In this thesis, we have tried to explain the regular and irregular(chaotic) dynamics of worm like micellar solutions on applying shear, through a detailed study of the equation of motion of a nematic order parameter tensor coupled to a hydrodynamic velocity field. We have assumed spatial variations only along one direction i.e. the gradient direction(1D model). The resulting phase diagram shows various interesting steady states or phases such as spatiotemporal chaos, temporal and spatiotemporal periodicities, and alignment of the director axis along the imposed flow field. The coupling of the orientational degrees of freedom of the order parameter with the hydrodynamic flow field holds the key to the appearance of dynamic shear bands in the system. We have solved numerically a set of coupled nonlinear equations to obtain the order parameter stress developed in the system; the magnitude of the order parameter tensor, the biaxiality parameter and the orientation of the director axis of the nemato gens under shear have also been studied in detail. To study the phase diagram obtained by time integration of the equation of motion mathematically, a stability analysis of the fixed point of motion for various parameter values has been performed so that the location of the chaotic-to-aligned phase boundary is verified. Also in the periodic region of the phase diagram, the stability of limit cycles is tested by analysing the fixed point of the corresponding Poincare map. Stability analysis of the periodic orbits leads to the observation that in the parameter space, there are regions of phase coexistence where chaotic or spatiotemporally intermittent behaviour coexists with periodic behaviour. When corrections in the imposed velocity field due to the order parameter stress were taken into account and the order parameter response was looked into at several points in the parameter space, the modified equations of motion were found to reproduce the earlier behaviour in all the different regimes if the value of a dimensionless viscosity parameter is taken to be such that the bare viscous stress overrides the order parameter stress. The phase boundaries are however different from the ones seen in the earlier model. However, for a choice of the viscosity parameter such that the order parameter stress and the bare viscous stress are comparable, we see two distinctly different attractors: a banded, periodic one that is common to both α1equalto 0, and not equal to 0 and a banded chaotic one for α1not equal to 0. Here, α1is a parameter that governs the nonlinearity in the stretching of the order parameter tensor along the direction of the applied shear. Quantitative analysis of the various chaotic attractors throws up not only positive Lyapunov exponents but also that the banded chaos is a “flip-flop” kind of chaos where the switching between two long-lived states of high and lows hear stress is chaotic, where as the behaviour in either of the two states is periodic, with either a single, isolated frequency or a bunch of harmonics. Also, the spatial correlation of the shear stress in the chaotic attractors is of much larger range than the temporal correlation, the latter being almost delta-function-like. On increasing the temperature of the system till it is above the isotropic–nematic transition temperature in the absence of shear, we find that under shear, similar attractors as those in the nematic case are observed, both for passive advection and for the full 1D hydrodynamics. This is an encouraging result since actual experiments are performed at a temperature for which the system is in the isotropic phase in the absence of shear. Thus for the 1D system, the parameter space has been explored quite extensively. Considering spatial variations only along the gradient axis of the system under shear is not enough since experiments have observed interesting behaviour in the vorticity plane in which Taylor velocity rolls were noted. Hence taking the system to 2D was necessary. Our numerical study of the 2D system under shear is incomplete because we came across computational difficulties. However, on shorter time scales we have seen a two-banded state with an oscillating interface and Taylor velocity rolls as well. The methodology used for the 2D study can also be used to reproduce the 1D results by the simple step of taking initial condition with no variation in the vorticity direction. This automatically ensures that no variation in the vorticity direction ever builds up because the equations of motion ensure that these variations in the system do not grow by themselves unless fed in at the start. Using this method, we were able to reproduce all the attractors found in the 1D calculation. Thus the 1D attractors have been observed using two different methods of calculation. Further work on the full 2D numerics needs to be done because we believe that spatiotemporally complex steady-state attractor s exist in the 2D system also for appropriate values of the parameters.

