Superdiffusive Cusp-Like Waves in the Mercuric Iodide Precipitate

Article pubs.acs.org/JPCA

Superdiffusive Cusp-Like Waves in the Mercuric Iodide Precipitate System and Their Transition to Regular Reaction Bands Mahmoud M. Ayass and Mazen Al-Ghoul* Department of Chemistry, American University of Beirut, Riad El-Solh, 1107 2020 Beirut, Lebanon S Supporting Information *

ABSTRACT: We report a two-dimensional (2D) reaction−diffusion system that exhibits a superdiffusive propagating wave with anomalous cusp-like contours. This wave results from a leading precipitation reaction (wavefront) and a trailing redissolution (waveback) between initially separated mercuric chloride and potassium iodide to produce mercuric iodide precipitate (HgI2) in a thin sheet of a solid hydrogel (agar) medium. The propagation dynamics is accompanied by continuous polymorphic transformations between the metastable yellow crystals and the stable red crystals of HgI2. We study the dynamics of wavefront and waveback propagation that reveals interesting anomalous superdiffusive behavior without the influence of external enhancement. We find that a transition from superdiffusive to subdiffusive dynamics occurs as a function of outer iodide concentration. Inner mercuric concentrations lead to the transition from the anomalous cusp-like to cusp-free regular bands. While gel concentration affects the speed of propagation of the wave, it has no effect on its shape or on its superdiffusive dynamics. Microscopically, we show that the macroscopic wave propagation and polymorphic transformations are accompanied by an Ostwald ripening mechanism in which larger red HgI2 crystals are formed at the expense of smaller yellow HgI2 crystals.

■

reaction−diffusion systems.38 Observations of reactions using vertical gel slabs allow cross-sectional visualization and lead to the fact that these aforementioned waves move within the precipitation band, which is also concurrently moving downward in the gel.21 Despite the complexity of these processes, reaction kinetics does not expose any evident autocatalysis in the mechanism. These aspects suggest that the wave propagation in the Al(OH)3 precipitation system is fundamentally different from the familiar reaction−diffusion waves seen, for example, in the BZ reaction.17 Under no advection, most of the previously mentioned systems produce propagating chemical waves resulting either in constant velocities39−41 or, in the case of diffusion-limited reaction, in velocities that are consistent with the normal diffusive profile; i.e., leading to a linear relation between the mean square displacement of particles with time. However, recent experiments have confirmed that when the system is exposed to modulations that augment diffusion, the occurrence of anomalous superdiffusive chemical waves is achieved. Such modulations include chaotic flow created by Faraday waves,39,42 azimuthal motion of counter-rotating vortices,43 and simple stirring-induced hydrodynamic turbulence.44 In this work, we study wave propagation in a heterogeneous system based on the precipitation reaction between mercuric chloride (HgCl2) and potassium iodide (KI) to form mercuric

INTRODUCTION Reaction−diffusion (RD) systems play a fundamental role in the formation of several interesting patterns in various disciplines.1−6 Particularly, RD systems with initially segregated reactants have been vastly studied theoretically,7−10 computationally,11−13 and experimentally.14−17 They are characterized by the presence of a propagating reaction band or front commonly observed in various systems.18−21 When the chemical reaction is carried in a gel matrix and involves the precipitation of a pair of initially separated coprecipitate species (frequently called outer and inner electrolytes),22 the leading chemical front might yield in its wake the so-called Liesegang bands,23 which are stationary periodic precipitation patterns with distinctive periodicity in both their band separation distance and time of formation.18,24−26 There exist cases where the diffusing outer electrolyte is capable of redissolving the original precipitate bands by forming a soluble complex, while the precipitation process continues ahead. Depending on the kinetics of precipitation and dissolution, a propagating stratum of bands or a single band of precipitate emerges. Several types of such precipitation systems with redissolution have been reported and include the Co(OH)2/Co(NH3)62+,27,28 HgI2/ HgI42−,29−31 Cr(OH)3/Cr(OH)4−,32−34 and Al(OH)3/Al(OH)4−21 systems. A recent study by Volford et al.,21 using AlCl3 as the inner electrolyte and NaOH as the outer electrolyte, reports a single traveling precipitation band of amphoteric aluminum hydroxide. When these patterns are viewed from above, they develop spiral and target waves much like the ones observed in thin films of the Belousov− Zhabotinsky (BZ) reaction35−37 and in other quasi-2D © 2014 American Chemical Society

Received: March 4, 2014 Revised: April 25, 2014 Published: May 9, 2014 3857

dx.doi.org/10.1021/jp502217z | J. Phys. Chem. A 2014, 118, 3857−3865

The Journal of Physical Chemistry A

Article

iodide (HgI2) precipitate, which upon excess iodide undergoes redissolution to form the complex [HgI4]2−: Hg 2 +(aq) + 2I−(aq) → β ‐HgI2(s)

(1)

β ‐HgI2(s) → α ‐HgI2(s)

(2)

α‐HgI2(s) + 2I−(aq) → [HgI4]2 − (aq)

