{"id":269,"date":"2018-12-11T10:16:42","date_gmt":"2018-12-11T10:16:42","guid":{"rendered":"http:\/\/msp13.epgpbooks.inflibnet.ac.in\/?post_type=chapter&p=269"},"modified":"2018-12-11T11:49:08","modified_gmt":"2018-12-11T11:49:08","slug":"dielectric-properties-lecture-8","status":"publish","type":"chapter","link":"https:\/\/ebooks.inflibnet.ac.in\/msp13\/chapter\/dielectric-properties-lecture-8\/","title":{"rendered":"Dielectric Properties Lecture 8"},"content":{"raw":"
\r\n
\"epgp<\/a>\r\n<\/span><\/div>\r\n\u00a0 \u00a0 <\/em><\/strong>\r\n\r\n \r\n\r\n \r\n\r\nLearning Outcomes:<\/em><\/strong>\r\n\r\n \r\n\r\nFrom this module students may get to know about the following:<\/em><\/strong>\r\n\r\n1.\u00a0\u00a0\u00a0\u00a0 <\/em>Derivation of Havrilik-Nigami equation for Dielectric relaxation<\/em>\r\n\r\n2.\u00a0\u00a0\u00a0\u00a0 <\/em>Detailed study of Molecular relaxation time<\/em>\r\n\r\n3.\u00a0\u00a0\u00a0\u00a0 <\/em>You will learn about Havriliak and negami dispersion and also Frohiloch\u2019s analysis<\/em>.\r\n\r\n \r\n\r\n8.1 Havrilik-Nigami equation for Dielectric relaxation:<\/strong>\r\n\r\n \r\n\r\nThe relaxation time is a function of temperature according to a chemical process defined by<\/span>\r\n\r\n<\/div>\r\n
\r\n\r\n\"\"\r\n\r\nin which \u00a0and \u00a0are constants. It is called the Arrhenius equation also.\r\n\r\n \r\n

There is no theoretical basis for dependence of x on T and in some liquids such as those studied by Davidson and Cole (1951) the relaxation time is expressed as<\/p>\r\n\"\"\r\n\r\nWhere Tc is a characteristic temperature for a particular liquid.\r\n\r\n \r\n

At \u00a0the relaxation time is infinity according to equation (2) which must be interpreted as meaning that the relaxation process becomes infinitely slow as we approach the characteristic temperature.<\/p>\r\n \r\n

The observed correspondence of T with viscosity is qualitatively in agreement with the molecular relaxation theory of Debye who obtained the equation<\/p>\r\n\"\"\r\n

The spread in these values arises due to the fact that the molecular relaxation time, and the relaxation time, obtained from the dielectric studies may be related in several ways. The inference is that for the first two liquids the units involved are much smaller where as for the third named liquid the unit involved may be the entire molecule. These details are included here to demonstrate the method employed to obtain insight into the relaxation mechanism from measurements of dielectric properties.<\/p>\r\n \r\n

8.2 MOLECULAR RELAXATION TIME<\/strong><\/p>\r\n \r\n

Molecular relaxation time may be expressed in terms of the viscosity of the liquid and the temperature as<\/span><\/p>\r\n\r\n<\/div>\r\n

\r\n\r\n\"\"\r\n

where a<\/em> is the molecular radius. The molecular relaxation time is assumed to be due to the inner\u00a0friction of the medium that hinders the rotation of polar molecules. Hence \u00a0is a function of viscosity. As an example of applicability of equation (4) we consider water that has a viscosity of 0.01 Poise at room temperature and an effective molecular radius of 2.2 x 10-10 m leading to a relaxation time of 2.5 x 10-11s. At relaxation, the condition \u00a0is satisfied and therefore which is in reasonable agreement with relaxation time obtained from dielectric studies. Equation (4) is however expected to be valid only approximately because the internal friction hindering the rotation of the molecule, which is a molecular parameter, is equated to the viscosity, which is a macroscopic parameter.<\/p>\r\n \r\n\r\nIt is well known that the viscosity of a liquid varies with the temperature according to an empirical law:\r\n\r\n \r\n\r\n\"\"\r\n\r\n \r\n\r\nin which c is a constant for a given liquid. Therefore eq. (4) may now be expressed as\r\n\r\n \r\n\r\n\"\"\r\n\r\n \r\n\r\nThe relaxation time increases with decreasing temperature, as found in many substances.\r\n\r\n \r\n\r\n8.3 FROHLICH'S ANALYSIS:<\/strong>\r\n\r\n \r\n

To understand Frohlich's model let us take the electric field along the + axis and suppose that the dipole can orient in only two directions; one parallel to the electric field, the other anti-parallel. Let us also assume that the dipole is rigidly attached to a molecule. The energy of the dipole is +w when it is parallel to the field, -w when it is anti-parallel, and zero when it is perpendicular. As the field alternates the dipole rotates from a parallel to the anti-parallel position or vice-versa. Only two positions are allowed. A rotation through 180\u00b0 is considered as a jump.<\/p>\r\n \r\n

Frohlich generalized the model which is based on the concept that the activation energies of dipoles vary between two constant values, w1 and w2. Frohlich assumed that each process obeyed the Arrhenius relationship and the relaxation times corresponding to w1and w2 are given by<\/span><\/p>\r\n\r\n<\/div>\r\n

