In this work, we generalized the (Bjerrum–Debye–Fuoss–MSA) double layer model to an ellipsoidal polyion (chondroitin sulfate) of (∣ Z s∣ e) structural charge, L s structural length, R minor axe and ( R 2 + L 2/4) 1/2 major axe. With L ⩽ L s. 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 + 1/2 Γ), 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 = L s, 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 C Na+ → 0). This result is contrary to Manning’s theory, for which α is a constant α M independent on the concentration C i : α M = b S/(∣ Z i ∣ L b), with b S = L s/∣ Z s∣ and L b = e 2/( εkT) is the Bjerrum length. However, our analysis shows that the rate of variation: (∂ α/∂ C i ) in a given range of concentration, depends on the structural parameter b S. Indeed, the critical Manning condition ( α −1 α M = 1, ⇒(∂ α/∂ C i ) ≈ 0), is compatible with the general following “rod-like model” approximation: ( 1 - α ) ≈ | Z i | ( π b S ) [ 4 π Z i 2 L b ] [ - b S ″ α ] [ α C i R 2 ] [ 1 - b S ″ α ] ; with b S ″ = α M - 1 only for some peculiar values of b S and L b (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 b S structural parameter of about 5.8 Å. This is the case of sodium chondroitin sulfate in water ( b S = 5.72 Å). The addition of dioxane increases L b, consequently, α is shifted from its Manning’s value. In order to verify this dioxane effect, we have compared experimental equivalent conductibilities Λ exp of sodium chondroitin sulfate in water ( no shift) and water–dioxane (60 wt%) mixture ( positive shift), to their theoretical values Λ M, Λ c th and Λ s th 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.
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