1. Introduction
Ceria (CeO
2) and zirconia (ZrO
2) are the main components of the three-way catalysts used in automobile pollutant abatement [
1,
2]. They enable a control of the partial pressure of oxygen near the catalyst surface during automotive emission [
3,
4] and are able to store and/or supply oxygen under fuel-lean and fuel-rich conditions, respectively, what is necessary for the conversion of nitrogen oxides, hydrocarbons and carbon monoxide [
5,
6]. AO
2 (A = Ce,Zr) releases oxygen during rich conditions and is converted to AO
2−x, whereas during lean conditions AO
2−x is oxidized back to AO
2 [
7,
8]. These properties are related to the ability of AO
2 system to promote migration/exchange of oxygen species in the reaction [
9,
10]. Due to high surface and bulk diffusivity, the oxygen atoms can be quickly transferred to the active sites form different parts of the catalysts [
11,
12]. The highest oxygen storage capacity and catalytic performance of ceria-zirconia materials is at approximately 40–60% of ceria content [
13,
14].
A powerful tool to study the various oxygen transport processes that may take place in a crystalline oxide is the
18O–
16O isotopic exchange technique [
15,
16,
17]. The oxygen isotopic exchange is a suitable method to analyze the diffusivity and interaction of molecular oxygen with a metal oxide. From the registered kinetic curves of oxygen species concentrations, the mechanisms of catalytic processes and heir rates in given catalyst are determined [
18,
19]. Three types of exchange mechanisms between gaseous dioxygen and lattice oxygen atom have been defined on oxide catalysts [
20,
21]: (1) homoexchange between adsorbed atoms itself, without participation of atoms form oxide; (2) simple hetero-exchange between one atom of adsorbed oxygen molecule and one atom of the solid oxide and (3) the multiple hetero-exchange when both atoms of adsorbed oxygen molecule are exchanged with solid oxide oxygen atoms. Information on the mechanism of exchange may be obtained from the kinetic curves of oxygen species partial pressures at the beginning of exchange reaction. The formation of
16O
18O species as a primary product from
18O
2 indicates that exchange takes place via a simple hetero-exchange, if
16O
2 species are formed firstly, it means multiple hetero-exchange mechanism [
22]. Isotopic exchange is generally carried out in a recycle, close reactor coupled to a mass spectrometer [
23,
24]. The evolution of the partial pressures of
16O
2,
16O
18O and
18O
2 during the exchange process are registered as a function of time in isothermal conditions. When the range of temperatures where the exchange process occurs is not well-known isothermal conditions could turn out to be tedious. In those cases, the method of temperature-programmed oxygen isotopic exchange is very useful. Using a linear and slow increase of sample temperature, the concentration of the adsorbing/reacting/desorbing molecules are registered [
25,
26]. The curves obtained in one temperature programing isotopic exchange experiment directly inform on the temperature window where the exchange takes place and give a complete view of the proportion of lattice oxygen atoms (surface and bulk) involved in the process depending of the temperature [
27]. Such experiment can permit comparison of different families of oxide samples in an easy way, comparing the evolution of the rate of exchange and the evolution of the number of exchanged atoms versus temperature [
28]. Increasing the temperature also enables us to rapidly reach an equilibrium between the
18O concentration in the gas phase and into the lattice and then to obtain data about the number of exchangeable atoms in the solid [
23].
In the temperature programmed isotopic exchange experiments a very important parameter is the surface area of the catalyst. The high surface area provides more active oxygen transfer from catalyst. In this work the influence of surface is of catalyst is investigated. Oxygen transfer form catalysts goes on through the bulk diffusion processes. The bulk diffusion processes in oxide catalysts where was started to investigate in our previous work Ref. [
29]. In this work the complete model of bulk diffusion during and exchange reactions in powder catalysts during temperature programmed oxygen isotopic exchange process is proposed and verification with experimental results is achieved.
2. The Model
Isotopic oxygen exchange on surfaces of catalysts goes on by homoexchange and by heteroexchange [
30,
31]. Homoexchange occurs without participation of surface oxygen. The heteroexchange takes place when oxygen atoms form the surface of oxide are involved into the process. The simple heteroexchange occurs when one oxygen atom in the molecule is replaced, in complex heteroexchange both oxygen atoms in molecule are replaced [
18,
22,
32]. Simple and complex oxygen isotopic heteroexchange reactions are listed in
Table 1.
