The electrostatic double layer

Interfacial Forces. Neighboring bubbles in a foam interact through a variely of forces which depend on the composition and thickness of liquid between them, and on the physical chemistry of their liquid—vapor interfaces. For a foam to be relatively stable, Ihe net interaction must be sufficiently repulsive at short distances to maintain a significant layer of liquid in between neighboring bubbles, Interfacial forces include ihe van der Waals inieracliun. the electrostatic double layer imeruclion. and disjoining pressure. 

Please note that the electrostatic double-layer force is fundamentally different from the Coulomb force. For example, if we consider two identical spherical particles of radius R you cannot take Eq. (6.1), insert the total surface charge as Qi and Q2, use the dielectric permittivity of water and expect to get a reasonable result. The main differences are the free charges (ions) in solution. They screen the electrostatic field emanating from the surfaces. 

In an aqueous medium, the electrostatic double-layer force is present. For distances x larger then the Debye length A it decays roughly exponentially F oc exp (—x/A >). 

One of the central questions in the stability of foams is why are liquid films between two adjacent bubbles stable, at least for some time In fact, a film of a pure liquid is not stable at all and will rupture immediately. Formally this can be attributed to the van der Waals attraction between the two gas phases across the liquid. As for emulsions, surfactant has to be added to stabilize a liquid film. The surfactant adsorbs to the two surfaces and reduces the surface tension. The main effect, however, is that the surfactant has to cause a repulsive force between the two parallel gas-liquid interfaces. Different interactions can stabilize foam films [570], For example, if we take an ionic surfactant, the electrostatic double-layer repulsion will have a stabilizing effect. 

In terms of potential, the electrostatic double-layer "wave" equation is... 

It is unfortunate that this macroscopic-continuum limitation is sometimes forgotten in overzealous application. The same limitation also holds in the theory of the electrostatic double layers for which we often make believe that the medium is a featureless continuum. Neglect of structure in double layers is equally risky, though, and even more common than in the computation of van der Waals forces. 

For the case of purely attractive forces (such as Lon-don-van der Waals forces) the length Sjr over which they act is a useful characteristic. An attractive force which acts over a distance which is much less than Sc will not contribute substantially to the overall rate. When repulsive forces (such as the electrostatic double-layer forces) are also present, they may effectively prevent particles from arriving at the collector, even when they act only over a very short distance. For this reason the decay length alone cannot characterize the relative importance of the joint effect of attractive and repulsive farces. Useful characteristics of their combined effect may be obtained by considering the total potential energy of interaction between the particle and the collector. 

The process of cell deposition in the presence of repulsive forces may be considered as a two-step sequence. First the cells move, primarily under the action of gravity, to a region very near to the surface. In order to move closer to the surface the particle must experience the energy barrier formed by the electrostatic double-layer repulsions and London attraction. Diffusion of cells over the energy barrier is the second step of the process. If the deposition rate is much smaller than the sedimentation rate the second step... 

It is well-known that free films of water stabilized by surfactants can exist as somewhat thicker primary films, or common black films, and thinner secondary films, or Newton black films. The thickness of the former decreases sharply upon addition of electrolyte, and for this reason its stability was attributed to the balance between the electrostatic double-layer repulsion and the van der Waals attraction. A decrease in its stability leads either to film rupture or to an abrupt thinning to a Newton black film, which consists of two surfactant monolayers separated by a very thin layer ofwater. The thickness of the Newton black film is almost independent of the concentration of electrolyte this suggests that another repulsive force than the double layer is involved in its stability. This repulsion is the result of the structuring of water in the vicinity of the surface. Extensive experimental measurements of the separation distance between neutral lipid bilayers in water as a function of applied pressure1 indicated that the hydration force has an exponential behavior, with a decay length between 1.5 and 3 A, and a preexponential factor that varies in a rather large range. 

The key link between ELS experiments and particle electrostatic properties is the theoretical model of colloidal electrohydrodynamics. The required model is considerably more complicated than the one needed in the interpretation of DLS data. DLS relies upon a relatively simple colloidal hydrodynamic model to relate the measured particle diffusivity to particle radius via the Stokes-Einstein Eq. (39). The colloidal electrohydrodynamic model for ELS must account for the complex physical/chemical/electrical structure of the particle surface as well as the distortion of the diffuse part of the electrostatic double layer due to the motion of the particle through the medium. 

