16912
J. Phys. Chem. 1996, 100, 16912-16920
Patterning Trends and Chaotic Behavior in Co2+/NH4OH Liesegang Systems R. Sultan* and S. Sadek Department of Chemistry, American UniVersity of Beirut, Beirut, Lebanon ReceiVed: April 1, 1996; In Final Form: June 11, 1996X
Co(OH)2 Liesegang patterns from Co2+ and NH4OH in gelatin are studied in thin glass tubes. The experiments are performed in the limit of large initial concentration difference (∆) between the interdiffusing electrolytes. Patterning trends are investigated as the initial concentration product σ ) [Co2+ ]0[NH4OH]20 is varied (we keep [NH4OH]0 constant and vary [Co2+]0 in the limit of large ∆). Time series of the total number of bands in the pattern show a random distribution apparently indicating the chaotic character of that variable. When the bands are counted in the bottom portion of the pattern (the last 4 cm measured from the last band), their number (denoted ν therein) decreases monotonically with time at all concentrations. The pattern thus regularizes itself in that region. The spacing between the bands is measured as a function of distance from the junction between the co-precipitate solutions, at different initial concentrations. The variation of spacing with distance is found to fit a quadratic equation. The variation of spacing with concentration at a given location is also studied and is found to involve an exponential term. Then a generalized equation expressing the spacing as a function of concentration and distance is determined. A three-dimensional plot is constructed to represent the obtained variations. Our results are then compared with previous experimental findings and theoretical formulations of Liesegang preciptation phenomena.
1. Background When co-precipitate ions interdiffuse in a one-dimensional gel medium, the precipitate forms in a spectacular pattern of parallel bands. This patterned precipitation is known as the Liesegang1 banding phenomenon after its discoverer, who first observed it in 1896. Since then, this behavior of precipitating systems in a gel matrix has been extensively studied.2,3 Several features of Liesegang band formation have been reported. Just like parallel bands are observed when the interdiffusion takes place in a test tube, concentric precipitate rings appear when it occurs along a radial direction originating at the center of a Petri dish.2-5 Liesegang bands display a rich structure in their shape, number, width, and general spatiotemporal distribution. The band shape and band location may be influenced by the gravitational field,6 and experiments have been carried out in tubes held in a horizontal position.7 The spacing between the bands normally increases as they move further away from the junction between the co-precipitate solutions (e.g. as they move down a tube held in a vertical position); but the opposite situation has been observed8-10 in several salt systems, thus manifesting the so-called “revert spacing” phenomenon. Furthermore, a specific band in a Liesegang pattern can develop structure at later times. Such a breakup of one band into several thinner ones is often called secondary banding and was found to accompany the rhythmic band formation in certain experiments and observed at later stages in others. In the latter case, the fine structures were believed to be a postnucleation phenomenon.6,12-14 Since the early observation of the phenomenon, there has been a great interest in the mechanism underlying the formation of precipitate patterns. Theories on precipitate patterning can be generally divided into two main categories: (i) prenucleation theories based on the Ostwald supersaturation-nucleation* To whom correspondence should be addressed. E-mail:
[email protected]. X Abstract published in AdVance ACS Abstracts, August 15, 1996.
S0022-3654(96)00958-6 CCC: $12.00
depletion cycle;15-20 these models involve the coupling of supersaturation to the competition between nucleation and droplet-growth kinetics; and (ii) postnucleation theories6,10,12 based on a mechanism mediated by surface tension,21 which emphasize the formation of a homogeneous colloid or a haze prior to patterning. This view was motivated by the observation of rhythmic patterns even in the absence of concentration gradients and as a result of the redistribution of an initially homogeneous sol. Recent models22 combine the two views and take all the mentioned factors into account. Other earlier models considering other factors such as adsorption, coagulation, and the diffusion wave of the formed electrolyte23 were advanced and are reviewed in ref 3. Quantitative characterization of Liesegang patterns is thoroughly found9,24-26,37 in the literature, notably in the work of Ross and his group.6,7,11 In this paper, we present a series of Liesegang experiments with CoCl2 and concentrated NH4OH as the working electrolytes in gelatin as the gel-forming medium, as suggested by Schibeci and Carlsen.29 This paper is mainly concerned with the investigation of the variation of number of bands and band spacing with some characteristic parameters. Interesting features are obtained, including the possibility of chaotic behavior. A similar study was carried out by Mu¨ller et al.7 on PbI2, although their characterization methods differ from ours. Ross and co-workers7,11,30 established that for large values of the initial concentration difference (∆) and the concentration product (σ) the patterns are deterministic, while they become increasingly stochastic as ∆ and σ are decreased to very small values. All the experiments described in this paper are performed in the limit of large ∆, varying the initial concentration of one electrolyte and keeping that of the other constant. The properties of the Co2+/NH4OH Liesegang system making it a suitable choice for our study are now summarized as follows. 1. The Liesegang pattern is manifested with relatively sharp, well-defined band boundaries and distinct band spacing, notably away from the two-electrolyte junction. © 1996 American Chemical Society
Co2+/NH4OH Liesegang Systems
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TABLE 1: Initial Concentration Parameters c0 ) [Co2+,2OH-]0 ) [Co2+]0, Concentration Difference ∆, and Concentration Product σa grams of CoCl2‚6H2O
c0 (M)
∆ (M)
σ (M3)
0.050 0.100 0.200 0.400 0.600 0.800 1.00 1.25 f 1.50 f 1.75 f 2.00
0.00841 0.0168 0.0336 0.0672 0.101 0.134 0.168 0.210 0.252 0.294 0.336
6.67 6.66 6.65 6.61 6.58 6.55 6.51 6.47 6.43 6.39 6.34
1.50 3.00 6.00 12.0 18.0 24.0 30.0 37.5 45.0 52.5 60.0
a The experiments corresponding to the sets marked with a star were performed at a later stage to fill the gap, as explained in the text. The initial concentration difference ∆ has an infinite value compared with the values (0.015-0.076 M) of ref 7. The parameter σ varies significantly since [Co2+]0 is varied at constant [NH4OH]0 ) 13.4 M.
