  We study the ternary Ohta-Kawasaki free energy that has been used to model
triblock copolymer systems. Its one-dimensional global minimizers are
conjectured to have cyclic patterns. However, some physical experiments and
computer simulations found triblock copolymers forming noncyclic lamellar
patterns. In this work, by comparing the free energies of the cyclic pattern
and some noncyclic candidates, we show that the conjecture does not hold for
some choices of parameters. Our results suggest that even in one dimension, the
global minimizers may take on very different patterns in different parameter
regimes. To unify the existing choices of the long range coefficient matrix, we
present a reformulation of the long range term using a generalized charge
interpretation, and thereby propose conditions on the matrix in order for the
global minimizers to reproduce physically relevant nanostructures of block
copolymers.
