NaOH

Article pubs.acs.org/Langmuir

Propagation Properties of the Precipitation Band in an AlCl3/NaOH System Takahiko Ban,*,† Yuichiro Nagatsu,‡ and Hideaki Tokuyama‡ †

Division of Chemical Engineering, Department of Materials Engineering Science, Graduate School of Engineering Science, Osaka University, Machikaneyamacho 1-3, Toyonaka, Osaka 560-8531, Japan ‡ Department of Chemical Engineering, Tokyo University of Agriculture and Technology, 2-24-16 Naka-cho, Koganei, Tokyo 184-8588, Japan S Supporting Information *

ABSTRACT: When inherently immobile solid particles collectively form precipitates in a reaction−diffusion system involving a redissolution reaction, a propagation phenomenon may occur in which a dynamic pattern of precipitation bands forms. This propagating precipitation phenomenon has been studied by many researchers. However, two completely different processesi.e., the reaction-diffusion of reactants and the crystal growth of products progress simultaneously in the system, thereby rendering the phenomenon complex. There are no well-established experimental laws for this propagating precipitation phenomenon, such as the spacing, time, and width laws associated with the well-known Liesegang phenomenon, which is static in the sense that precipitation bands form and remain at the same position. In fact, it has not been clarified which of the processes controls the propagation phenomenon. Accordingly, we have investigated the apparent diffusion coefficient associated with the dynamics of propagating precipitation band in an AlCl3/NaOH system for the case in which a large excess of outer electrolytes (i.e., OH−) diffuses into gel in which inner electrolytes (i.e.,Al3+) are homogeneously distributed. An isolated precipitation band of Al(OH)3 was formed horizontally in a test tube and propagated vertically in proportion to the square root of time. In our experimental results, we found that the apparent diffusion coefficient, Dp, possesses an exponential dependence on the initial concentrations of the outer electrolyte, and the inner electrolyte; the measured relation was Dp = D[Al3+]−0.6[OH−]0.6, where D = (0.63 ± 0.04) × 105 cm2/s. From our model equations based on the prenucleation theory, which take into account a redissolution reaction, we found that the dynamics of the reaction front of the outer and the inner electrolytes was an important factor in controlling the propagation of the precipitation band. In our simulation results, we obtained a similar dependence of the apparent diffusion coefficient on the electrolyte concentrations.

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INTRODUCTION Solid particles appear to move in a solid phase when a precipitation reaction is coupled with a redissolution process.1−5 This fascinating phenomenon occurs when a large excess of an electrolyte diffuses into an agar gel in which the other electrolyte is homogeneously dissolved; the former electrolyte is called the outer electrolyte, whereas the latter is called the inner electrolyte. The precipitate, once formed, is dissolved in the presence of excess outer electrolyte, and another precipitate forms ahead of the reaction front; in this way, the precipitation band visibly propagates. This propagating precipitation phenomenon has sparked a new research field in solid-phase dynamic pattern formation following the liquid-phase pattern formation, examples of which include the Belousov−Zhabotinsky reaction6,7 and the gasphase pattern formation observed in the CO oxidization process associated with platinum boards.8,9 More specifically, the dynamic pattern formation of precipitation bands due to © 2015 American Chemical Society

the redissolution of the precipitate has recently been reported. This phenomenon can be divided into two types: first, there exists chemical wave-like pattern formation in which the boundary interface of precipitation bands propagates in time with a t1 scaling on scales of more than several minutes;3−5 according to the second type, the boundary interface propagates with a t1/2 scaling on scales of several hours to several days.1,2 In the former case, precipitation formation in the AlCl3/NaOH system progresses three-dimensionally, and two main patterns−the double-armed spiral3,5 and turbulence types4 − have been reported; we refer to this precipitation formation as propagating precipitation waves. Ayass et al. reported that the propagation of precipitation patterns in the HgCl2/KI system could be categorized as superdiffusion.10 This Received: September 24, 2015 Revised: December 2, 2015 Published: December 31, 2015 604

