Unified Theoretical and Experimental View on Transient Shear Banding

Dense emulsions, colloidal gels, microgels, and foams all display a solidlike behavior at rest characterized by a yield stress, above which the material flows like a liquid. Such a fluidization transition often consists of long-lasting transient flows that involve shear-banded velocity profiles. The characteristic time for full fluidization ${\tau }_f$ has been reported to decay as a power law of the shear rate $\dot{\gamma }$ and of the shear stress $\sigma$ with respective exponents $\alpha$ and $\beta$. Strikingly, the ratio of these exponents was empirically observed to coincide with the exponent of the Herschel-Bulkley law that describes the steady-state flow behavior of these complex fluids. Here we introduce a continuum model, based on the minimization of a “free energy,” that captures quantitatively all the salient features associated with such transient shear banding. More generally, our results provide a unified theoretical framework for describing the yielding transition and the steady-state flow properties of yield stress fluids.

Figure 1

Stress-induced fluidization time ${\tau }_f$ versus reduced shear stress $\sigma -{\sigma }_c$ for Carbopol microgels at various weight concentrations: 0.5% (up-pointing triangle), 0.7% (down-pointing triangle), 1% (circle), and 3% (rectangle). Solid lines correspond to the best power-law fits of the various data sets ${\tau }_f\sim (\sigma -{\sigma }_c{)}^{-\beta }$ with exponent $\beta$ ranging from 2.8 to 6.2. Experimental conditions are listed in Supplemental Material Table S1 together with values of ${\sigma }_c$ and $\beta$ [17].

Figure 2

Stress-induced fluidization dynamics in (a),(b) theory and (c),(d) experiment on a 1 wt% Carbopol microgel in a smooth concentric cylinder geometry of gap 1 mm. (a),(c) Shear rate $\dot{\mathrm{\Gamma }}$ and $\dot{\gamma }$ versus time $t$ for a shear stress of $\mathrm{\Sigma }=1.1$ and $\sigma =41$ Pa, respectively. Insets: Velocity profiles $v$ normalized by the velocity of the moving plate $v_0$ as a function of the distance $y$ to the fixed plate normalized by the gap size $L$. Profiles taken at different times (symbol, time): (circle, 1100), (down-pointing triangle, $5.5\times {10}^4$), (square, $3.3\times {10}^5$), (up-pointing triangle, $6.6\times {10}^6$) in (a) and (circle, 1011 s), (down-pointing triangle, 6927 s), (square, 8193 s); (up-pointing triangle, 9522 s) in (c). (b),(d) Width $\delta$ of the fluidized shear band normalized by the gap width $L$ versus time $t$. The vertical dashed lines crossing (a),(b) and (c),(d), respectively, indicate the fluidization times $T_f$ and ${\tau }_f$.

Figure 3

Stress-induced fluidization time as a function of $m(\mathrm{\Sigma })$ defined by Eq. (2). (a) Theoretical predictions $T_f$. (b) Experiments from Fig. 1 where each dataset for ${\tau }_f$ was rescaled by the time ${\tau }_0$ shown in the inset as a function of the microgel concentration $C$ (see also Supplemental Material Table S1 [17]). Red lines show the predicted power law with exponent $-9/2$. The best power-law fits of the whole datasets yield exponents $-4.46\pm 0.10$ and $-4.69\pm 0.33$, respectively, for theory and experiments. The gray line in the inset is ${\tau }_0\sim C^4$.

Figure 4

Strain-induced fluidization time ${\tau }_f$ versus shear rate $\dot{\gamma }$ for a 1 wt % Carbopol microgel under the various experimental conditions listed in Supplemental Material Table S2 [17]. Inset: Theoretical prediction for $T_f$ versus $\dot{\mathrm{\Gamma }}$. Red lines show the predicted power law with exponent $-9/4$. The best power-law fits of the whole datasets yield exponents $-2.15\pm 0.10$ and $-2.45\pm 0.23$, respectively, for theory and experiments.