Motivated by the observation of rheochaos in sheared wormlike micelles
[Bandyopadhyay et al., Phys. Rev. Lett, 84 2022, (2000); Europhys. Lett. 56,
447 (2001); Pramana 53, 223 (1999)] we study the coupled nonlinear partial
differential equations for the hydrodynamic velocity and order parameter fields
in a sheared nematogenic fluid. In a suitable parameter range, we find
irregular, dynamic shear-banding and establish by decisive numerical tests that
the chaos we observe in the model is spatiotemporal in nature.Comment: Slight changes in text, references and Fig. 5 inset; 6 eps figures
  (figs 2,3,4 at lower resolution to reduce file size; full files available on
  request); accepted for publication in Phys Rev Let

A. K. Sood

Buddhapriya Chakrabarti

Chandan Dasgupta

Moumita Das

P. E. Cladis

P. G. de Gennes

R. Bandyopadhyay

R. Larson

S. Hess

Sriram Ramaswamy

English

arXiv

Motivated by recent observations of rheochaos in sheared wormlike micelles, we study the coupled nonlinear partial differential equations for the hydrodynamic velocity and order-parameter fields in a sheared nematogenic fluid. In a suitable parameter range, we find irregular, dynamic shear banding and establish by decisive numerical tests that the chaos we observe in the model is spatiotemporal in nature

Chakrabarti, Buddhapriya

Das, Moumita

Dasgupta, Chandan

Ramaswamy, Sriram

Sood, AK

Open Access Repository of IISc Research Publications

Spatiotemporal Rheochaos in Nematic Hydrodynamics

Crossref

Sood, A. K.

