Nonlinear Dynamics in Polymeric Systems - American Chemical Society

Chapter 8

Pattern Formation in a Complex Reaction System by Chemomechanical Coupling Downloaded by PENNSYLVANIA STATE UNIV on September 12, 2012 | http://pubs.acs.org Publication Date: November 18, 2003 | doi: 10.1021/bk-2004-0869.ch008

Stefan C. Müller Institut für Experimented Physik, Otto-von-Guericke-Universität Magdeburg, Postfach 4120, D-39016 Magdeburg, Germany

In an experimental study of the polyacrylamide-methylene blue-sulfide-oxygen (PA-MBO) sytem a variety of concen tration patterns have been observed. Initially, stripes or hexagons form. The latter transform to chevrons or white-eye patterns. Honeycombs are a third type of structure that deve lops from either white-eye or hexagonal patterns. For hexagons, the pattern wavelength shows a nearly proportional dependence on the height of the reaction layer. The motion of small tracer particles serve to monitor physical changes within the medium. There are three distinct phases of particle move ment. The first one is caused by hydrodynamic flows in the pre-gel solution, whereas the third, and possibly the second, phase arises from localized volume changes within the gel.

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© 2004 American Chemical Society

In Nonlinear Dynamics in Polymeric Systems; Pojman, J., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 2003.

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Downloaded by PENNSYLVANIA STATE UNIV on September 12, 2012 | http://pubs.acs.org Publication Date: November 18, 2003 | doi: 10.1021/bk-2004-0869.ch008

Introduction

During the last decades the investigation and analysis of concentration patterns forming in chemical non-equilibrium systems has attracted considerable attention. An overview of many interesting aspects of this field of research can be found in Réf. 1. The macroscopic order of these systems often arisesfromthe diffusive coupling of nonlinear reaction processes which can lead to the formation of stationary Turing patterns (2,5) or propagating waves (4,5). Turing patterns form in certain reaction-diffusion systems in which the inhibitor species diffuses faster than the activator species. An important motivation for their investigation is the possible involvement in biological morphogenesis (6). There is some evidence that concentration patterns of this kind can also generate shape and geometry by mechanical deformations. In 1996, the first experimental example in which propagating chemical waves lead to a periodic swelling of a gel matrix was reported (7). Another system in which these effects may be more profound is the polyacrylamide methylene blue-oxygen-sulfide (PA-MBO) system (8). The oscillating MB Ο (9,10) system involves the methylene blue catalyzed oxidation of HS" by molecular oxygen in an aqueous medium at pH« 12. The monomer methylene blue exists in the blue oxidized form MB , the colorless reduced form MBH and the MB* radical, an intermediate oxidation step. Under +

Figure 1. Hexagonal pattern in the PA-MBO system. Thickness of reaction layer, 1.7 mm; diameter ofpetri dish, 7 cm.


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atmospheric conditions only the M B ion is stable, whereas MBH and the radical are oxidized by the oxyen of the air. In the PA-MBO system, the MB Ο reaction is started in the presence of a polymerizing acrylamide/bisacrylamide pre-gel solution. In the course of several minutes, this complex reaction system forms transient hexagons, stripes, and zig-zag patterns that have typical wavelengths of 1-3 mm as reported by Watzl and Munster (8). An example of a hexagonal pattern is presented in Figure 1. Evidence for more complex patterns in this reaction system has been provided in Refs. 11-13 in an investigation that focused on later stages of pattern evolution. Some examples of this study are shown in Figure 2. The geometry and orientation of the fundamental stripe and hexagon patterns can be influenced by electric fields and by light (14), which is in qualitative agreement with results from reaction-diffusion models (75), and should support the hypothesis that the patterns primarily occur from Turing instabilities. However, in 1998, Kurin-Csôrgei et al. (16) presented experiments in which neither methylene blue nor sulfite was required for patterns to develop and they conclude that patterns form due to a hydrodynamical instability during the polymerization and gelation process. Subsequently, a model was suggested

Figure 2. Patterns in the PA-MBO reaction obtained for different initial concentrations of sodium sulfide: (A ) 32.7 mM, (B) 38.1 mM (early pattern), ( C) 38.1 mM (late pattern), (D) 40.0 mM, (E) 41.4 mM, and (F) 42.1 mM. Field of view, 16 χ 16 mm . (Reproduced with permissionfromreference 12. Copyright 2000 Wiley-VCH Verlag.) 2


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that is capable of generating Turing patterns in the absence of methylene blue and explains the patterned swelling of the gel matrix, due to an inhomogeneous spatial distribution of molecular oxygen (15). There is now clearly the need for additional investigations of this system in order to unravel the underlying mechanisms that may lead to a satisfactory explanation of the complexity of the observed patterns. In the following we present experimental data on the evolution of hexagonal patterns that contribute to the possible role of chemo-meehanical coupling in this system.

