Rotational invariance and Goldstone modes in nematic elastomers and gels

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Rotational invariance and Goldstone modes in nematic elastomers and gels

Peter Olmsted

To cite this version:

Peter Olmsted. Rotational invariance and Goldstone modes in nematic elastomers and gels. Journal

de Physique II, EDP Sciences, 1994, 4 (12), pp.2215-2230. �10.1051/jp2:1994257�. �jpa-00248127�

Classification Physics Abstracts

62.20D 61.40K 61.30C

Rotational invariance and Goldstone modes in nematic elastomers and gels

Peter D. Olmsted

(*)

Cavendish Laboratory, Madingley Road, Cambridge ^CB3 0HE, U.K.

(Received

13 April 1994, received in final form 19 July1994, accepted 29 July

1994)

Abstract. We investigate the symmetries of elastomers and gels cross-linked in _a nematic

state. The coupling between the local nematic order parameter and _an applied deformation

leads to _a class of uniform deformations which cost _no elastic energy, when accompanied by

a given rotation of the nematic director; this is _a specific realization of _a class of soft modes

originally

proposed, _{on symmetry} arguments, by Golubovid and Lubensky

[Phys.

Rev. Lett. 63

(1989)

1082]. The corresponding elastic theory has _a set of Goldstone modes which _possesses

singular

fluctuations. We describe several experimental signatures ^of these ideas, and elucidate the

physical

picture of these soft modes.

1. Introduction.

Polymer liquids crystals (PLC'S)

are

long-chain

macromolecules which _can order at low tem-

peratures ^into a nematic

phase

in which the

polymer adopts

a

prolate

_or oblate

conformation,

with the

polymer

backbone

preferring

to

align along

_or normal to _a direction i1, ^{the director.}

Main-chain PLC'S

adopt

_a

prolate configuration

in the nematic state, while side-chain PLC'S

adopt

_a

prolate

_or oblate

configuration, depending

_on the _nature of the

coupling

between the side-chain

mesogenic

unit and the flexible backbone. When these molecules _are crosslinked the

resulting

network _may

undergo

_a spontaneous ^{nematic transition} as the temperature is low-

ered,

accompanied

by

_a strain deformation due to the

crosslinking

constraints. The

resulting anisotropic networks,

which have been

synthesized by

several groups

[1-3],

have unusual elastic

properties

[4, 5], ^which are the

subject

of this discussion.

In _a series of _recent theoretical _papers Warner and co-workers [6-9] ^{studied the elastic} re-

sponse of elastomers

deep

in the nematic state.

They predict

_a barrier to director rotation under

strain,

with nematic instabilities when critical strains _are exceeded [6, 7]; ^{these transi-} tions have

recently

been observed [3]. ^More

unusually, they predict

_a soft elastic _response

ii-e-

with _no free energy

cost)

in certain

geometries

[8], as well _as similar anomalous _{responses to} (*) e-mail address:

[email protected]

© Les Editions de Physique 1994

applied

electric fields [9], ^{the latter of which has}

apparently

been _seen in earlier

experiments iii.

In this _{paper we}

analyze

these soft _responses from the

point

of view of

symmetries

_among the molecular

conformation,

molecular

orientation,

and the deformatiDn of crosslink

positions.

We find _a class of Goldstone modes in which _a uniform rotation of the director i1 may be

accompanied by

_any of _a continuum of

possible

uniform

deformations,

all at _no free _energy cost. These modes _{are a}

specific

realization of the soft

phonon

modes of

"anisotropic glasses", predicted

first

by

Golubovid and

Lubensky (GL)

[10]. ^{In the} present ^work _we include director

rotations _{as a}

degree

of freedom

(corresponding,

in the

language

of

GL,

to local rotations of the metric

tensor),

which act in concert with

phonon

fluctuations _to leave the free _energy

invariant. As with director fluctuations in nematic

liquids,

these modes _{are a} consequence of the broken rotational symmetry ^of the nematic state. These deformations include _a trivial

body

rotation

(also

_present in _a nematic

liquid)

_as well _as shears about _axes

orthogonal

to the nematic

director,

with

accompanying

extensions

dependent

_on the axis chosen and the

magnitude

of order in the

quenched

nematic state. The continuum elastic

theory explicitly displays

this

invariance,

and for

particular scattering geometries

_we

predict

anomalous director

fluctuations,

_as with nematic

liquids,

while for other

geometries

fluctuations _are

suppressed by

the network

elasticity. Similarly,

there _are soft

phonon fluctuations,

_as

predicted by

GL.

The outline of this _paper is _as follows. Section 2 presents a model free energy which

provides

a

simple

relation between

elasticity

and nematic order for Gaussian

chains,

and has been

extensively

studied

by

Warner and co-workers [6-9]. ^While most of the results here _are in fact

independent

of the

particular

elastic

model,

the Gaussian model

provides

_a useful and

simple setting

^within which to

explore

the

qualitative

behavior of nematic elastomers and

gels,

for both small and macroscopic deformationsr In section 3 _we discuss the

symmetries

^of this free energy. Then _we discuss the soft modes that follow from these

symmetries

and the symmetry-

breaking

into the nematic state, both from the

point

of view of

macroscopic

distortions

(Sect.

4)

and small fluctuations about the nematic state

(Sect. 5).