\r\n\r\n\"\"\r\n\r\n \r\n

The dipoles are distributed uniformly in the energy interval dw<\/em> and make a contribution to the dielectric constant according to \u00a0and. The analysis leads to fairly lengthy expressions in terms of the difference in the activation energy, <\/em>and the final equations are<\/p>\r\n \r\n\r\n\"\"\r\n\r\n\"\"\r\n\r\n<\/div>\r\n

\r\n\r\n\"\"\r\n

The temperature dependence of in the context of Frohlich's theory is often explained by assuming asymmetry in the energy level of the two positions of the dipole. At lower temperatures the lower well is occupied and the higher well remains empty. The number of dipoles jumping from the lower to higher well is zero. However, with increasing temperature the number of dipoles in the higher well increases until both wells are occupied equally at .Therefore there will be a temperature range, depending upon the energy difference, in which s\" increases strongly. According to this model the loss factor is<\/p>\r\n \r\n\r\n\"\"\r\n

where ( is the difference in the asymmetry of the two positions. Plots of \u00a0versus T for various values of V are shown in the range of V = 0 to 100 meV (fig. 2). By comparing the observed variation of vs T it is possible to estimate the average energy of asymmetry between the two wells.<\/p>\r\n\r\n<\/div>\r\n

\r\n\r\n\"\"\r\n\r\nFigure2. Temperature dependence of the relaxation strength calculated for various values of the asymmetry potential (w2 - w1)<\/span>\r\n\r\n \r\n\r\n8.4 HAVRILIAK AND NEGAMI DISPERSION:<\/strong>\r\n\r\n \r\n

HAVRILIAK and NEGAMI dispersion <\/span>is given to the more complicated structure in particular polymer materials. The complex plane plots of polymers obtained by isothermal measurements do not lend themselves to the simple treatment that is used in case of simple molecules. The main reasons for this difficulty are: (1) The dispersion in polymers is generally very broad so that data from a fixed temperature are not sufficient for analysis of the dispersion. Data from several temperatures have to be pooled to describe dispersions meaningfully. (2)<\/span><\/p>\r\n \r\n

The shapes of the plots in the complex plane are rarely as simple as that obtained with molecules of simpler structure rendering the determination of dispersion parameters very uncertain.<\/span><\/p>\r\n \r\n

In an attempt to study the -dispersion in many polymers, Havriliak and Negami have measured the dielectric properties of several polymers, -dispersion in a polymer is the process associated with the glass transition temperatures where many physical properties change in a significant way. In several <\/span>polymers the complex plane plot is linear at high frequencies and a circular arc at low frequencies. Attempts to fit a circular arc (Cole- Cole) is successful at lower frequencies but not at higher frequencies. Likewise, an attempted fit with a skewed circular arc (Davidson-Cole) is successful at higher frequencies but not at lower frequencies.<\/span><\/p>\r\n \r\n

The two dispersion equations, reproduced here for convenience, are represented by:<\/span><\/p>\r\n\r\n<\/div>\r\n

\r\n\r\n\"\"\r\n\r\n \r\n

To test the relaxation function given by equation (9) we apply successively the DeMoivre's theorem and rationalize the denominator to obtain the expressions<\/p>\r\n \r\n\r\n\"\"\r\n\r\n<\/div>\r\n

\r\n\r\n\"\"\r\n\r\n \r\n\r\nFrom equations (10) and (11) we note the following with regard to the dispersion parameter\r\n\r\n \r\n\r\n\"\"\r\n\r\n \r\n\r\nWe can <\/span>therefore evaluate <\/span>and \u00a0from the intercept of the curve with the real axis. To find the parameters and \u00a0we note that equations (10) and (11) result in the expression<\/span>\r\n\r\n<\/div>\r\n
\r\n\r\n\"\"\r\n

which provides a relation between the graphical parameter and the dispersion parameters,and. Again the relaxation time is given by the definition, and let us denote all parameters at this frequency by the subscript p. Havriliak and Negami (1966) also prove that the bisector of angle intersects the complex plane plot at. The point of intersection yields the value of by<\/p>\r\n \r\n\r\n\"\"\r\n\r\n \r\n

The analysis of experimental data to evaluate the dispersion parameters is carried out by the following procedure from the complex plane plots:<\/p>\r\n \r\n

The low frequency measurements are extrapolated to intersect the real axis from which is obtained. The high frequency measurements are extrapolated to intersect the real axis from which \u00a0is obtained. If data on refractive index is available then the relation \u00a0may be employed in specific materials.<\/p>\r\n \r\n\r\n\"\"\r\n\r\n<\/div>\r\n

Figure 3. Complex plane plot of according to H-N function. At high frequencies the plot is linear. At low frequencies the plot is circular.<\/p>\r\n \r\n

The Havriliak and Negami function is found to be very useful to describe the relaxation in amorphous polymers which exhibit asymmetrical shape near the glass transition temperature, TG. In the vicinity of<\/p>\r\n \r\n

TG the curves become broader as T is lowered. It has been suggested that the parameter represents a quantity that denotes chain connectivity and is related to the local density fluctuations. Chain connectivity in polymers should decrease as the temperature is lowered. The a-parameter slowly increases above TG which may be considered as indicative of this.<\/p>\r\n \r\n

