Deformation–induced orientational transitions in liquid crystals elastomer

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Deformation–induced orientational transitions in liquid crystals elastomer

P. Bladon, E. Terentjev, M. Warner

To cite this version:

P. Bladon, E. Terentjev, M. Warner. Deformation–induced orientational transitions in liquid crystals

elastomer. Journal de Physique II, EDP Sciences, 1994, 4 (1), pp.75-91. �10.1051/jp2:1994100�. �jpa-

00247951�

Classification

Physics

Abstracts

36.20C 61.308 61.40K

Deformation-induced orientational transitions in liquid crystals elastomer

P.

Bladon,

E.M.

Terentjev

and M. Warner

Cavendish

Laboratory, Madingley

Rcad,

Cambridge,

CB3

OHE,

Great-Britain

(Received

23 June

1993,

revised 29

September 1993, accepted

4

October1993)

Abstract. Solid

liquid crystals,

formed by

crosslinking polymeric

nematics into elastomers

are shown to

display

novel and

complex elasticity.

The internal

(nematic)

direction _can

experi-

ence a barrier to its rotation which

couples

to standard

elasticity.

We

investigate

this

elasticity by considering imposed

strains and demonstrate several

new orientational

phase

transitions, caused by the interaction between

applied

stress fields and bulk barriers to rotation.

1. introduction.

We find that

by stretching

_a

liquid crystalline

elastomer

obliquely

to its

optical

axis continuous and discontinuous transitions of the rotation of this axis _ensue. This _new

type

of nematic

transition has _now been observed

experimentally.

Polymer liquid crystals (PLC'S)

_are

long

chain molecules

composed

of _monomers

capable

themselves of

forming

nematic _{or more}

complex mesophases.

When concatenated _as

polymers they

combine the orientational order of nematics and the

entropy

driven

aspects

of

high poly-

mers. In

particular

such

polymers change

^their

shape

_on

orientationally ordering.

Given that

most

properties unique

to

polymers depend

_on their

shape,

it is clear that _new

properties

can

arise in these unusual materials.

Long

_ago de Gennes

[I] recognized

that the most dramatic effect of molecular

shape change coupling

to orientational order would be in elastomeric networks of PLC'S. Like conventional

elastomers these materials _can sustain _very

large

deformations which _cause molecular extension and

orientation,

but

conversely

spontaneous

alignment,

_{or a} nematic

phase

induced

by applied

stress, can lead to

spontaneous

^distortion or a

jump

in _a stress-strain relation.

More

recently

Abramchuk _{et al. [2]} and Warner et al. _{[3] have} constructed molecular models of these effects and obtained _an

understanding

of mechanical critical

points,

_memory ^{of crosslink-}

ing,

shifts in

phase equilibria

and stress-strain relations. Nematic and hence

anisotropic

chains

lead to a

straightforward

modification of the conventional Gaussian elastomer

theory.

The

theory

can be further

developed,

_as in conventional

networks,

to account for

junction point

fluctuations _[3] and other

effects,

but _to understand the

startling

_new

effects

visible in _ne- matic elastomers these elaborations _{are not necessary.} This

theory (and

later

theory

directed

towards biaxial effects

[4])

was

mostly

concerned with

phenomena

around the

spontaneous nematic-isotropic (N-I)

transition

including

the effect of

applied

stress _on the transition.

Experiments performed

after the initial molecular theories

[2,

3] have _seen the basic

effects,

for instance shifts in the

phase

transition

temperatures

^{with both}

crosslinking density

and

crosslinking

state

(I.e.

^nematic or

isotropic)

_[5], unusual stress-strain relations in the

region

of the N-I transition for the

elastomer, strong

^{deviations from classical}

stress-optical

laws and mechanical influences _on the

phase

transition itself [6].

Here we shall look at the

complex elasticity

^of the nematic

phase, including

the effect of

imposing

stress and strain _on elastomers

deep

in the nematic

phase,

with the

principal

stress-

strain _axes not coincident with the initial nematic direction. A

linear,

continuum

picture

has been sketched

by

de Gennes _[7] who described the

anisotropy

of such

media,

the

coupling

of

strain _to nematic order and the resistance to rotation. We shall find this rotational resistance to be

extremely

subtle and

dependent

_on

geometrical

constraints. A

preliminary

account of the

present

molecular

theory predicting

novel effects in the nematic

phase

has been

published recently _[8].

A molecular

theory,

in contrast to linear continuum

theory

_can describe non-linear

regimes

^where we find discontinuous transitions. The intimate

relationship

between rubber

elasticity

and

geometry

^allows us to

interpret

^these

effects, including

the role of constraints.

The

impetus

for this

theory

_comes from

experiments

_on nematic monodomains. The mis-

alignment

and interactions of

differently

oriented domains

partly

obscures the

underlying physics

of nematic solids. Indirect methods _[6]

could,

for

instance,

estimate the

spontaneous

mechanical 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 the

polydomain sample

to form _a monodomain.

More

recently

nematic monodomain

samples

have become available. Such

single crystal samples

_can be made

by crosslinking

in _a field oriented nematic melt

[9],

or

alternatively by

twc-stage crosslinking

and

stressing

the intermediate state

[10].

^{Internal stresses in the latter} method also

give instabilities,

but of _a different

type

^from those discussed in this paper.

