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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�
Classification
Physics
Abstracts61.30-61.30C-46.30C
Short Communication
On the elastic free energy for smectic-A liquid crystals
Robert
Holyst
andAndrzej
PoniewierskiInstitute of
Physical Chemistry,
Polish Academy of Sciences,Department III, Kasprzaka 44/52,
01-224 Warsaw, Poland
(Received
20July1992,
revised 19 November 1992,accepted
9 December1992)
Abstract. We
thoroughly
discuss the layer deformations in smectic-Aliquid crystals.
The invariant elastic free energy ispresented
in terms of the derivatives of the vector field normal tolayers
and thechange
of the distance between thelayers.
We compare our results with theprevious
works of de Gennes and of Grinstein and Pelcovits. It ispointed
out that anharmonicterms in the elastic free energy
might
beresponsible
for thechange
of the averagelayer spacing, layer spacing profile
and inparticular
for the tiltprofiles
in finite smectic systems.1 Introduction.
The smectic-A
liquid crystal phase
consists ofparallel, equidistant
two dimensionalliquid layers.
The average distance between thelayers (denoted d)
isroughly equal
to thelength
of a molecule in the
system.
Thelayers strongly fluctuates;
their deformations are conve-niently
describedby
the vert,icaldisplacement
oflayers,
u(~o,
voizo),
from the restposition
at
(~o,
voizo)
For convenience we assumed thatunperturbed layers
areperpendicular
to the z-axis. Since the seminal paperby
de Gennes [1] the smectic-A free energy for deformations has beenintensively
studied[2-5].
At the harmonic levelapproximation
thelong-wavelength
properties
of smectics are well describedby
thefollowing
free energydensity:
Here
Af~
is the twc-dimensionalLaplacian
withrespect
to ~o, vo, B is the smectic elastic constant associated withlayer compressions
andICI
is the elastic constant associated withlayer bending.
It was Grinstein and Pelcovits [4]~ whopointed
out that this free energydensity
is not invariant with
respect
to rotations and that anharmonic terms are necessary to preserve178 JOURNAL DE
PHYSIQUE
II N°1this invariance.
They proposed
thefollowing
free energydensity
which is invariant withrespect
to
global
rotations:F~p
=B
(°j[~~ jivu(r)12j
~
+
Ki Aiu(r))2j
(2)
where i7 =
(@/@~, @/@y, @/@z)
is the nablaoperator.
Here r=
(~,
y~z)
denotes apoint
in the deformedbody
and the minussign
results from our convention: r= ro +
kzu,
where thedisplacement
can be eithera function of ro or r. Another
expression including
anharmonic terms has beenproposed by
de Gennes [6] and it is also discussed in the Landau-Lifshitz book[7],
I-e-~
~ ~~~~~
~~~"~~°~~~~~
~
~~ ~~~"~~°~~~~
' ~~~
where i7
(~
=(@/@~oi @/@yo)
is two dimensional nablaoperator.
This free energy~however,
is not
completely
invariant withrespect
to rotations andonly
up to theeighth
order in smallangles.
In this paper we
thoroughly
discuss elastic deformations in the smectic-Aliquid crystal
and derive a moregeneral expression
for the free energydensity.
We recover both(2)
and(3)
as the first term of theexpansion
of thatexpression. Additionally
we note that anharmonic termsare
important
for determination of the averagelayer spacing
andeventually
for tiltprofiles
observed in thin
freely suspended
smecticliquid crystal
films [8~9].
For convenience we shalluse the continuous
description
of the smecticsystem.
The paper isorganized
as follows: in section 2 we discuss thepossible
deformations in smectics and the invariant form of the free energydensity
in terms of the vector normal to smecticlayers
and the distance between thelayers
measuredalong
the normal. In Section 3 we express the free energydensity
in terms of the verticaldisplacement
and obtain the Grinstein andPelcovits~
and also de Gennes freeenergy 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
thequantities
whichproperly
describe deformations in the smectic-Aphase.
There are two of them: a vectorquantity namely
the unit vector normal to thelayer, fi(r),
and a scalarquantity~
which measures the distance between thelayers along
the normal. We further assume that other
quantities
like the nematic order parameter and thedensity adjust
to thelayer
deformations. In the firstapproximation~ density changes
do notgenerate
new terms in the free energy, butonly
renormalize the smectic elastic constants [617].
Using
the two aforementionedquantities
we can write the free energydensity
in thesimplest
form invariant with respect to
global
rotations:F =
)B (~
~
~°
+
)Iii (17i1(r)[~
+I(2 (fi(r)
i7 xfi(r)[~
+o
~
2
(4j
+
I(3 [fi(r)
x(i7xfi(r))[~
where
do
is theunperturbed layer spacing.
Thedivergence
and the curloperators
are taken withrespect
to variables in the distorted state and thepoint
r isuniquely
related to some otherpoint
ro on theunperturbed layer.
Weneglect
thehigher
order terms in d-do
and in thederivatives of fi. The last three terms in this free energy have their
analogs
in nematics.Ki, K2
andK3
are the elastic constants forsplay,
twist and bend deformations offi, respectively.
Asone can see the
splay
of ficorresponds
to the bend of thelayers
while the bend of ficorresponds
to the
splay
of thelayers.
In our
description
ofsmectics,
weneglect dislocations,
which means that the total number oflayers
crossedalong
anypath going
from somepoint
A to anotherpoint
B is constant. This isequivalent
to the condition[6]:
?~)
"
°' (5)
where
d,
ingeneral, depends
on theposition
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 thelayer spacing
is constant which also eliminates bend from the free energy. Here weonly
note thatbend does not affect the
long wavelength properties
of thesystem. Combining equation (4)
and
equation (5)
weget
F =
B
(~
~~°
~
+
Ki (i7fi(r)[~
+I(3 ji7d
~
@i7d ) (6)
°
In the next section we express both
quantities
in terms of u(~o> voizo)
3. Wee energy in terms of the vertical
displacement.
