# Surface tension and capillary rise

The expression for the height rise in a capillary tube is well known, and the surface tension of the liquid air interface is involved. But as I understand the adhesion force between the water and glass molecules is responsible for this height rise. Then how can this expression be dependent only on the liquid air interface surface tension? Shouldn't the forces between the solid and liquid and liquid and liquid also be involved? Or is it so that the contact angle already is the result of all these forces, and that is why these do not appear in the expression?

• – user14250 May 15 '20 at 10:19

Your conjecture is correct: The rise of a liquid in a capillary is not just a function of the liquid-air surface tension but also the liquid-solid surface energy, AND this liquid-solid surface energy is present in the equation and its effect is represented by the contact angle parameter in the capillary rise equation.

Derivation of the capillary rise equation appears to be a bit involved, but there seems to be a good description here: Capillary Rise Equation Derivation

Note Figure 8.1 in the linked document: Even though it is the same liquid and same air for the two capillaries shown, in one capillary the rise is positive while in the other the rise is negative. The difference between the two capillaries? In one the contact angle is positive while in the other the contact angle is negative, presumably because the two capillaries are made of different materials so they have different liquid-solid surface energies.

You are right - the contact angle is indeed a function of the forces between the liquid and the wall. So when the capillary rise equation predicts the rise for a liquid with a given contact angle, it accounts for this effect.

So how does the contact angle relate to this energy? The Young-Dupré equation tells us that the relationship is:

$$\gamma(1+\cos\theta_c) = \Delta W_{SLV}$$

So given the energy $W_{SLV}$ per unit area for the liquid/solid interface, we can find $\theta$. As you can see, if the energy is equal to the surface tension of the liquid, then the contact angle will be 90°. When the energy is greater, the contact angle will be less than 90° - this corresponds to hydrophilic surfaces. For hydrophobic surfaces, it's the other way around.