Summary:<\/em><\/strong><\/p>\r\n \r\n

In this chapter Derivation of Havrilik-Nigami equation for Dielectric relaxation was done and detailed study of Molecular relaxation time was carried out. We have also learn about Havriliak and negami dispersion and also Frohiloch\u2019s analysis<\/em>.<\/p>\r\n \r\n\r\n\r\n\r\n
you can view video on Dielectric Properties Lecture 8<\/strong><\/td>\r\n\"\"<\/a><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n \r\n\r\nReferences:<\/em><\/strong>\r\n
    \r\n \t
  1. Werner K\u00e4nzig (1957). \"Ferroelectrics and Antiferroelectrics\". In Frederick Seitz, T. P. Das, David Turnbull, E. L. Hahn. <\/em>Solid State Physics <\/em>4. Academic Press. p. 5. <\/em>ISBN <\/em>0-12-607704-5.<\/em><\/li>\r\n \t
  2. M. Lines & A. Glass (1979). Principles and applications of ferroelectrics and related materials. Clarendon Press, Oxford.ISBN <\/em>0-19-851286-4.<\/em><\/li>\r\n \t
  3. See J.\u00a0 Valasek\u00a0 (1920).\u00a0 \"Piezoelectric\u00a0 and\u00a0 allied\u00a0 phenomena\u00a0 in\u00a0 Rochelle\u00a0 salt\". Physical\u00a0<\/em>Review 15:537.Bibcode:1920PhRv<\/em>...15..505.. <\/em>doi:10.1103\/PhysRev.15.505.and <\/em>J. Valasek\u00a0<\/em>(1921).\u00a0 \"Piezo-Electric\u00a0 and\u00a0 Allied\u00a0 Phenomena in Rochelle Salt\". Physical Review 17 (4):\u00a0<\/em>475.Bibcode:1921PhRv<\/em>...17..475V. <\/em>doi:10.1103\/PhysRev.17.475.<\/em><\/li>\r\n \t
  4. Chiang, Y. et al.: <\/em>Physical Ceramics, <\/em>John Wiley & Sons <\/em>1997, New York<\/em><\/li>\r\n \t
  5. Safari, Ahmad (2008). Piezoelectric and acoustic materials for transducer applications. Springer Science & Business Media. p. <\/em>21.ISBN <\/em>0387765409.<\/em><\/li>\r\n<\/ol>\r\n
    \r\n\r\n\u00a0 \u00a0 <\/em>References and Suggestive Readings<\/em><\/strong>\r\n
      \r\n \t
    1. \u00a0<\/em>A. S. Sidorkin (2006). Domain Structure in Ferroelectrics and Related Materials. Cambridge University Press. <\/em>ISBN 1<\/em>-904602-14-2.<\/em><\/li>\r\n \t
    2. Karin M Rabe, Jean-Marc Triscone, Charles H Ahn (2007). Physics of Ferroelectrics: A modern perspective. Springer. <\/em>ISBN 3<\/em>-540-34591-4.<\/em><\/li>\r\n \t
    3. Julio A. Gonzalo (2006). Effective Field Approach to Phase Transitions and Some Applications to Ferroelectrics. World Scientific.<\/em>ISBN 981<\/em>-256-875-1<\/li>\r\n<\/ol>\r\n\u00a0 \u00a0\u00a0<\/em>Web Links<\/em><\/strong>\r\n\r\n \r\n\r\n1.\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <\/em>A useful starter on ferroelectrics<\/em>\r\n\r\n2.\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <\/em>A ferroelectrics research group at Stony Brook University<\/em>\r\n\r\n3.\u00a0\u00a0\u00a0\u00a0\u00a0 <\/em>http:\/\/www.britannica.com\/science\/ferroelectricity<\/em>\r\n\r\n4.\u00a0\u00a0\u00a0\u00a0\u00a0 <\/em>http:\/\/www.slideshare.net\/researcher1234\/ferroelectric-and-piezoelectric-materials<\/em>\r\n\r\n<\/div>\r\n
      \r\n\r\n\u00a0\u00a0<\/em>\r\n\r\nAdditional Topics to be studied<\/em><\/strong>\r\n\r\n \r\n\r\n1) History of Ferroelectricity<\/span>\r\n\r\n \r\n\r\nHistory of Ferroelectricity<\/strong>\r\n\r\n<\/div>\r\n
      \r\n\r\n \r\n

