Modelling froth dynamics; persistence measures

The soap froth provides an idealised example of a cellular structure, which evolves or coarsens over laboratory time-scales and for which topological measures appear to provide an intuitive characterisation. However, froth is an intrinsically non-equilibrium system, and topological measures are not universally applicable to such processes. Recently, persistence has been proposed as a more general probe of non-equilibrium dynamics, where the froth is viewed as a two-phase system through construction of a virtual phase. We use a direct simulation method to investigate persistence for random 2-D (Voronoi) and hexagonal froths of size up to 2500 bubbles. We find that simulation results are qualitatively similar to those of experiment, with the normalised average area, 〈A∗(t)/A(0)〉, of persistent regions within a bubble at time t approaching an equilibrium value for a range of volume (or sampling) fraction values, φ, for the Voronoi froth. The case for the hexagonal is less clear, since exclusion of the defect (or defects) from the virtual phase leads to rapid decline in the average area of persistent bubbles. Simulation times required are very long, however, and evolution is slow for long-term survivors. Consequently, persistent behaviour is not demonstrated satisfactorily for the fraction of survivors, N∗(t)/N(t), in a random system of this size, although for the hexagonal with one or more seeded defects, there is some indication that decay depends on φ, for some colouring patterns. However, limiting slope values are probably not established.