1. KJM-MENA 3120 Inorganic Chemistry II Materials and Applications
Solid-State Electrochemistry
Fundamentals, Fuel Cells, Batteries
Week 1
Electrochemistry
Fundamentals
Defect chemistry
Diffusion and conductivity
Electrochemical cells
Truls Norby
Week 1 Introduction and Fundamentals
Week 2 Solid oxide fuel cells
Week 3 Batteries
Themes and applications not covered: Membranes, oxidation/corrosion, metallurgy, photoelectrochemistry…

2. Redox Chemistry and Electrochemistry
Redox reaction
Electrochemical reaction

4. In order to understand, analyse, and affect the conductivity in crystalline solids, we need to understand defect concentrations Defect chemistry

5. Brief history of defects
Early chemistry had no concept of stoichiometry or structure.
The finding that compounds generally contained elements in ratios of small integer numbers was a great breakthrough!
H2O CO NaCl CaCl2 NiO
Understanding that external geometry often reflected atomic structure.
Perfectness ruled. Variable composition (non- stoichiometry) was out.
However, variable composition in some intermetallic compounds became indisputable and in the end forced re-acceptance of non-stoichiometry.
But real understanding of defect chemistry of compounds mainly came about from the 1930s and onwards, attributable to Frenkel, Schottky, Wagner, Kröger…, many of them physicists, and almost all German!
Frenkel Schottky Wagner

6. Defects in an elemental solid (e.g. Si or Ni metal)
Notice the distortions of the lattice around defects
The size of the defect may be taken to be bigger than the point defect itself
Adapted from A. Almar-Næss: Metalliske materialer, Tapir, Oslo, 1991.

7. Defects in an ionic solid compound Example: Cation and anion vacancies in NiO Defects in ionic compounds are charged

8. Point defects nomenclature: Kröger-Vink notation
We will now start to consider defects as chemical entities
We need a notation for defects. Many notations have been in use. In modern defect chemistry, we use Kröger-Vink notation (after Kröger and Vink). It describes any entity in a structure; defects and “perfects”. The notation tells us
What the entity is, as the main symbol (A)
Chemical symbol
or v (for vacancy)
Where the entity is, as subscript (S)
Chemical symbol of the normal occupant of the site
or i for interstitial (normally empty) position
Its charge, real or effective, as superscript (C)
+, -, or 0 for real charges
or ., /, or x for effective positive, negative, or no charge
Note: The use of effective charge is preferred and one of the key points in defect chemistry.
We will learn what it is in the following slides

9. Effective charge The effective charge is defined as
the charge an entity in a site has
relative to (i.e. minus)
the charge the same site would have had in the ideal structure.
Example: An oxide ion O2- in an interstitial site (i)
Real charge of defect: -2
Real charge of interstitial (empty) site in ideal structure: 0
Effective charge: -2 – 0 = -2

10. Effective charge – more examples
Example: An oxide ion vacancy
Real charge of defect (vacancy = nothing): 0
Real charge of oxide ion O2- in ideal structure: -2
Effective charge: 0 – (-2) = +2
Example: A zirconium ion vacancy, e.g. in ZrO2
Real charge of defect: 0
Real charge of zirconium ion Zr4+ in ideal structure: +4
Effective charge: 0 – 4 = -4

11. Kröger-Vink notation – more examples
Dopants and impurities
Y3+ substituting Zr4+ in ZrO2
Li+ substituting Ni2+ in NiO
Li+ interstitials in e.g. NiO
Electronic defects
Defect electrons in conduction band
Electron holes in valence band

12. Kröger-Vink notation – also for elements of the ideal structure (constituents)
Cations, e.g. Mg2+ on normal Mg2+ sites in MgO
Anions, e.g. O2- on normal site in any oxide
Empty interstitial site

13. Kröger-Vink notation of dopants in elemental semiconductors, e.g. Si
Silicon atom in silicon
Boron atom (acceptor) in Si
Boron in Si ionised to B-
Phosphorous atom (donor) in Si
Phosphorous in Si ionised to P+

14. Protonic defects
Hydrogen ions, protons H+ , are naked nuclei, so small that they can not escape entrapment inside the electron cloud of other atoms or ions
In oxidic environments, they will thus always be bonded to oxide ions –O-H
They can not substitute other cations
In oxides, they will be defects that are interstitial, but the interstitial position is not a normal one; it is inside an oxide ion.
With this understanding, the notation of interstitial proton and substitutional hydroxide ion are equivalent.