A simplification to this for low surface potential where the electrostatic double layers only have weak overlap leads to an exponential expression for the force between two flat interfaces (in the case of a symmetric electrolyte and symmetric surfaces)... 

For similar micellas concentrations, the effect of electrolyte will be more severe for the anionic surfactant as it will suppress the electrostatic double-layer repulsive forces acting between the micelles. For nonionic surfactant, the repulsive force between micelles is a steric force rather than an electrostatic force, such that electrolyte has less of an effect. 

The ionic strength of a liquid bath affects the width of the diffuse layer and hence the range of the electrostatic double layer (EDL). In solutions containing high ionic densities, relatively small volumes of liquid contain enough counterions to balance the particle surface charge and the width of the diffuse layers becomes con arable or smaller than the range of the attractive van der Waals forces between the particle and the surface. Under these conditions... 

Even though the components of the total interaction potential between such complex adsorbents as solid carbons and a wide range of adsorbates can be grouped in many different ways 315,316], it is convenient and meaningful to consider only the London dispersion (induced dipole) forces and the electrostatic (double-layer) forces [620,621,76,77]. 

It is customarily assumed that the overall particle-particle interaction can be quantified by a net surface force, which is the sum of a number of independent forces. The most often considered force components are those due to the electrodynamic or van der Waals interactions, the electrostatic double-layer interaction, and other non-DLVO interactions. The first two interactions form the basis of the celebrated Derjaguin-Landau-Verwey-Overbeek (DLVO) theory on colloid stability and coagulation. The non-DLVO forces are usually determined by subtracting the DLVO forces from the experimental data. Therefore, precise prediction of DLVO forces is also critical to the determination of the non-DLVO forces. The surface force apparatus and atomic force microscopy (AFM) have been used to successfully quantify these interaction forces and have revealed important information about the surface force components. This chapter focuses on improved predictions for DLVO forces between colloid and nano-sized particles. The force data obtained with AFM tips are used to illustrate limits of the renowned Derjaguin approximation when applied to surfaces with nano-sized radii of curvature. 

First we consider the electrostatic (double layer) interaction between two identical charged plane parallel surfaces across a solution of symmetric Z Z electrolyte. The charge of a counterion (i.e., ion with charge opposite to that of the surface) is -Ze, whereas the charge of a coion is +Ze (Z = +1, +2,. ..) with e the elementary charge. If the separation between the two planes is very large, the number concentration of both counterions and coions would be equal to its bulk value, n, in the middle of the film. However, at finite separation, h, between the surfaces the two EDL overlap and the counterion and coion concentrations in the middle of the film, io and 2o> longer equal. Because the solution inside the film is supposed to be in electrochemical (Donnan) equilibrium with the bulk electrolyte solution of concentration q, we can write 20 0 or, alternatively,... 

SOLUTION 1) with this surface and contains the instruction -no edl, which turns off all electrostatic calculations, those necessary for the electrostatic double layer theory. It also defines the total number of L sites. There is no explicit mention of Langmuir isotherm because, as mentioned above, there is no difference between a Langmuir isotherm equilibrium constant and a single-site surface complex formation constant, which phreeqc understands. 

Electrokinetic properties of the virus, which we showed to be involved directly in the electrostatic double-layer contribution to virus... 

First measurements of the electrostatic double-layer force with the AFM were done in 1991 [9, 10]. The electrostatic double layer depends on the surface charge density (or the surface potential) and the ionic strength. A brief introduction to the theory of the electrostatic force is given in Chap. 4. The electrostatic double-layer force is in many cases responsible for the stabilization of dispersions. An AFM experiment can be regarded as directly probing the interaction between a sample surface and a colloidal particle (or the AFM tip). Since the AFM tip is relatively small, this interaction can be probed locally. The lateral spacial resolution can be of the order of few nanometers. 

With the SFA, the main predictions of the DLVO theory were verified. In particular the electrostatic double-layer force was analyzed for difierent salts and under... 

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