Structural Relaxation in Simple Yield Stress Materials Influences Their Rheology

Abstract

Simple yield stress materials are composed of soft particles, bubbles, or droplets with purely repulsive forces. The constituent elements are typically too large to undergo thermal fluctuations, suggesting that the internal structure of the material, and therefore the rheology, should not change over time. We explore the rheology of Carbopol, a prototypical simple yield stress material, and show that gradual structural relaxation of the material results in a small yet significant reduction in the dynamic yield stress. This relaxation process can lead to a non-monotonic creep deformation rate under constant stress, culminating in delayed fluidization of the material. These findings show that the yield stress is not merely a static material property but may be a function of the internal structure of the material.

Figures (5)

• Figure 1: Normal force relaxation in Carbopol under fixed compression without applied shear in a cone--plate rheometer. Data shown for two independently prepared samples.
• Figure 2: (a) Shear stress (left axis) and normal force (right axis) versus time for a Carbopol sample subjected to a constant shear rate of 0.02 s$^{-1}$. The shear stress, which approximates the yield stress at this low shear rate, gradually decreases as the material relaxes. (b) Normal force relaxation for a second Carbopol sample. A constant shear stress of 38 Pa is applied at the time intervals marked by the red and blue bands. (c) Resulting shear rates following the application of 38 Pa. In the first case (red triangles), the shear rate decreases to zero, indicating that 38 Pa is below the yield stress. In the second case (blue circles), after 5400 seconds of relaxation, the same applied stress produces a steady deformation rate, demonstrating that 38 Pa now exceeds the yield stress.
• Figure 3: Flow curves of Carbopol before and after relaxation. (a) Up-and-down shear rate sweep of a freshly loaded Carbopol sample. The inset shows the relaxation of normal force over time. A clear hysteresis is observed between the upward and downward sweeps, as structural relaxation progresses during the measurement. (b) The shear rate sweep is repeated on the same sample from (a). The up and down sweeps now closely overlap since structural relaxation has been completed.
• Figure 4: Yield stress relaxation is direction-dependent and reversible by mechanical agitation. (a) The material is initially sheared at a constant rate of 0.02 s$^{-1}$ for several hours to reach steady-state conditions. At $t = 0$, oscillatory shear (50 Hz, 0.2 rad amplitude) is applied for 5 minutes. Afterward, the original shear rate is resumed. The oscillatory perturbation induces a sharp increase in yield stress, which gradually relaxes over several hours. The inset shows the relative increase in yield stress as a function of the applied oscillation amplitude from multiple independent measurements. (b) A Carbopol sample is sheared at 0.02 s$^{-1}$ in the clockwise direction. At $t = 20,000$ s, shear is paused for 5 minutes and then resumed in the same direction, resulting in no significant change in yield stress. At $t = 30,300$ s, the shear direction is reversed to counterclockwise, producing a marked increase in yield stress that relaxes over approximately 2 hours.
• Figure 5: Non-monotonic shear rate and delayed yielding in Carbopol under constant stress. (a) A constant shear stress of 42 Pa is applied to a Carbopol sample immediately after loading and compression in the cone-plate system. The normal force (right axis) gradually decreases, indicating stress relaxation. Simultaneously, the shear rate (left axis) initially declines toward zero, suggesting the applied stress is below the yield stress. After approximately 1000 s, however, the shear rate begins to increase, indicating that the yield stress has relaxed below 42 Pa. (b) This non-monotonic shear rate response is reproduced using the oscillatory and reverse shearing protocols from Fig. 4. In the oscillatory case (black curve), the sample first reaches steady-state flow at a shear rate of 0.02 s$^{-1}$, then undergoes oscillatory shear at 50 Hz with a 0.2 rad amplitude for 5 minutes (see Fig. 4a). A constant shear stress of 34 Pa is subsequently applied, and the shear rate is tracked over time. In the reverse shearing case (pink and blue curves), the sample again reaches steady-state flow at 0.02 s$^{-1}$ in the clockwise direction. A constant shear stress of either 32 or 34 Pa is then applied in the counterclockwise direction, and the shear rate is recorded. In all cases, the applied stress is selected to lie above the relaxed yield stress (30 Pa) but below the transiently elevated yield stress induced by oscillatory shear or shear reversal (Fig. 4).