Soft condensed matter (SCM) systems are ubiquitous in nature. SCM systems contain mesoscopic structures in the size range 10 nm to 1 am that are held together by weak entropic forces. These materials are therefore easily perturbed by external fields such as shear, gravity and electric and magnetic fields and are novel systems for studying non-equilibrium phenomena. The elastic constants of these materials are ≈ 109 times smaller than conventional atomic fluids and hence it is possible to measure the viscoelastic response of these materials using commercial instruments such as rheometers. The relaxation time in SCM systems are of the order of milliseconds as compared to atomic systems where relaxation times are of the order of picoseconds. It is easy to study the effect of shear on SCM, as the shear rates attainable by commercial rheometers are of the order of the inverse of their relaxation times. The dynamics of SCM systems and their local rheological properties obtained using the method of probe diffusion can be quantified through dynamic light scattering experiments. The structure of SCM systems can be quantified using diffraction techniques such as small angle x-ray scattering. In this thesis we report experimental studies on the linear and nonlinear rheology and the dynamics of surfactant cetyltrimethylammonium tosylate (CTAT), which forms cylindrical wormlike micelles, studied using bulk rheology and dynamic light scattering (DLS) technique, respectively. We have also studied the phase behaviour of the ternary system formed by cetyltrimethylammonium 3-hydroxy-napthalene 2-carboxylate (CTAHN), sodium bromide (NaBr) and water using small angle x-ray scattering (SAXS). In Chapter 1, we discuss why SCM systems are suitable for studying non-equilibrium phenomena such as the effect of shear on the structure and dynamics of condensed matter. This is followed by a discussion on the chemical structure, phase behaviour and self assembling properties of the amphiphilic molecules in water. We then discuss the intermacromolecular forces such as van der Waals interaction, the screened Coulomb repulsion and hydrophobic and hydration forces. The systems that have been the subject of our experimental studies, viz. CTAT and CTAHN/NaBr/water have also been discussed in detail. This is followed by a theoretical background of linear and nonlinear rheology, dynamic light scattering and small angle x-ray scattering techniques. Next we describe the stress relaxation mechanisms in wormlike micelles. This is followed by a discussion on some standard techniques of nonlinear time series analysis, in particular the evaluation of the delay time L, the embedding dimension m, the correlation dimension ν and the Lyapunov exponent λ. We have also mentioned a few examples of experimental systems where chaos has been observed. We have also discussed in detail the various routes to chaos namely, the period-doubling route, the quasiperiodic route and the intermittency route. The concluding part of this chapter summarises the main results of the thesis. Chapter 2 discusses the experimental apparatus used in our studies. We have discussed the different components of the MCR-300 stress-controlled rheometer (Paar Physica, Germany). The rheo-small angle light scattering experiments and the direct visualisation experiments done using a home-made shear cell are also discussed. Next we describe the various experiments that can be done using a commercial rheometer. The frequency response and flow experiments have been discussed with some examples from our own work on entangled, cylindrical micelles. This is followed by a discussion on the various components of our dynamic light scattering (DLS) setup (Brookhaven Instruments, USA). Particle sizing of submicrometer colloidal spheres using our DLS setup has been discussed with an example of an angle-resolved DLS study of 0.05µm polystyrene colloids. Next we describe the various components of the SAXS setup (Hecus M. Braun, Austria). As an example application of SAXS we have quantified the structure of the lamellar phase formed by the surfactant CTAHN/water. We finally describe the sample preparation methods employed by us for the different experiments. Our nonlinear rheology experiments on viscoelastic gels of surfactant CTAT (cCT AT= 2wt%) in the presence of salt sodium chloride (NaCl) at various concentrations has been discussed in Chapter 3. We observe a plateau in the measured flow curve and this is attributed to a mechanical instability of the shear banding type. The slope of this plateau can be tuned by the addition of salt NaCl. This slope is due to a concentration difference between the shear bands arising from a Helfand-Fredrickson mechanism. This is confirmed by the presence of a “Butterfly” light scattering pattern in SALS experiments performed simultaneously with rheological measurements. We have carried out experiments at six different salt concentrations 10mM < cN aCl<1M, which yield plateau slopes (α) ranging from 0.07 < α < 0.4. We find that a minimum slope of 0.12, corresponding to a salt concentration of 25mM NaCl, is essential to see a “Butterfly” pattern indicating the onset of flow-concentration coupling at this α value. After this we turn our attention to stress/shear rate relaxation experiments. The remainder of this chapter is split in four parts. We show in Part-I that the routes to rheochaos in stress relaxation experiments is via Type-II intermittency. Interestingly in shear rate relaxation, the route is via Type-III intermittency. We also show that flow-concentration coupling is essential to see the route to rheochaos. This section also brings out the crucial role played by