Covariant elasticity for smectics A

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Covariant elasticity for smectics A

M. Kleman, O. Parodi

To cite this version:

M. Kleman, O. Parodi. Covariant elasticity for smectics A. Journal de Physique, 1975, 36 (7-8),

pp.671-681. �10.1051/jphys:01975003607-8067100�. �jpa-00208301�

COVARIANT ELASTICITY FOR SMECTICS A

M. KLEMAN

Laboratoire de

Physique

des Solides

(*),

Bât.

510,

Université de

Paris-Sud,

⁹¹⁴⁰⁵

Orsay,

^France

and O. PARODI

Groupe

^de

Dynamique

^{des Phases}

Condensées,

Laboratoire de

Cristallographie,

Université des Sciences et

Techniques

^du

Languedoc,

Place

Eugène-Bataillon,

³⁴⁰⁶⁰

Montpellier,

^France

(Reçu

^le

13 janvier 1975, accepté

^{le 5 mars}

1975)

Résumé. ²⁰¹⁴ On présente ^une théorie des déformations statiques ^d’un smectique A, ^dans l’hypo-

thèse où les variations en

épaisseur

des couches sont faibles, ^{mais en} incluant la possibilité ^de larges déformations de courbure. Les variables

indépendantes

^{utilisées sont} le directeur n(r) ^{et la} phase ^des couches 03A6(r)

(03A6(r)

est constant sur une couche

donnée).

Ces variables sont liées par ^une relation,

ce

qui

^{introduit un}

multiplicateur

^de Lagrange. ^{Le calcul tient} compte ^{de la}

perméation.

^Les

équations

d’équilibre ^{en volume et sur} les surfaces s’obtiennent très simplement à 1’aide des variables choisies,

et on peut ^{donner un sens}

physique

^au multiplicateur ^de Lagrange. ^{On étudie en}

application

^la

stabilité d’une structure circulaire cylindrique. ^{On trouve} que les courbes cylindriques ^{sont instables}

vis-à-vis de certaines déformations hélicoïdales et sinusoidales.

Abstract. ²⁰¹⁴ A

theory

of static deformations of smectics A is

presented,

taking ^{into account the} large ^curvature deformations of the layers, ^but assuming that the variation in the thickness of the

layers is small. The director n(r) ^{and the} layer

phase

^function 03A6(r) (03A6 ^{is constant on a} given layer)

are chosen as independent variables, ^but they ^{have to} be related by ^a relation, ^{to which} corresponds

a Lagrange multiplyer. Permeation is taken into account. With the

help

^{of these} variables, ^{the bulk}

and boundaries equilibrium

equations

^{can be} expressed

simply,

^{and the}

physical

meaning ^{of the} Lagrange

multiplyer

^is

straightforward.

The formulation is

applied

^{to the case of the} stability ^{of a}

cylindrical

structure. It is found that the cylinders ^{are unstable versus} certain helical and sinusoidal deformations.

Classification

Physics ^Abstracts

7.130

1. Introduction

(1).

^{- A number of} attempts ^have been made to describe the

elasticity

of smectic

phases

on a sound basis. The difficulties are of various kinds.

l.l Smectic

phases

^{are lamellar}

phases. They

can suffer _very

large

deformations of the

lamellae,

while the lamella thickness remains _very little disturb- ed. In that sense one must reach a

description

ⁱⁿ

which the radü of curvature can show _very

large variations,

while the local distortions of the thickness

can still be described in the framework of linear

elasticity,

^as

long

^as the local radii of curvature are

large compared

^{to the}

layer

^thickness.

(*) Laboratoire ^{associé au C.N.R.S.}

(1) This work has been partly done in order to satisfy ^the requi-

rements of a D.G.R.S.T. contract on defects in smectic phases.

1.2 Smectic

phases

^are

mesophases,

and may be described

by

^a director field

n(r) (n2 = 1).

^{In the}

ground

^state the directors which represent ^{the direc-} tion

of

the local molecule are

perpendicular

^{to the}

layers (inside

^the

layers),

^{and the}

layers

^are

planar.

The molecular

length

^is

equal

^{to the}

layer

^thickness

do.

For any deformation of the

layer,

^a

displacement

^of

the molecules

through

^the

layer

^{can be}

superimposed, independently.

^This process has been called

per-

meation

by

^Helfrich

[1].

As will be shown

later,

^the

only

^molecular

displacements through

^the

layers

which are to be considered are those for which ⁿ remains

perpendicular

^{to the}

layer.

