Self-assembly of magnetic balls: from chains to tubes

The self-assembly of spherical magnets (magnetic balls) is addressed theoretically. Minimal energy structures are obtained by optimization procedures as well as Monte Carlo computer simulations. Three typical shapes are obtained depending on the number of constitutive magnets N. In the regime of small N, chains are stable as dimers or trimers (i.e., N≤3), then rings become stable for (4≤N≤13) where dipole vectors adopt a vortexlike arrangement. A major finding concerns the stacking of rings as soon as N is large enough (N≥14). The number of stacked rings is found to increase as N^{2/3}, leading to a tubular structure at large N. All the relevant predicted shapes are experimentally reproduced by manipulating millimetric magnets.