(3)

where β-HgI2 is the yellow polymorph of mercuric iodide with an orthorhombic crystalline symmetry and α-HgI2 is the red polymorph with a tetragonal crystalline symmetry.45 The yellow form is the kinetically favored polymorph, thus appearing at early stages of evolution, but it is not thermodynamically stable (metastable) and would readily transform upon the slightest mechanical disturbance to the more stable and thermodynamically favored red form of the compound.46,47 There also exists a third orange polymorph that appears instantly upon interaction between the electrolytes. It is also, like the yellow form, kinetically favored and readily transforms to the red form upon the slightest touch. We observe our reaction in a two-dimensional setup (explained in the experimental section) in which a precipitation band with irregular cusp-like contours of the leading and trailing edge propagates radially from the center of the reactor. This band creates captivating precipitation patterns concomitant with the sequential polymorphic transformation between the yellow/ orange and red polymorphs of the crystalline HgI2. The alteration of initial outer and inner concentrations is shown in this study to have tremendous effect on the speed and the shape of the propagating band. One main result consists of the superdiffusive nature of the transport in this system without any external intervention to enhance diffusion. In the last section, we shed light on the microscopic dynamics accompanying the wave propagation.

Figure 1. Schematic illustration representing the 2D reactor used to carry out the diffusion reactions. (A) Two circular plexiglass plates; the first one is the base plate containing the gel mixture and the second one is the covering plate that has a smaller diameter, equipped with a delivery reservoir at its center in addition to 0.7 mm spacers. (B) Top view representation of the radially propagating irregular precipitation band. The two black bold lines from the center of the reactor represent the wavefront (df) and waveback (db) that are monitored with time to study the dynamics of the bands. (C) Side-view of the full experimental setup displaying the two plexiglass plates with their dimensions. The positions of both the gel mixture containing the inner electrolyte and the outer electrolyte are also indicated. The sketch on the right side represents the macro-lens camera connected to an iMac; in addition, the position at which the snapshots of the propagating bands were captured is also indicated.

equipped with a thermostat that maintains the temperature at 24.0 ± 0.2 °C. The gels are left for 2 h for their gelation and aging processes to complete. A simple aqueous solution of KI serves as the outer electrolyte of the system. The range of concentrations we used are as follows: for inner [Hg2+]0 = 0.10−0.24 M, and for outer [I−]0 = 1.0−6.0 M. Next, we proceed with the performance of the reaction by pouring the prepared KI solution into the reservoir of the reactor containing the solidified gel (initialization step). This will allow the outer to react with the gel layer only at its center and thus diffuse radially forming a 2D ring-shaped precipitation band. We monitor the reactions under a high-resolution digital camera (Cannon EOS 450D) connected to an iMac with a built-in remote shooting software, where the camera options can be altered according to needed specifications. The mode that we use for our analysis takes a clear focused shot of the setup, and the software is set to take a snapshot every 5 min. The reactions proceed for more than 24 h each; therefore, with our settings, we produce well above 300 snapshots of the reactions. Combining these frames produces a clear time evolution of the traveling reaction bands in our system. The experimental setup is represented in Figure 1C.

■

EXPERIMENTAL SECTION The materials used in our experiments include mercuric chloride (Fischer Chemical), potassium iodide (Merck), and agar gel (Bacto). We perform all our experiments in gel media. According to the required concentrations, we prepare stock solutions of both mercuric chloride (HgCl2) and potassium iodide (KI), using double distilled water. We prepare the inner electrolyte by dissolving agar powder at a 1% ratio of the volume in a beaker containing the required volume of HgCl2 solution and equipped with a magnetic stirring rod on a stirring/hot plate. When stirring/heating, we maintain the temperature of the mixture at a range within 80−90 °C for it not to reach a boiling level; in addition, to ensure that no errors in the concentration happen (due to evaporation), we cover the beaker with a glass plate. The solution continues to mix until it becomes sufficiently clear having no floating gel bits indicating the completion of the inner mixture. This mixture is transferred while hot to a circular, two-dimensional Liesegang reactor (Figure 1A,B) in which the reaction is to be carried out. The reactor is composed of a base plate that contains the gel mixture, which in turn is covered by a second plate of smaller diameter having a reservoir in its center, via which the hot inner mixture and the outer electrolyte (at a later stage) are delivered. The covering plate is equipped with spacers, of 0.7 mm thickness, that separate it from the base plate thus producing a thin sheet of gel layer in which the outer would diffuse radially. Next, we cover the plates and place them in a chamber

■

RESULTS AND DISCUSSION Wave Dynamics. We initiate the reactions by pouring the outer electrolyte (KI) into the central reservoir, which allows it to diffuse into the gel containing the inner electrolyte (HgCl2) to form a two-dimensional diffusion front. Behind this front, a gel region now forms in which both the reactants are available. 3858



