Orientation of nematic elastomers and gels by electric fields

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fields

E. Terentjev, M. Warner, P. Bladon

To cite this version:

E. Terentjev, M. Warner, P. Bladon. Orientation of nematic elastomers and gels by electric fields.

Journal de Physique II, EDP Sciences, 1994, 4 (4), pp.667-676. �10.1051/jp2:1994154�. �jpa-00247989�

J. Phys. II France 4

(1994)

667-676 _APRIL 1994, PAGE 667

Classification Physics Abstracts

36.20E 61.308 61.41

Orientation of nematic elastomers and gels by ^electric fields

E-M-

Terentjev,

M. Warner and P. Bladon

Cavendish Laboratory~ Madingley Road, Cambridge, CB3 OHE, U-K-

(Received 28 October 1993, received in final form 7

January1994)

Abstract. Solid liquid crystals, formed by crosslinking polymeric nematics into elastomers, display novel and complex elasticity. The internal nematic direction experiences _a barrier to its rotation which couples it to the deformations of standard elasticity. Electric fields acting

on anisotropic chains induce orientational torques, which compete with rubber elastic effects.

Outcome structures crucially depend _on the mechanical constraints applied to the sample. In set- ups with _{no or} few constraints, _an electric field rotates the nematic director without resistance, inducing also _a spontaneous shape change of _a rubber _or gel matrix. When certain strains

are prevented in the sample by external constraints, the magnitude of the elastic barrier is much higher than the electric contribution and

a very high electric field is required to create an

observable director rotation. In weakly anisotropic elastomers, for instance conventional rubbers which have been strained during crosslinking, the characteristic field will be considerably lower.

Experimental observations _on nematic gels support _our predictions.

1. Introduction.

The

impetus

for this

theory

_comes from electric field

experiments

_on nematic monodomains.

The first

experiment

on

(microscopic)

monodomains of elastomers _was carried out

by

Zentel

iii

where reorientation of the director under modest

applied

electric fields _{was seen} under _some circumstances but not under others.

Although

not

involving imposed

stresses or strains, these

experiments

are crucial to _our

understanding

of nematic rubber

elasticity

and _we shall

return to them at _some

length.

In

particular they

_appear to support an

unexpected

conclusion

of _our theoretical

model, namely

that elastic deformations _{can occur} without resistance in certain

geometries.

Later electric

experiments

on

polydomain

nematic elastomers

by

Barnes et al. _{[2] appear} to support these conclusions.

Many

initial mechanical

experiments

_were

performed

_on

polydomain samples (except

for

those of

Zentel).

The

misalignment

and interactions of domains

partly

obscure the

underlying physics

of nematic solids. Indirect methods _[3]

could,

for instance, estimate the spontaneous

shape

distortion of _an elastomer

undergoing

_a

thermally

induced

phase

transition. At the _same time it _was observed that _a universal value of

applied

stress _was

required

in order to induce

)OURN~L DE PH~SIQUE -T 4 N'4 APRIL 1994 25

the

polydomain sample

to form _a monodomain.

More

recently large

nematic monodomain

samples

have become available. Such

single crystal samples

_can be made

by crosslinking

in _a field oriented nematic melt [4], or

alternatively by

two-stage

crosslinking

^and

stressing

the intermediate state [5]. ^{de Gennes} [6]

envisaged

that

crosslinking

of

polymers

would endow these rather

special

nematics with _{a memory} of their initial director. Mechanical _or other fields that attempt to

modify

this direction would be

resisted

by

_a rotational barrier. This is indeed true for infinitesimal

distortions,

however _non- linear molecular models [7,8] show ^{that nematic} elastomers and their rotational barriers _are in fact _more

complex they

_can become "soft" under certain constraints,

offering

little resistance to director rotation and the associated strain deformations.

They

also exhibit instabilities and

phase

transitions when

subject

to

large

strains, I-e-

they

exhibit unusual behaviour found

only

in the

region

of great

non-linearity.

The transitions [7] induced

by application

of stress _are in

some sense the

opposite

^{of the} conventional Fredericks transition. A

simple

nematic is anchored

by

its surfaces and

subjected

to _a field in a

perpendicular

direction,

acting throughout

its bulk.

Here in nematic solids the resistance to rotation is associated with barriers in the bulk while the

driving

field

(stress)

is

applied

at the surface. The

predicted

nematic transition _over the

rotational barrier has been found in mechanical

experiments

of Mitchell et al. [9].

