Phenomenological Coefficients in Solid State Diffusion (an - - PowerPoint PPT Presentation

Phenomenological Coefficients in Solid State Diffusion (an introduction)

Graeme E Murch and Irina V Belova Diffusion in Solids Group School of Engineering The University of Newcastle Callaghan New South Wales Australia

G’day!

Collaborators: A B Lidiard (Reading and Oxford), A R Allnatt (UWO), D K Chaturvedi (Kurukshetra), M Martin (RWTH Aachen). Research supported by the Australian Research Council

• 1. Fick’s First Law and the Onsager Flux Equations.
• 5. How to make use of phenomenological coefficients:
• The Darken and Manning approaches.
• The Sum-Rule.
• 6. Some applications.

Talk Outline:

• 3. Allnatt’s Equation for the phenomenological coefficients

and the Einstein Equation.

• 2. The meaning of the phenomenological coefficients.
• 4. Correlation effects in phenomenological coefficients and

in tracer diffusion coefficients.

• 7. Conclusions.

Fick’s First Law (1855):

dx dC D J

i i i

− =

Because it does not recognize all of the direct and indirect driving forces acting on species i, Fick’s First Law is frequently insufficient as a condition for describing the flux. The actual driving force for diffusion is not the concentration gradient but the chemical potential gradient.

The Onsager (1934) Flux Equations of irreversible processes provide the general formalism through the postulate of linear relations between the fluxes and the driving forces: Lij : the phenomenological coefficients (independent of driving force) Xj : the driving forces

∑

=

j j ij i

X L J

Lij=Lji (reciprocity condition)

The A atoms respond only to the direct force qAE. The B atoms

• nly respond to the indirect force qAE and are then ‘dragged along’

by the A atoms. Consider a binary system AB. The Onsager Flux Equations are: Consider a hypothetical situation where A is charged and B is not, and the system is placed in an electric field E. The fluxes are then: JA= -LAA qAE

and JB= -LABqAE

The driving forces are then: XA = -qAE

and

XB = 0

JA = LAA XA + LAB XB JB = LBB XB + LAB XA

What are these phenomenological coefficients?

VkTt 6 R R L

j i ij

> ⋅ < =

(Allnatt 1982) Ri: the ‘collective displacement’ or displacement of the center-of-mass

• f species i in time t.

VkTt 6 R L

2 A AA

> < = VkTt 6 R R L

B A AB

> ⋅ < = E.g. If the moving A species does not interfere with the moving B species e.g. A and B do not compete for the same defects

• r A and B do not interact (i.e. different sublattices)

<RA·RB> = 0 and LAB = 0.

⇒

However, in most cases in solid-state diffusion the off-diagonal coefficients can be significant. They can be positive or negative.

VkTt 6 R R L

j i ij

> ⋅ < =

Allnatt’s (1982) equation for the Lij is a generalization of the Einstein (1905) equation for the tracer or self-diffusion coefficient:

t 6 r D

2 *

> < =

r = displacement of a tracer atom in time t The Einstein Equation is frequently used in Molecular Dynamics simulations, see Poster 31: Zhao et al., Poster 37: Leroy et al., Poster 38: Leroy et al., Poster 54: Plant et al., Poster 42: Chihara et al., Poster 39: Habasaki et al.

The relationship between the Einstein Equation and the Allnatt Equation can be appreciated if we consider a binary system of A* and A in which we allow the tracer A* concentration to be very low. Then we would have that:

VkT D L

* A A

* *

=

In solid-state diffusion, the motion of the atoms normally takes place by discrete jumps or hops (often called the ‘hopping model’).

I can hop too!

Defects such as vacancies provide the vehicles for atom motion. The hopping model is frequently used directly or indirectly in the modelling

• f solid state diffusion, see Poster 18: Maas et al., Poster 28: Sholl,

Poster 41: Kalnin et al., Poster 49: Radchenko et al.

In solid-state diffusion, the motion of the atoms normally takes place by discrete jumps or hops (often called the ‘hopping model’).

Dj*= fj (Z cv wj a2)

↑ ↑

correlated part uncorrelated part

Z: coordination number cv: vacancy concentration wj: exchange frequency of an atom of type j with a vacancy a: jump distance

fj: tracer correlation factor of atoms of type j. It is an

expression of the correlation between the directions of the successive jumps of a given atom of type j. It is usual then to partition diffusion coefficients such as the tracer diffusion coefficient in the following way:

I can hop too!

