Thermokinetic Origin of Luminescent Traveling Fronts in the H2O2

Article pubs.acs.org/JPCA

Thermokinetic Origin of Luminescent Traveling Fronts in the H2O2− NaOH−SCN−−Cu2+ Homogeneous Oscillator: Experiments and Model Albin Wiśniewski, Maciej T. Gorzkowski, Katarzyna Pekala, and Marek Orlik* Laboratory of Electroanalytical Chemistry, Faculty of Chemistry, University of Warsaw, Ul. Pasteura 1, 02-093 Warsaw, Poland S Supporting Information *

ABSTRACT: According to our original discovery, the oscillatory course of the Cu2+catalyzed oxidation of thiocyanate ions with hydrogen peroxide, in nonstirred medium and upon the addition of luminol as an indicator, can be a source of a novel type of dissipative patterns − luminescent traveling waves. The formation of these fronts, contrary to the patterns associated with the Belousov−Zhabotinsky reaction, cannot be explained in terms of coupled homogeneous kinetics and diffusion, and under isothermal conditions. Both experimental studies and numerical simulations of the kinetic mechanism suggest that the spatial progress of these waves requires mainly the temperature gradient in the solution, which affects the local chemical reaction rate (and thus the oscillation period), with practically negligible contribution from diffusion of reagents. As a consequence of this thermokinetic coupling, the observed traveling patterns are thus essentially the phase (or kinematic) waves, formed due to the spatial phase shift of the oscillations caused by differences in chemical reaction rates. The temperature gradient, caused by the significant heat effect of exothermic oxidation of thiocyanate by hydrogen peroxide, can emerge spontaneously as a local fluctuation or can be forced externally, if the control of progress of the luminescent waves is to be achieved.

1. INTRODUCTION Spatiotemporal patterns emerging in dissipative processes remain a subject of intensive studies as the possible models of various kinds of dynamic self-organization in both nonliving and living matter. Historically, the best known and most impressive example is the formation of concentric (as well as spiral) concentration waves in the Belousov−Zhabotinsky (BZ) reaction,1 while later discovered systems include, among others, chlorite−iodide−malonic acid (CIMA) oscillator,2 the ferrocyanide−iodate−sulfite system (FIS),3,4 the thiourea−iodate− sulfite (TuIS) reaction,5,6 the urea−urease reaction, with pH wave fronts,7 as well as hydrogen peroxide−sulfite systems producing sustained pH patterns. 8 Depending on the mechanisms of their formation, spatiotemporal patterns in aqueous solutions are usually classified as the trigger or the phase (or the kinematic) waves,9 although also other wave classifications have been proposed.10 Concentric or spiral patterns reported for BZ reaction in the thin-layer media are considered the trigger waves, in which excitable system, able to switch into oscillatory regime, is coupled with diffusion of intermediate species HBrO2, involved in an autocatalytic process.11 For the other group, different parts of the spatially extended system differ with the oscillation phase. The phase difference may occur for the same oscillation period or can be caused by its spatial inhomogeneity, due to different rates of oscillatory chemical reaction. The common feature of all types of phase waves is that they do not need mass transfer to occur. In presumably all reported cases, where the reactions occurred in aqueous media, the system could be assumed practically © 2013 American Chemical Society

isothermal, i.e., the temperature gradients were considered not essential for the formation of patterns, although they could contribute to their detailed morphology, either by changing local chemical kinetics or by causing local convection driven by induced density gradients in the gravitational field.12 In fact, the BZ reaction is a relatively weakly exothermic process, not even to compare with combustion process in gas phase, for which thermokinetic oscillations were described.13 Also, thermokinetic models were applied by Eiswirth and Ertl14 for explanation of pattern formation on catalytic surfaces. It seems that for the oscillatory processes in aqueous media the primary role of temperature gradients in the origin of chemical waves was not suggested so far. However, when the oscillatory process in aqueous media is significantly more exothermic than the BZ process, thermal gradients can be considered a potentially important contribution to the emergence of spatiotemporal instabilities. In one of our previous works,15 we proved that the Cu2+-catalyzed oscillatory oxidation of SCN− with H2O2, originally discovered and studied by Orbán16,17 et al. (but only under isothermal conditions of 25 ± 0.05 °C), characterizes with quite significant heat evolution, with the rate of temperature rise measured as 8 K during 10 min, i.e., for ca. 1 order of magnitude higher than for the BZ process (ca. 1.5 K within 10 min), for typical reactants concentrations. Quantitative interpretation of reReceived: August 30, 2013 Revised: October 8, 2013 Published: October 10, 2013 11155

dx.doi.org/10.1021/jp408713u | J. Phys. Chem. A 2013, 117, 11155−11166

The Journal of Physical Chemistry A

Article

surrounding air, while the graphite block, for enhanced thermostatic effect, was kept in contact with external water bath of constant temperature 22 °C. The thin layer of the solution introduced to this gouge was further covered by a fitted Plexiglas block, a part of which was preheated by hot water in order to transfer the corresponding temperature gradient to the solution. The studied solution was prepared ex tempore from stock solutions so that the initial concentrations of the reactants were: 0.051 M NaSCN, 0.045 M NaOH, and 0.34 M H2O2, with 5.5 × 10−4 M or 1.5 × 10−3 M luminol as a luminescent indicator. Next, two operations were performed practically simultaneously: (i) 0.048 mL of 0.01 M CuSO4 catalyst was injected to 1.95 mL of the studied solution, which was immediately stirred manually to homogenize the sample, in which the concentration of CuSO4 reached then 2.4 × 10−4 M; (ii) the Plexiglas cover was partially immersed into hot (90 °C) water in a separate container for 1 min, then removed and wiped clean. The studied solution was then placed in the Teflon or graphite reactor hole and covered by Plexiglas block, so that the upper layer of the solution touched its surface of different temperatures. The application of Teflon (DuPont) was due to both its chemical durability and a low thermal conductivity coefficient (0.25 W m−1 K−1; based on the information from the manufacturer), the latter property ensuring that the temperature gradients imposed and developing in the thin-layer of the solution would be not too quickly smoothed by the reactor body. In turn, the graphite body was expected to cause such an equilibration effect, due to much higher thermal conductivity coefficient: 120 W m−1 K−1 (based on the original manufacturer information). One should note that the best conducting metals, like Ag or Cu, were found unsuitable as the materials for the construction of the reactor body, due to their rapid interaction with the components of the oxidizing species involved in the studied process (as proved by visible changes of the color of their surfaces). The development of appearing luminescent spatiotemporal patterns was monitored in a dark room by taking the photographs (with the resolution of 5 megapixels), every 15 s, using Nikon D70 digital reflex camera fixed over the transparent Plexiglas cover (cf. Figure 1). For better visualization of the patterns, the images obtained for lower concentration of luminol were enhanced using commercial ACDSee 15 software (ACD Systems International Inc.). All pictures were modified in the same way, involving the appropriate adjustment of their histograms of exposition. Images obtained for higher concentration of luminol did not require that procedure. For comparison, many additional experiments were performed with solutions placed in vertically and horizontally oriented tubes of various diameters, heated locally from outside, in order to observe or to exclude the eventual effect of convection in the gravitational field. Although in vertical tubes of diameter greater than ca. 5 mm luminescent flows indicating buoyancy-driven convective flows could be observed, we did not observe any flow of such type in our thin-layer system, neither in the bulk nor on the surface, the motion of which was hindered by its contact with the Plexiglas cover. Therefore, we are convinced that the patterns described in this article are of nonconvective origin.

