Formation of droplets in microfluidic cross-junctions at small capillary numbers: Breakdown of the classical squeezing regime

Two decades of research on droplet formation in microchannels have led to the widely accepted view that droplets form through the squeezing mechanism when interfacial forces dominate over viscous forces. The initially surprising finding that the volume of the droplets is insensitive to the relative importance of these two forces is nowadays well understood from the constrained deformation of the droplet interface during formation. In this work, we show a lower limit of the squeezing mechanism for droplets produced in microfluidic crossjunctions. Below this limit, in the leaking regime, which was recently discovered for droplets produced in Tjunctions, the volume of the produced droplets strongly depends on the relative importance of interfacial and viscous forces, as captured by the capillary number. We reveal a fundamental difference in the mechanisms at play in the leaking regime between T- and cross-junctions. In cross-junctions, the droplet neck elongates substantially, and unlike the case of the T-junction, the magnitude of this elongation depends strongly on the value of the capillary number. This elongation significantly affects the final droplet volume in a low capillary number regime. Generalizing the classical squeezing law by lifting the original assumptions and incorporating both identified mechanisms of leaking through gutters and neck elongation, we derive a model for droplet formation and show that it agrees with our experiments.