Microscopically grounded constitutive model for dense suspensions of soft particles below jamming

We derive from particle-level dynamics a constitutive model describing the rheology of two-dimensional dense soft suspensions below the jamming transition, in a regime where hydrodynamic interactions between particles are screened. Based on a statistical description of particle dynamics, we obtain through a set of physically plausible approximations a nonlinear tensorial evolution equation for the deviatoric part of the stress tensor, involving the strain rate and vorticity tensors. This tensorial evolution equation involves singular terms usually not taken into account in phenomenological constitutive models, which most often assume a regular expansion in terms of the stress tensor. All coefficients appearing in the equation have known expressions in terms of the microscopic parameters of the model. The predictions of this microscopically grounded constitutive model have several qualitative features that are specific to the rheology of soft suspensions measured in experiments or simulations. The model shows a typical behavior of polymeric viscoelastic materials, such as normal stress differences quadratic in the shear rate $\dot{\gamma }$, as well as typical behaviors of suspensions of stiff particles, such as a particle pressure linear in $\dot{\gamma }$ and a zero-shear viscosity diverging at the jamming transition. The model also predicts a sharper shear thinning than other viscoelastic models at small shear rates, in qualitative agreement with experimental observations. Furthermore the shear thinning follows a critical scaling close to the jamming transition.

Figure 1

Schematic representation of the deformation of the first-neighbor shell below the jamming density.

Figure 2

Flow curves in simple shear (a) and planar elongational flow (b). (a) Dimensionless viscosity ${\eta }_{\mathrm{s}}$ as a function of dimensionless shear rate $\dot{\gamma }$ in logarithmic scale for several volume fractions below the jamming volume fraction ${\varphi }_{\mathrm{J}}$. In inset, scaled flow curves $\mathrm{\Delta }\varphi {\eta }_{\mathrm{s}}$ as a function of scaled shear rate $\dot{\gamma }/\mathrm{\Delta }\varphi$, evidencing the critical scaling of the viscosity when approaching jamming. (b) Extensional viscosity ${\eta }_{\mathrm{e}}$ as a function of extension rate $\dot{\epsilon }$. Inset: Trouton ratio ${\eta }_{\mathrm{e}}\left(\dot{\epsilon }\right)/{\eta }_{\mathrm{s}}\left(\dot{\gamma }\right)$ as a function of the scaled deformation rate, for $\dot{\epsilon }=\dot{\gamma }$. Trouton ratio takes its Newtonian value in the small rate limit, and for larger rates is predicted alternatively sub-Newtonian and super-Newtonian.

Figure 3

Viscosity ${\eta }_{\mathrm{s}}$ in simple shear as a function of the volume fraction $\varphi$ for several values of $\dot{\gamma }$ (solid lines) increasing from dark to light colors. The viscosity decreases close to ${\varphi }_{\mathrm{J}}$ for large $\dot{\gamma }/\mathrm{\Delta }\varphi$ values beyond the domain of validity of the model (dotted lines). With dashed lines, we show the approximation with the effective volume fraction ${\eta }_{\mathrm{Hard}}\left({\varphi }_{\mathrm{eff}}\right)$, with ${\varphi }_{\mathrm{eff}}$ given in Eq. (37). In the inset, an alternative form of the critical scaling, Eq. (35), $\dot{\gamma }{\eta }_{\mathrm{s}}=g_{\eta }(\mathrm{\Delta }\varphi /\dot{\gamma })$, with $g_{\eta }\left(x\right)=f_{\eta }(1/x)/x$.

Figure 4

Dimensionless first normal stress difference $N_1$ as a function of dimensionless shear rate $\dot{\gamma }$ for several volume fractions below the jamming volume fraction ${\varphi }_{\mathrm{J}}$, under simple shear flow.

Figure 5

(a) Macroscopic friction coefficient $\mu ={\mathrm{\Sigma }}_{12}/p$ as a function of the viscous number $J=\dot{\gamma }/p$ in simple shear, for several values of the dimensionless shear rate (Weissenberg number) $\dot{\gamma }$, with solid lines. With dashed lines, predictions from Eq. (46). In the inset the same data plotted as a function of $J/\dot{\gamma }$, showing that all shear rates collapse on the same curve, which is predicted to be exact at the lowest order in $J$ and $p$ by Eq. (46) (prediction shown in dashed line). (b) Volume fraction as a function of $J$, for the same values of the dimensionless shear rate $\dot{\gamma }$. The effect of softness is not visible, as it scales as $\dot{\gamma }$, which largest value here is $\dot{\gamma }={10}^{-4}$.

Figure 6

(a) Load curve in simple shear, ${\mathrm{\Sigma }}_{12}$, as a function of strain $\gamma$, for $\mathrm{\Delta }\varphi =0.001$ and several values of shear rate $\dot{\gamma }$, increasing from bottom to top. (b) Load curves for several $\mathrm{\Delta }\varphi$ values (increasing from dark to light) and several $\dot{\gamma }$, bundled by groups sharing the same scaled shear rate values $\dot{\gamma }/\mathrm{\Delta }\varphi$, increasing from bottom to top. Load curves collapse on a master curve in the small shear rate limit, showing the critical scaling of the dynamics at work.

Figure 7

Shear reversal at $\mathrm{\Delta }\varphi =0.01$ for several values of shear rate $\dot{\gamma }$, increasing from dark to light. In the main panel, the viscosity ${\eta }_{\mathrm{s}}$ scaled by its steady-state value ${\eta }^{\mathrm{st}}$ as a function of strain after reversal. In inset, the scaled normal stress difference $N_1/N_1^{\mathrm{st}}$.