``Pseudo-Casimir'' effect in liquid crystals

“Pseudo-Casimir” effect in liquid crystals A. Ajdari, Bertrand Duplantier, D. Hone, L. Peliti, Jacques Prost ... long-range forces analogous Casimir e...

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“Pseudo-Casimir” effect in liquid crystals A. Ajdari, Bertrand Duplantier, D. Hone, L. Peliti, Jacques Prost

To cite this version: A. Ajdari, Bertrand Duplantier, D. Hone, L. Peliti, Jacques Prost. “Pseudo-Casimir” effect in liquid crystals. Journal de Physique II, EDP Sciences, 1992, 2 (3), pp.487-501. �10.1051/jp2:1992145�. �jpa-00247645�

HAL Id: jpa-00247645 https://hal.archives-ouvertes.fr/jpa-00247645 Submitted on 1 Jan 1992

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Phys.

J.

II

France

2

(1992)

487-501

1992,

MARCH

487

PAGE

Classification

Physics

Abstracts

61.308

68.45

"Pseudo-Casilnir" Ajdari(I),

A.

(~)

Laboratoire

05,

Cedex

Peliti(~,**)

Prost(1)

and J.

10

Vauquelin,

rue

F-75231

Saday,

CE

Institut

de

Fondamentale,

Recherche

F-91191

Department of Physics, University of CaJifornia, Santa Barbara CA 93106, U-S-Ad'oltremare, Pad. 19, Dipartimento di Scienze Fisiche and Unitk INFM, Mostra Napoli, ItaJy 29

effect

show

by the presence electrodynamics.

in

result.

August 1991, accepted We

Abstract. nematics

We

Paris

Frante

Service

(Recdved

derive

the

that of

the

rigid

We

spatial

1991)

November

15

boundary walls give

discuss

different

behavior

this

of

imposed long-range

conditions rise

1-80125

to

caJculational interaction

on

the

forces

schemes for

smectics

fluctuations

director

analogous for and

the

to

the

derivation columnar

in

Casimir this

of

phases

in

geometries.

different

1.

L.

Th40rique (***), ESPCI,

Physico-Chimie

de

Hone(~),

D.

de Physique Th40rique, Gif-sur-Yvette Cedex, France

(~) (~) (~)

Duplantier(~,*),

B.

liquid crystals

in

effect

Introduction.

of the electromapletic field generate long-range forces between macroscopic conducting bodies. These fluctuations be of quantum or of thermal origin. as may electromapletic field corresponds a To each eigenmode of angular frequency w of the classical that, although the total remarked quantum zerc-point energy equal to hw/2. Casimir [I] first zerc-point energy of the electromagnetic field contained in a cavity bounded by conducting walls is divergent, its variation due to a displacement of the boundaries is finite and corresponds to weak, but between measurable attraction the walls. the In of two parallel, conducting a case plates separated by a distance d, Casimir showed that the interaction energy density per unit is by: given area

The

fluctuations

objects

such

E(d) The

presence

of h

witnesses

the

quantum

(*) Physique Th40rique CNRS. (**) Associato INFN, Sezione di Napoli. (***) URA 1382.

~ j~

i i

=

origin

of the

fluctuations.

JOURNAL

488

high

At are

origin.

It

regime (kBTd

this

In

temperature,

thermal

of

»

PHYSIQUE

DE

II

N°3

the efsect shows up in the classical regime, where same produces long-range interactions, akin to van der Waals hc), the energy density between two parallel conducting

fluctuations interactions.

given

walls is

by

Eld) where (R is

lliemann's

~)) ~((~

"

Ii'2) '

function.

2eta

analogous efsect takes place in anisotropic mesophases [2], when immersed bodies conthermal conditions they impose on strain orientational fluctuations, through the boundary liquid crystalline phases, which is the the surface. This is the case, for example, for nematic main subject of this paper, but also of smectic and columnar phases, which we shall touch upon briefly. There is however an important difserence with the electromagnetic case: if the geometry An

immersed

of the

ing

from

interaction. the

imposes

bodies

distortion

a

corresponding energetical

the

Such

mean-field

a

interaction

the

on

does

general

not

dominate

for

exist

field, the repulsion

director

average

will in

cost

the

of

case

result-

fluctuation-induced

the

uncharged

bodies

in

vacuum.

the

In

present

paper,

parallel plates,

two

nematic field is

in

consider

simplest

the

solvent,

nematic

a

to

the

wall

and

uniform

in

space,

geometry: namely, the boundary conditions

case

normal

with

("strong homeotropic anchoring~').

director normal

only

shall

we

immersed

situation

this

In

producing thereby

the

average

interaction.

