# Derjaguin formula

1. May 26, 2014

### LouisD

Hi everyone,
Decades ago, Derjaguin calculated formula which was able to predict the contact angle of a droplet on a solid substrate. The droplet and the solid substrate are separated by a thin liquid interlayer.
This formula was

cos(θ) = 1 + G(h0)/γ

where θ is the contact angle of the droplet on the solid substrate, h0 is the equilibrium distance between the droplet and the droplet (i.e the width of the thin liquid interlayer) and γ the surface tension of the liquid (water here).
The calculation of G(h0) leads to the following expression :

G(h0) = - A/(12.$\pi$.h0²)

with A the Hamaker constant of the system. This energy corresponds to the Lifshitz energy.
Here is my question : When you're calculating the Lifshitz energy between two materials (solid or liquid) you need to have between them a liquid or a gas. In this case, you just have air-water-solid substrate. I don't understand how it is possible to calculate the energy then, can somebody help me with that ? As I'm not a very good english speaker, I hope my question is clear otherwise don't hesitate to ask me to clarify any details.

For more detail one this, http://arxiv.org/pdf/1212.6583.pdf

Thanks you

2. Jun 11, 2014

### Greg Bernhardt

I'm sorry you are not generating any responses at the moment. Is there any additional information you can share with us? Any new findings?

3. Jun 13, 2014

### hilbert2

I needed the Derjaguin formula several times when I was writing my master's thesis on the behavior of thin liquid films on solid substrates. The formula we used was $\cos \theta = 1 + \frac{1}{\gamma}\int_{h*}^{\infty}\Pi (h)dh$, where $\theta$ is the contact angle, $\gamma$ is the surface tension, $h^*$ is precursor film thickness and $\Pi (h)$ is the disjoining pressure as a function of film thickness.

I'm not sure I really understand LouisD's question... What is Lifshitz energy? What do you mean by needing to have something between the two phases for which the energy is calculated?

4. Jun 26, 2014

### itaischles

I think I understand your question. The disjoining pressure in your case is made of van der Waals energy (specifically - London-Lifshitz dispersion forces). You said yourself that there is a thin film between the substrate and the liquid. In that case that is the medium which is used in the formula. The air plays no role here since it is far above the surface and you can probably assume infinite thickness for the liquid.