Permittivity calculation

Figure 10 Parameters used in Glueckauf s discontinuous model for electrolytes, and their comparison with static permittivity calculated as a function of distance from the ion centre. (A) cations (B) anions C is the centre of the water dipole A = the range where c = 1...

Figure 3. Imaginary component of the permittivity, calculated taking into account the KWW model (Equation (15)), obtained at different kww values (shown in the figure).

One of the methods used to study emulsions has been the use of dielectric spectroscopy. The permittivity of the emulsion can be used to characterize an emulsion and assign a stability (1,42,48—54). The Sjoblom group has measured the dielectric spectra using time-domain spectroscopy (TDS) technique. A sample is placed at the end of a coaxial line to measure total reflection. Reflected pulses are observed in time windows of 20 ns, Fourier transformed in the frequency range from 50 MHz to 2 GHz, and the complex permittivity calculated. Water or air can be used as reference sample. The total complex permittivity at a frequency (co) is given by ... 

The contribution to< M 2> (and hence to e(0)) from < AMj > is very small thus the static permittivity calculated in the simulation is well represented by the "local field modified reorientational term alone... 

Furthermore, the dielectric constant for solid solutions can be predicted as well, based on the assumption of a crystal structure identical to the individual components. In the example of NbTaOs, it was found that the average calculated permittivity (54) was identical to the arithmetic mean of the individual composing oxides, and a mean value for the dielectric constant was also found experimentally. It should be noted though that experimental values and calculated permittivities for crystals containing vacancies were found to be much lower than permittivities calculated for the perfect crystal. As crystal imperfections are often difficult to avoid in practice, this may be extrapolated to other material systems. 

N is the number of point charges within the molecule and Sq is the dielectric permittivity of the vacuum. This form is used especially in force fields like AMBER and CHARMM for proteins. As already mentioned, Coulombic 1,4-non-bonded interactions interfere with 1,4-torsional potentials and are therefore scaled (e.g., by 1 1.2 in AMBER). Please be aware that Coulombic interactions, unlike the bonded contributions to the PEF presented above, are not limited to a single molecule. If the system under consideration contains more than one molecule (like a peptide in a box of water), non-bonded interactions have to be calculated between the molecules, too. This principle also holds for the non-bonded van der Waals interactions, which are discussed in Section 7.2.3.6. 

The first modification is to simply scale the dielectric permittivity of free space (T( ) by a scale factorD to rn ediate or dam pen thelong range electrostatic interactions. Its value was often set to be between 1.0 and 7H.0, the macroscopic value for water. A value of D=2..5, so that u=2..5Ug, wasoften used in early CIIARMM calculation s. 

Also use constant dielectric for MM+ and OPLS calculations. Use the distance-dependent dielectric for AMBER and BlO-t to mimic the screening effects of solvation when no explicit solvent molecules are present. The scale factor for the dielectric permittivity, 8, can vary from 1 to 80. HyperChem sets 8 to 1.5 for MM-t. Use 1.0 for AMBER and OPLS, and 1.0-2.5 for BlO-t. 

The apphcation of microwave power to gaseous plasmas is also of interest (see Plasma technology). The basic microwave engineering procedure is first to calculate the microwave fields internal to the plasma and then calculate the internal power absorption given the externally appHed fields. The constitutive dielectric parameters are useful in such calculations. In the absence of d-c magnetic fields, the dielectric permittivity, S, of a plasma is given by equation 10 ... 

Various data sources (44) on plasma parameters can be used to calculate conditions for plasma excitation and resulting properties for microwave coupling. Interactions ia a d-c magnetic field are more compHcated and offer a rich array of means for microwave power transfer (45). The Hterature offers many data sources for dielectric or magnetic permittivities or permeabiHty of materials (30,31,46). Because these properties vary considerably with frequency and temperature, available experimental data are iasufficient to satisfy all proposed appHcations. In these cases, available theories can be appHed or the dielectric parameters can be determined experimentally (47). 

An alternative electrical method that has been used in the study of glass-ionomer cements has been the measurement of dielectric properties. Tay Braden (1981, 1984) measured the resistance and capacitance of setting cements at various times from mixing. From the results obtained, relative permittivity and resistivity were calculated. In general, as these cements set, their resistivity was found to fall rapidly, then to rise again. Both these results and the results of relative permittivity measurements were consistent with the cements comprising highly ionic and polar structures. 

FIGURE 26.23 Variables for calculating crossplane flow rates (permittivity). (Adapted from U.S. EPA, Requirements for Hazardous Waste Landfill Design, Construction, and Closure, EPA/625/4-89/022, U.S. Environmental Protection Agency, Cincinnati, OH, August 1989.)... 

