Spherical electric double layer

For studying the stability of colloidal particles in suspension (Chapter 13) or for determining the potential at the surface of particles (Chapter 12), one often needs expressions for potential distributions around small particles that have curved surfaces. Solving the Poisson-Boltzmann equation for curved geometries is not a simple matter, and one often needs elaborate numerical methods. The linearized Poisson-Boltzmann equation (i.e., the Poisson-Boltzmann equation in the Debye-Hiickel approximation) can, however, be solved for spherical electrical double layers relatively easily (see Section 12.3a), and one obtains, in place of Equation (37),... 

A popular representation of spherical micelles was devised by Hartley (26). As indicated in Fig. 1, the Hartley model of, e.g., an anionic micelle exhibits a spherical electric double layer composed of bulky, hydrated anionic heads of surfactant molecules and their counterions in the aqueous phase, while the hydrophobic tails, visualized as sticks, form a hydrocarbon-like micellar interior. Because of the high surface charge density of the micelle, there is only little electrolytic dissociation of counterions. The Hartley model explains the low conductivity of micellar solutions and the way surfactants work as detergents by solubilizing (i.e. incorporating) hydroi obic substrates. The model fails to explain certain NMR and fluorescence data that demonstrate some contact of... 

Yu Y-X, Wu J, Gao GH (2004) Density functional theory of spherical electric double layers and potential of colloidal particles in restricted-primitive-model electrolyte solutions. J Chem Phys 120 7223-7233... 

A. L. Loeb, J. Th. G. Overbeek, and P. H. Wiersema, The Electrical Double Layer Around a Spherical Particle, MIT Press, Cambridge, MA, 1961. 

The physicochemical forces between colloidal particles are described by the DLVO theory (DLVO refers to Deijaguin and Landau, and Verwey and Overbeek). This theory predicts the potential between spherical particles due to attractive London forces and repulsive forces due to electrical double layers. This potential can be attractive, or both repulsive and attractive. Two minima may be observed The primary minimum characterizes particles that are in close contact and are difficult to disperse, whereas the secondary minimum relates to looser dispersible particles. For more details, see Schowalter (1984). Undoubtedly, real cases may be far more complex Many particles may be present, particles are not always the same size, and particles are rarely spherical. However, the fundamental physics of the problem is similar. The incorporation of all these aspects into a simulation involving tens of thousands of aggregates is daunting and models have resorted to idealized descriptions. 

An illustration of the effect of micelle/nanoparticle volume fraction on contact line motion is found in [57]. They used 0.1 M NaCl solution to reduce the electrical double layer thickness surrounding the NaDS micelle. At a given number concentration of micelles, decreasing the size of each micelle decreases the volume fraction greatly, since the volume of each spherical micelle varies as the third power of the radius. Thus, the addition of electrolyte effectively reduced the micellar volume fraction in the aqueous medium. The authors found that the oil droplet that would otherwise become completely detached from the solid surface, came back to reattach itself to the solid when electrolyte was present. They rationalized this finding as being caused by the inability of the weakened structural disjoining forces to counteract the attraction of the oil drop to the solid surface. 

Before we proceed to the Gouy-Chapman theory of electrical double layers, it is worthwhile to note that relations similar to Equations (45) and (47) can also be derived for double layers surrounding spherical particles. The equation for surface charge density takes the form... 

Show that the one-dimensional Poisson equation for planar electrical double layers discussed in the text has the following analog in the case of spherically symmetric double layers ... 

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. 

If there are sufficiently strong repulsive interactions, such as from Ihe electric double-layer lorce. then the gas bubbles at the lop of u froth collect together without bursting. Furthermore, their interfaces approach as closely as these repulsive forces allow typically on the order of 100 nm. Thus bubbles on top of a froth can pack together very closely and still allow most uf the liquid to escape downward under the influence of gravity while maintaining their spherical shape. Given sufficient liquid, such a foam can resemble the random close-packed structure formed by hard spheres. 