no

on

of the

director We

shall

by adapting to the present case several techniques developed for the Casimir efsect. In the regime where anisotropic mesophases stable, thermal fluctuations dominate quantum effects. We shall only be concerned, are over therefore, with the analogue to the high-temperature limit of the Casimir effect (Eq. ii.2)). introduced The model and notation in section 2, where we show that longitudinal and are degrees of freedom separately to the effect. In section 3, we introduce contribute transverse transfer-matrix technique and we compute the free energy by exploiting the analogy with a the one-dimensional oscillator. A dynamic approach is expounded in section 4: we quantum introduce a formal dynamics by means of a Langevin equation, which allows to calculate directly the correlation functions of the exerted on the plates. Section 5 contains director and the stress the discussion: similarities and differences with the Casimir effect are pointed out, extensions mesophases are described, and a few cases where the effect we have described may to other be experimentally relevant reviewed. The splitting of longitudinal and modes is transverse are discussed in more detail in Appendix I, whereas approach based the direct counting of an on fluctuating modes and on the zeta regularization technique is reported in Appendix 2. show

2.

how

fluctuation-induced

the

interaction

may

be

calculated

Model.

thickness d, placed between situated two flat, parallel walls on respectively (Fig. I). We denote by I the nematic director, and by ( f I the unit vectors parallel to the z, y and z axes. The of energy of the system is the sum bulk contribution describing lib and of the surface the anchoring of nematic lis, the term a ordering on the walls. The bulk term is given by [3]: We the

consider

planes

nematic

a

and

z=0

lib

"

slab of

z=d

/ dzdy /~

dz 2

o

The

surface

contribution

is

Ki(div £)~

+

2

K2(£

rot

£)~

+

2

K3(£

)~j x

rot

(2.I)

given by 2ts

=

/ dzdy (- ~) (i 2

£)~,

(2.2)

N°3

EFFECT

PSEUDO-CASIMIR

IN

LIQUID

CRYSTAL

489

~

~

°

Fig.

drawing

Schematic

1.

of the

~

for

geometry

"Casimir"

the

nematic

case.

director tends to align along 0, the nematic the director tends to lie parallel to the the normal to the surface. On the other hand, if1 < 0, this unavoidable of complicated, in surface. The situation is made presence case, by the more shall preferred directions We only in the plane. anisotropy fields which tend to align £ along consider the first case, and take the strong anchoring limit, correspondinj to I cc. consider only director £ is uniform and parallel to k. If we In the state of lowest energy the where

the

integral

extends

both

to

walls.

If1

>

-

small

around

fluctuations

this

state,

have

we

In«> ny> i)

it We

shall

harmonic

by

denote

v

approximation

lib

=

twc-dimensional

a

obtains

one

/ dzdy /~ /

=

and

vector

(2.3)

(n>1). by I

the

)Ki(V

n)~

dz

+

three-dimensional

a

following expressions

for 2tb

n)~

)K2(V

x

n~(z,

z=d))

one.

In

the

and 2ts:

K3(0zn)~j

(2.4)

+ ,

o

ll~

=

(n~(z,

dzdy

Here

V is the

twc-dimensional

and

transverse

components:

nabla

y,

operator.

n

such

z=0)

=

ni

+

The

y,

field

may

n

(2.5) be

split

into

its

longitudinal

(2.6)

+ nt>

that V

By applying this decomposition

to

x

ni

(2A) ~lb

0;

=

we

=

V

nt

=

0.

(2.7)

obtain:

~li (nil +

~ltIntl

(2.8)

JOURNAL

490

PHYSIQUE

DE

II

N°3

where

/ ~~~~ ~ f dzdy j~

~~~~~ ~~in~i surface

The

field ni

and

transverse

and the

other

the

~~~~~~~~~i

~~~~ ~

~~'~~ '

dz

into

[(n~(v

two

transverse

form,

same

Therefore

nt.

(2.io)

+

of the

terms

one

~~(ozn~)2j

n~)2

x

one

may

one

involving

consider

the

the longitulongitudinal

separately.

fluctuations

Partition

3.

(2.5) splits

contribution

dinal

=

~~~~~

~~

function.

approximation, the partition function of nematic fluctuseparation of longitudinal and modes, we can first transverse consider only the longitudinal field ni, treating it as a scalar field #. This procedure can be therefore: justified by the projection operator technique discussed in Appendix I. We obtain

We

calculate,

now

in

ations

the

the

in

slab.

