2760
J. Phys. Chem. 1989, 93, 2760-2764
Experimental Study of the Chemical Waves in the Ce-Catalyzed Belousov-Zhabotlnskii Reaction. 2. Concentration Profilest Zsuzsanna Nagy-Ungvarai,**tStefan C. Muller,*John J. Tyson,*.s and Benno Hess* Max- Planck- Institut fur Erniihrungsphysiologie, Rheinlanddamm 201, 0-4600 Dortmund 1 , FRG, and Department of Biology, Virginia Polytechnic Institute and State University, Blacksburg, Virginia 24061 (Received: July 15, 1988; In Final Form: September 30, 1988)
Systematic measurements of wave profiles in the Ce-catalyzed Belousov-Zhabotinskii reaction were carried out in a wide concentration range of the reagents. The shape of the wave profiles strongly depends on the initial composition of the system. It shows a large variety from very sharp to very broad wave fronts, which was modeled by use of the revised set of rate constants of the Belousov-Zhabotinskii reaction. During the spatial pattern formation only a moderate amount of the catalyst takes part in the chemical reactions. The change in the catalyst concentration is 6-30% and always lies in the more reduced region.
Introduction
Over the past years much effort has been devoted to the study of spatio-temporal patterns formed in systems far from equilibrium.’-3 Although a lot of theoretical work has been done, only a few papers deal with the experimental determination of quantitative data of these systems. Most of the experimental studies are concerned with the velocity of chemical waves as a function Wood of the chemical composition of the reactant and Rosss measured for the first time concentration profiles in a one-dimensional arrangement along a line through the center of a circular wave in the ferroin-catalyzed Belousov-Zhabotinskii (BZ) reaction. In the same system, Muller, Plesser, and Hess9 and later Pagola and Vidallo reported on two-dimensional spectrophotometric measurements of spiral and concentric waves by digital video techniques. All of these concentration profiles were measured in a narrow range of initial concentration of the reagents” and show a similar, characteristic relaxation-type shape? a steep increase in the concentration of the oxidized form of the catalyst, followed by a slow decrease. However, by extending the concentration ranges, we find that the shape strongly depends on the initial conditions. In this paper we report on systematic measurements of wave profiles in the Ce-catalyzed BZ reaction in a wide concentration range as a function of the initial composition of the system and compare the results to numerical simulations based on an extended Oregonator model of the reaction. Experimental Section
The experimental procedure was the same as reported in the first part of our studies about Ce waves.I2 Malonic acid (Fluka), N a B r 0 3 , NaBr (Riedel de Haen), Ce(NH,J2(N0& (Aldrich), and sulfuric acid (Merck) were of analytical reagent grade. Laboratory distilled water was redistilled twice. All reactant solutions were filtered through 0.45-pm Millipore filters and thermostated to 25.0 f 0.1 OC. The solutions were then mixed to give the initial concentrations in a range as summarized in Table I. A volume of 3.63 m L of the mixture was placed in an optically flat Petri dish (diameter 6.8 cm) leading to a layer depth of 0.56 f 0.03 mm in the dish center, as derived by an optical calibration. Oxidized circular waves in the reduced solution layer evolved spontaneously around impurities or were triggered by a Ag or a hot Pt wire. The light absorption in the sample layer thermostated to 25.0 f 0.1 O C was measured by a UV-sensitive 2D spectr~photometer.’