Continuum theory of ferroelectric smectic C* elastomers

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Continuum theory of ferroelectric smectic C* elastomers

E. Terentjev, M. Waxner

To cite this version:

E. Terentjev, M. Waxner. Continuum theory of ferroelectric smectic C* elastomers. Journal de

Physique II, EDP Sciences, 1994, 4 (5), pp.849-864. �10.1051/jp2:1994169�. �jpa-00248006�

Classification Physics ^Abstracts

61.40K 61.30C 77.60

Continuuln theory of ferroelectric smectic C* elastomers

E. M.

Terentjev

and M. Warner

Cavendish Laboratory, Madingley Road, Cambridge CB3 OHE, England

(Received

15 October 1993, received in final form 21 December 1993, accepted 2 February

1994)

Abstract. The continuum free energy is derived for _a crosslinked network of smectic C and chiral smectic C* liquid crystal polymers under small strain and orientational distortions. The

coupling between elastic strain, the smectic C order parameter and the polarization

(induced

_or spontaneous in C*

phase)

is examined and several _new orientational and polarizational effects

are predicted. Using _group representation theory _we obtain all relevant invariants, describing

the coupling terms in the free _energy. It is shown, _on the other hand, that _some effects in conventional smectic C liquid crystals, for example the bistability and switching in ferroelectric C*, _no longer exist in corresponding elastomers. Possible experimental configurations _to observe

some of the predicted effects _are briefly discussed.

1. Introduction.

Liquid crystalline

elastomers have

recently

attracted much theoretical and

experimental

at, tention.

Crosslinking

of

polymer

chains creates a rubber with the

corresponding

translational

elasticity.

When such

crosslinking

is

performed

in _an

anisotropic

state, then memory of such _a such state becomes frozen in because the

topology

of

networking points

distribution is fixed. A

variety

of _new mater1al

properties

is

generated

_[1-3]. These

properties

stem from the

coupling

between mechanical

fields, easily

created and controlled in

rubbers,

and the

anisotropic phase

structure of

liquid crystals.

A considerable _amount of work has been done _on nematic elas- tomers,

including preliminary phenomenological

models [4, 5],

experiments

_[6-8] and detailed

molecular theories

[9-11].

Smectic elastomers represent a very different system ^{from the}

mechanica1point

of view be-

cause

they

have two characteristic

length scales,

the smectic

layer spacing

and the

(typically

much

larger) networking point separation. Layers

represent a more

rigid

component of the _sys- tem and

they

_may be rotated _or otherwise distorted

by

deformations of the

polymer

network.

Physically

the main

coupling

^between

layers

and the network is the

penalty

^for their relative translation. Let

U(r)

be the network

displacement

vector and

u(r)

the local

displacement

of

layers along

their normali. Then the

coupling

_r-

kT(N~ /R()(u

Uz_)~

penalizes

_an attempt to translate the

periodic

smectic lattice relative _to the

underlying network,

^with

N~

the crosslink-

JOURNAL DE PHYSIQUE it T 4,N' 5, ^MAY 1994 32

ing density

and

Ro

the radius of

gyration

of the network

strand, R(

r-

Np~

It is

important

to

emphasize

that the local

coupling (u _{Uz)~ generates}

the

coupling

between

layer

distortions and certain components of _a uniform strain

(lzz

^~"

)~, (lz~

^~" )~ and

(lzy

^~"

)~,

where

dz ox

by

ljj

₌

dUj/dxj,

with coefficients

proportional

to the _square of the

sample

size. This

effectively

locks uniform smectic distortions

affinely

with the strain I.

However,

the smectic variable

u(r)

determines the

phase

of _a

density

_wave at wavenumber _{qo> an} inverse

layer spacing,

^while

U(r)

measures

displacements

_{near zero} wavenumber. Therefore Uz and _{u are}

independent

variables

and do not count the _same

displacement

twice.

Together

with _an afline part,

u(r)

_may contain

independent, short,wavelength layer

deformations that

keep

the network

(crosslinking points distribution)

and the

sample shape

undistorted at

large.

We therefore consider two distortion fields of the network and of the

layer

system,

realizing

that _a

significant

part ^{of the}

layer

deformation _u is

rigidly

determined

by

the network deformation Uz _on

large length scales,

u ₌ Uz + vi Thus _vi is the

fluctuating

part ^of u.

Recently

_we have

presented

the continuum

theory

of smectic A and chiral smectic A* rubbers [12], ^based on group representation

theory, predicting

several _new mechanical and

piezoelectric

effects

generated by coupling

between I and Vui This paper reports ^the results of _a similar

analysis

of the smectic C and chiral

smectic C* elastomers

(for

convenience further _{on we} shall

drop

the

subscript

I at the smectic field _vi

).

Smectic C is _a

spec1al phase

_among all the

variety

of

liquid crystals.

Since the

discovery

of the ferroelectric behaviour of the chiral C* smectic

by Meyer

et al. [13] ^{this class of mater1als has} attracted great ^{industr1al and scientific} interest. Most

important properties, however, require

well-oriented

samples,

often with _an untwisted helicoidal structure. This orientation is achieved in conventional C* mater1als

by

surface

alignment,

electric _or

magnetic

fields _or

by

shear flow.

