Striped patterns in a thin droplet of a smectic C phase

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Striped patterns in a thin droplet of a smectic C phase

C. Allet, M. Kleman, P. Vidal

To cite this version:

C. Allet, M. Kleman, P. Vidal. Striped patterns in a thin droplet of a smectic C phase. Journal de

Physique, 1978, 39 (2), pp.181-188. �10.1051/jphys:01978003902018100�. �jpa-00208752�

STRIPED PATTERNS IN A THIN DROPLET OF A SMECTIC C PHASE

C.

ALLET,

^M.

KLEMAN

and P. VIDAL

Laboratoire de

Physique

^des

Solides,

Université

Paris-Sud,

⁹¹⁴⁰⁵

Orsay Cedex,

^France

(Reçu

^le

7 juillet 1977,

^{révisé le Il octobre}

1977, accepté

le 19 octobre

1977)

Résumé. ²⁰¹⁴ L’observation d’un système de rubans orientés dans une fine goutte ^de smectique ^C (Sm C)

déposée

^{sur un} substrat à ancrage unidirectionnel ^nous conduit à étudier la

géométrie

^des lignes de rang demi-entier dans les Sm C. Une

analyse

théorique ^{de ces} observations, s’appuyant ^sur

une comparaison ^{avec les} systèmes magnétiques ^en raison de la présence ^d’une paroi de Néel liée

aux disinclinaisons, nous

permet

^{de tirer} quelques conclusions sur les ordres de

grandeur

relatifs de quatre des dix coefficients de

rigidité

^{du Sm C étudié} (D.O.B.C.P.).

Abstract. ²⁰¹⁴ The observation of striped patterns ^{in a thin} droplet ^{of a smectic C}

phase

(Sm C) deposited ^{on a} substrate with unidirectional

anchoring

^{has led us to the} study ^{of the} geometry ^{of lines} of

half-integral

strength. ^A theoretical

analysis

^{of these} observations, ^{based on a} covariant formalism of the

elasticity

^{of Sm C} phases, ^and

using

^an

analogy

^with

magnetic

systems because of the presence of a Néel wall attached to disclinations, has enabled us to draw some conclusions about the relative orders of magnitude of four of the ten stiffness coefficients of the Sm C under study (D.O.B.C.P.).

Classification

Physics ^Abstracts

61.30 ^- 61. 70

1. Introduction. ^- D.O.B.C.P.

( 1)

^{is an}

elongated organic

molecule which is known

to

have two

liquid crystal modifications,

^{viz. a nematic}

phase

^{and a}

smectic C

phase separated by

^a first order transition Solid

60o’C SmC l12.S °C)Nem 66.5’oC

Isot.

This

phase

^is

optically

^biaxial

[1] :

^the

optical

^{axis d}

makes an

angle

^a of the order of 45° with the normal n

to the

layers.

^This

angle is,

^{as far as known measure-}

ments

tell,

invariant with

temperature.

^{When intro-} duced between two

glass plates

^treated

by evaporation

to

give

^one

(and

^the

same)

^{axis of} easy

alignment (anchoring direction)

^this

produces

patterns

display- ing

^two

planes

^of

symmetry,

^one

parallel

and the other

perpendicular

^{to that} easy axis : these

patterns

^are

clearly

^made

of layers perpendicular

^{to the}

glass plates,

at an

angle fi -

^{45° to the}

planes

^of

symmetry (Fig. 1).

Hence the

optical

^{axis and the}

anchoring

^{axis are}

practically parallel,

and it will be sufficient for ^our

FIG. 1. ^- Orientation of a Sm C phase ^{between two treated} glass plates. ^H indicates the anchoring ^{axis on} the substrates. In an actual

experiment ^the grain boundaries are decomposed in isoeccentric

parallel ellipses, ^{in focal} position ^with respect ^to hyperbolae ^located

in the mid-plane ^{of the} sample ^{and whose} asymptotes ^are perpen- dicular to the layers.

purposes ^to

identify

^{them as}

defining

^{a molecular}

orientation

(molecular axis) (2).