ported temperature variations allowed us further to assess the enthalpic effects of some crucial reaction steps.15 These observations and conclusions appear now to be essential for the explanation of the nature of, discovered also by us,18 luminescent waves in the thin-layer Orbán’s system, enriched with luminol as an indicator.19−21 In consequence, in the present article we report new experimental observations of luminescent traveling fronts in the H2O2−SCN−−OH−−Cu2+ system and propose the mechanism for their generation, proved by appropriate numerical modeling. In this analysis we also used our simplified kinetic mechanism22 of the process, which was earlier successfully applied to the explanation of the nontrivial, dependent on the indicator electrode material, potentiometric characteristics of this system.23 The elaborated by us mechanism of generation of chemical waves in H2O2−SCN−−OH−−Cu2+ system indicates the crucial role of temperature gradients in the solution, either formed spontaneously or imposed externally.

2. EXPERIMENTAL SECTION All reagents: pure for analysis (p.a.) NaSCN (SIGMA), p.a. NaOH (POCh, Poland), p.a. CuSO4·5H2O (Chempur), and p.a. 30% H2O2 (Chempur) were used without further purification. The formation and progress of luminescent fronts were observed in a horizontal thin-layer reactor depicted in Figure 1. The body of this reactor was made either from Teflon PTFE (DuPont) or Toyo Tanso Isotropic Graphite IG-11. In these rectangular prisms the quasi-rectangular (70 × 20 mm) and 1 mm deep gouge was carved as the reactor space. During the measurements, the Teflon block remained in contact with

Figure 1. Schematic view of the experimental setup with a horizontal thin-layer reactor for monitoring the luminescent chemical waves initiated by temperature gradient: (1) cuboidal Teflon block with a 1 mm deep gouge as the reaction space, (2) transparent Plexiglas cover as a source of temperature gradient transformed to the adjacent solution in a gouge. In other experiments, instead of Teflon, the graphite block of approximately 5 times larger height and remaining in contact with an external water bath (22 °C) was used. 11156



Article

Figure 2. Random birth and spatiotemporal evolution of luminescent waves in the thin-layer rectangular Teflon reactor from Figure 1. The layer of initially homogeneous solution with the oscillatory H2O2−NaSCN−NaOH−CuSO4 system + 5.5 × 10−4 M luminol was covered by transparent, isothermal Plexiglas block and photographed from the top every 15 s. Numbers at every image indicate the time (min:sec) measured from the addition of CuSO4 catalyst to the sample. Images with very weak, practically homogeneous luminescence, reported between the luminescent patterns are not shown. See Supporting Information for the time lapse animation showing the temporal evolution of those patterns.

3. RESULTS OF EXPERIMENTAL MEASUREMENTS The main experimental problem in the studies of formation of luminescent waves in the H2O2−SCN−−OH−−Cu2+ system was how to gain control over the places in which zones of strongest luminescence are being formed. Such a control would allow to obtain possibly reproducible generation of luminescent fronts in the investigated system and constitute a solid premise for the mechanism of its origin. As our preliminary experiments showed,18 in the thin-layer of a quiescent solution, the luminescent zones appeared initially in random locations (cf. Figures 1 and 3 in ref 18). However, it should also be noted that the initial random pattern was later repeating periodically several times (cf. again Figure 1 in ref 18), indicating that the randomly emerging patterns underwent further stabilization in a way resembling the principle of phase waves: the oscillatory system imitates the spatial progress of the wave as a result of self-induced and self-stabilized spatial gradient in the oscillation phase. It is thus clear that the mechanism of formation of luminescent spatiotemporal patterns in the Orbán’s system required further systematic studies, focused first on experimental search for the factors causing the inhomogeneous

distribution of luminescence in the permanently oscillating process, when the minima of the platinum electrode potential, used for monitoring of the oscillations, correlate with the bursts of maximum intensity of luminescence.18 On the basis of the established mechanism24,25 of the generation of this luminescence as caused by oxidation of luminol by H2O2 in the presence of catalytic Cu2+ ions, and in view of our recently found correlations of potentiometric response of the studied system with the concentration [[Cu(OH)3]−]/[[Cu(OH)2]−] ratio, it is reasonable to assume that the observed, both temporal and spatiotemporal variations of luminescence, are at least roughly proportional to the variations of the [Cu(OH)3]− concentration during the spontaneous oscillatory course of the reaction. Because of this involvement of Cu(II) ions in the intensity of luminescence, it was obvious to check if the local injection of an additional portion of a catalyst (CuSO4) solution (though limited by formation of a brown precipitate) could trigger the luminescent zone. However, our experiments showed that the increase in local CuSO4 concentration can only enhance the local luminescence, which did not spread along the reactor. Of particular importance is our final conclusion, drawn from 11157



