Breaking apart the surface tension of a water droplet

Question

How far apart do two vertical glass surfaces have to stand for a water droplet that is sticking to both of them to break into two?

I apologize in advance for the question possibly being too imprecise 😅.

Answer 1

Such a capillary bridge will break when either surface tension overcomes adhesion or when gravity overcomes them both. For water, we expect the bridge to break before it separates by a distance roughly equal to the diameter of a spherical drop having the same volume as the bridge.

As shown in this diagram from Wikipedia, for a water and glass bridge will have a concave shape.

Ignoring Gravity

For a drop of water with constant volume $V$, the contact radius $R$ and the waist radius $r_m$ of the bridge shrink as the plate separation $H$ increases until the bridge breaks. We can estimate a rough upper limit on $H$ by comparing the water surface energy ($2 \pi r_\mathrm{cyl} \sigma)$ of a cylindrical bridge to the surface energy ($4\pi r_{\mathrm{sph}}^2 \sigma$) of a spherical drop with the same volume. The surface tension $\sigma$ drops out and we find that converting to a spherical drop is energetically favourable where $H\gtrsim 2 r_{\mathrm{sph}} = (6V/\pi)^{1/3}\approx 1.2 V^{1/3}$.

We can get a more precise answer from this Wikipedia diagram (original from Petkov & Radoev) showing the relationship between where $R^*\equiv R/V^{1/3}$ and $H^*\equiv H/V^{1/3}$:

The different curves correspond to different contact angles $\theta$. For perfectly clean smooth silica glass, the contact angle for pure water can be close to $0^\circ$, but contact angles up to almost $60^\circ$ can be found online for less clean, less smooth, or otherwise different glass/water interfaces. We can see that for a cylindrical bridge ($\theta \sim 90^\circ$), our rough estimate of $H^*_\mathrm{max}\sim 1.2$ is pretty close. For the smallest contact angle ($\theta= 1^\circ$), $H^*_\mathrm{max}\sim 0.5$.

Effect of Gravity

Using dimensional analysis (as in "Gravity vs adhesion of a water droplet"), we expect gravity to break the bridge if the separation distance is

$$H\gtrsim \left(\frac{\sigma}{\rho g}\right)^{1/2}$$

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

$$H_\mathrm{max}\sim 3\,\mathrm{mm}$$

This means that to study really large capillary bridges, you'd need to book time on the vomit comet or the ISS.

Answer 2

The answer is that the droplet creates a minimal surface where the contact angle is tangential to this minimal surface.

Thus this question can not be answered before this contact angle of the glass is defined. The breaking will nevertheless happen in the saddle point of this shape.