Microstructural evolution of a model, shear-banding micellar solution during shear startup and cessation

We present direct measurements of the evolution of the segmental-level microstructure of a stable shear-banding polymerlike micelle solution during flow startup and cessation in the plane of flow. These measurements provide a definitive, quantitative microstructural understanding of the stages observed during flow startup: an initial elastic response with limited alignment that yields with a large stress overshoot to a homogeneous flow with associated micellar alignment that persists for approximately three relaxation times. This transient is followed by a shear (kink) band formation with a flow-aligned low-viscosity band that exhibits shear-induced concentration fluctuations and coexists with a nearly isotropic band of homogenous, highly viscoelastic micellar solution. Stable, steady banding flow is achieved only after approximately two reptation times. Flow cessation from this shear-banded state is also found to be nontrivial, exhibiting an initial fast relaxation with only minor structural relaxation, followed by a slower relaxation of the aligned micellar fluid with the equilibrium fluid's characteristic relaxation time. These measurements resolve a controversy in the literature surrounding the mechanism of shear banding in entangled wormlike micelles and, by means of comparison to existing literature, provide further insights into the mechanisms driving shear-banding instabilities in related systems. The methods and instrumentation described should find broad use in exploring complex fluid rheology and testing microstructure-based constitutive equations.

Figure 1

(a) Schematic diagram of the 1-2 Couette flow cell. Space resolution is provided by the slit. (b) Schematic representation of the synchronization of the SANS data collection with the transient deformation applied in the rheometer. The indicated cycle is repeated until statistically significant scattering is collected at each time bin.

Figure 2

(a) Steady flow curve measured by increasing the ramp of the shear rate (black squares) and stress values measured at 60 s after the start of transient experiments (colored symbols). (b) Transient shear stress versus strain for shear startup as indicated in (a). Regions I–III are defined by the onset ${\dot{\gamma }}_{1c}$ and end ${\dot{\gamma }}_{2c}$ of shear-banding flow.

Figure 3

Shear startup with $\dot{\gamma }=$ 2.2 s${}^{-1}$: The upper panel shows 2D SANS patterns in the $\vec{v}-\vec{\nabla }\vec{v}$ (1-2) plane as indicated, measured at the times indicated by the arrows; the middle panel shows $〈I(q,t)〉$ for these patterns; and the lower panel shows the shear stress versus time.

Figure 4

Shear startup with $\dot{\gamma }=$ 2.2 s${}^{-1}$: (a) stress response and (b) first normal stress difference. In (a) and (b) the measured rheology (solid line) is compared to that calculated (symbols) via the stress-SANS rule and measured data of $|A_f|$ and ${\varphi }_0$, given in (c) and (d), respectively. Solid lines in (c) and (d) are predictions using the stress-SANS rule and measured values of $\sigma$ and $N_1$.

Figure 5

Shear startup at $\dot{\gamma }$ = 22 s${}^{-1}$: The upper panel shows a schematic of the microstructure with corresponding SALS patterns [40], the middle panel shows 2D SANS patterns and the corresponding $〈I\left(q\right)〉$ for the inner ($r/H=0.2$) and outer ($r/H=0.8$) spatial positions, and the lower panel shows the shear stress versus time.

Figure 6

Shear startup at $\dot{\gamma }=22$ s${}^{-1}$: (a) shear stress and (b) first normal stress difference. The solid lines are rheological measurements and symbols are predictions via the stress-SANS rule (with $C_1=112$) and data of $|A_f|$ and ${\varphi }_0$, with measured values shown in (c) and (d), respectively. For (c) and (d) the lines are calculations of the respective order parameters using the stress-SANS rule and the measured rheology.

Figure 7

Shear startup at $\dot{\gamma }=22$ s${}^{-1}$: shear stress compared with predictions via the stress-SANS rule and the measured order parameters shown in Figs. 6 and 6 using the linear stress-SANS coefficient $C_1=11.2$ for the outer band for $t/\lambda >5$ and the nonlinear coefficient $C_1$= 112 for rest of the conditions, as shown in Fig. 6.

Figure 8

Stress relaxation experiments: The upper panel shows 2D SANS patterns and the corresponding $〈I\left(q\right)〉$ plots and the lower panel shows the evolution of (a) stress and (b) $|A_f|$ after shear cessation. Dashed lines in (a) are fits to a single-exponential decay for relaxation from $\dot{\gamma }=2.2$ s${}^{-1}$ and to two exponential decays for relaxation from $\dot{\gamma }=22$ s${}^{-1}$. Solid lines in (b) are fits to a single-exponential decay function $|A_f|\sim \exp (-t/\lambda )+|A_f{|}_{\infty }$, with an offset $|A_f{|}_{\infty }=0.007$.