Heitjans P., Karger J. (Eds.). Diffusion in Condensed Matter: Methods, Materials, Models
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5 Diffusion in Oxides 241
.
..
..
.
.
.
.
.
.
Fig. 5.19. Calculated steady state demixing profiles of component A in a mixed
oxide (A,B)O as a function of the applied voltage U (T = 1300
◦
C, x
0
A
=0.5,
γ = D
A
/D
B
= 2) [70].
diffusion coefficients are proportional to the cation vacancy fraction, x
V
.The
concentration profile of A (and B, x
A
+ x
B
= 1) can be obtained by inte-
gration of (5.46) and the overall mass balance for A. Details can be found
in [70]. Figure 5.19 shows steady-state demixing profiles calculated in this
way for different values of the applied voltage U . With increasing voltage the
anode-side of the oxide is more and more depleted by the faster component
A.
A typical result of a steady-state demixing experiment for (Co
1−x
Ni
x
)O
is shown in Fig 5.20. Enrichment of Co near the cathode side of the oxide
is found [73], as expected qualitatively due to the higher diffusivity of Co
compared to Ni [74]. For a quantitative modelling of the demixing (solid
line in Fig. 5.20) one also has to consider the cross terms in the transport
coefficient matrix L
ij
(i, j =A, B, h
•
). In particular the cross terms L
Ah
and
L
Bh
are important, as shown in the previous sections. Finally it should be
emphasized that the electric current during demixing is mainly conducted
by electron holes, because t
h
∼
=1andt
cation
≈ 10
−4
. Nevertheless, there is a
demixing process for the cations as long as they have different mobilities.
Demixing in a heterovalently doped oxide, (A
1−x
B
x
)O, is more compli-
cated. Due to Coulomb interactions between the defects (impurity-vacancy
pairs) the cross term L
AB
can no longer be neglected and might even de-
termine the complete demixing behavior. An example is Ga-doped CoO,
(Co
1−x
Ga
x
)O, in which Ga
•
Co
and V
A
form impurity-vacancy pairs,
{Ga
•
Co
, V
Co
}
[21]. These pairs have a negative excess charge and move to-
242 Manfred Martin
.
.
...
.
.
.
.
.
Fig. 5.20. Steady state demixing profile of Co in (Ni
1−x
Co
x
)O (T = 1444
◦
C,
x
0
A
=0.932, ∆z = 600 µm, t =17h); experiment; — theory.
wards the anode. Thus Ga is enriched at the anode side, independent of the
ratio of the diffusion coefficients of Co and Ga [71].
5.6 Conclusion
In this chapter we have outlined the basics of diffusion processes in oxides.
We have started in Sect. 5.2 with the so-called defect chemistry which de-
scribes all defect formation processes by quasi-chemical reactions and the
corresponding equilibria. In this way, a quantitative calculation of the defect
fractions in an oxide as a function of the intensive thermodynamic variables
temperature, T , and oxygen partial pressure, p
O
2
, is possible. The most im-
portant reaction is the incorporation of oxygen into the oxide which results
in the formation of a non-stoichiometric oxide, either with oxygen excess or
oxygen deficit. In the most simple cases, we obtain typical regimes in the
p
O
2
-dependence of the defect fractions. In each regime only two oppositely
charged defects dominate which exhibit a power-law dependence on p
O
2
with
atypicalexponent(Kr¨oger-Vink-diagram, see Fig. 5.4).
In Sect. 5.3, the basic diffusion processes in oxides, vacancy diffusion and
interstitial diffusion, have been considered. Self-diffusion and impurity diffu-
sion were discussed for oxides with dominating cation disorder and for oxides
with dominating oxygen disorder. Special emphasis was put on the role of
defect-defect interactions, such as the formation of impurity–vacancy pairs.
Chemical diffusion was treated in Sect. 5.4, and in Sect. 5.5 diffusion in ox-
ides exposed to external forces was analysed. In particular the latter section
shows that – apart from a fundamental interest – diffusion in oxides is of
great practical importance, e.g. for long-term degradation processes of oxides
in technical applications, such as fuel cells, sensors etc..
