I was thinking about how would capillary action change in a tube (classic example) and in a tube fitted inside another tube (considering water as the liquid involved).

Height of liquid column: enter image description here where:

$\gamma$ = liquid-air surface tension

$\theta$ = contact angle

$\rho$ = density of liquid

$g$ = gravity acceleration

$r$ = radius

I tried my best to draw the examples I'm interested in order to help my explanation.

I didn't consider the capillarity inside the smaller tube in both example #2 and #3 because I'd like to assume that $a/2$ in example #1 is close to $c$ in example #2 and #3 (drawings not to scale).

enter image description here

Since from what I understand the column height is given, among other things (most of which can't be changed, like liquid-air surface tension, contact angle, density of liquid and gravity acceleration), by the tube radius, I'd like to know if $c$ in example #2 can be considered as $a/2$ in example #1 to calculate column height using above formula.

Also I'd like to know how having beads of slightly smaller diameter than $c$ between the two tubes (example #3) would affect the column height.

If said beads were less dense than water, could they still improve column height or would they just form a floating mat on top of 1 unit thickness?

What'd be the column height of example #2 and #3 assuming $c$ as 1mm?

I'm quite sure that given the same reached height $h$ in example #2 and #3, $c$ of #2 has to be smaller than $c$ in #3.


1 Answer 1


If $c\ll R$ the radius of your outer tube, the total curvature is approximately $\cos \theta/c$, so you will get $$h = \frac{\gamma}{\rho g} \frac{\cos \theta}{c}.$$

Beads will usually lower the apparent surface tension, so you'll get a lower column, although the amount of that depends on their wetting properties and of their arrangement (packing), see e.g. http://www.sciencedirect.com/science/article/pii/S0021979785711502

  • $\begingroup$ Agree with the first part of the answer, but not with the statement "beads will usually lower the surface tension". The article you quote talks about lowering the surface tension with hydrophobic beads. Hydrophilic beads would presumably increase it. Need that caveat I think... $\endgroup$
    – Floris
    May 31, 2015 at 12:50
  • $\begingroup$ It's some time ago now that I answered and I haven't looked very deep into the paper again, but it seems to me that the particles were hydrophilic on the contrary. If you switch from some liquid-bead contact angle and 180° minus that angle it should give the same surface shape by symmetry, shouldn't it? Anyhow, I believe there's no simple documented answer to the dependence of surface tension on any particle type. $\endgroup$
    – Joce
    Jun 1, 2015 at 15:24
  • $\begingroup$ From the linked article: "The surface activity of polystyrene spheres, on the other hand, is high especially for spheres of diameters between 100 and 200 nm. The maximum in - Δγ is ca. 20 mN/m. The large difference in the surface activity between the two kinds of spheres is due to the difference in surface characters, i.e., highly polar and strongly hydrophobic for silica and polystyrene spheres, respectively." $\endgroup$
    – Floris
    Jun 1, 2015 at 17:44

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