A cellular automaton for the modeling of oscillations in a surface reaction

J. Chem. Phys. 121, 3206 (2004); https://doi.org/10.1063/1.1770455 121, 3206 © 2004 American Institute of Physics. A cellular automaton for the modeling of oscillations in a surface reaction Cite as: J. Chem. Phys. 121, 3206 (2004); https://doi.org/10.1063/1.1770455 Submitted: 02 March 2004 . Accepted: 14 May 2004 . Published Online: 30 July 2004 M. C. Lemos, and F. Jiménez-Morales ARTICLES YOU MAY BE INTERESTED IN Periodical forcing for the control of chaos in a chemical reaction The Journal of Chemical Physics 124, 014707 (2006); https://doi.org/10.1063/1.2141957 Reaction diffusion patterns in the catalytic CO-oxidation on Pt(110): Front propagation and spiral waves The Journal of Chemical Physics 98, 9977 (1993); https://doi.org/10.1063/1.464323 A cellular automaton for the modeling of oscillations in a surface reaction M. C. Lemos and F. Jime ´nez-Morales Departamento de Fı ´sica de la Materia Condensada, Universidad de Sevilla, P.O. Box 1065, Avenida Reina Mercedes s/n, 41080 Sevilla, Spain 共Received 2 March 2004; accepted 14 May 2004兲 The reaction of CO and O over a catalytic surface is studied with a cellular automata 共CA兲model. We extend the CA model proposed by Mai and von Niessen 关Phys. Rev. A 44 R6165 共1991兲兴 taking into account the variation of the temperature of the catalyst with the aim of analyzing the existence of oscillations in this reaction. The rate constants for different processes which govern the reaction are chosen in the Arrhenius form. Quasiperiodic, aperiodic, O-poisoned, and CO-poisoned regimes are observed depending on the temperature relaxation parameter. The results from the CA model presented are in agreement with several oscillatory behaviors which the catalyzed oxidation of CO exhibits. © 2004 American Institute of Physics. 关DOI: 10.1063/1.1770455兴 I. INTRODUCTION The oxidation of CO by O2over group VIII metal catalysts 共Pt, Pd, or Ir兲is one of the most intensely studied heterogeneous catalytic reactions.2–4 This reaction, as a good example of nonequilibrium systems, exhibits kinetic phase transitions, bistability, and associated hysteresis, and a rich variety of oscillatory behaviors, which vary from periodic to quasiperiodic and to aperiodic. Most models proposed to describe an oscillatory behavior in the catalyzed oxidation of CO involve nonlinearities of different natures. For example, models, where the reaction heats generated on the catalyst may be distributed over the surface much faster than they are dissipated in the ambient and, therefore, the temperature of the catalyst can be maintained different from its surroundings, have been considered. In these models, temperature oscillations are jointly originated with concentration oscillations. Thermokinetic oscillations arise due to the highly nonlinear dependence of the reaction rate upon the temperature. One of the first models to describe the appearance of temperature oscillations was proposed by Lagos, Sales, and Suhl.5On the basis of mean-field theory, they obtained the kinetic equations by using the singlet closure approximation and assuming that the surface is a square lattice. They formulated the kinetics of the reaction by an autonomous system of three nonlinear differential equations in terms of the CO and O coverages, and the surface temperature. Their numerical solutions are in qualitative agreement with experiments carried out by Dauchot and Van Cakenberghe.6The mechanism proposed by these experimenters to explain the appearance of thermokinetic oscillations is the following: 共a兲 Adsorption of O2is favored over that of CO by an increase in temperature of the catalyst. Thus, starting from the hot surface, O2has been mostly adsorbed. 共b兲As the surface cools down 共its surroundings are at a constant and different temperature兲, CO is progressively adsorbed and the reaction proceeds rapidly. 