Smectic B : long range translational order and static shear instability. Nonlinear effects

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Smectic B : long range translational order and static

shear instability. Nonlinear effects

I.S. Gersht, V.K. Pershin

To cite this version:

Smectic

B :

_{long range}

translational

order

and static shear

instability.

Nonlinear effects

I. S. Gersht and V. K. Pershin

Chelyabinsk

State

_University,

Lenin-Pr 77-43, 4S4 080

Chelyabinsk,

U.S.S.R.

(Received

on

_July

_{21, 1989,} revised on

_April

11, 1990, and

accepted

on

_May

_3,

₁₉₉₀₎

Abstract. 2014 Terms nonlinear with respect to the wave vector at translational oscillation

frequencies

have been taken into account for a

_layer

_system of uniaxial molecules and it has been

proved

that

_crystal

order can coexist with a shear

instability.

On this basis a new model of smectic B

_phase

has been

_proposed

where the structure

_{possessed long}

_range translational order in all directions. At the same time

interlayer dynamical

shear modulus decreases at lower

03C9ap

frequencies

of an

_applied

strain and

_equals

zero in the static case

_{(03C9ap ~ 0).}

Classification

Physics

Abstracts 61.30

Interlayer

shear modulus

_{C 1313}

is not zero and does not

_depend

on the

_frequency,

wap, of an

applied

strain in

ordinary layer crystals [1].

However,

experimental

studies

of layer

structures such as smectic B

liquid crystals

have shown

[2, 3]

that the shear modulus

C 1313

decreases at lower

_{W ap}

and tends to zero at

_{w ap -->}

0. At the same time

_long

_range

translational order is retained in all directions in _space in smectic B

_{[4-6]. According}

to theoretical

concepts

[7]

simultaneous coexistence of

_crystal

order and shear

_instability

contradicts the Landau-Peierls theorem

[8].

Hence,

these

_experimental

results should be correlated with the Landau-Peierls theorem. To solve this

_problem

terms nonlinear with

_respect

to wave vector K at translational oscillation

_frequencies

are taken into account _{in this paper and} a new model of smectic B is

proposed

on this basis. On the basis of this model

experimentally

observed structural

peculiarities

of smectic B are

_explained

and the

_crystal

smectic B

_phase

transition is described.

It should be noted that at

_present

not

_only

the existence of

_{translationally}

ordered smectic B has been

_proved

but a hexatic

[9] phase

has been found

[6]

that is also called

smectic B.

However,

hexatic

_phases

will not _{be considered in this paper. A}

_{phase having long}

range translational order in all directions will be called smectic B.

Let us consider a

_layered

structure formed

_{by long}

thin molecules normal to the

layers

and

characterized

_by

_{hexagonal crystalline}

_symmetry.

Let the Oz axis of the coordinate

system

be

1770

directed

_along

the

_long

axes of the molecules and the Ox axis coincide with the direction of

one of the vectors of fundamental translations within the

layer.

Let us write the elastic _energy of the

_system

as

_[10-12]

Here the first seven summands are _{the elastic energy of the atomic}

_{crystal [13].}

_Except

for the shear modulus that characterizes

_bending

of the

_{layers (C 3131)I (( C C3131 )I}

=

C 1313

=

C44)

all

notation is

standard,

e.g.

Ux

is the

_x-component

of the

_displacement

from

_equilibrium

position

and is a shear modulus within the

_layer.

The last

two summands are the corrections connected with the account of the effect of orientational

degrees

of freedom upon the translational ones, n is the director

_{«(( n) = {O, 0, 1 } ),}

K33

is the

Frank-constant,

UiZZ

=

a2ui /az2

(i

= _x,

y ).

It will be shown in this _paper that these

summands determine the discussed

_{peculiarities}

of correlational and elastic

_properties

of

smectic B. _

The last but one summand in

₍₁₎

characterizes translational deformations when all molecules tum

_through

one and the same

_angle.

The turn of the

director,

n, should be

followed

_{by layer bending}

because n is normal to the

_layers

in the

_equilibrium

state. This summand is written in view of

_using

the _{results of the paper}

_[14]

that deals with

magnetoelastic

interactions. It is shown in this paper that the presence of

magnetization

M

causes an additional summand in elastic energy. Because

of the

_generality

of the

_{developed symmetrical}

_approach

a substitution M ---> n can be fulfilled

and the last but one summand in

(1)

can be obtained with account of the

system symmetry.