Defects in Sm C

phases

^{have been}

described,

^{on the}

basis of

symmetry

arguments,

by Bouligand

^and

Kléman

[2]. They distinguish :

1)

m-lines : disclination lines whose rotation axis is

along

the normal n to the

layers ;

^{these defects form}

the well-known schlieren textures

(see

^{for ex. :}

Saupe [4]

and Demus

[3]).

2)

d-lines : dislocations of the

layered system.

(1) Bis-4’-n(decyloxybenzol)-2-chloro-1-4-phenylenediamine

The samples ^were kindly supplied ^{to us} by ^C. Germain, ^P. Keller,

L. Liébert and L. Strzelecki.

(2) ^{We are indebted to Professor F.} C. Frank for

pointing out

^to

us the difficulty ^of defining ^a unique direction of the molecule in a Sm C.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01978003902018100

182

3)

1-lines : disclinations whose rotation axis is

along

the normal N to the

plane

^of symmetry of the Sm C

phase ; (this plane

^is

perpendicular

^{to the}

layers

^and

contains the

optical axis).

We shall present ^{here what we believe to} be the first

experimental

evidence and first calculation of

straight

1-lines of

strength

±

1/2,

observed in a thin

droplet

of D.O. B. C.P.

deposited

^{on a}

glass-plate possessing

^an

easy axis of

alignment.

^{We shall}

present successively : a)

^{some remarks on the}

geometry

^{of these}

lines, b)

^the

experimental observations,

c)

^a calculation of the _energy of the observed texture,

including

^{a detailed}

study

^{of a Néel wall}

configuration

ⁱⁿ

partially

^twist

(S

^{= +}

1/2)

^disclina-

tions. This calculation uses a new covariant free energy

density

^for

strongly

^{curved Sm C}

phases

derived

by

^{one of us}

[5].

2.

Wedge

^and

twist

^1-lines

of half integral strength [2].

-

Figure

^2a

represents

^a

wedge

^{S = +}

1/21-line :

^the

axis of rotation N is

along

the line. One will notice that the

layer stacking

and the molecular axis distribution

present

^{each and}

independently

^a disclination of

strength S

^{= +}

1/2.

^{Because the} molecular axis is not fixed in a

unique

way

(see introduction),

it is better to

consider the

projection

^t of this axis on the

layer

^rather

than the axis itself. The distribution of t presents ^a S ⁼

1/2

disclination too, in contradiction with a current idea

(see

^{de Gennes}

[6])

^{that such a vector can}

suffer S ⁼ ± 1 disclinations

only (3).

Figure

^2b

represents

^a

(partially)

^twist

^line,

^{at an}

angle

^{u to} the rotation axis N. It is

partially

^{twist in a}

FiG. 2. - a) S ⁼ 1/2. The axis of rotation N is along ^the line ; b) Partially ^{twist S =} 1/2 line. On the right, ^{one of the} layers ^has

been developed ^to show the existence of a Néel wall in the t distri- bution.

peculiar

^{sense : the}

layer stacking

is still that of a

wedge ^line,

whereas the t distribution has a

partial

twist

character ;

^{let us also note}

that,

^{if one unwinds}

one of the

layers

^{on a}

plane,

the directions

t+

^{and t-} make an

angle

^{2 u and}

join through

^{a Néel wall}

separating

^{these two}

directions, corresponding

^{to the}

distribution of t in the

semi-cylindrical region.

A

(partially)

twist line can also be derived in which the t distribution remains of the

wedge type,

^whereas it is now the

layer

distribution which has a twist character. This _process is more

complex

^{than the}

former _one, because the dislocations of translation which have to be emitted

(or adsorbed) by

^{the line}

must now be discrete

[7]. ^However,

^{it is a}

possible

process, and it leads ^to the well-known focal lines of

layered

^media

[2].

^{We shall} not comment on them here.

Considerations similar to the above also

apply

^to

S ^{= -}

1/2lines.

In this _paper, we shall be concerned with S ⁼

± 1 /2

lines of the

type

^of

figure

^2b.

3. Observations. ^-

Strong anchoring

^{was achieved}

by evaporating

silicon monoxide on the

glass plates, according

^{to the}

technique

^{due to} Urbach et al.

[8].

This

strong anchoring

^{was essential to} obtain the textures described

below,

^which present ^a

practically

constant molecular direction ^on the surface

(4).

