Dynamics of polydisperse multiple emulsions in microfluidic channels

A. Tiribocchi, A. Montessori, M. Durve, F. Bonaccorso, M. Lauricella, and S. Succi

Phys. Rev. E 104, 065112 – Published 27 December 2021

Abstract

Multiple emulsions are a class of soft fluid in which small drops are immersed within a larger one and stabilized over long periods of time by a surfactant. We recently showed that, if a monodisperse multiple emulsion is subject to a pressure-driven flow, a wide variety of nonequilibrium steady states emerges at late times, whose dynamics relies on a complex interplay between hydrodynamic interactions and multibody collisions among internal drops. In this work, we use lattice Boltzmann simulations to study the dynamics of polydisperse double emulsions driven by a Poiseuille flow within a microfluidic channel. Our results show that their behavior is critically affected by multiple factors, such as initial position, polydispersity index, and area fraction occupied within the emulsion. While at low area fraction inner drops may exhibit either a periodic rotational motion (at low polydispersity) or arrange into nonmotile configurations (at high polydispersity) located far from each other, at larger values of area fraction they remain in tight contact and move unidirectionally. This decisively conditions their close-range dynamics, quantitatively assessed through a time-efficiency-like factor. Simulations also unveil the key role played by the capsule, whose shape changes can favor the formation of a selected number of nonequilibrium states in which both motile and nonmotile configurations are found.

Received 21 October 2021
Accepted 9 December 2021

DOI:https://doi.org/10.1103/PhysRevE.104.065112

Physics Subject Headings (PhySH)

Drop & bubble phenomena Microfluidics

Binary fluids Microbubbles

Lattice-Boltzmann methods Phase-field modeling

Fluid DynamicsPolymers & Soft Matter

Authors & Affiliations

A. Tiribocchi ^1,*, A. Montessori¹, M. Durve², F. Bonaccorso^1,2,3, M. Lauricella¹, and S. Succi^1,2,4

¹Istituto per le Applicazioni del Calcolo CNR, via dei Taurini 19, 00185 Rome, Italy
²Center for Life Nano Science@La Sapienza, Istituto Italiano di Tecnologia, 00161 Roma, Italy
³Department of Physics and INFN, University of Rome “Tor Vergata,” Via della Ricerca Scientifica, 00133 Rome, Italy
⁴Institute for Applied Computational Science, John A. Paulson School of Engineering and Applied Sciences, Harvard University, Cambridge, Massachusetts 02138, USA

^*Corresponding author: a.tiribocchi@iac.cnr.it; adrianotiribocchi @gmail.com

Article Text (Subscription Required)

Click to Expand

Supplemental Material (Subscription Required)

Click to Expand

References (Subscription Required)

Click to Expand

Issue

Vol. 104, Iss. 6 — December 2021

Reuse & Permissions

Access Options

Author publication services for translation and copyediting assistance advertisement