Large

monodomain elastomers lend themselves to the mechanical

experiments corresponding

to the

predictions

of _our

theory

_[8] and _are

explored

in detail in this _paper.

Already

Mitchell _et al.

[ll]

have

performed applied

strain measurements on monodomain rubbers far below their N-I transition. With _stress

applied perpendicular

to the

original

^director

they

find that at _a critical strain the director

jumps by x/2

to the

principal applied

stress direction.

In nematic elastomers mechanical

(stress-strain) fields,

nematic fields and

electromagnetic

fields _are all

coupled.

Nematic elastomers offer _a barrier _to the rotation of the current director away from the direction of the initial director. Internal rotational barriers and the concomi- tant resistance to strains

interacting

with these internal

degrees

^{of freedom} is reminiscent of the Cosserat

"couple-stress" elasticity _[12].

In

respect

of nematic elastomers the full linear

continuum

theory involving strains,

nematic director

gradients

and electric fields _{has been} de-

veloped recently _[13].

The non-linear

analysis

that follows shows that nematic elastomers and any rotational barriers _are in fact _more

complex

_a

variety

^of continuous and discontinuous

orientational transitions take

place

when

subject

to

large strains,

I.e.

they

exhibit unusual behaviour found

only

in the

region

^of

great non-linearity.

Such transitions induced

by appli-

cation 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. The director remains therefore uniform.

We extend classical rubber

elasticity theory

to the _case of networks formed from

intrinsically

anisotropic polymer

melts. Chain

shape clearly depends

_on the

magnitude

^of the nematic order and _can be calculated

by

_many methods. The

simplest

is to treat the chain _{as a} sequence of orientable

freely jointed

rods. This limits the extent of

anisotropy

to

1( ^< 31° where 1° is the effective

step length

in the

isotropic

state and

lj

^{is the value}

^along

^{the director in the}

nematic state. Often the

anisotropy

^induced

by

the nematic order is

greater

than

this,

for instance

1( ^~

101°

_{in main chain}

systems [14].

^An alternative is to use a worm-like chain model to calculate chain

shape.

^In this paper we can avoid the need to calculate the chain

anisotropy explicitly.

In _some

geometries

we find that the instabilities in _a strain

experiment depend only

on initial chain

anisotropy

and not that

stemming

from

applied

fields. In other

geometries

the instabilities

depend

_on the current chain

anisotropy,

that is the

equilibrium magnitude

and direction of order after strains have been

imposed. However, experiments

_are

often

performed deep

in the nematic state far from TNT where the

ordering

is

strong.

We _assume that the

magnitude

of the order

parameter

does _not

change,

but

simply

the

principal ordering

directions rotate to _{a new}

equilibrium

under the

application

of stress. Under this restriction all

instabilities, phase

transitions and soft deformations _can be described in terms of the initial

anisotropy

^{and the}

imposed

fields. The vital _measure of the network

anisotropy

at

formation,

(I(/1[)~/~,

^{is the}

spontaneous distortion, lc,

_on stress-free

cooling

of the network from its

isotropic phase

back to its formation temperature. This

quantity

can

easily

be measured

by

neutron

scattering [14],

or

macroscopically by observing

the

spontaneous

distortion

[11].

^Thus

our

predictions

_are

independent

of detailed

microscopic

theories and measurements of nematic

polymer shape.

We have assumed that

deformations, leading

to _new nematic

order,

_are

imposed

at the

same

temperature

as network formation. It is

frequently

the _case that the network is formed at _one

temperature, Tx,

and deformed at

another,

lower

temperature, To-

In

lowering

the

temperature

^{the network will}

spontaneously deform, altering

the initial network

anisotropy.

We show in

Appendix

that it is

simple

to account for this

spontaneous

deformation. All the results derived in this paper

hold,

but with _{a new} definition of the initial network

anisotropy,

which encompasses both the

crosslinking anisotropy

and the

spontaneous

deformation.

Due to _some

algebraic complexity

of calculations involved this _paper is

organized

as follows.

The next section outlines the basic

concepts

^{of the molecular}

theory

of Gaussian

anisotropic

networks and describes the

geometry

of strains and director orientation in considered char- acteristic _cases. Section 3

presents

^the

results, illustrating

transitions and instabilities that take

place.

After

that,

in section

4,

we include _some technical details and calculations and the

discussion of whether

approximations

_are involved. This

section, probably,

can be

skipped

at the first

reading

of the _{paper so} that elaborations do not mask the

qualitative understanding

of

underlying

effects _we describe here.

2. Formulation _of the

theory.

In this paper we use the

following general

scheme for notation. We _{use a}

superscript

^° to denote

quantities

in initial

(undeformed)

states. Hence the initial 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 ₊

I,j.

^When

^describing

^uniaxial tensors subscribes _i and

jj are

used to differentiate

betwin

_{the two} distinct _axes. In their

principal frame,

the

jj

component

is taken to lie

along

^{the I axis of that frame. As} an

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.

In _a nematic monodomain with director

n°

coincident with the I

axis,

we assume a

poly-

mer chain

sufficiently long

that _spans

R°

_{between network}

points have,

at the moment of

crosslinking,

_a Gaussian distribution:

Summation over repeated

indices has

been

assumed.

