About the singular behavior of the ionic condensation of sodium chondroitin sulfate: Conductivity study in water and water-dioxane mixture

In this work, we generalized the (Bjerrum-Debye-Fuoss-MSA) double layer model to an ellipsoidal polyion (chondroitin sulfate) of (∣Zs∣e) structural charge, Ls structural length, R minor axe and (R2 + L2/4)1/2 major axe. With L ⩽ Ls. Na+ counter ions are distributed on the contact (or condensed) layer and on the Debye layer (ionic atmosphere). Both layers are ellipsoidal equipotentials of, respectively, R and d minor axes and are concentric to the polyion. With d = (R + 1/2Γ), Γ is the Debye-MSA screen parameter. The equilibrium distribution of Na+ ions is derived from a “two states” statistical approach, leading to a general implicit expression for the rate of condensation (1 − α). The generality of this formula results from the fact that it takes into account the finite size of the polyion (L ≠ ∞ and R ≠ 0) and allows to calculate α for different conformations of the polyion: (ellipsoidal L ≠ 0, cylindrical: L = Ls, spherical: L → 0, and Manning's model: RL−1 → 0). The main conclusion of this model is that, α obeys to the Ostwald's principle of dilution (α → 1 when CNa+ → 0). This result is contrary to Manning's theory, for which α is a constant αM independent on the concentration Ci: αM = bS/(∣Zi∣Lb), with bS = Ls/∣Zs∣ and Lb = e2/(εkT) is the Bjerrum length. However, our analysis shows that the rate of variation: (∂α/∂Ci) in a given range of concentration, depends on the structural parameter bS. Indeed, the critical Manning condition (α−1αM = 1, ⇒(∂α/∂Ci) ≈ 0), is compatible with the general following “rod-like model” approximation:(1-α)≈|Zi|(πbS)[4πZi2Lb][-bS″α][αCiR2][1-bS″α];withbS″=αM-1only for some peculiar values of bS and Lb (i.e., dielectric constant: ε). In water at 25 °C (ε = 78.3), this singular behavior occurs for a range of a relative low or moderate concentration for some polyelectrolytes of bS structural parameter of about 5.8 Å. This is the case of sodium chondroitin sulfate in water (bS = 5.72 Å). The addition of dioxane increases Lb, consequently, α is shifted from its Manning's value. In order to verify this dioxane effect, we have compared experimental equivalent conductibilities Λexp of sodium chondroitin sulfate in water (no shift) and water-dioxane (60 wt%) mixture (positive shift), to their theoretical values ΛM, ΛcthandΛsth corresponding, respectively, to the Manning, cylindrical and spherical models. This comparison allows also, to explain the conformation “chosen” by the polyion, in order to minimizing the friction effects (due to: viscosity; ionic and dielectric relaxations) and therefore, to optimize its mobility by the shift of its rate of ionic condensation α.