X-ray scattering by edge-dislocations in the SA phase of mesomorphic side chain polyacrylates

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X-ray scattering by edge-dislocations in the SA phase of mesomorphic side chain polyacrylates

P. Davidson, B. Pansu, A. Levelut, L. Strzelecki

To cite this version:

P. Davidson, B. Pansu, A. Levelut, L. Strzelecki. X-ray scattering by edge-dislocations in the SA

phase of mesomorphic side chain polyacrylates. Journal de Physique II, EDP Sciences, 1991, 1 (1),

pp.61-74. �10.1051/jp2:1991139�. �jpa-00247500�

Classification

Physics

Abstracts

61.30 61.40 61.70

X-ray scattering by edge-dislocations in the SA phase of

mesomorphic. side chain polyacrylates

P.

Davidson,

B.

Pansu,

A:M. Levelut and L. Strzelecki

Laboratoire de

Physique

des

Solides(*),

Bitiments10, Universitd Paris-Sud,

914050rsay,

France

(Received 30May

1990,

accepted13

September 1990)

Rksumk. Les dichks de diffraction des rayons X par des

polymdres mdsomorphes

_en

peigne,

_en

phase

SA. Prksentent des trainkes diffuses _en forrne d'« ailes de

papillon

_». Nous montrons que

cette diffusion diffuse peut

s'expliquer

_par la prksence de dislocations-coin- En partant ^{de la}

description

des dislocations-coin donnde par De Gennes dans le cadre de la th60rie du continuum

dlastique

de la

phase

SA, nous avons calculd la transforrnde de Fourier du

champ

de dkformation.

Des expdriences de diffraction optique _sur des moddles de ddfauts ont aussi ktk effectudes afin de

reproduire

les dichks de diffraction des _rayons X. Los deux mkthodes _{montrent que cette} diffusion diffuse peut en effet bien

s'expliquer

_par la

prksence

de dislocations-coin. Leur densitk _a ktk

grossidrement

estimke I

quelques

108/cm2. La taille de leurs _{cmurs ne} devrait _pas

dkpasser quelques Angstr6ms. _D'aprds

l'allure du champ de ddformation

dlastique,

_on peut tirer _une longueur

typique

A ₌

(K/B)~'~=l,5A,

ce qui montre que la constante

klastique

B de

compression

des couches devrait dtre environ 100 fois

plus

klevke _en

phase

_SA

polymkrique

_» qu'en

phase

_{SA «} usuelle _»,

Abstract. The X-ray diffraction patterns ^of

mesomorphic

side chain polymers in the SA Phase

present diffuse streaks in

shape

of _« butterfly

wings

_». We show that this diffuse scattering may be due to the presence of edge dislocations. On the basis of _a previous

description

of edge dislocations within the framework of the elastic continuum

theory

of the SA

fihase given by

De Gennes, _we have calculated the Fourier transform of the deformation field.

Optical

diffraction

experiments

on sketches of defects have also been made to reproduce the X-ray scattering patterns. ^{Both methods show that this} diffuse

scattering

_may indeed be due to the presence of

edge

dislocations, Their

density

_may be

roughly

estimated to _some 108/cm~. ^{The size of their}

cores should be

only

a few

Angstr6ms,

From the

decay

of their elastic deformation field, _a

typical

length A

=

(K/B)~'~m1.5A

can be obtained which shows that the elastic constant B of compression of the layers should be about two- orders of magnitude

larger

in the

polymeric

SA

Phase

than in the conventional _one.

Inwoducfion.

The existence of

edge

dislocations in the SA

phase

of

ordinary

calamitic

liquid crystals

has

been known for _a

long

time. On the basis of the elastic continuum

theory

of the

(*)

Associk _au CNRS.

JOURNAL DE PHYSIQUE II T I, M I, JANVIER 1991

SA

phase,

De Gennes

[1, 2]

and Kleman et al.

[3, 4]

^have

already given

_a detailed theoretical and

experimental description

of such

edge

dislocations.

However, although

these

objects

could be observed

by optical microscopy,

to _our

knowledge,

their effects _on the

X-ray

diffraction patterns ^{have not} been

reported yet.

More

recently, mesormorphic

side chain

polymers

_were

produced

either

by

radical

polymerization

of

mesogenic

_{monomers or}

by grafting mesogenic

_{cores on} commercial backbones

[5, 6].

^Such

compounds

often exhibit SA

phases

^which sometimes

display

localized defects. For

instance,

localized defects related to the

crossing

of the

sublayers

of

mesogenic

cores

by

the backbones have been detected

through'their

contribution to the

X-ray

diffraction

pattern [7].

