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MATERIAL SCIENCE 2012-2013 What modules will be taught? 1. 2. 3. 4. 5. 6. Dielectric materials Magnetic materials Polymers and Ceramics Super-conductivity Optical Materials Thermoelectric materials ***Notable exception, we will not teach Semiconductor materials****** Books used for this course 1) 2) 3) 4) A Text book of solid state physics by S O Pillai Material Science and engineering an Introduction by W D Callister Jr Science of Engineering Materials by Srivastava and Srinivasan Elements of Material Science and Enginnering by Lawrence H Van Vlack BASICS OF DIELECTRIC MATERIALS Basic Questions • What is a dielectric? • Why there is any electrical effect if the insulators do not conduct electricity? • Why should a field induce a dipole moment in an atom if the atom is not a conducting sphere? What is a dielectric? 1. A non conducting or insulating substance that can sustain an electric field but does not conduct electric current. 2. Or an electrical insulator, a material with a low (compared with that of a metal) electrical conductivity. 3. Most generally, a dielectric is an insulator, a substance that is highly resistant to flow of electric current. Layers of such substances are commonly inserted into capacitors to improve their performance, and the term dielectric refers specifically to this application. 4. Can insulator be affected by electric field? • Faraday’s experimental observation: Capacitance of a capacitor is increased when an insulator is placed between the plates, the capacitance is increased by a factor if the insulator completely fills the space between the plates. • depends only on the nature of the insulating material and is called dielectric constant Q: What should be the dielectric constant of vacuum? Consider a parallel plate capacitor; then Capacitance Charge C 0 A d Q CV . Where A= Area of plates, d = Plate separation C= Capacitance Q= Charge on plate V= Voltage difference Now if we put a piece of insulating material like glass between the plates, we find that the capacitance increases. That means, the voltage decreases for the same charge. But voltage difference is the integral of the electric filed across the capacitor, therefore we conclude that, electric field is reduced even though the charges on the plates remain unchanged. Why should a field induce a dipole moment in an atom if the atom is not a conducting sphere? • Consider a single atom. • For spherically symmetric system; center of gravity of negative charges (electron cloud) coincides exactly with the location of the nucleus. Atom is unpolarised. • If we now apply an electrical field, the centers of charges (+ve and –ve) will be separated. The electron cloud will be pulled in the direction of the positive pole of the field, the nucleus to the negative one. The atom will be polarized. The effect of electrical field is that: 1. It induces electric dipoles in an unpolarized material 2. and tries to align them in the field direction. 3. The total effect of an electrical field on a dielectric material is called the polarization of the material. Review of some basic formula 1. Electric dipole: 2. Dipole Moment: p qr 3. Torque on the dipole exerted by an E-field pxE pE sin V p.E pE cos 4. Potential energy of dipole in an E-field 0, V pE ,V pE DKR-JIITN-2010-MS 5. Polarization: Defined as dipole moment per unit volume. If the number of dipoles per unit volume is N, and if each has moment p then polarization is given as (assuming that all the dipoles lie in the same direction) P Np Example: Suppose there are 3.34x1028 molecules per unit volume of water each having dipole moment 6x10-30 C-m. Solution: If all dipoles are oriented parallel to each other then ‘Polarization’ P = 3.34x1028 x 6x10-30 = 0.2004 C/m2 DKR-JIITN-2010-MS 6. ELECTRIC FLUX DENSITY (Dielectric displacement) AND POLARIZATION E0 Electric fluxdensity D 0 E0 P Where, D q = Electric flux density* A And P is called Polarization *According to Gauss law, Q E.dA encl 0 Polarization results in a reduction of the field inside the dielectric medium. Further, D E E P 0 0 r E 0 E P 0 E( r 1) P P Where, r ( r 1) 0 0E Here is known as electric susceptibility and r is known as relative dielectric constant of the medium. POLARIZABILITY Polarization of a medium is produced by field therefore, it is reasonable to assume that, p E Here ‘’ is known as polarizability of the molecule representing dipole moment per unit applied electric field The polarization can now be written as, P NE Thus, D 0 E P 0 E NE N D 0 (1 )E But, D 0 r E 0 N 0 r E 0 (1 )E N r 1 0 0 In all above expressions, N can be expressed in terms of density , molar mass M of the material and