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On the elastic free energy for smectic-A liquid crystals

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HAL Id: jpa-00247818

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Submitted on 1 Jan 1993

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On the elastic free energy for smectic-A liquid crystals

Robert Holyst, Andrzej Poniewierski

To cite this version:

Robert Holyst, Andrzej Poniewierski. On the elastic free energy for smectic-A liquid crystals. Journal

de Physique II, EDP Sciences, 1993, 3 (2), pp.177-182. �10.1051/jp2:1993119�. �jpa-00247818�

(2)

Classification

Physics

Abstracts

61.30-61.30C-46.30C

Short Communication

On the elastic free energy for smectic-A liquid crystals

Robert

Holyst

and

Andrzej

Poniewierski

Institute of

Physical Chemistry,

Polish Academy of Sciences,

Department III, Kasprzaka 44/52,

01-224 Warsaw, Poland

(Received

20

July1992,

revised 19 November 1992,

accepted

9 December1992

)

Abstract. We

thoroughly

discuss the layer deformations in smectic-A

liquid crystals.

The invariant elastic free energy is

presented

in terms of the derivatives of the vector field normal to

layers

and the

change

of the distance between the

layers.

We compare our results with the

previous

works of de Gennes and of Grinstein and Pelcovits. It is

pointed

out that anharmonic

terms in the elastic free energy

might

be

responsible

for the

change

of the average

layer spacing, layer spacing profile

and in

particular

for the tilt

profiles

in finite smectic systems.

1 Introduction.

The smectic-A

liquid crystal phase

consists of

parallel, equidistant

two dimensional

liquid layers.

The average distance between the

layers (denoted d)

is

roughly equal

to the

length

of a molecule in the

system.

The

layers strongly fluctuates;

their deformations are conve-

niently

described

by

the vert,ical

displacement

of

layers,

u

(~o,

voi

zo),

from the rest

position

at

(~o,

voi

zo)

For convenience we assumed that

unperturbed layers

are

perpendicular

to the z-axis. Since the seminal paper

by

de Gennes [1] the smectic-A free energy for deformations has been

intensively

studied

[2-5].

At the harmonic level

approximation

the

long-wavelength

properties

of smectics are well described

by

the

following

free energy

density:

Here

Af~

is the twc-dimensional

Laplacian

with

respect

to ~o, vo, B is the smectic elastic constant associated with

layer compressions

and

ICI

is the elastic constant associated with

layer bending.

It was Grinstein and Pelcovits [4]~ who

pointed

out that this free energy

density

is not invariant with

respect

to rotations and that anharmonic terms are necessary to preserve

(3)

178 JOURNAL DE

PHYSIQUE

II N°1

this invariance.

They proposed

the

following

free energy

density

which is invariant with

respect

to

global

rotations:

F~p

=

B

(°j[~~ jivu(r)12j

~

+

Ki Aiu(r))2j

(2)

where i7 =

(@/@~, @/@y, @/@z)

is the nabla

operator.

Here r

=

(~,

y~

z)

