Revisited Chaos in a DiffusionâPrecipitationâRedissolution Liesegang

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Revisited Chaos in a Diffusion-PrecipitationRedissolution Liesegang System Mustafa Saad, Abbas Safieddine, and Rabih Fayez Sultan J. Phys. Chem. A, Just Accepted Manuscript • DOI: 10.1021/acs.jpca.8b03229 • Publication Date (Web): 29 Jun 2018 Downloaded from http://pubs.acs.org on July 9, 2018

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The Journal of Physical Chemistry

Revisited Chaos in a Diffusion-Precipitation-Redissolution Liesegang System Mustafa Saad, Abbas Safieddine and Rabih Sultan* Department of Chemistry American University of Beirut P. O. Box 11-0236 Riad El Solh 1107 2020 Beirut, Lebanon Email: [email protected] Abstract Co(OH)2 Liesegang periodic precipitation systems exhibit oscillations in the number of bands due to band redissolution in high NH4 OH concentration. We revisit the problem considered earlier [Nasreddine and Sultan, J. Phys. Chem. A 1999, 103, 2934-2940.], by rigorously refining the experiments and the Chaos analysis. Chaos is established in this diffusion-precipitation-redissolution system, as is evident from the refined outputs of the Chaos analysis tools. A brief account of possible applications of Chaos in Liesegang systems is presented.

1

Introduction

Liesegang patterns of precipitate in gel1—4 have long fascinated the scientific community because of their beautiful display of colorful parallel discs or bands in a 1D tube (Figure 1.a), and concentric rings in a 2D Petri dish (Figure 1.b), as well as their rich underlying dynamics. Those patterns show a striking similarity with the bands observed in rocks5—10 (Figure 1.c) and hence establish an intimate link between a simple laboratory experiment (Figure 1.d), though with quite complex dynamics, and a large scale 1

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Figure 1: Liesegang patterns with analogies: a. A typical copper chromate (CuCrO4 ) pattern in a 1D tube. b. 2D concentric rings of Co(OH)2 in a Petri dish. c. The Liesegang stone mainly consisting of bands of the rhyolite mineral. d. A simulation of a rock pattern in 3D gel with Co(OH)2 surface shells.

commonly admired landscape. A Liesegang pattern in 1D exhibits a stratification of parallel bands of precipitate due to the coupling of ion diffusion to a precipitation reaction in a gel medium. The similarity between the Liesegang banding phenomenon and naturally occurring systems is not restricted to the geological scenery. An account of natural paradigms mimicking Liesegang patterns, sometimes involving a similar dynamical scenario, in Biology, Chemistry, Geology, Medicine and Engineering is presented in a review article (Ref.11 ). A more recent work12 reviews the classes of precipitation reactions including Liesegang banding and self-organization far from equilibrium. Some Liesegang systems experience a redissolution of the old formed bands (top of the tube) due to some complexing reaction.13,14 A typical such system is the Co(OH)2 precipitate from Co2+ and NH4 OH, which is subject to two chemical reactions 2


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taking place simultaneously, as follows:

Co2+ (aq) + 2NH4 OH(aq) → 2NH+ 4 (aq) + Co(OH)2 (s)

(1)

2+ − Co(OH)2 (s) + 6NH+ 4 (aq) + 4OH → Co(NH3 )6 (aq) + 6H2 O(l)

(2)

We note here that the possibility of formation of the Co(OH)2− 4 (yet of high formation constant Kf = 5.0 × 109 )15 is suppressed by the weak ionization of ammonia (Kb = 1.8 × 10−5 ). Thus the reaction:

2−

− Co(OH)2 (s) + 4NH4 OH (aq) → Co(OH)4 (aq) + 4NH+ 4 (aq) + 2OH (aq) (3)

has an equilibrium constant K = Kf Ksp (Kb )4 = 3.1×10−24 . Thus Co(NH3 )2+ 6 (of formation constant 1.3 × 105 )15 remains the dominant complex ion upon redissolution of the Co(OH)2 precipitate in excess NH3 solution1 . Co(OH)4− 6 is obviously much less likely to form. We studied this fascinating system extensively,13,16,17 and the present paper focuses on one selected study by Nasreddine and Sultan,17 wherein the total number of bands was shown to exhibit random oscillations due to the interplay between band formation (precipitation) and disappearance (redissolution). The study showed that the two events are strongly correlated, and that the oscillations are deterministic chaotic in nature. The overall pattern seemingly migrates down the tube, just like a relay between the coupled formation and disappearance processes. We revisit this dynamic system and reformulate the chaotic characterization through a refinement of the experiment and an improvement of the characterization tools. A major variant from the previous treatment17 is an interpolation method assuming a continuously coupled diffusion-precipitation-redissolution 1 Reaction

2 has K2 = Kf Ksp /(Kb )6 = 2.3 × 1019 . Ksp for Co(OH)2 = 5.9 × 10−15 .