In section 6 _we show how these symmetries must be present ⁱⁿ any model

of

_a nematic

gel

which _preserves rotational

symmetry

between the choice

of

director orientation and the network. This is

essentially

the argument of Golubovid and

Lubensky, applied

to nematic

gels

_{as a}

particular example. Finally,

_we

summarize in section 7.

2. Model free energy.

First _we describe the model free energy introduced

by

Warner and co-workers [6, 7]. ^The end-to-end

probability

distribution for _a

polymer

with _an

anisotropic

^Gaussian

conformation,

which describes _a nematic PLC state, ^is

P(R°, to)

=

(detto)~~/~ exp(-3R°tp~R° _/(2L)), (2.I)

where

R°

_{is the end-to-end distance and L the chain}

length.

The

eigenvalues

of the matrix to define the effective

(anisotropic)

step

lengths

^of the

polymer

at the moment of

crosslinking,

according

to

j~0 j~0

)~

~f~

j~

~)

o p

j

^Off'

The nematic order parameter

Q

is average of the second moment of the bond orientations _v,

Qnp

₌

~

~j _{(v«vp jbnp)p} 12.3)

~n~~

bonds

=

s(i~i~ j&«~), ^(2.4)

where S is

magnitude

of the nematic order and i1 is the director. The step

anisotropy to (Q)

is

a function of the nematic order parameter

Q,

and

depends

_on the model chosen for the

polymer chains,

such _as worm-like _or

freely-jointed

chains [5, 11]

and,

in the _case of side chain

LCP'S,

appropriate

couplings

between

side-groups

and backbone. Our results _are

independent

of the

particular

model for the

chains, requiring only

that the chains be

long enough

to be described

by

_an effective

anisotropic

Gaussian distribution. We _assume the nematic elastomer

comprises

crosslinked

polymers

with this

anisotropy,

_so that L is the strand

length

between

crosslinks,

and further _{assume a}

monodisperse

distribution of strand

lengths

L. We _assume the network to be either crosslinked while in the nematic state or,

equivalently,

after

having undergone

_a

spontaneous transition to a nematic state described

by

the distribution

P(R°, to ).

Now deform the cross-link

positions aflinely according

to R

= A

R°,

Imp

=

3Ra /3R). (2.5)

The

assumption

of

affinity

is reasonable for

wavelengths larger

than the _mean crosslink

spacing.

For deformations

iii

<

(Llio)~/~

the molecular

configurations

_are still

anisotropic Gaussians,

described

by

_{a new} step

anisotropy

I. The elastic free _{energy per} strand

(in

units of

kBT)

^of

the distorted _state is the

quenched

_{average over}

Ito,

F~i ₌

-(lnP(R,t))p~ (2.6)

=

(lY (AtoA~t~~)

In

(det tot~~))

,

(2.7)

where

Alp

+

>pa.

For _an

isotropic

state

(t

=

to WI,

with I the

identity)

this free _energy reduces to that of _a classical

incompressible

rubber.

The total free energy of the

homogeneous

system is

Tot ₌ F~i +

F»em(Q)

_±

n»t(#)

₊

Fcomp(#), (2.8)

where

Fnem(Q)

is the orientational entropy lost _upon

adopting

nematic order. This contribu- tion is

rotationally

invariant and

only depends

_on the

magnitude

of

Q,

and

depends

_on the model of chain chosen. Since _{we are} concerned with the most

general properties

^of this system,

we consider

changes

^of state which _preserve the

magnitude

of

Q,

and do _not consider

Fnem(Q)

further. Interactions between the network and _a

solvent,

if present, are contained in

lint(#)>

where # is the network volume fraction in the system, and

F~~mp(#)

^contains modifications of the elastic bulk

modulus,

whose form is still controversial

[12, 13].

^{We shall} see below that the forms of

tint(#)

and

F~~mp(#)

are irrelevant to this work.

This free _energy involves the strain tensor in the combination

AtoA~,

rather than

AA~,

_as

occurs in

isotropic

nonlinear

elasticity

_[14]. Corrections to Gaussian

elasticity (such

_as the

equivalent

of

Mooney-llivlin

terms [15]) ^{would involve}

higher

order invariants of

AtoA~

rather than AA~. _This reflects the broken rotational symmetry ^of the nematic state. In accord with this broken symmetry, ^{de Gennes} has

previously

noted _an invariance of the free _energy of _a nematic elastomer under simultaneous _rotations of the nematic order parameter ^{and the} strain field [4]; ^{in the next section} we show that this observation should be

supplemented by

_{a more}

general

condition.

3.

Symmetries

of the free _energy.

3.I ROTATIONAL INVARIANCE. Under the transformations

R°

-

VR°,

_R

-

UR, (3.1)

where U and V _are rotation

matrices,

the strain tensor and step

anisotropy

tensors transform

as

A _-

UAV~, _to

-

Vtov~,

_t

-

UtU~, _(3.2)

and the free _energy is invariant.

Hence,

separate rotations of the reference

(R°)

and current

(R)

coordinates leave the free _energy invariant. That

is,

the system is invariant under

O(3)

4#

O(3)

rotations, unlike _a nematic

liquid

in which there is _a

single

set of rotations under which both the director and the

spatial

coordinate _must transform

identically.

The _presence of two separate ^rotational

symmetries (separate

rotations of the states before and after

stretching)

is

responsible

for the soft modes.

