Relaxation of the topological T1 process in a two-dimensional foam

The so-called topological T1 process, during which bubbles within a foam exchange neighbours is studied. The Durand and Stone model (Phys. Rev. Lett., 97, 226101 (2006)) describes the growth of a film that is newly created during the T1 process, and also the evolution of surfactant concentration on this newly created film. Here some characteristic features of the Durand and Stone model (not previously described by Durand and Stone) are elucidated. In particular it is shown that the surfactant concentration on the newly created film is predicted to undergo an extremely rapid initial evolution, which occurs long before the film itself approaches anywhere near its final equilibrium length. Associated with this, the predicted length of the newly created film tends to exhibit an extremely rapid acceleration early on in its growth. An intermediate asymptotic analysis is developed to explain the above model predictions, by focussing on the regime when the film is several times larger than its initial length, but still several times smaller than its final length. A physical explanation is offered for these predictions in terms of slippage between material points instantaneously at the end of the newly created film, and the evolving location of the film endpoint itself: this slippage implies surfactant being transferred onto the newly created film from neighbouring films, overwhelming the amount of surfactant initially present. The implications of these predictions for the likely observations in an experimental study of the T1 process are discussed.