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scenario in time and hence in the space between the bands; in addition to a resetting of the observed oscillations relative to a base line defined by the best fit of the overall trend of number of bands (N) versus time (t). The plots of chaotic return maps and phase portraits are reinforced by more powerful computational tools, as can be clearly seen by mere comparison.

2

Experimental Section

A sample of gelatin powder (Difco) is added to a 0.10 M solution of CoCl2 .6H2 O (Aldrich) to yield 5% gelatin (w/w in H2 O), and heated with constant stirring to 90◦ C. The resulting viscous polymer is tranferred to a long tube of 4.0 mm diameter.

After all air bubbles are eliminated, the tube is placed in a water

jacket maintained at 20.0 ± 0.2◦ C by water circulation from a thermostat. After the equilibrium temparature is established, we add a solution of concentrated ammonium hydroxide (13.3 M NH4 OH) on top of the gel, marking the start of the reaction and hence initializing the time t = 0. A digital camera (Sony Cyber Shot DSC-QX30, operated by a Samsung J1 cell phone through the PlayMobile application) was mounted on a movable stand to descend with the down migrating pattern, and timer-programmed to take a picture of the entire pattern every three hours. Each photograph was digitized by first mapping the observed precipitate bands onto a two-dimensional graph of intensities using the software UN-ScanIt. This graph was then inspected to determine local minima and maxima associated with each peak present. The ten clearest peaks corresponding to specific bands were then identified and selected. The average intensity of those peaks was calculated and multiplied by 0.3. The resulting value was taken as a threshold, above which the peak intensity is considered a ’true’ band, and is thus counted. The result of this procedure is a time series (at three-hour increments)

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of number of bands (N) versus time (t). Note that the developed criterion is fundamental in the analysis, especially in the dissolution region (top of the pattern; see Figure 2.c). At late times, some very thin bands exist, and hence a rigorous criterion is needed to count or not count a band, for the sake of coherence and consistency. It is the dynamics of diffusion coupled to precipitation and redissolution inherent within the system, which will govern the observed chaotic oscillatory behavior.

3

Results and Discussion

We carry out two experimental runs, using 0.30 M Co2+ and 0.10 M Co2+ respectively, both with 13.3 M NH4 OH. The pattern of bands is shown is Fig. 2. We note here that if we use a lower concetration of NH4 OH (6.67 M), the

Figure 2: a. Co(OH)2 Liesegang pattern. [Co2+ ]0 = 0.10 M; [NH4 OH]0 = 6.67 M. The number of bands N increases monotonically with time t in what appears to be driven by diffusion. b. N versus t of the pattern in (a) yielding a typical diffusion curve. c. Co(OH)2 Liesegang pattern but with [NH4 OH]0 = 13.3 M.

redissolution reaction (Eq. 2) is suppressed; and the obtained pattern in this case displays a monotonic incerase of the number of bands with time as shown in the image of Fig. 2.a and the graph of Fig. 2.b. This situation contrasts with the seemingly migrating pattern (due to band formation coupled to redissolution) 5



seen in Fig. 2.c. Thus, the concentration of the outer electrolyte could be used as a control parameter that marks the onset of chaos, above a certain threshold value. Figure 3 displays the time series described in the previous section, for both the 0.30 M and 0.10 M Co2+ runs, both using 13.3 M NH4 OH. All computations and plots were performed using Mathematica.18 We directly

Figure 3: Chaotic oscillations of the number of bands N with time t. For both experiments, [NH4 OH]0 = 13.3 M. a. [Co2+ ]0 = 0.30 M; b. [Co2+ ]0 = 0.10 M .The net trend (smooth curve) passes through a maximum: first an increase followed by an overall decrease.

