Developable domains and commensurate structures in the twist grain boundary phases

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Developable domains and commensurate structures in the twist grain boundary phases

Yves Galerne

To cite this version:

Yves Galerne. Developable domains and commensurate structures in the twist grain boundary phases.

Journal de Physique II, EDP Sciences, 1994, 4 (10), pp.1699-1711. �10.1051/jp2:1994226�. �jpa-

00248071�

Classification Fhi.nil Ab.itiait.I

61.30 61.70

Developable domains and commensurate structures in the twist

grain boundary phases

Yves Galeme

Institut de Physique et Chimie de~ Matdriaux de Stra;bourg(*), Groupe des Matdnaux

Organique;.

23 _rue du Loe~s, B-P. 20 CR. 67037

Strasbourg

Cedex. France

jRe( en,ed ?~ Maich /994, _i-Pi en.e</ _m

finul

foiJ1i 4./lily 1994, aiiepte</ ^I-I ./ill_I /994)

Rdsumd. Le; phase~ ^TGB 16

joint~

de grains tordus) rdcemment ddcouvertes, et qui ^,ont

intermddiaires entre le~ phases chole~tdriques et smectiques, ont de~ propndtd~ phy~ique~

surprenantes. ^On peut par exemple ob~erver h la foi; le~ _textures typiques de; chole;tdrique~ et des

~tructure~ colomnaire~ dans le mime dchantillon TGBA, tandi~ que par ailleur,, de~ dchantillon~

de la phase TGBC peuvent prdsenter des

propridtd;

de commensurabilitd. Nous analy,ons ici _ce~

propridtds paradoxales

_en fonction de la

rigiditd

mdcanique a~socide h l'angle de rotation des bloc~

~mectiques. ^Nous en dddui;on; _une cla~sification

schdmatique

dan, laquelle )es phases TGBA

con~truisent de~ domaine~ ddveloppable~ mai~ _wnt incommen~urable~, et dans laquelle _au

contraire [es

phases

TGBC font des

conique~

focales et prdsentent une commensurabilitd d'ordre entier _avec

peut-dtre

des intermittences lifes h

l'dpais~eur

de l'dchantillon.

Abstract. The recently di,covered twist

grain boundary

(TGB) pha~es, intermediate between the chole~teric and the smectic phases, have surprising phy~ical properties. Both typical texture~ of the chole~tencs and columnar structures can be ob~erved _in the _same TGBA ~ample, while _on the other hand, TGBC sample~ exhibit _a commen~urate behavior. Here, _we analyse the~e paradoxical

propertie~

in terms of the mechanical rigidity aswciated with the rotation angle of the ~mectic

slabs. We deduce _a

simple

cla~sification in which the TGBA phases es~entially ^build up columnar

textures with

developable

domain, but _are incommen;urate, and in which conversely the TGBC

phases make focal _conic~ and show

a commensurate behavior of integer order, with

perhaps

intermittencies depending _on the ~ample thickne~s.

More than twenty years ago,

noticing

the formal

analogy

which exi~ts between

~uperconduc-

tors and smectic A

liquid crystals,

de Gennes

predicted

a

liquid crystal phase

similar _to the Abrikosov

(type-II) superconductors

Ii- In this

phase,

the twist distortion is

expelled

out of the smectic _structure in the _{same way as} the

magnetic

field is

expelled

^from the

superconduc-

tors, I-e-

by forming

a lattice of defect lines. Renn and

Lubensky [2]

then

proposed

the twist

grain boundary (TGB)

model for this

phase,

where (he lattice of defect lines is _an array of

(~) Unitd Mixte 380046-CNRS-ULP-EHICS.

1700 JOURNAL DE PHYSIQUE II N° 10

parallel

screw dislocation lines which condense the twist distortion

periodically

in

grain

boundaries and _rotate the smectic structure, slab

by

slab

(Fig.

). A few years

later,

the model is extended _to the tilted smectic

phases, distinguishing

the TGBA

phase

where the molecules

are

statistically

normal to the smectic

layers

^{from the TGBC}

phase

where the molecules _are tilted inside the

layers [3].

Both the TGBA

[4]

and TGBC

[5] phases

have been discovered

experimentally. They

demonstrate the

paradoxical properties

of a twist structure, with cholesteric colors,

containing

smectic

layers

as tested

by X-ray

diffraction.