A small amount

of crystalline

^{material was}

deposited

on the

glass plate

^{and the}

sample

introduced into the

oven

(Mettler FP5) previously

^{heated to} 160 °C. The material is then

liquid

^{and wets the surface. On}

decreasing

^the

temperature,

^one first observes at the nematic-smectic transition the appearance of fine

stripes, parallel

^{to the}

anchoring direction,

^whose width is

independent

^{of the}

sample

thickness. These transitional

stripes

^{are located on} the free surface of the

droplet.

^A

study

of them will be

presented

^{in a forth-}

coming

paper. Their existence has

already

^been

mentioned in another Sm C chemical

[9].

The range of existence of these transitional

stripes

is a few tenths of a

degree.

^Below

that, larger

^{and more}

contrasted

stripes

appear, which

push

the transitional

stripes

aside and grow

along

^{their own axes of elon-}

gation

^{in two}

practically orthogonal

directions.

By cycling

around the

temperature

^of

transition,

^{it is}

possible

^{to obtain a} very

regular pattern

^{of these two}

families,

^and

eventually

^{to have}

only

^one

family (Fig. 3).

^These

stripes

^are

definitely

^{different from the}

transitional

stripes : they

subsist down to the smectic-

crystal

transition without modification and can even be obtained when

heating

^the

crystal

^{to the smectic}

phase.

Each

stripe

^{contains two}

singular

^{lines of}

defects,

one on the

glass plate itself,

^{the other one near the free}

surface. Their

separation is,

ⁱⁿ

projection,

^{of the order}

of the half-width of the

stripe. Repeated

observations led ^us to the

following

conclusions

(1) The normal n to the layers ^{is a 2 n} symmetry ^{axis. Hence the} only disclinations made about n and involving ^a distribution of t have

integral strengths. ^The argument is different for N, ^{which is a} dyad

axis and can act as a rotation axis for t.

(4) ^{We thank M.} Boix for the preparation ^{of the} evaporated glass plates.

FiG. 3. ^- Crossed nicols. Two families of stripes ^at practically ^45°

to the anchoring ^{direction on the} glass-plate. Pure chemical. One will observe the weak undulation of the stripe pattern (see ^conclusion

of the _paper about that particular point).

Fie. 4. ^- Crossed nicols. ’T’he family ^of larger stripes ^is approxi- mately ^{at 45° to the} anchoring ^{direction on the} glass-plate. ^One

observes also transitional stripes (thinner stripes). Old chemical.

- when no

special

^{care is}

taken,

^{the two} families of

stripes

^are

equally

distributed

along

^two directions at an

angle

^{of about 45° to the} easy

axis,

- the

stripe repeat

distance increases monoto-

nically

^{with the}

specimen

thickness.

Although

^{we have}

not made

precise

measurements, ^a linear increase

seems to fit the

observations,

- for small thickness a

periodic

undulation of the

stripes

^is very often observed

(Fig. 3). Larger stripes

are less

regularly oriented,

orientation

becoming

very chaotic for _very

large stripes,

-

stripes

of small widths can present ^a

large

deviation from the 45°

direction,

^{but are in} any ^case very

straight,

- the surface

singularity

^lines

running along

^the

stripes

^leave faint tracks on the

glass plate

^when

passing

^{back to the nematic}

phase.

These tracks indicate a small variation of the _easy axis in the

vicinity

of those lines.

D.O.B.C.P. is a very labile

compound,

^{due to the}

presence of the two Schiff bases. We have observed an

old

sample

whose nematic-smectic transition

tempe-

rature was

approximately

^{105 °C}

(i.e.

^{5 °C lower than}

the _purer

material) ;

this transition

temperature

decreased

by

another 5 °C

during

the observations

(which

lasted half a

day).

^{The same} conclusions were

obtained as with the _purer

chemicals,

^{but the}

stripes

were

definitely larger

^{and more}

^chaotic,

^{for a similar}

thickness

(Fig. 4).

^{The ease} of observations in

pola-

rized

light

^with

large stripes

^{enabled us to} confirm the model

already

derived from the former

observations,

and

particularly

^to

study

^the

relationship

^beetween

the

top

^{and bottom}

singularity

^lines

(see Fig. 5). Top

lines

(and reciprocally

^bottom

lines)

^{finish at a}

point

where bottom

(and reciprocally top)

lines make ^a fork.