These

techniques

often fail for

liquid crystal polymers

due to their

high viscosity.

Crosslinked networks of such

polymers

offer _a wide prospects ^of

constructing

various orientations and

polarizational

_states in the smectic C*

phase by

^{mechanical fields} [3,

8, 14].

^{As in the} cases

of nematic and smectic A rubbers, the network strain tensor

ljj plays

_a vital role. The infinitesimal strain

ljj

is defined _as

A~

_&,j where the

Cauchy

strain tensor

A~

determines the

sample shape [9-11].

This _{paper uses group} representation

theory

to deduce the

complete

set of invariant terms in the free energy of smectic C and C*

elastomers, involving

translational and orientational variables and the

polarization.

In addition to the standard smectic, ^electrical and rubber elastic parts to the free _{energy, we have new terms}

unique

to smectic C elastomers

associated with _new

couplings

made

possible.

In the end of this _{paper we}

briefly

discuss

some

possibilities

of

experimental

observations of these

couplings including

spontaneous ^and

electrically

driven distortions.

Let _us note here that

liquid crystalline

elastomers

(as

all

rubbers)

_can withstand very

large

translational deformations and the

general theory

should not be limited to infinitesima1dis-

placement gradients,

_as would be the _case in conventiona1elastic solids. On the level of _a

continuum

model, however,

it is much

simpler

and _more illustrative to restrict ourselves

only

to the _case of small deformations. This _case retains all essential symmetry properties and many

predictions

can be

readily

made

simply by writing

^the continuum free _energy of deformation.

This paper is

organized

_as follows. Next section

gives

_a

general

discussion of smectic C and C*

phases

and _sets the

ground

^{for the}

subsequent

detailed

analysis.

In section 3 _we describe the continuum model of smectic and network

coupling deep

in the C*

phase, using

the

properties

of its

point

symmetry group C2

(in

all _{cases we} separate ^{chiral and non-chiral} contributions, which will survive in the

corresponding

centrosymmetric smectic C

elastomer).

Section 4

gives

the similar

analysis,

but _near the A-C

phase

transition, where the tilt

angle

(related

to smectic C order

parameter)

is small and _{one can} obtain

simplified

equations using

the

properties

of _more

symmetric

A

phase.

Section 5 considers _two

(out

of

many) examples

of

particular

effects that _are

predicted by

this continuum

theory:

spontaneous

shape change

of the

sample

at the A,C

phase

transition and

switching by

external electric field. The reader not

wishing

to follow the _group theoretic arguments ^of this _paper will find all the allowed

coupling

terms stated and discussed and should

simply skip

the _group representation

analysis.

For instance, at the

beginning

of section 3 the character of rotations is discussed

geometrically

and

then,

in

equations (6-9)

the

couplings

_are discussed in groups. The

appendix

interprets

geometrically

less obvious terms in

equations (8)

^and

(9).

The

coupling

contributions to the free _energy

density

_are summarized in

(10).

Likewise in section

4, equation (13) yields

the allowed terms,

they

are discussed

geometrically

in

(14-17).

The

examples

^discussed in section 5 and

appendix give

further

geometrica1feel

for _our results

(as

does Ref. [12] on smectic A

elastomers).

2. Ferroelectric smectic C* elastomer.

Smectic C

liquid crystals

have _a one-dimensional

density

modulation

(layers)

with the

optical

axis

(director n)

tilted with respect to the

layer

normal. This represents _a

significant

difference in

comparison

with the smectic A

phase,

where the

optical

axis direction _n is locked _to the local

layer

normal and is not _an

independent

continuum variable. Smectic A has the

point

symmetry

group Dmh

(or non-centrosymmetric

Dm for chiral smectic A*

mater1al).

Smectic C has _an additional

degree

of freedom _an azimuthal rotation of the tilted director around the

layer normal,

which is _a Goldstone mode at the transition from the

A-phase (from

which smectic C

typically appears).

The

C-phase

has _a two-component order parameter

f, representing

the

coupling

between the _wave vector of the

density modulation, (((no

and the director deviation,

&n, ^{from the I} axis of the

underlying

smectic A.

Assuming

that the

layer

structure and the nematic order do not

change significantly during

this

transition,

the two component ^order parameter may be written _as

f

r- [no x &n]:

~Z ~~Z~y,

fy

~Z~z

(I)

and is _a vector in the

layer plane, perpendicular

to &n

(see Fig.

I and [15] ^{for details of the} A-C

phase transition).

The

point

_group of the

resulting

smectic C

phase

is

C2h

and that of the chiral smectic C*

phase

is

C2.

It should be noted that _an attempt to create a continuum model of chiralsmectic C* rubbers has been made

previously

_[16]. That

work, however,

_appears

unsatisfactory

^for two

important

reasons:

(I)

the

polar

unit vector

fi (not

to be mixed with _our notation _p for _a local

polarization

axis!

along

the axis of _a helicoidal _structure is chosen _{as a} variable in the free _energy

density (coarse grain approximation).

