# Capillary bridges

As the lengthscales of a system are decreased of some orders of magnitude the relative importance of physical laws drastically changes. It is the case of capillary forces in water which arises when the scale reaches the millimetr.

The capillary bridge or meniscus or liquid ring is a little amount of liquid that is stable between two particles or more generally between two surfaces.

Surface tension and properties of the liquid-solid interface enable liquid to be stick on particles. Indeed, the effect of gravity is negligible if the capillary volume is sufficiently low (low Bond number).

When two spherical particles share a capillary bridge they experience a normal attractive force $F_{cap}$ that is a function of:

1. the particle radius, $R$
2. the capillary volume, $V$
3. the distance (gap), $s$
4. the liquid surface tension, $\gamma$
5. the contact angle, $\phi$

Several methods can be used to compute the capillary force:

1. numerical solution of the Young-Laplace equation (e.g. Lian et al, 1993)
2. using empirical/approximated relationships starting from Young-Laplace equation (e.g. Soulié et al., 2006; Willet…)
3. numerically solving the minimum energy problem (e.g. with Surface Evolver; Brakke, 1992)
4. using approximate theoretical solution starting from the minimum energy approach (Israelachvili, 1992; Rabinovich et al., 2005; Lambert et al., 2008)

The latter leads to a simple analyitical formula suitable for coupling with the Discrete Element Method

$F_{cap} = \frac{ 2 \pi R \gamma \cos \phi }{ 1+ \left[ s / 2 d \left ( s , V \right ) \right ] }$

with

$d \left ( s , V \right ) = \frac{ s }{ 2 } \left ( -1 + \sqrt{ 1 + \frac{ 2 V }{ \pi R s^2} } \right)$

All the geometrical variables are sketched in Figure 1.

In this case, the capillary attraction force must be eventually added to the classical repulsive force at the contact.

The effect of the water content on a granular material was studied with reference to some collapse test carried out in the laboratory and modelled with DEM.

Here are some results:

### References

Gabrieli, F., Lambert, P., Cola, S., & Calvetti, F. (2012). Micromechanical modelling of erosion due to evaporation in a partially wet granular slope. International Journal for Numerical and Analytical Methods in Geomechanics, 36(7), 918–943. https://doi.org/10.1002/nag.1038
Gabrieli, F., Artoni, R., Santomaso, A., & Cola, S. (2013). Discrete particle simulations and experiments on the collapse of wet granular columns. Physics of Fluids, 25(10). https://doi.org/10.1063/1.4826622
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