Article

This gel section, where precipitation and pattern formation occur, is referred to as the excitable region. In this region, there exists a zone, called the reaction front, at which the primary HgI2 precipitate forms. The primary precipitate mainly comprises the orange and yellow polymorphs, that are both metastable at ambient conditions, which in turn later transform to the red polymorph, the final and most stable structure of the crystals. Each of the yellow and orange polymorphs transform to the red form in their own manner. The orange crystals first form irregularly shaped red nuclei that spread and transform within hours, while preserving the external shape. Whereas the metastable yellow (yellowM) crystals form a red front, at the point of contact, which propagates rapidly to transform the whole crystal within seconds and significantly distort its structure.48,49 With time, the precipitation band of HgI2 propagates radially away from the center of the reactor, giving rise to a traveling wavefront. However, because of the excess outer iodide concentration, redissolution of the previously formed HgI2 band produces the colorless complex K2HgI4 thus giving rise to a propagating waveback. This interplay between precipitation and redissolution reactions results in the complex dynamics of the ring-like band studied in this system. Over a wide range of initial reaction concentrations of the inner and outer reactants (elaborated in upcoming sections) we observe the propagation of an irregularly shaped precipitation band with cusp-like borders from both sides (front and back). During its spatiotemporal evolution, the band undergoes an interesting transformation between the yellow/orange and red polymorphs of HgI2. This transformation occurs in a manner as if the yellow crystals are dissolving to yield needle-like structured α-red-HgI2 crystals. Figure 2 shows the evolution process with time of a single irregular band to form the cusplike pattern. The reason behind these abnormal structures seems to be due to the discontinuation of precipitate formation at certain points in the wavefront. During the propagation of the wavefront through the gel, these points expand into empty regions (precipitate-free). Thus, the wavefront is split up into segments called reaction zones, each of them being limited by regressing edges, along with two diagonal passive edges that limit the no precipitate regions from the diffusing band. The regressing edge and the two passive edges form a trapezoidal shape, which appears to be a very important qualitative property of the pattern. As diffusion continues, the length of the regressing edges decreases and therefore, approaching passive edges, they meet and annihilate each other, possibly because of the depletion of the reactants in their surroundings, thus producing a small triangular cusp at these annihilation points. After some time of diffusion, when the reactants are recovered, new wavefronts can develop anywhere along the passive edges, mostly at the wedge-like empty regions (precipitate-free). These wavefronts bring upon a different appearance of the cusp formation. They form in a bursting manner resulting in a curved shaped precipitation front as opposed to the aforementioned fronts that have linear features. Pair production and annihilation of traveling waves have been observed on several seashell patterns. Mathematical models based on nonlinear reaction−diffusion equations have reproduced this behavior.19 Similar cusp-like patterns, forming in gel media, were also encountered in the Cu(OH)218 and Al(OH)321 systems. Horizontal Speed of Cusp Formation. An important characteristic of the cusps lies in the speed of their formation. As the passive edges of the trapezoidal shaped structure

Figure 2. Top view snapshots of a section of the radially diffusing irregular precipitation band showing the process of cusp pattern development and formation with time. Initial conditions: inner, [Hg2+]0 = 0.21 M in 1% per volume agar gel; outer, [I−]0 = 3.0 M. Time after initialization: (A) 135, (B) 165, (C) 190, and (D) 200 min. Scale bar represents 0.5 cm (see Supporting Information).

approach each other horizontally, the length of the regressing edge decreases. Therefore, to predict the speed at which the cusps are forming, we measure the rate at which the regressing edge is disappearing. We perform these measurements on a range of different outer concentrations at constant inner concentration. For comparison of the range of concentrations, we pick out cusps forming in the same time interval in each setup. The alteration of the outer concentration shows a noticeable effect. The horizontal speeds of passive edges, leading to cusp formation, are calculated to be constant with time. In addition, when we compare different outer concentrations, the speed of the wavefront increases as a function of concentration. Figure 3 displays the speed of cusp formation at each outer concentration. It is significant that the higher outer concentration has a higher speed of cusp formation. We also measure speeds of cusps at different stages of the diffusion, of each specific setup, and conclude that at later stages of the reaction the new forming cusps are slower. This is reasoned to be due to the larger size of the cusp that forms at the later stage of the diffusion as a consequence of precipitation band broadening. The angle formed between the passive edge and the regressing edge brings forward important characteristics of the wavefronts. It is determined by two speeds: the perpendicular speed of the regressing edge in the reaction zone and the lateral speed of the passive edges. The angles are 3859



Article

Figure 3. Plot representing the speed of the cusp formation as a function of outer iodide concentration. Initial conditions: inner, [Hg2+]0 = 0.21 M in 1% per volume agar gel; outer, [I−]0 = 1.0, 2.0, 3.0, and 4.0 M.

found to be changing in all patterns forming and range between 158° and 162°. This indicates that the perpendicular speed of wavefront diffusion is not constant. Effect of Initial Outer Iodide Concentration. In this section, we investigate the effect of the initial outer concentration, [I−]0, on the wavefront and waveback dynamics. The alteration of the outer concentration with fixed inner concentration gives a variety of concentration gradients, which will lead to an important sum of information about the interaction. A reference setup with initial conditions is used (inner, [Hg2+]0 = 0.21 M in 1% agar gel at 24.0 ± 0.2 °C). The range of outer concentrations constitutes [I−]0 = 1.0, 2.0, 3.0, 4.0, 4.5, 5.0, and 6.0 M. We measure the displacement of the wavefront from the center of the reactor (df) as well as the distance from the center of the reactor to the waveback (db) with time elapsed. We then produce log−log plots (Figure 4) to analyze the velocities of the two aforementioned fronts, and consequently, we can obtain information about the dynamics of the width of the propagating band. These plots show a significant relation between the outer concentration and the velocities of both waves, in which the lowest concentration has the slowest velocities of propagation, and as the concentration increases, the velocities gradually increase as well. This is reasoned to be the result of larger concentration gradients leading to faster propagation of both waves. All of the waves typically display decreasing velocities with a sharp transition appearing at outer concentrations greater than 4.0 M as shown in Figure 4. Not only is their velocity higher but also they possess different features from the other concentrations. Their propagating precipitate bands themselves get significantly thinner and lose their cusp-like shape as the outer concentration increases. This behavior is depicted in Figure 5 showing a comparison of the width of the bands from four different outer concentrations. It is obvious that the width of the band produced by the outer 5.0 M is about one-third smaller than that produced by the outer 1.0 M.