3.2 STRAIN INVARIANCE. A nematic elastomer also _{possesses art} invariance under _a

change

of the strain tensor A, ^due to the broken rotational symmetry:

,

At(~~Uti~~~

(3.3)

~

ti/2vt-1/2~,

for

arbitrary

rotations U and V at fixed

to

and t. This is another _way of

stating

the _sym- metries of

equation (3.2).

In this

guise, however,

_we obtain _an intuitive

appreciation

of the

symmetries.

This invariance may be understood if _we recall that t

parametrizes

^the

anisotropy

of the distribution of end-to-end _vectors between crosslinks. For _an

isotropic

undeformed _sys- tem

(A

=

I,t

₌

tow I), equations (3.3) degenerate

to

rotations;

that

is,

_a rotated

isotropic

distribution remains

isotropic. However,

if _we

perform

_a rotation upon an

anisotropic

state without

changing

the

anisotropy

axis of the nematic

order,

the free _energy

generally

rises. This increase in free _{energy may} be avoided

by deforming

the

sample

in such _{a way} that the

identity

of

particular

cross-link _vectors

changes,

but the overall distribution remains

unchanged.

The

new state

is, semi-macroscopically,

identical to the

original

state

(I.e.

^{has the} same crosslink

distribution)

but has

changed

its

macroscopic shape.

To understand these statements _we

next rotate the nematic order parameter

(or to)

of _an undeformed elastomer and ask what

accompanying

deformations leave the free _energy invariant.

The free _energy of the undeformed system (A =

I,

t

=

to

is Fei ₌ lY

(totp~) /2=3/2,

which

we rewrite _as

Fei ₌

)lY (UwtoU~Uwtp~U~) (3.4)

Now _we

identify

the rotated version of the step

anisotropy,

^t~~ ₌

Uwtp~ U~.

This

corresponds

to a rotation of the nematic order parameter tensor

by

_w, and does not

change Fnem(Q). By comparison

with the elastic free _energy

(Eq. (2.7)),

the condition for _a strain to leave the free energy invariant under _a rotation of the nematic order parameter must be

AtoA~

=

UwtoU~

₌ t.

(3.5)

Note that

only

volume

preserving

strains

(det

A

=

I) satisfy

equation

(3.5),

_{as can} be _seen

by taking

^the determinant of both sides and

using

det Uw ₌ I.

However,

the free

energy'of

_a

swollen nematic rubber

(I.e.

a

gel)

differs from

equation (2.7) by

_a bulk modulus term F~~mp

ii)

and network-solvent interactions

l~nt(#).

Since these contributions _are invariant under _a vol-

ume

preserving deformation,

_our conclusions

apply

to both swollen and neat elastomers.

3.3 GENERAL _SOLUTION. The

general

solution to

equation (3.5)

is

A(#,w)

=

Uwt(/~U~Ujtj~/~, _(3.6)

where

Uj

is _an

arbitrary

rotation

by

_an

angle #.

This solution

applies

to _a

general (pos- sibly biaxial)

nematic state, with _any rotation _w of the nematic order parameter tensor

Q.

There _{are two} non-trivial _axes about which _to rotate _a uniaxial

Q,

and hence four parameters

(11, 42,wi,w2)I

for _a biaxial state there _are six parameters. ^For

#

_{= w we recover} the trivial

solution I

= Uw,

corresponding

to the

body

rotation discussed

by

de Gennes [4].

How do _we understand this result:

namely,

that _a rotation of the nematic director _may be

accompanied by

_any of _a continuum of non-trivial

deformations, parametrized by

the full rota- tion

group?

The _answer lies in the symmetry of the elastomer under simultaneous but separate rotations, combined with the broken symmetry of the nematic state. The _set

(I(#, w), w)

_com-

prises

the

long-wavelength

limits of the Goldstone modes for this system, ^and are the

analogs

of

spin

_waves in _a

ferromagnet

_or director _waves in _a nematic

liquid

_[16]:

long wavelength

deformations _away from the

broken-symmetry

state

which,

in the limit of infinite

wavelengths ii-e- uniform)

_{cost no energy.}

4.

Macroscopic

soft deformations.

Before

analyzing

the soft deformations in detail _we present _a cartoon of the

macroscopic

soft modes.

Figure

1 shows _an anisotropic ^network

undergoing

_a rotation and _a non-trivial dis- tortion. The rotation, of _course, leaves the free energy invariant, and is the

only operation

which leaves the free energy of _an

isotropic

network

unchanged.

Note

carefully

the nature of the non-trivial

deformation,

however. It is

performed

in

just

such _{a way} that the closer

crosslinks end up stretched further apart, ^{and the stretched crosslinks end} up closer

together.

The result is a state which has the _same distribution of crosslink end-to-end vectors, ^{and hence} the _{same energy} for the Gaussian model where each strand is _a

spring. However,

the director has rotated

by 7r/2,

and the

macroscopic shape

of the

sample

has

changed. (We ignore

^surface tension, ^which opposes such _a

deformation).

The _essence of the soft modes is that non-trivial

deformations _{can swap} the

"identity"

of crosslink positions in such _a way that the overall dis- tribution is

unchanged.

Such _{a swap can}

only

take

place

for

anisotropic (uniaxial

_or

biaxial)

distributions.

Now _we examine

quantitative examples.

We consider uniaxial

anisotropy,

to

=

iiI

+

(ijj ii)11, (4.I)

and

impose

_a rotation about the k axis

by

_w. We then let

Uj

be _a rotation about k _as well.

