Download Defects in ceramic structure 2

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 
v 
v 
v 
v 
Rules for defect reactions
Stoichiometric defect reactions
Defect reactions for compound crystals
Non-stoichiometric defect reactions
Extrinsic defect reactions
1 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 q  To generalise, for an MaXb compound, the following
relation has to be maintained at all times:
a ( X X +VX ) = b ( M M +V M )
M M +V M
a
=
X X +VX
b
q  For example, in Al2O3 :
M Al +V Al
2
=
X O +VO
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 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'' +VX••
or, perfect crystal ; ΔgS
ΔgS = the free energy change for
the formation of Schottky defect
q  Schottky defect reaction for Al2O3:
null = 2V Al''' + 3VO••
q  In general, for an MaOb oxide:
null = aV Mb– + bVOa+
Thermodynamics of Schottky defects
q  Assume that the number of ways of distributing cation
vacancies Vcat on (Ncat + Vcat) sites be Ω1, and
number of ways of distributing anion vacancies Van on
(Nan + Van) sites be Ω2.
q  The configuration entropy
ΔS = k lnΩ = k lnΩ1Ω2
€
( N cat + Vcat )! ( N an + Van)!
( N cat )!( Vcat )! ( N an)!( Van )!
where
Ω=
and
( N cat + ncat ) ( N an + nan) = 1
€
4 €
q  Finding minimum in free energy will yield
eq
Vaneq Vcat
(N
an
)(
eq
+ Vaneq Vcat
+ N cat
)
eq
% Δh − T ΔsS (
Vaneq Vcat
≈
= exp'− 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
[Va ] = [Vc ] = exp
where
€
[Vc ] =
$ Δh '
ΔsS
exp&− S )
% 2kT (
2k
Vcat
Vcat + N cat
and
[Va ] =
Van
Van + N an
Square brackets denote mole fractions of defects!
They are dimensionless!
€
€
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 Mx =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:
OOx =VO•• +Oi''
5 Thermodynamics of Frenkel defects
q  number of ways of distributing ni interstitials on N*
interstitial sites be Ω1, and
number of ways of distributing cation vacancies Vcat
on NT total sites be Ω2.
N *!
Ω1 =
( N * − ni )!n i!
Ω2 =
N T!
( NT − Vcat )!Vcat!
€
q  The configurational entropy is the same
ΔS = k lnΩ1Ω2
q  At equilibrium,
eq eq
%
(
%
(
Vcat
€ n i* ≈ exp'− ΔgF * ≈ exp TΔSF exp'− ΔhF *
& kT )
& kT )
NT N
kT
ΔgF = the free energy change for
the formation of Frenkel defect
v  Note that N* depends on crystal structure.
€
v  For example, for 1 mol NaCl, if the ions migrate to tetrahedral sites,
N* ≈ 2NAV.
6 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/cm3 and 187.8 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 ≈ 2NAV
# Δh &
Vcateq nieq
T ΔS F
=
exp
exp
%− F (
*
kT
NT N
$ kT '
eq eq
#
&
Vcat ni
6.6R
110 x1000
= exp
exp % −
(
23 2
R
2(6.02 x10 )
$ 8.314(500 + 273) '
Vcateq nieq = 1.957 x10 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/cm3 and 187.8 g/mol, respectively. State all necessary assumptions.
In case of Frenkel disorder,
number of cation vacant site = number of interstitial sites
Vcateq nieq = 1.957 x10 43 defects/mol 2
Vcateq = nieq = 4.43x1021 defects/mol
And the corresponding number of defects/cm3 is
! 6.5 $
Vcateq = nieq = 4.43x1021 #
& = 1.5x1020 defects/cm 2
" 187.7 %
7 Defects in Compound Crystals
Possible defects:
1.
2.
3.
4.
5.
vacant sites on each sub-lattice
ions/atoms on interstitial sites
impurity ions/atoms on each sub-lattice
unassociated electrons and holes
combination of these defects
Example:
q  Consider a crystal with a formula MaXb.
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:
#b&
a Xb
M ( g) !M!!
→ M Mx + % (VX(a/b) z + ze )
$a'
8 Example: Incorporating Si into SiO2
2
Si( g) !SiO
!!
→ SiSix + 2VO•• + 4 e #
Example: Incorporating Al into Al2O3
or,
x
2O3
Al ( g) !Al
!!
→ Al Al
+ 3 VO•• + 3e #
2
x
2O3
2Al ( g) !Al
!!
→ 2Al Al
+ 3VO•• + 6e #
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 MaOb 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 Metal excess, or oxygen deficient
(low oxygen partial pressure)
q  has the general formula Ma+δOb , or MaOb-δ.
q  places cation into interstitial site, or creating vacancy
in oxygen site.
q  using anion, typical defect reaction is:
OOx = 1 O2 ( g) +VO•• + 2e !
2
Redox reac5on OxO ⇒
VOx
1
O2 ( g) + VOx
2
VOx ⇒ VO• + e #
OxO ⇒
VO• ⇒ VO•• + e #
electrons are usually weakly
bonded; can be excited into
the conduction band
when species O2 escape as
natural, it leaves two electron
behind
1
O2 ( g) + VO•• + 2e #
2
€
€
€
10 Example: TiO2-y
2
OOx !TiO
!!
→ 1 O2 ( g) +VO•• + 2e #
2
+4
2Ti + 2e # = 2Ti +3
2TiTix +OOx = 2TiTi' + 1 O2 ( g) +VO••
2
Oxygen excess, or metal deficient
(high oxygen partial pressure)
q  has the general formula MaOb+δ , or Ma-δOb.
q  places oxygen into interstitial site, or creating vacancy
in cation site.
q  using anion, typical defect reaction is:
1 O2 ( g) = Oi'' + 2h•
2
Oix
This ionization creates
holes in valence band
This hole moves through
lattice and contribute to
electrical conductivity
11 Example: Fe1-xO
1 O2 ( g) !FeO
!!
→OOx +V Fe'' + 2h•
2
2Fe +2 + 2h• = 2Fe +3
x
2Fe Fe
+ 1 O2 ( g) = 2Fe •Fe +OOx +V Fe''
2
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 CaCl2 into NaCl
NaCl
'
CaCl 2 !2!!
→Ca•Na +VNa
+ 2ClClx
CaCl 2 !NaCl
!!
→Ca•Na +Cl i ' +ClClx
Between these two reactions, which one is more probable?
12 Example 2: Doping MgO with Al2O3
•
''
Al 2O3 !3MgO
!!→ 2Al Mg
+V Mg
+ 3OOx
Example 3: Doping Al2O3 with MgO
'
2O3
2MgO !Al
!!
→ 2Mg Al
+ 2OOx +VO••
Next Class
Lecture 19
Electrical and Ionic
Conduction in Ceramics
13