Article

Figure 3. Programmed birth and spatiotemporal evolution of luminescent waves from the upper top of the thin-layer Teflon rectangular reactor from Figure 1. The layer of initially homogeneous solution with the oscillatory H2O2−NaSCN−NaOH−CuSO4 system + 5.5 × 10−4 M luminol was covered by transparent Plexiglas block, preheated along its upper long edge, and photographed from the top every 15 s. Numbers at every image indicate the time (min:sec) measured from the addition of CuSO4 catalyst to the sample. Intermediate images with practically homogeneous luminescence distribution are not shown. The slow-down of subsequent luminescent fronts is reported. See Supporting Information for the time lapse animation showing the temporal evolution of those patterns.

numerous experiments, that the only operation that allows us to control the place from which the zone of enhanced luminescence starts and proceeds along the reaction space, is the local external heating. This also immediately suggests the eventual role of self-heating of the solution due to a course of relatively exothermic oxidation of thiocyanate ions with hydrogen peroxide. Below we collect the results of representative experiments in the absence and in the presence of external temperature gradients imposed in different sections of a thin-layer reactor. Figures 2−4 show results obtained for the Teflon reactor from Figure 1. For the case when the solution was initially homogeneous and no external temperature gradient was imposed (i.e., the solution was covered by Plexiglas cover of homogeneous temperature, Figure 2), the formation of spatiotemporal patterns occurred in random places, in clear analogy to our preliminary experiments in circular reactor.18 However, if the upper (Figure 3) or the left (Figure 4) zone of the solution was brought into contact with the Plexiglas cover preheated in respective places, the luminescent wave definitely started to develop from the heated-up areas. These results prove unambiguously that external heating is not decisive for

the formation of the waves, but facilitates their emergence in designed locations. Because of the expected role of temperature distribution in the propagation of luminescent waves, we made comparative experiments with the reactor body made from graphite, as also inert material, but of much better heat conductivity than Teflon, so the temperature gradients should be smoothened. As mentioned above, other well conducting materials, as e.g., silver, appeared inappropriate due to visible surface oxidation by combined effect of hydrogen peroxide and Ag2S formation. Results of experiments showed that in the case of graphite reactor, the luminescent patterns (requiring also higher concentration of luminol for better visualization) were not only much less pronounced, but their spatial evolution was very similar (i.e., random) for both initially isothermal conditions and preheated upper edge of the solution. Finally all inhomogeneities in the luminescence distribution vanished (cf. Figure 5). This means that the contact with graphite surface indeed facilitates the heat diffusion along the thin layer of the solution with the oscillatory process running, and our control on the birth and further progress of the luminescent wave is now largely lost. 11158



Article

Figure 4. Programmed birth and spatiotemporal evolution of luminescent waves from the left edge of the thin-layer rectangular Teflon reactor from Figure 1. The layer of initially homogeneous solution with the oscillatory H2O2−NaSCN−NaOH−CuSO4 system + 5.5 × 10−4 M luminol was covered by transparent Plexiglas block preheated at its left end and photographed from the top every 15 s. Numbers at every image indicate the time (min:sec) measured from the addition of CuSO4 catalyst to the sample. Intermediate images with no detectable development of luminescent spatiotemporal patterns are not shown. See Supporting Information for the time lapse animation showing the temporal evolution of those patterns.

Preliminary Interpretation of the Experimental Results. The first point that should be considered is whether the presence of luminol brings any distortion to the oscillatory kinetics in the studied system. Theoretically, such a possibility cannot be excluded since the oxidation of luminol, being a source of luminescence, engages not only dissolved oxygen

The Supporting Information available online includes the pictures from Figures 2−5 as the time lapse animations that show the evolution of those patterns in time scale, accelerated for the factor of 30, compared to real flow of time in the experiment. 11159



Article

(which itself, as a final, irreversibly forming product, appears to be rather not important for the oscillatory regime22), but also copper(II) species that is, in turn, crucial for that behavior as the component of the catalytic redox couple. However, our experiments with stirred batch reactors did not reveal noticeable effect of the presence of luminol on the period of the oscillations. The eventual additional local heat effect, coming from the oxidation of luminol, is difficult to assess. In the following analysis, we shall assume that luminol is a practically inert indicator of both temporal and spatial instabilities in the studied Orbán’s system. In view of our experimental results, the diagnosis of the origin of the reported luminescent waves requires further advanced mechanistic considerations which should include the role of temperature gradients. Unfortunately, simultaneous, noninvasive monitoring of the temperature distribution of the solution by infrared camera, as described in our previous paper,15 was now not possible since the solution surface remained always covered with the 1 cm-thick Plexiglas block, and the camera would then sense only the temperature of its external surface. Noteworthy, although the dipping of a hot wire into the solution with the Belousov−Zhabotinsky process also can trigger the wave in this zone, the temperature gradients are not necessary for that, and the waves can be induced also under isothermal conditions (typically upon local addition of AgNO3). Furthermore, the dynamics of the Belousov− Zhabotinsky reaction is in mechanistic details substantially different from our H2O2-based system: for BZ, the oscillations involve switching between the two quasi-steady states, and the BZ system is excitable, which characteristics were not reported for the presently studied Orbán’s system, both in original reports16,17 and our analysis of the simplified kinetic model.22 This means that our knowledge about the BZ system is not helpful for the explanation of the Orban’s system spatiotemporal dynamics, for which the new kinetic model with incorporated heat effects should be employed, in view also of the relatively high exothermicity of the studied oscillatory process.15 Of course, detailed analysis of all assumed effects for the process of such a complicated mechanism is rather difficult, the more so that temperature changes affect all kinetic and thermodynamic parameters of the system. Therefore, we limited our detailed analysis to the processes and effects which are, in our opinion, crucial for the observed wave phenomena. On the basis of such assumptions, we built-up an appropriate nonisothermal reaction-diffusion model and performed numerical calculations of the evolution of concentration profiles of kinetically important intermediates in time and space. The construction of the model and representative results of such calculations are presented in the next section.

