Wagner-Maxwell equation

In addition, the Wagncr-Maxwcll equation is able to adequately describe the dielectric property of ER fluids. Weiss [44] and Filisko [45] measured the dielectric constant and dielectric loss of ER fluids, and found that the Wagner-Maxwell equation is a suitable model to describe the dielectric property. The Wagner-Maxwell equation can also explain the frequency dependence of yield stress of ER fluids. As given in Figure 9 in Chapter 5, the frequency dependence of the yield stress and the dielectric constant follow a similar trend, further indicating that it is the Wagner-Maxwell polarization that controls the dielectric property of the ER suspension and then the ER effect. 

Maxwell-Wagner Type Theory. If a sphere with a dielectric constant 2 and conductivity k2 is suspended in a medium with a dielectric constant i and a conductivity kh the dielectric constant of the suspension may be expressed by the Maxwell equation (11),... 

Assuming (as it is reasonable) that for conditions in which the approximation ko 5> 1 is valid, the dynamic mobility also contains the (1 — Cq) dependence displayed by the static mobility (Equation (3.37)), one can expect a qualitative dependence of the dynamic mobility on the frequency of the field as shown in Figure 3.14. The first relaxation (the one at lowest frequency) in the modulus of u can be expected at the a-relaxation frequency (Equation (3.55)) as the dipole coefficient increases at such frequency, the mobility should decrease. If the frequency is increased, one finds the Maxwell-Wagner relaxation (Equation (3.54)), where the situation is reversed Re(Cg) decreases and the mobility increases. In addition, it can be shown [19,82] that at frequencies of the order of (rj/o Pp) the inertia of the particle hinders its motion, and the mobility decreases in a monotonic fashion. Depending on the particle size and the conductivity of the medium, the two latter relaxations might superimpose on each other and be impossible to distinguish. 

The complex dielectric constant of a suspension e of orientated ellipsoidal particles with the dielectric constant Cp at the particle volume fraction < ) dispersed in a continuous medium with a complex dielectric constant , can be calculated from the Maxwell-Wagner-Sillars equation [77] ... 

Numerical results from the above three type equations are compared by Banhcgyi [83]. The dielectric constant and loss of two-phase spherical particle mixture are calculated with the Maxwell-Wagner-Sillars equation, the Bottcher-Hsu equation, and the Looyenga equation using the parameters e i =2, p 8, S/m, CTp=10 S/m, and shown in Figure 23 against... 

Figure 23 The dielectric constant and loss vs. frequency calculated by using different models for spherical particle case at different particle volume fraction marked in the graphs, a) Maxwell-Wagner-Sillars equation b) Bdtlcher-Hsu equation c) Looyenga equation. Parameters are c , =2, p=8, Om=10 S/m, CTp=10 S/m. The particle volume fraction changes from 0.1 to 0.9 with the interval 0.2. Reproduced with permission from G. Banhegyi, Colloid Polym. Sci., 266(1988)11.

Dielectrie impedance is another method used to detect the phase separation. The permittivity and conductivity changes during the second-phase growing and interface formation. The effect is desalbed for Maxwell-Wagner-Sillars equation (1.1). 

Theoretical and semi-empirical equations were derived for gas emulsions (as well as for suspensions of non-conducting spherical particles and O/W emulsion), specifying Eq. (8.33) . A relation for coefficient B can be derived from Maxwell-Wagner equation [45,46]... 

Two other approaches have been taken to modelling the conductivity of composites, effective medium theories (Landauer, 1978) and computer simulation. In the effective medium approach the properties of the composite are determined by a combination of the properties of the two components. Treating a composite containing spherical inclusions as a series combination of slabs of the component materials leads to the Maxwell-Wagner relations, see Section 3.6.1. Treating the composite as a mixture of spherical particles with a broad size distribution in order to minimise voids leads to the equation ... 

Table 7 gives a summary of qualitative performances and problems encountered for simple shear and uniaxial elongational flows, using the Wagner and the Phan Thien Tanner equations or more simple models as special cases of the former. Additional information may also be found in papers by Tanner [46, 64]. All equations presented hereafter can be cast in the form of a linear Maxwell model in the small strain limit and therefore are suitable for the description of results of the linear viscoelasticity in the terminal zone of polymer melts. 

For the very simplified situation that the sphere behaves electrically as a pure capacitor, and the solution as a pure resistance, the relaxation can be described by a Maxwell-Wagner mechanism, with T = e e/K, see (1.6.6.321. Although some success has been claimed by Watillon s group J to apply this mechanism for a model, consisting of shells with different values of e and K, generally a more detailed double layer picture is needed. In fact, this Implies stealing from the transport equations of secs. 4.6a and b. generedizing these to the case of a.c. fields. 

Equation (3) shows that the real part of the Clausius-Mossotti factor goes to a low frequency co = 0) limiting value of (o -cr )/(o +2cr ), i.e. it depends on the conductivity of the particle and the medium. At high frequency ( oo) the limiting value is 6p-6m)l Sp+2 m) and the polarization is dominated by the permittivity of the particle and the medium. From Eq. (4), the imaginary part of the Clausius-Mossotti factor is zero at both low and high frequencies at the Maxwell-Wagner relaxation frequency /mw " it has a value of p- m)l pF2 m) - (Jp-(Jm)l... 

Recently, Pollack derived, by adapting a simple procedure, a Maxwell-Wagner type of equation for a highly elongated ellipsoid of revolution (18). Although his procedure is considerably different from those of Fricke and Sillars, the final form is essentially the same. He derived the following equations for the relaxation time ... 

Hanai s equation gives rise to dispersion curves that are broader than the Maxwell-Wagner equation (Takashima, 1989). Similar work had also earlier been done by Bmggeman (1935). 

This is the Maxwell—Wagner model of a capacitor with two dielectric layers. Even with only two layers, the equations are complicated with three layers, they become much... 

To study the effects of interaction of starch with silica, the broadband DRS method was applied to the starch/modified silica system at different hydration degrees. Several relaxations are observed for this system, and their temperature and frequency (i.e., relaxation time) depend on hydration of starch/silica (Figures 5.6 and 5.7). The relaxation at very low frequencies (/< 1 Hz) can be assigned to the Maxwell-Wagner-Sillars (MWS) mechanism associated with interfacial polarization and space charge polarization (which leads to diminution of 1 in Havriliak-Negami equation) or the 5 relaxation, which can be faster because of the water effect (Figures 5.8 and 5.9). 

Keywords dielectric relaxation, dielectric strength permittivity, dipole moment, polarization, relaxation, conductivity, relaxation time distribution, activation energy, Arrhenius equation, WLF-equation, Maxwell-Wagner polarization. 

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