Debye layer extension

Membrane-Induced Deionzation, Debye Layer Extension,... 

When charged colloidal particles in a dispersion approach each other such that the double layers begin to overlap (when particle separation becomes less than twice the double layer extension), then repulsion will occur. The individual double layers can no longer develop unrestrictedly, as the limited space does not allow complete potential decay [10, 11]. The potential v j2 half-way between the plates is no longer zero (as would be the case for isolated particles at 00). For two spherical particles of radius R and surface potential and condition x i <3 (where k is the reciprocal Debye length), the expression for the electrical double layer repulsive interaction is given by Deryaguin and Landau [10] and Verwey and Overbeek [11],... 

The double layer extension is determined by the electrolyte concentration and the valency of the counter ions, as given by the reciprocal of the Debye-Huckel parameter (1 /k) - referred to as the thickness of the double layer,... 

In order to describe the effects of the double layer on the particle motion, the Poisson equation is used. The Poisson equation relates the electrostatic potential field to the charge density in the double layer, and this gives rise to the concepts of zeta-potential and surface of shear. Using extensions of the double-layer theory, Debye and Huckel, Smoluchowski,... 

For the sake of simplicity, in what follows it will be considered that the double layer potential is sufficiently small to allow the linearization of the Poisson—Boltzmann equation (the Debye—Hiickel approximation). The extension to the nonlinear cases is (relatively) straightforward however, it will turn out that the differences from the DLVO theory are particularly important at high electrolyte concentrations, when the potentials are small. In this approximation, the distribution of charge inside the double layer is given by... 

Debye-Hiickel function that characterizes extension of double layer... 

Because of the compensation of the surface charge by the positively charged counterions, an exponential potential drop in the diffuse layer was discussed by Gouy (1910, 1917) and Chapman (1913). The potential at the particle surface is the Nernst potential if/0 the extension of the diffuse layer is equal to 1 Ik (Debye-Hiickel parameter). 

The investigations of Mikhailovskij are significant for the discussion of the limits of the macro-kinetic approach and their extension, which can be established by the analysis of the transition from Eqs (7.74) to Eq. (7.75). To do so, Mikhailovskij concluded, that the influence of the Debye atmosphere of counterions on the transport of a higher valency ion through the diffuse layer can be neglected at sufficiently low background electrolyte concentrations. This limit, of course, has to be defined for different valences of surfactant ions, different surface activity etc. 

Thus at a distance 1 Jk the potential has dropped by a factor of (1/e). This distance may be used as a measure of the extension of the double layer and is often loosely called the thickness of the double layer. According to the theoretical equations it has the value /k = [ekT/e l.CizlY and is identical with the parameter introduced in the Debye-Hiickel theory of electrolytes in which /k is identified with the radius of the ionic atmosphere. Of particular importance in colloid science is the fact that the thickness of the... 

The equations for the fluxes of and and the expression for the diffusion potential will be derived now for a filament on the basis of the assumptions discussed in Section 2.2. Across the length / of the proton-con-ducting filament a chemical potential difference FKh= —i Tln Ch(/)/ch(0) is assumed to be maintained by two synchronized chemical reactions which inject and take out H at z = 0 and z = 1 respectively. The K layer has the radial extension from r = ator = a-ft/ with the thickness d corresponding roughly to the Debye length. The concentration of in the K layer will... 

A new important result of these computations is the relation between the coupling coefficients and the permeability for thick double layers [cf Eq. (88b)]. The case of relatively thick double layers (fine porous media) was largely overlooked in the literature. Attention was generally focused on the opposite limit of vanishing Debye length, where the matching technique of inner and outer developments was extensively used (see Sec. I), which yielded the general relation (3) of Overbeek [12]. 

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 ... 

Fig. 1 compares the potential distribution at a semicon-ductor/electrolyte and a metal/electrolyte interface. The extension of the space charge layer in the semiconductor is a function of the mobile charge concentration in the bulk and can be approximated by a "Debye-length" L ... 

K is the Debye-Hiickd parameter, and /k is the extension of the double layer (double layer thickness) that is given by the expression,... 

As noted above, we find that this kinetics formulation is applicable to cases where the salt concentration is low, and deposition extensive, so that the "smeared charge approximation holds. Assuming that depositing DNA-SWCNTs form a homogenous layer on the surface implies that the surface charge will increase with deposited density Ps (in p.m of nanotubes per p.m of surface). As surface charge increases, surface potential relative to bulk solution will also increase. Within the linearized Debye-Hiickel approximation, the surface potential is related to surface charge as ... 

Integral Equation and Eield-Theoretic Approaches In addition to theories based on the direct analytical extension of the PB or DH equation, PB results are often compared with statistical-mechanical approaches based on integral equation or density functional methods. We mention only a few of the most recent theoretical developments. Among the more popular are the mean spherical approximation (MSA) and the hyper-netted chain (HNC) equation. Kjellander and Marcelja have developed an anisotropic HNC approximation that treats the double layer near a flat charged surface as a series of discrete layers.Attard, Mitchell and Ninham have used a Debye-Hiickel closure for the direct correlation function to obtain an analytical extension (in terms of elliptic integrals) to the PB equation for the planar double layer. Both of these approaches, which do not include finite volume corrections, treat the fluctuation potential in a manner similar to the MPB theory of Outhwaite. 

But biologists and biochemists do use the tools and concepts of physical chemistry. These have their roots in the slightly contaminated water of Debye-Hiickel theory and its extensions to include ion size and double layers. This is so for concepts like pH, buffers, pKa s, ion binding, interactions of ions with surfaces, membrane and zeta potentials, colloidal interactions that are deeply embedded in our collective psyche. The foundations of these have only been questioned over the last decade. The theories used... 

In the application to strong electrolytes serious doubt exists whether the application of the complete cq. (40) is permissible because it implies certain internal inconsistencies which have been analys ed most extensively by Kirkwood But Casimir extending Kirkwood s analysis has shown that these inconsistencies do not arise (remain very small) when the complete eq. (40) is applied to the double layer on a large plane interface or on a large particle if the electrolytic concentrations in the whole system remain so small that in the bulk of the solution the limiting laws of Debye and Huckel form a reasonable approximation. 

Moreover in the spherical double layer with 1/x large compared with the radius of the particle, the distribution of the potential is governed more by the factor 1/r in eq. (79) which comes from the spherical extension of the lines of force, than from double layer effects and therefore a slight incorrectness in the treatment of the double layer does not influence the distribution of the potential very much Table 1 in which the approximate theory of Debye and HUckel and the exact solution of eq. (40) by MULLER are compared for a case where 1/x = 5 a, illustrate this clearly. 

The first example (Fig. 5.70a) involves dipping a solid into an aqueous salt solution, thus, creating a soHd-liquid interface. Either cations or anions will be preferentially adsorbed on the smface, and the result is an excess surface charge. The coimter-charge is distributed in the zone of the solution adjacent to the interface the extent of this zone is determined by the Debye length (see Eq. (5.203)). It is inversely proportional to the square root of charge carrier concentration in the bulk solution. A rigid double-layer is formed in a concentrated solution in dilute solution a diffuse layer is formed with appreciable extension (typical numbers are several tens of nanometres). Such electrostatic effects are responsible for the kinetic stability of dispersed systems in colloid chemistry . 

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