Effects of volume conservation on Plateau border sag in foams

Abstract

In the sagging Plateau border model, borders which are loaded with the weight of the liquid that they contain are distorted normal to their length and in varying amounts along their length. Applied to a 3D foam structure, this would result in artificial variations in bubble size/volume, so a Surface Evolver model is developed for conserving the volume of bubbles adjacent to the sagging Plateau border. As a result of this modification to the sagging Plateau border model, the sag is reduced and the adjacent films bulge. An analytical solution for the case of conserved bubble volume shows good agreement with the Surface Evolver simulations. With increased loading the three-film structure collapses under its own weight–as is also the case for unconserved bubble volume. However, the loading that can be supported by the structure is greater with volume conservation than without. Finally, to extend these theoretical results and to obtain an outlook for the use of the sagging border model to simulate the limit of stability of foam structure, the onset of structural rearrangement or ‘convective (structural) instability’ is estimated from collapse in the Surface Evolver simulations.

Acknowledgements

BE was supported by an EC Marie Curie Fellowship (MEST-CT-2004-503750).

Notes

†Note that there is a prefactor error in the equation in 7. The prefactor given here is correct.

†Similarly, the estimate for c ₁₀ can be repeated for another periodic foam structure–e.g. the Weaire–Phelan structure 24, for which the average number of borders per face is 5.111 and the average film area is 1.818L ². This gives (L_f /L)_eff ≈ 0.711 and f ₂(L_f /L) ≈ 0.037. We redefine c ₁₀ = f ₂(L_f /L)/(3.35 cos α) based upon the geometry of an average Weaire–Phelan bubble (i.e. we adjust the prefactors in equation (30) and, as a result, equations (31) and (33); and again, c ₁₀ ≈ 0.022.