Spherical double layer

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. 

In 1954 Weiss32 used Bernal and Fowler s simplified solvation model,16 with an Inner Sphere of ionic coordination, i.e., a small spherical double layer around the ion of charge ze, followed by a sharp discontinuity at radius q, the edge of the Outer Sphere or Dielectric Continuum. He used a simple electrostatic argument to determine the energy to remove an electron at optical frequency from the Inner Sphere ... 

The repulsive force for two charged spherical double layers is obtained in the same way as that for two plates by considerii the force... 

Pictures like this are the basis of the analysis of ton exchange phenomena. 3.5e. Spherical double layers... 

Table 3.1. Ratio of exact surface charge to charge according to the DH approximation. Spherical double layer, (1-1) electrolyte. (Source Loeb et al., loc. cit. table 21). 

Summarizing, the far and near field differ in three respects. First they do so in range. Common double layer fields extend over distances of order x" in the absence of an external field such fields are radial for a spherical double layer, as shown In fig. 3.86,bl. On the other hand, the range of the far fields is of the order of the particle radius a, which for the case considered, means that they extend far beyond the double layer. In the second place they differ In magnitude, as already stated. Thirdly, the difference is that in the near field there exist local excess charges, whereas in the far field each volume element is electro-neutral. In mathematical language, p [r,0) = 0, where r and 6 are defined in fig. 3.87. Consequently, the Laplacian of the potential is also zero in the far field. 

Chan, B.K.C. Chan, D.Y.C. Electrical doublelayer interaction between spherical colloidal particles an exact solution. J. Colloid Interface Sci. 1983, 92 (1), 281-283 Palkar, S.A. Lenhoff, A.M. Energetic and entropic contributions to the interaction of unequal spherical double layers. J. Colloid Interface Sci. 1994, 165 (1), 177-194 Qian, Y. Bowen, W.R. Accuracy assessment of numerical solutions of the nonlinear Poisson-Boltzmann equation for charged colloidal particles. J. Colloid Interface Sci. 1998, 201 (1), 7-12 Carnie, S.L. Chan, D.Y.C. Stankovich, J. Computation of forces between spherical colloidal particles Nonlinear Poisson-Boltzmann theory. J. Colloid Interface Sci. 1994, 165 (1), 116-128 Stankovich, J. Carnie, S.L. Electrical double layer interaction between dissimilar spherical colloidal particles and between a sphere and a plate nonlinear Poisson-Boltzmann theory. Langmuir 1996,12 (6), 1453-61. 

To determine the distribution of electric potential, as modified by surface conductance, we again take the electric field to be spherically symmetric and to satisfy the Laplace equation. A thin spherical double layer shell is considered to surround the particle, and the conductivity of this shell is taken to have the mean value cr(. In reality the conductivity in the thin double layer varies continuously. Outside of the double layer shell the bulk conductivity is that of the electrolyte. This electrostatics problem is a straightforward one in which, from the Laplace equation, the solution for the potential is... 

For interacting identical spherical particles with spherical double layers, a similar calculation of the repulsive potential can be carried out (Overbeek 1972). Provided the thickness of the double layers is small compared with the particle size, the interaction between the double layers on the spherical particles can be assumed to be made up of contributions from infinitesimally small parallel rings, each of which can be considered as a flat plate (see Fig. 8.1.2). The energy of repulsion between the spherical double layers is then... 

For curved interfaces, Eq. (3) cannot be solved analytically. For spherical interfaces, the Debye-Hiickel approximation can be applied, provided that ze j/ < 25 mV. Alternatively, solutions may be obtained by numerical methods with the aid of a computer, as have been fully compiled for spherical double layers by Loeb et al. [7]. 

Consider an example of such calculation [36] for a spherically-symmetric diffusion double layer, formed on a non-conducting spherical particle of radius a. A spherical double layer of thickness dr contains the charge... 

The attraction between oppositely charged colloids can be understood and modeled using the DLVO theory [3-6]. The DLVO theory links the van der Waals attraction between particles with the electrostatic effects resulting from the presence of a double layer of counterions. A detailed theoretical discussion lies outside the scope of this chapter. One of the difficulties of the DLVO theory is that an exact analytical description of interaction of overlapping double layers is only known for flat, infinite parallel surfaces. For spherical double layers, approximations need to be made or numerical theoretical simulations need to be performed. 

Still neglecting the finite dimensions of the ions, we will now study the theory of a spherical double layer. 

For small particles, however, we see that the factor 1/r causes the electric potential to fall off more rapidly than the purely exponential expression found for the flat double layer. If indeed, the particle is small in comparison to the thickness of the double layer, then the charge in the surrounding ionic atmosphere, going from the particle surface to the bulk of the solution, must be distributed among spherical shells of increasing volume. This explains the characteristic difference between spherical and flat double layers, and shows that the equations for the spherical double layer are especially important if xa 1. 

