Figure 1

Zero-shear relaxation time ${\tau }_0$ (◻) and plateau modulus $G_0$ (●) as a function of temperature for a transient associative polymer network at 25 g/L.Reuse & Permissions

Figure 2

(Color online) Results from velocimetry measurements: (a) and (b) fluid velocity profiles across the gap $\left(x\right)$ of the couette geometry, measured at $T=20°\text{C}$ and (a) $\dot{\gamma }{\tau }_0=0.54$ and (b) 0.74 for a 30 g/L associative polymer solution $\left({\tau }_0=18\text{ms}\right)$; (c) transient velocity measurements $\left(\dot{\gamma }{\tau }_0=0.54\right)$ for four positions in the gap; (from top to bottom) (i) near the inner rotating cylinder $\left(x=1.9\text{mm}\right)$, (ii) and (iii) close to the interface between the shear bands ($x=1.6$ and $1.2\text{mm}$, respectively), and (iv) near the stationary outer wall $\left(x=0.4\text{mm}\right)$ [1].Reuse & Permissions

Figure 3

Dimensionless shear stress $\sigma /G_0$ versus dimensionless shear rate $\dot{\gamma }{\tau }_0$ for a transient polymer network at 25 g/L at three different temperatures (and thus three different relaxation times) (a) $5°\text{C}$ $\left({\tau }_0=72\text{ms}\right)$, (b) $10°\text{C}$ $\left({\tau }_0=37\text{ms}\right)$, and (c) $20°\text{C}$ $\left({\tau }_0=16\text{ms}\right)$. The vertical bars indicate the range of stress fluctuations for a given shear rate. The gray region indicates where shear banding and fluctuating stresses are found.Reuse & Permissions

Figure 4

(Color online) Typical transient stress response in the banded regime, at a steady-shear rate of $\dot{\gamma }\tau =1$, measured at $T=10°\text{C}$ $\left({\tau }_0=36\text{ms}\right)$. Arrows in the bottom panel indicate the two different events constituting the fracture-healing behavior; healing $(\searrow )$ and fractures $(\nearrow )$. Note that at short time scales $O\left({\tau }_0\right)$, at the start-up of the shear flow, the stress shows an overshoot, which is not visible here due to the longer sampling interval of 1 s $\left(\approx 30\cdot {\tau }_0\right)$.Reuse & Permissions

Figure 5

(Color online) (a) Distributions of the relative fluctuation in stress around their average. (b) Power spectra obtained by Fourier transformation of the raw stress signal. (c) Cumulative distributions of the stress drop $\Delta \sigma$ of fractures. (d) Cumulative distributions of the intervals $\Delta t$ between fractures; drawn lines are fits to Poisson distributions. All for various shear rates and relaxation times: $\dot{\gamma }{\tau }_0=1.0$ and ${\tau }_0=37\text{ms}$ (i.e., $T=10°\text{C}$, ●), $\dot{\gamma }{\tau }_0=1.6$ and ${\tau }_0=37\text{ms}$ (▼), $\dot{\gamma }{\tau }_0=2.0$ and ${\tau }_0=37\text{ms}$ (▲), $\dot{\gamma }{\tau }_0=1.0$ and ${\tau }_0=96\text{ms}$ (i.e., $T=3°\text{C}$, $+$), and $\dot{\gamma }{\tau }_0=1.0$ and ${\tau }_0=107\text{ms}$ (i.e., $T=2°\text{C}$, ◆). Inset in (c) shows the dependence on the decay time ${\tau }_i$ for the interval distribution between fractures as a function of applied shear rate.Reuse & Permissions

Figure 6

(Color online) Transient stress responses, plotted as the relative variations in stress around their average value, for four different temperature (indicated in the plot in $°\text{C}$) at $\dot{\gamma }{\tau }_0=1$.Reuse & Permissions

Figure 7

(a) Flow curves predicted by the microscopic constitutive relation [Eq. (3)] for various relaxation times ${\tau }_0$ (given in ms). Inset illustrates the metastable loop (dotted line), the steady-state tie line (horizontal plateau), and the definition of the predicted bandwidth $S$. (b) Experimentally determined bandwidth of the stress fluctuations, given as the difference between maximum and minimum stresses, relative to the average stress $\overline{\sigma }$. Drawn line is the prediction of our model. Note that the loop in the flow curve, and with that the shear banding behavior, disappears for ${\tau }_0⪡0.02\text{s}$.Reuse & Permissions

Figure 8

Characteristic time interval ${\tau }_i$ between fracture events as a function of overall applied shear rate, from fitting a single exponential decay to the data shown in Fig. 5d, for $T=10°\text{C}$ $\left({\tau }_0=37\text{ms}\right)$.Reuse & Permissions