This violates the

point

symmetry ^of the

phase. (ii) Following

the old arguments of de Gennes _[4]

only

the relative rotation of the network and of the smectic c-director is accounted

for,

while _{we now} know that

anisotropic

rubbers have much _more

subtle,

both

symmetric

and

antisymmetric elasticity ill,

12, 17]. As _a result of

(I), theory

_[16] obtains

some non-existent

effects,

for

example

terms like

p(p E)V# (Eqs. (1),(4)

of

[16]) predict

_a

polarization along

the helix axis _on

changing

the helix

pitch...

Other effects in [16] are known to have different

interpretation (for example,

standard

flexoelectricity).

In

addition,

the model [16] ^{is not}

applicable

to any uniform

(I.e. non-helical)

structure. As _a result of

(it)

that model fails _to describe effects of symmetric shear strains of the network and related

piezoelectric

effects

(we

shall _argue below that these effects _may be stronger ^{than relative}

rotations).

Therefore

we have to consider the task of

deriving

_a continuum

theory

^of smectic C and C* elastomers in its full _scope

here, heavily referring

to _a similar

analysis

and detailed discussion of smectic A system

[12].

4

~

8 n

..I.

_""

P

(a) (b)

Fig, I. Local geometry ^of _a side-chain smectic C elastomer

(a)

and orientation of relevant vectors in _a smectic C phase

(b).

0 is the tilt angle of the optical axis _n with respect ^{to the} layer normal _z, p is the direction of local polarization Pa in the layer plane

(also

the C2 symmetry

axis).

~ defines the

azimuthal angle of the c-director

([I

_x

n x

I])

in the layer plane.

We shall

develop

the continuum

description

of smectic C*

(or C)

elastomers from two different

positions: deep

inside the smectic C

phase

and at small order parameter, in this _case

utilizing

the

properties

of the A-C transition. Far from this transition, when the

magnitude

of the order parameter (f( ^is not

vanishingly

small and the helical

twisting

is

prevented by

_some

means, the

properties

of the material under infinitesimal deformations _are not different from

those in conventional monoclinic

crystals

and _are well-described in various textbooks [18]. ^For

example,

such "unwound" smectic C* has _a standard

piezoelectric effect, i~kPil)~

,

with

eight independent

coefficients _I

(the superscript

S denotes that

only

the

symmetric

part of

l~

is taken into

consideration, I[

=

)[l~

₊

lpi]).

Of these

eight

invariants, ^four contribute to the

change

P in spontaneous

polarization

Pa

(assume

that P~

points along

the

f

axis in the

Fig. 1):

Pylyy, Pylzz, Pylz~

and

Pyl(~. ₍₂₎

The other four

independent

_components of this

piezoeffect

_are induced in two

perpendicular

directions:

Pzl(~, Pz~(~

and

Pzl(~, Pzl(~. (3)

Note, that _even in the

helically

twisted

material,

the

piezoelectric polarization

induced in the I direction

along

the helix axis

(layer normal) by

the

symmetric

shear deformation is not

vanishing

after

averaging.

This effect is similar to the

piezoelectricity along

the helical axis in cholesteric elastomers [17] ^{and is absent in}

conventiona1liquid crystals.

When the smectic C* order parameter (f( ^{becomes small} near the

phase

transition to the A*-

phase

at

increasing

temperature, the

eight independent piezocoefficients corresponding

to terms in

equations (2)

and

(3)

_can be

expanded

in powers of

(f(. Comparison

with the

corresponding piezoeffect

in smectic A [12], To

(Pzl(~ Pyl(~),

tells that six of these coefficients vanish at

(f( -

0,

while the coefficients of

Pyl(~

and

P~l(~

become

equal

and of the

opposite sign.

Analysis

of representations of the group C2

(and

of its centrosymmetric counterpart

C2h) gives expansions

of relevant contributions to the free energy in terms of irreducible represen,

tations,

_see table II below for trivial

representations

ro.

Clearly

the number of

independent

terms in each category is very

high,

which is a natural consequence of the _very low symmetry

of the

phase.

There is _a total of144 invariants

arising

from the _cross

couplings

of P and

ljj

with orientational deformations of the smectic C*. In most realistic _cases of smectic C

phase, however,

it is not necessary ^to take all of these invariants into account. In the

example

above

we have _seen that

only

_one out of

eight

combinations

r-

Ill _was relevant at small order _param- eter (f( < I where _one is

tending

to the _case of smectic A* elastomers. One _can expect similar reduction for other terms _as well. The _reason is

that, typically,

the smectic C order _parame- ter is

proportional

to the tilt

angle

of the

optica1axis

and is small in _a wide

region

^of

phase

existence. In this _{case some} of the invariants under the C2

(or C2h)

_group operations, ^which

are

proportional

to

higher

_powers of (f( can be

neglected.

The

remaining major

effects _can be discovered using ^the

properties

of the _more

symmetric

_group

Dm (or Dmh) appropriate

to

smectic A*

(or A)

and the order parameter

f

which _{grows as we} evolve from the

corresponding A-phase.

It is done in the

fashion,

similar to the Landau

description

of the 2nd _order

phase

transitions when invariants in the

low,symmetry phase, containing

the

given

_power of the order parameter, are found _as triv1al representations in the

high,symmetry phase

^{of the free} energy

contributions, containing

^the

physical quantity

that is the order parameter in that

given

_power.