      It all began with a salt in La Rochelle, a small but important city at the south-west coast of France. Jehan Seignette, born in 1592, a militant protestant, succeded to run a pharmacy in spite of serious obstacles opposed to him by the clerics. One of his sons, Pierre, born in 1623, became a medical doctor at the University of Montpellier and his younger brother Elie, born in 1632, took over his father\u2019s business.\u00a0 In\u00a0 these\u00a0 years\u00a0 pharmacy\u00a0 consisted\u00a0 mainly\u00a0 in\u00a0 extracting\u00a0 plants\u00a0 and\u00a0 distilling\u00a0 essences.\u00a0Apparently purgatives such as \u201cfolia sennae\u201d introduced to Europe by Arab medical men in the early middle age, played an important role. Because of unpleasent side effects patients were very reluctant to take them. This was the reason Dr. Seignette suggested to his brother to look for some mineral drugs or to make some. Although \u201cmineralia\u201d were in use for curing various diseases in eastern countries already 2000 years B.C., Pierres idea was decisive. As the result of hard work Elie came out with a salt in approximately 1665, which he called \u201csel polychreste\u201d derived from the greek \u03c0\u03bf\u03bb\u03c5\u03c7\u03c1\u03b7\u03c3\u03c4\u03bf\u03c3<\/em>, which means a salt of various utilities. It was a real creation and the way he produced the salt was kept secret for ever it seems. Only 65 years later the French pharmacist and chemist Simon Boulduc in Paris found out by analysis that sel polychreste must be \u201csome soda\u201d. It is likely that Elie Seignette started from cr\u00e8me of tartrate (potassium hydrogen tartrate) he obtained from wines \u2013 so famous and abound in the Bordeaux region \u2013 and soda which makes the tartrate soluble in water. The \u201csel polychreste\u201d \u2013 or Rochelle salt as we know it today - conquered the market in France, especially in Paris and it was in widespread use for more than two centuries as a mild drug.<\/p>\r\n \r\n

      In the nineteenth century \u2013 nearly 200 years after its discovery - the physical properties of Rochelle salt began to excite interest. In 1824 David Brewster had observed the phenomenon of pyroelectricity in various crystals, among which was Rochelle Salt but perhaps the first systematic studies were those of the brothers Pierre and Paul-Jacques Curie in 1880. This classic work established unequivocally the existence of the piezoelectric effect and correctly identified Rochelle Salt and a number of other crystals as being piezoelectric. They also noticed that Rochelle salt was by far more active than quartz for instance and all the rest of the crystals they investigated. But the fascinating dielectric features of Rochelle salt escaped them. Thomas Alva Edison was maybe the first who used its piezoelectrical effect in a commercial application in 1899 \u2013 the phonograph. However, his invention was just a curiosity and far too expensive. At this time Rochelle salt was of pure academic significance. During World War I, however, physicists and electrical engineers showed an increasing interest in its physical properties mainlybecause of its unusually high piezoelectric moduli. At the beginning of the war 1914-18 A.M. Nicholson in the USA and Paul Langevin in France began to perfect independently an ultrasonic submarine detector. Their transducers were very similar: a mosaic of thin quartz crystals glued between two steel plates (the composite having a resonant frequency of about 50 KHz), mounted in a housing suitable for submersion. Working on past the end of the war, they did achieve their goal of emitting a high frequency \"chirp\" underwater and measuring depth by timing the return echo. The strategic importance of their achievement was not overlooked by any industrial nation, however, and since that time the development of sonar transducers, circuits, systems, and materials has never ceased.<\/p>\r\n \r\n

      Petrus Josephus Wilhelmus Debye, or Peter Debye as we know him today, professor of <\/span>theoretical Physics at the University of Z\u00fcrich had carefully observed the work on piezoelectricity and in 1912 he came up with an idea. To explain the results he knew he brought forth the hypothesis that a certain class of molecules <\/span>carry a <\/span>permanent electric dipole moment<\/em> in analogy to the magnetic moment of the atoms of paramagnetic substances. Following Langevin\u2019s theory of paramagnetism Debye gave the equation <\/span>(\u03b5-1)\/(\u03b5+2)=a+b\/T<\/em>, where <\/span>a<\/em> is proportional to the density of the substance and <\/span>b<\/em> to the square of the electric dipole moment. This relation was perfectly confirmed later in many cases.<\/span><\/p>\r\n \r\n

      Debye came to a further conclusion. According to his relation for a critical temperature <\/span>TK=b\/(1-a)<\/em> the dielectric constant reaches infinity. Therefore, he proposed <\/span>TK<\/em> to be the <\/span>analogue to the Curie temperature of a ferromagnet. For temperatures lower than <\/span>TK<\/em> a <\/span>permanent dielectric<\/em> polarisation ought to be expected even in the absence of an electric field. To his knowledge, he said, no such a phenomenon had been observed so far. The basic feature of <\/span>ferroelectricity<\/em> was anticipated, however! Erwin Schr\u00f6dinger in his \u201cHabilitations-Schrift\u201d submitted at the<\/span><\/p>\r\n\r\n<\/div>\r\n

      University of Vienna late in 1912, the year Debye published his \u201eVorl\u00e4ufige Mitteilung\u201c, went a step further. He elaborated on Debye\u2019s simple model and tried to extend it to solids. If this could be done successfully, Schr\u00f6dinger speculated, then all solids should become \u201cferroelektrisch\u201d at a sufficiently low temperature. So, in fact, the term ferroelectric or ferroelectricity was coined by Schr\u00f6dinger as early as 1912!<\/p>","rendered":"

      \n
      \"epgp<\/a>
      \n<\/span><\/div>\n

      \u00a0 \u00a0 <\/em><\/strong><\/p>\n

       <\/p>\n

       <\/p>\n

      Learning Outcomes:<\/em><\/strong><\/p>\n

       <\/p>\n

      From this module students may get to know about the following:<\/em><\/strong><\/p>\n

      1.\u00a0\u00a0\u00a0\u00a0 <\/em>Derivation of Havrilik-Nigami equation for Dielectric relaxation<\/em><\/p>\n