Figure 4. Natural log−log plots of the wavefront displacement (df in mm) and waveback displacement (db in mm) of the diffusing precipitation bands versus time (t in minutes). The top plot represents the wavefront dynamics, and the bottom plot represents the waveback dynamics. Initial conditions: inner, [Hg2+]0 = 0.21 M in 1% per volume agar gel. A range of different initial outer concentrations is tested: [I−]0 = 1.0, 2.0, 3.0, 4.0, 4.5, 5.0, and 6.0 M. The linear fits are performed on the points excluding those at short times. R2 > 0.991 for all fits.

The log−log plots in Figure 4 were linearly fitted to evaluate their slopes (α for the wavefront and β for the waveback). In a typical diffusion or diffusion-limited process the mean squared displacement of particles (d2) is a linear function of time (d2 ≈ t2α), where the exponent α is equal to 0.5, due to the fact that normally mixing is purely due to molecular diffusion as a result of underlying Brownian motion. However, our results shown in Figure 6 reveal that in the range of outer concentrations between 1.0 and 4.5 M, the slopes α and β varied between 0.55 and 0.73, indicating a superdiffusive behavior for the precipitation wavefront and redissolution waveback. This superdiffusive behavior is occurring without any external enhancement or modulation of motion. Moreover, a sharp 3860



Article

M, and as this concentration is exceeded, we enter a subdiffusive regime that is expected due to confinement of the particles in transient cages created by the gel network.1 It is also significant that the values of β are greater than the α values over most of the range of outer concentrations, and this fact has important implications on the time evolution of the width of the band. From Figure 4, both df and db are shown to exhibit the following time laws: df = atα and db = btβ, then the width can be defined as w = df − db. When β > α, the width increases in time to reach a maximum at tmax = (aα/bβ)1/(β−α), then it starts decreasing until the band is completely dissolved (w = 0). Effect of Initial Inner Mercuric Concentration. The inner mercuric concentration reveals a noteworthy effect on the actual structure of the propagating precipitation band. As the inner concentration decreases, the irregularly shaped band with cusp-like pattern (Figure 7A) continuously transforms to a ring-

Figure 5. Top view snapshots of a section of the radially diffusing irregular precipitation bands, taken at the same time after initialization (∼150 min). They express the difference in width as a function of outer iodide concentration. Initial conditions: inner, [Hg2+]0 = 0.21 M in 1% per volume agar gel; outer, [I−]0 = (A) 1.0, (B) 3.0, (C) 4.0, and (D) 5.0 M. Scale bar represents 0.5 cm.

Figure 7. Top view snapshots of a section of the radially diffusing precipitation bands showing the transition from an irregular cusped structure to a regular smooth structure, as a function of inner mercuric concentration. (A) Irregular cusp-like shaped precipitation band produced by initial conditions: inner, [Hg2+]0 = 0.21 M in 1% agar gel; outer, [I−]0 = 4.0 M. (B) Precipitation band at midway point of transition exhibiting features of both regular and irregular structures. Initial conditions: inner, [Hg2+]0 = 0.19 M in 1% agar gel; outer, [I−]0 = 4.0 M. (C) Regular ring-like shaped precipitation band produced by initial conditions: inner, [Hg2+]0 = 0.15 M in 1% agar gel; outer, [I−]0 = 4.0 M. The snapshots are taken at ∼400 min after initialization. All the precipitation bands express the polymorphic transformation of the HgI2 yellow and red crystals. Scale bars represents 0.5 cm.

like band with undistorted boundaries (Figure 7C). The transition between these two structures is particularly interesting (see Supporting Information). As our analysis shows, there exists an inflection region in the initial inner mercuric concentration above which only irregular cusp patterned bands form and below which one can only produce a regular smooth precipitation band. We test inner concentrations ranging from 0.10 to 0.24 M at a constant outer concentration of 4.0 M and constant gel percentage (1%). The inflection region is confirmed to exist in the range 0.17 to 0.19 M where the precipitation band exhibits both encountered

Figure 6. Plot representing the change of the slope values (α for wavefront and β for waveback), in the relation to log(d) ≈ (slope) log(t) (d is displacement in mm and t is time in minutes) with change of outer [I−]0 at constant inner [Hg2+]0 = 0.21 M and 1% agar. Dashed line indicates the value 0.5 at which the dynamics is consistent with the normal diffusive profile. Error bars represent the confidence range.

transition in the values of α and β to around 0.5 (normal diffusive regime) occurs as the outer concentration reaches 5.0 3861