Here _we extend

liquid crystalline

^rubber

elasticity theory

[7, _8] to take into account bulk orientational torques created

by

electric field. Our results show the presence of both "hard"

(I.e.

^{standard rubber}

elasticity)

and "soft

elasticity" regimes,

described in _our

previous

_papers,

with electric field _as well. All critical strains, fields and

angles

involved _are

presented

in _terms of intrinsic anisotropy ^{of the}

polymer

chains that

underly

the network before deformation,

I( Ii [.

A summary of _our results is: when

applying

_an electric

field,

in nematic rubber with all

or most of the components of the strain tensor I

externally constrained,

the director

undergoes

a continuous rotation towards the direction of field

(assuming

the positive ^dielectric anisotropy of the

material).

It is

important

to

emphasize

that there is _no threshold _even when the field is

applied perpendicular

to the nematic director in contrast to the conventional Fredericks

effect,

where the transition is at _a threshold in such _cases. The

magnitude

of director rotation

depends

_on the network parameters, dielectric

anisotropy

and the nematic order parameter ^of the rubber and is

expected

to be very small. In

liquid crystalline

elastomers with _{no or} few constraints _on the

shape change

of the

sample

electric fields switch the director orientation without _a

barrier, subject only

to viscous torques. ^As a result of such _a switch there _appears

a spontaneous deformation of the

underlying

rubber matrix that does not _cause any

change

in the free _energy. This is another appearance of _an

unexpected phenomenon

of the "soft

elasticity"

of unconstrained solid

liquid crystals,

described in [8].

2. Formulation of the model.

We should like _to address the interested reader to _our

previous

_{paper [7] on} this

subject,

where the detailed formulation of the

theory

is

given

and all relevant

approximations

_are discussed in _some detail. Here _we shall

briefly

recall the main concepts and equations ^used.

We _use the

following general

notation scheme. We _{use a}

superscript

^° to denote

quantities

in initial

(undeformed)

states. Hence the initial nematic director is denoted

by n°,

which evolves under _a deformation to _n. To describe tensors we use either suffix notation _{or a} double

underscore _e-g- I e

I,j.

When

describing

uniaxial tensors,

subscripts

_i and

jj are used _to

differentiate the

two principal

values. For

example,

the tensor used to describe the

anisotropic

step

length characterizing

the network chains in the initial

(undeformed)

state is 1°, ^{which has}

eigenvalues I(

^and

I[

in its

principal

frame. With this tensor the _{average square} form of the

N°4 NEMATIC NETWORKS IN ELECTRIC FIELDS 669

networking points positions (I.e,

^{end-to-end distance of}

polymer strands)

is

(R)R))

=

)Ll(.

The effective step

lengths

of the random

walk,

and thus the overall average

shape,

_are functions of the nematic order parameter

Qv

that includes both the

magnitude

and

(most importantly)

the current orientation of the director _n.

We

employ

the afline deformation

assumption,

which is

typical

for network

theory.

With it

the current network span to be defined _as

R,

=

l~jRj

with

I,j

the

macroscopic

deformation of the whole block of rubber. We consider deformations

I,j imposed

with respect to the initial

crosslinking

state. Since the shear modulus of rubber is around 10~

N/m~

and that for the volume

change

is

typically

^10~°

N/m~,

deformations of elastomers _{are at constant} volume

(to

within 10~~

accuracy),

that is:

Det[I,j]

= I. Such _a condition does not

apply

to the swollen nematic

gels,

however the conclusion about the "soft

response"

will be valid there _as well.

Taking

the usual

quenched

_average

Fej/kBT

₌

-(lnP(R))

_{p~~R°) one} obtains for the elastic free _{energy per} network strand [7]:

)

=

Tr[1°

_{16~' l~~}

_ii

_In

~~~ ⁱⁱ⁾

~ ~

~~~ ~

where

I,j

is the

anisotropic

step

length

tensor after the

deformation,

which in

general

_may differ from

I(

in both

magnitude

and orientation of its

principal

values [7]. Elastic free _energy

(ii,

obtained under _an assumption ^that chains between

crosslinking points

are

sufficiently long

and Gaussian, is

essentially model-independent

since it

depends only

_on observable parameters like the

sample shape

_or the order parameter. Another

assumption

has been that the

crosslinking

points themselves do not fluctuate. In fact the influence of

junction point

fluctuations has been

analysed

in _some detail

previously (see

_[12], for

example).

