Stable Bundle Size Configuration Diagrams

The main outcome of this theory is the stable bundle size k that is obtained by minimization oifik). The bundle size can be evaluated as a function of geometric parameters of single rods, vr and Lr, rod density, p, surface charge density, a, surface tension, /, solvation energy, Esoiv and the dissociation parameter, rj. 

The terms in square brackets are responsible for the parabolic shape. Between these two limits (complete phase segregation and dispersed rod limit), a cascade of transitions is seen between stable configurations. The main trends are that the aggregation number of stable bundles, k, decreases as a increases and y decreases. At sufficiently large a, k attains a value of 9. This bundle size allows 90% of the sidechains to 

All the major trends discussed in this section are in agreement with experimental findings on the impact of sidechain properties and sidechain grafting density on ionomer aggregation. 

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