From

equation (3.6)

_{we can} calculate the

following

nontrivial deformations

iii, w)

which leave the free _energy invariant:

1 0 0

A(4b,

_W)

" ° l~b(W)

(I

+

@f)

Sin

4b (4.2)

0 0

~Jp~(w)

Ii

^{o o}

A(#~,w)

= 0 ~Jc(w) ⁰

,

(4.3)

0 (1 +

fi@)

sin #~ ~Jp^~(w

' '

' ___

I

(a) (b)

Fig.

I. A cartoon of anisotropic gel undergoing

(a)

_a rotation and

(b)

a non-trivial deformation which does not change the free _energy

where

/~~(~°) " C°S~ _~° +

)

SIII~_~°

(4.4)

1

tti~

_{(W) =} cos~ _{W +}

)

_Siu~

W

(4.5)

II

tan16

₌

iL[~(w)(I fi~)

cosw sin _w

(4.6)

tan

#c

₌

~1~

(w)

I

@)

cos w sin _w.

(4. 7)

The deformations

corresponding

to

16

and #~ _are ^shears

accompanied by

contraction

along

and extension normal _to the initial director i1, _as shown in

(I)

and

(II)

of

figure 2,

and both involve rotations in the _{same sense as} that

applied

_to i1. The extension

required by

the two shears is

different,

and _{one can see} from

equations (4A)

and

(4.5)

that ~Jb(w) >

~tc(w).

^This

is reminiscent of nematic

liquid crystals

in

flow,

where the deformations shown

correspond

_to the

geometries

for

measuring

the Miesowicz viscosities qb and q~

[17].

^A

larger

extension is

possible

for

16, just

_as the

viscosity

_qb is smaller than _q~ and

yields

_a faster strain rate for _a

given

stress. Other values for

# correspond

to shears

along

different _axes, with the

magnitude

of the

accompanying

extensions

dependent

_on the chosen axis chosen. The _same continuum of solutions has been found

by

Warner et al. [8], ^{who fix} an extension

along

_a

particular

direction and minimize the free energy ^to find the director response

(fixed

uniaxial

stress).

In this

guise

the soft modes obtain _up to _a maximum strain which

depends

_on the direction chosen.

These

predictions

_may be tested in

(at least)

two ways. One _can

(A) impose

deforma- tions

along

different _axes and observe the

accompanying

rotation of the order

parameter,

or

(B)

rotate the order parameter

by,

for

example, rotating

an

aligning

electric

field,

and ob-

serve the deformation of the

sample

for different

imposed boundary

conditions.

Verifying

the

Z Z ~ n

r- i

j

t i

I

-J

(u (w

Fig.

2. Two deformations which leave the free energy invariant under

a rotation of the director fi by _w. The dashed box is the undeformed sample;

(I)

corresponds to

(#b, vb)

in the text and

(II)

corresponds to

(#c, pc).

quantitative

differences in strain is very

difficult,

since the differences between

pb(uJ)

and

p~(uJ)

are

slight. However,

_{one can}

verify

that _one of _a continuum of soft deformations

accompanies

a

single

rotation of the order parameter

by constraining

the

sample

in different

geometries.

As noted

by Terentjev,

et al. [9], the existence of soft deformation modes

helps

understand

experiments by

Zentel

[I],

^{who observed that oriented nematic elastomers}

^could, depending

_on

the

boundary

constraints, reorient and

change shape

under

applied

electric fields far _too small to have _an effect unless the response were small.

5. Continuum elastic

theory.

Finally,

_we discuss the elastic

theory

for small deformations

u(ro)

about the unstrained nematic state. We let

lap

₌

sap

+

dpua,

^{and consider} a rotation of the step

anisotropy (director)

t =

UwtoU$.

We consider _a state

deep

in the nematic

phase,

where fluctuations in the

magnitude

of

Q (and

thus

t)

_are

strongly suppressed.

^On symmetry

grounds

_we expect to find the

following

form for the elastic free _energy

density

_[18]:

ffluc _"

fel

₊

ffr, (~'~)

with

~' ~"~

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

~°~~

^~

^~~

^{~'~ ~ ~}

+

po(fi.

_e

.fi)~

+

pith(fix

_{e x}

fi)~

+

p2(fi.e

_x

fi)~ (5.2)

