Accounting for corner flow unifies the understanding of droplet formation in microfluidic channels

While shear emulsification is a well understood industrial process, geometrical confinement in microfluidic systems introduces fascinating complexity, so far prohibiting complete understanding of droplet formation. The size of confined droplets is controlled by the ratio between shear and capillary forces when both are of the same order, in a regime known as jetting, while being surprisingly insensitive to this ratio when shear is orders of magnitude smaller than capillary forces, in a regime known as squeezing. Here, we reveal that further reduction of—already negligibly small—shear unexpectedly re-introduces the dependence of droplet size on shear/capillary-force ratio. For the first time we formally account for the flow around forming droplets, to predict and discover experimentally an additional regime—leaking. Our model predicts droplet size and characterizes the transitions from leaking into squeezing and from squeezing into jetting, unifying the description for confined droplet generation, and offering a practical guide for applications.


Supplementary Note 2: Full solution for
The full solution of Eq. 2 (from the main text) and the equivalent equation . The first can be seen as a classical RC time, while the second arises from the rate at which the DP is supplied. The first one is negligible compared to the second one in the leaking regime (with Ca ≪ 1), while both are of the same order in the squeezing regime (where Ca is of order 1). Importantly, for the conditions studied here, the influence of both time scales is negligible and N ( ) reaches a fixed value after a short initial transient. Before demonstrating this, we comment on the physical interpretation of the RC time. In classical RC circuits, the RC time corresponds to the time that is required for the relaxation of the system after a change in the potential applied over the circuit. Here, it can be interpreted as the relaxation of the interfacial surface to the equilibrium shape after a change in applied pressure.
In the leaking regime, where the ratio between the two times scales,

Ca
, is small, the shape of 4 the interface almost instantaneously adapts to the rise in pressure due to the growth of the droplet. Hence, neck's shapes are close to the equilibrium shape in the leaking regime (see Fig.   4a in the main article).
The weak time-dependence of N ( ) is easily seen from an analysis of the time-dependent term

Supplementary Note 3: Experimental verification of time-independent
To experimentally verify the outcome of the above analysis, we estimated the instantaneous flow rate to the neck, N , by measuring the speed of the tip of forming droplets 1,2 using a highspeed camera. We here use the notion that the position of the tip, F , propagates at a rate equal to the sum of the flow rates of the neck and the DP F = N + D Eq. (2) or in dimensionless quantities where we normalized the position using F = F / and the time using * = C / 2 . We    12

Supplementary Note 6: Leaking regime for different channel aspect ratios
The leaking mechanism is a feature of the formation of droplets in channels with non-circular cross section. We hence expect this mechanism to also feature in rectangular channels with different aspect ratios. As the aspect ratio imposes the curvature of the interfaces during droplet formation as well as the size of the gutters, the quantitative dependence of droplet size on Ca and is expected to depend on the aspect ratio. Experiments performed in channels with aspect ratios / = 0.5 and / = 1 confirm that the general features are the same, while the quantitative features depend on aspect ratio, see Supplementary Figure 5 and Supplementary   Table 4.
A closer look at the leaking regime shows a lower slope in the log-log plot for the channel with / = 0.5, suggesting that some details of the mechanism of leaking (such as the linear relation between the curvature difference and the volume collected behind the forming droplet) depend on the aspect ratio. A more extensive analysis of the effect of channel aspect ratio and cross-sectional shape is beyond the scope of the present paper and part of future work.