Continuum theory of elasticity and piezoelectric effects in smectic A elastomers

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Continuum theory of elasticity and piezoelectric effects in smectic A elastomers

E. Terentjev, M. Warner

To cite this version:

E. Terentjev, M. Warner. Continuum theory of elasticity and piezoelectric effects in smectic A elas-

tomers. Journal de Physique II, EDP Sciences, 1994, 4 (1), pp.111-126. �10.1051/jp2:1994118�. �jpa-

00247943�

J. Phys. II France 4

(1994)

ill-126 _JANUARY 1994~ PAGE ill

Classification Physics Abstracts

61.40K 61.30C 77.60

Continuum theory of elasticity and piezoelectric elllects in smectic A elastomers

E-M-

Terentjev

and M. Warner

Cavendish Laboratory~ Madingley ltoad~ Cambridge CB3 OHE~ Great-Britain

(Received 30 July 1993~ accepted 6 October

1993)

Abstract. The general continuum free _energy is derived for _a liquid crystalline

(smectic)

elastomer under small strain and orientational distortion. Using _group representation theory _we obtain all invariants describing the coupling of translational and orientational degrees of free- dom. Possibilities of uniform rotation of the layer system under shear deformation _are outlined.

It is shown that in centrosymmetric materials

(smectic

A) the only polarizational _response is

flexoelectric, that is in the response ^to non-uniform orientational deformations. Chiral materials

(rubbers

of smectic A*) also have _a uniform piezoelectric effect in response ^to a shear deforma-

tionj _{or an} extension _at

an angle with the layer normal. Possible experimental configurations to observe

some of the predicted effects _are briefly discussed.

1. Introduction.

Liquid crystalline

elastomers _{are new} materials with various

exciting properties.

Unusual behaviour of semiflexible

polymer

chains, which is due to the competition between internal entropy ^and

packing

and

dispersion forces,

is combined with _a translational _memory

imposed by

the

networking

of these chains.

Thermotropic mesogenic

properties of such materials stem

typically

from the

rigid

rod-like chemical structures, which _are either

polymerized

into the main chain with flexible _{spacers, or} linked to this chain as side _{groups, or} both

ill. Crosslinking

of the

polymer

chains creates the rubber with

corresponding

translational

elasticity.

When such

crosslinking

is

performed

in _an

anisotropic

state, ^such a state becomes frozen in via the

topology

of

networking

points, generating a variety ^of new material properties

[1-3]. Altogether,

the _use of different chemistry, methods of

crosslinking

and also

by swelling

of the resultant rubber in chiral solvents has made it

possible

in recent years ^to create almost all essential

phases,

previously

obtained in low-molecular

weight liquid crystals.

As _a result of the interaction between orientational and translational

degrees

of freedom the _response of such systems to a

general non-symmetric

deformation is of _a

complex

_nature, described

by couple-stress elasticity

models

(see

_{(4] and} Refs.

therein). Recently

the continuum properties of

translationally

isotropic

(nematic

and cholesteric

monodomain)

elastomers have

been examined _[5] and unusual mechanical and

polarizational

effects have been predicted.

There _are several molecular models of nematic elastomers [3,

6-8]; experimental

research in

this _area also _{seems to} be very active and _a

variety

of

interesting

effects have been observed

[9-11].

Elastomers of smectic A

(and

its chiral

analogue

A* have also been obtained and

preliminary experimental investigations

_on them have been conducted

[I].

The mechanical properties ^of these elastomers _are different due _to additional symmetry

breaking (layering).

Therefore it is necessary ^to have _a systematic continuum

theory

to describe these materials. This article

uses group

representation theory

in _a formal

fashion,

similar to [5], to deduce the

complete

set of invariant terms in the free _energy of smectic elastomers,

involving

translational and orientational variables and the

polarization.

In the end of this paper we

briefly

discuss _some

possibilities

of

experimental

observations.

Let _us note here that

liquid crystalline

elastomers

(as

all

rubbers)

_can withstand very

large

translational deformations and the

general theory

should be in _no way limited to infinitesimal

displacement gradients,

as would be the _case in elastic solids. Indeed recent

experiments ill]

on monodomain nematic rubbers show that

large

strains drive _new nematic discontinuities and instabilities _not accessible in

simple

nematics and not described

by

continuum

theory.

These

experiments

also

clearly

indicate that

liquid crystal order,

in

particular

the

director,

is

separately

defineable

(but coupled to)

from rubber matrix in which it is imbedded. These nematic transitions _are known _to arise from the rotational

coupling

between the solid and the imbedded nematic. de Gennes

envisaged

this

coupling

[12] as well _as

possible problems

of director definition when the _cross link

density

is

high.

For small strains the rotational component ^{of the} deformation _can be

easily

extracted and the

coupling

written down. This is discussed

explicitly

in the continuum treatment of solid nematics [S]. ^{In solid smectics} we

have not

only

_an elastic

penalty

for relative rotations, but also for translations of the rubber matrix relative to the smectic

layers. Despite

the evidence that there _are dramatic _{new non-} linear effects in nematic

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

from the continuum free _energy of deformation. One _cannot estimate the values of

corresponding

material _constants other than

by

dimensional

analysis, although

_{we can} make _{some guesses}

by comparison

with related parameters in rubbers and

liquid

smectics.

