Also Double layer interaction

In filtration, the particle-collector interaction is taken as the sum of the London-van der Waals and double layer interactions, i.e. the Deijagin-Landau-Verwey-Overbeek (DLVO) theory. In most cases, the London-van der Waals force is attractive. The double layer interaction, on the other hand, may be repulsive or attractive depending on whether the surface of the particle and the collector bear like or opposite charges. The range and distance dependence is also different. The DLVO theory was later extended with contributions from the Born repulsion, hydration (structural) forces, hydrophobic interactions and steric hindrance originating from adsorbed macromolecules or polymers. Because no analytical solutions exist for the full convective diffusion equation, a number of approximations were devised (e.g., Smoluchowski-Levich approximation, and the surface force boundary layer approximation) to solve the equations in an approximate way, using analytical methods. 

The same graphical method can also be used to illustrate the nature of the double layer interaction free energy and to bring out a simple physical result which can be used to check numerical algorithms commonly used to calculate the interaction free energy. 

Other modifications to the theory of Anderson and Quinn [142] have been reviewed by Deen [146]. Malone and Quinn [147] modified the above theory to include the effect of electrostatic interactions on transport in microporous membranes. Smith and Deen [148] have also looked at these electrostatic or double layer interactions. More recently, Kim and Anderson [149] investigated the hindrance of solute transport in polymer lined micropores. Also, as briefly mentioned above, an excellent review of the theories presented for transport in microporous membranes has been given by Deen [146]. 

Not only do double layers interact with double layers, the metal of one sphere also interacts with the metal of the second sphere. There is what is called the van der Waals attraction, which is essentially a dispersion interaction that depends on r-6, and the electron overlap repulsion, which varies as r-12. These interactions between the bulk... 

Electrostatic and electrical double-layer interactions also create new opportunities in science and technology. We have already seen an example of this in a vignette in Chapter 1 on electrophoretic imaging devices, and another, on electrophotography, is described in the next... 

In addition to all these, it is also important to keep in mind that the results depend also on what types of surface equilibrium conditions exist as the double layers interact. For example, when two charged surfaces approach each other, the overlap of the double layers will also affect the manner in which the charges on the surfaces adjust themselves to the changing local conditions. As the double layers overlap and get compressed, the local ionic equilibrium at the surface may change, and this will clearly have an impact on the potential distribution and on the potential energy of interaction. 

Lyophilic colloids can also be desolvated (and precipitated if the electric double layer interaction is sufficiently small) by the addition of non-electrolytes, such as acetone or alcohol to aqueous gelatin solution and petrol ether to a solution of rubber in benzene. 

The electrostatic stabilization theory was developed for dilute colloidal systems and involves attractive van dcr Waals interactions and repulsive double layer interactions between two particles. They may lead to a potential barrier, an overall repulsion and/or to a minimum similar to that generated by steric stabilization. Johnson and Morrison [1] suggest that the stability in non-aqueous dispersions when the stabilizers are surfactant molecules, which arc relatively small, is due to scmi-stcric stabilization, hence to a smaller ran dcr Waals attraction between two particles caused by the adsorbed shell of surfactant molecules. The fact that such systems are quite stable suggests, however, that some repulsion is also prescni. In fact, it was demonstrated on the basis of electrophoretic measurements that a surface charge originates on solid particles suspended in aprotic liquids even in the absence of traces of... 

A lattice model for an electrolyte solution is proposed, which assumes that the hydrated ion occupies ti (i = 1, 2) sites on a water lattice. A lattice site is available to an ion i only if it is free (it is occupied by a water molecule, which does not hydrate an ion) and has also at least (i, - 1) first-neighbors free. The model accounts for the correlations between the probabilities of occupancy of adjacent sites and is used to calculate the excluded volume (lattice site exclusion) effect on the double layer interactions. It is shown that at high surface potentials the thickness of the double layer generated near a charged surface is increased, when compared to that predicted by the Poisson-Boltzmann treatment. However, at low surface potentials, the diffuse double layer can be slightly compressed, if the hydrated co-ions are larger than the hydrated counterions. The finite sizes of the ions can lead to either an increase or even a small decrease of the double layer repulsion. The effect can be strongly dependent on the hydration numbers of the two species of ions. 

Colloidal dispersions can be stabilized by attaching polymer chains to their surface [1-9]. When neutral polymer chains grafted on two parallel plates interpenetrate, a steric repulsion is generated. If the polymer chains grafted to the plates are charged, the double layer interaction between the two plates is also affected by the presence of the chains. 

