Chapter 13 - Densification and Grain Growth in Ionic Compounds

Chapter 13 - Densification and Grain Growth in Ionic Compounds

In a solid, atom transport occurs due to a chemical potential gradient caused by a change in capillary pressure

atom A

Driving force of sintering is caused by:

• differences in bulk pressure

• differences in vacancy concentration

• differences in vapour pressure

• differences in chemical potential

caused by differences in surface curvature of particles.

A

ion A⁺

AB ionic

ion B^-

In an ionic solid, there is an extra driving force:

• electrical potential gradient caused by the different diffusivities of different ions.

Diffusion and Sintering in Ionic Compounds

During diffusion, the material maintains its stoichiometry i.e. diffusion of molecules through the lattice.

During diffusion of a compound AB, the diffusion of each ionic species A⁺ and B^- occurs:

1. under a chemical potential gradient derived from a capillary pressure difference 2. under an electrical potential gradient derived from a difference in mobility

between the cation and the anion

Ambipolar diffusion takes place i.e. the cations and anions migrate in the same direction at the same rate due to their electrical interaction with each other.

E.g., if the cations A⁺ diffuses faster than the anions B^-, they will leave behind a net negative electric charge. This negative charge will slow down the cations and speed up the anions.

The driving force for diffusion of ions is the electrochemical potential gradient 

 = chemical potential gradient + electrical potential gradient

Diffusion flux of ionic species i is:

J_i = C_i_i = -C_iB_i_i = -C_iB_i[_i + Z_iF]

J_i = diffusion flux

C_i = molar concentration B_i = mechanical mobility

_i = chemical potential Z_i = effective charge

F = Faraday constant (96486.7 Coulomb / mole)

 = electrical potential ( means gradient)

 





















aF bF

X X

X X

M M

M M

RT μ D J C

RT μ D J C

For a compound M_aX_b with cation valence b, anion valence -a:

D_M = self-diffusion coefficient of M^b+

D_X = self-diffusion coefficient of X^a-

J_M and J_X are coupled due to the

requirement for electrical neutrality (Z_iJ_i = 0).

Diffusion takes place by flux of molecules M_aX_b

The diffusion flux of M_aX_b molecules can be derived from flux and stoichiometry constraints:

X M

X M

X M

X M

C C

C

J J

J

b a

b a

b 1 a

1 and

b 1 a

1









and the relationship: μ_{M X} μ_M μ_X

b

a    

 a b

Combining these equations with the ones for J_M and J_X gives:

This eqn. can be expressed in the form of Fick's first law:

D = effective self-diffusion coefficient of M_aX_b molecule

If there is a large difference in diffusivity between M^b+ and X^a- ions, the slower moving ion governs the effective diffusion coefficient D and the densification during sintering.

If sintering takes place by several diffusion paths e.g. lattice and g.b., the effective diffusivity is the sum of the contributions of each path.

D_i = effective diffusion coefficient of ion i

f = area fraction of path p

E.g. if both lattice and grain boundary diffusivity occur simultaneously and contribute to effective diffusivity:

l = lattice b = boundary

For a molecule M_aX_b: D_M = (D^l_Mf^l) + (D^b_Mf^b) D_X = (D^l_Xf^l) + (D^b_Xf^b)

The effective self-diffusion coefficient D of the molecule M_aX_b is given by

X M

X M

D D

D D D

a

b 



D^b_M = g.b. diffusivity of M D^l_M = lattice. diffusivity of M D^b_X = g.b. diffusivity of X D^l_X = lattice. diffusivity of X f^b = area path of g.b. diffusion f^l = area path of lattice diffusion a = no. of atoms M in formula M_aX_b b = no. of atoms X in formula M_aX_b

The contribution of g.b. diffusion increases as g.b. area increases i.e. grain size decreases.

The contribution of lattice diffusion is independent of grain size.

e.g. for an MX compound with D^b_X > D^b_M and D^l_X < D^l_M, D varies with grain size.

Sintering is controlled by the slowest moving species over its fastest path.

For lattice diffusion, D can vary with non-

stoichimetry addition because the concentrations of defects change.

E.g. an oxide MO with vacancies as major defects.

If Q_VM(activation energy of cation vacancy diffusion)

< Q_VX(activation energy of anion vacancy diffusion), the max D appears in the region with oxygen

deficiency.

• For lattice diffusion, D can vary with dopant addition because dopant addition changes the concentrations of defects.

