8 Dielectric Properties Lecture 7

K Asokan

Learning Outcomes:

From this module students may get to know about the following:

1.     Detailed study of equivalent circuit of the dielectric.

2.     The fundamental interfacial polarization.

3.     You will learn about the frequency dependence of.

In this lecture we will talk about the equivalent circuit of the dielectric.

A real dielectric may be represented by a capacitance in series with a resistance, or alternatively a capacitance in parallel with a resistance. We consider that this representation is successful if the frequency response of the equivalent circuit is identical to that of the real dielectric.

7.1     A SERIES EQUIVALENT CIRCUIT:

A capacitance in series with a resistance has series impedance given by

Where  is the capacitance without the dielectric. Since the two impedances are equal from the external circuit point of view we can equate equations (1) and (2). To obtain  and  as a function of frequency we equate the real and imaginary parts. This gives

Figure1. Equivalent circuits of a lossy dielectric

7.2 PARALLEL EQUIVALENT CIRCUIT:

A capacitance Cp in parallel with a resistance Rp may also be used as a equivalent circuit (fig. 1). The admittance of the parallel circuit is given by

7.3 SERIES-PARALLEL CIRCUIT:

Fig.1 also shows a series-parallel circuit in which a series branch having a capacitance and a resistance Rs is in parallel with a capacitance. We follow the same procedure to determine the real and imaginary parts of the complex dielectric constant. The admittance of the equivalent circuit is:

This result shows that the equivalent circuit yields  that is identical to the Debye criterion.

7.4 INTERFACIAL POLARIZATION:

Interfacial polarization, also known as space charge polarization, arises as a result of accumulation of charges locally as they drift through the material. In this respect, this kind of polarization is different from the three previously discussed mechanisms, namely, the electronic, orientational and atomic polarization, all of which are due to displacement of bound charges. The atoms or molecules are subject to a locally distorted electric field that is the sum of the applied field and various distortion mechanisms apply. In the case of interfacial polarization large scale distortions of the field takes place. For example, charges pile up in the volume or on the surface of the dielectric; predominantly due to change in conductivity that occurs at boundaries, imperfections such as cracks and defects, and boundary regions between the crystalline and amorphous regions within the same polymer. Regions of occluded moisture also cause an increase in conductivity locally, leading to accumulation of charges We consider the classic example of Maxwell-Wagner to derive the and characteristics due to the interfacial polarization that exists between two layers of dielectric materials that have different conductivity. Let and be the thickness of two materials that are in series. Their dielectric constant and resistivity are respectively and, with subscripts 1 and 2 denoting each material (Fig.2).

Figure 2. Dielectrics with different conductivities in series

When a direct voltage, V, is applied across the combination the voltage across each dielectric will be distributed, at t = 0, according to

Let us suppose that the condition set by expression (17) is satisfied by the components of the two layer dielectric. The admittance of the equivalent circuit (fig. 3) is given by

figure 3. Equivalent circuit for two dielectrics in series for interfacial polarization

Equation (25) gives the characteristics for interfacial polarization. It is identical to the Debye equation that is, the dispersion for interfacial polarization is identical with dipolar dispersion although the relaxation time for the former could be much longer. It can be as large as a few seconds in some heterogeneous materials. The relaxation spectrum given by equation (26) has two terms; the second term is identical to the Debye relaxation and at higher frequencies the relaxation for interfacial polarization is indistinguishable from dipolar relaxation.

7.5 FREQUENCY DEPENDENCE OF :

We are now in a position to represent the variation in the complex dielectric constant as a function of frequency, from  = 0 to . Fig. 4 shows the contribution of individual polarization mechanisms to the dielectric constant and their relaxation frequencies. As each process relaxes the dielectric constant becomes smaller because the contribution to polarization from that mechanism ceases. Beyond optical frequencies the dielectric constant is given by .

figure4. Frequency dependence of the real and imaginary parts of the dielectric constant (schematic).

The orientational polarization occurs in the radio frequency to microwave frequency range in dipolar liquids. However in polymers the dipoles may be constrained to rotate or move to a limited extent depending upon whether the dipole is a part of the main chain or side group. Correspondingly the relaxation frequency may be smaller, of the order of a few hundred kHz. Further, in solids there is no single vibration frequency but only a range of allowed frequencies, making the present treatment considerably simplified. The experimental results presented in the next chapter in a number of different polymers will make this evident.