Article

(1%) has relatively the highest velocity. While the speed of the front depends on the gel concentration, the slopes α and β are not dependent yielding a value close to 0.6. These tests are repeated several times to verify reproducibility of the results. Macro- and Microscopic Structures. During its propagation, the band undergoes an interesting conversion between the orange, yellow, and red polymorphs of HgI2, which we observe at a slow rate in the gel media allowing analysis of every proceeding step (see Supporting Information). In the early stages of diffusion, the precipitate primarily constitutes the orange and yellow polymorphs. First appearing is the orange form, which macroscopically does not adopt a well-defined structure. Furthermore, it only persists for a short insignificant period of the propagation since it completely transforms to the stable red polymorph (in some cases, we observe the orange form transform into the yellow form). The more interesting transformation is witnessed between the yellow and red polymorphs. Next the yellowM polymorph (β-HgI2) dominates the whole width of the precipitation band appearing as a mesh of tightly packed strands of crystals. As the propagation proceeds it also transforms gradually to give rise to the most stable form, the red polymorph (α-HgI2). The transformation occurs in a manner as if the yellow crystals are dissolving to yield needle-like structured α-red-HgI2 crystals. This transformation does not appear alone but is also accompanied by a morphological alteration of the crystalline forms. The β-yellowHgI2 has an orthorhombic crystal structure distribution of the Hg and I atoms, with the Hg having a coordination number of 2 + 4. However, the α-red-HgI2 has a tetragonal crystal structure, with Hg atom surrounded by 4 I atoms, thus forming sheets of HgI4 tetrahedra.49 The transformation from orange to red seemingly involves only the movement of Hg atoms in an invariant ccp matrix of I atoms. In contrast, the transformation from yellow to red indicates a complete rearrangement of the atoms.48,49 The yellow and red crystals have been reported extensively to crystallize by evaporation from several organic solvents.50 Yet we report a different technique of crystal growth, which adopts the formation of the HgI2 crystal structures in gel media. Figure 9 shows a time evolution of the precipitation band illustrating the structures of the red and yellow crystals at the intersection where polymorphic transformation occurs. To describe the structures of our crystals microscopically we implement the following procedure. We remove the precipitation band cautiously from the gel layer inside our reactor while preserving the structure and features of the crystals inside it. We then place it in a freeze-drying machine (Free Zone 2.5 Liter Bench-top Freeze-Dry System) to become a dried up gel mold containing the crystals, which makes the sample more convenient to analyze without inflicting any damage to its contents. Next, we study this sample under a scanning electron microscope (SEM), using an in-beam detector, to illustrate the microscopic features of the crystals we obtain. The samples we pick out for analysis are half way through the yellow/red transformation (shown in Figure 9). The images we capture display various attractive morphological structures of both polymorphs of HgI2. Figure 10 shows a panel of different images of several samples representing various parts of the precipitation band with false coloring to indicate the yellow and red regions. The yellow region appears as a webbing of packed crystal strands with each strand having branching structures that extend throughout its entire length. A closer view displays that

structures at different stages of the reaction. The band propagation initiates with a cusp pattern formation, yet at a later stage of the reaction−diffusion, the irregularity of the structure eventually fades away to reveal a regular ring precipitation band (Figure 7B). In addition, we never encounter a case in which the transition is from regular band to irregular band, the reaction always happens in a manner where irregularity stabilizes to form a final regular structure. All the concentrations below the inflection region show completely regular structures, yet all the concentrations above it display the cusp-like pattern formation of the precipitation band. It is noteworthy that the calculated values of slopes α and β ranged between 0.55 and 0.65 displaying anomalous superdiffusive dynamics before and after the aforementioned transition. This confirms that the inner concentration has an impact only on the structure of the precipitation band, and in at same time, the structure itself has no influence on the spatiotemporal dynamics of the reaction. Effect of Agar Gel Percentage. The effect of the concentration of the gel is also studied. For this purpose, we prepare a reference setup that serves as a control in all experiments with initial conditions as follows: inner concentration, [Hg2+]0 = 0.21 M in 1% of volume agar gel, and outer concentration, [I−]0 = 4.0 M, at 24.0 ± 0.2 °C. This setup produces the cusp-like patterns and exhibits anomalous superdiffusive dynamics. Therefore, the experiments to be performed are to detect any deviations from these characteristics. The gel percentages under investigation are 1%, 2%, and 3%. The linear fits to the log−log plots are reported in Figure 8. From the plots it is definite that the gel has an effect on the velocity of the wavefront, in that the highest gel percentage (3%) has the slowest velocity and the lowest gel percentage

Figure 8. Natural log−log plots of the displacement (df in mm) of the diffusing precipitation bands with time (t in minutes) for different gel concentrations. Initial conditions: inner, [Hg2+]0 = 0.21 M; outer, [I−]0 = 4.0 M. Three different agar gel percent per volume are tested: 1%, 2%, and 3%. The linear fits are performed on the points excluding those at short times. R2 > 0.997 for all fits. 3862



Article

Figure 9. Top view snapshots of the precipitation band evolution displaying the structural features of the red and yellow HgI2 polymorphs, in gel media, at the intersection at which the polymorphic transformation occurs. The yellow polymorph grows as a webbing of strands of crystals with tiny branching structures, and the red polymorphs grow as needle-like structures. Initial conditions: inner, [Hg2+]0 = 0.10 M in 1% per volume agar gel; outer, [I−]0 = 1.0 M. Time after initialization: (A) 100, (B) 400, (C) 700, and (D) 900 min. Scale bar represents 0.5 cm.