A first attempt, ^{due to Oseen}

[2],

^{insists on the}

analogy

between nematic and smectic

phases :

^his

theory

^uses the director as the fundamental variable.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01975003607-8067100

Oseen considers that the

layer

^{thickness cannot be}

modified _{in any}

displacement

^{of the}

layers,

^{and this}

leads to the condition

The

only

^term

surviving

^{in the} energy is therefore the

splay

^term

The

Lagrange

condition curl n ⁼ 0

implies

^that

Dupin cyclides (the shape

^taken

by

^the

layers

^{in the}

classical confocal domains first described

by

G. Friedel

[3])

^are solutions of the minimization

equations. Any

distorted medium has

only splay

energy and ^core energy

(along

the dislocations and

disclinations).

Permeation is of course forbidden in such a

theory

and molecules are restricted to stay inside the

layers.

The first

description

which includes a small varia- tion in the

layers’

thickness is due to de Gennes

[4].

De Gennes restricts his

theory

^to the distortions in the

vicinity

^of

planar

structures. The free _energy now

reads

where u is the

layer displacement

^{and z} the direction of the normal to the

layers

^{in their}

ground

^state.

Since n is

along

the normal to the disturbed

layers,

one

has,

^{in a linear}

theory

This free _energy has been used to discuss the distor- tions related to the _energy of

edge

^{and screw disloca-}

tions in

planar specimens [5, 6],

^{the free} energy of

symmetrical low-angle grain

boundaries

[7, 8]

^and

of

low-angle

^conical deformations of

layers [9]

similar to those which appear

along

the focal lines in the confocal domains. All these calculations assume

that the distortions of the

layers

^{are small with}

respect

^{to the}

planar

situation. In this

theory,

permea- tion does not appear

explicitly,

^{but it is}

implicit

^that

permeation

^{relaxes the stresses exerted on the fluid}

itself

by

^the

displacement

^{of the}

layers.

More

recently,

^{a new}

theory

of smectic

phases [10]

(referred

there after as

MPP)

^has

appeared

^{based on}

a

general analysis

^of

hydrodynamics

^{in terms of}

conserved

quantities. Apart

^from the classical

hydro- dynamic

^variables

(mass,

energy and

momentum)

a new variable is

introduced,

which characterizes the local state of symmetry. ^A

general property

^of conserved variables is that their fluctuation relaxa- tion time tends to

infinity

when the fluctuation wave- vector tends to zero. In that ^sense the new variables

can be considered as

quasi-conserved variables,

^and

are treated as the other classical

hydrodynamic

variables. In the case of smectics

A,

^{the new variable} is the

displacement

^{of the}

layers.

Quite simultaneously,

^{de Gennes}

[11] and

^MacMil-

lan

[12]

^have

given

^a

phenomenological theory

^for

the smectic A-nematic

phase

transition based _upon

Landau-Ginsburg equations.

^In these theories one uses

a complex

^order

parameter Y’

^{related to the}

modulation of the

density :

The

amplitude of §

vanishes in the nematic

phase

and characterizes the

degree

^{of smectic} order. The

phase

^{describes small} distortions of the

planar configuration.

Up

^{to now,} except

perhaps

^{for the Oseen}

model,

all these

approaches

considered

only

small deviations of the

planar

structure.

They

succeeded in

explaining

some

experimental

^{results such as}

Rayleigh [13]

or Brillouin

scattering

ⁱⁿ

planar configurations.

However non

planar configurations

^are of outstand-

ing importance

in smectics A. Confocal domains can

be formed with

only

small variations of the

layers interspacing. Hence,

^{even for}

largely

^curved

layers,

linear

elasticity qualifies

^{for the}

study of layer

^thickness

variation. In other words it is _necessary to have a

covariant

representation

of linear

elasticity. This

is the aim of the

present

paper.

In section 2 a covariant

expression

for the free- energy based on the

previous

considerations is

presented.

^This

expression

^has

already

^{been used}

by

one of us

[14]

^without any detailed

justification.

^In

section 3

equilibrium

conditions are obtained

through

a minimization

procedure. Though

the director _may appear ^{as an}

independent variable,

^{it is related to} the

gradient

^{of the}

phase

ç of the order parameter

through

^a constraint which is taken into account

by

a

Lagrange multiplyer.

^This

procedure,

^{which has}

already

been used

by

^{one of us}

[15],

allows for a

clear distinction between

permeation

pressure

(which

is the

thermodynamic conjugate

^of

cp)

^and torques.

In section

4,

^this

general theory

^is

applied

^{to the}

expression

^{of the}

free-energy

derived in section 2.

In this framework we discuss

planar

^and

homeotropic boundary conditions,

^the

possible

role of surface energy ^terms in thin

films,

the existence of

Dupin cyclides,

^and

finally

the order of

magnitude

^{of the}

variation of the

layer

thickness in confocal structures.