5 Diffusion in Oxides 243
Finally it should be mentioned that several important topics concerning
diffusion in oxides could not be discussed in detail in this chapter. Some of
them are:
Creep of oxides: Mass transport in oxides due to mechanical forces is
determined by the slowest moving species, which are the oxygen ions in oxides
with cation disorder and the cations in oxides with oxygen disorder. Thus,
measurements of the creep rate as a function of stress provide not only data
on the mechanical strength of an oxide but they yield also information on
the diffusivities of the minority defects (see e.g. [31]).
Solid state reactions between oxides: The formation of ternary ox-
ides, for example the spinel formation MgO + Al
2
O
3
→ MgAl
2
O
4
,pro-
ceeds by counter-diffusion of Mg
2+
and Al
3+
through the spinel MgAl
2
O
4
resulting in parabolic growth laws for the thickness of the spinel layer,
∆x(t)=(2k
p
t)
1/2
. The parabolic rate constant, k
p
, is a function of the
diffusion coefficients of both cations [75].
Oxidation of metals: Diffusion controlled growth of an oxide layer on
a metal is determined by ambipolar diffusion of cations and/or oxygen an-
ions and electronic defects through the oxide. As shown by Wagner [45], the
growth rate of the oxide layer is also determined by a parabolic rate law,
where the parabolic rate constant, k
p
, is given as a function of the diffusion
coefficients of cations and anions and the electronic transference number. If
the oxide is anisotropic the parabolic rate constant depends on the orienta-
tion of the oxide relative to the metal substrate. This case was found in a
recent investigation on the oxidation of the intermetallic, ordered compound
CoGa [76,77] where only Ga is oxidised and β-Ga
2
O
3
is formed.
5.7 Appendix
We consider a binary oxide A
1−δ
O with cation disorder. Using x
A
+ x
V
=1,
the thermodynamic factor ∂ ln a
A
/∂ ln x
A
can be rewritten as:
∂ ln a
A
∂ ln x
A
= −x
A
·
∂ ln a
A
∂x
V
Considering the equilibrium for the reaction A +
1
2
O
2
AO, K =1/(a
A
·
a
1/2
O
2
), the activity a
A
canbeexpressedintermsofa
O
2
:
∂ ln a
A
∂ ln x
A
=
1
2
·
x
A
x
V
·
∂ ln a
O
2
∂ ln x
V
As shown in Sect. 5.2.1, the vacancy fraction shows a simple power law de-
pendence on a
O
2
, x
V
=const·(a
O
2
)
1/m
,wherem is the typical exponent for
the dominating disorder type. For differently charged cation vacancies, V
α
A
,
m is given by m =2+2α, resulting in
244 Manfred Martin
∂ ln a
A
∂ ln x
A
=
1
2
·
x
A
x
V
· (2 + 2α) .
With this expression for the thermodynamic factor, the chemical diffusion
coefficient
˜
D in (5.30) can be written as
˜
D =
D
A
· x
A
x
V
· t
el
· (1 + α) .
Finally, we use the jump balance, D
A
x
A
= D
V
x
V
,andobtain
˜
D = D
V
· (1 + α) · t
el
.
If t
el
= 1 this expression simplifies to (5.31).
