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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 😅.

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  • $\begingroup$ Thanks! 😁 I did eventually conduct a small experiment, which gave me a distance of just under 3mm. But it’d be nice to have it confirmed with math & a proper scientific explanation. $\endgroup$ Aug 28, 2023 at 19:16
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    $\begingroup$ I'm also excited to see an answer. I expect the approach will be related to the variational approach used to analyze the Rayleigh instability point (i.e., at some point perturbations decrease the surface area rather than increasing it), but I'm not a pro at this approach. $\endgroup$ Aug 28, 2023 at 20:49

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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.

Concave capillary bridge schematic representation

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 $H^*\equiv H/V^{1/3}$: enter image description here

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.

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    $\begingroup$ I absolutely love the way you analyzed the question - thank you so much! ☺️ I can assure you that this question has a purpose, I'm designing a component in fusion 360 that interacts with water and I need to keep the water from sticking to both ends. $\endgroup$ Sep 3, 2023 at 11:00
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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.

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