Interface dynamics in shear-banding flow of giant micelles

S. Lerougea∗, M.A. Fardina, M. Argentinab, G. Gregoirea, O. Cardosoa

aLaboratoire Matiere et Systemes Complexes, Universite Paris-Diderot,UMR 7057 CNRS, 10 rue Alice Domon et Leonie Duquet, 75205 Paris Cedex 13

bInstitut Non Lineaire, Universite de Nice Sophia Antipolis, UMR 6618 CNRS,1361 Route des Lucioles, 06560 Valbonne, France

Received XXXXth Month, 200XAccepted XXXXth Month,200X

DOI: 10.1039/

We report on a non trivial dynamics of the inter-face between shear bands following a start-up offlow in a semi-dilute wormlike micellar systeminvestigated using a combination of mechanicaland optical measurements. During the buildingof the banding structure, we observed the stagesof formation, migration of the interface betweenbands and finally the destabilization of this in-terface along the vorticity axis. The mechanicalsignature of these processes has been indenti-fied in the time series of the shear stress. Theinterface instability occurs all along the stressplateau, the asymptotic wavelength of the pat-terns increasing with the control parameter typ-ically from a fraction of the gap width to aboutfour times the gap width. Three main regimesof dynamics are highlighted : a spatially stableoscillating mode approximately at the middleof the coexistence region flanked by two rangeswhere the dynamics appears more exotic withpropagative and chaotic events respectively atlow and high shear rates. The distribution ofsmall particles seeded in the solution stronglysuggests that the flow is three-dimensional. Fi-nally, we demonstrate that the shear-bandingscenario described in this paper is not specificto our system.

∗sandra.lerouge@univ-paris-diderot.fr

1 Introduction

Many complex fluids often show original non lin-ear responses when submitted to hydrodynamicforces even of low intensity. These non linearbehaviours resulting from the coupling betweenthe internal structure of the fluid and the floware usually associated with a new mesoscopicorganization of the system. In turn, the mod-ification of the supramolecular architecture ofthe fluid can affect the flow itself and, for ex-ample, generate shear localization effects gen-erally characterized by a splitting of the sys-tem into two macroscopic layers bearing differ-ent shear rates and stacked along the velocitygradient direction. This transition towards aheterogeneous flow has been reported in com-plex fluids of various microstructure such as ly-otropic micellar and lamellar phases [1, 2], tri-block copolymers solutions [4, 5], viral suspen-sions [6], thermotropic liquid crystal polymers[7], electro-rheological fluids [8], soft glassy ma-terials [9], granular materials [10, 11] or foams[12, 13, 14]. Among these systems, the shearbanding flow of reversible giant micelles is par-ticularly well documented [1]. The rheologi-cal signature of this type of flow has been ob-served for the first time in the pioneering workof Rehage et al [15]: the measured flow curve

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σ = f(γ) is composed of two stable branches re-spectively of high and low viscosities separatedby a stress plateau at σ = σp extending be-tween two critical shear rates γl and γh. Whenthe imposed shear rate γ is lower than γl, thestate of the system is described by the high vis-cosity branch which is generally shear-thinning :the micellar threads are slightly orientated andthe flow is homogeneous. For macroscopic shearrates above γl, the flow becomes unstable andevolves towards a banded state where the vis-cous and fluid phases coexist at constant stressσp. The modification of the control parameteris supposed to only affects the relative propor-tions 1 − αh and αh of each band according tothe lever rule γ = (1−αh)γl +αhγh that resultsfrom the continuity of the velocity at the inter-face. Above γh, the system is entirely convertedinto the fluid phase : the induced structures arestrongly aligned along the flow direction and thehomogeneity of the flow is recovered. This sce-nario has been predicted by Cates and cowork-ers more than ten years ago [16]. Recently, dif-ferent velocimetry techniques have been devel-opped to determine the velocity profiles in aplane velocity/velocity gradient resolved bothin time and space [17, 18, 19]. In particularSalmon et al demonstrated using dynamic lightscattering that the CPCl/NaSal system, exten-sively studied during the last decade, was fol-lowing the classical shear-banding organizationdescribed above [20].Besides, more complex banding pictures onslightly different systems also emerged from thecombination of macroscopic rheology togetherwith local measurements. Several groups es-tablished the existence of temporal fluctuationsof the local flow field using nuclear magneticresonance (NMR) [21, 19], particle image ve-locimetry (PIV) [22] or high frequency ultra-sonic velocimetry [23]. They also mentioned os-cillations of the interface position as functionof time, sometimes correlated with wall slip ef-fects. Moreover, flow birefringence experimentswhich give information on the averaged orienta-tion of the medium also show clear evidence oflocal ordering fluctuations in the banding struc-ture [24, 25, 22]. In some cases, these com-plex spatio-temporal evolutions are associated

with fluctuations in the shear stress time series[21, 19, 26, 27].Very recently, several theoretical studies basedon the derivation of the modified Johnson-Segalman equation using perturbation analysistempted to rationnalize these fluctuating be-haviours. They mainly focused on the stabilityof the shear-banded state in planar flow withrespect to small perturbations with wave vectorin the plane made by the flow and the vortic-ity directions [28, 29, 30]. For sufficiently thininterfaces, the banding state is found to be lin-early unstable, except for extremely low andhigh shear rates in the plateau region, with re-spect to perturbations with wave vector in theflow direction. The parameters driving the in-stability are jumps in normal stresses and shearrate accross the interface. The non linear analy-sis reveals a complex spatio-temporal dynamicsof the interface with a transition from travel-ling to rippling waves depending on the ratiobetween the thickeness of the interface and thelength of the cell. The authors also mentionedthe existence of a modulation of the interfacealong the vorticity direction with a much slowergrowth rate but a non negligible contribution tothe asymptotic state [31]. Subsequently, theseundulations generate velocity rolls stacked alongthe vorticity direction.In a previous study, we reported on the be-haviour of the interface in the shear-bandingflow of a sample made of cetyltrimethylammo-nium bromide and sodium nitrate in Couettegeometry [32]. Using the scattering propertiesof each band, we showed that the interface be-tween bands becomes unstable with the wavevector in the direction of the cylinders axis.In the present paper, we study the interface dy-namics of this same sample in a detailed way.We carefully examine the formation and desta-bilization of the interface together with the tem-poral evolution of the shear stress to identifythe mechanical signature of these different pro-cesses. We also explore the stress plateau in or-der to properly characterize the spatio-temporaldynamics: three main regimes were identified,with propagative events at low shear rates, spa-tially stable undulations for intermediate con-trolled parameter and finally chaotic pattern