Experimental

Following Ref. 8, we use the complete PA-MBO system, adding sodium hydroxide to further increase its pH value. The gel is left uncovered and kept under air at room tempreture. Consequently, the reaction system is semi-closed in the sense that the reactant oxygen is suppliedfromthe air, whereas sulfide is continuously consumed. A petri dish (diameter: 7.0 cm) serves as the reaction container for the PA-MBO medium. The blue patterns on a colorless background can be seen with the naked eye and are recorded by two-dimensional spectrophotometry for quantitative analyses. Preparation of the system: 0.24 ml N^'-methylene-bisacrylamide (2 g / 100 ml H 0), 0.13 ml NaOH (0.5 M), and 0.25 ml triethanolamine (30 g / 100ml H 0) are added to 3.17 ml acrylamide (20 g / 100ml H 0). A mixture of 0.29 ml methylene blue (0.318 g /100 ml H 0), 0.5 - 0.65 ml sodium sulfide (3.902 g Na S / 100ml H 0), and 0.87 ml sulfite solution (Na S0 ; 0.252 g / 100 ml H 0) is added. Before starting the polymerization with 0.13 ml ammonium persulfate (5 g / 25 ml H 0), double-distilled water is added to reach a final volume of 7.4 ml. Prior to all experiments, sodium sulfide is purified by refluxing commercially available Na S with Cu powder in ethanol under a nitrogen atmosphere for 1 h. 2

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Results and Discussion

Height-Dependence of Pattern Wavelength For reaction conditions with low initial [Na S] (30-40 mM), the PA-MBO system self-organizes patterns of blue dots (i.e., high concentration of MB ) that are arranged on hexagonal lattices (cf. Figure 2A). Together with irregular stripes and zig-zags (Figures 2E,F), hexagons are the fundamental patterns that 2

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98 develop during the early stages of the reaction. Moreover, hexagons exist in well-gelled media, whereas stripes and zig-zags are typically found in highly viscous, but not fully gelled systems ([Na S] > 40 mM). Our optical detection revealed no convincing evidence for hexagonal structures prior to gelation. The symmetry of the hexagonal patterns is typically affected by a small number of mismatches, such as penta-hepta defects. For a quantitative analysis, we calculated the two-dimensional power spectrafromthe raw data of individual video images. In this analysis, the digitized image data is Fourier-transformed from the (x,y)-space into the wave vector space (k ,k ). The inset of Figure 3A shows an example for the power distribution of a typical hexagonal PA-MBO pattern revealing six dots of high signal power that are arranged at a regular spacing on a circle. The radial distribution of the signal power in the main plot of Figure 3A shows a dominant wave vector at k = 0.5 mm . The corresponding wavelength of 2.0 mm denotes the average, minimal spacing of MB* dots in the real-space pattern. Hexagonal structures were investigated for reaction systems with layer heights in the range of 1.35 mm to 4.0 mm. Systems with thinner reaction layers always failed to generate optically detectable structures, which suggests the existence of a critical layer height of approximately 1 mm below which no patterns are formed. Visual inspection of the pattern shows that the wavelength of the structure increases with increasing layer heights, whereas their overall symmetry and appearance remains unchanged. The change in pattern wavelength with layer height was quantified by a power spectra analysis. The corresponding data in Figure 3B were obtained from experiments, in which the reaction systems were subjected to two different light intensities. Apparently, the pattern wavelength is nearly proportional to the layer height regardless of the intensity of the applied illumination. Up to now, we have not attempted to investigate patterns for layer heights above 4 mm. Because they will require the use of overall larger reaction vessels that assure a sufficiently large aspect ratio between the vertical and horizontal dimensions of the system. 2

