FIG. 4.

Schematic depiction of the interaction free energy method to calculate charge renormalization in macromolecules. The electrostatic potential distribution in an electrolyte-filled parallel-plate slit of height 2h is obtained by solving the non-linear Poisson-Boltzmann equation using constant charge boundary conditions, both with (left) and without the object (right). The average electrical potential ${⟨{\psi }^{\prime }\left(s\right)⟩}_{z^{*}}$ in the slit is evaluated over the dashed circular contour representing the “virtual” surface of an object placed at $z=z^{*}+R=h$ (right). Electrostatic interaction free energies, $F(z^{*})$, are calculated for an object of charge, Q, and radius, R, located at the mid-plane of the slit, at an inter-surface separation, $z^{*}=h-R$, using the $\psi (𝐫)$ distribution (left) and Eq. (9). Dashed vertical lines denote axes of cylindrical symmetry.