      2.\u00a0\u00a0\u00a0\u00a0 <\/em>Detailed study of Molecular relaxation time<\/em><\/p>\n

      3.\u00a0\u00a0\u00a0\u00a0 <\/em>You will learn about Havriliak and negami dispersion and also Frohiloch\u2019s analysis<\/em>.<\/p>\n

       <\/p>\n

      8.1 Havrilik-Nigami equation for Dielectric relaxation:<\/strong><\/p>\n

       <\/p>\n

      The relaxation time is a function of temperature according to a chemical process defined by<\/span><\/p>\n<\/div>\n

      \n

      \"\"<\/p>\n

      in which \u00a0and \u00a0are constants. It is called the Arrhenius equation also.<\/p>\n

       <\/p>\n

      There is no theoretical basis for dependence of x on T and in some liquids such as those studied by Davidson and Cole (1951) the relaxation time is expressed as<\/p>\n

      \"\"<\/p>\n

      Where Tc is a characteristic temperature for a particular liquid.<\/p>\n

       <\/p>\n

      At \u00a0the relaxation time is infinity according to equation (2) which must be interpreted as meaning that the relaxation process becomes infinitely slow as we approach the characteristic temperature.<\/p>\n

       <\/p>\n

      The observed correspondence of T with viscosity is qualitatively in agreement with the molecular relaxation theory of Debye who obtained the equation<\/p>\n

      \"\"<\/p>\n

      The spread in these values arises due to the fact that the molecular relaxation time, and the relaxation time, obtained from the dielectric studies may be related in several ways. The inference is that for the first two liquids the units involved are much smaller where as for the third named liquid the unit involved may be the entire molecule. These details are included here to demonstrate the method employed to obtain insight into the relaxation mechanism from measurements of dielectric properties.<\/p>\n

       <\/p>\n

      8.2 MOLECULAR RELAXATION TIME<\/strong><\/p>\n

       <\/p>\n

      Molecular relaxation time may be expressed in terms of the viscosity of the liquid and the temperature as<\/span><\/p>\n<\/div>\n

      \n

      \"\"<\/p>\n

      where a<\/em> is the molecular radius. The molecular relaxation time is assumed to be due to the inner\u00a0friction of the medium that hinders the rotation of polar molecules. Hence \u00a0is a function of viscosity. As an example of applicability of equation (4) we consider water that has a viscosity of 0.01 Poise at room temperature and an effective molecular radius of 2.2 x 10-10 m leading to a relaxation time of 2.5 x 10-11s. At relaxation, the condition \u00a0is satisfied and therefore which is in reasonable agreement with relaxation time obtained from dielectric studies. Equation (4) is however expected to be valid only approximately because the internal friction hindering the rotation of the molecule, which is a molecular parameter, is equated to the viscosity, which is a macroscopic parameter.<\/p>\n

       <\/p>\n

      It is well known that the viscosity of a liquid varies with the temperature according to an empirical law:<\/p>\n

       <\/p>\n

      \"\"<\/p>\n

       <\/p>\n

      in which c is a constant for a given liquid. Therefore eq. (4) may now be expressed as<\/p>\n

       <\/p>\n

      \"\"<\/p>\n

       <\/p>\n

      The relaxation time increases with decreasing temperature, as found in many substances.<\/p>\n

       <\/p>\n

      8.3 FROHLICH’S ANALYSIS:<\/strong><\/p>\n

       <\/p>\n

      To understand Frohlich’s model let us take the electric field along the + axis and suppose that the dipole can orient in only two directions; one parallel to the electric field, the other anti-parallel. Let us also assume that the dipole is rigidly attached to a molecule. The energy of the dipole is +w when it is parallel to the field, -w when it is anti-parallel, and zero when it is perpendicular. As the field alternates the dipole rotates from a parallel to the anti-parallel position or vice-versa. Only two positions are allowed. A rotation through 180\u00b0 is considered as a jump.<\/p>\n

       <\/p>\n

      Frohlich generalized the model which is based on the concept that the activation energies of dipoles vary between two constant values, w1 and w2. Frohlich assumed that each process obeyed the Arrhenius relationship and the relaxation times corresponding to w1and w2 are given by<\/span><\/p>\n<\/div>\n

      \n

      \"\"<\/p>\n

       <\/p>\n

      The dipoles are distributed uniformly in the energy interval dw<\/em> and make a contribution to the dielectric constant according to \u00a0and. The analysis leads to fairly lengthy expressions in terms of the difference in the activation energy, <\/em>and the final equations are<\/p>\n

       <\/p>\n

      \"\"<\/p>\n

      \"\"<\/p>\n<\/div>\n

      \n

      \"\"<\/p>\n

      The temperature dependence of in the context of Frohlich’s theory is often explained by assuming asymmetry in the energy level of the two positions of the dipole. At lower temperatures the lower well is occupied and the higher well remains empty. The number of dipoles jumping from the lower to higher well is zero. However, with increasing temperature the number of dipoles in the higher well increases until both wells are occupied equally at .Therefore there will be a temperature range, depending upon the energy difference, in which s” increases strongly. According to this model the loss factor is<\/p>\n

       <\/p>\n

      \"\"<\/p>\n

      where ( is the difference in the asymmetry of the two positions. Plots of \u00a0versus T for various values of V are shown in the range of V = 0 to 100 meV (fig. 2). By comparing the observed variation of vs T it is possible to estimate the average energy of asymmetry between the two wells.<\/p>\n<\/div>\n

