X-ray scattering by anisotropic elastic deformations around vacancies in ordered smectic phases

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X-ray scattering by anisotropic elastic deformations around vacancies in ordered smectic phases

E. Dubois-Violette, B. Pansu, P. Davidson, A. Levelut

To cite this version:

E. Dubois-Violette, B. Pansu, P. Davidson, A. Levelut. X-ray scattering by anisotropic elastic defor-

mations around vacancies in ordered smectic phases. Journal de Physique II, EDP Sciences, 1993, 3

(3), pp.395-405. �10.1051/jp2:1993139�. �jpa-00247841�

Classification

Physics

Abstracts

61.30 61.708 62.20D

X-ray scattering by anisotropic elastic deformations around vacancies in ordered smectic phases

E.

Dubois-Violette,

B.

Pansu,

P. Davidson and A. M. Levelut

Laboratoire de

Physique

des

Solides(*),

B£tknents10, Universit6 Paris-Sud,

914050rsay

Cedex, France

(Received 3

July

J992,

accepted

in

final form

J3 November J992)

Rksumk. Los

phases smectiques

B sont des

phases

ordonn6es h 3D, mais _{ce ne sont} pas des cristaux

parfaits.

Dans _ces

phases

it existe des cha~nes de moldcules dont l'ordre est facilement

perturbs

_par l'existence de lactlnes. Dans des

papiers pr6c6dents

_{nous avons} montr6 que la

signature

de _ces d6fauts _sur des clich6s de RX _est la

pr6sence

d'une ligne blanche diffuse. La

sym6trie hexagonale

du milieu

engendre

_un

champ

de d6formation

anisotrope

autour de

chaque

d6faut. Nous relions la forme de la _zone blanche diffuse _aux constantes

61astiques

du mat6riau.

Abstract Smectic B

phases

are 3D ordered

phases,

but

they

are not

perfect crystals.

In these

phases

there exist

strings

of molecules, the order of which is

easily

disturbed

by

the existence of vacancies. We have shown in

previous

_papers that _a

significant signature

of these defects _on X-ray diffraction pattems is the presence of _a white diffuse line. The hexagonal symmetry ^of the medium generates an

anisotropic

deformation field around each defect. We relate the

shape

of the white diffuse _zone to the elastic _constants of the material.

1. Introduction.

Smectic

liquid crystal [I] phases _present

_an

organization

With

layers

_{more or} less correlated. In smectic A

phases

the order within the

layers

is

liquid-like.

Smectic B

(SmB)

_or G

(SmG) phases

show _a

crystalline

3-dimensional

(3D)

order. The

layers

_are

weakly

correlated in the SmB

phase (hexagonal symmetry)

_or in the SmG

phase (monoclinic symmetry).

One consequence is the presence of _a

large

number of

stacking

faults. Another is the very low value of the shear elastic constant

Cm (related

to the

sliding

mode of the

layers

over one

another) compared

to the other

typical

elastic constants.

Contrary

to the SmA

phase,

in both the SmB and SmG

phases

there exists _an

organization

of the molecules in

strings.

In the SmB

phase,

the

strings

of molecules _are

perpendicular

to the

layers

whereas in the SmG

phase, they

are tilted.

X-ray

diffraction pattems ^{show white diffuse lines}

interpreted

_as induced

by

defects in these

strings

of molecules

[2].

A vacancy is created when _a molecule is

expelled

from _one

layer

to

(*) Associd _au CNRS.

the next

(Fig. I).

This disturbs the

string (defining

the _z

direction),

_{over a} distance 2 L related to the vacancy lifetime. This

length

is the diffusion

length

of the

compression

_wave

along

the

string during

this time. This time has been estimated in the _case of TBBA

[3]

to be of the order of 10~~~ _s. The

corresponding length

concems 5 to lo molecules. The concentration of these defects has been evaluated _to be of the order of 10-2

[2, 3].

In _a

previous

_paper

[3]

_we

analyzed

^this defect in _a lD

asymmetric

model

explaining

the presence of the white diffuse line at q~ =

0. We

only

took into _account the elastic defornlation

along

the

string

and

computed

the diffraction

pattem

^{of the 3D}

crystal

in the presence of these non-correlated lD defects

[3].

