Nonlinear dynamics of an interface between shear bands

Molecular Processes Leading to Shear Banding in Entangled Polymeric Solutions

The temporal and spatial evolution of shear banding during startup and steady-state shear flow was studied for solutions of entangled, linear, monodisperse polyethylene C3000H6002 dissolved in hexadecane and benzene solvents. A high-fidelity coarse-grained dissipative particle dynamics method was developed and evaluated based on previous NEMD simulations of similar solutions. The polymeric contribution to shear stress exhibited a monotonically increasing flow curve with a broad stress plateau at intermediate shear rates. For startup shear flow, transient shear banding was observed at applied shear rates within the steady-state shear stress plateau. Shear bands were generated at strain values where the first normal stress difference exhibited a maximum, with lifetimes persisting for up to several hundred strain units. During the lifetime of the shear bands, an inhomogeneous concentration distribution was evident within the system, with higher polymer concentration in the slow bands at low effective shear rate; i.e., γ˙<τR-1, and vice versa at high shear rate. At low values of applied shear rate, a reverse flow phenomenon was observed in the hexadecane solution, which resulted from elastic recoil of the molecules within the slow band. In all cases, the shear bands dissipated at high strains and the system attained steady-state behavior, with a uniform, linear velocity profile across the simulation cell and a homogeneous concentration.

Microstructural evolution and reverse flow in shear-banding of entangled polymer melts

The temporal and spatial evolution of shear banding under startup of shear flow was simulated for highly entangled, linear, monodisperse polyethylene melts of differing molecular weight, C₇₅₀H₁₅₀₂, C₁₂₀₀H₂₄₀₂, and C₃₀₀₀H₆₀₀₂, using a high-fidelity coarse-grained dissipative particle dynamics method. It was determined that shear stress was dominated by segmental orientation of entangled strands at low shear rates, but at a critical shear rate below the reciprocal of the Rouse time, flow-induced disentanglement resulted in the onset of chain tumbling that reduced the average degree of orientation, leading to a regime of decreasing shear stress, with a commensurate onset of increasing average chain extension imposed by the strong flow kinematics that ultimately drove the steady-state shear stress higher. During startup of shear flow, shear band development began immediately after the maximum in the first normal stress difference, where distinct fast and slow bands formed. The slow bands consisted of relatively entangled and coiled molecules, whereas the fast bands consisted of more disentangled and extended chains that experienced quasiperiodic rotation/retraction cycles. The simulation results often exhibited a generation of temporary reverse flow, in which the local fluid velocity was temporarily opposite to that of the bulk flow direction, at the onset of the shear-banding phenomena; this effect was consistent with earlier experiments and theoretical results. The physical mechanism for the generation of reverse flow during shear-band formation was investigated and found to be related to the recoil of the molecules comprising the slow band. Overall, the phenomenon of shear banding appeared to arise due to flow-induced disentanglement from orientational ordering and segmental stretching that affected individual chains to different degrees, ultimately resulting in regions of relatively coiled and entangled chains that evolved into a slow band, whereas the locally disentangled chains, experiencing quasiperiodic stretch-rotation cycles, formed a fast band. The transitional period resulted in a kinematic instability that generated the temporary reverse-flow phenomenon.

Transient shear banding in the nematic dumbbell model of liquid crystalline polymers

In the shear flow of liquid crystalline polymers (LCPs) the nematic director orientation can align with the flow direction for some materials but continuously tumble in others. The nematic dumbbell (ND) model was originally developed to describe the rheology of flow-aligning semiflexible LCPs, and flow-aligning LCPs are the focus in this paper. In the shear flow of monodomain LCPs, it is usually assumed that the spatial distribution of the velocity is uniform. This is in contrast to polymer solutions, where highly nonuniform spatial velocity profiles have been observed in experiments. We analyze the ND model, with an additional gradient term in the constitutive model, using a linear stability analysis. We investigate the separate cases of constant applied shear stress and constant applied shear rate. We find that the ND model has a transient flow instability to the formation of a spatially inhomogeneous flow velocity for certain starting orientations of the director. We calculate the spatially resolved flow profile in both constant applied stress and constant applied shear rate in start up from rest, using a model with one spatial dimension to illustrate the flow behavior of the fluid. For low shear rates flow reversal can be seen as the director realigns with the flow direction, whereas for high shear rates the director reorientation occurs simultaneously across the gap. Experimentally, this inhomogeneous flow is predicted to be observed in flow reversal experiments in LCPs.

