Elastic actions exchanged by eccentric cylinders in liquid crystals
Riccardo Rosso,
1
Epifanio G. Virga,
1
and Samo Kralj
2,3
1
Dipartimento di Matematica and CNISM, Università di Pavia, via Ferrata 1, I-27100 Pavia, Italy
2
Laboratory of Physics of Complex Systems, Faculty of Natural Sciences and Mathematics, University of Maribor, Koroška 160,
2000 Maribor, Slovenia
3
Condensed Matter Physics Department, Jožef Stefan Institute, Jamova 39, 1000 Ljubljana, Slovenia
Received 24 April 2006; revised manuscript received 2 October 2006; published 21 December 2006
Equilibria of a nematic liquid crystal confined between two eccentric cylinders are studied within a purely
director approach. A planar equilibrium configuration competes against a three-dimensional one. A stability
diagram is obtained in terms of both the ratio between the radii of the bounding cylinders and the distance
between their axes. It turns out that the nonplanar minimizer has a structure more complex than that envisaged
in the tensorial approach employed by McKay and Virga Phys. Rev. E 71, 041702 2005 and that the planar
configuration cannot be the absolute minimizer when the outer cylinder becomes a plane wall. The mechanical
actions transmitted by the nematic liquid crystal on both bounding cylinders are computed and compared with
other results available in the literature.
DOI: 10.1103/PhysRevE.74.061703 PACS numbers: 61.30.Dk, 64.70.Md
I. INTRODUCTION
In the past two decades, experimental techniques based on
the surface force apparatus made it possible to measure
forces between solid objects submerged in liquid crystals. In
their seminal paper, Horn et al. 1 introduced the notion of
structural forces exchanged by solid surfaces through an in-
tervening liquid crystal. These forces reveal the order modu-
lations occurring within a liquid crystal interposed between
rigid bodies, especially in the vicinity of their boundaries,
where the anchoring to a material substrate has the potential
to affect the surface ordering of the liquid crystal molecules
in direct contact with the foreign bodies.
Recently, structural forces have been viewed as special
cases of order forces 2, which result from more general
alterations of the molecular order, similar to those establish-
ing biaxial states in a defect core 3. These biaxial states
also arise in the presence of boundary frustration, when an-
tagonistic anchorings are enforced on surfaces brought near
to one another. Upon reducing the distance between the
bounding surfaces, order transitions can be induced in bulk
by bridging antagonistic, uniaxial states through a continuum
of biaxial states: one uniaxial order is thus destroyed, while
the other is being reconstructed. This phenomenon is often
referred to as order reconstruction 4 –7: it can be revealed
by the order force exchanged by surfaces with antagonistic
anchorings. Precisely, it has been shown that order recon-
struction weakens the repulsion between antagonistic anchor-
ings 8. This force, which in the absence of order recon-
struction would be repulsive at all separations between the
frustrating surfaces, and increasingly so when the separation
is steadily reduced, falls momentarily as a consequence of
order reconstruction when the repelling surfaces are a few
biaxial coherence lengths
b
’s apart with
b
in the range of
nanometers. This lack of monotonicity would induce a
snapping instability in a force-controlled experiment with an
ideal machine that could explore distances comparable with
b
. A similar behavior is also exhibited by the torque trans-
mitted from one anchoring surface to the other 2,8; actu-
ally, the transition predicted in a torque-controlled experi-
ment is neater, as the torque drops to zero at the transition
and this happens for separations larger than those required by
the force-driven transition.
These predictions were made for a twist cell and a first,
though indirect, experimental confirmation was later found
for the classical surface force apparatus 9. A geometry
similar to that of the actual experiment was considered in
10, where both force and torque transmitted between a cir-
cular cylinder and a flat wall were computed, under the as-
sumption of homeotropic anchoring on both surfaces, i.e.,
with molecules parallel to the surface unit normal. Elemen-
tary symmetry considerations show that this problem is
equivalent to that where two equal cylinders with parallel
axes are drawn close together. This study revealed a lack of
monotonicity in both the force and the torque diagrams—
attributed to curvature frustration rather than to order
reconstruction—and predicted a snapping transition similar
to that found experimentally 9, but for larger distances.
A previous study 11 had already been concerned with
this problem within the director theory of nematic liquid
crystals: the transmitted force computed for the planar equi-
librium solution—with the director everywhere in the plane
orthogonal to the cylinder’s axis—exhibited a monotonic be-
havior, diverging like 1/
h as the distance h between cylin-
der and wall decreases to zero. It was shown in 10 that
whenever h is smaller than a few times the radius R of the
cylinder the planar director solution is unstable against a
class of biaxial perturbations that render n escaped 12 in
the direction of the cylinder’s axis, so that the asymptotic
behavior of the transmitted force was computed in 11 for a
solution that is unlikely attained in reality when cylinder and
wall are very close to one another. Similarly, the study re-
ported in 10 is not exempt from criticism: the escaped so-
lution was computed within the order tensor theory and bi-
axial states were allowed alongside of the uniaxial ones, but
for computational ease the eigenframe of the order tensor
was constrained within a class of orientations, which, though
supported by intuition, were yet restricted. We wonder
PHYSICAL REVIEW E 74, 061703 2006
1539-3755/2006/746/06170314 ©2006 The American Physical Society 061703-1