Heitjans P., Karger J. (Eds.). Diffusion in Condensed Matter: Methods, Materials, Models

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5 Diﬀusion in Oxides 221

Fig. 5.8. Tracer diﬀusion

coeﬃcients of

Co and

Ga in (Co

1−x

)

1−δ

as a function of the dopant

fraction, x,atT = 1350

◦

and diﬀerent oxygen activi-

ties [21].

The modelling of both measured diﬀusion coeﬃcients, D

∗

and D

∗

,as

a function of the oxygen partial pressure, p

, and the dopant fraction, x,

yields the equilibrium constants K

and K

(see (5.16) and (5.17)) and

the diﬀusion coeﬃcients D

Co,i

and D

Ga,i

(i =1, 2) [21]. The results show

that there is essentially no binding between the dopant, Ga

•

, and singly

ionised vacancies, V



, while there is strong binding between Ga

•

and V



As a consequence, D

Co,1

and D

Co,P

are identical. D

Co,P

is two orders of

magnitude smaller than D

Co,1

and D

Co,2

and can therefore be neglected. In

summary, Ga is mobile by means of strongly bound pairs, {Ga

•

, V



}



,and

weakly bound pairs, {Ga

•

, V



}. Co is mobile by means of free vacancies,



and V



, and also by exchange with vacancies which are strongly bound

in pairs {Ga

•

, V



}. In the latter mechanism the vacancy performs jumps

around the Ga ion, thereby moving the Co ion but without dissociating the

pair.

The binding energy E

determined from the equilibrium constant K

z · exp(E

/RT ) turns out to depend slightly on temperature. Assuming

nearest-neighbour pairs (z = 12) we obtain a binding energy of about

40 kJ/mol, while we obtain values of about 50 kJ/mol if we assume next-

nearest neighbour pairs (z = 6). The corresponding Coulomb energies (as-

suming point charges) are 72 kJ/mol (z = 12) and 50 kJ/mol (z =6)which

might be an indication for the next-nearest neighbour pairs, as proposed by

theoretical calculations [23].

A detailed analysis of the diﬀusion coeﬃcients, D

Ga,i

and D

Co,i

, obtained

from the ﬁtting in terms of defect mobilities and correlation factors is impos-

sible because the (physical) correlation factors are not known. However, we

can draw some general conclusions.

– The diﬀusion coeﬃcients D

Co,1

and D

Co,2

describe the motion of the

solvent Co via free vacancies V



and V



, and can both be written

222 Manfred Martin

as a product of a vacancy diﬀusion coeﬃcient and a correlation factor.

If we assume that these correlation factors are identical to the geometric

correlation factor in an fcc lattice, f

=0.781, we obtain diﬀerent vacancy

diﬀusion coeﬃcients, D



and D



in Ga-doped CoO. This result is in

agreement with measurements of the charge of transport

in pure CoO

[24], and the modelling of these data in terms of the Onsager-Fuoss-theory

of liquid electrolytes shows also that the average mobility of the vacancies

changes with the oxygen activity.

– The diﬀusion coeﬃcients D

Ga,1

and D

Ga,2

describe the motion of the

solute Ga by means of pairs P

and P

, and both can be written in the

form D

Ga,i

=(1/6) ·s

·f

B,i

·ω

B,i

(i =1, 2), where s is the jump distance,

B,i

the correlation factor and ω

B,i

the impurity-vacancy exchange rate.

The experimentally obtained ratio D

Ga,1

Ga,2

is about ten. If we assume

that the exchange rates ω

Ga,1

and ω

Ga,2

are about the same for both

mechanisms we can conclude that the correlation factor f

Ga,2

must be

much smaller than f

Ga,1

. This is in qualitative agreement with strong

binding between Ga

•

and V



, compared to weak binding between Ga

•

and V



5.3.2 Diﬀusion in Oxides with Dominating Oxygen Disorder

Oxygen defects are now the majority defects, while cation defects are only

minority defects. Thus, oxygen diﬀusion will be much faster than cation dif-

fusion.

