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Defect Chemistry Composition Transport and Reactions in the Solid State; Part II Kinetics.

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Defect Chemistry : Composition, Transport, and
Reactions in the Solid State; Part 11: Kinetics**
By Joachim Maier*
Dedicuted to Professor Herrnann Schmalzried on the occasion of his 60th hirthdaj
Within the last few decades--though with its beginnings traceable to as early as 60 years
ago-a direction of research has developed almost unnoticed by the classical chemical disciplines, such that one can now recognize a chemistry Mithin the solid state that is analogous to
the long-familiar chemistry in the liquid state. It arises from those departures from the ideal
structure that are thermodynamically unavoidable, the point defects, and is referred to as
defect chemistry. It includes the description of ionic and electronic effects, and it considers
diffusion as a special step of the overall reaction. This area of chemistry enables one to describe
and treat in a unified way many apparently widely different phenomena such as ionic conduction in crystals, doping effects and p-n junctions in semiconductors, color centers in alkali
metal halides, image development in photography, passivation and corrosion of metals, the
kinetics of the synthesis and sintering of solid materials, the problems of rock formation during
the earth’s evolution, the mechanisms of gas sensors and high temperature fuel cells, the
performance of photosensitive electrodes, variations of the electron balance in high temperature superconductors, elementary processes of heterogeneous catalysis, nonequilibrium transitions and oscillations in semiconductors in electric fields, and inany more. In such phenomena
the equilibrium concentration of defects has an important double role: it not only determines
the disorder and the departures from the stoichiometric composition in the equilibrium state,
but also, together with the mobility as the kinetic parameter, is the key parameter concerning
the rates of physical processes, which are the main subject of the present discussion. Following
the account of the thermodynamic fundamentals in Part I. this second part of the review will
be concerned with kinetic processes. These are essentially the equilibration processes (physical
diffusion, electrical conduction, chemical diffusion, and reactions) that take place under the
influence of chemical and electric driving forces. As in Part I, here again the general characteristics of defect chemistry will clearly emerge, including the coexistence of ionic and electronic
charge carriers, and the importance of the spatial coordinates. The concluding section will
discuss some special characteristics of nonlinear processes that are too often overlooked in
solid-state studies.
1. Introduction
Part I of this review”] consisted of a detailed presentation
which showed that in thermodynamic equilibrium all solids
contain point defects whose nature and concentration depends not only on the material but also on the temperature,
the hydrostatic pressure, and the chemistry of the environment--for example, the oxygen partial pressure in the adjacent phase in the case of a binary oxide. Such defects may
take the form of extra “particles” (ions, electrons, etc.), missing particles, or foreign particles. In the same way as the H
and O H - ions or dissolved foreign species in water, these
defects determine not only the exact composition of the
phase but also its reactivity and in particular and the transport processes within it. Although Part I was essentially confined to equilibrium thermodynamics, we already saw there
that the concentrations of defects can be determined by measuring the ionic o r electronic conductivities, which result in
+
[*I
[**I
Prof Dr. .I.
Maier
Max-Planck-Institut fur Festkorperforschung
Heisenhergstrasse 1. D-W-7000 Stuttgart XO (FRG)
Part I: Thermodynamics appeared in the March issue of Angwundre
(%<,l??iC [ I ]
the ionic case from excess ions, vacancies, or foreign ions, or
in the electronic case from excess electrons or electron
holes.”] The conductivity is extremely sensitive to the defect
concentration, since it is normally only the defects that contribute, not the regular structural units. Moreover, as will be
shown later, the conductivity is the key quantity in understanding the dynamic processes in solids. In this article,
therefore, we shall start by considering the kinetic component of conductivity, which is the mobility of the point defects. or in chemical terminology the rate constant for the
“hopping reaction”. Next we will deal with the self-diffusion
of the particles, then with the kinetics of “true” chemical
reactions- -first, as a simple example, the homogeneous case
of the “dissolution” of oxygen in oxides, then as a phaseforming process the corrosion of a metal, and lastly a solidsolid reaction. In cases where such processes are diffusioncontrolled the relevant kinetic parameters are the diffusion
coefficients, which are closely related to the ionic and electronic conductivities. The concentrations of point defects are
also decisive for the rates of local chemical reactions, which
occur especially at surfaces (interfacial reactions, heterogeneous catalysis). The article will conclude by discussing nonlinear phenomena that are observed under conditions far
removed from equilibrium.
2. General Features of Solid-state Kinetics
In contrast to the situation in liquids the regular structural
units in solids are immobile, except for vibrations and rotations about their equilibrium positions. Nevertheless, there is
a degree of internal mobility due to point defects. Figure 1
illustrates this by taking vacant crystal sites as an example.
,1""1-1:":1
x o x x
x x x x
x x x x
x x x x
Fig. 1. Schematic representation of the migration of vacancies in a crystalline
phase. The mass transport is in the opposite direction.
At sufficiently high temperatures a neighboring atom o r ion
can move into the vacant site, and the resulting newly formed
vacancy then serves as a destination for a jump by another
neighbor. From an energetic point of view the new situation
does not differ from that existing previously. To be more
exact. there is no overall change in the standard free enthalpy
between initial and final state. However. it is necessary to
supply an activation energy corresponding to the bottleneck
(or, more correctly, the saddle point) for the jump process.
(A more sophisticated model has been developed for the case
of highly disordered material^.)^^] It is evident that the resulting mass transport can be described much more easily as the
migration of a vacancy, rather than as a complex superposition of the motions of all the particles involved. In a similar
way a particle occupying an interstitial lattice site also confers mobility, but here the direct hopping from one such
interstitial site to the nearest equivalent site will nearly always be less energetically favorable than, for example, the
jump of an interstitial particle towards an adjacent regular
particle which itself is then forced into another vacant interstitial site.
It is a characteristic feature of the solid state that the positional coordinates must always be taken into account. Strictly
speaking, all reactions in the solid state are heterogeneous, in
other words they are associated with a transport component.
So that we can approach the problem from a chemical standpoint. particle transport will be treated formally as a chemical reaction in the following discussion. If we represent such
a general reaction by Equation (I), this can include not only
gradients, will lead to a restoration of the equilibrium. The
behavior is depicted in Figure 2. If we express J as a power
series in (a/ax)j and neglect all higher terms beyond the
linear term, we can describe the kinetic behavior in the neighborhood of equilibrium by Equation (2). Here the factors /3,
are the corresponding kinetic coefficients. This mathematical d e ~ c r i p t i o n ' ~of] linear irreversible thermodynamic^'^. 61
is found in practice to be highly suitable in the case of the
transport behavior, but for chemical reactions a linear equation of this kind can only be used in situations very close to
equilibrium (Fig. 2). Before examining this in greater detail,
let us consider the special cases of pure electrical conduction
and pure diffusion (Fig. 2, riglit-, and left-hand branches,
respectively, under "particle transport"). Pure diffusion ocLinear irreversible Thermodynamics
particle transport o r reaction
/
ZFb(r') = zF'b(.~)
conventional
chemical reaction
\
A=B
particle transport
J
= -
p(6i6.r),ii
for ARG Q RT
J
=
reaction rate
(6i6.r) z F 4 = 0
diffusion
J
(b/6.Y) / I = 0
conduction
= - /1(6/6x) ,I
- --
J
=
- f r Z F ( 6 / 6 \ - )4
fl(8i6.r) c
I
I
J
=
flux density
i = current density
LT = '
Y Fz/j
0 E -RT
p
\
c
/
RT
c?F2
zF
Nernst-Einstein relation
the case of a conventional homogeneous chemical reaction
(.x=.x') but also that of a simple transport process (from x to
x') when A is identical to B. Under equilibrium conditions
there is no difference between the chemical potentials j on
the left- and right-hand sides of Equation ( 1 ) . Consequently,
for the case of particle transport, there must be no gradients
in the electrochemical potential (The electrochemical potential is the sum of a chemical potential term and an electrical
potential term, as explained in Part I). Conversely, the existence of gradients in the electrochemical potential means
that there are driving forces causing a particle flow J , which,
in the absence of external influences that may maintain the
Fig. 2. Diagram summarizing nonequilibrium behavior in solids close to equilibrium. The generalized representation includes both conventional chemical
reactions (.x = u') and particle transport (A =B) as special cases. Particle transport also includes the cases of pure diffusion (zA4 = 0) and pure electrical
conduction (Ap = 0).
curs in the trivial case where the particles or defects concerned carry no charges. The only remaining driving force is
then the gradient of the chemical potential p ; for sufficiently
low concentrations this is given"] by Equation (3). and thus
( a / a x ) p = (RT/c)(a/ax)c
(3)
529
the concentration gradient can also be conceived as the driving force. Equations 2 and 3 then lead to Fick’s law, in which
the diffusion coefficient D is proportional to pic. The other
special case of pure conduction occurs for very high defect
concentrations; the gradient of the chemical potential then
disappears, resulting in a current that is proportional to the
gradient of the electrical potential 4. This is obviously Ohm’s
law, with the conductivity CT proportional to 8. It follows
from this that the quantities D and CT are related through the
coefficient 8; for particles with charge number I? the relationship is given by Equation (4), which is the Nernst-Einstein
However, the potential difference applied to the specimen is
usually of the order of 1 V or less, and since the jump discm, compared with a specitance Ax is approximately
men thickness of about 10- cm or greater, the relevant potential difference A$ between adjacent sites is about l pV o r
Kinetics
particle transport or reaction
(4)
f
relation (see bottom to Fig. 2), F is the Faraday constant).
