Study of Transition Zones in the Carbon Monoxide Catalytic Oxidation on Platinum Using the Network Simulation Method

: A study of transition zones in the carbon monoxide catalytic oxidation over platinum is presented. After the design of a network model following the rules of the Network Simulation Method, it is run in a standard (free) software providing the fractional coverages of all species for di ﬀ erent values of carbon monoxide partial pressure, the main parameter that produces the change between a stationary or periodic response. The design of the model is explained in detail and no assumptions are made concerning the removing of oxidation fractional coverage. The illusory chaotic behavior associated with an inadequate time step in the numerical algorithm is studied. This work provides an explanation for the transition (bifurcation) between the stationary and the periodical response studies making use of Poincar é plane and phase-diagrams. The extinction of variable ﬂuctuation in the transition zone is analyzed to understand its relation with given values of transition partial pressures. Of particular interest is the small time span of the superﬁcial fractional coverage of carbon monoxide ﬂuctuation near the second transition partial pressure.


Introduction
The name "catalysis" was first used in 1836 by Berzelius for chemical reactions that involve the use of compounds, called catalysts, to accelerate these processes without being consumed [1]. A catalyst speeds up a reaction by lowering one or more activation energies or by introducing an alternative reaction with a lower activation energy; a catalyst should be selective. As a consequence, the activation energy for the desired product should be considerably diminished [2,3]. One of these processes, the catalytic oxidation of carbon monoxide on platinum, an example of air-metal interface oxidation with adsorption, is used in noble metal catalytic converters, which are common components in modern automobile engines. These systems are usually classified into two types: three-way and oxidation converters. However, from the point of view of the chemical reaction analysis, both are similar, a reason for which they are often considered the same type of chemical reaction.
A review of the literature reveals that there are four different approaches to modeling the chemical reactions that occur in these systems: the classical converter approach, the tanks-in-series approach, the more complex tanks-in-series approach and the Ford three-way converter model [4]. The classical Figure 1 shows the three steps of the Langmuir-Hinshelwood mechanism, represented by the following chemical equations [17].

Governing Equations
CO ads + O ads where * represents the surface hole effects.
where * represents the surface hole effects. The kinetic equations associated with the above chemical equation are [17]: The following additional chemical equations are introduced to consider the nature of the platinum surface in the catalytic process: where [Pt···Oads] represents oxidative species. The respective kinetic equations associated with the above chemical equation are: Carbon monoxide molecules can be attached to any part of the platinum surface to prevent oxygen chemisorption. Assuming the hypothesis described by Keren and Sheintuch [18], oxygen chemisorption over platinum, usually a dissociative process, is produced by a molecular bond break in a vacancy of crystalline structure on the surface. Carbon dioxide desorption, though much faster than the processes described above, influences the suppression of the other species [18]. It is assumed that these reactions are isothermal and that the partial reactive pressures, Pj, remain constant [18].
The kinetic equations, which are written with fractional coverages, θ, (more suitable than species concentrations because they are directly related with partial pressure and the cover surface) and dimensionless superficial density, y, yield the following balance equations [17]: The kinetic equations associated with the above chemical equation are [17]: The following additional chemical equations are introduced to consider the nature of the platinum surface in the catalytic process: where [Pt· · · O ads ] represents oxidative species. The respective kinetic equations associated with the above chemical equation are: Carbon monoxide molecules can be attached to any part of the platinum surface to prevent oxygen chemisorption. Assuming the hypothesis described by Keren and Sheintuch [18], oxygen chemisorption over platinum, usually a dissociative process, is produced by a molecular bond break in a vacancy of crystalline structure on the surface. Carbon dioxide desorption, though much faster than the processes described above, influences the suppression of the other species [18]. It is assumed that these reactions are isothermal and that the partial reactive pressures, P j , remain constant [18].
The kinetic equations, which are written with fractional coverages, θ, (more suitable than species concentrations because they are directly related with partial pressure and the cover surface) and dimensionless superficial density, y, yield the following balance equations [17]:  where  species  j=1  refers  to  carbon  monoxide  and  2 to oxygen, In this equation the time, t, is divided by τ s ox , the characteristic time of Equation (6) and equal to 1/k red ·X T . Thus, the variable t • is dimensionless. Now, ε ox can also be written as τ r /τ s ox , where τ r , the characteristic time of Equation (3), is equal to 1/k r ·X T . The fractional coverage of a species can be described as its superficial density, n s , and the ratio between activated surface and total surface. The superficial density, X, in turn, can be described as the subtraction between its maximum, X T , and the superficial ratio of catalyst blocked by the oxide.