Not all of the simple hetero-exchange reactions listed in
Table 1 result in a change in system composition. It may happen that an atom of the same type from the gas phase can be replaced with an atom of the same type from the surface (e.g.,
16O
g with
16O
s). In that case microscopically the oxygen exchange occurs, but the composition of system remains the same. In order to involve into the calculation only those cases of reaction which lead to compositional changes of system, in
Table 1 for simple heteroexchange reactions the probabilities are indicated. Through the probabilities that are included only those cases of exchange when composition of system occurs. Considering adsorption, in the proposed model it is assumed that adsorption is very fast process [
33,
34] and surface is covered by oxygen species with the same composition as composition in gas phase. The change in composition in the gas phase and on the surface is calculated according to the law of mass action well known in chemistry. Using this law in the case of simple heteroexchange the variation of concentrations in gas phase
n32,
n34 and
n36 of species
16O
16O
g,
16O
18O
g and
18O
18O
g, respectively, expressed in mol/m
3 is written as follows:
where:
kS is the rate constant of simple heteroexchange reactions and is expressed by Arrhenius law:
, where
AS and
QS are pre-exponential term and activation energy of simple heteroexchange.
R and
T are gas constant and temperature, respectively. The variables
c32,
c34 and
c36 are surface concentrations of species adsorbed on the surface
16O
16O
S,
16O
18O
S and
18O
18O
S, respectively, expressed in mol/m
2. The variation of surface concentration
c32,
c34 and
c36 are found from following relation:
where
V is volume of reactor and
S is surface area of catalyst. Only equation for
c32 is symmetrical to equation for
n34. Ratio
V/
S appears because of different dimensions of concentrations in gas phase
nij expressed in mol/m
3 and surface concentrations
cij expressed in mol/m
2. Later it will be shown that this ratio, which seems to be just technical parameter in fact plays very important role.
Using law of mass action in the case of complex heteroexchange the variation of concentrations in gas phase is written as follows:
where
kC is reaction rate constant of complex heteroexchange expressed by Arrhenius law
, where
AC and
QC are pre-exponential term and activation energy of complex heteroexchange. Due to the symmetry of complex heteroexchange reactions (that is not the case for simple heteroexchange), the variations of variables
c32,
c34 and
c36 can be written in very simple form.
From the mathematical point of view, it is very interesting to analyze obtained sets of Equations (1)–(3). For example, the Equation (3) keeping
n32,
n34 and
n35 as variables can be written in the following matrix.
The Equation (5) is symmetrical in various cross sections and show interesting mathematical regularities of system where the complex heteroexchange of isotopes takes place. For simple heteroexchange the Equation (1) obtains the following matrix form.
Equation (6) also is symmetric in various cross sections and show mathematical regularities in complex heteroexchange systems. More detail and deep analysis of those matrixes could give very useful additional information about the properties of the system.
In the case of homoexchange, when exchange occurs only between adsorbed molecules without participation of lattice atoms the reactions in which the change of system composition takes place are following [
20,
21].
The rate equations mathematically describing those reactions applying the mass action law are the next:
where
kO is rate constant of homoexchange expressed by Arrhenius law as
,
AO and
QO are preexponential term and activation energy of homoexchange, respectively. In the case of molar concentrations of
nij dimension of
kO is m
4/s mol.
Similar to Equations (5) and (6), if writing Equation (8) in matrix form keeping
n32,
n34 and
n35 as variables the following two equivalent matrixes can be written
Considering the process of bulk diffusion, when oxygen isotope atoms
18O penetrate in deeper layers of oxide, the concentration variation of oxygen atoms in one oxide layer
K as given is calculated by using the second Fick’s law expressed in finite increments. However, first it must be adapted to geometrical specifics of particles of powder catalysts. The particles of ceria and zirconia powder catalysts, which experimental results will be fitted by proposed model, have cubic-like geometrical form [
35,
36]. Describing mathematically the process of diffusion into bulk of powder particle it is necessary to assume the limitation of the depth. The Fick’s law expressed in finite increments means that the bulk is deleted into separate layers but in the cases of powder particles, when diffusion flux takes place from for all surfaces into the center of powder particle the area of layers decreases. To solve this problem, the cubic-like powder particles are virtually divided into four pyramids and in 2-d case each of them is divided into layers (see
Figure 1). The area of each
K layer can be found form the following relation:
where
is the area of surface layer,
h is the thickness of layer and
dox is the size of powder particle. The second Fick’s law expressed in finite increments, assuming decreasing areas of layers obtains the following form:
where
D is the bulk diffusion coefficient of oxygen atoms in oxide,
i = 16, 18 indicates the type of oxygen isotopes and coefficients
B(K) involve the changes of areas of layers and are expressed as:
It is assumed that diffusion of oxygen isotopes are balanced, i.e., 18O diffuses into the bulk and replaces 16O atoms which diffuses to the surface and the condition is fulfilled. Diffusion coefficient of both oxygen species is assumed the same.
