Two-dimensional spectrophotometry of spiral wave propagation in the Belousov-Zhabotinskii reaction

Physica 24D (1987) 71-86 North-Holland, Amsterdam

TWO-DIMENSIONAL SPECTROPHOTOMETRY THE BELOUSOV-ZHABOTINSKII REACTION

OF S P I R A L WAVE P R O P A G A T I O N IN

I. EXPERIMENTS AND DIGITAL DATA REPRESENTATION Stefan C. M O L L E R , Theo PLESSER and Benno HESS Max-Planck-lnstitut f~r Erniihrungsphysiologie, Rheinlanddamm 201, 4600 Dortmund, Fed. Rep. Germany

Received 9 June 1986

The spiral-shaped wave of chemical activity propagating in a 1 mm layer of an excitable solution of the Belousov-Zhabotinskii reaction is analysed quantitatively for a 5 min time interval in a 4.5 × 4.5 mm area. The spatial distribution of the reaction catalyst and indicator ferroin is measured by Ught absorption at 490 nm with a video- and computer-based two-dimensional spectrophotometer. Digital images with 10/~m spatial and 256 digital units intensity resolution are acquired at a frequency of 20 per minute. Initially, the spiral has a pitch of 1.18 ram, a revolution time of 17.2 s, and the wavefronts propagate outwards with a velocity of 69/~m/s. After 5 rain these quantities have changed by up to 5%. Three-dimensional graphic procedures and image analysis reveal that the wave profiles are symmetricclose to the spiral center and become highly asymmetric during their outward displacement. The core of the spiral has a radius of 0.35 mm. Its center is a singular site (diameter < 30/tin) at which variations in ferroin concentration are at least 10 times smaller than in the surrounding area of spiral propagation. With time, the wave profiles become smoother and show spatial inhomogeneities.

1. Introduction Spatial patterns that form in unstirred chemical solutions kept under conditions far from equilibrium result from the nonlinear coupling of chemical reaction with transport processes such as diffusion or convection. They display a remarkable richness of geometric forms. The phenomen o n of chemical wave propagation in the Belousov-Zhabotinskii reaction is one of the most prominent examples. First observations date back more than 15 years [1, 2], and ever since this spatial organization process has attracted m a n y experimentalists and theoreticians [3-7]. An especially interesting case is the formation of the spiral-shaped waves of chemical activity propagating in a quiescent but excitable medium of this reaction [8, 9]. On the basis of the available experimental material considerable efforts have been m a d e to propose various theoretical models for spiral formation that are overviewed in ref. 5.

Recently, a detailed experimental investigation of the spatial distribution of the reaction catalyst and indicator ferroin in a circular wave has been reported in terms of measurements of transmitted light intensities recorded on a one-dimensional photodiode array [10]. However, because of limited methodology, no comprehensive quantitative description has been put forward of the spatial distribution of ferroin concentration in a solution layer in which a spiral wave propagates. Spectrophotometric measurements with high spatial and temporal resolution are now feasible for two-dimensional patterns by using computerized video techniques and first results have been reported by the authors [11-13]. The technical aspects of the apparatus for 2D spectrophotometry of the spatial distribution of chemical species are presented in the following section. In section 3 the experimental procedure f o r initiating spiral-shaped waves in the Belousov-Zhabotinsldi reaction is specified. This

0167-2789/87/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

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S.C. Miiller et a L / Two-dimensional spectrophotornetry of spiral waves

section also comprises a description of the recorded image sequence and basic considerations concerning the evaluation of the digitally stored image data with a flexible package of software routines. Quantitative results, mostly in terms of concentrations of the oxidized form (ferriin) of the reaction catalyst and indicator, are reported in section 4. A vivid description of the characteristic features contained in a single spiral image is given and the singular properties of the core of the spiral wave are derived from a time sequence of images covering one full spiral revolution. From a longer lasting time sequence of images, it follows that the location of the core region of the spiral patterns remains stable in time but that remarkable changes become detectable after about 12 min, in that spatial inhomogeneities evolve which are superposed on the regular spiral shape. In the concluding remarks reference is made to a second paper in which the parameters of spiral propagation, as determined by graphical methods, are compared with results that are obtained by fitting spiral functions to the measured iso-concentration lines [14].

