# Flow phase diagrams for concentration-coupled shear banding

Fielding, S.M. and Olmsted, P.D. (2003) Flow phase diagrams for concentration-coupled shear banding. European Physical Journal E, 11. pp. 65-83. ISSN 1292-895X

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 $\dot{\gamma}$) and concentration, calculated within the two-fluid, non-local Johnson-Segalman (d-JS- $\phi$) model. We also give results for the macroscopic flow curves $\Sigma(\bar{\dot{\gamma}},\bar{\phi})$ for a range of (average) concentrations $\bar{\phi}$. 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, $\drm \Sigma/ \drm \bar{\dot{\gamma}}>0$. As the concentration is reduced, the width of the coexistence regime diminishes and eventually terminates at a non-equilibrium critical point $[\Sigma_{\rm c},\bar{\phi}_{\rm c},\bar{\dot{\gamma}}_{\rm c}]$. We outline the way in which the flow phase diagram can be reconstructed from a family of such flow curves, $\Sigma(\bar{\dot{\gamma}},\bar{\phi})$, measured for several different values of $\bar{\phi}$. 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 $r\equiv l/\xi$, 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.