In section 5 we show that

cylindrical configuration

are unstable versus an helical deformation.

2. A covariant

expression

^{for the}

free-energy.

^-

In the former

descriptions

of smectics

A, only

^small

deviations from the

planar

^structure

(parallel

^and

equidistant layers)

^were studied. The

displacement

of the

layers

^{was then the}

good

variable. If we want now to describe more

complex

structures, such ^as dislocations on

Dupin cyclides,

^these

displacements

are no

longer

small. One has therefore to find another

starting point, and,

^{as in the case of}

nematics,

^the

good

^{one is the}

phase

^{of the order} parameter. ^{Let us} define it now.

As was

pointed

^out

by

de Gennes the existence of

layers

^{results in a} modulation of the

density

where

is the order parameter. Here qs = 2

nldo

^where

do

is the lattice parameter ^{of the}

planar

^structure.

At a

given

temperature and far from the N --+ A ,

transition, tf 1

^is constant, ^{and the}

configuration

is

completely

^described

by

^the

phase (f’(r) :

^the

layers

are surfaces

and their

interspacing

^{is d =}

do/ grad

(p

.

^The

planar

structure, with Z-axis normal ^to the

layers

^corres-

ponds to

In smectics A a local minimum _energy is obtained when the director n is normal to the

layers,

^{and when}

the

layers spacing

^is

equal

^to

do,

i.e. when

Vep

^{= n.}

Therefore one

expects

^{in the}

free-energy

per unit- volume a term due to the

layers elasticity

where the tensor B has an axial symmetry ^{around n :}

Hence

To this _energy must be added Frank’s curvature

free-energy :

In the above we have taken _ç and n as

independent

variables.

However,

^the

free-energy

^is

changed

ⁱⁿ

an uniform rotation of n - which means that the initial

configuration

^is restored in a finite relaxation time. n is not an

hydrodynamic

^variable

^(a complete

discussion of this

point

^is

given

ⁱⁿ

MPP).

^{This does}

not mean

by

itself that an

equilibrium configuration

could not be achieved with a tilted n

(n. grad

p *

0),

the increase

of fl being

^balanced

by

^a decrease of

fF.

^{Let us now} discuss this

point.

^{Let v be the unit-}

vector of

Vp

^and

One would then have

Vôn is of the order of

1 /R

^{where R is a}

typical

^curvature

radius. Hence

Now,

far from the N --+ A and A - C transition temperatures, ^{B ~}

10’

_{cgs, K ~}

10-’

_cgs, which

means that

;’1. ^10-’

^{cm, which is a molecular}

length.

^{As a} consequence, ^{we can} expect ^{tilted n}

only

ⁱⁿ dislocation or disclination cores. In

fact,

as

BI,

^and

^Bl

^{are of the same}

^order,

^{one should}

rather

expect

^{a nematic structure in these cores.}

Since we are

dealing

^{with a}

macroscopic

^continuum

theory,

^{we can set}

Here e is the relative dilatation of the

layers.

^{It is}

easily

^seen,

using (2.5)

^that

The director is bound to be normal to the

layers,

and twist deformations are therefore

strictly

^forbidden.

Let us now look at bend deformations.

Using (2. 5),

one obtains

Or

This term does not involve the director curvatures.

B ₂ It must be

compared

^{to the}

layers free-energy, B 2 II ^le

Ve is of the order of

elR,

where R is

again

^a

typical

curvature radius.

Then,

since R is much greater than

A3

⁼

(K3/BII)1/2 ^{(~ 10-’ cm),}

^{one can}

^neglect

the bend term.

The

expression

for the elastic

free-energy

per unit- volume is now very

simple

This

expression

^{is very similar to} that introduced

by

^{de Gennes}

[4]

^{in the case} of the deformation of a

planar

^structure.

where u is the

displacement

^{of the}

layers.

^{With our}

notations,

^{we have for a distorted}

planar

^structure

and,

from eq.

(2.5),

On another hand this

expression

^{is valid}

only

^for

small layer

dilatations : the relative dilatation of the

layers is

As

long

^{as we} deal with the case of linear

elasticity,

we can

neglect

^{terms in}

^{e 3}

^{in the}

free-energy,

^and

put ^{e =} y. For stronger

deformations,

^{one should} have taken into account

higher

^{order terms in e}

in the

free-energy expansion.

^{This means that the}

form of the

free-energy

introduced

by

^{Bidaux et al.}

[7]

^{for low}

angle grain

boundaries is

strictly equivalent

^to eq.

(2.6)

^and

is,

as

pointed

^out

by

^the

authors,

^valid

only

^{for low}

angles.