By

adding

two terms

_proportional

to

( U2z X +

Uzy2)

we obtain a value of the shear modulus

C 3131

in the

_system

of uni-axial molecules

The result means that

_asymmetry

of shear moduli

_relatively

to the

_permutation

of indices is

provided by

the

_anisotropy

of the form of the molecules _{and the presence of} orientational

order

Here because the director is normal to the

layers

i.e.,

the function F =

F (n )

has a

minimum at n

_{= {0, 0, 1 }.}

Let us estimate the

_asymmetry

values of shear moduli _{in mesogeneous}

_crystals.

There is a

5 kcal/mole [15] (n

1 = {O, 0, 1 } :

n 2 1 n 1 )

in these

_{crystals. Substituting}

a molar volume

Vm

= 700

cm3/mole

[16]

and

supposing

that we

_get

_{C 3131}

-C 1313

e- 6 _x

10 8

dyn/cm2.

This value

corresponds satisfactorily

to the

_experimental

data

_{[ 17]}

according

to which

_{C 3131 -}

C ₁₃₁₃ = 11 _x

108

dyn/cm2

in

crystalline

TBBA

phase.

The last summand in

₍₁₎

accounts for the effect of

oriêntational

deformations on the

translational ones. This summand was discussed

_{[10, 11 ]}

when

studying

translational order in

columnar

_mesophase

formed

_by

disc-like

molecules.

It would be reasonable to assume in the structure in

_{question (that}

has

_{crystalline symmetry)}

that the ends of molecules in one

_layer

are

_rigidly

connected with the ends of molecules in the

_{neighbouring layers (e.g.,}

it is

_possible

in close

_packing).

It is _easy to see in this case that one of the effects caused

_by

bend deformation is the shear of

layers parallel

to each

_other,

_{i.e. ni} =

Uiz

+ 03C8 (UaB)

(i

= x, y ; a,

/3 = x,

y,

z). Substituting

this relation into the

_expression

of the bend deformation _energy

_Fb

= 1/2

_{K33 (n}

x rot n

)2

we

_get

2

the last summand in

₍₁₎

as well as some other corrections that are not

_important

for the

problem

in

_question

and are thus

_neglected.

Now let us

_get

the main relations _necessary to

_study

the elastic

_properties

and translational

order in the

_system.

Let us first calculate the

frequencies

of translational

oscillations,

w i(1

=

1, 2, 3

),

of the

_system.

From

(1)

we write the elastic energy connected with shear of

layers parallel

to each other

A

_{corresponding frequency}

is

_easily

obtained out of

_[12]

Hereafter M is the molecule mass,

f

is the lattice

_{spacing, l 3}

is the

_{layer spacing}

i is the wave vector

(i = x,

y,

z ).

using

the

_{expression [12, 13]}

for atomic

_{crystal frequencies, taking}

into account

₍₃₎

and the correction nonlinear with

_respect

to the wave vector

₍₅₎

we

_get

for the molecular

_system

considered [ 12,

18]

On the basis of

(6)-(8)

we can

_get

_{[10-12, 18]}

_expressions

for the mean _{square of relative}

1772

([Ux(O) -

Ux(L3)]2)

+ ([Uy(O) -

Uy(L3)]2)

as well as for the mean _square

_{displacements}

within the

_{layer U2) = (U2)}

_{+ ( U2y)}

and in the direction normal to the

_{layers ( U2z)}

where T is the

_{temperature ;}

0

0,

0z, -

7T

/ 2.

Relations

_(9)-(11)

allow us to

_study

the translational correlations in the

_system.

We shall

suppose the

layers

to be correlated if

where

_{Q (L3)}

=

exp(-

2

Tf2 X(L3)/l23).

Long

range translational order exists within the

layer

if

and in the direction normal to the

_layers

if

We use the

_expressions

obtained to solve the

_{problem given,}

that is to establish a

correlation between the translational order of smectic B and its

_stability

with

_respect

to the shear of

_{layers parallel}

to each other.

Let us first consider how this

_problem

is solved in terms

_{of existing}

models

_{[7, 19]}

of smectic B. The first « 2-dimensional » model

_[19]

is based on the

_assumption

that the

_inter-layer

shear

modulus C 1313

equals

zero. Here terms nonlinear

(proportional

to

0z 4

with

respect

to the wave

vector in the

_{frequencies (6),}

₍₇₎

are not taken into account.

The elastic

_properties

of the

_system

within this model are evident

_{C 1313}

= 0 for _any

frequency

of

_applied

strain.

To

_study

the correlational

_properties

of the _system let us substitute

(6)

and

(7)

into

(9)

and

(10).

We

_get

_{Q (L3)}

= 0 _at

any

L3,

U2> -->

oo. It means that the

layers

are not correlated and

translational order within the « 2-dimensional »

_layers

is

destroyed

in accordance with the

Landau-Peierls mechanism

_[8].