Here the quantity u is the mobility, which contains the same
information, expressed in electrial units, as does the diffusion coefficient expressed in thermal units.
If we now take a more detailed look at the kinetics (Fig. 3),
it becomes possible to specify the conditions under which the
treatment given here is valid. It is not necessary to restrict the
conditions to near-equilibrium. However, the simple rate
equations that are familiar to the chemist, and are applied in
the following discussion, are only valid if we assume low
concentrations from the outset. We again treat reactions and
transport in a unified way [Eq. (I)] with rates for both the
forward and reverse processes, as is usual in chemical kinetics. In the general case the free energy profile for the process
is assumed to be modified by an electrical potential (Fig. 3).
If the variation in the electrical potential from the initial state
to the final state is assumed to be linear, its effect is to alter
the transition threshold roughly by the amount f z F A $ / 2
and thus the usual chemical rate constants must be multiplied by exp( *zFA$/2RT). In the case of a chemical reaction the linear relationship between “flow rate” (i.e. reaction
rate) and driving force (ARC) only applies if A,G/RT < 1
(left-hand side of Fig. 3). a condition that is only satisfied
very close to equilibrum. In the case of particle transport
(right-hand side of Fig. 3) A and B are identical, and the
energy profile is symmetrical (apart from the superimposed
effect of an electric field, if present), so that ARCo= 0, since
one is dealing with a shift of a defect from one lattice site to
the nearest equivalent site. This involves passing through a
saddle point whose energy is higher than that of the initial
and final states. The difference between the initial (or final)
state and the transition state is the free enthalpy of migration. The reaction coordinate in this case is the positional
coordinate. In this sense, therefore, transport can perhaps be
regarded as the simplest of chemical reactions, namely a
positional rearrangement with a negligible standard affinity.
The Go profile for this case is defined not by a snapshot, but
by scanning the variation in free enthalpy as the defect successively occupies all the different positions. For the case of
pure diffusion (zA$ = 0), this leads directly to Fick’s first
law, in which D corresponds to the chemical rate constant,
that is it is proportional to expf - A G * / R T ) .Thus the linear
theory (as shown in Fig. 2) is found to be entirely valid in this
case. For pure electrical conduction we must initially consider a nonlinear law (in fact a sinh zFA4IRT law, see Fig. 3).
530
*
k=k,Z*
k = k, exp
-
AG *
RT
__
- -
k, = kG=k,
conventional
chemical reaction
particle transport
A(x)
J
=
+ A(.r‘)
k , exp
AG+
- cRT
[c
7FAb
(.r) e x p k-
0
c(x’) exp-
2RT
B
+
for IA,Gl
RT:
J x - A,G
\
I
I
A(;F+) = 0
diffusion
Ac = 0
conduction
V
/
Nernst-Einstein-relation
Fig. 3 . Diagram summarizing the kinetic treatment of nonequilibrium behavior at low defect concentrations. Chemical reactions, diffusion, and electrical
conduction appear as special cases (t = equilibrium concentration). I n the case
of transport, k, 16 proportional to A.r.
less. In contrast, the quantity F/RTis of the order of 10 V-’
even at 1000 K, and thus FA4/RT < 1. It is only necessary to
use the high field law for very high voltages o r very thin films,
o r for boundary layers at which there is a voltage drop of
cm).[81
typically 0.5 V over the Debye length (typically
Angca.. Chcm. Inr. Ed. Engl. 1993, 32, 528 - 542
In particular, in reactions carried out in electrochemical cells
the relationship between current and voltage follows the
Butler-Volmer equation[g1rather than Ohm's law (see Section 4.7). Ohm's law is obtained for near-equilibrium conditions by expanding the exponential function as a power series and truncating after the linear term. One can then treat
c as approximately constant and replace it by the equilibrium
concentration (Fig. 3). In addition to the Nernst-Einstein
relationship [Eq. (4)], it is found that the mobility corresponds essentially to the rate constant for the hopping reaction (see Fig. l ) , and a particular consequence of this is that
the temperature dependence of the mobility is determined by
the enthalpy of migration [Eq. ( 5 ) ] . In the above approxima-
cient.[lo.
141
The principles set out in this and the previous section will
now be illustrated by some representative examples.
4. Examples
4.1. Example 1: The Mobility in Electrically Conducting
Materials
AH ' / R T = A G t / R T -t AS'IR
+
tion it is necessary to apply an external voltage (i.e. A 4 0)
to obtain any measurable effect at all, but A$ has a negligible
effect on the migration threshold.
3. Chemical Diffusion
Although diffusion processes associated with compositional effects almost exclusively manifest themselves as
changes in the positions of neutral components of a system,
the defects that are involved carry electric charges, and consequently, such diffusion is always charge-coupled (ambipolar diffusion). For example, the diffusion of oxygen into
an oxide (with a change in the "oxygen content") usually
involves a coupled diffusion of 0'- ions and electrons, in
which the electron flux must be twice as great as the ion flux
and in the opposite direction."'.
The process can thus be
formally regarded as a diffusion of "0". It can be seen that
there is a parallel between the treatment of chemical diffusion and that of a chemical equilibrium."] Corresponding to
the chemical equilibrium conditions ((a/ax)b, = 0), we have
in this case the flow equations (Jk cc (a/ax)Pk)and the electroneutrality condition corresponds to the coupling of the
particle flows. Just as the potentials are linked to the control
parameters in a chemical equilibrium, here the gradients
must be so linked. The analysis shows that the oxygen flow
Jo resulting from the coupling is given by Equation (6),
and the proportionality constant relating it to the externally
applied driving force is thus a mixed-transport coefficient
(proportional to b ) . [ ' O . 'I Such mixed-transport coefficients
o r chemical diffusion coefficients (essentially the expression in parentheses on the right-hand side of Eq. (6)) play a
very important role in solid-state chemical kinetics and result in cases where n o redox effects occur, mostly from the
coupled, electrically neutral migration of different ionic species.I2. 21 If a current i is applied from an external source, the
internal currents n o longer cancel each other out. In this case
it is also possible to have pure electrical conduction (as is
usual in metals or solid electrolytes). In the general case the
A n p v Chin. In/. Ed. E ! I ~ /1993,
.
32. 528-542
flow of one of the components k is the sum of a conduction
term and a diffusion term (stoichiometry term). The conduction term is given by the external current multiplied by a
weighting factor, the fractional contribution of the conductivity crk to the total conductivity (r, which results in the
expression (ok/o)i.The diffusion term is determined by the
concentration gradient and the ambipolar diffusion coeffi-
Numerous examples of conductivity measurements in
which the ultimate purpose was to determine the equilibrium
concentration of defects have already been discussed in
Part I of this review. In measuring the conductivity one automatically excludes diffusion terms, and pure conduction
takes place. The electric fields used for such measurements
are kept low enough to ensure that the normal Ohm's law
behavior can be assumed in the calculations (see Fig. 3).
Various elegant D C and AC methods have been developedr2* 13, 14' for resolving the conductivity into its different components, such as electronic p-type or n-type conduction, or ionic conduction by vacancies or by interstitial ions.