The Network Model
From Equations (10) to (12), three coupled networks are formed, one per variable, following the indications given by González-Fernández and Alhama, and Sanchez-Perez (González-Fernández and Alhama, 2001, Sanchez-Perez, 2012, Sanchez-Perez et al., 2018). However, to facilitate your understanding, the steps for its design will be briefly explained. First, the equivalence between variable and electric voltage must be established (θ j (fractional coverage) ≡ V (electric voltage) and y (activated surface density) ≡ V).
Second, each sum of Equations (10) to (12) is considered an electric current, I j , that balances at a central node ( Figure 2). The time derivative of each variable, dθ j dt and dy dt , is implemented as a capacitor, C j , and the rest of the addends as controlled current sources, G j . So, the current I j for variable j = 1, according to Equation (10), can be separated into two controlled current sources, 1 (G 1,V ) and I 1,VI = θ 1 µ ox yθ 2 (G 1,VI ) with θ 1 , θ 2 and y being the voltages at the central node of each variable's network. The last term of the circuit ( Figure 2) is a resistor, R Inf , with supposedly infinite value to give stability to the circuit.
In this equation the time, t, is divided by τs ox , the characteristic time of Equation (6) and equal to 1/kred·XT. Thus, the variable t° is dimensionless. Now, ε ox can also be written as τ r /τs ox , where τ r , the characteristic time of Equation (3), is equal to 1/kr·XT.
The fractional coverage of a species can be described as its superficial density, n s , and the ratio between activated surface and total surface. The superficial density, X, in turn, can be described as the subtraction between its maximum, XT, and the superficial ratio of catalyst blocked by the oxide.

The Network Model
From Equations (10) to (12), three coupled networks are formed, one per variable, following the indications given by González-Fernández and Alhama, and Sanchez-Perez (González-Fernández and Alhama, 2001, Sanchez-Perez, 2012, Sanchez-Perez et al., 2018). However, to facilitate your understanding, the steps for its design will be briefly explained. First, the equivalence between variable and electric voltage must be established (θj (fractional coverage) ≡ V (electric voltage) and y (activated surface density) ≡ V).
Second, each sum of Equations (10) to (12) is considered an electric current, Ij, that balances at a central node ( Figure 2). The time derivative of each variable, and , is implemented as a capacitor, Cj, and the rest of the addends as controlled current sources, Gj. So, the current Ij for variable j = 1, according to Equation (    Finally, the model can be simulated with a standard circuit simulation code such as NgSpice or Pspice [27][28][29].
The software NgSpice [29] makes use of the most powerful computational algorithms required for the circuit simulation software to afford strong non-lineal and coupled mathematical models such as that described in this paper. These algorithms, based on the thesis of Nagel [36] and described in the NgSpice's manual [37], include trapezoidal integration [27], Gear´s fixed time methods [38] and the Runge-Kutta algorithm. The accuracy and efficiency of these methods are provided by reducing the local truncation error and the stability in the convergence of the numerical solution.

Simulation and Results
The first data used to characterize the reactions are the activation energies. The values of the dimensionless kinetic constants, D1, D2, μox and ε ox are 0.01, 0.0005, 5 and 0.0001, respectively. Other dimensionless constants used in Equations (10) and (11) are A1 and A2, with values 0.002475·P1 and 3, respectively [17].
To verify the model, we used the carbon monoxide partial pressure variable, P1, from several former experimental and numerical studies, 101.324 Pa [17,19,[39][40][41][42]. In the same way, the initial conditions for θ1, θ2 and y (0.9, 0.03 and 0.24, respectively) are applied. Figure 4 shows the periodic fluctuation of superficial fractional coverages and superficial density, which is similar to the results described in the above mentioned studies. Finally, the model can be simulated with a standard circuit simulation code such as NgSpice or Pspice [27][28][29].
The software NgSpice [29] makes use of the most powerful computational algorithms required for the circuit simulation software to afford strong non-lineal and coupled mathematical models such as that described in this paper. These algorithms, based on the thesis of Nagel [36] and described in the NgSpice's manual [37], include trapezoidal integration [27], Gear´s fixed time methods [38] and the Runge-Kutta algorithm. The accuracy and efficiency of these methods are provided by reducing the local truncation error and the stability in the convergence of the numerical solution.

