Modelling the passive breakup of a surfactant-contaminated droplet in a T-junction microchannel

A lattice Boltzmann method is used to explore the effect of surfactants on the unequal volume breakup of a droplet in a T-junction microchannel, and the asymmetry due to fabrication defects in real-life microchannels is modelled as the pressure difference between the two branch outlets ( $\Delta {P^\ast }$ ). We first study the effect of the surfactants on the droplet dynamics at different dimensionless initial droplet lengths ( $l_0^\ast $ ) and capillary numbers (Ca) under symmetric boundary conditions ( $\Delta {P^\ast } = 0$ ). The results indicate that the presence of surfactants promotes droplet deformation and breakup at small and moderate $l_0^\ast $ values, while the surfactant effect is weakened at large $l_0^\ast $ values. When the branch channels are completely blocked by the droplet, a linear relationship is observed between the dimensionless droplet length ( $l_d^\ast $ ) and dimensionless time ( ${t^\ast }$ ), and two formulas are proposed for predicting the evolution of $l_d^\ast $ with ${t^\ast }$ for the two systems. We then investigate the effect of the surfactants on the droplet breakup at different values of $\Delta {P^\ast }$ and bulk surfactant concentrations ( ${\psi _b}$ ) under asymmetric boundary conditions ( $\Delta {P^\ast } \ne 0$ ). It is observed that, as $\Delta {P^\ast }$ increases, the volume ratio of the generated droplets ( ${V_1}/{V_2}$ ) decreases to 0 in both systems, while the rate of decrease is higher in the clean system, i.e. the presence of surfactants could cause a decreased pressure difference between the droplet tips. As ${\psi _b}$ increases, ${V_1}/{V_2}$ first increases rapidly, then remains almost constant and finally decreases slightly. We thus establish a phase diagram that describes the ${V_1}/{V_2}$ variation with $\Delta {P^\ast }$ and ${\psi _b}$ .