Part V Sintering of Ionic Compounds

Part V Sintering of Ionic Compounds

In Parts II - IV, we looked at systems where:

• densification and grain growth usually occur by movement of atoms of a single component.

• vacancy concentration in a specific region is determined by capillary pressure due to the region's geometry.

In most ceramics (especially ionic compounds), vacancy concentration can be changed by adding dopants (additives). Therefore sinterability can change.

Depending on point defect concentration and temperature, the species that controls sinterability can also change.

Chapter 12 - Sintering Additives and Defect Chemistry

Chapter 13 - Densification and Grain Growth in Ionic Compounds

Chapter 12 - Sintering Additives and Defect Chemistry

Sintering additives are added to powders in order to:

1. enhance sinterability 2. control microstructure

E.g. addition of Ni to W causes: increase of grain boundary diffusivity of W grain boundary roughening transition

Usually, the roles of sintering additives are only known empirically i.e. we know that addition of additive A to ceramic B improves sinterability but we don't know why.

W W 0.4 wt % Ni

Samples sintered at 1200C for 20 min in H₂

This chapter will consider point defects formed by the addition of sintering additives* in ionic compounds with low defect concentrations.

If the concentration of point defects is low, we may assume that:

1. the matrix atoms and point defects form an ideal solution with no interaction between defects.

2. the concentration of matrix atoms is 1.

We can estimate the concentration of point defects caused by dopant addition.

If sintering is controlled by lattice diffusion, then the change in sinterability with dopant addition can be explained by this estimation.

*additives are usually called dopants if their concentration is low.

Three types of point defect: intrinsic, extrinsic and electronic.

• Vacancies:

-vacant atomic sites in a structure.

• Interstitials:

-"extra" atoms positioned between atomic sites.

Vacancy

distortion of planes

Interstitial

distortion of planes

Intrinsic Point Defects in Pure Elements

Boltzmann's constant (1.38 x 10 -23 J/atom-K) (8.62 x 10 -5 eV/atom-K)

 

N _v

N = exp - Q _v k T



  

 

No. of defects

No. of potential defect sites.

Activation energy

Temperature

Each lattice site is a potential vacancy site

• Equilibrium concentration varies with temperature Equilibrium Concentration:

Point Defects

Mixing entropy: Concepts of macrostate and microstate

macrostate microstate

(mole fraction) (arrangement configuration)

X_A=1

X_B=1 X_A=0.75, X_B=0.25

X_A=0.5, X_B=0.5

X_A=0.25, X_B=0.75

1 2 3 4

1 2 3 4 1 2 3 4

1 2 3 4 1 2 3 4

1 2 3 4 1 2 3 4

1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4

1 2 3 4

1 2 3 4

1 2 3 4 1 2 3 4

Atom of element A Atom of element B or vacancy

The mixing entropy is defined as a the number of microscopic configurations that result in the observed macroscopic description of the thermodynamic system.

S = k ln Ω

Where k is the Boltzmann constant 1.38x10^-23 J/K, Ω is the number of microstates

Mixing entropy

When a point defect (B atom, interstitial or vacancy) is formed:

① the enthalpy of the system increases: ΔH= n_B· Δ h.

② the entropy (mixing entropy) increases (Δ S=S₂-S₁).

Squeeze In: Δ h

Squeeze out: Δ h

Formation of defect Substitute B atom

Energy is required to form a point defect and so the enthalpy of the system increases. The mixing entropy also increases.

Mixing entropy and mixing free energy

The change of free energy in the system per mole:

ΔG= ΔH - TΔS.

Note:

Mixing entropy : ΔS= k (X_A ln X_A + X_B ln X_B)

ΔS

X_B

0 0.5 1

Incorporation of an impurity or vacancy increases both the first the second terms of the above equation.

 Incorporation of an impurity (vacancy) reduces the free energy of a solid system.

In other words,

• No material can be prepared without some degree of both chemical impurity and structural defects.

• Defects in a crystal increase entropy : automatic reaction.

Defect Chemical Reactions - Kröger-Vink Notation



Al Mg

Solute ion Site that defect occupies (in this case, a Mg lattice site)

Effective (relative) charge of defect = +1 (Al has +3 charge, Mg has +2)

V Mg 

Vacancy

Effective (relative) charge of defect = -2 (because Mg ion is absent)



Mg i

Interstitial site

Effective (relative) charge of defect = +2 (Interstitial site has 0 charge) Site that defect occupies (in this case, a Mg lattice site)

Solute ion

Frenkel Defect

Schottky Defect

Intrinsic Point Defects in Compounds

Ag i

x

Ag Ag V

Ag AgCl, in

e.g.  ^   e.g.nullV_Mg  V_O

“null” = creation of defects from perfect lattice

Frenkel Defect

For a simple metal oxide MO: M_M^X M⇄ _i^ V_M

If the number of defects is low compared to the number of lattice points:

  

F M

i K

kT V Δg

M  =



 

-

 =



 exp

 

atoms (with effective charge of +2)

 

g_F = formation free energy of Frenkel defect K_F = mass action equilibrium constant of Frenkel defect

The Frenkel defect does not create new lattice sites.

Schottky Defect

For a simple metal oxide MO: M_M O_O ⇄ V_MV_O^  M_B O_B

B = a place where a lattice site can form e.g. a grain boundary, surface or dislocation

The Schottky defect creates new lattice sites.