Despite

this

uncertainty

in

magnitudes

the existence of certain novel effects _can be

predicted

and

experimental

work will be needed to

verify

these effects.

In section 2 _we describe the essential character of solid smectics A and the form of the

coupling

to

displacement

and

polarization.

In sections 3 and 4 we use group

theory

to obtain the comlete _set of _terms that _are created when smectic A

polymer

is _cross linked. These _cross

couplings

_are in addition to the

underlying

smectic, rubber and

electromagnetic

terms that

are well-known. The

coupling

^free _energy is summarized in section S and the balance with the smectic and rubber contributions is discussed via estimates of the relative

magnitude

of

the terms. In section 6 we illustrate

examples

of the _new

physics

that the

coupling

in the

solid smectic offer. We minimize the free _energy with various

imposed

strain

fields, obtaining rotational, piezoelectric,

flexoelectric and

undulatory

_responses and

thereby

suggesting new

types ^of

experiments

and

possible applications

unique to

rubbery

smectics.

The reader uninterested in group

theory

could

easily

omit sections 3 and 4 _{on a} first

reading,

except

perhaps considering

_some of the interpretations of the terms there derived.

N°I THEORY OF SMECTIC A ELASTOMERS ₁₁₃

2. Smectic A elastomer.

Translational

deformqion

^{in the rubber is defined in} the usual way,

through

the second-rank

Cauchy

strain tensor A

(~)

_so ^{that the distance} R between the _two

given points

in the solid is transformed _as R'

=

I

_{.R. Zero} deformation

corresponds

to the unit _tensor

I;

_when distortions

are

small,

it _can be

expanded

in powers of

displacement gradients,

A~k G3 b,k + J~k. In the

general

_case there is _no

requirement

for J to be

symmetric,

since _a distributed

body-torque

may be present ^due to the

coupling

with

independent

orientational variables. One has to

keep

in

mind, however,

that when

I

_{is small and uniform its}

antisymmetric

part

corresponds

to the rotation of the

sample

_{as a} whole and _cannot contribute _as such to the elastic free _energy.

If the material is

incompressible,

_one has _a constraint:

Det[I]

= I. This is

essentially

the

case for rubber

undergoing general

distortions since the

Young

modulus is about 4 _or S orders of

magnitude

smaller than the bulk modulus. This condition of

incompressibility

fixes the

trace of

I

_{to be}

a

quadratic

form with cubic corrections,

Tr[I]

=

J~yJy~

₊

JzyJ~z

+ J~zJz~

J~~J~~ J~~Jzz Jy~Jzz

+

O(J~).

Thus _some terms

superficially

of linear order _are

essentially

quadratic if

they

involve

Tr[J].

We shall discuss here

only

the _case of uniform deformations, I-e- terms with

gradients, Vi,

_{will not be} considered.

Isotropic rubber,

of _course, has

only

two

independent

terms in the free _energy of

deformation,

determined

by

scalars

A(~

^and

^A,kAki (here

and below _{we assume} the summation _over the

repeated subscript indices).

Smectic A elastomers have the

point

symmetry _group

D~h

with the

principal

axis

along

the local director orientation _no and the

density

modulation in

layers perpendicular

to this axis. The convenient variable for this system is _a

layer displacement u(r) along

the

equilibrium

direction of _no, which we shall denote _as the I axis [13]. Smectic A is defined _{as a}

phase

in which the local director is

perpendicular

to the local

layer plane,

hence1~~ ₌

-8u/8x

and n~ =

-8u/8y.

Uniform in-plane

gradients (8u/8x)

and

(8u/8y)

correspond to the rotation of the

layers

system ^and do not contribute to its free _energy [13]. ^{When the material is}

chiral,

the

non-centrosymmetric

smectic

A*,

it has the

point

symmetry D~ _a

subgroup

of D~h. We shall

develop

_our model for this _case and then show which terms in the free _energy survive in classical

centrosymmetric

smectic A rubber. We _are

looking

^for the free _energy contribution that describes _an interaction between the deformation

I

_{and the}

_layer distortions,

i7 _u and i7~

u.

We also consider _the induced

polarization, P,

that may

couple

with deformations.

Together

these effects

produce

terms in the free _energy

density,

additional to the standard smectic and rubber free _energy:

F =

Frub(J)

₊

FsmA(i7u)

₊

b'J~J~~i7ku +k'J~~J~~i7ki7iu

+

V/"PmJ,j

₊

V~"~ PmJ,ji7klL

+

p'~"~~PmJ,ji7ki7llL

,

(1)

the last term

representing

^the flexoelectric effect.

The free energy density ^is an intrinsic characteristic of the system ^{and thus} must be invariant under the group elements of D~. The

(scalar)

invariants in the free energy that _are contained in

equation (I)

_can be calculated

using

group

theory.

The irreducible

representation

that

transforms _{as an} invariant scalar

(the

trivial

representation)

under D~ is To- Thus the number of

independent

invariant terms in

equation (I)

is

simply

the number of

to's

contained in the

group theoretical

representation

of the

corresponding

terms in

equation (I).

Other irreducible

representations

of D~ transform either _as traceless

symmetric

^n~~ rank tensors in the

layer

plane

(Tn), perpendicular

to the

principal

axis of D~, or as a vector

(Tz)

in the direction of this axis

(see

Tab. 1).

(~) We

use throughout this _paper the notation

Ii

to specify a tensorial structure of X

(not

to be mixed with the similar notation I to specify Cartesian coordinates).