As indicated in Equation (11) K > 1 the equation describes diffusion starting from the second layer K ≥ 2, K = 1 is the surface, first or adsorption layer where oxygen species form gas phase adsorbs (and desorbs). The equation is needed which could describe diffusion and oxygen exchange between surface (first) layer and second layer of catalysts. The mass action law in combination with Fick’s law we will use to build equation for diffusion between first and second monolayers.
The exchange reactions between oxygen species on the surface
18O
18O,
16O
18O and
16O
16O with oxygen atoms from second monolayer
16O
(K = 2) and
18O
(K = 2) are considered [
29]:
Using the mass action law and taking into account the gradient of concentrations the reaction rates of those reaction are expressed in following form:
where
kd is the reaction rate constant,
and
are atomic concentrations of oxygen of
16O and
18O on the surface
K = 1 and are calculated form concentrations of molecular species
16O
16O,
16O
18O and
18O
18O from the next relation:
Finally, the kinetics of composition in gas phase is calculated including all considered above processes simple and complex heteroexchange, homoexchange and diffusion:
The temperature programming exchange.
In the model the reaction rate constants kS, kC, kO, kd and diffusion coefficient D depend on temperature according to Arrhenius law , AO and QO are preexponential term and activation energy of. In temperature programming exchange process the temperature T depends on time linearly: , where T0 is initial temperature and b the rate temperature increase, t is the time.
3. Results and Discussion
In
Figure 2a,b the experimental and calculated dependencies of partial pressures of oxygen species
18O
2,
18O
16O and
16O
2 on temperature are presented. Results are calculated for ZrO
2 and CeO
2. Experimental results are taken form literature Refs. [
35,
36]. The temperature programmed oxygen isotopic exchange experiment was performed at following conditions: the temperature increase speed 1.6 °C/min, specific surface area
Sbet = 25 m
2/g for ceria and
Sbet = 25 m
2/g for zirconia, mass of powder catalyst
mcat = 0.25 g, initial pressure of
18O
2 48 mbar and volume of reactor
V = 12 cm
3. The surface concentration of oxygen in ceria depending on crystallite orientation varies from 0.97 nm
2 for (110) until 15.8 nm
2 for (111) [
35,
36] and in these calculations was taken as 1.09× 10
19 m
2. Similarly using data from Ref. [
36] for zirconia oxygen surface concentration was taken 1.09 × 10
19 m
−2. From the fitting of experimental curves presented in
Figure 2, the values of activation energies of simple and complex heteroexchange reactions, diffusion and its pre-exponential terms were found. Values are written in
Table 2.
The fitting results presented in
Figure 2 are quite good, taking into account the difficulty of the process. In TPIE experiments the temperature changes with time and curves in
Figure 2 represent not a simple dependency on temperature but the kinetic curves. Some deviations of calculated results form experimental ones in many cases occurs because of impurities, which always exist in real conditions and which are impossible to estimate in modeling. For zirconia
Figure 2a some deviation of calculated curves form experimental points occurs considering the temperature range where the process of exchange starts. Experimental results show that exchange process starts at a little lower temperatures tha, theoretical predictions. However, for ceria catalysts
Figure 2b the calculated curves very well correspond with experimental points in this interval of temperatures. For ceria a small deviation is observed at higher temperatures when steady state of process is reached, but only for
p34 curve, the curve
p32 show very good agreement in whole considered temperature interval. Comparing the temperature interval where the exchange process starts the lower temperature is for ceria catalyst.
