The Formation and Spatiotemporal Progress of the pH Wave Induced

Article pubs.acs.org/JPCB

The Formation and Spatiotemporal Progress of the pH Wave Induced by the Temperature Gradient in the Thin-Layer H2O2− Na2S2O3−H2SO4−CuSO4 Dynamical System Mikołaj Jędrusiak and Marek Orlik* Laboratory of Electroanalytical Chemistry, Faculty of Chemistry, University of Warsaw, Ul. Pasteura 1, 02-093 Warsaw, Poland ABSTRACT: The H2O2−S2O32−−H+−Cu2+ dynamical system exhibits sustained oscillations under flow conditions but reveals only a single initial peak of the indicator electrode potential and pH variation under batch isothermal conditions. Thus, in the latter case, there is no possibility of the coupling of the oscillations and diffusion which could lead to formation of sustained spatiotemporal patterns in this process. However, in the inhomogeneous temperature field, due to dependence of the local reaction kinetics on temperature, spatial inhomogeneities of pH distribution can develop which, in the presence of an appropriate indicator, thymol blue, manifest themselves as the color front traveling along the quasi-one-dimensional reactor. In this work, we describe the experimental conditions under which the above-mentioned phenomena can be observed and present their numerical model based on thermokinetic coupling and spatial coordinate introduced to earlier isothermal homogeneous kinetic mechanism.

1. INTRODUCTION Spatial and spatiotemporal dissipative patterns reported so far for initially homogeneous aqueous solutions are usually being explained in terms of appropriate coupling of diffusion and chemical reactions of a specific kinetic mechanism, i.e., with appropriate feedback loops, but only under isothermal conditions. This is the case for the model Turing patterns1 as well as for the excitable Belousov−Zhabotinsky system,2 including both its experimental characteristics and available theoretical kinetic models. More recently discovered systems which offer such instabilities, classified as evolution of target patterns, include, among others, chlorite−iodide−malonic acid (CIMA) oscillator,3 the ferrocyanide−iodate−sulfite system (FIS),4,5 the thiourea−iodate−sulfite (TuIS) reaction,6,7 the urea−urease reaction, with pH wave fronts,8 as well as hydrogen peroxide−sulfite systems producing sustained pH patterns.9 In all of these cases, the eventual nonuniform temperature distribution, either self-induced by enthalpic effects of the reaction or caused by the system’s interaction with surroundings, is not considered an essential cause for the development of spatiotemporal instabilities but at most as a factor that modifies the isothermally induced pattern, e.g., by inducing local convection due to creation of local density gradients. However, as we reported in our recent papers, the luminescent spatiotemporal phase waves in the H2O2− SCN−−Cu2+−OH− homogeneous Orbán oscillating system discovered by us,10,11 enriched with luminol, can be understood only in terms of the inhomogeneous spatial distribution of temperature that causes an appropriate spatial distribution of © XXXX American Chemical Society

the frequency of the oscillations through the thermokinetic coupling. For the sake of simplicity, one can explain the essential origin of the spatial patterns as being due to the spatiotemporal oscillatory regime estimated by A = Amax sin[ω(x)·t] dependence (Amax meaning the amplitude of the oscillation), when the bursts of maximum luminescence are ascribed, e.g., to the maxima of this simplified, sinusoidal dependence. Different parts of the system exhibit thus bursts of luminescence in different times, making an impression of the wave motion, with the difference only in the oscillation phase manifesting itself in this way. One of the premises for such a mechanism is the experimental observation that the velocity with which the region of inhomogeneously oscillating luminescence spreads along the reactor space correlates well with the thermal diffusivity coefficient of water, and not with the (lower for ca. 3 orders of magnitude) coefficients of molecular diffusion; the latter phenomenon can be even considered negligible within the time scale of a typical experiment. In our opinion, the scarcely considered thermokinetic origin of dissipative patterns in the liquid solutions could contribute to the possible mechanisms of morphogenesis of living matter, in addition to much better recognized and widely studied isothermal reaction−diffusion systems. In search of another process in which spatiotemporal and spatial dissipative patterns were not reported so far under isothermal conditions but could be induced by an inhomogeReceived: November 17, 2015 Revised: March 3, 2016

A

DOI: 10.1021/acs.jpcb.5b11274 J. Phys. Chem. B XXXX, XXX, XXX−XXX

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The Journal of Physical Chemistry B neous temperature field, we chose the H2O2−S2O32−−H+− Cu2+ system that is a well-known pH oscillator but operating only under flow conditions. In the batch reactor, only a single, broad peak of pH variation is observed shortly after preparing the reaction mixture.12 In this paper, we show how this single event of transient pH dynamics under batch conditions can become a source of a traveling front in a quasi-one-dimensional reactor upon an externally imposed inhomogeneous temperature field. The patterns were associated with the local variations of H+ ions and were detected using thymol blue as a pH indicator, which exhibits a color change from yellow to blue within the pH range 8.0−9.6.13

