Chapter 2. Dielectric Theories

Transcription

1 Chapter Dielectric Theories

2 . Dielectric Theories 1.1. Introduction Measurements of dielectric properties of materials is very important because it provide vital information regarding the material characteristics, electrical characteristics, microwave processing, design parameters required for many electronics applications etc. Dielectrics or dielectric materials are non-conducting materials (electrical insulators) which can store electric energy. The electrons in the molecules of dielectrics are tightly bound to the nuclei. Therefore, the charges in the molecules slightly shift their positions on application of the electric field. The applied electric field can distort the charge distribution of a dielectric atom/molecule in two ways: by stretching and by rotating [9]... Non-polar and Polar dielectrics Atoms combine together by sharing electrons to form a molecule. The redistribution of charges may or may not give rise to permanent electric dipole moment. Therefore, the dielectric molecules can be classified as polar molecules (have net dipole moment) and non-polar molecules (have zero dipole moment). Dielectrics can be formed of polar and non-polar molecules...1. Electric Dipole Moment Fig..1. An electric dipole

3 . Dielectric Theories 11 Two equal and opposite charges separated by a small distance forms an electric dipole (Fig..1). The strength of electric dipole is measured in terms of electric dipole moment. Electric dipole moment μ is a vector whose magnitude is the product of either charge and length of the dipole. It is directed along the axis of the dipole from q to +q [3]. The S.I. unit is of the electric dipole moment is Cm while the C.G.S. unit is debye (D). 1 D = 1-18 statc cm Cm. The total charge of an electric dipole is zero but as the charges are separated by some distance, the resultant electric field due to the dipole is not zero.... Non-polar Dielectrics Non-polar dielectrics are formed of non-polar molecules. In a non-polar molecule, the centres of gravity of the positively charged nuclei and negatively charged electron cloud are concentric in the absence of external electric field (Fig..). As there is no separation between the positive and the negative charges, such molecules do not have intrinsic dipole moment e.g. O, H, CO etc. Fig... Non-polar molecule when external electric field is zero When an external electric field is applied to a non-polar dielectric material, the positive and negative charges in the molecule experience forces along and against the direction of the applied electric field respectively. Due to these forces,

4 . Dielectric Theories 1 the positive and negative charges in a molecule move away from each other. There is a separation between positive and negative charges and their centres of gravity do not coincide. Fig..3. Non-polar molecule in an external electric field Thus, the external electric field will change the shape of the spherically symmetric molecule. The molecule now acquires an induced dipole moment μ (Fig..3). This is called as induced polarization. The induced dipole moment is directly proportional to the applied electric field [31]...3. Polar Dielectrics Polar dielectrics are composed of polar molecules. In polar molecules, the effective centres of the negatively charged electron cloud and the positively charged nuclei do not coincide even in the absence of the external electric field. There is a separation between the positive and negative charges. So, the molecule behaves as an electric dipole and possesses an intrinsic permanent dipole moment e.g. H O, hydroxyl group, DMSO etc. The electric dipole moments of these polar molecules are randomly oriented over a macroscopic dielectric sample (Fig..4) due to thermal agitations. Hence, the resultant electric dipole moment of the sample is zero in the absence of the external electric field.

5 . Dielectric Theories 13 Fig..4. Polar dielectric in the absence of external electric field When a uniform external electric field is applied to a polar dielectric, the polar molecules are stretched just like the non-polar molecules but the induced dipole moment is very small as compared with their permanent dipole moments. However, the positive and negative charges of the molecules experience equal and opposite forces due to the applied field (Fig..5) and a torque qdesin acts where is the angle between the direction of the dipole moment of a molecule and the applied electric field. Fig..5. Torque acting on a polar molecule in an external electric field The molecular dipoles rotate in the direction of the applied electric field. The macroscopic dielectric sample is said to be polarized and acquires a net electric dipole moment (Fig..6).

