PAPER
www.rsc.org/pccp | Physical Chemistry Chemical Physics
Precipitation patterns with polygonal boundaries between electrolytes Changwei Pan,a Qingyu Gao,*ab Jingxuan Xie,a Yu Xiac and Irving R. Epstein*b Received 7th April 2009, Accepted 3rd September 2009 First published as an Advance Article on the web 13th October 2009 DOI: 10.1039/b904445k Two-dimensional Liesegang patterns formed when the boundary between electrolytes is polygonal display a variety of patterns, such as dislocations (radial alleys of gaps), branches (anastomoses) and spirals, many of which can be found in nature. Each vertex of the polygon can produce a pair of dislocation lines or branch lines. The effect caused by a vertex decreases with the number of vertices. Double-armed spirals are observed in experiments with a pentagonal boundary. Hexagons, which begin to approach smooth circular boundaries, do not give rise to dislocations, but instead yield concentric precipitation rings. A simple model of nucleation growth enables us to simulate dislocations and spirals consistent with those seen in our experiments.
I.
Introduction
Pattern formation in chemically reacting systems is ubiquitous in nature,1 and the coupling between reaction and diffusion can give rise to a wealth of complex spatiotemporal patterns. Among the earliest phenomena to be studied were Liesegang rings, or bands,2 which arise when one electrolyte diffuses into another, resulting in a banded deposition of precipitation. This phenomenon has attracted the attention of scientists in many fields, including physics,3 chemistry,4,5 life sciences,6 and geosciences.7 Several dynamical interpretations have been proposed, most of which fall into two categories: pre-nucleation theories and post-nucleation theories. The former category, sometimes referred to as ion-product supersaturation theory, was first described by Ostwald8 in 1897. The theory postulates that ionic supersaturation occurs locally, causing the ion product to exceed its equilibrium value. The electrolytes then react to form the precipitation product (A + B - D), lowering the concentration of electrolytes near the site of deposition. The cycle is repeated, leading to the Liesegang pattern. This theory was later elaborated by Wagner,9 Prager,10 Zeldovitch et al.,11 and Smith,12 among others. In the post-nucleation theory,13 visible macroscopic precipitation patterns arise from an initial homogeneous sol of colloidal precipitate particles. The Liesegang patterns are formed through a Lifshitz–Slyozov instability,14,15 which favors aggregation resulting from the feedback among the colloid particles’ growth, diffusion and surface tension. Several sets of regularities have been found to govern Liesegang pattern formation. These are described by the spacing,16 time,17 width18–20 and Matalon–Packter21,22 law. The spacing law states that the ratio between adjacent ring positions tends to a constant as n increases: lim ðxn =xn1 Þ ¼ 1 þ p
n!1 a
ð1Þ
College of Chemical Engineering, China University of Mining and Technology, Xuzhou, 221008, China. E-mail:
[email protected] Department of Chemistry and Volen Center for Complex Systems, MS 015, Brandeis University, Waltham, MA 02454-9110, USA. E-mail:
[email protected] c Department of Chemistry, Boston University, Boston, MA 02215, USA b
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where xn and (1 + p) denote the position of the nth ring and the spacing coefficient, respectively. The time law relates xn to the time tn of its formation xn = B0tn1/2 + A0
(2)
where B0 and A0 are constants.17,23 These laws hold for situations in which the precipitation band is roughly planar.24 Liesegang precipitation patterns can display a number of forms depending on the geometry of the precipitation front.3 Planar fronts produced in test tubes25–30 and two-dimensional fronts generated from a circular initial condition31,32 have been well studied. Simple Liesegang precipitation patterns are subject to several instabilities that can lead to dislocation lines,18,32,33 branches (anastomoses),32,34,35 spirals,32,34,36–39 and dendrites.40–46 Such instabilities can be induced by the shape of the boundary of the central electrolyte, by light, and by electric fields. Here we investigate Liesegang instabilities when the central electrolyte is a regular polygon of solution that diffuses into a surrounding gel containing another electrolyte. We also carry out numerical simulations of the resulting precipitation patterns using a proposed nucleation-growth model.47–49
II. Experimental The gel containing the surrounding electrolyte, K2Cr2O7, was prepared by adding 9.90 g of gelatine to 100 ml of 0.1 wt% K2Cr2O7 (Fluka) solution and heating to 70–80 1C for 30 min. The mixture was continuously stirred and, after complete dissolution of the gelatin, the solution was poured into a Petri dish of diameter 8.9 cm, in the center of which a triangular, square, pentagonal or hexagonal rubber prism had previously been placed. After the solution cooled to room temperature, the Petri dish was transferred to a biochemical incubator at 23.5 0.5 1C. The rubber prism was removed after the gel coagulated, and AgNO3 (1.0 M) solution as the central electrolyte was poured into the groove left by the rubber prism. This electrolyte diffused into the gel and reacted with another electrolyte in the surrounding gel, K2Cr2O7. The formation of precipitate structures was monitored with a CCD camera (Panasonic WV-BP460/G) connected to a computer-controlled imaging system, using diffuse light from a Phys. Chem. Chem. Phys., 2009, 11, 11033–11039 | 11033
fluorescent lamp as the light source. The images collected were analyzed with Image-Pro Plus software.
