1. Lecture Notes III Oxygen ion conducting ceramics
Oxygen senors
Fuel Cells
Oxygen pumps
Heating elements
In this lecture we shall consider the electrical properties of some of the most important oxygen ion conducting oxides. As indicated on the slide these plays an important role both in sensors, fuel cells, oxygen pumps and heating elements. In later lectures we shall consider these applications in greater detail, but in this lecture we will focus on the defects present in these materials and how these influence their electrical properties.

2. Oxygen ion conductors: defect reactions
[1]
[2]
[3]
[4]
Let us first define the réactions leading to formation of defects in oxgen ion conductors:
Eq.1 – as discussed previously doping with oxides where the cation has a lower valence for instance of ZrO2 ,leads to the formation of oxygen vacancies as expressed here. These oxygen vacancies are necessary to have a high oxygen ion conduction. Please note that the vacancies ormed by doping only depend on the concentration of dopant and not on the oxygen pressure.
Eq.2 – at low oxygen pressures a reduction can take place, especially at high temperatures. In this process oxygen is removed from the lattice creating an oxygen vacancy and two electrons (these can be attached to cations which are reduced).
Eq.3 – at high oxygen pressure it is also possible that an oxidation can take place. This oxidation can be considered as an uptake of oxygen ions in oxygen vacancies forming normal oxygen ion sites in the lattice. For this reaction two electrons are needed and these are provided by neighboring cations where positive holes are formed. In this reaction we consider these positive holes as ”free entities” like the electrons in eq.2
Eq.4 – Interstitial oxygen ions can also be formed as Anti-Frenkel pairs, that is an oxygen ion is jumping into a interstitial position leaving an oxygen vacancy il the lattice.
Eq.5 – finally we have the intrisic ionization, where an electron is jumping from the valence band into the coduction band. In this reaction both a ”free” electron and a ”free” positive hole is formed
Eq.6 – this equation show the overall neutrality conditions obtained when all these reactions are included, all the concentrations of positive defects on one side and all the negative defect on the orther side in the equation. Note that h (concentration of holes) is designated by p, and that e(concentration of electrons) is designated by n. This neutrality conditions is easy to set up from the different defect reactions – for instance from Eq.2 it is clear that n=2[VO••].
[5]
[6]

3. Defect concentrations – p(O2)
Neutrality conditions:
p + 2[VO••] = n + 2[Oi″] + [MfM′]
Regions in Brouwer plot:
n = 2[VO••]
[MfM′] = 2[VO••]
p = [MfM′]
p = 2[Oi″]
Let us now calculate the dependence of the defect concentrations on the oxygen pressure as we did previously (See Lecture Note II slides 24 and 25 for VO••) for the single defects. First we need to establish the regions to be used in the construction of the Brouwer plot. This is generally done by considering the overall neutrality conditions shown again on this slide. Each regions then obey a simplified neutrality condition established by one member from each side of the overall neutrality equation. As shown on the slide this give rise to the regions mentioned on the slide –please note that the condition p = n is only a point, as the intrinsic ionization is independent of the oxygen pressure.

4. Calculation for region n = 2[VO••]
Eq. 2:
K(VO••) = [VO••] n2 p(O2)1/2 ;
[VO••] prop. to p(O2)-1/6 ; n prop. to p(O2)-1/6
Eq. 5:
Ki = n p
p prop. to p(O2) +1/6
As shown on this slide for the different reactions in this region it is quite simple to determine the proportionality between defect concentrations ond oxygen pressure from the equilibrium constants for the reactions in this region. First we insert the neutrality condition in the expression for K(VO••)by subsituting either n or [VO••] . This leads to a proportionality of -1/6 for the two defects, which also was obtained previously.
Inserting the proportionality for n in the expression for Ki then gives the proportionality of p of +1/6 and inserting the proportionality for [VO••] in the expression for KAF (equilibrium constant for formation of the anti-Frenkel defect pair) gives the oxygen pressure dependence of +1/6 for [Oi″].
The same type of calculations can easily be performed for the other regions defined on slide 3. These calculations will not be repeated here, but the proportionalities obtaiend are presented on the Brouwer plot on slide 5
Eq. 4:
KAF = [Oi″] [VO••]
[Oi″] prop. to p(O2)+1/6