One such method, based on electrodes with a selective blocking function, has been mentioned in Part I.['] and this will be
referred to again briefly in Section 4.3. Another method,
which does not involve measuring an external current, is the
concentration cell experiment. This enables one to separate
the ionic and electronic parts of the conductivity when these
are of similar
In general, if a galvanic cell
contains a solid electrolyte whose conduction is partly electronic, this acts as an internal short circuit, and consequently
the measured e.m.f. is only a fraction of the Nernst value
depending on the electronic contribution.["]
The most important of the kinetic parameters which affect
the conductivity is the mobility (u CK o/c), and the present
discussion centers on this quantity. We note once again that,
from the parallel with chemical kinetics, the mobility corresponds to the rate constant for the hopping reaction, apart
from a proportionality factor that is unimportant in this
context. The mobility term can be determined separately if
the charge carrier concentration is known from measurements of the Hall effect o r thermoelectric power, or is simply
fixed by the level of doping. Figure 4 shows values for the
mobilities of interstitial ions and vacancies in AgCI, determined from conductivity measurements on AgCl specimens
with different levels of Cd doping. The Arrhenius plot is seen
to be linear, as Figure 3 leads one to expect. The slope of
each line yields the corresponding migration enthalpy; in
agreement with computer simulations, this is found to be
appreciably greater for transport by vacancies than for
transport by interstitial ions.
In contrast, conduction by electronic charge carriers
through a band mechanism is only slightly temperature dependent. In this case raising the temperature increases the
scattering and thus slightly reduces the mobility. In the case
'
53 1
9/oc
15 50
100
-
200
LOO
1 -
-L
-5
-6
t
t
!
-
35
30
25
20
TO~T-’/K-‘
15
F I ~4.. Mobility of Frenkel defects in AgBr as a function of temperature 1181.
(Interstitial mechanism: top line, vacancy mechanism: bottom line).
of acoustic phonon scattering the temperature dependence is
typically of the form given in Equation (7). Figure 5 shows
mobility data for SnO,; at sufficiently high temperatures a
power law similar to Equation (7) applies. At lower temperatures the mobility is probably limited by polaron transport,
in which the polarization of the surrounding medium results
in a thermally activated charge transport mechanism. involving a localized hopping process. In superconducting states,
of course, ti is infinite.
Hall mobility/
-
cm2 v-’s-’
*O
t
2.0
given by Equation (8), and it therefore bears a simple relationship [Eq. (9)] to the diffusion coefficient D , of the mobile
defects.
2.5
3.0 3.5
-
4.0
~o~T-’/K-’
Fig. 5 . Mobility of excess electrons as a function of temperature for two different S n 0 2 samples, determined by Hall effect measurements [19].
Thus, whereas the diffusion coefficient of the defects, D,, is
proportional to their mobility u,, D02. is directly proportional to the ionic conductivity, and thus to the product ukck.
DoL can also be determined by a tracer diffusion experiment, in which one observes the mixing of two isotopes, such
as “ 0 and l80,by the interdiffusion of the corresponding
02-ions. In cases where the gas exchange is sufficiently
rapid, the most convenient method is to enrich one portion
of the natural 0, gas with ‘ * 0 2
then
, to interrupt the experiment after a certain time and measure the change in mass of
the gas phase. Alternatively, by using low temperatures one
can freeze the concentration profile and analyze it by secondary ion mass spectrometry (SIMS) or by some other
method. The ideal procedure would be to place the oxide
specimen in contact with a second oxide specimen with a
different isotope content, but this is not usually possible. The
change of concentration with time in a defined volume is
given by the difference between the inward and outward flow
rates, and therefore by the flux divergence (&/at = - aj/as),
and since also for tracer diffusion Fick’s first law is obeyed
we obtain Equation (10). This equation (Fick’s second law)
can be solved to determine DTrfor the appropriate boundary
and initial conditions.’221Except for some small corrections
(correlation factors), DTris identical to the self-diffusion coefficient mentioned previously, and from it one can calculate
Do,- and an ionic conductivity. Figure 6 shows some results
from tracer-SIMS diffusion measurements on the (La,Sr),CuO, system. These can be understood on the basis of the
defect chemistry described in Part I of this review (see Fig. 14
of Part I).”] The relationship between DT‘ and crlon is much
more complicated in the case of YBa,Cu,O,+,, as different
charge carriers are present (see Section 4.4). Figure 7 shows
the results of tracer measurements on isotopically labeled
iron in Fe,O, as a function of the oxygen partial pressure.
4.2. Example 2: Tracer Diffusion in Oxides
-8
As was shown previously, ionic conduction usually involves only one type of defect. The ionic conductivity is
proportional to the concentration of this defect species and
to its mobility u k or diffusion coefficient Dk.Let us suppose
that, in the absence of more detailed information about the
defect chemistry, we group together all the relevant ions
(oxygen ions in our example), regardless of whether they are
fixed or mobile (that is. at regular or defect sites). We can
then determine from oionan average mobility, or more correctly a self-diffusion coefficient, which is, of course, much
smaller than the mobility of the defect species.12]Its value is
532
it
1 -li\
-10
0
.
-.-.--
---
-x
0“ -14 Lg 0
cm2is
-16
--x
-
-18 -
-20
I
I
I
I
0.05
0.10
0.15
0.20
-7
I
-16
I
-lL
I
I
-12
-10
I
-8
k3 *cg
1
-6
-
I
-L
I
I
-2
0
expected from the equivalence of the ionic and electronic
currents [see Eq. (6)],contains a mixture of ionic and electronic transport terms. It can be seen that for very good
electronic conductors (a N ueon),the expressions for DT'
[Eq. (S)] and
are identical except for the thermodynamic
factor aln a,/alnc,. However, owing to the high oxygen concentration (in contrast to the low defect concentration) and
the strong ionic bonding, this Factor differs greatly from 1 (a,
is the oxygen activity). Assuming that the defect chemistry is
simple, the thermodynamic factor is approximately given by
Equation (13), in which c, and c, are the concentrations of
Fig. 7. Tracer diffusion results for 59Fe in magnetite Fe, ,04 1231. D* IS the
tracer diffusion coefficient.
The transition from an interstitial cation mechanism to a
cation vacancy mechanism can be clearly seen.[231
4.3. Example 3: Oxygen Uptake and Release in Oxides:
Chemical Diffusion and Surface Reaction
In the previous example no chemical effects are involved;
in other words there are no chemical gradients. The situation
alters radically when we consider the uptake or release of
oxygen in an oxide as a most simple example of a chemical
reaction. Leaving aside surface reactions, we are again concerned here with a diffusion process, but one that is associated with a change of composition, namely of the local oxygen
concentration. Our example is SrTiO,, which was previously
discussed in Part I of this review"] in connection with the
dependence of the electronic defect concentrations, and
therefore of the conductivity, on the oxygen partial pressure.
The data shown there (see Fig. 15, in ref. [I]) enable one to
read off the initial and final concentrations, and hence the
respective values of the conductivity, when the oxygen partial pressure is changed from P, to P 2 . However, to determine the rate of approach to the final state it is necessary to
consider the kinetics of the reaction [(Eq. (1 I)]. A knowledge
of
this rate is needed if, for example, the sensitive dependence of
the electrical conductivity on the oxygen content is to be used
as the basis of an oxygen sensor. Here, in contrast to Example 2, (Section 4.2) only the defects are involved in the diffusion. The relevant diffusion coefficient refers to the mobile
particles, and is very much greater than Do,-. It can easily be
seen that the diffusion of "0" into the material is an electrically neutral process involving the coupled migration of 0'ions and electrons (at low oxygen partial pressures the mechanism involves oxygen vacancies, V;, and conduction electrons, e', at high partial pressure, V; and holes, h' (for the
notation see also Part I,['] Section 2)). This chemical diffusion coefficient is given by an expression [(Eq. (12)] which, as
Angcii C'l~i~rrr/ I ? [ . Ed. Engl. 1993, 32. 528-542
the oxygen vacancies and of the conduction electrons, respectively. For a purely ionic conductor, in which the ionic
conductivity, and therefore also the concentration of ionic
defects, are much greater than the corresponding electronic
quantities, it follows from Equation (13) that B is identical
to the diffusion coefficient of the electronic defects (in this
case DJ.This means that the rate of diffusion is limited by
the electron flux. F o r an electronic conductor with an abundance of electrons, we have the opposite case in which d is
limited by the diffusion coefficient of the ionic defects D,.