A faint line

joins

^the

top

^and bottom lines at the

junc-

tion. These

properties

^{enabled us to} confirm that these lines are S ⁼

± 1/2 1-lines,

the bottom lines

being

S ⁼ +

1/2

lines and the top ^{lines S = -}

1/2

^lines.

The model is

presented figure

^{6. It} consists of half-

FIG. 5. ^- Topological relationship ^between top and bottom sin-

gularity ^{lines :} reciprocity ^{of the} relationship. ^The

faint,lines

^{at the}

junction (see text) ^{are not} indicated.

FIG. 6. ^- Model for the striped domains. One notices the Néel walls on the 1-lines.

cylinders of (partially)

^twist

^(S

^{= +}

1/2)

^lines

(u - 450) separated by (partially)

^{twist S = -}

1/2

Iines near the surface.

Undulating layers

^cover this whole structure,

so that the contact of the free surface with air is

homeotropic.

^{This is a} low-surface-tension

geometry,

as can be shown

by

^{the ease} with which

homeotropic

samples

^are achieved in free films

suspended

^over

holes. These

samples display

^the well-known Schlieren textures. It is to be observed that in the case of our free

droplet

^{this cannot be the} case, because the _presence of the S ^{= -}

1/2

^{lines near} the surface

imposes

^some

rigidity

^{on the free} rotation of the ^{vector t on} the surface.

4. Calculation of the Néel wall in a S=

+ 1/2

^{line. -}

We _present

^{here a} calculation of the Néel wall in the

184

half-cylinder

^{of a S =}

^{+ 1/2}

^line

(Fig. 2b), starting

from the

following

^free energy

density

where r is the radius of the

layer

^under

consideration,

0 is the

polar angle,

^and

(n/2 - úJ)

^{is the}

angle

^between

t and the

cylinder

^axis

(see Fig. ^{7). du/dn}

is the relative dilation of the

layers and B

^a coefficient of

compressi- bility.

FIG. 7. ^- Coordinates for the calculation of the Néel wall.

This free _energy

density

is obtained

by generalizing

the free _energy of ref.

[10],

^{which was written for small}

distortions with

respect

^{to the}

planar

lattice. The

Í2i,j

used in this reference

generalize

^{to a} contortion tensor

Kij

which describes the local rotation of the tri- hedron _n, t, ^N

In

specializing

^to

cylinders,

^we have assumed that the molecular axis does not twist from one

layer

^{to the}

next ; hence

A 33

^{= 0, and}

only

8 coefficients _appear in _eq.

(1)

instead of 10 in ref.

[10].

A full

discussion

^{of this tensor} and of the covariant free energy is

given

^{in Kléman}

[5] ;

^{here we are}

just

interested in the

cylindrical

^{case. We have}

adopted

the same notations as in the

Orsay Liquid Crystal

paper

[10]

for the stiffness coefficients in _eq.

(1).

^We

shall assume here that

the layers keep

^{a constant}

thickness in their

stacking,

^{so that}

du/dn

^{= 0.}

Minimizing f pf d v with

respect ^{to co} leads to the differential

equation

The

C;

^terms

integrate

^{to surface terms and do not}

play

any role.

Eq. (2)

^is

analogous

^to

equations

encountered in the

theory

^of

magnetic

^walls

(as

^{in fact our}

topo- logical

discussion

suggested) :

^this

analogy

^leads

us to compare ^{co to} the

angle

^{of the}

spin

^with

respect

to a fixed

direction,

^{and 0 to a} coordinate x perpen- dicular to the

wall;

this coordinate exists in the range - ^{oo x + oo.} The terms

involving

^the

coefficients

Bi

^and

B2

^{are most similar to}

exchange

terms, ^since

they

govern

gradients

^{of û) with} respect

to 0

(in

eq.

(1)) (as

^the

exchange

coefficient _governs

gradients

^{of ro with} respect ^{to x in the}

magnetic

^free

energy

density).

^{The terms}

involving

the coeffi- cients

A 12, A21, A

^{are most}

similar, analogously,

^to

magnetocrystalline anisotropy

^{terms. A} differential

equation

^like eq.

