This simulation study investigates the dependence of the structure of dry Nafion$^{tinytextregistered}$-like ionomers on the electrostatic interactions between the components of the molecules. In order to speed equilibration, a procedure was adopted which involved detaching the side chains from the backbone and cutting the backbone into segments, and then reassembling the macromolecule by means of a strong imposed attractive force between the cut ends of the backbone, and between the non-ionic ends of the side chains and the midpoints of the backbone segments. Parameters varied in this study include the dielectric constant, the free volume, side-chain length, and strength of head-group interactions. A series of coarse-grained mesoscale simulations shows the morphlogy to depend sensitively on the ratio of the strength of the dipole-dipole interactions between the side-chain acidic end groups to the strength of the other electrostatic components of the Hamiltonian. Examples of the two differing morphologies proposed by Gierke and by Gebel emerge from our simulations.