DOI: 10.1021/acs.langmuir.5b03571 Langmuir 2016, 32, 604−610

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Langmuir

the Liesegang phenomenon. Thus, the apparent diffusion coefficient of the propagating precipitation band is related to the diffusion flux of the outer electrolyte coupled to a precipitation reaction between the two electrolytes. The validation of theoretical models requires us to experimentally measure the dynamic properties of the reaction front between the outer and inner electrolytes, which cannot be directly measured. The continuous dynamics of the propagating precipitation band serves as a visually perceivable measure of the dynamics of the reaction front. In this paper, we investigate the dynamic properties of a single propagating precipitation band in the AlCl3/NaOH system. Because its dynamics is much simpler, we focus on the propagating precipitation band for long time scales rather than propagating precipitation waves for short time scales in order to elucidate its essential dynamic properties. The apparent diffusion coefficient of the propagating precipitation band was measured as a function of the outer and inner electrolyte concentrations. We also established a mathematical model based on the prenucleation theory, which includes a redissolution term. The experimental and simulated results show that the outer electrolyte promotes the propagation of the precipitation band, whereas the inner electrolyte inhibits its propagation.

result revealed that propagating precipitation waves do not have a uniform velocity. In the latter case, an isolated band in the Cr(NO3)3/NH4OH system1 or multiple bands in the CoCl2/NH4OH system2 form homogeneously in a plane, and propagate in the same direction as the diffusion of the outer electrolyte; the precipitation bands continue moving downward without degrading in proportion to the square root of time; we refer to this as a propagating precipitation band. The AlCl3/NaOH system housed in a radially symmetric, two-dimensional setup leads to a propagating precipitation ring that exhibits complex motion with limited feeding of the outer electrolyte; more specially, the front of the ring reverses its motion, while stationary bands are generated in its wake.11 Lagzi et al. qualitatively explained the dynamics on the basis of the phase-separation scenario of colloidal Al(OH)3. Furthermore, they formed a polygonal precipitation ring by controlling the pH field.12 On the other hand, the Liesegang phenomenon has been studied for a long time.13−28 When an electrolyte diffuses from an aqueous solution into a gel, a series of stationary precipitation bands appear at various positions in the gel. The Liesegang phenomenon consists of a regular discrete pattern with three scaling properties, which can be expressed in terms of time, spacing, and width laws. The position xn, which is the distance of the nth precipitation band from the interface of the gel and the aqueous solution, changes in proportion to the square root of time (called the time law). Furthermore, there exists a positive constant, α, such that xn + 1 = αxn (called the spacing law). wn + 1/wn is approximately constant, where wn is the width of the nth precipitation band (called the width law). There are two important theories that describe the Liesegang pattern: the prenucleation theory, which is based on the supersaturation effect,18−20 and the postnucleation theory, which is based on Ostwald’s ripening process of colloid particles.21−23 The prenucleation theory is more useful in understanding the Liesegang phenomenon because the mathematically rigorous works based on the prenucleation theory demonstrate the time and spacing laws.25−27 Ueyama and Mimura found that the production rate of the precipitate was a crucial factor in changing the precipitation pattern.28 Their simulation showed that the classic Liesegang bands changed to a spot or labyrinth pattern as the production rate increased. Furthermore, Toramaru et al. experimentally demonstrated that changing the gel concentration led to the formation of continuous tree-like crystal aggregates in a leadiodide Liesegang system.29 This pattern formation might be due to the increasing production rate of colloidal particles. Izsák and Lagzi performed numerical simulations to investigate the crossover from a propagating precipitation band to multiple discrete bands.30 The model is based on Ostwald’s ion-product supersaturation theory proposed by Büki et al.19 The simulation results showed that the propagating precipitation band is slower for a higher initial concentration of the inner electrolyte, whereas increasing the initial concentration of the outer electrolyte results in faster propagation. The multiple discrete bands repeatedly exhibited the cycle of creation and annihilation, subject to the spacing law. The dynamics of the reaction front in the Liesegang phenomenon was theoretically investigated.22−24 According to theoretical studies by Venzl and Antal, the effective diffusion coefficient of the reaction front depends on the concentration of the two electrolytes. The propagating precipitation band proceeds according to diffusion in a manner similar to that of

