Focal conic faceting in smectic-A liquid crystals

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Focal conic faceting in smectic-A liquid crystals

J. Fournier, Geoffroy Durand

To cite this version:

J. Fournier, Geoffroy Durand. Focal conic faceting in smectic-A liquid crystals. Journal de Physique

II, EDP Sciences, 1991, 1 (7), pp.845-870. �10.1051/jp2:1991113�. �jpa-00247560�

J

Phys

II France

(1991)

845-870 _JUILLET 1991, PAGE 845

Classification

Physics

Abstracts

61 30J 68 10C

Focal conic faceting in smectic-A Iiquid crystals

J B Foumier and G Durand

Laboratoire de

Physique

des solldes, bit 510, 91405

Orsay

Cedex, France

(Received

27 December1990, revised Ii March 199I,

accepted18

March1991)

Abstract. Smectic-A

nucleating

_in the isotropic

phase

_{presents vanous} bulk focal _conic defects close to the interface, within the _« bitonnets _» first descnbed by Fnedel and

Grandjean

_in 1910 We

expenmentally

study the

stability

of the _« bitonnets _in a

high-frequency

aligning electncal field

Although

the extemal shape appears out-of-equilibnum, the focal _conic texture _is found

stable We build _a model

explaining

the

general

interracial focal _conic texture it participates in

the min1mlzatlon of the interfacial energy by orienting ^the smectic

layers

almost

perpendicular

to

the interface, above _a critical _size

lo comparing

layer curvature and surface _energies Thls model

explains

new arborescent textures that _we observe inside

capillary

tubes We then discuss

equilibrium

shapes of _size L. We discuss three regimes L «

lo yields

3D-cristal-bke Wulf»

shapes

For L

m

lo, equilibnum shapes

must contain focal _comcs, although we cannot solve the general solution In the asymptotic limit L-cc, we

predict

fractal textures of dimension

~ d

~ 2 inside _«

optimal

configurations, sphencal as for 3D-liquids. Finally focal _comcs can be considered _as bulk facets _» which

pile

_up to decorate _« bitonnets and _more

general

interfacial

textures. Focal conics appear then _{as intnnsic} defects of Smectic-A materials _in contact with their

isotropic

phase

1. Introduction.

Smectic-A

(Sm-A)

_are lamellar

liquid crystalline phases [I]

In the

ground

_{state, an} ideal Sm-A

phase

consists in a uniform

piling

of

planar, parallel

and

equtdtstant layers

of molecular

thickness In practice, Sm-A

phases usually

present bulk distorted textures in which the

layers, although

_remaining

equidistant,

_can be

violently

curved These macroscopic

defects,

called

«focal

contcs ii were first studied

by

Friedel and

Grandjean [2] They

consist _m delimited bulk domains inside which the SnJ-A

layers

are curved _in the

vicinity

of two

disclination

lines,

_an

elltpse

and the

conjugated

confocal

hyperbola

Usually, focal

_{comcs are}

produced by

extemal constraints

forcing layer

curvature In most cases, the _{curvature is} induced to fulfill antogonistic orientational

boundary

conditions These antagonistic

«anchorings» generally

_{occur on} solid extnnsic substrates

[2],

or between

smectic

drops

and

amorphous liquid

_{matrixes, on} mixture interfaces [3]

However,

it has

recently

been shown that the surface energy anisotropy ^of interfaces _intrinsic to the

liquid

cnstalline material could also induce focal _conic defects

[4]

in a

particular planar

geometry,

in which the Sm-A

phase

_is sandwiched between air and the isotropic

phase

of the

matenal,

_a first-order textural

instability

creates free

floating

focal _conic networks The focal _conic

curvature is

produced by

the

antagonistic

^surface anisotropies ^{of the} two interfaces the Sm-

Alair

interface favors

parallel layer/surface

onentation and the intnnsic

Sm-Alisotropic

interface favors

perpendicular layer/surface

onentation.

More

generally,

it is

interesting

to discuss whether _a similar

instability

would _{occur, in} the absence of _any air

phase,

on intrinsic

Sm-Alisotropic

^interfaces of _various

shapes,

_i-e- to

study

the

problem

of the

equilibnum shape

of _a Sm-A volume nucleated inside the isotropic

phase.

Indeed,

it _is known _since Fnedel and

Grandjean's

studies _in

1910,

that the nucleation of Sm-A inside

Isotropic phase gives

_nse to _vanous

elongated shapes,

the _« Bdtonnets _»

[2],

decorated

by

_{more or} less

complicated

focal conic textures

However,

whether the

ongin

_» of these focal _{comics is} statical _or

dynamical,

_or whether

they

_are stable _or metastable has _never been

established.

In this paper, resuming the anisotropy of the

Sm-A/isotropic

interfacial energy as the

leading mechanism,

_we

study

the

equllibnum shapes

of Sm-A nucleated inside the

Isotropic phase

of the

liquid crystalline

matenal. After

recalling

in part ^{2 the focal} conic structure and the associated energy, we establish _m part

3,

the _«

Wuf-like

_»

elongated shapes

that would

correspond

to ideal

strictly

undistorted Sm-A

liquid crystals

nucleated inside the isotropic

phase Then,

_we discuss the associated

elastically

relaxed

configuration

and show that the

amplitude

of the relaxation should be infinitesimal.