2. The time scale for the appearance of the Co(OH)2 pattern (2-3 h) is convenient. 3. The pattern is suitable for prolonged visual observation: the core bands maintain their visual features for a long period of time (∼30 days) without being damaged. 4. The experiments are reproducible to a large extent. 5. To our knowledge, no quantitative study on this system has been reported. The patterns obtained display a variety of exotic morphological structures, and the band appearance ranges from flat to meniscus-like to bell-shaped. Helicoidal patterns are also obtained in tubes of relatively large diameter, resembling previous results reported in refs 4, 22, 23, 27, and 28. A detailed description of the morphological features we obtain in the Co2+/ NH4OH system is presented in ref 31. 2. Experimental Procedure A sample of solid cobalt chloride (CoCl2‚6H2O) was weighed to the nearest 0.01 g and transferred to a round-bottom flask containing 25.00 mL of distilled water and 1.50 g of gelatin. The mixture was heated with constant stirring for 5 min, during which the solution started boiling. The resulting gel was then immediately placed in a set of thin glass tubes (50 × 0.4 cm) until they became half full. The upper level of the gel in each tube was marked to indicate the interface between the coprecipitate solutions. The tubes were covered with parafilm paper and allowed to stand for 24 h at room temperature (22 ( 1 °C). Then 1.50 mL of concentrated ammonia was delivered using a graduated pipet on top of the solidified cobalt chloride gel. The tubes were covered again and allowed to stand. The above procedure was repeated for eight different initial CoCl2 concentrations using the same amount of gelatin. The varied concentrations are 0.05, 0.10, 0.20, 0.40, 0.60, 0.80, 1.00, and 2.00 g CoCl2‚6H2O/25.00 mL H2O. In all the experiments, the two electrolytes are subject to a wide initial concentration gradient.7,16 The initial molar concentrations of the electrolytes, along with the initial concentration difference (∆) and the concentration product (σ) are given in Table 1. We adopt the definition of Mu¨ller et al.7 for the parameters ∆ and σ as follows:
∆ ) 1/2[NH4OH]0 - [Co2+]0
(1)
σ ) [Co2+]0[NH4OH]20
(2)
where the subscripts zero denote initial concentrations.32
Immediately after the addition of the ammonia solution, a homogeneous blue solid formed at the junction of the two solutions. A few (3-4) hours later, Liesegang bands started to appear. The pattern (shown in Figure 1) consists of a set of concentric disks parallel to the surface of the diffusion front. The spacing between consecutive bands increased with increasing distance from the junction of the two electrolytes, an observation consistent with the normal trend for ring spacing. Note that although the shape of the bands varied at different locations in the tube,31 the overall final pattern is essentially a set of well-separated precipitate zones with color ranging from deep blue in the tubes containing high concentrations of CoCl2 to light blue in the tubes containing low concentrations. 3. Number of Bands The overall Liesegang Co(OH)2 pattern appears as a collection of parallel bands of intense blue color. A typical sample of the obtained pattern is displayed in Figure 1. Experiments were carried out in both large (diameter 1.5 cm) and thin (diameter 0.4 cm) tubes, but all the measurements were taken in the thin tubes. On a daily basis, the total number of bands along with the number of bands in the bottom portion of the pattern (i.e. the last 4 cm in each tube) were determined. The bands were counted while viewed through an ordinary magnifier. The initial stage of the precipitate formation exhibits a continuous precipitation zone near the interface beyond which discrete bands start to appear. During this early era (the first 3 days), the counting of the bands is the most difficult, because what appears as a turbid zone could include a small number of discrete bands with microscopic spacings, thus causing an error in the determination of the total number of bands N. As time advances however, the spacing between the bands increases, thus clearing up the turbidity of the pattern top and significantly reducing the size of the uncertainty (the average of which is estimated to be (1 band). The tube in Figure 1 represents a relatively late stage (t ) 19 days) showing the breakup of the continuous region into thin and narrowly spaced bands (Figure 1a), which can be distinguished more lucidly in Figure 1b (an enlargement of the upper part of the tube). The distances from the junction to the last band formed were measured with a ruler to the nearest 0.05 cm. The spatiotemporal evolution of the pattern is characterized by defining the following parameters, adopting the notation of Mu¨ller et al.:7
t ≡ time at which the measurements are taken N ≡ total number of bands in the pattern xn ≡ position of the nth band We also introduce a new parameter, ν, defined as follows:
