The energy of fcc and hcp foams

We present Surface Evolver evaluations of the difference in energy between face-centred cubic (fcc) and hexagonal close-packed (hcp) foams in the usual idealized model, for liquid fractions ranging from the dry to the wet limit. The difference vanishes in both limits, and favours hcp for all intermediate liquid fractions, as has been proven. The maximum relative energy difference is very small, of the order of 10-5. The asymptotic dependence on liquid fraction is non-analytic in both limits: we present explicit expressions in both cases, derived from first principles. They have been obtained from identifying node interactions (dry limit) and contact interactions (wet limit) as the respective sources for energy differences between fcc and hcp. The wet limit is well described by Morse-Witten theory which has proven to be very powerful for the analytic computation of the surface energy of slightly deformed bubbles.