In this chapter, we have derived a general criterion for the onset of linear instability to shear rate heterogeneity for the step stress protocol that is independent of fluid or model type, and depends only on derivatives of the shear rate in time. The criterion predicts the growth of shear rate perturbations whenever ∂2
t ˙γ/∂t˙γ > 0. We investigated the use of the criterion in three classes of material:
Polymeric fluids: we showed that both the RP and Giesekus models have quali- tatively similar responses of the shear rate ˙γ(t) to an imposed shear stress. In both cases, for values of the imposed stress nearest those on the weakest slope of the constitutive curve the shear rate rises dramatically by several orders of magnitude in a short time period. We showed that, consistent with the general criterion, tran- sient shear banding arose during the upwards curving, upwards sloping region of this shear rate rise. However, we also showed that while the magnitude of this transient shear banding is ‘significant’ (i.e., ˙γmax − ˙γmin larger than 5% of the shear rate at the same instant in time) in the RP model, it is not significant in the Giesekus model. This is because the rate of increase of ˙γ(t) is much larger in the RP model relative to that of the Giesekus model for parameters leading to comparable con- stitutive curves. (In fact, we never found ‘significant’ transient shear banding for any parameters in the Giesekus model.) We also showed that in the RP model, the system returns to (relative) homogeneity at steady state, regardless of the form of
noise inputted into the system (i.e., initial or continuous noise). In the Giesekus model with noise added continually, a small magnitude of shear banding arises dur- ing the regime of upwards curvature and slope of ˙γ(t); however, unlike that found in the RP model, this heterogeneity persists to steady state. We therefore conclude that the Giesekus model in our opinion is not appropriate for the description of the transient shear banding properties of entangled polymeric materials. However, the results of significant transient shear banding in the RP model are consistent with the experimental findings of entangled polymeric systems [16, 18, 19, 74–77, 161].
Soft glassy materials: The shear rate response to an imposed shear stress in the SGR model is qualitatively similar to that of the RP and Giesekus models. However, we did not explore the transient shear banding properties of the model in response to a step stress, since it has already been shown that shear bands do indeed arise during the upwards sloping, upwards curving region of the shear rate in time [117]. Rather, we investigated the creep: the progressive decrease of the shear rate in time), and the subsequent fluidisation: the sudden increase of the shear rate in time leading to steady flow in the SGR model. Our aim here was to determine relations for: the shear rate as a function of time during creep, and also for the ‘dip’ τdip and ‘fluidisation’ τf times (as a function of imposed stress Σ) at which the shear rate undergoes a minimum and an inflection in time, respectively. The dip and fluidisation times are relevant as they describe the time during which the system is linearly unstable to the formation of shear bands τdip < ∆t < τf, according to the general criterion above.
We first focussed on the behaviour of the creep regime that can exist for a long time before fluidisation occurs. Here, we showed that in the glassy phase of the model (x < 1) and for imposed stresses above the yield stress, the shear rate follows a power law in time with an exponent dependent on the noise temperature:
˙γ(∆t) ∼ t−1w ∆ttw −x
. We found this creep power law to be valid for ‘short’ times: ∆t tw. A different creep behaviour was found to exist at ‘long’ times ∆t tw, though we were unable to access this regime to determine the power law exponent. Next, we investigated the fluidisation behaviour of the model: this is the tran- sition from a solid-like response (creep — described above) to liquid-like flowing
behaviour. The solid-like creep regime ends as the shear rate undergoes a mini- mum at a time ∆t = τdip, before suddenly increasing over several orders of mag- nitude, passing through an inflection point at ∆t = τf before reaching a steadily flowing state. Recall that in between these two ‘minimum’ and ‘fluidisation’ times τdip < ∆t < τf, the shear rate satisfies ∂t2˙γ/∂t˙γ > 0. In the glass phase (x < 1) we found two regimes depending on whether the imposed stress larger or smaller than a critical stress Σc. Below the critical stress, we showed that the fluidisation time depends on the imposed stress via a power law: τf/tw ∼ (Σ − Σy)−α [where α ∼ O(1)]. Above the critical stress, the fluidisation time depends exponentially on the imposed stress: τf/tw ∼ e−F Σ. We also showed that, as long as a clear ‘dip’ time τdip could be found, it was proportional to the fluidisation time τdip∝ τf. Above the glass point (1 < x < 2) we showed that τf was proportional to the time at which Fielding et al. [62] predicted departure from the linear creep regime: τf ∼ Σ1/(1−x).
Glassy polymers: Finally, we investigated the rheological response in the glassy polymer model to an imposed shear stress in order to provide a comparison with re- cent numerics and experiments performed under a constant extensional load [58,88]. We showed that the shear rate, segmental relaxation time and polymer and solvent stresses respond qualitatively similarly in the shear geometry as in the extensional geometry. Strain hardening arises as the polymer takes an increasing fraction of the imposed stress, accompanied by a ‘dip’ in the segmental relaxation time indicating the re-vitrification of the solvent. We showed that this strain hardening inhibits the sudden increase of the shear rate in time and thus stabilises the system against shear banding according to the general criterion.
To summarise, we have shown that one should expect transient shear banding to arise generically during a step stress whenever the shear rate undergoes simultaneous positive slope and curvature in time. Until recently, it has not always been appreci- ated by the community that shear banding might arise during the time dependent response of complex fluids to a step stress.