Lattice Parameter Measurements from the Kossel Diagrams of the Cubic Liquid Crystal Blue Phases

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Lattice Parameter Measurements from the Kossel Diagrams of the Cubic Liquid Crystal Blue Phases

Richard Miller, Helen Gleeson

To cite this version:

Richard Miller, Helen Gleeson. Lattice Parameter Measurements from the Kossel Diagrams of the Cubic Liquid Crystal Blue Phases. Journal de Physique II, EDP Sciences, 1996, 6 (6), pp.909-922.

�10.1051/jp2:1996219�. �jpa-00248340�

J.

Phys.

II France 6

(1996)

909-922 JUNE 1996, PAGE 909

Lattice Parameter Measurements from the Kossel Diagrams of

the Cubic Liquid Crystal Blue Phases

Richard J, Miller and Helen F, Gleeson

(*)

Department

of

Physics

and

Astronomy,

Schuster

Laboratory,

Manchester

University,

Manchester M13 9PL, UK

(Received

6 June 1995, revised 19 December 1995,

accepted

20

February1996)

PACS.61.30.Eb

Experimental

determinations

ofsmectic,

nematic,

cholesteric,

and other structures PACS.42.25.Fx Diffraction and scattering

Abstract, The Kossel

diagram technique

has been used in the

study

of the blue

phases

of

liquid crystals

for _some years. In this paper

quantitative

information extracted from Kossel

diagrams

of blue

phases

_one

(BPI)

and two

(BPII)

is

presented.

The method used

provides

absolute measurements of the lattice parameters in these blue

phases

to _an accuracy of about

I%. The measurements reveal _a distortion in the BPI structure which is not obvious from

qual-

itative examination of the Kossel

diagrams

themselves. The

equality

of the cholesteric

phase

helical

pitch

at the transition _to BPI and the BPI helical

pitch

at the transition to BPII is _con- firmed. The

experimental technique

involved the _use of _a

high stability

temperature

controller,

reflection microscope,

high

resolution CCD _camera and

image digitization

system. The Kossel

diagrams

_were calibrated

using

_a

reflecting

diffraction

grating

and hence measurements of lattice parameters

required only

the _use of _a

single wavelength

of

light.

1. Introduction

The blue

phases iii

of

liquid crystals

_are

thermodynamically

distinct

phases

which _occur in materials with

high chirality. They

have been studied _now for _{some years} and the basic _reasons for their existence _are believed to be understood. Most of the blue

phases

have structures with

three dimensional

periodicities

of the order of visible

wavelengths, making Bragg

diffraction of visible

light

the most useful tool for their

study.

The Kossel

diagram technique

_was first

developed

for the

X-ray study

of

crystals

_[2].

Highly convergent

or

divergent light

_causes the

Bragg

condition to be fulfilled for several different

crystal planes

and the

resulting pattern

of diffracted

light gives

much information about the _structure of the

crystal.

In recent years ^t he Kossel

diagram technique

_[3] has been

adapted

for the

study

of the

periodic

blue

phases,

in

particular

for

determining

their

crystallographic

_{space groups [4].}

In work

by previous

authors [5] the lattice parameters of the blue

phases

have been de- termined

by measuring

the spectrum of

light

^{reflected from} or transmitted

through

the blue

phase

structures. In the work

presented

here the Kossel

diagrams

of the cubic blue

phases,

BPI and

BPII,

_are used _to

give

measurements of the lattice

parameters

of the

phases.

The Kossel

(*)Author

for correspondence

(e-mail: helen.gleeson@man.ac.uk)

@

Les

(ditions

_de

Physique

₁₉₉₆

diagram technique

has the

adilantage

_over

spectral

measurements since it allows simultaneous measurements to be made from _{a range} of different

crystal planes. Also,

all the measurements are made _on the _same monodomain

region

of the blue

phase sample.

The lattice

parameter

measurements of the blue

phases

_are

presented

_{as a} function of temperature and discussed for both BPI and BPII.

Using

the theoretical models of the cubic blue

phases [6],

the

temperature dependence

of the blue

phase

helical

pitch

is derived from the lattice

parameters

and

compared

to the

pitch

of the

adjacent

cholesteric

phase,

measured from the cholesteric reflection

spectra.

2~

Experimental

Method

2.I. THE KossEL DIAGRAM

TECHNIQUE.

The most convenient way ^to

provide highly

convergent light,

to

generate

Kossel

diagrams

from the small

sample

size

typically

used for

liquid crystals,

is via a

high magnification microscope objective.

Some of this incident

light.

interacting

with lattice

planes

within the

sample,

will then fulfil the

Bragg

condition and be diffracted while

light

from other directions will _pass unhindered

through

the

sample (Fig. 1).