However, there are many electronic conductors in which
ionic defects predominate, resulting in the mixed case given
by Equation(14). In the general case, assuming a simple
defect chemistry, d has a value intermediate between D, and
D, [Eq. (15)].[241In Equations (14) and (1 5 ) n and v are to be
understood as representing the dominant electronic and
ionic defect species, respectively. Measuring the variation of
the stoichiometry with time is a simple method of estimating
the chemical diffusion coefficient, which thus describes the
kinetics of the chemical reaction in the solid state (where this
is diffusion-limited). Such measurements can be simply performed by using gravimetric or conductometric methods.
Two alternative methods are described below.[".
The defect chemistry of nominally pure SrTiO, is very
often dominated by the presence of iron impurities. A proportion of the iron ions (normally F e 3 + ) are tetravalent,
depending on the oxygen partial pressure, and in interaction
with the normal defect chemistry. The associated transfer of
an electron from the valence band to the iron level in the
band gap can be described by Equation (1 6). The presence of
Fe4+ ions gives the oxide a reddish color, and their concentration can be measured in situ by optical absorption spectros~opy.[*'~It can be shown that here too the chemical
diffusion is, to a first approximation, governed by Fick's second law. The solution for small values of the time t. and the
533
-
]Redox Kinetics
I
1
I
high-temperature
thick samples
I
diffusion limitation
1
t.
I
00
I
4
I
c
-L
L
0
X-
c
1
low-temperature
thin samples
[I
1
I
I
-L
surface limitation
I
1
t=m
t
X -0
L
1
1.2 F
,+
0
40
80
-
120
thin
160
lo3 x t/min
-
into account other possible reactions proceeding in parallel.
The extent to which any one in this succession of migration
and reaction processes may become a rate-limiting step depends on the region within which the state parameters fall. In
fact it is found that at low temperatures the relative change
of (reduced) optical absorption increases linearly with time
for small values o f t , indicating that under these conditions
the rate-limiting step is not diffusion but the reaction at the
interface (Fig. 8, right-hand column). For this case the photographic images show uniform coloration, in contrast to
those for high temperatures in which color profiles corresponding to concentration profiles can be observed across
the sample (diffusion-limited case). Results such as these
allow quantitative determination of diffusion coefficients
and rate constants.[251
Another method, whose principle has been described by
Yokota," is based on earlier work by Wagner;r131its application to yellow PbO is described here.[141A sample of the
oxide with a defined composition (corresponding to the value of the external oxygen partial pressure Po,) is compressed
between two purely ionic conducting materials, sealed off
from the laboratory atmosphere by a glass coating, and polarized by a constant current (or constant voltage). In contrast to the example described above, an electric current term
is here superimposed on the chemical diffusion (as discussed
earlier). The ionic and electronic currents cancel each other
locally, except for the external current i. The variation with
time of the concentration profile is determined, to a first
approximation, by Fick's second law. As a consequence of
the boundary conditions the development of the profile is
symmetrical.[281 The driving force for the variation of the
composition within the material arises from the fact that
only the oxygen ions can cross the phase boundary between
the oxide and the purely ionic conductor. Electrons cannot
transfer through the phase boundaries either on one side
(negative pole) or on the other (positive pole). The electron
concentration profile that is thereby set up must, on electroneutrality grounds, be charge-compensated by an ionic
profile. In the steady state only ionic currents exist, even
Fig. 8. Kinetics of the uptake and release of oxygen in SrTiO,: theoretically
predicted profiles, and experimental data from optical absorption measurements (see text) [25]. c = defect concentration. I = spatial coordinate, A =
reduced absorption (arbitrary units).
fi
appropriate boundary and initial conditions, gives a
law,
from which d can be calculated (Fig. 8 left-hand column) (As
will be described in the next section, in this and many other
cases, internal trapping effects must also be included in the
analysis). In practice the overall process of oxygen uptake
and release is much more complicated [Eq. (17)]. Following
L
0
the arrival of the oxygen a t the surface by gas diffusion, it
must be adsorbed and must dissociate, then become ionized,
pass through the phase boundary, and migrate through the
space charge zones, before the chemical diffusion within the
material can actually begin. The suggested sequence shown
here does not show all the individual steps, nor does it take
534
5000
tls
10000
15000
20000
Fig. 9. The voltage of an electrochemical polarization cell consisting of PbO
between ZrO, electrodes, during polarization with a constant current and during depolarization (i.e., after switching off the current, see insert). In the steady
state the internal flow is purely ionic. In the transient state at sufficiently large
values of i the voltage follows an exponential law, and from the time constants
the chemical diffusion coefficient can be determined. T = 503 " C , i =
21.83 pAcm-2, d (polarization) = 2.3 x lO-'cm*s-', d (depolarization) =
2 . 2 ~1 0 - ' ~ m * s - ~ .
Angew. Chem. Ini. Ed. Engl. 1993, 32, 528-542
locally ; the ionic conductivity can therefore be separated
from the total conductivity (which determines the voltage
jump at the start of the experiment), and the chemical diffusion coefficient can be determined from the variation of the
profile with time. It can be shown that for small values o f t
the change in the voltage is a linear function of
and for
large values of t it is a linear function of exp( - t / z ) ; here 6
is inversely proportional to the time constant z,which itself
is proportional to the square of the sample thickness. The
depolarization after switching off the current behaves
similarly (Fig. 9).[lo, 141 The same method can also be
used to determine ionic conductivities and chemical diffusion coefficients in efficient electronic conductors such as
YBa,Cu,O,+, (Fig.
fi,
for the chemical diffusion coefficient [Eq. (15)] gives good
agreement with the observed temperature dependence at
high temperatures, but at lower temperatures the predictions
are in error by several orders of magnitude. The trapping
reaction shown in Equation (16) is exactly the type of internal reaction that leads to a local particle production through
the effects of diffusion, even when there is local equilibrium.
It can be shown that Equation (1 5 ) must then be modified by
including a trapping factor,“ 71 whose value can be calculated
from a knowledge of the defect chemistry. The absolute calculation of d then gives an excellent quantitative fit to the
observed temperature dependence (Fig. 12). The underlying
-5 r
lg (0
‘.
t
0.5
-8
1.0
1.6
14
103T-’IK-’
-
-
2.0
1.5
1 0 3 ~T/K
Fig. 12. Temperature dependence of the chemical diffusion coefficient for oxygen in SrTiO,. When dynamic trapping effects are included the theory accounts
d withsatisfactorily for the data without recourse to a fitting parameter:
out trapping, - . - d with trapping, - - - self diffusion coefficient of V i i ~this curve
was calculated. symbols are measured values; according to ref. [25].
~
Fig. 10. The Steady-State voltage as a function of temperature for an electrochemical polarization cell consisting of YBa,Cu,O,+, between ZrO, electrodes. The slope yields the ionic conductivity of YBa,Cu,O,+, [26].
4.4. Example 4: Modification of Chemical Diffusion
by Trapping Effects
The normal relationships need to be radically modified
when. in addition to the diffusion process itself, there are
reaction processes, especially those associated with the trapping of electronic charge carriers by ionic
Localized variations of concentration with time can then also arise
from the source and sink terms associated with internal shifts
in the equilibria of the trapping reactions, as indicated in
Figure 11. These effects require modification not only of the
Oi’(.r’) + h’(u’)
O;(x’)
Fig. 1 1 . Diffusion in a solid accompanied by changes in valency represented
here as a combined reactiondiffusion process.
~
reason for this is that the process described by Equation (16)
provides mobile electronic charge carriers by dissociation of
the trapped states for example electron holes from Fe4+, so
that a different (oh.% oFe4+
e 0) effective concentration has
to be considered.
