Electrochemistry High Temperature Concepts

Transcription

1 Electrochemstry Hgh Temperature Concepts Ngel Sammes POSTECH South Korea 2 nd Jont European Summer School on Fuel Cell and Hydrogen Technology

2 OK Let s Get Started

3 Introducton to Electrochemstry Electrochemstry s: OR Generaton of electrcty by realzng the energy from a spontaneous chemcal reacton usng electrcty to force a chemcal reacton to occur Electrochemstry s all about the concepts of oxdaton and reducton.

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5 Oxdaton and Reducton Oxdaton s Reducton s the loss of electrons an ncrease n oxdaton state the addton of oxygen the loss of hydrogen 2 Mg + O 2 2 MgO magnesum s losng electrons the gan of electrons a decrease n oxdaton state the loss of oxygen the addton of hydrogen MgO + H 2 Mg + H 2 O Mg 2+ n MgO gans electrons

6 In Electrochemstry we get Redox Reactons Oxdaton and reducton always occur together n a chemcal reacton. For ths reason, these reactons are called redox reactons. Although there are dfferent ways of dentfyng a redox reacton, the best s to look for a change n oxdaton state: 2 Fe I - 2 Fe 2+ + I 2 2 H 2 O 2 H 2 + O 2 2 AgNO 3 + Cu 2 Ag + Cu(NO 3 ) 2 HCl + AgNO 3 AgCl + HNO 3

7 More Defntons Oxdzng Agent the substance n a chemcal reacton whch causes another speces to be oxdzed. the oxdzng agent always gets reduced n the reacton. Reducng Agent the substance n a chemcal reacton that causes another speces to be reduced. the reducng agent always gets oxdzed!

8 Oxdaton Numbers In order to keep track of what loses electrons and what gans them, we assgn oxdaton numbers.

9 Oxdaton and Reducton A speces s oxdzed when t loses electrons. Here, znc loses two electrons to go from neutral znc metal to the Zn 2+ on.

10 Oxdaton and Reducton A speces s reduced when t gans electrons. Here, each of the H + gans an electron and they combne to form H 2.

11 Oxdaton and Reducton What s reduced s the oxdzng agent. H + oxdzes Zn by takng electrons from t. What s oxdzed s the reducng agent. Zn reduces H + by gvng t electrons.

12 Half-Reacton Method Consder the reacton between MnO 4 and C 2 O 4 2 : MnO 4 (aq) + C 2 O 4 2 (aq) Mn 2+ (aq) + CO 2 (aq)

13 Half-Reacton Method Frst, we assgn oxdaton numbers MnO 4 + C 2 O 4 2- Mn 2+ + CO 2 Snce the manganese goes from +7 to +2, t s reduced. Snce the carbon goes from +3 to +4, t s oxdzed.

14 Oxdaton Half-Reacton C 2 O 4 2 CO 2 To balance the carbon, we add a coeffcent of 2: C 2 O CO 2

15 Oxdaton Half-Reacton C 2 O CO 2 The oxygen s now balanced as well. To balance the charge, we must add 2 electrons to the rght sde. C 2 O CO e

16 Reducton Half-Reacton MnO 4 Mn 2+ The manganese s balanced; to balance the oxygen, we must add 4 waters to the rght sde. MnO 4 Mn H 2 O

17 Reducton Half-Reacton MnO 4 Mn H 2 O To balance the hydrogen, we add 8 H + to the left sde. 8 H + + MnO 4 Mn H 2 O

18 Reducton Half-Reacton 8 H + + MnO 4 Mn H 2 O To balance the charge, we add 5 e to the left sde. 5 e + 8 H + + MnO 4 Mn H 2 O

19 Combnng the Half-Reactons Now we evaluate the two half-reactons together: C 2 O CO e 5 e + 8 H + + MnO 4 Mn H 2 O To attan the same number of electrons on each sde, we wll multply the frst reacton by 5 and the second by 2.

20 Combnng the Half-Reactons 5 C 2 O CO e 10 e + 16 H MnO 4 2 Mn H 2 O When we add these together, we get: 10 e + 16 H MnO C 2 O Mn H 2 O + 10 CO e

21 Combnng the Half-Reactons 10 e + 16 H MnO C 2 O Mn H 2 O + 10 CO e The only thng that appears on both sdes are the electrons. Subtractng them, we are left wth: 16 H MnO C 2 O Mn H 2 O + 10 CO 2

22 Voltac Cells In spontaneous oxdaton-reducton (redox) reactons, electrons are transferred and energy s released.

23 Voltac Cells We can use that energy to do work f we make the electrons flow through an external devce. We call such a setup a voltac cell.

24 Voltac Cells A typcal cell looks lke ths. The oxdaton occurs at the anode. The reducton occurs at the cathode.

25 Voltac Cells Once even one electron flows from the anode to the cathode, the charges n each beaker would not be balanced and the flow of electrons would stop.

26 Voltac Cells Therefore, we use a salt brdge, usually a U-shaped tube that contans a salt soluton, to keep the charges balanced. Catons move toward the cathode. Anons move toward the anode.

27 Voltac Cells In the cell, then, electrons leave the anode and flow through the wre to the cathode. As the electrons leave the anode, the catons formed dssolve nto the soluton n the anode compartment.

28 Voltac Cells As the electrons reach the cathode, catons n the cathode are attracted to the now negatve cathode. The electrons are taken by the caton, and the neutral metal s deposted on the cathode.

29 Electromotve Force (emf) Water only spontaneously flows one way n a waterfall. Lkewse, electrons only spontaneously flow one way n a redox reacton from hgher to lower potental energy.

30 Electromotve Force (emf) The potental dfference between the anode and cathode n a cell s called the electromotve force (emf). It s also called the cell potental, and s desgnated E cell.

31 Cell Potental Cell potental s measured n volts (V). 1 V = 1 J C

32 Standard Reducton Potentals Reducton potentals for many electrodes have been measured and tabulated.

33 Standard Hydrogen Electrode Ther values are referenced to a standard hydrogen electrode (SHE). By defnton, the reducton potental for hydrogen s 0 V: 2 H + (aq, 1M) + 2 e H 2 (g, 1 atm)

34 Standard Cell Potentals The cell potental at standard condtons can be found through ths equaton: E cell = E red (cathode) E red (anode) Because cell potental s based on the potental energy per unt of charge, t s an ntensve property.