Avogadro's number NA as N N A M Thus dielectric constant can be written as: N r 1 0 N A r 1 ( ) 0M However, experiments show that though above equations hold good in gases but not for liquids and solids i.e. in the condensed physical systems. Section II : Summary • Dipoles in solid dielectrics; Polarization. • Polarization is dipole moment per unit volume: • A relation between E & P: • Connection between the Polarization P and the Electrical Displacement D • Polarizability N r 1 0 p= q·E P = p· N V P 0 E D 0E P or D = ε0 (+1) E p =E =P / N E V Experiments show that last equation hold good in gases but not for liquids and solids i.e. in the condensed physical systems. So we need some corrections in the formulae for solids. Local Field and Clausius - Mosotti Equation • Here, we are looking the effect of external field on atoms and molecules in a solid or liquid system. • What an atom "sees" as local electrical field or the local field Eloc or EL to be the field felt by one particle (mostly an atom) of the material. • we may express EL as a superposition of the external field E0 and some field Emat introduced by the surrounding material of an atom. • EL = E0 + Emat CLAUSIUS MOSOTTI RELATION LOCAL FIELD Eloc E0 E1 E2 E 3 E0 = External field E1 = Field due to polarization charges lying on the surface of the sample. E0 E1 E2 = Field due to polarization charges lying on the surface of Lorentz sphere. E3 = Field due to other dipoles lying within the Lorentz sphere. Lorentz sphere E2 Central dipole Calculation of various fields: Depolarizing field E1: E1 P 0 This field depends on the geometrical shape of the external surface. Above equation is for a simple case of an infinite slab. Field for a standard geometry is given as NP E1 0 Here N is known as depolarizing factor. The values of N for other regular shapes are given below: Shape Sphere Thin slab Thin slab Cylinder Cylinder Axis any normal in plane Longitudinal Transverse N 1/3 1 0 0 ½ Calculation of E2: Surface area dA of the sphere lying between and +d is given as dA 2r 2 sin d Charge on the surface dA would be dq P cos (2r 2 sin d ) Field due to this charge at the centre of the sphere would be dE dE dq 40 r 2 Field in the direction of applied field would be dE2 dE cos dq cos 40 r 2 Field due to charges on the entire cavity thus would be, E2 dE2 0 dq cos 2 4 r 0 0 0 P 2r 2 sin cos 2 d 40 r 2 E2 P 3 0 Calculation of E3: The field due to other dipoles in the cavity may be calculated by using the equation 2 1 3( p.r )r r p E 40 r5 The result depends on crystal structure of the solid under consideration. However for highly symmetrical structure like cubic it sum up to zero. Thus E3 0 (In other structure E3 may not vanish and it should be included in the equation). Thus Eloc would be Eloc E0 E1 E2 E 3 Eloc P P E0 0 3 0 2P E0 3 0 P E 3 0 Eloc = EL= Lorentz field, E is known as Maxwell field. Now the polarization would be given as P NEL P N P N E N ( E ) 3 0 3 0 N P (1 ) N E 3 0 N E N Again P (1 ) N E P N 3 0 1 3 0 Now 1 D 0 r E 0 E P NE 0 r E 0 E N 1 3 0 r 1 N r 2 3 0 CLAUSIUS MOSOTTI RELATION N r 1 N 0 (1 ) 3 0 N / 0 ( r 1) N (1 ) 3 0 3 ( r 2) N (1 ) 3 0 Reconsider CLAUSIUS MOSOTTI relation, r 1 N r 2 3 0 Molar mass r 1 M N M ( ) ( ) r 2 3 0 Since, Therefore , NM Density NA r 1 M N A ( ) r 2 3 0 M MOLAR POLARIZABILITY EXAMPLE: An elemental dielectric material has r = 12 and it contains 5x1028 atoms/m3. Calculate its electronic polarizability assuming Lorentz field. SOLUTION: Using CLAUSIUS MOSOTTI relation, r 1 N r 2 3 0 12 1 5 10 28 12 2 3 8.85 10 12 4.17 1020 Fm2 Section III : Polarization Mechanisms 1. Types of Dielectrics • Polar • Non Polar 2. Electronic polarization: 3. Ionic polarization: 4. Orientation (Dipolar) polarization : • interface polarization Types of Dielectrics: • Polar dielectrics: – Materials having permanent dipole moments – Net dipole moment– Not zero – Many natural molecules are examples of systems with a finite electric dipole moment (permanent dipole moment), since in most types of molecules the centers of gravity of the positive and negative charge distributions do not coincide. – Ex. Water Dipole moment of water molecule. • Non Polar dielectrics: • Net dipole moment – zero, (in the absence of E) • centers of gravity of the positive and negative charge distributions coincide with each other. • Ex. O2, N2 and Nobel gases Polarization Mechanisms • Dielectric Polarization is nothing but the displacement of charged particles under the action of the electric field to which they are subjected. • Therefore this displacement of the electric charges results in the formation of electric dipole moment in atoms, ions or molecules of the material. – There are essentially