Contact angle and Marangoni force

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. Fig. 56). 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. Fig. 57):

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

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

(505)\[\boxed{\cos(\theta^{eq})=\frac{\sigma_{Gs}-\sigma_{Ls}}{\sigma_{LG}}}\]
../../../_images/Contact-Angle.png

Fig. 56 Contact angle

../../../_images/Deriving_Young.png

Fig. 57 Derivation of Young-Dupré Law

Boundary condition for diffuse interface

Wall free energy

In the case of phase-field theory, the interface is diffuse (see Fig. 58) and the minimization is now carried out with an additional free energy: the wall free energy \(\mathscr{F}+\mathscr{F}_{w}\) where

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

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:

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

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:

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

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

../../../_images/Diffuse_Contact-Angle.png

Fig. 58 Contact angle for diffuse interface

Marangoni force

Difference of normal stress tensor between liquid and gas

Normal stress
(509)\[\begin{split}(\overline{\overline{\boldsymbol{T}}}_{l}-\overline{\overline{\boldsymbol{T}}}_{g})\cdot\hat{\boldsymbol{n}}&=\boldsymbol{\nabla}\cdot\bigl\{\sigma(c)\bigl[\overline{\overline{\boldsymbol{I}}}-\hat{\boldsymbol{n}}\otimes\hat{\boldsymbol{n}}\bigr]\bigr\}\\ &=-\underbrace{\sigma(c)\kappa\hat{\boldsymbol{n}}}_{\text{Normal}}+\underbrace{{\color{red}\boldsymbol{\nabla}_{S}\sigma}(c)}_{\text{Tangential}}\end{split}\]

where

  • \(\overline{\overline{\boldsymbol{T}}}_{\Phi}=-p_{\Phi}\overline{\overline{\boldsymbol{I}}}+\eta_{\Phi}(\boldsymbol{\nabla}\boldsymbol{u}+\boldsymbol{\nabla}\boldsymbol{u}^{T})\): stress tensor with \(\Phi=l,g\)

  • \(\hat{\boldsymbol{n}}\): normal vector

  • \(\kappa=\boldsymbol{\nabla}\cdot\hat{\boldsymbol{n}}\): interface curvature

  • \(\boldsymbol{\nabla}_{S}\,\hat{=}\,\boldsymbol{\nabla}-\hat{\boldsymbol{n}}(\hat{\boldsymbol{n}}\cdot\boldsymbol{\nabla})\): gradient along surface

Normal force

In Eq. (509), if \(\sigma\) is a constant then the normal force is the capillary force:

(510)\[\boldsymbol{F}_{c}=-\delta_{\Sigma}\sigma\kappa\hat{\boldsymbol{n}}\]
Marangoni force

In Eq. (509), if \(\sigma\) is a function of composition \(\sigma(c(\boldsymbol{x},t))\) or temperature \(\sigma(T(\boldsymbol{x},t))\), then

(511)\[\boldsymbol{F}_{s}=\boldsymbol{F}_{c}+{\color{red}\boldsymbol{F}_{M}}=\delta_{\Sigma}(-\sigma\kappa\hat{\boldsymbol{n}}+{\color{red}\boldsymbol{\nabla}_{S}\sigma})\]

Usual function of \(\sigma\) as function of \(c\) or \(T\)

Linear
(512)\[\sigma(c) =\sigma_{ref}+\frac{d\sigma}{dc}(c-c_{ref})\]

where \(d\sigma/dc=\sigma_{c}<0\)

Logarithm
(513)\[\sigma=\sigma_{ref}\left[1+\beta\ln(1-\frac{c}{c_{\infty}})\right]\]

Marangoni force in the phase-field framework

Surface tension force
(514)\[\boldsymbol{F}_{s}=\delta_{d}[-\sigma\kappa_{\phi}\boldsymbol{n}_{\phi}+\boldsymbol{\nabla}_{s}\sigma]\]
Capillary and Marangoni forces
\[\begin{split}\boldsymbol{F}_{c}&=\mu_{\phi}\boldsymbol{\nabla}\phi\\\boldsymbol{F}_{M}&=\frac{3W}{2}\left[\left|\boldsymbol{\nabla}\phi\right|^{2}\boldsymbol{\nabla}\sigma-(\boldsymbol{\nabla}\phi\cdot\boldsymbol{\nabla}\sigma)\boldsymbol{\nabla}\phi\right]\end{split}\]

Proof for \(\boldsymbol{F}_{M}\)

In (514) use definition of

\[\begin{split}\boldsymbol{\nabla}_{s}&\hat{=}\boldsymbol{\nabla}-\boldsymbol{n}_{\phi}(\boldsymbol{n}_{\phi}\cdot\boldsymbol{\nabla})\\ \boldsymbol{n}_{\phi}&\hat{=}\frac{\boldsymbol{\nabla}\phi}{\left|\boldsymbol{\nabla}\phi\right|}\\ \delta_{d}&\hat{=}\frac{3W}{2}\left|\boldsymbol{\nabla}\phi\right|^{2}\end{split}\]

Use definition of

\[\begin{split}\boldsymbol{F}_{M}\,&=\,\delta_{d}\boldsymbol{\nabla}_{s}\sigma\\ \,&=\,\frac{3W}{2}\left|\boldsymbol{\nabla}\phi\right|^{2}[\boldsymbol{\nabla}\sigma-\boldsymbol{n}_{\phi}(\boldsymbol{n}_{\phi}\cdot\boldsymbol{\nabla}\sigma)]\\ \,&=\,\frac{3W}{2}\left|\boldsymbol{\nabla}\phi\right|^{2}\left[\boldsymbol{\nabla}\sigma-\frac{\boldsymbol{\nabla}\phi}{\left|\boldsymbol{\nabla}\phi\right|}\left(\frac{\boldsymbol{\nabla}\phi}{\left|\boldsymbol{\nabla}\phi\right|}\cdot\boldsymbol{\nabla}\sigma\right)\right]\\ \,&=\,\frac{3W}{2}\left[\left|\boldsymbol{\nabla}\phi\right|^{2}\boldsymbol{\nabla}\sigma-(\boldsymbol{\nabla}\phi\cdot\boldsymbol{\nabla}\sigma)\boldsymbol{\nabla}\phi\right]\end{split}\]

with chain rule for \(\boldsymbol{\nabla}\sigma=(\partial\sigma/\partial c)\boldsymbol{\nabla}c\)

(515)\[\boldsymbol{F}_{M}=\frac{3W}{2}\frac{\partial\sigma}{\partial c}\left[\left|\boldsymbol{\nabla}\phi\right|^{2}\boldsymbol{\nabla}c-(\boldsymbol{\nabla}\phi\cdot\boldsymbol{\nabla}c)\boldsymbol{\nabla}\phi\right]\]

References

Section author: Alain Cartalade