these strands carry small octahedrally (square bipyramidal) structured crystals with sizes ranging between 3 and 5 μm. The red part, however, shows many yet very similar structures. The fully grown crystals macroscopically appear as needles, yet under the SEM the microscopic structure reveals that each needle is made of several octahedra stacked on top of each other, in a sideways fashion, forming sawtooth structure. Other needles appeared to have pointy extremities, like pyramids, owing to the fact that they are constructed in a similar way as the aforementioned needles, yet the octahedra are stacked perfectly on top of each other in a head-to-head manner. We also observe several single octahedral crystals that are not stacked to form a needle structure. These octahedra are very similar to the ones appearing in the yellow form yet have sizes ranging between 50 and 90 μm, which makes them nearly 20 times larger. Such a behavior brings upon a confirmation to the fact that Ostwald ripening51,52 is taking place in the gel during the transformation between yellow and red crystals, whereby larger crystalline α-HgI2 forms at the expense of the smaller βHgI 2 . The Ostwald ripening mechanism involves the reconstruction of octahedral Hg(II) in the layers from the orthorhombic β-HgI2, which in fact, involves the breaking and forming of bonds, proceeded by a dissolution−reformation process. All this also involves a morphological change of the crystals in the two phases, where smaller octahedral structures are preferred in the β-polymorph, whereas larger octahedra are preferred in the α-polymorph. Videos are supplied in the Supporting Information displaying the complete evolution of three different types of precipitation bands. The video pertaining to the regular precipitation band expresses the polymorphic transformation in a well-defined way. From the beginning of the diffusion the yellow polymorph

Figure 10. Panel showing several SEM images captured of various regions of the precipitation band. False coloring added on the images to indicate the real colors of the crystals. Initial conditions: inner, [Hg2+]0 = 0.10 M in 1% per volume agar gel; outer, [I−]0 = 1.0 M. (A) A webbing of tightly packed crystal strands with branching structures, in the yellow region. Scale bar represents 100 μm. (B) A magnified image of a yellow region showing the small, undeveloped octahedral shaped crystals carried by the branching crystal strands. Scale bar represents 20 μm. (C,D) These images display the intersection edge between the yellow and red regions of the precipitation band. The yellow region appears as the tightly packed crystal strands, while the large crystals protruding out of the yellow region are the red needles with their various forms. Scale bars in panels C and D represent 100 and 200 μm, respectively. (E) A red needle crystal structure formed by a side-to-side overlap of large octahedrally shaped crystals. Scale bar represents 50 μm. (F) A large octahedrally shaped crystal in the red region. Scale bar represents 20 μm.

appears and dominates the entire width of the band. As the diffusion continues the needles of the red polymorph start appearing from under the yellow mesh but behind the wavefront. As described earlier, the width of the precipitation band increases as it diffuses further. At some point of the diffusion the band is three-quarters composed of red needles and the rest is yellow (yet we observe cases where the entire precipitation band is dominated by the red polymorph). Afterward, a rather fascinating transition occurs in which the polymorphic transformation occurs in an oscillatory manner. As the red polymorph covers three-quarters of the band, we realize that the yellow to red transformation stops, yet diffusion continues to produce new yellow crystals at the leading wavefront and the red needles start disappearing due to the 3863