,

This model

_corresponds

to

_experimental

data

_{[2, 3] according}

to which

_{C 1313 --> 0}

when

wap -->

0. At the same time it contradicts

_fairly

reliable

_experimental

data

_proving

that

C1313 #

0 when

_{lù ap # 0}

_[2,

_{3, 20,}

_21]

and there is

_long

_range translational order within the

layer.

On this basis the conclusion can be made that this model is incorrect for smectic B.

In terms of the alternative «

_{crystalline »}

smectic B model

[7]

it is assumed that

C 1313’ though

very

small,

does not

_equal

zero.

_Here,

as well as in the first

model,

terms nonlinear with

respect

to the wave vector are not taken into account.

It is evident that C ₁₃₁₃ does not

_depend

on _Cùap and conditions

_{( 12)-(14)}

are held in this case

experimental

results

_according

to which :

i)

dispersion

curve of the transverse wave

propagating normally

to the

_layers

is

_{substantially changed}

at the

_{crystal-smectic}

B

transition

_{[22] (it closely}

resembles a

_{parabola w -}

K2Z

in smectic

_{B) ; ii)}

C ₁₃₁₃ decreases with

decreasing

_{wap[2, 3, 21, 22]}

]

(C 1313 ~

108

dyn/cm2

if

_wap~

108

Hz,

C 1313 ~ 106

dyn/cm2

if

wap ~103

Hz

_[2,

_{3, 20,}

_{22]) ; iii) C 1313 equals}

zero in the static case

_{(wap ->}

_{0 ) [2,}

_3].

There exists a

_point

of view

_[2,

_3]

that the

_{frequency dependence}

of

_{C 1313}

can be

_explained

in terms of the

_{« crystalline »}

model if the effect of mobile defects would be taken into account. It is

_suggested

that the

_{experimentally}

measured shear modulus

_{C 1313}

differs from

zero and coincides with the ideal lattice shear modulus at

_{sufficiently high}

_lIJap.-

With

decreasing

Wap

the shear modulus decreases due to the effect of mobile defects. If

lIJ ap -+

0 then

C 1313 -->

0 due to emission and climb of

_edge

dislocations. It is shown in the

paper

[21] that

the real

part

RB

of the shear

_impedance

increases

essentially

with

_increasing

frequency v

= lIJ

_ap/2

’TT from 5 to 85

_MHz,

its alteration

_{RB -}

_RA

is rather low at the smectic B-smectic A transition if v = 85 MHz. These results must indicate in accordance to

[2,

3,

21

] that

there are defects in smectic B the relaxation time _{T B} of which is

_comparable

with

the relaxation time in smectic

A,

TB/ TA 10.

Arising

out of this statement, a conclusion

about the existence of

_only

local translational order in smectic B contradicts the results of

[4-6].

Disputability

of the

_point

of view discussed here is also evident from reference

_[22].

It is shown in this paper that

C 1313

increases with

increasing

frequency v

in smectic B and can be even

_greater

than in a

crystal

at

_greater

_{frequencies v - 0.1}

THz.

Let us now

_analyse

a conclusion

_{[2, 3]}

that the conversion into zero of the statical shear

modulus is due to emission and climb of

_edge

dislocations. We note that the estimation

_[2]

of the

_ratio,

_{El (T/2),}

_{(E =}

C ₁₃₁₃

Q2

_l3

is cohesive

_energy)

_is,

_obviously,

not

_quite

correct. An ideal lattice shear modulus which _may be of order

108

_dyn/CM2

according

to the

point

of view discussed in and to the results from

_{[20, 22]}

is to be introduced into this ratio. Then

E/(T/2) ~ 1.

The condition

_{E T/2}

does not,

obviously, correspond

to the

_phase

considered because the ideal lattice shear modulus is to be zero in this case

_(i.e.,

C ₁₃₁₃ = 0 at _any lIJ

ap,

layers

are not

correlated,

crystalline

order within the

layer

is

disturbed).

If E >

_T/2

then the motion of

_edge

dislocations may lead to

_plastic

strain at rather

_great

stresses 6 > _ao

_{only. C 1313}

decreases

_rapidly

with

_increasing

a in this case. Such a behaviour

of

_{C 1313}

has not been observed in

_experiments.

Let us now, assume that

_C1313

= 0 and nonlinear

(proportional

to

0j

corrections in the

frequencies (6), (7)

are not zero.

_According

to

₍₃₎

the shear modulus

_{C3131 characterizing}

bend of

_layers

can be other than zero in this case due to _{the presence of} orientational order.

