  Double emulsion formation in a hierarchical flow-focusing channel is
systematically investigated using a free energy ternary lattice Boltzmann
model. A three dimensional formation regime diagram is constructed based on the
capillary numbers of the inner ($Ca_i$), middle ($Ca_m$) and outer ($Ca_o$)
phase fluids. The results show that the formation diagram can be classified
into periodic two-step region, periodic one-step region, and non-periodic
region. By varying $Ca_i$ and $Ca_m$ in the two-step formation region,
different morphologies are obtained, including the regular double emulsions,
decussate regimes with one or two alternate empty droplets, and structures with
multiple inner droplets contained in the continuous middle phase thread.
Bidisperse behaviors are also frequently encountered in the two-step formation
region. In the periodic one-step formation region, scaling laws are proposed
for the double emulsion size and for the size ratio between the inner droplet
and the overall double emulsion. Furthermore, we show that the interfacial
tension ratio can greatly change the morphologies of the obtained emulsion
droplets, and the channel geometry plays an important role in changing the
formation regimes and the double emulsion sizes. In particular, narrowing the
side inlets or the distance between the two side inlets promotes the conversion
from the two-step formation regime to the one-step formation regime.