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EXPERIMENTAL SECTION

Agar (guaranteed grade, Wako) was added to pure water at 3% w/w by stirring on an 80 °C hot plate. The solution was removed from the hot plate, and a predetermined amount of AlCl3·6H2O (guaranteed grade, Wako) was added to the solution and completely dissolved by stirring for a few minutes. Ten milliliters of this mixture was transferred to a test tube, capped, and left at room temperature for 48 h. Next, 10 mL of NaOH solution was added to the formed gel, and the reaction was subsequently observed. The pattern formation of the propagating precipitation band was recorded by a digital camera for 6 days. Mathematical Model. The outer electrolyte OH− reacts with the inner electrolyte Al3+, which is homogeneously distributed in the gel beforehand; the colloidal product Al(OH)3 forms on the basis of the following reaction: OH− + Al3+ → Al(OH)3. When the concentration of the colloidal product Al(OH)3 exceeds the supersaturated concentration, the precipitate Al(OH)3 is formed at a rate proportional to the difference between the colloid concentration and the saturation concentration. When excess OH− reacts with the precipitate, complex formation occurs, and the precipitate is dissolved. A sharp layer of Al(OH)3 was formed through the precipitation and complex formation according to the reactions shown below:

Al3 + + 3OH− ⇆ Al(OH)3 (colloid)

(1a)

Al(OH)3 (colloid) ⇆ Al(OH)3 (precipitate)

(1b)

Al(OH)3 (precipitate) + OH− ⇆ [Al(OH)4 ]−

(1c)

We added a redissolution term associated with the presence of excess outer electrolytes to the prenucleation theory.18 Our model then becomes

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∂a ∂ 2a = Da 2 − k1amb − k 2ad ∂t ∂x

(2a)

∂b ∂ 2b = Db 2 − k1amb ∂t ∂x

(2b)

∂c ∂ 2c = Dc 2 + k1amb − P(c , d) ∂t ∂x

(2c)

∂d = P(c , d) − k 2ad ∂t

(2d) DOI: 10.1021/acs.langmuir.5b03571 Langmuir 2016, 32, 604−610

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Langmuir where a, b, c, and d represent the concentrations of OH−, Al3+, colloidal Al(OH)3, and precipitation Al(OH)3, and Da, Db, and Dc represent the diffusion coefficients of OH−, Al3+, and Al(OH)3, respectively. k1 and k2 represent the rate constants for the forward reactions in eqs 1a and 1c, respectively, and m represents the reaction order. In the simulation, the back reactions of eqs 1a−1c were ignored. P(c,d) represents a precipitation term that includes the effect of supersaturation, subject to the following relation: ⎧ 0 if c < css and d = 0 ⎫ ⎪ ⎪ ⎬ P(c , d) = ⎨ ⎪ ⎪ ⎩ q(c − cs) if c ≥ css or d > 0 ⎭

(3)

where cs and css represent the saturation and supersaturation concentrations, respectively. q represents the rate constant of precipitate formation. Considering the stoichiometric coefficients of the reaction relations of OH− and Al3+, setting the reaction order m to 3 reflects the actual state. However, setting the reaction order to 1 did not qualitatively alter the simulation results based on the influence of the reaction order described below. Thus, in order to simplify the equation, the results of the simulation with m = 1 are mainly shown below. Time, distance, and concentration are dimensionless in the above nondimensional model.

Figure 2. Relationship between the position of the propagating precipitation band and the square root of time. The position of the propagating precipitation band position changed linearly, and the slope flattened as the concentration increased. [NaOH] = 2.5 M and [gel] = 3% w/w.