note the striking contrast between the random oscillations observed, and the

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purely diffusive nature of the N versus t profile obtained in the lower NH4 OH concentration regime (6.67 M; seen in Fig. 2.b). The smooth curve is a best fit of the N versus t oscillatory variation. This curve passes through a maximum, and hence illustrates the competition between the precipitation reaction and band formation on one hand (Eq. 1), and the redissolution reaction (Eq. 2) on the other. At early times, the precipitation is dominant, whereas at later times, redissolution becomes overwhelming, as indicated by the net decrease in the overall number of bands, yet maintaing the oscillatory behavior. We now examine the chaotic nature of the oscillations. One variant from the previous treatment,17 is an interpolation technique we adopt here, given the slowness of the process of band formation, which shadows the smoothness of the dynamics. The interpolation is carried out by decreasing the time interval to 0.1 hour. Because the time interval of 0.1 hour does not correspond to the actual time of measurement, this method ensures the continuity of the process by counting fractions of bands. Figure 4 shows the power spectra of the time series for the 0.30 M and 0.10 M runs respectively. We observe that the two power spectra show a notable similarity, revealing a deterministic chaotic regime. The power spectra show the elevated baseline typical of a random system but also the presence of significant periodicities, consistently with the two inherent features of random dissolution and periodic precipitation. The next-amplitude maps and the 3D phase portraits are depicted in Figs. 5 and 6 respectively. A strange attractor trajectory is clearly seen, most notably in the 3D phase-space plots of Fig. 6. We now investigate the oscillatory behavior relative to the fitted curve obtained in Fig. 3. We subtract the value in the best fit from the actual intensity (number of bands). We call the obtained quantity Q. The results for the time series and the reconstituted next-amplitude maps and phase portraits are displayed in Figs. 7 and 8. The phase portraits in frames d) of Figs. 7 and 8 exhibit essentially the same type of trajectory as the ones in Fig. 6, thus 7



revealing the robustness of this dynamical system, and confirming the chaotic nature of the oscillations. Chaotic behavior in Liesegang systems can be generated through various other routes. The use of a non-constant diffusion source such as a constantly fed unstirred reactor (CFUR)19 can yield a non-conventional Liesegang pattern with unpredictable band formation trend. Onset of irregular band formation with non-conformity to the power laws20—22 was observed in multi-precipitate systems.23 Das et al.24 obtained alternation of yellow and red HgI2 precipitate polymorphs under the effect of light and electric fields. Strange rhythmicity of band multiplets with ascending number of bands within each group was obtained in a three-component Co(OH)2 -Ni(OH)2 -Mg(OH)2 system.25 Aperiodicity in the distribution of Liesegang bands was conjectured and demonstrated26 in a two-precipitate system using the competitive particle growth (CPG)27 model. Finally, the precipitation-redissolution scenario via complex formation was simulated by various reaction-diffusion models28,29 . The present study consolidated the belief that such diffusion-precipitation-redissolution Liesegang systems exhibit deterministic chaos in the dynamical oscillations of the number of bands. Acknowledgement 1 This study was supported by a research grant of the University Research Board (URB) of the American University of Beirut. The authors thank Abdur-Rahman Al-Hamaly for his help in the camera setup and picture automation.

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Figure 4: Power spectra of the chaotic oscillations in Fig. 3. For both experiments, [NH4 OH]0 = 13.3 M. a. [Co2+ ]0 = 0.30 M; b. [Co2+ ]0 = 0.10 M .The two spectra are essentially similar, revealing deterministic chaos.

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Figure 5: Return maps for the time series plotted in Fig. 3. Strange attractor trajectories are typical of chaotic oscillations. For both experiments, [NH4 OH]0 = 13.3 M. a. [Co2+ ]0 = 0.30 M; b. [Co2+ ]0 = 0.10 M.

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Figure 6: 3D Phase portraits for the time series in Fig. 3. N’and N ” are the first and second derivatives of the variable N respectively. The loop-shaped attractors reveal a chaotic dynamics. For both experiments, [NH4 OH]0 = 13.3 M. a. [Co2+ ]0 = 0.30 M; b. [Co2+ ]0 = 0.10 M.

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Figure 7: a. Time series of the 0.30 M experiment, relative to the fitted curve of Fig. 3.a. b. Next-amplitude map. c. Phase portrait displayed as separate points. d. Phase portrait with the points joined in the right time sequence. Q’ and Q” are first and second derivatives of the interpolation function Q respectively.

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Figure 8: a. Time series of the 0.10 M experiment, relative to the fitted curve of Fig. 3.b. b. Next-amplitude map. c. Phase portrait displayed as separate points. d. Phase portrait with the points joined in the right time sequence. Q’ and Q” are first and second derivatives of the interpolation function Q respectively.

