Permittivity, of particles

Figure 45 Static permittivity of particle suspensions vs. time, recorded during sedimentation. Measurements are performed in the bottom of the sedimentation cell, and at different pH conditions, (a) Alumina particles (b) alumina particles coated with EHEC. (From Ref 182.)...

Even though the atomic polarisability is a microscopic property of the atoms and molecules, it can be related to the macroscopic dielectric properties, i.e. to the relative permittivities of particles and medium, via the Clausius-Mossotti equation ... 

Heuristic Fxplanation As we can see from Fig. 22-31, the DEP response of real (as opposed to perfect insulator) particles with frequency can be rather complicated. We use a simple illustration to account for such a response. The force is proportional to the difference between the dielectric permittivities of the particle and the surrounding medium. Since a part of the polarization in real systems is thermally activated, there is a delayed response which shows as a phase lag between D, the dielectric displacement, and E, the electric-field intensity. To take this into account we may replace the simple (absolute) dielectric constant by the complex (absolute) dielectric... 

The symbols and are, respectively, the laplacian operators for a single nucleus and a single electron. The variable is the distance between nuclei a and / , Vai the distance between nucleus a and electron i, and the distance between electrons i and j. The summations are taken over each pair of particles. The quantity e is equal to the magnitude of the electronic charge e in CGS units and to e/(47reo) / in SI units, where eo is the permittivity of free space. 

Unlike charges attract and like charges repel each other, so there is a high concentration of counterions attracted to the particle surface whilst co-ions (those with the same sign charge as that of the surface) are repelled. Thermal motion, i.e. diffusion, opposes this local concentration gradient so that the counterions are in a diffuse cloud around the particle. Of course particles which have a like charge will also repel each other but the interaction of the particle surfaces will be screened by the counterion clouds between the particles. The interaction potential is a function of the surface potential, i]/o, and the permittivity of the fluid phase, e = r80, where r is the relative permittivity.12,27... 

Although the necessary labor is increased, no new concepts are required to extend the results above for a homogeneous ellipsoid to a coated ellipsoid. We denote by el the permittivity of the inner or core ellipsoid with semiaxes flj, c, e2 is the permittivity of the outer ellipsoid with semiaxes a2,b2,c2. This coated ellipsoidal particle is in a medium with permittivity em. As in the preceding section, we introduce ellipsoidal coordinates , tj, f ... 

In this equation, V2 = d2/dx2 + d2/dy2 + d2/dz2 denotes the Laplacian operator of cartesian second derivatives, p(r) is the charge density in a spherical shell of radius r and infinitesimal thickness dr centered at the particle of interest (see diagram), k is the effective dielectric constant, and e0 is the permittivity of free space (8.854 x 10 12 in SI units). The energy of interaction / , of ions of charge z,c with their surroundings,... 

Please note that the electrostatic double-layer force is fundamentally different from the Coulomb force. For example, if we consider two identical spherical particles of radius R you cannot take Eq. (6.1), insert the total surface charge as Qi and Q2, use the dielectric permittivity of water and expect to get a reasonable result. The main differences are the free charges (ions) in solution. They screen the electrostatic field emanating from the surfaces. 

In our early work33 [50] the constant field model was applied to liquid water, where the harmonic law of particles motion, corresponding to a parabolic potential, was actually employed in the final calculations of the complex permittivity. In this work, qualitative description of only the libration band was obtained, while neither the R-band nor the low-frequency (Debye) relaxation band was described. Moreover, the fitted mean lifetime x of the dipoles, moving in the potential well, is unreasonably short ( ().02 ps)—that is, about an order of magnitude less than in more accurate calculations, which will be made here. 

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