Double-layer Free Energy Research Articles

Mobile Surface Charge Can Immobilize the Air/Water Interface.

Recent Surface Force Apparatus measurements on thin film drainage as a bubble approaches a surface are reinterpreted in terms of a new model for the air/water interface. In this model, surface charge at the interface can be convected and can diffuse along the surface as the film drains. This creates surface tension gradients, since surface tension includes a charge-dependent contribution from the double-layer free energy. Although this electrocapillary effect is relatively small, we show here that the gradients are large enough to create Marangoni effects that influence the hydrodynamic flow in real systems. The new model can account for observed changes in hydrodynamic boundary condition from mobile initially to immobile during early stages of film drainage, and back to partially mobile as drainage progresses. Experimental film profiles for a millimeter-size bubble in 1 mM KCl solution driven toward a mica surface at 27 μm/s are reasonably well described with this mobile surface charge model. At longer times, there are still features of the experimental data that remain to be explained, which suggests further modeling is warranted.

Electrical properties of a charged surface in a general electrolyte solution

The electrical properties of a planar charged surface in a general electrolyte solution are examined. Analytical expressions for the electrical potential, surface excess of co-ions, double-layer free energy, and entropy are derived. A perturbation method is proposed for the resolution of the Poisson–Boltzmann equation governing the electrical potential distribution. The analysis extends the classic Gouy–Chapman's result to an arbitrary a:b electrolyte solution. The present approach yields a system of linear ordinary differential equations, which can be solved to an arbitrary order, in principle. The performance of the present perturbation solution is satisfactory for conditions normally encountered in practice. For example, if the second-order solution is adopted, deviations in the electrical potential and thermodynamic properties examined are less than 5%.

Approximate Analytical Expressions for the Properties of an Electrical Double Layer with Asymmetric Electrolytes: Cylindrical and Spherical Geometries

Approximate analytical expressions for the electrical potential distribution, surface excess of co-ion, and double-layer free energy and entropy for both cylindrical and spherical surfaces are derived. The present analysis extends previous results in that asymmetric electrolytes are considered. The mathematical treatment of the Poisson-Boltzmann equation is based on a semi-empirical fourth-order polynomial function, along with a parameter reflecting the degree of asymmetry of electrolytes and a parameter measuring the effect of curvature of a surface. The result of numerical simulation reveals that the deviation of the present approximate result from the corresponding exact value for the double-layer properties under consideration is about 4%. The performance of the expression obtained by following a perturbation algorithm is also examined. It is found that for a relative thin to medium-thick double layer, the performance of two approaches are comparable; for a relative thick double layer, the present analysis yields a much better result.

Thermodynamic analysis of polydispersity in ionic micellar systems and its effect on small-angle neutron scattering data treatment

The authors analyze small-angle neutron scattering (SANS) data of sodium dodecyl sulfate (SDS) and sodium bis(2-ethylhexyl) sulfosuccinate (AOT) ionic micellar solutions taking into account the effect of size polydispersity on the particle form factor. The intermicellar structure factor is computed by a generalized one-component macroion theory (GOCM) assuming a constant fractional charge. The model fittings to SANS data for different concentrations allow us to extract the free energy parameters of micelle formation and growth, the size distribution function of micelles, and the minimum micelle size which is consistent with the fully stretched hydrocarbon tail lengths of the surfactant molecules. The critical micellar concentration (cmc) is predicted from the free energy parameters correctly. Combining the free energy of micelle formation with the double-layer free energy around the averaged micellar surface calculated by the nonlinear Poisson-Boltzmann equation, they obtain the hydrophobic free energy of micellization which is in quantitative agreement with the literature value. These analyses confirm the applicability of the ladder model of micellar growth in salt free ionic micellar solutions at moderate concentrations. The degree of polydispersity in size is about 11% for 2% SDS solution at 40/degrees/C and 17% for 1% AOT at 22.6/degrees/C.

Direct measurement of the interaction between phosphatidylglycerol bilayers in aqueous electrolyte solutions

