Microdroplets spatially resolved ellipsometry

The velocity U (, t) in Eq. 5.8b is merely the velocity of Eq. 5.2. It results from the stationary Navier-Stokes equation written at a given time t. This is allowed because the characteristic times for the building-up of velocity profiles of the Navier-Stokes equation are much shorter than the ones of the spreading dynamics. 

In the flat part of the film, where the curvature term is negligible, Eq. 5.8a can be understood as a diffusion-like equation  

This means that in Eq. 5.9 the profile f (x, t) depends on the variable x /t and not independently on x and t these films are referred to as diffusive films. In the mesoscopic range of thickness, and for disjoining pressure controlled by van der Waals 

the factor f /3 j corresponds to a Poiseuille flow, with no slip at the solid. Therefore, the validity of Eq. 5.10a is restricted to mesoscopic films. However, for molecular ones, the proportionality between D(f) and dU/d is retained. Note that as the droplet shape f(x, t) evolves slowly in time, U(f) 9f/9x evolves slowly in space therefore the profile f (x, t) gives qualitative information on the shape of the disjoining pressure ellipsometric droplets profiles are not merely striking pictures. Let us comment further on short-range contributions. 

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The spontaneous spreading of thin microdroplets on solid substrates is slow and can be conveniently monitored using spatially resolved ellipsometry. On the other hand, extended thin wetting films can be spin-coated on solids or deposited on liquid substrates and thereafter studied under a polarized microscope for nanometric thicknesses or a Brewster angle microscope for molecular ones. X-ray reflectometry provides complementary information. 

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