~ The parallel, highly stable light beam of a Cermax LX 300 UV lamp (Kontron) filtered by two optical filters (Schott, UV-IL and UG 11; resulting wavelength 344 nm; filter bandwidth 9 nm) illuminated the sample layer. The 2D distribution of the transmitted light through a 10 X 10 mm2 layer area was imaged by a HamDedicated to Professor Richard M. Noyes on the occasion of his 70th birthday. Max-Planck-Institut fur Erniihrungsphysiologie. 8 Virginia Polytechnic Institute and State University. +
*
0022-3654/89/2093-2760$01.50/0
TABLE I: Composition of the Reactant Solution after Completion of the Bromination Reaction of Malonic Acid (MA) to Bromomalonic Acid (BrMA) in the Process
Br03- + 2Br-
reactant
Br03HzS04 MA
BrMA Ce(1V)
+ 3MA + 3H’
= 3BrMA
+ 3H20
series A concn stand. range, M concn, M 0.12-0.57 0.3 0.16-0.91 0.41 0.01-0.366 0.03
series B concn stand. range, M concn, M 0.2-0.6 0.3 0.2-1.0 0.41 0.04-0.5 0.12
0.09
0.09
0
0
0.003-0.012
0.006
0.003-0.012
0.006
amatsu video camera system with a Nikon UV Nikkor photolens on a 512 X 512 array of picture elements (pixels) with 256 digital unit intensity resolution. Selected images could be stored on the magnetic disk of a Perkin-Elmer 3230 computer at a frequency up to 30 frames min-’ in the form of intensity values Z(x,y,t) of the pixels at space coordinates x,y and time t . A bias value of 23 gray levels was subtracted from each of the stored intensity data. The pixel noise of about f 4 gray levels was reduced by calculating moving averages of 3 X 3 pixel areas. Intensity values that are located along a line through the wave center were presented as intensity profiles. For conversion of the intensity data into concentrations, Ce(1V) absorption spectra were taken that yielded an extinction coefficient of 4700 M-’ cm-’ at 344 nm. Using this value, concentration profiles could be established based on the two-dimensional version of the Beer-Lambert law: 1 C(X,Y,t) = log [Io(x,Y)/I(x,YJ)l Slopes of the concentration profiles were determined from the derivative of spline fits of the measured c ( x y , t ) data.I3 The concentration of Ce(IV) and therefore the light absorption in the initial (“reduced”) solution layer are a function of the composition of the solution. In order to obtain an appropriate intensity in this initial solution layer, the incident intensity of the monitoring light beam had to be changed from experiment to experiment. This changes the value of Zo(x,y),which was taken (1) Field, R. J.; Burger, M., Eds.; Oscillations and Traveling Waves in Chemical Systems; Wiley-Interscience: New York, 1985. (2) Showalter, K.; Tyson, J. J. J . Chem. Educ. 1987, 64, 742. (3) Ross, J.; Muller, S . C.; Vidal, C. Science 1988, 240, 460. (4) Luther, R. Z . Electrochem. 1906, 12, 596. (5) Field, R. J.; Noyes, R. M. J . Am. Chem. SOC.1974, 96, 2001. (6) Kuhnert, L.; Krug, H.-J. J . Phys. Chem. 1987, 91, 730. (7) Showalter, K. J . Phys. Chem. 1981, 85, 440. (8) Wood, P. M.; Ross, J. J . Chem. Phys. 1985, 82, 1924. (9) Muller, S . C.; Plesser, Th.; Hess, B. Science 1985, 230, 661. (10) Pagola, A,; Vidal, C. J . Phys. Chem. 1987, 91, 501. (1 1) Winfree, A. T. Science 1972, 175, 634. (12) Nagy-Ungvarai, Zs.; Tyson, J. J.; Hess, B. J. Phys. Chem. 1989, 93,
707.
(13)
Muller, S . C.; Plesser, Th.; Hess, B. Physica D
0 1989 American Chemical Society
1987, 24, 71
and 87.
