First optical observation of Kossel diagrams in chiral smectic phases

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First optical observation of Kossel diagrams in chiral

smectic phases

H.-S. Kitzerow, G. Heppke, H. Schmid, B. Jérôme, P. Pieranski

To cite this version:

First

_optical

observation of Kossel

_diagrams

in chiral smectic

phases

H.-S. Kitzerow

_(1),

G.

_Heppke

_(1),

H. Schmid

_(1),

B. Jérôme

₍₂₎

and P. Pieranski

₍₂₎

(1)

Iwan-N.-Stranski-Institut, Sekr. ER 11, Technische Universität Berlin, StraBe des 17.

Juni 135, 1000 Berlin 12, F.R.G.

(2)

Laboratoire de

Physique

des Solides, Bât. 510, Université de Paris-Sud, Faculté des _Sciences, 91405

Orsay

Cedex, France

(Received

on March _{19, 1990,}

accepted

on

_May

_17,

₁₉₉₀₎

Résumé. 2014 Au moyen d’un

microscope polarisant

en

_{configuration conoscopique}

on étudie pour

la

_première

fois les

_diagrammes

de Kossel des

_phases

chirales SmC* et SmM*. Dans le cas de la

phase

SmC* on observe un seul anneau de Kossel

correspondant

à la

_{périodicité}

_p/2

de la

composante

tensorielle |m| = 2

de la structure hélicoïdale du

_paramètre

_{d’ordre 03B5(r).}

La

variation du pas p

(T)

en fonction de la

_{température,}

obtenue à

_partir

des

_diagrammes

de Kossel,

est en accord avec des mesures faites

indépendamment

par réflexion de

Bragg

en incidence normale aux couches

_smectiques.

Le

_diagramme

de Kossel de la

_phase

SmM * _comporte aussi un

seul anneau central. En

_plus

d’anneau de Kossel, dans le cas de la

phase

SmC* on observe des

spirales d’Airy.

Abstract. 2014

By

means of

_{optical microscopy,}

Kossel

diagrams

of chiral smectic

phases

are

observed for the first time. Due to

_{Bragg reflection,}

these

diagrams

occur in the focal

_plane

of the

microscope

lens if the

_sample

is illuminated with _convergent monochromatic

light.

The Kossel

diagram

for the SmC*

phase

2014 observed for the chiral

_compound

CE3

_{(BDH) 2014}

consists of a

central

_{ring corresponding}

to the

_periodicity

_p/2

of

_the

_{|m| =}

2 mode of the tensor order

parameter

03B5(r)

describing

the helical structure. The _temperature

_dependence

of the

_pitch

p(T)

determined

_by

the Kossel method is in _agreement with

_{spectroscopic investigations}

of the

Bragg

scattering

in back reflection. The Kossel

_diagrams

of the unknown chiral smectic structure

SmM * _{consist also of one central}

ring ;

no additional

_reciprocal

lattice vectors are observed in the

visible

_wavelength

range for this

phase.

In addition to the Kossel

diagrams,

we _report on the observation of

Airy spirals

in the SmC*

phase.

Classification

Physics

Abstracts 07.60P - 61.30

1. Introduction.

Smectic

_{liquid crystals}

formed

_by

rodlike molecules are characterized

_by

an orientational order of these molecules and

_{by layer}

structure

_[1].

The

_density

wave

perpendicular

to the

layers

can be described

_by

a wavevector _q,

_with

_{1 ql 1}

=27r/f,

where the

layer

spacing

f

is of the order of several

À

_(1-2

molecular

_lengths).

For some smectic

_phases,

the director

d (r ) indicating

the

_preferred

orientation of the molecular

_long

axes is tilted

by

an

_angle

T with

_respect

to the wavevector _{qt. In} the _presence of chiral

molecules,

the tilted smectic

phases

SmC,

SmI and SmF form a helix structure

_(Fig.

_1).

In terms of a tensor order

parameter

e(r),

the helix structure of the SmC*

_{phase corresponds}

to the

_following

Fourier series :

where non-zero coefficients e+ m or _{e - ID} describe a

_righthanded

and a lefthanded

helix,

respectively. Depending

on the

material,

the

pitch p

= 2 7r

/

1 qo 1

can exhibit a value between several hundreds of nm

(for highly

chiral

systems)

and

infinity (for

non-chiral

systems).

