Gouy-Chapman length

The maximum brush height occurs at ZB 0.1 a. For smaller ZB, the Gouy-Chapman length becomes larger than the brush height. Hence, counter-ions are free to leave the brush giving rise to its relaxation back to the reduced extension of a quasi-neutral brush. 

Fig. 4 Brush height versus Bjerrum length average end-point height (squares) and Gouy-Chapman length (thick line). The dashed line gives (ze) of a corresponding uncharged brush...

For water at 298 K, Eq = 78.5 gives Lb = 7.14 A. The second length that we introduce is the Gouy-Chapman length ... 

Systems with a negative surface charge density have more biochemical relevance, but equations for the positive case are shown to illustrate the asymmetry in the solution due to inclusion of a 2 1 electrolyte. Note that in Eqs. [41] and [42], the Gouy-Chapman length appears with a factor representing the largest covmterion valence at the sirnface, as indicated explicitly in the 2 z electrolyte case of Eq. [35]. 

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 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 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 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... 

We will see that this generalization works well for charged cylinders. To estimate the surface potential from the apparent Gouy-Chapman length within the PGC approximation, we simply solve Eq. [321] for w and use Eq. [316]. The apparent PGC surface charge density is obtained from the analog of... 

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