Yield-stress transition in suspensions of deformable droplets

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. S1. Overlap free energy before and after the yielding transition. Color maps of the free energy of overlaps f overlaps defined in the main text, for different values of body force f before ((a)-(b)) and after ((c)-(d)) the yielding transition . White contour lines correspond to f overlaps = 0.001. For low values of the bodyforce f ((a)-(b)) droplet contacts percolate between the two walls. Beyond the transition the pattern of droplet contacts no longer percolate in the flow gradient direction.

ADDITIONAL RESULTS
In the following paragraphs we provide some additional evidence to support the results presented in the main text.

Morphology and flow behaviour near the yield-stress transition
As mentioned in the main text (see Fig. 1 and related discussion) the yielding transition is accompanied by a morphological transition, from a non-flowing, disordered FIG. S2. Plug-like flow behaviour. Fluid velocity profile (as a function of the channel width z/L), for f = 4.0 × 10 −10 , beyond the yielding transition. The flow appears to be pluglike, with the velocity of the droplets (solid circles) being less than the overall fluid velocity, showing that the flow has a permeative component.
state to a flowing, ordered one. In the former, droplets overlaps create a percolating network whereas after yielding the pattern of droplet contacts no longer percolate along the flow gradient direction. This change can be appreciated by looking at the color maps of the free energy of overlaps (computed in steady state), for different values of the body-force f , which are displayed in Fig. S1. Moreover, in the flowing state the flow becomes plug-like (see Fig. S2).
To better understand the morphological differences between the droplets configurations before and after the yielding transition we computed the distribution of the area of the Voronoi cells calculated by starting from the droplet centres of mass. The results, for two configurations before (f = 0.5 × 10 −6 ) and after the transition (f = 5.5 × 10 −6 ), are displayed in Fig. S3(a) and (b), respectively. Although no defects are present in the Voronoi tessellation in either cases , the Voronoi cells sensibly deform in the non-flowing configuration (panel (a)) while they arrange in a more regular shape in the flowing state (panel (b)). This behaviour is quantified by the distribution of the area of the Voronoi cells in the two cases (panel(c)). Elastic energy (∼ K) measuring droplet deformation (dark blue curve) and overlap free energy (∼ ϵ) measuring droplet response to contacts (purple curve). (c) Major (dark blue) and minor (light blue) axis (dmax and dmin, respectively) computed as eigenvalues of the intertial tensor (see text) normalized with respect to the nominal droplet radius R. (d) Droplet deformation computed as the relative difference of the major and minor axis (dmax − dmin)/(dmax + dmin). The vertical line at iteration t = 4.13 × 10 6 marks the time at which a slipping events occurs.

Stick-slip motion
Close to the transition point f c the dynamics of the system is characterized by the stick-slip motion of the droplets, namely an intermittent regime where the droplets alternate stationary lapses -with the droplets sticking close to each others-and bursts of motion where the accumulated stress relaxes, leading the system of droplets to move and arrange in a new stationary configuration. These observations can be rationalized by looking at the time series of some dynamical observables shown in Fig. S4. Panel (a) shows the total throughput Q as a function of time. After the initial settling dynamics, the throughput drops orders of magnitude, signalling that the system only features permeative flows, typical of the solid-like phase. At time t = 4.13 × 10 6 , the layers of droplets in the bulk slip along those anchored to the boundary (see movie S4), resulting in a steep increase of the throughput flow. It is interesting to compare this behavior with the free energy evolution in panel (b). Along the preceding stationary lapse, both the elastic free energy F def (dark blue line) and the overlap free energy F overlap (purple line) are first constant and begin to sensibly decline as the slipping event is approached. This occurs as the result of small rearrangements of the droplets in the channel responsible for the release of the excess stress. As the droplets begin to move, both terms rapidly increase again, both because the moving droplets in the bulk tend to push on each other causing overlaps, and because the motion of the droplets is also accompanied by their deformation.
To quantify the droplet deformation we measure the major and minor axis of the droplets, d max and d min , respectively. These can be calculated, by computing the square root of the two eigenvalues of the tensor of inertia: droplet can be measured as the relative difference of the major and minor axis (d max − d min )/(d max + d min ). The behavior of both the droplet axis and deformation, for a droplet in the bulk, is shown in panels (c) and (d), respectively. It is worth noting that right before the slipping event the droplet deformation is first drastically reduced, then it increases again as the system starts to flow.

Single Droplet
As mentioned in the main text, more insights into the fundamental difference in behaviour between the conserved and non-conserved model can be gained by analysing in more detail the behaviour of a single droplet at a solid wall under an external forcing, and with neutral wetting boundary conditions ( Fig. S5(a)). In the conserved model the droplet sticks to the wall and requires a finite forcing to start moving, as showed by the plot of the y-component of the droplet velocity as a func-tion of the body-force f , in Fig. S5(b). In the nonconserved model, instead, evaporation and condensation provide also in this case another pathway for contact line motion, and the droplet drifts along the wall for any value of the forcing. All parameters are the same as in the multiphase simulations except for the droplet radius which is R = 8.5. and the simulation box which is of size L = 96.

Polydisperse suspensions
In the main text we explored the effect of polydispersity on the yielding transition by considering a bi-disperse suspension (where the droplet radius of one component is twice as large as that of the other). Analogously to what observed in the monodisperse case, for low values of the bodyforce, the suspended droplets are jammed and immobile (panels (a) and (c) of Fig. 4 of the main text). As the bodyforce is increased, a transition to a flowing state is observed (panels (b) and (c) of Fig. 4 of the main text). The transition is accompanied by a morphological rearrangement with the droplets of the two species separating in different region of the channel, with the large droplets in the centre of the channel, and the small ones close to the walls, effectively creating a lubricating layer. The time evolution of the system, after the yielding transition, is shown in Fig. S6.