Debye-Gouy-Chapman length

Inserting expressions for the surface potential from Eqs. [89] and [28] into Eq. [91] along with the definition of the Debye-Gouy-Chapman length in Eq. [27], we obtain the following relationship between apparent and actual Gouy-Chapman lengths ... 

The PGC expression for the Debye-Gouy-Chapman length follows from Eqs. [149] and [153] ... 

Figure 17 The error in the PGC solution of Eq. [152] according to Eq. [158] for a cylinder as a function of the scaled radius Ki,a for several values of the scaled surface charge density a /ao curves obtained using the exact (Eq. [154]) and approximate (Eq. [155]) Debye-Gouy-Chapman lengths are grouped by arrows.

Figure 24 The surface potential of a negatively charged sphere of radius 20 A (or cylinder of radius 10 A) in 0.01 and 0.1 M 1 1 electrolytes (kdO, = 0.7 and 2.0, respectively) as a function of surface charge density obtained from the PGC solution of Eq. [153] based on the Debye-Gouy-Chapman length of Eq. [154] (solid lines) and Eq. [155] (dashed lines) as well as from the NLDH expression of Eq. [181] in which exact (dotted lines Eq. [180]) and approximate (dotted-dashed lines Eq. [182]) values for parameter ( were used. The exact potential values obtained using a finite-difference method (Eq. [389]) are shown as circles values from Eq. [250] are shown as squares.

The apparent Debye-Hiickel (ADH) potential written in terms of the apparent Gouy-Chapman length is... 

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 apparent Gouy-Chapman length for which the Debye-Hiickel potential asymptotically matches the PB profile, subject to the accuracy of Eq. [152], as shown for a sphere in Figure 18, is... 

According to the Gouy-Chapman model, the thickness of the diffuse countercharge atmosphere in the medium (diffuse double layer) is characterised by the Debye length k 1, which depends on the electrostatic properties of the... 

Of course, when the volume concentration of mobile charges is sufficiently high that the Debye length is comparable with the ionic radius of the mobile ion(s), a combination of the Helmholtz and Gouy-Chapman models is required. This is achieved by assuming that the measured Cdi value is a series combination of that due to the Gouy-Chapman model (Cgc) and that due to the Helmholtz model (Ch), i.e. 

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 theoiy outlined above is a takeoff on the Debye Huckel theory of ionic solvation. In the electrochemistry literature it is known as the Gouy-Chapman theory. The Debye screening length is seen to depend linearly on ff and to decrease as (z+n+ +z zi ) /2 with increasing ionic densities. For a solution of monovalent salt, where z+ = z = 1 and = = n, this length is given by... 

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