35 Cell Potentals For the oxdaton n ths cell, E red = 0.76 V For the reducton, E red = V

36 Cell Potentals E cell = E red (cathode) E red (anode) = V ( 0.76 V) = V

37 Oxdzng and Reducng Agents The greater the dfference between the two, the greater the voltage of the cell.

38 Electrochemcal Cells We are nterested n the process that affects the transport of charge at the nterface Electrode charge s carred by electrons Electrolyte charge s carred by ons An electrochemcal cell typcally conssts of two electrodes

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40 How does the relate to G? G = -nfe Where: G = free energy n = number of moles of electron F = Faraday s constant (9.65 x 10 4 J/V mol) E o = standard redox potental If at nonstandard state: G = -nfe

41 Gbbs Free Energy A cell n whch the overall cell reacton s NOT n equlbrum can generate electrcal work The cell potental s related to the magntude of ths work Thus the E cell s also known as the EMF For spontaneous reacton, the cell potental should be postve

42 Electrochemcal Cells under Nonstandard Condtons Remember, the standard cell potental, E s measured under standard condtons (298 K, 1 atm, and 1.0 M). If we are not under standard condtons, we have to solve for E under nonstandard condtons Use the Nernst Equaton (where Q s the electrochemcal cell reacton): RT Ecell = E θ cell ln(q) nf

43 Cells at Equlbrum The mnmum on the curve s at equlbrum Equlbrum The Gbbs Free Energy s zero, and thus the cell cannot do external work Thus: Q = K; the equlbrum constant for the θ cell reacton nfe ln K = RT cell

44 Cells at Equlbrum Thus we can get the standard potental of the cell

45 Structure of the Double Layer Interface Role Electrode Knetcs are governed by the potental dfference across a thn layer adjacent to the electrode surface Ths layer s called the double layer Potental dfference across the thn layer s about 0.1V Large electrc feld (10 6 V/cm)

46 Structure of the Double Layer Interface Role Large drvng force for the electrode reacton Because of the large electrc feld we wll have charge separaton n the double layer Electroneutralty wll not apply to the double layer regon At equlbrum no current s appled (use thermodynamcs) When have current devate from equlbrum Dfference between the potental and the equlbrum potental s called the OVERPOTENTIAL

47 Structure of the Double Layer Interface Role The surface overpotental s gven by: η s = φ φ o Where : η s s the surface overpotental φ s the potental due to the current φ 0 s the equlbrum potental

48 Structure of the Double Layer When we apply a potental to an electrode, the charges that accumulate at the surface attract opposte charges from the electrolyte We have a dstrbuton of charges to balance There are a number of dfferent models to determne ths effect (of the double layer): Helmholtz model Gouy and Chapman model Stern model

49 Helmholtz model Developed n 1879 Smplest Two parallel layers of charges separated by solvent molecules Dstance (d) represents the outer Helmholtz layer Fxed dstrbuton of layer Electrode d Electrolyte Solvent

50 Gouy-Chapman Model Ψ 0 - Dstance from Surface Dffuson plane (λ) + + x

51 Gouy-Chapman Model Assumed Posson-Boltzmann dstrbuton of ons from surface ons are pont charges ons do not nteract wth each other no fxed charges Assumed that dffuse layer begns at some dstance from the surface (λ)

52 Stern Model Ψ ζ Ψ Dffuson layer Stern Plane Shear Plane Gouy Plane d + + Bulk Soluton x

53 Stern Model Ths combnes the Helmholtz and the Gouy-Chapman Some of the charge s fxed (d regon) and some s dffuse (or spread out) The total length of the boundary s gven by the fxed regon plus the dffuse regon

54 Consequences of the Double-Layer Speces outsde the Helmholtz regon are too dstant to react. The drvng force for the reacton s the potental drop across the Helmholtz regon, rather than the potental drop across the whole double layer. Concentraton at the bulk s dfferent to the concentraton at the surface of the electrode.

55 Consequences of the Double-Layer When we study the knetcs (next) we need only the ntrnsc effect of knetcs (need to elmnate the effect of the double layer): Add a non-reactng supportng electrolyte to the soluton (f lqud); Ths can ncrease the C G-C, then the overall capactance can be approxmated by C H.

56 Electrode Knetcs In ordnary (by that I mean n a typcal reacton) we express the progress of a reacton by plottng the reacton coordnate versus the energy

57 Electrode Knetcs Now, let us consder one elementary step electrochemcal reacton: kc Where: + O + e O + s the oxdzed speces R s the reduced speces k c s the cathodc reacton rate constant k a s the anodc reacton rate constant k a R

58 Electrode Knetcs A more negatve potental (more postve energy) tends to promote reducton At progressvely more negatve potental, the energy of the oxdzed speces s ncreased φ 3 : reducton s favored φ 1 : oxdaton s favored φ 2 : equlbrum potental; no net reacton takes place

59 Electrode Knetcs Consder the case where we start an experment at the potental φ 1 and we reduce t to φ 2. The actvaton energy for the frst process (E act1 ) s hgher than that for the second process (E act2 ) We can express the actvaton energy for the second process as a functon of the frst: G = G + βnf( φ ) 1 C 2 C1 2 φ

60 Electrode Knetcs Where β s the symmetry factor (transfer coeffcent) whch represents the fracton of energy that has been used to reduce the actvaton energy of the reacton. Smlarly, the actvaton energy for the anodc process (whch ncreases) can be expressed by: G a = G ( 1 β ) nf( φ φ ) 2 a1 2 1 n s the number of electrons transferred n the reacton (n s 1 most of the tme, unusual to have more than 1 n an elementary step)

61 Electrode Knetcs The form of our knetc expresson s the same as that for chemcal reactons (Arrhenus relatonshp) Where: k = G k exp RT k s a rate constant (cm/s) G s the free energy of actvaton

62 Electrode Knetcs The rate of electrochemcal reacton s drectly proportonal to the current densty r = nf = k ' c exp G RT r s the reacton rate (mol/s cm 2 ) s the current densty (A/cm 2 ) c s the reactant concentraton (mol/cm 3 )

63 Electrode Knetcs For the general anodc reacton agan: O + + e kc k R We can substtute G a a = G ( 1 β ) nf( φ φ ) 2 a1 2 1 Into r = nf = k ' c G exp RT

64 Electrode Knetcs And assumng that we have a reference electrode (e drop the subscrpts) we get: r a = a nf = k ' a c R exp G a (1 RT β ) nf φ

65 Electrode Knetcs We can redefne the reacton constant to nclude the actvaton energy at our reference potental: Ths s from substtutng k = k exp Into the last equaton, gvng: G RT r a = a nf = k a c R (1 exp β ) nf RT φ

66 Electrode Knetcs Smlarly for the cathodc reacton: r c = c nf = k c c o exp β nf RT φ The NET current densty (= a = c ) s the dfference between the anodc and cathodc current denstes r = r a = r c = nf = k a c R exp ( 1 β ) nfφ RT k c c o βnfφ exp RT

67 Electrode Knetcs At equlbrum the net current densty s ZERO However, the rates of the anodc and cathodc reacton are NOT ZERO The magntude of both ( a and c ) are the same Ths s called the EXCHANGE CURRENT DENSITY ( o )

68 Electrode Knetcs If we desgnate the equlbrum potental as φ o then ( β ) 0 1 nfφ = k c exp k c a R c o nf RT exp 0 βnfφ RT 0 0 φ Takng the logs of the equaton above and we get: = RT nf ln k k c a RT nf ln C C R o

69 Electrode Knetcs Now, f we substtute ths equaton 0 φ = nto RT nf ln k k c a RT nf ln C C R o r = r a = r c = nf = k a c R exp ( 1 β ) nfφ RT k c c o βnfφ exp RT

70 Electrode Knetcs We Get (assumng that we use the defnton of overpotental) o η = φ φ s where, η s the surface potental, φ s the s 0 potental due to the current, and φ equlbrum potental s the nf = k a c R exp ( 1 β ) RT nf ηs + RT nf ln k k c a + RT nf ln C C o R k c c o βnf exp RT η s + RT nf ln k k c a + RT nf ln C C o R