Table I. Irreducible

representations

and characters for the _group D~ with

operations: unity

I,

_{infinite rotation around the} I axis

Cj

a~id rotation

by

180° around the axis in the

layer plane

_u2.

Doo I C~ _1L2

ro I I I

- scalar

Fz I I -I _-

I,

lL

ri 2 2 _cos # ⁰ -

I, j

rn 2 2 _cos n# 0 _- n~~-rank _{tensor in}

I, j

V 3 1 + 2 _cos # -I

(ri

e

rz)

_- P, i7, etc.

Since the

principal

axis of the point group D~ is the local director

orientation,

the

displace-

ment vector u transforms like the irreducible representation Tz. The transformation

properties

of the strain tensor, ^induced

polarization

and the

gradient

operator _are not affected

by

the

symmetries

of the system ^{and thus} are determined

by

the _{proper power} of the vector represen- tation,

V,

that has components

along

^land in the

layer plane. Obtaining

the representation ^for i7k i7iu

(Tab. III),

_one has to take into account that two

(out

of

four)

irreducible

representations

contained in it

(Tz

and

Ti) correspond

to functions

(8~u/8z~)

and

((8~u/8x8z), (8~u/8y8z)) respectively.

The first of these derivatives _must be discarded _at this stage as it represents

non-uniformity

in the

compression

of

layers

and is small

(this

statement _may be reconsidered in

highly

^swollen

elastomers,

where the

layer separation

is _{not so}

rigid

and additional related

effects _are

anticipated).

Each of the derivatives in the second group is

exactly equal

to _zero due to the smectic

layers connectivity

_[14]. We take the

liberty

to repeat here the basic arguments of [14]:

Consider the uniform system of flat

layers

with their normal

io _along

_{the I}

_axis;

_I I no. Let these

layers

be

arbitrarily distorted,

_{u =}

u(r),

but without dislocations.

Then,

if _one draws _an

arbitrary

closed contour C _across this system, ^the sum of

projections

of the local

layer

normal

I(r)

on this contour is _zero:

f~ I(r)

_dl

= o. This

integral

transforms into the

integral

over

the _area A surrounded

by

C:

J

~ curl

I(r) _dS,

_{which tells}

us that curl

I

e o.

Recognizing

that

I

m

io

₊

_bk(r)

= Cte.

i7iu(r),

_we obtain that

(8~u/8x8z)

= o and

(8~u/8y8z)

= o.

3. Elastic

coupling.

As _a result of the discussion above the number of

physically

relevant irreducible

representations

is reduced and _we should consider

only

the

remaining representations

in the table III. The basic rules for the

representations algebra

^for D~ _are

explained

in the table II. It is

straightforward

to

apply

this

algebra

to

expressions

in the table III and equation

(I)

and obtain the relevant irreducible

representations:

1(~i7ku

- 3To e Tz e 4Ti e

(higher

rank

tensors) (2)

J

(i7ku

- To e

2Tz

e

2Ti

e

(higher

rank

tensors)

J(i7ki7iu

- To e

3Tz

e

2fi

e

(higher

rank

tensors)

J(i7ki7iu

-

Toe2Tie. (higherranktensors)

N°I THEORY OF SMECTIC A ELASTOMERS lls

Table II.

Multiplication

table for irreducible representations of D~. In the

diagonal

el- ements the _upper entry

gives

^the

symmetric

_{square [T~],} the bottom the

antisy~n~netric

square

(T~)

of the

corresponding

representation T.

Dco Fo Fz

Fo

j

Fz Fi F2

Fz Fz ~°

Fi F2

°

Fi Fi Fi ~~

/

^~° _F3

ill Fi

F2 F2 F2 F3 ill Fi ~~ _~ ~°

rz

Table III.

Representations

for the main fields in smectic A* elastomers

(group

D~

).

j jS

i~

_{(V~j F2}

fl~ Fl _fl~2F0

j~~ j~~

^~

(~~~z' ~~~z)

_{~ ~~~'~~~~~}

~y yy

IA jv2 j

ri e r~

ji>)z, -I>tz _I

_e

lty

vii ^i~iii~ _ii iii

~ ~r~

ii j~ l~lilll~l 'i

», ii

+

11

where

iS

_and

i~

are,

respectively, symmetric

and

antisymmetric

parts _{of J~~ and} we are

looking

for trivial

representations

_as

possible

contributions to the free energy

density.

