Screening length, Debye

Fig. 1. Electron temperature and density regions for plasmas (7—9) where the numbers and the diagonal lines represent (—) the Debye screening length,...

Also shown in Figure 1 are the Debye screening length and Debye sphere size. For gaseous plasmas, A)-, 1 (11). SoHd-state plasmas or... 

In the second group of models, the pc surface consists only of very small crystallites with a linear parameter y, whose sizes are comparable with the electrical double-layer parameters, i.e., with the effective Debye screening length in the bulk of the diffuse layer near the face j.262,263 In the case of such electrodes, inner layers at different monocrystalline areas are considered to be independent, but the diffuse layer is common for the entire surface of a pc electrode and depends on the average charge density <7pc = R ZjOjOj [Fig. 10(b)]. The capacitance Cj al is obtained by the equation... 

K = Debye screening length 4 = model parameter n = osmotic pressure p = density T = tortuosity factor <1> = swelling ratios... 

Using the equation, very strong concentration effects in small systems have been calculated. For instance, if the macroaqueous phase contains 1 M NaCl and 1 /rM NaTPB, the concentration of this electrolyte in the micro-organic phase at partition equilibrium is 1390/rM [14] This approach is valid if the phases in small systems are thick enough (> 1 /rm), in comparison to the Debye screening length, to fulfill the electroneutrality conditions. 

Chazelviel treated departures from electroneutraltiy in growth from binary solution by dividing the electrolysis cell into zones [40]. In the bulk of the cell he assumed electroneutraltiy, while he allowed departures from electroneutraltiy in a region larger than the Debye screening length. A scale x defines the extent of the transport-induced space charge. 

Because the inverse Debye length is calculated from the ionic surfactant concentration of the continuous phase, the only unknown parameter is the surface potential i/io this can be obtained from a fit of these expressions to the experimental data. The theoretical values of FeQx) are shown by the continuous curves in Eig. 2.5, for the three surfactant concentrations. The agreement between theory and experiment is spectacular, and as expected, the surface potential increases with the bulk surfactant concentration as a result of the adsorption equilibrium. Consequently, a higher surfactant concentration induces a larger repulsion, but is also characterized by a shorter range due to the decrease of the Debye screening length. 

The function on the right hand side of Eq. (34) consists of a series of elliptic integrals, which depend not only on the unknown electrostatic force but also on the surface charge densities, q and on the interface and protein surface, respectively, and on the inverse Debye screening length (1/K). 

As the ionic strength increases, the Debye screening length decreases, resulting in interparticle electrostatic interactions of shorter range. The theoretical surface potential can be further related to the net protein charge Q (Home et al, 2007) ... 

Thus, 3>n is only a function of the inverse Debye screening length To, defined by... 

In the model, Xp is determined by a temperature dependent electrophoretic mobility factor [123] which contains the viscosity of the solvent as well as its relative permittivity, Xc °, the radius of the polymer chain and the Debye screening length 1D. The following equation holds for the case that electrolyte and polyelectrolyte are in the same concentration range ... 

Figure J8 Photoionization cross-section of the sodium ground state for different Debye screenings as a function of the ejected electron energy. Xp is the Debye screening length in au. Reprinted with permission (license number 2041490188918) from [252].

Here q and q2 are the ion charges in units of electron charge e, and the so-called Debye screening length... 

Flory-type free energy calculations show that the root mean square end-to-end distance of a polyelectrolyte increases linearly with the chain length at infinite dilution and without added salt [40]. Using the above perturbation theory, scaling relations at finite densities are obtained. The influence of the interaction with other polymer chains, counterions, and added salt is captured in the Debye screening length xT1. 

Fig. 4 Root mean square end-to-end distance of flexible polyelectrolyte chains as a function of chain length for lB/b=0.5. The Debye screening length decreases from top to bottom (tc=0.05, 0.1, 0.2, 0.4, 0.8). The slopes of the straight lines are 1 and 3/5, respectively...

Here, T is the absolute temperature, e is the bulk dielectric constant of the solvent, P is the number of phosphate charges, k is the inverse of the Debye screening length, kB is the Boltzmann constant, qna is the renormalized charge, the interaction between the charges is screened Debye-Hiickel potential, and — j b is the distance between a pair of charges labeled i and /... 

In an important work, Ramanathan analyzed the colloidal and the micellar limits of a charged sphere [36]. In the latter case, the Debye screening length is larger than the radius of the sphere. In the colloidal limit, the reverse is true, namely, the radius of the sphere is larger than the Debye screening length. 

The situation is complicated for a short polyion. In short, we mean a polyion whose length is much smaller than the Debye screening length, L 0, L — oo, L = o(k 1), the total work in assembling the polyion charge and the ion atmosphere is [40]... 

The radial distribution function of polymer-counterion as a function of distance from the polyion surface exhibits two peaks. These peaks are interpreted as reflecting two populations of counterions a diffuse Debye-Htickel cloud at a distance characteristic of the Debye screening length, and the condensed layer near the polymer surface where the condensed counterions reside [43]. 

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