Formation and flow behavior of micellar membranes in a T-shaped microchannel

Understanding the formation and instability behavior of membranes is of fundamental interest and practical relevance to various biotechnological applications and self-assembly systems. Surfactant micellar membranes serve as a simple model system when surfactant molecules self-assemble into micellar structures under flow, but observing such process in real time is a major challenge due to limitations in spatiotemporal resolutions. We use a simple T-shaped microchannel to capture the formation and flow behavior of an ionic surfactant micro-micellar-membrane (μMM) when an aqueous stream of organic salt sodium salicylate (NaSal) meets a stream of cationic surfactant cetyltrimethylammonium bromide (CTAB). The μMM is shown to grow and become unstable depending on the flow rate, as characterized using micro-particle image velocimetry, fluorescence microscopy, flow birefringence, and bulk rheometry. We propose a simple model that accounts for the flow, elasticity and inertia of the μMM to analyze its flow behavior. Our experimental protocol can be easily replicated in conventional laboratories without the need of utilizing sophisticated equipment such as synchrotron small angle X-ray scattering and micro-electronics circuits. Our combined experimental and modeling results can be extrapolated to provide new insights to study the flow behavior and thermodynamic phases of lipid membranes, membrane proteins, and biological membranes.

Binary mixtures of rod-like colloids under shear: microscopically-based equilibrium theory and order-parameter dynamics

This paper is concerned with the dynamics of a binary mixture of rod-like, repulsive colloidal particles driven out of equilibrium by means of a steady shear flow (Couette geometry). To this end we first derive, starting from a microscopic density functional in Parsons-Lee approximation, a mesoscopic free energy functional whose main variables are the orientational order parameter tensors. Based on this mesoscopic functional we then explore the stability of isotropic and nematic equilibrium phases in terms of composition and rod lengths. Second, by combining the equilibrium theory with the Doi-Hess approach for the order parameter dynamics under shear, we investigate the orientational dynamics of binary mixtures for a range of shear rates and coupling parameters. We find a variety of dynamical states, including synchronized oscillatory states of the two components, but also symmetry breaking behavior where the components display different in-plane oscillatory states.

A flow visualization and superposition rheology study of shear-banding wormlike micelle solutions

In this paper, we use rheometry and flow visualization to study the dynamics of the interface between shear bands in a wormlike micellar solution sheared between concentric cylinders, i.e., in a Taylor-Couette (TC) cell, and to evaluate the stress diffusion coefficient and the stress correlation length in the Johnson-Segalman model. Two wormlike micellar solutions are studied: an aqueous solution of CTAB-NaNO3 and a solution of CPCl-NaSal in brine. These systems are highly elastic, exhibit Maxwellian behavior in linear viscoelasticity experiments, and shear banding in nonlinear experiments [S. Lerouge, et al., Soft Matter, 2008, 4, 1808-1819, M. A. Fardin, et al., Soft Matter, 2012, 8(39), 10072-10089, P. Ballesta, et al., J. Rheol., 2007, 51, 1047]. A large, custom-built, computer controlled TC cell allows us to rotate both cylinders independently and to visualize the flow in the r-z plane using a CCD camera. At low shear rates, the flow is stable and the fluid appears homogeneous throughout the gap between the cylinders. Above a critical shear rate, a shear banding transition occurs. This manifests itself in the formation of two distinct bands in the r-z plane, with an interface between the two bands. For sufficiently high ramp speeds, multiple steps of interface evolution are identified, as noted by Radulescu, Lerouge, and others [O. Redulescu, et al., Europhys. Lett., 2003, 62, 230, S. Lerouge, et al., Soft Matter, 2008, 4, 1808-1819]. We quantify the interface travel using direct visualization and use this measure, as well as superposition rheometry [P. Ballesta, et al., J. Rheol., 2007, 51, 1047], to determine the stress diffusion coefficient D and the stress correlation length ζ in the Johnson-Segalman model. These parameters are evaluated at different temperatures, shear rates, and gap sizes. We find that the stress diffusion coefficient and the stress correlation length exhibit a strong dependence on the gap of the Taylor-Couette cell for both shear-banding systems. For the CTAB-NaNO3 system, we report a linear dependence of the stress diffusion coefficient on temperature for the parameter range considered. In addition, we find that for this system, the stress diffusion coefficient is independent of shear rate. For the CPCl-NaSal system, we observe the same color changes in the sample reported by others on extended light exposure; however, we find that different histories of light exposure do not affect the measured stress diffusion coefficient.