ν ≡ number of bands in the last 4 cm portion of the pattern This is done to search for conditions under which we can obtain deterministic patterning variables. It is to be noted that this constitutes a novel characterization, not considered in previous works on Liesegang systems. Figure 2a-h shows the variation of the total number of bands (N) with time for a set of initial concentrations. In all the plots, a complete scatter is observed, suggesting a sequence of uncorrelated events. No clear dependence of the total number of bands on time can be inferred. As a result, no trend in going from a given concentration to another is apparent. This unpredictable random behavior is particularly interesting and could be an indication of a situation of macroscopic chaos. The erratic distribution in the time
16914 J. Phys. Chem., Vol. 100, No. 42, 1996
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Figure 1. Co(OH)2 Liesegang bands at an advanced stage (time t ) 19 days). [Co2+]0 ) 0.400 g/25.0 mL water, [NH4OH]0 ) 13.4 M, gelatin concentration ) 1.50 g/25.0 mL water. (a, left) Overall view of the pattern: the middle and the lower bands are sharp and distinct. The upper bands indicated by the arrows have a concave (meniscus-like) shape. (b, right) Large view of the upper portion of the tube showing the detailed structure of the first bands. This closer look allows a separate and clear depiction of each band.
evolution of the total number of bands N observed in Figure 2 supports the chaotic character of that variable. Note that the possible experimental error due to the initial continuous zone of precipitate (discussed above) is significant only at the early stage of the pattern. Furthermore, this error has little influence here since the random behavior is obtained at all times including the later stages, where the band count is tremendously easier. We note further that the continuous precipitate zone is obtained in other studies7 where the number of bands constitutes a variable of interest and was also predicted theoretically.18,19 To test our results, we investigate the variation of the number of bands with concentration at an advanced time (20 days). Call the stoichiometric (initial) concentration of salt c0, i.e. c0 ) [Co2+,2OH-]0. The plot of N versus c0 at t ) 20 days is depicted in Figure 3 and shows a notable resemblance to the results of Mu¨ller et al.7,34 on PbI2. The novel feature in our study is the time series for the total number of bands not considered in previous studies. Kai et al.11 showed that a correlation exists between the tendency toward a stochastic behavior in Liesegang systems and the decrease of the initial concentration parameters (∆ and σ) to very small values. The striking difference here is that the random character of the N-versus-t plots is seen even at high Values of σ and ∆. This was further confirmed by the results of three additional experiments on the large σ limit indicated by a star in Table 1 (note that ∆ is also large in all the experiments). Scatters similar to those of Figure 2 were obtained. The characterization of deterministic chemical chaos suggested by time series charts (like the ones seen in Figure 2) requires a careful control of the experimental conditions and a
detailed analysis of the data records.35,36 A systematic characterization of the apparent chaos obtained here is currently under consideration and is the subject of a following paper. In a search for other time-dependent variables involving the number of bands, we choose to count the number of bands in a region that is remote from the first band, namely, the bottom portion of the tube. We take the last 4 cm of the pattern measured from the last band and measure the number of bands in that portion (denoted ν). Figure 4 shows a plot of ν versus time for different initial salt concentrations (c0). In contrast with the previous behavior, a family of well-defined curves is obtained. Clearly, ν decreases with time for a given concentration. Thus, we see that the pattern regularizes itself as we move away from the first band. Furthermore, for the range of concentrations represented in Figure 4a, ν increases as c0 increases at constant time, indicating a strong effect of the initial salt concentration on the features of the pattern. In other words, the higher concentration curves lie above the lower concentration curves, revealing some kind of a trend. This trend is however reversed as we go beyond c0 ) 0.10 M (Figure 4b), revealing the complex behavior encountered in these Liesegang systems. This switch in the trend as we go to higher concentrations is also illustrated in Figure 5, where we plot ν versus concentration at a fixed time (t ) 13 days). As expected, a maximum is noted in the plot. 4. Band Spacing While the tubes were covered and left undisturbed, the formation of the precipitate pattern evolved, spanning a
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J. Phys. Chem., Vol. 100, No. 42, 1996 16915
Figure 2. Time series charts for the total number of bands ((1 band) in the pattern at different concentrations: (a) c0 ) 0.00864 M, (b) c0 ) 0.0168 M, (c) c0 ) 0.0336 M, (d) c0 ) 0.0672 M, (e) c0 ) 0.101 M, (f) c0 ) 0.134 M, (g) c0 ) 0.168 M, (h) c0 ) 0.336 M. At all concentrations, the obtained points are totally scattered, suggesting a chaotic behavior.
