Shear Banding of Colloidal Glasses: Observation of a Dynamic First-Order Transition

We demonstrate that application of an increasing shear field on a glass leads to an intriguing dynamic first-order transition in analogy with equilibrium transitions. By following the particle dynamics as a function of the driving field in a colloidal glass, we identify a critical shear rate upon which the diffusion time scale of the glass exhibits a sudden discontinuity. Using a new dynamic order parameter, we show that this discontinuity is analogous to a first-order transition, in which the applied stress acts as the conjugate field on the system’s dynamic evolution. These results offer new perspectives to comprehend the generic shear-banding instability of a wide range of amorphous materials.

Figure 1

Deformation map of colloidal glasses at volume fraction $\varphi =0.60$. The flow is homogeneous at low shear rates (a) and inhomogeneous beyond the critical shear rate ${\dot{\gamma }}_c\sim 6\times 10^{-5}{\mathrm{s}}^{-1}$ (b). The figures show height-dependent particle displacements at shear rates $\dot{\gamma }=3\times 10^{-5}{\mathrm{s}}^{-1}$ (a) and $\dot{\gamma }=1\times 10^{-4}{\mathrm{s}}^{-1}$ (b). Each cross represents a particle. Symbols in (b) indicate average flow profile after $\gamma \sim 0.3$ (circles), 0.8 (squares), and 0.95 (stars), demonstrating stable shear bands. Dashed horizontal lines (b) delineate the shear bands. Inset: Average nonaffine displacement (during $\mathrm{\Delta }t=2\min$, left axis) and structural distortion (eigenvalue ratio of Minkowski tensor $W_1^{20}$ of Voronoi volumes, see Ref. [22], right axis) versus strain demonstrate steady state after $\gamma \sim 0.3$.

Figure 2

Mean-square displacements of the particles. Upper left inset: Mean-square displacement in the upper (circles) and lower shear bands (dots) at ${\dot{\gamma }}^{*}=2$. Main panel: Mean-square displacements as a function of rescaled time for the applied shear rates ${\dot{\gamma }}^{*}=0.3$ (blue), 0.6 (cyan), 1.2 (green), 2 (magenta), and 5.6 (red). Lower right inset: Strain correlations of low shear band for $t=70$ (left) and 350 s (right).

Figure 3

Dynamic order parameter and phase diagram. (a) Dynamic order parameter as a function of observation time. $K$ is a linear measure of the system’s dynamic evolution. (b) Histogram of order parameter values for increasing observation times. The emerging bimodal distribution indicates dynamic phase coexistence. (c) Corresponding dynamic phase diagram: Mean order parameter as a function of applied strain rate. The dashed lines delineate boundaries of the shear-banding regime. Inset shows the dynamic order parameter at continuously increasing applied shear rate, for particles in the upper (red squares) and lower regions (blue dots). Arrow demarcates sudden change of order parameter.

Figure 4

Glass structure and density. (a) Pair correlation function of particles in the low- (blue dots) and high-shear bands (red squares). Inset: Angle-resolved $g(r)$ along the shear-shear gradient plane, extension ($\pi /4$, diamonds) and compression direction ($3\pi /4$, circles), for the low- (blue) and high-shear bands (red). Arrows indicate decrease of $g(r)$ in the extension direction. (b) Dilation parameter ${\alpha }_{\min }$ versus shear rate determined from the minimum of the mean-square difference of $g(r)$ (inset). Dashed lines are guides to the eye.