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Polarization in Dielectrics | Applied Physics - II | DielectricsAyush Agarwal

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3515_Ch 6_Dielectric Properties of Materials_ M A IslamDr. Mohammad Aminul Islam

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- 1. Dielectrics By:Mayank Pandey VIT University, Vellore
- 2. introDuction Dielectrics are the materials having electric dipole moment permantly Dipole: A dipole is an entity in which equal positive and negative charges are separated by a small distance.. DIPOLE moment (µele ): The product of magnitude of either of the charges and separation distance b/w them is called Dipole moment. µe = q . x coul – m +q -q X All dielectrics are electrical insulators and they are mainly used to store electrical energy. It stores with minimum dissipation power). Since, the e- are bound to their parent molecules & hence, there is no free charge Ex: Mica, glass, plastic, water & polar molecules…
- 4. Important terms in dielectrics 1) Electric intensity or electric field strength Def:- The force per unit charge “dq” is known as electric field strength (E). Where “dq” is point charge , E is electric field, F is force applied on point charge “dq”. E= F/dq = Q / 4πεr2 where “ε” is permittivity. What is permittivity? It is a measure of resistance that is encountered when forming an electric field in a medium. “ In simple words permittivity is a measure of how an electric field effects and is effected by a dielectric medium”.
- 5. a) ε (permittivity of medium):- How much electric field generated per unit charge in that medium. b) ε0 (permittivity of space) :- The electric field generated in vacuum. It is constant value ε0=8.85 x 10-12 F/m. Imp points:1) More electric flux exist in a medium with a high permittivity(because of polarization). 2) Permittivity is directly related to “ Susceptibility” which is a measure of how easily a dielectric polarize in a response of an electric field. “ permittivity relates to a materials ability to transmit an electric field” ε = εr . ε0 = (1+χ ) ε0 Relative permittivity Susceptibility
- 6. 2) Electric Flux density or Electric displacement Vector:The electric flux density or electric displacement vector “D” is the number of flux line’s crossing a surface normal to the lines, divided by the surface area. D = Q/ 4π r2 where , 4π r2 is the surface area of the sphere of radius “r”. 3) Dielectric Parameters :a) Dielectric constant(εr ):- It is defined as the ratio of permittivity of medium(ε) to the permittivity of free space(ε0). εr = ε / ε0 b) Electric dipole moment (μ):- The product of magnitude of charges & distance of separation is known as electric dipole moment (μ ). μ = Q.r
- 7. c) Electric Polarization :- The process of producing electric dipoles by an electric field is called polarization in dielectrics. “ In simple words polarization P is defined as the dipole moment per unit volume averaged over the volume of a cell” P = μ / volume d) Polarizability :- When a dielectric material is placed in an electric field, the displacement of electric charge gives rise to the creation of dipole in the material . The polarization P of an elementary particle is directly proportional to the electric field strength E. P∝E P = αE α → polarizability constant The unit of “α” is Fm2
- 8. Electric susceptibility:• The polarization vector P is proportional to the total electric flux density and direction of electric field. Therefore the polarization vector can be written P = ε 0 χe E P χe = ε0E ε 0 (ε r − 1) E = ε0E χe = ε r −1
- 9. Various polarization processes: 1. Electronic polarization 2. Ionic polarization 3. Orientation polarization 4. Space charge polarization
- 10. 1) Electronic Polarization:- When an EF is applied to an atom, +vely charged nucleus displaces in the direction of field and ẽ could in opposite direction. This kind of displacement will produce an electric dipole with in the atom i.e, dipole moment is proportional to the magnitude of field strength. This displacement between electron and nucleus produces induced dipole moment & hence polarization. This is called electronic polarization. It increases with increase of volume of the atom. This kind of polarization is mostly exhibited in Monatomic gases . It occurs only at optical frequencies (1015Hz). It is independent of temperature. He α e = ____ × 10-40 F − m 2 Ne Ar 0.18 0.35 1.46 Kr Xe 2.18 3.54
- 11. Expression for Electronic Polarization :Consider a atom in an EF of intensity ‘E’ since the nucleus (+ze) and electron cloud (-ze) of the atom have opposite charges and acted upon by Lorentz force (FL). Subsequently nucleus moves in the direction of field and electron cloud in opposite direction. When electron cloud and nucleus get shifted from their normal positions, an attractive force b/w them is created and the separation continuous until columbic force F C is balanced with Lorentz force FL, Finally a new equilibriums state is established.