denotes a

point

in the deformed

body

and the minus

sign

results from our convention: r

= ro +

kzu,

where the

displacement

can be either

a function of ro or r. Another

expression including

anharmonic terms has been

proposed by

de Gennes [6] and it is also discussed in the Landau-Lifshitz book

[7],

I-e-

~

~ ~~~~~

~~~"~~°~~~~~

~

~~ ~~~"~~°~~~~

' ~~~

where i7

(~

=

(@/@~oi @/@yo)

is two dimensional nabla

operator.

This free energy~

however,

is not

completely

invariant with

respect

to rotations and

only

up to the

eighth

order in small

angles.

In this paper we

thoroughly

discuss elastic deformations in the smectic-A

liquid crystal

and derive a more

general expression

for the free energy

density.

We recover both

(2)

and

(3)

as the first term of the

expansion

of that

expression. Additionally

we note that anharmonic terms

are

important

for determination of the average

layer spacing

and

eventually

for tilt

profiles

observed in thin

freely suspended

smectic

liquid crystal

films [8~

9].

For convenience we shall

use the continuous

description

of the smectic

system.

The paper is

organized

as follows: in section 2 we discuss the

possible

deformations in smectics and the invariant form of the free energy

density

in terms of the vector normal to smectic

layers

and the distance between the

layers

measured

along

the normal. In Section 3 we express the free energy

density

in terms of the vertical

displacement

and obtain the Grinstein and

Pelcovits~

and also de Gennes free

energy densities as

special

cases. A short summary is contained in section 4.

2. Deformations in smectic-A

phase.

First of all we have to

identify

the

quantities

which

properly

describe deformations in the smectic-A

phase.

There are two of them: a vector

quantity namely

the unit vector normal to the

layer, fi(r),

and a scalar

quantity~

which measures the distance between the

layers along

the normal. We further assume that other

quantities

like the nematic order parameter and the

density adjust

to the

layer

deformations. In the first

approximation~ density changes

do not

generate

new terms in the free energy, but

only

renormalize the smectic elastic constants [61

7].

Using

the two aforementioned

quantities

we can write the free energy

density

in the

simplest

form invariant with respect to

global

rotations:

F =

)B (~

~

+

)Iii (17i1(r)[~

+

I(2 (fi(r)

i7 x

fi(r)[~

+

o

~

2

(4j

+

I(3 [fi(r)

x

(i7xfi(r))[~

where

do

is the

unperturbed layer spacing.

The

divergence

and the curl

operators

are taken with

respect

to variables in the distorted state and the

point

r is

uniquely

related to some other

point

ro on the

unperturbed layer.

We

neglect

the

higher

order terms in d-

do

and in the

(4)

derivatives of fi. The last three terms in this free energy have their

analogs

in nematics.

Ki, K2

and

K3

are the elastic constants for

splay,

twist and bend deformations of

fi, respectively.

As

one can see the

splay

of fi

corresponds

to the bend of the

layers

while the bend of fi

corresponds

to the

splay

of the

layers.

In our

description

of

smectics,

we

neglect dislocations,

which means that the total number of

layers

crossed

along

any

path going

from some

point

A to another

point

B is constant. This is

equivalent

to the condition

[6]:

?~)

"

°' (5)

where

d,

in

general, depends

on the

position

r. Condition

(5)

eliminates the twist term from the free energy, but not the bend termj in de Gennes book it is assumed [6] that the

layer spacing

is constant which also eliminates bend from the free energy. Here we

only

note that

bend does not affect the

long wavelength properties

of the

system. Combining equation (4)

and

equation (5)

we

get

F =

B

(~

~

~

+

Ki (i7fi(r)[~

+

I(3 ji7d

~

@i7d ) (6)

°

In the next section we express both

quantities

in terms of u(~o> voi

zo)

3. Wee energy in terms of the vertical

displacement.

In

smecticsi

fi and d are not

independent

and can be

expressed

in terms of the vertical dis-

placement

field u

(ro)

The vector normal to the

layer

at

point

r

= r

(ro)

is

simply given by:

n =

l~o ~~~

, ~~~

l + V

~ u

where ro

"

(~o,

vo,

zo)

is the coordination

point

in the

unperturbed system.

The distance

between

layers

measured

along

fi is

d =

dz (ilij, (8j

where

dz

is the distance between

layers

measured

along

the z-axis- We

find,

to the lowest order in the derivatives of u, that

dz

=

do (1+ ))

(9)

o

which

together

with

equations (7, 8) gives

d in terms

ofu, (~

+

RI

d "

do

~

~ (10)

~

In order to

represent

the free energy

given by equation (6)

in terms of u, we have to transform the

point

r on a

perturbed layer

to the

point

ro on the

unperturbed layer

and the

corresponding

nabla

operators

as follows:

z#z0,

(5)

180 JOURNAL DE

PHYSIQUE

II N°1

" Yo,

z = zo + u

(zo>

Yo

zo)

and

~ IO

au

~

~jtt lo'

~~~~

fizo

)

au

"

~ ~~iu 1'

~~~~

~ fiZo

t

"

~

~ou £

~~~~

ozo

Combining equations (7, 10-13)

we recover the condition

given by equation (5)

for the constant number of

layers. Finally

we obtain the

following expression

for the free energy

density:

F = B

(~

~

~°)

+

I(i i7 f~fii)~

+

It3 lji7f~d j~i7 (~

d

,

(14)

°

~ ~

~

where

hi

"

-i7f~ u/~.