In
smecticsi
fi and d are notindependent
and can beexpressed
in terms of the vertical dis-placement
field u(ro)
The vector normal to thelayer
atpoint
r= r
(ro)
issimply given by:
n =
l~o ~~~
, ~~~
l + V
~ u
where ro
"
(~o,
vo,zo)
is the coordinationpoint
in theunperturbed system.
The distancebetween
layers
measuredalong
fi isd =
dz (ilij, (8j
where
dz
is the distance betweenlayers
measuredalong
the z-axis- Wefind,
to the lowest order in the derivatives of u, thatdz
=do (1+ ))
(9)
o
which
together
withequations (7, 8) gives
d in termsofu, (~
+RI
d "
do
~
~ (10)
~
In order to
represent
the free energygiven by equation (6)
in terms of u, we have to transform thepoint
r on aperturbed layer
to thepoint
ro on theunperturbed layer
and thecorresponding
nabla
operators
as follows:z#z0,
180 JOURNAL DE
PHYSIQUE
II N°1" Yo,
z = zo + u
(zo>
Yozo)
and~ IO
au~
~jtt lo'
~~~~fizo
)
au"
~ ~~iu 1'
~~~~
~ fiZo
t
"
~
~ou £
~~~~
ozo
Combining equations (7, 10-13)
we recover the conditiongiven by equation (5)
for the constant number oflayers. Finally
we obtain thefollowing expression
for the free energydensity:
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
acoupling
betweencompression
and undulation oflayers,
it does not affect thelong-wavelength properties
ofthesystem. Ignoring
the bend term and anharmonic contribution to thesplay term,
we find anapproximate
form of FF = B
°~°
~
i +
I<i (Ai~lL(ro)) (15)
~~~~~~ro))
~
~
The free energy
density given by equation (14)
orequation (15)
is invariant withrespect
to rotations. Forexample, rotating
thesystem
aroundy-axis by
theangle
we find:u=z0
c~i
--1-z0tani. (16)
For such
spurious
deformation we find fromequation (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 theexpansion, losing
however the rotational invariance of F.Alternatively,
the free energydensity
can be
expressed
in terms of Vu.Using equations (6), (7),
and(11)-(13)
we findF =
B
l +
Iii (i7fi(r)[~ (17)
~/l
2(°"/°z (V"(r)(~/2)
~
where
(-ou lox, -oulay,
Ifi"/fiZ) fi(r)
"(18)
~~
~ ~~/
~zivu(r)
i~/2j
and we have
neglected
the bend term. Oneeasily
checks that F= 0 when
u =
z(I
cos @) ~ sin @.(19)
In the lowest order
expansion
influ/fiz [i7u[~/2
we recover the Grinstein and Pelcovitsexpression (see Eq. (2)).
The transformationproperties
of u result from the fact that it is defined as the verticaldisplacement
of thelayer
from its restposition.
4. Conclusion
Summary.
The
simplest
form of the free energydensity
invariant withrespect
toglobal
rotations isgiven by equation (15)
orequivalently by equation (17)
and in theexpanded
formby equations (3)
and
(2), respectively.
The correct variabledescribing
deformations ofliquid layers
is the verticaldisplacement
of thelayer
from its restposition.
This variable transforms under rotationaccording
toequations (16)
or(19).
In the case of a smectic withpositional
order inside thelayers,
which canfreely
slideagainst
eachother,
it is moreappropriate
to use two variables fordescribing
itsdeformations, namely
a verticaldisplacement
and a two dimensional straintensor
[10].
The anharmonic terms are
important
notonly
becausethey
areresponsible
for the undula- tionalinstability
under the influence of external stress[6, 7],
but also becausethey
can shift theequilibrium positions
of smecticlayers.
It is well known from thetheory
of the anharmonicoscillator and the
theory
of atom vibrations in solids that third order terms in the elastic energy shift the centers of vibrations. Nearinterfaces,
due to thechange
of fluctuations [5] one canexpect
thelayer spacing profile. Finally,
since the distance betweenlayers deq
is related to thelength
of the molecule L and the tiltangle # by
thefollowing equation:
d~q = L cos
# (20)
we can
expect
a tiltprofile
near interfaces as well[8, 9].
We defer studies of thisfascinating problem
to a future work.Acknowledgements.
This work was
supported
inpart by
a KBNgrant.
References
[ii
DE GENNESP-G-,
J.Phys. Colloq.
France 30(1969)
C4-65, [2] CAtLLEA.,
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 PELCOVITSR-A-, Phys.
Rev. A 26(1982)
915.[5] HOLYST
R,,
TWEET D-J- and SORENSENL-B-, Phys.
Rev, Lett, 65(1990)
2153;182 JOURNAL DE
PHYSIQUE
II N°1HOLYST R.,
Phys.
Rev. A 42(1990)
7511.[6] DE GENNES
P-G-,
ThePhysics
ofLiquid Crystals (Oxford University Press,
London,1974)
pp. 281-300.
[7] LANDAU L-D- and LIFSHITz
E-M-, Theory
ofElasticity (Pergamon Press,
Thirdedition, 1986)
pp. 172-177.
(8j HEINEKAMP
S.,
PELCOVITSR-A-,
FONTES E., CHENE-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 SORENSENL-B-, Phys.
Rev. Lent.65
(1990)
2157.[10] SACHDEV S. and NELSON