Measuring composition changes in gas phase during TPIE process the important parameter is the ratio between volume of reactor and total surface of catalyst V/S which is involved into calculations through Equations (2) and (4). This ratio influences significantly the partial pressures of oxygen species, including the steady state regime. This ratio depends on mass of catalysts and specific surface area through the relation S = mcat·Sbet. This relation also means that the same influence in the kinetic curves have both parameters mass of catalyst and specific area of catalyst. Moreover, because in Equations (2) and (4) the ratio V/S exist, the volume of catalyst also significantly affects the shapes kinetic curves and the values of partial pressures when steady state is reached. At the steady state if when mass of catalysts is relatively high p34 < p32 and when catalyst mass is low, situation is in opposite: p34 > p32. However, it is important to consider the influence of process parameters on the shapes of kinetic curves and partial pressures of species at steady state regime.
In
Figure 3 the calculated curves of partial pressures of oxygen species at steady state regime as a dependency on ratio
S/
V are presented. The shapes of curves
p34 and
p32 significantly differs. The curve
p34 pass maximum while the curve
p32 goes up in whole interval of
S/
V. The position of maximum in curve
p34 corresponds with position of cross section point of pressure curves of
p32 and
p36. The curves
p32 and
p34 cross each other and after the amount of species
16O
2 at steady state becomes higher than
18O
16O with further increase of
S/
V. The point at which curves
p32 and
p34 cross each other depends on total pressure: it shifts to lower values of
S/
V when total pressure decreases. It is seen in
Figure 2 (dot lines).
Experimental kinetic curves of oxygen species partial pressures registered during TPIE may have quite different shapes. For example, the curve
p34 in
Figure 2b has a maximum, but the curve
p34 not. Many experimental results show that the maximums can be broad or narrow, high or low, or sometimes they are not formed at all. It depends on type of catalyst. In order to clarify this situation and to find some regularities the calculations were performed by varying activation energies of simple and complex heteroexchange. The influence of
QS and
QC (activation energies of simple and complex exchange) is analyzed in
Figure 4a the calculated curves obtained by changing
QC and (b) the curves by changing
QS are presented. From
Figure 4a it is seen that with increase of
QC the temperature at which the molecules
16O
2 start to form shifts to higher temperatures and this shift at the beginning is very significant and later reduces (see 4 and 5 curves). However, the temperature of formation of
16O
18O species almost does not depend on
QC but the shape of
p34 curves significantly depends on
QC. At low values of
QC the maximum is not observed in those curves, but it appears and increases with the increase of
QC. In opposite it is for
p32 curves, at low
QC the broad but not well-expressed maximum is seen but with increase of
QC it disappears. The influence of
QS which presented in
Figure 4b is different. The temperature of formation of
16O
18O species is very sensitive on
QS and it is not so sensitive for formation of
16O
2 molecules, especially at higher values of
QS (see 4 and 5 curves of
p32). Maximums in
p34 curves are well expressed at lower values of
QS and disappear at high values of
QS. In opposite it is for
p32 curves, no maximums are seen at low values of
QS and broad maximums appear at high values of
QS. From
Figure 4a,b it is seen that
QS and
QC values determine which oxygen species with increase of temperature will be formed first. Depending on
QS and
QC values with increase of temperature the oxygen species
16O
18O can start to form first and
16O
2 after or in opposite,
16O
2 first and
16O
18O after. The cases when both oxygen species start to form together at the same temperature also can be observed in
Figure 4.
The results presented in
Figure 4 were obtained at relatively high ratio of
S/
V, in the case when steady state pressures as
p32 >
p34. The case of relatively low ratio
S/
V when steady state pressures are
p32 <
p34 is considered in
Figure 5. In
Figure 5a the influence of
QC is shown. It is seen that curves of
p34 almost are not affected by changes of
QC, but curves
p32 are significantly affected by
QC. At low values of
QC curves
p32 have big and broad maximums which decrease and finally disappear with increase of
QC. In
Figure 5a it is also seen that at low values of
QC with increase of temperature species of
16O
2 are formed firstly and
16O
18O after. However, the situation reversely changes at high values of
QC:
16O
18O are formed first and then
16O
2. The influence of
QC in the case when steady state pressures fulfil condition
p32 < p34 is presented in
Figure 5b. Now, the change of
QS significantly affects both curves. At higher values of
QS the maximums in curves
p32 start to appear and become broader with increase of
QS. However, it is interesting to note that when maximum appears, the temperature of formation of
16O
2 does not change any more with further increase of
QS. The curves of
p34 continuously shift to higher temperatures with increase of
QS but the shape of curves remain the same and no maximums appear. At low values of
QS with increase of temperature the species of
16O
18O are formed firstly and then
16O
2, but at high values of
QS situation reversely changes: species
16O
2 first and
16O
18O after.