Due

harmonic

to

the

zi

Due

One

one

for

each

#(q,z)

=

in

(z,y) plane, Zi

factors

parallel

the

the

(- ]

wavevector

is the

f D#(q, z) (-( exP

Fourier

q

(~ii~i

~s

+

to

id)

(3.1) independent

into

contributions

(z, y) plane.

elastic

have

parameters

may

be

zi(q) '~~~~~

J~~~
Kd(#> lo

=

=

I/kBT;

(z, y) plane,

(-liq)

(3.2) ,

and

we

ii~(oz<)~j

have

defined

(3.3)

k;

=

K;/kBT,

(3.4)

I =1, 2, 3.

form

f d#od#i

the

the

exP

by kBT:

exp

I-

iii'((' D>(z) xP1-

satisfies

#~(q>

+

rescaled

in the

cast

+

)kiq2<2

j~ dz

been

=

Equation (3.2)

=

=d)))

(#~(q> z=0)

#(z, y,z) along

of

transform

I

The

exp

independent

fig The

/ D>

has:

zi(q) where

invariance

translation

to

Zi(q),

=

(#I

+

~ dz

if)

Kd(#i, #o),

ii~>~~'~

+

>~~°~~~i1

(3.5)

~~~~

equation

~Kd(#> lo)

(£$ ~

=

3

kiq~#~j

Kd(#> lo)>

(3.7)

EFFECT

PSEUDO-CASIMIR

N°3

analogous

to

Schr6dinger equation

the

LIQUID

IN

CRYSTAL

oscillator.

one-dimensional

for the

491

The

initial

condition

reads:

Ko(<>
Kd(#,#o)

expanded

be

may

in the

(3.8)


&(<

=

form

m

Kd(#> lo)

(3'9)

£e~~~(~~~'~>~ltp(i)ltj(10)>

"

p=0

(j)

'~~~~~

' ~~ and

where

~p's

the

(t> 'C3

=

eigenfunctions

are

q

3

quantum

of the

) ())

~p(#)

(3.10)

q,

=

They

oscillator.

harmonic

are

given by

e-fl~~~'~Hp(@#)>

=

(3.ii)

where

~~

Hp(z)

and

p-th

is the

Equation (3.5)

~~ ~~~

the

form

~

zi(q)

The

takes

(i~ik3)~q>

polynomial.

Hermite

now

=

£e-Wq~P+1'2>d fd#oe-~~"2~~(#o))

=

integrals

zt(~)

can

(Ref. [4],

evaluated

be

p~~i )

(9) '

e-~~~'~

(7.373.2) p.837)

formula

(3.13)

/d#ie-~~?'2~~(#1))

-~~~~

~ii~l,~"

=

and

give

(li II

j

~

(3.14)

~

Performing the

sum

Zi(q)

obtain

we

e~~~~'~

~~

j

i

=

flq

~

The is

longitudinal

of the

contribution therefore

modes

the free

e~~°~~

flq

+ 1

energy

per

+ 1

to

2

~~

~~

unit

area

(3.15)

of the

nematic

slab

given by

~

~~~ kBT

~

/ j

2

kBT ~

2

j

~$2

~q

~~'~~~

(2x)2"~ d2q (2x)2

~ ~~

j

d2q (2x)2

2x ((flq)1( fl~ + I )j

~

x

j

flq-I)~ ~

fl~

+

j

~

_z~

~

(3.16)

JOURNAL

492

Taking

strong anchoring Emit (I

the

now

~~~ / (~12"~

~

PHYSIQUE

DE

cc)

-

~~~

~

N°3

obtain

we

/

II

~12 ~~

~x)

~~~~~ (3.17)

~

The

first

Both We

is

term

independent

d, is a contribution divergent for (q(

of

terms

are

interested

are

interaction

contribution

bulk

a

(~~ ~~ ~~

~~~

~

the

in

between

the

third It

=

~

~~

(R(3)

where

gration The

contribution

of the

problem.

this

fluctuation-induced

yield

to

(I)

qd

(3.18)

~~~~~ ~~

~)

'

modes

transverse

substitution

the

to

convergent,

is

of Ki Thus, walls is given by

the

kBT

~~

interaction

obtained

thus

~

~~~

is

obviously

elastic

the

to

of Ki

with

the

instead

K2

between

attraction

The

walls.

have

we

moved

the

inte-

upper

infinity.

to

longitudinal ones, up equation (3,18), with the

'-2 exp

with

cope

the

'~~

and, since the integral

1.202,

=

limit

to

represents

°

~~

how

below

and

evaluated

Ii

dq In

q

finite,

is

explicitly

be

/~

discuss

shall

which

may

~~~'

[email protected]

We

cc.

term,

surface

the

term,

walls.

The second energy density of the system. between the nematic and the tension

the free

to to

-

~~~~

~

jc3

is

energy

The

K2.