4. NUMERICAL CALCULATIONS 4.1. Construction of the Numerical Model. In the detailed construction of the numerical procedure, we employed our simplified 9-variable mechanism used earlier to explain both oscillatory potentiometric responses22 under isothermal conditions and associated temperature variations15 for the H2O2− SCN−−OH−−Cu2+ system, obtained by implementation of respective reaction enthalpies into the model. In that latter work, in spite of the modeling of the increase in solution temperature caused by these enthalpic effects, all the kinetic rate constants were assumed insensitive to temperature. This

Figure 5. Random birth, evolution, and final decay of weak luminescent waves in the thin-layer of the H2O2−NaSCN−NaOH− CuSO4 system + 1.5 × 10−3 M luminol in the graphite reactor, facilitating the smoothing of the local temperature gradients: (top) initially isothermal sample, (middle) sample preheated along the upper edge, and (bottom) sample preheated along the left edge with Plexiglas cover. See Supporting Information for the time lapse animation showing the temporal evolution of those patterns. 11160



Article

simplification was then justified by a relative short modeled time, within which the temperature rise was quite modest, equal to ca. only 2.5 K (maximum value) for the homogeneous case (cf. Figure 10 in ref 15). Since now it is necessary to model the essential effect of imposed local temperature gradients, that exceeded those caused by self-heating of the solution, also the Arrhenius-type dependences have to be implemented, at least for the reaction step(s) with the highest activation energy, i.e., with the rate constant(s) most sensitive to temperature. Taking into account both reaction enthalpies and the Arrhenius dependence of the rate constant(s) on temperature creates a thermokinetic coupling in the model system. The spatial progress of both the concentration and the temperature fronts was modeled in the one-dimensional space, which under real experimental conditions could correspond to a reactor in the form of a long tube (7 cm length, as in the experiment) and of a relatively small inner diameter, with the gradients formed at its left side. An explicit finite difference method was employed as the calculation algorithm. According to principles of this method the total reaction space (L) was divided into M spatial steps Δx, and the total model time tmax into N temporal steps Δt meeting the numerical stability condition26 DΔt/(Δx)2 < 0.5, where D can be either the diffusion coefficient of molecules/ions (typically 10−6−10−5 cm2 s−1 in aqueous media) or the thermal diffusivity coefficient DT (typically 10−3 cm2 s−1 in water). Since for liquid media DT > D, the stability of numerical calculations required that the DTΔt/(Δx)2 < 0.5 condition had to be met. The dynamical variables (concentrations of respective intermediates), the differential equations, and the kinetic rate constants were already specified in ref 22. In particular, for clarity of presentation, the enumeration of reaction steps, collected in Table 1, is the same as used in Table 1 of ref 22. The following further assumptions were accepted in the detailed construction of the model: (i) Based on our earlier thermochemical calculations from ref 15, we transferred to the present model the estimated enthalpy effects of reaction steps (3) and (9): ΔH3 = −600 kJ mol−1, ΔH9 = −1500 kJ mol−1. (ii) Knowing that the effect of temperature on rate constants is strongest for those reactions that are characterized with the highest activation energy, we assumed that the (simplified) reaction step H 2O2 + SCN− → OSCN− + H 2O

Table 1. Reaction Steps and Associated Rate Constants Used for Numerical Calculations of Concentration Variations in the Simplified Model of the Oscillatory Oxidation of SCN− with H2O2, at 25 °Ca step number 1 2 3

4 5

6

7 8 9 10 11 12

i

kM8 = 1 × 103 M−1 s−1 kM9 = 1 M−1 s−1 kM18 = 3 × 103 M−1 s−1 (forward reaction); kM19 = 2 × 106 M−1 s−1 (backward reaction) kM20 = 1 × 105 M−1 s−1 k6′ = 6.93 × 10−3 s−1 k8′ = 5 × 104 M−1 s−1 kM21 = 2 × 103 M−1 s−1 kM3 = 1 × 10−2 s−1 kM4 = 20 M−1 s−1 kCu21 = 100 M−1 s−1

in which cp denotes the specific heat capacity of the solution (approximated as that of pure water: 4.18 kJ kg−1 K−1), ΔHi and vi are the enthalpy and the chemical reaction rate of given ith individual reaction step (cf. Table 1), while DT [m2 s−1] is the above-mentioned thermal diffusivity coefficient

(1)

DT =

k ρc p

(4)

with k meaning the thermal conductivity coefficient [W m−1 K−1] and ρ the fluid density [kg m−3] For the assumed parameters and homogeneous initial concentrations, the model system always demonstrated uniform oscillatory behavior. Calculations were performed both for model systems perfectly isothermal (298 K) and for the systems with thermokinetic coupling between the reaction enthalpies and activation energy included, under conditions of local initial perturbations of the [Cu(OH)3]− concentration or temperature, of different spatial dependence. During further simulations, in the full version of the model, first the diffusion progress of the reactants (eq 2) and of temperature (first term of eq 3) was calculated in time step Δt. Then, on the basis of this newly obtained spatial concentrations array, the local progress of chemical reactions and the associated increase of local temperature, occurring due to the enthalpic effects, was computed (second term in eq 3). The calculated increase in temperature was subsequently used to calculate a new,

(2)

∑ ΔHivi

kM7 = 5 × 102 M−1 s−1

The numbering of rate constants corresponds to notation used in refs 17 and 22. The reaction scheme (OSCN− → OOSCN−) means that fast, practically stoichiometric transformation of these species was assumed. Species in square brackets are important only for the reaction stoichiometry and are not involved in the rate determining steps (see refs 17 and 22 for additional explanation). As the crucial dependence in the present simulations, the rate constant of step (1) was assumed to vary with temperature (see text for the justification).