∑

∞ =

> θ < + =

1 m ) m (

cos 2 1 f

The tracer correlation factor can be expressed in terms of the cosine of the angle between the ‘first’ jump and all subsequent jumps of a given atom (the tracer):

• 1.0
• 0.5

0.5 1.0 1 2 3 4 5 6 7 8 9 10 11 Cos(θA

m)

m

Example of the convergence

• f the cosine between the first

tracer A jump and the m’th tracer A jump. .

Phenomenological coefficients can be partitioned in a similar way to diffusion coefficients:

Lij= fij (j) (Z cv wj a2 N cj / 6 V k T) ↑ ↑

correlated part uncorrelated part

fij

(j) : collective correlation factor. It is an expression of the

correlation between the directions of successive jumps

• f the centers-of-mass of the species present.

The collective correlation factors can be expressed in terms of the cosine of the angle between the ‘first’ jump and all subsequent jumps of the same species (diagonal factor) or another species (off-diagonal factor)

∑

∞ =

> θ < + =

1 m ) m ( ii ii

cos 2 1 f

Diagonal collective correlation factor:

∑ ∑

∞ = ∞ =

> θ < + > θ < =

1 m ) m ( BA A A B B 1 m ) m ( AB ) A ( AB

cos n C n C cos f

Off-diagonal collective correlation factor (binary case only):

• 0.4
• 0.2

0.2 0.4 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Cos(θAA

m )

m

Example of the convergence of the cosine between the first collective jump and the m’th collective jump (of the same species A).

The phenomenological coefficients are extremely difficult to measure directly in the solid state.

• First Strategy: Find relations between the

phenomenological coefficients and the (measurable) tracer diffusion coefficients: If we want them, how do we proceed?

Example 1: The Darken Relations (1948) :

kT / D C L

* i i ii =

Lij=0

Example 2: The Manning Relations (1971) for the random alloy:

⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ + =

∑

k * k k * i i * i i ii

D C M D C 2 1 kT D C L

∑

=

k * k k * j j * i i ij

) D C M ( kT D C D C 2 L

The Manning Relations can also be derived on the basis of two ‘intuitive’ assumptions without recourse to the random alloy model (Lidiard 1986). They have also been derived for binary ordered structures (Belova and Murch 1997)

• Second Strategy: Find relations between the phenomenological

coefficients themselves in order to reduce their number.

⇒ ‘Sum-Rules’ ⇒ ‘Sum-Rules’

Initial vacancy-atom jump Possible vacancy jumps after time t

Schematic illustration for the origin of the ‘Sum-Rule’. Vector summation

• f

= 0

V j j M 1 i i j ij

c c Aw w / w L =

∑

=

e.g. For the random binary system AB:

The ‘Sum-Rule’ for the phenomenological coefficients in a multicomponent random system is (Moleko and Allnatt 1986):

AB A B 2 B B V BB AB B A 2 A A V AA

L w w kT a w c Nc L , L w w kT a w c Nc L − = − =

In the binary system there is then only one independent phenomenological coefficient, not three. (In the ternary random system there are three independent phenomenological coefficients, not six.) For the binary system, the ‘Sum-Rule’ is:

Analogous ‘Sum-Rule’ expressions have since been derived for:

• Diffusion via divacancies in the random alloy

(Belova and Murch 2005).

• Diffusion via dumb-bell interstitials in the random alloy

(Sharma, Chaturvedi, Belova and Murch 2000).

• Diffusion via vacancy-pairs in strongly ionic compounds

(Belova and Murch 2004).

• Diffusion via vacancies in substitutional intermetallics

(Belova and Murch 2001, Allnatt, Belova and Murch 2005).

• Diffusion via vacancies in the five-frequency impurity

diffusion model (Belova and Murch 2005).

Application of the Onsager flux equations and the Sum-Rule (binary alloy):

B AB A AA A

X L X L J + =

A AB B BB B

X L X L J + = ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ∂ γ ∂ + ∂ ∂ − = ∂ μ ∂ − = c ln ln 1 x C C kT x X

i i i i

The Onsager flux equations are:

φ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − =

B AB A AA I A

C L C L kT D φ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − =

A AB B BB I B

C L C L kT D

The intrinsic diffusion coefficients (found from the Kirkendall shift and the interdiffusion coefficient) are:

x C D J

i I i i

∂ ∂ − =

Local lattice reference frame

⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ γ ∂ + = φ

B

c ln ln 1

φ is the ‘thermodynamic factor’:

AB B BB A AB A AA B I B I A

L c L c L c L c D D − − ≡

B A I B I A

w w D D =

Measurement of the ratio of the intrinsic diffusivities directly thus gives the ratio of the exchange frequencies. There are no complicating correlation factors. The ratio of the intrinsic diffusion coefficients is: Application of the Sum-Rule then gives (Belova and Murch 1997):

If we had simply used the Darken relations (where all

• ff-diagonal phenomenological coefficients are put equal to

zero) we would then have obtained:

B * A * I B I A

D D D D =

(This is a very rough approximation)

I B A I A B

D c D c D ~ + =

This general equation for the interdiffusion coefficient in a binary alloy is, in effect, an extension of the Einstein Equation (1905).