The matrix 1)~

of

ffective

step

lengths defines the

chain

shape parallel and to the director n°

for

a uniaxial _phase,

that

is

_(R)Rj) = )Ll( where L

is

_the chain ontour length. The effective step

random walk, and thus the _overall

Q

^{o o}

Q~j

"

-Q (3n)n( 6,j)

_% o

-Q/2

o

~ 0 0

-Q/2

in _a

principal

frame. We shall find occasion to discuss both the

magnitude Q

and

principal

direction _n of this order

parameter.

The inverse effective

step length

matrix in

(I)

is:

1/1(

^{0 0}

jl[)~~

₌

l/1[6,j

₊

(I/I( I/I [)n[nj

_{+ 0}

_1/1[

₀

₁₂₎

o o

ill(

with

eigenvalues if I[

and

1/1(

^{in its}

^principal

^frame

^dependent

on the order

parameter Q°.

We shall confine ourselves to uniaxial formation _states

(in

the

twc-step

^method

[io]

^of

achieving

monodomains this need not

always

be the

case).

The effective

step lengths

_are related to the

mean square chain size

by ((Rj)~)

=

)Llj

and

((R[)~)

=

(Ll[.

In

highly

ordered main-chain melts ljj ^»

li [14].

We

employ

the affine deformation

assumption,

_an

assumption

that

pervades

network

theory.

Junction

point

fluctuations _are

damped by connectivity compelling junctions

to deform in

geometric proportion

to the whole. Junction

point

fluctuations _are

easily

handled and alter the elastic free energy

by

the trivial

multiplicative

factor

(1-

2

Ii),

where

#

is the

junction point functionality.

This modification has been demonstrated for nematic elastomers in reference

[3].

It is not essential to the _new

phenomena

_we introduce here.

Using

the affine deformation

assumption

we define the _current network span ^to be

R,

=

I,jRj

with

A,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

lo~ N/m~

and that for the volume

change

is

typically ^lo~° N/m2,

deformations of elastomers _are at constant volume

(to

within

lo~~ accuracy),

that is:

Det[I,j]

= I. In

general

the current

temperature

_may be different from that of the formation state.

Span probabilities

_are

governed by

_a

distribution, FIR), differing

from

Po(R°)

in

equation (1)

in that the

(l~)~~

tensor,

describing

the current chain

shape, depends

_on the current

ordering

tensor

Q. Taking

the usual

quenched

_average

Fej/kBT

=

-(In FIR))

_{p~(no) one} obtains for the elastic free _{energy per} network strand:

~

=

~

(lik)~~(RkRi)o

₌ ^~

(lik)~~(lkjRjh~R))o

BT

2L 2L

e

~Tr[1°

16~'

l~~ 11

,

(3)

where _we have used

(R)Rj)

₌

)Ll(

from

equation (I).

There is also _an additional term which

arises from the normalization of

P(R). Including

this

term,

the free energy is:

~'

=

Tr[1°

_1°~

_{l~~ ii}

_-in

^~~~~~ ₍₄₎

kBT

2 ~ " ~ "

et[~]

The resultant elastic free energy will

depend

_on the nematic order at formation

through

1° and

on the nematic order in the current state

through l~~,

that is

F~i

"

Fei(Q°, Q).

In turn

Q

will

= = =

depend

_on the

imposed

I

through

both

magnitude Q

and direction _n of the current nematic order. It is the _purpose of this _paper to

specify

how

Q

and _n

depend

_on the

imposed

strain.

As _a result of

applying I, minimizing

the free _energy will

yield

_new

Q

and _n,

differing

from

Q°

and

n°,

and hence _{a new}

equilibrium shape

I. Thus the current

director,

_n, is in

general

not coincident with the initial

n°, except

if I has the _same

principal

frame _as

n°

and is _a uniaxial extension. In this latter _case

ljj

=

I, ii

₌

I/fi

_and

~'

=

~~

l~

_{+ 2}^~~

is)

kBT li

Allowing

I in

is)

to relax to its

equilibrium (I.e. zerc-stress) value,

that is

taking 0F~i/0A

₌

o, gives

the

spontaneous shape change

Am ₌

(ljjl §/I(li)~~~

^{due to a}

^change

^{in the current}

nematic order with respect to its initial state [3].

One

particular

_case of this is essential. If _one heats the nematic rubber _so that the current state is

isotropic,

its

step length

tensor is

=

16,k.

The

spontaneous

distortion is then Am ₌

ill /lj)~/~,

^{which will in}

^general

^{be less than} one, a

flattening,

if the

crosslinking

state was

prolate

uniaxial. This is _a useful

experiment

since it is _a direct _measure of the chain

anisotropy

and is

pivotal

in what follows. The inverse

experiment

^would be the

cooling

the

initially isotropic

rubber down to the nematic state with

Q

_"

Q°.

This

gives

the

spontaneous

distortion

lc

₌ I

/lrr

>

I,

_an

elongation

of the

sample

that can be

easily

measured. In

general

_we

envisage

performing elasticity experiments

at the formation

temperature,

or at lower

temperatures

where Am >

1,

I.e. further chain

ordering

_{causes a}

spontaneous elongation

of the

sample.

