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Injected Drops with Surfactants: A Priori Scaling for Predicting Impeded Drop Necking

Vidya Venkataramani1, Fang Jin2, and Kathleen J. Stebe1. (1) Chemical and Biomolecular Engineering, Johns Hopkins University, Maryland Hall, Rm 221, 3400 North Charles St, Baltimore, MD 21218, (2) Department of Mathematics, University of California, Irvine, 476 Rowland Hall, Irvine, CA 92697

Soluble surfactant effects on the evolution of buoyant viscous drops injected into an external viscous fluid are studied numerically. Without surfactants, drops first emerge from the orifice as roughly spherical caps, subsequently elongate under the influence of gravity, then enter a necking regime in which surface tension drives snap-off in the drop neck rapidly for the small capillary numbers Ca studied here. As the necks form, the surface dilatation rate is positive near the drop apex, and highly negative in a small region above the neck. When surfactants are present, they adsorb as the drop expands, and collect rapidly as the surface contracts above the neck. In the neck region, they change the ensuing dynamics if their rate of removal by mass transfer (i.e. by desorption or bulk diffusion) is slow compared to the prevailing surface contraction rate. If, however, the rate of surfactant removal is rapid compared to the rate of surface contraction, the drop detachment dynamics are not altered by the presence of surfactants.

In prior work, we have identified thresholds for drop detachment for adsorption-desorption controlled surfactants (1) and for diffusion controlled surfactants (2). In the adsorption-desorption controlled limit, over a broad range of equilibrium surfactant coverages (x), it was shown that drops form more complex necks, or fail to form necks at all, if the rate of desorption was very slow compared to the prevailing convection rate, as characterized by a Biot number (Bi). A diverse set of neck shapes ranging inverted, and symmetrical necks were reported, as well neck/ no-neck thresholds. Here, we explain these maps of necking behaviors in terms of bounding values for the fastest rate of supply to the interface by surfactant adsorption, A=Bi/1-x, and the slowest rate of removal by surfactant desorption, D=Bi/x. In the sorption controlled limit, we show that the regimes of complex neck shapes can be attributed to regions where A is rapid enough to supply surfactant to the interface, and D is slow enough to impede its removal from the contracting neck region. Simple scaling arguments and thresholds are developed based on these concepts.

In the diffusion limited regime, drops failed to form necks only for very high equilibrium surfactant coverages x and only over a narrow range of diffusion rates, characterized by a dimensionless quantity, Y, which is the ratio of the diffusion rate to the injection rate. Otherwise, rapid diffusion fluxes created prevented strong surface tension changes from occurring. These rapid fluxes away from the surface are not predicted well by a priori scaling analysis because of two effects. First, the drop dilutes the space around it as it expands. Second, the sublayer concentration adjacent to the interface becomes very high over a region close to the surface as the neck contracts. This local elevation in concentration creates a rapid flux away from the neck, preventing surfactant accumulation and the concomitant occurrence of decreased surface tensions. We calculate numerical values for threshold desorption fluxes for the limited regime in which surfactants alter the drop dynamics.

We also report new studies in the mixed control limit in which both adsorption-desorption and diffusion control the surfactant transport. The supply and removal of surfactant to and from the interface is limited by the slower of the two mass transport mechanisms, i.e., adsorption-desorption and diffusion. Supply by diffusion or adsorption can rate limiting. However, (since diffusion fluxes away from the neck are rapid as argued above) desorption controls the rate of surfactant removal. A priori scaling arguments are applied to organize the observed regimes of drop behavior in this case.

(1) F. Jin, N. R. Gupta and K. J. Stebe, “The detachment of a viscous drop in a viscous solution in the presence of a soluble surfactant,” Phys. Fluids 18, 022103 (2006).

(2) F. Jin and K. J. Stebe, “The effects of diffusion controlled surfactant on a viscous drop injected into a vicous medium,” Phys. Fluids 19, 1 (2007)