Generalizing

the model of reference

[4]

for Sm-A

plates,

_we construct _a

configuration containing

_a

single

focal _conic defect which _is

significantly

less

energetical,

above _a

typical

_size, than the previous Wulf solution.

Equihbnum shapes

must then contain focal _comics. In

part 4,

we

experimentally study

the

stability

of the _« bitonnets _» focal _comics. We show that

although

the extemal

shape

_is

probably

out of

equilibnurn, according

to a

lD-crystal

behavior _», the bulk focal conic _can be considered _{as an}

equilibrium

texture. In consequence, in part ⁵ we consider interfaces of

arbitrary shapes.

We

establish,

_m the _case of

large interfaces,

_a

general procedure

to build focal conic textures

re1axlng

the amsotropic interfacial _energy. We then discuss the _case of the

bitonnets _» In

part 6,

we

study asymptotic equihbnum shapes

in the limit of

large

Sm-A volumes and show that it must

correspond

to

spheres

with _a radial _«

Apollonius

_»

[5]

network of focal _comics.

Finally,

_we show in conclusion that focal _{comics can} be

geometncally

considered _as the _« volumic

facets

_» of Sm-A

equilibnurn shapes, by analogy

with the surface facets of usual

3D-crystals.

A

partial

_summary of this work _was

recently presented

_{in a}

workshop [6].

2. Focal conic bulk defects.

An _« ideal Sm-A

phase,

without any

defects,

_consists in _a one-dimensional

(lD) piling

of 2D

liquid layers,

at the _same time

planar, parallel

and

equidistant (Fig I)

Inside each

layer

of molecular thickness _a

(a

30

h

_{for usual}

therrnotropic Sm-A),

rod-like molecules _{are on} the average onented in the direction of the normal n to the

layers (nematic ordenng

of director _n, n~

=

I).

The

specific properties

of smectlc

phases

_are

mainly

due to the fact that

they

are low- dimensional

crystals (lD-crystals

and

2D-liquids)

For

instance,

unlike

3D-crystals

which

always

consist _in domains of uniform onentation

separated by

_grain

boundanes,

real Sm-A

phases

_{are never quite} ideal but do

usually

show macroscopic bulk

defects,

the so-called

focal

comcs, in which the Sm-A

layers

_{are no}

longer planar

but

violently

curved. This property is

charactenstic of low-dimensional

crystals,

which _can exbJbit

large

bulk distortions of pure

curvature

strictly leaving

the

lD-crystal

undilated In the _case of

3D-crystals, however,

all

types

^{of bulk} distortions dilate the 3D-lattice.

The free energy

density f~j

associated with the bulk elastic distortions of Sm-A

phases

contains two terms

[1]

:

fel(~l

"

(K(Cl(~)

⁺

^C2(~l)~

⁺

jBE(~l~ ⁽¹¹

N 7 FOCAL CONIC FACETING IN SMECTIC-A

LIQUID

CRYSTALS 847

al llllllllllllllllllllllllllll

~

lllllllllllllllllllllllllllll lllllll

Fig.

Schematlc representation ^of an « ideal Sm-A

phase

The first term descnbes the curvature energy

(cj

and _c~

being

the local

principal

curvatures of the Sm-A

layers,

K 10~^~

cgs)

the second term descnbes the energy associated with

layer

dilation

(e

stands for the relative

layer

thickness _vanation,

B~10~cgs).

Note that

B

K/A

^~ defines _a penetration »

length

A

iii,

which compares vith _a few molecular

lengths

far from second-order

phase

transitions.

With

simple

dimensional arguments, ^let us first recall

why

_pure curvature defects do

actually

_{occur in} Sm-A. We consider _{a one}

length problem

in which the Sm-A

layers

are

curved with _a

typical

curvature radius R~L inside _a volume

~L~ (Fig. 2a)

The total

integrated corresponding

_energy writes

(K/L~) L~~

_KL and grows

linearly

with the _size L of the domain Note that if R goes to zero inside the

volume,

a

Log

L factor appears because of the curvature

divergence. However,

such _a modest factor will not

change

the

discussion

qualitatively,

_as shown later on

Thls, indeed,

_is a remarkable point ^that curvature volumic distortions behave _{as «} lines ». Therefore _«

focal

_{conic » pure} curvature

defects,

of

' ,'> ",~"

), r'b _J

$, IQ

", ~,, ~'r

~' .~

"( ," [ _,~t

,-1'),'

-" T/ ^'

~

N~

a) b)

Fig. 2.-Elastic distortions _{in a one}

length

~L

problem a)The

Sm-A

layers

are curved

l

IL

inside a volume

L~. b) They

are also

generally

dilated if _no precaution is taken

energy

roughly KL,

_are less

energetical

than grain

boundaries,

whose energy

obviously

scales

L~

_{In the case of}

3D-crystals, grain

boundanes _are created to separate domains

having

different orientations With _our

previous argument,

smectics will

prefer

curvature to grain boundanes to connect such different domains. In practice, it is often the _case that _an

antagonistic anchoring

of the

layers

_on the

boundary plates

induces _a macroscopic

layer

curvature l

IL

_{on a}

length

L.