5. Oxygen ion conductors: Brouwer plot
low pressure
high pressure
This Brouwer plot show the straight log [defect concentration] vs log p(O2) lines obtained for the diffent defects in the four regions. This diagram is constructed in the following way:
Starting from the left, the line for log n vs log p(O2) is drawn arbritarily with a slope of -1/6. As soon as this line is fixed the lines for log n can be drawn in the other regions with their apropriate slopes an an extension to this first line.
2) Once the log n line is established in the first region the position of the log [VO••] line is easily determined from the neutrality condition:
n = 2[VO••], and once this line is establised the lines for log [VO••] in the other regions are also fixed.
3) Starting from the right we can now place the log p line at the same level as the log n line as we know that log n = log p. It is therefore also possible place the log n lines in the other regions with their appropriate slopes.
4) From the neutrality condition for the region to the right we can also determine the the position of the log [Oi″] in this region and in the other regions starting from from the right
5) Note that log [MfM′] is constant independent on the oxygen pressure.
Note also that for the n = 2[VO••] region n (electrons) are dominating,and he oxygen conductor is a n-conductor in this oxgen pressure range.
For the regions to the right it is however p (positive holes) which are dominating, and the material becomes a p-conductor in these high pressure ranges.
In the intermidiate pressure range, however, VO•• id dominating and the material is therefore an oxygen ion conductor in this pressure range.
Ion conductor
n-conductor
p-conductor

6. Conductivity plot σtotal = σion+ σn+ σp ti = 1
From the Brouwer plot it is now possible to construct the conductivity plot. Previously (Lect. Notes 1, slide 36) we have derived an expression for the total conductivity, which as shown on this slide is the sum of the ionic,electronic and the positive hole conductances. This total conductance was also in slide 36 shown to be:
σtotal = 2e[VO••]µ(VO••) + enµ(n) + enµ(p),
where e is the electronic charge, [VO••], n and p are the concentration of the oxygen vacancies, electrons and positive holes respectively, and the µ-values the mobilities of the three types of defects.
As the mobilities of the electrons and the positive holes are perhaps 1000 times bigger than the mobility of the oxygen vacancies, the n-conduction will dominate at low oxygen pressure and p-conduction at high pressure, and ionic conduction will only dominate in the medium pressure range, where ti = 1.In this range the oxygen ion conduction is of course constant as he vacancy concentration is determined by the dopant concentration. This range is called the “electrolytic domain”.
Note that the slopes for the oxygen pressure dependence of the different defects in the Brouwer plot are maintained in the conductivity plot. This is easy to prove by inserting the oxygen pressure dependencies in the equation above.
Note also that there are regions with mixed conduction, i.e. both with ionic and respectively electronic- or positive hole conduction. In these regions ti decreases from 1 to zero. These regions are however quite small because of the high mobilities af the electrons and positive holes compared to the mobility of the oxygen vacancies. Finally the region between the oxygen pressures where the transport number of vacancies, or that of electrons and positive holes, becomes 0,5, is designated as the “ionic domain”.
The domain boundary Pn indicates the p(O2) where ti = tn = 0,5, or σi = σn.
The domain boundary Pp indicates the p(O2) where ti = tp = 0,5, or σi = σp, whereas
PO is the boundary (not indicated in the plot) where tn= tP = 0,5 (ti = 0) or σn = σp.
Pn and Pp are important as they show the oxygen pressure at which the p and n conduction take over from the ionic conduction. This is important in many appications of oxygen ion conductors, where the conductor can short circuit at these pressures – see sensors and fuel cells in a later Lecture Note.
Transport number: ti + tn + tp = 1

7. Influence of temperature
Conductvity: ionic and n and p conduction
Domain boundaries
Let us first look at how the ionic and the n - and p conduction depend on temperature. In Lecture Notes I we discussed the the formula for the total conductivity – see slide 8, whereas the formula for the temperature dependence is discussed in slide 9.
The oxygen pressure for the domain boundaries discussed in slide 6 also depend on the tempreature. We shall discuss this in slide 11 and onward.

