  Conventional wetting theories on rough surfaces with Wenzel, Cassie-Baxter,
and Penetrate modes suggest the possibility of tuning the contact angle by
adjusting the surface texture. Despite decades of intensive study, there are
still many experimental results that are not well understood because
conventional wetting theory, which assume an infinite droplet size, has been
used to explain measurements of finite-sized droplets. In this study, we
suggest a wetting theory that is applicable to any droplet size based on the
free energy landscape analysis of various wetting modes of finite-sized
droplets on a 2D textured surface. The key finding of our study is that there
are many quantized wetting angles with local free energy minima; the
implication of this is remarkable. We find that the conventional theories can
predict the contact angle at the global free energy minimum if the droplet size
is 40 times or larger than the characteristic scale of the surface roughness.
Furthermore, we confirm that the pinning origin is the local free energy minima
and obtain the energy barriers of pinning as a function of geometric factors.
We validate our theory against experimental results on an anisotropic rough
surface. In addition, we discuss wetting on a non-uniform rough surface with a
rough central region and flat edge. Our findings clarify the extent to which
the conventional wetting theory is valid and expand the physical understanding
of wetting phenomena of small liquid drops on rough surfaces.