Invariants contained in the

product

of the

symmetric

part of J with the uniform

layer

deformation _are

~~~

~ ~~~ ~

~ ~~~

~~

' ~~~

Tr(J]

~~

rz

-[J~~Jyy

+

J~~Jzz

+

JyyJzz J(( J(j J(j]

^~~

82 82

One _sees in the last _term that the

incompressibility

constraint turns _a term

superficially

of

O(J)

into _one of

O(J~).

The first _term arises

physically

from _a

penalty

_on ^{the relative distortion}

of the smectic

layers

and the rubber matrix in which

they

_are

embedded,

_+~

(Jzz 8u/8z)~.

The square ^terms in this _can be

thought

of

belonging

to

(and renormalizing)

the rubber and smectic free

energies.

The _cross term has _a

negative

_si&n.

The

single

invariant with the antisymmetric part ^of J _can be re-written

using

the notations for _{rotation vectors}

u~~ =

)e~~kJ~k

^{and fl,} = e~~zi7~~Jz:

lJ)~

~"

+

J)~

^~"

la

(u~

fl) (4)

ax 8Y

For infinitesimal deformations the

antisymmetric

part of the strain tensor

i~ corresponds

to a trivial

geometrical body

rotation _over the

angle

_u~

[at larger

deformations the combination

J~ arcsin(tan J~)

r~

O(J~)

_causes

changes

in the free

energy].

In this _case,

however,

_we have two

independent

^{variables that} characterize the uniform

body

rotation:

J~

_and

Vi

_u. It is clear that there is _an energy

penalty

_[12] when these two rotations do not coincide

(I.e.

the system of

layers

rotates with respect to the

sample body),

_a

penalty proportional

to (u~

fl)~

in the

leading

order. Invariant

(4)

represents, in

fact,

the _cross term of this _energy

penalty,

while the two square terms, u~~ and fl~

belong

to rubber and smectic parts ^{of the free} energy.

Realizing

that neither of these parts can

depend separately

_on the

angle

of

corresponding

rotation, _we have to be consistent and

always

use the

j6int

contribution to the free _energy of _{a smectic} A

elastomer,

(u~i fl)~

+

l~))

⁺

~))

₂

_~>lz)

₊

_{>)z ~")}

₊

21(>lz)~

+

(>)z)~l (5)

~

v

~

x v

which represents a real

physical

effect.

There _are two invariants, one symmetric ^and _one

antisymmetric,

associated with non-uniform

layer

modulation:

Note that both of these two terms _are chiral and _are absent in

centrosymmetric

smectic A elas- tomers. In fact the second term represents _a much

higher

order effect in J.

Symmetry analysis

finds invariants

using

both the

antisymmetric

and

symmetric

parts ^of

I,

_but

_only

_{the differ-}

ence between the effect of

i~

_and

a trivial

body

rotation [in ^this case:

J)~ arcsin(tan J(~)

_r~

O(J]~)] corresponds

to a

physical change

ⁱⁿ the system. ^In

discussing experiment

_we examine these _terms

further, suggesting

_{a way} to

probe

the first term and

discarding

the second.

4. Piezo- and

flexoelectricity.

Dielectric

polarization

_may be induced

by

various kinds of deformation in smectic A*

(or A)

elastomers. Two components ^of this

polarization along

the I axis

(Pz)

and in the

layer plane (Pi correspond

to irreducible parts Tz and Ti of the _vector

representation (Tab. I).

Multiplying

these

by representations

of the mechanical deformations in the system,

equations (2),

_we obtain the

following

expansions

(in

which

To-terms

represent

possible

invariants in the

free

energy):

P,J)k

_- To ill 3Tz e 2Ti e

P,J)k

_- 2To e Tz e

P,J)~i7iu

_{- STO} e 7Tz _e 3Ti e

P,J)~i7iu

_- 4To e 3Tz e 2Ti e

P,J)ki71i7mu

- SToe.

P,J)~i71i7mu

- 2Toe.

(7)

The term _r~

voPi

_{in the free}

energy

(I), independent

of the

layer deformation, gives

two

independent piezoelectric

invariants with

Pi

and _one with Pz:

p

jS

_p ~S

~ ~z ~ ~z

~~~

~~~~z ll~~)z

"

(~l

_'UJ)

llz~)y

% l'zUJz

(8)

N°1 THEORY OF SMECTIC A ELASTOMERS 117

where the terms with

J~

_represent,

as

usual,

_a

higher

order effect (r~

PJ~).

All these three terms are chiral and _are absent in the

centrosymmetric

smectic A. The

piezoelectric

term

Pi J~

in

equation (8)

is very

important during

the transition to _a ferroelectric smectic C*

phase.

It will be discussed in _more detail below in connection with _{a new} electroclinic effect in smectic A*; another _paper will be devoted to smectic C and C* elastomers.

When there is _a uniform deformation of the smectic

layers [terms

r~ vPJi7u in the free energy

(1)],

the chiral smectic A* elastomer exhibits _a rich

piezoelectric

effect. Invariants in the free energy with the

polarization

induced

along

the

layer

normal _are:

pz

l~"jS

~ ^yz

^~"jS

~ ~z

X y

Pz

I)J)~ _("J(~)

e

Pz[fl

_{x u~]zj} Pz

("J)~

% Pzu~z

(" (9)

X Z Z

Different combinations of deformation fields create

polarization

in the

layer plane:

P~

~("J~ )J(~) Py ~)Jyy _("J(~)

=

(Pi i~ _fl)j

X X

(P~J(~ P~J(~) (" ^P~ (" P~)) ^{Tr[I]; P~ ("} P~)) ^Jzz

Z X Y X

and

(P~J)~ P~J)~)

^~"

=

(Pi

_u~)

(" ^P~

^~" +

P~)) J)~

= (u~

[Pi

_x

fl]) (lo)

°Z _{Z X}

where,

_as

before,

the terms with

Tr[J]

and with

J~

_{represent the}

higher

order effects (r~ PJ~i7u and

PJ~i7u respectively).

It is well-known that non-uniform deformations in

anisotropic

media generate ^{the sc-called} flexoelectric effect. This effect is present in all

liquid crystals, including centrosymmetric phases,

because the initial symmetry of the material becomes

significantly

broken

by

the de-

formation

gradient

field. Smectic elastomers have in addition the translational strain

field, interacting

with the

polarization

and director

gradients. Following

terms _r~

pPJ17i7u

_con-

tribute to the flexoelectric part ^{of the free} energy

(I)

with the

polarization

induced

along

the

layer

normal:

Pz

(2>i~$

>~~

Ill >»[)

~ ~~~

^~

~~) ^~~~~'

^~