共c兲The CO is eliminated by the LangmuirHinshelwood 共LH兲mechanism and the hot surface resulting from that exothermic reaction is again covered primarily and rapidly with O2. Lagos et al. did not consider the role played by desorption of reactives and they neglected the temperature dependence of the LH mechanism. Only the adsorption of O2was assumed to be temperature dependent. It seems that the conditions for the occurrence of oscillations in this model5are far from the experimental conditions, because those assumptions made such model less realistic.7,8 However, the simple structure of the model of Lagos et al. is amenable to a simple bifurcation analysis. Stability analysis of the model has also revealed that oscillatory behavior is obtained only for certain values of the parameter that controls the decay of temperature fluctuations of the surface. An interesting result obtained from their5mathematical analysis is that, in the oscillatory state, the average yield of the reaction product may be greater than the yield in the corresponding steady state; in this way, efficiency of catalyst is improved. In another paper, Lagos, Simoes, and Godoy9generalize the kinetic model previously developed by them,5incorporating an experimentally observed ‘‘slow’’ suboxide kinetic step. Their results are in agreement with experimental data: stable, oscillatory, and quasiaperiodic regimes are obtained. The catalyzed oxidation reaction of CO was also studied by Ziff, Gulari, and Barshad10 共ZGB model兲. These authors studied the reaction via Monte Carlo simulations on a square lattice and found three zones with regard to the fraction of CO molecules in the gas phase, yCO . For low values of yCO , yCO⬍y1⫽0.389, the CO pressure is low compared to the O2 pressure and, as a consequence, the catalyst is poisoned by O2. For y1⬍yCO⬍y2⫽0.525, the CO adsorption competes with oxygen adsorption, and the CO2production increases with CO pressure. Therefore, there is a reactive steady state where CO2molecules are continuously produced. The transition at y1is a second-order 共continuous兲kinetic phase transition. For yCO⬎y2the O2pressure is low compared to the CO pressure, and the catalyst is poisoned by CO. The kinetic phase transition at y2is of first order, because the CO2production rate changes discontinuously. Luque11 studied the CO oxidation reaction on the basis JOURNAL OF CHEMICAL PHYSICS VOLUME 121, NUMBER 7 15 AUGUST 2004 32060021-9606/2004/121(7)/3206/6/$22.00 © 2004 American Institute of Physics of ZGB model. He derived the kinetic equations by starting from a master equation, following the generalized GlauberIsing model. Since the distribution of adsorbed particles on the surface can affect the transition probabilities in the adsorption and desorption processes, the spatial correlations were considered, and larger clusters that involved up to eight sites were introduced. A set of five nonlinear kinetic equations was obtained for the densities of CO(nCO) and atoms O(nO), and pairs CO-CO(nCO-CO), CO-O(nCO-O), and O-O(nO-O) considering a doublet closure approximation. In addition to the three zones obtained by Ziff, Gulari, and Barshad, a bistability zone was found: one reactive stable state producing CO2and another in which the catalyst is poisoned by CO; depending on initial conditions, the reaction will evolve towards one of the stable stationary states. On the basis of the results obtained by Lagos, Sales, and Suhl5and in order to study the oscillatory behavior of the reaction, Luque’s model11 was extended by us by adding an equation for the rate of change of the surface temperature.12 The results demonstrate that the nonlinearity introduced in the model gives rise to multiple stable stationary states and oscillations for certain values of the temperature relaxation parameter and other operating condition parameter. In the bistability zone, one reactive state producing CO2and another in which the catalyst is poisoned by CO are found. The new reactive state can be either a stable node or an oscillatory state, and appears only for certain values of the temperature relaxation parameter. Thus, as in other models, this parameter plays an important role in the onset of oscillations. In addition to the use of mean-field theory and Monte Carlo 共MC兲simulation for the study and modeling of heterogeneous catalytic systems, recently the technique of cellular automata13 共CA兲is another possibility to simulate the oscillatory behavior of the oxidation of CO on metallic surfaces with the advantage that the CA approach, faster than MC simulations, may become equivalent to the latter. A cellular automaton consists of a discrete number of cells or sites which evolve in discrete time steps following the same deterministic 共or probabilistic兲rules. These simple transition rules can be easily programmed. In the course of the temporary evolution each cell can only adapt values from a finite set. The rules for the