This diffracted

light

^will form part ^of a cone of

light

diffracted from _a set of

planes

in the

sample

whose axis is the

reciprocal

lattice vector of the

planes

and whose _cone

angle

^is

given by

the

Bragg

condition. Where the Kossel lines _are observed in reflection the

objective

also collects the _cone of diffracted

light

which _comes to a focus in the back focal

plane.

Here it

produces

_a

circle, ellipse

_or line

depending

_upon the orientation of the _cone axis. Since the incident

convergent light will,

in

general,

^fulfill the

Bragg

condition for _more than _{one set} of

planes

in the

sample

the back focal

plane image

consists of sets of

circles, ellipses

and lines.

These

depend directly

_on the

wavelength

^{of the} incident

light

and the orientation and size of

Back focal

plane

Objective

~

Convergent

_,

monochromatic

',

~~~~~~~~~

light

~- _'

~ jet

^Oi

Planes

~

1n tile

sample

/ ,

/ ,

/ ,

/ ,

/

_' ^"

Fig.

I. Schematic

diagram illustrating

the Kossel

diagram technique.

The Kossel

diagram

is

generated by

the

focusing

of

parallel light

reflected from the

sample.

N°6 LATTICE PARAMETERS FROIVI BLUE PHASE KOSSEL DIAGRAl/IS 911

p

s

o

R~dius ₌ 21kol

k~

Fig.

2. Construction used to determine the _cone of

light

diffracted from the

reciprocal

lattice vector T in the

sample (see text).

Two

example

incident and diffracted wavevectors _are

shown,

ko and kr

respectively.

the

reciprocal

lattice vectors in the

sample.

The

image produced by

them is called _a Kossel

diagram.

Theoretical Kossel

diagrams

_may be constructed via _a

fairly simple geometrical technique

_[7].

Consider the lattice of

points

in

reciprocal

_space

corresponding

to the sets of

planes

in real space, each of which is

given by

_a

reciprocal

lattice vector _T with b~iller indices

[hki].

From the

Bragg scattering

condition it may be _seen that

jrj

_m

2j~jsino, (1)

where _~c is the wavevector of the

light

in the

sample

and 9 is the

Bragg scattering angle.

Hence

only

those

reciprocal

lattice vectors which describe

points

in

reciprocal

_space within _a

sphere, S,

of radius 2(~c( centred _on the

origin 0,

_{may cause}

Bragg

diffraction of the incident

light (Fig. 2).

^{The incident}

light

diffracted from the

reciprocal

lattice vector forms _{a cone~}

C,

described

by

the

origin

of the

sphere

and the intersection of the

sphere

with _a

plane P, perpendicular

to the

reciprocal

lattice vector. This

plane

contains the

point

in

reciprocal

_space described

by

the

reciprocal

lattice _vector. The Kossel

line,

viewed in the back focal

plane

of the

microscope,

is then

given by

the

projection

of the intersection of S and P onto _a

plane perpendicular

to the

viewing

direction

(Fig. 3).

From this construction it may be _seen that

light leaving

the

sample

at _a

polar angle

_o to the

viewing

direction is focused to a

point

on

the back focal

plane

such that the radial distance of the

point

from the centre of the

image

is

proportional

to sin _a. This

relationship

is considered in _more detail in the results section.

In order _to achieve _as

large

a range of incident

angles

of

light

_as

possible

it is necessary ^to choose _an

objective

for the

microscope

with _a

large

numerical aperture, ^iv. Via Snell's law this relates

directly

to the

largest angle, A,

of

convergent light

in the

sample,

relative to the

central axis of the

optics,

such that

N = n sin

A, (2)

where _n is the refractive index of the

sample. Liquid crystals typically

have _a refractive index

iePing

dkecfion

Kossel

diagram

in

~4eving plane

~fitl ~

' ~

~

Reciprocal

lattice Nwtor

21kol

a

Fig.

3.

Projection

of Kossel lines

on the surface of the

sphere,

in

reciprocal

_space, onto the

viewing plane

to construct _a Kossel

diagram.

Note that

points

_on the