For a solid such as YBa,CuO,+, in which, as explained in
Section 3.1.3 of Part I of the review, the differently charged
defects O;, Oi, and 0; must be taken into account, a more
general treatment is needed. These defect species are again
connected by redox reactions, but can also themselves contribute to the migration process.[14,l 7 I For this general case
Equation (12) is replaced by Equation (18). Here n2 and oI
are the respective contributions of 0; and 0 ;to the conductivity, and s is an analogous quantity that is proportional
to the diffusion coefficient for 0;. Equation (18) can be
put into a generalized form for all types of charge carriers.[14. 1 7 1
flow equations involving the diffusion coefficients, but usually also of the formulas for evaluating these by the electrochemical methods. The details will not be discussed here, but
can be found in the literature.[14.‘’I
The Fe-doped SrTiO, material discussed in the last section
serves as a relatively simpe application. The usual expression
A n g m Chem In1 Ed. Engl 1993, 32. 528-542
For YBa,Cu,O,+, also, one can similarly determine the
ionic conductivity and the chemical diffusion coefficient by
a Wagner-Hebb polarization experiment,[261and the selfdiffusion coefficient of the oxygen ions by tracer experiments.[2’1 However, to understand the significance of the
quantities measured and their interrelationships, a more detailed consideration is necessary, which takes into account
535
the redox interacti~ns."~.
''I In particular, the tracer diffusion coefficient is no longer proportional to the ionic conductivity, nor is the ionic conductivity in the polarized state
necessarily the same as the normal ionic conductivity, owing
to the dissociation processes. Furthermore, neutral defects
play an important role in determining the behavior of the
tracer and chemical diffusion coefficients. A simple statistical model based on the defect model in Part I (Section 3.1.3)
predicts a thermodynamic factor for YBa,Cu,O,+, which is
good agreement with the experimental results.[261
For solid-state reactions that are diffusion-controlled,
which is very often the case for specimens of macroscopic
dimensions (thickness L), since J x L-' the most important
kinetic parameter is nearly always the chemical diffusion
coefficient. A simple example is the oxidation of a metal M,
to its oxide, which was discussed many years ago by Carl
The reaction begins at the gas-metal interface
with the formation of a thin oxide layer. If the molar volume
of the oxide in this layer differs significantly from that of the
this usually results in the formation of islands or
cracks allowing the reaction to progress further by gaseous
diffusion (Fig. 13, left). If, on the other hand, a dense layer
is formed (Fig. 13, right), further reaction is only possible if
t
place through a migration of oxygen vacancies and conduction electrons in the same direction, whereas for ZnO the
probable mechanism involves a migration of zinc ions (in the
interstitial lattice) and electrons in the same direction (since
this is an n-type semiconductor).[34]The effective flow rate
of the metal M through the oxide. which determines the rate
of reaction, can be determined from Equation (6). This leads
to Equation (19). As the flow geometry is known, the rela-
tionship can also be expressed in integrated form [Eq. (19),
second part]. Here the brackets indicate an average value,
since there is usually an appreciable variation in composition, and therefore in conductivity, through the thickness L
of the oxide layer. The chemical potential of the metal has its
highest (and maximum possible) value on the metal side of
the interface, and its lowest value on the gas side; that of the
oxygen has its lowest (and minimum possible) value on the
metal side, and its highest value on the gas side.
The four main factors influencing the flow rate can be
identified (1. 2, 3, and 4 below) from Equation (19). For a
high rate the driving force, which is the (average) gradient of
the thermodynamic potential for this component (ApM/L).
must be as large as possible (there is a linear relationship
between the flow rate and the driving force). Since there is no
gradient on the metal side, this requirement is fulfilled when
1) the oxygen partial pressure is as high as possible, and
2) the layer thickness is as small as possible. However, a high
rate also requires that the specific kinetic coefficient (the
expression in parentheses) should be sufficiently large; thus
the electronic conductivity 3) and the ionic conductivity
4) must both be sufficiently large. (If neutral 0 atoms can
migrate, as is possibly the case in YB~,CU,O,+,. then Do;,
represented by s in Eq. (18), also becomes important.) Since
the flow rate J , and therefore ultimately the resulting rate of
increase of the oxide layer thickness (i),
are both inversely
proportional to the thickness (i cc L - I), this yields the wellknown square root law [Eq. (20)].[3'1
(20)
passivation
growth into
metal
gas phase
Figure 14 shows that these predictions are confirmed in
the case of the oxidation of zinc. The slope is a measure of the
I.
there is a chemical diffusion of either the metal (involving
migration of M + and e- in the same direction) or the oxygen
(migration of 0 2 -and 2 e - in opposite directions). In the
former case the oxide grows outward into the gas phase, and
in the latter case it grows inward into the metal. If either the
ionic or the electronic conductivity is negligibly small at the
relevant temperature, as IS the case for A1,0, at room temperature, the defect layer has a passivating function.[331
Thus, corrosion of the type described above is only possible
if an oxide phase with a mixed conduction mechanism exists.
For example, in the case of SnO,, corrosion would take
536
It
20
0
'
"/
Zn
l+l.Orn/oAII
Fig 14. Progress of corrosion in zinc (Zn + 1/20, + ZnO) wlth and without
doping (Am = increase in mass, which IS proportional to the thickness of the
oxide layer; one-dimensional geometry I S assumed) 1351. k, a ( u 7 , + . ) ) ci,". ,
x [Zn;']; T = 390'C. (m/,Al. mol O h Al).
Aflxew. Chem. Inr. Ed. Engl. 1993. 32, 528- 542
effective rate constant, which is proportional to (cionce
/c).
Since electronic conduction predominates in ZnO, this expression reduces simply to (cion).The steep temperature
dependence of the ionic conductivity readily explains why
the rate of growth of the oxide layer thickness is extremely
sensitive to the temperature. The effects of homogeneous
doping are clearly evident in Figure 14. In ZnO the ionic
conductivity is probably due mainly to zinc ions in the interstitial lattice, although oxygen vacancies are an alternative
possibility; in any case, however, the charge carriers are positively charged ionic defects. In either case doping the metal
(as in Part I[']), and therefore also the oxide, with lithium
(Li;") causes an increase in the ionic charge carrier concentration. This results, as expected, in much faster corrosion.
The effect of the relatively small change in the activity of the
metal is insignificant here. Doping with aluminum, which
leads to the formation of Al;, (or AIT"), has the opposite
effect, giving a considerable degree of corrosion protection.
Solid-solid reactions can be approached in a similar way.
An example is the reaction whereby a spine1 is formed from
the oxides of two
(in this case the mechanism
may also involve an ambipolar diffusion of the different
cationic species). Figure 15 shows that a t large layer thicknesses the growth follows a
law, whereas for small thicknesses the growth is linear. For small values of L the diffu-
fi
Amlmgcm-2
0.2
um. This is expressed by Equation (21), where cat represents
dAR G"/d[cat] = 0
+ dAG */d[cat]
the catalyst. The commonest examples of heterogeneously
catalyzed reactions are those in which a solid influences a
reaction in the gas state. The usual mechanism involves the
adsorption of gas molecules at the surface followed by a
reaction in the surface region. The effect of the catalyst in
speeding up the reaction relies firstly on the increase in the
effective particle concentration. However, in addition to the
presence of adsorption centers, it is essential that rapid diffusion should be possible at the surface. Therefore it is not
surprising that mixed conductors, a class that includes many
transition metal oxides, often have very good catalytic properties.
A surface o r interface can be treated as an extended defect,
which is characterized by a surface tension; alternatively it
can be regarded as an ensemble of interacting point defects.
possessing a chemical standard potential different from that
of the
I n this way, based on the Langmuir theory,
the adsorption of a gas molecule G onto the catalyst can be
described by Equation (22). in which V,, and G,,, are surface
defects. This might, for example, represent the adsorption of
N, onto the iron catalyst in the Haber-Bosch ammonia synthesis. By applying the law of mass action to Equation (22)
we obtain the Familiar Langmuir equation [Eq. (23)], in
I
tlmin
-
Fig. 15. Growth of the spinel layer (ZnAl,O,) in the reaction between the
oxides ZnO and AI,O, (one-dimensional geometry) [2b]. T = 1350 C.
sion through the thickness is so rapid that the rate of growth
i is determined by the surface reaction, and is independent
of L, and thus L(t) is a linear function. From the foregoing
discussion it is clear that, even for the purely diffusion-controlled case. the predicted
law is based on a number of
idealized assumptions (e.g., that space charge effects are negligible, that the interface is planar, and that one-dimensional
geometry applies). It would be wrong to suppose that the
above approach can lead to a complete and accurate treatment of a real reaction that takes place entirely in the solid
state (with complications such as multiple phases, complex possibly fractal - grain size distributions and geometries, the
appearance of new phase boundaries, etc.), even though various attempts at such analyses have been reported. Nevertheless, useful information can be obtained, qualitatively at
least. about important parameters, for example the requirements of intimate contact, small grain size, large driving
force, or high temperature.
fi
4.6. Example 6: Catalysis at the Surface and in the
Interior of Solids
By its presence in a reacting system a catalyst alters the
free activation energy but not the position of the equilibri
Angcii . Clwu. I n / . Ed. Engl. 1993. 32. 528 -542
which 0 is the fraction of the surface sites that are occupied.r371In many cases the catalyst increases the reaction rate
to such an extent that the adsorption becomes the rate-determining step. Since we are dealing with a typical chemical
reaction, similarly as in the case of dissociation. recombination, and desorption that might follow, the rate can be adequately described by usual kinetic relationships (for example,
forward rate ,KPGO)under nonequilibrium conditions. Only
when the conditions are very close to equilibrium does this
relationship reduce to one of proportionality to the free reaction enthalpy, as is required by linear irreversible thermodynamics (see Figs. 2, 3).