Returning

the value of Am to

(5) gives

the elastic free _{energy as a} function of I and 1°:

F~i IQ)

"

F~j ii ,1,1°)

which represents an addition to the Landau-de Gennes free _energy of the nematic

ordi$qiJirtic

_{in the current order}

_{parameter Q}

_{if the}

crosslinking

is done in the

isotropic state, quartic

and

quadratic

in

Q

if the

crosslinking

_was

performed

in the nematic state see [3]

for _a detailed discussion. The free _energy

F~i(I ,1,1°)

then is minimized with

respect

to

Q

in

order _to

totally

solve the

problem.

^~

We _are interested here in the _case where

an

imposed

deformation I has

principal

_axes not coincident with

n°, _{thereby creating}

_an

equilibrium

state with _n at some

angle

6 with respect to

n°.

_In

separate

_papers we shall consider

applied (uniaxial)

stress and electric

fields,

both of which lead to novel _counter intuitive effects. We shall determine 6 _{as a} function of the

magnitude

of deformation and the orientation of the frame of the

imposed

deformation

A,j

with

respect

to

n°.

_{To do this}

we shall

proceed essentially

_as

above, minimizing F~i

with respect to any free components of and with

respect

to

Q

and

6,

which characterize

Q.

We consider several

types

of deformations in

increasing

order of

complexity taking

two

principal

_axes of distortion to define the

(z, z) plane.

For these distortions _we show below that the director rotates about the axis and remains in the

(z, z) plane,

therefore _{one can}

effectively

operate ^with

(2

_x

2) anisotropic

matrices. Each

type

^of deformation

produces

characteristic

instabilities. Some

produce

"soft

elasticity" phenomenon

hitherto unknown in rubber theories

n

.~

.~2

f

,., , ,

a)

n

~

e _,

' j~~

' v

b) _c)

Fig.

I.

Alignment

geometry ⁱⁿ the

(~,z) plane

for

(a)

constrained extension

[Sect. 3.I]; (b) imposed

extension experiment

by

Mitchell et al.

[Sect. 3.2]; (c) simple

shear

[Sect. 3.3].

_no and _{n are}

initial and _current directors

respectively;

unit vectors _u

(u,

_v for

simple shear)

define the

principal

axes of deformation.

and which _we discuss in _a separate _paper. ^{We list here} each type ^of deformation and

analyze

them later in separate sections.

(A)

Uniaxial extension at an

angle

_a with

respect

to I

(coincident

with the initial director

n°), figure

la. In terms of the unit vector u of the direction of

extension,

the deformation

tensor at a constant volume is

A~

=

(1/@)6,j

_{+ (A}

lilt)1t,ltj.

This is

simple conceptually

but difficult _to

apply

in

practice

since biaxial _{stresses are} needed to create uniaxial deformation in _a

misaligned

but

naturally

uniaxial

system.

The

simplicity

of I however allows _{one to see} the hallmark of instabilities

occurring

in _more

complex geometrfis.

_{We find}

a

jump

in the

director at a critical strain

applied

at a =

x/2.

For _{a <}

x/2,

the director _moves

continuously

away from

n°

as I increases.

(B)

Extension

applied

at _an

angle

_a ^with

respect

to

n°,

but normal transverse relaxation

permitted (giving

in

general

_a biaxial

I).

Relaxation via

simple

shear strains is

prohibited by symmetry, figure

1b. This

geometry iithat employed experimentally by

Mitchell et al.

[ll].

We

again

^find a

discontinuity

in the direction of _n with

I,

but _{now over a} range of _a around

x/2.

As _one reduces _a below

x/2 eventually

these transitions terminate at _a critical

point.

There is mechanical

hysteresis

where the transitions _are discontinuous.

(C) Simple

shear at an

angle

a with respect to the initial

director, figure

lc. In terms

of unit vectors

along

the direction of

shear,

_u, and

along

the

gradient,

_v, the deformation is

l~

=

6,j

+

lltiuj.

At _a critical _a,

dependent only

on the initial network

anisotropy,

the rotation of _n with I is continuous until _a critical value of I is attained. This critical I likewise

depends only

_on initial

anisotropy.

3, Results.

Henceforth _{we use a} coordinate

system (z,z,y)

based _{on u e} rotated

by

_a about

§

from the

original (principal)

^frame

of1°.

_{This will be}

a

principal frame,

for

instance,

for

imposed

uniaxial strain and _a

simple

frame in which _to consider shear. The tensors 1° and

l~~

_have

to be rotated

by angles

_a. and A

= a 6 from their

principal

frames. Second rank

(2

_x

2)

tensors, for instance 1°, ^{will be} characterized

by

their _mean 1°

=

(1( +1§)/2

and

anisotropy

61°

=

(lj -1[)/2

^{of their}

^principal

^{values and}

^similarly

^for

^l~~

^{and I. Details of} calculations

are to be found in the

following

section 4.

3.I UNIAXIAL EXTENSION.

Using

the

expressions

for 1° and

l~~ given by equation (2)

with the

corresponding

orientation of the director in each _case, and

taking

the uniaxial form 1 =

~

$~

one obtains for the elastic

part

^of the free energy

(with ly

₌

lilt)

" o

I/

^'

where the coefficients of the

angularly varying

terms

depend simply

on

1°,

^and

angle

_a. The variation of

F~i

with the relative orientation of the current

director, A,

is

through

^the

^Cl

^and

C2

terms.

