Surface variations of the density and scalar order parameter and the elastic constants
of a uniaxial nematic phase
V. M. Pergamenshchik* and S. Z
ˇ
umer
†
Institute of Physics, Prospect Nauki 46, Kiev 252022, Ukraine
and University of Ljubljana, Jadranska 19, 1000 Ljubljana, Slovenia
Received 22 June 1998
The elastic constants K
24
and K
13
of a spatially restricted nematic phase are found to essentially depend on
behavior of the density and orientational order parameter at the surface. The cancellation of the effective
constant K
13
, recently revealed by Faetti and Riccardi J. Phys. II 5, 1165 1995, is obtained as a particular
case of a constant and arbitrary ; whereas a spatial-dependent violates this cancellation and restores a
finite K
13
term. S1063-651X9951802-3
PACS numbers: 61.30.Gd, 64.70.Md
I. INTRODUCTION
More than a quarter of a century ago the elastic theory of
a uniaxial nematic liquid crystal had taken the final form in
the series of papers 1 by Nehring and Saupe. In this theory,
the nematic phase is described by the director nx, which is
a unit vector pointing along the average direction of the long
molecular axes in the vicinity of the point x. The director
deformations associated with nonvanishing director deriva-
tives n are assumed to be sufficiently weak, i.e., =l
M
n
1, where l
M
is the molecular length of order of the inter-
action range. This approach, however, essentially presup-
poses that the local symmetry in the vicinity of any spatial
point inside the nematic body is the symmetry of infinite
nematic medium. By virtue of this symmetry, leading terms
in the deformation free energy FE appear to be quadratic in
. The studies of the recent decade have shown that incorpo-
rating spatial boundedness into the elastic approach is not
trivial and does not reduce to just considering a surface ten-
sion.
First, surface was shown to induce an additional elastic
term F
1
linear in n whose density vanishes in the bulk 2.
Then, the leading part up to terms O (
2
) of the deforma-
tion FE of a nematic liquid crystal contained in the volume V
actually takes the form
F =
1
2
K
F
-K
24
F
24
+K
13
F
13
+F
1
, 1
where =1,2,3, and the standard infinite-medium quadratic
FE terms F
are given by 1
F
11
=
dV “• n
2
, F
22
=
dV n•“ n
2
,
F
33
=
dV n“ n
2
;
2
F
24
=
dV “• “• n - n•“ n ,
F
13
=
dV “• n• “• n .
The infinite-medium elastic constants K
and K
13
can be
calculated provided the pairwise interaction
G „n( x' ), n( x), x' -x… between two infinitesimal nematic vol-
umes centered at the points x' and x is known, while K
24
=( K
11
+K
22
+2 K
13
)/4 1. The form of F
1
will be discussed
somewhat below. The scalar G depends on the vectors n'
=n( x' ), n, and r=x' -x only through the scalar combina-
tions =n–n' , =r–n, ' =r–n' , and r =| r| , i.e., G
=G ( , , ' , r ).
Second, the K
24
and K
13
terms in Eq. 2 are total diver-
gences and in a restricted body can be written as surface
integrals with the density linear in n . In spite of this, in
three dimensions the K
24
and K
13
terms do not reduce to a
surface tension anchoring3,4 and, possessing a unique
ability to gain the FE for finite deformations, are an impor-
tant source of pattern formation see reviews 5,6. For in-
stance, it was found that both the K
24
and K
13
terms are
responsible for the stripe domains in thin nematic films 7,8.
Third, the very possibility of having a nonzero K
13
re-
quires justification. The problem derives from the important
result by Faetti and Riccardi 9 revealed that the sum F
1
-K
24
F
24
+K
13
F
13
=-
1
4
( K
11
+K
22
) F
24
, and thus the term
F
13
is cancelled out. Recently, this cancellation was shown
to be dictated by the FE symmetry 10. In this situation, the
problem of status of the K
13
term has turned into a search for
possible additional sources thereof hidden in subsurface phe-
nomena. Presently, the only such source of nonzero K
13
con-
sidered in the literature 10–12 is nondeformational, the so-
called homogeneous part of the nematic FE giving rise to the
intrinsic anchoring. However, in Ref. 3 where this source
was pointed out, the derivative-dependent terms and, in par-
ticular, the term apparently similar to the K
13
term, were
shown to be much smaller than the anchoring. Thus, this
source cannot provide a non-negligible value of K
13
.
Nonetheless, the result K
13
=0 obtained in 9,10 might
be inconclusive for another reason recently considered by
Pergamenshchik 4. Indeed, it assumes an unrealistic ideal
surface where the density and order parameter constant
everywhere in the nematic body abruptly vanish. However,
in the general case of a nonideal surface where and are
spatially dependent the value of K
13
can change 4. Physi-
cally, substantial surface variations of were suggested to be
essential for anchoring related phenomena
*Electronic address: pergam@victor.carrier.kiev.ua
†
Electronic address: slobodan.zumer@fmf.uni-lj.si
RAPID COMMUNICATIONS
PHYSICAL REVIEW E MARCH 1999 VOLUME 59, NUMBER 3
PRE 59 1063-651X/99/593/25314/$15.00 R2531 ©1999 The American Physical Society