Pre-empting Plateau: The nature of topological transitions in foam

When the area of a face in a dry foam approaches zero in some quasistatic processes, Plateau’s rules dictate that there must be an instability. This is more subtle than generally supposed. We argue that it is generally pre-empted, that is, the instability arises before an unstable multiple vertex is formed. Experiments and calculations which simulate Plateau’s wire frame experiments support this view. Copyright c © EPLA, 2007 Introduction. – The idealised model of a dry foam consists of faces (films) meeting in edges or lines (Plateau borders). One of Plateau’s celebrated rules [1] specifies that the symmetric tetrahedral junction of four such lines is the only possible vertex in an equilibrium structure. Hence if a vertex with more than four lines is formed in any quasistatic process (by variation of boundary constraints, or by coarsening, for example), there is an immediate topological transition to a Plateau-allowed configuration [2]. The two most elementary ways in which this scenario can arise are by the vanishing of a triangular face or an edge, provoking topological changes which are the inverse of each other. Figure 1 illustrates two configurations which may be transformed into each other by such changes. More complicated changes may arise in practice but they may be regarded as compounds of these two processes [2]. This much is conventional wisdom in the physics of foams. However we argue here that a significant aspect of the vanishing-face transition has been overlooked, originally by Plateau himself. It appears that in general the area of a face cannot proceed to zero in such a quasistatic process, and the transition is necessarily pre-empted. By this we mean that the system becomes unstable at some earlier point, with non-zero face area. The area of the face then shrinks to zero spontaneously and the topological transition ensues. That this has not been generally realised may be because of undue reliance on the analogy with the simpler two-dimensional foam, where no such pre-emptive effect exists in general. We have recently become aware of a single instance of its observation [3]. Soap films spanning the frame of a triangular prism. – To begin with, we consider a specific case, corresponding to one of the simplest demonstrations with Plateau’s wire frames: the triangular prism. For a small axial ratio c/a, this displays a central triangular face, as shown in configuration I of fig. 1. As c/a is gradually increased, the face area decreases and the regime of interest may be explored. We have done this in preliminary experiments with plastic frames of variable axial ratio, as supplied by Cochranes of Oxford Ltd., and dilute solution of the commercial dish-washing detergent Fairy Liquid. When such a frame with axial ratio c/a= 0.42 was withdrawn from the solution, we obtained configuration I of fig. 1. As we then increased the ratio c/a (without re-introducing the frame into the solution) at a certain point slightly below c/a= 0.5 the triangular face was observed to spontaneously collapse and there was a transition to the alternative configuration II. Figure 2 illustrates schematically the contrast between this behaviour and the topological change in 2d foam. Figure 1 was generated using the Surface Evolver [4]. This software enables us to accurately locate the critical value of axial ratio c/a for the ideal dry foam, which we find to be 0.487. As shown in fig. 3, for values of c/a between 0.413 and 0.487 there are two stable minima of total surface area,