Long-Lived Neighbors Determine the Rheological Response of Glasses

Glasses exhibit a liquidlike structure but a solidlike rheological response with plastic deformations only occurring beyond yielding. Thus, predicting the rheological behavior from the microscopic structure is difficult, but important for materials science. Here, we consider colloidal suspensions and propose to supplement the static structural information with the local dynamics, namely, the rearrangement and breaking of the cage of neighbors. This is quantified by the mean squared nonaffine displacement and the number of particles that remain nearest neighbors for a long time, i.e., long-lived neighbors, respectively. Both quantities are followed under shear using confocal microscopy and are the basis to calculate the affine and nonaffine contributions to the elastic stress, which is complemented by the viscoelastic stress to give the total stress. During start-up of shear, the model predicts three transient regimes that result from the interplay of affine, nonaffine, and viscoelastic contributions. Our prediction quantitatively agrees with rheological data and their dependencies on volume fraction and shear rate.

Figure 1

Mean number of long-lived neighbors $n(\gamma )$ as a function of strain $\gamma$ obtained from (main figure) confocal microscopy and (inset) rheology for shear rate $\dot{\gamma }=0.0028{\mathrm{s}}^{-1}$, volume fraction $\varphi =0.565$, and the large particles ($R_2=780\mathrm{nm}$). The solid lines represent fits [Eq. (1)], for which the first few data points (open circles) have been disregarded in the case of the rheological data.

Figure 2

(a) Slices in the velocity-vorticity plane rendered from confocal microscopy images at strains $\gamma =0.072$, 0.164, and 0.277 (left to right). The squared nonaffine displacement of each particle during lag time $\mathrm{\Delta }t$, quantified by $D_{\min }^2(\gamma ,\mathrm{\Delta }t)/(2R{)}^2$, is indicated by the color of the spheres. (b) Ensemble average $⟨D_{\min }^2(\gamma ,\mathrm{\Delta }t)⟩/(2R{)}^2$ versus $\gamma$. The solid and dashed lines represent (almost identical) fits with cubic and quadratic order, respectively [Eq. (2)].

Figure 3

Normalized transient total stress ${\sigma }_{\text{tot}}(\gamma )/{\sigma }_{\text{tot}}(\gamma \to \infty )$ as a function of strain $\gamma =\dot{\gamma }t$ during start-up for (a) large particles and (b) small particles with different shear rates $\dot{\gamma }$ (main figure) and volume fractions $\varphi$ (inset) with $\dot{\gamma }$ and $\varphi$ as indicated [39]. Solid lines represent fits, dotted and dashed lines represent the elastic ${\sigma }_{\mathrm{el}}$ [Eq. (5)] and viscous ${\sigma }_{\mathrm{visc}}$ [Eq. (6)] contributions, respectively. Blue circles in (a) indicate ${\sigma }_{\mathrm{el}}$ as determined from the individual $n({\gamma }_i)$ (data points in Fig. 1) without using Eq. (1) (each symbol represents the average of four data points). The different regimes are indicated in (a).