evolution of a cell only depend on the state of the cells in the local neighborhood. At each time step the value of each site is updated. This feature allows parallel processing and results a substantial reduction in the computational time. Physical, chemical, and biological systems with many discrete elements and with local interactions can be modeled using CA. A survey of recent literature indicates that the study of surface chemical reactions with CA is possible. So, there is a study of the ZGB model by Mai and von Niessen.1They introduce a CA approach which describes the steps of adsorption and reaction correctly and obtain a good quantitative agreement with the ZGB model. Later, Tambe, Jayaraman, and Kulkarni,14 following Mai–von Niessen CA model, formulate CA transition rules for the catalytic oxidation of CO incorporating the Eley-Rideal 共ER兲step, that is, the reaction between CO in the gas phase with the atomic adsorbed oxygen. In this paper, we extend the CA proposed by Mai and von Niessen1in order to analyze the existence of thermochemical oscillations in the case of CO oxidation. In this way, we can compare the results obtained by us in a previous article12 with the ones computed here via CA simulations. In the following section, the cellular automaton model of Mai and von Niessen is extended by adding an equation for the rate of change of the surface temperature. The results of our simulations are described in Sec. III and discussed in the light of mean-field solutions obtained in a previous paper. The conclusions and future developments of CA model presented here are summarized in the last section. II. THE MODEL The surface of the catalyst is represented as a twodimensional square lattice with periodic boundary conditions. The assumption is that the molecules of carbon monoxide CO共ads兲and the oxygen atoms O共ads兲both occupy single active sites on the lattice, where ‘‘ads’’ indicates that the species is adsorbed on the surface. The carbon monoxide and oxygen molecules, CO共gas兲and O2(gas), collide randomly on the surface and when there is an available site they are adsorbed. Thus, gaseous CO共gas兲and O2(gas) can competitively be adsorbed on the same active sites. Two adjacent sites are needed for the oxygen molecule O2(gas) to be adsorbed. Upon adsorption, the O2(gas) dissociates into two O atoms, each one residing on a separate surface site. When a molecule is adsorbed on the surface, it becomes fixed, so that superficial diffusion is ignored in this model. If a CO共ads兲 molecule is a neighbor of an O共ads兲atom, the reaction occurs and the product molecule CO2is desorbed as soon as it is formed. The model assumes that the reaction takes place according to three basic steps: CO共gas)→CO共ads), 共1兲 O2共gas兲→2O共ads), 共2兲 CO共ads)⫹O共ads)→CO2共gas兲.共3兲 The reaction between dissociatively adsorbed oxygen and adsorbed CO, process 共3兲, is known as the LH mechanism. The rate constants ki, for each one of three kinetic mechanisms named above (i⫽1, adsorption of CO; i⫽2, adsorption of O2;i⫽3, desorption of CO2), are chosen in the Arrhenius form: ki⫽Aiexp 冉 ⫺Ei kBT 冊 , where Aiare frequency factors, kBis the Boltzmann constant, Tdenotes the temperature of the surface, and Eiare activation energies of each one of the three elemental processes. These energies correspond to the action of the substrate and are assumed to be coverage independent. With the aim of studying the oscillatory behavior of the reaction, the model must be supplemented by the addition of an equation describing the variation of the surface temperature due to the adsorption and reaction processes: CdT dt ⫽⫺L共T⫺TB兲⫹兺 i⫽1 3 ⌬Hipi, 3207J. Chem. Phys., Vol. 121, No. 7, 15 August 2004 A cellular automaton for the modeling of oscillations where Cis the heat capacity of the catalyst, Lis the heat transfer coefficient 共Lis approximately equal to thermal conductivity multiplied by a geometric factor兲,TBis the room temperature, ⌬Hiare the reaction heats of the processes 共1兲– 共3兲, including also a suitable geometric factor, and piare the number of processes of type iwhich take place per unit area. The previous equation can be written as dT dt ⫽⫺ ␥ 共T⫺TB兲⫹兺 i⫽1 3 hipi,共4兲 where ␥ ⫽L/Cis the dissipative factor or the relaxation rate of Tto TBand hi⫽⌬Hi/C. Thus, the linear term ⫺ ␥ (T ⫺TB) describes the dissipation heat from the catalyst surface towards its surroundings, while 兺i⫽1 3hipiis the net heat generated on the