~~~

^~

~~)

^~~~ ~~~~

and

P~

(>izt

+

>izt)

₊

_P~ ^>iz _Ill

₊

>izt)

a2q~ a2q~

~~~~~

_~

~~~~~ 8X~

~

y~~

'

~~

~~lz$

_~

_~lz~)

_~

_{~~ ~~lz~)}

_~

~tz $)

'

a2q~ a2q~

(P~>tz

₊

P~>iz) (w

₊

p) ₍₁₂₎

where,

_as

before,

the terms with

J~

_and with

Tr[J]

represent the

higher

order effects. It is inconsistent to

keep

them in the free _{energy on} this level of

approximation

for these _terms would be

proportional

to the fourth and

higher

_powers of

perturbation

in the system, at small J.

5.

Coupling

free energy.

Summarizing

results of previous sections _we write down the

complete

expression for the _cou-

pling'terms

in the free _energy of the chiral smectic A* elastomer.

Keeping only

the

leading

terms in each category, we have from

equations (3-6)

and

(8-12)

~ ~ ~~~~~

~

~ ~~

~~~~~

~ ~~~

~~j

^~

^~~~~

^~~~

+b~i>~~>yy

+ >~~>zz +

>yy>zz iii iii iii] jj

~~~~ ~~~~ ~YY~$

^~lY~

~~l ~~j

+

Po(P~>j~ Py>i~)

+

P(P~>(z Py>lz))

+ P'

P~ ) _{Py ~j)}

_>zz

~

~°~~ ~~

_~

~)

~~~ ~

~~~~ ~~l~ $

~"

~

~" ~)

~~'~~~~lz

_~

_~~~lz~ ~~

_~

~)

