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?
Surface Tension – Understanding Capillary Rise
capillary-actionfluid-staticssurface-tension
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Surface tension is a quite confusing subject, especially viewed from a purely mechanical point of view. It appears whenever you have an interface between a condensed phase say $A$ and another immiscible fluid phase $B$.
Thus the first thing to note is that surface tension has always to do with an interface. The surface tension coefficient often denoted $\gamma_{A,B}$ will tell how "costly" it is, in term of energy, for such an interface to exist.
Now the reason why it is costly to have such an interface is ultimately due to the effective adhesion forces between the molecules in each phase. To simplify a bit, there are two principles at play:
(1) In a quite good approximation, molecules interact with van der Waals (vdW) interactions which are always attractive (in vacuum). Furthermore, the vdW forces are the strongest with molecules of the same kind.
(2) In a dense phase of certain molecules, the cohesive energy density is higher than for the same molecules in a more dilute phase.
These two rules have two implications:
If phases $A$ and $B$ comprises the same molecules but have very different densities (e.g. liquid water/water vapour interface), then by the rule (2) there is a big loss in cohesive energy density for each piece of interface created between the two phases. From a mechanical point of view, it is fine to say that molecules in the liquid phase are simply pulled stronger towards the liquid phase than the gas phase.
If phases $A$ and $B$ are two condensed phases comprising different molecules, then by the rule (1), it is also costly to generate an interface between $A$ and $B$.
This leads to the property that the surface tension coefficient $\gamma_{AB}$ is always positive.
Now, in most real cases, multiple interfaces are involved at the same time. Most of the time three interfaces. This is the case for the meniscus you mention but also for the insects walking on water.
To discuss the insect example, one needs to guess whether its legs are wetting or not. If they were, then it is likely that it could not walk on water as it would be preferable for it to actually sink in water. It must have quite a lot of short straight hairs on the legs to induce a hydrophobic effect effectively "repelling" water and inducing only a single contact point with water and then one only needs to care about the deformation of the water/air interface.
Now, regarding the direction of the force, one needs to discriminate two things:
the total interaction between a phase $A$ and a phase $B$
the surface tension interaction between phase $A$ and $B$
While the former accounts for all possible forces between the phases, the latter is only concerned with the shape of an interface and acts by definition tangentially to the interface.
For example, in the first example you mention, this is a mixture of both:
First, the liquid wets the rope which more or less implies a strong adhesion with it, second the liquid exerts a tension related to the $\gamma_{air/soap}$ interface which acts along the interface air/soap but perpendicular to the interface rope/soap; that's mainly because we consider ourselves in a case of ultra-ideal wetting. Thus what it says is that Nature prefers gaining a bit of energy by extending the interface air/soap a bit rather than gaining a much bigger amount of energy by detaching the rope or whatever object you might use from the soap film.
Try the same experiment with a tube made of GoreTex, I am not sure you would get the same outcome.
Surface tension is a phenomenon which occurs irrespective of whether a solid surface is in the vicinity. It is the result of the discontinuity in molecular attractive forces present at the free surface. The translates into a situation in which the surface behaves as if there is an elastic membrane embedded within the surface. The membrane force acts locally in the direction tangent to the surface, and its value per unit length is equal to the surface tension. If the fluid surface is curved (as in the capillary case), the membrane can support a pressure differential across the surface (sort of the way a balloon supports a pressure difference from inside to outside).
When the fluid surface makes contact with a solid surface (as at the capillary wall), there are three phases in mutual contact (solid, liquid, and gas). The contact angle between these three phases is a unique function of the solid and liquid involved.
The force balance in your book takes into account both these effects (i.e., the pressure difference resulting from surface tension, and the surface tension force acting at the capillary wall at the contact angle).
Best Answer
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.