2. EXPERIMENTAL SECTION 2.1. Materials. All reactants, H2O2 (POCh, Poland, 30%, pure for analysis), Na2S2O3·5H2O (Chempur, Poland, pure for analysis), H2SO4 (95%, POCh, pure for analysis), CuSO4·5H2O (POCh, pure for analysis), and thymol blue (POCh, pure for analysis), were taken without further purification. Triply distilled water further deionized by MilliPore filters was used. The following stock solutions were prepared: 0.26 M H2O2, 0.025 M Na2S2O3, 0.0033 M CuSO4, 0.068 M H2SO4, and a saturated (0.24 mM) solution of thymol blue. A solution of hydrogen peroxide was prepared daily ex tempore to avoid its decomposition. 2.2. Apparatus. Measurements of the dynamic instabilities in the H2O2−S2O32−−H+−Cu2+ system were performed under batch conditions, in two types of the reactors. One of them, which served to study only temporal instabilities in a homogeneous system, was a conventional vessel in which the solution of a volume of 75 mL was placed and continuously stirred during the potentiometric measurement. The second type of reactor was designed for the studies of the traveling fronts in the quiescent thin-layer system. This thin-layer reactor was made of a Teflon block (the dimensions of the reaction solution space were 2 × 20 × 70 mm3). As a cover of the Teflon reactor, the transparent polycarbonate block (of dimensions 10 × 30 × 90 mm3) in a single piece or cut for several equal pieces was used. In the former case, the initial continuous temperature distribution (which in our opinion could be roughly estimated by the T(x) ∼ exp(−x) dependence) was produced along the entire block, while in the latter case every piece was initially preheated to a different temperature, and the set of such prepared connected elements produced a staircase profile of temperature with the heat transport between them slowed down due to discontinuity of that construction. Progress of chemical reaction was recorded by Nikon D90 camera from above, as shown in Figure 1. The course of the chemical reaction in the stirred solution was monitored potentiometrically. The potential of the platinum net electrode was measured vs the saturated silver chloride electrode using the CHI660 Electrochemical Workstation (CH Instruments). On the basis of earlier literature data (cf. Figure 7 in ref 12), the decrease in Pt electrode potential should correlate with the increase in solution pH, but we measured also the solution pH, since the shapes of the temporal dependences of both quantities, carrying essential kinetic information, are not exactly a mirror reflection of each other. For pH measurements, the combined pH-sensitive glass electrode type ERH-11A (Hydromet, Poland), connected to a computer-assisted CPI-505 pH-meter (Elmetron, Poland), was employed. All of these potentiometric and pH measurements were performed in an open batch reactor with thermostated

Figure 1. Schematic view of the experimental setup. (A) Horizontal thin-layer Teflon reactor (1), with a single piece of polycarbonate cover used as a source of external temperature gradient (2), and reflex camera; (B) polycarbonate cover composed of sticking separate segments.

water jacket, with the solution continuously stirred using the OP-912/3 magnetic stirrer (Radelkis, Hungary). The temperature of the reacting mixture was controlled by MK70 (MLW, GDR) thermostat. In turn, the formation and the progress of the color front in the thin-layer reactor was observed visually due to the addition of thymol blue indicator and photographed using the digital camera. For better visualization of the patterns, the contrast of the images was slightly improved using the Corel Photopaint ver. X6 (Corel Corp. © 2012) correction tools.

3. RESULTS AND DISCUSSION 3.1. Basic Calorimetric Characteristics of the H2O2− S2O32−−H+−Cu2+ System. Since the spatial distribution of temperature is considered by us now as the main reason for the evolution of traveling fronts, it is important to check whether the oxidation of thiocyanates by hydrogen peroxide is exothermic enough to produce its own temperature effects that could significantly interfere with the externally applied temperature gradients. In the case of the H2O2−SCN−−OH−− Cu2+ oscillator formerly studied by us, the relatively high exothermicity of the oxidation of thiocyanates by hydrogen peroxide (8 K temperature increase within 10 min14) appeared to be an important contribution to the self-organization of the system into thermally driven, sustained luminescent patterns. Hence, in order to take control of the induction and evolution of the luminescent patterns, it was then necessary to impose a sufficiently steep, external temperature gradient. Thus, it was justified to verify if an analogous interaction can operate also for the presently studied H2O2−S2O32−−H+−Cu2+ system. The calorimetric experiment, carried out for the permanently stirred 50 mL solution of a typical composition, used in most of our experiments, i.e., 0.1 M H2O2 + 0.01 M Na2S2O3 + 0.001 M H2SO4 + 2.5 × 10−6 M CuSO4, revealed, however, only a ca. 0.8 K increase in temperature over a 8 min long experiment. It is thus clear that, in order to induce the spatial or spatiotemporal patterns of thermokinetic origin in the present system, on one hand, it is necessary to impose an externally produced temperature gradient and, on the other hand, the own enthalpic B


