# Calculating the equilibrium distance

1. Jul 8, 2006

### Clausius2

Hi there,

Let's imagine a sphere touching a plane. Then, I may consider valid the law for adhesion coming from VDW forces:

$$F=\frac{Aa}{6h^2}$$

where A is the Hammaker constant, a is the sphere radius and h is the gap between the two bodies. My question is, is there a way of calculating the equilibrium distance $$h_o$$ where the force reaches its maximum? What cannot be is that as h tends to zero F tends to infinity. There must be some point of equilibrium for very small h in which maybe repulsion forces start to take place. Is that right?

2. Jul 9, 2006

### Astronuc

Staff Emeritus
I guess I'd like to see the derivation of this formula.

F clearly cannot go to infinite.

What bothers me is "h is the gap between the two bodies." How does one achieve adhesion with a gap between bodies?

Adhesion should be a function of surface roughness, in addition to VDW. Surface roughness would affect the effective surface area involvedin VDW. And there could also be a softness (hardness) component.

3. Jul 10, 2006

### Gokul43201

Staff Emeritus
Typically, in Hammaker models, the separation between particles is a center-to-center spacing and the global maximum of the inter-particle force is at "contact". By contact, one roughly means that the particles are close enough (of the order of an angstrom between surfaces) that repulsion from surface electrons equals the attractive VDW force.

Clearly, the (Hammaker) force equation provided above models only part of the forces present in the system, since it has no horizontal tangent at finite separation (ie: it provides no finite equilibrium separation).

Last edited: Jul 10, 2006
4. Jul 10, 2006

### Clausius2

It is obtained integrating the interaction energy (see fundamentals of adhesion, Lee). You're right about the roughness, h models an average gap between both surfaces.

Yeah, I did a little of research and I found that the equilibrium distance is usually of order of angstroms. What bothers me now is that I couldn't assume Continuum flow of air in such a small gap.

Thanks guys.

5. Jul 10, 2006

### Gokul43201

Staff Emeritus
Just to satisfy my curiosity, what is the system you are trying to model?

6. Jul 10, 2006

### Clausius2

No problem. Actually, it is for my thesis. About Electromechanics of Particles, in particular I'm looking now at a micro particle near a wall at Low Reynolds Numbers. It's funny.