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References 1. Liesegang, R. E. Chemische Fernwirkung. Lieseg. Photograph. Arch. 1896, 37, 305-309. 2. Liesegang, R. E. Chemische Fernwirkung. Lieseg. Photograph. Arch. 1896, 37, 331-336. 3. Liesegang R. E. Über einige Eigenschaften von Gallerten. Naturwiss. Wochenschr. 1896, 11, 353-362. 4. Henisch, H. K. Crystals in Gels and Liesegang Rings; Cambridge University Press: Cambridge, U.K., 1988. 5. Liesegang, R. E. Geologische Diffusionen; Steinkopff: Dresden, 1913. 6. Liesegang, R. E. Die Achate; Steinkopf: Dresden-Leipzig: 1915. 7. Fractals and Dynamic Systems in Geoscience; Kruhl, J. H., Ed.; SpringerVerlag: Berlin, 1994. 8. Growth, Dissolution and Pattern Formation in Geosystems; Jamtveit, B., Meakin, P., Eds.; Kluwer: Dordrecht, 1999. 9. Ortoleva, P. Geochemical Self-Organization; Oxford University Press: New York, 1994. 10. Msharrafieh, M.; Al-Ghoul, M.; Zaknoun, F.; El-Rassy, H.; El-Joubeily, S; Sultan, R. Simulation of Geochemical Banding I: Acidization-Precipitation Experiments in a Ferruginous Limestone Rock. Chemical Geology 2016, 440, 42. 11. Sadek, S.; Sultan, R. In Precipitation Patterns in Reaction-Diffusion Systems; Lagzi I. Ed.; Research SignPost Publications: Kerala, 2011, Chapter 1, pp. 1-43. 14


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12. Nakouzi, E.; Steinbock, O.; Self-Organization in Precipitation Reactions Far from Equilibrium. Sci. Adv. 2016, 2, e1601144. 13. Sultan, R. Propagating Fronts in Periodic Precipitation Systems with Redissolution. Phys. Chem. Chem. Phys. (PCCP) 2002, 4, 1253-1261. 14. Volford, A.; Izsák, F.; Ripszám, M.; Lagzi, I. Pattern Formation and SelfOrganization in a Simple Precipitation System. Langmuir 2007, 23, 961964. 15. WebAssign, Cengage Learning, www.webassign.net/trochemistry2/a-2c.pdf (accessed June 1, 2018). 16. Sultan, R.; Sadek, S. Patterning Trends and Chaotic Behavior in Co2+ /NH4 OH Liesegang Systems. J. Phys. Chem. 1996, 100, 16912-16920. 17. Nasreddine, V.; Sultan, R. Propagating Fronts and Chaotic Dynamics in Co(OH)2 Liesegang Systems. J. Phys. Chem. A 1999, 103, 2934-2940. 18. Mathematica, version 7.0, software for technical computation; Wolfram Research: Champaign, IL, 2009. 19. Das, I.; Pushkarna, A.; Bhattacharjee, A. Dynamic Instability and LightInduced Spatial Bifurcation of HgI2 and External Electric Field Experiments in Two-Dimensional Gel Media. J. Phys. Chem. 1991, 95, 3866-3873. 20. Jablczynski, C. K. Mémoires Présentés à la Société Chimique. Les Anneaux de Liesegang. Bull. Soc. Chim. Fr. 1923, 11, 1592-1602. 21. Antal, T.; Droz, M.; Magnin, J.; Rácz, Z.; Zrinyi, M. Derivation of the Matalon-Packter Law for Liesegang Patterns. J. Chem. Phys. 1998, 109, 9479-9486.

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22. Shreif, Z.; Mandalian, L.; Abi-Haydar A.; Sultan, R. Taming Ring Morphology in 2D Co(OH)2 Liesegang Patterns. Phys. Chem. Chem. Phys. 2004, 6, 3461-3466. 23. Msharrafieh, M.; Sultan, R. Dynamics of a Complex Diffusion-PrecipitationRedissolution Liesegang Pattern. Chem. Phys. Lett. 2006, 421, 221-226. 24. Das, I.; Pushkarna, A.; Bhattacharjee, A. New Results on Light-Induced Spatial Bifurcation and Electrical Field Effect on Chemical Waves In the HgCl2 -KI System in Gel Media. J. Phys. Chem. 1990, 94, 8968-8973. 25. Msharrafieh, M.; Sultan, R. Patterns with High Rhythmicity Levels in Multiple-Salt Liesegang Systems, ChemPhysChem 2005, 6, 2647-2653. 26. Sultan, R.; Ortoleva, P. Periodic and Aperiodic Banding in Two Precipitate Post-Nucleation Systems. Physica D 1993, 63, 202-212. 27. Feinn, D.; Scalf, W.; Ortoleva, P.; Schmidt, S.; Wolff, M. Spontaneous Pattern Formation in Precipitating Systems. J. Chem. Phys. 1978, 69, 2739. 28. Al-Ghoul, M.; Sultan, R. Front Propagation in Patterned Precipitation 1. Simulation of a Migrating Co(OH)2 Liesegang Pattern. J. Phys. Chem. 2001, 105, 8053-8058. 29. Lagzi, I. Liesegang patterns: Complex formation of precipitate in an electric field. Pramana - J. Phys. 2005, 64, 291-298.

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Figure 9: TOC. This is NOT a Figure.

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Revisited Chaos in a DiffusionâPrecipitationâRedissolution Liesegang

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