A state when

_{C 1313}

= 0 and all the other coefficients _are other than _{zero at} the

frequencies

(6)-(8)

is discussed as a model of smectic B in this _paper.

We discuss translational order in the direction normal to the

_{layers. Taking}

into account

C33 » C 3131 [17, 20]

and

_{integrating (11)}

we

_get

Assuming

that at least order of

_{magnitude C3131 - C 1313 - 108}

_dyn/cm2

is not

_changed

at the

crystal-smectic

B transition

_{(i.e., assuming}

that

_{C3131 ~ 108}

_dyn/cm2)

and

_substituting

T/k

= 400

K,

C33

=

1.2

_x

101°

dyn/cm2 [17], f

=

5 Â

_{we get}

/(U7 _

1.6

 «l3

"’" 30

À.

This result

_corresponds

to

_experimental

data

_[23],

Q -

1-2 Â

and

_according

to

₍₁₄₎

proves that the

system

considered has

_crystalline

order in the

_arrangement

of

_layers.

On the basis of the

above-stated

a conclusion can be made that the elastic

stability

of

smectic B

_relatively

to the bend of

_layers

as well as the existence of

_long

_{range translational}

1774

Now we

_analyse

the _system

_stability

with

_respect

to shear of

_{layers parallel}

to each other. It

can be seen from

₍₄₎

that the

_system

is unstable

_against

linear

_(Ui,,

=

0,

i = _x,

y )

deformations

_{F(Ui =}

₀₎

=

F(Ui

=

Pi Z) (Pi

=

Cst.)

in the case considered. Nonlinear deformations

_{(Uizz =F 0)}

lead to the elastic _energy

increase, i.e.,

cause

_restoring

forces. It is

evident that such deformations take

_place

if the

_applied

strain is

_{time-dependent.}

For the

_quantitative

_description

of « nonlinear »

restoring

forces from

(4), (5)

the effective

interlayer

shear modulus can be introduced

where p =

M/(l2 l3

is the

density.

It can be seen from this

_expression

that the

_system

is unstable with

_respect

to static linear deformations.

_However,

due to the influence of

nonlinear effects the

_interlayer

shear modulus becomes other than zero in the case of

nonlinear

_{(in particular, dynamical)}

deformations. It increases with

_increasing

wave vector

(« nonlinearity »)

0j

and/or frequency

Cdap

of the

applied

strain.

Using (16)

it is

_possible

to estimate the Frank

_{constant, K33,}

is smectic B.

_Substituting

C’f - 7

x

10g

dyn/cm2

at 2

_{71’ / Kz}

= 5 145

Â

[20]

into

(16)

we

_get

_{K33 =}

5 x

10- 2 dyn. Using

the results

_[21],

_{C ef 1313}

= 1.7 _x

107

dyn/cm2

_at

Cdap/271’

= 15

MHz,

we

_get

K33 =

3.4 x

10 - 2

_dyn.

As

_{expected [15], K33}

is much

_greater

_(by

about 4 orders in smectic

_B)

in smectic

phases

than

_typical

values

_{K33 ~ 10- 6 dyn}

in the nematic

phase.

It should be noted at the same time that the

_{experimental frequency dependence}

of

C 1313 [2,

3,

21,

22]

is not

_always

in

_conformity

with the linear law

_(16).

This may

be,

perhaps,

explained

if nonlinear

terms

of

_higher

order with

_respect

to the wave vector would be taken

into account. It is

_{possible [2,}

3,

21]

that it is associated with relaxation processes

(which

are

caused

_by

the effect of mobile

_defects,

_{viscosity, etc.). Experimental}

methods where the effect of relaxation _processes is excluded consist in

_studying

the elastic

_properties

with

respect

to nonlinear static deformations. The influence of the motion of

_edge

dislocations on the value

of C ₁₃₁₃ at low can be excluded if a sufficient low _{stress, u,} and

_samples

with

_sufficiently

_great

layer

dimensions,

L,

are used. If C ₁₃₁₃ does not

_depend

on Land u then the motion of

_edge

dislocations _may be

_neglected.

Now we discuss translational order within the

_layer

and

_interlayer

correlations. After

integrating (10)

we

_get

where

_{Substituting K33}

= 5 x

10 - 2

_dyn,

T/k = 400 K,

f=5Á, l3=30Á

into

₍₁₇₎

and

_{using [17]}

_{Cll=2.6xl010dyn/cm2,}

C 66

= 4.9 _x

10 9

dyn/cm2

_we

_get

U2> =

0.3 À

f.

This result is in

agreement

with

experimental

data

_[23]

_{and proves the existence of}

_long

range translational order within the

layer.

To describe

_interlayer

correlations we _compare

_(9),

(10).