(2)

^{has as}

solutions,

^{either one wall}

separating

the range of x

(here 0)

^{in two domains}

magnetized along

easy

directions,

^{or a}

periodic

^dis-

tribution of walls. In each _case, the

boundary

^condi-

tions are

dco/dx

^{= 0 for the} easy

directions,

^{which are}

defined as those which minimize the

magneto- crystalline anisotropy

energy.

Let us

apply

^this

analogy

^{here : the} magneto-

crystalline anisotropy

energy is

proportional

^{to :}

and the extrema are obtained for

i. e. for :

a)

col ⁼ 0

(the

molecular axis is therefore in a

right

section of the

cylinder) ;

b)

(02 ⁼

03C0/2 (t along

^the

cylinder ^axis,

i.e. here the molecular axis is

parallel

^{to the}

anchoring

^{direction on}

the

glass plate) ;

The minima are obtained for

d2fm

> 0 dw2

It is _necessary here

to

recall that

by

virtue of free energy

thermodynamic stability conditions,

^{A is the}

only

coefficient which is liable to be

negative.

^This

appears in the

thermodynamic inequality :

It can be shown that this condition has as a conse-

quence that

According

^to eq.

(5),

mi and _ro2 are minima if A - 2

A 12

^{and A -}

2 A21

^are

positive.

^{This must}

occur

simultaneously.

^{If so, A is}

positive

^and C03 is

necessarily

^{a maximum}

of , fm,

if it exists.

On the other

hand,

^{if A is}

negative,

^{w1 and} ro2 ^are

maxima,

_ro3 exists and is the

only

^minimum

of fm.

A 0 : One looks for a first

integral

^of eq.

(2) (in

^{which we} shall make

B, = B2

⁼

B) satisfying

^the

boundary

^condition

dm/d0

^{= 0 for ro =} ro3. This

yields :

o) oscillates either in the _range

( -

(03, ⁺

w3), passing through

^{w =}

^0,

^or outside this _range,

passing through

0) =

:t Te/2.

It is

clear,

^a

priori,

^{that the}

larger

range will have the

larger

energy and

corresponds

^to unstable walls. The situation in which A 0 does not

qualify

^{to build}

Néel walls on S ⁼ +

1/2

^twist

disclinations,

^{since (J)}

takes the values w ⁼ ±

n/2

^{and (J) = 0 in these defects.}

A > 0 : One finds different first

integrals, according

to whether the chosen minimum is (J) ⁼ 0

(fm

⁼

A 12/2)

or m =

n/2 ( fm

⁼

A21/2).

^The

experimental

^{results in}

D.O.B.C.P. seem to indicate that the reasonable choice is m =

n/2 (which might

indicate that

A 12

^{is much}

larger

^than

A21

ⁱⁿ

D.O.B.C.P.) (5).

^{Let us}

study

^this

case in more detail. The first

intégral

^{is :}

The second term is

positive

for all values

ofm, requiring

Also,

^the

thermodynamic inequality (eq. (7)) ^{requires :}

Eq. (9)

^therefore

integrates

^{to :}

A12 + A21 - A

where m ⁼

A12 + A21 - A

is a

positive quantity A12 - A21

which is smaller than

unity (one

^can

put : m

⁼

sin2 v)

and the constants of

integration

have been chosen so

that 0 ⁼

03C0/2

^{for m = 0}

(at

^the apex of the semi-

cylindrical layers

^{in the S =}

1/2 line).

The definite

integral

^{in eq.}

(12)

^{is an}

elliptic integral

of the third kind

II(1; wlv) (Abramowitz

^and

^Stegun [11]).

By analogy

^{with the}

magnetic

^{case, we let 0 vary in}

the _range

(-

cc, +

cc), knowing

^{that the}

only physical

range is

(0, n).

The distribution of co is

periodîc

^{with 0 if}

71(1 ; colt» always

has finite values for finite m when (0 varies in the _range

( -

^{00, +}

(0).

^{If on}

the

contrary N(1 ; wlv)

becomes infinite for co ⁼

n/2,

there is

only

^one wall. This is indeed the case whatever v

might be,

^{and we can} ascertain that there ^is

only

^{one ir}

rotation

of t,

^at most, ^on the

semi-cylindrical layers of

the S ⁼

1/2

^line.