The Journal of Physical Chemistry, Vol. 93, No. 7, 1989 2761
Ce-Catalyzed Belousov-Zhabotinskii Reaction TABLE II: Parameters of the Concentration Profiles of the Ce Waves
[BrOy], M
initial concentrations [Ce(IV)], [H+], [MA], M M M
0.57 0.12 0.3 0.3 0.3 0.3 0.3 0.3 0.6 0.2 0.3 0.3 0.3 0.3 0.3 0.3
0.006 0.006 0.012 0.003 0.006 0.006 0.006 0.006 0.006 0.006 0.012 0.003 0.006 0.006 0.006 0.006
0.41 0.41 0.41 0.41 0.91 0.16 0.41 0.41 0.41 0.41 0.41 0.41 1.0 0.2 0.41 0.41
[BrMA], M
amplitude A[Ce(IV)], mM
max slope, mM/mm
halfwidth, mm
0.09 0.09 0.09 0.09 0.09 0.09 0.09 0.09 0 0 0 0 0 0 0 0
0.80 0.40 1.15 0.40 1.15 0.31 0.60 0.65 0.68 0.32 0.77 0.28 1.27 0.17 0.55 0.80
3.63 3.34 4.48 3.34 5.59 2.85 6.08 2.41 3.90 2.29 6.16 2.85 6.27 1.88 6.95 3.80
2.00 0.88 2.12 1.08 2.04 0.74 0.40 2.36 2.10 1.20 1.28 1.12 1.56 0.46 0.34 5.44
0.03 0.03 0.03 0.03 0.03 0.03 0.366 0.01 0.12 0.12 0.12 0.12 0.12 0.12 0.5 0.04
symmetry parametef at at wavehalffront width bottom 0.67 0.26 0.30 0.35 0.69 0.28 0.54 0.27 0.96 0.43 0.68 0.62 1.03 0.53 0.55 0.75
0.36 0.13 0.15 0.18 0.40 0.19 0.26 0.18 0.30 0.27 0.26 0.42 0.44 0.53 0.30 0.35
OThe symmetry parameter is defined as the ratio of oxidizing and reducing segments of the wave profiles. reduced) layer.
x-fold incr in concn
BrOC Ce(IV) H+ MA
4.75 4 5.69 36.6
Br0,Ce(I\ '1
3 4 5 20
x-fold incr in amplitude Series A
obsd
theor
1.7 0.9 1.35 0.5 2.0
0.4 0.03 0.2 0.15 0.75
0.28 0.15 0.11 0.17 0.33
0.061 0.0025 0.0084 0.034 0.084
0.8 1.5
0.025 0.2
0.13 0.25
0.0045 0.020
fox
1.3
* [Ce(IV)] in the more excitable (more
series A
TABLE 111 Effect of Change of Component Concentrations on the Amplitude and Maximum Slope of the Ce(IV) Concentration Profiles
varied component
[Ce(lv)l,b mM obsd calcd
series B
x-fold incr in max slope
2 2.88 3.71 0.92
1.09 1.34 1.96 2.52
2.13 2.75 7.47 0.69
1.70 2.16 3.34 1.82
Series B H+
MA
as the light intensity after transmission through the covered Petri dish containing a 0.56-mm layer of a cerous sulfate solution in sulfuric acid. Therefore, the determination of absolute concentration values in the concentration profiles requires a calibration for each measurement. We performed this calibration procedure in selected experiments; in all other cases the concentration profiles were calibrated in terms of concentration changes. The calibration curves (log Zvs Ce(1V) concentration) of the selected experiments are parallel lines, showing that the same A log Zvalues correspond to the same A[Ce(IV)] values, as to be expected from the Beer-Lambert law.
Results The physical and chemical properties of the cerium catalyst allowed us to study the Ce waves in a wider concentration range and for a longer time than is possible in case of the ferroin waves. One of the reactant concentrations was varied over the range shown in Table I while all other concentrations were maintained constant. Selected wave profiles showing the effect of the change in the different concentrations are shown in Figure 1. In the wide concentration range used the profiles of the cerium waves show a considerable variety in their shape, similar to the variety of redox potential or [Br-] vs time curves measured in stirred solutions of the system.14 The wave profiles are not always as uniform as in the narrow concentration range used for studying the chemical waves in the ferroin-catalyzed BZ reaction. Amplitude and frequency, slope and width of the oxidized area, concentration of the oxidized and reduced form of the catalyst, and the symmetry parameter in a wave front depend in a characteristic way on the (14) 8649.
