Surface charge density diffuse double layer

Let us redefine the problem. We start from the premises that the diffuse part of the double layer is fully relaxed but that the surfactant adsorption is not relaxed at all. In other words, the surfactant charge cr is fixed. We assume it to be negative, as in Fig. 1. Let us say that Na ions are the counterions in the Stem layer, and let their surface charge density in this layer be Then the question is what happens to the sum... 

Derive the general equation for the differential capacity of the diffuse double layer from the Gouy-Chapman equations. Make a plot of surface charge density tr versus this capacity. Show under what conditions your expressions reduce to the simple Helmholtz formula of Eq. V-17. 

We shall use the familiar Gouy-Chapman model (3 ) to describe the behaviour of the diffuse double lpyer. According to this model the application of a potential iji at a planar solid/electrolyte interface will cause an accumulation of counter-ions and a depletion of co-ions in the electrolyte near the interface. The disposition of diffuse double layer implies that if the surface potential of the planar interface at a 1 1 electrolyte is t ) then its surface charge density will be given by ( 3)... 

The surface charge density of the diffuse part of the double layer is given by the Gouy-Chapman equation ... 

So far, we have used the Maxwell equations of electrostatics to determine the distribution of ions in solution around an isolated, charged, flat surface. This distribution must be the equilibrium one. Hence, when a second snrface, also similarly charged, is brought close, the two surfaces will see each other as soon as their diffuse double-layers overlap. The ion densities aronnd each surface will then be altered from their equilibrinm valne and this will lead to an increase in energy and a repulsive force between the snrfaces. This situation is illustrated schematically in Fignre 6.12 for non-interacting and interacting flat snrfaces. 

The charge density, Volta potential, etc., are calculated for the diffuse double layer formed by adsorption of a strong 1 1 electrolyte from aqueous solution onto solid particles. The experimental isotherm can be resolved into individual isotherms without the common monolayer assumption. That for the electrolyte permits relating Guggenheim-Adam surface excess, double layer properties, and equilibrium concentrations. The ratio u0/T2N declines from two at zero potential toward unity with rising potential. Unity is closely reached near kT/e = 10 for spheres of 1000 A. radius but is still about 1.3 for plates. In dispersions of Sterling FTG in aqueous sodium ff-naphthalene sulfonate a maximum potential of kT/e = 7 (170 mv.) is reached at 4 X 10 3M electrolyte. The results are useful in interpretation of the stability of the dispersions. 

Stern assumed that a Langmuir-type adsorption isotherm could be used to describe the equilibrium between ions adsorbed in the Stern layer and those in the diffuse part of the double layer. Considering only the adsorption of counter-ions, the surface charge density cr, of the Stern layer is given by the expression... 

The diffuse double layer is generated by the anions trapped in the potential well. The surface charge density , due to their distribution between 0 and dE, is given by ... 

H. Ohshima, Diffuse double layer interaction between two spherical particles with constant surface charge density in an electrolyte solution, Colloid Polymer Sci. 263, 158-163 (1975). 

The Poisson-Boltzman (P-B) equation commonly serves as the basis from which electrostatic interactions between suspended clay particles in solution are described ([23], see Sec.II. A. 2). In aqueous environments, both inner and outer-sphere complexes may form, and these complexes along with the intrinsic surface charge density are included in the net particle surface charge density (crp, 4). When clay mineral particles are suspended in water, a diffuse double layer (DDL) of ion charge is structured with an associated volumetric charge density (p ) if av 0. Given that the entire system must remain electrically neutral, ap then must equal — f p dx. In its simplest form, the DDL may be described, with the help of the P-B equation, by the traditional Gouy-Chapman [23-27] model, which describes the inner potential variation as a function of distance from the particle surface [23]. 

A double layer consists of a thin immobile Stern layer and a diffuse layer. The boundary between the two layers is known as the plane of shear. In this article, the particle surface is referred to as this plane. Thus, the surface charge density is contributed by the charge on the particle surface plus the charge in the Stern layer. In the diffuse layer, the flux of ionic species i is given by... 

Gouy length — The width of the diffuse double layer at an electrode depends on a number of factors among which the charge density q on the surface of the metal and the concentration c of electrolyte in the solution are paramount. Roughly speaking, the charge in, and the potential of, the double layer falls off exponentially as one proceeds into the solution from the interface. 

The diffuse double layer charge over a surface of area S can be taken by integration of the charge density over the surface (defined by S2). 

When ions specifically adsorb at the interface, the excess surface charge density is divided into three parts, the charges in the two diffuse parts of the double layer, q and q, and the charge due to the specifically adsorbed ions, q° [25]. The electroneutrality condition of the entire interfacial region is... 

The surface charge density for the diffuse double layer is thus given by <7o + <7i, and this is the quantity which must replace <7 in equations (7) to (10). In spite of the slight difference in the significance of <7, equation (10) may, therefore, still be regarded as applicable, provided the solution is dilute. The sign of <7i is probably always opposite to that of <7o, and so <7o + <7i is numerically less than charge density on the surface of the solid. 

The potential ([i is called the potential of the outer Helmholtz plane. The diffuse double layer starts at the outer Helmholtz plane, where the potential is ( ). It is this value of the potential, rather than (j), that must be used in Eqs. 14G and 15G, to relate the surface charge density and the diffuse-double-layer capacitance to potential. 

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