Article pubs.acs.org/JPCA
Chemical Waves in Heterogeneous Media Mahmoud M. Ayass,† Mazen Al-Ghoul,† and István Lagzi*,‡ †
Department of Chemistry, American University of Beirut, P.O. Box 11-0236, Riad El-Solh 1107 2020, Beirut, Lebanon Department of Physics, Budapest University of Technology and Economics, P.O. Box H-1521, Budafoki út 8, Budapest, Hungary
‡
ABSTRACT: Formation of precipitation patterns and wave propagation in excitable media have attracted considerable scientific interest in the context of nonlinear chemical kinetics because of a new approach to micro and nanofabrication, in addition to some biological aspects. All precipitation patterns share common morphological characteristics, namely the formed patterns are stationary and no dynamical patterns can be observed in these classical precipitation systems (e.g., Liesegang phenomenon). However, it has been recently shown that in several circumstances dynamic patterns (chemical waves) can exist in purely inorganic precipitation systems similar to the well-known and studied (excitable) waves in Belousov−Zhabotinsky reaction. In this study, we show how to fine-tune the pattern characteristics in precipitation systems, such as the wavelength and the pattern morphology by changing the concentrations of the reagents, and we demonstrate chemical waves on a moving 3D spherical precipitation layer. We show that such precipitation waves have anomalous transport property, specifically superdiffusive nature, and it can be controlled by the initial concentration of the inner electrolyte. Moreover, we present several precipitation systems in which chemical wave propagation inside a moving precipitation layer can emerge. This observation points out the generality and robustness of similar behavior in diffusion-precipitation systems.
1. INTRODUCTION In nature and in laboratories, patterns can usually appear in a self-organizing manner far from their thermodynamic equilibrium by coupling their transport properties (advection, diffusion, etc.) with chemical reactions of the involved species.1,2 These patterns can be categorized as being spatially steady or unsteady and temporally static or dynamic.1−3 An extensively studied area of nonlinear chemistry is the wave propagation in an excitable media.4 Belousov−Zhabotinsky (BZ) reaction provides a typical example to investigate several phenomena such as wave propagation and chemical chaos.4,5 In two-dimensional media, excitation waves can produce rotating spiral waves,1−4 and at a higher dimension (3D), patterns such as the so-called scroll waves are formed.6−9 The main motivation for the study of such three-dimensional excitation waves originates from biology, where rotating vortex structures in the heart are found to cause cardiac arrhythmia and ventricular fibrillation.10,11 However, it is important to note that the scroll wave formation is expected to appear as one large pattern invading the entire medium, thus making it difficult to control and design waves in such a special geometrical constrain (e.g., at curved surface) in 3D. Nevertheless, trials were carried out to design and observe BZ waves on a curved (stationary) surface loaded with a catalyst.12,13 The oldest and most spectacular patterns formation phenomenon reported are the Liesegang patterns (or precipitation patterns: PP).14,15 Most of them emerge in the wake of a moving reaction front. In a usual experimental setup, the Liesegang pattern occurs due to a precipitation reaction between two chemical species, one of them diffuses into a gel (outer electrolyte) from outside and another species is © 2014 American Chemical Society
homogeneously distributed in the gel matrix (inner electrolyte).15 The produced pattern can consist of a set of precipitation bands or rings (depending on the geometry of the system), which are perpendicular to the diffusion flux of the invading (outer) electrolyte.14,15 All PPs reported and published are static and stationary in the sense that the formed precipitation objects (bands, rings, or more complex ones) stay at the given position, where they are formed. Seemingly “moving” structures can be observed if the complex formation of precipitate is possible in excess of the outer electrolyte.16−20 However, at a given spatial location, the amount of precipitate grows up to a given amount and it decreases due to complex formation, and locally there appears to be no dynamic change in the concentration of the precipitate.21 Until now it has been accepted that there is no dynamic pattern formation in precipitation systems, and