Mode-resolved charge renormalization combined with anisotropic charge regulation

Develop a mode-resolved charge-renormalization framework for anisotropically charged, charge-regulating spheres that retains the non-diagonal mode couplings generated by both the renormalization operator and the capacitance operator, while specifying an appropriate infinite-dilution prescription beyond cell-model schemes.

Background

The paper derives a closed-form theory for electrostatic interactions between anisotropically charged, charge-regulating spheres within the linearized Poisson–Boltzmann approximation. Its charge-regulation framework is expressed through a multipole-space capacitance operator, which can couple distinct charge modes. The discussion proposes composing this theory with charge renormalization to extend its quantitative reach to low-salt and high-charge regimes where linearization becomes inaccurate.

The unresolved extension concerns the simultaneous treatment of mode-dependent charge renormalization and anisotropic charge regulation. Because fine multipolar features experience different screening and saturation behavior, the renormalization operator is expected to be mode-resolved and potentially non-diagonal when mode couplings are retained. The paper also notes that an appropriate prescription at infinite dilution is not supplied by the cell-model schemes considered in related work, leaving the combined construction unresolved.