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1 MME 467: Ceramics for Advanced Applications Lecture 18 Defects in Ceramics 2 Ref: Barsoum, Fundamentals of Ceramics, Ch6, McGraw-Hill, 2000 Prof. A. K. M. B. Rashid Department of MME, BUET, Dhaka Topics to discuss... Defects reactions v Rules for defect reactions v Stoichiometric defect reactions v Defect reactions for compound crystals v Non-stoichiometric defect reactions v Extrinsic defect reactions 1

2 Defect Reactions q Each defect can be treated as chemical entities and treat them in a manner referred to as defect chemistry. q Formation of various defects and interactions of various defects may be conceptualised in terms of mass action equilibria by means of defect equations. Rules for defect reactions 1. Mass Balance v Mass cannot be created or destroyed. Vacancies have zero mass. 2. Electroneutrality, or charge balance v Charge cannot be created or destroyed. 3. Preservation of regular site ratio v Ratio between the numbers of regular cation and anion sites must remain constant and equal to the ratio of the parent lattice. v Thus if a normal lattice site of one component is created or destroyed, the corresponding number of normal sites of the other constituent must be simultaneously created or destroyed so as to preserve the site ratio of the compound. 2

3 q To generalise, for an M a X b compound, the following relation has to be maintained at all times: ( ) = b( M M +V M ) a X X +V X M M +V M X X +V X = a b q For example, in Al 2 O 3 : M Al +V Al X O +V O = 2 3 Note that, this does not mean that the number of atoms or ions has to maintain that ratio, but only the number of sites. Stoichiometric Defect Reactions Crystal chemistry (cations/ anions ratio) does not change. No mass is transferred across the crystal boundary. Scho%ky Defect Vacancy vacancy pair Important defects in this category include: 1. Schottky defects 2. Frenkel defects Frenkel Defect Vacancy interstitial pair 3

4 Schottky defects q In schottky defects, electric-charge-equivalent numbers of vacancies are formed on each sub-lattice. q For any compound MX (M +2, X 2 ), the Schottky defect reaction is: null =V M '' +V X or, perfect crystal ; Δg S Δg S = the free energy change for the formation of Schottky defect q Schottky defect reaction for Al 2 O 3 : null = 2V ''' Al + 3V O q In general, for an M a O b oxide: null = av M b + bv O a+ Thermodynamics of Schottky defects q Assume that the number of ways of distributing cation vacancies V cat on (N cat + V cat ) sites be Ω 1, and number of ways of distributing anion vacancies V an on (N an + V an ) sites be Ω 2. q The configuration entropy ΔS = k lnω = k lnω 1 Ω 2 where and Ω = ( N + V cat cat)! ( N an + V an )! ( N cat )!( V cat )! ( N an )!( V an )! ( N cat + n cat ) ( N an + n an ) = 1 4

5 q Finding minimum in free energy will yield V eq eq an V cat eq ( N an + V an ) V eq cat + N cat ( ) V an eq eq V cat % = exp Δh S T Δs ' S N an N cat & kt ( * ) Product of cation and anion vacancy concentrations is a constant that depends only on temperature q When Schottky defects dominate, then where [ V a ] = [ V c ] = exp Δs S [ V c ] = V cat V cat + N cat 2k exp $ Δh ' S & ) % 2kT ( and [ V a ] = Square brackets denote mole fractions of defects! They are dimensionless! V an V an + N an Frenkel Defects q A cation removed from its normal site to an interstitial site to form an interstitial vacancy pair q For any trivalent cation (M +3 ), the Frenkel defect reaction is: M M x =V M ''' + M i q For anti-frenkel defect, an anion is removed form an interstitial vacancy pair q For oxygen ion (O 2 ), the Anti-Frenkel defect reaction is: O O x =V O +O i '' 5

6 Thermodynamics of Frenkel defects q number of ways of distributing n i interstitials on N* interstitial sites be Ω 1, and number of ways of distributing cation vacancies V cat on N T total sites be Ω 2. Ω 1 = Ω 2 = N *! ( N * n i )!n i! N T! ( N T V cat )!V cat! q The configurational entropy is the same ΔS = k lnω 1 Ω 2 q At equilibrium, V eq eq cat n i N T N * % exp Δg ( F ' * exp TΔS F & kt ) kt exp % Δh ( F ' * & kt ) v Note that N* depends on crystal structure. Δg F = the free energy change for the formation of Frenkel defect v For example, for 1 mol NaCl, if the ions migrate to tetrahedral sites, N* 2N AV. 6