Oxygen Self-Diﬀusion

The determination of the self-diﬀusion coeﬃcient of oxygen is accomplished

by using one of the two stable oxygen isotopes,

Oor

O. The sample

is annealed in an isotope-enriched atmosphere and the diﬀusion proﬁle is

generally determined by secondary ion mass spectrometry (SIMS) [25] (see

Chap. 1, Sect. 1.4.1). Mathematically the experimental setup corresponds

to the well-known inﬁnite source solution [26]. However, if exchange of the

oxygen isotope between the atmosphere and the oxide is not suﬃciently fast

there will be no equilibrium for the isotope at the surface. Most often, the

rate of isotope exchange at the oxide surface is assumed to be proportional

to the isotope concentrations in the gas and the solid, c

and c

, resulting in

−D

∗

∂c

∂x



x=0

= k · (c

− c

) (5.22)

where D

∗

is the oxygen tracer diﬀusion coeﬃcient and k the surface ex-

change coeﬃcient. The solution of the diﬀusion equation for a semi-inﬁnite

The charge of transport will be discussed in more detail in Sect. 5.5.2

5 Diﬀusion in Oxides 223

medium subject to the boundary condition in (5.22) and having an initial

concentration c

is [26]

c(x, t)=c

+(c

− c

) ·



erfc



√

∗



− exp



∗



· erfc

√

∗

+ k

∗



. (5.23)

The parameters D

∗

and k are then obtained by ﬁtting (5.23) to the experi-

mental proﬁle. A detailed analysis of the conditions where D

∗

and k can be

determined unambiguously can be found in [27].

Two typical examples of oxygen diﬀusion in oxides of the ABO

per-

ovskite structure are shown in Fig. 5.9 [28]. Both examples belong to sam-

ples in the solid solution series between Sr-doped lanthanum cobaltate,

0.8

0.2

CoO

3−δ

, and Sr-doped lanthanum manganate, La

0.8

0.2

MnO

3−δ

The oxides are doped with SrO to increase the fraction of oxygen vacan-

cies (see (5.18)). In the manganese-rich sample, La

0.8

0.2

0.8

0.2

3−δ

the oxygen diﬀusion proﬁle extends over a few microns and can be deter-

mined by SIMS depth proﬁling (Fig. 5.9a). In Sr-doped lanthanum cobaltate,

0.8

0.2

CoO

3−δ

, however, oxygen diﬀusion is much faster and the penetra-

tion depth is about two orders of magnitude larger (Fig. 5.9b). For such large

penetration depths the SIMS depth proﬁling technique is no longer applica-

ble; the SIMS line-scanning technique [25] is used instead. Here the SIMS

analysis is performed on a section perpendicular to the sample surface. By

ﬁtting (5.23) to the proﬁles, the oxygen diﬀusion coeﬃcient and the surface

exchange coeﬃcient in both materials were determined. It turns out that in

0.8

0.2

CoO

3−δ

the oxygen diﬀusion coeﬃcient is about ﬁve orders of mag-

nitude larger than in La

0.8

0.2

0.8

0.2

3−δ

, while the surface exchange

coeﬃcient is about 2 orders of magnitude larger in the former oxide than in

the latter.

In general, oxygen diﬀusion can proceed in the same way as described for

cation diﬀusion in (5.19), namely by means of oxygen vacancies and oxygen

interstitials. As discussed in [28] there is, however, no evidence for oxygen

interstitials in ABO

perovskite oxides, mainly for geometrical reasons. Thus

the oxygen tracer diﬀusion coeﬃcient is simply given by

∗

= f

· D

· [V

••

] (5.24)

where f

=0.69 is the geometrical correlation factor for oxygen tracer dif-

fusion in the oxygen sublattice of the ABO

structure [29] and D

the

self-diﬀusion coeﬃcient of oxygen vacancies. In several other perovskites

the combination of oxygen diﬀusion coeﬃcients and measured vacancy frac-

tions shows that the self-diﬀusion coeﬃcient of oxygen vacancies, D

,ex-

hibits only small variations between diﬀerent perovskites [29]. Thus, the

224 Manfred Martin

(a) (b)

Depth ( m)

c’(x)

Depth ( m)

c’(x)