- 12. E x +Ze R R No field fig(1) In the presence of field fig (2) fig(2) represents displacement of nucleus and electron cloud and we assume that the –ve charge in the cloud uniformly distributed over a sphere of radius R and the spherical shape does not change for convenience.
- 13. Let “σ” be the charge density of the sphere − Ze σ= 4 3 πR 3 - Ze represents the total charge in the sphere. Thus the charge inside the sphere is 4 q e ⇒ σ . π .x 3 3 − ze 4 ⇒ 4 .π .x 3 .πR 3 3 3 ( = − ze 3 x 3 R ) - - - - - (1) 1 qe .q p 1 − ze.x 3 − z 2e 2 x ( ze ) = Now Fc = . 2 = - - - - - (2) 2 3 3 4πε 0 x 4πε 0 x R 4πε 0 R “Where Fc is a coulomb force acting between nucleus and electron”
- 14. Force experienced by displaced nucleus in EF of Strength E is FL = Eq = ZeE -----(3) Fc = Fl − z 2e 2 x 4πε 0 R 3 = ZeE - - - - - (4) − zex =E 4πε 0 R 3 − zex − zex = 4πε 0 R 3 αe α e = 4πε 0 R 3 ∴αe = 4πε0 R 3 Hence electronic Polarisability is directly proportional to cube of the radius of the atom.
- 15. 2) Ionic polarization The ionic polarization occurs, when atoms form molecules and it is mainly due to a relative displacement of the atomic components of the molecule in the presence of an electric field. When a EF is applied to the molecule, the positive ions displaced by X1 to the negative side electric field and negative ions displaced by X2 to the positive side of field. The resultant dipole moment µ = q ( X1 + X2)..
- 16. + + + Electric field _ anion cat ion _ _ + + _ x1 x2 + _ _ + _ + _
- 17. Restoring force constant depend upon the mass of the ion and natural frequency and is given by Where, x1 & x2 are displacement of +ve and –ve ions. “M” is the mass of -ve ion and “m” is the mass of +ve ions. ω0 is the angular frequency of respective ions 2 F = eE = m.w0 x or eE x= 2 m.w0 ∴ x1 + x2 = eE 1 1 [ + M] 2 m w0
- 18. Therefore, the induced dipole moment is given by:2 ∴ µionic e E 1 1 = e( x1 + x2 ) = 2 [ m + M ] w0 or α ionic µionic e 1 1 = ⇒ 2 [m + M] E w0 2 Hence, αionic is the ionic polarisation a)This polarization occurs at frequency 1013 Hz (IR). b)It is a slower process compared to electronic polarization. c)It is independent of temperature.
- 19. 3) Orientation Polarization It is also called dipolar or molecular polarization. The molecules such as H2,N2,O2,Cl2,CH4,CCl4 etc, does not carry any dipole because centre of positive charge and centre of negative charge coincides. On the other hand molecules like CH3Cl, H2O,HCl, ethyl acetate (polar molecules) carries dipoles even in the absence of electric field. How ever the net dipole moment is negligibly small since all the molecular dipoles are oriented randomly when there is no EF. In the presence of the electric field these all dipoles orient them selves in the direction of field as a result the net dipole moment becomes enormous. a) It occurs at a frequency 106 Hz to 1010Hz. b) It is slow process compare to ionic polarization. c) It greatly depends on temperature.
- 20. Expression for orientation polarization 2 N .µ orie .E Po = N .µ orie ⇒ = N .α o .E 3kT 2 µ orie αo = 3kT ∴α = α elec + α ionic + α ori = 4πε o R 3 + This is called Langevin dielectrics. e2 2 w0 [ 1 M + 1 m ] 2 µ ori + 3kT – Debye equation for total Polaris ability in
- 21. Frequency Dependence of Dielectric Properties Polarisation : When an ac field is applied to a dielectric material is a function of time, it follows the equation Where, a) P is the maximum polarisation attained due to prolong application of the electric field, and b) τr the relaxation time for the particular polarisation process, is the average time between molecular collision (in case of liquid), during the application of the electric field. The relaxation time is a measure of the polarisation process and is the time taken for a polarisation process to reach 67% of the maximum value.
