Sticky-disk limit of planar $N$-bubbles

We study planar $N$-clusters that minimize, under an area constraint, a weighted perimeter $P_\varepsilon$ depending on a small parameter $\varepsilon>0$. Specifically we weight $2-\varepsilon$ the boundary between the interior chambers and $1$ the boundary between an interior chamber and the exterior one. We prove that as $\varepsilon\to 0$ minimizers of $P_\varepsilon$ converge to configurations of disjoint disks that maximize the number of tangencies, each weighted by the harmonic mean of the radii of the two tangent disks. We also obtain some information on the structure of minimizers for small $\varepsilon$.