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RESULTS AND DISCUSSION A typical propagating precipitation band is shown in Figure 1. A single precipitation band was formed and propagated down-

involved redissolution obeys to the time law associated with the well-known Liesegang phenomenon. Hence, the apparent diffusion coefficient of the precipitation band can be defined by the following equation:

x=

2Dpt

(4)

where x [cm] represents the distance of the propagating precipitation band from the interface between the gel and the solution of the outer electrolyte, and t [s] and Dp [cm2/s] correspond to the time and apparent diffusion coefficient, respectively. Using this equation, the apparent diffusion coefficient was determined from the slope of the measurement time points shown in Figure 2. Figure 3 shows a log−log plot of

Figure 1. A typical experimental example of precipitation band propagation in agar gel. The time points are (a) 2, (b) 25, (c) 73, and (d) 136 h. Arrows indicate the position of the propagating precipitation band. [Al3+] = 0.1 M, [NaOH] = 2.5 M, and [gel] = 3% w/w. Figure 3. (a) Relationship between the apparent diffusion coefficient of the propagating precipitation band and the outer electrolyte concentration. The slope of the line is 0.61 ± 0.02. (b) Relationship between the apparent diffusion coefficient of the propagating precipitation band and the inner electrolyte concentration. The slope of the line is −0.60 ± 0.03.

ward without degradation. Repetitive precipitation and redissolution reactions caused the band to propagate. The gel color changed to yellow owing to the alkaline substance that remained after the propagating precipitation band passed. The time-course of the band position from the gel−solution interface was experimentally measured by changing the concentration of the inner electrolytes from 0.05 to 0.25 M. The relationship between the position of the propagating precipitation band and the square root of time is shown in Figure 2. At all concentrations, the precipitation band propagated in proportion to the square root of time. For concentrations higher than 0.2 M, immobile precipitation bands were formed after 50 h. This may be considered as the formation of static Liesegang bands owing to the depletion of the outer electrolytes. The behavior of the band in a system that

the relationship between the apparent diffusion coefficient and the concentration of the outer electrolyte, in which the apparent diffusion coefficient is the average value of three measurements, and the error bars represent the standard deviations. The apparent diffusion coefficient increased as the concentration increased (Figure 3a). Next, the apparent diffusion coefficient was measured as a function of the concentration of the inner electrolyte. In contrast to the propagation dynamics using the outer 606

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Langmuir electrolyte, the apparent diffusion coefficient deceased with an increase in the concentration of the inner electrolyte (Figure 3b). The following relation was established from the experimental results:

Dp = Da0 rab0 rb

(5)

where a0 and bo represent the initial concentrations of the outer and inner electrolytes. The value of ra = 0.61 ± 0.02 in the initial concentrations of the outer electrolyte was determined from the experimental results for a constant concentration of the inner electrolyte; on the other hand, we obtained rb = −0.60 ± 0.03 for a constant concentration of the outer electrolyte. The absolute values of both exponents experimentally determined by eq 5 are almost identical; on the other hand, ra is opposite in sign to rb. Therefore, D in eq 5 should be independent of the concentrations of both electrolytes, and should be represented in the same units as the diffusion coefficient. If the apparent diffusion coefficient Dp is divided by [Al3+]−0.6[OH−]0.6, its value should be constant. The value of D is plotted in Figure S1. We obtain the constant value: D = (0.63 ± 0.04) × 105 cm 2/s . According to the theoretical studies by Venzl, the position of the reaction front, X is described as23 ⎛ a − b0 ⎞ X = 2 Dt erf −1⎜ 0 ⎟ ⎝ a0 + b0 ⎠

Figure 4. Simulated concentration distributions using eqs 2 and 3. The red and black curves in the upper panels represent the concentration distributions of the outer electrolyte (a) and the inner electrolyte (b), respectively, whereas the red and black curves in the lower panels represent the concentration distributions of the colloid (c) and the precipitate (d), respectively. Panels a and b show the concentration distributions at t = 20, and at t = 400, respectively. An isolated band propagates, and its position is equal to the position of the reaction front of the product. The values of the parameters are m = 1, Da = 0.001, Db = 0, Dc = 0.001, cs = 0.2, css = 0.20, k1 = 50, k2 = 0.5, q = 50, a0 = 10, and b0 = 1. Time, distance, and concentration are dimensionless.