- 24. Dielectric loss When an ac field is applied to a dielectric material, some amount of electrical energy is absorbed by the dielectric material and is wasted in the form of heat. This loss is known as dielectric loss. The dielectric loss is the major engineering problem. a) In an ideal dielectric, the current leads the voltage by an angle of 90 degree. b) But in the case of a commercial dielectric, the current does not exactly leads the voltage by 90 degree. It leads by some other angle q less than 90 degree. The angle f = 90 – q, is known as the dielectric loss angle.
- 25. Relation between current and voltage in dielectrics ω ε0 ε r ’ E0 I I ϕ ϕ J ϕ θ 90 0 ω ε0 εr’’ E0 V V E0 For a dielectric having capacitance C and voltage V applied to it at a frequency f Hz, the dielectric power loss is given by, P = VI cos θ Since, I = V/ Xc where Xc is the capacitive reactance and is equal to 1/jωC. Therefore,
- 26. Since θ is very small, sin ɸ = tan ɸ and P= j V2 ɷ Ctan ɸ, where tan ɸ is said to be the power factor of the dielectric. The power loss depends only on the power factor of the dielectric as long as the applied voltage, frequency and capacitance are kept constant. The dielectric loss is increased by the following factors : 1) High frequency of the applied voltage. 2) High value of the applied voltage. 3) High temperature. 4) Humidity A) The dielectric losses in the radio frequency region are usually due to dipole rotation. B) The dielectric losses at lower frequencies are mainly due to dc resistivity. C) The dielectric losses in the optical region are associated with electron and they are known as optical obsorption.
- 27. Conductance: Electrical conductance measures how easily electricity flows along a certain path through an electrical element. The SI derived unit of conductance is the siemen. Because it is the reciprocal of electrical resistance (measured in ohms), historically, this unit was referred to as the mho. Resistance: The electrical resistance of an electrical conductor is the opposition to the passage of an electric current through that conductor; the inverse quantity is electrical conductance, the ease at which an electric current passes. The SI unit of electrical resistance is the ohm (Ω). Impedance : Electrical impedance, or simply impedance, describes a measure of opposition to alternating current (AC). Electrical impedance extends the concept of resistance to AC circuits, describing not only the relative amplitudes of the voltage and current, but also the relative phases. When the circuit is driven with direct current (DC) there is no distinction between impedance and resistance; the latter can be thought of as impedance with zero phase angle.
- 28. Susceptance (B) : It is the imaginary part of admittance. The inverse of admittance is impedance and the real part of admittance is conductance. In SI units, susceptance is measured in siemens. The general equation defining admittance is given by Y= G + jB where, Y is the admittance, measured in siemens (a.k.a. mho, the inverse of ohm).G is the conductance, measured in siemens. j is the imaginary unit, and B is the susceptance, measured in siemens. The admittance (Y) is the inverse of the impedance (Z). or
- 29. Where, Z = R + jX Z is the impedance, measured in ohms R is the resistance, measured in ohms X is the reactance, measured in ohms. Note: The susceptance is the imaginary part of the admittance. The magnitude of admittance is given by:
- 30. Dielectric permittivity If a varying field V(t) is applied to a material, then the polarization and induced charge Q are related as where ϵ* is the complex dielectric constant. The frequency dependent complex dielectric permittivity is given by Where ϵ s and ϵ α are the low and high frequency dielectric constants respectively. ɷ =2πf is the angular frequency, τ is the time constant. The ε* is given by where ϵ ’ is the relative permittivity or dielectric constant, ϵ" is the dielectric loss
- 31. The dielectric permittivity ϵ * is related to complex impedance Z* by ɷ is the angular frequency and C0 = t/Aɷϵ 0 is the capacitance of the free space The real and imaginary parts of the dielectric permittivity are calculated using the above equations, pellet dimensions and the measured impedance data.