2 2 2

~~~°~~

^~

^~~~~~~~~~~

^~

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

where _w

= fi _x

6fi,

fl

=

jV

x u, and _e

=

j _(Vu

₊

Vu~).

The first two terms of fey were first

given

in _a

phenomenological theory by

de Gennes [4], ^{who noted the invariance of the}

theory

under simultaneous

rigid body

rotations. The final five terms of

fei

_are the invariants allowed

~

i n

,

~~

~f _'

$~

~, _lf

,'

,' ,_I

' I I '

Fig.

3. Geometry for calculations of scattering from director fluctuations.

for _a uniaxial medium. Here _we have

included,

for

generality,

the coefficients which include

compressibility (~o

^and

~i),

which _are absent for _an

incompressible

elastomer. The Frank free energy

ffr penalizes

non-uniform director rotations

6fi,

and Ki, K2, and K3

respectively

_govern

splay,

bend and twist fluctuations [17].

For the

incompressible

Gaussian nematic elastomer _an expansion ^of

equation (2.7)

to har- monic order in _u and _uJ

yields

the

following

coefficients

(see Appendix A):

al =

pLkBT(ljj ⁱⁱ )~(ljjli)~~ (5.4)

a2 =

2pLkBT(1(-1()(ljjli)~~ (5.5)

po = pi =

2pLkBT (5.6)

p2 =

pLkBT(ljj+li)~(ljjli)~~ (5.7)

~o = ~i = cc.

(5.8)

Since _al and _{a2 vary as} the strand

density

_pL, the

coupling

between orientation and strain fluctuations is much weaker for _a

"floppier"

network. The

coupling

is stronger for _{a more}

anisotropic

network.

As _a result of the

coupling

between

elasticity

and nematic

order,

director fluctuations

acquire

a "mass" and

destroy

the

turbidity

of the nematic

liquid,

except ^for

particular

fluctuations which _can take

advantage

of the soft modes.

Similarly,

there is _a set of soft

phonons

_[10] which

can take

advantage

of the soft modes.

5.I DIRECTOR FLUCTUATIONS. For director fluctuations

accompanied by

_a soft mode _we expect a

singular long wavelength

_response, which should be detectable

by depolarized light scattering

_[20]. We may use the

equipartition

theorem to

integrate

out the strain fluctuations from

equation (5.2)

and find the wavevector

dependence

of director fluctuations. We consider

a geometry ^where Q-it _{= cos} b and

6n~(q).Q=

sin bcoslfi

(see Fig. 3)

and find

(see Appendix B)

the

following:

~~~~'~~~~~~~

a~(b)

+

q2/~~~~~+

_Ki

_sin~b)

^~

_ay(b)

+ q2

~~~o~~~+

K2

sin~bl'

^~~'~~

where the "effective masses"

a~(b)

and

oy(b)

_are

displayed

in

Appendix

B.

For _a

general scattering

_geometry the effective _{masses are non-zero,}

reflecting

the

coupling

between strain and director _rotation which

destroys

the strong

scattering ((6n6n)

~-

q~~)

characteristic of _a nematic

liquid crystal. However,

for

particular geometries

^{the effective}

masses vanish and _{we recover} behavior characteristic of _a nematic

liquid.

For b

= 0, ^which

corresponds

to bend

fluctuations,

the two terms of

equation (5.9)

add _to

give

((6fi(q)(~)bend

_"

~~jj (5.10)

K3q

For b

=

7r/2,

which correspond to _a combination of twist and

splay fluctuations,

the fluctuations

are

~~~~~~~~~~~~'~~'~~'~~

~~~~~~

^~

~~~i~~~'

^~~'~~~

where

1fi ⁼ 0 is _a pure

splay

fluctuation and _1fi

=

7r/2

is _a pure twist fluctuation. Hence bend fluctuations _are massless and should scatter

strongly,

and fluctuations with _q in the _x-y

plane

should also scatter

strongly,

_as

long

_as

they

have _a component ^of

splay.

The massless bend and

splay

fluctuations _are

accompanied by

the deformations shown in

figure

2, ^which

correspond

to the _{q -} 0 limit of deformations which induce _pure

splay (I)

and bend

(II).

For _{a pure} twist

(6n

1 q, fi I

q)

the fluctuations _are massive. This is sensible within the soft mode

picture,

since the _{q -} 0 limit of _a

twist-producing

deformation

u(q)

is of order q~ ^{and hence} cannot be _a soft mode.

However,

_an admixture of twist and

splay

is

massless,

as

long

as it contains acme

splay

component. A mixture of bend and

splay, however,

is not massless. To

summarize,

_we find the

following

soft director fluctuations which _appear in any nematic

gel

with soft _response:

q _ii fi pure bend fluctuations

(5.12)

q I

fi,

_q

/

6fi

splay,

_or

splay

and tvlist fluctuations.

(5.13)

5.2 PHONON _SPECTRUM. If _we

integrate

out the director fluctuations from

equation (5.2)

we recover an elastic

theory

for _a uniaxial

gel which,

based _on symmetry, has five elastic

constants [21]. ^{As shown}

by GL,

the

imposition

^of a spontaneous ^uniaxial deformation of

crosslink

positions, together

with the symmetry ^of equation

(3.2), implies

the

vanishing

of _one of these elastic constants

(corresponding

to

(fi

_{e x}

fi)~),

and the

corresponding phonons

_are soft [10]. ^That

is,

the elastic _{constant p2 is} renormalized

by

director fluctuations:

~~

_~~ ~y2

4~i

^{~' ~~ ~~~}

The

corresponding phonons

_are

soft,

and must be stabilized in _a continuum

theory by higher

order

gradient

terms. The soft

phonons,

first noted

by

GL

[10],

are

u(q)

I

h,

_q

ii h discotic

undulations, (5.15)

u(q)

ii

fi,

_q I fi smectic-A-like undulations

(5.16)

and have the fluctuation spectrum

(iu(q)i~)

~-

q~~,

rather than the

q~~

^{behavior of} conven-

tional

phonons.

These

phonons

_are the companions ^ofthe soft director fluctuations of

equations

(5.12, 5.13).