Article

(6) Panfilov, A.; Keldermann, R.; Nash, M. Drift and Breakup of Spiral Waves in Reaction−Diffusion−Mechanics Systems. Proc. Natl. Acad. Sci. U.S.A. 2007, 104, 7922−7926. (7) Gálfi, L.; Rácz, Z. Properties of the Reaction Front in an A + B → C Type Reaction-Diffusion Process. Phys. Rev. A 1988, 38, 3151. (8) Koza, Z.; Taitelbaum, H. Motion of the Reaction Front in the A + B → C Reaction-Diffusion System. Phys. Rev. E 1996, 54, R1040. (9) Mansour, A. A.; Al Ghoul, M. Scaling and Crossover Dynamics in the Hyperbolic Reaction-Diffusion Equations of Initially Separated Components. Phys. Rev. E 2011, 84, 026107. (10) van Baalen, G.; Schenkel, A.; Wittwer, P. Asymptotics of Solutions in A + B → C Reaction−Diffusion Systems. Commun. Math. Phys. 2000, 210, 145−176. (11) Cornell, S.; Droz, M.; Chopard, B. Role of Fluctuations for Inhomogeneous Reaction-Diffusion Phenomena. Phys. Rev. A 1991, 44, 4826. (12) Larralde, H.; Araujo, M.; Havlin, S.; Stanley, H. E. Reaction Front for A + B → C Diffusion-Reaction Systems with Initially Separated Reactants. Phys. Rev. A 1992, 46, 855. (13) Taitelbaum, H.; Yen, A.; Kopelman, R.; Havlin, S.; Weiss, G. H. Effects of Bias on the Kinetics of A + B → C with Initially Separated Reactants. Phys. Rev. E 1996, 54, 5942. (14) Horvát, S.; Hantz, P. Pattern Formation Induced by IonSelective Surfaces: Models and Simulations. J. Chem. Phys. 2005, 123, 034707. (15) Koo, Y. E.; Li, L.; Kopelman, R. Reaction Front Dynamics in Diffusion-Controlled Particle-Antiparticle Annihilation: Experiments and Simulations. Mol. Cryst. Liq. Cryst. 1990, 183, 187−192. (16) Léger, C.; Argoul, F.; Bazant, M. Z. Front Dynamics During Diffusion-Limited Corrosion of Ramified Electrodeposits. J. Phys. Chem. B 1999, 103, 5841−5851. (17) Tinsley, M. R.; Collison, D.; Showalter, K. Propagating Precipitation Waves: Experiments and Modeling. J. Phys. Chem. A 2013, 117, 12719−12725. (18) Hantz, P. Pattern Formation in the NaOH + CuCl2 Reaction. J. Phys. Chem. B 2000, 104, 4266−4272. (19) Meinhardt, H. The Algorithmic Beauty of Sea Shells; Springer: New York, 2009. (20) Sultan, R.; Sadek, S. Patterning Trends and Chaotic Behavior in Co2+/NH4OH Liesegang Systems. J. Phys. Chem. 1996, 100, 16912− 16920. (21) Volford, A.; Izsák, F.; Ripszám, M.; Lagzi, I. Pattern Formation and Self-Organization in a Simple Precipitation System. Langmuir 2007, 23, 961−964. (22) Müller, S. C.; Ross, J. Spatial Structure Formation in Precipitation Reactions. J. Phys. Chem. A 2003, 107, 7997−8008. (23) Liesegang, R. Ueber Einige Eigenschaften Von Gallerten. Naturwiss. Wochenschr. 1896, 10, 353−362. (24) 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. (25) Lagzi, I. Controlling and Engineering Precipitation Patterns. Langmuir 2012, 28, 3350−3354. (26) Hantz, P. Regular Microscopic Patterns Produced by Simple Reaction−Diffusion Systems. Phys. Chem. Chem. Phys. 2002, 4, 1262− 1267. (27) Al-Ghoul, M.; Sultan, R. Front Propagation in Patterned Precipitation. 1. Simulation of a Migrating Co(OH)2 Liesegang Pattern. J. Phys. Chem. A 2001, 105, 8053−8058. (28) Badr, L.; Sultan, R. Ring Morphology and pH Effects in 2D and 1D Co(OH)2 Liesegang Systems. J. Phys. Chem. A 2009, 113, 6581− 6586. (29) Das, I.; Pushkarna, A.; Agrawal, N. R. Chemical Waves and Light-Induced Spatial Bifurcation in the Mercuric Chloride-Potassium Iodide System in Gel Media. J. Phys. Chem. 1989, 93, 7269−7275. (30) Das, I.; Pushkarna, A.; Bhattacharjee, A. New Results on LightInduced Spatial Bifurcation and Electrical Field Effect on Chemical Waves in the Mercury (II) Chloride-Potassium Iodide System in Gel Media. J. Phys. Chem. 1990, 94, 8968−8973.

dissolution of the waveback. As this continues, a period is reached where the precipitation band is dominated by the yellow polymorph again, and the cycle repeats itself.

■

CONCLUSIONS We study a heterogeneous two-dimensional reaction−diffusion system embracing the precipitation reaction between mercuric chloride and potassium iodide to produce mercuric iodide crystals diffusing in agar gel medium. The propagation is accompanied by polymorphic transformations between the different crystals of mercuric iodide. Altering initial reaction conditions lead to the formation of cusp-like patterns of the precipitation front. The inner concentration displayed a major impact on the structure of the precipitation band in which at high concentrations the band forms irregular cusp patterns of the wavefront, yet as the concentration is decreased the irregularity fades away to uncover a regular ring-like band. The outer concentration imposed a significant effect on the dynamics of the reaction in which the diffusivity of the propagating band was altered from being anomalously superdiffusive at low concentrations to being subdiffusive at higher concentrations (5 M and greater). The concentration of the gel exhibits no influence on cusp pattern formation or on the anomalous diffusion of the precipitation band. On the microscopic level, the situation is also complex. SEM images display the morphology of the crystal structures and sizes of both yellowM and red polymorphs and reveal the existence of an Ostwald ripening mechanism accompanying the polymorphic transformation.

■

ASSOCIATED CONTENT

* Supporting Information S

Four videos are provided displaying the propagation of three different types of precipitation bands. An extra file is provided explaining the contents of each video. This material is available free of charge via the Internet at http://pubs.acs.org.

■

AUTHOR INFORMATION

Corresponding Author

*(M.A.-G.) Phone: +961-1-350000 ext. 3999. Fax: +961-1365217. E-mail: [email protected]. Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS This work is supported by the Lebanese Council for National Scientific Research (LCNSR), the University Research Board, American University of Beirut.

