Nonlocal effects in the shear banding of a thixotropic yield stress fluid

We observe a novel type of shear banding in the rheology of thixotropic yield-stress fluids that is due to the coupling of both nonlocality and thixotropy. The latter is known to lead to shear banding even in homogeneous stress fields. Yet, in the presence of nonlocal effects such bands appear smoothed by the continuous shear rate variation across the shear band. Here, we introduce a simple nonlocal model for the shear banding (NL-SB), and we implement it for the analysis of sub-mm rheo-MRI velocimetry measurements of a milk microgel suspension in a cone-and-plate geometry. The proposed NL-SB model enables quantifying the cooperativity length, yielding values in the order of the aggregate size in the microgel.

Figure 1

Two-dimensional confocal images of the acid milk microgel suspension before rheology or rheo-MRI experiments. Samples were dyed with rhodamine B and imaged using a Leica SP8 Dive multiphoton confocal imaging microscope. The scale bar represents $50\mu \mathrm{m}$.

Figure 2

Comparison between up (black) and down (red) hysteresis loops obtained from bulk rheology measurements of an acid milk microgel suspension (empty circles), and from the $\lambda$-model (thick lines) with parameters chosen to match the experimental data. The critical shear stress ${\tau }_c$, obtained from viscosity bifurcation experiments, and the critical shear rate ${\dot{\gamma }}_c$ are indicated as thin dot-dashed lines. The down-sweep curve of the $\lambda$ model is unstable ($d\tau /d\dot{\gamma }<0$) for $\dot{\gamma }<{\dot{\gamma }}_c$ (red dashed line).

Figure 3

Fitting results for the ${}^1\mathrm{H}$ rheo-MRI velocity measurements at increasing applied shear rates, for the acid milk microgel suspension, obtained using the NL-SB model: (a) normalized velocity profiles, (b) $\xi$ values, and (c) ${\dot{\gamma }}_{z=0}$ values. Each rheo-MRI velocity profile was corrected for slippage by extrapolation of the last five pixels at the plate, as in Refs. [19, 20, 24].

Figure 4

Comparison between viscosity profiles calculated from 1H rheo-MRI measurements (symbols) and fitted velocities obtained using the NL-SB model (lines) at increasing shear rate values: $1{\text{s}}^{-1}$ (squares), $3{\text{s}}^{-1}$ (circles), $5{\text{s}}^{-1}$ (triangles), $7{\text{s}}^{-1}$ (inverted triangles), $20{\text{s}}^{-1}$ (diamonds), and $40{\text{s}}^{-1}$ (stars).

Figure 5

Theoretical velocity profiles $v\left(z\right)$ obtained with the NL-SB model for different values of the correlation length $\xi$ at (a) $l=1.0$ and (b) $l=0.5$, taking into account nonlocal effects only or coexistent nonlocal and shear-banding effects, respectively.

Figure 6

Comparison between (a) low-to-high and (b) high-to-low hysteresis flow curves of the milk microgel suspension.

Figure 7

Reproducibility of hysteresis flow curves of a milk microgel suspension when performed from (a) low-to-high and (b) high-to-low shear rates. (c) Effect of preshear on the rheological response of the microgel suspension.

Figure 8

Viscosity bifurcation measurements for the acid milk microgel suspension in a cone-and-plate geometry (cone angle $1^{\circ }$, radius 25 mm). Before imposing a constant shear stress, the sample was presheared at $200{\mathrm{s}}^{-1}$ for 30 seconds.

Figure 9

Fitted 1H rheo-MRI velocity profiles at increasing applied shear rates, for the acid milk microgel suspension, obtained using the NL-SB model for a shear-independent cooperativity length $\xi$. For each case only velocity profiles acquired in a certain shear rate range were considered for the fitting: $\dot{\gamma }=\left(1\text{–}40\right){\mathrm{s}}^{-1}$ (Case A), $\dot{\gamma }=\left(7\text{–}40\right){\mathrm{s}}^{-1}$ (Case B), and $\dot{\gamma }=\left(1\text{–}6\right){\mathrm{s}}^{-1}$ (Case C).