III.
Model
We used the nucleation-growth model of ref. 47–49 to simulate the Liesegang patterns, assuming the chemical reaction to be 2AgNO3 (aq) + K2Cr2O7 (aq) - Ag2Cr2O7 (colloid) - Ag2Cr2O7 (s)k + 2KNO3 (aq) Ag2Cr2O7 (aq) produced prior to the creation of rings is free to diffuse throughout the gel. If the local concentration of free silver dichromate, C, reaches the threshold value, C*, nucleation occurs, followed by aggregation into immobile precipitate D. Denoting the concentration of Ag+ cation as A, and the concentration of dichromate ion as B, the resulting system of partial differential reaction–diffusion equations is: @A ¼ r2 A 2RyðA2 B KÞ ð3aÞ @t @B DB 2 r B RyðA2 B KÞ ¼ @t DA
ð3bÞ
@C DC 2 ¼ r C þ RyðA2 B KÞ CyðC C Þ CNðC;DÞ @t DA ð3cÞ @D ¼ CyðC C Þ þ CNðC;DÞ ð3dÞ @t
where t is the dimensionless reaction time (t = tDA/L2); L is the characteristic size of the pattern; r2 is the Laplacian operator; K is the solubility product of silver dichromate; and DA, DB, DC are the diffusion coefficients of A, B, and C, respectively. y(A2B K) is a step function for the reaction such that y(A2B K) = 1 when A2B K Z 0, otherwise y(A2B K) = 0; y(C C*) is an analogous step function for precipitation. Using the two step functions here implies that both the reaction and precipitation processes are much faster than the diffusive transport, so that these chemical processes occur instantaneously relative to the time scale for diffusion. The reaction kinetics function R is given by R = d [cr + (1 cr)r] exp(s2)
(4)
where d denotes the increment in C and is calculated from the real root of the equation (A 2d)2 (B d) = K; the stoichiometric parameter s is given by ref. 47 as s = (A 2B)/(A + 2B); the coefficient cr, 0 r cr r 1, determines the degree of stochasticity of the reaction (cr = 1 corresponds to deterministic reaction, and cr = 0 corresponds to a fully probabilistic model); and r, a uniformly distributed random number in [0,1], provides the stochastic fluctuation. The function N(C, D) = 1 if there is any precipitate at the current grid point, or, alternatively, if there is no precipitate at the current location but precipitate is present at any of its neighbouring grid points and C exceeds an aggregation threshold (D*). In all other cases, N(C, D) = 0.
Fig. 1 Liesegang patterns formed with a triangular boundary between electrolytes. (a) Snapshot taken 62 h after initiation. Dashed lines indicate positions of dislocations. (b) Magnification of the white dashed square in (a). (c) Simulated snapshot of Liesegang pattern at t = 1000 (dimensionless time units), A = 1.0, B = 0.01, DA : DB : DC = 1.0 : 0.75 : 0.07, K = 1.1 1012, C* = 0.01, D* = 0.009815, cr = 0.80.
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To integrate the above partial differential equations, the two-dimensional medium of 100 100 dimensionless spatial units was divided into 600 600 grid points. Zero-flux boundary conditions were used in all calculations. The initial conditions were A(x, t = 0) = C(x, t = 0) = D(x, t = 0) = 0. The electrolyte B was homogeneously distributed in the two-dimensional medium. The reagent A was then added to a polygonal region at the center of the medium. The parameter values (dimensionless time units) employed were: A0 = 1.0, B0 = 0.01 = DA : DB : DC = 1.0 : 0.75 : 0.07, K = 1.1 1012, C* = 0.01, and D* (triangle) = 0.009815, D* (square) = 0.00983, D* (pentagon/single-armed spiral) = 0.00985, D* (pentagon/double-armed spirals) = 0.00990. The Euler method with time step (Dt = 0.005) and space grid-step (Dx = 0.25) was employed on a 600 600 grid to integrate the differential equations.
IV.
