ~t'}cos~_1b

IA.6)

from which _{one can} in

principle

obtain _an

equilibrium

solution with

nonvanishing

tilt

angle

of

rh,

sin em e cos en

coslb #

0

(Fig.1).

There is

Ci

symmetry and _a

polar

axis

(along

the helical

axis)

defined

by

qoeijk

IA V)fir(fir V)flip

=

-q(

_jib sin

en

sin em sin

en

xm

IA.7)

if the tilt

angles en

and

em

_{are non-zero} and if i1 and & _{are not}

coplanar

with

fi (in

the

coplanar

_case

en

=

7r/2 em

and fix rh is

untilted,

since sin

enxm

₊

(cos~ en -sin~ em )~/~

₌

0). However,

due to the smallness of the biaxial order

M,

all the _terms

contributing

to ~t' ⁱⁿ

equation (A.6)

_may

generally

be

neglected

in the _case discussed here

leading

to the untilted rh structure,

only.

Appendix

B.

Hydrodynamic description.

In the bulk of this

manuscript

_we have discussed the conditions for the _appearance of cholesteric structures with _a

polar

axis

parallel

to the helical axis

(Cm

_symmetry

globally). Naturallly

the

question

arises how the

macroscopic

^and

hydrodynamic properties

^of such _a

phase

differ from those of conventional cholesteric structures

(Dm

symmetry

globally).

We will concentrate in the

following

_on

global properties exclusively,

I-e- _on

length

scales

large compared

to the

pitch

for which _a

"coarse-grained"

[16]

hydrodynamic description applies.

Such _an

analysis

has been

given by Lubensky

[17] ^for

nonpolar

cholesterics. We modified this

description by incorporating

into the

hydrodynamic equations

reversible

dynamic

terms

[18-24]

^that exist

only

for the various types ^of

polar

cholesteric

phases

considered here. These

polar

terms

(including

in

particular

the material tensor g~j ^{or its}

anisotropic

part

ga) provide

_an additional

coupling

between the _stress tensor and the molecular field of the director. Since

they

_are of

higher

order in the

gradients

than the

non-polar coupling

terms,

they

do not

play

_a

significant role, generally,

and it _seems to be hard to

distinguish experimentally

_a conventional from _an unconventional

cholesteric _structure

by

their _{presence or} absence. Another subtle

polar

effect

(a dissipative dynamic coupling

between the stress tensor and _a temperature

gradient

_as well _{as an} electric

field)

_was mentioned in the

appendix

of reference [25].

Static

polar coupling effects,

such _as

(longitudinal) piezoelectricity

and

pyroelectricity,

_were discussed

by

the authors for

both,

low molecular

weight liquid crystals [24-26]

and

liquid crystalline

elastomers

[15, 27].

^{These effects exist}

only

in the

polar

structures discussed in this paper and _are absent in the conventional cholesteric structures. Of _course, to determine

directly

whether the systems studied in references [5-7]

belong

to the class of

polar cholesterics,

it would be

highly

desirable _{to measure a} P E

hysteresis loop

in order to find _out the

precise

electric

properties

of the

ground

state of these cholesteric

liquid crystalline

elastomers.

Clearly

the _same type ^of

experiment

is also _necessary for the low molecular

weight

systems

investigated

in references [8, 9].

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