Scaling dynamics of particles confined at fluid-fluid interfaces

Particle-laden fluid interfaces exhibit complex linear viscoelastic behavior resulting from collective particle dynamics confined to two dimensions. While interfacial rheology has been widely used to characterize such systems, a consistent framework for comparing relaxation behavior across different particle-laden interfaces remains limited. In this work, we investigate the interfacial shear rheology of hydrophobically modified silica particle monolayers at air-water and oil-water interfaces via small amplitude oscillatory shear measurements. Dynamic moduli obtained at different particle surface concentrations are combined into master curves via time-surface concentration superposition, and the viscoelastic response is analyzed in terms of relaxation time spectra obtained using the parsimonious spectral approach. The resulting spectra are well described by a truncated double power-law form analogous to the Baumgärtel-Schausberger-Winter spectrum originally developed for bulk viscoelastic materials. We refer to this representation as a two-dimensional BSW (2dBSW) spectrum and demonstrate that it captures the dominant relaxation features of the interfacial networks studied here. To further examine the scope of this approach, published interfacial rheology data for particle-laden interfaces with varying particle attributes and subphase conditions are reanalyzed within the same framework. Despite wide variations in particle type and interfacial environment in these publications, the relaxation spectra collapse onto a common self-similar form consisting of two power law columns of relaxation modes, with the longest relaxation time reflecting the strength of interparticle attractions. This apparent universality suggests that the linear viscoelastic response of particle-laden interfaces is governed by generic network features, making 2dBSW a useful and transferable description of their linear viscoelasticity.