An elegant example of how the defect chemistry influences
reaction rates is the HCI elimination reaction of tcrt-butyl
chloride, which has been investigated by Simkovich and
Wagner.1381Acid- and base-catalyzed homogeneous elimination reactiom are well known in organic chemistry. As described in Part I, doping AgCl with CdCI, increases the concentration of vacancies that act as bases. It is possible that
here it facilitates the detachment of a proton (with subsequent or simultaneous elimination of the chlorine ion). The
experiments show that the rate constant is, in fact, proportional to the cadmium concentration.
More recent experiments with AgCl heterogenously doped
by contact with A1,0, have yielded a deeper understanding.1391As expected from the analogy between heterogeneous and homogeneous doping as explained in Part I, the
rate of elimination of HCI from tert-butyl chloride is in531
creased considerably by the presence of AgCI: AI,O, dispersions. Figure 16 shows the pressure increase measured during the reaction. Evidently the increased Va, concentration
0.41
cI
conduction bands, respectively. Equation (24) describes the
transfer of an electron from the valence band into the redox
level, and Equation (25) that of an electron from the redox
level into the conduction band. The sum of these two equa-
electrolyte
composite( v =composite
silane-treated
0 2 2 . N=7 I
I
gas phase
( ( p = O Z Z . N.71
I
0.3
0.2
( P - P o 1 I Po
01
0
- 0.1
0
20
10
-
30
tlmin
40
50
Fig. 16. Fractional pressure increase as a measure of the progress of the reaction during the elimination of HCI from trrt-butyl chloride. showing the catalytic effect of silver chloride heterogeneously doped with AI2O3(p = volume
fraction of AI,O, in the catalyst, N = number ofcatalyst pellets). The effects of
pure A1,0, and pure AgCl are both very much smaller (despite the greatly
increased specific surface area of the former) [39]. T = 423 K. particle
size = 0.06 pm.
has the effect of a basic catalyst, facilitating the adsorption
of tert-butyl chloride and the detachment of a proton. As in
the discussion of the boundary effects on the conductivity
(see Part I), data on the conductivities of such composite
electrolytes"] yield quantitative information about rate constants as functions of volume fraction, particle size, temperature, and interface chemistry, but restricted space does not
allow a detailed description here.
Defect chemistry in the bulk is of special interest in relation to electrocatalysis. In a hydrogen-oxygen fuel cell a
considerably higher efficiency is attained than by direct combustion to water, since the chemical energy is converted directly into electrical energy instead of through heat energy.
In addition to the importance of the electrolyte (for example,
homogeneously doped ZrO, as a solid electrolyte oxygen
vacancies as charge carriers see Fig. 12 of Part I"]), the electrodes too play a vital role. If conventional electrodes are
used, then the required reaction can only take place at the
three-phase junction between the electrode (electron source
or sink), the electrolyte (an 0' or H conductor), and the
gas phase (0,-H, reservoir), as in the upper diagram of
Figure 17. If instead a mixed conductor is used for the electrodes with, for example, O2 and e - as charge carriers, as
is the case for (La,Sr)CoO,, then oxygen can migrate
through the solid by chemical diffusion, and oxygen (or hydrogen) can enter at any point on the electrode surface
(Fig. 17, lower).
In conclusion it must be mentioned that homogeneous
catalysis can also be observed within the solid itself. An
example is the more rapid transfer of electrons from the
valence band into the conduction band, resulting in the formation of an electron-hole pair (or the reverse process of
recombination), that may take place when states within the
band gap (recombination centers) are also involved. The
process is given, in the structure elements notation, by Equations (24)-(26), where VB and CB represent the valence and
-
-
538
+
- 0
conductor
Fig. 17. With conventional metal electrodes, oxygen uptake can only take
place at a three-phasejunction.With an oxidemixed conductor as theelectrode,
oxygen can enter at any point on its surface.
tions is the band-band transfer reaction [Eq. (26)]. In the
building elements notation the sum is the pair formation
reaction [Eq. (27)] that is already familiar from Part I.
nil
+- e ' +
h'
(27)
4.7. Example 7: The Electrochemical Transfer Reaction
The transfer of a charge carrier from the s u r f x e into the
bulk is neither a conventional reaction (A B, x = x' nor a
pure transport process (A = B, x =# x'). Over the reaction
distance there exists a strong electric field as well as a difference in the standard potential. Here, therefore, the analysis
of an electrochemical reaction, as represented in Figure 3,
must be applied in its most general form [Eq. (l)]. If the
current is limited by this transfer step, as is very often the
case when the diffusion in the bulk and the sorption at the
surface are both rapid, the current-voltage curve takes the
form described by the Butler-Volmer equation.''] Figure 18
shows a current-voltage characteristic measured for CaOdoped CeO, ]'4I.
At high overvoltages the logarithm of the
current, and not the current itself, is a linear function of the
overvoltage (i.e. Tafel law instead of Ohm's law). A direct
proportionality law only applies close to zero voltage ( U + 0).
The resistance corresponding to the ohmic regime is proportional to the exchange current density, which is a measure of
the reversibility of the electrode reaction, and can also be
interpreted in terms of the defect chemistry.
+
Anxen. Chem. Int. Ed. Engl. 1993, 32, 528-542
u
200
400
600
0
overvoltage/rnV
-
Fig. 18. The steady-state current-voltage characteristic for CaO-doped CeO,.
The kinetic behavior is entirely determined by the phase boundary transfer step,
resulting in a markedly nonlinear characteristic 1401.
this might be the multiplication of bacteria, or the growth of
a city, as well as impact ionization in a solid (in which an
electron already present gives up some energy to form a new
electron-hole pair).[411In such a process the increase in concentration of the product causes an acceleration of the reaction, and the system explodes or produces an allergic response. Here there is a positive feedback mechanism, which
drives the system further from equilibrium. It can very easily
be shown that Equation (28) then holds. (If the back reaction
becomes important, its rate is proportional to [A]’, and in
the neighborhood of equilibrium a negative feedback becomes dominant [Eq. (29)], as one expects.) Such a destabi-
6[A]’/6[A] > 0
6[A]’/6[A] < 0
4.8. Example 8: Nonlinear Effects
It has been amply demonstrated here that in transport
processes there is normally a linear relationship between rate
and driving force (see Fig. 2). Nevertheless, it has been mentioned at several points (e.g., in Section 4.7) that this no
longer applies for large driving forces; such behavior is especially common for field effects, and even more so for chemical reactions. In this section, however, the emphasis will be
on the type of nonlinear effects that are characteristic of
complex systems under conditions far from equilibr~m.[~‘]
Even for some quite simple reaction schemes, including
those encountered in defect chemistry, conditions leading to
nonlinear behavior can arise very easily.
A necessary criterion for equilibrium in a system is that
there is no entropy production within the system. In the
nonequilibrium case the entropy production is always positive for any process, in accordance with the second law of
thermodynamics. Consequently, any (slight) departure from
equilibrium results in a retroactive force (negative feedback);
this tends to restore the equilibrium state, which is thereby
stabilized. Often, however, there are external conditions that
hinder the system from attaining its equilibrium state; an
example is the previously described case of a specimen
through which a current is passed and in which the electron
current is blocked by ionic conductor contacts. If a constant
voltage is applied in this case, then a steady state becomes
estabIished, in which a (reduced) ion current flows through
the circuit (cf. the measurement of the ionic part of the conductivity in YBa,Cu,O,+,, as mentioned in Section 4.3). If
the applied voltage is sufficiently small, so that the conditions are close to equilibrium, the entropy production is, of
course not zero, but at a minimum according to the principle
of Glansdorff and Prig~gine.[~’]
It can also be shown that a
steady state close to equilibrium is stable. However, under
conditions far from equilibrium we have the interesting situation that a steady state is no longer necessarily stabIe, and
such states can then undergo changes in a variety of ways.