Recently,

new

surprising physical properties

have been announced in the TGB

phases. Developable

domains

typical

of the columnar

systems ^{have been}

optically

observed in the TGBA

phase

of

1-[4-(n-hexadecyloxy)phenylcar- bonyloxy)-4-phenyl]-2-[4-(2-jS)-octyloxysulfinylphenyl)]ethyne [6],

which _at first

sight

_seems

to be

impossible

in _a

layered phase

^and a little bit earlier, commensurate structures have been

shown

by

means of

X-ray

diffraction measurements in well

aligned

TGBC

samples

of 3-

fluoro-4-[1-methylheptyloxy]-4'-[4"-alkoxy-2", 3"-difluorobenzoyloxy]tolane [7]. Though

the

commensurate

phases

are classical and

easily

understood when

they

_mean a rational ratio

between _two

lengths

^of about the molecular

size,

because

they just

need then _a lock-in

coupling

in the range of the molecular

interactions, they

are not trivial at all when

they

involve

lengths

several orders of

magnitude larger.

In the _case of reference [7], ^the two

lengths

concerned _are the helicoidal

period along

^the

z-axis, Ao

^l ~m, and the thickness of the

smectic slabs

f~

loo _nm, which both _are much

larger

than the molecules. The

explanation

of the

commensurability

is therefore _a

really tricky problem

even when the ferroelectric

polarizations

of the TGBC

phase

are taken into _account.

Here _we propose ^to discuss the

developable

domains and the commensurate structures

observed in the TGB

phases,

in _terms of mechanical

properties.

The mechanibal

properties

of the TGB

phases

are

essentially govemed by

^the elastic constants associated with the four

typical lengths

involved in these

phases

: d the

layer

thickness, Ao ^{the helicoidal}

period, f~

the thickness of the smectic slabs and

f~

the distance between the dislocation lines in the

grain

boundaries. The first _two distances and their

corresponding

elasticities~ _are classical and in well separate ranges, the

layer

thickness d

being

mubh stiffer than the helicoidal

period

Ao,

They

_are thus unable _to interfere. The other _two distances _are of the _same order of

Screw Grain Smectic

dislocations boundaries layers

Fig. I. General _view of the TGB structure (from Ref. (4)]. The helicoidal axis is along the z-axis.

magnitude

and their associated elasticities do not have definite relative values, _so that _one could observe that in _some TGB

phases, f~

^{is softer than}

f~

^{while in other} ones it is the _reverse.

In the

following,

we shall examine the two extreme cases, case

(a)

where

f~

^is a much _more

adjustable length

than

f~,

and

case16)

where. on contrary,

f~

^{is much stiffer than}

i~,

and almost

completely

^determined

by

the temperature ^{T. The} intermediate situations _are

discussed

by

^Barois

[8].

Developable

domains in TGBA

phases.

Let _us first examine the

physical origin

of the

developable

domains observed in the TGBA

phases. Generally,

the

developable

domains _are obtained in

phases

^{made of} hard and

infinitely long

^{columns of molecules}

organized

in close

packing

^order

[9].

The

continuity

of the columns

lead~ _to the conservation law

div pt ₌ ^o

('

where t is the unit _vector tangent to the columns and _p is their

density

measured in _{a transverse}

plane.

Because the columns are hard and

disposed

in _a

close-packing

order without hole~

between them,

they

have _a constant

density

_p, and

equation

(I) reduces to

div t ₌ 0

(2)

This _means that the columnar systems cannot

undergo splay

deformations. Because of close-

packing,

_moreover~ the columns are

organized

in _a definite

lattice, hexagonal

or centered

rectangular,

and

they

are

perpendicular

to common

planes,

so that

they

cannot twist around t

[9],

I-e-

t curl t ₌ 0

(3)

The above _two

equations (2)

and

(3)

~how that the bend is the

only

distortion allowed in the

columnar structures.

They

are characteristic of the

developable

domains

[10].

As

recently

discussed

[6],

the TGBA

phases

_may be _{seen as}

built-up

of microcolumns carved in the smectic

planes by

^the screw dislocations

(Fig. 2).

These microcolumn~ which

only

exist in the

grain boundary regions,

_merge back in

layers

inside the smectic slabs.

They

are therefore discontinuous. Let _us first look at their

physical properties

in the _ca~e of

hypothesis

la) where the

length f~

is

supposed

to be much stiffer than

f~.

The

layer spacing

d

being

also

quite

stiff the twist

angle

AH

=

2 sin~ ^'

(d/2 f~)

is _{a constant}

quantity independent

of the stresses

acting

in the

sample.

The lateral dimensions of the microcolumns,

(llcos

(~6/2)

and

f~,

are therefore well defined while their

length ~f~/2

_may

easily

be

elongated.

The

density

of the microcolumn~ inside of the

grain boundary

_p

= cos (A 6/2

)Ii,]

d, then remains

equal

to a constant

independently

of the _stresses.

