document: When incompatible three component polymer chains are tethered at a junction point, the resultant star molecules of the ABC type are in a very frustrated field in bulk. That is, their junction points cannot be aligned on two-dimensional planes but on one-dimensional lines, as schematically shown in Fig. 1. Furthermore, when the chain length difference is not so large, the array of junction points tends to be straight and long one. Consequently each domain with mesoscopic sizes becomes cylinders, and their cross sections could be conformed by polygons [28,29]. This is because three interfaces, A/B, B/C and C/A are likely to be flat since there exist no junction points at interfaces and therefore chain entropy contribution to the free energy of structure formation is considerably small comparing with regular block and graft copolymer systems. As a matter of fact, Dotera predicted several tiling patterns by the diagonal bond method, a new Monte Carlo Simulation [30], while Gemma and Dotera pointed out that only three regular tilings, i.e., (6.6.6), (4.8.8) and (4.6.12) are permitted for three-branched molecules proposed as the “even polygon theorem” [31].
extractive_keyphrases: chain entropy cylinders diagonal bond method monte carlo simulation polygons polymer chains star molecules
abstractive_keyphrases: even polygon theorem three-branched molecules three interfaces, a/b, b/c and c/a