viewing plane

_are related to points _on

the

sphere by

the sine of the

angle

_o.

of the order of 1.6.

Hence, using

a 100x oil immersion

objective

with _a numerical aperture ^of 1.3

gives

_a maximum _cone

angle

inside the

liquid crystal

of about 55°.

The refractive indices of the

sample

_were determined

using

an Abb4 refractometer _{[8] in} both the cholesteric and blue

phases.

This allowed determination of the lattice

parameters

^from the Kossel

diagram images

_as well _as the helical

pitch

from the cholesteric

phase

reflection

spectrum.

^The refractometer allows measurement of both the

ordinary

and

extraordinary

refractive indices in the cholesteric

phase.

The accuracy of the refractive index measurements were of the order of +0.002 with _a

temperature stability

of 0.I °C.

2.2. THE KossEL DIAGRAM APPARATUS. As discussed

above,

Kossel

diagrams

_are viewed

in the back focal

plane

of the

objective

when the

sample

is illuminated with monochromatic

light.

In these

experiments

_an Ion Laser

Technology

450ASL tuneable _argon ion laser _was used

as a

light

_source. This _was

coupled

to the reflection _arm of the

microscope using

_a short

length

of fibre

optic

bundle. The laser _was tuneable and exhibited two

strong

emissions at 514.5 _nm and 488.0 _{nm as} well _as ,veaker

emissions,

at

wavelengths including

476.5 _nm and 457.9 _nm.

The _power output in the

strong

^{lines could be varied from 3 mW} to 20 mW but

generally only

3 mW _was needed to

produce

_a clear

image.

These

wavelengths

_are

ideally

suited to

viewing

the blue

phase

Kossel lines since

they

_are at the short end of the visible

wavelength

_range. The

speckle pattern,

due to the coherence of the laser

light,

_was

effectively

removed

by vibrating

the fibre

optic

bundle. The fibre

optic

cable _gave the added

advantages

of

depolarizing

the

light

and

allowing

_easy

coupling

of the

light

_source to the

microscope

reflection _arm.

N°6 LATTICE PARAMETERS FROM BLUE PHASE KOSSEL DIAGRAMS 913

CCD video

camera

Adjustable

lens holder and lens

Fixed

polarizer

Optical

fibre holder

Optical

fibre

Microscope

from laser

reflection _arrn

Microscope

focus

adjustment

Insulating

steel extension tube Oil immersion

objective Objective heating

collar

Heating stage

Fig.

4. Schematic

diagram

of the

microscope configuration,

used for

taking pictures

of the Kossel

diagrams, showing

the apparatus ^built to

image

the back focal

plane

of the

objective.

An

optical

fibre is used to

couple

the laser to the microscope reflection _arm.

A Bertrand

lens, normally

used to obtain _an

image

of the back focal

plane,

_was not available for the

microscope. Hence,

it _was necessary ^{to construct} some

equipment

which would

image

the back focal

plane

at the video _camera. This Kossel

diagram imager

was

designed

to

replace

the trinocular head and fixed

analyser

of the

microscope (Fig. 4).

At the bottom of the

imager

is _a base which both holds _a

polarizer

and attaches to the

microscope.

Above

this,

three 35 _cm steel posts

support

^{the CCD} camera situated at the centre of _a metal

plate.

The _camera used

was a Hitachi Denshi KP-Ml black and white CCD _camera since this is small and

light

but at the _same time

gives high

resolution

(640 by

480

pixels).

A second

plate holding

_a

simple

biconvex lens is free to _move

along

the steel posts on

bearings

and its

position

is controlled

by

a

long

steel rod

tapped

with _{a screw} thread. The

focusing

of the _camera is then controlled

by

simply turning

the

tapped

steel rod. An aperture of about I _cm diameter _was

placed

in front of this lens to

help

cut out

stray light

from the

image.

The _use of _an oil immersion

microscope objective requires placing

the

objective

in thermal

contact with the

sample. Hence,

