can anyone help me in understanding what is Debye length and Debye shielding with proper explanation?
Answers
Electrostatic Potential, Cation Condensation, and Anion Exclusion
The negative charge of clay mineral basal surfaces is screened by cation adsorption and anion exclusion near the clay mineral surface in a region known as the electrical double layer (EDL). Measurements of anion exclusion and electrophoretic mobility in aqueous dispersions of clay mineral particles indicate that the EDL has a thickness on the order of several nanometers with a strong dependence on ionic strength (Sposito, 1992). The EDL can be conceptually subdivided into a Stern layer containing inner- and outer-sphere surface complexes and a diffuse layer containing ions that interact with the surface through long-range electrostatics (Henderson and Boda, 2009; Lee et al., 2010).
The diffuse layer composition and structure is the matter of intensive research because it governs many macroscopically observed phenomena including swelling, osmosis, and particle aggregation (Leroy et al., 2006). A simplified yet powerful description of the diffuse layer can be achieved in the framework of the modified Gouy–Chapman model, which uses the Poisson equation and the Boltzmann distribution and makes the following three simplifying hypotheses: that the solution phase is a uniform continuum characterized solely by its dielectric permittivity ε (78.3 × 8.85419·10−12 F m−1 for water at 298 K); that the surface charge σ (in m−2) is uniformly distributed in the interfacial plane; and that the potential of mean force Wi(x) applying to an ion i at a distance x from the surface is determined only by the mean electrostatic potential at the same position, ψ(x). Additionally, the modified Gouy–Chapman model postulates that the ions cannot approach the surface closer than a distance a. If clay mineral basal surfaces are modeled as infinite planes, the Poisson equation can be written as follows:
(1.14)d2ψ(x)dx2=−1ε∑iziFci(x)
and the Boltzmann equation is given by:
(1.15)ci(x)=ci0exp(−ziFψ(x)RT)
where ci(x) and ci0 are the volumetric concentrations of ion i at a distance x and at infinite distance from the surface (mol m−3), F is Faraday's constant (96 485 C mol−1), R is the ideal gas constant (8.314 J K−1 mol−1), T is absolute temperature (K), and the electrostatic potential is conventionally defined as zero at an infinite distance from the surface. In the case of an isolated plane, Eqn (1.14) can be solved analytically for simple electrolyte compositions. In the case of a 1:1 electrolyte such as NaCl, the electrostatic potential is related to the ionic strength (cNa0 = cCl0 = 1000I )1 through the following equation:
(1.16)Fψ(x)RT=4arctanh[tanh(Fψ(a)4RT)×exp(−κ(x−a))]
where κ is the inverse of the Debye length (κ−1):
(1.17)κ=2F21000IεRT
and the electrostatic potential at the distance of closest approach is given by:
(1.18)Fψ(a)RT=−2arcsinh(F2|σ|2κεRTNA)
Equations (1.16) and (1.18) are strictly valid only for 1:1 electrolytes. Other analytical solutions to Eqn (1.14) exist for n:n symmetric electrolytes with n > 1 (e.g., CaSO4), for 2:1 electrolytes (e.g., CaCl2) and for 2:1:1 mixed electrolytes (e.g., CaCl2/NaCl) (Chen and Singh, 2002). In practice, however, Eqns (1.16) and (1.18) are often used regardless of the type of electrolyte.
The electrostatic potential and Na/Cl concentration profiles obtained from Eqns (1.16) and (1.18) are shown in Figure 1.4 for T = 298 K, σ = 6.2·1017 m−2 = 0.62 nm−2 = −0.1 C m−2 (a value characteristic of Mt basal surfaces; see Table 1.1) and I = 0.1 or I = 0.01. The characteristic thickness of the diffuse layer (i.e., the length scale associated with the decay of ψ(x)) depends on ionic strength and is related to the Debye length through the exponentially decaying term of Eqn (1.16). The characteristic distance associated with cation adsorption is significantly shorter than the characteristic thickness of the diffuse layer, a phenomenon known as counterion condensation (Sposito, 2004).