Phenomenological Coefficients in Solid State Diffusion …diffusion.uni-leipzig.de/powerpoint_presentations/pdf/murch.pdf · Phenomenological Coefficients in Solid State Diffusion

Phenomenological Coefficients in SolidState Diffusion

(an introduction)

Graeme E Murch and Irina V Belova

Diffusion in Solids GroupSchool of Engineering

The University of NewcastleCallaghan

New South WalesAustralia

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):

dxdCDJ i

ii −=

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 theflux.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

jiji XLJ Lij=Lji(reciprocity condition)

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

VkTt6RR

L jiij

>⋅<= (Allnatt 1982)

Ri: the ‘collective displacement’ or displacement of the center-of-massof species i in time t.

VkTt6RL

2A

AA><

=VkTt6

RRL BAAB

>⋅<=

E.g.

If the moving A species does not interfere with the moving B speciese.g. A and B do not compete for the same defectsor 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.

VkTt6RR

L jiij

>⋅<=

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

t6rD

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 binarysystem of A* and A in which we allow the tracer A* concentration to be very low.

Then we would have that: VkTDL

*

AA ** =

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 modellingof 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 partZ: coordination numbercv: vacancy concentrationwj: exchange frequency of an atom of type j with a vacancya: 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 tracerdiffusion coefficient in the following way:

I can hop too!

∑∞

=>θ<+=

1m

)m(cos21f

The tracer correlation factor can be expressed interms 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

0.5

1.0

0 1 2 3 4 5 6 7 8 9 10 11

Cos(θAm)

m

Example of the convergence of the cosine between the first tracer A jump and them’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 of the centers-of-mass of the species present.

The collective correlation factors can be expressed interms 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<+=

1m

)m(iiii cos21f

Diagonal collective correlation factor:

∑∑∞

=

∞

=>θ<+>θ<=

1m

)m(BA

AA

BB

1m

)m(AB

)A(AB cos

nCnCcosf

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

-0.4

-0.2

0

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(θAAm )

mExample of the convergence of the cosine between the first collective jump and them’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/DCL *iiii = Lij=0

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

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛+=

∑k

*kk0

*ii

*ii

ii DCMDC21

kTDCL

∑=

k

*kk0

*jj

*ii

ij )DCM(kTDCDC2

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 jumpPossible vacancy jumps after time t

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

Vector summation of = 0

Vjj

M

1iijij ccAww/wL =∑

=

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):

ABA

B2

BBVBBAB

B

A2

AAVAA L

ww

kTawcNc

L,Lww

kTawcNc

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 beenderived 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):

BABAAAA XLXLJ +=

AABBBBB XLXLJ +=⎟⎠⎞

⎜⎝⎛

∂γ∂

+∂

∂−=

∂μ∂

−=cln

ln1x

CCkT

xX i

i

ii

The Onsager flux equations are:

φ⎟⎟⎠

⎞⎜⎜⎝

⎛−=

B

AB

A

AAIA C

LCLkTD

φ⎟⎟⎠

⎞⎜⎜⎝

⎛−=

A

AB

B

BBIB C

LCLkTD

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

xCDJ iI

ii ∂∂

−=

Local lattice reference frame

⎟⎟⎠

⎞⎜⎜⎝

⎛∂

γ∂+=φ

Bclnln1

φ is the ‘thermodynamic factor’:

ABBBBA

ABAAABIB

IA

LcLcLcLc

DD

−−

≡

B

AIB

IA

ww

DD

=

Measurement of the ratio of the intrinsic diffusivitiesdirectly 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 off-diagonal phenomenological coefficients are put equal to zero) we would then have obtained:

B*

A*

IB

IA

DD

DD

=

(This is a very rough approximation)

IBA

IAB DcDcD~ +=

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

00B

AA J

xCD~J −=∂

∂−= Lab. reference

frame

The interdiffusion coefficient:

VkTtRR

L jiij 6

>⋅<=

After application of the Allnatt Equation for the Lij :

φBA

BAAB

cNtc)RcRc(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 DIAg/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’O2P”

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 22

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

PO2PO2

PO2PO2

(–)(Applied Electric Field)

(+)