15. Electroneutrality
One of the key points in defect chemistry is the ability to express electroneutrality in terms of the few defects and their effective charges and to skip the real charges of all the normal structural elements
 positive charges =  negative charges
can be replaced by
 positive effective charges =  negative effective charges
 positive effective charges -  negative effective charges = 0

16. Electroneutrality
The number of charges is counted over a volume element, and so we use the concentration of the defect species s multiplied with the number of charges z per defect
Example, oxide MO with oxygen vacancies, acceptor dopants, and defect electrons:
If electrons dominate over acceptors, we can simplify:
Note: These are not chemical reactions, they are mathematical relations and must be read as that. For instance, in the above: Are there two vacancies for each electron or vice versa?

17. Defect chemical reaction equations
Defect chemical reaction equation rules:
Conserve mass
Conserve charge
Conserve the structure (ratio of sites)

18. Electronic defects

19. Intrinsic electronic ionisation
Three equivalent reaction equations:
Consider charges, electrons and sites:
Simpler; skip sites:
Simplest; skip valence band electrons:

20. Valence defects – localised electrons and holes
Example: Fe2O3

21. Doping of semiconductors
In covalently bonded semiconductors, the valence electrons will strive to satisfy the octet rule for each atom.
As example, we add P or B to Si.
Si has 4 valence electrons and forms 4 covalent bonds.
Phosphorous P has 5 valence electrons. When dissolved in the Si structure it thus easily donates its extra electron to the conduction band in order to become isoeletronic with Si.
Boron B has 3 valence electrons. When dissolved in the Si structure it thus easily accepts the lacking electron from the valence band in order to become isoeletronic with Si.

22. Doping of Si with P (donor) or B (acceptor)

23. Point defects

24. Frenkel disorder in NiO

25. Schottky disorder in NiO
new structural unit
O2-
or, equivalently:

26. Oxygen deficiency
“Normal” chemistry:
The two electrons of the O2- ion are shown left behind
Defect chemistry:
More realistic picture, where the two electrons are delocalised on neighbouring cations

27. Oxygen deficiency
The two electrons of the O2- ion are shown left behind
The two electrons are loosely bonded since the nuclear charge of the former O2- ion is gone. They get a high energy close to the state of the reduced cations…the conduction band. The vacancy is a donor.

28. Ionisation of the oxygen vacancy donor
Electrons excited to conduction band delocalised over entire crystal, mainly in orbitals of reduced cation

29. Oxygen deficiency – overall reaction

30. Defect reactions involving foreign elements Substituents Dopants

31. Foreign elements; some terminology
Foreign elements are often classified as
impurities – non-intentionally present
dopants – intentionally added in small amounts
substituents – intentionally substituted for a host component
(we tend to call it all doping and dopants)
They may dissolve interstitially or substitutionally
Substitutionally dissolved foreign elements may be
homovalent – with the same valency as the host it replaces
heterovalent – with a different valency than the host it replaces.
Also called aliovalent
Heterovalent metals
Higher valent metals will sometimes be denoted Mh (h for higher valent).
Lower valent metals will sometimes be denoted Ml (l for lower valent).

