Layer-network coupling in smectic elastomers

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Layer-network coupling in smectic elastomers

T. Lubensky, E. Terentjev, M. Warner

To cite this version:

T. Lubensky, E. Terentjev, M. Warner. Layer-network coupling in smectic elastomers. Journal de

Physique II, EDP Sciences, 1994, 4 (9), pp.1457-1459. �10.1051/jp2:1994210�. �jpa-00248054�

J. Phys. II £Fance 4

(1994)

1457-1459 _SEPTEMBER 1994, PAGE 1457

Classification Physics Abstracts

61.40K 61.30C 77.60

Short Communication

Layer-network coupling in smectic elastomers

T.C.

Lubensky (~),

^E-M-

Terentjev (~)

and M. Warner _(~

(~) ^Cavendish Laboratory,

Madingley

Road,

Cambridge

CB3 OHE, England

(~) Physics Department, University of Pennsylvania, Philadelphia, PA 19104, U.S.A.

(Received 20 May 1994, accepted 18 July1994)

Abstract. The complex coupling between elastic strains and smectic layer distortions in smectic A elastomers is re-analysed in terms of the penalty for the relative translations of layers with respect to the matrix in which they _are embedded. This modifies the starting point of _a recent theory of the elastic free _energy of smectic A elastomers

(Terentjev

E.M, and Warner M., J. Phys. II £Fance 4

(1994) ill-126),

but leaves most of the analysis itself and the conclusions unchanged. In the large, layers _move affinely with the elastic matrix, but there

are layer fluctuations possible _{on a} scale set by the _mean size of chains in the network. There is _a rigid constraint that uniform shear strains of the network produce corresponding uniform

rotations of the smectic layer system.

New

mesophase solids,

in the form of rubber networks that _are nematic _or smectic, have remarkable elastic

properties

because of their

liquid crystalline

internal

degrees

of freedom.

These

degrees

of freedom _are

coupled

to elastic strains and lead to novel forms of mechanical

instability,

^soft

modes, piezo-

and

ferrc-electricity,

and _new smectic structures.

Descriptions

of these _new

phenomena

have been both molecular and continuum in nature for nematic

elastomers,

and limited _to continuum models for smectics. Because elastomers _are

capable

of very

large deformations,

_{many new} effects _appear in _a

highly

non-linear

regime

and _a molecular

approach

is

preferable.

In order to estimate the size of effects within continuum

theory,

and to _see what

length

scales _{emerge, a} molecular

approach

is also desirable.

However,

since there is _no molecular

description

of smectic elastomers _at this time, the continuum

theory

_may

help

to understand the

qualitative

^effects and

underlying physics

of such _a system.

The purpose of this note is to

clarify

the nature of the

coupling

between smectic

layer

distortions and distortions of the

polymer

network in which these

layers

_are embedded. It will _turn out to _a

large

extent that

layer

distortions are affine with those of the

network,

small

fluctuations _on top of uniform strains

being possible. By examining layer displacements

relative to the matrix in which

they

_are

embedded,

_we fix the

lengths

scales involved in the continuum

1458 JOURNAL DE PHYSIQUE II N°9

analysis.

^A recent

analysis

_[1] ^based _on ^relative strains rather than

displacements

is shown _to have _some

shortcomings, though

most of the

predictions

of _new effects for smectic elastomers still stand.

Relative translations

coupling.

If _we let the _z

displacement (in

the direction of its

normal)

of the smectic

layer

at the

point

_r be

U(r),

then there is

clearly

_a

penalty

for its motion with respect to the _z component ^{of the} local network

displacement,

_Vz

jr).

For small relative

displacements

this must

penalty

_must be

harmonic with _a free _energy cost of:

Fd ₌ A

/ _{drivzlr) Ulr)l~ Ii)}

The coefficient A cannot be estimated within continuum

theory,

but the dimensional

analysis

allows the relevant

magnitudes

^and

length

scales to emerge:

equation (1)

describes _an effect based _on the entropy ^of a

rubbery network,

which in continuum model has _a short distance cut-off at

Ro

the

spatial

extent of

elastically

active strands. Hence the characteristic energy scale is kBT

and, taking

into account the number of strands _per unit volume

N~,

_{we can} write

A _m

aN~kBT/R( ₍₂₎

where _a

depends

on the smectic order parameter ^{and vanishes in} a

high-temperature homoge-

neous

phase,

and where

N~kBT

_{m ~ is} the shear modulus of the elastomer

(for simplicity

in

the

isotropic state).

Since chain

configurations

in the

rubbery

state _are random walks _we have

N~

~w

(lR()~~,

with

being

the step

length

of such _a walk.

Equation (1)

suggests that smectic

layers

_can fluctuate in

position by

_~w Ro which _may be

considerably larger

than da.