Taking

the minimal free _energy, at

0F/0A

₌

o,

^and

inserting

A,

6A,

1° and

61°,

_one

obtains for the

equilibrium

orientation:

tan 2A

=

~(A( I)@sin

2a

~~~ ^~ ~~~~~

~/~)

₊

(If 1)(12

+

1/1)

_cos 2a

(7)

where

lc

=

(lj/1§)~/~ ^specifies

^the

^anisotropy

^{of the initial strand} conformation.

The rotation 6 of _n is shown _{as a} function of the

imposed

I at constant a in

figure

2a.

Figure

2b shows

6(a)

for various

(fixed)

I. When I is

imposed

at an

angle

a =

x/2

an internal barrier

prevents

_any ^{rotation of} n until I >

i~.

The

system

^{then breaks}

symmetry

^and

jumps

to 6

=

x/2.

For smaller _{a <}

x/2

there is _no

degeneracy

in the direction of rotation and the director is

continuously dragged

around towards _o.

One would

expect

that the discontinuous

jump

^{in the director} orientation is due

purely

to a

symmetry argument

^{that there is} a

degeneracy

at _a

=

x/2.

In fact _we shall _see in the next subsection that the _reason for such

discontinuity

is _more subtle and it may

persist

for _some

region

of finite _a <

x/2.

The

discontinuity

in

6(a)

is _seen in _a different

guise

in _some of the

other

geometries,

below.

The _curves for different _{o cross} and this is more

easily

seen in

figure

2b

where,

for _a

given

I <

lc,

the

6(a)

_curves have maxima, which

disappear

with _a

singularity

_at the critical

extension ₌

lc.

90

(£')

~

(/J)

2

~

~

60

~ i ^Ch

~ ~

~° 2

2

~D

30 3

,"'

l-o 12 1.4 16 18 2 0

~ ,,"

1

~ ~ ~

« (de_g)^{~ ° ~ °}

Fig.

2. Director orientation under extension.

(a)

Plot 0 _us.

~,

at fixed

angles

_{o: curve}

(I)

a = 80

deg, (2)

89 deg,

(3)

90

deg. (b)

Plot 0 _{us. o} for fixed extensions: _curve

(1)

~

= l.2,

(2)

and

(3)

are for ~ just below and above the critical value ~c ₌ 31~/~~,

(4)

~

= 2. In all _cases initial anisotropy

~(

=

l(/1(

= 3.

3.2 IMPOSED _EXTENSION. The

geometry

^of

imposed

extension

lzz

₌ I for _an

experiment

with

arbitrary

_a

allowing

transverse normal relaxation

only

is shown in

figure

16 where

l~z

=

lz~

= o.

(More complicated setups

^could be

envisaged,

for instance the biaxial relaxometer frame with rollers

[15]

so that

l~~

becomes fixed _as

well).

When _a

#

o _or

x/2,

shear relaxation is in

general

needed. The

experimental _geometry

restricts such shear _{strains to a}

(hopefully)

small

region

^of non-uniform deformation _near the

clamps (shown

in

Fig. lb).

The bulk of the

sample

_can

only

relax in

l~~,

with

ly

₌

I/l~~l

fixed

by incompressibility.

Again rotating

tensors to the

(I, I, j) frame,

_one obtains for the elastic energy, now with

l~~

appearing,

~ _~~~~

~

~~~~~

~

~~~~~"

_~

l])12

^~~~

The coefficients _a

f

arise from

writing1°

=

~

()

^and

^l~~

=

~

~

and

depend

on the

~ c z

j

_e

angles

_a and A. The coefficient _r

=

I(/ly

and is

equal

to

unity

if the

magnitude

of the order

parameter

remains constant.

Given that _we fix

only I,

we minimize the free _energy

(8)

with

respect

to

l~~

to obtain a

condition for

>~~.

bel~~

₊

cfll$~ ~

_~

~~~

This

quartic yields

the

equilibrium l~ (A, I, a)

_{as a} function of A

= a

6,

I-e- _{as a} function of the as

yet

undetermined

angle

6 of the director.

Minimizing

_over choices of the director

angle 6,

which appears in coefficients

d,

_e and

f,

that is

0F/0A

=

o,

_we obtain

261°ll~~

_{sin 2a}

~~~ ~~

(lo)

1°(12 1]~)

+

61°(l~ +1(~

_C°S 2°

Equilibrium

strain relaxation and orientation _are

specified by equations (9)

and

(lo).

We consider some

specific

_cases.

a ~ .'

j.--

_--

/ c,1'

,

~ ^_~'

p'd

_"L~

~ /j'

(

~'i

)",~

~' )li

~ '~

l,(

'ij

jj

^~j

~~

/~,

o

~~,

./jj ^.'

,(~(,'

_j ^'

_- "_."

, ~ ~ ,~ ~ ~ ~

l-o I-I 12 1.3 1.4 1.5 1.6

1

~

a) b)

Fig.

3.

a)Director

orientation under imposed strain, 0 _us. ~, is shown for initial anisotropy

~(

= 1°

ii

It (

= 3. The different _curves

correspond

to different

(fixed)

values of _o. Curve

(a)

_o

= 90

deg,

(b)

87.5

deg, (c)

85

deg, (d)

82.5

deg. b)

Plot of the transverse strain under imposed strain, ~~~ _~s. ~, for the _same values of

a and

I( It (

as above. The upper and lower boundaries of the hysteresis

region

are ~c, and ~c,<,

respectively.

(I) Specializing

to _{a =}

x/2,

_one has _c

=

o,

a =

I[

and b

=

I(.

^The

^{quartic (9)}

^{is trivial and}

the transverse dimensions become