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Analysis of the Motion of Colloidal Particles The inspection of the reaction system with an optical microscope reveals the presence of small particles that carry out a systematic motion within the medium. These particles (diameter « 30 μπι) cannot be readily dissolved or broken apart by treatment in an ultrasonic bath. It is likely that they are similar or identical to colloidal sulfur that forms in the reaction. The impact of these colloidal particles on the reaction dynamics in



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Figure 3. (A) Power spectrum of the signal intensity as obtained from a typical hexagonal pattern in the PA-MBO medium. The dominant wavelength is λ j = l/kj = 2.0±0.3 mm. The inset shows the square modulus of the pattern's two-dimensional Fourier transform. (B) Wavelength of hexagonal patterns vs. height of the reaction layer under white-light illumination of 17 W/m (circles) and 30 W/m (crosses). (Adapted with permission from reference 13. Copyright 2002 Oldenbourg Wissenschafisverlag.)

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In Nonlinear Dynamics in Polymeric Systems; Pojman, J., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 2003. f

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Figure 4. Sequence of snapshots:frameA shows a homogenous phase (note that the black speckles are talcum particles used to trace motion on the gel). Transition to hexagonal patterns is shown inframesB-D, followed by the formation of 'white-eye ' structures in E. Frames F-H show the subsequent transformation of the white-eye structures to honeycomb patterns. The arrows indicate the direction of particle motion in a single patch of the pattern. Temporal sequence after prepanng the system: A) 9 min, B) 9.5 min C)10 min, D) 13 min E) 17 min, F) 24 min, G) 26 min, H) 30 min.



101 the PA-MBO system is uncertain. Here, we use them as convenient tracers that reveal insights into the mechanical and/or hydrodynamic changes in the system. In the following experiments, each particle is resolved by approximately 5x5 pixels. Its movement is followed by computational particle tracking. Figure 4 shows snapshots of a selected small area taken at different stages, starting from spatial homogeneity and leading via hexagons and white-eyes to honeycomb structures. During the early phases of the reaction, the particles perform a collective motion within the still liquid, layer (Figure 4A) and we could not detect any indication for macroscopic concentration patterns. This behavior ceases in the course of the reaction and gives way to a more organized motion illustrated in Figure 4B,C as well as in Figure 5A. The underlying data for this second phase is acquired 10 min after initiation of the reaction and, at this time, the system has reached a highly viscous, possibly gel-like state. Simultaneously, hexagonal patterns emerge that reflect the spatial patterning in the concentration of M B described above. The particles are now moving towards the center of the bluish patches, one of which is captured in the image of Figure 5A (particle speed, 5-10 μπι/s). This motion lasts for 1-2 min after which all particles cease to move. At this time visual inspection yields the first clear indications for surface deformations of the reaction layer. +

Figure 5. Measured particle traces superimposed onto the optically detected concentration pattern. Arrows mark the direction ofparticle movement during the early phase of the pattern formation (A), and during the transition to honeycomb patterns (B). Field of view: 1.7 χ 1.6 mm . (Adapted with permission from reference 13. Copyright 2002 Oldenbourg Wissenschafisverlag.) 2

The third, distinct phase of particle motion seems to be associated with secondary changes in the concentration pattern. The transition of hexagons to more complex structures, such as the "white-eye", honeycomb or chevron-like



102 patterns is shown in Figures 2B-D, respectively (12). The transition from hexagons to honeycombs occurs through an intermediate "white-eye" state (Figure 2B and 4E) that slowly evolvesfromthe initial pattern in a "one-patchto-one-eye" fashion. During this continuous change no significant motion of the particles is detected. Subsequently, the "white-eye" patches undergo a transition in which the blue rings of the white eyes grow in size (cf. frames F and G of Figure 4) and give rise to the honeycomb-like pattern shown in Figure 2C (and also Figure 4H). This transition propagates rapidly through the entire system and often nucleates at defects of the original pattern. The front of this propagating transition can change the location of individual concentration patches and thereby reduce the number of defects (for details, see Ref. 11). It is during this phase that the position of the colloidal particles in the gel matrix is again affected. Figure 5B shows our measurements of the trajectories of numerous particles that now movefromthe center of the concentration patch outwards and, thus, retrace the motion observed during the second phase (Figure 5A). After this outward motion is completed, all particles essentially remain immobilized. The macroscopic pattern of MB* concentration shows no significant changes any more except for a gradual fading of the bluish color. The surface deformations generated during the second and third phases of particle reorientation are permanent and withstand repetitive drying and swelling with water. Profilometric analyses of the patterned surface of a dried PA-MBO gel yielded a height difference of 3 μπι between the indentations, the locations of which corresponded to the patches of the original concentration pattern.