Field, R. J.; Koros, E.;Noyes, R.M.J. Am. Chem. SOC.1972, 94,
L
'
I
05
0
2
4
6
8
w 0
x
2
4
6
8
(mm)
Figure 1. Concentration profiles of trigger waves along space coordinate x in solutions with the following initial concentration. Series A: varied concentrations are (a) 0.57 M BrOC, (b) 0.12 M Br03-, (c) 0.012 M Ce(IV), (d) 0.003 M Ce(IV), (e) 0.91 M H+, (f) 0.16 M H+, (g) 0.366 M MA, and (h) 0.01 M MA, constant concentrations have the values 0.3 M Br03-, 0.03 M MA, 0.09 M BrMA, 0.006 M Ce(IV), and 0.41 M H2S04. Series B: varied concentrations are (a) 0.6 M BrO,-, (b) 0.2 M BrOC, (c) 0.012 M Ce(IV), (d) 0.003 M Ce(IV), (e) 1.0 M H2S04, (0 0.2 M H2S04,(9) 0.5 M MA, and (h) 0.04 M MA; constant concentrations have the values 0.3 M Br03-, 0.12 M MA, no BrMA, 0.006 M Ce(IV), and 0.41 M H2S04.
initial concentrations of the reagents. The most important parameters of the wave profiles are collected in Table 11.
2162
The Journal of Physical Chemistry, Vol. 93, No. 7, 1989
41
Nagy-Ungvarai et al.
a
60
40
-4 1
6l n
4
2
e
T
a
T
b
A
20
0
Figure 2. Concentration gradients of trigger waves along space coordinate x derived from (a) Figure lg, series B, and (b) Figure la, series B.
All five initial reactants influence the parameters of the profiles; however, the degree of their effect is different as we have shown for the two most important parameters, amplitude and maximum slope, in Table 111. H+, Ce(IV), and Br03- have a large effect, but in [MA] very big changes are necessary for a noticeable effect in the wave parameters. Solutions with and without an initial BrMA content (series A and B in Figure 1, respectively) are similar in many respects; they differ from each other mainly in their shape. In the profiles of solutions without any initial BrMA content the segment of increasing Ce(IV) concentration can be subdivided into two parts: a fast and a following slow decrease in [Ce(IV)] (most pronounced in Figure la,h, series B). These two parts cannot be clearly distinguished in wave fronts of solutions with an initial BrMA content. The slow increase in [Ce(IV)] completely disappears at high MA concentrations in both series A and B. In Figure 2 we plotted the derivatives of the spline fits of the measured concentration profiles for two typical cases without and with a slow Ce(IV) increase. The concentration gradient curve in Figure 2a derived from the concentration curve in Figure lg, series B, has an almost symmetric positive peak whereas the positive peak in Figure 2b has a shoulder corresponding to the biphasic [Ce(IV)] increase of the respective concentration curve in Figure la, series B. As can be seen from the amplitude data of Table 11, only a moderate amount (ca. 6-30%) of the catalyst is converted in the chemical reactions in a wave front. Varying the initial conditions changes not only the Ce(IV) conversion but also its position on the total catalyst concentration scale (cf. [Ce(IV)] in the excitable layer). The catalyst conversion in a wave front always lies in the more reduced half of the total catalyst concentration. We represent these data in Figure 3 as percentage changes of the [Ce(IV)]. The minimum value of the [Ce(IV)] corresponds to the concentration of the excitable layer or in most cases the more reduced part in a wave (Table 11); the maximum one corresponds to the highest [Ce(IV)] in the first wave.
Modeling In the first part of our studies about waves in the Ce-catalyzed BZ reaction12 we were able to reproduce in quantitative detail our experimental measurements of wave velocities using a slight modification of the Oregonator model of the reaction. In this paper we test the model further by comparing computed and measured wave profiles. The model is described in detail in ref 12, so we
Figure 3. Change in [Ce(IV)] in percent of the initial (total) [Ce(IV)] for the first wave in solutions identical with those in Figure 1, series A, (b), (a), (e), (h), (g), (d) and ( 4 , respectively.