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at high shear rates. The asymptotic wave-length is found to increase with the mean shearrate whereas the variation of the amplitude ofthe interface profile is non-monotonic. Thethree-dimensional character of the flow field isstrongly suggested by the organization of smalltracers embedded in our solution into stripesstacked along the vorticity direction. Finally,we show that this interface instability is not par-ticular to our sample but is also present in theCPCl/NaSal system studied, among others, inreferences [20, 22]. Our results are discussed bythe light of the recent litterature on the stabilityof shear banded flows [31] and in the context ofpurely elastic instabilities [33, 34].

2 Experimental details

2.1 Materials

The micellar sample under investigation is madeof a classical surfactant, the cetyltrimethylam-monium bromide (CTAB) at 0.3M mixed witha mineral salt, the sodium nitrate (NaNO3) at0.405M, in distilled water. Both the surfactantand the salt are supplied by Acros Organics andare used without further purification. The tem-perature is fixed at T=28◦C. At the concentra-tion chosen here (around 11% wt), far from theisotropic/nematic transition at rest, the solu-tion is semi-dilute and made of highly entan-gled wormlike micelles forming an elastic net-work. In the linear regime, this network be-haves as a purely Maxwellian element with asingle relaxation time τR = 0.23 ± 0.02s anda plateau modulus G0 = 238 ± 5Pa. Undersimple shear flow, the fluid is known to showstrong non linear behaviour with the existenceof a stress plateau characteristic of the shear-banding transition [35].

2.2 Setup

2.2.1 The Couette cell

Our experiments are performed in a PerspexCouette device built in our lab and specially de-signed for the observations of the velocity gradi-ent - vorticity (~v, z) plane (Fig. 1). The sample

is placed in a gap e of 1.13 mm between two con-centric cylinders. The inner rotating cylinderhas a radius R1=13.33 mm and a height h = 40mm. It is of Mooney-Couette type with a cone-shaped lower part. The outer cylinder has aconstant thickness of 2 mm and is surroundedby water which keeps the sample at constanttemperature. The small tank, which allows thewater circulation, is equiped with two glass win-dows, one for the radial incident beam and theother one for the direct observation of the gap.This configuration of the tank reduces the dis-tortions of the image due to refraction effects onthe successive cylindrical interfaces.The cell is closed by a small plug which limitsthe destabilization processes of the free surfaceof the fluid at high strain rates. This allowsto reach more easily the second stable branchwithout forming foam or bubbles, what oftenoccurs with semi-dilute micellar samples [36].We checked that the interfacial instability weobserve between shear bands is not affected bythe presence of the plug. A home-made solventtrap is also used to limit evaporation of the sam-ple.

2.2.2 The rheo-optical device

Our transparent Couette cell is fitted to a stress-controlled rheometer (Physica MCR 500) alsoworking in strain-controlled mode via a soft-controlled feedback loop, allowing to recordthe mechanical and optical responses simulta-neously. A He-Ne laser (λ = 633 nm) propagat-ing along the velocity gradient axis is used aslight source for the visualizations of the gap.By means of a plano-convex cylindrical lens(f = 4mm), the slightly divergent beam iswidened behind the focal point to form a laserlight sheet. The thickness of the sheet (1.5 mm)is determined by the diameter of the incidentbeam into the flat surface of the lens. The in-tensity distribution in the light sheet being ofGaussian shape, the distance between the focalpoint and the input window of the tank is suchas the high intensity region of the sheet coversall the height of the cell. Finally, a digital cam-era with resolution 720× 576 pixels records thescattered intensity at 90◦ at a rate of 25 frames

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Figure 1: (Color online)(a) Experimental setup for the observation of the gap in the plane (~v, z).(b) Top view of the Couette cell and of the measurement configuration.

per second, giving a view of gap in the plane(~v, z). The field of observation is centered athalfway of the cell, and varies from 0.5 to 1.7cm according to the chosen magnification. Weapplied a numerical algorithm to each frame inorder to detect the interface. The wavelengthof the dominant mode is computed by directFourier transform. We also deduce the evolu-tion with time of the amplitude of the interfaceprofile.

3 Results and discussion

3.1 Mechanical rheology

3.1.1 Characterization of the stationarystate

The steady-state shear stress as function of theshear rate obtained under strain-controlled con-ditions is shown in the semi-logarithmic plot infigure 2. The observed constitutive behaviouris in accordance with the non linear rheologyof most of giant micelles systems [1] : the flowcurve is made up of two increasing branches sep-arated by a stress plateau at σp = 147 ± 0.5Pa extending from γl = 4.4 ± 0.4s−1 to γh =97±5s−1, characteristic of a shear-banding tran-sition. A closer inspection of this graph deservesfurther comments. First, the stress plateaupresents a significant positive slope. Assuming

that the total stress tensor T only depends onthe radial coordinate [37], the momentum bal-ance∇T = 0 implies that, in the Couette geom-etry, the shear stress varies in the gap betweenthe cylinders as:

σ(r) =Γ

2πhr2= σ(R1)

R21

r2(1)

=2R2

2

R21 +R2

2

σR2

1

r2(2)

where Γ is the torque measured on the axis ofthe inner cylinder, h is the height of the cell,σ(R1) the local stress at the rotor, and σ thestress measured by the rheometer. As demon-strated in reference [2] with the hypothesis thatthe interface between bands is stable at a givenlocal stress [38, 39, 40], this non homogeneityof the shear stress generates a stress increment∆σ ≈ 2eσ∗p/R1 where σ∗p = 2R2