During

the last

decade,

the

theory

and the

_application

of chiral smectic

_phases

have found remarkable

interest,

since their ferroelectric

_properties

allow one to realize very fast bistable

electrooptical

devices

[2].

Fig.

1. - Schematic

_{representation}

for the structure of the cholesteric

(a)

and the chiral smectic

phase

SmC*

(b, c).

The director d

_representing

the

_long

molecular axes is

_{given by d (r )}

=

(sin

T cos

_{(qo. r ),}

sin T sin

_(qo.

_r),

cos

_T).

In the cholesteric

phase (a),

d is

perpendicular

to the helix axis, whereas the SmC*

_{phase (b)}

shows a tilt

angle

T with 0 T : 90°. In addition to the orientational order, the SmC*

In this _paper, a new alternative to

_investigate

the

_pitch

in smectic helix structures is

presented.

It is shown that the Kossel method which is a well known

_{scattering technique}

to

investigate

solid

_crystals

_[3],

colloidal

_crystals

_[4],

_{liquid crystalline}

blue

_phases

_[5]

and cholesteric

_phases

_[6],

can be also

applied

to

_investigate

helicoidal smectic

_phases.

So

far,

four different methods were used to

_investigate

the

_pitch

in these

_{phases. According}

to the method

_{by Martinot-Lagarde [7],}

the

_sample

is

contained

between two

_glass

slides. A uniform orientation of the helix axis

_parallel

to the surface is obtained

_by

slow

_cooling

of the

sample

into the smectic

_phase

while a

_magnetic

field is

applied.

With this

_preparation,

a

parallel, straight fringe

pattern

occurs where the distance between the

_fringes

is

_equal

to the

pitch

[8].

If the

_pitch

is

_larger

than 1 jjum, it can be determined

_{directly by}

observation in a

polarizing microscope.

Another method to determine the

_spacing

between the lines in a

_{sample prepared}

in the same _{way is the diffraction}

_method,

where the

_sample

is illuminated with a laser beam and the

pitch

is obtained from the distance between the interference maxima of the diffraction

pattern.

This

method,

initially

used to

_investigate

cholesteric

_{phases [9]}

was

_applied

to chiral smectic

_{phases by}

S. A. Rôzaùski et al.

_{[10, 11].}

The

Cano-method,

which is very common to determine the

_pitch

in cholesteric

_phases

_[12],

was also used to

_investigate

smectic helix structures

_[10].

In a

wedge shaped sample

with a

strong

anchoring

of the molecules so that the wavevector _{qo is orientiented}

_{perpendicular}

to

the

surface,

the helix structure is distorted

according

to the conditions that the elastic free

energy is minimized and that - at the same time -

everywhere

the

_sample

thickness is

_equal

to half of the

_pitch

times an

_integer.

For each

_change

of this

_integer

value

_by

_one, a disclination line occurs which can be observed in a

_{polarizing microscope.}

The

_pitch

can be obtained from the distance of these disclination lines.

Finally,

selective reflection which is well known from cholesteric

_{phases [13],}

was also observed for chiral smectic structures in the visible

_[14-16]

and in the infrared

_wavelength

range

[10,

17,

18].

This

phenomenon

was

_{predicted theoretically by}

D. W. Berreman

_[19]

and the

_theory

was

_developed

also

_by

other authors

_[20-22].

K. Hori

[14]

found that indeed two

selective reflection bands can be observed in the SmC*

_phase,

which

_correspond

to the terms

with the coefficients _e-t

_{1 and}

_{e+ 2} in

_{equation (1), respectively.}

The

_{respective wavelengths}

are related to the

_pitch

p

by

the

Bragg

condition

Here

_he

_{spatial periodicity d}

can be either d = _p

(corresponding

to the term with

lm 1 =;1

in the tensor order

_parameter)

or d =

p /2

(corresponding

to

1 m 1 = 2).