In this chapter, we give exact expressions and various approximate expressions for the force and potential energy of the electrical double-layer interaction between two parallel similar plates. Expressions for the double-layer interaction between two parallel plates are important not only for the interaction between plate-like particles but also for the interaction between two spheres or two cylinders, because the double-interaction between two spheres or two cylinders can be approximately calculated from the corresponding interaction between two parallel plates via Deijaguin s approximation, as shown in Chapter 12. We will discuss the case of two parallel dissimilar plates in Chapter 10. 

Equation (9.197) as combined with Eq. (9.201) is the required expression for the potential energy of the double-layer interaction per unit area between two parallel similar plates with constant surface charge density cr. From the nature of this linearization, the obtained potential distribution (Eq. (9.187)) is only accurate near the plate surface and thus the interaction energy expression (9.197) is also only accurate for small plate separations h. Indeed, as /z —> 0, Eq. (9.197) gives the following correct limiting expression [15, 16] ... 

In Chapter 11, we derived the double-layer interaction energy between two parallel plates with arbitrary surface potentials at large separations compared with the Debye length 1/k with the help of the linear superposition approximation. These results, which do not depend on the type of the double-layer interaction, can be applied both to the constant surface potential and to the constant surface charge density cases as well as their mixed case. In addition, the results obtained on the basis of the linear superposition approximation can be applied not only to hard particles but also to soft particles. We now apply Derjaguin s approximation to these results to obtain the sphere-sphere interaction energy, as shown below. 

Double Layer Interactions and Interfacial Charge. Schulman et al (42) have proposed that the phase continuity can be controlled readily by interfacial charge. If the concentration of the counterions for the ionic surfactant is higher and the diffuse electrical double layer at the interface is compressed, water-in-oil microemulsions are formed. If the concentration of the counterions is sufficiently decreased to produce a charge at the oil-water interface, the system presumably inverts to an oil-in-water type microemulsion. It was also proposed that for the droplets of spherical shape, the resulting microemulsions are isotropic and exhibit Newtonian flow behavior with one diffused band in X-ray diffraction pattern. Moreover, for droplets of cylindrical shape, the resulting microemulsions are optically anisotropic and non-Newtonian flow behavior with two di-fused bands in X-ray diffraction (9). The concept of molecular interactions at the oil-water interface for the formation of microemulsions was further extended by Prince (49). Prince (50) also discussed the differences in solubilization in micellar and microemulsion systems. 

In concentrated suspensions many body interactions between the colloidal particles determine the effective colloid-colloid interaction. Beresford-Smith and Chan (1983) [37] showed that in that case the effective colloid-colloid interaction can nevertheless be described by an effective pair interaction energy to characterise the electrical double layer interaction. This pair interaction energy also has a screened Coulomb form just as in the classical DLVO theory but the Debye screening parameter k now depends on the intrinsic coxmterion concentration and the concentration of added electrolyte in the system. This makes the effective pair energy dependent on the volume fraction of the particles (see general discussion of the paper of Beresford-Smith and Chan [38]. 

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. 

Note that the theory predicts ion-correlation attraction not only across water films with overlapping EDL, but also across oily films intervening between two water phases. In the latter case, is not zero because the ions belonging to the two outer double layers interact across the thin dielectric (oil) film. The theory for such a fihn predicts that 11, 3 is negative (attractive) and strongly dependent on the dielectric permittivity of the oil film can be comparable by magnitude with n i = 0 in this case. 

In addition to ensuring that conditions are such that nucleation and growth of particles to colloidal size are controlled, it is also essential that an appropriate stabilisation mechanism is operative. In the case of colloids formed in aqueous media from electrolytes this stabilisation usually arises from double-layer interaction, while in non-aqueous media it is necessary to employ some form of steric stabilisation. 

A recent development in double-layer interaction theory, known as the surface-regulation approach (3i), also predicts interaction potentials that are intermediate in value. Data required for application of the regulation approach are not available. 

Dielectrophoresis has also been used to manipulate macromolecules such as DNA, viruses, proteins, and carbon nanotubes. The term colloids will be used here to generally describe a particle between 1 and 1,000 nm. At this scale we need to take into consideration additional parameters that will affect the efficiency and application of dielectrophoresis. The first is Brownian motion, or the random chaotic movement of molecules, which will introduce another destabilizing variable if we were to trap colloids. Second, electrostatic effects at the surface of colloids, created by the electrical double layer, will influence particle-particle interactions. Factors such as hydrodynamic drag, buoyancy, electrothermal effects, and a particle s double layer interactions need to be considered when applying dielectrophoresis to colloids. 

The predictions based on a constant C and a constant a are also compared in Fig. 3a-c. The constant a assumption (d = — 8 mC/m ) leads to a very close prediction for the electrical conductance however, this model overrates the hydro-dynamic conductance at the high-concentration range and underrates it very much at the low-concentration rate. The constant C assumption underrates the electrical conductance when the double layers interact, but overrates the hydrodynamic conductance at the same... 

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