Conclusions A remarkable variety of semi-stationary concentration structures have been shown to form in the PA-MBO system. Irregular stripes and hexagons are the simplest and apparently fundamental patterns. Optical detection reveals that hexagons form at a time that is not easily discriminatedfromthe gelation time of the acrylamide gel. Our data yield clear evidence for hydrodynamic flows during the early phases of the reaction (Figure 4A). Additional information is provided by our measurements of the pattern wavelength versus layer height (Figure 3B). The results suggest a proportional dependence and show that there is a critical layer height below which no concentration patterns are formed. Both findings appear to be in good agreement with the assumption of a surface-tension driven Marangoni instability. Although it is, hence, tempting to conclude that the patterns resultfromfluid flow, it remains puzzling that the hydrodynamic motion observed lacks spatial patterning and shows an insufficient temporal stability. Moreover, the height dependence of the pattern wavelength could also be caused



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by the influx of oxygenfromthe ambient atmosphere into the reaction layer. The underlying reaction-diffusion processes have to be taken into account, because they control the effective oxygen concentration and therefore the polymerization as well as the MBO reaction as a function of the layer height (17). The first, spatially organized motion of the tracer particles correlates with the evolving concentration patches of the hexagonal pattern. Up to now, we are not able to answer the question whether this motion is driven by hydrodynamic flows in the highly viscous pre-gel solution or by mechanical deformations of the gelled medium. However, it is should be emphasized that at this stage the particle motion is always directed towards the center of the evolving patches of high M B concentration. This finding is remarkable because hydrodynamic convection cells are expected to involve inward as well as outward flow of the fluid. Certainly, a precise determination of the gel point during pattern evolution is desirable. Since the gel point of the PA-MBO system depends on the intensity of illumination, its measurement using the methods proposed in (18,19) is especially difficult and further efforts in this direction are under way. Mechanical instabilities of polymer systems usually arise in shrinking or swelling gels. A classic example is the formation of patterns on the surface of ionized acrylamide gels that swell due to an osmotically driven inflow of water (20-22). This inflow leads to mechanical stress within the gel and shear bending of the solvent-exposed gel layer. Despite the complex reactivity of the PA-MBO system, this type of mechanical instability seems to persist in our experiments and provides a possible explanation for the sudden, often front-like transition to honeycomb patterns. In this process, the reduction of mechanical stress is affected by the spatially periodic concentration patterns. Consequently, one can interpret the outward displacement of particles during the third phase of motion as a truly chemo-mechanical phenomenon. +

Acknowledgments. The author thanks O. Steinbock, E. Kasper and K. Tsuji for fruitful discussions and E. Slamova for providing the photograph in Fig. 1.

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16. Kurin-Csörgei, K.; Orban, M.; Zhabotinsky, A. M . ; Epstein, I. R. Chem. Phys. Lett. 1998, 295, 70-74. 17. Münster, A. F. Chem. Phys. Lett. 1999, 311, 102-104.

18. Parreno, J.; Piérola, I. F. Polymer 1990, 31, 1768. 19. Lyubimova, T.; Caglio, S.; Gelfi, C.; Ringhetti, P. G.; Rabilloud, T. Electrophoresis 1993, 14, 40-50. 20. Tanaka, T.; Sun, S.-T.; Hirokawa, Y.; Katayama, S.; Kucera, J.; Hirose, Y.; Amiya, T. Nature 1987, 325, 796-798. 21. T. Hwa, T.; Kardar, M. Phys. Rev. Lett. 1988, 61, 106-109. 22. Suzuki, Α.; Yoshikawa, S.; Bai, G. J. Chem. Phys. 1999, 111, 360-367.


Nonlinear Dynamics in Polymeric Systems - American Chemical Society

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