Thermodynamic approach to rheology of complex fluids: Flow-concentration coupling B. García-Rojas, 1 F. Bautista, 2 J. E. Puig, 2 and O. Manero 1, * 1 Instituto de Investigaciones en Materiales & Facultad de Química, UNAM, A.P. 70-360, México, Distrito Federal 04510, Mexico 2 Departamentos de Física & Ingeniería Química, Universidad de Guadalajara, Boulevard M. García Barragán 1451, Guadalajara, Jal 44430, Mexico Received 18 February 2009; revised manuscript received 29 April 2009; published 22 September 2009 In this work, the generalized Bautista-Manero-Puig BMP model derived from extended irreversible ther- modynamics EIT is used to analyze the coupling of stress with concentration in complex fluids. It is shown that this model is consistent with previous analyses that predict mechanical and thermodynamic instabilities in the shear-banding regime. In particular, for simple shear flows, the model presented here predicts the structure factor in the plane of shear and the onset of instabilities in the gradient-vorticity plane. Furthermore, the model predicts distinctive features of the models of Brochard–de Gennes and Schmitt et al. as particular cases. For finite stress relaxation time, the generalized BMP model allows the prediction of transient structures normal to the vorticity axis. Instabilities are predicted in the regions of high viscosity, which suggest that the induction of a more viscous phase in a shear-thickening solution can lead the system to instability, in this case, the layering is predicted perpendicular to the vorticity direction. These transient structural patterns within the shear- thickening region correspond to spinodal phase separation. When the mechanical and thermodynamic insta- bilities are uncoupled, the model predictions agree with experiments and with the transient-gel model of Brochard and de Gennes. DOI: 10.1103/PhysRevE.80.036313 PACS numbers: 47.20.-k, 47.50.-d, 83.80.Qr, 83.10.-y I. INTRODUCTION In recent years, much attention has focused on complex fluids, in which a certain internal structure of the fluid is strongly affected by a flow field see, for example, a general review of material instability in complex fluids, by Goddard 1. Experimentally, the research has been done through the application of scattering techniques to nonequilibrium phe- nomena under shear. Birefringence and dichroism combined with rheomechanical data have also been used for sensitive detection of the spatial anisotropy arising from composition fluctuations and molecular alignment in complex flows. Hence the prediction of the associated structure factor is a key issue of theoretical modeling. Analogously to critical liquid-gas behavior of simple liq- uids, shear flow causes a thermal shift in the critical point 2. The flow of complex fluids, like wormlike micelles, polymer and surfactants solutions, or emulsions, can modify or induce phase transitions, such as the de-mixing transition found in high molecular weight polymer solutions 3. One explanation of these changes in the critical point is based on a coupling of stress with order parameter fluctuations near the phase transition 4. Shear flow suppresses those fluctua- tions with characteristic times larger than reciprocal of the shear rate. Generally speaking, in a simple liquid near the gas-liquid transition, such slow modes tend to be longer wavelength fluctuations characteristic of critical behavior, and hence, the suppression of these fluctuations shifts the critical point to lower temperatures. By contrast, in the de- mixing of polymer solutions, stress enhances critical fluctua- tions and raises the de-mixing temperature 5. Complex fluids show a variety of flow phenomena such as flow-induced phase transitions and instabilities 6–9. These systems are characterized by a nonmonotonic relation be- tween the shear stress,  12 , and the shear rate,  ˙ 12 10–12, where the subscripts 1, 2, and 3 indicate the velocity, veloc- ity gradient, and vorticity directions, respectively. As indi- cated elsewhere, this kind of flow curve exhibits a constant shear stress with at least two values of shear rate 13–18. Consequently, the flow is nonhomogeneous and each phase is supporting a different shear rate. In the banded regime, changes in shear rate essentially alter the proportion of the low and high viscosity bands and the initially homogeneous flow becomes mechanically unstable. The kinetics of formation of inhomogeneous flows in the plateau stress range has distinctive features. The characteris- tic time for the sigmoidal evolution of the stress toward a steady-state plateau was found to be much longer than the reptation time. According to Porte et al. 19, such kinetics is typical of a nucleation-and-growth process, usually associ- ated with first-order phase transitions. In some systems, the nucleation-and-growth process leads to a kind of spinodal instability 20. The models derived from a criterion of complete me- chanical instability, corresponding to a negative slope in the flow curve and in which the medium separates into a fluid phase of high shear rate and a viscous phase of low shear rate the so-called shear-banding instability, sometimes disagree with experimental data. It has been observed that the fluid phase often nucleates in the region of shear rates smaller than the critical shear rates that bound the negative slope region, i.e., before the criterion of mechanical instability is reached. In this regard, both approaches, mechanical instability and nonequilibrium phase transition, are not incompatible. To incorporate nonequilibrium phase transitions and rheo- logically driven instabilities, some theories have been for- warded, such as that of Schmitt et al. 21. These authors proposed a classification scheme for the instabilities arising * Corresponding author; manero@servidor.unam.mx PHYSICAL REVIEW E 80, 036313 2009 1539-3755/2009/803/03631312 ©2009 The American Physical Society 036313-1