Chaotic dynamics in shear-thickening surfactant solutions

– We report the observation of dynamical behaviour in dilute, aqueous solutions of a surfactant CTAT (cetyl trimethylammonium p-toluenesulphonate), below the overlap concentration c⋆. At these concentrations, CTAT forms cylindrical micelles and shows a pronounced shear thickening transition above a concentration-dependent critical shear rate γ̇c. An analysis of the time-series of the stress relaxations at controlled shear rates in the shear-thickening regime shows the existence of correlation dimensions greater than two and positive Lyapunov exponents. This indicates the existence of deterministic chaos in the dynamics of stress relaxation at these concentrations and shear rates. The observed chaotic behaviour may be attributed to the stick-slip between the shear induced structure (SIS) formed in the sheared surfactant solution and the coexisting dilute phase. At still higher shear rates, when the SIS spans the Couette, there is a transition to higher-dimensional dynamics arising out of the breakage and recombination of the SIS. Introduction. – It is well known that surfactant molecules, under appropriate conditions, can assemble reversibly in solution to form various supramolecular structures. In very dilute solutions, at concentrations above the critical micelle concentration, these molecules selfassemble to form spherical micelles. Increasing the surfactant concentration results in the formation of cylindrical structures, which overlap and entangle to form clear viscoelastic solutions of giant wormlike micelles. At still higher concentrations, the surfactant solutions form mesophases exhibiting nematic, hexagonal or lamellar ordering. It is possible to pass from one phase to the other by simply changing the surfactant and cosurfactant concentrations, the salinity or the temperature of the system. The cylindrical structures that are formed at very low concentrations c (c < c, c is the overlap concentration), get entangled on shearing at a shear rate γ̇ > γ̇c to exhibit a pronounced increase in viscosity [1, 2, 3]. At γ̇ < γ̇c, the surfactant solutions exhibit Newtonian flow behaviour. Recently, direct imaging studies of light scattered by the shear-thickening CTAB/ NaSal and TTAA/ NaSal have shown the existence of unstable shear induced structures/ phases (SIS/ SIP) [2, 3], which fracture at very high shear rates, giving rise to shear thinning [3]. Direct imaging of the SIS using freeze-fracture electron microscopy shows that these structures span hundreds of microns and have sponge-like textures[4]. Typeset using EURO-TEX 2 EUROPHYSICS LETTERS In this paper, we present an experimental study on the nonlinear dynamics of stress relaxation in dilute solutions of the surfactant CTAT, at concentrations below the overlap concentration c. At these concentrations, CTAT forms cylindrical micelles [5] and undergoes a shear-thickening transition when sheared at γ̇ > γ̇c, which is followed by shear-thinning at higher values of γ̇ [1]. The phase diagram and flow behaviour of aqueous solutions of CTAT in the linear and nonlinear regimes have been investigated extensively by Soltero et al. [5]. Previous investigations of the flow behaviour of shear-thickening solutions of CTAT [1] using rheometry and neutron scattering have shown the coexistence of a highly viscoelastic shear-induced phase (SIP) with a viscous regime made up of short aggregates. Oscillations in the apparent viscosity (and hence the stress) of dilute CTAT at controlled shear rates in the shear thickened regime have been observed by Gamez-Corrales et al. [1]. Observations of viscosity oscillations in CTAB/NaSal [2] and TTAA/NaSal [3] in water under controlled stress conditions have been explained in terms of the growth and retraction of the SIS that form in the sheared solution. In a recent paper [6], we have established the existence of deterministic chaos [7] in the stress relaxation of shear-thinning CTAT solutions at concentrations c > c. The oscillations in the normal and viscoelastic stresses were explained in terms of the stick-slip between shear bands [8] that form in the plateau region of the flow curve. Here, we extend our experiments to more dilute solutions of CTAT (c = 11mM, zero-shear viscosity η◦ ∼ 2mPas, c ⋆ ∼ 15mM) and show that the occurence of a stress plateau in the flow curve of the surfactant solution is not a prerequisite for the existence of dynamical behaviour in its stress relaxation. In the following sections, we will demonstrate the existence of deterministic chaos in the stress relaxation of shear-thickened CTAT, possibly due to the stick-slip between the SIS with the coexisting dilute phases. To our knowledge, this is the first observation of deterministic dynamics in the stress relaxation of shear-thickening surfactant systems. The observed chaotic behaviour is followed by an increase in complexity of the dynamics of stress relaxation at higher shear rates due to the fracture of the SIS, which is accompanied by the formation of large vortices in the sheared solution [3]. Sample Preparation and Experimental Apparatus. – CTAT, purchased from Sigma Chemicals, Bangalore, India was used to prepare the samples by dissolving requisite amounts of CTAT in doubly distilled and deionised water. The sample containers were sealed and kept at 25◦C and shaken frequently to accelerate homogenisation. By measuring the viscosities of CTAT solutions prepared at different concentrations as a function of the shear rate γ̇, and extrapolating the values to γ̇=0, we have estimated the overlap concentration c of CTAT in water to be ∼ 15mM. The rheological measurements were made using a Rheolyst AR1000N stress-controlled rheometer at T=25◦C, in a concentric cylinder geometry with an conical end (outer cylinder diameter = 30mm, inner cylinder diameter = 27.66mm, immersed height = 25.8mm). Experimental Results. – Figure 1 shows the flow curve of 11mM CTAT at T=25◦C. The flow curve shows all the distinct regimes seen in TTAA/NaSal by Hu et al. in stress-controlled experiments [3], viz., a Newtonian region of constant viscosity (Regime I) followed by a shear thickening regime where the SIS nucleates heterogeneously from the inner wall of the Couette cell (Regime II). In regime II, the shear rate is seen to decrease with stress for a controlled stress measurement, which supports the idea of coexistense between the SIS and the dilute phase at these shear rates [1]. Next, we observe a shear-thickened regime where the stress increases almost linearly with shear rate (Regime III). In this regime, the SIS forms a percolating cluster that undergoes