^On the other

hand, edge

dislocations have

already

been

directly

observed in

closely

related systems

(mesomorphic

_« combined _»

polymers

which possess

mesogenic

_cores

both in the backbone and _as

side-chains) by

electron

microscopy through

^direct

imaging

of the

layers [8, 9].

^This _paper presents the

X-ray

diffraction

pattern

of _a

mesomorphic

side chain

polyacrylate

in the

S~ phase.

This

pattern displays

_some diffuse

scattering

that _we think _to be due to the presence of

edge

dislocations in this

phase.

Our

arguments

_are derived

fr6m

_both

optical

simulation

(section 2)

and direct

computation (section 3)

of the Fourier transform of the elastic deformation field of _an

edge

dislocation in _a smectic

phase given by

De Gennes

[I].

These

comparisons

will lead to _a discussion about the orders of

magnitude

of the different elastic _constants of the

polymeric

_SA

phase

and about the

density

and

origin

of such

edge

dislocations. '~

1.

Experimental.

The

polymer

studied

belongs

to a

hom610gous

series of the

following

chemical formula _:

)CH-CH~M

C=O

~~CH2hOCO~OCO-~CN

where _m

Represents

the

polymerization degrbe (=30).

These

polymirs

_{will be} called

P~ according

to the

number,

of

methylene

_groups in the spacer.

The

synthesis

and characterization

together

with _a structural

study

of the SA

-Phase

have

already been,presented

in _a

previous

_paper

[10]. Hereafter,

when

needed,

_we shall call the SA

phase displayed by

these

mesomorphic

side chain

polymers

_:

polymeric

SA

phase

in

contrast to the _« conventional SA

phase

of small mesogens.

Very

well

aligned Samples

of

polymer

_{P~ could} be obtained

by Slowly cooling

them

(5° C/hr)

from the

isotropic phase

into the SA

phasi

in _a

magnetic

field of 1.7 T.

By

_very well

aligned Samples,

_{we mean} that the

mosaicity

_was

comparable

to the resolution of _our Set-up.

Only fairly

well

aligned Samples

of

polymer Pi~

and P~ ^{could be}

produced

in this way and because of this

Slight disorientation,

the diffuse

Scattering

could not be observed _aS

clearly

_aS in the _case of

polymer _P~.

The

Samples

_were contained in Lindemann

capillary

tubes of1.5 _mm diameter.

The

polymorphism

of

polymer

_P~ iS the

following [10]

_: G15 °CSA l10

~CI,

where G stands for

glassy

smectic A state and I for

isotropic liquid.

A fixed film-fixed

sample X-ray apparatus already

described

[I Ii vias

used to

perform

the

X-ray

diffraction

experimentii

A

punctually

focussed monochromatic

X-ray

beam of

wavelength

A CuKa

=1.541h

was

produced by

reflection _{on a}

doubly

curved

pyrolytic

graphite

monochromator. The diffracted

X-rays

were collected _{on a}

cylindrical

film _{at a}

distance R

=

60 _mm from the

sample

which _was

placed

in _{an oven} heated

by

_an air stream.

The temperature control _was within _± I °C.

Figure

la shows the diffraction

pattern displayed by

_an

aligned sample

of

polymer

P~ at room temperature and

figure

16 shows the _same diffraction

pattern overexposed. Except

Fig.

la. X-ray diffraction pattern ^of

polymer

P8 in the SA Phase. (a) Bragg reflections _; (b) Wide

angle

diffuse crescents

(c)

_« butterfly

wings

_». H is the

magnetic

field direction. (One lst order

Bragg

reflection is missing because it falls in the beam stop.)

Fig,16.-Overexposed X-ray

diffraction pattern of

polymer

P8 in the SA

Phase

which makes the

butterfly

wings » clearly visible.

(2)

Fig,

lc. -

of

the X-ray

iffraction

attern. Bragg

reflections

;

(b)

for the usual

blurring

with

temperature,

this

X-ray

diffraction

pattern

^remains

mostly unchanged

in the whole temperature range of the SA

Phase. Figure

lc is _a schematic

representation

^of this

pattern.

^Three

types

^{of elements} can be detected in

figure

I _:

Along

the

meridian,

a series of resolution limited reflections

(a)

shows that the

sample

is in _a smectic

phase

with _a

periodicity equal

to 48

h.