We shall define the

angular

^width of the wall as twice the value of 0 for ro ⁼

n/4. According

^to

[11] (see

p.

600, Fig. 17.11,

¹⁹⁷²

edition), 77(1 ; w/v)

^is

practi- cally independent

^{of v and}

equal

^to

unity

^{for co =}

n/4.

Hence the

angular

width of the wall is

The

dependence

^on

A,

^{which is} very

small,

^is

neglected

here. One notices that for

A 12 = A21,

^{the width}

becomes infinite.

û) does not take

exactly

the values

n/2,

³

n/2,

^for

0 ⁼

0,

n,

except

^if

AOO

^is very small. Our

experi-

mental

AOO

^is

probably

^not very

small,

^{since we}

observe small deviations of the _easy directions on the

glass plate.

^If

AOo

^{were much}

larger

^{than n, on the}

other

hand,

the deviations would be _very

large.

However,

^{we cannot conclude}

clearly

^without

knowing

the surface

anchoring

^energy

Ws.

^{It is}

probable

^that

this

anchoring

energy is

large (see

^eq.

⁽¹⁶⁾

^{in which}

sense this is

understandable),

^{but not}

large enough

^to

overcome the bulk

torques

^on the molecular axes due to a

large AOo.

^{Let us} also notice

that,

^{in our}

experi-

ment, ^the

optical

^{contrast is much more} accentuated when one uses old chemicals : in this case it is reaso-

nable to think that the

exchange

coefficient B is lowered with respect ^to purer

chemicals,

whereas the

anisotropy coeffiçients

^{are not}

changed

^much

(we

^still

get ^{a Sm C}

phase

^{with a molecular axis at 450 to the}

layer normal).

^{Hence we}

expect

^{a smaller}

AOo

^{for old}

chemicals,

^which

implies

^a

vanishing

deviation of the easy axis on the substrate if

AOO

^{is small}

enough.

This will also increase the contrast of the 1-lines. This is what we

observe,

and the fact that the

stripes

^{do not}

leave tracks on the substrate in the case of old che- micals.

Still in

analogy

^with

magnetic ^walls,

^one expects the _energy of the Néel wall to be

proportional

^to

J B(AI2 - ^{A21). Taking}

^{into account the}

^r-depen-

(1) ^{There are} good ^{reasons to} believe that A 12 ^{is much} larger

than A 21 in ^{Sm C} phases ^which display broken fan textures (cf.

Kléman [5]).

186

dence

(eq. (1)),

this leads

finally

^{to an} energy per unit

length

^{of a}

half-cylinder

of radius R :

where rc

^{is a core} radius of the order of a

layer

^thick-

ness. In the

following

^{we shall use} the notation

Let us now estimate the surface _energy contribution at the surface.

Writing d

^{for a}

typical

surface dis- clination

thickness,

^and

WS

for the value of the

anchoring

^energy

^(i.e.

the energy per

cm2

_necessary to tilt

the

surface molecules towards an

homeotropic orientation),

^one should have

approximately,

^balanc-

ing

bulk contributions and surface

anchoring

energy

on the

substrate,

and this is also

approximately

the surface _{energy per}

half-cylinder. do

^{is small}

(comparable

^{to a}

layer

thickness for

example)

^if

Ws

^is

large enough.

^{It is a core}

energy, whence our notation

Wc. Finally,

^the energy of a S = +

1 /2 line

^{on the}

glass plate

is of order

where A’ is a

quantity

^{in which}

only

^the

anisotropy

coefficients

A, A 12’ A21

^{are involved.}

5. Estimation of the width of the

stripes.

^{- Accord-}

ing

^{to our model}

of figure ^6,

^{one can}

^estimate

^the energy of a

stripe

pattern

by adding :

- the energy of the

singularity

^{lines. For the}

S ⁼

1/2 lines,

^use eq.

(17).

^{For the S = -}

1/2 lines,

one bas an

expression

^{of the}

type of eq. (15). Finally,

one writes

- the elastic _energy stored in the

compressed

^or

dilated

layers.

^{We shall} here consider two

pàrts :

a)

^the

elastic

energy

W+

^{of the}

layers

^{located between}

the free surface and the

layer limiting

^the

semi- cylinders

^and

b)

^the

^’elastic

energy W of the

layers

below this

limiting layer.