2. Apparatus The main components for computerized digital spectrophotometry in two dimensions are schematically drawn in fig. 1 (see also [11]). The light source is a stabilized 150 W high-pressure mercury lamp. An expanded parallel beam of white light is produced by passing through a series of lenses. A specific wavelength is selected by an interference filter. In this instance, a filter with peak transmittance at 490 nm and 8 nm band width at 10% peak transmittance is selected for maximum ferroin absorption (see section 3). Neutral density filters are inserted into the light path for adjustment of the intensity of the light beam. Its diameter can be modified with an iris diaphragm. After appropriate optical adjustment and deflection by an optically fiat mirror, a vertically directed beam of 2 cm diameter with a spatial homogeneity better than + 1.5% per cm enters a thin layer (ap-

proximately 1 mm deep) of the reactant mixture in the object plane. The petri dish containing the sample layer is constructed from a Pyrex glass plate of 6.8 cm diameter, the optical homogeneity of which was checked with this apparatus. A shallow cylinder of Pyrex glass (12 mm long) was glued to the bottom plate. Before use the dish was siliconized by a procedure described in [15] in order to reduce the undesirable influence of microscopic scratches and inhomogeneities at the boundary of the dish. Another mirror and a photolens system with close-up attachments (Nikon-Nikkor, focal length 85 mm) serve for imaging a small square section of the light transmitted through the object plane on the CdSe photoconductive target of a TV camera (Hamamatsu C-1000) equipped with a vidicon tube (Hamamatsu N983). The tube has a spectral response from 200 to 750 nm and an image raster resolution of 512 × 512 picture elements (pixels). For these investigations the imaged area is 5.1 x 5.1 mm2. Consequently, the raster resolution corresponds to a spatial resolution of 10/~m per pixel. (Due to systematic errors in the imaging characteristics of the target the vertical extent of the recorded area is about 2% larger than the horizontal extent. This leads to a small geometric distortion of the image data.) The optical components and the TV equipment are mounted on a vibration isolated optical table. The output signal of the control unit of the camera system (fig. 1) is routed to a black and white monitor for direct observation and to the analog/digital converter in the video frame buffer (VTE, Digital Video GmbH, Braunschweig, FRG) which converts the video signal into one out of 256 intensity levels (grey levels). A bias Value of 20 digital units is preset for the nonilluminated target and has to be subtracted if the measured grey levels are evaluated in terms of light intensities. The grey level of each of the pixels is stored as one byte in the digital memory array of the video frame buffer. In the experiments presented here slightly reduced frames of 450 × 450 pixels (4.5 x 4.5 mm 2 areas) were recorded only. Thus,

73

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boundary effects of the target are excluded and the storage space per image is reduced. The translation of grey levels into colors is facilitated by color.look-up tables stored in the buffer. Pseudocolor versions of the images can be displayed on the color monitors. Some basic considerations for the use of the color technique are given in refs. 13, 16.

The TV equipment is integrated in a multiuser computer department based on a Perkin Elmer 3230 machine with 4 megabytes of main memory. The quantitative acquisition and digital storage of the 2D transmission data on magnetic disk is feasible at a maximum frequency of 30 frames per minute. This frequency is limited by the time constant of the camera target as well as the time

74

S.C. Miiller et al./ Two-dimensional spectrophotometry of spiral waves

necessary for transferring the data from the buffer to the disk. Both are in the order of 2 s. For these experiments an acquisition frequency of 20 frames per minute was chosen. A detailed description of the apparatus is given in [11]. The software used for evaluation of the image data is briefly described in section 3.4 and more comprehensively discussed in [13].

3. Experiments 3.1. Materials Solutions were prepared with reagent grade chemicals and distilled water. The three initial reactant solutions were sodium bromate in sulfuric acid, sodium bromide in water, and malonic acid in water. A 25 mM solution of ferroin, the tris (1,10-phenantroline) ferrous sulfate complex, was prepared by dissolving stoichiometric amounts of phenantroline and ferrous sulfate in 25 mM sulfuric acid. All solutions were filtered through 0.44 /~m Millipore filter and stored in separate containers. Concentrations were calculated from weights of dissolved chemicals. The extinction coefficient of the reduced (red) state of the catalyst was determined from absorbance measurements of a 0.5 mM solution in a 2 mm cuvette and was found to be 1.06 × 104 M -1 cm -1 at 490 nm. At this wavelength the absorption of the oxidized (blue) form of this indicator and reaction catalyst, ferriin, is only about 1% of that of ferroin and can be neglected within the experimental error. The values are in good agreement with the results in

