Gravity vs adhesion of a water droplet

Question

How much volume can a water droplet that is attached to a vertical glass surface have before it begins to roll downwards?

Answer 1

The answer depends on the purity of the water, the temperature of the water, the temperature of the glass, the type of glass and its surface texture and cleanliness. It could depend on the relative humidity. Sounds like a fun experiment!

We can also make some rough theoretical estimates. Using dimensional analysis, we expect the critical volume where the drop will start to move to be of order $$V_c\sim \left(\frac{\sigma}{\rho g}\right)^{3/2}$$ where $\sigma$ is the surface tension, $\rho$ is the density of the drop, and $g\approx 9.81\,\mathrm{m/s^2}$ is the acceleration due to gravity. For water at $20$°C, $\sigma\approx 72.8\,\mathrm{N/m}$ and $\rho\approx 1000\,\mathrm{kg/m^3}$, giving

$$V_c\sim 20\,\mathrm{\mu L}$$

More precise modelling requires knowing the contact angles of the bottom and top of the drop which tell us about the water-glass adhesion, e.g. see "On the ability of drops or bubbles to stick to non-horizontal surfaces of solids". A study of "Drops at Rest on a Tilted Plane" gave values in the range of $10-20\,\mathrm{\mu L}$ for the 90° critical angle for pure water on glass substrates treated in different ways. I'd estimate slightly smaller values ($\sim 5\,\mathrm{\mu L}$) from a quick experiment with my hand mirror, but it is pretty warm and humid.

This all assumes a drop of pure liquid water at room temperature and pressure on clean common window glass, but the results are reasonably robust to small variations.

Water surface tension varies by only about 20% from 0 to 100°C, and liquid water's density varies by only about 4%. Adding salts increases the surface tension and decreases adhesion. A saturated aqueous NaCl solution has surface tension about 15% higher. The contact angle increases from 40° for pure water to 55° for saturated aqueous NaCl solution on microscope cover glass, or from 48° to 76° for commercial glass. (The bigger the angle, the bigger the ratio of surface tension to adhesion.)

Surfactants have a more significant effects, however, with common detergents lowering surface tension by up to a factor of 3, so the glass needs to be very well rinsed after cleaning. Since detergent is extremely hydrophilic, if a thin layer is left on the glass, this could also effectively increase the adhesion since the water sticks to the detergent and the detergent sticks to the glass.