(6)

−1

where erf represents the inverse error function, and D represents the diffusion coefficient of the electrolytes. On the other hand, the apparent diffusion coefficient of the propagating precipitation band in this study is described experimentally as Dp = Da0 rab0 rb . If the dynamics of the propagating precipitation band reflects the dynamics of the reaction front, we obtain ⎛ a − b0 ⎞ a0 ra/2b0 rb/2 ≈ erf −1⎜ 0 ⎟ ⎝ a0 + b0 ⎠

The concentrations of these reactants smoothly decrease toward the reaction front, nearly reaching zero at the cross point. In the region in which the reaction front of the outer electrolyte and inner electrolyte passed through, the concentration of the colloid particle is equal to the saturation concentration. The concentration smoothly decreases from ahead of the reaction front, reaching zero (lower panel of Figure 4a). The concentration distribution of the precipitation is broad to some extent, and the peak’s location is equal to the location of the reaction front. The concentration shows a steep gradient at the front of the peak, slowly decreases at the back, and reaches zero at both the front and back. The precipitate follows the reaction front (Figure 4b). Although some parameters used in the simulation constitute unrealistic values, this has no qualitative effect on the dynamics of the propagating precipitation band. Our model produces a propagation pattern of the isolated precipitate that is similar to that of the experiment. Our simulation states that the reaction front of the outer and inner electrolytes controls the dynamics of the propagating precipitation band. eqs 2a and 2b play key roles in producing the precipitate precursor, i.e., the material supply for colloidal growth. The term kab in eq 2c behaves as a source term of colloidal particles. In turn, the spatiotemporal changes in kab are determined by eqs 2a and 2b. Thus, colloidal growth strongly depends on eqs 2a and 2b, and these are called the master equations, whereas eqs 2c and 2d are called the slave equations.27 A simulation was performed by changing the concentration of the inner electrolyte while holding the concentration of the outer electrolyte constant. The position of the precipitate was

(7)

a0rb0−r

The inverse error function varies as (r ≈ 0.13 − 0.87) as a0 or b0 changes from 0.001 to 10. Thus, the exponents obtained in this study may correspond to the concentration range expressed by r = 0.3. Before moving on to the model simulation of the propagating precipitation band, we introduce a few results concerning the formation of static precipitation bands obtained by the simulation based on the proposed mathematical model. When the supersaturation concentration css is higher than the saturation concentration cs in the absence of the redissolution term, i.e., k2 = 0, the simulation shows that Liesegang bands form (Supporting Information Figure S2). The gap between the formed bands was widened with an increase in the supersaturation concentration. Furthermore, we confirmed that the simulation results satisfied the time law and the spacing law. In the presence of the redissolution term, we obtained a propagating precipitation band in the simulation. The typical simulation results of the propagating precipitation band are shown in Figure 4. The upper panels show the concentration distributions of the outer and inner electrolytes, whereas the lower panels show the concentration distributions of the colloid particles and the precipitate. The outer electrolyte diffuses from the origin and its concentration decreases as x increases, consuming the inner electrolyte (upper panel of Figure 4a). 607

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Langmuir expressed as a function of t1/2, and the results are shown in Figure 5. The precipitate propagation shows a linear relation-

Figure 7. Relationship between the exponent rb of eq 5 and the diffusion coefficient ratio of the electrolytes. The red and black curves indicate values at reaction orders of m = 1 and m = 3, respectively. The values of the parameters are Da = 0.001, Dc = 0.001, cs = 0.2, css = 0.20, k1 = 50, k2 = 0.5, and q = 50.