orientational ordering of the nematics during rheochaos using SALS measurements performed simultaneously with rheological measurements. In part-II, we study the spatio-temporal dynamics of the shear induced band en route to rheochaos. Our direct visualisation experiments show that the complex dynamics observed in stress/shear rate relaxation measurements during the route to rheochaos is a manifestation of the spatio-temporal dynamics of the high shear band. In part-III, we describe the results of our stress/shear rate relaxation measurements at a fixed shear rate/stress with temperature as the control parameter and thereby control the micellar length. We see the Type-II intermittency route to rheochaos in stress relaxation measurements and the Type-III intermittency route to rheochaos in shear rate relaxation measurements. We conclude this section by showing the results of linear rheology measurements carried out at different temperatures. We estimate the mean micellar length ¯L, reptation time τrepand the breaking time τbreak. We show that L¯ increases by ≈ 58%, as the sample goes through the route to rheochaos. In Part-I of this chapter we had only qualitatively discussed the correlations between the measured time series of stress and the VH scattered intensity during the Type-II intermittency route to rheochaos. In part-IV we have attempted to quantify the correlations between the two time series using the technique of linear and nonlinear Granger causality. We have also studied the phase space dynamics of the two time series using the technique of Cross Recurrence Plots. We show that there exists a causal feedback mechanism between the stress and the VH intensity with the latter having a stronger causal effect. We have also shown that the bivariate time series share similar phase space dynamics using the method of Cross Recurrence Plots. In chapter 4, we have studied the dynamics of wormlike micellar gels of surfactant CTAT using the DLS technique. We report an interesting result in the dynamics of these systems: concentration fluctuations in semidilute wormlike-micelle solutions of the cationic surfactant Cetyltrimethylammonium Tosylate (CTAT) at wavenumber q have a mean decay rate α qz, with z -̃1.8, for a wide range of surfactant concentrations just above the overlap value c∗. The process we are seeing is thus superdiffusive, like a L´evy flight, relaxing on a length scale L in a time of order less than L2 . The rheological behaviour of this system is highly non-Maxwellian and indicates that the micelle-recombination kinetics is diffusion-controlled (DC) (micelles recombine with their original partners). With added salt (100mM NaCl) the rheometric behaviour turns Maxwellian, indicating a crossover to a mean-field (MF) regime (micelles can recombine with any other micellar end). The concentration fluctuations, correspondingly, show normal diffusive behaviour. The stress relaxation time, moreover is about twenty times slower without salt than with 100mM NaCl. Towards the end of this chapter, we propose an explanation of these observations based on the idea that stress due to long-lived orientational order enhances concentration fluctuations in DC regime. In the previous chapter we had studied the dynamics of wormlike micellar gels of pure CTAT 2wt% and found superdiffusive relaxation of concentration fluctuations due to a nonlinear coupling of long-lived stress and orientational fluctuations to the con- centration. In chapter 5 we present results from dynamic light scattering experiments to quantify the diffusive motion of polystyrene (PS) colloids in the same system. This chapter is split in two parts. In Part-I, we discuss dynamics of PS particles of radius 115 nm and 60 nm in CTAT 2wt%. The radius of the colloidal spheres is comparable to the mesh size ξ = 80 nm of the wormlike micellar network and hence we are probing the network dynamics. We find that ∆r2(t) is wavevector independent at small and large lag times. However at intermediate times, we find an anomalous wavevector dependence which we believe arises from the rapid restructuring of the gel network. This anomalous wavevector dependence of ∆r2(t) disappears as the temperature is increased. In Part-II we discuss the dynamics of PS particles of radius 25 nm and 10 nm, smaller than ξ, in CTAT 1wt% & 2wt%. We once again find an anomalous wavevector dependence of ∆r2(t) at intermediate times for the 2wt% sample. Surprisingly, at large times the particle motion is not diffusive, rather ∆r2(t) saturates. We do not have a clear understanding of this as yet. Also for the 10 nm particle, the motion at small lag times is superdiffusive. The motion of these particles is probably influenced by the superdiffusion of concentration fluctuations observed in pure CTAT 2wt% system (chapter 4). In chapter 6, we report the observation of an intermediate mesh phase with rhom- bohedral symmetry, corresponding to the space group R¯3m, in the ternary system consisting of CTAHN/NaBr/water. It occurs at lower temperatures between a random mesh phase (LDα ) and a lamellar phase (Lα) on increasing the surfactant concentration φs. The micellar aggregates, both in the intermediate and random mesh phases, are found to be made up of a two-dimensional network of rod-like segments, with three rods meeting at each node. SAXS studies also show the presence of small angle peaks corresponding to ad−spacing of 25 nm. Freeze fracture electron microscopy results shows that this peak may correspond to the presence of nodule like structures with no long-range correlations. The thesis concludes with a summary of main results and a brief discussion of the scope for future work in Chapter 7.