Separation-driven coalescence of droplets: An analytical criterion for the approach to contact

Recent microfluidic experiments by Bremond, Thiam & Bibette (Phys. Rev. Lett., vol. 100, 2008, paper no. 024501), along with simulations by Yoon et al. (Phys. Fluid, vol. 19, 2007, paper no. 102102) and near-contact experiments and simulations by Manica et al. (Langmuir, vol. 24, 2008, pp. 1381-1390), have demonstrated that two droplets can coalesce as they are separating rather than upon their collision. We analyse the experimental microfluidic flow configuration for the approach to contact with a two-dimensional model: we apply a lubrication analysis followed by the method of domain perturbation to determine the droplet deformation as a function of time. We find the approximate shape for the deformed droplet at the time of contact. In particular, for droplets of radius R, moving apart according to h₀(t) = h₀(0) + αt², where 2h₀(t) is the separation distance, we define a non-dimensional arameter A = 4 C μ R²α^1/2/πγ[h₀ (0)]^3/2, where μ is the viscosity of the continuous phase; γ is the interfacial tension; and C depends on the viscosity ratio between the droplets and the continuous phase. Our model suggests that there exists a critical value A_crit = 16/3^3/2 ≈ 3.0792, below which separation is unlikely to facilitate the coalescence of the droplets. The predictions are in good agreement with available experimental data.