Foam Evolution in Two Dimensions

Abstract

Foams are examples of a large class of systems endowed with a cellular structure pattern: they consist of a tiling of space with compact polygonal (two dimensions) or polyhedral (three dimensions) domains, separated by what we will consider structureless boundaries [1, 2, 3, 4]. Besides the intrinsic interest in the properties of foams (e.g. their viscoelasticity), which make them so useful in a large variety of applications [5], foams have been the focus of intensive research within the last two decades, as a paradigm of cellular structure evolution [6, 7, 8, 9]. This evolution is driven by the fact that the boundaries between bubbles, simple soap films, are endowed with surface tension and are typically curved. This curvature, which betrays pressure differences between neighbouring bubbles, induces a transfer of gas between them. As a result, some bubbles shrink while others grow, so that the average bubble size in the foam grows with time, and the total film area decreases (see Fig. 1). Thus a foam should be viewed as a system which is out of equilibrium, and evolving towards it as it lowers more and more the energy of its boundaries. A remarkable fact about this evolution is that it is characterized by statistical properties which are independent of scale: distributions such as the number of sides of polygonal bubbles do not change as a result of the evolution. Furthermore, this evolution is universal: systems disparately different in their microscopic structure such as polycrystalline materials [10], monolayers of lipids on water [11, 12], and magnetic domains [13] which also posess a cellular structure, display essentially the same behavior. In fact, the interest in foams is due in no small part to C. S. Smith [14], a metallurgist at the University of Chicago, who had the original insight of using foams as caricatures of polycrystalline metals.