characteristic period of time. Typically after 20 days, the bands in the tube containing the least amount of cobalt chloride (for a constant NH4OH concentration) reached the bottom of the tube. The spacing between two consecutive rings broadens as the distance from the interface increases (i.e. as we go further down in the tube), as seen in Figure 1. This increase in spacing with distance is the normal trend for a Liesegang pattern, as opposed to a situation observed in certain systems where the spacing becomes narrower with increasing distance: the socalled “revert spacing” phenomenon. During the patternforming process the tubes were kept in a vertical position. At time t ) 20 days, the spacing between consecutive bands for each concentration was measured. At large times, the bands in the upper portion of the pattern develop a concave shape
maintaining one straight edge (as pointed out by the arrows in Figure 1a; see also the details in Figure 1b and Figure 6). Under these conditions, the spacing is measured from the straight edge of one band to the straight edge of the next one (Figure 6). Our aim is to determine a relation for the variation of band spacing with distance at a given initial (Co2+,2OH-) concentration (or a given σ) and then investigate the correlation between this bandspacing law and the initial concentration parameters. The tube was laid horizontally on a luminous table, and the spacing ∆xn was measured (to the nearest 0.05 mm here) using a magnified graduated scale (Optique Oculaire). Adopting again the notation of ref 7, ∆xn is defined by the relation
∆xn ) xn+1 - xn
(3)
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Figure 3. Dependence of the total number of bands N on initial concentration c0 at time t ) 20 days, showing a sharp increase to a maximum followed by a smooth decrease. This profile reproduces the behavior of Liesegang systems obtained in previous studies.
Figure 5. Variation of ν with the stoichiometric salt concentration c0 at a fixed time (t ) 13 days). We see that ν increases sharply with concentration, passes through a maximum, and then decreases smoothly, illustrating the behavior seen in Figure 4, whereby the trend is reversed beyond c0 ) 0.10 M.
Figure 4. Time evolution of the number of bands in the last 4 cm portion of the pattern (denoted ν), showing a clear trend whereby ν decreases monotonically with time. The family of curves at different stoichiometric salt concentrations (indicated) reveals a “pattern regularization” in the lower part of the tube. (a) The higher concentration curves lie above the lower concentration ones. (b) The trend of Figure 4a is reversed.
Figure 6. Enlarged picture of Co(OH)2 precipitate bands in a thin tube at time t ) 19 days. The bands look like concave meniscuses though maintaining the upper edge straight. The band spacing ∆xn is measured from the straight edge of one band to the straight edge of the other. The initial concentrations are the same as in the caption of Figure 1.
where n is the band number ranging from 1 to N. Let jxn denote the mean position between two consecutive bands defined by
jxn ) (xn + xn+1)/2
(4)
With this, the spacing law at a given concentration can be inferred from a plot of ∆xn versus jxn. In the following discussion, all the measurements (at all concentrations) are taken
at t ) 20 days. Figure 7 shows a plot of band position xn versus band number (n). Clearly xn is a monotonically increasing function of n. The curves corresponding to the different concentrations show a trend whereby the lower curves correspond to the higher concentrations and vice versa, and in the right order (see Figure 7a). Crossing between curves is also obtained (Figure 7b). These results are similar to those obtained by Mu¨ller et al.7 on PbI2, although the shapes of the curves are not the same, thus reflecting the different physicochemical properties characterizing each system.
Co2+/NH4OH Liesegang Systems
J. Phys. Chem., Vol. 100, No. 42, 1996 16917
Figure 7. Variation of band spacing with band number for different concentrations indicated. The curves are smooth and satisfy the simple spacing law of ref 26. The general trend is that the higher concentration curves lie below the lower concentration curves, although crossing between curves is also obtained (frame b).