6 . Dielectric Theories 14 Fig..6. A polar dielectric sample in an external electric field The complete alignment of the dipole moments in the direction of the applied electric field is not possible due to the thermal agitations. The polar dielectrics have permittivity greater than non-polar dielectrics due to the additional polarization due to orientation. The orientation polarization decreases with increase in temperature. Therefore, the permittivity of polar dielectrics falls more rapidly than that of the non-polar dielectrics with increase in temperature..3. Dielectric Polarization When a dielectric material is placed in an electric field, electric dipoles are formed (in non-polar dielectrics) or alignment of existing dipoles takes place in the direction of the applied field (in polar dielectrics). This process is called dielectric polarization P and the material is said to be polarized. This causes redistribution of charges. In many dielectric materials, the polarization is directly proportional to the applied electric field. Such materials are called as linear dielectrics. Therefore, for linear dielectrics, P E (.1)

7 . Dielectric Theories 15 where is constant called polarizability of the material. Dielectric polarization produces an internal electric field in a molecule due to the redistribution of the charges. The internal electric field is opposes the applied electric field so the resultant electric field inside the dielectric is reduced. Thus, polarization has a significant effect on the electric field inside the dielectric. A dielectric material can store electric energy by means of polarization. The relation between macroscopic polarization P and microscopic dipole moments μ i of the molecules in a volume V is [3] P 1 V i (.) The different polarization mechanisms are the distortion polarization (electronic polarization and atomic polarization), dipole/ orientation polarization and space charge polarization Electronic polarization The electronic polarization arises when an electric field is applied to a dielectrics formed of symmetric molecules. A symmetric molecule does not have an intrinsic permanent dipole moment. When an electric field is applied, there is a displacement of the electron cloud relative to the nucleus and the dipole moment arises. The molecules are polarized due to the induced dipole moments. The electrons respond rapidly to the field changes. The response time is of the order of 1-15 s which comparable to the time period of optical frequencies.

8 . Dielectric Theories Atomic polarization Atomic polarization occurs in the molecules made of atoms of different electro-negativities. The centre of gravity of the electron clouds is displaced towards the more electronegative atom so the atoms in the molecules have opposite charges. When an electric field is applied, the equilibrium positions of the atoms changes and polarization takes place. The displacement of the atoms takes place in a very short span of the order of 1-13 s to 1-1 s comparable to the time period of the infrared light Orientation/Dipole polarization When centres of gravity of positive and negative charges in a molecule made of different atoms do not overlap, the molecule has a permanent electric dipole moment even though no electric field is applied. The dipole moments are randomly oriented in a dielectric sample so the effective dipole moment of the sample is zero. When an external electric field is applied, torque acts on the dipoles which cause the dipoles to align in the direction of the applied field. This gives rise to the orientation or dipole polarization. Orientation polarization is a function of molecular size, viscosity, temperature and frequency of the applied electric field. The time taken by the molecules to relax is 1-1 s to 1-1 s which is comparable to the time periods of microwaves Space charge polarization The charges bound to atoms or charge carriers or molecules of a dielectric which can displace in a dielectric material but cannot be neutralized are called space charges. Some dielectrics may contain space charges. If an electric field is applied, the space charges move and the dielectric material get polarized. Such polarization

9 . Dielectric Theories 17 is known as space or surface-charge polarization. It is the only polarization mechanism in which macroscopic charge transportation takes place. If the space charge polarization takes place due to hopping of ions, electrons or holes, then it is called as hopping polarization. On the other hand, if there is separation of positive and negative mobile charges under the influence of electric field, then it is known as interfacial polarization. These four polarization mechanisms are characterized by respective polarizabilities: electronic polarizability ( e ), atomic polarizability ( a ), orientation polarizability ( o ) and surface charge polarizability ( s ). The resultant polarizability () of a dielectric is [33]: (.3) e a o s The net polarization of a dielectric is the vector sum of the polarizations arising due to the polarization mechanisms discussed above. In alternating electric fields, a phase shift may occur between the polarizing field and the resulting polarization. Thus, polarization is a complex quantity..4. Dielectric relaxation theories The relative permittivity (or dielectric constant) is a measure of the extent to which the electric charge distribution in a material can be distorted or polarized due to the application of an electric field [34]. It depends upon polarizability of a material and the frequency of the applied field. It is one of the dielectric parameters that determine how a material can interact with electromagnetic waves. The dielectric relaxation theories can be broadly classified as Theories of static permittivity and Theories of dynamic permittivity.