71 = nfk 1 β c Electrode Knetcs Rearrangng ths equaton gves: k β a c 1 β o c β R exp ( 1 β ) nf βnf ηs exp ηs RT RT Ths s the general knetcs expresson for the frst order elementary step gven n O + + e k k c a R

72 Electrode Knetcs The concentraton of the reactants are at the surface of the electrode The cathodc and anodc knetc constants can be evaluated at equlbrum from the exchange current densty k = o and k = c a nfc 0 o Where the superscrpt (0) represents equlbrum condtons o nfc 0 R

73 Electrode Knetcs Substtutng the knetc constants nto the general expresson gven earler, we get: ( ) = s s R R o o o RT nf RT nf c c c c η β η β β β exp 1 exp 0 1 0

74 Butler-Volmer Equaton Redefnng the transfer coeffcents for the anodc and cathodc components as: α a = (1 β ) n α = βn c And assumng the concentraton at the surface s equal to the concentraton at the bulk whch wll be the case under equlbrum condtons, then we get

75 Butler-Volmer Equaton We get α F α F = exp a η exp c η o s s RT RT Ths s known as the Butler-Volmer Equaton Thus there are three varables (α a, α c and 0 ) that need to be determned to use a Butler- Volmer equaton

76 Butler-Volmer Equaton B-V equatons gve a good representaton of expermental data for many systems The exchange current densty s a strong functon of temperature When the exchange current densty s very large, the reactons are sad to be REVERSIBLE

77 Butler-Volmer Equaton When TWO reactons take place smultaneously, on the same electrode surface, we can use the BV equaton for both of them We wll have to determne the ndvdual parameters for both reactons

78 Lnear Form of the Butler-Volmer Equaton One of the dsadvantages of the BV equaton s that the overpotental cannot be expressed mplctly To study ths, two approxmatons have been made: Small surface overpotental Large surface overpotental

79 Lnear Form of the Butler-Volmer Equaton When the overpotental s very small, the exponental term n the BV equaton α af exp ηs RT αcf exp η RT = o s Can be expanded usng the Maclaurn seres, neglectng some of the terms n the seres: = o ( α + α ) F a RT c η s

80 αnf βnf = 0 exp( η) exp( η) RT RT

81 Lnear Form of the Butler-Volmer Equaton Ths s a lnear form of the BV equaton The current densty s a functon of only one parameter ( o and the transfer coeffcents can be defned as one constant) It s used to model systems operatng at low current denstes Often used when the overpotental s 10 mv or less

82 Tafel Equaton If the overpotental s large and postve, the second term n the BV equaton can be neglected Thus: α af αcf exp ηs exp η RT RT = o s α af = o exp ηs RT If the overpotental s large and negatve, then the frst term can be gnored: α = cf o exp ηs RT

83 η B s A Tafel Equaton These are known as TAFEL EQUATIONS Take the logs of α af = o exp ηs RT B log A = The constant B s called the TAFEL SLOPE Use of the Tafel approxmaton depends On the error that can be tolerated 2.303RT It s generally used when the overpotental Is at least 50 to 100 mv α af The Tafel slope vares between 30 to 300 mv/decade = = 2.303RT α F a log o

84 Tafel Equaton Values of the exchange current densty and the transfer coeffcent are obtaned expermentally Plot overpotental versus log(). The slope of the lne wll gve the transfer coeffcent, and the ntercept wll gve the exchange current densty We wll come back to ths later so please try and remember!!!

85 Tafel Slope slope = αf/rt slope = (1 α)f/rt ln I E n ln{i n } E 85 c a b d

86 Drvng force for conducton Lorentz Force Law

87 Drvng force for conducton current densty J = σ Ε electrcal feld electrcal conductvty By the way: also Ohm s Law!

88 Terms for electronc conducton Resstance Resstvty V = I x R ρ = RA/l Conductance G = σl/a Conductvy J = σe conductvty charge per electron σ = n µ e q e e 1/[Ωm] or S/m number of electrons electron moblty

89 Classfcaton Materal class Conductors Sem- Conductors Insulators

90 Ionc Conducton Ths s due to the crystal structure Electronc conducton s prmarly due to the electronc band gap Can vary from hghly onc conducton (almost no electronc) to quas-metallc

91 Defects n Non-stochometrc Bnary Compounds We have: Intrnsc defects Extrnsc defects (dopants/mpurty) Intrnsc defects fall nto two man categores (see later): Schottky Defects Frenkel Defects

92 Potental Gradents as Drvng Force Usually drvng force for dffuson s the chemcal potental of the partcles Thus drvng force for transport of electrcal charge s the electrcal potental gradent Let us consder transport of partcles under force F 2 j (moles or partcles/cm sec) = c (moles or partcles/cm 3 ). v ( cm / s)

93 Potental Gradents as Drvng Force Drft velocty v s proportonal to the drvng force, F B s the proportonalty factor (Beweglchket) t s the average drft velocty per unt drvng force and dp F = v = B Where P s the potental dx F

94 Potental Gradents as Drvng Force Thus, dp j = cv = cb dx Now, we can say µ s where, µ = µ Where, a = c c o o + kt ln related to chemcal a conc reference conc actvty a

95 Potental Gradents as Drvng Force Thus, dµ = dx Thus, Now, kt j D d ln c dx = c = B B kt kt c dc dx dµ dc = BkT dx dx = Dff Coeft. dc Thus, j = D dx Ths s only vald for neutral = partcles

96 Dffuson Before we go any further let us look at the concept of dffuson

97 The Sold State Sold State Electrochemstry can be splt nto 2 man felds: Ioncs Propertes of the electrolyte Electrodcs Electrode reactons

98 Bref hstory of structure, stochometry, and defects Early chemstry had no concept of stochometry or structure. The fndng that compounds generally contaned elements n ratos of small nteger numbers was a great breakthrough! Understandng that external geometry often reflected atomc structure. Perfectness ruled. Non-stochometry was out. Intermetallc compounds forced re-acceptance of nonstochometry. But real understandng of defect chemstry of compounds s less than 100 years old.

99 Introducton Classcal chemstry and crystallography gave an dealzed pcture of the composton and crystal structure of norganc compounds. It was not untl the 1930's when Wagner and Schottky (1930) showed, through statstcal thermodynamc treatment of mxed phases that crystal structures are not deal. Some lattce stes wll be empty (vacant) and extra atoms may occupy the ntersttal space between the atoms on the lattce stes. The empty lattce stes are termed vacances and the extra atoms, ntersttal atoms. C. Wagner and W. Schottky, Theore der geordneten Mschphasen, Z. Phys. Chem., Vol 11, 1930, p 163

100 Introducton Followng Wagner and Schottky all crystallne solds wll at any temperature contan vacances and extra atoms and wll as such exhbt devatons from the deal structure. Furthermore, all norganc compounds may n prncple have varable composton and thus be nonstochometrc. These devatons or mperfectons are called defects. The reason for ths s that by conventon the deal structure s used as the reference state, and any devaton from ths deal state s termed a defect.