The most impressive examples of this are found in living
organisms (usually involving chemical reactions).
A kinetic peculiarity that can very easily result in a system
becoming nonlinear, and therefore interesting, is autocatalysis (see Fig. 19, top). The process shown here (top row) represents, quite generally, a growth reaction; for example,
Angew. Chc.m. In!. Ed. Engl. 1993, 32, 528-542
lizing relationship between flow and driving force (e.g., between current and voltage) often occurs in solids when the
current density in the system falls with increasing voltage
(negative differential conductivity). For example, the impact
ionization already mentioned acts in this way.[431I n the second row of Figure 19 the “growth reaction” is accompanied
by a competing “decay reaction”. Whether or not the species
X “survives”, to use the biological term, depends not only on
the rate constants but also on the “food supply” [A]. If the
stationary solution [XI,, (zero or finite) is plotted as a function of [A], it is found that there is a characteristic quantitative change corresponding to a nonequilibrium phase transition. Where there are several species X, the question of
whether or not a given species survives is determined entirely
by the rate constants and the food concentration, which is
assumed to remain constant and is thus inexhaustible. On
the other hand, if [A] is not inexhaustible there is a competition situation in which the survival criteria become ever more
severe as the process continues, as analyzed by E i g e r ~ . [If~ ~ ]
the surviving species now undergo modification, for example
through the effects of radiation (mutation in the biological
case), these new variants may either replace the old ones or
die out. It can be seen that under nonlinear conditions, even
very simple reaction sequences can lead to effects analogous
to the “biological” phenomena of growth, selection, and
mutation. A slightly more complicated reaction scheme can
lead to reactions that oscillate with time; examples include
the Lotka-VoIterra and Belousov-Zhabutinski reacti~ns.[~’l
The analysis also explains certain spatial effects, especially
spatially periodic patterns such as the dissipative structures
in the Belousov reaction, or patterns observed in biologial
systems.[421Here the spatial coordinate enters into the analysis due to the fact that diffusion is now an important part of
the overall process. Instabilities can destroy the spatial symmetry in such cases. An increasing divergence from equilibrium, passing successively through a number of bifurcation
points, can eventually lead to a fractal pattern characterized
by a deterministic chaos situation (in which the end result
cannot be predicted).
The above discussion obviously has important implications for defect chemistry. Even a small degree of complexity
in defect reactions is enough to lead to such effects. Moreover, the reactions are usually linked to diffusion processes.
I n semiconductor physics especially, effects of the kind de539
autocatalytic reaction
A+X
autocatalytic +
decay reaction
A+X
X
-
Chemical kinetics far from equilibrium
2X
growth. positive leedback
k,
e
2X
A
growth or death
Z
[A] = const
no selection pressure
W, = k , [ A ]
~
-
autocatalytic +
decay selection pressure
( + perturbation)
nonequilibrium phase
transformation bifurcation
k,
A+X
2X
X --- z
[A] # const
+
competition
selection
mutation
w, = . / ( I )
Lotka-Volterra reaction scheme
A+X
-
X+Y
Y-z
Ix- y
or
structurally
unstable
oscillation
--
A -X
2 X t Y
B+X
Brusselator
reaction scheme
-2X
2Y
1x1
3x
Y+D
limit cycle oscillation
I /9
(CO + 0 ---“ CO. Pt,,, ---” Pt,,)
catalytic CO oxidation
x
symmetry breaking.
deterministic chaos
----l-above reaction scheme
+ diffusion
compartmentation.
dissipative structure in space
(e.g X ( x ) X Z X ( x ’ )
x
Fig. 19. Summary of chemical kinetics under conditions far from equilibrium. showing that even quite simple reaction sequences can result in typical nonlinear
phenomena, such as kinetic phase transformations, dissipative structures. and deterministic chaos (see text). B = bifurcation parameter, I = site.
scribed have been
A typical autocatalytic reaction is the impact ionization mentioned earlier. Here, for
example, an electron whose energy lies in the conduction
band causes the formation of a new electron-hole pair as in
Equation (30) (the reverse process of Auger recombination),
so that the overall effect is an electron-catalyzed growth in
the electron concentration. The departure from equilibrium
can also often be controlled by an external voltage (overvoltage). Figure 20 shows a nonequilibrium phase transition
observed in gold-doped silicon when the applied voltage exceeds a critical value. This corresponds exactly to the third
diagram in Figure 19. Here the relevant growth and decay
I
mechanisms are impact ionization [Eq. (30)]and band-band
e’
+ h’+O
recombination [Eq. (31)]. Since the dependent variable in
this case is the stationary electron concentration, the effect
observed is a (voltage-induced) insulator-conductor transition. Figure 21 shows current oscillations observed with different values of the applied voltage for doped germanium at
high electric field strengths, and simultaneous irradiation in
the far IR region. As the field is increased the oscillation
period is doubled, and eventually deterministic chaos is
reached. This is apparent from the “phase portraits” (Zvs. I ) ,
conductor
.
i/
typ. 1 0 ~ - 1 0 ~ ~ / c r n
k-
Fig. 20. Nonequilibrium phase transition (from insulator to conductor) in Audoped silicon, controlled by varying the applied voltage in the nonlinear region
[43]. [ e - ] = concentration of stationary electrons. k ’ = rate constant of the
impact ionization.
540
(31)
Id)
-
Fig. 21. Periodic structures and chaos phenomena in germanium. controlled by
varying the applied voltage in the nonlinear region [45].
Angrw. Chem. Ini. Ed. Engl. 1993, 32. 528-542
from the time evolved I ( t ) , and from the plot of the current
minima as a function of the voltage.[451 Finally, Figure 22
shows images from a nonequilibrium structure in p-type ger-
layer effects to be incorporated into the overall theory, and
enable diffusion to be treated as a special case of an elementary chemical reaction step. In particular, every solid-state
chemical reaction must begin at a boundary and extend into
the interior by chemical transport.
4) In view of the above, defect chemistry, which is the
chemistry of the solid state, is an essential foundation stone
of modern materials research. This review has aimed to
present the case for defect chemistry, and the examples described are a testimony to its capabilities.
Received: April 1. 1992 [A 896 IE]
German version: Angew. Clzem. 1993. 105. 558
Fig. 22. ('urrent tilaments obscrvcd by the EBIC (electron-beam-induced current) technique in p-type gcrmanium with two different voltages [46].
manium, recorded with a scanning electron microscope by
using the EBIC (electron-beam-induced current) techn i q ~ e . [The
~ ~ bright
]
lines (current filaments) represent zones
in which the electron concentration has changed. These are
only observed under nonequilibrium conditions, and accordingly disappear when the current is switched off. Spatial (and
also temporal) periodic structures and deterministic chaos
behavior can also be observed under suitable conditions during the catalytic oxidation of CO on platinum surfaces.[471
The study of nonlinear effects in solids, especially with regard to ionic conduction phenomena, and questions of the
morphology can be expected to reveal many more surprises
in the future.
5. Summary
The most important points that have been discussed qualitatively in the two parts of this article, the thermodynamic
and the kinetic, are the following:
1) All solids with ordered structures exhibit a finite concentration of point defects in thermodynamic equilibrium.
Although these are of no consequence for some properties,
they play a key role in a wide range of phenomena of importance in chemistry, especially transport and reactivity. With
regard to properties such as these, the behavior of a solid
cannot be predicted without a prior knowledge of its detailed
composition and therefore of the state parameters that control it, for example the oxygen partial pressure in the case of
an oxide.
2) Defect chemistry, which is the study and quantitative
description of the nature and concentration of such defects
and of their interactions, is a logical extension of the solution-state chemistry of the relevant species, of which the
most familiar example is the chemistry of aqueous solutions.
In an analogous way defect chemistry provides the basis for
the thermodynamic and kinetic treatment of solid-state
chemistry.
