Breaking Anchored Droplets in a Microfluidic Hele-Shaw Cell

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Breaking Anchored Droplets in a Microfluidic Hele-Shaw Cell

Gabriel Amselem, P. T. Brun, François Gallaire, and Charles N. Baroud

Phys. Rev. Applied 3, 054006 – Published 12 May 2015

Abstract

We study microfluidic self-digitization in Hele-Shaw cells using pancake droplets anchored to surface-tension traps. We show that above a critical flow rate, large anchored droplets break up to form two daughter droplets, one of which remains in the anchor. Below the critical flow velocity for breakup, the shape of the anchored drop is given by an elastica equation that depends on the capillary number of the outer fluid. As the velocity crosses the critical value, the equation stops admitting a solution that satisfies the boundary conditions; the drop breaks up in spite of the neck still having finite width. A similar breaking event also takes place between the holes of an array of anchors, which we use to produce a 2D array of stationary drops in situ.

Received 1 December 2014

DOI:https://doi.org/10.1103/PhysRevApplied.3.054006

Authors & Affiliations

Gabriel Amselem¹, P. T. Brun^2,*, François Gallaire², and Charles N. Baroud^1,†

¹LadHyX and Department of Mechanics, Ecole Polytechnique, CNRS, 91128 Palaiseau Cedex, France
²LFMI, EPFL, Lausanne, Switzerland

^*Present address: Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA.
^†baroud@ladhyx.polytechnique.fr

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Issue

Vol. 3, Iss. 5 — May 2015

Subject Areas

Fluid Dynamics

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Images

Figure 1
(a) Schematic of an anchor of diameter $d$ and height $e$ , trapping a drop of radius $R$ (no external flow); the chamber height is $h$ . (b) Sketch of the microfluidic device: droplets are produced with a flow focuser and trapped on an anchor centered in the observation chamber, $w = 2.5 mm$ . (c) Typical evolution of the droplet neck width (see inset) as the outer flow velocity $U$ is increased. When the critical velocity is passed, the neck never reaches a stationary state; instead, it decreases until it matches the channel height ( $h = 49 μ m$ ), where it becomes unstable by Rayleigh-Plateau instability. (d) For a given microchannel, a trapped droplet either escapes from its anchor at a critical value of the capillary number (grayed area, $R ≲ 600 μ m$ ) or breaks on the anchor ( $R ≳ 600 μ m$ ), leaving a smaller droplet behind. Experimental points for droplets of FC-40 in glycerol-water mixtures of different viscosities: $μ = 0.93 cP$ (dark gray), $μ = 1.6 cP$ (medium gray), and $μ = 3.3 cP$ (black). SDS is used as a surfactant at a concentration of 2% in all cases.
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Figure 2
(a) Evolution of the fitting parameter $ℓ_{f}$ with the value of $Ca / h^{2}$ , for different channel heights, trap diameters, and droplet fluid viscosities. The cloud of data points confirms that $ℓ^{- 2} \sim Ca / h^{2}$ . (b) Photograph of an anchored droplet under flow and system of coordinates in which the interface $r (s)$ is derived. $s$ is the arclength along the interface, and $r^{'} (s)$ is its tangent. (c) From left to right: Droplet shape for increasing outer fluid velocities. Orange dotted lines are the best-fitting elastica shape. (d) Three droplets of different viscosities ( $μ_{1} = 1.2, μ_{2} = 4.1, μ_{3} = 24 cP$ ) but similar volumes, for the same value of $μ U / γ h^{2}$ : the droplet shape is independent of the droplet viscosity.
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Figure 3
(a) Branches of solutions of the elastica shown in the plane $(1 / \bar{ℓ^{2}}, {\bar{κ}}_{0})$ , each corresponding to a given value of $\bar{d}$ . The stable part (unstable) is denoted with a black line (gray line). For a given $\bar{d}$ , the critical value $1 / \bar{ℓ^{⋆ 2}}$ at which the drop breaks is indicated by a star corresponding to the turning point of the branch of solutions. Yellow line shows the interpolation of a discrete number of turning points. Inset: Shape of the anchored drop at the critical value of $1 / \bar{ℓ^{⋆ 2}}$ . (b) Comparison between experimental values of $1 / \bar{ℓ^{⋆ 2}}$ at which the drops break with the theoretical prediction from part (a).
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Figure 4
(a) Remaining area (dimensionless) as a function of $\bar{d} = d / R$ . Dots: experiments. Line: theoretical prediction. Insets: Two droplets of different sizes left on the same anchor ( $d = 150 μ m$ ) by two different drops. (b) An array made up of 1568 individual drops produced through self-digitization. The aqueous solution contains fluorescein to aid in the visualization. We use square anchors with side $d = 130 μ m$ in a channel of height $h = 35 μ m$ . Scale bar: 2 mm.
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