Publications of the IAS Fellows

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Spatiotemporal rheochaos in nematic hydrodynamics
Buddhapriya Chakrabarti1[a],,∗ Moumita Das,,† Chandan Dasgupta1,, Sriram Ramaswamy1,, and A.K. Sood‡
Department of Physics, Indian Institute of Science, Bangalore 560012 INDIA
1Condensed Matter Theory Unit, JNCASR, Bangalore 560064 INDIA
Motivated by recent observations of rheochaos in sheared wormlike micelles, we study the coupled
nonlinear partial differential equations for the hydrodynamic velocity and order parameter fields in a
sheared nematogenic fluid. In a suitable parameter range, we find irregular, dynamic shear-banding
and establish by decisive numerical tests that the chaos we observe in the model is spatiotemporal
in nature.
PACS numbers: 61.30.-v,95.10.Fh,47.50.+d
Wormlike micellar solutions of cetyltrimethylammo-
nium tosylate (CTAT) show rheochaos [1, 2, 3]: for a suit-
able range of concentrations and temperatures, the stress
vs. shear-rate curve in [1, 3] shows a plateau signalling
a rheological phase-coexistence and, in the plateau re-
gion, the time-series of the shear and normal stresses
at constant shear-rate show deterministic chaos. This is
remarkable because it means that the rheological equa-
tion of state [4], that is, the relation between stress and
shear-rate, even for macroscopic experimental samples,
has huge fluctuations about the mean [5]. A realistic
description of the complex dynamics of micellar systems
as well as of other rheological oscillations [6, 7] may re-
quire variables specific to the system in question [8, 9].
The studies of [10, 11, 12], however, are significant in
that they find temporal rheochaos in the dynamics of
the passively advected alignment tensor alone. They use
the well-established equations of hydrodynamics for a
nematic order parameter, outside the regime of stable
flow alignment, but consider only spatially homogeneous
states [13] and hence cannot explore the relation of the
observed chaos to shear-banding [14].
In this paper we study numerically the equations of
the traceless symmetric order parameter for a sheared ne-
matogenic system, allowing for spatial variation. In the
parameter range termed “Complex” in the phase diagram
of the studies [11, 12] of the spatially homogeneous situa-
tion, we find spatiotemporal chaos. We give evidence for
this in Fig. 2 for the dynamic instability of shear bands,
Fig. 5 for the spatial distribution of “stress drops”, Fig.
1(b) for the local phase portrait in the chaotic regime,
and, decisively, Fig. 6 for the number of positive Lya-
punov exponents as a function of (sub)system size. We
now present our model in detail and show how these re-
sults arise.
Models of complex rheology [4] generally postulate an
equation of motion for the stress in addition to that for
the velocity field. Our work was motivated in part by
the failure [15] to find chaos in the the popular Johnson-
Segalman (JS) [16] model for the coupled dynamics of the
stress tensor and the hydrodynamic velocity field. Since
this model is linear in the stress, and contains bilinear
products of velocity-gradient and stress, it is natural to
ask if nonlinearities in the stress are the missing ingre-
dient. The JS equation is most naturally thought of as
arising from the underlying dynamics of a local alignment
tensor, the traceless, symmetric nematic order parameter
Q, measuring in this case the alignment of the micellar
worms. The stress σOP [17] due to Q is defined via a
free-energy functional F [Q] [12, 18, 19, 20]. Nonlineari-
ties arise naturally in nematodynamics from terms of or-
der higher than Q2 in F . It is these that lead to the rich
dynamical phase diagram, including a variety of routes
to chaos, in the work of [10, 11, 12].
The alignment tensor obeys the equation of motion
∂Q
∂t
+ u ·∇Q = τ−1G+ (α0κ + α1κ ·Q)ST
+ Ω ·Q−Q ·Ω, (1)
where I is the unit tensor, u is the hydrodynamic veloc-
ity field, κ ≡ (1/2)[∇u + (∇u)T ] and Ω ≡ (1/2)[∇u −
(∇u)T ] the shear-rate and vorticity tensors respectively,
τ is a bare relaxation time, α0 and α1 are parameters
related to flow alignment, the subscript ST denotes sym-
metrisation and trace-removal, and
G ≡ −(δF/δQ)ST = −[AQ−
√
6B(Q ·Q)ST + CQQ : Q
+ Γ1∇2Q + Γ2(∇∇ ·Q)ST ] (2)
is the molecular field conjugate to Q, for a Landau-de
Gennes [18] free-energy functional F with upto quartic
terms in Q, with phenomenological coefficients A, B, C,
Γ1, Γ2, and the simplest bilinears in ∇Q. In mean-
field theory, the isotropic-nematic transition occurs at
A = A∗ = 2B
2/9C. As in [10, 11, 12], we impose plane
Couette flow with u = yγ˙xˆ, ignoring modifications of
this flow by order-parameter stresses, and treating γ˙ as
a control parameter, but, unlike in that work, we allow
spatial inhomogeneity in the y direction.
As in [12], we rescale time by the relaxation time τ/A∗
at the isotropic-nematic transition, Q as well by its mag-
nitude at that transition, and distances by the diffusion
length made from Γ1 and τ/A∗. The ratio Γ2/Γ1 of
the Frank constants is therefore a free parameter which
we have set to 1 in our study. We choose A = 0 and
α1 = 0 throughout, and λk = −(2/
√
3)α0 and γ˙ with
2−0.5 0 0.5 1
0
0.5
1
 