[10]. 3.2. Sample preparation and spiral wave initiation A quiescent, excitable solution of the BelousovZhabotinskii reaction was obtained by preparing a mixture of 48 mM sodium bromide, 340 mM sodium bromate, 95 mM malonic acid, and 378 mM sulfuric acid from the three initial solutions in a final volume of 4.4 ml. The orange-brown color of the solution indicates the generation of

molecular bromine. Several minutes later, after complete evolution of bromine, this color disappeared. Subsequently, 0.7 ml of 25 mM ferroin solution was added with a final concentration of 3.5 mM. After repeated filtering a volume of 3.6 ml was poured into the siliconized dish (diameter 6.8 cm) which just prior to the experiment had been carefully cleaned from dust particles. The resulting solution layer with a depth of 1 mm was kept at an ambient temperature of 24 + I°C. A circular wave, perceptible by the blue color of ferriin, was triggered by immersing the tip of a hot platinum wire with a diameter of 0.3 mm and a temperature of about 200°C into the solution layer with a micro-manipulator. When the diameter of the expanding circular wave had reached about 10 mm, a small section of the wave front was disrupted with a gentle blast of air ejected from a 500/~1 Eppendorf pipette (diameter of disposable tip = 0.5 mm). A region in the dish was chosen in which no bubbles of carbon dioxide had formed. The formation of a pair of spiral shaped waves with opposite sense of rotation started near the "open" ends of the disrupted wave fronts. Immediately afterwards, the dish was covered with a dust-free glass plate leaving an air gap of 11 mm in height. This prevents the formation of convective patterns caused by the evaporative cooling of the layer surface [17, 18] and reduces drifts of the patterns due to hydrodynamic flows. One to two minutes were then required for the complete evolution of regular spiral waves. 3.3. The recorded image sequence The Belousov-Zhabotinskii reagent was placed in the petri dish immediately after addition of ferroin to the initial reaction mixture. Initiation of the spiral pattern by the procedure described above was finished after 3 min. Another 2 min passed by before the position of the (now covered) dish had been carefully adjusted with micrometer screws in order to move the clockwise turning spiral into the 4.5 × 4.5 mm 2 area for image recording. During this time all disturbances resulting from the ini-

S.C. Mfiller et al./ Two-dimensional spectrophotomet~, of spiral waves

tiation procedure had disappeared. At this point the automatic recording of a time sequence of images with a 450 x 450 pixel frame was started. One hundred images were taken in a period of 5 rain. The time intervals between subsequent frames varied between 2.5 and 3.5 s with an average of 3 s and were measured with an accuracy of _+0.1 s. No data were recorded during the following 5 min. Then a sequence of 25 images with the same acquisition frequency was recorded showing the spiral pattern at a late stage of the experiment. 3.4. Evaluation of image data After storage, the 125 recorded images occupy 25 megabytes on magnetic disk. In order to handle this large amount of data in an efficient way the software package GRIPS (Graphic Raster Interactive Processing System) is used which was developed in this institute and consists of seventy commands. A brief description of the most frequently used operations follows. Their main purpose is to illustrate specific details of the observations and to emphasize quantitative relations which cannot be detected by simple visual inspection of the images. All data "manipulations" are linear or logarithmic and no complicated transformations of the original measurements are performed. 1) Data acquisition consists of commands for storage of one video frame in the video buffer by "freezing" an image, for transfer of the frame to the memory of the computer and on magnetic disk, and for adding a new entry to an image database for later retrieval. 2) For data presentation the stored data are transferred to the video frame buffer and displayed on a black-and-white or a color monitor (see fig. 1). Flexible routines are available for producing pseudo-color images in which different intensity levels are represented by different colors. This technique, that may considerably enhance the visibility of structural details, is described in [13, 16]. Text, markers, and simple line drawings can be superposed at specific grey levels.

75

3) Data segments are extracted by storing the space coordinates x, y and the intensity values I(x, y, t) of those pixels of a time sequence that are located within a specific area or along a specific curve (e.g. lines, squares, circles etc.) on a separate file which is subsequently used in other computer programs for data presentation and analysis. Examples are plots of intensity and concentration profiles, the fitting of theoretical model parameters to iso-intensity lines, or the application of high level graphic packages for three-dimensional perspective representations. 4) Fundamental numerical operations such as addition, subtraction, multiplication, and division can be performed pixel by pixel for any pair of images. Important applications are the subtraction of bias values and the multiplication of shading corrections derived from reference images. With a spatial homogeneity of the illuminating light field better than 1.5% in the observation area, the latter turned out to be negligible for this spiral sequence. Multiplicative correction factors had to be introduced, however, to account for temporal intensity fluctuations. These were calculated by normalization procedures specified in section 4.2. Image histograms are calculated for obtaining grey level averages, standard deviations, and other statistical parameters. They are used to estimate global chemical changes and light source fluctuations. The inherent pixel noise which results in a standard deviation of about 5 grey levels in an image of uniform intensity is effectively reduced by shrinking the image or, preferentially, by calculating moving averages. Areas of 3 × 3 pixels as basic units usually suffice for satisfactory noise reduction. The increase of the signal to noise ratio by a factor of three implies a remarkable improvement of the image quality, while the setback in spatial resolution remains small compared with the relevant length scale of the patterns. 5) The transformation of the measured intensity distribution I(x, y, t) into the corresponding ferroin concentrations ?(x, y, t) is calculated with the spatio-temporal version of the Lambert-Beet