Along

the

equator

and

perpendicular

to

it,

two wide

angle

diffuse crescents

(b)

show that the

layers

are

liquid-like

and that the

mesogenic

_{cores are}

perpendicular

to the

layers.

Therefore,

the smectic

phase

is of the

S~ type.

The third order of reflection _on the smectic

layers

is the strongest one and this reveals _a

peculiar

electron

density profile along

the normal to the

layers.

This

point

_was discussed at

length

in reference

[10].

The last elements

(c)

of the

X-ray

diffraction pattern are diffuse _areas

starting

from the 3rd order

Bragg

spots ^and

extending

in _a direction

parallel

to the equator. ^Rather strong ^close

o

~#

o

~

~ o

o

z' '

~ O

q~o ^o"

Q [

if

~x

~

o o

.

.

O

.

,

~

Fig.

2.-Microdensitometric

recording

of the

X-ray (iffraction

pattern, The numbers refer to the

isointensity

fines. (The four last _ones

(6-9)

_are

meaningless

because they are affected

by

the wide angle diffuse

crescent.)

_s~ and s~ are the components of the diffusion vectors

along

the _x and _z axes

respectively. (a) Bragg

reflections

(b)

Wide angle diffuse crescents

(c) butterfly wings

_».

to the

Bragg

spots, their

intensity quickly

decreases and

they

widen out as

they

extend away from the

Bragg spots

so

that, altogether, they

have the

shape

of

butterfly wings

». This overall

shape

is due _to the existence of _a darker

region (I) separated

from the

background (2) by

two

parabola looking

_curvqs C and C'

(Fig. lc).

In order to obtain _more

quantitative results,

_we

performed

_a twb dimensional microden- sitometric

recording

of the

X-ray

diffraction

pattern. Figure

2 shows the

isointensity

lines of the

X-ray

diffraction pattern. The first

isointensity

lines

(1-5)

_are closed

loops

_non

symmetrical

with respect to the axis X'

parallel

to the

equator and, )ding through

the 3rd

Bragg

spot. ^This asymmetry ^{is due} to the

Lorentz-polarizitioil

_factor

_[12]

_{which decreases}

with

increasing scattering angle.

In the _case of wavevectors neither

parallel

_nor

peijendicular

to the

meridian,

the corrections for

Lorentz-polarization

factors

iecome complicated

and _we did not carry them _out. However this factor varies

slowly _except

at the

origin

of

reciprocal

space.

(In particular,

its influence may ^be less than that of _a

posiible _slight mosaicity.)

The next

isointensity

lines

(6-9)

_are

strongly

influenced

by

the wide

angle

diffuse

rings

and

therefore their

shape

is

meaningless.

Furthermore,

the diffuse _elements

(c) only

exist in the

vicinity

of the 3rd order

Bragg

spots and similar elements cannot be _seen close to the lst and 2nd order

Bragg _spots.

From that respect, ^the diffuse elements

(c)

_seem to follow the structure factor of the smectic

layers

I,e, the Fourier transform of the electron

density profile along

the normal to the

layers [lo].

This behaviour is that awaited for defects _or fluctuations

affecting

the _mean structure

[12]. (It

should be nofied that the

signal

_over noise ratio of the

butterfly wings

»

(c)

is rather small. If

hypothetical

_«

butterfly wings

» attached to the first

Bragg spots existed,

their

intensity

would be smaller than that of those attached to the third order

Bragg spots-and

such diffuse elements

might

well be lost in the

background.)

In the

following sections,

_we show that these _«

butterfly wings

_{» may} be due to the _presence of

edge

dislocations in the lamellar

phase.

In

particular

we shall try

lo

_{account for the}

shape

of

curves C and C' and for the

shape

of the

isointensity

lines of the microdensitometric

recordings.

2.

Optical

simulations.

In order Jo ^check our

assumption

that the «

butterfly wings present.in t~e X-raj

diffraction

pattern displayed by polymer

_P~ in its

S~ phase

_are due to

edge,

dislocations in the smectic

layers,

_we

have compared

the

X-ray patterns

to

optical

_ones

performed

on

optical gratings simulating

_arrays of smectic

layers

with _a defect.

(This procedure _which

consists in

comparing

the

X-ray patterns wijh optical

_ones obtained from assumed sketches of defects has

already

been

employed

in Ref.

[7].)

The first step

cpnsists

in

drawing

such arrays with the

help

of d computer. ^The

original drawingi

_as the _one shown in

figure

3 have _an

approximative

size of 20 _{cm x} 20 _cm. The

period

in between the

layers

is

typically

5 _mm. The distorsion of the

layers

due to the presence of _one dislocation is calculated with formula

(3) (see

in the

Appendix)

deduced from

the

elastic model of- De Gennes

[I].