The

result,

obtained under the _same assumption about Gaussian

chains,

is that fluctuations of the

crosslinking points

are

correlated

through

the network

connectivity

and renormalize the free _energy. This renormal- ization

requires

the

replacement

of the

prefactor

the number of

elastically

active network strands N~

by

the number of

microscopic

constraints,

NT,

where

NT

_"

N~(1-

_2/~b) with # the crosslink

functionality [the

number of crosslinks in the

network, N~,

is

N~

₌ 2N~

Ii

whence

NT

"

N~(#/2 -1)

_]. At lowest

possible

~b ^# 3 this reduces the total elastic free _energy

by

_a factor 3 and may be essential for _some estimates. In

general

the

quantitative importance

of

junction

point fluctuations

rapidly

decreases with the

increasing functionality #

of these points in the network.

Electric fields

couple

with the dielectric

anisotropy

of the

material,

the

corresponding

part in the free _energy is, _as usual [10],

bfe

₌

)eobea(E n)~.

Electric fields _can be

applied

with _or without constraints _on the

responsive

strains. In _some

geometries

a response is

possible

for low

fields, figure

la. Other _more constrained

geometries

^lead to

unphysically high

fields

required

to

produce

_a noticeable director rotation,

figure

16. These theoretical

predictions explain

_some of the many

contradictory experimental

results of director orientation in nematic elastomers.

It is clear that the

equilibrium

n must be in the

plane

of I

ii-e-

^of

n°)

and E. It is

sufficiently general

to consider _a

(2

x

2) problem

^{in the}

(I, ii plane

which is based _on the direction of electric field E

along

the I axis and the initial nematic

director,

^{n°. In} this permanent ^frame all relevant tensors have their

principal

frames rotated around the ji axis

by

the

corresponding angles:

1°

by

the

angle

_a, for

example.

With the rotation of _n

by

the

angle b,

the

major

principal

direction of also rotates and makes the

angle

A

= o b with the direction of electric field

E((I.

As _a

result,

all tensors have

off-diagonal

elements in

(I, I) only

and _none

connecting

_ji with either I _or I. Thus in the free _energy

ii

there is _a

single

term

I(~l(~ ^/[~

^and

Tr[1°

16~' l~~

ii

where _now the tensors can be

thought

of _as

(2

x

2)

tensors of the

remaining

~

fi

_E ^~

~

a

n° ^/

~

a)

E ^~

i ~

j

~f ^O

n

~

b)

Fig. I. Alignment geometry ⁱⁿ the

(z,

z) plane for

(a)

molecular switching by _an electric field in the freely suspended sample [Sect.

3.I]; (b)

electric field acting _{on a} constrained sample [Sect. 3.2].

no and _{n are} initial and current directors respectively; E is the electric field.

elements

(I, ii.

It is convenient to introduce the

parametrization

in the

(I, ii

^frame:

c b ' e

f

^~~~

with coefficients

depending

_on the

corresponding

rotation

angles,

a = 1° + b1° _cos 2a b

= 1° b1° _cos 2a _{c =}

-b1°

_{sin 20}

(3)

d = l~l ₊

bl~~

cos 2A _e

= l~l

bl~~

cos 2A

f

₌

^-bl~~

sin 2A

(parameters fl,

b1° _are the _mean and

anisotropy

of the

corresponding principal

values:

fl

=

(1°

+1[)/2

and b1°

= (1(

-1[)/2

and

similarly

for

l~~).

3. Calculations and results.

A

simple comparison

of electrical

energies

in _an anisotropic

body

with stored elastic _en-

ergies

ⁱⁿ an elastomer suggests ^{that electric fields should be} irrelevant: Ueje~tr;~/Ueja~t;~ +~

eobeE~ /NrkBT,

^{where be} is the

anisotropy

^{of the DC dielectric} response of the elastomer and Nr is the number of

microscopic

constraints in the network per unit volume

(see

the discus- sion in the

previous

section that NY ₌

N~(1- 2/#)

with N~ the number of

elastically

active network strands

). NrkBT

is of the order of _an elastic modulus of _a

rubber,

_say 10~

J/m~.

For the above ratio to be of order

unity

we need E

~J

NrkBTleobe

~J

10~

V/m.

This

large

N°4 NEMATIC NETWORKS IN ELECTRIC FIELDS ₆₇₁

field is founded _{on a} low estimate of elastic modulus and _a very

high

estimate

(be

~J

10)

^of

anisotropy.