101 Perfect Crystal Our course n defects takes the perfect structure as startng pont. Ths can be seen as the deally defect-free nteror of a sngle crystal or large crystallte gran at 0 K.

102 Some smple classes of oxde structures Formula Caton:anon coordnaton Type and number of occuped nterstces MO 6:6 1/1 of octahedral stes MO 4:4 1/2 of tetrahedral stes M 2 O 8:4 1/1 of tetrahedral stes occuped M 2 O 3, ABO 3 6:4 2/3 of octahedral stes MO 2 6:3 ½ of octahedral stes AB 2 O 4 1/8 of tetrahedral and 1/2 of octahedral stes fcc of anons NaCl, MgO, CaO, CoO, NO, FeO a.o. Znc blende: ZnS Ant-fluorte: L 2 O, Na 2 O a.o. Spnel: MgAl 2 O 4 Inverse spnel: Fe 3 O 4 hcp of anons FeS, NS Wurtzte: ZnS, BeO, ZnO Corundum: Al 2 O 3, Fe 2 O 3, Cr 2 O 3 a.o. Ilmente: FeTO 3 Rutle: TO 2, SnO 2

103 We shall use 2-dmensonal structures for our schematc representatons of defects Elemental sold Ionc compound

104 Perfect vs defectve structure Perfect structure (deally exsts only at 0K) No mass transport or onc conductvty No electronc conductvty n onc materals and semconductors; Defects ntroduce mass transport and electronc transport; dffuson, conductvty New electrcal, optcal, magnetc, mechancal propertes Defect-dependent propertes

105 Kröger-Vnk notaton for 0-dmensonal defects Pont defects Vacances Intersttals Substtutonal defects Electronc defects Delocalsed electrons electron holes Valence defects Trapped electrons Trapped holes Cluster/assocated defects Kröger-Vnk-notaton c A s A = chemcal speces or v (vacancy) s = ste; lattce poston or (ntersttal) c = charge Effectve charge = Real charge on ste mnus charge ste would have n perfect lattce Notaton for effectve charge: postve / negatve x neutral (optonal)

106 Perfect lattce of MX, e.g. ZnO 2+ Zn Zn x Zn Zn 2- O O x O O v x v

107 Vacances and ntersttals // v Zn Zn v O // O

108 Electronc defects / e / Zn Zn h Zn Zn O O

109 Foregn speces / Ag Zn Ga Zn / N O F O L

110 Defects are donors and acceptors E H E c Ga Zn x v O v O v O / Ag Zn x v Zn / v Zn // v Zn E v

111 Ionc Conducton Ths s due to the crystal structure Electronc conducton s prmarly due to the electronc band gap Can vary from hghly onc conducton (almost no electronc) to quas-metallc

112 Defects n Non-stochometrc Bnary Compounds We have: Intrnsc defects Extrnsc defects (dopants/mpurty) Intrnsc defects predomnantly fall nto two man categores: Schottky Defects Frenkel Defects

113 Lattce Defects The concept of a perfect lattce s adequate for explanng structure-nsenstve propertes (esp. for metals). But, to understand structure-senstve propertes, t s necessary to consder numerous lattce defects. Practcally all mechancal propertes are structure-senstve propertes. (almost) structure-nsenstve elastc constants Meltng ponts densty Specfc heat coeffcent of thermal expanson structure-senstve Electrcal conductvty Semconductng propertes Yeld stress Fracture strength Creep strength

114 Types of Imperfectons Vacancy atoms Intersttal atoms Substtutonal atoms Dslocatons Edges, Screws, Mxed Pont defects Lne defects Gran Boundares Area/Planar defects Stackng Faults Ant-Phase and Twn Boundares

115 Length Scale of Imperfectons pont, lne, planar, and volumetrc defects Vacances, mpurtes dslocatons Gran and twn boundares Vods Inclusons precptates

116 Pont Defects Vacances: vacant atomc stes n a structure. dstorton of planes Vacancy Self-Intersttals: "extra" atoms n between atomc stes. dstorton of planes selfntersttal

117 Pont Defects Self-ntersttal: atom crowded n holes Vacancy: a vacant lattce ste It s not possble to create a crystal free of vacances. About 1 out of 10,000 stes are vacant near meltng. Self-ntersttals are much less lkely n metals, e.g.,, as t s hard to get bg atom nto small hole - there s large dstortons n lattce requred that costs energy. Thermodynamcs (temperature and countng) provdes an expresson for Vacancy Concentraton: (see handout) N = exp Q v k B T N v atom Vac Q v =vacancy formaton energy k B = 1.38 x J/atom-K = 8.62 x 10 5 ev/atom-k k B /mole = R = cal/mol-k Defects ALWAYS cost energy!

118 Pont Defects n Ceramcs Caton Intersttal Caton Vacancy Anon Vacancy

119 Frenkel and Schottky Defects: pared anons and catons Electronc neutralty must be mantaned n crystal. Defects must come n pars to mantan Q=0. Caton-vacancy + Caton-ntersttal = Frenkel Defect ( Q=0) In AX-type crystals, Caton-vacancy + Anon-vacancy = Schottky Defect ( Q=0) Schottky Frenkel 119

120 Schottky Dsorder Gven by: M X M + O Where, X O K S = V '' M.. O [ '' ][..] V V M + V O + ( MO) defect

121 Frenkel Dsorder Ths s less common, except for the latter case of Ant-Frenkel (AF) dsorder Frenkel dsorder s gven by M Where, Ant O X M X O + V Where, X Frenkel: + V X K K F AF = [ ][ ].. '' M V O = M ''.. + V.. O '' M [ ][ ] ''.. O V + V M O

122 Temperature Dependence of Schottky Defect Here [ ] [ ] V '' = V.. = n = N exp f M O s 2kT N s = number of Schottky defects/m 3 T = temperature N = number of catons/anons/m 3 ΔH f = enthalpy of formaton H

123 MF Me Temperature Dependence of Schottky Defect For alkal and lead haldes ΔH f 2.14 x 10-3 T m For oxdes, there s no such approxmaton Influence of dopant (eg MF 2 n MO, and Me 2 O n MO). '' M X + 2F + V 2 M O M 2 O 2Me. M + O X O + V.. O

124 Steady-state Dffuson: J ~ gradent of c Concentraton Profle, C(x): [kg/m 3 ] Cu flux N flux Concentraton of Cu [kg/m 3 ] Concentraton of N [kg/m 3 ] Poston, x Fck's Frst Law: D s a constant! The steeper the concentraton profle, the greater the flux!