We now follow a gradual and systematic approach to correlate band spacing, distance, and concentration. The spacing ∆xn was plotted against the distance from the junction, i.e. the average position of the nth band at time t ) 20 days. The resulting plots at different concentrations are shown in Figure 8. For a given [Co2+,2OH-]0 ≡ c0, the dependence of spacing ∆xn on average band position jxn is found to fit a quadratic equation of the form
∆xn ) axjn2 + bxjn + c
(5)
The coefficients a, b, and c are determined by a least squares curve-fitting method. The curves in Figure 8 also show a trend in the dependence of band spacing on initial salt concentration c0 at constant position. However, we note that this trend is broken when we go down to very low concentrations. In the overall sequence of Figure 8, the dashed curves (corresponding to c0 ) 0.00841 M and 0.0168 M) should normally lie below the solid curves of Figure 8a. This observation reproduces previous results on band spacing in Liesegang systems, namely, that the patterns become less deterministic and sometimes totally unpredictable at low concentration gradients or low concentration products7,30 (the latter condition is verified here). Note also that the curves in Figure 8 change their concavity as c0 increases, an interesting behavior revealing the rich structure inherent in these systems. The curves of Figure 8b show a gap zone between the two upper curves (corresponding to c0 ) 0.168 M and 0.336 M),
Figure 8. Band spacing (∆xn) versus distance (xj) from the junction between the two electrolytes, at different concentrations indicated. The points correspond to the experimental results.The solid curves represent best fits (found to be quadratic) obtained in the computer. The band spacing increases as we move away from the interface. Generally at a given location, the spacing is larger for higher concentrations. (a) The trend is not followed at low concentrations. The dashed curves (corresponding to the lower concentrations) should normally lie below the two solid curves. These observations are consistent with previous results obtained in Liesegang systems,7,11,30 wherein the patterns become less deterministic at low concentrations. (b) Spacing versus distance in the high-concentration domains. The trend is followed rigorously here. The gap between the two upper curves simply represents regions of unexplored concentrations.
reflecting a relatively wide range of unexplored concentrations. Because we are seeking a relation between band spacing and concentration, it is necessary to fill this gap, and hence perform a few additional experiments. Measurements were indeed made for c0 ) 0.210, 0.252, and 0.294 M at time t ) 20 days. The whole set of fitted curves (after including the results of the last three experiments) is shown in Figure 9, and as we can see, the new plots fall exactly in their predicted positions, providing decisive evidence of the apparent trend. The next step is to use the latter results, i.e. the band spacing/ location plots at concentrations 0.0336, 0.101, 0.168, 0.210, 0.252, 0.294, and 0.336 M (Figure 9), to extract the variation of band spacing with concentration at constant position. This was done for jxn ) 6, 8, 10, 12, and 14 cm, and the variation of band spacing with concentration for a fixed position is depicted in Figure 10. The curves fit a linear combination involving an exponential term of the form
∆xn ) h + gc0 + fc02 + dec0
(6)
To complete the study of the variation of band spacing with (i) distance from the junction and (ii) initial salt concentration,
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Figure 9. Family of computer-fitted curves showing the dependence of band spacing on distance from the junction for different concentrations shown. Three additional experiments are performed for three different concentrations c0 ) 0.210, 0.252, and 0.294 M in the region of the gap seen in Figure 8b. The curves fall exactly in their expected positions, confirming the observed trend.
Figure 11. Dependence of band spacing on location (distance from the junction) and concentration. The three-dimensional plot is obtained from a computer fit of data points guided by eqs 7 and 8. This image was produced on Mathematica. Planes M and N contain the bottom and top curves of Figure 9, respectively. Planes O and P contain the bottom and top curves of Figure 10, respectively.
where jxn has been replaced by jx for simplicity. The fitting was performed in the computer, and the resulting three-dimensional plot is depicted in Figure 11. To check the validity of the obtained fit, we examine the shapes of the curves in the border planes. The curves in planes M and N correspond to the bottom and the top curves of Figure 9, respectively (note the change in concavity), while the curves in the border planes O and P correspond to the bottom and the top curves of Figure 10, respectively. Hence, this realization provides a good check on our method and supports the validity of the surface plot. Any point on the surface gives the spacing between the Co(OH)2 bands at a given concentration and a given distance from the junction. Figure 10. Dependence of band spacing at a fixed location on stoichiometric salt concentration c0. The different curves correspond to jx ) 6, 8, 10, 12, and 14 cm. The spacing at a given position increases with concentration in a functional form involving an exponential term (given by eq 6).