present here only certain essential facts. We solve the partial differential equation system
where x = [HBr02]/[HBr02],,,,
z = [Ce(Iv)l/[Cel,,f,
[HBr02],,f = k5HA/2k4 [Celrcf= (ksHA)’/2k&jB
t = time/time scale, time scale = l/kjB
s = space/space scale, space scale = (k5HAD)’i2/kjB 6
= kjB/kSHA, = 2k,k4/kZkS, ~ - 5= 2k,k_S(kjB/ksk@A)’
and
H = [H2S04], A = [Br03-], B = [MA]
+ [BrMA]
Furthermore u = u(x,z) =
- Z)
(C
+ [(C
2X(2
+ K+Z)
-Z)’
+ 8 ~ - 5 ~ +( 2K+Z)]’i2
with c = [Cel,,,~/[Cel,,f,
K+
= k+k~HA/2k&jB
The stoichiometric coefficient h we take to be 1.5, as in ref 12. The rate constants k2, k3, k4, k5, k+, k6, k+, and kj are given in Table I11 of ref 12. With these values of rate constants and the standard concentrations given in Table I, series A, we find that [HBr02],f = 1.2 X
M
[Ce],,, = 0.125 M time scale = 21 s space scale = 1.8 mm t
= 0.01, q = 0.0002,
K+
= 0.01,
K+
= 300
In estimating the space scale we have used D = 1.5 X
cm2
The Journal of Physical Chemistry, Vol. 93, No. 7, 1989 2763
Ce-Catalyzed Belousov-Zhabotinskii Reaction
1.5 r
TABLE I V Parameter Values Used in the Simulations of Experimental Series A' space time expt. c K - ~ K+ c scale, mm scale, s 2.5 21 la 0.0053 0.0028 572 0.0133 lb 0.025 0.0625 120 0.30 1.1 21 1.8 21 IC 0.01 0.01 300 0.096 1.8 21 300 0.024 Id 0.01 0.01 2.7 21 le 0.0045 0.0004 664 0.0097 116 0.32 1.1 21 If 0.026 0.43 lg 0.038 0.144 80 0.18 0.5 6 2.2 25 Ih 0.0083 0.0069 360 0.04
b 8
"In all cases, q = 0.0002 and h = 1.5.
s-I for the diffusion coefficient of H B r 0 2 . Equation 1 was solved numerically by using a finite-difference approximation to the diffusion operator (a2/ds2) and by first-order explicit Euler integration of the resulting set of ordinary differential equations. A trigger wave was allowed to circulate around a ring of adjustable circumference. The circumference of the ring was chosen to give wavelengths comparable to the observed periodicities in Figure 1, series A, Le., between 3 and 8 mm. The spatial step size was 30 Mm and temporal step size was 20 ms. These step sizes, though larger than the values used in ref 12, are sufficiently small to calculate wave forms of [Ce(IV)] within an accuracy of a few percent. Simulations were carried out with the parameter values listed in Table IV, and the calculated concentration profiles of [Ce(IV)] are illustrated in Figure 4, which should be compared with Figure 1, series A. The calculated wave forms are remarkably similar to the observed forms. Also, the amplitude of the wave front, [Ce(IV)lmax- [Ce(IV)Imi,, agrees quite well with the observed amplitude. However, the calculated values of [Ce(IV)lmin,which should be close to [Ce(IV)] in the resting excitable medium (eleventh column in Table 11), are consistently smaller than the observed values by 3- to 30-fold. We tried to correct this discrepancy by increasing q (which raises [Ce(IV)Imin)and decreasing K~ (which raises [Ce(IV)lmax), but we were unable to find parameter values which reproduced accurately both the absolute Ce concentrations and the wave speed.
Discussion Since chemical waves in the BZ reaction represent a coupling between diffusion and an autocatalytic process, changes in the Ce wave fronts are strongly connected to changes in the autocatalytic reaction, as well as other chemical processes influencing it. Ce(IV) is a product of the autocatalytic reaction sequence
+ Br03- + H+ = 2BrO2' + HzO BrOz' + cat"' + H+ = cat("+L)++ HBrOZ 2 H B r 0 2 = HOBr + Br03- + H+ HBrOz
(R5) (R6) (R4)
The autocatalytic reaction was studied under the conditions of oscillatory or excitable reaction mixtures ([Ce(IV)]