5.1 ESTIMATION or

W+. -

^{We shall assume that}

the

layer limiting

^the

semi-cylinders

^{has a sinusoidal}

h.

^{2 nx -} duse a resu lto btained . d.in th h

shape 2

^cos

R ’

and use a resu t obtained in thé ^case of Sm A

mesophases (see

^ref.

[6]) :

^the

displacement

^of

the

layers

^{above the}

limiting layer

varies like ,

where

À is a

penetration length (~ (K/B) 1 /2 by analogy

with Sm

A).

The elastic _energy stored between z ⁼ 0 and z = d

is,

per

repeat

distanoe R :

Most of this _energy can be relaxed

by

^the presence of

straight edge

dislocations

parallel

^{to the}

stripes.

^We

shall comment on that later on.

5.2 ESTIMATION OF W_ . - Let us consider one of the

semi-cylinders.

^The

limiting layers

^{suffer a dis-}

placement

whose maximum is localized at the

S= -1/2

disclinations : the

amplitude

^{of this}

displacement

^{is of}

the order of R - b. It is feasible to estimate the elastic energy stored in the

cylinder by using

^{a treatment}

of Sm

A-type, involving only

^a

coefficient B

^{and a}

penetration length,

^{as above. For this}

calculation,

^{it is}

necessary ^{to use} the free _energy of a deformed smectic

cylinder (see

Kléman and Parodi

[12],

eq.

(5.8)

and

(5.9)).

^One obtains the

following dependence

where a is a numerical coefficient of the order

of unity.

This is a

large

energy, due to the _presence of the

quantity

in the denominator. We now show that it

can be reduced

by

^{a factor -}

À/R f

^one relaxes the distortions

by

the introduction

of edge

dislocations all

of

the same

sign.

The

problem

^is

analogous

^to that of the

comparison

of

energies

^{of a}

crystal beam,

either bent

elastically,

^or

bent

plastically (Fig. 8)

with dislocations of

Burgers

vector b.

In the first case

(Fig. 8a),

^the energy of the beam is of the order of

Eh 3/R 2 (see

^for

example

^{Landau and}

Lifshitz, Elasticity)

where E is

Young’s

^{modulus. To}

bend it

plastically

needs the introduction of ^a

density

of dislocation lines n ⁼

1/bR [13], homogeneously

located on

layers separated by

distances h’ -

JbR.

Each of these sub-beams has an elastic _energy

hence,

the total elastic _energy of the

plastically

^bent

is ^~

Eh3 /D/A

the p lastic ener is that of the beam is ^~

EH3- (Rb ). The plastic

energy is that ofthe

dislocations,

^i.e.

h2

FIG. 8. ^- Model for the plastic relaxation of a bent beam.

These are two

rough

ways of

estimating

^{the energy of}

the beam in

figure 8b,

^and

they

^{lead to the same order}

of

magnitude.

^Hence

Rb/h2

is the reduction factor.

Let us now

apply

this result to the smectic

phase.

Such a

phase

^{has an effective}

Young modulus,

E ⁼

K/h2 (see

^ref.

[14]),

^{hence the} argument ^{can be}

repeated.

^In

computing

the reduction

factor,

^{we shall}

make h ⁼ R and b = À

(since À

^is of the order of a

layer thickness,

^{i.e. a}

Burgers vector).

The reduction factor is therefore of the order of

À/R,

^{which we insert}

in _eq.

(20), yielding :

A similar argument ^can be made for

W+ .

^{The radius}

of curvature of the

layers being R2jb,

^this

gives

^a

reduction factor

À.b / R 2.

^This

yields :

i.e. a

practically negligible ’energy,

^{inasmuch as, the}

predominant

^{tenu in the} upper

region

is the surface

tension,

^{to which we now tum our} attention :

- a free surface _energy. Call _y the surface

tension ;

one

gets :

Let us discuss the

relative

^{values of y and}

^As.

^For

Sm A

phases,

^one

has B

107

dynjcm2, À ~

³⁰

^Á.

Let

^us

adopt

^{here the same} numerical values. Hence

BÀ ~

3

dyn/cm.