• Impurities can also change defect concentrations and D.

• dopants can also cause secondary effects (e.g. changes in g.b. diffusivity or g.b.

energy) which can affect densification.

The effect of a dopant on densification is evaluated by experiment.

Electrostatic Potential Effect on Interfacial Segregation

• In pure ionic compounds, thermally generated defects are present and must maintain electrical neutrality.

• Frenkel defects can form "internally" within the lattice, but Schottky defects need a site to which atoms can be moved e.g. a surface, grain boundary or dislocation.

• This site is called a source or sink for defects. Defects are emitted or absorbed at this site.

Schottky Defect



 

V_Mg V_O null

e.g.

• Defect sources/sinks are assumed to be perfect i.e. there is no limit to the rate at which defects can be formed or removed.

• At thermal equilibrium, the

concentration of defects in the lattice reaches a constant value.

g_VMg g_VO.





M gsurf

''

VMg

''

Osurf





VO

Ionic Space Charge

• Consider Schottky defect formation in a compound MX

• This has a formation energy g_s.

• The Schottky reaction can be separated into cation and anion vacancy formation reactions e.g.

• The Schottky formation energy can be separated into formation energies for each defect i.e. g_s = g_VMg + g_VO.

• The individual formation energy is the energy needed to bring the ion to the surface.

• Individual defect energies can differ from each other e.g. in an oxide with a close- packed anion sublattice, the cation vacancy formation energy < oxygen vacancy formation energy.

• This causes the concentration of ions on the surface to differ from those in the bulk. The surface (grain boundary, dislocation) becomes non-stoichiometric and has a net electrical charge. This charge is compensated by an adjacent space-charge layer.



 

V_Mg V_O null

e.g.







   

 _{surf Mg}^'' _O^x _surf^'' _O

x

Mg Mg V and O O V

Mg

-ve space charge +ve surface charge

• The vacancy formation energies are g_VMg and g_VO^.

• Imagine that g_VMg< g_VO. Then >

• If g_VMg < g_VO, then will increase and will decrease.

• increases with increasing and decreases with decreasing Intrinsic potential (pure material)

  

  

surf O

'' surf x

O

1 S ''

Mg surf

'' Mg surf

x Mg

V O

V O

O

V M g

V M g

M g

K

K





























Adding these two reactions gives:







   



 _O^x _{surf surf}^'' _Mg^'' _O

x

Mg O Mg O V V

Mg

The surface ions are equivalent to the lattice ions, so:

 

 

Mg V V V

V

null    ^  ^  K



  

• The concentration of each type of vacancy will vary with distance x in the space-charge layer.



  

 

 



  

     

     



 



  













  







kT

x x e

kT

x e x

2 exp g

V

2 g

exp V

O Mg

'' V O

V Mg

(x) = spatially varying potential

(x) = zero at the surface, _ in the bulk

In the bulk,

 

 





4 1

At the surface,

  ^{ }   ^{ }



 



 

 









 

 



x kT

x kT^VMg _O^{'' VO}

Mg

exp g 0

V and g

exp 0

V

effective charge = 2

effective charge = 2

N.b. for compounds with other effective charges e.g. NaCl, these equations will change slightly.

-ve space charge +ve surface charge

There is an electrostatic potential difference between the surface and the interior.

Extrinsic potential (doped material)

• Changes in bulk defects cause changes in the surface charge.

• e.g. doping MgO with Al₂O₃

• Al₂O₃ 2Al^_Mg + V''_Mg + 3O^x_O

• An increase in will cause Mg^_surf ions to go back into the bulk.

• The surface charge becomes negative (due to excess O^2- ions at the surface).

• In the bulk:

   

 





M gsurf

''

VMg

''

VMg

''

Osurf





VO

      ^{ }











  







 ^





kT

x e

x g 2

exp 2 Al

V_Mg 1 _Mg ^VMg

The potential difference between surface and bulk is:

 

2 ln g

Mg V_Mg

kT kT

e_   ^ 

• Note that Al ions segregate at the surface region.

• This is another form of solute segregation.

• E.g. Sc³⁺ and Mg²⁺ have similar ionic radii

(0.072 and 0.075 nm). Sc shouldn't segregate at g.b.s.

• But Sc_Mg^ affect the ionic space charge and are segregated in the space-charge region just like Al_Mg^.

• Sc doping can cause solute drag-limited grain growth.

• Changes in T, pO₂ and [dopant] can affect the segregation of dopants