8. Total conductivity σtotal = 2e[VO••](VO••) + enn + epp
σtotal = σion + σn + σp
σtotal = 2e[VO••](VO••) + enn + epp
Note: mobility of electronic defects much bigger than for ions
The total conductivity is thus the sum of the three contributions, the conductivities of electrons, postive holes and the oxygen ions.
In this slide is also shown the transport numbers, i.e. the ratio of the current carried by a carrier. The sum of the numbers for the three species is of cource 1.
As the mobility for the electronic carriers is much bigger than the moboility of the ions, even small concentration of these carriers will give an overall electronic conductivity.
Transport numbers:
tion+ tn+ tp = 1

9. Dependence on temperature
Both carrier concentration and mobility are thermally
activated.
Arrhenius equation describe tthe temperature dependence
of both ionic and electronic conduction:
σ = σ0exp(-Q/kT)*
Where:
σ0 factor depending on temperature, Q activation energy
k Boltzmann constant
T absolute temperature
As both the carrier concentration and the mobilty f the carriers are thermally activated processes the conduction will forllow the Arrhenius expression shown in this slide. Plotting log (σ) versus 1/T should therefore give a straight line from which the activation energy can be determined from the slope.
The formula given en red is however only approximative- the correct formaula is given in blact at the bottom of the slide. The approximative formula is howeve often used, but the difference between a log (σ) vs 1/T and a log (σ) + log (T) vs 1/T plots is not significant ads log(T) is approximately constant over a appreciable tempeature range.
*correct formula is: σ T = σ0exp(-Q/kT)

10. Typical oxygen conductors
This slide show a log (σi) vs 1/T plot for typical oxygen conductors formed by doping. As discussed in the previous slide straight lines with a characteristic slope should be obtained in this plot. This is indeed the case for most of the oxides exept for the Bi – Y oxide which apparently undergo a phase transition at about 650 C. Many of the oxides shown here are used as oxygen conductors in many applications, which we shall consider later.

11. Influence of temperature on domain boundaries
In slide 6 we defined the oxygen pressures for following domain boundaries:
P0 where σn = σp;
Pp where σp = σion, or ti = 0.5
Pn where σn = σion, or ti = 0.5
It is easy to show that these boundaries depend on the temperature from the three fondamental equations for the temperature dependence of respectively σi,σn and σp in the ionic domain (slide 6) :
σion = σiono exp (-Eion/kT);
σp = σpo exp(-Ep/kT) p(O2)+1/4;
σn = σno exp(-En/kT) p(O2)-1/4.
Playing around with these equations and using the geneal conditions for the domain boundaries can be obtained:
lnP0 = -2(En – Ep)/(kT) + 2ln ( σno / σpo );
lnPp = +4(Ep - Eion)/(kT) + 4ln(σiono / σpo );
lnPn = -4(En – Eion)/(kT) + 4ln(σno / σiono ).
Plotting these equations in an Arrhenius plot gives the curves shown on the next slide.

12. Domain boundaries of stabilized zirconiaPp
Ionic domain P0 Pn
The equations derived in slide 11 thus give a straight line in this Arrhenus plot. It will be noted that the ionic domain between Pp and Pn becomes smaller with increasing temperature and it dissapear completely at a characteristic temperature. Ionic conduction is therefore not possible above this temprature, which of course is important for the application of oxygen ion conductors in sensors and fuel cells, as will be discussed later.

13. Practice: Calculate oxygen ion conductivity
Calculate ionic conductivity of Zr(Y)O2-x containing 8 mole % YO1.5, (ZrO0,92Y0,08O1.92
Charge of electron: 1,6 x coulomb
Density : d = 5,52 g/cm3
Mobility of oxygen ions: 5,1x10-5 cm2/V sec
σion = 2e [VO••] µ(VO••)
[VO••] = x d NA/M
x deviation from stoichiometric compostition
NA Avogadros number = 6,02 •1023
M molecular weight (Zr = 40, Y = 39, O = 16)

14. Answers to practice: Calculate oxygen ion conductiviy
First we calculate the molecular weight for the compound:
Zr: 40 x 0,92 = 36,80
Y: 39 x 0,08 = 3,12
O:16 x 1,92 = 30,72
Total ,64
Then se calculate the concentration of the oxygen vancancies (numbers per cm3) from the formula given, which gives: 3,8 x 1021
and finally we can calculate the conductivity from the mobility (given), the concentration oaf vacancies (calculated) and the electronic charge (given). The result should be: 6,2 x 10-2 ( Ωcm)-1 (= S/cm).

