Nematic distortions in adjustable geometries: repartitioning and instabilities

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Nematic distortions in adjustable geometries:

repartitioning and instabilities

D. Williams, A. Halperin

To cite this version:

D. Williams, A. Halperin. Nematic distortions in adjustable geometries: repartitioning and insta- bilities. Journal de Physique II, EDP Sciences, 1994, 4 (10), pp.1633-1638. �10.1051/jp2:1994109�.

�jpa-00248066�

Classification

Physics

Abstracts

64.70M 82.70G 62.10

Short Communication

Nematic distortions in adjustable geometries: repartitioning

and instabilities

D.R.M. Williams

(~,*)

and A.

Halperin _(~)

(~)

Laboratoire de

Physique

de la Matikre Condens4e

(**), Collkge

de France, 75231 Paris Cedex 05, France

(~)

Laboratoire Ldon

Brillouin, C.E.N.-Saclay,

91191 Gif-sur-Yvette Cedex, France

(Received

30 November 1993, revised 25

ApriJ

1§94,

accepted

30

August 1§94)

Abstract. Nematic distortion _{can occur} in constraint free systems, where the geometry is

adjustable

and the nematic is free _to

repartition

itself among various

regions.

In such _cases the nematic distortion

gives

rise to _a term in the free energy which

depends

_on the dimensions of the system. This _can

produce

_a mechanical force due to nematic distortions. In turn this leads to _a rich

phenomenology including

_a

variety

of instabilities. Different scenarios _are found when the

distortions _are due to surface

alignment

and to external fields. These effects _are relevant _to the

understanding

of _a variety of systems among them nematic

gels

and nematic systems studied

using

the force measurement apparatus.

The force measurement

apparatus (FMA) developed by

Israelchvili has been used with

great

effect _to

study

_a

_variety

of

simple

and

complex

fluids

iii.

While it has _a

great potential

for the

study

of

liquid crystals,

there has

only

been _a small effort in this direction

[2].

^Such

experiments

involve _a novel situation in

comparison

to

typical

studies of nematic

distortions,

which

usually

involve constrained systems: the

geometry,

^the dimensions and the

anchoring

conditions _are fixed. A

typical example

is

splay

in _a nematic confined to _a

rigid capillary imposing perpendicular anchoring

_[3]. In marked contrast, these constraints _are not enforced

within the FMA. The nematic fluid is free to

spatially repartition

itself between different

regions

so as to attain _a state of minimum free _energy. While these effects _are

easily

observable within the

FMA,

similar

phenomena

_are

expected

to _occur in other

systems

such _as nematic

gels [4],

nematics in porous media _etc. As _we shall _see, the

repartitioning gives

rise to a mechanical force. In certain _cases,

instabilities,

reminiscent of second order

phase

transitions _are

expected.

(*)

Present Address:

University

of

Michigan,

Ann

Arbor,

MI 48109-1120, U.S.A.

Permanent Address: Institute of Advanced

Studies,

Research School of

Physical

Sciences and

Engineering,

The Australian National University, Canberra.

(**)

Unitd Associde _no. 792 du CNRS.

1634 JOURNAL DE

PHYSIQUE

II N°10

This letter is concerned with theoretical

aspects

^{of these}

phenomena.

The

repartitioning

effects

are due to the

long

_range orientational order in the nematic. The distortion of this order

gives

rise to _a free _energy

penalty

whose

magnitude

is _a function of the dimensions of the

system.

Thus,

^while the nematic distortion is _a bulk

phenomenon,

it is somewhat reminiscent of surface tension

[5]. However,

the nematic distortions

give

rise to _a richer behaviour. As _we shall _see, the

corresponding

term in the free _{energy can} be either _{a convex or a concave} function of the dimensions of the

system.

^{In marked} contrast, the surface tension contribution to the free energy is

always

_{a concave} function of the system ^size.

Furthermore,

electric and

magnetic

fields

typically

have weak effects _on the surface tension while

giving

rise to

important

^effects

on the nematic distortions. In

particular,

the

application

of such fields _can

produce

_a nematic distortion in _a

previously

undistorted

sample,

and this distortion lowers the free energy. This

phenomenology

is of interest from _a

variety

^of

perspectives.