■

REFERENCES

(1) Grzybowski, B. Chemistry in Motion: Reaction-Diffusion Systems for Micro-and Nanotechnology; Wiley Online Library: New York,2009. (2) Kapral, R.; Showalter, K. Chemical Waves and Patterns; Kluwer Academic Publishers: Dordrecht, The Netherlands, 1995. (3) Murray, J. D. Mathematical Biology; Springer: New York, 2002; Vol. 2. (4) Cross, M. C.; Hohenberg, P. C. Pattern Formation Outside of Equilibrium. Rev. Mod. Phys. 1993, 65, 851. (5) Deutsch, A.; Dormann, S. Cellular Automaton Modeling of Biological Pattern Formation: Characterization, Applications, and Analysis; Springer: New York, 2005. 3864



Article

(31) Das, I.; Pushkarna, A.; Bhattacharjee, A. Dynamic Instability and Light-Induced Spatial Bifurcation of Mercuric Iodide and External Electric Field Experiments in Two-Dimensional Gel Media. J. Phys. Chem. 1991, 95, 3866−3873. (32) Hilal, N.; Sultan, R. Density Oscillations in Precipitate Domains of a Propagating Cr(OH)3 Ring. Chem. Phys. Lett. 2003, 374, 183− 186. (33) Sultan, R.; Panjarian, S. Propagating Fronts in 2D Cr(OH)3 Precipitate Systems in Gelled Media. Phys. D 2001, 157, 241−250. (34) Zrinyi, M.; Galfi, L.; Smidroczki, E.; Racz, Z.; Horkay, F. Direct Observation of a Crossover from Heterogeneous Traveling Wave to Liesegang Pattern Formation. J. Phys. Chem. 1991, 95, 1618−1620. (35) Zaikin, A.; Zhabotinsky, A. Concentration Wave Propagation in Two-Dimensional Liquid-Phase Self-Oscillating System. Nature 1970, 225, 535−537. (36) Field, R. J.; Koros, E.; Noyes, R. M. Oscillations in Chemical Systems. II. Thorough Analysis of Temporal Oscillation in the Bromate−Cerium−Malonic Acid System. J. Am. Chem. Soc. 1972, 94, 8649−8664. (37) Taylor, A. F. Mechanism and Phenomenology of an Oscillating Chemical Reaction. Prog. React. Kinet. Mech. 2002, 27, 247−325. (38) Berenstein, I.; Muñ uzuri, A. P.; Yang, L.; Dolnik, M.; Zhabotinsky, A. M.; Epstein, I. R. Breathing Spiral Waves in the Chlorine Dioxide−Iodine−Malonic Acid Reaction-Diffusion System. Phys. Rev. E 2008, 78, 025101. (39) Fernández-García, G.; Pérez-Muñuzuri, V. Superdiffusive Wave Front Propagation in a Chemical Active Flow. Eur. Phys. J. Spec. Top. 2008, 165, 169−174. (40) Vanag, V. K.; Epstein, I. R. Inwardly Rotating Spiral Waves in a Reaction-Diffusion System. Science 2001, 294, 835−837. (41) Wood, P. M.; Ross, J. A Quantitative Study of Chemical Waves in the Belousov−Zhabotinsky Reaction. J. Chem. Phys. 1985, 82, 1924−1936. (42) von Kameke, A.; Huhn, F.; Fernández-García, G.; Muñuzuri, A.; Pérez-Muñuzuri, V. Propagation of a Chemical Wave Front in a QuasiTwo-Dimensional Superdiffusive Flow. Phys. Rev. E 2010, 81, 066211. (43) Paoletti, M.; Nugent, C.; Solomon, T. Synchronization of Oscillating Reactions in an Extended Fluid System. Phys. Rev. Lett. 2006, 96, 124101. (44) Noszticzius, Z.; Bodnar, Z.; Garamszegi, L.; Wittmann, M. Hydrodynamic Turbulence and Diffusion-Controlled Reactions: Simulation of the Effect of Stirring on the Oscillating Belousov− Zhabotinskii Reaction with the Radicalator Model. J. Phys. Chem. 1991, 95, 6575−6580. (45) Jeffrey, G. A.; Vlasse, M. Crystal Structures of the Red, Yellow, and Orange Forms of Mercuric Iodide. Inorg. Chem. 1967, 6, 396−399. (46) Hostettler, M.; Birkedal, H.; Schwarzenbach, D. The Structure of Orange HgI2. I. Polytypic Layer Structure. Acta Crystallogr., Sect. B: Struct. Sci. 2002, 58, 903−913. (47) Hostettler, M.; Birkedal, H.; Schwarzenbach, D. The Yellow Polymorphs of Mercuric Iodide (HgI2). Helv. Chim. Acta 2003, 86, 1410−1422. (48) Hostettler, M.; Birkedal, H.; Schwarzenbach, D. Polymorphs and Structures of Mercuric Iodide. Chimia 2001, 55, 541−545. (49) Hostettler, M.; Schwarzenbach, D. Phase Diagrams and Structures of HgX2 (X = I, Br, Cl, F). C. R. Chim. 2005, 8, 147−156. (50) Kleber, W.; Raidt, H.; Leupold, K. O. Ein Beitrag Zur Züchtung Von Quecksilber (II)−Jodid−Einkristallen. Krist. Tech. 1968, 3, 65− 78. (51) Al-Ghoul, M.; Ammar, M.; Al-Kaysi, R. O. Band Propagation, Scaling Laws and Phase Transition in a Precipitate System. I: Experimental Study. J. Phys. Chem. A 2012, 116, 4427−4437. (52) Greenwell, H. C.; Bindley, L. A.; Unwin, P. R.; Holliman, P. J.; Jones, W.; Coveney, P. V.; Barnes, S. L. In Situ Monitoring of Crystal Growth and Dissolution of Oriented Layered Double-Hydroxide Crystals Immobilized on Silicon. J. Cryst. Growth 2006, 294, 53−59.

3865


Superdiffusive Cusp-Like Waves in the Mercuric Iodide Precipitate

Recommend Documents