Flow-induced structures observed in a viscoelastic reverse wormlike micellar system by magnetic resonance imaging and NMR velocimetry

Organogel formed by lecithin reverse wormlike micelles in the isotropic concentrated regime exhibits flow induced micro-heterogeneities detected by rheo-NMR and NMR-velocimetry.

Multi-scale simulation of non-equilibrium phase transitions under shear flow in dilute polymer solutions

In this paper we propose a multi-scale two-fluid model that couples a coarse-grained microscopic method to the two-fluid framework for studying multi-phase fluids under shear flow.

Nonlinear viscoelasticity of entangled wormlike micellar fluid under large-amplitude oscillatory shear: role of elastic Taylor-Couette instability

The role of elastic Taylor-Couette flow instabilities in the dynamic nonlinear viscoelastic response of an entangled wormlike micellar fluid is studied by large-amplitude oscillatory shear (LAOS) rheology and in situ polarized light scattering over a wide range of strain and angular frequency values, both above and below the linear crossover point. Well inside the nonlinear regime, higher harmonic decomposition of the resulting stress signal reveals that the normalized third harmonic I_{3}/I_{1} shows a power-law behavior with strain amplitude. In addition, I_{3}/I_{1} and the elastic component of stress amplitude σ_{0}{E} show a very prominent maximum at the strain value where the number density (n_{v}) of the Taylor vortices is maximum. A subsequent increase in applied strain (γ) results in the distortions of the vortices and a concomitant decrease in n_{v}, accompanied by a sharp drop in I_{3} and σ_{0}{E}. The peak position of the spatial correlation function of the scattered intensity along the vorticity direction also captures the crossover. Lissajous plots indicate an intracycle strain hardening for the values of γ corresponding to the peak of I_{3}, similar to that observed for hard-sphere glasses.

Interface instabilities and chaotic rheological responses in binary polymer mixtures under shear flow

The numerical results of RP–FH model reveal another possible cause of the rheochaos: a vortex structure emerges within the central band.

Instabilities in wormlike micelle systems. From shear-banding to elastic turbulence

Shear-banding is ubiquitous in complex fluids. It is related to the organization of the flow into macroscopic bands bearing different viscosities and local shear rates and stacked along the velocity gradient direction. This flow-induced transition towards a heterogeneous flow state has been reported in a variety of systems, including wormlike micellar solutions, telechelic polymers, emulsions, clay suspensions, colloidal gels, star polymers, granular materials, or foams. In the past twenty years, shear-banding flows have been probed by various techniques, such as rheometry, velocimetry and flow birefringence. In wormlike micelle solutions, many of the data collected exhibit unexplained spatio-temporal fluctuations. Different candidates have been identified, the main ones being wall slip, interfacial instability between bands or bulk instability of one of the bands. In this review, we present experimental evidence for a purely elastic instability of the high shear rate band as the main origin for fluctuating shear-banding flows.

Nonaxisymmetric instability of shear-banded Taylor-Couette flow

Recent experiments show that shear-banded flows of semidilute wormlike micelles in Taylor-Couette geometry exhibit a flow instability in the form of Taylor-like vortices. Here we perform the nonaxisymmetric linear stability analysis of the diffusive Johnson-Segalman model of shear banding and show that the nature of this instability depends on the applied shear rate. For the experimentally relevant parameters, we find that at the beginning of the stress plateau the instability is driven by the interface between the bands, while most of the stress plateau is occupied by the bulk instability of the high-shear-rate band. Our work significantly alters the recently proposed stability diagram of shear-banded flows based on axisymmetric analysis.