The _mean

refractive index n in

_{equation (2)}

appears due to refraction at the

_sample

surface,

the

_angle

à describes the direction of

_light

incidence with

_respect

to the helix axis. The

_{Bragg peak}

corresponding

to d =

p was found to exhibit a _very weak

_intensity

for

_light

incidence

_along

the

reciprocal

lattice vector _qo

_[14].

In

_conclusion,

several methods which were

_developed

in order to determine the

_pitch

in cholesteric

_{phases [9,}

_12,

_13]

were also

_applied

to

_study

chiral smectic

_phases

_[9-11,

_10,

_14-18].

Thus,

it seemed of interest to see whether also the Kossel method is suitable to

_investigate

smectic

_phases,

since this method is well known for blue

_{phases [5]}

and for cholesteric

_phases

[6].

2. Kossel

_diagrams.

Kossel

_diagrams

occur due to

_{Bragg scattering}

if

_divergent

or

_convergent

monochromatic radiation is scattered

_by

_{single crystals.}

These

_diagrams

were detected in the year 1934

_by

W.

obtained

_{by using}

a

single

_crystal

of copper as the anode in an

_X-ray

tube. In this

_experiment,

the

_crystal

was a source of

_divergent

radiation which was scattered

_by

the cubic structure of this

_{crystal. According}

to the

_Bragg

_condition,

the radiation reflected

_by

a

_family

of

_planes

described

_by

the Miller indices

_(hkf)

forms a cone. This Kossel cone is

_{perpendicular}

to the

planes

(hkl )

and exhibits an

_aperture

_angle

_à,

given by

the relation

where k is the wavevector of the incident radiation

_with

_{1 k 1}

= 2 Ir

/A.

Consequently,

W.

Kossel et al.

_[3]

observed dark and

_bright

lines on a

_photographic

plate,

each of these Kossel lines

_being

a cone section of a Kossel cone and thus

representing

the orientation and

interplanar

distance for the

_respective

set of

_planes

_(hkf).

Today,

the

_principle

of this method is used in order to

_{investigate crystal}

structures with

convergent

electron beams

_[23],

and for colloidal

_{crystals using}

visible

_{light [4]. Recently,}

it was shown that Kossel

_diagrams

can be observed also in

liquid crystals, namely

in the cubic blue

_phases

BPI

_[5]

and BPII

_[24]

as well as in the cholesteric

phase [6].

In these

applications,

the

_crystal

structures are illuminated with

convergent

monochromatic radiation

by

an external source. For the visible

wavelength

range, refraction at the

_sample

surface has to be taken into

account as in

equation (2),

and thus

3.

_Experiment.

In order to

_investigate

Kossel

_diagrams,

a

_{metallurgic microscope}

PME

_(Olympus)

was

equipped

with an oil immersion

_{objective exhibiting}

a

_large

numeric

_aperture

_(1.3).

The

sample (Fig. 2a)

was illuminated with

convergent

monochromatic

_light

and observed in back reflection. As shown for blue

_{phases [5],}

Kossel

_{diagrams occurring}

_{in the visible range} can be observed

_using

this

_{equipment. They}

occur in the focal

_plane

of the

microscope

lens and can be observed

_by

means of a Bertrand lens or

_{just by removing}

the eye

_piece

of the

_microscope.

The

_samples

were

_prepared

in two different ways. The substance was contained between a thin

_glass

slide and the flat end of a

_{glass cylinder}

which was

_transparent

in one

_type

of

_sample

preparation

or coated with an aluminium

_layer

in the other

_type

of

_samples.

In the first case,

only

the

_light

which is reflected due to

_{Bragg scattering}

can be

_observed,

whereas in the latter case also transmitted

_light

can be seen due to reflection at the second interface of the

_sample.

In both cases, the

sample

was illuminated with

_{linearly polarized light}

and the

_analyzer

was crossed with

_respect

to the

_plane

of

_polarization

of the incident beam.

For

_{chiral liquid crystal}

structures,

Bragg

scattering

can be observed in back reflection if the

respective reciprocal

lattice vector _{qo is} oriented

_along

the direction of

observation,

i.e.

perpendicular

to the

_sample

surface in our

_experimental

_setup.