Fig. 5.9. Typical

O diﬀusion proﬁles, showing the normalized isotope frac-

tion, c



(x)=(c(x) − c

)/(c

− c

), against depth, together with the ﬁtted

curves according to (5.23) [28]. (a) determined by SIMS depth proﬁling of a

0.8

0.2

0.8

0.2

3−δ

sample (

O anneal at 1000

◦

C for 3840 s). (b) deter-

mined by SIMS linescanning of a La

0.8

0.2

CoO

3−δ

sample (

O anneal at 1000

◦

for 1675 s).

large diﬀerence of the observed oxygen diﬀusion coeﬃcients is predomi-

nantly due to the diﬀerence in the oxygen vacancy concentrations between

0.8

0.2

0.8

0.2

3−δ

and La

0.8

0.2

CoO

3−δ

Cation Diﬀusion

In oxides with dominating oxygen disorder cation defects are only minor-

ity defects and consequently cation diﬀusion is much slower than oxygen

diﬀusion. Cation diﬀusion is nevertheless important since the slowest mov-

ing species determine many fundamental processes, such as sintering [30],

creep [31] or internal friction [32].

An important example is yttria-stabilized zirconia, (Zr

1−x

2−x/2

(YSZ), which exhibits very high oxygen ion conductivity and is therefore used

as electrolyte material in high-temperature applications [33]. While there is a

considerable amount of data available on the oxygen transport (e.g. [34,35]),

only little is known about the cation transport in YSZ [36, 37]. Figure 5.10

shows recent results for the diﬀusion coeﬃcients of Y and Zr in single crys-

talline YSZ [38]. The diﬀusion coeﬃcient of yttrium was measured using the

radioactive isotope

Y, while the diﬀusion coeﬃcient of Zr was obtained by

implanting the stable isotope

Zr, annealing at elevated temperatures and

subsequent SIMS analysis.

From Fig. 5.10 it can be seen that Zr diﬀusion becomes slower with in-

creasing Y-content. This is due to the fact that the dopant yttrium deter-

mines the fraction of oxygen vacancies, [Y



]=2[V

••

], which again deter-

mines via the Schottky equilibrium (see (5.1)) the fraction of cation vacan-

cies. Thus cation diﬀusion should be slower the higher the dopant fraction, as

5 Diﬀusion in Oxides 225

5.00 5.50 6.00 6.50 7.00

-19

-18

-17

-16

-15

-14

-13

-12

Zr in YSZ-10

Y in YSZ-10

Y in YSZ-11

Sc in YSZ-11

Zr in YSZ-32

²/

]

1/T [10000 K

-1

]

2000 1900 1800 1700 1600 1500 1400

T [K]

Fig. 5.10. Temperature dependence of the tracer diﬀusion coeﬃcients of Zr, Y and

Sc in yttria-doped zirconia, YSZ, containing 10, 11 and 32 mol% Y

[38].

observed. As expected, comparison of the self-diﬀusion coeﬃcients of oxygen

and cations in YSZ shows that D

is about 5 orders of magnitude larger than

cation

Similar results were found for doped lanthanum gallate,

1−x

1−y

3−(x+y)/2

(LSGM), which has a higher oxygen ion con-

ductivity than YSZ and is therefore a candidate for solid oxide fuel cells work-

ing at intermediate temperatures [39]. Here oxygen diﬀusion coeﬃcients [40]

and cation self- and impurity diﬀusion coeﬃcients [41, 42] have been mea-

sured. In contrast to YSZ there are two cation sublattices in the ABO

perovskite LSGM, an A- and a B-sublattice. In the perovskite structure (see

Fig. 5.11), a direct jump of A-cations within the A-sublattice is possible,

while a direct jump of B-cations within the B-sublattice is impossible due

to the oxygen ion located between two nearest neighbour B-sites. Thus, for

diﬀusion of B-cations curved diﬀusion pathways or jumps to next-nearest

neighbour sites must be considered. Atomistic simulations of migration en-

ergies in lanthanum gallate [43] yield much higher migration energies for

diﬀusion of B-cations than for diﬀusion of A-cations. The measured tracer

diﬀusion coeﬃcients of La, Sr and Mg [41] and the impurity diﬀusion coef-

ﬁcients for Fe, Cr and Y [41] are, however, very similar and show identical

activation energies, although Mg, Fe and Cr should occupy B-sites while La,

Sr and Y occupy A-sites. These observations may be an indication for anti-

site disorder in the perovskite LSGM, i.e. a small fraction of B-cations may

occupy A-sites. This would be suﬃcient to enable diﬀusion of B-cations with

similar diﬀusion coeﬃcients as A-cations. A more detailled diﬀusion model

226 Manfred Martin

Fig. 5.11. Structure of the perovskite

ABO

considers a defect cluster consisting of an A-site vacancy, a B-site vacancy

and at least one oxygen vacancy [41]. This cluster is mobile as an entity, i.e.

