Owens - Wendt Model for Surface Energy of Solid - Liquid Interface

[Dario56]

Owens - Wendt model is used for calculating surface energy on liquid - solid interface and it is given by following equation: $$ \gamma_{sl} = \gamma_s + \gamma_l -2(\sqrt {\gamma_l^d \gamma_s^d} + \sqrt {\gamma_l^p \gamma_s^p}) $$

So, if we use liquid and solid of known surface energy as well as their components (dispersive and polar contributions) we can calculate surface energy of the interface.

It is stated that model has 2 assumptions:

1) Total surface energy of any individual component (solid and liquid) is a sum of polar and dispersion contributions

2) Dispersion and polar interactions between solid and liquid on the interface contribute to decrease of surface energy of the interface as geometric mean of individual contributions

Given these assumptions:

1) How is the equation of this model derived (equation written in the question)?
2) Why is there number 2 multiplying geometric mean contributions for decreasing surface energy of the interface since formula for geometric mean doesn't include that number?

 

[TeethWhitener]

How is the equation of this model derived (equation written in the question)?

Start with Young's equation:
$$\gamma_{sl}=\gamma_{s}-\gamma_{l}\cos{\theta}$$
where $\theta$ is the contact angle. This is the equilibrium expression of the solid-liquid, solid-gas, and liquid-gas interfacial tensions.
Defining the work of adhesion as
$$W_{ab}=\gamma_a+\gamma_b-\gamma_{ab}$$
and plugging into Young's equation gives us the Young-Dupre equation:
$$W_{sl}=\gamma_l(1+\cos{\theta})$$
This leads into your second question:

Why is there number 2 multiplying geometric mean contributions for decreasing surface energy of the interface since formula for geometric mean doesn't include that number?

We can imagine a single layer of molecules at the liquid-gas (or solid-gas) interface as being attracted by the liquid (solid) with surface tension $\gamma_l$ (or $\gamma_s$). If we then think about the solid-liquid interface, we see that the attraction of the molecules at the interface for the bulk phase (so liquid-liquid attraction or solid-solid attraction) is fighting against the attraction of the molecules for the other phase. So instead of $\gamma_{sl}$ being additive ($\gamma_{sl}=\gamma_s+\gamma_l$), the interfacial tension is lessened by the attraction by the interfaces: $\gamma_{sl}=(\gamma_s-f_{sl})+(\gamma_l-f_{sl})$. Since the liquid and solid each have an interfacial tension that's being lessened by mutual attraction, we get a factor of two.
The rest is basically empirical. As you pointed out, the lessening enters into the equation as the geometric mean of the two tensions: $f_{sl}=\sqrt{\gamma_s\gamma_l}$. In fact, this was basically Good's equation in a nutshell. You'll notice that substituting $\gamma_s=\gamma_s^d+\gamma_s^p$ and $\gamma_l=\gamma_l^d+\gamma_l^p$ gives
$$\gamma_{sl}=\gamma_s+\gamma_l-2\sqrt{(\gamma_s^d+\gamma_s^p)(\gamma_l^d+\gamma_l^p)}$$
instead of
$$\gamma_{sl} = \gamma_s + \gamma_l -2(\sqrt {\gamma_l^d \gamma_s^d} + \sqrt {\gamma_l^p \gamma_s^p})$$
Part of the discrepancy is explained by assuming (as Fowkes did) that dispersive forces only interact with dispersive forces and polar forces only interact with polar forces. I'm not particularly comfortable with that, but it's what was done in the original papers. Of course that only gets you partway there and doesn't really explain the separation of the square roots. I'm not really sure what drove this decision (and I don't have access to Owens and Wendt's original paper right now). Probably it fit the data better. Maybe someone else can provide some more insight.