there are no similarities between patterns in excitable media and patterns emerging in precipitation reactions. This is because precipitation systems are simple, static, usually contain only two inorganic salts, and the formation of the precipitate is mainly dominated by local nucleation and growth processes. It is thereby hard to imagine that dynamic waves can exist in precipitation (heterogeneous) systems similar to those in excitable (homogeneous) systems. Recently, it has been shown that similar self-organized chemical waves can exist in a precipitation reaction−diffusion system.22−26 The spontaneous appearance of traveling waves (spirals and target patterns) Received: September 12, 2014 Revised: November 25, 2014 Published: November 25, 2014 11678
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precipitation reaction with aluminum ions produced a white, insoluble precipitation layer at the gel interface (Al3+(aq) + 3OH−(aq) → Al(OH)3 (s)). Subsequently, this layer moved due to the complex formation (redissolution) of aluminum hydroxide in excess of hydroxide ions (outer electrolyte) producing the soluble aluminum complex (Al(OH)3 (s) + OH−(aq) → [Al(OH)4]− (aq)). On the basis of these events, a thin precipitation layer evolved in the gel, where diffusion was solely the transport of the chemical species. The thickness of the layer was ∼10−100 μm depending on time of evolution and the concentration of the inner electrolyte. It was found that the thickness increased with time and concentration of the inner electrolyte. A few minutes after the initialization, nucleation of chemical waves could be observed inside the precipitation layer with the wave fronts being perpendicular to the planar chemical front of the outer electrolyte moving downward in the gel disk (in the Petri dish). The measured wavelengths were found to be dependent on the initial concentration of the inner electrolyte (Figure 2). The initial concentration of the outer
inside a thin and moving precipitation front is reported in aluminum hydroxide and mercuric iodide precipitation systems.22,26 A coexistence of moving precipitation layer with “excitable” traveling waves inside the precipitation layer is observed. We report here several precipitation systems, where this behavior is observed, and which indicate that the dynamic wave propagation is not related to a specific precipitation system and could be a common behavior in such heterogeneous systems. Using this unique property that the waves and spirals are formed inside a well-defined location (inside a precipitation layer with usual thickness of 10−50 μm),22 it will be shown that, these dynamic structures can be designed inside arbitrarily oriented and moving precipitation surfaces. Additionally, we found that the transport property of such waves exhibits anomalous behavior, namely the position of the front does not scale either with the time (which would provide finite velocity/ constant velocity) or with the square root of the time (which would provide diffusive behavior).
2. EXPERIMENTAL SECTION We carried out experiments in three chemical systems (aluminum hydroxide, zinc hydroxide and mercuric (II) iodide). We used, as outer electrolytes, sodium hydroxide (NaOH, Sigma-Aldrich) for the aluminium hydroxide and zinc hydroxide systems, and potassium iodide (KI, Sigma-Aldrich) for the mercuric iodide system; and aluminum chloride (AlCl3· 6H2O, Reanal), zinc chloride (ZnCl2, Reanal) and mercuric chloride (HgCl2, Sigma-Aldrich) as inner electrolytes. Both electrolytes were initially separated: inner electrolyte was placed into the gel (disk or sphere), and the outer electrolyte was a simple solution. An agarose (Reanal) solution 1% with a given amount of the inner electrolyte was prepared, which was heated up to 90 °C and stirred continuously until it became clear. The hot solution was poured either in a Petri dish or into a glass bubble to produce a gel ball, the thin glass was removed prior the experiments. After the gelation process (∼2 h), experiments were initiated by pouring the solution of outer electrolyte on top of the gel disk or by placing the produced agarose gel sphere (ball) containing the inner electrolyte in the solution of the outer electrolyte. Figure 1 shows the experimental setup. Figure 2. Chemical wave formation in aluminum hydroxide system in Petri dishes using different initial concentration of the inner electrolyte (aluminum chloride). The pattern formation was observed from the top. The initial concentration of the outer electrolyte (sodium hydroxide) is 2.5 M.