we combine the results of the fits represented by eqs 5 and 6 to construct a three-dimensional plot. The result is a twodimensional surface where every point has the coordinates (xjn, c0, ∆xn), corresponding to the Cartesian coordinates x, y, and z, respectively. The three-dimensional fit is guided by the dependence of the coefficients in eq 5 on c0 and the coefficients in eq 6 on jxn as follows:
∆xn ) a(c0)xjn2 + b(c0)xjn + c(c0)
(7)
∆xn ) h(xjn) + g(xjn)c0 + f(xjn)c02 + d(xjn)ec0
(8)
Equations 7 and 8 provide a guidance for the determination of a generalized equation expressing the dependence of band spacing on distance from the junction and stoichiometric salt concentration:
∆xn ) ∆xn(xjn, c0)
(9)
Combining the expressions of ∆xn(xj) (eq 7) and ∆xn(c0) (eq 8), we seek a curve fit of the form
∆xn ) Rxj2ec0 + βxj2c02 + γxj2c0 + δxjec0 + xjc02 + ζxjc0 + ηxj2 + θxj + λc0 + µc02 + ξec0 + π (10)
5. Discussion The variety in structure obtained in this study reveals the rich though complex dynamics inherent in these Liesegang systems. Similar patterning trends were investigated by Mu¨ller et al.7 in studies on PbI2, but no chaotic behavior was reported. Liesegang patterns are characterized in several ways and via measurements of various parameters. The two main variables considered here are the band spacing and the number of bands. While the former variable was found to follow regular trends, the latter manifested a random nature in its time sequence. Ring-spacing laws were worked out for different systems, formulating the dependence of spacing on either band number or band location. Here we study spacing as a function of distance from the junction taken as the mean distance between the two spaced bands. Then we study the effect of concentration on band spacing. The three-dimensional picture obtained in Figure 11 provides a basis for the prediction of the band spacing for a given concentration and at a given distance from the interface. The earliest (and perhaps most famous) spacing law proposed by Jablczynski24 predicts a linear dependence of band spacing on distance. According to it, the plot of band spacing versus band location yields a straight line passing through the origin. However, this dependence could not be verified in a certain number of Liesegang studies. Different spacing laws were obtained in different systems. Mathur25 reports an exponential spacing law for the periodic precipitation of lead iodate. In a later article,37 he presents a theory based solely on the diffusion profiles of the co-precipitate ions, which accounts for his
Co2+/NH4OH Liesegang Systems observed exponential law. He shows that Jablczynski’s relation is obtained only in the limit of small xn, i.e. for the first few rings: a small xn allows the expansion of the exponential function and the neglection of higher order terms. He also found that even over domains where a linear relation applies, the Jablczynski plot fails to pass through the origin. Furthermore, Jablczynski’s equation cannot account for revert spacing observed in many experiments.8-10,38 Kanniah et al.9 tested the various spacing laws on Co(II) oxinate, AgI, and PbCrO4. They proposed an elaborate law based on the theory of preferential adsorption, which could explain the revert spacing observed with the latter two salt systems. The transition from revert to direct spacing was attributed to the shift in the charge on the precipitate surface from positive to negative (or vice versa) due to preferential adsorption of the ions. Note that the exponential law can (to a certain extent)9,37 explain the revert spacing phenomenon, while the linear law cannot. Our quadratic fit could be an intermediary form bewteen the linear and exponential dependences. A possible reason for the disruption of the normal spacing law observed here lies in the nonconstant diffusion source of the outer electrolyte (NH4OH). In the long time regime, the assumption that the diffusion source is constant (a key point in Jablczynski’s model) is no longer valid. As a result, the linear relation no longer applies. Since our measurements were taken at t ) 20 days, deviations from linearity are to be expected. Note however that the curves in Figure 9 become nearly linear in the region of the first bands, i.e. closer to the interface (see the curve appearance at the left where jxn is relatively small). These observations are again consistent with Mathur’s theory.37 In their study on PbI2, Mu¨ller et al.7 found that the linear spacing law applies only for systems with large N (corresponding to large σ or ∆). Although the latter condition is satisfied in our Co(OH)2 experiments, we see that the linear dependence of band spacing on position is disrupted as jxn increases. From what preceded, we see that the phenomenon is of such a wide diversity that no universal law could yet be obtained as a generalization. A survey of the various spacing laws in Liesegang systems is found in ref 39. Several factors (such as the identities and concentrations of the diffusing ion and its coprecipitate in the gel, the salt stoichiometry, the nature of the gel and its concentration, the solubility product of the salt, and the tube geometry) play a role in deciding on the spacing law obeyed. Jablczynski’s model allows