mechanisms: three basic kinds of polarization 1. Electronic polarization: also called atomic polarization. An electric field will always displace the center of charge of the electrons with respect to the nucleus and thus induce a dipole moment. e.g noble gases. Polarization Mechanism 2. Ionic polarization: In this case a (solid) material must have some ionic character. It then automatically has internal dipoles, but net dipole moment is zero. The external field then induces net dipoles by slightly displacing the ions from their rest position. Ex. simple ionic crystals like NaCl. 3. Orientational polarization: Some time called “Dipolar polarization”; Here the (usually liquid or gaseous) material must have natural dipoles which can rotate freely. In thermal equilibrium, the dipoles will be randomly oriented and thus carry no net polarization. The external field aligns these dipoles to some extent and thus induces a polarization of the material. Ex. is water, i.e. H2O in its liquid form. NOTE: • Some or all of these mechanisms may act simultaneously. • Atomic polarization is always present in any material and thus becomes superimposed on whatever other mechanism there might be. • All three mechanisms are essential for basic consideration and calculations. ************************************************************************************** However interface polarization is also found in materials: Surfaces, grain boundaries, interface boundaries may be charged, i.e. they contain dipoles which may become oriented to some degree in an external field and thus contribute to the polarization of the material. – There is simply no general way to calculate the charges on interfaces nor their contribution to the total polarization of a material. Interface polarization is therefore often omitted from the discussion of dielectric properties. SOURCES OF POLARIZABILITY 1. Electronic Polarizability E 0 2. Ionic Polarizability E 0 3. Dipolar or orientational Polarizability E 0 1. ELECTRONIC POLARIZATION: Volume of the atom is, 4 V R3 3 Where, R = Radius of spherically symmetric atom E 0 If z be the atomic number then charge/ volume of atom would be 3 ze 4 R 3 In presence of field E Force on the charges F1 ZeE This leads to the separation of charges. Coulomb force between separated charge would be F2 Ze X Field produced by displaced charges on nucleus Ze X Charge enclosed in the sphere of radius d 4o d 2 Ze 4 3 Z 2e 2 d Ze 4 3 3Ze d d 2 2 3 4o d 3 40 R 3 4o d 3 4R In the equilibrium position, the two forces , F1 and F2 are equal, thus Z 2e2d ZeE 40 R 3 Zed E 40 R 3 40 R 3 E d Ze This is equilibrium separation between charges, which is proportional to the field. Now the induced electric dipole moment would be 40 R 3 E pe Zed Ze( ) Ze pe 40 R 3 E pe 40 R 3 E But according to the definition of polarizability, pe e E Comparing, e 40 R 3 (e = electronic polarizability) Thus, Electronic Polarization can be given as Pe 0 ( r 1) E N e E r 1 N e 0 Where N is number of atoms/ m3. 2. IONIC POLARIZATION: pi i E Polarization Mechanisms • Dielectric Polarization is nothing but the displacement of charged particles under the action of the electric field to which they are subjected. • Therefore this displacement of the electric charges results in the formation of electric dipole moment in atoms, ions or molecules of the material. – There are essentially mechanisms: three basic kinds of polarization 1. Electronic polarization: also called atomic polarization. An electric field will always displace the center of charge of the electrons with respect to the nucleus and thus induce a dipole moment. e.g noble gases. Polarization Mechanism 2. Ionic polarization: In this case a (solid) material must have some ionic character. It then automatically has internal dipoles, but net dipole moment is zero. The external field then induces net dipoles by slightly displacing the ions from their rest position. Ex. simple ionic crystals like NaCl. 3. Orientational polarization: Some time called “Dipolar polarization”; Here the (usually liquid or gaseous) material must have natural dipoles which can rotate freely. In thermal equilibrium, the dipoles will be randomly oriented and thus carry no net polarization. The external field aligns these dipoles to some extent and thus induces a polarization of the material. Ex. is water, i.e. H2O in its liquid form. Torque=p X E 3. DIPOLAR POLARIZATION Without field with field Consider a molecule which carries a permanent dipole moment p (like water molecule) is placed in an electric field. The potential energy of the dipole would be: U p.E pE cos The electric field will apply a torque which will rotate the dipole in the direction of the EF. The