Abstract
The onset and growth of rigid clusters in a two-dimensional (2D) suspension
in shear flow are studied by numerical simulation. The suspension exhibits the
lubricated-to-frictional rheology transition but is studied at stresses above
the levels that cause extreme shear thickening. At large solid area fraction,
$\phi$, but below the jamming fraction, we find that there is critical $\phi_c$
beyond which the proportion of particles in rigid clusters grows sharply, as
$f_{\rm rig} \sim (\phi-\phi_c)^{\beta}$ with $\beta=1/8$, and at which the
fluctuations in the net rigidity grow sharply, with a susceptibility measure
$\chi_{\rm rig} \sim |\phi-\phi_c|^{-\gamma}$ with $\gamma = 7/4$. By applying
finite size scaling, the correlation length, arising from the correlation of
rigid domains, is found to scale as $\xi \sim |\phi-\phi_c|^{-\nu}$ with $\nu =
1$. The system is thus found to exhibit criticality, with critical exponents
consistent with the 2D Ising transition. This behavior occurs over a range of
stresses, with $\phi_c$ increasing as the stress decreases, consistent with the
known increase in jamming fraction with reduction of stress for
shear-thickening suspensions.