,

The microcolumns _are contained inside the smectic

layers

of both slabs before and behind the

grain boundary.

The unit vector t tangent to them is therefore

perpendicular

to the smectic

directors of the _two slabs, _nj and _n~. It is written _as

t ₌ in x n~)/sin AH

(4)

This _vector, which is defined

discretely,

_may be

generalized

into _a continuous field.

Assuming

the

derivability

of this _vector field, _we get :

div t ₌

(n,

curl _{n n} curl

n~)

bin he

,

1702 JOURNAL DE PHYSIQUE II N° lo

Fig. 2. ^-

The

in the (x, y)-plane of a grain

boundary. The full and

lines represent the layers of _the smectic slabs _before and behind the _grain boundary,

respectively. The mixed _{dashed lines}

sketch the screw

dislocations which cut the layers in microcolumns.

which shows, since curl n=0 in the smectic

phase,

that

equation (2)

is satisfied:

div _t

=

0. Let _{us now} note that the

grain

boundaries _are

locally plane

and

perpendicular

to the helicoidal axis, and therefore to the microcolumns

(Fig.

2). This condition which

corresponds

to the minimum of the

grain boundary

surfaces, I-e- _to the minimum of their energy, indicates that the circulation of the _vector t

along

a closed

path

contained inside the

grain boundaries,

is null. It results that

equation (3)

is also satisfied, and therefore that the t-field of the TGBA

phases verifying hypothesis (a),

^is

organized

in

dei'elopable

domaiits.

This is _a

surprising

consequence for _a system made of microcolumns. However, the microcolumns _{are not}

independent

and

they

_can be associated from smectic slab _to smectic slab in

infinitely long

columns.

Figure

3 shows the microcolumns in _two successive

grain

boundaries observed

along

the z-axis, I-e- in _a

plane parallel

to the

diagonal

axes

xj and x~ of their

respective rectangular

lattices. The front microcolumns

(open

circlesl _are

rotated around the z-axis

(cross) by

^{AH relative} to the back _ones

(dot~),

the _rows of

microcolumns

parallel

to the

xj-axi~ becoming

^then

parallel

to the

x~-axis.

In this way, the microcolumns

along

the

x~-axis

coincide in both

grain boundaries,

while the other rows

X, o

~

o

,

o

o

o ^.

.

~

o

o ^.

o ^. .

~

o _x

Fig.

3. Slice~ of microcolumns _{in two} successive grain boundaries, observed along the,z-axis. The microcolumns _are sketched _as open circles when they belong to the first grain boundary, and _as dots when

they

_are in the second _one. The _two

neighboring

slice~ of microcolumns _are rotated by AH from _one another around the z-axi,, marked by _{a cross} j+) in the figure. In this rotation, the _{xj-axis comes} along the

xj-axi~.

It result, that the columns located _on the xj-axi~ coincide from _one dice to the other _one. The other rows of column;, parallel to the _{xj -axi~,} become parallel to the x~-axi~ ^{after the} rotation, keeping the

same di~tance from the z-axi~. These column~ therefore do not coincide from _one slice to the other _one.

but appear to be tranAated along _x~ by _a vector smaller than the intercolumnar distance.

parallel

to the

x~-axis

_are

just

shifted from _one another

by

_a translation

along

the

x~-axis

with _an

amplitude

smaller than the intercolumnar distance. We may thus consider that

the microcolumns _are

joined together

and form continuous columns

undergoing

small

deviations inside the smectic slabs. On the

whole,

the column~ appear then _to be

slightly

twisted around the _z-axis. However, the

position

of the z-axis is not determined _{in a}

unique

manner in the TGBA

phases

and _can

change

^from one

grain boundary

to the other _one. The

twisting

of the columns _as described above, is therefore somewhat

arbitrary.

Nevertheless, this

picture

is useful _to show that continuous columns may be tracked in the TGBA

phase

^{in the}

case of the

hypothesis (a),

and

helps

understand the

phy~ical origin

of the

developable

domains observed.

Inside the

developable

surfaces and their immediate

vicinity,

I-e- inside the _{cores, or} the

« eyes _», of the

developable domains,

the columnar

ordering

cannot be achieved

[9].

Most

probably~

in the _case of the TGBA

phase,

these small

regions

are restricted to one smectic slab

only.

A little bit farther around the eyes, the columns _are

strongly

bent.