For instance the two component order parameter

(

of smectic C transforms

according

to the irreducible

representation rj

of the _group Dm

(or Dmh)

of smectic A. Likewise the _wave vector of the

density

modulation

k((no

transforms _as

ri~°°J

_{and the director}

_deviation,

_&n, from the I axis in smectic C transforms _as

r)~°°~ Taking

combinations and _powers of these and other

quantities

_we must, within

Dm, identify

the

resulting

invariants of the C

phase.

This

analysis using

smectic A _{as a} basis is

meaningful,

^of _course,

only

_so

long

_as it is

possible

to

expand

in powers of the order parameter, I.e. _near 2nd order transitions which

happens

to be the _case for the smectic C

phase.

Such

analysis

for the

liquid

smectic C*

(or C) phase

is

presented,

for

example,

in [15, 19]. ^{In elastomers} we have to examine

additiona1effects, imposed by

translational deformations. In the limit of small tilt

angle

9 of the

director,

the free _energy

density

is much

simpler

to

analyse, using

smectic A _{as a} basis. At the _same time such _an

analysis

retains all essential symmetry

properties

^{and effects of the smectic C*}

phase.

We shall examine this limit in _more detail in section 4.

3. Free _energy of deformations

(C2).

Note,

first of

all,

that all effects described _in [12] ^{for smectic A*}

(or A)

elastomers remain valid below the transition into the

corresponding

C

phase,

with the difference that

they

become

split

into many

independent

invariants in the low symmetry

phase,

_see for instance the

piezoelectric

terms above. In this section _we shall _use the full low symmetry C2 ^first

recalling

the

underlying liquid

smectic C* free _energy [19] ^{in the absence of}

crosslinking,

and then

giving

^the form of

the rubber elastic free _energy

(at

small

deformations)

for _an elastomer with C2 symmetry.

The monoclinic

chira1group

C2 of two-fold rotation about the axis

f

is

cyclic (I.e. abelian)

and has

only

one-dimensiona1irreducible

representations (see

Tab. 1):

triv1alro>

which _corre-

sponds

to a vector component

along

the

principal

axis direction

(which

is thus also the direction of

polarization,

_p, in the

layer plane),

and

ri>

which

changes sign

under 180° rotation about

this axis and

corresponds

to _a vector component ^{in the}

plane including

the

layer

normal land the director tilt &n,

figure

I.

Any

vector transforms _as V

= ro e 2rj under this

group's operations.

Hence the chiral smectic C*

phase

has _an

equilibrium polarization

P~ in the

layer plane,

which

corresponds

to a trivial component ro in the vector

representation

and is

perpendicular

to the director

(in

_contrast, the non-chiral _group

C2h

has

only

a scalar trivial

representation and, therefore,

regular

smectic C is

non-polar

and uniform in

equilibrium). Normally

^the bulk

polarization

is

Table I. Characters of irreducible

representations

of the _group C2 with operations:

unity I

_{and rotation}

by

180° around the

ji

axis in the

layer plane

_c2.

C2 _c2

ro

i _-

scalar,

P~

rj

i -i _-

I,

_ni

V 3 -1

-P,R,V,fl

V 4§ V 9 1 _- 1 ₌

l~

₊

l~

Table II.

Expansions

of _terms

containing I, fl, f

and P in the free energy

density

in _terms of irreducible

representations

of the group

C2.

d

_[V4§

Vi 4ro

e

2ri

AA

jv

~

vj ro

e

2ri

(fl)~

_{[[V2]2] 13ro e}

_8ri

fill

_[V4§

_Vi

_{© V}

_8ro

_e

_10ri

flvfl

[V4§

Vi

© V 4§ V 28ro e

26ri

PI~Q

_{V 4§}

_IV

_{4§V]4§ V 28ro e}

_26ri

Pflvfl

_V

©

[V

4§

Vi

© V 4§ V

80ro

e

82ri

not observable due to the

twisting

in the smectic C*

phase:

at the

phase

transition from the A*

phase (Dm fl

C2 _group

transformation)

the Lifshits

invariant, [fz(dry/dz) fy(dfz/dz)]

9~(d#/dz),

induces _an equilibrium twist of this

polarization

when

moving along

the

layer

r-

normal direction ?.

A convenient variable _to describe deformations in the C

phase

is the rotation vector with three components:

layer

distortions fix ₌

du/dy, fly

₌

-du/dx

and azimuthal rotation of the director

flz

₊

#,

^{where ? is} chosen _{as an}

equilibrium layer

normal and

u(r)

is the local

displacement

of

layers along

I. Thus flz _- ri represents the azimuthal rotation of the director in the

layer plane. By choosing

£

along

_ni + c we can

identify fix

+

flc

_-

rj

and

fly

e

flp

_-

ro (Tab. I).

The smectic C elastic free _{energy can} be then derived

(see _[19, 20]) by looking

^{for the} trivial representations contained in the term

proportional

to

(VQ)~,

since

this free _{energy cannot}

depend

_on the

angles

of uniform rotation themselves. There _are certain constraints _on

layer deformations, following

from the definition of Q:

dflz/dz

=

any /dz

₌

0,

dflz lox

=

-ally/dy. T~king

into account these constraints _one has

only

the

following

relevant irreducible

representations

^Vfl e V © V -

3ro

e 3rj

(in

_a brief

appendix

_we discuss how these constraints reduce the number of

invariants). Therefore, (Vfl)~

_-

12ro

e....