125 Steady-State Dffuson Steady State: concentraton profle not changng wth tme. Apply Fck's Frst Law: If Jx)left = Jx)rght, then Result: the slope, dc/dx, must be constant (.e., slope doesn't vary wth poston)! J x = D dc dx dc = dc dx dx left rght

126 Steady-State Dffuson Rate of dffuson ndependent of tme J ~ dc dx C 1 C 1 Fck s frst law of dffuson f lnear dc dx C 2 C 2 J = D dc dx x 1 x D dffuson coeffcent 2 x C x = C x 2 2 C x 1 1

127 Non-Steady-State Dffuson Concentraton profle, C(x), changes w/ tme. To conserve matter: Fck's Frst Law: Governng Eqn.:

128 Non-Steady-State Dffuson: another look Concentraton profle, C(x), changes w/ tme. Rate of accumulaton C(x) C t dx = J x J x+dx C t dx = J x (J x + J x x dx) = J x x dx Usng Fck s Law: C t = J x x = x D C x Fck s 2nd Law If D s constant: Fck's Second "Law" c t = x D c x D c 2 x 2

129 Non-Steady-State Dffuson: C = c(x,t) concentraton of dffusng speces s a functon of both tme and poston Fck's Second "Law" c t D c 2 x 2 Copper dffuses nto a bar of alumnum. C s B.C. at t = 0, C = C o for 0 x at t > 0, C = C S for x = 0 (fxed surface conc.) C = C o for x =

130 Non-Steady-State Dffuson Cu dffuses nto a bar of Al. C S C(x,t) Fck's Second "Law": c t D c 2 x 2 C o Soluton: "error functon erf (z) = 2 2 e y dy 0 π z

131 Potental Gradents as Drvng Force Usually drvng force for dffuson s the chemcal potental of the partcles Thus drvng force for transport of electrcal charge s the electrcal potental gradent Let us consder transport of partcles under force F 2 j (moles or partcles/cm sec) = c (moles or partcles/cm 3 ). v ( cm / s)

132 Potental Gradents as Drvng Force Drft velocty v s proportonal to the drvng force, F v = B F B s the proportonalty factor (Beweglchket) t s the average drft velocty per unt drvng force and F = dp dx Where P s the potental

133 Potental Gradents as Drvng Force Thus, dp j = cv = cb dx Now, we can say µ s where, µ = µ Where, a = c c o o + kt ln related to chemcal a conc reference conc actvty a

134 Potental Gradents as Drvng Force Thus, dµ dx = Thus, Now, kt j D d ln c dx = c = B B kt kt c dc dx dµ dc = BkT dx dx = Dff Coeft. dc Thus, j = D dx Ths s only vald for neutral = partcles

135 Smplfed Model for One-Dmensonal Dffuson From, j = D dc dx And, let us consder a 1D model where the partcles jump between parallel planes separated by dstance s.

136 Drunkard s walk from G. GAMOW

137 Random Dffuson Let us consder jumps of the atoms are random We have, after a startng pont, the dsplacement of a dffusng atom R n (after n jumps) R n = s 1 + s s n j= 1 To obtan a value for the magntude of the sum, smply square R n = n s j

138 Random Dffuson Thus, R 2 n = 2 Rn n = s + j= 1 2 j 2 n 1 n j= 1 k = j+ 1 s j s k If, as above, the jump vectors are equal, e s 1 = s 2.= s j = s (as for the cubc system) And f they are random and uncorrelated, then the 2 nd term on the RHS wll approach zero for large numbers of jumps, as on an average s j and s k have an equal chance of beng +ve or ve.

139 Random Dffuson Thus, R 2 n Thus, n = j= 1 R n 2 s j = = ns ns 2 = ns 2 = 6Dt mean dsplacement Thus, the mean dsplacement s proportonal to the of the number of jumps X ndvdual jump dstance By combnng the equaton above and we can express the random dffuson n terms of D Thus, 2 2 (r refers to random walk) R n or, = R n ns = = 6D t 6D t r r t s the tme whch the mean square Dsplacement takes place

140 R x 2 n Random Dffuson Let us consder the dsplacement n a sngle dmenson (eg x-drecton) From smple geometry, we have = 3x 2 where x s the mean square dsplacement n each orthogonal drecton (for a cubc system) = 2 2D t r Ths gves the mean dffuson length n one drecton n a 3D cubc crystal

141 Dffuson Mechansms Vacancy Mechansm

142 Dffuson Mechansms Intersttal Mechansm

143 Dffuson Mechansms Intersttalalcy Mechansm

144 Other Mechansms These nclude: Crowdan Rng Mechansm (more common n metals)

145 Dffuson n Compounds: Ionc Conductors Unlke dffuson n metals, dffuson n compounds nvolves second-neghbor mgraton. Snce the actvaton energes are hgh, the D s are low unless vacances are present from non-stochometrc ratos of atoms. e.g., NO There are Schottky defects O N O N O N O N O N O N 2+ O 2 N O N 2+ O 2 N O 2 + O N N 2+ O N O N O N O N O N O The two vacances cannot accept neghbors because they have wrong charge, and on dffuson needs 2nd neghbors wth hgh barrers (actvaton energes). 145

146 Dffuson n Compounds: Ionc Conductors D s n an onc compound are seldom comparable because of sze, change and/or structural dfferences. Two sources of conducton: on dffuson and va e - hoppng from ons of varable valency, e.g., Fe 2+ to Fe 3+, n appled electrc feld. e.g., onc In NaCl at 1000 K, D Na+ ~ 5D Cl,whereas at 825 K D Na+ ~ 50D Cl! Ths s prmarly due to sze r Na+ = 1 A vs r Cl =1.8 A. e.g., oxdes In uranum oxde, U 4+( O 2 ) 2, at 1000 K (extrapolated), D O ~ 10 7 D U. Ths s mostly due to charge,.e. more energy to actvate 4+ U on. Also, UO s not stochometrc, havng U 3+ ons to gve UO 2-x, so that the anon vacances sgnfcantly ncrease O 2- moblty. e.g., sold-solutons of oxdes (leads to defects, e.g., vacances) If Fe 1-x O (x=2.5-4% at 1500 K, 3Fe 2+ -> 2Fe 3+ + vac.) s dssolved n MgO under reducng condtons, then Mg 2+ dffuson ncreases. If MgF 2 s dssolved n LF (2L + -> Mg 2+ + vac.), then L + dffuson ncreases. All due to addtonal vacances. 146

147 Ceramc Compounds: Al 2 O 3 Holes for dffuson Unt cell defned by Al ons: 2 Al + 3 O 147

148 Summary: Structure and Dffuson Dffuson FASTER for... open crystal structures lower meltng T materals materals w/secondary bondng smaller dffusng atoms catons lower densty materals Dffuson SLOWER for... close-packed structures hgher meltng T materals materals w/covalent bondng larger dffusng atoms anons hgher densty materals 148

149

150 Temperature Dependence of Attempt Frequency ω For atoms to jump, they have barrers Large part of the barrer s stran energy requred to DISPLACE neghborng atoms to create a suffcently large openng to allow atoms to jump See fgure over showng potental energy barrer

151 Potental Energy of atom dffusng n a sold

152 Temperature Dependence of Attempt Frequency ω Potental heght s ΔH m (actvaton energy to jump) Each atom vbrates n ts poston and durng a fracton of tme gven by Boltzmann dstrbuton factor (exp(- ΔH m /RT)) It possesses suffcent energy to overcome the barrer Thus, ωα exp H M RT

153 Temperature Dependence of Attempt Frequency ω Zenner (1951/52) consdered the system/atom n ts ntal equlbrum condton and n ts actvated state at the top of the potental barrer He found: G ω = ν exp RT M S = ν exp R H exp RT v s the vbratonal frequency (assumed to be approx. the Debye frequency of Hz M M