In the case of chiral smectic

phases

this orientation _{of qo}

_corresponds

to an

_alignment

of the molecules

_{perpendicular}

to

the

_sample

surface. In order to obtain such a

_{homeotropic alignment}

of the

_liquid

_crystal,

the

inner

_glass

surfaces were coated with lecithin. For the cholesteric

_phase,

however,

a

_planar

(parallel) alignment

of the molecules at the

_sample

surface is

_required

in order to

_get

an

orientation _{of qo}

_{perpendicular}

to the surface. This

_alignment

could be induced

_by

shear

occurring

when the

_{glass cylinder}

was rotated with

_respect

to the cover slide in the

_sample

described above. The

_sample

thickness was

_adjustable

between about 5 _)JLm and 50 _Fim. The

temperature

was controlled

_by

means of Peltier elements

_using

an AC

_bridge

and measured

Fig.

2. -

Experimental

setup :

(a)

Kossel

_{diagrams (Fig. 3)}

and

_{conoscopic figures (Fig. 6)}

were

observed in a

_{polarizing microscope}

when

_illuminating

the

_sample

with convergent monochromatic

light. (b)

Reflection _spectra were measured

_using

a diode array spectrometer. The

_sample

was

For

_comparison

of the results of the

_microscopic

observation with

_{spectroscopic}

data of the

Bragg

scattering,

also the reflection

_spectra

were measured

(Fig. 2b).

For this purpose, the

sample

was illuminated with

white,

circularly polarized light

and the

_circularly

_polarized

part

of the reflected

_light

with the same handedness was

_subjected

to

_{spectral analysis by}

a diode

array

spectrometer

PR-702A

_{(Photo Research).}

The

_angular

_dependence

of the selective reflection was studied

_{by adjusting}

the

_{angle 0}

between the direction of

_light

incidence and the surface normal of the

_sample.

The

_position

of the detector was

_adjusted

as

_well,

so that the reflection condition was fulfilled for each measurement.

The material under

_{investigation}

was the well known

_compound

(+)-4-n-Hexyloxyphenyl-4-(2-methylbutyl)biphenyl-4’-carboxylate (CE3)

which was received

by

British

_Drug

House

_(BDH)

and used without further

_{purification.}

This

compound

was studied

_{previously by}

K. Hori

_{[14, 15]}

and

_by

P. Pollmann et al.

[16].

CE3 exhibits a smectic SmC*

_phase

as well as a cholesteric

phase,

both

phases

showing Bragg

reflection in the visible

wavelength

range. The transition

temperatures

reported

in reference

[14]

are : Cr 69.0 °C SmC* 80.0 °C N* 164.5 °C Iso.

Additionally,

we

_investigated

the

_{following pyrimidine}

derivative

This

_compound

was

_{synthesized recently by}

D. Lôtzsch

[27].

The

_compound

(2)

shows also selective reflection of visible

_light

for a chiral smectic

_phase.

_{However, X-ray}

_{investigations}

of this

_{phase by}

D. Demus et al.

_[28]

revealed a new smectic _structure, called

_SmM*,

which is different from the known smectic helix structures.

_According

to reference

_[27],

the transition

temperatures

are Cr 113.0 °C

_(SmJ

95.3 °C SmM*

_{102.0 °C)}

SmC* 154.5 °C Iso.

4. Results and discussion.

Kossel

_diagrams

were observed in the

_phase

SmC*

_{(Fig. 3)}

as well as in the cholesteric

_phase

of the chiral

_compound

CE3. The Kossel

_diagrams

for both

_phases

consist of a

_{single ring}

in the center of the

_{diagram indicating}

the existence of a

_reciprocal

lattice vector _{qo oriented}

along

the direction of observation. The diameter of the

_Kossel

_ring

decreases

_continuously

with

_{increasing wavelength A}

of the incident monochromatic

_light.

No Kossel line is observed if this

_wavelength

exceeds the value A * where the diameter of the central Kossel

ring

becomes zero. This

_{wavelength A}

* was measured as a function of

_temperature.