In this model the

important

_parameter

(without dimension)

is the ratio

Aid,'A it

a

typical length

which

depends

_on the elastic constants of

splay Ki

and

compression

E of the smectic

layers

=

(Ki/E)~'~

and d is the smectic

period.

The

drawings

are then reduced _on slides

by

_a factor 25. We thus obtain

optical gratings

the

period

of which is around

200~um.

When illuminated

by

an

expanded

He-Ne laser beam

(A

= 0.6328 ~um

),

such

gratings produce Bragg _spots

at

angles lying

in the _range of a few

lo=3

_rad. A

simple optical

system of lenses

performs

the

focussing

of this diffraction

pattern

on a

photographic

film in _{a camera.} The distance between the

Bragg

spots on the~ film is

typically

_a few millimeters. In

figure 3,

we

reproduce

two

drawings (A/d=0.I

and

a)

a

')

b)

b~)

Fig.

3.

Computer drawings a), b)

and

corresponding optical

diffraction patterns a'), b'). a),

a')/A/d

0.03

b), b~)/x/d

=

= 0.1,

A

Id

=

0.03)

and the

corresponding

diffraction

patterns

that we have obtained. Several

Bragg

spots are visible and _some diffuse

intensity

_appears around these

spots.

This diffuse

intensity

is located in _areas

starting

from these

spots

and

spreading

in the direction

perpendicular

to the

normal _to the

layers.

We have tried to

reproduce

the

«butterfly wings» (curves

C and

C') experimentally

observed in the

X-ray scattering pattern

^{of the} «

polymeric S~ phase.

However in the

optical

diffraction

experiments,

_{we are} limited

by

the noise

probably

due to the

speckle

of the laser. Curves C and C' _can

only

be

guessed

on the

over-exposed optical

patterns. ^This shows that the noise is

particularly

low in the

X-ray experiments.

Indeed _we have

computed

in the

appendix

the diffuse

intensity

scattered

by

_an

edge

dislocation when

moving along

_a

straight

line

parallel

to the s~ axis and at a distance _s~

= 0.05 d~ from the

Bragg

_spot, the diffuse

intensity

falls off

by

about 4 _or 5 orders of

magnitude

from s~ = d~ to s~ = d~ ₊ 0,1d~ A

particularly

low

signal/noise

ratio is therefore needed to

comfortably

observe _curves C and C'. To

improve

the

comparison

between the

X-ray

pattern and the

optical

_{ones, we} have also

performed

_a two dimensional microdensitometric

recording

of the

optical patterns

and then drawn

by _computer

the

isointensity

lines.

Figure

4

shows _an

example

of these _curves in the _case where _A

Id

=

0.03 and

e/d

=

0.3

(e

^is the thickness of the

layer).

The

e/d

ratio represents the transmission

profile

of the

layer.

It

plays

the _same role _as the electron

density profile

for

X-rays

and it has been

arbitrarily

chosen to

reproduce

the

«butterfly wings

around strong

Bragg spots.

^The

optical isointensity

lines

(Fig. 4)

are in

agreement

^{with both the}

isointensity

lines of the

X-ray

pattern ^microden-

O

°

ll o o

O O

o O

Q ~ ~

)

J3°,

^~

0 _o

oo

f~s

~ ~

~

j~

^{~ Cb c~ °}

)

(Cl ~i~

C~~

O

) °

Q Q

O O

~ Q

~ Q

~

O

~

O

~

Fig. 4.-Microdensitometric recording of the

optical

diffraction pattern for

Id

= 0.03. (a)

Bragg

reflections _; (c) _«

butterfly

wings _».

sitometry

and with those

computed

in the

appendix using

the model of De Gennes. We have thus verified that the

positions

of the maxima of the

isointensity

lines _are. located _on

parabolae. By comparing

such

parabolae

and those of

figurelb,

the value

Id

= 0.030 _± 0.015 _can be estimated in _a

semiquantitative

way. This method. _can then also be used to

analyse

the

microdensitometry

of the

X-ray

pattern.