Notation
a
i
chemical activity of component i
a
O
2
chemical activity of oxygen
A metal A in the binary oxide AO or the ternary oxide (A,B)O
A
res
residual activity
A
sect
activity during sectioning
B metal B in the ternary oxide (A,B)O
c
i
concentration of species i
D
i
diffusion coefficient of species i
D
A
self-diffusion coefficient of A
D
A
i
interstitial self-diffusion coefficient
D
V
vacancy self-diffusion coefficient
˜
D chemical diffusion coefficient
e
electron
f
0
geometrical correlation factor
f
B
impurity correlation factor
F Faraday constant
h
•
electron hole
j
i
diffusion flux of species i
k surface exchange coefficient
k
B
Boltzmann constant
K equilibrium constant
L
ij
transport coefficient
p pressure
p
O
2
oxygen partial pressure
p
∗
O
2
oxygen partial pressure at the stoichiometric point (δ =0)
R universal gas constant
S
q
p
structure element with excess charge q in sublattice p
t time
T temperature
5 Diffusion in Oxides 245
t
el
electronic transference number
V
m
molar volume
x position
x
i
molar fraction of species i (identical to [i])
z position in the moving reference frame
z
i
formal charge of species i
z
i,eff
effective charge of species i
δ non-stoichiometry of the oxide A
1−δ
O
η
i
electrochemical potential of species i
µ
i
chemical potential of species i
µ
O
2
chemical potential of oxygen
σ electrical conductivity
Φ electrical potential
ω
i
jump rate of a vacancy in jump of type i
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6 Diffusion in Metallic Glasses
and Supercooled Melts
Franz Faupel and Klaus R¨atzke
6.1 Introduction
A wide variety of materials ranging from oxide glasses to polymers can so-
lidify as glasses rather than crystals. While diffusion plays an important role
in all media at elevated temperatures, diffusion in glasses is of particular
interest. The metastability of these materials makes them prone to various
rearrangement processes such as structural relaxation, phase separation and
crystallization. In the supercooled liquid state diffusion is very important
in connection with the glass transition, which appears to be a kinetic phe-
nomenon [1]. With the recent discovery of novel bulk-glass-forming alloys,
exhibiting high stability against crystallization in the supercooled state [2,3],
the dynamics of the glass transition could also be studied in metallic sys-
tems, which are the paradigm of dense random packing. Stimulated by the
development of bulk metallic glasses and advances in computer simulation,
recent progress in the understanding of the diffusion mechanisms in metallic
glasses and supercooled melts has been considerable. For recent reviews the
reader is referred to [4, 5]
In this chapter, after discussing some general aspects of diffusion in non-
crystalline matter, which are not related to a particular class of materials, we
focus on metallic glass-forming systems. Metallic glasses are so-called ‘fragile
glasses’. Unlike silica and other strong inorganic glasses fragile glasses exhibit
a glass transition in a narrow temperature range, where marked changes in
physical properties can be observed. Emphasis is placed on results from ra-
diotracer measurements. Moreover, we frequently refer to results from com-
puter simulations which provide valuable insight into the atomic dynamics.
The following section briefly recalls important characteristics of diffusion in
crystalline solids. Normal liquids well above the melting point are treated in
Sect. 6.3. In this context, the free-volume model and the Stokes-Einstein rela-
tion are introduced. The transition from liquid-like continuous small motion
to thermally activated hopping is the subject of Sect. 6.4. In this connec-
tion the mode-coupling theory and a recent extension to medium-assisted
hopping in the glassy state are discussed. The concept of medium-assisted
hopping implies strongly collective diffusion and is applied to the interpreta-
tion of isotope effect measurements in metallic glasses in Sect. 6.5. Relaxation
phenomena that are pertinent to the understanding of diffusion near the glass
250 Franz Faupel and Klaus R¨atzke
transition temperature are examined in Sect. 6.4, too. Furthermore, conse-
quences of a distribution of activation energies are considered. In Sect. 6.5
atomic transport in conventional metallic glasses is critically reviewed, and
measurements of the pressure and mass dependence of diffusion are presented.
These measurements point to highly cooperative hopping processes in fully
relaxed metastable glasses. The effect of excess volume, quenched in from the
liquid state, is also addressed. Sect. 6.6 covers diffusion in supercooled melts
of the novel bulk-glass-forming alloys. Of particular interest are the changes
in the atomic dynamics concomitant with the liquid-to-glass transition. It
will be shown that from the microscopic point of view diffusion in the deeply
supercooled liquid state is solid-like, taking place by thermally activated col-
lective processes. Microscopic viscous flow via atomic collisions is observed to
set in well above the caloric glass transition temperature in full accord with
the mode coupling scenario. In this section we also discuss very recent diffu-
sion and isotope effect data in the equilibrium melt of a particularly stable
metallic glass-forming alloy. Diffusion measurements from the glassy to the
equilibrium liquid state have not been possible so far. It is demonstrated that
even in the equilibrium melt the atomic dynamics are highly collective and
still depart strongly from the hydrodynamic regime of uncorrelated binary
collisions. A summary is given in Sect. 6.7.
