Gouy-Chapman relation

Isolated sphere with charge Q and a surface potential Identified as which obeys Coulomb s law (0= Anae ell). Using Stokes law, v=QE/Qnr)a, eq. 14.3.5] Is then Immediately obtained. This oversimplified situation applies only when the double layer contains very few Ions, as In media of extremely low dielectric permittivity. In sec. 3.11 It was verified that under those conditions screening Is negligible so that the Gouy-Chapman relations between charges and potentials reduce to that of Coulomb. Hence, the result Is as expected. 

The charge in the diffuse double layer, given by the Gouy-Chapman relation (Verwey and Overbeek, 1948) as modified by Stern, is ... 

Since the interface behaves like a capacitor, Helmholtz described it as two rigid charged planes of opposite sign [2]. For a more quantitative description Gouy and Chapman introduced a model for the electrolyte at a microscopic level [2]. In the Gouy-Chapman approach the interfacial properties are related to ionic distributions at the interface, the solvent is a dielectric medium of dielectric constant e filling the solution half-space up to the perfect charged plane—the wall. The ionic solution is considered as formed... 

Nakagaki1U) has given a theoretical treatment of the electrostatic interactions by using the Gouy-Chapman equation for the relation between the surface charge density oe and surface potential /. The experimental data for (Lys)n agrees very well with the theoretical curve obtained. 

When the ITIES is polarized with a potential difference 0, there is a separation of electrical charge across it. According to the Gouy-Chapman theory, the charges in the aqueous and organic diffuse layers are related to the potential drops and A0 in the respective layers by the equations... 

The Gouy-Chapman theory relates electrolyte concentration, cation valence, and dielectric constant to the thickness of this double layer (see Equation 26.2). This theory was originally developed for dilute suspensions of solids in a liquid. However, experience confirms that the principles can be applied qualitatively to soil, even compacted soil that is not in suspension.5... 

The Gouy Chapman diffuse layer model has been shown to describe adequately the electrostatic potential produced by charges at the surface of the membrane [137]. For a symmetrical background electrolyte, a and i// are related by ... 

According to the Gouy-Chapman theory the surface charge density gp [C nr2] is related to the potential at the surface y [volt] (Eq. (vi) in Fig. 3.2)... 

The dashed line in the complex in (4.21) and (4.22) indicates an outer-sphere (o.s.) surface complex, Kos stands for the outer-sphere complex formation constant and kads [M 1 s 1] refers to the intrinsic adsorption rate constant at zero surface charge (Wehrli et al., 1990). Kos can be calculated with the help of a relation from Gouy Chapman theory (Appendix Chapter 3). 

For simplicity, we will consider the case in which surface charge and potential are positive, and that only anions adsorb. Furthermore, the potential drop in the Gouy-Chapman layer will be assumed to be small enough that its charge/potential relation can be linearized. The V o/oo/pH relationship can then be derived parametrically, with the charge in the Gouy-Chapman layer cr4 as the parameter. The potential at the plane of anion adsorption can then be calculated and substituted in Equation 28 to give ... 

Ionic surfactants are electrolytes dissociated in water, forming an electrical double layer consisting of counterions and co-ions at the interface. The Gouy-Chapman theory is used to model the double layer. In conjunction with the Gibbs adsorption equation and the equations of state, the theory allows the surfactant adsorption and the related interfacial properties to be determined [9,10] (The Gibbs adsorption model is certainly simpler than the Butler-Lucassen-Reynders model for this case.). 

This is the expression we were looking for. What does it tell us For one thing, that the potential decays exponentially as the distance from the electrode increases pig. 6.64(b)], Further, as the solution concentration c0 increases, k increases and falls more and more sharply. This potential-distance relation [Fig. 6.64(b)] is an important and simple result from the Gouy-Chapman model. It forms a valuable basis for thinking about the interaction of the diffuse charges around what are called colloidal particles (see Section 6.10.2). 

What are the implications of Eq. (6.132) The Stem synthesis of the two models implies a synthesis of the potential-distance relations characteristic of these two models [Fig. 6.66(b)] a Z/ncar variation in the region from. v = 0 to the position of the OHP according to the Helmholtz-Perrin model (see Section 6.6.2), and an exponential potential drop in the region from OHP to the bulk of solution according to the Gouy-Chapman model (see Section 6.6.4), as shown in Fig. 6.67. 

Before we proceed to the Gouy-Chapman theory of electrical double layers, it is worthwhile to note that relations similar to Equations (45) and (47) can also be derived for double layers surrounding spherical particles. The equation for surface charge density takes the form... 

What is the diffuse layer, and what is its relation to the Gouy-Chapman theory of electrical double layers ... 

The Gouy—Chapman treatment combines the Poisson equation in onedimensional form, which relates the electrical potential (x) to the charge density at x, and the Boltzman distribution of ions in thermal motion [6,18]... 

While the linear adsorption isotherms of Figure 4 are illustrative only, they are not inconsistent with reality. The simplest theory of the electrical double layer, the Gouy-Chapman approximation, predicts that if the pH is not far from the isoelectric point, the charge represented by counter ions in the diffuse double layer is related to the surface potential as follows (4, 52, 86) ... 

This simple equation is, however, only valid for R Xp- If the radius is not much larger than the Debye length we can no longer treat the particle surface as an almost planar surface. In fact, we can no longer use the Gouy-Chapman theory but have to apply the theory of Debye and Hiickel. Debye and Hiickel explicitly considered the electric double layer of a sphere. A result of their theory is that the total surface charge and surface potential are related by... 

The opposite effect, the inhibition of PKC, has also been studied. It was concluded that inhibition by mono- and divalent cations is related to the reduction of surface potential of the studied PS/PC liposomes by ion binding according to the Gouy-Chapman theory. Later, the same authors studied the inhibitory effect of the local anesthetics tetracaine and procaine on PKC in liposomes [10]. The local anesthetics significantly reduced the negatively charged surface potential, T, of phospholipid bilayers. The anesthetics were even capable of changing the surface potential to... 

Finally, it may be noted that, since lim, -, p = 0, o-Ac= may be related to through Eq, [3] (which reduces to the Gouy—Chapman equation when p = 0). Substituting this relation into [9] ... 

This model is based on the Gouy-Chapman theory (diffuse double-layer theory). The theory states that in the area of the boundary layer between solid and aqueous phase, independently of the surface charge, increased concentrations of cations and anions within a diffuse layer exists because of electrostatic forces. In contrast to the constant-capacitance model, the electrical potential does not change up to a certain distance from the phase boundaries and is not immediately declining in a linear manner (Fig. 14 a). Diffusion counteracts these forces, leading to dilution with increasing distance from the boundary. This relation can be described physically by the Poisson-Boltzmann equation. 

Many more-sophisticated models have been put forth to describe electrokinetic phenomena at surfaces. Considerations have included distance of closest approach of counterions, conduction behind the shear plane, specific adsorption of electrolyte ions, variability of permittivity and viscosity in the electrical double layer, discreteness of charge on the surface, surface roughness, surface porosity, and surface-bound water [7], Perhaps the most commonly used model has been the Gouy-Chapman-Stem-Grahame model 8]. This model separates the counterion region into a compact, surface-bound Stern" layer, wherein potential decays linearly, and a diffuse region that obeys the Poisson-Boltzmann relation. 

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