Anomalies of the temperature dependence of electrostriction in blue phases

(1)

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Anomalies of the temperature dependence of electrostriction in blue phases

V. Dolganov, O. Lourie, G. Heppke, H.-S. Kitzerow

To cite this version:

V. Dolganov, O. Lourie, G. Heppke, H.-S. Kitzerow. Anomalies of the temperature dependence of electrostriction in blue phases. Journal de Physique II, EDP Sciences, 1993, 3 (7), pp.1087-1096.

�10.1051/jp2:1993185�. �jpa-00247884�

(2)

J.

Phys.

II France 3 (1993) 1087-1096 _JULY 1993, PAGE 1087

Classification Physics Abstracts

61.30

Anomalies of the temperature dependence of electrostriction in

blue phases

V. K.

Dolganov II),

O. R. Lourie

II),

G.

Heppke (2)

^and ^H.-S. ^Kitzerow

(2)

(')

_Institute of Solid State

Physics,

Russian Academy of Sciences, 142432,

Chernogolovka,

Moscow distr., Russia

(2) Iwan-N.-Stranski-Institut, Sektretariat ER II, Technische Universitit Berlin, Strasse des 17. Juni 135, D-1000 Berlin 12,

Germany

(Received 29 October 1992, accepted in

final form

30 Maich 1993)

Abstract. We investigated the field-induced Shift of the blue phase diffraction bands. The

temperature

dependence

of the electrostriction tensor components is determined for the first time.

In the blue

phase

BPI _at _constant field strength E fl [001], _a change of the temperture ^leads to a

change

of both the unit cell volume and its shape. The deformation of the unit cell

displays

_a

pretransitional

behaviour _on

approaching

^the ^BPI _- ^BPII

phase

transition temperature.

1. Introduction.

In the

liquid-crystalline

blue

phases

BPI and BPII

[I]

the local

ordering

of molecular orientation forms _a 3D

periodic

structure,

having

^the symmetry ^of

crystals.

BPI exhibits _a

body

centered cubic lattice

O~,

while BPII is described

by

_a

simple

cubic _structure

O~.

So

far,

essential progress has been achieved in the theoretical

description

and

experimental study

of blue

phases.

In the framework of the Landau

theory

on

phase

transitions and of the disclination model

[2-4]

it has been shown that for substances with _a small cholesteric

pitch

the cubic

structures,

O~

_and

O~,

_can be

thermodynamically

stable between the

isotropic liquid

and the

cholesteric

phase.

The structures have been determined

experimentally

and their

optic

characteristics have been examined in detail.

A series of

electrooptic

effects associated with the

change

of

crystal

orientation and local director in the lattice

[5-9],

electrostriction

[9-1Ii

and

phase

transitions

[10-15]

have been discovered in _an electric field.

Temperature dependences

of the cell parameters, ^the ^scalar order parameter s

[16, 17],

and the

optical activity,

have been measured for the temperature range of blue

phases.

It should be noted that these alterations do _not exhibit

pretransitional

behaviour for the BPI-BPII transition. For instance, _e _can be

regarded

as an order

parameter only

for the transition _to the

isotropic liquid,

I-e- the temperature

dependence

for _~ has _no

singularities

in the

vicinity

of the BPI-BPII transition. As yet,

only

^few ^attention has been

(3)

devoted to the temperature

dependence

of the electrostriction tensor

f.

This _tensor describes the

dependence

of _a field-induced strain k of the blue

phase

lattice _on the

applied

electric field E _:

Uij ~

Yijki l~t l~i II)

From the function _y

ii

(T)

it has been concluded

[18]

that the BPI lattice is not stable _near the BPI-BPII

phase

transition.

Here, _we present ^for ^the ^first ^time ^the temperature

dependence

of all coefficients of the electrostriction _tensor for BPI in _two mixtures with

negative

dielectric

anisotropy.

^Both

mixtures

give

dependence

of

f

differs

distinctly

for the orientations E

fl [001

and E

fl [01 ].

This indicates _an essential

change

in the electrostriction

anisotropy.

For both mixtures the deformation of the unit cell at

El [001] displays

_a

pretransitional

behaviour upon

reaching

the BPI-BPII

phase

transition

temperature.

2.

Experimental.

Two separate mixtures, A and B, have been

investigated.