Additionally,

the

_{Bragg wavelength}

_Ào

of a

_{macroscopic sample}

was measured in back

reflection

₍₀

_{= 5°)}

_using

the

_{spectrometer.}

_Comparison

shows

_{(Fig. 4)}

that the values

À * obtained

_by

the Kossel method are about 15 to 20 nm

higher

than the values

Ào

measured

_{spectroscopically.}

This can be

_{explained by}

the linewidth of the

_Bragg

reflection. The

_{wavelength A}

o

represents

the maximum

intensity

of reflected

light

whereas the value

À * obtained

_by

the Kossel method

_corresponds

to the maximum

_wavelength

where the

intensity

is not zero.

_Indeed,

the difference between these

_{wavelengths corresponds}

to the

Fig.

3. -

Kossel

_diagrams

of the SmC*

_phase

observed with CE 3 for t = 70.1 °C for different

Fig.

4. -

a) Temperature dependence

of the

_Bragg

_{wavelength A0}

measured in back reflection

(diamonds)

and of the

_wavelength

A *

(squares)

where the central Kossel

ring

vanishes when the

wavelength

of the incident monochromatic

_light

is increased. The two different types of squares

corresponding

to

_repeated

measurements show that the results obtained

by

the Kossel method are

_good

Comparison

of the values A *

_{and À 0}

measured in this work with the data

_{given by}

K. Hori

[14]

shows that these

_{wavelengths correspond}

to the

_spatial

_periodicity

_{d = p /2}

in

equation

(2).

Consequently,

the

_pitch

p of the helix structure is related to the

_wavelength

A*

_by

where AA -- 15-20 nm is the

_spectral

halfwidth of the selective reflection band of the chiral smectic

_phase.

The reflected

_light

is

_{right circularly polarized, indicating}

that the observed Kossel line

_represents

a non-zero coefficient e, 2 of the

_respective

term in the

_expansion

of the

tensor order

_parameter

_(equation

_1).

As

_expected,

the values À.

_{and A 0}

for the

_Bragg

_wavelength

show the same

_dependence

on the

_temperature

_(Fig.

_4a).

A

_plot

of the

_reciprocal

values versus

_temperature

_(Fig.

_4b)

shows

that the

_pitch

does not

_{diverge completely}

at the transition

_temperature

SmC *-N *,

indicating

that this transition is of first order. This behaviour is in

_agreement

with the observations

by

P.

Pollmann and K. Schulte

[16]

that the transition

temperature

is shifted if the pressure is

varried. This latter behaviour indicated that the transition SmC *-N * is also connected with a discontinuous

_change

of the volume.

In order to _{compare the}

_Bragg

_scattering

of the smectic C*

_phase

and the cholesteric

_phase

more

_precisely,

the

_{angular dependence}

of the selective reflection was

_investigated

in CE3 for

two

_temperatures

_{(65.4 °C}

and

_{81.6 °C)}

where these two

_phases

show the same

_Bragg

wavelength.

As

_{expected by}

the

_{Bragg condition,}

this

_wavelength

decreases in both cases with

increasing angle 0

of

_light

incidence with

_respect

to the surface normal

_{(Fig. 5a).}

However,

the

_dependence

is

_slightly

different for both

_phases,

which can be

_{explained by}

the difference of the effective refractive index n in both

_phases.

Due to refraction at the

_sample

surface,

the

angle à

between the wavevector _{qo and the} direction of

_light

incidence inside the

_sample

is related to the

_{angle 0}

measured outside the

_{sample by}

Using

this

relation,

equation (2)

becomes

Accordingly,

a

_straight

line is

expected

if

(À / À 0)2

is

plotted

versus

sin2 (0) (Fig. 5b) :

From

_figure

5b the values

_nN*

= 1.724 and

nsmc*

= 1.585 were obtained for the refractive indices of the cholesteric and the smectic

_{phase, respectively.}

When

_{investigating}

the Kossel

_diagrams

on

_samples

with a mirror at the second

interface,

Airy spirals (Fig.

6)

were observed in addition to the Kossel line. These

_{conoscopic figures}

occurring

due to rotation of the

_plane

of

_polarization

are known for chiral solids such as

quartz

[25]

and such

_spirals

were also observed in cholesteric

_{phases [26].}

For a

_quartz

plate

observed in transmission with a circular

_analyzer,

an

_{Airy spiral}

with two branches can be

observed,

whereas

_{Airy spirals}

with four branches occur if two

_quartz

_plates

with the same

thickness and

_opposite

handedness are observed between crossed linear

polarizers.