They

therefore

undergo large elongations

that the softnels of A and

f~

cannot

yield completely. Edge-dislocation

lines have then to be introduced in both the helicoidal and smectic slab _structures

(Fig. 4),

each _one of the

edge-dislocations

in the helicoidal structure, called

X-lines~ being

associated to

Ajf~ edge-dislocations

ⁱⁿ the smectic slab

ordering~ geometrically embracing

them. The

density

of the

x-lines

in the curved

regions,

IA

o R~

essentially depends

on R the local radius of curvature, I-e- the distance _to the

corresponding

_eye. Their energy

density

_per volume unit _w>, is of the order of

w>

K/A,j

R

(5)

where K is the twist elastic _constant of the TGBA

phase

around the z-axi~. The

edge-

dislocations in the lattice of the smectic slabs

correspond

to the

suppression

of

grain

boundaries,

and of the _screw dislocations that

they

contain.

They

^therefore have _an energy

density

smaller than the elastic energy which has

produced

them,

Kq(,

where qjj=

2

r/Ag.

The energy of these

Ao/f~ edge-dislocation

lines associated with _one

X-line

is thus smaller than

Kq( _A~(~

K, and may be

integrated

in the

expression

(5) ^without

changing

its order of

magnitude.

So~ the energy

density

of the

edge-dislocations

^{involved in the} curved

regions

is much smaller than the twist energy

density

of the structure itself Kq(j~ as soon a~

/,

_', ^'

'

~ '

, '

' ' i

'~ '

i /

~i

/ /

' '

'

/ /

/ j /~

/~ /~

-j

~-1

(

i l' _/

) ' ~ '

i

' '

1'

~

" '

-- ' l'

1704 JOURNAL DE

PHYSIQUE

II N° 10

R >

Ao.

^This energy may therefore be

neglected,

which shows that~ in _case (a), the TGBA

structure is able to bend without

difficulty

and _to build up

developable

domains.

Let _{us now}

quickly

take the

opposite hypothesis, (b)

^{instead of}

(al.

Since in the

hypothesis (b), f~

can vary more

easily

than

f~,

the twist

angle

AH and the

density

of the microcolumns _p

are no

longer

fixed

quantities.

It results in

particular

from

equation (4),

that div t # 0, and therefore that the t-field cannot be

organized

in

developable

domains. In

fact,

since in this

hypothesis

the smectic slabs have _a definite thickness

i~, they

constitute

parallel layers

ⁱⁿ the system ^{which should therefore build}

focal

canics.

Commensurability

in TGBC

phases.

We _now examine the

possibility

of commensurate structures in

well-aligned samples

of the

TGBC

phases

_as

experimentally

studied with

X-ray

diffraction

by

Navailles _et al.

[71.

The

sample

has _an

approximately

uniform thickness D and its

glass plates

are treated in such _a way that the z-axis of the TGBC _structure is

perpendicular

to them. This orientation is obtained with

a

planar anchoring

of the molecules _onto the

glass plates,

rubbed

along

the x-axis to

impose

the direction of the molecules. It results that the

sample

contains _an

integer

number _fi of helicoidal

half-pitches

A/2

(Fig. 5)

D

=

nA/2.

(6)

Naturally,

the

pitch

A measured in such _a

sample

is

slightly

different from the undistorted

pitch

A~~(T) ^{that could be measured in} a free

sample.

In order to minimize the elastic energy of the

distorsion, A ^' is _as close _as

possible

to

A~j(T)~

^' and therefore _n is

given by

:

~~ ~~~

/jT)

~ ~~~

If the temperature T is

changed,

or if the

sample

thickness D is _not

really

uniform, the number of

half-pitches

also

changes jumping by integer

values. This is achieved in the

sample by

means of

simple

dislocation lines in the helicoidal order~ I-e-

by

_means of

X-lines [I Ii

of

strength

1/2

[12].

The observation of

X-lines

of

strength

1/2 is natural in the TGBA

phase

z

x

Fig.

5.-Cut of the

sample

_in the ix, z)-plane. The dashed line; parallel to the

plates

sketch the helicoidal

half-pitches

of the TGB _structure.

where the

halfpitch

A/2 is the

period

of the system. ^{It is less evident in the TGBC}

phase

because there, the tilt of the molecules which _occurs in _a

plane containing

the helicoidal _z

axis,

makes the

period

to become

equal

to A. The observation of

X-lines

of

strength

1/2 _may however be understood if _one

imagines

that the

sign

of the

anchoring pretilt

^is

changed

_on one

glass plate

at the level of the

X-lines (Fig. 6a).

The molecules around these

places

where the

anchoring

tilt

changes, continuously join

the other _ones

through

rotations

along

the smectic-C

cone,

keeping

a constant tilt

angle

inside their smectic

layers. They

become thus

locally

perpendicular

to the helicoidal axis in the

vicinity

of the

X-lines

_so that, except ^for the

angular

shift of the molecules referred _to their smectics

layers,

the situation is the _{same as} in the TGBA

phase (Fig. 6b).

