  We present a study of the spreading of liquid droplets on a solid substrate
at very small scales. We focus on the regime where effective wetting energy
(binding potential) and surface tension effects significantly influence steady
and spreading droplets. In particular, we focus on strong packing and layering
effects in the liquid near the substrate due to underlying density oscillations
in the fluid caused by attractive substrate-liquid interactions. We show that
such phenomena can be described by a thin-film (or long-wave or lubrication)
model including an oscillatory Derjaguin (or disjoining/conjoining) pressure,
and explore the effects it has on steady droplet shapes and the spreading
dynamics of droplets on both, an adsorption (or precursor) layer and completely
dry substrates. At the molecular scale, commonly used two-term binding
potentials with a single preferred minimum controlling the adsorption layer
height are inadequate to capture the rich behaviour caused by the near-wall
layered molecular packing. The adsorption layer is often sub-monolayer in
thickness, i.e., the dynamics along the layer consists of single-particle
hopping, leading to a diffusive dynamics, rather than the collective
hydrodynamic motion implicit in standard thin-film models. We therefore modify
the model in such a way that for thicker films the standard hydrodynamic theory
is realised, but for very thin layers a diffusion equation is recovered.