The first _one

(A)

is _a mixture of the

chiral

compound

S811

(Merck, Germany,

30 fb

by weight)

with the nematic mixture EN18

(Chisso Corp. Japan,

70

wt.-%).

The second _one

(B)

is _a mixture

analogous

to that used earlier

[18]

a chiral-nematic mixture with the chiral

compound

^BNX-16

(Vilnius University,

Lithuania

(81

wt.

-fb))

^and ^the ^nematic

liquid crystal

1650D

II

8.9 _wt.

-fb).

Both mixtures have

negative

dielectric

anisotropy.

The mixture A forms the three blue

phases BPI,

BPII, and BPIII

[19], showing

the

following temperature

ranges. ^BPI 41.80-42.12

°C,

BPII 42.12-42.29 °C,

BPIII _: 42.29-42.4 °C. The mixture B forms the blue

phases

BPI and BPII. The

temperature

ranges are for BPI _: 37.64-38.025 °C, for BPII 38.025-38.47 °C.

The

sample

_was contained between two

glass

slides coated with a transparent

conducting layer. Mylar

_spacers of 17 _~Lm thickness _were used. The temperature was controlled with _an accuracy of _± 0.005 °C. In order _to

investigate

the behaviour of blue

phases

in _an electric

field,

sinusoidal

voltages

with _a

frequency

of 4 kHz _were utilized. Diffraction lines _were measured in the

light

transmission

spectrum.

Samples

with

[001]

and

[0

II directions normal _to the

glass plates

_were

prepared. Samples

of BPI with the cell

plane parallel

to the

glass plates

were obtained

by applying

sinusoidal

voltage (m

45 V ) to the

sample.

After _a slow decrease of the

field,

_a uniform

[001

orientation

was obtained. A

[011]

orientation in BPI

samples

was achieved

by

slow

cooling

from BPII.

The orientation

[011] fl

E is not stable and

eventually

the

sample

_may

acquire

the orientation

[001 ^fl

E. The time constant of reorientation

depends

_on the

applied

^field

strength

and _on the

temperature.

Ât Û _~ ^{40 V} ând T~ ^T ~ 0.03

°C,

the orientation

[011]

^fl E _was stable for several hours. At _a

higher

temperature, ^the reorientation time decreased

drastically,

and it _was not

possible

_to make measurements for the lattice orientation

[011]

fl E in the close

vicinity

of the BPI-BPII

phase

^transition

temperature.

3. Results.

Beyond

the

sample absorption region

the BPI transmission spectrum ^for ^the

[001]

faces

parallel

_to the

glass plates

^is

presented by

a

single

band

A~.

^The ^band ^involves one reflection

(002)

and four reflections

loll ),

coincidijin wavelengths

^for a

body

centered cubic lattice

O~ (Aw~

₌ d _n sin _@,

=

90 °;

A~jj

=

2 d _n sin _@,

= 45° _;

Aw~

₌

A~jj ).

^Without an

electric field the

spectral position

of reflections shows _a

temperature dependence

which is

typical

for blue

phases (Figs.

la,

2a)

_: _a

short-wavelength

shift of reflections

(compression

of the unit

cell)

is observed with

increasing

temperature. Deformation of the cubic _structure of blue

phases

in _an electric field

(electrostriction)

leads _to

splitting

of the bands A~~~ and

(4)