In our

Fig.

5. -

Angular dependence

of the selective reflection of CE 3 for the chiral smectic

_phase

SmC*

( *, t = 65.4 °C )

and for the cholesteric

_{phase (+, 81.6 °C ): a) Bragg wavelength}

versus

_angle

0 of

_light

incidence.

_{b) Square}

of the reduced

_{Bragg wavelength}

versus

sin2

0for the same data as

Fig.

6. -

Airy spiral

observed in the SmC*

_phase

of CE 3 for t = 60.1 °C, A = 430 nm

_{«,: k 0)}

-reflection at the second interface. This leads to the same

_{conoscopic figure}

which is observed for two chiral

_plates

of

_opposite

handedness in transmission.

It should be stressed that we observed for the first time

_{conoscopic figures}

which show the

phenomena

of

_optical

rotation

_{dispersion (Airy spiral)}

and of

_Bragg

_{scattering (Kossel ring)}

simultaneously.

According

to the fact that the

_optical

rotation

_{changes sign}

at the

_Bragg

wavelength [13],

the

_{Airy spirals}

exhibit

opposite

handedness for

_{wavelengths larger}

and for

wavelengths

smaller than A _o,

_{respectively.}

However,

the

_shape

of the

_{Airy spirals}

was found

to

_depend

also on the

_sample

thickness.

Finally,

we wish to

_report

on the observation of Kossel

_diagrams

in the unknown smectic

phase

SmM*

_occurring

in the substance

_(2).

This

_{monotropic phase}

occurs at

_temperatures

between 95.3 °C and 102.0 °C

_{[27, 28].}

In the SmM*

_phase,

we observed a Kossel

_diagram

consisting

of a

_single

central

_ring,

as in the SmC*

_phase

of CE3. This Kossel line

_corresponds

to a

_reciprocal

lattice vector

_along

the

_viewing

direction. Since no other Kossel lines were

observed,

it can be concluded that this unknown

_phase

exhibits a helix structure

_{only along}

one

_{reciprocal lattice}

vector but no

_{periodicities along}

other directions which are of the order of the

_wavelength

of visible

_light.

5. Conclusions.

Phase in

_compound

₍₂₎

consist of one circle in the center. This Kossel

_ring

_represents the mode

_with

_{lm 1 =}

2 in the tensor order

parameter

(Eq. (1))

and the diameter of this

_ring

corresponds

to a

_spatial

_periodicity

of half of the

_{pitch (d}

_{= p /2).}

_According

to

_previous

observations,

the occurrence of a second Kossel

_ring

can be

_expected

which

_corresponds

to

the term

_with

_{1 m 1 =}

1 of the tensor order

_parameter

and a

_spatial

_periodicity

of d =

p.

However,

this Kossel line should occur

_only

for a _very

_large

_aperture

_angle

of the Kossel cone. It has not been detected so far.

The determination of the

_{wavelength A}

* where the diameter of the Kossel

ring

becomes zero on

_{increasing wavelength}

allows one to determine the

pitch

of the helical structure.

Although

the accuracy of this method is limited

by

the half width of the selective reflection

peak,

the values are

_good

_reproducible

and the

_temperature

_dependence

of the

_pitch

can be

checked

_{easily by}

this method. Additional information on the

_optical

rotation

_dispersion

might

be obtained from

_{Airy spirals}

observed in the same

_experimental

_setup.

A more detailed

_analysis

of the

_{shape of}

these

_{conoscopic figures}

_{is in progress.}

Acknowledgements.

The authors would like to thank D. Lôtzsch for

_making

the

_{compound (2)}

available to us and for useful _{discussions. We also express} our thanks to BDH Chemicals Ltd. for

_supplying

us with the

_compound

CE 3. This work was

_{supported by}

the Deutsche

Forschungsgemeinschaft

(Sonderforschungsbereich « Anisotrope

Fluide

»)

and

by

the Centre National de la Recherche

Scientifique.

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