A model of pattern formation by precipitation

Chernavskii, D.S.; Полежаев, А. А.; Müller, Stefan

doi:10.1016/0167-2789(91)90115-p

Cited by 50 publications

(52 citation statements)

References 22 publications

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“…Liesegang rings are periodic precipitations formed due to the diffusion of one electrolyte in to the gel and reacting with another. Many researchers have attempted to give explanation by various theories [28][29][30][31][32]. Recently, George and Varghese [33] have proposed a new scenario; they treated the formation of the periodic ring system as a moving boundary problem.…”

Section: Diffusion and Inhibition Studymentioning

confidence: 98%

Characterization and inhibitive study of gel-grown hydroxyapatite crystals at physiological temperature

Parekh

Joshi

Vaidya

2008

Journal of Crystal Growth

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Section: Diffusion and Inhibition Studymentioning

confidence: 98%

Characterization and inhibitive study of gel-grown hydroxyapatite crystals at physiological temperature

Parekh

Joshi

Vaidya

2008

Journal of Crystal Growth

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“…The results are understood by expanding and simulating a model of formation of precipitation patterns [14]. We explicitly observe that the origin of helices/helicoids is not to be found in the fluctuations and asymmetry of the initial-or boundary conditions [15,16]. Instead, the growth of unstable modes, the dynamics of the front, and the bulk fluctuations (noise) combine to yield the helices.…”

mentioning

confidence: 99%

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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“…We assume that the final precipitate form and all charged species do not diffuse while all neutral species undergo diffusion in the absence of convection [39][40][41][42] . This is based on the experimental fact that the precipitation front propagates from the region rich in hydroxide to the region rich in nickel ions 7 .…”

Section: Theoretical Methodsmentioning

confidence: 99%

Band Propagation, Scaling Laws, and Phase Transition in a Precipitate System. 2. Computational Study

Mansour

Al-Ghoul

2015

J. Phys. Chem. A

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In this second paper, we introduce a chemical kinetic model that investigates the dynamics of the experimental Ni(2+)/NH3-OH(-) Liesegang system characterized by a pattern of β-nickel hydroxide bands led by a growing pulse of α-nickel hydroxide. The model is based on a system of reaction-diffusion equations describing the precipitation reaction and dissolution of the nickel hydroxide polymorphs by ammonia. The hydroxide ions are assumed to be static whereas ammonia serves as a diffusing "vehicle" that supplies the hydroxide ions along the precipitation zone, and these ions in turn react with the static Ni(2+) ions. The precipitation-diffusion equations are coupled to nucleation, polymorphic transition, and growth rate equations, each of which is characterized by a critical constant specific to the solid phase dynamics. In the proposed model, priority is given to polymorphic transition rather than nucleation. This implies that the critical constants must be subject to a constraint different than that derived for the Lifshitz-Slyozov instability encountered in classical Liesegang patterns. Numerical simulations confirm the validity of our model and the derived constraint. The pulse position and width are found to scale in time as t(α) with α ≃ 0.5, in agreement with the experimental results. Finally, the mass of the bands is shown to oscillate in time, suggesting competition between growth and polymorphic transition on one side and dissolution on the other.

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A model of pattern formation by precipitation

Cited by 50 publications

References 22 publications

Characterization and inhibitive study of gel-grown hydroxyapatite crystals at physiological temperature

Characterization and inhibitive study of gel-grown hydroxyapatite crystals at physiological temperature

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

Band Propagation, Scaling Laws, and Phase Transition in a Precipitate System. 2. Computational Study

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