Results and discussion
With a polygonal boundary between electrolytes, the rate of dilution resulting from diffusion of the central electrolyte from a vertex and from an edge differ because of their different curvatures. During the Ag+ diffusion that leads to the formation of precipitation bands, the boundary curvatures lead to a non-uniform spatial distribution of Ag+. Lower concentrations of Ag+ arise in the neighbourhood of a vertex than near the middle of an edge. The Matalon–Packter21 law gives the relation between the spacing coefficient and the concentration of invading electrolyte (p E 1/A0). If one takes the effect of curvature into consideration, the spacing coefficient (p) corresponding to a vertex should be larger than that corresponding to an edge, resulting in different interband intervals, band dislocations and other phenomena such as branches and spirals.
Fig. 2 Experimental and simulated ring position vs. ring number with triangular boundary. (a) Experimental results; (b) simulated curves. Curves I and III show distance to the midpoint of an edge of the triangle, curves II and IV show distance from a vertex.
Fig. 3 Time law (eqn (2)) for triangular boundary between electrolytes. (a) Experimental curves I for rings emanating from an edge, and II for rings emanating from a vertex. (b) Simulated curves III (edge) and IV (vertex). Table 1
Parameters of the spacing (eqn (1)) and time (eqn (2)) laws
Parameters
Boundary shape
Experiment
Simulation (dimensionless units)
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B0 (edge bands)/cm h 1 + p (edge bands) B0 (vertex bands)/cm h1/2 1 + p (vertex bands) B0 (edge bands) 1 + p (edge bands) B0 (vertex bands) 1 + p (vertex bands)
0.411 1.097 0.340 1.121 6.172 1.092 5.904 1.128
0.020 0.007 0.010 0.007 0.011 0.004 0.009 0.005
Square boundary 0.343 1.096 0.332 1.106 5.933 1.049 5.743 1.060
0.020 0.010 0.010 0.008 0.008 0.010 0.007 0.003
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Fig. 1 shows Liesegang patterns formed with a triangular boundary. The dashed lines in Fig. 1a track the breaks in the otherwise continuous rings; Fig. 1b is an enlargement of the dashed frame in Fig. 1a. The ring dislocations begin to appear in the first visible ring. When a dislocation becomes large enough, the ring breaks, forming two branches. Two lines of dislocations emerge from each vertex of the triangle. In Fig. 1c, we show a simulation of this experiment. We can see that the radially aligned dislocations correspond to the vertices of the triangle. The dislocations become more obvious as we move further from the boundary, because the bands moving out from the edges and the vertices of the triangle obey different spacing and time laws. From examination of Fig. 1a, the bands extending from a vertex are spaced further apart, and the region between pairs of dislocation lines contains fewer bands than from an edge. The simulations in Fig. 1c also show larger band intervals from dislocations, in agreement with the above discussion. This behavior, which is plotted in Fig. 2, is seen in both the experiments (Fig. 2a) and the simulations (Fig. 2b). For all dislocated rings in the experiment, the distance xn to the midpoint of the edge of the triangle is smaller than that corresponding to the vertex of the triangle, which is consistent
Fig. 6 Minimum radius at which dislocation occurred in simulations. DA : DB : DC = 1.0 : 0.75 : 0.07, K = 1.1 1012, C* = 0.01, D* = 0.00983, cr = 1.00 and 0.20.
with the simulations. After band branching begins, the distance of bands emerging from the vertex of the triangle increases from the edge of the triangle. Fig. 3 shows the analysis of the time law for the bands with a triangular boundary. The t1/2 dependence is clear. In both experiment and simulation, the slope of the line corresponding to propagation from the edges exceeds that for the vertices, though the difference is considerably greater in the
Fig. 4 Liesegang patterns with square boundary between electrolytes. (a) 61 hours into an experiment. (b) At 1000 (dimensionless time units) in a simulation, DA : DB : DC = 1.0 : 0.75 : 0.07, K = 1.1 1012, C* = 0.01, D* = 0.00983, cr = 0.80.
Fig. 5 Experimental Liesegang patterns with pentagonal and hexagonal boundaries. (a) Double-armed spirals with a pentagonal boundary. One arm is highlighted in red. (b) Circular Liesegang rings formed with a hexagonal boundary.
