Constant surface potential model

FIGURE 9.4 Reduced potential energy V (h) — Kl64nkT)V h) as a function of scaled plate separation Kh for the constant surface charge density model calculated with Eq. (9.125) for a = 0, 0.1, and 1 in comparison with V h) = (K/64nkT)V h) for the constant surface potential model calculated with Eq. (9.141). 

Comparison is made with the results for the two conventional models for hard plates given by Honig and Mul [11]. We see that the values of the interaction energy calculated on the basis of the Donnan potential regulation model lie between those calculated from the conventional interaction models (i.e., the constant surface potential model and the constant surface charge density model) and are close to the results obtained the linear superposition approximation. 

Fig. 10. The repulsive force between two plates at constant surface potential and various NaCl concentrations. The following parameters are employed in the calculations Po = —0.05 V, e1 = 10, E11 = 80, 8 = 10 A. NaCl concentration (1) 0.001 M (2) 0.01 M. The solid lines are for the force predicted by the new model and the dashed lines are for the force predicted by the Gouy-Chapman theory.

At constant surface potential, the force provided by the new model is smaller than predicted by the Gouy-Chapman theory. In contrast, at constant surface charge density, the dielectric constant and the thickness of region I have effects opposite to those at constant surface potential. 

When the surface charge is generated through dissociation equilibrium, the difference between the forces calculated using the new model and the Gouy-Chapman theory is much larger than for constant charge density or constant surface potential. 

Until we discovered the constancy of the surface potential from the uniaxial stress results, like most other people, I had been more interested in constant surface charge models. If you do not know how the valency of a macroion varies with the external conditions, it is reasonable to assume it to be constant unless given evidence to the contrary. Given the evidence that y/0 70 mV is roughly constant for the n-butylammonium vermiculite system, what other consequences follow from this In particular, what happens if we apply the coulombic attraction theory with the constant surface potential boundary condition ... 

Equation (9.136) (or Eq. (9.137)) is a transcendental equation for which can be solved numerically. By substituting the obtained value into Eq. (9.86), which holds irrespective of the type of double-layer interaction (constant surface potential or constant surface charge density models), we can calculate the interaction force P h). 

In this chapter, we discuss two models for the electrostatic interaction between two parallel dissimilar hard plates, that is, the constant surface charge density model and the surface potential model. We start with the low potential case and then we treat with the case of arbitrary potential. 

As in the case of two interacting soft plates, when the thicknesses of the surface charge layers on soft spheres 1 and 2 are very large compared with the Debye length 1/k, the potential deep inside the surface charge layer is practically equal to the Donnan potential (Eqs. (15.51) and (15.52)), independent of the particle separation H. In contrast to the usual electrostatic interaction models assuming constant surface potential or constant surface... 

Relatively complete elaborations for the cylinder model have been given by for instance, Anderson and Koh and Levine et al. K In these two theories the solution Is assumed to contain (1-1) electrolytes with =u. Both theories fail to account for conduction behind the slip plane, and both solve the electrokinetic equations, taking double layer overlap into account. Anderson and Koh assume this overlap to take place at fixed surface charge (which, because of the implicit rigid particle model of the cylinder wall, comes down to fixed tr =cT ), whereas Levine et al. do so for constant surface potential (essentially fixed Anderson and Koh also considered capUlaries of other... 

Fig. 4 Comparison of the model improvements on the Derjaguin approximation to the exact numerical computational results of the full Poisson-Boltzmann equation for two spheres with the scaled radius Rk — 0.1 and Rk — 15 and constant surface potential [j/ ez/(k-gT) = 1. The scaled energy, G h), on the vertical axis is defined by G(h) = (/,)/jsM...

The Eq. (3.33) is based on the Gouy-Chapman model (i.e. the Stem layer is ignored) and on charge regulation with constant surface potential (CP). It applies to small surface potentials ( i/ o, l < 25 mV, to thin double layers (xa 1) and to small surface distances (h x,). Equation (3.33) provides only rough estimates for moderate and thick double layers (xa < 5). Yet, Sader et al. (1995) showed that by replacing the termxi +X2 withxi +x2+2 /i(i.e. twice the particle centre distance ri2), the HHF-equation is applicable to arbitrary values of h. 

In most cases, only relatively simple approximations for ridi are needed to capture the essential physics of double-layer interaction forces. Such approximations are typically valid for small surface charges where linearization of the Poisson-Boltzmann equation is acceptable. Under these conditions and assuming univalent electrolytes, examples of constant surface potential and constant surface charge models for fldi are given by the following ... 

Figure 6-12. Model for Ihe Calculation of the van der Waals potential experienced by a single T6 molecule on a Tfi ordered surface. Each molecule is modeled as a chain of 6 polarizable spherical units, and the surface as 8-laycr slab, each layer containing 266 molecules (only pan of the cluster is shown). Tire model is based on X-ray diffraction and dielectric constant experimental data. The two configurations used for evaluating the corrugation of the surface potential are shown. Adapted with permission front Ref. [48].

Morishima et al. [75, 76] have shown a remarkable effect of the polyelectrolyte surface potential on photoinduced ET in the laser photolysis of APh-x (8) and QPh-x (12) with viologens as electron acceptors. Decay profiles for the SPV (14) radical anion (SPV- ) generated by the photoinduced ET following a 347.1-nm laser excitation were monitored at 602 nm (Fig. 13) [75], For APh-9, the SPV- transient absorption persisted for several hundred microseconds after the laser pulse. The second-order rate constant (kb) for the back ET from SPV- to the oxidized Phen residue (Phen+) was estimated to be 8.7 x 107 M 1 s-1 for the APh-9-SPV system. For the monomer model system (AM(15)-SPV), on the other hand, kb was 2.8 x 109 M-1 s-1. This marked retardation of the back ET in the APh-9-SPV system is attributed to the electrostatic repulsion of SPV- by the electric field on the molecular surface of APh-9. The addition of NaCl decreases the electrostatic interaction. In fact, it increased the back ET rate. For example, at NaCl concentrations of 0.025 and 0.2 M, the value of kb increased to 2.5 x 108 and... 

The electrified interface is generally referred to as the electric double layer (EDL). This name originates from the simple parallel plate capacitor model of the interface attributed to Helmholtz.1,9 In this model, the charge on the surface of the electrode is balanced by a plane of charge (in the form of nonspecifically adsorbed ions) equal in magnitude, but opposite in sign, in the solution. These ions have only a coulombic interaction with the electrode surface, and the plane they form is called the outer Helmholtz plane (OHP). Helmholtz s model assumes a linear variation of potential from the electrode to the OHP. The bulk solution begins immediately beyond the OHP and is constant in potential (see Fig. 1). 

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