Derivation of the Matalon-Packter law for Liesegang patterns

Antal, Tibor; Droz, Michel; Magnin, J.; Rácz, Zoltán; Zrı́nyi, Miklós

doi:10.1063/1.477609

Cited by 135 publications

(208 citation statements)

References 27 publications

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“…This assumption will be used throughout the two-dimensional simulations. The front, characterized by the spatio-temporal properties of kab (rate of production of Cs), has been studied in detail [17,18]. It is narrow and moves diffusively (its position is given by x f = 2D f t where D f can be expressed through D and b 0 /a 0 ).…”

Section: Supplementary Information a Detailed Model Descriptionmentioning

confidence: 99%

“…how the reaction product, C, turns into precipitate). While the front properties have been thoroughly studied and understood both theoretically [17,18] and experimentally [19,20], the dynamics of precipitation is more debated [13,18]. The competing pre-and post-nucleation views can be combined [18,21], and we shall use a simple version [14] based on the Cahn-Hilliard equation with noise added [22][23][24].…”

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confidence: 99%

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Probability of the Emergence of Helical Precipitation Patterns in the Wake of Reaction-Diffusion Fronts

et al. 2013

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Helical and helicoidal precipitation patterns emerging in the wake of reaction-diffusion fronts are studied. In our experiments, these chiral structures arise with well defined probabilities PH controlled by conditions such as e.g., the initial concentration of the reagents. We develop a model which describes the observed experimental trends. The results suggest that PH is determined by a delicate interplay among the time-and length-scales related to the front and to the unstable precipitation modes and, furthermore, the noise amplitude also plays a quantifiable role.PACS numbers: 68.35.Ct Helices and helicoids are present from nano-to macroscale (ZnO nanohelices [1], macromolecules and inorganic crystals with a helical structure [2, 3], precipitation helices [4][5][6], fiber geometry of heart walls [7]). Formation of these fascinating structures generally follows two routes. First, templates with chiral symmetry (e.g., oragogel fibers) may exist in the system, and the symmetry is just transcribed to a structure (e.g., inorganic materials [8]) at a larger scale. Second, spontaneous symmetry breaking may occur through the self-assembly of achiral building blocks into a helical/helicoidal structure, as e.g. in case of crystals with chiral morphology [2,9].Theoretically, the symmetry-breaking route is more interesting. Universal aspects may emerge and the robust features of this self-organization process may be important for applications as well. Indeed, control over creating helical structures would make engineering (in particular, the bottom-up design of micro-patterns [10]) more flexible since chiral morphology of materials are known to affect their physical (electronic) properties [6,11].In order to develop insight into the genesis of helices/helicoids, we investigate an emblematic example of pattern formation, namely the formation of precipitation patterns in the wake of reaction-diffusion fronts [12,13]. The motivation for this choice comes from the observation that helicoidal structures have an axis, and the correlations are simple in the plane perpendicular to the axis. This suggests that building the perpendicular correlations in the wake of an advancing planar front may be a simple and natural mechanism of creating helices/helicoids. Additional motivation comes from the existence of a large body of knowledge in the related Liesegang phenomena [12,13]. It allows the use of well-established experimental and theoretical approaches, thus making it easier to develop a novel view on the formation of helical structures.Our main results concern the probabilistic aspects of the symmetry-breaking route. We determine the probability P H of the emergence of single helices/helicoids in Liesegang-type experiments as the conditions such as the initial concentration of inner or outer electrolytes, or the temperature are changed. P H is found to be well reproducible and large (P H > 0.5 for some parameter range). The results are understood by expanding and simulating a model of formation of precipitation patterns [14]...

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Section: Supplementary Information a Detailed Model Descriptionmentioning

confidence: 99%

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confidence: 99%

Probability of the Emergence of Helical Precipitation Patterns in the Wake of Reaction-Diffusion Fronts

et al. 2013

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“…There is a third result which has not been discussed here but will be important in a later application when the front will figure as a source of particles in the description of the formation of Liesegang bands. Namely, the front leaves behind a constant concentration of C particles [23,20].…”

Section: Beyond Mean-fieldmentioning

confidence: 99%

“…A prototype of this pattern-forming process is the Liesegang phenomena [18] ‡ where a chemical reactant (called inner electrolyte) is dissolved in a gel matrix and a second reactant (outer electrolyte) diffuses in and reacts with the inner electrolyte. Under certain conditions, the dynamics of the reaction product generates a family of high-density precipitate zones whose properties obey generic laws [20] which have been understood in terms of a simple phase separation scenario [21].…”

Section: Introductionmentioning

confidence: 99%

Reaction–diffusion fronts with inhomogeneous initial conditions

Bena

Droz

Martens

et al. 2007

J. Phys.: Condens. Matter

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Abstract.Properties of reaction zones resulting from A + B → C type reaction-diffusion processes are investigated by analytical and numerical methods. The reagents A and B are separated initially and, in addition, there is an initial macroscopic inhomogeneity in the distribution of the B species. For simple two-dimensional geometries, exact analytical results are presented for the time-evolution of the geometric shape of the front. We also show using cellular automata simulations that the fluctuations can be neglected both in the shape and in the width of the front.

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“…This is the so-called regular banding situation, which has been recently explained [12] using the phase separation in the presence of a moving front as the underlying mechanism. Briefly, the reaction front, which moves diffusively, leaves behind a constant concentration c 0 of the reaction product, that we shall conventionally name hereafter C particles [13][14][15]. At a coarsegrained level, the dynamics of the C particles (that can diffuse, and are also attracting each other) can be described by a Cahn-Hilliard equation [16][17][18] with a source term corresponding to the moving reaction front.…”

Section: Introductionmentioning

confidence: 99%

Guiding fields for phase separation: Controlling Liesegang patterns

et al. 2007

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Liesegang patterns emerge from precipitation processes and may be used to build bulk structures at submicron lengthscales. Thus they have significant potential for technological applications provided adequate methods of control can be devised. Here we describe a simple, physically realizable patterncontrol based on the notion of driven precipitation meaning that the phase-separation is governed by a guiding field such as, for example, a temperature or a pH field. The phase-separation is modeled through a non-autonomous Cahn-Hilliard equation whose spinodal is determined by the evolving guiding field. Control over the dynamics of the spinodal gives control over the velocity of the instability front which separates the stable and unstable regions of the system. Since the wavelength of the pattern is largely determined by this velocity, the distance between successive precipitation bands becomes controllable. We demonstrate the above ideas by numerical studies of a 1D system with diffusive guiding field. We find that the results can be accurately described by employing a linear stability analysis (pulled-front theory) for determining the velocity -local-wavelength relationship. From the perspective of the Liesegang theory, our results indicate that the so-called revert patterns may be naturally generated by diffusive guiding fields.

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Derivation of the Matalon-Packter law for Liesegang patterns

Cited by 135 publications

References 27 publications

Probability of the Emergence of Helical Precipitation Patterns in the Wake of Reaction-Diffusion Fronts

Probability of the Emergence of Helical Precipitation Patterns in the Wake of Reaction-Diffusion Fronts

Reaction–diffusion fronts with inhomogeneous initial conditions

Guiding fields for phase separation: Controlling Liesegang patterns

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