(+) (–)

Steady-State Condition: Ji – vci N= 0, i = A,B

⎥⎦

⎤⎢⎣

⎡−−

⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡−μ−μ∇−μ−μ∇ −

vNcvNc

LLLL

)Eq()Eq(

B

A1

BBAB

ABAA

BVB

AVA⇒

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

),Eq(L)Eq(LJ BVBABAVAAAA −∇−∇−−∇−∇−= μμμμ

),Eq(L)Eq(LJ AVAABBVBBBB −∇−∇−−∇−∇−= μμμμ

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:

kTE)qq(cc

wc

wc

acvNc

ddc BABA

B

B

A

A

V

AA −+⎟⎟

⎠

⎞⎜⎜⎝

⎛+

−=

12ξ

⇒

Eqacw

vNkT)( AVA

VA +−=−∇ 2μμ

Eqacw

vNkT)( BVB

VB +−=−∇ 2μμ

Steady-state demixed profile of Co in (Co,Mg)O in an oxygen potential gradient (Experimental data: Schmalzried et al. 1979).

0.30.35

0.40.45

0.50.55

0.60.65

0.7

0 0.2 0.4 0.6 0.8 1ξ (reduced distance)

Co

com

posi

tion

Experimental DataAnalytical Result

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

wCo/wMg = 5cV

2/cV1 = 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 0.2 0.4 0.6 0.8 1

ξ (reduced distance)

Co

com

posi

tion

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

.XLJXLXLJXLXLJ

ZZZZ

AABBBBB

BABAAAA

=

+=

+=where for the internal forces we have: EqX

EqXEqX

ZZ

BBB

AAA

−=

+−∇=

+−∇=

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):

AIAA cNDJ ∇−= B

IBB cNDJ ∇−=

where

⎟⎟⎠

⎞⎜⎜⎝

⎛

+++−+−φ

=ZZABBBAA

ZAABBAAZZ2ABBBAA

BA

ZIA LL2LL

)c/)cLcL(LLLLcNc

kTcD

⎟⎟⎠

⎞⎜⎜⎝

⎛

+++−+−φ

=ZZABBBAA

ZBABABBZZ2ABBBAA

BA

ZIB LL2LL

)c/)cLcL(LLLLcNc

kTcD

and

φ: thermodynamic factor

Application of the Sum-Rule to the ratio of the intrinsic diffusion coefficientsgives:

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

A/DIB = wA/wB.

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

)ww(w)ww(w

DD

ZAB

ZBAIB

IA

++

= where wZ is the anion vacancy exchange frequency.

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

A=DIB.

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 rateof cation interdiffusion).

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

The general expression forcouple (AZ-BZ):

in a strongly ionic interdiffusion D~

( )κ

++−φ+−=

Z

IBA

IABZZBA

2ABBBAA

2Z

BA

ZBA

c)dcdc(Lcc)LLL(c

cc2NkTq)qq(D~

B

AB

A

AAIA c

LcLd −=

A

AB

B

BBIB c

LcLd −=

where

ZZ2ZABBABB

2BAA

2A LqLqq2LqLqκ +++=

The interdiffusion coefficient:

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

VkTt6RR

L jiij

>⋅<=

( )( )( ) ><+>+<

>−−<φ=

22ZBA

BAABBAAB

ZBA

ZZ

RRR

RpRcRcRcccc

LkTD~

iZZi

)(iiZ

i cLcLc

p +=02

⇒

There are two limiting cases to consider:

( )φ

>−<=

ZBA

BAABccNtcRcRcD~

6

2

I. When the anions are much more mobile than the cations (Belova and Murch 2005) :

( )( )( ) >+<

>−−<φ= 2

2

BABA

BAAABBBAABVaZ

RRctc

RwcRwcRcRccaNcD~

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 consistentdiffusion 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.

)f1(Lf

cNc)cc(kTD~

0

AB0

BA

BA

−+

=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, oxides, 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 sameapproximation level for strongly ionic compounds when the cations are much more mobile than the anions:

φ+

=*BB*AA

*B*A

DcDcDDD~ (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.

φσ

=kTf

CqDDD~

dc0

ion2

*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!