Contact angles of surfactant solutions on heterogeneous surfaces

Using Gibbs' adsorption equation and a literature isotherm, a new general model to predict the contact angle of surfactant solutions on (smooth or rough) chemically heterogeneous surfaces is constructed based on the Cassie equation. The model allows for adsorption at the liquid–vapor, solid–liquid, and solid–vapor interfaces. Solid–vapor adsorption is allowed in order to model the autophobic effect on hydrophilic surfaces. Using representative values for the coefficients which describe adsorption at each interface, model predictions for contact angles as a function of f parameters (area fractions) and surfactant concentration are made for heterogeneous surfaces made up of different materials. On smooth surfaces, the f parameters serve as weighting factors determining how to combine the effects of surfactant adsorption on each material to predict the behavior on the heterogeneous surface. Due to the non-linear nature of the model, the inclusion of a small amount of hydrophobic material has a greater effect on a predominantly hydrophilic material than vice versa, explaining the result seen in literature that a small amount of hydrophobic contamination (such as oil) significantly increases contact angle on a hydrophilic surface. The fact that even a small amount of heterogeneity can greatly change experimental results could lead to incorrect experimental conclusions about surfactant adsorption if a surface were wrongly assumed to be homogeneous. Model predictions rapidly become more complex as the number of differently wettable materials present on the surface increases. Also, an approximately equal weighting of different materials generally leads to more complex behaviors compared to heterogeneous surfaces composed largely of a single material. Rough heterogeneous surfaces follow previous results for surfactant wetting of rough homogeneous surfaces, leading to an amplification/attenuation of surfactant effects for penetrated/unpenetrated wetting, and further increasing the complexity of predictions. These potential complexities point to the importance of characterizing the heterogeneities of any surface under consideration. With proper characterization, the model described in this paper will allow for prediction of contact angles on all types of heterogeneous surfaces, and design of surfaces for specific interactions with surfactant solutions.