Note that these _are transverse

phonons;

the

only

transverse

phonons

which _are not soft _are those in the plane normal to the uniaxial direction fi. This is because the

only

deformations in this

plane

which _can leave the free _energy invariant _are trivial rotations. How- ever, if the system

undergoes

_a spontaneous ^biaxial

transition,

all

planes

_possess non-trivial soft deformations. Hence all _transverse

phonons

_are

soft

in _a spontaneous biaxial nematic elastomer _or

gel.

^Such a system _emerges, for

example,

in side-chain LCP'S in which both the backbone and side-chains _are

mesogenic,

and

prefer

to be

orthogonal

_[22].

This continuum free energy

applies

for

wavelengths larger

than the crosslink spacing; _q <

q~~ ci

(ljjL)~~/~,

^where

^typically

^L ~- 200

1.

_{On smaller}

lengthscales

nematic fluctuations

are unconstrained

by

the crosslinks and should behave _as in _a nematic

liquid, (6n6n) ~-1/q~.

At

longer lengths

the deformation is affine and the

approximations leading

to the

theory

_are sensible.

6. Non-Gaussian

elasticity

and

universality.

Our discussion has used _a model Gaussian nematic elastomer _to

explore

the Goldstone modes in _a nematic network. In this section _we show

how,

based _on

arguments by

Golubovid and

Lubensky

_[10], the Goldstone modes _are not restricted to this

particular model,

but

apply

to any model of _a nematic

gel

for which the choice of direction for the director is not correlated with the distribution of crosslinks. This argument ^relies

heavily

_on the work of GL

[10],

so we first review their argument ^{and then discuss the extension} to

possible

classes of models for

nematic elastomers.

In their

language,

^the

isotropic

state is characterized

by

_a distribution of _mass

points (x~),

such that the system ^{is invariant under rotations} x~ -

Ux~.

^This

corresponds,

in _our model of

a nematic

gel,

to _an

isotropic

distribution of crosslink end-to-end vectors in the

isotropic

state.

Next,

allow the distribution of _mass

points

to

undergo

_a spontaneous ^{uniaxial distortion to} a new set of

positions (R°), given by

Ro(x~)

₌

A.x~, (6.i)

where A is

analogous

to the

"square

root" of the step

anisotropy to

in the nematic

gel,

and the uniaxial distortion is in fact the nematic transition. The free _energy of the system ^{in the}

nematic state is now invariant under two sets of rotations:

Ro(x~)

_-

R(x~)

₌

ueRo(u~x~), (6.2)

where Uo is _a

rigid-body

rotation of the

network,

and the rotation

U~

_expresses the freedom of the nematic director to choose _any

direction, independent

of the

original

distribution of _mass

points (crosslink positions),

_as in

equation (6.I).

This symmetry ^under simultaneous separate rotations is the _same set of

symmetries

found in the model for the Gaussian

elastomer, (Eq.

(3.I)).

Now _we examine the continuum elastic

theory

^{of the} nematic state

constructed,

_as in

equation (5.2),

from uniaxial invariants of the

symmetric

strain tensor,

~"~

2

iRj

^~

^'R$ 1'

^~~'~~

where

u(R°)

= R R° _{is the}

phonon

variable. Let _us consider the behavior of the strain tensor under the _set of

symmetries (6.2),

^which must leave the free energy ^invariant.

Letting

R be the symmetry transformation

(6.2),

_we find

E =

((UOAU~A~~ I)

+

(UOAU~A~~ I)~j ^(6.4)

t

~#~ [AJ"A~~ A~~~J"A~)

,

(6.5)

where

J$p

^{is the} Levi-Civita tensor, and _we have

expanded

the rotation matrices

(Uo

₌

^e°»~"

to lowest order in

4

and @. Summation _over

repeated

indices is

implied.

The

rigid body

rotation has _no contribution _to the

symmetric strain,

_as it should not, but there is _a non-trivial

symmetric

strain which

depends

_on the

arbitrary

rotation

4.

If A is _an

isotropic

dilation _or compression, ^{then the} strain _e vanishes.

Similarly,

_e vanishes if

4

is _a rotation about the direction of

uniaxiality. However,

if

4

is _a rotation about _one of the

oblique directions,

which rotates the

director,

_{a non-zero} strain _em is

induced,

where I

denotes directions

orthogonal

to fi. Since the elastic free _energy must, on symmetry

grounds,

be invariant under

(6.2),

these components ^of strain must not appear in the continuum free energy. Hence the elastic _{constant p2} for the invariant

(fi

_{e x}

_fi)~

_must vanish. If _we

enlarge

the continuum elastic free _energy to include both director fluctuations and elastic

strain,

_as

in

equation (5.2),

the

vanishing

of _p2 is

accomplished by

the "renormalization"

by

director

fluctuations,

_as in

equation (5.14). Moreover,

this renormalization _{occurs to} all orders in strain and director fluctuations.

As described in section 4, the

key

to

understanding

the soft modes is the realization that

an

anisotropic

distribution _may be turned into the _same

anisotropic distribution,

in _a different

principle frame, by

both _a trivial rotation (@

above),

and

by

_a deformation which

exchanges

the identities of members within the distribution while

leaving

the statistics of the distribution

unchanged.

This symmetry

only

exists

if,

in

fact,

there is _no correlation between the distribu- tion of crosslinks

(or

_mass

points)

and the direction of

uniaxiality

chosen _upon

undergoing

the

nematic transition.

Now _{we return} to _our discussion of the nematic

gel.

