6 Dielectric Properties Lecture 5

K Asokan

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Learning Outcomes:

 

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

1.      The fundamental of Dielectric Theories of Liquids and Solutions.

2.      You will learn about Onsager’s theory for static dielectric constant.

3.      Detailed study of Debye’s theory for static dielectric constant.

 

In the last few lectures we have discussed about the dielectric properties of solids. In this lecture we will talk about the dielectric properties of the liquids.

 

5.1 DIELECTRIC THEORIES OF LIQUIDS AND SOLUTIONS:

 

The state of aggregation of molecules in a continuum depends on the type of chemical bond, molecular geometry, mutual effect between atomic groups, nature of complexes etc. A system of electric charges of molecules in the neighborhood involves in the process of molecular interactions. The spatial arrangement of electrically charged atoms and molecules in the system is perturbed by the influence of physical conditions. These facts on the basis of certain theories describe the bulk Properties of the substances that exist in a physical state.

 

The dielectric constant depends on how polarizable a material is and the frequency of the applied field. The fall of polarizability is related to the decrease of dielectric constant and occurrence of absorption of electrical energy constituting dielectric dispersion, this behavior is

 

shown by the frequency dependent dielectric loss. The static and dynamic dielectric mechanisms govern the behavior of dipolar liquids and liquid mixtures. Debye found the phenomenon of dielectric dispersion occurring in liquids containing polar molecules.

 

In general, the aim of proposed theoretical models is to describe the dielectric behavior of liquids by assessing the electric dipole moments and tracing their origin. The dipole moments either permanent or induced do depend on the external field and atomic and molecular structure of the dielectric substance. The dipole moment is a characteristic quantity of atomic and molecular polarization. The polarization is found to maintain equilibrium for a constant or a very slowly varying external field. The dielectric constants appropriate to such kind of time independent fields are termed static dielectric constant with zero or negligible dielectric loss.

 

5.2 Debye’s theory for static dielectric constant

 

The Debye’s theory for the static dielectric constant and dipole moment incorporates the Langevin method of finding mean moment and Boltzmann law of distribution of moments about an applied field. A basic relation for the static dielectric constant in the usual notations, is

Here  includes the ionic and electronic polarizability terms and the last term includes the dipolar term. This relation, when, is reduced to the well known Clausius-Mosotti equation for the optical refraction

extremely valid for gases, non-polar liquids and solids but not for dense liquids as it fails to reproduce the static dielectric constants. However, the dipole moment calculations from the static dielectric data show that the Debye equation holds good. Approximately to dilute solutions but not so to polar liquids. The inadequacy of equation (1) for polar liquids is due to the neglect of Lorentz inner field and local directional forces exerted by the neighboring molecules. In order to overcome the inadequacy Lars-onsager suggested a better approximation by taking into account the ‘internal field’, due to long range interactions and improved the treatment.

 

5.3 Onsager’s theory for static dielectric constant:

 

The Debye theory is satisfactory for gases, vapors of polar liquids and dilute solutions of polar substances in non-polar solvents. For pure polar liquids the value of calculated from Debye equation does not agree with the dipole moments calculated from measurements on the vapor phase where the Debye equation is known to apply.

 

Onsager attributed these difficulties to the inaccuracy caused by neglecting the reaction field. The field which acts upon a molecule in a polarized dielectric may be decomposed into a cavity field and a reaction field which is proportional to the total electric moment and depends on the instantaneous orientation of the molecule. The mean orientation of a molecule is determined by the orienting force couple exerted by the cavity field upon the electric moment of the molecule. The approach of Debye, though a major step in the development of dielectric theory, is equivalent to the assumption that the effective orienting field equals the average cavity field plus the reaction field. This is inaccurate because the reaction field does not exert a torque on the molecule. The Onsager field is therefore lower than that considered by Debye by an amount equal to the reaction field.

 

Onsager considered a spherical cavity of the dimension of a molecule with a permanent dipole moment  at its center. It is assumed that the molecule occupies a sphere of radius r, i.e.  and its polarizability is isotropic. The field acting on a molecule is made up of three components:

 

(1)  The externally applied field E along Z direction.

(2)  A field due to the polarization of the dielectric.

(3)  A reaction field R due to the dipole moment , of the molecule itself.

 

The three components combined together give rise to the Lorentz field which was shown to be

 

All the quantities on the right side of this equation are constants and we can make the substitution

 

The reaction field equation is derived by assuming that the dipole was rigid, i.e, its dipole moment was constant. This is only an approximation because the reaction field increases the dipole moment, the increase being R. This in turn will increase the reaction field to Rm and equation (6) will be modified as

 

Kirkwood gives an alternate expression for the modified dipole moment  on the assumption that the total moment of the molecule consists of a point dipole at the center of a sphere of radius a of dielectric constant unity, as opposed to n2 implied in equation (15)

 

At the beginning of this section it was mentioned that a part of the internal field is due to the reaction field R. When the dipole is directed by an external field, the average value of the reaction field in the direction of E is where the symbols enclosing signifies average value, assuming that the reaction field follows the direction of, instantaneously. Since Rm has the same direction as,at any instant,  does not contribute to the directing torque. We have to, therefore, apply a correction to the internal field as

 

We note here that the left side of equation (22) is not zero because, the relationship,does not hold true for polar substances at steady fields.