Figure 5. Relationship between the position of the precipitate and the initial concentration of the inner electrolytes (b0). The precipitate propagates in proportion to the square root of time. The values of the parameters are the same as those in Figure 4, excluding the initial concentration, b0.

calculation, the precipitation dynamics changed from a propagating precipitation band to multiple discrete bands, the behavior of which is shown in Figure S4. A single precipitation band with a steep front and broad back-propagated in an early stage. Another thin precipitation band was then suddenly formed at its front. At this stage, the peak of the first propagating precipitation band decreased and disappeared as the second precipitation band grew. Next, a new precipitation band was formed in addition to the second one, and the old band decreased as the new precipitation band grew. One band was repeatedly formed after another, and gradually separated from the old one. Finally, several thin precipitation bands with similar widths were formed. Note that during the cycle of production, growth, dissolution, and disappearance, the multiple thin precipitation bands appeared not to move continuously, unlike the isolated propagating precipitation band. We found that the emergence position of the new precipitation band was equal to that of the reaction front. Furthermore, we confirmed that the space between precipitation bands increased as the supersaturation concentration increased for a given saturation concentration. The formation dynamics of the multiple discrete precipitation bands corresponds to the moving Liesegang bands in the Co2+/NH4OH reaction system.2 Finally, we performed a simulation in which the reaction rate of redissolution k2 and the rate constant of precipitate formation q were changed independently. We found that the reaction rate of precipitate formation q and the rate constant of redissolution k2 had no effect on the dynamics of the propagating precipitation band, although these reaction rates had an effect on the concentration and shape of the propagating precipitation band (See Figures S5 and S6). The velocity of the propagating precipitation band was independent of these reaction rates.

ship to t1/2, similarly to the experimental results. The velocity of precipitate propagation slowed as the initial concentration of the inner electrolyte increased. The apparent diffusion coefficient of the precipitate, Dp,cal., can be determined using eq 4, and the results are shown in Figure 6. The apparent diffusion coefficient increased as the

Figure 6. (a) Relationship between the apparent diffusion coefficient of the precipitate and the initial concentration of the outer electrolyte. (b) Relationship between the apparent diffusion coefficient of the precipitate and the initial concentration of the inner electrolyte. The values of the parameters are the same as those in Figure 4, excluding the initial concentrations, a0 and b0.

initial concentration of the outer electrolyte increased, whereas it deceased as the initial concentration of the inner electrolyte increased, similarly to the results of the experiment. Based on simulation results, the exponent rb was determined using eq 5, where it depends on various parameters. Herein, only the influence of the ratio of the diffusion coefficient of the inner electrolyte to the diffusion coefficient of the outer electrolyte on the exponent rb is presented in Figure 7. The value of rb decreased as the diffusion coefficient of the outer electrolyte increased. A similar tendency was obtained at m = 3. The value of rb at m = 3 was lower than that at m = 1. The influence of the ratio of supersaturation concentration to saturation concentration on the exponents ra and rb is shown in Figure S3. Both exponents decreased as this ratio increased. When the difference between the saturation concentration cs and supersaturation concentration css was varied in the

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CONCLUSION We performed an experiment concerning the propagating precipitation band using the AlCl3/NaOH system, and performed simulations using our model equation, which was obtained by adding a redissolution term to the prenucleation theory. The dynamics of the propagating precipitation band may be applicable to an autonomous classification of functional fine particles.31,32 In a test tube containing agar gel in which 608