the calculation of the number of bands that are formed at a given time. This and other spacing laws generally predict a monotonous increase of the total number of bands with time. Contrary to this expectation, we find here that for a given concentration the number of bands varies erratically with time, suggesting a situation of chaos. We believe that the random variation of the number of bands is not stochastic in nature, neither is it due to temperature fluctuations. An explanation of this behavior could be the dissolution of bands near the end of the tube in an unpredictable way. The latter may be attributed to several possible scenarios and deserves further investigation. One interpretation is the influence of the “third” electrolyte resulting from the reaction, namely, NH4Cl.23 Near the flux-free bottom of the tube, the concentration of NH4Cl accumulates and becomes in excess, shifting the precipitation reaction to the left,40 thus causing the redissolution of the last Co(OH)2 bands. In other words, the excess NH4Cl sets up a diffusion wave in the opposite direction. A comprehensive analysis of this and other features of rhythmic precipitation was presented in the context of the “diffusion wave theory” by Wo. Ostwald.23 Another possible mechanism that may cause
J. Phys. Chem., Vol. 100, No. 42, 1996 16919 band dissolution is the competitive growth between the precipitate particles, coupled to the no-flux boundary conditions at the end of the tube. This mechanism causes the larger particles to grow at the expense of the smaller ones: the socalled Lifshitz-Slyousov instability.21 Note that we visually observed the various stages of a band dissolution at the bottom of the pattern.31 The chaotic behavior is therefore inherent in the dynamics of the reaction-diffusion processes. The concentration dependence of the overall number of bands at a fixed time reproduces previous profiles obtained for PbI2 (Figure 3). This similarity eliminates the possibility of attributing the obtained time-chart scatters to experimental error. A chaotic behavior similar to the one apparently obtained here was predicted to characterize the number of bands in a twoprecipitate postnucleation model.41 It is to be noted that this latter result was obtained by calculations based on the “competitive particle growth” (CPG)12 model, thus supporting the view that the chaotic behavior obtained here could perhaps be mediated by a similar mechanism. The interest in precipitate patterning phenomena is growing notably in geology42 because of the applicability of their underlying dynamics in modeling “geochemical self-organization”.43 Liesegang himself44,45 pointed out the similarities between the features of the laboratory experiments and the banding observed in rocks. The strata in a rock could result from the precipitation of minerals as the encounter of coprecipitates is driven by the diffusive flow of the infiltrating water.46 The modeling20 of the banded deposition of goethite (an iron oxide) behind a pyrite (iron sulfide) front in a porous rock rich in oxidative water is based on a bifurcation of the Ostwald cycle. Powerful computational techniques are nowadays used to simulate the diverse patterning features displayed by geological systems (see ref 43 and citations therein). In many cases, these studies were motivated by precipitation experiments carried out in the laboratory, along with various theoretical models set forth to explain the exotic patterns that these experiments exhibit. Acknowledgment. This work was supported by a Bobst start-up research grant (American University of Beirut). The authors would like to thank the referees for their thorough review and valuable suggestions. References and Notes (1) Liesegang, R. E. Lieseg. Phot. ArchiV. 1896, 21, 221. Liesegang, R. E. Naturewiss. Wochenschr. 1896, 11, 353. (2) Hedges, E. S. Liesegang Rings and Other Periodic Structures; Chapman and Hall: London, 1932. (3) Stern, K. H. Chem. ReV. 1954, 54, 79. A Bibliography of Liesegang Rings, 2nd ed.; Government Printing Office: Washington, DC, 1967. (4) Kai, S.; Mu¨ller, S. C.; Ross, J. Science 1982, 216, 635. (5) Krug, H.-J.; Jacob, K.-H.; Dietrich, S. In Fractals and Dynamic Systems in Geoscience; Kruhl, J. H., Ed. Springer-Verlag: Berlin, 1994. (6) Kai, S.; Mu¨ller, S. C.; Ross, J. J. Chem. Phys. 1982, 76, 1392. (7) Mu¨ller, S. C.; Kai, S.; Ross, J. J. Phys. Chem. 1982, 86, 4078. (8) Mathur, P. B.; Ghosh, S. Kolloid Z. 1958, 159, 143. (9) Kanniah, N.; Gnanam F. D.; Ramasamy, P. Proc. Indian Acad. Sci., Chem. Sci. 1984, 93, 801. (10) Flicker, M.; Ross, J. J. Chem. Phys. 1974, 60, 3458. (11) Kai, S.; Mu¨ller, S. C.; Ross, J. J. Phys. Chem. 1983, 87, 806. (12) Feinn, D.; Scalf, W.; Ortoleva, P.; Schmidt, S.; Wolff, M. J. Chem. Phys. 1978, 69, 27. (13) Feeney, R.; Ortoleva, P.; Strickholm, P.; Schmidt, S.; Chadam, J. J. Chem. Phys. 1983, 78, 1293. (14) Lovett, R.; Ortoleva, P.; Ross, J. J. Chem. Phys. 1978, 69, 947. (15) Ostwald, Wi. Lehrbuch der Allgemeinen Chemie, 2. Aufl., Band 2, 2. Teil: Verwandtschaftslehre, Engelmann: Leipzig, 1896-1902; p 778. (16) Prager, S. J. Chem. Phys. 1956, 25, 279. (17) Wagner, C. J. Colloid Sci. 1950, 5, 85. (18) Dee, G. T. Phys. ReV. Lett. 1986, 57, 275. (19) Dee, G. T. Physica D 1986, 23, 340.