15. What determine the ionic condutivity
Several factors are important:
Host oxide
Type and concentration of dopant;
Temperature;
First of all the host oxide (its structure and the composition) determine the overall electrical properties. However no pure oxides show ionic conduction except at very high temperatures where oxygen vacancies are formed by an intrinsic Anti-Frenkel reaction. In this discussion we shall therefore focus on oxides where oxygen vacancies are formed by doping with oxides with aliovalent cations and where high concentrations of vacancies can be obtained. A large number of such oxides was presented i slide 10. The conductivity which can be obtained with a given host oxide therefore clearly depend on the type and the concentration of the dopants. Finally the temperature of course play an important role, as the migration of the ions is a thermally activated process (slide 9). In many cases, the defects are, especially at low temperatures, bound in defect clusters which inhibit the free movements of the defects and thus decreases the conductivity. The dissociation of these clusters also depend on the temperature.

16. Host Oxides/dopants
Fluorite Oxides – structure fcc (face centered cubic)
Many of the oxides showing high oxygen ion conduction belong to the fluorite oxides – that is oxides which crystallises in the face centered structure, the fcc structure. The unit cell for this structure is shown on the figure. For this structure it is characteristic that the oxygen ions are sitting in the corners of a cube, which again is placed in the face centered cube formed by the cations (metal). There is thus ample space in the oxygen cube for accepting interstitial ions, for instance interstitial oxygen ions in oxides with excess oxygen (UO2+x for instance).
Examples on such fluorite oxides are given in the slide. These oxides can accomodate quite large concentrations of aliovalent dopants as Y2O3 and CaO and they form solid solutions with these dopants over quite large composition ranges. Both the type of oxide as well as the type of dopant determine the actual conductivity obtained.
Examples: ZrO2, ThO2, CeO2 doped with Y2O3, CaO

17. Free defects vs bound defects
Let us first consider the situation when the defects are free to move in the lattice; we can also say that the defect in this case are non-interacting and randomly distributed in the lattice.
Apparently this take place when the concentration of defects and thus of the dopant is sufficiently low, and that the temperature is sufficiently high. The limits between free and bound defects strongly depent on host oxide and type of dopant.

18. Activation energy for conduction of free defects
σion T = C [VO••] exp ( - ΔHm/kT)
Already in slide 9 we saw that the conduction generally can be expressed by an equation of the type:
σ = A exp( - Ea/kT), where A is a constant, Eathe activation energy, and k the Boltzmann constant - note that this is the corret formula.
For the ionic conduction we found the general equation (slide 8):
σion = 2 e [VO••] µ(VO••), where e is the electronic charge, [VO••] the concentration of oxygen vacancies and µ(VO••) the mobility of oxygen ions jumping from an oxygen site into an oxygen vacancy.
Now the mobility can be expressed thermodynamically as:
µ = (B/T) exp (-ΔGm/kT), B is a constant containing among other terms a geometric factor (structure dependence),the jump attempt frequencyand the jump distance, ΔGm which is the Gibbs free enegy for the jump, and T the absolute temperature. It is also well known that:
ΔGm= ΔHm – TΔSm, where ΔHmis the enthalpy of the jump and ΔSm the entropy of the jump, which is considered as temperature independent. Itroducing these terms into the above equation for the mobility gives a new mobility expession of the form: µ = (B′ / T) exp(-ΔHm/kT) with a new constant containing the entropy term, B′.
It is now possible to establish the final equation for the oxygen ion conduction to be:
σion T = C[VO••] exp ( - ΔHm/kT), where C is a general constant containing some further constants, for instance the number of anoin sites per unit volume.
At low concentrations and at relatively high temperatures plotting log (σion T) versus 1/T will therefore produce a straight line with a characteristic slope of - ΔHm/k.
Plotting log (σion