3) As well as leading to a universally valid treatment of the
chemistry of ions in the condensed state, the approaches
developed in defect chemistry also provide a unified description of ionic and electronic charge carriers, allow boundary
Anqcir.. CJwm. In,. Ed. Ennl. 1993. 32. 528 -542
[I] Part I: J. Maier. Angew. Chem. 1993, 105,333;AngeM,.Chem. lnt. Ed. Engt.
1993, 32. 313.
[2] Some recommended textbooks dealing with the field are: a) H.
Schmalzried, SolidState Reactions, Verlag Chemie, Weinheim, 1981 ; b) H.
Schmalzried, N. Navrotsky, Festkorperthermodvnamik, Verlag Chemie.
Weinheim, 1975; c) K. Hauffe, Reaktionen in und an festen Stoflen.
Springer, Berlin, 1986; d) F. A. Kroger, Chemistry of lmprr/ect Crystals.
North Holland, Amsterdam. 1964; e) H. Rickert, Electrochemutry o/
S0lid.s. An Introduction, Springer, Berlin 1988; f) Also strongly recommended is the review by A. B. Lidiard, Handbnrh der Physik 1957.20.246.
[3] K. Funke. Solid State h i e s 1986, 18/19, 183.
[4] Strictly speaking Equation (2) should, for greater accuracy, also take into
account the electrochemical potential gradients of other particles (see also
ref. [S]).
[51 L. Onsager, Phys. Rev. 1931, 37, 405; ibid. 1932, 38. 2265.
[6] S. R. de Groot, Thermodynamics of Irreversible Processes, North Holland,
Amsterdam, 1951; I. Prigogine, Etude thermodynamique des ph&nomenes
irrewrsiblrs, Dunod, Paris, 1947.
171 As explained in Part I [I] for low concentrations the chemical potential [I
is a linear function of the logarithm of the concentration.
(81 J. J. O'Dwyer, The Theory ofElecrrica1Conduction and Breukdoun. Clarendon, Oxford, 1973.
[9] J. 0. M. Bockris, A. K. Reddy, Modern Electrochemirfry, Plenum. New
York, 1970.
[lo] C. Wagner. Atom Movements. American Society for Metals, Cleveland,
1951, p. 153; Progr. Solid. Stale Chem.. Yo/. 6 (Ed.: F. Reiss), Pergamon.
Oxford, 1971.
Ill] I. Yokota, J. Phys. Soc. Jpn. 1961, 16, 2213.
[12] H. Schmalzried, 2. Phys. Chem. N . E 1962, 33, 1 1 1 .
[13] C. Wagner. Proc. 7th Meeting In!. Committee Etectrochem. Thermodvn.
Kinetics, London 1955, 7 , 301; 1. Riess. Solid State lonics 1992, 51, 219.
[14] J. Maier, Z . Phys. Chem. N . E 1984, 140, 191.
1151 H. Schmalzried, 2. Phys. Chem. N . k: 1963, 38, 87.
[16] The modifications to the interpretation and calculation of results from
these methods that become necessary when the ionic defects can undergo
internal valency changes are described in refs. 114, 171.
[17] J. Maier, Mat. Res. Soc. Svmp. Proc. 2f0, Materials Research Society,
Pittsburgh, 1991, p. 499; J. Maier, G. Schwitzgebel, Phys. Starus Solrdi B
1982, 113, 535; J. Amer. Ceram. Soc. in press.
[18] D. Muller, Phys. Stalus Solidi 1965, 1312, 775.
1191 C . G. Fonstadt, R. H. Rediker, J. Appl. Phys. 1971.42, 2911
[20] E J. Opila, G. Pfundtner. J. Maier, H. L. Tuller, B. J. Wuensch, Mater. Sci.
Eng. B, in press.
[21] S. J. Rothnian, J. L. Routbart, 1. E. Baker, Ph-ys. Rev. B 1989, 40, 8852.
[22] Mathematical solutions to a variety of diffusion problems are given in: J.
Crank, The Mathematics of Dffusion, University Press, London, 1987. A
more detailed and comprehensive treatment based on the isomorphic heat
conduction equation can be found in: H. S. Carslaw and J. C. Jaeger,
Conduction of Heat in Solids, Clarendon Press, Oxford, 1959.
[23] R. Dieckmann, H. Schmalzried. Ber. Bunseriges. Phys. Chem. 1977, 81,
344.
[24] See, for example: L. Heyne in So/idE(ec!rolyies. (Ed.: S. Geller), Springer,
Berlin, 1977.
[25] T. Bieger, J. Maier, R. Waser, Solid State lonics 1992, 53-56, 578; T.
Bieger, J. Maier, R. Waser, Sensors Actuators 1991, 7 . 763.
[26] G. Pfundtner, J. Maier. Proc. Bilateral German-Soviet Symp. HTSC,
Karlsruhe, September 1990, 598.
[27] H.-J. Hagemann, Dissertation, Technische Hochschule Gachen, 1980;
K. A. Miiller, W. Berlinger, R. S. Rubin, Phys. Rev. 1969, 186. 361 ; K. A.
Muller; T. von Waldkirch, W Berlinger, B. W. Faughnan, SolidState Commun. 1971, 9, 1071.
[28] J. Maier, J. Phv.r. Chem. Solids 1985, 46, 197.
[29] Trapping effects can be treated by including trapping reactions in the
diffusion process [14, 171, as well as by the inclusions of Onsager coupling
terms 1301.
541
[30] M. Martin, H. Schmalzried, Solid State Ionics 1986, 20, 75; H. D.
Wiemhofer, [hid. 1990. 40141, 530.
[31] C. Wagner. Z . Angew. Chem. 1936, 49, 735.
[32] N. B. Pilling, R. E. Bedworth, J: Inst. Met. 1923, 29, 529.
[33] Another extreme example is provided by the batteries used in cardiac
pacemakers, in which a lithium/iodine contact forms Li1 as a solid elec-
trolyte. This results in a (self-repairing) electrochemical cell. For a long
operational life the electronic contribution to the conductivity should be
negligible, since otherwise rapid self-discharging would occur.
[34] M. H. Sukhar, H. L. Tuller, Adv. Ceram. 1984, 7 , 71.
[35] C. Gensch, K. Hauffe, Z. Phys. Chem. 1951, 196, 427.
[36] Surfaces and interfaces are not the main topic of this article, and it would
be inappropriate to go into more detail here.
[37] A statistical analysis shows that the activity of the occupied fraction
of the surface sites can be expressed approximately as Ojl - 0; from a
naive standpoint this reflects, in an idealized model, the structure elements V,, and G,, (which correspond to the terms (1 - 8 ) and 0, respectively).
[38] G. Simkovich, C. Wagner, J. Catal. 1962, 1, 521.
542
[39] P. Murugaraj, J Maier. Solid Slate Ionics 1989, 32/33. 993; J. Maier, P.
Murugaraj, ibid. 1990, 40141, 463.
[401 D. Y Wang, A. S. Nowick, J. Elecfrochem.Soc. 1979,126, 1155; ibid. 1979,
126,1166.
[41] G. Nicolis, I. Prigogine, Self-Orgunizaton in Non-Equilibrium Systems.
Wiley, New York, 1972; W. Ebeling, Slrukrurbildung bei irreversiblen
Prozessen, Teubner, Leipdg, 1976; H. Haken, Synergerik, Springer, Berlin,
1983.
[42] A. Gierer, M. Meinhardt, Lect. Math. Life Sci. 1974, 7 , 163.
[43] E. Scho11, Non-Equilibrium Phase Transilions in Semiconductors, Springer,
Berlin, 1987.
(441 M. Eigen, Naturwissenschafkn 1971, 58, 465.
I451 S. W. Teitsworth, R. M. Westervelt, E. E. Haller, Phys. Rev.L e f t . 1983,51,
825.
1461 K. M. Mayer, R. Gross, J. Parisi, J. Peinke, R. P. Huebener, Solid S/ate
Commun. 1987, 63, 55.
[47] M. P. Cox, G. Ertl, R. Imbihl, Phys. Rev. Let!. 1985,15,1725; see also the
description of research in Jahrbuch der Max-Planck-Gesellschaft MPG,
1991, p. 364.
Angew. Chem. Inl. Ed. Engl. 1993,32, 528-542
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