Self-organization in the flow of complex fluids (colloid and polymer systems). Part 2: Theoretical models

Flow induced transitions in complex fluids are usually accompanied by changes in the internal media structure and the flow symmetry. In this review paper, we discuss the theoretical models and approaches that have been used for the analysis of different types of flow instabilities and flow patterns. The main attention is focused on the basic fluid models which reveal vortex and banding flow structures at high shear rates. The Oldroyd-B fluid is one of such models. The Reynolds and the Weissenberg (or Deborah) numbers are the parameters governing its flow behavior. For this model, the secondary flow patterns arising in viscometric flows of different geometries at the bifurcation point are described. Complex fluids which are able to exist in multiple states can form coexisting bands of different structures with different rheological properties and flowing with different shear rates at the same shear stress. Shear banding is typical for fluids demonstrating non-monotonous flow curves described by such models as the diffusive Johnson-Segalman fluid model, for example. Recent progress in exploring this phenomenon is discussed.

Azimuthal instability of the interface in a shear banded flow by direct visual observation

The stability of the shear banded flow of a Maxwellian fluid is studied from an experimental point of view using rheology and flow visualization with polarized light. We show that the one-layer homogeneous flow cannot sustain shear rates corresponding to the end of the stress plateau. The high shear rate branch is not found and the shear stress oscillates at the end of the plateau. An azimuthal instability appears: the shear induced band becomes unstable and the interface between the two bands undulates in time and space with a period τ, a wavelength λ and a wave vector k parallel to the direction of the tangential velocity.

A coupled map lattice model for rheological chaos in sheared nematic liquid crystals

A variety of complex fluids under shear exhibit complex spatiotemporal behavior, including what is now termed rheological chaos, at moderate values of the shear rate. Such chaos associated with rheological response occurs in regimes where the Reynolds number is very small. It must thus arise as a consequence of the coupling of the flow to internal structural variables describing the local state of the fluid. We propose a coupled map lattice model for such complex spatiotemporal behavior in a passively sheared nematic liquid crystal using local maps constructed so as to accurately describe the spatially homogeneous case. Such local maps are coupled diffusively to nearest and next-nearest neighbors to mimic the effects of spatial gradients in the underlying equations of motion. We investigate the dynamical steady states obtained as parameters in the map and the strength of the spatial coupling are varied, studying local temporal properties at a single site as well as spatiotemporal features of the extended system. Our methods reproduce the full range of spatiotemporal behavior seen in earlier one-dimensional studies based on partial differential equations. We report results for both the one- and two-dimensional cases, showing that spatial coupling favors uniform or periodically time-varying states, as intuitively expected. We demonstrate and characterize regimes of spatiotemporal intermittency out of which chaos develops. Our work indicates that similar simplified lattice models of the dynamics of complex fluids under shear should provide useful ways to access and quantify spatiotemporal complexity in such problems, in addition to representing a fast and numerically tractable alternative to continuum representations.

Banded spatiotemporal chaos in sheared nematogenic fluids

We present the results of a numerical study of a model of the hydrodynamics of a sheared nematogenic fluid, taking into account the effects of order-parameter stresses on the velocity profile but allowing spatial variations only in the gradient direction. When parameter values are such that the stress from orientational distortions is comparable to the bare viscous stress, the system exhibits steady states with the characteristics of shear banding. In addition, nonlinearity in the coupling of extensional flow to orientation leads to the appearance of a new steady state in which the features of both spatiotemporal chaos and shear banding are present.

Conductivity measurements in a shear-banding wormlike micellar system

Shear banding in the cetylpyridinium chloride/sodium salicylate micellar system is investigated using electrical conductivity measurements parallel to the velocity and parallel to the vorticity in a cylindrical Couette cell. The measurements show that the conductivity parallel to the velocity (vorticity) increases (decreases) monotonically with applied shear rate. The shear-induced anisotropy is over one order of magnitude lower than the anisotropy of the N(c) nematic phase. The steady-state conductivity measurements indicate that the anisotropy of the shear induced low-viscosity (high shear rate) phase is not significantly larger than the anisotropy of the high viscosity (low shear rate) phase. We estimate that the micelles in the shear induced low viscosity band are relatively short, with a characteristic length to diameter ratio of 5-15. The relaxation behavior following the onset of shear is markedly different above and below the first critical value γ1, in agreement with results obtained by other methods. The transient measurements show that the overall anisotropy of the sample decreases as the steady state is approached, i.e., the micellar length/the degree of order decrease.