catalyst surface in the reaction steps. The model can give rise to oscillatory behavior for certain values of the set of parameters 兵 Ai,Ei,hi, ␥ 其 , room temperature being at TB⫽300K. The kinetics of the reaction is described in terms of the density of adsorbed particles of CO and O, nCO and nO, the production of CO2,R, that is, the density of pairs CO-O which are formed and leave the surface immediately after its formation, and, finally, the surface temperature T, which differs from the temperature of the gas. Let us emphasize the difference between the ambient temperature TB, which is essentially constant in time, and the surface temperature Tof the catalyst. The model involves the surface temperature as a dynamical variable. The variation of this surface temperature, being subject to Eq. 共4兲, can lead to thermal instabilities because the dependence of the different reaction steps 共1兲–共3兲upon the temperature at Arrhenius law is the main feedback mechanism generating oscillations. We follow Mai and von Niessen in the cellular automaton procedure. The CA uses a square lattice with N⫽L⫻Lcells. To overcome problems with the stoichiometry of the chemical reaction, the lattice is divided into Margolus blocks containing 2⫻2 cells as elementary cells. All the cells in the Margolus block are simultaneously updated following a probabilistic rule. After one sweep through the lattice, the Margolus block is shifted by one cell to the right, then down, then left, and then up to get all the different configurations. In the Mai–von Niessen model the transition probabilities of the CA depend on the initial state of the cells, on the statistical weights of the individual configurations, which are taken as classical, and on the mole fractions of the gaseous reactants, yCO and yO, where yO⫽1⫺yCO . CA transition rules are shown in Fig. 1. For each initial state, the sum of all transition probabilities is normalized to 1. In our model, yCO is defined as the relative rate of potentially efficacious collisions of CO molecules on the surface; that is, yCO⫽k1/(k1 ⫹k2), meanwhile yO⫽k2/(k1⫹k2). According to the ZGB model, the probability of reaction between pairs CO-O, process 共3兲, is assumed to be unity. III. RESULTS Initially the problem was complicated, since describing the reaction requires many parameters. In this paper, we want to compare the cellular automata simulations with the results previously analyzed by us, starting from the kinetic equations obtained by using a mean-field approximation type. According to experimental data, it seems that oscillations occur under the conditions where oxygen adsorption is the main mechanism. Therefore, we assume that the adsorption of CO and the reaction between O and CO on the surface are unactivated processes, E1⫽0, E3⫽0; the adsorption of O2is the only activated process. We make a minor simplification considering that the reaction heat for the adsorption of O is twice the value of that of the CO and the reaction heat for CO2is negligible, and so h2⫽2h1and h3⫽0. We have fixed the values A1⫽0.05, A2⫽4⫻105,A3⫽1, E1 ⫽0, E2/kB⫽6000, E3⫽0, h1⫽150, h2⫽300, and h3⫽0共in arbitrary units兲, which have been used in Ref. 12, where the temperature relaxation parameter ␥ is the control parameter. FIG. 1. Cellular automata transition rules for the CO⫹O2reaction. Adenotes a CO molecule, Ban oxygen atom, and 夝any occupied site. The transition probabilities of CA depend on the initial state of the cells, on the statistical weights of the individual configurations, which are taken as classical, and on the mole fractions of the gaseous reactants, yCO and yO, where yO⫽1⫺yCO . In the model, yCO is defined as yCO⫽k1/(k1⫹k2), where ki ⫽Aiexp(⫺Ei/kBT). For each initial state, the sum of all transition probabilities is normalized to 1. 3208 J. Chem. Phys., Vol. 121, No. 7, 15 August 2004 M. C. Lemos and F. Jime ´nez-Morales The simulation was started with an empty lattice of 256 ⫻256 sites with periodic boundary conditions. In most cases the regimes have been reached after 2000 time steps. However, for the transitions from aperiodic regime to quasiperiodic behavior and from this to poisoning of the surface by CO, 12000 and 6000 time steps were necessary, respectively. To assure the validity of our simulations we have used four different series of random numbers in each simulation, yielding essentially the same results. For each value of ␥ , the simulation consisted of 2000 iterations over the entire lattice. After one sweep through the lattice, the Margolus block is shifted by one cell to the right, then down, then left, and finally up. Therefore, an iteration 共time step兲consists of four sweeps over the complete lattice. After each iteration, the number of adsorption processes of CO and O, and the number of Langmuir-Hinshelwood events were counted and the surface temperature was modified according to Eq. 