2σp/(R21 + R2

2)is the local stress at the moving wall associatedwith the onset of the plateau. We find ∆σ ≈ 27Pa whereas the effective increase of the stresscomputed from the flow curve is about 50 Pa ,indicating that the slope of the stress plateau isnot fully explained by the curvature effects.Second, the coexistence region is composed oftwo distinct parts, the small change of variationoccuring around a shear rate γ+ = 54 ± 3s−1

as illustrated in figure 2b. Each part can bedescribed by a power law σ = Aγα with the fol-lowing parameters: A = 130Pa.sα, α = 0.08 for

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γl ≤ γ < γ+ and A = 100Pa.sα, α = 0.14 forγ+ ≤ γ ≤ γh. This evolution is reproducibleand we shall see later that the slight breakingoff of the slope in the stress plateau at γ+ isconcomittent with the appearance of a peculiarregime of spatio-temporal dynamics of the inter-face between bands, suggesting that the flow dy-namics can play a role in the additionnal slopeof the stress plateau.Third, as mentioned in section 1, the accessto the second branch of the flow curve is of-ten made delicate because of flow instabilitiesthat generate bubbles and finally the expulsionof the sample from the gap of the cell. The plugon the top of our Couette device reduces thefree surface of the fluid and shifts the instabili-ties thresholds towards higher shear rates. Thenit becomes possible to provide a good approxi-mation of the second critical shear rate γh andto describe the flow behaviour of the inducedphase. We observe a deviation to the power lawadjusting the second part of the stress plateauaround γ = 97 ± 5s−1 and we assimilate thisvalue with the shear rate γh corresponding tothe upper limit of the stress plateau. This esti-mation is corroborated by the direct visualiza-tions of the sample under shear (see section 3.2),which show that all the gap is filled with the in-duced phase above 97s−1. The value of the stan-dard deviation is estimated from these opticalobservations and mainly takes into the accountthe fact that the precise shear rate for whichthe interface becomes perfectly flat and effec-

Figure 2: (a) Experimental steady-state flowcurve measured under strain-controlled condi-tions. The sampling of the rate sweep is 180sper data point. (b) Enlargement of σ(γ) overγ = 20− 150s−1.

tively touches the fixed wall is difficult to deter-mine. From γh to 110 s−1, the second branch of

Figure 3: (Color online) Snapshots of the plane(r, z) of the Couette cell illuminated by a ra-dial laser sheet : (a) The imposed shear rateis 105 s−1 and lies in the shear-thinning regionof the second branch. The induced phase com-pletely fills the gap and appears homogeneous.(b-d) Irregular patterns of the induced structureat γ = 130s−1 in the shear-thickening regimealong the second branch. (e) For comparison, aview of the gap at 2 s−1, along the low viscositybranch.

the flow curve follows a power law regime withA = 61Pa.sα and α = 0.26, indicating that theinduced phase is strongly shear-thinning, a fea-ture already highlighted by Salmon and cowork-ers on the classical CPCl/NaSal(6.3%) systemusing both global rheology and local velocime-try measurements [20]. In this small range ofshear rates, the induced phase appears homoge-neous (see figure 3.a). Above 110s−1, the sampleadopts a shear-thickening behaviour : in thatcase, the snapshots (fig.3.b-d) give a visual im-pression of strong spatial disorder in the gap ofthe cell, typical of random flow, this disordernucleating systematically from the lower andupper boundaries of the inner rotating cylin-der before spreading over all the height of thecell. This picture together with the increaseof the torque as function of the shear rate is

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reminiscent of the state of elastic turbulenceextensively studied by Groisman and Steinberg[41] on dilute viscoelastic polymer solutions, theshear-thickening evolution being interpreted interms of an increased flow resistance. Since sucha phenomenon is driven by normal stresses ef-fects, our observations tend to suggest that theinduced structures are viscoelastic. Nonethe-less, the window of shear rates that can be ex-plored above 110 s−1 is tiny since the flow israpidly perturbed by bubbles, making a system-atic study of the second branch difficult.

3.1.2 Time-dependent experiments

Figure 4 displays the evolution of the shearstress as a function of time towards its steadystate after a sudden start-up of flow for threedifferent shear rates along the coexistence zoneand on different time scales. The temporalstress response recorded with a sampling of0.1s, is typical of a sample undergoing a transi-tion of shear-banding type [42, 43, 24, 44], withan overshoot followed by a relaxation to anapparently stable value in stretched exponen-tial or damped oscillations depending on theapplied shear rate (Fig.4.a). The subsequentbehaviour is a slow increase of the shear stressto its steady state value with a relative stressincrement of about 2.5% and can be related tothe small undershoot already observed on othersurfactant systems [42, 43, 24, 44] (Fig.3.b).

A closer inspection of this part of the stress dy-namics reveals different features depending onthe applied shear rate. For most of the strainrates ranging from γl to 45s−1, we observed gen-erally two distincts variations as shown in figure5.a at γ = 30s−1 : the slow stress growth to-wards the steady state follows at first an affinelaw with a slope p and then a monoexponen-tial increase with a characteristic time τg, thecrossover between these two regimes occuring ata time named τ1. For some shear rates, the mo-noexponential growth is prolounged by a slightdecrease towards steady-state (see fig.5.a). Theparameters p and τg computed from the fittingprocedure are gathered in figure 5.b. Except atlow shear rates, the slope p is approximately

Figure 4: (Color online) Time-dependent re-sponse of the shear stress σ(t) after the start-upof flow for several imposed shear rates along thestress plateau γ = 7s−1 (yellow line), γ = 30s−1

(orange line), γ = 70s−1 (red line) (a) At shorttimes. (b) On a longer time-scale to capturesteady-state.

constant whereas the time τg is of the orderof 10-15s. Let us mention that for some im-posed shear rates, the kink at τ1 is not alwaysso marked as in figure 5.a and in a few cases, thestress profile might also be fitted by a sigmoıdalshape. Moreover, the time τ1 and the fittingparameters have been found to depend on theflow history : τ1, p, τg and the time characteriz-ing the steady state increase with the time dur-ing which the sample stays at rest between twoconsecutive tests. The study of the effect of thislag time could maybe bring information on therelaxation of the induced structures and is leftfor a future work. In the present investigation,two consecutive tests have been systematicallyperformed with a fixed lag time of the order oftwo minutes, namely much larger than the owntime of the system and enough to ensure the re-producibility between the different tests.

