Simulation and analysis of foam structure

Abstract

We use the method of bond-orientational order parameter analysis on X-ray tomographic data to investigate the interal structure of an experimental sample of ~25 000 microbubbles. By employing appropriate cutoff distances in the parameter space, we see that over the 7 days of the experiment the amount of ordering in the sample increases signficantly. In line with previous experiments and simulations, we see a preference for face-centred cubic (fcc) ordering over hexagonal close-packed (hcp). We present a simple geometrical argument concerning the ideal shapes of bubbles at an arbitrary liquid fraction between the dry and wet limits. By applying the appropriate transformation to an fcc bubble at a given liquid fraction, we obtain a ‘trial’ hcp bubble of the same surface area. This surface can be relaxed, proving that the hcp structure has lower energy than the fcc. We perform Surface Evolver simulations of fcc and hcp bubbles over the full range of liquid fractions. The trend observed confirms our proof: the energies are equal at the wet and dry limits, and for intermediate liquid fractions the surface area of the hcp bubbles is very slightly lower. The Z-cone model is a mathematical formulation which provides analytic approximations to the energy of a bubble as it is deformed. We verify its accuracy for some fundamental test cases: a bubble compressed between parallel plates, a bubble confined to a cube, and a bubble confined to a regular dodecahedron. We see that the energies predicted by the model are accurate. We apply the model to the case of a bubble in an fcc foam, and see once again that the predicted values of liquid fraction and energy match those obtained from computer simulation. For the fcc case, we obtain from the model an interaction potential similar to that reported in previous simulations. We derive expressions for the osmotic pressure and, hence, a liquid fraction profile from the cone model expressions which agree with experimental data. Furthermore, we obtain a relationship between liquid fraction and surface liquid fraction which, again, matches experimental data well. We extend the model to deal with the body-centred cubic structure, resulting in excellent agreement between the model and simulation over the full range of liquid fractions. We investigate the variation of energy with liquid fraction close to the critical liquid fractions at which nearest neighbours and next-to-nearest neighbours are lost. At each point we see logartithmic terms in the variation of energy; however, the forms are different. We present the results of experiments and simulations concerning the interaction between soap films and fibres: a fibre in the plane of a film, and a fibre in a Plateau border. In each case we see that our simulation predicts the lengths of films and Plateau borders involved. In the latter case we can calculate the force necessary to unpin the fibre from the Plateau border. Finally, we present simulations concerning pairs of fibres, modelled as infinitely long rigid cylinders, bridged by a liquid drop, for the case of a small contact angle. We see that the drop acts to pull the fibres together, and that, for certain drop volumes, they preferentially orient at an angle which is neither parallel nor perpendicular. We see similar behaviour for a slightly increased contact angle.