Reply to “Comment on ‘Self-assembly of magnetic balls: From chains to tubes’ ”

The authors of the Comment [Phys. Rev. E 91, 057201 (2015)] propose compact round clusters as, energetically, better candidates than stacked rings found in Messina et al. [Phys. Rev. E 89, 011202 (2014)] (forming open tubes) at a pretty large number of constitutive magnets, typically for $N\gtrsim 1300$. Our new findings show that elongated rodlike structures can even outmatch the reported structures in Friedrich et al. [Phys. Rev. E 91, 057201 (2015)] and in Messina et al. [Phys. Rev. E 89, 011202 (2014)] from typically $N\gtrsim 460$.

Figure 1

Typical rodlike structures made up of straight chains. A top view as well as two side views are provided in order to properly visualize the characteristics of the microstructures. (a) $N=767$ with a ($3\times 3$)-row base, (b) $N=1289$ with a ($3\times 4$)-row base, and (c) $N=2093$ with a ($4\times 4$)-row base.

Figure 2

Reduced energy per magnet $u_N$ for different rodlike structures, see Fig. 1 for the corresponding microstructures. Open tubes as considered in the original paper [1] are also shown. The constant value $u_{\infty }^{\mathrm{plane}}=-2.759$ corresponds to an infinite triangular lattice with parallel dipoles pointing in the dense direction [1].