7 Worked Example 6.1 Estimate the number of Frenkel defects in AgBr (rocksalt structure) at 500 ºC. The enthalpy of formation of the defect is 110 kj/mol, and the entropy of formation is 6.6R. The density and molecular weight are 6.5 g/cm 3 and g/mol, respectively. State all necessary assumptions. Let us assume that [1] the Frenkel disorder occurs on the cation sub-lattice [2] silver ions go into the tetrahedral sites. Then number of interstitial sites = 2 x number of lattice sites 2N AV V eq eq cat n i N T N = expt ΔS F * kt V eq eq cat n i 6.6R = exp 2(6.02x ) exp # Δh & F % ( $ kt ' R exp # 110x1000 & % ( $ 8.314( ) ' V eq cat n eq i =1.957x10 43 defects/mol 2 Worked Example 6.1 Estimate the number of Frenkel defects in AgBr (rocksalt structure) at 500 ºC. The enthalpy of formation of the defect is 110 kj/mol, and the entropy of formation is 6.6R. The density and molecular weight are 6.5 g/cm 3 and g/mol, respectively. State all necessary assumptions. In case of Frenkel disorder, number of cation vacant site = number of interstitial sites V eq cat n eq i =1.957x10 43 defects/mol 2 V eq cat = n eq i = 4.43x10 21 defects/mol And the corresponding number of defects/cm 3 is V cat! eq = n eq i = 4.43x $ # & =1.5x10 20 defects/cm 2 " % 7

8 Defects in Compound Crystals Possible defects: 1. vacant sites on each sub-lattice 2. ions/atoms on interstitial sites 3. impurity ions/atoms on each sub-lattice 4. unassociated electrons and holes 5. combination of these defects Example: q Consider a crystal with a formula M a X b. q If valence of M is z, then the valence of X will be -(a/b)z. q Reaction with its surroundings when M is added to its normal cation site: M (g) M a X!!! b M x + # b M % $ a & (V (a/b)z X + z e) ' 8

9 Example: Incorporating Si into SiO 2 Si(g) SiO!!! 2 Si x Si + 2V O + 4 e# Example: Incorporating Al into Al 2 O 3 Al (g) Al 2 O!!! 3 Al x Al + 3 V 2 O + 3 e# or, 2Al (g) Al 2 O!!! 3 2Al x Al + 3V O + 6 e# Non-stoichiometric Defect Reactions q Composition of crystal changes due to this defect because mass is transferred across the boundary of the crystal. q For a general M a O b oxide compound, two nonstoichiometric defects can occur: 1. Metal excess, or oxygen deficient 2. Oxygen excess, or metal deficient q In both of these cases, a/b ratio is changed. 9

10 Metal excess, or oxygen deficient (low oxygen partial pressure) q has the general formula M a+δ O b, or M a O b-δ. q places cation into interstitial site, or creating vacancy in oxygen site. q using anion, typical defect reaction is: O x O = 1 O 2 2 (g) +V O + 2 e! Redox reac5on V O x electrons are usually weakly bonded; can be excited into the conduction band when species O 2 escape as natural, it leaves two electron behind O x O 1 2 O g x 2( ) + V O V x O V O + e # V O V O + e # O x O 1 2 O g 2( ) + V O + 2 e # 10

11 Example: TiO 2-y x TiO O 2 O!!! 1 O 2 2 (g) +V O + 2 e# 2Ti e # = 2Ti +3 2Ti x x Ti +O O = 2Ti ' Ti + 1 O 2 2 (g) +V O Oxygen excess, or metal deficient (high oxygen partial pressure) q has the general formula M a O b+δ, or M a-δ O b. q places oxygen into interstitial site, or creating vacancy in cation site. q using anion, typical defect reaction is: 1 2 O 2 (g) = O i '' + 2h O i x This ionization creates holes in valence band This hole moves through lattice and contribute to electrical conductivity 11

12 Example: Fe 1-x O 1 2 O 2 (g) FeO!!! O x '' O +V Fe + 2h 2Fe h = 2Fe +3 2Fe x Fe + 1 O 2 2 (g) = 2Fe Fe +O x '' O +V Fe Extrinsic Defect Reactions q Defects created by impurities. q Usually substitute host ions of the same or nearest electronegativity, even if the sizes of the ions differ. So cations substitute for cations and anions for anions. For example, in NaCl, Ca and O would be expected to occupy Na and Cl sites, respectively. Example 1: Incorporating CaCl 2 into NaCl CaCl 2 CaCl 2 2NaCl ' x!!! Ca Na +V Na + 2Cl Cl NaCl!!! Ca Na +Cl ' x i +Cl Cl Between these two reactions, which one is more probable? 12

13 Example 2: Doping MgO with Al 2 O 3 Al 2 O 3 3MgO ''!!! 2Al Mg +V Mg + 3O O x Example 3: Doping Al 2 O 3 with MgO 2MgO Al 2O! 3 '!! 2Mg Al + 2O O x +V O Next Class Lecture 19 Electrical and Ionic Conduction in Ceramics 13