3. RESULTS AND DISCUSSION (i). Aluminum Hydroxide System. A solution of 2.5 M sodium hydroxide was used as the outer electrolyte which was poured on top of the gel or in which the agarose ball was dropped. The hydroxide ions diffused into the gel, and
electrolyte has practically no effect on the wavelength and pattern morphology; however, the effect of the inner electrolyte concentration (aluminum chloride) is pronounced. We observed more wave nucleation centers and decreasing wavelength at higher aluminum ion concentrations. Consequently, no distinct wave fronts can be observed at higher inner electrolyte concentration, even at the beginning of the experiment, resulting in fused spotty pattern with coarsening dynamics in time (Figure 2c). In Figure 3, we represent the quantitative evaluation of the dependence of the wavelength on the inner electrolyte concentration. It is noteworthy that there is a threshold concentration for the aluminum ion below which no spontaneous pattern formation can be observed. However, if we perturb the precipitation layer with a needle (simply
Figure 1. Sketch of the experimental setup. The outer and inner electrolytes are: sodium hydroxide, potassium iodide (outer); and aluminium chloride, zinc chloride and mercuric chloride (inner). Pattern formation occurs inside the precipitation layer. 11679
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Figure 5. Pattern formation inside a moving and spherically symmetric precipitation layer in an agarose sphere. The initial concentration of the inner (aluminum chloride) is (a) 0.23 M, (b) 0.50 M, and (c) 0.54 M, respectively. The initial concentration of the outer electrolyte (sodium hydroxide) is 2.5 M. The precipitation front moves toward the center.
Figure 3. Dependence of the wavelength of the chemical waves on the initial concentration of the inner electrolyte (aluminum chloride) in aluminum hydroxide system. Figures show the pattern structure after 12 min of the start of experiments. Initial concentration of the outer electrolyte (sodium hydroxide) is 2.5 M. The size of the figures is 2 cm × 2 cm.
electrolyte. Analyzing the wavefront position (r) in time we can infer the dynamics of the wave/front propagation. Finite front velocity (r ∼ t) is common in chemical systems where the diffusion of the species is coupled to their autocatalytic reaction(s), e.g., BZ systems and autocatalytic fronts.1−3,27,28 Diffusive front propagation (r ∼ t0.5) can be observed if the phenomenon is driven by the diffusion of an invading chemical species such as in Liesegang systems.14,15 In general, we can write this relation in a more general form: r ∼ tα, where exponent α evinces the dynamics of the transport processes. If α = 1, it corresponds to finite front propagation and if α = 0.5, it corresponds to diffusive front propagation. By definition, if this exponent α is lower than 0.5, the transport process is called subdiffusive, which is encountered in highly heterogeneous media where the porous networks tend to trap particles, like in gels29 and in metal organic frameworks (MOFs).30 On the other hand, if the exponent α is greater than 0.5, the process is called superdiffusive. Such processes are usually encountered in systems involving enhanced motion, such as Faraday waves31 and stirring-induced hydrodynamic turbulence.32 Figure 6
pinching the layer), single circular wave evolves through the precipitation layer. This behavior is reminiscent of excitable chemical waves in the BZ reaction. However, the main difference between these two systems is that the BZ patterns are produced in homogeneous media, while in our case precipitation introduces heterogeneity into the picture, thereby, increasing the complexity of the system. Wave propagation is always produced in a thin layer that is perpendicular to the diffusion flux of the outer electrolyte (hydroxide ions). We can evolve these precipitation-complex formation reactions in 3D in a hydrogel sphere containing inner electrolyte by submerging it in a solution of the outer electrolyte. There would be an evolution of a thin precipitation layer with self-organizing patterns similar to the planar setup (in a Petri dish); however, the main difference lies in the fact that this layer is itself a sphere moving toward the center of the gel ball (Figure 4). Depending on the initial concentration of
Figure 4. Propagation of the chemical waves (spirals and targets) inside a moving and spherically symmetric precipitation layer in an agarose sphere in aluminum hydroxide system. The initial concentration of the inner (aluminum chloride) and outer (sodium hydroxide) electrolytes are 0.23 and 2.5 M, respectively. The precipitation front moves toward the center. The time interval between the pictures is 2 min.