19. Activation energies for conduction of bound defects
Dopants with +3 cations, e.g. Y3+, in host with +4 cations, e.g. ZrO2
Defect cluster: (YZr′ VO••)•
σionT = C exp (- (ΔHm + ΔH(A•))/kT)
In the case of Y3+ doped ZrO2 the defect cluster shown on the slide can dissociate into free defects following the reaction:
(YZr′ VO••)• (designated A•) = YZr′ + VO•• (A= associate)
Using law of mass action we obtain:
K(A•)= [YZr′] [VO••] / [A•]
and the condition of electroneutrality is: [YZr′ ] = [VO••] + [A•]
If we assume full defect association then
[A•] >> [VO••] and [YZr′ ] = [A•], which leads to:
K(A•)= [VO••] which can be expanded into:
[VO••] = (1/W)exp (-ΔH(A•)/kT), where W is the number of orientations of the defect cluster and ΔH(A•) the enthalpy of association of the defect cluster.
By substitution into the general expression for the ionic conduction (previous slide) the equation shown on the slide is obtained.The activation energy Ea is therefore in this case a sum of the enthalpies of migration and cluster formation respetively.
Plotting log (σion T) versus 1/T at relatively low temperatures a straight line with the slope –(ΔHm + ΔH (A•))/k should also be btained in this case. The slope will however be different as compared to higher temperatures as the activation energy also contain the association energy of the defect clusters. If this plot is extended over a larger temperature range (see plot on slide), then a break in the slope will be observed around some characteristic temperature (~800 K has been proposed in the litterature for CaO doped fluorite oxides) above which the defects are essentially free. In this temperature range the slope decreases indicating a decrease in the activation energy.

20. Activation energy for conduction of bound defects
Dopants with +2 cations, e.g. Ca2+, in host with +4 cations, e.g. ZrO2
Defect cluster: (CaZr″ VO••)x
σionT = CM1/2 C1 exp((- (ΔHm+ ΔH(Ax)/2)/kT)
Dissociation of defect cluster:
(CaZr″ VO••)x (designated Ax)= CaZr″ + VO••
Application of mass action law:
K(Ax) = [CaZr″ ] [VO••] / [Ax]
Neutrality condition:
[VO••] = [CaZr″ ]
Substitution in eq for K(Ax):
[VO••]2 = K(Ax) [Ax]
At full association:
[Ax] >> [VO••] and [Ax] is close to CM (= total dopant concntration).
K(Ax) can be expanded to:
K(Ax) = (1/W) exp(-ΔH(Ax)/kT), where W again is the number of orientations of the cluster and ΔH(Ax) the enthyalpy for the formation of the defect cluster.
Inserting this expression into the equation for the concentration of gives:
[VO••] = CM1/2 C1 exp( - ΔH(Ax)/2kT), where C1 is a new constant.
Finaly the expression shown on the slide for the oxygen ion conductivity can be obtained.
The activation energy then also in this case contain the enthalpy of association of the defect cluster , but this time only half of this enthalpy. Plotting log (σionT) versus 1/T should thus give a straight line with a characteristic slope as discussed previously. The slope should also in this case change in a certain temperature range ( ~ 800K) above which the oxygen vacancies are essentially free.

21. Comparison of activation energies for free and bound defects
Free defects ΔHm
(CaZr″ VO••)x ΔHm + ΔH(Ax)/2
(YZr′ VO•• )• ΔHm + ΔH(A•)
In the previous slides we derived the activation energies for the three cases shown on the slide. Let us now compare the energies obtained for the two types of defect clusters.
First we can estimate the binding energy for the clusters using the general relationship: ΔH(A) = Q1Q2 / εr, where the two Q-values are the effective charges of the defects in the cluster, ε the dielectric constant of the material and r the separation of the two defects.
For (CaZr″ VO••)x we obtain: ΔH(Ax) = 4e2 / εr, and
for (YZr′ VO••)• we obtain: ΔH(A• ) = 2e2 / εr.
and ΔH(Ax) ~ 2 ΔH(A• ), which is quite surprising considering that we are talking about two different defect clusters with different dopants.There are however other factors than the defect charges which must be taken into conderation and it is also a question whether the dielectric constant and the distance between the defects are the same in the two cases. In the next slide we shall discuss some results which indicate that the binding energy of such defect clusters strongly depend on the mismatch between he dopant and the host cations.