Now,

_a

difficulty

arises _since

curving arbitranly

the

layers

will also

arbitrarily

dilate them

(Fig. 2b).

Let _us then compare curvature distortions with

dilative distortions _in the _{same one}

length problem (Be

_~)

L~~

KL gives e

AIL,

which

corresponds

to the _maximum

layer

thickness variation _in the volume

~

L~,

if _we

require

that dilation _energy should not exceed curvature energy. Since A is

molecular,

_~ must vanish for macroscopic curvature; ^{tlus of} course comes from the fact that dilation energy scales

L and pure ^curvature

only

scales L. Since macroscopic ^curvature ~

L cannot suffer _{» a}

dilation greater than _e

~

AIL (e.g.

_e

~10~~

_{for L ~10 ~m, far from} second-order

phase transitions),

it is a very

good

approximation to _use the

geometrical description,

I-e to _assume

strictly

_e

=

0, excluding layer

dilation

[7].

We _now recall the focal _conic structure

[8], using

our previous

arguments

^which

general

structure can we build with families of curved but

equidistant layers

? If _one starts with any

general (undulated)

surface and try to build

parallel

and

stnctly equidistant surfaces,

the process will

rapidly

_stop because

singularities,

where the

layers

_{are no more}

defined,

will appear ^on surfaces

[8] (Fig. 3).

This _is not

acceptable

because the energy

L~

_{of such surface}

singulanties

_compares with that of _grain boundaries. To build structures of linear _energy

L,

the rules must be revisited _in the

following

_{way :} which

general

structure can we build if

we curve the Sm-A

layers

with the constraints of

keeping

them

stnctly equidistant

and

creating

no more than point or line

singularities

? For point

singularity,

the solution _is tnvial

and consists in

folding

the

layers

on concentnc

spheres,

the _« disclination _» point

singularity

being

the _{conunon center} of the

spheres (Fig 4a).

In the _case of line

singularities,

it can be shown

[8] that,

_m the most

general

_case,

they

must be _an

ellipse (El

and the

conjugated

«

confocal

_»

hyperbola (Hi

in the

perpendicular plane (Fig. 4b)

The term confocal _» refers to the property that the focus of _{one conic} coincides with the summit of the other and _vice

,

s I

I ,

I

3~~

aj

Fig 3 -An

arbitrary

undulated layer S cannot indefinitely _gJve birth to

parallel equidistant layers

The loci «i and «~

(only

_{«j is} represented) build surface

singulanties

where the layer curvature will diverge

N 7 FOCAL CONIC FACETING IN SMECTIC-A LIQUID CRYSTALS 849

M

t t

' j

' ' , ' j

,

H

a)

b)

Fig

4 Pure curvature Sm-A defects The locus of the layer curvature centers defines discbnations

singularities,

reducing

a) to a point ID

= 0) m the

sphencal piling

textures, b) to a pair of confocal

ellipse (E)

and

hyperbola (HI

lines

(D

=

I _in the _« focal _{conic texture.}

versa These confocal _conic

singulanties

_are Volterra disclination lines of the

(nematically

onented)

smectic

ordenng

The

ordenng (isotropic

_or

nematic)

_in the _core of these lines _{is not} known. At least the smectic

ordenng

does melt.

The bulk Sm-A

layers

_are built _m such _a way that their local normal _n, which _is also the nematic

director,

lies _on the fictitious lines

joining

_any

point

M _on the

ellipse

to any

point

M' _on the

hyperbola (Fig 4b).

The

simplest

focal conic has _a revolution

symmetry

and

corresponds

to the

degenerated

_{case in} which the

ellipse

becomes _a circle and the

hyperbola

becomes the

perpendicular conjugated straight

line

(the

revolution

axis)

_passing

through

the

center of the circle The Sm-A

layers

are then concentnc ton folded around the circle and

intercepting

the

straight

line with _an

angular discontinuity

when the radii of the ton become

larger

than the circle radius. The

general ellipse-hyperbola

focal _conic structure can be

easily

denved

by

continuous deformation from the

previous

_{case, i-e} the Sm-A

layers

take the

general shape

of the so-called

Dupm's cycfides corresponding

to ton-like surfaces whose

« radii _» increase _in the direction of the

ellipse

focus _not

occupied by

the

hyperbola

A difficult

problem, already

discussed

by

_many authors

[2, 3, 5, 8, 9],

concems the

filling of

space with focal conic defects. The

pnncipal problem

_is to avoid _strains between

neighbouring

focal comics The main results can be summanzed _as follows one can define conical domains

as the focal _conic texture limited inside _{a cone} whose apex O lies _on the

hyperbola

and whose base is the

ellipse (Fig. 5a)

It _{is a} strong

geometncal property

of _« confocal _{» conics} that such

cones are

always

_cones of revolution. Since the molecules _are

parallel

to the _cone generatnces

everywhere

_on the cone, no disclination _{strains anse} if the _{cones are} associated

tangentially along

their

generatrices [8] (Fig 5b)

Such _cones define _a radial distnbution around their

common apex O and the interstices between them _can be

filled,

without strains, ^with

spherical

layers [9]

also centered _on O. The further

filling

associations

depend

_on the

particular boundary

conditions and

boundary shapes.