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The Journal of Physical Chemistry B effect of the reaction should not significantly interfere with even a moderate external temperature gradient. 3.2. Effect of Temperature on Temporal Dynamics of the Pt Electrode Potential. On the basis of earlier works, it is known that, upon addition of CuSO4 catalyst to the batch reactor containing an acidic (H2SO4) solution of hydrogen peroxide and sodium thiosulfate, the reaction dynamics, monitored through the variation of the potential of either a platinum or glass indicator electrode, is revealed through the single broad peak of the respective quantity, preceded by a relatively short induction period. One should note here that in our experiments we used a concentration of the copper(II) catalyst ca. 10 times lower than in earlier works,12 in order to slow down the reaction kinetics to the level allowing us to monitor the system dynamics more precisely than for the relatively fast concentration variations in the samples of original composition. For the purposes of our studies, it was crucial to investigate the effect of temperature on the above dynamics, since its presence is an essential premise for the onset of spatiotemporal instability in the inhomogeneous temperature field. In our experiments, the stock solutions of hydrogen peroxide, sodium thiosulfate, sulfuric acid, and copper(II) sulfate were poured into the batch reactor and its content was thoroughly stirred. The initial concentrations of the reactants in the reactor were 0.1 M H2O2, 0.01 M Na2S2O3, and 0.001 M H2SO4. After the temperature of the solution was established, the instability was initiated by addition of such an amount of the stock solution of CuSO4 that its concentration reached the value 2.5 × 10−6 M and at the same time the measurements of the potentiometric and pH responses of the system have been started. These measurements were repeated for several temperatures, ranging from 288 to 313 K. Figure 2 shows representative results of these experiments, for several selected temperatures. Comparison of parts A and B of Figure 2 confirms that the decrease in the Pt electrode is roughly equivalent to the increase in the solution pH. Analysis of E−t and pH−t dependences from Figure 2 shows that the effect of temperature on the induction period and on the width of the region with the transient decrease of the electrode potential (essentially equivalent to an increase in pH) is different. The induction period only slightly decreases with increasing temperature, while the region of the low electrode potential narrows then relatively significantly. Thus, one can expect that the inhomogeneous spatial temperature distribution will switch the system to higher pH and its return to a lower value in various places in different times, giving rise to spatiotemporal patterns, revealed as a color distribution in the presence of an appropriate acid−base indicator. 3.3. Estimation of the Formal Activation Energy of the Reaction. Since modeling of the effect of temperature requires the estimation of the effective (formal) activation energy of the studied process, we interpreted the dependence of the induction time on temperature, as reported in Figure 2, in terms of the relevant Arrhenius model. The induction times, measured as the time required for the first steep change of the system’s characteristics, are close to each other both for electrode potential or for pH measurements, as their correlation, close to ideal linear dependence, shown in Figure 3A, proves. At this point, it is important to note that in our further considerations we used the data of temporal dynamics of the Pt electrode potential E, in spite of an apparently more direct

Figure 2. Effect of temperature on temporal dynamics of (A) Pt electrode potential and (B) solution pH of the continuously stirred sample with a composition of 0.1 M H2O2 + 0.01 M Na2S2O3 + 0.001 M H2SO4 + 2.5 × 10−6 M CuSO4. Potential−time and pH dependences were measured at the temperatures indicated at the respective curves.

correlation of the studied phenomena with pH, not E, variations. We did it for several reasons: first, at times longer than the induction period, the dynamics of the electrode potential (being presumably the actual mixed potential) revealed more abrupt variations than the dynamics of pH, allowing thus the significant changes of the system’s state to be detected more precisely (cf. Figure 2). Second, due to limitations of the laboratory equipment, pH variations could have been measured with only 1 s resolution, while the Pt electrode potential was recorded every 100 ms, allowing thus to monitor also steep variations in the system’s state. The next argument comes from the model system’s dynamics (described later), in which pH variations are steeper than in the experiment, and thus the model pH courses become easier to compare with the experimental E−t dependences. Finally, if one assumes pH ≈ 8 as the critical value at which the switch of the color of the thymol blue indicator takes place, the change of the system’s state occurs at similar times, regardless of whether the pH or Pt electrode potential is considered. In consequence of this decision, for the purpose of the estimation of the Arrhenius activation energy, we assumed that the reciprocal induction period (1/tind), determined from the Pt C


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The Journal of Physical Chemistry B

Figure 4. Arrhenius plot for the H2O2−S2O32−−Cu2+−H+ system, based on the temperature dependence of the induction period tind (expressed here in seconds), preceding the transient decrease in Pt electrode potential (or increase in the solution pH). Initial concentrations of reactants: 0.1 M H2O2, 0.01 M Na2S2O3, 0.001 M H2SO4, 2.5 × 10−6 M CuSO4. Linear regression parameters: ln(1/tind) = −6563.93 (±502.89) × 1/T + 18.67 (±1.67), r2 = 0.876.

