Counterion Condensation, the Gibbs Equation, and Surfactant Binding: An Integrated Description of the Behavior of Polyelectrolytes and Their Mixtures with Surfactants at the Air–Water Interface

By applying the Gibbs equation to the bulk binding isotherms and surface composition of the air–water (A–W) interface in polyelectrolyte–surfactant (PE–S) systems, we show that their surface behavior can be explained semiquantitatively in terms of four concentration regions, which we label as A, B, C, and D. In the lowest-concentration range A, there are no bound PE–S complexes in the bulk but there may be adsorption of PE–S complexes at the surface. When significant adsorption occurs in this region, the surface tension (ST) drops with increasing concentration like a simple surfactant solution. Region B extends from the onset of bulk PE–S binding to the end of cooperative binding, in which the slow variation of surfactant activity with cooperative binding means that the ST changes relatively little, although adsorption may be significant. This leads to an approximate plateau, which may be at high or low ST. Region C starts where the binding in the bulk complex loses its cooperativity leading to a rapid change of surfactant activity with the total concentration. This, combined with significant adsorption, often leads to a sharp drop in ST. Region D is where precipitation and redissolution of the bulk PE–S complex occur. ST peaks may arise in region D because of loss of the solution complex that matches the value of the preferred surface stoichiometry, which seems to have a well-defined value for each system. The analysis is applied to the experimental systems, sodium polystyrene sulfonate–alkyltrimethylammonium bromides and poly(diallyldimethyl chloride)–sodium alkyl sulfates, with and without the added electrolyte, and includes data from bulk binding isotherms, phase diagrams, aggregation behavior, and direct measurements of the surface excess and stoichiometry of the surface. The successful fits of the Gibbs equation to the data confirm that the surfaces in these systems are largely equilibrated.