Irreversible nanogel formation in surfactant solutions by microporous flow

Self-assembly of surfactant molecules into micelles of various shapes and forms has been extensively used to synthesize soft nanomaterials. Translucent solutions containing rod-like surfactant micelles can self-organize under flow to form viscoelastic gels. This flow-induced structure (FIS) formation has excited much fundamental research and pragmatic interest as a cost-effective manufacturing route for active nanomaterials. However, its practical impact has been very limited because all reported FIS transitions are reversible because the gel disintegrates soon after flow stoppage. We present a new microfluidics-assisted robust laminar-flow process, which allows for the generation of extension rates many orders of magnitude greater than is realizable in conventional devices, to produce purely flow-induced permanent nanogels. Cryogenic transmission electron microscopy imaging of the gel reveals a partially aligned micelle network. The critical flow rate for gel formation is consistent with the Turner-Cates fusion mechanism, proposed originally to explain reversible FIS formation in rod-like micelle solutions.

Two-dimensional perturbations in a scalar model for shear banding

We present an analytical study of a toy model for shear banding, without normal stresses, which uses a piecewise linear approximation to the flow curve (shear stress as a function of shear rate). This model exhibits multiple stationary states, one of which is linearly stable against general two-dimensional perturbations. This is in contrast to analogous results for the Johnson-Segalman model, which includes normal stresses, and which has been reported to be linearly unstable for general two-dimensional perturbations. This strongly suggests that the linear instabilities found in the Johnson-Segalman can be attributed to normal stress effects.

Shear-induced dynamic polarization and mesoscopic structure in suspensions of polar nanorods

We investigate the spatiotemporal behavior of sheared suspensions of rodlike particles with permanent dipole moments. Our calculations are based on a self-consistent hydrodynamic model including feedback effects between orientational motion and velocity profile. The competition between shear-induced tumbling motion and the boundary conditions imposed by plates leads to oscillatory alignment structures. These give rise to a spontaneous time-dependent polarization generating, in turn, magnetic fields. This novel shear-induced effect is robust against varying the boundary conditions. The field strengths are of a measurable magnitude for a broad parameter range.

Spatiotemporal nematodynamics in wormlike micelles en route to rheochaos

We show through polarized light scattering experiments the spatially inhomogeneous orientational dynamics for shear-thinning wormlike micellar gels (cetyltrimethylammonium tosylate+sodium chloride+H2O ) en route to rheochaos. For shear rates in the plateau of the flow curve, we see alternating bright and dark birefringent stripes stacked along the vorticity. The orientational order in adjacent bands is predominantly oriented at +45 degrees and -45 degrees to the flow (v) in the (v,nablav) plane, respectively. We have made an attempt to correlate the observed orientational ordering in terms of the two-dimensional Taylor-like velocity rolls in a gradient banding fluid. The bands show spatial motion along the vorticity, and the orientation dynamics of the interface delineating adjacent bands completely correlates with the temporal dynamics of the stress. Furthermore, the observed spatial dynamics of the interfaces of the rolls depends crucially on the gap width of the Couette cell.

Shear banding and rheochaos in associative polymer networks

We present experimental evidence of an instability in the shear flow of transient networks formed by telechelic associative polymers. Velocimetry experiments show the formation of shear bands, following a complex pattern upon increasing the overall shear rate. The chaotic nature of the stress response in transient flow is indicative of spatiotemporal fluctuations of the banded structure. This is supported by time-resolved velocimetry measurements.

Non-brownian microrheology of a fluid-gel interface

We use stroboscopic video microscopy to study the motion of a sheared fluid-gel interface. Mechanical noise plays a role analogous to temperature, but with a low-frequency breakdown of linear response consistent with an underlying instability. We relate the fast motion of the interface to the rheological properties of the gel, laying the foundation for a non-Brownian optical microrheology.

