Derivation of the mechanical and thermodynamic potentials from the generalized BMP model under shear-banding flow

Abstract

In this work, we demonstrate that the thermodynamic potential calculated from the steady-state flow curve using the definition of free energy from irreversible thermodynamics, and the mechanical potential derived from the generalized BMP constitutive equation provide the same information in the unstable and metastable regions of the flow curve. The contribution of normal stresses in both potentials as well as its weight on the position of the stress plateau are explicitly exposed. The plateau stress is univocally defined by the location of the critical shear rates corresponding to the minima in the potential. This demonstration is carried out using experimental data of wormlike micellar solutions for various concentrations and temperatures, including regions close to the non-equilibrium critical point. A method to accurately determine the non-equilibrium critical point (or the critical temperature) in a direct form is provided here. Bifurcation points are defined, notably one located at the high shear-rate branch of the flow curve separating the regions of one real solution and three real solutions. The first normal stress difference exhibits three real solutions as sources of elastic instabilities in the high shear band. The contribution of the second normal-stress difference in both the mechanical and thermodynamic potentials is clearly exposed. These results demonstrate that the non-equilibrium phase transition and the mechanical instability as sources of the banded flow are essentially two manifestations of the same reality.