Summary:

In this chapter a detailed description of equivalent circuit of the dielectric was given. In addition the fundamental of interfacial polarization was studied. We also learned about the frequency dependence of.

    References:

1. Bunget, I., & Popescu, M. (1984). Physics of solid dielectrics. Amsterdam: Elsevier.
2. Kao, K. (2004). Dielectric phenomena in solids with emphasis on physical concepts of electronic processes. Amsterdam: Academic Press.
3. Kasap, S. (2006). Principles of electronic materials and devices (3rd ed.). Boston: McGraw-Hill.

    References and Suggestive Readings

1. Hanai, T., and K. Sekine. “Theory of dielectric relaxations due to the interfacial polarization for two-component suspensions of spheres.” Colloid and Polymer Science 264.10 (1986): 888-895.
2. Bertram, Brian D., and Rosario A. Gerhardt. “Frequency-Dependent Dielectric Properties and Percolation Behavior of Alumina-Silicon Carbide Whisker Composites.”

     Web Links

1.      http://www.sciencedirect.com/science/article/pii/0304388680900091

2.      http://link.springer.com/article/10.1007%2FBF00367589

3.      http://chemwiki.ucdavis.edu/Physical…Polarizability 

4.        http://chemwiki.ucdavis.edu/u_Materi…ezoelectricity

Additional Topics to be studied

Dielectric relaxation due to interfacial polarization

It is well known theoretically as well as experimentally that suspensions of particles in a continuous medium show dielectric relaxations due to interfacial polarization. Maxwell and Wagner proposed a dielectric theory of the interracial polarization for a dilute suspension of spherical particles. Afterwards Hanai developed a dielectric theory of interracial polarization for concentrated suspensions on the basis of the Maxwell-Wagner theory. The dielectric relaxations predicted from the theories were discussed experimentally by many workers. The limiting values of the permittivities and conductivities at high and low frequencies in regard to the dielectric relaxations were discussed for a variety of emulsions. The frequency dependence of the permittivities and the conductivities was also discussed in detail for W/O emulsions and suspensions of ion exchange resin gel beads in water. Furthermore the dielectric relaxations for the concentrated suspensions of spheres covered with a shell were formulated N 105 and were successfully applied to the observations of polystyrene microcapsules. All the examples showed that the theory developed by Hanai for concentrated suspensions is in satisfactory agreement with the observed results as compared with the Maxwell-Wagner theory derived for dilute suspensions. At the present stage of the development of theories, it is desired to formulate and discuss the dielectric relaxation behavior of a concentrated suspension containing two kinds of dispersed particles; the suspension of this type is termed a two-component suspension hereinafter. As regards dielectric theories for such two-component suspensions, Grosse proposed an equation of the Bruggeman-Hanai type extended to two component suspensions. Since conductivities are left out of theoretical consideration, dielectric relaxations due to the interfacial polarization cannot be discussed with his equation. Recently Boned and Peyrelasse derived some theoretical formulas of complex permittivities for the case of multicomponent ellipsoidal suspensions. Their discussion is of great use especially for such dilute ellipsoidal or spheroidal suspensions. Boyle also derived theoretical equations for the permittivity and the conductivity of suspensions of an oriented dispersed phase of spheroidal shape applicable to higher concentrations. No attempt has so far been made to formulate the complex permittivity of two- or multicomponent suspensions.

Extension of the Maxwell-Wagner theory to a dilute two-component suspension of spheres

Maxwell and Wagner presented a dielectric theory of interracial polarization for a dilute suspension of spherical particles. Without loss of generality of the formulation, their theoretical formula can be extended to a suspension of dispersed particles of two kinds, henceforward termed j- and k-spheres, as the following:

                                                            (1)

where ɛ*, ɛa*, ɛj*, and ɛk* denote the complex relative permittivity of the suspension, the continuous medium, the suspended j- and k-spheres, and ɸj and ɸk mean the volume fractions of the j- and k-spl~eres, respectively. Asterisked permittivities ɛ*’s are written as ɛ* = ɛ + x/(jωɛv) in terms of relative permittivity ɛ, electrical conductivity x, angular frequency ω, the permittivity of vacuum ɛv, and imaginary unit j.

                                                                                                                (2)

This Equation (1) is transformed to an explicit form with respect to e* as