The capacitance determined from the initial slopes of the charging curve is about 10/a F/cm2. Taking the dielectric permittivity as 9.0, one could calculate that initially (at the OCP) an oxide layer of the barrier type existed, which was about 0.6 nm thick. A Tafelian dependence of the extrapolated initial potential on current density, with slopes of the order of 700-1000 mV/decade, indicates transport control in the oxide film. The subsequent rise of potential resembles that of barrier-layer formation. Indeed, the inverse field, calculated as the ratio between the change of oxide film thickness (calculated from Faraday s law) and the change of potential, was found to be about 1.3 nm/V, which is in the usual range. The maximum and the subsequent decay to a steady state resemble the behavior associated with pore nucleation and growth. Hence, one could conclude that the same inhomogeneity which leads to pore formation results in the localized attack in halide solutions. 

Calculations using the experimentally determined permittivities of silver and gold show, however, that this model is not adequate as it stands the predicted enhancement effects are too small and are maximal at the wrong wavelength. 

The first (exponential) term represents repulsion between electron orbitals on the atoms. The second term can be seen to be opposite in sign to the first and so represents an attraction—the weak van der Waals interaction between the electron orbitals on approaching atoms. The adjustable parameters can sometimes be calculated using quantum mechanics, but in other systems they are derived empirically by comparing the measured physical properties of a crystal, relative permittivity, elastic constants, and so on, with those calculated with varying parameters until the best fit is obtained. Some parameters obtained in this way, relevant to the calculation of the stability of phases in the system SrO-SrTiC>3, are given in Table 2.3. 

Dielectric constant (DE) values are reported as permittivity with the symbol e or K The polymer cylindrical donuts were used for the measurement of DE on a Hewlett-Packard 8510 automated network analyzer. The analyzer is capable of measuring 401 data points over a frequency band of 500 MHz to 18.5GHz. Typically Sll and S21 values, which correspond to reflection and transmission, respectively, are measured and then these values are used to calculate the permittivity and permeability. 

The fourth term is a polarisation term. Here E(z) = di/z/dz is the electric field at position z. In previously published SCF results for charged bilayers, this last term is typically absent. It can be shown that the polarisation term is necessary to obtain accurate thermodynamic data. We note that all qualitative results of previous calculations remain valid and that, for example, properties such as the equilibrium membrane thickness are not affected significantly. The polarisation term represents relatively straightforward physics. If a (united) atom with a finite polarisability of erA is introduced from the bulk where the potential is zero to the coordinate z where a finite electric field exists, it will be polarised. The dipole that forms is proportional to the electric field and the relative dielectric permittivity of the (united) atom. The energy gain due to this is also proportional to the electric field, hence this term is proportional to the square of the electric field. The polarisation of the molecule also has an entropic consequence. It can be shown that the free energy effect for the polarisation, which should be included in the segment potential, is just half this value... 

In Eqs. (16) and (17), a and P refer to atoms of the solute and solvent, respectively q is the permittivity of free space, Qa and Qp are atomic charges, and Rap is the distance between atoms a and p. The parameters eap, aap, Aap, Bap and Cap can either be assigned by fitting to experimental data or can be the arithmetic or geometric means of literature values for the individual atom types.10,65,66 The atomic charges are commonly determined by requiring that they reproduce the calculated molecular electrostatic potentials.10 In order to provide better descriptions of the solvent s structure, Eqs. (16) and (17) are generally extended to include solvent-solvent intermolecular interactions. 

A similar conclusion arises from the capacitance data for the mercury electrode at far negative potentials (q 0), where anions are desorbed. In this potential range, the double-layer capacitance in various electrolytes is generally equal to ca. 0.17 F Assuming that the molecular diameter of water is 0.31 nm, the electric permittivity can be calculated as j = Cd/e0 = 5.95. The data on thiourea adsorption on different metals and in different solvents have been used to find the apparent electric permittivity of the inner layer. According to the concept proposed by Parsons, thiourea can be treated as a probe dipole. It has been cdculated for the Hg electrode that at (7 / = O.fij is equal to 11.4, 5.8, 5.1, and 10.6 in water, methanol, ethanol, and acetone, respectively. 

The nonlinear part of the susceptibility was introduced into the quasi-linear finite-difference scheme via iterations, so that at any longitudinal point, the magnitude of E calculated at the previous longitudinal point was used as a zero approximation. This approach is better than the split-step method since it allows one to jointly simulate both the mode field diffraction on irregular sections of the waveguide and the self-action effect by introducing the nonlinear permittivity into the implicit finite-difference scheme which describes the propagation of the total field. 

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