The formal derivation of the

equilibrium

^condi-

tions will be done in the next section. Two

slightly

different

approaches

could be used :

(i)

^The

only

variable is

cp(r).

^Then

and, using

the notation (p,i ^-

a (P _~xi

^{and the usual}

summation convention over

repeated indices,

^the

free-energy

per unit-volume is

given by :

In this

expression, pf

^{is not a}

quadratic

function of

~,i and (pji, and this leads to non linear

equations.

The second

approach

^is

simpler.

(ii)

^One

keeps n(r)

^and

({J(r)

^as

non-independent

variables : the

free-energy

per unit-volume is

again

with the constraints

We now need to use

Langrange multiplyers

^when

formulating

^the

equilibrium conditions,

^{but these}

equilibrium

conditions will take a much

simpler

form than in the first

approach.

Before

deriving

^the

equilibrium conditions,

^{it is} worthwhile to compare this

expression

of the free- energy with that used

by

^{de Gennes}

[ 11 ] and

^McMil-

lan

[12]

^near the N H A transition temperature.

De Gennes introduced a

Landau-Ginsburg

^free-

energy per unit-volume

for small deformations of the

planar

structure. Here

The z-axis is taken normal to the

layers

^{of the} unper- turbed

configurations,

and ôn is the director

projec-

tion on the

X,

^Y

plane. ( 1 j M)

^{is a tensor with axial}

symmetry :

It is

easily

^seen

^that, using (2.11),

eq.

(2.10)

^can

be rewritten as

This

expression,

^{which was} first introduced

by McMillan,

is covariant. Far from the transition

temperature 1 If¡ 1

^can be considered as a constant.

We then find (2.12) :

which is

exactly

eq.

(2. 2a),

^{with the}

correspondence

3.

Equilibrium

conditions. ^- 3 .1 MINIMIZATION OF THE FREE-ENERGY. ^- We will use the second

approach

defined in section 2. The

free-energy

per

unit-volume, pf,

^{is a function}

with the constraints

given by

eq.

(2.9)

^{which can be}

rewritten as

where

Pi

^{is the}

projection

operator ^{on the}

layers :

The

equilibrium

conditions will be obtained

by

^a

minimisation of the total

free-energy

with

Here ,ui and 1

^are

Lagrange multipliers.

Let us now

apply

^{the usual}

procedure.

^The

posi-

tion r of an

elementary

^{volume of the}

^fluid,

^the

phase cp(r)

and the director

n(r)

^{are assumed to be inde-}

pendent variables,

^{which is correct as far as we use}

f instead of f

^{as a}

specific free-energy. Starting

^from

an

equilibrium configuration,

^{let us}

perform

^the

following

infinitesimal transformation :

Then the

specific free-energy f

^becomes

And the new total

free-energy

^is

given by :

From eq.

(3. 5),

^{one obtains}

A standard calculation

gives

where

Then,

^{with the}

help

^of eq.

(3. 9)

^{and after a}

partial integration,

eq.

(3. 7)

^can be rewritten as

Here

bfs

^are surface terms, uii ^{is the stress tensor, h} the molecular field

acting

^{on the}

director, and g

the

permeation

^force

(thermodynamic conjugate

^to

the

layers displacement). Using

^the

following

^notations

One finds

Here ^v is the outer

normal, p

^{is the} pressure with the usual definition

and the vector S has components

The bulk

equilibrium

conditions are therefore

Note that the

Lagrange multiplier A.

^has

disappeared

from _eq.

(3.14).

From eq.

(3.14c)

^one

gets

^{0 :}

3.2 THERMODYNAMIC POTENTIAL AND EQUILIBRIUM.

- We will now show that _eq.

(3.14a),

which allows for the _pressure

determination,

^can be rewritten as

where y

is the usual

specific thermodynamic potential

Let us start from _eq.

(3.12a).

^Then

using

^eq.

(3.13)

one derives

Using

^now eq.

(2.9),

^{one has}

since 0 is normal to the director n. Then _eq.

(3.13)

can be rewritten as

Or, using

eq.

(3.16)

Taking

^{into account} eq.

(3.14b)

^and

(3.14c),

^one

has,

^{for the}

equilibrium

^conditions

Another

interesting point

is that the stress tensor,

as defined

by

eq.

(3.12a),

^{is not}

symmetrical.

