  In a fixed domain of $\Bbb{R}^N$ we study the asymptotic behaviour of optimal
clusters associated to $\alpha$-Cheeger constants and natural energies like the
sum or maximum: we prove that, as the parameter $\alpha$ converges to the
"critical" value $\Big (\frac{N-1}{N}\Big ) _+$, optimal Cheeger clusters
converge to solutions of different packing problems for balls, depending on the
energy under consideration. As well, we propose an efficient phase field
approach based on a multiphase Gamma convergence result of Modica-Mortola type,
in order to compute $\alpha$-Cheeger constants, optimal clusters and, as a
consequence of the asymptotic result, optimal packings. Numerical experiments
are carried over in two and three space dimensions.
