  Aqueous foams and a wide range of related systems are believed to coarsen by
gas diffusion between neighboring domains into a statistically self-similar
scaling state, after the decay of initial transients, such that dimensionless
size and shape distributions become time independent and the average grows as a
power law. Partial integrodifferential equations for the time evolution of the
size distribution for such phase separating systems can be formulated for
arbitrary initial conditions, but these are cumbersome for analyzing data on
non-scaling state preparations. Here we show that essential features of the
approach to the scaling state can be captured by an exactly-solvable ordinary
differential equation for the evolution of the average bubble size. The key
ingredient is to characterize the the bubble size distribution approximately,
using the average size of all bubbles and the average size of the critical
bubbles, which instantaneously neither grow nor shrink. The difference between
these two averages serves as a proxy for the width of the size distribution. To
test our model, we compare with data for quasi-two dimensional dry foams
created with three different initial amounts of polydispersity. This allows us
to readily identify the critical radius from the average area of six-sided
bubbles, whose growth rate is zero by the von~Neumann law. The growth of the
average and critical radii agree quite well with exact solution, though the
most monodisperse sample crosses over to the scaling state faster than
expected. A simpler approximate solution of our model performs equally well.
Our approach is applicable to 3d foams, which we demonstrate by re-analyzing
prior data, as well as to froths of dilute droplets and to phase separation
kinetics for more general systems such as emulsions, binary mixtures, and
alloys.