it _was necessary ^to ensure that almost _no thermal

gradient

exists between the

objective

and the

sample by controlling

the

objective

temperature. ^The

temperature

^{of the}

objective

_was matched to that of the

sample using

a brass

heating collar,

which fitted around the

objective.

The

temperature

of the brass

heating

collar _was controlled

independently

of the

sample using

_a Linkam Tb~S90

temperature

controller with _a

stability

of 0.01 °C. The

objective

_was also insulated from the

microscope body by

a thin steel extension

tube. The

temperature

read-out of the Linkam controller _was calibrated

against

that of the

sample heating stage by

the _use of _a

sample phase

transition. The

temperature gradient

_across

the

sample

could then be held to within 0.04 °C due to the _{errors on} the calibration. The

sample heating stage

itself has been described in detail elsewhere [9] and had _a

temperature stability

within 0.01 °C.

2.3. PITCH MEASUREMENTS _{IN THE} CHOLESTERiC PHASE. For

comparison

with the lat-

tice parameter measurements made in the blue

phases

the

pitch

of the

adjacent

cholesteric

phase

_was measured in the

region

of the cholesteric _to blue

phase

transition. The

pitch

_was

determined,

from the selectii~e reflection

spectrum

^{and refractive indices of the} cholesteric

phase,

to an accuracy of +3 nm. This is _{a couimon}

technique

and has been described in detail elsewhere

[10].

2.4. DAT.~ CAPTURE AND ANALYSIS. Video

pictures

^of the Kossel

diagrams,

taken

by

the

CCD _{camera, were}

digitized using

a DIPIX

Technologies

P360F frame

grabber

_[11] and

image

processing

^software _[12] both installed _{onto a} 486 IBM

compatible personal computer.

The

digitized pictures produced

have _a resolution of 640

by

480

pixels.

The software allows many

images

of the Kossel

diagrams

to be taken and

averaged

to reduce noise and also allows direct

access to the

light intensity

information contained within the

images.

The

images

could also

be

processed

to

improve

^their contrast and

brightness

in order to

emphasise

certain features in the

diagrams.

Having

obtained the Kossel

diagram images they

must then be

processed

to obtain the lattice parameter information. As discussed above the Kossel

diagram image

_may contain

circles, ellipses

and

straight lines,

each of ~N-hich is

generated by

_a set of

reflecting planes

in the

sample.

It may be _seen, from Section 2.

I,

that the centre of _a circle _or

ellipse gives

the

projection

of the end of the relevant

reciprocal

lattice vector onto the

viewing plane,

in

reciprocal

_space.

For _a

straight

Kossel line this centre may be deduced

by taking

_a

perpendicular

from the Kossel line to the centre of the

diagram.

For the circle and

straight

^Kossel lines the

angle

of the relevant

reciprocal

lattice vector to the

viewing plane

is obvious. In the _case of the

ellipse, ho,vever,

the

angle

is

given by

the ratio of the

major

and minor _axes of the

ellipse. Hence,

the

reciprocal

lattice,,ector

given by

an

ellipse

_may

only

be deduced

easily

when both

major

and minor _axes may be _seen.

This

technique,

_as described _so

far, only/ gives

the

reciprocal

lattice vectors in terms of distances measured _on the Kossel

diagram image. Hence,

_a

scaling

^factor is

required

to convert these distances to

"lengths"

in

reciprocal

_space. This

scaling factor,

_{~y, may} be calculated from the numerical aperture, ^{N. If the} radius of the I<ossel

diagram image

is then

given by

r, it may be shown that

'f "

~), 13)

where k is the modulus of the wavevector of the monochromatic

light

in _{a vacuum.} A _more accurate method could

require fitting

_an

ellipse

to each Kossel line.

N°6 LATTICE PARAMETERS FROIVI BLUE PHASE KOSSEL DIAGRAMS 915

la) 4CB

-

h

/

h

/

C4H9

16) CE2

~§ /

~

,,

f~CH2 I

/

I

/

~ /