Let _{us come} back _to the energetics ^{of focal} comics. Since the curvature

diverges

_near the

disclination

lines,

it is _{now no}

longer

obvious that _a focal _conic domain of size

L

(size

of the

ellipse) corresponds

to an energy KL, The

length f (the

smectic coherence

length)

of the _cores around disclination lines _{appear as a} natural cut-off for the curvature

H

I'

E

~

a) b)

Fig

5.

Space filling

with Sm-A _pure curvature defects a) Focal conic limited to a revolution cone of

axJs OC The

oblique

base of the _{cone is an}

ellipse (El

of center C The

hyperbola (Hl

_goes from the focus F to the asymptote OC b) Focal _conic texture between two

glass plates (from

Y

Bouligand

_[8])

The _{cones are} tangent

along

their _comon generatnces OO~ The bulk interstices between neighbouring

focal _{conic are}

probably

filled [9] with

sphencal layers

centered _on O and O'

energy

integration. Although

the

problem

_{is no}

longer

_a

one-length problem,

it has been shown

[10]

that the _core size

f only

_appears in a

slowly

_{varying corrective} factor ln

(L/f ).

The focal _{conic energy is} the _sum of three terms : the energy W~ of ^the

integrated

bulk curvature and the

energies ll~

and W~ ^{of the} respective

ellipse

and

hyperbola

_cores of

disclination. In the _case of _a

degenerated

focal conic of

revolution,

the bulk term

W~,

integrated

in _a

cylindncal

domain of revolution whose base is the circle of radius _r, is written

[10]

W~ ^{Kr In}

(r/f) Traditionally [I],

the _core dischnation of the

ellipse,

which is

topologlcally

melted

everywhere,

_gives a contnbution of order K per unit

length (curving

_an

angle

_{unity over a} distance

f

must compare with

melting).

Th1s gives

ll~ Ki~,

in which

i~ represents

^the

ellipse length, However,

_{a more precise}

description

should take into account, for _vanous excentncities, the

dependence

of the

proportionality

coefficient

along

the line The _case of the

hyperbola

disclination line _{is more}

delicate,

it has been shown

[I ii

that both the _core dischnation and _its energy should vanish far from the

ellipse

_since the

angular discontinuity

becomes infinitesimal and _can be

elastically

relaxed. It _is therefore _a usual

approximation [5]

to count

only

the

sign#icative

_part of the

hyperbola,

whose

length

_compares

with that of the

ellipse.

The energies of the _two disclinatlon lines _are then

roughly proportional

The total energy,

integrated

inside _an

oblique cylindncal

domain

having

the

hyperbola

direction and the

ellipse base,

_can then be written _as W~

Kf~ (a

+

p

in

ji~/fj)

,

(2j

the

proportionality

coefficients _a and

p depending

_on the excentncity

[10].

Formula

(3)

should be

regarded

_as valid _in the l1mlt of _very

long hyperbolae

If the

hyperbola length

compares with the

ellipse length,

it can be convenient

[4]

to treat them _as

independent vanables,

I.e. to add _a term

~

Ki~ proportional

to the

hyperbola length,

The previous

general

results _on focal _comcs will be useful _to discuss the part

played by

bulk distortions _in the

equilibnurn shapes

of Sm-A nucleated inside _its

isotropic phase

3.

Equifibdum shapes.

Surface energy

amsotropies play

_an

important

^role m

determining equllibnum shapes.

Observing

Sm-A

phases,

Fnedel first

guessed [12],

from the

shape

of the smectic germs

N 7 FOCAL CONIC FACETING IN SMECTIC-A

LIQUID

CRYSTALS 851

(«bitonnets»),

the

tendency

of the

layers

to orient normal _to the

Isotropic/Smectic-A

interface. He also noticed the

tendency

of smectic

layers

to

align parallel

to air

drop

surfaces (« gouttes ^fi

gradins »)

He _was however

puzzled,

in the first _case, that _a surface _energy

mlnimurn could exist _{in a}

non-crystalline

direction. Later _on,

X-ray experiments [13]

demonstrated the

tendency

of the Sm-A

layers

to

align parallel

to _air interfaces More

recently, quantitative

temperature measurements of the

Sm-Alair

surface _{energy were}

performed

_on

equilibnum

facets for free

drops laying

_{on a} solid substrate

[14].

The

Sm-Alair parallel alignment corresponds

to the

expected crystalline

behavior.

Conversely,

_{as was}

quantitatively

shown

recently by

the textural

instability

of reference

[4],

interfaces with the tsotropic

phases

of Sm-A matenals do

present

^the opposite

tendency

: the smallest surface

energy

corresponds

to the

perpendicular layerlinterface

onentation The expenment ^of

reference

[4]

^studied the behavior of _a Sm-A

plate

sandwiched between two

parallel

interfaces

Sm-Alair

_{on one} side and

Sm-A/Isotropic

_on the other. Above _a thickness threshold of the Sm-A

plate,

_a textural

instability

_occurs,

creating

free

floating hexagonal

networks of focal _conics, whose bulk curvature is induced to fulfil the two

antagonistic

interfacial

anisotropies

In the absence of any air

interface,

_a

frustration

_persists

[15]

: a closed Sm-A volume,

completely

immersed inside the isotropic

phase,

cannot have both the

preferred

ideal bulk

ground

state

~parallel planar equidistant layers)

and the

preferred perpendicular layers/inter-

face onentation

everywhere along

the

boundary.

