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Module 2: Defect Chemistry and Defect Equilibria
Introduction
Introduction
Materials in general consist of defects which can be divided into a variety of categories such as point defects or 0-D defects, line defects or 1-D defects and 2-D or surface defects. These defects play an important role in determining the properties of ceramic materials and in this context, the role of point defects is extremely important. In this module, we will learn about various point defects, the role of stoichiometry i.e. cation and anion excess and deficit, the role of foreign atoms on the defect chemistry. Subsequently, we will adopt a simple thermodynamic basis for calculating their concentration in equilibrium and then will extend the Gibbs-Duhem relation for chemical systems to the defects in ceramics considering them to be equivalent to the dilute solutions, an approximation which is fairly valid. This will lead us to the determination of defect concentrations as a function of partial pressure of oxygen which is an important exercise to establish the defect concentration vs pO (^2)
diagrams, called Brower’s diagrams.
The Module contains:
Point Defects
Kroger-Vink Notation in a Metal Oxide, MO
Defect Reactions
Defect Structures in Stoichiometric Oxides
Defect Structures in Non-Stoichiometric Oxides
Oxygen Deficient Oxides
Dissolution of Foreign Cations in an Oxide
Concentration of Intrinsic Defects
Intrinsic and Extrinsic Defects
Units for Defect Concentration
Defect Equilibria
Defect Equilibria in Stoichiometric Oxides
Defect Equilibria in Non-Stoichiometric Oxides
Defect Structures involving Oxygern Vacancies and Interstitials
Defect Equilibrium Diagram
A Simple Procedure for Constructing at Brower's Diagram
Extent of Non-Stoichiometry
Comparative Behaviour of TiO 2 and MgO vis-à-vis Oxygen Pressure
Electronic Disorder
Examples of Intrinsic Electronic and Ionic Defect Concentrations
Summary
Suggested Reading:
Nonstoichiometry, Diffusion and Electrical Conductivity in Binary Metal Oxides (Science & Technology of Materials), P.K. Kofstad, John Wiley and Sons Inc.
Physical Ceramics: Principles for Ceramic Science and Engineering, Y.-M. Chiang, D. P. Birnie, and W. D. Kingery, Wiley-VCH
Introduction to the Thermodynamics of Materials, David R. Gaskell, Taylor and Francis.
Module 2: Defect Chemistry and Defect Equilibria
Kroger-Vink Notation in a Metal Oxide (MO)
2.2 Kroger–Vink notation in a metal oxide, (MO)
Kroger-vink notations are typically used to depict the atomic defects with charges. Following tables provide the most common notations.
Regular Sites
M (^) m: normal or regular occupied metal or cation site
O (^) o : normal or regular occupied oxygen or anion site
Point Defects (a • (dot) means a positive charge and a ' (prime) means a
negative charge)
Oxygen (anion) vacancy V^ O
Metal (cation) vacancy V^ M
Oxygen (anion) interstitial O^ i
Metal (cation) interstitial M^ i
Vacant interstitial site V^ i
Foreign cation M^ f
Foreign cation on regular metal site
M (^) fm
Foreign cation on interstitial site
M (^) fi
A normal cation or anion in an oxide with zero effective charge
M (^) Mx^ or O (^) Ox
Charged oxygen vacancy: (^) V (^) O•^ or V (^) O••
Charged metal vacancy (^) V (^) M'^ or V (^) M''
Charged metal or oxygen interstitial M (^) i••^ and O (^) i ''
Neutral cation and anion vacancies V (^) Mx^ or V (^) Ox
electrons and holes e' or h •
Module 2: Defect Chemistry and Defect Equilibria
Defect Reactions
2.3 Defect Reactions
Rules for writing defect reactions
Ratio of regular cation and anion sites is always constant.
Mass balance to be preserved.
Electrical neutrality is to be always preserved.
Both ionic and electronic defect compensations are possible determined by the energetics.
We will assume complete ionization of defects.
Figure 2.2 Frenkel Defect
This defect can form inside the crystal.
It forms where cations are appreciably smaller then anions.
Defect reaction is written as
0 V (^) M''^ + M (^) i••
In cases where anions form the disorder, then it is called as Anti-Frenkel. The corresponding defect reaction in that case would be
0 V 0 ••^ + O (^) i''
Examples of compounds showing this defect are AgBr type compounds such as AgBr, AgI etc.
2.4.3 Intrinsic Ionization
Thermal creation of electron hole pair and is depicted by
Module 2: Defect Chemistry and Defect Equilibria
Defect Structures in Non-Stoichiometric Oxides
2.5 Defect Structures in Non - Stoichiometric Oxides
Mainly of two types
i. Oxygen deficient (or excess metal)
ii. Metal deficient (or excess oxygen)
Nonstoichiometry necessitates presence of point defects and extent of non-stoichiometry determines the concentration of Defects.
In such oxides, electrical neutrality is preserved via the formation of point defects and electronic changes.
Intrinsic ionization is always a possibility.
Module 2: Defect Chemistry and Defect Equilibria
Defect Structures in Non-Stoichiometric Oxides
2.5.2 Metal Deficient Oxides
Formation of either metal vacancies or oxygen interstitials (excess oxygen)
Formation occurs typically at the surface.
The following cases are possible:
2.5.2.1 If metal deficiency is dominating defect then
Depicted as metal deficient oxide M (^) 1-y O ( y is the extent of non-stoichiometry)
Possible defect reaction is that of electronic compensation.