~ ~~

~~ ^~~~ ~iy

~ ~~~

~~

~

~ ~~~

~~

^{~ ~~~}

~iy

~~~~

Other terms, discussed in the previous sections _on the basis of symmetry

only,

will start to contribute to the free _{energy at}

larger deformations,

J

+~ I. Chiral terms, ^which are present

only

in the

non-centrosymmetric

smectic A*

elastomer,

_are marked

by

tildes _on their coefficients

(I.e.

these material constants vanish

exactly

in non-chiral smectic

A).

The

coupling

free _energy

density (13)

should be used

together

with the standard smectic A free _energy,

~B ~" ~

+

~K ~~(

+

~~j ^,

and the free energy of anisotropic rubber. At

2 8z 2 ax

8y

infinitesimal deformations this last contribution in smectic A

(as

well _as in

nematic)

elastomers takes the form

(see

Tab. III for invariants of

[J~]~):

~ol~~z+jQ2i~[~~zz+(Q3 (i~[~j)~+(Q12((~~=)~+(~~z)~j+jQ22~~~+~(~~~)~+~~~j ^(~~)

Five coefficients

Qi

_are

required

for this uniaxial system ^{in the}

general

_case.

Incompressibility

constraint demanded for

rubber, Tr[J]

r~

J~

(see

Sect.

2) effectively

eliminates two terms with

Q2

and

Q3.

The three components ^of extension _are related to each

other,

Jzz m -J~~

J~~.

As _a result, the linear elastic free _energy of _an

incompressible

rubber takes the form:

Qi(>~~

₊

>»)~

₊

joi~i(>iz)2

⁺

^(>(z)21

⁺

)Q~~i>i~

⁺

2(>i~)2

+

>i~i (is)

Two terms

mJ

(J~)~

have been discussed in section 3 after

equation (4) they

_are part ^of the energy

penalty (u~i fl)~

for the relative rotation of

layers

with respect to the

sample

body.

N°1 THEORY OF SMECTIC _A ELASTOMERS ₁₁₉

They

_are

specific

for smectic elastomers and

analogous

to rotational

coupling

terms in nematic elastomers. The discussion after

(3)

shows that there is _a

contribution,

additional to that in nematics and

only

found in smectic

elastomers,

to the

J(~

term.

Of all the components ^of

I

_{the I-strain,}

_Jzz,

_has

a

special significance

since it

directly

effects the

layer spacing

_u if _we make the ailine deformation assumption usual for elastomers. We would expect ⁱⁿ the continuum free _energy

(over

distances

corresponding

to _a few

layers

_or

more)

that this assumption holds since it is determined

by

the

networking

points separation

length

scale. We _can

decompose

the

layer displacement

field _u into _a uniform

layer

strain

uo(z)

_{= ~foz,} ailine in the

large,

I-e- Jzz _{= ~fo>} and small deviations from it _vi

(x~y, z).

These deviations from aflineness _are

penalized by

_a term of the form

~bi Jzz

^~"

,

which gen-

2 8z

erates the _cross term

Jzz~"

_in

_(13).

_{With the ailine assumption this bi term}

_collapses

_to

~

8z

~bi

^~~

,

a

simple (and small)

renormalization of the

corresponding

smectic contribution.

2

~8z~

Thus the assumption of ailineness has

essentially

eliminated the first _term in the

coupling

free energy

density (13).

The remainder of

(13)

after the substitution _u

= Jzzz + vi

(x,y, z)

is _a mixture of terms of

quadratic

and cubic

order,

for instance the b2 term is

O(J~)

₊ O J~

~"~

8z Note that these arguments ^{about the ailineness} of deformations

have,

in

fact,

e~n _lrea~y

applied

after equation

(4).

We demonstrated there that there would be _no difference between infinitesimal

J(~

and au

lax

in the

fully

ailine

situition,

_{the term}

(u~

fl)~

^is

essentially

the result of

fixing

the coefficient before the _cross term

(4)

and it reflects the

(nonlinear) penalty

for _a deviation from such full ailineness. At the _same time the system is allowed to have nonuniform

layer

undulations _vi

(x,

_y,

z),

_so

long

_as

lull

is smaller than _a

layer thickness,

since

on such short

length

scale _we do not expect the

assumption

of ailine deformations _to hold

any more.

Again,

this discussion of

length

scales and the freedom to

independently

define _a

vi

(x,

_y,

z) coupled

to deformations of the matrix is

analogous

to that

required

in solid nematics [12] ^{and is discussed in section 3.}

~

There is _an additional

nonlinearity

in the B ~"

term of

purely geometrical origin.

This 2

8z

is _an

effect,

_common in conventional smectic A

liquid crystals

[13]

tilting

_a fixed number of

layers

while not otherwise

changing

the I-dimension of _a

sample

forces the

layer spacing

to

decrease. A _measure of the

angle

of tilt is _ni

=

-i7iu,

the

projection

of the director onto the

@ plane. Applying

this to _{our case} of almost affine elastomer _we have

B

Jz-

+

(~ _~n()

^~ _{+.. m}

^{~B ~"~ ~} BJzzn (

₊

Bn§ ₍₁₆₎

2 _Z 2 2 82 2 8

where the last _term is

commonly neglected

in conventional smectics _as it represents a

higher

power of _a small effect. In the _case of elastomers this

(nonlinear)

term has to be

kept along

with the ba term in the free _energy

density (13)

because it bounds

layer

rotation.

It is clear that Jzz and ~"

are

intimately

related in solid smectics and it is relevant to compare the _energy costs

o/)hese

distortions in the two materials when

they

_are

uncoupled.

In

simple

smectics the

layer

compression ^{modulus B is of the order of}

(1-5)

_x

10~J/m~.

For side chain smectic

polymer

melts _one would expect ^{values of} the _same order,

perhaps slightly

lower because the chain backbone could

conceivably

weaken the

layer

correlation. In main chain

polymer

smectics the fact that _monomers

along

_a chain _are

spatially

correlated _over

a persistence

length,

one would expect ^B to be enhanced by the number of _monomers in _a

persistent length.

When the

underlying

nematic order is

high,

the induced

persistence length

can be _very

large

_as neutron

scattering experiments

_on nematic main chain

polymers

attest.

The shear modulus of rubbers is in the _range

lo~-lo~J/m~.

Thus in many cases we would expect that the elastomer

undergoes

_a deformation _on the

background

of _a

generally

_more

rigid

construction the system ^of

layers

with fixed

separation.

One of the immediate consequences of this will be the _{a very}

high

value of _an effective

(observable)

rubber elastic modulus

Qi

in

comparison

with other moduli in

equation (lS).

The

layer

bend constant K is

analogous

to the

splay

constant Kii in nematics.

Again,

_we expect ^it to be unenhanced when _we go from

simple

to

polymer

side chain smectics. On the other hand it is known

(see [lS]

and Refs.

therein)

that for main chain

polymers

Kii

(and

hence here

K)

should be enhanced

by

factors

depending

_on the number of _monomers in _a

persistence length.

The smectic

length

scale

mJ

@@

is then

probably

invariant to

polymerization

of whatever type and _{we can} take _{as a}

guide

values derived from

simple

smectics. On the other hand the ratio of the rubber to smectic moduli

(or Qa/B)

_may become _even smaller for main chain

polymer

networks than the

already

small value

suggested

^from

simple

smectics. The constants b (b~, ba, b2) ^all act to

penalize

the relative

displacement

of

layers

with respect to the

rubbery

matrix

they

_are embedded in. We

accordingly

expect ^b to be of the order of the

rubber shear

modulus, lo~-lo~J/m~.

6. Possible observable effects.

Consider _a flat sheet of the chiral smectic A* elastomer, the situation most

frequently

found in _an

experimental

_set-up. There _{are two} basic

possibilities: (I)

smectic

layers parallel

to the

sample plane (this plane

is defined _as

if,

since _we have chosen for

consistency

the

layer

normal

along I),

and

(it) layers perpendicular

to the

sample

boundaries in the sc-called

"bookshelf

geometry" (in

this _{case we} denote the

sample plane

_as

@), figure I(I,it).

There _are two basic types ^of

polymer liquid crystals

with

mesogenic

units in the main chain _or in the side _groups. It is quite obvious that the main chain

polymer

will have _an

extremely

strong

anchoring

of the director

parallel

to the flat surface of _a

sample.

Therefore _one would expect that _a main chain smectic A

polymer

will exist in the geometry

(it)

almost

exclusively.

One would have _more freedom with

alignment

^of _a side chain

polymer liquid crystal,

where the surface

anchoring acting

_on the

mesogenic

_{groups can} be varied

by experimental

conditions.

Still,

the backbone chain retains _some

anisotropy

of its

packing

^{and therefore} the geometry

(I)

seems to be

preferential

for side chain

polymer

systems.

Consider two basic types of deformation of such flat

sample:

A uniaxial extension in the

plane, figure lA,

and B

simple shear, figure

lB [in ^both cases the

figure

shows the view from the top _on the

corresponding sample sheet].

Let _us consider _an unswollen rubber, that is deformations _{are at constant} volume. The strain _tensor

I

_{takes the form for the}

cases

(I.A)

and

(it.B),

for

example (the

other two

geometries

are

completely analogous):

lh

o o

(x)

o o 0

(x)

J("~)=

_o

-)1h

o

(y); I(~"~J=

_{o o o}

_(y)

o o

-)1h (z)

o 1h o

(z)

For

simplicity

_we consider the _case of

imposed

uniaxial

(A)

_or shear

(B)

strain. In _a real experiment one would impose a strain in one dimension and the response in the other two would not

necessarily

be identical because of the intrinsic

anisotropy

of the material. An

exception

is, for

example, (it.A)

where strain is

imposed along

the

anisotropy

axis I.

N°1 THEORY OF SMECTIC A ELASTOMERS 121

AZ

, , , ,

d x

~

w'

(j)

AX

d

~

(

ii

,r £ '~, r £ '~,

,

(A) (B)

Fig.

I. Possible geometry for the observation of piezoelectric ^{effects in} the smectic A* elastomer.

(I) and (it) two characteristic layer arrangements in _a flat sample sheet;

(A)

and

(B)

two types of

imposed afline deformation in the plane of the sample.

6.I UNIFORM _ROTATIONS AND PIEzOEFFECTS. The

leading

terms in the main

expression

for the free _energy

density (13)

_are those with bs, ba and

(chiral)

_PO The ba term

penalizes

the relative rotation of

layers

and the

sample,

it involves the

antisymmetric

_part of the strain tensor

I.

_One

way ^to examine the effect of these terms is to consider siiriple shear deformation,

figure lB,

which has both

symmetric

^and

antisymmetric

components. This is _an obvious geometry to induce _a uniform rotation of smectic

layers,

_{ni "} -i7iu,

subject

to the fixed

layer separation

constraint.

Apparently

shear is the best way to create a monodomain

sample

out of

a mosaic smectic elastomer. In this case,

however,

the barrier related _to

grain

boundaries and dislocations motion has to be _overcome and there would be _a

corresponding

threshold strain.

A

simple

method to examine the effect of

symmetric

shear strain is _an uniaxial extension in

a

direction,

^different from that of the coordinate _axes. Consider the

sample

in the geometry

(iii [with

its

plane pi

and

subject

it to a uniaxial extension 1h in this plane at _an

angle

_a

with respect to the I

(layer normal)

direction. In the frame

@

_we have _a component ^of shear

J(~

₌

(A

sin 2a _as well _as

diagonal elements, Jzz

=

(A

⁺

_~A

cos 2a,

J~~

=

~A ~A

_cos2a

4 4

JOURNAL DE PHYSIQUE >I T 4 N' JANUARY lq94 ~

and

J~~

₌ A. Let _us first consider the main mechanical effect in response to the

symmetric

2

shear deformation. The relevant terms in the free _energy

density

_are

(we

shall _assume ailine

iaYer C°mPressi°n I

=

>zz)

~ au

8uj~

₁

8u)~ 8u)~

~

