Wet to dry self-transitions in dense emulsions: From order to disorder and back

One of the most distinctive hallmarks of many-body systems far from equilibrium is the spontaneous emergence of order under conditions where disorder would be plausibly expected. Here, we report on the self-transition between ordered and disordered emulsions in divergent microfluidic channels, i.e., from monodisperse assemblies to heterogeneous polydisperse foamlike structures, and back again to ordered ones. The transition is driven by the nonlinear competition between viscous dissipation and surface tension forces as controlled by the device geometry, particularly the aperture angle of the divergent microfluidic channel. An unexpected route back to order is observed in the regime of large opening angles, where a trend towards increasing disorder would be intuitively expected.

Figure 1

Droplet assemblies within a microfluidic channel with a divergent opening angle $\alpha ={45}^{\circ }$ for two different inlet channel Capillary numbers (a) $\text{Ca}=0.04$, (c) $\text{Ca}=0.16$. (b) Clearly shows the ordered, hexagonal packing typical of wet-state emulsions, while (d) the foamlike structure which results in a neat distortion of the Delauney triangulation (blue solid lines connecting the centers of neighboring droplets) and the associated Voronoi tesselation (dotted polygons enclosing the droplets) as well. The red lines are isocontour lines drawn for ${\rho }_{\min }^N<{\rho }^N<{\rho }_{\max }^N$ being ${\rho }^N$ the local phase field. The lines are superimposed to a density field. The thickness of the red isocontour line has been widening in order to better visualize the droplet contours.

Figure 2

(a)–(d) ($\text{Ca}\sim 0.04$) Push-and-slide mechanism of the outcoming droplet. The yellow-triangle droplet comes out of the channel, pushes the orange-dotted one which, in turn, takes the place of its nearest-neighbor droplet (the red-star one). (e)–(h) ($\text{Ca}\sim 0.16$) Droplet pinch-off process. The dotted-orange droplet undergoes a transversal stretching due to the squeezing between the outcoming droplet and the red-star drop. The stretched droplet finally reaches a critical elongation and thinning under the confinement of the neighbor drops before pinch-off. (i)–(n) Experimental sequence of the breakup mechanism at $\text{Ca}\sim 0.08$ (see [38]). The experimental and numerical critical Capillary numbers above which droplets pinch off can be observed are $\text{Ca}\ge \sim 0.04$ and $\text{Ca}\ge \sim 0.05$, respectively.

Figure 3

Equivalent droplet diameter distributions for each pair of Capillary number and opening angle of the divergent channel $[\text{Ca}=0.04$ (dashed line), $\text{Ca}=0.1$ (dotted line), and $\text{Ca}=0.16$ (full line)]. The insets report snapshots of the droplet fields for different values of the Capillary number (Ca increases from top to bottom). The equivalent droplet diameter is the diameter of the circular droplet with the same area of the deformed droplet and can be computed as $D_e=\sqrt{4(A_d/\pi )}$ being $A_d$ the area of the droplet.

Figure 4

(a) Equivalent droplet diameter distributions for two couples of Capillary numbers and opening angles of the divergent channel, namely, $\text{Ca}=0.04, \text{Ca}=0.16$ and $\alpha ={45}^{\circ }, \alpha ={60}^{\circ }, \alpha ={90}^{\circ }$. (b) Droplets' field within the main channel for the case $\text{Ca}=0.16$ and $\alpha ={90}^{\circ }$.

Figure 5

(a)–(g) Droplets' field at the outlet of the injection nozzle for $\alpha ={90}^{\circ }$ and $\text{Ca}=0.16$ with the normalized vector field superimposed. Even at high Capillary numbers, the sharp deceleration, clearly evidenced by the velocity profiles reported in (h) taken in two distinct sections, within the nozzle (open circles) and at a downstream section, favors the transversal displacement of the outcoming droplets, which preferentially slide on the neighboring droplets rather than squeezing them. The velocity field is scaled with the maximum flow velocity at the inlet channel. The two sections, at which the velocity profiles are evaluated are at $x=190lu$ (inside the injection channel) and $x=340lu$ (within the main channel). The arrow within the plot indicates the average velocity drop between the inlet and the main channel. The axis of the plot is $u/u_M$ (normalized magnitude of the velocity) versus $y$ (crossflow coordinate).

Figure 6

Nondimensional dispersity ($\tilde{\delta }$) as a function of the opening angle $\tilde{\alpha }$ for different values of Ca. Each dispersity set, $\delta (\text{Ca},\tilde{\alpha })$, follows a Gaussian trend, with mean and variance depending on the Capillary number. Fitting function: $\tilde{\delta }\left(\tilde{\alpha }\right)=e^{-{\left[(\tilde{\alpha }-{\alpha }_M)/\left(2\text{Ca}\right)\right]}^2}$.

Figure 7

Droplet generation via the internal periodic boundary conditions. Populations of the dispersed and the continuous phase are copied from section (B) back to section (A) and, at the same time, a standard streaming and collision process occurs within the bulk domain.