solution pH), with only a slight effect on the duration of the induction period, we performed the thin-layer experiments, under conditions of the inhomogeneous temperature distribution, in the following way. To a separate vessel, the stock solutions of hydrogen peroxide, sodium thiosulfate, and sulfuric acid were poured; a few droplets of saturated thymol blue solution were added to ensure its concentration that allowed to follow visually the formation and progress of color fronts. The content of the vessel was then immediately and carefully homogenized by intensive stirring. In order to prevent the premature onset of the instability, only shortly before the end of stirring, the stock solution of CuSO4 was added using the micropipette. In such a sample, the initial concentrations of particular species were, as before, 0.1 M H2O2, 0.01 M Na2S2O3, 0.001 M H2SO4, 2.5 × 10−6 M CuSO4, and ca. 10−5 M thymol blue. The final reaction mixture prepared in this way was then immediately poured into the Teflon thin-layer reactor depicted in Figure 1. Simultaneously, during the preparation of the sample, in the polycarbonate cover the initial temperature gradient was prepared by immersion of one of the cover’s ends (for ca. 1 min) into the boiling water, placed in a vessel, the large part of which was filled up with hot steam. In this way, the lower part of the cover was heated up to a higher temperature than the upper part, and along the entire cover, a smooth spatial temperature distribution should develop which, as mentioned in the Experimental Section, could be roughly estimated by the T(x) ∼ exp(−x) dependence. The cover was then immediately put into contact with the thin layer of the reaction mixture. It was observed that initially the entire solution was yellow (indicating a pH below ca. 8.0), but within ca. 40 s, meaning that the induction period elapsed, the whole reaction area became bluish, indicating thus the global rise of the solution pH. Within the next few minutes, it was observed that the yellow region reappears again at the lefthand end of the reactor, i.e., preheated to higher temperatures, and it further expands toward the right side, thus penetrating cooler regions of the solution. Selected images illustrating the progress of this thermokinetic front of pH are depicted in Figure 5.

Figure 3. (A) Dependence between the induction times measured on the basis of temporal dynamics of the Pt electrode potential (cf. Figure 2A) and pH (cf. Figure 2B) for the temperature range 288−313 K. Linear regression parameters: Induction period (Pt) = (6.582 ± 2.856) + (1.126 ± 0.118) × induction period (pH), r2 = 0.928. (B) Dependence between the duration time of the decreased Pt electrode potential (cf. Figure 2A) and enhanced pH (cf. Figure 2B). Compared to Figure 2, the present plots include additional data, determined for T = 288 K. Linear regression parameters: Time (Pt) = (−32.241 ± 39.771) + (1.164 ± 0.118) × time (pH), r2 = 0.933.

electrode potential dynamics, is an appropriate measure of the rate constant of the slowest kinetic step (krds), which is a usually accepted assumption in kinetic studies (cf., e.g., ref 15). Accordingly, the natural logarithm of the reciprocal induction period was plotted versus the reciprocal solution temperature, yielding points which fairly satisfactorily could have been estimated by linear dependence (see Figure 4 and its caption for the linear regression parameters). The Arrhenius activation energy, obtained from multiplication of the minus slope of this line by the universal gas constant, yielded a value of Ea = 54.5 kJ/mol which is typical of the kinetic characteristics of the reactions, the rate of which increases for a factor of 2−3 with the increase in temperature for 10 K, under typical ambient conditions (empirical van’t Hoff rule). 3.4. Thermally Driven Expanding Front in the ThinLayer Reactor. In view of the temperature effect on the system’s dynamics, visualized in Figure 2, which shows the acceleration of the processes through mainly the narrowing of the transient decrease in Pt potential (or increase in the D


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this case also stepwise, i.e., occurred homogeneously under every cuboidal piece of cover in the subsequent portions of the solution. This observation strongly suggested the essential role of the imposed temperature gradient in the evolution of the color front, with the negligible role of diffusion of the temperature profile in time, as being the process only slightly affecting the detailed distribution of the color zones. Finally, one should note that theoretically it should be possible to observe not only an expanding yellow front but also a narrow blue stripe traveling along the thin-layer reactor. For that, however, it would be necessary to find such three, precisely controlled temperatures that at an appropriate time the system in the cooler region remains still in the initial yellow phase, in the warmest regionalready returned back to the final yellow phase, while in the middle region of a moderate temperaturein the transient state of the high pH blue zone. We were however not able to tune the experimental parameters to the values causing the expected phenomenon, since it was not possible to achieve a precise control of the spatial temperature distribution in the thin-layer system. 3.5. Numerical Modeling of Thermokinetic Patterns in the H2O2−S2O32−−H+−Cu2+ System. In the first step of the development of the theoretical construction, it is necessary to recognize the essential factors underlying the reported spatiotemporal phenomena. In our previous model11 explaining the luminescent traveling fronts in the H2O2−SCN−−Cu2+− OH− system as the phase waves, we showed that the entire phenomenon could be explained in terms of thermokinetic coupling between the kinetics of the oscillatory process and spatial distribution of temperature, caused both by the contribution from the relatively high enthalpic effect of the entire reaction and the externally applied temperature inhomogeneity. Accordingly, under a simplifying assumption of the thermally isolated system, the spatiotemporal dynamics of the initial spatial temperature distribution was modeled in terms of the dependence involving both the diffusion of temperature (treated as a pseudoparticle) and the contribution from the enthalpic effect of appropriate reaction steps

