Low Charge Density Case

3 SPHERICAL SOFT PARTICLE 4.3.1 Low Charge Density Case 

Consider a spherical soft particle consisting the particle core of radius a covered by an ion-penetrable layer of polyelectrolytes of thickness d. The outer radius of the particle is thus given hy b = a + d (Fig. 4.4). Within the surface layer, ionized groups of valence Z are distributed at a constant density N. The Poisson-Boltzmann equations (4.1) and (4.2) are replaced by the following spherical Poisson-Boltzmann equations for electric potential r being the distance from the center of the particle  

We first treat the case in which the fixed-charge density ZeN is low. Then Eqs. (4.50) and (4.51) are linearized to give 

If we take the hmit b — a Oin Eq. (4.57), keeping the product Nd constant, that is, keeping the total amount of fixed charges a = ZeNd constant, then Eq. (4.57) becomes 

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Note that the following exact expression for the electrostatic interaction between two porous spheres (spherical polyelectrolytes) for the low charge density case has been derived [5,6] (Eq. (13.46)) ... 

To conclude this section let us note that already, with this very simple model, we find a variety of behaviors. There is a clear effect of the asymmetry of the ions. We have obtained a simple description of the role of the major constituents of the phenomena—coulombic interaction, ideal entropy, and specific interaction. In the Lie group invariant (78) Coulombic attraction leads to the term -cr /2. Ideal entropy yields a contribution proportional to the kinetic pressure 2 g +g ) and the specific part yields a contribution which retains the bilinear form a g +a g g + a g. At high charge densities the asymptotic behavior is determined by the opposition of the coulombic and specific non-coulombic contributions. At low charge densities the entropic contribution is important and, in the case of a totally symmetric electrolyte, the effect of the specific non-coulombic interaction is cancelled so that the behavior of the system is determined by coulombic and entropic contributions. 

The experimental results showed no significant difference in adsorption rate with respect to molecular weight for the high charge density case. However, at low charge density the... 

The hexacarbonylmetalates(-II) of group 4 can be synthesized by reducing the tetrachlorides with potassium in the presence of a cryptand. In the case of hafnium, the starting material is a tertiary-phosphine-carbonyl complex of the zerovalent metal (see equations 52 and 53). The cryptand/(metal cation) system, characterized by a large mass and a low charge density, presumably has a stabilizing effect on the carbonyl anion. Reactions of equations (52) and (53) are carried out with CO at atmospheric pressure. ... 

The nonlinear equations (4.76) and (4.77) have not been solved analytically except in the low potential case (i.e., the low fixed-charge density case). In this case, the solution to Eqs. (4.76) and (4.77) is... 

Equation (6.6) cannot be solved analytically but its approximate solution for the case of dilute suspensions has been obtained by Imai and Oosawa [3,4]. They showed that there are two distinct cases separated by a certain critical value of the surface charge density cr or the total surface charge Q, that is, case 1 low surface charge density case and case 2 high surface charge density case, as schematically shown in Fig. 6.2. For case 1, there are two regions I R [Pg.135]

We compare the exact numerical solution to the Poisson-Boltzmann equation (6.6) and the approximate results, Eq. (6.37) for case 1 (low surface charge density case) and Eq. (6.50) for case 2 (high surface charge density case) in Fig. 6.3, in which the scaled surface potential jo = zeij/JkT is plotted as a function of the scaled... 

We apply the Derjaguin s approximation (Eq. (12.3)) to the low-potential approximate expression for the plate-plate interaction energy, that is, Eqs. (9.53) and (9.65), obtaining the following two formulas for the interaction between two similar spheres 1 and 2 of radius a carrying unperturbed surface potential ij/f, at separation H at constant surface potential, V (H), and that for the constants surface charged density case, V (//) ... 

The bromine molecule is an example of an electrophilic species that has a low charge density. In this case, the covalent bond between the bromine atoms may be polarised by being in close proximity to a charged centre. Any dipole that is induced would only be small, and so the corresponding charge density must be low. 

In strong polyelectrolytes with low charge density, the polymer is coiled. In the case of a weak PEL such as PEI, PAC, or chitosan as a natural polymer, the complex formation is influenced by the pH dependence of the polymer charge. In such cases, the complexes are formed between coils of PC and coils of PA. 

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