Herschel-Bulkley Shearing Rheology Near the Athermal Jamming Transition

We consider the rheology of soft-core frictionless disks in two dimensions in the neighborhood of the athermal jamming transition. From numerical simulations of bidisperse, overdamped particles, we argue that the divergence of the viscosity below jamming is characteristic of the hard-core limit, independent of the particular soft-core interaction. We develop a mapping from soft-core to hard-core particles that recovers all the critical behavior found in earlier scaling analyses. Using this mapping we derive a relation that gives the exponent of the nonlinear Herschel-Bulkley rheology above jamming in terms of the exponent of the diverging viscosity below jamming.

Figure 1

(a) ${\eta }_p\equiv p/\dot{\gamma }$ vs $\varphi$ for data below ${\varphi }_J$ in the linear rheology region. Open symbols are for the harmonic interaction, solid symbols are for Hertzian. (b) Same data as in (a) but plotted vs ${\varphi }_J-\varphi$ with ${\varphi }_J=0.8433$.

Figure 2

(a) ${\eta }_p\equiv p/\dot{\gamma }$ vs $\varphi$ for data above and below ${\varphi }_J$ for harmonic soft-core particles. (b) Same data as in (a) but plotted vs ${\varphi }_J-{\varphi }_{\mathrm{eff}}$. Dashed line in (a) and solid line in (b) is the fit to the model of Eqs. (2, 6). Symbols in (b) correspond to the legend given in (a).

Figure 3

(a) ${\eta }_p\equiv p/\dot{\gamma }$ vs $\varphi$ for data above and below ${\varphi }_J$ for Hertzian soft-core particles. (b) Same data as in (a) but plotted vs ${\varphi }_J-{\varphi }_{\mathrm{eff}}$. Dashed line in (a) and solid line in (b) is the fit to the model of Eqs. (2, 6), using the same values of $A$, ${\varphi }_J$, and $\beta$ as found for the harmonic interaction model. Symbols in (b) correspond to the legend given in (a).

Figure 4

(a) Scaling collapse of energy as in Eq. (8), using values ${\varphi }_J$, $y_E$, and $z\nu =\beta +y_p$ obtained from our fit to the ${\varphi }_{\mathrm{eff}}$ model. Two branches of the scaling function correspond to $\varphi$ above and below ${\varphi }_J$. (b) Plot of $E/|\delta \varphi {|}^{y_E}-c_E$ vs $x\equiv \dot{\gamma }/|\delta \varphi {|}^{z\nu }$ for $\varphi >{\varphi }_J$. Dashed line is a fit to the large $x$ data, giving a power-law behavior with exponent $q_E\approx 0.55$; solid line is a fit to the small $x$ data, giving a power-law behavior with exponent $b\approx 0.37$. Horizontal short dashed line is the value $c_E$; data below this line satisfy $(E-E_0)/E_0<1$.