154 Resultng Analyss of Dffuson Coeffcent D r D D From the above: 2 ( Sd + SM Dr = αaoν exp R Typcally, D = D exp( Q / o Thus, Q = H r = αa νn 2 o and Q = H d, frozen M d o + H 2 Sd + SM and Do = αaoν exp R If N s a constant and ndependent of M S exp R M ) exp RT ) exp ( H H RT M d + H RT s the pre - exponental factor, and Q s the actvaton energy d = D o M ) temperature (e frozen n) exp H RT M

155 Oxygen Vacancy Dffuson n Oxygen Defcent Oxdes Here, oxygen vacances predomnate Thus, H.. S.. V o 1/3 V o H + M + S 1 M 3 2 1/ 6 exp 3 Dr = αaoν Po2 exp 4 R RT Dr ncreases as oxygen partal pressure decreases H.. V o and, Q = + H M 3 If concentraton of oxygen vacances determned by lower valent mpurtes '' [ A ] 2 SM H M Dr = αaoν M exp exp R RT and get Q = ΔH, thus get transton from ntrnsc to extrnsc...see over M

156 Dffuson Coeffcent for Oxygen Dffuson by Vacancy Mechansm n Oxygen Defcent Sold As T decreases, defect nteractons become more mportant

157 Intersttal Dffuson Here dffuson occurs by solute atoms jumpng from one ntersttal ste to another. Much smpler The expresson for ntersttal dffuson s smpler that that for substtutonal. 2 F / RT D = αa pν exp m p = number of nearest ntersttal stes a = lattce parameter, α = geometrc factor ν = vbratonal frequency F = free energy per mole for jumpng m

158 Intersttal Dffuson Thus only contans one free energy term Not dependant upon presence of vacances Because, F 2 + Sm / R Qm / RT D = αa pν exp exp Sm = entropy change of the lattce Q = work assocated wth jumpng across m = Q T S the actvaton energy

159 Measurement of Intersttal Dffuson Coeffcents The Snoek Effect The study of ntersttal dffuson by nternal-frcton. In a BCC lke ron, ntersttal atoms st on the centers of the cube edges or at the centers of the cube faces. See fgure over An ntersttal atom at ether x or w would le between two ron atoms algned n a <100> drecton.

160 Nature of the stes that ntersttal carbon atoms occupy n BBC Fe

161 Measurement of Intersttal Dffuson Coeffcents The Snoek Effect The occupancy of one of these, pushes apart the two solvent atoms (a and b) An atom at x or w ncreases the length of the crystal n the [100] drecton. An atom at y or z n the [010] and [001] respectvely Thus, f an external force s appled to the crystal (as s the case here), so that t produces a state of tensle stress parallel to the [100] axs, t wll stran the lattce and those stes wth axes parallel to [100] wll have ther openngs enlarged (closed n axes normal to stress).

162 n n K s s n = Measurement of Intersttal Dffuson Coeffcents The Snoek Effect p p If appled stress s small, and stran s small, the number of excess solute atoms per unt volume that are n ntersttal stes s small. Thus the number of stes s drectly proportonal to the stress = Ks n = addtonal number of a proportonalty constant tensle stress solute atoms n preferred postons,

163 Measurement of Intersttal Dffuson Coeffcents The Snoek Effect Each of the addtonal solute atoms adds a small ncrement to the length. The total stran of the metal conssts of two parts: Normal elastc stran (ε el ) Anelastc stran (ε an ) whch s caused by the movement of solute atoms nto stes wth axes parallel to the stress axs ε = ε + ε el an

164 Measurement of Intersttal Dffuson Coeffcents The Snoek Effect When a stress s suddenly appled, the elastc component can be consdered to develop nstantly. The anelastc stran, however, s tme dependent, and does not appear nstantly. The sudden applcaton of a stress places the solute atoms n a non-equlbrum dstrbuton Equlbrum now corresponds to an excess of solute atoms, n p, n stes wth axes parallel to the stress

165 Measurement of Intersttal Dffuson Coeffcents The Snoek Effect Equlbrum occurs due to thermal movement eventually The net effect of the stress s to cause a slghtly greater number of jumps However, at equlbrum ths number wll be the same. The rate at whch the number of addtonal atoms n preferred stes grows, depends on the number of the excess stes that are stll unoccuped

166 D D D Dffuson along Gran Boundares and Free Surfaces Q s Atom movement not only n bulk of crystals Can occur along surfaces and gran boundares Expermental measurements have shown that surface and gb forms are: S b S o = = D D b and S o o e e D Q Q b o b S / RT / RT are the constants of the actvaton energy the dffuson coeffents

167 Dffuson along Gran Boundares and Free Surfaces It has been shown that dffuson s more rapd along gbs than n the nteror. Free surface rates are larger than both Surface dffuson s very mportant n metallurgcal processes However, gbs are also very mportant as they form a network, and there are more of them They also cause large errors n calculatons of dffuson n crystals

168 Dffuson along Gran Boundares and Free Surfaces When we measure the dffuson coeffcent of a polycrystal, the value s the combned effect of volume and gran boundary. What s obtaned s an apparent dffusvty (D ap ) The dffuson s not smply the summaton. Gran Boundary dffuson s faster than bulk. However, as the gran boundares fll up, loss of dffuson occurs

169 Dffuson along Gran Boundares and Free Surfaces See fgure over Ths represents a dffuson couple composed of metals A and B Both are polycrystallne Gran boundares only on RHS Arrows show A nto B To study ths we remove thn layers and analyze at a dstance (dx)

170 Combned effect of gb and bulk dffuson

171 The problem s complex. For a gven rato of D gb /D b the relatve number of A atoms that reach dx s a functon of gran sze. The smaller the gran sze, the greater the total gran-boundary are avalable, and thus the more sgnfcant they become. Fg shows gb and bulk dffuson for Ag n Ag

172 Bulk and gb dffuson for Ag

173 Dffuson along Gran Boundares and Free Surfaces Both types of dffuson show a straght lne relatonshp on the log D-1/T system For gran boundary dffuson, the equaton of the lne s (D b )D gb = e -20,200/RT For bulk dffuson t s (D l )D b = e - 45,950/RT Thus: Dffuson easer along gb Dfferent effects of T on gb and bulk

174 Dffuson along Gran Boundares and Free Surfaces Thus, at hgh temperatures dffuson through the bulk overpowers the gb dffuson. At low temperatures, gb domnates.

175 Dffuson Coeffcent of Pont Defects Sometmes we can consder dffuson of vacances themselves The vacancy can theoretcally jump to any one of the occuped nearest neghbors 2 Thus, Dv = αaoωn N s the fracton of occuped stes In dlute solutons, N s approx 1 Thus, D s NOT dependent on Nd Thus, Dr N = DV Nd General Form of the Equaton Dr N = Dd Nd D = defect dffuson coeffcent d

176 Overvew Now, for a fxed partcle,, wth charge Z e dφ F = Ze = ZeE dx φ s the electrcal potental dφ E = dx The flux of j = c B F Concentraton Moblty

177 Thus, j = u = = ( Z e) Overvew Thus, Z ec 2 = charge moblty = Therfore, Z ej = current densty x flux x charge B B E c E = Z ec u Z eb E = conductvty x E

178 Overvew Where, Total electrcal conductvty, σ, s the sum of the partal conductvtes, σ of dfferent charge carrers σ σ = = Zecu σ = conductvty (s/cm) where, t = σ σ

179 Charge Carrers n Ionc Compounds Now, ( ) elec on p n elec a c on p n a c p n a c p p n n a a c c p n a c t t t t t t t t t σ t σ t σ t σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ + = + = + = = = = = = = = 1 and,,, or, Das Bld kann zurzet ncht angezegt werden.

