Constant surface potential model Double layer interaction

If the dissociation of the ionizable groups on the particle surface is not complete, or the configurational entropy Sc of adsorbed potential-determining ions depends on N, then neither of ij/o nor of cr remain constant during interaction. This type of double--layer interaction is called charge regulation model. In this model, we should use Eqs. (8.35) and (5.44) for the double-layer free energy [ 11-13]. 

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 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 ... 

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). 

It is noted that the molecular interaction parameter described by Eq. 52 of the improved model is a function of the surfactant concentration. Surprisingly, the dependence is rather significant (Eig. 9) and has been neglected in the conventional theories that use as a fitting parameter independent of the surfactant concentration. Obviously, the resultant force acting in the inner Helmholtz plane of the double layer is attractive and strongly influences the adsorption of the surfactants and binding of the counterions. Note that surface potential f s is the contribution due to the adsorption only, while the experimentally measured surface potential also includes the surface potential of the solvent (water). The effect of the electrical potential of the solvent on adsorption is included in the adsorption constants Ki and K2. 

The first theoretical description of the double layers assumed that the ions interact via a mean potential, which obeys the Poisson equation.2 Such a simple theory is clearly only approximate and sometimes predicts ionic concentrations in the vicinity of the surface that exceed the available volume.3 There were a number of attempts to improve the model, by accounting for the variation of the dielectric constant in the vicinity of the surface,4 for the volume-exclusion effects of the ions,5 or for additional interactions between ions and surfaces, due to the screened image force potential,6 to the van der Waals interactions of the ions7 with the system, or to the change in hydration energy when an ion approaches the interface.8... 

There is a range of equations used describing the experimental data for the interactions of a substance as liquid and solid phases. They extend from simple empirical equations (sorption isotherms) to complicated mechanistic models based on surface complexation for the determination of electric potentials, e.g. constant-capacitance, diffuse-double layer and triple-layer model. 

First, assume that the surface charge on the membrane particles does not interact with the mobile protons (no proton release or uptake). An ion step will result in an increase in the double-layer capacitances of the particles and consequently in a decrease of the surface potentials fr, because the charge densities remain constant. The ISFET will measure a transient change in the mean pore potential. As a result of the potential changes, an ion redistribution will take place and the equilibrium situation is re-established. The theoretical maximum ion step response is the change in the mean pore potential. This is comparable with the Donnan model where the theoretical maximum is determined by the change in the Donnan potential at the membrane solution interface. 

When two surfaces in an electrolyte environment approach one another, their double layers overlap and several situations may arise. In the case of oxide surfaces, the interaction may itself influence the degree of dissociation of surface groups, such that neither the surface potential nor the surface charge remains constant. A charge regulation model may then be more appropriate [23]. The constant charge and the constant potential model allow, however, both upper and lower limits for the strength of the interaction to be estimated. For most of the model calculations performed, it is assumed that... 

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