32. Ni1-xO doped substitutionally with Li2O
Li+ and Ni2+ are similar in size, so Li+ may substitute Ni2+. This will constitute acceptor-doping with effectively negative dopants. (This is utilised in Li-doped NiO for p-type conducting electrodes for fuel cells, batteries etc.)
Ni1-xO contains nickel vacancies and electron holes.
The doping may thus be compensated by consuming Ni vacancies
or – better - by producing electron holes. This is an oxidation reaction and requires uptake of oxygen

33. ZrO2-y doped substitutionally with Y2O3
Y3+ will form effectively negative defects when substituting Zr4+ and thus acts as an acceptor. It must be compensated by a positive defect.
ZrO2-y contains oxygen vacancies and electrons
The doping is thus most relevantly written in terms of forming oxygen vacancies:

34. ZrO2-y doped substitutionally with Y2O3
Note: Electrons donated from oxygen vacancy are accepted by Y dopants; no electronic defects in the bands.

35. Hydration – dissolution of protons from H2O
Water as source of protons. Equivalent to other oxides as source of foreign elements.
Example: Hydration of acceptor-doped MO2, whereby oxygen vacancies are annihilated, and protons dissolved as hydroxide ions.
The acceptor dopants are already in, and are not visible in the hydration reaction in this case

36. Ternary and higher compounds
With ternary and higher compounds the site ratio conservation becomes a little more troublesome to handle, that’s all.
For instance, consider the perovskite CaTiO3. To form Schottky defects in this we need to form vacancies on both cation sites, in the proper ratio:
And to form e.g. metal deficiency we need to do something similar:
(But oxygen deficiency or excess would be just as simple as for binary oxides, since the two cations sites are not affected in this case …)

37. Doping of ternary compounds
The same rule applies: Write the doping as you imagine the synthesis is done: If you are doping by substituting one component, you have to remove some of the component it is replacing, and thus having some left of the other component to react with the dopant.
For instance, to make undoped LaScO3, you would probably react La2O3 and Sc2O3 and you could write this as:
Now, to dope it with Ca2+ substituting La3+ you would replace some La2O3 with CaO and let that CaO react with the available Sc2O3:
The latter is thus a proper doping reaction for doping CaO into LaScO3, replacing La2O3.

38. Defect structure Equilibrium coefficients Electroneutrality
Allows determination of all defect concentrations
For 2-3 defects: Analytical solutions
For 3 or more defects: Numerical solutions necessary

39. Frenkel defects in NiO Defect formation Equilibrium coefficient
Electroneutrality
Insert and solve:

40. Intrinsic semicondutor
Intrinsic electronic exitation
Equilibrium coefficient
Electroneutrality, insert, solve:

41. Oxygen deficient oxide
Brouwer diagram
Oxygen deficient oxide
Oxygen vacancy formation
Equilibrium coefficient
Electroneutrality, insert, solve:

42. Defect structure with 3 defects; Y-substituted ZrO2-y
Total electroneutrality
Total solution;
Analytic or numerical or
Simple limiting Brouwer cases

43. LiFePO4
Li deficiency

44. LiFePO4 Brouwer diagram vs aLi

45. LiFePO4 Brouwer diagram vs [D]

46. LiFePO4 concentration and conductivity of holes vs 1/T

47. Computational methods in defect chemistry
Example reaction
Energy calculation
Static lattice energy or
Quantum mechanical (ab initio, Density Functional Theory (DFT))
Entropy estimate or calculation
Defect concentration
Electroneutrality
Solve for Fermi level μe and all concentrations at T, pi…

50. Transport Diffusion and ionic conductivity (Electronic conductivity)

51. Solid-state crystalline electrolytes – diffusion mechanisms
Interstitial
Vacancy
Random diffusion is driven by thermal energy – it is thermally activated:

52. Random diffusion – atomistic parameterisation
A matter of jump rate and jump distance
Random diffusion: Travel far, get «nowhere»
Parameterising Dr

53. Transport in an electrical field
Nernst-Einstein relation
Random diffusivity Dr
Thermal energy kT
Mechanical mobility B («beweglichkeit»)
Electrical force F in an electrical field E: F = -zeE
Net drift velocity v
Small perturbation of random diffusion
B is a linearised transport coefficient that applies for small perturbations (compared to thermal energy)