10 . Dielectric Theories Theories of static permittivity The polar dielectric materials when placed in a steady or slowly varying electric field can maintain equilibrium under all types of polarization. The dielectric permittivity in such type of time independent electric fields is termed as static dielectric constant or static permittivity ( o ). Here the dielectric loss is zero or negligible. Suppose that a parallel plate capacitor is charged with charge on one plate is +q and on the other plate is q (in statc). If the medium between the plates is the vacuum, then the magnitude of electric field intensity is given by E vacuum 4q (.4) If a dielectric material of dielectric constant is introduced between the plates of the capacitor then the electric field intensity reduces to 4q E (.5) The surface density of the oppositely signed charges on the surface of the dielectrics is known as polarization P. It is the total charge passing through unit area within the dielectric parallel to the plates []. The electric displacement D is 1 P q 1 (.6) D 4q Rearranging and using equation (.5) in above equation:

11 . Dielectric Theories 19 D E From (.5) and (.6), the above equation can be modified as and uniform then D E 4P (.7) 1 4P E (.8) If the field across the plates, separated by distance d, of the capacitor is V E d or V Ed (.9) The capacitance C is related to the total charge Q as Q C (.1) V From (.5), (.9) and (.1), the capacitance of parallel plates each of area A with a dielectric medium of dielectric constant is C A 4d (.11) The measurement of capacitance is important to know the static dielectric constant. To calculate the microscopic polarization in terms of macroscopic polarization, the inner field F is to be calculated because the actual field experienced by a molecule is generally not the same as the applied macroscopic field E.

12 . Dielectric Theories Fig..7. Microscopic cavity inside a polarized dielectric Consider a dielectric placed in an electric field E. To calculate the microscopic polarization in terms of the macroscopic polarization, assume a microscopic spherical region (cavity) surrounding the molecule but larger than the molecule (Fig..7). The interaction between the molecular dipoles within the cavity can be calculated. The field inside the cavity, inner field F, will be the resultant of the applied field E, the field due to the induced charges on surface of the cavity and the field due to the molecular dipoles inside the cavity. According to Lorentz, the dipoles inside the spherical region produce zero field. Therefore, the inner field arises only due to the external contribution such as the applied field and field due to the induced charges on surface of the cavity but not due to the polarization inside the spherical region. It is given by []: F E 3 (.1) Debye theory of static permittivity The interpretation of dielectric properties of a material can be done on the basis of the permanent and induced electric dipole moments of the atoms and molecules which compose the material. The induced polarization is

13 . Dielectric Theories 1 P induced N F V (.13) where N is the Avogadro s number, is the dipole polarizability of the atoms and molecules and V is the molar volume. The polar molecules are randomly oriented in the absence of an applied electric field. The polarization of a system of rigid polar molecules was first calculated by Debye [35] using the equation for the permanent polarization: P permanent N cos (.14) where N is number of molecules per unit volume and cos is the mean value of the cosine of the angle of inclination between a dipole and the applied electric field. The dipole moments are distributed about the applied electric field in accordance of the Boltzmann s distribution law from which Debye derived the equation for the permanent polarization in polar molecules: N F P permanent 3VkT (.15) where k is the Boltzmann constant and T is the absolute temperature. The effective polarization is P P induced P permanent By using (.13) and (.15) in the above equation: P N F (.16) V 3kT

14 . Dielectric Theories Combining (.8), (.1) and (.16), the Debye equation for static dielectric constant is obtained: 1 4N 3V 3kT (.17) The Debye equation has been successfully used to find the static dielectric constant of many polar gases and polar liquids. However, it fails in case of the dense fluids because of the inadequacy in the Lorentz inner field. In dense fluids, the permanent dipoles may lose their orientation freedom due to association and steric hindrance. Therefore, in case of the dense fluids, the near fields cannot be neglected due to the interactions of the permanent dipoles with their surroundings. For non-polar molecules, the electric dipole moment is zero. The equation (.17) reduces to the Clausius-Mossotti formula which is useful to obtain the static dielectric constant of only non-polar molecules.: 4N 3V 1 (.18) Onsager s theory of static permittivity Debye used the Lorentz model to calculate local fields but it fails in case of dense fluids because the close range interactions in dense fluids cannot be ignored. Onsager proposed a new model for dipolar dielectrics [33,36]. His model is restricted to spherical polar molecules. He assumed a spherical cavity of molecular size embedded in a continuous and homogeneous dielectric material with a single dipolar molecule as a point dipole at its centre. The radius a of the cavity can be obtained as:

15 . Dielectric Theories 3 4 Na a 3 4N 1 3 (.19) The internal field or the local field F in the cavity is made up of two parts: the cavity field G produced in the empty cavity by the applied field E and the reaction field R set up in the cavity due to the polarization induced by the molecular dipole in its surroundings. The cavity field G is 3 G 1E (.) And, the reaction field R is given by R 1m 1a 3 (.1) The molecular dipole moment m is the vector sum of the permanent and the induced dipole moments: m F (.) The average cos of orientations of molecules to the external electric field is calculated using Boltzmann statistics. The final Onsager equation for static dielectric constant is n n 4N n 9kT (.3) where n is refractive index. The above expression is better than the Debye expression to find the static dielectric constants of dense fluids.