result

contribution

total

analogous is

therefore

of the

that

to

nematic

modes

i~

~

1C2

to

to

(3.19)

~~~~~fl'

'i

of the

analogous

attractive.

introduction equation (3.17) is removed by the of cutoff A in the integral over q. This cutoff corresponds to the shortest wavelength of upper an parallel to the wall, and is of the order of the inverse molecular fluctuations in the directions explicit physical interpretation. On the other hand, no such cutoff has size. It has therefore an calculation, on the wavelength of fluctuations in the z direction. We do been imposed, in our not expect this slight inconsistency to modify our final result (3.19). described and the the phenomenon we have The analogy that we have highlighted between problem by electrodynamics analyze the Casimir effect in quantum [I] suggests to present is, authors of distinguish reference [6] methods developed for the Casimir effect two 6]. The evaluation of method, based on the direct mode-summation broad classes of approaches: the infinite over energy eigenvalues of the zerc-point modes, and local formulations, in which sums The

divergence

exanfines

one

stress

tensor,

discussed direct We

the

is close is

discuss

in

mode in

next

terms

two

of

propagation expressed in

philosophy

of the the

be

can

reported in

first

constrained

which

evaluation

calculation

of the

to

the

of

field

virtual

terms

Appendix

2.

section

method

a

based

on

the

zeta

direct

and

considers method

the

vacuum

just attempted a regularization technique. This

methods.

mode-summation

taking advantage of the

sum,

quanta

propagators.

of

The We

evaluation

have

we

have

also

of the

stress

tensor.

N°3

Dyna~nic

4.

Let

approach.

in

Langevin

Waals

493

this

between

present a

CRYSTAL

the walls by the nematic exerted section the definition of the stress on them, and show how it can be directly calculated by an approach based on equation, sinfilar to the method originally used by Lifshitz [7] to discuss van der

discuss

We

LIQUID

IN

EFFECT

PSEUDO-CASIMIR

forces.

consider

us

fluid, enclosed in

a

(e.g., either corresponding free

field i7

longitudinal

the

a or

transverse

reads:

energy

F

V, whose local ordering is described by a scalar component of the nematic field n). The

volume

the

/

d/F(?i7).

(4.i)

=

v

dimensional Here fl is the three boundary conditions, i7 0.

nabla

shall

We

operator.

=

Let

consider

us

element

/

~$

di

=

=

The

first

virtual

hand,

does

i7

The

+

of V due to

/

no

~$



bi7 +

0(Viz)

s=av

the

equilibrium vanish

more

/ d§

Therefore SF

=

stress

be

each

surface

written:

SF

/ (d§

~~ ~~

d§ bif(fli7).

nfinimum

of the

elastic

surface

S,

whereas

it

vanishes

approximated

on

actual

surface

bi=

=

/ s=av

actual

be

can

bi7 +

is

state

the

on

~$ bi.

by

T

"strong anchoring"

displacement biof

a

may

a

the

0q~

free

energy. the on

by:

(4.3)

0.

+

/ d§

0(Vq~)

s

the

have

d§ bif(fli7)

bi7 + fli7

Defining

of F

variation

b(0V). Therefore, bi7

0V +

surface

~$ because

vanishes

term

other

the

expansion

of V.

~~

0(Viz)

v

virtual

b(fli7)

/ ~~~ On

a

boundary 0V

dS of the

SF

effect of

the

to

suppose

bif.

(4A)

s

$4,

T

(4.$)

S we

have

therefore

The ncompressible

first

nematics.

This description bath, between

implying

the

We shall

is

well

adapted

to

our

problem, in

which

the walls

are

immersed in

a

nematic

that walls.

now

relate

the above

pression

to

correlation the

7i

=

x

functions

/ dzdy /~ o

of

the

dz [(Vq~)~ + (0zq~)~]

(4.7) ,

JOURNAL

494

@@,

where

K

nabla

operator,

=

I

(2)

=

PHYSIQUE

DE

longitudinal (transverse) modes,

for

'C3~ stress

the

on

wall

at

z

we

V is

the

twc-dimensional

?

have

the

taken

(j'

I'(fiz9')~)

"

l/2

~_

(4.8)

d.

=

reads

h

=

Tzz

where

N°3

and h

The

II

with

average

respect

((fiz9')~

to

+

thermal

the

(4.9)

(V9')~j fluctuations.

simply calculated within a dynamic can averages approach. Although the introduction of a dynanfic equation is strictly speaking unnecessary, it simplifies considerably the calculations, and it has a physical appeal, since it clarifies the director. of the fluctuations We fact that the stress we are computing originates in the thermal thus introduce Langevin equation describing a model dynamics of our system: a The

appearing

thermal

7j~ q(F, t)

where

is

Gaussian

a

(q(f t)) dynamics

The

of the

system,

of the

model

The

but

(in

i7(I,t)

which

field is

does

relations

above

the

G(I,t; i~, t')

are

interested) given by

then

q7(I, t)

where

sufficient

Green's

is the

=

f di~dt'G(I,

function