The forces considered in this letter

are due to static distortions. It is thus useful to contrast them to the weaker forces

arising

because of fluctuations in the nematic order [6]. ^{The three} cases

exemplify

different forms of finite size effects. Of

greater importance

is the relevance of these effects to the

understanding

of

experiments involving

nematics in constraint free

systems.

The FMA is of

particular

interest.

In _a

typical experiment,

_a fluid is confined between two

molecularly

smooth mica surfaces while in contact with _a reservoir. A

compressive

force is

applied

and the

separation

between the

plates

is measured thus

yielding

a force-distance

profile iii. However,

the apparatus ^{also allows for} measurement of the

birefringence

^of the

sample

and for the

application

of _an electric field and of shear

iii. Finally,

cleaved mica

imposes anchoring parallel

to the surface at _a definite

angle

from _a

crystallographic

axis. The

anchoring

behaviour _may be modified

by deposition

of _an

appropriate

surfactant

monolayer.

These

options

enable _a

variety

of

experiments

_on nematics [7]. ^{In all of these} cases one should allow for the role of distortion driven

repartitioning

of the

nematic between the slit and the reservoir. As _we shall _see, this _can

modify

the force-distance

profile

_as well _as

give

rise to instabilities. Our initial discussion _concerns nematic distortions due to the combination of

anchoring

conditions and the

imposed _geometry.

Similar situations

may occur when _a nematic penetrates into _{a porous} medium. The

subsequent

discussion focuses

on slit

geometries

in which the distortion is due _to mismatched

anchoring

conditions _or the

application

of _an external electric field. While these considerations _are motivated

primarily by

their relevance _to FMA

experiments,

_we also discuss the field induced

collapse

of

liquid

crystalline

lamellar

mesogels.

For didactic _{reasons, our} initial discussion _{concerns a} somewhat artificial

system.

^{It is} cho-

sen because of its rich behaviour traceable to the

spatial

variation of the distortion free energy

density.

The system ^{consists of} a nematic fluid confined to _a

wedge

obtained

by cutting

a

radial sector out of two concentric

cylinders (Fig. la).

The radial walls

impose parallel

_an-

choring. Perpendicular anchoring

is

imposed by

the inner and outer

cylindrical

surfaces. For

simplicity

the surface _energy associated with all the surfaces is assumed to be _zero. The

wedge

is

specified by

four parameters. ^A constant

height, L,

an inner

radius,

_r;, an outer

radius,

ra, and _an apex

angle,

6. The director

field,

_n, in this

cylindrical

_system

obeys

V _n

=

1IT.

Accordingly,

the distortion free energy

density

is

K~/2r~

and the distortion free _{energy per} unit

length

is

F/L

=

(6/2ir)(K~ /2) f)°dr2irr(V n)~

=

(6K~/2)In(ra IT,)

where

K~

is the

splay

elastic constant. In the

following we'use

_the dimensionless free _energy,

f

₌

2F/LK~,

and the volume of the

wedge is,

V

=

(r( r) )L6/2.

We first discuss the behaviour of _a

single adjustable wedge.

Since in this _case the volume is constant _we consider two scenarios where either _{r, or}

ra are held fixed. In each _{case we} seek the

equilibrium configuration

of the system. When r; is

fixed,

6 and _{ra are} allowed to vary and

f

=

(6/2)In(1+ 2V/6Lra).

The

corresponding

equilibrium

state is

specified by

⁶ = 0 and _{ra = cc.} In this

configuration

the nematic fluid avoids the

inner, strongly splayed region.

In the

opposite

case, when _ra is constant while 6 and

R _ro

(al _"=-,_ _.--;.