Above 45s−1, the stress growth becomes veryslow and it is really difficult to extract a clearvariation law. It is not possible to define τ1either because the kink is buried in the shearstress fluctuations or because the variation lawsof the slow stress growth towards the steadystate differ from those below 45s−1. In the nextsection, we shall try to identify the processes as-sociated with these particular evolutions and weshall see that, whatever the averaged shear ratemay be, the small undershoot in the shear stressappears as the mechanical signature of the in-

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Figure 5: (Color online) a) View of the smallundershoot in the stress response during thestartup of flow at 30s−1. For t < τ1, the stressincreases linearly whereas above τ1, the varia-tion can be fitted by a monoexponential growth.b) Slope p (�) of the linear law and character-istic time τg (◦) of the exponential growth com-puted from the fitting procedure.

terface instability between bands.Figure 6 shows that τ1 increases with the meanshear rate once it exceeds 10s−1 which corre-sponds approximately to the spinodal point inthe underlying flow curve [42, 43]. At low shearrates, we observe an upturn towards higher val-ues of τ1 due to the existence of metastablestates.Finally we can note that the stress signal fluc-tuates around its steady-state value, the ampli-tude of the fluctuations growing with the im-posed shear rate from 0.05% at the beginningof the plateau to 0.3% above 60s−1, while thetemporal fluctuations of the controlled shearrate never exceed 0.05%. Such variations inthe fluctuations of σ(t) are probably related tothe different regimes of spatio-temporal dynam-ics of the interface observed all along the stressplateau (see section 3.2.2).

3.2 Optical results

3.2.1 Observation of the banding struc-ture

Figure 7 illustrates the main stages of the kinet-ics of formation of both the banding state andthe interfacial instability for a controlled shearrate of 30s−1. Let us recall that the scatter-ing signal is gathered simultaneously with thetemporal stress evolution allowing thus a pre-

Figure 6: Comparison between the times τ1(full symbols) extracted from the stress signaland the time τ2 (open symbols) characterizingthe beginning of the destabilization of the inter-face.

cise correlation of the structural and mechan-ical responses. The sample is initially at restand does not scatter the laser light (photo 1).At the onset of the simple shear flow, the en-tire gap becomes turbid, the maximum of scat-tered intensity being reached around 0.5s whenthe stress overshoot occurs (photo 3). Birefrin-gence measurements have demonstrated thatthis non linear elastic response is associatedwith a strong orientation and stretching of themicellar threads with respect to the flow direc-tion [44]. The observed turbidity results fromthe stretching of the micellar network that gen-erates concentration fluctuations along the flowdirection [45] as suggested by small angle lightscattering experiments under shear [25, 22]. Ac-cording to Kadoma and coworkers, a semi-dilutesurfactant solution presents heterogeneties inmicelles concentration at the scale of a few meshsizes with regions of different densities of entan-glements [46]. The application of the flow fieldpreferentially deforms the regions of low densityof entanglements and produces the formation ofclusters of mesoscopic size, first organized par-allel to the flow direction. At this time, all thenew phase is nucleated but not arranged into amacroscopic band.The building of the banding structure startswith the relaxation of the stress overshoot : theturbidity first diminishes in a homogeneous way.This may be due either to the relaxation or to

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Figure 7: (Color online) Photos of the gap of the Couette cell in the plane (r, z) after a startupexperiment at 30s−1. The left and right sides of each picture corresponds respectively to theinner and the outer cylinders. The height of the field of observation is centered at halfway of themeasuring cell and limited to about 16 mm in order to keep a reasonable spatial resolution forthe post-processing. At time t = 0, the sample is at rest.

the breakup of the stretched micellar threadsbetween the clusters. Then from 1.8s, one canobserve the formation of a diffuse interface thatbegins to migrate from the fixed wall to its sta-tionnary position in the gap (see photos 6 to9). The corresponding behaviour in the shearstress response is the sigmoıdal relaxation or thedamped oscillations depending on the magni-tude of the averaged shear rate. The migrationis performed first rapidly with an approximatespeed of 0.05 mm.s−1 and then more slowlyabove t=5s. This process is accompagnied by asharpening of the interface, the profile of whichis flat on the total height of the Couette cell.

Moreover, the band located near the fixedcylinder does not present any particular scat-tering properties as the sample at rest. How-ever, this band is birefringent (see figure 8),suggesting that it contains the initial entangledmicellar network orientated by the local flowfield. On the other hand, the induced band re-mains turbid, the scattered intensity level be-ing uniform along the direction of the cylin-der axis. The orientation and the stretching ofthe micron-sized domains created at the onsetof flow could generate a form dichroısm related

to the anisotropy of the scattered light. It hasbeen shown recently that appearance of flow-induced dichroısm in wormlike micellar systemsis responsible for the turbidity [47, 48]. Thisform effect has to be studied in a careful way: it appears necessary to probe uniquely theinduced band using small angle light scatteringexperiments under shear to determine quantita-tively the characteristic size of the clusters andthe way of which they interact. This shall helpto establish the nature of the induced structureand is left for a future work. The induced bandis also strongly birefringent as illustrated in fig-ure 8, reflecting a prononced orientation stateof the micellar threads in the clusters. Let usnote that form dichroısm is usually associatedwith form birefringence. However, the experi-ment between crossed polarizers does not allowto discriminate between form and intrinsic ef-fects.From t '8s, the front adopts a sharp profileand continues to move at a speed varying from0.007 to 2.10−4 mm/s (see the inset in fig.9) upto a time τ2 ' 29s where it seems to have al-most reach its equilibrium position associatedwith the plateau value. During this slow dis-

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Figure 8: Snapshot of the top of the Couettecell in the plane (~v, r) placed between crossedpolarizer and analyzer and illuminated using anextended white light source. The letters A andB indicate the moving and the fixed cylindersrespectively. The radius of the inner cylinder is24.5 mm giving a gap of 1.5 mm and the heightof the cell is 30 mm. The main axis of the po-larizer is orientated at approximately 20◦ withrespect to the flow direction. The applied shearrate is 30s−1.

placement, the interface position evolves expo-nentially (see figure 9) and the shear stress in-creases almost linearly as function of time (seefigure 5.a). Let us mention that the period ofthe oscillations modulating the exponential evo-lution in figure 9 is due to the imperfect rotationof the inner cylinder (the default of coaxialityinherent to the the apparatus is of the order of20 µm).