16 . Dielectric Theories 4 However, according to the both, Debye and Onsager expressions, the inner field increases without limit on increasing the external field E. Also, both of these theories do not take into account the local forces between the neighboring dipoles but the dielectric constant is sensitive to these forces Kirkwood s and Frohlich s theory Kirkwood took into account the local forces between the neighboring dipoles i.e. the short range dipole-dipole interaction [37]. He assumed a spherical cavity, of the dielectric material, containing dipoles and the field inside the cavity is smaller than the outer field by a factor of 3. Also, he assumed a strong directional coupling between each fixed configurations of the dipoles. The dipole moment of the cavity M with one molecule in a fixed orientation and others are allowed to take random configurations is made of two components: moment induced by the molecule and moment arising due to local ordering associated with a structural model. It is given by * M (.4) 1) 9 The static permittivity is related to M as: which yields: 4M 1 (.5) VE ( 1)( 3 1) 4N * 3VkT 4N 3VkT g (.6)

17 . Dielectric Theories 5 where V is the molar volume, * is the moment of the sphere with a central molecule is in fixed alignment and the other molecules in the sphere are partially aligned due to the structural factors. The volume of the sphere should be large enough so that the dielectric material outside it can be treated as electrically homogeneous from the point of view of the dipole. But, it should be small as compared with the dielectric volume. In the equation (.6), *=g where g is the Kirkwood correlation parameter which measures the intermolecular angular correlation in a material. The Kirkwood model assumes an unpolarizable point molecular dipole but some induced dipole moment arises. So, it is necessary to take into account the induced dipole moment in the molecule. If the field acting on the molecule is taken to be the Onsager cavity field then the Kirkwood equation can be modified as: ( 1)( 3 1) 4N V g 3kT (.7) where is the induced polarizability of the permanent molecular dipole. The above equation is inadequate to explain the contribution of distortion polarization. Also, it does not reduce to the Onsager s equation when g = 1. Frohlich modified the Kirkwood equation by taking into account the total local field F and the distortion polarization as: )( n ) 4N ( n ) V ( n g 9kT (.8) This is the Kirkwood-Frohlich equation [38,39]. It is identical to the Onsager s equation when g = 1.

18 . Dielectric Theories Theories of dynamic permittivity It deals with the kinetic properties of the molecules of dielectrics. The phenomenon of dielectric relaxation takes place when a dielectric material is exposed to an alternating electric field. When a dielectric material is placed in an alternating electric field, the molecular dipoles have to change their orientations in order to follow the field. At low frequencies, the dipoles can follow the field i.e. the orientation polarization can attain the equilibrium so the dielectric permittivity will be high. As the frequency of the applied field is increased, motion of the dipoles also increases. A momentary delay or phase lag between the applied electric field and orientation of the dipoles develops due to the intermolecular forces. The momentary delay or the lag in the polarization is called dielectric relaxation. Due to this lag, the dielectric material draws energy from the source of electricity which is dissipated as heat. There is less energy storage and increase in dielectric losses. So, the dielectric permittivity goes on decreasing and the dielectric loss goes on increasing with increase in frequency of the applied electric field. When the frequency of the applied field is sufficiently high, a peak occurs in the dielectric loss. Here the molecular dipoles rotate as fast as they can. The energy is transferred to the material and lost to the field at the fastest rate. The frequency corresponding to the dielectric loss peak is called the relaxation frequency. Above the relaxation frequency, the time period of the applied electric field is smaller than the relaxation time of the molecular dipoles. So, the orientations of the molecular dipoles are not influenced by the applied field and they remain