~~~

the

are

of the

KfI~G

with

the

boundary

Fourier

y,

z

with

transformation

=

0,t;

?,t')

respect

solution

of this

~

ensure

have

we

so

dynamical behavior equilibrium properties

actual the

considered.

far

>,t')q(>, t'),

(4.12)

equation (4.10) and

evolution

I)b(t

G(z,y,

=

to

and t

z, y

z

satisfies:

t'),

(4.13)

system

~~~~'~ ,~

of

equations

KwGqw

=

h,t;

I,t') =

(4.14)

0.

yields:

fi2

(iW7 + Kq~)Gqw

The

the

that

(4.I1)

conditions

G(z, A

ones

6(F-

=

describe

to

t;

t')b(F- /).

2~kBTb(t

=

necessarily

not

are

we

(4.io)

n(F> t)>

=

(q(I, t)q(I, t'))

0;

just defined

have

we

K?~i7

noise, satisfying

white

=

be

formula

in this

=

b(z

z'); ~~ ~~~

reads:

sinh(«h)]~~

l-[«K -[«K sinh(«h)]~~

sinh sinh

[«(z'- h)] sinh(«z), [«(z h)] sinh(«z'),

if

z

<

if

z

>

z'; z';

~

~~

PSEUDO-CASIMIR

N°3

EFFECT

LIQUID

IN

CRYSTAL

495

Where ~

We

can

the

evaluate

now

the

that

account

last

stress

the

on

vanishes

term

2)

ij7

=

z

plane, using equation (4.9) and boundary conditions (4.14):

h

=

into

now

/ / () ~

~~~~ Tzz

=

order

an

obtain

to

infinite

~~"~~'~'~~z=h

~~~~~~

(4,16)

integrating

£Y

the

for

stress

ATzz

distance:

and

z',

over

obtain:

we

oth(«~h)j

~/~'

_~~

=

finite

a

Tg.

Tzz

We

back to the

original length scale,

£Y

value of h,

£Y

_~~

(4.19) ,

corresponding

equation (3.18)

to

or

some

the

corresponding

value for

algebra,

the

(4.20) that ATzz

account

must

also be rescaled,

(R(3) ~)i>

=

to

after

lx ~~~~~~~'

-~ (?

ATzz

subtract

we

obtain,

thus

and taking into

obtain

we

G,

coth(«+h)

~'~

Going

of

18)

'~~~~

«(+q; +w).

a*

where

°

q

form

~~/"(

=

~

In

the explicit

account

into

1(4

[l~ (j ~~

Taking

taking

of the

because

j~ dz'

~~'~~~

+

analogous

one

for

(4.21) the

modes.

transverse

Discussion.

5.

discussed in this paper is obviously analogous to the Casimir effect, isotropic case (xi K3) the Frank elastic energy is identical to K2 with ii (£t) Playing the role of the electric (mapletic) field. The boundary conditions correspond to a field constrained between grounded conducting plates. It is therefore surprise that in the isotropic case equation (3.19) coincides with equation (1.2). no It might then totally superfluous to derive the same expression with three different appear techniques. However, while in the electromagnetic problem there is no small scale (ultraviolet) cutoff, there is one obvious one in the spectrum of nematic molecular size. As fluctuations: ultraviolet divergent terms, which must be dropped altogether in the electrca consequence, theory), have a definite physical meaning in magnetic problem (in the spirit of renormalization nematics. For example, the first term of equation (3.17) corresponds to a bulk free energy, and the second to a surface tension. They scale like kBTqfd and kBTqf respectively, where qc is of The

phenomenon

have

we

Tone observes that in the the electromagnetic energy,

the one

order

of

may

wonder

an

"

=

inverse

molecular

if there

is

any

Since

size.

kBTqcd~~

the

able to reveal its existence, the Euler-MacLaurin Therefore it is important to attack the problem the dynamical approach is to allow us to evaluate It

involves

only

one

cutoff-dependent

term

Tg,

Casimir-like

Although

term.

the

summation

with

scales

like

kBTd~~,

regularization would zeta the dynamical approach or

different

directly

which

interaction

the

expresses

techniques. force

applied

the

pressure

The on

be

un-

could.

advantage

the

due to

boundaries.

short-scale

of

JOURNAL

496

fluctuations. the

via

energies We

will

It

before:

be the

should can

try

now

To

effect

why there

understand

to

simplest picture that we can give is be thought of by the fluctuations can be

may

seen

particles the

as

a

larger

of size ideal

outer

gas

the

that

Here

is

n

gap

particle

if the

has

nkBT

=

up

to

a

a

interpretation, there is no correct d dependence has excluded '§~article" size, but one this

of

quantizes

field

is

The feel

(5.I) an

extension

expression

is

that

it is clear

the

of the

occurrence

particles of larger size. Indeed, if all multiple integers of d are also

f

I ~

~~°

d~

the

smallest

Riemann

(R

function

accept

that

in

down

trace

(5.2)

2

interplay of short and long scales. equation (5.I) by retaining only

obtained

can

of

spectrum

a

modes,

the

harmonically fluctuating equal to their wavelength. excluded from it, and the walls closer together: The

+

for the

room

been

The

exclusion

The

mediated

() )) c<

1

the

long scales.

interaction

factor.

+ nt

ni

to

and

-Arzz.

=

is

Indeed, hunches on the numerical factor too. argument provides some (Ki/K3)Q~d longitudinal linear size for the along it have size d must a z, fields, in the (z, y) plane. We have therefore, with obvious the transverse

and (K2/K3)~'~d for notations:

In

mixing short physics: the

number

each

numerical

sizable, since it

and

Since each particle has "particles". density scales like d~~ This

density of excluded direction, their

number

the

colloid

as

p

proportional to d in equivalent to (4.21), Actually the same

term

no

from

gas

which

pressure

a

is

numerically

is

term

depletion force. a particles, whose size between the plates are tends to bring the walls

of ideal

collection

N°3

anchoring energy, essentially of anchoring been has never source energy smooth surface; reported small anchoring

this

borrowed

II

anisotropy

surface

knowledge,

our

tension

surface to

always be large on of inhomogeneities.

should

it

the

contribute

also

anisotropy.

cutoff

considered

noting that

worth

It is

kBTqf.

of order

PHYSIQUE

DE

we

excluded:

one

the

gap

then

has

(R(3)

I ~

k~

(5.3)

~~

~=j

simplified picture

This

smectics

as

simultaneous

Here

u

is the

vectors

in the

and

u

pression

columnar

and

to

smectic

existence

us

to

consider

phases

in

the

geometries

of first

second

and

(f

~s

=

7ic

=

~

d~/1(()) f (v d~/

+

at

order

qz

scales

The

like

liq~>

A~ ~~

~

time

same

figure

2.

liquid crystals common

feature

al v~«)~j

+ »

such is the

(5.4)

lvU

particles"

and in

other

Their

~i~~I)~

U)~

"excluded

of

elasticities:

+

displacement of the layers along the V are perpendicular to the column curvature.

the

allows

normal

direction

axis

The

must

columnar

z.

now

phases

~kBT/(lid2), -(~ikBT)/(d~), -(kBT)/(1('~d5/2) ~(~lkBT)/(d~),

have like

>i

two a

(ii)~]

z;

and

twc-dimensional

lengths lj and 13 compare anisotropic shape, since very

(q/13)~'~ case

a).

cas~

b)~

~)

