Double layer Debye-Hiickel length

When Di > i>2, the effective Debye—Hiickel length X (which now depends on ip(x)) is larger than that obtained for the Poisson—Boltzmann equation. Consequently, the diffuse double layer is larger in the vicinity of a charged surface, as predicted earlier.4 7 9 However, when V2 > Vi (small counterions), X < X and the diffuse double layer is compressed. The effect is proportional to the ionic strength and is, in general, small for typical electrolyte concentrations, since n(v — v[Pg.337]

When the electrolyte concentration is increased, the range of the double layer decreases dramatically (the Debye-Hiickel length decreases) and the magnitude of the surface potential also decreases. In the linear approximation, ]f(x) = iJj(c E)cxp( — (x-dB)/X) (for x>dB) and the second right-hand-side term of Eq. (48) becomes ... 

The thickness of the double layer is usually given as being approximately 1.5k "S where k is the Debye-Hiickel length ... 

The surfaces of thylakoid membranes carry charges associated with the proteins. Hence, diffuse double layers of ions on the membrane surfaces exist in which the concentration of ions deviates from that in the bulk phase. This phenomenon is particularly relevant to the interthylakoidal space in grana stacks since the spacing between thylakoids is of the order of the Debye-Hiickel length. Moreover, it becomes the more prominent the lower the concentration of salt. 

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... 

Equation (1.9) is the linearized Poisson-Boltzmann equation and k in Eq. (1.10) is the Debye-Htickel parameter. This linearization is called the Debye-Hiickel approximation and Eq. (1.9) is called the Debye-Hiickel equation. The reciprocal of k (i.e., 1/k), which is called the Debye length, corresponds to the thickness of the double layer. Note that nf in Eqs. (1.5) and (1.10) is given in units of m . If one uses the units of M (mol/L), then must be replaced by IQQQNAn, Na being Avogadro s number. 

Debye-Hiickel parameter k (the Debye length), which has the dimension of length, serves as a measure for the thickness of the electrical double layer. Figure 1.5 plots the... 

Debye Length A parameter in the Debye—Hiickel theory of electrolyte solutions, k-1. For aqueous solutions at 25 °C, k = 3.288y7 in reciprocal nanometers, where I is the ionic strength of the solution. The Debye length is also used in the DLVO theory, where it is referred to as the electric double-layer thickness. See also Electric Double-Layer Thickness. 

A charged particle immersed in a liquid containing an electrolyte is surrounded by the electrical diffuse double layer. The thickness of the electrical double layer is given by the Debye length 1/k (k = Debye-Hiickel parameter). For a general electrolyte composed of N ionic mobile... 

The extension of the double layer, or more precisely of the diffuse layer, is indicated by the Debye length 1/k, where k is called Debye-Hiickel parameter, which can be calculated from the ionic strength I of the solution ... 

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... 

Note that the ionic valency, z includes the sign of the ion charge. For example, S04 has z = -2 and Ca " has z = +2. The atomic valency, on the other hand, refers to the number of possible bonds an atom can form with other atoms and is always positive. The term K (as opposed to is called the Debye-Hiickel parameter. The Debye length, is often referred to as the thickness of the double layef even though the region of varying potential is of the order of 3/k to 4/k (Hunter, 1993). The Stem layer is, in most cases, much smaller than the diffuse layer and is of the order of the counter-ion diameter. 

The thickness of the diffuse double layer or Debye length p corresponds to the radius of the ionic cloud of the Debye-Hiickel theory as given by Equation 1.22b. For the electrode/ electrolyte interface, this is the distance at which the potential drops by 1/e of its total change within the diffuse double layer. The same relation (Equation 1.44) is important... 

Thus, the situation of a semiconductor is similar to the diffuse double layer described by the Debye-Hiickel theory. In the case of the semiconductor, the charge carriers are electrons for an n-type semiconductor and positive holes for a p-type semiconductor. The width of the space charge layer is given by the Debye length p. For the semiconductor, the expression 2Nj I of Equation 1.22b has to be replaced by the concentration of charge carriers within the semiconductor, i.e., for an n-type semiconductor by the concentration of electrons, which equals approximately that of the donors as shown in Equation 1.179. 

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