In a similar way we may understand another remarkable point ro be derived from M ii 11 e r s tables. For a flat double layer we found that the form of the electric potential curve is radically changed by an increase in the ionic charge. M ii 11 e r s data however, show, that for a spherical particle (with jco 1) the valency of the ions in the solution has only a minor influence upon the decline of the electric potential in the diffuse layer. Hence, also in this respect the Debyc-Hiickel theory is a much better approximation for the spherical double layer field than for the flat double layer, once we wish to apply this theory to cases where the potential is no longer small. 

Potential energy of interaction of two spherical double layers. 

For a spherical double layer, the solution using the Debye-Huckel approximation, will be ... 

In general, both parameters, surface charge and potential, are regulated. For Carnie s canonical intermediate case, i.e., for Aj = 0, the interaction energy between two spherical double layers obeys ... 

The basic theory of gel formation from colloidal particles has been formulated by Thomas and McCorkle (228), who show that the Verwey-Overbeek theory for the interaction of two spherical double layers around adjacent spherical colloidal particles leads to isotropic flocculation. New particles can be attached more readily to the ends of a chainlike floe where the repulsion energy barrier is at a minimum. It is this type of aggregation that converts a sol to a gel at a certain point by forming an infinite network of chains of particles throughout the sol volume. (See also Chapter 3.)... 

The numerical results also demonstrate that saturation effect at high potential is a general feature of spherical double layers. Because the potential diminishes with distance from the surface, linearization becomes appropriate and the decay is exponential, viz. 

EXAMPLE 23.3 The spherical double layer. Let s compute the electrostatic potential ipir) as a fimction of the radial distance from a charged sphere in a salt solution. The sphere has radius a, net charge Q, and a uniform surface charge density a Q/ 4na ). The sphere is in a monovalent salt solution that has a Debye length 1/k. The potential changes only in the radial direction so (dtfj/dO) = 0 and id ip/d4> ) = 0. For spherical coordinates (see Equation (17.33)), the linearized Poisson-Boltzmann Equation (23.6) becomes... 

This equation can be transformed into a formula describing the interaction between curved surfaces, such as that between two spherical double-layers. This is carried out by using the Derjaguin equation. The latter connects the force between two spherical double-layers and the interaction energy per unit area, Vr, of two plane interacting double-layers. It is assumed that both the spherical and the plane double-layers carry the same surface charge density, which leads to the following ... 

Why and how do these molecules manage to assemble, within seconds, perfectly spherical double-layered aggrcgate.s The why is associated with thermodynamics, and addresses the question of why these gigantic structures, and not others, are energetically favored the how has more to do with the kinetic pathway of formation-through which intermediates and at which rates the giant vesicles form. 

Full evaluation of equation (2.4) thus requires knowledge of the charge distribution at the electrode - electrolyte interface, a problem that has been explored in various works.For example, Dickinson and Compton recently used numerical modelling to solve the Poisson - Boltzmann equation, which describes the electric field in an electrolyte solution under thermodynamic equilibrium, for hemispherical electrodes. The simulations revealed a transition between two classical limits a planar double layer as predicted by the Gouy - Chapman model and the spherical double layer associated with a point charge (Coulomb s Law). This is illustrated in Fig. 2.2, in which the dimensionless charge density, Q ( FrqjRTEQEg) is plotted as a function of the dimensionless hemispherical electrode radius,... 

Clearly, Eq. [338] represents an approximation to the actual interaction energy since (1) the rings become progressively less parallel as p and hence H increases, (2) this ignores contributions from that part of the larger sphere beyond the smaller sphere radius as well as those from the backsides of both spheres, and (3) there are pressure contributions on the dividing plane beyond min not considered at all. These concerns are minimized if the system meets two conditions (1) the closest spacing between spheres is much less than the smaller radius, and (2) the thickness of both spherical double layers is small ... 

R. Natarajan and R. S. Schechter,/. Colloid Interface ScL, 113,241 (1986). The Solution of the Nonlinear Poisson-Boltzmann Equation for Thick, Spherical Double Layers. 

S. A. Palkar and A. M. Lenhoff, /. Colloid Interface Sci, 165, 177 (1994). Energetic and Entropic Contributions to the Interaction of Unequal Spherical Double Layers. 

Gonzalez-Tovar, E. and M. Lozada-Cassou. 1989. The spherical double layer A hypernetted chain mean spherical approximation calculation for a model spherical colloid particle. Journal of Physical Chemistry 93 (9) 3761-3768. 

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