3. Discussion.

The elastic deformation field of _an

edge

dislocation in _a SA

Phase

_was

givbn by

De Gennes and

by

Klkman

[1-4]

in the

following

form

~~~~~

~

~ar ~

~~~~ ~ ~~~~

where

u~(x)

is the

displacement field, (see Fig. 5a)

and A is a

length

defined

by

=

(Ki/E)~'~

where

Ki

and E _are the elastic _constants of

splay

and

compression

of the

layers,

which take part ^{of the free} energy. The

Burgers

vector of the

edge

dislocation is chosen here

equal

to the smectic

period

d. The

amplitude

scattered

by

this

displacement

field will be

proportional

to its Fourier transform. The

§etails

of this calculation _are

given

in

Appendix

and the diffuse

scattering amplitude

is

given by

:

pF

_~

~2

_~

~2

A'

s~,

_~

+ 8s~

~ ~ ~

~

~' _{~ ~}

d s~ 4 _ar

(3s~)

d ₊

(4

_ar

Ads~)

where s~ is the

x-component

of the diffusion vector and

8s~

is the difference between the _z-

component

of the diffusion vector and that of the

Bragg

reflection of

integer

order

p.F~p/d)

is the structure factor of _a smectic

layer.

The diffuse

scattering intensity I(s)

is

proportional

to the square of this

amplitude.

l~

^P

I

4 ^' _i

1 ,

3 _,

j

11

~ _'

---l---'U2

, J

, J

i ', ,,

',, «" X

~

~

i d

Fig.

5a. -Scheme of _an

edge

dislocation in _a smectic phase d is the smectic

period,

un(x) is the

displacement along

Oz of the n-th

layer.

The elastic distortion field is located inside _a domain limited

by

a

parabola

P, its curvature is A

=

(B/K)~'~

where K and B _are

respectively

the elastic constants of

splay

and

compression

of the

layers.

With this

result,

let _us first account for the

shape

of the

butterfly wings-»

this

shape

is due to the contrast between _a darker

region (I)

and _a clear _one

(2)

limited

by

two _curves C and C'

(see Fig. lc).

For s~ constant, we have _{a crossover} when the two terms in the

denominator of

I(s)

_are

equal,

that is when

8sj

=

4

ar~A~s) (equation

of _curves C and

C~).

^For

region (2)

in which

8s)

is

larger

than this

value,

the

intensity

falls off

sharply.

In _a

semiquantitative

_{way, a} fit of _curves C and C' with

parabolae provides

_a value A

=

1.50 _± 0.25

h.

The

experimental isointensity

lines _can also be

compared

to the closed _ones shown in

figure

5b obtained

by directly computing

the diffuse

scattering (see Appendix).

This

computation

indicates that the. diffuse

scattering

is then located around the

Bragg

spots ⁱⁿ

areas

spreading perpendicularly

to the

layers.

The

isointensity

lines _are closed lines

starting

from the

Bragg

spots ^{and which} are in first

approximation symmetrical

around the

8s~

axis

(normal

to the smectic

layers)

and the s~ axis. These lines _are

given by equations

of

the

following

_type

3s~

=

(As~

Es

))~'~

where s~ is the component ^{of the}

scattering

vector s

along

the _x direction and

3s~

is the difference between the _z

component

^of s and that of the

Bragg spot

around which the diffuse

scattering

is studied. Each line

prqsents

a maximum of

3s~

as a function of _s~ at _a

point ^M

of coordinates

(s~)~

and

(3s~)~.

The set of all maxima is _a

parabola

of

equation

_:

(~SZ)M

~ ~ _"

N~

_h

_(Sx)if

This method

provides

_a value of'A A

=

1.2 _± 0.6

h

_{which fits}

quite

well

with'the previous

one deduced from the

shape

of the _«

butterfly wings

».

The

butterfly wings

_{» are} visible

inly

in the

vicinity

of the 3rd order

Bragg

spots. ^This can

be

explained by

the

prehence it

_{the diffuse}

_{scatterinj amplitude}

_{of the term}

_{pF ~p/d)}

_where

p is the order of the

Bragg spoi

arobnd which the diffuse

intensity

is

analysed.'Indeed

the

structure factor

F~p/d)

is maximum for p = 3 and very small for _p

=

2 _{as can} be _{seen on} the

intensity

of the

Bragg spots [10] and',

_on the other

hand,

the factor _p in the diffuse

scattering amplitude

could

explain why

the

butterfly wings

_{» are} not visible around the lst

Bragg spot.

dsz

' ' '

'

' '

'

' '

, ,

,

ds~

, '

, ,

' '

,' '

'

Fig.

5b.- Calculated

isointensity

-lines around _a Bragg spot ⁱⁿ reduced coordinates

(ds~,ds~).