Adding

the

layer compression

term and

grouping

those combinations that transform into each other

by integration by

parts we can write the smectic C _energy of deformations

(see [19])

~~ ²

~2~

²

~2~

²

~2~

²

i~

dz ^~

~~~

dp~

^{~ ~~~} dc2 ^{~ ~}

~dpdc)

^~

~

jd~uj ^d#

^~

jd~uj ^d#

^~

jd~u jai

~ dc~

dp

^{~ ~}

dp2 dp

^{~ °}

dpdc

dz ^~

~~ ~~~

~

~

~~ ~$

~

~

~~ ~~

~

~

~~~ ~~

^~~~

where _we have returned to the

original

concept of

layer displacement

^field

u(r)

^{and azimuthal}

rotation

angle #(r) (in

this section the order parameter is assumed _to be of the order of

unity).

Cartesian coordinates d and

ji correspond

to the usual £ and

faxes,

chosen

along

the director

projection

_ni

" c and the local

polarization

_p

(see Fig. I). Equation (4)

contains

only

_non-

chiral _terms and is the _same in smectic C and C*

phases.

As shown in the table

II, (l~)~

_-

13ro

e

8rj

and hence there _are thirteen

independent

linear rubber elastic

moduli, corresponding

to the

following

terms in the free _energy

Frub

_"

IQ

_~~jS

jS.

2 'J _ij ki

(j

₎₂

(jS

₎₂

(j

₎₂

(jS

₎₂

(jS

₎₂

(jS

₎₂

zz zc cc pp cp zp

lzzlf~, lzzlcc> lzzlpp, lf~lcc> lf~lpp, _lpplcc> I)~lf~ (5)

where

subscripts

_c and _p

correspond,

_as in

equation (4),

to the directions in the

layer plane along

_ni and the local spontaneous

polarization Pa, respectively.

In most cases the constraint of the rubber

incompressibility

is present

(see

the discussion in

[12]).

This condition reads

Det[A~]

₌ I, which fixes the trace of the small deformation tensor

l~, Tr[ljj]

- 0. As _a result of this constraint the number of

independent

terms

r-

l~ in the elastic _energy of

incompressible

rubber is twelve. At the _same time the apparent ^{number of linear rubber} elastic invariants is

only nine,

because the

incompressibility

constraint eliminates all terms in

(5) containing

lzz

by grouping

their moduli in linear combinations. It is

interesting

to compare invariants

(5)

with the

corresponding

^rubber elastic _energy in the smectic A elastomer

[12],

^{which has}

only

three apparent invariant terms

(see

also the next

section).

A _new feature in smectic C elastomers

is,

^of _course, the

coupling

between deformations of the

network, ljj,

and deformations of the smectic C structure, Vfl. As _we have discussed in the

Introduction,

uniform

layer

rotations

flp

and

flc

_are

rigidly

locked to the network strain

components lzc and

lzp by

the constraint _on relative

displacements (u-Uz

)~. ^Nonuniform

layer rotation, represented by Vfl,

_may be determined

by

small non,affine smectic distortions and

produces independent

effects

(the

_same,

incidentally, applies

to the component flz +

#,

which is

a

purely

orientational

field). Considering only

infinitesimal

symmetric strains, l~

-

4ro

e 2ri

and relevant components ^{of Vfl} -

3ro

e 3ri _we obtain

eighteen independent

invariant terms in the

coupling

free _energy,

corresponding

to all

possible products arising

^from:

and

~~~~~ ^~

~~~~~~

~ ~~~P ^~ ~~P~ ^~

~i~

~

~i

~

~~

with _a different coefficient for each

product

that arises

(the incompressibility

constraint, lzz _m -)cc

lpp,

makes the apparent number of

independent

invariants

fifteen).

We discuss

(6)

in _more detail in the

appendix.

It is

important

to note that all these terms are chiral and

are absent in

centrosymmetric

smectic

C,

where the

only

linear

coupling

between smectic and rubber distortion fields is related to the uniform

layer compression

_{I +}

(au /dz):

izzi; >ccl; >ppi; >fci (7)

[these

terms are present, of _course, in the chiral mater1al too in addition to

(6);

the first _term here does not survive in _an

incompressible system].

There _are

independent

effects

describing

the

coupling

between the strain and the relative

layer rotations, flc

and

flp.

This

phenomenon

_represents the

penalty

for deviations from affine

deformation pattern and _are similar _to those in smectic A elastomers

(see

the discussion in Introduction and in

[12]),

~~~&~ ~~ ~~~&~ ~~ ~~~T~

^~~

~~~&

^{~~ ~~~}

~cP~l ~zP~

(as usual,

the term

containing lzz

is

effectively

eliminated in _an

incompressible system).

It is

interesting that,

unlike in the smectic A case,

layer

rotation _can be

generated

not

only by

shear

deformations,

but also

by

extensions and

compressions, given by diagonal

elements of

~;j. Finally,

there is _a

coupling

between shear deformations and the relative azimuthal rotation of the

director,

flz _e

#,

unique to these tilted smectic elastomers

jS _~, jS _~ (g)

cp , zp

The character of these terms and the

geometrical

_reason for the absence

of1(~#,

for

example,

is discussed in the

appendix,

as is the

coupling

of

l~

_to

_#.

