Double-layer interaction

When two double layers overlap a repulsive pair potential develops which leads to a repulsive pressure. Dispersed like-charged colloids hence repel each other 

The interparticle separation dependence of double layer repulsion is approximately exponential [26] 

The quantity B can be expressed in terms of the surface charge density Oc of the interacting colloids [26] 

The surface potential of a charged colloidal particle typically varies from 10 to 100 mV leading to B values in the same range as given above. 

The double layer interaction between two Kke-charged colloidal particles is sketched in Fig. 1.1 (upper dashed curve). 

When two particles approach each other so that their EDLs overlap, they experience a repulsive or attractive interaction that results from the electro-static and osmotic forces between the ions and the surfaces. Consider, for instance, two particles with identical surface charge and potential. Then, the overlap of the diffuse layers means 

In order to calculate the interaction energy, it is necessary to assume a model for the charge regulation. Camie and Chan (1993) proposed a linear approach for the 

The set of basic equations is completed by the following boundary condition  

This equation holds tme for small permittivity of the particles (sp ,) If this restriction is not fulfilled, the potential distribution inside the particles has to be described as weU. According to Camie and Chan (1993), the error in using Eq. (3.28) for particles with a relatively large permittivity Sp is relevant only for constant charge regulation with small particle sizes. However, in the case of a rather rough surface, the penetration of the electric field inside the particle cannot be ignored even for small ratios ep/cm (Dukhin and Lyklema 1989). 

For an isolated sphere with linear charge regulation, the boundary condition may be written as  

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More sophisticated approaches to describe double layer interactions have been developed more recently. Using cell models, the full Poisson-Boltzmann equation can be solved for ordered stmctures. The approach by Alexander et al shows how the effective colloidal particle charge saturates when the bare particle charge is increased [4o]. Using integral equation methods, the behaviour of the primitive model has been studied, in which all the interactions between the colloidal macro-ions and the small ions are addressed (see, for instance, [44, 45]). 

The situations would be totally different when the two surfaces are put in electrolyte solutions. This is because of formation of the electrical double layers due to the existence of ions in the gap between solid surfaces. The electrical double layers interact with each other, which gives rise to a repulsive pressure between the two planar surfaces as... 

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. 

At negative potentials in alkaline solutions, adsorbed NA retains K+ ions, as demonstrated by Auger spectroscopy, Figure 5-B. This retention of K+ ions is due to interaction of K+ with the pendant carboxylate moiety and greatly exceeds the amounts expected simply from diffuse double-layer interactions. Potential-dependence of K+ retention is essentially absent for compounds incapable of potential-dependent carboxylate pendancy (pyridine, picolinic acid, isonicotinic acid and 2,6-pyridine dicarboxylic acid). 

Electrostatic double layer interaction, 12 5 Electrostatic effects, in organic separations, 21 660... 

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. 

The double layer interaction energy is given In terms of the eluant dielectric constant e by (37). 

Even though this equation is difficult to solve, many approximate methods have been used. Equation (A.3) is, however, interesting for what it tells us about the double-layer interaction. It can be rearranged in the form... 

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... 

The total interaction between the two metal spheres can therefore be classified into two parts (1) the surface, or double-layer, interaction determined by the Gouy-Chapman potential t f0e"Krand (2) the volume, or bulk, interaction —Ar-6 + Br 12. The interaction between double layers ranges from indifference at large distances to increasing repulsion as the particles approach. The bulk interaction leads to an attraction unless the spheres get too close, when there is a sharp repulsion (Fig. 6.131). The total interaction energy depends on the interplay of the surface (double layer) and volume (bulk) effects and may be represented thus... 

Sols and Gels. The essence of the behavior characteristic of the colloidal state is that double-layer interactions are as significant as bulk interactions. In other words, surface interactions are on a par with volume interactions. This condition can therefore be realized in all systems where the surface-to-volume ratios are high, i.e., at submicroscopic dimensions. 

The term tertiary electroviscous effect is applied to the changes in the conformation of poly electrolytes that are caused by //t/ramolecular double-layer interactions. It is customary to extend this definition to include all effects in which the geometry of the system is altered as a result of double-layer interactions. 

The Electrical Double Layer and Double-Layer Interactions... 

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. 

Why is it that the force of double-layer interactions for curved surfaces cannot be derived using osmotic pressure arguments as is done in the case of planar double layers ... 

Kruyt, H. R. (Ed.), Colloid Science, Vol. 1. Irreversible Systems, Elsevier, Amsterdam, Netherlands, 1952. (A classic reference on colloids. Chapters 4 and 6 by Overbeek, cited below, discuss the electrochemistry of the double layer and double-layer interactions.)... 

Chapters 11 and 12 in the present edition focus exclusively on the theories of electrical double layers and forces due to double-layer interactions (Chapter 11) and electrokinetic phenomena (Chapter 12). Chapter 11 includes expressions for interacting spherical double layers, and both chapters provide additional examples of applications of the concepts covered. 

The observed equilibrium thickness represents the film dimensions where the attractive and repulsive forces within the film are balanced. This parameter is very dependent upon the ionic composition of the solution as a major stabilizing force arizes from the ionic double layer interactions between any charged adsorbed layers confining the film. Increasing the ionic strength can reduce the repulsion between layers and at a critical concentration can induce a transition from the primary or common black film to a secondary or Newton black film. These latter films are very thin and contain little or no free interlamellar liquid. Such a transition is observed with SDS films in 0.5 M NaCl and results in a film that is only 5 nm thick. The drainage properties of these films follows that described above but the first black spot spreads instantly and almost explosively to occupy the whole film. This latter process occurs in the millisecond timescale. 

The modulation frequency is typically in the range from 100 Hz to 3 kHz, and thus much lower than the resonance frequencies of the cantilever and the scanner. This enables better control of the forces exerted on the sample. The z-mod-ulation amplitude can be varied between 10 nm and 1 pm to ensure that that the tip is retracted from the surface. Shear forces are reduced permitting investigation of soft samples because of the small duration of the tip-surface contact, between 10 3 and 10 4 s. Pulse force mode SFM has been used to map adhesion of heterogeneous polymers in dependence of temperature and molecular weight as well as map electrostatic double-layer interactions [158-160]. 

Ideally, lyophobic sols are stabilised entirely by electric double-layer interactions and, as such, present colloid stability at its simplest. 

For the case of two spherical particles of radii a and a2, Stern potentials, iftdi and i//d2, and a shortest distance, H, between their Stern layers, Healy and co-workers195 have derived the following expressions for constant-potential, V, and constant-charge, Fr, double-layer interactions. The low-potential form of the Poisson-Boltzmann distribution (equation 7.12) is assumed to hold and Kax and xa2 are assumed to be large compared with unity ... 

In another method, Roberts and Tabor201 measured the electric double layer repulsion between a transparent rubber sphere and a plane glass surface separated by surfactant solution. As the surfaces were brought together, the double-layer interaction caused a distortion of the rubber surface which was monitored interferometrically. 

Macromolecular solutions are stabilised by a combination of electric double layer interaction and solvation, and both of these stabilising... 

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. 

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