while the local dynamics of temperature, being a consequence of heat evolution in an exothermic process and the heat exchange with the surroundings, was expressed by the equation 1 ∂T ∂ 2T = DT 2 − ∂t cp ∂x

rate constant(s) kM5 = 7.5 × 10−4 M−1 s−1 (at 25 °C)

a

corresponding to (1) in Table 1, is a kinetic bottleneck for the overall process. The rate constant of (1) is equal to 7.5 × 10−4 M−1 s−1 at ca. 298 K.17 The relevant activation energy was now assumed as a reliable value Ea,1 = 80 kJ mol−1. For all other steps (i = 2 to 12) the activation energies, as presumably much lower than the Ea,1 value, were assumed negligible, and thus set to zero in the calculations. (iii) The transport of chemical species was assumed to proceed according to the Fick’s laws for one-dimensional linear diffusion ∂c ∂ 2c =D 2 ∂t ∂x

reaction scheme H2O2 + SCN− → (OSCN− → OOSCN−) + H2O 2 OOSCN− → OOS(O)CN− + (OSCN− → OOSCN−) OOSCN− + OOS(O)CN− + [OH−] → (OSCN− → OOSCN−) + SO42− +HOCN 2 OOS(O)CN− + [OH−] → OS(O)CN− + SO42− + HOCN OS(O)CN− + OOS(O)CN− + H2O ⇄ 2 OS(O)CN• + [2OH−] OS(O)CN• + Cu(SCN)2− + [3 OH−] → OS(O)CN− + Cu(OH)3− + [2 SCN−] HO2Cu(OH)2− → Cu(OH)2− + HO2• Cu(OH)2− + HO2− → Cu(OH)3− + OH• + OH− OS(O)CN− + HO2• → SO3•− + HOCN HO2Cu(OH)2− + [2 SCN−] → Cu(SCN)2− + HO2• 2 HO2• → H2O2 + O2 Cu(OH)3− + HO2− → HO2Cu(OH)2− + OH−

(3) 11161



Article

enhanced rate constant of step (1), employed in the iterative calculation of the progress of chemical reactions in the next time step. 4.2. Results of Numerical Calculations and Diagnosis of the Origin of Patterns. Results of calculations are shown in the order of increasing degree of complexity of the model, in order to demonstrate which physical and chemical factors are sufficient for the onset of luminescent waves and which change only the quantitative characteristics of the phenomenon. In the following it will be assumed that the intensity of the luminescence is proportional to the concentration of the catalytic [Cu(OH)3]−ions, the spatial and temporal distribution of which in the reactor will be discussed. 4.2.1. Concentration Fronts of [Cu(OH)3]− Ions in an Isothermal System. In the first step of the modeling the perfectly isothermal (T = 298 K) system was assumed, i.e., all reaction enthalpies, the activation energy and the thermal conductivity coefficient were all assumed as equal to zero. It should be kept in mind that the model system, according to its experimental characteristics, is always in an oscillatory state. In order to check whether concentration fronts can develop and travel along so defined system, at the left edge of the model reactor, in the first ten (numbered 0−9, of altogether 405) discrete Δx spatial cells, the local [Cu(OH)3]− concentration was increased to 1 × 10−7 M, while in the remaining 395 cells it was set to 1 × 10−8 M, at the beginning of the simulations. Figure 6A clearly shows that these perturbations produced a local enhancement of [Cu(OH)3]− concentrations on a background of permanently oscillating concentration profile of those ions, spreading however on a very limited distance along the reactor and quickly decaying. This means that under assumed isothermal conditions the oscillatory system appeared to be stable against applied spatial perturbation, i.e., the traveling wave of [Cu(OH)3]− (and thus of the luminescence) was not born. This is a model proof that the isothermal system does not generate traveling waves, in line with experimental premises that only the temperature gradients are necessary for this phenomenon to occur. Anticipating the question, if the assumed amplitude of perturbation was not too small for the onset of the instabilities, we performed other simulations for the initial [Cu(OH)3]− concentration in the cells 0−9 enhanced up to 1 × 10−5 M and even 1 × 10−3 M (the latter being rather unreliable since in real system the brown precipitate of copper compounds is then formed). Figure 6B proves that even for such high initial concentration difference, reaching 3 or 5 orders of magnitude, the traveling front of Cu(II) species was also not formed in the model system. Instead, a steep initial concentration gradient around the border between the spatial cells 9 and 10 was created, which was soon smoothed, and the reactor space became divided into two parts differing with the oscillatory period and thus the oscillation phase. If it was possible to report this in real experiments, it would mean the alternate bursts of luminescence in the left (narrow) and right (relatively wide) zones of the solution. Temporal phenomena of such course were indeed observed by us if concentrations of CuSO4 solutions in different parts of the reactor differed initially for ca. 2 orders of magnitude (10−4 and 10−2 M). 4.2.2. Concentration Fronts of [Cu(OH)3]− Ions in a Nonisothermal System with Nonzero Activation Energy. In view of the model calculations described in the previous section it is clear that in search of the concentration front propagation one should study further the role of the temperature gradient

Figure 6. Effect of the [Cu(OH)3]− initial concentration step at the left edge of the model one-dimensional reactor with the oscillatory process running at isothermal (T = 298 K) conditions. Numbers in the insets indicate the actual model time in seconds. The total length of the model reactor (7 cm) was divided into 405 spatial cells Δx. (A) The local enhancement of [Cu(OH)3]− concentration (1 × 10−7 M) was initially introduced to the first 10 cells (0−9), while the initial concentration of [Cu(OH)3]− in the remaining cells 10−404 was equal to 1 × 10−8 M. This difference quickly becomes insignificant, and the system returns to practically homogeneous oscillatory state. Only the first 21 spatial cells (numbered 0−20) are shown in the plot. (B) The local, significantly enhanced [Cu(OH)3]− concentration (1 × 10−3 M) was initially assumed in the first 10 cells, while the initial concentration of [Cu(OH)3]− in the remaining cells 10−404 was equal to 1 × 10−8 M; for clarity, the concentration profiles in only 51 (0−50) cells are shown. The left- and right-hand-sides of the system oscillate with different periods, but no traveling front is formed. Other initial parameters for parts (A) and (B): [[Cu(OH)2]−]0 = 1 × 10−12 M; [[HO2Cu(OH)2]−]0 = 1 × 10−8 M; [[Cu(SCN)2]−]0 = 2.46 × 10−4 M; [HO2•]0 = 1 × 10−8 M; [OOSCN−]0 = 1 × 10−6 M; [OOS(O)CN−]0 = 1 × 10−8 M; [OS(O)CN−]0 = 1 × 10−8 M; [OS(O)CN•]0 = 1 × 10−8 M. Constant equilibrium concentrations: [H2O2] = 0.3425 M, [HO2−] = 0.0375 M. Diffusion coefficients of all chemical species were equal to 5 × 10−6 cm2 s−1.