Since M_B and O_B are the same as M_M and O_O, we can also write: null ⇄ V_MV_O^

The concentrations of these defects are expressed as:

null = the perfect lattice

  

V Δg

V  =



 

-

 ^ = exp ^gS = formation free energy of Schottky defect K_S = mass action equilibrium constant of

Schottky defect

Electronic Defects - Energy Band Structure in Solids

• In isolated atoms, each electron shell and subshell has its own separate (discrete) energy . Each energy state in a subshell has the same energy.

• In solids, the electrons are affected by the electrons and nuclei of neighbouring atoms.

• The energy states in each subshell split into different states, with slightly different energies.

• These split states form energy bands.

• Gaps may exist between energy bands (band gaps).

At 0K, three types of band structure are possible:

E_f = Fermi energy, the energy corresponding to the highest filled energy state at 0K in a metal.

For a semiconductor or insulator, E_f is approximately in the middle of the band gap.

1. In metals, the highest energy band is partly filled. Electrons can easily be excited to higher energy states and the metal can conduct electricity.

2. In a semiconductor, the lower energy band (valence band) is completely filled. The upper energy band (conduction band) is completely empty.

Electrons have to be excited across the band gap E_g in order to be able to conduct electricity. An electron hole is left behind in the valence band.

3. In an insulator, the band gap is much larger than in a semiconductor.

• At 0K, all electrons are in the lowest possible energy levels, under constraint of the Pauli exclusion principle (up to 2 electrons per energy level).

• At T > 0K, some electrons have enough energy to jump into the conduction band.

• An intrinsic electronic defect consists of a free electron in the conduction band and a free electron hole in the valence band.

• The defect chemical reaction is:

null e' + h⇄ ^ e' = electron; h^ = hole

The probability P(E) of an electron occupying an energy level E is:

 

 

 

P

- f

= 

exp 1

1

 

2

C V

f

E

E E 

 ^EV = energy level of valence band E_C = energy level of conduction band

If the concentrations of free electrons and holes are low:

   

   



 - -



=

=



 - -



 =

=



kT E E

N h n

p

kT E E

N e n

n

V f

V h

f C

C e

exp exp

n_e = no. electrons / unit volume in conduction band

n_h = no. electron holes / unit volume in valence band

N_C = conduction band density of states (density of electron energy states in the conduction band per volume of crystal).

N_V = valence band density of states (density of electron hole energy states in the valence band per volume of crystal)

m_e* = effective mass of electron

m_h* = effective mass of electron hole h = Planck constant

2 3

2

2 3

2

2 2 2 2



 



= 



 



= 





h πm kT N

h πm kT N

h V

e C

At a given temp., the product of [e'][h^] is constant:

  

g K

kT h E

e

np  =



 



 



 

-

 =

= ^ exp

K_i = mass action constant of electronic defect

E_g = E_C - E_V

Formation of Point Defects in Compounds by Additives

Charge neutrality must be maintained through formation of vacancies.

Mg²⁺

Fe²⁺

1 2

3

NiO in MgO Al₂O₃ in MgO

Fe_1-xO (x 0.05)

Defect Chemical Reactions - Examples

Fe x

O Fe

3 2

Mg x

O Mg

3 2

x O x

Mg

V O

3 Fe

2 O

Fe 3.

V O

3 Al

2 O

Al 2.

O Ni

NiO

1.

 





 













Mass, site (cation / anion ratio) and charge balance must be maintained.

(Ni +2 charge, Mg +2 charge)

(Al +3 charge, Mg +2 charge)

(Fe +2 or +3 charge)

1. mass balance – the reaction cannot create or lose mass (n.b. vacancies have no mass);

2. site balance – the cation / anion ratio of the solvent crystal lattice must be maintained (n.b. vancancies create lattice sites);

3. charge balance – overall electrical neutrality of the crystal lattice must be maintained.

MgO

MgO

For the general equation: L₂O₃  2 L^_M + V''_M + 3 O^x_O

[V''_M ]  ½ [L^_M]

• Addition of L₂O₃ to MO increases [V''_M ] and hence the lattice diffusion coefficient D_M of M.

• If lattice diffusion of M controls the sintering of this material, sinterability is expected to increase with addition of L₂O₃.

MO

Oxidation and Reduction Reactions.

Ionic solids can react with the ambient atmosphere.

The ambient gas (e.g. O₂, H₂) is another type of solute species.

Reduction

O^x_O  ½ O₂(g) + V^_O + 2 e'

Oxidation

n.b. the two electrons from the O^2- ion are left behind in the solid

½ O₂(g) + V^_O  O^x_O + 2 h^

n.b. two electron holes are formed in the solid to balance the effective charge on both sides of the equation

e.g. for oxidation of MgO; ½ O₂(g)  O^x_O + V''_Mg + 2 h^

We can write oxidation reactions even if no oxygen vacancies are present.

New lattice sites are created on the MgO crystal lattice.

Electronic versus Ionic Compensation of Solutes

An aliovalent solute can be charge compensated by ionic defects or by electrons and holes.

Ionic compensation

2 Nb₂O₅  4 Nb^_Ti + 10 O^x_O + V''''_Ti

Electronic compensation

2 Nb₂O₅  4 Nb^_Ti + 8 O^x_O + O₂ (g) + 4 e'

Dominant at high [Nb], high pO₂ and low temp

Dominant at low [Nb], low pO₂ and high temp

Solutes can also be compensated by a combination of ionic and electronic compensation.

TiO₂

TiO₂