- 32. Electric modulus The complex electric modulus is defined by the reciprocal of the complex Permittivity. where M* is the complex modulus, ϵ* is the dielectric permittivity, M' is the real and M" is the imaginary parts of modulus. a) The complex electric modulus spectrum represents the measure of the distribution of Ion energies or configuration in the structure and it also describes the electrical relaxation and microscopic properties of Ionic glasses b) The modulus formalism has been adopted as it suppresses the polarization effects at the electrode/electrolyte interface Hence, the complex electric modulus M (ω) spectra reflects the dynamic properties of the sample alone.
- 33. For parallel combination of RC element, the real and imaginary parts of the modulus are given by
- 34. Applicability of electric modulus 1) The applicability of the electric modulus formalism is investigated on a Debye-type relaxation process, the interfacial polarization or MaxwellWagner-Sillars effect. Debye-type relaxation:- Debye relaxation is the dielectric relaxation response of an ideal, non interacting population of dipoles to an alternating external electric field. It is usually expressed in the complex permittivity of a medium as a function of the field's frequency: Variants of the Debye equation a) Cole–Cole equation b) Cole–Davidson equation c) Havriliak–Negami relaxation d) Kohlrausch–Williams–Watts function (Fourier transform of stretched exponential function)
- 35. Maxwell-Wagner-Sillars effect : In dielectric spectroscopy, large frequency dependent contributions to the dielectric response, especially at low frequencies, may come from build-ups of charge. This, so-called Maxwell–Wagner–Sillars polarization (or often just Maxwell-Wagner polarization), It occurs either at inner dielectric boundary layers on a mesoscopic scale, or at the external electrode-sample interface on a macroscopic scale. In both cases this leads to a separation of charges (such as through a depletion layer). The charges are often separated over a considerable distance (relative to the atomic and molecular sizes), and the contribution to dielectric loss can therefore be orders of magnitude larger than the dielectric response due to molecular fluctuations.
- 36. 2) Electric modulus, which has been proposed for the description of systems with ionic conductivity and related relaxation processes, presents advantages in comparison to the classical approach of the real and imaginary part of dielectric permittivity. 3) In composite polymeric materials, relaxation phenomena in the low-frequency region are attributed to the heterogeneity of the systems. 4) For the investigation of these processes through electric modulus formalism, hybrid composite systems consisting of epoxy resin-metal powder-aramid fibers were prepared with various filler contents and their dielectric spectra were recorded in the frequency range 10 Hz-10 MHz in the temperature interval 30150 oC.
- 37. 5) The Debye, Cole-Cole, Davidson-Cole and HavriliakNegami equations of dielectric relaxation are expressed in the electric modulus form. 6) Correlation between experimental data and the various expressions produced, shows that interfacial polarization in the systems examined is, mostly, better described by the Davidson-Cole approach and only in the system with the higher heterogeneity must be used. “Heterogeneity is a problem that can arise when attempting to undertake a meta-analysis. Ideally, the studies whose results are being combined in the meta-analysis should all be undertaken in the same way and to the same experimental protocols: study heterogeneity is a term used to indicate that this ideal is not fully met.”
- 38. Internal fields or local fields Local field or internal field in a dielectric is the space and time average of the electric field intensity acting on a particular molecule in the dielectric material. In other words, the field acting at the location of an atom is known as local or internal field “E”. The internal field Ei must be equal to the sum of applied field plus the field produced at the location of the atom by the dipoles of all other atoms. Ei = E + the field due to all other dipoles
- 39. Evaluation of internal field Consider a dielectric be placed between the plates of a parallel plate capacitor and let there be an imaginary spherical cavity around the atom A inside the dielectric. The internal field at the atom site ‘A’ can be made up of four components E1 ,E2, E3 & E4.
- 40. + + + + + + + + + ++ _ _ _ _ _ _ _ + + + + + + _ + A _ _ Spherical Cavity + _ + + + + _ _ _ _ + + + _ _ _ _ _ _ _ _ _ _ E Dielectric material
- 41. Field E1: E1 is the field intensity at A due to the charge density on the plates E1 = D ε0 D = ε0 E + P ε0 E + P E1 = ε0 E1 = E + P ε0 ..........(1)
- 42. Field E2: E2 is the field intensity at A due to the charge density induced on the two sides of the dielectric. −P E2 = ...........(2) ε0 Field E3: E3 is the field intensity at A due to the atoms contained in the cavity, we are assuming a cubic structure, so E3 = 0.
