“PseudoCasimir” 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. “PseudoCasimir” effect in liquid crystals. Journal de Physique II, EDP Sciences, 1992, 2 (3), pp.487501. �10.1051/jp2:1992145�. �jpa00247645�
HAL Id: jpa00247645 https://hal.archivesouvertes.fr/jpa00247645 Submitted on 1 Jan 1992
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Phys.
J.
II
France
2
(1992)
487501
1992,
MARCH
487
PAGE
Classification
Physics
Abstracts
61.308
68.45
"PseudoCasilnir" Ajdari(I),
A.
(~)
Laboratoire
05,
Cedex
Peliti(~,**)
Prost(1)
and J.
10
Vauquelin,
rue
F75231
Saday,
CE
Institut
de
Fondamentale,
Recherche
F91191
Department of Physics, University of CaJifornia, Santa Barbara CA 93106, USAd'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 longrange
conditions rise
180125
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,
PhysicoChimie
de
Hone(~),
D.
de Physique Th40rique, GifsurYvette Cedex, France
(~) (~) (~)
Duplantier(~,*),
B.
liquid crystals
in
effect
Introduction.
of the electromapletic field generate longrange 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 zercpoint energy equal to hw/2. Casimir [I] first zercpoint 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 longrange 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
meanfield
a
interaction
the
on
does
general
not
dominate
for
exist
field, the repulsion
director
average
will in
cost
the
of
case
result
fluctuationinduced
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 hightemperature 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 transfermatrix technique and we compute the free energy by exploiting the analogy with a the onedimensional 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
fluctuationinduced
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
PSEUDOCASIMIR
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
=
twcdimensional
a
obtains
one
/ dzdy /~ /
=
and
vector
(2.3)
(n>1). by I
the
)Ki(V
n)~
dz
+
threedimensional
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
twcdimensional
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
PSEUDOCASIMIR
N°3
analogous
to
Schr6dinger equation
the
LIQUID
IN
CRYSTAL
oscillator.
onedimensional
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
efl~~~'~Hp(@#)>
=
(3.ii)
where
~~
Hp(z)
and
pth
is the
Equation (3.5)
~~ ~~~
the
form
~
zi(q)
The
takes
(i~ik3)~q>
polynomial.
Hermite
now
=
£eWq~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
flqI)~ ~
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
fluctuationinduced
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 modesummation broad classes of approaches: the infinite over energy eigenvalues of the zercpoint 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.
modesummation
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
PSEUDOCASIMIR
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
~$
d§
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
twcdimensional
?
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';
~
~~
PSEUDOCASIMIR
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 EulerMacLaurin Therefore it is important to attack the problem the dynamical approach is to allow us to evaluate It
involves
only
one
cutoffdependent
term
Tg,
Casimirlike
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.
shortscale
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
twcdimensional
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)
PSEUDOCASIMIR
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
smecticA
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 longrange attraction discussed in this The analysis paper. consideration of the modification of the fluctuation of this problem requires spectrum due to nonuniform distribution the of the order induced by dislocations. parameter of fluctuationinduced An interesting forces wetting. Wetting of surconsequence concerns faces by smectic layers has been observed at the isotropicair 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. (smecticisotropic) interface), smecticair 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. isotropicair interface close to the smecticisotropic phase transition. K
=
isotropic and smectic is worth noting that
~K
This
result
Acknowledgements.
gratefully acknowledges
DH
visit, and
his
DMR8906783.
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 PhysiccChimie 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
PSEUDOCASIMIR
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) =12,
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~)
=
"
j20
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 EulerMacLaurin 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 LV, Zh. Eksp. Theor. Phys. 96 (1989) 632 [Sov. Phys. JETP
of smectics
Mikheev
PSEUDOCASIMIR
N°3
EFFECT
IN
LIQUID
CRYSTAL
501
References
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Casindr
H. B.
G.,
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Kon.
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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., AlsNielsen 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.
[11]
87.
E.
175.
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3135.
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57
(1986)