The

arguments

^above

imply

that _any

model of _a nematic

gel

which _possesses this symmetry, ^Gaussian _or

generalized

to include

higher

moments of the strain deformation

A,

must possess a set of soft deformation modes. In

fact,

_any nonlinear elastic

theory

of _a nematic

gel

must be constructed from invariants of

A,

to, and t which respect

symmetries (3.1, 3.2) [such

as lh

(>to A~)

^"

t~~'),

with _n and _m

arbitrary]

and it is

straightforward

to show that all such invariants lead to the _same set of deformations which leave the free _energy of the undeformed state

unchanged, (Eq. (3.6)).

This

implies

that the continuum elastic

theory (Eq. (5.2)), always

has the relation

pf

= p~

al /(401)

_among

its elastic constants.

Our model for the nematic

gel

has the Goldstone modes because _we have assumed that the director and the distribution of crosslinks in the

isotropic

state are uncorrelated.

Indeed,

we characterized the isotropic state

by

_a Gaussian distribution of end-tc-end vectors, ^{and did}

not include _any correlations between crosslinks _or

entanglements.

While it is conceivable that correlations _among crosslinks could _preserve this symmetry, it is

probable

that

topological entanglements destroy

this symmetry and create

quenched

disorder in the network. This would _emerge in the nematic state as

quenched

random stresses, a characteristic of disordered

amorphous

systems

[10, 23],

and it is

possible

that this disorder could

destroy

the

long-range

order of the nematic state [10]. ^{The work}

by

Golubovid and

Lubensky

suggests ^the

following interesting

and

important

consequences of

quenched

random _stresses:

(I)

the

already

soft

phonons

_are

predicted

to

soften

^from

(u(q)~)

~-

q~~

to ~-

q~~

and

(2)

within _a harmonic

approximation, long-range

nematic order is

destroyed

at dimensions d < die = 3. The estimate of the lower critical dimension die _" 3 is,

however,

_a

highly

uncertain

approximation [10],

^and

remains _an open question.

JOURNAL DE PHYSIQUEII T 4, N' 12, DECEMBER 1994 83

7.

Summary.

We have examined the

symmetries

of _an elastomer crosslinked in _a nematic state,

assuming

the network

comprises monodisperse

strands which

obey anisotropic

Gaussian statistics. For

a director rotation

through

_an

angle

_{uJ, a} continuum of non-trivial deformations

A(#,

_uJ) leaves the free _energy invariant; the set (uJ,

A(#,uJ)) comprises

the Goldstone modes of the broken symmetry state. These modes should also be present ⁱⁿ a swollen

elastomer,

since

they

_are

volume-preserving.

The elastic

theory

for small deformations is

non-trivial,

and certain scat-

tering geometries

should

yield

anomalous

scattering

from director fluctuations.

Entirely

analc-

gously,

there is _a set of soft transverse

phonons

_[10] whose fluctuations scale _as

((u(q)(~)

~-

q~~

rather than

q~~ Interestingly,

for _a spontaneous ^biaxial transition all _transverse

phonons

_are soft.

Any

additional corrections to Gaussian

elasticity,

which also

obey

the symmetry that the nematic director is uncorrelated from the crosslink

distribution,

must possess the _same soft director and

phonon

modes.

Acknowledgments.

The author is

grateful

to M.

Warner,

E.

Terentjev

and P. Bladon for _many conversations and for

introducing

him to their model system; and to C. Nex and M. Cates for

helpful

discussions.

Appendix

A.

Reduction to continuum elastic

theory.

Here _we outline the reduction of the non-linear elastic energy

(Eq. (2.7)),

_to the continuum elastic

theory

free energy

(Eq. (5.2)).

We consider _a state with _a non-trivial step

anisotropy given by

equation

(4.I).

For temperatures well below the nematic transition temperature ^the

magnitude

of the

uniaxiality

is fixed

by

the nematic entropy, _{so we} consider fluctuations which include small elastic strains and rotations of the uniaxial direction. For small deformations

u(r)

_away from the

isotropic

state,

lap

₌

sap

+

dpua. (A.I)

A rotation of the director

corresponds

to a rotation of the step

anisotropy

tensor:

t =

e~»~"toe~~»~" (A.2)

=

to

+ hi + A2 +

,

(A.3)

where

hi =

uJ~(J"to toJ") (A.4)

A2 = uJ~uJ~

[J"J"to

+

toJ"J"

2

J"toJ"] (A.5)

The rotation is

by

an

angle

uJ

through

the axis _w and

e~»~"

_{is the rotation}

matrix,

where

J$p

is the Levi-Civita

alternating

tensor.

Next _we

specialize

to an

incompressible

_system.

Incompressibility

is maintained

by prohibit- ing volume-changing deformations;

that

is,

_we

require

detA = I

(A.6)

= + lhvu ₊

((lhvu)~ lhvuvuj (A.7)

+

((lhvu)~

^{+ 2}

^lh(Vu)~

^{3 lhvu lh}

_(Vu)~j

where the second

equation

is _an exact expansion of the determinant of A _as

given by equation (A.I).