The

X-lines

of

strength

1/2 _can therefore also exist in the TGBC

phase. They

just

need in addition the energy associated with the rotation of the molecules

along

the

smectic C _cone, which is

composed

of the

coupling

_energy of the molecular tilt direction with the helicoidal

axis,

and of the nematic-like elastic energy of the distortion. Both these

energies

are weaker than the association energy of _two

X-lines

of

strength

1/2 into _one of

strength

I,

since

they

are not observed to associate

effectively.

One could _now try to minimize the

distorted volume and

consequently

its energy,

by driving

the

X-lines

_nearer to the upper

plate (Fig. 6).

This

displacement

of the

X-lines

should however be limited because of the

repulsive

interactions of the

X-lines

with their

images through

the

plates~

which arise from the local

compressions

^of the cholesteric

layers.

On the whole, the

X-lines

of

strength

1/2 should find their

equilibrium position

^{somewhere between the middle of the}

sample

and the

plate

where the

anchoring pretilt

is reversed.

The

glass plates

of the

sample

do not

only impose

the molecular

anchoring

which leads _to the

quantization

of the helicoidal

half-pitches~ they

also mark the limits of the smectic slabs.

The

sample

^{thus contains} an

integer

^number _n~ ^{of smectic slabs}

D

= n~

i~. ₍₈₎

It results that the thickness of the smectic slabs

f~

^{is modified from its ideal} value

f~(T),

and that their number in the

sample,

_n~, ^like the helicoidal

1/2-pitches,

is

quantified.

n~ can thus

change by integer

values

only.

These

changes

are marked

by edge-dislocation

lines which _can be observed

optically although they

are much

lighter

than the

X-lines [13].

z

z

X x

a) b)

Fig. 6. Molecular organization of the TGBC pha~e around _a X-line of strength 1/2~ parallel to the _y- axi~. a) General _view. b) In the

vicinity

of the X-line.

1706 JOURNAL DE PHYSIQUE II N° lo

From

equations (6)

and

(8),

we

immediately

deduce that the ratio

r of the helicoidal

1/2-pitch

over the thickness of the smectic slabs, is _a rational number

I =

/2

f~ ii~fii

,

(9)

and therefore that~ because of the

anchoring

conditions, the _two

lengths

A and

f~

are

systematically

commensurate.

In fact~ the results of the

X-ray

measurements are more

~pecific

than that

[7]. They yield only integer

values for _i~ which in other words, ^{indicates that} the

commensurability

is of

integeJ.

order. In order to

analyze

this

surprising

result, _we consider

again

the two extreme

cases for the elastic

properties

of the TGB

phases.

We

begin

first

by taking hypothesis (b)

in

which the thickness of the smectic slabs

f~

^{is considered} to be hard while the

angle

between

neighboring

slabs AH is soft. It results from this

hypothesis

that the number of the smectic slabs n~ will

adjust

close to its ideal value

D/f~(T),

and in _a similar _{manner as} the helicoidal

pitch,

will be

given approximately by

~~ ~~~

f~(T)

^~ 2 ~~~~

So,

in

hypothesis

(b)~ continuous

changes

of the temperature or of the

sample

thickne~s make n~

jump by

^one-unit steps

only.

These

jumps

are materialized in the

sample by edge-dislocation

lines of

Burgers

vector

equal

to

unity, bordering regions

^with a constant number of smectic slabs n~. n~ thus increases

gradually

in the

sample,

in _a similar way as n

(Fig. 7a),

with the result that the ratio _r varies in _a broken staircase _manner

(Fig. 7b).

More

precisely,

_I varies

between the _two limits

(D/f~

1/2 II (2 D/A

o +

1/2)

and

(D/f~

₊

1/2)/ (2

D/A

ii

1/2), taking nq

x

n

n+I

a)

r

x bj

Fig. 7. a) Schematic top ^{view of} a TGB sample oriented _as indicated in the _text. Its thicknes~ D increa~es

continuously

from the left _to the

right along

the x-axis. It results that the number of helicoidal

half-pitches

goes from _fi to n + when pa~sed _a X-line (thick[ and that the number of the smectic slabs (indicated at the top ^{of the}

figure)

increases

by

_one unit each time that

an

edge-dislocation

line (thin) is cros~ed. In the hatched

regions,

the ratio _r is integer (= _q). b) Variations of the ratio _i

along

the x-axis.

The ratio _r is integer (= q) ⁱⁿ some steps ^{of the broken} staircase.

constant values within the interval Ar

=

n~/n~

centered at

Ag/2 f~

and

independent

of D.