~~~~~8Y~~~

_8Y

2~~~~

_8Y

~8~

8Y ^~

+POJ(~P~

+

PJ(~JzzP~

+

Pj ₍₁₇₎

2soXi

where the last _{two terms} in the mechanical part

originate

from the fixed

layer separation

constraint. It is

possible

to solve the full Euler

problem

for

(17)

and find the rotation of layers

8u/8y

in the

plane

of the

sample.

It is _more

instructive, however,

to examine characteristic

limiting

_cases,

keeping

in mind the relative

magnitudes

of parameters,

b/B

mJ

10~~ Consider

angles

_a < 54° _so that both deformations Jzz and

J(~

are

positive.

The first four

(mechanical)

terms of equation

(17)

_can be

symbolically

rewritten _as

(simply putting

Jzz and

J(~

_mJ

A)

F

~ 1

@~~~~2~~~4~~

~

where C

mJ

Ab/B,

D

mJ

(b/B A),

E

mJ I and X

=

8u/8y.

F is dominated

by

the linear and

quartic

terms when (1h( <

C~/~.

_Given D

= o at A

= lho ₌

b/B,

this

requires

1h be in the interval

[ho (b/B)~/~,

ho ₊

(b/B)~/~), whereupon

X

mJ

-(b/B)~/~ (B/b)~/~(A ho)

For extension 1h below this

interval, including negative

values, ^the

quadratic

dominates in

determining

the minimum F and X

mJ

-(b/B)A/(Ao A).

For small

A,

A <

b/B

_we have X

mJ A for the rotation of

layers.

For A

g -ho

this rotation saturates at X

mJ

b/B.

For

larger positive

^A

(extension

1h _>

b/B),

the

quadratic

and

quartic

terms dominate and _we have X

mJ

D

mJ (1h

lho)~/~

These qualitative delineations of

regions

of A

by

the

scaling

behaviour of rotation au

lay

_can

clearly

be

bridged by

_{an exact} solution which involves _a

simple

cubic

equation.