Actually,

for _some

compounds, experimental _pattems

reveal _some

broadening

of the white

zone for

large

_q~

[2] (Fig. 2).

We show in this paper that the

shape

of the diffuse white line is related _to the

complete

3D elastic defornlation field

coupled

to the defects considered

~ymmetric

this time. In section 3 _we

give

_an exact solution for this field in _an infinite medium

j j i j j

j i j

j i j ~~

2L

clj _j

_~

i j j i

i i j i i

i i i i i

i i i i i

i i i i i

Fig.

I. Schematic

representation

of the defect, The

ellipses

symbolize the molecules which

are

displaced

around the vacancy, along the three directions away from their

positions

in the ideal crystal.

Fig,

2, X-ray diffraction pattem ^{of the SmB}

phase

of compound 40.8 (fixed crystal

experiment,

the

arrow shows the direction of the _z axis). Notice the

broadening

(half

angular

width

=

10-~ rad) of the white _zone

passing through

the

origin

of the

reciprocal

_space.

in _tennis of all the elastic constants in SmB

phases

: an

important point

^is the very low value of

Cm compared

to the other constants. Then, ⁱⁿ section

5,

we restrict the solution in the infinite medium _{to a} domain defined

by

the finite range of the defect. Section 6 is devoted to the

computation

^of the scattered

intensity

in the frame of small

displacements

in _a continuous medium.

2. Diffuse

scattering intensity.

The scattered

amplitude A(s) (where

_s is the

scattering

vector of modulus _s

= 2 sin o/A and

2 o is the

scattering angle)

^is

easily

calculated with _use of the

following

classical

[4] expansion

for small

displacements U~.

_s

U~

« I _:

A(s)

=

£ f e~'"~'°~~~~~= £ f e~'"~'~~ _(l

_{+ 2 ins}

_{.U~) (I)}

n n

where r~ is the

position

of the n-th molecule in the

perfect crystal.

The diffuse

amplitude

is driven

by

the last _ternl of

equation (I).

The continuous

displacement U(r)

can be

expanded

in Fourier modes _{as ;}

u

(r)

₌

u~

e~ ~'"~'~

d~q

,

(2)

which leads for the diffuse

scattering intensity I(s)

in the continuous limit _{to :}

I(s)= £ ^f2 is u~(~ (3)

t

where q = s

-ic*.

In the

vicinity

of _a

Bragg peak, only

_one term of this _sum

really

contributes. Around the

origin

of

reciprocal

_space, this

gives

for the diffuse

intensity Io(s):

Io (s)

₌

f~

_q

u~

^~

, q = s

(4)

We shall _now deternline the elastic defornlation field induced

by

the vacancy.

3. Deformation field in _an infinite medium.

We first consider the

displacement

^{field induced in} an infinite medium

by

_an

anisotropic

local dilatation in the _z

direction, parallel

to the molecular direction. The elastic forces

satisfy

the

following equations

_:

F~ _F~

^{= 0}

=

0

(5)

F~

=

W8~ _8~ 8].

A dilatation

corresponds

to a

negative

value of W.

In _a

hexagonal phase

the elastic forces

[5]

read _:

~x ~~'~ ~~~~ ~)) ^~~~~

^{~ ~}

_{~) ~~~)}

a~Uz

~

+

(C13 ~12~ (6)

3X 3~ ^~

~~~~ ~~~

~~~ ^{~~ ~~}

Fy

~

~~

'~ ~ _~~~

~~~

~

~~~~

~

~~~~

~~)

^~

a~Uz

+

(C13 ~12~ (7)