N° 7 ANOMALIES OF THE TEMPERATURE DEPENDENCE OF ELECTROSTRICTION ¹⁰⁸⁹

11

~

b

I,

nlT

~~~

~

4 I

~

+ .

u '

~ ~ @

~

D

@ D

-20

~ ~h

~ mm

42.0

42.1 42.042.I

Fig. I.-a)

spectrum : (.) Ao

~

(A

o~~ + 2 A ~j j )/3

;

b) emperature ependence do~~

do~~ 2 _d~~ ( _*),

U

= 39 ixture .

(5)

1-O

,, a

I

'

) ,"'~~'"',

^"

,"' /~

°

$

_i J i ," ^/ ^°

I j II

/

'

o o

z~ _, I _I

l$°'~

j

l~

^' _, _o ^°

i ' ,

j II

o o

4i ~

,' ^~

~ 3

a o

' 2

16 on° 4

14

400 450 500

Wa~ekngth, _n3n

Fig.

3. Transmission spectrum ⁱⁿ an electric field for the orientation El [001]. (1), (2), (3) BPI. (I) T

= 41.99 °C, (2) T

=

42.07 °C, (3) T

=

42.12 °C, (4) BPII. Mixture A.

(short-wavelength

band in

Fig. 3),

which is in agreement ^with ^the

tetragonal

symmetry ^of ^the

cell

[19]

in _an electric field

El [001].

The temperature

dependences

^for

A~~~(T)

^and

A~j_j

(T)

differ

considerably (Figs.

la,

2a).

To separate ^the ^effects ^related to electrostriction and temperature ^variation we have

plotted

the shifts of the

(002)

and

(0

II reflections relative _to

their

positions

^without a field in

figures16,

2b (A~~~ =

A~~~(E)

A~;

A~jj

₌

A~jj(E)

A~).

^To ^calculate A~~~ and

A~ij

^for ^T _~ T~

(Fig. 4)

the value of A~ was determined

by

linear

extrapolation

of the

temperature dependence Ao(T).

^The lattice constants of the deformed lattice _are

given by dwj(E)

=

d ₊ _a,

dj~~(E)

=

d ₊ b, where d is the lattice _constant of the undisturbed cubic lattice d~~j ^and

dim

are the lattice constants

parallel

and

perpendicular

to the field direction. The shift of the

(002)

reflection is caused

by

_a

change

of the

interplanar spacing

_d~~j

~ °°2 ⁼

~~

₊ _~ _n

(2~

The shift of the

(011)

reflections _occurs due to variations of both the lattice _constant

d~jj

and the diffraction

angle

_@ojj

doll

_~

) _Ii

₊

_~~i~ ₍₃₎

Sin 0011

-) _Ii _~i~~ ₍₄₎

A ojj ⁼ 2

dojj

n sin @ojj =

(d

+

b)

_n

~

dim

n

(5)

For the field direction

[001

the shift of the

(002

reflection is

proportional

to the

change

of the lattice parameter

parallel

to the field

direction,

the shift of

Aoii

is

proportional

to the

change

of the lattice

parameter

^normal to the field direction. If the volume of the unit cell is

preserved (a

= 2

b)

_we get ^from

equations (2)

and

(5)

_:

~002

_" ^~ ~011

(6)

In this _case, the _sum

(Aw~

+ 2 A

ojj

)/3

should be constant (= A

o),

I-e- Aoo~ + 2

Aoj

j ⁼ 0.

Figures

I and 2

present

^the ^values ^of

(Aw~

+ 2

Aojj )/3

and

Aw~

+ 2

Aojj

obtained from the

experimental

values of A

oo~ and A

oj j.

Proceeding

from the data of

figures

I and

2,

_we conclude

that within the

experimental

_accuracy of _our measurements the deformation of the unit cell does _not affect its volume.

(6)

N° 7 ANOMALIES OF THE TEMPERATURE DEPENDENCE OF ELECTROSTRICTION 1091

Tuming

to the electrostriction

coefficients,

_our

experimental

observation of the functions

A~~i(E)

=

fj~~ijE~ yields

effective coefficients

fj~~ij

^which are linear combinations of the components

y,~~~

^defined ⁱⁿ

equation (I).

_yj

iij ⁼

f

joojj ^can be calculated either from the shift of the

(002) reflection, f

joey ⁼ Aoo~/A_o

E~,

_or

(due

_to the constant volume of the unit

cell)

from the

splitting

between the reflection bands A

= Aoo~

Aojj, fjoojj

₌ ^~ ^~

~

(Figs. 5, 6).

The 3

Ao

^E

loo

u

6,

_oo

mm

BPI BAIT

50

[

~ ^A~

6

Do ₆

a

~ T~

0

42.0 42.I

T,°C

Fig. ^4. -Temperature

dependence

of the

splitting

A between the _two reflections (002) and (011), in BPI _: (o) U 42 V, (D) U

=

39 V, (/L) U

=

30 V. Mixture A.

o