It _is then of _some interest to examine which

compromise would be chosen _in this frustrated situation In

particular,

we wonder whether

intnnsic focal _conics should be present or not at

equilibnum

what would be the

equilibrium

«

shape

_» of _a Sm-A closed volume

having

nucleated inside the

isotropic phase

?

We first recall _some well-known

simple

results _on

crystal equilibnum shapes

The

amsotropic

surface energy

y(0, WI

of

crystals,

function of the relative

bulk/surface

onentation

usually

defined

by

the zenithal and azimuthal

angles

0 and _q, leads to

equilibnum

shapes

with

facets and,

_more

generally, elongations,

which differ

strongly

from the

sphencal shapes corresponding

to y isotropic. ^In the

simplest

model, _to determine the

equilibnum shapes,

_one must minimize the surface energy for _{a given} fixed total

volume,

with respect to the surface

shape,

_i-e solve Wulrs

problem

_:

iyds+aV=mln. ⁽³⁾

The

corresponding

vanational formulation leads to the well-known

Wufs

construction

[16]

In such _a

description,

for

3D-crystals,

the bulk is

stnctly

assumed to be «ideal _{», i e.} the

eventuality

of bulk elastic

distortions,

which involve _too

high

_{energies, is} not considered

Therefore,

_{since y is} the

only physical quantity

in the

problem,

_no charactenstic

length

appears and Wulrs solutions _are homothetic

(the

solution for _a given volume V~ _is deduced from the _one

corresponding

to a volume V

by

_a

simple homothetie,

_i.e. has the _same

«shape»). Usually,

in the _case of

3D-crystals,

the function

y(0, WI

presents denvative discontinuities _near the onentations close _to

crystalline planes.

These discontinuities _are

responsible

for the existence of

facets

in

equilibrium shapes However,

real

crystal

facets do not

correspond

to

equilibnum

but _are rather due to

dynamic

effects

[17],

_{except in very} few

cases as in solid helium

[18]

To discuss the _case of

Sm-A,

let _us first establish the

corresponding

Wulrs solution In agreement with reference

[4],

we assume y to be anisotropic, the minimal value

(y~

corresponding

_to

layers perpendicular

to the

Sm-A/Isotropic interface,

and the maximal value

(yjj corresponding

to

layers parallel

to it In addition

(and

also

by

lack of

infonnation),

_we

assume

y(0

to be _a smooth function of the

angle

0 between the normal to the Sm-A

layers

and the normal to the interface. From uniaxial symmetry, y is

independent

of _q. Wulrs

solution must then be _a defined homothetical

shape

of

revolution,

the normal to the

monocrystalhne layers' piling being parallel

to the axis of

elongation

in order to _increase the

lengths

of the low

energetical

surface

parts (Fig. 6a)

Let _us call L the

length

of the

long axisj

I-e- the

scaling length.

Due to the homothetical property, the total surface of the solution is S

=

aL~,

_where

a

[y]

is a pure ^constant

(a

~

l

).

The total energy

W~

^{of Wulrs solution}

contains

only

_a surface term.

Integrating

the interfacial _energy

yields

W~

₌ a

fL~ ₍₄₎

where

f

is _a constant

angular

_mean value of _y. Wulrs solution

explores

the function

y

(0 by locally changing

the surface orientations with

respect

to the fixed bulk _axis. In

fact,

the

interfacia1energy

_{is a} function of the

surface/volume

_respective onentation which _can also be modified

by locally changing

the bulk _axis directions with respect ^{to the}

surface,

_i.e.

introducing

bulk distortions. In the _case of

3D-crystals

which _are very «

stiff»,

it _is a very

good

approximation to

neglect

these

degrees

of freedom

However,

Smectics which _are

only ID-crystals (and 2D-liquids)

_are much _« softer and _can exhibit _a very different behavior.

,

C~

~

R

a) bj

Fig

6 -a)Wulrs solution

corresponding

to an assumed «ideal _»

strictly

undistorted Sm-A bulk

b)

Previous

configuration elastically

relaxed ~6@ As _{soon as} L _is macroscopic, 8@ _is found

infinitesimal

Solving

the

general problem including

bulk

distortions,

_i-e

iyds+ fddv+aV=min (5)

is a difficult

problem,

the solutions of which _are unknown In

pnnciple, fd

must describe all the

possible discrepancies

with respect to the ideal bulk

equilibrium ground

state

iii,

_{i e.}

fd(r j

=

fel (r j

₊

f j (r j _weq1 (61

The first term is the elastic free energy

density

associated with _curvature and dilation of the

N 7 FOCAL CONIC FACETING IN SMECTIC-A

LIQUID

CRYSTALS 853

Sm-A

layers (see

Part

2)

and the second _one descnbes _an eventual

me1tlng

of the Sm-A order

(e.g

_near discfination lines _{since we are} not concerned with dislocations which _are

microscopic

defects)

Although

the solutions of equation

(5)

are

unknown,

it _is

possible

_to make _{a semi-} quantitat1ve discussion. Let _{us see} how Wulrs

solution,

which

stnctly

assumed

rigid layers,

now

adapts

if _we reintroduce the finite Sm-A

elasticity.