~~~ case

the

(5.5)

d(.

One

thus finds:

(5.6)

PSEUDO-CASIMIR

N°3

Fig. 2. (c and d).

The

Schematic

determination

draining

of the

of

prefactor

the

EFFECT

LIQUID

IN

CRYSTAL

497

a~

b)

C)

d)

geometries

relevant

beyond

is

for

the

(a

smectic-A

scope

of

this

and

b)

analysis,

and

columnar

but

the

phases

argument

leading to (5.4) suggests that (R(2), (R(4), (R(5/2) and (R(5) should respectively come into play. We have checked by explicit calculations that this is indeed true for cases a) and c). These the most interesting geometries, since they lead to forces stronger than van der Waals. For are smectics

we

obtain

indeed:

ATzz Since any

Al is

a

adjustable

measurable

parameters.

quantity,

-~j(R(2). prediction

this

Force

IS.?)

=

could

be

apparatuses

measurement

tested

[8] been

are

experimentally well

suited

in

without

principle

performed [9]. However, the imply the existence of dislocation loops. Creation (upon compression) or annihilation (upon dilation) of these loops gives rise to oscillatory forces, whose minimum sits on an background. It is not clear whether attractive this background corresponds to the long-range attraction discussed in this The analysis paper. consideration of the modification of the fluctuation of this problem requires spectrum due to non-uniform distribution the of the order induced by dislocations. parameter of fluctuation-induced An interesting forces wetting. Wetting of surconsequence concerns faces by smectic layers has been observed at the isotropic-air interface of some mesogenic compounds [10]. In the case of a disjoiniqg pressure arising from van der Waals forces the growth of the wetting layer is known to follow a (T Tc)~l'~ law, where Tc is the bulk transition With suitable boundary conditions (e.g., strong (weak) anchoring at the temperature. (smectic-isotropic) interface), smectic-air the growth should follow a (T Tc)~Q~ law. Indeed, the the energy density per unit of smectic layer reads in such a case area for