6.2 Characteristics of Diffusion in Crystals
Unlike diffusion in amorphous matter, transport in crystalline solids is well
understood, at least as far as self-diffusion and diffusion of trace impurities
are concerned (see Chap. 1). Small solute atoms occupy interstitial sites and
perform thermally activated jumps between equilibrium positions (Chap. 1,
Fig. 1.8). The temperature dependence of interstitial diffusion obeys the well
known Arrhenius law (Chap. 1). Diffusion coefficients for interstitials ap-
proach values in liquids near the melting point (
∼
=10
−10
m
2
/s). A representa-
tive value for the pre-exponential factor D
0
in metals is 10
−6
m
2
/s.
Large impurities substitute for solvent atoms in the lattice. It is now well
established that diffusion of these substitutional solutes and self-diffusion in
most crystalline solids generally proceeds via vacancies (Chap. 1). For a va-
cancy mechanism (Chap. 1, Fig. 1.9) one again ends up with an Arrhenius
law. It has to be pointed out, however, that the activation enthalpy now con-
tains the additional contribution for the formation of a vacancy, which is of
the same order of magnitude as the migration enthalpy (Chap. 1, (1.56)).
Consequently, substitutional diffusion is many orders of magnitude slower
than interstitial diffusion. Moreover, an additional term arising from the for-
mation entropy enters into D
0
(Chap. 1, (1.57)). D
0
is typically of the order
of 10
−4
m
2
/s. As discussed in Chap. 1, a correlation factor f comes into play
for indirect diffusion of tracer atoms via vacancies. If there is no interaction
between tracer atoms and vacancies, specifically if the tracer is a self isotope
6 Diffusion in Metallic Glasses and Supercooled Melts 251
of the host, f is an additional geometry constant in D
0
which can be esti-
mated as f
∼
=1−2/Z (Chap. 1, (1.58)). We note that f is of the order of unity
in close packed crystals. Impurity-vacancy binding can significantly lower the
correlation factor for impurities provided these are strongly diluted [6].
In view of the great success of the vacancy model for diffusion in crystals,
many investigators also invoked vacancy-like defects to explain diffusion in
glasses and even in liquids. While there is general agreement now that solid-
like vacancies do not exist in liquids (Sect. 6.3), their role in metallic glasses
is still a controversial issue in literature (Sect. 6.5.2).
6.3 Diffusion in Simple Liquids
We have seen in the preceding section that diffusion in crystals occurs by very
infrequent large jumps from one lattice point to a neighboring site. The other
extreme is a dilute gas, where all molecules fly freely between frequent binary
collisions. According to the kinetic theory of gases in the Chapman-Enskog
formulation [7], the diffusivity is determined by the mean free path λ and the
collision frequency ν or the mean velocity
v according to
D =
1
3
νλ
2
=
1
3
vλ . (6.1)
In air at standard temperature and pressure (
v = speed of sound, λ = 300
molecular diameters) (6.1) yields D
∼
=10
−5
m
2
/s for an average molecule.
The properties of liquids are somewhere in between those of crystalline
solids and gases. Liquids are almost as compact as solids and they exhibit a
pronounced short-range order, which is reflected in the pair correlation func-
tion g(r) (see Fig. 6.1). The latter describes the probability of finding an atom
at a distance (r, r +dr)fromagivenatom.ng(r)dV is the average number
of atoms in the volume dV at a distance r,wheren is the number density.
The Fourier transform of g(r), the structure factor S(q), can be measured
with x-rays and neutrons (see Chap. 3). If the atoms move, the correlation
between two given atoms decays. The so-called Van Hove space-time correla-
tion function G(r, t)=G
s
(r, t)+G
d
(r, t) is related by Fourier transformation
to the dynamical structure factor, which is accessible to neutron scattering.
The self and distinct parts G
s
and G
d
represent the probability that a given
particle or some other atoms, respectively, will be at (r, t)whenthegiven
atom is at r =0att =0.
Diffusion in liquids [8] is much faster than in solids because of the more
open structure. A typical melting-point value of D is 10
−9
m
2
/s. Unlike in
solids, the temperature dependence of D is weak. The early experimental
findings that diffusion rates in liquids could be fitted to an exponential de-
pendence on 1/T over short temperature ranges and the compact structure
of liquids led to an identification of the mechanism with that of vacancy dif-
fusion in solids. However, there is evidence now from various experimental