Marangoni effects on drop deformation in an extensional flow: The role of surfactant physical chemistry. I. Insoluble surfactants

The shape of a drop centered in an axisymmetric extensional flow is determined by the viscous stresses that deform the drop and surface tension γ′ that resists the deformation. The ratio of these stresses is given by the capillary number, Ca. When Ca is small enough, the drop attains a steady shape. However, above a threshold value, Cacr, the drop elongates continuously, and no steady shape is attained. When surfactants are present on the drop interface, the surface tension is determined by the surface concentration profile, which varies throughout the deformation process. Initially, the surface tension is given by γeq′, in equilibrium with the uniform surface concentration Γeq′. When the flow is initiated, surfactant is swept toward the drop tips, reducing the surface tension there, and altering the interfacial stress balance tangentially through Marangoni stresses and normally through the Laplace pressure. In this paper, the effects of an insoluble surfactant on drop deformation are studied. In previous work, either a surface equation of state for the surface tension γ′ that is linear in the surface concentration Γ′ was used, an approximation that is valid only for dilute Γ′, or Γ′ sufficiently dilute for the linear approximation to be valid were studied. In this paper, a nonlinear surface equation of state that accounts for surface saturation and nonideal interactions among the surfactant molecules is adopted. The linear framework results are recovered for Γ′ that are sufficiently dilute. As Γ′ is increased, the effects of saturation and surfactant interactions are probed at constant initial Γeq′ and at constant initial γeq′. Finally, the case of strong intersurfactant cohesion is treated with a first-order surface phase transformation model. At moderate surface concentrations, these nonlinear phenomena strongly alter the steady drop deformations and Cacr relative to the uniform surface tension and linear equation of state results.