_as the smectic

layers

_are not

everywhere perpendicular

to the

interface,

_a surface torque bends the

layers symmetrically

and dnves the system towards _an

elastically

relaxed

configuration (Fig. 6b)

Let _us call R

=

~IL

_{(~1 ~} l the revolution radius of Wulrs

solution,

_{Ho a mean}

angle

at the surface and &0 the _{« mean »}

bending angle

An elastic relaxation _{&@ creates} both curvature of

mean » radius _p

R/&0

and dilation of _{« mean}

amplitude

_e

R(&0)/L.

The correspon-

ding

_energies,

integrated

in the volume V

=

ply L~,

with

p l,

scale then _as

W~~

₂

~K(

^~~_R

)~pL~~KL(&0)~ ⁽⁷⁾

W~

^{B ~~~ ~}

p

L

~ EL ~(&0 )~

(8)

2 L

As L _is

always

much

larger

than A, the

leading

_{term is W~}

W~

«

W~) Assumlng

_y to have

roughly

the form

y(0 )

_y~ +

by cos~ _0,

_the

energy of Wulrs solution _is lowered

by

_an

amount

Aw

«L2 _by jcos2

Ho + &o

j cos2 ooj

+

~

(

_{EL 3(}

_{&o11 (9j}

~1

Minimization of

equation (9)

