Coarsening of three-dimensional grains in crystals, or bubbles in dry foams, tends towards a universal, statistically scale-invariant regime

We perform extensive Potts model simulations of three-dimensional dry foam coarsening. Starting with 2.25 million bubbles, we have enough statistics to fulfill the three constraints required for the study of statistical scale invariance: first, enough time for the transient to end and reach the scaling state; then, enough time in the scaling state itself to characterize its properties; and finally, enough bubbles at the end to avoid spurious finite size effects. In the scaling state, we find that the average surface area of the bubbles increases linearly with time. The geometry bubble shape and size and topology number of faces and edges , as well as their correlations, become constant in time. Their distributions agree with the data of the literature. We present an analytical model universal, up to parameters extracted from the simulations for a disordered foam minimizing its free energy, which agrees with the simulations. We discuss the limitations of the simulations and of the model