Figure 5. Spatiotemporal evolution of a decreased pH front along the thin layer of the solution in the Teflon reactor covered with the polycarbonate cuboid, the left edge of which was preheated in boiling water. With thymol blue as a pH indicator, blue and yellow regions correspond to relatively high (above 9.6) and low (below 8.0) pH, respectively. Subsequent photos were taken approximately every 8 s, and the time scale is given in [min:sec]. The first three images (t = 0, 0:14, 0:28) show an initially homogeneous yellow solution which slowly turns bluish on the fourth image (t = 0:43) and, starting from the next slide, the yellow zone that appears at the left, warmer side of the reactor, expands along its entire length. Initial concentrations of reactants: 0.1 M H2O2, 0.01 M Na2S2O3, 0.001 M H2SO4, 2.5 × 10−6 M CuSO4, and ca. 1 × 10−5 M thymol blue as an indicator. An initial temperature difference between the opposite ends of the polycarbonate cover was approximately equal to 10 K (i.e., from 310 to 300 K). A slight cooling effect of the surroundings was expected. Eventually, the whole area of the reactor turns yellow, which is not depicted here.

One should note that, in spite of the presence of the thymol blue indicator only in the thin-layer reactor, the induction period of the reaction is close to that reported for the same temperature in a stirred batch reactor (compare the time scales in Figure 2 and Figure 5). This means that in spite of the relatively high, compared to [H+] varying from ca. 10−8 to ca. 10−5 M, concentration (ca. 10−5 M) of thymol blue which is a weak organic acid (pKa1 = 2.0, pKa2 = 8.813), this indicator does not detectably interfere with acid−base equilibria of the main reaction system. The above-given experimental procedure produces a temperature profile in the form of a presumably quasi-exponential decay gradient, smoothly varying along the entire solution layer. In another experiment, we used the cover cut into several cuboid pieces (as shown in Figure 1B), each of which was preheated separately to a different temperature, which after assembling and put into contact with the solution produced in its thin layer a staircase temperature profile. For the same initial concentrations of reactants, as indicated in the caption to Figure 2, the essential results of all experiments were qualitatively the same; i.e., the existence of a temperature gradient caused the spatial separation of the solution into the blue and yellow zones, with the heat transport in the latter case being slower than for the continuous polycarbonate phase. This limitation in the heat transport between the neighboring cuboic lids was confirmed by the observation that the border between the blue and yellow zones of the solution coincided with the position of the separation between the lids. The reactor of such a construction operates thus like the system with a practically steady-state staircase temperature distribution, at least within the time scale of the experiment lasting for ca. 3 min. In consequence, the reported progress of the color front was in

∂T 1 = DT∇2 (T ) − ∂t cp

∑ ΔHivi i

(1)

where DT is a thermal diffusivity coefficient, cp is the heat capacity of the solution, and ΔHi and νi are the enthalpy and the reaction rate of the ith reaction step, respectively. From the quantitative point of view, it was also important to note that DT, equal to ca. 10−3 cm2 s−1 in aqueous media, is ca. 2−3 orders of magnitude higher than the coefficients of molecular diffusion, which allowed us to consider the latter phenomenon negligible in the interpretation of observed luminescent patterns.11 We keep this assumption also in the present model which corresponds to the realistic time scale of the experiment of ca. 3 min, when the progress of molecular diffusion should not exceed ca. 0.5 cm (i.e., the maximum extent of the diffusion profile estimated as 6 × (π × D × t)1/2), with the reactor length equal to 7 cm. In our opinion, in the model that is oriented on the reproduction of the essential source of the pattern formation in the H2O2−S2O32−−H+−Cu2+ system, further simplifications to the temperature dynamics, described by eq 1, can be made. First, sinceaccording to section 3.1the experimentally measured temperature effect of the oxidation of thiosulfates by hydrogen peroxide is relatively low, it is possible to assume all E


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The Journal of Physical Chemistry B ΔHi = 0. In other words, in our calculations, the only source of initial temperature distribution will be the externally imposed temperature gradient, and then eq 1 simplifies to ∂T = DT∇2 (T ) ∂t