Usual values for surface tensions in

nematics

^{are 30}

dyn/cm

^{to 100}

dyn/ cm2.

^Hence

BÀ _y. It is also

expected

^that y will _vary with the

purity

^{of the}

sample (Landau

^and

Lifshitz,

Statistical

Physics). Finally,

^{we shall}

neglect W+ (and a fortiori W*) +

ⁱⁿ

comparison

^with

Wsurf.

We have to look for the minimum of

for R + d ⁼ D

being

^{constant. Let us} first differen- tiate with

respect

^{to the}

amplitude

of deformation.

One

gets

ô is therefore of the order of a hundredth of R. Let us now introduce this values of ô in W and minimize with respect ^to

R, keeping

^{in mind R + d = D. One shall}

neglect

^terms of the order of

ô’ compared

^to

^{R 2.}

^One

obtains :

which minimizes to

A

rough

estimate of the solution is obtained

by making

^the

exponential

^{and the}

logarithmic

^factors

both

equal

^to

unity.

^This

yields

the linear

dependence

6. Conclusions. ^- This _paper shows the first

example

^{of a}

complete description

^{of a new}

type

^of defect

specific

^{to a Sm C}

phase,

^both

experimentally

and

theoretically, using

^{a new covariant}

elasticity.

This defect involves the _presence of a Néel wall.

Another

important

feature of the

analysis

is that the

stability

of curved

layers

under mechanical stress

requires

^the presence of a

density

^of

edge

dislocations of like

sign.

^But in this conclusion we want to remark

on three other aspects, ^not

developed

^{in the text.}

i)

^INFLUENCE OF IMPURITIES. ^- Most

liquid crystal

chemicals are very labile

materials,

^which appears

clearly

in the variation of the transition

temperature.

But the stiffness coefficients should _vary also. It is apparent ^{here that}

and

both decrease with the

impurity

content, ^which

implies

^a decrease in the

exchange

coefficient B.

The

decrease

of AOo explains

the decrease of the

anchoring

effects at the substrate.

_

Also it is reasonable to

expect

that B decreases when the

impurity

^content increases. But the increase in R

implies

^that y should diminish much faster

(than B 2 Â,3),

^{which can be due to}

segregation

^of

impurities

on surfaces. This seems to be in correlation with the formation of clusters

(on

the substrate and at the free

surface)

ⁱⁿ

impure

^materials.

ii)

UNDULATIONS ALONG THE STRIPES

(in

pure ^mate-

rials).

^{This is}

certainly

^{an elastic}

instability, analogous

to that one invoked in

[15]

^to

explain

the transfor- mation of

oily

streaks into chains of focal

domains, relaxing W*+.

Since it is a small _energy, the

periodicity

is

large.

^{We have not tried to} calculate it.

iii)

^{REVERSAL OF THE NÉEL WALL} CHIRALITY. ^- It is clear from

figure

⁶

that,

^the

chirality

^{in a}

stripe

being given,

^the

adjacent stripes

^{must have the same}

chirality,

^{unless one} introduces a

strong supplemen-

tary

^defect

along

^the

stripes. Correspondingly,

^a

188

change

^of

chirality along

^a

stripe

will induce a

change

of

chirality

ⁱⁿ

adjacent stripes.

^{Zones of}

opposite

chiralities are therefore

separated by

^lines transverse to the

stripes,

^{which can}

eventually

^form

loops by joining

two transverse lines

by

^two

strong longitudinal

defects. When

cooling

^the

specimen

from the nematic

phase,

disclination

loops

often subsist and

play

^{such a}

role :

they

delimitate

regions

^of

opposite

chiralities.

Acknowledgments.

^{- We wish to} thank Profes-

sor Sir Charles Frank and Dr. Y.

Bouligand

^for

stimulating

discussions.

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[3] DEMUS, D., Krist. Tech. 10 (1975) ^933.

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[5] KLÉMAN, M., Points, lignes, parois ^dans les fluides anisotropes

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[6] ^DE GENNES, ^P. G., ^The Physics of Liquid Crystals (Oxford University Press, Oxford) ^1974.

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[8] URBACH, W., BOIX, ^{M. and} GUYON, E., Appl. Phys. ^{Lett. 25} (1974) 479.

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