16920 J. Phys. Chem., Vol. 100, No. 42, 1996 (20) Sultan, R.; Ortoleva, P.; DePasquale, F.; Tartaglia, P. Earth Sci. ReV. 1990, 29, 163. (21) Lifshitz, I. M.; Slyouzov, V. V. J. Phys. Chem. Solids 1961, 19, 35. (22) Chernavskii, D. S.; Polezhaev, A. A.; Mu¨ller, S. C. Physica D 1991, 54, 160. (23) Ostwald, Wo. Kolloid Z. 1925, (R. Zsigmondy-Ergaenzungsband zu Band 36), 380. (24) Jablczynski, C. K. Bull. Soc. Chim. Fr. 1923 33, 1592. (25) Mathur P. B.; Sanghi, C. L. J. Indian Chem. Soc. 1959, 36, 683. (26) Gnanam, F. D.; Krishnan, S.; Ramasamy, P.; Laddha, G. S. J. Colloid Interface Sci. 1980, 73, 193. (27) Hatschek, E. Proc. R. Soc. London Ser. A 1921, 99, 496. (28) Hatschek, E. Biochem. J. 1929, 14, 418. (29) Schibeci R. A.; Carlsen, C. J. Chem. Educ. 1988, 65, 365. (30) Ross J. In Fronts, Interfaces and Patterns; Bishop A. L., Campbell, L. J., Channell, P. J. Eds.; Elsevier: Amsterdam, 1984 (Physica D, 12, 303). (31) Sadek, S. M.Sc. Thesis, American University of Beirut, 1995. Sadek, S.; Sultan, R. In preparation. (32) Equations 1 and 2 should normally be expressed in terms of the concentration of hydroxide ion (co-precipitate of Co2+) in the form ∆ ) 1/2[OH-]0 - [Co2+]0, σ ) [Co2+]0[OH-]20. However NH4OH is not completely dissociated (Kb ) 1.78 × 10-5) and thus [NH4OH]0 * [OH-]0. Even though, it is appropriate to write ∆ and σ in terms of [NH4OH]0, since the reaction Co2+ + 2NH4OH f Co(OH)2 + 2NH4+ is stoichiometric with K ) 5.35 × 104 (calculated from Kb and Ksp ) 5.92 × 10-15 (ref 33) for Co(OH)2). (33) CRC Handbook of Chemistry and Physics, 75th ed.; CRC Press: Boca Raton, FL, 1994-1995.
Sultan and Sadek (34) Compare Figure 3 with Figure 4 of ref 7. (35) Marek, M.; Schreiber, I. Chaotic BehaVior of Deterministic DissipatiVe Structures; Cambridge University Press: Cambridge, 1991; and references therein. (36) Swinney H. L.; Roux, J. C. In Nonequilibrium Dynamics in Chemical Systems; Vidal, C., Pacault, A., Eds.; Springer: Berlin, 1984; p 124. (37) Mathur, P. B. Bull. Chem. Soc. Jpn. 1961, 34, 437. (38) We recently observed shades of Ag2SO4 showing revert spacing. (39) Henisch, H. K. Crystals in Gels and Liesegang Rings; Cambridge University Press: Cambridge, 1988. (40) In the reaction CoCl2(aq) + 2NH4OH(aq) h Co(OH)2(s) + 2NH4Cl(aq), an excess of the electrolyte NH4Cl shifts the equilibrium to the left. As the reaction quotient [NH4Cl]2/([CoCl2] [NH4OH]2) exceeds a certain value, the precipitate Co(OH)2 dissolves. This happens in our system near the bottom of the tube (i.e. with the last Co(OH)2 bands) where [NH4Cl] builds up because of the no-flux boundary condition. (41) Sultan R.; Ortoleva, P. Physica D 1993, 63, 202. (42) Kruhl, J. H. Fractals and Dynamic Systems in Geoscience; Springer-Verlag: Berlin, 1994; and references therein. (43) Ortoleva, P. Geochemical Self-Organization; Oxford University Press: New York, 1994. (44) Liesegang, R. E. Geologische Diffusionen; Steinkopff: Dresden, 1913. (45) Liesegang, R. E. Die Achate; Steinkopff: Dresden-Leipsig, 1915. (46) Ortoleva, P., Ed. Self-Organization in Geological Systems; Proceedings of a workshop held June 1988, University of California, San Diego, Earth Sci. ReV. 1990, 29, 1
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