According to the model proposed by Rad-ulescu et al [49], the front propagation towardsthe final equilibrium position r∗ begins whenthe interface fully sharpens and is controlled bythe stress diffusion coefficient D included in theoriginal Johnson-Segalman model [53]. In thisframework, the solution of the equation of mo-tion of the interface gives:

r(t)− r∗ = (r(0)− r∗)e−t/τ ′(3)

with

τ ′ =τ 2ReKG0γ√DτRηlγlδγ

(4)

whereKG0τRηlγl

' 0.3 [49], δγ is the width of

the stress plateau and ηl the viscosity of thesample just before the transition. In our case,

Figure 9: (Color online) Position of the in-terface as function of time during its slow mi-gration towards its stationary position in thegap of the Couette cell. This evolution is well-described by a single exponential decay (redline). The inset illustrates the position of theinterface from t=0 s until a time slightly upperthan τ2.

the characteristic time τ ′ extracted from theinterface displacement as function of time isequal to 6.2±0.3 s and the diffusion coefficientD computed from this last equation is foundto be 7.10−11m2.s−1, giving a correlation lengthς =

√DτR ' 4µm. This value is larger by

three orders of magnitude than the diffusion co-efficient estimated on the same sample at 30◦Cby following the stress relaxation after a smallstep between two values of the shear rate in thecoexistence regime [49]. The model predicteda stress relaxation on three well distinct timescales and supposed that the slowest relaxationstage was associated with the travel of the thininterface towards its stationary position. How-ever this treatment ignored the phenomenon ofdestabilization of the interface which seems tobe in fact the slowest process in the transientstress as we will see just below. Hence, thestrong deviation between the values of D ob-tained here and in ref.[49] could certainly beexplained by the growth of the interface insta-bility. Let us note that recently, stress diffusioncoefficients of the order of 10−12 − 10−11m2.s−1

have been computed for CPCl/NaSal solutionsat two different concentrations using superposi-tion rheology together with ultrasonic velocime-try [50]. Correlation lengths deduced from such

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Figure 10: Photograph of the plane (r, z), thefield of observation being extended on the totalheight of the Couette cell, namely 40mm.

values of D are also in agreement with recentmeasurements in straight microchannel on thesame systems [51].At τ2, we observe the first signs of destabiliza-tion of the interface along the cylinder axis.Then from τ2 the instability grows with timeand finally saturates around t = 60−70 s wherethe interface shows undulations with a well-defined wavelength and a finite amplitude. Theperiodic pattern spreads over on all the height ofthe inner rotating wall as displays in figure 10.The amplitude of the interface profile is mini-mum on the edges of the inner cylinder.Let us note that the boundary conditions havebeen changed in order to test their effects onthe existence of the instability. Our sample wassheared with a classical inner cylinder, with orwithout the plug, and in a partially filled celland in each case, we observe the destabiliza-tion of the interface between bands. For themoment, we do not examine the way in whichthe undulations of the interface are quantitavelyaffected by a modification of the boundary con-ditions, of the gap thickness, the height and thecurvature of the cell.

In order to identify the part of the stress dy-namics corresponding to the appearance and thedevelopment of the interface instability, we com-pared the time τ1 at which the kink betweenthe linear and exponential regimes occurs dur-ing the slow growth of the shear stress towardssteady state and the time τ2 at which the directvisualizations reveal the emergence of the insta-bility (see figure 6). The correlation betweenboth times is satisfying on the range γ & γl to40 s−1 taking into account the standard devia-tion on the measured values.Just above the threshold γl, the proportion ofthe induced band is extremely low and the res-olution of our optical device does not allow for

the detection of the interface profile while, from45s−1 and as mentioned in section 3.1.2, it isno more possible to define τ1 and we do notobserve any particular changes during the slowstress growth around t = τ2. The existence ofa complex dynamics of the interface above 45s−1 (see section 3.2.2) could be responsible ofthese modifications. However, if the differentregimes are not detectable in the stress signalat high shear rates, the instability still growsduring the undershoot.Hence, the undershoot in the stress curves asfunction of time (Fig.4.b) appears as the me-chanical signature of the interface instabilityalong the cylinder axis. The evidence of sucha characteristic in the transient stress profileshas been highlighted in other systems [43, 42]suggesting that this instability is not particularto our system. We shall show in the conclusionthat the CPCl/NaSal system, intensively stud-ied by different groups during the last decade[42, 19, 20], also presents such an instability.In the next section, we focus on the spatio-temporal dynamics of the interface all along thestress plateau.

3.2.2 Phenomenology of the interfacedynamics

We follow the interface dynamics for a givenstep shear rate by collecting images of the gap inthe plane (r, z) at a frame rate of 25 images persecond. From each frame, we extract the inter-face profile and reconstruct the spatio-temporaldiagram summarizing its evolution as functionof the time and space coordinates. This proce-dure is reproduced for various shear rates alongthe coexistence plateau and on a time scale al-lowing to capture both the transient regime andthe asymptotic behaviour. Figure 11 gathersthe different patterns that we have been able toidentify. The grey levels materialize the posi-tion of the interface in the gap, black and whiteregions being associated respectively with theminima and maxima of the interface amplitudeviewed from the moving cylinder. The z coordi-nate corresponds to the direction of the cylinderaxis, the origin being chosen at halfway of thecylinders.

