Condition for a liquid to stick to a surface

Here is a way to think of this problem which I hope will help get you started. I'll do the setup and you'll finish it.

Draw yourself a picture of a droplet of liquid in profile, sitting atop a solid surface, surrounded by air. The so-called "triple junction" is where solid, liquid, and air meet at the edge of the droplet. We will focus on the junction on the right-hand edge of the droplet.

Now zoom in on that triple junction and draw a dot right at the junction. you can now draw in three vectors with their origins at the dot. One points to the right and runs parallel to the solid surface, one points to the left and runs parallel the solid surface, and the third is tangent to the curve of the liquid droplet and points up into the air.

Your homework (see wikipedia for "contact angle") is to associate those three vectors with the solid-air, liquid-air, and solid-liquid surface energies. Then you can see 1) where the contact angle of the droplet at the triple junction comes from, 2) why air enters into the picture, and 3) whether the droplet wants to spread across the solid surface or curl up into a ball.

Defining wettability in terms of surface energy is just an alternative way to defining it in terms of cohesive and adhesive forces.

Using these two approaches, one based on energy and the other based on forces, is common in physics. For instance, we can say that a body falls on the ground because it is attracted by the gravitational force or we can say that it falls because it minimizes its potential energy. Both approaches lead to the same result.

Similarly, if, as part of a chemical reaction, an electron moves from atom A to atom B, we can say that it is due to a stronger force of attraction by atom B or we can say that it is due to a lower energy state provided by atom B.

So, saying that "a fluid will stick to a surface if the surface energy between fluid and solid is smaller than the sum of surface energies between solid air and fluid air", is roughly equivalent to saying that the adhesive forces between the fluid and the solid are stronger than the cohesive forces within each of the two.

We can, simplistically, say, that for a molecule of a fluid to form a "bond" with a molecule of a solid, both have to give up "bonds" with molecules of their own and this is going to happen only if the energy of a new "bond" is lower then the energy of the old "bonds", which would be the case, when the adhesive forces are, on balance, stronger than the cohesive forces.

The terms "fluid air" and "solid air" are apparently used to describe surface energy of a solid and surface energy of a fluid (surface tension), associated with cohesion forces, in the absence of a contact with other solids or fluids.