Summarizing

the discussion in this section _we should write

symbolically

linear _cross terms in the total free _energy of deformations in _a smectic C* elastomer

Fd ₌

m,jkil(Vkfli

₊

b,jl(i

₊

I,jkl(flk. (10)

Here the

corresponding

terms describe: strain-curvature

coupling (6)

with 17

independent

coefficients m~ki,

strain-layer compression coupling (7)

with 4 coefficients

b~,

strain-rotation

coupling (8),(9)

with 8 coefficients

I,jk.

The

coupling

between the

antisymmetric

part ^{of shear} strain and the azimuthal rotation of the director in the

layer plane

is

briefly

discussed in the

appendix.

It has to be

emphasized

_once

again

that

only

non-afline

layer

rotations, Vui _as mentioned in the

introduction,

_are relevant for this free _energy: uniform

gradients

of _{u are}

rigidly

locked with the

corresponding

_components of strain in _a

macroscopic

network.

Evidently

the total number of invariant _terms in the free _energy of smectic C

(C*

elastomer is

already

_very

high

and will become _even

higher

when

considering polarizational effects,

_see table II. It is difficult _to separate ^the

major,

most

significant

effects. It is

worth, therefore, examining

the _case of

relatively

small order parameter (f( < I. We shall examine this limit in _more detail in the next section when

dealing

with the

PAS(

and

PISVf

_terms since the 108 invariants

provided

within C2 symmetry

by Plsfl

_and

PISVfI

effects _are

clearly unmanageable.

4.

Expansion

at small tilt

angles (Dm).

At (f( - 0 _we arrive at the _case of smectic A

(or

chiral smectic

A*)

elastomer [12]. ^{We take} the

liberty

to re-write here the relevant parts ^of the

coupling

elastic free _energy,

depending

_on

1 and non-afline part Vui

(we

remind that the

subscript

I is _common for all entries of _u and

fi _dropped

_for

convenience).

We address the reader to that _paper for all details and discussion.

~ ~q~ ~lt

~~

~~~~

~ ~~

~~~~~~j

^~~~

)~

~2q~

21t~j

+

kj2

_{(~XX ~YY}

dX~g

^~~~~~~~ ~~~

+ Po

(P~l(~ Pyl(~)

₊

P(P~l(~ Pyl(~) ~"

⁺

^P'lzz P~ ) _{Fyi) (I I)}

Z X

where tildes mark the chiral terms which _are absent in

centrosymmetric

mater1als. The _con- straint _on relative

displacements (u

= Uz + vi

(r),

_see the discussion in

Introduction) essent1ally

eliminated several terms in

(11)

and transformed relative rotation

coupling

into

lsviui

_In-

stead _one arrives at the effective _mass for the field _vi

(r)

and the nonlinear

contribution,

similar to that for the Helfrich-Hurault effect [19].

At infinitesimal deformations the rubber

elasticity

part of the free _energy,

corresponding

to the uniaxial symmetry ^of smectic A is

~~~~~~

~ ~YY~~ ^~

~~~~~~~z)~

~

(~~z)~]

^~

~l22~$z

~

~(~~y)~

~ ~~yj

(~~)

[compare

with

(5)

^for the system

deep

in the smectic C

phase].

Three rather than five coeffi- cients

Qi

_{[12] are}

required

for this uniaxial system because of the

incompressibility

constraint

demanded for rubber.

We remind the reader that in the limit (f( < I the

analysis

is to be carried out in the smectic A* reference state with the symmetry Dm

(or

Dmh in

centrosymmetric mater1als), regarding

all smectic C effects _as

perturbations.

These additional

phenomena

are associated with the addition of _an extra vector into the

problem, namely

the smectic C order parameter

f,

_a vector in the

layer plane

and hence associated with the

representation

rj of the _group

Dm, see table III. Let _us

consider,

for

simplicity, only

^uniform or

helically

twisted

(along I) confijurations ^(so

^{that V ~}

^d/dz

^-

^{rz). Then,}

^for infinitesimal deformations

(I,e. discarding

all I components ^{for the} reasons

already

^discussed in Sect.

2)

_we have the

following

relevant representations

(see

_[12] for illustration of the

algebra

of these

representations):

I(fk

-

ro

e rz e 3rj e

(higherranktensors)

I(Vzfi

-

ro

e

rz

e 3rj e

(higherranktensors) I(fk(1

_- 3ro e

(higherranktensors)

Pkl(fi

- 4ro _e

(higherranktensors)

Pkl(Vzfm

_- 4ro e

(higherranktensors)

,

(13)

and the

single

combination with the structure

Pol~P

was included in

equation (11)

above.

Corresponding

terms,

containing

also

(du/dz)

_-

ro>

_can be written for _every invariant in

(13).

Such terms,

however,

represent a

higher

order effect with the _same symmetry ^and _may often be

neglected.

We _now look at the thirteen invariants

generated by (13).

