Curvature, Clouds and Capillarity

One of the fundamental results from the Laplace-Young derivation is that the pressure on the concave side of a curved liquid-vapor surface is greater than that on the convex side. Lord Kelvin immediately saw the impact a curved surface would have on the equilibrium vapor pressure (when comparted to a flat surface).

For example, consider a spherical droplet, which puts the liquid on the concave side of the interface (which is at higher pressure than the convex side). Essentially being an applied pressure, this will in turn increase the vapor pressure above the surface (see previous lecture on how external pressure increases the escape tendency of the liquid, leading to the Kelvin equation:

ln(P_drop/P_bulk) = 2γM/rρRT

One example of its meteorophysicochemical consequences: Moist air rises naturally from the surface of the earth and will experience at some altitude a combination of pressure and temperature that makes condensation favorable (in terms of chemical potentials, μ^G > μ^L). The natural tendency then is the spontaneous formation of microdroplets; however, the vapor pressure is so huge for small radii that the opposite force is to evaporate immediately). Eventually this enhanced evaporative effect will be overcome by spontaneous coagulation into larger droplets, which is further aided by solid aerosols in the atmosphere (serving as nucleation sites). All of this, of course, leads to cloud formation, whose different types are partly dictated by the external pressure on the system of microdroplets.

For liquid droplets deposited on, say, glass, the internal cohesive forces struggle against possible adhesive forces with the surface. This interplay can be seen clearly as a function of the contact angle the liquid-gas interface makes with the surface. Angles close to 0° correspond to liquids that are considered good wetters while those close to 180° are nonwetting. Something we should all strive for methinks.

Contact angles are seen again in the phenomenon of capillarity, the spontaneous rising of a liquid with appreciable adhesive forces with a hollow tube (or other porous medium). Simple physics leads to the formula h=(2γ/ρgr)cosθ_c, which can easily be utilized for measuring the surface tension of a simple fluid by measuring its capillarity height. Examples of this phenomenon can be seen all around us.