with

respect

to &0

yields

_:

~~~2 ~2 ~2

~~

4

p

^{~~~ ~}

~° Gl Gl

^~~~~

in which _we have introduced both A

(K/B)"~

_and

lo K/Ay

The constant A _is known _to

compare ^to a few molecular

lengths

far from second-order

phase

transitions and the order of

magnitude

of

to

_is known _{as a} result of the

instability

studied in reference

[4],

1-e-

to

01-1 _~m for usual Sm-A

phases,

with

K~10~~

_{cgs and}

_{by 01cgs}

_It follows that

above microscopic molecular _sizes, &0 _is

infinitesimal

and this relaxation irrelevant _{as m} the

case of

3D-crystals.

This _is due to the fact that this symmetnc ^{solution which} can be obtained

continuously

from Wulrs _one must involve dilation of the

lD-crystal.

Dilation energy which scales

L~

_{must be of} infinitesimal

amplitude

to compare with interfacial energies

scaling

~L~

_{To find}

configurations definitely

less

energetical

than Wulrs

solution,

_one must introduce

only

non-dilative bulk distortions This _is indeed

possible,

as we show _now, in the

case of low-dimensional

crystals

like _smectics and

corresponds

to the pure curvature of focal

conic defects.

We shall _now exhibit _a

particular configuration

with focal _conic bulk distortions of energy

W~c

^{lower than the} energy

Ww

of Wulrs solution. We choose _a

particular configuration

which has the _same homothetic extemal surface than Wulrs solution

and,

in

addition,

_we

place

_a

revolution focal _conic

(circle

_on the surface and

conjugated straJght

line _on the revolution

axis)

inside the

bulk, according

to

figure

7a Note that the symmetnc

up-side

down focal _conic

is not introduced _as will be discussed _in

part

⁵

W~c

contains a surface term

lI~

and _a volume term W~ ^{As the extemal}

shape

is chosen

homothetically

identical to Wulrs _one, the surface energy is

ll~

= a y~c

L~ ll~

is _now

signyicatively

lower than Wulis surface energy since the

layer

onentation fits better the

extremity

near the circle and _is

quite unchanged everywhere

C~

,

, E

i

i

u i i

aj b)

Fig

7 -Model

configurations

based _on Wulrs solution with one revolution focal _conic included.

a)

Perspectlve

view of the dischnatlon lines b) Cross section

through

the revolution _axis The Sm-A layers _{are now quasi}

perpendicular

to the

Sm-A/Isotropic

interface _near the circle E. H should be tilted to allow for another focal _conic to relax the other end.

else

(Fig 7b),

_{1-e- y~c~}

f stnctly. Therefore,

_since the focal _{conic energy} W~ is

negligible compared

to

L~

_{for L}

large enough,

_{i-e- W~}

e(L~),

there exists _a charactenstic

length to

above which the solution

including

the focal conic is

energetically

lower To deternune

to,

let _us introduce

explicitly

the focal conic _energy W~ of order

(Cte)

KRln

(R/f)

₊

(Cte) KR,

as

previously

discussed _in part ^2. The total _{energy can} then be written _as

WFC~ «y~cL~+ _(vi

₊

_{v~in (L/fij}

_KL

_(iii

where _vi and

v~~l

_are dimensionless numbers.

Therefore,

L»

to _{yields WFC~ Ww,}

to being

_given

by

to

~

j

^~°

₍₁₂₁

Y YFC

where v~ = vi +

v21n ((off) _l;

And again, we find back the

typical

size

lo ~K/Ay,

of

order

01-1~m,

_companng the amsotropy ^{of the} surface _energy with the Sm-A

layer

curvature constant. Note that _in other _contexts with fixed

boundary shapes,

such _a

characteristic

length

due to the

competition

between surface and bulk energies was first mentioned _in reference

[9]

^{and then}

developed

_in reference

[3]

^The main difference here _is that _we discuss the intnnsic _case of

nucleation,

_in which both surface and bulk texture are free

parameters.

We have therefore found _a

particular configuration

_{containing one} focal _conic, which is

energetically

lower than Wulrs solution. This indeed is

enough

to ensure that

equilibnum shapes

must contain focal _comics above the

typical

_size

to. However,

the

general equllibnum shape

solution _is

probably

much different from this

simple

_one, In

fact,

because of

to,

the homothetical

property disappears,

and _one should expect, for each volume

V,

_one

particular

solution characterized both

by

its extemal surface

shape

and its bulk focal

conic

texture. One should then also

expect

first-order transitions between different classes of

bt 7 FOCAL CONIC FACETING IN SMECTIC-A

LIQUID

CRYSTALS 855

solutions

involving

different focal _conic arrangements, and all

corresponding hysteresis phenomena.

For instance, in the vicinity of

lo,

_an

instability

s1mllar to that of reference

[4]

should _occur to create the first focal _conic from Wulrs solution.

Unfortunately,

the value

lo

0.I-I _~m

prevents

^from

observing

it

directly

under

optical

_microscopy Note that _in the geometry of reference

[4],

it is because of the presence of the _air interface that the

instability

threshold _is raised _{up to a} macroscopic value

However,

_as known _since

1910,

focal _{conics are} indeed

always

observed inside nucleated

shapes

We _now

present

an

experiment

demonstra-

ting

^their

stab1llty

4.

Experimental study

of

shape

and texture

stability.

Nucleation of Sm-A inside the

isotropic phase

of the

liquid crystalline

material gives rise to the so-called Sm-A _« bdtonnets _»

[2]

^In

figure

8 _we have

reproduced

the

original

_« bitonnet _»

drawings by

Friedel and

Grandjean.

From these

drawings,

it does not seem at once obvious that these bitonnets

correspond

to the

equilibrium shapes

that could be

expected

from _our

model. To discuss _{in more} detail the agreement ^with our

model,

it is _{convenient to}

« separate ^the

stability

of the

surface shape

itself from the

stability

of the bulk texture for _a given surface

shape.

Tins will be indeed

possible, provided

the results of the

following experimental study

of the _« bitonnet

stability

Because Sm-A _are still

ID-crystals,

it is not

obvious, by analogy

with

3D-crystals,

that the observed

shapes correspond

to

equihbnum.

It

is well known for instance that actual

crystal

facets _in nucleated

shapes

_are due to

dynamical

processes

~growth instabilities)

than

correspond

to the _ones

expected

for

equilibrium shapes

One _cannot either exclude _a

possible dynamical

_ongin for the focal _comics.

Moreover,

_as focal

conic textures could be metastable systems, their persistance in stabilized «bitonnets _» of

finite life-time _is not _a

proof

of their

stability.

We _now

present

an

expenment

^{which studies} the bitonnet _»

stability using

an AC electrical field E to act _on the bulk focal _conic texture of the bitonnets

Spatial

field

gradients

VE also

play

an important part in

controlling

the

« bitonnet _» locations