Results are presented of force measurements between deposited bilayers of dimyristoylphosphatidyl glycerol (DMPG) at T greater than Tm, and distearoylphosphatidyl glycerol (DSPG) at T less than Tm. Below a bilayer separation of 100 nm, a repulsive double-layer force is measured, which can be explained through the combined screening and binding effect of the counterions in electrolyte solutions of NaCl, HCl, CaCl2, or mixtures of these. The binding of cations to bilayers in the fluid phase (DMPG) appears to be greater than to bilayers in the gel phase (DSPG). At shorter range, below approximately 3 nm, an attractive interaction is measured in solutions containing CaCl2, which was found to be slightly stronger than the theoretically expected van der Waals interaction. No hydration force was observed to exist in solutions containing CaCl2. In NaCl solutions, the measured interbilayer force can completely be accounted for by the electrostatic repulsion, down to a bilayer separation of at least 2 nm, below which no accurate measurements were possible anymore. Parallel measurements on PG monolayers show that the contraction of a DMPG monolayer following addition of CaCl2 is significantly greater than what is predicted from the change in the double-layer free energy alone. This indicates that changes in the lateral interactions between the lipid headgroups probably involve Ca2+-bridge binding and/or a possible dehydration of the lipid headgroups through Ca2+ binding. The results shed new light on both the interbilayer and intrabilayer interactions of PG and identify the possible factors responsible for the morphological behavior of PG aggregates.

Spatially varying polarization in water. A model for the electric double layer and the hydration force

The formalism derived in the preceding paper is used to model the response of aqueous solutions to a spatially-varying applied electric field. It leads to a generalized theory of the electric double layer which explicitly takes into account the microscopic structure of the solvent. As the solvent polarization is allowed to vary depending on both the macroscopic electric field and the specific interactions at the surface, the result is more freedom in the structure of the electric double layer. We consider generalized expressions for the double-layer free energy, as well as the interaction of two double layers which include specific surface polarization. Such interaction consists of both classical double-layer repulsion and the strong, short-range ‘hydration force’.

Double layer interaction energy of a concentrated system of charged parallel colloidal plates in aqueous electrolyte

A concentrated system of charged plate-like colloidal particles, arranged in parallel layers at a common separation and immersed in a uni-univalent aqueous electrolyte is considered. The electric double layer free energy and force are derived under the assumptions that the plate charge is due to the adsorption or desorption of potential determining (p.d.) ions according to the Nernst equation and that the potential distribution in the diffuse layers satisfies the Poisson-Boltzmann (P.B.) equation. The simple case where the p.d. ion is one of the ion species of the 1–1 electrolyte is considered. In addition to exact expressions in terms of elliptic integrals, approximate formulae suitable at large separation and large potentials are obtained. Numerical results are given to illustrate the difference between this “concentrated sol” and the corresponding “dilute sol”, that is a pair of parallel plates in a large volume of electrolyte. The effective thickness of the double layers depends significantly on whether the p.d. ion is adsorbed or desorbed. In the case of adsorption, depletion of p.d. ions from the dispersion medium causes the double layer thickness and hence the double layer repulsion to be greater in the concentrated system, as compared with the dilute system. Comparison with some experiments of Barclay and Ottewill shows the same general trend. The opposite effect of a smaller double layer repulsion is obtained in the desorption case.

Approximate methods of determining the double-layer free energy of interaction between two charged colloidal spheres

We investigate several approximate methods of finding the double-layer interaction free energy, according to the Derjaguin-Landau-Verwey-Overbeek theory, of two charged spheres with centers at distance R apart in electrolyte solution. The spheres may have unequal radii a 1 and a 2 and unequal surface potentials. An appropriate approximation at reasonably large values of κ(R − a 1 − a 2), where κ is the Debye-Huckel constant, is that of linear superposition (L.S.A.). The potential in the neighborhood of the plane equidistant from the two centers is equated to ψ 1(r 1) + ψ 2(r 2), where r 1 and r 2 are distances from the centers and ψ 1 and ψ 2 are the potentials due to the two spheres considered as isolated systems. By using the theoretical work of Gronwall, La Mer, and Sandved or the numerical results of Loeb, Wiersema, and Overbeek for single sphere potential distributions, the method can be applied to higher surface potentials, outside the linear Debye-Hu¨ckel range. A simple general form of the interaction energy at large separations suitable for all radii and potentials is obtained. An integral equation method, recently developed to solve the Debye-Hu¨ckel equation for two identical, charged particles, is extended to the case of two particles having both unequal radii ( a 1, a 2) and unequal potentials. This is applicable when κa 1 and κa 2 ≥ 5 and surface potentials are small. It gives the correct limiting form at large separations and also reduces to a formula which is derived from the method of Derjaguin at small separations. An extension to larger potentials is suggested. We also examine the conditions under which two spherical particles, having equal radii but unequal surface potentials of the same sign, show electric double-layer attraction in the Debye-Hu¨ckel range. Numerical values of interaction free energy and force are compared with those directly computed for pairs of spheres of equal radius a and equal potential. The proposed extension to larger potentials of the interaction formula by the integral equation method gives overall satisfactory agreement with available numerical results for κa ∼ 5. As an alternative, it is possible to combine the L.S.A. at large separations with the Derjaguin formula at small separations and so obtain good values for both force and interaction energy at all separations and potentials.