“Everything flows?”: elastic effects on startup flows of yield-stress fluids
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It is now 30 years since Barnes and Walters published a provocative paper in which they asserted that the yield stress is an experimental artifact. We now know that the situation is far more complicated than understood at the time, and that the mechanics of the solid material prior to yielding must be considered carefully. In this paper, we examine the response of a well-studied “simple” yield-stress material, namely a Carbopol gel that exhibits no thixotropy, and demonstrate the significance of the pre-yielding behavior through a number of elementary measurements.
KeywordsRheology experiments Yield stress Viscoelasticity Flow curves Kelvin-Voigt Maxwell fluid
In 1985, Howard Barnes and Ken Walters published a provocative paper entitled “The yield stress myth?” (Barnes and Walters 1985), in which they asserted that the yield stress is an experimental artifact, and notably that all fluids will show viscous (indeed, Newtonian) behavior at sufficiently small stresses. They stated that “the yield stress hypothesis, which has hitherto been a useful empiricism, is no longer necessary, and … fluids which flow at high stresses will flow at all lower stresses, i.e., the viscosity, although large, is always finite and there is no yield stress.”1 This assertion by two very prominent rheologists caused a flurry of discussion and publication, with substantial parsing of the meaning of the words “yield stress;” i.e., is the yield stress a material property or a useful approximation for materials that exhibit a large reduction in viscosity over a narrow shear stress range? Barnes and Walters supported their assertion with data obtained using a constant-stress rheometer that showed a Newtonian regime at stresses lower than the apparent yield stress, and Barnes subsequently showed similar data on a number of different materials (Barnes 1999; Roberts and Barnes 2001), including Carbopol.
The concept of a yield-stress fluid was popularized by Bingham, who included such fluids in the context of yielding in many classes of materials in his 1922 book Fluidity and Plasticity (Bingham 1922). Barnes (1999) has written a comprehensive review of the history of the study of yielding, in which he places the common yield-stress fluids currently being studied in the context of phenomena like creep in metals and plastics. Modern interest in yield-stress fluids largely dates from work by Oldroyd (1947) and Prager (Hohenemser and Prager 1932; Prager 1961) that put the description of such materials into an invariant continuum formulation that can be applied to flows in complex geometries. Both Oldroyd and Prager assumed that there is a transition between a solid and a fluid at a critical value of a stress invariant, typically taken to be a yield surface defined by the von Mises criterion (Prager 1961). Prager assumed that no deformation was possible on the “solid” side of the yield surface. Oldroyd assumed that the material is an incompressible elastic (Hookean) solid before yielding, with a stress proportional to the strain, and a viscous material thereafter, with a stress that is linear in the rate of deformation. Most subsequent investigators have assumed that the solid has an infinite modulus, in which case no deformation is possible prior to yielding, and the assumption of linearity after yielding has been generalized to include power-law behavior and even viscoelasticity. The Oldroyd–Prager formulation, with a discontinuous transition between solid and liquid, is at the heart of the yield-stress controversy initiated by Barnes and Walters.
Møller et al. (2009a) showed that the apparent Newtonian viscosity observed by Barnes at stresses below the apparent yield stress was not a true viscosity but was in fact an experimental artifact whose value depends on the waiting time prior to measurement (i.e., the elapsed time between initiating the deformation and recording the measurement), increasing with a power-law dependence on the waiting time; the exponents were between ½ and 1, depending on the material. What is in fact being observed is a response of the unyielded material that gives a ratio of stress to shear rate that is independent of the imposed stress, hence appearing to be a constant viscosity.
Is the yield stress a flow to no-flow transition? i.e., as per Barnes and Walters (1985), is there viscous flow below the yield stress?
Can the yield stress be inferred by extrapolation of the flow curve to zero shear rate?
Can the yield stress be inferred from startup experiments?
Are non-linear oscillatory shear measurements a better way to infer the yield stress?
Extrapolation to zero shear rate: flow curves
Flow to no-flow transition
Startup: experiments at a constant shear rate
The effect of viscoelasticity: pre-yielding mechanics
Of course, a material cannot be roughly a Maxwell fluid in one class of deformations and roughly a Kelvin–Voigt solid in another unless there is a hidden variable in a more general formulation that interpolates between the behaviors. This is the case in the kinematic hardening model used by Dimitriou et al. (2013), for example, in which the “back stress” evolves dynamically and affects the mechanics. The back stress can be viewed as a “lambda parameter” (e.g., Denn and Bonn (2011)) in simple shear flow and causes the location of the yield surface to adjust, depending on the deformation state, as in the general framework of the evolution of the yield stress surface for elastoviscoplastic solids that was developed by Naghdi and Srinivasa (1992). The kinematic hardening model can be shown to be roughly Maxwellian for small deformations at a constant shear rate and to be Maxwellian for the difference τ − τ y close to yielding, so it reflects the behavior seen in Fig. 5. Dimitriou et al. (2013) have shown via a numerical simulation at constant stress that the model predicts behavior qualitatively like that shown in Fig. 4.
This short article is intended to highlight the significance of the description of the pre-yielded material in considering the mechanics of yield-stress fluids. For the simple yield-stress fluid considered here, the transition appears to be based on a critical strain, with the possibility of dissipative deformations in a viscoelastic solid that make the critical stress under transient conditions deformation dependent. It is clear experimentally that the appearance of a Newtonian fluid regime at stresses below the yield stress is an artifact that would be observed with the simplest viscoelastic solid representation, namely a Kelvin–Voigt solid. We have not addressed the likely failure of the Oldroyd–Prager formalism following yielding, but there is convincing evidence that a viscoelastic fluid description is necessary for materials like the Carbopol studied here. Indeed, Fraggedakis et al. (2016) have employed both kinematic hardening and a viscoelastic model by Saramito (2007) to describe the kinematics and settling dynamics of a spherical particle through a Carbopol gel.
Finally, we note that for the ideal (non-thixotropic) yield-stress fluid studied here, the transition in a plot of total stress versus strain in finite-amplitude oscillatory shear gives a value of the yield stress that is consistent with the yield stress obtained by extrapolation of the flow curve and the value obtained in a startup experiment, with the added information of the yield strain. This method has the advantage of eliminating artifacts associated with startup flows or extrapolation of the flow curve. Any of these methods properly used, however, can give a reliable value of the yield stress.
Barnes has described the paper as having been presented at the Fourth International Congress on Rheology in 1984 in a number of publications, but the paper does not appear in the Congress proceedings.
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- Bonn D, Paredes J, Denn MM, Berthier L, Divoux T, Manneville S (2015) Yield stress materials in soft condensed matter. arXiv preprint. arXiv:1502.05281
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