# Capillary Action and Young-Laplace

1. Feb 2, 2013

### EtherealPanMan

Hello all. This is my first post and I wasn't exactly sure where to put it, so I apologize if it could be in a better place.

Ok... here is my issue. I am currently enrolled in a Surfaces/Interfaces course. Capillary action eludes me. I do not understand WHY capillary action occurs in relation to interface pressure differences. I have attached the slide in question.

My biggest confusion is the line "because of the pressure difference...". What about the pressure difference causes the liquid to rise up the tube? It seems counter intuitive. If the surface is curved as shown, this implies that the pressure of the air is greater than the pressure of the liquid (ΔP wrt to liquid is negative). It seems like the greater pressure would push DOWN on the water and cause it to go down the tube....

Any explanation of this would be VERY helpful! Cheers!

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2. Feb 2, 2013

### Jano L.

Your idea is understandable, but this is not how surface tension works. Unfortunately the slide does not give very accurate explanation either. Here is one way to understand it:

When the pipe is immersed in water, water comes into contact with glass. In the end, the water will assume such position that the whole system is in equilibrium (does not move).

Now, water "loves" glass if the latter is clean enough (no fat on it). Some water near the wall gets higher to touch the glass and thus increase the total area of contact with it. That is why the boundary water-air is not plane but forms a meniscus. Initially, this meniscus has unknown shape (it is improbable that it is spherical).

But when the meniscus is formed, instantly the pressure in the water just beneath the surface tends to decrease. This is because the force of the atmospheric pressure pushing surface down from above is partly balanced by the capillary forces due to glass, acting on the water on the circumference of the water surface, pulling water up.

As a result, the water in the pipe experiences two pressures from above and from below, and that from below is slightly higher. For this reason, water starts rising up.

Roughly speaking, the water rises as long as the pulling force of the glass is stronger than the force of gravity pulling the water down. The equilibrium height of the water surface is such that the two forces are equal. If $r$ is radius of the sphere which meniscus is a part of, we have equation of equilibrium

magnitude of decrease of pressure force from above = weight of the column of water above the normal water level

$$\frac{2\sigma}{r} \pi a^2 = \rho g h \pi a^2$$

Using $r = a/ \cos \theta$, you can find the equilibrium height $h$ for given pipe.

This analysis is correct for water; it rises and has convex surface (U).

If the liquid is mercury, the surface tension force will push the mercury down at the contact rim, so the surface will set down and have concave surface ( U upside down).

3. Feb 4, 2013

### Andy Resnick

The Laplace equation (ΔP = 2σκ) relates the interface curvature κ to a pressure jump across the interface, and is derived from conservation of energy. Young's equation represents mechanical equilibrium at a three-phase contact line (where the solid and both fluid phases meet). The slide appears to connect the two in an unclear manner.

Capillary rise (or the converse, depending on the contact angle), meaning the height of a fluid column, is also a force balance- a balance of the weight of fluid and 'surface tension force' from Young's equation.

Does this help?