 

We may approximate equation (22) with regard to specific conditions as follows:

(1) For non-polar materials. We then obtain

which is the Lorenz-Lorentz relation.

 

(2)  For polar gases at low pressures the following approximations apply

 

which is the Debye equation for polar gases.

 

If we view the Onsager’s equation (22) as a correction to the Debye equation then it is of interest to calculate the magnitude of the modified reaction field Rm and the modified dipole moment  . Table 1 gives the appropriate data and the calculated values. Rm has a magnitude of the order of 109 V/ m. To obtain the significance of this field we compare the electric field that exists due to the dipole along its axis and along the perpendicular to the axis. The potential due to a dipole at a point with co-ordinates () as

 

The field due to the dipole has two components given by

you can view video on Dielectric Properties Lecture 5

    References:

  1. Onsager, Lars. “Electric moments of molecules in liquids.” Journal of the American Chemical Society 58.8 (1936): 1486-1493.
  2. Martin, Anna J., Gerhard Meier, and Alfred Saupe. “Extended Debye theory for dielectric relaxations in nematic liquid crystals.” Symposia of the Faraday Society. Vol. 5. Royal Society of Chemistry, 1971.
  3. Kirkwood, John G. “The dielectric polarization of polar liquids.” The Journal of Chemical Physics 7.10 (1939): 911-919.
  4. Fröhlich, H. “General theory of the static dielectric constant.” Transactions of the Faraday Society 44 (1948): 238-243.
  5. Nee, Tsu‐Wei, and Robert Zwanzig. “Theory of dielectric relaxation in polar liquids.” The Journal of Chemical Physics 52.12 (1970): 6353-6363.
  6. Nee, Tsu‐Wei, and Robert Zwanzig. “Theory of dielectric relaxation in polar liquids.” The Journal of Chemical Physics 52.12 (1970): 6353-6363.

    References and Suggestive Readings

  1. Hasted, John Barrett. Aqueous dielectrics. Chapman and Hall, 1973.
  2. Chandler, David, and Hans C. Andersen. “Optimized cluster expansions for classical fluids. II. Theory of molecular liquids.” The Journal of Chemical Physics 57.5 (1972): 1930-1937.
  3. Powles, J. G. “Dielectric relaxation and the internal field.” The Journal of Chemical Physics 21.4 (1953): 633-637.
  4. Cossi, Maurizio, and Vincenzo Barone. “Time-dependent density functional theory for molecules in liquid solutions.” The Journal of chemical physics115.10 (2001): 4708-4717.

    Web Links

  1. http://shodhganga.inflibnet.ac.in/bitstream/10603/1283/6/06_part%201.pdf
  2. http://shodhganga.inflibnet.ac.in/bitstream/10603/771/11/11_part%201.pdf
  3. http://pubs.acs.org/doi/pdf/10.1021/cr60082a002
  4. http://nvlpubs.nist.gov/nistpubs/jres/53/jresv53n4p229_A1b.pdf
  5. http://iopscience.iop.org/article/10.1088/0370-1328/72/4/307/pdf;jsessionid=38E534EF717E1BF2D605694961006BEA.c4.iopscience.cld.iop.org.

   Additional Topics to be studied

 

Dielectric Dispersion

 

In physics, dielectric dispersion is the dependence of the permittivity of a dielectric material on the frequency of an applied electric field. Because there is a lag between changes in polarization and changes in the electric field, the permittivity of the dielectric is a complicated function of frequency of the electric field. Dielectric dispersion is very important for the applications of dielectric materials and for the analysis of polarization systems.

 

This is one instance of a general phenomenon known as material dispersion: a frequency-dependent response of a medium for wave propagation.

 

When the frequency becomes higher:

 

1. dipolar polarization can no longer follow the oscillations of the electric field in the microwave region around 1010 Hz;

2. ionic polarization and molecular distortion polarization can no longer track the electric field past the infrared or far-infrared region around 1013 Hz, ;

3.  electronic polarization loses its response in the ultraviolet region around 1015 Hz.

 

In the frequency region above ultraviolet, permittivity approaches the constant ε0 in every substance, where ε0 is the permittivity of the free space. Because permittivity indicates the strength of the relation between an electric field and polarization, if a polarization process loses its response, permittivity decreases.