76

S. C. Mfiller et al. / Two-dimensional spectrophotomet~ of spiral waves

law

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where e = 1.06 × 104 M -1 cm -1 is the molar extinction coefficient of ferroin at 490 nm (see section 3.1), d = 1 mm the depth of the reaction layer, and I o the intensity of the spatially homogeneous light beam after transmission through the covered dish containing a 1 mm layer of distilled water. It is assumed that the wave patterns do not display any spatial inhomogeneities in the direction normal to the bottom of the dish. This implies that the observed structures extend over the entire depth of the layer. The concentration of ferriin, c(x, y, t), follows from mass conservation according to

c(x, y , t ) = C t o t - g ( x , y , t ) ,

(2)

where ctot = 3.5 mM is the ferroin concentration in the initial reaction mixture. The determination of absolute concentration values c or ~ is subject to errors caused by intensity fluctuations of the monitoring light beam as well as global chemical changes and imperfections in the correction procedures, whereas concentration changes Ac or A? within one image can be measured with much higher accuracy. The concentration patterns will be mostly discussed in terms of ferriin concentration changes,

Ac(x, y, t) = c(x, y, t) - Cref(t),

(3)

where Cree(t) denotes an appropriately chosen reference value for each image (see section 4.2). 6) The list of important commands is concluded with two procedures for image transformations. The expansion or "zooming" of image sections leads to an enhancement of the visibility of local effects. Binary images are obtained by logical operations by which grey level thresholds or ranges are set for revealing the geometric characteristics of specific intensity levels.

4.1. Quantitative description of the spiral-shaped wave A snapshot of the original recording of the investigated spiral wave, as displayed on a TV monitor, is shown in fig. 2A. The photographed image represents an early stage of the experiment and was taken during the first recorded spiral revolution. The grey levels approximately cover the range from 132 to 224 with an average of 175. After subtracting the 20 units preset for zero intensity and normalizing the intensity distribution to an average grey level of 128 (for reasons given in section 4.2) the 450 x 450 pixels of the 4.5 mm × 4.5 mm area have grey levels between 92 and 168. Decreasing the pixel noise by a moving average of 3 x 3 pixel areas slightly reduces the variation of grey levels to.an interval from 96 to 165. In this section we mainly present the quantitative information contained in the noise reduced version of this single image. A two-fold expansion of the central part of the normalized original image leads to the picture shown in fig. 2B. Those pixels having intensities in a narrow band of two grey levels half-way between the minimum and maximum intensities of transmitted light, are displayed in full white. Fig. 2C shows the same image section, expanded after application of the 3 x 3 pixel moving average to the normalized measurement. Due to the noise reduction the coarseness, as compared with picture 2B, has visibly diminished and, more obviously, the contours of the two adjacent intermediate intensity levels (in white) are much sharper. The averaging procedure thus proves to be an appropriate method for substantial improvement of the image quality. This advantage outweighs by far the slight loss of spatial resolution which, with 10 /~m in the original measurements, is much bglow the characteristic length scale of the wave pattern. For the following evaluations a three-dimensional coordinate system is defined with the two

S.C. Mi~ller et al. / Two-dimensional spectrophotometrv of spiral waves

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spatial coordinates x and y in the plane of the solution layer, having their origin in the lower left corner of the observation area (see fig. 2A). The third c o o r d i n a t e is the intensity I ( x , y, t) of the pixels in units of a reference value /ref or, after c o n v e r s i o n according to eqs. (1) and (2), their ferriin concentration c( x, y, t ). I n fig. 3 an intensity profile along the x coordinate at Yo = 1.98 m m (passing through the spiral center, see section 4.2) is plotted as a dotted line after application of a moving average. The same profile in terms of the corresponding ferriin concentration d a t a is plotted as a full line. The ferriin c o n c e n t r a t i o n is scaled as such that the wave amplitudes in intensity and concentration units are almost the same. The ferriin (and the corres p o n d i n g ferroin) concentrations vary by 0.28 m M while the average ferriin level in the image is a b o u t 3.2 m M . The shape of the concentration variations differs only slightly from that of the m e a s u r e d wave profiles of transmitted light. Therefore, the main features of the spiral structure can be discussed either in terms of intensity or concentration. N o t e the difference between the shape of the individual waves inside and outside the central region around x 0 = 2.25 m m (see below). M o r e details of the 2D concentration distribution in the spiral pattern are visualized by applying the pseudo-color technique. In plate 1 seven c o n c e n t r a t i o n ranges can be distinguished by different colors. T h e y correspond to intervals of equal width (0.04 m M each) within the modulation range

