Z-cone model for the energy of an ordered foam

Abstract

We develop the Z-Cone Model, in terms of which the energy of a foam may be estimated. It is directly applicable to an ordered structure in which every bubble has Z identical neighbours. The energy (i.e. surface area) may be analytically related to liquid fraction. It has the correct asymptotic form in the limits of dry and wet foam, with prefactors dependent on Z. In particular, the variation of energy with deformation in the wet limit is consistent with the anomalous behaviour found by Morse and Witten [Europhysics Letters, 1993, 22, 549] and Lacasse et al. [Physical Review E, 54, 5436], with a prefactor Z/2.

Structure and energy of liquid foams

“…This expression agrees very well with data obtained from corresponding Surface Evolver simulation for small deformations [70,75] (see Fig. 16 later).…”

“…(10) [70,75]. For example, in the case of periodic foam structures with N neighbouring bubbles analytical calculations using the Z-cone model [75,76] show that the relative surface excess is given by…”

“…it depends on the number of neighbours of a bubble. Moreover, even though it contains a harmonic term, it is "logarithmically soft" for small deformations [70,75]. Approximations of the type of potential corresponding to Eq.…”

“…Approximations for the osmotic pressure in the wet limit may be obtained using models of the interaction potentials, such as the ones discussed at the beginning of this section [6,69,75,77].…”

Structure of marginally jammed polydisperse packings of frictionless spheres

“…It is also highly relevant for many practical problems ranging from storage and industrial packing to the properties of soft materials such as emulsions, foams, or granular materials [3][4][5][6][7][8][9][10][11]. Amorphous packings are particularly difficult to understand due to the complexity of disordered nonequilibrium structures.…”