共4兲. The data output show the temperature of the catalyst surface, T, the coverages fractions of CO and O, nCO and nO, and the number of pairs CO-O which are formed per unit area, R, in each iteration over the entire lattice. Oscillating regimes are obtained in the transition region where the surface changes from a complete coverage by oxygen to a complete coverage by CO molecules. The poisoning of the surface of the catalyst by CO or O occurs at room temperature, TB⫽300K. Depending on the temperature relaxation parameter ␥ , we obtain a region where the catalyst is not poisoned: 0.005⭐ ␥ ⭐0.27. In this range there is either an aperiodic regime between the limits 0.005⭐ ␥ ⭐0.09 or a quasiperiodic behavior for 0.10⭐ ␥ ⭐0.27. In both cases, the surface temperature fluctuates around a value greater than ambient temperature, favoring in this way the adsorption of O2and decreasing the adsorption of CO. This fact shows that the CO2production is favored. On the other hand, for 0⬍ ␥ ⭐0.004, the surface is quickly poisoned by atoms of O at room temperature and, therefore, the reaction between CO molecules and O atoms is not possible. The same situation happens for ␥ ⭓0.28, where the surface is covered slowly with CO, thus producing poisoning of the catalyst by this reactant. Our CA model is able to describe a richer variety of behaviors 共quasiperiodic, aperiodic, and two poisoned states兲 than within the mean-field approximation. There, and for all values of ␥ , there was a state corresponding to complete coverage with CO at room temperature, which is a stable node. Besides this state, a new solution was obtained via mean-field approximation when the values of ␥ varied. This new state can be either a periodic oscillatory state in the range 0.10⭐ ␥ ⭐0.15 共the temperature fluctuates around a greater value than ambient兲or a stable node for 0.03⭐ ␥ ⭐0.09, corresponding to a reactive steady state producing CO2at a greater temperature than room temperature. Quasiperiodic, aperiodic, and O-poisoned states were never obtained by mean-field approximation. To study qualitatively the system behavior, we use the Poincare ´’s map, also known as the iterative map. If the behavior is periodic with a single frequency, the Poincare ´’s map will show a single point. On the other hand, if the behavior is aperiodic, the points will appear on the map in a scattered way, not following any recognized line and filling up the iterative map. Finally, if the behavior is quasiperiodic, the points densely populate an invariant closed curve. Figure 2 shows the time series of T,nCO ,nO, and Rfor two values of ␥ . It is also shown the corresponding iterative map for each variable. For ␥ ⫽0.20, the Poincare ´’s map shows a quasiperiodic behavior: the points are put on a same closed curve perfectly defined. However, for ␥ ⫽0.01, the aperiodic behavior can be observed. Quasiperiodic behavior of the oscillations of the catalyst temperature, coverage fractions of CO and O, and CO2production for ␥ ⫽0.20 are shown in Fig. 3. Different frequencies of the quasiperiodic regime can be observed. Between 0.10⭐ ␥ ⭐0.27, corresponding to quasiperiodicity, the oscillation amplitude and the size of the iterative map increase as ␥ increases. This is shown in Fig. 4, where it can be seen that the average production of CO2is favored as ␥ increases. At ␥ ⫽0.28, the quasiperiodic regime disappears and the surface remains poisoned by CO at TB⫽300K. Finally, Fig. 5 shows the four regimes which are found in the CA model. For ␥ ⫽0.004, the catalyst is saturated by adsorbed oxygen after a transient time. The state obtained for ␥ ⫽0.07 is an aperiodic state where the temperature of the surface fluctuates around a greater value than room temperature and a small production of CO2is obtained due to that the surface is not saturated by any of the reactive species. For ␥ ⫽0.15, the oscillatory behavior is quasiperiodic and the amplitude of the oscillations is greater than the previous state 共 ␥ ⫽0.07兲, thus