~~~

~

~~~ ~~~~ ^~~~

~

~~ ~~

and

~z ~~

~~ ~_~~~

~~~

~

~~

~~ ~ ~ _~~~

z

~~~ ~

a~U~

a~U~

a~U~

a~Uy

~

~~" ~"~ W

^~

fi

^~

Q

^~

_w

a~U~ a2U~ a~Uz

~

~~

~~ ~

~~ ax az

~

dy

az

~

~~

~~ ~

az~

^~~~

with

C~~

₌

(C

ii C

i~).

2

Calculations _can be

perforated

with the classical method of the Green functions

[6], but,

_as

we are

essentially

interested in the

X-ray scattering, equations (6), (7)

and

(8)

can be

directly

solved in Fourier space with _:

q=qi

i+qzi (9)

u

= U~~ ⁺

U~ i ( lo)

where

(

and

t,

_are the unit vectors in the radial and _z directions of the Fourier space. The Fourier

components

^{of the} defornlation _{are :}

i

(c13

+

cm) q)

_qi

Uqi

~ + ~'~

(11)

2 arD

iqz(Cii qi

₊

_CM

_~~~

Uq~ ^" W

(12)

2 WD with

l~"

(Cll~44)~~+(Cll~33~~(3~~~13~44)~~~~+~33~44~~

~~d

~~ ~~~+~~.

Equations (11)

and

(12)

_may be transfornled with _use of the relation _:

l~

~

(Cll C44)(~~

₊

~2~~)(~~

₊

~3~~), ₍₁₃₎

where a~ and a~ are the roots of _:

~~Cll~44+ (C(3+~~13~44~~ll~33)a+~33~44~0. (14)

One obtains _:

~41

"

W ~~t

~13

+ C

~

~ "~

C~ (a~

l

~ ~2 ~~ +

q~

_a~

q)

_~

[

li~

₌ ^{W ~~~}

~~

~

j ~"

^{~~ ~}

_'j ~" ₍₁₆₎

~ ~ "~

ll

~44(~2

_{~3 a2 ~z + ~t a3 ~z + ~t}

The Fourier transfornls

(F )

^of

equations (15)

and

(16)

_are obtained with _use of the property :

~

_F

_jg(q)j

= F

ii

2 wq;

g(q)1

and of the

following

result _:

~

(2

_ar)~

~j)

+

q( ₎ I(z~

+

p

~)~'~

where _r

=

pp°

₊

zz°, p°

and z° _are the unit vectors in the radial and _z directions of the direct

space and

p~=x~+y~.

_{This Fourier transform is}

_easily

_{deduced from}

F[I/q'~]

_with

q'~

=

q'(

+

q')

_{after a}

_rescaling q'(

=

aq).

The

resulting

defornlation field which also has _a

cylindrical

symmetry :

U#Upp°+U~Z°,

then reads :

l~

^a3

₍₁₇₎

(~13

+

~44~

_~

~

~

2 312 2

~)~~~

~f~ = + W

~ _~ ~

~~

~~(a~ a2)

_{(Z + ~2 ~ ~~ ~ ~~ ~} and

z

la2

^{C ii}

^CM

^{a3 C ii}

^CM

~

_~

~~'~ "C11 ~44(~3 ~2)

_{(Z~ + a2} P~)~~~ (Z~ + a3 P ~)~~~

~~~~

Before

exploiting

these results in order to compute ^{the diffuse}

scattering intensity,

_we shall first discuss the order of

magnitude

of the elastic constants.

4. Elastic constants.

In SmB

phases

the

layers

_can

glide

_{more or} less

easily

_over each other. This property is characterized

by

the elastic constant

Cm,

which is much smaller than the other elastic

constants. The coefficient

Cm

has been measured

by

different

techniques

in several SmB

samples.

For the SmB

phase

of

compound 40.8,

results obtained from mechanical shear

experiments [7] give Cm

m 5 _x

10+~

_{cgs. This result is in}

_agreement

with estimations from dislocation motions

[8]

and ultrasonic studies

[9].

The value of the other elastic _c

ypstants

are much

larger, typically

of the order of

10~-10"

cgs

[7, 10].

Precise measurements of these elastic constants have been

perforated [11]

but for another related

compound (50.8).

Far from the

SA-S~ transition,

the authors

give

_: C

ii ^m 1.63 _x

10~°

_{cgs, C}

i~ ^m 1.52 _x

10"

cgs,

C~~

_m ^1.61 x

10"

_{cgs. The}

_parameter

_{which will appear as}

_pertinent

_{is the ratio}

a ₌

Cm/C

ii.

This ratio

a has _a very low value

m 10~

~-10~

~ and will be used _{as a} small

expansion parameter

in the

following.