~~~ ie~~~~ /j' _~"

~~~

~~~

/j

Inserting

the

equilibrium

values of the l's into

(8) yields

Although

the final strains _are not uniaxial

(l~~ # lyy) (11)

is reminiscent of the classical uniaxial result. For _a

=

x/2

the solutions of

(lo)

for the

equilibrium

director orientation 6 _are

6 = or

x/2.

The direction 6

= becomes unstable when

0~F/06~

~_~ ^-

o,

which _occurs at I

=

lc.

We thus

predict

_a

jump

in 6 from _{o -}

x/2

at I ₌

lc.

The

turning point

in F at 6

=

x/2

first becomes _a minimum at I ₌

@,

_a

significantly

smaller distortion. As _one

might expect

for _a discontinuous

phase

transition there is _a

hysteresis.

On

releasing

strain after

achieving

the

jump

^from 6

= 0 to 6

=

x/2

at

lc,

we

predict

that the system ^{should remain} with 6

=

x/2

until I

=

@

_is

achieved, whereupon

it

jumps

back to

= o. The

hysteresis region

^{is thus I} E

(@, lc).

(ii)

Strain

imposed

at _a

# x/2: contrary

to

expectations

based _{on case}

(A)

the discontinuities and

hysteresis

_we

predict

continue to be

displayed

for _a range of _a <

x/2. Figure

3a shows

6,

the direction of _{n, as} I is increased. The _upper and lower transition

deformations,

_now denoted

by lc<

and

lc,<,

_are ^{indicated and} are marked _as solid lines for the _o

=

x/2

_case.

For

angles

_a <

x/2, lc,

diminishes

rapidly

and the

hysteretic region

_narrows until _a reaches

a critical value when the transition becomes continuous.

Explicit

conditions for this critical

point

_are

lengthy

and

complicated

to solve

analytically. Figure

3b shows how the associated

relaxing

strain

l~~

varies with

I, suffering

the _same discontinuities and

hysteresis

_as 6.

The

experiments

of Mitchell et al.

[ll]

_were

performed

at so

°C,

about 70° below the N- I transition. We expect our

assumption

of the

constancy

^{of the}

magnitude Q

of the order

parameter

to

apply

_very well to their

experiments.

Mitchell et al. indeed _see the

discontinuity

in 6 _we discuss above. The thermal

history

of the

sample

must be accounted for if the

crosslinking

temperature

is not that of the mechanical

experiment.

We discuss this in

Appendix.

Figure

3b shows that the value of the

jump

in

l~~

and 6

depend sensitively

_on the

alignment

a. To

analyze

Mitchell _et al.'s

experiments

one needs _a

precise

value of _a and

ideally

_measure-

ments of

lc, (a)

and

lc« (a),

the

region

of

hysteresis

in

6,

for _a

variety

of

angles

_a. Since _at these low

temperatures

our

predictions

_are functions

only

of the _one

separately

measurable

quantity lc

and is not

dependent

_{on any}

parameters,

^detailed

quantitative comparison

^of

theory

and

experiment

^{would be}

highly

desirable.

3.3 SIMPLE _SHEAR. In the

(I,I,j)

frame with I

along

the shear direction

1 ₌ +

6,j

+

lit,uj (12)

Substituting equations (19)

into

(4)

_one

again

obtains _a free _energy of the form

fi

=

61~~ [Cl

_cos ^2A +

2C2

sin

2A]

+

C3 (13)

B

where the coefficients

Ck depend

_{on o,} I and the

anisotropy. Again

_{we can} minimize the free energy with

respect

to the director

angle

6

by taking 0F/06

= o

whereupon

the relative

angle

1h = a 6 is

given by

~~~ ~~

~~~ ~~~

~

12

if ~~) ^~~(~i ~~ i(~ ^{~~~~) ~~~~}

^~ _sin

_2a]

^~~~

When the initial state is

isotropic, lc

₌

I,

_we find that tan 2A

=

-2/1

the director follows the

principal

extensional component of the shear _as

expected.

For _a nematic monodomain with

anisotropy I( /1[

₌ 3 at

formation,

the director orientation

6(1)

is shown in

figure

4. For

large enough

initial

angle

_a

= a* between unit vectors _u and _no there is _a

discontinuity

in

6(1),

another manifestation of that found in subsections

(A)

and

(B).

The character of the director rotation

6(1)

and its discontinuities _can be established _ana-

lytically

from

(14), analysis

that _we present ⁱⁿ the _next section. A convenient _measure of the

anisotropy

is A

=

(I] I)/(I]

₊

I)

₊

(lj I[)/(lj

₊

§).