54. Flux density and current density
Net flux density
Net current density
Charge mobility u = |ze|B
Electrical conductivity σ = |ze|cu
Nernst-Einstein
Ohm’s law
Notice that conductivity contains concentration – the other terms don’t

55. Defects and constituent ions
Example: Vacancy mechanism

56. Electronic conductivity
Pure non-polar solid

58. Thermodynamics of electrochemical cells

59. Cell voltage and Gibbs energy
An electrochemical cell can do or receive electrical work
Given by the Gibbs energy change of the process
Gibbs energies are extensive properties
Cell voltages are intensive properties

60. The Nernst equation
Gibbs energy change varies with reaction quotient Q:
which yields the Nernst equation:
At equilibrium, ΔG = 0 and E = 0:

61. Nernst equation example half cell and full cell
½ O2(g) + 2e- = O2-
Important to know know how to express activities and what the standard states are for
Gases 1 bar a = p/p0
Aqueous solutions m ≈ 1 M a = c/c0
Condensed phases Pure condensed phase a = X
H2 + ½ O2(g) = H2O(g)
T = 298 K, and using log10 instead of lne

63. Electrochemical cells

64. Transport; Overpotential losses
Electrochemical cell; H+ ion conductor in O2+H2O vs H2+H2O
Φ(V)
I > 0 (Fuel cell)
I < 0 (Electrolysis)
ηi=IRi
I = 0 (OCV)

65. Electrolyte resistance and ionic conductivity
ηelectrolyte = ηi = I Ri
Resistance (ohm)
Area Specific Resistance (ASR) (ohm cm2):
Conductivity (S/cm)
C is volume concentration in 1/cm3 or mol/cm3
U is charge mobility in cm2/sV
Intensive vs extensive properties; names and units….?

66. Types of ionic conductors
Liquid – good dynamics, high conductivities
Aqueous – Grotthuss (free proton) and vehicle mechanisms
Acid H3O+
Alkaline OH-
Molten salts CO32-
Ionic liquids – complex and/or organic ions
Liquid in solid matrix – medium to high conductivities
Porous ceramic
Polymer
Solid-state electrolytes – low to medium conductivities
H+, O2-, Li+

67. Conductivity, resistance, IR-loss
Product of charge, charge mobility, and concentration
s can be a constituent or a defect
I > 0 (Fuel cell)
ηi=IRi

68. Electrode kinetics

69. Electrode kinetics Oxz + ne- = Redz-n Redz-n = Oxz + ne-
Example: Hi,metal = H+i,oxide + e-metal
Energy
Reaction coordinate
Products
Reactants
ΔGf0
ΔGb0
ΔG0

70. An oxidation reaction Forward (anodic) and backward (cathodic) rates
Equilibrium:

71. An electrochemical oxidation
Reaction rates are affected by the electrode voltage U (=V2-V1) and symmetry factor β of the barrier:

72. Standard conditions and general conditions
Equilibrium standard potential U0 and standard rate constant k0
Equilibrium (open circuit) voltage Ueq under other conditions; exchange current density i0
Combination with the Nernst equation yields
Note that both reactant and product activities increase i0. Why?

73. Butler-Volmer equation
Current density i vs overpotential η = U – Ueq
For small overpotentials, |η| <<RT/nF;

74. Butler-Volmer equation; Tafel analysis
For large positive overpotentials, η >>RT/nF
For large negative overpotentials, -η >>RT/nF

76. Summary of week 1
Redox, Electrochemistry, Solid-state electrochemistry
Fundamentals
Defects
Types, nomenclature, thermodynamics
Equilibrium coefficients, electroneutrality
Transport
Point defects
Diffusion,
Electronic defects
Charge mobility
Conductivity
Electrochemical cells
Thermodynamics
Bulk (IR) losses
Electrode kinetics