energy of the dipole is minimum when it is aligned with the field (-pE) and it is maximum when it is antiparallel to the EF(+pE)According to Boltzmann distribution, no. of molecules with energy U at equilibrium temperature T would be: n n 0 e U kT n e n0 pE cos kT Let n(θ) be the number of dipoles per unit solid angle at θ, we have n( ) n0 e pE cos kT The number of dipoles in a solid angle dW n0 e pE cos kT Note: Here dW is calculated as follows: dW n0 e pE cos kT 2 sin d Dipole moment of dipoles making angle with the field (along x-axis) is p x p cos Therefore, the dipole moment along the field within angle dW n0 e pE cos kT 2 sin d ( p cos ) Now, average dipole moment (Total dipole moment divided by total no. of dipoles) can be written as n e pE cos kT 0 p 2 sin ( p cos )d 0 n e 0 0 pE cos kT 2 sin d n e pE cos kT 0 p 2 sin ( p cos )d 0 n e pE cos kT 0 2 sin d p p 0 e sin cos d pE cos kT sin d 0 0 Let e pE cos kT pE cos x kT Therefore, a cos x and and, pE a kT a sin d dx a a Limits Substituting all above, the integral becomes, a p 1 p a x e xdx a a x e dx a p 1 [ xe x e x ] aa e a e a 1 a x a a p a a [e ] a e e p ea ea 1 a a p e e a (Langevin Function) Or, p 1 coth( a) L (a ) p a From the above equation, a p pL(a ) L (a ) Thus polarization would be, Po NpL(a) Ps L(a) Ps = Saturation polarization a pE kT Po Ps L(a) Where, pE a kT CASE 1: When a is very high (at low temperature) i.e. a >> 1, L(a) = 1 Po = Ps CASE 2: When a is very low (at high temperature) i.e. a <<1 1 a a3 1 a coth( a) ... a 3 45 a 3 1 1 a 1 L(a) coth( a) a a 3 a a L(a ) 3 a Po Ps 3 Np 2 E Po 3kT Np 2 E Orientation polarization Po 3kT And in terms of orientation polarizability P0 N o E p2 o 3kT 1 o T Thus orientation polarizability is inversely proportional to T. Total polarization P Pe Pi Po N e E N i E N o E P 0 ( r 1) E N ( e i ) E N o E 0 ( r 1) N ( e i ) N o Np 2 0 ( r 1) N ( e i ) 3kT In general, therefore, we may write total polarizability as e i o or p2 ei 3kT temperature independent or p2 e i 3kT Substituting into ClausiusMosotti relation, we have r 1 M N A p2 M ( ) ( ei ) r 2 3 0 3kT Polar substances N A ei 3 0 N A p2 slope 9 0 k Non-polar substances Clausius-Mosotti relation (in terms of refractive inex) may alternatively be written as r 1 n 2 1 N Lorentz-Lorenz Relation 2 r 2 n 2 3 0 If the material consists of different types of molecules then Clausius-Mosotti relation may be written as r 1 1 N i i r 2 3 0 Where Ni is the no. of molecules per unit volume and i is polarizability of ith kind of molecule. Section III : Polarization Mechanisms 1. Types of Dielectrics • Polar • Non Polar 2. Electronic polarization: 3. Ionic polarization: 4. Orientation (Dipolar) polarization : • interface polarization Types of Dielectrics: • Polar dielectrics: – Materials having permanent dipole moments – Net dipole moment– Not zero – Many natural molecules are examples of systems with a finite electric dipole moment (permanent dipole moment), since in most types of molecules the centers of gravity of the positive and negative charge distributions do not coincide. – Ex. Water Dipole moment of water molecule. • Non Polar dielectrics: • Net dipole moment – zero, (in the absence of E) • centers of gravity of the positive and negative charge distributions coincide with each other. • Ex. O2, N2 and Nobel gases SOURCES OF POLARIZABILITY 1. Electronic Polarizability E 0 2. Ionic Polarizability E 0 3. Dipolar or orientational Polarizability E 0 Orientational polarization: Some time called “Dipolar polarization”; Here the (usually liquid or gaseous) material must have natural dipoles which can rotate freely. In thermal equilibrium, the dipoles will be randomly oriented and thus carry no net polarization. The external field aligns these dipoles to some extent and thus induces a polarization of the material. Ex. is water, i.e. H2O in its liquid form. 1. Molecules that have permanent dipole moment show orientational polarization. 2. One atom is always more electronegative than the other. Eg in H Cl, Cl is more electronegative than H so the shared electrons shift more towards Cl atom. So a HCl molecule has a net dipole moment with dipole directed from Cl to H. These are AB type molecules. 3. Molecules like A-A have no permanent dipole moment. Eg H2 , O2 4. Let us now look at AB2 type molecules. They can have net dipole moment(each bond has a dipole moment) if the molecule has no centre of symmetry. In general a molecule ABCD… has dipole moment if it has no centre of symmetry. Important thing to remember if that each bond is an individual dipole and total dipole moment of the molecule is the vector sum of all the dipoles due to all the bonds. Dipole moment=0 Dipole moment nonzero