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Figure 11: Spatiotemporal evolution of the position of the interface in the gap of the Couette cellduring a step shear rate from rest to (a) γ = 6.5s−1, (b) γ = 13s−1, (c) γ = 30s−1, (d) γ = 50s−1,(e) γ = 70s−1. The position of the interface in the gap is given in grey scale, the origin beingtaken at the moving wall. The z axis represents the spatial coordinate along the cylinder axis.Plots (f) and (g) correspond to applied shear rates of 13 and 50s−1 at a temperature of 30◦.

Figure 11.a displays the spatiotemporal se-quence at 6.5s−1. After a transient of 24±4sincluding the construction, sharpening and mi-gration of the interface, the pattern exhibits, atleast at large scale, a well-defined wavelengthλ = 0.5 ± 0.1 mm, namely approximately halfof the gap width. However, an inspection atsmaller scale reveals a complex dynamics : glob-ally, the pattern oscillates alternatively towardsthe top and the bottom of the Couette cell witha temporal period T = 17 ± 1s, which seemsto be dissociated both of the period of rotationof the rotor (11.4s in that case) and of the timeconstant of the feed-back loop. Moreover, wavespropagating towards the bottom of the cell are

clearly visible (see the top of the diagram). Letus note that propagative events towards the topof the cell simultaneously occur at other heights(not shown here), a feature already observed onthe same system at a slightly different temper-ature [32]. At this shear rate, we computed anaveraged phase velocity vφ = 18µm.s−1 with astandard deviation of 2µm.s−1. This value istwo orders of magnitude lower than the tangen-tial velocity of the moving cylinder.This type of behaviour has been observed in therange of shear rates comprised between γl and8s−1. vφ is found to increase with the macro-scopic shear rate and reaches 34±1µm.s−1 atthe upper limit of domain (a) whereas T dimin-

11

Figure 12: (Color online) Illustration of theextension of each regime of interface dynamicsalong the coexistence plateau.

ishes but still in a decorrelated way with therotation of the inner wall. Let us note howeverthat, just above the threshold γl, the propor-tion of the induced band is so reduced that theresolution of our CCD device forbids the quan-titative description of the interface dynamics.The subsequent behaviour of the interface is il-lustrated in figure 11.b. In this range of shearrates extending from 8 to 13s−1, the dynamicalevolution of the interface is mainly character-ized by a decrease of the initially selected wave-length a few tens of second after the destabi-lization of the flat interface. This change of thewavelength is materialized by the developmentof new maxima (white zones) in the interfaceprofile, as observed for example at 13s−1 (Fig.11.b) around t ' 50s. This effect is even moremarked in figure 11.f for the same solution butat a slightly different temperature : in this situ-ation, we detect that the spatial frequency dou-bles. After this transient stage, the interfaceadopts a spatially stable profile with a wave-length λ = 1.48± 0.03mm.Besides, the oscillation of the pattern towardsthe top and the bottom of the cell, typicalof the previous dynamical regime, persists lo-cally at short times before being finally damped.From the diagram at 13s−1, the pseudo-periodT of the damped oscillations is estimated to12.8±0.5s. For comparison, one rotation of theinner cylinder is performed in 5.7s.Figure 11.c shows the dynamical behaviour en-

countered on a large part of the stress plateaufrom 13 to 45s−1. In that case, the most ampli-fied wavenumber in the initial stages of the in-stability growth is also the asymptotically dom-inant mode. The experiment at 30s−1 has beencarried out on a duration of one hour and wechecked that there was no coarsening of theblack and white regions : the interface keeps aspatially stable profile on very long times witha wavelength λ = 3.11 ± 0.07mm. A closer in-spection of the diagram reveals that the am-plitude of the interface profile is modulated incourse of time : an oscillation of the minima(black zones) with a characteristic time of 5 s isclearly visible in figure 11.c whereas the positionof the maxima does not evolve at this particu-lar frequency. This phenomenon, detectable forshear rates ranging from 10 to 50s−1, is also il-lustrated on the photographs in figure 13 focus-ing on a minimum during a time scale of about 5s. It is tempting to interpret this modulation ofamplitude as a possible signature of instabilitywith a wave vector orientated along the veloc-ity. In the framework of the diffusive Johnson-Segalman model in the planar case, a linear sta-bility analysis of a 1D shear-banding base flowrecently predicted that the interface was unsta-ble with respect to modes with wavevector alongthe velocity for a given range of interface sizes[28, 29]. However, this assumption is difficultto defend in our case since the frequency of themodulation does not vary significantly with themacroscopic shear rate. Besides, the couplingwith the default of coaxiality intrinsic to theCouette cell when both frequencies are compa-rable, might affect the oscillation of the min-ima and makes the interpretation really compli-cated. Further experiments, for example basedon observation of the interface in the (~v, ~∇v)plane have to be performed in order to under-stand the origin of this modulation.The following regime (cf. Fig.11.d) is approx-imately confined between 45 and 55s−1 and ischaracterized by a increase of the initially se-lected wavelength. In particular, at 50s−1, thegrowth of the asymptotic mode is associatedwith a wavelength twice as large. This changeof dominant mode occurs after about 50 to60 s and beyond, the interface profile remains

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Figure 13: (Color online) a) Snapshots of onehollow of the interface profile at 30s−1 in per-manent regime on a period ∆τ ' 5 s. b) Corre-sponding behaviour extracted from the spatio-temporal diagram at 30s−1.