^{As was}

pointed

^out

by MPP,

^{one can}

always

^{define an}

equi-

valent

symmetrical

^{stress. However} from this non-

symmetrical

stress-tensor, ^{one can derive a} torque balance

equation relating

^the

antisymmetrical

part of the stress-tensor, ^the torque ^{due to} the molecular field and the surface torques. ^{In order to do that}

we need

i)

^{to derive a} space

isotropy relationship relating

^the

partial

derivatives of the

free-energy

and

expressing

the invariance of the local

free-energy

in an overall infinitesimal rotation of the

physical

system ^and

ii)

^to

give

^a

physical interpretation

^{of the}

surface term

bfs

defined in _eq.

(3.12d).

3.3 SPACE ISOTROPY RELATION. ^- Let us

perform

an infinitesimal rotation.

Then,

ⁱⁿ eq.

(3.6),

^{we have}

where : eijk ^{is the}

antisymmetrical

^{rank-3 tensor}

and 80 the infinitesimal rotation vector. In such a

rotation

pf

^{must be}

invariant,

if there is no external torque

coming

^for

example

^{from a}

magnetic

^field.

From eq.

(3.19)

^and

(3.9),

^{we have}

We therefore have

This

expression

^must vanish for _{any value} of 8w.

Hence we find the

space-isotropy

^relation

It will be

easily

^seen that this relation is satisfied

by

the

free-energy

^defined

by

eq.

(2.8).

3.4 SURFACE TERMS. ^- Let us now look at the

physical meaning

^of

bfs

^defined

by

eq.

(3.12d).

^There

are two terms. The first

is

clearly

^{related to}

permeation.

From eq.

(3.13),

S has two components. ^{The first} component ^is

from _eq.

(2.8).

^{It will be}

important

^for

layers parallel

to the surface. Let us assume n ⁼ v and e >

0,

^which

means that we have a dilatation of the

layers.

^Then

the total

free-energy

is lowered for

positive ’cp,

^i.e.

by

nucleation of new

layers

^{on the}

surface,

^which tend to relax the

layers’

dilatation. On the other

hand,

for

negative

^e

(contraction

^{of the}

layers)

this relaxa- tion _process is obtained

by

^a surface destruction of

layers.

The second component

is dominant when the

layers

^{are normal to} the surface.

It tends to make the

layers glide along

the surface in order to relax their

bending.

^,

It must be

emphasized

here that S takes into account

only

^{the bulk}

configuration

^{effects on} the interface.

Depending

^{on the} true nature of the surface there

are other terms that take into account surface effects.

There is a surface

permeation

^force

S,

characteristic of the

surface,

^{and the}

equilibrium

will be achieved for

Let us now look at the second term

It will be shown that this

corresponds

^{to a surface}

torque. ^{8n is} the variation of the unit-vector n in an

infinitesimal rotation defined

by

^{8w :}

Then and

where

r

clearly

appears ^to be a surface torque ^{exerted on} the direction as a result of bulk effects. Here

again

^we

must take into account the external surface torque ^r characteristic of the

physical

^{nature of the true}

surface,

and the

equilibrium

conditions will then be

given by

The

interesting point

is that the

knowledge

^{of the}

equilibrium configuration

allows the determination of S on

r,

^{and hence}

of S

and

T.

3.5 TORQUE ^BALANCE EQUATION. ^- Let us start from _eq.

(3.20)

^{which we can write}

Using

eq.

(3.12a)

^and

(3.13)

^{we have}

From eq.

(2. 5),

Using

eq.

(3.12c)

^and

(3.22),

^{we find from} eq.

(3.23)

that :

This is the torque ^balance

equation.

^{We must}

emphasize again

that this

equation

^{is valid}

only

^if

6, h and t are derived from a

purely

^{elastie free-}

energy, .i.e.

including

^no effect of external fields such as

magnetic

^or electric fields.

From eq.

(3.23)

^the

origin

^{of the} asymetry ^{of the}

stress-tensor can be

easily

^{seen. In an}

isotropic

^medium

there is no elastic

body

torque. ^{On the other}

hand,

in a

liquid crystal,

the elastic

body

torque is the result of the torque ^{due to} the action of the molecular

field h on the director and of the surface torques

acting

^on the director.

4.

Application

^{to Frank}

free-energy.

^{- 4. 1 GENE-}

RAL FORMULATION. ^- Let us now

apply

^the

previous

results to the case when the curvature free _energy

fF is,

^{as was discussed in section}

2,

^{reduced to}

We shall assume that

K,

^is

independent

^{of the}

density,

which is the standard

incompressibility hypothesis

made for smectics A. This

hypothesis

^is

corroborated,

as far as we

know, by

^the

high

velocities measured in first sound

experiments.

^The pressure p however still _appears in the

equations

^we have obtained : p is now the

Lagrange multiplier

^{relative to the}

constraint of

incompressibility (2)

As

explained ^above,

^{we have}

dropped

^{in the Frank}

force _energy the term relative to

twist, strictly

^for-

bidden on the basis of the existence of

layers,

^and

the term relative to

bend,

since this latter term is

so small

compared

^{to the term of}

splay.