More

precisely,

_{we can} obtain _an estimation of the coefficients

a~ and a~ in tennis of _a. To the lowest order in _{a, one} has :

~

~ll ~il ~33~~(3

l~ll ^{~33 ~~(3}

2 ~ c

~

~2

_ty

~2

ii 11 ~~~~

~44 ~33 ~li ~33 ~il

~~

~ll Cll~33~~(3

^~

~ll~33~~(3

JOURNAL DE PHYSIQUE11 T3, N'3, MARCH 1W3 16

X-ray

diffraction

pattems

have been

mainly perforated

_on the SmB

phase

^of 40. 8 for which the coefficient _a has been estimated. On the contrary, the other elastic constants

appearing

in the

expressions

of a~ and a~

(Eq. (19))

have _not yet ^{been measured. In order} to get an

estimation,

C ii

C~~ C(~ _C~~

C

ii we have used the ratios

~ ^m

0. I, and

~ ^m

8 _as evaluated

~ l1 ~

l1 ~

33 ~

l 3

from the measurements in the SmB

phase

of 50.8

[11].

Then the order of

magnitude

of

a~ and a~ are a~ m

few10~

_and

a~ m10~

'-10~~

5. Defects of finite size.

As the vacancy has _a finite lifetime the elastic deformation field

only

extends _{over a} finite range. Let 2L be the characteristic

length

in the _z direction which _can be

estimated,

_as

explained

in

[3],

to a few molecular

lengths

from coherent _neutron

scattering experiments [12].

Then the

boundary

condition

taking

this feature into account reads _:

Uz(p =0,z=L)=0. (20)

A solution

satisfying

both the elastic

equations (5-8),

and the above

boundary

condition

(20)

is _:

~~

^~~~

~ _~~2 P~

^{~~ ~}

^{~~ ~~ ~~3} P~

^{~~ ~}

^{~~ ~~}

_~~

~~

where

~~4wC)icm(a~~a~)

In order to deternline the radial extension of the

distortion,

_we sketch the

plot

of the _curve

Uz

₌ 0 in the

(p,

_z

) plane (Fig. 3).

In the limit of small _a, the

region

between the _curve and the

plane

_z

=

0 is very small. Then _{we can}

neglect

^this zone and

approximate

the volume _{to an}

ellipsoid,

^the surface of which is

given by

:

z~ _{+ a~ p}^~

=

L~ (22)

z

+L

p

Fig.

3. Curve U~ 0

defining

the extension of the deformation field. Its

global shape

is

ellipsoidal

except near the

plane

_{z =} 0. The coordinates of

point

^E are p (E ) L/

land

z(E)

Lla(/~,

In the SmB case where aj ^' is small, the

ellipsoid

is

elongated

in the _z direction.

Here it is

important

to stress

that,

due to the 3 dimensional nature of the

elasticity,

_a

distortion

extending

_{over a} distance L in the _z direction also extends in the transverse direction

over a distance L/

/.

_{Then the} defornlation field is limited to the

ellipsoidal

^{volume scaled}

by

the vacancy lifetime. Therefore _we shall restrict the

displacement U(p, z)

to this volume

and take _:

(C

i~ +

CM)

_p

lj

^{a~ a~ a~ a~}

U~

= + W

~ ~ ~~~ ~ _{~ ~ ~~~} ~

~ "~

il

~44 (~3 ~2)

_{(Z + a2 P}

₎

L (Z + a3 P L

(23)

6.

X-ray intensity

scattered

by

the defects.

As

explained

in section

2,

the diffuse

scattering intensity (Eq. 4)

is related to the Fourier transform of U which we shall limit to the above

ellipsoid.

The exact

computation

of this Fourier transform is

difficult,

but its behaviour _can be deduced from that of _a

specific displacement

field inside _a

spherical domain,

with _use of _an

anisotropic

linear

rescaling.