      \n

      \"\"<\/p>\n

      Figure2. Temperature dependence of the relaxation strength calculated for various values of the asymmetry potential (w2 – w1)<\/span><\/p>\n

       <\/p>\n

      8.4 HAVRILIAK AND NEGAMI DISPERSION:<\/strong><\/p>\n

       <\/p>\n

      HAVRILIAK and NEGAMI dispersion <\/span>is given to the more complicated structure in particular polymer materials. The complex plane plots of polymers obtained by isothermal measurements do not lend themselves to the simple treatment that is used in case of simple molecules. The main reasons for this difficulty are: (1) The dispersion in polymers is generally very broad so that data from a fixed temperature are not sufficient for analysis of the dispersion. Data from several temperatures have to be pooled to describe dispersions meaningfully. (2)<\/span><\/p>\n

       <\/p>\n

      The shapes of the plots in the complex plane are rarely as simple as that obtained with molecules of simpler structure rendering the determination of dispersion parameters very uncertain.<\/span><\/p>\n

       <\/p>\n

      In an attempt to study the -dispersion in many polymers, Havriliak and Negami have measured the dielectric properties of several polymers, -dispersion in a polymer is the process associated with the glass transition temperatures where many physical properties change in a significant way. In several <\/span>polymers the complex plane plot is linear at high frequencies and a circular arc at low frequencies. Attempts to fit a circular arc (Cole- Cole) is successful at lower frequencies but not at higher frequencies. Likewise, an attempted fit with a skewed circular arc (Davidson-Cole) is successful at higher frequencies but not at lower frequencies.<\/span><\/p>\n

       <\/p>\n

      The two dispersion equations, reproduced here for convenience, are represented by:<\/span><\/p>\n<\/div>\n

      \n

      \"\"<\/p>\n

       <\/p>\n

      To test the relaxation function given by equation (9) we apply successively the DeMoivre’s theorem and rationalize the denominator to obtain the expressions<\/p>\n

       <\/p>\n

      \"\"<\/p>\n<\/div>\n

      \n

      \"\"<\/p>\n

       <\/p>\n

      From equations (10) and (11) we note the following with regard to the dispersion parameter<\/p>\n

       <\/p>\n

      \"\"<\/p>\n

       <\/p>\n

      We can <\/span>therefore evaluate <\/span>and \u00a0from the intercept of the curve with the real axis. To find the parameters and \u00a0we note that equations (10) and (11) result in the expression<\/span><\/p>\n<\/div>\n

      \n

      \"\"<\/p>\n

      which provides a relation between the graphical parameter and the dispersion parameters,and. Again the relaxation time is given by the definition, and let us denote all parameters at this frequency by the subscript p. Havriliak and Negami (1966) also prove that the bisector of angle intersects the complex plane plot at. The point of intersection yields the value of by<\/p>\n

       <\/p>\n

      \"\"<\/p>\n

       <\/p>\n

      The analysis of experimental data to evaluate the dispersion parameters is carried out by the following procedure from the complex plane plots:<\/p>\n

       <\/p>\n

      The low frequency measurements are extrapolated to intersect the real axis from which is obtained. The high frequency measurements are extrapolated to intersect the real axis from which \u00a0is obtained. If data on refractive index is available then the relation \u00a0may be employed in specific materials.<\/p>\n

       <\/p>\n

      \"\"<\/p>\n<\/div>\n

      Figure 3. Complex plane plot of according to H-N function. At high frequencies the plot is linear. At low frequencies the plot is circular.<\/p>\n

       <\/p>\n

      The Havriliak and Negami function is found to be very useful to describe the relaxation in amorphous polymers which exhibit asymmetrical shape near the glass transition temperature, TG. In the vicinity of<\/p>\n

       <\/p>\n

      TG the curves become broader as T is lowered. It has been suggested that the parameter represents a quantity that denotes chain connectivity and is related to the local density fluctuations. Chain connectivity in polymers should decrease as the temperature is lowered. The a-parameter slowly increases above TG which may be considered as indicative of this.<\/p>\n

       <\/p>\n

      Summary:<\/em><\/strong><\/p>\n

       <\/p>\n

      In this chapter Derivation of Havrilik-Nigami equation for Dielectric relaxation was done and detailed study of Molecular relaxation time was carried out. We have also learn about Havriliak and negami dispersion and also Frohiloch\u2019s analysis<\/em>.<\/p>\n

       <\/p>\n\n\n\n
      you can view video on Dielectric Properties Lecture 8<\/strong><\/td>\n\"\"<\/a><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