Nonlocal effects in flows of wormlike micellar solutions

The flow curve of wormlike micelles usually exhibits a stress plateau sigma* separating high and low viscosity branches, leading to shear-banded flows. We study the flow of semidilute wormlike micellar systems in a confined geometry: a straight microchannel. We characterize their local rheology thanks to particle image velocimetry. We show that flow curves cannot be described by a simple constitutive equation linking the local shear stress to the local shear rate. We demonstrate the existence of nonlocal effects in the flow of wormlike micellar systems and make use of a theoretical framework allowing the measurement of correlation lengths.

Complex dynamics of shear banded flows

Many complex fluids undergo a flow induced transition to a state of coexisting bands of differing viscosities and internal structuring. This effect, which is called "shear banding", is widely observed in wormlike micellar surfactants, onion surfactants, colloidal suspensions and polymer solutions. According to a rapidly accumulating body of experimental evidence, shear bands often exhibit complex dynamics, which can be either oscillatory or chaotic in nature. This can be seen in the unsteady response of the bulk rheological signals, and in the motion of the interface between the bands. After giving a brief overview of this experimental evidence, we review in some detail recent efforts to address it theoretically.

Vorticity structuring and velocity rolls triggered by gradient shear bands

We suggest a mechanism by which vorticity structuring and velocity rolls can form in complex fluids, triggered by the linear instability of one-dimensional gradient shear banded flow. We support this with a numerical study of the diffusive Johnson-Segalman model. In the steady vorticity structured state, the thickness of the interface between the bands remains finite in the limit of zero stress diffusivity, presenting a possible challenge to the accepted theory of shear banding.

Evidence for three-dimensional unstable flows in shear-banding wormlike micelles

We report on an experimental study of the shear-banding phenomenon in the concentrated wormlike micellar system CTAB at 20wt.% in D2O . Time-resolved velocity profiles are recorded using ultrasonic velocimetry simultaneously to global rheological data. Our results confirm the studies performed previously by Fischer and Callaghan [Phys. Rev. E 64, 011501 (2001)]. Time averaged velocity profiles display an unsheared "nematic gel." In the range of applied shear rate, the flow field exhibits very fast temporal fluctuations. Suspicions for the presence of three-dimensional flow are evidenced and possible causes for a three-dimensional instability are discussed together with the coupling of wall slip to bulk dynamic.

Granger causality and cross recurrence plots in rheochaos

Our stress relaxation measurements on wormlike micelles using a Rheo-SALS (rheology + small angle light scattering) apparatus allow simultaneous measurements of the stress and the scattered depolarized intensity. The latter is sensitive to orientational ordering of the micelles. To determine the presence of causal influences between the stress and the depolarized intensity time series, we have used the technique of linear and nonlinear Granger causality. We find there exists a feedback mechanism between the two time series and that the orientational order has a stronger causal effect on the stress than vice versa. We have also studied the phase space dynamics of the stress and the depolarized intensity time series using the recently developed technique of cross recurrence plots (CRPs). The presence of diagonal line structures in the CRPs unambiguously proves that the two time series share similar phase space dynamics.

Properties, main applications and perspectives of worm micelles

This tutorial review deals with one of the most remarkable forms of surfactant aggregates, described as having a flexible, elongated cylindrical shape. Three structural scale lengths are pertinent to the flexibility and mobility of worm micelles: the cross-sectional radius, r(cs), the overall (contour) length, L, and the persistence length, l(p). The diversity of l(p) values in amphiphilic systems is demonstrated as well as the relation between L and l(p). The review also discusses the viscoelasticity of worm micelles and the relaxation mechanisms underlying this dominant property. Many aspects of viscoelasticity--such as non-linearity, shear banding, flow-induced phase transition, rheochaos--are only shortly described. The prevailing application of worm micelles, namely as fracture fluids and drag reducing agents are discussed in detail, stressing the effect of variations in the surfactant molecular structure on the efficacy of worm micelles. The vague possibility of using "smart" worm micelles in the foreseeable future is tersely outlined.