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Langmuir Al3+ was homogeneously dissolved, when OH− diffused from the top of the gel, an isolated precipitation band propagated in the direction of OH− diffusion, and the boundary interface of the propagating precipitation band was proportional to the square root of time. The propagating precipitation band obeyed the same time law that is observed in the static Liesegang phenomenon. Furthermore, the apparent diffusion coefficient of the propagating precipitation band decreased as the inner electrolyte concentration increased, whereas it increased as the outer electrolyte concentration increased. These findings can be explained by focusing on the behavior of the reaction front. Increasing the concentration of the inner electrolytes inhibits excess diffusion of the outer electrolytes into the gel, leading to a decrease in the velocity of the reaction front propagation; the propagation of the precipitation band subsequently slows down. On the other hand, an increase in the concentration of the outer electrolyte facilitates diffusion of the outer electrolyte into the gel, which may have enhanced the reaction front propagation (and consequently the precipitation band propagation). Our model shows that the transition from the propagating precipitation band to the multiple discrete bands depends on the difference between the saturation and supersaturation concentrations. In order to apply our model to the propagating precipitation waves reported by Volford et al.,3 it should be modified to produce a uniform velocity of propagating precipitation patterns, which requires the autocatalytic production of the precipitate.5

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(6) Zaikin, A. N.; Zhabotinsky, A. M. Concentration Wave Propagation in Two-Dimensional Liquid-Phase Self-Oscillating System. Nature 1970, 225, 535−537. (7) Ouyang, Q.; Flesselles, J.-M. Transition from Spirals to Defect Turbulence Driven by a Convective Instability. Nature 1996, 379, 143−146. (8) Jakubith, S.; Rotermund, H. H.; Engel, W.; Von Oertzen, A.; Ertl, G. Spatiotemporal Concentration Patterns in a Surface Reaction: Propagating and Standing Waves, Rotating Spirals, and Turbulence. Phys. Rev. Lett. 1990, 65, 3013. (9) Rotermund, H. H.; Engel, W.; Kordesch, M.; Ertl, G. Imaging of Spatio-Temporal Pattern Evolution during Carbon Monoxide Oxidation on Platinums. Nature 1990, 343, 355−357. (10) Ayass, M. M.; Lagzi, I.; Al-Ghoul, M. Three-Dimensional Superdiffusive Chemical Waves in a Precipitation System. Phys. Chem. Chem. Phys. 2014, 16, 24656−24660. (11) Lagzi, I.; Pápai, P.; Rácz, Z. Complex Motion of Precipitation Bands. Chem. Phys. Lett. 2007, 433, 286−291. (12) Volford, A.; Izsák, F.; Ripszám, M.; Lagzi, I. Systematic Front Distortion and Presence of Consecutive Fronts in a Precipitation System. J. Phys. Chem. B 2006, 110, 4535−4537. (13) Liesegang, R. E. Ueber einige Eigenschaften von Gallerten. Naturw. Wochenschr. 1896, 11, 353−362. (14) Matalon, R.; Packter, A. The Liesegang Phenomenon. I. Sol Protection and Diffusion. J. Colloid Sci. 1955, 10, 46−62. (15) Gálfi, L.; Rácz, Z. Properties of the reaction front in an A+ B→ C type reaction-diffusion process. Phys. Rev. A: At., Mol., Opt. Phys. 1988, 38, 3151−3154. (16) Antal, T.; Droz, M.; Magnin, J.; Rácz, Z. Formation of Liesegang Patterns: A Spinodal Decomposition Scenario. Phys. Rev. Lett. 1999, 83, 2880−2883. (17) Nabika, H.; Sato, M.; Unoura, K. Liesegang patterns engineered by a chemical reaction assisted by complex formation. Langmuir 2014, 30, 5047−5051. (18) Wagner, C. Mathematical Analysis of the Formation of Periodic Precipitations. J. Colloid Sci. 1950, 5, 85−97. (19) Keller, J. B.; Rubinow, S. I. Recurrent Precipitation and Liesegang Rings. J. Chem. Phys. 1981, 74, 5000−5007. (20) Büki, A.; Kárpáti- Smidróczki, E.; Zrínyi, M. Computer Simulation of Regular Liesegang Structures. J. Chem. Phys. 1995, 103, 10387−10392. (21) Feeney, R.; Schmidt, S. L.; Strickholm, P.; Chandam, J.; Ortoleva, P. Periodic Precipitation and Coarsening Waves: Applications of the Competitive Particle Growth Model. J. Chem. Phys. 1983, 78, 1293−1311. (22) Venzl, G. Pattern formation in precipitation processes. I. The theory of Competitive Coarsening. J. Chem. Phys. 1986, 85, 1996− 2005. (23) Venzl, G. Pattern formation in precipitation processes. II. A Postnucleation Theory of Liesegang Bands. J. Chem. Phys. 1986, 85, 2006−2011. (24) Antal, T.; Droz, M.; Magnin, J.; Rácz, Z.; Zrinyi, M. Derivation of the Matalon-Packter Law fow Liesegang Patterns. J. Chem. Phys. 1998, 109, 9479−9486. (25) Hilhorst, D.; van der Hout, R.; Peletier, L. A. The Fast Reaction Limit for a Reaction-Diffusion. J. Math. Anal. Appl. 1996, 199, 349− 373. (26) Ohnishi, I.; Mimura, M. A Mathematical Aspect for Liesegang Phenomena. Proceedings of Equadiff-11 2005, 11, 343−352. (27) Hilhorst, D.; van der Hout, R.; Mimura, M.; Ohnishi, I. A Mathematical Study of the One-Dimensional Keller and Rubinow Model for Liesegang Bands. J. Stat. Phys. 2009, 135, 107−132. (28) Ueyama, D.; Mimura, M. Simulation Analysis for Precipitation Pattern of Liesegang Type. (in Japanese). RIMS Kokyuroku 2006, 1522, 136−143. (29) Toramaru, A.; Harada, T.; Okamura, T. Experimental Pattern Transitions in a Liesegang system. Phys. D 2003, 183, 133−140.