this

purpose,

and

experiments

geometry of the experiment

involves

in

smectics

F

have

indeed

boundaries,

curved

=

fd

+

~~~',

which

(5.8)

JOURNAL

498

where

(R(2)/32x,

and f phases.

PHYSIQUE

DE

II

N°3

(T

Tc) is the difference in the bulk free energy between the gives the proposed law for d upon minimization. It symmetric boundary conditions possible also (e.g., rigid boundary are /1): in conditions require the anisotropic part of the interfacial tension to be larger than the behavior this is qualitatively changed. The layer is wetting finite the transition at case since the attraction due to the smectic be temperature, fluctuations compensated by the can der Waals disjoining This could explain the finite value of smectic layers at the van pressure. isotropic-air interface close to the smectic-isotropic phase transition. K

=

isotropic and smectic is worth noting that

~K

This

result

Acknowledgements.

gratefully acknowledges

DH

visit, and

his

DMR-8906783.

Appendix Splitting

and

and

a

here

with

and

some

more

«,

grant

Th40rique of the

number

ESPCI

for

a

Let

defined

=

splitting

introduce

us

fluctuating field n into its longitulongitudinal and parallel projectors Pi

of the

the

of:

means

fl

the

care

components.

Pin;

=

nt

(Al.I)

Ptn.

"

by:

(Pi)ap where

under

modes.

transverse

ni are

Fundation

Laboratoire de Physicc-Chimie pleasant working atmosphere.

during

provided by ESPCI

support

Science

the

longitudinal

transverse

Pt by

They

financial

National

I.

of

discuss

We

dinal

and

of the

support

thanks

LP

hospitality

kind

most

hospitality and

the

financial

the

2tb

V~~0a0p

Hamiltonian

The

1, 2.

=

#

/ dzdy /~

(Pt)ap

=

(bulk plus surface) dz

()(Pin)Ai(Pin)

V~~0a0p,

Sap can

+

then

be

(Al.2) in

written

j(Ptn)At(Ptn)j

the

form

(Al.3) ,

o

At and At

where on

we

are

suitable

and,

operators,

say,

periodic

fluctuating field n. By performing the functional integral, and exploiting obtain the following expression for the free energy:

boundary

conditions

are

imposed

the

F

By taking

the

Fourier

tranform

=

kBT)Tr in the

(PIAIP,

(z, y) plane, 2

~

In

~~~~

+

we

/ L ('nw~n ~

the

property

PtAtPt)

P~ =

P of the

projectors,

(Ai.4)

obtain

(Al.5)

+ In w~

~"

'

"

Since they are both scalar and w(n are the eigenvalues of At and At operators, obtain if there independent result would Pi TrPt I, this is the same two one were Hamiltonian Hamiltonian 2tt> and with the 2t,, and one to the scalar fields, one subject to the surface term 7is. same

where and

Tr

w(n

=

#

N°3

PSEUDO-CASIMIR

IN

EFFECT

LIQUID

CRYSTAL

499

Appendix 2. regularization.

Zeta

discuss

We Casimir

effect

boundary

appendix

in this

regularized by

means

is, 6, 11].

conditions

For

the

on

a

based

method

the

assume

summation

=

q2(o) We

direct

the

on

zeta regularization technique, already simplicity, let us first consider a scalar situated 0 two parallel plates at z

of the

Hamiltonian

to

2t[i7]

=

be

q~(d)

=

fluctuating modes,

over

in the

known

context

field i7, subject and z = d:

to

(A2.i)

o.

=

of the

vanishing

given by

/ dzdy /~

)K(Viz)~

dz

K3(0zi7)~j

+

(A2.2) ,

o

(= K2) for longitudinal (transverse) modes. where K Ki along the (z, y) plane and expand in eigenmodes along the expression for the energy:

We

=

~

The

free

energy

is

L~d

[email protected]

given by: F

If the

functional

~c~~ii~qn i~

=

integral

is

-kBT In

=

kBT~~ =

we

have

introduced

the

/ ~j

2

shorthand

transform

the

~l

~

(A2.3)


(A2A)

(A2.5)

