Surface potentials, charged spheres

Case (c) sphere 1 maintained at constant surface potential and sphere 2 at constant surface charge density... 

In view of this equation the effect of the ionic atmosphere on the potential of the central ion is equivalent to the effect of a charge of the same magnitude (that is — zke) distributed over the surface of a sphere with a radius of a + LD around the central ion. In very dilute solutions, LD a in more concentrated solutions, the Debye length LD is comparable to or even smaller than a. The radius of the ionic atmosphere calculated from the centre of the central ion is then LD + a. 

Consider a homogeneous, isotropic sphere that is placed in an arbitrary medium in which there exists a uniform static electric field E0 = E0ez (Fig. 5.3). If the permittivities of the sphere and medium are different, a charge will be induced on the surface of the sphere. Therefore, the initially uniform field will be distorted by the introduction of the sphere. The electric fields inside and outside the sphere, Ej and E2, respectively, are derivable from scalar potentials 0,(r, 6) and 02(r, 8)... 

Figure 4. Calculated dependence of the ratio of the surface charge density to the surface excess on the electrostatic surface potential. Curve A is for the infinite flat plate case. Curve B is for a sphere with a 1000 A. radius...

This simple equation is, however, only valid for R Xp- If the radius is not much larger than the Debye length we can no longer treat the particle surface as an almost planar surface. In fact, we can no longer use the Gouy-Chapman theory but have to apply the theory of Debye and Hiickel. Debye and Hiickel explicitly considered the electric double layer of a sphere. A result of their theory is that the total surface charge and surface potential are related by... 

Fig. 1.1 Radial distribution of potential for a metal sphere of radius R carrying a positive charge Q, illustrating the contributions of the outer potential and the surface potential. The inner potential is constant inside the sphere...

The effect of relaxing the condition of electroneutrality in terms of the electrical potential of a charged phase must first be considered. We choose a single-phase system containing 10 10 mol of an ionic species with a charge number of +1. The phase is assumed to be spherical with a radius of 1 cm and surrounded by empty space. Essentially, all of the excess charge will reside on the surface of the sphere. The electrical potential of the sphere is given by... 

Fig. 6.3. The calculated reduced surface potential = e(ri)/kT versus the logarithm of the amphiphile concentration C (M) with no salt added for a spherical, cylindrical and planar aggregate. The surface charge density has been chosen as fixed at a8 = 0.228 Cm- 2. The radii of the sphere and the cylinder are 1.8 nm...

Fig. 6.4. The calculated reduced surface potential 4> = e(r()/kT versus the inverse radius for a sphere and a cylinder. Amphiphile concentration 45 mM and surface charge density <7S = 0.228 Cm-2...

Although each SCM shares certain common features the formulation of the adsorption planes is different for each SCM. In the DDLM the relationship between surface charge, diffuse-layer potential, d, is calculated via the Gouy-Chapman equation (Table 5.1), while in the CCM a linear relationship between surface potential, s, is assumed by assigning a constant value for the inner-layer capacitance, kBoth models assume that the adsorbed species form inner-sphere complexes with surface hydroxyls. The TLM in its original... 

At low surface potentials Eq. (5.5) reduces to Eq. (5.2). Secondly, an inner layer exists because ions are not really point charges and an ion can only approach a surface to the extent allowed by its hydration sphere. The Stern model specifically incorporates a layer of specifically adsorbed ions bounded by a plane termed the Stern plane (see Figure 4.3). In this case the potential changes from ip° at the surface, to ip(d) at the Stern plane, to ip = 0 in bulk solution. 

For calculating the electrostatic potential of a charged metallic sphere, the charge on the sphere can he considered to reside at its center. What is the charge (in both Coulombs and moles of electrons) on a 1.0-cm-radius sphere so that the electrostatic potential at the surface of the sphere is 1.0 V ... 

In which z is the valence of the ions surrounding the sphere, e the electronic charge and /o the surface potential. 

AH rate calculations presented above have used the double-layer interaction given by Eq. [p3 as applied to approach at constant surface potential. Suppose the charge density had been held constant instead. How would this change affect the rate of deposition The answer is illustrated in Fig. 5. When the sphere and plate are many Debye lengths apart, the... 

Integrating Eq. (13) is executed over the sphere surface. Since the potentials on the surfaces of the spheres are equal inside and outside, we can use either potentials representations. If the surface charges are constant, integration leads to calculations of the spherical functions integrals over the total surfaces of the spheres. After integrating Eq. (13), taking into account the expansions (Eq. 5), we obtain for the free energy... 

The exact solution for interaction of a system of small spherical particles in an electrolyte is obtained. On the basis of the exact solutions, closed formulae for calculating ion-electrostatic energy of two spheres are derived. Our zero approximation corresponds to results of other authors in simple cases, and generalizes ones in the range of small values where the parameter kH < 2. In this paper we consider the case when surface charges are given, but the problem of spherical particles interaction with given surface potentials can be solved similarly. 

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