Considering the steady state pressures of oxygen species, it is important to note that additionally to ratio
S/V, there is another parameter which influences the steady state pressures and modifies the kinetic. This parameter is the initial surface concentration of oxygen
COX. In the model this parameter is involved through the initial value of surface concentration
c32. Parameter
c32 is variable but at the beginning at
t = 0 it equals to
COX:
c32 (
t = 0) =
COX. Surface concentration of oxygen depends on the type of catalyst, method of preparation and also orientation of oxide crystallite grains [
35,
36]. The influence of oxygen surface concentration is shown in
Figure 6 where the partial pressure curves calculated for different surface concentration of oxygen
COX are presented. It is seen that in contrast with curves presented in
Figure 4 and
Figure 5, the curves of
Figure 6 at steady state regime differs. Since
COX depends on crystallite surface orientation curves presented in
Figure 6 can also be considered as presentation of the influence of crystallite surface orientation catalysts. Parameter
COX affects both, the temperature of formation of oxygen species and the pressure at steady state regime.
In the heterogeneous exchange, the process of diffusion is very important and influences the kinetic curves. The influence of bulk diffusion process is shown in
Figure 7 where the calculated kinetic curves of partial pressure of oxygen specie
16O
2 as a dependency on temperature (which increases with time in TPIE) are presented. The curves are calculated at different values of diffusion activation energy at constant pre-exponential factor
Adif = 1.90·10
−24 m
2s
−1. All other parameters were kept as constants and were the same as in calculations presented in in
Figure 2b. It is seen that diffusion significantly influences the shape of kinetic curves in the transition period and at steady state. The partial pressure at steady state of
16O
2 decreases with increase of
Qdif. This result is logic, because the increase of
Qdif means the decrease of diffusivity. At higher diffusivity more amount of
16O
2 appear in gas phase because of diffusion of oxygen atoms from the bulk of catalyst. In order to see this effect in more detail the dependence of
16O
2 partial pressure at steady state regime is presented in
Figure 8. The considered interval is narrow and change of pressure is small, nevertheless, the nonlinear dependence of steady state partial pressure on diffusivity can be seen.
In order to check the validity of model the concentration distribution in deeper layers, the concentration deps profiles of atoms
16O and
18O were calculated. Results are presented in
Figure 9, where depth profile curves are calculated for different values of
Qdif. Obtained curves are typical diffusion curves and shows the correctness of calculations. However, the bulk diffusion on powder catalysts is not so simple as discussed above presenting
Figure 1 and Equation (12) and depth profile curves at higher diffusivity can be more complex. It is seen that with increase of diffusivity (decrease of
Qdif) the concentration of
16O at the surface layers decreases and is replaced by oxygen
18O coming from gas phase. It means what more oxygen
16O must appear in gas phase forming oxygen species
16O
2 and
16O
18O.
Above the bulk diffusion was considered, but the surface diffusion is also very important in catalysis. It depends on homogeneity of surface. In the case of homogeneous surface, the exchange takes place directly on surface of oxide, the surface diffusion does not change surface composition of species and mathematically there is no need to write new equation because of absence of new variables. Concentration gradients on the surface in that case are not formed and it is not possible to consider surface mobility of atoms. Situation is significantly different in the case of nonhomogeneous surfaces, e.g., when noble metal nanoparticles are formed on the surfaces of oxides such as in M/CeO
2 (M-noble metal, e.g., Pt, Pd, Au) catalysts. Such catalysts are used in order to reduce oxygen exchange temperature [
23] (which on noble metals is less) what is very important in automotive catalysts. In that case, because of spillover [
12,
37,
38] the oxygen concentration gradients are formed toward metal nanoparticles. The model of surface and bulk diffusion because of oxygen spillover was proposed in our previous works [
33,
34]. In these studies the surface diffusion was combined with bulk diffusion and finally two dimensional diffusion model was created for consideration of nonhomogeneous catalysts such as M/Ce
xZr
(1−x)O
2. It was found [
28,
39] that process of bulk diffusion becomes more important with increase of content of Zr. For pure ceria CeO
2 samples bulk diffusion is weak and dominates surface diffusion.