B A A

J x C D ~ J − = ∂ ∂ − =

• Lab. reference

frame

The interdiffusion coefficient:

VkTt R R L

j i ij

6 > ⋅ < =

After application of the Allnatt Equation for the Lij :

φ

B A B A A B

c Ntc ) R c R c ( D ~ 6

2 >

− < =

Belova and Murch 1998

⇒

With direct access to the ratio of the atom-vacancy exchange frequencies, one can also use a diffusion kinetics theory to gain access to the tracer correlation factors:

The Ag-Cd system:

a) The ratio DI

Ag/DI Cd (= wAg/wCd)

as a function of cCd at 873K. (Iorio et al. 1973); b) Corresponding tracer correlation factors using the Moleko et al. (1986) Self consistent diffusion kinetics formalism.

⇒ Cd is more correlated (more jump reversals) in its motion than Ag.

Demixing of A and B cations in (A,B)O in an oxygen potential gradient (gives a gradient of cation vacancies):

< P’

O2

P”

O2

wA > wB

(A is blue and B is pink)

After Segregation

Initially JA+JB

Cation Vacancies

P’

O

P’

O

P”

O

P”

O

JV

v

J

2 2 2 2

V

Velocity v ~ 10 -10 ms-1

Demixing of A and B cations in (A,B)O in an electric field.

wA > wB

(A is blue and B is pink)

Velocity v

After Segregation

Initially

JB Cation Vacancies JV JA

P

O2

P

O2

P

O2

P

O2

(–)

(Applied Electric Field)

(+) (+) (–)

Steady-State Condition: Ji – vci N= 0, i = A,B ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ − − ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ = ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ − μ − μ ∇ − μ − μ ∇

−

vN c vN c L L L L ) E q ( ) E q (

B A 1 BB AB AB AA B V B A V A

⇒

Analysis of demixing of A and B cations in a mixed oxide (A,B)O in an oxygen potential gradient and electric field:

Onsager Flux Equations: μi: chemical potential of component i (A, B or V (vacancies)

E: Electric field qi: charge on component i

), E q ( L ) E q ( L J

B V B AB A V A AA A

− ∇ − ∇ − − ∇ − ∇ − = μ μ μ μ

), E q ( L ) E q ( L J

A V A AB B V B BB B

− ∇ − ∇ − − ∇ − ∇ − = μ μ μ μ

Application of the Sum-Rule gives simply that: We now assume random mixing of the two cations. The demixed steady-state composition profile of, say, A, is given by:

kT E ) q q ( c c w c w c a c vN c d dc

B A B A B B A A V A A

− + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + − = 1

2

ξ

⇒

E q a c w vNkT ) (

A V A V A

+ − = − ∇

2

μ μ E q a c w vNkT ) (

B V B V B

+ − = − ∇

2

μ μ

Steady-state demixed profile of Co in (Co,Mg)O in an

• xygen potential gradient (Experimental data: Schmalzried et al.

1979).

0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.2 0.4 0.6 0.8 1 ξ (reduced distance) Co composition

Experimental Data Analytical Result

Initial composition: cCo = 0.51 The fitting parameter is wCo/wMg.

wCo/wMg = 5 cV

2/cV 1 = 1.43

Steady-state demixed profile of Co in (Co,Ni)O in an electric field (Experimental data: Martin 2000).

The fitting parameter is wCo/wNi.

0.89 0.9 0.91 0.92 0.93 0.94 0.95 0.96 0.2 0.4 0.6 0.8 1

ξ (reduced distance) Co composition

qEa/kT=0.676 Initial composition: cCo = 0.93

(Voltage = 50 mV, sample thickness = 600 μm) ⇔ (qEa/kT = 0.676)

wCo/wNi= 5.0

Analysis of interdiffusion in a strongly ionic diffusion couple AZ-BZ:

• The cations A and B diffuse via vacancies on the cation sublattice.
• The anions Z diffuse via vacancies on the anion sublattice.

