The kinetics of re-equilibration of the anionic surfactant sodium dodecylbenzene sulfonate at the air-solution interface have been studied using neutron reflectivity. The experimental arrangement incorporates a novel flow cell in which the subphase can be exchanged (diluted) using a laminar flow while the surface region remains unaltered. The rate of the re-equilibration is relatively slow and occurs over many tens of minutes, which is comparable with the dilution time scale of approximately 10-30 min. A detailed mathematical model, in which the rate of the desorption is determined by transport through a near-surface diffusion layer into a diluted bulk solution below, is developed and provides a good description of the time-dependent adsorption data. A key parameter of the model is the ratio of the depth of the diffusion layer, H(c), to the depth of the fluid, H(f), and we find that this is related to the reduced Péclet number, Pe*, for the system, via H(c)/H(f) = C/Pe*(1/2). Although from a highly idealized experimental arrangement, the results provide an important insight into the "rinse mechanism", which is applicable to a wide variety of domestic and industrial circumstances.
We present a mathematical model to describe the distribution of surfactant pairs in a multilayer structure beneath an adsorbed monolayer. A mesoscopic model comprising a set of ordinary differential equations that couple the rearrangement of surfactant within the multilayer to the surface adsorption kinetics is first derived. This model is then extended to the macroscopic scale by taking the continuum limit that exploits the typically large number of surfactant layers, which results in a novel third-order partial differential equation. The model is generalized to allow for the presence of two adsorbing boundaries, which results in an implicit free-boundary problem. The system predicts physically observed features in multilayer systems such as the initial formation of smaller lamellar structures and the typical number of layers that form in equilibrium.
In this paper, we consider the straining flow of a weakly interacting polymer–surfactant solution below a free surface, with the bulk surfactant concentration above the critical micelle concentration. We formulate a set of coupled differential equations describing the concentration of monomers, micelles, polymer, and polymer–micelle aggregates in the flow. We analyse the model in several asymptotic limits, and make predictions about the distribution of each of the species. In particular, in the large-reaction-rate limit we find that the model predicts a region near the free surface where no micelles or aggregates are present, and beneath this a region where the concentration of surfactant is constant, across which the concentration of aggregates increases until all the free polymer is consumed. For certain parameter regimes, a maximum in the concentration of the polymer–micelle complex occurs within the bulk fluid. In the finite-reaction-rate limit, micelles, and aggregates are present right up to the free surface, and the plateau in the concentration of surfactant in the bulk is no longer present. Results from the asymptotic theory compare favorably with full numerical solutions.
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