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

код для вставкиСкачатьDefect 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. 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