AZ BZ

. X L J X L X L J X L X L J

Z ZZ Z A AB B BB B B AB A AA A

= + = + =

where for the internal forces we have:

E q X E q X E q X

Z Z B B B A A A

− = + −∇ = + −∇ = μ μ

The Onsager Flux Equations are (we assume that qA = qB = -qZ):

Application of the electro-neutrality conditions and the Gibbs-Duhem relation gives for the intrinsic diffusion coefficients

(e.g. Belova and Murch 2004):

A I A A

c N D J ∇ − =

B I B B

c N D J ∇ − =

where

⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + + + − + − φ =

ZZ AB BB AA Z A AB B AA ZZ 2 AB BB AA B A Z I A

L L 2 L L ) c / ) c L c L ( L L L L c Nc kTc D

⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + + + − + − φ =

ZZ AB BB AA Z B AB A BB ZZ 2 AB BB AA B A Z I B

L L 2 L L ) c / ) c L c L ( L L L L c Nc kTc D

and φ: thermodynamic factor

Application of the Sum-Rule to the ratio of the intrinsic diffusion coefficients gives:

• 1. For the limiting case wZ >> wA (wB), (anion mobility is relatively high):

DI

A/DI B = wA/wB.

(this is the same result as for the binary alloy)

) w w ( w ) w w ( w D D

Z A B Z B A I B I A

+ + =

where wZ is the anion vacancy exchange frequency.

• 2. For the limiting case wZ << wA (wB), (anion mobility is relatively low):

DI

A=DI B.

The almost immobile anion sublattice requires that the fluxes of the cations A and B are equal and opposite. ⇒ The mobility on the anion sublattice no longer determines the rate

• f cation interdiffusion).

⇒ no net cation vacancy flux and no marker shift in interdiffusion.

The general expression for couple (AZ-BZ): in a strongly ionic interdiffusion

D ~

( )

κ + + − φ + − =

Z I B A I A B ZZ B A 2 AB BB AA 2 Z B A Z B A

c ) d c d c ( L c c ) L L L ( c c c 2 N kT q ) q q ( D ~

B AB A AA I A

c L c L d − =

A AB B BB I B

c L c L d − =

where

ZZ 2 Z AB B A BB 2 B AA 2 A

L q L q q 2 L q L q κ + + + =

The interdiffusion coefficient:

Using Allnatt’s (1982) equation for the phenomenological coefficients:

VkTt 6 R R L

j i ij

> ⋅ < =

( )( ) ( )

> < + > + < > − − < φ =

2 2 Z B A B A A B B A A B Z B A ZZ

R R R R p R c R c R c c c c L kT D ~

i ZZ i ) ( ii Z i

c L c L c p + =

2

⇒

There are two limiting cases to consider:

( )

φ > − < =

Z B A B A A B

c c Ntc R c R c D ~ 6

2

I. When the anions are much more mobile than the cations

(Belova and Murch 2005) :

( )( ) ( ) >

+ < > − − < φ =

2 2 B A B A B A A A B B B A A B Va Z

R R c tc R w c R w c R c R c c a Nc D ~

II. When the cations are much more mobile than the anions

(Belova and Murch 2005) : These equations for the interdiffusion coefficients in a strongly ionic compound are, in effect, extensions of the Einstein Equation (1905).

We consider further the case where the anions are much slower than the cations, e.g. in silicates, glasses, transition metal oxides. We apply the Sum-Rule and make use of the accurate self consistent diffusion kinetics theory of Moleko et al. (1989): If LAB were to be neglected (this is the Darken approximation) it would be equivalent to implying that interdiffusion is impossible.

) f 1 ( L f c Nc ) c c ( kT D ~

AB B A B A

− + =

φ

Belova and Murch 2004: f0 : the geometric tracer correlation factor (depends on lattice only)

• Expt. : J.J. Stiglich, Jr. et al. (1973).

Example: Extraction of LAB from the interdiffusion coefficient in (CoO-NiO). Direct access is now possible to LAB in transition metal oxides,

• xides, in silicates, in glasses etc, i.e. whenever the anion

mobility is low compared with the cations.

Some other results that can be obtained at the same approximation level for strongly ionic compounds when the cations are much more mobile than the anions:

φ + =

* B B * A A * B * A

D c D c D D D ~

(This is the Nernst-Planck Equation)

The testing of these equations in, say, a silicate, would require measurements of the tracer diffusion coefficients, the interdiffusion coefficient, the ionic conductivity and the thermodynamic factor.

φ σ = kT f C q D D D ~

dc ion 2 * B * A

and

Conclusions:

• The Onsager flux expressions and Allnatt’s equation for the

phenomenological coefficients can rightly be considered generalizations of Fick’s First Law and the Einstein Equation

• The Onsager flux expressions and Allnatt’s equation,

together with the Sum-rule, bring substantial simplifications to many chemical diffusion problems.

Vielen Dank!