a1
 
 C → A 
0.6 0.7 0.8 0.9 1 1.1
−0.4286
0.4714
 
a1
  T 
−0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1
0
0.5
1
 
a1
 
 C 
−2 −1.5 −1 −0.5 0 0.5
0
2
4
 a0 
 
a1
 
 A 
FIG. 1: Plots showing a1(x0, t) vs a0(x0, t) for tumbling
(panel T), chaotic (C), onset of aligning (C→A) and align-
ing (A) regimes.
values as specified in the figures. In the orthonormal ba-
sis [11, 12] T0 =
√
3/2(zˆzˆ)ST , T1 =
√
1/2(xˆxˆ − yˆyˆ),
T2 =
√
2(xˆyˆ)ST , T3 =
√
2(xˆzˆ)ST , T4 =
√
2(yˆzˆ)ST ,
we expand Q =
∑
i aiTi and work with the equations
of motion for the five components ai which follow from
(1). These equations are numerically integrated using
a 4th order Runge-Kutta scheme with a fixed time step
(∆t = 0.001). For all the results quoted here a sym-
metrized form of the finite difference scheme involving
nearest neighbors is used to calculate the gradient terms.
We have checked that our results are not changed if
smaller values of ∆t are used. We have further checked
that the results do not change if the grid spacing is
changed and more neighbors to the left and right of a par-
ticular site in question are used to calculate the deriva-
tive. This gives us confidence that the results quoted
here do reflect the behavior of a continuum theory and
are not artifacts of the numerical procedure used. We use
boundary conditions with the director being normal to
the walls. With this, we discard the 1st 6×106 timesteps
to avoid any possibly transient behavior. We monitor the
time evolution of the system for the next 5 × 106 time
steps (i.e. t=5000), recording configurations after every
103 steps. We have carried out the study with system
sizes ranging from 100 to 5000. In the time-series analy-
sis for the Lyapunov spectrum, we run the simulation till
t = 20, 000, for a spatial system size of 5000, recording
data at spatial points at gaps of 10.
We focus on the parameter region labelled ‘C’ or ‘Com-
plex’ in [11, 12], where the order-parameter dynamics
shows temporal chaos. We begin by verifying that the
inclusion of spatial variation does not alter the phase di-
agram radically. To this end, we construct local phase
portraits (plotting different components of the order pa-
rameter against each other) as a function of the param-
eters entering into the model, the shear rate γ˙ and the
tumbling parameter λk. Shown in Fig. 1 are the lo-
cal phase portraits for a particular point x0 for various
shear-rates γ˙, obtained by holding the value of the tum-
bling parameter fixed at λk = 1.25. We have checked that
the character of the phase portrait (space-filling or oth-
erwise) remains intact upon going from one space point
to another though there is no phase coherence between
two such portraits. A closed curve corresponding to a
limit cycle is seen in the tumbling region of parameter
space (denoted by ‘T’) in the figure, while in the ‘C’ re-
gion of the phase space it is space filling. When one goes
away from the ‘C’ region of the phase space to the region
where the director aligns with the flow the points reduce
to those on a line and eventually in the aligning regime
where the director has already aligned with the flow it
is represented by a point. This assures us that the local
dynamics in the spatially extended case is similar to that
of the ODEs of [11, 12].
The contribution of the alignment tensor to the devi-
atoric stress is [12, 19, 20] σOP ∝ α0G − α1(Q · G)ST
where G, defined in (2), is the nematic molecular field,
and the total deviatoric stress is σOP plus the bare vis-
cous stress which is a constant within the passive con-
vection approximation. We therefore look at σOP alone.
The spatiotemporal nature of the chaos [21] in this sys-
tem is seen strikingly, if qualitatively, in the space-time
plots of the shear stress (the xy component of σOP ), ref-
ered to as Σ henceforth, in Fig. 2. Two other figures are
instructive, showing the periodic tumbling regime (Fig.
3) and a transition from spatiotemporal chaos to sta-
ble flow alignment (Fig. 4). The parameter values at
which these are seen, furthermore, correspond well with
the phase diagram of [11, 12]. Our method therefore
reveals nicely the rich spatiotemporal structure of the
phase diagram of this system. In Fig. 4, several distinct
events, such as the persistence, movement, and abrupt
disappearance of shear bands can be seen, whereas in
Fig. 3 the structure is simply periodic. We find that the
typical length scale at which banding occurs is a fraction
of the system size. In order to characterize better the
disorderly structure of the shear bands in Fig. 2, we look
at the distribution of band sizes or spatial “stress drops”
as follows. At a given time (say ti), we define a threshold
Σ0 = 0.8, a little above the global mean 〈Σ(y, t)〉y,t, and
map the spatial configuration to a space-time array of
±1: Σ˜ = sgn(Σ−Σ0). Fig. 5 shows the histogram of the
spatial length of intervals corresponding to the +state.
We have considered configurations extending over 2500
spatial points, and the statistics is summed over configu-
rations sampled at 5000 times (i.e. i = 1, 5000). We see
that the distribution of band lengths in the spatiotempo-