Simulation and Results
The first data used to characterize the reactions are the activation energies. The values of the dimensionless kinetic constants, D 1 , D 2 , µ ox and ε ox are 0.01, 0.0005, 5 and 0.0001, respectively. Other dimensionless constants used in Equations (10) and (11) are A 1 and A 2 , with values 0.002475·P 1 and 3, respectively [17].
To verify the model, we used the carbon monoxide partial pressure variable, P 1 , from several former experimental and numerical studies, 101.324 Pa [17,19,[39][40][41][42]. In the same way, the initial conditions for θ 1 , θ 2 and y (0.9, 0.03 and 0.24, respectively) are applied. Figure 4 shows the periodic fluctuation of superficial fractional coverages and superficial density, which is similar to the results described in the above mentioned studies.   [17,18,43].
Following the indication given by Sanchez-Perez [44], in the first zone, up to 99.992 Pa, the studied variables have stationary values, that is, the converter is not operative. Within this zone, the value of superficial fractional coverage of carbon monoxide increases as its partial pressure increases ( Figure 5) which is the opposite to the behaviour of the superficial fractional coverage of oxygen ( Figure 6). Obviously, the number of carbon monoxide molecules inside the gas phase depends on its partial pressure, and the surface they occupy on the converter follows the same relation. For this reason, the superficial fractional coverage increases as the partial pressure increases. The superficial fractional coverage of oxygen depends on the behaviour of the carbon monoxide. This reasoning is also valid for the superficial density, Figure 7.
In the second zone, from 99.992 to 186.651 Pa, the studied variables fluctuate, that is, the converter is operative. Of note is the relation between the superficial fractional coverages of oxygen and carbon monoxide, θO/θCO, that is, nO s /nCO s is 0.36/0.48. This is the same as the inverse relation between oxygen and carbon atomic weights, 3/4. Therefore, there is a relation between the volume ratio of the two species and the respective surface they occupy, when fluctuation starts. The explanation lies in the optimal distribution of atoms on the converter surface, considering that the carbon monoxide molecule, before binding with the second oxygen atom, adheres to the surface through the carbon atom. Fluctuation increases its amplitude as the carbon monoxide partial pressure increases. Finally, the amplitude reaches a level that stops the fluctuation. In the third zone, above 186.651 Pa, the studied variables become stationary, that is, the converter is not operative.  . Amplitude values at stationary oxygen superficial fractional coverage versus carbon monoxide partial pressure [44].   Figure 5. Amplitude values at stationary superficial fractional coverage of carbon monoxide versus carbon monoxide partial pressure [44].
Following the indication given by Sanchez-Perez [44], in the first zone, up to 99.992 Pa, the studied variables have stationary values, that is, the converter is not operative. Within this zone, the value of superficial fractional coverage of carbon monoxide increases as its partial pressure increases ( Figure 5) which is the opposite to the behaviour of the superficial fractional coverage of oxygen ( Figure 6). Obviously, the number of carbon monoxide molecules inside the gas phase depends on its partial pressure, and the surface they occupy on the converter follows the same relation. For this reason, the superficial fractional coverage increases as the partial pressure increases. The superficial fractional coverage of oxygen depends on the behaviour of the carbon monoxide. This reasoning is also valid for the superficial density, Figure 7.   To understand the superficial fractional coverages and the superficial density behaviour inside the first zone, we choose a partial pressure belong to this zone, 39.997 Pa. Figure 8 shows the evolution of these variables; note that the variables tend asymptotically towards a constant value. As the carbon    To understand the superficial fractional coverages and the superficial density behaviour inside the first zone, we choose a partial pressure belong to this zone, 39.997 Pa. Figure 8 shows the evolution of these variables; note that the variables tend asymptotically towards a constant value. As the carbon  Figure 7. Amplitude values at stationary superficial density versus carbon monoxide partial pressure [44].
In the second zone, from 99.992 to 186.651 Pa, the studied variables fluctuate, that is, the converter is operative. Of note is the relation between the superficial fractional coverages of oxygen and carbon monoxide, θ O /θ CO , that is, n O s /n CO s is 0.36/0.48. This is the same as the inverse relation between oxygen and carbon atomic weights, 3/4. Therefore, there is a relation between the volume ratio of the two species and the respective surface they occupy, when fluctuation starts. The explanation lies in the optimal distribution of atoms on the converter surface, considering that the carbon monoxide molecule, before binding with the second oxygen atom, adheres to the surface through the carbon atom. Fluctuation increases its amplitude as the carbon monoxide partial pressure increases. Finally, the amplitude reaches a level that stops the fluctuation. In the third zone, above 186.651 Pa, the studied variables become stationary, that is, the converter is not operative.
To understand the superficial fractional coverages and the superficial density behaviour inside the first zone, we choose a partial pressure belong to this zone, 39.997 Pa. Figure 8 shows the evolution of these variables; note that the variables tend asymptotically towards a constant value. As the carbon monoxide partial pressure approaches the transition value between the first and the second zone, the superficial fractional coverages and the superficial density start to fluctuate, Figure 9.
superficial fractional coverages and the superficial density start to fluctuate, Figure 9.
The closer to the second zone, the greater the time needed for the process to stop (Figure 9d). Figure 10 shows the phase diagram, in which it is possible to distinguish the slow annihilation rate. Note that a very small time step is needed in the numerical algorithm, 10 −4 s, compared with the total time of several seconds, to attain sufficient accuracy. An unsuitable selection of the time step depicts an incorrect solution, such as chaotic behaviour.  The closer to the second zone, the greater the time needed for the process to stop (Figure 9d). Figure 10 shows the phase diagram, in which it is possible to distinguish the slow annihilation rate. Note that a very small time step is needed in the numerical algorithm, 10 −4 s, compared with the total time of several seconds, to attain sufficient accuracy. An unsuitable selection of the time step depicts an incorrect solution, such as chaotic behaviour.  The closer to the second zone, the greater the time needed for the process to stop (Figure 9d). Figure 10 shows the phase diagram, in which it is possible to distinguish the slow annihilation rate. Note that a very small time step is needed in the numerical algorithm, 10 −4 s, compared with the total time of several seconds, to attain sufficient accuracy. An unsuitable selection of the time step depicts an incorrect solution, such as chaotic behaviour. The same study was made for the second zone. As shown in Figure 4, for a partial pressure of 101.324 Pa inside this zone, the variables fluctuate periodically. Figure 11a shows the periodical fluctuation close to the transition zone between the second and the third zones. Of note is the small span elapsed as superficial fractional coverage of carbon monoxide values change. As the carbon monoxide partial pressure approaches the transition value between the second and the third zone, the superficial fractional coverages and the superficial density stop fluctuating, Figure 11b. Figure 12 shows the phase diagram, in which it is possible to distinguish the slow annihilation rate. Note that only a very small time step in the numerical algorithm is necessary for sufficient accuracy; hence, an unsuitable selection of the time step depicts an incorrect solution, such as chaotic behaviour. The same study was made for the second zone. As shown in Figure 4, for a partial pressure of 101.324 Pa inside this zone, the variables fluctuate periodically. Figure 11a shows the periodical fluctuation close to the transition zone between the second and the third zones. Of note is the small span elapsed as superficial fractional coverage of carbon monoxide values change. As the carbon monoxide partial pressure approaches the transition value between the second and the third zone, the superficial fractional coverages and the superficial density stop fluctuating, Figure 11b. Figure 12 shows the phase diagram, in which it is possible to distinguish the slow annihilation rate. Note that only a very small time step in the numerical algorithm is necessary for sufficient accuracy; hence, an unsuitable selection of the time step depicts an incorrect solution, such as chaotic behaviour. The comments concerning the first zone are applicable to the third zone, Figure 13. The comments concerning the first zone are applicable to the third zone, Figure 13. The comments concerning the first zone are applicable to the third zone, Figure 13.  Figure 14 shows a more detailed study, using a phase plane, of the superficial fractional coverage of carbon monoxide at its partial pressure of 140 Pa (second zone), which is clearly periodical. Figure  15, a phase space, represents a detail of the change of superficial fractional coverage on its right side. Once again, the short time step used in the numerical algorithm necessary to obtain a clear image is of note, as is used in the later figures. The single point in Figure 16, a Poincaré plane for its period (1.599 s), confirms the periodicity of the superficial fractional coverage fluctuation of carbon monoxide.   Figure 14 shows a more detailed study, using a phase plane, of the superficial fractional coverage of carbon monoxide at its partial pressure of 140 Pa (second zone), which is clearly periodical. Figure 15, a phase space, represents a detail of the change of superficial fractional coverage on its right side. Once again, the short time step used in the numerical algorithm necessary to obtain a clear image is of note, as is used in the later figures. The single point in Figure 16, a Poincaré plane for its period (1.599 s), confirms the periodicity of the superficial fractional coverage fluctuation of carbon monoxide.  Figure 14 shows a more detailed study, using a phase plane, of the superficial fractional coverage of carbon monoxide at its partial pressure of 140 Pa (second zone), which is clearly periodical. Figure  15, a phase space, represents a detail of the change of superficial fractional coverage on its right side. Once again, the short time step used in the numerical algorithm necessary to obtain a clear image is of note, as is used in the later figures. The single point in Figure 16, a Poincaré plane for its period (1.599 s), confirms the periodicity of the superficial fractional coverage fluctuation of carbon monoxide.    Figure 10.
We now analyse the boundary between zones. Figure 17 shows the relation between superficial fractional coverage of carbon monoxide and the superficial density for a carbon monoxide pressure belonging to the first zone close to the transition to the second zone. The initial conditions have been changed to depict the fact that these values have no influence. The explanation of this transition clearly involves an equilibrium point in the dynamical system, and a similar explanation is applicable to the transition between the second and third zones.   Figure 10.
We now analyse the boundary between zones. Figure 17 shows the relation between superficial fractional coverage of carbon monoxide and the superficial density for a carbon monoxide pressure belonging to the first zone close to the transition to the second zone. The initial conditions have been changed to depict the fact that these values have no influence. The explanation of this transition clearly involves an equilibrium point in the dynamical system, and a similar explanation is applicable to the transition between the second and third zones.  Figure 10.
We now analyse the boundary between zones. Figure 17 shows the relation between superficial fractional coverage of carbon monoxide and the superficial density for a carbon monoxide pressure belonging to the first zone close to the transition to the second zone. The initial conditions have been changed to depict the fact that these values have no influence. The explanation of this transition clearly involves an equilibrium point in the dynamical system, and a similar explanation is applicable to the transition between the second and third zones.