It is _easy to note that

x(L3) 4 U2).

Hence,

according

to

₍₁₂₎

and

₍₁₃₎

it follows that the

_layers

are correlated

both in the case discussed and in all other cases when there is

_crystalline

order within the

The result obtained - coexistence of

crystalline

order and static shear

_instability

- is based

on the account of nonlinear effects. If

_{C 1313}

= 0 then nonlinear terms

_provide

_{the convergence}

of the

_{integrals (9), (10).}

It indicates that « nonlinear »

_restoring

forces

_{provide interlayer}

correlations and stabilize 3-dimensional

_crystalline

order within the

_layers.

Let us consider the

_{crystal-smectic}

B transition. This transition is

_{accompanied by}

the

entropy

jump, ASCB,

it is the first order transition

_{[16, 24].}

It is

_important

that

4SCB

is as a rule several times

_larger

than the total of

_entropy

_{jumps corresponding}

to further

smectic

_{B-isotropic liquid}

transitions. The first

_stage

_{of melting (a crystal-smectic}

B

_transition)

« consumes » a

_{greater part}

of

_melting

heat. At the same time it is evident that the

_system

symmetry

change

at the

_{crystal-smectic}

B transition is « weaker » than that at the smectic

B-isotropic liquid

transition.

_(The

former transition can be

accompanied by

small

changes

of the

angle

between the fundamental translational vectors

_[17],

the latter one is characterized

_by

translational and orientational

_{disarrangement).}

Hence,

it follows that the

_entropy

_jump

at the

_{crystal-smectic}

B transition is not determined

by

the

system symmetry

change.

If this transition is considered as the first

_stage

of

melting

(i.e.,

of the

_{crystal-isotropic liquid transition)}

then a more

general

conclusion can be made

that the

_{system symmetry}

_change

not

_only

does not determine the

_entropy

_jump

at

_melting,

but does not make a main contribution to it either.

Now we discuss how

_4SCB

can be calculated. It can

_be,

_clearly,

made while

_using

the

«

_{crystalline »}

formalism. Calculations should be carried out with account of orientational and

intramolecular

_degrees

of freedom. It is a rather difficult

_problem.

_However,

we

_might,

apparently,

use the theorem of

_{equipartition}

of _energy

_[25]

and assume that an

_equal

heat

amount of the transition

_corresponds

to each

_degree

of freedom. Then the

_entropy

_jump

can

be connected with

_the

_change

of elastic moduli

_[8,

_12,

_18]

Here So

is the

_entropy

_jump

of the 1-dimensional atomic chain at the isothermal

change

of its elastic modulus from

_C33(Cr)

to

_{C 33 ( B ),}

P =

C /k

is a number of

_degrees

of

_freedom,

C is

, the

heat

_capacity,

_C33(Cr)

and

_{C 33 ( B )}

are the elastic moduli in the

_crystal

and smectic

_B,

respectively. Substituting [17]

C33(Cr)

= 1.4 x

101° dyn/cm2,

_{C 33 ( B )}

= 1.2 _x

101° dyn/cm2

and

_C

= 120

cal/mole.K [16]

into

(18)

for TBBA we

_get

_{ASCB =}

9

cal/mole.K.

This result is in

satisfactory

agreement

with

_experimental

data

_[24]

_according

to

_which,

for

_example,

ASCB =

10

_cal/mole.K

for TBBA.

Notice that the relation

₍₁₈₎

does not contradict the

_general

definition

_{[8, 25]}

of

_entropy

according

to which it is

_{expressed by}

the distribution function of the

_system.

The fact is that lower elastic moduli lead to

_greater

_{interparticle}

distances and thermal fluctuations. The

change

of the distribution function

and, hence,

of the

_{system entropy}

is due to this.

To

_summarize,

a new model of smectic B based on the account of nonlinear effects is

proposed

in this paper.

According

to the model smectic B _{possesses real}

_long

_range

translational order in all directions. At the same time the

_interlayer

shear modulus decreases

with

_{decreasing frequency}

of an

_applied

strain and

_equals

zero in the static case. A conclusion

is made that the

system symmetry

change

is not a factor

_determining

an

_entropy

jump

at the

crystal-smectic

B transition. This

_jump

is connected with the

_change

of elastic moduli. It should be noted that the main idea of _{this paper}

(i.e.,

the fact that nonlinear effects are to be taken into account when the

system

is characterized

_by

shear

_instability)

has been used in

[12, 18]

for

_studying

correlational and elastic

_{properties of partially}

ordered smectic

_phases.

A

supposition

can be made that nonlinear effects should be

always

taken into account in the

1776

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