As noted

before, although

the bend term associated

~

with

K3 gives

a

coupling

between

compression

and undulation of

layers,

it does not affect the

long-wavelength properties

ofthe

system. Ignoring

the bend term and anharmonic contribution to the

splay term,

we find an

approximate

form of F

F = B

°~°

~

i +

I<i (Ai~lL(ro)) (15)

~~~~~~ro))

~

~

The free energy

density given by equation (14)

or

equation (15)

is invariant with

respect

to rotations. For

example, rotating

the

system

around

y-axis by

the

angle

we find:

u=z0

c~i

--1

-z0tani. (16)

For such

spurious

deformation we find from

equation (15)

that F = 0.

Expanding equation (15)

in

i7f~u)~

we recover the de Gennes free energy

density (Eq. (3))

in the lowest order of the

expansion, losing

however the rotational invariance of F.

Alternatively,

the free energy

density

can be

expressed

in terms of Vu.

Using equations (6), (7),

and

(11)-(13)

we find

F =

B

l +

Iii (i7fi(r)[~ (17)

~/l

2

(°"/°z (V"(r)(~/2)

~

(6)

where

(-ou lox, -oulay,

I

fi"/fiZ) fi(r)

"

(18)

~~

~ ~~

/

~z

ivu(r)

i~

/2j

and we have

neglected

the bend term. One

easily

checks that F

= 0 when

u =

z(I

cos @) ~ sin @.

(19)

In the lowest order

expansion

in

flu/fiz [i7u[~/2

we recover the Grinstein and Pelcovits

expression (see Eq. (2)).

The transformation

properties

of u result from the fact that it is defined as the vertical

displacement

of the

layer

from its rest

position.

4. Conclusion

Summary.

The

simplest

form of the free energy

density

invariant with

respect

to

global

rotations is

given by equation (15)

or

equivalently by equation (17)

and in the

expanded

form

by equations (3)

and

(2), respectively.

The correct variable

describing

deformations of

liquid layers

is the vertical

displacement

of the

layer

from its rest

position.

This variable transforms under rotation

according

to

equations (16)

or

(19).

In the case of a smectic with

positional

order inside the

layers,

which can

freely

slide

against

each

other,

it is more

appropriate

to use two variables for

describing

its

deformations, namely

a vertical

displacement

and a two dimensional strain

tensor

[10].

The anharmonic terms are

important

not

only

because

they

are

responsible

for the undula- tional

instability

under the influence of external stress

[6, 7],

but also because

they

can shift the

equilibrium positions

of smectic

layers.

It is well known from the

theory

of the anharmonic

oscillator and the

theory

of atom vibrations in solids that third order terms in the elastic energy shift the centers of vibrations. Near

interfaces,

due to the

change

of fluctuations [5] one can

expect

the

layer spacing profile. Finally,

since the distance between

layers deq

is related to the

length

of the molecule L and the tilt

angle # by

the

following equation:

d~q = L cos

# (20)

we can

expect

a tilt

profile

near interfaces as well

[8, 9].

We defer studies of this

fascinating problem

to a future work.

Acknowledgements.

This work was

supported

in

part by

a KBN

grant.

References

[ii

DE GENNES

P-G-,

J.

Phys. Colloq.

France 30

(1969)

C4-65, [2] CAtLLE

A.,

C,R. Acad. Sci, Ser B 274

(1972)

891,

[3] GUNTHER L., IMRY Y, and LAJzEROWICz J,,

Phys,

Rev, A 22

(1980)

1733, [4] GRINSTEIN G, and PELCOVITS

R-A-, Phys.

Rev. A 26

(1982)

915.

[5] HOLYST

R,,

TWEET D-J- and SORENSEN

L-B-, Phys.

Rev, Lett, 65

(1990)

2153;

(7)

182 JOURNAL DE

PHYSIQUE

II N°1

HOLYST R.,

Phys.

Rev. A 42

(1990)

7511.

[6] DE GENNES

P-G-,

The

Physics

of

Liquid Crystals (Oxford University Press,

London,

1974)

pp. 281-300.

[7] LANDAU L-D- and LIFSHITz

E-M-, Theory

of

Elasticity (Pergamon Press,

Third

edition, 1986)

pp. 172-177.

(8j HEINEKAMP

S.,

PELCOVITS

R-A-,

FONTES E., CHEN

E-Y-,

PINDAK R. and MEYER R-B-,

Phys-

Rev. Lett, 52

(1984)

1017.

[9j TWEET D-J-, HOLYST

R-,

SWANSON B-D-, STRAGIER H, and SORENSEN

L-B-, Phys.

Rev. Lent.

65

(1990)

2157.

[10] SACHDEV S. and NELSON

D-R-,

J.

Phys.

C 17

(1984)

5473.

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