(2)

One can consider further simplifications. For DT = 10−3 cm2 s−1, the diffusion profile of temperature, within 3 min of the experimental observation of the color spatiotemporal front, may reach a distance of ca. 4.5 cm, i.e., not exceeding the total length of the reactor (7 cm). Assuming that the heat capacity of the polycarbonate cover is high enough to prevent its significant cooling, and based on the observation of the color front progressing in the solution being in contact with separated cover lids, we postulate that, within the time scale of the experiment, for the purposes of the explanation of the essential source of the phenomenon, one can assume a constant in the time scalar field of temperature: ∂T =0 ∂t

(3)

The validity of this assumption will be additionally verified below by numerical calculations. In order to introduce further the essential source of the traveling front, i.e., the thermokinetic coupling between the local chemical reaction rate and temperature, we assumed the Arrhenius-type dependence for the rate-determining step (rds): k rds(T ) = A e−Erds / RT

(4)

At this step of the construction of the model, one has to refer to the particular chemical reaction mechanism for the H2O2− S2O32−−H+−Cu2+ system. Among available literature suggestions,16 we found the simplified model by Orbán et al.17 appropriate for our calculations, with the flow parameter k0 set to zero, as our model describes the batch reactor. Also, in order to ensure optimum concordance between our experimental and numerical results, shown later in Figure 6, we appropriately (cf. Table 1) modified two parameters (rate constants k1 and k1′) of the original model. The premise for the modification of the k1′ constant was to keep the product k1′[Cu2+] the same as in the earlier work:17 since there [Cu2+] = 10−5 M and k1′ = 3 × 107 M−3 s−1, but now [Cu2+] = 2.5 × 10−6 M, the actual k1′ was modified by a factor of 4, i.e., to a value of 1.2 × 108 M−3 s−1. In the next step, the k1 value was adjusted in order to achieve the maximum concordance with experimental results at T = 298 K (cf. Figure 6). The five essential dynamic variables are a = [H2O2] ≡ [A], b = [S2O32−] ≡ [B], c = [HOS2O32−] ≡ [C], d = [OH−] ≡ [D], and h = [H+] ≡ [H]. Formally, the spatiotemporal dynamics of every concentration (ci) is strictly described by the system of the partial differential equations (PDEs) of a general form ∂ci = f (c1 , c 2 , ..., c5) + Di∇2 (ci) ∂t

Figure 6. Model pH−time series for the homogeneous, isothermal H2O2−S2O32−−H+−Cu2+ system plotted for various temperatures in the range 288−313 K, indicated at corresponding curves. Initial model concentrations of species (equal to initial ones in a real batch stirred reactor): a ≡ [A] = 0.1 M, b ≡ [B] = 0.01 M, c ≡ [C] = 0 M, d ≡ [D] = 0 M, h ≡ [H] = 0.002 M, and [Cu2+] = 2.5 × 10−6 M = const. Kinetic parameters and temperatures: (A) calculations performed for literature17 values of k1 = 0.019 M−1 s−1 and k1′ = 3 × 107 M−3 s−1 and temperatures of (1) 310 K, (2) 307 K, (3) 305 K, (4) 302 K, (5) 300 K, (6) 297 K, and (7) 295 K; (B) calculations performed for adjusted by us values of k1 = 0.06 M−1 s−1 and k1′ = 1.2 × 108 M−3 s−1 and temperatures of (1) 310 K, (2) 307 K, (3) 305 K, (4) 302 K, (5) 300 K, (6) 297 K, (7) 295 K, (8) 292 K, and (9) 290 K.

(5)

with f(c1, c2, ..., c5) being the kinetic terms. However, due to the above-mentioned assumed negligible contribution from the molecular diffusion of species (i.e., Di = 0), all kinetic equations are simplified to the form appropriate for homogeneous systems da = −k1ab − k1′abd[Cu 2 +] − 3k 2ac dt

db = −k1ab − k1′abd[Cu 2 +] − k 3bc dt

(7)

dc = k1ab + k1′abd[Cu 2 +] − k 2ac − k 3bc dt

(8)

dd = k1ab + k1′abd + k 3bc − k −4dh + k4[H 2O] dt

(9)

dh = 3k 2ac − k −4dh + k4[H 2O] dt

(10)

with [H2O] being practically constant and assumed equal to the molar concentration of pure water. In this mechanism, we consider the reaction step A1 the slowest and thus rate determining. This assumption is corroborated by the lowest value of the rate constant k1,

(6) F


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Table 1. Reaction Steps, Corresponding Rate Laws, and (Original17 and Modified by Us) Rate Constants for the Model of the Batch Thiosulfate−Hydrogen Peroxide−Cu2+ System17a step A1

reaction

H 2O2 + S2O32 − → HOS2O3− + OH− (occurring along parallel noncatalytic and catalytic (Cu2+) pathways)

