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