Indeed,

each

part

of both components ^{of the}

displacement

field _can be deduced from _a

spherical displacement

field _:

~~~~~

~?

by

the

rescaling x~

~ a~

x~ (resp a~)

,

y~

_- a~

y~ (resp a~)

,

z~

~ z~

(24)

Let _us take this field limited to a

spherical

domain of radius R. Its Fourier transform reads _:

i~ ^U°(r) e~~"~'~ d~r

=

~

(~ ^{(l E(2 arqR} )) (25)

o q

E(qR),

the Fourier transfornl of _a

homogeneous sphere

of radius

R,

is _:

E(2 arqR)

₌ 3 ^~'~ ~~

""~

~~

""~~

^{~°~ ~~}

^"~~~ (26) (2 arqR)

One

recognizes

in

expression (25)

two contributions. The first _one ^~

(~

^{is the Fourier}

q

transfornl of in _an infinite medium. The second _one

corresponds

to the correction induced

r

by

the finite size of the

perturbation.

In the SmB _case, the

anisotropic

size effect _can be deduced from

equation (26)

with _use of the

following rescaling

_:

qj

-

~~

,

q(

-

~~

,

q)

-

qj ₍₂₇₎

~2 ~2

We _assume that the

intensity,

_as in the

spherical

case, is still

given by

the

product

of the

Fourier transfornl of the field in _an infinite medium with the form factor of the finite domain.

This

assumption

is

strictly

valid for a~ = 0. After _some

straightforward

calculation _one obtains the diffuse

scattering intensity

_:

~

lwq)( ^(C

^{ii C i~}

^CM

q(

₊

_Cm q)

²

Io (q)

₌

f

~

~ ~ ~ ~

(l E(2 arqL )) (28)

~ "

(~

II

~44 (~t

+ ~2 ~z

)(~t

+ ~3 ~z

)

where _@~

=

q)

₊

q( _la~.

1, ¹⁼⁵

a =loo

0.

0.

0

B

0.

0.

qz,

Qi"°.~

0.1 0.2 0.3 0.4 0.5

1, 1=5

a =loo

o

o

o.

B

o.

0.

~ ~ ~ ~ ~ ~ ~ ~

~qz. ^qi=1

1, 1=5

a =loo

o

0.

B

o

0.

~ ~ ~ ~ ~ qz,

qi=2

Fig.

4. Behaviour of

Io(q~)

for a~ = 100, L

= 5 _c and for different values of q~.

Io(q~)

^is

given

in arbitrary ^unit.

Starting

^from zero at q~ = 0, it reaches _a

plateau

for larger q~. The limit of the white

region

corresponds

to

point

B determined by

I~~/2.

The diffuse

scattering intensity

vanishes _{for qz}

=

0,

this

gives

the white line _on the diffraction pattem. ^{The white diffuse} zone is therefore localized around the

plane

_qz

=

0,

but for the

measurements on

40.8,

it gets ^broader for

larger

_q~. This _can be

interpreted using equation (28).

The behaviour of

Io(q~)

for different values of q~ is shown in

figure

4. The

intensity

vanishes _{at q~}

= 0. Then it increases for

increasing

_q~

reaching

_a

plateau

with _some

oscillations. The border of the white _zone

corresponds

to _a

strong

variation of the scattered

intensity.

It may be characterized

by

the

point

^B in

figure

4 at half value of the maximum

intensity.

The

position

of this

point

in the

reciprocal

_space

gives

the relation

q~(q~ ) defining

the border of the white _zone. A

typical

variation of

q~(q~ )

is

given

in

figure 5,

where _one notices the _{cross over} between the two

following regimes

_:

~°r i°W values Of qi q= L- ^' ~~~~

for

large

values of q~ qz = rq~ /

/. ₍₃₀₎

The cross over between the _two

regimes (29, 30) happens

for qi "

r~'L~' /.

_The

interesting

result is that _we

predict

_a

broadening

of the white _{zone as} observed

experimentally

and that the

strength

of this effect is

directly

related to the elastic _constants. The value of

r

(r

=

1.3

)

is deduced from the theoretical _curves

plotted

for different values of the parameter a~. Let _us compare this theoretical result _to the

experimental

diffraction

pattem performed

_on

40.8. In

figure

5 _we also show the

experimental

results

(half intensity width)

obtained from microdensitometric

recordings

of the

X-ray

diffraction

pattems.

^{The half}

angular spread

of the

white _zone is evaluated to about

10~'rad

which

gives,

with _use of

equation (30),

q~

o 4

/

-l'j'

'

/

,

/

~~'

$%'~'

Q_I

~j/

0.

o 2

q~

L~'°C,°2~2°

= IO c, fi~= ^loo Experimentol points

L

= 5 c, o~= ^loo

Fig.

5. Calculated coordinates

q~(q~

) of the border of the white diffuse zone for L