A2

3H 2O2 + HOS2 O3− → 2SO4 2 − + 3H+ + 2H 2O

A3

HOS2O3−

A4

H 2O ⇄ H+ + OH−

+ S2O3

2−

→ S4 O6

2−

+ OH

−

rate laws

rate constants

v1 = k1[H 2O2 ][S2O32 −] + k1′[Cu 2 +][OH−][H 2O2 ][S2O32 −]

k1 = 0.06 M−1 s−1 (original value: 0.019 M−1 s−1), k1′ = 1.2 × 108 M−3 s−1 (original value: 3 × 107 M−3 s−1)

v2 = k 2[H 2O2 ][HOS2O3−]

k2 = 1 M−1 s−1

v3 =

k 3[HOS2O3−][S2O32 −]

v4 = k4[H 2O] +

−

v−4 = k −4[H ][OH ]

k3 = 50 M−1 s−1 k4[H2O] = 10−3 M s−1, k−4 = 1011 M−1 s−1

Step A1 is considered to be non-elementary. The concentration of Cu2+ species is assumed constant in time and equal to 2.5 × 10−6 M. All rate constants correspond to 298 K.

a

compared to k2 and k3, as well as by our numerical experiments in which the incorporation of the Arrhenius dependence (eq 4) only to the expression for k1 allowed the essential expected dynamics of the model system to be kept. Furthermore, since in its two parallel pathways k1 ≪ k1′, or activation energy E1 ≫ E1′, reaction A1 will reveal a much stronger sensitivity to temperature variations than parallel reaction with the rate constant k1′. Thus, in model calculations, only the rate constant k1 corresponding to the relevant reaction step will be affected by temperature, according to eq 4. Accordingly, eq 6 attains the following form da = −(A e−Ea / RT )ab − k1′abd[Cu 2 +] − 3k 2ac dt

than the dependences depicted in Figure 6B, satisfactory linear correlations, with practically unit slope and close to zero intercepts, between both the experimental and theoretical induction periods (Figure 7A) and the experimental and theoretical duration times of high pH state (Figure 7B). Having the essential kinetic properties of the model system confirmed in this way, we performed the model calculations of the spatiotemporal progress of the pH wave in the spatially distributed system with the imposed steady state (cf. eq 3), arbitrarily chosen temperature distribution. Kinetic eqs 7−11 were numerically solved for an exemplary temperature profile (cf. Figure 8B) within the range 298−310 K, for the set of rate constants adjusted by us. Figure 8A illustrates the model progress of the low pH front that exhibits a striking similarity with the experiment, including the total time (ca. 5 min) which the model system requires to cover the entire span of reaction space with the yellow zone of decreased pH. This confirms, independently of the experimental results with separated segments of the heated cover, that the time scale of the experiment is essentially determined by the coupling between the local reaction kinetics and imposed temperature, the temporal diffusion of which along the reactor can thus be considered negligible. There are only certain differences regarding the times of the onset and the progress of the yellow wave which in the model starts at 100 s, while in real experiments it occurs after ca. 60 s following the preparation of the reaction mixture. This discrepancy is an obvious consequence of the simplifications underlying the kinetic reaction scheme which cause the model temporal dynamics of pH to not perfectly match a more complex, asymmetrical shape of this dependence reported in the experiment, and moreover, the pH at its maximum is temperature-dependent in the experiment, while in the kinetic model it reaches always a similar value, close to ca. 11 for every assumed T.

(11)

where the pre-exponential Arrhenius factor A constant, equal to 6.87 × 107 M−1 s−1, was estimated on the basis of known values of k1 (at 298 K) and the activation energy, determined as described above. Numerical calculations were performed for two types of the H2O2−S2O32−−H+−Cu2+ system: (i) the homogeneous batch one, of a given, uniform solution temperature and (ii) the spatially distributed, one-dimensional, batch one, with the steady state, assumed distribution of the solution temperature. Results of the first group of calculations, showing the temporal evolution of the model system’s pH, are presented in Figure 6. For both sets of model kinetic data, the pH−t curves reveal striking qualitative but only semiquantitative similarities to the experimental characteristics of the system, shown in Figure 2. Qualitative agreement means that the induction period is only slightly dependent on temperature within its studied range, while the time window of enhanced pH state shrinks significantly with increasing temperature. Quantitative disagreement manifested itself in the overestimation of the induction period for the factor of ca. 3, and the duration of enhanced pH state for the factor of ca. 5, if the original17 set of rate constant values was used. This is the point where, in order to obtain satisfactory agreement with the experimental duration of that region for the same temperature, we appropriately adjusted the k1 (and k1′) rate constants to the values indicated in Table 1, as indicated above. Although one could theoretically consider also modifications of k2 and k3 rate constants, our analysis showed that only an increase in k1 and k1′ produces the expected decrease in both characteristic times; the increase in k2 would cause further undesirable increase in the induction period, while the increase in k3 would also undesirably elongate the duration of the enhanced pH state. Figure 7 shows, even more distinctly