Fig. 2. A) Digital image of the spiral wave propagating in a 1 mm layer of an excitable solution of the BelousovZhabotinskii reaction, recorded at 490 nm by means of light absorption by ferroin 2 min after spiral initiation. For initial concentrations see text. The image consists of 450 x 450 picture elements having digital intensity units between 132 and 224 out of 256 possible values. The (x, y)-coordinate system in the observation plane is indicated. B) Twofold expansion of the central section of image A. The grey level distribution is normalized (see text) and varies from 92 to 168. An intermediate intensity interval is shown in white. C) Same expansion as in image B, after application of a moving average of 3 x 3 pixel areas to the normalized version of image A.

78

S.C. Miiller et a l l Two-dimensional spectrophotomet~ of spiral waves

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Fig. 3. Profiles of transmitted light intensity (dotted line) and of ferriin concentration (full line). The curves are extracted from fig. 1A after application of a 3 x 3 pixel moving spatial average. They pass in the x-direction through the center of the spiral core with Y0 = 1.98 mm (see figs. 7 and 8). The intensity is given in units of the average intensity of the image, Irer. Concentration variations are calculated by eqs. (1) to (3).

of the redox state of the catalyst ferroin exhibited in the averaged version of fig. 2A. In plate 2 special attention is given to the central area of the spiral after expansion by a factor of five. Ten colors are used, each for a concentration interval of 0.028 mM, except for the level shown in bright

red. This color is reserved for the narrow concentration range corresponding to that at the center of the spiral. How its location can be determined, is discussed in the following section. This zoomed color picture shows that concentration gradients are especially steep at all sides of the spiral tip. A rough estimate of these gradients derived from the profile of fig. 3 leads to values of at least 1.5 m M / m m . A combined representation of the 2D pixel distribution and the profiles along specified directions is possible with a software package for three-dimensional graphics [19]. In fig. 4A about one half of the spiral is placed in the (x, y, c)coordinate system at a specific view site and looked at from a specific view point. The concentration profile along the cut through the pattern in x direction at an intermediate value of the y coordinate is clearly visible. The 3D impression of this graphic presentation is enhanced due to the fact that the brightness of the individual picture elements is retained and thus remains to be proportional to the numerical value of the concentration coordinate. Consequently, there is a twofold pre-

Fig. 4. Three-dimensional half-tone representation of the spatial distribution of ferriin concentration c. The 150 x 150 data points are taken from a threefold shrinked version of the image in fig. 1A. The pictures are composed of dotted concentration profiles along the x coordinate, similar to that shown in fig. 3 but each consisting of only 150 data points. They are plotted for about half of (A) and for all 150 values of the y-coordinate (B). The applied graphics procedure retains the measured brightness of each pixel.

3 Plate I.

Pseudo-color representation of the spiral pattern of fig. 2A in terms of ferriin concentration after application o f a 3 pixel moving average. Seven equidistant concentration intervals covering the entire range of ferriin concentration mo tions (0.28 m M ) are shown in different colors from dark blue to yellow.

Plate 2.

Central section o f the spiral of plate 1 after zooming by a factor of 5. The ferriin concentration interval is divided int colors. An exception is the red color which is assigned to the level at the center of the spiral core. This is located in the rf of the white circle (compare fig. 7).

Plate 3.

Three-dimensional representation o f the spiral pattern derived from a threefold shrinked version of fig. 2A. The picture sists o f 150 line drawings of profiles in the x-direction. The view point is rotated by an angle o f about 180 ° with resp~ the angle chosen in fig. 4 such that the front o f the spiral tip can be seen.

Plate 4.

Central area o f fig. 8 expanded to full screen by a factor o f four. The inner part of the core is shown in pseudo-colors outer boundary of the core region is indicated by the circle.

80

S.C. Mfiller et al./ Two-dimensional spectrophotomet~ of spiral waves

of hidden lines - is applied to the central region of the same image. A front view of the spiral tip is given in fig. 5A. The horizontal lines of the grid show that the shape of the wave is symmetric close to its tip, whereas a distinct asymmetric shape is characteristic for the wave profile at the front edge of the grid, located about 1 mm to the left of the tip. The side view of the same object in fig. 5B illustrates the steepness of the concentration descent f r o m the highest level at the spiral crest to the concentration trough right in front of the tip. 4.2. The core o f the spiral