~~~ ,' '',

~

l

0"I

~'~

,,~

°~

~~~~~

~ p

j',

_{B pj j}

m'~l~

₁₀ ''

>, ^° 0.1 75

j o_o

1' ,~

0. I BPI

~'i (

BPII

a DO

~ ^~

~~*~,

^~ ^°

~ '

'i,y

o_o

~''

0.0 ~

' ''

'

~ +

i

₊ ⁺

/~

~+ ,

~ ^~

* *, _~ ~

*

l

_*_"0

* '#

»j

0 ^*

-10. 0

42.0 42.I T,°C 37.9 38.0

T,°C

Fig.5. Fig.

6.

Fig. 5.-

Temperature dependence

of the electrostriction coefficients in BPI _: yjii "

ijooii,

(o) U

=

42 V, ID) U 39 V, (/L) U 30 V _; (.) _y

j~ii (+)

fjoii

j, U 39 V (*) _y2323. BPII _: (Q) yii _ii, U

= 39 V, mixture A.

Fig. ^6.-

Temperature dependence

of the electrostriction coefficients in BPI: yjii "

ijooii,

lo) U

=

45 V, (D) U

= 39 V, (.)

y/j(j

(+) fjoi~~, ^U ^{39 V} (*) _y2323. BPII (Q ) _yjiti, U

= 39 V,

mixture B.

(7)

electrostriction coefficient

§jojjj, characterizing

the variation of the

interplanar spacing doj

j for the orientation E

fl [011 ],

_was determined in BPI from the shift of the diffraction band

Aojj

in the transmission

spectrum

:

fjojjj

= AA

ojj/Aojj ^E~ (Figs. 5, 6).

The electrostriction coefficient

fjoojj

ⁱⁿ ^BPII was found from the shift of the diffraction band

Awj

ⁱⁿ

samples

with the orientation

[001] fl

E

(Figs. 5, 6).

4. Discussion.

The most remarkable feature of the data obtained is _a drastic increase in the

splitting

A

(Fig. 4)

and in _y

ii

(Figs.

5,

6)

_on

approaching

the

phase

transition temperature. ^There ^is an essential difference in the behaviour of

fjoojj

and

fjojjj.

The electrostriction coefficient

fjoojj diverges

close _to the transition temperature, ^while

f

joyj

depends only slightly

on temperature. ^Since ^the volume of the unit cell is

preserved (yjj~~

= yj _{ii j}

), only

_two of the three electrostriction 2

tensor

components (in

cubic structures yjjjj, yjj~~ and y~~~~) are

independent

_yjjjj

=

fjoojj,

_y~~~~

=

fjojjj fjoojj [20].

An

anomalously

drastic increase of _yj

ii

IT)

_occurs

only

4

in BPI. At the

phase

transition BPI-BPII, the

sign

of _y

j ii

changes

and its absolute value above the transition temperature ^is ^smaller. ^For

BPII,

_y

iij does not

depend

_on the temperature. ^The shift of the diffraction bands in _a substance with _a

large

cholesteric

pitch,

observed in paper

[21],

is

likely

to be associated with the

change

of electrostriction. The values of

y~~~~

presented

in

figures 5,

6 _were obtained from the relation y~~~~ =

fjojjj fjo~jj.

In 4

contrast to yj _ii _j, y~~~~ does not exhibit _a critical behaviour close to the BPI-BPII transition.

This result indicates that the lattice becomes soft with respect to field-induced deformations

along

^the axes of the cubic lattice

(which

_are described

by

_yjjjj

),

^but ^remains stable _as far _as

shear deformations are concemed

(which

_are described

by

^the

off-diagonal

elements of

k).

The difference in the behaviour of _y

j jjj and y~~~~

(Figs. 5, 6)

^results in a

strong change

^of

the electrostriction

anisotropy

1~ =

(yj

iii

yjj~~)/2

_y~~~~ in BPI

(Figs.

7,

8).

The absolute value of

1~ increases _on

approaching

the

phase

o-o

xx

_~_~ ^~ x~~ ^xx

~

~§

-l.0

~

~ x

x

~

~~'°

'~

^~~

n~ ^~x

-2.0 ^~

42.0 42.I

T,°C

37.9 38.0

T,°C