This

analysis

breaks down well before the coefficient of the

quartic changes sign

at Jzz ₌

2/3

due _to

higher

order contributions in the smectic term B

~")~

When 8z

(b/B)

is small

(as

_we have

argued

above that it

might be),

the various

scaling regimes

can

be isolated in _an experiment. A similar

analysis

_can be

performed

at

large angle

of uniaxial extension _a > SS°

(in

the direction closer to the

layer plane)

where the results will be quite different because of the

change

in the relative

sign

of

Jzz

and

J(~.

What this linear

theory

does not do is

predict

the rotations achieved

by large

strains.

In chiral materials there will be _a standard linear

piezoelectric effect,

induced

by

the shear

J(~,

described

by

equation

(8)

and the material constant PO- This effect has the _{same sym-} metry as the electrodinic effect in conventional smectics A*

[16],

^{when the}

polarization

P~ is

induced

by

the molecular tilt n~ with respect to the layer normal. Our effect in elastomers is _a true

piezoelectricity

in _a non-centrosymmetric material. The relevant terms in the free _energy for this _{case are}

POP~J(~

+

~Pj.

An additional second order contribution to this

polar-

5oXi

ization will _come from the term

PJ(~P~ (",

^{since there} is _a

layer compression

associated with the considered geometry ^of uniaxial extension.z

Altogether

the induced uniform polarization

perpendicular

to the

sample

sheet is

P~

=

-~eoXiAsin(2a) ^Po

₊

_AP(1+ ^cos20)j ₍₁₈₎

4 4

N°1 THEORY OF SMECTIC A ELASTOMERS 123

This effect should be

easily

observable and may represent the most important property ^of chiral smectic A* elastomers. The

dimensionality

of

piezoelectric

coefficients _PO and P is that of

polarization,

^C

/m~,

and therefore _one would expect these constants to be of the _same order of

magnitude

_as the piezoelectric parameter _pp in

corresponding

ferroelectric smectic C* materials [17]. ^{A crude estimate of} coefficients _PO and P is related to the effective

dipole

moment per

monomer, po ~

lo~~~ Cm and the size of this _monomer

mJ 20

I

_in

_liquid crystalline polymers.

Thus _we obtain Po, ^P _r~ 10~~-lo~~

C/m~, although

it is clear that _many

specific

molecular effects _may

significantly

alter this estimate.

6.2 UNDULATING _soLuTIoNs. Some of the _most

interesting

nonuniform effects in _conven-

tional smectics A _are the Helfrich-Hurault

instability

and its variants [13]. ^Consider

again

the

nonlinearity

described

by equation (16).

The

BJzzn[

term leads to _a reduction in energy

by setting

_{up an}

undulating

structure so

long

_as Jzz > o, that is the

layers

_are dilated. For this

one needs the geometry

(I.A)

with 1h _< o _or

(it.A)

with A _> o. It is not

surprising

that in a

solid smectic A the mechanical strain _can induce this

instability.

There

might

be _an

important

difference with

liquid

smectics: it is

expected

in the Helfrich-Hurault

instability

of conventional systems ^{that dilatation} is first accomodated

by layer

undulation but

ultimately

the

propagation

of dislocations will increase the number of

layers by

the

appropriate

amount _so that the system will become uniform

again.

In _an elastomer the _appearance of _an extra

layer

would cost elastic distortional _energy of the matrix and is

presumably

much

suppressed.

One would

accordingly

expect permanent undulations to be created

by

strain. Since the

period

of undulation is close to the _wave

length

of

light (see

_[13] there _are obvious device

potentialities

where strains _can

switch

light

via induced

gratings.

Considering

^further the two

layer

dilatational

geometries

above with the

diagonal I [Fig. 1A],

it is apparent ^{that the} po and _p2 terms in the

coupling

free energy

(13)

_can also become active.

Writing

out the tilt _n

(

in

full,

the free _energy

density

is

~

~

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

_~

_{~~j~l~ ~~} ^~~~

_~

~~ ^~

~

+

i»o>zz

_~1~(>~~ +

>»)i

Pz

(l~

+

()~ ⁽¹⁹⁾

where for the _case

(I.A)

_we have

J~~

= A and Jzz ₌

J~~

₌

-)A. Normally

the terms linear in

undulating

variables vanish when

integrating

F

through

^all _space ^with the condition that _{vi =} o _on both sample ^boundaries

along

z. An

example

would be

b2Tr[I]~"~

With

bz

vi "

asin(kzz)sin(kx)sin(ky),

the first three terms in

(19) yield,

_on

integration through

_a

volume

V,

_{an energy}

U =

VBa~ k)

+

~k~ zzk~j

(20)

16 B

with

@fi

+~ 20

I,

a standard smectic scale. U is minimized

by taking

the lowest value for kz

= 7r

Id,

^where d is the

sample

thickness [see

Fig~~j(I)].

^The

instability

_occurs at

Jzz

=

~~fifi

with in

plane

_wave vector k

=

~ ~

The

polarization

will still further d

d ~~

reduce the _energy U if it too adopts a modulation: Pz

= p

sin(kz z) sin(kz) sin(ky).

An estimate of the amplitude _p of this polarization wave can be made

taking

into account the

polarization