19 . Dielectric Theories 7 randomly oriented. The orientation polarization disappears and both the dielectric permittivity and the dielectric loss fall off. The variation in dielectric permittivity and dielectric loss with the variation in frequency over a broadband of frequencies of the applied electric field is called dielectric dispersion. Also, the dielectric permittivity of a dielectric falls with increase in temperature because rise in temperature causes increase in disorder of the molecular dipoles. The interactions between a macroscopic material and electromagnetic fields can be generally described by Maxwell s equations:. D (.9). B (.3) D H J (.31) t E B t (.3) where D is the dielectric displacement, is the charge density, H is the magnetic field, J is the current density, E is the electric field and B is the magnetic induction. For small electric field strengths, D can be expressed as: D * E (.33)

20 . Dielectric Theories 8 * where is the complex relative permittivity. A dielectric materials polarization does not respond instantaneously to the applied electric field. There is a phase difference between the applied electric field E(t) and the resulting dielectric displacement D (t). So, permittivity is represented as a complex function of angular frequency () of the applied electric field. The complex relative permittivity can be expressed as: * ( ) ( ) j ( ) (.34) The real part () is the dielectric constant or the permittivity which measures the energy stored in the dielectric material and the imaginary part () is the dielectric loss. The complex permittivity is related to the complex refractive index according to the Maxwell s equations as: * * (n ) (.35) Therefore, dielectric relaxation spectroscopy can be treated as the continuation of the optical spectroscopy but at lower frequencies [3]. When an electric field is applied to a polar substance, the polar molecules will align in the direction of the applied filed. When the electric field is switched off, the molecules do not return to the random orientation immediately. The time taken by dipoles to adopt the random orientation when the applied electric field is switched off is called relaxation time. The relation between relaxation time (), angular frequency for relaxation ( c ) and relaxation frequency (f c ) is: 1 1 (.36) c f c

21 . Dielectric Theories 9 The relaxation time is the most important parameter in dielectric spectroscopy since rotational motion of each dipole represents a characteristic relaxation time. It depends upon the nature of the functional group, volume of the molecule, temperature etc. The functional groups capable of hydrogen bonding have a strong influence on the relaxation time. So, the relaxation properties of mixtures provide an insight into the nature of the mixing process. If the solute and solvent mix well at molecular level a single relaxation time is exhibited at the average position. On the other hand, if the solute and solvent do not mix well at the molecular level, two distinct relaxation times are observed and they do not differ very much from those in the pure forms. This suggests that there is formation of aggregates which are micro assemblies of like molecules. The molecular relaxation times influence the dielectric parameters which are critical in the rate of heating in a microwave cavity. The relaxation time may involve a whole molecule or a functional group attached to a large molecule. It depends upon the intermolecular forces and the molecular size. The relaxation time varies inversely with temperature since all type of molecular movements become faster at higher temperatures. The dielectric relaxation behavior of polar liquids can be described with different models such as Debye model, Cole-Cole model, Cole-Davidson model and Havriliak-Negami model Debye model Debye model is suitable for the materials that exhibit a single relaxation time. The complex permittivity at angular frequency is given by the Debye equation [35]:

22 . Dielectric Theories 3 * 1 j (.37) where is the high frequency dielectric constant, is the static dielectric constant and is the macroscopic relaxation time. The complex permittivity can be split into real part () represents the permittivity and the imaginary part () known as dielectric loss and given as: (.38) 1 ( ) 1 (.39) Fig..8. Complex permittivity spectrum (Debye model) The variation in and with frequency is shown in Fig..8. Here, the loss peak is symmetric. Cole and Cole [4] showed that for Debye relaxation, the plot of on horizontal axis and on the vertical axis yields a semicircle (Fig..9) centered at

23 . Dielectric Theories 31, and of radius. This can be shown by rearranging the equations (.38) and (.39) resulting in the equation of a circle as: (.4) The equations (.38) and (.39) shows that the dielectric loss becomes maximum when the angular frequency is 1 or 1. Therefore, ( ) max (.41) Fig..9. Cole-Cole plot for Debye type relaxation If all the points of and fall on the semicircle, the material exhibits Debye relaxation. The Debye theory assumes that the molecules are spherical in shape so there is no influence of axis of rotation of the molecule on the value of * in an external field. The Debye model is capable of analyzing the relaxation processes in gases and some liquids. In complex systems, the molecules may have different shapes and there may be interaction between the molecules. In such systems, some dipoles relax with one