180 Nernst-Ensten Relaton Now, D = But, u = = ktb by calculatng ths Z eb Z ec u and E = σ E wth kt kt Then, D = ktb = u = σ 2 2 Ze cz e Ths s the Nernst-Ensten Relaton Consder 1D system wth a seres of parallel planes separated by a dstance s

181 Nernst-Ensten Relaton Assume homogeneous Volume concentraton n the planes s c Equal probablty of jumpng to neghborng planes Number of partcles whch jump from plane 1 to 2, and 2 to 1per unt tme s equal and opposte And = 1/ 2ωc s

182 Nernst-Ensten Relaton And, = = + = kt H kt Z ese kt Z ese s c j kt Z ese H k S m m m reverse exp k S exp where, 2 exp 2 exp 2 1/ Thus, 2 exp exp m ν ω ω ν ω

183 Nernst-Ensten Relaton Wth no electrc feld, the actvaton energy s ΔH m When apply and electrc feld, E, jump frequency n postve drecton s ncreased, and n negatve drecton s decreased In forward drecton, the actvaton energy s REDUCED to: H m In the negatve drecton, t s ncreased to H + 1/ 2Z ese m 1/ 2Z ese

184 Electrc Feld on Mgraton Energy

185 j Nernst-Ensten Relaton = 1/ The net partcle flux s the dfference between the number of jumps n the forward and reverse drectons 2c where, ω s ( ω ω ) forward forward reverse S = ν exp k m exp H m ZeSE 2 kt

186 Nernst-Ensten Relaton When, Z ese << 2kT We can wrte that If, D czee Then, j = D kt Now, we know that j = Thus, = 1/ 2ωs Z ec D B E = 2 B kt = c u = j E u = 1/ 2ωs σ E = Z e kt Z e 2 c ZeE kt kt = σ c Z e 2 2

187 Nernst-Ensten Relaton Or Rearranged ( Z e) 2 c D σ = kt

188 Ionc Conducton: related to fuel cells Molten salts and aqueous electrolytes conduct charge when placed n electrc feld, +q and q move n opposte drectons. The same occurs n solds although at much slower rate. Each on has charge of Ze (e = 1.6 x amp*sec), so on movement nduces onc conducton σ = nµze Conductvty D va the Ensten equatons: Hence σ onc = nz 2 e 2 k B T D = nz 2 e 2 k B T s related to moblty, µ, whch s related to µ = ZeD / k B T D o e Q / RT log 10 σ onc ~ ln nz 2 e 2 k B T D o Q 2.3RT So, electrcal conducton can be used determne dffuson data n onc solds. e.g., What conductvty results by Ca 2+ dffuson n CaO at 2000 K? CaO has NaCl structure wth a= 4.81 A, wth D(2000 K)~10-14 m 2 /s, and Z=2. n Ca 2+ = 4 cell cell (4.81x10 10 m) = x1028 / m 3 σ = nz 2 e 2 k B T D ~ 1.3x10 5 ohm cm

189 For Ionc Conductvty to occur

190 Conductvty n MO Electronc Conductvty Here, σ el = σ n + σ p = enµ n + epµ p Now, E g E g = 2( E s c)( ev ) = sze of band gap, Es = Eat / equv,c = constant Drft or carrer mobltes

191 Energy Band

192 Intrnsc As T ncreases, electrons can be excted across the forbdden band nto conducton band ntrnsc onzaton Usng classcal statstcs n = N c exp E c kt E F p = N v exp E F kt E v

193 Intrnsc N c and N v are number of avalable states/densty of states n conducton and valence bands E c s the lowest energy level n conducton band E v s hghest level n the valence band E F s the Ferm level

194 Intrnsc Now, the defect equlbrum s O and, = e K Eg Thus, K = np = NcNv exp kt If number of electrons and holes are equal n = p = + h = np semconductor), then K. 1/ 2 = E ( N N ) 1/ 2 g c v exp 2kT (as n an ntrnsc

195 Intrnsc Thus, σ elec = σ n + σ p = enµ n + epµ p = e ( ) 1/ 2( + ) N N µ µ c v Often wrtten as n p exp E g 2kT σ elec = CONST.exp E g 2kT

196 Intrnsc Thus, ntrnsc electronc conductvty INCREASES wth DECREASING energy gap

197 Extrnsc Ths s summarzed n the Fgure over The Donor Effect The onzaton of a Donor D x may be D x = D. + e Thus, K D = [ ]. D n [ ] x D and N D = total number of donors = [ ] [ ]. x D + D

198 Doped System

199 η Ω s the resstance to onc movement (gven as the conductvty of the electrolyte materal; ths s typcally gven as approxmately 1S/cm at 1000 o C) We also must consder the onc transport number of the materal (t ) whch s gven as: t = σ /σ total (approxmately 1 for doped- ZrO 2 )

200 Methods for Measurng Partal Ionc and Electronc Conductvtes We wll look at the specfc measurements of partal conductvtes, before lookng at how to measure onc conductvty n general. There are a number of methods that one can consder: Hebb-Wagner Polarzaton Method (electronc or onc) Short Crcutng Method Smultaneous Measurement of electronc and onc conductvty Tubandt/Httorf Method EMF Measurements Other lesser technques

201 General Technques

202 Advantages of EIS

203 Assume a Black Box Approach

204 So, What s EIS???

205 Complex Plane

206 The (RC) Crcut - Remember Constant phase elements (CPE) may be regarded as non-deal capactors defned by the constants Y and n, and ther mpedance s gven accordng to The CPE s very versatle ( a very general dsperson formula ): If n = 1, the CPE represents an deal capactor If n = 0, the CPE represents a resstor Z Q = [ ( ) ] n Y jω 1 If n = -1, the CPE represents an nductor If n = 0.5 the CPE represents a Warburg element 100 Peak frequency: ω 0 = (RC) -1 -X / Ω 50 0 n = 1 n = 0.9 n = R / Ω Constant phase element

207 Impedance Spectroscopy n Sold State Ioncs What: A technque for studyng the conductvty of onc conductors, mxed conductors, electrode knetcs and related phenomena Features: Elmnates the need for non-blockng electrodes The mpedance due to gran nterors, gran boundares and dfferent electrode propertes can be measured ndependently How: A small AC voltage (e.g. 10 mv 1 V) s mposed on the sample over a wde range of frequences (e.g. 1 MHz 0.1 Hz), and the complex mpedance s measured