spatially stable with an asymptotic wavelengthλ = 4.2± 0.2mm, namely about four times thegap width. Figure 11.g provides an other illus-tration of the regime (d) at a temperature of30◦C.For γ ≥ 55s−1, we observed a pronounced modi-fication of the interface dynamics (cf. Fig.11.e).The emergence of this complex dynamics coin-cides with the change in the slope of the stressplateau at γ+ mentioned in section 3.1.1. Thefirst signs of destabilization of the interface aredetectable around 50 to 60 s, the length scaleof the instability being of the order of 2 timesthe gap width at this moment. Let us notethat this initial wavelength does not vary signif-icantly with the imposed rate of strain. After atransient slightly lower than 200s, the amplitudeof the interface profile saturates while the wave-length continuously evolves in course of time. Infact, two neightbouring hollows have a tendencyto merge. When the length scale between twominima increases, several other hollows close tothese minima nucleate and finally merge againwith hollows of longer life time. This processeshappen on all the duration of the experiment('25 min) : the system does not seem to tendtoward a stationnary situation and the spatio-temporal diagram strongly suggests a chaoticdynamics.When the applied shear rate is incrementedalong this region of the stress plateau, the in-terface dynamics speeds up, the life time of the

Figure 14: (a) Asymptotic wavelength λ (•)versus γ. The lack of experimental points above50s−1 is explained by the impossibility to definean asymptotic mode in regime (e). (b) Ampli-tude A of the interface profile (�) as function ofγ.

minima being reduced and the number of unionand nucleation events increasing notably. As inthe first regime, it was not possible to describethe dynamics at the vicinity of the threshold γhbecause of the extremely low value of the am-plitude of the interface profile.

Figure 12 summarizes the different regimesof interface dynamics along the coexistenceplateau. On a great part of the stress plateau,the asymptotic pattern is stable in space (re-gions b, c and d), whereras the dynamics be-comes complex when one approaches the thresh-olds (regions a and e), namely when the inter-face is close to the walls.From the spatio-temporal diagrams, we ex-tracted the wavelength and the amplitude ofthe dominant mode at long times and plottedtheir evolutions with the control parameter γ(cf. Fig.14). The asymptotic wavelength in-creases with γ and begins to saturate when oneenters in regime (d). When γ rises above γ+, wecannot define an asymptotic wavelength. How-ever, it is still possible to determine the orderof magnitude of the ”local” wavelength in thisregime : the values are spaced out between 3and 4 mm and globally decreases as the shearrate approaches γh. In the range of strain rateswhere λ is accessible, this latter does not seemto follow a simple scaling law with γ. As forthe amplitude of the interface, it follows a non-monotonic behaviour and presents a maximumfor γ between 40 and 50 s−1. Let us remark that

13

the maximum value Amax ' 120µm is reachedwhen the two bands are in equal proportions inthe gap (see figure 14 and 15).In figure 15, we plotted the proportion of theinduced band predicted by the lever rule αh =(γ − γl)/(γh − γl). First, as already noted atanother temperature [32], the values computedfrom the latter equation with γl = 4.4s−1 andγh = 97s−1 (black line) are underestimated,even if we take into account the uncertainty onthe critical shear rates (red dotted lines). Sec-ond, the experimental data do not seem to fol-low a linear behaviour with the applied shearrate contrary to the observations of Salmon etal on CPCl/NaSal [20]. Such deviation couldperhaps be explained by the undulated profileof the interface along the z direction. Besides,the existence of wall slip can not be excludedand could also justify that the measured pro-portion of the induced band is lower than theexpected value. Let us finally recall that wehave considered that the turbid band was asso-ciated with the high shear rate band : for themoment, this hypothesis remains still to demon-strate by simultaneously measuring the local ve-locities and the optical properties. This type ofexperiments have been performed very recentlyon various CPCl/NaSal samples and they indi-cate a strong correlation between shear, bire-fringence and turbidity banding. The authorsalso mention a breakdown of the lever rule [52].

4 Conclusion

In this paper, we have highlighted a complexshear-banding scenario in which the steadybanded state is characterized by an interfacebetween bands undulating along the cylinderaxis. Recent theoretical papers dealing withthe stability of an initially 1D gradient bandedbase flow in the framework of the diffusiveJohnson-Segalman model [28, 29, 30, 31] havedemonstrated that for sufficiently thin inter-faces, the 1D state is unstable with respect tomodes with wavevector along both the flowand the vorticity directions. Let us comparesome of the predictions of the model in ref. [31]

Figure 15: (Color online) Proportion αh of theinduced band (◦) as function of γ. The blackline represents the predictions of the lever rulewith γl = 4.4s−1 and γh = 97s−1. The dashedred lines show the extreme values of αh takinginto account the error bars on the critical shearrates.

with our results. First, the instability alongthe vorticity direction is predicted to growmuch slower but in the asymptotic state, theamplitudes of the oscillations in each of thesedirections are of the same order of magnitude[31]. Taking into account the values of theamplitude gathered in figure 14, the same typeof effect along the velocity direction should bedetectable. However, for the moment, we justdetect a temporal modulation of the amplitude,the origin of which is not established andthat does not seem to be compatible withundulations in the flow direction. To answerthis question, we plan to modify our Couettedevice in order to visualize the interface profilein the plane of a laser sheet perpendicular tothe cylinder axis.Second, the comparison between the transientevolution of the shear stress and the kineticsof the instability allowed us to identify theslow stress growth towards steady state as themechanical signature of the interface insta-bility. This behaviour is quite well capturedby the model. This tends to point out thatsamples exhibiting such a feature in theirstress time series are liable to undergo aninterface instability. Figure 16 illustrates forexample the steady state banding structureof a solution of CPCl/NaSal (6.3%) in 0.5M

14

NaCl brine at a mean shear rate of 9 s−1.The interface between bands clearly undulates

Figure 16: Interface profile in a plane (r, z)for a sample of CPCl/NaSal (6.3%) in 0.5 MNaCl brine at T=21.5 ◦C. The mean shear rateis γ = 9s−1 and the snapshot is taken after thesteady banded state is achieved.