~~~

~

/

~~~

C2H5

.,

~

~'~C~H~

~~3

Fig.

5. Chemical structures of the

compounds

used in the mixture for this _paper. These compounds

were

provided by

Merck UK Ltd.

3. Materials

The material used in this work _was chosen because it exhibits both of the cubic blue

phases (BPI

and

BPII)

with _a

reasonably

useful

temperature

_{range (+~ I}

°C), Bragg

reflections in the visible

light

_range and at _a convenient

temperature

_{(+~ 45}

°C).

It consisted of 20.1+ 0.2 mole

percentage

^of CE2 in 4CB. Both of these

components

were

provided by

Merck UK Ltd

[13]

and their

naming

convention has been used. The chemical structures of these

compounds

_are

shown in

Figure

5.

4~ Results and Discussion

Kossel

diagram images

_,vere taken at _a range of different

temperatures

across BPI and BPII of the mixture of CE2 and 4CB discussed above. The

wavelength

of the argon ^ion laser _was tuned to 514.5 _nm.

Figures

6 and 7 show

examples

of Kossel

diagram images

taken of BPI and BPII

respectively.

Each of the

figures

show two different

crystal

orientations which _were identified with the aid of theoretical Kossel

diagrams [14].

The theoretical Kossel

diagrams

for

approximately equal

lattice

parameters

_are shown

alongside

the

images

in each of the

figures picking

out the

reciprocal

lattice vectors associated with each line. In these

diagrams only

those lines which _{appear on} the top surface of the

sphere

in

reciprocal

_{space are} shown since the

microscope objective

_can

only

collect

light

diffracted towards it. These

images

are of

comparable quality

to those found in other,vork

[15].

Examining Figure

6a in _more

detail,

Kossel lines associated with the

reciprocal

lattice vectors

(011), (101), (l10)~ (T01), (T10), (002)

and

(020)

_may be

clearly

_seen,

though (01i)

and

(0il)

are

barely

visible. This is

because,

for these lines _to be _seen

clearly,

the

light

incident _on the

crystal

must be

travelling

^{towards the}

objective.

These lines

do, however,

_seem to appear as

darker lines

against

a

background

of faint and diffuse scattered

light.

In

Figure

fib the _same

problem

occurs with

seeing

the

(0T1), (T01), (i10)

and

(1T0)

lines.

Figure

7 shows Kossel

diagrams

taken from BPII with

viewing

directions

along [011]

and

ill I].

In these

diagrams

all the

expected

lines _can be

clearly

_seen.

Figure

7a shows

(010)

and

(001)

lines while

Figure

7b shows these lines

plus (100).

In Section 2.I _a

geometrical technique

_was discussed for

generating

theoretical Kossel dia- grams and this

geometry

^{has been used in}

calculating

values for the

reciprocal

lattice vectors,

presented

below. This involved

considering

the

projection

^of the intersection of _{a cone} and

la)

(l10) (200)

~

(002) (l10)

(011) (011)

j00) (101)

(b)

001)

(l10)

(oil)

(oil) (no)

Fig.

6. Two Kossel

diagram images

taken in BPI at 514.5 _nm, 0.29 °C below the BPI to BPII

phase transition, for different crystal orientations.

(a)

[011] viewing ^direction.

(b) [l12] viewing

direction. Theoretical Kossel

diagrams corresponding approximately

to the

images

_are also shown to aid identification of the different lines. The dashed circle in the schematic

diagrams

shows the limit of the field of view in the

images.

sphere

onto a

viewing plane. However,

this

experiment

^{has used}

optics

^with a

large

numerical

aperture

(+~

1.3)

_so it is

possible

that distortions _are

present

in the back focal

plane

of the

objective

which will affect the results. In order to check for such distortions the

optics

of the

system

were calibrated

using

a

reflecting

diffraction

grating.

^{The laser beam} was

aligned

to pass

through

the

optical system

and hit the diffraction

grating

at normal incidence. The Kos- sel

diagram generated

from this consisted of

a series of

spots corresponding

to the diffraction maxima. The

objective

collected

light

from diffraction maxima up ^to the sixth order. The

angle, 9m,

at which the

light

from the m-th order enters the

objective

is well defined and

given by

~ = non sin

9m

₌

@, (4)

where ~ is the

wavelength

of

light,

d is the

spacing

of lines _on the

grating,

_non is the refractive index of the immersion oil used with the

objective

and _~ is _a parameter ^which from Snell's law is

independent

of the refractive index of the medium. There is then found to be _a

good

linear

relationship

between the

positions

of the diffraction

grating

Kossel

diagram peaks,

in

pixels,

N°6 LATTICE PARAMETERS FROM BLUE PHASE KOSSEL DIAGRAMS 917

(a)

,/~

~°~°~ )

^"

' j

'

~

,~ (001)

~

^'

lb)

loio)

/~~~~,

' ',

/

~

(001)

_,'

i i

1

/ ,

'~ ~"~

(100)

~

Fig.

7. Two Kossel

diagram images

taken in BPII at 514.5 _nm, 0.10 °C below the BPII to

isotropic

phase transition, for different orientations of the crystal structure.

(a)

[011] viewing direction.

(b) [iii] viewing

direction. Theoretical Kossel