increasing the averages of the densities of CO and O, the yield of CO2, and the surface temperature. For ␥ ⫽0.28, the surface is covered slowly with CO, thus producing poisoning of the catalyst by this reactant. It can be also observed that the amplitude of the oscillations starts to decrease slowly until the catalyst surface is poisoned suddenly by CO molecules at ambient temperature. IV. CONCLUSIONS In short, we have presented a CA model, extension from the Mai–von Niessen model,1that includes the temperature as a dynamic variable with the purpose to model the oscillatory behavior of reaction of CO. The nonlinear dependence of the rate constants on the temperature given by the Arrhenius law introduced in the CA model gives rise to quasiperiodic and aperiodic behaviors for certain values of the temperature relaxation parameter. As in others models, this parameter plays an important role in the onset of oscillatory regimes. We think that the flexibility of the CA model presented here allows us to continue studying this surface’s chemical reaction. Future works will analyze the effects of the addition of the Eley-Rideal step, the inclusion of defects on the surface, and the Langmuir-Hinshelwood as an activated process, among others topics. 3209J. Chem. Phys., Vol. 121, No. 7, 15 August 2004 A cellular automaton for the modeling of oscillations FIG. 2. Time series of the surface temperature, T共in Kelvin兲, dimensionless coverage fractions of CO and O, nCO and nO, and the production of CO2,R 关defined as the number of process of type 共3兲which take place per unit area兴, for two values of the control parameter ␥ . The time series of the CA simulations show aperiodic and quasiperiodic regimes. 共a兲For ␥ ⫽0.01, the regime is aperiodic as can be observed in ‘‘filled space’’ from the Poincare ´’s maps. 共b兲For ␥ ⫽0.20, the Poincare ´’s maps show a quasiperiodic behavior. FIG. 3. Temporal behavior of the surface temperature 共in kelvins兲, densities of CO and O, and production of CO2for ␥ ⫽0.20. This state corresponds to the region where the system behavior is quasiperiodic. FIG. 4. Development of the oscillations of the reaction product CO2and the associated Poincare ´’s maps for different values of ␥ . As can be seen, the oscillation amplitude and the size of the Poincare ´’s map increase as ␥ increases, thus favoring the average yield of CO2. 3210 J. Chem. Phys., Vol. 121, No. 7, 15 August 2004 M. C. Lemos and F. Jime ´nez-Morales ACKNOWLEDGMENTS This work was partially financed by the Spanish Government 共Project No. BFM2003-03986/FISI兲and the European Fund for Regional Development. 1J. Mai and W. von Niessen, Phys. Rev. A 44, R6165 共1991兲. 2L. F. Razon and R. A. Schmitz, Catal. Rev. - Sci. Eng. 28,89共1986兲. 3S. K. Scott, Chemical Chaos 共Clarendon Press, Oxford, 1991兲. 4M. M. Slin’ko and N. I. Jaeger, in Oscillating Heterogeneous Catalytic Systems, Studies in Surface Science and Catalysis Vol. 86, edited by B. Delmon and J. T. Yates 共Elsevier, Amsterdam, 1994兲. 5R. E. Lagos, B. C. Sales, and H. Suhl, Surf. Sci. 82, 525 共1979兲. 6J. P. Dauchot and J. Van Cakenberghe, Nature 共London兲, Phys. Sci. 246, 61 共1973兲; Jpn. J. Appl. Phys., Suppl. 2, 533 共1974兲. 7R. Dagonnier, M. Dumont, and J. Nuyts, Surf. Sci. 95, L217 共1980兲. 8R. E. Lagos, B. C. Sales, and H. Suhl, Surf. Sci. 95, L223 共1980兲. 9R. E. Lagos, T. Simoes, and A. L. Godoy, Physica A 257,401共1998兲. 10R. M. Ziff, E. Gulari, and Y. Barshad, Phys. Rev. Lett. 56,2553共1986兲. 11 J. J. Luque, Phys. Rev. A 42, 3319 共1990兲. 12M. C. Lemos, J. J. Luque, and F. Jime ´nez-Morales, Phys. Rev. E 51, 5360 共1995兲. 13T. Toffoli and N. Margolus, Cellular Automata Machines 共MIT, Moscow, 1987兲. 14S. S. Tambe, V. K. Jayaraman, and B. D. Kulkarni, Chem. Phys. Lett. 225, 303 共1994兲. FIG. 5. Results obtained from cellular automaton simulations as a function of the control parameter ␥ . The graphs show the temporal behavior of the system. 共a兲For ␥ ⫽0.004, the surface is poisoned quickly by atoms of O. For ␥ ⫽0.07, there is an aperiodic state. 共b兲For ␥ ⫽0.15, the behavior of the system is quasiperiodic and, finally, for ␥ ⫽0.28, the surface is poisoned by CO. In the poisoned states, the catalyst surface remains at room temperature, TB ⫽300 K, while in the oscillatory regimes the surface temperature fluctuates around a temperature greater than TB. 3211J. Chem. Phys., Vol. 121, No. 7, 15 August 2004 A cellular automaton for the modeling of oscillations