without dissociation by means of four, correlated jumps of A- and B-cations

and oxygen ions. As a result, A- and B-cations have identical diﬀusivities.

5.4 Chemical Diﬀusion

So far we have considered self- and impurity diﬀusion processes in chemi-

cally homogeneous oxides without any concentration gradients. We will now

consider diﬀusion in concentration gradients, which is called chemical diﬀu-

sion. It is well known, from irreversible thermodynamics, that the real driving

force for isothermal mass transport of a component i is not its concentration

gradient but the gradient of its electrochemical potential, η

= µ

+ z

·F ·Φ,

where µ

is the chemical potential and Φ the electric potential [44] (z

is the

charge number and F, Faraday’s constant)

. The resulting ﬂux equation for

component i is

= −L

·∇(µ

+ z

· F · Φ) (5.25)

where L

is the so-called Onsager transport coeﬃcient

. For the sake of sim-

plicity, we will consider only a binary oxide AO with dominating cation dis-

order. The results can easily be transferred to oxides with dominating oxygen

disorder. We start with an oxide which is equilibrated at elevated tempera-

tures and at a certain oxygen partial pressure, p

(1)

. If the partial pressure of

the surrounding atmosphere is increased to p

(2)

an oxidation process takes

place. Oxygen is incorporated into the crystal by adding new lattice sites to

the crystal and producing cation vacancies and electron holes (see (5.8)). In

the case of reduction, lattice planes are annihilated and oxygen is released

For neutral particles (z

= 0) the driving force for mass transport is the gradient

of the chemical potential, as already discussed for metals in Sect. 1.3.2 of Chap. 1.

Cross coeﬃcients L

are neglected in this section but will be considered in

Sect. 5.5.

5 Diﬀusion in Oxides 227

to the atmosphere. To preserve local electrical neutrality during the result-

ing transport processes of cations and electronic defects their ﬂuxes must be

coupled:

+ j

•

=0 . (5.26)

From (5.26) we can calculate the internal electrical potential gradient, ∇Φ

(the so-called Nernst ﬁeld), and insert the resulting expression into the ﬂux

equation for the cations in (5.25). With the equilibrium A + 2h

•

 A

between metal, A, cations, A

, and electron holes, h

•

, the ﬂux of cations

can be written as

= −L

· t

·∇µ

(5.27)

where t

= L

/(4L

+ L

) is the electronic transference number. Equa-

tion (5.27) is the so-called Wagner formula for chemical or ambipolar diﬀu-

sion [45]. If the thermodynamic driving force, ∇µ

, is written in terms of the

concentration gradient, ∇µ

=(∂µ

/∂c

) ·∇c

,weobtain

= −L

· t

· (∂µ

/∂c

) ·∇c

= −

D ·∇c

(5.28)

which deﬁnes the chemical diﬀusion coeﬃcient,

D. Due to the ﬂux coupling,

+ j

= 0, the vacancy ﬂux can also be written in terms of the chemical

diﬀusion coeﬃcient:

= −

D ·∇c

. (5.29)

Finally, we write L

in terms of the diﬀusion coeﬃcient D

and the con-

centration c

, L

= D

· c

/RT (Einstein relation) and make use of

= µ

+ RT ln a

,wherea

is the activity of A. The result for

D is

D = D

· t

∂ ln a

∂ ln x

. (5.30)

Thus, the chemical diﬀusion coeﬃcient,

D, is the product of the cation self-

diﬀusion coeﬃcient, D

, the electronic transference number, t

, and the ther-

modynamic factor, ∂ ln a

/∂ ln x

. Equation (5.30) applies also to oxides with

dominating oxygen disorder if D

, a

and x

are substituted by D

, a

and

The chemical diﬀusion coeﬃcient,

D,ofanoxideA

1−δ

O determines the

equilibration kinetics of δ after a change of the external oxygen partial pres-

sure. It can be determined by measuring δ directly, e.g. via thermogravimetry

or by measuring a quantity which is proportional to δ, such as the electronic

conductivity. These so-called relaxation experiments have been used to mea-

sure

D in various oxides such as CoO with dominating cation disorder [46],

or (La,Mn)CoO

[47] and (La,Sr)CrO

[48] with dominating oxygen disorder.