The initial rotation

6(1

_-

0). Figure

4 shows this to be

positive

at small _a, I.e. when the direction of shear is close _to the initial nematic orientation

n°. Approaching

a critical value of

a = ac the director _can

undergo

a

negative

rotation _as I rises from _zero. This first

happens

at

1 i

~

llj ^I[

°~ 2 ^~°~

~~~

2 ^~°~

l( +1[

^~~~~

For _a < ac we have 6

initially positive,

that is

d6/dl (,_~>

o. For the value

lc

₌

3~/~

_shown

in

figure

⁴ we have _{ac =} 30°. For _a > ac the initial rotation is

negative.

In the

region (ac,

a*

the director rotation

again

becomes

positive

^for

large enough

shears. The transition between

negative

^and

positive

^{rotation 6 becomes}

^sharper

as a increases toward a*.

c

~

t

co

e

C, d

120

2.0 -1.5 -1.0 -0.5 0.5 1.5 2.0

~

Fig.

4. Director orientation under shear, 0

us. ~, at initial anisotropy

~(

=

l( ^It (

= 3 for fixed

angles

_a:

(a)

_a

= 0

deg, (b)

_oc

= 30

deg, (c)

and

(c')

_a just below and above the critical a*

= 37.2 deg,

(d)

_a

= 45

deg, (e)

90

deg.

For _a < _ac

[between

_curves

(a)

and

(b)]

the

sign

of the initial rotation

0(~

«

l)

is

positive.

When _a is between _oc and o*

[curve8 (b)

and

(c)]

^{this initial} rotation is

negative

but it

eventually changes sign

at

increasing

k

Finally,

at _a > a*

[between

_curves

(c')

and

(e)]

the

director

always

rotates

against

the shear-induced rotation of the rubber matrix.

The critical a* where the director rotation

angle 6(~) displays

a

discontinuity

is

given by

a*

=

cos~~ @l(16)

with

corresponding

critical shear

~ ~ ~

((i

A2)~/2

)

~~~~

4. Calculations,

We present ^here more technical details omitted in the

previous

section. At the _same time _we shall discuss when results _are exact and when

they

involve

approximations.

Cases will be in the order of

previous

sections.

It is

clear,

for I _> I in

figure la,

for

general

shear I in

figure

lc and for the stress and

applied

electric field _cases, that the

equilibrium

_n must be in the

plane

of I

(I.e.

^of

n°)

_and

u.

It is

sufficiently general

to consider _a

(2

x

2) problem

in the

(z, z) plane.

With the rotation of

n

by

the

angle 6,

the

major principal

direction of I also rotates. Seen

in,

_say, the frame of

I,

matrices 1° and have

off-diagonal

elements in

(z, z) only

and _none

connecting j

with either I

or I. Thus in the free energy

(4)

^there is _an isolated term

I(~l(/ly.

In

Tr[1°

16~'

l~~ ii

the tensors can now be

thought

of _as

(2

x

2)

tensors of the

remaining

^elements

(z, z).

I and

l~~

have

principal

frames rotated

respectively by angles

_a and 6 _away from that

of1°.

The current conformations

depend

_on the _current order

parameter Q (which

_may be much

larger

than

Q°

if the

temperature

^has

dropped

since

fabrication).

In

general,

since the _axes of I _are not coincident with those

of1°

_{the distortion forces the network chains and hence the}

Phase

itself to be

biaxial,

with _a

seiond

_order

_parameter

_{X. In its}

principal

^frame I then has

three distinct elements

[, _ly

and

lz depending

on

Q

and X.

Strictly speaking

the total free energy, the _sum of the nematic and elastic contributions

Ftotai

₌

F~~rr

+

F~i

must be minimized with

respect

to

Q,

X and 6 for each

I,

_{a or} E

imposed. Experience

shows that X is

generally

small [4]. ^For many

geometries

_we show that the

optimal angle

6 does not

depend

_on

Q

and X

(although F~i

and

Fn~rr do).

We shall

throughout simply

minimize

F~i

with respect ^to the

angle

6 of the _new

ordering,

but not with

respect

to the _new

magnitudes Q

and X of the order.

This

corresponds

to

temperatures

well below the N-I transition where the order saturates and

no further

changes

_are induced

by applied

fields. We shall indicate in the detailed derivations

the _cases where this is exact, ^and discuss the

procedure

where it is

approximate.

Since it is sufficient to consider _a

(2

_x

2) problem,

_{we can}

always

write

1[~

^{in the form of}

equation (2), replacing lz

and

[

with _ljj and

ii respectively (the

third component of the tensor

for _can of _course be different from

ii ).