22. Binding energies of defect clusters
M2O3 - dopants
In this plot the binding energy (in eV) is plottet against the ionic radius of typical M3+ dopant cations in CeO2. The defect clusters formed are therefore of the type: (MCe′ VO••)• and the enthalpy of association correspond thus to ΔH(A•) discussed in slide 19. As you see from this plot there is a mimum in the binding energy for Gd3+ which in fact quite closely correspond to the point where rGd ~ rCe, that is where the radius of the dopant cation is close to that of the host. The binding energy of the cluster thus clearly depent on he mismatch (and on the resulting relocation of the ions in the lattice), the higher this mismatch is the higher is the binding energy. Generally gadolinia doped oxides therefore have a higher conductivity than for instance yttria doped oxides

23. Dependence on defect concentrations
In this plot the conductivity is plottet versus dopant concentrations at constant temperature (> 800 K) for various dopants in zirconia. For all dopant a maximum is observed. The reason for this maximum in conductivity is the increase in conductivity due to increasing oxygen vacancy concëntration with increasing dopant concentration until a concentration is reached where the defects start to interac. After this characteristic composition the oxygen vacancies will thus be less free to move which produces a decrease in the conductivity.
Generally the composition range is divided in two parts:
The dilute solution range below the composition of the maximum in the conductivity and the concentrated range above this composition.The limiting compostion between the two ranges is generally considered to be x = 0.08 (in Zr1-xMxO2-x). The dilute solution is therefore defined to be in the range 0<x<0,08 and the concentrated range as x>0,08. The concentration giving the maximum conductivity therefore lies in the concentrated range.

24. Conductivity data: Ce(Y)O2-x
Let us finally look at some typical conduction data for each of the two groups discussed priviously represented by the two types of dopants MfM′ and MfM″ respectively.
First we shall consider yttria doped ceria: Ce(Y)O2-x .The plot shows the isorhermal conductivity ( T = 454 K) as a function of the Y2O3 concentration starting from the dilute range and up into the concentrated range. The curve showing a maximum is the ionic conductivity versus mole % of Y2O3 (arrow pointing to the left) – this curve crrespond to the curves shown in slide 23 - , whereas the curve showing a minimum is the activation energy versus mole % of Y2O3.
Comparing the two curves it is clear that a maximum in the conductivity is coupled with a minimum in the activation energy.
In slide 19 we introduced the defect clusters in yttria doped oxides to be (YCe′ VO••)• , but in fact for the formation of one VO•• two YCe′ are required, of which one will be used for the cluster formation and the other one will de free. Now the dopant ions can be considered to be randomly distributed in the cation lattice –and furthermore these can be considered to remain in these positions due to the very low diffusion of cations. in these oxides. The overall defect structure is thus determined from the random dstribution of YCe′ .From the curves it is clear that the increase in the conducion in the dilute range is due to a decrease in the activation energy. To explain this decrease it is assumed that the association energy of the defect clusters is reduced by electrostatic interaction between the ”freed” migrating VO•• , and surrounding (YCe′ VO••)• and YCe′ - this decrease can be understood un terms of a pulling by the YCe′ -ions in the VO•• , which then will be less tightly bound in the clusters.
At concentrations above the maximum conductivity larger defect complexes are formed, which bind the oxygen vacancies more firmly and thus increases he activation energy.

25. Conductivity data: Ce(Ca)O2-x
High temperatures
The plot shows the ionic conductivity for Ce(Cax)O2-x as a function of the mole% CaO added at different temperatures in the high temperatire region. As discussed previously the oxygen vacancies are not bound in defect clusters and they can therefore moove freely at these high temperatures.The general equation for ionic conduction under these conditions was derived in slide 18 to be:
σion T = C[VO••] exp ( - ΔHm/kT), where C is a constant and ΔHm is the enthalpy of migration of free oxygen vacancies. As one oxygen vacancy is formed by each CaCe″ the concentration of oyxygebn vacancies therefore correspond to the concentration of the dopant (x), and a straight line relationship between the conductivity and mole % of CaO should therefore be obtained. This is also what is observed in the range 1 – 8 mole % CaO (x = 0.01 – 0.08). Furthermore, from the conductivity values at fixed compositions at the different temperatures it is possible to determine the activation energy. In this case a constant activation energy of 0.62 eV was obtained for x< This value therefore correspond to the activation energy for the migration of free vacancies in this system.
At higher dopant concentrations the conductivity reach a maximum, after which conductivity again decreases with increasing dopant concentrations. The reason for this decrease is that the oxygen vacancies are tied up in larger defect associates, which restrict their movement.

