The vortex-driven dynamics of droplets within droplets

Understanding the fluid-structure interaction is crucial for an optimal design and manufacturing of soft mesoscale materials. Multi-core emulsions are a class of soft fluids assembled from cluster configurations of deformable oil-water double droplets (cores), often employed as building-blocks for the realisation of devices of interest in bio-technology, such as drug-delivery, tissue engineering and regenerative medicine. Here, we study the physics of multi-core emulsions flowing in microfluidic channels and report numerical evidence of a surprisingly rich variety of driven non-equilibrium states (NES), whose formation is caused by a dipolar fluid vortex triggered by the sheared structure of the flow carrier within the microchannel. The observed dynamic regimes range from long-lived NES at low core-area fraction, characterised by a planetary-like motion of the internal drops, to short-lived ones at high core-area fraction, in which a pre-chaotic motion results from multi-body collisions of inner drops, as combined with self-consistent hydrodynamic interactions. The onset of pre-chaotic behavior is marked by transitions of the cores from one vortex to another, a process that we interpret as manifestations of the system to maximize its entropy by filling voids, as they arise dynamically within the capsule.

Here we provide further details about the numerical method and the simulation parameters.
The equations for the order parameter φ i (Equation 2 of the main text) and the Navier-Stokes equation (Equation 3 of the main text) are solved by using a hybrid lattice Boltzmann (LB) method [1], in which Equation 2 is integrated via a finite-difference predictor-corrector algorithm and Equation 3 via a standard LB approach.
As reported in the main text, simulations are peformed on a rectangular lattice with size ratio Γ = Lz Ly ranging from 0.16 to 0.22. More specifically, Γ = 0.167 (L y = 600, L z = 100) for the core-free droplet, Γ = 0.2 (L y = 600, L z = 120) for the single-core emulsion and Γ = 0.21 (L y = 800, L z = 170) for the two-core and higher complex emulsions. Periodic boundary conditions are set along the y-axis and two flat walls along the zaxis, placed at z = 0 and z = L z . Here no-slip conditions hold for the velocity field (i.e. v z (z = 0, z = L z ) = 0) and neutral wetting for the order parameters φ i . The latter ones are achieved by setting The first one guarantess density conservation (no mass flux through the walls) while the second one imposes the wetting to be neutral. Like in previous works [2,3], the pressure gradient ∆p producing the Poiseulle flow is modeled through a body force (force per unit density) added to the collision operator of the LB equation at each lattice node.
Thermodynamic parameters have been chosen as follows: a = 0.07, k = 0.1, M = 0.1 and = 0.05 (a value larger than 0.005 is enough to prevent droplet merging). These values fix the surface tension and the interface width to σ = 8ak 9 0.08 and ξ = 2 2k respectively. Also, the dynamic viscosity η of both fluid components is set equal to 5/3. Such approximation, retained for simplicity, may be relaxed by letting η depends on φ [4,5]. Lattice spacing and integration timestep have been kept fixed to ∆x = 1 and ∆t = 1, while droplet radii are chosen as follows: R = 30 for the corefree droplet, R i = 15 and R O = 30 for a single-core emulsion, and R i = 17 and R O = 56 for emulsions containing more than one core. Here R i is the radius of the cores while R O is the one of the surrounding shell.

SUPPLEMENTARY NOTE 2: VELOCITY PROFILE UNDER POISEUILLE FLOW
In Supplementary Figure 1 we report, for example, the typical steady-state velocity profile observed in a twocore emulsion for different values of the pressure gradient. They are averaged over space and time, i.e. the channel length and approximately 3×10 5 time steps at the steady state. The curves are compatible with a parabolic profile expected in an isotropic fluid with the same viscosity, and remain essentially unaltered for the other multi-core emulsions considered in this work.
However, substantial modifications occur when instantaneous configurations are considered. In Supplementary  Figure 2 we show, for instance, the instantaneous velocity profile observed in core-free (a), one-core (b), two-core (c) and three-core (d) emulsions calculated along a cross section of the channel where internal cores temporarily accumulate. While in (a) the parabolic profile is only weakly disturbed by the droplet interface, in (b)-(d) it is significantly modified by local bumps and dips caused by internal cores. Such distortions wash out when these profiles are averaged over space and time.

SUPPLEMENTARY NOTE 3: STRUCTURE OF THE VELOCITY FIELD IN FOUR, FIVE AND SIX-CORE EMULSIONS
In Supplementary Figure 3  (b) five and (c) six cores. In all cases, the interior structure of the field exhibits significant deviations (heavier as the number of cores increases) from the double-vortex pattern of a core-free emulsion.
As discussed in the paper, under Poiseuille flow a fourcore emulsion, originally designed as in Fig.2d of the main text, only temporarily survives in a state of the form 1, 2, 3|4 (Supplementary Figure 3a), since the effective area fraction occupied by three cores is larger than 0.35 in half emulsion. This causes a crossing of a drop (3 in Supplementary Figure 3a) driven a heavy flux pushing it downwards, thus leading to the long-lived nonequilibrium state 1, 2|3, 4 (see Fig.3h of the main text). In this state, couples of drops display a planetary-like motion within each half of the emulsion and no further crossing occurs.
Increasing the number of inner drops, such as in five and six-core systems (Supplementary Figure 3b,c), favours multiple crossings between the two regions of the emulsions, a process generally driven by a pre-chaotic flows resulting from the complex coupling between velocity and phase field. This is why only short-lived states are observed in these systems.