Axial Capillary Forces

Abstract

This chapter presents classical techniques to compute the shape of a meniscus using the energetic method and its numerical implementation using Surface Evolver. It provides the numerical solution of the Laplace equation for axially symmetric configuration and some useful analytical approximations (circular or toroidal approximation, parabolic approximation). Formal equivalence between these approaches is given and results are provided as rules of thumbs for the designer.

Appendix

1.1 Capillary Force Developed by a Meniscus at Equilibrium

Let us consider the meniscus depicted in Fig. 2.20. At equilibrium, the sum of forces acting on any slice of the meniscus must be equal to zero. On each face, capillary forces can be split into two contributions: the so-called tension force $F_T$, due to the action of surface tension along the tangent to liquid-gas interface, and the Laplace or (also called) capillary force $F_L$ originating from the pressure acting on the face.

Fig. 2.20

Equilibrium of a meniscus slice comprised between $z$ and $z+dz$. It can be shown that the capillary force computed at height $z$ exactly balance the the capillary force computed at height $z+dz$

The force exerted on the bottom face of the slice is equal to:

$$\begin{aligned} \bar{F}(z)=\left( \underbrace{\pi r^2 \Delta p}_{F_L} - \underbrace{2 \pi r \gamma \cos u}_{F_T}\right) \bar{1}_z \end{aligned}$$

(2.46)

while the force acting on the top face of the slice is given by:

$$\begin{aligned} \bar{F}(z+\mathrm{{d}}z)&= \left( -\pi (r+\mathrm{{d}}r)^2 \Delta p + 2 \pi (r+\mathrm{{d}}r) \gamma \cos (u+\mathrm{{d}}u)\right) \bar{1}_z\nonumber \\&= \left[ -\pi r^2 \Delta p + 2 \pi r \gamma \cos u+2\pi (\underbrace{-r \mathrm{{d}}r \Delta p +r \gamma \sin u \mathrm{{d}}u +\gamma \cos u \mathrm{{d}}r}_{I}))\right] \bar{1}_z\nonumber \\ \end{aligned}$$

(2.47)

The underbraced expression $I$ can be shown to be equal to zero by expressing the Laplace law (Eq. (2.24)):

$$\begin{aligned} \Delta p&=2H \gamma \nonumber \\&=\left( -\frac{r^{\prime \prime }}{(1+r^{\prime 2})^{3/2}}+\frac{1}{r(1+r^{\prime 2})^{1/2}}\right) \gamma \end{aligned}$$

(2.48)

$$\begin{aligned}&=\left( \frac{\mathrm{{d}}u}{\mathrm{{d}}s}+\frac{\cos u}{r}\right) \gamma \end{aligned}$$

(2.49)

$$\begin{aligned}&=\left( \frac{\mathrm{{d}}u}{\mathrm{{d}}r}\sin u+\frac{\cos u}{r}\right) \gamma \end{aligned}$$

(2.50)

leading to:

$$\begin{aligned} I=-r \mathrm{{d}}u \sin u \gamma -\mathrm{{d}}r \cos u \gamma +r \gamma \sin u \mathrm{{d}}u +\gamma \cos u \mathrm{{d}}r=0 \end{aligned}$$

(2.51)

Consequently, the forces $\bar{F}(z)+\bar{F}(z+\mathrm{{d}}z)$ balance, and the capillary force given by $F(z)$ can be computed at any value of $z$. This means that in the case of two solids linked by a liquid meniscus, the force can be computed on the top component or on the bottom component. For the sake of convenience, it can also be computed at the neck in case the latter exists (it may not exist if the extremum radius of the meniscus corresponds to one of both wetting radii).

1.2 Equivalence of Formulations

A lot of work has been reported on capillary forces modeling (see for example [1, 4–6, 9, 12–14]), based on the energetic method (i.e. derivation of the total interface energy) or a direct force computation from the meniscus geometry, the latter being either determined exactly through the numerical solving of the so-called Laplace equation or approximated by a predefined geometrical profile such as a circle (i.e. toroidal approximation) or a parabola. The energetic approach is usually quite clear on its approximations: the liquid-vapor interface energy is sometimes neglected in order not to compute the exact shape of the meniscus, but an exact solution can be found if the lateral is computed for example by mean of a finite element solver such as Surface Evolver. At the contrary, literature results are not so clear as far as the other method is concerned. For example, some authors neglect the so-called tension term with respect to the Laplace term . This sometimes pertinent assumption has led many author authors to add the tension term to the result obtained by deriving the interface energy, i.e. to mix both methods. A recent by one of the authors [15] contributed to clarify this situation by showing that the capillary force obtained by deriving the interfacial energy is exactly equal to the sum of the Laplace and tension terms. The equivalence is considered with three qualitative arguments, and an analytical argument is developed in the case of the interaction between a prism and a place. Experimental results also contributed to show this equivalence.

Mathematically, the equivalence between the energetic approach and the direct formulation based on the Laplace and the tension terms can be shown:

$$\begin{aligned} F=F_{\mathrm{L }}+F_{\mathrm{T }}=-\frac{\mathrm{{d}}W}{\mathrm{{d}}z} \end{aligned}$$

(2.52)

where $F_\mathrm{L }$ and $F_\mathrm{T }$ are given by  (2.46), $W$ by  (2.1). $z$ is the separation distance between both solids.

As it is shown that both approaches are equivalent, it means that the energetic approach already involves the tension term and the Laplace term on an implicit way. Consequently, the energetic approach as proposed by Israelachvili (see (2.8)) includes both terms, even if, for zero separation distance, the pressure term usually dominates the tension one. For axially symmetric configurations, the method based on the Laplace equation will be preferred because it can be easily numerically solved.