The dynamics of a topological change in a system of soap films

Abstract

The study of soap films spanning wire frames continues to provide insight into both static and dynamic properties of foams. Experiments show that sufficiently small triangular faces shrink spontaneously, with their area varying with time as $t^{0.8}$. The growth of the Plateau border that emerges after the collapse of the triangular face is initially linear, followed by an oscillatory relaxation to the equilibrium length. Its initial growth rate decreases with the viscosity of the liquid. We also consider wire frames in the shape of n-sided prisms with regular polygonal ends, $n\ge 3$, and employ experiments and computer simulations to examine the stability of soap film configurations within them. Spontaneous shrinking of small faces occurs in four and five-sided prismoidal frames, while this instability is suppressed for $n\ge 6$.

Introduction

Plateau’s celebrated rules describe the topology and geometry of a system of soap films [1]. In particular they state that three films meet symmetrically in a line, called a Plateau border, at angles of 120°, and that four Plateau borders meet symmetrically in a vertex at the tetrahedral angle of ${\text{cos}}^{-1}(-1/3)\simeq 109.47°$ [2]. No other configurations are allowed in a dry foam. The 120° rule reflects the equilibrium of the three surface tension vectors at the intersection of three soap films. The rule that not more than four of these intersection lines can meet in a vertex was only proven in 1976 with full mathematical rigour [3] (the origin of this proof dates back to the 19th century [4]). The symmetry of the vertex then follows from the symmetry of the adjoining intersections of the films.

However, it was recently demonstrated, both experimentally and computationally, that Plateau’s rules, albeit necessary, are not sufficient for stability [5]. In particular it was shown that small triangular faces are not stable, with the result that the soap films will undergo spontaneous topological transitions that remove such faces, once the face area has decreased below a critical value.

In this paper we extend our previous work [5] in several directions. We employ experiments to help us to understand the dynamics of this instability, which is probably important in the rheology of foams, and static (or quasi-static) simulations using the Surface Evolver to establish equilibrium film shapes in different frame shapes and sizes, providing a benchmark configuration for our experiments. Our experiments concern both the dynamics of the vanishing triangular face and the growth of a new Plateau border. We find that the bulk viscosity of the soap solution slows down the shrinking of the face only if the viscosity is increased (by adding glycerol) by at least a factor of 10. For the growth of the new Plateau border we find that the bulk viscosity plays a much more prominent role. Finally we report new computations with regard to equilibrium configurations in frames with six, seven, eight and nine sides.