Need help understanding dynamic and static contact angles

Question

I've run into a conceptual road block. I'm coming to you guys because I think my adviser is getting annoyed with me. The concept involves a meniscus being pulled up a cylinder. I understand that the main force "pulling" on the meniscus at the contact line is due to the difference between the static and dynamic contact angles. This force per unit length takes the form:

$F=\gamma (Cos(\theta_{static})-Cos(\theta_{dynamic}))$

The concept that describes determining the static and dynamic contact angles have been given to me in terms of minimizing the interface between the liquid and air when the meniscus has reached a known height. In the words of someone much more knowledgable than me:

"The fluid interface only knows about the equilibrium contact angle as a boundary condition. For any circle and any height there is a minimum energy configuration of the fluid surface. All of these solutions have a different angle. You find the equilibrium by finding the minimum energy solution that also intersects the solid at the contact angle."

For some reason, I am unable to see where I can explore these concepts to determine a relationship between the dynamic and contact angles. Can anyone offer elucidation as to these concepts?

Thanks

Answer 1

I think you are confusing two things here:

1) On one hand you have the effect of capillarity, where a liquid near a solid will adjust the contact angle according to the balance of surface tensions between the liquid, gas and solid. This happens according to the static contact angle $\theta_{static}$ and only over a range of the capillary length $\sqrt{\frac{\rho g}{\gamma}}$. See e.g. wetting at mit.edu or the wiki on capillary action

2) On the other hand you have the dynamic contact angle which occurs due to the balance of viscous forces, which scale as $\mu u$ and capillary force, which scale as $\gamma \left(\cos\theta_{static}-\cos\theta_{dynamic}\right)$ (note that both forces are per unit length here!). This occurs when you drag a plate or a cylinder out of a liquid bath at a non-zero speed, $u$.

The balance I write here is only conceptually correct. If you want the true calculation of the dynamic contact angle you need a more involved mathematical derivation which leads to the so-called Cox-Voinov model (a decent explanation by one of the current leading scientists in this field can be found here). The Cox-Voinov model reads $\theta_{dynamic}^3=\theta_{static}^3+9\frac{\mu u}{\gamma}ln\frac{x_{max}}{x_{min}}$. Where $x_{min}$ and $x_{max}$ are a small and a large-scale cut-off length respectively associated with some molecular scale and the size of your droplet or the capillary length (whichever is smaller).

Answer 2

Still working on this? One interesting experiment to explain these ideas is that of a water droplet resting on a flat plate. The contact angle all around is equal and a function of the surface properties of the plate. Now pick up one end of the plate slightly such that the droplet moves slowly downhill. The contact angle on the high and low ends are changed (with the low end angle increased and high end angle descreased) because of the dynamic effect.