24 . Dielectric Theories 3 characteristic relaxation time, some with other. If there is a continuous distribution of relaxation times then it is known as non-debye behavior. It cannot be explained by single Debye equation. Following models can be used to explain the non-debye behavior Cole-Cole model In certain polar dielectrics, such as long chain molecules, a broader loss peak is observed and the maximum value of is lower than that predicted by equation (.41). So the plot of vs. will be distorted. Cole-Cole showed that the plot will be a semi-circle but with its centre below the axis (Fig..1). Fig..1. Cole-Cole plot for Cole-Cole model K.S Cole and R.H. Cole modified the Debye equation for the complex dielectric constant as [4]: * 1 1 ( j ) (.4) where is the relaxation time distribution parameter. The value of parameter is such that < 1, increases with increase in internal degrees of freedom of

25 . Dielectric Theories 33 molecules and decreases with increase in temperature. The Cole-Cole equation describes the symmetrical distribution of relaxation times. The value of is a measure of broadening of the loss curve (width of distribution). Equation (.4) gives the Debye relaxation when is. The dielectric permittivity and loss terms in equation (.4) can be obtained as below: 1 1 ( ) 1 sin (.43) 1 1 (1 ) 1 sin ( 1 ) cos (.44) 1 (1 ) sin Fig..11. Complex permittivity plot for Cole-Cole model The plot of vs. frequency f is shown in Fig..11. In this case also, the loss curve is symmetric as in the Debye relaxation. Near the relaxation frequencies, the value of decreases more slowly with frequency in the Cole-Cole relaxation as compared with the Debye relaxation. As the value of increases, the loss curve

26 . Dielectric Theories 34 broadens and the loss peak becomes smaller than the Debye relaxation. The distribution of relaxation times is larger for the larger value of. The relaxation time corresponding to the loss peak is the mean or principal relaxation time Cole-Davidson model From the experimental results for certain materials, Davidson and Cole found that the Cole-Cole arc is not symmetrical but a skewed arc. This may be because as the chain length increases, the molecules become less rigid and can relax in more than one way. The different functional groups and also may be whole molecule may rotate. The relaxation time for the functional groups is relatively smaller than that for a whole molecule. They suggested a model, known as Cole- Davidson model, to study the asymmetric distribution of relaxation times. The equation for Cole-Davidson relaxation model is [41]: * 1 j where the shape parameter varies such as < Equation (.45) converts to Debye equation when is 1. The real and imaginary parts of the complex permittivity in equation (.45) can be expressed as: cos cos (.46) cos sin (.47) where arctan( ) (.48) Fig..1 shows a plot of vs.for the Cole-Davidson model. The arc cuts the axis at and where, corresponds to the low frequency and high frequency ends respectively.

27 . Dielectric Theories 35 Fig..1. Cole-Cole plot for Cole-Davidson model The arc becomes a semicircle as while it converts to a straight line making an angle with the axis as. The arc becomes more and more asymmetric as the value of decreases while it becomes more and more symmetric as temperature increases. Skewed arc behavior can be explained in terms of cooperative phenomenon and multiple relaxation processes [4, 43]. The variation in complex permittivity with the frequency of applied electric field is plotted in Fig. (.13). It can be seen that the loss curve is asymmetric near the relaxation frequency.. Fig..13. Complex permittivity plot for Cole-Davidson behavior

28 . Dielectric Theories Havriliak-Negami model The dielectric behavior of more complicated molecular structures, such as polymers, cannot be studied by applying the above discussed models because the dispersion in polymers is generally very broad and the shape of the arc in the Cole- Cole plot are rather complicated. Havriliak and Negami suggested a model to study the dielectric properties of polymers. The Havriliak-Negami expression is [4]: * 1 j 1 (.49) The exponent are shape parameters where describes the width and describes the asymmetry of the loss peak respectively. The values of the shape parameters are such that < 1 and < 1. The equation (.49) can describe Debye relaxation ( =, = 1), Cole-Cole relaxation ( = 1) and Cole-Davidson relaxation ( = ). A relaxation process can be completely described by the five parameters andwhich can be extracted by applying the Havriliak- Negami relaxation model. The real and imaginary parts of the complex permittivity can be separated as: 1 / ( 1 ) 1 1 sin cos( ) (.5) 1 sin( ) / (1 ) 1 1 sin (.51)

