Modeling Adsorption of Cationic Surfactants at Air/Water Interface without Using the Gibbs Equation Chi M. Phan,* ,† Thu N. Le, † Cuong V. Nguyen, † and Shin-ichi Yusa ‡ † Department of Chemical Engineering, Curtin University, Perth, WA 6845, Australia ‡ Department of Materials Science and Chemistry, University of Hyogo, 2167 Shosha, Himeji, Hyogo 671-2280, Japan * S Supporting Information ABSTRACT: The Gibbs adsorption equation has been indispensable in predicting the surfactant adsorption at the interfaces, with many applications in industrial and natural processes. This study uses a new theoretical framework to model surfactant adsorption at the air/water interface without the Gibbs equation. The model was applied to two surfactants, C14TAB and C16TAB, to determine the maximum surface excesses. The obtained values demonstrated a fundamental change, which was verified by simulations, in the molecular arrangement at the interface. The new insights, in combination with recent discoveries in the field, expose the limitations of applying the Gibbs adsorption equation to cationic surfactants at the air/water interface. ■ INTRODUCTION Soluble surface-active agents are critical components in many industrial and natural processes. With a small concentration in the liquid phase, these chemicals can excessively concentrate at the interface and markedly change the physical properties of the interface. In the literature, the interfacial adsorption of surfactant is often calculated from bulk concentrations rather than direct measurement. 1 The theoretical analysis of soluble surfactant adsorption at the interfaces has been essentially built around the Gibbs adsorption equation, 1-3 which might be considered as an “accepted dogma” 4 in the literature. However, experimental observations of cationic surfactants at the air/water interface, including NMR 5 and surface potential, 6 have contradicted the theoretical prediction from the Gibbs adsorption equation. Similarly, a study on thin solid film has exposed some limitations of Gibbs analysis. 7 Previously, we have successfully modeled C16TAB (cetyl- trimethylammonium bromide) adsorption at the air/water interface without the Gibbs adsorption isotherm. 8 This study applies the model to C14TAB (myristyltrimethylammonium bromide) to quantify the influence of surfactants length on the interfacial excesses. More significantly, the results are then combined with molecular simulations to reveal new insights into the interface zone and re-examine the applicability of the Gibbs adsorption isotherm. ■ THEORETICAL MODEL The theoretical model was developed for dynamic adsorption of surfactants at the air/water interface, which is more comprehensive than the equilibrium adsorption. The dynamic model, however, inherently covers the equilibrium state (since all variables return to their equilibrium values at equilibrium). Most importantly, the dynamic adsorption can be verified quantified at different bulk concentrations and thus verify the consistency and reliability of the modeling results. The adsorption process of surfactant at the air/water interface is shown in Figure 1. In this model, the subsurface liquid plane is defined as the limit of the diffusion zone for surfactant. 9,10 Within the interfacial zone, the surfactant molecule movement (both rotational and lateral motions) is partially restricted, due to the balances/interactions with water, counterion and other surfactant molecules. The diffusion process is limited by CnTA + diffusion coefficient (since Br - has much higher diffusion coefficient). In Figure 1, the air/water interface was left empty to emphasize that the theory is applicable regardless of the molecular arrangement within this zone. The dynamic mass transfer can be modeled by the Ward and Tordai equation: 11,12 ∫ π τ τ Γ = − − t D c t c t () 2 { ( ) d( )} t b 0 s (1) where D is the diffusion coefficient, τ is a dummy variable of integration, and Γ(t) and c s (t) are transient surface excess and subsurface plane concentration, respectively. Since the interfacial zone is finite, the adsorbed concen- trations are related by the Langmuir isotherm: Received: November 20, 2012 Revised: March 11, 2013 Published: March 18, 2013 Article pubs.acs.org/Langmuir © 2013 American Chemical Society 4743 dx.doi.org/10.1021/la3046302 | Langmuir 2013, 29, 4743-4749