Yield-stress transition in suspensions of deformable droplets

Abstract

Yield-stress materials, which require a sufficiently large forcing to flow, are currently ill-understood theoretically. To gain insight into their yielding transition, we study numerically the rheology of a suspension of deformable droplets in 2D. We show that the suspension displays yield-stress behavior, with droplets remaining motionless below a critical body-force. In this phase, droplets jam to form an amorphous structure, whereas they order in the flowing phase. Yielding is linked to a percolation transition in the contacts of droplet-droplet overlaps and requires strict conservation of the droplet area to exist. Close to the transition, we find strong oscillations in the droplet motion that resemble those found experimentally in confined colloidal glasses. We show that even when droplets are static, the underlying solvent moves by permeation so that the viscosity of the composite system is never truly infinite, and its value ceases to be a bulk material property of the system.

Fig. 1.. Yielding transition in the conserved model.

(A and B) Color map of ϕ = ∑_i ‍ ϕ_i for f < f_c [f = 2.0 × 10⁻⁶ in (A)] and f > f_c [f = 4.0 × 10⁻⁶ in (B)], for the conserved model. Black and red regions correspond to ϕ = 0 and ϕ = 2, respectively. (C) Free energy of overlaps (see text) as a function of body-force f. The insets of (C) show clusters of contacting droplets, resulting from a density-based spatial clustering analysis on the free energy of overlaps (see the Supplementary Materials). Different colors correspond to different clusters. Left and right inset correspond to the configuration shown in (A) and (B), respectively. Movies of the dynamics corresponding to (A) and (B) can be seen in movies S1 and S2, respectively (see the Supplementary Materials).

Fig. 2.. Flow behavior in the conserved and nonconserved models.

(A and B) Average droplet velocity (A) and throughput flow (B) for the conserved model. The inset of (A) shows the mean droplet speed close to criticality and the result of the fit (dashed line) with the function ⟨v⟩ ∝ (f − f_c)^β, with β ≃ 0.54. The inset of (B) shows the effective viscosity η_eff as a function of the body-force f. (C and D) Average droplet velocity (C) and throughput flow (D) for the nonconserved model. The inset of (C) shows the area of three nearby droplets versus time for f = 1.0 × 10⁻⁶.

Fig. 3.. Oscillations near the yielding transition.

(A) Throughput flow versus time, for the conserved model, for different values of f near f_c = 3.15 × 10⁻⁶. (B) Plot of the variance of the oscillations as a function of f.

Fig. 4.. Yielding of bidisperse suspensions.

(A and B) Color map of ϕ = ∑_i ‍ ϕ_i for f < f_c [f = 2.0 × 10⁻⁶ in (A)] and f > f_c [f = 6.0 × 10⁻⁶ in (B)]. (C) Average droplet velocity as a function of the body-force f.

Fig. 5.. Yielding phase diagram.

(A) Phase diagram as a function of body-force f and surface tension γ. Orange squares, flowing systems; purple circles, nonmoving states. (B) Phase diagram as a function of f and system size L plane. Red squares, flowing systems; blue circles, nonmoving states.