29 . Dielectric Theories 37 where arctan cos 1 1 sin (.5) The Cole-Cole plot and Complex permittivity spectrum is for Havriliak- Negami model are shown in Fig..14 and Fig..15 respectively. Fig..14. Cole-Cole plot for HN behavior Fig..15. Complex permittivity plot for HN model

30 . Dielectric Theories Dielectric parameters related to structural changes There is no exact theory to investigate the correlation between the dielectric parameters and molecular interactions as well as the structural changes in mixtures. The available theories with some assumptions can yield a trend concerning the molecular interactions and structural changes Kirkwood correlation factor Kirkwood introduced a correlation factor g [37] to describe the correlations between the orientations of a molecular dipole and its neighbors by taking into account the short range interactions. For pure liquids, the Kirkwood correlation factor can be estimated by using the Kirkwood-Frohlich equation: 4 N g 9kTM (.53) where N is the Avogadro s number, is the dipole moment in the gaseous phase, is the density, k is the Boltzmann constant, T is the absolute temperature, M is the molecular weight, is the static dielectric constant and is the dielectric constant at high frequency. The deviation of value of g from unity is a measure of short range molecular ordering due to molecular interactions. There is influence of temperature, concentration, chain length etc. on the value of g. The equation (.53) is to be modified for the binary mixture of polar liquids, say A and B, to give the effective Kirkwood correlation factor g eff. The modified Kirkwood equation for binary mixtures is [38] 4 N 9kT A A M A X A B B M B X B g eff m m m m m m (.54)

31 . Dielectric Theories 39 where X A and X B are the mole fractions of liquids A and B in the binary mixture respectively. The suffixes A, B and m represents liquid A, liquid B and binary mixture respectively. The deviation of the Kirkwood correlation factor (g or g eff ) from unity is a measure of molecular association of a molecule with its neighbors. It can be interpreted as: 1. If g eff > 1, the molecules tend to arrange themselves with parallel dipole moments.. If g eff < 1, the molecules tend to order with antiparallel dipole moments. 3. If g eff = 1, there is no specific correlation or there is non-association between the molecular dipoles..5.. Thermodynamic parameters The relaxation processes in dielectrics may be treated as transition of a dipole between two minima of energy separated by a potential barrier. Let K be the number of times per unit time a dipole acquires sufficient energy to cross two minima of energy separated by a potential barrier. According to Eyring [34], kt G K exp (.55) h RT where G is the difference in free molar enthalpy between the excited and the ground state, R is gas constant. According to laws of thermodynamics [45], G=H-TS where H is molar enthalpy of activation and S is molar entropy of activation. Therefore, the relaxation process is analogous to the chemical rate

32 . Dielectric Theories 4 process [46]. Also, the microscopic relaxation time is related to K as K=1/.Therefore, Rearranging and taking log, h H S exp exp kt RT R (.56) H S h ln( T ) ln RT R k (.57) From equation (.59), if H and S are temperature independent, the plot of log 1 (T) -1 against 1/T is a straight line with negative slope from which H can be calculated. There is no physical significance of S calculated in this way, but the order of magnitude of H can give some idea about the molecular energy involved in the relaxation process..6. Summary Different theories related to dielectrics such as Polar and Non-polar dielectrics, Dielectric polarization mechanisms are discussed. The vector sum of different polarizations gives the resultant polarization. The dielectric relaxation theories such as the theories of static permittivity and theories of dynamic permittivity are discussed. The Debye theory of static permittivity is useful to find the static dielectric constant of low density fluids but fails in case of dense fluids. Onsager modified the Debye s equation so that it can be applied in case of dense fluids. But, both these theories are inadequate as the effect of short range molecular interaction is not considered. The Kirkwood-Frohlich

33 . Dielectric Theories 41 equation for static dielectric constant was derived by assuming the effect of the local forces. The theories of dynamic permittivity explain the kinetic properties of molecules when exposed to an alternating electric field. Different models such as Debye model, Cole-Cole model, Cole-Davidson model, Havriliak-Negami model, which can be suitably applied to explain the dielectric relaxation behavior of polar liquids, are discussed. The Debye model can be applied to the materials having single relaxation time. The remaining models are useful to explain the relaxation behavior of the materials having distribution of relaxation times. The Kirkwood correlation factor and thermodynamic parameters which can throw light on molecular interaction and structural changes in mixtures are discussed.