Percolation in suspensions and de Gennes conjectures

Dense suspensions display complex flow properties, intermediate between solid and liquid. When sheared, a suspension self-organizes and forms particle clusters that are likely to percolate, possibly leading to significant changes in the overall behavior. Some theoretical conjectures on percolation in suspensions were proposed by de Gennes some 35 years ago. Although still used, they have not received any validations so far. In this Rapid Communication, we use three-dimensional detailed numerical simulations to understand the formation of percolation clusters and assess de Gennes conjectures. We found that sheared noncolloidal suspensions do show percolation clusters occurring at a critical volume fraction in the range 0.3–0.4 depending on the system size. Percolation clusters are roughly linear, extremely transient, and involve a limited number of particles. We have computed critical exponents and found that clusters can be described reasonably well by standard isotropic percolation theory. The only disagreement with de Gennes concerns the role of percolation clusters on rheology which is found to be weak. Our results eventually validate de Gennes conjectures and demonstrate the relevance of percolation concepts in suspension physics.

Figure 1

Spanning probability $\langle \mathrm{\Pi }\rangle$ (a) and spanning strength $\langle P\rangle$ (b) as a function of volume fraction $\varphi$ for different system sizes $L$.

Figure 2

A spanning cluster in a sheared bounded suspension ($\varphi =0.32$; $L=30a$). Only particles belonging to spanning cluster are represented.

Figure 3

Average ($•$) and maximum ($\circ$) coordination numbers in spanning clusters as a function of volume fraction ($L=20a$).

Figure 4

Characteristic spanning times $\langle {\tau }_s\rangle (•)$ and waiting times $\langle {\tau }_w\rangle (\circ )$ as a function of volume fraction ($L=20a$; ${\varphi }_{*}=0.29$). Times are rescaled by the inverse shear rate ${\dot{\gamma }}^{-1}$.

Figure 5

Rescaled mean cluster size $\langle S\rangle L^{-\gamma /\nu }$ (a) and rescaled strength $\langle P\rangle L^{\beta /\nu }$ (b) as a function of $(\varphi -{\varphi }_{\text{cr}})L^{1/\nu }$. The best data collapse is obtained for ${\varphi }_{\mathrm{cr}}=0.415$, $\gamma =1.91$, $\nu =1.02$, and $\beta =0.18$. Scales are arbitrary since they depend on $L$ (here, $L$ is dimensional with $a=0.1$).

Figure 6

Suspension relative viscosity ${\eta }^r$ as a function of strain ($\varphi =0.34$, $L=20a$). A gray circle marks the presence of at least one spanning cluster.