ASSOCIATED CONTENT

* Supporting Information S

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.langmuir.5b03571. Additional figures as described in the text (PDF)

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AUTHOR INFORMATION

Corresponding Author

*Tel:+81-6-6850-6625; Fax: +81-6-6850-6625; E-mail: ban@ cheng.es.osaka-u.ac.jp. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors thank Mr. Katsuyuki Suzuki at Doshisha University for constructive criticism. This study was financially supported by the Hosokawa Powder Technology Foundation.

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REFERENCES

(1) Zrínyi, M.; Gálfi, L.; Smidróczki, E.; Rácz, Z.; Horkay, F. Direct Observation of a Crossover from Heterogeneous Traveling Wave to Liesegang Pattern Formation. J. Phys. Chem. 1991, 95, 1618−1620. (2) Nasreddine, V.; Sultan, R. Propagating Fronts and Chaotic Dynamics in Co(OH)2 Liesegang Systems. J. Phys. Chem. A 1999, 103, 2934−2940. (3) Volford, A.; Izsák, F.; Ripszám, M.; Lagzi, I. Pattern Formation and Self-Organization in a Simple Precipitation System. Langmuir 2007, 23, 961−964. (4) Volford, A.; Lagzi, I.; Molnár, F., Jr.; Rácz, Z. Coarsening of Precipitation Patterns in a Moving Reaction-Diffusion Front. Phys. Rev. E 2009, 80, 055102. (5) Tinsley, M. R.; Collison, D.; Showalter, K. Propagating Precipitation Waves: Experiments and Modeling. J. Phys. Chem. A 2013, 117, 12719−12725. 609

DOI: 10.1021/acs.langmuir.5b03571 Langmuir 2016, 32, 604−610

Article

Langmuir (30) Izsák, F.; Lagzi, I. Simulation of a Crossover from the Precipitation Wave to Moving Liesegang Pattern Formation. J. Phys. Chem. A 2005, 109, 730−733. (31) Putnis, A.; Fernandez-Diaz, L.; Prieto, M. Experimentally Produced Oscillatory Zoning in the (Ba, Sr)SO4 Solid Solution. Nature 1992, 358, 743−745. (32) Zhang, Y.; Ban, T.; Shioi, A. Spontaenous Particle Formation with Fine Structures in Gelatin. J. Chem. Eng. Jpn. 2006, 39, 587−595.

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