Among

the steps ^{in this}

staircase,

_some

correspond

to an

integer

value _q of _r. These

integer

steps ^{do exist}

effectively,

_on both sides of the

X-lines~ provided

that the

jumps

Ar at the

X-lines

are

larger

than I, and that the variations of the

sample

thickness AD which drive the

I-

variations~ _are

larger

than

Ao/2.

The last condition is

easily

fulfilled since D is

usually

not

defined _to better than _~m and that Ao l ~m. The other and _more

interesting condition,

Arm I, is

equivalent

to

n~ m

n~

,

or

Ajm4Df~. _(ll)

If this condition is satisfied, the broken staircase in

figure

7b

comprises

at least _an

integer

value. If not, there is

just

a

probability

that _an

integer

value _q falls inside the interval hi-. The

sample

then has in

general

lost its property of

commensurability

of

integer

order. If the

experimental

conditions _are

changed

from this situation, _e.g.

by modifying

the temperature, both

ii and

fb change,

so that _r may

again

catch an

integer

^{value inside its} Ar interval. In such

a

sample,

_one could therefore observe iiiieimittenc.ies of

commensurability

and incommen-

surability

of

integer

order.

Another consequence of

hypothesis 16)

is that AH, the rotation

angle

_per smectic slab, can

easily

be

adjusted

to fit the twist

imposed by

the

anchoring

conditions onto the

plates

_:

nr ₌ fi~ ^AH.

j12)

This condition~ which may also be written _as he rfi.~ shows that he is commensurate to _r.

The

X-ray

diffraction patterns

performed

in the (x~ y

)-plane

on a TGBC structure

satisfying hypothesis (b)

should therefore

display rings

of

equidistant Bragg

spots. ^In

general,

these

Bragg

spots are too numerou~ to be resolved.

They

_merge ^into a continuous

ring [14].

However, if the diffraction is

performed

in _a

region

of

integer

_r (= _q), the rotation per block,

he

=

~

~, _j13)

2 _q

only

leads _to ? q

Bragg

spots ^which are

easily

resolved since _{q is} small

experimentally

q, lo (see Ref.

[7]). Practically,

the

X-ray

beam cannot be focussed _{on a}

single region

of

integer

_r. Continuous diffraction

rings arising

from

regions

of other values of _r

necessarily superimpose

onto the 2 _q

Bragg spotsl

One should then observe the

Bragg

spots ^with a

relatively

_poor contrast. In fact, the situation is _a little bit _more involved because the

sample

should

adopt

the chevron _structure when cooled down into the TGBC

phase,

instead of the

bookshelf structure

depicted

in

figure

5. The mechanism for the formation of this _structure should be the _{same as} in the well-known smectic C and smectic C *

phases.

The

thinning

of the smectic

layers

on

cooling

induces the

buckling

of the

layers

which break about the middle of the

sample, leading

to the chevron-like _structure. The helicoidal z-axis in

figure

5 is therefore

broken and tilted _on both sides

by

the chevron

angle.

This tilt

angle,

and

particularly

its azimuthal direction, _{are not} well-defined

quantities. They

can vary from

place

to

place

in the

sample.

The diffraction rings

produced by

the different

regions

of the

sample

are therefore

differently

tilted in the

reciprocal

_space, and _{are not}

exactly superimposed.

The contrast of the 2 _q

Bragg

spots in their

reciprocal plane,

should

consequently

be

larger

in _a chevron

sample

than in _a bookshelf

sample.

This mechanism _can

easily explain

the

large

contrast which is

observed

experimentally,

^without

invoking

_any ml.iiisic

commensurability

_as in refer-

ence [15].

1708 JOURNAL DE PHYSIQUE II N° lo

We may now examine the

experimental

results

given

in reference

[7]

in more detail. With _an

average

pitch

A~ ^l.5 ~m, q =

Ao/2 f~

lo, and _a

sample

^{thickness D, lo} ~m, the condition (I I) for

regions

with

integer

ratios _{q to}

systematically

exist within the

sample

is

approximately

^{satisfied. The} commensurate behavior is thus

effectively

^observed on the

X-ray

diffraction patterns ^with

q-jumps

from _an

integer

value to the next one, driven

by

the

continuous drift of

An/2f~

with temperature. ^These

q-jumps

however _occur with _some

hysteresis mainly arising

from the

systematic delay

of _on the free helicoidal

pitch An(T),

and related to the

delayed

motion of the

X-lines

when

changing

temperature. ^Such a

delay

in the motion of

X-lines

is classical in

liquid crystal phases

such _as the cholesteric _or smectic C *

phases.

It is the consequence of the weak tension _or energy per unit

length

of the _X- lines, unsufficient _to free them

completely

from their

anchoring points.