rally chaotic regime is fairly broad and roughly exponen-
tial in shape, and the time-averaged spatial correlation
function of the stress also decays on a length scale of
about 5 bands.
The spatially averaged shear stress shows irregular os-
cillations in time, but analysis of this signal based on the
techniques of [22] does not give reliable results for esti-
3FIG. 2: Space-time behavior of the shear stress in the chaotic
regime, γ˙ = 3.678 and λk=1.25. Slice taken from a system of
size 5000
mating the correlation dimension ν mainly because the
chaos that we observe is quite high dimensional (embed-
ding dimension [22, 23] m ≥ 10). A very long data train
is then required for the analysis of the spatially averaged
time series to yield a correct value of the correlation di-
mensions. We however establish conclusively that the os-
cillatory behavior of the stress and the first normal stress
difference shows signatures of spatiotemporal chaos. In-
stead of trying to implement the correlation-dimension
method for our spatially extended problem, we study the
Lyapunov spectrum [22, 24]. Further, instead of studying
systems of ever-increasing size, we look at subsystems of
size Ns in a given large system of size N , i.e., at space
points j in an interval i0 < j < i0 + Ns − 1 where i0 is
an arbitrary reference point. For spatiotemporal chaos
we expect to find that the number of positive Lyapunov
exponents grows systematically with Ns, which is pre-
cisely what Fig. 6 shows. For both figures in Fig. 6 ,
we carry out the procedure with two different reference
points i0 and find essentially the same curves. Further-
more, it has been reported in many studies of spatiotem-
porally chaotic systems [25] that when calculating the
subsystem Lyapunov spectrum for increasing subsystem
size Ns, one finds that the Lyapunov exponents of two
consecutive sizes are interleaved, i.e. the ith Lyapunov
exponent λi for the sub-system of size Ns lies between the
ith and (i+1)th Lyapunov exponent of the subsystem of
size Ns+1. A direct consequence of this property is that
with increasing subsystem size Ns, the largest Lyapunov
exponent will also increase. This is clearly seen in Fig. 6
(b).
In summary, we have shown that the stress field in
the equations of sheared, nonlinear nematodynamics, in
a subrange of the parameter space where flow align-
ment does not obtain, shows spatiotemporal chaos. Our
numerical study of this model, which extends that by
[10, 11, 12] for the purely temporal behavior, shows ir-
regular, dynamic shear banding, and finds that the num-
FIG. 3: Space-time behavior of the shear stress in the tum-
bling regime, γ˙ = 5.0 and λk = 0.9. Slice taken from a system
of size 5000.
FIG. 4: Space-time behavior of the shear stress in the chaotic
to aligning regime, γ˙ = 4.1 and λk=1.25. Slice taken from a
system of size 5000.
∆y
N +
(∆y
)
0 25 500
4
8x 10
4
0 20 40
0
1
∆y
<Σ
(y,t
)Σ(y
+∆
y,t
)> y
,t
e−(∆y/5.5834)
1.6819
FIG. 5: Spatial distribution of “stress drops” (correspond-
ing to residence intervals in which the shear stress is above
a threshold Σ0 = 0.8) in the Chaotic regime (Fig. 3). In-
set shows the spatial autocorrelation function averaged over
t=5000 (circles) with the best fit to the form exp[(−∆y/ξ)α].
40 5 10 150
70
N
s
N λ
+
0 5 10 150
0.4
N
s
λ la
rge
st
(a) 
(b) 
FIG. 6: Number of positive Lyapunov exponents (top panel)
and the largest Lyapunov exponent (bottom panel) as func-
tions of subsystem size Ns, for γ˙ = 3.678, λk = 1.25, Em-
bedding dimension for the time series of each space point is
10. Diamonds and circles correspond to data taken from two
different regions of space (i0=51 and i0=101 respectively, see
text).
ber of positive Lyapunov exponents as well as the largest
of these grow systematically with system size. Since we
do not know what parameter values in our model corre-
spond to the experiments, a detailed comparison to data
is not possible at this stage. However, it seems likely to
us that the nonlinear relaxation of the order parameter,
together with the coupling of nematic order parameter
to flow, are the key ingredients for rheological chaos in a
variety of problems. Related issues such as the effect of
two- and three-dimensional variation, hydrodynamic flow
beyond passive advection, the routes to spatiotemporal
chaos and, of course, the role of other degrees of freedom
such as variable micelle length and lifetime [9] remain to
be explored. Meanwhile, we look forward to quantitative
experimental studies of spatiotemporal rheochaos [26].
We thank G. Ananthakrishna and R. Pandit for very
useful discussions, and SERC, IISc for computational fa-
cilities. MD acknowledges support from CSIR, India, and
CD and SR from DST, India through the Centre for Con-
densed Matter Theory.
∗ Electronic address: buddho@physics.umass.edu
† Electronic address: moumita@physics.iisc.ernet.in
‡ Electronic address: asood@physics.iisc.ernet.in
[a] Present address: Department of Physics, University of
Massachusetts, Amherst, MA 01003, USA.
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Spatiotemporal rheochaos in nematic hydrodynamics

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