208 Real Impedance Spectra X / MΩ cm R / MΩ cm The spectrum can be ftted by usng:

209 CONDUCTIVITY GRAINS 0.6 -X / MΩ cm R / MΩ cm Gran Boundares n Ionc Conductors

210 Key requrements for the SOFC electrolyte: Good onc conducton No electronc conducton Control of concentraton and moblty of onc and electronc speces

211 Conductvty as a functon of T

212 Conductvty n Zrcona (as an example)

213 Stablzed Zrcona

214 Zrconates (zrcona based materals, Y or Sc doped zrcona) ScSZ Structure: Cubc fluorte Oxygen conductvty Electronc conductvty Mole% Y 2 O 3 or Sc 2 O 3 Fergus, Journal of Power Sources, Avalable onlne (2006)

215 Favored Materals (SOFC) Cathode (Ar Electrode) (La 1-x Ca x )MnO 3 (Perovskte) (La 1-x Sr x )(Co 1-x Fe x )O 3 (Perovskte) (Sm 1-x Sr x )CoO 3 (Perovskte) (Pr 1-x Sr x )(Co 1-x Mn x )O 3 (Perovskte) Anode (H 2 /CO Electrode) N/Zr 1-x Y x O 2 Compostes Electrolyte (Ar Electrode) Zr 1-x Y x O 2 (Fluorte) Ce 1-x R x O 2, R = Rare Earth Ion (Fluorte) B 2-x R x O 3, R = Rare Earth Ion (Defect Fluorte) Gd 1.9 Ca 0.1 T 2 O 6.95 (Pyrochlore) (La,Nd) 0.8 Sr 0.2 Ga 0.8 Mg 0.2 O 2.8 (Perovskte) Interconnect (between Cathode and Anode) La 1-x Sr x CrO 3 (Perovskte)

216 Desgn Prncples: O 2- Conductors Hgh concentraton of anon vacances necessary for O 2- hoppng to occur Hgh Symmetry provdes equvalent potentals between occuped and vacant stes Hgh Specfc Free Volume (Free Volume/Total Volume) vod space/vacances provde dffuson pathways for O 2- ons Polarzable catons (ncludng catons wth stereoactve lone pars) polarzable catons can deform durng hoppng, whch lowers the actvaton energy Favorable chemcal stablty, cost and thermal expanson characterstcs for commercal applcatons

217 Phase Transtons n ZrO 2 Room Temperature Monoclnc (P2 1 /c) 7 coordnate Zr 4 coord. + 3 coord. O 2- Hgh Temperature Cubc (Fm3m) cubc coordnaton for Zr tetrahedral coord. for O 2-

218 Effect of Dopants: ZrO 2, CeO 2 Dopng ZrO 2 (Zr 1-x Y x O 2-x/2, Zr 1-x Ca x O 2-x ) fulflls two purposes Introduces anon vacances (lower valent caton needed) Stablzes the hgh symmetry cubc structure (larger catons are most effectve) We can also consder replacng Zr wth a larger caton (.e. Ce 4+ ) n order to stablze the cubc fluorte structure, or wth a lower valent caton (.e. B 3+ ) to ncrease the vacancy concentraton. Compound r 4+ Specfc Free Conductvty (Angstroms) 800 ºC Zr 0.8 Y 0.2 O S/cm Ce 0.8 Gd 0.2 O S/cm δ-b 2 O S/cm (730 C) B 2 O 3 s only cubc from 730 ºC to t s meltng pont of 830 ºC. Dopng s necessary to stablze the cubc structure to lower temps.

219 Here we have the case of an Oxygen Defcent Oxdes wth Lower Valent Dopant Catons eg ZrO ZrO Y wth Y 0 Y = V ' Zr.. O or Ca + 2Y ' Zr '' Zr Look at the case of Y ' Zr Defect equlbrum between vacances and electrons wll be O X O = V.. O + 2e ' + 1/ 2O 2 K VO = [.. V ] O [ X O ] O n 2 Po 1/ 2 2 Electroneutralty wll be : 2 [..] [ ' V = Y ] + n O Zr

220 Oxygen Defcent Oxdes wth Lower Valent Dopant Catons There wll be 2 condtons: If 2[Vö] n >> [Y Zr ] The foregn catons do not affect the defect equlbrum The electrons and oxygen vacancy concentratons are gven by ther own equlbrum and are proportonal to Po 2-1/4 (see earler) If 2[Vö] [Y Zr ] >> n Oxygen vacancy concentraton s determned and fxed by the dopant content Ths s the Extrnsc Regon

221 Oxygen Defcent Oxdes wth Lower Valent Dopant Catons The concentraton of the mnorty defect, n, s gven by: ( ) 1/ 2[ ] ' 1/ 2 1/ 4 n = 2KVO YZr Po2 Now, n and p are related by K = np p ncreases n the extrnsc regon as Po 2 ncreases See Fgure over

222 Conc. Of defects as a functon of oxygen partal pressure n an oxygen defcent oxde contanng oxygen vacances

223 SOFC anode: one phase vs. two phase materals. Mxed conductvty Cermets: N-YSZ Perovskte-related structures: Doped ttanates (SrTO 3 ), chromtes, vanadates, ferrtes; cerates are stable n reducng atmospheres Cubc fluorte structures: Zr and Ce-based materals (Y-T-Zr oxdes; YTZ) Pyrochlores: Gd 2 T 2 O Spnel related materals (Mg 2 TO 4 ) Bsmuth oxde

224 ) CERMETS-mxture of metals and oxde ceramcs Functon of Metal Phase: Works as a catalyst Possesses electronc conductvty Functon of Ceramc Oxde Phase: Preserve porous structure Elmnates mass transport lmtatons Matches TEC Why N? Hgh catalytc actvty Low cost

225 ) Perovskte oxdes for anodes operatng on hydrocarbon fuels : LSCM (La 0.75 Sr 0.75 Cr 0.5 Mn 0.5 O 3 ) 1. Stable; form sngle phase 2. Conductvty n reducng atmosphere s low ( S/cm) Mn 4+ /Mn 3+ Mn 3+ /Mn 2+ LSCM No coke formaton LSCM +Cu Ar LSCM+Cu+Pt CH 4 H 2 Anode Overpotentlal at 800oC n dry CH 4 Goodenough et al. Sold State Ioncs 177 (2006)

226 Oxygen reducton on the LSM cathode LIMITING STEPS LSM 1.Gas dffuson 2. Adsorpton 3. Charge transfer reacton (1) 4. Surface dffuson 5. Charge transfer reacton (2) 6. Incorporaton of oxygen ons nto electrolyte lattce YSZ At hgh temperatures >900 o C YSZ and LSM can react to form pyrochlore La 2 Zr 2 O 7 or/and perovskte SrZrO 3 Chen et al. Journal of Power Sources 123 (2006) 17-25

227 Anode Reactons

228 Cathode Reactons

229 Open Crcut Voltage (OCV)

230 Cell Voltage

231 Cell Voltage

232 OCV as a Functon of Hydrogen Partal Pressure

233 I-V Characterstcs of a Cell

234 The Losses

235 I-V Characterstcs of a Cell

236 Operaton of a SOFC

237 Sngle Cell performance for Selected H 2 /H 2 0 Ratos at 800 o C