Creation of holes
Conduction due to holes i.e. a p- type conductor
Examples of oxides showing this characteristics are MnO, NiO, CoO, FeO etc.
2.5.2.2 If metal deficiency is dominating defect then
Oxides depicted as MO (^) 2+x
Oxygen interstitials can form due to following reaction
P-type conductor
Example can be an oxide like UO (^) 2.
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Module 2: Defect Chemistry
Concentration of Intrinsic Defects
2.7 Concentration of Intrinsic Defects
Let us consider the formation of Frenkel defects in a halide, MX, i.e.
M (^) M + X (^) X V (^) M ' + M (^) i•^ + X (^) X
Change in the free energy (ΔG) upon formation of 'n' Frenkel defect pairs at an expense of ΔG (^) f energy per pair
where ΔS (^) C is the change in configurational entropy and is positive. Equilibrium concentration of defects is found by minimizing ΔG w. r. t. n i.e. the concentration at which free energy is minimum.
Change in entropy is given by
(2.2)
where W is the number of ways in which defects can be arranged. Now, as per the defect reaction shown above, number of Frenkel pairs (n) would lead to the formation equal number of interstitials (n (^) i ) as well as vacancies (n (^) v ) i.e.
Assume that total number of lattice sites = N Number of ways to arrange the vacancies, W (^) v is
Ways to arrange the interstitials (assuming that N lattice sites are equivalent to N interstitial sites), Wi are
Total number of possible configurations
So, entropy change will now be
OR
or
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For large values of N, Sterling’s approximant can be applied which leads to
and total free energy change is
Figure 2.3 Equilibrium Vacancy
Concentration
Now, if vacancies were stable defects, then at certain concentration, the free energy change has to be minimum, as shown in the figure. Hence, at equilibrium, we can safely write that
Now at equilibrium,
We can also assume since number of vacancies is much smaller than number of lattice sites in absolute terms.
This results in
Now we know that ΔG (^) f = ΔH (^) f - TΔS (^) v where ΔH enthalpy of Frenkel defect formation and ΔS (^) v = vibrational entropy change.
Hence Equation (2.12) further simplifies to
Module 2: Defect Chemistry and Defect Equilibria
Intrinsic and Extrinsic Defects
2.8 Intrinsic and Extrinsic Defects
2.8.1 Intrinsic behavior
Defect which can be determined from the intrinsic defect equation and is temperature dependent, increasing with increasing temperature.
2.8.2 Extrinsic behavior
Extrinsic defects are defects caused by impurities consisting of aliovalent cations.
Defect concentration depends upon impurity concentration which is constant and independent of temperature. Only at very high temperatures, intrinsic behavior again dominates, and the cross-over temperature depends upon the defect formation energy.
2.8.3 Example
Defect formation energies for some ceramic materials are
Here, one can see the relation with the melting point that melting point of MgO is ~2825°C while it is ~801°C for NaCl. So, at any given temperature NaCl will have much larger defect concentration than MgO. However, at the same homologous temperature, defect concentrations can be quite similar.
Interestingly, while the highest achievable purity level in MgO is 1 ppm, in NaCl, it is 50 ppm. Typically, these impurities consist of aliovalent cations which give rise to defects, called extrinsic defects. Thus the concentration of extrinsic defects is much greater than intrinsic defect concentration in MgO. As a result, defects in NaCl are likely to be intrinsic but MgO is most likely to contain extrinsic defects.
Module 2: Defect Chemistry and Defect Equilibria
Units for Defect Concentration
2.9 Units for Defect Concentration
Defect concentration fraction, n/N , is nothing but the ratio of number of defects, n, relative to number of occupied lattice sites N i.e. defect concentration fraction. The denominator should actually be n+N but since, N>>n, it can be approximated as n+N ~ N. Commonly used units for concentration is #/cm 3 or cm -
Typical defect concentration in ceramics ~ 1 ppm.
So, if the density of atoms in a solid ~10 23 cm -3^ , 1 ppm concentration would be equivalent to 10 17 cm -^. Conversion of mole fraction to number per unit volume can be the following:
No. of formula units per unit volume =
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which leads to K = K 0 exp (-ΔH 0 /RT), where K 0 = ΔS 0 and R is the gas constant.
Alternatively,
This is an important outcome as it shows that we can treat the defects in a solid as solutes in a solvent.
Module 2: Defect Chemistry and Defect Equilibria
Defect Equilibria in Stoichiometric Oxides
2.11 Defect Equilibria in Stoichiometric Oxides
The defects which we usually consider in stoichiometric oxides are Schottky and Frenkel defects and
following paragraphs so analysis for both these kinds of defects for an oxide MO.
2.11.1 Schottky Defects
Defect reaction in an oxide MO is written as
Equilibrium constant for this reaction is K (^) S = [ ] [V (^) M '']
Here square brackets i.e. [ ] are used for concentration. Equilibrium constant can be also be expressed as
where ΔG (^) S is the molar free energy of defect formation and is ΔH (^) S - TΔS (^) S , where ΔH (^) S is the enthalpy for defect formation and ΔS (^) S is the entropy change which is mainly vibrational in nature and can be assumed to be constant. This leads to
If Schottky defects dominate, then
[ ] (2.24)
Here, as one can see, defect concentrations are independent of pO 2.
2.11.2 Frenkel defects
For an oxide MO
which leads to
OR