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Fig. 5. Three-dimensional surface grid plots of the spatial distribution of ferriin concentration c derived from a threefold shrinked 1.5 × 2.1 mm area of fig. 1A containingthe spiral tip. The tip is shown from the front (A) and from the side (B). sentation of grey levels: in terms of their location in the 3D perspective and in terms of their pixel intensity. Fig. 4B shows the complete 3D image of the spiral from the same view point. In plate 3 a slightly different version of this 3D graphics technique is used. The concentration profiles in x-direction that, in fig. 4, were plotted as dotted intensity-modulated lines are now replaced by full lines of equal intensity. With the view point chosen, this image "in gold" offers a frontal look at the spiral tip. In fig. 5, still another 3D version-realized by a surface-grid with removal

After analysis of one single "snapshot" of the spiral wave we proceed to the description of properties that can be derived from the recorded time series of images. First we specify the normalization procedure for the correction of fluctuations ot the incident light intensity that turns out to be an important improvement for the following evaluations. In fig. 6A the temporal evolution of the spatial average of the 450 × 450 pixel images is shown for the first 5 minutes (100 images) of the experiment. The curve marked 1% represents the time evolution of an intensity level up to that the integral of the intensity distribution accumulates 1% of the 450 × 450 picture elements ot the full image. The curve marked 99% represents the time evolution of an intensity level up to that the integral accumulates 99% of the picture elements of the full image. All the three curves document the fluctuations of the light source as well as a small overall drift due to chemical changes in the sample layer. Normalization of each image with its average intensity value and numerical adjustment to an intermediate value of 128 grey levels leads to fig. 6B. The upper and lower limits are now almost straight, smooth curves showing that, indeed, the individual images of the sequence can be discussed and compared among each other in a consistent way. Their slightly converging slopes reflect changes in the chemistry of the solution layei only. The normalization procedure implies a loss

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of precise information about the global chemical change which finally leads to the fully oxidized state of the sample. The six images of the first revolution of the spiral wave were transformed to images of ferriin concentration by eqs. (1) and (2) and overlayed in one image. Fig. 7a is a photograph of the first image of this initial sequence. In the five following pictures (fig. 7b-f) the concentration values of the subsequently recorded images are added pixel by pixel to this image. In each of these consecutive steps an appropriate bias was subtracted in order to fulfill the condition that the resulting con-

81

centration levels be contained in the interval of 0 to 255 digital units available for display on a TV screen. The successive application of the additive procedure leads, without distortion of the relative grey level distribution, to the full-revolution result of fig. 7f. The wave crests of the contributing single images can be recognized as spiral shaped bright bands. They differ somewhat in brightness and spacing due to the fact that the time intervals between consecutive images are not precisely equal and the set of six images covers slightly more than one spiral revolution. The differences in brightness are reduced when this additive procedure is replaced by a so-called logical overlay. In fig. 8, only the maximum of the six intensity values at any given site in the six images of the first spiral revolution is retained for display on the screen. It represents the 2D envelope of the maximum of the ferriin concentration data. Both in figs. 7f and 8 the spiral-shaped bands merge into a dark spot inside which the variations of ferriin concentration remain significantly below those found at all outer sites of the observation area. This dark spot is the core of the spiral centered at x0=2.25 mm and y0 =1.98 mm. Its radius appears to be different in the two overlay methods. This is caused by the choice of contrast for optimum photographic reproduction. The properties of the spiral core are characterized in a more quantitative manner by graphically superposing a large number of consecutively measured concentration profiles as shown in fig. 9. They all pass through the core center and are plotted along the x-direction, covering two and a half revolutions (15 profiles). During this relatively short period of time effects due to drift in space can be neglected, while for longer time intervals such drifts have usually to be taken into account (see below). The upper and lower envelopes of the profiles show that at the center of the spiral core (x o = 2.25 mm, Y0 = 1.98 ram) the variation of ferriin concentration is less than 10% of that outside the core area where the medium is excited to full ferriin amplitudes of

82

S.C. Mfiller et aL / Two-dimensional spectrophotometry of spiral waves

Fig. 7 . Sequence of additively superposed spiral patterns derived from images during the first recorded revolution of the spiral. Starting at observation time zero (a) the following images present the consecutive additive overlay of the concentration distribution m e a s u r e d at t = 2.9 s (b); t = 5.9 s (c); t = 9.0 s (d); t = 12.0 s (e); t = 15.2 s (f). The photographs present a slightly reduced area of the original frames (3.7 x 3.7 mm2). In the last picture there is an area centered at x 0 = 2.25 m m and Y0 = 1.98 m m - the core of the spiral - in which the ferriin concentration varies much less than in the surrounding area.