This effect could be

reinforced in the TGBC

phases

which _seem to be

particularly

viscous

[I?].

We _now

briefly analyse

the consequences in the _case of

hypothesis (a).

In this _case,

f~

can

easily change

under _an

applied

stress

contrarily

to AH which

keeps

^close to its

unperturbed

value A

6~~jT).

The conditions _on the direction of the molecules at the surfaces, are

the _{same as} in the

previous hypothesis,

so that both

equation II ?)

which, results from the molecular

anchoring

onto the

plates,

^and

equation (6),

which expresses the

quantification

of the helicoidal

pitch

inside the

~ample,

^{remain valid. A~ for the conditions} on the smectic slabs,

they

now result from the

rigidity

of AH, and

equation (10)

has to be

replaced using

equation (12), by

the

approximation

n~ =

Jnt

i

+

(14)

A°o

2

This

equation

shows that each

X-line

in a

sample

of variable thickness, generates about

r/A6jj simple

dislocation-lines in the

ordering

of the smectic slabs, _or

equivalently

_one

dislocation-line of

Burgers

vector

equal

to

rd/A6~~,

and that

they

are connected and

superimposed,

_so that the ratio _I= r/AH remains _{constant to} within _a small interval hr.

Using equation II

2) and

taking

into _account the fact that n~ is determined _to within _one unit

through equation

(14), _we see that Aili

= A (AH

)/~6

=

An~/n~

₌

I/fi~,

and therefore that the interval _Ar

= In AID _« I. The chance that _I falls _{on an}

integer

value is thus small _now. We may conclude that in the _case of

hypothesis (a),

the

sample

does _not in

general

exhibit any

commensurate behavior of

integer

order.

Conclusions.

We may

finally

draw _a

simple panel

^for the

macroscopic properties

of the TGB

phases.

according

to the relative

rigidities

of the intermediate distances

f~

and

f~

characteristic of these

phases.

In _case

(al

where

f~

is soft

compared

to

f~,

the

lengths

A and

f~

remain

independent

of

one another, and do _not exhibit any commensurate behavior of

integer

order, ^{but the} structure

is then made of microcolumns which have the remarkable static property ^of

building

up

textures with

developable

domains. Let _{us stress} here that the microcolumn~ do not

yield

all the

hydrodynamical properties usually

found in the columnar systems. to the TGB

phases.

For instance.

applied

shear _stresses

produce

flows

perpendicular

to the microcolumns instead of

parallel

to them

[6].

This

paradoxical

behavior arises from the

grain

boundaries which _are

perpendicular

to the columns and, since

they

cut them~ form _easy

gliding planes

in the structure.

In the _{reverse case}

16)

where

f~

is hard

compared

to

f~,

the TGB structures _{can no}

longer

be

considered _as built up of microcolumns.

They

_are made of

parallel

smectic slabs. and their

textures form focal conics. On the other hand, the well-oriented

samples

^{of thickness smaller}

than

AIM

_f~~ exhibit _a commensurate behavior of

integer

order

arising

from the

anchoring

conditions. If the thickness of the

samples

is

larger

than this limit, the

commensurability

of

integer

order appears

only

with intermittencies driven

by

the

experimental

conditions of

temperature

^and pressure.

One may now ask how the

lengths f~

and

f~,

which have been measured to

belong

to the same range of

magnitude

in all the TGB

phases,

_can have

rigidities

different

enough

to lead to the _so different _cases (a) and

(b)

discussed above. The

compression

coefficient B~ ^{and the}

z

Fig.

8. -Cut _view of bent smectic dabs in the local plane defined by the helicoidal _z-axis and the screw-dislocations (full _wavy line~). The smectic plane~ (thin straight lines) remain parallel to _z during the deformation of the slabs.

~ ~

~~lf

j

~

⁺

~

6~ h~

+

/

ai bi

Fig.

9. Schematic repre,entation of the molecular orientations in neighboring smectic slabs of the TGBC phase. ^{The circles of radiu~} ~ ~ketch the

generatrice~

of the ~mectic C _cone~ in the neighboring

;labs (above ^{and below the} horizontal line re~pectivelyi ^the actual molecular orientations being denoted

by

dots. The _crosses at the _center of the circles _are the normals _to the _smectic layers in the corresponding slabs. a) At rest, the molecule~

are tilted in the plane

containing

the helicoidal z-axi~ and the normal _to the smectic layers. ^{The normals} to the layers in the _two

adjacent

slabs make the angle AH. b)

Angular

stres,e~

applied to the sample _can change the angle between the normal~ _to the layers in the two neighboring ^slabs up to A0~jj while keeping the angle between the plane~ of the z-axis and the molecules equal _to A0~ on rotating the molecules along the smectic C _{cones in} opposite directions.

