Investigations of liquid surface rheology of surfactant solutions by droplet shape oscillations: Theory

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Abstract
A theoretical analysis is presented for the shape oscillations of drops suspended in air. For drops of surfactant solution, the oscillation frequency is basically determined by the surface tension; the free-damping constant depends on the surface viscoelasticities. Different types of surfactant mass transfer at the droplet surface produce different surface rheological behaviors. Analytical approximate solutions for free-oscillation frequency and damping constant are derived by a perturbation method as functions of surface compositional elasticity, surface dilatational viscosity, and surface shear viscosity. These solutions are verified with exact numerical solutions. The existence of a second oscillation mode due to surface elasticity is illustrated. The phase relationships between the external driving forces and the droplet shapes for forced oscillations are discussed. It is found that at the 90° phase shift, the driving frequency and the slope of the phase diagram are equivalent to the free-oscillation frequency and damping constant.