Surfactants are found in many natural and industrial processes. The addition of surfactants to an emulsion reduces the interfacial tension of the surface of the drops, and affects the drop deformation and the resulting rheology. Prior studies on surfactant-covered drops have focused primarily on the drop deformation and modes of drop breakup in shear and extensional flows, with little focus on the rheology of the system. In this work, we numerically study the rheology of emulsions in the presence of surfactants in a broad range of kinematics by solving the boundary-integral (BI) equation for the evolution of the drop surface, coupled with the convection-diffusion equation for the distribution of the surfactant on the surface of the drops. The general approach to constitutive modeling of non-Newtonian liquids based on the Oldroyd equation with variable coefficients, which was developed in our prior work (Martin et al. 2014), is applied to this system. This approach involves numerically calculating the rheological functions for two canonical steady flows--simple shear and planar extension at arbitrary flow intensities--and using these functions to find the values of the five Oldroyd parameters as functions of an instantaneous invariant of the flow, chosen as the excess energy dissipation rate due to the presence of drops. This constitutive modeling scheme, which was originally applied to predict the rheology in dilute emulsions of surfactant-free deformable drops (Martin et al. 2014), is extended in this work to account for the presence of surfactant. The resulting constitutive model is applied to various test cases, different from simple shear and planar extension, to evaluate the effectiveness of the model in predicting the emulsion stress for an arbitrary history of deformation of a material element, both Lagrangian steady and unsteady, and to compare the results to the precise BI calculations. Our model achieves excellent stress predictions for planar-mixed flows and uniaxial extension, with more deviation from the exact results for uniaxial compression at large capillary numbers. Another type of kinematics considered is the flow in the ball bearing between two eccentric rotating spheres, where the rheological response of a small surfactant-covered drop is evaluated along a material trajectory. The results confirm the validity of our general scheme of constitutive modeling based on the generalized Oldroyd equation, with variable coefficients fitted to two steady base flows, for a variety of complex, non-Newtonian fluids.
As well as the application of the generalized Oldroyd model to the case of emulsions with surfactant, the behavior of the droplets and the resulting surfactant distribution are evaluated based on three different surface equations of state: the linear, Langmuir, and Frumkin equations. The chosen equation of state provides a relationship between the surfactant concentration on the surface of the drop to the interfacial tension. This work expands on previous work of Pawar and Stebe (1996) and Bazhlekov et al. (2006) to determine how the selected equation of state influences the drop deformation and the resulting rheology of the emulsion in the presence of surfactants.
1. Martin, R., A. Zinchenko, and R. Davis, â??A generalized Oldroydâ??s model for non-Newtonian liquids with applications to a dilute emulsion of deformable drops,â? J. Rheol. 58: 759-778 (2014).
2. Pawar, Y., and K. Stebe, â??Marangoni effects on drop deformation in an extensional flow: The role of surfactant physical chemistry. I. Insoluble surfactants,â? Phys. Fluids 8: 1738-1751 (1996).
3. Bazhlekov, I. B., P. D. Anderson, and H. E. H. Meijer, â??Numerical investigation of the effect of insoluble surfactants on drop deformation and breakup in simple shear flow,â? J. Colloid and Interface Sci. 298: 369-394 (2006).