We are then left with the

following equations

-- .

---- I,j ^{-1 .. -1}

There are no

simple

solutions of the

equation

^for

equilibrium

^{div S = 0}

and,

ⁱⁿ

particular,

^{as we}

shall _see, the

Dupin cyclides

of the classical confocal textures are not, ⁱⁿ

general,

solutions of this

equation.

However,

^before

going

^to

applications

^{in which we}

shall take an

approximate equation

^for

pf

^and

div S ⁼

0,

^a few remarks on eq.

(4) (3

^to

6)

^should

be made.

4.2 HOMEOTROPIC BOUNDARY CONDITIONS. ^-

Consider a

sample

^limited

by

^a

boundary perpendi-

cular to the molecules

(homeotropic sample).

^The

boundary

conditions

involving

^the

stability

^{of the}

layers

^read

(see

^section

3.4)

i.e.,

^{for v}

along

^{n : e = 0.}

(1) ^{Let us} notice that the condition div u ⁼ 0 is debatable in a

stratified médium ; div u ⁼ 0 is a condition which is natural in a true 3. D liquid, ^{but the} question arises whether a smectic phase ^is

from that point ^{of view a true} liquid, ^{or a 2.D} liquid, for which the

incompressibility condition reads rather

div( P. u) ^{= 0.}

A discussion of these topics ^is given in reference [16] ^{for the case}

of a membrane.

This represents ^a

layer

^{of constant thickness}

equal

to the

equilibrium

^thickness

do.

Also,

^the

boundary

conditions

involving

^{the sta-}

bility

of the fluid read

where F is an

applied force, necessarily

^{normal to the}

boundary.

One notices that the

particular

^{form of}

pfF

^{does not}

play

^a role in the

boundary

conditions in

homeotropic samples. According

^{to the set of}

eq.

(4.3 ^to 6)

^{and div S =}

0,

^{one can} therefore fix at will the values of div n on the

boundary,

^{i.e. fix}

the mean curvature

1 1

It is _easy to show that if _{u1 + U2} is fixed on one

boundary

L, ^the distribution of the director n is

completely

determined. If one considers the

neighbor- ing layer

^L’

(at

^an infinitesimal distance

do), applica-

tion of Stokes theorem to the vector S

(on

^{a volume}

limited

by

^{two small areas on L and}

L’,

^viz.

dSL

and

dSL,

^and

by

^{the common normals to}

dSL

^and

dSL,)

^{enables us to calculate}

E’,

^since

(div n) L’

^{is known}

by

^the

knowledge

^{of L’. The} process ^can be continued

on

adjacent layers L",

^etc...

4. 3 THIN HOMEOTROPIC SAMPLES. ^- If the homeo-

tropic sample

^{is of}

vanishing

^thickness

(one

^{or a}

few

layers thick),

^{and if} one assumes that such a

thin film is

supported by

^a

(closed) loop

^{of wire of}

a

given shape,

the surface _energy terms, up ^{to now}

completely disregarded,

^will

play

^a role in the deter- mination of the

shape

^of the surface that the

layers

will take. It is reasonable ^to assume that the surface terms consist of the sum of a surface tension A

(inde- pendent

^{of the}

curvature),

^{a term}

proportional

^to

((j 1

⁺

62)2,

^viz.

B(ul

⁺

(j 2)2,

^{and a term}

proportional

to the

gaussian

curvature ai 62

(3).

^{But one can show}

that the term in _{(j 1 (j 2} is irrelevant in a

truly liquid layer, by

^the

following

argument, taken from refe-

rence

[16] :

^{assume that} Ul Q2 is an

hydrodynamic variable ;

^one then has to consider as a

physical

fluctuation the

layer

fluctuations which

keep

Ul (J 2 constant. Under such a constraint the

layer

^{surface 1}

is transformed to a surface E’

applicable point by point

^{on 1}

(ex :

^a

plane

transformed to a

developable).

Such a

peculiar

deformation can

always

^be

^relaxed,

if necessary, ^{to a} surface f of smaller

splay

energy than

l’,

^{in a}

(short)

^finite

time, by

viscous relaxation.

Hence _{Q1 Q2} is not an

hydrodynamic

^variable.

If this is true, ^one is therefore

left,

^{for the}

equation determining

^the

shape

^{of the}

surface,

^{with the}

Laplace equation

(3) ^{Such a term} appears in the Frank and Oseen expressions ^of

the free energy of ^a mesophase (the K4 ^{term in Frank’s} original free energy).