S.C. Miiller et a l l Two-dimensional spectrophotometry of spiral waves

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x(mm)

Fig. 9. Superposition of 15 ferriin concentration profiles such as shown in fig. 3. The profiles were calculated from consecutive images to which a moving 3 × 3 pixel average was applied. They cover about two and a half spiral revolutions from the beginning of recording and pass through the center of the spiral core in x-direction at Y0 = 1.98 ram. The modulation in ferriin concentration at the core center (x 0 = 2.25 ram) is less than 10% of that of the full amplitude waves exciting all outer areas of the investigated section.

about 0.28 mM. The concentration at the singular point in space is 0.05 mM above that of the minimum and 0.23 mM below that of the maximum of the ferriin amplitude. With a diameter of 3 0 / x m or less, the core center occupies a volume

,

240

,

300

(s)

Fig. 10. Temporal evolution of experimentally determined spiral parameters. (A) Distance A between waves outside the core region measured for an intermediate grey level at the outer (O) and inner (©) slope of the waves. Each point was obtained by averaging the distances in x- and y-direction measured for 6 consecutive images. (B) Velocity v of wave propagation, derived from the same data as used in (A) and the time intervals between consecutive images. (C, D) Deviation of the spiral center from its initial location ~o, )70. The coordinates Xo, Yo were determined for 18 superposed image sets such as shown in fig. 7f.

smaller than 10 -7 ml. The transition from this quasi-stationary site to the fully developed wave takes place in a circular area with a radius of about 0.35 mm. This radius is considered as the experimental definition of the outer boundary of the spiral core. The most noticeable changes in the dynamic behaviour are observed, however, in a much smaller area around the spiral center. The envelopes of the profiles indicate that the waves

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S. C. Miiller et al. / Two-dimensional spectrophotometry of spiral waves

Fig. 11. The spiral structure at t = 720 s shown in intensity units. The contrast chosen for display on the TV screen is much higher than that in fig. 2 in order to show the patches on the spiral crest that are only faintly visible in (A). The boundaries of the patches are digitally enhanced in (B).

attain 90% of their maximum ferriin concentration with respect to the value at the center already in an area with a radius of 0.2 mm. The spatial symmetry of the core region is further illustrated in plate 4 in which the central area of the overlay in fig. 8 and the grey level variation inside the core region is transformed into four colors. They cover intervals of equal width in the range from the spiral center up to the points where 90% of the full wave amplitude are reached (compare fig. 9). The black and white contrast of the remaining grey-level interval is digitally enhanced. The estimated boundary of the core region is indicated by the circle with a radius of 0.35 mm.

4.3. Time evolution of the spiral pattern Each spiral image recorded during the first 5 minutes was evaluated in terms of the distance between wave fronts or the spiral pitch, A, and the velocity of wave propagation, v, measured half-way between the wave maximum and minimum in x- and y-direction outside the core. The

coordinates of the spiral center, x 0 and Y0, were derived for each full spiral revolution by the superposition technique demonstrated in fig. 7. The time dependent parameters obtained by these local methods are shown in fig. 10. In a following publication these will be compared with results obtained from global methods by fitting spiral functions to the experimental spirals [14]. During the 300 s under consideration the spiral pitch (measured at the steep front) increases by 2.5% from initially 1.18 mm to 1.2 mm. It is interesting to note that the pitch measured between the steep outer wave fronts is 2% smaller than that measured at the same grey level between the smooth inner slopes of the wave profiles. This may be related to the different distances from the core region. The average propagation velocity is 68.5 ~tm/s and remains constant within an experimental error of +2.5%. The coordinates of the spiral center are subject to a small, somewhat erratic drift in space, both in x- and y-direction which is less than 70/~m or about 6% of the pitch during the 300 s of recording. The revolution period was estimated from the angular positions of the spiral tip with respect to the spiral center. It

S. C. Miiller et al. / Two-dimensional spectrophotometry of spiral waves 1.5

85

ature gradients at the layer surface may be important.

1.25

5. Concluding remarks 0.75

0.5 x(mm)

Fig. 12. Intensity profile passing through the current location of the center of the core, extracted from fig. 1I along the x-direction. There is a pronouncedflatteningof the wave crests as compared to fig. 3.

is 17.2 s for the first revolution and 17.9 s for the revolution after 300 s. Another time-dependent feature is the shape of the intensity or concentration profiles. Quite generally, there is a decrease in wave amplitudes in the course of time which is reflected by the converging trend of the 1% and 99% curves in fig. 6B. This effect is accompanied by the small increase of the average distance between wave fronts shown in fig. 10A. At a late stage, about 12 min after the start of image recording, we observed the formation of bright patches on top of the continuously broadening crests of the revolving spiral wave (fig. 11). This phenomenon was found to occur quite generally during the "aging" of a spiral wave. The patches are very faint and almost invisible (fig. l l A ) . Therefore, their contours are digitally emphasized in white (fig. llB). An intensity profile passing in x-direction through the core center of the image is plotted in fig. 12. The changes in the shape of the waves become clear by comparison with fig. 3. The bright patches on top of the wave crests can be barely recognized as a small increase in intensity around x = 1.8 mm and x = 3.7 mm. The reason for the occurrence of this additional patterning process is not yet understood, but the effect of convective processes due to small temper-

