Charge regulation model

5 ALTERNATIVE EXPRESSION FOR THE ELECTRIC PART OF THE FREE ENERGY OF DOUBLE-LAYER INTERACTION 

The electric part Eei of the free energy of double-layer interaction (Eq. (5.4)) can be expressed in a different way on the basis of the Debye charging process in which all the ions and the particles are charged simultaneously from zero to their full charge. Let /I be a parameter that expresses a stage of the charging process and varies from 0 to 1. Then F x can be expressed by 

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]. 

Namely, if there are Vmax binding sites for ions of valence Z on the surface of particle i (/ = 1, 2) and we assume 1 1 binding of the Langmuir type, then the double-layer free energy is given by 

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I. M. Metcalfe and T. W. Healy, Charge-regulation modeling of the Schulze-Hardy rule and related coagulation effects, Faraday Discuss. Chem. Soc. 90 335 (1990). 

Which boundary condition is more realistic depends on the materials used. In addition, the electrolyte and the speed of the approach might have an influence. Prica and coworkers measured force curves between 2irconia, which showed a constant charge behavior [84]. Also the force between two surfaces coated with densely packed carboxylic groups followed constant charge conditions [94]. In other cases, constant potential conditions are more appropriate. Most cases, however, lay between the two extremes. Then often, a charge regulation model is applied [156]. 

Information about the number of sites per unit area for each site is needed to calculate the surface charge. Because of the complexity of the charge-regulation model, many experiments under different conditions (pH, ionic strength, etc.) are needed in order to... 

C. Application of charge regulation model for determination of equilibrium constants of surface reactions... 

Another problem rests in the difference between the surface potential that appears in a charge regulation model and the experimentally determined eiectro-... 

But again, the Ogo obtained in this way cannot be directly compared to the charge density that appears in the charge regulation model. 

Such a set of reactions responsible for charging the surface constitutes a charge regulation model. Writing down the mass action law for each of the reactions,... 

Particular implementation of a charge regulation model depends largely on the form of Eq. (39). In the so-called low potential, or Debye, approximation, the latter... 

T o give some more ideas of how the choice of a potential-to-charge relation can affect the behavior of a charge regulation model, Fig. 4 compares spatial distributions of the potential within cylindrical cavities of different sizes under different... 

FIG. 4 Radial distribution of potential inside cylindrical pores with charge-regulated walls. The charge-regulation model used in the calculation was based on the reaction set... 

FIG. 5 (a, c) Surface charge density and (b, d) surface potential as functions of pH and pore size. (a. b) Linear potential-to-charge relation (Debye-Huckel) (c, d) nonlinear potential-to-charge relation (Poisson-Boltzmann). The charge regulation model used in the calculation was based on reaction set (34) with the following model parameters ... 

C. Application of Charge Regulation Model for Determination of Equilibrium Constants of Surface Reactions... 

The charge regulation model can be used successfully for interpretating pH poten-tiometric titration data, thus enabling the determination of intrinsic equilibrium constants of surface reactions and the prediction of surface composition under various conditions. The following example demonstrates this. 

Consider the titration of reactive silanols at the surface of a porous silica gel [25]. Typical titration data are shown in Fig. 7 by discrete points. One may want to approximate these data by a titration curve calculated according to the charge regulation model based on the minimal reaction set... 

FIG. 8 Surface speciation of phosphate-doped silica gel (S sp 187m /g, loading of phosphate 1.2 mmol/g), retrieved from titration data by applying the charge regulation model based on the reaction set... 

Since the HHF-solulion is based on the Debye-Hiickel approximation, it can be combined with a linear charge regulation model (Eq. (3.27)). That yields the more general expression (Camie et al. 1994a here combined with the mentioned modification of Sader et al. 1995) ... 

The singularity method further assumes that the linearised PBE (Eq. (3.17)) holds tme and that a linear charge regulation model (Eq. (3.27)) can be applied. Most equations in Sect. 4.6.2 are given for normalised expressions of the double layer parameters. These are defined as follows ... 

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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