Fig

7

Fig.

8.

Fig.

7.- Temperature

dependence

of the electrostriction

anisotropy (yjjjj yjj~~)/2

_y~~~~ in BPI.

Mixture A.

Fig. 8. -Temperature dependence of the electrostriction

anisotropy

(yjjjj-yjj~~)/2 _y~~~~ in BPI.

Mixture B.

(8)

For both mixtures the

dependence

_yjj

ii

(T)

can be described

by

the function _yj

iii

(T)

=

A(T* T)~ '.

The temperature ^T* ^is ⁱⁿ ^close

vicinity

of the

phase

transition temperature :

T* T~ _m ^0.05-0.08 °C, which is

by

_an order of

magnitude

less than the

corresponding

value for the

phase

transition from _a nematic _or cholesteric

phase

to the

isotropic liquid.

Note that T* determined in this way indicates the

high-temperature

limit of

divergence

for yj _ii _j, since the latter may have _a

temperature-independent

_part. In this _case, the difference T* T~ ^will ^be of _a smaller value.

Therefore,

the temperature ^variation ^of yjj _ii indicates _a

pretransitional

behaviour for the BPI-BPII transition in both mixtures.

Since the electrostriction is the result of _a

competition

between electric and elastic

forces,

let

us consider the connection between these

properties

in _more detail in the

following paragraph.

In

general,

the

dependence

of the free energy on an extemal field E and _on deformations k of the blue

phase

lattice is described

by [22]

AF

=

A,~a

_{u,~ u~i} _x,~~i

E,

_{E~ E~}

Et

_{p~~~i E~} _E~ _u~i

(7)

Here,

the elastic constant tensor

I

characterizes the stiffness of the lattice,

I

is the non-linear dielectric

susceptibility

tensor, and the

elasto-optic

tensor

#

_describes

a

coupling

term which connects the free energy to both k and E. The electrostriction tensor

f

which is

investigated

in

our paper is connected to these

macroscopic properties by [22]

Yi jki ^~

( 7r ~,j,in, ~'nmti (8)

where

(

_is _the _tensor _of

_elasticity

or

compliance coefficients,

I-e- the

reciprocal

tensor to

I. Physically, equation (8)

_means that the

magnitude

of the electrostrictive effect is

large

if the lattice is

mechanically

soft and/or if the

coupling

between the electric field and the strain in the free energy

expansion (I.e.

the third term in

Eq. (7))

is

large.

In order _to

explore

these relations in _more

detail,

_we have _to express the individual components ^of ^the electrostriction _tensor

by

the components ^of ^the

elasto-optic

tensor and the elastic _constant _tensor. For cubic structures, the _tensors

(

_and

#

_have _three

independent

components, ^which can be used for

expressing

Y _I, Y _122, ~fld Y2323

Yllll

"~ (~llll~'llll

₊

~~l122~'l122), (9)

Yl122 "

) _(~llll _~'l122 +~l122~'llll +~l122~'l122), (10)

Y2323 ~

)

~2323 ~'2323

The

splitting

between the

reflections,

A

= A

w~ A

~jj ⁼

(a

b _n,

depends

on the difference between the electrostriction components ^A = y _{j j} y_{j j}_~~

) E~

A

o. For

(

_y

j j j y _~~

)

we

obtain the

simple

relation _:

Yllll Yl122 "

) _(~'till ~~'l122)(~llll ~~l122). (l~)

The _tensor

(

_is

_reciprocal

_to _the _elastic _constant _tensor

I.

_Thus, _for

a cubic _structure the

relation between the

independent

components ^of ^the tensors

(

_and

I

_has _the _form

:

