Theoretical study on sequential splitting of droplets flowing through fractal tree-shaped microchannel networks

A theoretical model for describing the sequential splitting of droplets flowing through the fractal tree-shaped microchannel network with arbitrary branch level is developed to explore the mechanisms underlying the hydraulic imbalance on the high-throughput droplets production. Accordingly, detailed droplet splitting characteristics are presented, including the droplet velocities in branches, droplet distribution coefficient, and monodispersity of droplets production. It is found that the uniformity of droplet mainly depends on two key dimensionless parameters, namely Λ₁ and Λ₂, where Λ₁ is determined by the initial working condition involving the initial lengths and viscosity of the continuous and discrete phases, the width of the 0th level channel, and the initial capillary number, and Λ₂ is concerned with the outlet pressure p_{out, [2}^n-1_] and the pressure drops of the continuous and discrete phases at the 0th level channel. The monodispersity of droplets goes down with the decreasing Λ₁ and the increasing Λ₂. Based on the Λ₁ and Λ₂, the optimal design strategies contributing to enhancing the droplet generation uniformity are recommended, including increasing Ca, enlarging the length of continuous phase flow at the main channel, and increasing the lengths of each level channels. Furthermore, the tree-shaped microchannel networks with higher branch level (n) have stronger ability to resist asymmetric disturbances. In particular, in our work, the width fractal dimension Δ = 2 is adopted as a proper key structure parameter, so as to ensure sequential breakup of droplet at each T-junction under symmetric condition.