Flow phase diagrams for concentration-coupled shear banding

After surveying the experimental evidence for concentration coupling in the shear banding of wormlike micellar surfactant systems, we present flow phase diagrams spanned by shear stress Sigma (or strain rate (gamma) over dot) and concentration, calculated within the two-fluid, non-local Johnson-Segalman (d-JS-phi) model. We also give results for the macroscopic flow curves Sigma(gamma radical anion, (phi) over bar) for a range of (average) concentrations (phi) over bar. For any concentration that is high enough to give shear banding, the flow curve shows the usual non-analytic kink at the onset of banding, followed by a coexistence "plateau" that slopes upwards, dSigma/dgamma radical anion > 0. As the concentration is reduced, the width of the coexistence regime diminishes and eventually terminates at a non-equilibrium critical point [Sigma(c), (phi) over bar (c), gamma radical anion (c)]. We outline the way in which the flow phase diagram can be reconstructed from a family of such flow curves, measured for several different values of (phi) over bar. This reconstruction could be used to check new measurements of concentration differences between the coexisting bands. Our d-JS-phi model contains two different spatial gradient terms that describe the interface between the shear bands. The first is in the viscoelastic constitutive equation, with a characteristic (mesh) length l. The second is in the (generalised) Cahn-Hilliard equation, with the characteristic length xi for equilibrium concentration-fluctuations. We show that the phase diagrams (and so also the flow curves) depend on the ratio r = l/xi, with loss of unique state selection at r = 0. We also give results for the full shear-banded profiles, and study the divergence of the interfacial width (relative to l and xi) at the critical point.