i=iliif

e 8

~-"Ii-°II

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

~~~ (Cl _, i

~ i i i i i, i

"""""°~-~~Z°-~-~ Gi'iif if<,','<'<

i i

Fig.

I. The director

configurations

in the various

geometries.

The director _n is

aligned along

the

molecules, represented by

bars.

(a)

A double nematic wedge.

(b)

A nematic between two

plates

in the twisted geometry, seen side-on. The nematic molecules twist out of the

plane

from top to bottom.

(c)

A nematic in _a slit after

undergoing

the Frederiks transition,

again

_seen side-on.

r; are free to

adjust,

_we find

f

₌

(-6/2)In(1 2V/6Lra).

The

equilibrium

state

corresponds

to 6

= 2ir and _{r; =}

(r( VLlir)~/~

A richer behaviour is found when two

coupled systems

are considered _{so as} to allow the nematic fluid _to

repartition

between the two

adjustable wedges.

We discuss the three _cases in which

(6,r;), (r;,ra)

and

(6,ra) respectively

_are held

constant. As _we shall _see, the three systems differ

radically

with

regard

to their

stability.

In the first _{case we} denote the two

adjustable

outer radii

by

_ra and

Ra.

For this

system f

=

6[In(Ra IT,)

+

In(ra IT;)]

and elimination of _ra

using

V ₌

(6L/2)[(R( r))

₊

(r( r))]

leads to a free _{energy convex} with respect to

Ra, f

₌ 61n

~, ()

+ 2

$) ^~~j.

_This

r; r;

system ^{is unstable: in}

equilibrium

all the nematic fluid is found in _one

wedge.

This situation is reminiscent of the

instability,

due to surface

tension,

of _two

coupled droplets is, _8].

In this

case the

larger droplet

"eats" the smaller

droplet

_{so as} to lower the total surface _energy. In marked contrast neutral

equilibrium

is found when _r; and _{ra are} fixed while the _apex

angles,

6 and 8 _are free to vary, so that

f

=

(6

₊

8)In(ra _{IT; ).} Finally,

when _ra and 6 _are held fixed the system exhibits stable

equilibrium. Denoting

the inner radii of the two

wedges by

_r; and

R;

_we find _{a concave}

f

₌ 61n

~f (- _~~~

+ 2

$) _{~~j leading}

to _a stable

equilibrium

Lr~

_r~

at

R;

= r;. The differences between the three cases are all traceable to the constraints and the

r~~ spatial dependence

of the free energy

density.

While different functional forms _may obtain in other _cases, the rich behavioural

repertoire

is _a

general

trait of unconstrained

liquid crystalline systems.

Nematic distortions _can also be

produced by combining

surface

alignment

with

antagonistic

fields. These situations involve

qualitatively

distinct effects. To illustrate this

point

consider

a

system consisting

^of a distorted nematic in two

coupled

slits. The nematic fluid is free to

repartition

between the two slits whose

heights, Hi

and

H2,

_are

adjustable.

When the distortion is induced

by

mismatched

anchoring

the

system

is found to be stable. On the other

hand,

when the distortion is driven

by

_an external field the

system

can be either neutral _or unstable. These differences _occur because the free energy of _a nematic slab distorted

by

_a field is lower in

comparison

with the free energy of the undistorted _state. In marked _contrast the free _energy of _a nematic distorted because of

boundary

conditions is

higher.

We first discuss

distortions in the absence of _a field. The

simplest

^field free distortion is pure twist. It _occurs when both _upper and lower

plates impose parallel

but

orthogonal anchoring I-e-,

the director

1636 JOURNAL DE

PHYSIQUE

II N°10

n at the interfaces of the two

confining plates

is

parallel

to the surface and _{n at} the upper

boundary

is

perpendicular

to _n at the lower _one

(Fig. lb).

The

angle

made

by

the director with easy direction _on the lower

plate

is

6(z)

=

(ir/2H)z. Accordingly~

the free _{energy per}

area A is

Ft

=

(AKt /2) f/ _dz(d6/dz)~

=

(ir~ /8)AKt /H.

Here

Kt

is the twist elastic constant and _z is the

height

above the lower

plate.

For two

coupled slits,

^each of basal _area

A,

the total

volume, V,

is

conserved,

V ₌

A(Hi

+

H2)

where

Hi

and

H2

_are the widths of the _two slits. The total free energy, ^F =

(ir~/8)AKt _[Hp~

₊

(VIA Hi )~~j,

has _a minimum at

Hi

_"

H2

+

VIA indicating

stable

equilibrium

at this

symmetric

