Electrostatic double-layer interaction between stacked charged bilayers

The inapplicability of the DLVO theory to multilayered anionic bilayers is found in terms of the co-ion-valence dependence of the lamellar repeat distance. Most of the added salt is expelled from the interlamellar space to the bulk due to the Gibbs-Donnan effect on multiple bilayers with the bulk. The electrostatic double-layer interaction is well expressed by the formula recently proposed by Trefalt. The osmotic pressure due to the expelled ions, rather than the van der Waals interaction, is the main origin of the attractive force between the bilayers.

Figure 1

(a) SAXS intensities without background ($J\left(q\right)-B_{\mathrm{G}}$), over a wide $q$ range in NaCl (red), ${\mathrm{Na}}_2{\mathrm{SO}}_4$ (green), and ${\mathrm{Na}}_3{\mathrm{PO}}_4$ (blue) solutions and pure water (black). The added ${\mathrm{Na}}^{+}$ concentration is $\rho Z$ = 30 mM. Solid lines are the results of fitting. (b) Deduced electron density profiles of DOPS bilayers in the three saline solutions. Dashed lines are located at $\pm 19.9\AA$.

Figure 2

Water layer thickness $d_{\mathrm{w}}$ as a function of the ionic strength $I$ in each saline solution [NaCl (red circle), ${\mathrm{Na}}_2{\mathrm{SO}}_4$ (green square), ${\mathrm{Na}}_3{\mathrm{PO}}_4$ (blue cross)]. Colored solid lines represent the water layer thicknesses where $U\left(d_{\mathrm{w}}\right)$ [Eq. (10)] takes a minimum. Dash-dotted, dashed, and dotted lines are the plots of Eq. (11) for NaCl, ${\mathrm{Na}}_2{\mathrm{SO}}_4$, and ${\mathrm{Na}}_3{\mathrm{PO}}_4$ solutions, respectively.

Figure 3

Potential energies per unit area with $[U\left(d_{\mathrm{w}}\right)$, solid lines] and without $[U_{\mathrm{DL}}\left(d_{\mathrm{w}}\right)$, dashed lines] van der Waals interaction when the ionic strength is $I=300$ mM. NaCl (red), ${\mathrm{Na}}_2{\mathrm{SO}}_4$ (green), and ${\mathrm{Na}}_3{\mathrm{PO}}_4$ (blue). The arrows locate the minima. The inset shows the close-up around the minimum for co-ion valence 1. The minima were located at 25.8 and 26.0 $\AA$ for $U\left(d_{\mathrm{w}}\right)$ and $U_{\mathrm{DL}}\left(d_{\mathrm{w}}\right)$, respectively.

Figure 4

Molecular fractions of co-ions in the bulk phase to the total co-ion added as functions of $I$. NaCl (red), ${\mathrm{Na}}_2{\mathrm{SO}}_4$ (green), and ${\mathrm{Na}}_3{\mathrm{PO}}_4$ (blue). Over 99% of added coions are expelled from the aqueous phase between bilayers.