Then the

only _part

of the free _energy

(4)

that

depends

on the current orientation _n

(or

_on the

angle

A _{= o} 6

=

cos~~(n u)

if _one instead _measures

angles

^from the

principal

frame of

I)

is the second term below:

fi

=

£Fo(Q, lyy)

+

~~~ )lTr[1° ^{(nl)~' (nl)] (18)}

B B 2 ljj 1 ^{~ " "}

The additional

component

to the free _energy,

Fo,

contains the

1(~[l(/ly] factor, independent

of the frame rotations _a, 6 but

dependent

_on

Q°, Q

and X.

Fo

also contains

In(Det[1°]/Det[ ii

which has _no

explicit 1,

6 _{or a}

dependence

and

only depends

_on

Q

and X.

Calculations _are

performed

in _a coordinate

system

^based on the distortion field I. On rotation

by

_a and A

= a 6 from their _own

principal

frames to the frame of

I,

I and

l~~

become ^~°

1° =

fli

₊ 61°

1°~~

^°

~~f1~°~

^°

(19)

sin o cos o sin a

The _mean of the

principal

elements remains invariant _on

rotation,

the

anisotropies 61°

and

61~~

transform with

angular

factors to

non-diagonal

forms. The coefficients _a

through f, arising

when 1° and

l~~

are written _as 2 _x 2 matrices

[see

^after

(8)],

_are

a = 1° + 61° _cos 20 b ₌ 1° 61° _cos 2a _c

=

-61°

_{sin 2a}

(21)

d = l~l ₊

61~~

cos 2A _e

= I-1

61~~

cos 21h

f

=

-61~~

_{sin 21h}

r =

I(/ly

4.I UNIAXIAL EXTENSION.

Inserting

in

(18)

the above elements

of1°

_and

l~~

_{and the}

simple

uniaxial form for I _one obtains for the coefficients

Ck

in

(6)

Cl

_"

21°161

_{+ 61°}

[l~

+

(61)~]

_cos

2a; C2

= 61°

[i~ (61)~]

sin 2a

C~

₌

2@(1° _[i~

₊

(61)~]

+

261°161cos 2a) (22)

The _mean and

anisotropy

^{of I} are I

=

(I

₊ I

II) _/2

_{and 61}

=

(1- lilt)/2.

The variation with the director

angle

_o 6 is

purely through

the

Cl

and

C2

terms of

(6)

and these terms have _{a common}

prefactor

of

61~~

Free _energy is minimized at 1h

satisfying

tan 21h ₌

C2/Ci

and thus _we have the

resulting equation (7)

which does not involve

l~~

_and

hence does not involve

Q

_or X. In fact

inspection

of

equation (18)

shows that this is

a~general

result. In

(18)

^the

dependence

_on the order

parameters Q

and X is restricted to the first term of the free energy and the

prefactor

of the trace which in turn contains

only

6 and _a.

If all

components

of the the deformation I _are

imposed,

that is there is _no freedom _to

relax,

then _{we can} minimize this free energy with

respect

to 6 to obtain the orientation of _n without

considering

the

self-consistency problem (the minimization)

in

Q

and X.

Accordingly

the results of section 3.I do not involve _any

approximation

of

Q

and X.

4.2 IMPOSED _EXTENSIONS

(AT

o =

x/2).

Take _o

=

x/2

_{as an} illustration of the _process.

With the

explicit

forms of coefficients _a

f

_now

given

in

(21),

the

collapse

of the

quartic equation (9)

when the coefficient

c(a

=

x/2)

_e o _{can now} be

clearly

_seen. Coefficients d and

e become

simple

functions of 6: d

= I-1

61~l

cos 26 and _e

=

@

₊

_61~~

cos 26. Since F is _a function of _cos

26,

~~~~~~~

°"°

~ _~°~

~0(~~26)

~~~~

This vanishes at

@

~~

"

q

^"

^{~f' (~~)}

(compare

with

lc appearing

in

Eq. (7)).

Since F

depends

_on cos 26 it is clear that 6

=

x/2

is

also _a

turning point

and _at I

=

lc,

it is _a minimum.

The

stability

of the other

turning point

is established from

(23) by setting 0F/0(cos 26)

= o

at 6

=

x/2;

it

yields

~

fi~

~

(il)~

^{~~'~ ~~~~}

This value

lc,,

is smaller than the first

instability

at

lc,

associated with 6

= 0:

1/6

lc<,

₌

lc, (~) (26)

ill

Note

that,

_as

expected,

the limits to

stability

~~, and

lc,,

involve

components

^{of the} current

tensor of the chain

shape

and hence the current values of order parameters

Q

and X. Self-

consistency

demands

that~one

_{also minimizes with respect to the}

magnitudes

of

Q

and X.

This

will,

in addition to the above determination of the orientation of _n

(I.e. 6)

of the tensor

l~~,

also fix the

magnitudes

_ljj and

li (more accurately

denoted

by lz

and

l~)

ⁱⁿ the above.

imposing

strains I not

aligned

with

n°

when _near the N-I transition

presumably

has _a

profound

effect _on the

ordi

_of

a nematic elastomer. There is then _no alternative but to

optimize F~i

with

respect

to

Q

and X. Bladon and Warner _[4] have _seen this in _an extreme form when

describing

uniaxial

compressions

to the

isotropic phase

of _an elastomer _near the N-I transition

using

the worm-like chain model for the nematic

polymer.

There is

ultimately

_a transition to _a biaxial nematic state with _n

perpendicular

to the

compression

axis and

large

values of

Q

and X created.

At lower temperatures where the order is

already high

_we

adopt

a

simplifying assumption:

with

Q° large

_any further increase caused

by

I is minimal and the most that _can be achieved is a rotation

(by

amount

6)

of

l~~

_{with respect to the axis}

of1°.

_Thus

we have numerical

equalities

of the

principal

frame

quantities

_{ljj =}

I(

^and

li

_" I

(

_{with the ratio}

r =

I(/ly

₌ 1.