state.

As noted

above,

the behaviour encountered for field induced distortions _is

markedly

different.

This is because the field favors the distorted _state whose free energy is thus lower

[9].

^For

concreteness consider

plates imposing perpendicular anchoring

conditions

(Fig. 1c).

The

electric field is

applied perpendicular

to the slit but favors

alignment parallel

to the

plates, I.e.,

the nematic has _a

negative

dielectric

anisotropy.

A

simple expression

for the free _{energy per} unit _area,

F,

is obtained via the "one constant"

approximation,

when all the elastic constants

of the nematic _are assumed to be

equal. Ignoring

terms

independent

of

H,

F is

given by

F =

(K(~~ ff

_dz

_(~ (fl)~ sin~6(z)

,

where K is the nematic elastic _constant. Here

(

is

z

a '~coherence

length"

_[9] related to the electric field

strength

E via E

=

(~~(Klea)~/~,

where ea is the dielectric

anisotropy. 6(z)

_measures the

angle

made

by

^the director with the normal to the

plates

at

height

_z. The first term allows for the distortion of the director while the

second reflects the

coupling

to the electric field. At very small

fields,

before the transition takes

place,

there is _no distortion

I.e., 6(z)

= 0 and F

= 0. The transition takes

place

at the critical field E

=

Ec

for which H ₌

ir(.

A distortion _{occurs as soon as} the field is increased above

Ec

_so that

(

<

Hlir

and the free _energy is lowered to F ₌ 11

/2)K(~~u(H/().

Here

u(h)

is _a

negative, monotonically decreasing

function of h ₌

H/(.

It _can be written

implicitly

in terms of

elliptic

functions

[10]

^{but for} our purposes it is sufficient _to consider _two limits

[11]

u(h)

=

-2(hlir -1)~

for h _mlr and

u(h)

= -1+

4h~~ -16h~~e~~

for h » ir. Consider first _a

system

^{made of} two slits of

equal width, 11.

Both _are

subjected

to _a

barely

subcritical field

I,e.,

E _<

Ec

and E _m

Ec.

This

system

^{is unstable} since _a

slight repartitioning

of the nematic fluid will

bring

the thicker

slit,

of width

Hi,

into the

supercritical regime,

E _>

Ec.

The nematic in this slit will

undergo

^distortion with _an

accompanying lowering

^of its free _energy.

Hi

will grow

further since

increasing Hi

lowers the elastic

penalty

thus

enabling _stronger

distortions. In turn, this

provides

for better

alignment

with the field and further reduction of the free _energy.

We _now present a

quantitative

discussion of this effect for _a

system

made of _two

"supercritical

slits"

I.e.,

in both E >

Ec,

and the nematic is

already

distorted. In

particular,

consider Slits of _area A and of

heights 11(1- d)

and

11(1+ d)

where 11 is the _mean

height

and

211d

the

height

difference. The free _{energy per} unit volume is

f

₌

ii d) Ii

+

ii

₊

d) f2

where

f~

^{is the} free _{energy per} unit volume whose

limiting

forms _were

given

above. The weak and

strong

^field

limits for the total free energy are then

~

~~ ^~

j~ ~~~ ~~~'

^{~~ ~} ~ ~

Ii)

f

₌

-4K(~

h~ e~ cosh dh h » 1r

where

I

e

11/(.

The free _energy, in both the

strong

^{and weak field limits is} an inverted

potential

well about d

= 0. The

system

^is thus unstable to any

perturbation I.e.,

once the field is switched _on the

larger

reservoir grows at the expense of the smaller.

Paradoxically,

the effect in the

strong

^field limit is much weaker. This is because the distortion is

mostly

confined to two

boundary layers,

of thickness

(, adjacent

to the walls. The weak

instability

reflects the

exponentially

weak

penetration

of the distortion into the bulk. A neutral

equilibrium

is found in the limit of _very

strong fields,

when this effect is

negligible.

^Our

analysis neglects

_one effect:

as one of the

layers

thins

out,

^it

will, eventually,

become distortion-free because

Ec

-~

H~~

This will enhance the

instability.

The

instability

_can be

strongly suppressed by gravity.

This efsect

depends

on the orientation of the

system

^with respect to the

gravitational

field and _may be eliminated

by adopting

a horizontal orientation.

Our discussion thus far focused _on finite

systems.

^In the

following

_we consider _a system

involving