Conclusions
A clear insight into the catalytic converter function, especially as regards the different zones of carbon monoxide partial pressure, is gained by means of the proposed model in this paper whose design is based on the Network Simulation Method. The model assumes no simplifications in the governing equations and is run in a standard circuit simulation software with negligible computing time. The incorrect selection of the time step in the numerical simulation has a remarkable influence on the results, introducing an illusory chaotic behaviour when it is not small enough.
The simulations of the model also provide an explanation for the transition between the stationary and the periodical response, an issue that is partially discussed by other authors. Each transition zone, a bifurcation, brings up an additional numerical difficulty in the selection of the parameters of the computing solver and demands a special understanding of the numerical tool to avoid misapplications that give rise to a limit cycle instead of chaotic behaviour and the opposite. In addition, the start and extinction of the fluctuations, which are also relevant from the point of view of the technical control of the catalysis, are analysed to understand their relation with the carbon monoxide partial pressures, the control parameter of the system.
The methodology applied to the study of this chemical process, based on the use of the Poincaré plane and phase-diagrams, is suitable in the analysis of chaotic systems. Despite the difficulty of choosing the location (or the characteristic time) for the Poincaré plane, this analytical tool is quite effective since it reveals the nature of the system. The numerical simulations are successfully compared with experimental results from the literature.
It is worth mentioning the small time span in which the variables fluctuate near the extinction bifurcation, where the system stops working. The aforementioned two features, the time step and time span, introduced in this work are not mentioned in the available literature, which makes it possible to avoid the illusory chaotic behaviour.