= 5 c and

10 _c and for a~ =

20 and 100. One fits _on these _curves the parameter r l.3

defining

the

reghne

for large values of q~ q~ =

rq~/ /.

a~ ~100. This leads _to the estimated ratio _:

Cm ~lcijc~~-C(~

$~~~ C(1

If _{one uses} the values of C

ii,

C~~

^and C

i~ of

50.8,

_one obtains

~j"

^{10~ ~ which is in}

^good

ii

agreement ^{with the}

experimental

estimation

[7- lo].

There is no

experimental points

for low

enough

_q~ which prevents a

precise

determination of the

length

of the defects to be

perfornled.

Nevertheless their total

length

2L _can be estimated _as

conceming

at least lo molecules.

Besides one can evaluate the elastic energy of the defect with _use of the defornlation field

given

in

equations (21)

and

(23).

This energy

depends

_on the

strength

W of the vacancy. The

C ii

C~~ C(~

dilatation

Vo

^induced

by

the defect scales _as W

=

Vo

~ With the

assumption

ii

that

Vo

is of the order of _a molecular volume _one finds that the elastic energy of _one defect is of the order of kT. This conf1rnls the thernlal

origin

of the defect.

7. Conclusion.

We have

given

a theoretical

interpretation

of _a very

peculiar

feature of the

experimental X-ray

diffraction

pattem

^of some SmB

phases

_: the existence of _a white diffuse line close to the

origin

of the

reciprocal

_space. It is

interpreted

_as due to the presence of uncorrelated transient defects linked _{to a} disturbance of the

packing

in the molecule

strings. Although

this defect

mainly

concems a finite

portion

of

length

2 L in the _z

direction,

it also induces _a distortion in the other

directions

through

the 3D

elasticity

of the

hexagonal phase.

This

brings

about the

broadening

of the white _zone for

large

_q~. The

resulting anisotropic

elastic defornlation field leads to the observed diffusion

shape. Although

_our calculation is done in the frame of

small,

continuous and static

defornlations,

it still reveals the

importance

of the elastic

anisotropy

of the medium.

We have shown that the

shape

of the white _zone

depends

_on the _two

ratios,

_a

=

~j"

^and

11

lcii ^{c~~ c13}

p =

~

(m

the assumption a small

compared

to

p). p

is known from

[I I]

in

C11

50. 8 and _{we assume} that this ternl is

strongly

related _to the

crystalline

structure and therefore should be

nearly

the _same for all

compounds exhibiting

a SmB

phase

with

hexagonal

_compact

array. Then the

comparison

of theoretical results and

experimental

data in SmB

phases

provides

_an evaluation of

~~~

in

good

agreement ^{with the mechanical} measurements. It is

Cjj

worth

noticing

that

X-ray experiments

thus

give

infornlation _on the elastic constants via the distortion

generated by

the defects. However _one has to

keep

in mind

that,

in _a

general

case, the

assumption

of _a much smaller than p _may be ruled out

preventing

a

straightforward

derivation of C

j

j/C

m. This is

certainly

the _case for the SmG

phase

of several

compounds,

such

as

TBBA,

where C

ii/C~~

is one or two orders of

magnitude larger

than in 40. 8 _: the

previous

model would then

predict

_a

large broadening

^of the white _zone.

However, experimentally

this

zone remains of constant thickness and

thus,

the I -dimensional treatment of

[3]

_{seems more}

appropriate.

References

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liquid crystal phases

(World Scientific,

Singapore,

1988).

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Phys.

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Technique

de la

Radiocristallographie

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Physical Properties

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Crystals

(Oxford, London, 1957).

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Phys.

Status Solidi 30 (1968) 645 KRONER E., Z.

Physik

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