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experiments. In Table 1 we present the parameters of the spacing law and time law (eqn (1) and (2)) for the triangular and square boundaries. The simulation qualitatively reproduces the experimentally observed differences between the vertex and edge bands. With a circular boundary between electrolytes, the distribution of precipitation bands is isotropic. Dislocations can only arise from perturbations or concentration gradients in the reaction regions. With the asymmetry introduced by the
polygonal boundaries employed here, differences in spacing coefficients between regions near the edge and regions near the vertices naturally give rise to dislocations and subsequent branch formation as seen in Fig. 1. The dislocation behavior with a square boundary is similar to, but not as sharp as, that observed with a triangular boundary. A typical experimental result is shown in Fig. 4a. The simulation in Fig. 4b qualitatively reproduces the observed behavior. The time law with a square boundary is
Fig. 7 Simulated Liesegang spirals with a pentagonal boundary between electrolytes and schematic illustration of spiral formation. (a) Singlearmed spiral, C* = 0.01, D* = 0.00985, cr = 0.20, DA : DB : DC = 1 : 0.75 : 0.02. Time = 240 (dimensionless time units). (b) Double-armed spirals, C* = 0.01, D* = 0.0099, cr = 0.20, DA : DB : DC = 1 : 0.75 : 0.02. Time = 200 (dimensionless time units). (c) Schematic illustration of single-armed spiral formation. (d) Schematic illustration of double-armed spiral formation.
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similar to that found with a triangular boundary, but, as Table 1 shows, the difference in slope between the side bands and vertex bands is somewhat smaller, leading to a more subtle set of dislocation phenomena. In both experiment and simulation, the band intervals from vertices exceeded those from edges as the dislocation bands appeared (Fig. 4). With a pentagonal boundary, a remarkable new phenomenon, double-armed spirals (Fig. 5a), is observed. One of the two spiral arms is highlighted in red to guide the eye. Single-armed spirals have previously been reported in two-dimensional Liesegang band experiments,32,34 and helical bands have been observed in a three-dimensional cylindrical geometry.38 Spiral and helical bands are thought to result from dislocations and anastomosis of parallel bands. The precipitation patterns found with the hexagonal boundary (Fig. 5b) begin to approach the situation with a circular boundary. Here we failed to observe dislocations under our experimental conditions. In our simulations we found dislocations with all four polygonal boundaries (triangle, square, pentagon, hexagon) with parameters C* = 0.01, D* = 0.00983. The minimum radius at which dislocations arise increases as the number of vertices increases; that is, it is more difficult to break the rings when the number of sides increases, or, equivalently, the angle between them decreases. This behavior is illustrated in Fig. 6. The computation also indicates that ring rupture becomes more facile as cr decreases, i.e., as the strength of the stochastic fluctuations grows. Differences in the spacing and time laws for precipitation bands extending from vertices and from edges, respectively, lead to dislocations and other complex phenomena when polygonal boundaries are present. In the model, enhanced stochasticity and easier nucleation, e.g., by decreasing cr and C* or the difference between C* (nucleation threshold) and D* (aggregation threshold) favor band dislocations, anastomoses and spiral formation. Al-Ghoul and Sultan50 varied the thresholds (C* and D*) to delineate how these parameters affect the formation of uniform pulses versus Liesegang bands. Here, a single-armed spiral, formed by migration of a dislocation between adjacent rings, is shown in Fig. 7a. On further increasing D* to approach C*, we obtain the doublearmed spirals seen in Fig. 7b. As we illustrate schematically in Fig. 7c and d, if one starts from concentric circular bands (target patterns) and a dislocation causes a shift to an adjacent band along one radius, single-armed spirals are obtained. If the bands move to adjacent rings along two radii, one directed outward and the other inward, double-armed spirals form. Thus the evolution from Liesegang targets to spirals can be understood in terms of dislocation migrations.
V.
Conclusion
The polygonal electrolyte boundaries studied here lie between the more widely studied conditions of planar and circular bands. The anisotropy of our boundaries generates such interesting phenomena as dislocations, branches and spirals as a result of the different spacing and time laws for bands emanating from polygonal edges and vertices. As the number of edges increases, the behavior of the edge- and 11038 | Phys. Chem. Chem. Phys., 2009, 11, 11033–11039
vertex-generated bands begins to converge, reducing the effects of dislocations as we move from the triangular to the hexagonal boundary condition. With the larger polygons, the dislocation lines are weaker and less numerous, and spirals may develop, for example from one or two dislocation lines under the pentagonal boundary condition. The evolution from dislocation to branching (anastomosis) is related to stochastic fluctuations in both reaction and diffusion. It may be useful to apply stochastic cellular automaton algorithms41 to simulate the branching behavior. By carefully tailoring the boundary between the inner and outer electrolytes, it should be possible to design and control the formation of complex Liesegang patterns.
Acknowledgements This work was supported in part by the U.S. National Science Foundation (CHE-0615507), and by NSFC (Grant 20573134), NCET (Grant 05-0477) and JSNSF (BK2007037). The authors are indebted to Dr Qun Wang and Mr Kailong Zhang for assistance with the calculations.
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