Figure 6. Dependence of the exponent α (obtained from the plot of log r ∼ α log t) on the initial concentration of the inner electrolyte (aluminum chloride) in aluminum hydroxide system. The initial concentration of the outer electrolyte (sodium hydroxide) is 2.5 M. The α values higher than 0.5 indicate the superdiffusive nature of the propagating chemical waves.
the inner electrolyte, various types of pattern morphology can be obtained similarly to the planar case, mainly, chemical waves (spirals, double spirals, Figure 5a) and spotty patterns (Figure 5b). If the concentration of the inner electrolyte is high, it can occasionally cause cracks on the surface of the agarose sphere and, consequently, the outer electrolyte invades through these cracks and produces more precipitate inside them.23,24 Another interesting behavior is the velocity of the precipitation waves and their propagation, which is always perpendicular to the diffusion flux of the invading outer
shows the dependence of the exponent α on the inner electrolyte concentration (aluminum chlorite). The exponent was obtained from the log−log plot (log r versus α log t), where the slope of the linear curve provides the exponent α. In the present aluminum hydroxide system, α varies between 0.58 and 0.75 and increases with the inner electrolyte concentration. These findings are similar to the results which have been recently obtained in the mercuric iodide system.26 The 11680
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concentration of the inner electrolyte, and ∼1 M for the outer electrolyte.
explanation of this interesting phenomenon is currently not known but can be originated from the fact that the precipitation layer consists of different forms of precipitates and one of them can selectively control the mass transport of electrolytes through the membrane built from precipitate.25 Our results have a potential to validate those assumptions used in ref25 and can improve the model with more natural setting. (ii). Other Systems: Zinc Hydroxide and Mercury(II) Iodide Systems. In order to explore the robustness of chemical wave formation in precipitate systems, we investigated several inorganic salt systems, in which precipitation and complex formation occur similar to the aluminum hydroxide system. We select the zinc hydroxide system, where the initially formed precipitate Zn(OH)2 can be redissolved in excess of hydroxide ion:
4. CONCLUSION We reported several heterogeneous systems (aluminum hydroxide, zinc hydroxide, and mercuric iodide), in which chemical wave propagation can be observed as a result of diffusion and chemical reactions between two inorganic salts producing an insoluble precipitate concomitant to redissolution due to complex formation in excess of outer electrolyte. Interplay between the precipitation, complex formation and diffusion processes produces a thin moving precipitation layer in which wave propagation (spiral, double spiral and target pattern) is perpendicular to the diffusion flux of the outer electrolyte. We are thus able to recreate wave propagation in a curved precipitation layer inside a sphere made of a hydrogel. Our findings also prove the hypothesis that chemical wave propagation (similar to the BZ system) can exist in diffusionprecipitation systems, and this behavior is much more common than is previously proclaimed. In fact, the main difficulty to observe these waves lies in the bulk formation of the precipitate, which might simply overlap or annihilate each other.33,34 Consequently, systems with precipitation and redissolution by complex formation are perfect candidates to observe and study these heterogeneous chemical waves. We also showed that chemical waves in a reaction−diffusion system exhibit anomalous (superdiffusive) behavior without any external forcing. This is a unique observation in a system, where reactions of chemical species are coupled to their molecular diffusion. It is hard to imagine that similar precipitation processes can be responsible for pattern formation in living system; however, it should be mentioned that seashell pattern formation involves precipitation in animate systems.35 Our simple chemical systems could provide new insight into understanding pattern formation in both living and nonliving systems.
Zn 2 +(aq) + 2OH−(aq) → Zn(OH)2 (s) Zn(OH)2 (s) + 2OH−(aq) → [Zn(OH)4 ]2 − (aq)
On the other hand, we studied another chemical system, where the complex formation is not driven by pH, to show that pH is not a driving force for such phenomena. Mercury(II) iodide can also form a soluble complex in excess of iodide ion: Hg 2 +(aq) + 2I−(aq) → HgI2(s)
HgI2(s) + 2I−(aq) → [HgI4]2 − (aq)
We carried out experiments in these systems in Petri dishes replacing aluminum salt with zinc and mercury salts as inner electrolytes, and using hydroxide and iodide ions as outer electrolytes, respectively. Figure 7 shows the chemical wave
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AUTHOR INFORMATION
Corresponding Author
*(I.L.) E-mail:
[email protected]. Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS This work is supported by the Lebanese Council for National Scientific Research (LCNSR), the University Research Board, American University of Beirut, and the Hungarian Research Fund (OTKA K104666).
Figure 7. Chemical wave formation in (a) zinc hydroxide and (b) mercury(II) iodide systems in Petri dishes. The pattern formation was observed from the top. The initial concentration of the outer electrolytes is (a) 2.5 M (sodium hydroxide) and (b) 1.0 M (potassium iodide).
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REFERENCES
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