In eqn,

~

notation

£~ / (~~2' eqn

=

k q2 + ii~

~~~'~~

(]~) ~,

(A2.7)

with rescaled Frank (Eq. (3A)), and where a constant contribution constants §y normal12ation. We are therefore proceeding to a direct summation modes, identified by the quantum numbers q and n. The sum (A2.5) is divergent and may be regularized by setting

((S) where

following

obtain

we

~

where

1i~l

+ ~3

obtaining

/ Di7 e~~'~B~

formally performed, F

z

Fourier

the

take

can

direction,

=

has over

dropped fluctuating

been the

/ f(eqn)~'>

(A2.8)

the real part of s is sufficiently large so that the sum converges. to the whole (finite) complex plane. One then sets'

Then

((s)

is

analytically

continued

f~ q

n

In eqn

=

)((s)~> >=o

(A2.9)

JOURNAL

500

((s)

where

analytically

is the

j~ q2

<(s) =1-2,

continued

function.

have:

We

is

positive integer.

a

only

differentiate

the

but

the

has

zeta

continued

function

(z~

/

~

explicitly

for

z

~~

because

-2n

=

z

p.

complex plane.

Let

vanishes

for

and

I

=

z

=

at

this

purpose

it is

~~~~

(~

~'~l'

For

remark

us

differentiating ((s)

in

factor.

(9.513.3),

formula

at

interested

are

we

the

to

simple pole

a

Riemann

~~'

1)

vanishes

above

~~~~

if

)~~~)

which

d

Therefore,

~~~~~

the

~

K

(R(z)

representation (Ref. [4],

the

,~

analytically

be

may

function

2eta

(A2,10)

"

I)

((s)

form

this

~~'~'

2~2'

4x(s

Riemann

i j~)

=

"

j-20

In

N°3

~jj~'~~ ~

ij~i~j'

j3

+

PHYSIQUE II

DE

s

that

the

-2n, where n 0, we need =

convenient

to

uie

1072)

i~~

~~~~ ~

T(-n)

equation is symmetric in the interchange expression is the same for z +3. Then

z

The

cc.

=

I

-

z.

~~~'~~~

expression We

between

interested

are

braces in

z

=

in

-2,

=

l~~~~~

~~

~l ~sll~
"

=

-&
(A2,12)

,=o

Substituting

this

result

into

equation (A2.10) yields

~~~

lx ~~f)

~~~~~>

(A2.13)

,~~

corresponding to equation (3.18) or to the analogous one for The interesting property of this approach is that no explicit on

q

on

nor

section

the

does

3

z

quantum

not

number In

appear.

n.

reference

The

ill]

asymmetry it is

shown

that

that

the

transverse

cutoff we

the

has

have zeta

been

modes.

imposed,

discussed

at

the

regularization

neither end

of

method

yields the same result as a method in which the sum over the discrete quantum number n is regularized by the imposition of an exponential cutoff. Indeed, it may be seen that all such regularization method yield equivalent results, realized it is that the physics of the once problem imposes to subtract the free energy of unconstrained fluctuations in a region having volume and the the plate area. same same The result obtained by a method close to the original method used for the same may be Casimir effect in electrodynamics, namely, by means of the Euler-MacLaurin summation forintroduction of a cutoff at large values mula. In this case the expressions are regularized by the calculation. of (q(, which is let to infinity at the end of the We shall not discuss this method, involved than the have just expounded, and since the since it is algebraically more we one asymmetry Note

added

We learn surface 69

in in

that

the

cutoff

procedure

is

more

difficult

to

circumvent.

proof: the

tension

(1989) 358].

in:

case

in the geometry of figure 2a has been worked out for arbitrary L-V-, Zh. Eksp. Theor. Phys. 96 (1989) 632 [Sov. Phys. JETP

of smectics

Mikheev

PSEUDO-CASIMIR

N°3

EFFECT

IN

LIQUID

CRYSTAL

501

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ii]

Casindr

H. B.

G.,

Proc.

Kon.

Ned.

Akad.

Wet.

51

(1948)

793.

[2] Ajdari A., Peliti L. and Prost J., Phys. Rev. Lett. 66 (1991) 1481. [3] de Gennes P.-G., The Physics of Liquid CrystaJs (Oxford: Oxford Univ. Press, 1974). (San Diego: Products [4] Gradshteyn I. S. and Ryzhik I. M., Tables of Integrals, Series and

Acadelnic

Press, 1987). [5] [6] [7] [8] [9] [10]

Reuter Plunien

M.

and

G.,

W.,

Dittrich

Miller

B.

and

Eur.

Greiner

J.

Phys. 6 (1985) W., Phys. Reps.

33. 134

(1986)

M., Sov. Phys. JETP 2 11 956) 73. Ninham B. W., Rev. Parker J. L., Christenson H. K. and Richetti P., K4kichelf P., Parker J. L, and Ninham B. W., Ocko B. M., Braslau A., Pershan P. J., Als-Nielsen G. and Lifshitz

Sci.

Svaiter

N. F.

and

Svaiter

B.

F., J.

Math.

Phys.

32

(1991)

Iustrum.

60 (1989) (1990) 252.

Nature

346

Deutsch

M., Phys.

94.

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87.

E.

175.

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3135.

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