Wagner Dispersion

Could this be behavior in terms of the Maxwell-Wagner dispersion, which would arise through conductivity in the double layer near the polyanion. In support of this, the dielectric constant falls as the frequency increases (Fig. 2.79). 

Maxwell-Wagner Dispersion.—Macroscopic heterogeneities of the complex permittivity must always result in an apparent overall dielectric dispersion, even if the above-discussed orientation effect cannot occur. We may investigate this for the case of spherical particles of radius a and complex permittivity eg which are suspended in a medium with complex permittivity eg. It follows from electrostatic theory that the presence of one such sphere is equivalent to a dipole moment given by... 

Another model of experimental interest concerns the case of a highly conductive shell around practically non-conductive material. It may be applied to macromolecules or colloidal particles in electrolyte solution which usually have counterion atmospheres so that the field may displace freely movable ionic charges on their surfaces. The resulting dielectric effect turns out to be equivalent to a simple Maxwell-Wagner dispersion of particles having an apparent bulk conductivity of... 

H. Fricke, The Maxwell-Wagner dispersion in a suspension of ellipsoids, J. Phys. Chem. 57, 934-937 (1953). 

Maxwell—Wagner dispersion is due to a conductance in parallel with a capacitance for each dielectric, so that the interface can be charged by the conductivity. With zero conductivity in both dielectrics, there is no charging of the interface from free charge carriers. If side one of the dielectric is without conductivity (oi =0 and Ri = °o), then Cpext at very low frequencies becomes equal to Ci. At very high frequencies, the conductivities are without influence. 

The models of Figures 4.30—4.32 will exhibit Maxwell—Wagner dispersion. The anisotropy of Figure 4.32 disappears at high frequencies because the capacitive membranes are short-circuited. For example, when anisotropy is caused by air in the lungs, the anisotropy may persist at virtually all frequencies. 

The Maxwell-Wagner Dispersion in a Suspension of Ellipsoids, J. Phys. chem. 57, 934—937. 

The Maxwell-Wagner dispersion effect due to conductance in parallel with capacitance for two ideal dielectric materials in series Rj Cj - Rj Cj can also be represented by Debye dispersion without postulating anything about dipole relaxation in dielectric. In the ideal case of zero conductivity for both dielectrics (R, — , R —> ), there is no charging of the interfaces from free charge carriers, and the relaxation can be modeled by a single capacitive relaxation-time constant. 

While the results reported in this section deal with dispersions in the frequency range 3-60 MHz, which seem indeed to be interpretable in terms of heterogeneity effects ( Maxwell-Wagner effects ) [12], it should be noted that dispersions in polyelectrolyte-solution systems have also been observed at lower frequencies [13-16], and that these dispersions have been interpreted in a different manner. In one case [15] the frequency range of the observed dispersions overlaps somewhat with the Maxwell-Wagner dispersion range of the clay-solution aggregates described here. 

Dominant contributions are responsible for the a, fi, and y dispersions. They include for the a-effect, apparent membrane property changes as described in the text for the fi-effect, tissue structure (Maxwell-Wagner effect) and for the y-effect, polarity of the water molecule (Debye effect). Fine structural effects are responsible for deviations as indicated by the dashed lines. These include contributions from subcellular organelles, proteins, and counterion relaxation effects (see text). 

Table III attempts to summarize at what level of biological complexity the various mechanisms occur. Electrolytes display only the y-dispersion characteristic of water. Biological macromolecules in water add to the water s Y-dispersion a 6-dispersion. It is caused by bound water and rotating side groups in the case of proteins, and by rotation of the total molecule in the case of the amino acids and, in particular, proteins and nucleic acids add further dispersions in the 6 and a-range as indicated. Suspensions of cells free of protein would display a Maxwell-Wagner 6-dispersion and the Y-dispersion of water.

The dielectric properties of tissues and cell suspensions will be summarized for the total frequency range from a few Hz to 20 GHz. Three pronounced relaxation regions at ELF, RF and MW frequencies are due to counterion relaxation and membrane invaginations, to Maxwell-Wagner effects, and to the frequency dependent properties of normal water at microwave frequencies. Superimposed on these major dispersions are fine structure effects caused by cellular organelles, protein bound water, polar tissue proteins, and side chain rotation. 

Wilcock A, Tebbens J, Fuss F, Wagner J, Brewster M. Spectrophotometric analysis of electrochemically treated, simulated, dispersed dyebath effluent. Text Chem Color 1992 24 29-37. 

Maranzano, B. J. and Wagner, N. J. 2002. Flow-small angle neutron scattering measurements of colloidal dispersion microstructure evolution through the shear-thickening transition. J. Chem. Phys. 117 10291-10302. 

A general expression for the electroconductivity of two-phase systems with a random arrangement of spherical particles of the disperse phase is given by Wagner... 

Provided that G > Gp (for liquid foams x of solutions > x of air) we obtain Eq. (39) from Eq. (41) by substitution of k instead of G. In contrast to Wagner s formula, Odelevsky s formula holds for all concentrations of the disperse phase (gas) and for all types of gas-filled systems gaseous emulsions (d < 0.74), spherical (0.74 < d< 0.9) and polyhedral ( > 0.9) foams. It requires isotropy of the matrix structures and equal diameters of the disperse phase inclusions. Therefore, the dependence of the ratio of the foam to the solution electroconductivity on the degree of foaming in the general form is given by equation... 

If one imagines the structure of a liquid foam as a system of cubical gas bubbles (Fig. 19) and that the electrical current is directed upward, the horizontal walls of bubbles (perpendicular to the current direction) do not participate in electroconductivity and 2 of die 6 walls of each cube do not contribute to conduction. Then we have kIk = 4/6 K = 2/3 K, i.e. Manegold s formula is applicable. On the other hand, this equation coincides with Wagner s and Odolevsky s equations, which is to be expected since both these relations are also based on a cubical model of the disperse system. 

Maxwell-Wagner polarization (20-23) arises in heterogeneous specimens containing domains of different conductivity and/or dielectric constant. This phenomenon can be distinguished by low-frequency impedance dispersions that extend over several decades of frequency. In addition, the magnitude and frequency dependence of Maxwell-Wagner polarization is related to spatial fluctuations of dielectric... 

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