implemented in the model system. Evidently in any inhomogeneous temperature field the oscillatory process must occur with different rates along the reactor. The sensitivity of the reaction kinetics to temperature is realized by introducing, to the kinetic mechanism, the nonzero activation energy of the slowest reaction step (1), here assumed as equal to 80 kJ mol−1. At this stage of modeling we do not yet allow explicitly for the heat transfer along the reactor, and with this assumption, it is not worthy to consider a trivial case of the temperature step in 11162



Article

the model reactor, as it must simply produce two zones with different oscillations frequency, i.e., alternate bursts of luminescence creating the illusion of traveling fronts, but being in fact only two-compartment phase waves. Consequently, this case does not reflect the onset and the propagation of the traveling luminescent wave along the real reactor preheated at its left edge. More sophisticated and worth of analysis is the linear temperature gradient imposed along the entire reactor, which can be understood as either simply the model initial condition (albeit difficult for exact experimental realization) or, alternatively, as the idealized stage of evolution of the initial temperature step, under implicit then assumption of allowed heat transfer along the reactor. In this case, a continuous variation of the oscillation period along the reactor length takes place, and the respective formation of traveling kinematic waves is illustrated in Figure 7. The advantage of this simulation is that it clearly indicates that modeling of propagating traveling fronts starting from the preheated local zone of the reactor requires that also a heat transfer along this

reactor has to be implemented in the model, as described in the next section. 4.2.3. Concentration Fronts of [Cu(OH)3]− Ions in a Nonisothermal System with Nonzero Activation Energy and Heat Diffusion. According to eq 4, the diffusion of a heat zone along the reactor is controlled by the thermal diffusivity coefficient DT. Since in aqueous solutions DT ≫ D (diffusion coefficient of chemical species), the front of heat wave must overtake that of diffusing particles. Also, in order to make the model even more reliable, one has to include enthalpic effects of two reaction steps, ΔH3 and ΔH9, in line with eq 4. Figure 8 shows representative results of calculations of the developments of the [Cu(OH)3]− concentration wave (A) and temperature profile (B), induced by the initial temperature step, which shows the spatial progress of the heating of the solution from the left edge of the reactor in the relevant experiment (cf. Figure 4).

Figure 8. Effect of the initial single-stepped temperature gradient on the wave propagation in the model reactor of a length of 7 cm, divided into 99 spatial cells Δx. (A) The development of the concentration profile of [Cu(OH)3]− as a function of time for the initial temperature step: T = 308 K imposed in the spatial cells 0−19, and T = 298 K in the spatial cells 20−98 of the model reactor, with initially homogeneous distribution of the [Cu(OH)3]− concentration = 1 × 10−8 M along the entire reactor; (B) the corresponding temporal development of temperature profiles. For both parts, solid lines show the temporal evolution (with final decay) of the first wave/ temperature front, while the dashed lines are of the subsequent front. Thermal diffusivity coefficient DT = 1 × 10−3 cm2 s−1. Activation energy of the reaction step (1) Ea,1 = 80 kJ mol−1. Reaction enthalpies ΔH3 = −600 kJ mol−1 and ΔH9 = −1500 kJ mol−1. Other parameters are the same as those in Figure 6.

Figure 7. Effect of the linear temperature gradient (0.02 K/spatial cell), imposed according to the model Tx = (306.08 − 0.02x) dependence, with the cell number x varying from 0 to 404, in the absence of the perturbation of initial [Cu(OH)3]− concentration. (A) Spatial distribution of [Cu(OH)3]− concentration along the entire reactor, for the times (in sec) indicated in the inset; (B) the corresponding temporal dependence of the oscillating [Cu(OH)3]− concentration in selected cells of numbers indicated in the inset, showing that the speed of the wave propagation decreases with time. Initial concentration of [Cu(OH)3]− = 1 × 10−8 M. Activation energy of the reaction step (1) Ea,1 = 80 kJ mol−1. Other parameters are the same as those in Figure 6. 11163



Article

results (local enhancement of only the [Cu(OH)3]− concentration does not trigger the waves, either, even for thermokinetic coupling with a heat transfer assumed). Evidently sufficient local temperature gradients must develop for the onset of these particular phenomena. Regarding the experimental systems, if the initially isothermal solution is left for prolonged time with no external heating applied (cf. Figure 2 in this article and Figure 1 in ref 18), local fluctuations of the reaction rate, caused also by thermal interaction with surroundings, may appear sufficient to cause a spatial distribution of the oscillation phase and thus to produce the impression of the traveling luminescent wave. Such an interaction with the presumably warmer surrounding might explain the onset of the luminescent zone at the edge of the thin layer system, as Figure 1 in ref 12 illustrates. Accordingly, in a final stage of the present study we refer to the phenomenon of a random birth of luminescent patterns, caused by a numerically randomly generated distribution of initial temperature in a model system, for assumed maximum amplitudes. Our previous studies (cf. Figure 9 in ref 15) proved that the temperature of the (stirred) solution can increase for as much as 8 K after 10 min of the oscillatory process associated with the SCN− oxidation with H2O2. In the case of quiescent thin layer system, studied now, this will be an approximate average