4. CONCLUSIONS Following our earlier discovery of luminescent phase waves in the H2O2−SCN−−OH−−Cu2+ system,11 the color traveling pH fronts, reported by us here for the H2O2−S2O32−−H+−Cu2+ dynamical system, constitute one more example of spatiotemporal patterns caused by thermokinetic coupling of the nonlinear kinetics with the inhomogeneous temperature field. Both of these systems exhibit however different detailed mechanisms of the thermally driven pattern formation. In the former one, the system was permanently oscillatory, and the inhomogeneity in temperature distribution caused appropriate G


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Figure 7. Comparison of the experimental and model effect of temperature on (A) the induction period and (B) the duration time of enhanced pH state for adjusted k1 and k1′ rate constants. Parameters of linear regression: (A) y = 1.023 (±0.076)x + 2.695 (±2.449), r2 = 0.878; (B) y = 0.843 (±0.031)x + 18.093 (±8.410), r2 = 0.967. The y and x symbols represent the abscissa and ordinate, respectively, in the proper coordinate system. Initial concentrations of reactants in the experimentthe same as those for Figure 2. Initial concentrations of model speciesthe same as those for Figure 6. Calculations and measurements were made for temperatures ranging from 288 to 313 K, with 1 K intervals.

Figure 8. (A) Graphical representation of calculations of the onset and spatiotemporal progress of the decreased pH wave along the onedimensional reactor, under the assumption of the model steady state, arbitrarily chosen temperature distribution (∂T/∂t = 0) shown in part B. The progress of the pH wave in part A was calculated on the basis of eqs 7−11, thus without taking into account both molecular and heat diffusion. The model 1-D reactor of 7 cm length was divided into 13 segments, and the distribution of pH was calculated every 15 s, with the total time equal to 300 s, i.e., close to the duration of the experiment visualized in Figure 5. Regions of the diagram in part A denote the following: (I) pH ≤ 8.0; (II) 8.0 < pH < 9.6; (III) pH ≥ 9.6, corresponding to analogous color variations of the thymol blue indicator.

spatial dispersion of the frequency of oscillations, assisted with bursts of luminescence at those places where the oxidation of luminol was most intensive. This dispersion gave rise to the impression of light wave propagation, being in fact only the phase waves. The presently studied H2O2−S2O32−−H+−Cu2+ system does not exhibit sustained oscillations under batch conditions, but only a single transient increase in solution pH, following the induction period. Since the duration of high pH state, contrary to the induction period, appeared to be strongly dependent on temperature, creation of a temperature distribution along the quasi-one-dimensional thin-layer reactor resulted in the onset and spatial expansion of the low pH front, visualized, in the presence of thymol blue indicator, as the yellow zone starting from the region of enhanced temperature and progressing toward the blue homogeneous zone of

relatively high pH. In the theoretical approach, the construction of the kinetic model was focused on the maximum concordance between the shape and quantitative characteristics of the temporal variations of solution pH at different temperatures. In discussions of the detailed assumptions of the model, it was indicated that for the explanation of the essential source of color traveling fronts in the studied system it was not necessary to invoke explicitly the molecular diffusion of reacting species and even of the (much faster) diffusion of temperature, as the phenomenon reported would be observed also for the steady state, e.g., linear or exponential gradient of temperature distribution along the solution. We outlined also the theoretical possibility for the formation of a narrow blue stripe traveling H


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(15) Nogueira, P. A.; Oliveira, H. C. L.; Varela, H. Time Evolution of the Activation Energy in a Batch Chemical Oscillator. J. Phys. Chem. A 2008, 112, 12412−12415. (16) Schiller, J. E. A New Reaction of Cyanide with Peroxide and Thiosulfate at pH 7−9. Inorg. Chem. 1987, 26, 948−950. (17) Orbán, M.; Kurin-Csörgei, K.; Rábai, G.; Epstein, I. R. Mechanistic Studies of Oscillatory Copper(II) Catalyzed Oxidation Reactions of Sulfur Compounds. Chem. Eng. Sci. 2000, 55, 267−273.

through a yellow zone in the same system, but sophisticated parameters of the temperature distribution which are required to create respective thermokinetic coupling leave this variant of the experiment for the future investigations. Our recent results open possibilities for further studies of the little-studied structure-forming effect of a scalar temperature field combined with specific chemical kinetics in liquid solutions.

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AUTHOR INFORMATION

Corresponding Author

*Phone: +48-22-55 26345. Fax: +48-22-55-26434. E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS M.J. acknowledges financial support from the research grant 120000-501/86-DSM-110200, from the Faculty of Chemistry, University of Warsaw.

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REFERENCES

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The Formation and Spatiotemporal Progress of the pH Wave Induced

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