In

reality

fields

might

need to be

higher

than this in order to _{see some} effects. This

negative expectation

_was indeed fulfilled

by

the initial

experiments

of Finkelmann _et al.

[I ii.

The _responses when

actually

seen

by

Zentel and Barnes _et al. _{were to} fields modest

compared

to the above.

Two types ^of

experiments

with electric fields have been

performed: (al

Zentel

[ii

and later Barnes, Davis and Mitchell _[2]

applied

_a field _{to a}

piece

of elastomer small

enough

that it _was

freely suspended

in _a

surrounding

solvent and _was unconstrained

by

the

glass plates, figure

la.

(b)

Zentel also considered _a small piece ^{of nematic} elastomer

rigidly

^confined between transparent

glass

electrodes _so

that,

_on

application

of _a

field,

deformation is

restricted, figure

16.

3.I FREELY _{SUSPENDED SAMPLE.} It would _appear from

experiments

that in this _case the

nematic director rotation observed is

rapid

and takes

place

before _any

possible body

rotation.

An

anisotropic body

suffers _a torque under _an electric

field, acting

to

align

n with

E,

and this would _{create a} viscous

drag, hindering

rotation of the

sample

_{as a} whole. The time scales for this _{process can} be

calculated,

but here _we shall

simply

record _an apparent absence of

body

rotation in

experiments,

that is, _we shall take l~z

= lz~. Recall that in

(I),

_{Fej was} the free energy per elastic strand in the network. Since _{we are now}

comparing

^electrical and elastic energy densities _we must divide

through by )NrkBT

to get

~/~j

~ ⁼

rl(~

₊

adl(~

₊

bet(~

₊

(ae

₊

bd)1(~

₊

2cfl(~

₊

2(a f

+

cd)lzzlz~

₊

2(ce

+

bf)l~~lz~

+

2cflzzl~~ )~)~))

^{cos~ A}

⁽⁴⁾

T B

where _r

=

1(~/[~.

Let _us take _a

simple

geometry ^with a =

x/2,

that is E is

applied perpendicular

to

n°,

which will illustrate that these elastomers _are

apparently

soft. Let _us also _assume that the

magnitude

of the nematic order parameter remains constant, that is

lj

= ljj and

I[

= ii- In

equation

(4),

^cos~ A _-

sin~

_{b and}

c = 0. This free energy is of the form F

=

g(I, b) Rsin~

_b.

We minimize

g(I, b)

with respect to all

possible

deformations

l~~, ^l~~,

lzz

id _lz~ (= l~z) subject

to fixed

density (Det[I]

₌

ii

and at fixed b. The result is

l~~

₌ I

if

+ 3 +

(1(

_{1) cos} 2b

~~~

/~/lf

₊

61(

_{+ +}

(1(

1)2 cos 2b ^'

31(

₊ 1

(1( ii

_cos 2b

