Nonmonotonic Stress Relaxation after Cessation of Steady Shear Flow in Supramolecular Assemblies

Stress relaxation upon cessation of shear flow is known to be described by single-mode or multimode monotonic exponential decays. This is considered to be ubiquitous in nature. However, we found that, in some cases, the relaxation becomes anomalous in that an increase in the relaxing stress is observed. Those observations were made for physicochemically very different systems, having in common, however, the presence of self-associating units generating structures at large length scales. The nonmonotonic stress relaxation can be described phenomenologically by a generic model based on a redistribution of energy after the flow has stopped. When broken bonds are reestablished after flow cessation, the released energy is partly used to locally increase the elastic energy by the formation of deformed domains. If shear has induced order such that these elastic domains are partly aligned, the reestablishing of bonds gives rise to an increase of the overall stress.

Figure 1

Illustration of the experimental protocol and structural changes of the investigated living polymers (system A) and polymer gels (systems B and C) under shear and during relaxation. Large shear magnitudes lead to bond breakage resulting in stored energy (1). Bond reformation leads to work performed on the polymers and causes local elastic inhomogeneities, indicated by orientationally correlated local segments marked by yellow lines, resulting in increasing bulk stress (2), which eventually relaxes thermally until the equilibrated networks of living polymers or polymer gels are reestablished (3).

Figure 2

Stress relaxation experiments with the supramolecular systems A (top), B (middle), and C (bottom). At intermediate times stresses increase. The shear stress is normalized to the steady-state shear stress, and shear rates applied prior to cessation of flow are listed in the legends.

Figure 3

Increase of fraction of formed bonds following flow cessation, $1-f$, with time along with the stress evolution for systems A, B, and C, respectively, in red, blue, and black at initial shear rates of 7, 10, and $4{\mathrm{s}}^{-1}$. The circles are experimental stress data, lines are the model fits, and dashed lines are the associated $1-f$ values. For system A in red, the time axis has been multiplied by 10 so that all data fit into the same plot. Inset: Variation of the inverse effective stress decay time $({\tau }_{\mathrm{eff}}{)}^{-1}$ with time (see text). Horizontal lines indicate the inverse decay times $1/{\tau }_{\mathrm{l}}$ to be reached at very long times.