diagrams corresponding approximately

to the

images

_are

also shown to aid identification of the different lines. The dashed circle in the schematic

diagrams

shows the limit of the field of view in the

images.

and the

parameter

_~ oc sin

9m (Fig. 8).

This linear fit shows that the

geometrical

construction considered in Section

2.I,

and illustrated in

Figures

2 and

3,

is valid for the

high

numerical

aperture optical system

used here. This

technique

also allows the numerical aperture,

N,

of the

optical system

to be measured

directly

from the

edge

of the Kossel

diagram image.

The

numerical aperture was found to be 1.23 + 0.01.

Using

the method discussed in Section 2.4 it is

possible

to measure the absolute value of the

reciprocal

lattice vector for certain Kossel lines shown in these

images.

Measurements

were made

using

this method for several of the Kossel lines and the

layer spacings

associated with these lines _are illustrated in

Figure

9. These measurements _were made _on the

(011), (0il), (01i), (020)

and

(002)

lines in BPI

diagrams,

similar to that shown in

Figure

6a.

In BPII the

(010)

and

(001)

lines _were measured from

diagrams

similar to

Figure

7a. The refractive index is

important

in

determining

the

layer spacings

of these

phases. Using

the Abb4 refractometer this _was found to be 1.598 at 514.5 _nm and

effectively

constant over the whole blue

phase temperature

_range.

Figure

9 also illustrates the transition

temperatures

^of the mixture

(+0.01 °C).

~ Peak

positions

G Residuals 6

m 4

u ~

i

2

.~

-~---~- -i---§- 'i

H

m ~

j

~f

~ i~

u

n- _4

-6

-1.5 -1 -o.5 o o.5 1.5

~ = m1/d

Fig.

8. Calibration

plot

for the Kossel

diagram images generated by

_a

reflecting

diffraction

grating.

This shows the

position

of the diffraction

peaks

_on the

image

_{w. j} where ~ is

proportional

to the sine of the diffraction

angle (see text).

This also sho~vs the residuals of the _curve fit.

~

~

§

₊

§ ~~~ ~~~(

$

i~ $

$ f ~~( )~~ll)/(01-1)

~ 8 . BPII (010)

e

fl

_{A BPII (001)}

~ .~ j

8

fl

BPI ~pjj ^C~

i~

fl .I

b ~

g

I

"

i * ~ # j

~ # @

47.0 47.2 47.4 47.6 47.8 48.0 48.2 48.4

Temperature

fC

Fig.

9. The layer

spacings

associated with various

reciprocal

lattice vectors in blue phases _one and two. The transition temperatures are illustrated

by

vertical lines and _were measured to +0.01 °C.

The lattice

parameter

measurements from Kossel lines made in this _{way are}

highly

accurate.

Measurements made _on the radii of the Kossel line

ellipses typically

had _errors of the order of

2%

for the

(020)

and

(002)

lines and

0.5%

for

(011), (0T1)

and

(01T)

lines in BPI. For BPII the radii of the

(001)

and

(010)

Kossel lines _were measured to

1%.

Errors _on the final results _were

deduced

using

the statistical

propagation

of _errors formula. These _were found to be about

1%

for the

(011), (0T1)

and

(01T)

lines in

BPI,

while for

(020)

and

(002)

lines in BPI and

(001)

and

(010)

lines in BPII the _error is about

0.5%.

The reduction in error,

particularly

for the

(020)

and

(002)

lines in BPI is

surprising

and deserves _more consideration. The

propagation

of _errors formula

depends

_on the

partial

differential of _a function with respect ^{to its various}

N°6 LATTICE PARAMETERS FROM BLUE PHASE KOSSEL DIAGRAMS 919

terms. If the

major

axis of _a Kossel line

ellipse

is

given by

a then it may be shown that

where _r is the modulus of the

reciprocal

lattice vector, n is the refractive index of the blue

phase,

k is the modulus of the _wave vector of the

light

in free _space and

~y is defined in

equation (3). Hence,

when _{r m} 2nk the

partial

differential in

equation (5)

is

small,

_as is the

case for the

(020)

and

(002)

lines in BPI shown

here,

and the _error in the

reciprocal

lattice

vector

only depends weakly

_on the _error in the measurement of _a from the Kossel

diagram.

However,

for the other Kossel lines measured here

equation (5)

is not small and the _{errors on} the

reciprocal

lattice vectors _are

correspondingly larger.

The _errors

quoted

for measurements from the Kossel

diagrams

_are based _on the confidence in

measuring

the _centres of the Kossel

line _cross sections rather than the widths of the lines.

The

graph

in

Figure

9 shows _a distortion of the BPI cubic structure exhibited

by

the differ-

ence in

layer spacings

between that associated with the

(011)

reflection and the _ones associated with the

(011)

and

(01T)

reflections. These

suggest

^{that the blue}

phase

cubic structure has been

compressed by 2.6% along

the

[011]

^{direction and stretched}

by

the _same amount in the

[011]

^and

[0T1]

directions. Work

by previous

authors

[16]

^{has shown that with such} a distortion

the

(020), (110)

and

(i10)

Kossel lines do _{not cross at} the _same

point,

_as

they

do in

Figure

6a.

However,

theoretical Kossel

diagrams (Fig. 10)

for distortions in the

[011]

^{direction of}

2.6%, 10%

and