~~~

vi~/lf

₊

61f~+

¹⁺

(lf

1)2 cos 2b ^'

~~~

vi~/lf

₊

~~~+ l~~~il~

11² cos 2b

~~~ _~~~

where l~

m

(I) Ii [)~

is the characteristic spontaneous deformation of the elastomer _on

cooling

it down from the

isotropic phase

[7,

12],

an

independently

measured parameter.

Inserting

these components ^of deformation into

(4) gives identically 2F/NrkBT

₌ 3

Rsin~

_{b the elastic} part of the free energy is

independent

of the

deformation,

_as found in _our

previous

_{paper on} the

"soft

elasticity"

_{response [8],} and

independent

of b. Therefore there is _no resistance to director rotation and

corresponding shape change (5)

^when an electric field is

applied

to _a

mechanically

unconstrained elastomer. For the _same trivial _{reason as} in the elastic _case [8] ^the

magnitude

of the nematic order parameter

Q

does not

change during

^this

switching

of the director between b

= 0 and b

= a

(x/2).

This is because the elastic free _energy remains at its absolute minimum and hence _any

change

in

Q

would _{cause an} increase of F. Therefore the

assumption

that the

current step

length

tensor

I(b)

is

merely

1° rotated

by

the

angle b, [compare

with

Eqs. (ii

and

(3)]

is not an

approximation

at all.

If _one considers nematic

gels (highly

swollen networks that _can

easily

be

compressed

_so that the condition

Det[I]

= I is not valid

anymore)

the _same trivial arguments

apply

to illustrate that the solution

(5)

will still be valid.

Any change

in the

sample

^volume _{causes an} elastic

response from the

gel,

at the _same time there exists _a

special

type of

deformation, given by volume-concerving

strains

(5),

that is "soft" and _preserves the elastic energy ^{at its} absolute

minimum. So there is _{no reason} for

mechanically

unconstrained _nematic

gels

to alter their volume and the _response of such _a system to _an electric field will be the _{same as} that of

elastomers.

The free _energy is minimized

by adopting

the director rotation b

=

x/2,

that _{is at} director

n

parallel

to electric field

E,

with the

shape change given by (5):

j

j3/2,

_j

j~)

zz c i xx ~c3/2

This

surprising

new "soft

elasticity"

result

(see

also [8]) ^is a

qualitative explanation

for the unusual

experimental

results of Zentel

[ii

^{and Mitchell} et al. [2], who observed this molec- ular

switching

^and the related

shape change. (We

have

presented

above in this section the estimates,

showing

that had _any standard rubber elastic resistance been present, ^{the electric} energy would be too small _{to overcome} it. Therefore the _very fact of

positive

results in

experi-

ments

[ii

and _[2] supports the concept of the "soft

elasticity"

in

liquid crystalline elastomers).

Further

experiments

_on unconstrained monodomain nematic elastomers

subjected

to modest electric fields would be most

interesting

_{as a} test of this unusual soft _response

prediction.

Other _more constrained

geometries give

_a "hard" response with

corresponding

thresholds and discontinuities _at

high

fields.

3.2 SAMPLE CONSTRAINED BY CELL BOUNDARIES. There _are, of _{course, many ways} to

create mechanical constraints _{on a}

sample

of rubber.

Qualitatively they

all should exhibit similar types of

phenomena.

For definiteness _we shall stay close to the geometry

apparently

used in the

experiment [ii

and sketched in

figure

16.

When the

sample

of nematic elastomer is swollen _or

compressed

so that it is

effectively clamped

between the two electrodes

(the

cell

boundaries)

_some components ^of strain _are pro-

hibited. In the geometry shown in

figure

16 these _{are an} extension

along

the I axis and _a shear,

parallel

to it, lzz + i and lz~

= 0,

respectively.

Three other components l~~,

l~~

and l~z _are free _to relax

during

the

attempted

director reorientation, ^driven

by

electric field E. We still _assume that the rubber materia1is

effectively incompressible giving l~~

₌ I

/l~~.

The free energy can be written

using

the _same definitions

(3)

for elements _a, b,c and

d,

_e,

f. Recalling

the

imposed

constraints _we write down:

~)

=

rl(~

₊ ad ₊

bet(~

₊

ael(~

_{+ 2a}

_fl~z

_{+ 2cel~~l~z}

NT BT

+

2cfl» )~)~))

^{cos~ A}

⁽⁷⁾

r B

Again making

the

simplifying assumption

that is _a rotated version

of1°,

_the minimization

N°4 NEMATIC NETWORKS IN ELECTRIC FIELDS ₆₇₃

with respect to the

remaining

components ^of strain

gives:

~~~

)~ e(ab~ 2)j~~~

'

~~~

~~~)

^~~~

where ab- c~ ₊

Det[1°]

₌

1(1[.

^{Note that, under assumption about the constant nematic order} parameter,

Det[l~~]

+ de

f~

= i

/(ab c~). Substituting

these components of deformation into the elastic part of the free _energy

(7),

_we obtain that

2Fej /NrkBT

₌

1(~

+

21(~. Finally, writing

all parameters in their

explicit

forms _we have

2F

If

_{+ i +}

(I( ii

_cos 2a

NrkBT ~f

+ I +

(~f I)

cos

2(ct b)

^~

~

~l( ~llif~~~i) ^~~i)~~ _b)j

^~~~

^~~°~~~°

~~ ~~~

where

l~

=

(I(/1[)~

^{and the} parameter R ₌

eobeaE~/(NrkBT),

the ratio of characteristic electric and

ru~ber

_elastic

_energies.