26. Conductivity data for Ce(Ca)O2-x
Low temperatures – 500 K
In this plot is shown: Upper – logaritmic plot of conductivity versus mole% of CaO; Lower – activation energy versus % CaO. Both plots at 500 K.
Let us first consider the type of defect present at low temperatures for this system: In slide 20 we saw that the defect cluster at low temperature is a neutral cluster of the type (CaCe″ VO••)x . The concentration of free vacancies, which can migrate and thus contribute to the conduction, are thus formed in a dissociation n equilibrium with
The corresponding equation for the oxygen ion conduction was in slide 20 derived to be:
σion T = CM1/2 C1 exp((- (ΔHm + ΔH(Ax)/2)/kT), where CM is the dopant concentration, C1 a constant, ΔHm the enthalpy of migration of free oxygen vacancies and ΔH(Ax) the association enthalpy for the defect clusters.The linear relationship of log σ versus log CM ith a slope of ½ is clearly obtained.
From the lower plot we observe that ΔH(Ax) = 0.93 and constant independent of dopant concentration when x> From the high temperature results (slide 25) we found that the enthalpy of migration o free vacancies was about 0.62 eV and the enthalpy of association of these clusters must therefore be about ( 0.93 – 0.62)x2 = 0,62 eV, that is of the same magnitude as the migration enthalpy for the free vacancies.

27. Practice
The conductivity for Ce(Ca)2-x can be read from slide 25 at different temperatures and for different dopant concentrations. For 8 mole% CaO we find for the four different temperatures:
973 K: σion 102 = 3,13 ; σion = (Ωcm)-1
1073 K: σion =
1173 K: σion =
1273 K: σion =
At these hight temperatutes the VO•• are free. Make an Arrhenius plot ( log(σion T) versus 1/T) using these conductivity values and the activation energy for the migration of the oxgen vacancies.
As shown in slide 18, the conductivity can in this case be expressed by the equation:
σion T = C[VO••] exp ( - ΔHm/kT), where ΔHm is the activation energy (enthalpy) for the migration of the oxygen vacancies and C is a constant..

28. Answers to practice
From the equation mentioned in slide 27, we will get a straight line if ln(σion T) is plotted versus 1/T. This plot is shown in this slide. ΔHm can then be determined from the slope as shown. The value obtained for ΔHm is quite close to the litterature valie cited earlier.
That the Boltzman constant is equal to is quite easy to derive from the fundamental unit for k, which is erg deg-1, and form the fact that 1 eV = erg.

29. Content Lecture Notes III: Oxygen ion conducting ceramics
(Numbers indicate the slide number).
Oxygen ion conductors: defect reactions – 2
Defect concentrations – 3
Calculation for region n = 2 [VO••] – 4
Brouwer plot – 5
Conductivity plot – 6
Influenec of temperature – 7
Total conductivity – 8
Dependence on tem,perature – 9
Typical oxygen conductors – 10
Influence of temperature on domain boundaries – 11
Domain boundaries stabilized zirconia – 12
Practice – 13
Answers to practice- 14
What determine ionic conductivity – 15
Host Oxides/dopants – 16
Free defects vs bound defects – 17
Activation energy for conduction free defects – 18
Activation energies bound defects (Y3+dopant) – 19
Activation energies bound defects (Ca2+ dopants) – 20
Comparison of activation energies – 21
Binding energies of defect clusters – 22
Dependence on dopant concentrations – 23
Conductivity data Ce(Y)O2-x – 24
Coductivity data Ce(Ca) O2-x (high temperature) – 25
Conductivity data Ca(Ca)O2-x(low temperature) – 26
Practice – 27
Answers to practice - 28