On the Factor in Gibbs Equation for Ionic Surfactants

The adsorption layer of soluble surfactants can be described by the well-known Gibbs equation. It states that the surface excess Γ is given by the slope of the isotherm according to  Γ = −1/(mRT)∂σ_e/∂ln a. The rigorous thermodynamic treatment requires a factor m = 2 for 1:1 electrolyte in the case of the absence of indifferent electrolyte. This aspect is addressed in the following contribution. For this purpose, a soluble C₁₂ and an insoluble cationic amphiphile C₂₀ was designed and synthesized. Both contain the same head group which possess a sufficiently high hyperpolarizability for surface second-harmonic generation (SHG) measurements. SHG is highly surface specific, and polarization dependent measurements allow a determination of the orientational order within the adsorption layer. The intensity of the SHG signal is proportional to the square of the number density N, and with a calibration based on the SHG signal of the Langmuir monolayer of the insoluble compound the absolute number density can be determined. The corresponding (π,A) isotherm of the insoluble monolayer shows only a liquid expanded region, and experimental errors are minor since for both experiments the head group which is in an identical physical environment is responsible for the SHG signal. Furthermore, complications arising from local field corrections are eliminated. In our case, the factor m was determined to be m = 1.28. These findings are discussed in detail.