along the vertical axis, the wavelength and theamplitude of the oscillation being respectively4 mm and 120µm. We can also remark thatthe mechanical behaviour of the CTAB/KBrsystem is compatible with the existence of aninterface instability [44]. In this system, athree-bands structure has been observed usingthe flow birefringence technique. In that case,the birefringence signal is averaged on thetotal height of the cell. Hence the intermediateband in this particular banding structurecould presumably result from the undulationsof the interface along the vertical direction.This means that scattering or birefringencedata resulting from an averaging along thevertical axis in Couette geometry could leadto misleading interpretation and should beexamined with care. Let us mention thatthe slow growth of the mechanical stress alsooccurs in the shear-banding flow of onions [2] ortelechelic polymers [5], making these complexfluids potential candidates for the interfaceinstability.Third, besides similar orders of magnitude forthe wavelength and amplitude in the final state,at first sight, the evolution of the asymptoticwavelength with the average shear rate seemsto be at least qualitatively reproduced by themodel (see the inset in figure 2 of ref. [31]).Nevertheless, it would be necessary to computeit for smaller interface width to properly testthe agreement on a much larger range of controlparameter.Fourth, one of the crucial point described bythe model is the structure of the flow field inthe plane (r, z). The scenario proposed by the

author is as follows : the origin of the interfaceinstability comes from the existence in theunperturbed flow of discontinuities across theinterface in shear rate and in the normal stressparallel to the interface [28, 54, 55], the effectsof these two driving terms being intertwinedand not clearly understood yet. When theinterface is subjected to a small perturbation, itbecomes inclined, its normal n being no longerradial. In order to keep the total velocityand the total traction (S.n, where S denotesthe unpertubed polymer stress and the stressperturbation) at the interface continous, aperturbation of the flow field must develop,mass conservation inducing recirculation whichenhances the initial perturbation of the inter-face. In turn, Taylor-like vortices stacked alongthe vorticity direction form in the gap of theCouette cell, the size of the rolls scaling withthe gap width. The centers of the rolls arelocalized at the vicinity of the interface, in theregions where this one is tilted.In order to test roughly the three-dimensional

Figure 17: External view of the Couette cellunder ambient illumination at 20 min after thestart-up of flow at γ = 30s−1. Glass spheresof mean size 30 µm are seeded at a concentra-tion of 1%, giving to the sample a milky aspect.The particles are organized into white stripesstacked along the vertical axis.

character of the flow field, we added smallglass spheres (Sphericel, Potters industries)of mean radius 30µm and density 1.1 at a

15

concentration of 1%. We checked that themechanical properties of our sample were notaffected by the particles. During a start-upof flow experiment at γ = 30s−1, we observedthat the glass beads tend to collapse into whitestripes stacked along the vertical direction (seeFig. 17), the wavelength of the pattern reachingapproximately 2.5 mm when the steady-stateis achieved. Of course this basic experimentdoes not allow to conclude about the structureof the flow field but it strongly suggests thatrecirculations occur in the sample.Let us also mention that streaming velocitygradient along the vorticity direction inducedby an interface has also been reported inmolecular-dynamics simulations of the coex-isting paranematic and nematic liquid-crystalphases under shear flow [56].Despite of the number of our observationscaptured by the model in [31], an alternativemechanism can be invoked to explain thevortex flow and the interface instability : thepurely elastic instability in curved geometryat low Reynolds and Taylor numbers [57, 33].This instability, observed in dilute viscoelasticpolymer solutions [59] is due to the stretchingof the molecules along the streamlines. Thisstretching along curved streamlines leads tonegative normal stress difference N1 = σθθ−σrr,producing a volume force (hoop stress), whichacts in a direction opposite to the centrifugaleffects. Above a critical Weissenberg number,the hoop stress overcomes the centrifugal forceand the azimuthal base flow becomes unstablewith the development of Taylor-like vorticesfollowing a complex dynamics depending onthe viscosity of the fluid and on the appliedshear rate [58]. We emphazised in section 3.1.1that the induced phase is elastic. Hence, onecan presumably assume that such an instabilitycan potentially be triggered at least in thisphase, generating vortices that destabilize theinterface between bands. It should be noticedthat a similar argument has been advanced asa possible origin of three-dimensional flow inthe concentrated wormlike CTAB/D2O system[60]. Contrary to the model in ref [31], thismechanism needs curvature of the shearing cell.A relevant way to test the hypothesis of the

elastic instability is to play with the curvatureof the Couette cell and eventually to look atthe system in a straight channel.To summarise, we have presented a completerheo-optical study of the dynamics of theshear-banding flow in a semi-dilute micellarsystem. We have shown that our sample wasnot following the classical scenario reportedin reference [17]. The interface between shearbands is found to be unstable with respect towavevector along the vorticity direction, themechanical signature of this instability beingcharacterized by a small undershoot in thestress response. This behaviour is extremelyrobust and does not seem typical to our sample.Except in the vicinity of the thresholds wherethe evolution of the interface is not accessible,we have identified a complex spatio-temporaldynamics all along the coexistence plateauusing the shear as controlled parameter : weobserved propagative events at low shear rates,stable oscillating modes at intermediate shearrates and finally chaotic patterns for the higheststrain rates. Let us mention that this complexdynamics is very similar to that observed inthe damped Kuramoto Sivashinski [61]. Thislatter has been first introduced as a model forthe transition to spatio-temporal intermittency.In reference [32], the equation for the shearbanding dynamics has been guessed because ofthe small aspect ratio of the cell and becauseof the natural translational symmetry of thecylinder. We are now working in deriving thismodel equation from fluid mechanics principles,assuming a simple strain rate/stress functionsimilar to the flow curve measured in thepresent article [62].Moreover, the organization into stripes of smallparticles embedded in the solution suggeststhat the flow is three-dimensional with Taylor-like velocity rolls stacked along the vorticityaxis. Further work based on particle imagevelocimetry experiments are considered to fullydetermine the velocity profiles in a radial plane[63].The nature of the induced band remains anopen problem. The observation of random flowalong the second branch indicates that the newstructures are elastic while the turbidity points

16

out the existence of a micrometric length scalein the system. Small angle light scatteringexperiments under shear performed selectivelyin the induced band could bring informationon the characteristic size responsible for thisturbidity. However it will be presumablyinadequate to determine the organization ofthis ”phase” at smaller scale.

AcknowledgementsThe authors thank J.L Counord for the build-ing of the optical device, J.P. Decruppe forfruitful discussions and the ANR JCJC-0020for financial support.

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