The presented study of spiral wave propagation in the Belousov-Zhabotinskii reaction provides, for the first time, detailed experimental data on the two-dimensional concentration distribution of the reaction catalyst ferroin/ferriin. An unprecedented resolution with respect to space, time, and concentration is achieved by application of modern video and computer techniques. These allow for the fast acquisition of data from a system displaying highly complex chemical dynamics. The technical aspects of the method have been described and an analysis of a time sequence of two-dimensional spiral data in terms of one chemical c o m p o n e n t - t h e catalyst and indicator ferr o i n - h a s been given. The main results are, independent of any model assumptions: (1) The location of the spiral center is determined within an accuracy better than 30 ~tm. It has singular properties in that at this site the ferroin concentration is quasi-stationary. (2) The spiral core is contained in a disk around this center with a radius of 0.35 mm inside of which the oxidation/ reduction cycle gradually attains the full amplitude that is characteristic for wave propagation outside the core. (3) The wave profiles are symmetric close to the spiral center and asymmetric outside the core. (4) Local determination of spiral parameters yield a spiral pitch of 1.18 mm, a propagation velocity of 68.5 /xm/s and a revolution period of 17.2 s. These numbers change very little during the first 5 rain of observation. (5) The location of the spiral center is subject to a small drift of less than 10 /~m/min. (6) After 12 min, the aging process leads to the formation of patchlike oxidized areas in the regions of high ferriin c o n c e n t r a t i o n s - the crest of the waves. In the subsequent paper [14] the same sequence of digital images of the spiral will be further

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S. C. Mfdler et a L / Two-dimensional spectrophotometry of spiral waves

analyzed by fitting spiral functions to the measured data and by quantifying concentration gradients.

Acknowledgements We thank Mr. U. Heidecke for experimentation in the laboratory, Mr. K. Dreher and Mr. R. Schiiller for hardware and system software support, Ms. G. Schulte for photography, and Ms. B. Plettenberg for typing the manuscript.

References [1] H.-G. Busse, J. Phys. Chem. 73 (1969) 750. [2] A.N. Zaikin and A.M. Zhabotinskii, Nature 225 (1970) 535. [3] J.J. Tyson, Lect. Notes Biomath., vol. 10 (Springer, Berlin, 1976). [4] A.T. Winfree, The Geometry of Biological Time (Springer, New York, 1980).

[5] Oscillations and Travelling Waves in Chemical Systems, R.J. Field and M. Burger, eds. (Wiley, New York, 1985) [6] Springer Series in Synergetics, H. Haken, ed., Volumes: 5 (1979), 6 (1980), 12 (1981), 27 (1984), 28 (1984), 29 (1985). [7] H.-G. Busse and B. Hess, Nature 244 (1973) 203. [8] A.M. Zhabotinsky, Investigation of homogeneous chemical auto-oscillating systems, Thesis (in Russian) (Institute of Biological Physics of the Academy of Science, USSR, Pushchino, 1970). [9] A.T. Winfree, Science 175 (1972) 634. [10] P.M. Wood and J. Ross, J. Chem. Phys. 82 (1985) 1924. [11] S.C. Mi~ller, Th. Plesser and B. Hess, Anal. Biochem. 146 (1985) 125. [12] S.C. MiJller, Th. Plesser and B. Hess, Science 230 (1985) 661. [13] S.C. Mfiller, Th. Plesser and B. Hess, Naturwissenschaften 73 (1986) 165. [14] S.C. Miiller, Th. Plesser and B. Hess, Physica 24D (1987) 87. [15] T. Mantatis, E.F. Fritsch and J. Sambrook, Molecular Cloning (Cold Spring Harbor Laboratory, 1982), p. 437. [16] G.M. Murch, Compt. Graph. Forum 4 (1985) 127. [17] J.K. Platten and J.C. Legros, Convection in Liquids (Springer, Berlin, 1984). [18] S.C. Miiller, Th. Plesser and B. Hess, Ber. Bunsenges. Phys. Chem. 89 (1985) 654. [19] Graphics Compatibility System (GCS), Waterways Experiment Station, Vicksburg, Miss, USA; The Scientific Graphics Utilities, National Center for Atmospheric Research, Boulder, Colorado, USA.