Figure D3.5.8 (A) Three possible models for systems (spherical particle) with an electrical double layer. (B) Corresponding electrical potential as a function of the separation distance.

In the years 1910-1917 Gouy2 and Chapman3 went a step further. They took into account a thermal motion of the ions. Thermal fluctuations tend to drive the counterions away form the surface. They lead to the formation of a diffuse layer, which is more extended than a molecular layer. For the simple case of a planar, negatively charged plane this is illustrated in Fig. 4.1. Gouy and Chapman applied their theory on the electric double layer to planar surfaces [54-56], Later, Debye and Hiickel calculated the potential and ion distribution around spherical surfaces [57],... 

The calculation of the interaction energy, VR, which results from the overlapping of the diffuse parts of the electric double layers around two spherical particles (as described by Gouy-Chapman theory) is complex. No exact analytical expression can be given and recourse must be had to numerical solutions or to various approximations. 

The history of PB theory can be traced back to the Gouy-Chapmann theory and Debye-Huchel theory in the early of 1900s (e.g., see Camie and Torrie, 1984). These two theories represent special simplified forms of the PB theory Gouy-Chapmann theory is a one-dimensional simplification for electric double-layer, while the Debye-Huchel theory is a special solution for spherical symmetric system. The PB equation can be derived based on the Poisson equation with a self-consistent mean electric potential tj/ and a Boltzmann distribution for the ions... 

In the simplest example of colloid stability, suspension partides would be stabilized entirely by the repulsive forces created when two charged surfaces approach each other and their electric double layers overlap. The repulsive energy VR for spherical particles, or rigid droplets, is given approximately as ... 

A quantitative treatment of the effects of electrolytes on colloid stability has been independently developed by Deryagen and Landau and by Verwey and Over-beek (DLVO), who considered the additive of the interaction forces, mainly electrostatic repulsive and van der Waals attractive forces as the particles approach each other. Repulsive forces between particles arise from the overlapping of the diffuse layer in the electrical double layer of two approaching particles. No simple analytical expression can be given for these repulsive interaction forces. Under certain assumptions, the surface potential is small and remains constant the thickness of the double layer is large and the overlap of the electrical double layer is small. The repulsive energy (VR) between two spherical particles of equal size can be calculated by ... 

The interaction between two spherical colloids can be transformed by the Derjaguin approximation [29] to the interaction between two flat surfaces (see Appendix A). The net osmotic pressure in an electric double layer is the difference between the internal force, F n, and the external or bulk force, Fex, and is related to the force between two colloids Posm = F n — Fex/a, where a is the area. 

HLC have suggested that the solvent dipoles near the colloidal particles are preferentially aligned. This effect is well known in theories of the electrical double layer. One simple way of accounting for this effect is through the use of a Stern layer of low dielectric constant near the colloidal particles. It is difficult to calculate this correction for spherical particles. As a result, HLC considered a hard sphere fluid between two hard walls and with a region of low dielectric constant near the walls. They found that Eq. (62) should be generalized to... 

Fig. 14.5. Experimental rejection (o) and theoretical prediction of the critical pressure for filtration of BSA in 0.001 M NaCI solution at pH 9 at a membrane of mean pore diameter 84 nm. Rejection is high below the critical pressure as electrical double layer repulsion prevents the protein (effective spherical diameter 6nm) from entering the membrane pores. As the critical pressure is approached, hydrodynamic forces increase and drive the...

Application of the electric double layer theory to soil minerals at a quantitative level is difficult because soil mineral surfaces at the microscopic scale are not well defined, that is, they are neither perfectly spherical nor flat, as the double layer requires. However, application of the double layer theory at a qualitative level is appropriate because it explains much of the behavior of soil minerals in solution, for example, dispersion, flocculation, soil permeability, and cation and/or anion adsorption. When equilibrium between the counterions at the surface (near the charged surface) and the equilibrium solution is met, the average concentration of the counterions at any... 

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