The translational deformation acts to alter the state of smectic C* order

mainly

iia the first two contributions in

(13):

aoiiz>iz iy>izi

and &i

I([ _>lz

₊

()

>lz) ⁽¹⁴⁾

Table III. Irreducible

representations

and characters for the _group

Dm

with

operations:

unity I,

_{infinite rotation around the I axis}

_Cj

_{and rotation}

_by

_{180° around the axis in the}

layer plane

_u2.

r0

1 1 _-

rz I I I _-

I,

_U

rj 2 2 _cos

#

0 _-

£, f, f

rn

2 2 _cos

n#

0 _-

n~~,rank

_{tensor in}

£, f

V 3 1 ₊ -1 e _-

P, V,

etc.

These two terms _are what the 8 and 28 invariants of

lsfl

_and

l~vfl

_under

C2 (see

Tab.

II) degenerate

into when

approaching

the A,C transition. The term with &i is chiral and _can be found

only

in C* elastomers

(here

and below _we will

designate

chiral coefficients with

tilde).

The first _term in

(14)

represents _a "mechanoclinic

effect",

similar _to the electrodinic effect in smectic A*

by

Garoff and

Meyer

_[21]. In this _case spontaneous ^shear deformations _appear with the onset of smectic C order

(a

non-chiral

effect)

_or,

alternatively,

_{one can} create such order and all related

phenomena by shearing

the smectic A rubber

along

the direction of the

layer

normal ?.

Invariants second-order in (f( ^{take the form:}

°22[f~~zz

₊

2fzfyl(~

+

f(lyyj

, all

(f(~lzz °00(f(~~k[I (15)

(the

condition of

incompressibility, Det[I]

= I, ^transforms into

Tr[1]

_-

0, effectively

elimi-

nating

the last

term).

These _terms

reprisent

_{another component} of the mechanodinic effect discussed above. There _are spontaneous extensions and shear in the

layer plane

in the smectic C

elastomer,

but these deformations _are

proportional

to the _square of the order parameter (f(

and _are therefore small.

Of the four invariant

(ro)

terms,

describing

the mixed

piezoelectric effect,

r-

Plf

in

(13),

three

correspond

to the

polarization

in the

layer plane (along

Pa for

diagonal

elements of I and

perpendicular

to P~ ^{for the}

in-plane

shear

deformation)

and _one

(Pjj)

to the

polarizatii

along

the

layer

normal:

?22il~z

(fz~zz

+

fy~~y)

~ ~Y

(fz~~y

+

fy~yy)]

?J2(~zfz

₊

~yfy)~zz

VJJI~z

(fz~~z

~

fY~~z)

Po~(Pziz+Pyiy)Trl& ₍₁₆₎

(the

last _term is

effectively

eliminated in _an

incompressible system).

All these invariants

are chiral and _are present

only

in the smectic C* mater1al.

They

_{represent a new} uniform

piezoelectric effect, arising

after the transition into the smectic C*

phase

due to the additional symmetry

breaking, represented by

its order parameter

f.

When there is _a

helica1twisting

in the smectic C*

elastomer, f

=

f(z),

_an

imposed

defor- mation generates ^{the mixed}

piezoelectric

flexoelectric effect. Four

corresponding

^non,chiral

invariants of

Plvf

in

(13)

take the

explicit

form:

v12

lPz II

PY

II _{>zz viiPz} Ill _>iz II

>iz)

V02

(l~z) _l~y~~) ^~kii (~~)

(where,

_as

usual,

the last term is absent in _an

incompressible system).

In order to

bring

these

expressions

closer to _{a more} familiar notation, let _us call 9 the tilt

angle

of the director with respect to the

layer

normal and

#

the

angle

of its azimuthal

rotation in the

plane.

Then

fz

=

-)

sin 29 sin

#

and

fy

₌

)

sin

29cos# (we

_may

always

assume that 9 <

1).

A uniform helix

corresponds

to

#

_{= qz.} Substitution of this convention

into the equations

(14-17)

shows that the second term in

(14)

is

just

the first one,

multiplied by

_q.

Similarly

all

independent

invariants in

(17)

are the _ones in

(16)

with _a factor _q

(in

the

corresponding

^non-chiral smectic C the

equilibrium

_wave number _q

= 0,

leaving only

the invariants

(14)

first term,

(15)

and

(17)

in the free

energy).

On the other

hand,

unless _we consider non-uniform

deformations,

the main effects of

coupling

in chiral smectic C* elastomers

are

simply

the first _term in

equation (14)

and the

piezoelectric

terms

(16),

with

coefficients, dependent

_on the _{wave vector q} of the helical

twisting (if any)

in the material.

5.

Implementations

of the

theory.

In the _two

preceding

sections _we have outlined several _new effects in smectic C elastomers and their chiral C* modifications.

Obviously

there _are many

possible experimental

situations where these effects _may be observed _or

utilized,

the _scope of this _paper does not allow _us to

explore

all of them. Here _we shall illustrate the present continuum

theory

of smectic rubbers _on

just

two

examples

^the

phase

transition behaviour and the effect of _an external electric field.

5.I SPONTANEOUS DEFORMATIONS BELOW A-C _PHASE TRANSITION.

Liquid crystalline

elastomers _are

unique

systems where the simultaneous

coupling

of _a translational deformation

field,

orientational distortions and

polarization

takes

place.

We have _seen in the section 3

that when _a system ^with

sufficiently

low symmetry is

considered,

the

resulting

continuum free energy may be quite

complicated.

