Microscopic Theory for the Rheology of Jammed Soft Suspensions

We develop a constitutive model allowing for the description of the rheology of two-dimensional soft dense suspensions above jamming. Starting from a statistical description of the particle dynamics, we derive, using a set of approximations, a nonlinear tensorial evolution equation linking the deviatoric part of the stress tensor to the strain-rate and vorticity tensors. The coefficients appearing in this equation can be expressed in terms of the packing fraction and of particle-level parameters. This constitutive equation rooted in the microscopic dynamic qualitatively reproduces a number of salient features of the rheology of jammed soft suspensions, including the presence of yield stresses for the shear component of the stress and for the normal stress difference. More complex protocols like the relaxation after a preshear are also considered, showing a smaller stress after relaxation for a stronger preshear.

Figure 1

Particle shear stress $\sigma$ and normal stress difference $N_1$ in explicit dimensionless forms, for different protocols. (a) Stationary flow curves for different packing fractions $\varphi ={\varphi }_J+\mathrm{\Delta }\varphi$ with $\mathrm{\Delta }\varphi =0.01$, 0.02, 0.03, 0.04 (bottom to top). Inset: pressure $p(\varphi )$. (b) Transient response at a constant shear rate $\dot{\gamma }$ after a preshear at low shear rate ${\dot{\gamma }}_{\mathrm{ps}}={10}^{-5}$, showing an overshoot on the shear stress $\sigma$ for $\dot{\gamma }{\tau }_0=2\times {10}^{-3}$, $5\times {10}^{-3}$, ${10}^{-2}$ at $\varphi ={\varphi }_J+0.01$. (c) Schematic representation in the plane $(N_1/2,\sigma )$ of the transient response shown in panel (b), highlighting the fast dynamics of $S$ and the slow dynamics of $\theta$. Particle configurations corresponding to the colored dots indexed by ${\gamma }_i$ are schematically drawn with the same color code. (d) Stress relaxation at $\dot{\gamma }=0$ after a preshear at shear rate ${\dot{\gamma }}_{\mathrm{ps}}{\tau }_0=2\times {10}^{-3}$, $5\times {10}^{-3}$, ${10}^{-2}$, showing that a stronger preshear eventually leads to a lower value of the shear stress. (e) Schematic representation in the plane $(N_1/2,\sigma )$ of the relaxation shown in panel (d), which occurs at constant $\theta$. Data are obtained by numerical integration of Eq. (14), using a linear force $f(r)=f_0(r-2a)/a$ (for $r<2a$) to evaluate the coefficients.