Roughness and scaling in cellular patterns: analysis of a simple model

A foam can be decomposed into successive layers of cells at the same topolo gical distance to an origin, which is either an arbitrary cell or a basal p lane. The shape of these layers (profile and thickening) indicates the degr ee of randomness of the cellular pattern. To support this idea, we analyse the layer shapes in 2D rectangular models of foam. As confirmed by numerica l simulations, the fluctuations in the direction normal to the layers are s elf-affine on a significant range of scales, with values of the exponents c ompatible with the KPZ universality class: zeta similar or equal to 0.5 for the roughness exponent and z similar or equal to 1.5 for the dynamic expon ent measuring the increase of the intralayer correlation length. These fluc tuations are not sufficient, however, to affect the dominant behaviour of t he number of cells per layer, found to saturate in cylindrical geometry, an d to increase linearly in concentric geometry.