- 44. Field E4: 1) This is due to polarized charges on the surface of the spherical cavity. dA = 2π . pq.qR dA = 2π .r sin θ .rdθ dA = 2π .r sin θdθ 2 Where dA is Surface area between θ & θ+dθ…
- 45. 2) The total charge present on the surface area dA is… dq = ( normal component of polarization ) X ( surface area ) dq = p cos θ × dA dq = 2πr p cos θ . sin θ .dθ 2
- 46. 3) The field due to this charge at A, denoted by dE4 is given by 1 dq dE4 = 4πε0 r 2 The field in θ = 0 direction 1 dq cos θ dE4 = 4πε 0 r2 1 dE4 = (2πr 2 p cos θ . sin θ .dθ ) cos θ 2 4πε 0 r P dE4 = cos 2 θ . sin θ .dθ 2ε 0
- 47. π 4) Thus the total field E4 due to the charges on the surface of the entire cavity is E4 = ∫ dE4 0 π P =∫ cos 2 θ. sin θ.dθ 2ε0 0 P = 2ε0 π cos 2 θ. sin θ.dθ ∫ 0 let..x = cos θ →dx = −sin θ θ d P = 2ε0 − 1 x 2 .dx ∫ 1 − P x 3 −1 − P −1 −1 = ( )1 ⇒ ( ) 2ε0 3 2ε0 3 P E4 = 3ε0
- 48. The internal field or Lorentz field can be written as Ei = E1 + E2 + E3 + E4 p p p Ei = ( E + ) − + 0 + εo εo 3ε o p Ei = E + 3ε o
- 49. Classius – Mosotti relation : Consider a dielectric material having cubic structure , and assume ionic Polarizability & Orientational polarizability are zero.. αi = α0 = 0 polarization..P = Nµ P = Nα e Ei ......where., µ = α e Ei P where., Ei = E + 3ε 0
- 50. P = Nαe Ei P P = Nα e ( E + ) 3ε 0 P P = Nα e E + Nα e 3ε 0 P P − Nα e = Nα e E 3ε 0 Nα e P (1 − ) = Nα e E 3ε 0 Nα e E P= ...................(1) Nα e (1 − ) 3ε 0
- 51. We known that the polarization vector P = ε 0 E (ε r − 1)............(2) from eq n s (1) & (2) Nα e E = ε 0 E (ε r − 1) Nα e (1 − ) 3ε 0 1− Nα e Nα e E = 3ε 0 ε 0 E (ε r −1) 1= Nα e Nα e E + 3ε 0 ε 0 E (ε r − 1) 1= Nα e Nα e + 3ε 0 ε 0 (ε r − 1) 1= Nα e 3 (1 + ) 3ε 0 ε r −1 Nα e 1 = 3 3ε 0 (1 + ) ε r −1 Nα e ε r − 1 = ...... → Classius Mosotti relation 3ε 0 εr + 2 Where “N” is the number of dipoles per unit volume
- 52. Ferro electric materials or Ferro electricity Ferro electric crystals exhibit spontaneous polarization i.e. electric polarization with out electric field. Ferro electric crystals possess high dielectric constant. Each unit cell of a Ferro electric crystal carries a reversible electric dipole moment. Examples: Barium Titanate (BaTiO3) , Sodium nitrate (NaNO3) ,Rochelle salt etc..
- 53. Piezo- electricity The process of creating electric polarization by mechanical stress is called as piezo electric effect. This process is used in conversion of mechanical energy into electrical energy and also electrical energy into mechanical energy. According to inverse piezo electric effect, when an electric stress is applied, the material becomes strained. This strain is directly proportional to the applied field. Examples: quartz crystal , Rochelle salt etc., “Piezo electric materials or peizo electric semiconductors such as Gas, Zno and CdS are finding applications in ultrasonic amplifiers.”