Conclusions
A clear insight into the catalytic converter function, especially as regards the different zones of carbon monoxide partial pressure, is gained by means of the proposed model in this paper whose design is based on the Network Simulation Method. The model assumes no simplifications in the governing equations and is run in a standard circuit simulation software with negligible computing time. The incorrect selection of the time step in the numerical simulation has a remarkable influence on the results, introducing an illusory chaotic behaviour when it is not small enough.
The simulations of the model also provide an explanation for the transition between the stationary and the periodical response, an issue that is partially discussed by other authors. Each transition zone, a bifurcation, brings up an additional numerical difficulty in the selection of the parameters of the computing solver and demands a special understanding of the numerical tool to avoid misapplications that give rise to a limit cycle instead of chaotic behaviour and the opposite. In addition, the start and extinction of the fluctuations, which are also relevant from the point of view of the technical control of the catalysis, are analysed to understand their relation with the carbon monoxide partial pressures, the control parameter of the system.
The methodology applied to the study of this chemical process, based on the use of the Poincaré plane and phase-diagrams, is suitable in the analysis of chaotic systems. Despite the difficulty of choosing the location (or the characteristic time) for the Poincaré plane, this analytical tool is quite effective since it reveals the nature of the system. The numerical simulations are successfully compared with experimental results from the literature.
It is worth mentioning the small time span in which the variables fluctuate near the extinction bifurcation, where the system stops working. The aforementioned two features, the time step and time span, introduced in this work are not mentioned in the available literature, which makes it possible to avoid the illusory chaotic behaviour.

Conflicts of Interest:
The authors declare no conflict of interest.