17 lo JOURNAL DE PHYSIQUE II N° lo

curvature

elasticity

coefficient

K~

of the smectic slabs _are related to

f~ by

the relation

f~ ~~, _typical

_{of the}

_layered

_{systems. We}

may consider

(Fig. 8)

that the

bending

of the B~

l~

smectic slabs is

only weakly coupled

to the

splay

of the smectic

layers,

unlike what is

proposed

in reference

[16],

and therefore that the _curvature energy of the smectic slabs

mainly

arises from the increased

length

^of the screw-dislocation

lines,

I. _e. from the _excess disordered

volume that

they

_generate when bent. The

compression

coefficient B~ ^{arises from the}

repulsion

interaction which exists between dislocation lines of the _same nature and

sign.

Both the elastic coefficients,

K~

and

B~,

thus result from intrinsic

properties

of the screw-dislocation lines.

They

should therefore be rather similar in all the TGB

phases,

as is

experimentally

confirmed from the similar measurements obtained for

f~.

So, the

changes

in the relative

rigidities

which

are needed to lead to the different _cases

(a)

and

(b)

should

mainly

arise from the

rigidity

associated with

f~,

_or

equivalently

with AH. We may notice here that AH should be much softer in the TGBC

phases

than in the TGBA

phases,

because of the tilt

angle

~ of the molecules

inside the smectic

layers

which adds its

elasticity

in series _to the

elasticity

^{associated with} the rotation

angle

AH of the smectic slabs. More

precisely,

the molecules _on both sides of the

grain

boundaries _{can rotate} in

opposite

directions

along

the smectic C _cone.

They

thus leave their

preferred

orientation in the

plane

of the helicoidal z-axis and of the normal to the smectic

layers~

to increase the

angle

AH that

they naturally

make in _two

adjacent

slabs

(Fig. 9a).

_{up to}

A0~jj (Fig.

9b). This mechanism,

naturally

restricted

by

the condition

(AH~jj-AH(

~2~, (15)

suggests ^{that the TGBA}

phases

and TGBC

phases

with small tilt

angles,

should

mainly correspond

to case (a), and the TGBC

phases

with

large

tilt

angles

to case

(b).

This conclusion is

clearly

consistent with the observations

reported

in references

[6]

and

[7].

Let us also notice

that the

proposed

mechanism of molecular rotations

along

the smectic C _cone

already

evoked

above _to

explain

the existence of the

X-lines

of

strength

1/2 in the TGBC

phases,

is

corroborated with the observations

by

Isaert

[12]

that the

edge-dislocations

in the smectic slab

ordering

_{are more} visible in the TGBC

pha~es

than in the TGBA _ones. These observations may be

explained

in the

following

manner. The stresses

produced by

the

edge-dislocation

lines

apply

not

only

to the slab thickness

f~,

but also _onto the

angle

AH

through equation

(12)~ so

that the molecules around the lines _are rotated

along

the smectic C _cone, and get closer to the (x,

y)-plane (Fig.

9b). The

birefringence

is thus

locally

^increased and the

optical

indices

modified.

making

the

edge-dislocation

lines _more visible in the TGBC

phases

than in the

TGBA

phases

where such _a rotation mechanism _cannot exist.

Let _us notice _to conclude that the classification

proposed

here for the

macroscopic properties

of the TGB

phases

is very schematic. However _we think that it _can stimulate

experiments

to

test the

validity

and

generality

of the above discussions. The intermediate situations which would

probably

arise then, should be

particularly interesting

to

~tudy.

References

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[3] Renn S. R. and Lubensky T. C.~ Mol Ci_i,<t Liq Ci_i<u. ²⁰⁹ (1991) 349 Renn S. R. and Lubensky T. C.. Flit<.< Ret-. A 45 (1992) 953.

j4] Goodby J. W.~

Waugh

M. A., Stein S. M., Chin E.. Pindak R. and Patel J. S.. Nature (London) 337 (1989) 449

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M. A., Stein S. M., Chin E.. Pindak R. and Patel J. S.. /. Am. Chem Sac 11

(1989) 81 19.

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Twieg

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[6] Ribeiro A. C., Dreyer A.~ Oswald L., Nicoud J.-F., Soldera A.~ Guillon D. and Galerne Y.,

6~Colloque

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Physics

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Liquid Crystals

(Clarendon, Oxford, 1993).

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II 3] Isaert N.. Girold C., Nguyen H. T., Navailles L. and Barois P., Fourth International Conference _on Ferroelectric Liquid Crystals, Tokyo, Japan (1993).

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