If the oxide is a good semiconductor (t

= 1) and the majority defects are

cation vacancies, V



(with negative excess charge α =2,1,0),andelectron

holes, h

•

, (or oxygen vacancies, V

•

, and electrons, e



) the formula for

D in

(5.30) simpliﬁes to (see appendix)

228 Manfred Martin

D = D

· (1 + α) . (5.31)

Thus, from the measured chemical diﬀusion coeﬃcient,

D, the self-diﬀusion

coeﬃcient, D

, of the dominating vacancies can be calculated, if their excess

charge is known. This is the normal procedure, e.g. mostly adopted in oxides

with oxygen disorder where vacancies V

••

(α = 2) dominate. If, on the other

hand, the vacancy self-diﬀusion coeﬃcient is known the excess charge of the

vacancies can be calculated. As will be shown in Sect. 5.5.1, both the vacancy

self-diﬀusion coeﬃcient, D

, and the chemical diﬀusion coeﬃcient,

D,can

be obtained simultaneously by performing tracer self-diﬀusion experiments

during chemical diﬀusion. Then, from both diﬀusion coeﬃcients the excess

charge α can be obtained via (5.31).

In the general case of a mixed conductor, t

= 1, the chemical diﬀusion

coeﬃcient may show a strong dependence on the oxygen partial pressure for

two reasons: (i) the electronic transference number, t

, depends on the oxygen

partial pressure. (ii) if the stability ﬁeld of the oxide contains the stoichio-

metric point, δ(p

∗

) = 0, the thermodynamic factor and also the chemical

diﬀusion coeﬃcient exhibit a maximum at this oxygen partial pressure. This

case is found, e.g., in BaTiO

[49].

Another technique that has been used recently for the measurement of

chemical diﬀusion coeﬃcients in Fe-doped SrTiO

, which is a mixed conduc-

tor, uses the optical absorption of the sample [50]. In this way, time- and

position-resolved concentration proﬁles of oxygen can be determined from

which the chemical diﬀusion coeﬃcient is evaluated.

5.5 Diﬀusion in Oxides Exposed to External Forces

If an oxide is exposed to external thermodynamic forces, e.g. an oxygen po-

tential gradient or an electric potential gradient, defect ﬂuxes are induced

which again cause ﬂuxes of the chemical components. As before, it is rea-

sonable to distinguish between dominating oxygen disorder and dominating

cation disorder.

In oxides where the oxygen ions are much more mobile than the cations,

essentially only oxygen is driven through the oxide

. For pure oxygen ion

conductors this situation corresponds to an electrolyte in a solid oxide fuel

cell (applied oxygen potential gradient) or an electrochemical oxygen pump

(applied electric potential gradient). For mixed conductors this situation cor-

responds to oxygen permeation cells. A detailed analysis of these cases is,

however, beyond the scope of this chapter and can be found, e.g., in [51].

The (driven) motion of the slower cations is, however, a possible origin of

long-term degradation processes, such as creep or kinetic demixing (see also

Sect. 5.5.1).

5 Diﬀusion in Oxides 229

In oxides with dominating cation disorder external forces act on the mobile

cations. The implications will be considered in more detail in the following

sections.

5.5.1 Diﬀusion in an Oxygen Potential Gradient

Chemical Diﬀusion in an Oxygen Potential Gradient

If an oxide A

1−δ

O of thickness ∆z is exposed to an oxygen potential gradient,

established, e.g., by diﬀerent gas mixtures on both sides of the disc, diﬀerent

concentrations of cation vacancies, c

(1)

and c

(2)

, are established on both sides

of the disc (according to (5.9)). As a consequence, a vacancy ﬂux, j

−

D·∇c

(see (5.29)), occurs from the high- to the low-oxygen potential side.