Consecutive

isointensity

lines _are drawn for

intensity

ratios of10. The maxima s~(s~) ^{of these}

isointensity

lines _are located _{on a_} parabola drawn in dashed lines.

Therefore,

_{as was}

already.

shown

by optical modelling,

the _«

butterfly wings

»

(c)

_may well be due to the

scattering by edge

dislocations

present

^{in the} «

polymeric S~ phase.

The

description given by

De Gennes

[1,

2] ^has been extended

by

Klkman

[3]

in order _to include the bend distortion which _occurs close to the dislocation _core. Our calculation

neglects

such _a

type

of distortion and is

only

valid in the limit

A)sj

_{« I, A~}

=

(K~/E)~'~, K~

^{is the bend elastic constant. The fact that} curves C and C' indeed fit to

parabolae

within 10- 20 §la shows that this

approximation

is

justified

_;

otherwise,

_curves C and C" would tend to

straight

lines.

(By

the _way, it also shows that the influence of the

Lorentz-polarization

factor is not very

important

in this range of

s).

The _range of _s considered here

implies

that A~ should be less than _a few

Angstroms.

_In other

words,

the dislocation _core should be very

small,

less than 10

I.

_Such

a situation has

already

been described since

edge

dislocation _cores of less than 10

h

_{diameter have been observed}

_by

_electron

_microscopy

_{in Germanium}

_[13].

On the other

hand,

the fact that the diffuse

scattering

in the

polymeric S~ phase

_can be detected in _a

region

of _s~ of the order of _a few

hngstr6ms _implies

_{that A is of that order of}

magnitude compared

to A

=

20h

_{in the}

case of the «conventional

S~ phase.

Now,

let _us comment these two orders of

magnitude

_(A

= 1.50 _± 0.25

h

_{and A}

~ less than _a

few

hi _only

very few measurements of the elastic constants of the

mesophases

of side chain

polymers

exist in the literature

they only

_concem the nematic

phase

_: P. Fabre _et al.

[14, 15]

have measured

by

the Frbedericksz transition

method,

the values of the

splay (Ki)

and bend

(K~)

^elastic constants of _some

mesomorphic

side chain

polysiloxanes

in the nematic

phase.

Their measurements show

that~Ki

and

K~

^{of this}

«polymeric

nematic

phase

_were of the

same order of

magnitude

_as those of the conventional nematic

phase. Later,

these results

were confirmed

by Rupp

et al,

ii 6]

in _a

study

of _some

mesomorphic

side chain

polyacrylates closely

related _to

pblymers P~.

As far _as the elastic constant E is

concerned,

_no measurement

could be found in the literature.

However, by inspecting

the

profiles

of the reflections _on

(he layers

with _a

high

resolution

X-ray set-up,

^Nachaliel et al, _were able to show that E should be

roughly

two orders of

magnitude larger

in the

«polymeric» S~ phase (for

from the

S~/N transition)

than in the _« conventional

S~ phase [17].

The orders of

magnitude

that _we found for and A

strongly suggest

^{that the} constant E for

a

polymeric»

SA

Phase

^should be about two orders of

magnitude larger

than that of _a conventional SA

Phase.

This _can be understood

by considering

the effect of the backbones which _are confined in

sublayers squeezed

between

adjacent sublayers

of

mesogenic

_cores.

Apart

from their _own

rigidity,

such backbones

sublayers, by preventing

the

mesogenic

_cores

of _a

given layer

to insert into the

neighbouring

_ones, lessen the

permeation

effect

[18, 2]

^and therefore increase the elastic constant B.

We may now

give

_an estimate of both the dislocation

density

and the ratio of the distorted volume to the total _one.

By considering

both the

intensity

and the

angular

extension of the diffuse

scattering compared

to those of the

Bragg reflections,

we

roughly qualitatively

evaluate the

experimental

ratio of the

integrated intensity

scattered

by

the defects to the

intensity

reflected

by

the

layers

_:

I~~~~~~j~n~~~~~~~~/I~~~~~ ⁼

10~~

_{as an} order of

magnitude. Now,

let _us consider that the dislocation lines _are infinite in the _y direction

(along

the

X-ray beam).

The resolution of _our

set-up

is

Aq~

=

Aq~

= 2 _ar

10~3 h~ _Therefore,

we have _a domain of diameter 103

h

_{in which}

we can detect that the

mesogenic

_cores

coherently

diffract into the

Bragg _spots.

Let _{us assume} that _we have N defects in this domain. We estimate the dimensions of a defect to be

roughly

500

h along

_z and 100

h _along

x.

