Onsager-Manning-Oosawa Condensation Phenomenon and the Effect of Salt

Making use of results pertaining to Painlevé III type equations, we revisit the celebrated Onsager-Manning-Oosawa condensation phenomenon for charged stiff linear polymers, in the mean-field approximation with salt. We obtain analytically the associated critical line charge density and show that it is severely affected by finite salt effects, whereas previous results focused on the no salt limit. In addition, we obtain explicit expressions for the condensate thickness and the electric potential. The case of asymmetric electrolytes is also briefly addressed.

Figure 1

Contour lines for $\sigma$ and $\mu$ (and hence for $\lambda$) in the charge-salt plane, as deduced from Eqs. (7, 8). The thick dashed curve shows the locus of critical charges ${\xi }_c$ and separates the iso-$\sigma$ curves for $\xi <{\xi }_c$ (from left to right, the curves correspond to $\sigma =0.2$, 0.4, 0.6, and 0.8) from the iso-$\mu$ curves shown—on the right-hand side—for $\xi >{\xi }_c$ (from bottom to top, $\mu =0.08$,0.1, 0.13, and 0.2).

Figure 2

$\sigma$ versus bare charge. In the main graph and in the inset, three values of $\kappa a$ have been chosen. From left to right, $\kappa a={10}^{-2}$, ${10}^{-3}$, and ${10}^{-6}$. For $\kappa a\to 0$, $\sigma \to \xi$ (see the dashed first bisectrix) while $\mu \to 0$. The associated critical charges are indicated by the arrows (also in the inset). The inset shows $\mu$ versus $\xi$, and the dashed curves correspond to approximation (12). The vertical line indicates $\xi =1$.