~/ ^~~~'

,

~

'

)) )/ j _ll~

^'

~ _~

a) '~i c) _oh e)

Fig

8

Typical

bfitonnet

shapes

with their focal _conic textures

corresponding

to the nucleation of Sm-A inside its Isotropic phase (from ^{G Fnedel and F.}

Grandjean

[2])

The

experimental set-up

is the

following.

_{we use} the

compound

10CB

(4-n-decyl- 4~-cyano-biphenyl)

which

undergoes

_a first-order Sm-A~

Isotropic phase

transition at

50.5 °C Our cell consists _in two honzontal

glass plates

between which two nickel _wires of

50 _~m diameter _are

placed

_one in front of the

other,

the extremities

being separated by

200 _~m. No other spacers than the _{wires are} used. The _{wires are} connected to _a

frequency

generator dehvenng

a sinusoidal tension m the range 0-200 V and

frequency fin

the range ^0-

l0kHz The cell, filled with the

liquid crystal material,

is

placed

inside _a temperature controlled _oven with 10-2 °C

temperature

^{resolution We observe the} preparation ^with a

video _camera attached to a

polarizing

_microscope We start, at _zero

voltage, by observing

the isotropic

phase.

Then _we

carefully

decrease the temperature ^{below the Sm-A} ~

Isotropic

transition. We observe the _appearance of the Sm-A bitonnets

nucleating

_very often _on the

glass substrates, probably

because of the occasional presence of

impunties favoring

the Sm-A

germination

^We now follow this

procedure

_{: we}

apply

_a

voltage

V ~100 V and vanable

frequency f

For low

frequencies f

_~ 500 Hz _or DC

frequencies,

_we observe _an electro-

hydrodynamical

flow

probably

due to _ionic convection To cancel such

effects,

_we fix

f

~10 kHz well above the

charge

relaxation

frequency.

We first observe that the _« biton-

nets » are more or less attracted towards the nearest electrode This

phenomenon, probably

caused

by

the field

gradients,

_{appears as a} convenient way to dnve the _« bitonnets out of the

glass

substrate toward the

isotropic liquid. Switching

^{off the} electrical

field,

free

floating

«bitonnets _{» are now}

easily recognized

since

they

dnft

slowly

inside the

isotropic liquid.

They closely correspond

to the

description

_given

by

Friedel and

Grandjean [2] they

_present

various external

shapes

decorated with bulk

focal

_conics

having

their disclination lines _{more or} less virtual

(out

of the

physical

Sm-A

volume). They

_are

elongated

_{in a} direction which _is

parallel

to the average

layer piling

direction inside the bulk. This _is

easily

verified

by searching

for the _maximum

birefnngence darkening

between crossed nicols. Due to the

layer

curvature

of the focal _comics, there _{is no} total extinction. The extemal _« bitonnet

»

shapes

have _{more or} less the revolution

symmetry

^{around the}

elongation axis,

but

they

often

present

^lateral

irregulanties,

the so-called «sailfies»

[2], corresponding

to local

rapid

vanations of the

surfacelaxls

distance. The extemal

shape strongly depends

on the _« bitonnet

»

history

those which unstuck from the

glass plates

have _a free

floating shape

somehow related to their initial

shape

Now,

raising

the temperature, it is

possible

to decrease their size down to a few microns and then to let them grow again. In this _case, their extemal

shape

_{is more}

regular

and strong « saillies _are absent.

However,

it often

happens

that _a small bitonnet

collapses

inside a

bigger

_one,

_creating

_a saillie localized at the impact ^level.

Usually,

such saillies _» take _{more or} less _a revolution symmetry ^after a bulk texture rearrangement ^It therefore

appears, as will be discussed

later,

that the extemal

shape

is most

probably

out of

equilibnum

We _now isolate _a 50 _~m

large

and 100 _~m

long

_« bitonnet _», containing

clearly

visible focal _comics, between the two electrodes

(Fig. 9a)

and

apply

_various

voltages

_{at a}

frequency f

10 kHz We observe that the bitonnet _{» is} onented

by

the electncal field _in such _{a way} that _its

long

axis becomes

parallel

to the field lines.

However,

the _main effect of the electrical field _{is to suppress} the focal _conic domains For low tensions, in the range 10-30

V, only

the

large

focal _conic domains

(with

sizes 50 ~Lm)

compared

with the «bitonnet» size _are

suppressed Increasing

the tension, smaller domains

disappear

and for _a 200 V

voltage,

all focal _comcs

diseappear (Fig. 9b).

The bitonnet presents now a

uniformly

onented ideal undistorted bulk

~perfect

_extinction between crossed

nicols),

the normal _{conunon to} the

layers

been

parallel

both to the field lines and the «bitonnet» _axis In presence of the electrical

field,

the external surface of the

« bitonnet _{» is} not modified

(the

surface _« saillies _» do not

relax). However,

if the _« bitonnet _» had not

exactly

_a revolution symmetry, ^{the latter}

is «

instantly

_»

obtained,

_i-e- within _one video frame

(~

20

ms).

Then,

to achieve the experiment, we

suddenly

switch-off the electrical field The focal

comics reappear

«instantly

_», I-e also within _one video frame

(~20ms),

inside the bulk

(Fig. 9c) According

to our

optical resolution,

the focal conic texture

reproduces

a pattern very similar to the previous one

Before

discussing

the effect of the electrical field _on the bulk texture, which _is the important

point

of the

expenment,

_we ^first come back to the «saillies» of the «bitonnets» to

bt 7 FOCAL CONIC FACETING IN SMECTIC-A

LIQUID

CRYSTALS 857

h

~

,j

jc

It

S_ ~ '

~/r

,~

/.~

[

~

_i

aj bi ci

Fig

9.

Vanishing

of tile focal _conic texture inside _an Sm-A _« bitonnet _» loo _+ni

long,

_{using a} HF electrical field E. The black

shapes

are the Z 50 _~m nickel _vnre electrodes

al

Before

applyrng

E.

b)

E

is

applied,

tile focal comics all vanish.

cl

E is switched

off, they

_«

instantly

_{» reappear}

quantitatively

discuss their

persistance.

^In a true

3D-liquid,

_a sinular sailhe of curvature radius

~

R would induce _a

Laplace

_pressure

&p y/R

between the inside volume and the rest of the bulk. This pressure &p would then create _a

reacting

flow

re1axlng

the «sailhe»

(Fig, 10a).

The relaxation life thne

~ r~D is defined

by

the balance between the viscous forces and the pressure forces.

Calling

_~ the viscosity, ^the flow

velocity

_v

R/r3D

is then

given by

~Av~Vp

The

spatial

denvatives

taking place

on the _same

length ~R,

this

gives

v

y/~,

and then

r~D

~'~ (13)

Y

t j ,

'

t t

'

, t

i ,

,

, ,

a) b)

Fig

^lo. _-« Saiflie _» relaxation _via the

Laplace

pressure induced flow

(arrows) a) Simple 3D-liquid

case

b)

Sm-A _case the permeation

phenomenon

slows down the relaxation process across the ID-

crystaflme

direction