Microscopic Theory of Two-Step Yielding in Attractive Colloids

Attractive colloids display two distinct amorphous solid phases: the attractive glass due to particle bonding and the repulsive glass due to the hard-core repulsion. By means of a microscopic mean field approach, we analyze their response to a quasistatic shear strain. We find that the presence of two distinct interaction length scales may result in a sharp two-step yielding process, which can be associated with a hysteretic stress response or with a reversible but nonmonotonic stress-strain curve. We derive a generic phase diagram characterized by two distinct yielding lines, an inverse yielding and a critical point separating the hysteretic and reversible regimes. Our results should be applicable to a large class of glassy materials characterized by two distinct interaction length scales.

Figure 1

Phase diagram in the $(\widehat{T},\widehat{\phi })$ plane at fixed range $\widehat{\sigma }=0.062$, in the absence of strain. The full blue (black) line is the dynamical transition at which the repulsive (attractive) glass appears. On the green line, the two glass states merge in a single glass. The glass-glass coexistence region is in shaded gray. The two green dots indicate higher-order singularities of the Franz-Parisi potential. The red dashed line indicates the value of $\widehat{T}=0.485$ studied in the following, and the red crosses indicate the three state points studied in Fig. 2.

Figure 2

Stress-strain curves in logarithmic scale for several densities $\widehat{\phi }$, at $\widehat{T}=0.485$ and $\widehat{\sigma }=0.062$. (a) Coexistence of the two glasses displaying distinct yielding points leading to a two-step yielding process for the attractive glass. (b) The coexistence only happens at $\gamma >0$, leading to a hysteresis loop at intermediate $\gamma$, as indicated by the arrows. (c) The hysteresis loop closes leading to a smooth and reversible but nonmonotonic stress-strain curve with a single yielding point.

Figure 3

Yielding phase diagram in the $(\gamma ,\widehat{\phi })$ plane obtained by following the evolution in $\widehat{\phi }$ of the yielding points defined in Fig. 2. The lines ${\gamma }_Y^a$ and ${\gamma }_Y^{\mathrm{inv}}$ delimitate the coexistence region and merge in a critical point. Outside the coexistence region, only one glass exists. The dynamical transitions, at which the ${\gamma }_Y$ vanish, are, respectively, located in ${\widehat{\phi }}_d^r=6.4697$, ${\widehat{\phi }}_d^a=6.5272$, ${\widehat{\phi }}_{\mathrm{sp}}=6.9093$. Vertical lines indicate the densities for which the stress-strain curve is reported in Fig. 2.