Due to the ﬂux coupling, j

+ j

= 0, cations are driven in the opposite

direction. When vacancies and cations arrive at the oxide surfaces, reduction

and oxidation of the oxide occur at the low- and high-oxygen potential side,

respectively:



+2h

•

+AO

reduction

−−−−−−→

←−−−−−−

oxidation

(g) . (5.32)

Thus, lattice planes are removed from the low oxygen potential side and

added to the high oxygen potential side. As a result, the crystal surfaces move

relatively to the immobile oxygen sublattice to the side of the higher oxygen

activity. The crystal displacement and the vacancy concentration proﬁle can

be calculated by solving the diﬀusion equation [52]. After a short transient

period the crystal moves with a constant velocity. A steady-state solution

can be calculated by transforming from the laboratory reference frame

to a

moving coordinate system (coordinate z) which is ﬁxed at one surface. The

steady-state vacancy fraction proﬁle in the moving system, x

(z), is linear

in position, z, to a very good approximation, and the steady state velocity,

v,isgivenby

= a + b · z, v=

D · b (5.33)

with

a = x

(1)

,b=

(2)

− x

(1)

∆z

. (5.34)

Experiments with the model system CoO exposed to an oxygen potential

gradient conﬁrm the shift of the crystal surfaces relatively to the immobile

oxygen sublattice [52].

Since oxygen is essentially immobile, the laboratory reference frame is identical

to the lattice reference frame.

230 Manfred Martin

Tracer Diﬀusion in an Oxygen Potential Gradient

The motion of cation tracers (being chemically identical to the cation A or

being an impurity) in an oxide which is exposed to an oxygen potential gradi-

ent is inﬂuenced by the directed vacancy ﬂux in (5.29) in two respects: Firstly,

the tracer diﬀusion coeﬃcient is proportional to the vacancy fraction which

varies linearly with position. Thus, tracer diﬀusion takes place with a linearly

position dependent tracer diﬀusion coeﬃcient. Secondly, the tracer ions are

moved by the drifting vacancies. This drift ﬂux of the tracer particles, which

is over and above their normal Brownian motion, reﬂects the interaction of

the tracer ions with the vacancies. In summary, the tracer ﬂux consists of

two parts: the ﬁrst part, j

diﬀ

tracer

, describes Brownian motion, as in the case

of a homogeneous crystal, but with the diﬀerence that the diﬀusion coeﬃ-

cient is position dependent. The second part is a drift term, j

drift

tracer

,whichis

proportional to the vacancy ﬂux, as will be shown below.

Tracer Self-Diﬀusion in an Oxygen Potential Gradient

In this case the tracer ions A

∗

are chemically identical to the normal cations

A, but in contrast to them they are distinguishable. Thus, the tracer diﬀusion

coeﬃcient, D

∗

= f

· D

· x

, contains the geometrical tracer correlation

factor, f

, and the ﬁrst part of the ﬂux of the tracer particles has the form

diﬀ

∗

= −f

· D

· x

·∇c

∗

. (5.35)

Since the tracer particles are chemically identical to the normal cations they

are moved by the directed vacancy ﬂux in the same way as the normal cations.

However, only a fraction x

∗

of the total amount of A exists as tracer. There-

fore the drift ﬂux of the tracer has the form:

drift

∗

= −x

∗

· j

. (5.36)

The source solution for this diﬀusion problem is given by [53]

∗

(z,t)=

· b · t

· exp



−

2a + b · (z + z

)

· f

· b

· t



· I



(a + bz)

1/2

· (a + bz

)

1/2

· f

· b

· t



(5.37)

where M is the total amount of tracer per unit area, z

the initial position of

the tracer source, and I

a Bessel function of order zero. In contrast to the

source solution for a constant diﬀusion coeﬃcient (Gaussian) the maximum

of the curve shifts with increasing time to the side of higher oxygen potential,

and the proﬁle becomes more and more asymmetric. The initial tracer source

position can be marked by inert markers. Its position in the moving system

is z

marker

= z

+ v · t. The position of the maximum, z

max

(t), relative to the

marker position is then given by