If

Ap

⁼

0,

^then 61 + U2 ^{= 0} and therefore the

layer

is a minimal surface. Since ^{e =} 0

(4.7),

^{the bulk}

energy of the

layer vanishes,

^and the surface area

spanned by

the closed curve is that which minimizes the areas enclosed

by

^{the curve.}

If

L1p ::/= 0,

ai + 02 takes a constant value over the surface area span

by

^{the curve.}

4.4 SOLUTIONS WITH CONSTANT CURVATURE LAYERS.

- It seems

interesting, according

^{to the}

previous considerations,

^{to look for} solutions of div S ⁼ 0

consisting

^of

layers,

of which all have constant cur- vature. Such solutions would

always imply

strong

singularities.

^{Let us}

apply

Stokes theorem to the vector S for a volume

consisting

^{of two elements of}

surface

dS,

^and

dS2

^{on two different}

layers,

^linked

on their boundaries

by

the lines normal to the set of successive

layers

^between

dSi

^and

dSz.

^{One finds}

or,

introducing

^the

layer

thicknesses

dl and d2

This conservation law holds

along

the lines normal

to the set of

layers.

^{It means}

than,

^{when d -}

do,

dS becomes infinite.

The

simplest

^{solution to our}

problem

^is

evidently

when the surfaces of constant curvatures are

spheres

and

cylinders.

^{In that} case, the

layers

^are

parallel,

and one finds

according

^to eq.

(4.10)

.for a

cylinder

^e

= a/r

for a

sphere

^{e =}

b/r2 ,

where a and b are constants.

Inspection

of the free energy shows

that,

^{in the}

bulk, pf d V

is minimized for a ⁼ b ⁼ 0. But solutions with a and b different from zero must not be

disregarded.

4.5 PLANAR BOUNDARY CONDITIONS. - If the

sample

^{is limited}

by planar boundaries,

^{i.e. such} that the director is in the

plane

^{of the}

boundaries,

which

implies

^{that the}

layers

^are

perpendicular

^to

the

boundary,

^the

boundary

conditions read :

- For the

layers

This

equation

^means that the intersection of each

layer

^{with the}

boundary

^{is a line}

orthogonal

^{to the}

set of lines div n ⁼ Const. in the considered

layer.

- For the

layers again

This

equation

expresses the fact that a torque ^{has to} be exerted in the direction

parallel

^to the intersection

of the

layer

^{with the}

boundary

^to compensate ^{for the} surface torque.

- For the fluid

There is a force exerted on the surface

boundary

^on

the molecules.

Except

^{for the case when the}

boundary

is

planar,

this force does not reduce to a normal pressure.

4.6 DUPIN CYCLIDES. ^- Are

Dupin cyclides

^solu-

tions of the

equations

^of

equilibrium

^{div S = 0 ?}

Dupin cyclides

^{in a} confocal domain are

parallel

surfaces. One therefore has

which reads

If one now expresses div ^{S = 0 as a} function of

e

and g

^one notices that one obtains an

equation

^of

the

type

where 0 is a linear operator

acting

^on g. Since a

is a function

of g only (and

^{not of r : e is a constant}

on a

given layer),

eq.

(4.16)

has solutions

only

^if

0(g)

^{is a} function of _g. This occurs

if g

^{is an}

eigen-

function of the operator ^{0. We therefore are led to} infer that

Dupin cyclides

^are solutions of div S ⁼ 0

only

^{for a discrete set} of values of _g. Such a result has in fact been reached

by

^Kleman

[16],

^{who has}

shown

that,

^{for a confocal} domain defined

by

^{its focal}

curves,

only

^one

Dupin cyclide obeys

^the

equilibrium equation.

^It is also shown in reference

[16]

^{that this}

cyclide

^{has no} cusp

points

and is of the first

species (in

^{the sense of}

Bouligand). Hence,

^{in a}

given

^confocal

domain,

^{all the}

layers

^are

(slightly)

different from

Dupin cyclides,

except ^one.

4.7 ORDER OF MAGNITUDE OF e. - A crude eva-

luation of E can now be

given :

^{div n} is of order

1/R,

the mean radial curvature ;

Pij Vj

^{div n} is of order

1/RL

^{where L is a} characteristic transverse

length

for the variation of

R,

^{and div 0} is therefore of order

Kl/ R 2

^{L. In the same} way diB

(Ben)

^{is of order}

Be/R.

Hence the

equilibrium

condition leads to

where

Ac

⁼

(K1/B)1/2

is the coherence

length.

For a smectic

A, Ac

is of the order of a molecular

length,

except ^{near the} upper transition temperature.

