Breaking one into three: surface-tension-driven droplet breakup in T-junctions

TL;DR

This work addresses how to control droplet breakup in microfluidic T-junctions to achieve three-daughter droplets via a surface-tension-driven mechanism. It combines geometry-guided experiments and gutter-flow modelling with COMSOL simulations to reveal a lateral breakup regime that arises from capillary pressure imbalances across a confined cross-section with $h>w_i>w_o$. A confinement parameter $eta$ predicts lateral breakup onset, while a two-step low-$Ca$ pocket-inflation model accounts for pocket growth and pinch-off; at higher $Ca$, a conventional central breakup can re-emerge, with the transition scaling near $(Ca_o ar{L}_o)^* oughly 1.9$ and depending on geometry. The findings provide a practical design rule to tailor droplet size and composition on demand, expanding the toolkit for droplet-based microfluidics and enabling new functions in microreactors and analytical platforms.

Abstract

Droplet breakup is an important phenomenon in the field of microfluidics to generate daughter droplets. In this work, a novel breakup regime in the widely studied T-junction geometry is reported, where the pinch-off occurs laterally in the two outlet channels, leading to the formation of three daughter droplets, rather than at the center of the junction for conventional T-junctions which leads to two daughter droplets. It is demonstrated that this new mechanism is driven by surface tension, and a design rule for the T-junction geometry is proposed. A model for low values of the capillary number $Ca$ is developed to predict the formation and growth of an underlying carrier fluid pocket that accounts for this lateral breakup mechanism. At higher values of $Ca$, the conventional regime of central breakup becomes dominant again. The competition between the new and the conventional regime is explored. Altogether, this novel droplet formation method at T-junction provides the functionality of alternating droplet size and composition, which can be important for the design of new microfluidic tools.

Figures (6)

• Figure 1: (a) Geometry of the T-junctions used throughout our study: both aspect ratio $h/w_o$ and width ratio $w_i/w_o$ are larger than unity. The eye shows the observation perspective during experiment.(b) Time sequence of a lateral breakup process for a short (blue) and a long (red) droplet. Inset shows the interface at the moment of rupture (red arrow), captured at a frame rate of 50,000 fps.
• Figure 2: Capillary instability responsible for lateral breakup (a)Presentation of key droplet parameters with a top view of a T-junction through which a droplet splits. (b) Cross sectional view in the outlet channel (indicated by black arrowheads in (a)) showing three different gutter radius. These three conditions can hold for the same droplet passing the junction at different times. When $R_g >R_g^*$, the necking starts. (c) Colormap representing $\beta$ in a diagram representing the aspect ratio $h/w_o$ as a function of the width ratio $w_i/w_o$. Blue region corresponds to $\beta<1$, i.e. geometries that does not allow the occurrence of lateral break-up; Green regions correspond to $\beta>1$, i.e. geometries prone to exhibit lateral breakup. Each marker corresponds to one of the geometries experimentally tested (see Table.\ref{['tab:geo']}). A black symbol correspond to an absence of lateral breakup observed, and a red marker corresponds to a presence of a lateral breakup.
• Figure 3: (a) Modeling of the pocket development for one arm of the T junction, with the eye showing the observation perspective during experiment. During the inflation of the lateral pocket, the droplet can be divided into three regions with a length of $L_i(t)$,$L_p(t)$ and $L_o(t)$. The coordinate $\eta$ starts from the junction and is parallel to the outlet channel. In the parts of length $L_i(t)$ and $L_o(t)$, gutters are maintained, and the gutter flows are represented by grey arrows. (b) Relative rear cap velocity $v^{'}_{r} /v_{r}$ (see definition in the inset) once the droplet has entered the junction versus the droplet capillary number $Ca$ before having penetrated the junction. Inset shows the kymograph from which droplet cap trajectory is measured (yellow line), whose slope before and after the vertical dotted line gives $v_r$ and $v^{'}_{r}$ respectively. (c) Result of step 1: schematic plot of equation (\ref{['eq:gutter']}) using junction coordinate S. Each blue curve represents the cubic power of the gutter radius along the droplet, from $F_i$ at the rear interface in the inlet to $F_o$ at the front interface in the outlet. Three such gutter radius profiles are shown corresponding to three time points, i.e., three droplet locations, as the droplet is advancing inside the channel. At t1: the droplet is about to enter the junction; t2: part of the droplet passes the junction, but the gutter radius inside the outlet channel is below $R_g^*=w_o/2$; t3: the droplet further advances, and $R_g=R_g^*$ is met at the red square; from this stage on, further advancing will cause $R_g>R_g^*$ inside the outlet channel, onset of necking, a pocket forms and inflates.(d) Result of step 2: temporal evolution of the pocket predicted by the model (for geometry $w_o/h=14/62$, $w_i/h=30/62$, with initial droplet length of $L/w_o = 8$ and capillary number of $Ca = 0.005$). The vertical axis corresponds to the $\eta$ coordinate (non-dimensionalized with $w_i$), and the length of $L_p(t)$ and $L_o(t)$ are drawn in color and grey respectively as a function of non-dimensional time (horizontal axis); The colormap represents the normalized local pocket depth $z(s,t)/h$ with the simulation stopped when $z(s,t)/h$ goes below $(w_o/h)/2 \approx 0.11$.
• Figure 4: Breakup moment of six breakup events with increasing $Ca$. The $Ca$ value from top to bottom is 0.016, 0.025, 0.056, 0.088, 0.129 and 0.177. The breakup location in the right outlet channel is shown with black arrowheads. The scale bar represents 30 $\mu$m.
• Figure 5: (a) Time sequences of example breakup events for the same droplet length under three flow conditions. The breakup regime shifts from lateral to central breakup from low to high $Ca$. The scale bar represents 30 $\mu$m. (b) Kymograph of light intensity along the central part of the outlet channel (white dashed line), for two droplets of the same size but with different regimes; The grey value is turned into colors in the colormap. At time zero, both droplets enter the junction, the trajectory of the front interface forms a straight line. After the rear interface arrives at the junction (inset), the change of intensity due to the light scattering from the interface of the pocket is captured (white arrowheads), which always starts from the junction as predicted. At the end of each process, new interface(s) is formed at the locations indicated by the black arrows. (c) The breakup transition regime map of $Ca_o$ versus $\bar{L}_{o}$ for geometry A (Figure \ref{['fgr:static']}c), where $Ca_{o}=Ca(w_{i}/w_{o})/2$ and $\bar{L}_{o} = L(0)(w_{i}/w_{o})/w_{o}/2$. Blue and red circles represent lateral and central breakups respectively. The area of each circle is proportional to the characteristic breakup time, defined from when droplet rear interface reaches the junction to the final breakup moment. The black curve represents the function $Ca_{o} = b \bar{L}_{o}^{-1}$, where $b=1.9$ in this case (geometry A). Inset: $d_p$, the pinch off lateral distance from the junction center (in um) versus the product of $Ca_{o}\bar{L}_{o}$. (d) Logarithmic representation of the breakup transition regime map for $Ca_o$ vs. $\bar{L}_{o}$ for four geometries (A, B, D, E) with different $\beta$ values, represented by four colors. The filled and hollow markers represent lateral (LB) and central breakups (CB) respectively. Dashed lines of slope $-1$ are represented to guide the eyes.