(These

values _are obtained from the

decay

of the elastic distortion field

alonj

the _z and _x

axes.)

Inside _a

defect,

the

mesogenic

_cores diffract

coherently

into the

butterfly wings

_».

However,

_two different

defects do not scatter

coherently.

From

this,

it follows that the

intensity

ratio is

given by

N(500 x100)~/(10~)~ =10~~

which

gives

a number of N =4

x10~~

dislocation line in _a

domain infinite

along

the

X-ray

beam and of radius

1000h.

_{With the dimensions of the}

defect mentioned

above,

this leads to _a ratio of distorted volume to the total _one of

V~~i~~~/V~=2 x10~~

Such _a ratio is >of _a correct order of

magnitude

to

give

rise _{to a}

comparable

small

ingle

diffuse

scattering [19].

The number N

corresponds

to a dislocation

density

of

= 4 _x

10~

dislocation _per

cm2.

This _can be

compared

to the

10~-10~

dislocations _per

cm2

in

good crystals

and _to the

101°-1012

dislocations in _a laminated metal

[20].

It _can also be

compared

to the 109 _screw dislocations which _can be found in _some

lyotropic

lamellar

phases [21]. Therefore,

it _can be _seen that the

edge

dislocation

density present

in the

«polymeric» S~ phase

is _a

fairly large

_one, all the _more since the

«polymeric»

S~ phase

is _a viscoelastic medium in which the defects should heal.

(I,e,

two dislocations of

opposite Burgers

vectors may move and cancel each

other.) However,

this

edge

dislocation climb can be difficult in the _case of _a

polymeric

»

S~ phase

because its

viscosity

is 103

larger

than that of _a conventional

S~ phase _[14]

and also because it involves _a

permeation

_process

which is

expected

to be

difficult. Indeed, _keeping

a

sample

for _a

iveek just

below the

cliaring

temperature

does not

appreciably change

the diffraction

pittern.

Let _us

alto

recall that

polymer P~

^does not present a Nematic

phase

in which

pretransitional

smectic fluctuations could appear. Therefore _a

large- density

of defects may be awaited at the

I/S~ transition,

which do not vanish for the _reasons mentioned above and _are frozen in under T~.

Conclusion.

The

X-ray

diffraction patterns of the

S~ phase

of _some

mesomorphic

side chain

polymers display

diffuse streaks in

shape

of

butterfly wings

_». It has been shown that this diffuse

scattering

_may be due to the presence of

edge

dislocations in the

«polymeric S~ phase.

Indeed,

the

presdnce

of

edge

dislocations

ip

the

S~ phase

of related

compounds (mesomorphic

combined

polymers)

has

alreadj

been established

by

electron

microscopy. Thfis,

in _our case, we

estiiaate

_in

edge

dislocation

density

of about

10~/cm2

and the size of their _cores should be less than 10

hngstr6rii.

_{From the}

_study

of their elastic

deformajion field,

it has been

inferrid

_{that A should be}

_roughly

_1.5

hngstr6ms

_{and that the elastic}

constini

_{E of}

compression

of the

layers

should be about two orders of

q~agnitude larger

ⁱⁿ the

polymeric

_»

S~ phase

than

in

_the

« conkentional

ine.

_{Of course,}

a direct

rheological measiri of _{E_Would}

be needed to confirm this estimation.

However,

the

problem

of

obthinirig

_a well oriented monodomain free of defects _may be difficult _to solve.

The

question

arises to understand

why ^iuch

effects of

edge

dislocations on the

X-ray scattering

patterns ^have not yet ^been observed in the

S~ phases

of other side chain

polymers.

Several _{reasons may} be mentioned :

first,

the

regions

distorted

by

different

edge

dislocations do not scatter

coherently,

therefore _a

fairly large density

of dislocations is needed to affect the

X-ray

diffraction

patterns. Moreover,

other

types

^{of disorder}

[11]

may also

give

_some

scattered

intensity

in the _same

region

of

scattering

vector and may thus hide the

butterfly wings

which have _a rather low

signal"

_over noise ratio.

Finally,

the. dislocation

density might

also

depend

_on the

polymerization degree

and it would therefore be

interesting

to

study

its influence.

Acknowledgements.

we

would like _to thank J.

Charvolin,

M.

Klbman,

J. Prolt for fruitful

discussions,

M.

Keldi,

S. Loeul who took

part

^{of the}

experiments

and F.

Augier,

^{A. Saint} Martin for technical

help.

Appendix.

