Of Gouy-Chapman

The charge of the diffuse EDL part (x > Xj) can be described by the equations of Gouy-Chapman theory, but with the value tj/j rather than /o ... 

APPENDIX B Derivation of the Main Equation of Gouy-Chapman Theory... 

Figure 10. Electrical double layer models. Top right (a) typical type of potential vs. composition plot for a charged surface compared to (b) constant capacitance model. Top left Two double-layer models, (a) diffuse double layer, (b) part parallel plate capacitor and part diffuse layer.. Bottom left Stem layer model. Incorporation of adsorbed ions to surface. From Hiemenz and Rajagopalan (1997) Bottom right Comparison of Gouy-Chapman and Stem-Grahame models of the electrical double layer. From Davis and Kent (1990).

Equation 5 the fundamental equation of Gouy-Chapman theory. When ze f << kT, eq 5 can be linearized. The resulting equation is... 

R. O. James and G. A. Parks, Characterization of aqueous colloids by their electrical double-layer and intrinsic surface chemical properties. Surface and Colloid Science 12 119 (1982). Perhaps the most complete review of the triple layer model from the perspective of Gouy-Chapman-Stem-Graham e double layer theory. 

Fig. 1. Electric potential function, y (0, in a diffiuse charge layer of an electrochemical double layer according to the theory of Gouy-Chapman. Curves are given for Z = 8. z — 4, z = 2 and z z = ve y ve jkT, h -- xx). Dotted lines ...

We have hitherto reasoned as if the potential were constant during all changes in the electrolyte concentration. This would, b.e right if we could describe the double layer completely by the theory of Gouy-Chapman. But as we have seen ( 5 of. Chapter VII), especially for larger concentrations some correction has to be made in respect of the dimensions and the specific adsorption of the-iOns, and a possible form of this cor-... 

The theory of Gouy-Chapman becomes inadequate when k and/or large. The theory... 

The sum of interfacial potential and distribution potential is termed the iimer or Galvani potential, designated by . It will be discussed below that the observed transmembrane potential difference, Em, is due either to AV, AU, or to AV plus AU, as defined above. It should be remembered that there are two components of (=V + U). They are concerned with only the distribution potential. If compounds, such as phospholipids and interface-active agents, are preferentially adsorbed at the interface, the so-called adsorption potential (V) may also develop. As shown in Fig. 2, both adsorbed fixed charge species and dipoles may contribute to the observed potential. The nature and origin of the adsorption (or interfacial) potential can be discussed in terms of the classical EDL theory of Gouy-Chapman-Stern-Graham (7,15-19]. 

The applicability of Eq. (5.24) breaks down for weakly charged particles kflT 1) because the free energy of the EDL is then strongly related to the (apparent) surface potential y/Q. For spherical particles with a thin double layer ku 1) of Gouy-Chapman type, Deijaguin (1940) derived a criterion for the suspension stability that can be expressed as ... 

The electrostatic double-layer force can be calculated using the continuum theory, which is based on the theory of Gouy, Chapman, Debye, and Hiickel for an electrical double layer. The Debye length relates the surface charge density of a surface to the electrostatic surface potential /o via the Grahame equation, which for 1 1 electrolytes can be expressed as... 

A number of new approximate, but accurate, analytical results are also presented here the most significant ones are (1) the extension of Gouy-Chapman theory to mixed electrolyte solutions whereby an effective counterion valence is introduced, (2) two approximate potential profiles for curved surfaces (one of them new) are generalized to include the presence of mixed electrolytes, (3) the apparent surface charge density for curved surfaces for which the Debye-Hiickel potential asymptotically matches the Poisson-Boltzmann profile, and (4) a unified treatment of two interacting charged surfaces. 

First, we find the interaction energy between two (vmequal) surfaces within the Debye-lTuckel approximation. To simplify the notation, consider two surfaces at x = R with charges densities a( R) = a in equilibrium with a bulk electrolyte. (For the remainder of this section, convenience and correspondence with previous work requires that we work with charge densities instead of Gouy-Chapman lengths.) We solve the DH equation in the region between the surfaces... 

The double layer description of Gouy-Chapman may be used from the outer Helmholtz plane on, and this is in principle completely known because the total amount of adsorbed cations is an experimentally determinable quantity. Together with the value of x this is enough to determine the distribution of charge and potential completely. 

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