Chaotic orientational behavior of a nematic liquid crystal subjected to a steady shear flow
Go
¨
tz Riena
¨
cker,
1
Martin Kro
¨
ger,
1,2,3
and Siegfried Hess
1,2,
*
1
Institut fu ¨r Theoretische Physik, Technische Universita ¨t Berlin, Fakulta ¨t II, Hardenbergstrasse 36, D-10623 Berlin, Germany
2
Institute for Theoretical Physics, University of California Santa Barbara, California 93106-4030
3
Polymer Physics, Material Sciences, ETH Zu ¨rich, CH-8092 Zu ¨rich, Switzerland
Received 12 April 2002; published 29 October 2002
Based on a relaxation equation for the second rank alignment tensor characterizing the molecular orientation
in liquid crystals, we report on a number of symmetry-breaking transient states and simple periodic and
irregular, chaotic out-of-plane orbits under steady flow. Both an intermittency route and a period-doubling
route to chaos are found for this five-dimensional dynamic system in a certain range of parameters shear rate,
tumbling parameter at isotropic-nematic coexistence, and reduced temperature. A link to the corresponding
rheochaotic states, present in complex fluids, is made.
DOI: 10.1103/PhysRevE.66.040702 PACS numbers: 61.30.Gd, 61.30.Cz
A nematic liquid crystal LC subjected to a steady shear
flow can either go to a stationary flow aligned state or re-
spond with a time dependent molecular orientation depend-
ing on the magnitude of the tumbling parameter 1–3.
Both flow alignment and time dependent orientation, fre-
quently referred to as ‘‘tumbling’’ behavior are observed in
thermotropic, lyotropic, and polymeric LC’s 4 In the tum-
bling regime, however, the dynamics are more complex than
the Ericksen-Leslie director theory can describe. The second
alignment tensor is needed to characterize the molecular ori-
entation. Detailed theoretical studies 5, based on the solu-
tions of a generalized Fokker-Planck equation 6,7, revealed
that in addition to the tumbling motion, wagging and kayak-
ing types of motions, as well as combinations thereof occur.
Recently, also chaotic motions were inferred from a moment
approximation to the Fokker-Planck equation leading to a
65-dimensional dynamical system 8 for uniaxial particles.
While we consider uniaxial particles in this note, one may
notice that for long triaxial ellipsoidal non-Brownian par-
ticles chaotic behavior had also been predicted in Ref. 9.
Here we report on our discovery 10 that a closed nonlinear
relaxation equation for the alignment tensor, being equiva-
lent to a five-dimensional dynamical system and strongly
related to the full Fokker-Planck equation, leads to a chaotic
behavior for particular values of the tumbling parameter and
in certain ranges of the shear rate. Both the frequency dou-
bling route, as in Ref. 8, and the intermittency route to
chaos are found for the simpler system. Due to the coupling
between the alignment and stress tensor a relationship is
given below, one may attempt to model the time dependent
and also chaotic rheological behavior seen in the recent ex-
periments on micellar materials 11, dense lamellar phases
12, and dense suspensions 13—and discussed in more
general theoretical considerations on rheochaos 14—by
variants of the dynamic system to be characterized in this
paper. An illustrative example is given in Ref. 15, where
equations similar to the one to be discussed below were used
to describe the effect of nonchaotic shear thickening.
The quantity , specified below, is the derivative of a
Landau-de Gennes free energy with respect to the align-
ment tensor, it contains terms of first, second, and third order
in a. The equation stated here was first derived within the
framework of irreversible thermodynamics 16, where the
relaxation time coefficients
a
0 and
ap
are considered as
phenomenological parameters. It had been shown in Refs.
6,17 that
a
and
ap
are proportional to the Ericksen-Leslie
viscosity coefficients
1
and
2
, respectively. The basic
equation used here can also be derived, within certain ap-
proximations, from a Fokker-Planck equation for the orien-
tational distributions function that contains a torque associ-
ated with the molecular field proportional to a 6,7,18. Then
a
and the ratio -
ap
/
a
can be related to the rotational
diffusion coefficent and to a nonsphericity parameter associ-
ated with the shape of a particle. Equation 1 is applicable to
both the isotropic and the nematic phases. Limiting cases
that follow from this equation are the pretransitional behav-
ior of the flow birefringence 19,20 in the isotropic phase
( a) is approximated by its term linear in a] and the
Ericksen-Leslie theory in the uniaxial nematic phase. In the
latter case, the Ericksen-Leslie viscosity coefficients
1
and
2
are proportional to
a
and
ap
, respectively, and
=-
2
/
1
. Equation 1 has been applied to the study of
the influence of a shear flow on the isotropic-nematic phase
transition 19,20, and discussed intensively in recent, in par-
ticular, experimental works, see e.g., Refs. 3,4 and refer-
ences cited therein. *Corresponding author.
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PHYSICAL REVIEW E 66, 040702R2002
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