Decorated phases in triblock copolymers: zeroth- and first-order analysis

We study a two-dimensional inhibitory ternary system characterized by a free energy functional which combines an interface short-range interaction energy promoting micro-domain growth with a Coulomb-type long-range interaction energy which prevents micro-domains from unlimited spreading. Here we consider a scenario in which two species are dominant and one species is vanishingly small. In this scenario two energy levels are distinguished: the zeroth-order energy encodes information on the optimal arrangement of the dominant constituents, while the first-order energy gives the shape of the vanishing constituent. This first-order energy also shows that, for any optimal configuration, the vanishing phase must lie on the boundary between the two dominant constituents and form lens clusters also known as vesica piscis.