Hence the components ^of

u(r)

must

satisfy

V _u

=

((lYVuVu lYVu)~j

⁺

^(A.8)

We _now

expand

the elastic free _energy

(2.7)

to

quadratic

order in the small

quantities u(r)

and _{uJ, to} obtain

F~i Ci lY

(Vu tp~Ai)~to(Vu tp~Ai _)tp~

₊

_{lY(Vuvu) (tp~Ai}

_)~

_(A.9)

2

where _we have used the identities

lYA2tp~

=

jlY [Aitp~Aitp~)

and

lYtp~

hi

= 0. The

strain may be

separated

into

symmetric

and

antisymmetric

parts; Vu

= e +

fl~J",

where

e =

((Vu

₊

Vu~)

and fl ₌

jV

x u.

Using

the definition of to

(4.I), recognizing

that the director fi is the axis of

uniaxiality I,

and hence that _{w x} fi

= 6fi ₊

O(6fi~),

_we

finally

arrive at the continuum free _energy

(Eq. (5.2)),

with the elastic constants

(5.4-5.8).

Appendix

B.

Scattering

from director fluctuations.

Here _we sketch the calculation for the

scattering

from director fluctuations. The

anisotropy

of the dielectric tensor fag ^is determined

by

the local nematic order parameter,

through

the

relation

cap(r,t)

₌

16~p

+

fifoap(r,t)

₊

(B.I)

Omitted terms

correspond

to

quantities

built from other tensorial

quantities,

such _as the ori- entational order tensor of the

polymer

backbone in the _case of side-chain LCP'S. We _assume in this work that the

mesogenic

unit dominates the dielectric

properties

of the medium. Be-

cause e is _a

homogeneous

scalar, the fluctuations of the k

#

0 Fourier components of fag are

proportional

to the fluctuations of the director

(6E«P(k) 6E>p(-k))

=

fif~ (6Q«p(k,t) 6Q>p(-k,t)), (B.2)

where

6Qap

_"

S(fia6fip

+

fip6fi~). (B.3)

The constant fif~ _may be determined

experimentally

and the

angle

brackets denote _a thermal

average. The differential cross-section per unit solid

angle

for the elastic

scattering

of

light through

wavevector k is

proportional

to the fluctuations of the fluid dielectric _tensor [20,

24],

da QJ~

I

_16gr2~4 ~~~"fl~~~

~~~P(~~))

_{flu fi~fiA}

#I (B,4)

~4 161~2~4

~~~~ (~'fi~(~~'fi'~)

~ 2

fi'fi fi.fi'~(§fi.fi'§fi.fi)

+

fi.fi'~(§fi.fi)) (B 5)

where

fi

and

fi'

_are,

respectively,

the

polarization

unit vectors of the incident and scattered

light,

and _uJ is the

frequency

of the

light.

The thermal _averages

(6fia6fip)

_are calculated vlith the aid of the continuum free _energy

(Eq. (5.I)).

To

perform

this calculation _{we use} the geometry ^of

figure

3. First _we write the

phonon

field

u(r)

_as

u(r)

= u~k +

uyf

+

uzi, (B.6)

where the director fi is

along

the I axis. After Fourier

transforming,

the free _energy

density (5.I)

becomes:

2

fflu~(u,6fi)

=

~q~u(q).G(b).u(-q)

4

li~Al(b)~(~l'6fi(~~) A2(b)"z(~)6fi~(~~)

₊

~'~'l

+(6fi~(q)(~ [al

₊

q~(K3 ^cos~b

+ Ki

sin~b)) (B.7) +(6fiy(q)(~ [ai

₊

_q~(K3 ^cos~b

₊

K2 sin~b))

,

where

G~~(b)

₌

(al

_{+ a2 +}

p2) ^cos~b

₊

4(pi

₊

~i) ^sin~b (B.8)

Gyy(b)

₌

(al

_{+ a2 +}

p2) ^cos~b

₊

2pi sin~b (B.9)

Gzz(b)

=

(ai

_a2 +

p2)

+

2(ai

₊

2po) ^cos~b (B.10)

G~z(b)

=

Gz~(b)

=

(p2

+

2~o)

sinbcosb

(B.ll)

Ai(b)

₌

(ai

+

~a2)cosb _(B.12)

2

A2(b)

₌

(ai ja2)sinb, ^(B.13)

and all other elements of

G(b)

vanish. Note that 6fi

=

(6fi~, 6fiy, 0).

In this free _{energy al} is the bare "mass" for uniform director fluctuations. To obtain this expression we have used the relation between elastic constants which follows from the soft modes in the system, _p2

all (4ai

₌ 0. Next _we

integrate

out the

phonon degrees

of freedom

u(q)

to

yield

_an effective free _energy

governing

director fluctuations. The

coupling

^of

phonon

and director

degrees

of freedom softens of the effective _{mass ai,}

leading

to the soft

(Goldstone)

modes. We find:

2

fn~m(6fi)

₌

(6fi~(q)(~ [a~(b)

+

q~(K3 ^cos~b

+ Ki

sin~b)) (B.14) +(6fiy(q)(~ [ay(b)

+

q~(K3 ^cos~b

+ K2

sin~b))

,

(B.15)

where the effective _{masses are}

a~(o)

= ai

A(o) ^G-i (o) A(o) (B.16)

ay(o)

= ai

jl~(j~, ^(B.17)

and

A(b)

=

(Al (b), -A2(b)).

These

simplify

to

2ai

sin~b

_cos2b

2~o

(~o

+

al)

+

ja( _4(pi

₊

_~ci)(ai

₊

2po)j

°~~~~

G(~(b) Gm(b)Gzz(b) ^~~'~~~

°~~~~ ~~))~~~

~~'~~~