Images

Figure 1
Equilibrium configurations of three double emulsions with different sizes of the inner drops. (a), (b) $R_{1} = 10$ , $R_{2} = 20$ ( $h = 2$ , $A_{f} ≃ 0.14$ ); (c), (d) $R_{1} = 10$ , $R_{2} = 30$ ( $h = 3$ , $A_{f} ≃ 0.28$ ); and (e), (f) $R_{1} = 10$ , $R_{2} = 40$ ( $h = 4$ , $A_{f} ≃ 0.47$ ). Only a portion of the microchannel is shown. Symbol ${| i, j 〉}_{h = n}$ (where $i, j, n$ are integer numbers) indicates a state in which two cores (such as 1 and 2) of different size are positioned next to each other, while $h$ is the polydispersity index. Snapshots with equal values of $h$ [such as (a) and (b), (c) and (d), (e) and (f)] only differ for the initial position of the cores. The radius of the external droplet has been kept fixed to $R_{e} = 60$ . Colors represent the values of the order parameter $ϕ$ , ranging from 0 (black) to $≃ 2$ (yellow). This applies to all figures.
Reuse & Permissions
Figure 2
Nonequilibrium steady states observed in polydisperse two-core emulsions subject to a Poiseuille flow. Here $Re ≃ 2$ and $Ca ≃ 0.53$ . The left column represents states whose initial conditions shown in Figs. 1, 1, and 1 are of the form ${| 1, 2 〉}_{h = 2, 3, 4}$ , while the right column stems from initial configurations of the form ${| 2, 1 〉}_{h = 2, 3, 4}$ . Red arrows indicate the velocity field computed in the frame of reference of the external droplet. If $h = 2$ (a), (b) one can identify two regimes: (i) one in which the large core gets stuck at the leading edge of the emulsion while the small one is captured by the upper fluid vortex, and (ii) a further one in which the cores exhibit a periodic motion triggered by the fluid vortex. In both cases, the whole emulsion is dragged rightwards by the flow. White arrows indicate the direction of motion of the cores. If $h = 3$ (c), (d) and $h = 4$ (e), (f), internal drops attain a nonmotile configuration, either akin to that shown in (a) or consisting of two cores stuck at the front of the emulsion. Magenta arrows in (c) represent the direction of two counterclockwise vortices formed within the larger core.
Reuse & Permissions
Figure 3
Time evolution of the displacement $Δ y_{c m}$ (left column) and $Δ z_{c m}$ (right column) of the internal cores. The top panels refer to states of $| 1, 2 〉$ and $| 2, 1 〉$ with $h = 2$ , while the bottom ones concern those with $h = 3$ . If $h = 2$ , cores may either get transported by the fluid and move unidirectionally, or they may display a persistent periodic motion confined within a region of the emulsion and triggered by the fluid vortex. If $h = 3$ , the internal cores are generally dragged by the flow unidirectionally, either remaining sufficiently far from each other or accumulating at the front of the emulsion.
Reuse & Permissions
Figure 4
Time evolution of the distance $| Δ r_{c m} |$ between the centers of mass of the cores, starting their motion from states of the type $| 1, 2 〉$ (a)–(c) and $| 2, 1 〉$ (b)–(d) and for different values of $h$ . Horizontal lines indicate the distance $d = R_{1} + R_{2} + l$ below which cores interact, since opposite interfaces belonging to different drops sustain a temporary film of fluid. Assuming an interface width $ξ$ approximately equal to four to five lattice sites, one has $l = 2 ξ$ . Thus in (a) and (b) $d = 40$ ( $R_{1} = 10$ , $R_{2} = 20$ , $l = 10$ ) for $h = 2$ (red horizontal line), $d = 42$ ( $R_{1} = 12$ , $R_{2} = 20$ , $l = 10$ ) for $h = 1.67$ (green horizontal line), and $d = 48$ ( $R_{1} = 18$ , $R_{2} = 20$ , $l = 10$ ) for $h = 1.11$ (blue horizontal line). Note that here $R_{2}$ is fixed at 20 and $R_{1}$ increases. In (c) and (d) $d = 50$ ( $R_{1} = 10$ , $R_{2} = 30$ , $l = 10$ ) for $h = 3$ (red horizontal line), $d = 52$ ( $R_{1} = 12$ , $R_{2} = 30$ , $l = 10$ ) for $h = 2.5$ (green horizontal line), $d = 55$ ( $R_{1} = 15$ , $R_{2} = 30$ , $l = 10$ ) for $h = 2$ (blue horizontal line), and $d = 64$ ( $R_{1} = 24$ , $R_{2} = 30$ , $l = 10$ ) for $h = 1.25$ (purple horizontal line). Here $R_{2}$ is fixed at 30 and $R_{1}$ increases.
Reuse & Permissions
Figure 5
Here we plot the ratio $Λ = T_{int} / T_{tot}$ for different values of $h$ . The quantity $T_{int}$ represents the time (in simulation units) in which two cores are in close contact and sustain a thin film of fluid, an effect occurring when $| Δ r_{c m} | < d$ , while $T_{tot} = 10^{6}$ is the total simulation time. Note that in systems starting from a $| 1, 2 〉$ configuration [such as those in Figs. 1 in which drop 1 is smaller than drop 2], the ratio $T_{int} / T_{tot}$ is high for low values of $h$ (i.e., when drops have a comparable size and occupy a large portion of the emulsion) and diminishes as $h$ increases (i.e., the size of droplet 1 lessens while that of drop 2 remains fixed). On the contrary, systems starting from a $| 2, 1 〉$ configuration display an approximately constant and high value of $Λ$ .
Reuse & Permissions
Figure 6
The top row shows a selection of nonequilibrium steady states of a two-core emulsion with $h = 2$ (initially prepared in the state $| 1, 2 〉$ ) observed by varying the capillary number of the shell while keeping the Reynolds number fixed. At high values of Ca (a) the external droplet exhibits a highly squeezed projectilelike shape which essentially prevents the internal motion of the cores. Decreasing Ca (b), (c), the shell acquires a fingerlike configuration in which the cores separate and comove with the external droplet. Further diminishing Ca (d), (e), the shell takes on a rounded shape in which cores may either rotate periodically or migrate toward separate regions of the emulsion. The bottom row (f)–(j) shows the velocity field computed in the frame of reference of the shell.
Reuse & Permissions

Physical Review E

covering statistical, nonlinear, biological, and soft matter physics