Near the focal

lines,

^{L can be} very small

but,

except in the _core, if is greater ^than

Â..

^Then 8 gg

Âc/R.

Taking Â,,, - 10-7, R ~ 10-4,

^{we have} 8 gg

10-’.

On the other

hand,

^for

cholesterics, Ac is

^{of the}

order of the

pitch

qo, which means that s is no

longer

small. This can

explain

^the strong deformations of focal structures that have been observed on

cholesterics.

5. The limit of small s.

Stability

^of

cylindrical

solutions. ^- 5.1

QUASI-PLANAR

^{LAYERS. - In the}

limit of small _a, the small distortions from a

planar

situation are described

by

^{the same}

equations

^as

those used

by

^{de Gennes}

[4].

One considers a function (p

given by

lp.

^{= z -}

u(r)

where u is a finite

quantity,

which varies

slowly.

^Its

derivatives are small

quantities.

^One

finds,

^when

keeping

the second order terms in those derivatives

from which one obtains E and n.

hence one obtains the same force energy ^as de Gennes’

but one notices

that,

^{if one}

keeps

third order terms in the B term

(as

has been done

by

^Clark

and

Meyer [17]),

^it is necessary ^to include terms of the

same order in the

K,

^term.

5. 2 CYLINDRICAL SAMPLES. ^- Consider now a

stacking

^of

layers

in concentric

cylinders.

^{We want}

to

investigate

^the

stability

^{of such a}

system

^{in the}

vicinity

^of the solution e ⁼ 0

(layers

^{of constant}

thickness).

^{Let us define}

Calculation of e and n up ^to the second order in u

yields :

where

div

where

Building

^{now the free} energy up ^to the second

order,

^{we notice a} remarkable différence with _eq.

(5.4).

The term in B is

basically

^{of the same form}

but the term in

Ki

^{now contains}

quadratic

^{terms of a}

complete

^{different nature. We have}

This

quantity

^{can be}

negative

^and

gives

^{rise to}

instabilities of the

cylindrical solutions,

^{for some} types of fluctuations of ^u. In order to

investigate them,

^we first write the extra free _energy

b(pf ), drop- ping

^{the terms which add} up ^to surface terms since their _presence would

just

shift the _energy

by

^{a cons-}

tant amount of little

physical meaning

^as

long

^{as the}

boundaries conditions are not taken into account.

We have

u

obeys

^the

Euler-Lagrange equation

We

investigate

différent solutions of

(5.9),

^whose

difference with

(5.4)

^is

striking.

5.2 .1 Solutions

depending only

^{on 0 and z. -}

They

are

necessarily

^{of the} type

where

do

^{is the}

layer

^{thickness at}

equilibrium,

^{and S}

is an

integer

^{in order for u to} be consistent with the

periodicity

^{of the}

layers

^{after 0 has}

changed by

^an

angle equal

^{to 2 n, u therefore}

represents

^{a kind of} dislocation

along

the axis of the

cylinder :

^the

layers spiral

about the axis

(at

^constant

z),

^{but if one now}

considers the variation

along

^{z, the}

spiral

^rotates

about the axis with a

pitch equal

^{to p}

= 2 ^n/q.

^This

helical variation means that an extra

layer

^{of thickness}

do

^{is added}

along

the axis after a variation of z

equal

to p.

Eq. (5.10)

represents

only

^the

beginning

^{of the}

instability ;

^{but the}

symmetry

^{which is}

displayed by

this

equation clearly

indicates that a helical distortion of the core from the

cylindrical

disclination we have started from will take

place.

Let us calculate the variation

b(pf )

in energy due to

(5.10).

This variation is

always negative.

^We

find

(per

^unit

length

^of

line)

where rc

^{is a core} radius and R a

(large)

cut-off radius.

This

expression

^{is valid as}

long

^{as the terms of third}

order which we have

dropped

^are

negligible.

^{This is}

certainly

^{true if}

do q « 1. q

^is therefore limited.

Exp. (5.11)

^would suggest ^{that S is as}

large

^as

possible,

but the core

always gives

^a

positive

contribution to the _energy

which,

when balanced with

(5 .11 ),

^should

fix the

optimal

^{values of}

S. q

is related to that same

balance and also to the size of the

specimen.

^Accord-

ing

^{to a} calculation

by

^Durand

[18] concerning

^the

effects of fluctuations of size 2

nlq along

^a

layer,

^{it is}

reasonable to

relate q

^{to the}

specimen

^size

by

^the

relationship q2 Â,,

^{R - 1.}

5.2.2 Solutions

oscillating

^{with z. -}

They

^are

of the form

where m is an

integer.

^{We have}