The effects of surfactants on drop deformation and breakup

Abstract

The effects of surface-active agents on drop deformation and breakup in extensional flows at low Reynolds numbers are described. In this free-boundary problem, determination of the interfacial velocity requires knowledge of the distribution of surfactant, which, in turn, requires knowledge of the interfacial velocity field. We account for this explicit coupling of the unknown drop shape and the evolving surfactant distribution. An analytical result valid for nearly spherical distortions is presented first. Finite drop deformation is studied numerically using the boundary-integral method in conjunction with the time-dependent convective–diffusion equation for surfactant transport. This procedure accurately follows interfacial tension variations, produced by non-uniform surfactant distribution, on the evolving interface. The numerical method allows for an arbitrary equation of state relating interfacial tension to the local concentration of surfactant, although calculations are presented only for the common linear equation of state. Also, only the case of insoluble surfactant is studied.The analytical and numerical results indicate that at low capillary numbers the presence of surfactant causes larger deformation than would occur for a drop with a constant interfacial tension equal to the initial equilibrium value. The increased deformation occurs owing to surfactant being swept to the end of the drop where it acts to locally lower the interfacial tension, which therefore requires increased deformation to satisfy the normal stress balance. However, at larger capillary numbers and finite deformations, this convective effect competes with ‘dilution’ of the surfactant due to interfacial area increases. These two different effects of surface-active material are illustrated and discussed and their influence on the critical capillary number for breakup is presented.

Droplet behavior in the presence of insoluble surfactants

“…7. Our results in supercritical flow resemble the results obtained by Li and Pozrikidis 15 and Stone and Leal 16 in subcritical flows. For Peϭ0, the droplet deformation decreases with increasing ‫,*⌫ץ/*ץ‬ i.e., with increasing ␤.…”

“…In such a case where the Marangoni stresses are large, the droplet deformation becomes dilution dominated and the droplet deforms less than an equivalent clean droplet. 10,[13][14][15][16] For higher viscosity ratio droplets, the convection is small compared to the diffusion ͑low Pe͒ and the surfactant is more uniformly distributed. 8 Therefore, the effect of surfactant is smaller for higher viscosity ratios.…”

Microscale tipstreaming in a microfluidic flow focusing device

“…Modeling this process is not trivial since a detailed model for the problem involves simultaneous solutions of the Stokes equations in each liquid together with velocity and stress boundary conditions at the interface. [25][26][27][28] The normal and shear stress boundary conditions couple the velocity field with variations in the surface tension along the interface, and additional equations are also needed in order to couple surface tension with surface concentration, and surface concentration with diffusion, adsorption, and desorption processes for mass transfer of molecules from the bulk onto the interface.…”

“…These numerics are based upon a boundary integral formulation that was extended to include surfactant dynamics by Stone and Leal 26 and Milliken et al 27 While Eggleton and co-workers focused on elongational flow, recent work by Bazhlekov et al 28 using similar numerical methods has also shown similar behavior in droplets subject to shearing flows, also using insoluble surfactants. It is worth noting that this recent work reproduces numerically an experimental result reported by Grace 29 showing that below a critical viscosity ratio, surfactants lead to a sharp reduction in the critical capillary number for breakup of sheared droplets, and that this critical capillary number is independent of viscosity ratio below the critical value.…”

Pore-scale mass and reactant transport in multiphase porous media flows

“…Normalizing velocities by U = (ρ w − ρ nw )gR 2 /µ w and length scales by the average pore radius R, the stress jump associated with surface tension at the interface between the two fluids becomes (Stone & Leal 1990;Pozrikidis 1992;Manga & Stone 1993) 11) where the superscript * refers to a dimensionless variable, x is located at the interface between the two fluids, n is the outward normal to the interface, ∇ s is the gradient along the interface, Bo = ρgR 2 /γ is the Bond number and λ = µ nw /µ w . Finally, we solve for the conservation of enthalpy for the three phases…”