The free energy

(9) clearly

shows that there is _a rubber elastic resistance to the director rotation and _we therefore expect the distortions to be small. It is

appropriate

then _to

expand (9)

in powers of b < 1:

~~)

~ ^cS 3 + C2

(l~, al

b~ ₊ C3

(l~, a)

9~ ₊

C4(l~, a)

b~

r B

-R(cos~

_{a +} b sin 2a b~ _cos

2a) (10)

where

coefficients, determining

the elastic _{response, are}

given by

~ ~

(ll

_1)~

sin~

_2a

~

[lf

+ I +

(lf ii

_cos

20]2

~

(l(

_1)~ sin

2a[6(1(

_{11 +}

6(1(

_{+ 1) cos} 2a ₊

(lf ii ^sin~ 2a]

~

[l(

+ I +

(I( Ii

_cos

20]3

and _{a more}

complicated expression

for

C4.

Init1al rotation of the director is

given by

lowest- order terms in the free _energy

expansion (10).

At

oblique angles

_a between E and n° _we obtain

~ R

eobeaE~

~ ~

"

~ 2C2

6NrkBT(lf

1)2 sin 2a ~~~ ^{~ ~} ~~~ ~~ ~°~ ~°~~

(ll)

This

expression

indicates that the

magnitude

of the continuous director rotation

b(E)

is indeed very small for finite nematic order parameter

([l~ Ii

_~J

ii

and

typical

rubber elastic and

electric

energies,

b

~J Ueje~tric/Ueja~t;c < I.

Equation (I Ii,

obtained from the balance of lowest

order terms in the free _energy

expansion,

^fails in the

regions

_{a -} 0 and _{a -}

x/2

when

next-order terms must be accounted for. The

region

of

validity

of

(iii

is from _a

=

a(

to

a = x

/2 a(,

^where

~

eobeaE~

°~ _~'

6NrkBT(1( -1)2

When the electric field is

applied

almost

parallel

to the initia1director orientation, _a <

a(

< I, the rotation

angle

is,

qualitatively,

~00 OJ~£

~

~

R+3

j

c

When the electric field is

applied

^almost

perpendicular

to the initial

director,

_{a >}

x/2 a],

the situation is different and it renders _an

unexpected

result. In the

limiting

_{case a}

= x

/2

both

coefficients C2 and C3 in the free _energy

expansion (10) disappear

and the first

non-vanishing

contribution to the elastic part ^of the free _energy is

given by

the fourth-order term, which takes the

simple

form in this limit:

C4(a

=

7r/2)

=

((Ii

1)~

(12)

The

boundary

of the linear

regime (iii

at _{a -}

x/2

is therefore determined

by

the balance C2

~

C4b~

_m

C4(R/2C2)~

which results in the estimate for

o(:

~~

~'

~144(I-1)4 ^144N~~~~)f

₁₎₄ ^~~~~

Let _us

emphasize that,

unlike in _a Fredericks effect in

ordinary liquid crystals,

this _perpen- dicular geometry does not offer _a threshold for the director

orientation,

the

angle

^of director

rotation increases

continuously

with field:

~~°°

~'

~4 3Nr~~)I)~ _ii

₂ ^~~~

Note the characteristic

change

of the

scaling,

b

~J E in this _case.

Therefore _{we may} conclude that when the

sample

of nematic rubber is not allowed _to deform

freely

to preserve the

highest-entropy configuration

of

anisotropic

chains _as

they

rotate in the

field,

the resisted uniform rotation of the director is continuous and

proportional

to E~

(to

E in the almost

perpendicular geometry). However,

in order to observe such _a rotation, a very

large

electric field is

required:

E

~J

[NrkBTleobea]~/~ ()

lj ⁽¹⁵⁾

1

Such _a

high

electric field is

unlikely

to be achieved in _an experiment. It arises from the "hard

response"

of the nematic elastomer to _a deformation

/reorientation

when the freedom of strains to relax is limited

by

_some

imposed

constraints. This _case is in _a

sharp

contrast with the result of the

previous subsection,

where _a low electric field _was shown to be sufficient _to generate the

considerable director rotation and associated

shape change

of the free

floating sample piece

of nematic elastomer.

4. Summary and conclusions.

In the unconstrained _case the great freedom of _a chain

(even

with the

incompressibility

_con- straint, _ever present in

rubbers)

_means that its

initially

anisotropic Gaussian distribution _can be distorted _{at no} cost to the entropy ^{and the free} energy does not rise _as the director

(the

signature

^{of this}

anisotropic distribution)

rotates. In _{[8] we} have _proven that in this "soft

elasticity" regime

^effects are

purely entropic.