20%

show that the

changes

in the Kossel

diagram

_are

highly

non-linear.

Moreover,

the three theoretical Kossel lines appear ^to still _cross at the _same

point

for _a distortion of

2.6%.

Hence,

such _a small distortion is not inconsistent with the Kossel

diagram

shown in

Figure

6a.

This

distortion,

in the absence of electric

fields,

is most

likely

^due to _some

pinning

effect of the

periodic

structure at the

sample

cell

glass

surfaces and _a consequent distortion from the

natural cubic structure. It is _not

possible

to say whether _a similar distortion is

occurring

in BPII due to _a lack of information about _any reflections other than the

(001)

and

(010).

The helical

pitch

in the blue

phases

_{may now} be considered. In BPI the

(002)

reflection _near the middle of the

phase,

at 47.33

°C, gives

_a

layer spacing

of 173.8 + 0.6 _nm. The

accepted

structure of BPI

(space

_group

14132)

has unit cell size ~vhich

corresponds

to a full turn of the blue

phase

helix

[6],

while the

layer spacing corresponds

to a half of this unit cell size.

Hence,

in BPI the helical

pitch

is

given by

347.6 +1.2 _{nm at} 47.33 ^°C.

Similarly,

in BPII

(space

_group

P4232)

the unit cell size

corresponds

to a half turn of the helix.

Hence,

at 47.78 °C the

layer

spacing

of the reflection

gives

a helical

pitch

of 377.0 +1.8 _nm. The helical

pitch

_across the two blue

phases

is sho~vn in

Figure

ii. This also shows the helical

pitch

measured in the cholesteric

phase

of this material deduced from the selective reflection spectra. as described in Section 2.3.

From these data it _can be _seen that the helical

pitch changes

little from the cholesteric

phase

to

BPI,

_across the

phase

transition. In BPI the

pitch

decreases with

increasing

temperature

which reflects _a

shrinking

of the unit cell size. Previous authors [17] have found _an

equality

between the helical

pitch

in the cholesteric

phase,

at the transition to

BPI,

and the helical

pitch

in

BPI,

at the transition to BPII. Within _errors this

equality

_may be _seen in

Figure

ii.

At the BPI to BPII transition the

pitch

increases

by

about

10%

and then continues to decrease with

increasing

temperature _up to the

clearing point.

The decrease in

pitch

with

increasing temperature,

within the blue

phases,

_may be

explained qualitatively by considering

director

configurations

in the theoretical models of the blue

phases.

The blue

phases

exist because of _a balance between the increase in free _energy due disclinations and the decrease in free _energy due to the _so called double twist tubes

[6].

^{The free} energy of the double twist tubes is reduced

by

_an increase in the

chirality

_or

shortening

of the

pitch

and

hence

by

_a reduction in the unit cell size.

However,

at the _same time _a reduction in the unit

(a)

2.6fb

(110)

(020)

(iio)

16) 10fb

(c) 20fb ,./ ".,

:. x

Y~/z Y~z

Fig.

10. Theoretical Kossel

diagrams illustrating

the non-linear effect of distortions of

(a) 2.6%,

16) ^{10% and}

(c)

20% in the [011] direction of the blue

phase

_one lattice.

cell size _{causes an} increase in the

gradients

of the director field around the disclinations and _an

increase in the free _energy associated with the disclinations. This balance fixes the size of the unit cell.

However,

the free energy ^cost of

forming

the disclinations reduces _as the

temperature increases,

due to the

changing

order

parameter, shifting

the balance towards _a smaller unit cell size and shorter

pitch.

This

argument

may also

explain

the increase in the

pitch

from BPI to BPII.

Since,

_as mentioned

above,

the BPII _structure contains twice the

length

of disclination lines per unit volume _as

BPI,

the increased _amount of disclinations in BPII shifts the balance

discussed above towards

longer pitch.

Now

considering

the

equality

of the cholesteric

phase pitch

at the transition to BPI and the BPI

pitch

at the transition to BPII. The models show that both the cubic blue

phases

contain the _same ratio of disclination lines _to double twist tubes but that the

packing

^of these is much

denser in the BPII _structure.

Hence,

the _same critical

pitch

is

required

in both

systems

^for

N°6 LATTICE PARAMETERS FROM BLUE PHASE KOSSEL DIAGRAMS 921

x BPI

pitch

from

(020)

o BPI

yitch

^from

(002)

D BPI

pitch

from

(010)

o BPII

pitch

from

(001)

.

ChPitch j~

E Cholesteric

phase j

f

#

BPI BPII .9

( it

,e

fl

Z

_~f

^z

~

if~~

46.0 46.5 47.0 47.5 48.0 48.5

Temperature

/°C

Fig.

ii. The helical

pitch

measured in the cholesteric

phase

and blue

phases

_{as a} function of

temperature. Different

points

in the blue

phases,

at the _same temperature, represent measurements derived from different lattice _vectors

(see key).

the double twist structures to _overcome the free energy ^cost of

forming

disclinations.

However,

BPII forms _at the

higher temperature

since the denser

packing

^{of double} twist tubes and disclination lines

