Debye length/thickness

N is the number of charged groups per particle of hydrodynamic radius R e is the electron charge K is the reciprocal of Debye length thickness n is the viscosity of the medium. 

Since the potential decays roughly exponentially in a double layer, the double layer is only a few Debye lengths thick. If the double layer is very small on the relevant diffusional scales of the system, i.e. if the Debye length is much shorter than the size of the electrode, then the radius of the outside of the double layer is coincident with the electrode radius to a good approximation. 

Problem 10.2 The Debye length - thickness of the double layer... 

The region of the gradual potential drop from the Helmholtz layer into the bulk of the solution is called the Gouy or diffuse layer (29,30). The Gouy layer has similar characteristics to the ion atmosphere from electrolyte theory. This layer has an almost exponential decay of potential with increasing distance. The thickness of the diffuse layer may be approximated by the Debye length of the electrolyte. 

In most electrochemical systems, the double layer is very thin (1—10 nm). The thickness is characterized by the debye length, X,... 

Thermal diffusivity Temperature sensitivity Temperature difference Thickness of tube Aspect ratio, relation of Cp/Cy Fluid dielectric constant Wall zeta potential Dimensionless temperature Friction factor, Debye length Mean free path Dynamic viscosity Kinematic viscosity Bejan number Density... 

We can see from this equation that the potential / at the point r = 0 has the value that would exist if there were at distance 1/k a point charge -zj or, if we take into account the spherical symmetry of the system, if the entire ionic atmosphere having this charge were concentrated on a spherical surface with radius 1/k around the central ion. Therefore, the parameter = 1/k with the dimensions of length is called the ejfective thickness of the ionic atmosphere or Debye radius (Debye length). This is one of the most important parameters describing the ionic atmosphere under given conditions. 

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... 

The thickness of these layers is in the range of <2h = 0.4 to 0.6 nm for the compact layer, space charge layer, and c(d = 1 to 10 nm for the diffuse layer. The thickness of space charge layer, dac, and the thicknesses of diffuse layer, (fd, depend on the concentrations of mobile charge carries Tii in the semiconductor and in the aqueous solution, respectively. The Debye length, Ljy, may be used as a measure of the thickness of these two respective layers as shown in Eqn. 5-61 ... 

Further, the thicknesses of the diffuse and space charge layers depend on the potentials Ma and i sc across the respective layers for the space charge layer the thickness, dgc, is expressed, to a first approximation, by dsc = 2Lox (eA /kT)- [Memming, 1983]. The Debye length, Ld, is about 100 nm in usual semiconductors with impurity concentrations in the order of 10 cm and is about 10 nm in dilute 0.01 M ionic solutions. 

The total potential A/ across the electrode interface may be expressed approximately by using the thickness of space charge layer approximated by the Debye length Ld.9c, the thickness of compact layer du, and the thickness of... 

The thickness of depletion and deep depletion layers may be approximated by the effective Debye length, Lo, ff, given in Eqn. 5-70 Ld, is inversely proportional to the square root of the impiuity concentration, In ordinary semiconductors... 

One of the main assumptions of the Donnan partition model is that two well-defined phases (polymer and solution) exist and the electrostatic potential presents a sharp transition between them. This approximation is fulfilled when the typical decay length of the electrostatic potential (Debye length) is much shorter than the film thickness. The other limiting situation is that where all the redox sites are located in a plane and thus the Debye length is larger than the film thickness. This situation can be described by the surface potential model ... 

We shall now consider what happens when the film thickness is of the order of the Debye length. In such a situation, no analytical expressions can be derived and numerical calculations should be used [125]. The real situation could be even more complicated, since an ill-defined film thickness can exist, like the example in Figure 2.6. We can use the molecular theory to obtain a self-consistently determined electrostatic potential profile across the interface as was shown in Figure 2.7 (see... 

On the basis of this description, a relationship between the two lengths 8 and K can be established. Different 5 values are obtained by gradually increasing the amount of micelles and fitting the force profiles. The evolution of 5 as a function of the calculated Debye length is plotted in Fig. 2.8. The thickness 5 increases linearly with The inherent coupling between depletion and doublelayer forces is reflected by this empirical linear relationship which is a consequence of the electrostatic repulsion between droplets and micelles. The thickness 5 may be conceptually defined as a distance of closer approach between droplets and micelles and thus may be empirically obtained by writing ... 

The disjoining pressure vs. thickness isotherms of thin liquid films (TFB) were measured between hexadecane droplets stabilized by 0.1 wt% of -casein. The profiles obey classical electrostatic behavior. Figure 2.20a shows the experimentally obtained rt(/i) isotherm (dots) and the best fit using electrostatic standard equations. The Debye length was calculated from the electrolyte concentration using Eq. (2.11). The only free parameter was the surface potential, which was found to be —30 mV. It agrees fairly well with the surface potential deduced from electrophoretic measurements for jS-casein-covered particles (—30 to —36 mV). 

As expected, the D-H theory tells us that ions tend to cluster around the central ion. A fundamental property of the counterion distribution is the thickness of the ion atmosphere. This thickness is determined by the quantity Debye length or Debye radius (1/k). The magnitude of 1/k has dimension in centimeters, as follows ... 

In addn, for an ionized gas to be called a plasma, it must have an equal number of pos and neg charges for, by definition, a plasma has no net charge. Regions termed "sheaths , having large (net charges) do develop at the plasma boundaries. Such sheaths are to the plasma what the surface is to a solid or liquid, and their thickness is of the order of the "Debye length ... 

Here the same scaling as in (4.1.1), (4.1.2) has been employed. Once again, e is the square of the ratio of the Debye length to the dimensional thickness of the compartment (L). 

The thickness of the Donnan phase layer rD was equated to the Debye length so that wD/w = rD/r = l/r /pCs. However, experimental values of wD/w calculated from measurements of SP and SN in cellulose-KCl by means of a combination of Eq. (42), the corresponding equation for the counterion and the proper version of Eq. (37), namely SpS = w2D, did not agree very well with the aforesaid theoretical rD values 116). Recently, this difficulty has been resolved by showing 110) that Eqs. (41) and (42) become identical under the proper conditions (namely at high Cs), upon setting... 

Which thickness do we have to use This depends on the relevant parameter. If we are for instance, interested in the density of a water surface, a realistic thickness is in the order of 1 nm. Let us assume that a salt is dissolved in the water. Then the concentration of ions might vary over a much larger distance (characterized by the Debye length, see Section 4.2.2). With respect to the ion concentration, the thickness is thus much larger. In case of doubt, it is safer to choose a large value for the thickness. 

We can observe electro-osmosis directly with an optical microscope using liquids, which contain small, yet visible, particles as markers. Most measurements are made in capillaries. An electric field is tangentially applied and the quantity of liquid transported per unit time is measured (Fig. 5.13). Capillaries have typical diameters from 10 fim up to 1 mm. The diameter is thus much larger than the Debye length. Then the flow rate will change only close to a solid-liquid interface. Some Debye lengths away from the boundary, the flow rate is constant. Neglecting the thickness of the electric double layer, the liquid volume V transported per time is... 

Buck, 1981). The thickness of the space charge ka 1 is called the Debye length. It depends on temperature, on the dielectric constant e of the membrane, and on the valency and concentration of the binding sites. 

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