       <\/p>\n

      References:<\/em><\/strong><\/p>\n

        \n
      1. Werner K\u00e4nzig (1957). “Ferroelectrics and Antiferroelectrics”. In Frederick Seitz, T. P. Das, David Turnbull, E. L. Hahn. <\/em>Solid State Physics <\/em>4. Academic Press. p. 5. <\/em>ISBN <\/em>0-12-607704-5.<\/em><\/li>\n
      2. M. Lines & A. Glass (1979). Principles and applications of ferroelectrics and related materials. Clarendon Press, Oxford.ISBN <\/em>0-19-851286-4.<\/em><\/li>\n
      3. See J.\u00a0 Valasek\u00a0 (1920).\u00a0 “Piezoelectric\u00a0 and\u00a0 allied\u00a0 phenomena\u00a0 in\u00a0 Rochelle\u00a0 salt”. Physical\u00a0<\/em>Review 15:537.Bibcode:1920PhRv<\/em>…15..505.. <\/em>doi:10.1103\/PhysRev.15.505.and <\/em>J. Valasek\u00a0<\/em>(1921).\u00a0 “Piezo-Electric\u00a0 and\u00a0 Allied\u00a0 Phenomena in Rochelle Salt”. Physical Review 17 (4):\u00a0<\/em>475.Bibcode:1921PhRv<\/em>…17..475V. <\/em>doi:10.1103\/PhysRev.17.475.<\/em><\/li>\n
      4. Chiang, Y. et al.: <\/em>Physical Ceramics, <\/em>John Wiley & Sons <\/em>1997, New York<\/em><\/li>\n
      5. Safari, Ahmad (2008). Piezoelectric and acoustic materials for transducer applications. Springer Science & Business Media. p. <\/em>21.ISBN <\/em>0387765409.<\/em><\/li>\n<\/ol>\n
        \n

        \u00a0 \u00a0 <\/em>References and Suggestive Readings<\/em><\/strong><\/p>\n

          \n
        1. \u00a0<\/em>A. S. Sidorkin (2006). Domain Structure in Ferroelectrics and Related Materials. Cambridge University Press. <\/em>ISBN 1<\/em>-904602-14-2.<\/em><\/li>\n
        2. Karin M Rabe, Jean-Marc Triscone, Charles H Ahn (2007). Physics of Ferroelectrics: A modern perspective. Springer. <\/em>ISBN 3<\/em>-540-34591-4.<\/em><\/li>\n
        3. Julio A. Gonzalo (2006). Effective Field Approach to Phase Transitions and Some Applications to Ferroelectrics. World Scientific.<\/em>ISBN 981<\/em>-256-875-1<\/li>\n<\/ol>\n

          \u00a0 \u00a0\u00a0<\/em>Web Links<\/em><\/strong><\/p>\n

           <\/p>\n

          1.\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <\/em>A useful starter on ferroelectrics<\/em><\/p>\n

          2.\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <\/em>A ferroelectrics research group at Stony Brook University<\/em><\/p>\n

          3.\u00a0\u00a0\u00a0\u00a0\u00a0 <\/em>http:\/\/www.britannica.com\/science\/ferroelectricity<\/em><\/p>\n

          4.\u00a0\u00a0\u00a0\u00a0\u00a0 <\/em>http:\/\/www.slideshare.net\/researcher1234\/ferroelectric-and-piezoelectric-materials<\/em><\/p>\n<\/div>\n

          \n

          \u00a0\u00a0<\/em><\/p>\n

          Additional Topics to be studied<\/em><\/strong><\/p>\n

           <\/p>\n

          1) History of Ferroelectricity<\/span><\/p>\n

           <\/p>\n

          History of Ferroelectricity<\/strong><\/p>\n<\/div>\n

          \n

           <\/p>\n

          It all began with a salt in La Rochelle, a small but important city at the south-west coast of France. Jehan Seignette, born in 1592, a militant protestant, succeded to run a pharmacy in spite of serious obstacles opposed to him by the clerics. One of his sons, Pierre, born in 1623, became a medical doctor at the University of Montpellier and his younger brother Elie, born in 1632, took over his father\u2019s business.\u00a0 In\u00a0 these\u00a0 years\u00a0 pharmacy\u00a0 consisted\u00a0 mainly\u00a0 in\u00a0 extracting\u00a0 plants\u00a0 and\u00a0 distilling\u00a0 essences.\u00a0Apparently purgatives such as \u201cfolia sennae\u201d introduced to Europe by Arab medical men in the early middle age, played an important role. Because of unpleasent side effects patients were very reluctant to take them. This was the reason Dr. Seignette suggested to his brother to look for some mineral drugs or to make some. Although \u201cmineralia\u201d were in use for curing various diseases in eastern countries already 2000 years B.C., Pierres idea was decisive. As the result of hard work Elie came out with a salt in approximately 1665, which he called \u201csel polychreste\u201d derived from the greek \u03c0\u03bf\u03bb\u03c5\u03c7\u03c1\u03b7\u03c3\u03c4\u03bf\u03c3<\/em>, which means a salt of various utilities. It was a real creation and the way he produced the salt was kept secret for ever it seems. Only 65 years later the French pharmacist and chemist Simon Boulduc in Paris found out by analysis that sel polychreste must be \u201csome soda\u201d. It is likely that Elie Seignette started from cr\u00e8me of tartrate (potassium hydrogen tartrate) he obtained from wines \u2013 so famous and abound in the Bordeaux region \u2013 and soda which makes the tartrate soluble in water. The \u201csel polychreste\u201d \u2013 or Rochelle salt as we know it today – conquered the market in France, especially in Paris and it was in widespread use for more than two centuries as a mild drug.<\/p>\n