Let _us consider the

simplified

geometry of _a smectic A elastomer with all

layers

undistorted and

perpendicular

to the I axis. The second order A-C

phase

transition in conventional

liquid crystals

is described in [15] ⁱⁿ detail, here _we shall find main effects that _are due _to the

underlying

elastomer network. The free _energy

density

of such

a

simplified

uniform system is

F =

a9~

₊ c9~

jip9(P~

sin

# Py

cos

#) ao9(1(~

sin

#

+

1(~

cos

#)

2 4

~V0(l~x~(z ~Y~~z)

~

Q12[(~~z)~

~

(~~z)~j

^~

~(~~

_~

_{~~) (~~)}

Xi

where _we have taken

fz

_m -9 sin

#

and

fy

m

9cos#

,

see

figure

I; the coefficient _a is temperature

dependent,

_{a =}

ao(T Tc). Equation (18)

contains

only

the main terms, ^which

are second order in the smectic C order parameter (f( ₌ ^9.

Straightforward

minimization with respect to the unconstrained fields I and P leads to the renormalization of apparent ^material constants:

ji[

=

jip aoPo/Q12i al

= ao

XiipPol x[

_{= xi}

[I

xi

i~ _/Q12j

^~~ and

Q[~

₌

Qj2

_{Xi i~~. A} second order

phase

transition will take

place

at a lower temperature ⁱⁿ comparison with the

corresponding

uncrosslinked

polymer

smectic: a'

= a

a(/Q12 x[ij. And, finally,

there _are spontaneous deformation and

polarization (in

chiral

system)

induced

by

the

arising

smectic C order:

~, ~,

>lz

"

go ~°~4 >(z

"

go ~i~4 (19)

12 12

Pz =

x[ji[9

sin

#

,

Py

=

-x[ji[9

_cos

#

,

while the

optimized

free energy is

independent

_on the azimuthal

angle #,

F ₌

)a'9~

+

(c9~,

I-e- the Goldstone mode is

preserved

in such

mechanically

unconstrained smectic C elastomer.

Equations (19)

show that spontaneous

polarization

P~ is created in the

layer plane

_perpen, dicular to the tilted director orientation

(I.e. Pal if,

_as

expected

from the symmetry

arguments).

There is _a spontaneous ^shear deformation in the

plane containing

the

layer

normal and the current

optical

axis of the

material,

_see

figure

I. Both these fields _are

linearly proportional

to the order parameter _near the transition. Similar

procedure

of minimization with respect

to other components of the deformation tensor

[using

expressions

(15)

and

(16)] gives

these

deformations,

also based _on the I- _n

plane,

_{as a} second order effect:

~~Y

~ _(°22

V22/~iX~)°~

Sin

2#

22

~~~~ ~YYj

~ _(°22

COS

2#

+ V22/~~X~ )9~

22

~~~~ ~ ~~~~

~(Ql 2Q22)

^{~~~~ ~~~ ~~~~ ~~~~~~~~~~~ ~~~~}

The spontaneous

shape change (19)

^and

(20)

^{should be}

clearly

_{seen on}

experiment.

One of the obvious

applications

of such effect is the control of the A-C transition

point by

external stresses, which introduce mechanical constraints _on the

sample (in

the

limiting

_case of

sample

with

totally clamped shape

the transition will not take

place

at

all). Application

of _some

stresses may break the rotational symmetry of smectic A in the

layer plane

and therefore

eliminate the Goldstone mode

degeneracy

with respect to azimuthal

angle #.

We have considered here _a

mechanically

unconstrained

sample.

In chiral mater1al the _re-

sulting

smectic C*

phase

will also be

helically

twisted [15] around the

layer

norma1with the

wave

number, similarly

renormalized due to the

coupling

with the network.

Accordingly

the spontaneous

shape change

of the

sample

will take _a curious

helica1form, resembling

_{a screw}

thread,

albeit modified

by

the

complexities

of induced torsional deformations. _We do _{not pur-}

sue this here. The inverse experiment is also

interesting:

the

unwinding

of _an unconstrained smectic C* elastomer

by

the

application

of electric field.

5.2 SWITCHING BY ELECTRIC FIELD. One of the features of the ferroelectric _smectic C*

phase

of

conventiona1liquid crystals,

most

important

for

applications,

is the property of bista-

bility

with respect to the

azimutha1switching

of its

polarization (see

_[22], for

example).

In

infinite

samples

there is _no energy

difference,

_{or a} barrier between the two

configurations

in

figure

2 with

polarization

_up and down

respectively.

The _same is true for _some bounded

geometries,

the main

example

is SSFLC

(surface

stabilized ferroelectric

liquid crystal) align-

ment [22]. ^{Hence the}

switching

between these _{two states}

(by

_an external electric

field,

for

example)

is controlled

only by

viscous forces associated with the

corresponding

bulk reorien- tation.

When the orientational order is frozen into the

topology

of _a

polymer

network, this situation may

change.

In _a

mechanically

unconstrained

sample

the free _energy does not

depend

_on the azimuthal

angle #

^and thus the

polarization (and

the

optical

axis with

it)

will be rotated

by

_an

applied

electric field with

only

viscous resistance. We

predict

that this will generate