Isotropic <--> Anisotropic Surface Geometry Transitions Induced by Adsorbed Surfactants at Water/Vapor Interfaces

It is well known that adsorbates at a water/vapor interface change surface geometry through altered surface tension. Yet detailed theoretical studies of surface geometry in the presence of adsorbates are relatively sparse and many applications focus upon ensemble average surface geometric characteristics. Here we demonstrate that different interpretations of surface geometry emerge when considering the distributions of the geometric descriptors of surface curvature and orientation as a function of adsorbed surfactant concentration and sterics. At low surface coverage, the tributyl phosphate (TBP) sorbed water/vapor surface has an increased presence of ridges that are defined by principle curvatures $\kappa_1$ and $\kappa_2$ of opposite signs yet close in magnitude. As the TBP surface coverage increases, the difference in principle curvatures slowly increases. However there is a distinct transition of the surface geometry, where the ridge-like features become much more pronounced, having sides whose orientation is normal to a flat interfacial plane. Thus as the TBP surfactant is added to the surface, the surface curvatures become significantly anisotropic in terms of the difference in magnitude of $\kappa_1$ and $\kappa_2.$ We label this an \textit{isotropic $\rightarrow$ anisotropic} geometric transition. Comparing the surface geometry as a function of carbon tail length of the alkyl phosphate surfactant reveals that smaller surfactants also anisotropically enhance surface curvatures on the surface, and that adsorbed alkyl tails to the surface stabilize and increase the symmetry of surface waves along the two principle curvature axes. We label this an \textit{anisotropic $\rightarrow$ isotropic} geometric transition. These results reflect the opportunity to incorporate more realistic distributions of surface geometry within the collective understanding of statistical theories of surfaces, including capillary wave theory.