In fact, Figure 8 is crucial for the confirmation of a thermokinetic mechanism of the wave phenomena in the system considered. Under present model assumptions, the thermokinetically induced inhomogeneity in the concentration of [Cu(OH)3]− is now not damped and even proceeds along the reactor, simulating in this way the traveling of the luminescent wave. A comparison of the solid and dotted profiles in Figure 8A shows that the second wave moves slower than the first one, due to, increasing with time, phase shift of the oscillations along the reactor. One should note that this tendency was observed also in the experiment (cf. Figure 4). Also, the speed of the wave propagation in the above model (6−12.6 mm min−1) is similar to that observed in the experiment (8.5−9.4 mm min−1, cf. Figure 4). Furthermore, comparison of parts A and B of Figure 8 clearly shows that the spatial progress of the peak of [Cu(OH)3]− concentration is determined by the spatial development of the temperature profile, which expands as a function of time. In other words, the initial local heating of the left-hand-side of the reactor induces the luminescent wave that penetrates only this region of the solution that underwent further heating. Because of this heat expansion, the front of every subsequent wave reaches farther than the previous one. Beyond this heated region (cf. Figure 8: for the cell numbers 55−98 at t = 400 s, and 75−98 at t = 1050 s), the oscillatory course of the reaction occurs with the same frequency. One should emphasize that these model conclusions are quite well concordant with our experimental observations in which fronts of luminescence were periodically born and traveled along the reactor up to the narrow zone in its righthand-side, when periodic bursts of homogeneous luminescence occurred (see enclosed time lapse animations as Supporting Information). Furthermore, the heat wave in the maximum model time of t = 1250 s (cf. Figure 8) should penetrate the region of the order of several multiples of (DTt)1/2 = 1.11 cm, which is comparable with the 7 cm total length of the reactor. For comparison, the thickness of the diffusion layer of the chemical species is then equal only to (Dt)1/2 = 0.079 cm, what clearly indicates that the diffusion transport is not decisive for the progress of the luminescent wave. Concluding, the actually suggested thermokinetic mechanism of the wave propagation has to involve not only the Arrheniustype coupling between the reaction rate and local temperature but requires also the expansion of the heated-up zone of the solution as the condition for the sustained progress of the luminescent wave. As follows from the previous section, the model progress of the luminescent wave, shown in Figure 8A, can be considered a phenomenon analogous to that illustrated in Figure 7, where thermal diffusion was not yet included, but the linear temperature gradient, instead of temperature step (cf. Figure 7) was already imposed. In fact, thermal diffusion changes the initial temperature step into a quasi-linear temperature distribution, and this is the main factor controlling the progress of luminescent waves in a heated-up area. 4.2.4. Random Birth and Interaction of the Concentration Fronts of [Cu(OH)3]− in a Nonisothermal System with an Activation Energy and Heat Diffusion. Intuitively one can suppose that if the externally imposed temperature difference is as high as, e.g., 10 K, the self-heating of the solution due to exothermic process is not a decisive factor. In the opposite case, these effects have to be taken into account, also in view of both experimental data (local injection of CuSO4 under isothermal conditions does not trigger the traveling wave) and simulation

Figure 9. Development of inhomogeneous concentration profiles of [Cu(OH)3]− as a function of time, in seconds (indicated in the insets), for the initial random distribution of the solution temperature, of the maximum amplitude: (A) 1 K and (B) 10 K. Initial concentration of [Cu(OH)3]− = 1 × 10−8 M. Other parameters are the same as those in Figures 6 and 8. 11164



Article

the oscillation phase caused by them would remain, so the wave patterns would persist in spite of an already homogeneous temperature field. Concluding, the described mechanism of the formation of luminescence pattern is presumably the first example of thermokinetic coupling in aqueous solution, causing the wave phenomena in oscillatory systems.

temperature increase of the entire sample, with local temperature differences not exceeding (at most) a few degrees, and only if the solution is placed in the reactor made of material of poor thermal conductivity (like Teflon, cf. Figure 2). Figure 9 shows the results of our calculations for two cases assumed as extreme ones, regarding the maximum amplitude of random temperature variations in the solution: from ΔTmax = 1 K (Figure 9A) to ΔTmax = 10 K (Figure 9B). The calculations involved the full version of the model, i.e., with implemented enthalpic effects of two reaction steps, the activation energy of the rate-determining step, and thermal diffusion of heat in the solution. The existence of several neighboring maxima of [Cu(OH)3]− concentration corresponds to spots of luminescence observed in the experiment (cf. Figure 2), and the gradually expanding and overlapping diffusion profiles reflect also the experimentally observed collisions of neighboring luminescent zones. Furthermore, calculations show that if the temperature gradients are relatively high (as for Figure 9B), after certain decrease in their amplitude due to thermal diffusion, they practically stabilize at a certain noticeable level, which effect is evidently caused by the self-heating of the solution due to exothermic reaction steps. The only essential discrepancy between the above calculations and the experiment is only of quantitative nature, i.e., the model waves visualized by concentration profiles in Figure 9 are moving along the reactor faster (Figure 9A, 90 mm min−1; Figure 9B, 20 mm min−1) than the experimental ones (ranging from 5 to maximum measurable limit of 30 mm min−1). This is presumably due to both difficult estimation of distribution of solution temperature in real experiment and, to some extent, unavoidalble simplifications in the construction of the thermokinetic model.

■

ASSOCIATED CONTENT

S Supporting Information *

For better visualization of the dynamics of the experimentally reported luminescent spatiotemporal patterns, from their photographic images shown in Figures 2−5, the respective time lapse animations (as .gif type files) were produced. This material is available free of charge via the Internet at http:// pubs.acs.org.

■

AUTHOR INFORMATION

Corresponding Author

*(M.O.) E-mail: [email protected]. Fax: +48 22 822 59 96. Tel: +48 22 822 02 11, ext. 245. Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS This scientific work was partly financed through the research Grant No. N204 242134 from the Ministry of Science and Higher Education, Poland, for the years 2008−2011, for the research project.

■

REFERENCES

(1) Zhabotinsky, A. M. Concentrations Autooscillations; Nauka: Moscow, Russia, 1974 (in Russian). (2) Epstein, I. R. Chemistry: Chemically Generated Spatial Structures. Nature 1984, 307, 692. (3) Lee, K. J.; McCormick, W. D.; Ouyang, Q.; Swinney, H. L. Pattern Formation by Interacting Chemical Fronts. Science 1993, 261, 192−195. (4) Lee, K. J.; McCormick, W. D.; Pearson, J. E.; Swinney, H. L. Experimental Observation of Self-replicating Spots in a ReactionDiffusion System. Nature 1994, 369, 215−218. (5) Horváth, J.; Szalai, I.; De Kepper, P. An Experimental Design Method Leading to Chemical Turing Patterns. Science 2009, 324, 772−775. (6) Horváth, J.; Szalai, I.; De Kepper, P. Pattern Formation in the Thiourea-Iodate-Sulfite System: Spatial Bistability, Waves, and Stationary Patterns. Phys. D 2010, 239, 776−784. (7) Wrobel, M. M.; Bánsági, T., Jr.; Scott, S. K.; Taylor, A. F.; Bounds, C. O.; Carranzo, A.; Pojman, J. A. pH Wave-Front Propagation in the Urea−Urease Reaction. Biophys. J. 2012, 103, 610−615. (8) Szalai, I.; Horváth, J.; Takács, N.; De Kepper, P. Sustained SelfOrganizing pH Patterns in Hydrogen Peroxide Driven Aqueous Redox Systems. Phys. Chem. Chem. Phys. 2011, 13, 20228−20234. (9) Epstein, I. R.; Pojman, J. A. An Introduction to Nonlinear Chemical Dynamics. Oscillations, Waves, Patterns, and Chaos; Oxford University Press: New York, 1998. (10) Aliev, R. R. Oscillation Phase Dynamics in the BelousovZhabotinsky Reaction. Implementation to Image Processing. J. Phys. Chem. 1994, 98, 3999−4002. (11) Tyson, J. A Quantitative Account of Oscillations, Bistability, and Traveling Waves in the Belousov-Zhabotinsky Reaction. In Oscillations and Travelling Waves in Chemical Systems; Field, R. J., Burger, M.,Eds;, Wiley-Interscience: New York, 1985 (Russian translation, Mir: Moscow, 1988); pp 117−166.

