Theory of Ionomer Bundle Formation

The calculation of the equilibrium bundle size employs a mean field approach, which can be rationalized in several steps. In the first step, the system is divided into noninteracting cells of equal size, each containing a single bundle in the core and an outer electrolyte shell that accommodates dissociated and hydrated protons. In the second step, the ensemble of cells is replaced by a single effective cell with cylindrical geometry, concentric with the bundle in the core. The length of the cell is assumed to be the same as the rod length Lr. The cell radius rc is a function of the density p of rods 

Rod-like ionomer lonomer bundles Unit cell with bundle 

The model assumes that the space between rods inside a bundle is electrolyte-free. Therefore, acid dissociation could only occur at the bundle surface. The degree of acid dissociation in the bundle determines the charge density at the surface of the bundle, assumed to be continuous. 

In the case of only one type of ionic species in the solution, namely, dissociated protons, electrostatic interactions would drive the system toward a dispersed state, if considered as the sole type of interaction. However, backbone hydrophobicity constitutes a significant driving force for aggregation, reverting this trend. 

In the treatment sketched above in abbreviated form, the free energy density of a -bundle is given by 

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