Mechanisms of Diffusion II. Ionic Crystals L5 11/3/03-1-

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1 Mechanisms of Diffusion II. Ionic Crystals 3.05 L5 11/3/03-1-

2 Charges on point imperfections Point imperfections in ionic crystals are generally electrically charged L5 11/3/03 - (a) Unit cell in perfect crystal; charge-neutral. (b) Removal of a cation to a surface step, leaving a unit cell with a cation vacancy. Net charge associated with defective cell is 1. (c) Formation of anion interstitial; net charge is 1.

3 Kröger-Vink notation X = what is at the site (El or V) X Z Y where Y= what site is defective (El or i) Z = effective charge at the site ( = +; '= ) Examples Intrinsic defects: Schottky and Frenkel Defects involving impurities 3.05 L5 11/3/03-3

4 Schottky disorder Charge-compensating anion vacancies + cation vacancies e.g. in MgO Formation: null = V Mg + V O Equilibrium: K S = [V Mg Charge neutrality: [V Mg ] = [V O ] ] [V O ] = exp G S f kt 3.05 L5 11/3/03-4

5 Frenkel pair Charge-compensating vacancy-interstitial pair e.g. cation Frenkel pair in LiF Formation: Equilibrium: Li Li = V Li K F = [V Li + Li i ] [Li Li ] = exp Charge neutrality: [V Li ] = [Li Li ] G F f kt 3.05 L5 11/3/03-5

6 Self-diffusion in KCl Analogous to self-diffusion in metals with and f G f S SS f H [ V S K ]= exp = exp exp kt k kt K =1 K [ V K ] giving for the self-diffusivity on the cation sites m f m + S S f H CV + H S D K = ga f exp S CV exp k kt 3.05 L5 11/3/03-6

7 Extrinsic defects Isovalent impurities e.g. CaO in MgO MgO Incorporation: CaO Ca Mg + O O Equilibrium: K S = [V Mg ] [V O ] = exp G S f Charge neutrality: [V Mg ] = [V O ] kt 3.05 L5 11/3/03-7

8 Extrinsic defects, cont d Aliovalent impurities e.g. CaO in ZrO Incorporation: CaO ZrO Ca Zr + V O + OO 4 Equilibrium: K S = [V Zr ] [V O ] = exp G S f kt 4 Charge neutrality: 4[V Zr ] + Ca [ Zr ] = [V O ] 3.05 L5 11/3/03-8

9 Cation diffusion in KCl with Ca Impurity incorporation CaCl Schottky equilibrium K S = [ V K Neutrality condition KCl Ca K + V K + Cl Cl ][V Cl ] = exp G S f kt [V Cl ] + [Ca K ] = [ V K ] 3.05 L5 11/3/03-9

10 Diffusion in KCl with Ca, cont d This leads to 1 4[ V K ] [Ca K ] pure [ V K ]= [Ca K ] and to two regimes: Intrinsic: Small impurity concentration or high T [ V K ] pure >> [Ca K ], then [ V K ]= [ V K ] pure Extrinsic: High impurity concentration or low T V K << [Ca K ], then [ V K ]= [Ca K ] [ ] pure 3.05 L5 11/3/03-10

11 Diffusion in KCl with Ca, cont d This leads to this behavior for the cation diffusivity ln [ V K ] ln D K Intrinsic H S f k Intrinsic m H CV f + H S k ln [ Ca K ] Extrinsic Extrinsic m H CV k 1 T 1 T +=JE L=?=?O??A JH=JE 3.05 L5 11/3/03-11

12 e.g., cation diffusion in FeO (note various valence states of Fe are possible) FeO can be oxidized to make a cation deficient oxide Fe 1 x O e 0 neutral oxygen atom 3+ Oxygen atom approaches, attracts two electrons from Fe and gets ionized. e 3+ Oxygen-deficient crystal. Note cation vacancy and two ferric cations. oxidation reaction: 3.05 L5 11/3/ O = O x x O + V Fe + h Fe, where h Fe = Fe Fe Fe Fe

13 Oxidation reaction: 1 x O = O O + V Fe Equilibrium constant for this reaction: K eq = + h Fe [ [ V Fe ] p O h Fe ] = exp G kt Charge neutrality: [ V Fe ] = [h Fe ] Cation vacancy concentration: [ G 3kT V ] = (1 4) 13 exp ( p ) Fe O 16 Diffusion in impure FeO will have three regimes: 1. High T, low oxygen pressure, dominated by Schottky defect equilibria. High oxygen pressure, dominated by oxidation reaction 3. Low T, low oxygen pressure, dominated by extrinsic impurities 3.05 L5 11/3/03-13

14 Oxidation reaction: 1 O = O x O + V Fe + h Fe Equilibrium constant for this reaction: K eq = [ ] [ V Fe ] h Fe p O = exp G kt Charge neutrality: [ ] = h Fe V Fe [ ] Cation vacancy concentration: [ V Fe ] = 14 ( ) 13 exp G ( ) 16 3kT p O Diffusion in impure FeO will have three regimes: 1. High T, low oxygen pressure, dominated by Schottky defect equilibria. High oxygen pressure, dominated by oxidation reaction 3. Low T, low oxygen pressure, dominated by extrinsic impurities 3.05 L5 11/3/03-14-

15 Cation diffusivity Arrhenius plot Intrinsic ln D Fe H f CV + k m H CV diffusivity varies as (P O ) 1/6 H 3 + k m H CV Extrinsic m H CV k Position of middle segment will depend on oxygen pressure, hence this region will not be observable at low oxygen pressures or high impurity contents 1 T 3.05 L5 11/3/03-15

16 Microstructural processes Densification of powder compacts by sintering Creep deformation at high temperatures Grain growth Solid solid transformation kinetics, including oxidation of metals Electrical conduction Ionic conductors for chemical and gas sensors, solid electrolytes, fuel cells Sensors emissions HC NO x 3.05 L5 11/3/ " # air-to-fuel ratio

17 3.05 L5 11/3/03-17-

A great many properties of crystals are determined by imperfections.

Defect in ceramics A great many properties of crystals are determined by imperfections. Electrical conductivity Diffusion transport imperfection Optical properties Rate of kinetic process Precipitation