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In this paper, we explore a novel approach to predict equilibrium adsorption properties from experimental dynamic surface tension (DST) data and the known rate-limiting adsorption kinetics mechanism, an approach that has never been pursued in the DST literature. Specifically, we develop a new methodology to predict the equilibrium surface tension versus surfactant bulk solution concentration (ESTC) behavior of nonionic surfactants from experimental DST data when the adsorption kinetics rate-limiting mechanism is diffusion controlled. The new methodology requires the following three inputs: (1) experimental DST data measured at a single surfactant bulk solution concentration, Cb, (2) the diffusion coefficient of the surfactant molecule, D, and (3) a single equilibrium surface tension data point, to predict the entire ESTC curve applicable over a wide range of surfactant bulk solution concentrations which are less than, or equal to, Cb. We demonstrate the applicability of the new methodology by predicting the ESTC curves of the two alkyl poly (ethylene oxide) nonionic surfactants C12E4 and C12E6, and validate the results by comparing the predictions with (a) equilibrium surface tension measurements, (b) surface-expansion measurements, and (c) pendant-bubble dynamic surface tension measurements for t

Traditionally, surfactant bulk solutions in which dynamic surface tension (DST) measurements are conducted using the pendant-bubble apparatus are assumed to be quiescent. Consequently, the transport of surfactant molecules in the bulk solution is often modeled as being purely diffusive when analyzing the experimental pendant-bubble DST data. In this Article, we analyze the experimental pendant-bubble DST data of the alkyl poly (ethylene oxide) nonionic surfactants, C12E4 and C12E6, and demonstrate that both surfactants exhibit "superdiffusive" adsorption kinetics behavior with characteristics that challenge the traditional assumption of a quiescent surfactant bulk solution. In other words, the observed superdiffusive adsorption behavior points to the possible existence of convection currents in the surfactant bulk solution. The analysis presented here involves the following steps: (1) constructing an adsorption kinetics model that corresponds to the fastest rate at which surfactant molecules adsorb onto the actual pendant-bubble surface from a quiescent solution, (2) predicting the DST behaviors of C12E4 and C12E6 at several surfactant bulk solution concentrations using the model constructed in step 1, and (3) comparing the predicted DST profiles with the experimental DST profiles. This comparison reveals systematic deviations for both C12E4 and C12E6 with the following characteristics: (a) the experimental DST profiles exhibit adsorption kinetics behavior, which is faster than the predicted fastest rate of surfactant adsorption from a quiescent surfactant bulk solution at time scales greater than 100 s, and (b) the experimental DST profiles and the predicted DST behaviors approach the same equilibrium surface tension values. Characteristic (b) indicates that the cause of the observed systematic deviations may be associated with the adsorption kinetics mechanism adopted in the model used rather than with the equilibrium behavior. Characteristic (a) indicates that the actual surfactant bulk solution in which the DST measurement was conducted, most likely, cannot be considered to be quiescent at time scales greater than 100 s. Accordingly, the observed superdiffusive adsorption behavior is interpreted as resulting from convection currents present in a nonquiescent surfactant bulk solution. Convection currents accelerate the surfactant adsorption process by increasing the rate of surfactant transport in the bulk solution. The systematic nature of the deviations observed between the predicted DST profiles and the experimental DST behavior for C12E4 and C12E6 suggests that the nonquiescent nature of the surfactant bulk solution may be intrinsic to the experimental pendant-bubble DST measurement approach. To validate this possibility, we identified generic features in the experimental DST data when DST measurements are conducted in a nonquiescent surfactant bulk solution, and the DST measurements are analyzed assuming that the surfactant bulk solution is quiescent. An examination of the DST literature reveals that thes...

How does one design a surfactant mixture using a set of available surfactants such that it exhibits a desired adsorption kinetics behavior? The traditional approach used to address this design problem involves conducting trial-and-error experiments with specific surfactant mixtures. This approach is typically time-consuming and resource-intensive and becomes increasingly challenging when the number of surfactants that can be mixed increases. In this article, we propose a new theoretical framework to identify a surfactant mixture that most closely meets a desired adsorption kinetics behavior. Specifically, the new theoretical framework involves (a) formulating the surfactant mixture design problem as an optimization problem using an adsorption kinetics model and (b) solving the optimization problem using a commercial optimization package. The proposed framework aims to identify the surfactant mixture that most closely satisfies the desired adsorption kinetics behavior subject to the predictive capabilities of the chosen adsorption kinetics model. Experiments can then be conducted at the identified surfactant mixture condition to validate the predictions. We demonstrate the reliability and effectiveness of the proposed theoretical framework through a realistic case study by identifying a nonionic surfactant mixture consisting of up to four alkyl poly(ethylene oxide) surfactants (C(10)E(4), C(12)E(5), C(12)E(6), and C(10)E(8)) such that it most closely exhibits a desired dynamic surface tension (DST) profile. Specifically, we use the Mulqueen-Stebe-Blankschtein (MSB) adsorption kinetics model (Mulqueen, M.; Stebe, K. J.; Blankschtein, D. Langmuir 2001, 17, 5196-5207) to formulate the optimization problem as well as the SNOPT commercial optimization solver to identify a surfactant mixture consisting of these four surfactants that most closely exhibits the desired DST profile. Finally, we compare the experimental DST profile measured at the surfactant mixture condition identified by the new theoretical framework with the desired DST profile and find good agreement between the two profiles.