Flow phase diagrams for concentration-coupled shear banding

After surveying the experimental evidence for concentration coupling in the shear banding of wormlike micellar surfactant solutions, we present flow phase diagrams spanned by shear stress (or strain-rate) and concentration in the two-fluid, non-local Johnson-Segalman (d-JS-phi) model. We also present macroscopic flow curves for a range of (average) concentrations. For any concentration 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. As the concentration is reduced, the width of the coexistence regime diminishes, then terminates at a non-equilibrium critical point. We outline the way in which the flow phase diagram can be reconstructed from a family of such flow curves measured for several different average concentrations. This reconstruction could be used to check new measurements of concentration differences between the coexisting bands. Our d-JS-phi model contains two spatial gradient terms describing 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 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 at the critical point.