Modeling of droplet breakup in a microfluidic T-shaped junction with a phase-field model

A phase-field method is applied to the modeling of flow and breakup of droplets in a T-shaped junction in the hydrodynamic regime where capillary and viscous stresses dominate over inertial forces, which is characteristic of microfluidic devices. The transport equations are solved numerically in the three-dimensional geometry, and the dependence of the droplet breakup on the flow rates, surface tension and viscosities of the two components is investigated in detail. The model reproduces quite accurately the phase diagram observed in experiments performed with immiscible fluids. The critical capillary number for droplet breakup depends on the viscosity contrast, with a trend which is analogous to that observed for individual isolated droplets in hyperbolic flow.

Figure 1

Left, examples of simulations of two-phase flow dynamics in microfluidic devices: (a) droplet breakup in a symmetric T-junction; (b) and (c), droplet formation in a T-junction with cross flow; (d) droplet formation in a flow-focusing cross junction. Right, frames showing the modeled three-dimensional flow of a droplet in a symmetric T-junction with square cross section.

Figure 2

Visualization of spurious currents for a static droplet relaxed on three different cubic meshes: (a) $\mathrm{\Delta }x=C$, (b) $\mathrm{\Delta }x=2C∕3$, (c) $\mathrm{\Delta }x=C∕2$, $C=0.1$. The pressure, concentration, and velocity fields have reached the stationary state, and parasite currents persist in the proximity of the interface region indicated by the concentric circles. The spurious currents reflect the fourfold symmetry of the lattice, and depend on the size of the mesh. Measured in units of the reference velocity $\mathrm{Re}∕\mathrm{We}$, the maximum spurious velocity decreases from $4\times {10}^{-3}$ (a) to $2\times {10}^{-3}$ (b), and $3\times {10}^{-4}$ (c).

Figure 3

Variation of deformation (a) and tilt angle (b) as a function of capillary number for a droplet in simple shear. The straight lines show the $\mathrm{Ca}⪡1$ limits derived by Taylor [27] and Chaffey and Brenner [30] $(\lambda =1)$. The simulations were performed on a $80\times 60\times 60$ box with the top and bottom faces moving with opposite velocities parallel to the longer side. $C=\mathrm{\Delta }x=0.05$.

Figure 4

Droplet flowing in a capillary with square cross section of width $w$; the (normalized) droplet excess speed is plotted against the capillary number for different viscosities: $\lambda =1∕32$ $(엯)$, $\lambda =1∕8$ $(▵)$, and $\lambda =1∕2$ $(◻)$. $U$ is the droplet speed while the $V$ is average flow speed of the continuous phase (see inset). The undeformed radius of the droplet is $0.55w$. The simulation mesh is made by $125\times 25\times 25$ points, $C=\mathrm{\Delta }x=0.04$.

Figure 5

Bottom, $({\epsilon }_0,{\mathrm{Ca}}_T)$ diagram summarizing the conditions for breaking drops at a symmetric T-junction for $\lambda =1∕8$ (left) and $\lambda =1$ (right). Empty symbols correspond to nonbreaking droplets, filled symbols to breaking droplets; diamonds correspond to simulations preformed with $C=1∕15$, while squares refer to simulations with $C=1∕20$. The solid curves show the separating boundary from Eq. (21), with the viscosity dependent coefficients $\alpha (1∕8)=1$ and $\alpha \left(1\right)=0.5$. $15\times 15$ and $20\times 20$ grid points were used for the square cross section, with $\mathrm{\Delta }x$ equal to $1∕15$ or $1∕20$, respectively. $\mathrm{Re}$ varied from $0.1$ to $1$. Top, the frames in (a) and (b) show the motion of droplets with different elongation and the same capillary number, corresponding to the points labeled $a$ and $b$ in the phase diagram for $\lambda =1∕8$.