5. CONCLUSIONS At the end of these experimental and model analysis of the luminescent fronts in H2O2−SCN−−OH−−Cu2+ oscillator it is useful to make a final statement regarding our first report on those phenomena, which involved a thin layer circular reactor. We conclude that the stochastic birth and periodic repetition of initial luminescent patterns in time, reported there (cf. Figures 1 and 3 in ref 18) can be classified as a special case of the phase (or kinematic) waves. They were formed due to the thermokinetic mechanism, causing the different frequencies, and thus different phases of the oscillations in various points of the system. Initial stochastic fluctuations of the reaction kinetics, due to interaction with warmer surrounding, caused these local concentration and temperature gradients. Thus, the system, after the initial period, is set into spatial distribution of different phases of the oscillations, represented by spatial distribution of luminescence. The oscillatory course of the process causes the periodic repetition of the initial set of luminescent patterns. Moreover, since essentially the same pattern of luminescence repeats several times, one infers that the formed temperature gradients are asymptotically (but only as long as the reaction proceeds) not damped, but remain sustained. On the basis of our calculation results, we conclude that this effect, although against the tendency of thermal diffusion to smooth those gradients, is in fact possible in the studied system and is mainly due to the exothermic reaction steps which in this way stabilize the patterns. An interesting theoretical aspect of this stabilization, but confirmed also by our calculations, is that even if temperature gradients finally vanished, the differences in 11165



Article

(12) Pojman, J. A.; Epstein, I. R. Convective Effects on Chemical Waves. 1. Mechanisms and Stability Criteria. J. Phys. Chem. 1990, 94, 4966−4972. (13) Griffiths, J. F. Thermokinetic Oscillations In Homogeneous Gas-Phase Oxidation. In Oscillations and Travelling Waves in Chemical Systems; Field, R. J., Burger, M., Eds.; Wiley-Interscience: New York, 1985 (Russian translation, Mir: Moscow, 1988); pp 569 − 607. (14) Eiswirth, M.; Ertl, G. Pattern Formation on Catalytic Surfaces. In Chemical Waves and Patterns; Kapral, R., Showalter, K., Eds.; Kluwer Academic Publishers: Dordrecht, The Netherlands, 1995; pp 447− 483. (15) Pekala, K.; Wiśniewski, A.; Jurczakowski, R.; Wiśniewski, T.; Wojdyga, M.; Orlik, M. Monitoring of Spatiotemporal Patterns in the Oscillatory Chemical Reactions with the Infrared Camera: Experiments and Model Interpretation. J. Phys. Chem. A 2010, 114, 7903− 7911. (16) Orbán, M. Oscillations and Bistability in the Cu(II)-Catalyzed Reaction Between H2O2 and KSCN. J. Am. Chem. Soc. 1986, 108, 6893−6898. (17) Luo, Y.; Orbán, M.; Kustin, K.; Epstein, I. R. Mechanistic Study of Oscillations and Bistability in the Cu(II)-Catalyzed Reaction between H2O2 and KSCN. J. Am. Chem. Soc. 1989, 111, 4541−4548. (18) Pekala, K.; Jurczakowski, R.; Lewera, A.; Orlik, M. Luminescent Chemical Waves in the Cu(II)-Catalyzed Oscillatory Oxidation of SCN− Ions with Hydrogen Peroxide. J. Phys. Chem. A 2007, 111, 3439−3442. (19) Sattar, S.; Epstein, I. R. Interaction of Luminol with the Oscillating System Hydrogen Peroxide-Potassium ThiocyanateCopper Sulfate-Sodium Hydroxide. J. Phys. Chem. 1990, 94, 275−277. (20) Orbán, M. Private communication, 1998. (21) Zhao, C.; Zheng, J.; Xie, J.; Liu, H.; Gao, Q. Investigations of Different Chemiluminescent Peaks in H2O2−SCN−−Cu2+−OH−− Luminol Flow System. Luminescence 2009, 26, 130−135. (22) Wiśniewski, A.; Pekala, K.; Orlik, M. Kinetic Model of the H2O2−SCN−−OH−−Cu2+ Oscillator and Its Application to the Interpretation of the Potentiometric Responses of Various Inert Electrodes Monitoring the Reaction Course. J. Phys. Chem. A 2010, 114, 183−190. (23) Pekala, K.; Jurczakowski, R.; Orlik, M. On the Interpretation of the Potentiometric Response of the Inert Solid Electrodes in the Monitoring of the Oscillatory Processes Involving Hydrogen Peroxide. J. Solid State Electrochem. 2010, 14, 27−34. (24) Isaacs, N. S. Physical Organic Chemistry. Exercises; PWN: Warsaw, Poland, 1974 (in Polish). (25) White, E. H.; Zafiriou, O.; Kägi, H. H.; Hill, J. H. M. Chemiluminescence of Luminol: The Chemical Reaction. J. Am. Chem. Soc. 1964, 86, 940−941. (26) Crank, J. The Mathematics of Diffusion; Clarendon Press: Oxford, U.K., 1975.

11166


Thermokinetic Origin of Luminescent Traveling Fronts in the H2O2

Recommend Documents