Which surfactants reduce surface tension faster? A scaling argument for diffusion-controlled adsorption

Consider the example of surfactant adsorbing from an infinite solution to a freshly formed planar interface. There is an implicit length scale in this problem, the adsorption depth h, which is the depth depleted to supply the interface with the adsorbed surfactant. From a mass balance, h can be shown to be the ratio of the equilibrium surface concentration Γeq to the bulk concentration C∞. The characteristic time scale for diffusion to the interface is τD=h2/D, where D is the diffusivity of the surfactant in solution. The significance of this time scale is demonstrated by numerically integrating the equations governing diffusion-controlled adsorption to a planar interface. The surface tension equilibrates within 1–10 times τD regardless of bulk concentration, even for surfactants with strong interactions. Dynamic surface tension data obtained by pendant bubble method are rescaled using τD to scale time. For high enough bulk concentrations, the re-normalized surface tension evolutions nearly superpose, demonstrating that τD is indeed the relevant time scale for this process. Surface tension evolutions for a variety of surfactants are compared. Those with the smallest values for τD equilibrate fastest. Since diffusion coefficients vary only weakly for surfactants of similar size, the differences in the equilibration times for various surfactant solutions can be attributed to their differing adsorption depths. These depths are determined by the equilibrium adsorption isotherms, allowing τD to be calculated a priori from equilibrium surface tension data, and surfactant solutions to be sorted in terms of which will reduce the surface tension more rapidly. Finally, trends predicted by τD to gauge what surfactant properties are required for rapid surface tension reduction are discussed. These trends are shown to be in agreement with guiding principles that have been suggested from prior structure–property studies.