           <\/p>\n

          In the nineteenth century \u2013 nearly 200 years after its discovery – the physical properties of Rochelle salt began to excite interest. In 1824 David Brewster had observed the phenomenon of pyroelectricity in various crystals, among which was Rochelle Salt but perhaps the first systematic studies were those of the brothers Pierre and Paul-Jacques Curie in 1880. This classic work established unequivocally the existence of the piezoelectric effect and correctly identified Rochelle Salt and a number of other crystals as being piezoelectric. They also noticed that Rochelle salt was by far more active than quartz for instance and all the rest of the crystals they investigated. But the fascinating dielectric features of Rochelle salt escaped them. Thomas Alva Edison was maybe the first who used its piezoelectrical effect in a commercial application in 1899 \u2013 the phonograph. However, his invention was just a curiosity and far too expensive. At this time Rochelle salt was of pure academic significance. During World War I, however, physicists and electrical engineers showed an increasing interest in its physical properties mainlybecause of its unusually high piezoelectric moduli. At the beginning of the war 1914-18 A.M. Nicholson in the USA and Paul Langevin in France began to perfect independently an ultrasonic submarine detector. Their transducers were very similar: a mosaic of thin quartz crystals glued between two steel plates (the composite having a resonant frequency of about 50 KHz), mounted in a housing suitable for submersion. Working on past the end of the war, they did achieve their goal of emitting a high frequency “chirp” underwater and measuring depth by timing the return echo. The strategic importance of their achievement was not overlooked by any industrial nation, however, and since that time the development of sonar transducers, circuits, systems, and materials has never ceased.<\/p>\n

           <\/p>\n

          Petrus Josephus Wilhelmus Debye, or Peter Debye as we know him today, professor of <\/span>theoretical Physics at the University of Z\u00fcrich had carefully observed the work on piezoelectricity and in 1912 he came up with an idea. To explain the results he knew he brought forth the hypothesis that a certain class of molecules <\/span>carry a <\/span>permanent electric dipole moment<\/em> in analogy to the magnetic moment of the atoms of paramagnetic substances. Following Langevin\u2019s theory of paramagnetism Debye gave the equation <\/span>(\u03b5-1)\/(\u03b5+2)=a+b\/T<\/em>, where <\/span>a<\/em> is proportional to the density of the substance and <\/span>b<\/em> to the square of the electric dipole moment. This relation was perfectly confirmed later in many cases.<\/span><\/p>\n

           <\/p>\n

          Debye came to a further conclusion. According to his relation for a critical temperature <\/span>TK=b\/(1-a)<\/em> the dielectric constant reaches infinity. Therefore, he proposed <\/span>TK<\/em> to be the <\/span>analogue to the Curie temperature of a ferromagnet. For temperatures lower than <\/span>TK<\/em> a <\/span>permanent dielectric<\/em> polarisation ought to be expected even in the absence of an electric field. To his knowledge, he said, no such a phenomenon had been observed so far. The basic feature of <\/span>ferroelectricity<\/em> was anticipated, however! Erwin Schr\u00f6dinger in his \u201cHabilitations-Schrift\u201d submitted at the<\/span><\/p>\n<\/div>\n

          University of Vienna late in 1912, the year Debye published his \u201eVorl\u00e4ufige Mitteilung\u201c, went a step further. He elaborated on Debye\u2019s simple model and tried to extend it to solids. If this could be done successfully, Schr\u00f6dinger speculated, then all solids should become \u201cferroelektrisch\u201d at a sufficiently low temperature. So, in fact, the term ferroelectric or ferroelectricity was coined by Schr\u00f6dinger as early as 1912!<\/p>\n","protected":false},"author":3,"menu_order":9,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":["dr-k-asokan"],"pb_section_license":""},"chapter-type":[],"contributor":[58],"license":[],"class_list":["post-269","chapter","type-chapter","status-publish","hentry","contributor-dr-k-asokan"],"part":3,"_links":{"self":[{"href":"https:\/\/ebooks.inflibnet.ac.in\/msp13\/wp-json\/pressbooks\/v2\/chapters\/269","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/ebooks.inflibnet.ac.in\/msp13\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/ebooks.inflibnet.ac.in\/msp13\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/ebooks.inflibnet.ac.in\/msp13\/wp-json\/wp\/v2\/users\/3"}],"version-history":[{"count":5,"href":"https:\/\/ebooks.inflibnet.ac.in\/msp13\/wp-json\/pressbooks\/v2\/chapters\/269\/revisions"}],"predecessor-version":[{"id":293,"href":"https:\/\/ebooks.inflibnet.ac.in\/msp13\/wp-json\/pressbooks\/v2\/chapters\/269\/revisions\/293"}],"part":[{"href":"https:\/\/ebooks.inflibnet.ac.in\/msp13\/wp-json\/pressbooks\/v2\/parts\/3"}],"metadata":[{"href":"https:\/\/ebooks.inflibnet.ac.in\/msp13\/wp-json\/pressbooks\/v2\/chapters\/269\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/ebooks.inflibnet.ac.in\/msp13\/wp-json\/wp\/v2\/media?parent=269"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/ebooks.inflibnet.ac.in\/msp13\/wp-json\/pressbooks\/v2\/chapter-type?post=269"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/ebooks.inflibnet.ac.in\/msp13\/wp-json\/wp\/v2\/contributor?post=269"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/ebooks.inflibnet.ac.in\/msp13\/wp-json\/wp\/v2\/license?post=269"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}