J. Non-Newtonian Fluid Mech. 164 (2009) 66–75 Contents lists available at ScienceDirect Journal of Non-Newtonian Fluid Mechanics journal homepage: www.elsevier.com/locate/jnnfm Modeling the thixotropic behavior of structured fluids Paulo R. de Souza Mendes Department of Mechanical Engineering, Pontifícia Universidade Católica-RJ, Rua Marquês de São Vicente 225, Rio de Janeiro, RJ 22453-900, Brazil article info Article history: Received 1 June 2009 Received in revised form 6 August 2009 Accepted 10 August 2009 Keywords: Viscoplastic fluid Thixotropy Structure parameter abstract A novel approach for modeling the mechanical behavior of thixotropic viscoplastic fluids is presented. Non-monotonic flow curves, stress overshoot during microstructure breakdown flows at constant shear rate, and viscosity bifurcation are some of the common aspects of structured fluids that are predicted by the new model. It involves two evolution equations, one for the stress and the other for the structure parameter. Simple ideas are employed to describe the microstructure, and, as a result, a model with a clear physical basis is obtained. In addition to the flow curve, which by construction is exactly predicted, it is shown that the model is able to predict correctly the behavior observed in the usual rheometric transient flows, among which abrupt changes in shear rate (microstructure buildup or breakdown experiments) and abrupt changes in shear stress (viscosity bifurcation experiments). The model is frame-indifferent and applicable to complex flows. © 2009 Elsevier B.V. All rights reserved. 1. Introduction Structured fluids are found in a wide range of human activities. Most suspensions, emulsions, and foams are structured fluids, and some examples are: personal care products, cosmetics, different foods, nanocomposites, paints, inks, cements, adhesives, greases, natural muds, drilling muds, crude oils, gels, and mining, coal and metal slurries. Structured fluids exhibit non-Newtonian mechanical behavior. At small stress levels, their microstructure often confers them an elastic behavior. In this case, beyond a certain stress threshold, usu- ally called yield stress, a major microstructure collapse occurs which causes dramatic drops in viscosity and elasticity. While under constant stress conditions for some long enough period of time, the microstructure of a structured fluid usually acquires a stable configuration, which is the result of the equilib- rium between the microstructure buildup and breakdown rates. If the microstructure changes do not occur instantaneously after a stress change, the structured fluid is said to be time-dependent. A time-dependent fluid is said to be thixotropic if its viscos- ity decreases/increases with time as it undergoes a shear rate increase/decrease, and if in addition these viscosity changes are reversible. On the other hand, a time-dependent fluid is said to be antithixotropic if its viscosity increases/decreases with time as it undergoes a shear rate increase/decrease, and if in addition these viscosity changes are reversible. E-mail address: pmendes@puc-rio.br. Most structured fluids that exhibit a yield stress are time- dependent, especially in the small stress range. Some extent of irreversibility in the microstructure changes is also often observed, but modeling and characterizing the mechanical behavior of irre- versible structured fluids are rather difficult tasks (e.g. [20]). Actually, as far as their mechanical behavior is concerned, even thixotropic fluids are far from being thoroughly understood [15]. Barnes [4] published a detailed review of thixotropy, where he described the phenomenon, discussed numerous examples, sum- marized its history, and gave an overview of the state of the art. In this review, Barnes [4] pointed out that most of the then avail- able theories only described the viscous thixotropic phenomenon, and that only a few attempted to describe viscoelastic effects. He grouped the viscous theories into three different categories: first those that employ the so-called structure parameter, usually , a scalar quantity that typically varies in the interval [0, 1] and represents an indirect measure of the level of structuring (these are usually called structural kinetics models); second those that use some direct information of the microstructure, usually called microstructural models; and third those just based on viscosity-time data. As an example of viscoelastic model, Barnes [4] cited the one set forth by Acierno et al. [1–3], although these workers focused on low-density polyethylene rather than on thixotropic fluids. They proposed a multimode Maxwell-type differential equation set for stress whose relaxation times and shear moduli depend on a struc- ture parameter. Mujumdar et al. [17] also gave a thorough discussion of the thixotropy literature, including a quite complete comparison between the various structural kinetics models then found in the literature. These models consist essentially of an evolution equation 0377-0257/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.jnnfm.2009.08.005