N°4 NEMATIC NETWORKS IN ELECTRIC FIELDS 675

It should be

emphasized

that this unusual result is not

simply

that _we have _a

body

rotation about the ji ^axis

(the

form of I

given

above is different in section 3.I _we

specifically

assumed the

symmetric

form of

deformations,

l~z _m

lz~).

It would _seem that with sufficient freedom to

relax, macroscopic

distortions _can be achieved

by rotating anisotropic

chains until

they

_are

aligned

with the external electric field.

Evidently

such _an exact cancellation of _any resistance to _an external torque ^is a feature of this

specific theory

based _on Gaussian chains and afline

deformations

approximations.

On

practice

_one would observe _some small _resistance caused

by

other effects in _more

sophisticated

models of rubber

elasticity, involving entanglements

and

correlations, probably

_some small resistance will be created.

Experimental

evidence from nematic elastomers

subjected

to electric fields _11,2] suggests that these effects _are indeed small.

This is _an observation of central importance to models of conventional

elastomers,

where the

complications

of

entanglements

and correlations have been discussed for _over 50 years. The

magnitude

of the threshold field

required

in the soft

regime

_{11, 2]}

(where

_our conventional

approach

to rubber

elasticity predicts

that _zero field _is

required)

will be _an indicator of

just

how

significant

_more subtle effects in

elastomers,

such _as

entanglements

and

correlations, actually

are.

We

reemphasize

that _{we are}

dealing

with uniform director

only,

also in _contrast with the conventional Fredericks effect. The

possible

creation of non-uniformities due to a surface

anchoring

would

introduce,

of _course,

an additional _source of rotational resistance.

In the series of last _papers [7, 8] we have shown that nematic solids _are

qualitatively

_new types ^of

liquid crystal

and elastomeric media. Their

novelty

derives from the

coupling

of their orientational and translational

degrees

^of

freedom,

_a

coupling

that is absent in conventional

liquid crystals

and in conventional elastic media. The

coupling

arises because nematic

ordering

in

polymers

leads to _a molecular

shape change,

and in elastomers molecular

shape change

leads to macroscopic

shape change.

Since these

macroscopic shape changes couple

to stress, so we

have the

coupling

of stress

indirectly

to nematic order.

We have

predicted

_new

phenomena: (a)

when _a moderate electric field is

applied

to the _me-

chanica1ly

unconstrained nematic

elastomer,

the director _n rotates with little _{or no} resistance towards the field

(assuming

the

positive

^dielectric anisotropy

boa).

There is _a spontaneous

shape change

^of the

sample

associated with such internal reorientation.

(b)

A hard elastic

response of the

sufficiently

constrained elastomer

essentially

prevents ^the

possibility

of obser- vation of any reaction to the

applied

field. We predict that in all situations the director _n will

continuously

rotate with the

field,

either with b

~J

E~ _or b

~J E

depending

on the

misalignment

a of the field with the init1al director, ^but that the

magnitude

of this rotation must be too small for realistic electric fields.

An

interesting possibility

exists to make this "hard

regime"

observable. If _one creates a

nematic elastomer with _an

extremely

small order parameter

(small

chain

anisotropy),

the characteristic field

required

to achieve _a considerable rotation will become much lower. In

equation (15)

two entries _are

proportional

to the nematic order parameter, boa

~J

Q

and (ljj

Iii ii

~J

Q.

As _a result the

magnitude

of this characteristic field decreases _as

Q~/~

at

Q

_- 0 and _{may come} into _an accessible _range. Then

by recording

the

slope

of

b(E2)

_{one can}

measure the ratio of electric and rubber elastic

energies

in the system. ^{One should} note also

that at very small order parameter ^the upper

boundary

of the linear

regime, a( (13)

becomes

large

and therefore the

dependence b(E)

will be

always

linear _as it is

given by (14).

One _{way to}

create such

weakly anisotropic

rubber would be to crosslink _a

regular isotropic polymer

chains

in the _presence of external

orienting

electric _or

magnetic

field. The _memory about its direction will be frozen in the network

topology,

while the

(controllable) magnitude

of

poling

^field will

determine the

degree

of

resulting anisotropy.

Acknowledgments.

This work _was

supported by

^SERC UK

(PB

and

EMT)

and

by

Unilever PLC

(MW).

We would also like to thank Peter

Olmsted,

Geoff Mitchell and Rudolf Zentel for valuable and

interesting

discussions.

References

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