Statistical Mechanics of Two-dimensional Foams

The methods of statistical mechanics are applied to two-dimensional foams under macroscopic agitation. A new variable the total cell curvature is introduced, which plays the role of energy in conventional statistical thermodynamics. The probability distribution of the number of sides for a cell of given area is derived. This expression allows to correlate the distribution of sides (“topological disorder”) to the distribution of sizes (“geometrical disorder”) in a foam. The model predictions agree well with available experimental data. Introduction. – Foams and related physical systems (like emulsions, biological tissues, or polycrystals) are ubiquitous, and serve as a paradigm for a wide range of physical phenomena and mathematical problems [1–4]. One of them deals with the general topological and geometrical properties of cellular materials [5–12]. In this connection, Quilliet et al. [5, 6] studied recently the topological features of two-dimensional (2D) soap froths under slow oscillatory shear. Such macroscopic strain induces rearrangements within the foam, and the number of sides of every cell evolves in time through local topological changes (T 1 events) [1,13,14]. Nevertheless, Quilliet et al. reported the existence of an equilibrium state after few cycles, characterized by a stationary probability distribution of the number of sides per cell (topological disorder). They also showed that the width of this distribution of sides is strongly correlated to the distribution of bubble sizes (geometrical disorder) within the foam. These results suggest that the macroscopic state of a homogeneously sheared foam can be adequately described using the ideas and formalism of statistical thermodynamics. Indeed, the pioneering work of Edwards on granular matter [15] has shown how the powerful arsenal of statistical physics can be extended to athermal systems. This method has proven its applicability to other fields as well [16]. Because there is no thermal averaging due to Brownian motion, this approach requires the presence of a macroscopic agitation (analogue to an effective temperature) allowing the system to explore its entire phase space. Various attempts have been made in the past to describe the geometrical and topological properties of 2D foams using the concepts of statistical thermodynamics [17–22]. However, these former theoretical approaches rely on strong assumptions: either they use an ad-hoc interaction potential between bubbles [17, 18], involve (rather than deduce) empirical laws correlating size and side distributions [19–21], or ignore some geometrical constraints (for instance, only the mean bubble area is specified, not the individual bubble areas [19, 20, 22]). Some of these models are based on the maximum entropy (information theory) formalism [19,20], which has been subject to controversy [23]. Other models invoke minimisation of the energy [21], or a combination of both principles [22] to describe the state of a foam. However, it has been established [12, 24–27] that different arrangements (topologies) of a large number of bubbles do not really affect the energy (see discussion below). Furthermore, none of these models can account for the correlations between topological and geometrical disorders reported by Quilliet et al. [5, 6]. In this letter we set up a framework for describing the equilibrium state of a two-dimensional foam, basing our development on analogies with conventional statistical mechanics. As for other athermal systems [15, 16], we show that the energy is not relevant to describe the macroscopic state of a foam. Instead, a more appropriate state variable is introduced: the total cell curvature. We establish the function of state which is minimized for a finite cluster of bubbles at equilibrium. This thermodynamic potential