In this

appendix,

_we calculate the

intensity

scattered

by

_an

edge

dislocation in _a smectic _array

as shown -in

figure

5a. The

period

of the smectic

layers

in the Oz direction is d. The _mean

position

of the n-th

layer

is

given (for

n _~

0) by z~(x)

=

nd ₊

u~(x) ⁽¹⁾

u~(x)

is the

displacement along

Oz of the n-th

layer.

Some

particular

values _{are :}

Un(+CC)"d/2, Un(°)~dm, Un(~CC)~0.

The variations of u~ ^as a function of _x have been described

by

^{de Gennes}

[1] using

_an elastic model. It involves the square ^{root A} of the ratio of the elastic _constants

Ki

^and E. This model

gives

_:

u

(x)

=

~ +

~

~~

~iqx_~ ^ndAq~

(~)

~ 4 4

aT

iq

or in _an

equivalent

formulation

3U~ d ~2

~

fi

(3)

S

4

fi /$

These

expressions

^{show that the}

typical length

wich characterizes the distortion is

(n

Ad )~'~ ^{for the n-th}

layer.

The elastic distortion field is thus located inside _a domain limited

by

a

parabola. Expression (3)

has been used to

make, by

computer, the basic

drawings

used in the

optical

simulations.

The

_scattered

intensity

also

depends

on

p(z)

which

represents

the electronic

density

in

X-ray scattering

and the transmission factor in the

optical

simulation. In this latter _case

p(z)

is

equal

to I when _z is located between z~

e/2

and z~ +

e/2

where z~ is the

position

of the n-th

layer

and _e is the thickness of _a

layer.

It is _zero elsewhere.

In both _cases, the scattered

amplitude A(s)

is therefore

given by

2 _ins nd ^~ i ₂

1ws

~ °~

2 _{ins x} 2 _{ins u} (xi

A

(S)

=

I

_e ^~

~

dZ _p

(Z)

_e ~' dX _e ~ e ^{~ ~}

(4)

n

-

~

-

w

2 _ins nd

)~

°~ 2 ins _x 2

ins u (xi

A

(s)

=

F

(s~) jj

e ^~ dx _e ~ e ^{~ ~}

(5)

is

depends ^on the distribution of thelectronic

density.

In the

case

of _optical

scattering it

is

related

_to the thickness e of the

liyer,

_that is _the

profile.

In the ssumption of

mall u~(x),

A

(s) is the sum

of

escribes the

usual

_Bragg spots

due to

the^mectic The _second one gives the

diffuse cattering amplitude

A'(s)

and ^can be ^xpressed as

:

A'(sj, s~)

_ar ^~~

~~~~~~

dq jj

e~~~~"~~~~^~~~~~ dx e~~~e~~"~~~

(6)

2 q

S~ dF

(S~)

_{n(2 ins d 4}

w 2~ds2j

i

~ ^{~ ~}

S~

s~

dF(s~)

_i

e~~ ^"~~~~

~ ~

~ 2~~2

~

2~~~2 ~~~

x 2 _cos

(2

_ars~

d)

_e ~ + e ^~

For small values of _r

=

4 _ar ^~ Ads this

expression

reduces _to s~

dF(s-)

A'(s~ s~)

~

(8)

~~

(l

_{cos 2 ars~}

d) (I

_r +

r~

(1

₂ ^{cos 2 ars~}

d)

This shows that the diffuse

amplitude

is located around the

Bragg spots (of

order p, p

integer)

s~ =

p/d+ 3s~.

Incorporating _8s~

in

equation (8),

_one obtains _:

A'

(s~

^{~ + 8 s~}

~~

_~

'~

~~

~~~

(9)

d s~ 4 _ar

~(8s~)~ d~

+

(4

_ar ^~A

ds()~

In the above

equation,

let _{us assume} that s~

F(s=)

does not vary around the

Bragg

spots ^and focuss _on the variation of the

intensity

due to the presence of the defect.

Following

the

previous assumption,

the

isointensity

lines

corresponding

to the diffuse

scattering

_{aroung a}

Bragg spot

are

represented

in

figure

5b. Each line

8s~

_as a function of

s~ shows _a maximum

(8s~)~

which is located _{on a}

parabola

described

by

(8s~)~

=

2 ar

/

_A

_{(s~)[ (10)}

The _curvature of this

parabola depends only

on A.

Taking

into account the term

s~

F(s~)

which in the

optical

diffraction is

proportional

to sin

(ares=),

where _e is the tllickness of each

layer,

will disturb the symmetry ^{of the}

isointensity

lines around the s~ axis.

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