Capillary nanowaves on surfactant-laden liquid films with surface viscosity and elasticity

Thermal motions of molecules can generate nanowaves on the free surface of a liquid film. As nanofilms are susceptible to the contamination of surfactants, this work investigates the effects of surfactants on dynamics of nanowaves on a bounded film with a finite depth, using both molecular dynamics simulations and analytical theories. In molecular simulations, a bead-spring model is adopted to simulate surfactants, where beads are connected by the finite extensive nonlinear elastic potentials. Fourier transforms of the film surface profiles $h(x,t)$ extracted from molecular simulations are performed to obtain the static spectrum $|h_q|_{\mathrm{rms}}$ and temporal correlations of surface modes $<h_q(0)h_q^*(t)>$. It is shown that the spectral amplitude is increased for the contaminated liquid surface compared to the clean surface because surfactants can decrease surface tension. A higher concentration of surfactants on the surface not only decreases the surface tension but also causes elastic energy to the free surface, as the scaling of spectral amplitude with wavenumbers changes from $|h_q|_{\mathrm{rms}}\sim q^{-1}$ to $|h_q|_{\mathrm{rms}}\sim q^{-2}$ for modes with large wavenumbers. Regarding the temporal correlations of surface modes, it is observed that the presence of surfactants leads to a slower decay, which, however, cannot be predicted by only considering the decreased surface tension. Based on the Boussinesq Scriven model for surface viscosity, a linear stability analysis of Stokes flow for films with arbitrary depth is conducted and the obtained dispersion relation considering surface viscosity can justify the simulation results.