an infinite reservoir

coupled

to _a slit

imposing parallel anchoring

_as described in the

preceding paragraph.

Both _are

subjected

to _an electric field E >

Ec

^oriented

along

the normal to the

bounding

surfaces _{so as} to induce _a Frederiks transition within the slit. The free _energy

density

of the nematic reservoir is lower than that of the distorted nematic since the

alignment

with the field is attained with _no distortion

penalty.

This

system

is thus unstable and the nematic is

expected

to drain out of the slit. This effect _can lead to _a field induced

collapse

of

a

nematic,

lamellar

mesogel _[12].

This

physical

network consists of

well-aligned glassy

sheets

bridged by

main chain

liquid crystalline polymers (LCPS)

and swollen

by

_a nematic solvent.

It _can be

produced

from _a lamellar melt of ABA triblock

copolymers

with LCP B blocks with the

glassy

A domains

imposing perpendicular anchoring.

The

resulting physical

network is

swollen

by

a nematic solvent to

produce

_a

gel

^which is immersed in _a reservoir of nematic solvent. The lamellar

mesogel produced

is thus _a stack of slits _as described earlier.

However,

these slits _are modified

by

the presence of

loops

and

bridges

^formed

by

the LCP B blocks.

These oppose the

draining

of the nematic solvent. The

shrinking

of the

mesogel

leads to an

increased concentration

resulting

ⁱⁿ an interaction

penalty.

In _a 8

solvent,

when _monomer-

monomer interactions _are not

important

the dominant

penalty

is due to the deformation of

the LCPS. For

simplicity

_we limit the discussion to the _case of A blocks whose

glass

transition

temperature, Tg,

^is

higher

than the nematic transition

temperature

^of the LCP B

blocks, TNI.

We further limit ourselves to a 8 nematic solvent. In this _case the swollen

mesogel

consists of

a stack of ABA "slits" of width H m Rjjo> ^the span of the undistorted LCP in the direction of _n. The nematic solvent within the

gel undergoes

a distortion _upon

application

of _a

strong enough

field

I.e.~ (

< R

jolir.

The free _{energy per} chain is

Fcha;n

_"

K(~~EH(H/1r( _1)~

₊

_kTR(o /H~. (2)

The first term~

allowing

for the field induced

distortion,

is

indistinguishable

from the form obtained for _{a pure} nematic. This

simple approximation

is

justified

because the LCP volume fraction is low and because the

mesogenic

_monomer ^is

typically

chosen _{as a} solvent. In such _a

case the elastic constants and the electric

anisotropy

of the solution _{are very} close to the values

characterizing

the neat nematic. Minimization of

Fchain Perturbatively

about H

=

(1r yields

H

=1r((1

+

fl)

where

fl

_m

kTR(~ /(KE()

^m

(~/()R(~ ^/E.

^Consider

^{symmetric triblocks,}

^when

the

polymerization degrees

of the A and B

blocks~ NA

and

NB

are such that

NA

_-z

NB/2.

The

area per chain in the lamellar melt above

TNI

is E _m

~~N~/3~

where N

=

NA

+

NB (12]

^while

Rjjo

=

~N(/~

E is frozen

by

the

quench

to below

Tg.

^{Thus for}

(

m

Rjjo

^{we have}

fl

_m

N(/~ ^/N~/3

and the chain

elasticity

^is too weak to oppose the

collapse.

The chemical

potential

of nematic

liquid crystals

includes _a term due to

possible

nematic distortions. This term is reminiscent of the contribution of surface tension in that it is _a function of the

sample

size.

However~

in marked

distinction~

the distortion term _can be either

an

increasing

_{or a}

decreasing

function of the

sample

size.

Furthermore, through coupling

with external fields the distortion _can decrease the free _energy of the

system.

^This results in _a

variety

of nemato-mechanic effects.

Among

them _are instabilities such _as the field induced

collapse

of nematic

gels.

These effects _are

important

for the

understanding

of the behaviour of

nematic

liquid crystals

in constraint free

systems~

such _as the force measurement

apparatus.

1638 JOURNAL DE

PHYSIQUE

II N°10

Acknowledgements.

DRMW _was

partly

funded

by

Donors of the Petroleum Research

Fund,

administered

by

the American Chemical

Society

and the NSF under

grant

^No.'s DMR-92-57544 and DMR-91-17249.

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