The

layer

and network contributors to equation

(1)

_are:

Ujr)

=

Uj0)

_{+ r~w, +}

ujr) j3)

I§(r)

=

I§(0)

+

r,~j,

+

v~(r)

where

U(0)

and

I§(0)

are the _mean

layer

and _mean network

displacements,

_w~ is the uniform part of T7~U

(the gradient

of

layer distortions)

and

u(r)

is the

fluctuating

remainder, ~z~ is the

(z,

_I)~~ component ^of the tensor ~ of uniform distortions of the

network,

and _v is the

fluctuating

part. ^Summation over

repeated

indices I

= 1, 2,^{3 is}

implied. Inserting (3)

into free _energy

(1)

one obtains:

)

=

V[Vz(0) U(0)]~

+

V.V~/~(w~

~z~)~ +

/

_dr

_{[u(r) vz(r)]~ (4)}

(with

V the volume of the

system).

The first term

implies

that there _can be _no uniform relative

displacement

of

layers

and

network,

that is there is _no Goldstone mode of the relative translation in _our system. The

second, extremely large

term constrains uniform

layer

strains to

exactly

^follow the uniform elastic strains of the network. The third term is _a

penalty

on

layer fluctuations,

additional to those

acting

_on the

layers

of conventional smectics, and correlates

layer

fluctuations with mechanical fluctuations of the network.

Effect of the w~-~z~

coupling

on nematic

aspects

of the free energy.

Underlying

the smectic order is the nematic order of the component ^{rods and hence that of} the chains which the rods _compose. There is

another,

nematic, contribution to the free _energy

N°9 LAYER-NETWORK COUPLING IN SMECTIC ELASTOMERS 1459

which _occurs when the

network,

which fixes the

configuration

of

chains,

is rotated relative _to the nematic director [2, 3]. ^{Since in smectic A}

phases

the director is

perpendicular

to

layers,

director rotation about _a

perpendicular

axis _can be described

by

the rotation of the

layers

themselves.

Thus the de Gennes' _term for the free _energy

density penalising

nematic

rotation,

_{T7 iv,} relative to that of the

matrix, (3Vz /3x~)~,

is b

((3Vz /3x~ )~

T7,Uj

, where A denotes the

antisymmet-

ric part ^of a distortion

(giving rotations). Setting

T7,u _{= w,} +

3u/3x,

and

writing

the matrix rotation about _{x as}

~)~

⁺

(3vz lay 3vy/3z)

and about _{y as}

-~)~ (3vz lax

3v~

/3z),

2 2

~

the energy can be

expressed

_as b

~)~

+

(3vz lay 3vy/3z)

_wy au

lay (plus

the simi-

2

lar term for rotation about

y). According

to the second term in

equation (4)

_we must

identify

the uniform part w, of

3U/3x,

with

~z,, leaving

^behind ~)~

(the symmetric part)

and the

fluctuating

_parts. Whence the _energy

penalty

becomes

that

is,

layer fluctuations couple to ~S, _the symmetric part

of

the strain in the_rubber.

The coupling

(5) does not _eadto

new

ffectssince

~S uniform

and au _/3x~

fluctuating. Expanding out, one _obtains

b J

dr(A),)~ + (au

lax,

- (T7,vz)~)~, where _the

coupling

_between~S

and the

spatially

_{varyingtermsreduces}

to a

_surfaceontribution and is

dropped. (Thus in

[1] the

on-fluctuating parts of

the

network-layer ^uplings are not as

in

_the

first _part

of

_quation (17), _but

are the _ri#d

coupling of the uniform

distortions

urvivors

of

equation (5)

above are a renormalisation of

the matrix

_rubber

[by b(~(~)~] _and constraints on are absent in ventional smectics.

The remainder

of

(1] should

be read with the

layer ^ctuations

u(r) in

mind,

as is

discussed

(page l19). The

coefficient

b

is

also

iscussed in [1](page 120) andby ^rguments similar to

those above fixing A,

suggested

that is in the range 10~ - 10~J/m~.

In

summary we conclude

that

^uniform smectic

are

^rigidly locked to

sponding

network _strains, while

undulating eformations take a

complex

form

and

involve

ouplings between

the

^atrix and layer istortions.

These couplings have a _profound

effect on

the

critical phenomena of ^stomeric smectics _and we shall

Acknowledgements.

This research has been

supported by

Unilever PLC.

References

[1] Terentjev E.M, and Warner M., J. Phys. II France 4

(1994)

ill.

[2] ^de Gennes P-G-, in Polymer Liquid Crystals, A. Ciferri, W-R- Krigbaum and R-B- Meyer Eds.

(Academic,

New York,1982); G-R- Acad. Sm. Ser. B281

(1975)

101.

[3] ^Bladon P., Terentjev E.M. and Warner M., J. Phy3. II France 4

(1994)

75.