Contact angle

Law of Young-Dupré

Law of Young-Dupré

When a liquid (phase \(L\)) and a gas (phase \(G\)) are in contact with a solid wall, there is an equilibrium angle \(\theta^{eq}\) corresponding to the equilibrium of three capillary forces \(\vec{\sigma}_{Ls}\) the capillary force between Liquid and solid, \(\vec{\sigma}_{Gs}\) the capillary force between Gas and solid, and \(\vec{\sigma}_{LG}\) the capillary force between Liquid and Gas (see Fig. Contact-Angle-Concept). The contact angle \(\theta^{eq}\) can be expressed with the three surface tensions by the Young-Dupré law. Its derivation (see [1]) can be performed with the work \(d\mathcal{W}\) of a small displacement \(dx\) (see Fig. Young-Dupre-Derivation):

(502)\[d\mathcal{W}=(\sigma_{Gs}-\sigma_{Ls})dx-\sigma_{LG}dx\cos(\theta^{eq})\]

At equilibrium \(d\mathcal{W}=0\) and we obtain:

(503)\[\boxed{\cos(\theta^{eq})=\frac{\sigma_{Gs}-\sigma_{Ls}}{\sigma_{LG}}}\]

Fig. 51 Contact angle

Fig. 52 Derivation of Young-Dupré Law

Boundary condition for diffuse interface

Wall free energy

In the case of phase-field theory, the minimization is now carried out with an additional free energy: the wall free energy \(\mathscr{F}+\mathscr{F}_{w}\) where

(504)\[\mathscr{F}_{w}=\int_{\partial V}[(\sigma_{Gs}-\sigma_{Ls})p(\phi)+\sigma_{Ls}]d(\partial V)\]

where \(d(\partial V)\) is the wall surface, and \(p(\phi)\) is the interpolation polynomial \(p(\phi)=\phi^{2}(3-2\phi)\). The variation \(\delta(\mathscr{F}+\mathscr{F}_{w})/\delta\phi\) yields:

(505)\[\int_{\partial V}\biggl[\zeta\boldsymbol{\nabla}\phi\cdot\hat{\boldsymbol{n}}+\underbrace{(\sigma_{Gs}-\sigma_{Ls})}_{\text{use Young-Dupré law}}p^{\prime}(\phi)\biggr]d(\partial V)=0\]

where \(\hat{\boldsymbol{n}}\) is the normal vector at the wall boundary. After using the Young-Dupré law, we obtain the boundary condition for the phase-field \(\phi\) at the solid surface:

(506)\[\boxed{\zeta\boldsymbol{\nabla}\phi\cdot\hat{\boldsymbol{n}}=-\sigma_{LG}\cos(\theta^{eq})p^{\prime}(\phi)}\]

Remark: if \(\theta^{eq}=90{^\circ}\) then \(\boldsymbol{\nabla}\phi\cdot\hat{\boldsymbol{n}}=0\)

Fig. 53 Contact angle for diffuse interface

References

[1]

Pierre-Gilles de Gennes and Francoise Brochard-Wyart and David Quere. Capillarity and Wetting Phenomena: Drops, Bubbles, Pearls, Waves. Springer-Verlag, 2004. doi:https://doi.org/10.1007/978-0-387-21656-0.

Section author: Alain Cartalade