requires greater

curvature of the director in the

intervening

_space, which is allowable due to the lower elastic _constants at the

higher

temperature.

In other work

[18]

^the

temperature dependence

of the lattice

parameters

^{in BPI} are

generally

found to be much

stronger

^{than in BPII.}

However,

_no

significant

difference _was found here.

Also this work confirms the

discontinuity

in the helical

pitch

^{between BPI and BPII found} elsewhere. This

discontinuity

has been associated with _a contraction in the unit cell size from BPI to BPII

of12% [19]

ⁱⁿ

differing

materials.

However,

in _common with other work

[I],

here the unit cell size _was found to contract

by

about

45%

from BPI to BPII.

5. Conclusions

In this _paper

quantitative

lattice

parameter

measurements of the cubic blue

phases

have been taken

directly

from their Kossel

diagrams.

It is clear that this

technique

of

measuring

lattice

parameters

^and

pitch

allows _more detailed

comparisons

between the blue

phase

structures found in different materials than _was

previously possible.

The _use of this

high

resolution Kossel

diagram imaging system

^allowed the observation of distortions

(2.6% along

the

[011]

direction)

in the BPI structure, in the absence of _an

applied field,

which _were not apparent from

qualitative

examination of the Kossel

diagrams.

The distortion _was considered to be due to _a surface

pinning

effect. The existence of this distortion is

important

in

considering

the effect of external electric fields and thin cells _on blue

phase

structures since lower fields _may be needed for _a

phase

transition to _an electric field induced blue

phase. Indeed,

in _a very thin cell

(<

I

~tm),

it may be

possible

_create

enough

distortion in the blue

phase

structure to allow

observation of blue

phases normally

_seen

only

with _an

applied

electric _or

magnetic

field.

The

accepted

structures of the cubic blue

phases

were used to relate the lattice parameter measurements to helical

pitch

in the blue

phases.

The cholesteric

phase pitch

at the transition

to BPI _was found _to

equal

the BPI

pitch

at the transition _to

BPII, confirming

the results of

previous

authors. It _was

proposed

that this

equality

is due to the

equal

ratios of disclina- tion lines to double twist tubes in the

accepted

models of BPI and BPII. The temperature

dependence

of the blue

phase pitch

_was

qualitatively explained

in terms of the balance in free

energies

of the double twist structure and the blue

phase

disclinations.

In future work this

apparatus

^and

technique

will be

applied

to

determining

^{of the}

dependence

of the measured distortions _on

sample

thickness and to _some of the

currently

unknown blue

phase

structures.

Acknowledgments

The authors would like to thank the referees for _{a very}

thorough

and considered

appraisal

of this work. Also William Deakin for useful

discussions,

the SERC

(now EPSRC)

for

provision

of

funding during

the _course of this work

(RJl/I)

and the

Royal Society

for _an

equipment grant

for the CCD _camera and

image digitization

system.

References

[1] Crooker P.

P., Liq. Cryst.

5

(1989) 751; Stegemeyer

H. et

al., Liq. Cryst.

1

(1986) 3;

Belyakov

V. A. and Dmitrienko V.

E.,

Sov.

Phys. Usp.

28

(1985)

535.

[2] Kossel

W.,

Loeck V. and

Voges H.,

Z.

Phys.

94

(1935)

139.

[3] Cladis P.

E.,

Garel T. and Pieranski

P., Phys.

Rev. Lett. 57

(1986) 2841;

JAr6me B. and Pieranski

P., Liq. Cryst.

5

(1989) 799; Heppke

G. _et

al.,

J.

Phys.

France 50

(1989)

549.

[4] Burns G. and Glazer A.

M., "Space Groups

For Solid State Scientists"

(Academic

Press,

London, 1978).

[5] Meiboom S. and Sammon

M., Phys.

Rev. A 24

(1981) 468;

Johnson D. L. et

al., Phys.

Rev. Lett. 45

(1980) 641; Heppke

G. et

al.,

^J.

Phys.

France 50

(1989)

549.

16] Dubois-Violette E. and Pansu

B.,

Mol.

Cryst. Liq. Cryst.

165

(1988)

151.

[7] Pieranski P. _et

al.,

J.

Phys.

France 42

(1981)

53.

[8] Model

60/HR, Bellingham

&

Stanley Ltd., Polyfract Works, Longfield Rd., Tunbridge Wells, Kent,

UK.

[9] Miller R. J. and Gleeson H.

F.,

Meas. Sci. Tech. 5

(1994)

904.

[10]

^{Miller R.}

J.~

PhD

Thesis, University

of

Manchester, Manchester,

UK

(1994).

[11]

^DIPIX

Technologies Inc.,

1050 Baxter

Rd., Ottawa, Ontario,

Canada.

[12]

^{Accuware version}

2.4,

Automated Visual

Inspection,

2519 Palmdale

Court,

Santa

Clara, CA95051,

USA.

[13] ^{Merck UK}

Ltd.,

^Merck

House, Poole, Dorset,

UK.

[14]

^Kitzerow

H.-S.,

PhD

thesis, University

of

Berlin, Berlin, Germany (1989).

[15]

^{J4r6me B. and Pieranski}

P., Liq. Cryst.

5

(1989)

799.

[16] ^{Cladis P.}

E.,

Garel T. and Pieranski

P., Phys.

Rev. Lett. 57

(1986)

2841.

[17]

^{Johnson D.}

L.,

Flack J. H. and Crooker P.

P., Phys.

Rev. Lett. 45

(1980)

641.

[18]

^{Crooker P.}

P., Liq. Cryst.

5

(1989)

751.

[19]

^Her

J.,

Rao B. B. and Ho J. T.,

Phys.

Rev. A 24

(1981)

3272.