~~~~~

(~

II II

l~)~l~~~~

~ ~ l122)

~~~~

(9)

~~~~~

(~llll

~

122~~~~~ll

+ ~

~l122)

~~~~

s ~~~

l

(15)

~ 4 ~2323

From

(12)-(14)

^the

following

relation must be fulfilled :

Yiiii-yjj~~=

I

_~'nil-Pj122

~~ ~~~~l

~l122 (16)

The relation yj _{ii j} =

2 _y

j j~~

imposes

limitations _on the components ^{of the}

elasto-optic

tensor

fi.

_It _follows _from

equations (9), (10)

and from the relation _yjjjj

=-2yjj~~,

that

Pi

jjj ⁼ 2 P

jj~~. This relation leads _to the electrostriction coefficients _:

Yllll ~

£~'till(~llll ~~l122) (17)

From

equations (11), (13)-(15), (17)

and

(18)

_we can obtain the components ^of ^the electrostriction _tensor

~llll ~

~

~[ill

~~~~~~~

_{~ ~~~~~}

(i~~

Yl122 ~

~llll

~~

"~~~~~i ~i122) ^~20)

y~~~~ =

~~~~~

(21)

~ "~

2323

Equations

^similar to

(19)-(21)

have been obtained

previously by

Stark and Trebin

[23] using

the

assumption

that _an electric field _causes

only

a

change

of the orientational order

(P

ii j ⁼ 2 P

j~~

).

However, ⁱⁿ the present _paper the

equations (17)-(21)

have been derived from the condition of conservation

ofthe

unit cell volume under its deformation in _an electric field. It should be noted that the relation

(16)

is valid without the condition of _constant unit cell

volume,

whereas the relations

(19), (20)

_are valid

only

under the condition _yjjjj

=

~ Yl122, °~

~'II

II ^~ ~

~'l122.

Using

the

equations (19)-(21),

we can now discuss how the electrostriction coefficients _are affected

by

the components of

fi

_and

I.

_The _relation

yj _ii_j =

2 _y

jj~~ leads _to _a

relationship

between the

elasto-optic

coefficients

(P

ii11 ⁼ 2 P

jj~~), ^but ^it ^does ^not ^lead to a

relationship

between the

elasticity

tensor

components. Signs

and values A,~~i are

only

limited

by

the

crystal stability

condition. The

crystal

deformation energy A~~~i u~~ u,t should be

positive

at any

values of

u~~. This

imposes

certain restrictions _on _A~~~i

~2323~0, ~llll*~l122~°, ~llll +~~l122~°' (~~)

Electrostriction tensor

components

^have ^been ^calculated

numerically by

_means of the Landau- de Gennes

theory [23, 24]. However,

_a numerical

comparison

of absolute values of A~~a ^with ^the

experimental

values cannot be done, since the final results obtained in

[23, 24]

involve Landau

coefficients,

for which _no exact data _are available _at present. ^The theoretical

(10)

calculations neither

predict

the

sign

of the temperature

dependence

of f nor

yield

in

quantitative

values for the temperature coefficients of

f.

A decrease of

f

with

increasing

temperature

[24],

or an increase of f ^with temperature

[23]

have been

predicted

in BPI and BPII. In conclusion, the theoretical considerations available _as yet

give

_no hints how to

explain

the unusual temperature

dependence

of the electrostriction tensor observed in _our

experiments especially,

it has not been discussed whether yj _{ii j} may grow

anomalously

in

vicinity

of the

BPI-BPII

phase

However,

in

previously published experimental

works

[25-27]

the temperature

dependence

of the shear modulus has been measured for blue

phases [25-27].

The variation of the shear

modulus in BPI _near the BPI

- BPII

phase

transition

temperature

amounts

approximately

30 fb (T~ ^T m 0.15

°C) [27].

_y~~~~

(Figs. 5, 6)

is

changed by

about the _same

magnitude.

One may thus suppose that the basic contribution to the

temperature dependence

of

f

_comes from the temperature

dependence I _IT). According

to

equation II 9),

an increase of _yj

ii j, while

approaching

the

phase

transition temperature

(Figs. 5, 6),

_can be associated with _a

decreasing

value of the difference A

jjjj A

jj~~. It is

important

to note that this difference is involved in

the conditions for elastic

stability

of the

crystal (see

relations

(22)).

As A

jii A

jj~~ tends _to zero, the

O~ (BPI)

lattice becomes unstable relative to spontaneous deformations. Estimates of the mean-square fluctuation values of the cubic lattice _near the BPI-BPII transition temperature

[18]

_are in agreement ^with ^such an

interpretation

of the cause for the BPI

- BPII

phase

transition. There _are other evidences of temperature

dependent dynamical

_processes in BAIT

[28, 29]

_as the transition to BPIII is

approached.

5. Conclusion.

We have determined the temperature

dependence

for all the three

independent

components ^of

the electrostriction tensor, y~~~i, ^for ^two ^mixtures ^with ^the

negative

dielectric

anisotropy.

The temperature

dependences

for yjjjj and y~~~~ in BPI differ

essentially

which leads to

deformation of the unit cell with

varying

temperature. ^The coefficient yiiii and the

electrostriction

anisotropy

1~

display

_a

pretransitional behavior,

when

approaching

the BPI- BPII

phase

transition temperature. ^Such a behaviour may be

explained by

the temperature

behaviour of the

elasticity

module,

and,

as a result, the BPI lattice may become unstable

relative _to spontaneous deformations. In this _case, the BPI- BPII

phase

transition and

pretransitional

behaviour _can be described

using

the components ^of ^the ^elastic ^constant ^tensor

I

as an order parameter.

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